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0704.1424

The Amadeus project at Dafne

The AMADEUS project at DAΦNE Catalina. Curceanu LNF – INFN, Via. E. Fermi 40, 00044 Frascati (Roma), Italy On behalf on the AMADEUS Collaboration 1. The AMADEUS scientific case The change of the hadron masses and hadron interactions in the nuclear medium and the structure of cold dense hadronic matter are hot topics of hadron physics today. These important, yet unsolved, problems will be the research field of AMADEUS [1] (Antikaonic Matter At DAFNE: Experiments with Unraveling Spectroscopy). AMADEUS will search for antikaon-mediated deeply bound nuclear states, which could represent, indeed, the ideal conditions for investigating the way in which the spontaneous and explicit chiral symmetry breaking pattern of low-energy QCD occur in the nuclear environment. In a few-body nuclear system the isospin I=0 K N interaction plays an important role: it can favor the existence of discrete nuclear bound states of K in nuclei, while contracting the nucleus, thus producing a cold dense nuclear system, named “deeply bound kaonic nucleus” or “kaonic nuclear cluster”. Many important impacts then follow: - such compact exotic nuclear systems might get formed with binding energies so large (~100 MeV) that their widths turn out rather narrow, since the Σπ decay channel is energetically closed and, additionally, the Λπ channel is forbidden by isospin selection rule; - high-density cold nuclear matter might be formed around K⎯, which could provide information concerning a modification of the kaon mass and of the K N interaction in the nuclear medium; - empirical information could be obtained on whether kaon condensation can occur in nuclear matter, with implications in astrophysics: neutron stars, strange stars. - nuclear dynamics under extreme conditions (nuclear compressibility, etc) could be investigated. The hypothesis of the possible existence of deeply bound kaonic nuclear states was already formulated in 1986 by Wycech [2] but is only a few years old in the structured form of a phenomenological model formulated by Akaishi and Yamazaki [3]. The existence or not of such deeply bound systems is presently matter of vivid discussions among theoreticians and experimentalists. First experimental indications have been produced at KEK [4-6], LNF [7], GSI [8] and BNL [9]. Lately, the experimental situation became more complicated, since one of the two systems seen at KEK by E471, the S0(3115), was not re-seen in E570 with a larger statistics, as preliminary presented at the IX International Conference on Hypernuclear and Strange Particle Physics, Mainz 10-14 October 2006. In the same time, the interpretation of the experimental results is matter of lively discussion, as resulting from the talks given at the previously mentioned Workshop. The new proposal, AMADEUS at DAΦNE, has the goal to perform, for the first time, a systematic and complete spectroscopic study of deeply bound kaonic nuclei, both in formation and in the decay processes. Moreover, AMADEUS aims to perform other types of measurements as: elastic and inelastic kaon interactions on various nuclei, obtaining important information for a better understanding of the undergoing processes. 2. AMADEUS at DAΦNE The φ-factory DAΦNE at LNF is a double-ring e+e- collider, designed to operate at the center of mass energy of the φ-meson, whose decay delivers almost monochromatic K+K- pairs with a momentum of 127 MeV/c. A peak luminosity in excess of 1x10 32 cm-2 s-1 is presently obtained in the FINUDA run. Such a source of kaons is intrinsically clean, a situation unattainable with a hadron machine, which normally has an intense pion background. The planned upgrade of DAΦNE will reach a peak luminosity in excess of 10 33 cm-2 s-1 [10], which, of course, makes it an ideal machine to study stopped K- induced reactions to search for deeply bound antikaon nuclear clusters. This new facility will deliver an integrated luminosity of about 10 fb-1 per year and equipped with a dedicated 4π detector KLOE [11] complemented with the AMADEUS apparatus will become the top level scientific center to study kaonic nuclei using K- induced processes at rest, as proposed in the original work of Akaishi and Yamazaki [3]. This represents a complementary approach to other experimental studies, planned at GSI with the FOPI detector and at J-PARC, E15 experiment, to be seen as part of a global strategy to attack these important open problems of low-energy QCD. In Figure 1 the location of AMADEUS setup within KLOE detector is shown. Figure 1: The AMADEUS setup within KLOE. For the integration of the AMADEUS setup within KLOE a solution which is presently under study is to use a toroidal target placed around the beam pipe and surrounding the interaction region. The beam pipe can be a thin-walled aluminum pipe with carbon fiber reinforcement. A degrader which might be an “active” one, i.e. a scintillator (or scintillating fiber) detector is placed around the pipe just in front of the target. This detector is essential, delivering an optimal trigger condition by making use of the back-to-back topology of the kaons generated from the φ- decay. The AMADEUS collaboration considers as well the implementation of an inner tracker, to get more information about the formation region (for better background suppression) of the deeply bound states. Since the KLOE collaboration envisages the use of an inner vertex detector this might open the way to a unique inner vertex detector to be used by both groups. The possible setup is shown in Figure 2. two TPC sections with triple GEM and x-y readout on both sides kaon trigger made of 2+3 scintillating fibers layers, inside a vacuum chamber half-toroidal cryogenic target cell vacuum chamber thin-walled beam pipe Figure 2: A possible layout of AMADEUS within KLOE 3. AMADEUS scientific program The AMADEUS first phase program foresees the investigation of the most basic antikaon - mediated clusters, namely: - kaonic dibaryon states ppK- and pnK-, produced via 3He (stopped K, n/p) reactions; - kaonic 3-baryon states ppnK- and pnnK-, produced via 4He (stopped K-, n/p) reactions. The first goal of AMADEUS is to clearly settle the question about the existence of antikaon-mediated bound nuclear clusters. To this aim a systematic and complete campaign of spectroscopic studies on light and medium nuclei looking for nuclear clusters both in formation and in decay will be performed The program is based on the following objectives: (i) To perform precision measurements in order to determine the quantum numbers (spin, parity, isospin) of all states, including excited ones, in addition to their binding energies and decay widths. Masses of kaonic clusters are obtained by missing mass analysis, measuring proton and neutron distributions from the formation process. In KLOE the proton spectra can be obtained with 1 -2 MeV precision. For the neutron spectra the achieved precision is of the order of 2-4 MeV. (ii) As all the states of kaonic nuclei are quasi -stationary, important information on their structure is contained in their total and partial decay widths. Until now, the experimental values on the total decay width are under discussion and no information on partial decay channels is available. Total decay widths, accessible via the formation process, can be resolved in AMADEUS at the 1 -3 MeV level for proton spectra, and at a few MeV, still under study, for neutron spectra. The partial decay widths may be resolved at the level from 2 to 10 MeV depending on the decay channel. (iii) Detailed structure information can be extracted from a Dalitz analysis of three-body decays of kaonic nuclei, as was pointed out recently by Kienle, Akaishi and Yamazaki [6]. Acknowledgements We gratefully acknowledge the very good cooperation with the DAΦNE and KLOE teams. Part of this work was supported by “Transnational access to Research Infrastructure”, TARI – LNF activity within the HadronPhysics I3 project, Contract No. RII3-CT-2004-506078. References [1] AMADEUS LOI, www.lnf.infn.it (Nuclear Physics – SIDDHARTA) [2] S. Wycech, Nucl. Phys. A450 (1986) 399c. [3] Y. Akaishi and T. Yamazaki, Phys. Rev. C 65 (2002) 044005. [4] T. Suzuki et al., Phys. Lett. B 597 (2004) 263. [5] M. Iwasaki et al., arXiv: nucl-ex/0310018. [6] M. Iwasaki (KEK PS-E471 Collaboration), Proceedings EXA05, Editors A. Hirtl, J. Marton, E. Widmann, J. Zmeskal, Austrian Academy of Science Press, Vienna 2005, p.191. [7] M. Agnello et al., Phys. Rev. Lett. 94 (2005) 212303. [8] N. Hermann (FOPI Collaboration), Proceedings EXA05, Editors A. Hirtl, J. Marton, E. Widmann, J. Zmeskal, Austrian Academy of Science Press, Vienna 2005, p.61. [9] T. Kishimoto et al., Nucl. Phys. A 754 (2005) 383c. [10] http://www.lnf.infn.it/acceleratori/dafne/NOTEDAFNE/G/G-68.pdf [11] KLOE: A General Purpose Detector for DAFNE, The KLOE Collaboration, LNF-92/019 (IR). [12] P. Kienle, Y. Akaishi and T. Yamazaki, Phys. Lett. B 632 (2006) 187. http://www.lnf.infn.it/acceleratori/dafne/NOTEDAFNE/G/G-68.pdf On behalf on the AMADEUS Collaboration Figure 2: A possible layout of AMADEUS within KLOE Acknowledgements We gratefully acknowledge the very good cooperation with the DANE and KLOE teams. Part of this work was supported by “Transnational access to Research Infrastructure”, TARI – LNF activity within the HadronPhysics I3 project, Contract No. RII3-CT-2004-506078.

0704.1425

Measurements and analysis of the upper critical field $H_{c2}$ on an underdoped and overdoped $La_{2-x}Sr_xCuO_4$ compounds

Measurements and analysis of the upper critical field H c2 of underdoped and overdoped La2−xSrxCuO4 compounds D. H. N. Dias and E. V. L. de Mello Instituto de F́ısica, Universidade Federal Fluminense, Niterói, R. J., 24210-340, Brazil J. L. Gonzalez and H. A. Borges Dept. de F́ısica, Pontif́ıcia Universidade Católica de Rio de Janeiro, R. Marques de São Vicente 225, R. J., 22453-900 Brasil. A. Gomes Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. A. Loose FU-Berlin. Berlin. Germany A. Lopez Universidade do Estado de Rio de Janeiro, Rio de Janeiro, Brazil. F. Vieira and E. Baggio-Saitovitch Centro Brasileiro de Pesquisas F́ısicas, Rio de Janeiro, Brazil. (Dated: November 21, 2018) The upper critical field Hc2 is one of the many non conventional properties of high-Tc cuprates. It is possible that the Hc2(T ) anomalies are due to the presence of inhomogeneities in the local charge carrier density ρ of the CuO2 planes. In order to study this point, we have prepared good quality samples of polycrystalline La2−xSrxCuO4 using the wet-chemical method, which has been demonstrated to produce samples with a good cation distribution. In particular, we have studied the temperature dependence of the upper critical field, Hc2(T ), through magnetization measurements on two samples with opposite average carrier concentration (ρm = x) and nearly equal critical temperatures, namely, ρm = 0.08 (underdoped) and ρm = 0.25 (overdoped). The results close to Tc do not follow the usual Ginzburg-Landau theory and are interpreted by a theory which takes into account the influence of the inhomogeneities. PACS numbers: 74.72.-h, 74.80.-g, 74.20.De, 02.70.Bf I. INTRODUCTION High critical temperature superconductors (HTSC) display many non-conventional properties which remain to be explained by a concise physical picture[1, 2, 3]. This state of affairs might be due to the fact that, differ- ently from low temperature superconductors, these ma- terials have a large degree of intrinsic inhomogeneities. Although its origin is unknown, there are many evidences from different experiments that they do not have a homo- geneous doping level[4, 5]. For instance, recent Scanning Tunneling Microscopy (STM)[6, 7, 8, 9] measurements have detected strong variations in the density of states as the tip travels over a clean and sharp surface. Also, neu- tron diffraction data have revealed a complex structure of the charge distribution that has become known as ”stripe structure”[10]. Nuclear Quadruple Resonance (NQR)[11] and Angle Resolve Photo-Emission (ARPES)[12] have also detected inhomogeneities in the local environment and in the charge distribution. Based on these experimental evidences we argue that this behavior and the inhomogeneities are possibly due to a phase separation transition (PST) connected with the anomalies seen at the upper pseudogap temperature. Such PST would bring the system to a disordered charge distribution state, with the formation of islands or re- gions of distinct doping levels[13]. Since any PST de- pends on the chemical mobility, this approach may shed some light on the reason why some compounds appear to be more homogeneous or, at least, do not display any gross inhomogeneity[14, 15], although the phase di- agrams of cuprates seem to be universal. Therefore, we think it is possible to formulate a unified theory for the HTSC despite of their different degrees of disorder. In this paper we explore this possibility by showing that the anomalies of the upper critical field Hc2(T ) as function of the temperature T are in agreement with charge carriers inhomogeneities in two samples with com- pletely different average doping levels: one underdoped and other in the far overdoped region of the super- conducting phase diagram. In order to achieve this goal this work is threefold: i) we prepared samples of http://arxiv.org/abs/0704.1425v2 La2−xSrxCuO4 (LSCO), ii) we made several sets of Hc2(T ) measurements and iii) perform a theoretical in- terpretation of the data. The non conventional fea- tures of Hc2(T ), like a positive curvature and absence of saturation at low temperatures, are well known from many previous experiments[16, 17, 18]. We concentrat in two samples in the underdoped and overdoped dop- ing levels, and measured Hc2(T ) to detect qualitative doping dependent behavior. We interpreted the results through a theory that takes into account the different contribution from stripes of different local charge den- sity. These calculations are based on the Cahn-Hilliard (CH) theory of phase separation[19, 20] for compounds with ρm ≤ 0.20[2, 13] and on a Gaussian charge fluc- tuation around the average for ρm ≥ 0.20. In both cases the resulting inhomogeneous systems are studied with a Bogoliubov-deGennes approach to a disordered superconductor [13]. Indeed there are several different approaches to deal with the inhomogeneities in HTSC, like the method of Ghosal at al[21] of a local disorder in the chemical potential, Nunner et al[22] that deals with out of plane chemical disorder, and Cabo et al[23] that introduced an in plane Gaussian disorder around de av- erage doping, to mention just a few of what can be found in the literature. So far, all these approaches succeeded in explain some HTSC features, what shows that the in- homogeneities are important, but only new and refined experiments will be able to determine the correct way to deal with them. II. SAMPLE PREPARATION Several polycrystalline samples belonging to the La2−xSrxCuO4 system were prepared by the wet- chemical method according to reference[24, 25]. Pure (99,99%) oxide and carbonate compounds, namely La2O3, CuO, and SrCO3 were dried at 150 0C and weighted with adequate stoichiometrical proportions. The powders were dissolved into 50 ml of ultra pure acetic acid (CH3COOH) and the final solution was dried and after heated at 9000C during 24 hours in flowing oxygen. After that the powders were quenched at room temper- ature, then the mixtures were reground and pressed into pellets. Finally the pre-sintered samples were sintered at 10500C during 50 hours. Fig.(1) shows the x-rays diffraction performed in both samples prepared according the wet-chemical method. The x-ray diffraction (XRD) patterns were obtained in a (XPert PRO PANalytical) powder diffractometer using CuKα radiation (λ = 1.5418Å). Data were collected by step-scanning mode (200 ≤ 2θ ≤ 900) and 2 s counting time in each step at room temperature. Orthorhombic (Bmab) and tetragonal (F4mmm) structure space groups were assumed in the Rietveld analysis for the 0.08 and 0.25 strontium percent samples respectively. Once we have characterized the sample, we have per- formed magnetic measurements by a SQUID magnetome- 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 25 % Sr 8 % Sr FIG. 1: X-rays diffraction spectra at room temperature for both samples used in the experiment. The principal peaks were identified according to Rietveld analysis. ter, as described in the next section. III. EXPERIMENTAL PROCEDURE 5 10 15 20 25 -8,0x10-5 -6,0x10-5 -4,0x10-5 -2,0x10-5 18 19 20 21 22 23 24 25 26 27 28 Tc(3kOe)=21.15 Tc(1kOe)=22.6 K 1 kOe 3 kOe 7 kOe 10 kOe T (K) 25 % Sr FIG. 2: (Color on line) The zero-field cooling magnetic mo- ment at some applied magnetic fields as a function of tem- perature for the sample with 25% of strontium. The insert shows how the values of Tc were determined for two values of the applied field. Magnetization curves as a function of temperature were obtained with a SQUID magnetometer in a conven- tional DC mode. The measurements were performed in zero field-cooling conditions with moderate applied fields ranging between zero and two Tesla. Fig.(2) contains sets of M(T) curves for the sample La1.75Sr0.25CuO4 (25 % of strontium) is presented. In the inset we show details of how Tc(H) were obtained for two selected fields, namely 3kG and 1 kG. By taking the measured values of Tc(H), we obtain the plot shown in Fig.(3), from which we can extrapolate the value of Tc(Hc2 = 0). For the present case, we obtain Tc(Hc2 = 0) = 24.1K and for the under- doped sample, we get Tc(Hc2 = 0) = 31.1K. A similar set of curves was obtained for the 8% strontium sample. In this manner we got the value of Tc(Hc2 = 0) for both samples. 18 19 20 21 22 23 24 25 26 27 12000 16000 20000 T (K) FIG. 3: (Color on line) Values of Tc(H) calculated as shown in inset of Fig.(2) ( for La1.75Sr0.25CuO4). The value of Tc(Hc2 = 0) is taken where the curve extrapolates to zero. In this case we get Tc(Hc2 = 0) = 24.1K, value that will be used further on (Fig.(7)). On the other hand, the critical temperatures Tc(H = 0) for both samples were determined from many sets of resistivity data, by taking the maximum of the first derivative of the resistivity vs temperature curves, namely Tc(8%Sr) = 22.8K and Tc(25%Sr) = 19.9K. From these data, the widths of the superconducting transitions were estimated at half-maximum of the first derivative, as displayed in Fig.(4). For the sample with 25% of strontium the ∆Tc was about 4.5 K while in the 0.08% sample the transition was wider pointing out to the presence of strong inhomogeneities in the sample. The presence of these disorder will be important in the discussions presented in the next sections. 10 20 30 40 0 10 20 30 40 ∆ΤC=4.5 K FIG. 4: Resistivity as function of temperature. The insert shows the first derivative of the data and the maximum is taken as the superconductivity transition temperature for this sample at Tc(25%Sr) = 19.9K IV. PST RESULTS AND DISCUSSION As discussed in the introduction, there are many exper- imental evidences showing that some cuprates are highly inhomogeneous in their charge distribution while others are not, but all of them display the same phase diagram. To deal with this non trivial problem, we have intro- duced the idea of a PST that, depending on the mobility of the ions, can generates various degrees of decomposi- tion. This phase segregation process can form patterns on the sample, as the stripes[10] or patchwork[6], gener- ating islands with different values of the charge density, or can merely form small fluctuations around an average doping level. Applying a Bogoliubov-deGennes (BdG) theory to these systems we were able to calculate the local superconducting pairing amplitude at a given site ”i” in a cluster ”l”. Thus, in our calculations, a given sample with average or mean charge density ρm may be composed of local regions with local densities ρ(l). Here we show results on a 14× 14 matrix, that is, 14 clusters on a stripe form (each stripe has 14 sites) (l = 1to14) and 196 sites (i runs from 1 to 196). In general, regions with larger charge densities or dop- ing levels usually may become superconducting, and those with very low doping level are insulators and never become superconducting. This anomalous behavior is de- tected by the superconducting pairing amplitude ∆(l, T ) which, as the temperature is decreased, starts at Tc(l), increases and saturates at low temperatures. This allows us to define a local superconducting temperature Tc(l). Here we exhibit simulations on a square mesh 14× 14 that display stripe inhomogeneities similar to the exper- imental results[10], but derived from a CH phase separa- FIG. 5: (Color on line) Temperature evolution of the local pairing amplitude ∆(i, T ) at each stripe on a square of 14×14, that is, with 196 sites ”i”, for different samples. Because of the inhomogeneities, the sites in the left have ρ(i) ≈ 0 and the ones in the right ρ(i) ≈ 2ρm. As the systems are cooled down, more regions become superconducting and they percolate at Tc(ρm). These percolation threshold temperatures for each compound are indicated in their respective panel. The panel with ρ = 0.22 does not have the stripe structure, because it has a random distribution of doping values, which are sorted following a Gaussian distribution. tion theory applied to the LSCO series[13]. According to the CH results[19, 20], the square mesh phase separates into a bimodal distribution of charge. For underdoped samples the phase separation is essentially total and leads to two halves where the 7 stripes at the left are character- ized by local doping ρ(l) ≈ 0 and the 7 at the right side have ρ(l) ≈ 2ρm as showed in the top panel of Fig.(5) (for ρ = 0.05). As ρm increases, the charge fluctuations also increases, changing the properties of a compound into metallic, and superconducting at low temperatures. Thus, for compounds with 0.06 < ρm < 0.20, at low tem- peratures, as one can see in Fig.(5), there are also stripes with non-vanishing densities at low density regions. No- tice that the values of ∆ are constant in a given stripe, in all its sites ”i”, and that is why we may plot ∆(i, T ) at each site, although that is meaningful only at a given stripe ∆(l, T ). As the density of charge carriers increases, it produces the ∆(i, T ) at the very low doping sites, i.e., in the left region of Fig.(5). When this occurs at low temperatures, it is possible that the system becomes su- perconductor at Tc(ρm) by the percolation of all the local superconducting regions[26, 27] and it can hold a dissipa- tionless current. The percolation among the local regions where ∆(l, T ) in non-zero may occur either directly or by Josephson coupling and the associated temperature, or the superconducting temperature Tc(ρm), is also shown in the panels of Fig.(5). Notice that the compound with ρ = 0.22 does not have the stripe structure, it has just a Gaussian distribution of doping values, since it is larger than the PST threshold of ρ ≈ 0.2. Above Tc(ρm), depending on the value of ρm, the com- pounds may be formed by mixtures of superconducting, insulator and normal domains and above the pairing for- mation temperature (the onset temperature which some- times is called the lower pseudogap), they are a disor- dered metal with mixtures of normal (ρ(l) > 0.05) and insulator (ρ(l) ≤ 0.05) regions. For ρm ≥ 0.20 the charge disorder is practically zero, with a small fluctu- ation around ρm. From these calculations, we identify Tonset(ρm) as the highest temperature (Tc(l)) which in- duces a ∆(l, T ) in a given compound which is easily seen in the panels of Fig.(5). Tonset may be also identified with the onset of Nernst signal[13]. To make clear how the values of Tonset of a given compound are obtained, we show the larger values of Tc(i) and Tc(l) in Fig.(6) for the sample with ρ = 0.15. FIG. 6: (Color on line) To explain the concept of local super- conducting temperature and specially how certain regions de- velops a non-vanishing pairing amplitude, we plot the ∆(i, T ) or ∆(l, T ) for each site ”i” of our two dimensional 14 × 14 array. The onset temperatures values for which these pairing amplitudes develop for each stripe are clearly indicated in the figure, and the highest value, namely, T = 90K, is taken as the lower pseudogap temperature of this compound. V. Hc2 RESULTS AND CALCULATIONS It is well known that the Ginzburg-Landau (GL) up- per critical field of a homogeneous superconductor is a linear function of the temperature near Tc and falls to zero at this temperature[28]. This behavior is not ob- served by our measurements, showing another departure from conventional properties. As we can see in Fig.(7), the Hc2(T ) experimental points for both samples are lin- ear only near and below T/Tc ≤ 0.9. As the tempera- ture increases it performs an upturn curvature falling to zero further beyond Tc. Consequently, we need some new ideas or theories to interpret these results. From these Hc2(T ) curves, we can estimate that the upper critical field goes to zero at about 31.1 K, which is substantially higher than the value Tc = 22.8K of the 8 %-Sr sample (from resistivity measurements as discussed in section II). Similarly, Hc2(T ) falls to zero at 24.1K for the 25 %-Sr sample which has a Tc = 19.9K. Thus we see that Hc2(T ) vanishes at temperatures 8-22% larger than Tc. This effect, the nonzero value of Hc2 above Tc, is quite unusual for a normal superconductor but it was also observed in LSCO and Bi-2201 cuprates by the group of Wang et al[29]. We will show below that this result may be explained as a consequence of the intrinsic disorder in HTSC, namely the presence of regions with different local dopings and distinct superconducting local temperatures Tc(l). In order to provide an interpretation to these results, we applied a generalization of the GL Hc2(T ) expression following along the lines described by Caixeiro et al[28]: The GL upper critical field near Tc of a homogeneous superconductor may be written as Hc2(T ) = Tc − T . (T < Tc) (1) At a temperature T , we take each superconducting re- gion in the sample characterized by a ρ(l) as the source that generates a ∆(l) to produce a magnetic response and displays a local Hc2, provided that T ≤ Tc(l). As discussed, these regions can be in stripe or others forms, but the important point is that they are characterized by a region of fairly constant density ρ(l) that at T ≤ Tc(l) may shield the applied magnetic field. Thus each of such given local region has a local superconducting tempera- ture Tc(l) and will contribute to the upper critical field with a local linear upper critical field H l (T ) near Tc(l) according to the usual GL approach. This is justified be- cause each region has a constant density, like a type II low temperature superconductor, and should posses its own Hc2 that is expected to vanish linearly at Tc(l). Consequently the total contribution of the local su- perconducting regions to the whole sample upper critical field is the sum of all the H l (T )’s, Hc2(T ) = Tc(l)− T Tc(l) (T ). (T < Tc(l) ≤ Tonset(ρm)) (2) WhereW is the total number of superconducting regions, stripes or islands with its local Tc(l) ≥ T . The maxi- mum value of Tc(l) is the pseudogap temperature iden- tified above as the Tonset(ρm). For the LSCO series a coherence length of ξab(0) ≈ 22Å is used, in accordance with the measurements[28]. This value of ξab(0) leads to Hc2(0)=Φ/2πξ (0)=64T. Due to the limitations of the GL approach, we expect the result of this equation to be accurate only near and above the system Tc. Fig.(7a) shows both the Hc2 results of the generalized GL calculations together with the experimental values for underdoped La1.92Sr0.08CuO4 compound. In the cal- culations on this compound, we used a maximum tem- perature of superconducting formation, Tonset ≈ 90K FIG. 7: Experimental points (connected by a thin line) against the reduced temperature t = T/Tc and the calcu- lated curve (thick line) of Hc2 considering inhomogeneous samples with a stripe distribution of local superconducting temperature Tc(l) for ρ = 0.08 and a similar calculation with a Gaussian distribution for ρ = 0.25. (the maximum Tc(l) for this sample) from pseudogap estimates[5] and from our calculations shown in Fig.(5). This is the reason why the calculated curve falls to zero at large values of the reduced temperature t = T/Tc. The measured critical temperature is, by the first derivative of the resistivity, Tc(ρm) = 22.8K. Thus, using no ad- justable parameters, only values taken from experiments, we are able to obtain very reasonable agreement with the Hc2 experimental values and, more importantly, a clear explanation why it does not vanish at Tc: at tem- peratures just above Tc there are some superconducting islands that do not percolate, leading to a finite resis- tivity, but they are still large enough to produce a clear magnetic response. The sum of such local magnetic re- sponses is clearly seen in our experiments and in other Hc2 measurements[29]. The magnetic contributions from non percolated islands above Tc was also measured in the form of an anomalous magnetization[23, 30] for under- doped compounds. Such results were interpreted within the framework of the critical state model on a charge disordered superconductor made up of islands[31], very close to the above approach. Fig.(7b) also shows the data and the calculations on the overdoped La1.75Sr0.25CuO4. Perhaps the PST line or upper pseudogap vanishes at ρm = 0.20[2], what is in agreement with many experiments that indicate more ordered behavior to overdoped than the under- doped compounds[13, 32]. Thus, we considered just a small Gaussian variation in the local charge density which yielded also a small variation in ∆Tc(ρm) ≈ 18%. This calculation, with small fluctuations instead of large stripe like variations, is in agreement with the Fermi liquid behavior of overdoped samples. Accordingly, the vari- ations of ∆Tc(l) are very similar to the compound with ρm = 0.22, showed in the last panel of Fig.(5). As a con- sequence, the calculated Hc2(T ) curve of the overdoped sample falls to zero just 18% above Tc, while the under- doped vanishes at a much larger temperature. VI. CONCLUSION We observed that the measuredHc2(T ) curves for both underdoped and overdoped La2−xSrxCuO4 compounds display several non-conventional features like the positive curvature and non-vanishing values above Tc. The original GL approach to Hc2 near Tc fails to repro- duce these behaviors. However it is possible to describe qualitatively well the observed behavior by a generaliza- tion of the GL theory that takes into account the intrinsic charge inhomogeneities. As an additional step towards an unified description, we assumed that such disorder was originated from a phase separation transition, possi- bly near the upper pseudogap temperature. The calcula- tions were done in connection with the BdG formalism to obtain the distribution of local superconducting temper- ature Tc(l), that is the onset of local pairing amplitude, on clusters with local charge density ρ(l). The measurements yield stronger non-conventional be- havior for the underdoped sample which may be an indi- cation of a larger degree of inhomogeneity, in agreement with other experiments[8, 32]. The different degrees of disorder were taken into account by the phase separa- tion and this unified approach reproduced well the Hc2 results. Thus we conclude that the observed unusual fea- tures associated with Hc2 for both samples are consis- tent with the presence of charge inhomogeneities in the La2−xSrxCuO4, which depending on the value of x or ρm, appear either in the form of stripes or in the form of small fluctuation around the average doping level. J. L. González and A. Lopez acknowledge financial sup- port from Faperj. E. V. L. de Mello is grateful to CNPq for partial financial support. [1] T. Timusk and B. Statt. Rep. Prog. Phys. 62, 61 (1999). [2] J. L. Tallon and J. W. Loram, Physica C 349, 53 (2001). [3] Patrick A. Lee, Naoto Nagaosa, Xiao-Gang Wen, Rev. Mod. 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0704.1426

Thermal Hadronization and Hawking-Unruh Radiation in QCD

rev. 10. 6. 07 BNL-NT-07/18 BI-TP 2007/06 Thermal Hadronization and Hawking-Unruh Radiation in QCD P. Castorinaa, D. Kharzeevb and H. Satzc a) Dipartimento di Fisica, Università di Catania, and INFN Sezione di Catania, Via Santa Sofia 64 I-95123 Catania, Italia b) Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA c) Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany Abstract We conjecture that because of color confinement, the physical vacuum forms an event horizon for quarks and gluons which can be crossed only by quantum tunneling, i.e., through the QCD counterpart of Hawking radiation by black holes. Since such radiation cannot transmit information to the outside, it must be thermal, of a temperature de- termined by the chromodynamic force at the confinement surface, and it must maintain color neutrality. We explore the possibility that the resulting process provides a com- mon mechanism for thermal hadron production in high energy interactions, from e+e− annihilation to heavy ion collisions. Keywords: event horizon, color confinement, Hawking-Unruh radiation, multihadron production PACS: 04.70Dy, 12.38Aw, 12.38Mh, 12.40Ee, 25.75Nq, 97.60Lf http://arxiv.org/abs/0704.1426v2 1 Introduction The aim of this paper is to develop a conceptual framework for a universal form of thermal multihadron production in high energy collisions. Our work is based on two seemingly disjoint observations: • Color confinement in QCD does not allow colored constituents to exist in the physi- cal vacuum, and thus in some sense creates a situation similar to the gravitational confinement provided by black holes. • Numerous high energy collision experiments have provided strong evidence for the thermal nature of multihadron production, indicating a universal hadronization tem- perature TH ≃ 150− 200 MeV. We want to suggest that quantum tunnelling through a color event horizon, as QCD counterpart of Hawking-Unruh radiation from black holes, can relate these observations in a quite natural way. The idea that the color confinement of quarks and gluons in hadrons may have a dual description in terms of a theory in curved space–time is not new. Both gravitational confinement of matter inside a black hole [1] and the de Sitter solution of the Einstein equations with a cosmological constant describing a “closed” universe of constant cur- vature [2] have been proposed as possible descriptions of quark confinement. Soon it became clear that asymptotic freedom [3] and the scale anomaly [4,5] in QCD completely determine the structure of low–energy gluodynamics [6]. This effective theory can be con- veniently formulated in terms of the Einstein–Hilbert action in a curved background. At shorter distances (inside hadrons), the effective action has the form of classical Yang–Mills theory in a curved (but conformally flat) metric [7]. The “cosmological constant” present in this theory corresponds to the non–perturbative energy density of the vacuum, or the “gluon condensate” [8]. It is worthwhile to also mention here the well–known conjectured holographic correspon- dence between the large N limit of supersymmetric Yang–Mills theory in 3+1 dimensions and supergravity in an anti–de Sitter space–time sphere, AdS5 × S5 [9]. This example illustrates the possible deep relation between Yang–Mills theories and gravity; however, a conformal theory clearly differs from the examples noted above, in which the scale anomaly (describing the breaking of conformal invariance by quantum effects) was used as a guiding principle for constructing an effective curved space–time description. Let us assume that color confinement indeed allows a dual description in terms of the gravitational confinement of matter inside black holes. What are the implications of this hypothesis for hadronic physics? Hawking [10] showed that black holes emit thermal radiation due to quantum tunneling through the event horizon. Shortly afterwards, Unruh [11] demonstrated that the presence of an event horizon in accelerating frames also leads to thermal radiation. It was soon conjectured that the periodic motion of quarks in a confining potential [12], or the acceleration which accompanies inelastic hadronic collisions [13–15], are associated with an effective temperature for hadron emission. Recently, a QCD–based picture of thermal production based on the parton description of high energy hadronic collisions has been proposed [16, 17]. The effective temperature T is in this case determined either by the string tension σ, with a relation , (1) or, in the gluon saturation regime, by the saturation momentum Qs describing the strength of the color fields in the colliding hadrons or nuclei, with T ≃ (Qs/2π). Turning now to the second observation, we recall that over the years, hadron production studies in a variety of high energy collision experiments have shown a remarkably universal feature. From e+e− annihilation to p− p and p− p̄ interactions and further to collisions of heavy nuclei, with energies from a few GeV up to the TeV range, the production pattern always shows striking thermal aspects, connected to an apparently quite universal temperature around TH ≃ 150 − 200 MeV [18]. As a specific illustration we recall that the relative abundance of two hadron species a and b, of masses ma and mb, respectively, is essentially determined by the ratio of their Boltzmann factors [19], R(a/b) ∼ exp−{(ma −mb)/TH}. (2) What is the origin of this thermal behaviour? While high energy heavy ion collisions involve large numbers of incident partons and thus could allow invoking some “thermal- isation” scheme through rescattering, in e+e− annihilation the predominant initial state is one energetic qq̄ pair, and the number of hadronic secondaries per unit rapidity is too small to consider statistical averages. The case in p− p/p− p̄ collisions is similar. This enigma has led to the idea that all such collision experiments result in the formation of a strong color field “disturbing” the physical vacuum. The disturbed vacuum then recovers by producing hadrons according to a maximum entropy principle: the actually observed final state is that with the largest phase space volume. While this provides an intuitive basis for a statistical description, it does not account for a universal temperature. Why don’t more energetic collisions result in a higher hadronization temperature? A further piece in this puzzle is the observation that the value of the temperature deter- mined in the mentioned collision studies is quite similar to the confinement/deconfinement transition temperature found in lattice studies of strong interaction thermodynamics [20]. While hadronization in high energy collisions deals with a dynamical situation, the ener- gy loss of fast color charges “traversing” the physical vacuum, lattice QCD addresses the equilibrium thermodynamics of unbound vs. bound color charges. Why should the resulting critical temperatures be similar or even identical? We shall here consider these phenomena as reflections of the QCD counterpart of the Hawking radiation emitted by black holes [10]. These ultimate stellar states provide a gravitational form of confinement and hence, as already noted, their physics was quite soon compared to that of color confinement in QCD [1, 2], where colored constituents are confined to “white holes” (colorless from the outside, but colored inside). It should be emphasized from the outset that in contrast to the original black hole physics in gravitation, where confinement is on a classical level complete, in QCD confinement refers only to color-carrying constituents; thus, e.g., photons or leptons are not affected. In black hole physics, as noted above, it was shown that the event horizon for systems undergoing uniform acceleration leads to quantum tunnelling and hence to thermal radia- tion [11]. Our aim here is to show that such Hawking-Unruh radiation, as obtained in the specific situation of QCD, provides a viable account for the thermal behavior observed in multihadron production by high energy collisions. Furthermore, in the process we also want to elucidate a bit the common origin of the “limiting temperature” concepts which have arisen in strong interaction physics over the years. We begin by reviewing those features of black hole physics and Hawking radiation which are relevant for our considerations, and then discuss how they can be implemented in QCD. In particular, we show that modifications of the effective space-time structure, in a perturbative approach as well as in a non-perturbative treatment based on a large-scale dilaton field, lead to an event horizon in QCD. Following this, we present the main conceptual consequences of our conjecture. • Color confinement and the instability of the physical vacuum under pair produc- tion form an event horizon for quarks, allowing a transition only through quantum tunnelling; this leads to thermal radiation of a temperature TQ determined by the string tension. • Hadron production in high energy collisions occurs through a succession of such tunnelling processes. The resulting cascade is a realization of the same partition process which leads to a limiting temperature in the statistical bootstrap and dual resonance models. • The temperature TQ of QCD Hawking-Unruh radiation can depend only on the baryon number and the angular momentum of the deconfined system. The former could provide a dependence of TQ on the baryon number density, while the angular momentum pattern of the radiation allows a centrality-dependence of TQ and elliptic flow. • In kinetic thermalization, the initial state information is successively lost through collisions, converging to a time-independent equilibrium state. In contrast, the stochastic QCD Hawking radiation is “born in equilibrium”, since quantum tun- nelling a priori does not allow information transfer. 2 Event Horizons in Gravitation and in QCD 2.1 Black holes A black hole is formed as the final stage of a neutron star after gravitational collapse [24]. It has a mass M concentrated in such a small volume that the resulting gravitational field confines all matter and even photons to remain inside the event horizon R of the system: no causal connection with the outside is possible. As a consequence, black holes have three (and only three) observable properties: mass M , charge Q and angular momentum J . This section will address mainly black holes with Q = J = 0; we shall come back to the more general properties in section 4. We use units of h̄ = c = 1. The event horizon appears in a study of the gravitational metric, which in flat space has the form ds2 = g dt2 − g−1 dr2 − r2[dθ2 + sin2 θdφ2], (3) using units where c = 1. The field strength of the interaction is contained in the coefficient g(r), g(r) = , (4) leading back to the Minkowski metric in the large distance limit r → ∞. The vanishing of g(r) specifies the Schwarzschild radius R as event horizon, R = 2GM. (5) It is interesting to note that the mass of a black hole thus grows linearly with R, analogous to the behaviour of the confining potential in strong interactions: M(R) = (2G)−1 R. Classically, a black hole would persist forever and remain forever invisible. On a quan- tum level, however, its constituents (photons, leptons and hadrons) have a non-vanishing chance to escape by tunnelling through the barrier presented by the event horizon. Equiv- alently, we can say that the strong force field at the surface of the black hole can bring vacuum fluctuations on-shell. The resulting Hawking radiation [10] cannot convey any information about the internal state of the black hole; it must be therefore be thermal. For a non-rotating black hole of vanishing charge (denoted as Schwarzschild black hole), the first law of thermodynamics, dM = TdS (6) combined with the area law for the black hole entropy [25], leads to the corresponding radiation temperature TBH = . (8) This temperature is inversely proportional to the mass of the black hole, and since the radiation reduces the mass, the radiation temperature will increase with time, as the black hole evaporates. For black holes of stellar size, however, one finds TBH <∼ 2 × 10 −8 ◦K, which is many orders of magnitude below the 2.7 ◦K cosmic microwave background, and hence not detectable. It is instructive to consider the Schwarzschild radius of a typical hadron, assuming a mass m ∼ 1 GeV: Rhadg ≃ 1.3× 10−38 GeV−1 ≃ 2.7× 10−39 fm. (9) To become a gravitational black hole, the mass of the hadron would thus have to be compressed into a volume more than 10100 times smaller than its actual volume, with a radius of about 1 fm. On the other hand, if instead we increase the interaction strength from gravitation to strong interaction [1], we gain in the resulting “strong” Schwarzschild radius Rhads a factor , (10) where αs is the dimensionless strong coupling and Gm 2 the corresponding dimensionless gravitational coupling for the case in question. This leads to Rhads ≃ ; (11) with the effective value of αs ∼ O(1) we thus get Rhads ∼ O(1) fm.∗ In other words, the confinement radius of a hadron is about the size of its “strong” Schwarzschild radius, so that we could consider quark confinement as the strong interaction version of the gravitational confinement in black holes [1, 2]. We had seen that the mass of a black hole grows linearly with the event horizon, M = (1/2G)R, so that in gravitation 1/2G plays the role of the string tension in strong inter- action physics. The replacement GM2 → αs here leads to σ ≃ m ≃ 0.16 GeV2, (12) if one uses the mentioned effective saturation value αs ≃ 3 [26]. The value of αs ∼ 1 thus gives a reasonable string tension as well as a reasonable radius. 2.2 Quasi-Abelian case The appearence of an event horizon occurs in general relativity through the modification of the underlying space-time structure by the gravitational interaction. Such modifica- tions have also been discussed for other interactions. In particular, it was noted that in electrodynamics, non-linear in-medium effects can lead to photons propagating along geodesics which are not null in Minkowski space-time; this can even lead to photon trap- ping, restricting the motion of photons to a compact region of space [27]. Thus, an effective Lagrangian L(F ) depending on a one-parameter background field, F = FµνF µν , results in a modified metric gµν = ηµνL′ − 4FαµF αν L′′, (13) where the primes indicate first and second derivatives with respect to F . Hence g00 = L′ − 4FL′′ = 0 (14) defines the radius of the compact region of the theory, i.e., the counterpart of a black hole [27]. QCD is an inherently non-linear theory, with the physical vacuum playing the role of a medium [28]. The general structure of the effective Lagrangian in a background field F , compatible with gauge invariance, renormalization group results [3] and trace anomaly, has the unique form [29] LQCD = g2(gF ) µνǫ(gF ). (15) ∗In fact, some studies [26] indicate that at large distances, the strong coupling freezes at αs ≃ 3; in that case the corresponding radius becomes Rhad ≃ 1 fm. Here ǫ(gF ) is the dielectric “constant” of the system in the presence of the background field; the F -dependence of ǫ(F ) effectively turns the QCD vacuum into a non-linear medium. On a one-loop pertubative level we have ǫ(gF ) ≃ 1− β0 , (16) where β0 = (11Nc − 2Nf)/48π2, with Nc and Nf specifying the number of colors and flavors, respectively. Using this form in the formalism of [27] leads to gt changing sign (i.e., gt = 0) at gF ∗ = Λ2 exp{−4π/β0g2}, (17) indicating a possible horizon at r∗ ∼ 1/ gF ∗ [30]. It is clear that this line of argument can at best provide some hints, since we used the lowest order perturbative form of the beta-function, even though at the horizon perturbation theory will presumably break down. Nevertheless, we believe that it suggests the possibility of an event horizon for QCD; the crucial feature is the asymptotic freedom of QCD [3], which leads to ǫ < 1 and allows g00 to vanish even without the external medium effects required in QED. 2.3 Non-Abelian case Indeed, a different and more solid suggestion that in QCD there is an event horizon comes from studying the theory on a curved background. For gluodynamics, such a program is discussed in [7]. Classical gluodynamics is a scale-invariant theory, but quantum fluctu- ations break this invariance, with the trace of the energy-momentum tensor introducing non-perturbative effects, associated with the vacuum energy density ǫvac. It was shown [6] that low-energy theorems can be used to determine the form of the effective Yang-Mills Lagrangian in a curved but conformally flat metric gµν(x) = ηµνe h(x), (18) where the dilaton field h(x) is coupled to the trace of the energy momentum tensor, θµµ. The resulting action has the form eh(∂µh) 2 − 1 (F aµν) 2 + e2h(ǫvac − 1 θµµ|pert). ; (19) here ǫvac is the absolute value of the energy density of the vacuum and mG the dilaton mass; the trace of the energy-momentum tensor has been separated into perturbative and non-perturbative contributions, θµµ = θ µ|pert+ < θµµ >= θµµ|pert − 4ǫvac. (20) The crucial point for our considerations is that the first term of eq. (19) can be written eh(∂µh) 2 = R −g, (21) defining R as the Ricci scalar of the theory. Hence eq. (19) has the structure of an Einstein-Hilbert Lagrangian of gluodynamics in the presence of an effective gravitation, (F aµν) 2 + e2h(ǫvac − 1 θµµ|pert). ; (22) where G is now given by . (23) The relation 1/2G → σ between G and the string tension conjectured above then leads . (24) On the other hand, the string tension is just the energy density of the vacuum times the transverse string area, σ = ǫvacπr2T . (25) Combining relations (24) and (25), we have ≃ 0.4 fm, (26) using mG ≃ 1.5 GeV for the scalar glueball mass. Eq. (26) thus gives us the transverse extension or horizon of the string. From eqs. (24) or (25) we can obtain a further consistency check. Given the glueball mass and the string tension σ ≃ 0.16 GeV2, we find for the vacuum energy density ǫvac ≃ 3 σ m2G ≃ 0.013 GeV4 ≃ 1.7 GeV/fm . (27) This is the value for pure gluodynamics; since the energy density is related to the trace of the energy–momentum tensor by the relation (20), and θµµ = (F aµν) 2 ≃ −b g (F aµν) 2, (28) with the coefficient b = 11Nc−2Nf of the β-function, we can estimate that for three-flavor ǫvacQCD = 11Nc − 2Nf ǫvac = ǫvac ≃ 0.01 GeV4, (29) which is in perfect agreement with the original value of the gluon condensate [8] ≃ 0.012 GeV4; (30) note that ǫvacQCD = 27/32 〈(αs/π)G2〉). 3 Hyperbolic Motion and Hawking-Unruh Radiation In general relativity, the event horizon appeared as consequence of the geometrized grav- itational force, but its occurrence and its role for thermal radiation was soon generalized by Unruh [11]. A system undergoing uniform acceleration a relative to a stationary ob- server eventually reaches a classical turning point and thus encounters an event horizon. Let us recall the resulting hyperbolic motion [34]. A point mass m subject to a constant force F satisfies the equation of motion 1− v2 = F, (31) where v(t) = dx/dt is the velocity, normalized to the speed of light c = 1. This equation is solved by the parametric form through the so-called Rindler coordinates, x = ξ cosh aτ t = ξ sinh aτ, (32) where a = F/m denotes the acceleration in the instantaneous rest frame of m, and τ the proper time, with dτ = 1− v2dt. If we impose the boundary condition that the velocity at t = 0 vanishes, we have ξ = 1/a and x(t=0) = 1/a. The resulting world line is shown in Fig. 1. It corresponds to the mass m coming from x = ∞ at t = −∞ at with a velocity arbitrarily close to that of light, decelerating uniformly until it comes to rest at the classsical turning point xH = −(1/a), t = 0. Subsequently, it accelerates again and returns to x = ∞ at t = ∞, approaching the speed of light. For given a, the light cone originating at a distance xH = 1/a away from the turning point of m defines a space-time region inaccessible to m: no photon in this region can (classically) ever reach m, in much the same way as photons cannot escape from a black hole. Here the acceleration is crucial, of course; if m stops accelerating, it will eventually become visible in the “hidden region”. mass m region hidden event horizon Figure 1: Hyberbolic motion The metric of such a accelerating system becomes in spherical coordinates [31] ds2 = ξ2a2dτ 2 − dξ2 − ξ2 cosh2 aτ(dθ2 + sin2 θdφ2), (33) which we want to compare to the black hole metric (3). Making in the latter the coordinate transformation [32] , (34) where the surface gravity κ is given by , (35) we obtain for r → R the black hole form ds2 = η2κ2dt2 − dη2 −R2(dθ2 + sin2θdφ2). (36) When we compare eqs. (33) and (36), it is evident that the system in uniform acceleration can be mapped onto a spherical black hole, and vice versa, provided we identify the surface gravity κ with the acceleration a. The vacuum through which m travels is, for a stationary observer, empty space. On a quantum level, however, it contains vacuum fluctuations. The accelerating mass m can bring these on-shell, using up a (small) part of its energy, so that for m the vacuum becomes a thermal medium of temperature . (37) Consider such a fluctuation into an e+e− pair, flying apart in opposite directions. One electron is absorbed by the mass m, the other penetrates into the “hidden region” and can never be detected by m (see Fig. 2. Since thus neither an observer on m nor a stationary observer in the hidden region can ever obtain access to full information, each will register the observed radiation as thermal (Einstein-Podolsky-Rosen effect [33]). In other words: the accelerating mass m sees the vacuum as a physical medium of temperature TU , while a stationary observer in the hidden region observes thermal radiation of temperature TU as a consequence of the passing of m. mass m Figure 2: Unruh radiation We here also mention that the entropy in the case of an accelerating system again becomes 1/4 of the event horizon area, as in the black hole case, so that also here the correspondence remains valid [35]. In the case of gravity, we have the force F = ma = G , (38) on a probe of mass m. With R = 2 GM for the (Schwarzschild) black hole radius, we have a = 1/(4 GM) for the acceleration at the event horizon and hence the Unruh temperature (37) leads back to the Hawking temperature of eq. (8). In summary, we note that constant acceleration leads to an event horizon, which can be surpassed only by quantum tunnelling and at the expense of complete information loss, leading to thermal radiation as the only resulting signal. 4 Pair Production and String Breaking In the previous section, we had considered a classical object, the mass m, undergoing accelerated motion in the physical vacuum; because of quantum fluctuations, this vacuum appears to m as a thermal medium of temperature TU . In this section, we shall first address the modifications which arise if the object undergoing accelerated motion is itself a quantum system, so that in the presence of a strong field it becomes unstable under pair production. Next we turn to the specific additional features which come in when the basic constituents are subject to color confinement and can only exist in color neutral bound states. As starting point, we consider two-jet e+e− annihilation at cms energy e+e− → γ∗ → qq̄ → hadrons. (39) The initially produced qq̄ pair flies apart, subject to the constant confining force given by the string tension σ; this results in hyperbolic motion [14] of the type discussed in the previous section. At t = 0, the q and q̄ separate with an initial velocity v0 = p/ p2 +m2, where p ≃ s/2 is the momentum of the primary constituents in the overall cms and m the effective quark mass. We now have to solve Eq. (31) with this situation as boundary condition; the force F = σ, (40) is given by the string tension σ binding the qq̄ system. The solution is x̃ = [1− 1− v0t̃+ t̃2] (41) with x̃ = x/x0 and t̃ = t/x0; here the scale factor 1− v20 γ (42) is the inverse of the acceleration a measured in the overall cms. The velocity becomes v(t) = (v0/2)− t̃ 1− v0t̃+ t̃2 ; (43) it vanishes for t̃∗ = ⇒ t∗ = v0 γ, (44) thus defining x(t∗) = 1− (v20/4) as classical turning point and hence as the classical event horizon measured in the overall cms (see Fig. 3). xx(t )*xQ Figure 3: Classical and quantum horizons in qq̄ separation Eq. (45) allows the q and the q̄ to separate arbitrarily far, provided the pair has enough initial energy; this clearly violates color confinement. Our mistake was to consider the qq̄ system as classical; in quantum field theory, it is not possible to increase the potential energy of a given qq̄ state beyond the threshold value necessary to bring a virtual qq̄ pair on-shell. In QED, the corresponding phenomenon was addressed by Schwinger [36], who showed that in the presence of a constant electric field of strength E the probability of producing an electron-positron pair is given by P (M, E) ∼ exp{−πm2e/eE}, (46) with me denoting the electron mass and e denoting the electric charge. This result is in fact a specific case of the Hawking-Unruh phenomenon, as shown in [16]. In QCD, we expect a similar effect when the string tension exceeds the pair production limit, i.e., σ x > 2m (47) where m specifies the effective quark mass. Beyond this point, any further stretching of the string is expected to produce a qq̄ pair with the probability P (M,σ) ∼ exp{−πm2/σ}, (48) with the string tension σ replacing the electric field strength eE . This string breaking acts like a quantum event horizon xq = 2 m/σ, which becomes operative long before the classical turning point is ever reached (see Fig. 3). Moreover, the resulting allowed separation distance for our qq̄ pair, the color confinement radius xQ, does not depend on the initial energy of the primary quarks. There are some important differences between QCD and QED. In case of the latter, the initial electric charges which lead to the field E can exist independently in the physical vacuum, and the produced pair can be simply ionized into an e+ and an e−. In contrast, neither the primary quark nor the constituents of the qq̄ pair have an independent ex- istence, so that in string breaking color neutrality must be preserved. As a result, the Hawking radiation in QCD must consist of qq̄ pairs, and these can be produced in an in- finite number of different excitation states of increasing mass and degeneracy. Moreover, the qq̄ pair spectrum is itself determined by the strength σ of the field, in contrast to the exponent m2e/E in eq. (46), where the value of E has no relation to the electron mass me. Hadron production in e+e− annihilation is believed to proceed in the form of a self-similar cascade [37, 38]. Initially, we have the separating primary qq̄ pair, γ → [qq̄] (49) where the square brackets indicate color neutrality. When the energy of the resulting color flux tube becomes large enough, a further pair q1q̄1 is excited from the vacuum by two-gluon exchange (see Fig. 4), γ → [q[q̄1q1]q̄]. (50) Although the new pair is at rest in the overall cms, each of its constituents has a transverse momentum kT determined, through the uncertainty relation, by the transverse dimension rT of the flux tube. The slow q̄1 now screens the fast primary q from its original partner q̄, with an analoguous effect for the q1 and the primary antiquark. To estimate the qq̄ separation distance at the point of pair production, we recall that the thickness of the flux tube connecting the qq̄ pair is in string theory given by [41] r2T = 2k + 1 , (51) where K is the string length in units of an intrinsic vibration measure. Lattice studies [43] show that for strings in the range of 1 - 2 fm, the first string excitation dominates, so that we have rT = c0 , (52) with c0 ≃ 1 or slightly larger. Higher excitations lead to a greater thickness and eventually to a divergence (the “roughening” transition). From the uncertainty relation we then have . (53) With this transverse energy is included in eq. (47), we obtain for the pair production separation xQ σxq = 2 m2 + k2T ⇒ xq ≃ m2 + (πσ/2 c20) ≃ ≃ 1 fm, (54) with σ = 0.2 GeV2, m2 ≪ σ, and c0 ≃ 1. Once the new pair is present, we have a color-neutral system qq̄1q1q̄; but since there is a sequence of connecting string potentials qq̄1, q̄1q1 and q1q̄, the primary string is not yet broken. To achieve that, the binding of the new pair has to be overcome, i.e., the q1 has to tunnel through the barrier of the confining potential provided by q̄1, and vice versa. Now the q excerts a longitudinal force on the q̄1, the q̄ on the q1, resulting in a longitudinal acceleration and ordering of q1 and q̄1. When (see Fig. 4) σx(q1q̄1) = 2 m2 + k2T , (55) Figure 4: String breaking through qq̄ pair production the q̄1 reaches its q1q̄1 horizon; on the other hand, when σx(qq̄1) = 2 m2 + k2T , (56) the new flux tube qq̄1 reaches the energy needed to produce a further pair q2q̄2. The q̄2 screens the primary q from the q1 and forms a new flux tube qq̄2. At this point, the original string is broken, and the remaining pair q̄1q2 form a color neutral bound state which is emitted as Hawking radiation in the form of hadrons, with the relative weights of the different states governed by the corresponding Unruh temperature. The resulting pattern is schematically illustrated in Fig. 4. To determine the temperature of the hadronic Hawking radiation, we return to the original pair excitation process. To produce a quark of momentum kT , we have to bring it on-shell and change its velocity from zero to v = kT/(m 2 + k2T ) 1/2 ≃ 1. This has to be achieved in the time of the fluctuation determined by the virtuality of the pair, ∆τ = 1/∆E ≃ 1/2kT . The resulting acceleration thus becomes ≃ 2 kT ≃ 2πσ/c0 ≃ 1 GeV, (57) which leads to ≃ 160− 180 MeV (58) for the hadronic Unruh temperature. It governs the momentum distribution and the relative species abundances of the emitted hadrons. A given step in the evolution of the hadronization cascade of a primary quark or antiquark produced in e+e− annihilation thus involves several distinct phenomena. The color field created by the separating q and q̄ produces a further pair q1q̄1 and then provides an acceleration of the q1, increasing its longitudinal momentum. When it reaches the q1q̄1 confinement horizon, still another pair q2q̄2 is excited; the state q̄1q2 is emitted as a hadron, the q̄2 forms together with the primary q a new flux tube. This pattern thus step by step increases the longitudinal momentum of the “accompanying” q̄i as well as of the emitted hadron. This, together with the energy of the produced pairs, causes a corresponding deceleration of the primary quarks q and q̄, in order to maintain overall energy conservation. In Fig. 5, we show the world lines given by the acceleration q̄i → q̄i+1 (qi → qi+1) and that of formation threshold of the hadrons q̄iqi+1 and the corresponding antiparticles. hadron radiation quark acceleration Figure 5: Quark acceleration and hadronization world lines The energy loss and deceleration of the primary quark q in this self-similar cascade, together with the acceleration of the accompanying partner q̄i from the successive pairs brings q and q̄i closer and closer to each other in momentum, from an initial separation qq̄1 of s/2, until they finally are combined into a hadron and the cascade is ended. The resulting pattern is shown in Fig. 6. hadrons Figure 6: Hadronization in e+e− annihilation The number of emitted hadrons, the multiplicity ν(s), follows quite naturally from the picture presented here. The classical string length, in the absence of quantum pair for- mation, is given by the classical turning point determined in eq. (45). The thickness of a flux tube of such an “overstretched” string is known [41]; from eq. (51) we get R2T = 2k + 1 ln 2K, (59) where K is the string length. From eq. (45) we thus get R2T ≃ s (60) for the flux tube thickness in the case of the classical string length. In parton language, the logarithmic growth of the transverse hadron size is due to parton random walk (”Gribov diffusion” [42]); this phenomenon is responsible for diffraction cone shrinkage in high– energy hadron scattering. Because of pair production, the string breaks whenever it is stretched to the length xq given in eq. (54); its thickness rT at this point is given by eq. (51). The multiplicity can thus be estimated by the ratio of the corresponding classical to quantum transverse flux tube areas, ν(s) ∼ R s, (61) and is found to grow logarithmically with the e+e− annihilation energy, as is observed experimentally over a considerable range. We note here that in our argumentation we have neglected parton evolution, which would cause the emitted radiation (e.g., q̄1q2 in Fig. 4) to start another cascade of the same type. Such evolution effects result eventually in a stronger increase of the multiplicity. The formation of a white hole does not affect the production of hard processes at early times (e.g., multiple jet production), which is responsible for an additional growth of the measured multiplicity. A further effect we have not taken into account here is parton saturation. At sufficiently high energy, stronger color fields can lead to gluon saturation and thus to a higher temper- ature determined by the saturation momentum [16]. The resulting system then expands and hadronizes at the universal temperature determined by the string tension. It interesting to compare the separation of two energetic light quarks, as we have consid- ered here, with that of two static heavy quarks Q and Q̄. From quarkonium studies it is known that 2(MD −mc) ≃ 2(MB −mb) ≃ 1.2 GeV, (62) where MD(MB) and mc(mb) are the masses of open charm (beauty) mesons and of the corresponding charm (beauty) quarks, respectively. The energy needed to separate a heavy QQ̄ pair thus is independent of the mass of the heavy quarks, indicating that the string breaking involved here is really a consequence of the vacuum, through qq̄ pair excitation. With σxQ ≃ 1.2 GeV ⇒ xQ ≃ 1.2 fm (63) we find that the resulting separation threshold for pair excitation agrees well with that found above in eq. (54). Lattice QCD studies lead to similar results. Up to now, we have considered hadron production in e+e− annihilation, in which the virtual photon produces a confined colored qq̄ pair as a “white hole”. Turning now to hadron-hadron collisions, we note that here two incident white holes combine to form a new system of the same kind, as schematically illustrated in Fig. 7. Again the resulting string or strong color field produces a sequence of qq̄ pairs of increasing cms momentum, leading to the well-known multiperipheral hadroproduction cascade shown in Fig. 8. We recall here the comments made above concerning parton evolution and saturation; in hadronic collisions as well, these phenomena will affect the multiplicity, but not the relative abundances. In the case of heavy ion collisions, two new elements enter. The resulting systems could now have an overall baryon number, up to B = 400 or more. To take that into account, we need to consider the counterpart of charged black holes. Furthermore, in heavy ion collisions the resulting hadron production can be studied as function of centrality, and peripheral collisions could lead to an interaction region with an effective overall angular momentum. Hence we will also consider rotating black holes. In the next section, we then summarize the relevant features of black holes with Q 6= 0, J 6= 0. γ h h (a) (b) Figure 7: “White hole” structure in e+e− annihilation (a) and hadronic collisions (b) hadrons Figure 8: Hadronization in hadron-hadron collisions 5 Charged and Rotating Black Holes As mentioned, an outside observer the only characteristics of a black hole are its mass M , its electric charge Q, and its spin or angular momentum J . Hence any further observables, such as the event horizon or the Hawking temperature, must be expressable in terms of these three quantities. The event horizon of a black hole is created by the strong gravitational attraction, which leads to a diverging Schwarzschild metric at a certain value of the spatial extension R. Specifically, the invariant space-time length element ds2 is at the equator given by ds2 = (1− 2GM/R) dt2 − 1− 2GM/R dr2, (64) with r and t for flat space and time coordinates; it is seen to diverge at the Schwarzschild radius RS = 2GM . If the black hole has a net electric charge Q, the resulting Coulomb repulsion will oppose and hence weaken the gravitational attraction; this will in turn modify the event horizon. As a result, the corresponding form (denoted as Reissner- Nordström metric) becomes ds2 = (1− 2GM/R +GQ2/R2) dt2 − 1 1− 2GM/R +GQ2/R2 dr2. (65) For this, the divergence leads to the smaller Reissner-Nordström radius RRN = GM (1 + 1−Q2/GM2), (66) which reduces to the Schwarzschild radius RS for Q = 0. The temperature of the Hawking radiation now becomes [24, 39] TBH(M,Q) = TBH(M, 0) 1−Q2/GM2 1−Q2/GM2 ) 2 ; (67) its functional form is illustrated in Fig. 9. We note that with increasing charge, the Coulomb repulsion weakens the gravitational field at the event horizon and hence de- creases the temperature of the corresponding quantum excitations. As Q2 → GM2, the gravitational force is fully compensated and there is no more Hawking radiation. 0.5 1.0 Q / GM T (M,Q) / T (M,0) BH BH Figure 9: Radiation temperature for a charged black hole In a similar way, the effect of the angular momentum of a rotating black hole can be incorporated. It is now the centripetal force which counteracts the gravitational attraction and hence reduces its strength. The resulting Kerr metric must take into account that in this case the rotational symmetry is reduced to an axial symmetry, and with θ denoting the angle relative to the polar axis θ = 0, it is (at fixed longitude) given by ds2 = 1− 2GMR R2 + j2 cos2 θ dt2 − R 2 + j2 cos2 θ R2 − 2GMR + j2 dr2 − (R2 + j2 cos2 θ) dθ2. (68) The angular momentum of the black hole is here specified by the parameter j = J/M ; for a = 0, we again recover the Schwarzschild case. The general situation is now somewhat more complex, since eq. (68) leads to two different divergence points. The solution RK = GM (1 + 1− j2/(GM)2 ) (69) defines the actual event horizon, corresponding to absolute confinement. But the resulting black hole is now embedded in a larger ellipsoid RE = GM (1 + 1− [j2/(GM)2] cos2 θ), (70) as illustrated in Fig. 10. The two surfaces touch at the poles, and the region between them is denoted as the ergosphere. Unlike the black hole proper, communication between the ergosphere and the outside world is possible. Any object in the ergosphere will, however, suffer from the rotational drag of the rotating black hole and thereby gain momentum. We shall return to this shortly; first, however, we note that the temperature of the Hawking radiation from a rotating black hole becomes TBH(M,J) = TBH(M, 0) 1− j2/(GM)2 1− j2/(GM)2 . (71) For a non-rotating black hole, with j = 0, this also reduces to the Hawking temperature for the Schwarzschild case. ergosphere black hole Figure 10: Geometry of a rotating black hole To illustrate the effect of the ergosphere, imagine radiation from a Schwarzschild black hole emitted radially outward from the event horizon. In the case of a Kerr black hole, such an emission is possible only along the polar axis; for all other values of θ, the momentum of the emitted radiation (even light) will increase due to the rotational drag in the ergosphere. This effect ceases only once the radiation leaves the ergosphere. Since the amount of drag depends on θ, the momentum of the radiation emitted from a rotating black hole, as measured at large distances, will depend on the latitude at which it is emitted and increase from pole to equator. Finally, for completeness, we note that for black holes with both spin and charge (denoted as Kerr-Newman), the event horizon is given by RKN = GM (1 + 1− [Q2/GM2]− [j2/(GM)2]), (72) and the radiation temperature becomes [24, 39] TBH(M,Q, J) = TBH(M, 0, 0) 1− (GQ2 + j2)/(GM)2 1− (GQ2 + j2)/(GM)2 ) 2 + j2/(GM)2 . (73) The decrease of TBH for Q 6= 0, J 6= 0 expresses the fact that both the Coulomb repulsion and the rotational force counteract the gravitational attraction, and if they win, the black hole is dissolved. The dependence of a black hole on its basic properties M,Q, J is very similar to the dependence of a thermodynamic system on a set of thermodynamic observables. The first law of thermodynamics can be written as dE = TdS + φdQ+ ωdJ, (74) expressing the variation of the energy with entropy S, charge Q and spin J ; here φ denotes the electrostatic potential per charge and ω the rotational velocity. The corresponding relation in black hole thermodynamics becomes dM = TBHdSBH + ΦdQ + ΩdJ, (75) where the entropy SBH is defined as the area of the event horizon, SBH = π(R2KN + j . (76) The temperature is given by eq. (73), and 4πQRRN , Ω = specify the electrostatic potential Φ and the rotational velocity Ω. The considerations of this section were for spherical black holes. As seen above, such objects are in fact equivalent to uniformly accelerating systems. An application to actual high energy collisions involves a further assumption. Thermal Hawking-Unruh radiation arises already from a single QQ̄ system, as seen above in the discussion of e+e− anni- hilation. If we treat the systems produced in heavy ion collisions as black holes of an overall baryon number and/or an overall spin, we are assuming that the collision leads to a large-scale collective system, in which each accelerating parton is affected by totality of the other accelerating partons. This assumption clearly goes beyond our event horizon conjecture and, in particular, it need not be correct in order to obtain thermal hadron production. 6 Baryon Density and Angular Momentum 6.1 Vacuum Pressure and Baryon Repulsion We now want to consider the extension of charged black hole physics to color confinement in the case of collective systems with a net baryon number. In eq. (67) we had seen that the reduction of the gravitational attraction by Coulomb repulsion in a charged black hole modifies the event horizon and hence in turn also the temperature of Hawking radiation. The crucial quantity here is the ratio Q2/GM2 of the repulsive overall Coulomb force, Q2/R2, to the attractive overall gravitational force, GM2/R2, at the horizon. In QCD, we have a “white” hole containing colored quarks, confined by chromodynamic forces or, equivalently, by the pressure of the physical vacuum. If the system has a non- vanishing overall baryon number, the baryon-number dependent interaction will also affect the forces at the event horizon. The simplest instance of such a force the repulsion between quarks due to Fermi statistics, but more generally, there will be repulsive effects of the type present in cold dense baryonic matter, such as neutron stars. The resulting pressure will modify the confinement horizon and hence lead to a corresponding modification of the Hawking-Unruh temperature of hadronization. By using the conjectured correspondence between black hole thermodynamics and the thermodynamics of confined color charges, we translate black hole mass, charge and grav- itational constant into white hole energy, net baryon number and string tension, {M,Q,G} ↔ {E,B, 1/2σ}. (78) Hence eq. (67) leads us to the relation TQ(B) = TQ(B = 0) 1− 2σ B2/E2 1− 2σB2/E2 ) 2 ; (79) for the dependence of the hadronization temperature on the ratio of net baryon number B and energy E, with TQ(B = 0) given by eq. (58). Its functional form is the same as that illustrated in Fig. 9. It would be interesting to test the prediction (79) against experimental data; one could identify B with the net baryon number per unit rapidity dNB/dy and E with the total transverse energy per unit rapidity dET/dy. The reduction of the hadronization temper- ature with baryon number could thus occur in two ways. A sufficient decrease of the collision energy, e.g. from peak SPS to AGS energy, will strongly reduce dET/dy, while dNB/dy is not affected as much. This leads to the known decrease of T (µB) with increas- ing µB [44], and it will be interesting to see if the form (79) agrees with the observed behaviour. A second, novel possibility would be to consider hadrochemistry as a func- tion of rapidity. At peak SPS energy, dNB/dy remains essentially constant out to about y = 2, while dET/dy drops by more than a factor of two from y = 0 to y = 2 [45]. A similar behaviour occurs at still lower collision energies. Hence it would seem worthwhile to check if an abundance analysis at large y indeed shows the expected decrease of the hadronization temperature. 6.2 Angular Momentum and Non-Central Collisions The dependence of Hawking radiation on the angular momentum of the emitting sys- tem introduces another interesting aspect for the “white hole evaporation” we have been considering. Consider a nucleus-nucleus collision at non-zero impact parameter b. If the interaction is of collective nature, the resulting interaction system may have some angular momentum orthogonal to the reaction plane (see Fig. 11). In central collisions, this will not be the case, nor for extremely peripheral ones, where one expects essentially just individual nucleon-nucleon collisions without any collective effects. If it possible to consider a kinematic region in which the interacting system does have an overall spin, then the resulting Hawking radiation temperature should be correspondingly reduced, as seen in eq. (71). The effect is not so easily quantified, but simply a reduction Figure 11: Rotating interaction region in non-central AA collision of the hadronization temperature for non-central collisions would quite indicative. Such a reduction could appear only in the temperature determined by the relative abundances, since, as we shall see shortly, the transverse momentum spectra should show modifications due to the role of the ergosphere. We next turn to the momentum spectrum of the Hawking radiation emitted from a ro- tating white hole. As discussed in section 4, such radiation will exhibit an azimuthal asymmetry due to the presence of the ergosphere, which by its rotation will affect the momentum spectrum of any passing object. At the event horizon, the momentum of all radiation is determined by the corresponding Hawking temperature (71); but the passage of the ergosphere adds rotational motion to the emerging radiation and hence increases its momentum. As a result, only radiation emitted directly along the polar axis will have momenta as specified by the Hawking temperature; with increasing latitude θ (see Fig. 12)a, the rotation will increase the radiation momentum up to a maximum value in the equatorial plane. spectators spectators ergosphere Figure 12: Transverse plane view of a non-central AA collision Hawking radiation from a rotating source thus leads for nuclear collisions quite naturally to what in hydrodynamic studies is denoted as elliptic flow. It is interesting to note that both scenarios involve collective effects: while in hydrodynamics, it is assumed that non-central collisions lead to an azimuthally anisotropic pressure gradient, we have here assumed that such collisions lead to an overall angular momentum of the emitting system. Concluding this section we emphasize that the results obtained here for the Hawking temperatures of systems with finite baryon density or with an effective overall spin depend crucially on the assumption of collectivity. If the different nucleon-nucleon interactions in a heavy ion collision do not result in sufficiently collective behavior, the corresponding modifications of Tq do not apply. In the case of black holes with spin, we moreover have no way to relate in a quantitative way centrality and overall spin. Both cases do show, however, that such extensions lead to qualitatively reasonable modifications. 7 Temperature and Acceleration Limits We had seen that the underlying confinement dynamics of high energy hadron collisions and e+e− annihilation led to a limit on the acceleration (or the corresponding deceleration) in the self-similar hadronization cascade - a limit which can be specified in terms of the string tension. In turn, this led to a limiting Unruh hadronization temperature . (80) We emphasize that a Hawking-Unruh temperature as such can a priori have any value; it is the universal limit on the acceleration that leads to a universal temperature for the emitted hadron radiation. In the study of strongly interacting matter, temperature limits are well-known and arise for an ideal gas of different composite constituents (“resonances” or “fireballs” of varying mass M), if the composition law provides a sufficiently fast increase of the degeneracy ρ(M) with M . If the number of states of a constituent of mass M grows exponentially, ρ(M) ∼ M−a exp{bM}, (81) with constants a and b, then the grand canonical partition function for an ideal gas in a volume V Z(T, V ) = (2π)3 dMρ(M) d3p exp{− p2 +M2 /T} , (82) diverges for T > TH ≡ 1/b, (83) so that the Hagedorn temperature TH [18] constitutes an upper limit for the temperature of hadronic matter. In the dual resonance model [46], the resonance composition pattern is governed by linearly rising Regge trajectories, α′M2n = n+ α0, n = 1, 2, ..., (84) in terms of the universal Regge slope α′ ≃ 1 GeV−2 and a constant (of order unity) specifying the family (π, ρ, ...). For an ideal resonance gas in D − 1 space and one time dimension, one then obtains [47] (D + 1), b = 2π , (85) leading to the temperature limit . (86) In string theory, the Regge resonance pattern is replaced by string excitation modes, retaining the same underlying partition structure, with α′ = 1/2πσ relating Regge slope and string tension. Hence we get for the corresponding limiting temperature. For D = 4, this coincides with eq. (1) from [16] and agrees within 20 % with the Unruh temperature (80) determined by the lowest string excitation alone (c0 = 1 in eq. (58)). Prior to the dual resonance model, Hagedorn had determined the level density ρ(M) of fireballs composed of fireballs, requiring the same composition pattern at each level [18]. The resulting bootstrap condition leads to [48] a = 3 b = r0 (2 ln 2− 1) ]−1/3 ≃ r0, (88) where r0 measures the range of the strong interaction. With r0 ≃ 1 fm, we thus get TH ≃ 0.2 GeV for the limiting temperature of hadronic matter. If we identify r0 with the pair production separation xq obtained in eq. (54), we get and hence again agreement with the hadronic Unruh temperature (80). Hadronic matter as an ideal gas of constituents with self-similar composition spectra (“resonances of resonances” or “fireballs of fireballs”) thus leads to an upper limit of the temperature, because the level density of such constituents increases exponentially. What does this have to do with the limiting acceleration found in the qq̄ cascade of e+e− annihilation? To address this problem, it is useful to recall the underlying reason for the exponential increase of the level density in the dual resonance model and the bootstrap model. The common origin in both cases is a classical partition problem, which in its simplest form [40] asks: how many ways ρ(M) are there to partition a given integer M into ordered combinations of integers? As example, we have for M = 4 the partitions 4, 3+1, 1+3, 2+2, 2+1+1, 1+2+1, 1+1+2, 1+1+1+1; thus here ρ(M = 4) = 8 = 2M−1. It can be shown that this is generally valid, so that ρ(M) = exp{M ln 2}. (90) For a “gas of integers”, T0 = 1/ ln 2 would thus become the limiting temperature; the crucial feature in thermodynamics is the exponential increase in the level density due to the equal a priori weights given to all possible partitions. Returning now to the quark cascade in e+e− annihilation, we note that the form we have discussed above is a particular limiting case. We assumed that the color field of the separating qq̄ excites in the first step one new pair from the vacuum; in principle, though with much smaller probability, it can also excite two or more. The same is true at the next step, when the tunnelling produces one further pair: here also, there can be two or more. Thus the e+e− cascade indeed provides a partition problem of the same kind. What remains to be shown are the two specific features of our case: that the dominant decay chain is one where in each step one hadron is produced, which provides the constant deceleration of the primary quark and antiquark. The statistical bootstrap model as well as the dual resonance model lead to self-similar decay cascades, starting from a massive fireball (or resonance), which decays into further fireballs, and so on, until at the end one has light hadrons. In Fig. 13a we illustrate such a cascade for the case where the average number k̄ of constituents per step in the decay (or composition) partition pattern M → M11 +M12 + ... +M1k; M11 → M21 +M22 + ...+M2k; ... (91) is k = 3. In the statistical bootstrap model, k̄ can be can be determined [49]; it is found that the crucial feature here is the power term multiplying the exponential increase in eq. (81). For a < 5/2, the distribution in k is given by F (k) = (ln 2)k−1 (k − 1)! , (92) so that the average becomes k̄ = 1 + 2 ln 2 ≃ 2.4. (93) The dominant decay (∼ 70%) is thus into two constituents, with 24 % three-body and 6 % four-body decays. While in general the fireball mass M could decrease in each step by M/k, i.e., by an amount depending on M , the case a < 5/2 is found to be dominated by one heavy and one soft light hadron, M → M1 + h1; M1 → M2 + h2; ... (94) where hi denotes final hadrons; the pattern is shown in Fig. 13b. Moreover, the three- and four-body decays also lead to one heavy state plus soft light hadrons. The decay thus provides a uniform decrease of the fireball mass by the average hadron mass or transverse energy. (a) (b) Figure 13: Fireball decay patterns We therefore conclude that the hadronization pattern we had obtained for e+e− anni- hilation is indeed also connected to the same partition problem as the one leading to exponential level densities. 8 Stochastic vs. Kinetic Thermalization In statistical mechanics, a basic topic is the evolution of a system of many degrees of freedom from non-equilibrium to equilibrium. Starting from a non-equilibrium initial state of low entropy, the system is assumed to evolve as a function of time through collisions to a time-independent equlibrium state of maximum entropy. In other words, the system loses the information about its initial state through a sequence of collisions and thus becomes thermalized. In this sense, thermalization in heavy ion collisions was studied as the transition from an initial state of two colliding beams of “parallel” partons to a final state in which these partons have locally isostropic distributions. This “kinetic” thermalization requires a sufficient density of constituents, sufficiently large interaction cross sections, and a certain amount of time. From such a point of view, the observation of thermal hadron production in high energy collisions, in particular in e+e− and pp interactions, is a puzzle: how could these systems ever “have reached” thermalization? Already Hagedorn [50] had therefore concluded that the emitted hadrons were “born in equilibrium”. Given an exponentially increasing resonance mass spectrum, it remained unclear why collisions should result in a thermal system. Hawking radiation provides a stochastic rather than kinetic approach to equilibrium, with a randomization essentially provided by the quantum physics of the Einstein-Podolsky- Rosen effect. The barrier to information transfer due the event horizon requires that the resulting radiation states excited from the vacuum are distributed according to max- imum entropy, with a temperature determined by the strength of the “confining” field. The ensemble of all produced hadrons, averaged over all events, then leads to the same equilibrium distribution as obtained in hadronic matter by kinetic equilibration. In the case of a very high energy collision with a high average multiplicity already one event can provide such equilibrium; because of the interruption of information transfer at each of the successive quantum color horizons, there is no phase relation between two successive production steps in a given event. The destruction of memory, which in kinetic equili- bration is achieved through sufficiently many successive collisions, is here automatically provided by the tunnelling process. So the thermal hadronic final state in high energy collisions is not reached through a kinetic process; it is rather provided by successively throwing dice. 9 Conclusions We have shown that quantum tunnelling through the color confinement horizon leads to thermal hadron production in the form of Hawking-Unruh radiation. In particular, this implies: • The radiation temperature TQ is determined by the transverse extension of the color flux tube, giving , (95) in terms of the string tension σ. • The multiplicity ν(s) of the produced hadrons is approximately given by the increase of the flux tube thickness with string length, leading to ν(s) ≃ ln s, (96) where s denotes the cms collision energy. Parton evolution and gluon saturation will, however, increase this, as will early hard production. The universality of the resulting abundances is, however, not affected. • The temperature of Hawking radiation can in general depend on the charge and the angular momentum of the emitting system. The former here provides a baryon- number dependence of the hadronization temperature and predicts a decrease of TQ for sufficiently high baryon density. The latter provides the basis for the possibility of elliptic flow and of a dependence of TQ on the centrality of AA collisions. • The limiting temperature obtained in the statistical bootstrap and the dual reso- nance or string model arises from a self-similar composition pattern leading to an exponentially growing level density. We find that the underlying partition problem also leads to the cascade form obtained for hadron emission in high energy collisions, so that the dynamic and the thermodynamic limits have the same origin. • In statistical QCD, thermal equilibrium is reached kinetically from an initial non- equilibrium state, with memory destruction through successive interactions of the constituents. In high energy collisions, tunnelling prohibits information transfer and hence leads to stochastic production, so that we have a thermal distribution from the outset. We close with a general comment. In astrophysics, Hawking-Unruh radiation has so far never been observed. The thermal hadron spectra in high energy collisions may thus indeed be the first experimental instance of such radiation, though in strong interaction instead of gravitation. Acknowledgements D. K. is grateful to E. Levin and K. Tuchin for useful and stimulating discussions. H. S. thanks M. Kozlov for helpful discussions and a critical reading to the text. The work of D. K. was supported by the U.S. Department of Energy underContract No. DE-AC02- 98CH10886. References [1] E. Recami and P. Castorina, Lett. Nuovo Cim. 15 (1976) 347. [2] A. Salam and J. Strathdee, Phys. Rev. D18 (1978) 4596; see also C.J. Isham, A. Salam, and J. Strathdee, Phys. Rev. D3 (1971) 867. [3] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973). [4] J. R. Ellis, Nucl. Phys. B 22, 478 (1970); R. J. Crewther, Phys. Lett. B 33, 305 (1970); M. S. Chanowitz and J. R. Ellis, Phys. Lett. B 40, 397 (1972); J. Schechter, Phys. Rev. D 21, 3393 (1980). [5] J. C. Collins, A. Duncan and S. D. Joglekar, Phys. Rev. D 16, 438 (1977); N. K. Nielsen, Nucl. Phys. B 120, 212 (1977). [6] A. A. Migdal and M. A. Shifman, Phys. Lett. B 114 (1982) 445. [7] D. Kharzeev, E. Levin and K. Tuchin, Phys. Lett. B 547, 21 (2002); Phys. Rev. D 70, 054005 (2004) [8] M. A. 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C64 (2001) 024901 (heavy ions); P. Braun-Munziger, K. Redlich and J. Stachel, (heavy ions). [20] See e.g., M. Cheng et al., Phys. Rev. D 74 (2006) 054507, for the latest state and references to earlier work. [21] T. D. Lee, Nucl. Phys. B264 (1986) 437. [22] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042. [23] J. Dias de Deus and C. Pajares, hep-ph/0605148 [24] See e.g., Li Zhi Fang and R. Ruffini, Basic Concepts in Relativistic Astrophysics, World Scientific, Singapore 1983. [25] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333. [26] R. Alkofer, C. S. Fischer and F. J. Llanes-Estrada, Phys. Lett. B6111 (2005) 279. [27] M. Novello et al., Phys. Rev. D61 (2000) 045001. [28] T. D. Lee, in Statistical Mechanics of Quarks and Hadrons, H. Satz (Ed.), North Holland Publishing Co., Amsterdam 1980. [29] H. Pagels and E. T. Tomboulis, Nucl. Phys. B143 (1978) 453; P. Castorina and M. Consoli, Phys. Rev. D 35 (1987) 3249; L. B. Abbott, Nucl. Phys. B185 (1981) 189; L. 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Lüscher, G. Münster and P. Weisz, Nucl. Phys. B180 (1981) 1. http://arxiv.org/abs/hep-ph/0605148 http://arxiv.org/abs/nucl-th/0206073 http://arxiv.org/abs/hep-th/0606018 [42] V. N. Gribov, arXiv:hep-ph/0006158. [43] G. Bali, K. Schilling and C. Schlichter, Phys. Rev. D 51 (1995) 5165. [44] F. Becattini et al., Phys. Rev. C 64 (2001) 024901 [45] See e.g., F. Sikler et al. (NA49), Nucl. Phys. A661 (1999) 45c. [46] S. Fubini and G. Veneziano, Nuovo Cim. 64A (1969) 811; K. Bardakci and S. Mandelstam, Phys. Rev. 184 (1969) 1640. [47] K. Huang and S. Weinberg, Phys. Rev. Lett. 25 (1970) 895. [48] W. Nahm, Nucl. Phys. 45 B (1972) 525. [49] S. Frautschi, Phys. Rev. D3 (1971) 2821. [50] R. Hagedorn, Thermodynamics of Strong Interactions, CERN 71-12, 1971. http://arxiv.org/abs/hep-ph/0006158 Introduction Event Horizons in Gravitation and in QCD Black holes Quasi-Abelian case Non-Abelian case Hyperbolic Motion and Hawking-Unruh Radiation Pair Production and String Breaking Charged and Rotating Black Holes Baryon Density and Angular Momentum Vacuum Pressure and Baryon Repulsion Angular Momentum and Non-Central Collisions Temperature and Acceleration Limits Stochastic vs. Kinetic Thermalization Conclusions

0704.1427

Photo-assisted shot noise in Coulomb interacting systems

PHOTO-ASSISTED SHOT NOISE IN COULOMB INTERACTING SYSTEMS A. CRÉPIEUX, M. GUIGOU, A. POPOFF, P. DEVILLARD, AND T. MARTIN Centre de Physique Théorique, CNRS Luminy case 907, 13288 Marseille cedex 9, France We consider the fluctuations of the electrical current (shot noise) in the presence of a voltage time-modulation. For a non-interacting metal, it is known that the derivative of the photo- assisted noise has a staircase behavior. In the presence of Coulomb interactions, we show that the photo-assisted noise presents a more complex profile, in particular for the two following systems: 1) a two-dimensional electron gas in the fractional quantum Hall regime for which we have obtained evenly spaced singularities in the noise derivative, with a spacing related to the filling factor and, 2) a carbon nanotube for which a smoothed staircase in the noise derivative is obtained. 1 Introduction In mesoscopic systems, the measurement of shot noise makes it possible to probe the effective charges which flow in conductors. This has been illustrated experimentally and theoretically when the interaction between electrons is less important 1 or when it is more relevant 2,3. Additional informations can be obtained through the photo-assisted noise when an AC bias is superposed to the DC bias. Experimentally, photo-assisted noise has been measured in diffusive wires, diffusive junctions and quantum point contacts4. For normal metals, the noise derivative displays steps5 at integer values of the ratio ω0/ω, where ω is the AC frequency and ω0 is related to the DC voltage. We naturally expect that this behavior is modified in Coulomb interacting systems. The present work deals with two specific one-dimensional correlated systems: a Hall bar in the fractional quantum Hall regime, for which charge transport occurs via two counter- propagating chiral edges states, and a carbon nanotube to which electrons are injected from a Scanning Tunneling Microscope (STM) tip. 2 Photo-assisted noise in the fractional quantum Hall regime The first system we consider is a two-dimensional electrons gas in the fractional quantum Hall regime which is described by the Hamiltonian H = H0 + HB. The kinetic term H0 = (h̄vF /4π) r=R,L ds∂sφr(t)) 2 describes the right and left moving chiral excitations along the edge states (φR and φL are the bosonic fields), and HB = A(t)Ψ R(t)ΨL(t) + h.c. describes the transfer of quasiparticles from one edge to the other. ΨR(L) = FR(L)e νφR(L)(t)/ 2πa where FR(L) is a Klein factor, a, the short-distance cutoff and ν, the filling factor which characterizes the charge e∗ = νe of the backscattered quasiparticles. The backscattering amplitude between the edge states has a time dependence due to the applied voltage V (t) = V0 + V1 cos(ωt): A(t) = Γ0e dtV (t) = Γ0e iω0t+i sin(ωt) = Γ0 ei(ω0+pω)t , (1) http://arxiv.org/abs/0704.1427v1 where ω0 ≡ νeV0/h̄ and ω1 ≡ νeV1/h̄. We have made an expansion in term of an infinite sum of Bessel functions Jp of order p, which is a signature of photo-assisted processes The backscattering current noise correlator is expressed with the help of the Keldysh contour: S(t, t′) = η=±1〈TK{IB(tη)IB(t′−η) exp(−i K dt1HB(t1))}〉/2 where IB(t) = iνeA(t)Ψ R(t)ΨL(t)− h.c. is the current operator. We are interested in the Poissonian limit only, so in the weak backscattering case, one collects the second order contribution in the tunnel amplitude A(t). The main purpose of this work is to analyze the double Fourier transform of the noise S(Ω1,Ω2) ∝ dt′S(t, t′) exp(iΩ1t+ iΩ2t ′) when both frequencies Ω1 and Ω2 are set to zero. At zero temperature, the shot noise exhibits divergences 7 at each integer value of the ratio ω0/ω which are not physical since they appear in a range of frequencies where the perturbative calculation turns out to be no more valid. At finite temperature, the photo-assisted noise reads: S(0, 0) = (e∗)2Γ20 2π2a2Γ(2ν) )2ν (2π )2ν−1 (ω0 + pω)β ν + i (ω0 + pω)β , (2) where Γ is the Gamma function and β = 1/kBT . −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 Figure 1: Noise derivative in the fractional quantum Hall regime as a function of ω0/ω = νV0/h̄ω for (left) ν = 1 at temperatures: kBT/h̄ω = 0.01 (solid line) and kBT/h̄ω = 0.1 (dashed line) and for (right) ν = 1/3 at temperatures: kBT/h̄ω = 0.05 (solid line) and kBT/h̄ω = 0.15 (dashed line). We take ω1/ω = eV1/h̄ω = 3/2. We have tested the validity of our result by setting ν = 1 which corresponds to a non- interacting system. The derivative of the noise according to the bias voltage exhibits staircase behavior as shown on Fig. 1 (left). Steps occur every time ω0 is an integer multiple of the AC frequency. This is in complete agreement with the results obtained be Lesovik and Levitov for a Fermi liquid 5. For a non-integer value of the filling factor (ν = 1/3), the shot noise derivative exhibits evenly spaced singularities as seen on Fig. 1 (right), which are reminiscent of the tun- neling density of states singularities for Laughlin quasiparticles. The spacing is determined by the quasiparticle charge νe and the ratio of the bias voltage with respect to the AC frequency. Photo-assisted transport can thus be considered as a probe for effective charges at such filling factors, and could be used in the study of more complicated fractions of the quantum Hall effect. 3 Photo-assisted noise in carbon nanotube We consider the following setup: an STM tip close to a carbon nanotube connected to leads at both extremities. A voltage applied between the STM and the nanotube allows electrons to tunnel in the center region of the nanotube. As a result, charge excitations propagate along the nanotube towards the right and left leads. This system is described by the Hamiltonian H = HN +HSTM +HT. The nanotube is a non-chiral Luttinger liquid dx vjδ(x) Kjδ(x)(∂xφjδ) 2 +K−1 (x)(∂xθjδ) , (3) where x is the position along the nanotube, φjδ and θjδ are non-chiral bosonic fields and Kjδ is the Coulomb interactions parameter for each charge/spin, total/relative sectors jδ ∈ {c+, c−, s+, s−}. We take Kc− = Ks+ = Ks− = 1, and we assume that Kc+ depends on position 9 as depicted on Fig. 2a. The velocities satisfy vjδ(x) = vF/Kjδ(x). The electrons in the metallic STM tip are assumed to be non-interacting. For convenience, the electron field cσ in the STM tip can be described in terms of a semi-infinite Luttinger liquid with Coulomb interactions parameters all equal to one. The tunnel Hamiltonian between the STM tip and the nanotube at position x = 0 is HT(t) = rασ Γ(t)Ψ rασ(0, t)cσ(t) + h.c. where r corresponds to the branch index, α to the mode index and σ to the spin. The fermionic fields for electrons in the nanotube and in the STM tip are respectively defined by: Ψrασ(x, t) = Frασ e ikFrx+iqFαx+iϕrασ(x,t)/ 2πa and cσ(t) = fσ e iϕ̃σ(t)/ 2πa where a is the ultraviolet cutoff of the Luttinger liquid model, Frασ and fσ are Klein factors, kF is the Fermi momentum and qF is the momentum mismatch associated with the two modes α. In the presence of a voltage modulation superimposed on the constant DC voltage, V (t) = V0+V1 cos(ωt), the tunnel amplitude becomes Γ(t) = Γ p=−∞ Jp (ω1/ω) exp(i(ω0+pω)t), where Jp the Bessel function on order p, ω0 ≡ eV0/h̄ and ω1 ≡ eV1/h̄. The calculation of noise is thus analogous to the one which applies to the fractional quantum Hall effect, except that here only electrons tunnel in the nanotube. We thus obtain 10: S(0, 0) = 4Γ2e2 cos((ω0 + pω)τ) (1 + ν) arctan + (−bc+)nKc+ arctan 2avF τ a2+(nLKc+)2−(vF τ)2 a2+(nLKc+)2−(vF τ)2 a2+(nLKc+)2 2avF τ a2+(nLKc+)2 +(−bc+)nKc+ ) , (4) where bc+ = (Kc+ − 1)/(Kc+ + 1) is the reflexion coefficient at the nanotube contacts x = ±L/2 and ν = jδ(1/Kjδ +Kjδ)/8. The “standard” way 5 to display the results for photo-assisted transport is to consider the noise derivative as a function of voltage: in particular, this allows to compare the results with the non-interacting case where the noise derivative exhibits a staircase variation. In Fig. 2b, we plot the numerically computed noise derivative as a function of the ratio ω0/ω in the presence of Coulomb interactions (Kc+ = 0.2). For ωL/ω = 0.1 (full line), we are in the limit where the wave packet spatial extension is smaller that the nanotube length. In this regime (except for a small region close to the origin) the vast majority of the voltage scale lies in the regime where ω0 > ωL. The noise derivative differs from the single electron behavior, in the sense that the sharp steps and plateaus expected in this case are absent. Instead, because of Coulomb interactions effects, the noise derivative is smoothed out, but there is a clear reminiscence of the step positions: the slope of dS(x, x; Ω = 0)/dω0 increases abruptly at the location of the steps. We attribute the smoothing to the tunneling density of states on the nanotube which is modified by the Coulomb interactions in the nanotube. For ωL/ω = 1.2 (dashed line), we are in an intermediate regime for which electron wave packets are comparable to the nanotube length. For ωL/ω = 6 (dashed-dotted ������������ ������������ ������������ ������������ ������������ ������������ ������ Nanotube K (x) L / 2 −L / 2 0 1 2 3 4 /ω = 0.1 / ω = 1.2 / ω = 6 Figure 2: a) Picture of the system and spatial variation of the Coulomb interactions parameter Kc+; b) Shot noise derivative in the nanotube for Kc+ = 0.2, ω1/ω = 2, ωc/ω = 100 and several values of ωL/ω. line), we are in the limit where electron wave packets are larger than the nanotube length, and as a consequence, the finite size effects dominate over the Coulomb interactions effects and a stepwise behavior in dS(x, x; Ω = 0)/dω0, which is typical of non-interacting metals, can be identified. 4 Conclusion Photo-assisted noise is affected by Coulomb interactions in one-dimensional systems. In the fractional quantum Hall effect, the photo-assisted noise shows evenly spaced singularities with a spacing related to the filling factor. As a consequence, photo-assisted noise measurement in such a system could be used to extract fractional charge. In carbon nanotube, Coulomb interactions affect the height and shape of the steps in the differential noise when ωL/ω is small. On the contrary, when ωL/ω ≥ 1, finite size effects play an important role and attenuate the Coulomb interactions effects. References 1. M. Reznikov et al., Phys. Rev. Lett. 75, 3340 (1995); A. Kumar et al., ibid. 76, 2778 (1996); G. B. Lesovik, JETP Lett. 70, 208 (1999); M. Büttiker, Phys. Rev. Lett. 65, 2901 (1990); M. Büttiker, Phys. Rev. B 45, 3807 (1992); C.W.J. Beenakker and H. van Houten, ibid. 43, 12066 (1991); T. Martin and R. Landauer, ibid. 45, 1742 (1992). 2. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 72, 724 (1994); C. de C. Chamon et al., Phys. Rev. B 51, 2363 (1995); P. Fendley et al., Phys. Rev. Lett. 75, 2196 (1995). 3. L. Saminadayar et al., Phys. Rev. Lett. 79, 2526 (1997); R. de-Picciotto et al., Nature 389, 162 (1997). 4. R. J. Schoelkopf et al., Phys. Rev. Lett. 80, 2437 (1998); A. A. Kozhevnikov et al., ibid. 84, 3398 (2000); L. H. Reydellet et al., ibid. 90, 176803 (2003). 5. G. B. Lesovik and L. S. Levitov, Phys. Rev. Lett. 72, 538 (1994). 6. M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998) ; G. Platero and R. Aguado, ibid. 395, 1 (2004). 7. A. Crépieux, P. Devillard, and T. Martin, Phys. Rev. B 69, 205302 (2004). 8. R. Egger and A. Gogolin, Eur. Phys. J. B 3, 781 (1998). 9. I. Safi and H. J. Schulz, Phys. Rev. B 52, 17040 (1995); D. L. Maslov and M. Stone, ibid. 52, 5539 (1995); V. V. Ponomarenko, ibid. 52, R8666 (1995). 10. M. Guigou, A. Popoff, T. Martin, and A. Crépieux, cond-mat/0611627. http://arxiv.org/abs/cond-mat/0611627 Introduction Photo-assisted noise in the fractional quantum Hall regime Photo-assisted noise in carbon nanotube Conclusion

0704.1428

A unified projection formalism for the Al-Pd-Mn quasicrystal Xi-approximants and their metadislocations

A unified projection formalism for the Al-Pd-Mn quasicrystal Ξ-approximants and their metadislocations M. Engel and H.-R. Trebin Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany email: [email protected] June 23, 2021 Abstract The approximants ξ, ξ′ and ξ′n of the quasicrystal Al-Mn-Pd display most in- teresting plastic properties as for example phason-induced deformation processes (Klein, H., Audier, M., Boudard, M., de Boissieu, M., Beraha, L., and Duneau, M., 1996, Phil. Mag. A, 73, 309.) or metadislocations (Klein, H., Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82, 3468.). Here we demonstrate that the phases and their deformed or defected states can be described by a simple projection formalism in three-dimensional space - not as usual in four to six dimensions. With the method we can interpret microstructures observed with electron microscopy as phasonic phase boundaries. Furthermore we determine the metadislocations of lowest energy and relate them uniquely to exper- imentally observed ones. Since moving metadislocations in the ξ′-phase can create new phason-planes, we suggest a dislocation induced phase transition from ξ′ to ξ′n. The methods developed in this paper can as well be used for various other complex metallic alloys. 1 Introduction A large number of periodic and quasiperiodic phases have been observed in the ternary Al-Pd-Mn system (Klein, Durand, and Audier 2000). Apart from the stable icosahedral phase (i-phase) there is a stable decagonal phase (d-phase) with 1.2 nm periodicity and a metastable d-phase with 1.6 nm periodicity, found by Tsai et al. (1991). The orthorhombic Ξ-approximants (ξ, ξ′ and ξ′n) of the i-phase form a class of closely related periodic phases and can be viewed as approximants of this 1.6 nm d-phase, as they have the following features in common with the d-phase: (a) They are arrangements of columns of Mackay-type clusters and intermediary atoms (Sun and Hiraga 1996). These clusters consist of about 52 atoms, placed on concentric shells of icosahedral symmetry as shown for the ξ′-phase by Boudard et al. (1996). (b) The columns are in registry, i.e. the compounds also have a layer structure. The clusters contain about 80% of the atoms. Therefore a coarsened structural de- scription makes sense where only the projections of the cluster columns along the column lines are marked. They lead to two-dimensional tilings, which are characteristic for the respective approximant phase. The vertices of the tiles correspond to the projections of the cluster columns. The tilings contain flattend hexagons, which for the ξ-phase are aligned parallel (fig- ure 1(a)), for the ξ′-phase are staggered in two orientations (figure 1(b)). In the ξ′-phase isolated combinations of a pentagon and a banana-shaped nonagon (figure 1(c)) are ob- served, along which the orientation of the hexagons is inverted and which are able to move by flips. Klein et al. (1996) therefore have termed them phason-lines, following the nota- tion of related defects in quasicrystals. In bending experiments these phason-lines order into periodically aligned phason-planes with 1, 2, . . . , n rows of hexagons in between (fig- ure 1(d) and (e)), forming periodic superstructures. Here we propose to name these phases ξ′2-, ξ 3-, . . ., ξ n+1-phases, the reason for the index-shift being given later. Most observed is the ξ′2-phase, which is also known as Ψ-phase (Klein 1997). The ξ′n-phases can be considered striped defect lattices. The stripes, which are the phason-planes, can bend and move, varying their distances (Beraha et al. 1997). Klein et al. (1999) have observed dislocations in the stripe pattern and have called them metadis- locations. They are the characteristic textures of partial dislocations in the basic tiling. In this article we are developing a simplified projection formalism to describe all the Ξ-phases. It is derived from the hyperspace method of the quasiperiodic phases. As a consequence, phasonic degrees of freedom can exist in the Ξ-phases which allow move- ments of the phason-planes. We determine those hyperlattice Burgers vectors which label the partial dislocations of lowest energies and relate them to the Burgers vectors of the metadislocations. Indeed, these are the only ones which are observed. The charming fea- ture of our formalism is that the simplest model is working in three space so that all steps are easily imaginable. 2 Geometrical models for i-Al-Pd-Mn and its approx- imants The atomic positions for a quasicrystal can be described by decorating the lattice points of a periodic hypercrystal with atomic surfaces and marking the points where these are inter- sected by a planar cut space (cut-method). An equivalent method is the strip-method (Katz and Duneau 1986). Approximants are constructed by using inclined cut spaces (Duneau and Audier 1994). For a detailed explanation see also Gratias, Katz, and Quiquandon (1995). 0.78 nm Figure 1: Tilings of four different approximant phases of the 1.6 nm d-phase. (a) Parallel alignment of hexagons in the ξ-phase. (b) Staggered arrangement of hexagons in the ξ′- phase. Two hexagon rows are marked grey. (c) The hexagons H occur in two different orientations. The other tiles, the pentagon P and the nonagon N can only be observed in combination, called phason-line. The edge length of the tiles is: t6D = 0.78 nm. (d) In the ξ2-phase there is one row of hexagons between two phason-lines, also called phason-planes. (e) In the ξ3-phase there are two rows of hexagons between neighbouring phason-planes. The arrangement of hexagons is flipped at a phason-line as highlighted in the figure. 2.1 Sixdimensional cut-method for the i-phase and approximants A six-dimensional hyperspace with orthogonal basis vectors e6D1 , . . ., e 6 of length l 0.645 nm was used by Katz and Gratias (1994) to model the i-phase of Al-Pd-Mn. The hyperspace was decorated with three different atomic surfaces positioned on different nodes in a face-centred lattice, namely • even nodes: n0 = {(z1, . . . , z6), zi ∈ Z | i zi = 0 mod 2}, • odd nodes: n1 = {(z1, . . . , z6), zi ∈ Z | i zi = 1 mod 2}, • even body-centre nodes: bc0 = {(z1 + 12 , . . . , z6 + ), zi ∈ Z | i zi = 0 mod 2}. The cut space is spanned by three vectors, whose components in the above basis are (τ = 1 5 + 1) is the golden mean): a6Di = (1, τ, 0,−1, τ, 0), b6Di = (τ, 0, 1, τ, 0,−1), c6Di = (0, 1, τ, 0,−1, τ). (1) Beraha et al. (1997) applied the cut-method for the three-dimensional atomistic descrip- tion of the phases ξ and ξ′ as approximants of the icosahedral phase. In order to position each single atom correctly, they had to modify the atomic surfaces slightly. It turned out that the centre volumes of the n0 atomic surfaces, which lie again on a six-dimensional face-centred lattice, correspond to the cluster centre atoms. This way the model is reduced to three-dimensional tilings for the cluster positions only. For the ξ′-phase Beraha et al. have derived the vectors that span a unit cell of the inclined cut space. We add the vectors for the inclined cut spaces of the ξ-phase and the ξ′n-phases: a6Dξ = (0, 1, 1,−1, 0, 1), b6Dξ = (5, 1, 1, 1, 1,−1), c6Dξ = (0, 0, 1, 1,−1, 1) (2a) a6Dξ′ = (0, 2, 1,−2, 1, 2), b6Dξ′ = b6Dξ , c6Dξ′ = (0, 0, 1, 1,−1, 1) (2b) a6Dξ′n = (0, 2, 1,−2, 1, 2), b = b6Dξ , c = (0, 0, 2n+ 1, 2n,−2n− 1, 2n) (2c) The b6Dξ -vector is the same for all the phases, because it marks the periodicity in the tenfold direction of the d-phase, which coincides with the column line of the Mackay-type clusters. By projecting in direction of b6Dξ , two-dimensional tilings like those in figure 1 can be obtained. The edge length of the tiles is t6D = 1 τ + 2 l6D = 0.78 nm. It can be calculated by projecting all connection vectors e6Di + e j of neighbouring n0-sites onto the tiling plane. The shortest projections have length t6D. Since the six vectors a6Dξ , c ξ , a ξ′ , c ξ′ , a and c6Dξ′n = 2nc ξ′ − a6Dξ′ + 2a6Dξ lie in a three-dimensional subspace of the six-dimensional hyperspace, a description of the tilings in a three-dimensional hyperspace is possible by the cut-method. The model can be simplified by using a simple cubic lattice in three-space, as will be shown in detail in section 2.4. We can arrive at this conclusion directly after substituting the tiling of hexagons, pentagons and nonagons by a tiling of rhombs. 2.2 Rhombic substitution tiling The new tiles are the thin and thick Penrose rhombs which in general are spanned by the vectors of a regular five-star. The interior angles of the tiles are multiples of 36◦. As shown in figure 2, a hexagon of the original tiling is substituted by a thick rhomb, and a nonagon/pentagon combination is substituted by a combination of a thin and a thick rhomb.1 Therefore we will refer to the thin rhombs as phason-lines. A row of hexagons in staggered orientation is substituted by a row of alternating thick rhombs, and a phason- plane is substituted by a combination of a row of alternating thick rhombs and a row of thin rhombs, which again will be called phason-plane. Hence the number of rows of thick rhombs between two phason-planes in the new tiling for the ξ′n-phase is exactly n. Thus the tilings for the column positions of the Ξ-phases emerge as approximants of the Penrose tiling. However, only rhombs of three orientations show up. To span these only three prongs of the five-star are required. This is another argument why we can apply a projection formalism in a three-dimensional hyperspace which was mentioned above and is elaborated further below. To model the lattices of all Ξ-phases, their phasonic degrees of freedom and their metadislocations we found it suitable to resort to a modified cut and projection method. To our knowledge it has not been applied yet in the literature. It makes use of atomic hypervolumes and requires a short section. 2.3 Atomic hypervolumes for a geometrical description in hyper- space The method will be explained by the example of the well known one-dimensional Fibonacci- chain: The two-dimensional hyperspace is partitioned into equal unit cells, called atomic volumes (in general atomic hypervolumes) as shown in figure 3(a).2 For the construction of an approximant, two different lines are needed: the cut line E (in general cut plane or cut space) and the projection line E (in general projection plane or projection space, also named physical space). Those cells that are cut by E are selected (marked grey). The middle point of those cells is projected onto the projection line.3 The projection leads to two different intervals on E, either a small one (S), when the neighbouring selected cells meet vertically, or a large one (L), when they meet horizontally, forming the tiles of the Fibonacci-chain. 1A similar tiling has been presented by Klein et al. (1996). They used thick rhombs to model the ξ- and ξ′-phase and small hexagons for the phason-lines. However no description in hyperspace was presented, explaining the arrangement of the tiles. 2As a generalisation several overlapping atomic volumes could be used, as long as the number of times a point is covered is constant for the hyperspace. This way the closeness condition (Frenckel, Henley, and Siggia 1986) of the original cut-method is automatically fulfilled. But we will only make use of the method of atomic volumes for the canonical case of a hyper-cubic lattice Zn and the unit cells as atomic volumes. 3Any other choice is possible too, as long as it is the same point for each cell, since a different selection only leads to a global translation of the tiling. 1.26 nm (c) (d) (b)(a) Figure 2: New tilings can be created by substituting the original tiles with thin Penrose rhombs RN, called phason-lines, and thick Penrose rhombs RB. The ξ-phase (a) and the ξ′-phase (b) are built only with the thick rhombs, while for the ξ′2-phase (c) and the ξ phase (d) also the thin rhombs are needed. Between two phason-planes in the ξ′n-phase there are n rows of alternating thick rhombs (marked grey). The new tiles are shown in (e). The edge length of the tiles is: t5D = 1.26 nm. The orientation of E determines the shape of the tiles. In the example of the Fibonacci- chain, it determines the ratio of S to L. In contrast, the orientation of E fixes the arrange- ment of the tiles. Since different approximants of the same quasiperiodic phase are built from the same tiles, but with different arrangement of the tiles, the only difference in the construction is the varying orientation of E . A rational slope of the cut line leads to a periodic tiling, while an irrational slope generates a quasiperiodic tiling. The special case of E and E being identical is the original cut-method as mentioned above. A geometrical restriction on the orientation of E is imposed by the fact, that projected tiles must not overlap. In the example of the Fibonacci-chain this means, that the slope of E must be positive. For a counterexample when it is not fulfilled see figure 3 (b). With this method we now construct the tilings of the cluster projections. In the original cut-method the shape of the atomic surfaces depends on the orientation of the cut space. For the Fibonacci-chain the atomic surfaces are the projections of the unit cells on the orthogonal complement of E . Because we will deal with cut spaces with spatially varying orientation in this paper, we would need a varying shape of the atomic surfaces as well. The method of atomic hypervolumes avoids these problems, as the atomic hypervolumes themselves do not vary. 2.4 Three- and five-dimensional hyperspace for approximants of Al-Pd-Mn Because the rhombic tilings for the Ξ-phases are approximants of the Penrose tiling they can be described in a five-dimensional hyperspace with basis {e5D1 , . . . , e5D5 }, the hyper- cubic lattice Z5 and the unit cells as atomic hypervolumes. The projection plane of the Penrose tiling is spanned by the vectors a5Dp = sin(2π 0 ), sin(2π 1 ), sin(2π 2 ), sin(2π 3 ), sin(2π 4 c5Dp = cos(2π 0 ), cos(2π 1 ), cos(2π 2 ), cos(2π 3 ), cos(2π 4 and the projection matrix is obtained by writing these vectors normalised in the rows: π5D‖ = sin(2π 0 ) sin(2π 1 ) sin(2π 2 ) sin(2π 3 ) sin(2π 4 cos(2π 0 ) cos(2π 1 ) cos(2π 2 ) cos(2π 3 ) cos(2π 4 . (4) The projections f i = π i of the five-dimensional basis vectors give a regular five-star as drawn in figure 4(a). The rhomb tiles are the projection of two-dimensional faces of the five-dimensional unit cells. Each is spanned by two vectors of the five-star. The edge length of the rhomb tiles can be calculated from the edge length of the pentagon/hexagon/nonagon-tiles as shown in figure 2(e): t6D = 2 cos(72◦) t5D = τ t5D. This gives: t5D = 1 τ + 2l6D = 1.26 nm, leading to the length of the basis vectors in the five-dimensional hyperspace: l5D = τ τ + 2 l6D = 1.99 nm. A basis {a5D, c5D} for the cut plane of a periodic approximant can be determined with a basis of the unit cell of the tiling. This is done in figures 4(b)-(e) for various approximant Figure 3: (a) Construction of an approximant of the Fibonacci-chain by the method of atomic volumes. A cut line E and a projection line E are required (see text). (b) If the slope of E is negative, the projected tiles overlap. This case is geometrically not allowed. In the bottom left and top right corner a small part of the quasiperiodic Fibonacci-chain is drawn. (e)(a) (d) f 3f 4 f5 − f 2 4f 1 + f 2 + f 5 f5 − f 2 6f 1 + f 2 + f 5 f5 − f 2 Figure 4: (a) The projected basis vectors f i lie on a regular five-star. The three rhombs of the tilings in figure 2 can be constructed with f 1, f 2 and f 5. Unit cells of approximants and their bases are shown: the ξ-phase (b), the ξ′-phase (c), the ξ′2-phase (d) and the ξ′3-phase (e). phases: a5Dξ = (0, 0, 0, 0, 1), c ξ = (1, 0, 0, 0, 0), (5a) a5Dξ′ = (0,−1, 0, 0, 1), c5Dξ′ = (1, 0, 0, 0, 0), (5b) a5Dξ′n = (0,−1, 0, 0, 1), c = (2n, 1, 0, 0, 1). (5c) We now want to derive the transition matrix T 6D5D from the five- to the six-dimensional hyperspace which maps the basis vectors a5D and c5D on a6D and c6D. The first column of T 6D5D is fixed by the relation c ξ = T ξ . The other columns follow from symmetry considerations. The fivefold rotation of the hyperspaces that fixes the projection plane is for the five- and six-dimensional model: R5D =  0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0  , R6D =  1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 −1 0  . (6) Observe, that R6D does not change the vector b6Dξ . The symmetry operation in the two hyperspace must match: R6DT 6D5D = T 5D. This determines the (i + 1)-th column of T 6D5D via c6Dξ = [R 6D]−i ◦ T 6D5D ◦ [R5D]i c5Dξ (7) resulting in: T 6D5D =  0 0 0 0 0 0 −1 −1 1 1 1 0 −1 −1 1 1 1 0 −1 −1 −1 −1 1 1 0 1 −1 −1 0 1  . (8) After fixing a coordinate system for the atomic hypervolumes in the five-dimensional hyperspace, the approximant phases can appear in five different orientations corresponding to a cyclic permutation of the basis vectors. Each orientation of the phases ξ′ and ξ′n is characterised by the orientation of the thin rhombs. By fixing their orientation, a lower- dimensional hyperspace can be used. Since the third and fourth components of the basis vectors in (5) vanish, a description in a three-dimensional hyperspace with cubic lattice Z3 (lattice parameter l5D = l3D) and the unit cell as atomic volume is possible by omitting these components. The transition matrix is simply: T 5D3D =  1 0 0 0 1 0 0 0 0 0 0 0 0 0 1  . (9) For the Ξ-phases this three-dimensional model leads to the same tilings as the five-dimensional model, but only one orientation of the thin rhombs and two orientations of the thick rhombs can show up. The projection matrix π3D‖ is derived from π ‖ again by omitting the third and fourth components:4 π3D‖ = sin(2π 0 ) sin(2π 1 ) sin(2π 4 cos(2π 0 ) cos(2π 1 ) cos(2π 4 2 + τ −1 2 + τ τ−1 1 . (10) Bases for the cut planes in the three-dimensional model are given by a3Dξ = (0, 0, 1), c ξ = (1, 0, 0), (11a) a3Dξ′ = (0,−1, 1), c3Dξ′ = (1, 0, 0), (11b) a3Dξ′n = (0,−1, 1), c = (2n, 1, 1). (11c) 2.5 Relationships of the hyperspaces So far the three- and five-dimensional hyperspace have been introduced only phenomeno- logically for the description of tilings of the approximant phases. Now we want to show 4This projection is not orthogonal, while the projection π5D‖ is orthogonal. That is a consequence of the omission of two hyperspace dimensions. how they are related to the original six-dimensional hyperspace. This is meant as a clari- fication of hyperspace geometry. Mathematically, the formation of the approximants from the icosahedral quasicrystal proceeds in two steps: (i) The configuration in direction of a fivefold rotation axis b is changed, while the configuration perpendicular to this direction in the plane spanned by a and c is unaltered, leading to a pentagonal or decagonal quasicrystal. This step cannot be described simply by a reorientation of the cutplane. Additionally a relaxation of the atoms in direction of b is necessary as shown by Beraha et al. (1997). (ii) The configuration in the plane spanned by a and c is changed, while the configuration in direction of b is unaltered. This step is accurately described by a reorientation of the cutplane as used in this paper. In step (i) a fivefold symmetry of the icosahedral quasicrystal is conserved. The fivefold rotation R6D (equation (6)) operates trivially on a two-dimensional subspace U1 spanned by (1, 0, 0, 0, 0, 0), (0, 1, 1, 1, 1,−1) and as a true rotation on the four-dimensional orthogo- nal complement U2 spanned by (0, 0, 1, 1,−1, 1), (0,−1, 0, 1,−1,−1), (0,−1,−1, 0, 1,−1), (0, 1,−1,−1, 1, 0). So for quasicrystals and approximants the lattice vector b is projected from U1, and the vectors a and c are projected from U2. The restriction of Z6 onto U2 leads to a non-cubic four-dimensional lattice (the A4 root lattice) that could in principle be used for the construction of the Penrose tiling and its approximant tilings. By adding a one-dimensional complementary space ∆, these tilings can be constructed from the Z5-lattice. This is well-known and the usual way to describe the Penrose lattice. The relation of the five-dimensional hyperspace to the three- dimensional hyperspace is described in section 2.4. Figure 5 summarizes all the hyperspace relationships. 3 Phasonic degrees of freedom Besides standard elastic (phononic) degrees of freedom, quasicrystals have additional de- grees of freedom (Socolar, Lubensky, and Steinhardt 1986), which originate from the fact that the cut space is embedded in hyperspace. Local excitations of these so-called pha- sonic degrees of freedom, correspond to continuous displacements of the cut space along the direction orthogonal to it in hyperspace. The direction of the excitation is understood as the direction of the displacement. The number of the phasonic degrees of freedom for a n-dimensional hyperspace and a d-dimensional cut space is equal to n− d. As will be shown, phasonic degrees of freedom are also possible for approximant phases, although then they are not any more continuous degrees of freedom. For convenience we nonetheless continue using this notation. U1 ⊕ U2 (6D) U1 (2D) U2 (4D) U2 ⊕∆ (5D) 3D Span(b) (1D) Span(a, c) (2D) Span(a, b, c) (3D) T 6D5D T 5D3D hyperspaces physical spaces Figure 5: Relationship of the hyperspaces and the physical spaces. The five-dimensional space is built on top of the four-dimensional space U2 perpendicular to the fivefold two- dimensional plane U1. It is not a (full) subspace of the original six-dimensional space. Solid arrows indicate projections (subspace relationship). Dashed arrows indicate inclusions. T 6D5D |U2 is an inclusion and T 6D5D |∆ = 0 the zero mapping. Figure 6: Local excitation of the phasonic degree of freedom for an approximant of the Fibonacci-chain. The excitation rearranges the sequence of the tiles. 3.1 Restriction to excitable degrees of freedom In the example of the Fibonacci-chain, a local excitation of the phasonic degree of freedom is shown in figure 6. Because of the displacement the cut line E now selects different atomic volumes. In the tiling this leads to a local rearrangement of the tiles, called phasonic jumps. In general the excitations are considered small and distributed over a wide tiling region, so that the orientational deviation from the planar cut space is small. A special case has to be examined separately: If the cut plane of an approximant is parallel to boundaries of atomic volumes, it is possible that a local excitation of phasonic degrees of freedom leads to overlapping tiles. If, for example in the Fibonacci-chain, the cut line of the approximant has slope 0, a local excitation always creates regions of the cut line having positive as well as negative slope. The regions of negative slope lead to overlapping tiles, as shown in figure 3(b) earlier. In such a case, accordingly, local displacements of the cut plane are geometrically not allowed and the phasonic degree of freedom is termed not excitable. With regard to the approximants of the 1.6 nm d-(Al-Pd-Mn) in the five-dimensional model, the phasonic degrees of freedom of the Ξ-phases are not excitable in the directions of e5D3 and e 4 . So for a full description of these phases the three-dimensional hyperspace with basis {e3D1 = e5D1 , e3D2 = e5D2 , e3D3 = e5D5 } and only one phasonic degree of freedom is sufficient. This last phasonic degree of freedom is excitable in the ξ′n-phases, leading to new phenomena. But it is not excitable in the ξ- and the ξ′-phase. (The cut plane of the ξ′- phase, for example, is parallel to the e3D1 -axis). So the phases ξ and ξ ′ behave like normal periodic crystals, having only phononic degrees of freedom. It is a pleasant aspect of the three-dimensional hyperspace that the construction for- malism can easily be visualised. Thus in the next sections we can present the cut plane n1 × n2 n2 (b) n1 × n2 Figure 7: (a) Two different cut planes with normal vectors n1 and n2. They have the line g with direction vector n1 × n2 in common. (b) A transition between the two phases is realised by adjusting the orientation of the cut plane continuously (see text). for phase boundaries and dislocations as two-dimensional curved surfaces. 3.2 Phasonic phase boundaries Different approximant phases are characterised by different orientations of the cut space. As a feature of the phasonic degrees of freedom, a spatially dependent continuous transfor- mation from one orientation of the cut space to another is possible, leading to a phasonic phase boundary. Let two phases be characterised by the normal vectors n1 and n2 of their cut planes E1 and E2 (figure 7(a)). For calculating the cut plane E of the phasonic phase boundary we choose the coordinate system in a way, that the line g = E1 ∩E2 with direction n1×n2 runs through the origin. In the limit far away from g on the side of phase 1 the new cut plane E contains the vector v1 = n1 × (n1 × n2), while on the other side E the vector v2 = n2 × (n1 × n2). By using a transition function f : R→ [0, 1] with properties f(x) = 0, lim f(x) = 1 and f(0) = 1/2 (12) the cut plane for the phase boundary E as shown in figure 7(b) is parametrised by: n1 × n2 ‖n1 × n2‖ f(t)v2 + (1− f(t))v1 ‖f(t)v2 + (1− f(t))v1‖ ; s, t ∈ R . (13) Klein et al. (1996) have taken a transmission electron micrograph of a phasonic phase boundary (figure 8(a)), without interpreting it as such. The phason-lines can be seen as dark contrasts. Phase 1 on the left side is one of the phases ξ(α), 0 ≤ α ≤ 1 with α ≈ 0.8, by which we denote an intermediate phase of a ξ- and a ξ′-approximant. Its cut plane is spanned by aξ(α) = (0,−α, 1) and cξ(α) = (1, 0, 0). Special cases are: ξ(0) = ξ and ξ(1) = ξ′. Phase 2 on the right side is the ξ′2-approximant. The tiling for this phasonic phase boundary from ξ(0.8) to ξ′2 is calculated as explained above and shown in figure 8(b). For the transition function we used: f(x) = 0 , x ≤ 0 arctan(x) , x > 0. In the region of the ξ(0.8)-phase (x < 0) the cut plane is flat, since in the ξ(α)-phases the phasonic degree of freedom is not excitable, just as in the ξ- or the ξ′-phase. In the region of the ξ′2-phase (x > 0) excitations of the phasonic degree of freedom show as bendings of the phason-planes. As a direct consequence of the hyperspace description the orientation of the phasonic phase boundary is completely determined by the involved approximant phases. On the other side, by observing this orientation we can identify the approximant phases, labelled by the parameter α. 4 Dislocations: Energetic consideration In quasicrystals and approximants dislocations do exist and are characterised by a unique Burgers vector b, which now is a vector of the hyperlattice. A dislocation is accompanied by a phonon- and a phason-strain field, due to the phononic part b‖ = π‖b and the phasonic part b⊥ = π⊥b (π⊥ projects on the orthogonal complement of E), respectively, of the Burgers vector. If the phasonic part vanishes, the dislocation is a classical one and can be described without the hyperspace methods. Since the lattice constants of periodic approximants are large, classical dislocations have gigantic Burgers vectors with huge phonon-strains. They are energetically unfavourable and will not be observed. By extending the linear theory of elasticity to hyperspace (approximating the phasonic degree as continuous), the line energy E of a dislocation grows quadratically with increasing lengths ‖b‖‖ and ‖b⊥‖.5 Assuming isotropy in the phononic and phasonic part, E can be expressed as: E = cphon‖b‖‖2 + cphas‖b⊥‖2 + ccoupl‖b‖‖‖b⊥‖. (15) Besides a phononic contribution with elastic constant cphon and a phasonic contribution with cphas, a coupling term is present with ccoupl. For estimating the different contributions in (15) the elastic constants in the approx- imant phases are assumed to be comparable to those of i-Al-Pd-Mn. For the icosahedral phase the phononic elastic constants have been determined by Amazit et al. (1995) from sound-propagation. The phasonic elastic constants have been measured with neutron and x-ray scattering experiments by Letoublon et al. (2001). From the experimental values it 5There is an ongoing discussion whether the elastic energy grows linear (locked state) or quadratically (unlocked state) in the phasonic strain, see the review by Edagawa (2001). However at higher temperatures, when dislocations can form and move, the unlocked state is entropically stabilised. 6In several papers the coupling term is neglected, although its appearance is a consequence of the linear theory of elasticity. Figure 8: Phasonic phase boundary between the phase ξ(0.8) on the left side and the phase ξ′2 on the right side. (a) Transmission electron micrograph. The darker spots correspond to phason-lines (Klein et al. 1996). (b) Calculated tiling for the three-dimensional model (see text). can be concluded, that cphas is a few orders of magnitudes smaller than cphon. Furthermore computer simulations (Koschella et al. 2002) suggest that ccoupl has about the same value as cphas. So the largest contribution to E comes from the phononic part. In the tiling picture this means that it is energetically more favourable to rearrange parts of the tiling than to deform tiles. 5 Dislocations in the ξ′n-phases The construction method for a dislocation in hyperspace has been discussed e.g. by Bohsung and Trebin (1989). A dislocation with Burgers vector b forces upon the cut plane a dis- placement field u with a phononic and a phasonic part. In a first approximation the displacement field can be distributed isotropically in physical space and becomes in spher- ical coordinates: u(r, ϕ) = b. (16) 5.1 Metadislocations in the three-dimensional model The steps involved in the construction of a dislocation are visualised in figure 9. The distorted cut plane has a strain jump of b3D along a radial line as shown in figure 9(a). The middle points of the atomic volumes selected by the cut plane form a staircase surface as shown in figure 9(b). This surface is projected with π3D‖ onto the projection plane (figure 9(c)). Up to now only the phason-strain of the dislocation has been taken regard of and results in a rearrangement of the tiles. The gap in the tiling is caused by the jump line in the cut plane. In a last step the phonon-strain is introduced (figures 9(d), (e)), closing the gap and deforming the tiles. In (figure 9(f)) a Burgers circuit is constructed. The sides of the tiles correspond to projected basis vectors f i of the hyperlattice. In the example the lines A, C and B, D cancel each other, only E is left and is lifted to the Burgers vector b3D = (−2, 1,−3) = −2e3D1 + e3D2 − 3e3D3 . Consider now an arbitrary dislocation with b3D = (b1, b2, b3), bi ∈ Z. Its phononic component b‖ = π τ + 2(b2 − b3) τ−1(b2 + b3) . (17) must be small for low dislocation energy, as shown in the last section. This is achieved, if b2 = b3 and ≈ −τ . The best approximating rational values for τ are given by frac- tions Fm of successive Fibonacci numbers (Fm)m∈N = (1, 1, 2, 3, 5, 8, 13, 21, . . .) defined by Fm+1 = Fm−1 + Fm with start values F1 = F2 = 1. They have the following properties, which can be proved by induction: (i) Fmτ + Fm−1 = τ m (18a) (ii) Fm+1 − Fmτ = (−τ)−m (18b) (iii) Fm = 5(τm − (−τ)−m) (18c) (b) (c) (d) (e) 3f2 3f3 b3D = (−2, 1,−3) Figure 9: Construction of a dislocation in the three-dimensional model. (a) The displaced cut plane E of the ξ′3-phase around the dislocation core. (b) Staircase surface of those atomic volumes, that are cut by E . (c) Projection of the staircase surface onto the projection plane E. (d) The new tiling around the dislocation core with rearrangements of the tiles due to phason-strain. (e) The gap in the tiling is closed by phonon-strain. (f) The Burgers vector b3d can be obtained by performing a Burgers circuit. Therefore the best candidates for Burgers vectors of low energy dislocations are b3D =  Fm−1−Fm  with b‖ = 1 Fm−1 − τ−1Fm and length: ‖b‖‖ = τ−m 10 l3D = τ−m · 1.26 nm. (20) With (8) and (9) we get the 6D Burgers vector: b6D = T 6D5D T 3D = (0, 0,−Fm−2, Fm−1, Fm−2, Fm−1). (21) The phasonic component for the six-dimensional hyperspace can be calculated with (18): ‖b⊥‖ = ‖b6D‖2 − ‖b‖‖2 = 2 [(Fm−1)2 + (Fm−2)2] (l6D)2 − τ−2m(l3D)2 = τm−3 10 l3D = τm−3 · 1.26 nm, (22) yielding the strain accommodation parameter ζ, i.e. the ratio of the phasonic and the phononic component of the Burgers vector (Feuerbacher et al. 1997): = τ 2m−3. (23) Tilings for dislocations with m = 4 and m = 5 are shown in figures 10(a) and (c). The dislocation cores appear as triangles which can be used for Burgers circuits. Phason-planes come in from the left side of the tiling, ending at the flat side of the triangle, their number being equal to that of the tiles there, which is 2Fm. Whereas the phononic part of the Burgers vector is much smaller than the side length l3D of the rhombs, the phasonic part is so large that it leads to a bending of the phason-planes over a huge area of the tiling. The arrangement of the phason-lines can be viewed as a metastructure which, in the simple case of a dislocation free ξ′n-phase is a striped centred rectangular lattice. A par- tial dislocation in the tiling leads to a dislocation in the striped metastructure, named metadislocation. A dislocation with a three-dimensional Burgers vector as in (19) is called metadislocation of type m. In contrary to the dislocation constructed in hyperspace, a metadislocation is an ordinary dislocation in the metastructure with a two-dimensional Burgers vector. The Burgers vector bmeta of a type m metadislocation is oriented in the direction of the lattice vector cξ′n = π . Since a lattice cell contains two phason-planes, a type m metadislocation has Fm inserted rows of lattice cells in this direction, leading to a Burgers vector bmeta = Fmπ 2n+ τ−1 of length ‖bmeta‖ = l3DFm(2n+ τ−1) = Fm(2n+ τ−1) · 1.26 nm. (25) The most frequent dislocations are the type 4 metadislocations in the ξ′2-phase. They have been observed by electron microscopy (Klein et al. 1999), see figure 10(b). From our theory we derive the lengths ‖b‖‖ = 0.184 nm and ‖b⊥‖ = 2.04 nm (ζ = 11.1) for the dislocation and ‖bmeta‖ = 17.5 nm for the metadislocation, which is larger by two orders of magnitude. For comparison we present the metadislocation also in the hexagon/pentagon/nonagon-tiling (figure 10(d)). Transmission electron micrographs of type 3, 5 and 6 metadislocations have been published, too, by Klein and Feuerbacher (2003). 5.2 Metadislocations in the five-dimensional model Other kinds of metadislocations reported by Klein and Feuerbacher (2003) also show a large scale arrangement of phason-lines, but require the five-dimensional hyperspace for description, since phason-lines occur in more than one orientation. Here we will discuss two different types of metadislocations: 1. Metadislocations with Burgers vector of the form: b5D = (Fm,−Fm−1, Fm, 0, 0): The phononic component is: b‖ = π 10(−τ)−m cos(72◦) sin(72◦) . (26) with length ‖b‖‖ = τ−m · 1.26 nm. The Burgers vector is rotated by 72◦ compared to the metadislocation in (19). A tiling of such a metadislocation with m = 3 is shown in figure 11(a). Phason-lines now appear in two different orientations. Fm rows of tilted phason-lines ending at the dislocation core cannot be explained within the three-dimensional model. 2. Metadislocations with Burgers vector of the form: b5D = (0, Fm−1, Fm−1, 0, Fm): The phononic component is: 10(−τ)−m−1 cos(−72◦) sin(−72◦) . (27) with length ‖b‖‖ = τ−m−1 · 1.26 nm. A tiling with m = 5 is shown in figure 11(b). This time Fm−1 rows of tilted phason-lines end at the dislocation core. For transmission electron micrographs of the two discussed metadislocations see fig- ure 11(c) and (d). In the first figure the gap extending to the left in the tiling has been closed by the motion of the phason-planes. In the second figure the metadislocation seems to have moved up, while building a sack-shaped area of ξ′-phase below. On the sides of this area, five phason-planes are cut into halves by a region of ξ′-phase. (c) (d) Figure 10: (a) Tiling of a type 4 metadislocation in the ξ′4-phase. (b) Transmission electron micrograph of a type 4 metadislocation in the ξ′2-phase, courtesy by H. Klein and M. Feuerbacher. (c) Tiling of a type 5 metadislocation in the ξ′4-phase. (d) For comparison the calculated hexagon/pentagon/nonagon tiling of a type 4 metadislocation in the ξ′4- phase is shown. 20b3D‖ 20b3D‖ Figure 11: Tilings of metadislocations in the ξ′1-phase with Burgers vectors (a) b (2,−1, 2, 0, 0) and (b) b5D = (0, 3, 3, 0, 5). The phason-strain field is not distributed isotropic. The phononic component of the Burgers vector is drawn magnified 20 times, since it is too small otherwise. (c) and (d) show transmission electron micrographs of these two metadislocations, courtesy by H. Klein and M. Feuerbacher. (a) (b) Figure 12: (a) A tiling of a metadislocation in the ξ′-phase with six inserted phason-planes. In the dislocation tail a strip of ξ′3-phase is created. (b) Cut plane for the metadislocation in the ξ′-phase. The phason-strain field is not relaxed isotropic, but concentrated in the dislocation tail, where the phason-lines are observed. 6 Metadislocations in the ξ′-phase Although phasonic degrees of freedom are not excitable continuously in the ξ′-phase, metadislocations with phasonic components, which cause a noncontinuous local phason- strain do exist and have been observed experimentally by Klein and Feuerbacher (2003). They can be described in the three-dimensional hyperspace. To minimise the dislocation energy, their Burgers vectors b3D are the same as in the ξ′n-phases (19). For a tiling see figure 12(a) in the case m = 4. Similar to metadislocations in the ξ′n-phases, a tail of six in- serted phason-planes ends at the flat sides of the dislocation core. Towards the dislocation core the phason-planes converge, while far away they run parallel. Contrary to metadislo- cations in the ξ′n-phases the phason-strain now is not distributed isotropically around the dislocation core, but concentrated in the region, where phason-lines are observed. The shape of the cut plane for the metadislocation in the three-dimensional model is shown in figure 12(b), where the coordinate system is rotated, so that the cut plane Eξ′ of the ξ′-phase lies horizontally. Because of its small phononic component, the Burgers vector is almost perpendicular to Eξ′ . We place the dislocation in the origin and the strain jump of b3D along the y-axis, along a line in the middle of the strip of phason lines. Inside the dislocation tail and far away from the core the strain field is u3D(x, y) = |x| − b3D. (28) The parameter λ controls the width of the dislocation tail. To simulate a convergence of the dislocation tail λ = λ(y) must be designed decreasing with distance from the dislocation core. Outside of the dislocation tail, the cut plane is parallel to Eξ′ . The region with the phason lines can be viewed as a very thin strip of newly created ξ′n-phase, embedded in surrounding ξ′-phase. We suggest, that these dislocations move in the direction of their tail, because for the movement perpendicular to it the entire tail would have to be dragged along. The movement is pure climb, because it is perpendicular to the burgers vector and the dislocation line. There are indications (Mompiou, Caillard, and Feuerbacher 2004), that in the i-Al-Pd-Mn quasicrystal dislocations can move by pure climb. Furthermore the most often observed dislocations in i-Al-Pd-Mn have the same burgers vectors as the metadislocations in the Ξ-phases (Rosenfeld et al. 1995). Finally it should be mentioned that Feuerbacher and Caillard (2004) also observed other dislocations in the ξ′-phase with Burgers vectors that do not lie perpendicular to the vector b6Dξ′ . These cannot be pictured as two-dimensional tilings, but require a true three-dimensional description in physical space. 7 Conclusion In this paper we have introduced a simple projection formalism in five-dimensional hyper- space for the description of the i-(Al-Pd-Mn) approximants ξ, ξ′ and ξ′n as two-dimensional tilings. The tilings correspond to the projections of clusters columns. In most cases the formalism can even be restricted to a three-dimensional hyperspace. The tilings are gen- erated by cuts and projections through a hyperspace which is partitioned into atomic hypervolumes. We have shown that phasonic degrees of freedom in form of continuous displacements of the cut space do exist in these phases and can be either excitable or not. They play a fundamental role for phasonic phase boundaries as well as dislocations. In the case of the ξ- and ξ′-phase no phasonic degrees of freedom are excitable, while in the ξ′n-phases there is exactly one excitable phasonic degree of freedom, which is connected to the bending of the phason-planes. Nevertheless metadislocations in the ξ′-phase can exist, creating phason-planes and forming a thin strip of ξ′n-phase in their tails while moving by climb. In this way, a consec- utive motion of metadislocations through the ξ′-phase induces a phase transformation to a ξ′n-phase, making the phasonic degree of freedom excitable. The same metadislocations continue to exist in the ξ′n-phases. It is possible that phasonic degrees of freedom and dislocations with phasonic com- ponents also occur in various other complex metallic alloys, that have connections to quasiperiodic phases. Thus restrictions for ordinary dislocations, that are imposed by the large unit cells, can be overcome. References Amazit, Y., Fischer, M., Perrin, B., and Zarembowitch, A., 1995, Proccedings of the 5th International Conference on Quasicrystals, edited by C. Janot and R. Mossieri (Singapore: World Scientific), p.584. Beraha, L., Duneau, M., Klein, H., and Audier, M., 1997, Phil. Mag. A, 76, 587. Bohsung, J., and Trebin, H.-R., 1989, Introduction to the Mathematics of Quasicrystals, edited by M.V. Jarić (London: Academic Press), p.183. Boudard, M., Klein, H., de Boissieu, M., Audier, M., and Vincent, H., 1996, Phil. Mag. A, 74, 939. Duneau, M., and Audier, M., 1994, Lectures on quasicrystals, edited by F. Hippert and D. Gratias (Les Ulis: Les Editions de Physique), p. 283. Edagawa, K., 2001, Mater. Sci. Eng., A309, 528. Feuerbacher, M., and Caillard, D., 2004, Acta Mater., 52, 1297. Feuerbacher, M., Metzmacher, C., Wollgarten, M., Urban K., Baufeld, B., Bartsch, M., and Messerschmidt, U., 1997, Mater. Sci. Eng., A233, 103. Frenkel, D.M., Henley, C.H., and Siggia, E.D., 1986, Phys. Rev. B, 34, 3649. Gratias, D., Katz, A., and Quiquandon, M., 1995, J. Phys. Cond. Matter, 7, 9101. Katz, A. and Duneau, M., 1986, J. Phys. France, 47, 181. Katz, A. and Gratias, D., 1994, Lectures on quasicrystals, edited by F. Hippert and D. Gratias (Les Ulis: Les Editions de Physique), p. 187. Klein, H., 1997, PhD Thesis, Institut National Polytechnique de Grenoble, France. Klein, H., Audier, M., Boudard, M., de Boissieu, M., Beraha, L., and Duneau, M., 1996, Phil. Mag. A, 73, 309. Klein, H., Durand-Charre, M., and Audier, M., 2000, J. All. Comp., 296, 128. Klein, H., and Feuerbacher, M., 2003, Phil. Mag., 83, 4103. Klein, H., Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82, 3468. Koschella, U., Gähler, F., Roth, J., and Trebin, H.-R., 2002, J. All. Comp., 342, 287. Létoublon, A., de Boissieu, M., Boudard, M., Mancini, L., Gastaldi, J., Hennion, B., Caudron, R., and Bellissent, R., 2001, Phil. Mag. Lett., 81, 273. Mompiou, F., Caillard, D., and Feuerbacher, M., 2004, Phil. Mag., 84, 2777. Rosenfeld, R., Feuerbacher, M., Baufeld, B., Bartsch, M., Wollgarten, M., Hanke, G., Beyss, M., Messerschmidt, U., and Urban, K., 1995, Phil. Mag. Lett., 72, 375. Socolar, J.E.S., Lubensky, T.C., and Steinhardt, P.J., 1986, Phys. Rev. B, 34, 3345. Sun, W. and Hiraga, K., 1996, Phil. Mag. A, 73, 951. Tsai, A.P., Yokoyama, Y., Inoue, A., and Masumoto, T., 1991, J. Mater. Res., 6, 2646. Introduction Geometrical models for i-Al-Pd-Mn and its approximants Sixdimensional cut-method for the i-phase and approximants Rhombic substitution tiling Atomic hypervolumes for a geometrical description in hyperspace Three- and five-dimensional hyperspace for approximants of Al-Pd-Mn Relationships of the hyperspaces Phasonic degrees of freedom Restriction to excitable degrees of freedom Phasonic phase boundaries Dislocations: Energetic consideration Dislocations in the 'n-phases Metadislocations in the three-dimensional model Metadislocations in the five-dimensional model Metadislocations in the '-phase Conclusion References

0704.1429

Light stops in the MSSM parameter space

October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla Modern Physics Letters A c© World Scientific Publishing Company LIGHT STOPS IN THE MSSM PARAMETER SPACE A.V.GLADYSHEV1,2, D.I.KAZAKOV1,2, M.G.PAUCAR1 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, 6 Joliot-Curie, Dubna, Moscow Region, Russian Federation [email protected], [email protected], [email protected] 2Institute of Theoretical and Experimental Physics, 117218, 25 Bolshaya Cheremushkinskaya, Moscow, Russian Federation Received (Day Month Year) Revised (Day Month Year) We consider the regions of the MSSM parameter space where the top squarks become light and even may be the LSP. This happens when the triple scalar coupling A becomes very big compared to m0. We show that in this case the requirement that the LSP is neutral imposes noticeable constraint on the parameter space excluding low m0 and m1/2 similar to constraint from the Higgs mass limit. In some cases these constraints overlap. This picture takes place in a wide region of tanβ. In a narrow band close to the border line the stops are long-lived particles and decay into quarks and neutralino (chargino). The cross-section of their production at LHC via gluon fusion mechanism in this region may reach a few pb. Keywords: Supersymmetry; long-lived particles. PACS Nos.: 12.60.Jv, 14.80.Ly 1. Introduction Preparing for SUSY discovery at LHC one faces the problem of well defined predic- tions since the variety of models and scenarios open up a wide range of possibili- ties 1,2,3,4,5,6. Due to the absence of a SUSY golden mode, one has to explore the parameter space looking for high cross-sections, low background processes, typical missing energy events, etc. All these possibilities are realized within some mechanism of SUSY breaking (mSUGRA and gauge mediation are the most popular) and de- pend on particular choice of a region in SUSY parameter space 7,8,9,10,11,12,13,14. Besides commonly accepted benchmark points which are widely discussed in the literature 15,16,17,18 there still exist some exotic regions in parameter space where unusual relations hold and one can expect interesting phenomena. In particular, in Ref. 19 we considered the so-called coannihilation region in mSUGRA parameter space where one can have long-lived staus which can decay at some distance from the collision point or even fly through detector. This region of m0 − m1/2 plane http://arxiv.org/abs/0704.1429v1 October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla 2 A.V.Gladyshev, D.I.Kazakov, M.G.Paucar exists for all values of tanβ and moves towards higher values of m0 and m1/2 with increase of the latter. However, the area where one can have long-lived particles is very narrow (for each tanβ) and needs severe fine-tuning. Here we explore another region of parameter space which appears only for large negative scalar triple coupling A0 and is distinguished by the light stops. On the border of this region, in full analogy with the stau coannihilation region, the top squark becomes the LSP and near this border one might get the long-lived stops. Below we discuss this possibility in detail and consider also its phenomenological consequences for the LHC. 2. Constraints on the MSSM Parameter Space for Large Negative Values of A In what follows we consider the MSSM with gravity mediated supersymmetry break- ing and the universal soft terms. We thus have the parameter space defined by m0,m1/2, A, tanβ and we take the sign of µ to be positive motivated by contribu- tion to the anomalous magnetic moment of muon 20,21,22,23,24,25,26,27. Imposing the constraints like: i) the gauge couplings unification 28,29,30, ii) neutrality of the LSP 31,32, iii) the Higgs boson and SUSY mass experimental limits 33,34,35, iv) radiative electroweak symmetry breaking, we get the allowed region of parameter space. Projected to the m0 −m1/2 plane this region depends on the values of tanβ and A. In case when A is large enough the squarks of the third generation, and first of all stop, become relatively light. This happens via the see-saw mechanism while diagonalizing the stop mass matrix m̃2tL mt(At − µ cotβ) mt(At − µ cotβ) m̃ where tL = m̃ (4M2W −M Z) cos 2β, tR = m̃ (M2W −M Z) cos 2β. The off-diagonal terms increase with A, become large for large mq (that is why the third generation) and give negative contribution to the lightest squark mass defined by minus sign in eq.(1). m̃21,2 = m̃2tL + m̃ (m̃2tL − m̃ 2 + 4m2t (At − µ cotβ) . (1) Hence, increasing |A| one can make the lightest stop as light as one likes it to be and even make it the LSP. The situation is similar to that with stau for small m0 and large m1/2 when stau becomes the LSP. For squarks it takes place for low m1/2 and low m0. One actually gets the border line where stop becomes the LSP. The region below this line is forbidden. It exists only for large negative A, for small A it is completely ruled out by the LEP Higgs limit. October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla Light Stops in the MSSM Parameter Space 3 100 200 300 400 h0, mass LEP limits [GeV] t c1® ~t® c ~t NLSP LSP ~t 1t® b = - 800 [GeV] mass LEP limits mass LEP limits 200 400 600 800 [GeV] t c1® 1t® b h0, mass LEP limits = - 1500 [GeV] mass LEP limits ~t NLSP Fig. 1. Allowed region of the mSUGRA parameter space for A0 = −800,−1500 GeV and tan β = 10. At the left from the border stau is an LSP, below the border stop is the LSP. The dotted line is the LEP Higgs mass limit. Also shown are the contours where various stop decay modes emerge. It should be noted that in this region one gets not only the light stop, but also the light Higgs, since the radiative correction to the Higgs mass is proportional to the log of the stop mass. The stop mass boundary is close to the Higgs mass one and they may overlap for intermediate values of tanβ. We show the projection of SUSY parameter space to the m0 −m1/2 plane in Figs. 1 and 2 for different values October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla 4 A.V.Gladyshev, D.I.Kazakov, M.G.Paucar 200 400 600 800 1000 1200 m0[GeV] h0 mass LEP limits = - 2500 [GeV] t® ° t c1® ~t® c 1t® b ~t NLSP 400 800 1200 1600 [GeV] ~t® c 1t® b h0 mass LEP limits = - 3500 [GeV] ~t NLSP t c1® Fig. 2. Allowed region of the mSUGRA parameter space for −2500,−3500 GeV and tan β = 10. At the left from the border stau is an LSP, below the border stop is the LSP. The dotted line is the LEP Higgs mass limit. Also shown are the contours where various stop decay modes emerge. of A and fixed tanβ. To calculate it we use the ISAJET v.7.64 code 36. One can see that when |A| decreases the border line moves down and finally disappears. On the contrary, increasing |A| one gets larger forbidden area and the value of the stop mass at the border increases. Changing tanβ one does not influence the stop border line, the only effect is the October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla Light Stops in the MSSM Parameter Space 5 300 600 900 1200 1500 [GeV] m = 172.7 [GeV] A = -3500 [GeV] Tanb=10 Tanb=20 Tanb=30 Tanb=40 Fig. 3. Stau and stop constraints in the m0 − m1/2 plane for A0 = −3500 GeV and different values of tanβ. shift of stau border line. It moves to the right with increase of tanβ as shown in Fig.3, so that the whole forbidden area increases and covers the left bottom corner of the m0 −m1/2 plane. It should be mentioned that the region near the border line is very sensitive to the SM parameters, a minor shift in αs or mt and mb leads to noticeable change of spectrum as can be seen from comparison of different codes at 37,38,39. The other constraint that is of interest in this region is the relic density one. Given the amount of the Dark matter from WMAP experiment 40,41 one is left with a narrow band of allowed region which goes along the stau border line, then along the Higgs limit line and then along the radiative symmetry breaking line. In the case of light stop when it is almost degenerate with the lightest chargino and neutralino, when calculating the relic density one has to take into account not only the annihilation of two stops, but also the coannihilation diagrams. There are two side processes: stop chargino annihilation and stop neutralino annihilation. Calculating the relic density with the help of MicrOmegas package 42,43 one finds that again it is very sensitive to the input parameters, however, since the stop border line is very close to the Higgs one, the relic density constraint may be met here fitting A and/or tanβ. October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla 6 A.V.Gladyshev, D.I.Kazakov, M.G.Paucar 3. Phenomenological Consequences of the Light Stop Scenario The phenomenology of the discussed scenario is of great interest at the moment since the first physics results of the coming LHC are expected in the nearest future. Light stops could be produced already during first months of its operation 44,45. The main diagrams of pair stop production (as well as other type of squarks) are presented in Fig. 4. A single stop production via weak interactions is also possible. Fig. 4. Main stop production diagrams at LHC. Since stops are relatively light in our scenario, the production cross sections are quite large and may achieve tens or even hundreds of pb for mt̃ < 150 GeV. The cross sections and their dependence on the stop mass for different values of |A| are shown in Fig. 5. As one expects they quickly fall down when the mass of stop is increased. The range of each curve corresponds to the region in the (m0 − m1/2) plane where the light stop is the next-to-lightest SUSY particle, and the Higgs and chargino mass limits are satisfied as well. One may notice, that even for very large values of |A| when stops become heavier than several hundreds GeV, the cross sections are of order of few per cent of pb, which is still enough for detection with the high LHC luminosity. Being created the stop decay. There are several different decay modes depending on the stop mass. If stop is heavy enough it decays to the bottom quark and the light- est chargino (t̃ → bχ̃± ). However, for large values of |A0|, namely A0 < −1500 GeV the region where this decay takes place is getting smaller and even disappear due to mass inequality mt̃ < mb + mχ̃± (see right bottom corner in Fig. 2). In this case the dominant decay mode is the decay to the top quark and the lightest neu- tralino (t̃ → tχ̃01). Light stop decays to the charm quark and the lightest neutralino (t̃ → cχ̃0 ) 46. The latter decay, though it is loop-suppressed, has the branching ratio 100 %. In Fig. 6 we show different allowed decay modes for different values of |A0| = 800, 1500, 2500, 3500 GeV as functions of the m1/2 parameter. The values of m0 were chosen in the middle of allowed regions in Figs 1 and 2, namely m0 = 250, 450, 650, 1000 GeV. One can see that for small values of m1/2 we are very close October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla Light Stops in the MSSM Parameter Space 7 200 400 600 800 1000 1200 [GeV] m>0 = -800 [GeV] = -1500 [GeV] = -2500 [GeV] = -3500 [GeV] Fig. 5. Cross sections of the pair stop production as a function of the stop mass. Different curves correspond to different values of A0 parameter (A0 = −800,−1500,−2500,−3500 GeV). to the neutralino–stop border line and the only allowed decay mode is t̃ → cχ̃01. With the increase of the m1/2 the stop mass becomes larger which opens the possibility of new decay modes (t̃ → bχ̃± , and later t̃ → tχ̃0 The bottom part of Fig. 6 shows the stop lifetimes for different values of |A0|. The biggest lifetime corresponds to the t̃ → cχ̃0 decay. Breaks on the curves correspond to switching on the new decay mode. As one can see the lifetime cold be quite large in a wide area of the A0−m1/2 parameter space, even for heavy stops if |A0| is very 4. Conclusion In this letter we have demonstrated that there is a possibility of stop next-to-lightest supersymmetric particle. For large negative values of the trilinear soft supersym- metry breaking parameter A0 there exist a narrow band along the line mt̃ = mχ̃0 in the m0 − m1/2 plane where the cross section of stop pair production is quite large at the LHC energy and stops have relatively large lifetime. This may give interesting signatures, like secondary vertices inside the detector, or even escaping the detector. Another interesting possibility is a formation of so-called R-hadrons (bound states of supersymmetric particles). This may happen if stops live longer than hadronisation time. October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla 8 A.V.Gladyshev, D.I.Kazakov, M.G.Paucar -4000 -3000 -2000 -1000 1/2 [G t ~ [1x10-13, 1x10-18, 1x10-25] [sec] t ~ [1x10-18, 1x10-24] [sec] t ~ [1x10-24, 1x10-25] [sec] LSP~t ~t® c t c1® 1t® b 250 500 750 1000 2500 = -800 [Gev] = -1500 [Gev] = -2500 [Gev] = -3500 [Gev] [GeV] 1X10-13 1X10-15 1X10-17 1X10-19 1X10-21 1X10-23 Fig. 6. Different decay modes of stops and corresponding lifetimes. Experimental Higgs and chargino mass limits as well as WMAP relic density limit can be easily satisfied in our scenario. However, the strong fine-tuning is re- quired. Moreover, it is worth mentioning that light stops are favoured by the baryon asymmetry of the Universe. October 30, 2018 14:33 WSPC/INSTRUCTION FILE lightstops˙mpla Light Stops in the MSSM Parameter Space 9 Acknowledgements Financial support from RFBR grant # 05-02-17603 and grant of the Ministry of Education and Science of the Russian Federation # 5362.2006.2 is kindly acknowl- edged. References 1. H.P. Nilles, Phys. Rept. 110, 1 (1984). 2. H.E. Haber and G.L. Kane, Phys. Rept. 117, 75 (1985). 3. H.E. Haber, Introductory Low-Energy Supersymmetry, Lectures given at TASI 1992, (SCIPP 92/33, 1993), hep-ph/9306207. 4. D.I. Kazakov, Beyond the Standard Model, Lectures at the European School of High- Energy Physics 2004, hep-ph/0411064. 5. A.V. Gladyshev and D.I. Kazakov, Yadernaya Fizika 70, No.7, 1 (2007). 6. A.V. Gladyshev and D.I. Kazakov, Supersymmetry and LHC, Lectures given at 9th International Moscow School of Physics and 34th ITEP Winter School of Physics, hep-ph/0606288 7. L.E. 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0704.1430

The blue plume population in dwarf spheroidal galaxies: genuine blue stragglers or young stellar population?

Astronomy & Astrophysics manuscript no. aa c© ESO 2019 August 20, 2019 The blue plume population in dwarf spheroidal galaxies: genuine blue stragglers or young stellar population?⋆ Y. Momany1, E.V. Held1, I. Saviane2, S. Zaggia1, L. Rizzi3, and M. Gullieuszik1 1 INAF: Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy e-mail: yazan.almomany,enrico.held,marco.gullieuszik,[email protected] 2 European Southern Observatory, A. de Cordova 3107, Santiago, Chile e-mail: [email protected] 3 Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA e-mail: [email protected] Received December 23, 2006; accepted April 3, 2007 ABSTRACT Aims. Blue stragglers (BSS) in the Milky Way field and globular/open clusters are thought to be the product of either primordial or collisional binary systems. In the context of dwarf spheroidal galaxies it is hard to firmly disentangle a genuine BSS population from young main sequence (MS) stars tracing a ∼ 1 − 2 Gyr old star forming episode. Methods. Assuming that their blue plume populations are made of BSS, we estimate the BSS frequency (FBSSHB ; as normalized to the horizontal branch star counts) for 8 Local Group non star-forming dwarf galaxies, using a compilation of ground and space based photometry. Results. (i) The BSS frequency in dwarf galaxies, at any given MV , is always higher than that in globular clusters of similar lumi- nosities; (ii) the BSS frequency for the lowest luminosity dwarf galaxies is in excellent agreement with that observed in the Milky Way halo and open clusters; and most interestingly (iii) derive a statistically significant FBSSHB −MV anti-correlation for dwarf galaxies, similar to that observed in globular clusters. Conclusions. The low density, almost collision-less, environments of our dwarf galaxy sample allow us to infer (i) their very low dynamical evolution; (ii) a negligible production of collisional BSS; and consequently (iii) that their blue plumes are mainly made of primordial binaries. The dwarf galaxies FBSSHB −MV anti-correlation can be used as a discriminator: galaxies obeying the anti-correlation are more likely to possess genuine primordial BSS rather than young main sequence stars. Key words. Galaxies: dwarf – globular clusters: general – blue stragglers – stars: evolution 1. Introduction In the context of Galactic Globular clusters studies, blue strag- glers (BSS: a hotter and bluer extension of normal main se- quence stars) represent the highest manifestation of the inter- play between stellar evolution and stellar dynamics (Meylan & Heggie 1997). The origin of BSS is sought as either primordial binaries coeval with the cluster formation epoch, or to a continu- ous production (in successive epochs) of collisional binaries due to dynamical collisions/encounters experienced by single/binary stars throughout the life of the cluster. Ever since their identifi- cation in (Sandage 1953), BSS have been subject of many pho- tometric and a handful of spectroscopic (Ferraro et al. 2006, and references therein) studies. Nevertheless, BSS remain quite dif- ficult to understand in the light of a single comprehensive sce- nario. Indeed, it is sometimes necessary to invoke both the pri- mordial and collisional mechanisms to explain the BSS distribu- tion in the same cluster. For example the bimodal distribution of BSS in 47Tuc (Ferraro et al. 2004; highly peaked in the cluster center, rapidly decreasing at intermediate radii, and finally ris- ing again at larger radii) can be interpreted as evidence of two formation scenarios at work: primordial binaries at the cluster Send offprint requests to: Y. Momany ⋆ Based on archival ESO and HST data. periphery (where it is easier for them to survive) and collisional binaries at the center (where it is easier to form). A spectroscopic survey by Preston & Sneden (2000) of Milky Way field blue metal-poor stars suggested that over 60% of their sample is made up by binaries, and that at least 50% of their blue metal-poor sample are BSS. Piotto et al. (2004) pre- sented a homogeneous compilation of ∼ 3000 BSS (based on HST observations of 56 globular cluster), and derived a signif- icant and rather puzzling anti-correlation between the BSS spe- cific frequency and the cluster total absolute luminosity (mass). That is to say that more massive clusters are surprisingly BSS deficient, as if their higher collision rate had no correlation with the production of collisional BSS. Another puzzling observable is that the BSS frequency in Milky Way (MW) field is at least an order of magnitude larger than that of globular clusters. Recently De Marchi et al. (2006) presented a photometric compilation for Galactic open clusters with −6 ≤ MV ≤ −3, and confirmed an extension of the BSS frequency-MV anti-correlation to the open clusters regime. In an attempt to explain these observa- tional trends, Davies et al. (2004) envisage that while the num- ber of BSS produced via collisions tends to increase with clus- ter mass, becoming the dominant formation channel for clus- ters with MV ≤ −8.8, the BSS number originating from pri- mordial binaries should decrease with increasing cluster mass. Accounting for these two opposite trends and binary evolution, http://arxiv.org/abs/0704.1430v1 2 Y. Momany et al.: The blue plume population in dwarf spheroidal galaxies: Davies et al. (2004) models are able to re-produce the observed BSS population, whose total number seems independent of the cluster mass. Color-magnitude diagrams (CMD) of typically old dwarf spheroidal galaxies like Ursa Minor (Feltzing et al. 1999 and Wyse et al. 2002) show the presence of a well-separated blue plume of stars that very much resembles an old BSS population, as that observed in globular and open clusters. However, in the context of dwarf galaxies one cannot exclude that blue plume stars may include genuinely young main sequence (MS) stars, i.e. a residual star forming activity (e.g. Held 2005, and refer- ences therein). The BSS-young MS ambiguity is hard to resolve, and has been discussed before for Carina (Hurley-Keller et al. 1998), Draco (Aparicio et al. 2001), and Ursa Minor (Carrera et al. 2002). In order to investigate this ambiguity, in this paper we measure the BSS frequency in the dwarf spheroidal galaxy Leo II and collect BSS counts in 8 other galaxies. Dwarf spheroidals/irregulars in which there is current or recent (≤ 500 Myr) star formation are not considered. For example, the Fornax dwarf is known to possess a young population of ∼ 200 Myr (e.g. Saviane et al. 2000) with young MS stars brighter than the horizontal branch (HB) level. Therefore Fornax cannot be con- sidered in this paper. The Sagittarius dwarf spheroidal, on the other hand, shows an extended blue plume, yet it does not ex- ceed the HB level, and is included in the study. We also consider the case of the Carina dwarf and re-derive its BSS frequency. As discussed in Hurley-Keller et al. (1998) and Monelli et al. (2003), Carina shows evidence of star formation in recent epochs (∼ 1 Gyr). However, Carina represents the first case in which the MS-BSS ambiguity in dwarf galaxies has been addressed: a BSS frequency was derived and compared with that in Galactic glob- ular clusters. It is also a case in which the “youngest MS stars” do not exceed the HB luminosity level; i.e. Carina does not pos- sess a significant recent star formation rate. As we shall see in Sect. 3.1, the Carina BSS frequency will be most useful for com- parison with other dwarf galaxies of similar luminosities. The BSS frequency is therefore collected for Sagittarius, Sculptor, Leo II, Sextans, Ursa Minor, Draco, Carina, Ursa Major and Boötes and these compared internally, and externally (with that reported for Galactic halo, open and globular clusters). This allows us to address the dependence of the BSS frequency on environment from a wider perspective. 2. BSS frequency data points The dwarf galaxies we study in this paper span a large range of distances (∼ 25 Kpc for Sagittarius to ∼ 200 Kpc for Leo II). This basically precludes the availability of one homogeneous and large-area imaging data-set reaching 1 − 2 magnitudes be- low the old MS turn-offs (see however the recent HST/WFPC2 archival survey of dwarf galaxies by Holtzman et al. 2006). Thus estimating the BSS frequency for dwarf galaxies, unfor- tunately, must rely on a compilation from various sources. We present new reductions of archival imaging from ESO/NTT and HST/WFPC2 for Leo II, and ESO/2.2m BVI Pre-Flames 1◦× 1◦ WFI mosaic for Sagittarius1. These were reduced and calibrated following the standard recipes in Held et al. (1999) and Momany et al. (2001, 2002 and 2005). For the remaining dwarf galaxies we estimate the BSS frequency from either public photometric catalogs (Sextans by Lee et al.2003) or published photometry kindly provided by the authors (Ursa Minor by Carrera et al. 1 Excluding the inner 14.′ × 14.′ region around M54. Fig. 1. Left panels display the HST CMDs of Leo II, upon which we highlight the BSS and HB selection boxes and 2.5, 1.2 and 1 Gyr, [Fe/H]= −1.3 isochrones from Girardi et al. (2002). Right upper panel displays the ESO/NTT diagram of Leo II along with 1.0 and 0.8 Gyr isochrones highlighting the extension of the ver- tical clump sequence (dashed lines mark the MS phase while continuous lines track the post-MS evolution). Also plotted are the RR Lyrae stars from Siegel & Majewski (2000) and one (as- terisk) of the 4 anomalous Cepheids (the remaining 3 are outside the NTT field). Lower right panel displays the ESO/2.2m CMD of Sagittarius highlighting the Galactic contamination and the extension of its blue plume. 2002, Draco by Aparicio et al. 2001, Sculptor by Rizzi et al. 2003, Ursa Major by Willman et al. 2005, Boötes by Belokurov et al. 2006 and Carina by Monelli et al. 2003). All the photomet- ric catalogs extend to and beyond the galaxy half light radius; i.e. we cover a significant fraction of the galaxies and therefore the estimated BSS frequency should not be affected by specific spatial gradients, if present. The only exception is that relative to Sagittarius. With a core radius of ∼ 3.7◦, the estimated BSS fre- quency of our 1◦ square degree field refers to less than 3.5% areal coverage of Sagittarius, or a conservative ∼ 6% of the stellar populations. Nevertheless, our Sagittarius catalog is one among very few wide-field available catalogs of Sagittarius that reach the BSS magnitude level with an appropriate completeness level, and it is worthwhile to employ it in this BSS analysis. A delicate aspect of performing star counts is estimating the Galactic foreground/background contribution in the covered area. To compute it in a homogeneous way we made use of the Trilegal code (Girardi et al. 2005) that provides synthetic stel- lar photometry of the Milky Way components (disk, halo, and bulge). Star counts were performed on the simulated diagrams (using the same selection boxes) and subtracted from the ob- served HB and BSS star counts. We calculate the specific fre- quency of BSS (normalizing the number of BSS to the HB) as: FBSSHB = log(NBSS/NHB). We remind the reader that uncertain- Y. Momany et al.: The blue plume population in dwarf spheroidal galaxies: 3 ties in the (i) photometric incompleteness correction, (ii) fore- ground/background subtraction, (iii) possible overlap between old and intermediate age stellar population around the HB level, and (iv) confusion between BSS and normal MS stars, are all unavoidable problems and affect the present and similar stud- ies. The reported error bars account for the propagation of the Poisson errors on the star counts, but mostly reflect the depen- dence on the uncertainty in properly defining the HB and BSS selection boxes. 2.1. Dwarf galaxies with a non-standard BSS population The Leo II dwarf might be a prototype of a galaxy whose blue plume properties differ from the classical BSS sequence found in globular clusters. Figure 1 displays the HST/NTT CMDs of Leo II upon which we highlight the BSS and HB selection boxes. The Leo II BSS specific frequency was estimated from the HST diagram, whose coverage is comparable with the galaxy half light radius, about ∼ 2 arcmin. The most notable feature is the detection of a vertical extension in correspondence of the red HB region. Stars forming this sequence are usually called verti- cal clump stars (VC, see Gallart et al. 2005). These are helium- burning stars of few hundred Myr to ∼ 1 Gyr old population whose progenitors are to be searched in the blue plume. Indeed, the relatively large area covered by the NTT shows a CMD with a well defined VC sequence that can be matched by ∼ 1 Gyr isochrones. Since in the context of dwarf galaxies one cannot exclude the presence of an extended star formation, the detec- tion of VC stars (as is also the case for Draco, see Aparicio et al. 2001) would suggest that the blue plume population may well hide a genuinely young main sequence. This possibility will be further investigated in a detailed reconstruction of the star for- mation history of Leo II (Rizzi et al in prep.). A second diagnostic in support of a non-standard BSS se- quence lies in the very extension of the Leo II blue plume. The luminosity function of blue stragglers in globular clusters has been found to increase from a luminosity cutoff at MV ∼ 1.9 toward the ancient MS turn-off at MV ∼ 4.0 (Sarajedini & Da Costa 1991, Fusi Pecci et al. 1992). In the case of Leo II, the cutoff luminosity should correspond to V ≃ 23.8, whereas we observe it at brighter magnitudes (V ≃ 23.0 and V ≃ 22.7 in the HST and NTT diagrams respectively), i.e. the BSS sequence in Leo II is more extended than that in globulars. A third diagnostic is that relative to the ratio of BSS to blue HB stars (7, 000 ≤ Teff ≤ 11, 500 K: the lower temperature limit marks the blue border of the RR Lyrae instability strip while the upper temperature limit signs the horizontal branch truncation in dwarf spheroidal galaxies, as noted in Momany et al. 2004, 2002). This diagnostic has been used by Hurley-Keller et al. (1998) in favor of a genuine ∼ 1 Gyr MS population in Carina, and similarly can be applied for Leo II. Basically, for Galactic globular clusters Preston et al. (1994) find a BSS to blue HB ratio of ∼ 0.6. This is however much lower than what we estimate for Leo II: using the 4 HST/WFPC2 catalogs we find a BSS to blue HB ratio of ∼ 9.2. A fourth diagnostic is the BSS to VC ratio that we estimate to remain constant (∼ 8.1) for each of the 4 WFPC2 chips, as well as for the the entire WFPC2 catalog. This particular observable does not necessarily unveil the true nature of the blue plume in Leo II, it however confirms a tight correlation between the BSS and VC populations. Lastly, we recall the discovery of four intermediate age anomalous Cepheids in Leo II (Siegel & Majewski 2000). These are explained as due to either extremely low metallicity variables or to mass transfer (and possibly coalescence) in a close binary systems. However, as discussed by Siegel & Majewski (2000), should the anomalous Cepheids in Leo II be due to mass transfer binaries, then the number of blue stragglers (in the HST field) is predicted to be ∼ 0.5 − 5. This is much lower than the observed number of BSS: corrected for incompleteness, we estimate a to- tal of 175 BSS stars in the HST CMD. Thus, accounting for: (i) the presence of VC stars; (ii) the extension and (iii) number of the BSS population in Leo II makes it more likely that Leo II has experienced an extended star formation history. Similar conclu- sions can be applied to Sagittarius (possessing an extended blue plume whose cutoff luminosity is brighter than MV ∼ 1.9, as il- lustrated in Fig. 1) and Draco (possessing VC stars, as discussed in Aparicio et al. 2001). Despite all these arguments, it remains difficult to rule out the BSS interpretation. In this regard, it is important to note that the strongest evidence put in favor of a recent star formation episode in Leo II (i.e. the detection of VC stars) is a double-edge sword. Indeed, VC stars have been detected in globular clusters and investigators needed not to invoke a recent star formation in these systems: the presence of the VC sequence could and has been interpreted as due to evolved-BSS. As an example, we consider the case of M80. Ferraro et al. (1999) derive a ratio of the BSS to evolved-BSS (or VC) of ∼ 7. This is very close to the BSS to VC ratio of ∼ 8 that we estimate in Leo II. Mighell & Rich (1996) were the first to suggest that the VC sequence in Leo II can be due to the evolved BSS in the helium-burning phase. Should this be the case then one need not to explain the absence of gas (Blitz & Robishaw 2000) fueling a recent star formation in Leo II; since there should not be any. 2.2. Dwarf galaxies showing a standard BSS population For few dwarf galaxies there is no hint for the presence of VC stars (see the CMDs of Sextans by Lee et al. 2003 and Ursa Minor by Carrera et al. 2002). Although foreground contamina- tion might contribute in veiling the VC population, in general, the absence of VC or other intermediate age indicators seemed to indicate a rather normal BSS population in these galaxies. One interesting anomaly, however, has been detected in the spatial distribution of the BSS population in Sextans. Lee et al. (2003) find that brighter BSS in Sextans are more strongly con- centrated towards the galaxy center, while fainter BSS are lack- ing in the central regions and follow the distribution of old stars in the outer regions. In the context of globular clusters a sim- ilar trend is often attributed to the higher occurrence of colli- sional binaries in higher density environments (i.e. the center) normally producing brighter and bluer BSS. However, the colli- sional rate in dwarf spheroidals like Sextans is much lower than that in globular clusters (see Sect. 4), and therefore dynamical evolution in Sextans cannot account for a higher production of collisional binaries. Thus, leaving aside this particular distribu- tion anomaly, the overall blue plume properties in galaxies like Sextans, Ursa Minor, Ursa Major and Boötes have been inter- preted as the old BSS population. 3. BSS frequency: analysis 3.1. BSS frequency in dwarf galaxies and globular clusters We now address the BSS frequency for our dwarf galaxies sam- ple and make an internal comparison. For a wider perspective, we also compare the overall BSS frequency in dwarf galaxies with that observed in other stellar systems. 4 Y. Momany et al.: The blue plume population in dwarf spheroidal galaxies: Fig. 2. The FBSS vs MV diagram for globular clusters (Piotto et al. 2004) open clusters (De Marchi et al. 2006) and dwarf spheroidal galaxies. The horizontal line shows the mean BSS frequency for Milky Way field stars (Preston & Sneden 2000). Figure 2 displays the FBSS vs MV diagram for our dwarf galaxy sample together with the data-points of Piotto et al. (2004) and De Marchi et al. (2006) for globular and open clus- ters, respectively. Of the original open cluster sample we only plot clusters for which ≥ 2 BSS stars were found. To the globular cluster sample we add the BSS frequency of ωCen (as derived by Ferraro et al. 2006). The anomalies in ωCen are multi-fold and span a multiple MS, sub-giant and red giant branches (see Bedin et al. 2004) and peculiar chemical enrichment (Piotto et al. 2005), that are often used in favor of a dwarf galaxy origin. Moreover we estimate the BSS frequency in 2 peculiar systems: (i) NGC1841 (Saviane et al. 2003) the LMC most metal-poor and most distant (∼ 10 Kpc from the LMC bar) globular cluster that is also young and incompatible with the LMC halo rota- tion; and (ii) NGC2419 (Momany et al. in prep.) a massive MW cluster at ∼ 90 Kpc from the Galactic center. The three data- points are based on deep HST/ACS, WFPC2 and ACS archival data, respectively. Figure 2 clearly shows that, regardless of their specific peculiarities, ωCen, NGC1841 and NGC2419 are consistent with the general globular clusters FBSSHB − MV anti- correlation. Before turning our attention to the BSS frequency in dwarf galaxies, we first comment on the case of NGC6717 and NGC6838 (the two faintest globular clusters with the highest FBSSHB frequency). Located at (l, b)=(13 ◦,−11◦) and (50◦,−5◦), the two globular clusters can be subject to significant bulge/disk contamination that was not accounted for in the Piotto et al. analysis. Trilegal simulations showed in fact that a consider- able number of Galactic young MS stars would overlap with the clusters BSS sequences and this can account for their rather high BSS frequency with respect to globulars of similar MV . Allowing for the exclusion of NGC6717 and NGC6838, it results immediately that the lowest luminosity dwarfs (Boötes and Ursa Major) would possess a higher FBSSHB than globular clus- ters with similar MV . Most interestingly, their F HB is in fact fully compatible with that observed in open clusters. This com- patibility between dwarf galaxies and open clusters may suggest that there exists a “saturation” in the BSS frequency (at 0.3−0.4) for the lowest luminosity systems. Thus, the relatively high FBSSHB of Boötes and Ursa Major adds more evidence in favor of a dwarf galaxy classification of the 2 systems. Indeed, although their lu- minosities is several times fainter than Draco or Ursa Minor, the physical size of the two galaxies (r1/2 ≃ 220 and 250 pc respec- tively) exceeds that of more massive galaxies like Ursa Minor (r1/2 ≃ 150 pc). Another interesting feature is the significant difference be- tween the BSS frequency of Carina with that derived for dwarf galaxies with similar luminosity, i.e. Draco, Ursa Minor, Sextans, Sculptor and Leo II. Although it is only a lower limit2, 2 It is hard to account for the BSS population originating from the older and fainter MS turn-off. Y. Momany et al.: The blue plume population in dwarf spheroidal galaxies: 5 the “BSS” frequency for Carina is of great help in suggesting a threshold near which a galaxy BSS frequency might hide some level of recent star formation. The aforementioned 5 galaxies however have a lower BSS frequency, a hint that these galaxies possess a normal BSS population rather than a young MS. This confirms previous conclusions for Sextans (Lee et al. 2003) and Ursa Minor (Carrera et al. 2002), but is in contradiction with that of Aparicio et al. (2001) for Draco. However, the Aparicio et al. conclusion was mainly based on the detection of the VC stars, a feature that, as we argued, remains an ambiguous indi- cator. Indeed, Fig. 2 shows that the BSS frequency in Draco is very close to that of Ursa Minor, a galaxy acceptably known to possess an old BSS population. Lastly, leaving aside the extreme dynamical history of Sagittarius and allowing for the uncertainties (due to the heavy Galactic contamination and the relatively small sampled popula- tions) it turns out that its blue plume-HB frequency is (i) lower than that of a recently star-forming galaxy like Carina, and most interestingly; (ii) in good agreement with the expected BSS fre- quency as derived from the FBSSHB − MV anti-correlation for the 7 remaining galaxies in our sample. Added to the clear absence of MS stars overlapping or exceeding the Sagittarius HB luminos- ity level (see Fig. 1), we suggest that the Sagittarius blue plume is a “normal” BSS sequence. As a matter of fact, Sagittarius is probably the nearest system with the largest BSS population: over 2600 BSS stars in the inner 1◦ × 1◦ field. To summarize, from Fig. 2 one finds that FBSSHB in dwarf galaxies is (i) always higher than that in globular clusters, (ii) very close, for the lowest luminosity dwarfs, to that observed in the MW field and open clusters, (iii) the Carina specific FBSSHB frequency probably sets a threshold for star-forming galaxies, and most interestingly, (iv) shows a hint of a FBSSHB − MV anti- correlation. 3.2. A FBSSHB − MV anti-correlation for dwarf galaxies ? We here explore the statistical significance of a possible FBSSHB − MV correlation. The linear-correlation coefficient (Bevington 1969) for the 8 galaxies (excluding Carina) data-points is 0.984. The corresponding probability that any random sample of un- correlated experimental data-points would yield a correlation coefficient of 0.984 is < 10−6. Given the greater uncertainties associated with the Sagittarius BSS frequency, one may be in- terested in the correlation coefficient excluding the Sagittarius data-point. In this case, the resulting correlation coefficient re- mains however quite high (0.972) and the probability that the 7 remaining data-points would randomly correlate is as low as 10−4. Thus, the statistical significance of the FBSSHB − MV anti- correlation in non star-forming dwarf galaxies is quite high. We follow the methods outlined in Feigelson & Babu (1992) and fit least-squares linear regressions. In particular, the intercept and slope regression coefficients were estimated through 5 lin- ear models (see the code of Feigelson & Babu for details) the average of which gives (a, b)=(0.699 ± 0.081, 0.070 ± 0.010) and (a, b)=(0.631± 0.120, 0.062± 0.014) including and exclud- ing the Sagittarius data-point, respectively. The reported errors were estimated through Bootstrap and Jacknife simulations so as to provide more realistic a and b errors. However, to firmly establish this FBSSHB − MV anti-correlation one needs to increase the dwarf galaxies sample, in particular at the two luminosity extremes. Unfortunately there are not many non star-forming dwarf galaxies with −13.3 ≤ MV ≤ − 10.1 (c.f. table 14 in Mateo 1998), and few exceptions may come from Fig. 3. The BSS frequency as a function of the half light radius (panel a), the central surface brightness (panel b), the velocity dispersion (panel c) and the stellar collision factor (panel d). See text for details. deeper imaging of galaxies like And I and And II. On the other hand, more Local Group dwarf galaxies are being discovered in the low luminosity regime (−8.0 ≤ MV ≤ −5.0). Deeper imag- ing of recently discovered galaxies like Com, CVn IIm, Her and Leo IV (Belokurov et al. 2007) and Willman 1 (Willman et al. 2006) are needed to estimate their BSS frequency. The impor- tance of these low-luminosity galaxies is easily understood once we exclude Boötes and Ursa Major from the BSS frequency cor- relation analysis. In this case, the correlation coefficient for the 5 remaining data-points is found to drop to 0.901 having a prob- ability of random correlation as high as 1.5 × 10−1. Thus, a final word on the FBSSHB − MV anti-correlation must await for more data-points at both luminosity extremes. 4. Discussion and Conclusions For a sample of 8 non star-forming dwarf galaxies, we have tested the hypothesis that the blue plume populations are made of a genuine BSS population (as that observed in open and globular clusters) and estimated their frequency with respect to HB stars. Should this assumption be incorrect (and the blue plume pop- ulation is made of young MS stars) then one would not expect an anti-correlation between the galaxies total luminosity (mass) and the blue plume frequency, but rather a correlation between the two. Instead, and within the limits of this and similar analy- sis, we detect a statistically significant anti-correlation between FBSSHB and MV . A similar anti-correlation has been reported for Galactic open and globular clusters. A positive detection of vertical clump stars does not provide a clear-cut evidence in favor of a recent star formation episode in a dwarf galaxy. This is because the vertical clump population has been detected in globular clusters and one cannot exclude 6 Y. Momany et al.: The blue plume population in dwarf spheroidal galaxies: that these are evolved-BSS. Thus, the main difference between the blue plume (in non star-forming dwarf galaxies) and the BSS sequence (in globulars clusters) is that regarding their number. Should a dwarf galaxy “obey” the FBSSHB − MV anti-correlation displayed by our sample then its blue plume population is prob- ably made of blue stragglers. Do dwarf galaxies harbor a significant population of col- lisional binaries ? The answer is no. This relies on the intrin- sic properties of dwarf galaxies and the consequent difference with those of globular clusters. Indeed, it is enough to recall that the central luminosity density of a dwarf galaxy (e.g. Ursa Minor: 0.006 L⊙ pc −3 at MV = −8.9) is several orders of mag- nitudes lower than that found in a typical globular cluster (e.g. NGC7089 ∼ 8000 L⊙ pc −3 at MV = −9.0). This implies that the collisional parameter of dwarf galaxies is very low, and un- ambiguously point to the much slower dynamical evolution of dwarf galaxies. To further emphasize this last point, we search for FBSSHB dependencies on other dwarf galaxies structural param- eters. Figure 3 plots the BSS frequency as a function of the half light radius (panel a), the central surface brightness (panel b), the velocity dispersion (panel c) and the stellar collision factor (panel d)3. Panels a to c plot the FBSSHB as a function of observed globular/dwarf galaxies quantities. Panel a shows that globular clusters and dwarf spheroidals form 2 quite distinct families. This is further confirmed in panel b, although the central sur- face brightness distribution might suggest a FBSSHB connection of the two. Panel c shows a correlation between FBSSHB and the cen- tral velocity dispersion for globular clusters. This reflects the known globular cluster fundamental plane relations, as shown by Djorgovski (1995). Despite the similarities in their velocity dispersion, dwarf galaxies form a separate group from globular clusters, showing systematically higher FBSSHB . In panel d we show FBSSHB as a function of a calculated quan- tity: the stellar specific collision parameter (log Γ⋆: the num- ber of collisions per star per year). More specifically, following Piotto et al. (2004), for both globular clusters and dwarf galax- ies we estimate log Γ⋆ from the central surface density and the system core size. The mean collisional parameter of the 9 stud- ied galaxies is ≃ −19. The lowest value is that for Sagittarius with log Γ⋆ ≃ −20.2, and this is due to its very extended galaxy core. Compared with the mean value of −14.8 for the globular clusters sample (see also the lower panel of Fig. 1 in Piotto et al. 2004), the estimated number of collisions per star per year in a dwarf spheroidal is 10−5 times lower. This almost precludes the occurrence of collisional binaries in dwarf galaxies, and one may conclude that genuine BSS sequences in dwarf galaxies are mainly made of primordial binaries. Not all primordial binaries, now present in a dwarf galaxy, turn into or are already in the form of BSS. In particular, it is the low exchange encounter probabilities in environments like the Galactic halo or dwarf galaxies that guarantees a friendly environment and a slower consumption/evolution of primordial binary systems. The BSS production (via evolution off the MS of the primary and the consequent mass-transfer to the secondary that may become a BSS) is still taking place in the present epoch 3 For globular clusters we make use of the Trager et al. (1995) and Pryor & Meylan (1993) tables, whereas for dwarf spheroidals we use the tables from Mateo (1998) and updated velocity dispersions from recent measurements (Sculptor: Tolstoy et al. 2004, Sextans: Walker et al. 2006), Carina: Koch et al. 2007, Boötes: Muñoz et al. 2006, Ursa Minor: Muñoz et al. 2005, Ursa Major: Kleyna et al. 2005, Sagittarius: Zaggia et al. 2004, Draco: Muñoz et al. 2005; Carina: Muñoz et al. 2006b). and this can explain the high frequency of primordial BSS in dwarf galaxies as well as the Galactic halo. Lastly, it is interesting to note how the BSS frequency in the low-luminosity dwarfs and open clusters (log(NBSS/NHB) ∼ 0.3 − 0.4) is very close to that derived for the Galactic halo (log(NBSS/NHB) ∼ 0.6) by Preston & Sneden. The latter value however has been derived relying on a composite sample of only 62 blue metal-poor stars that are (i) distributed at different line of sights; (ii) at different distances; and most importantly, (iii) for which no observational BSS-HB star-by-star correspondence can be established. Thus, allowing for all these uncertainties in the field BSS frequency (see also the discussion in Ferraro et al. 2006), it is safe to conclude that the observed open clusters- dwarf galaxies BSS frequency sets a realistic, and observational upper limit to the primordial BSS frequency in stellar systems. Acknowledgements. We thank Alvio Renzini and Giampaolo Piotto for useful discussions that helped improve this paper. We are also grateful to Belokurov V., Willman B., Carrera R., Monelli M. and Aparicio A. for kindly providing us their photometric catalogs. References Aparicio, A., Carrera, R., & Martı́nez-Delgado, D. 2001, AJ, 122, 2524 Bedin, L. 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E., & Vandenberg, D. A. 1996, ASP Conf. Ser. 98: From Stars to Galaxies: the Impact of Stellar Physics on Galaxy Evolution, 98, 328 Trager, S. C., King, I. R., & Djorgovski, S. 1995, AJ, 109, 218 Tolstoy, E., et al. 2004, ApJ, 617, L119 Walker, M. G., Mateo, M., Olszewski, E. W., Pal, J. K., Sen, B., & Woodroofe, M. 2006, ApJ, 642, L41 Willman, B., et al. 2005, ApJ, 626, L85 Willman, B., et al. 2006, arXiv:astro-ph/0603486 Wyse, R., Gilmore, G., Houdashelt, M., Feltzing, S., Hebb, L., Gallagher, J. & Smecker-Hane, T. 2002, New Astronomy, 7, 395 Zaggia, S., et al. 2004, Mem. Societa Astronomica Italiana Supplement, 5, 291 http://arxiv.org/abs/astro-ph/0603486 Introduction BSS frequency data points Dwarf galaxies with a non-standard BSS population Dwarf galaxies showing a standard BSS population BSS frequency: analysis BSS frequency in dwarf galaxies and globular clusters A FBSSHB- MV anti-correlation for dwarf galaxies ? Discussion and Conclusions

0704.1431

Generalized characteristic polynomials of graph bundles

GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE Abstract. In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0, 1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae. 1. Introduction One of classical invariants in graph theory is the characteristic polynomial which comes from the adjacency matrices. It displays not only graph theoretical properties but also algebraic prospectives, such as spectra of graphs. There have been many meaningful gener- alizations of characteristic polynomials [5,12]. In particular, we are interested in one found by Cvetkovic and alt. as a polynomial on two variables [5], FG(λ, µ) = det(λI − (A(G)− µD(G))). The zeta functions of finite graphs [1, 2, 8] feature of Riemanns zeta functions and can be considered as an analogue of the Dedekind zeta functions of a number field. It can be expressed as the determinant of a perturbation of the Laplacian and a counterpart of the Riemann hypothesis [19]. Bartholdi introduced the Bartholdi zeta function ZG(u, t) of a graph G together with a comprehensive overview and problems on the Bartholdi zeta functions [1]. He also showed that the reciprocal of the Bartholdi zeta function of G is ZG(u, t) 1− (1− u)2t2 )εG−νG I − A(G)t+ (1− u)(DG − (1− u)I)t Kwak and alt. studied the Bartholdi zeta functions of some graph bundles having regular fibers [11]. Mizuno and Sato also studied the zeta function and the Bartholdi zeta function of graph coverings [13,15]. Recently, it was shown that the Bartholdi zeta function ZG(u, t) can be found as the reciprocal of the generalized characteristic polynomials FG(λ, µ) with a suitable substitution [9]. The aim of the present article is to find computational formulae for generalized char- acteristic polynomials FG(λ, µ) of graph bundles and its applications. For computational formulae, we show that if the fiber of the graph bundle is a Schreier graph, the conjugate class of the adjacency matrix has a representative whose characteristic polynomial can be 2000 Mathematics Subject Classification. 05C50, 05C25, 15A15, 15A18. Key words and phrases. generalized characteristic polynomials, graph bundles, the Bartholdi zeta functions. This research was supported by the Yeungnam University research grants in 2007. http://arxiv.org/abs/0704.1431v1 2 DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE computed efficiently using the representation theory of the symmetric group. We also pro- vide computational formulae for generalized characteristic polynomials of graph bundles G ×φ F where the images of φ lie in an abelian subgroup Γ of Aut(F ). To demonstrate the efficiency of our computation formulae, we calculate the generalized characteristic poly- nomials FG of some Kn-bungles G × φ Kn. Consequently, we can obtain the generalized characteristic polynomials FK1,m×Kn(λ, µ) of K1,m ×Kn which is a standard model of net- work with hubs. Its adjacency matrix, known as a “kite”, is one of important examples in matrix analysis. The outline of this paper is as follows. First, we review the terminology of the generalized characteristic polynomials and show that the number of spanning trees in a graph is the partial derivative (at (0, 1)) of the generalized characteristic polynomial of the given graph in section 2. Next, we study a similarity of the adjacency matrices of graph bundles and find computational formulae for generalized characteristic polynomial FG(λ, µ) of graph bundles in section 3. In section 4, we find the generalized characteristic polynomial of K1,m ×Kn and find the number of spanning trees of K1,m ×Kn. 2. Generalized characteristic polynomials and complexity In the section, we review the definitions and useful properties of the generalized charac- teristic polynomials and find the number of spanning trees in a graph using the generalized characteristic polynomials. Let G be an undirected finite simple graph with vertex set V (G) and edge set E(G). Let νG and εG denote the number of vertices and edges of G, respectively. An adjacency matrix A(G) = (aij) is the νG×νG matrix with aij = 1 if vi and vj are adjacent and aij = 0 otherwise. The degree matrix D(G) of G is the diagonal matrix whose (i, i)-th entry is the degree dGi = degG(vi) of vi in G for each 1 ≤ i ≤ νG. The complexity κ(G) of G is the number of spanning trees in G. An automorphism of G is a permutation of the vertex set V (G) that preserves the adjacency. By |X|, we denote the cardinality of a finite set X . The set of automorphisms forms a permutation group, called the automorphism group Aut(G) of G. The characteristic polynomial of G, denoted by Φ(G;λ), is the characteristic polynomial det(λI − A(G)) of A(G). Cvetkovic and alt. introduced a polynomial on two variables of G, FG(λ, µ) = det(λI − (A(G) − µD(G))) as a generalization of characteristic polynomials of G [5], for example, the characteristic polynomial of G is FG(λ, 0) and the characteristic polynomial of the Laplacian matrix D(G)− A(G) of G is (−1)νGFG(−λ, 1). In [9]], it was shown that the Bartholdi zeta function ZG(u, t) of a graph can be obtained from the polynomial FG(λ, µ) with a suitable substitution as follows. ZG(u, t) 1− (1− u)2t2 )εG−νG tνGFG − (1− u)2t, (1− u)t FG(λ, µ) = (1− µ2)εG 1− µ2 1− µ2 The complexities for various graphs have been studied [13,16]. In particular, Northshield showed that the complexity of a graph G can be given by the derivative f ′G(1) = 2(εG − νG) κ(G) GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES 3 of the function fG(u) = det[I − u A(G) + u 2 (D(G)− I)], for a connected graph G [16]. By considering the idea of taking the derivative, we find that the complexity of a finite graph can be expressed as the partial derivative of the generalized characteristic polynomial FG(λ, µ) evaluated at (0, 1). Theorem 2.1. Let FG(λ, µ) = det(λI − (A(G)− µD(G))) be the generalized characteristic polynomial of a graph G. Then the number of spanning trees in G, κ(G), is |(0,1), where εG is the number of edges of G. Proof. Let Bλ,µ = λI − (A(G) − µD(G)) = ((bλ,µ)ij) and let B λ,µ = ((bλ,µ) ij) denote the matrix Bλ,µ with each entry of k-th row replace the corresponding partial derivative with respect to µ. Then (det Bλ,µ) = sgn(σ) ((bλ,µ)iσ(i)) (bλ,µ) iσ(i)) = (det Bkλ,µ). Since (b0,1) ij = d i δij + aij(δkj − 1) = diδij − aij + aijδki, |(0,1) = det (B0,1 + aij(δki)). Let (Mk)ij = aijδki and let (C0,1)ij be the cofactor of bij in B0,1. Then det (B0,1 +M (B0,1 +M k)kj(−1) k+j(C0,1)kj = dk (C0,1)kk. Since det B0,1 = 0, (C0,1)ij = κ(G) for all i and j [3]. Hence, |(0,1) = dk (C0,1)kk = κ(G) dk = 2 εGκ(G). Thus, κ(G) = |(0,1). 3. Generalized characteristic polynomials of graph bundles Let G be a connected graph and let ~G be the digraph obtained from G by replacing each edge of G with a pair of oppositely directed edges. The set of directed edges of ~G is denoted by E( ~G). By e−1, we mean the reverse edge to an edge e ∈ E( ~G). We denote the directed edge e of ~G by uv if the initial and the terminal vertices of e are u and v, respectively. For a finite group Γ, a Γ-voltage assignment of G is a function φ : E( ~G) → Γ such that φ(e−1) = φ(e)−1 for all e ∈ E( ~G). We denote the set of all Γ-voltage assignments of G by C1(G; Γ). 4 DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE Let F be another graph and let φ ∈ C1(G; Aut(F )). Now, we construct a graph G×φ F with the vertex set V (G ×φ F ) = V (G) × V (F ), and two vertices (u1, v1) and (u2, v2) are adjacent in G ×φ F if either u1u2 ∈ E( ~G) and v2 = v φ(u1u2) 1 = v1φ(u1u2) or u1 = u2 and v1v2 ∈ E(F ). We call G × φ F the F -bundle over G associated with φ (or, simply a graph bundle) and the first coordinate projection induces the bundle projection pφ : G ×φ F → G. The graphs G and F are called the base and the fibre of the graph bundle G ×φ F , respectively. Note that the map pφ maps vertices to vertices, but the image of an edge can be either an edge or a vertex. If F = Kn, the complement of the complete graph Kn of n vertices, then an F -bundle over G is just an n-fold graph covering over G. If φ(e) is the identity of Aut(F ) for all e ∈ E( ~G), then G ×φ F is just the Cartesian product of G and F , a detail can be found in [10]. Let φ be an Aut(F )-voltage assignment of G. For each γ ∈ Aut(F ), let ~G(φ,γ) denote the spanning subgraph of the digraph ~G whose directed edge set is φ−1(γ). Thus the digraph ~G is the edge-disjoint union of spanning subgraphs ~G(φ,γ), γ ∈ Aut(F ). Let V (G) = {u1, u2, . . . , uνG} and V (F ) = {v1, v2, . . . , vνF }. We define an order relation ≤ on V (G× as follows: for (ui, vk), (uj, vℓ) ∈ V (G × φ F ), (ui, vk) ≤ (uj, vℓ) if and only if either k < ℓ or k = ℓ and i ≤ j. Let P (γ) denote the νF × νF permutation matrix associated with γ ∈ Aut(F ) corresponding to the action of Aut(F ) on V (F ), i.e., its (i, j)-entry P (γ)ij = 1 if γ(vi) = vj and P (γ)ij = 0 otherwise. Then for any γ, δ ∈ Aut(F ), P (δγ) = P (δ)P (γ). Kwak and Lee expressed the adjacency matrix A(G ×φ F ) of a graph bundle G ×φ F as follows. Theorem 3.1 ( [10]). Let G and F be graphs and let φ be an Aut(F )-voltage assignment of G. Then the adjacent matrix of the F -bundle G×φ F is A(G×φ F ) = γ∈Aut(F ) P (γ)⊗A( ~G(φ,γ))  + A(F )⊗ IνG , where P (γ) is the νF × νF permutation matrix associated with γ ∈ Aut(F ) corresponding to the action of Aut(F ) on V (F ), and IνG is the identity matrix of order νG. For any finite group Γ, a representation ρ of a group Γ over the complex numbers is a group homomorphism from Γ to the general linear group GL(r,C) of invertible r×r matrices over C. The number r is called the degree of the representation ρ [18]. Suppose that Γ ≤ Sn is a permutation group on Ω. It is clear that P : Γ → GL(r,C) defined by γ → P (γ), where P (γ) is the permutation matrix associated with γ ∈ Γ corresponding to the action of Γ on Ω, is a representation of Γ. It is called the permutation representation. Let ρ1 = 1, ρ2, . . . , ρℓ be the irreducible representations of Γ and let fi be the degree of ρi for each 1 ≤ i ≤ ℓ, where f1 = 1 and i=1 f i = |Γ|. It is well-known that the permutation representation P can be decomposed as the direct sum of irreducible representations : ρ = ⊕ℓi=1miρi [18]. In other words, there exists an unitary matrix M of order |Γ| such that M−1P (γ)M = (Imi ⊗ ρi(γ)) (2) GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES 5 for any γ ∈ Γ, where mi ≥ 0 is the multiplicity of the irreducible representation ρi in the permutation representation P and i=1mifi = νF . Notice that m1 ≥ 1 because it represents the number of orbits under the action of the group Γ. It is not hard to show that (M ⊗ IνG) −1A(G×φ F )(M ⊗ IνG) = Imi ⊗ ρi(γ)⊗ A( ~G(φ,γ)) + A(F )⊗ IνG . For any vertex (ui, vk) ∈ V (G × φ F ), its degree is dGi + d k , where d i = degG(ui) and dFk = degF (vk). Then, by our construction of D(G× φ F ), the diagonal matrix D(G×φ F ) is equal to IνF ⊗D(G) +D(F )⊗ IνG and thus (M ⊗ IνG) −1D(G×φ F )(M ⊗ IνG) = IνF ⊗D(G) +D(F )⊗ IνG . Therefore, the matrix A(G×φ F )− µD(G×φ F ) is similar to Imi ⊗ ρi(γ)⊗ A( ~G(φ,γ))− Ifi ⊗ µD(G) + (A(F )− µD(F ))⊗ IνG. By summarizing these, we obtain the following theorem. Theorem 3.2. Let G and F be two connected graphs and let φ be an Aut(F )-voltage assignment of G. Let Γ be the subgroup of the symmetric group Sn. Furthermore, let ρ1 = 1, ρ2, . . . , ρℓ be the irreducible representations of Γ having degree f1, f2, . . . , fℓ, respectively. Then the matrix A(G×φ F )− µD(G×φ F ) is similar to Imi ⊗ ρi(γ)⊗ A( ~G(φ,γ))− Ifi ⊗ µD(G) + (A(F )− µD(F ))⊗ IνG, where mi ≥ 0 is the multiplicity of the irreducible representation ρi in the permutation representation P and i=1mifi = νF . � A graph F is called a Schreier graph if there exists a subset S of SνF such that S −1 = S and the adjacency matrix A(F ) of F is s∈S P (s). We call such an S the connecting set of the Schreier graph F . Notice that a Schreier graph with connecting set S is a regular graph of degree |S| and most regular graphs are Schreier graphs [7, Section 2.3]. The definition of the Schreier graph here is different that of original one. But, they are basically identical [7, Section 2.4]. Clearly, every Cayley graph is a Schreier graph. Theorem 3.3. Let G be a connected graph and let F be a Schreier graph with connecting set S. Let φ : E( ~G) → Aut(F ) be a permutation voltage assignment. Let Γ be the subgroup of the symmetric group SνF generated by {φ(e), s : e ∈ E( ~G), s ∈ S}. Furthermore, let ρ1 = 1, ρ2, . . . , ρℓ be the irreducible representations of Γ having degree f1, f2, . . . , fℓ, respectively. Then the matrix A(G×φ F )− µD(G×φ F ) is similar to Imi ⊗ γ∈Aut(F ) ρi(γ)⊗ A( ~G(φ,γ))− Ifi ⊗ µ(D(G) + |S|IνG) + ρi(s) ⊗ IνG 6 DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE where mi ≥ 0 is the multiplicity of the irreducible representation ρi in the permutation representation P and i=1mifi = νF . � Proof. Let F be a Schreier graph with a connecting set S. Then the adjacency matrix of F is A(F ) = s∈S P (s). Hence, for any voltage assignment φ : E( ~G) → Aut(F ), one can see that A(G×φ F ) = γ∈Aut(F ) P (γ)⊗ A( ~G(φ,γ)) P (s)⊗ IνG. Let Γ be the subgroup of SνF generated by {φ(e), s : e ∈ E( ~G), s ∈ S}. Since F is a regular graph of degree |S|, one can see that D(G×φ F ) = IνF ⊗ (D(G) + |S|IνG). Now, one can have that the matrix A(G×φ F )− µD(G×φ F ) is similar to Imi ⊗ γ∈Aut(F ) ρi(γ)⊗ A( ~G(φ,γ))− Ifi ⊗ µ(D(G) + |S|IνG) + ρi(s) ⊗ IνG It is easy to see that the following theorem follows immediately from Theorem 3.3. Theorem 3.4. Let G be a connected graph and let F be a Schreier graph with connecting set S. Let φ : E( ~G) → Aut(F ) be a permutation voltage assignment. Let Γ be the subgroup of the symmetric group SνF generated by {φ(e), s : e ∈ E( ~G), s ∈ S}. Furthermore, let ρ1 = 1, ρ2, . . . , ρℓ be the irreducible representations of Γ having degree f1, f2, . . . , fℓ, respectively. Then the characteristic polynomial FG×φF (λ, µ) of a graph bundle G× φ F is Ifi ⊗ [(λ+ µ|S|)IνG + µD(G)] − γ∈Aut(F ) ρi(γ)⊗A( ~G(φ,γ))− ρi(s) ⊗ IνG where mi ≥ 0 is the multiplicity of the irreducible representation ρi in the permutation representation P and i=1mifi = νF . � Let F = Kn be the trivial graph on n vertices. Then any Aut(Kn)-voltage assignment is just a permutation voltage assignment defined in [7], and G ×φ Kn = G φ is just an n-fold covering graph of G. In this case, it may not be a regular covering. Now, the following comes from Theorem 3.2. Corollary 3.5. Let G be a connected graph and let F = Kn. The characteristic polynomial FGφ(λ, µ) of the connected covering G φ of a graph G derived from a permutation voltage assignment φ : E( ~G) → Sn is FG(λ, µ)× Ifi ⊗ [(λ+ r)IνG + µD(G)]− ρi(γ)⊗A( ~G(φ,γ))− ρi(s) ⊗ IνG where mi ≥ 0 is the multiplicity of the irreducible representation ρi in the permutation representation P and i=1mifi = n. � GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES 7 Next, we consider the characteristic polynomial depending on two variable of graph bun- dles G×φ F where the images of φ lie in an abelian subgroup Γ of Aut(F ) and the fiber F is r-regular. In this case, for any γ1, γ2 ∈ Γ, the permutation matrices P (γ1) and P (γ2) are commutative and DF = rIνF . It is well-known (see [3]) that every permutation matrix P (γ) commutes with the adja- cency matrix A(F ) of F for all γ ∈ Aut(F ). Since the matrices P (γ), γ ∈ Γ, and A(F ) are all diagonalizable and commute with each other, they can be diagonalized simultaneously. i.e., there exists an invertible matrix MΓ such that M Γ P (γ)MΓ and M Γ A(F )MΓ are di- agonal matrices for all γ ∈ Γ. Let λ(γ,1), . . . , λ(γ,νF ) be the eigenvalues of the permutation matrix P (γ) and let λ(F,1), . . . , λ(F,νF ) be the eigenvalues of the adjacency matrix A(F ). M−1Γ P (γ)MΓ = λ(γ,1) 0 . . . 0 λ(γ,νF )  and M−1Γ A(F )MΓ = λ(F,1) 0 . . . 0 λ(F,νF ) Using these similarities, we find that (MΓ ⊗ IνG) P (γ)⊗A( ~G(φ,γ)) + A(F )⊗ IνG (MΓ ⊗ IνG) λ(γ,i)A( ~G(φ,γ)) + λ(F,i)IνG Recall that (M ⊗ IνG) −1D(G×φ F )(M ⊗ IνG) = IνF ⊗D(G) + rIνF ⊗ IνG = (D(G) + rIνG). By summarizing these facts, we find the following theorem. Theorem 3.6. Let G be a connected graph and let F be a connected regular graph of degree r. If the images of φ ∈ C1(G; Aut(F )) lie in an abelian subgroup of Aut(F ), then the matrix A(G×φ F )− µD(G×φ F ) is similar to (λ(F,i) − rµ)IνG + λ(γ,i)A( ~G(φ,γ))− µD(G) Notice that the Cartesian product G × F of two graphs G and F is a F -bundle over G associated with the trivial voltage assignment φ, i.e., φ(e) = 1 for all e ∈ E( ~G) and A(G) = A( ~G). The following corollary comes from this observation. Corollary 3.7. For any connected graph G and a connected r-regular graph F , the matrix A(G× F )− µD(G× F ) of the cartesian product G× F is similar to 8 DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE (λ(F,i) − rµ)IνG+ (A(G)−µD(G)) In particular, if G is a regular graph of degree dG, then the matrix A(G× F )− µD(G× F ) of the cartesian product G× F is λ(F,i) + λ(G,j) − (r + d)µ where λ(G,j) (1 ≤ j ≤ νG) and λ(F,i) (1 ≤ i ≤ νF ) are the eigenvalues of G and F , respectively. � Now, we consider the case the images of φ ∈ C1(G; Aut(F )) lie in an abelian subgroup of Aut(F ). A vertex-and-edge weighted digraph is a pair Dω = (D,ω), where D is a digraph and ω : V (D) E(D) → C is a function. We call ω the vertex-and-edge weight function on D. Moreover, if ω(e−1) = ω(e), the complex conjugate of ω(e), for each edge e ∈ E(D), we say that ω is symmetric. Given any vertex-and-edge weighted digraph Dω, the adjacency matrix A(Dω) = (aij) of Dω is a square matrix of order | V (D) | defined by aij = e∈E({vi},{vj}) ω(e), and the degree matrix DDω is the diagonal matrix whose (i, i)-th entry is ω(vi). We define FDω(λ, µ) = det (λI − (A(Dω)− µDDω)) . For any Γ-voltage assignment φ of G, let ωi(φ) : V ( ~G) E( ~G) → C be the function defined ωi(φ)(v) = degG(v), ωi(φ)(e) = λ(φ(e),i) for e ∈ E( ~G) and v ∈ V ( ~G) where i = 1, 2, . . . , νF . Using Theorem 3.6, we have the following theorem. Theorem 3.8. Let G be a connected graph and let F be a connected regular graph of degree r. If the images of φ ∈ C1(G; Aut(F )) lie in an abelian subgroup of Aut(F ), then the characteristic polynomial FG×φF (λ, µ) of a graph bundle G× φ F is F ~Gωi(φ) (λ+ rµ− λ(F,i), µ). Now, the following corollary follows immediately from Corollary 3.7. Corollary 3.9. For any connected graph G and a connected r-regular graph F , the char- acteristic polynomial FG×F (λ, µ) of the cartesian product G× F is FG(λ+ rµ− λ(F,i), µ). GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES 9 In particular, if G is a regular graph of degree dG, then the characteristic polynomial FG×F (λ, µ) of the cartesian product G× F is λ+ (r + d)µ− λ(G,j) − λ(F,i) where λ(G,j) (1 ≤ j ≤ νG) and λ(F,i) (1 ≤ i ≤ νF ) are the eigenvalues of G and F , respectively. � 4. Generalized characteristic polynomial of K1,m ×Kn In this section, we find the generalized characteristic polynomial of K1,m ×Kn and find the number of spanning trees of K1,m ×Kn. As we mentioned in introduction, K1,m ×Kn is a typical model for networks with hubs thus, we will count its spanning trees. Since Kn features many nice structures, we discuss the generalized characteristic polynomial of graph bundles with a fiber, Cayley graph. Let A be a finite group with identity idA and let S be a set of generators for A with the properties that S = S−1 and idA 6∈ S, where S −1 = {x−1 | x ∈ Ω}. The Cayley graph Cay(A, S) is a simple graph whose vertex-set and edge-set are defined as follows: V (Cay(A, S)) = A and E(Cay(A, S)) = {{g, h} | g−1h ∈ S}. From now on, we assume that A is an abelian group of order n. Let G be a graph and let φ : E( ~G) → A be an A-voltage assignment. Notice that the left action A on the vertex set A of Cay(A, S) gives a group homomorphism from A to Aut(Cay(A, S)). Let P be the permutation representation of A corresponding to the action. Then the map φ̃ : E( ~G) → Aut(Cay(A, S)) defined by φ̃(e) = P (φ(e)) for any e ∈ E( ~G) is an Aut(Cay(A, S))-voltage assignment. We also denote it φ. Notice that every irreducible representation of an abelian group is linear. For convenience, let χ1 be the principal character of A and χ2, . . . , χn be the other n− 1 irreducible characters of A. Now, by Theorem 3.3, we have that the matrix A(G×φ Cay(A, S))− µD(G×φ Cay(A, S)) is similar to |S|(1− µ))IνG + A( ~G)− µD(G) (χi(S)− |S|µ) IνG + χi(γ)A( ~G(φ,γ))− µD(G) where χi(S) = χi(s) for each i = 2, 3, . . . , n. By Theorem 3.8, we have that the characteristic polynomial FG×φCay(A,S)(λ, µ) of a graph bundle G× φ Cay(A, S) is FG(λ+ |S|(µ− 1), µ)× F ~Gωi(φ) (λ+ |S|µ− χi(S), µ), where ωi(φ) : V ( ~G) E( ~G) → C be the function defined by ωi(φ)(v) = degG(v), ωi(φ)(e) = χi(φ(e)) 10 DONGSEOK KIM, HYE KYUNG KIM, AND JAEUN LEE for e ∈ E( ~G) and v ∈ V ( ~G) where i = 2, 3, . . . , n. Let Kn be the complete graph on n vertices. Then Kn is isomorphic to Cay(A,A− {idA}) for any group A of order n. Since χ1(A− {idA}) = n− 1 and χi(A− {idA}) = −1 for each i = 2, 3, . . . , n, we have FG×φKn(λ, µ) = FG(λ+ (n− 1)(µ− 1), µ)× F ~Gωi(φ) (λ+ (n− 1)µ+ 1, µ). Moreover over if φ is the trivial voltage assignment, then FG×Kn(λ, µ) = FG(λ+ (n− 1)(µ− 1), µ)× FG(λ+ (n− 1)µ+ 1, µ) Let G be the complete bipartite graph K1,m which also called a star graph. Notice that K1,m is a tree and hence every graph bundle K1,m × φ F is isomorphic to the cartesian product K1,m × F of K1,m and F . It is known [9] that for any natural numbers s and t FKs,t(λ, µ) = (λ+ tµ) s−1(λ+ sµ)t−1 [(λ+ sµ)(λ+ tµ)− st] and hence FK1,m(λ, µ) = (λ+ µ) m−1 [(λ+ µ)(λ+mµ)−m]. Now, we can see that FK1,m×φKn(λ, µ) = FK1,m(λ+ (n− 1)(µ− 1), µ)× FK1,m(λ+ (n− 1)µ+ 1, µ) = [λ+ nµ− (n− 1)]m−1 {[λ + nµ− (n− 1)][λ+ (m+ n− 1)µ− (n− 1)]−m} ×[λ + nµ+ 1](m−1)(n−1) {[λ+ nµ+ 1][λ+ (m+ n− 1)µ+ 1]−m} Now, by applying Theorem 2.1, we have that the number of spanning trees of K1,m ×Kn κ(K1,m ×Kn) = n n−2(m+ n+ 1)n+1(n+ 1)(m−1)(n−1). References [1] L. Bartholdi, Counting pathes in graphs, Enseign. Math., 45 (1999), 83–131. [2] H. Bass, The Ihara.Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992), 717–797. [3] N. Biggs, Algebraic Graph Theory, 2nd ed Cambridge University Press, London, 1993. [4] R. Brualdi and H. Ryser, Combinatorial matrix theory, Cambrige Univ. Press, Cambridge, 1991. [5] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Academic Press, NewYork, 1979. [6] Y. Chae, J. H. Kwak, and J. Lee, Characteristic polynomials of some graph bundles, J. Korean Math. Soc. 30 (1993), 229–249. [7] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, New York 1987. [8] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 18 (1966), 219–235. [9] H. K. Kim and J. Lee, A generalized characteristic polynomial of a graph having a semifree action, Discrete Mathematics, to appear. [10] J. H. Kwak and J. Lee, Characteristic polynomials of some graph bundles II, Linear and Multilinear Algebra 32 (1992), 61–73. [11] J. H. Kwak, J. Lee, M. Y. Sohn, Bartholdi zeta functions of graph bundles having regular fibers , European Journal of Combinatories 26 (2005), 593–605. [12] R. Lipton, N. Vishnoi and Z. Zalcstein, A Generalization of the Characteristic Polynomial of a Graph, CC Technical Report; GIT-CC-03-51, http://citeseer.ist.psu.edu/642697.html [13] H. Mizuno and I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B, 89 (2003), 17–26. http://citeseer.ist.psu.edu/642697.html GENERALIZED CHARACTERISTIC POLYNOMIALS OF GRAPH BUNDLES 11 [14] H. Mizuno and I. Sato, Zeta functions of graph coverings, J. Combin. Theory Ser. B, 80 (2000), 247–257. [15] H. Mizuno and I. Sato, Bartholdi zeta functions of graph coverings, J. Combin. Theory Ser. B, 89 (2003), 27–41. [16] S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B, 74 (1998), 408–410. [17] C. Oliveira, N. Maia de Abreu and S. Jurkiewicz, The characteristic polynomial of the Laplacian of graphs in (a, b)-linear classes, Linear Algebra and its Applications, 356(1) (2002), 113–121. [18] J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York, 1977. [19] H. Stark, A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996), 126–165. Department of Mathematics, Kyungpook National University, Taegu, 702-201 Korea E-mail address : [email protected] Mathematics, Catholic University of Taegu, Kyongsan, 712-702 Korea E-mail address : [email protected] Department of Mathematics, Yeungnam University, Kyongsan, 712-749, Korea E-mail address : [email protected] 1. Introduction 2. Generalized characteristic polynomials and complexity 3. Generalized characteristic polynomials of graph bundles 4. Generalized characteristic polynomial of K1,mKn References

0704.1432

Some invariants of pretzel links

SOME INVARIANTS OF PRETZEL LINKS DONGSEOK KIM AND JAEUN LEE Abstract. We show that nontrivial classical pretzel knots L(p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n- pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same. 1. Introduction Let L(p1, p2, . . . , pn) be an n-pretzel link in S 3 where pi ∈ Z represents the number of half twists as depicted in Figure 1. In particular, if n = 3, it is called a classical pretzel link, denoted by L(p, q, r). If n is odd, then an n-pretzel link L(p1, p2, . . . , pn) is a knot if and only if none of two pi’s are even, a pretzel knot is denoted by K(p1, p2, . . . , pn). If n is even, then L(p1, p2, . . . , pn) is a knot if and only if only one of the pi’s is even. Generally the number of even pi’s is the number of components, unless pi’s are all odd. Since pretzel links have nice structures, they have been studied extensively. For example, several polynomials of pretzel links have been calculated [13, 15, 22, 28]. Y. Shinohara calculated the signature of pretzel links [34]. Two and three fold covering spaces branched along pretzel knots have been described [4, 19]. For notations and definitions, we refer to [2]. A link L is almost alternating if it is not alternating and there is a diagram DL of L such that one crossing change makes the diagram alternating; we call DL an almost alternating diagram. One of the classifications of links is that they are classified by hyperbolic, torus or satellite links [2]. First we show that classical pretzel links are prime and either alternating or almost alternating. Menasco has shown that prime alternating knots are either hyperbolic or torus knots [24]. It has been generalized by Adams that prime almost alternating knots are either hyperbolic or torus knots [1]. It is known that no satellite knot is an almost alternating knot [17]. Thus, we can classify classical pretzel knots completely by hyperbolic or torus knots. Let L be a link in S3. A compact orientable surface F is a Seifert surface of L if the boundary of F is L. The existence of such a surface was first proven by Seifert using an algorithm on a diagram of L, named after him as Seifert’s algorithm [33]. The genus of a link L can be defined by the minimal genus among all Seifert surfaces 2000 Mathematics Subject Classification. 57M25, 57M27. Key words and phrases. pretzel links, Conway polynomial, Seifert surfaces, genus, basket number. The first author was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-351-C00005). The second author was supported in part by Com2MaC-KOSEF(R11-1999-054). http://arxiv.org/abs/0704.1432v1 2 DONGSEOK KIM AND JAEUN LEE p1 p2 p3 . . . pn−1 pn Figure 1. An n-pretzel link L(p1, p2, . . . , pn) of L, denoted by g(L). A Seifert surface F of L with the minimal genus g(L) is called a minimal genus Seifert surface of L. A Seifert surface of L is canonical if it is obtained from a diagram of L by applying Seifert’s algorithm. Then the minimal genus among all canonical Seifert surfaces of L is called the canonical genus of L, denoted by gc(L). A Seifert surface F of L is said to be free if the fundamental group of the complement of F , namely, π1(S 3 −F) is a free group. Then the minimal genus among all free Seifert surfaces of L is called the free genus for L, denoted by gf (L). Since any canonical Seifert surface is free, we have the following inequalities, g(L) ≤ gf(L) ≤ gc(L). There are many interesting results about the above inequalities [5, 8, 21, 26, 29, 32]. Gabai has geometrically shown that the minimal genus Seifert surface of n-pretzel links can be found as a Murasugi sum using Thurston norms and proved that the Seifert surfaces obtained by applying Seifert’s algorithm to the standard diagram of L(2k1 + 1, 2k2 + 1, . . . , 2kn + 1) and L(2k1, 2k2, . . . , 2kn) are minimal genus Seifert surfaces [12]. There is a classical inequality regarding the Alexander polynomial and the genus g(L) of a link L: G. Torres showed the following inequality, 2g(L) ≥ degree∆L − µ+ 1(1) where ∆L is the Alexander polynomial of L and µ is the number of components of L [36]. R. Crowell showed that the equality in inequality (1) holds for alternat- ing links [8]. Cimasoni has found a similar inequality from multi-variable Alexander polynomials [6]. In fact, we can find the genera of oriented n-pretzel links from the in- equality (1) and the Conway polynomial found in section 3, i.e., we will show that the equality in inequality (1) holds for all n-pretzel links with at least one even crossing. For pretzel links L(2k1, 2k2, . . . , 2kn) with all possible orientations, Nakagawa showed that a genus and a canonical genus are the same [28]. The idea of Nakagawa [28] can be extended to arbitrary n-pretzel links, i.e., we can show that these three genera g(L), gf(L) and gc(L) are the same. Some of Seifert surfaces of links feature extra structures. Seifert surfaces obtained by plumbings annuli have been studied extensively for the fibreness of links and sur- faces [10–12, 14, 25, 29, 31, 35]. Rudolph has introduced several plumbed Seifert sur- faces [30]. Let An ⊂ S 3 denote an n-twisted unknotted annulus. A Seifert surface F is a basket surface if F = D2 or if F = F0 ∗α An which can be constructed by plumbing SOME INVARIANTS OF PRETZEL LINKS 3 Figure 2. An alternating diagram of L(q,−1, r). An to a basket surface F0 along a proper arc α ⊂ D2 ⊂ F0 [30]. A basket number of a link L, denoted by bk(L), is the minimal number of annuli used to obtain a basket surface F such that ∂F = L [3, 16]. As a consequence of the results in section 4 and a result [3, Corollary 3.3], we find the basket number of n-pretzel links. The outline of this paper is as follows. In section 2, we mainly focus on the classical pretzel links L(p, q, r). We find Conway polynomial of n-pretzel links in section 3. In section 4, we study the genera of n-pretzel links. In section 5, we compute the basket number of n-pretzel links. 2. Classical pretzel links L(p, q, r) 2.1. Almost alternating. One can see that L(p, q, r) is alternating if p, q, r have the same signs. Since every alternating link (including any unlink) has an almost alter- nating diagram, we are going to show that every nontrivial pretzel link has an almost alternating diagram. Since the notation depends on the choice of +,− crossings, it is sufficient to show that L(−p, q, r) has an almost alternating diagram where p, q, r are positive. In particular, one might expect that L(−1, q, r) is almost alternating, but surprisingly it is also alternating. Theorem 2.1. For positive integers p, q and r, L(−1, q, r) is an alternating link and L(−p, q, r) has an almost alternating diagram. Proof. One can see that L(q,−1, r) is isotopic to L(q−2, 1, r−2) as shown in Figure 2. For the second part, see Figure 3. � Theorem 2.2. All nontrivial pretzel knots K(p, q, r) are either torus knots or hyper- bolic knots. Proof. The key ingredient of theorem is that prime alternating (almost alternating) knots are either hyperbolic or torus knots [24, Corollary 2] ( [1, Corollary 2.4], respec- tively). Since every pretzel knot has an almost alternating diagram by Theorem 2.1, we need to show that all nontrivial classical pretzel knots are prime. Since no two of p, q, r are even, there are two cases : i) all of them are odd, ii) exactly one is even. 4 DONGSEOK KIM AND JAEUN LEE Figure 3. An almost alternating diagram of L(p,−q, r). i) p ≡ q ≡ r ≡ 1 (mod 2). For this case, we can use the genus of K = K(p, q, r). Suppose K = K1#K2. Since a Seifert surface of K is the punctured torus, it has genus 1 as described in the left top of Figure 2. But 1 = g(K) = g(K1)+g(K2). Thus one of g(K1) or g(K2) has to be 0, i.e., one of Ki is trivial. Therefore K cannot be decomposed as a connected sum of two nontrivial knots. ii) Suppose that p is even i.e., p = 2l, and q, r are odd. Then it is easy to see that the left two twisting parts form a prime tangle (except when |p| = 2l and |q| = 1). The right part is an untangle, but since r is odd, we can use a result of Lickorish [23, Theorem 3] to conclude that K(2l, q, r) is prime. For the above exceptional cases, we can assume that |r| = 1 because we can choose |q| ≥ |r|. So all possible cases are K(2l,±1,∓1), K(2l, 1, 1) and K(2l,−1,−1). But the first one is the unknot and the other two can be deformed to K(p, q, r) of all odd crossings, i.e, K(2l,−1,−1) = K(2l − 1, 1, 1) and K(2l, 1, 1) = K(2l + 1,−1,−1). This completes the proof. � 2.2. Prime torus pretzel knots. The primary goal of this section is to decide which classical pretzel knots are torus knots. For convenience, the (m,n) torus link is denoted by T(m,n). One can see that all 2-string torus links are alternating. C. Adams has conjectured that only (3, 4) and (3, 5) torus knots are almost alternating [1]. One can see that K(−2, 3, 3) is the (3, 4) torus knot and K(−2, 3, 5) is the (3, 5) torus knot. Since the branched double cover of a torus link is a Seifert fibred space with the base surface S2 and at most three exceptional fibers, and the branched double cover of a nontrivial n-pretzel link is a Seifert fibred space with n exceptional fibers, there will be no torus knot of the form K(p1, p2, · · · , pn) for n ≥ 4 and |pi| ≥ 2. To find all torus knots, we use the Jones polynomials of K(2l, q, r) because the genera of pretzel knots tell us that no K(p, q, r), with p, q, r all odd, is a torus knot except the unknot and trefoil, and it is known that K(p,−1,1) is the unknot and K(±1,±1,±1) are trefoils, which are the only torus knots of genus 1. Remark that the genus of an (m,n) torus knot is (m − 1)(n − 1)/2. The Jones polynomial of an (m,n) torus link (m ≤ n) is given by equation (2) if m is odd, by equation (3) if 4 ≤ m is even, and by equation (4) if m = 2 and n is even. This is due to the original work by Jones [18] but still there is no combinatorial proof for these formulae. SOME INVARIANTS OF PRETZEL LINKS 5 −t(m−1)(n−1)/2[tm+n−2 + tm+n−4 + · · ·+ tn+1 − tm−1 − · · · − t2 − 1],(2) −t(m−1)(n−1)/2[tm+n−2 + tm+n−4 + · · ·+ tn − tn−1 − · · · − t2 − 1],(3) −t(n−1)/2[tn − tn−1 + tn−2 − · · · − t3 + t2 + 1].(4) Using a formula for the Jones polynomials of n-pretzel knots in [22], we find the following lemma. Since the Jones polynomial of the mirror image L of L can be found by VL(t) = VL(t −1), we may assume q, r are positive integers. Lemma 2.3. Let l, q, r be positive integers. Let k = 2l + q + r. VK(2,1,r) = t (r+1)/2−(2+1)(tr+2+1 − 2tr+2 + 2tr+1 − · · ·+ 2t3 − t2 + t− 1), VK(2l,q,r) = t (q+r)/2−(2l+1)(tk − 2tk−1 + 3tk−2 − 4tk−3 + · · · − 3t2 + t− 1), if l ≥ 1, VK(2l,−q,r) = −t (−4l−3q+r)/2(tq+r − tq+r−1 + · · · − t+ 1) if q > 1, VK(−2,1,r) = −t (r+1)/2(tr+2 − tr+1 + tr − · · ·+ t3 − t2 − 1), VK(−2,3,3) = −t 3(t5 − t2 − 1), VK(−2,3,5) = −t 4(t6 − t2 − 1), VK(−2,3,r) = −t (3+r)/2(t3+r−2 − tr + · · · − t2 − 1) if r ≥ 7, VK(−2,q,r) = −t (q+r)/2(−tq+r−1 + 2tq+r−2 − · · · − t2 − 1) if q, r ≥ 5, VK(−2l,q,r) = −t (q+r)/2(at∗ + · · · ± t∓ 1) if l, q, r > 1. By comparing Jones polynomials of pretzel knots in Lemma 2.3 and Jones polyno- mials of torus knots in equation (2), (3) and (4), we find the following theorem. Theorem 2.4. The following are the only nontrivial pretzel knots which are torus knots. 1) K(p,±1,∓1) are unknots for all p. 2) K(±1,±1,±1) are (2,±3) torus knots. 3) K(±2,∓1,±r) are (2,±r ∓ 2) torus knots. 4) K(∓2,±3,±3), K(∓2,±3,±5) are (3,±4), (3,±5) torus knots, respectively. Proof. We only need to consider K(2l, q, r). We can see that K(2,−1,r) can be de- formed to K(0,1,r − 2) by a move shown in Figure 2. The coefficient of t1 and the second leading coefficient of the Jones polynomial of a torus knot are zero, but by Lemma 2.3 these are possible only for K(−2, 3, 3), K(−2, 3, 5) and their mirror im- ages. But the number of terms in the Jones polynomials of these knots is 3, and only (3, n) torus knots have this property. By comparing the terms of the highest degree, we conclude that K(∓2,±3,±3) and K(∓2,±3,±5) are the remaining non-alternating torus knots. � 6 DONGSEOK KIM AND JAEUN LEE 2.3. Minimal genus Seifert surfaces. When one applies Seifert’s algorithm to a diagram of a link L, in general one may not get a minimal genus Seifert surface. In fact, Moriah found infinitely many knots which have no diagram on which Seifert’s algorithm produces a minimal genus surface [26]. But it is known that a minimal genus Seifert surface can be obtained from an alternating diagram by applying Seifert’s algorithm [27] and more generally, alternative links [20]. We prove that the Seifert surface obtained by applying Seifert’s algorithm to the diagram in Figure 4 of a pretzel knot K(p, q, r) is a minimal genus Seifert surface. Since K(2l, q, r) and its mirror image are alternating, without loss of a generality, we only need to find Alexander polynomials of K(−2l, q, r) and K(−2l, q,−r). Lemma 2.5. Let l, q, r be positive integers. ∆K(−2l,q,r)(t) = t −(q+r)/2(ltq+r − (2l − 1)tq+r−1 + · · · − (2l − 1)t+ l), ∆K(−2l,q,−r)(t) = t −(q+r−2)/2(tq+r−2 − 2tq+r−3 + · · · − 2t+ 1). Proof. One can prove inductively the lemma by the following recurrence formulae which come from the skein relations, and the formulae for the Alexander polynomial of the (2, p) torus links. ∆T(2,±p)(t) = t (1−p)/2(tp−1 − tp−2 + · · · − t+ 1) if p is odd, ∆T(2,±p)(t) = t (1−p)/2(−tp−1 + tp−2 + · · · − t+ 1) if p is even, ∆K(−2,q,±r)(t) = ∆T(2,q)(t)∆T(2,r)(t) + (t −1/2 − t1/2)∆T(2,q±r)(t), ∆K(−2l,q,±r)(t) = ∆K(−2(l−1),q,±r)(t) + (t −1/2 − t1/2)∆T(2,q±r)(t). Theorem 2.6. The surface obtained by applying Seifert’s algorithm to the pretzel knot K(p, q, r) as in Figure 4 is a minimal genus Seifert surface, if 1/|p|+1/|q|+1/|r| ≤ 1. Proof. We consider two cases : i) all of p, q, r are odd, ii) exactly one of p, q, r is even. For the first case, the first Seifert surface in Figure 4 is clearly a minimal genus since its genus is 1 unless K(p, q, r) is the unknot. But it can not be the unknot by the hypothesis. For the second case, we can consider K(−2l, q,±r), K(−2l, q,±r) or their mirror images, where l, q, r are positive. Their canonical Seifert surfaces are given in Figure 4. To prove these surfaces are minimal genus Seifert surfaces, first we find 2g(K(−2l, q,±r)) ≥ q+r−1±1 using the Alexander polynomials of K(−2l, q, r) and K(−2l, q,−r) given in Lemma 2.5 and inequality (1). But the genus of the second Seifert surface in Figure 4 is (q+r)/2, and the third surface in Figure 4 is (q+r−2)/2. It completes the proof. � By combining Theorem 2.4 and Theorem 2.6, we find the following corollary. Corollary 2.7. The genus of K(p, q, r) is as follows. 1) K(p,±1,∓1), K(±2,∓1,±3) have genus 0 for all p. SOME INVARIANTS OF PRETZEL LINKS 7 , ±p ±q 2l ±p 2l ∓r Figure 4. Minimal genus Seifert surfaces of the pretzel knots K(p, q, r). 2) K(p, q, r) has genus 1 if p ≡ q ≡ r ≡ 1 (mod 2) and we are not in case 1). 3) K(±2,∓1,±r) has genus (|r − 2| − 1)/2. 4) K(∓2l, q, r) has genus (|q|+ |r|)/2 if q, r have the same sign and we are not in any of the previous cases. 5) K(∓2l, q, r) has genus (|q| + |r| − 2)/2 if q, r have different signs and we are not in cases 1), 2) or 3). For classical pretzel links, one can see that L(2l1, 2l2, 2l3) has genus 0. For L(2l1, 2l2, r), we are going to see more interesting results for the genus because there is a freedom to choose orientations of the components. But, Lemma 2.5 remains true for arbitrary integers q, r, so we can find the following corollary. Corollary 2.8. The genus of the link L(2l1, 2l2, r), where |l1| ≥ |l2| , l1, l2 > 0(unless we indicate differently) and r ≥ 0, is as follows. 1) L(2l1, 2l2,±r) has genus 0 if r ≡ 0 (mod 2) and l1, l2 are nonzero integers. 2) L(±2,±2l2,∓1) has genus (|2l2 − 2| − 2)/2. 3) K(∓2l1,∓2l2,∓r) has genus (|l2| + |r| − 1)/2 if we are not in one of the previous cases. 4) K(∓2l1,∓2l2,±r) has genus (|l2| + |r| − 3)/2 if we are not in any of the previous cases. 5) K(∓2l1,±2l2,∓r) has genus (|l2|+ |r| − 3)/2 if we are not in case 1). 8 DONGSEOK KIM AND JAEUN LEE o1 o2 o3 . . . o2k o2k+1 e1 o2 o3 . . . o2k o2k+1 e1 o2 o3 . . . o2k−1 o2k (8, 9) o1 o2 o3 . . . o2k−1 o2k Figure 5. all oriented n-pretzel knots L(p1, p2, . . . , pn) 6) K(∓2l1,±2l2,±r) has genus (|l2|+ |r|−1)/2 if we are not in case 1) and |l1| > |l2|, or has genus (|l2|+ |r| − 3)/2 if we are not in case 1) and |l1| = |l2|. Proof. We follow the proof of Theorem 2.4 and Theorem 2.6 carefully ; if r = ±1, the link will have two representatives by the move we used in the proof of Theorem 2.1, we get the result, with a note that we have a freedom to choose an orientation of the component which goes through two even crossing boxes. � 3. Conway polynomials of n-pretzel links To find the polynomial invariants of n-pretzel links, we will use a computation tree : a computation tree of a link polynomial PL is an edge weighted, rooted binary tree whose vertices are links, the root of the tree is L, two vertices L1, L2 are children of a vertex Lp if PLp = w(Lp(1))PL1 + w(Lp(2))PL2 , and w(Lp(i)) is the weight on the edge between Lp and Li. One can see that the link polynomial PL can be computed as follows, Lp∈P(Lv) w(Lp(i))PLv , where L is the set of all vertices of valence 1 and P(Lv) is the set of all vertices of the path from the root to the vertex Lv. In general, it is easy to find PL if we repeatedly use the skein relations until each vertex Lv becomes an unlink. Moreover, one can replace links by other for a convenience of the computation. For instance, J. Franks and R. F. Williams used braids to find a beautiful result on Jones polynomial [9]. SOME INVARIANTS OF PRETZEL LINKS 9 To compute Conway Polynomials of n-pretzel links, we will use a new notation for n-pretzel links which will be used for vertices of a computation tree. We called a rectangle in Figure 5 a box and the link moves in the same direction in a box if it has the orientation as in the second box from the left of the diagram (6) of Figure 5, in the opposite directions if it has the orientation as in the first box from the left of the diagram (6) in Figure 5. If we have a box for which two strings move in the opposite directions and we use the skein relation at this box, then the resulting links have either less number of the boxes or less number of crossings. One can see that an opposite direction can be happened only for a box with even number of crossings (but this is not sufficient) except in the case that n is even and all the pi’s are odd (we will handle this case separately). Suppose we have at least one even crossing box. We may assume that it is p1 = 2l1. Let us remark that the Conway polynomial vanishes for split links. The following is our new notation for n-pretzel links. From a given n-pretzel link L with an orientation O, we can represent L by a vector in (Z×Z2) n such as (pǫ11 , p 2 , . . . , p n ), where ǫi = 1(−1) if the link moves in the same(opposite, respectively) direction in the box corresponding to pi with respect to the given orientation O. Write p1i = pi. First we find the following recursive formula, ∇L(pǫ11 ,p 2 ,...,p ,...,p . . . ˆ∇T . . .∇T − liz∇ 2 ,..., ,...,p where the term under ˆ is deleted. By repeatedly using above formulae, we can make a computation tree that there is no negative ǫi for the representative at each vertex of valence 1. Then, we can expand (. . . , pi, . . .) into (. . . , pi ± 1(= p i), . . .) and (. . . , pi ± 2, . . .) with suitable weights on edges, 1 or ±z where |pi| > |p i|. We can keep on expanding at the crossings until all the entries in the vectors of vertices of valence 1 are either 0 or ±1. At this stage, if it has more than two 0’s then we stop the expansion and change the vertex to zero because it is a split link. If it has only one zero, it is a composite link of T(2,pi)’s. Otherwise, we change the vector to an integral value m, the sum of the signs of entries in the vector. In fact, it is the closed braid of two strings represented by σm1 . Therefore, we can compute the Conway polynomial of a link L using this computation tree and the Conway polynomial of closed 2-braids. 3.1. Conway polynomial of n-pretzel knots. The general figures of n-pretzel knots are given in Figure 5 (the right-top one is a two components link) where e1 = 2l, oi = 2ki + 1. We can see that there is at most one box in which the knot moves in opposite directions. But for a two component link, all boxes might move in opposite directions for the orientation which is not in Figure 5. Counterclockwise from the top-right, we get representatives, (o−11 , o 2 , . . . , o 2k ), (o1, o2, . . . , o2k), (o 1 , o 2 , . . . , o−12k+1), (e 1 , o2, o3, . . . , o2k+1) and (e1, o2, o3, . . . , o2k). By using a computation tree for these representatives, we find Theorem 3.1. For convenience, we abbreviate ∇T(2,n) by ∇n throughout the section. 10 DONGSEOK KIM AND JAEUN LEE Theorem 3.1. Let e′1 = sign(e1)(|e1| − 1), o i = sign(oi)(|oi| − 1), α = i=2 sign(oi) and β = sign(e1). The Conway polynomials of n-pretzel knots in Figure 5 are ∇L(o1,o2,o3,...,on) = (n−1)/2 2i,(5) ∇L(e1,o2,o3,...,on) = ∇o2∇o3 . . .∇on [1− lz[− ]],(6) ∇L(e1,o2,o3,...,on) = ∇o2∇o3 . . .∇on [∇e′1 +∇e1 [− β + α ]],(7) ∇L(o1,o2,o3,...,on) = (n+1)/2 2i−1,(8) ∇L(o1,o2,o3,...,on) = ∇o1∇o2 . . .∇on [∇ i=1 sign(oi) ],(9) where for L(o1, o2, o3, . . . , on) we have two possible orientations because it is a two components link, so we get (8) for (o−11 , o 2 , . . . , o 2k ) and (9) for (o1, o2, . . . , o2k). Proof. We will only prove (6) but one can prove the other by a similar argument. In the computation tree, we use skein relation at crossings until vertices of valence 1 in the computation tree up to this point will be (c1, c2, . . . , cn) where ci is either 0 or ±1. Since the Conway polynomials of split links vanish, we may assume there are no than one 0’s. The first term in the parenthesis comes from the case where all |ci| are 1 because it is again the (2, α) torus link horizontally. It is a two component link with linking number −α/2, so its Conway polynomial is −(α/2)z. For the case where only one ci = 0, the values on edges to the vertex will contribute exactly ∇o′ and the vertex is the composite link of (2, oj) torus knots j = 2, . . . , n except i. � 3.2. Conway polynomials of n-pretzel links. Since we have already handled links of all odd crossings, we assume that n-pretzel links have at least one even crossing box. Let L(p1, p2, . . . , pn) be an n-pretzel link and let s be the number of even pi’s. Then it is a link of s components. The Conway polynomial ∇L depends on the choice of the orientation of L. There are 2s−1 possible orientations of L. But one can easily see that the link always moves in the same direction in all boxes of odd crossings for arbitrary orientation. For further purpose, we will calculate the Conway polynomial of the pretzel link with the following orientations. For the existence of such orientations, we will prove it in Lemma 4.2 : if n − s is even, then there exists an orientation O of L such that the link L moves in the opposite directions in all boxes of even pi. If n − s is odd, then there exists an orientation O of L such that the link L moves in the opposite directions in all boxes of even pi except one pt but without loss of a generality we assume that p1 = pt. Theorem 3.2. Let L(p1, p2, . . . , pn) be a pretzel link with the above orientation O. Let pei = 2li be all even and poj = 2kj +1 be all odd. Let s be the number of even pi’s SOME INVARIANTS OF PRETZEL LINKS 11 and let α = i=1 sign(poi) and β = sign(p1). Let p i = sign(pi)(|pi| − 1). If n− s is even, then the Conway polynomial of L(p1, p2, . . . , pn) is (−li)]z ∇poi )[− ∇p′oi ] + [ j=1,j 6=i (−lj)]z If n− s is odd, then the Conway polynomial of L(p1, p2, . . . , pn) is (−li)]z ∇poi)∇p1 [− α + β ∇p′oi ] + [ j=2,j 6=i (−lj)]z Proof. It is clear by choosing (p−1e1 , p , . . ., p−1es , po1 , . . ., pon−s) and (pe1, p , . . ., p−1es , po1 , . . ., pon−s), respectively. � More generally, we get the following results by taking (p−1e1 , p , . . ., p−1et , pet+1, . . ., pes , po1, . . ., pon−s) for a representative of L(p1, p2, . . . , pn) induced by an orientation Theorem 3.3. Let pei = 2li be all even and poj = 2kj + 1 be all odd. Let s be the number of even pi. Let t be the number of even pi in the corresponding boxes in which the link moves in the opposite direction, say pei where i = 1, 2, . . ., t. and let j=1 sign(poj) and β = i=t+1 sign(pei). Let p i = sign(pi)(|pi| − 1). Then the Conway polynomial of L(p1, p2, . . . , pn) with the orientation O is (−li)]z ∇poi)( ∇pej )[− α + β i=t+1 ∇p′ei ∇p′oj ] + [ j=1,j 6=i (−lj)]z 4. Genera of n-pretzel links We will consider the genus of an n-pretzel link with at least one even crossing box. Let FL be a Seifert surface of an n-pretzel link L. For the rest of the section, let χ(FL) be the Euler characteristic of FL, V be the number of Seifert circles, E be the number of crossings and F be the number of the components of L. 4.1. Genera of n-pretzel knots with one even pi. We divide into two cases : i) n is odd, ii) n is even. For the first case: n is odd, we can see that the degree of ∇K(e1,o1,o2,...,on) is degree(∇oi) = 2 + (|oi| − 1), and the coefficient of this leading term is −lα/2 from Theorem 3.1. 12 DONGSEOK KIM AND JAEUN LEE Suppose α is nonzero. Then the Seifert surface F obtained by applying Seifert’s algorithm to the diagram in Figure 5 is a minimal genus surface. The genus of the Seifert surface FK is g(FK) = [2− χ(FK)] = (2− V + E − F ) [2− (|e1|+ n− 2) + (|e1|+ |oi|)− 1] = (|o1| − 1)] degree ∇K(e1,o1,o2,...,on). Suppose α = 0. This means that we have the same number of positive and negative twists on odd twists. If we look at the Conway polynomial in equation 6, we drop exactly one in degree with new leading coefficient 1. It is sufficient to show that the degree of the following term is negative. Remark that ∇oi = ∇−oi . −lz[− ] = −l[0 + ] = −l[ sign(oi)(∇|oi| −∇|oi|−2) ∇|oi| = −l[ (sign(oi) + ∇|oi|−2 ∇|oi| )] = −l[ ∇|oi|−2 ∇|oi| We hope to find a minimal surface of this genus. For the first case, the sign of an n-pretzel is (±,±, . . . ,±, even,∓,∓, . . . ,∓). The rule is to use the move from the outmost pair. Then the moves in Figure 6 will increase V by two but will not change E, F (= 1); thus we get a surface with one less genus. If we represent the move by the Conway notation for algebraic links [7], it is either (. . . ,−a, . . . , b, . . .) ⇒ (. . . , (−1)(−a+1), . . . , (b−1)(1), . . .) or (. . . , a, . . . ,−b, . . .)⇒ (. . . , (1)(a−1), . . .,(−1) (−b+ 1),. . .) where the sign sum of the oi’s between a, b has to be zero. For the general case, if we only look at the signs of the odd twists from o1, we can find a pair oi, oj such that we can apply the move we described above. The resulting diagram satisfies the same hypothesis with strictly smaller twisted bands. Inductively we get a well-defined sequence of moves which makes the desired diagram on which Seifert’s algorithm will produce a minimal genus surface. Figure 6 shows the effect on V,E. This completes the case i). For the second case, n is even, we can see that the degree of ∇K(e1,o1,o2,...,on) is 1 + degree(∇e1) + degree(∇oi) = |e1|+ (|oi| − 1), and the coefficient of the leading term is −sign(e1)(α + β)/2 from Theorem 3.3. Suppose α+β is nonzero. Then the Seifert surface F obtained by applying Seifert’s algorithm to the diagram in Figure 5 is a minimal genus surface. The genus of the Seifert surface FK is SOME INVARIANTS OF PRETZEL LINKS 13 o3on−1 ∼= Figure 6. How to modify a diagram in Figure 5 to find a minimal genus diagram of L(p1, p2, . . . , pn). g(FK) = [2− χ(FK)] = (2− V + E − F ) [2− (n) + [|e1|+ (|oi|)]− 1] = [|e1|+ (|o1| − 1)] degree∇K(e1,o1,o2,...,on). Suppose α + β = 0. This means that we have the same number of positive and negative twists. As we did before we drop exactly one in the degree of the Conway polynomial in equation 7 with new leading coefficient 1. All arguments are the same if we change the term in parentheses in the equation as follows. [∇e′1 +∇e1(− β + α )] = ∇e1 [− β + α We can find a minimal surface of this genus by the same method as shown in Figure 6 if we handle the even crossing box together. This gives us the following theorem. Theorem 4.1. Let K(p1, o2, o3, . . . , on) be an n-pretzel knot with one even p1. Let α i=2 sign(oi) and β= sign(p1). Suppose |p1|, |oi| ≥ 2. Let (|oi| − 1). 14 DONGSEOK KIM AND JAEUN LEE Figure 7. Boundary orientation of L(p1, p2, . . . , pn). Then the genus g(K) of K is g(K) = (δ + 2) if n is odd and α 6= 0, δ if n is even and α = 0, (|p1|+ δ) if n is even and α + β 6= 0, (|p1|+ δ)− 1 if n is even and α + β = 0. 4.2. Genera of n-Pretzel links. Intuitively, if we have more even pi’s with opposite directions, then we will have a surface of smaller genus. So we want to choose an orientation which forces all the even pi’s to move in the opposite directions, but this may not be possible for all cases. Lemma 4.2. Let L(p1, p2, . . . , pn) be an n-pretzel link and let s be the number of even pi’s. If n− s is even, then there exists an orientation of L such that the link L moves in opposite directions in all boxes of even pi. If n − s is odd and a given pt is even, then there exists an orientation of L such that the link L moves in opposite directions in all boxes of even pi’s except the one corresponding to pt. Proof. If all pj between two even pi and pk are odd, the number of these pj ’s odd (mod 2) will decide the boundary orientation as depicted in Figure 7. If the number of odd crossing boxes is even, we can orient the link such that the link moves oppositely in all boxes of even crossings. Otherwise there is just one box for which the link moves in the same direction. So starting from pt will complete the proof. � Let us denote the orientation we choose in Lemma 4.2 by O′. From Theorem 3.2, we can do almost the same comparison by using equation (1). But we have to be careful to use (1) for links. Since it was defined for oriented links, we can interpolate it as follows. g(L) = minO{min{genus of F(L,O) | FL,O is a Seifert surface with the orientation O}}. where the first O runs over all possible orientations of L. So (1) gives us an inequality on the second minimum of the fixed orientation O and ∇(L,O). SOME INVARIANTS OF PRETZEL LINKS 15 We divide into two cases : i) n− s is even, ii) n− s is odd. For the first case, n− s even, we can see that the degree of ∇L(p1,p2,...,pn) is degree(∇pmi ) + 1 = s+ (|pmi | − 1) + 1, and the coefficient of this leading term is −α/2 from Theorem 3.2. Suppose α is nonzero. Then the Seifert surface F obtained by applying Seifert’s algorithm with the fixed orientation O′ is a minimal genus surface of (L,O′). Let us find the genus of the Seifert surface F(L,O′). 2g(FL) = 2− χ(FL) = 2− (V − E + F ) = 2− (n− s) + ( (|pmj | − 1)) + [ |pij |+ (|pmi|)]− s = 2 + (|pki| − 1) = degree(∇L)− s + 1. For the rest of the cases of the arguments are parallel to the argument for knots. Next, we explain how pt will be chosen for the rest of the article. Remark 4.3. First, we look at the minimum of the absolute value of pei over all even crossings. If the minimum is taken by the unique pei or by pei’s of the same sign, we choose it for pt. If there are more than two pei’s with different signs and the same absolute value, then we look at the value α, the sign sum of odd crossings. If it is neither 1 nor −1, then we pick the positive one for pt. If α = 1(−1), pick the negative(positive) one for pt. For the second case, n− s odd, we find pt as described the above. For the last two cases, we will drop the genus by 1. Denote the orientation we chose here by O1. Lemma 4.4. For an arbitrary orientation O, degree∇(L,O) ≥ degree∇(L,O1). Proof. If we count tO, the number of even crossings in which the link moves in the opposite directions with respect to O, we can see that tO ≤ tO1. If we look at the Conway polynomial in Theorem 3.3, we have that i) we can ignore the second term, ii) increasing t by 1 will change the degree of the second term by −(|pi| − 2), and by hypothesis, |pi| ≥ 2. � Theorem 4.5. Let L(p1, o2, . . . , os, es+1, . . . , en) be an n-pretzel link with at least one even pi. Let α = i=2 sign(poi) and β = sign(pt). Suppose |oi|, |ej| ≥ 2. Let pt be the integer described in Remark 4.3. Let l be the number of even pi’s. Let (|oi| − 1). Then the genus g(L) of L is 16 DONGSEOK KIM AND JAEUN LEE g(L) = δ + 1 if n− s is even and α 6= 0, δ if n− s is even and α = 0, (|pt|+ δ) if n− s is odd and α + β 6= 0, (|pt|+ δ)− 1 if n− s is odd and α + β = 0. Proof. It follows from Theorem 3.2, 3.3 and Lemma 4.4. � 5. The basket numbers of pretzel links First let us recall a definition of the basket number. Let An ⊂ S 3 denote an n- twisted unknotted annulus. A Seifert surface F is a basket surface if F = D2 or if F = F0 ∗αAn which can be constructed by plumbing An to a basket surface F0 along a proper arc α ⊂ D2 ⊂ F0. A basket number of a link L, denoted by bk(L), is the minimal number of annuli used to obtain a basket surface F such that ∂F = L. For standard definitions and notations, we refer to [30]. Throughout the section, we will assume all links are not splitable, i.e., Seifert surfaces are connected. Otherwise, we can handle each connected component separately. For the basket number and the genus of a link, there is a useful theorem. Theorem 5.1 ( [3]). Let L be a link of l components. Then the basket number of L is bounded by the genus and the canonical genus of L as, 2g(L) + l − 1 ≤ bk(L) ≤ 2gc(L) + l − 1. Since we have found that a minimal genus surface of a pretzel link L of genus g(L) can be obtained by applying Seifert algorithm on a diagram of L, i.e., g(L) = gc(L), we find that the basket number of a pretzel link L is 2g(L)+l−1, i.e., bk(L) = 2g(L)+l−1. Theorem 5.2. Let K(p1, o2, o3, . . . , on) be an n-pretzel knot with one even p1. Let α i=2 sign(oi) and β= sign(p1). Suppose |p1|, |oi| ≥ 2. Let (|oi| − 1). Then the basket number bk(K) of K, bk(K) = δ + 2 if n is odd and α 6= 0, δ if n is even and α = 0, |p1|+ δ if n is even and α+ β 6= 0, |p1|+ δ − 2 if n is even and α+ β = 0. Theorem 5.3. Let L(p1, o2, . . . , os, es+1, . . . , en) be an n-pretzel link with at least one even pi. Let α = i=2 sign(poi) and β = sign(pt). Suppose |oi|, |ej| ≥ 2. Let pt be the integer described in Remark 4.3. Let l be the number of even pi’s. Let (|oi| − 1). SOME INVARIANTS OF PRETZEL LINKS 17 Then the basket number bk(L) of L, bk(L) = δ + l + 1 if n− s is even and α 6= 0, δ + l − 1 if n− s is even and α = 0, |pt|+ δ + l − 1 if n− s is odd and α+ β 6= 0, |pt|+ δ + l − 3 if n− s is odd and α+ β = 0. Acknowledgments The author would like to thank Cameron Gordon for helpful discussion, valuable comments on this work. The TEX macro package PSTricks [37] was essential for typesetting the equations and figures. References [1] C. Adams, Almost alternating links, Topology and its applications 46 (1992), 151–165. [2] C. Adams, The knot book, W.H. Freeman and Company. 1994. [3] Y. Bae, D. Kim and C. Park, Basket, flat plumbing and flat plumbing basket numbers of links, preprint, arXiv:GT/0607079. [4] R. Bedient, Double branched covers and pretzel knots, Math. Sem. Notes Kobe Univ., 11(2) (1983), 179–198. [5] M. Brittenham, Free genus one knots with large volume, Pacific J. Math. 201(1) (2001), 61–82. [6] D. Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv., 79 (2004), 126–146. [7] J. Conway, An enumeration of knots and links, and some of their algebraic properties, Compu- tational problemas in abstract algebra, Pergamon Press. 1969. [8] R. Crowell, Genus of alternating link types, Ann. of Math., 69 (1959), 258–275. [9] J. Franks and R. Williams, Braids and the Jones polynomials, Trans. AMS, 303(1) (1987), 97–108. [10] D. Gabai, The Murasugi sum is a natural geometric operation, in: Low-Dimensional Topology (San Francisco, CA, USA, 1981), Amer. Math. Soc., Providence, RI, (1983), 131–143. [11] D. Gabai, The Murasugi sum is a natural geometric operation II, in: Combinatorial Methods in Topology and Algebraic Geometry (Rochester, NY, USA, 1982), Amer. Math. Soc., Providence, RI, (1985), 93–100. [12] D. Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc. 59(339) (1986), 1–98. [13] M. Hara, Q-polynomial of pretzel links, Tokyo J. of Math., 16(1) (1993), 183–190. [14] J. Harer, How to construct all fibered knots and links, Topology 21(3) (1982), 263–280. [15] M. Hara, Y. Nakagawa and Y. Ohyama, The Conway potential functions for pretzel links and Montesinos links, Kobe J. of Math., 6(1) (1989), 1–21. [16] C. Hayashi and M. Wada, Constructing links by plumbing flat annuli, J. Knot Theory Ramifi- cations 2 (1993), 427–429. [17] J. Hoste, M. Thistlethwaite and J. Weeks, The First 1701936 Knots, Math. Intell. 20 (1998), 33–48. [18] V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126(2) (1987), 335–388. [19] F. Hosokawa and Y. Nakanishi, On 3-fold irregular branched covering spaces of pretzel knots, Osaka J. Math., 23(2) (1986), 249–254. [20] L. Kauffman, Combinatorics and knot theory, Contemp. Math 20 (1983), 181–200. [21] M. Kobayashi and T. Kobayashi, On canonical genus and free genus of knot, J. Knot Theory Ramifications 5 (1996), 77–85. 18 DONGSEOK KIM AND JAEUN LEE [22] R. Landvoy, The Jones polynomial of pretzel knots and links, Topology and its applications 83 (1998), 135–147. [23] W. Lickorish, Prime knots and tangles, Trans. AMS 267(1) (1981), 321–332. [24] W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), 37–44. [25] H. Morton, Fibred knots with a given Alexander polynomial, in: Knots, Braids and Singularities (Plans-sur-Bex, 1982), Enseign. Math., Geneva, (1983), 205–222. [26] Y. Moriah, On the free genus of knots, Proc. of AMS 99(2) (1987), 373–379. [27] K. Murasugi, On the genus of the alternating knot, I. II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. [28] Y. Nakagawa, On the Alexander polynomials of pretzel links L(p1, . . . , pn), Kobe J. Math., 3(2) (1987), 167–177. [29] T. Nakamura, On canonical genus of fibered knot, J. Knot Theory Ramifications 11 (2002), 341–352. [30] L. Rudolph, Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids, Topology Appl. 116 (2001), 255–277. [31] L. Rudolph, Quasipositive annuli (Constructions of quasipositive knots and links IV.), J. Knot Theory Ramifications 1(4) (1992), 451–466. [32] M. Sakuma, Minimal genus Seifert surfaces for special arborescent links, Osaka J. Math. 31(4) (1994), 861–905. [33] H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571–592. [34] Y. Shinohara, On the signature of pretzel links, Topology and computer science(Atami, 1986), 217–224, Kinokuniya, Tokyo, 1987. [35] J. Stallings, Constructions of fibred knots and links, in: Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 55–60. [36] G. Torres, On the Alexander polynomail, Ann. of Math., 57(1) (1953), 57–89. [37] T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp. princeton.edu/pub/tvz/. Department of Mathematics, Yeungnam University, Kyongsan, 712-749, Korea E-mail address : [email protected] Department of Mathematics, Yeungnam University, Kyongsan, 712-749, Korea E-mail address : [email protected] ftp://ftp 1. Introduction 2. Classical pretzel links L(p,q,r) 2.1. Almost alternating 2.2. Prime torus pretzel knots 2.3. Minimal genus Seifert surfaces 3. Conway polynomials of n-pretzel links 3.1. Conway polynomial of n-pretzel knots 3.2. Conway polynomials of n-pretzel links 4. Genera of n-pretzel links 4.1. Genera of n-pretzel knots with one even pi 4.2. Genera of n-Pretzel links 5. The basket numbers of pretzel links Acknowledgments References

0704.1433

Exact retrospective Monte Carlo computation of arithmetic average Asian options

Exact retrospective Monte Carlo computation of arithmetic average Asian options Benjamin Jourdain1 and Mohamed Sbai 1 Abstract Taking advantage of the recent literature on exact simulation algorithms (Beskos et al. [1]) and unbiased estimation of the expectation of certain functional integrals (Wagner [23], Beskos et al. [2] and Fearnhead et al. [6]), we apply an exact simulation based technique for pricing continuous arithmetic average Asian options in the Black & Scholes framework. Unlike existing Monte Carlo methods, we are no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling. Numerical results of simulation studies are presented and variance reduction problems are considered. Introduction Although the Black & Scholes framework is very simple, it is still a challenging task to efficiently price Asian options. Since we do not know explicitly the distribution of the arithmetic sum of log-normal variables, there is no closed form solution for the price of an Asian option. By the early nineties, many researchers attempted to address this problem and hence different approaches were studied including analytic approxima- tions (see Turnball and Wakeman [20], Vorst [22], Levy [15] and more recently Lord [16]), PDE methods (see Vecer [21], Rogers and Shi [18], Ingersoll [11], Dubois and Lelievre [5]), Laplace transform inversion methods (see Geman and Yor [10], Geman and Eydeland [8]) and, of course, Monte Carlo simulation methods (see Kemna and Vorst [13], Broadie and Glasserman [3], Fu et al. [7]). Monte Carlo simulation can be computationally expensive because of the usual statistical error. Variance reduction techniques are then essential to accelerate the convergence (one of the most efficient techniques is the Kemna&Vorst control variate based on the geometric average). One must also account for the inherent discretization bias resulting from approximating the continuous average of the stock price with a discrete one. It is crucial to choose with care the discretization scheme in order to have an accurate solution (see Lapeyre and Temam [14]). The main contribution of our work is to fully address this last feature by the use, after a suitable change of variables, of an exact simulation method inspired from the recent work of Beskos et al. [1, 2] and Fearnhead et al. [6]. In the first part of the paper, we recall the algorithm introduced by Beskos et al. [1] in order to simulate sample-paths of processes solving one-dimensional stochastic differential equations. By a suitable change of variables, one may suppose that the diffusion coefficient is equal to one. Then, according to the Girsanov theorem, one may deal with the drift coefficient by introducing an exponential martingale weight. Because of the one-dimensional setting, the stochastic integral in this exponential weight is equal to a standard 1Project team Math Fi, CERMICS, Ecole des Ponts, Paristech, supported by the ANR program ADAP’MC. Postal address : 6-8 av. Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2. E-mails : [email protected] and [email protected] http://arxiv.org/abs/0704.1433v3 integral with respect to the time variable up to the addition of a function of the terminal value of the path. Under suitable assumptions, conditionally on a Brownian path, an event with probability equal to the normalized exponential weight can be simulated using a Poisson point process. This allows to accept or reject this Brownian path as a path solution to the SDE with diffusion coefficient equal to one. In finance, one is interested in computing expectations rather than exact simulation of the paths. In this perspective, computation of the exponential importance sampling weight is enough. The entire series expansion of the exponential function permits to replace this exponential weight by a computable weight with the same conditional expectation given the Brownian path. This idea was first introduced by Wagner [23, 24, 25, 26] in a statistical physics context and it was very recently revisited by Beskos et al. [2] and Fearnhead et al. [6] for the estimation of partially observed diffusions. Some of the assumptions necessary to implement the exact algorithm of Beskos et al. [1] can then be weakened. The second part is devoted to the application of these methods to option pricing within the Black & Scholes framework. Throughout the paper, St = S0 exp σWt + (r − δ − represents the stock price at time t, T the maturity of the option, r the short interest rate, σ the volatility parameter, δ the dividend rate and (W )t∈[0,T ] denotes a standard Brownian motion on the risk-neutral probability space (Ω,F ,P). We are interested in computing the price C0 = E e−rTf αST + β of a European option with pay-off f αST + β assumed to be square integrable under the risk neutral measure P. The constants α and β are two given non-negative parameters. When α > 0, we remark that, by a change of variables inspired by Rogers and Shi [18], αST + β has the same law as the solution at time T of a well-chosen one-dimensional stochastic differential equation. Then it is easy to implement the exact methods previously presented. The case α = 0 of standard Asian options is more intricate. The previous approach does not work and we propose a new change of variables which is singular at initial time. It is not possible to implement neither the exact simulation algorithm nor the method based on the unbiased estimator of Wagner [23] and we propose a pseudo-exact hybrid method which appears as an extension of the exact simulation algorithm. In both cases, one first replaces the integral with respect to the time variable in the function f by an integral with respect to time in the exponential function. Because of the nice properties of this last function, exact computation is possible. 1 Exact Simulation techniques 1.1 The exact simulation method of Beskos et al. [1] In a recent paper, Beskos et al. [1] proposed an algorithm which allows to simulate exactly the solution of a 1-dimensional stochastic differential equation. Under some hypotheses, they manage to implement an acceptance-rejection algorithm over the whole path of the solution, based on recursive simulation of a biased Brownian motion. Let us briefly recall their methodology. We refer to [1] for the demonstrations and a detailed presentation. Consider the stochastic process (ξt)0≤t≤T determined as the solution of a general stochastic differential equation of the form : { dξt = b(ξt)dt+ σ(ξt)dWt ξ0 = ξ ∈ R where b and σ are scalar functions satisfying the usual Lipschitz and growth conditions with σ non vanishing. To simplify this equation, Beskos et al. [1] suggest to use the following change of variables : Xt = η(ξt) where η is a primitive of 1 (η(x) = Under the additional assumption that 1 is continuously differentiable, one can apply Itô’s lemma to get dXt = η ′(ξt)dξt + η′′(ξt) d< ξ, ξ >t b(ξt) σ(ξt) dt+ dWt − σ′(ξt) b(η−1(Xt)) σ(η−1(Xt)) ′(η−1(Xt)) ︸ ︷︷ ︸ a(Xt) dt+ dWt So ξt = η −1(Xt) where (Xt)t is a solution of the stochastic differential equation dXt = a(Xt)dt+ dWt X0 = x. Thus, without loss of generality, one can start from equation (2) instead of (1). Let us denote by (W xt )t∈[0,T ] the process (Wt+x)t∈[0,T ], by QWx its law and by QX the law of the process (Xt)t∈[0,T ]. From now on, we will denote by (Yt)t∈[0,T ] the canonical process, that is the coordinate mapping on the set C([0, T ],R) of real continuous maps on [0, T ] (see Revuz and Yor [17] or Karatzas and Shreve [12]). One needs the following assumption to be true Assumption 1 : Under QWx , the process Lt = exp a(Yu)dYu − a2(Yu)du is a martingale. According to Rydberg [19] (see the proof of Proposition 4 where we give his argument on a specific example), a sufficient condition for this assumption to hold is -Existence and uniqueness in law of a solution to the SDE (2). -∀t ∈ [0, T ], a2(Yu)du <∞, QX and QWx almost surely on C([0, T ],R). Thanks to this assumption, one can apply the Girsanov theorem to get that QX is absolutely continuous with respect to QWx and its Radon-Nikodym derivative is equal to = exp a(Yt)dYt − a2(Yt)dt Consider A the primitive of the drift a, and assume that Assumption 2 : a is continuously differentiable. Since, by Itô’s lemma, A(W xT ) = A(x) + a(W xt )dW a′(W xt )dt, we have = exp A(YT )−A(x)− a2(Yt) + a ′(Yt)dt Before setting up an acceptance-rejection algorithm using this Radon-Nikodym derivative, a last step is needed. To ensure the existence of a density h(u) proportional to exp(A(u) − (u−x) ), it is necessary and sufficient that the following assumption holds Assumption 3 : The function u 7→ exp(A(u)− (u−x) ) is integrable. Finally, let us define a process Zt distributed according to the following law QZ (W xt )t∈[0,T ]|W xT = y h(y)dy where the notation L(.|.) stands for the conditional law. One has = C exp a2(Yt) + a ′(Yt)dt where C is a normalizing constant. At this level, Beskos et al. [1] need another assumption Assumption 4 : The function φ : x 7→ a 2(x)+a′(x) is bounded from below. Therefore, one can find a lower bound k of this function and eventually the Radon-Nikodym derivative of the change of measure between X and Z takes the form = Ce−kT exp φ(Yt)− k dt The idea behind the exact algorithm is the following : suppose that one is able to simulate a continuous path Zt(ω) distributed according to QZ and let M(ω) be an upper bound of the mapping t 7→ φ(Zt(ω))− k. Let N be an independent random variable which follows the Poisson distribution with parameter TM(ω) and let (Ui, Vi)i=1...N be a sequence of independent random variables uniformly distributed on [0, T ]× [0,M(ω)]. Then, the number of points (Ui, Vi) which fall below the graph {(t, φ(Zt(ω))− k); t ∈ [0, T ]} is equal to zero with probability exp φ(Zt(ω))− k dt . Actually, simulating the whole path (Zt)t∈[0,T ] is not necessary. It is sufficient to determine an upper bound for φ(Zt)− k since, as pointed out by the authors, it is possible to simulate recursively a Brownian motion on a bounded time interval by first simulating its endpoint, then simulating its minimum or its maximum and finally simulating the other points2. For this reason, one needs the following assumption for the algorithm to be feasible : Assumption 5 : Either lim sup φ(u) < +∞ or lim sup φ(u) < +∞. Suppose for example that lim sup φ(u) < +∞. The exact algorithm of Bekos et al. [1] then takes the following form : Algorithm 1 1. Draw the ending point ZT of the process Z with respect to the density h. 2. Simulate the minimum m of the process Z given ZT . 3. Fix an upper bound M(m) = sup{φ(u)− k;u ≥ m} for the mapping t 7→ φ(Zt)− k. 4. Draw N according to the Poisson distribution with parameter TM(m) and draw (Ui, Vi)i=1...N , a sequence of independent variables uniformly distributed on [0, T ]× [0,M(m)]. 5. Fill in the path of Z at the remaining times (Ui)i=1...N . 2In their paper, the authors explain how to do such a decomposition of the Brownian path. 6. Evaluate the number of points (Vi)i=1...N such that Vi ≤ φ(ZUi)− k. If it is equal to zero, then return the simulated path Z. Else, return to step 1. This algorithm gives exact skeletons of the process X , solution of the SDE (2). Once accepted, a path can be further recursively simulated at additional times without any other acceptance/rejection criteria. We also point out that the same technique can be generalized by replacing the Brownian motion in the law of the proposal Z by any process that one is able to simulate recursively by first simulating its ending point, its minimum/maximum and then the other points. Also, the extension of the algorithm to the inhomogeneous case, where the drift coefficient a in (2), and therefore the function φ, depend on the time variable t, is straightforward given that the assumptions presented above are appropriately modified. 1.2 The unbiased estimator (U.E) In finance, the pricing of contingent claims often comes down to the problem of computing an expectation of the form C0 = E (f(XT )) (3) where X is a solution of the SDE (2) and f is a scalar function such that f(XT ) is square integrable. In a simulation based approach, one is usually unable to exhibit an explicit solution of this SDE and will therefore resort to numerical discretization schemes, such as the Euler or Milstein schemes, which introduce a bias. Of course, the exact algorithm presented above avoids this bias. Here, we are going to present a technique which permits to compute exactly the expectation (3) while assumptions 4 and 5 on the function a which appears in the Radon-Nikodym derivative are relaxed. Using the previous results and notations, we get, under the assumptions 1 and 2, that C0 = E f(W xT ) exp A(W xT )−A(x) − a2(W xt ) + a ′(W xt )dt . (4) In order to implement an importance sampling method, let us introduce a positive density ρ on the real line and a process (Zt)t∈[0,T ] distributed according to the following law QZ (W xt )t∈[0,T ]|W xT = y ρ(y)dy. By (4), one has C0 = E ψ(ZT ) exp φ(Zt)dt where ψ : z 7→ f(z) e A(z)−A(x)− (z−x)2 2πρ(z) and φ : z 7→ a 2(z)+a′(z) . We do not impose ρ to be equal to the density h of the previous section. It is a free parameter chosen in such a way that it reduces the variance of the simulation. In his first paper, Wagner [23] constructs an unbiased estimator of the expectation (5) when ψ is a constant, (Zt)t∈[0,T ] is an R d−valued Markov process with known transition function and φ is a measurable function such that E |φ(Zt)|dt < +∞. His main idea is to expand the exponential term in a power series, then, using the transition function of the underlying Markov process and symmetry arguments, he constructs a signed measure ν on the space Y = n=0([0, T ]× Rd)n+1 such that the expectation at hand is equal to ν(Y). Consequently, any probability measure µ on Y that is absolutely continuous with respect to ν gives rise to an unbiased estimator ζ defined on (Y, µ) via ζ(y) = dν (y). In practice, a suitable way to construct such an estimator is to use a Markov chain with an absorbing state. Wagner also discusses variance reduction techniques, specially importance sampling and a shift procedure consisting on adding a constant c to the integrand φ and then multiplying by the factor e−cT in order to get the right expectation. Wagner [25] extends the class of unbiased estimators by perturbing the integrand φ by a suitably chosen function φ0 and then using mixed integration formulas representation. Very recently, Beskos et al. [2] obtained a simplified unbiased estimator for (5), termed Poisson estimator, using Wagner’s idea of expanding the exponential in a power series and his shift procedure. To be specific, the Poisson estimator writes ψ(ZT )e cpT−cT c− φ(ZVi) where N is a Poisson random variable with parameter cP and (Vi)i is a sequence of independent random variables uniformly distributed on [0, T ]. Fearnhead et al. [6] generalized this estimator allowing c and cP to depend on Z and N to be distributed according to any positive probability distribution on N. They termed the new estimator the generalized Poisson estimator. We introduce a new degree of freedom by allowing the sequence (Vi)i to be distributed according to any positive density on [0, T ]. This gives rise to the following unbiased estimator for (5) : Lemma 1 — Let pZ and qZ denote respectively a positive probability measure on N and a positive probability density on [0, T ]. Let N be distributed according to pZ and (Vi)i∈N∗ be a sequence of independent random variables identically distributed according to the density qZ , both independent from each other conditionally on the process (Zt)t∈[0,T ]. Let cZ be a real number which may depend on Z. Assume that |ψ(ZT )|e−cZT exp |cZ − φ(Zt)|dt ψ(ZT )e −cZT 1 pZ(N)N ! cZ − φ(ZVi) qZ(Vi) is an unbiased estimator of C0. Proof : The result follows from ψ(ZT )e −cZT 1 pZ(N)N ! cZ − φ(ZVi) qZ(Vi) ∣∣∣(Zt)t∈[0,T ] = ψ(ZT )e cZ − φ(Zt)dt pZ(n)n! pZ(n) = ψ(ZT ) exp φ(Zt)dt Using (7), one is now able to compute the expectation at hand by a simple Monte Carlo simulation. The practical choice of pZ and qZ conditionally on Z is studied in the appendix 4.1. As pointed out in Fearnhead et al. [6], this method is an extension of the exact algorithm method since, under assumptions 3, 4 and 5, the reinforced integrability assumption of Lemma 1 is always satisfied. Indeed, suppose for example that lim sup φ(u) < +∞ and let k be a lower bound of φ, mZ be the minimum of the process Z and MZ an upper bound of {φ(u)− k, u ≥ mZ}. Then, taking cZ = MZ + k in Lemma 1 ensures the integrability condition : |ψ(ZT )|e−(MZ+k)T e |MZ+k−φ(Zt)|dt |ψ(ZT )|e−(MZ+k)T e MZ+k−φ(Zt)dt |ψ(ZT )|e− φ(Zt)dt and hence, one is allowed to write that C0 = E ψ(ZT )e −(MZ+k)T 1 pZ(N)N ! MZ + k − φ(ZVi) qZ(Vi) Better still, the random variable ψ(ZT )e −(MZ+k)T 1 pZ(N)N ! MZ+k−φ(ZVi ) qZ(Vi) is square integrable when pZ is the Poisson distribution with parameterMZT + k and qZ is the uniform distribution on [0, T ] since we have then ψ(ZT )e −(MZ+k)T 1 pZ(N)N ! MZ + k − φ(ZVi) qZ(Vi)  = E ψ2(ZT ) 1− φ(ZVi) MZ + k ψ2(ZT ) The last inequality follows from the square integrability of f : whenever one is able to simulate from the density h, introduced in the exact algorithm, by doing rejection sampling, there exists a density ρ such that ψ, which is equal to f(ZT ) h(ZT ) ρ(ZT ) up to a constant factor, is dominated by f and so is square integrable. The square integrability property is very important in that we use a Monte Carlo method. We see that, whenever the exact algorithm is feasible, the unbiased estimator of lemma 1 is a simulable square integrable random variable, at least for the previous choice of pZ and qZ . Remark 2 — One can derive two estimators of C0 from the result of Lemma 1 : f(ZiT ) T )−A(x)− 2πρ(ZiT ) e−cZT pZ(N i)N i! cZ − φ(ZiV i f(ZiT ) T )−A(x)− 2πρ(ZiT ) pZ(N i)N i! cZ − φ(ZiV i T )−A(x)− 2πρ(ZiT ) pZ(N i)N i! cZ − φ(ZiV i 2 Application : the pricing of continuous Asian options In the Black & Scholes model, the stock price is the solution of the following SDE under the risk-neutral measure P = (r − δ)dt+ σdWt (8) where all the parameters are constant : r is the short interest rate, δ is the dividend rate and σ is the volatility. Throughout, we denote γ = r − δ − σ . The path-wise unique solution of (8) is St = S0 exp(σWt + γt) . We consider an option with pay-off of the form αST + β where f is a given function such that E αST + β < ∞, T is the maturity of the option and α, β are two given non negative parameters3. Note that for α = 0, this is the pay-off of a standard continuous Asian option. The fundamental theorem of arbitrage-free pricing ensures that the price of the option under consideration C0 = E e−rTf αST + β At first sight, the problem seems to involve two variables : the stock price and the integral of the stock price with respect to time. Dealing with the PDE associated with Asian option pricing, Rogers and Rogers and Shi [18] used a suitable change of variables to reduce the spatial dimension of the problem to one. We are going to use a similar idea. αS0 + βS0 e−σWu−γudu eσWt+γt. We have that ξt = αS0e σWt+γt + βS0 eσ(Wt−Wu)+γ(t−u)du = αS0e σBt+γt + βS0 eσBs+γsds where we set Bs = Wt −Wt−s, ∀s ∈ [0, t]. Clearly, (Bs)s∈[0,t] is a Brownian motion and thus the following lemma holds Lemma 3 — ∀t ∈ [0, T ], ξt and αSt + β Sudu have the same law. As a consequence C0 = E e−rT f(ξT ) By applying Itô’s lemma, we verify that the process (ξt)t≥0 is a positive solution of the following 1-dimensional stochastic differential equation for which path-wise uniqueness holds dξt = βS0dt+ ξt(σdWt + (γ + ξ0 = αS0. We are thus able to value C0 by Monte Carlo simulation without resorting to discretization schemes using one of the exact simulation techniques described in the previous section. In the case α = 0, one has to deal with the fact that ξt starts from zero which is the reason why we distinguish two cases. 3The underlying of this option is a weighted average of the stock price at maturity and the running average of the stock price until maturity with respective weights α and βT . 2.1 The case α 6= 0 We are going to apply both the exact algorithm of Beskos et al. [1] and the method based on the unbiased estimator of lemma 1. We make the following change of variables to have a diffusion coefficient equal to 1 : log(ξt) dXt = ( + βS0 e−σXt)dt+ dWt X0 = x with x = log(αS0) C0 = E e−rTf(eσXT ) The following proposition ensures that assumption 1 is satisfied. Proposition 4 — The process (Lt)t∈[0,T ] defined by Lt = exp e−σYt) dYt − e−σYt)2dt is a martingale under QWx . Proof : Under QWx , (Lt)t∈[0,T ] is clearly a non-negative local martingale and hence a super-martingale. Then, it is a true martingale if and only if E (LT ) = 1. Checking the classical Novikov’s or Kamazaki’s criteria is not straightforward. Instead, we are going to use the approach developed by Rydberg [19] (see also Wong and Heyde [27]) who takes advantage of the link between explosions of SDEs and the martingale property of stochastic exponentials. Let us define the following stopping times : τn(Y ) = inf t ∈ R+ such that e−σYu du ≥ n with the convention inf{∅} = +∞. The stopped process (Lt∧τn(Y ))t∈[0,T ] is a true martingale under QWx since Novikov’s condition is fulfilled. According to the Girsanov theorem, one can define a new probability measure QnX , which is absolutely continuous with respect to QWx , by its Radon-Nikodym derivative = LT∧τn(Y ). Hence 1{τn(Y )>T} 1{τn(Y )>T}LT∧τn(Y ) Since (τn(Y ))n∈N is a non decreasing sequence, we can pass to the limit in the right hand side We get X (τn(Y ) > T ) = EQWx 1{τ∞(Y )>T}LT∧τ∞(Y ) where τ∞(Y ) denotes the limit of the non decreasing sequence (τn(Y ))n∈N. Under QWx , (Yt)t∈[0,T ] has the same law as a Brownian motion starting from x so τ∞(Y ) = +∞ ,QWx almost surely, and consequently = lim X (τn(Y ) > T ) . On the other hand, the Girsanov theorem implies that, under QnX , (Yt)t∈[0,T∧τn(Y )] solves a SDE of the form (11). To conclude the proof, it is sufficient to check that trajectorial uniqueness holds for this SDE. Indeed, the law of (Yt)t∈[0,T∧τn(Y )] under Q X is the same as the law of (Yt)t∈[0,T∧τn(Y )] under QX . Hence X (τn(Y ) > T ) = QX (τn(Y ) > T ) −→ QX (τ∞(Y ) > T ) . Clearly, + βS0 e−σYu du <∞, QX almost surely, so = QX (τ∞(Y ) > T ) = 1 as required. In order to check trajectorial uniqueness for the SDE (11), we consider two solutions X1 and X2. We have that d(X1t −X2t ) = t − e−σX dt ⇒ d|X1t −X2t | = sign(X1t −X2t ) t − e−σX |X1t −X2t | = sign(X1s −X2s ) t − e−σX ds ≤ 0. The last inequality follows from the fact that x 7→ e−σx is a decreasing function. Finally, almost surely, ∀t ≥ 0, X1t = X2t which leads to strong uniqueness. ✷ Consequently, thanks to the Girsanov theorem, we have = exp e−σYt) ︸ ︷︷ ︸ a(Yt) dYt − e−σYt)2dt  . (12) Set A(u) = a(x)dx = γ u+ βS0 (1− e−σu). Then = exp A(YT )−A(x)− a2(Yt) + a ′(Yt)dt The function u 7→ exp A(u)− (u−Y0) = exp u+ βS0 (1− e−σu)− (u−Y0) is clearly integrable so we can define a new process (Zt)t∈[0,T ] distributed according to the following law QZ (Wt)t∈[0,T ]|WT = y h(y)dy where the probability density h is of the form h(u) = C exp A(u)− (u− Y0) with C a normalizing constant. (13) Remark 5 — Simulating from this probability distribution is not difficult (see the appendix 4.2 for an appropriate method of acceptance/rejection sampling). We have = C exp (a2(Yt) + a ′(Yt))dt Set φ(x) = a2(x)+a′(x) e−σx)2−βS0e−σx . A direct calculation gives φ(x) = if 2γ ≥ σ2 log( 2βS0 σ2−2γ ) otherwise. Set k = infx∈R φ(x). Finally, we get = Ce−kT exp φ(Yt)− k dt We check that φ(x) = φ(x) = +∞. Hence we can apply the algorithm 1 to simulate exactly XT and compute C0 = E e−rTf(eσXT ) Monte Carlo. On the other hand, using (12) we get C0 = E e−rTf(eσW T ) exp A(W xT )−A(x) − a2(W xt ) + a ′(W xt )dt and we can also use the unbiased estimator presented in the previous section to compute this expectation. Remark 6 — We also applied the exact algorithm based on a geometric Brownian motion instead of the standard Brownian motion which seems more intuitive given the form of the SDE (10). The algorithm is feasible because we can simulate recursively a drifted Brownian motion and therefore a geometric Brownian motion by an exponential change of variables. The results we obtained were not different from the first method. 2.1.1 Numerical computation For numerical tests, we consider the case f(x) = (x−K)+ which corresponds to the European call option with strikeK. Using the exact simulation algorithm presented above, we can simulate the underlying αST + β Stdt at maturity (see Figure 1). Then, all we have to do is a simple Monte Carlo method to get the price of the option under consideration. Using the unbiased estimator, we get C0 = E e−rT (eσZT −K)+ eA(ZT )−A(x)− (ZT −x) 2πρ(ZT ) e−(MZ+k)T pZ(N)N ! MZ + k − φ(ZVi) q(Vi) where (Zt)t∈[0,T ], ρ,MZ , k, pZ and qZ are defined as in section 1.2. In order to ensure square integrability, we choose pZ to be a Poisson distribution with parameterMZT +k and qZ to be the uniform distribution on [0, T ]. For the density ρ, a good choice is to consider the density that we use to simulate from the distribution h by rejection sampling. We test these exact methods against a standard discretization scheme with the variance reduction tech- nique of Kemna and Vorst [13]. As pointed out by Lapeyre and Temam [14], the discretization of the integral by a simple Riemannian sum is not efficient. Instead, we use the trapezoidal discretization. In the sequel, we will denote this method by Trap+KV. The table 1 gives the results we obtained for the following arbitrary set of parameters : S0 = 100, K = 100, r = 0.05, σ = 0.3, δ = 0, T = 1, α = 0.6 and β = 0.4. The computation has been made on a computer with a 2.8 Ghz Intel Penthium 4 processor. We intentionally choose a large number of simulations in order to show the influence of the number of time steps when using a discretization scheme. 10 53 97 140 183 227 270 313 357 400 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 PSfrag replacements αST + β Exact Simulation of the underlying : αST + β Figure 1: Histogram of 105 independent realizations of αST + β Stdt for α = 0.6 and β = 0.4 compared with the lognormal distribution of ST . Method M N Acceptance rate Price C.I at 95% CPU 10 11.46 [11.43, 11.48] 5 s Trap+KV 20 106 - 11.46 [11.43, 11.49] 9 s 50 11.47 [11.44, 11.5] 21 s Exact Simulation - 106 24% 11.46 [11.43, 11.5] 81 s U.E (cP =MZ , cZ =MZ + k) - 10 6 - 11.46 [11.43, 11.49] 17 s U.E (cP = cZ = 1/T ) - 10 6 - 11.46 [11.43, 11.49] 6 s Table 1: Price of the option (9) using a standard discretization technique and exact simulation methods. Empirical evidence shows that the exact simulation method is quite slow. This is mainly due to the fact that the rejection algorithm has a little acceptance rate (24% according to table 1). Using a geometric Brownian motion instead of a standard Brownian motion did not improve the results. Also, simulating recursively a Brownian path conditionally on its terminal value and its minimum is time consuming. The unbiased estimator is more efficient, especially when we can avoid the recursive simulation of the Brownian path. To do so, we choose for pZ a Poisson distribution with mean cPT where cP is a free parameter. If we assume that the integrability condition in lemma 1 holds, then we can write that C0 = E e−rT (eσZT −K)+ eA(ZT )−A(x)− (ZT −x) 2πρ(ZT ) ecPT−cZT cZ − φ(ZVi) Regarding the dependence of the exact simulation method with respect to the parameters α and β, it is intuitive that whenever α >> β, the method performs well since the logarithm of the underlying is not far from the logarithm of the geometric Brownian motion on which we do rejection-sampling. The table 2 confirms this intuition. We see that we cannot apply the algorithm for small values of α and then let α→ 0 to treat the case α = 0. α + β 0.3 0.4 0.5 0.6 0.7 Acceptance Rate 0.003% 0.47% 5.66% 24.43% 53.85% Table 2: Influence of the parameter α on the acceptance rate of the exact algorithm. 2.2 Standard Asian options : the case α = 0 and β > 0 A standard Asian option is a European option on the average of the stock price over a determined period until maturity. An Asian call, for example, has a pay-off of the form ( 1 Sudu−K)+. With our previous notations, it corresponds to the case α = 0, β = 1 and f(x) = (x−K)+. The change of variables we used above is no longer suitable because it starts from zero when α = 0. Instead, we consider the following new definition of the process ξ eσ(Wt−Wu)+γ(t−u)du ξ0 = S0. Obviously, the two variables ξT and Sudu have the same law. Hence, the price of the Asian option becomes C0 = E e−rT f e−rTf(ξT ) Remark 7 — The pricing of floating strike Asian options is also straightforward using this method. It is even more natural to consider these options since it unveils the appropriate change of variables as we shall see below. Let us consider a floating strike Asian call for example. We have to compute C0 = E Sudu− ST Using S̃t = Ste δt as a numéraire (see the seminal paper of Geman et al. [9]), we immediately obtain that C0 = EP −δT ( 1 du− 1 where P is the probability measure associated to the numéraire S̃t. It is defined by its Radon-Nikodym derivative = eσWT− Under P , the process Bt =Wt − σt is a Brownian motion and we can write that C0 = EP −δT ( 1 eσ(Bu−BT )+(r−δ+ )(u−T )du − 1 −δT ( 1 eσ(Wu−WT )+(r−δ+ )(u−T )du− 1 ξT − S0 where ξt is the process defined by (14) but with γ = r − δ + σ . We see therefore that the problem simplifies to the fixed strike Asian pricing problem. Let us write down the stochastic differential equation that rules the process (ξt)t∈[0,T ]. Using Itô’s lemma, we get { dξt = ξ0−ξt dt+ ξt σdWt + (γ + ξ0 = S0. Note that we are faced with a singularity problem near 0 because of the term ξ0−ξt . We are going to reduce its effect using another change of variables. Using Itô’s lemma, we show that C0 = E e−rTf where Xt = log(ξt/ξ0) solves the following SDE dXt = σdWt + γdt+ e−Xt−1 X0 = 0. Lemma 8 — Existence and strong uniqueness hold for the stochastic differential equation (16). Proof : Existence is obvious since we have a particular solution Xt. The diffusion coefficient being constant and the drift coefficient being a decreasing function in the spatial variable, we have also strong uniqueness for the SDE (see the proof of Proposition 4). ✷ Because of the singularity of the term e −Xt−1 in the drift coefficient, the law of (Xt)t≥0 is not absolutely continuous with respect to the law of (σWt)t≥0. That is why we now define (Zt)t≥0 by the following SDE with an affine inhomogeneous drift coefficient : dZt = σdWt + γdt− Z0 = X0 = 0. The drift coefficient exhibits the same behavior as the one in (16) in the limit t → 0 in order to ensure the desired absolute continuity property. It is affine in the spatial variable so that (Zt)t≥0 is a Gaussian process and as such is easy to simulate recursively. Lemma 9 — The process sdWs + t (18) is the unique solution of the stochastic differential equation (17). Proof : Using Itô’s Lemma, we easily check that Zt given by (18) is a solution of (17). Again, constant diffusion coefficient and decreasing drift coefficient ensures strong uniqueness. ✷ Remark 10 — For the computation of the price C0 = E e−rT (S0e XT −K)+ of a standard Asian call option, the random variable e−rT (S0e ZT − K)+ provides a natural control variate. Indeed, since ZT is a Gaussian random variable with mean γ T and variance σ , one has e−rT (S0e ZT −K)+ = S0e −r)TN −Ke−rTN (d) where N is the cumulative standard normal distribution function and d = log(S0/K)+ Notice that in Kemna and Vorst [13], the authors suggest the use of the control variate S0 exp σWt + γt dt which has the same law than e−rT ZT −K σWt + γt dt is also a Gaussian variable with mean T and variance σ In order to define a new probability measure under which (Zt)t≥0 solves the SDE (16), one introduces Lt = exp e−Zs − 1 + Zs dWs − e−Zs − 1 + Zs Because of the singularity of the coefficients in the neighborhood of s = 0, one has to check that the integrals in Lt are well defined. This relies on the following lemma Lemma 11 — Let ǫ > 0. In a random neighborhood of s = 0, we have |Zs| ≤ cs −ǫ and |Xs| ≤ cs where c is a constant depending on σ,γ and ǫ. Since ∀ǫ > 0, ∀z ≤ cs 12−ǫ, e−z − 1 + z ≤ Cs−4ǫ, we can choose ǫ < 1 to deduce that Lt is well defined. Proof : We easily check that the Gaussian process (Bt)t∈[0,T ] defined by Bt = ∫ (3t) 13 sdWs is a standard Brownian motion. Thanks to the law of iterated logarithm for the Brownian motion (see for example Karatzas and Shreve [12] p. 112), there exists t1(ω) such that ∀t ≤ t1(ω), |Bt(ω)| ≤ t 4ω is an element of the underlying probability space Ω. Therefore, ∀t ≤ (3t1(ω)) 3 , |Zt(ω)| = (ω) + ∣∣ ≤ σ Taking c = max( σ ) yields ∀t ≤ (3t1(ω)) 3 ∧ 1, |Zt(ω)| ≤ ct On the other hand, recall that Xt = log(ξt/ξ0) = log eσWt+γt e−σWu−γudu . So, using the law of iterated logarithm for the Brownian motion, we deduce that there exists t2(ω) such that ∀t ≤ t2(ω), 0 ≤ eσWt(ω)+γt e−σWu(ω)−γudu ≤ 1 −ǫ−γudu. Denote g(t) = 1 −ǫ−γudu and let us investigate the order in time near zero of this function. We have that +γt = 1 + σt −ǫ +O(t1−2ǫ)∫ t −ǫ−γudu = t+ −ǫ +O(t2−2ǫ) hence g(t) = 1 + (σ + −ǫ +O(t1−2ǫ), so Xt(ω) ≤ log (g(t)) ∼ −ǫ, which ends the proof for Xt. ✷ Proposition 12 — (Lt)t∈[0,T ] is a martingale and, consequently, for all g : C([0, T ]) → R measurable, the random variables g((Xt)0≤t≤T ) and g((Zt)0≤t≤T )LT are simultaneously integrable and then g((Xt)0≤t≤T ) g((Zt)0≤t≤T )LT Proof : The proof is similar to the proof of Proposition 4. We have already shown existence and strong uniqueness for both SDE (16) and (17). Showing that the stopping time τn(Y ) = inf t ∈ R+ such that e−Ys − 1 + Ys ds ≥ n , with the convention inf{∅} = +∞, have infinite limits when n tends to +∞, QX and QZ almost surely, follows from the previous lemma. One has LT = exp e−Zt − 1 + Zt dZt − e−Zt − 1 + Zt e−Zt − 1 + Zt + γ − Zt Set A(t, z) = 1− z + z2 − e−z . The function A : ]0, T ]×R→ R is continuously differentiable in time and twice continuously differentiable in space. So, we can apply Itô’s Lemma on the interval [ǫ, T ] for ǫ > 0 : A(T, ZT ) = A(ǫ, Zǫ) + e−Zt − 1 + Zt dZt − 1− Zt + Z − e−Zt 1− e−Zt Using the lemma 9, we let ǫ→ 0 to obtain A(T, ZT ) = e−Zt − 1 + Zt dZt − 1− Zt + Z − e−Zt 1− e−Zt LT = exp A(T, ZT )− φ(t, Zt)dt where φ is the mapping φ(t, z) = e−z − 1 + z − z 1− e−z e−z − 1 + z e−z − 1 + z + γ − z . (19) By (15) and Proposition 12, we get C0 = E e−rTf(S0e ZT ) exp A(T, ZT )− φ(t, Zt)dt . (20) Since for each t > 0, lim φ(t, z) = +∞ and lim φ(t, z) = −∞, it is not possible to apply the exact algorithm. One can use the unbiased estimator, at least theoretically, if there exists a random variable cZ measurable with respect to Z such that eA(T,ZT )−(r+cZ)T |f(S0eZT )|e |cZ−φ(t,Zt)|dt Unfortunately, this reinforced integrability condition is never satisfied : Lemma 13 — Assume that f is a non identically zero function. Let pZ and qZ denote respectively a positive probability measure on N and a positive probability density on [0, T ]. Let N be distributed according to pZ and (Ui)i∈N∗ be a sequence of independent random variables identically distributed according to the density qZ , both independent conditionally on the process (Zt)t∈[0,T ]. Then the random variable eA(T,ZT )−rT f(S0e pZ(N)N ! −φ(Ui, ZUi) qZ(Ui) is non integrable. Proof : By conditioning on Z, one has ∆ := E eA(T,ZT )−rT |f(S0eZT )| pZ(N)N ! |φ(Ui,ZUi )| qZ (Ui) eA(T,ZT )−rT |f(S0eZT )|e |φ(t,Zt)|dt eA(T,ZT )−rT |f(S0eZT )|e |φ(t,Zt)|dt One can easily show that, ∀z < 0 and ∀t ∈ [T , T ], φ(t, z) ≥ φ(z) where φ(z) = e−z − 1 + z − z2 e−z − 1 + z e−z − 1 + z + γ+ − 2 z Since φ(z) ∼ σ2T 2 , there exists c < 0 such that for all z < c, φ(z) ≥ e−2z σ2T 2 . Hence, ∆ ≥ E eA(T,ZT )−rT |f(S0eZT )|e e−2Zt1{Zt<c}dt eA(T,ZT )−rT |f(S0eZT )|e− 2σ2T e e−2Ztdt Using Jensen’s inequality we get ∆ ≥ E eA(T,ZT )−rT |f(S0eZT )|e− 2σ2T exp We have seen in the proof of lemma 11 that Zt = t where (Bt)t≥0 is a standard Brownian motion. So, conditionally on ZT , Ztdt is a gaussian random variable and hence ∆ = +∞. We are in a situation where eA(T,ZT )−rT |f(S0eZT )|E [∣∣∣ 1pZ(N)N ! −φ(Ui,ZUi ) qZ (Ui) ∣∣∣(Zt)t∈[0,T ] is non inte- grable while eA(T,ZT )−rT |f(S0eZT )| pZ (N)N ! −φ(Ui,ZUi ) qZ(Ui) ∣∣∣(Zt)t∈[0,T ] ]∣∣∣ is integrable since e−rT |f(S0eZT )| exp A(T, ZT )− φ(t, Zt)dt <∞. Then, a natural idea would consist in considering, for a given n ∈ N∗, the random variable eA(T,ZT )−rT |f(S0eZT )|E ∣∣∣∣∣∣ pZ(Nj)Nj ! −φ(U ji , ZUj ∣∣∣∣∣∣ ∣∣∣(Zt)t∈[0,T ] where (Nj)1≤j≤n are independent variables having the same law as N and i )i∈N∗ 1≤j≤n are independent sequences having the same law as (Ui)i∈N∗ , both independent conditionally on the process (Zt)t∈[0,T ]. The following general result tells us that this is not sufficient to circumvent integrability problems. Lemma 14 — Let Y and Z be two real random variables and g : R → R a given measurable function. Assume that g(Z)E (Y |Z) is integrable while g(Z)E (|Y | |Z) is non integrable. Then, when (Yi)1≤i≤n is a sequence of independent random variables having the same law as Y , ∀n ∈ N∗, the random variable g(Z)E i=1 Yi| |Z is non integrable. Proof : Denote by e, e1 and en three functions satisfying ∀z ∈ R, e(z) = E (Y |Z = z) , e1(z) = E (|Y1| |Z = z) and en(z) = E (∣∣∣∣∣ ∣∣∣∣∣ |Z = z On the one hand, since |g(z)| |e(z)|PZ(dz) <∞ and |g(z)| e1(z)PZ(dz) = +∞ , where PZ is the law of Z, we have that |g(z)| e1(z)1{e1(z)≥2|e(z)|}PZ(dz) = +∞. On the other hand, ∀z ∈ R, en(z) ≥ (∣∣∣∣∣ ∣∣∣∣∣ 1{∀2≤j≤n,Yj≥0}|Z = z (∣∣∣∣∣ ∣∣∣∣∣ 1{∀2≤j≤n,Yj<0}|Z = z Y +1 |Z = z P (Y1 ≥ 0|Z = z)n−1 + E Y −1 |Z = z P (Y1 < 0|Z = z)n−1 e1(z)+e(z) P (Y1 ≥ 0|Z = z)n−1 + e1(z)−e(z)2 P (Y1 < 0|Z = z) e1(z) 1{e1(z)≥2|e(z)|}P (Y1 ≥ 0|Z = z) e1(z) 1{e1(z)≥2|e(z)|}P (Y1 < 0|Z = z) ≥ e1(z) 1{e1(z)≥2|e(z)|} Hence, E g(Z)E (∣∣ 1 i=1 Yi ∣∣ |Z |g(z)|en(z)PZ(dz) = +∞. ✷ There is still hope yet. In the proof of Lemma 13, we saw that integrability problems appear when Zt takes large negative values so that φ(t, Zt) tends rapidly towards +∞. Since lim φ(t, z) = −∞, one possible issue is to split the function φ(t, Zt) into a positive part and a negative part. The first term can be handled by the exact simulation technique whereas the second term, which as we shall see in the following section presents no integrability problems, can be handled by the unbiased estimator technique. 2.2.1 An hybrid pseudo-exact method We rewrite (20) in the following form C0 = E eA(T,ZT )−rTf(S0e ZT )e φ−(t,Zt)dte− φ+(t,Zt)dt . (22) Let pZ and qZ denote respectively a positive probability measure on N and a positive probability density on [0, T ]. Let N be distributed according to pZ and (Ui)i∈N∗ be a sequence of independent random variables identically distributed according to the density qZ , both independent conditionally on the process (Zt)t∈[0,T ]. Note that, since eA(T,ZT )−rTf(S0e ZT )e |φ−(t,Zt)|dte− φ+(t,Zt)dt = eA(T,ZT )−rT f(S0e ZT )e− φ(t,Zt)dt is integrable, one has C0 = E eA(T,ZT )−rT f(S0e pZ(N)N ! φ−(Ui, ZUi) qZ(Ui) φ+(t,Zt)dt . (23) Remark 15 — There is no hope that this estimator is square integrable. Indeed, one can show as in Lemma 13 that E −(t,Zt)) = +∞ since (φ−(t, z))2 is of order z4 for large positive z. The idea then is to apply the exact simulation technique to simulate an event with probability e− φ+(t,Zt)dt. Since for each t > 0, lim φ+(t, z) = +∞, one needs to bound from above φ+(t, z), uniformly with respect to t ∈ [0, T ], for z > c where c < 0 is a given constant. Thanks to the following lemma, it is possible to do so but only uniformly with respect to t ∈ [ǫ, T ] for all ǫ > 0 : Lemma 16 — For all 0 < t ≤ T , φ+(t, z) ≤ γ ∀c < 0, sup z∈[c,0] φ+(t, z) ≤ e −c − 1 + c (1 + γ+t) + (e−c − 1)2 2σ2t2 Proof : Let z > 0. It is useful to distinguish two cases according to the sign of γ : 1. γ ≥ 0 We rewrite φ in the following form φ(t, z) = e−z − 1 + z − z 1− e−z 2 − (z ∧ 1)2 2σ2t2 (e−z − 1)2 − (z ∧ 1)2 2σ2t2 First note that e−z−1+z− z2 ≤ 0, 1−e (e−z−1)2−(z∧1)2 2σ2t2 ≤ 0. Moreover, 2 − (z ∧ 1)2 2σ2t2 if γt ≤ 1 otherwise Consequently, φ+(t, z) ≤ γ 2. γ ≤ 0 Now we rewrite φ in the following form φ(t, z) = e−z − 1 + z − z e−z − 1 + z (e−z − 1)2 − z2 2σ2t2 1− e−z It is then easy to show that φ+(t, z) ≤ 1 Note that 1 . Hence, gathering the two cases yields the first part of the lemma. Let now z ∈ [c, 0] for a given negative constant c. We rewrite φ in the following form φ(t, z) = e−z − 1 + z (1 + γ+t) + (e−z − 1)2 2σ2t2 σ2t2︸ ︷︷ ︸ ≥0 for z<0 1− e−z − γ− e −z − 1 + z σ2t︸ ︷︷ ︸ ≤0 for z<0 Since ∂z e−z − 1 + z (1 + γ+t) + (e−z − 1)2 2σ2t2 1− e−2z − 2z + tγ+(1− e−z) is negative for all z < 0, one has that z∈[c,0] φ+(t, z) ≤ e −c − 1 + c (1 + γ+t) + (e−c − 1)2 2σ2t2 This lemma suggests to apply the exact algorithm on [ǫ, T ] for a fixed positive threshold ǫ. It remains to handle the time interval [0, ǫ[. Thanks to the following lemma, we that φ+(t, Zt) can be approximately bounded from above for small t, almost surely, by a function of t. The idea is then to extend the exact simulation algorithm by simulating an inhomogeneous Poisson process. Of course, this hybrid method is no longer exact since the positive threshold for which the upper bound holds is random. Lemma 17 — For all η > 0, there exists a random neighborhood of t = 0 such that φ+(t, Zt) ≤ −η (24) where c = max( σ Proof : We rewrite (19) this way φ(t, z) = 1− e−z e−z − 1 + z 1− z + z2 − e−z − 1 (e−z − 1 + z)(e−z − 1− z) and make the following Taylor expansions 1− z + z − e−z − 1 (e−z − 1 + z)(e−z − 1− z) z3 +O(z4) 1− e−z e−z − 1 + z z +O(z2). On the other hand, we have seen in the proof of lemma 11 that there exists a random neighborhood of zero such that Zt ≤ ct −η where c = max( σ ). We conclude that, in a random neighborhood of zero, φ+(t, Zt) ≤ 2.2.2 Numerical computation For numerical computation, we are going to use the following set of parameters : S0 = 100, K = 100, σ = 0.2, r = 0.1, δ = 0 and T = 1. To fix the ideas, let us consider a call option. The price C0 writes as follows C0 = E eA(T,ZT )−rT ZT −K φ−(Ui, ZUi) φ+(t,Zt)dt where N ∼ P(cp) and (Ui)i≥1 is an independent sequence of independent random variables uniformly dis- tributed in [0, T ]. The parameter cp > 0 is set to one in the following. We give a description of the hybrid method we implement : Algorithm 2 On the time interval Ij := [ 1. Simulate Z T , Z T and a lower bound mj for the minimum of (Zt)t∈Ij (use the fact that Zt = t where (Bt)t≥0 is a standard Brownian motion). 2. Find M j > 0 such that ∀t ∈ Ij , φ+(t, Zt) ≤M j (use Lemma 16). 3. Simulate an homogeneous spatial Poisson process on the rectangle Ij × [0,M j] and accept (respectively reject) the trajectory simulated if the number of points falling below the graph (φ+(t, Zt))t∈Ij is equal to (respectively different from) zero. Carry on this acceptance rejection algorithm until reaching a time interval IJ for a chosen J ∈ N∗. On the remaining time interval [0, T ], use the same acceptance/rejection algorithm but with an inhomogeneous spatial Poisson process this time (use Lemma 17). In table 3, we give the price obtained by our method for different values of the positive threshold ǫ = T The number M of Monte Carlo simulations is equal to 105 and the true price is equal to 7.042 (computed using a Monte Carlo method with a trapezoidal scheme and a Kemna-Vorst control variate technique). Price CPU ǫ = T 6.9394 7s ǫ = T 6.9590 10s ǫ = T 6.9703 13s ǫ = T 6.9952 17s ǫ = T 7.0423 21s Table 3: Price of the Asian call using the hybrid-pseudo exact method. Clearly, the method is not yet competitive regarding computation time. Nevertheless, unlike the usual discretization methods, it is not prone to discretization errors. 3 Conclusion In this article, we have applied two original Monte Carlo methods for pricing Asian like options which have the following pay-off : (αST + β Stdt −K)+. In the case α 6= 0, we applied both the algorithm of Beskos et al. [1] and a method based on the unbiased estimator of Wagner [23] and more recently the Poisson estimator of Beskos et al. [2] and the generalized Poisson estimator of Fearnhead et al. [6]. The numerical results show that the latter performs the best. The more interesting case α = 0, which corresponds to usual continuously monitored Asian options, can not be treated using neither the exact algorithm, nor the method of exact computation of expectation but we investigate an hybrid pseudo-exact method which combines the two techniques. More generally, this hybrid method is an extension of the two exact methods and can be applied in other situations. From a practical point of view, the main contribution of these techniques is to allow Monte Carlo pricing without resorting to discretization schemes. Hence, we are no longer prone to the discretization bias that we encounter in standard Monte Carlo methods for pricing Asian like options. Even though these exact methods are time consuming, they provide a good and reliable benchmark. References [1] A. Beskos, O. Papaspiliopoulos, and Gareth O. Roberts. Retrospective exact simulation of diffusion sample paths. Bernoulli, 12(6), December 2006. [2] A. Beskos, O. Papaspiliopoulos, Gareth O. Roberts, and Paul Fearnhead. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. To appear in the Journal of the Royal Statistical Society, Series B. [3] M. Broadie and P. Glasserman. Estimating security price derivatives using simulation. Management Science, 42(2):269–285, 1996. [4] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E Knuth. On the lambert W function. Advances in Computational Mathematics, 5:329–359, 1996. [5] F. Dubois and T. Lelievre. Efficient pricing of asian options by the pde approach. Journal of Compu- tational Finance, 8(2), 2004. [6] Paul Fearnhead, O. Papaspiliopoulos, and Gareth O. Roberts. Particle filters for partially observed diffusions. Working paper. Lancaster University., 2006. [7] M. Fu, D. Madan, and T. Wang. Pricing continuous asian options : a comparison of monte carlo and laplace transform inversion methods. Journal of Computational Finance, 2(2), 1999. [8] H. Geman and A. Eydeland. Domino effect. Risk, pages 65–67, April 1995. [9] H. Geman, N. El Karoui, and J.C. Rochet. Changes of numéraires, changes of probability measure and option pricing. J. Appl. Probab., 32(2):443–458, 1995. [10] H. Geman and M. Yor. Bessel processes, asian option and perpetuities. Mathematical Finance, 3(4), 1993. [11] J.E. Ingersoll. Theory of Financial Decision Making. Rowman & Littlefield, 1987. [12] I. Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus. Springer-Verlag New-York, second edition, 1991. [13] A. Kemna and A. Vorst. A pricing method for options based on average asset values. Journal of Banking and Finance, 14(1):113–129, 1990. [14] B. Lapeyre and E. Temam. Competitive Monte Carlo methods for pricing asian options. Journal of Computational Finance, 5(1), 2001. [15] E. Levy. Pricing european average rate currency options. Journal of International Money and Finance, 11(5):474–491, October 1992. [16] R. Lord. Partially exact and bounded approximations for arithmetic Asian options. Journal of Com- putational Finance, 10(2), 2006. [17] D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer-Verlag Berlin Heidelberg, 1991. [18] L. C. G. Rogers and Z. Shi. The value of an Asian option. J. Appl. Probab., 32(4):1077–1088, 1995. [19] T. H. Rydberg. A note on the existence of unique equivalent martingale measures in a markovian setting. Finance and Stochastics, 1(3):251–257, 1997. [20] S. Turnball and L. Wakeman. A quick algorithm for pricing european average options. Journal of Financial and Quantitative Analysis, 16:377–389, 1991. [21] J. Vecer. A new pde approach for pricing arithmetic asian options. Journal of Computational Finance, 4(4), 2001. [22] T. Vorst. Prices and hedge ratios of average exchange rate options. International Review of Financial Analysis, 1(3):179–193, 1992. [23] W. Wagner. Unbiased Monte Carlo evaluation of certain functional integrals. J. Comput. Phys., 71(1):21–33, 1987. [24] W. Wagner. Monte Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stochastic Anal. Appl., 6(4):447–468, 1988. [25] W. Wagner. Unbiased multi-step estimators for the Monte Carlo evaluation of certain functional inte- grals. J. Comput. Phys., 79(2):336–352, 1988. [26] W. Wagner. Unbiased Monte Carlo estimators for functionals of weak solutions of stochastic differential equations. Stochastics Stochastics Rep., 28(1):1–20, 1989. [27] B. Wong and C. C. Heyde. On the martingale property of stochastic exponentials. J. Appl. Probab., 41(3):654–664, 2004. 4 Appendix 4.1 The practical choice of p and q in the U.E method The best choice for the probability law p of N and the common density q of the variables (Vi)i≥1 is obviously the one for which the variance of the simulation is minimum. In a very general setting, it is difficult to tackle this issue. In order to have a first idea, we are going to restrict ourselves to the computation p(N)N ! g(Vi) q(Vi) where g : [0, T ] → R. Lemma 18 — When g is a measurable function on [0, T ] such that 0 < |g(t)|dt < +∞, the variance of p(N)N ! g(Vi) q(Vi) is minimal for qopt(t) = |g(t)| |g(t)|dt 1[0,T ](t) and popt(n) = |g(t)|dt |g(t)|dt Proof : Minimizing the variance in (7) comes down to minimizing the expectation of the square of p(N)N ! g(Vi) q(Vi) F (p, q) = E (p(N)N !)2 g2(Vi) q2(Vi) g2(t) p(n) (n!)2 Using Cauchy-Schwartz inequality we obtain a lower bound for F (p, q) F (p, q) = g2(t) p(n)n! p(n) ≥ g2(t) q(t)dt |g(t)|dt = exp |g(t)|dt We easily check that this lower bound is attained for qopt and popt. The optimal probability distribution popt is the Poisson law with parameter |g(t)|dt. This justifies our use of a Poisson distribution for p. 4.2 Simulation from the distribution h given by (13) Recall that h(u) = C exp A(u)− (u−X0) = C exp (1− e−σu)− (u−X0) where C is a normalizing constant. The expansion of the exponential e−σu at the first order yields h(u) ≈ C exp u− (u−X0) = C exp (u− (X0 + T (γ+βS0)σ )) This suggests to do rejection sampling using the normal distribution with mean X0 + T (γ+βS0) variance T as prior. Unfortunately, for a standard set of parameters, this method gives bad results. Even a second order expansion of e−σu which also modifies the variance does not work. In order to get round this problem, we evaluate the mode u∗ of h. We have h′(u∗) = C ∗ −X0 (1− e−σu )− (u ∗ −X0)2 So, h′(u∗) = 0 if and only if ∗ −X0 which writes σ(u∗ −X0 − T )eσ(u ∗−X0− γσT ) = TβS0e −σX0−γT . The function x 7→ xex is continuous and increasing on [0,+∞[ and so is its inverse which we denote by W . Since TβS0e −σX0−γT ≥ 0, we deduce that h is unimodal and that its mode satisfies γT +W βS0Te −γT−σX0 + σX0 The functionW is the well-known Lambert function, also called the Omega function. It is uniquely valued on [0,+∞[ and there are robust and fast numerical methods based on series expansion for approximating this function (see for example Corless et al. [4]). Numerical tests showed that performing rejection sampling using a Gaussian distribution with variance T and mean u∗ instead of X0 + T (γ+βS0) gives plain satisfaction. In table 4, we see that for arbitrary choice of the parameter α , the acceptance rate of the algorithm is always high (of order 70%) and that the computation time is low. Nb of simulations Acceptance rate Computation time 0.2 61% 3s 0.5 106 68% 3s 0.8 80% 2s Table 4: Acceptance rate of the rejection algorithm of simulating from the distribution h in (13) with S0 = 100, σ = 0.3, T = 2 and r = 0.1. 1 Exact Simulation techniques 1.1 The exact simulation method of Beskos et al. BPR 1.2 The unbiased estimator (U.E) 2 Application : the pricing of continuous Asian options 2.1 The case =0 2.1.1 Numerical computation 2.2 Standard Asian options : the case = 0 and > 0 2.2.1 An hybrid pseudo-exact method 2.2.2 Numerical computation 3 Conclusion Bibliography 4 Appendix 4.1 The practical choice of p and q in the U.E method 4.2 Simulation from the distribution h given by (13)

0704.1434

Rotationally-invariant slave-boson formalism and momentum dependence of the quasiparticle weight

arXiv:0704.1434v2 [cond-mat.str-el] 2 Aug 2007 Rotationally-invariant slave-boson formalism and momentum dependence of the quasiparticle weight Frank Lechermann,1, 2, ∗ Antoine Georges,2 Gabriel Kotliar,3, 2 and Olivier Parcollet4 1I. Institut für Theoretische Physik, Universität Hamburg,Jungiusstrasse 9,20355 Hamburg, Germany 2Centre de Physique Théorique, École Polytechnique, 91128 Palaiseau Cedex, France 3Serin Physics Laboratories, Rutgers University, Piscataway, NJ, USA 4Service de Physique Théorique, CEA/DSM/SPhT-CNRS/SPM/URA 2306 CEA Saclay, F-91191 Gif-Sur-Yvette, France We generalize the rotationally-invariant formulation of the slave-boson formalism to multiorbital models, with arbitrary interactions, crystal fields, and multiplet structure. This allows for the study of multiplet effects on the nature of low-energy quasiparticles. Non-diagonal components of the matrix of quasiparticle weights can be calculated within this framework. When combined with cluster extensions of dynamical mean-field theory, this method allows us to address the effects of spatial correlations, such as the generation of the superexchange and the momentum dependence of the quasiparticle weight. We illustrate the method on a two-band Hubbard model, a Hubbard model made of two coupled layers, and a two-dimensional single-band Hubbard model (within a two-site cellular dynamical mean-field approximation). PACS numbers: 71.10.-w,71.10.Fd,71.30.+h,74.25.Jb I. INTRODUCTION AND MOTIVATIONS A. General motivations The method of introducing auxiliary bosons in order to facilitate the description of interacting fermionic sys- tems is an important technique in theoretical many-body physics. In this regard, the so-called slave boson (SB) approach is a very useful tool in dealing with models of strongly correlated electrons. Slave boson mean-field the- ory (SBMFT), i.e., at the saddle-point level, is the sim- plest possible realization of a Landau Fermi liquid (for a review, see e.g. Ref.1). Within SBMFT, a simplified description of the low-energy quasiparticles is obtained, while high-energy (incoherent) excitations are associated with fluctuations around the saddle point. In particu- lar, two essential features are captured by SBMFT: (i) the Fermi surface (FS) of the interacting system (sat- isfying Luttinger’s theorem) is determined by the zero- frequency self-energy, which is in turn determined by the Lagrange multipliers associated with the constraints and (ii) the quasiparticle (QP) weight Z is determined by the saddle-point values of the slave bosons. Hence SBMFT is a well-tailored technique when attempting to understand the low-energy physics emerging from more sophisticated theoretical tools, such as for example dynamical mean- field theory (DMFT), which deals with the full frequency dependence of the self-energy. In this paper, we are concerned with the construction of a slave-boson formalism which is able to deal with the two following problems. 1) In multi-orbital models, handle an arbitrary form of the interaction hamiltonian, not restricted to density- density terms, and possibly including interorbital hop- pings or hybridizations. We aim in particular at describ- ing correctly the multiplets (eigenstates of the atomic hamiltonian), but we also want to be able to work in an arbitrary basis set, not necessarily that of the atomic multiplets (and of course, to obtain identical results, in- dependent of the choice of basis). 2) Describe situations in which the QP weight is not uniform along the Fermi surface, but instead varies as a function of the momentum, i.e., Z=Z(k). There are clear physical motivations for addressing each of these issues. The first one is encountered when- ever one wants to deal with a specific correlated material in a realistic setting (see e.g. Ref. 2). Usually, more than one band is relevant to the physics (e.g. a t2g triplet or eg doublet for transition metal-oxides, or the full 7- fold set of f -orbitals in rare-earth, actinides and their compounds). The second issue is an outstanding one in connection with cuprate superconductors. In those materials, a strong differentiation in momentum space is observed in the “normal” (i.e., non-superconducting) state, especially in the underdoped regime (for a review, see e.g. Ref. 3). For momenta close to the nodal re- gions, i.e., close to the regions where the superconduct- ing gap vanishes, reasonably long-lived QPs are found. In contrast, in the antinodal directions, the angle-resolved photoemission (ARPES) spectra reveal only a broad line- shape with no well-defined QPs. The nature of the in- cipient normal state in the underdoped regime (i.e., the state achieved by suppressing the intervening supercon- ductivity) has been a subject of debate. One possibil- ity is that QPs would eventually emerge at low-enough temperature in the antinodal region as well, but with a much smaller QP weight ZAN≪ZN. Another possibility is that coherent QPs simply do not emerge in the antin- odal region. Anyhow, there is evidence from ARPES and other experiments3 that the QP weight (whenever it can be defined) has significant variation along the FS and is larger at the nodes. Since the QP weight sets the scale for the coherence temperature below which long- lived QPs form, a smaller Z means a smaller coher- ence temperature. Hence, if the temperature is higher http://arxiv.org/abs/0704.1434v2 than the coherence scale associated with momenta close to the nodes, and larger than the one associated with the antinodes, QPs will be visible only in the nodal re- gions. At this temperature, the FS will thus appear as being formed of “Fermi arcs”, as indeed observed experi- mentally4. Important differences between the nodal and antinodal region in the superconducting state have also been unraveled by recent experiments, in particular from Raman scattering which revealed two different energy scales with different doping dependence, associated with each of these regions5. Momentum-space differentiation of QP properties is therefore a key feature of cuprate su- perconductors, but it is also an issue which is particularly difficult to handle theoretically. As we now explain, these two issues are actually closely related one to the other. In a general multi-orbital model, the self-energy is a matrix Σαβ (α, β are orbital indices). Except when a particular symmetry dictates otherwise, this matrix has in general off-diagonal (interorbital) com- ponents and these off-diagonal components may have a non-zero linear term in the low-frequency expansion, hence yielding non-diagonal components of the matrix of QP weights defined as: . (1) On the other hand, a momentum dependent QP weight Z(k) means that, in real-space, Zij=Z(Ri−Rj) depends on the separation between lattice sites (a momentum- independent Z means that Zij=Z δij is purely local). Hence, in both cases, one has to handle a QP weight which is a matrix in either the orbital or the site in- dices. The connection becomes very direct in the frame- work of cluster extensions of DMFT (for reviews, see e.g. Refs [6–9]). There, a lattice problem is mapped onto a finite-size cluster which is self-consistently coupled to an environment. This finite-size cluster can be viewed as a multi-orbital (or molecular) quantum impurity problem, in which each site plays the role of an atomic orbital. Re- cently, numerical solutions of various forms of cluster ex- tensions to the DMFT equations for the two-dimensional Hubbard model have clearly revealed the phenomenon of momentum-space differentiation10–12. Developing low- energy analytical tools to interpret, understand, and gen- eralize the results of these calculations is clearly an im- portant and timely issue. The slave-boson methods de- veloped in the present work are a step in this direction. Obviously, the existence of off-diagonal components of the Ẑ-matrix is a basis-set dependent issue. A proper choice of orbital basis can be made, which diagonalizes this matrix. In certain cases, this basis is dictated by symmetry considerations, while in the absence of sym- metries, the basis set in which Z is diagonal cannot be guessed a priori. For instance, in a two-site cluster or two-orbital model in which the two sites play equivalent roles, even and odd combinations diagonalize not only the Ẑ-matrix, but in fact the self-energy matrix itself for all frequencies (see Sec. III). In such cases, it may be favorable to work in this orbital basis set, and deal only with diagonal QP weights. However, performing the rotation into this orbital basis set will in general trans- form the interacting hamiltonian into a more complicated form. For example, starting from a density-density inter- action, it may induce interaction terms which are not of the density-density type (i.e., involve exchange, pair hop- ping, etc.). For these reasons, it is essential to consider slave-boson formalisms which can handle both arbitrary interaction terms, and non-diagonal components of the QP weight matrix: these two issues are indeed connected. The formalism presented in this article builds on earlier ideas of Li, Wölfle, Hirschfeld and Frésard13,14 (see Ap- pendix. A), in which the SB formalism is formulated in a fully rotationally-invariant manner (see also Refs [15,16] in the framework of the Gutzwiller approximation), so that the orbital basis set needs not be specified from the beginning, and the final results are guaranteed to be equivalent irrespectively of the chosen basis set. B. Some notations In this paper, we shall consider multi-orbital models of correlated electrons with hamiltonians of the form: H = Hkin + Hloc[i] , (2) with Hkin = εαβ(k) d kαdkβ . (3) In these expressions, α, β label electronic species and run from 1 to M (i.e., M is twice the number of atomic or- bitals in the context of a multi-orbital model of electrons with spin: α=(m,σ),σ=↑, ↓). The k vector runs over the Brillouin zone of the lattice, whose sites are labeled by i (in the context of cluster-DMFT, i will label clusters and runs over the superlattice sites, thus k runs over the reduced Brillouin zone of the superlattice, see Sec. III C). The first term in (2) is the kinetic energy: εαβ(k) is the Fourier transform of the (possibly off-diagonal) hoppings and does not contain any local terms (i.e., εαβ(k)=0). Hloc contains both the one-body local terms and the in- teractions, assumed to be local. A general form for Hloc is55: Hloc = ε0αβd αdβ + Uαβγδ d βdδdγ . (4) Fock states form a convenient basis set of the local Hilbert space on each site. They are specified by se- quences n=(n1, . . . , nM ), with nα=0, 1 (we consider a single site and drop the site index): |n〉 = · · · |vac〉 . (5) In the following, {|A〉} will denote an arbitrary basis set of the local Hilbert space, specified by its components on the Fock states: |A〉 = 〈n|A〉 |n〉 , (6) while |Γ〉 will denote the eigenstates of the local hamil- tonian, i.e., the ‘atomic’ multiplets such that: Hloc|Γ〉 = EΓ |Γ〉 . (7) C. Slave bosons for density-density interactions: a reminder When the orbital densities nα are good quantum num- bers for the local hamiltonian Hloc, i.e., when the eigen- states of the latter are labeled by nα, a very simple slave boson formalism can be constructed which is a direct multi-orbital generalization of the 4-boson scheme intro- duced by Kotliar and Ruckenstein17,18. While this is standard material, we feel appropriate to briefly remind the reader of how this scheme works, in order to con- sider generalizations later on. We thus specialize in this subsection to a local hamiltonian of the form: Hloc = ε0αn̂α + Uαβ n̂α n̂β , (8) so that the multiplets are the Fock states |n〉 themselves, with eigenenergies: ε0αnα + Uαβ nα nβ . (9) To each Fock state, one associates a boson creation op- erator φ†n. Furthermore, auxiliary fermions f α are intro- duced which correspond to quasiparticle degrees of free- dom. The (local) enlarged Hilbert space thus consists of states which are built from tensor products of a QP Fock state, times an arbitrary number of bosons. In con- trast, the physical Hilbert space is generated by the basis set consisting of the 2M states which contain exactly one boson, and in which this boson matches the QP Fock state. Thus, the states representing the original physical states (5) in the enlarged Hilbert space, in a one-to-one manner, are the following (“physical”) states: |n 〉 ≡ φ†n|vac〉 ⊗ |n〉f . (10) The underlining in |n 〉 allows one to distinguish between the original Fock state of the physical electrons |n〉, and its representative state in the enlarged Hilbert space. In this expression, |n〉f stands for the QP Fock state: |n〉f ≡ · · · |vac〉 . (11) It is easily checked that a simple set of constraints uniquely specifies the physical states among all the states of the enlarged Hilbert space, namely: φ†nφn = 1 (12) nφn = f αfα , ∀α . (13) The first constraint imposes that only states with a single boson are retained, while the second one insures that the fermionic (QP) and bosonic contents match. Obviously, the saddle-point values of the slave bosons will have a simple interpretation, |φn|2 being the probability associ- ated with the Fock space configuration n. The operator: d†α = 〈n|f †α|m〉φ†nφm f †α (14) is a faithful representation of the physical electron cre- ation operator on the representatives (10), namely: d†α|n 〉 = 〈n′|d †a |n〉 |n′ 〉 , (15) in which, in fact, the r.h.s is either zero (if nα= 1) or composed of just a single state (with n′α= 1 and otherwise β for β 6=α). This expression of the physical electron operators is not unique however: obviously, one can for example multiply this with any operator acting as the identity on the physical states. This is true as long as the constraint is treated exactly. When treated in the mean- field approximation however, (i.e., at saddle point), these equivalent expressions will not lead to the same results. In fact, (14) suffers from a serious drawback namely it does not yield the exact non-interacting (Uαβ=0) limit at saddle-point. Instead, the expression: 〈n|f †α|m〉 [∆̂α]− 2φ†nφm [1− ∆̂α]− 2 f †α (16) with ∆̂α[φ] ≡ nφn , (17) turns out to satisfy this requirement, while having ex- actly the same action as (14) when acting on physical states. This choice of normalization is actually very natu- ral given the probabilistic interpretation of |φn|2: the ex- pression [∆̂α] −1/2φ†n (resp. φm [1−∆̂α]−1/2) is actually a probability amplitude, normalized over the restricted set of physical states such that nα=1 (resp. nα=0). Hence the combination of boson fields in (16) is a transition probability between the statem withmα=0 and the state n with nα=1. Anyhow, whether the simplest expression (14) or the normalized expression (16) is chosen for the physical op- erator, the relation between the physical and QP single- particle operators is of the form: dα = r̂α[φ] fα (18) It is important to note that the orbital index carried by the physical operator is identical to that of the QP oper- ator. An immediate consequence is that the self-energy at the saddle-point level is a diagonal matrix in orbital space Σαβ=δαβΣα, which reads: Σα(ω) = Σα(0) + ω , (19) with Zα = |rα|2 (20) Σα(0) = λα/|rα|2 − ε0α . (21) In these expressions, rα is evaluated at saddle-point level, and λα is the saddle-point value of the Lagrange multi- pliers enforcing the constraint (13). The expression (20) of the QP weight is an immediate consequence of (18): at saddle-point level, rα becomes a c-number and (18) implies that the physical electron carries a spectral weight |rα|2. Hence, in order to describe within SBMFT situations in which the QP weight is a non-diagonal matrix, one must disentangle the orbital indices carried by the physical electron and those carried by the QP degrees of freedom. These operators will then be related by a non-diagonal matrix: dα = R̂αβ [φ] fβ . (22) This is precisely what the formalism exposed in this ar- ticle achieves. The physical significance of such a non- diagonal relation is that creating a physical electron in a given orbital may induce the creation of QPs in any other orbital. Thinking of orbital as real-space indices (within e.g. cluster-DMFT), this means that the cre- ation of a physical electron on a given site induces QPs on other sites in a non-local manner, corresponding to a momentum-dependent Z(k). D. Difficulties with naive generalizations to the multi-orbital case Let us come back to the general multi-orbital inter- action (4). In order to motivate the fully rotationally- invariant formalism exposed in the next section, let us point out some difficulties arising when attempting to generalize the simple SB formalism of the previous sec- tion. The central difference between the general interaction (4) and the density-density form (8) is that the atomic multiplets |Γ〉 are no longer Fock states. Thus, it would seem natural to associate a slave boson φΓ to each of the atomic multiplets. Indeed, Bünemann et al.19 (see also [15]) have proposed generalized Gutzwiller wave func- tions in which a variational parameter (a.k.a a probabil- ity |φΓ|2) is associated with each atomic multiplet (see also Ref. [20] and the recent work of Dai et al.21 in the SB context). A slave-boson formulation requires a clear identification of the physical states within the enlarged Hilbert space. A natural idea is to define those in one- to-one correspondence with the atomic multiplets, as: |Γ〉 ?= φ†Γ|vac〉 ⊗ 〈n|Γ〉 |n〉f . (23) The local part of the hamiltonian has a simple repre- sentation on these physical states Hloc= However, a major difficulty is that there is no simple constraint implementing the restriction to these physi- cal states, and such that it is quadratic in the fermionic (QP) degrees of freedom (which is essential in order to yield a manageable saddle point). In particular, it is eas- ily checked that the apparently natural constraint21: f †αfα 〈Γ|n̂α|Γ〉φ†ΓφΓ (24) is actually not satisfied by the states (23) as an operator identity56. Further difficulties also arise when attempt- ing to derive an expression for the physical creation op- erators. These difficulties stem from the fact that two atomic multiplets having particle numbers differing by one unit cannot in general be related by the action of a single-fermion creation. One might also think of defining the physical states in correspondence to the Fock states, as: |n〉 ?= |n〉f ⊗ 〈Γ|n〉φ†Γ|vac〉 (25) which do satisfy the following quadratic constraint: f †αfα = 〈Γ|n̂α|Γ′〉φ†ΓφΓ′ . (26) However, another difficulty then arises. Namely, it is not possible to write the local interaction hamiltonian purely in terms of bosonic degrees of freedom, which is the whole purpose of SB representations. In particular, the obvious expression Hloc= ΓφΓ which had the correct action on states (23) no longer works for states (25) since it leaves unchanged the fermionic content of them. After some thinking, one actually realizes that these naive generalizations are all faced with the same problem, namely that they do not embody the crucial conceptual distinction between physical and QP degrees of freedom. Both (23) and (25) assume a priori a definite relation be- tween the physical and QP content of a state. The key to a successful SB formalism is therefore to disentangle physical and quasiparticle degrees of freedom, and letting the variational principle at saddle point decide which re- lationship actually exists between those. We shall see however in Sec. III A that, provided the lo- cal hamiltonian has enough symmetries, the rotationally- invariant formalism of the present article does correspond to assigning at saddle point a probability to each atomic configuration (multiplet) |Γ〉, hence establishing contact with the previous works of Refs. [19,21]. Yet for less sym- metric hamiltonians, the general formalism of the present article is requested. II. ROTATIONALLY-INVARIANT SLAVE-BOSON FORMALISM A. Physical Hilbert space and constraints In order to construct a SB formalism in which physi- cal and QP states are disentangled, we shall associate a slave boson φΓn to each pair of atomic multiplet |Γ〉 and QP Fock state |n〉f . More generally, we can work in an arbitrary basis set |A〉 of the local Hilbert space, not nec- essarily that of the atomic multiplets, and consider slave bosons φAn. As we shall see, the formalism introduced in this article is such that two different choices of basis sets are related by a unitary transformation and there- fore lead to identical results. In particular, one could also choose the physical Fock states |m〉d as the basis set A, and work with slave bosons φmn which form the com- ponents of a density matrix connecting the physical and QP spaces. It is crucial however to keep in mind that the first index (A) refers to physical-electron states, while the second one (n) refers to quasiparticles. A priori, a slave boson φAn can be introduced for any pair (A, n). However, in this paper, we shall restrict our- selves to phases which do not display an off-diagonal su- perconducting long-range order, and hence one can re- strict the φAn’s to pairs of states which have the same total particle number on a given site (the local hamilto- nian Hloc commutes with αdα). The formalism is easily extended to superconducting states14,15,22 by lift- ing this assumption and modifying appropriately the ex- pressions derived in this section. In the following, we consider basis states A which are eigenstates of the local particle number (denoted by NA), and hence a φAn is introduced provided α nα=NA. The representation of such a basis state in the enlarged Hilbert space is defined as: |A〉 ≡ 1√ An|vac〉 ⊗ |n〉f . (27) In this expression, DA denotes the dimension of the sub- space of the Hilbert space with particle number identical to that of A, i.e., DA≡ D(NA)= . This insures a proper normalization of the state. As before, the “under- line” in |A〉 allows to distinguish this state, which lives in the tensor product Hilbert space of QP and boson states, from the physical electron state |A〉. Having decided on the physical states, we need to iden- tify a set of constraints which select these physical states out of the enlarged Hilbert space in a necessary and suf- ficient manner. It turns out that the following (M2 + 1) constraints achieve this goal: AnφAn = 1 (28) An′φAn 〈n|f αfα′ |n′〉 = f †α fα′ , ∀α .(29) The first constraint is obvious and requires that the physical states are single-boson states. It is easy to check that the physical states satisfy the second set of con- straints (29), but a little more subtle to actually prove that this set of constraints is sufficient to uniquely select the physical states (27) in the enlarged Hilbert space. The detailed proof is given in Appendix B. Let us em- phasize that the order of primed and unprimed indices in (29) is of central importance. B. Representation of the physical electron operators We now turn to the representation of the physical elec- tron creation operator on the representatives (27) of the physical states in the enlarged Hilbert space. We need to find an operator which acts on these representatives ex- actly as d †α acts on the physical basis |A〉. Namely, given the matrix elements 〈A|d †α|B〉 such that d †α |B〉 = 〈A|d †α|B〉 |A〉 , (30) we want to find an operator d†α (in terms of the boson and QP operators) such that d†α |B〉 = 〈A|d †α|B〉 |A〉 . (31) 1. Proximate expression As in the case of the density-density interactions dis- cussed above (Sec. IC), the answer is not unique. We first construct the generalization of expression (14) to the present formalism (i.e., ignore at first the question of the proper operators to be inserted in order to recover the correct non-interacting limit). The following expression is shown in Appendix C to satisfy (31): β,AB,nm 〈A|d †α|B〉〈n|f β |m〉√ NA(M −NB) AnφBm f β . (32) We note that NA=NB+1 in this expression can take the values 1, . . . ,M . Hence, we see that within this formalism, the physical and QP operators are indeed related by a non-diagonal transformation (22): dα = R̂[φ]αβ fβ (33) with the R̂-matrix corresponding to (32) given by (R̂∗αβ denotes the complex conjugate of R̂αβ): R̂[φ]∗αβ = AB,nm 〈A|d †α|B〉〈n|f β |m〉√ NA(M −NB) AnφBm . (34) The action of (32) on physical states, and the proof that it satisfies (31) are detailed in Appendix C. 2. Improved expression The simple expression (32), although having the cor- rect action on the physical states, suffers from the same drawback than (14) in the case of density-density inter- actions. Namely, at saddle-point level (i.e., with the con- straint satisfied on average instead of exactly), the non- interacting limit is not appropriately recovered. Thus, one needs to generalize the improved expression (16) to the present rotationally-invariant formalism. However, care must be taken to do so in a way which respects gauge invariance (i.e., the possibility of making an ar- bitrary unitary rotation on the QP orbital indices, see Sec. II C). We consider the following operators, bilinear in the bosonic fields: AnφAm〈m|f αfβ |n〉 (35) AnφAm〈m|fβf α|n〉 , (36) which can be interpreted as particle- and hole- like QP density matrices (note that when the constraint is satis- fied exactly: ∆̂ αβ=δαβ−∆̂ αβ). We then choose to modify the R-matrix in the following manner (see Appendix C): R̂[φ]∗αβ = AB,nm,γ 〈A|d †α|B〉〈n|f †γ |m〉φ AnφBmMγβ , (37) with Mγβ ≡ ∣∣∣∣∣ ∆̂(p)∆̂(h) + ∆̂(h)∆̂(p) )]− 12 ∣∣∣∣∣ .(38) We chose to let the QP density matrices enter the M - matrix in a symmetrized way in order to respect equiva- lent treatment of particles and holes. Expression (37) can be shown to be gauge-invariant, and turns out to yield the correct non-interacting limit at saddle point. How- ever, although it yields a saddle point satisfying all the appropriate physical requirements, it is not fully justified as an operator identity. C. Gauge invariance As usual in formalisms using slave particles, a gauge symmetry is present which allows one to freely rotate the QP orbital indices, independently on each lattice site. Physical observables are of course gauge-invariant. Let us consider an arbitrary SU(M) rotation of the QP op- erators: f †α = Uαβ f̃ β (39) This rotation induces a corresponding unitary transfor- mation U(U) of the QP Fock states |n〉f . This unitary transformation is characterized by the fact that the ex- pectation value of fα in its Fock basis is an invariant tensor: it is the same in every basis. Therefore (summa- tion over repeated indices is implicit everywhere in the following): 〈n|f †β|m〉 = Uββ′U(U) nn′〈n′|f β′ |m ′〉 U(U)mm′ (40) n|f †αfβ|m = Uαα′U ββ′U∗(U)nn′ n′|f †α′fβ′ |m U(U)mm′ (the second expression can be deduced from the first us- ing closure relations). We can now check that if the slave bosons transforms like φAn = U(U)nn′ φ̃An′ (42) then the constraints and the expressions of the phys- ical electron operator (either (32) or (37)) are gauge- invariant. Namely, the R-matrix obeys the following transformation law: R̂[φ]αβ = R̂[φ̃]αβ′ Uββ′ (43) and therefore the physical electron operator is invariant: dα = R̂[φ̃]αβ f̃β = R̂[φ]αβfβ (44) D. Change of physical and quasiparticle basis sets It is clear that the basis |A〉 of the local Hilbert space (i.e., the physical basis states) can be chosen arbitrarily in this formalism. Indeed, making a basis change from |A〉 to |Ã〉, all the expressions above keep an identical form provided the bosons corresponding to the new basis are defined as: 〈A|Ã〉φ†An (45) As mentioned above, it is often convenient to use the eigenstates |Γ〉 of Hloc as a basis set. Changing the basis states associated with quasiparti- cles is a somewhat trickier issue. Up to now, we have worked with Fock states |n〉f . A different basis set |Q〉f can be used, provided however the unitary matrix 〈Q|n〉 is real, i.e., 〈Q|n〉=〈n|Q〉. Indeed, the matrix element 〈Q|n〉 appears in the transformation of the physical states and of the constraint, while 〈n|Q〉 appears in the trans- formation of the physical electron operator. When this matrix elements are real, new bosons can be defined in the transformed QP basis according to: 〈Q|n〉φ†An , (〈Q|n〉 = 〈n|Q〉) (46) In particular, when the local hamiltonian is a real sym- metric matrix, the same linear combinations of Fock states which define the atomic multiplets |Γ〉 can be used for QPs, and bosons φΓΓ′ can be considered. This is sometimes a useful way of interpreting the formalism and the results at saddle point (see Sec. III A). E. Expression of the hamiltonian, free energy and Green’s function In this section, we derive the expression of the hamil- tonian in terms of the slave boson and QP fermionic vari- ables. We then construct the free-energy functional to be minimized within a mean-field treatment, and express the Green’s function and self-energy at saddle point. We recall that the full hamiltonian (2) reads, in terms of the physical electron variables: H=Hkin + iHloc[i] with Hkin= αβ εαβ(k) d kαdkβ the intersite kinetic energy and Hloc the local part of the hamiltonian on a given site i, with general form (4). It is easily checked that the following bosonic operator is a faithful representation of Hloc on the representatives of the physical states in the enlarged Hilbert space: H loc = 〈A|Hloc|B〉 AnφBn (47) If the basis |Γ〉 of atomic multiplets is used, this simplifies down to: H loc = ΓnφΓn (48) Using the bosonic R-operators relating the physical elec- tron to the QP operators, yields the following expression of the kinetic energy: H kin = αα′ββ′ [R̂†]αα′εα′β′(k)R̂β′β f kαfkβ (49) A mean-field theory is obtained by condensing the slave bosons into c-numbers 〈φAn〉 ≡ ϕAn. The constraints are implemented by introducing Lagrange multipliers: λ0 associated with (28) and λαα′ ≡ [Λ]αα′ associated with (29). The saddle point is obtained by extremalizing, over the ϕAn’s and the Lagrange multipliers, the following free-energy functional: Ω[{ϕAn}; Λ, λ0] = (50) = − 1 tr ln 1 + e−β(R †(ϕ)ε(k)R(ϕ)+Λ) ABnn′ ϕ∗An′ δnn′δAB λ0 + δnn′〈A|Hloc|B〉 Λαβ〈n|f †αfβ |n′〉 ϕBn (51) The saddle-point equations, as well as technical aspects of their numerical solution, are detailed in Appendix D. Finally, we derive the expressions of the Green’s func- tions Ĝ, the self-energy Σ̂ and the QP weight Ẑ at sad- dle point. For the QPs, the one-particle Green’s func- tion Gf,αβ(k, τ − τ ′)≡−〈f †kα(τ)fkβ(τ ′)〉 reads (in matrix form): f (k, ω) = ω −R †(ϕ) ε(k)R(ϕ) −Λ (52) and hence the physical electron Green’s function reads (we drop the ϕ dependence for convenience): d (k, ω) = [R †]−1G−1f R = ω (RR†)−1 − [R†]−1ΛR−1 − ε(k) , (53) while the non-interacting Green’s function is (including the one-body term present in Hloc): d0 (k, ω) = ω11− ε 0 − ε(k) . (54) The physical self-energy is thus: Σd(ω) ≡ G−1d0 −G 1− [RR†]−1 + [R†]−1ΛR−1 − ε0 . (55) So that the matrix of QP weights is obtained in terms of the R̂-matrix at saddle point as: Z = RR† . (56) This generalizes (20) to non-diagonal cases. It is easily checked that these expressions of the physical quantities Gd,Σd and Z are indeed gauge-invariant. III. ILLUSTRATIVE RESULTS In the following, we apply the above formalism to three different model problems in strongly correlated physics. First, we consider two popular models, namely the two- band Hubbard model on a three-dimensional (3D) cubic lattice, and a “bi-layer” model, coupling two Hubbard 3D cubic lattices . Finally a two-site cluster (cluster-DMFT) approximation to the single-band Hubbard model on a two-dimensional (2D) square lattice is investigated. Hence these models have in common that they all involve two coupled orbitals (associated, in the cluster-DMFT (CDMFT) framework, to the dimer made of two lattice sites). The present formalism is of course not restricted to two-orbital problems, however such models provide the simplest examples where the power of the method may be demonstrated. A. Two-band Hubbard model The Hubbard model involving two correlated bands, without further onsite hybridization or crystal-field split- ting, serves as one of the standard problems in condensed matter theory. In contrast to the traditional single-band model, the formal interaction term in Eq. (4) now gener- ates in the most general fully SU(2) symmetric case four energy parameters, i.e., the intraorbital Hubbard U and the interorbital Hubbard U ′ as well as the two exchange couplings J ,JC . Thus the present atomic hamiltonian TABLE I: Eigenstates |Γ〉 of the SU(2) rotationally-invariant two-band Hubbard model. Spin values and energies are given for the eigenstates. The last column shows the slave bosons for the description of the eigenstates in the SBMFT formalism. No. Eigenstate |Γ〉 SΓ SzΓ EΓ φΓn 1 |00, 00〉 0 0 0 φ1,|00,00〉 2 | ↑ 0, 00〉 1 0 φ2,|↑0,00〉 3 |0 ↓, 00〉 1 0 φ3,|0↓,00〉 4 |00, ↑ 0〉 1 0 φ4,|00,↑0〉 5 |00, 0 ↓〉 1 0 φ5,|00,0↓〉 6 | ↑ 0, ↑ 0〉 1 1 U ′ − J φ6,|↑0,↑0〉 (| ↑ 0, 0 ↓〉+ |0 ↓, ↑ 0〉) 1 0 U ′ − J φ7,|↑0,0↓〉, φ7,|0↓,↑0〉 8 |0 ↓, 0 ↓〉 1 -1 U ′ − J φ8,|0↓,0↓〉 (| ↑ 0, 0 ↓〉 − |0 ↓, ↑ 0〉) 0 0 U ′ + J φ9,|↑0,0↓〉, φ9,|0↓,↑0〉 10 1√ (| ↑↓, 00〉 − |00, ↑↓〉) 0 0 U − JC φ10,|↑↓,00〉, φ10,|00,↑↓〉 11 1√ (| ↑↓, 00〉+ |00, ↑↓〉) 0 0 U + JC 11,|↑↓,00〉, φ11,|00,↑↓〉 12 | ↑↓, ↑ 0〉 1 U + 2U ′ − J φ12,|↑↓,↑0〉 13 | ↑↓, 0 ↓〉 1 U + 2U ′ − J φ13,|↑↓,0↓〉 14 | ↑ 0, ↑↓〉 1 U + 2U ′ − J φ 14,|↑0,↑↓〉 15 |0 ↓, ↑↓〉 1 U + 2U ′ − J φ 15,|0↓,↑↓〉 16 | ↑↓, ↑↓〉 0 0 2U + 4U ′ − 2J φ16,|↑↓,↑↓〉 reads Hloc = U nα↑nα↓ + U n1σn2σ′ n1σn2σ + J 2σ̄d1σ̄d2σ 1↓d2↓d2↑ + d 2↓d1↓d1↑ . (57) The kinetic energy shall contain only intraband terms for a basic tight-binding (TB) model for s-bands on a 3D simple cubic lattice with lattice constant a. Thus the corresponding hamiltonian is written as Hkin = − α=1,2 iασdjασ , (58) with the eigenvalues εα(k) = − µ=xyz cos(kµa) , (59) where tα denotes the hopping parameter for orbital α=1,2. For convenience, we set a=1. The factor 1/3 in eq. (58) is to normalize the total bandwidth to Wα=4tα. Because of the cubic symmetry, U ′=U−2J may be used, and furthermore we set J=JC . This model is similar to the one considered by Bünemann et al.19 using a gener- alized Gutzwiller approximation (see also [15]). Our sim- pler TB description exhibits in principle perfect nesting, however this issue is not relevant at the present level. The 3D two-band Hubbard model is studied to make contact with the named previous work and in order to establish the connection between the Gutzwiller and slave-boson points of view. When working in the SU(2) rotationally-invariant case, the 24=16 atomic eigenstates |Γ〉 of the local hamil- tonian (57) serve as the appropriate atomic basis (see Ta- ble I), however also the simpler Fock basis (or any other) may be used. Of course, in the Fock basis, a more com- plicated energy matrix must be used in the saddle-point equations (see Appendix D). It should be clear from Tab. I that there are 20 nonzero slave-boson amplitudes φΓn for the current problem. The S z=0 triplet as well as the three singlets are described with two φΓn, respec- tively. In principle, even more φΓn may be introduced in the beginning of the iteration cycle to minimize Ω, but at convergence those one will come out to be strictly zero. Of course, in high-symmetry situations there is still some redundancy within the set of the 20 SBs. For instance, for equal bandwidth at half filling (see Fig. 1), all the one- and three-particle SBs are identical, as well as the zero- and four particle SB. Moreover the Sz = ±1 triplet SBs are equal because of the degeneracy. The two SBs describing the Sz=0 triplet are also identical, with a mag- nitude φ (t,0)n (t,±1)n/ 2. Also the bosons describing one specific singlet have the same absolute value, however they carry the multiplet phase information, i.e., have plus or minus sign. In conclusion, in the orbitally degenerate case, the SB amplitudes at saddle point are of the form: ϕΓn = 〈n|Γ〉 yΓ , (60) in which the matrix element 〈Γ|n〉 is entirely determined by Hloc and yΓ is a (coupling-dependent) amplitude, de- pending only on the eigenstate Γ. This is more clearly in- 0 0.5 1.0 1.5 2.0 U/(2t) φ1, |00,00> φ2, |10,00> φ10, |11,00> φ6, |10,10> φ11, |11,00> FIG. 1: (color online) Inequivalent Slave-boson probabili- ties |φΓn|2 for the two-band Hubbard model at half-filling for equal bandwidth and J/U=0.2. Note that φ10,|↑↓,00〉 and φ11,|↑↓,00〉 describe part of the singlet states, hence their over- all amplitude is scaled by 1/ terpreted when atomic states are also used as basis states for QPs (Sec. II D). Indeed, Eq. (60) means that: ϕΓΓ′ = δΓΓ′ yΓ (61) Hence, in this highly symmetric case, the saddle point is indeed of the diagonal form considered in Refs. [19,21]. Once the symmetry is lowered, more SBs become inequiv- alent and this relation does not hold anymore: there are off-diagonal components even when the basis of atomic states is used for both physical and QP states. In this context, the present formalism becomes essential. Dif- ferent bandwidths for each orbital, together with a finite doping away from half-filling lead for instance to two dif- ferent absolute values for the two SBs associated with the singlets formed by the two doubly-occupied Fock states (as seen at the end of this paragraph in Fig.6). Since no interorbital hybridization is applied in this section, the Ẑ-matrix is diagonal. We consider first the simple case of equal bandwidths t1=t2=0.5 (note that in all our applications, t sets the unit of energy), thus Z11=Z22=Z. Figure 2 shows the variation of Z for different ratios J/U in the half-filled case (n=2). The critical coupling Uc for the Mott transition with J=0 ob- tained from this slave-boson calculation is in accordance with the result of the analytical formula given by Frésard and Kotliar18. It is seen that an increased J lowers the critical U and moreover changes the transition from sec- ond to first order. Note that in this regard, Fig. 2 depicts Z up to the spinodal boundary, i.e., the true transition (following from an energy comparison) is expected to be at slightly lower Uc. One can also observe the nonmono- tonic character for the evolution of the critical Z at this boundary when increasing J/U . We plot in Fig. 2 addi- tionally the results when restricting the atomic hamilto- nian to density-density terms only, in order to check for the importance of the then neglected spin-flip and pair- hopping terms. For larger J/U the critical Z from the 0 1 2 3 4 U/(2t) rotationally-invariant density-density FIG. 2: (color online) Influence of J on the Mott transition in the two-band Hubbard model at half fill- ing (n=2) for equal bandwidth. From right to left: J/U=0, 0.01, 0.02, 0.05, 0.10, 0.20, 0.45. latter description is larger compared to the rotationally- invariant one and moreover it is monotonically grow- ing. The latter feature strengthens the first-order char- acter in the density-density formulation for growing J/U , whereas for rotationally-invariant interactions this char- acter is strongly weakened in that regime. Although for J/U=0.45 the jump of Z is quite small, the transition is however still first order in the present calculation. Fur- thermore, there appears to be a crossover between the two approaches concerning the reachable metallic spin- odal boundary when increasing J/U . At quarter filling (n=1) a continuous transition is ob- tained for all the previous interactions (see Fig. 3). Com- pared to the half-filled case, the density-density approx- 0 2 4 6 8 U/(2t) rotationally-invariant density-density J/U=0.45 J/U=0.20 FIG. 3: (color online) Influence of J on the Mott transi- tion in the two-band Hubbard model at quarter filling (n=1) for equal bandwidth. The first three combined curves for the two types of interactions (from left to right) belong to: J/U=0, 0.05, 0.10. Arrows indicate the labelling for the two larger J/U ratios. 0 0.2 0.4 0.6 0.8 1.0 J/(2t) U = 1 U = 2 U = 3 FIG. 4: (color online) Influence of J for fixed U at n=2 (solid lines) and n=1 (dotted-dashed lines) for equal bandwidth and full SU(2) symmetry. The vertical dotted lines mark the limit we set for J , respectively. imation appears to be less severe for small J/U , but leads to some differences compared to the rotationally- invariant form for large J/U . Note that for J/U=0.45, U ′−J in the local hamiltonian (57) becomes negative. Thus a corresponding change of the ground state may lead to the resulting nonmonotonic behavior for Uc then observed in Fig. 3. The critical U for J=0 is smaller than at half filling and with increasing J/U the transition is shifted to larger Uc (with the above named exception for J/U large). Hence J has a rather different influence on the degree of correlation for the two fillings. While for n=2 the Hund’s rule coupling substantially enhances the correlations, seen by the decrease in Z, for n=1 the oppo- site effect may be observed. This is also demonstrated in Fig. 4 which displays the influence of J for fixed values of U comparing half filling with quarter filling. The strong decrease in Z upon increasing J was recently shown to be 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 filling J/U = 0 J/U = 0.1 J/U = 0.2 FIG. 5: Filling dependence of Z for selected values of J/U within the equal-bandwidth two-band model with full SU(2) symmetry (U=1.75). important for the physical properties of actinides, in par- ticular regarding the distinct properties of δ-Plutonium and Curium23. For each U shown in Fig. 4, the density- density limiting value U/3 was used as an upper bound for J . However, for U=2 and U=3 the system shows already a first-order transition at half-filling below the latter limit. The QP residue Z is shown as a function of filling n in Fig. 5 for U=1.75 and three ratios J/U . For J=0 it is observed that Z(n) exhibits two minima, both located at integer filling. The minimum at n=1 is deeper, cor- responding to a lower value for Uc in the quarter-filled case. Because of the filling-dependent effect of J seen in Fig. 4, the nonmonotonic character of Z(n) is lifted for growing J/U . The two-band Hubbard model was already extensively studied in the more elaborate DMFT framework in in- finite dimensions. Such investigations reveal the same qualitative change of the critical U for different integer fillings24,25, of course with some minor quantitative dif- ferences. Also the reduction24,26–29 of Uc and the onset of a first-order Mott transition24,26,27,29 for finite J/U at half-filling is in accordance. Concerning the latter effect, the trend of weakening the first-order tendency for large J/U is also reproduced and there is some discussion26,27 about the possibility of even changing back to a contin- uous Mott transition in that regime. The increasing Uc with growing J at quarter filling was also found by Song and Zou29. Finally, in Fig. 6 a comparison between the equal- bandwidth and the different-bandwidth cases at noninte- ger filling n=1.5 is displayed (J/U=0.2). ForW1=W2 the model does not show a metal-insulator transition because of the doping. Also the filling of both bands is iden- 0 1 2 3 4 U/(2t) 0 1 2 3 4 FIG. 6: (color online) Comparison of the two-band model for W1=W2 (left) and W1=2W2 (right) at noninteger filling n=1.5 and J/U=0.2. The ratio x is plotted for the singlet states coupling the doubly occupied Fock states (see Tab I), demonstrating that ϕΓΓ′ is no longer diagonal in this case. tical and constant with increasing U (ns1=ns2=0.375), and as stated earlier the SBs are still of the form given by eq. (60). However, when breaking the symmetry be- tween the two bands by considering different bandwidths, the model behaves qualitatively rather differently. The individual band fillings are not identical anymore, favor- ing the larger-bandwidth band for U=0. With increas- ing U the system manages to drive at least one band insulating by transferring charge from the broader into the narrower band, until the latter is filled with one elec- tron30,31. Hence Z2 of the narrower band becomes zero at an orbital-selective Mott transition (OSMT)30–35. This asymmetric model has also a more sophisticated SB de- scription, since for instance the SBs of the singlets built out of the respective doubly-occupied Fock states have now different amplitudes. B. The Hubbard bilayer Next, we consider a model consisting of two single- band Hubbard models (two “layers”), coupled by an in- terlayer hopping V . This rather popular model has al- ready been subject of various studies36–39. For simplicity and in order to make connection to the previous section, each layer is described here by a 3D cubic lattice, with an onsite repulsion U and an intralayer bandwidth Wα (α = 1, 2), possibly different for the two layers. Hence the local hamiltonian for this problem reads Hloc = U α=1,2 nα↑nα↓ + V 1σd2σ + d 2σd1σ 1σd1σ′d 2σ′d2σ , (62) where the last term describes a possible spin-spin interac- tion between the layers. However, for simplicity, we only 0 1 2 3 U/(2t) V = 0 V = 0.25 FIG. 7: (color online) Half-filled bi-layer with equal band- width for V=0 and V=0.25. For V=0 the filling per spin within the two bands is identical (ns1=ns2=0.5), whereas for finite V the the symmetry-adapted bonding/antibonding states have different filling, denoted n+,n−. 0 0.5 1.0 U/(2t) 0 0.5 1.0 1.5 2.0 = 0.75 = 0.50 = 0.25 FIG. 8: (color online) QP residues Zi and symmetry-adapted fillings n+,n− for the half-filled bilayer. Left: W2/W1=0.5 and V=0. Right: various bandwidth ratios and V=0.1. In the right part, the curves for smaller Z and n are associated with the lower-bandwidth band. present in this article results with J=0. Our choice of kinetic energy is equivalent to the one in the last section, i.e., given by Eqs. (58,59). In the presence of V , an off-diagonal self-energy Σ12(ω) is generated. Furthermore, away from half-filling (n1+n2=2), this self-energy is expected to have a term linear in ω at low frequency, and hence Z12 6=0. We note that, when the bandwidths are equal (W1=W2), the bi- layer model can be transformed into a two-orbital model by a k-independent rotation to the bonding-antibonding (or +,−) basis. In the latter basis, there is no hybridiza- tion but instead a crystal-field splitting (=2V ) between the two orbitals. The couplings of the two-orbital hamil- tonian are given by (in the notation of the previous sec- tion, and for J=0): Ueff=U eff=Jeff=U/2. When the bandwidths are different however, the interlayer hopping cannot be eliminated without generating non-local inter- dimer interactions. Due to the reduced symmetry of the present model in comparison to the two-band Hubbard model from the previous section, the number of nonzero SBs φAn equals now 36 (we use here the Fock basis for |A〉). We first con- sider the simplest case of a half-filled system (n1=n2=1) with equal bandwidths W1=W2 (and J=0). Results for the intralayer QP weight and the orbital occupancies of the bonding and antibonding bands are given in Fig. 7. It is seen that the Mott transition is continuous for V=0 but becomes discontinuous in the presence of an inter- layer hopping V 6=0. For V=0.25 the spinodal boundary of the metallic regime is reached for U∼2.055. These re- sults are consistent with findings in previous works36–40 within the DMFT framework. Still focusing on the half-filled case, we display in Fig. 8 the QP weight as a function of U for different bandwidth ratios W2/W1. When V=0, one has two independent 1.5 1.6 1.7 1.8 1.9 2.0 filling -0.03 -0.02 -0.01 FIG. 9: (color online) Doped bilayer with equal bandwidth and V=0.25 for U=2.054 (<Uc). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 U/(2t) -0.20 -0.15 -0.10 -0.05 FIG. 10: (color online) Bilayer at fixed doping (n=1.88) with equal bandwidth and V=0.25. Full green (gray) lines: fillings n+,n−, dashed dark lines: QP weights Z+, Z−. The vertical dotted line marks the critical U at half filling. Mott transitions in each layer, i.e., an OSMT scenario, at which Z vanishes continuously. In the presence of a non-zero V , this is replaced by a single discontinuous transition for both orbitals. This is consistent with pre- vious findings on the OSMT problem32. We now consider the effect of finite doping away from half filling. Fig. 9 displays the diagonal (Z11=Z22) ele- ments as well as the now appearing Z12 element of the QP weight matrix as a function of doping for U<Uc. Ad- ditionally shown are the symmetry-adapted QP weights Z+,− (occupations n+,−) which follow from diagonalizing the Ẑ (∆̂(p)) matrix. Since Z12 is small in this case, the Z+,− are rather similar to Z11=Z22 and merge with the latter at half-filling. On the other hand, the polarization of the (+,−) bands is still increasing. As it is seen in Fig. 10 the off-diagonal component Z12 becomes increasingly important for larger U (>Uc) in the doped case. It follows that in this regime the QP weights Z+,− for the bonding/antibonding bands have rather dif- ferent magnitude/behavior. Whereas Z− is monotoni- cally decreasing, Z+ turns around and grows again (as U = 0 U = 3.5 R Γ X M Γ FIG. 11: (color online) QP bands of the doped bilayer model (n=1.88) with equal bandwidths and V=0.25. The domi- nately filled band is the bonding one, respectively. also is the filling of the bonding band). Hence, this model is a simple example in which a differentiation between QP properties in different regions of the FS occur. Fig. 11 shows the QP (+,−)-bands in the noninteracting and in- teracting case (U>Uc), exhibiting strong orbital polariza- tion and different band narrowing close to the insulating state. For very small doping and large U a transition to a new metallic phase is found, which will be discussed in detail in a forthcoming publication41. Finally, we have also investigated a case in which the interlayer (interorbital) hopping does not have a lo- cal component (V=0), but does have a non-local one V=t12 6=0, treated in the band term of the hamiltonian. Hence the corresponding energy matrix reads here ε(k) = −2 t11 t12 t12 t22 µ=xyz cos(kµa) , (63) with the choice t11=t22=0.5 and t12=0.25, as well as a=1. In that case, a continuous Mott transition within 0 1 2 U/(2t) FIG. 12: (color online) Half-filled bilayer, with equal band- widths and V=0, but with a non-local interlayer hybridization an OSMT scenario can be recovered, with, interestingly, a sizeable value of the off-diagonal Z12 (Fig. 12). At the transition Z11=Z22=Z12≡Zc holds, i.e., the Ẑ-matrix has a zero eigenvalue, associated with the (antibonding) in- sulating band. Note however that no net orbital polar- ization appears with V being purely non-local. C. Application to the momentum-dependence of the quasiparticle weight within cluster extensions of In this section, we finally consider the implications of the rotationally invariant SB technique for the Mott tran- sition and the momentum-dependence of the QP weight, in the framework of cluster extensions of DMFT. For simplicity, we consider a CDMFT approach to the two-dimensional Hubbard model with nearest-neighbor hopping t and a next-nearest neighbor hopping t′, based on clusters consisting of two sites (dimers), arranged in a columnar way on the square lattice (see Fig. 13). The “local” hamiltonian on each dimer is formally identical to the one introduced in the previous section for the bi- layer model, i.e. Eq. (62), with the value V=−t of the inter-‘orbital’ hybridization. The interdimer kinetic en- ergy matrix reads (we set again a=1): ε11(K) = ε22(K) = −2t cosKy (64) ε12(K) = ε 21(K) = −t ei 2Kx − 2t′ 1 + ei 2Kx cosKy in which K denotes a momentum in the reduced Bril- louin zone (BZ) of the superlattice: Kx ∈ [−π/2,+π/2], Ky ∈ [−π,+π]. Note again that in SB calculations, the intradimer t has to be treated separately from the rest of the kinetic energy within Hloc. It is easy to check that when putting back −t into the offdiagonal elements of the above kinetic-energy matrix, the eigenvalues just correspond to the one of a single band: ε(k) = −2t(cos kx + cos ky)− 4t′ cos kx cos ky (65) PSfrag replacements 1 2 FIG. 13: Square lattice in the 2-site CDMFT picture. 0 0.5 1.0 1.5 2.0 U/(2t) 0 0.5 1.0 1.5 2.0 FIG. 14: (color online) Half-filled two-dimensional Hubbard model within 2-site CDMFT. Left: QP weights and band fillings, right: static dimer self-energy Σ. in the full BZ of the original lattice kx,y ∈ [−π,+π]. In the following, we set t=0.25 and consider successively t′=0 and t′= −0.3 t (a value appropriate to hole-doped cuprates). Note that, in this article, we do not consider a bigger cluster than the 2-site dimer, even in the pres- ence of t′. Hence, the cluster self-energy will only con- tain Σ11 and Σ12 components, i.e., has a spatial range limited to the dimer. As a result, no renormalization of the effective t′ is taken into account. This is of course an oversimplification (particularly in view of the demon- strated physical importance11,12 of Σ13 close to the Mott transition). Larger clusters will be considered within the present SBMFT in a further publication. The goal of the present (simplified) study is to make a point of princi- ple, namely that the SB formalism can indeed produce a momentum-dependent Z(k). As the cluster symme- try of the problem at hand is identical to the bilayer model from the last section, the number of nonzero SBs amounts again to 36. Figures 14, 15 and 16 summa- rize the main findings, at half-filling and as a function of doping, respectively. Let us first concentrate on the case 0.75 0.80 0.85 0.90 0.95 1.00 filling per site FIG. 15: (color online) Doped two-dimensional Hubbard model (t′=0) within 2-site CDMFT, for U=2.195 (Uc∼2.197). 0.0 0.5 1.0 1.5 2.0 2.5 U/(2t) 0.5 1.0 1.5 2.0 2.5 -0.02 -0.01 FIG. 16: (color online) Two-dimensional Hubbard model within 2-site CDMFT at fixed doping n=0.94. Left: t-only model, right: t-t′ model (t′=−0.3t). The vertical lines denote the critical U at half filling, respectively. t′=0. Obviously, the Mott transition at half-filling in this case occurs in a manner which is very similar to the bi- layer model with a finite interlayer hybridization studied in the previous section (Fig. 7): a first-order transition is found. The static part of the self-energy Σ11=Σ22 equals U/2, while Σ12 (which has no frequency dependence at half-filling within SBMFT) has a more complicated nega- tive amplitude close to the transition. Note that we only discuss the paramagnetic solution, though, of course, the system is in principle unstable against antiferromagnetic order for any U . Upon doping, a finite value of Z12 is generated. The behavior of Z12 is rather similar to the case of the bilayer model, except for the change of sign (Figs. 15, 16). Hence its amplitude is again significantly enhanced for U>Uc, i.e., the Z+,− values tend to mani- festly deviate from each other. This is therefore signalling an increasingly nonlocal component of Z(k) as the Mott insulating state is approached at strong coupling. Again, further studies in the latter regime at small doping will be published soon41. Including the effect of a nonzero nearest-neighbor hop- ping t′ 6=0 turns out to lead to significant differences. Al- though the first-order character of the transition remains stable, the critical U is significantly lower (Fig. 14). The static components of the self-energy behave rather simi- larly to the t-only case, with some minor quantitative dif- ferences. There is a small negative Z12 with a maximum amplitude ∼0.02, remaining nonzero also at the Mott transition (∼0.01). The main difference in comparison to t′=0 is that here, in the doped case, Z12 changes sign from negative to positive close to the insulating regime for U>Uc+δ (with δ>0) (see Fig. 16). Thus the degree of correlation of the effective (bonding-antibonding) bands is inverted. These differences have to be interpreted with caution however, since again the hopping range on the lattice is larger than our cluster size, and definitive con- clusions will have to be drawn from a study involving Σ13 as well. Nonetheless, keeping with the simplified treatment based on a 2-site cluster, we now describe the resulting momentum dependence of the QP weight Z(k) for the t-t′ model. The matrix elements Σ11 and Σ12 of the cluster (physical) self-energy matrix Σc are obtained from the SB amplitudes at saddle point according to (55). The self-energy is then periodized on the whole lattice, in the form42,43: Σlat(k, ω) = Σ11(ω) + Σ12(ω)(cos kx + cos ky) . (66) The interacting FS is defined as follows µ− ε(k)− Σlat(k, ω=0) = 0 . (67) For our case, using Eqs. (65, 66), this reads µ− Σ11(0) + Σ12(0) (cos kx + cos ky) +4t′ cos kx cos ky = 0 . (68) Hence the FS deforms in a nontrivial way in the presence of Σlat, when including t ′ in the present 2-site CDMFT description. The QP weight Z(k) can be derived from Σlat according to: Z(k) = Σlat(k, ω) ]−1∣∣∣∣∣ , (69) which leads here to: Z(k) = −1]11 + −1]12(cos kx + cos ky) Z211 − Z212 Z11 − Z12(cos kx + cos ky) A contour plot of this function is displayed in Fig. 17. Note that it varies only according to (cos kx+cos ky). Be- cause the interacting FS involves both t and t′, and hence 0.195 0.196 0.197 0.198 0.199 0.201 0.202 0.203 0.204 0.205 PSfrag replacements (0,0) (π, π) 0.278 0.280 0.282 0.284 0.286 FIG. 17: (color online) Interacting Fermi surface (solid lines) for the CDMFT treatment of the 2D t-t′ Hubbard model with t′=−0.3t and U=2.5 at n=0.94 (per site). The color contours show the variation of Z(k) (smallest at antinodes). both lattice harmonics (cos kx + cos ky) and cos kx cos ky (for t′ 6=0), it cuts through different contour lines of Z(k). This results in a QP weight which varies on the FS. Fig- ure 17 shows Z(k) for k close to the interacting FS. Albeit the momentum variation is quantitatively quite small, the key qualitative effect of Z being different on different part of the FS is indeed found. It is seen that the QPs along the nodal direction, i.e., along (0,0)-(π, π), have slightly larger Z than the ones in the the antinodal direction ((0,0)-(0, π)). Hence these results are indeed in qualitative agreement with ARPES measurements on cuprates. Note that to get nodal points to be more co- herent that antinodal ones in this 2-site scheme, Z12>0 is actually crucial. Our results provide, to our knowledge, the first ex- ample of a SB calculation which can address the issue of the momentum dependence of the QP weight. We believe that the too small variation of Z along the FS found here is due to the oversimplified 2-site description in which Σ13 is neglected. We intend to consider improvements on this issue using the present SBMFT in a forthcoming work. Finally, let us make contact with previous work on the two-dimensional Hubbard model. Of course, this model has been intensively studied with a variety of methods such as: quantum Monte-Carlo44,45, exact diag- onalization46,47, path-integral renormalization group48, functional renormalization group49 and various quan- tum cluster methods (dynamical cluster approxima- tion50, cluster extensions of dynamical mean-field the- ory7, and variational cluster perturbation theory8). We shall not attempt here a detailed comparison between the rotationally-invariant SB method (which anyhow is a mean-field technique tailored to address low-energy is- sues) with the results of these numerical methods over the whole phase diagram (note in particular that we have not yet investigated long-range ordered phases, such as antiferromagnetism or superconductivity). Rather, we would like to point out that some recent numerical studies using the above methods10–12,49 have indeed re- vealed the emergence of momentum-space differentiation in the two-dimensional Hubbard model. We hope that the rotationally-invariant SB method will help under- stand qualitatively the low-energy physics emerging from these results. IV. CONCLUSION AND PERSPECTIVES In this paper, we extended and generalized the rotationally invariant formulation of the slave-boson method13,14. Our formulation achieves two goals: (i) extending the slave-boson method in order to accommodate the most general crystal fields, interactions and multiplet struc- tures and (ii) the development of a technique which can describe QP weights and Fermi liquid parameters which vary along the Fermi surface. The key aspect of the formalism is to introduce slave boson fields which form a matrix with entries labeled by a pair of a physical state and a QP state (within an ar- bitrary choice of basis set). As a result, a density matrix is constructed instead of just a probability amplitude for each state. While the first objective (i) could also be achieved by generalizing appropriately the Gutzwiller approxima- tion15,16, we find the slave-boson approach to be some- what more flexible, in the sense that it is a mean field theory which can in principle be improved by computing fluctuations around the saddle point. Our application to the two band model seems promising. While further work is needed to benchmark the accuracy of the rotationally- invariant slave-boson method against exact quantum im- purity solvers, it is clear that already in the single site multiorbital DMFT setting, our method has numerous advantages. It obeys the Luttinger theorem even in the presence of multiplets, and can accommodate full atomic physics information. Furthermore, the off-diagonal el- ements of the matrix of QP weights can be calculated within this method, while the standard slave-boson or Gutzwiller approximations (using probability amplitudes instead of a density matrix) cannot achieve this goal. Our technique achieves the second objective (ii) via a detour, namely the use of cluster extensions of dynami- cal mean-field theory in order to reduce the lattice to a multisite (molecular) impurity problem, to which we ap- ply our rotationally-invariant slave-boson method as an impurity solver. Because the intersite matrix elements of the QP weight can be calculated, it leads on the lat- tice to a momentum-dependence of the QP residue Z(k). We successfully demonstrated this point, in the frame- work of a 2-site CDMFT study of the single-band 2D Hubbard model. We did find that the QP weight at the nodes is somewhat larger than at the antinodes, al- though the magnitude of this effect is expected to in- crease within a more realistic study involving a larger cluster (e.g., a square plaquette), which is left for fu- ture work. A major challenge is the direct extension of our slave-boson approach to the lattice, without resorting to the cluster-DMFT detour. In this context, we men- tion that other slave-boson techniques, which introduce magnetic correlations through the use of link variables to decouple the superexchange J term1, can be interpreted in terms of a k dependent self-energy. However, within such schemes, the derivative of the self-energy with re- spect to frequency is momentum independent (in contrast to the static part), yielding a k independent QP residue. Hence, our approach goes beyond these methods, at least in conjunction with the cluster-DMFT approach. We hope that having an economic impurity solver based on SBs will allow us to study larger cluster sizes than feasible with other methods, and most importantly help us un- derstanding the low-energy physics emerging from these cluster dynamical mean-field theories. Finally, we limited our study to slave bosons which do not mix the particle number. The extension to full charge-rotational invariance and superconductivity is possible (see Refs. [14,22] in the single-orbital case), and will be presented in a separate paper. In this con- text, the slave boson method will incorporate the SU(2) charge symmetry and its extension away from half fill- ing considered by Wen and Lee51 and the rotationally- invariant slave-boson formalism can serve as a power- ful tool for interpreting the low-energy physics emerging from plaquette-CDMFT studies of this issue. Acknowledgments We are grateful to Pablo Cornaglia and Michel Ferrero for very useful discussions and remarks. As this work was being completed, we learned of a parallel effort by Michele Fabrizio52, in the framework of the Gutzwiller approximation. In particular, the form of the constraints advocated in this work matches our constraints (28,29) in SB language. A.G. also thanks him for discussions. This work has been supported by the “Chaire Blaise Pas- cal” (région Ile de France and Fondation de l’Ecole Nor- male Supérieure), the European Union (under contract “Psi-k f-electrons” HPRN-CT-2002-00295), the CNRS and Ecole Polytechnique. G.K. is supported by the NSF under Grant No. DMR 0528969. APPENDIX A: SINGLE-ORBITAL CASE AND CONNECTION WITH PREVIOUS WORK Here, we briefly consider the single-orbital case (M=2), which also allows to make contact with Refs. [13, 14]. These authors introduced in this case a rotationally- invariant formalism, with the calculation of response functions associated with the saddle point as their main motivation. For M=2, the following local basis set can be considered (whether or not Hloc is diagonal in this basis): N = 0 : |0〉 , N = 1 : |σ〉 = d †σ |0〉 (A1) N = 2 : |D〉 = d †↑d ↓ |0〉 . Hence, we introduce the following bosons (not mixing sectors with different particle numbers, i.e., not consid- ering superconducting states): φ00 ≡ φE , φσσ′ , φ↑↓ ≡ φD . (A2) Up to normalizations, the bosons p σσ′ introduced in Ref. 13 correspond to φ σσ′ . In contrast, the stan- dard Kotliar-Ruckenstein17 formalism introduces only two bosons p†σ in the one-particle sector. The represen- tatives (27) of the physical states read here: |0〉 = φ†E |vac〉 |σ〉 = σ′ |vac〉 (A3) |D〉 = φ†Dd ↓ |vac〉 , and the constraints (28,29) read: 1 = φ EφE + σσ′φσσ′ + φ DφD (A4) f †αfα = φ DφD + φ†σαφσα (A5) ↑f↓ = σ↓φσ↑ (A6) ↓f↑ = σ↑φσ↓ . (A7) Not including, for simplicity, the square-root normaliza- tions in (37), needed however in order to insure a correct U=0 limit at saddle point, the ‘simplest’ expression (32) of the electron creation operators read: ↑βφE + (−1) Dφ↓β ] f β (A8) ↓βφE − (−1) Dφ↑β ] f β . (A9) Apart from the motivations of Ref. [13] (associated with fluctuations and response functions), the usefulness of the rotationally-invariant scheme in the single-orbital case can be demonstrated on a toy model consisting of a one- band Hubbard model with a magnetic field, purposely written in the Sx direction (i.e., in the form h d ↑d↓+h.c, analogous to a hybridization). Although the direction of the field should not matter, a direct application of the standard Kotliar-Ruckenstein formalism is impossible in that case. The rotationally-invariant formalism can be shown to lead to the correct saddle point, independently of the spin-quantization axis. APPENDIX B: DERIVATION OF EQ. (29) In this section, we show that the physical states of the form (27) are exactly those selected by the constraints (29) and (28). First, it is easy to check that states of the form (27) do satisfy these constraints. Indeed, let us act on the state |C〉 ≡ 1√ Cm|vac〉 ⊗ |m〉f and with (29). The l.h.s leads to: f †α fα′ |C〉 = Cm|vac〉 ⊗ f α fα′ |m〉f 〈m′|f †α fα′ |m〉φ Cm|vac〉 ⊗ |m ′〉f . When acting with the r.h.s, only the term A=C and n=m gives a non-vanishing contribution, hence: An′φAn 〈n|f αfα′ |n′〉|C〉 = 〈n|f †αfα′ |n′〉φ Cn′ |vac〉 ⊗ |n〉f . We now prove that (29) are sufficient conditions, which is a bit more difficult. Since (28) excludes states with more than one boson, it is enough to consider a general state of the form: |C;W 〉 ≡ Wpq φ Cp|vac〉 ⊗ |q〉f . (B2) and to show that (29) implies Wpq ∝ δpq. Acting on this state with each term in the constraint (29) yields for the l.h.s: f †α fα′ |C;W 〉 = Wpq φ Cp|vac〉 ⊗ f αfα′ |q〉f Cp|vac〉 ⊗ |r〉f Wpq〈r|f †αfα′ |q〉 .(B3) Let us now act with the r.h.s. Only the terms with A=C and n=p contribute, leading to: An′φAn 〈n|f αfα′ |n′〉|C;W 〉 Wpq φ Cn′ |vac〉 ⊗ |q〉f 〈p|f αfα′ |n′〉 Cp|vac〉 ⊗ |r〉f Wqr〈q|f †αfα′ |p〉 ,(B4) where the last expression comes from a change of indices n′ → p, q → r, p→ q. We see that the constraint is satisfied provided that the following identity holds, for all orbital indices αα′ and all states p, r: Wpq〈r|f †αfα′ |q〉 = Wqr〈q|f †αfα′ |p〉 . (B5) Let us first look at the case α=α′, which reads: rαWpr = pαWpr . (B6) Hence Wpr=0 unless pα=rα for all α, so that: Wpq = wp δpq . (B7) Substituting this into (B5), we obtain: wp 〈r|f †αfα′ |p〉 = wr 〈r|f †αfα′ |p〉 . (B8) Thus, if r and p are related by a move of a QP from one state to another (a transposition of two occupation numbers), then wp=wr. Moreover, two Fock states in the same sector HN of the Hilbert space are related by a permutation of the occupied states, which can be de- composed in a product of transpositions. Hence, wp is a constant for p ∈ HN , and Wpq∝δpq as claimed. APPENDIX C: PHYSICAL CREATION OPERATOR 1. Proximate expression First let us note that there is a systematic route to find the expression for d, which consists in writing the operator as : d†α = 〈A|d †α|B〉 |A〉〈B| n∈HA,m∈HB 〈A|d †α|B〉φ AnφBmX nm , (C1) withXfnm=|n〉f 〈m|f in usual Hubbard notations. Xfnm is obviously not just a one-particle operator f †, even when restricted to the sectors of interest in the above formula, since the states n and m can differ in many places. How- ever, because any transposition of two QPs, when acting on a physical state, can be replaced by a corresponding operation on bosons using the constraint (29), and be- cause any product of bosonic operators which cannot be reduced to a quadratic form will produce a state which is out of the physical subspace, the physical operator must in the end take the form: d†α = CAnBm(α, β) φ AnφBm f β . (C2) One can solve for the coefficients CAnBm(α, β), requesting proper action on the physical states. In this section however, we restrict ourselves to proving that Eq. (32) does the job, i.e., that d†α = β,AB,nm 〈A|d †α|B〉〈n|f β |m〉√ NA(M −NB) AnφBm f β (C3) satisfies d†α |B〉 = 〈A|d †α|B〉 |A〉 . (C4) We start by proving the formula : 〈n|f †β|p〉 f β |p〉 = (N + 1) |n〉 , (C5) where the sum over p runs over the basis of the subspace HN (states with N QPs) of the Fock space. First, we have in general : 〈n|f †β |p〉 f β|p〉 = n′∈HN+1 an′ |n′〉 , (C6) but, because f † is the creation operator (it connects one basis state to only one another) : an′ = β,p∈HN 〈n|f †β|p〉〈n ′|f †β|p〉 ∝ δnn′ β,p∈HN ∣∣∣〈n|f †β |p〉 We now use the fact that the tensor is invariant (it has the same expression in every basis) and use the notations introduced for Eq. (40): U is a uni- tary transformation of the one QP states and U is the corresponding transformation in the Fock states. 〈n|f †β |m〉=Uββ′U∗nn′〈n′|f β′ |m′〉 Umm′ and we have : ∣∣∣〈n|f †β |m〉 β,m∈HN β′m′∈HN n′∈HN+1 β′′m′′∈HN n′′∈HN+1 Uββ′U ββ′′U∗nn′Unn′′ ×Umm′U∗mm′′〈n′|f ′〉〈n′′|f †β′′ |m n′,n′′∈HN+1 U∗nn′Unn′′〈n′|f β′ |p〉〈n ′′|f †β′ |p〉 β,p∈HN ∣∣∣〈(Un)|f †β |m〉 Moreover, any couple of elements of the basis of the Fock state can be connected by a U transformation (with a U that permutes the one QP basis state), therefore an≡a is a independent of n. a can then be determined by sum- ming (C5) over n : n∈HN+1 n∈HN+1 ∣∣∣〈n|f †β |p〉 n∈HN+1 βfβ|n〉 n∈HN+1 (N + 1) , (C9) leading to a=N + 1. This completes the proof of (C5). It is now simple to compute the action of (32) : acting on |C〉≡ 1√ Cp|vac〉⊗ |p〉f with this operator, only the term B=C,m=p contributes, and we get : d†α|C〉 = DC(NC + 1)(M −NC) A,n∈HNC+1 〈A|d †α|C〉φ An|vac〉 ⊗ p∈HNC 〈n|f †β |p〉 f NC + 1 DC(M −NC) A,n∈HNC+1 〈A|d †α|C〉φ An|vac〉 ⊗ |n〉 〈A|d †α|C〉 |C〉 , (C10) which is identical to (31). 2. Improved expression In this section we present arguments for the improved formula used in this paper. First, it is useful to define the “natural orbitals” (NO) basis as the basis which diago- nalizes the quasiparticle and quasihole density matrices corresponding to the average constraint, which is given αβ [φ] ≡ φ∗AnφAm〈m|f †αfβ |n〉 (C11a) αβ [φ] ≡ φ∗AnφAm〈m|fβf †α|n〉 (C11b) φ∗AnφAn − ∆̂ αβ [φ] . Let us denote by ξλ, |λ〉 the eigenvalues and eigenvectors of those matrices : ξλ〈α|λ〉〈λ|β〉 , |λ〉 = 〈α|λ〉 |α〉 , (C12) which is equivalent to use the NO quasiparticle operator λ such that : 〈λ|α〉f †α , 〈ψ λψµ〉 = δλµξλ . (C13) To be fully explicit, we can consider the particular basis transformation (in the notations of the section above) f †α=Uαλψ λ which rotates to the NOs, and the corre- sponding rotation on the bosons: φAn=U(U)nn′ΩAn′ . The rotation matrix is: Uαλ = 〈α|λ〉 , (C14) and in the NO basis: Ω∗AnΩBm〈m|ψ λψµ|n〉 = δλµ Ω∗AnΩAnnλ = δλµξλ({ΩAn}) , (C15) αβ [φ] = Uαλ ξλ [U †]λβ . (C16) The idea is to generalize the Kotliar-Ruckenstein nor- malization factor in the NO basis, where the QP density being diagonal, its probabilistic interpretation is more transparent. Hence, the improved expression of d reads: d†α = λ,AB,nm 〈A|d †α|B〉 〈n|ψ λ|m〉√ ξλ({ΩAn})(1− ξλ({ΩAn})) AnΩBm ψ (C17) Note that the formal square-root normalisation, i.e., NA(M −NB), does not appear in this representa- tion. We can now rotate back to the generic basis we started from and use the gauge invariance, leading to : d†α = AB,nm,βγ CAnBm(α, β)φ AnφBm 〈β|λ〉〈λ|γ〉 ξλ(1 − ξλ) AB,nm,βγ CAnBm(α, β)φ AnφBm 〈β|[∆̂ (p)∆̂(h)]− 2 |γ〉 f †γ ,(C18) with CAnBm(α, β) = A|d†α|B n|f †β|m .(C19) Hence this yields the following form for the R-matrix: R[φ]∗αβ = AB,nm,δ CAnBm(α, δ)φ AnφBm 〈δ|[∆̂ (p)∆̂(h)]− 2 |β〉 . (C20) In the actual implementation of the saddle point calcu- lations, the explicit use of both the quasiparticle and the quasihole density matrices has been utilized (i.e., not us- ing their relation). Although at convergence the different representations yield the same values, writing the equa- tions via both, particle and hole density matrix, appears to be necessary within the minimization cycle. This is due to the fact that the derivatives with respect to the slave bosons have to be symmetric, however when using only the particle density matrix (or its eigensystem de- composition) for instance the derivative with respect to the empty boson vanish, although this one exists in the KR case. In the end an even more symmetrized form, i.e., 1 (∆̂(p)∆̂(h)+∆̂(h)∆̂(p)) was used for the square root in eq. (C18). Thus defining the following matrix: Mγβ = ∣∣∣∣∣ (∆̂(p)∆̂(h) + ∆̂(h)∆̂(p)) ∣∣∣∣∣ , (C21) the electron operators are written as d†α = CAnBm(α, γ)φ AnφBmMγβ f R∗αβf β (C22) CAnBm(α, γ)φ BmφAnMβγ fβ Rαβfβ , (C23) and correspondingly independently written the elements of the R,R†-matrices read R[φ]αβ = AB,nm,γ CAnBm(α, γ)φ BmφAn M̂βγ (C24) R†[φ]αβ ≡ R[φ]∗βα = AB,nm,γ CAnBm(β, γ)φ AnφBm M̂γα .(C25) APPENDIX D: DETAILS ON THE SADDLE-POINT EQUATIONS AND THEIR NUMERICAL SOLUTION The saddle-point equations for T=0 are obtained by performing the partial derivatives with respect to all the variables, i.e., condensed slave-boson amplitudes and La- grange multipliers: AnϕAn (D1) = 〈f †αfβ〉 − A,nn′ n|d†αdβ |n′ ϕAn (D2) Am + λ0ϕ m|d†αdβ |n′ Cn′ (D3) Bm + λ0ϕCm n|d†αdβ |m ϕCn (D4) with 〈f †αfβ〉 = f̃kj 〈α|νkj〉 〈νkj |β〉 . (D5) The εkj are the eigenvalues (with band index j) of the QP matrix (R†(ϕ)ε(k)R(ϕ) +Λ) with corresponding eigen- vector |νkj〉, while f̃kj denotes the occupation number of the state |νkj〉 for a given total number of particles, to be evaluated by standard k-integration techniques (e.g. tetrahedron method, Gaussian smearing, etc.). 1. Some slave-boson derivatives a. Eigenvalues and R matrices. The derivatives of the eigenvalues with respect to the slave bosons, i.e., may be performed pertubatively: ε(k)R + Λ )∣∣∣∣ νkj ε(k)R +R†ε(k) ∣∣∣∣ νkj ∣∣∣∣∣∣ 〈β|ε(k)R ε(k)|α〉∂R̂αβ ∣∣∣∣∣∣ 〈νkj |α〉 〈β|ε(k)R|νkj〉 +〈νkj |R†ε(k)|α〉 ∂R̂αβ 〈β|νkj〉 .(D6) The therefore needed explicit expressions for the deriva- tives of the R,R†-matrices read as follows (using eqs.(C24,C25)): ∂R̂αβ AB,nn′,γ CAnBn′(α, γ)ϕ δCmAn M̂βγ + ϕAn ∂M̂βγ ∂R̂αβ AB,nn′,γ CAnBn′(α, γ)ϕAn δCmBn′M̂βγ + ϕ ∂M̂βγ analogous for [R̂†]αβ b. The M matrix. As it is seen, the derivatives in- volve the derivative of the M matrix (C21). This deriva- tive is computed as follows. Lets first write M as M̂γβ = 〈γ|K−1/2|β〉 . (D7) What we are looking for is the derivative of K−1/2 with respect to the SBs. In order to get access to this quantity we use the identity −1/2)K−1/2 +K−1/2(∂ϕK −1/2) = ∂ϕK −1 (D8) ⇔ XK−1/2 +K−1/2X = Y , (D9) with X=∂ϕK −1/2 and Y=∂ϕK −1. We then apply P which transforms K to its eigensystem. This yields L + LX′ = Y′ , (D10) where the prime denotes that the quantities de- fined above are expressed in that eigensystem, and L=P†K−1/2P. Since in the eigensystem K−1/2 is di- agonal, i.e., L is, the last equation can be written in components and X′ determined: X ′ijLj + LiX ij = A ij ⇔ X ′ij = Li + Lj . (D11) Backtransforming to X=PX′P†=∂ϕK −1/2 yields the desired derivative of K and subsequently of M. To perform the described computation we need to know ∂ϕK −1 in eq. (D8), however this quan- tity may be straightforwardly calculated when start- ing from the identity KK−1/2K−1/2=1, resulting in −1=−K−1(∂ϕK)K−1. 2. Mixing In order to solve the saddle-point equations, a method to deal with a system of nonlinear equations F as a func- tion of the variables (slave bosons, Lagrange multipliers) x has to be utilized: F(x) = 0 (D12) In the present work we tested several quasi-Newton tech- niques (e.g. Broyden53, modified Broyden54, etc.) to handle this numerically. Thereby from a starting guess for x the variables are updated via x(m+1) = x(m) + J F(m) , (D13) since we want F(m+1) to be zero to linear order. The jacobian J is here defined as follows Jij ≡ − (D14) and is not calculated exactly (this would involve sec- ond derivatives and would lead to the Newton-Raphson method) but is computed at each step m via formulae which dictate several constraints on how J should evolve. 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0704.1435

On the failure of subadditivity of the Wigner-Yanase entropy

On the failure of subadditivity of the Wigner-Yanase entropy Robert Seiringer Department of Physics, Jadwin Hall, Princeton University P.O. Box 708, Princeton, NJ 08542, USA Email: [email protected] April 3, 2007 Abstract It was recently shown by Hansen that the Wigner-Yanase entropy is, for general states of quantum systems, not subadditive with respect to de- composition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting appli- cations in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample. In 1963, Wigner and Yanase [4] introduced the entropy-like quantity SWY(ρ,K) = 1 Tr [ρ1/2,K]2 = Tr ρ1/2Kρ1/2K − Tr ρK2 (1) for density matrices ρ of quantum systems, with K some fixed self-adjoint op- erator. They showed that SWY is concave in ρ [4, 5] and, for pure states, subadditive with respect to decomposition of the quantum system into two sub- systems. More precisely, if |ψ〉 is a normalized vector in the tensor product of two Hilbert spaces, H1 ⊗H2, and K1 and K2 are self-adjoint operators on H1 and H2, respectively, then WY(|ψ〉〈ψ|,K1 ⊗ 1+ 1⊗K2) ≤ SWY(ρ1,K1) + SWY(ρ2,K2) , (2) where ρ1 = TrH2 |ψ〉〈ψ| and ρ2 = TrH1 |ψ〉〈ψ| denote the reduced states of the subsystems. Recently, it was shown by Hansen [2] that this subadditivity fails for general mixed states. Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. c© 2007 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. http://arxiv.org/abs/0704.1435v1 This leaves open the question whether the Wigner-Yanase entropy is sub- additive for pure states with respect to decompositions into more than 2 sub- systems. If true, this property would have interesting consequences concerning density matrix functionals used in quantum chemistry, as will be explained be- low. We shall show, however, that this property does not hold, in general. Let ρ = |ψ〉〈ψ| be a pure state on a tensor product of N Hilbert spaces, i=1 Hi, and let Ki be self-adjoint operators on Hi. For simplicity we use the same symbol for the operators on H which act as the identity on the remaining factors. Subadditivity of SWY would mean that −SWY(|ψ〉〈ψ|, iKi) = − 〈ψ | iKi|ψ〉 TrHiρiK i − TrHiρ i Kiρ , (3) where ρi is the reduced density matrix of |ψ〉〈ψ| on Hi. Assume now that all the Hi are equal to the same H1, say, and that also all the Ki are equal, i.e., Ki acts as K on the i’th factor for some fixed operator K on H1. Ineq. (3) together with concavity of SWY would thus imply that − 〈ψ | iKi|ψ〉 2 ≥ TrH1γK2 − TrH1γ1/2Kγ1/2K , (4) i6=j KiKj ≥ (TrH1γK) 2 − TrH1γ1/2Kγ1/2K , (5) where γ = i ρi denotes the one-particle density matrix of |ψ〉〈ψ|. This repre- sents a correlation inequality, bounding from below two-particle terms in terms of one-particle terms only. As explained in [1], the validity of (4) for continuous quantum systems in the case whereK is the characteristic function of a ball of arbitrary size and location would imply that the ground state energies of Coulomb systems like atoms and molecules could be bounded from below by a density-matrix functional introduced by Müller [3]. For N = 2 this follows from the result in [4]. In the following, we shall show that, in general, (4) fails to hold for N = 3, and hence for all N ≥ 3. We choose the simplest nontrivial three-particle Hilbert space, C2 ⊗C2 ⊗C2, and pick a basis {| ↑〉, | ↓〉} in C2. We choose K = | ↑〉〈↑ |, ψ(↑, ↑, ↑) = 2√ ψ(↑, ↑, ↓) = ψ(↑, ↓, ↑) = ψ(↓, ↑, ↑) = 4√ ψ(↑, ↓, ↓) = ψ(↓, ↑, ↓) = ψ(↓, ↓, ↑) = 1√ ψ(↓, ↓, ↓) = 0 . (6) 1This particular counterexample was found with the aid of the computer algebra software Mathematica. 〈ψ|ψ〉 = 1 22 + 3 ∗ 42 + 3 ∗ 1 iKi|ψ〉 = 3 ∗ 22 + 2 ∗ 3 ∗ 42 + 1 ∗ 3 ∗ 1 32 ∗ 22 + 22 ∗ 3 ∗ 42 + 1 ∗ 3 ∗ 1 and hence the left side of Ineq. (4) equals ≈ 0.126942 . (8) The one-particle density matrix γ is given by the 2× 2-matrix 37 16 16 18 whose square root equals 1/2 ≈ 5.85827 1.63729 1.63729 3.91399 . (10) Hence the right side of (4) is 37− (5.85827)2 ≈ 0.146221 > 0.126942 . (11) This shows that Ineq. (4) fails in general for N > 2, and hence the Wigner- Yanase entropy is not subadditive with respect to the decomposition of pure states into more than 2 subsystems. We note that the same counterexample can also be constructed for contin- uous quantum systems, where K equals the characteristic functions of some measurable set B. One simply takes B and Ω to be two disjoint sets, each with volume one, and sets ψ(x1, x2, x3) = 2 if all 3 particles are in B 4 if 2 particles are in B and 1 in Ω 1 if 1 particle is in B and 2 in Ω 0 otherwise. This leads to the same counterexample as above. Similarly, one can construct a counterexample for fermionic (i.e., antisym- metric) wavefunctions which, after all, is the case of interest in [1]. Simply take (x, y) as the coordinates of one particle, choose the wave function to be the product of (12) for the x variables and a Slater-determinant for the y variables, which is non-zero only if all the y’s are in some set Λ. IfK denotes multiplication by the characteristic function of B × Λ, this leads to the same counterexample as before. References [1] R.L. Frank, E.H. Lieb, R. Seiringer, H. Siedentop, Müller’s exchange- correlation energy in density-matrix-functional theory, in preparation [2] F. Hansen, The Wigner-Yanase entropy is not subadditive, J. Stat. Phys. 126, 643–648 (2007). [3] A.M.K. Müller, Explicit approximate relation between reduced two- and one- particle density matrices, Phys. Lett. A 105, 446–452 (1984). [4] E.P. Wigner, M.M. Yanase, Information content of distributions, Proc. Nat. Acad. Sci. USA 49, 910–918 (1963). [5] E.P. Wigner, M.M. Yanase, On the positive semidefinite nature of a certain matrix expression, Canad. J. Math. 16, 397–406 (1964).

0704.1436

Symmetries and the cosmological constant puzzle

Symmetries and the cosmological constant puzzle A.A. Andrianov a,b, F. Cannata c,d, P. Giacconi c, A.Yu. Kamenshchik c,d,e, R. Soldati c,d a V.A. Fock Department of Theoretical Physics, Saint Petersburg State University, 198904, S.-Petersburg, Russia b Departament Estructura i Constituents de la Materia, Universitat de Barcelona, 08028, Barcelona, Spain c Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, 40126 Bologna, Italia d Dipartimento di Fisica, Universitá di Bologna e L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, 119334 Moscow, Russia Abstract We outline the evaluation of the cosmological constant in the framework of the standard field- theoretical treatment of vacuum energy and discuss the relation between the vacuum energy problem and the gauge-group spontaneous symmetry breaking. We suggest possible extensions of the ’t Hooft-Nobbenhuis symmetry, in particular, its complexification till duality symmetry and discuss the compatible implementation on gravity. We propose to use the discrete time-reflection transform to formulate a framework in which one can eliminate the huge contributions of vacuum energy into the effective cosmological constant and suggest that the breaking of time–reflection symmetry could be responsible for a small observable value of this constant. 1 Introduction The so called cosmological constant problem has two quite different aspects, which are not always clearly distinguished in the literature. One of these aspects is genuinely classical or even geometrical in its origin. The corresponding question could be formulated as follows: the classical cosmological constant, which can be introduced into Einstein–Hilbert action and Einstein equations, should be equal to zero and if it should, why? In other words: does a symmetry exists which forces the vanishing of this constant? The second aspect is, instead, purely quantum-theoretical. Independently of the presence of the classical cosmological constant, the vacuum fluctuations of the quantum fields give the contribution to the energy- momentum tensor which behaves as a cosmological constant, i.e. has the equation of state p = −ρ, where p is pressure and ρ is energy density. Naturally, this constribution is ultraviolet divergent. In the quantum field theory without gravity the problem is resolved by choosing the normal (Wick) ordering of creation and annihilation operators. This procedure is justified by fact that one measures the differences between energy levels, and not their absolute values. However, it is just the absolute value of terms in the energy-momentum tensor which stays in the right-hand side of the Einstein equations and in the presence of gravity the Wick’s normal–ordering loses its validity. A generally accepted procedure of the renormalization of vacuum energy does not exist, while the näıve cutoff imposed on the integration in the four-momentum space at the Planck scale gives huge values. How can one cope with them? One of possible approaches consists in the search of symmetries, which prohibit the existence of the cosmological constant and eliminate it on both the levels: classical and quantum. It is well known that supersymmetry suppresses divergences due to the compensating role of fermions and bosons. However, there are some difficulties at the application of the supersymmetric models to both the cosmological constant problem http://arxiv.org/abs/0704.1436v3 and to the correct description of the particle physics phenomenology, including the Higgs boson mass problem. Recently arguments were provided to force the vanishing of cosmological constant even limiting oneself to the bosonic sector [1]. The formulation of this symmetry requires to give meaning to the space–time coordinates complexification (see also [2], [3], where a similar transformation was proposed but in six dimensions). Here we would like to try to fold together the idea of complexification and the idea of compensation in a some new way. The structure of the paper is as follows: in Sec. 2 we outline the evaluation of the cosmological constant in the framework of the standard field-theoretical treatment of vacuum energy and discuss the relation between the vacuum energy problem and the gauge-group spontaneous symmetry breaking. In Sec. 3 we study the proposal [1] and its possible extensions, starting from the formalism developed in Sec. 2, in particular, its complexification allows to introduce a suitable duality symmetry. Its effect on gravity is also discussed and possible ways to implement a correct gravity interaction are outlined. Finally, in Sec. 4 we propose to use the discrete time reflection to formulate a framework in which one can eliminate the huge contributions of vacuum energy into the effective cosmological constant and, moreover, we suggest that the breaking of the T symmetry could be responsible for a small observable value of this constant. 2 Renormalization of vacuum energy density and spontaneous symmetry breaking Recall that summing the zero–point energies of all the normal modes of some field component of mass m up to a wave number cutoff K ≫ m yields a vacuum energy density 〈 ρ(m) 〉 = 1 4πk2dk (2π)3 k2 +m2 = + ln 2 +O . (1) If we trust in general relativity up to the Planck scale Mp, we might take K ≃Mp = (8πGN )−1/2 , which would give 〈 ρ 〉 ≈ 2−10 π−4G−2N = 2× 10 71 GeV4 . (2) But it is known that the observable value of the effective cosmological constant is less than about 10−47 GeV4 , and, sometimes one writes that a huge fine-tuning seems to be at work. However, it is necessary to be careful with such statements 1 because naively cutoffed expression (1) does not cor- respond to the cosmological constant. Within the same regularization one can calculate the vacuum pressure, 〈 p(m) 〉 ≡ 〈 1 j 〉 = (1−m∂m)〈 ρ(m) 〉 . (3) When comparing (1) with (3) one finds that the quartic divergences behave like radiation p = ρ/3, the quadratic ones as a perfect fluid with the equation of state p = −ρ/3 and the logarithmic divergences reproduce the cosmological constant equation of state p = −ρ. Evidently first two components are not entirely Lorentz-invariant but are determined in the rest frame of the Universe. On the other hand, as was pointed out by Zeldovich [4] the vacuum expectation of the energy-momentum tensor should be Lorentz-invariant and that means that it should be proportional to the metric tensor. That implies that the pressure is equal to the energy density taken with the opposite sign, or in other words, it means that that vacuum energy-momentum tensor must behave as a cosmological constant. The Lorentz- invariant part can unambiguosly separated by averaging different components of the energy-momentum tensor over Lorentz transformations and it does not include any radiation background thereby starting from quadratic divergences only (see similar arguments in [5]). Still the vacuum energy remains huge as compared to the energy density related to observable cosmological constant. 1We are grateful to A.A. Starobinsky, who has attracted our attention to this problem. Meantime, in [4] it was shown that requiring the elimination of all the divergences due to some general renormalization procedure, equivalent to introducing a spectral function of some kind, one automatically deduces that the finite part of the energy-momentum tensor have a Lorentz-invariant form. Considering the spectral function not as a renormalization tool, but as giving a real particle specrtrum, one can have a general restrictions on the particle physics models, providing the cancellation of the ultraviolet divergences in the energy-momentum tensor not only on the Minkowski, but also on the de Sitter background [6]. Here we discuss how various forms of regularization can be used to control the UV divergences. We can try to regularize the zero–point energy density in terms of the dimensional regularization [7]: namely, 〈 ρ 〉 = 1 (2π)1−n dn−1k (k2 +m2) (2π)1−n 2π(n−1)/2 Γ[ (n− 1)/2 ] dk kn−2(k2 +m2) 2 . (4) Unfortunately there is no strip in the complex n–plane in which the above integral is well defined, so that dimensional regularization is not appropriate in order to give a meaning to the zero–point energy. Alternatively we could also define the zero–point energy density in the path–integral formalism [7], which turns out to be quite convenient in view of its generalization to the curved space. Consider the classical action S[φ ] = d 4x [ ∂µφ∂ µφ−m2 φ2 ] , (5) and define the kinetic invertible operator, Kx := (�x +m2 − iε), (6) and its Feynman propagator , GF (x− y) ≡ − K−1xy = (2π)4 (k2 −m2 + iε)−1 exp{ik · (x− y)} . (7) Then we find the generating functional Z[ J ] := D[φ ] exp i S [φ ] + i d 4x J(x)φ(x) := Z[ 0 ] exp d 4y J(x)GF (x− y)J(y) Now, in order to end up with a dimensionless generating functional, we can formally write Z [ 0 ] = N (det ||µ−20 K ||)−1/2 = N exp [ 1/2Tr ln (µ 20 K−1) ] , (9) where N is an irrelevant numerical normalization constant that we shall omit in the sequel, whereas µ0 is an arbitrary wave number or momentum scale. A first possibility is to understand the formal relationships (9) in terms of the ζ–function regularization [9] that yields ln Z[ 0 ] = (2π)4 µ−20 (−k2 +m2) . (10) After changing the integration variable k0 = ik4 we find ln Z[ 0 ] = i Γ(s− 2) (m/µ0) , (11) so that we can eventually write Z[ 0 ] = eiW = exp −i L4 〈 ρ 〉eff , 〈 ρ 〉eff = ln (m2/µ 20 )− 3/2 . (12) We see that the ζ–function regularization drives to a result for the zero–point vacuum energy density which turns out to be IR logarithmically divergent and positive when the infrared cutoff µ0 ∝ L −1 is removed. It seems that the ζ–function regularization is not adequate in treating the cosmological constant problem as it re-directs the problem from UV to IR region. Turning back to eq. (9), a second possibility is to use the ultraviolet cutoff regularization of the large wave number field modes: namely, ϑ(Q− K) ln(K/µ20) ∫ Q−m2 q ln [ (q +m2)/µ20 ] dq (13) where k0 = ik4 , k E = k + k24 , Q ∼M 2p , so that we eventually obtain 〈 ρ 〉eff = 128π2 Q2(2 ln [Q2/µ20 ]− 1)− 4Qm2(ln [Q2/µ20 ]− 1) + 2m4 ln (m2/µ20)− 3m4 in a satisfactory agreement with eq. (1) up to a redefinition of the large wave number cutoff. Then one could fit the Lorentz invariant part of (1) to (13) choosing lnQ/µ20 = 1/2 as for this choice the contribution ∼ Λ4 also vanishes in (13) . To be consistent with the standard model of particle physics we have to take care of spontaneous symmetry breaking in the field theory. In this framework the charged scalar field potential takes the form (with µ2 > 0, λ > 0) V (φ) = V0 − µ2φ†φ+ λ(φ†φ)2 . (15) The classical minimum of this potential occurs at the constant field values φ†φ = µ2/2λ so that it is convenient to parameterize the scalar field φ by writing φ(x) = U(x) v + σ(x) . (16) We can now make a gauge transformation in order to eliminate U(x) from the lagrangian – unitary gauge – in such a way that Vmin = V0 − . (17) According to the review [8] it is apparently suggested that the classical potential should vanish at φ = 0, which would give V0 = 0, so that some classical negative contribution to the zero-point energy density would be there. In the electroweak theory, if we assume an Higgs boson mass mH = µ 2 ≃ 150 GeV, this would give ρ0 ≃ −(150 GeV) 4/16λ , which even for λ as small as α2 would yield | ρ0 | ≃ 10 12 GeV 4, larger than the observed value by a factor 10 59. Of course we know of no reason why V0 or Λ must vanish, and it is quite possible that V0 or Λ cancels the term −µ 4/λ (and higher order corrections), but this example neatly shows how un–natural is to get a reasonably small effective cosmological constant. In general, if we turn to the shifted field σ(x) in the unitary gauge, we obtain the Lagrange density for the shifted field L[σ ] = 1 ∂µσ ∂ µσ − 1 (2µ2)σ2 ∓ µ λσ3 − λ σ4 − V0 + . (18) Accordingly, the zero–point energy density in the symmetry broken phase appears to be 〈 ρ 〉 = 〈 ρ 〉div + V0 − . (19) It is clear that we can easily remove the divergent part of the zero–point energy density of the scalar field σ(x) after the introduction of a realmirror free real scalar field ϕ(x) with a classical Lagrange density related to that one of eq. (18) L [ϕ ] = 1 µϕ− 1 (2µ2)ϕ2 , (20) together with a ghost pair of scalar fields. These mirror and ghost fields are supposed not to interact directly to the SM fields. At the classical level the ghost fields are described by anticommuting, real Grassmann algebra valued, field functions η(x) = η†(x), η̄(x) = η̄ † with Lagrange density LGP = − i ∂µη̄ ∂ µη + 2iµ2 η̄ η . (21) We have Π η = + i ∂0η̄ , Π η = − i ∂0η , (22) so that correspondingly HGP = dx [ η̇Π η + ˙̄ηΠ η − LGP ] = − i ∂0η̄ ∂0η +∇η̄ · ∇η + 2µ2 η̄ η . (23) Now one can quantize the ghost pair with the help of canonical anti–commutation relations . As a consequence, after Fourier decomposition of ghost fields we eventually obtain HGP = i dp p0 [ η̄ †(p)η (p)− η †(p)η̄ (p) + i (L/2π)3 ] , (24) so that we get the ghost pair negative contribution to the zero point energy density in the large wave number cutoff regularization with the Planck mass 〈 ρ 〉GP = − (2π)−3 4πp2dp (p2 + 2µ2) 2 . (25) Thus the above mirror–symmetry, which does not mix the standard model multiplets, would admittedly resolve the cosmological constant problem in the scalar Higgs sector only. It can be extended onto the entire field content of the standard model, at the expense of introducing more ghost fields with wrong spin-statistics relation. Evidently, gravity will mix the standard model fields with their related mirror– replicæ and ghost–pair, leading eventually to the breaking of the spin-statistics relation and even unitarity in the standard model world. A rather sophisticated proposal will be formulated in the next section to skip that nasty mixing and unitarity loss. 3 ’t Hooft-Nobbenhuis symmetry and cosmological constant In this section we explore the symmetry against the change of the full metric sign by continuation of real space-time variables to complex values – in the original proposal [1] to imaginary ones – first in the flat Minkowski space-time, ηµν = diag ‖ +,−,−,− ‖ ; xµ 7→ −iy µ , y µ = y µ ∗ ; ∂µ 7→ i∂µ ; kµ 7→ ikµ . (26) Moreover, for a real scalar field φ(x) we shall set : φ(x) = φ(−iy) 7→ φ̃(y) = φ̃∗(y) ; d4x 7→ d4y ; (27) We stress that φ̃(y) 6= φ(−iy) since φ̃(y) is evidently real, whereas φ(−iy) is in general complex. Therefore, the ’t Hooft-Nobbenhuis transformation is not merely an analytic continuation, as it involves an essential change of the functional base–space. It is of course analogous to what we do when we make the transition to the Euclidean formulation, e.g. φ(t) 7→ φ(τ = −it), in which we perform a simultaneous mapping of one functional space – a subspace of the complex function space, to another one spanned by real functions φ̃(τ) of the Euclidean–time coordinate τ . In so doing, one finally comes to a theory of scalar tachyon – this is the reason why t’ Hooft and Nobbenhuis actually neglect masses – namely, µφ−m2φ2 φ4 7−→ −Ly ; Ly ≡ µφ+m2φ2 φ4 . (28) with a repulsive quartic self-interaction, the issue of the vacuum stability against small time-dependent perturbations keeping admittedly open. We also remark that if the space-time is not flat, then the continuation towards complex coordinates makes the metric also complex. However, since the background metric is the solution of Einstein equation, it can not be näıvely defined by analytic continuation. Rather, one has to perform the corresponding mapping of the metric functional base–space and then solve the Einstein equations with a transformed new energy–momentum tensor, because the matter fields are in turn suitably mapped. As a result, one might expect the very same background metric, if at least the classical matter distribution remains unchanged, what is far from being obvious for massive interacting matter fields. Our complementary proposal: extended duality symmetry Once that the ’t Hooft–Nobbenhuis proposal necessarily involves the mapping of the functional base– space, one could naturally include into this mapping also the analytic continuation of the field variables. To this concern, one could treat all the field amplitudes and coordinates, but the metric, on the same footing: e.g. for real scalar fields, φ(x) = φ(−iy) 7→ iϕ(y) ; ϕ(y) = ϕ∗(y) ; Ly(ϕ) = µϕ+m2ϕ2 ϕ4 . (29) This transformation does realize a link between two scalar field theories without and with spontaneous symmetry breaking . The latter one has its classical minima at 〈ϕ〉 = ±mλ−1/2 . After the field amplitude shift ϕ 7→ 〈ϕ〉+ ϕ , the effective lagrangian reads Ly(ϕ) = µϕ− 2m2ϕ2 ϕ4 ∓m λϕ3 . (30) Moreover, for vector gauge fields we shall write Aµ(x) 7→ i Vµ(y) ; D xµ = ∂ xµ + iAµ(x) 7→ iD yµ = i [ ∂ yµ + i Vµ(y) ] ; Fµν(x) 7→ −Gµν(y) (31) to preserve covariant derivatives. The above transformations just leave both the Maxwell’s lagrangian and the action invariant. Finally, for massless fermions with Yukawa coupling g ψ(x) 7→ ψ(−iy) 7→ Ψ(y) ; ψ̄(x) 7→ − iψ̄(−iy) 7→ Ψ̄(y) = Ψ† γ0 (32) Lx = ψ̄ (i 6∂ − 6A− g φ)ψ 7−→ Ly = Ψ̄ (i 6∂ − 6V − g ϕ)Ψ (33) so that the spontaneous symmetry breaking mechanism allows to generate the correct fermion masses. We remark, had one included the bare fermion masses, then they would become imaginary with the above rules. Therefore the transformed fermions would be unstable, having the imaginary bare part and the real part generated by spontaneous symmetry breaking. Vacuum energy under the extended duality symmetry Suppose that the vacuum energy is compensated to zero in the original theory. This compensation can be described in an effective theory style by introducing a number of shadow fields – in analogy with the Pauli-Villars regularization scheme – with the same coupling constants but different masses. For instance, a real scalar field φ is supplemented by N shadow fields ϕj , j = 1, 2, . . . , N with masses Mj , the same quartic coupling constant and the same effective lagrangian but with a positive or negative weight of their contribution into the effective action Γ Γ = Γ(φ,m, λ)− (−1) pjΓ(ϕj ,Mj , λ); pj = 1, 2; (34) where we keep the same notation φ, ϕj to indicate the classical mean field variables in the Legendre functional transform. We stress that they are combined, at the level of the effective action, in order to exactly compensate the vacuum energy contributions. All of them are bosons, although some of them behave as ghosts in the leading quasi–classical approximation. Within the framework of quantum field theory, one can interpret the part of this set with negative sign (even pj) as originating from the evolution backward in time with anticausal prescription for propagators, i.e. with replacing +iε into −iε in (6),(7) – see an example in [11]. Concerning the zero–point energy, one normally has three types of leading divergences – see eq. (14) ∼ Q 2 ∼ Qm2 ∼ m4 log (Q/m2) and with the help of a number of shadow fields one exactly cancels the divergences if (−1) pj = 1, (−1) pjM2j = 0, p0 = 1 M0 ≡ m, (−1) pjM4j = 0 . (35) We assume of course the preliminary mass and coupling constant renormalization. One can also assume that the shadow world consists of sufficiently heavy particles in order to reduce their influence as much as possible on the physics accessible in the standard model real world. Then the minimal number of such fields is equal to five. The first sum rule can be interpreted as a “conservation law” of a number of matter sub-worlds evolving forward and backward in time in a certain accordance with no-time origin of our universe [13]. On the one hand, once the light shadow fields has been accepted, one can restrict himself to solely one species with negative sign of its effective action – compare with [11]. On the other hand, the cancellation of quartic and quadratic divergences has to be resorted to Planck’s scale physics, where the very notion of low energy fields with their Lagrangians of canonical dimension four is admittedly questionable. A self–consistent treatment at low energies must deal then with light mass scales and relatively light shadow fields (as compared to the Planck mass) and therefore only with the last relation, which involves the fourth powers of the shadow masses. After the duality transformation φj 7→ ϕj and the resolution of spontaneous symmetry breaking by shifting each field in 〈ϕj〉 = ±Mj/ λ , one finds the classical vacuum energy density 〈 ρ 〉cl = (−1) pj M2j 〈ϕj〉 〈ϕj〉4 = − 1 (−1) pjM4j = 0 . (36) Thus, quite remarkably, after the extended duality transformation the scalar field vacuum energy density remains equal to zero, whereas the masses are generated, both for fermions and for gauge bosons, thanks to the Higgs mechanism. Certainly all the standard model fields must be replicated in the shadow sectors, if one provides the zero cosmological constant. If those shadow fields do not interact with each other, it cannot be conceivably embedded into a minimal supersymmetry. On the contrary, if one starts from a minimal exact supersymmetry, this dressing by shadow fields and the subsequent extended duality transformation might lead to a spontaneous symmetry breaking for supersymmetry with zero cosmological constant in the outcome. Hints for gravity Suppose that shadow fields interact with our world only through gravity and therefore they belong to the dark side of the universe. Then we could exploit the shadow fields as a part of the matter in the universe and not merely just like regularizing fields. Since after the extended duality transformation we change the sign of derivatives but not of the metric, we have Rµν(x) 7−→ −Rµν(y) (37) Sg = − 16πGN d4x [R(x)− 2Λ) ] 7−→ 1 16πGN d4y [R(y) + 2Λ ] . (38) There is no invariance, as the sign of cosmological constant is unchanged albeit gravity becomes anti– gravity. The possible solutions are: 1. anti–gravity in the symmetric phase GN < 0 (repulsion supports this phase) is replaced by true gravity in the spontaneous symmetry broken phase, although one has to check classical solutions ; 2. gravity is induced solely by matter and therefore the overall sign of the gravitational action remains the same under the extended symmetry transformation and the spontaneous symmetry breaking, albeit the compensation mechanism, if it is exact, does select out a vanishing coefficient in front of the scalar curvature ; 3. there is a coupling to scalar fields: namely, Sg = − j (x) R(x) 7−→ j (y) d4y R(y) + . . . = 16πGN d4y R(y) + . . . (39) so that 16πGN . (40) If gravity is not principally induced by matter fields, then in the latter case there is no prescribed relation between the individual gravitational scalar couplings Bj . In such a circumstance, one could adjust them to support essential invariance under the extended duality transformation and a tiny cosmological constant might be generated via vacuum polarization. 4 Cosmological constant, time arrow and T violation First of all, let us remark that according to recent observational results such as the discovery of the cosmic acceleration [14] it is reasonable to think that the real value of the cosmological constant is not strictly zero. Indeed, the so called ΛCDM cosmological model based on the presence of the cosmological constant has acquired the status of the standard cosmological model. Thus, the first “classical” aspect of the cosmological constant does not seem to be problematic anymore and the classical cosmological constant can have any value, being one of the fundamental constants. The control of vacuum fluctuations is really important. The idea of the (almost) complete cancellation of the vacuum fluctuations seems very attractive because it permits to resolve both the cosmological and quantum field theoretical problems, connected with its treatment. Our idea is very simple. We are inspired by two facts. 1. the classical equations of motion are invariant in respect to the time inversion. 2. gravity being reparametrisation-invariant theory, does not have a time [13]. Indeed, at least for the closed cosmological models the Hamiltonian of the theory is equal to zero and the näıve notion of time looses sense. An effective time arises in the process of interaction with matter and due to the breakdown of the gauge (reparametrisation) invariance due to the gauge fixing choice – there is ample literature devoted to this topic [15]. Hence, we suggest the following postulate: the vacuum state evolves time-symmetrically according to the evolution operator W (t) = T e−iHt T−1 e−iHt + e−iHt T e−iHt T−1 where H is the Hamiltonian and T is the operator of time inversion. If the theory is invariant with respect to the time inversion operation, i.e. THT−1 = H, (42) then W (t) = I, which corresponds effectively to zero energy of the vacuum state. In some respect this situation can be described in terms of negative (mirror) matter - some kind of shadow matter. The presence of two replicas of fields, having opposite signs of the vacuum energy results in the complete cancellation of vacuum energy in the same sense as the mirror energy reflection of ref. [11] (see also [12]). Much before a similar idea was elaborated by Linde [16] – see also [17, 18], where the idea of the second negative energy world was put forward. Certain hints from Superstring theory for shadow matter with negative vacuum energy were established in [19]. Thus, its contribution to the effective cosmological constant vanishes. Therefore our prescription (41) is equivalent to subtraction of the ground state energy only if time reversal holds (42). In this it does not coincide with earlier proposals [16, 11, 17, 18]. All written above assumes the exact time invariance of the fundamental physical theory. However, one could invoke a suitable small breaking of the time symmetry. Indeed, the violation of the CP invariance is an experimental fact, and the conservation of the CPT symmetry implies unavoidably the breakdown of the time symmetry. Such type of breakdown could occur even spontaneously as was suggested by Tsung Dao Lee in his seminal paper [20]. So, we are lead to suspect the existence of a connection between a small T (or CP) symmetry violation and a small observable value of the cosmological constant. In a way our approach reminds that of the mirror world or mirror particles – see [21] and references theirein. The mirror symmetry is as well known as the symmetry with respect to spatial reflections or parity P symmetry. However, the absence of significant interactions between mirror particles and normal ones is imposed by the phenomenology and not by general principles like in the case of T reflection. The problems arising in application of the spontaneous T symmetry breaking to cosmology and ways of their solution were considered in [22]. It is important to emphasize, that if the connection between the T violation and the cosmological constant value does indeed exist, then it could be not connected with the presence of the standard CP- breaking terms in the CKM matrix, since we are interested in the vacuum expectation energy diagrams, to which those terms do not give a contribution. It would be rather connected with more subtle scheme, which could explain the small scale of the observable cosmological constant. The idea of the presence of fields evolving backward in time and co-existing with “normal” fields evolving forward in time was used in many different contexts. First of all, one should cite the works by Wheeler and Feynman on time symmetric electrodynamics [23] together with the so called transactional interpretation of quantum mechanics [24]. We should emphasize once again that there are no particles moving backward in time in our forward-in-time-world. The only influence which this time reversed world makes on us is just the presence of vacuum energy in the right-hand-side of the Einstein equations. Moreover, the appearances of such known observable quantum fluctuation effects like the Casimir effect could not be influenced by the energy reversed world as well because their observability is based on their interaction with normal particles, which provides boundary conditions responsible for these effects. Such an interaction breaks the time symmetric evolution (41) which as we have suggested is valid only for vacuum state. Notice that the idea that the direction of time can be connected with the existence of a cosmological term was first put forward by M.P. Bronstein in the context of Friedmann cosmology [25]. Acknowledgement We are grateful to A.A. Starobinsky and G. Venturi for fruitful discussions. The work of A.A. was supported by Grants SAB2005-0140; RFBR 05-02-17477 and Programs RNP 2.1.1.1112; LSS-5538.2006.2. A.K. was partially supported by RFBR 05-02-17450 and LSS-1157.2006.2. References [1] G.’t Hooft and S. Nobbenhuis, Class. Quantum Grav. 23 (2006) 3819. [2] G. Bonelli and A. Boyarsky, Phys. Lett. B 490 (2000) 147 . [3] R. Erdem, Phys. Lett B 621 11 (2005) 11; ibid. B 639 (2006) 348. [4] Ya.B. Zeldovich, JETP Lett. 6 (1967) 316; Sov. Phys. - Uspekhi 11 (1968) 381. [5] E.Kh. Akhmedov, hep-th/0204048. [6] A.Y. Kamenshchik, A. Tronconi, G.P. Vacca and G. Venturi, Phys. Rev. D 75 (2007) 083514. [7] N.D. Birrel, P.C.W. Davies, Quantum fields in curved space, Cambridge Univ. Press, Cambridge, U.K. (1982). [8] S. Weinberg, Rev. Mod. Phys. 61, No. 1 (1989) 1. [9] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. [10] T. Kugo, I. Ojima, Suppl. Progr. Theor. Phys. 66 (1979) 1. [11] D.E. Kaplan, R. Sundrum, JHEP 0607, 042 (2006) [arXiv:hep-th/0505265]. [12] H. T. Elze, hep-th/0510267 . [13] B.S. DeWitt, Phys. Rev. 160 (1967) 1113. [14] A. G. Riess et al., Astron. J. 116 (1998)1009; S. Perlmutter et al., Astroph. J. 517 (1999) 565. [15] A.O. Barvinsky, Phys. Rept. 230 (1993) 237. [16] A.D. Linde, Phys. Lett. B 200 (1988) 272. [17] F. Henry-Couannier, Int. J. Mod. Phys. A 20 (2005) 2341; gr-qc/0410055. [18] J.W. Moffat, Phys. Lett. B 627 (2005) 9; hep-th/0507020. [19] A.A. Tseytlin, Phys. Rev. Lett. 66 (1991) 545 . [20] T.D. Lee, Phys. Rev. D 8 (1973) 1226. [21] L.B. Okun, Talk at ITEP Meeting on the Future of Heavy Flavor Physics, Moscow, Russia, 24-25 Jul 2006, hep-ph/0606202. http://arxiv.org/abs/hep-th/0204048 http://arxiv.org/abs/hep-th/0505265 http://arxiv.org/abs/hep-th/0510267 http://arxiv.org/abs/gr-qc/0410055 http://arxiv.org/abs/hep-th/0507020 http://arxiv.org/abs/hep-ph/0606202 [22] A.D. Dolgov, Phys. Rept. 222 (1992) 309; Lectures at 9th Int. Moscow School of Physics and 34th ITEP Winter School of Physics, Moscow, Russia, 21 Feb - 1 Mar 2006, hep-ph/0606230. [23] J.A. Wheeler, R.P. Feynman, Rev. Mod. Phys. 17 (1945) 157; ibid., 21 (1949) 425. [24] J.G. Cramer, Rev. Mod. Phys. 58 (1986) 647. [25] Matvei P. Bronstein, Phys. Z. Sowjetunion, 3 (1933) 73. http://arxiv.org/abs/hep-ph/0606230 Introduction Renormalization of vacuum energy density and spontaneous symmetry breaking 't Hooft-Nobbenhuis symmetry and cosmological constant Cosmological constant, time arrow and T violation

0704.1437

The property of kappa-deformed statistics for a relativistic gas in an electromagnetic field: kappa parameter and kappa-distribution

The κ parameter for a relativistic gas arXiv:0704.1437v1 The property of κ -deformed statistics for a relativistic gas in an electromagnetic field: κ parameter and κ -distribution Guo Lina, Du Jiulin* and Liu Zhipeng Department of Physics, School of Science, Tianjin University, Tianjin 300072, China Abstract We investigate the physical property of the κ parameter and the κ -distribution in the κ -deformed statistics, based on Kaniadakis entropy, for a relativistic gas in an electromagnetic field. We derive two relations for the relativistic gas in the framework of κ -deformed statistics, which describe the physical situation represented by the relativistic κ -distribution function, provide a reasonable connection between the parameter κ , the temperature four-gradient and the four-vector potential gradient, and thus present for the case 0≠κ one clearly physical meaning. It is shown that such a physical situation is a meta-equilibrium state, but has a new physical characteristic. PACS: 05.90.+m; 05.20.Dd; 03.30.+p Keywords: generalized statistics; relativistic gas * Corresponding author: [email protected] In recent years, some new statistics have been put forward to generalize the classical Boltzmann–Gibbs (BG) one, such as the nonextensive statistics based on Tsallis entropy (q-entropy)[1], the so-calledκ-deformed statistics based on theκ-entropy given by Kaniadakis [2] besides others. We mention that the concept ofκ-distribution was introduced about four decades ago and also discussed recently by Leubner [3], which was actually equivalent to the nonextensive q-distribution, but different from Kaniadakis one. These statistics are studied all dependent on one parameter. For instance, the nonextensive statistics depends on the parameter q different from unity and it will recover BG one if we take the parameter to be unity; theκ-deformed statistics depends on the parameterκ different from zero and it will recover BG one if we take the parameter to be zero. Then it is naturally important question for us to ask what should these parameters stand for and under what physical situations should these statistics be suitable for the statistical description of a system. For the nonextensive statistics, we know that the parameter q different from unit is related to the temperature gradient, T∇ , and the long-range potentials, ϕ , of the systems such as self-gravitating system [4,5] and plasma system [6], by ϕ∇−∇ )~ qT 1( . Thus it can be reasonably applied to describe thermodynamic properties of the systems under an external field when they are in the nonequilibrium stationary-state. And this characteristic for the nonextensive statistics has recently received the experiment support by the helioseismological measurements [7]. As potential astrophysical examples with regard to the above theoretical investigation, a new application of the nonextensive theory for both dark matter [8], stars [9] and hot plasma [3, 10] density distributions in clustered astrophysical structures was investigated. In this situation, the potential gradient is parallel to the temperature gradient, thus fitting perfectly well into the discussions [4-7]. For theκ-deformed statistics, we actually know little about the physical property of the parameterκdifferent from zero and the corresponding statistics although it has been studied by Kaniadakis and his co-authors in many papers. In this letter, we study the physical properties in the case of parameterκ≠ 0 for a relativistic gas under an electromagnetic field. The general form of theκ-entropy proposed by Kaniadakis can be written by theκ -distribution function, f , [2] as 321 )()( afaSafK += κ , (1) being , which can have different expressions after having given specific choices for the parameters a ∫−= ffvdfS κκ ln )( 3 i with i=1,2,3, where ai may depend on the position x. For instance, a particular case of K(f) is ( ) ( ) 2 1 2 1 S d v f f κ κ κ κ κ = − −  + −  ∫ 1 κ− , (2) which recovers to the Boltzmann-Gibbs entropy in the limitκ→0, where z is a positive real parameter that needs to be specified. Based on Kaniadakis entropy, theκ-deformed statistics can be defined by the so-calledκ-exponential andκ-logarithm functions as ( ) ( ) 2 2exp 1f f f κ κ κ= + + , (3) ( )ln = . (4) As one may check, we have expκ(lnκ(f) )= lnκ(expκ(f )) = f and the above functions could reduce to the standard exponential and logarithm ones if we letκ→0. Actually, thisκ -framework can lead to a class of one parameter deformed structures with interesting mathematical properties[11]. We consider a relativistic gas containing N point particles of mass m enclosed in a volume V, under the action of an external four-force field Fµ. The temporal evolution of the relativistic distribution function f(x, p) is governed by the following relativisticκ -transport equation [11,12]: ( )fp f mF C f µ µ κ ∂ + = , (5) where µ = 0, 1, 2, 3, the particles have four-momentum = in each point of the space-time, with their energy E/c = µpp ≡ ),/( pcE ),( rctxx =≡ µ 2c22 m+p , and denotes the differentiation with respect to time-space coordinates and the relativisticκ-collisional term. According to the generalized H theorem [11,12], the relativistic version ofκ-distribution function in this framework approach to ( )1 ,tc− ∂ ∇µ∂ = ( ) ( ) 2 2, exp 1f x p B B κ θ κ θ κ= = + + θ , (6) where B may depend on x, and µβαθ pxx )()( += (7) with ( )xα a scalar and µβ a four-vector, arbitrary space and time-dependent parameters. And the relativisticκ-collisional term vanishes and Eq.(5) becomes p f mF ∂ + = . (8) The distribution (6) can also be obtained by maximizing K(f) under the right constraints. Let us now consider the external four-force field as the Lorentz one, namely, ( ) ( ) ( ), /F x p Q mc F x pµν ν= − (9) with Q the charge of the particle and F µν the Maxwell electromagnetic tensor. In this case, Eq.(8) is 1 0 p f Qc F p µ ν µ − ∂∂ − = (10) or, equivalently, it can be written as pFQcfp . (11) And theκ-distribution function, Eq.(6), [11, 12] reads , exp u p c QA x U f x p B −  − +  =    u p c QA x U u p c QA x U k T k T µ µ µ µ − −      − + − +      = + +         , (12) where kB is Boltzmann constant, Uµ is the mean four-vector velocity of the gas with , T(x) is the temperature field, u is the Gibbs function per particle, and A2cUU =µµ µ(x) is the four-vector potential. In the limitκ→0, it reduces to the well known relativistic expression [13] correctly, , exp u p c QA x U f x p B −  − +  =    (13) From Eq.(12), we have ( ) ( ) u p c QA x U u p c QA x U f x p B k T k T µ µ µ µ µ µκ κ κ κ − −      − + − +      = + +         , (14) ( )12 2 2 u p c QA U f B B µκ κ κκ = + f κ . (15) Substitute Eq.(15) into Eq(11) and after a series of calculations, we get the equation ( ) ( ) 2 2 2 2 2 2 22 1 2 2 22 2 B B B p B p B p B up U p B uc QA k T k T k T µ κ µ κ µ κ µ µ κ µUµ µ µ µ µ κ −∂ + ∂ − ∂ − ∂ ( ) ( ) 2 2 2 21 1 B B B p B p U p B p U c QA U p B c QA U k T k T k T µ κ µ µ κ µ µ µ κ µ µ µ µ µ µ µ κ κ κ− −+ ∂ + ∂ + ∂ 2 2 2 2 2 2 2 2 2 2 22 2 22 2 p T p T p Tup U uc QA Uu B B B T T Tk T k T k T µ µ µµ µ µ µ µ µ µκ κ κκ κ κ −∂ ∂ ∂ − + + ( ) ( ) ( ) ( ) 2 2 2 2 11 2 2 2 2 2 2 2 22 2 22 p T p U p T p T c QA Up U c QA U B B B T T Tk T k T k T µ µ µ µ µµ µ µ µ µ µµ µκ κ κκ κ κ −−∂ ∂ ∂ − − − 2 21 12 2 2 2 2 2 2 B B B c Qp A U U p T c Qp A UQ u B B F p B k T c k T T k T k T µ µ µ µ µ µ µ µ µκ κ µν κ νκ κ κ − − ∂ ∂  − − −        2 2 2 22 2 B B B B p T p U c Qp A U p T c QA U c Qp A U T k T k T T k T k T µ µ µ µ µ µ µ µ1 µ µ µ µ µ µ µκ κκ κ − − −∂ ∂ ∂ ∂ + + µ 2 2 2 22 2 B B B B p T U p T p U Uu Q Q B F p B F p T k T c k T T k T c k T µ µ µ µ µ µ µ µκ µν κ µν ν νκ κ 2 2 2 22 2 B B B B p T c QA U U c Qp A U UQ Q B F p B F p T k T c k T k T c k T µ µ µ µ µ µ µ µ µ µκ µν κ µν ν νκ κ − −∂ ∂ − + 0= .(16) We consider that Eq. (16) is the identically null for any arbitrary p and the sum of the coefficients of every power for p in this equation must be zero. Thus, when we consider the coefficients for the forth-power terms of p in Eq. (16), we obtain the relation,  ∂ ∂  =        . (17) The coefficients of the third power of p is 2 2 1 2 2 2 2 22 2 2 B uU B U c QA U B Tk T k T k T µ µκ κ µ µ κ µ µ µ µ − ∂ + ∂ + 2 2 2 2 2 222 2 T U c QA U T U c Q A U T T k Tk T µ µ µ µ µκ κκ κ − −∂ ∂ ∂ − + µ T U UQ T k T c k T µ µ µκ µνκ + 0= . (18) Substitute Eq.(17) into Eq.(18), we get ( ) 0T A F Uν µνµ µ µ∂ ∂ − = (19) The coefficients of the second power of p read ( ) ( ) 2 2 2 2 2 2 22 1 2 2 22 B B B B B B uc QA U B c QA U k T k T k T κ κ κ µ κ µ µ µ µ µ µ κ − −∂ + ∂ − ∂ + ∂ 1 µ ( ) ( ) 22 2 2 112 2 2 2 2 2 2 2 22 2 22 c QA UT T Tuc QA Uu B B B T T Tk T k T k T µµ µ µµκ κ κκ κ κ −−∂ ∂ ∂ − + − 2 2 21 12 2 2 2 2 2 2 B B B c Q A U U T c Q A UQ u B B F B k T c k T T k T k T µ µ µ µκ κ µν κκ κ κ − − ∂ ∂  − − −        µ µ∂ 2 2 2 22 2 B B B T c QA U c Q A U T Uu Q T k T k T T k T c k µ µ µ µ µκ κκ κ − −∂ ∂ ∂ + + µµν 2 2 2 22 2 B B B T c QA U U c Q A U UQ Q B F B F T k T c k T k T c k T µ µ µ µ µκ µν κκ κ − −∂ ∂ − + 0 µµν = . (20) Substitute Eqs.(17) and (19) into Eq.(20), we find ( ) ( ) 2 222 2 k T A F U µ µκ ∂ = ∂ − µ  , (21) which easily leads to the relation µµ κ FAQTkB −∂=∂ . (22) Furthermore, using the electromagnetic tensor expression, F A Aµν νµ ν= ∂ − ∂ µ , we can write Eqs.(19)and (22), respectively, as ( )( ) 0=∂∂ µνµ AT (23) νµ κ AQTkB ∂=∂ , or νµκ AQTkB ∂∂= . (24) where and are the relativistic expressions of the temperature gradient and the external potential gradient, respectively. From Eq.(24) we find that the parameter κ is different from zero if and only if the quantity Tµ∂ is not equal to zero, thus showing that 0≠κ represents the physical situation of the relativistic gas with a time-spatial inhomogeneous temperature field and under the electromagnetic field. This is a nonequilibrium stationary-state. But such a nonequilibrium relativistic gas should have one additional characteristic: from Eq.(23) we find that the temperature four-gradient Tµ∂ must be vertical to the four-potential gradient . µν A∂ If take κ = 0, one has = 0, which just is the physical situation represented by the standard distribution (13). The non-relativistic versions can be written directly from Eqs.(23) and (24) by 0=∇⋅∇ ϕT ; ϕκ ∇=∇ QTkB , (25) which are in agreement on the relations derived recently for the parameter κ 0≠ in a nonequilibrium system [14]. Thus Eqs.(23) and (24) extend these classical relations to the special relativistic case. In summary, we obtain two relations, Eqs.(23) and (24), of the parameter κ in the κ -deformed statistics for a relativistic gas in an electromagnetic field. They describe the physical situation represented by the relativisticκ-distribution function (12) and provide a reasonable connection between the parameter κ , the temperature four-gradient and the four-vector potential gradient, so presenting the case of the parameter 0≠κ one clearly physical meaning. Such a physical situation is a nonequilibrium stationary-state, but has a new characteristic given by Eq.(23). Additionally, from Eq.(17) it follows that when B is a constant also the temperature turn out to be constant, which is independent on the parameter . Therefore the equilibrium state emerges also in the case of a relativistic gas described by the -distribution based on the entropy K(f) in presence of an external field. Conversely when B depends on x, the system admits the above nonequilibrium stationary-state. From the view of this point, the physical situation described by the κ -distribution may be called a metaequilibrium state. Additional remarks: Most recently, Plastino et al reported a general discussion on the generalized entropy and the maximum entropy approach [15]. Acknowledgements This work is supported by the project of “985” program of TJU of China and also by the National Natural Science Foundation of China, No. 10675088. References [1] C.Tsallis, J. Stat. Phys. 52(1988)479. [2] G. Kaniadakis, Physica A 296 (2001) 405. [3] M.P.Leubner, Astrophys. J. 604(2004)469. [4] J.L.Du, Europhys.Lett. 67(2004)893. [5] J.L.Du, Astrophys.Space Sci. 305(2006)247. [6] J.L.Du, Phys.Lett.A 329(2004)262. [7] J.L.Du, Europhys.Lett. 75(2006)861. [8] M.P.Leubner, Astrophys. J. 632(2005)L1. [9] J.L.Du, New Astron. 12(2006)60. [10] S.Shaikh, A.Khan and P.K.Bhatia, Z. Naturforsch. 61A(2006)275. [11] G. Kaniadakis, Phys.Rev.E 66(2002)056126; Phys.Rev.E 72(2005)036108. [12] R. Silva, Eur.Phys. J. B 54(2006)499; cond-mat/0603177. [13] R. Hakim, Phys. Rev. 162(1967)128; H. D. Zeh, The Physical Basis of the Direction of Time, Spring-Verlage, Berlin-Heidelberg,1992. [14] L.N. Guo and J. L. Du, Phys. Lett. A 362(2007)368. [15] A.R. Plastino, A. Plastino, and B.H. Soffer, Phys. Lett. A 363(2007)48.

0704.1438

A KK-monopole giant graviton in AdS_5 x Y_5

UG-FT-216/07 CAFPE-86/07 FFUOV-07/03 PUPT-2231 April 2007 A KK-monopole giant graviton in AdS5 × Y5 Bert Janssena,1, Yolanda Lozanob,2 and Diego Rodŕıguez-Gómezc,3 a Departamento de F́ısica Teórica y del Cosmos and Centro Andaluz de F́ısica de Part́ıculas Elementales Universidad de Granada, 18071 Granada, Spain b Departamento de F́ısica, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain c Department of Physics, Princeton University, Princeton, NJ 08540, U.S.A. ABSTRACT We construct a new giant graviton solution in AdS5 × Y5, with Y5 a quasi-regular Sasaki-Einstein manifold, consisting on a Kaluza-Klein monopole wrapped around the Y5 and with its Taub-NUT direction in AdS5. We find that this configuration has minimal energy when put in the centre of AdS5, where it behaves as a massless particle. When we take Y5 to be S 5, we provide a microscopical description in terms of multiple gravitational waves expanding into the fuzzy S5 defined as an S1 bundle over the fuzzy CP 2. Finally we provide a possible field theory dual interpretation of the construction. 1E-mail address: [email protected] 2E-mail address: [email protected] 3E-mail address: [email protected] http://arxiv.org/abs/0704.1438v3 1 Introduction As it is well-known, giant gravitons are stable brane configurations with non-zero angular momen- tum, that are wrapped on (n− 2)- or (m− 2)-spheres in AdSm×Sn spacetimes and carry a dipole moment with respect to the background gauge potential [1, 2, 3, 4]. They are not topologically stable, but are at dynamical equilibrium because the contraction due to the tension of the brane is precisely cancelled by the expansion due to the coupling of the angular momentum to the back- ground flux field. These spherical brane configurations turn out to be massless, conserve the same number of supersymmetries and carry the same quantum numbers of a graviton. Giant graviton configurations were first proposed as a way to satisfy the stringy exclusion principle implied by the AdS/CFT correspondence [1]. The spherical (n− 2)-brane expands into the Sn part of the geometry with a radius proportional to its angular momentum. Since this radius is bounded by the radius of the Sn, the configuration has associated a maximum angular momentum. The (m − 2)-brane giant graviton configurations [2, 3], on the other hand, expand into the AdSm part of the geometry, and they do not satisfy the stringy exclusion principle. For a discussion on the degeneracy of these two types of giant gravitons and the point-like graviton, we refer for instance to [5] and references therein. The construction of giant gravitons has also been generalised to AdS5 × Y5 spacetimes, where Y5 is a Sasaki-Einstein manifold. In [6] and [7] a D3-brane wrapped around the angular S 3 of the AdS5 and moving along the Reeb vector of the Sasaki-Einstein space was considered, yielding a dual giant graviton. A generalisation of giant graviton configurations preserving 1/4 or 1/8 of the supersymmetries has also been considered by Mikhailov [8] in the AdS5 × T 1,1 spacetime. In this paper we find a new giant graviton configuration in AdS5 × Y5, which consists of a Kaluza-Klein (KK) monopole with internal angular momentum, wrapping the Y5 part of the geometry and with Taub-NUT direction in the AdS5 part. This solution has distinguishing features with respect to the previous giant graviton solutions constructed in the literature. First, the monopole does not couple to the 4-form potential of the background and the configuration is therefore not at a dynamical equilibrium position. Still it is stable because by construction it is wrapped around the entire Y5. Secondly, it has a fixed size L, the “radius” of the Y5, independent of the momentum of the configuration. In fact the energy of the monopole depends only on its position in the AdS5 part of the spacetime, and it is minimised when the monopole sits at the centre of AdS5, where it behaves as a massless particle. In this sense this new giant graviton configuration does not provide a realisation of the stringy exclusion principle. However, its mere existence is sufficiently surprising to motivate a closer look at the configuration. Furthermore, the fact that the giant graviton is built up from a Kaluza-Klein monopole, could lead to interesting view points in the context of the AdS/CFT correspondence. The organisation of this paper is as follows. In section 2 we present the Kaluza-Klein monopole giant graviton solution. We start by introducing our probe monopole and then construct an action suitable to describe it. We then calculate the energy of the configuration and show that when the monopole sits at the centre of AdS5 it behaves as a massless particle. In section 3 we move to consider the microscopical description of this configuration in terms of expanding gravitational waves. Given that the fuzzy version of an arbitrary Sasaki-Einstein manifold is not known, we particularise to the case in which Y5 = S 5. The fuzzy 5-sphere that we consider is defined as an S1 bundle over the fuzzy CP 2. This fuzzy manifold has been successfully used in the microscopical description of 5-sphere giant gravitons [9, 10], and of the baryon vertex with magnetic flux [11]. In these examples the fibre structure of the S5 plays a crucial role in the construction. Finally in section 4 we present a candidate description of our configuration in the field theory side. We end with some conclusions in section 5. 2 A new giant graviton solution 2.1 The Kaluza-Klein monopole probe Consider the AdS5 × Y5 spacetime, with Y5 a quasi-regular five-dimensional Sasaki-Einstein man- ifold. All these Sasaki-Einstein manifolds have a constant norm Killing vector, called the Reeb vector. For the cases we are interested in, the U(1) action of the Reeb vector is free and the quotient space is (at least locally) a four-dimensional regular Kähler-Einstein manifold M4 with positive curvature. In that case the metric on Y5 can (at least locally) be written as a U(1) fibre bundle over the M4, ds2Y = ds M + (dψ +B) 2, (2.1) where ds2M is the metric on the M4 and the Killing vector k µ = δ is the Reeb vector. The Kähler form on M4 is related to the fibre connection B via ωM = dB.4 The AdS5 × Y5 background contains as well a non-vanishing 4-form RR-potential. Using the U(1) decomposition above, the metric of AdS5 × Y5 can be written as ds2 = −(1 + r )dt2 + (1 + r dΩ22 + (dχ+A) ds2M + (dψ +B) , (2.2) where we have used global coordinates in the AdS part and written the angular S3 contained in AdS5 as a U(1) fibre over S 2. A and B stand for the connections of the S3 and Y5 fibre bundles respectively. In these coordinates, the fibre directions χ and ψ are clearly globally defined isometry directions. Consider now a KK-monopole wrapped on the Y5, with Taub-NUT direction χ and propagating along ψ. This will be our KK-monopole probe. In order to study the dynamics of this monopole we start by constructing an action suitable to describe it. The effective action describing the dynamics of the Type IIB Kaluza-Klein monopole was constructed in [13]. Like the Type IIA NS5-brane, to which it is related by T-duality along the Taub-NUT direction, the action for the monopole is described by a six-dimensional (2, 0) tensor supermultiplet, which contains a self-dual 2-form Ŵ+ and 5 scalars {X i, ω, ω̃}. The self-dual 2-form is associated to the (S-duality invariant) configuration of the monopole intersecting a D3- brane, wrapped on the Taub-NUT direction. The worldvolume scalars ω and ω̃ are associated with the intersections of D5- and NS5-branes respectively, and form a doublet under S-duality. Finally the scalars X i (with i = 1, 2, 3) are the embedding scalars, that describe the position of the monopole in the transverse space. Note that although the worldvolume of the monopole is six-dimensional, its position is specified by only three embedding scalars. This is because the Taub-NUT direction is considered to be transverse, but being an isometry direction it does not yield a dynamical degree of freedom. The KK-monopole action takes in fact the form of a gauged sigma model, where the degree of freedom corresponding to the Taub-NUT direction is gauged away [14]. Due to the presence of the self-dual two-form Ŵ+ , there is no straightforward covariant formulation of the action (see for example [15]). However, like in the case of the five-brane [16, 17], it is possible to give an approximation, expanding the action to quadratic order in the self-dual two-form. In our case, the situation is actually simpler. As our KK-monopole probe is wrapped around Y5, the U(1) fibre direction ψ is contained in its worldvolume. Therefore it is possible to effectively compactify the monopole over the fibre direction and to consider the (much simpler) action for a wrapped KK-monopole. Moreover, momentum charge along this U(1) fibre direction can easily be induced by switching on an appropriate magnetic flux in the worldvolume. The field content of the wrapped monopole is given by the five-dimensional (1, 1) vector multiplet, which contains 5 scalars and one vector, and is the dimensional reduction of the six- dimensional (2, 0) tensor multiplet. The self-duality condition becomes a Hodge-duality condition 4For an extensive summary on the properties of Sasaki-Einstein manifolds we refer to [12]. between the vector and a two-form, which does not appear explicitly in the action.5 In this way an action can be constructed to all orders in the field strength. In practise, the action of the wrapped monopole is most easily constructed from the action of the Type IIA KK-monopole, as the latter has the six-dimensional (1, 1) vector multiplet as its worldvolume field content [18]. Af- ter T-dualising along a worldvolume direction, the resulting action describes a Type IIB monopole wrapped along the T-duality direction, with an effectively five-dimensional worldvolume. As our KK-monopole probe is wrapped on the S1 fibre direction of the Y5, its spatial worldvolume becomes effectively R×M4. 2.2 The action for the wrapped monopole The starting point is the action for the Type IIA Kaluza-Klein monopole constructed in [18], which we compactify along a worldvolume direction and T-dualize. The resulting action describes a Type IIB Kaluza-Klein monopole which is wrapped on the T-duality direction and has, effectively, a five-dimensional worldvolume. The T-duality direction appears in the action as a new isometric direction, whose Killing vector we denote by kµ. On the other hand the Killing vector associated with the Taub-NUT direction is denoted by ℓµ. The explicit action is given by S = −T4 d5σ e−2φkℓ2 | det(DaXµDbXνgµν + eφk−1ℓ−1Fab)| P [ikiℓN (7)]− P ∧ F − 1 [k(1) ∧ F ∧ F + . . . , (2.3) where the scalars k and ℓ are the norm of kµ and ℓµ respectively, k(1) denotes the 1-form with com- ponents kµ and (iℓikΩ)µ1...µn = ℓ ρkνΩνρµ1...µn . In this action the pull-backs into the worldvolume are taken with gauge covariant derivatives µ = ∂aX µ − k−2kν∂aXνkµ − ℓ−2ℓν∂aXνℓµ, (2.4) which ensure local invariance under the isometric transformations generated by the two Killing vectors δXµ = Λ(1)(σ)kµ + Λ(2)(σ)ℓµ . (2.5) In this way the embedding scalars corresponding to the isometry directions are eliminated as dynamical degrees of freedom and the action is given by a gauged sigma model of the type first considered in [14]. The two-form field strength F is defined as F = 2∂V (1) + P [ikiℓC(4)] , (2.6) where the worldvolume vector field V (1) is the T-dual of the vector field of the Type IIA monopole (or, alternatively, the dimensional reduction of the self-dual two-form Ŵ+). While in the Type IIA monopole the vector field is associated to D2-branes wrapped on the Taub-NUT direction, in the IIB case it is associated to D3-branes, wrapped on both Killing directions. The action of the Type IIA monopole contains as well a worldvolume scalar associated to strings wrapped on the Taub-NUT direction. This field gives, upon T-duality, a worldvolume scalar ω which forms a doublet under S-duality with the T-dual ω̃ of the component of the IIA vector field along the T-duality direction. These two scalars are necessary in order to compensate for the two degrees of freedom associated to the two transverse scalars that have been eliminated from the action through the gauging procedure. This scalar doublet does however not play a role in our construction and has therefore been set to zero in our action above. The action (2.3) should then be regarded as a truncated action suitable for the study of the wrapped monopole in the AdS5 × Y5 background. 5This is very similar to the M5-brane case. The unwrapped M5-brane contains a self-dual 2-form in its world- volume, whereas the M5-brane wrapped on the eleventh direction (the D4) depends on an unconstrained five- dimensional vector field. In the Chern-Simons part of the action we find a coupling to N (7), the tensor field dual to the Taub-NUT Killing vector ℓµ, considered as a 1-form. The contraction iℓN (7) is the field to which a KK-monopole with Taub-NUT direction ℓµ couples minimally (see [18]). In (2.3) this field is further contracted with the second isometric direction kµ, indicating that the monopole is wrapped along this direction. More importantly for our construction below, the second coupling in the CS action involves the momentum operator P [k(1)/k2], associated to the isometric direction with Killing vector kµ. Therefore, it is possible to induce momentum charge in this isometric direction, with an appropriate choice of F . As we will show below, we will make use of this coupling to let the monopole propagate along the isometric direction ψ. Finally, the dots indicate couplings to other Type IIB background fields which do not play a role in our construction. 2.3 The giant graviton solution Let us now particularise the action (2.3) to our probe KK-monopole. We take our monopole wrapped on the transverse Y5. Therefore the fibre direction of the decomposition of Y5 as a U(1) fibre bundle over M4 is identified as the isometric worldvolume direction in (2.3), and M4 as the effective four-dimensional spatial worldvolume. The Taub-NUT direction is taken along the S1 fibre direction of the S3 contained in AdS5. Therefore, we have explicitly kµ = δ , ℓµ = δµχ. (2.7) With this choice of Killing directions the contribution of the 4-form RR-potential of the AdS5×Y5 background to the action vanishes. This is so because both couplings to C(4), in (2.6) and in (2.3), involve directions along Y5, plus the Taub-NUT direction, χ, which lives in the AdS5 part of the spacetime. Furthermore, in order to induce momentum charge in the ψ direction we choose the world- volume vector field V proportional to the curvature tensor of the Y5 fibre connection B, such F = ∗F , F ∧ F = 2n2ΩM , (2.8) where ΩM is the volume of M4 and the Hodge star is taken with respect to the metric on this manifold6. With this Ansatz F satisfies trivially the Bianchi identities. Then, through the second coupling in the Chern-Simons part of the action (2.3), we have that k−2k(1) ∧ F ∧ F = n2TW k−2k(1) , (2.9) where we have used the fact that the tension of the wrapped monopole is related to the tension of the point-like object carrying momentum charge (the gravitational wave) through ΩMT4 = TW . Therefore, with this Ansatz for F , we are dissolving in the worldvolume n2 momentum charges in the ψ direction. Notice that the instantonic nature of (2.8) guarantees that the equations of motion for F are satisfied. A second remarkable property of the Ansatz (2.8) is that the determinant of (P [g]ab +Fab) is a perfect square [9, 11], such that the Born-Infeld part of the action (2.3) gives rise to S = −T4 dt dΩM |gM |, (2.10) which after integration over M4 gives rise to the following Hamiltonian . (2.11) 6The integral above is non-zero because it is the product of two integrals, F , over non-trivial two-cycles in M4 (see for example [19]). Since F = 2πn due to Dirac quantisation condition, n represents the winding number of D3-branes wrapped around each of the two cycles. For our construction we have chosen the same winding number in both cycles in order to preserve the self-duality condition (2.8). The energy of the configuration is therefore a function of the radial coordinate r of AdS5 and is clearly minimised when r = 0, that is, when the monopole is sitting at the centre of AdS5. Moreover, for this value of r the energy is given by . (2.12) Therefore, the configuration that we have proposed behaves as a giant graviton: it has the energy of a massless particle with momentum Pψ but clearly has some finite radius L, as it is wrapped around the entire Y5. Since it saturates a BPS bound it is a solution of the equations of motion. Finally, we should note that there is no dynamical equilibrium between the brane tension and the angular momentum, as in the traditional giant graviton configurations of [1, 2, 3, 4]. However the stability of the configuration is still guaranteed due to the fact that it wraps the entire transverse space. 3 A microscopical description in terms of dielectric gravi- tational waves It is by now well-known that the traditional giant graviton configurations of [1, 2, 3, 4] can be described microscopically in terms of multiple gravitational waves expanding into (a fuzzy version of) the corresponding spherical brane by Myers’ dielectric effect [20]. In particular, the M5-brane giant graviton configurations of the AdS4 × S7 and AdS7 × S4 spacetimes have been described in terms of multiple M-waves expanding into a fuzzy 5-sphere that is defined as an S1 bundle over a fuzzy CP 2 [9]. A non-trivial check of the validity of this description is that it agrees exactly with the spherical brane description in [1, 2] when the number of gravitons is very large. In this spirit one would expect that the KK-monopole giant graviton configuration constructed in the previous section would be described microscopically in terms of dielectric Type IIB gravita- tional waves expanding into a fuzzy Y5. The fuzzy version of general quasi-regular Sasaki-Einstein manifolds is however not known. Therefore we will restrict to the case in which Y5 coincides with the 5-sphere. In this case we will see that the waves expand into a fuzzy 5-sphere, defined as an S1 bundle over a fuzzy CP 2, i.e. the same type of fuzzy manifolds involved in the description used in [9] (see also [10, 11]). Consider now a number N of coinciding gravitational waves in the AdS5 × S5 background. The action describing multiple gravitational waves in Type IIB was constructed in [21, 22] and it contains the following couplings: SW = − TW dτ STr Eµν − Eµi(Q−1 − δ)ijEjkEkν (3.1) dτ STr −P [k−2k(1)] − iP [(iX iX)iℓC(4)]− 12P [(iX iX) 2iℓikN (7)] + . . . where Eµν = Gµν − k−1ℓ−1(ikiℓC(4))µν , Gµν = gµν − k−2kµkν − ℓ−2ℓµℓν , Qµν = δ ν + ikℓ [X µ, Xρ]Eρν , ((iX iX)iℓC4)λ = [X ρ, Xν]ℓµCµνρλ. (3.2) (iℓikΩ)µ1...µn = ℓ ρkνΩνρµ1...µn , This action is valid to describe waves propagating in backgrounds which contain two isometric directions, parametrised in the action by the Killing vectors kµ and ℓµ. The wave action is actually a gauged sigma model in which the embedding scalars associated to the Killing directions are projected out. The physical meaning of kµ is that it corresponds to the propagation direction of the gravitational waves, while ℓµ is an isometry direction inherited from the T-duality operation involved in the construction of the action (see [21] and [22] for more details). Although the only non-zero term in the Chern-Simons action in the AdS5 × S5 background is the dipole coupling to C(4), it is worth calling the attention to the quadrupole coupling to N (7). Indeed, this coupling shows that the waves can expand via a quadrupole effect into a monopole with Taub-NUT direction parametrised by lµ and further wrapped around the kµ direction. This monopole will then act as the source of iℓikN Let us now use the action (3.1) to describe microscopically the KK-monopole of the previous section in the AdS5×S5 background. In this caseM4 = CP 2 and ds2M stands for the Fubini-Study metric on the CP 2 (see for instance [23]). The identification of the isometry directions of (3.1) is then obvious. In order to account for the momentum in the worldvolume of the monopole, we identify the propagation direction of the waves with the S5 fibre direction ψ, while the extra isometry will be identified with the Taub-NUT direction χ of the monopole: kµ = δ , ℓµ = δµχ. (3.3) With this choice of Killing vectors it is clear that the contribution of C(4) to the action vanishes, both in the BI and in the CS parts. Therefore, any dielectric effect will be purely gravitational [24, 25, 26]. Furthermore, we will have the gravitational waves expand into the entire five-sphere, whose fuzzy version we choose to describe as an S1 bundle over the fuzzy CP 2. Therefore, we take the non-commutative scalars in (3.1) to parametrise the fuzzy CP 2 base of the S5. The fuzzy CP 2 has been extensively studied in the literature. For its use in the giant graviton context we refer to [9, 10], where more details on the construction that we sketch below can be found. CP 2 is the coset manifold SU(3)/U(2) and can be defined as the submanifold of R8 determined by the constraints xixi = 1 , j,k=1 dijkxjxk = xi , (3.4) where dijk are the components of the totally symmetric SU(3)-invariant tensor. A fuzzy version of CP 2 can then be obtained by imposing the conditions (3.4) at the level of matrices (see for example [27]). We define a set of coordinates X i (i = 1, . . . , 8) as X i = (2N − 2)/3 , (3.5) where T i are the SU(3) generators in the N -dimensional irreducible representations (k, 0) or (0, k), with N = (k + 1)(k + 2)/2. The first constraint in (3.4) is then trivially satisfied through the quadratic Casimir (2N − 2)/3 of the group, whereas the rest of the constraints are satisfied for any N (see [27, 9] for the details). The commutation relations between the X i are given by [X i, Xj] = i f ijk (2N − 2)/3 Xk, (3.6) with f ijk the structure constants of SU(3) in the algebra of the Gell-Mann matrices [λi, λj ] = 2if ijkλk. Substituting the Ansätze (3.5) and (3.3) in the action (3.1), we find S = −TW dτ STr 3L6r2 32(N − 1)X , (3.7) up to order N−2. Here we have dropped those contributions to det Q that vanish when taking the symmetrised trace, and ignored higher powers of N which will vanish in the large N limit.7 7These terms cannot be nicely arranged into higher powers of the quadratic Casimir without explicit use of the constraints. Taking the symmetrised trace we arrive at the following Hamiltonian 32(N − 1) , (3.8) which, in the large N limit, is in perfect agreement with the Hamiltonian for the spherical KK- monopole, given by (2.11). To see this we should recall that in the macroscopical description the momentum charge of the configuration is the number of waves dissolved in the worldvolume, and is therefore given by n2. In the microscopical description the momentum charge is given directly by the number of microscopic waves, N . Therefore in the large N limit N and n2 must coincide. The Hamiltonian in (3.8) is also a function of the radial coordinate r of AdS5 and it is minimised at r = 0 where it takes the value E = Pψ/L, thus corresponding to a giant graviton configuration. 4 A possible interpretation in the dual field theory In this section we try to give a possible field theoretic interpretation of the giant graviton con- figuration that we have studied, along the lines of [3] (see also [28, 29, 30, 31]). We will discuss the giant graviton configuration in the AdS5 × S5 background, but we will later speculate on a possible generalisation to other Sasaki-Einstein spaces. We have learnt in the previous sections that the fibre direction in the S3 contained in AdS5 plays a crucial role in the construction of the giant graviton configuration, as it is identified with the Taub-NUT direction of the monopole. Therefore, it is useful to work in the global patch for It is then natural to consider the dual field theory as living in R×S3, where there is a conformal coupling to the curvature. The bosonic piece of the action reads dt dΩ3 Tr µΦa + Φ∗aΦa + [Φa,Φ , (4.1) where L is the radius of the S3 and the Φa (with a = 1, 2, 3) are the complexification of the 6 adjoint real scalars X i of N = 4 SYM. After defining Φa = Xa+ iXa+3, only an SU(3) subgroup of the original SO(6) R-symmetry group remains explicit. Regarding S3 as an S1 bundle over S2 it seems a consistent truncation to assume that the Φa do not depend on the fibre coordinate. Actually, this will be the field theory analogue of the fact that this direction corresponds to the Taub-NUT direction of the monopole on the gravity side. Taking adapted coordinates to the U(1) fibration we have S = 2π dΩ2 Tr − ∂tΦ∗a∂tΦa − 4Φ∗a∆S2Φa + Φ∗aΦa + [Φa,Φ , (4.2) where ∆S2 is the Laplacian in the 2-sphere. We can then expand the scalars in spherical harmonics Φ a on the two-sphere. Given that we will be interested in the lowest energy modes, we will truncate all of them except the massless mode Φ a , which corresponds to the constant mode on the S 2. Furthermore, we consider the following Ansatz for the gauge and SU(3) dependence of our fields Φ(0)a = e if(t)Ma ⊗ Ja, (4.3) where f(t) is an arbitrary function of time, the traceless matrix Ma is defined as Ma = diag M − 1 , · · · ,− M − 1 , (4.4) and the Ja’s are SU(2)-generators in a k-dimensional representation. Since the total rank of the gauge group is N we should have that kM = N . Indeed, in this branch the gauge group breaks into SU(M). The gauge transformations which are left are those of the form Φa → gΦag†, g = diag(g̃k, · · · , g̃k) , (4.5) where the g̃ki are SU(2) gauge transformations of dimension k. The action then reduces to S = 8π2 C2(k)M M − 1 − (f ′)2v2 + v , (4.6) where v2 = δabvavb is to be interpreted as a non-dynamical parameter whose value will determine the minima of the potential. In addition, C2(k) is the Casimir of the SU(2) k-dimensional repre- sentation. In this action f(t) is a cyclic variable and therefore its conjugate momentum, p, will be conserved. The Hamiltonian is given by H = p2 (M − 1) 32π2v2C2(k)M 8π2MC2(k)v (M − 1)L2 , (4.7) which has a minimum for v2 = (M−1) 32π2C2(k)M pL. Remarkably the on-shell energy is precisely the dispersion relation . (4.8) Therefore, the configuration (4.3) can be seen as a massless particle. Furthermore, out of the original SU(3) rotating our Φa just an SU(2) survives, given that with our Ansatz the Φa become a vector of SU(2). Thus, the moduli space reduces to SU(3)/U(2), which is precisely the symmetry of CP 2 as a coset space, which is in turn the manifold wrapped by our KK-monopole. Given that our construction of the wrapped KK-monopole works not just in the S5 case, but also in more generic spaces, we expect a similar field theory description for the dual of a generic Sasaki-Einstein space. In supporting this claim, let us notice that the potential term did not play any role in the S5 case, because with the Ansatz we assumed, it vanishes. In the generic case, we will assume a similar Ansatz for our fields, namely Xα = e if(t)M⊗Gα , (4.9) where now for simplicity we take the same M matrix as before but with all the v’s identical. The Gα are the generators of the global symmetry group G. Given this Ansatz, we expect that the superpotential does not play any role, not even in the most generic Y p,q case. In addition, since we take AdS in the global patch, the field theory will be defined in R × S3, so we will always have the conformal coupling to the curvature. Just this term, together with the kinetic energy, is enough to reproduce a dispersion relation of the form E ∼ p. In the general case we can also regard the S3 as an S1 bundle over S2, and take our fields independent of the U(1). This is the field theory counterpart of the presence of the Taub-NUT direction in the gravity side. In addition, out of the full global symmetry group G we will just keep the subgroup g compatible with our Ansatz,8 so we would expect a moduli space of the form G/g. Let us consider for example the conifold. In this case the global symmetry group is SU(2)×SU(2). Therefore we have to first reduce it to the diagonal SU(2) and then take the conifold scalars A and B to be eif(t)M. This leaves an [SU(2)×SU(2)]D/U(1) moduli space, which is the symmetry of 2 2-spheres, and this is in turn what one would get from the gravity side. Finally, we would like to note that our construction is quite generic. From the gravity point of view we just require that the momentum of the KK-monopole wrapping the five-dimensional space is taken along a U(1) fibre direction, and that its Taub-NUT direction is along the S1 in the decomposition of the S3 ⊂ AdS5 as an S1 bundle over S2. In the field theory side our requirements are also quite generic. The existence of the Taub-NUT direction is reflected on the fact that the SCFT is defined in R×S3 and S3 is taken as S1 over S2. In addition, our description is not sensitive to the superpotential, which we believe is the counterpart to the fact that the KK-monopole wraps the whole 5-dimensional manifold. Then, we are left with the kinetic term and the conformal coupling to curvature, which is enough to ensure the right dispersion relation. Since in general our Ansatz will reduce the original global symmetry, the moduli space will be G/g, which we believe will correspond in general to the symmetry of the 4-dimensional base which the KK-monopole wraps. 8Note that g may involve discrete subgroups such as Zk 5 Conclusions In this letter we have constructed a new type of giant graviton solution in AdS5 × Y5, with Y5 a quasi-regular Sasaki-Einstein manifold. This solution consists on a Kaluza-Klein monopole with internal momentum, wrapped around the entire Y5 and with Taub-NUT direction along the AdS5 part. Although the dynamics of this monopole can be described using the effective action for the Type IIB Kaluza-Klein monopole constructed in [13], this action is only known to quadratic order in the self-dual 2-form of its six-dimensional (2, 0) tensor multiplet field content. However, given that Y5 can be decomposed as a U(1) bundle over a four-dimensional Kähler-Einstein manifold M4, it is possible to use the action for a monopole wrapped on a U(1) direction to describe it. This action, having the field content of the five-dimensional (1, 1) vector multiplet, is known to all orders. Moreover, it is possible to induce momentum charge along the U(1) direction through a suitably chosen worldvolume vector field with non-zero instanton number. Using the action for a U(1) wrapped monopole we have shown that the energy of the configuration depends on its radial position in the AdS space and behaves as a massless particle when put in the origin, while having the size of the Y5. Given that the spherical monopole carries a non-vanishing momentum charge there should be a microscopical description of the same configuration in terms of expanding gravitational waves. This description would involve however the fuzzy version of Y5, which is not known in general. Therefore, we have restricted to the case in which Y5 = S 5, and let the multiple dielectric gravita- tional waves expand into a fuzzy 5-sphere. The fuzzy 5-sphere built up by the gravitational waves is constructed as an Abelian fibre bundle over a fuzzy CP 2, a construction that has been used before in the study of the traditional giant gravitons and the baryon vertex with magnetic flux. The configuration thus obtained turns out to exactly agree in the limit of large number of waves with the effective KK-monopole description. We believe there are several reasons why this new giant graviton solution has not been found earlier in the literature. First of all, since it has no relation with the stringy exclusion principle it is not straightforward to find the corresponding state in the CFT side. Moreover, as we have shown in section 4, our scalar field configuration breaks the R-symmetry group in a rather peculiar way, making explicit the U(1) fibre structure of the S3. Secondly, the fact that it is built up from a Kaluza-Klein monopole and not from a more ordinary type of brane, makes our construction more involved. An interesting question to answer would be whether the KK-monopole giant graviton solution is supersymmetric or not. This is however difficult to answer, on the one hand due to the form of the Killing spinors in the particular coordinate system that we are using and, on the other hand, due to the fact that the kappa-symmetry for the Kaluza-Klein monopole is not known. Yet, the fact that the configuration is massless implies that it saturates a BPS bound, which hints to the fact that it probably preserves some fraction of the supersymmetry. We would like to leave this problem for future investigations. Acknowledgements We wish to thank S.Benvenutti, B. Craps, D. van den Bleeken, T. van Proeyen and A.V. Ramallo for the useful discussions. The work of B.J. is done as part of the program Ramón y Cajal of the Ministerio de Educación y Ciencia (M.E.C.) of Spain. He is also partially supported by the M.E.C. under contract FIS 2004-06823 and by the Junta de Andalućıa group FQM 101. The work of Y.L. has been partially supported by the CICYT grant MEC-06/FPA2006-09199 (Spain) and by the European Commission FP6 program MRTN-CT-2004-005104, in which she is associated to Universidad Autónoma de Madrid. D.R.G. is supported by the Fullbright-M.E.C. fellowship FU-2006-0740. References [1] J. McGreevy, L. Susskind, N. Toumbas, JHEP 0006 (2000) 008, hep-th/0003075. [2] M. Grisaru, R. Myers, Ø. Tafjord, JHEP 0008 (2000) 040, hep-th/0008015. [3] A. Hashimoto, S. Hirano, N. Itzhaki, JHEP 0008 (2000) 051, hep-th/0008016. [4] S.R. Das, A. Jevicki, S.D. Mathur, Phys. Rev. D63 (2001) 044001, hep-th/0008088. [5] I. Bena, D.J. Smith, Phys. Rev. D71 (2005) 025005, hep-th/0401173. [6] D. Martelli, J. Sparks, Nucl.Phys. 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Naqvi, JHEP 0703 (2007) 034, hep-th/0612032. http://arxiv.org/abs/hep-th/0003075 http://arxiv.org/abs/hep-th/0008015 http://arxiv.org/abs/hep-th/0008016 http://arxiv.org/abs/hep-th/0008088 http://arxiv.org/abs/hep-th/0401173 http://arxiv.org/abs/hep-th/0608060 http://arxiv.org/abs/hep-th/0608093 http://arxiv.org/abs/hep-th/0010206 http://arxiv.org/abs/hep-th/0411181 http://arxiv.org/abs/hep-th/0505073 http://arxiv.org/abs/hep-th/0606264 http://arxiv.org/abs/hep-th/0603021 http://arxiv.org/abs/hep-th/9806169 http://arxiv.org/abs/hep-th/9706117 http://arxiv.org/abs/hep-th/9701037 http://arxiv.org/abs/hep-th/9606118 http://arxiv.org/abs/hep-th/9712115 http://arxiv.org/abs/hep-th/9802199 http://arxiv.org/abs/hep-th/0411264 http://arxiv.org/abs/hep-th/9910053 http://arxiv.org/abs/hep-th/0205254 http://arxiv.org/abs/hep-th/0303183 http://arxiv.org/abs/hep-th/0102200 http://arxiv.org/abs/hep-th/0212250 http://arxiv.org/abs/hep-th/0509228 http://arxiv.org/abs/hep-th/0103023 http://arxiv.org/abs/hep-th/0111222 http://arxiv.org/abs/hep-th/0403110 http://arxiv.org/abs/hep-th/0609173 http://arxiv.org/abs/hep-th/0612032 Introduction A new giant graviton solution The Kaluza-Klein monopole probe The action for the wrapped monopole The giant graviton solution A microscopical description in terms of dielectric gravitational waves A possible interpretation in the dual field theory Conclusions

0704.1439

Critical Behavior of a Trapped Interacting Bose Gas

Critical Behavior of a Trapped Interacting Bose Gas ∗ T. Donner1, S. Ritter1, T. Bourdel1, A. Öttl1, M. Köhl1,2†, T. Esslinger1 1Institute of Quantum Electronics, ETH Zürich, CH–8093 Zürich, Switzerland 2Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom †To whom correspondence should be addressed; E-mail: [email protected]. The phase transition of Bose-Einstein condensation is studied in the critical regime, when fluctua- tions extend far beyond the length scale of thermal de Broglie waves. Using matter-wave interference we measure the correlation length of these critical fluctuations as a function of temperature. The diverging behavior of the correlation length above the critical temperature is observed, from which we determine the critical exponent of the correlation length for a trapped, weakly interacting Bose gas to be ν = 0.67±0.13. This measurement has direct implications for the understanding of second order phase transitions. Phase transitions are among the most dramatic phe- nomena in nature, when minute variations in the con- ditions controlling a system can trigger a fundamental change of its properties. For example, lowering the tem- perature below a critical value creates a finite magne- tization of ferromagnetic materials or, similarly, allows for the generation of superfluid currents. Generally, a transition takes place between a disordered phase and a phase exhibiting off-diagonal long-range order which is the magnetization or the superfluid density in the above cases. Near a second-order phase transition point the fluctuations of the order parameter are so dominant that they completely govern the behavior of the system on all length scales [1]. In fact, the large scale fluctuations in the vicinity of a transition already indicate the onset of the phase on the other side of the transition. Near a second-order phase transition, macroscopic quantities show a universal scaling behavior which is characterized by critical exponents [1] that depend only on general properties of the system, such as its dimen- sionality, symmetry of the order parameter or the range of interaction. Accordingly, phase transitions are classi- fied in terms of universality classes. Bose-Einstein con- densation in three dimensions, for example, is in the same universality class as a three-dimensional XY mag- net. Moreover, the physics of quantum phase transitions occurring at zero temperature can often be mapped on thermally driven phase transitions in higher spatial di- mensions. The phase transition scenario of Bose-Einstein conden- sation in a weakly interacting atomic gas is unique as it is free of impurities and the two-body interactions are precisely known. As the gas condenses, trapped bosonic atoms of a macroscopic number accumulate in a single ∗This is the author’s version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in SCIENCE, 315, (16 March 2007), doi:10.1126/science.1138807. quantum state and can be described by the condensate wave function, the order parameter of the transition. However, it has proven to be experimentally difficult to access the physics of the phase transition itself. In partic- ular, the critical regime has escaped observation as it re- quires an extremely close and controlled approach to the critical temperature. Meanwhile, advanced theoretical methods have increased the understanding of the critical regime in a gas of weakly interacting bosons [2, 3, 4, 5]. Yet, the theoretical description of the experimental situ- ation, a Bose gas in a harmonic trap, has remained un- clear. FIG. 1: Schematics of the correlation function and the cor- relation length close to the phase transition temperature of Bose-Einstein condensation. Above the critical temperature Tc the condensate fraction is zero (see right) and for T ≫ Tc the correlation function decays approximately as a Gaussian on a length scale set by the thermal de Broglie wavelength λdB. As the temperature approaches the critical temperature long-range fluctuations start to govern the system and the correlation length ξ increases dramatically. Exactly at the critical temperature ξ diverges and the correlation function decays algebraically for r > λdB (Eq 1). We report on a measurement of the correlation length of a trapped Bose gas within the critical regime just above the transition temperature. The visibility of a matter- wave interference pattern gives us direct access to the http://arxiv.org/abs/0704.1439v1 first order correlation function. Exploiting our experi- mental temperature resolution of 0.3 nK (corresponding to 2×10−3 of the critical temperature) we observe the di- vergence of the correlation length and determine its crit- ical exponent ν. This direct measurement of ν through the single particle density matrix complements the mea- surements of other critical exponents in liquid Helium [6, 7, 8], which is believed to be in the same universality class as the weakly interacting Bose gas. In a Bose gas the physics of fluctuations of the or- der parameter is governed by different length scales. Far above the phase transition temperature, classical ther- mal fluctuations dominate. Their characteristic length scale is determined by the thermal de Broglie wavelength λdB and the correlation function can be approximated by 〈Ψ†(r)Ψ(0)〉 ∝ exp(−πr2/λ2dB) with r being the separa- tion of the two probed locations [9] (Fig 1). Nontriv- ial fluctuations of the order parameter Ψ close to the critical temperature become visible when their length scale becomes larger than the thermal de Broglie wave- length. The density matrix of a homogeneous Bose gas for r > λdB can be expressed by the correlation function [10, 11] 〈Ψ†(r)Ψ(0)〉 ∝ 1 exp(−r/ξ), (1) where ξ denotes the correlation length of the order pa- rameter. The correlation length ξ is a function of temper- ature T and diverges as the system approaches the phase transition (Fig 1). This results in the algebraic decay of the correlation function with distance 〈Ψ†(r)Ψ(0)〉 ∝ 1/r at the phase transition. The theory of critical phenomena predicts a divergence of ξ according to a power-law ξ ∝ |(T − Tc)/Tc|−ν , (2) where ν is the critical exponent of the correlation length and Tc the critical temperature. The value of the critical exponent depends only on the universality class of the system. While for non-interacting systems the critical expo- nents can be calculated exactly [1, 12], the presence of interactions adds richness to the physics of the system. Determining the value of the critical exponent using Lan- dau’s theory of phase transitions results in a value of ν = 1/2 for the homogeneous system. This value is the result of both a classical theory and a mean field ap- proximation to quantum systems. However, calculations initially by Onsager [13] and later the techniques of the renormalization group method [1] showed that mean field theory fails to describe the physics at a phase transition. Very close to the critical temperature – in the critical regime – the fluctuations become strongly correlated and a perturbative or mean-field treatment becomes impossi- ble making this regime very challenging. We consider a weakly interacting Bose gas with den- sity n and the interaction strength parameterized by the s-wave scattering length a = 5.3 nm in the dilute limit n1/3a ≪ 1. In the critical regime mean-field theory fails because the fluctuations of Ψ become more domi- nant than its mean value. This can be determined by the Ginzburg criterion ξ > λ2dB/( 128π2a) ≃ 0.4µm [14, 15]. Similarly, these enhanced fluctuations are re- sponsible for a nontrivial shift of the critical tempera- ture of Bose-Einstein condensation [2, 3, 4, 17]. The critical regime of a weakly interacting Bose gas offers an intriguing possibility to study physics beyond the usual mean-field approximation [18] which has been observed in cold atomic gases only in reduced dimensionality be- fore [19, 20, 21, 22]. In our experiment, we let two atomic beams, which originate from two different locations spaced by a dis- tance r inside the trapped atom cloud, interfere. From the visibility of the interference pattern the first order correlation function [23] of the Bose gas above the criti- cal temperature and the correlation length ξ can be de- termined. Separation (µm) 1.004 T 1.019 T 1.058 T 0.0 0.5 1.0 1.5 2.0 FIG. 2: Spatial correlation function of a trapped Bose gas close to the critical temperature. Shown is the visibility of a matter wave interference pattern originating from two regions separated by r in an atomic cloud just above the transition temperature. The gray line is a Gaussian with a width given by the thermal de Broglie wavelength λdB which changes only marginally for the temperature range considered here. The experimental data show phase correlations extending far be- yond the scale set by λdB . The solid line is a fit proportional e−r/ξ for r > λdB. Each data point is the mean of on average 12 measurements, the error-bars are ±SD. We prepare a sample of 4 × 106 87Rb atoms in the |F = 1,mF = −1〉 hyperfine ground state in a mag- netic trap[24]. The trapping frequencies are (ωx, ωy, ωz) = 2π × (39, 7, 29)Hz, where z denotes the vertical axis. Evaporatively cooled to just below the critical tempera- ture the sample reaches a density of n = 2.3× 1013 cm−3 giving an elastic collision rate of 90 s−1. The tempera- ture is controlled by holding the atoms in the trap for a defined period of time during which energy is transferred to the atoms due to resonant stray light, fluctuations of the trap potential or background gas collisions. From absorption images we determine the heating rate to be 4.4± 0.8 nK/s. Using this technique we cover a range of temperatures from 0.001 < (T − Tc)/Tc < 0.07 over a time scale of seconds. For output coupling of the atoms we use microwave fre- quency fields to spin-flip the atoms into the magnetically untrapped state |F = 2,mF = 0〉. The resonance con- dition for this transition is given by the local magnetic field and the released atoms propagate downwards due to gravity. The regions of output coupling are chosen sym- metrically with respect to the center of the trapped cloud and can be approximated by horizontal planes spaced by a distance r [23]. The two released atomic beams inter- fere with each other. For the measurement we typically extract 4 × 104 atoms over a time scale of 0.5 s which is approximately 1% of the trapped sample. We detect the interference pattern in time with single atom reso- lution using a high finesse optical cavity, placed 36mm below the center of the magnetic trap. An atom entering the cavity mode decreases the transmission of a probe beam resonant with the cavity. Due to the geometry of our apparatus, only atoms with a transverse momentum (px, py) ≃ 0 are detected which results in an overall de- tection efficiency of 1% for every atom output coupled from the cloud. From the arrival times of the atoms we find the visibility V(r) of the interference pattern [25]. From repeated measurements with different pairs of mi- crowave frequencies we measure V(r) with r ranging from 0 to 4 λdB where λdB ≃ 0.5µm. With the given heat rate a segmentation of the ac- quired visibility data into time bins of ∆t = 72ms length allows for a temperature resolution of 0.3 nK which cor- responds to 0.002Tc. The time bin length was chosen to optimize between shot-noise limited determination of the visibility from the finite number of atom arrivals and sufficiently good temperature resolution. For the analysis we have chosen time bins overlapping by 50%. Figure 2 shows the measured visibility as a function of slit separation r very close to the critical temperature Tc. We observe that the visibility decays on a much longer length scale than predicted by the thermal de Broglie wavelength λdB. We fit the long distance tail r > λdB with Eq 1 (solid line) and determine the correla- tion length ξ. The strong temperature dependence of the correlation function is directly visible. As T approaches Tc the visibility curves become more long ranged and similarly the correlation length ξ increases. The obser- vation of long-range correlations shows how the size of the correlated regions strongly increases as the tempera- ture is varied only minimally in the vicinity of the phase transition. Figure 3 shows how the measured correlation length ξ diverges as the system approaches the critical tempera- 10-3 10-2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Reduced temperature (T-T FIG. 3: Divergence of the correlation length ξ as a function of temperature. The red line is a fit of Eq 2 to the data with ν and Tc as free parameters. Plotted is one data set for a specific temporal offset t0. The error bars are ± SD, according to fits to Eq 1. They also reflect the scattering between different data sets. Inset: Double logarithmic plot of the same data. ture. Generally, an algebraic divergence of the correlation length is predicted. We fit our data with the power law according to Eq 2, leaving the value of Tc as a free fit parameter, which has a typical relative error of 5× 10−4. Therefore our analysis is independent of an exact cali- bration of both temperature and heating rate provided that the heating rate is constant. The resulting value for the critical exponent is ν = 0.67 ± 0.13. The value of the critical exponent is averaged over 30 temporal offsets 0 < t0 < ∆t of the analyzing time bin window and the error is the reduced χ2 error. Systematic errors on the value of ν could be introduced by the detector response function. We find the visibility for a pure Bose-Einstein condensate to be 100% with a statistical error of 2% over the range of r investigated. This uncertainty of the visi- bility would amount to a systematic error of the critical exponent of 0.01 and is neglected as compared to the sta- tistical error. The weak singularity of the heat capacity near the λ-transition [1] results in an error of ν of less than 0.01. Finite size effects are expected when the correlation length is large [16, 26] and they may lead to a slight un- derestimation of ν for our conditions. Moreover, the har- monic confining potential introduces a spatially varying density. The phase transition takes place at the center of the trap and non-perturbative fluctuations are thus ex- pected within a finite radius R [5]. Using the Ginzburg criterion as given in [14] we find R ≈ 10µm, whereas the rms size of the thermal cloud is 58µm. The longest dis- tance we probe in our experiment is 2µm which is well below this radius R. So far, in interacting systems the critical exponent ν has been determined for the homogeneous system. The λ-transition in liquid Helium is among the most accu- rately investigated systems at criticality. One expects to observe the same critical exponents despite the fact that the density differs by ten orders of magnitude. In the measurements with liquid Helium the critical exponent of the specific heat α has been measured in a spaceborne experiment [8]. Through the scaling relation α = 2− 3ν the value of the critical exponent ν ≃ 0.67 is inferred being in agreement with theoretical predictions [27, 28]. Alternatively, the exponent ζ ≃ 0.67 (which is related to the superfluid density ρs = |Ψ|2 instead of the order parameter Ψ) can be measured directly in second sound experiments in liquid Helium [6, 7]. Due to an argument by Josephson [29] it is believed that ν = ζ, however, a measurement of ν directly through the density matrix has so far been impossible with Helium. The long-range behaviour of the correlation function of a trapped Bose gas in the critical regime reveals the behavior of a phase transition in a weakly interacting sys- tem. Our measured value for the critical exponent does not coincide with that obtained by a simple mean-field model or with that of an ideal non-interacting Bose gas. The value is in good agreement with the expectations of renormalization group theory applied to a homogeneous gas of bosons and with measurements using strongly in- teracting superfluid Helium. [1] J. Zinn Justin, Quantum Field Theory and Critical Phe- nomena (Oxford University Press, 1996). [2] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë, D. Vau- therin, Phys. Rev. Lett. 83, 1703 (1999). [3] P. Arnold, G. Moore, Phys. Rev. Lett. 87, 120401 (2001). [4] V. A. Kashurnikov, N. V. Prokof’ev, B. V. Svistunov, Phys. Rev. Lett. 87, 120402 (2001). [5] P. Arnold, B. Tomasik, Phys. Rev. A 64, 053609 (2001). [6] L. S. Goldner, N. Mulders, G. Ahlers, J. Low Temp. Phys. 93, 131 (1993). [7] M. J. Adriaans, D. R. Swanson, J. A. Lipa, Physica B 194, 733 (1993). [8] J. A. Lipa, J. A. Nissen, D. A. Stricker, D. R. Swanson, T. C. P. Chui, Phys. Rev. B 68, 174518 (2003). [9] M. Naraschewski, R. J. Glauber, Phys. Rev. A 59, 4595 (1999). [10] K. Huang, Statistical Mechanics (John Wiley & Sons, 1987). [11] In the general d-dimensional case 〈Ψ†(r)Ψ(0)〉 ∝ r−(d−2+η) exp(−r/ξ) with η being a critical exponent. The calculated and measured values of η are on the order of 10−2 and therefore we neglect η in our analysis. [12] For a noninteracting gas the critical exponent can be calculated to be ν = 1 for homogeneous systems and ν = 1/2 for harmonically trapped systems. [13] L. Onsager, Phys. Rev. 65, 117 (1944). [14] S. Giorgini, L. P. Pitaevskii, S. Stringari, Phys. Rev. A 54, R4633 (1996). [15] The numerical coefficient may be different for a trapped gas and has been omitted in reference [16]. [16] K. Damle, T. Senthil, S. N. Majumdar, S. Sachdev, Eu- rophys. Lett 36, 7 (1996). [17] N. Prokof’ev, O. Ruebenacker, B. Svistunov, Phys. Rev. A 69, 053625 (2004). [18] Q. Niu, I. Carusotto, A. B. Kuklov, Phys. Rev. A 73, 053604 (2006). [19] T. Stöferle, H. Moritz, C. Schori, M. Köhl, T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004). [20] B. Paredes et al., Nature 429, 277 (2004). [21] T. Kinoshita, T. Wenger, D. S. Weiss, Science 305, 1125 (2004). [22] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, J. Dalibard, Nature 441, 1118 (2006). [23] I. Bloch, T. W. Hänsch, T. Esslinger, Nature 403, 166 (2000). [24] A. Öttl, S. Ritter, M. Köhl, T. Esslinger, Rev. Sci. In- strum. 77, 063118 (2006). [25] T. Bourdel et al., Phys. Rev. A 73, 043602 (2006). [26] J. A. Lipa et al., Phys. Rev. Lett. 84, 4894 (2000). [27] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, Phys. Rev. B 63, 214503 (2001). [28] E. Burovski, J. Machta, N. Prokof’ev, B. Svistunov, Phys. Rev. B 74, 132502 (2006). [29] B. D. Josephson, Phys. Lett. 21, 608 (1966). [30] We would like to thank G. Blatter, F. Brennecke, A. Kuklov, G. Shlyapnikov, M. Troyer, and W. Zwerger for insightful discussions, SEP Information Sciences, OLAQUI and QSIT for funding. T.B. acknowledges funding by an EU Marie Curie fellowship under contract MEIF-CT-2005-023612.

0704.1440

Tiling models for metadislocations in AlPdMn approximants

7 Tiling models for metadislocations in AlPdMn approximants M. Engel∗ and H.-R. Trebin Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany November 28, 2018 Abstract The AlPdMn quasicrystal approximants ξ, ξ′, and ξ′n of the 1.6 nm decagonal phase and R, T , and Tn of the 1.2 nm decagonal phase can be viewed as arrange- ments of cluster columns on two-dimensional tilings. We substitute the tiles by Penrose rhombs and show, that alternative tilings can be constructed by a simple cut and projection formalism in three dimensional hyperspace. It follows that in the approximants there is a phasonic degree of freedom, whose excitation results in the reshuffling of the clusters. We apply the tiling model for metadislocations, which are special textures of partial dislocations. 1 Introduction A quasiperiodic structure can be described as a cut through a hyperlattice decorated with atomic surfaces. As a consequence the atoms are arranged in a finite number of different local environments frequently leading to a substructure of highly symmetric clusters. The cluster positions can again be modelled by a simpler decoration consisting in the simplest case of exactly one atomic surface (for the cluster centre) per hyperlattice unit cell. The displacement of the cut space (phasonic displacement) is a discrete degree of freedom, called phasonic degree of freedom. It can be excited locally, leading to a rearrangement of the clusters by correlated atomic jumps. This view is supported by recent diffraction data of coherent phason modes in i(cosahedral)-AlPdMn (Coddens et al. 1999, Francoual et al. 2003) and by in situ observations of phason jumps via high-resolution transmission electron microscopy (HRTEM) in d(ecagonal)-AlCuCo (Edagawa et al. 2002). The cut formalism cannot be applied directly if there is a gradient in the phasonic displacement. This is examplified by the extreme case of a cut space running inbetween ∗Author for correspondence. Email: [email protected] http://arxiv.org/abs/0704.1440v1 all atomic surfaces and not touching any of them. Hence we resort to another method: We substitute the atomic surfaces by atomic hypervolumes (Engel and Trebin 2005) of the same dimension as the hyperspace. For the construction two different spaces are needed: Those atomic hypervolumes that are cut by the (possibly deformed) cut space ECut are selected. The centre of each selected atomic hypervolume is projected onto the projection space EProj. By shearing the cut space, i.e. by introducing a linear phasonic displacement, periodic approximants are created. Phasonic degrees of freedom also can exist in these and play a fundamental role for phase transitions (Edagawa et al. 2004). Here we discuss linear defects, metadislocations, in the phasonic degree of freedom for approximants of the AlPdMn system. This system is especially adequate for the exam- ination of phasonic degrees of freedom since a stable i-phase, a stable 1.2 nm d-phase, a metastable 1.6 nm d-phase (which is assumed to be a solid solution of Mn in d-AlPd (Steurer 2004)), and a large variety of approximants have been observed in the phase diagramm (Klein et al. 2000). All of them, as well as several binary AlPd and AlMn quasicrystals and approximants are related structurally. 2 Tiling models A hyperspace model for i-AlPdMn has been proposed by Katz and Gratias (1993). It uses a six dimensional face-centred hyperlattice F 6D with lattice constant 2l6D = 1.29 nm and serves as a starting point since to our knowledge all newer, more complicated hyperspace models are refinements. It has been shown, that the approximants ξ and ξ′ (Beraha et al. 1997) of the 1.6 nm d-phase and the approximants R and T (Beraha et al. 1998) of the 1.2 nm d-phase are described on an atomistic level by a shear in this model. However the authors had to introduce an additional mirror symmetry to assure full tenfold symmetry. The main building units are Mackay-type clusters, whose centres are projected from the hyperspace by using one atomic surface per hyperlattice unit cell. It is a subset of a triacontahedron, deflated by τ = 1 5+1) with respect to the canonical triacontahedron, which is the projection of the hypercube with edge length l6D on the orthogonal complement of EProj. The distance of the cluster centres is the shortest projection of neighboring F sites ei ± ej multiplied by the deflation factor: t6D = 15 τ + 2l6D ≃ 0.78 nm. The relation of the approximants to the i-phase is given by two consecutive shears of ECut in the hyperspace: The first shear changes the cluster arrangement in direction of a fivefold axis e1. Together with the introduction of a mirror plane this results in a decagonal quasicrystal. The clusters are then aligned in columns parallel to the tenfold axis, so that the structures can be described by two-dimensional tilings, which are the projections in the column direction. The second shear rearranges the columns perpendicular to e1. We will now consider the tilings for the 1.6 nm and the 1.2 nm phases separately. Neighboring clusters in the 1.6 nm d-phase and its approximants lie on planes per- pendicular to e1. Thus the tile length is t 6D as defined above. The ξ- and the ξ′-phase are built from flattened hexagons (H) arranged in parallel and in alternated orientation respectively. By introducing additional rows of pentagons (P) and nonagons (N) between PSfrag replacements 1.6 nm d-phase 1.2 nm d-phase D U N H P D’ S’ B’ H’ AR AR’OR OR’ Figure 1: Calculated tilings for various approximants of the AlPdMn i-phase. (a) 1.6 nm phases: The tiling of the d-phase is the Tübingen Triangle Tiling (TTT). (b) 1.2 nm phases: The D’ centers lie on a τ 2 inflated TTT. (c) The tiles can be substituted with Penrose rhomb tiles. The substitution is different for the 1.6 nm phases and the 1.2 nm phases. each n− 1 rows of alternated Hs in the ξ′-phase we obtain the ξ′n-phases. Such a PN-row is called a phason plane, which is justified since it is elongated in the e1 direction. As we will see, phason planes play an important role for the cluster reshuffling resulting from an excitation of the phasonic degree of freedom. Further tiles, the decagon (D) and an U-shaped tile (U) are observed in the 1.6 nm d-phase (Fig. 1 (a)). Neighboring clusters in the 1.2 nm d-phase and its approximants lie on two planes staggered perpendicular to e1 with distance 10l6D ≃ 0.41 nm. The resulting tile length is t′6D = 1 10τl6D ≃ 0.66 nm. The R- and the T -phase are built from elongated hexagons (H’) arranged in parallel and in alternated orientation respectively. The Tn-phases are created by introducing into the T -phase additional rows of boat-shaped tiles (B’) between each n rows of alternated H’s. A B’-row is again called a phason-plane. For the 1.2 nm d-phase additionally a decagon (D’) and a star-shaped tile (S’) are needed (Fig. 1 (b)). By substituting the H, P, N, H’, and B’ tiles with acute rhombs (AR), (AR’) and obtuse rhombs (OR), (OR’) as shown in Fig. 1 (c) we obtain new tilings for the approximants, which can be interpreted as approximants of the Penrose-tiling. An H is substituted by an OR, while a phason plane corresponds to a combination of an AR-row and an OR- row. So the ξ′n-phase has n OR-rows inbetween neighboring AR-rows. Similarly the Tn- phase has n AR-rows inbetween neighboring phason planes, represented by OR-rows. The rhombs occuring in the new tilings for the Ξ-approximants (ξ, ξ′, ξ′n), as well as for the T - approximants (R, T , Tn) both only need three of the five basis vectors of the Penrose-tiling to be constructed. Therefore the tilings can be modelled in a simple three-dimensional hyperspace with the Z3-lattice and lattice constant l3D = τ τ + 2l6D ≃ 1.99 nm. The projection matrices are (si = sin(2π ), ci = cos(2π s0 s1 s4 c0 c1 c4 s0 s2 s3 c0 c2 c3 , (1) leading to an edge length of the tiles: t3D = 1 10l3D ≃ 1.26 nm. There is one atomic hypervolume per unit cell, which is just the unit cell, and one phasonic degree of freedom. This three-dimensional hyperspace is the simplest model for a phasonic degree freedom besides the Fibonacci-chain. 3 Metadislocations In the formalism of atomic hypervolumes a dislocation can be introduced into a tiling by a generalised Voltera process (Engel and Trebin 2005). It is uniquely characterised by a translation vector of the hyperlattice, the Burgers vector b (here: b3D = (b1, b2, b3), bi ∈ Z), that splits up into a phononic component b‖ = π‖b (deforming the tiles) and a phasonic component b⊥ (rearranging the tiles). The latter can only be calculated from the full six- dimensional Katz-Gratias model. If it is not zero, such a dislocation is a partial dislocation. By (i) extending the linear theory of elasticity to the hyperspace, (ii) approximating the phasonic degree as continuous, and (iii) assuming isotropy in the strain fields, the line PSfrag replacements 1.6 nm d-phase 1.2 nm d-phase Figure 2: Tilings of the ξ′3- (left) and the T3-phase (right) with m = 4 metadislocations. 2Fm new phason planes are inserted from the left, ending at the triangular shaped dislo- cation core. energy E of a dislocation is expressed as: E = cphon‖b‖‖2 + cphas‖b⊥‖2 + ccoupl‖b‖‖‖b⊥‖. (2) Besides a phononic contribution with material constant cphon and a phasonic contri- bution with cphas, a coupling term is present with ccoupl. According to experiment we assume cphon ≫ cphas ≈ ccoupl. Since stable dislocations are those with the lowest energy, we have to minimise ‖b‖‖. We will discuss this in parallel for dislocations in the Ξ- and the T -approximants. The minimization yields b2 = b3 in both cases. Furthermore we have b1 = −τ−1b2 for b3DΞ and b1 = τb2 for b3DT . Here we approximate τ−1 by the fractions Fm−1/Fm and τ by the fractions Fm+1/Fm respectively. (Fm)m∈N are the Fibonacci num- bers with start values F1 = F2 = 1. Finally the Burgers vectors of stable dislocations are: Ξ = (Fm−1,−Fm,−Fm) and b T = (Fm+1, Fm, Fm). Interestingly they correspond to the same six-dimensional Burgers vectors: Ξ = b T = (0, 0,−Fm−2, Fm−1, Fm−2, Fm−1). (3) Hence it suffices to consider both cases together for the rest of our calculations. The phononic component b‖ = b Ξ = b is perpendicular to the phason-planes (in the vertical direction in Fig. 1). We get ‖b‖‖ = τ−mt3D and ‖b⊥‖ = τm−3t3D. Substituting this into (2), we have E = [cphonτ −2m+3 + cphasτ 2m−3 + ccoupl] τ −3(t3D)2. There is a minimum for cphon/cphas = τ 4m−6. This determines the Burgers vector with lowest energy for given values for the material constants cphon and cphas. However it has to be noted that these are not necessarily identical for the Ξ- and the T -approximants. Tilings of the ξ′3- and the T3-phase with m = 4 dislocations have been calculated (Fig. 2). They show large rearrangements of the tiles due to the phasonic component b⊥ and negligible deformations of the tiles due to the smaller phononic component b‖. The dislocations are also dislocations in the metastructure of the phason planes. Therefore Klein et al. (1999), who discovered these dislocations in HRTEM images of the ξ′2-phase, named them metadislocations. The fact, that the experimentally most often observed metadislocations are those withm = 4 suggests cphon/cphas = τ 10 ≃ 123. We do not know of Burgers vector determinations or observations of metadislocations in the T -approximants, but dislocations with the Burgers vectors (3) are also the ones most often observed in the i-phase (Rosenfeld et al. 1995). 4 Discussion and conclusion In the ξ′n- and the Tn-phases the phasonic degree of freedom is related to the movement of the phason planes. Since there are no phason planes in the ξ-, ξ′-, R-, and T -phase, the phasonic degree of freedom cannot be excited locally. However metadislocations can exist in the ξ′- and the T -phase, but not in the ξ- and the R-phase. It can be shown, that there is no consistent way to introduce dislocations with phasonic components in the latter phases. A motion of the metadislocation (like the motion of any dislocation in a quasicrystal or large unit cell approximant) is necessarily accompanied by diffusion in the form of tile rearrangements. The motion is possible by climb in direction of the phason planes or by glide perpendicular to them. During the climb motion new phason planes are created (or dissolved) behind the dislocation core. A large number of metadislocations moving through the ξ′- or T -phase could even lead to a phase transformation to the ξ′n- or Tn-phases making the phasonic degree of freedom continously excitable. This is affirmed by HRTEM images of phase boundaries between the ξ′- and the ξ′2-phase formed by metadislcations (Heggen and Feuerbacher 2005). On the other side, glide motion does not change the number of phason planes. At least in the ξ′- and the T -phase glide motion seems unprobable, since the phason planes running out of the dislocation core would have to be dragged along, while climb motion only needs a re- construction of the tiling near the dislocation core. (Similar arguments leading to the same conclusion, as well as newer experimental work are presented in Feuerbacher and Heggen (2005).) We have to note, that there are no direct observations of metadislocation motion in approximants yet, although in the i-phase dislocations with identical Burgers vectors have been shown by in-situ observations to move by climb (Mompiou et al. 2004). References Beraha, L., Duneau, M., Klein, H., and Audier, M., 1997, Phil. Mag. A, 76, 587. Beraha, L., Duneau, M., Klein, H., and Audier, M., 1998, Phil. Mag. A, 78, 345. Coddens, G., Lyonnard, S., Hennion, B., and Calvayrac, Y., 1999, Phys. Rev. Lett., 83, 3226. Edagawa, K., Mandal, P., Hosono, K., Suzuki, K., and Takeuchi, S., 2004, Phys. Rev. B, 70, 184202. Edagawa, K., Suzuki, K., and Takeuchi, S., 2002, J. All. Comp., 342, 271. Engel, M., and Trebin, H.-R., 2005, Phil. Mag., in press. Feuerbacher, M., and Heggen, M., 2005, Phil. Mag., this volume. Francoual, S., Livet, F., de Boissieu, M., Yakhou, F., Bley, F., Letou- blon, A., Caudron, R., and Gastaldi, J., 2003, Phys. Rev. Lett., 91, 225501. Katz, A., and Gratias, G., 1993, Journ. Non-Cryst. Solids, 153-154, 187. Klein, H., Durand-Charre, M., and Audier, M., 2000, J. All. Comp., 296, 128. Klein, H., Feuerbacher, M., Schall, P., and Urban, K., 1999, Phys. Rev. Lett., 82, 3468. Heggen, M., and Feuerbacher, M., 2005, Phil. Mag., this volume. Mompiou, F., Caillard, D., and Feuerbacher, M., 2004, Phil. Mag., 84, 2777. Rosenfeld, R., Feuerbacher, M., Baufeld, B., Bartsch, M., Wollgarten, M., Hanke, G., Beyss, M., Messerschmidt, U., and Urban, K., 1995, Phil. Mag. Lett., 72, 375. Steurer, W., 2004, Z. Kristallogr., 219, 391. Introduction Tiling models Metadislocations Discussion and conclusion

0704.1441

Heisenberg antiferromagnet with anisotropic exchange on the Kagome lattice: Description of the magnetic properties of volborthite

arXiv:0704.1441v3 [cond-mat.str-el] 6 Aug 2007 Heisenberg antiferromagnet with anisotropic exchange on the kagomé lattice: Description of the magnetic properties of volborthite T. Yavors’kii,1 W. Apel,2 and H.-U. Everts3 Department of Physics and Astronomy, University of Waterloo, 200 University Avenue W, Waterloo, N2L 3G1, Canada. Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, D-30167 Hannover, Germany. (Dated: September 12, 2021) We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest- neighbour exchange couplings on the kagomé net, i.e., with coupling J in one lattice direction and couplings J ′ along the other two directions. For J/J ′ >∼ 1, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J ′ < 1/2 and a large manifold of canted spin states for J/J ′ > 1/2. To include quantum effects self-consistently, we investigate the Sp(N ) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N ) symmetric model depends on a parameter κ that measures the importance of quantum effects. Our numerical calculations reveal that in the κ-J/J ′ plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N ) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J ′ ≫ 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition. PACS numbers: 75.10.Jm,75.30.Kz,75.50.Ee I. INTRODUCTION In the ongoing search for novel states of condensed mat- ter, frustrated antiferromagnets have played a key role (for a recent review, see Ref. 1). Among the many sub- stances that have been investigated experimentally and the numerous spin models that have been studied theo- retically, those in which the magnetic ions occupy the ver- tices of corner-sharing frustrating entities have attracted particular attention in this context. The best known ex- amples are the kagomé antiferromagnet (KAF), consist- ing of corner sharing triangles, and the pyrochlore anti- ferromagnet, consisting of corner sharing tetrahedra (see Fig. 1). The main distinction between the KAF, the pyrochlore antiferromagnet and other frustrated and unfrustrated AAAA AAAA AAAAAAAA (a) (b) FIG. 1: kagomé lattice (a), pyrochlore lattice (b) magnets is the large ground-state degeneracy of the for- mer: classical Heisenberg antiferromagnets with nearest- neighbour interactions on corner-sharing lattices have a large ground-state degeneracy, which in the above two ex- amples even leads to a finite ground-state entropy (see, e.g., Ref. 2 and references therein). Quantum effects may lift this degeneracy, and, indeed, in numerical stud- ies of small cells of the spin 1 KAF, an exponentially large number of very low-lying quantum states has been observed3,4. It has been suggested that this abundance of low-lying states can be understood in a description of the low-energy physics of the quantumKAF as spin liquid consisting of nearest-neighbour spin singlets5,6. However, a complete picture of the ground state and of the excita- tions of the KAF is still missing. Further theoretical, but also experimental studies with emphasis on the quantum properties of the KAF are therefore highly desirable. In this last respect, the mineral volborthite is a very promis- ing candidate. It has been the subject of several recent experimental investigations7–10. The magnetic lattice of this natural antiferromagnet consists of the S = 1/2 spins of Cu2+ ions that are located on the vertices of well sep- arated planar kagomé-like nets. A monoclinic distortion of the lattice leads to a slight difference between the ex- change couplings along one lattice direction (J) and the two other directions (J ′)(see Fig. 2). Since neither signs of long-range order nor signs of a spin-gapped singlet ground-state were found in experiments on volborthite, the substance seems to be a good candidate for the ob- servation of the low-energy features that are thought to http://arxiv.org/abs/0704.1441v3 PSfrag replacements [i, j] 〈i, k FIG. 2: (Color online) Anisotropic kagomé model. The cou- pling J ′ and the nearest neighbour distance will be set equal to unity in the calculations. δ1(‖ êx), δ2, and δ3 are the three primitive lattice vectors of the kagomé net. be typical for kagomé type antiferromagnets1. Whether and to what extent the different exchange cou- plings along different lattice directions of the kagomé net of volborthite influence the low-energy physics of the sys- tem is presently unknown. In the present paper, we study this question on the basis of the model Hamiltonian HAKAF = J [i,j] SiSj + J 〈k,i〉 SkSi . (1) The symbols [i, j] and 〈k, i〉 denote, respectively, bonds between nearest-neighbour sites on the horizontal chains (a, b) and bonds between the middle sites (c) and the sites a, b, see Fig. 2. Since the physics of this model de- pends only on the ratio J/J ′ of the exchange constants, we set J ′ = 1 in the sequel. We will consider the spatially anisotropic kagomé antiferromagnet (AKAF), Eq. (1), in the full range of J , 0 < J < ∞ since this is of theoretical interest: one expects to see quantum phase transitions as J is increased. It is of particular interest to find out whether there is a transition from two-dimensional mag- netic states to a set of decoupled chains with free spins on the axes between the chains for large values of J . The paper is organised as follows. In Sec. II, we consider the model (1) in the classical limit. At this level, we find no sign of a transition from the two-dimensional magnet to a set of decoupled chains as J increases to infinity. Nonetheless, the ground-state degeneracy, as well as the spin wave spectrum are found to change qualitatively as the anisotropy of the model varies. In Sec. III, we consider a generalisation of the SU(2) symmetric model (1) to the Sp(N ) symmetric version11,12 and describe its properties in the large-N limit, where a mean-field treatment of the model is adequate. We obtain a de- tailed description of how possible ground states of the model depend on the coupling J and on the spin length S. A fairly rich phase diagram with a ferrimagnetic phase for small J , long-range ordered and short-ranged incommensurate phases for intermediate values of J , and a decoupled-chain phase for large J emerges. Parts of these results have been published previously, see Ref. 13. In Sec. IV, we devise trial quantum ground states of the original S = 1/2 model. We chose the states such that they are exact eigenstates of HAKAF, if the couplings on the upward pointing triangles of Fig. 2 are switched off, and we then treat these couplings perturbatively. In the limit J −→ ∞ this yields an effective Hamiltonian for the spins on the c sites which represents an anisotropic triangular antiferromagnet. The conclusions of Starykh and Balents14 about the ground state of this effective model lead us to conjecture the existence of a quantum phase transition in the AKAF for large J . In Sec. V, we summarise and discuss our results. In two Appendices, we present technical details of the counting procedure for the classical ground-states, and of the Ginzburg-Landau type procedure that allows us to determine the bound- aries in the phase diagram analytically. II. CLASSICAL AND SEMICLASSICAL ASPECTS Similar to other isotropic spin models on lattices with triangular elementary cells, the classical ground states of HAKAF, Eq. (1), are spin configurations, which satisfy the condition that for each elementary triangular plaquette of the lattice, Fig. 2, the energy is minimal. For J = 0, this yields a ferrimagnetic state with the chain spins aligned in one direction and the middle spins point- ing in the opposite direction, so that the total magneti- sation is M = N▽S (N▽: number of downward pointing triangles, N▽ = Ns/3 where Ns is the number of sites of the system). We illustrate this situation in Fig. 3. Ac- cording to the Lieb-Mattis theorem, the exact quantum ground state (GS) of the model HAKAF also has total spin Stot = N▽ S for J = 0, see Ref. 15, i.e., for J = 0, the quantum GS is ferrimagnetic too. By continuity, one expects the quantum GS to remain ferrimagnetic for suf- ficiently small finite J . This will be confirmed by our considerations of the large-N limit of the Sp(N ) version of our model (see the analytical and numerical work in Sects. III, III C and Appendix B) and by the block spin perturbation approach (Sec. IV). Classically, the ferri- FIG. 3: (Color online) Ferrimagnetic state for J = 0, i.e., when there is no coupling between chain spins, cf. Fig. 2. ωα(q) FIG. 4: (Color online) Spin-wave frequencies ωα(q), α = 1, 2, 3 for J = 0.4; the contour at the top of the plot marks half the Brillouin zone. magnetic state remains stable up to J = 1/2. The ex- citation spectrum of the ferrimagnetic state obtained in linear spin-wave (LSW) approximation is shown in Fig. 4. The analytic expressions for these three frequency sur- faces ωα(q), α = 1, 2, 3, are obtained as solutions of a third-order secular equation and are too lengthy to be presented here. However, one can easily assure oneself that the dispersion of the gapless mode is quadratic at the origin. Thus, one has the typical mode structure of a ferrimagnet here with one ferromagnetic mode and two optical modes, see, e.g., Ref. 16. As J increases towards 1/2, the ferromagnetic frequency surface looses its disper- sion and turns into a plane of zero modes, one zero mode for each wave vector in the magnetic Brillouin zone (BZ), at J = 1/2. The gap of the lower optical mode closes at this value of J in the centre of the BZ and the dispersion of this mode becomes linear for small wave vectors as for an antiferromagnetic spin-wave mode. At J = 1/2, the classical GS configuration changes from the unique ferrimagnetic state to an ensemble of degener- ate canted coplanar states. These states are characterised by two variables: the angle θ, which the middle spin of a given triangular plaquette forms with the two chain spins of the same plaquette (see Fig. 5), and the two valued chirality χ = ±1, which denotes the direction in which the spins turn as one moves around the plaquette in the mathematically positive sense. For J ≥ 1/2, the requirement that the energy of any of the elementary triangular plaquettes of the lattice Fig. 2 be minimal is θ = arccos(−1/(2J)), (θ > 0). The differ- ent degenerate canted states arise from different possibil- FIG. 5: (Color online) Canted spins of the AKAF at J > 1/2. ωα(q) FIG. 6: (Color online) Same as Fig.4 for J = 0.6. ities to assign positive or negative chiralities to the pla- quettes of the lattice. We show in the Appendix A that for the general case of θ 6= 2π/3 (J 6= 1), the number of spin configurations, NanisoGS does not grow exponentially with the number of sites. Rather, NanisoGS < 2 where α < 3. This implies that the ground-state en- tropy per spin of the classical AKAF vanishes in the thermodynamic limit. In this respect, the anisotropic model differs qualitatively from the isotropic KAF in the classical limit, which has an extensive entropy per spin. In the limit J → 1, the anisotropic model ap- proaches the isotropic KAF. Hence, one expects that for the anisotropic model there is an extensive number of low-lying excited states that become degenerate with the GS in the isotropic limit. As in the case of the isotropic KAF, the spin-wave Hamil- tonian is in linear order independent of the particular classical GS that has been chosen as the starting point of the expansion, Ref. 17. This implies that lowest-order quantum fluctuation do not select one or a group of clas- sical GSs as true GSs, i.e., the possible ordering effects of quantum fluctuations are not captured by the linear spin-wave (LSW) approximation. Figs. 6, 7 show the spin-wave frequency surfaces for J = 0.6 and for J = 3. It is easy to show analytically that, as is illustrated in these figures, the plane of zero frequency modes persists for all values of J greater than 1/2. The surfaces for J < 1/2 and for J > 1/2 join smoothly at J = 1/2. Thus, in the ωα(q) FIG. 7: (Color online) Same as Fig.4 for J = 3. LSW approximation, the transition from the ferrimag- netically ordered state to the canted spin states appears to be of second order. For J ≫ 1, the nonzero frequen- cies gradually loose their dispersion perpendicular to the strong-J direction and take the shape of the spin-wave spectrum of antiferromagnetic chains parallel to this di- rection. However, no sign of a further transition from the canted spin states to a set of decoupled spin chains is found in this semiclassical approach. In the next section, we will consider the symplectic Sp(N ) generalisation of the antiferromagnetic model HAKAF in the large-N limit. This approach, which was first proposed by Read and Sachdev, Refs. 11,12, as a method to study frustrated antiferromagnets, has the benefit of including the or- dering effects of quantum fluctuations self-consistently. It is of particular interest for spin models with two or more competing exchange couplings in the different lat- tice directions or over different lattice distances such as the present model, the J1-J2-J3 model 11, the Shastry- Sutherland antiferromagnet18 and the anisotropic trian- gular antiferromagnet19. For these models, it has pro- vided an unbiased selection of possible GSs that may or may not be ordered depending on the value of a param- eter κ, which is connected with the spin length S (see below). III. MEAN FIELD Sp(N ) APPROACH A. Brief review of the method For a general antiferromagnetic Heisenberg model with a positive interaction matrix Jij , Jij Si · Sj , (2) the Sp(N ) generalisation reads HSp(N ) = − 2N (J jβ)(Jγδb j) . (3) Here, . . . is the 2N × 2N generalisation of the 2× 2 antisymmetric tensor , (5) and bαi with α = 1, . . . , 2N are the Sp(N ) boson anni- hilation operators. (Here and in the sequel, we closely follow the notation of Ref. 12; in particular, summation over repeated upper and lower indices is implied.) Thus, J αβb†iαb jβ is the generalisation of the creation operator jβ for a singlet on the bond (i, j). For the special case N = 1, one finds (J αβb†iαb jβ)(Jγδb j) = −2Si · Sj + nbinbj/2 + δijnbi , where nbi = b i (7) is the boson number operator at site i and where Si = b i /2 (8) is the usual SU(2) spin operator at site i. (τ are the Pauli matrices). Then, if one imposes the constraint that the number of bosons is the same for all lattice sites, nbi ≡ nb, the HamiltonianHSp(1) is the familiar SU(2) in- variant antiferromagnetic Heisenberg Hamiltonian (plus some constants) with nb = 2S. In the subsequent exposition, we shall consider a Hamil- tonian of the form (3) in the large-N limit following the strategy of Refs. 11,12. Depending on the values of the couplings Jij and of κ, the GS of HSp(N ) may either break the global Sp(N ) symmetry and exhibit LRO or it may be Sp(N ) symmetric with only SRO. Breaking of the Sp(N ) symmetry will happen through condensation, i.e., by macroscopic occupation of one of the Bose fields bα. To allow for this, we introduce the parametrisation bmσi = b̃m̃σi with α = (mσ), m = 1, · · · ,N , m̃ = 2, · · · ,N and σ =↑, ↓. The field xσi is proportional to the condensate amplitude, 〈bmσi 〉 = N δm1 xσi . Aiming at a mean field treatment of the HamiltonianHSp(N ), which becomes ex- act in the large N limit, we decouple the quartic part by the Hubbard-Stratonovich technique with complex fields Qij = −Qji and with Lagrange multipliers λi that en- force the local constraints (7). The variables Qij which are defined on nearest neighbour bonds of the lattice are expectation values of the bond singlet creation opera- tors in the GS, Qij = 〈 σσ′ ε j mσ′〉 and are to be determined self-consistently from the mean field type Hamiltonian HMF = Jij |Qij |2 − Qijεσσ′ Nxσi xσ b̃m̃σi b̃ + h.c. N|xσi |2 + im̃σ b̃ i − nb The variational ground state energy, EMF, of HMF in the large-N limit is obtained by diagonalising the bosonic part of HMF, integrating over the 2(N − 1)Ns bosonic fields b̃m̃σi in the action associated with HMF. One ob- tains: Jij |Qij |2 − Qijεσσ′x j + h.c. ωµ(k;Q, λ) + |xσi |2 − 1− κ . (11) Here ωµ(k;Q, λ) are the positive eigenvalues of the bosonic part of HMF, and κ = nb/N is kept fixed in the limiting procedure11,12. The parameter κ is a measure for the importance of quantum fluctuations: by vary- ing κ from small to large values, one drives the system from the regime dominated by quantum fluctuations to the classical regime, i.e., from the disordered into the or- dered region. Finally, the GS is obtained by finding the saddlepoint of EMF in the space of the variables Qij and xσi subject to the constraints ∂EMF(Q, λ)/∂λi = 0 . (12) In addition to the GS itself, the spin-spin correlation function Gij = 〈Si ·Sj〉 in the GS is an important piece of information. In particular, by considering its behaviour in the limit |i − j| −→ ∞, one can distinguish between LRO and SRO. According to Sachdev12, to obtain Gij in the Sp(N ) symmetric approach, the SU(2) invariant ex- pression Si ·Sj must be replaced by the Sp(N ) invariant expression 4N 2 (b j − J αγJβδb j). (13) Within the mean field approach, Gij can then be calcu- lated straightforwardly. B. The anisotropic kagomé antiferromagnet 1. Choice of mean-field variables We wish to apply the procedure described above to the AKAF represented by the Hamiltonian (1). To render the problem of finding the eigenvalues ωµ in Eq. (10) and of optimising EMF tractable, we have to restrict the number of variables Qij and λi. We do so by demand- ing that the mean field Hamiltonian HMF for the spinon operators b(†) is symmetric under transformations of the projective symmetry group (PSG) that is related to the symmetry group of the spin Hamiltonian HAKAF (Eq. 1) (see Ref. 20). We include two translations, a rotation by π and a mirror axis orthogonal to the preferred direc- tion of the exchange constants (J). Thus generalising the treatment of Wang and Vishwanath to our model, we find eight mean-field states with different symmetries. Seven of them have flux in the sense of Ref. 21 in various cells of the lattice. Following the arguments in Ref. 21, we ex- clude all flux-carrying states and end up with the solution (cf. Fig. 8) P1,2,3 = Q1,2,3, Q3 = Q2, and λb = λa. PSfrag replacements Q1 FIG. 8: (Color online) Arrangement of mean field parameters: Q1 ≡ Qab, Q2 ≡ Qbc, and Q3 ≡ Qca denote the intra triangle bonds, P1 ≡ Qba′ , P2 ≡ Qcb′ and P3 ≡ Qac′ denote the inter triangle bonds. λa, λb, and λc are the Lagrange multipliers needed to implement the constraints on the sites a, b, and c. In order to check the flux-argument in Ref. 21, we have explicitely studied the solution P1,2,3 = −Q1,2,3 and found that it is always of higher energy (For J = 1, this agrees with the result of Ref. 12). Thus, the expression Eq. (11) can now be cast into the = J |Q1|2 + 2 |Q2|2 − (2λa + λc)(κ+ 1) ωµ(k) 1 + |xµ(k)|2 , (14) where the condensate is written in diagonalized form and ωµ(k) are the three positive solutions of det D̂(ω) = 0 . (15) Here, D̂(ω) = Λ̂− ωÎ Q̂ Q̂† Λ̂+ ωÎ , (16) Λ̂= diag(λa, λc, λa) , (17) 0 Q̃2(k) −JQ̃1(−k) −Q̃2(−k) 0 Q̃3(k) JQ̃1(k) −Q̃3(−k) 0  , (18) and Q̃a(k) = eiδak/2 − e−iδak/2 , a = 1, 2, 3, δ1,2,3, see Fig. 2. 2. Technical details of the numerical extremalisation Determination of the ground state of the AKAF in the considered approximation has been reduced to minimiza- tion of the Eq. (14) with respect to two variables Q1 and Q2, subject to the Lagrange constraints with respect to two parameters λa and λc. Being apparently trivial, the optimization procedure turns out to be quite involved technically. First, we find it crucial to consider at least two differ- ent chemical potentials. Other than for the spatially isotropic KAF, J = 1, we were not able to find a non- trivial solution if we used a single λ, λa = λb = λc. If λa and λc are different, [Λ̂, Q̂] 6= 0, the Lagrange multipliers enter the expressions for the frequencies ωµ non-trivially, other than in the case of a global uniform chemical po- tential (cf. Ref. 12). In turn, the Lagrange constraints cannot be satisfied semi-analytically, and require a nu- merical treatment. Second, we choose to work directly in the thermodynamic limit Ns → ∞ of the model (14) by performing a numerical self-adapting integration over the BZ. In this limit, the singularities can be integrated, and symmetry breaking is signalled by the appearance of a finite value of the condensate amplitude xµ(k) at a cer- tain wavevector k = qord, which characterises the type of magnetic order. We mention here that the extremalisa- tion of a mean-field energy of the type of Eq. (14) can also be achieved by solving the pertinent stationarity condi- tions numerically for finite systems, i.e., for finite Ns, see e.g., Ref. 22. Then, the type of magnetic order has to be detected by calculating the structure factor. Third, we see that the Eq. (14) has a minimum with respect to the physical bond parameters Q1 and Q2 only after the elimination of the chemical potentials. In the full Q− λ space we face an extremalization problem. Technically, we find it convenient to use a polar coordi- nate parametrisation for the variablesQ1, Q2 and λa, λc: Q1 = Q cos(α), Q2 = Q sin(α) , (20) λa = Λ sin(β), λc = Λcos(β) . (21) We perform an optimization with respect to the variables Q,Λ, α, β, as well as condensate densities xµ(k) in accord with the following algorithm (J and κ are kept fixed). i. We fix the angles α, β and the amplitude Q, and first exploit the stationarity condition for EMF with re- spect to Λ. It is convenient to write the corresponding equation in the following form: [2 sin(β) + cos(β)] (κ+ 1) |xµ(k)|2 ∂Λωµ(k) ∂Λ ωµ(k) , (22) where Ω = 8π2/ 3 is the volume of the unit cell. One finds that Q and Λ enter the Eq. (22) only via the ratio ξ = Λ/Q. The requirement that the frequencies must be positive, ωµ(k) ≥ 0, defines a lower limit ξmin(α, β) for ξ: the frequencies ωµ(k) are positive for ξ > ξmin(α, β); for ξ = ξmin(α, β), the lowest mode ωµ0 vanishes at some point(s) k0 in the BZ. When this happens, the corre- sponding condensate density xµ0(k0) can be put non- zero, if this is necessary to satisfy Eq. (22). It is im- portant to note that in order to determine the actual value of ξmin(α, β) (as well as those of Q, α and β) it suffices to only consider Eq. (22) at xµ(k) = 0, irrespec- tive of whether there is condensate, ωµ0(k0) = 0, or not, ωµ(k) 6= 0 for all k, µ. We solve the Eq. (22) for ξ numerically in two steps. First, we determine ξmin(α, β): we decrease ξ from large positive values until the condition ωµ0(k0) = 0 signals that ξ = ξmin(α, β). Second, we set xµ(k) ≡ 0 and at- tempt to satisfy Eq. (22) in the interval ξ ≥ ξmin(α, β). To this end, we set Λ = ξQ in Eq. (14) and vary ξ to de- termine the extremum of EMF (i.e., Eq. (22)). We find that the extremum is a maximum. If this maximum oc- curs for some ξ > ξmin(α, β), then Eq. (22) is satisfied with xµ(k) = 0. If, however, EMF(α, β, ξQ,Q) decreases monotonously as we lower ξ down to ξ = ξmin(α, β) , then the Eq. (22) cannot be solved with xµ(k) = 0. In this case, a finite condensate density xµ0 (k0) 6= 0, is re- quired, in order to “compensate” for too large a value of the lhs. of Eq. (22). This fixes both ξ = ξmin(α, β) and the value xµ0 (k0) (cf. sects III B and IV B of Ref. 12). ii. Having determined the value of ξ, we notice that the function EMF(α, β,Λ, Q) is quadratic in Q and bounded from below, which allows an analytical deter- mination of Q as the position of the minimum. iii. Finally, knowing the values of Λ and Q, we pro- ceed by a numerical extremalization of EMF with respect to the angles. The calculations show that EMF as a func- tion of the angle β possesses a maximum, and a minimum as a function of the angle α after β has been eliminated. Thus, the variational energy EMF is bounded from below in the variables Q1 and Q2, as expected. iv. We iterate this procedure (i)-(iii) until conver- gence is achieved. C. Numerical results of the Sp(N) formalism The results of the Sp(N ) approach in the large-N limit are summarised in the zero temperature phase diagram of the AKAF, Fig. 9. The central part of the phase dia- gram is occupied by the incommensurate (IC) phase with LRO at sufficiently small 1/κ. The phase boundary that separates the region with SRO from the region with LRO was found by checking whether for a given pair of J and 1/κ the lowest branch of the one spinon spectrum ωµ(k) has zeros in the BZ or not, i.e., whether there will be condensate at one or several points in the Brillouin zone or not. As one might expect, LRO is maximally sup- pressed by quantum fluctuations for J = 1, which is the case of maximal frustration. For J = 0, the exact quantum ground state of the AKAF is ferrimagnetic (FM) according to the Lieb- Mattis theorem15. In this state, the expectation value Q1 which measures the singlet weight on the horizontal bonds vanishes. As shown in Fig. 10, our Sp(N ) calcu- lations recover this exact result and extend it to a finite 0 0.2 0.4 0.6 0.8 1 PSfrag replacements J/(J + 1) IC/SRO IC/LRO DC/SRO FIG. 9: (Color online) Phase diagram of HAKAF as obtained in the Sp(N ) approach. Symbols and lines, respectively, de- note numerical and analytical results for the phase boundaries (see text, Subsec. III B 2 and Appendix B). Quantum fluctu- ations increase along the vertical axis. LRO: Long Range Order; SRO: short range order; FM: ferrimagnet; IC: incom- mensurate phase; DC: decoupled chains. At J = 1, the results of Ref. 12 are recovered. Incommensurate order (see Fig. 12) occurs between the boundaries of the ferrimagnetic phase (×) and of the decoupled chain phase (∗). 1/κ=1 1/κ=1.4 1/κ=3 1/κ=5 1/κ=7 1/κ=10 0 0.2 0.4 0.6 0.8 1 PSfrag replacements J/(J + 1) FIG. 10: (Color online) Mean field parameter Q1 as function of the anisotropy. interval 0 ≤ J ≤ Jferri(κ), which narrows as 1/κ in- creases. The parameter Q2, which measures the singlet weight on the diagonal bonds, is independent of J in this interval; its value decreases as 1/κ increases (see Fig. 11). Remarkably, the FM state retains its LRO in its entire region of existence. As J is increased beyond Jferri(κ), Q1 increases in the manner of an order parameter at a second order phase transition. At the same time, the parameter Q2 begins to decrease, and eventuallly it drops to zero at some J = JDC(κ). Thus, the large-N approach predicts the existence of a decoupled-chain phase in the region above the phase boundary JDC(κ). Q2 decreases to zero con- tinuously so that the phase transition at JDC(κ) appears to be of second order again. Both LRO and SRO phases may be characterised by an ordering wave vector qord = 2kmin, where kmin is that wave vector at which the one-spinon excitation spectrum ωµ(k) has its minimum. The static spin structure factor S(q) develops a peak at qord. In Fig. 12, we display the x-component of the ordering vector qxord = q ord(J) 1/κ=1 1/κ=1.4 1/κ=3 1/κ=5 1/κ=7 1/κ=10 0 0.2 0.4 0.6 0.8 1 PSfrag replacements J/(J + 1) FIG. 11: (Color online) Mean field parameter Q2 as function of the anisotropy. 1/κ=1 1/κ=1.4 1/κ=3 1/κ=5 1/κ=7 1/κ=10 0 0.2 0.4 0.6 0.8 1 PSfrag replacements J/(J + 1) FIG. 12: (Color online) Ordering wave vector qxord as function of the anisotropy. ord = 0). At the kagomé point J = 1, |qxord| = 4π/3 is independent of the value of κ. For 1/κ <∼ 3, the behaviour of qxord as a function of J is as expected: as J increases, it increases monotonously until the phase boundary JDC(κ) is reached and remains constant inside the DC phase. However, for 1/κ >∼ 3 the function qordx (J) develops a minimum at J ≈ 1.5, which becomes more pronounced as 1/κ increases. In Sec. III B 2 we emphasised that contrary to previ- ous applications of the large-N approach to spin models on kagomé and anisotropic triangular lattices11,12,19, we found it essential to consider two chemical potentials λa and λc here, one for the spins on the horizontal lattice lines (λa) and one for the middle spins (λc). We display the values of these parameters as functions of J in Fig. 13. We have no physical explanation for the behaviour of λa, λc as functions of J and κ but it is gratifying to see that λa = λc at J = 1 independent of κ in accordance with earlier work12. As indicated above, along with numerical study of Eq. 11, we performed extensive analytical calculations, both to 1/κ=1 1/κ=1.4 1/κ=5 0 0.2 0.4 0.6 0.8 1 PSfrag replacements J/(J + 1) FIG. 13: (Color online) Lagrange multipliers λa, λc (chemical potentials) as functions of the anisotropy. corroborate the numerics and to obtain additional insight into the problem. Details of the analytical techniques are presented in Appendix B. Here we state that we were able to analytically determine Sp(N ) phase boundaries between the SRO and LRO DC phase, between the DC and IC phase, and between the FM and IC phase, see Fig. 9. Moreover, our analytical calculations allowed us to explicitly confirm the existence of LRO inside the FM phase and immediately to the right of the FM-IC phase boundary. Likewise, the regions with SRO and LRO in- side and immediately to the left of IC-DC phase bound- ary were determined analytically. This was achieved by evaluating in these regions the Sp(N ) generalisation of the spin-spin correlation function 〈Si,u ·Sj,v〉 of the model defined by expression (13). (u, v = a, b, c denote the sites of the triangular cells i and j of the model, see Fig. 8). On the right-hand side of the FM-IC boundary and in- side the FM phase, we find for large distances between the cells, |rj − ri| ≫ 1, 〈Si,u · Sj,v〉 ∼ Su Sv , (23a) where |x3(kmin)|2 N▽ (λc + λa) w = a, b −λa w = c (23b) and u, v = a, b, c denote the sites of the triangu- lar cells i and j of the model, see Fig. 8. Here, |x3(kmin)|2/N▽ is the condensate density at kmin = (−π, 0), |x3(kmin)|2/N▽ = κ, see Eq. (B8). On the FM-IC transition line and inside the FM phase, where Eqs. (23) are valid, the parameters λa and λc are not independent but can be expressed in terms of a single parameter δ, see Eqs. (B4), (B12). The sign pattern on the right-hand side of Eq. (23) and the ordering wave vector qord = 2kmin = (−2π, 0) are indeed the proper- ties one expects to find for the long-distance behaviour of the spin-spin correlation function of a ferrimagneti- cally ordered state. Since |x3(kmin)|2/N▽ remains finite for arbitrarily small values of κ, the mean-field Sp(N ) approach predicts that this order persists in the ex- treme quantum limit of our model, 1/κ ≫ 1. Together with Eqs. (23), the fact that the condensate density |x3(kmin)|2/N▽ remains constant inside the FM region, see Eq. (B8), implies that the magnetisation of the FM phase remains constant up to the FM-IC phase boundary. The same behaviour of the magnetisation of a ferrimag- netic phase has previously been observed in an exact- diagonalisation study of a one-dimensional kagomé-like antiferromagnet23. At the FM-IC phase boundary, the magnetisation becomes spatially modulated with an in- commensurate wave vector qord = 2kmin. On the left-hand side of the IC-DC boundary and in- side the DC phase we find the following large distance behaviour of the spin-spin correlation function: 〈Si,c · Sj,c〉∼ cos [2kmin (ri − rj)] 1 + q21 |x3(kmin)|2 + |x3(−kmin)|2 3 (kmin) ,(24a) 〈Si,u · Sj,v〉∼ 0 for u, v 6= c, c . (24b) Here, q1 and λa denote the saddle-point values of these variables obtained from Eqs. (B31), (B32). q2 is a func- tion of q1, determined by Eq. (B36) or by Eq. (B44) depending on whether 1/κ < 1/κs or 1/κ > 1/κs (κs = 0.181, see Fig. 22). ω 3 (kmin) is the value of the second-order expansion coefficient of the lowest spinon frequency ω3(k), cf. Eqs. (B20), (B21c), at its min- imum, and 2kmin is the ordering wave vector imme- diately to the left on the IC-DC phase boundary and inside the DC phase; it is determined by Eq. (B35). |x3(kmin)|2/N▽ = |x3(−kmin)|2/N▽ are the condensate densities at the wave vectors ±kmin. As is shown in Ap- pendix B, ω 3 (kmin) remains finite for 1/κ > 1/κs and hence |x3(±kmin)|2/N▽ vanishes. Thus, 〈Si,c ·Sj,c〉 ∼ 0, i.e., there is no LRO in this region. By contrast, for 1/κ < 1/κs both, |x3(±kmin)|2/N▽ and ω̄(2)3 (kmin) van- ish when the IC-DC phase boundary is approached from the left. However their ratio, which determines the spin- spin-correlation function, Eqs. (24), remains finite in this limit according to Eq. (B42). Thus, for 1/κ < 1/κs, Eqs. (24) show that while the chain spins Sa, Sb remain disordered, there is long-range IC order between the mid- dle spins Sc along the IC-DC phase boundary and inside the DC phase for sufficiently large κ. The middle spins occupy the sites of a triangular lattice. Remarkably, the correlations between these spins predicted by Eqs. (24) are compatible with the spin pattern Sj,c = S [cos(2kminrj)êx + sin(2kminrj)êy] (25) that would obtain if the middle spins Sc were classical spins coupled by a classical Heisenberg model with ex- change constant J̃ along one lattice direction and cou- plings J̃ ′ along the other two directions with a ratio J̃ ′/J̃ such that incommensurate order with wave vector 2kmin would be established. This persistence of long range or- der in the DC phase of the AKAF distinguishes our result from the result obtained by Chung et al., Ref. 19, in their large-N Sp(N ) treatment of the anisotropic triangular antiferromagnet: there the DC phase consists of uncor- related linear spin chains. Qualitative considerations of the finite-N corrections to the mean-field Sp(N ) result led the authors of Ref. 19 to the conclusion that instead of the DC phase there is spin-Peierls order in the large–J region of their model. In the next section, we will present a different approach, a block-spin perturbation theory, to get further insight into the properties of the AKAF for the physical spin-1/2 case. FIG. 14: (Color online) The kagomé lattice as a triangular lattice of downward pointing triangles. The coupling strength is J on the horizontal bond and unity on the other two bonds. IV. BLOCK-SPIN PERTURBATION APPROACH The basic idea of the block-spin perturbation theory is to calculate the states of small clusters of a given lattice exactly and to treat the coupling between these clusters perturbatively. The basic building blocks of the kagomé lattice are triangles. Thus it is natural to consider the trimerised kagomé lattice in which the spins on the down- ward pointing triangles are assumed to be strongly cou- pled whereas the coupling on the bonds of the upward pointing triangles are assumed to be weak, see Fig. 14. (Clearly, the exchange of the roles of the upward and the downward pointing triangles will not affect the fur- ther development to be presented in the current section.) The Hamiltonian for this trimerised model reads H(J, γ) = H▽(J) + γH△(J) , 0 ≤ γ ≤ 1 , (26) where H▽(J) (H△(J)) denote those terms in the Hamiltonian (Eq. 1) that act on the bonds of the down- ward (upward) pointing triangles. We will determine approximate GSs of this trimerised model in different ranges of J in a perturbation expansion w.r.t. γ. The hope is that the results will provide some qualitative insight into the GS properties of the non-trimerised model H(J, 1) which is our original model Eq. (1). The same strategy has previously been applied sucessfully to frustrated spin models by several authors5,24–26. Obviously, the GSs of the unperturbed Hamiltonian H(J, 0) are products of GSs of the individual downward pointing triangular plaquettes. The GSs of a single plaquette and the corresponding energies are 1. for J < 1: |α〉 = 1√ |↑↑↓〉 − |↓↑↑〉 |↑↓↑〉 − |↓↑↑〉 , (27a) PSfrag replacements |α(ᾱ)〉: ∣β(β̄) ↑ (↓)↑ (↓) ↑ (↓) FIG. 15: Ground-states of triangular plaquettes. Heavy lines depict singlets. The coupling strength is J on the horizontal bond and unity on the other two bonds. |ᾱ〉 = 1√ |↓↓↑〉 − |↑↓↓〉 |↓↑↓〉 − |↑↓↓〉 , (27b) εα = εᾱ = −1 + J/4 ; (27c) 2. J > 1: |β〉 = 1√ |↑↓↑〉 − |↑↑↓〉 , (28a) |↓↑↓〉 − |↓↓↑〉 , (28b) εβ = εβ̄ = −3/4J . (28c) Here, the ket vectors denote the spin state of the plaquette in the Sz basis. The three arrows inside the |cba〉 symbol denote from left to right the spin direction at the sites c, b and a of the plaquettes in Fig. 15. The states |α〉 (|ᾱ〉) and |β〉 ( ) have total z-spin 1/2 (−1/2). They can be depicted graphically as shown in Fig. 15. From these plaquette states, the zeroth order GSs of the Hamiltonian H(J, γ) will be constructed. We treat the cases J < 1 and J > 1 separately. 1. J < 1: Since the states |α〉, |ᾱ〉 are the GSs of the individual downward pointing plaquettes in this case, the states |A(M)〉 = iǫ{M} jǫ{N▽−M} |ᾱj〉 , (29) are here the zeroth order GSs of H(J, γ). The set {M} is a subset of M out of the N▽ downward pointing tri- angles of the 3N▽-site kagomé lattice; the subscripts i, j denote the position of individual triangular plaquettes in the lattice of these plaquettes which is also triangular, see Fig. 14. The zeroth order energy eigenvalues associated with the states |A(M)〉 do not depend on M : = N▽(−1 + J/4) . (30) Hence, there are in total 2N▽ degenerate zeroth order GSs |A(M)〉. The single plaquette states |α〉, |ᾱ〉 satisfy the conditions for the validity of the Lieb-Mattis theorem, Ref. 15: after a canonical transformation which rotates the spins on the sites a and b by π around the z-axis |↑〉 → i|↑〉, |↓〉 → −i|↓〉, and which leaves the spins on the site c fixed the coefficients of all basis states on the right sides of Eqs. (27a, 27b) become positive (+1/ 6). As a consequence, all the GSs |A(M)〉 satisfy the conditions for the validity of the Lieb-Mattis theorem. For J = 0 it follows from this theorem that the total magnetisation of the exact quantum GS |Φexact〉 of the Hamiltonian HAKAF must be an eigenstate of the total magnetisation m̂tot = (Szi,a + S i,b + S i,c) (31) with eigenvalue mtot = N▽/2, i.e., |Φexact〉 must be a ferrimagnetic state. By continuity, one expects this to be the case not only for J = 0, but up to a certain fi- nite value of J . This suggests that the state |A(M=0)〉, c.f. Eq (29), is the appropriate zeroth order GS in this case and that the degeneracy of the states |A(M)〉 is lifted by the perturbation H△ in favour of the state |A(0)〉. To confirm this, we determine the creation energy of a flipped plaquette in first order in γ, i.e., the difference of the energy of the state with one plaquette spin flipped relative to the ferrimagnetic state, and the energy of the ferrimagnetic state: δ(1)EA(M=1) = EA(1) − EA(0). (32) A simple calculation yields δ(1)EA(M=1) = γ(1− J) , (33) i.e., to first order, |A(M=0)〉, the ferrimagnetic GS is stable w.r.t. a flip of a single plaquette spin, as long as J < 1. As a further check on the stability of the state |A(M=0)〉, we calculate the dispersion of the excitation energy of a propagating single flipped plaquette spin. For this purpose, we need to determine the overlap matrix el- ements between the state with a flipped plaquette spin at the site j and states with a flipped spin at one of the neighbouring sites, tj,j±δ1 = 〈ᾱj |〈αj±δ1 |γJSjSj±δ1 |αj〉|ᾱj±δ1〉 = (34a) tj,j±δ2,3 = 〈ᾱj | αj±δ2,3 ∣γSjSj±δ2,3 |αj〉 ∣ᾱj±δ2,3 (34b) Here, δν , ν = 1, 2, 3, are the primitive lattice vectors of the kagomé net, see Fig. 2; they connect the sites of the plaquette lattice. Then, by diagonalising the ensuing transfer Hamiltonian Htrans = γ J ( |j + δ1〉〈j|+ |j − δ1〉〈j| ) (|j + δ2〉〈j|+ |j − δ2〉〈j|) (|j + δ3〉〈j|+ |j − δ3〉〈j|) ,(35) where |j〉 denotes the state with a flipped plaquette spin at site j, we obtain for the kinetic energy of this excita- tion: ε(k) = J cos(kx)− cos( ) cos( . (36) Adding the energy for the creation of a single flipped pla- quette spin, we find for the total energy of the excitation in the limit of small wave vector k ω(k) = − J)k2x + k2y +O(k4) . (37) Obviously, the ferrimagnetic state |A(M=0)〉 becomes unstable against a propagating flipped plaquette spin already at J = 1/4, i.e., much earlier than suggested by the excitation energy of a static flipped spin (see Eq. (33)). We remark that this bound is independent of the actual magnitude of the perturbation parameter γ and therefore, the qualitative result may survive in the limit γ = 1. 2. J > 1: In this region, the states |B(M)〉 = iǫ{M} jǫ{N▽−M} with eigenenergy = N▽(−3J/4) . (39) are the zeroth order eigenstates of H(J, γ). These states consist of free spins on the c-sites and of spin-singlet dimers that cover every second bond of the horizontal chains of the lattice. We wish to answer the question of whether the 2N▽-fold degeneracy of these states, which results from the degrees of freedom of the free spins, is lifted by the perturbation γH△; in other words, we want to find out whether the middle spins remain decoupled from the chain spins. We proceed as in case (i). We compare in a perturbation expansion w.r.t. γ the en- ergy of the state |B(0)〉 with the energy of |B(1)〉, i.e. with the state with one plaquette spin flipped relative to |B(0)〉. We denote this difference by δ(1)EB(M = 1) = EB(1) − EB(0). Surprisingly, we find that the matrix el- ements 〈B(M)|H△|B(M)〉 vanish for any choice of M . There is no first order correction to the energy E δ(1)EB(M = 1) = 0. Moreover, we observe that the off-diagonal matrix elements 〈B′(M)|H△|B(M)〉, where |B′(M)〉 and |B(M)〉 contain identical numbers of states but differ in their distribution over the N▽ down- ward pointing triangles, also vanish. This implies that, in contrast to case (i), a flipped plaquette spin cannot hop to a neighbouring site in a first order process. Coupling between the spins on the c-sites occurs only in second order in γ. It is succinctly described by an effective spin Hamiltonian for the c-site spins, which are at the same time total spins of the downward pointing plaquettes (see Fig. 15): Heff = Szi S Sxi S .(40) Here, Sαi , α = x, y, z, denote plaquette spin operators; i is the position of a downward pointing plaquette on the triangular lattice formed by these plaquettes. The exchange couplings J and J⊥ are given as second order matrix elements of H△: 〈Bi↑, i′↑|H△|X〉 〈X |H△|Bi↑, i′↑〉 2 εB − εX 〈Bi↑, i′↓|H△|Y 〉 〈Y |H△|Bi↑, i′↓〉 2 εB − εY (41a) 〈Bi↓, i′↑|H△|X〉 〈X |H△|Bi↑, i′↓〉 2 εB − εX , (41b) and i′ ≡ i+ δν . Here, the states |Biσ, i′σ′〉 are zeroth order GSs, Eq. (38), whose spin patterns are identical on all sites except for the sites i and i′ where the z- components of the spins take the values σ and σ′, re- spectively; |X〉 and |Y 〉 are excited states of H▽. Of course, since the SU(2) symmetry of the original Hamil- tonian H(J, γ) must be conserved in the derivation of Heff , the expressions Eqs. (41) must yield identical re- sults, J ≡ Jδν . Non-zero contributions to J and J⊥ are obtained if either the same term SiSi′ of H△ acts in both matrix elements of the numerators of Eq. (41) (two-block contributions) or the terms SiSk, SkSi′ act in the left and right elements, respectively, where the plaquette geometry must be as shown in Fig. 16 (three-block contributions). In contrast to the case of the isotropic KAF studied by Zhitomirsky26, the three-block PSfrag replacements FIG. 16: Configurations of ▽ blocks contributing to the in- terblock couplings J and J⊥δν . Double dashed lines indicate that the same term element of H△ acts twice between the ▽ blocks at sites i and i′ (see also text). contributions do not produce three-spin interactions in the present case. Rather, they contribute to the ex- change interactions J and J⊥ of the HamiltonianHeff , Eq. (40). The evaluation of the expressions (41) yields Jδ1= γ 1 + 1 1 +O(J−1) Jδ2 = Jδ3 = 1 + 1 1 + O(J−1) . (43) Obviously, these results are useful for J ≫ 1. There, Heff represents a spin 1/2 Heisenberg Hamiltonian on the triangular lattice of the c-sites with a coupling along the δ1 direction that is strong (O(γ2/J)) in com- parison to the couplings in the two other directions (O(γ2/J2)). This limiting case of the anisotropic trian- gular Heisenberg antiferromagnet (ATHAF) has recently been analysed by Starykh and Balents with field theo- retical methods14. These authors find that in the limit of strong anisotropy, K ≡ Jδ1/Jδ2 → ∞, the GS of the model Eq. (40) is a collinearly ordered antiferromag- net (CAF) as depicted in Fig. 17. Since the ordering wave vector qCAF = (π, π/2) of this phase does not evolve continuously from the ordering wave vector qIC of the incommensurate (IC) spiral phase of the ATHAF, FIG. 17: (Color online) Collinear antiferromagnetic state (CAF) on the triangular lattice14. FIG. 18: (Color online) Tentative ground-state of the anisotropic kagomé antiferromagnet in the limit J ≫ 1. Dou- ble lines: dimers between the spins on the end points. (qIC = (qx(K), 0) with −3π/2 ≤ qx(K) ≤ −π for 1/2 ≤ K ≤ ∞), they conclude that the IC phase and the CAF phase must be separated by a quantum phase transition. For the trimerised anisotropic kagomé model, Eq. 26, these results have the following implications: i) While in the limit of strong anisotropy J ≫ 1 there is long-range collinear antiferromagnetic order among the c-site spins, the a- and b-site spins are paired in singlets, see Fig. 18. ii) This picture of the GS of the trimerised anisotropic kagomé model Eq. (26) differs from the result obtained in the Sp(N ) approach insofar as for sufficiently large κ the Sp(N ) approach predicts long range IC order among the c-site spins up to arbitrarily large values of J . Thus, if the picture of a CAF phase for large anisotropy persists in the non-trimerised limit H(J, γ = 1) of the model Eq. 26, one would expect a quantum phase transition between the IC phase and the CAF phase of the AKAF similarly as for the ATHAF. In closing this section, we remark that the calculation that led to the effective Hamiltonian Heff , Eq. 40, i.e., to the coupling between the c spins in the strongly anisotropic limit, shows clearly that this coupling arises from quantum fluctuations of the a and b spins. V. SUMMARY AND DISCUSSION In this work, we have studied the ground state (GS) phase diagram of the quantum Heisenberg antiferromagnet on the kagomé lattice with spatially anisotropic exchange (AKAF). The model is relevant for a description of mag- netic properties of volborthite, which is a natural reali- sation of a spin 1/2 antiferromagnet consisting of weakly coupled slightly distorted kagomé layers. A small mono- clinic distortion along one of the three lattice directions causes the exchange coupling along this direction, J , to differ from the couplings in the other two directions, J ′, which we set equal to unity, cf. Fig. 2. We have inves- tigated the problem in the full range of the anisotropy, 0 ≤ J ≤ ∞, using three different approximate methods: the classical and semiclassical approach, the mean-field Sp(N ) approach, and a block-spin perturbation theory. The case J = 1 is the much studied isotropic kagomé antiferromagnet (KAF). Exact diagonalisation studies of this model3,4 are available. Their results speak conclu- sively in favour of a spin liquid ground state1. This view is supported by block-spin approaches5,6. Conflicting re- sults have been found in Refs. 27–31, where various va- lence bond crystal (VBC) states are proposed as ground states of the KAF. However, a recent comparison of the exact spectrum of the 36-site sample of the KAF against the excitation spectra allowed by the symmetries of these states, casts doubts on their validity32. Within the whole anisotropy range, the case J = 0 is spe- cial, since it allows for an exact characterisation of the quantum GS as ferrimagnetic (FM) with a total mag- netisation of M = S Ns/3 for a system of Ns spins of magnitude S. In the classical picture, this state corre- sponds to a unique staggered layout of spins with a non- zero net magnetisation of the lattice unit cell (cf. Fig. 3). In the classical limit, the ferrimagnetic ground state sur- vives up to J = 1/2. For J > 1/2, the “chain” spins (i.e., spins coupled by J) begin to tilt gradually towards the middle (remaining) spins (see Fig. 5). This allows for a formation of a large degenerate manifold of canted spin states. In contrast to the isotropic case J = 1, where the degeneracy grows exponentially with the system size Ns, its growth is weaker: 2 Ns for J 6= 1. This im- plies that there must be an increasingly large number of classical low energy configurations as J approaches unity. In the linear semiclassical approximation, the spin-wave spectrum has one zero-frequency mode for each point of the magnetic Brillouin zone (BZ). The spectrum is identical for the different canted states for all J > 1/2. Thus, in this order of the semiclassical approximation, no order-by-disorder mechanism appears that would se- lect one particular state or a particular group of states from the manifold of canted states as true ground states. In the limit J → ∞, the frequency spectrum of non-zero modes gradually takes the shape of the spectrum that one would expect for a set of uncoupled antiferromag- netic spin chains parallel to the strong-J direction. No qualitative change from the set of canted spin states to the set of decoupled chains at a finite value of J is found. We have further explored the nature of the phases at var- ious J exploiting the mean field (MF) Sp(N ) approach, that incorporates the effect of quantum fluctuations not only perturbatively, but self-consistently. The strength of quantum fluctuations is controlled by a parameter κ, which is the analogue of the spin value S in the origi- nal SU(2) symmetric model. In fact, for N = 1, when the Sp(1) symmetric model is equivalent to the SU(2) model, κ = 2S. For general N , this last identity does not hold, but κ is still a measure for the importance of quantum fluctuations that are strong for κ ≪ 1 and weak for κ ≫ 1. In the MF Sp(N ) approach, the nature of the phases that occur can be read from the values of the mean field parameters Q1 and Q2 and from the spec- trum of the bosonic spinon excitations. While the mean field parameters Q1 and Q2 (cf. Fig. 8) are the GS ex- pectation values of singlet bond operators, the structure of the spinon spectrum, ωµ(k;Q, λ), determines the ex- istence or non existence of long-range order (LRO): If the spectrum becomes gapless at some wavevector qord, a Bose condensate will form and a modulated structure with the wavevector 2qord will acquire LRO. As was to be expected, the phase diagram of the AKAF obtained by the MF Sp(N ) approach contains an incom- mensurate (IC) phase in the vicinity of the isotropic point J = 1 which is ordered for sufficiently large κ according to this approach, see Fig. 9. Qualitatively, we may gauge the value of κ against the spin length S by looking at the line J = 1 of the phase diagram which is the location of the Sp(N ) analogue of the isotropic SU(2) symmet- ric kagomé model: since, as we have argued above, the SU(2) model is disordered for S = 1/2, we may con- clude from Fig. 9 that the value of 1/κ that corresponds to S = 1/2 must be greater than two. Somewhat sur- prisingly, the FM phase remains long-range ordered for arbitrarily small κ. This may reflect the fact that in the SU(2) version of the model, the FM phase is ordered even for the smallest physical spin value S = 1/2. A new feature of the phase diagram is the prediction of a decou- pled chain (DC) phase for large enough J , which has no classical analogue. In this phase, the chains of strongly coupled a- and b-site spins show no magnetic order. The c-site spins which are interspersed between these chains and which occupy the sites of triangular sublattice are decoupled from the chain spins. However, they may or may not exhibit long range order among themselves de- pending on the magnitude of κ. Remarkably, the spin- spin correlations, whose asymptotics were obtained ana- lytically, are compatible with the spin-spin correlations of an anisotropic classical Heisenberg antiferromagnet on the triangular lattice whose exchange couplings differ in one direction from those in the other two directions. In order to tackle the problem of the GSs of the AKAF from a third corner, we have used a block-spin pertur- bation theory. This method has the advantage of being applicable directly to the spin 1/2 version of the model. In applying this approach, one has to initially group the spins of the model in clusters. For the kagomé lattice, it is natural to choose the spins around either the upward or the downward pointing triangles as clusters of strongly coupled units and to consider the coupling between these clusters, γ, as the small expansion parameter. Thus one trimerises the original model (see Fig. 14) and in so do- ing, one breaks the translational invariance of the original model. In the zeroth order of this expansion, two regions can be distinguished by the eigenenergies of the individ- ual trimers: J < 1 and J > 1. For sufficiently small J , one recovers the FM state as the GS in first order w.r.t. γ. For J > 1, there are no first order corrections to the en- ergy. Following an earlier application of the block-spin technique to the isotropic KAF26, we determine for J > 1 in second order in γ an effective HamiltonianHeff for the block-spins which can be identified as the middle spins of the original model and that occupy the sites of a trian- gular lattice. Heff is a Heisenberg Hamiltonian with a coupling Jδ1 of the order of γ 2/J along the δ1 direction (cf. Fig. 2) and couplings Jδ2 = Jδ3 of the order of γ along the other two directions. The calculations that lead to these results show clearly that the couplings between the c-spins of the AKAF are due to fluctuations of the sin- glets between the a- and b-spins into excited states. In a very recent field theoretical study, Starykh and Balents14 arrive at the conclusion that for Jδ1 ≫ Jδ2,δ3 , the ground state of the anisotropic triangular antiferromagnet rep- resented by Heff is a collinearly ordered spin state, see Fig. 17. Then, together with the singlet dimers between the a- and b-spins of the downward pointing triangles, the state depicted in Fig. 18 emerges as the candidate ground state of the AKAF in the limit of large anisotropy J ≫ 1: while nearest neighbour spins on the strongly coupled a-b chains form singlets and decouple magnetically from the spins on the c sites, the latter order in a collinear antifer- romagnetic structure. This structure cannot be obtained by a continuous deformation of the spiral IC structure that is predicted by the Sp(N ) approach and is believed to prevail for sufficiently large κ in the region of moder- ate anisotropy. As a consequence of the trimerisation, the state depicted in Fig. 18 breaks the translational symme- try of our original model, Eq. (1). If this state survives as the ground state of the non-trimerised model, i.e., when the expansion parameter γ approaches unity, then, ow- ing to their incompatible symmetries, the spiral IC phase and the large J phase of our model must be separated by a quantum phase transition. Acknowledgments One of the authors (HUE) acknowledges a useful discus- sion with F. Mila. The work at the University of Water- loo was supported by the Canada Research Chair (Tier I, Michel Gingras). We thank M. Gingras for a critical reading of the manuscript and numerous helpful sugges- tions. APPENDIX A: GROUND STATE DEGENERACY FOR GENERAL J We first derive the constraint on the chiralities that leads to the reduction in the number of degenerate ground states for general J relative to the special case J = 1. Let χ1, · · · , χ6 be the chiralities of the six triangles sur- rounding one of the hexagons of the kagomé lattice, and let φ1, · · · , φ6 denote the angles that define the direc- tions of the spin vectors on the six corners of the hexagon, see Fig. 19. Then, as is seen in Fig. 19, the following relations between the angle φ1, and the angles φ2 · · · , φ6 are an immediate consequence of these definitions: φ2=φ1 − θ χ2 , (A1a) φ3=φ1 − θ (χ2 + χ3) , (A1b) φ4=φ1 − θ (χ2 + χ3)− (2π−2θ) χ4 , (A1c) φ5=φ1 − θ (χ2 + χ3 + χ5)− (2π−2θ) χ4 , (A1d) φ6=φ1 − θ (χ2 + χ3 + χ5 + χ6)− (2π−2θ) χ4, (A1e) and φ6 = φ1 + (2π − 2θ)χ1 . (A1f) From the last two of these relations it follows that the chiralities χ1, · · · , χ6 are constrained by the sum rule χ2 + χ3 + χ5 + χ6 − 2χ1 − 2χ4 = 0 . (A2) For the isotropic kagomé system, J = 1, θ = 2π/3, one finds instead of the constraint (A2) the sum rule χj = n where n = 0, 1, 2 (A3) which is obviously less restrictive than (A2). Next, we present the arguments that lead to the estimate NanisoGS (N▽) N▽ with α < 3 (A4) for the number NanisoGS (N▽) of classical GSs of an anisotropic kagomé AF with N▽ downward pointing tri- angles (the number of sites is 3N▽). Any planar config- uration of a cell of the kagomé lattice can be constructed by decorating the successive rows of up and down point- ing triangles with chirality values χ = ±1 starting with the first row. We consider only square cells with rows with N▽ downward pointing triangles. Then, each row consists of 2 N▽ triangles, see Fig. 20. Obviously, there are 22 N▽ ways to decorate the first row. Disregarding certain exceptions, which will be dis- cussed below, one can, for a given configuration of the first row, choose the chirality of an arbitrary triangle of the second row to be either +1 or −1. After this choice has been made, the constraint (A2) fixes the chi- ralities of all the remaining triangles of the second row PSfrag replacements FIG. 19: Chiralities around heaxagonal plaquette FIG. 20: Example of a chirality distribution; dark and light shaded triangles represent positive and negative chirality, re- spectively. Chirality configurations in boxes fix the chirality distribution of the row above them uniquely. An empty cir- cle inside a triangle indicates that its chirality can be chosen freely to be positive or negative. uniquely. Proceeding in this manner from row to row one would generate 22 N▽ · 2 N▽ distributions of chi- ralities over the N▽ downward pointing triangles of the cell. For finite lattice cells, the requirement of periodic boundary conditions imposes further constraints on the number of possible chirality distributions in these cells, but the effect of these constraints will become negligible in the thermodynamic limit N▽ → ∞. However, there is a further reduction of the number of possible chirality distributions: For a given distribution in a row it is not always possible to find two distributions for the successive row which both satisfy the constraint (A2). If in a row the lower half of a hexagon of the next row is decorated by chiralities in the manner − + − or + − + (see boxes in Fig. 20), then the chiralities of the next row are fixed uniquely. This reduces the number of possible chirality distributions. Obviously, this reduction of the number of possible chirality distributions survives in the thermo- dynamic limit so that the exponent in (A4) is less than N▽, the value one would have expected without this reduction. We have calculated the number of distribu- tions for cells of up to N▽ = 13× 13 and have found the value α ≃ 2.18 for the constant in the expression (A4), see Fig. 21. As we have mentioned above, the sum rule (A3) which applies for the isotropic kagomé AF is less restrictive than the sum rule (A2). Consequently, the number of chirality distributions in the isotropic model33, N isoGS ∼ 1.18333N▽ (A5) is larger than in the anisotropic model. Since the tran- sition from the anisotropic model to the isotropic model happens through a continuous variation of the coupling constant J , there should be a continuous transition be- tween the numbers of GS configurations in these two cases. Presumably, this transition implies that the den- sity of low-energy states of the anisotropic model in- creases exponentially with an exponent ∼ N▽ so that for J → 1 a sufficient number of states collapses to the GS to bring about the transition between the laws (A4) and (A5). 2 4 6 8 10 12 14 linear fit minPSfrag replacements FIG. 21: (Color online) Number of chirality distributions, NanisoGS , of cells of up to N▽ = 13 × 13. Dotted line: min = N▽ ln 2 (lower bound); dashed line: max = 3 N▽ ln 2 (upper bound); full line: ln(NanisoGS ) = 0.65 + 2.18 N▽ ln 2 (linear fit to the numerical results). APPENDIX B: PHASE BOUNDARIES The FM phase and the DC phase are chacterised by the vanishing of the parameters Q1 and Q2, respec- tively. Our numerical results in section III C show that at the respective phase boundaries, Q1 and Q2 decrease to zero like order parameters at second order phase tran- sitions. This suggests that we expand the mean field energy EMF, Eq. (14), w.r.t. either Q1 or Q2 in the manner of a Landau-Ginzburg (LG) expansion and de- termine the phase boundaries and the properties of the FM and the DC phase from this expansion. We write EMF /(N▽N ) = e(α)LG(Qα) where LG(Qα) = eα + rα |Qα| 2 + gα |Qα|4 +O(|(Qα|6) . (B1) The coefficients eα, rα and gα are functions of the vari- ables κ and J , of the parameters λa, λc and of Q2, Q1 for α = 1, 2, respectively. The saddle point of e LG(Qα) w. r. t. λa, λc and Qβ, β 6= α, determines the physical values of these parameters. For e LG(Qα) to qualify as a bona fide Landau-Ginzburg energy describing a second order phase transition with Qα playing the role of an or- der parameter, the coefficients gα have to be positive at the saddle point. For g1, i.e., inside and on the bound- ary of the FM phase, this follows from the numerical result: Q1 is found to remain zero for all J ≤ JF (κ). By contrast, we have no numerical results for J ≥ JDC(κ), i.e., inside and on the boundary of the DC phase. There- fore, we need to show by analytic means that g2 > 0. 1. The FM phase and the FM-IC phase boundary Since, as we have just remarked, we know that g1 > 0, the remaining task is to determine the coefficients e1 and r1 of e LG. To this end, we have to expand the mean field energy EMF, Eq. (14), w.r.t. Q1 which amounts to expanding the frequencies ωµ(k) w.r.t. Q1. As can be in- ferred from the expressions (16), (18) the frequencies de- pend on Q1 only through the combination ε 2 = J2|Q̃1|2. Therefore, we write the expansion in the form ωµ(k; ε) = ω µ (k) + ε 2ω(1)µ (k) +O(ε4) ω̄(0)µ (k) + ε̄ 2ω̄(1)µ (k) +O(ε̄4) with λ+ = (λa + λc)/2 , ω̄ µ = ω µ /λ+ , ω̄ µ (k) = λ+ ∂ε2 ωµ(k; ε)|ε=0 and ε̄ = ε/λ+. Here, the introduction of the “dimensionless” quantities µ , ε̄ looks like an unneccessary complication but it will help to keep expressions further below simple. Setting Q1 = 0 in the matrix D̂(ω), Eq. (16), and solving Eq. (15) for ω we find 1 (k) = wF (k) + δ , (B3a) 2 (k) = 1− δ , (B3b) 3 (k) = wF (k) − δ . (B3c) δ = λ−/λ+ with λ− = (λc − λa)/2 (B4) wF (k) = 1− q̄22[sin2(s2/2) + sin2(s3/2)] (B5) q̄2 = |Q2|/λ+ (B6) and sa = δak , a = 2, 3 (see Fig. 8) . From our numerical results, Fig. 13, we know that λc > λa and hence δ > 0. Therefore, ω̄ 3 (k) < ω̄ 1,2(k), and hence, if the minimum of ω̄ 3 (k) vanishes at the point kmin in the Brillouin zone, ω̄ 1,2(kmin) will be fi- nite. Thus, since condensate can only occur when one of the frequencies ω̄ µ , µ = 1, 2, 3 vanishes there may be a finite condensate density |x3(kmin)|2, but the densities |x1|2 and |x2|2 will certainly be zero. With these remarks and with the above results for ω̄ µ we find from Eq. (14) e1/λ+ = 2λ+q̄ 2 − (3− δ)(κ+ 1) 1 (k) + ω̄ 2 (k) + ω̄ 3 (k) 3 (kmin) |x3(kmin)|2/N▽ . (B7) Stationarity of e1 w.r.t. λ−, λ+, and q̄ 2 (which is equiv- alent to stationarity w.r.t. λa, λc), and Q 2 requires the following three conditions to be fulfilled: ∂e1/∂λ− = 0 : |x3(kmin)|2 = κ ; (B8) ∂e1/∂λ+ = 0 : 2λ+q̄2 2 − 3 κ− 1 +E2(q̄2) + wF (kmin) = 0 ; ∂e1/∂q̄ 2 = 0 : 2λ+ − [K2(q̄2)−E2(q̄2)]− wF (kmin) = 0 ; (B10) with K2(q̄2) = ds3 wF (k) E2(q̄2) = ds3 wF (k) . (B11) According to Eq. (B8), condensate must be present in the FM region. This requires that ω̄ 3 (kmin) vanishes. From Eq. (B3c) it is seen that kmin = (−π, 0), so that 3 (kmin) = 0, if wF (kmin) = 1− 2q̄22 = . (B12) Within the FM region and on the FM-IC boundary (i.e., for Q1 = 0) the saddle-point values of q̄2, λ+, and λ− are then determined as functions of κ by the Eqs. (B9), (B10) and (B12). Remarkably, within this re- gion these quantities are independent of the value of the exchange constant J . The solution of these equations shows that 0 ≤ q̄2 ≤ 2/3 for 0 < κ < ∞, cf. Figs. 11, 13. The FM-IC phase boundary is the solution of r1(κ, J) = 0 (cf. Eq. (B1), where r1 = ∂e LG/∂Q (B13) with e LG (EMF) from Eq. (14). We obtain s2 + s3 Ω(1)(k) k→kmin s2 + s3 3 (k) (B14) with Ω(1)(k) = −ω̄(1)1 (k) − ω̄ 2 (k)− ω̄ 3 (k). To obtain the expansion coefficients ω̄ µ (k) which appear in the last equation, we solve Eq. (15) to first order in the expansion w.r.t. ε̄2. We find Ω(1)(k)= wF (k)(wF (k) + 1− 2δ) wF (k) + 1 4 sin2 (2wF (k) + 1− δ) (wF (k) + 1)(1− δ) (wF (k)2 − δ2) (B15) k→kmin s2 + s3 3 (k) = − 1 1 + δ 1− δ . (B16) With these results Eq. (B14) can, in the thermodynamic limit, be cast into the form − I3(q̄2) 1 + δ 1− δ = JF (κ) , (B17) where I3(q̄2) = ds3 2 sin Ω(1)(k) . (B18) Then, with q̄2 = q̄2(κ) and λ± = λ±(κ) as obtained from Eqs. (B9), (B10) and (B12), the condition r1 = 0 is an equation for the FM-IC phase boundary J = JF (κ) which yields the graph shown in Fig. 9. As we have mentioned above, inside the FM region, i.e., for J < JF (κ), the saddle-point values of the quantities q̄ and λ± and hence of Q2, λa, λc and |x3(kmin)| are independent of the exchange coupling J , i.e., they retain the values they attain on the FM-IC phase boundary, cf. Figs. 11, 2. The DC phase and the IC-DC phase boundary Proceeding in exact analogy to the development in the previous subsection we now expand EMF/(N▽ N ) in powers of |Q2|2. However, instead of working with the variables Q1, Q2, λa, λc we work with q1, Q2, λa, q2 here, where J |Q1| , (B19a) |Q2|√ . (B19b) The replacement of |Q1| is purely a matter of conve- nience. By contrast, the replacement of variables Q2, λc, which according to the numerics vanish simultaneously as J approaches the IC-DC phase boundary, by the pair Q2, q2 leaves us with only one vanishing variable, since, as will be seen below, q2 remains finite throughout. a. Expansion of e We write ωµ(k) = ω µ (k) + ω µ (k)Q 2 + ω µ (k)Q 2 +O(Q62) (B20) and determine the coefficients ω µ , n = 1, . . . 4, by solv- ing Eq. (15) for ω iteratively. We obtain 1 (k) + ω 2 (k) = 2λa wDC(k) , ω 3 = 0 , (B21a) 1 (k) + ω 2 (k) = − 1− cos kx cos ky wDC(k) , (B21b) 3 (k) = q22 λa C(k)2 −D(k)2 . (B21c) Here, C(k) = 1 − q22 1 − cos kx cos ky wDC(k)2 , (B22) D(k) = q1 sin k x q22 cos kx − cos ky wDC(k)2 , (B23) wDC(k) = 1− q21 sin2 kx . (B24) The coefficients ω µ (k), µ = 1, 2 , 3, will only be needed in the determination of the coefficient g2 of the fourth order term of e LG(Q2) which will be discussed later. We will first concentrate on the determination of the zeroth order term, e2, and of the coefficient r2 of the second order term of e LG(Q2). Under the assumption that g2 is positive, this will provide us with an expression for the IC-DC phase boundary. With the above expressions for ω 1 + ω 2 and ω ν = 0, 1, we obtain for the coefficients of the Landau Ginzburg energy from Eqs. (14), (B1) e2(q1, λa) = λ2a q − 2λa 1 + κ− 1 wDC(k) (B25) r2(q1, q2, λa, |x3(kmin)|2) = (κ + 1) 1− cos kx cos ky wDC(k) 3 (k) + |x3(kmin)|2 ω 3 (kmin) . (B26) These are valid for arbitrary values of the parameters q1, λa, q2, and |x3(kmin)|. In the next subsection, we will calculate their saddle point values for given Q2 and thus fix the parameters. Here, we have only allowed for the existence of a condensate component |x3(kmin)|2. This is justified since, as Eqs. (B20) and (B2) show, ω3 < ω1,2 for sufficiently small Q2 so that conceivably ω3(k) may vanish at some point kmin in the the Brillouin zone, while ω1(k) and ω2(k) remain finite at kmin, and hence a finite condensate density |x3(kmin)|2 may occur at this point. b. Saddle point, phase boundary Next we need to determine the saddle point of e LG(Q2) in the space of the variables q1, λa, q2, and |x3(kmin)|. First, the saddle point values of q1 and λa are obtained as expansions in powers of Q2, a + λ 2 +O(Q42) , (B27a) 1 + q 2 +O(Q42) , (B27b) where λ a , q 1 are the solutions of ∂λae2=0 , (B28a) ∂q1e2=0 . (B28b) Since the first derivatives of e2 vanish at λa = λ a , q1 = 1 , Eqs. (B28), we have e2 = e 2 + e 2 +O(Q62), (B29) r2 = r 2 + r 2 +O(Q42) (B30) Here, e 2 and r 2 are the expressions (B25) and (B26) with λa and q1 replaced by λ a and q 1 . The fourth order term of e2, Eq. (B29), and the second order term of r2 contribute only to the fourth order term of e which will be determined later. Therefore, we postpone the presentation of explicit expressions for λ a , q 1 and the ensuing expressions for e 2 and r 2 until later. With e2 from Eq. (B25), Eqs. (B28) yield the equations 1 )− 1, (B31) 1 ) − E(q , (B32) which determine the saddle point values q 1 and λ ( K and E are the elliptic integrals of the first and the second kind.) Next we seek the extremum of e LG w.r.t. q2. Since e2 is independent of q2 we, neglecting terms of order Q 2, have 0=∂q2 r q32 λa κ + 1 − I1(q1, q2) C(kmin) |x3(kmin)|2 3 (kmin) . (B33) I1(q1, q2) = ∫ π/2 [C(k)2 −D(k)2]1/2 (B34) (In these expressions and in the sequel, we use an abbre- viated notation: λa, q1 and q2 denote the zeroth order quantities λ a , q 1 and q 2 .) kmin is the location of the minimum of ω 3 (k), min = 0 ; kxmin ,−π ≤ kxmin ≤ − (B35) From (B21c) and (B35) it follows that ω 3 (kmin) = 0, q22 = (1 − q21)/2. (B36) As a function of q2 the integral I1(q1, q2) increases monotonously, 1 = I1(q1, 0) ≤ I1(q1, q2) ≤ I1 (q1) for 0 ≤ q2 ≤ (1 − q21)/2 . (B37) We have defined I1(q1) := max I1(q1, q2) = I1 (1 − q21)/2 (B38) As is seen in Fig. 22, the graphs of the functions κ = κ(q1), Eq. (B31), and of I1 = I1(q1) intersect at q1s ≃ 0.708, κs ≃ 0.181. Therefore, in solving Eq. (B33) for q2, two cases have to be considered separately: i. q1 > q1s, κ > κs. In this case, a solution ex- ists only, if the last term in parentheses in Eq. (B33) is positive. This requires that ω 3 (kmin) = 0 because, as has been discussed before, |x3(kmin)| and hence the ra- tio |x3(kmin)|2/ω(2)3 (kmin) would vanish otherwise. The 0.95 1.05 1.15 1.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements q1q1s κ(q1) + 1 I1(q1) FIG. 22: (Color online) I1 and κ+ 1 as functions of q1. condition ω3(kmin) = 0 implies that q 2 = (1− q21)/2, cf. Eq. (B36). Using this result and Eq. (B32) to eliminate q2 and λa from Eq. (B26) we find 2 = 2 1− JDC(κ) , (B39) where JDC(κ)= (κ+ 1) (3− q21)/2 + Ĩ2(q1, (1− q21)/2) (1 − q21) q21 π 4[K(q1)−E(q1)] (B40) Ĩ2(q1, q2) = C(k)2 −D(k)2 (B41) is the IC-DC phase boundary for κ > κs, i.e., in the region where the ratio |x3(kmin)|2/ω(2)3 (kmin) is finite. According to the discussion at the end of section III C, cf Eq. (24), this is the region where LRO prevails along the decoupled chains, cf. Fig. 9. In the development leading to Eq. (B40) for the phase boundary, we have not needed the solution of Eq. (B40) explicitly, but we note it here for completeness: C(kmin) |x3(kmin)|2 3 (kmin) |x3|2 1+3q21 1−q21 =1 + κ− I1(q1) > 0. (B42) These relations show that while |x3(kmin)| = 0, the ratio |x3(kmin)|2/ω(2)3 (kmin) remains finite. ii. q1 < q1s, κ < κs. In this case, we must have I1(q1, q2)) < I1 (1 − q21)/2 , (B43) (see Eq. (B38)). Consequently q22 < (1 − q21)/2 so that 3 (kmin) > 0 and hence no condensate can develop, |x3|2 = 0. Then, Eq. (B33) yields the equation I1(q1, q2) = 1 + κ (B44) which replaces Eq. (B36) and determines q2 as a function of q1, q2 = q2(q1). Then, proceeding as in case (i) one finds for the IC-DC phase boundary in the region κ < κs JDC(κ)= (κ+ 1)(1 + q22) + Ĩ2(q1, q2) q21 π 4 [K(q1)−E(q1)] . (B45) Here, q1 = q1(κ) from Eq. (B31) and q2 = q2(κ) from Eq. (B44)(with q1 = q1(κ)). We note here that inside the DC phase, i.e., for J > JDC(κ), where Q2 = λc = 0, the saddle-point values of q1 and λa/J and hence of Q1 are independent of J , cf. Eqs. (B31), (B32). Hence the graphs of Q1 and λa for J < JDC and for J > JDC join smoothly at J = JDC , cf. Figs. 10, 13. Furthermore, it follows from Eq. (B42), that the ratio (|x3(±kmin)|2/N▽)/(λa q22ω 3 (kmin)), which occurs in the amplitude of the spin-spin correlation func- tion, cf. Eq. (24a), is also independent of J inside the DC phase and retains the value that it has attained at the IC-DC transition line. c. Stability of the phase boundary In deriving the phase boundary from the condition r 0 we have tacitly assumed that the coefficient g2 of the fourth order term in the LG expansion, Eq. (B1), is pos- itive. In the remaining part of this appendix we will sketch the steps which lead to the conclusion that this is indeed the case. Expanding in the expression (B1) for e LG the coefficients e2 and r2 w.r.t. the second order contributions to q1 and λa, q 1 and λ a , cf. Eqs. (B27) we obtain LG = e 2 + r 2 + (g2 + g 2 +O(Q62) , (B46) where 1 + ω 2 + ω (B47) is the contribution to the fourth order term of e LG that arises from the fourth order terms of the frequencies ωµ in the sum in Eq. (14) whereas the contribution to e of the expansion of e2 and r2 is ∂2q1e2|0 ∂q1∂λae2|0 ∂λa∂q1e2|0 ∂2λae2|0 ∂q1r2|0 ∂λar2|0 . (B48) (In Eq. (B48) the notations ∂2q1e2|0 etc. indicate that af- ter the derivatives have been taken the variables q1, λa etc. have to be replaced by their zeroth order values q a etc.) The evaluation of the contribution (B47) is straight- foward: the coefficients ω µ , µ = 1, 2, 3, were obtained by solving Eq. (15) for ω iteratively to fourth order. As the explicit expressions are rather lengthy and contain no direct information, we refrain from presenting them here. The sum over k that is required in Eq. (B47) was done numerically. g2 was obtained in the form g̃2(q1) , (B49) where g̃2(q1) is a function of q1 alone which is always positive so that g2 > 0 throughout. Remarkably, no explicit dependence on the coupling constant J appears in these results. The evaluation of g′, Eq. (B48) requires the knowledge of explicit expressions for q 1 and λ a . These are obtained by expanding e2 to first order in q 1 and λ a , inserting the results into the expression (B1) for e LG and requiring that the terms of order Q22 satisfy the extremum condi- tions w.r.t. q1 and λa: 0 = q e2|0 + λ(2)a ∂q1∂λae2|0 + ∂q1r2|0 ,(B50a) 0 = q 1 ∂q1∂λae2|0 + λ(2)a ∂2λae2|0 + ∂λar2|0 .(B50b) The solution of these equations reads = −M̂ ∂q1r2|0 ∂λar2|0 , (B51) M̂−1 = ∂2q1e2|0 ∂q1∂λae2|0 ∂λa∂q1e2|0 ∂2λae2|0 . (B52) Inserting these results into Eq. (B48) one finds g′2 = − ∂q1r2|0 ∂λar2|0 ∂q1r2|0 ∂λar2|0 . (B53) While the second derivatives of e2 are obtained straight- forwardly from Eq. (B25) the derivatives ∂q1r2|0 and ∂λar2|0 have to be calculated separately for the region q1 < q1s, where there is no condensate, |x3(kmin)|2 = 0, and for the region q1 > q1s, where |x3(kmin)|2 > 0. Fi- nally, the result for g′ can be cast into the form g′2 = xq xλ (B54) where M̂ ′ = 4q21 Λ(κ+ 1− Λ) 1− q21 −Λ (2− q21)Λ− κ− 1 (B55) Λ ≡ 1 [K(q1)−E(q1)] (K, E : elliptic integrals) (B56) xq = q 1(Λ− κ− 1) + 1− q21 q1∂q1 Ĩ2(q1, q2) q2=q2(q1) −Θ(q1 − q1s) 1− q21 (κ+ 1− I1(q1)) , xλ = ( + 1)(κ+ 1)− 1 Ĩ2(q1, q2) q2=q2(q1) . (B57) Here Θ is the step function; the integrals I1(q1) and Ĩ2(q1, q2) have been defined above, cf. Eqs. (B38) and (B41), respectively. After numerical evaluation of these integrals, we find that g′2 = g 2(q1) is positive for all values of q1. 1 G. Misguich and C. Lhuillier, Frustration in Two- Dimensional Quantum Antiferromagnets (World Scien- tific, Singapore, 2004), chap. 5. 2 R. Moessner, Can. J. Phys. 79, 1283 (2001). 3 P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and P. Sindzingre, Phys. Rev. B 56, 2521 (1997). 4 C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre, Eur. Phys. J. B 2, 501 (1998). 5 F. Mila, Phys. 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Mattis, J. Math. Phys. 3, 749 (1962). 16 S. Brehmer, H.-J. Mikeska, and S. Yamamoto, J. Phys. Condens. Matter 9, 3921 (1997). 17 A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45, 2899 (1992). 18 C. H. Chung, J. B. Marston, and S. Sachdev, Phys. Rev. B 64, 134407 (2001). 19 C. H. Chung, J.B.Marston, and R. H. McKenzie, J. Phys. Condens. Matter 13, 5159 (2001). 20 F. Wang and A. Vishwanath, Phys. Rev. B 74, 174423 (2006). 21 O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Euro- phys. Lett. 73, 278 (2006). 22 G. Misguich, B. Bernu, and C. Lhuillier, J. Low Temp. Phys. 110, 327 (1998). 23 C. Waldtmann, H. Kreutzmann, U. Schollwöck, K. Maisinger, and H.-U. Everts, Phys. Rev. B 62, 9472 (2000). 24 V. Subrahmanyam, Phys. Rev. B 52, 1133 (1995). 25 C. Raghu, I. Rudra, S. Ramasesha, and D. Sen, Phys. Rev. B 62, 9484 (2000). 26 M. E. Zhitomirsky, Phys. Rev. B 71, 214413 (2005). 27 P. Nikolic and T. Senthil, Phys. Rev. B 68, 214415 (2003). 28 J. B. Marston and C. Zeng, J. Appl. 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0704.1442

On the Nature of Ultra-Luminous X-ray Sources from Optical/IR Measurements

Black Holes: from Stars to Galaxies – across the Range of Masses Proceedings IAU Symposium No. 238, 2006 V. Karas & G. Matt, eds. c© 2006 International Astronomical Union DOI: 00.0000/X000000000000000X On the Nature of Ultra-Luminous X-ray Sources from Optical/IR Measurements Mark Cropper1, Chris Copperwheat1, Roberto Soria2,1, Kinwah Wu1 1Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 2Centre for Astrophysics, Harvard Smithsonian Astrophysical Observatory, 60 Garden St, Cambridge, MA 02138, USA Abstract. We present a model for the prediction of the optical/infra-red emission from ULXs. In the model, ULXs are binary systems with accretion taking place through Roche lobe overflow. We show that irradiation effects and presence of an accretion disk significantly modify the optical/infrared flux compared to single stars, and also that the system orientation is important. We include additional constraints from the mass transfer rate to constrain the parameters of the donor star, and to a lesser extent the mass of the BH. We apply the model to fit photometric data for several ULX counterparts. We find that most donor stars are of spectral type B and are older and less massive than reported elsewhere, but that no late-type donors are admissable. The degeneracy of the acceptable parameter space will be significantly reduced with observations over a wider spectral range, and if time-resolved data become available. Keywords. black hole physics – X-rays: galaxies – X-rays: stars – accretion, accretion discs – binaries: general 1. Introduction Ultra-luminous X-ray sources (ULXs) are non-nuclear X-ray sources in nearby galax- ies with inferred luminosity >few×1039 ergs s−1. This luminosity exceeds the Edding- ton luminosity of a 20M⊙ black hole (BH) (an observational overview is available in Fabbiano, 2004). While these objects are generally agreed to be binary systems, the na- ture of their constituents is still controversial. Their emission could be as a result of sub-Eddington accretion rates onto intermediate mass black holes (IMBH) with masses ∼ 200− 1000M⊙, (Colbert & Mushotzky, 1999), super-Eddington accretion onto stellar- mass BH (Begelmann 2002, King 2001) or Eddington accretion onto BH with masses in the range ∼ 50− 200 M⊙ (Soria & Kuncic 2006). Recently, reasonably secure optical counterparts for these systems have been identified, mostly using HST observations. This has opened a new channel of investigation into the nature of ULX. The optical/infrared emission is derived from the irradiated mass donor star and disk, so it is essential to model these appropriately in both the spectral and time domain if system parameters such as the mass and radius of the mass donor star and the mass of the BH are to be constrained. This paper describes such a model for the optical/infrared emission, and summarises some of the constraints that can be derived from its application to optical/infrared data. The objectives of this work are (a) to provide constraints on the possible optical counterparts of ULXs, eliminating those candidates which are inconsistent with the predicted colours/variability; (b) to determine the characteristics of the ULX constituent parts as accurately as possible; (c) to constrain the origin of ULXs and (d) to make predictions for future observations. More detailed expositions can be found in Copperwheat et al. (2005) and Copperwheat et al. (2006). http://arxiv.org/abs/0704.1442v1 2 Cropper et al. Figure 1. The variation in intensity B(τ ) with τ = 2/3 for (right) an irradiated O5V star and (left) a disk for a BH mass of 150M⊙. The plot shows the view onto the orbital plane with the labelled distances in units of R⊙. Note that the intensity scales are logarithmic for the disk and linear for the star. 2. The model The compact object in the model is a BH of mass in the range 10−1000M⊙. The mass donor star fills its Roche lobe, and accretion takes place through Roche lobe overflow into an accretion disk. We assume the disk to be a standard thin disk, tidally truncated at a radius 0.6 of that of the distance to the L1 point. The mass donor star evolves according to the isolated star evolutionary tracks of Lejeune & Schaerer (2001). We ignore the effects of mass transfer on the star. We assume also that mass transfer is driven by the nuclear evolution of the mass donor. The model includes the Roche geometry, gravity and limb darkening, disk shadowing, radiation pressure according to the prescription of Phillips & Podsiadlowski (2002), the evolution of the companion and system orientation effects (inclination and binary phase). The X-ray irradiation is assumed to be isotropic. The irradiation of the disk and star is handled according to a formulation byWu et al. (2001) which is based originally on the grey stellar irradiation model of Milne (1926) and in- corporates the different opacities of the irradiated surface to hard and soft X-rays (the X-ray hardness ratio is an input parameter). The effective temperature of the irradiated star or irrdiated disk is a superposition of the irradiated and natural temperatures i.e. Teff = Bx(2/3) + T unirr where Bx(τ) is derived in Copperwheat et al. (2005). Figure 1 provides an example of the irradiated disk and star. Using this model we can predict the different contributions of the constituents of the ULX as a function of (for example) BH mass and donor star spectral type, as shown in Figure 2. For high-mass BHs, the disk is large, and hence disk emission dominates. In addition, irradiation effects are much larger for late-type stars. We make predictions of the optical/infrared flux. We find that there is generally better system parameter discrimination at infrared wavelengths. We also predict the variability timescale. Because of the axial symmetry of our disks, these are dominated by the (modified) ellipsoidal variations from the Roche-lobe-filling donor stars. The variability timescale is typically days, dependent on BH mass. The photometric predictions can be compared to observations of ULX counterparts in different wavebands, and acceptable model parameter regimes determined from χ2 fitting to the observations. If only single-epoch photometric observations are available in 2 or 3 bands the model fits are underconstrained, leading to degenerate solutions. An example is shown in Figure 3(left). Nevertheless, even with limited photometric data much of the possible parameter space can be eliminated, and with additional wavelength coverage particularly in the infrared, degeneracies can be reduced further. We add a further constraint for the χ2 fitting using the mass transfer rate, which is The Nature of ULXs from Optical/IR data 3 10 100 1000 BH Mass (Msun) Irr. Star Unirr. Star Irr. Star + Disk O5V star, disk model #2 (ξ=0.1)(c) 10 100 1000 BH Mass (Msun) Irr. Star Unirr. Star Irr. Star + Disk G0I star, disk model #2 (ξ=0.1)(d) Figure 2. The V band absolute magnitudes for an un-irradiated and irradiated O5V star and accretion disk (left) and for a G0I supergiant with disk (right), shown as a function of BH mass. Here Lx = 10 40 ergs s−1, cos i = 0.5 and the star is at superior conjunction. determined by the evolution of the stellar radius (from the evolutionary tracks) compared with the evolution of the Roche lobe radius (Wu, 1997, Ritter, 1988). We measure the mass transfer rate from the X-ray luminosity assuming an accretion efficiency η = 0.1 appropriate for a BH, and select only those secondary stars which are evolving in radius on nuclear timescales in such a way as to provide the measured mass transfer rate. 3. Fits to data We have gathered HST and VLT photometric data available up to mid-2006 on the most luminous ULX counterparts, where we can be reasonably certain that mass transfer is driven by Roche lobe overflow. The input data for M81 X-6, NGC 4559 X-7, M101 ULX- 1, NGC 5408 ULX, Holmberg II ULX, NGC 1313 X-2 (C1) and NGC5204 ULX are given in table 1 of Copperwheat et al. (2006) and for M51 X5/9 in Copperwheat (2007). We fitted our model to these photometric data, using the additional mass transfer constraint from X-ray measurements. Figure 3 (right) shows an example of the allowed parameter space in the donor star mass vs BH mass plane from the χ2 fitting. We also produced similar plots for the donor star radius and donor star age, which we determined from the mass-radius relation from the evolutionary tracks. More details can be found in Copperwheat et al. (2006). These fits provide the current spectral type of the mass donor star. By tracing to earlier times along the evolutionary tracks, the ZAMS spectral type can be predicted assuming the mass loss has not significantly altered the evolution. 4. Main outcomes The optical/infrared emission from our binary ULX model is significantly different from models assuming unirradiated companion stars and no disks, hence it is not adequate to assume standard colours from single stars to determine the mass donor characteristics in ULXs. The model fits to the currently available data do not provide strong constraints on the BH mass, mainly because of the unknown orientation of the system. Depending on inclination, upper or lower limits can, however, be set. For example, the BH mass in NGC 5204 is < 240 M⊙ for cos i = 0.5, while that in NGC 1313 X-1 is < 100 M⊙. The reason is that the signature of a disk can be almost entirely hidden for cos i = 0.0, in which case little information can be obtained on the size of the primary Roche lobe. The mass, radius and age of the donor star are, however, more tightly constrained. 4 Cropper et al. 3.02.01.00.80.60.4 Wavelength (µm) M1=30Msun, M2=8Msun Stellar age=30Myr, cos(i)=0.0 M1=50Msun, M2=16Msun Stellar age=10Myr, cos(i)=0.5 M1=1000Msun, M2=70Msun Stellar age=1Myr, cos(i)=0.5 M1=600Msun, M2=110Msun Stellar age=1Myr, cos(i)=0.0 10 100 1000 BH Mass (Msun) Figure 3. (left) Absolute magnitude as a function of wavelength for different combinations of BH (M1) and donor star (M2) and inclination i compared to photometric measurements for NGC5408 ULX, indicating the degeneracy of possible solutions when no mass transfer constraints are applied. (right) Acceptable parameter space for the mass donor star projected onto the donor mass/BH mass plane when the mass transfer constraint is included. Contours are at 68, 90 (solid line), 95 and 99% confidence levels. Typical ages range from 107 − 108 years, with typical ZAMS masses 5 − 10 M⊙, with some up to 50 M⊙. In general we find that the mass donor stars are less massive and older than generally quoted in the literature from less comprehensive modeling – this is to be expected given the effects of irradiation. We find that none of the systems are found to contain late-type mass donor stars, whatever disk component is admitted. The preference for donor stars of spectral type B is interesting. The high mass transfer rates and modest donor star masses require that ULX lifetimes are short (this is true also for higher mass donors which in any case have short lifetimes). The duration that the donor star has been in contact with its Roche lobe is an important parameter for ULXs. If it has been in contact for some Myr, especially in the case of B-type ZAMS stars, then binary evolution models will be required. The diagnostic capability of our model for the BH mass and donor star characteristics will improve significantly as more filter bands, and particularly time-resolved data become available. Acknowledgements R. Soria acknowledges support from a Marie Curie Fellowship from the EC. References Begelmann, M.C. 2002, Astrophys. J. Lett. 568, L97 Colbert, E. J. M. & Mushotzky, R. F. 1999, Astrophys. J. 519, 89 Copperwheat C., Cropper M., Soria R. & Wu K. 2005, Mon. Not. Roy. Astr. Soc. 362, 79 Copperwheat C., Cropper M., Soria R. & Wu K. 2006, Mon. Not. Roy. Astr. Soc. submitted Copperwheat C. 2007, PhD Thesis University of London, submitted Fabbiano G. 2004, Rev. Mex. A. A. (Serie de Conferencias) 20, 46 King A. R. 2002, Mon. Not. Roy. Astr. Soc. 335, L13 Lejeune T. & Schaerer D. 2001, Astron. & Astrophys., 366, 538 Milne E.A. 1926, Mon. Not. Roy. Astr. Soc. 87, 43 Phillips S. N. & Podsiadlowski P. 2002, Mon. Not. Roy. Astr. Soc. 337, 431 Ritter H. 1988, Astron. & Astrophys. 202, 93 Soria, R.S. & Kuncic, Z. 2006, Adv. Space. Res. (Cospar special issue) submitted Wu, K. 1997 Accretion Phenomena and Related Outflows; IAU Colloquium 163., ASP Conference Series Vol. 121; ed. D. T. Wickramasinghe; G. V. Bicknell & L. Ferrario (1997), p.283 Wu K., Soria R., Hunstead R.W. & Johnston H.M. 2001, Mon. Not. Roy. Astr. Soc. 320, 177 Introduction The model Fits to data Main outcomes

0704.1443

Dephasing due to a fluctuating fractional quantum Hall edge current

DEPHASING DUE TO A FLUCTUATING FRACTIONAL QUANTUM HALL EDGE CURRENT T. K. T. NGUYEN1,2, A. CRÉPIEUX1, T. JONCKHEERE1, A. V. NGUYEN2, Y. LEVINSON3, AND T. MARTIN1 1 Centre de Physique Théorique, CNRS Luminy case 907, 13288 Marseille cedex 9, France 2 Institute of Physics and Electronics, 10 Dao Tan, Cong Vi, Ba Dinh, Hanoi, Vietnam 3 Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel The dephasing rate of an electron level in a quantum dot, placed next to a fluctuating edge current in the fractional quantum Hall effect, is considered. Using perturbation theory, we first show that this rate has an anomalous dependence on the bias voltage applied to the neighboring quantum point contact, because of the Luttinger liquid physics which describes the fractional Hall fluid. Next, we describe exactly the weak to strong backscattering crossover using the Bethe-Ansatz solution. 1 Introduction The presence of electrical environment influences the transport through a quantum dot: its energy level acquires a finite linewidth if the environment has strong charge fluctuations. Several experiments, performed with a quantum dot embedded in an Aharonov-Bohm loop, probed the phase coherence of transport when the dot is coupled to a controlled environment, such as a quantum point contact (QPC)1. Charge fluctuations in the QPC create a fluctuating potential at the dot, modulate its electron level, and destroy the coherence of the transmission through the dot 2. Theoretical studies for describing this dephasing have been developped 3,4, and were applied to a quantum Hall geometry 5, and to a normal metal-superconductor QPC6. In this work, we consider the case of dephasing from a QPC in the fractional quantum Hall effect (FQHE) regime 7. QPC transmission can be described by tunneling between edge states which represent collective excitations of the quantum Hall fluid. It is interesting because the transport properties deviate strongly from the case of normal conductors 8,9,10: for the weak backscattering (BS) case, the current at zero temperature may increase when the voltage bias is lowered, while in the strong BS case, the I(V ) is highly non linear. It is thus important to address the issue of dephasing from a Luttinger liquid. 2 Dephasing in the fractional quantum Hall regime The system we consider is depicted in Fig. 1. The single level Hamiltonian for the dot reads HQD = ǫ0c †c, where c† creates an electron. This dot is coupled capacitively to a point contact in the FQHE. The Hamiltonian which describes the edge modes in the absence of tunneling is: H0 = (h̄vF /4π) dx[(∂xφ1) 2 + (∂xφ2) 2], where φ1 and φ2 are the chiral Luttinger bosonic fields, which are related to the electron density operators ρ1(2) by ∂xφ1(2)(x) = πρ1(2)(x)/ http://arxiv.org/abs/0704.1443v1 ��������������� ��������������� ��������������� ��������������� ������������ ������������ ������������ ������������ ���PSfrag replacements IB IB a) b) Figure 1: Schematic description of the setup: the quantum dot (top) is coupled capacitively to a quantum point contact in the FQHE regime: a) case of weak backscattering, b) case of strong backscattering. By varying the gate potential of the QPC, one can switch from a weak BS situation, where the Hall liquid remains in one piece (Fig. 1a), to a strong BS situation where the Hall liquid is split in two (Fig. 1b). In the former case, the entities which tunnel are edge quasiparticle excitations. In the latter case, between the two fluids, only electrons can tunnel. We consider the weak BS case, and then we use a duality transformation8,11 to describe the strong BS case. The tunneling Hamiltonian between edges 1 and 2 reads Ht = Γ0e iω0tψ+2 (0)ψ1(0) + h.c. where we have used a Peierls substitution to include the voltage: ω0 = e ⋆V/h̄ (e⋆ = νe is the effective charge, ν is the filling factor). The quasiparticle operator is ψi(x) = e νφi(x)/ 2πα (the spatial cutoff is α = vF τ0, with τ0 the temporal cutoff). The Hamiltonian describing the interaction between the dot and the QPC reads Hint = dxf(x)ρ1(x), with f(x) a Coulomb interaction kernel, which is assumed to include screening by the nearby gates f(x) ≃ e2e−|x|/λs/ x2 + d2, where d is the distance from the dot to the edge, λs is a screening length. The dephasing rate, expressed in terms of irreducible charge fluctuations in the adjacent wire, is written as 3,4,5: τ−1ϕ = dxf(x) dx′f(x′)〈〈ρ1(x, t)ρ1(x′, 0) + ρ1(x′, 0)ρ1(x, t)〉〉 . (1) The equilibrium contribution to the dephasing rate corresponds to the zero order in the tunneling amplitude Γ0: (τ−1ϕ ) (0) = 4π2h̄2 dxf(x) dx′f(x′) ∂2xx′G 1 (x− x ′, t) , (2) where the bosonic Green’s function is G i (x − x′, t1 − t2) = 〈φi(x, t 1 )φi(x ′, tη22 ) − φ2i 〉. The coefficients η,η1,2 = ± identify the upper/lower branch of the Keldysh contour. There is no contribution to first order in the tunneling Hamiltonian, while the non-equilibrium contribution corresponding to the second order in Γ0 exists: (τ−1ϕ ) (2) = − ν 4π2h̄4 2(2πα)2 dxf(x) dx′f(x′) η,η1,η2,ǫ ×eiǫω0(t1−t2)eνG (0,t1−t2)eνG (0,t1−t2) ∂2xx′G 1 (x− x ′, t) + ν[∂xG 1 (x, t− t1)− ∂xG 1 (x, t− t2)][∂x′G ′,−t1)− ∂x′G−ηη21 (x ′,−t2)] . (3) The dephasing rate depends on the geometry of the set up via the length scales d, λs, and α. The assumption of strong screening λs ∼ α = vF τ0 is made (f(x) ≃ 2e2αδ(x)/d). Inserting 1 (x, t) = − ln sinh[π[(x/vF −t)((η+η′)sgn(t)−(η−η′))/2+iτ0]/h̄β] sinh[iπτ0/h̄β] , where β = 1/kBT , in the dephasing rate gives: (τ (0) = 4e4τ20 ν/πh̄ 3βd2 and, (τ−1ϕ ) (2) = e4Γ20 π2h̄4v2F d ν2τ2ν0 Γ(2ν) )2ν−1 ω0h̄β ν + i ω0h̄β . (4) Note that (τ−1ϕ ) (2) = (eτ0/d) 2SI(0), with SI(0) = dt〈〈I(t)I(0)〉〉 the zero-frequency BS noise. The non-equilibrium contribution of the dephasing rate is proportional to the zero- frequency noise9,10,11,12 in the quantum Hall liquid. At zero temperature, the non-equilibrium dephasing rate given by Eq. (4) leads to (τ−1ϕ ) (2) ∝ |ω0|2ν−1 and depends on the QPC bias with the exponent 2ν − 1 < 0, in sharp contrast with the linear dependence obtained by Levinson 3. 0 0.2 0.4 0.6 0.8 1 β = 10 β = 5 β = 50 0 0.5 1 1.5 2 0 1 2 3 4 5 β = 10 β = 10 β = 100 β = 100 β = 50 β = 50 Figure 2: (Left) Dephasing rate, plotted in units of e4Γ20τ0/π 2h̄4v2F d 2, as a function of the filling factor for weak BS (full line) and strong BS (dashed line) at QPC bias eV = 0.1. The star, diamond and circle points correspond to the Laughlin fractions ν = 1/m, m odd integer. (Right) Dephasing rate as a function of QPC bias with ν = 1/3 for weak BS (full line) and strong BS (dashed line). The insert is the ratio of non-equilibrium contribution in dephasing rate between the arbitrary screening and strong screening multiplied by (α/d)2 as a function of d/λs. In the left of Fig. 2, we plot the dependence of the non-equilibrium contribution of the dephasing rate on the filling factor ν for weak BS and strong BS for several temperatures (β = 5, 10, 50) at fixed QPC bias. ν is considered as a continuous variable, while it has physical meaning only at Laughlin fractions 7. For the strong BS case, the dephasing rate increases when the ν increases. For weak BS and 1/β ≪ eV , the dephasing rate has a local maximum at ν < 1/2, the position of which depends on temperature: when the temperature increases, it gets closer to ν = 1/2. The rate at ν = 1 is smaller than that at ν = 1/3. This result demonstrates that for two different filling factors, we can have comparable dephasing rates. For weak BS and 1/β > eV , the dephasing rate increases when the filling factor increases. In the right of Fig. 2, the dependence of the dephasing rate on the QPC bias voltage is plotted for ν = 1/3 and several temperatures. In the case of strong BS, the dephasing rate increases when the bias increases. For 1/β ≪ eV , the dephasing rate saturates, whereas for 1/β > eV , the dephasing rate increases when eV increases, but it increases from a finite value (not shown), which is proportional to the temperature. Things are quite different at weak BS. At high temperatures, the dephasing rate decreases when we increase eV : this behavior is symptomatic of current and noise characteristic in a Luttinger liquid. In the low temperature case 1/β ≪ eV , for small eV , the lower the temperature, the bigger the dephasing rate and the faster it decreases when we increase eV . 3 General formula for the decoherence rate The charge fluctuations are directly related to the current fluctuations along the edges which are identical to the fluctuations of the tunneling current. The tunneling current fluctuations were computed non pertubatively using Bethe-Ansatz techniques13. We can therefore invoke current conservation at the point contact to derive a general formula for the decoherence rate, which describes the crossover from weak to strong BS: (τ−1ϕ ) (2) = (e3τ20 /d 2)(V Gdiff−I)ν/(1−ν) where Gdiff = ∂V I is the differential conductance and I is the current 14. This expression allows us to describe the crossover in the dephasing rate from the weak to the strong BS regime. Remarkably, it is possible to go beyond the strong screening limit, and one can compute Eq. (3) for an arbitrary Coulomb kernel f(x). The result can be displayed in terms of the ratio between the arbitrary screening dephasing rate and the strong screening dephasing rate: (τ−1ϕ ) (τ−1ϕ ) (eα)2 dxf(x) . (5) If the Coulomb interaction kernel f(x) is chosen as suggested before, the dephasing rate at arbitrary λs has an analytical expression: F = (πd/2α) 2[E0(d/λs)+N0(d/λs)], where E0(d/λs) and N0(d/λs) are the Weber and the Neumann functions of zero order. F is plotted in the insert of Fig. 2: F is infinite in the absence of screening. However, the presence of metallic gates always imposes a finite screening length. F decreases with d/λs and approaches 1 when λs is close to the spatial cutoff α. The dephasing rate increases when the screening decreases. 4 Conclusion We have established a general formula for the dephasing rate of a quantum dot located in the proximity of a fluctuating fractional edge current. For strong screening, we have shown that the dephasing rate is given by the tunneling current noise, for both weak and strong BS. For weaker screening, the spatial dependence of the density-density correlation function has to be taken into account, but we have shown explicitly that the long range nature of the Coulomb interaction can be included as a trivial multiplicative factor. The fact that the dephasing rate decreases with increasing voltage can be reconciled with the fact that the charge noise is directly related to the BS current noise in the FQHE. There it is known, and seen experimentally, that when the bias voltage dominates over the temperature, both the tunneling current and noise bear a power law dependence ∼ V 2ν−1 with a negative exponent. The fact that at low temperatures, the dephasing rate for filling factors can be lower than that of the integer quantum Hall effect comes as a surprise and is a consequence of chiral Luttinger liquid theory. References 1. A. Yacoby et al., Phys. Rev. Lett. 74, 4047 (1995); A. Yacoby et al., Phys. Rev. B 53, 9583 (1996); E. Buks et al., Phys. Rev. Lett. 77, 4664 (1996); R. Schuster et al., Nature 385, 417 (1997). 2. E. Buks et al., Nature 391, 871 (1998); D. Sprinzak et al., Phys. Rev. Lett. 84, 5820 (2000). 3. Y. Levinson, Europhys. Lett. 39, 299 (1997). 4. I.L. Aleiner, N.S. Wingreen, and Y. Meir, Phy. Rev. Lett. 79, 3740 (1997). 5. Y. Levinson, Phys. Rev. B 61, 4748 (2000). 6. R. Guyon, T. Martin, and G.B. Lesovik, Phys. Rev. B 64, 035315 (2001). 7. D.C. Tsui et al., Phys. Rev. Lett. 48, 1559 (1982); R.B. Laughlin, ibid. 50, 1395 (1983). 8. C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 (1992). 9. C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 72, 724 (1994). 10. C. Chamon, D.E. Freed, and X.G. Wen, Phys. Rev. B 51, 2363 (1995). 11. C. Chamon, D.E. Freed, and X.G. Wen, Phys. Rev. B 53, 4033 (1996). 12. T. Martin in Les Houches Summer School session LXXXI, edited by E. Akkermans, H. Bouchiat, S. Guéron, and G. Montambaux (Elsevier, 2005). 13. P. Fendley and H. Saleur, Phys. Rev. B 54, 10845 (1996). 14. P. Fendley et al., Phys. Rev. Lett. 75, 2196 (1995); Phys. Rev. B 52, 8934 (1995). Introduction Dephasing in the fractional quantum Hall regime General formula for the decoherence rate Conclusion

0704.1444

Fracture of complex metallic alloys: An atomistic study of model systems

Fracture of complex metallic alloys: An atomistic study of model systems Frohmut Rösch and Hans-Rainer Trebin Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Peter Gumbsch Institut für Zuverlässigkeit von Bauteilen und Systemen, Universität Karlsruhe, Kaiserstr. 12, 76131 Karlsruhe, Germany Fraunhofer Institut für Werkstoffmechanik, Wöhlerstr. 11, 79108 Freiburg, Germany Abstract Molecular dynamics simulations of crack propagation are performed for two extreme cases of complex metallic alloys (CMAs): In a model quasicrystal the structure is de- termined by clusters of atoms, whereas the model C15 Laves phase is a simple periodic stacking of a unit cell. The simulations reveal that the basic building units of the struc- tures also govern their fracture behaviour. Atoms in the Laves phase play a comparable role to the clusters in the quasicrystal. Although the latter are not rigid units, they have to be regarded as significant physical entities. 1 Introduction Complex metallic alloys are intermetallic compounds with large unit cells containing from tens up to thousands of atoms. Often, distinct local arrangements of atoms – clusters – can be viewed as building units. Both, the cluster diameter and the lattice constant imply length scales which should be reflected in physical properties. CMAs frequently combine interesting properties like high melting point, high temperature strength, and low density. However, possible applications are often limited by extreme brittleness at low or ambient temperature. To enlighten the role of clusters and periodicity in fracture, we perform molecular dynamics simulations of two extreme cases of CMAs: An icosahedral quasicrystal and a C15 Laves phase. As we are interested in the general qualitative features of the structures, we use three-dimensional model systems consisting of about five million atoms and model potentials (Lennard-Jones). This deliberate choice in the past often helped to reveal fundamental aspects of fracture (see e.g. Abraham (2003)). The quasicrystal can be viewed as a CMA with an infinitely large unit cell, such that no periodicity is present and clusters are the main feature of the structure. On the other hand, the C15 Laves phase has 24 atoms in the cubic unit cell and no clusters. The structure of the Laves phases is determined by periodicity but already quite complex, such that complicated deformation mechanisms might emerge (see e.g. Chisholm et al. (2005)). 2 Models and method The three-dimensional model quasicrystal used in our numerical experiments has been pro- posed by Henley and Elser (1986) as a structure model for icosahedral (Al,Zn)63Mg37. It is built up from the prolate and oblate rhombohedra of the three-dimensional Penrose tiling. As we do not distinguish between Al and Zn type atoms, we term its decoration icosahedral binary model (for details see e.g. Rösch et al. (2005)). In the upcoming figures the Al and Zn type atoms (A atoms) are displayed as grey (online: red) balls, whereas Mg type atoms (B atoms) are shown in black (online: blue). The shortest distance between two A atoms is denoted r0 which corresponds to about 2.5 Å. Inherent in the structure are Bergman-type clusters, which also can be viewed as basic building units. As no reliable “realistic” effective potentials are available for the fracture of CMAs, the interactions are modelled by simple Lennard-Jones pair potentials (see Sec. 1, Rösch et al. (2004), and Rösch et al. (2005)). These potentials keep the model stable even under strong de- formations or introduction of point defects and have been used in our group together with the icosahedral binary model to simulate dislocation motion (Schaaf et al. (2003)) and even shock waves (Roth (2005)). Very similar potentials have shown to stabilize the icosahedral atomic structure (Roth et al. (1995)). By the choice of these model potentials and model structures we qualitatively probe the influence of structural aspects of the investigated compounds with- out being specific to a special kind of material. Model potentials are often used in fracture simulations and have led to useful insight into fundamental mechanisms (see e.g. Abraham (2003)). The minimum of the Lennard-Jones potential for the interactions between atoms of different kind is set to twice the value of that for atoms of the same type. However, the conclusions drawn from our simulations remain essentially unaffected by setting all binding energies equal, which again indicates that we are mainly probing structural effects. A fundamental building unit of the simulated quasicrystal – the prolate rhombohedron – in a slightly deformed way forms the cubic C15 Laves phase A2B by periodic arrangement. But in this structure no clusters of the quasicrystal are present. Because of the close struc- tural relationship of the C15 Laves phase to the quasicrystal model, we use the same model potentials. Our samples have dimensions of approximately 450r0×150r0×70r0 and contain about five million atoms. Periodic boundary conditions are applied in the direction parallel to the crack front. To simulate mode I fracture, we first determine potential cleavage planes. According to Griffith (1921) crack propagation becomes possible, when the elastic energy is sufficient to generate two new fracture surfaces. Thus, potential cleavage planes should be those with low surface energies. In the quasicrystal these are specific twofold and fivefold planes (see Rösch et al. (2005)). There atomically sharp seed cracks are inserted. The samples are uniaxially strained perpendicular to the crack plane up to the Griffith load and then relaxed to obtain the displacement field of the stable crack at zero temperature. Subsequently, a temperature of about 10−4 of the melting temperature is applied. The sample is further loaded by linear scaling of the corresponding displacement field for this temperature. The response of the system then is monitored by molecular dynamics techniques. The radiation emitted by the propagating crack is damped away outside an elliptical region to prevent reflections (see Rösch et al. (2005) and Gumbsch et al. (1997)). 3 Results and discussion In the model quasicrystal we investigated how clusters and the plane structure influence crack propagation. A paper on this detailed study has recently been published (see Rösch et al. (2005)), where brittle fracture without any crack tip plasticity was reported. The following results from that paper indicate that the clusters determine the brittle cleavage fracture of the model quasicrystal: First, circumvention or intersection of clusters slows down the cracks. Second, the fracture surfaces show characteristic height variations giving rise to an overall roughness on the cluster scale (see Fig. 1, left). Lines along which the clusters are located are also visible in the height profiles. Third, the crack intersects fewer clusters than a planar cut with low surface energy does (see Fig. 2). Another observation of the simulations is that the plane structure also influences fracture. Cracks located perpendicular to twofold and fivefold axes fluctuate about a constant height. Thus, the roughness of the crack surfaces can be assigned to the clusters, whereas constant average heights of the fracture surfaces reflect the plane structure of the quasicrystal. Now we compare simulation results of the quasicrystal to those of the C15 Laves phase. For this structure we also observe brittle failure. But the fracture surfaces at low loads – if at all – only are rough on an atomic scale. This becomes apparent from Figs. 1 and 3. In Fig. 3 only atoms near a (111) fracture surface are displayed. The seed crack shown on the top propagated in [21̄1̄] direction. The material perfectly cleaved (Fig. 3, bottom), which resulted in smooth fracture surfaces. In Fig. 1 sections of geometrically scanned fracture surfaces of the icosahedral model quasicrystal (left) and of the C15 model Laves phase (middle, right) are compared. The cleavage planes are located perpendicular to a twofold (left) and a [010] axis (middle, right). The crack propagated along a twofold (left) and a [101] axis (middle, right). The colour coding in the left image is adjusted to the cluster diameter (online: from blue to red). When colour coded like the quasicrystal, the fracture surfaces of the C15 Laves phase lack any roughness (middle, online: only green). After adjusting the colour coding, atomic rows become visible (right). So, fracture surfaces of the quasicrystal are rough on the cluster scale, whereas those of the Laves phase only are rough on an atomic scale. Thus, atoms in the Laves phase play a comparable role to the clusters in the quasicrystal – they determine the overall roughness of the fracture surfaces. The atomistic view of cleavage on a (011) plane in Fig. 4 reveals an interesting effect1: If the seed crack there would be continued, the lines would terminate the upper and lower halves of the sample. However, as can be seen in the time sequence of Fig. 4, this is not the case: The dynamic crack instead takes a zig-zag like route. Entire atomic rows alternately move upwards and downwards. This leads to rather symmetric upper and lower fracture surfaces, the creation of which also requires a comparable amount of energy. This rather symmetrical creation of fracture surfaces is favoured, even though a planar cut would lead to surfaces with lower total energy. Thus, the actual fracture path cannot be predicted by a simple energy criterion. Such a behaviour also was observed in B2 NiAl (see e.g. Gumbsch (2001)) and is a consequence of lattice trapping (Thomson et al. (1971)), which – similar to the Peierls barrier for dislocation motion – allows overloads for cracks that do not result in crack propagation. The increased load for a crack to propagate will therefore not necessarily lead to fracture surfaces of lowest energy. The discrete nature of matter is responsible for these observations. The fracture path is strongly 1A similar behaviour is observed for the orientation shown in Fig. 1 (middle, right). influenced by the arrangement of atoms near the crack front, which depends on the initial cleavage plane as well as the crack propagation direction. 4 Conclusions and outlook In conclusion, the simulation results of the two extreme cases of model CMAs indicate that the basic building units of the structures govern also their physical properties. The role of atoms in the Laves phase is – from a certain point of view – played by the clusters in the quasicrystal. Although these are not rigid units, they are significant physical entities. To further enlighten the fracture processes in the C15 Laves phase, we currently perform simulations with force-matched (Ercolessi and Adams (1994), Brommer and Gähler (2006)) effective embedded atom method potentials for NbCr2 (Rösch et al. (2006)). Although we expect that the overall qualitative behaviour (e.g. the roughness of the fracture surfaces) is already represented well by our simple model potentials, results certainly will differ quantita- tively for diverse materials, i.e. interactions. Especially, the lattice trapping mentioned above strongly depends on the potentials used. Future studies will concentrate on material specific simulations on systems of CMAs, in which compounds with very different unit cell sizes and local arrangements exist. New effects are expected e.g. when the size of the unit cell and cluster diameters become comparable. Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft under contract number TR 154/20-1 is gratefully acknowledged. References F. F. Abraham, Advances in Physics 52 (8), 727 (2003). M. F. Chisholm, S. Kumar, and P. Hazzledine, Science 307, 701 (2005). C. L. Henley and V. Elser, Phil. Mag. B 53 (3), L59 (1986). F. Rösch, Ch. Rudhart, J. Roth, H.-R. Trebin, and P. Gumbsch, Phys. Rev. B 72, 014128 (2005). F. Rösch, Ch. Rudhart, P. Gumbsch, and H.-R. Trebin, Mat. Res. Soc. Symp. Proc. 805, LL9.3 (2004). G. D. Schaaf, J. Roth, and H.-R. Trebin, Phil. Mag. 83 (21), 2449 (2003). J. Roth, Phys. Rev. B 71, 064102 (2005). J. W. Roth, R. Schilling, and H.-R. Trebin, Phys. Rev. B 51 (22), 15833 (1995). A. A. Griffith, Philos. Trans. R. Soc. Lond. Ser. A 221, 163 (1921). P. Gumbsch, S. J. Zhou, and B. L. Holian, Phys. Rev. B 55 (6), 3445 (1997). P. Gumbsch, Mater. Sci. Eng. A 319-321, 1 (2001). R. Thomson, C. Hsieh, and V. Rana, J. Appl. Phys. 42 (8), 3154 (1971). F. Ercolessi and J. B. Adams, Europhys. Lett. 26 (8), 583 (1994). P. Brommer and F. Gähler, Phil. Mag. 86 (6-8), 753 (2006). F. Rösch, H.-R. Trebin, and P. Gumbsch, Int. J. Fracture 139 (3-4), 517 (2006). Figures Figure 1: Sections of typical fracture surfaces of an icosahedral model quasicrystal (left) and of a C15 model Laves phase (middle, right). The colour coding in the left and middle picture is adjusted to the cluster diameter (online: from blue to red). For details see text. The side length of the squares is about 14 nm. Figure 2: Clusters cut by the crack in a model quasicrystal: Only the smaller parts of those clusters are displayed that were divided by the crack. Obviously, the dynamic crack (right) intersects fewer clusters than the low energy seed crack (left). The cleavage plane is located perpendicular to a fivefold axis, the crack propagated in twofold direction. Figure 3: View inside a crack of the C15 model Laves phase. Perfect brittle cleavage fracture is observed. a) b) c) d) Figure 4: Fracture of a C15 model Laves phase: Atomic configurations in the vicinity of a propagating crack (time sequence). The fracture surface is located perpendicular to the [011] direction, the crack propagates along the [100] direction (from left to right). Introduction Models and method Results and discussion Conclusions and outlook References

0704.1445

Deformed Wigner crystal in a one-dimensional quantum dot

Deformed Wigner crystal in a one-dimensional quantum dot Yasha Gindikin and Vladimir A. Sablikov Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow District, 141190, Russia �e spatial Fourier spectrum of the electron density distribution in a �nite 1D system and the distribution function of electrons over single-particle states are studied in detail to show that there are two universal features in their behavior, which characterize the electron ordering and the deformation of Wigner crystal by boundaries. �e distribution function has a δ -like singularity at the Fermi momentum kF . �e Fourier spectrum of the density has a step-like form at the wavevector 2kF , with the harmonics being absent or vanishing above this threshold. �ese features are found by calculations using exact diagonalization method. �ey are shown to be caused by Wigner ordering of electrons, a�ected by the boundaries. However the common Lu�inger liquid model with open boundaries fails to capture these features, because it overestimates the deformation of the Wigner crystal. An improvement of the Lu�inger liquid model is proposed which allows one to describe the above features correctly. It is based on the corrected form of the density operator conserving the particle number. I. INTRODUCTION One-dimensional (1D) quantum dots a�ract a great deal of a�ention as appealing model objects to study the e�ects of electron-electron (e-e) interaction, which is principally im- portant for 1D electron systems.1,2 �e interest to the many- electron state in bounded 1D systems is presently increased due to the recent progress in magnetotunnelling spectroscopy studies of such structures.3–5 As a consequence of e-e inter- action, electrons form the strongly correlated state, which is referred to as Lu�inger liquid. Main distinctive features of a Lu�inger liquid are the absence of fermionic quasiparticles, which manifests itself in the absence of the Fermi step in the momentum distribution function, the power-law behavior of spectral functions near the Fermi energy with interaction- dependent exponents,6,7 and Wigner-like correlations of elec- trons.8 �ese properties were established for ideal, i.e. boundless 1D systems. However, whether they hold for realistic meso- scopic systems of �nite length is not obvious. Indeed, the presence of boundaries may strongly a�ect the electron state and excitations since 1D electron correlation functions are known to decay as a power of distance, hence there is no characteristic length. Many observations performed on meso- scopic structures containing a �nite 1D system do not con�rm the theoretical predictions made for ideal systems. Just recall the interaction-driven conductance renormalization9 in an in�nite 1D system that does not actually occur because of the contacts,10–12 the spin polarization13,14 that should not exist ac- cording to the Ma�is-Lieb theorem, the ‘0.7’ anomaly,15 many facts of the nonuniversality of conductance quantization,16–18 which existing theories fail to explain, and so on. It is commonly believed that electrons in a bounded 1D sys- tem form the Lu�inger liquid. �e boundary e�ect consists in the change of electron correlations near the ends which are described by additional boundary exponents. �is conclusion was made in several theoretical works using the bosonization approach.19–21 Unfortunately, this approach is based on a num- ber of model assumptions, which are not well justi�ed. �ese are (i) the extension of the linear electron spectrum to in�nite negative energy, which results in a violation of the conserva- tion laws,22,23 (ii) the linearization of the electron spectrum, which can lead to a striking departure from the properties of a real system with quadratic dispersion relation,24 and (iii) neglecting some parts of the 2kF components of the electron densities in the e-e interaction Hamiltonian. �e goal of our paper is to investigate the electron state in 1D quantum dots beyondmodel assumptions. For this purpose the exact diagonalization method is employed to calculate nu- merically the electron density distribution in a �nite 1D wire and the distribution function of electrons over single-particle states. We have found that there are two unexpected features. First, the distribution function has a δ -like singularity at the Fermi energy at the background of a smooth dependence on the energy. �e second feature relates to the Fourier spec- trum of the spatial distribution of the electron density. �e Fourier spectrum has a step-like form at the wavevector 2kF . Above this threshold, the harmonics are absent or vanishing. �ese properties are universal in the sense that they do not de- pend on the e-e interaction strength, interaction radius, wire length, mean electron density. We argue that these features are caused by the Wigner ordering of electrons. We compare the obtained results with the calculations carried out within the frame of the common Lu�inger liquid approach to �nd that it fails to capture the above results. �e Lu�inger liquid theory also gives a singularity of the distribution function at the Fermi energy, but its form is incorrect. �e Fourier spec- trum of the electron density distribution turns out to have an incorrect singularity at 2kF and not to vanish above this value. We have clari�ed that this discrepancy appears because the Lu�inger liquid theory does not describe correctly the deformation of the Wigner crystal by the boundaries. �e reason of this shortcoming lies in the violation of the parti- cle number conservation within the bosonization approach used. We �nd a way to remedy the Lu�inger liquid theory by introducing an improved expression for the electron density operator. �e proposed approach allows one to describe cor- rectly the above features of the distribution function and the Fourier harmonics of electron density. �e formulation of problem and exact diagonalization re- sults are presented in section II. �e results of bosonization with zero boundary conditions are given in section III. Sec- tion IV contains a simple model of Wigner ordering, as well as the interpretation of the results. Section V includes the review of bosonization approach for the case of zero bound- ary conditions, the construction of the density operator and the calculation of the observables with the particle number conservation taken into account. Technical details pertaining to the exact diagonalization can be found in Appendix A. II. EXACT DIAGONALIZATION STUDY OF 1D CORRELATED STATE A. Statement of problem Consider N spinless electrons in a 1D quantum box with zero boundary conditions for the many-electron wavefunc- tion, Ψ|x=0 = Ψ|x=L = 0 . (1) �e Hamiltonian is H = − V (xi − x j ) + U (xi ) , (2) whereV (x) is the e-e interaction potential, andU (x) is the po- tential of positively charged background, which is considered as jelly. Exact diagonalization method25 reduces to �nding the Hamiltonian matrix in the appropriate basis Φp and solving the eigenvalue problem by the standard methods of computa- tional linear algebra.26 As a result we obtain themany-electron wave function Ψ(x1, ..,xN ), expanded in this basis, Ψ(x1, ..,xN ) = apΦp(x1, ..,xN ) , (3) and the spectrum. It is convenient to choose the basis func- tions in the form of the Slater determinants Φp(x1, ..,xN ) = ������� ψα1 (x1) · · · ψαN (x1) ψα1 (xN ) · · · ψαN (xN ) ������� , (4) built from the eigenfunctions of the non-interacting single- particle Hamiltonian, ψq(x) = . (5) �e quantum number q ∈ N, labelling single-particle states, is analogous to momentum in translationally invariant systems, and will be called so for brevity. �e many-particle state, la- belled by the vector p = (α1, ..,αN ), is obtained by occupying the single-particle states ψαi , i = 1 . . .N . We adhere to the ordering convention that αi < αk for i < k . �e Hamiltonian matrix elements are given in Appendix A. �e interaction potential is chosen in the form V (x1 − x2) = Ve2/(2ϵd) exp(−|x1 − x2 |/d), with d being the interaction ra- dius, ϵ the permi�ivity of the medium. �is allows us to �nd the matrix elements analytically to cut the calculation time. Using such form of the potential is not essentially a limitation, because in our calculations we are able to vary the parameters V and d in a broad range to explore both the short-range and long-range (on the system length scale) interaction cases. Now let us express the observables we are going to �nd via the coe�cients of expansion (3). �e momentum distribution function is de�ned as n(q) = 〈Ψ| c+qcq |Ψ〉 , (6) where cq is electron destruction operator. Since ��Φp〉 is the eigenvector of c+qcq , n(q) = |ap |2θ p , (7) where the function θqp equals one if the many-particle state Φp has the single-particle stateψq occupied, and zero otherwise. �e average value of the particle density operator δ (x − xi ) (8) equals ρ(x) = 〈Ψ| ρ |Ψ〉 = Ψ∗(x ,x2, ..xN )Ψ(x ,x2, ..xN )dx2...dxN . Using (3), one gets ρ(x) = p1,p2 a∗p1ap2γp1p2 (x) , (10) where for p1 = p2 = (α1, ..,αN ) γp1p1 (x) = |ψαi (x)| 2 , (11) γp1p2 (x) = (−1) k1+k2ψ ∗αk1 (x)ψαk2 (x) (12) for the case when p1 and p2 have only two di�erent occupied statesψαi in positions k1 and k2, respectively; γ = 0 otherwise. �e cosine Fourier-transform of the density is ρ(q) = ρ(x) cos dx , q ∈ N , (13) and sine Fourier-transform is zero, since according to Eqs. (11), (12), ρ(x) contains only cosine harmonics. B. Results Below we present the results obtained for N = 12 electrons. �e system length is L = 333aB , the interaction radius d = n ( q ) 0 4 8 1 2 1 6 2 0 2 40 0 . 2 0 . 4 0 . 6 0 . 8 FIG. 1. Momentum distribution function, calculated for the system of N = 12 electrons. �e value of q = 12 corresponds to kF . 33aB , with aB being e�ective Bohr’s radius, V = 3.6. �e corresponding value of the RPA parameter rs = (2naB )−1 is 13.9, the estimate of the Lu�inger liquid interaction parameter according to д = (1 +Vq=0/π~vF )−0.5 gives 0.3. �e distribution function n(q) over the single-particle states ψq is shown in Fig. 1. Far from the Fermi surface, the form of the n(q) curve is smoothed by the interaction, which is familiar from the Lu�inger model. However, right at the Fermi surface there appears an unexpected δ -type singularity, with the value of the distribution function being close to 1 at this point. �e presence of the singularity was checked by changing the number N of electrons from 3 to 20, varying the length parameters and the interaction strength by the two orders of magnitude. �e result proved to be perfectly stable against the change in the system parameters (L,d,V ,N ). Hence, the δ -singularity in the momentum distribution n(q) at q = kF is a universal property of �nite 1D systems. Its origin is explained in section IV. Electron density ρ(x) in the ground state is an oscillatory function, the amplitude of which decays o� the boundaries. To analyze the ordering in the electron state, consider the Fourier- transform of electron density ρ(q), which is shown in Fig. 2. For comparison, the results for non-interacting electrons are also provided. For free electrons, ρ(q) has a step-like structure, with ρ(q) = −0.5 for 0 < q ≤ 2kF , and ρ(q) = 0 for q > 2kF . We emphasize that such threshold behavior holds even for a strongly interacting system, where ρ(q) remains very close to zero for q > 2kF . Electron-electron interaction modi�es the values of density harmonics only for 0 < q ≤ 2kF . �e harmonics ρ(q) for 0 < q < 2kF are suppressed by the interaction. �e q = 2kF harmonic of the density is enhanced by interaction, reaching the values comparable to one half of the background density. �is re�ects the well-known fact that e-e interaction leads to the strong electron correlations on the scale of average inter-particle distance, or in other words, to the Wigner-type ordering in the system. � 1 . 5 � 0 . 5 0 0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 I n t e r a c t i o n o nI n t e r a c t i o n o f f FIG. 2. �e Fourier-transform of electron density. �e value of q = 24 corresponds to 2kF . �e presented results are exact, since they are based on a precise many-particle wave function. In the next section we compare them to the results of the Lu�inger liquid theory. III. BOUNDED LUTTINGER LIQUID THEORY One of the most advanced analytical theories of a strongly correlated electron state in 1D quantum dots is a Lu�inger liquid theory, based on the bosonization with zero (open) boundary conditions.19–21 In this theory the spatial distribu- tion of electron density and the distribution of electrons over single-particle states are expressed through the Green func- tion of chiral fermions G+(x ,y) = 〈ψ++ (y)ψ+(x)〉 via n(q) = dxdyG+(x ,y)eiq(y−x ) , (14) and ρ(x) = −e−2ikF xG+(−x ,x) + c .c .�e Green function is G+(x ,y) = (eβ − 1) (д+1)2 4д +1 eβ − ei L (x−y) (cosh β − cos 2πxL )(cosh β − cos ] д−1−д cosh β − cos 2π (x−y)L ] (д+1)2 cosh β − cos 2π (x+y)L ] 1−д2 with the dimensionless cuto� parameter β ≈ N −1. �e momentum distribution function is presented in Fig. 3. A comparison with exact diagonalization shows that this re- sult is qualitatively incorrect. �e momentum distribution function, calculated within bosonization, also has a singular- ity at q = kF , but instead of a single δ -peak at q = kF , n(q) deviates from a smoothed step in a �nite band around kF , where the form of the curve is close to the derivative of the δ -peak.27 n ( q ) 0 . 1 0 . 2 0 . 3 0 . 4 B o xR i n g FIG. 3. Momentum distribution in the Lu�inger model with zero (�lled circles) and periodic (empty circles) boundary conditions, the interaction parameter д = 0.3. � 2 0 � 1 0 2 k F FIG. 4. �e Fourier-transform of electron density in the Lu�inger model with zero boundary conditions, the interaction parameter д = 0.3. Electron density, calculated according to (15), equals ρ(x) = 1 − 2 2 sinhд(β/2) cos(2kFx − 2f (x)) [cosh β − cos(2πx/L)]д/2 with f (x) being f (x) = arctan sin(2πx/L) eβ − cos(2πx/L) . (17) Its Fourier-transform is presented in Fig. (4). �e qualitative error of this result is that ρ(q) does not vanish at q > 2kF , but, on the contrary, grows rapidly as q → 2kF + 0. In other words, a whole branch appears in the region q > 2kF , which is absent in the exact solution. �us bosonization breaks down for the momenta close to kF and multiples of it. �is scale corresponds to the mean inter-particle distance. Hence, the bosonization with zero boundary conditions incorrectly treats the short-ranged elec- tron correlations, responsible for the formation of the ordered, Wigner-like state in 1D quantum dots. In the next section we demonstrate that the bosonization results describe the deformed Wigner crystal. IV. MODEL OF WIGNER ORDERING Consider a simple model that takes into account theWigner ordering from the very beginning. In this model, the many- electronwavefunctionΦ(y1, ..,yN ) is represented by the Slater determinant Φ(y1, ..,yN )= ������� ϕ(y1−x1) · · · ϕ(y1−xN ) ϕ(yN −x1) · · · ϕ(yN −xN ) ������� , (18) built on the single-particle wavefunctions ϕ(x) = π−1/4a−1/2 exp(−x2/2a2), localized at positions xk , k = 1 . . .N . �e wavefunction ‘width’ a is assumed to be smaller than the distance between the particles L/N , so the wavefunctions do not overlap and form an orthonormal set. �e Fourier-transform of the density is ρ(q) = e− π 2q2a2 . (19) �e momentum distribution function n(q) can be expressed through the electron Green function, similarly to Eq. (14), n(q) = dxdyG(x ,y)ψ ∗q (x)ψq(y) . (20) �e Green function G(x ,y) = 〈ψ+(y)ψ (x)〉 is related to the one-particle density matrix ρ(x ,y) = Φ∗(y, z2, .., zN )Φ(x , z2, .., zN )dz2..dzN , (21) via G(x ,y) = N ρ(x ,y).28 Substitute this into Eq. (20) to get n(q) = N ����∫ dy Φ∗(y, z)ψq(y)����2 , (22) where z = (z2, .., zN ). Using Eq. (18), we �nally obtain n(q) = ����∫ dy ϕ∗(y − xk )ψq(y)����2 2q2a2 First consider the case of a Wigner crystal, i.e. when xk = (k − 12 ) N . �e Fourier-transform of the density ρ(q) is non- zero only for multiples of 2kF , ρ(n · 2kF ) = (−1)nNe−a 2k2Fn . (24) �e momentum distribution function n(q) equals n(q) = (−1)mδq,mN a2π 2 . (25) n ( q ) 0 . 2 0 . 4 0 . 6 0 . 8 2 k Fk F FIG. 5. Momentum distribution function in the model of localized electrons. n ( q ) 0 5 1 0 1 50 0 . 2 0 . 4 0 . 6 0 . 8 FIG. 6. Momentum distribution function, calculated by exact diag- onalization for the system of N = 5 electrons. �e value of q = 5 corresponds to kF . Fig. 5 shows that n(q) is fully consistent with the results of exact diagonalization, including the δ -singularity at the Fermi surface. Moreover, the additional δ -singularities at multiples of kF , predicted by the model, do exist in the exact solution for stronger interaction (д ≈ 0.1), see Fig. 6. �is model proves that the origin of the singularities of n(q) is the ordering of electrons in bounded 1D systems. In in�nite 1D systems, the ordering is destroyed by �uctuations, and short-range electron correlations manifest themselves only in dynamic response to external perturbation.22 If the system is �nite, the boundaries pin the charge density waves, giving rise to Friedel oscillations, the amplitude of which is enhanced by e-e interaction. �is results in the increase of the weight of the state with q = kF , which is re�ected in the momentum distribution function, as well as in the high value of 2kF - harmonic of density. �e conclusion about the important role of short-range electron correlations in bounded 1D systems is con�rmed by the calculation of momentum distribution of electrons in a ring, i.e. in a system without boundaries. In this case, the singularity of the momentum distribution disappears, as can be seen from Fig. (3). n ( q ) 0 . 2 0 . 4 0 . 6 0 . 8 FIG. 7. Momentum distribution of the deformed Wigner crystal. � 2 0 � 1 0 2 k F FIG. 8. �e Fourier-transform of electron density for the deformed Wigner crystal. Now let us squeeze electrons down to the center of the system by introducing a displacement δxk ∝ −(k − N2 ) �e momentum distribution function and the density Fourier- spectrum, calculated according to (19), (23) for this case, are presented in Figs. 7, 8. �e curves are clearly identical to those from the previous section. �is con�rms our suggestion that bosonization describes the electron state that is deformed in comparison with the exact solution. �e origin of such deformation is discussed in the next section. V. BOSONIZATIONWITH PARTICLE NUMBER CONSERVATION A. Formalism Let us brie�y review the theory of the bounded Lu�inger liquids, following19–21. Decompose the electron �eld operator ψ (x) = cqψq(x) (26) into the �eldsψr (x) of the so-called r -fermions (r = ±), ψ (x) = eikF xψ+(x) + e−ikF xψ−(x) , (27) where ψr (x) = − irkx . (28) �e �elds are not independent, since ψ+(x) = −ψ−(−x) , (29) so we will deal withψ+(x) alone. �e la�er has the property ψ+(L) = ψ+(−L) , (30) i.e. it satis�es periodic boundary conditions on the interval [−L,L]. Hence, it can be bosonized in a conventional way. �e main assumptions of bosonization are the lineariza- tion of r -fermion spectrum near the Fermi surface and its extrapolation to in�nity. �ey allow one to solve the model exactly. Consider the r -fermion density operator, ρ(q) = : c+k+qck := c+k+qck − δq,0 c+k ck for q , 0. Zero harmonic ρ(q = 0) is the number of particles operator ∆N . �e density operator obeys the commutation relation [ρ(q), ρ(−q′)] = −δq,q′ , (32) which allows one to introduce bosons b+q = i ρq , q > 0 . (33) �e r -fermion �eld operator has the following boson repre- sentation, ψ+(x) = − U ei(ϕ(x )+ L ∆N ) , (34) where the bosonic phase ϕ equals ϕ(x) = e−iqx−α |q | ρq , (35) U stands for the ladder operator, α denotes the ultraviolet cuto�, which by the order of magnitude equals k−1F . �e r -fermion spatial density is related to bosonic phase ρ+(x) =: ψ++ (x)ψ+(x) := , (36) and ρ−(x) = ρ+(−x). To obtain the density operator of real electrons, one has to expressψ (x) in terms ofψr (x). Using Eq. (27), one gets ρ(x) = ρlw(x) + ρCDW(x) , (37) where the �rst term is the long-wave component of the den- sity, ρlw(x) = ρ+(x) + ρ−(x) , (38) and the second term is the charge density wave (CDW) com- ponent ρCDW(x) = e−2ikF xψ++ (x)ψ−(x) + h.c . (39) which describes short-range electron correlations. In terms of bosonic phase, the electron density operator equals ρ(x) = cos(2kFx − 2φ(x) − 2f (x)), (40) where φ(x) = 12 (ϕ(−x) − ϕ(x)), and function f (x) is an addi- tional phase due to zero boundary conditions, f (x) = [ϕ(x),ϕ(−x)] = arctan sin(2πx/L) eβ − cos(2πx/L) , (41) with dimensionless cuto� β = πα/L ≈ N −1. �e kinetic energy H0 = vF k : c+k ck : (42) is bosonized using Kronig’s identity to give H0 = vF q : b+qbq : + (∆N )2 . (43) �e interaction part of the Hamiltonian equals dxdy ρ(x)V (x − y)ρ(y) , (44) where ρ(x) is given by Eq. (40). In the model considered,19–21 the interaction operator is simpli�ed by retaining only the direct dxdy (ρ+(x)ρ+(y) + ρ−(x)ρ−(y))V (x − y) (45) and cross terms dxdy (ρ+(x)ρ−(y) + ρ−(x)ρ+(y))V (x − y) . (46) �e cross term contains the non-local contribution ρ+(x)ρ−(y) = ρ+(x)ρ+(−y), which makes the diagonalization of the Hamiltonian impossible, unless the interaction poten- tial is assumed short-ranged, that is of the form Vδ (x − y). Under this assumption, the direct and cross terms transform dx ρ2+(x) , (47) dx ρ+(x)ρ+(−x) . (48) Combining the terms, and using bosonic representation (36) of the density operator, we arrive at the following model Hamiltonian: H = H0 +Vd +Vc = (vF + b+qbq + q + bqbq (vF + )(∆N )2. �e Hamiltonian is diagonalized by Bogoliubov transforma- b̃q = e iSbqe −iS , (50) where q − bqbq . (51) �e Hamiltonian is diagonal if e2ξ = (1 + V /πvF )−1/2. �e interaction parameter д ≡ e2ξ belongs to (0, 1) for repulsive interaction, and equals unity for free electrons. �e diagonal form of the Hamiltonian is qv(q) b̃+q b̃q + vN (∆N )2 , (52) v(q) = vF +V /2π cosh 2ξ , vN = vF + . (53) �e bosonic representation (34) of the �eld operator, the den- sity operator (40) and quadratic Hamiltonian (52) are su�cient to obtain the Green function (15) with all the ensuing prob- lems for the momentum distribution (14) and density (16). �e root of these problems lies in the density operator (40), commonly used in the bosonization approach. B. Density operator �e bosonization approach as presented above violates the particle number conservation. To see this, just notice that the integral of the density �uctuation (the second term in Eq. (16)) over the system length is not zero. �is is a well- known problem, which exists not only in the case of zero boundary conditions, but also in a standard bosonization on the ring.22 �e problem arises at the level of the electron density operator (40), the CDW component of which does not conserve the number of particles in an isolated system. �e physical reason of the violation of the particle number conservation is that the CDW component of (40) includes the response of the in�nite positron sea, which is not completely eliminated when using the approximate relation (27). In the case of zero boundary conditions, the situation is reacher because now there appear problems even with the long-wave component of the density. Indeed, Eq. (16) of the common Lu�inger liquid theory does not give a correct tran- sition to the case of non-interacting electrons in the box. �e density of free electrons ρfree(x) = sin(2kF + πL )x 2 sin πxL contains an additional term 1/2L, missing in (16). �is term, being integrated over the length of the system, gives an extra charge of e/2. �us the microscopic theory leads to the density operator that violates the particle number conservation. We will obtain the correct operator, following the harmonic-�uid approach by Haldane.29 Introduce the phase θ (x) that increases by π each time x passes the location xk of a k-th particle. �e particle density operator (8) becomes ρ(x) = ∂xθ δ (θ (x) − kπ ) = e2imθ (x ) . (55) According to Eq. (40), θ (x) = kFx − φ(x) − f (x) . (56) �us the density operator takes the form ρ(x) = ) cos(2kFx − 2φ(x) − 2f (x)), where we retain onlym = 0,±1 harmonics, and halve the am- plitude of the CDW component to obtain the correct transition to the non-interacting case.22 �e density operator of Eq. (57) has the form of a full dif- ferential, which guarantees that the integral of the density �uctuation over the length of the system is zero, i.e. the num- ber of particles is conserved. �e long-wave part of the oper- ator contains an additional term −∂x f /π that gives the 1/2L component, missing in (16), since in the bulk of the system f (x) ≈ π4 − C. Hamiltonian and observables �e interaction part (44) of the Hamiltonian, having been calculated with the density operator of Eq. (57), gets additional terms of the form30 dx ∂xφ∂x f = iV (b̃+2q − b̃2q). (58) By shi�ing bosons d2q = b̃2q + i we obtain a diagonalized full Hamiltonian H + H1, H + H1 = qv(q)d+qdq . (60) �e bosonic phase (35) is transformed into ϕ(x) = ϕ0(x) + ϕ1(x), where ϕ0(x) is linear in new bosons, ϕ0(x)= (ceiqx−se−iqx )dq + (ce−iqx−seiqx )d+q with c and s being, respectively, cosh ξ and sinh ξ . �e func- tion ϕ1(x) is the new phase, speci�c to the case of zero bound- ary conditions, ϕ1(x) = Af (x), A = Vд2/2πvF . As a result, the �eld operator (34) acquires the factor of exp(iAf (x)), and the Green function (15) transforms as Gnew(x ,y) = G+(x ,y)eiA(f (x )−f (y)) . (62) �e average value of the density equals 〈ρ(x)〉 = − (1 −A) sinhд(β/2) 21−д/2π sin(2kFx − 2(1 −A)f (x)) [cosh β − cos 2πxL ]д/2 Note that the additional phase (1 −A)f (x), which appears in Eq. (63), changes the period of density oscillations. Since ∂x f < 0 everywhere in the system except narrow regions near the ends, the Wigner crystal is squeezed by the boundaries. �e deformation is determined by the coe�cient (1−A). In the common Lu�inger liquid theory A = 0. Our approach with corrected expression (57) for the density operator gives A > 0. Hence, restoring total neutrality results in the reduction of the Wigner crystal compression. �e distribution function n(q) and the Fourier spectrum of the density calculated with the use of the above expression for A coincide with the numerical calculations quite well. How- ever this formula is justi�ed for weak interaction (1 − д � 1). We propose a generalized expression for any д taking into account that A is known for three limiting cases: i) For д = 1, A = 0 to provide the correct transition to the case of non-interacting electrons. ii) For weak interaction, A should be proportional to V /2πvF , in agreement with our model. iii) For strong interaction д → 0, A → 1 to provide the transition to the limiting case of the Wigner crystal with strictly periodic density. �e simplest choice of A = 1 − д satis�es these require- ments and proves to be highly successful, as is demonstrated below. Figs. 9, 10 show the momentum distribution function and the density Fourier-transform, calculated according to Eqs. (62), (63). �ey are seen to agree nicely with the exact results of section II. Fig. 11 shows the spatial distribution of the density, calcu- lated according to Eqs. (16), (54), (63). It is seen that according to the standard bosonization the electron density maxima al- most coincide with those of free electrons, even for strong n ( q ) 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 FIG. 9. Momentum distribution function, the violation of the particle number conservation is �xed. 0 2 k F FIG. 10. �e Fourier-transform of electron density, the violation of the particle number conservation is �xed. interaction. In contrast, according to Eq. (63) the electron loca- tions are shi�ed towards the periodic positions as Wigner or- dering prescribes at strong interaction. �us the bosonization approach that does not respect the particle number conserva- tion leads to the picture of the state with electrons squeezed to the center of the system. VI. CONCLUDING REMARKS In this work, the ground state of interacting electrons in a 1D quantum dot was investigated using the exact diagonal- ization. An unexpected δ -like singularity was discovered in the momentum distribution function at the Fermi energy. A threshold behavior was found for the spatial Fourier-spectrum of electron density, with the step at 2kF . �ese e�ects are sta- ble against the change in the system length, interaction radius, the number of electrons, and the interaction strength. �us we suggest that they are inherent in �nite 1D electronic systems. We proposed a simple model which proved that these e�ects originate from the formation of the Wigner molecule in a 1D quantum dot. Comparison of exact results with the Lu�inger 0 0 . 5 10 FIG. 11. �e spatial distribution of the density, calculated according to Eqs. (16) (dashed line), Eq. (54) (thin solid line), Eq. (63) (solid line). model with zero boundary conditions shows that the la�er does not correctly describe both the momentum distribution near the Fermi energy and the density Fourier-spectrum. �e problem is that bosonization overestimates the deformation of the Wigner crystal caused by the boundaries by introduc- ing the excessive positive charge into the 1D system, which a�racts electrons to the center of the system. �is is a result of using the density operator that violates the number of par- ticles conservation. We derived the density operator devoid of the mentioned shortcoming, corrected the Hamiltonian, and calculated the observables to �nd a nice agreement with the exact results. We emphasize that our results are speci�c to �nite 1D sys- tems with boundaries. �e above features of the momentum distribution function and density Fourier-spectrum are both kept with increasing the system length. As far as the spec- trum remains discrete, the δ -singularity of n(q) refers to a single point q = kF . Formally, the features survive even at L → ∞. However, this is not equivalent to the thermody- namic limit, which also requires that dephasing processes be taken into account. In the correct thermodynamic limit the features disappear together with the boundary e�ects. ACKNOWLEDGMENTS �is work was supported by Russian Foundation for Basic Research (project No. 05-02-16854), Russian Academy of Sci- ences (programs “�antum Nanostructures” and “Strongly Correlated Electrons in Semiconductors, Metals, Supercon- ductors, and Magnetic Materials”), RF Ministry of Education and Science, and Russian Science Support Foundation. Appendix A: Hamiltonian matrix elements �e Hamiltonian matrix elements Hp1p2 = Φ∗p1HΦp2 dx1..dxN (A1) are calculated by substituting Eqs. (2), (4) into Eq. (A1), expand- ing the Slater determinants, and using the orthogonality prop- erty of single-particle functions (5) during integration.31–34 �e matrix element of the kinetic energy equals Tp1p2 = ~2π 2 k2i δp1p2 . (A2) For the matrix elements of the pair e-e interaction, four situations are possible, depending on p1 and p2. 1. If p1 = p2 = (α1, ..,αN ), i.e. diagonal matrix elements are calculated, then Vp1p1 = i>j=1 Vαiα j ,αiα j , (A3) where Vαiα j ,αkαl =∫ ϕ∗αiα j (x1,x2)V (x1 − x2)ϕαkαl (x1,x2)dx1dx2, ϕαiα j (x1,x2) = ����ψαi (x1) ψα j (x1)ψαi (x2) ψα j (x2) ���� . (A5) 2. If p1 and p2 are identical except two states, numbered k1 and k2, respectively, i.e. if p1 = (α1, ..αk1 , ..αN ), p2 = (α1, ..βk2 , ..αN ), then Vp1p2 = (−1) k1+k2 Vαk1αi ,βk2αi , (A6) 3. If p1 and p2 are identical except four states, numbered ki , i = 1 . . . 4, i.e. if p1 = (α1, ..αk1 , ..αk2 , ..αN ), p2 = (α1, ..βk3 , ..βk4 , ..αN ), then Vp1p2 = (−1) Vαk1αk2,βk3 βk4 . (A7) 4. If p1 and p2 contain more than four not coincident states αi , then Vp1p2 = 0 . (A8) Matrix elements of the interaction between electrons and positive background are as follows. 1. If p1 = p2 = (α1, ..,αN ), then Up1p1 = fαiαi , (A9) where fαiαk = ψ ∗αi (x)U (x)ψαk (x)dx . (A10) 2. If p1 and p2 are identical except two states, numbered k1 and k2, respectively, i.e. if p1 = (α1, ..αk1 , ..αN ), p2 = (α1, ..βk2 , ..αN ), then Up1p2 = fαk1 βk2 . (A11) 3. Otherwise,Up1p2 = 0. �e matrix element of the pair e-e interaction (A4) equals Vl j,km = [f (l , j,k,m) − f (l , j,k,−m)− f (l , j,−k,m) + f (l , j,−k,−m) − f (l ,−j,k,m)+ f (l ,−j,k,−m) + f (l ,−j,−k,m) − f (l ,−j,−k,−m)− f (j, l ,k,m) + f (j, l ,k,−m) + f (j, l ,−k,m)− f (j, l ,−k,−m) + f (j,−l ,k,m) − f (j,−l ,k,−m)− f (j,−l ,−k,m) + f (j,−l ,−k,−m)], (A12) where f (l ,n,k,m) = д(l + k,n +m), and д(p,q) = 0 for odd (p + q), while for even (p + q) д(p,q) = (e−a(−1)p − 1) 1 + π a2 pq (1 + π 2a2 p 2)(1 + π 2a2 q 1 + π δp,−q , (A13) with a = L/d . Matrix element (A10) equals fln = N (д(l + n, 0) − д(l − n, 0)) . (A14) 1 L.P. Kouwenhoven, D.G. Austing, and S. Tarucha, Rep. Prog. Phys. 64, 701 (2001). 2 S.M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 (2002). 3 G.A. Fiete, J. Qian, Ya. Tserkovnyak, and B.I. 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Löwdin, Phys. Rev. 97, 1474 (1955). Deformed Wigner crystal in a one-dimensional quantum dot Abstract I Introduction II Exact diagonalization study of 1D correlated state A Statement of problem B Results III Bounded Luttinger liquid theory IV Model of Wigner ordering V Bosonization with particle number conservation A Formalism B Density operator C Hamiltonian and observables VI Concluding remarks Acknowledgments A Hamiltonian matrix elements References

0704.1446

Curvature in Synthetic Differential Geometry of Groupoids

Curvature in Synthetic Differential Geometry of Groupoids Hirokazu Nishimura Institute of Mathematics, University of Tsukuba Tsukuba, Ibaraki, 305-8571, Japan October 28, 2018 Abstract We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particu- lar, the classical Bianchi identity is deduced from its combinatorial one. 1 Introduction The notion of curvature, which is one of the most fundamental concepts in dif- ferential geometry, retrieves its combinatorial or geometric meaning in synthetic differential geomety. It was Kock [4] who studied it up to the second Bianchi identity synthetically for the first time. In particular, he has revealed the com- binatorial nature of the second Bianchi identity by deducing it from an abstract In [4] Kock trotted out first neighborhood relations, which are indeed to be seen in formal manifolds, but which are no longer expected to be seen in microlinear spaces in general. Since we believe that microlinear spaces should play the same role in synthetic differential geometry as smooth manifolds have done in classical differential geometry, we have elevated his ideas to a microlinear context in [8]. However we were not so happy, because our proof of the second Bianchi identity there appeared unnecessarily involved, making us feel that we were somewhat off the point, though the proof was completely correct. Recently we got accustomed to groupoids, which encouraged us to attack the same problem once again. Within the framework of groupoids, we find it pleasant to think multiplicatively rather than additively (cf. Nishimura [11]), which helps grasp the nature of the second Bianchi identity firmly. Now we are to the point. What we have to do in order to deduce the classical second Bianchi identity from the combinatorial one is only to note some commutativity on the infinitesimal level, though groupoids are, by and large, highly noncom- mutative. Our present experience is merely an example of the familiar wisdom in mathematics that a good generalization reveals the nature. http://arxiv.org/abs/0704.1446v1 2 Preliminaries 2.1 Synthetic Differential Geometry Our standard reference on synthetic differential geometry is Chapters 1-5 of Lavendhomme [5]. We will work internally within a good topos, in which the intended set R of real numbers is endowed with a cornucopia of nilpotent in- finitesimals pursuant to the general Kock-Lawvere axiom. To see how to build such a good topos, the reader is referred to Kock [1] or Moerdijk and Reyes [7]. Any space mentioned in this paper will be assumed to be microlinear, unless stated to the contrary. We denote by D the set {d ∈ R | d2 = 0}, as is usual in synthetic differential geometry. Given a group G, we denote by AG the tangent space of G at its identity, i.e., the totality of mappings t : D → G such that t0 is the identity of G. We will often write td rather than t(d) for any d ∈ D. As we will see shortly, AG is more than an R-module. Proposition 1 For any t ∈ AG and any (d1, d2) ∈ D(2), we have td1+d2 = td1td2 = td2td1 so that td1 and td2 commute. Proof. By the same token as in Proposition 3 of §3.2 of Lavendhomme [5]. As an easy corollary of this proposition, we can see that t−d = (td) for we have (d,−d) ∈ D(2). Proposition 2 For any t1, t2 ∈ AG, we have (t1 + t2)d = (t2)d(t1)d = (t1)d(t2)d for any d ∈ D, so that (t1)d and (t2)d commute. Proof. By the same token as in Proposition 6 of §3.2 of Lavendhomme [5]. As an easy corollary of this proposition, we can see, by way of example, that (t1)d1d2 and (t2)d1d3 commute for any d1, d2, d3 ∈ D, for we have (t1)d1d2(t2)d1d3 = (d2t1)d1(d3t2)d1 = (d3t2)d1(d2t1)d1 = (t2)d1d3(t1)d1d2 Proposition 3 For any t1, t2 ∈ AG, there exists a unique s ∈ AG such that sd1d2 = (t2)−d2(t1)−d1(t2)d2(t1)d1 for any d1, d2 ∈ D. Proof. By the same token as in pp.71-72 of Lavendhomme [5]. We will write [t1, t2] for the above s. Theorem 4 The R-module AG endowed with the above Lie bracket [·, ·] is a Lie algebra over R. Proof. By the same token as in our previous paper [10]. 2.2 Groupoids Groupoids are, roughly speaking, categories whose morphisms are always invert- ible. Our standard reference on groupoids is MacKenzie [6]. Given a groupoid G over a base M with its object inclusion map id : M → G and its source and target projections α, β : G → M , we denote by B(G) the totality of bisections of G, i.e., the totality of mappings σ : M → G such that α ◦ σ is the identity mapping on M and β ◦ σ is a bijection of M onto M . It is well known that B(G) is a group with respect to the operation ∗, where for any σ, ρ ∈ B(G), σ ∗ ρ ∈ B(G) is defined to be (σ ∗ ρ)(x) = σ((β ◦ ρ)(x))ρ(x) for any x ∈ M . It can easily be shown that the space B(G) is microlinear, provided that both M and G are microlinear, for which the reader is referred to Proposition 6 of Nishimura [10]. Given x ∈ M , we denote by AnxG the totality of mappings γ : D n → G with γ(0, ..., 0) = idx and (α ◦ γ)(d1, ..., dn) = x for any (d1, ..., dn) ∈ D n. We denote by AnG the set-theoretic union of AnxG’s for all x ∈ M . In particular, we usually write AxG and AG in place of AxG and AG respectively. It is easy to see that AG is naturally a vector bundle over M . A morphism ϕ : H → G of groupoids over M naturally gives rise to a morphism ϕ∗ : AH → AG of vector bundles over M . As in §3.2.1 of Lavendhomme [5], where three distinct but equivalent viewpoints of vector fields are presented, the totality Γ(AG) of sections of the vector bundle AG can canonically be identified with the totality of tangent vectors to B(G) at id, for which the reader is referred to Nishimura [10]. We will enjoy this identification freely, and we dare to write Γ(AG) for the totality of tangent vectors to B(G) at id. Given X,Y ∈ Γ(AG), we define a microsquare Y ∗X to B(G) at id to be (Y ∗X)(d1, d2) = Yd2 ∗Xd1 for any (d1, d2) ∈ D Given γ ∈ An+1G and e ∈ D, we define γie ∈ A nG (1 ≤ i ≤ n+ 1) to be γie(d1, ..., dn) = γ(d1, ..., di−1, e, di, ..., dn)γ(0, ..., 0, e , 0, ..., 0)−1 for any (d1, ..., dn) ∈ D n. For our later use in the last section of this paper, we introduce a variant of this notation. Given γ ∈ An+2G and e1, e2 ∈ D, we define γi,je1,e2 ∈ A nG (1 ≤ i < j ≤ n+ 2) to be e1,e2 (d1, ..., dn) = γ(d1, ..., di−1, e1, di, ..., dj−2, e2, dj−1, ..., dn)γ(0, ..., 0, e1 , ..., e2 , 0, ..., 0)−1 Given γ ∈ A2G, we define τ1γ ∈ A 2G to be τ1γ (d1, d2) = γ(d1, 0) for any (d1, d2) ∈ D 2. Similarly, given γ ∈ A2G, we define τ2γ ∈ A 2G to be γ (d1, d2) = γ(0, d2) for any (d1, d2) ∈ D 2. Given γ ∈ A2G, we define Σγ ∈ A2G to be (Σγ)(d1, d2) = γ(d2, d1) for any (d1, d2) ∈ D Any γ ∈ A2G can canonically be identified with the mapping e ∈ D 7→ γ1e ∈ AG, so that we can identify A2G and (AG)D. As is expected, this identification enables us to define γ2 − γ1 ∈ A 2G for γ1, γ2 ∈ A 2G, provided that γ1(0, ·) = γ2(0, ·). Similarly, we can define γ2 − γ1 ∈ A 2G for γ1, γ2 ∈ A 2G, provided that γ1(·, 0) = γ2(·, 0). Given γ1, γ2 ∈ A 2G, their strong difference γ2 − γ1 ∈ AG is defined, provided that γ1 |D(2)= γ2 |D(2). Lavendhomme’s [5] treatment of strong difference − in §3.4 carries over mutatis mutandis to our present context. We note in passing the following simple proposition on strong difference −, which is not to be seen in our standard reference [5] on synthetic differential geometry. Proposition 5 For any γ1, γ2, γ3 ∈ A 2G with γ1 |D(2)= γ2 |D(2)= γ3 |D(2), we − γ1) + (γ3 − γ2) + (γ1 − γ3) = 0 2.3 Differential Forms Given a groupoid G and a vector bundle E over the same space M , the space Cn(G,E) of differential n-forms with values in E consists of all mappings ω from AnG to E whose restriction to AnxG for each x ∈ M takes values in Ex satisfying the following n-homogeneous and alternating properties: 1. We have ω(a · γ) = aω(γ) (1 ≤ i ≤ n) for any a ∈ R and any γ ∈ AnxG, where a · γ ∈ AnxG is defined to be γ)(d1, ..., dn) = γ(d1, ..., di−1, adi, di+1, ...dn) for any (d1, ..., dn) ∈ D 2. We have ω(γ ◦Dθ) = sign(θ)ω(γ) for any permutation θ of {1, ..., n}, where Dθ : Dn → Dn permutes the n coordinates by θ. 3 Connection Let π : H → G be a morphism of groupoids over M . Let L be the kernel of π with its canonical injection ι : L → H . It is clear that L is a group bundle over M . These entities shall be fixed throughout the rest of the paper. Thus we have an exact sequence of groupoids as follows: 0 → L A connection ∇ with respect to π is a morphism ∇ : AG → AH of vector bundles over M such that the composition π∗ ◦ ∇ is the identity mapping of AG. A connection ∇ with respect to π shall be fixed throughout the rest of the paper. If G happens to be M ×M (the pair groupoid of M) with π being the projection h ∈ H 7→ (α(h), β(h)) ∈ M ×M , our present notion of connection degenerates into the classical one of infinitesimal connection. Given γ ∈ An+1G, we define γi ∈ AG (1 ≤ i ≤ n+ 1) to be γi(d) = γ(0, ..., 0, d , 0, ..., 0) for any d ∈ D. As in our previous paper [11], we have Theorem 6 Given ω ∈ Cn(G,AL), there exists a unique d∇ω ∈ C n+1(G,AL) such that ((d∇ω)(γ))d1...dn+1 {(ω(γi0))d1...bdi...dn+1((∇γi)di) −1(ω(γidi))−d1... bdi...dn+1(∇γi)di} (−1)i for any γ ∈ An+1G and any (d1, ..., dn+1) ∈ D 4 A Lift of the Connection ∇ to microsquares Let us define a mapping A2G → A2H , which shall be denoted by the same symbol ∇ hopefully without any possible confusion, to be ∇γ(d1, d2) = (∇γ )d2(∇γ 0 )d1 for any γ ∈ A2G. It is easy to see that Proposition 7 For any γ ∈ A2G and any a ∈ R, we have ∇(a · γ) = a · ∇(a · γ) = a · Corollary 8 For any γ1, γ2 ∈ A 2G, we have ∇(γ2 − γ1) = ∇γ2 − ∇γ1 provided that γ1(0, ·) = γ2(0, ·); ∇(γ2 − γ1) = ∇γ2 − ∇γ1 provided that γ1(·, 0) = γ2(·, 0). Proof. This follows from the above proposition by Proposition 10 of §1.2 of Lavendhomme [5]. Proposition 9 For any t ∈ A1G, we define εt ∈ A 2G to be εt(d1, d2) = t(d1d2) Then we have (∇εt)(d1, d2) = (∇t)(d1d2) for any d1, d2 ∈ D. Proof. It suffices to note that (∇εt)(d1, d2) = (∇(d1t))(d2) = (d1∇t)(d2) = (∇t)(d1d2) Theorem 10 For any γ1, γ2 ∈ A 2G with γ1 |D(2)= γ2 |D(2), we have − γ1) = ∇γ2 Proof. Let d1, d2 ∈ D. We have (∇(γ2 − γ1))(d1d2) = (∇ε )(d1, d2) [By Proposition 9] = (∇((γ2 − ))(d1, d2) [By Proposition 7 of §3.4 of Lavendhomme [5]] = ((∇γ2 − ∇γ1)− ∇τ2γ1)(d1, d2) [By Corollary 8] = ((∇γ2 − ∇γ1)− )(d1, d2) (d1, d2) [By Proposition 7 of §3.4 of Lavendhomme [5]] = (∇γ2 −∇γ1)(d1d2) [By Proposition 9] Since d1, d2 ∈ D were arbitrary, the desired conclusion follows at once. 5 The Curvature Form Proposition 11 For any γ ∈ A2G, there exists a unique t ∈ A1L such that ι(td1d2) = ((∇γ 0)d1) −1((∇γ1d1)d2) −1(∇γ2d2)d1(∇γ for any d1, d2 ∈ D. Proof. Let η ∈ A2H to be η(d1, d2) = ((∇γ 0 )d1) −1((∇γ1d1)d2) −1(∇γ2d2)d1(∇γ for any d1, d2 ∈ D. Then it is easy to see that η(d, 0) = η(0, d) = idα(η(0,0)) Therefore there exists unique t′ ∈ A1H such that t′d1d2 = η(d1, d2) Furthermore we have π(η(d1, d2)) = π(((∇γ20 )d1) −1)π(((∇γ1d1)d2) −1)π((∇γ2d2 )d1)π((∇γ 0 )d2) = ((γ20)d1) −1((γ1d1)d2) −1(γ2d2)d1(γ = γ(d1, 0) −1(γ(d1, d2)γ(d1, 0) −1)−1γ(d1, d2)γ(0, d2) −1γ(0, d2) = idα(η(0,0)) Therefore there exists a unique t ∈ A1L with ι(t) = t′. This completes the proof. We write Ω(γ) for the above t. Now we have Proposition 12 The mapping Ω : A2G → A1L consists in C2(G,AL). Proof. We have to show that Ω(a · γ) = aΩ(γ) (1) Ω(a · γ) = aΩ(γ) (2) Ω(Σγ) = −Ω(γ) (3) for any γ ∈ A2G and any a ∈ R. Now we deal with (1), leaving a similar treatment of (2) to the reader. Let d1, d2 ∈ D. We have ι(Ω(a · γ))d1d2 = ((∇(a · γ)20)d1) −1((∇(a · γ)1d1)d2) −1(∇(a · γ)2d2)d1(∇(a ·1 γ)10)d2 = ((∇γ20 )ad1) −1((∇γ1ad1)d2) −1(∇γ2d2)ad1(∇γ 0 )d2 = ι(Ω(γ))ad1d2 = ι(aΩ(γ))d1d2 Now we deal with (3). We have ι(Ω(Σγ))ι(Ω(γ))d1d2 = {((∇γ10 )d2) −1((∇γ2d2)d1) −1(∇γ1d1)d2(∇γ 0)d1}{((∇γ 0)d1) −1((∇γ1d1)d2) −1(∇γ2d2)d1(∇γ 0)d2} = idα(γ(0,0)) Thie completes the proof. We call Ω the curvature form of ∇. Proposition 13 For any γ ∈ A2G, we have Ω(γ) = Σ∇Σγ Proof. As in the proof of Proposition 8 of §3.4 of Lavendhomme [5], let us consider a function l : D2 ∨D → H given by l(d1, d2, e) = (∇γ )d2(∇γ 0)d1Ω(γ)e for any (d1, d2, e) ∈ D 2∨D. Then it is easy to see that l(d1, d2, 0) = (∇γ)(d1, d2) and l(d1, d2, d1d2) = (Σ∇Σγ)(d1, d2). Therefore we have (Σ∇Σγ −∇γ)e = l(0, 0, e) = Ω(γ)e This completes the proof. Now we deal with tensorial aspects of Ω. It is easy to see that Proposition 14 Let X,Y ∈ Γ(AG). Then we have ∇(Y ∗X) = ∇Y ∗ ∇X Now we have the following familiar form for Ω. Theorem 15 Let X,Y ∈ Γ(AG). Then we have Ω(Y ∗X) = ∇[X,Y ]− [∇X,∇Y ] Proof. It suffices to note that Ω(Y ∗X) = Σ∇Σ(Y ∗X) −∇(Y ∗X) [By Proposition 13] = ∇Σ(Y ∗X) − Σ∇(Y ∗X) [By Proposition 6 of §3.4 of Lavendhomme [5]] = ∇(Σ(Y ∗X) −X ∗ Y )− (Σ∇(Y ∗X) −∇(X ∗ Y )) [By Proposition 5] = ∇(Y ∗X − Σ(X ∗ Y ))− (∇(Y ∗X) − Σ∇(X ∗ Y )) [By Proposition 6 of §3.4 of Lavendhomme [5]] = ∇(Y ∗X − Σ(X ∗ Y ))− (∇Y ∗ ∇X − Σ(∇X ∗ ∇Y )) [By Proposition 14] = ∇[X,Y ]− [∇X,∇Y ] [By Proposition 8 of §3.4 of Lavendhomme [5]] 6 The Bianchi Identity Let us begin with the following abstract Bianchi identity, which traces back to Kock [4], though our version is cubical, while Kock’s one is simplicial. Theorem 16 Let γ ∈ A2G. Let d1, d2, d3 ∈ D. We denote points β(γ(0, 0, 0)), β(γ(d1, 0, 0)), β(γ(0, d2, 0)), β(γ(0, 0, d3)), β(γ(d1, d2, 0)), β(γ(d1, 0, d3)), β(γ(0, d2, d3)) and β(γ(d1, d2, d3)) by O, A, B, C, D, E, F and G respectively. These eight points are depicted figuratively as the eight vertices of a cube: ւ ↓ ւ ↓ ↓ ւ ↓ ւ Theorem 17 For each pair (X,Y ) of adjacent vertices X,Y of the cube, PXY denotes the following arrow in H, while PY X denotes the inverse of PXY : POA = (∇γ 0,0)d1 POB = (∇γ 0,0)d2 POC = (∇γ 0,0)d3 PAD = (∇γ PAE = (∇γ PBD = (∇γ PBF = (∇γ PCE = (∇γ PCF = (∇γ PDG = (∇γ d1,d2 PEG = (∇γ d2,d3 PFG = (∇γ d1,d3 For any four vertices X,Y, Z,W of the cube rounding one of the six facial squares of the cube, RXY ZW denotes PWXPZWPY ZPXY . Then we have PAOPDAPGDRGFBDRGECFRGDAEPDGPADPOAROCEAROBFCROADB = idO Proof. Write over the desired identity exclusively in terms of PXY ’s, and write off all consective PXY PY X ’s. Now we are ready to establish the second Bianchi identity in familiar form. Theorem 18 We have d∇Ω = 0 Proof. Let γ, d1, d2, d3, O,A,B,C,D,E, F,G be the same as in the previous theorem. By the very definition of Ω, we have ROADB = Ω(γ 0)−d1d2 ROBFC = Ω(γ 0)−d2d3 ROCEA = Ω(γ 0)d1d3 Now we have the following three calculations: PAOPDAPGDRGDAEPDGPADPOA = PAORAEGDPOA = ((∇γ 0,0 )d1) −1Ω(γ1d1)d2d3(∇γ 0,0 )d1 PAOPDAPGDRGECFPDGPADPOA = PAOPDAPGDPEGPCERCFGEPECPGEPDGPADPOA = PAORAEGDPEAPCERCFGEPECPAERADGEPOA = ROCEAPCOPECPAERAEGDPEAPCERCFGEPECPAERADGEPEAPCEPOCROAEC = Ω(γ20 )d1d3((∇γ 0,0)d3) −1{((∇γ −1(∇γ )d3Ω(γ )d2d3((∇γ −1(∇γ Ω(γ3d3)d1d2{((∇γ −1(∇γ )d3Ω(γ )−d2d3((∇γ −1(∇γ 0,0)d3Ω(γ 0)−d1d3 = Ω(γ20 )d1d3{((∇γ 0,0)d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3}Ω(γ 0)−d1d3 [By Proposition 2, cf. Figure (4)] = ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 [By Proposition 2, cf. Figure (5)] C ((∇γ −1(∇γ )d3Ω(γ )−d2d3((∇γ −1(∇γ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ3d3)d1d2 ↓ ↓ Ω(γ )d1d2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ −1(∇γ )d3Ω(γ )−d2d3((∇γ −1(∇γ )d1 C O ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ20)−d1d3 ↓ ↓ Ω(γ 0)−d1d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 O PAOPDAPGDRGFBDPDGPADPOA = PAOPDARDGFBPADPOA = ROBDAPBORBDGFPOBROADB = Ω(γ30)d1d2{((∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0 )d2}Ω(γ 0)−d1d2 = ((∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0 )d2 [By Proposition 2, cf. Figure (6)] O ((∇γ 0,0 )d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 −−−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ30)−d1d2 ↓ ↓ Ω(γ 0)−d1d2 −−−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 O Therefore we have (d∇Ω(γ))−d1d2d3 = Ω(γ10)−d2d3{((∇γ 0,0)d1) −1Ω(γ1d1)d2d3(∇γ 0,0)d1} Ω(γ20)d1d3{((∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2} Ω(γ30)−d1d2{((∇γ 0,0)d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3} = {((∇γ 0,0)d1) −1Ω(γ1d1)d2d3(∇γ 0,0)d1}{((∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2} {((∇γ 0,0)d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3}Ω(γ 0)−d2d3Ω(γ 0)d1d3Ω(γ 0)−d1d2 [By Proposition 2, cf. Figures (7)-(12)] = {((∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2}{((∇γ 0,0)d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3} {((∇γ 0,0)d1) −1Ω(γ1d1)d2d3(∇γ 0,0)d1}Ω(γ 0)d1d3Ω(γ 0)−d2d3Ω(γ 0)−d1d2 [By Proposition 2, cf. Figures (13)-(15)] = {PAOPDAPGDRGFBDPDGPADPOA}{PAOPDAPGDRGECFPDGPADPOA} {PAOPDAPGDRGDAEPDGPADPOA}ROCEAROBFCROADB = idO O ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ10)−d2d3 ↓ ↓ Ω(γ 0)−d2d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 O O (∇γ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ10)−d2d3 ↓ ↓ Ω(γ 0)−d2d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0)d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 O O ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ10)−d2d3 ↓ ↓ Ω(γ 0)−d2d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 O O ((∇γ 0,0 )d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 −−−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ20)d1d3 ↓ ↓ Ω(γ 0)d1d3 −−−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d2) −1Ω(γ2d2)−d1d3(∇γ 0,0)d2 O O ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ20)d1d3 ↓ ↓ Ω(γ 0)d1d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 O O ((∇γ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ Ω(γ30)−d1d2 ↓ ↓ Ω(γ 0)−d1d2 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0 )d3) −1Ω(γ3d3)d1d2(∇γ 0,0)d3 O O ((∇γ 0,0)d1) 0,0)d1 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0)d2) −d1d3 0,0)d2 ↓ ↓ ((∇γ 0,0)d2) −d1d3 0,0)d2 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0)d1) 0,0)d1 O O ((∇γ 0,0)d1) 0,0)d1 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0)d3) )d1d2(∇γ 0,0)d3 ↓ ↓ ((∇γ 0,0)d3) )d1d2(∇γ 0,0)d3 −−−−−−−−−−−−−−−−−−−−−−−−−→ 0,0)d1) 0,0)d1 O O Ω(γ10)−d2d3 −−−−−−−→ Ω(γ20)d1d3 ↓ ↓ Ω(γ 0)d1d3 −−−−−−−→ Ω(γ10)−d2d3 O This completes the proof. References [1] Kock, A.: Synthetic Differential Geometry, London Mathematical Society Lecture Note Series, 51, Cambridge University Press, Cambridge, 1981. [2] Kock, A.:Differential forms with values in groups (preliminary report), Cahiers Top. Geom. Diff., 22 (1981), 141-148. [3] Kock, A.:Differential forms with values in groups, Bull. Austral. Math. Soc., 25 (1982), 357-386. [4] Kock, A.:Combinatorics of curvature, and the Bianchi identity, Theory and Applications of Categories, 2 (1996), 69-89. [5] Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996. [6] Mackenzie, K. C. H.:General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge Uni- versity Press, Cambridge, 2005. [7] Moerdijk, I. and Reyes, G. E.: Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991. [8] Nishimura, H.:Another curvature in synthetic differential geometry, Bull. Belg. Math. Soc. Simon Stevin, 7 (2000), 161-171. [9] Nishimura, H.:The first Bianchi identity in synthetic differential geometry, Journal of Pure and Applied Algebra, 160 (2001), 263-274. [10] Nishimura, H.:The Lie algebra of the group of bisections, Far East Journal of Mathematical Sciences, 24 (2007), 329-342. [11] Nishimura, H.:Another coboundary operator for differential forms with val- ues in the Lie algebra bundle of a group bundle, Mathematics ArXiv, math.DG/0612067 http://arxiv.org/abs/math/0612067 Introduction Preliminaries Synthetic Differential Geometry Groupoids Differential Forms Connection A Lift of the Connection to microsquares The Curvature Form The Bianchi Identity

0704.1447

Can Gravity Probe B usefully constrain torsion gravity theories?

Can Gravity Probe B usefully constrain torsion gravity theories? Éanna É. Flanagan, Eran Rosenthal Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853 (Dated: draft of May 8, 2007; printed October 9, 2018 at 9:18) In most theories of gravity involving torsion, the source for torsion is the intrinsic spin of matter. Since the spins of fermions are normally randomly oriented in macroscopic bodies, the amount of torsion generated by macroscopic bodies is normally negligible. However, in a recent paper, Mao et al. (gr-qc/0608121) point out that there is a class of theories, including the Hayashi-Shirafuji (1979) theory, in which the angular momentum of macroscopic spinning bodies generates a significant amount of torsion. They further argue that, by the principle of action equals reaction, one would expect the angular momentum of test bodies to couple to a background torsion field, and therefore the precession of the Gravity Probe B gyroscopes should be affected in these theories by the torsion generated by the Earth. We show that in fact the principle of action equals reaction does not apply to these theories, essentially because the torsion is not an independent dynamical degree of freedom. We examine in detail a generalization of the Hayashi-Shirafuji theory suggested by Mao et al. called Einstein- Hayashi-Shirafuji theory. There are a variety of different versions of this theory, depending on the precise form of the coupling to matter chosen for the torsion. We show that for any coupling to matter that is compatible with the spin transport equation postulated by Mao et al., the theory has either ghosts or an ill-posed initial value formulation. These theoretical problems can be avoided by specializing the parameters of the theory and in addition choosing the standard minimal coupling to matter of the torsion tensor. This yields a consistent theory, but one in which the action equals reaction principle is violated, and in which the angular momentum of the gyroscopes does not couple to the Earth’s torsion field. Thus, the Einstein-Hayashi-Shirafuji theory does not predict a detectable torsion signal for Gravity Probe B. There may be other torsion theories which do. I. INTRODUCTION AND SUMMARY A. Theories of gravity with torsion General relativity (GR) is in good agreement with all current experimental data from laboratory tests, the So- lar System1 and binary pulsars [2]. However there is good motivation to consider modifications and extensions of GR: low energy limits of string theory and higher dimen- sional models usually involve extra, long range universal forces mediated by scalar fields, and in addition the ob- served acceleration of the Universe may be due to a mod- ification of GR at large distances. One may hope that new and highly accurate experiments, such as Gravity Probe B [3], will enable one to test for deviations from One natural framework in which to generalize GR is to allow the connection Γ to be a nonsymmetric inde- pendent dynamical variable instead of being determined by the metric. The covariant derivative of a vector vµ is defined in the usual way as ∇µvν = ∂µvν + Γνµλvλ. (1.1) If one retains the assumption that the connection is met- ric compatible, ∇µgνλ = 0, then it can be shown that the connection is determined uniquely by the metric and the 1 An exception is the Pioneer anomaly [1], which remains contro- versial. torsion tensor S λµν ≡ Γλ[µν]. (1.2) One obtains Γλµν = −K λµν (1.3) where the first term is the Levi-Civita connection deter- mined by the metric and K λµν = −S λµν − Sλνµ − Sλµν (1.4) is called the contorsion tensor. A spacetime equipped with a metric gµν and a torsion tensor is called a Riemann-Cartan spacetime. The Riemann tensor R of the full connection (1.3) is related to the usual Rie- mann tensor R̃ of the Levi-Civita connection by + ∇̃ρK µλν − ∇̃λK ρν +K K σρν −K µρσ K σλν , (1.5) where ∇̃µ is the Levi-Civita derivative operator, and our convention for the Riemann tensor is given by Eq. (B5). The action for theories of gravity in this framework has the generic form S[gµν ,Γ µν ,Ψ] = SG[gµν ,Γ µν ] + Smatter[gµν ,Γ µν ,Ψ] , (1.6) where SG and Smatter are the gravitational and matter ac- tions and Ψ collectively denotes the matter fields. There is an extensive literature on theories of gravity of this type; see the review articles [4, 5, 6, 7]. http://arxiv.org/abs/0704.1447v2 http://arxiv.org/abs/gr-qc/0608121 It is often useful to re-express these theories using the tetrad formalism [8]. In this formalism the independent variables are taken to be a tetrad of four linearly in- dependent vector fields e µa (x), and a tetrad connection ω abµ = −ω baµ defined by ~∇~ea~eb = e µa ω cµ b ~ec. (1.7) Here tetrad indices a run from 0 to 3 and are raised and lowered using the Minkowski metric ηab ≡ diag(−1, 1, 1, 1). The two sets of variables, gµν ,Γλµν and e µa , ω µ are related by gµν = ηabe ν , (1.8a) Γλµν = e eaν,µ + ω , (1.8b) where the dual basis of one-forms eaµ is defined by e µa e µ = δ a. (1.9) The action of the theory in terms of the tetrad variables S[e µa , ω µ ,Ψ] = SG[e a , ω µ ] + Smatter[e a , ω µ ,Ψ]. (1.10) Normally the theory is invariant under local Lorentz transformations Λ ba = Λ a (x) of the tetrad e µa → Λ ba e , (1.11a) ω aµ b → Λac Λ db ω cµ d − Λac,µΛ cb , (1.11b) together with the corresponding transformations of any fermionic matter fields. There are three different categories of theories involv- ing torsion: • Theories in which torsion is an independent dynam- ical variable and the field equations for torsion are algebraic, for example the Einstein-Cartan theory [4] in which the gravitational action is proportional to the Ricci scalar. In these theories the torsion vanishes in vacuum. • Theories in which the torsion tensor is an indepen- dent dynamical variable and a propagating degree of freedom, for example Refs. [4, 6, 7, 9, 10, 11, 12]. The source for torsion is the tensor δSmatter δω abµ For the standard, minimal coupling of torsion to matter, this tensor is a measure of density of funda- mental or intrinsic spin, and thus is very small when averaged over macroscopic distances in unpolarized matter [4, 6]. (For non-standard couplings it is con- ceivable that this tensor could be non-negligible.) 2 The connection (1.8b) is automatically metric compatible by virtue of the antisymmetry of ω abµ on a and b. • Theories in which the torsion is not an indepen- dent dynamical variable, but is specified in terms of some other degrees of freedom in the theory, for example a scalar potential [13, 14] or a rank 2 ten- sor potential [7, 15, 16]. A particular special case of the third category are the so-called teleparallel theories [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In these theories the only dynamical variable is the tetrad e µa , the tetrad con- nection ω abµ is not an independent variable. In addition the local Lorentz transformations (1.11a) are not a sym- metry of the theory. The tetrad therefore contains 6 extra physical degrees of freedom which are normally gauged away by the local Lorentz symmetry. In linear perturba- tion theory about flat spacetime these extra degrees of freedom act like a antisymmetric, rank 2 tensor poten- tial for the torsion; see Sec. II below for more details. In teleparallel theories the torsion is defined to be S λµν = e λa (e ν,µ − eaµ,ν) . (1.12) The form of this equation is invariant under coordinate transformations but not under the local Lorentz transfor- mations (1.11a). It follows from this definition and from Eq. (1.8b) that the tetrad connection ω abµ vanishes, and it follows that the Riemann tensor Rabµν ≡ eaλebσRλσµν = ω abν ,µ − ω abµ ,ν + ω aµ c ω cbν − ω aν c ω cbµ (1.13) also vanishes. Thus in teleparallel theories the curvature (1.5) of the full connection vanishes, and so on the right hand side of Eq. (1.5) the contorsion terms must can- cel the curvature term. Hence, if the spacetime metric is close to that predicted by general relativity, so that R̃ ∼ M/r3 at a distance r from a mass M , then the torsion must be of order S ∼ K ∼ M/r2. Therefore, teleparallel theories generically predict a non-negligible torsion for macroscopic, unpolarized bodies, unlike con- ventional torsion theories. B. Constraining torsion with Gravity Probe B The prevailing lore about torsion theories has been that they are very difficult to distinguish from general rel- ativity, since the torsion generated by macroscopic bod- ies is normally negligibly small for the reasons discussed above [7, 33]. However, a recent paper by Mao, Tegmark, Guth and Cabi (MTGC) [34] points out that teleparal- lel theories are an exception in this regard. They sug- gest that Gravity Probe B (GPB) might be an ideal tool to probe such torsion theories. In particular they argue that since the angular momentum of macroscopic bod- ies generates torsion, one would expect that the angular momentum of test bodies such as the GPB gyroscopes would couple to the Earth’s torsion field, by the princi- ple of “action equals reaction”. MTGC also review the literature on the equations of motion and spin precession of test bodies in torsion the- ories [14, 16, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. They argue that because there is some disagreement in this lit- erature, and because the precise form of the coupling of torsion to matter is not known, it is reasonable to assume that test bodies fall along geodesics of the full connection (called autoparallels), and to assume that the spin of a gyroscope is parallel transported with respect to the full connection. They introduce a theory of gravity called the Einstein-Hayashi-Shirafuji (EHS) theory, a general- ization of an earlier teleparallel theory of Hayashi and Shirafuji [18], and compute the constraints that GPB will be able to place on this theory for their assumed equations of motion and spin transport. In this paper we re-examine the utility of GPB as a probe of torsion gravity theories. We agree with the general philosophy expressed by MTGC that the pre- cise form of the coupling of torsion to matter is some- thing that should be tested experimentally rather than assumed a priori. However, while it is conceivable that there could exist couplings which would predict a de- tectable torsion signal for GPB, we show that teleparallel theories and the EHS theory in particular do not. We start by discussing the action equals reaction prin- ciple. This appears to be a robust and very generic ar- gument, indicating that the angular momentum of a test body should couple to torsion in theories where spinning bodies generate torsion. However, in fact there is a loop- hole in the argument, and in particular it does not apply to the EHS theory, as we show in detail in Sec. IV be- low. The nature of the loophole can be understood using a simple model. Consider in Minkowski spacetime the following theory of two scalar fields Φ1 and Φ2 and a particle of mass m S = − (∇Φ1)2 + (∇Φ2)2 dλ(m+ qΦ1) . (1.14) Here λ is a parameter along the worldline and q is a scalar charge. In this theory the particle generates a Φ1 field but not a Φ2 field, and correspondingly it feels a force from the Φ1 field but not the Φ2 field, in accordance with the action equals reaction idea. However, consider now the theory in non-canonical variables Φ̃1 = Φ1, Φ̃2 = Φ1+Φ2. In terms of these variables the particle generates both a Φ̃1 field and a Φ̃2 field, but feels a force only from the Φ̃1 field, in violation of action equals reaction. Thus, we see that the action equals reaction principle can only be applied to the independent dynamical variables in the theory, which diagonalize the kinetic energy term in the action. In the EHS theory, the torsion and metric are not independent dynamical variables; see Sec. IV below. C. The Einstein-Hayashi-Shirafuji theory In the remainder of this paper we examine in detail the EHS theory suggested by MTGC. In this theory the defining relation (1.12) between torsion and tetrad for teleparallel theories is replaced by S λµν = e λa (e ν,µ − eaµ,ν) , (1.15) where parameter σ lies in the range 0 ≤ σ ≤ 1. For σ = 1 this reduces to the teleparallel case, while for σ = 0 the torsion tensor vanishes. Thus, the EHS theory interpo- lates between GR at σ = 0, in which the torsion van- ishes but the Riemann tensor is in general different from zero, and the Hayashi-Shirafuji teleparallel theory[18] at σ = 1, in which the Riemann tensor vanishes but the torsion tensor is in general different from zero. The only dynamical variable in the gravitational sector of this theory is the tetrad e µa , and, as for the teleparallel theories, the theory is generally covariant but not invari- ant under the local Lorentz transformations (1.11a) of the tetrad. The action for the theory can be written as S[e µa ,Ψ] = SG[e a ] + Smatter[e a , ω µ ,Ψ], (1.16) where Ψ denotes the matter fields and, in the second term, ω abµ denotes the tetrad connection obtained from the torsion tensor (1.15) via Eqs. (1.3), (1.4) and (1.8b). The matter action Smatter is not specified by MTGC, so there are different versions of the EHS theory depend- ing on the form of the coupling to torsion chosen in this matter action. The gravitational action is given by a ] = a1tµνλt µνλ + a2vνv ν + a3aνa (1.17) Here a1, a2 and a3 are free parameters with dimensions of mass squared (we use units with ~ = c = 1), and the tensors tµνλ, v ν and aν are defined to be the irreducible pieces of the torsion tensor (1.15), but with the factor of σ removed: vµ = σ −1S λµλ , (1.18a) σ−1ǫµνρσS σρν , (1.18b) tλµν = σ −1Sν(µλ) + (gνλvµ + gνµvλ)− gλµvν .(1.18c) Also in Eq. (1.17) g denotes the metric determinant, where gµν is given in terms of the tetrad by Eq. (1.8a). For fixed a1, a2 and a3, this gravitational action is in- dependent of the parameter σ; this parameter enters the theory only through the dependence of the matter action on the torsion tensor (1.15).3 3 The parameters defining the theory are therefore (a1, a2, a3, σ). MTGC use a different set of parameters (c1, c2, c3, κ, σ) which are not all independent. The relation between the two sets of parameters can be derived using the identity of Appendix B and is a1 = σ 2c1 − 4/(3κ), a2 = σ 2c2 + 4/(3κ), a3 = σ 2c3 − 3/κ. In Appendix B we show that the gravitational action can be rewritten in terms of the Ricci scalar R({}) of the Levi-Civita connection as a ] = −g [d1R({}) + 4d2vνvν + 9d3aνaν ] , (1.19) where d1 = −3a1/8, d2 = (a1 + a2)/4, d3 = a3/9− a1/4. We shall refer to the three-dimensional space parameter- ized by (d1, d2, d3) as the gravitational-action parameter space. As mentioned above, there are different versions of the EHS theory, depending on the form of the matter action Smatter chosen; MTGC do not specify a matter action. Consider now what is required in order to predict the signal seen by GPB. The experiment consists of an Earth- orbiting satellite carrying four very stable gyroscopes, and the measured quantity is the time dependence of the angles between the spins of the gyroscopes and the direc- tion to a fixed guide star. To compute this quantity in an arbitrary Riemann-Cartan spacetime, it is sufficient to know the equations of motion and of spin transport for a spinning point particle4. These equations can be computed in principle for any matter action. MTGC as- sume that the matter action is such that the trajectory of the spinning point particle is either an autoparallel (a geodesic of the full connection) or an extremal (a geodesic of the metric), and that its spin is parallel transported with respect to the full connection. D. Requirements necessary to ensure physical viability of the theory In this paper we constrain the parameter space of the EHS theory by imposing a set of physical requirements. To simplify the analysis we first linearize the EHS theory with respect to a flat torsion-free spacetime, and then im- pose physical requirements on the linearized theory. The linearized theory is completely characterized by two ten- sor fields: a symmetric field hµν , and an antisymmetric field aµν . In terms of these fields the torsion tensor is given by S λµν = σ(h [ν,µ] − 2aλ [ν,µ] )/2, and the metric is given by gµν = ηµν + hµν . We impose three types of requirements. First, we re- quire that the theory have no ghosts, i.e., that the Hamil- tonian of the theory be bounded from below. This re- quirement rules out most of the three-dimensional pa- rameter space of the gravitational action SG. The re- maining viable subdomain of the parameter space con- sists of two intersecting two-dimensional planes. Second, we further require that the theory have a well posed initial value formulation. This means that if the 4 One also needs to know the trajectories of photons, but these are determined by gauge invariance to be just the null geodesics of the metric, as in GR [4]. physical degrees of freedom are specified on an initial spacelike hypersurface, the future evolution of these de- grees of freedom is uniquely determined. Now, for many theories some of the degrees of freedom are nonphysical and are associated with a gauge symmetry. For exam- ple, in classical electrodynamics the field equations for the vector potential Aµ are invariant under the gauge transformation Aµ → Aµ + ϕ,µ. These field equations therefore do not predict a unique evolution for the vector potential, and correspondingly consist of a set of under- determined partial differential equations. Nevertheless, classical electrodynamics has a well posed initial value formulation, because the degrees of freedom whose evo- lution cannot be predicted are pure gauge. A similar situation arises in linearized EHS theory. There, the gravitational action is invariant under certain symmetries of the dynamical variables (not diffeomor- phisms), and correspondingly the field equations form a set of an underdetermined partial differential equations. Therefore the theory can have a well posed initial value formulation only if the undetermined degrees of freedom are pure gauge. This can be the case only if the matter action is also invariant under the symmetries. However, the equations of motion and spin precession postulated by MTGC do not respect these symmetries, and they should inherit such a property from the matter action. We conclude therefore that the initial value formulation is ill-posed. This argument applies in most of the re- maining portion of the parameter space of the theory. The subdomain that is not excluded by this argument and by the requirement of no ghosts consists of a single line in the three dimensional space. Third, in this remaining subdomain, the linearized gravitational action reduces to that of GR; it depends on the tetrad only through the metric gµν . In particu- lar this means that the torsion tensor is completely un- determined in vacuum, and so for a generic matter ac- tion, the motion of test bodies cannot be predicted. The corresponding inconsistency of the Hayashi-Shirafuji the- ory in this limit has been previously discussed in Refs. [18, 23, 25, 26, 27, 29, 30, 31]. This inconsistency is avoided if one constructs a special matter action in which the unpredictability of the torsion tensor is associated with a gauge symmetry of the theory, in this case the predications of the EHS theory coincide with the predic- tions of GR. Finally, our argument that the initial value formulation is ill posed can be evaded by modifying the coupling of the torsion tensor to matter in the theory. Rather than postulating the equations of motion and spin precession used by MTGC, we instead assume that the coupling of torsion to matter is the standard, minimal coupling described in Refs. [4, 6]. For this coupling, the matter action is invariant under the symmetries of the gravita- tional action discussed above, in a portion of the parame- ter space, and so the theory has a well posed initial value formulation in which the undetermined degrees of free- dom are interpreted as gauge degrees of freedom. This Autoparallel Extremal Standard Matter Coupling Sector of parameter space D0 = {d2 6= 0, d3 6= 0} Ghosts Ghosts Ghosts D1 = {d2 6= 0, d3 = 0} I.V.F. I.V.F. I.V.F. D2 = {d2 = 0, d3 6= 0} I.V.F. I.V.F. Consistent but no GPB torsion signal (action=reaction violated) D3 = {d2 = 0, d3 = 0} Inconsistent Inconsistent/GR Inconsistent TABLE I: A summary of the status of the Einstein-Hayashi-Shirafuji theory [34] in different sectors of its parameter space. The rows of the table are these different sectors; the parameters d2 and d3, which appear in the gravitational part of the action, are defined in Eq. (1.19) in the text. There are different versions of the Einstein-Hayashi-Shirafuji theory depending on the precise form chosen of the coupling of the torsion tensor to matter fields. These different versions are the columns of the table. “Autoparallel” means that it is assumed that the matter coupling is such that freely falling bodies move on geodesics of the full connection, while “Extremal” means they move on geodesics of the Levi-Civita connection determined by the metric. These were the two cases considered by Mao et al. [34]. “Standard Matter Coupling” means the standard, minimal coupling of torsion to matter fields [4, 6], which in general gives rise to motions of test bodies that is neither autoparallel nor extremal. The meanings of the various entries in the table are as follows. “Ghosts” means that some of the degrees of freedom in the theory are ghostlike at short distances, signaling an instability that rules out the theory. “I.V.F.” means that the theory does not have a well posed initial value formulation, and so is ruled out. “Inconsistent” means that the theory does not predict the value of the torsion tensor, so the motion of test bodies cannot be predicted, while “GR” means that the theory reduces to general relativity. Finally “no GPB torsion signal” means that there is no torsion-induced coupling between the Earth’s angular momentum and that of the Gravity Probe B gyroscopes; there is only a coupling between the fundamental spins of the Earth’s fermions and of those in the gyroscopes, which gives a negligible signal as those spins are randomly oriented. is the interpretation suggested in the original paper by Hayashi and Shirafuji [18]. For this case, we again ex- amine the linearized theory for the fields hµν and aµν . We find that hµν satisfies the same equation as the met- ric perturbation in GR, while aµν satisfies a wave equa- tion (with a suitable choice of gauge) whose source is obtained from the intrinsic spin density of matter. As discussed earlier, this implies that for a macroscopic ob- ject for which the spins of the elementary particles are not correlated over macroscopic scales, aµν will be neg- ligible. Hence the spacetime of the linearized theory is completely characterized by the metric alone, and so its predictions coincide with those of GR5 and there will be no extra signal in GPB. A summary of the status of the EHS theory in various different cases discussed above is given in Table I. To summarize, there are no cases in which the EHS theory gives a detectable torsion signal in GPB. However, it is nevertheless possible that other torsion theories in the other categories discussed in Sec. I A, with suitable choice of matter coupling, could predict a detectable sig- nal. Various possibilities for non-minimal couplings are discussed by Shapiro [6]. It would be interesting to find a torsion theory that predicts a detectable torsion signal for GPB; such a theory would be an example to which the theory-independent framework developed by MTGC 5 This is despite the fact that the torsion tensor is generically nonzero. The torsion tensor is not an independent degree of freedom in this limit, it is given in terms of the metric by the first term in Eq. (2.3b). (a generalization of the parameterized post-Newtonian framework to include torsion) could be applied. E. Organization of this paper This paper is organized as follows. In Sec. II we derive the dynamical variables and the action of the linearized EHS theory. In Sec. II B we study the action for the an- tisymmetric field aµν , temporarily setting the symmetric field hµν to zero. We show that this theory has ghosts on a subdomain of the parameter space. Section II E and Appendix A extend this result to the complete linearized theory, including the symmetric field hµν , thereby ruling out a subdomain of the parameters space. We then focus on the complementary subdomain and show that it is in- variant under certain symmetries. Section III reviews the necessary requirements for a well posed initial-value for- mulation. In Sec. III A we use these requirements to rule out a subdomain of the parameter space that has an ill- posed initial value formulation. The remaining portion of the parameter space is discussed in Sec. II F Finally, section IV considers the EHS theory with the standard matter-torsion coupling. In a certain portion of parameter space this theory has a well-posed initial value formulation and no ghosts, but we show that the devi- ations of its predictions from those of GR are negligible for unpolarized macroscopic bodies. Final conclusions are given in Sec. V. II. LINEARIZATION ABOUT FLAT, TORSION-FREE SPACETIME OF THE EINSTEIN-HAYASHI-SHIRAFUJI THEORY A. Action and variables of linearized theory To linearize the EHS theory we first decompose the tetrads e µa and dual one-forms e µ into background tetrads and one-forms and perturbations: e µa = b a + δc a , (2.1a) eaµ = b µ + δe µ. (2.1b) We assume that the background tetrads b µa are constants for which the metric (1.8a) is the Minkowski metric, gαβ = ηαβ , and for which the torsion (1.15) is vanishing, S λµν = 0. Thus, to zeroth order, the spacetime is flat and torsion-free. Throughout this paper we will work to leading order in the tetrad perturbations. Hereafter, un- less we explicitly state otherwise, Greek indices are raised and lowered with ηµν and Latin indices with ηab. From the definition (1.9) of the dual basis applied to both the full tetrads and the background tetrads, we find that δeaµb a = −δc νa baµ. Thus we can take δeaµ to be the fundamental variable of the theory. We next convert this quantity into a spacetime rank 2 tensor using the background tetrad, and take the independent symmetric and antisymmetric pieces. This yields the definitions hµν ≡ 2δeb(µb ν)ηab, (2.2a) aµν ≡ δeb[µb ν]ηab. (2.2b) The formulae (1.8a) and (1.15) for the metric and torsion now yield gµν = ηµν + hµν , (2.3a) S λµν = hλ[ν,µ] − 2a [ν,µ] . (2.3b) The linearized gravitational action SlinearG can now be ob- tained by substituting the expressions (2.3) for the met- ric and torsion into the action (1.19) and expanding to quadratic order in hµν and aµν . The resulting action can be written schematically in the form SlinearG = SS [hµν ] + SC [hµν , aαβ ] + SA[aµν ] , (2.4) where SS is a quadratic in the symmetric tensor hµν , SA is quadratic in the antisymmetric tensor aµν , and SC contains the cross terms. B. The antisymmetric term in the action We now focus on the antisymmetric term SA in the ac- tion, ignoring for the moment the other two terms. For this theory we derive two kinds of results. First, we con- strain the parameter space (d1, d2, d3) by imposing the requirement of no ghosts, and second we derive symme- try transformations under which the action SA (with spe- cific parameters) is invariant. Later in Sec. II E we will extend some of these results to the complete linearized action SlinearG . The antisymmetric action SA[aµν ] is constructed from the antisymmetric field aµν in Minkowski spacetime, and consists of terms that are product of derivatives of aµν , of the form (∂λaµν) 2. Actions of this type also arise in gravitational theories with a non-symmetric metric and have been extensively studied. See Refs. [45, 46] for a discussion of the existence of ghosts in theories of this type. There are only three linearly independent terms of the form (∂λaµν) 2, namely aµ ,σσ , aµν,λa µν,λ , aµν,λa µλ,ν . From these terms one can construct only two functionally independent actions, since the identity a ,σµσ − aµλ,σaµσ,λ = ∂λ(aµλa ,σµσ − aµλ,σaµσ) (2.5) shows that a linear combination of the terms is a diver- gence which can be converted to a surface term upon integrating and thereby discarded. Therefore, the most general action of this type can be written as d4x[d2a aµσ,σ + d3a a∗µσ,σ] , (2.6) where a∗µν ≡ −ǫµνρσaρσ/2 (here ǫµνρσ is the Levi Civita tensor of a flat spacetime), and d2,d3 are free parameters. Indeed an explicit calculation using Eqs. (1.18), (1.19) and (2.3) shows that SA is given by the expression (2.6), where the parameters d2 and d3 are those defined after Eq. (1.19). Next, we specialize to a particular Lorentz frame, and rewrite the action in terms of the vectors E and B defined by Ei = a0i and Bi = ǫijkajk, where i, j, k run from 1 to 3. This gives aµσ,σ = (Ė+∇ ×B)2 − (∇ ·E)2 , (2.7) where an overdot denotes differentiation with respect to time. The corresponding expression for a a∗µσ,σ has the same form, except for the substitutions E → B and B → −E. We now consider the case where both d2 and d3 are nonzero. In this case Hamiltonian density corresponding to the action (2.6) takes the form (πB + 2d3∇×E)2 + d3(∇·B)2 − d3(∇×E)2 (πE − 2d2∇×B)2 + d2(∇·E)2 − d2(∇×B)2 , (2.8) where πE and πB denote the momenta conjugate to E and B. C. Ghosts We now constrain the values of the coefficients d2 and d3 by demanding that 3x be bounded from below. Let us start by considering the coefficient d3, and sup- pose first that d3 < 0. At a given point in space, keeping the values of E, B and πE fixed, we can make HA arbi- trarily negative by choosing πB to be arbitrarily large. The same is clearly true for the integrated Hamiltonian 3x. Next suppose that d3 > 0. By fixing B, ∇ ·E, πE and adjusting the value of πB to keep the value of πB + 2d3∇×E fixed, we can make HA arbitrarily neg- ative, this time by choosing an arbitrarily large value for ∇×E. By applying analogous considerations to d2 we reach the conclusion that the theory defined by HA has ghosts on the domain D0 = {(d1, d2, d3)| d2 6= 0, d3 6= 0} (2.9) in parameter space. Similar analyses can be found in Refs. [45, 46] for non-symmetric gravity theories, and in Ref. [47] for teleparallel gravity theories. D. Symmetries We now consider the domain in parameter space not excluded by the above analysis, which consists of the two 2D regions D1 = {(d1, d2, d3)| d2 6= 0, d3 = 0} , (2.10a) D2 = {(d1, d2, d3)| d2 = 0, d3 6= 0} , (2.10b) together with the line D3 = {(d1, d2, d3)| d2 = 0, d3 = 0} . (2.10c) The antisymmetric action SA vanishes identically on D3, so in this subsection we will not consider D3 any further. We are interested in the symmetries of the antisym- metric action SA on the domains D1 and D2. From the formula (2.6) for the action we see that these two do- mains are isomorphic to one another under the duality transformation aµν → a∗µν or E → B, B → −E. (2.11) Therefore it is sufficient to focus on one of the domains, say D1. From Eqs. (2.6) and (2.7), the antisymmetric action on this domain is SA |D1 = d2[(Ė+∇×B)2 − (∇ · E)2] d4x . (2.12) Consider now the initial value problem for E and B. Suppose that we are given sufficient initial data on some constant time hypersurface, and that we wish to deter- mine the time evolution of E(t) and B(t) using the action (2.12). Note that this action is independent of the longi- tudinal part of B, which means that the evolution of this longitudinal part can be prescribed arbitrarily, indepen- dent of the initial data. Therefore, the field equations for E and B must form a set of an underdetermined partial differential equations. As discussed in the introduction, this can only be consistent if the undetermined degrees of freedom can be interpreted as being gauge degrees of freedom. The action (2.12) is invariant under the symmetry aµν → aµν + ǫµναβχα,β , (2.13) where χα(x) is an arbitrary vector field. This can be seen from the fact that the action (2.12) is given by the first term in Eq. (2.6), which depends on aµν only through its divergence a ,νµν . Similarly on the domain D2 the anti- symmetric action (2.6) is invariant under the symmetry aµν → aµν + χ[µ,ν]. (2.14) These are the symmetries that are responsible for the in- determinacy in the evolution equations. We will study in later sections the conditions under which these sym- metries can be interpreted as gauge symmetries, thus al- lowing the theory to have a well posed initial value for- mulation. As discussed in the introduction, the gauge symmetry interpretation requires the matter action to be invariant6 under the symmetries (2.13) and (2.14). E. The complete linearized action Up to now we have ignored the pieces SS and SC of the complete linearized action (2.4), and have studied a reduced theory depending only on the antisymmetric field aµν described by the action SA alone. We showed that this reduced theory has ghosts if both d2 and d3 are nonzero, i.e., on the domain D0. In Appendix A this result is generalized to the complete linearized theory, in- cluding the symmetric field hµν , showing that the com- plete theory also has ghosts in the domainD0. Essentially we show that that whenever the Hamiltonian 3x is unbounded from below, then the corresponding Hamilto- nian of the full linearized theory is also unbounded from below. The symmetries (2.13) and (2.14) of the reduced the- ory on the domains D1 and D2 also generalize to the complete linearized theory. This can be seen as follows. Since the symmetries only involve the antisymmetric field aµν , the only additional term in the complete action (2.4) whose invariance needs to be checked is the cross term 6 A general discussion of the problems that arise when the gravi- tational action is invariant under a symmetry not shared by the matter action can be found in Leclerc [31]. SC [hµν , aρσ]. This cross term can be written as hαβ,γaµν,ρP αβγµνρ d4x , where Pαβγµνρ is a tensor constructed from the Minkowski metric. Integrating by parts and discarding a surface term yields SC = − hαβaµν,ργP αβγµνρ d4x . At least two of the indices on aµν,ργ must be contracted with one another, and since aµν is antisymmetric it fol- lows that only divergence terms of the form a σµσ, γ can appear. These divergence terms are invariant under the symmetry (2.13). For the symmetry (2.14), we compute explicitly the cross term SC specialized to the domain D2. Since d2 = 0, the only term in the general action (1.19) that can contribute to this cross term is the aνa ν term in- volving the square of the axial piece of the torsion; the Ricci scalar term depends only on hµν . Using the defini- tion (1.18b) of this axial piece together with the formula (2.3b) for the torsion in terms of hµν and aµν gives λνρσaνσ,ρ. (2.15) Since this depends only on the antisymmetric field aµν it does not generate any cross terms, and we conclude that the cross term SC vanishes identically on the domain D2. Therefore the complete action is invariant under the symmetry (2.14) on D2. F. The general relativity limit of the gravitational action So far we have considered the domains D0, D1 and D2 of the gravitational action parameter space. We now fo- cus on the remaining domain D3. From the definitions (2.10c) and (1.19) we find that in this domain the grav- itational action reduces to that of general relativity, so it is invariant under local Lorentz transformations of the tetrad e µa → Λ ba (x)e . This invariance guarantees that the left hand side of the Euler-Lagrange equation eaµgρν = δSmatter eaµgρν , is a symmetric tensor [8]. Consistency now requires the right hand side to be symmetric. However, the torsion tensor is not invariant under local Lorentz transforma- tions. Therefore, a generic matter action that couples between the torsion tensor and matter fields would break the local Lorentz symmetry. This matter action produces a nonsymmetric right hand side for the Euler-Lagrange equation, thereby rendering the theory inconsistent. The corresponding inconsistency of the Hayashi-Shirafuji the- ory in this limit has been previously discussed in Refs. [18, 23, 25, 26, 27, 29, 30, 31]. The inconsistency is avoided if the matter action is invariant under local Lorentz transformations. In this case the torsion ten- sor is undetermined by the EHS theory, and the theory reduces to GR. III. INITIAL VALUE FORMULATION OF THE THEORY In this section we focus on the domains D1 and D2 that are not ruled out by the existence of ghosts, and examine in more detail the conditions under which the theory on these domains has a well posed initial value formulation. Suppose that we specify as initial data {aµν , hαβ} and {ȧµν , ḣαβ} on some initial constant time hypersurface, and ask whether the evolution of the fields for all sub- sequent times is uniquely determined. Now the action (2.6) is invariant under certain symmetries which allow us to generate new solutions that correspond to the same initial data. These symmetries consist of, first, diffeomor- phisms xµ → xµ−ξµ(x) under which the fields transform hµν → hµν + 2ξ(µ,ν) , (3.1a) aµν → aµν + ξ[ν,µ] , (3.1b) and second, the symmetries (2.13) on D1 and (2.14) on D2. By invoking one of these transformations in a space- time region to the future of the initial data hypersurface, we can generate new solutions for the field-equations that correspond to the same initial data. Therefore the evo- lution of the fields hµν and aµν cannot be uniquely pre- dicted. For the theory to have a well-posed initial value formulation it is necessary that all of these transforma- tions correspond to gauge symmetries, which means that all observables should remain invariant under these trans- formations. Then the failure of the theory to uniquely predict hµν and aµν is merely associated with the unpre- dictability of nonphysical degrees of freedom. We now focus on the observables that will be mea- sured by the GPB experiment. If we use the equations of motion and spin transport for test bodies postulated by MTGC, then these observables are not invariant un- der the symmetries (2.13) and (2.14), as we now show. Thus the initial value formulation is ill posed for these postulated equations of motion. A. Observations with Gravity Probe B We focus on one of the four GPB gyroscopes, and represent it as a particle with trajectory zα(τ) where τ is proper time, and with spin sα(τ). We let the 4- momentum of the photons from the distant fixed guide star be kα. Let θ be the angle, as measured in the frame of the gyroscope, between its spin and the direction to the guide star. Then we have cos θ = ~s · ~k (~u · ~k) , (3.2) where ~s·~k = gµνsµkν . This can be seen from the formulae for these vectors in the rest frame of the gyroscope: ~u = (1,0), ~k = ω(1,−n) and ~s = (0, s), where n is a unit vector in the direction of the star. The equations of motion and spin precession postu- lated by MTGC are = 0 , = 0 , (3.3) where uµ = dzµ/dτ is the 4-velocity and ≡ uµ∇µ (3.4) is the covariant derivative operator along the worldline with respect to the full connection ∇µ. In other words, the particle travels on a geodesic of the full connection (an autoparallel curve). MTGC also consider the possi- bility that the equation of motion is ≡ uν∇̃νuµ = 0, (3.5) where ∇̃ν is the Levi-Civita connection determined by the metric, so that the particle travels along a geodesic of the metric (an extremal curve). As mentioned by MTGC this possibility is theoretically inconsistent since by Eqs. (3.3) the orthogonality of uµ and sµ is not maintained during the evolution. Nevertheless, we shall also con- sider this possibility below. Finally, as mentioned above, photons follow null geodesics of the metric: kµ∇̃µkν = 0. (3.6) We now apply the operator D/Dτ to the formula (3.2) for the angle θ, and use the autoparallel equations of motion (3.3) together with ∇µgµλ = 0. This gives d[cos θ] (~u · ~k) ~s · ~k ~u · ~k . (3.7) The measurable accumulated change in cos(θ) in the in- terval τ1 → τ2 is therefore ∆[cos(θ)] = (~u · ~k) ~s− ~s · ~u · ~k (3.8) Next, we examine how the change in angle (3.8) trans- forms under the symmetry transformations (2.13) and (2.14). For this purpose it is sufficient to consider the motion of the gyroscope in a flat, torsion-free spacetime, since we are working to linear order. We use Lorentzian coordinates where = 0, which implies that for an initially static gyroscope uµ, sµ and kµ are all constants, so that ∆[cos(θ)] = 0. Consider now the effect of the transformations (2.13) or (2.14). Denoting the transformed quantities with primes we find that = 0, Γ µν = 0 + δΓ From Eqs. (3.3) we find that z′µ(τ ′) = zµ(τ) + δzµ(τ) and s′µ(τ ′) = sµ(τ) + δsµ(τ), where both δz(τ) and δsµ are O(χ). It turns out that the precise expressions for these quantities are not required for our calculation. No- tice that for a fixed distant star the field kµ(x) (before the transformation) is approximately constant in a neighbor- hood of the gyroscope, in the sense that kµ,ν = 0. Now, Eq. (3.6) implies that kµ(x) remains invariant under the transformations. Therefore the derivative of k′µ(x) along z′(τ ′) is given by u′αk′β . (3.9) Substituting Eq. (3.9) and s′µ(τ ′) into Eq. (3.7), and re- taining only terms which are O(χ) (so we can drop the distinction between τ and τ ′) we obtain d[cos(θ′)] (~u · ~k) −~s · ~u · ~k . (3.10) From Eqs. (1.3), (1.4) and (2.3) the change δΓ in the connection coefficients is = σηβλǫ µλρσχσ,ρα , (3.11) for the symmetry (2.13) and = (σ/2)(χ ) (3.12) for the symmetry (2.14). We substitute these transformation rules into Eq. (3.10) and then substitute the result into Eq. (3.8). Re- calling that χα are arbitrary functions of the coordinates, we find that by invoking the transformations (2.13) or (2.14) we can set ∆[cos(θ)] to have an arbitrary value. Thus, the observable angle is not invariant under the symmetry transformations, and hence they cannot be in- terpreted to be gauge transformations. [In particular this implies that the matter action must be non-invariant]. Consequently the initial value formulation of the theory is ill posed. We now repeat this analysis for extremal worldlines satisfying Eq. (3.5). Equation (3.10) acquires an addi- tional term − ~s · (~u · ~k)2 ~k · D~u = − ~s · (~u · ~k)2 on the right hand side. In addition the change to ex- tremal worldlines alters the quantity δuµ, but this does not appear in the final formula (3.10). As before we find that the initial value formulation is ill posed. Finally, MTGC also consider the possibility of an ex- tremal worldline together with the following equation for the spin precession = 0. (3.13) Here the antisymmetric tensor sαβ is related to the parti- cle spin through sµ = ǫµνρσuνsρσ. This relation guaran- tees that the orthogonality condition sµuµ = 0 is satisfied throughout the motion of the particle.7 By examining the transformation of ∆[cos(θ)] under the symmetries (2.13) and (2.14) for an extremal worldline for which the law for the spin precession is given by (3.13), we find as before that the initial value formulation is ill posed. IV. EINSTEIN-HAYASHI-SHIRAFUJI THEORY WITH STANDARD MATTER COUPLING The analysis so far suggests that in order to obtain a consistent theory, one needs to choose a matter action which is invariant under the symmetries (2.13) or (2.14) of the gravitational action. This would allow those sym- metries to be interpreted as gauge symmetries and allow the theory to have a well posed initial value formula- tion. In this section we show that the standard, mini- mal coupling of matter to torsion [4, 6] does respect the symmetry (2.14), and so one does obtain a consistent linearized theory on the domain D2 by choosing this cou- pling.8 For the special case of σ = 1, this matter action is the one suggested in the original Hayashi-Shirafuji paper [18]. Following the analysis of Hayashi and Shirafuji, we show that the predictions of this linearized theory coin- cide with those of linearized GR for macroscopic sources with negligible net intrinsic spin. The standard Dirac action in an Einstein-Cartan spacetime is SD = −gLD, where e µa (ψ̄γ aDµψ −Dµψγaψ)−mψ̄ψ , (4.1a) Dµ = ∂µ − σbcgρνe b ∇µe ρc , (4.1b) ∇µe ρc = ∂µe ρc + Γρµνe νc , (4.1c) σbc = [γb, γc], (4.1d) 7 As noted by MTGC, normally one also demands a “transver- sality” condition sαβu β = 0. However, by Eqs. (3.5) and (3.13) this condition can not be maintained along an extremal world- line. This also implies that the norm of sµ is not constant along the evolution. 8 Note that this implies in particular that the standard matter coupling does not predict either extremal curves or autoparallel curves for the motions of test bodies, since those cases are not invariant under the symmetry (2.14). and γb are Dirac matrices with the representation given in Ref. [48] that satisfy γaγb + γbγa = −2ηab. Also ψ̄ denotes the adjoint spinor defined by ψ̄ = ψ†γ0, where † denotes Hermitian conjugation. The torsion tensor enters this action only through the covariant derivative in Eq. (4.1b). As is well known, this action can be written as the usual torsion-free action together with a coupling of the fermion to the axial piece (1.18b) of the torsion tensor [6]. Using the definitions (1.4) and (1.18) we obtain e µa (ψ̄γ aD̃µψ − D̃µψγaψ)−mψ̄ψ aµψ̄γ 5γbψ . (4.2) Here D̃µ = ∂µ− iσbcgρνe νb ∇̃µe ρc /4, ∇̃µe ρc = ∂µe ρc + e νc , and γ 5 = iγ0γ1γ2γ3, where we use the convention ǫ0123 = Now the first line in the Dirac action (4.2) depends only on the metric, and in particular it is independent of aµν , so it is trivially invariant under the transformation (2.14). The second line depends on aµν only through the axial piece aµ of the torsion, which is given by Eq. (2.15), and which is also invariant under (2.14). Therefore the entire action (4.2) is invariant under the symmetry. We conclude that on the domain D2 we have a consis- tent theory with a well posed initial value formulation, in which the symmetry (2.14) is a gauge symmetry. The self-consistency of this theory for σ = 1 was previously discussed by Leclerc [32]. From Eqs. (1.19) and (2.10b), the complete action for the theory is S[eaµ,Ψ] = R({}) + 9d3aµaµ +SD[e µ, ψ, ψ̄] + . . . , (4.3) where κ̂ = 1/(2d1). Here the ellipses denote additional terms in the standard model of particle physics that are not coupled to the torsion; the only term that couples to the torsion is the Dirac action for the fermions (the “minimal coupling” scheme of Refs. [4, 6]). The linearized equation of motion for eaµ obtained from this action gives equations for hµν and aµν : �h̄µν + h̄ ρ(µ,ν) ηµν h̄ = κ̂T(µν), (4.4a) �aµν + 2a ρ[µ,ν] T[µν] . (4.4b) Here � = ηαβ∂α∂β , h̄µν ≡ hµν − ηµνh ρρ , Tµν is the non- symmetric energy-momentum tensor defined by T νµ ≡ δSmatter eaµ, (4.5) and Tµν is its leading order term in a perturbative ex- pansion (i.e. Tµν is independent of hµν and aµν). The source terms in Eqs. (4.4) obey the conservation laws = 0 and T = 0, by virtue of the invariance of the matter action under the transformations (3.1) and (2.14). By using these transformations we can impose the gauge conditions h̄ ,νµν = 0 and a µν = 0, thereby reducing the field equations to wave equations: �h̄µν = −2κ̂T(µν), (4.6a) �aµν = T[µν] . (4.6b) The first of these is the usual linearized Einstein equa- tion. Next we examine the antisymmetric piece of the stress energy tensor which acts as a source for aµν in Eq. (4.6b). One can show [18] that this antisymmetric piece is related to the divergence of the spin density tensor by T[βα] = σb ab ,µ , (4.7) where the subscript b νc indicates evaluation at the back- ground values of the tetrad. Recall that the spin density is defined by δSmatter[e ν ,Ψ] δω abµ where in this definition the matter action is considered to be a functional of the independent variables efα , ω and Ψ. The matter action can be brought to this de- sired form by substituting Dµ = ∂µ − i4σ bcgρµω the expression for LD in Eq. (4.1a). Relation (1.7) guar- antees that this expression for Dµ equals to our original expression in Eq. (4.1b). Equations (4.6b) and (4.7) imply that aµν is sourced only by the intrinsic spin density of matter. As we have discussed, integrating Eq. (4.6b) for a macroscopic object for which the spins of the elementary particles are not aligned over a macroscopic scale gives a negligible aµν [18], and consequently the predictions of the linearized theory coincide with those of GR. Thus there is no extra torsion-related signal predicted for GPB for this theory. The lack of an experimental signature of torsion may seem strange, since the torsion tensor is non-vanishing even in the limit where one can neglect intrinsic spin. As discussed in the introduction, the explanation is that the torsion is not an independent dynamical degree of freedom. More specifically, to linear order, macroscopic bodies give rise to a metric perturbation hµν in the same way as in GR, and then the torsion is simply defined to S λµν = hλ[ν,µ] (4.8) [c.f. the first term in Eq. (2.3b)]. This definition has no dynamical consequence, since only the axial piece (1.18b) of torsion couples to matter, and the expression (4.8) has no axial piece. V. CONCLUSIONS Preliminary results from Gravity Probe B will be an- nounced in April 2007. The primary scientific goals of the experiment are to verify the predictions of general relativity for geodetic precession and for dragging of in- ertial frames [3]. However the mission is also potentially useful as a probe of modifications of general relativity. One class of theories of gravity that GPB could poten- tially usefully constrain are theories involving a dynam- ical torsion tensor. Mao et. al. suggested a particular class of torsion theories that they argued would predict a measurable torsion signal for GPB [34]. We have shown that this particular class of theories is internally consis- tent in only a small region of its parameter space, and in that consistent region does not predict any signal for GPB. There may exist other torsion theories which could be usefully constrained by GPB. It would be interesting to find such theories. Acknowledgments This research was sponsored in part by NSF grant PHY-0457200. APPENDIX A: GHOSTS IN THE LINEARIZED THEORY In this Appendix we show that the Hamiltonian of the complete linearized EHS theory (2.4) is unbounded below whenever the corresponding Hamiltonian of the antisym- metric term (2.6) is unbounded from below. This allows us to deduce the existence of ghosts in the complete the- ory from their existence in the reduced theory (2.6). It is sufficient to show this property for the case of finite dimensional, quadratic dynamical systems. We consider a system with N degrees of freedom whose Lagrangian and Hamiltonian are given by q̇mq̇nKmn − V (qm) , (A1) pmpn(K + V (qm) . (A2) Here qm denotes the generalized coordinates and pm = Kmnq̇n denotes the conjugate momenta, where the in- dices m,n run from 1 to N, and Kmn = Knm. Recall that the antisymmetric action SA was obtained by ig- noring some of the dynamical variables of the linearized EHS theory. In a discrete theory this corresponds to writing a reduced Lagrangian Lr that depends only on some of generalized coordinates qi, where i runs from 1 to M , M < N . The reduced theory has the following Lagrangian and Hamiltonian q̇iq̇jkij − U(qi) , (A3) p̃ip̃j(k + U(qi) . (A4) Here p̃i = kij q̇j , where kij = Kij for i, j = 1...M and U(q1, ....qM ) = V (q1....qM , 0..0), (A5) i.e., the potential U(qi) is obtained from V (qm) by setting qm = 0 for m =M + 1...N . Now suppose that the potential term U(qi) of the re- duced Hamiltonian Hr is unbounded from below. It fol- lows immediately from the definition (A5) that V (qm) is also unbounded from below, and so the complete Hamil- tonian (A2) is unbounded from below. Suppose next that the kinetic term of the reduced Hamiltonian Hr is unbounded from below. By virtue of Eq. (A4) this means that at least one of the eigenvalues of (k−1)ij must be negative. Denoting the eigenvalues of the matrix kij by λ(i), 1 ≤ i ≤ M , we find that there exists an eigenvalue l for which λ−1 < 0. This implies that there exists an M dimensional eigenvector w i for which w j kij = λ(l) < 0. We now construct an N di- mensional vector defined by ηm = (w 1 ...w , 0...0). By definition, this vector satisfies ηmηnKmn = λ(l) < 0, im- plying that Kmn has a negative eigenvalue. This means that the complete Hamiltonian (A2) is unbounded from below. APPENDIX B: ALTERNATIVE FORM OF GRAVITATIONAL ACTION In this Appendix we derive identity −gR({})d4x = tµνλt ν − 6aνaν , (B1) where R({}) is the Ricci scalar of the Levi-Civita con- nection, and tµνλ, v µ and aµ are the irreducible pieces (1.18) of the torsion tensor with the factor of σ removed. Combining this identity with the formula (1.17) for the gravitational action of the EHS theory yields the alter- native form (1.19) of that action. The idea is to introduce a new torsion tensor S̄ λµν ≡ e λa (e ν,µ − eaµ,ν) . (B2) This is just the torsion tensor (1.15) of the EHS theory but specialized to σ = 1, i.e., it is the torsion tensor of the Hayashi-Shirafuji teleparallel theory[18]. From Eqs. (1.15) and (1.18) it is related to the fields tµνλ, v µ and aν by S̄νµλ = (tλµν − tλνµ) + (gλµvν − gλνvµ) + ǫλµνρaρ . We denote the corresponding metric compatible con- nection by Γ̄αβγ and the corresponding Riemann tensor by R̄ . 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Note that this reference uses a different metric sig- nature.). http://arxiv.org/abs/gr-qc/0608121

0704.1448

Structures in the Universe and Origin of Galaxies

Structures in the Universe and Origin of Galaxies Structures in the Universe and Origin of Galaxies V. A. Rantsev-Kartinov INF RRC "Kurchatov Institute", Moscow, Russia, [email protected] Abstract The analysis of images (of obtained in various ranges of the lengths of waves) of luminous objects in the universe by means of a method of multilevel dynamic contrasting led author to the conclusions: a) the structures of all observable galaxies represents a complicated constructions which have the tendency to self-similarity and made of separate (basic) blocks, which are a coaxially tubular structures and a cartwheel-like structures; b) the majority of observable objects in the universe are luminous butt-ends of almost invisible (of almost completely transparent) of filamentary formations which structures are seen only near to their luminous butt-ends; c) the result of analysis of images of cosmic objects show the structure of many pairs of cooperating galaxies point to opportunity of their formation at butt-ends generated in a place of break of the similar filament; d) the interacting galaxies (M 81 and M 82) show they are butt-ends of sawed off of two branches of a treelike filament and their interaction is coming out through this filament; e) as our Universe is in dynamics the processes of formation of stars, galaxies and their congestions can go and presently by means of a fracturing of filaments with a corresponding diameters and of the time for their such formation is necessary much less, than along existing standard model. 1. Introduction. Research by author of a skeletal structures of the Universe (SSU) began from the analysis of images of various types of plasma by means of a method multilevel dynamic contrasting (ММDC), developed and described earlier [1a, b]. The analysis of images by ММDC is carried out by imposing of various computer maps of contrasting on the image of plasma obtained by the various methods and in any spectral ranges. Some results of the given analysis of a modern database of images of space objects here are given. It is shown, that the topology of the revealed space structures is identical to those which have been already found out and described earlier in a wide range of physical environments , the phenomena and scales [1-2]. The basic role in SSU is connected with its separate blocks in the form of coaxially-tubular blocks (CTB). These CTB have complex multi-layered structure of the telescopic enclosed tubes which lateral walls represent a weaving of similar filaments of smaller diameter, with the central cord. Except for that, these blocks inside have also radial connections. Extended filaments of these structures are collected of almost identical CTB, which are flexibly connected among themselves as in joints of a skeleton. It is assumed such joints may be realized due to stringing of the individual CTB on common flow of the magnetic field which penetrates the whole such filament, and itself the CTB are the interacting magnetic dipoles with micro-dust skeletons, which are immersed into plasma. 2. Observations of cartwheel structures in laboratory and cosmic plasma Here, we will try to connect laboratory experiments and space by considering a short gallery of cartwheel-like structures, which are probably the most inconvenient objects for universally describing the entire range of observed space scales. In laboratory electric discharges [1] and their respective dust deposits [2] the cartwheels may be located either on the butt-ends of a tubes, or on an «axle-tree» filament, but they may be separate blocks also (the smallest cartwheels are of diameter less than 100 nm (see Figs. 2 and 3 in Phys. Lett. A, 269, 363-367 (2000)). So, similar structures of different scales are found in the following typical examples: (i) big icy particles of a hail (Fig. 1A), (ii) a fragment of tornado (Fig. 1B), (iii) a supernova remnant. Fig. 1. The cartwheel-like structures at different length scales are presented. A) Big icy particles of hail of diameter 4.5 cm (a), 5 cm (b), and 5 cm (c). The frame in the left lower part of the image (a) is contrasted separately to show an elliptic image of the edge of the radial directed tubular structure. The entire structure seems to contain a number of similar radial blocks. A distinct coaxial structure of the cartwheel type is seen in the central part of image (b). Image (c) shows strong radial bonds between the central point and the «wheel». B) Top section: A fragment of the photographic image of a massive tornado of estimated size of some hundred meters in diameter. Bottom section: A fragment of the top image shows the cartwheel whose slightly elliptic image is seen in the center. The cartwheel seems to be located on a long axle-tree directed down to the right and ended with a bright spot on the axle’s edge (see its additionally contrasted image in the left corner insert on the bottom image). C) «A flaming cosmic wheel» of the supernova remnant E0102-72, with «puzzling spoke-like structures in its interior», which is stretched across forty light-years in Small Magellan Cloud, 190,000 light-years from Earth [13]. The radial directed spokes are ended with tubular structures seen on the outer edge of the cartwheel. The inverted (and additionally contrasted) image of the edge of such a tubule (marked with the square bracket) is given in the left corner insert (note that the tubule’s edge itself seems to possess a tubular block, of smaller diameter, seen on the bottom of the insert). Fig. 2. a) The «cartwheel» structure in tokamak TM-2 plasma (minor radius 8 cm, toroidal direction is horizontal) Diameters of larger and smaller ring-shaped structures on a common axle are ~2.2 cm and ~1 cm, respectively. Image (positive) is taken in visible light by a streak camera with time resolution < 1 µs (original picture is taken from Vinogradova N.D., Razumova K.A. Int. Conf. Plasma Phys., Culham, U.K., 1965). b) The TEM image (magnification 34,000) of a small fragment of dust particle (1.2 µm diameter) extracted from the oil used in the vacuum pumping system of tokamak T-10. The tubule, whose edge with the distinct central rod is seen in the lower left part of the figure, is of ~70 nm diameter and ~140 nm long. Diameter of slightly inhomogeneous cylinder on the left side of the tubule is ~10 nm. The radial bonds between the side-on cylinder and the central rod are of ~ 10 nm diameter. c) Another fragment of the same dust particle. The cartwheel (a coaxial two-ring structure, D ~ 100 nm) is declined with respect to image’s plane and located on a thick rod (which probably “works” as an axle of the cartwheel). Note that the cosmic wheel’s skeleton (FIGURE 1C) tends to repeat the structure of the icy cartwheel (FIGURE 1A) up to details of its constituent blocks. In particular, some of radial directed spokes are ended with a tubular structure seen on the outer edge of the cartwheel. Moreover, in the edge cross-section of this tubular structure, the global cartwheel of the icy particle contains a smaller cartwheel whose axle is directed radially (see left lower window in FIGURE 1A). Thus, there is a trend toward self-similarity (the evidences for such a trend in tubular skeletons found in the dust deposits). Fig. 3. The schematic image of structures such as "cartwheel" is given here. Thus, the cosmic wheel's skeleton tends to repeat the structure of the cartwheel itself up to details of its constituent blocks as in the icy cartwheel (see Fig. 1A). 3. Observations of "electric torch-like structures“ in laboratory and cosmic plasma One of the new phenomena which has been found out at the analysis of images of plasma, were a rectilinear dark filamentary structures which butt-end can shine as open butt-end of optical fibers in such ranges of lengths of waves which correspond with temperature of researched plasma. Such the CTS have been described and have received the name" electric torch-like structures" (ETSs) [1c, d]. width 0.35 cm width 1.75 cm Fig. 4. a) The image of a luminescence of tokamak T-6 of plasma in seen light, at exposure ~ 1 µ s. height of a picture corresponds to ~ 8 cm. In the left bottom corner the tubular structure such as The ETS with diameter ~1 cm is visible and directed almost along a diagonal of figure. b) Right section: The image (width 1.75 cm) of the left-hand side of the denser and hotter core of the vertically aligned long plasma column (the Z-pinch axis in shown with a dashed-and-dotted line) in the electric discharge of Z-pinch type. The image (positive) is taken in the visible light with time exposure 2 ns nearly at the moment when discharge’s magnetic field squeezes the hot plasma column and partly strips a skeletal network from ambient luminous plasma (for an example of a strongly stripped skeleton in the same Z-pinch. Left section: The magnified, 0.35 cm wide window reveals the «hot spot» to be the edge of the filament which is close enough to a brighter core. A B C Fig. 5. The"electric torch-like structures“ (ETLS) in Z-pinch plasmas, electronic temperatures ~ 300 eV, density ~ 1018 cm-3, time of exposure ~ 2 ns. Heights of these images are ~ 1 mm, diameters of onductor of electromagnetic signals. Therefore, the open end of a dendrite electric circuit or a local observable ETLSs are ~ 0.25 mm. It seems that the straight blocks of skeletons may work as a guiding system for (and/or) a disruption of such a circuit (e.g., its sparkling, fractures, etc.) may self-illuminate its and to make its observable. The similar phenomena are observed in space plasma. Many luminous objects in the Universe represent such luminous butt-end of the CTS. The butt-ends of such open optical paths can correspond to sizes of stars, planetary nebulas, or galaxies and their congestions. Fig. 6. The torch-like structure of the filament of ~ 0.1 light year in diameter is seen in the fragment of the optical image of the Crab nebula [3]). The light from the filament’s edge anisotropically illuminates the ambient gas. Regardless the cylindrical straight body of the filament is visible due to internal radiation source or external illumination, the visible continuity of the filament is not destroyed by the partly opaque surrounding medium (seen in the left upper corner of the window) and may be traced in the full image outside the window. Fig. 7. The electric torch- like structure in the 2,400 light- year-wide core of the spiral galaxy NGC 1512 [4], which is 30 million light-years away and 70,000 light years across. (a) Color-composite image of the core, (b) similar image in the 2200 Å light, (c) contrasted image of a part of the image «b» (diameter of straight filament with a bright spot on its edge, seen in the upper right, is ~ 60 light-years). 4. Revelation of a Coaxially-Tubular Blocks of the SSU. The CТS are basis of the SSU as this type of blocks composes an overwhelming part of it. It is possible to show, that almost all luminous objects in the Universe are butt-ends of the above mentioned blocks of either scale. The CTS revealed in the Universe tend to self-similarity that leads to fractality of built by them and by observed structures. It is easy to demonstrate it by the examples of a structure of blocks of various scales of galaxies. The central fragment very large spiral galaxy where are precisely revealed the CTS of various scales (from 1017 cm up to 4 1018 cm) as an example is produced in the next figure. Fig. 8. a) Spiral galaxy NGC 1808. Constellation: Columba [5]. Distance: 40 million light years. b) A central portion of this galaxy. c) The width of figure corresponds ~ 7 1018 cm. The arrow specifies a direction of an axis of a galaxy. In the center of this galaxy it is revealed: the CТS (cylinder) in diameter ~ 2.3 1018 cm ; a telescopic putted tubular block (by diameter ~1.7 1018 cm ) on its axis; a dark filament (in diameter ~ 5 1017 cm) inside of them on axis of galaxy. Continuation in space above a galaxy of a central filament is a dark filament, leaving on the center same filament, but the greater diameter. The similar structure from telescopic enclosed the CТS is revealed to the left of the axis. It has of blue the CТS in the center of the similar structure built from tubes with a bright blue luminescence of their butt-end. In the center inside this structure it is precisely looked through bright white the CТS which center leaves dark thin filament which butt-end above all structure shine a blue luminescence. d) The magnified part of the window "с“. 5. Revelation of skeletal structures of the cooperating galaxies The CTS are most representative in the common structure of the Universe. Their participation in its dynamics leads to observation and definition of role of these structures in every possible space accident. It is possible to show, all cooperating galaxies represent interactions of the similar CTSs in the form of a breaks and/or collisions of the CTSs of a corresponding sizes. The image processed by means of the ММDC and received in the field of ultraviolet on next Fig. 5 is given, which represents the image of two cooperating galaxies, UGC 06471 and UGC 06472, and corresponding schematic of its representations in a window on the right. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/04/image/d The galaxy UGC 06471 and UGC 06472 Fig. 9. The galaxy UGC 06471 and UGC 06472 [6]. It is revealed, that the given object represents interaction of a three CTS. It is possible to assume, that two galaxies in the foreground are butt-end of a break of one the CTS (from the analysis of identity having chopped off on their external environments). The third CTS could be the reason of such break, as a result of its collision with a joint of first two. More thin analysis of this group of the CTS shows their topological identity. These blocks represent telescopic the enclosed tubes with radial connections. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/04/image/d (http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/22/image/a ) Fig. 10. Two interacting galaxies NGC 7318 A and NGC 7318 B [7] which are in constellation Pegasus on distance 270 million light years from the Earth are given here. These galaxies represent mutually - perpendicular butt-ends of a break of one CTS with diameter ~ 3 1022 cm. The structure of this break is precisely traced and its plan is given in a window below. This structure has fractured in a place of its fold at folding up twice. The multi-layered design of an internal structure of break- up of this CTS is very well seen. These galaxies were born as a result of a given filament fracturing. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/22/image/a A B Fig. 11. A) Here w have the cooperating galaxies NGC 7319A and NGC 7319B [7] of the same constellation Pegasus which also are by result of a fracturing of the CTS with diameter ~ 3 1022 cm. B) Here the result of the MMDC analysis of this image is presented. Butt-ends of the fracturing are located mutually perpendicularly. Dark filament on the right shows possible points of interface of the fracturing. From under index “b”, at the left, the outline of a dark filament which could be the reason of fracturing of this CTS is seen. C) This is a scheme of the image. C Fig. 12. a) The images of o cooperating galaxies 038/4039 of NGC 4 constellation Corvus [8] are submitted. b) Here the result of the MMDC analysis of galaxies image is presented. Identity of structures of these two galaxies as topological, so and in the sizes is visible. Moreover, in structure of the top galaxy a radial spokes which connects of a central coaxially- tubular part of galaxy with tubes located on lateral surface of a filament with the greater size are looked through. It is seen, what interface details of the lower galaxy have the same sizes as at the upper galaxy external coaxial of the top galaxy shows that the coaxially-tubular structures from which it consist have the same topology as the topology of whole galaxy. These two galaxies as a punch and a matrix have the same topology and the size. That is, their birth is caused by a breach of one CTS because their corresponding elements can be combined precisely. c) This is schematic representation of the given image. Fig. 13. Here two cooperating galaxies - NGC 2207 and IC 2163 are submitted [9]. The width of figure corresponds ~ 4 1023 cm. The analysis of the image (by means of MMDC) is applied for build a united network of the niverse. In that case all objects of this network are in direct connection. Such communication is revealing structure of interaction. For simplification of perception the schematic image of the given interaction of galaxies below is given. It is visible, that the given galaxies represent butt-ends of the broken CTS (the larger galaxy, NGC 2207, is on the left; the smaller one, IC 2163, is on the right). The first of them has the same size which is characteristic for spiral galaxies ~ 2 1023 cm Proceeding from the submitted scheme, it is possible to tell that the script of observable process can appear absolutely other, than it is represented now by astronomers. The large-scale CTS, incorporating among themselves can shown for the neighboring galaxies especially strongly when they belong to one treelike filament, being butt-ends of its branches. Sometimes this is able to reveal, and demonstrate. A B C Fig. 14. A) They are the g galaxies М81 and М82 to opinion of contemporary astronomers, according to the im 6. Cartwheel-Like Structures of the Universe cooperatin of the constellation « the Big she- bear» [10]. B) The MMDC has allowed revealing their structural interaction. М81 is a butt-end of an internal and acting part of a treelike filament with diameter ~ 3.5 1023 cm, and М82 a butt-end of its lateral cut down branch. To the right of the М82 it is looked through parallel to an axis of the basic filament the dark CTS which, obviously, has cut off a branch, having created the given galaxy. The bright luminous object located before М82 and hardly below of it, presentation of the image. Contrary age, obviously, these galaxies did not collide. lays on an axis of this dark structure. C) This is a schematic r The CWS are the in the Universe, and also hey are the most typical blocks of f which are difficultly to vealed by us topology of fractals spontaneously most interesting and complex observable blocks the observable fractal structures ot confuse with any another. If such structure is well oriented in a flatness of a shearing, then (at condition of a corresponding statistics) the structure clearly becomes apparent because the basic massif of points of this structure is fitted to a rim of a wheel, to its axis and radial spokes, making (on the square) half of area of whole wheel. It allows to identify precisely its under such circumstances. It is theoretically difficult to explain topology CWS by means of magnetic hydrodynamics. The mechanism [1a,b] of construction of the re gathering at formation of electric breakdown at the presence of elementary blocks of a dust, which have tendency to forming structures (for example, as carbon nanotubes or a similar structures but of other elements and chemical compounds) have been earlier considered. The sequence of generations CWS right up to the size ~ 1023 cm already has been shown earlier [1a, b]. The result of the analysis by means of ММDC of maps of redshifts is given below. A B C D Fig. 15. A) A fragment of distribution of the galaxies ( distances L up to 2.5 billion light-years away) 3.0° thickness in the South Galactic Pole strip. B) The image of map of 20,000 galaxies (for redshifts Z < 0.3, i.e. at of slice is centered at declination - 45° redshift in the red points in the colored image at http://www.astro.ucla.edu/~wright/lcrs.html [11]. The left border of the cone crosses the left hand side of the figure at a distance ~ 1.5 109 light years. Thickening of the spots with subsequent smoothing of the image are used here for correlation analysis the result of which gives a circle and a straight radial filaments - connections similar to a spokes. C) Here the increased image of South Pole of a redshift map after carrying out of the correlation analysis is given. At the left it is clearly seen the structure of type of cartwheel. D) Here the increased image of window in image C is given. It is clearly seen the components of cartwheel structure, the diameter of which is about 1.5 1027 cm (1.5 109 of light years). http://www.astro.ucla.edu/~wright/lcrs.html Fig. 16. Here have given the revealed Universe structure on the Southern part mentioned above databases, but not on 3.0 degree a cut but on its full set, i.e., its 9 degree a variant. Fig. 17. In center it is given image of the CWS with scale ~ 1.5 1027 cm which is result of analysis by means of the ММDC of a map redshift [11], at the left it is given image of the CWS of galactic scale (~ 1023 cm) [12]; on the right it is given image of the CWS of an explosion of supernova (~ 4 1019 cm) [13]. Here topological identity of observable large-scale structures of the Universe distinguished almost on 8 orders of size is precisely traced. It is well visible, that on an exit of radial spokes through a rim of the CWS it is formed structures similar to it itself. Through them the given structure is intertwined into the general network of the Universe. 7. CONCLUSIONS a) Our Universe has general skeletal structure which is made of separate coaxially-tubular blocks and blocks of type cartwheels. b) It is self-similar on any scales, and, h ) All luminous objects observable in the Universe are or by free butt-ends of the corresponding ence, fractal. sizes of coaxially-tubular blocks (of telescopic putted in each other), or by breaks of filaments which is assembled of these blocks. of formation of stars, galaxies and their d) As our Universe is in dynamics the processes congestions can go and presently by means of a fracturing of filaments with a corresponding diameters and of the time for their such formation is necessary much less, than along existing standard model. This is evidently shown on an examples of revealed structures at the analyses of the images cooperating galaxies and a part of structure of the Universe showing the same topology which been revealed by us on a wide range of scales and the phenomena earlier. The identity of -7 28 observable topology now is shown from 10 сm up to 10 cm, i.e., on 35 orders of size. It is possible to assume, that if we will be able to analysis of structure of substance at moving aside reduction of scales there we shall discover the same topology. That it means and what will be a result for us - will show time. References 1. A.B.Kukushkin, V.A.Rantsev-Kartinov, a) Phys.Lett. A, 306, 175-183; b) Science in Russia, 1, 42-47, (2004); c) Laser and Particle Beams, 16, 445-471,(1998); d) Proc. 17-th IAEA Energy Conference, Yokohama, Japan, 3, 1131-1134, (1998); e) Rev.Sci.Instrum., 70, 1387-1391,( 1999); f) “Advances in Plasma Phys. Research”, (Ed. F. Gerard, Nova Science Fusion Publishers, New York), 2, 1-22, (2002). B.N.Kolbasov, A.B.Kukushkin, V.A.Rantsev-Kartinov, et.set., Phys. Lett. A:a) 269, 363-367, (2000); b) 291, 447-452, (2001); c) Plasma Devices and Operations, 8 , 257-268, (2001). http://imgsrc.hubblesite.org/hu/db/2005/37/images/a/formats/full_tif.tif 4. 001/16http://hubblesite.org/newscenter/archive/2 5. http://imgsrc.hubblesite.org/hu/db/1998/12/images/b/formats/1280_wallpaper.jpg 6. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/04/image/d 7. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/22/image/a /releases/1996/36/image/a8. http://hubblesite.org/newscenter/newsdesk/archive 9. http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/41/image/ 10. http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/08/image/d 11. Shectman, S. A. et. al. The Las Campanas redshift survey. Astrophys. J. 470, 172- 188 (1996) 12. http://imgsrc.hubblesite.org/hu/db/1996/36/images/a/formats/web_print.jpg 13. http://chandra.harvard.edu/photo/snrg/index.html#object1 http://hubblesite.org/newscenter/archive/2001/16 http://imgsrc.hubblesite.org/hu/db/1998/12/images/b/formats/1280_wallpaper.jpg http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/04/image/d http://imgsrc.hubblesite.org/hu/db/1998/12/images/b/formats/1280_wallpaper.jpg http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/41/image/

0704.1449

The classification ofseparable simple C*-algebras which are inductive limits of continuous-trace C*-algebraswith spectrum homeomorphic to the closed interval [0,1]

THE CLASSIFICATION OF SEPARABLE SIMPLE C*-ALGEBRAS WHICH ARE INDUCTIVE LIMITS OF CONTINUOUS-TRACE C*-ALGEBRAS WITH SPECTRUM HOMEOMORPHIC TO THE CLOSED INTERVAL [0,1] GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Abstract. A classification is given of certain separable nuclear C*- algebras not necessarily of real rank zero, namely, the class of sep- arable simple C*-algebras which are inductive limits of continuous- trace C*-algebras whose building blocks have spectrum homeo- morphic to the closed interval [0, 1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated. 1991 Mathematics Subject Classification. 46L35, 46L06. Key words and phrases. K-theory, classification, C*-algebras, inductive limits, real rank one. Date: September 2005. http://arxiv.org/abs/0704.1449v1 2 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU 1. Introduction It is shown in [23] that an important class of separable simple crossed product C*-algebras are approximately subhomogeneous. Recall that a C*- algebra is said to be subhomogeneous if it is isomorphic to a sub-C*-algebra ofMn(C0(X)) for some natural number n and for some locally compact Haus- dorff space X. An approximately subhomogeneous C*-algebra, abbreviated ASH algebra, is an inductive limit of subhomogeneous algebras. This article contains a partial result in the direction of classifying all simple ASH algebras by their Elliott invariant. The first result on the classification of C*-algebras not of real rank zero was the classification by G. Elliott of unital simple approximate interval al- gebras, abbreviated AI algebras (see [12]). This result was extended to the non-unital case independently by I. Stevens ([30]) and K. Thomsen ([34]). Also, an interesting partial extension of this result to the non-simple case was given by K. Stevens ([32]). It is worth mentioning that all these algebras are what are referred to as approximately homogeneous algebras, abbreviated AH algebras, and that the most general classification result for simple AH algebras was obtained by Elliott, Gong and Li in [16]. One of the first isomorphism results for ASH algebras was the proof by H. Su of the classification of C*-algebras of real rank zero which are inductive limits of matrix algebras over non-Hausdorff graphs; see [33]. The classifica- tion of ASH algebras was also considered in [19], [27] and [29]. (This list of contributions is intended to be representative rather than complete for the classification of ASH algebras.) An important work on the classification of ASH algebras not of real rank zero, and in fact one of the first ones, is due to I. Stevens ([31]). The main result of the present paper is a substantial extension of Stevens’s work, to the class consisting of all simple C*-algebras which are inductive limits of continuous-trace C*-algebras with spectrum homeomorphic to the closed in- terval [0, 1] (or to a finite disjoint union of closed intervals). In particular, the spectra of the building blocks considered here are the same as for those CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 3 considered by Stevens. The building blocks themselves are more general. The isomorphism theorem is proved by applying the Elliott intertwining argument. Inspired by I. Stevens’s work, the proof proceeds by showing an Existence Theorem and a Uniqueness Theorem for certain special continuous trace C*- algebras. (As can be seen from the proofs, it is convenient to have a special kind of continuous trace C*-algebra as the domain algebra in both these theorems. By special we mean having finite dimensional irreducible repre- sentations and such that the dimension of the representation, as a function on the interval, is a finite (lower semicontinuous) step function.) The present Existence Theorem, Theorem 5.1, differs in an important way from that of [31], Theorem 29.4.1. In fact Theorem 29.4.1 of [31] is false, as is shown in Section 5.1 below. The proof of the present Existence Theorem is an eigenvalue pattern per- turbation, as shown in Section 5, which is similar to the approach used in [31]. (Indeed, once the statement of Theorem 29.4.1 of [31] is corrected, the argument given in [31] does not need to be essentially changed.) The proof of the present Uniqueness Theorem is different from the one in [31]. It uses the finite presentation of special continuous trace C*-algebras that was given in [17] and [18]. Also the present Uniqueness Theorem has the advantage that both the statement and the proof are intrinsic, i.e., there is no need to say that the building blocks are hereditary sub-C*-algebras of interval algebras as in [31]. In order to apply the Existence and Uniqueness Theorems, it is necessary to approximate the general continuous trace C*-algebras appearing in a given inductive limit decomposition by special continuous trace C*-algebras, as described in [18], Theorem 4.15. This is admissible since in [18] (and also more generally in [17]), it is shown that these special C*-algebras are weakly semiprojective, i.e., have stable relations. (A result of T. Loring, Lemma 15.2.2, [24], allows one to conclude that the original inductive limit decom- position can be replaced by an inductive limit of special continuous trace 4 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU C*-algebras.) An important step of the proof of the isomorphism theorem is the pulling back of the invariant from the inductive limit to the finite stages. The invari- ant has roughly two major components: a stable part and a non-stable part. The pulling back of the stable part is contained in [12] or [31] and is performed in the present situation with respect to the unital hereditary sub-C*-algebras. The intertwining which is obtained at the level of the stable invariant will approximately respect the non-stable part of the invariant on finitely many elements, as pointed out in [31]. To be able to apply the Existence Theo- rem it is crucial to ensure that the non-stable part of the invariant is exactly preserved on finitely many elements (actually, just a single element). It is possible to obtain an exact preservation of the non-stable invariant on finitely many elements because one can change the given finite stage algebras in the inductive limit decomposition in such a way that a non-zero gap arises at the level of the affine function spaces; see Section 8 below. It is this non-zero gap that will ultimately guarantee (after passing to subsequences in a convenient way) the exact intertwining on finte sets of the non-stable invariant, as shown in Section 9. It is worth mentioning that in the pulling back of the stable invariant, we must ensure, at the same time that the maps at the affine func- tion space level are given by eigenvalue patterns. This is necessary in order to apply the Existence Theorem and is possible by the Thomsen-Li theorem. Now all the hypotheses of the Elliott intertwining argument are fulfilled and in this way the proof of the isomorphism Theorem 3.1 is completed. I. Stevens’s description of the range of the invariant is also extended to include the case of unbounded traces (Theorem 3.2). To conclude, the class of simple inductive limits of continuous-trace C*- algebras under consideration is compared with the class of simple AI alge- bras. CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 5 2. The invariant The invariant is similar to the invariant I. Stevens has used in [31], usually summed up as the Elliott invariant, namely, (K0(A),AffT +A, Aff ′A), where K0(A) is a partially ordered abelian group, AffT +A is a partially ordered vec- tor space consisting of linear and continuous functions defined on the cone of traces T+A, Aff′A is a certain special subset of AffT+A. The special subset Aff ′A is the most important part of the invariant for our purposes, and in an informal way it might be said to be the non-stable part of the AffT+A. Formally, the special subset Aff′A is the convex set obtained as the closure of {â ∈ AffT+A| a ≥ 0, a ∈ Ped(A) and ||a|| ≤ 1} inside AffT+A, with respect to the topology naturally associated to a full projection. Here â is the linear and continuous function defined by the positive element a from the Pedersen ideal by â(τ) = τ(a) where τ ∈ T+A. As shown in [31], Remark 30.1.1 and Remark 30.1.2, the information given by Aff′A is equivalent with that given by the trace-norm map, which is a lower semicontinuous function µ : T+A→ R, µ(τ) = ||τ || and ∞ if τ is unbounded. It is a crucial fact that the trace-norm map is equivalent to the dimension function in the case of a building block algebra, cf. Section 4 below. The dimension function of a building block (i.e. the function that assigns to each point in the spectrum of the building block the dimension of the irreducible representation) can be viewed as a lower semicontinuous function on the ex- treme traces normalized on minimal projections in primitive quotiens and hence we can compare it with functions from AffT+A. Then the subset Aff′A is the closure of the set of all affine functions smaller than the dimension function. Conversely by taking the supremum over all elements of Aff′A we recover the dimension function in the case of the building blocks. 3. The results Using the invariant described above it is possible to prove a complete iso- morphism theorem, namely, 6 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Theorem 3.1. Let A and B be two non-unital simple C*-algebras which are inductive limits of continuous-trace C*-algebras with spectrum homeomorphic to [0, 1]. Assume that 1. there is a order preserving isomorphism ψ0 : K0(A) → K0(B), 2. there is an isomorphism ψT : AffT + → AffT+B, such that ψT (Aff ′A) ⊆ Aff ′, 3. the two isomorphisms are compatible: ψ̂0([p]) = ψT ([̂p]), [p] ∈ K0(A). Then there is an isomorphism of the algebras A and B that induces the given isomorphism at the level of the invariant. A description is given of the range of the invariant. More precisely, the following theorem is proved: Theorem 3.2. Suppose that G is a simple countable dimension group and V is the cone associated to a metrizable Choquet simplex. Let λ : V → Hom+(G,R) be a continuous affine map which takes extreme rays into ex- treme rays. Let f : V → [0,+∞] be an affine lower semicontinuous map, zero at zero and only at zero. Then (G, V, λ, f) is the invariant of some simple non-unital inductive limit of continuous-trace C*-algebras whose spectrum is the closed interval [0, 1]. 4. Special continuous trace C*-algebras with spectrum the interval [0,1] In this section we will introduce some terminology. A very important piece of data that we shall consider is a map that assigns, to each class of irreducible CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 7 representations, the dimension of a representation from that class. Roughly speaking, the dimension function can be thought of as the non-stable part of the invariant when restricted to the building blocks. Definition 4.1. Let A be a C*-algebra and let  denote the spectrum of A. Then the dimension function is the map from  to R ∪ +∞, π 7→ dim(Hπ), where by dim(Hπ) we mean the dimension of the irreducible representation It was shown in [18], Theorem 4.13, that the dimension function is a com- plete invariant for continuous trace C*-algebras with spectrum the closed interval [0, 1]. Also concrete examples were constructed for each given di- mension function, cf. Section 7 of [18]. Therefore given a lower semicontinuous integer valued (i.e., a “dimension function”) which is finite-valued and bounded we can exhibit a continuous trace C*-algebra  C0(An) C0(An) C0(An) . . . C0(An) C0(An) C0(An−1) C0(An−1) . . . C0(An−1) C0(An) C0(An−1) C0(An−2) . . . C0(An−2) . . . C0(An) C0(An−1) C0(An−2) . . . C[0, 1]  ⊆Mn ⊗ C[0, 1]. whose dimension function is the given function. Here An ⊆ An−1 ⊆ · · · ⊆ [0, 1] and each Ai is an open subset of [0, 1]. Moreover any trace on such an algebra is of the form tr ⊗ ν, where tr is the usual trace normalized on minimal matrix projections and ν is a finite measure on [0, 1]. The extreme traces are parameterized by t ∈ [0, 1], and are given as (tr ⊗ δt)t∈[0,1], where 8 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU δt is the normalized point mass at t. Then the trace norm map is equal to the dimension function when restricted to the extreme traces. To see that the trace norm map is equivalent to the special subset Aff ′() of the affine function space AffT+() we repeat the proof of I. Stevens from [31], Remark 30.1.1 and Remark 30.1.2. Inspired by a construction of I. Stevens in [31] we make Definition 4.2. A continuous-trace C*-algebra whose spectrum is [0, 1] will be called a special continuous-trace C*-algebra if its dimension function is a finite-valued finite step function: there is a partition of [0, 1] into a finite union of intervals such that the dimension function is finite and constant on each such subinterval. Remark 4.1. Let A be a continuous trace C*-algebra with spectrum [0, 1] and with dimension function d : [0, 1] → N ∪ {+∞}. There exists a projection- valued function that if composed with the rank function gives rise to the dimension function d. To see this first we notice that because the Dixmier- Douady invariant of A is trivial, the C*-algebra A is a continuous field of elementary C*-algebras over [0, 1], where the fibers are hereditary sub- C*-algebras of the algebra of compact operators. Then take the unit of the hereditary sub-C*-algebra in each fiber. In this way we construct a projection-valued function which is lower semicontinuous. By composing this constructed projection-valued function with the rank function we get the dimension function d. Remark 4.2. A priori our definition for a special sub-C*-algebra is more general than I. Stevens’s definition. As it is shown in [18], any special sub-C*-algebra in our sense is isomorphic to a special sub-C*-algebra in I. Stevens’s sense. Remark 4.3. It was shown in [18] that special continuous trace C*-algebras are finite presented and weakly semiprojective. Also a stronger result was proven in [8], namely that special continuous trace C*-algebras are strongly semiprojective. CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 9 5. Balanced inequalities and the Existence Theorem The proof of the isomorphism theorem 3.1 is based on the Elliott inter- twining argument. Among the main ingredients of this procedure are the Existence Theorem that will be described below as well as the Uniqueness Theorem that is presented in Section 6. It is worth noticing that for the Existence Theorem and the Uniqueness Theorem we require that the inequalities are balanced, i.e., independent of the choice we make for the normalization of the affine function space. We normalize the affine function spaces with respect to a full projection. Even though we fix a projection in the domain algebra for both the Existence The- orem and the Uniqueness Theorem, this choice does not make any difference when we apply the theorems to obtain an approximate commuting diagram. As was pointed out to us by Andrew Toms, we only need to consider a com- patible family of projections when we go through the whole proof, provided that a corresponding projection is chosen in the codomain algebra. In fact, we can state the theorems without mentioning the choices of the projections as long as their K0 -classes are compatible with respect to the K0 -map under consideration even though they exist and some choices of them will be used during the proof. To be able to focus on the new aspects of the present Existence Theorem as opposed to the Existence Theorem for unital continuous trace C*-algebras proved by Elliott in [12], we will both state the theorem and prove it in terms of so-called eigenvalue pattern maps. In our situation an eigenvalue pattern map is a positive unital map from C([0, 1]) to C([0, 1]) which is a finite sum of *-homomorphisms from C([0, 1]) to C([0, 1]). Using the Gelfand theory each such *-homomorphism is given by a continuous function from [0, 1] to [0, 1]. As follows from the intertwining of the invariant and will be explained below, Section 9, one can always obtain a (non-necessarily compatible) eigenvalue patterns maps. The proof of the Existence Theorem is obtained by perturbing an eigen- value pattern map between the affine function spaces in a such a way that it 10 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU defines an algebra map between the building blocks. Theorem 5.1. Let A be a special building block and by dA denote the dimension function of A. Let a finite subset F contained in AffT+A, and ǫ > 0 be given. There is f ′ ∈ Aff ′A such that for any special building block B with dimension function dB, and maps k : D(A) → D(B) and T : AffT +A → AffT+B verifying the conditions 1. k has multiplicity Mk, 2. T is given by an eigenvalue pattern and has the property T (f ′) ≤ dB, 3. k and T are exactly compatible, i.e., k̂([r]) = T ( ˆ[r]), there is a homomorphism ψ : A→ B such that k = ψ0 and ||(T − ψT )a||k̂(p) ≤ ǫ||a||p̂, a ∈ F. Remark 5.2. Recall that AffT+A is a Banach space with a norm given by ||f ||p = sup{|f(τ)| | τ(p) = 1, τ ∈ T+A}, where f ∈ AffT+A and p is a fixed full projection of A. In addition, using the norm we just defined, AffT+A is identified with C([0, 1]). This identification allows us to compare in the supremum norm the dimension function and elements of AffT+A. Also the norm of AffT+B is defined with respect to a projection from B which is Murray-von Neumann equivalent to k(p). Since our inequalities at the level of the affine function spaces are balanced, which is the only theorem that makes sense, in particular they are independent of the choice of the projection p. Proof. The idea of the proof is to choose in a clever way a function f ′ and then change within the given tolerance the eigenvalue functions that appear in the CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 11 eigenvalue pattern T so that the image of the dimension function dA under the new eigenvalue pattern is smaller than or equal the dimension function of the algebra B, as desired. Let ǫ > 0 and a finite set F ⊂ AffT+A be given. As already mentioned it is a crucial step how f ′ is chosen. There is no loss in generality if we assume that the dimension function dA has only one discontinuity point, t0 ∈ [0, 1]. Figure 1. Dimension function dA. Choose f ′ to be a continuous function such that f ′(t) = dA(t) for t ∈ [0, t0−δ]∪ [t0+δ, 1], f ′(t) ≤ dA(t) for t ∈ [0, 1], and f ′(t0) = dA(t0), where δ ≤ . Hence f ′ is a continuous function defined on the interval [0, 1] which approximate dA, namely f ′ is equal to dA except on a small neighbourhood around the discontinuity point. 12 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Figure 2. Graph of f ′. Next we proceed by showing how to change the eigenfunctions such that a desired eigenvalue pattern is obtained. We will carry out this procedure in a very special case, namely all the eigenfunctions are assumed to be the identity function. Figure 3. Eigenfunction λ. In the above picture we have the original eigenvalue function λ which is the identity map. We define a new eigenvalue function as the picture shows below, Figure 4. More precisely the new eigenvalue function λ̂ : [0, 1] → [0, 1], λ̂(t) = t for t ∈ [0, t0− δ)∪ (t0+ δ+ δt0, 1], λ̂(t) = t0− δ for t ∈ [t0− δ, t0+ δ], CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 13 the linear map λ̂(t) = t0 − δ + (t − t0 − δ) 2δ+δt0 for t ∈ [t0 + δ, t0 + δ + δρ], where δt0 is a strictly positive number such that t0 + δ + δt0 ≤ 1. Figure 4. Eigenfunction λ̂. A short computation or a geometric argument shows that the difference ||λ− λ̂||∞ = 2δ. Moreover the dimension function dA evaluated on the perturbed eigenvalue λ̂ is smaller then f ′ evaluated on the given eigenvalue λ dA(λ̂(t)) ≤ f ′(λ(t)). Hence by hypothesis 2 we have dA ◦ λ̂ ≤ f ′ ◦ λ ≤ dB. Here we say that one dimension function is smaller than another one if the relation holds pointwise. The change of the eigenvalues is small because of the choice of δ ||(Tλ − T )(a)||k̂(p) = ||a ◦ (λ̂i − λi)||k̂(p) 14 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU sup{|a ◦ (λ̂i − λi)(τ)| | τ(k(p)) = 1, τ ∈ T ∣∣∣∣a ◦ (λ̂i − λi) )∣∣∣∣ | τ(p) = Mk, τ ∈ T Mk||a ◦ (λ̂i − λi)||p̂ ≤ 2δM k ||a||p̂ ≤ ǫ||a||p̂, a ∈ F. To obtain the inequality above we used the linearity of the function a◦(λ̂i− λi) and that an extreme trace τ in T +A has the property that τ(k(p)) = 1 if and only if τ(p) =Mk. We claim that the argument for the special case shown above can be ex- tended to the case of piecewise linear eigenfunctions which is known to be equivalent to the general case of continuous eigenfunctions that arise in the inductive limits of interval algebras (see for instance [12]). 5.1. An exact inequality is necessary between the non-stable part of the invariant. As mentioned in the introduction, the Theorem 29.4.1 of [31] is false. To prove the Existence Theorem it is fundamental to have an exact inequality between the non-stable part of the invariant at the level of the affine function space, i.e., T (f) ≤ dB for some continuous affine function f ≤ dA. A weaker inequality is required in the statement of the Existence Theorem of [31], Theorem 29.4.1, i.e., T (f) ≤ dB(1+δ) for some small δ > 0. Therefore it is possible to construct a counterexample to the I. Stevens Ex- istence Theorem. This counterexample is already assuming that the positive linear map T is given by an eigenvalue pattern. To reduce the proof of The- orem 29.4.1 of [31] to an eigenvalue pattern problem, one needs an extra assumption in hypothesis 2, for instance a positive gap η > 0 in the other side of the inequality described above T (f) + η ≤ dB(1 + δ). CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 15 Next we describe the counterexample. Let dA be the lower semicontinu- ous function defined on [0, 1] which is equal to 2 on the subintervals [0, 1/2) and (1/2, 1], and equal to 1 at 1/2. Let ǫ0 be such that 0 < ǫ0 < 1/4 and F = {a1(t) = t}. Let f be a continuous function which approximates dA. Since they can not be equal everywhere around 1/2, we can assume that f(t) < 2 = dA(t) for all t in (1/2− η, 1/2+ η), where η > 0 can be chosen as small as needed. Let δ > 0 be given. There exists a positive integerMk such that 2Mk−1 Then choose T to be defined by Mk eigenvalue functions (λi)i=1,...,Mk , all be- ing the identity functions, λi(t) = t, for all i = 1, . . . ,Mk. Next choose B to be a continuous trace C*-algebra with dimension function constant equal to 2Mk − 1. Note that the hypothesis 2 of the Existence Theorem 29.4.1 from [31] holds T (f)(t) = f ◦ λi(t) ≤ 2Mk ≤ (1 + δ)dB(t). Now we claim that among all perturbations of T which are within the given ǫ0 with respect to the finite set F , the particular one P which is given by the continuous eigenfunctions (µi)i=1,...,Mk that have the property µi(t) = 1/2 for t ∈ (1/2− η, 1/2 + η), is the smallest in the sense that the value of P (dA) is the smallest. Here it is important to notice that because ǫ0 < 1/4 it forces that (µi)i(t) = λ(t) = t for t close to 0 and 1 including 0 and 1. In particular we have (µi)(0) = λi(0) = 0. Therefore P (dA)(0) = dA(µi(0) = 2Mk > 2Mk − 1 = dB(0). 16 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Therefore we cannot perturb the eigenfunctions to obtain a compatible eigenvalue pattern and the Existence Theorem as stated in [31] cannot be proved. 6. Uniqueness Theorem It is important to notice that the conclusion of the Existence Theorem is part of the hypothesis of the Uniqueness Theorem; this makes sense since all inequalities are balanced (i.e. independent of the choice of projection with respect to which the normalization is done). Theorem 6.1. Let A be a special continuous-trace C*-algebra, F ⊂ A a finite subset and ǫ > 0. Let B be a special continuous-trace C*-algebra and ψ, ϕ : A→ B be maps with the following properties: 1. ϕ0 = ψ0 : K0(A) → K0(B), 2. ψ and ϕ have at least the fraction δ of their eigenvalues in each of the d consecutive subintervals of length 1 of [0, 1], for some d > 0 such that for r̂i the functions equal to 0 from 0 to , equal to 1 on [ i+1 , 1] and linear in between, for each 0 ≤ i ≤ d, ||(ϕT − ψT )(r̂i)||K(p) < δ||r̂i||p, with respect to the norm of AffT+B, Then there is an approximately inner automorphism of B, f , such that ||(ψ − fϕ)(a)|| < ǫ, a ∈ F Proof. Because of the isomorphism theorem 4.13 from [18], there is no loss of generality to assume that our building blocks are in a very special form CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 17  C0(A1) C0(A1) C0(A1) . . . C0(A1) C0(A1) C0(A2) C0(A2) . . . C0(A2) C0(A1) C0(A2) C0(A3) . . . C0(A3) . . . C0(A1) C0(A2) C0(A3) . . . C[0, 1]  Notice that the cancellation property holds for the unital sub-C*-algebra of A and any projection of A is Murray-von Neumann equivalent to a projection inside of the unital sub-C*-algebra. Therefore the cancellation property holds for A. A similar argument shows that the cancellation property holds for any continuous-trace C*-algebra with the spectrum the closed interval [0, 1]. Since ϕ0 = ψ0, we can assume that ϕ(p) = ψ(p), where p is the unit of the sub-C*-algebra C([0, 1]) of A. In other words the restrictions of the maps to the unital subalgebra share the same unit. The stable part of the Elliott invariant (i.e., the K0 group and the affine function space AffT+) of A and of C([0, 1]) is the same. Let us restrict the two maps ϕ and ψ to the unital sub-C*-algebra C([0, 1]). The image of C([0, 1]) under ϕ and ψ is up to a unitary a full matrix algebra over the interval. Then using assumptions 1 and 2 we notice that the hypotheses of the Elliott Uniqueness Theorem ([12], Theorem 6), are fulfilled. Hence we get a partial isometry V of B (a unitary inside of the full matrix sub-C*algebra of B) such ||ϕ(fAi ⊗ enn)− V ψ(fAi ⊗ enn)V ∗|| ≤ ǫ, i ∈ {1, . . . , n}. We want this relation to hold for the case when the domain is A. We follow a strategy already present in the case of full matrix over the interval. An 18 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU important data that we will use is that the domain algebra A has a finite presentation. In fact we will use the concrete description of this presentation that was given in [18], Section 8. The set of generators consists of elements of the form fAi ⊗ ein which are certain positive functions tensor the matrix units. For each i let ui be a continuous function defined on [0, 1] which is equal to 1 on Ai except near the end points of each open subinterval of Ai and 0 otherwise. One can think of ui as an approximate unit of the functions fAi, i ∈ {1, . . . , n} and later estimates depend on the size of the subset of Ai where ui is not equal to 1. Define ϕ(ui ⊗ eni) ∗V ψ(ui ⊗ eni). Vψ(fAi ⊗ eni)V ϕ(uk⊗enk) ∗V ψ(uk⊗enk))ψ(fAi⊗eni)( ψ(ul⊗eln) ∗V ∗ϕ(ul⊗enl)) = = ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)( ψ(ul ⊗ eln) ∗V ∗ϕ(ul ⊗ enl)) = = ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)( ψ(ul ⊗ enl)V ∗ϕ(ul ⊗ enl)) = = ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)ψ(ui ⊗ ein)V ∗ϕ(ui ⊗ eni) = = ϕ(un ⊗ enn)V ψ(fAi ⊗ enn)V ∗ϕ(ui ⊗ eni). Now we have that ϕ(fAi ⊗ eni) = ϕ(un ⊗ enn)ϕ(fAi ⊗ enn)ϕ(ui ⊗ eni). CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 19 Therefore ||ϕ(fAi ⊗ eni)− Vψ(fAi ⊗ eni)V ∗|| = ||ϕ(un ⊗ enn)(ϕ(fAi ⊗ enn)− V ψ(fAi ⊗ enn)V ∗)ϕ(ui ⊗ eni)|| ≤ ≤ ||ϕ(un ⊗ enn)||ǫ||ϕ(ui ⊗ eni)|| i.e. it can be made as small as needed. We want to argue that V gives rise to a partial isometry. Let us calculate V∗V = ψ(ul ⊗ enl) ∗V ∗ϕ(ul ⊗ enl) ϕ(ui ⊗ eni) ∗V ψ(ui ⊗ eni) = ψ(ul ⊗ eln)V ∗ϕ(ul ⊗ enl) ϕ(ui ⊗ ein)V ψ(ui ⊗ eni) = Assuming that each ui is equal to 1 on the open intervals Ai except small neighbourhood around the end points of Ai we get ψ(ui ⊗ ein)V ∗ϕ(ui ⊗ enn)V ψ(ui ⊗ eni) which is very close to ψ(ui ⊗ ein)ψ(ul ⊗ enn)ψ(ui ⊗ eni) = ψ(ui ⊗ eii) which is the value of the projection-valued map of the hereditary sub-C*- algebra generated by ψ(A) inside B. In other words V∗V is as close as we want to be a projection. It is important to notice that this is true if we are not in a small neighbourhood of the singularity points of the dimension function of the hereditary sub-C*-algebra generated by ψ(A) (i.e. whenever 20 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU ui = 1). Similarly VV∗ is almost equal to the i=1 ϕ(ui ⊗ eii) if we are not in a small neighbourhood of the singularity points of the dimension function of the hereditary sub-C*-algebra generated by ϕ(A). Notice that any singularity point y0 of the dimension function of the hereditary sub-C*-algebra generated by ϕ(A) or ψ(A) has the property that there is an eigenfunction λi such that λi(y0) is a singularity point of the dimension function dA ofA. In addition λi is uniform continuous function from [0, 1] to [0, 1]. Hence small neighbourhoods of y0 correspond to small neighbourhoods of some singularity point of dA. From the polar decomposition V = W|V| we get a partial isometry W . We claim that W still intertwines approximately the two maps ϕ and ψ, i.e., ||ϕ(fAi ⊗ eni)−Wψ(fAi ⊗ eni)W ∗|| < 3ǫ, ||W∗ϕ(fAi ⊗ eni)W − ψ(fAi ⊗ eni)|| < 3ǫ. This is true because ||ϕ(fAi ⊗ eni)−Wψ(fAi ⊗ eni)W ∗|| = = ||ϕ(fAi⊗eni)−Vψ(fAi⊗eni)V ∗+W|V|ψ(fAi⊗eni)|V|W ∗−Wψ(fAi⊗eni)W ∗|| ≤ ≤ ||ϕ(fAi⊗eni)−Vψ(fAi⊗eni)V ∗||+||W|V|ψ(fAi⊗eni)|V|W ∗−Wψ(fAi⊗eni)W ∗|| ≤ ≤ ǫ+ |||V|ψ(fAi ⊗ eni)|V| − ψ(fAi ⊗ eni)|| ≤ ≤ ǫ+ |||V|ψ(fAi ⊗ eni)|V|− |V|ψ(fAi ⊗ eni)+ |V|ψ(fAi ⊗ eni)−ψ(fAi ⊗ eni)|| ≤ ≤ ǫ+ ǫ+ ǫ = 3ǫ. and similarly we get the other desired inequality. Hence we have constructed a family of partial isometriesW from the hered- itary sub-C*-algebra generated by ϕ(A) to the hereditary sub-C*-algebra CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 21 generated by ψ(A). In addition W induces an isomorphism between the two above mentioned hereditary sub-C*-algebras. In particular it implies that the two hereditary sub-C*-algebras have the same dimension function. Next we will show how to approximateW with a unitary in the unitazation of the codomain algebra. Let us start by applying Theorem 4.12 of [18] to the projection-valued function corresponding to the hereditary sub-C*-algebra generated by ϕ(A). Hence we get a decomposition, possibly infinite, in terms of functions each of which is projection-valued of rank 1 on a certain open subset of [0, 1] and zero otherwise. Notice that the discontinuity points of the dimension func- tion of the hereditary sub-C*-algebra generated by ϕ(A) correspond to the discontinuity points of the functions appearing in the decomposition and the open sets are increasing in a suitable sense. Next we apply Lemma 6.2 for each point at singularity in the interval [0, 1], or, in other words, to each function appearing on the decomposition. Thus, we have a family of unitaries that preserves the continuity of the continuous elements of the hereditary sub-C*-algebra ϕ(A) and at the same time has the property that it still intertwines the two maps. In the following lemma the hereditary sub-C*-algebras H1 and H2 are as- sumed to be continuous bundles over [0, 1] (for more details about continuous bundles of C*-algebras see [20]). If A is a continuous bundle of C*-algebras over [0, 1] then At stands for the fiber of A over t. Lemma 6.2. Let H1 and H2 be hereditary sub-C*-algebra of M2(C[0, 1]) with the same spectrum [0, 1] and identical dimension function equal to 1 on the closed interval [0, t0] and equal to 2 on the half-open interval (t0, 1], t0 ∈ (0, 1). Let W = (W (t))t∈[0,1] be a family of partial isometries indexed by the points of [0, 1]. For each t ∈ [0, 1], Wt : M2(C) → M2(C) such that W (t)W (t) the unit of H t1 and W (t) ∗W (t) = the unit of H t2. Then there exists a family 22 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU W⊥ of partial isometries indexed by [0, 1] such that W + W⊥ is a unitary inside of M2(C[0, 1]) and (W +W ⊥)t(f)(t) = Wt(f)(t) for any continuous function f ∈ H1 and t ∈ [0, 1]. Proof. Diagrammatically the dimension function of H1 and H2 can be pictured as follows. Figure 5. Dimension function of H1 and H2. We construct the family W⊥ = (W⊥t )t∈[0,1] as follows. Fix a t in [0, 1], t ≤ t0. W (t) is a partial isometry on some dimension-one subspace ofM2(C). Hence Wt(M) = c(M)Mt where c(M) is a constant depending on M and Mt is a projection matrix in M2(C). Let W t = c(M)(I2 − Mt). Notice that Wt +W t is a unitary operator on M2(C). If t > t0 then W t = 0. The family of unitaries (Wt +W t )t∈[0,1] is continuous except at the point t0. Our work below shows that this family can be modified to be continuous overall [0, 1]. Extend (Wt)t∈[0,t0] to be a continuous family (W t )t∈[0,1] of partial isometries on dimension-one subspaces of M2(C). W and lim t→t0,t>t0 (Wt −W t ) are two partial isometries on the same dimension one subspace of M2(C), hence they CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 23 differ by a constant of absolute value one, i.e. W⊥t0 = c limt→t0,t>t0 (Wt −W Define the continuous family of unitaries (Ut)t∈[0,1] to be Ut = Wt +W t ≤ t0 and Ut = W t + c(Wt −W t ) if t > t0. Hence the continuous family of unitaries Wt is given by Ut and (Ut(f)(t) = Wt(f)(t) for any continuous function f ∈ H1 and t ∈ [0, 1]. 7. Inductive limits of special continuous trace C*-algebras Next let us show that the Existence Theorem and the Uniqueness Theorem presented above can be applied, i.e., that the hypotheses of the theorems can be fulfilled. As a first step in this direction let us show that an inductive limit of continuous-trace C*-algebras with spectrum [0, 1] (or disjoint unions of closed intervals) is isomorphic to an inductive limit of special continuous- trace C*-algebras. The basic tools in establishing this step are the fact that special continous trace C*-algebras are semiprojective (cf. [18], Theorem 6.5) and a result by T. Loring ([24], Lemma 15.2.2) which for the convenience of the reader we state below: Suppose that A is a C*-algebra containing a (not necessarily nested) se- quence of sub-C*-algebras An with the property that for all ǫ > 0 and for any finite number of elements x1, . . . , xk of A, there exist an integer n such {x1, . . . , xk} ⊂ǫ An. 24 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU If each An is weakly semiprojective and finitely presented, then A ∼= lim (Ank, γk) for some subsequence of (An) and some maps γk : Ank → Ank+1. Proposition 7.1. Let A be a simple inductive limit of continuous-trace C*- algebras whose building blocks have their spectrum homeomorphic to [0, 1]. Then A is an inductive limit of direct sums of special continuous-trace C*- algebras with spectrum [0, 1]. Proof. In Proposition 5.4 and Theorem 6.5 of [18] it is proved that the class of special continuous trace C*-algebras with spectrum [0, 1] are finitely pre- sented and have weakly stable relations. Each building block from the in- ductive limit decomposition of A can be approximated by special continuous trace C*-algebras (cf. Theorem 6.14 of [18]). Then A satisfies Loring’s hy- pothesis where the sequence of semiprojective algebras is given by the special algebras from the approximation of the building blocks. Thus the Loring’s lemma implies that A is an inductive limit of special continuous trace C*- algebras. � 8. Getting a non-zero gap at the level of affine function spaces To be able to exactly intertwine the non-stable part of the invariant it is useful to know that the dimension function of any building block Am or Bm is taken by the homomorphism φm,m+1 respectively ψm,m+1 into a function smaller than or equal to the dimension function of Am+1 or Bm+1 such that a non-zero gap arises. In other words we want to exclude the possible cases when the dimension function is taken into the next stage dimension function such that equality holds at a point or at more points. We shall show this in the following lemma. Recall that because of Proposition 7.1, the algebras that we want to classify can be assumed to be inductive limits of special CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 25 continuous trace C*-algebras with spectrum [0, 1], i.e., A ∼= lim (An, φnm) and B ∼= lim (Bn, ψnm), where An, Bn are special continuous trace C*-algebras. Lemma 8.1. Let A = lim (An, φnm) be a simple C*-algebra, where each An is a special continuous trace C*-algebra with spectrum the closed interval [0, 1] and the dimension function assumed to be a finite-valued bounded function. Then there exists δ1 > 0, a subsequence (Ani)ni≥0 of (An)n and a sequence of maps φi : Ani → Ani+1 such that 1. A ∼= lim (Ani, φnimi), 2. (φn1n2)T (P̂An1) + δ1 < P̂An2 , where the inequality holds pointwise, (φnm)T is the induced map at the level of the affine function spaces, PAn1 and PAn2 are the units of the biduals of An1 and An2, and P̂An1 and P̂An2 denote the corresponding lower semicontinuous functions. Proof. Let A be equal to lim An with maps φn,m : An → Am. The plan is to keep the same building blocks and to change slightly the maps with respect to some given finite sets such that the desired property holds. To do this we use the property that the building blocks that appear in the inductive limit decomposition are weakly semiprojective. Assume that the dimension function of φ12(A1) equals the dimension func- tion of A2 at some point or even everywhere and let ǫ > 0, F1 ⊂ A1 be given. Because the largest value of the dimension function of the hereditary sub-C*-algebra generated by φ12(A1) inside A2 is attained on an open subset U of [0, 1], let us construct another dimension function as follows: shrink one of the open intervals of the open set U to get U ′ and in exchange enlarge the interval adjacent to that discontinuity point. U ′ is constructed in a such a way that is as close as necessary to the given U . In this manner we find a sub-C*-algebra B which is as close as we want to the hereditary sub-C*-algebra generated by φ12(A1) inside of A2. Next we use that A1 is weakly semiprojective to find another *-homomorphism ρ1 : A1 → B which is close within the given ǫ on the given finite set F1. 26 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Then there exists some open interval between the dimension function of A2 and the dimension function of the B. This open interval corresponds to a non-zero ideal I1 inside of A2. Now the image of I1 in the inductive limit is also a non-zero ideal. Since the inductive limit is simple, it implies that the ideal is the whole algebra. We know that there are full projections in the inductive limit. Therefore there is a finite stage in the inductive limit of the ideals coming from I1 that has a full projection. Assume that the finite stage is inside of Ak. This means that at that stage the image of the ideal I1 is Ak. Pick a strictly positive element a1 in I1. Then the image of a1 in Ak will be strictly positive at each point from [0, 1], k > 1. This shows that the image of the dimension function dB inside the dimension of Ak has a gap of at least 1 everywhere in [0, 1]. Because of the normalizations of the affine function, this gap of size 1 will correspond to some strictly non-zero δ1. To complete the proof we relabel B as An1, Ak as An2 etc. Corollary 8.2. Let A = lim (An, φn,m) be a simple C*-algebra. Then there ex- ists a sequence (δi)i≥1, δi > 0, a subsequence of algebras (Ani)i≥1 of (Ai)i≥1 and a sequence of maps φ : Ani → Ani+1 such that: 1. A ∼= lim (Ani, φni,mi), 2. φTni,ni+1(P̂Ani) + δi < P̂Ani+1 . Proof. Follows by successively applying the previous lemma. � 9. Pulling back of the isomorphism between inductive limits at the level of the invariant Step 1 The intertwining between the stable part of the invariant With no loss of generality we assume that the building blocks have the CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 27 following concrete representation  C0(A1) C0(A1) C0(A1) . . . C0(A1) C0(A1) C0(A2) C0(A2) . . . C0(A2) C0(A1) C0(A2) C0(A3) . . . C0(A3) . . . C0(A1) C0(A2) C0(A3) . . . C[0, 1]  One can distinguish a full unital hereditary sub-C*-algebra  0 0 0 . . . 0 . . . 0 . . . 0 0 . . . C[0, 1] . . . C[0, 1] . . . 0 . . . C[0, 1] . . . C[0, 1]  The unital hereditary sub-C*-algebra has the same stable invariant (i.e., K0, AffT + and the pairing) as the given C*-algebra. Moreover the unital hereditary sub-C*-algebra is a full matrix algebra over the closed interval [0, 1]. Using this fact we derive an intertwining between the stable invariant, as is shown in [31] or originally in [12]. It is important to mention the method of normalizing the affine function spaces. Pick a full projection p1 ∈ A1. Normalize the affine space AffT with respect to p1. Next consider a image of p1 in A2 under the map at the dimension range level, call it p2. Normalize AffT +A2 with respect to p2. Note that the map which is induced at the affine level is a contraction. Continue in this way so that we obtain an inductive limit sequence at the level of the 28 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU affine spaces, with all the maps being contractions: AffT+A1 → AffT +A2 → · · · → AffT Let p∞ denote the image of p1 in the inductive limit A and denote by q∞ a representative of φ0(p∞) in B. Then there exists q1 ∈ B1 such that the image of q1 is q∞ in the inductive limit. Normalize the AffT +B1 with respect to q1, AffT+B2 with respect to a image of q1 in B2 and so on. Hence we obtain another inductive limit of affine spaces with contractions maps AffT+A1 → AffT +A2 → · · · → AffT AffT+B1 → AffT +B2 · · · → AffT As already mentioned above, we pull back the invariant for the unital hered- itary sub-C*-algebras (i.e. full matrix algebras or the stable invariant). This will give rise to an exact commuting diagram at the K0-level, an approximate commuting diagram at the affine function spaces level and an exact pairing. The compatibility can be made exact as shown in [11] by noticing that , be- cause of simplicity, non-zero positive elements in both K0 and AffT + are sent into strictly positive elements and then normalize the affine function spaces in a suitable way. To summarize, we now have a commutative diagram C[0, 1] −→ C[0, 1] −→ . . . −→ (AffT+A,Aff ′A) ↓ τ1 ր τ 1 ↓ τ2 ր τ 2 ր l C[0, 1] −→ C[0, 1] −→ . . . (AffT+B,Aff ′B) where AffT+Ai and AffT +Bi are identified with C([0, 1]) and each finite stage algebra Ai and Bi is assumed to have only one direct summand. For us it is very important to study the pulling back of the non-stable part CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 29 of the invariant. Step 2. The intertwining of the non-stable part of the invariant As I. Stevens mentioned in [31], at this moment we know that the non- stable part of the invariant is only approximately mapped at a later stage into the non-stable part of the invariant. To be able to apply the Existence Theorem 5.1, one needs to check that hypothesis 2 can be ensured. Otherwise, a counterexample can be given to the Existence Theorem, as shown in Section 5.1 above. The special assumption from the hypothesis of the isomorphism theorem, φT (Aff ′A) ⊆ Aff ′B, as well as Corollary 8.2 will be used to prove the above mentioned claim. By applying Corollary 8.2 to the given inductive limits A = lim (An, φn,m), B = lim (Bn, φn,m) we get two sequences (δi)i≥1, δi > 0 and (δ i)i≥1, δ i > 0 respectively, and two subsequences of algebras such that after relabeling, we can assume that φii+1(P̂Ai) + δi < P̂Ai+1 , ψii+1(P̂Ai) + δi < P̂Ai+1 , ψii+1(P̂Ai) + δi < P̂Ai+1 and ψii+1(P̂Ai) + δi < P̂Ai+1 for all i ≥ 1. Reworking the intertwining of the stable invariant for the new sequences of algebras and the new maps that have gaps δi we obtain the following intertwining C([0, 1]) −→ C([0, 1]) −→ . . . −→ (AffT+A,Aff ′A) ↓ τ1 ր τ 1 ↓ τ2 ր τ 2 ր l C[0, 1] −→ C[0, 1] −→ . . . (AffT+B,Aff ′B) As a consequence of the Thomsen-Li theorem, which in the present case states that the closed convex hull of the set of all unital *-homomorphisms of C([0, 1]) in the strong operator topology is exactly the set of positive of unital operators on C([0, 1]), we can assume that all the maps φii+1, ψii+1, τi, τ i are 30 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU given by eigenvalue patterns. Because each such map takes the unit, say p̂, into the unit, K̂(p), it follows that each map is an average of the eigenvalues, i.e., φi,i+1(f) = , etc. Let P̂A1 be the image in the affine function space of the unit in the bidual of A1. Take a continuous function f smaller than P̂A1. It is important to say that there are no extra conditions on f , i.e., f can be any element of the special set AffT′A1. Then there exists δ1 > 0 such that φ12(P̂A1) + δ1 < P̂A2. Since φ12(f) ≤ φ12(P̂A1) we have φ12(f + δ1) ≤ φ12(P̂A1 + δ1) < P̂A2. Since φT (Aff ′A) ⊆ Aff ′B, it follows that there exists a large N and ǫN ≤ δ1 such that τN ◦ φN−2N−1 ◦ . . . φ12(f + δ1) < P̂BN + ǫN . It is important to say that a different choice for f will give rise to pos- sibly different N . This is not a difficulty because we can always pass to subsequence. Equivalently we have τN ◦ φN−2N−1 ◦ . . . φ12(f) + δ1 < P̂BN + ǫN . Using δ1 ≥ ǫN we conclude τN ◦ φN−2N−1 ◦ . . . φ12(f) < P̂BN , which is the desired strict inequality from the hypothesis 2 of the Existence Theorem 5.1. CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 31 10. The Isomorphism Theorem To complete the proof of the Isomorphism Theorem 3.1 for the algebras Ai = A and lim Bi = B, we have to construct an approximate commu- tative diagram at the algebra level in the following sense, as was defined by Elliott in [11],“for any fixed element in any Ai (or Bi), the difference of the images of this element along two different paths in the diagram, starting at Ai (or Bi) and ending at the same place, converges to zero as the number of steps for which the two paths coincide, starting at the beginning, tends to infinity.” At this stage because of Step 2 of the previous section, Section 9, we can ap- ply the Existence Theorem to generate a sequence of algebra homomorphisms ν1, ν2, . . . and ν 2, . . . such that ||τi(f)−νi∗(f)|| ||f || ||τ ′i(f)−ν i∗(f)|| ||f || f ∈ Fi and g ∈ Gi, where νi∗, ν i∗, are the induced afine maps by algebra maps νi, ν i, and Fi and Gi are finite sets. After relabeling the indices of the inductive limit systems we now have a (not necessarily approximately commutative) diagram of algebra homomor- phisms −→ A2 −→ . . . −→ A ↓ τ1 ր τ 1 ↓ τ2 ր τ −→ B2 −→ . . . −→ B that induces an approximately commutative diagram at the level of the in- variant. This will be done with respect to given arbitrary finite sets Fi ⊂ Ai and Gi ⊂ Bi. To make the diagram approximately commuting we modify the diagonal maps by composing with approximately inner automorphisms and this will 32 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU be done with respect to a given arbitrary finite sets Fi ⊂ Ai and Gi ⊂ Bi with dense union in A and B respectively. Here we notice that we can apply the Uniqueness Theorem to the data ob- tained from the the Existence Theorem because our inequalities are balanced. For every ǫ > 0 we find an increasing sequence of integers 1 = M0 < L1 < M2 < L2 < . . . and unitaries (UMi+1) ∈ A , (V ni )n ∈ B such that for f ∈ FMi and g ∈ GLi we have ||UMi+1τ (τMi(f))U − φMiMi+1(f)|| ||f || ||VMi+1τLi(τ (g))V ∗Li+1 − φLiLi+1(g)|| ||g|| In other words passing to suitable subsequences of algebras, it is possible to perturb each of the homomorphisms obtained in the Existence Theorem by an approximately inner automorphism, in such a way that the diagram becomes an approximate intertwining, in the sense of Theorem 2.1, [11]. Therefore, by the Elliott approximate intertwining theorem (see [11], The- orem 2.1), the algebras A and B are isomorphic. 11. The range of the invariant In this section we prove Theorem 3.2 which answers the question what are the possible values of the invariant from the isomorphism theorem 3.1. It is useful to notice that the invariant consists of two parts. One part is the stable part, i.e., K0, AffT +, λ : T+ 7→ S(K0) which was shown by K. Thomsen in [34] to be necessary if one wants to construct an AI-algebra, and the other part which one may call the non-stable part, namely Aff′ or equivalently, as shown in [31], Remark 30.1.1 and Remark 30.1.2, the trace norm map. It is the non-stable part of the invariant that one needs to investigate in its full generality. Next the definition of the trace norm map is introduced. CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 33 Definition 11.1. Let A be a sub-C*-algebra of a C∗-algebra B. The trace norm map associated to A is a function f : T+(A) → (0,∞] such that f(τ) = ||τ |A||, ∞ if τ is unbounded. Recall that: Definition 11.2. T+(A) is the cone of positive trace functionals on A with the inherited w*-topology. Remark 11.1. The trace norm map is a lower semicontinuous affine map (being a supremum of a sequence of continuous functions). Remark 11.2. The dimension range can be determined using the values of the trace norm map f , the simplex of tracial states S and dimension group G. A formula for the dimension range D is: D = {x ∈ G/v(x) < f(v), v ∈ S, v 6= 0} I. Stevens has constructed a hereditary sub-C*-algebra of a simple (uni- tal) AI-algebra which is obtained as an inductive limit of hereditary sub-C*- algebras of interval algebras, and has as a trace norm map any given affine continuous function; cf.[31], Proposition 30.1.7. Moreover she showed that any lower semicontinuous map can be realized as a trace norm map in a spe- cial case. Our result is a generalization to the case of unbounded trace norm map when restricted to the base of the cone. It is worth mentioning that our approach gives another proof in the case of any lower semicontinuous map as a trace norm map. Still our approach is using the I. Stevens’s proof for the case of continuous trace norm map. Theorem 3.2 Suppose that G is a simple countable dimension group, V is the cone associated to a metrizable Choquet simplex. Let λ : V → Hom+(G,R) be a continuous affine map and taking extreme rays into extreme rays. Let f : V → [0,∞] be an affine lower semicontinuous map, zero at zero and only at zero. Then [G, V, λ, f ] is the Elliott invariant of some simple non-unital 34 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU inductive limit of continuous trace C*-algebras whose spectrum is the closed interval [0, 1] or a finite disjoint union of closed intervals. Proof. The proof is based on I. Stevens’s proof in a special case and consists of several steps. Step 0 We start by constructing a simple stable AI algebra A with its Elliott in- variant: [(G,D), V, λ]. We know that this is possible (see [30]). By tensoring with the algebra of compact operators we may assume A is a simple stable AI algebra. Step 1 We restrict the map f to some base S of the cone T+(A), where the cone V is naturally identified with T+(A). Since any lower semicontinuous affine map f : S → (0,+∞] is a pointwise limit of an increasing sequence of con- tinuous affine positive maps, (see [2]), we can choose f = limfn, where fn are continuous affine and strictly positive functions. Moreover by considering the sequence of functions gn = fn+1 − fn if n > 1 and g1 = f1 we get that: gn = f Step 2 Next we use the results of Stevens ([31], Prop. 30.1.7), to realize each such continuous affine map gn as the norm map of a hereditary sub-C*-algebra Bn (which is an inductive limit of special algebra) of the AI algebra A obtained at Step 0. Consider the L∞ direct sum ⊕Bi as a sub-C*-algebra of A. The trace norm map of the sub-C*-algebra ⊕Bi of A is equal to i=1 gn = f . CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 35 To see that ⊕Bi is a sub-C*-algebra of A we use that A is a stable C*- algebra: ⊕Bi = . . .  ⊆ A⊗K ∼= A. Next denote with H the hereditary sub-C*-algebra generated by ⊕Bi inside of A. To prove that the trace norm map of H is f is enough to show that the norm of a trace on ⊕Bi is the same as on H. It suffices to prove that an approximate unit of the sub-C*-algebra ⊕Bi is still an approximate unit for the hereditary sub-C*-algebra H. We shall prove first that the hereditary sub-C*-algebra generated by ⊕Bi coincides with the hereditary sub-C*-algebra generated by one of its approx- imate units. Let (uλ)λ be an approximate unit of ⊕Bi. Denote by U the hereditary sub-C*-algebra of H generated by {(uλ)λ}. We want to prove that U is equal with H. Since (uλ)λ is a subset of ⊕Bi we clearly have U ⊆ H. For the other inclusion, one can observe that for all b ∈ ⊕Bi : b = lim uλbuλ. Now each uλbuλ is an element of the hereditary sub-C*-algebra generated by (uλ)λ and hence b ∈ U . Therefore ⊕Bi ⊂ U which implies H ⊆ U . We conclude thatH = U and hence the trace normmap ofH is f . Therefore 36 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU H is a simple hereditary sub-C*-algebra of an AI algebra with the prescribed invariant. � Remark 11.3. The approximate unit (uλ)λ of ⊕Bi is still an approximate unit for the hereditary sub-C*-algebra U . To see why this is true let us consider the sub-C*-algebra of A defined as follows: {h ∈ A | h = lim uλh}. This sub-C*-algebra of A is a hereditary sub-C*-algebra. Indeed let 0 ≤ k ≤ h with h = lim uλh. We want to prove that k = lim Consider the hereditary sub-C*-algebra hAh of A which clearly contains h (because h2 = lim huλh). Therefore k ∈ hAh. Since h = lim uλh we obtain that uλ is an approximate unit for hAh. In particular k = lim and hence {h ∈ A |h = lim uλh} is a hereditary sub-C*-algebra of A. Since U is the smallest hereditary containing (uλ)λ we get that U ⊆ {h ∈ A |h = lim and uλ is an approximate unit for U . 12. Non-AI algebras which are inductive limits of continuous-trace C*-algebras In this section we present a necessary and sufficient condition on the invari- ant for the algebra to be AI. We shall use this in the next section to construct an inductive limit of continuous trace C*-algebras with spectrum [0, 1] which is not an AI algebra. With [G, V, λ, f ] as before we observe that for an AI algebra with Elliott invariant canonically isomorphic to the given invariant the following equality CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 37 always holds: f(v) = sup{v(g) : g ∈ D}, where D is the dimension range. This is seen by simply using the fact that any AI algebra has an approximate unit consisting of projections. Therefore a sufficient condition imposed on the invariant in order to get an inductive limit of continuous trace C*-algebra with spectrum [0, 1] but not an AI algebra is f(v) 6= sup{v(g) : g ∈ D}. This condition is also necessary. Namely assume that we have f(v) = sup{v(g) : g ∈ D} and we have constructed a simple C*-algebra A which is an inductive limit of continuous trace C*-algebras with spectrum [0, 1] and with the invariant canonically isomorphic with the tuple [G, V, λ, f ]. Consider D = {x ∈ G : v(x) < f(v), v ∈ S, v 6= 0}, where S is a base of the cone V . For the tuple [G,D, V, S, λ] we can build (via the range of the invariant for simple AI algebras, [30]) a simple AI-algebra B with the invariant naturally isomorphic with the given tuple. Note that the trace norm map which is defined starting from the tuple [K0(B), D(B), T +B, λB] is exactly f because of the equality f(v) = sup{v(g) : g ∈ D} and B is an AI algebra. It is clear that B is an inductive limit of continuous trace C*-algebras with spectrum [0, 1] and hence by the isomorphism theorem 2.1 we conclude that A isomorphic to B. Hence A is a simple AI algebra as desired and we have proved the following theorem: 38 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU Theorem 12.1. Let A be a simple C*-algebra which is an inductive limit of continuous-trace C*-algebras whose spectrum is homeomorphic to [0, 1]. A necessary and sufficient condition for A to be a simple AI algebra is f(v) = sup{v(g) : g ∈ D}. 13. The class of simple inductive limits of continuous trace C*-algebras with spectrum [0, 1] is much larger than the class of simple AI algebras To see this consider the simple AI algebra necessarily not of real rank zero with scaled dimension group (Q,Q+) and cone of positive trace functionals a 2-dimensional cone; see [30]. Then the set of possible stably AI algebras, or equivalently the set of possible trace norm maps, may be represented as the extended affine space shown in the following schematic diagram: Figure 6. Each off-diagonal point in the diagram is the trace norm map of one of I. Stevens’s algebras. The boundary points of the first quadrant are removed (dotted lines) and the points with infinite coordinates are allowed. The di- mension range is embedded in a canonical way in the extended affine space as the main diagonal consisting of the points with rational coordinates. The two bold lines represent the cases of inductive limits of continuous CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 39 trace C*-algebras with unbounded trace norm map (points on these two lines have at least one coordinate infinity). If the point is off the diagonal and in the first quadrant, by Theorem 12.1 we get that the corresponding C*-algebra is an inductive limit of continuous trace C*-algebras which is not AI-algebra. It is clear that the size of the set of points off the diagonal is much larger then the size of the set of points on the diagonal. (For instance in terms of the Lebesgue measure.) This picture shows that the class of simple AI algebras sits inside the class of inductive limits of continuous trace C*-algebras in the same way that the main diagonal sits inside the first quadrant. References [1] C. A. Akemann, G. K. Pedersen, Ideal perturbation of elements in C*- algebras, Math. Scand. 41 (1997), 117–139. [2] E. M. Alfsen, Compact convex sets and boundary integrals, Springer- Verlag, New York, 1971. [3] L. G. Brown, Stable isomorphism of hereditary subalgebras of C*-algebras, Pacific J. Math. 71, (1977), 335–348. [4] L. G. Brown, P. Green and M. A. 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Mygind, Classification of certain simple C*-algebras with torsion in K1, Canad. J. Math. 53 (2001), 1223–1308. [30] I. Stevens, Simple approximate circle algebras, II, Operator Algebras and their Applications, Fields Institute Communications, Vol. 20, American Mathematical Society, Providence, RI, 1998, 97–104. [31] I. Stevens, Hereditary subalgebras of certain simple non real rank zero C*- algebras, Lectures on Operator Theory, Fields Institute Communications, Vol. 13, American Mathematical Society, Providence, RI, 1999, 209–241. [32] K. Stevens, The classification of certain non-simple approximate inter- val algebras, Ph. D. thesis, 1994; Fields Institute Communications, 20, American Mathematical Society, Providence, RI, 1998, 105–148. [33] H. Su, On the classification of C*-algebras: Inductive limits of matrix algebras over non-Hausdorff graphs, Memoirs Amer. Math. Soc. Number 547 114 (1995). [34] K. Thomsen, Inductive limits of interval algebras: The tracial state space, Amer. J. Math. 116 (1994), 605–620. [35] J. Villadsen, The range of the Elliott invariant, J. Reine Angew. Math. 462 (1995), 31–55. Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4 E-mail address : [email protected] Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4 E-mail address : [email protected] 1. Introduction 2. The invariant 3. The results 4. Special continuous trace C*-algebras with spectrum the interval [0,1] 5. Balanced inequalities and the Existence Theorem 5.1. An exact inequality is necessary between the non-stable part of the invariant 6. Uniqueness Theorem 7. Inductive limits of special continuous trace C*-algebras 8. Getting a non-zero gap at the level of affine function spaces 9. Pulling back of the isomorphism between inductive limits at the level of the invariant 10. The Isomorphism Theorem 11. The range of the invariant 12. Non-AI algebras which are inductive limits of continuous-trace C*-algebras 13. The class of simple inductive limits of continuous trace C*-algebras with spectrum [0,1] is much larger than the class of simple AI algebras References

0704.1450

U B V R I Photometry of Stellar Structures throughout the Disk of the Barred Galaxy NGC 3367

Accepted for publication in The Astronomical Journal, Apr 3, 2007; to appear in the July 2007 issue U B V R I Photometry of Stellar Structures throughout the Disk of the Barred Galaxy NGC 3367 J. Antonio Garćıa-Barreto, Héctor Hernández-Toledo, Edmundo Moreno-Dı́az Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Apartado Postal 70-264, México D.F. 04510 México Tula Bernal-Maŕın and A. Lućıa Villarreal-Castillo Facultad de Ciencias, Universidad Nacional Autónoma de México, México D.F. 04510 México ABSTRACT We report new detailed surface U,B, V, R, and I photometry of 81 stellar structures in the disk of the barred galaxy NGC 3367. The images show many different structures indicating that star formation is going on in the most part of the disk. NGC 3367 is known to have a very high concentration of molecular gas distribution in the central regions of the galaxy and bipolar synchrotron emission from the nucleus with two lobes (at 6 kpc) forming a triple structure similar to a radio galaxy. We have determined the U,B, V, R, and I magnitudes and U−B ,B−V , U−V , and V −I colors for the central region (nucleus), a region which includes supernovae 2003 AA, and 79 star associations throughout NGC 3367. Estimation of ages of star associations is very difficult due to several factors, among them: filling factor, metallicity, spatial distribution of each structure and the fact that we estimated the magnitudes with a circular aperture of 16 pixels in diameter, equivalent to 6′′.8 ∼ 1.4 kpc. However, if the colors derived for NGC 3367 were similar to the colors expected of star clusters with theoretical evolutionary star tracks developed for the LMC and had a similar metallicity, NGC 3367 show 51% of the observed structures with age type SWB I (few tens of Myrs), with seven sources outside the bright surface brightness visible disk of NGC 3367. Subject headings: galaxies: stellar structures — galaxies: individual (NGC 3367) — galaxies: starburst—galaxies: photometry 1. Introduction It is a continuous topic of study how a galaxy evolves with time (Larson 1977; van der Kruit 1989; Pfenniger 2000). Integrated photometry of star clusters and star associations in a galaxy can provide fundamental insight on the star-formation history (Bica et al., 1996). U,B, V colors of galaxies depend mainly on their history of star formation (Searle, Sargent & Bagnuolo 1973; Larson & Tinsley 1978). The fact that morphologically peculiar galaxies show bluer http://arxiv.org/abs/0704.1450v1 – 2 – U −B colors in the two color diagram suggests that galaxies with peculiar appearance have experienced anomalous star formation histories characterized by recent burst. Models with recent bursts of star formation (SFR) predict colors that in general lie off the normal relation for models with monotonically decreasing SFR (Larson & Tinsley 1978), for example, a very recent strong burst results in an excessively blue U−B for a given B−V (Larson & Tinsley 1978). Integrated photometry of star clusters and star associations in a galaxy would provide fundamental insight on the star-formation history (Bica et al., 1996). NGC 3367, the disk galaxy with a stellar bar of our study, is noteworthy because of observational characteristics, some of them are: 1) it shows a bipolar synchrotron outflow from the nucleus with a compact unresolved source with deconvolved diameter less than 65 pc, and two large lobes straddling the nucleus with a total, projected extent on the sky, from the north-east to the south-west lobe of about 12 kpc (an image like a radio galaxy, but from, a nearby late disk galaxy); only the south-west lobe shows polarized emission, suggesting that this lobe is out of the plane of the disk and is closer to the observer than the other lobe (Garćıa-Barreto et al. 1998; Garćıa-Barreto, Franco & Rudnick 2002); 2) the axis of the ejected outflow is highly inclined with respect to the axis of rotation of the disk of the galaxy (Garćıa-Barreto et al. 1998; Garćıa-Barreto, Franco & Rudnick 2002); 3) there is a lot of molecular gas mostly concentrated in the central 9′′ with an unresolved source with M(H2) ∼ 5.9× 10 8 M⊙; this molecular mass suggests an optical extinction value Av(l) ∼ 75 magnitudes toward the nucleus (Garćıa-Barreto et al. 2005) (as it has been detected through its CO(J=1-0) emission with the OVRO mm Array1); its total molecular (H2) mass within 11.4 kpc diameter is M(H2) ∼ 2.6× 10 9 M⊙ (Garćıa-Barreto et al. 2005); 4) it shows a bright southwest optical structure (resembling a “bow shock”) with a collection of Hα knots at a galactocentric radius of about 10 kpc from the nucleus (Sandage 1961; Garćıa-Barreto, Franco & Carrillo 1996a; Garćıa-Barreto et al. 1996b), 5) it has crooked arms in its northwest side, and 6) it seems to be isolated with no similar-diameter galaxy in its proximity; NGC 3367 lies, behind the Leo group of galaxies, at a distance of 43.6 Mpc, with a closest companion lying at a projected distance of ≈ 563 kpc or 18 optical diameters away (Garćıa-Barreto, Carrillo & Vera-Villamizar 2003). NGC 3367 is an SBc(s) barred spiral galaxy with a stellar bar structure of diame- ter ≈ 32′′ ∼ 6.7 kpc oriented at a position angle (P.A.) ≈ 65 − −70◦. The disk is in- clined with respect to the plane of the sky at ∼ 30◦ (Garćıa-Barreto & Rosado 2001). NGC3367 has an X ray luminosity much larger than any normal spiral galaxy (Gioia et al. 1990; Stocke et al. 1991; Fabbiano, Kim & Trinchieri 1992). Its radio continuum emission at a 4′′.5 angular resolution shows weak radio continuum emission extended throughout the disk (Garćıa-Barreto et al. 1998). From Fabry Perot observations of the Hα emis- sion it is deduced that the kinematic major axis lies projected on the disk at a P.A. of 51◦ (Garćıa-Barreto & Rosado 2001). Its optical spectrum shows moderately broad Hα+[NII] lines with FWHM∼ 650 km s−1, but its emission of Hβ is stronger than its emission of [OIII]λ5007Å and there is weak emission of He II λ4686Å suggesting the ex- istence of WR stars (Véron-Cetty & Véron 1986; Véron, Gonçalves & Véron-Cetty 1997; Ho, Filippenko & Sargent 1997). NGC 3367 has been clasiffied as an HII nucleus (from the optical line ratios; see (Ho et al., 1997a) and normal content of atomic hydrogen with 1The Owens Valley Radio Observatory millimeter array is operated by Caltech and it is supported by the National Science Foundation grant AST 99-81546 – 3 – MHI ∼ 7× 10 9 M⊙ (Huchtmeier & Seiradakis 1985). Table 1 lists relevant information for NGC 3367 coming from the literature. Previous U,B, V, I phometry of NGC 3367 has been obtained for the whole galaxy with magnitudes reported in RC3 (de Vaucouleurs et al., 1993) and compiled in the NASA Extragalactic Data Base (NED)2 as listed in Table 1; previously reported colors are as follows: U − B = −0.16, B − V = 0.55, B − I = 1.45. Larson & Tinsley (1978) have discussed different factors that would affect the U,B, V colors, among them are reddening (galactic and intrinsic), gaseous emission lines and nonthermal emission, chemical composition and initial mass function (IMF). The color B − V of NGC 3367 agrees with the color expected from a late type disk galaxy (Allen 1976; Larson & Tinsley 1978); however the U −B color of NGC 3367 certainly lie above the mean value of normal disk galaxies and the position in the U−B vs B−V (Larson & Tinsley 1978) is more in agreement with the statistical colors of galaxies from the Arp Atlas of peculiar galaxies 3. These colors of NGC 3367 suggest that this late type barred galaxy is currently having many regions with recent star formation. Although galactic reddening is a major source of uncertainity in the U,B, V colors, for this galaxy it is not a major concern given its lalitude, b = +58◦. We report our magnitude estimates for the different structures corrected for Galactic extinction. Gaseous emission lines, different chemical composition and different initial mass function may contribute to the uncertainities of the expected colors (Larson & Tinsley 1978). This paper reports new U,B, V, R, I photometry of 81 individual structures throughout NGC 3367 using the 84 cm optical telescope in San Pedro Mártir, Baja California, México (Richter, et al. 2005). For the data reduction and analysis we used the IRAF4 task phot with an aperture radius of 8 pixels (3′′.41 ∼ 715 pc). The existence of a close connection between violent galactic dynamical phenomena is consistent with theoretical expectations that high velocity collisions and shock fronts should be effective in compressing gas to high densities and triggering rapid star formation. Section 2 reports the observations, §3 reports the results and §4 gives the discussion and finally §5 describes the conclusions. 2. Observations and Data Reduction The observations were carried out using the 84 cm optical telescope at the Observatorio Astronómico Nacional at San Pedro Mártir, Baja California, México on 2005, March 8 and 9. We used a CCD camera Site3 with a 1024×1024 pixels; each pixel size is equivalent to 2NED is operated by the Jet Propulsion Laboratory, California Institute of Technology under contract with the National Aeronautics and Space Administration. 3As a comparison the U − B and B − V of an early type (elliptical) galaxy are U − B ≥ +0.55, and B − V ≥ +0.80 (Larson & Tinsley 1978). 4The IRAF package is written and supported by the IRAF programming group at the National Optical Astronomy Observatories (NOAO) in Tucson, Arizona. NOAO is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under cooperative agreement with the National Science Foundation (NSF). – 4 – 0′′.426 and we did not use any binning. Attached to the camera was a filter box. A journal of the photometric observations is given in Table 2. Column (1) give a characteristic; columns (2)-(6) give the filters; rows indicate the data for each column, seeing, central wavelength and total width of the filter used, and the number of frames times the integration time (in seconds) of each of the images (M67 and NGC 3367) in each filter. Images were debiased, trimmed, and flat-fielded using standard IRAF procedures. First, the bias level of the CCD was subtracted from all exposures. A run of 10 bias images was obtained per night, and these were combined into a single bias frame which was then applied to the object frames. The images were flat-fielded using sky flats taken in each filter at the beginning and at the end of each night. In the case of estimating the magnitudes for the structures throughout the disk in NGC 3367, we have made careful analysis utilizing different IRAF tasks, in particular, qphot and phot using different aperture diameters for bright stars in the field of NGC 3367 in order to estimate the uncertainities in the final magnitudes at different filters before estimating the final magnitudes reported for the different structures in NGC 3367. The magnitudes reported here for the structures within NGC 3367 were estimated using an aperture radius of 8 pixels (∼ 3′′.4 ∼ 715 pc). Photometric calibration was achieved by nightly observations of standard stars of known magnitudes from the “Dipper Asterism” M67 star cluster (Guilliland et al., 1991; Chevalier & Ilovaisky 1991). A total of 9 standard stars with a color interval −0.1 ≤ (B − V ) ≤ 1.4 and a similar range in (V − I) were observed. The main extinction coefficients in B, V , R and I as well as the color terms were calculated according to the following equations (Larsen 1999): U − u0 = αU + βU(u− b)0 B − b0 = αB + βB(b− v)0 V − v0 = αV + βV (b− v)0 R− r0 = αR + βR(v − r)0 I − i0 = αI + βI(v − r)0 where B, V , R and I will be the estimated magnitudes, b, v, r and i are the instrumental (and airmass-corrected) magnitudes. α and β are the transformation coefficients for each filter. In a first iteration, a constant value associated with the sky background was subtracted from the images by using an interactive procedure that allows the user to select regions on the frame free of galaxies and bright stars. Notice that uncertainities in determining the sky background may be the dominant source of uncertainities in the estimation of the magnitudes, color and surface brightness profiles. The most energetic cosmic-ray events were automatically masked using the IRAF task cosmicrays and field stars outside the bright visible disk of NGC 3367 were removed using the IRAF task imedit when necessary. Within the galaxy itself, care was taken to identify superposed stars. A final step in the basic reduction involved registration of all available frames for each galaxy and in each filter to within ±0.1 pixel. This step was performed by measuring centroids for foreground stars on the images and then performing geometric transformations using IRAF tasks geomap and geotran. All final images were convolved with a two dimensional gaussian function in such a way as to have a similar final resolution equivalent to the largest one given by the V filter seeing, that is, full width at half maximum, fwhm∼ 2′′.2. – 5 – 2.1. Uncertainities in the Photometry 2.1.1. As a result of the Method An estimation of the uncertainities in our photometry involves two parts: 1) The pro- cedures to obtain instrumental magnitudes and 2) the uncertainty when such instrumental magnitudes are transformed to a standard system. For 1), notice that the magnitudes pro- duced at the output of the IRAF routines qphot and phot differ from each other as a result of different methods used in those procedures. Since we also have applied extinction correc- tions to the instrumental magnitudes in this step, our estimation of the uncertainities are mainly concerned with these corrections and the estimation of the airmass. After a least square fitting, the associated uncertainities to the slope for each principal extinction coeffi- cient are; δ(kU) ∼ 0.038, δ(kB) ∼ 0.038, δ(kV ) ∼ 0.035, δ(kR) ∼ 0.025 and δ(kI) ∼ 0.025. An additional uncertainity δ(airmass) ∼ 0.005 from the airmass routines in IRAF was also considered. For 2), the zero point and first order color terms are the most important to consider. After transforming to the standard system, by adopting our best-fit coefficients, the formal uncertainities from the assumed relations for α were 0.05, 0.05, 0.04, 0.04 and 0.04 in U , B, V , R and I and 0.04, 0.03, 0.03 and 0.04 for β. To estimate the total uncertainity in each band, it is necessary to use the transformation equations and then propagate the errors. Total typical uncertainties are 0.1, 0.1, 0.1, 0.09 and 0.1 in U , B, V , R and I bands, respectively. The estimated total magnitudes in this work were compared against other estimations reported in the literature. This has been done for the standard stars and for NGC 3367 in common with other works. 2.1.2. Standard Stars as calibrators Our estimated CCD magnitudes for the nine stars in M67 are listed in Table 3 along with magnitudes and cross reference identification numbers reported in the literature (Guilliland et al., 1991; Chevalier & Ilovaisky 1991). A plot of our CCD estimated magnitudes versus those reported by Guilliland et al., (1991) and Chevalier & Ilovaisky (1991) are shown in Figure 1 with no significant deviations, a continuous line with slope 1 is also plotted. These results suggest, a σ ∼ 0.12, as the typical magnitude uncertainity. This is in agreement with our prior uncertainity estimations. – 6 – Fig. 1.— Comparison between our estimated magnitudes and those from Gilliland et al. (1991) [U , B, V and R bands] and Chevalier & Ilovaisky (1991) [I band] for 9 stars in common taken as standards 3. Results 3.1. Total Magnitudes and Colors In this study, we have estimated total magnitudes for U,B, V, R, I computed from differ- ent circular apertures chosen interactively to assure that they are large enough to contain the whole galaxy and still small enough to limit the uncertainities due to bad estimation of the flux from the background and any field stars. Finally we chose an aperture with a diameter of 140′′ equivalent to 2′.33; this is slightly larger than the 2′.02 major diameter as given in NED. There is, however, a foreground star within the disk of NGC 3367 which we did not remove before estimating the global magnitudes and colors for NGC 3367. The star is object 82 in Tables 7 and 9 and it lies about 29′′ to the north-west of the nucleus; we did not attempt to remove it because it is surrounded with emission from stellar structures in the disk and thus it makes it difficult to estimate the intensity level to replace the star with; star ID 82 turns out to be a weak star with magnitudes U ∼ 17.9 B ∼ 17.9 V ∼ 17.4 R ∼ 16.9 I ∼ 16.2. Total magnitudes from NGC 3367 were estimated in each band by using the phot routines in IRAF. Our final estimated magnitudes for NGC 3367 were corrected for Galactic ex- tinction (but not for intrinsic extinction); they are U = 11.70 ± 0.1, B = 11.91 ± 0.1, V = 11.52 ± 0.1, R = 11.05 ± 0.1, and I = 10.43 ± 0.1. The colors of NGC 3367 are U −B = −0.21, B − V = 0.39, B − I = 1.48. – 7 – Since the aperture to estimate the magnitudes for the stellar structures throughout NGC 3367 must be small in order to include only star associations or small structures, it was decided to use an aperture with radius 8 pixels equivalent to 3′′.41 ∼ 715 pc and use IRAF task phot. One compared the estimated magnitudes, with an aperture of 8 pixels, for the bright stars in the field south west of NGC 3367 with those listed in Table 5 (with aperture radius of 26 pixels). The magnitude differences vary according to the star and the filter. For star ID 86 the difference in empirical magnitudes are ∆U(8 − 26) = 0.07, ∆B(8 − 26) = 0.09, ∆V (8 − 26) = 0.10, ∆R(8 − 26) = 0.09, and ∆I(8 − 26) = 0.17; while for star ID 91 the differences in magnitudes are ∆U(8 − 26) = 0.06, ∆B(8 − 26) = 0.08, ∆V (8 − 26) = 0.11, ∆R(8 − 26) = 0.10, and ∆I(8 − 26) = 0.18, fair is to say that the magnitudes with aperture radius of 8 pixels have been corrected for Galactic extinction. Magnitudes differences for each filter for both stars are similar indicating indeed that using a smaller aperture radius gives an artificial weaker object by the amount given in each filter but the differences are within the values for the uncertainities of our method. The estimated magnitudes for the different structures throughout NGC 3367 (stellar associations) can be taken as reliable and are listed in Table 6. Table 7 lists the estimated magnitudes for the bright stars and a background galaxy in the field of NGC 3367. Table 8 lists the different colors U −B, B− V and V − I for the stellar structures throughout NGC 3367, while table 9 lists the colors of the stars and the small galaxy in the field of NGC 3367. See Tables 6 for comments on individual objects. Our estimated magnitudes for the whole galaxy in U , B, V and I bands and those reported in the RC3 Catalogue (de Vaucouleurs et al., 1993) are in good agreement. The Galactic extinction towards NGC 3367 in the different filters are: AU = 0.15, AB = 0.12, AV = 0.09, AR = 0.08, and AI = 0.05 (Schlegel, Finbeiner & Davis 1998). 3.2. UBV RI Photometry of Star Associations and structures It is a complicated problem to carry out photometry of stellar structures located within the disk of a galaxy, partly because of the strongly varying background, and partly because the structures are not perfect point sources (in other words, there must be a varying filling factor) nor circles (as it was our aperture). For disk galaxies, the colors U − B, B − V , V − I are closely related to spectral type and show that there exist remarkable differences in stellar content (Searle, Sargent & Bagnuolo 1973). The colors of structures in a given galaxy are most certainly affected by internal reddening and chemical enrichment. In a particular galaxy, as in our study, the colors of different structures would indicate the contributions of the component stars in star clusters, very young star clusters embedded in HII region or compact HII regions, stellar associations without gas, associations embedded in HII regions and star clusters and associations with traces of emission [e.g. (Bica et al., 1996)]. It was decided in our photometric study of structures in NGC 3367 to estimate the color magnitudes of different structures using IRAF task phot with an aperture with radius 8 pixels (3′′.4) which translates, in NGC 3367, to a radius with linear scale of 715 pc. No aperture correction were applied to our estimated magnitudes of the different structures. The photometry of all structures was obtained with the IRAF task phot which takes an area of the sky, well off the extention of the disk of the galaxy in study, as a background to be – 8 – compared with. In order to have an estimation of how good were our derived magnitudes for the stellar structures throughout NGC 3367, we first estimated the magnitudes of bright field stars to the south west of NGC 3367 (objets ID’s 85 and 90 in Table 7) utilizing two IRAF tasks, qphot, and phot; each of these stars is relatively isolated and thus we could use apertures with different radii in order to estimate their magnitudes in the five filters. Table 4 gives our estimated magnitudes for the different filters for two bright stars in the south west of the field of NGC 3367 utilizing IRAF task qphot using an aperture radius of 23 pixels, equivalent to 9′′.80, and a width of ring (in order to estimate the brightness of the sky) of pixels, equivalent to width of ring of 2′′.13. Additionally IRAF task phot was used with three different circular apertures, one with a radius of 20 pixels, equivalent to 8′′.52, other with radius 23 pixels, equivalent to 9′′.80 and another one with 26 pixels, equivalent to 11′′.08. The estimated magnitudes for the five filters, U , B, V , R, and I, are listed in Table 5. The largest uncertainity observed in Table 5 using three different apertures amounts to ∼ 0.05 mag in U , B, V , and R and it is ∼ 0.13 mag in I for both stars. Comparison between estimated magnitudes using IRAF tasks qphot and phot for these field stars may be estimated comparing the results listed in Tables 4 and 5. The difference amounts in the worst case to 0.12 mag indicating that indeed IRAF task phot obtains a brighter value in each filter with the largest aperture; however the difference in magnitudes using different apertures with IRAF task phot is at worst ∼ 0.03 in the I filter between using an aperture’s radius of 20 pixels and an aperture’s radius of 26 pixels. The structures were chosen from the images in the I and U filters by eye at several loca- tions of the disk of the galaxy with emphasis in choosing regions at different positions within NGC 3367, for example: regions at larger distances than the bright south-west semicircle (more than 10 kpc from the center, easily observed in images from the Palomar Sky Surveys), from the stellar bar’s ends, from structures along and on spiral arms, from structures just beyond the stellar bar, etc. in order to have enough of them as to be able to compare their colors and estimate their ages. It included structures on the outside of the intense semicircle structure with bright surface brightness to the west of the disk. Table 6 lists the positions, α(J2000.0) and δ(J2000.0) 5 of the different structures throughout NGC 3367, where we have estimated their magnitudes in the U,B, V, R and I broadband filters corrected for Galactic extinction. Table 8 is similar to Table 6 and lists the colors U−B, B−V , and V −I corrected for Galactic extinction of each stellar structure in NGC 3367 along with their estimated SWB type. We have estimated the colors of the area in front of the nucleus; the estimated mag- nitudes were not corrected for intrinsic extinction; a visual extinction has been estimated, from blue spectroscopy, to be only AV ∼ 0.9 Cid Fernandes et al. (2005). True magnitudes must be brighter since it is known that there is a lot of molecular gas in the central region of NGC 3367 (Garćıa-Barreto et al. 2005). At the distance of NGC 3367 (43.6 Mpc) the absolute magnitude is MV = -13.24 taking into account the Galactic optical extinction. As 5The peak of the I image was anchored (in spatial position) to the peak of the high resolution ra- dio continuum emission at 8.4 GHz (Garćıa-Barreto, Franco & Rudnick 2002) to the position α(J2000) = 10h 46m 34s.956 δ(J2000) = +13◦ 45′ 02′′.94. Although this is a logical supposition, it needs to be verified since in other galaxies the peak of the radio continuum emission does not coincide with the compact optical nucleus as a result of high extinction. – 9 – a comparison, the compact nucleus in the barred spiral galaxy NGC 4314, a LINER with a circumnuclear structure (Garćıa-Barreto et al. 1991), has the following magnitude and col- ors within an aperture of 6′′, 290 pc: V = 14.07, U −B = 0.51, and B−V = 1.11 (Benedict 1980), MV = -10.93 (at 10 Mpc taking into account the galactic optical extinction); the compact nucleus of NGC 4314 is reported to have a blue magnitude of B = 16.6 mag per square arc second (Lynds, Furenlid & Rubin 1973). Figure 2 shows the U −B versus B − V color color diagram for the different structures throughout the disk of NGC 3367. As a comparison, Figure 3, shows the corresponding U−B versus B−V diagram for the Large Magellanic Cloud (Bica et al., 1996) stellar associations embedded in HII regions (NA in the nomenclature of Bica et al.), star clusters (C), very young clusters embedded in HII region or compact HII region (NC), stellar associations without gas (A) and a supernovae remanent. Previuos U,B, V photometry of the SMC and LMC star clusters indicated that young clusters in the SMC were concentrated along its bar, while old clusters showed no concentration to the bar (van den Bergh 1981); while the young clusters in the LMC were in and around the bar (van den Bergh 1981). The difference between the color color diagrams between the clusters in the magellanic clouds and the structures in NGC 3367 is noteworthy: the LMC has many very young clusters with colors equivalent to the SWB 0 type (Searle, Wilkinson & Bagnuolo 1980; Bica, Claŕıa & Dottori 1992; Bica et al., 1996) 6 (see section on ages below) and might also be the result of filling factor (radius of the aperture used, since the LMC is much closer than NGC 3367), intrinsic extinction and different chemical enrichment in NGC 3367. 6The location of star clusters of different ages, in the U − B versus B − V color diagram delineate an average sequence known as SWB 0 to SWB VII (Searle, Wilkinson & Bagnuolo 1980). – 10 – Fig. 2.— The color color diagram of different structures throughout NGC 3367 is shown. Filled hexagon is the compact nucleus (ID 1), open squares are the bright star associations embedded in HII regions to the west of the on the bright semicircle rim of the disk (ID 2 and 3), open hexagons are the bright stellar associations embedded in HII regions east of the end of the bar (ID 4 - 6), filled squares are stellar associations to the northeast and southwest of the bar (ID 37 and 38 respectively), asterisk is the region which includes supernovae 2003aa (ID 42), open triangles are stellar associations in the disk, filled triangles are stellar associations beyond the semicircle bright end of the western side of the disk (ID 52 and 53). SWB type (Searle, Wilkinson & Bagnuolo 1980; Bica et al., 1996), SWB 0 have estimated ages between 0 and 107 yrs, SWB I between 107 and 3 × 107 yrs, SWB II between 3 × 107 and 7 × 107 yrs, SWB III between 7× 107 and 2 × 108 yrs, SWB IVA between 2× 108 and 4× 108 yrs, SWB IVB between 4× 108 and 8× 108 yrs, SWB V between 8× 108 and 2× 109 yrs, SWB VI between 2× 109 and 5× 109 yrs, and SWB VII between 5× 109 and 1.6× 1010 – 11 – -0.5 0 0.5 1 SWB II SWB III SWB IVB SWB 0 SWB I SWB IVA SWB V SWB VI SWB VII Fig. 3.— The color color diagram of different stellar associations in the Large Magellanic Cloud (Bica et al., 1996) is shown as a comparison. In their nomenclature, open polygons are stellar associations embedded in HII regions (NA, U−B in the interval from -1.1 to -0.55 and B−V in the interval -0.4 to 0.2), filled polygons are stellar associations without gas (A, U − B in the interval from -0.9 to -0.5 and B − V in the interval -0.1 to 0.4), open squares are very young clusters embedded in HII regions or compact HII regions (NC, U −B in the interval from -1.05 to -0.3 and B − V in the interval -0.3 to 0.65), and open triangles are star clusters (C, U −B in the interval from -0.95 to +0.5 and B − V in the interval -0.25 to 0.95) (Bica et al., 1996). 3.3. U B V R I Images This section shows various images of NGC 3367. Figure 4 is the image in the U filter in a grey scale and contours, proportional to surface brightness, showing the different structures throughout NGC 3367. – 12 – Fig. 4.— Optical image, of NGC 3367, is shown in the broadband filter in the Ultraviolet in greyscale and contours. Contours are in mag arcsec−2, and approximately correspond to mU ∼ 23.6, 22.6, 21.8, 21.4, 21.1, 20.7, 20.4, 19.6, 18.8, 18.1, and 17.4. – 13 – 77 78 131243 Right Ascension (J2000)10 46 40 36 Fig. 5.— North east quadrant of the U image showing with circles the positions of different stellar structures; the circle is ∼ 6′′.8 in diameter which, at the distance of NGC 3367, corresponds to a linear scale of 1.430 kpc. – 14 – Right Ascension (J2000) +13 45 00 Fig. 6.— As for Fig. 5 but showing the Southeast quadrant. – 15 – Right Ascension (J2000) Fig. 7.— As for Fig. 5 but showing the Southwest quadrant. – 16 – Fig. 8.— As for Fig. 5 but showing the Northwest quadrant. – 17 – Figures 5, 6, 7, and 8 show the four different quadrants of NGC 3367 with a circle indicating the position of the structure observed and its number, according to table 6, with the circle having the diameter of the aperture utilized for the photometry, that is ∼ 6′′.8 which at the distance of NGC 3367 translates into a linear scale of ∼ 1.4 kpc. Figure 9 shows the I image of the galaxy with a larger area in order to indicate the field stars and the background spiral galaxy. Figures 10, 11 and 12 show the U −B, B−V and V − I color images of NGC 3367. Fig. 9.— Optical image of NGC 3367 in the broadband I filter is shown in greyscale and contours; contours are in mag arcsec−2 and approximately correspond to m∼20.2, 19.2, 18.7, 18.4, 18.2, and 18. Circles indicate the positions of the stars in the field of NGC 3367, except for circle in the upper left corner that indicates the position of stellar structure ID 75; each circle is 8 pixels in radius or ∼ 3′′.4. Stars 86 and 91 were used to estimate the uncertainities in our photometry when using IRAF tasks qphot (with one aperture) and phot with three different circular apertures. ID 90 is a background (disk) galaxy. – 18 – RIGHT ASCENSION (J2000) 10 46 40 38 36 34 32 30 13 46 15 45 45 44 45 Fig. 10.— U − B color image of the disk galaxy NGC 3367: Greyscale is in arbitrary units in such a way that white areas indicate brighter ultraviolet emission. – 19 – RIGHT ASCENSION (J2000) 10 46 40 38 36 34 32 30 13 46 15 45 45 44 45 Fig. 11.— B − V color image of the disk galaxy NGC 3367. – 20 – RIGHT ASCENSION (J2000) 10 46 40 38 36 34 32 30 13 46 15 45 45 44 45 Fig. 12.— V − I color image of the disk galaxy NGC 3367. 4. Discussion 4.1. Spatial distribution of structures with different SWB type There are 42 stellar structures with age type SWB I 7 out of 81 observed which cor- respond to 51%. A natural question is, where is their spatial location? Structure ID 1 corresponds to the central innermost stellar association which includes the nucleus of the galaxy; ID 2 and 3 are two stellar associations embedded in bright HII regions on the western bright surface brightness edge of the disk at about 10 kpc from the center; ID 4 and 5 lie on an inner eastern bright surface brightness area which seems to be a spiral arm or part of an inner ring (just beyond the edge of the stellar bar) at about 4.5 kpc from the center; ID 6 7Approximate intervals of ages to different SWB classes are as follows (Searle, Wilkinson & Bagnuolo 1980; Bica et al., 1996): SWB 0 have ages between 0 and 107 yrs, SWB I have ages between 107 and 3×107 yrs, SWB II have agesbetween 3 × 107 and 7 × 107 yrs, SWB III have ages between 7 × 107 and 2 × 108 yrs, SWB IVAhave ages between 2× 108 and 4× 108 yrs, SWB IVB have ages between 4× 108 and 8× 108 yrs,SWB V have ages between 8×108 and 2×109 yrs, SWB VI have ages between 2×109 and 5×109yrs, and SWB VII have ages between 5 × 109 and 1.6 × 1010 yrs (Chiosi, Bertelli & Bressan 1988; Girardi & Bica 1993; Anders & Fritze-v. Alvensleben 2003). – 21 – Fig. 13.— Histogram of number of structures studied in this work versus their B − V color. Notice that 61 out of 81 of the structures have B − V color less than 0.5 (blue). and 48 lie on another structure on an eastern spiral arm, parallel to the arm where ID 4 and 5 lie, at about 8 kpc from the center; ID 7 and 8 lie on the inner most spiral arm north-west of the center; ID 14 lies on the crooked spiral arm north west of the center at about 6 kpc; ID 18 lies at about 10 kpc east of the center just outside of the second spiral arm; ID 19 - 22, 25, 27 all lie on the southern spiral arm; ID 30 lies on a circular surface brightness structure south-west of the south-west end of the stellar bar; ID 31 - 35 and 40 all lie on the western spiral arm; ID 43 lies on an interarm on the eastern region of the disk at about 8.5 kpc; ID 57 lies on an interarm region west of the nucleus at about 8 kpc. Structures ID’s 52 - 55, 58, 59, 62 - 74, and 80 lie outside the bright surface brightness semicircle at distances more than about 13 kpc. The features are detected and shown in Figs. 4, 5, 6, 7, and 8. It is important to emphasize that in the central region (ID 1), the western sources (ID 4 - 6) and the eastern sources (ID 2, 3), there is a large velocity dispersion of ionized gas, detected through Hα (Garćıa-Barreto & Rosado 2001) and in the molecular gas CO (Garćıa-Barreto et al. 2005). In summary, most of the structures with age type SWB I lie close or on spiral arms. Perhaps the most important result of the spatial distribution of structures with age type SWB I is that several lie outside of the eastern bright surface brightness disk, ID 58 and 59 to the south, ID 65, 66, 69, 70, to the east while ID 53, 71 - 74 lie to the western side and ID 75 lies well outside the disk to the north-eastern end. The colors of structures ID 58, 59, 71 - 74 and 75 suggest that they are newly formed stellar structures and yet they lie at distances beyond the bright visible disk of NGC 3367, that is beyond 10 kpc from the center. – 22 – 4.2. Estimated ages of the structures throughout NGC 3367 It is difficult to speak about ages of the different structures throughout NGC 3367 mainly because, as we have seen, different structures could indicate star associations em- bbeded in HII regions, very young star cluster embedded in HII regions, star associations not associated with gas, star associations in or around the nucleus, and other combinations. Among some of the problems with estimating the magnitudes of different structures are: fill- ing factor, spatial distribution of structure within the aperture, and the fact that we used a circular aperture. We realize that with our circular aperture diameter, of 6′′.8 ∼ 1.4 kpc, for estimating magnitudes and colors one can not detect individual star clusters in this galaxy. However, the study of star clusters with U,B, V, R, and I photometry and colors, and their derived ages in the LMC (Searle, Sargent & Bagnuolo 1973; Searle, Wilkinson & Bagnuolo 1980) might be taken as an excellent reference to compare the U,B, V, R, and I colors of the different structures throughout NGC 3367 and estimate their ages if only one were to assume that the colors were equivalent, the chemical abundance were similar, have the same metal- licity as in the LMC (which of course there is no reason to be an absolute valid assumption) and more important we might use the SWB ages if one ignores the internal reddening. ID 42, which includes supernova 2003AA, (asterisk in Figs. 2 and 4) has colors similar to SWB II. Structures ID 52 and ID 53, beyond the bright western high surface brightness rim of NGC 3367 (filled triangles in Fig. 2) have equivalent colors to clusters SWB IVB and V. ID 37 and ID 38 are the structures north-east and south-west respectively of the stellar bar (filled squares in Fig. 2) have colors similar to SWB VI, indicating old stellar populations, but this result needs to be taken with caution since there the colors have not been corrected for intrinsic extinction as a result of the probable existence of molecular gas (CO, H2) and dust at least to the southwest side of the bar (Garćıa-Barreto et al. 2005). Figure 12 shows the histogram of the number of structure throughout NGC 3367 versus the color B − V where there is a large number of structures with blue colors, B − V ≤ 0.5. Figure 13 shows the histogram of age groups according to SWB types (assuming that the models for star clusters in the LMC also apply to the structures and angular resolution observed in NGC 3367) where 64% of the structures observed have age types smaller or equal than SWB type II or less than 70 Myrs. Figs. 13 and 14 suggest that a large number of structures throughout NGC 3367 are relatively blue and young structures. This number of young structures would probably increase as one were able to somehow correct for intrinsic extinction, and different metallicity. 5. Conclusions To conclude, we have obtained several images in the broadband filters U,B, V, R, and I in order to estimate the optical magnitudes and colors of 81 different stellar structures throughout the disk of NGC 3367. The structures were chosen from the images in the I and U filters by eye at different locations of the disk of the galaxy in order to have enough of them as to be able to compare their colors as a function of their spatial location. It included structures to the north-east, south and west, well outside of the semicircle structure with bright surface brightness. Our estimated magnitudes for the NGC 3367, after correcting for Galactic extinction are U = 11.70±0.1, B = 11.91±0.1,V = 11.52±0.1, R = 11.05±0.1, and I = 10.43±0.1. NGC – 23 – Fig. 14.— Histogram of number of structures studied in this work versus age group, SWB type. Notice that 86% (70 out of 81) of the structures have SWB types I, II or III. This number would increase as a result of intrinsic extinction. 3367 global colors, after being corrected for Galactic extinction are U−B = −0.21, B−V = 0.39, B−I = 1.48. All images except the image in V were convolved with an appropiate two dimensional gaussian function in order to have a final of 2′′.2 full width at half maximum, since the seeing was different for each filter and was about 1′′.7 to 2′′.2. The 81 stellar structures reported in this paper were selected mainly from the U and the I images based on the location in the disk in order to have as many as possible from different positions in NGC 3367. We have made careful analysis of the calibration methods to estimate magnitudes and have compared different IRAF tasks, in particular, qphot and phot utilizing different apertures in order to estimate the uncertainities in the final magnitudes at different filters with 9 stars in M 67 as our standards. In the case of estimating the magnitudes for the structures throughout the disk in NGC 3367, we have made careful analysis utilizing different IRAF tasks, in particular, qphot and phot using different aperture diameters for the bright stars in the field of NGC 3367 in order to estimate the uncertainities in the final magnitudes at different filters before estimating the final magnitudes reported for the different structures in NGC 3367. The magnitudes reported here for the structures in NGC 3367 were estimated using an aperture radius of 8 pixels (∼ 3′′.4 ∼ 715 pc). We were able to estimate the optical magnitudes U,B, V, R, and I and colors U−B, B− V, U − V, and V − I for 81 different stellar structures throughout the disk in NGC 3367. It is difficult to estimated the ages of the structures in NGC 3367 given our circular aperture diameter, equivalent to 6′′.8 ∼ 1.4 kpc. However if their colors were taken as representative as the colors of star clusters (studied by many other authors with better angular resolution, – 24 – appropiate chemical abundance and current theoretical star evolutionary tracks) in the LMC many have colors equivalent to cluster types SWB I, and SWB II suggesting that many are young associations with only tens of millions of years of age. We found 41 structures out of 81 (50%) with age type SWB I. Several of those structures lie near or in spiral arms; some lie outside of the bright surface brightness visible disk (for example the POSS image). In order to understand better the large scale environment of NGC 3367 it would also be very important to determine the large scale spatial distribution and kinematics of atomic hydrogen (HI) and new observations with high angular resolution of the emission of the molecular gas (CO, H2) in the center of NGC 3367. Acknowledgements JAGB and HMHT thank the staff of the Observatorio Astronomico Nacional at San Pedro Mártir, Baja California, México for the help in the observations. HMHT acknowl- edges support from grant CONACyT (México) 42810. JAGB acknowledges partial financial support from DGAPA (UNAM, México) grant IN107806-2. The authors would like to thank the anonimous referee for his (her) comments and suggestions on how to improve this paper. – 25 – REFERENCES Allen, C.W. 1976, Astrophysical Quantities, 3rd. Edition (London: The Athalone Press) Anders, P. & Fritze-v. Alvensleben, U. 2003, A&A, 401, 1063 Benedict, G. F. 1980, AJ, 85, 513 Bica, E., Claŕıa, J. J., & Dottori, H., 1992, AJ, 103, 1859 Bica, E., Claria, J. J., Dottori, H., Santos, J. F. C., Jr., & Piatti, A. E. 1996 ApJS, 102, 57 Chevalier, C., & Ilovaisky, S.A., 1991, A&AS, 90,225 Chiosi, C., Bertelli, G., & Bressan, A. 1988, A&A, 196, 84 Cid Fernandes, R., González Delgado, R. M., Storchi-Bergmann, T., Martins, L. 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Véron-Cetty, M.-P., & Véron, P. 1986, A&AS, 66, 335 – 27 – Véron, P., Gonçalves, A.C. & Véron-Cetty, M.-P., 1997, A&A, 319, 52 This preprint was prepared with the AAS LATEX macros v5.2. – 28 – Table 1: General Properties of the barred disk galaxy NGC 3367 Characteristic Value Reference Hubble Type (RC3) SB(rs)c 3 Nucleus spectral type Sy 2-like, HII 1,2 U mag 11.89 ± 0.15 3 B mag 12.05 ± 0.14 3 V mag 11.50 ± 0.14 3 I mag 10.60 ± 0.07 4 Vsys(HI) 3030 ± 8 km s Vsys(opt) 2850 ± 50 km s Vsys(FP a) 3032 ± 3 km s−1 6 Vsys(CO) 3035 ± 5 km s α(J2000) 10h 46m 34.95s 8 δ(J2000) +13◦ 46′ 02.9′′ 8 log LB/L⊙ 10.68 9 Photometric major axis P.A. 109◦ ± 4 10 Photometric inclination 6◦ ± 1 10 Major kinematic axis P.A. 51◦ ± 3 6 Inclination of disk (from kinematics) 30 Stellar bar P.A. 65◦ ± 5 11 Stellar bar diameter 32′′ ∼ 6.7 kpc 11 M(HI) 7× 109 M⊙ 5 M(H2), r=27 ′′ 2.6× 109 M⊙ 7 M(H2), r=4 ′′.5 5.9× 108 M⊙ 7 Visual extinction, Av/2 toward nucleus b ∼ 75 mag 7 Total Radio Continuum (20 cm) flux density 119.5 mJy 12 Sum flux density (20 cm) from extended lobes 39 mJy 8 Projected distance of each synchrotron lobe ∼6 kpc 8 Position angle of inner radio continuum synchroton emission ∼ 46◦± 6 8 Position angle of synchrotron lobes ∼ 30◦ ± 20 8, 13 References: 1) Véron-Cetty & Véron (1986), 2) Ho, et al. (1997a), 3) de Vaucouleurs et al. (1993) RC3, 4) Nasa Extragalactic Database (NED), 5) Huchtmeier & Seiradakis (1985), 6) Garćıa-Barreto & Rosado (2001), 7) Garćıa-Barreto et al. (2005), 8) Garćıa-Barreto, Franco & Rudnick (2002), 9) Tully (1988), 10) Grosbøl (1985), 11) Garćıa-Barreto et al. (1996a,b), 12) Condon et al. (1998), 13) Garćıa-Barreto et al. (1998) a From Fabry-Perot Hα optical interferometry; b Maloney (1990). – 29 – Table 2: Journal of observations, average seeing and exposure time. U B V R I Seeing ∼ 2′′.0 ∼ 2′′.1 ∼ 2′′.2 ∼ 1′′.80 ∼ 1′′.70 ∆λ (Å) 600 950 1400 400 1600 λcentralefective (Å) 3540 4330 5750 6340 8040 M67 8×30s 8×30s 8×15s 8×5s 8×5s NGC 3367 5×600s 5×600s 5×600s 8×300s 8×300s Table 3: Our estimated UBVRI photometry of stars in M67 with IRAF phot, and those observed by other groups Our ID ID1,2 U(ours) U1 B(ours) B1 V (ours) V 1 R(ours) R1 I(ours) I2 1 48,14c 13.79 13.76 13.75 13.74 13.17 13.16 12.84 12.84 12.50 12.49 2 16,15c 12.92 12.90 12.85 12.86 12.27 12.27 11.94 11.95 11.59 11.57 33 49,17c 13.85 13.84 13.80 13.78 13.21 13.20 12.88 12.86 12.53 12.50 4 28,18c 13.36 13.37 13.35 13.37 12.89 12.91 12.63 12.65 12.32 12.33 5 52,19 13.84 13.84 13.79 13.80 13.21 13.22 12.88 12.89 12.53 12.53 6 27,20c 13.37 13.40 13.35 13.35 12.78 12.78 12.46 12.46 12.10 12.12 7 13,21c 12.62 12.65 12.59 12.60 12.13 12.14 11.86 11.87 11.54 11.57 8 37,22c 13.68 13.70 13.41 13.43 12.61 12.63 12.16 12.17 11.72 11.74 9 44,23 13.69 13.72 13.66 13.65 13.09 13.09 12.77 12.77 12.42 12.43 1 (Guilliland et al., 1991); 2 (Chevalier & Ilovaisky 1991); 3 Position of star #3 is: α(B1950.0)=08h 48m 37s.5, δ(B1950.0)=+11◦ 57′ 23′′.4; as a comparison, the position and magnitude reported in the USNO-B Star Catalog (Monet et al. 2003), for a star probably associated with star #3 is: α(B1950.0)=08h 48m 37s.95, δ(B1950.0)=+11◦ 57′ 57′′.9 magB1 = 13.98, magB2 = 13.54 Table 4: UBVRI estimated photometry of bright stars in the field of NGC 3367 with IRAF task qphot, aperture (apr), and width of ring (wor) apr = 23 pixels (9′′.8), wor = 5′′ ID α(J2000.0) δ(J2000.0) U B V R I 85 10h 46m 27s.56 +13◦ 43′ 46′′.0 13.91 13.51 12.87 12.47 12.22 90 10h 46m 30s.60 +13◦ 42′ 28′′.6 13.13 13.00 12.68 12.37 12.12 – 30 – Table 5: UBVRI Comparative Photometry of bright stars in the field of NGC 3367, using IRAF task phot with three aperture (apr) radii: apr 20 pixels ∼ 8′′.5, 23 pixels ∼ 9′′.8 and 26 pixels ∼ 11′′.1 apr = 20 pixels (8′′.5) ID α(J2000.0) δ(J2000.0) U B V R I 85 10h 46m 27s.56 +13◦ 43′ 46′′.0 13.87 13.46 12.82 12.43 12.12 90 10h 46m 30s.60 +13◦ 42′ 28′′.6 13.09 12.95 12.63 12.32 12.02 apr = 23 pixels (9′′.8) 85 10h 46m 27s.56 +13◦ 43′46′′.0 13.87 13.46 12.81 12.42 12.11 90 10h 46m 30s.60 +13◦ 42′28′′.6 13.09 12.95 12.62 12.31 12.01 apr = 26 pixels (11′′.1) 85 10h 46m 27s.56 +13◦ 43′46′′.0 13.86 13.45 12.81 12.42 12.09 90 10h 46m 30s.60 +13◦ 42′28′′.6 13.09 12.95 12.62 12.31 11.99 – 31 – Table 6. UBV RI Photometry, corrected for Galactic extinction, of structures throughout NGC 3367 using IRAF task phot using an aperture with radius 8 pixels ∼ 3′′.41 ∼ 715 pc ID α(J2000.0) δ(J2000.0) U B V R I 11 34s.9492 45′ 02′′ .82 14.84 15.39 14.92 14.39 13.92 2 31s.71 45′ 00′′ .0 16.49 17.28 17.05 16.82 16.45 3 31s.70 44′ 54′′ .1 16.45 17.24 17.00 16.79 16.43 4 36s.62 45′ 09′′ .2 16.65 16.91 16.47 16.03 15.50 5 36s.45 45′ 05′′ .0 16.64 16.92 16.47 16.02 15.48 6 37s.46 45′ 11′′ .7 16.89 17.31 17.10 16.79 16.31 7 34s.66 45′ 16′′ .3 16.98 17.27 16.84 16.42 15.82 8 33s.96 45′ 10′′ .2 16.73 17.05 16.62 16.20 15.62 9 36s.62 45′ 18′′ .6 17.09 17.24 16.79 16.36 15.84 10 36s.44 45′ 15′′ .3 17.16 17.24 16.72 16.26 15.67 11 35s.68 45′ 22′′ .2 17.58 17.71 17.24 16.77 16.18 12 36s.10 45′ 28′′ .4 17.42 17.63 17.22 16.83 16.27 13 35s.73 45′ 28′′ .0 17.56 17.72 17.31 16.91 16.36 14 33s.40 45′ 20′′ .8 17.51 17.83 17.48 17.08 16.51 15 32s.84 45′ 13′′ .2 17.82 17.93 17.49 17.06 16.43 16 32s.29 45′ 05′′ .2 17.75 18.02 17.68 17.30 16.71 17 35s.67 44′ 39′′ .9 17.22 17.49 17.09 16.69 16.14 18 37s.91 45′ 03′′ .3 18.58 19.09 18.70 18.29 17.64 19 37s.47 44′ 29′′ .8 18.05 18.53 18.33 17.93 17.48 20 36s.47 44′ 26′′ .0 17.52 17.90 17.49 17.08 16.58 21 36s.02 44′ 27′′ .1 17.76 18.02 17.57 17.15 16.60 22 36s.03 44′ 24′′ .3 17.32 17.83 17.47 17.07 16.63 23 35s.20 44′ 23′′ .6 17.42 17.66 17.32 16.95 16.43 24 35s.30 44′ 23′′ .7 17.38 17.62 17.30 16.93 16.42 25 34s.17 44′ 25′′ .9 17.57 17.84 17.43 17.06 16.50 26 33s.71 44′ 26′′ .6 17.66 17.94 17.59 17.22 16.68 27 33s.42 44′ 38′′ .6 17.35 17.62 17.25 16.87 16.28 28 33s.40 44′ 44′′ .5 17.54 17.67 17.24 16.81 16.21 29 33s.31 44′ 48′′ .2 17.45 17.59 17.15 16.75 16.12 30 32s.84 44′ 45′′ .7 17.53 17.88 17.50 17.15 16.54 31 31s.68 45′ 07′′ .7 17.78 18.26 17.99 17.66 17.10 32 32s.06 45′ 20′′ .3 17.45 17.89 17.58 17.27 16.79 33 32s.16 45′ 22′′ .4 17.43 17.87 17.58 17.27 16.80 34 31s.93 45′ 24′′ .3 17.47 17.91 17.68 17.36 16.88 35 32s.02 45′ 30′′ .7 17.49 18.04 17.78 17.46 17.02 36 35s.45 45′ 14′′ .6 17.61 17.57 16.91 16.43 15.76 373 35s.68 45′ 06′′ .2 17.61 17.25 16.40 15.84 15.20 384 34s.15 45′ 00′′ .5 17.50 17.29 16.48 15.93 15.25 39 34s.69 45′ 50′′ .0 18.69 18.96 18.63 18.24 17.75 40 33s.63 45′ 51′′ .4 18.35 18.72 18.45 18.12 17.70 41 36s.32 45′ 42′′ .3 18.16 18.31 17.96 17.62 17.12 425 36s.74 45′ 32′′ .9 18.14 18.29 17.85 17.42 16.91 43 37s.03 45′ 28′′ .1 17.88 18.16 17.72 17.28 16.77 44 37s.39 45′ 35′′ .4 18.14 18.32 17.99 17.62 17.09 45 34s.42 45′ 31′′ .9 17.73 17.83 17.45 17.07 16.52 46 33s.23 45′ 26′′ .9 17.64 17.82 17.41 17.02 16.48 47 35s.01 44′ 47′′ .3 17.67 17.62 17.05 16.59 15.94 48 37s.02 44′ 59′′ .1 17.58 17.86 17.44 17.00 16.46 49 36s.37 44′ 48′′ .5 17.82 17.88 17.37 16.90 16.31 50 35s.62 44′ 49′′ .1 17.65 17.75 17.20 16.72 16.09 51 35s.82 44′ 54′′ .9 17.38 17.49 16.97 16.52 15.89 52 31s.03 44′ 49′′ .6 19.42 19.60 19.01 18.58 17.79 53 32s.06 44′ 31′′ .5 19.21 19.67 19.05 18.61 17.81 54 38s.19 44′ 38′′ .1 19.04 19.34 19.01 18.61 17.92 55 38s.32 44′ 44′′ .2 19.32 19.56 19.12 18.66 18.02 56 36s.95 44′ 37′′ .0 18.51 18.63 18.18 17.76 17.13 57 32s.44 44′ 47′′ .7 17.96 18.23 17.83 17.42 16.84 58 36s.22 44′ 09′′ .0 19.70 19.93 19.48 19.07 18.30 59 36s.60 44′ 10′′ .4 19.42 19.81 19.43 19.00 18.29 60 37s.12 45′ 44′′ .9 18.90 19.02 18.71 18.28 17.77 61 37s.55 45′ 44′′ .3 18.62 18.84 18.63 18.23 17.86 62 35s.56 46′ 04′′ .5 19.68 18.84 18.63 18.23 17.86 63 34s.39 46′ 04′′ .5 19.61 19.80 19.63 19.27 18.96 64 33s.87 46′ 06′′ .0 19.81 19.90 19.85 19.36 19.12 65 32s.70 45′ 53′′ .1 19.50 19.69 19.46 19.18 18.70 66 39s.05 44′ 52′′ .2 19.57 19.87 19.32 18.96 18.21 67 39s.39 45′ 04′′ .5 20.07 20.12 19.63 19.08 18.49 68 39s.47 45′ 08′′ .3 20.10 20.23 19.86 19.29 18.60 69 38s.72 45′ 01′′ .7 19.96 20.07 19.56 19.16 18.18 70 39s.10 44′ 53′′ .7 19.62 19.85 19.35 18.97 18.23 71 31s.10 44′ 18′′ .4 20.42 20.70 20.31 19.89 18.76 72 30s.98 44′ 13′′ .9 20.51 20.92 20.48 20.17 18.88 73 31s.07 44′ 07′′ .1 20.14 21.24 20.66 20.33 19.16 74 30s.30 45′ 19′′ .5 20.61 21.01 21.34 20.58 19.43 75 43s.42 46′ 10′′ .0 20.90 21.30 20.79 20.02 20.73 76 36s.69 45′ 08′′ .2 16.73 16.99 16.55 16.12 15.62 77 35s.94 45′ 10′′ .6 17.46 17.31 16.59 16.09 15.44 78 33s.99 44′ 46′′ .8 18.01 18.02 17.36 16.84 16.15 79 36s.44 45′ 38′′ .6 18.25 18.28 17.93 17.56 17.04 80 39s.07 45′ 05′′ .7 20.31 20.21 19.53 19.13 18.27 81 35s.07 45′ 21′′ .7 18.27 18.20 17.60 17.12 16.41 1Centermost region (nucleus + part of bulge) of the galaxy; 2Positions of peak of emission, in afilter where the intensity was strongest, α(J2000.0)=10h 46m; δ(J2000.0)=+13◦ ; 3northeast region of the stellar bar; 4southwest region of the stellar bar; – 32 – 5position of peak which includes supernova 2003AA type Ia (Swift, Pugh & Li 2003; Filippenko & Chornock 2003). – 33 – Table 7. UBV RI Photometry of stars and a background galaxy in the field of NGC 3367, corrected for Galactic extinction, using IRAF task phot using an aperture with radius 8 pixels ∼ 3′′.41 ∼ 715 pc ID α(J2000.0) δ(J2000.0) U B V R I 821 33s.10 45′ 11′′ .4 17.91 17.94 17.37 16.86 16.21 83 28s.56 45′ 54′′ .7 19.66 19.04 18.13 17.52 17.04 84 42s.53 44′ 33′′ .5 18.20 17.94 17.19 16.75 16.36 852 27s.56 43′ 46′′ .0 13.93 13.54 12.91 12.51 12.26 86 29s.34 44′ 12′′ .5 21.85 21.73 20.29 19.40 17.93 87 29s.78 43′ 27′′ .9 21.64 20.99 20.09 19.29 18.45 88 39s.28 44′ 08′′ .0 20.63 20.34 19.63 18.65 17.32 893 28s.62 42′ 41′′ .5 20.63 20.34 19.13 18.43 17.56 904 30s.60 42′ 28′′ .6 13.16 13.03 12.73 12.41 12.17 1dim star within NGC 3367; 2bright star south west in the field of NGC 3367; this star was taken as a test star in order to estimate the uncertainities when obtaining its magnitudes with different aperture radii; 3small galaxy with weak surface brightness to the south south west in the field of NGC 3367; 4bright star south south west in the field of NGC 3367; this star was taken as a test star in order to estimate the uncertainities when obtaining its magnitudes with different aperture radii. – 34 – Table 8. U-B, B-V, V-I Colors of Stellar Structures in NGC 3367. ID U − B B − V V − I SWB type1 12 -0.55 0.47 1.00 I 2 -0.79 0.23 0.60 I 3 -0.79 0.24 0.57 I 4 -0.26 0.44 0.97 I 5 -0.28 0.45 0.99 I 6 -0.42 0.21 0.79 I 7 -0.29 0.43 1.02 I 8 -0.32 0.43 1.00 I 9 -0.15 0.45 0.95 II 10 -0.08 0.52 1.05 III 11 -0.13 0.47 1.06 II 12 -0.21 0.41 0.95 II 13 -0.16 0.41 0.95 II 14 -0.32 0.35 0.97 I 15 -0.11 0.44 1.06 III 16 -0.27 0.34 0.97 I 17 -0.27 0.40 0.95 I 18 -0.51 0.39 1.05 I 19 -0.48 0.20 0.84 I 20 -0.38 0.41 0.91 I 21 -0.26 0.51 0.97 I 22 -0.51 0.36 0.84 I 23 -0.24 0.34 0.89 I 24 -0.24 0.32 0.88 I 25 -0.27 0.41 0.93 I 26 -0.28 0.35 0.91 I 27 -0.27 0.37 0.97 I 28 -0.13 0.43 1.03 II 29 -0.14 0.44 1.03 II 30 -0.35 0.38 0.96 I 31 -0.48 0.27 0.89 I 32 -0.44 0.31 0.79 I 33 -0.44 0.29 0.78 I 34 -0.44 0.23 0.80 I 35 -0.55 0.26 0.76 I 36 +0.04 0.66 1.15 VII 373 +0.36 0.85 1.20 VI 384 +0.21 0.81 1.23 VII 39 -0.27 0.33 0.88 I 40 -0.37 0.27 0.75 I 41 -0.15 0.35 0.84 II 425 -0.15 0.44 0.94 II 43 -0.28 0.44 0.95 I 44 -0.18 0.33 0.88 II 45 -0.10 0.38 0.93 III 46 -0.18 0.41 0.93 II 47 +0.05 0.57 1.11 IVB 48 -0.28 0.42 0.98 I 49 -0.06 0.51 1.06 IVA 50 -0.10 0.55 1.11 III 51 -0.11 0.52 1.08 III 52 -0.18 0.59 1.22 I 53 -0.44 0.60 1.24 I 54 -0.30 0.33 1.09 I 55 -0.24 0.44 1.10 I 56 -0.12 0.45 1.05 III 57 -0.27 0.40 0.99 I 58 -0.23 0.45 1.18 I 59 -0.39 0.38 1.14 I 60 -0.12 0.31 0.94 III 61 -0.22 0.21 0.77 II 62 -0.01 0.17 0.75 III 63 -0.19 0.17 0.67 II 64 -0.09 0.05 0.73 III 65 -0.19 0.23 0.76 II 66 -0.30 0.55 1.11 I 67 -0.05 0.49 1.14 IVA 68 -0.13 0.37 1.26 III 69 -0.11 0.51 1.38 III 70 -0.23 0.50 1.12 I 71 -0.28 0.38 1.55 I 72 -0.41 0.44 1.60 I 73 -1.10 0.58 1.50 I 74 -0.40 -0.33 1.91 II 75 -0.40 0.51 0.36 I 76 -0.26 0.44 0.94 I 77 +0.15 0.72 1.15 VII 78 -0.01 0.66 1.21 VII 79 -0.03 0.35 0.89 IVA 80 +0.10 0.68 1.26 VII 81 +0.07 0.60 1.19 IVB 1SWB type (Searle, Wilkinson & Bagnuolo 1980; Bica et al., 1996). Each type suggest an approximate interval of age to different SWB classes as follows: SWB 0 have ages between 0 and 107 yrs, SWB I have ages between 107 and 3×107 yrs, SWB II have ages between 3×107 and 7 × 107 yrs, SWB III have ages between 7 × 107 and 2 × 108 yrs, SWB IVA have ages between 2 × 108 and 4 × 108 yrs, SWB IVB have ages – 35 – between 4×108 and 8×108 yrs, SWB V have ages between 8 × 108 and 2 × 109 yrs, SWB VI have ages between 2 × 109 and 5 × 109 yrs, and SWB VII have ages between 5 × 109 and 1.6 × 1010 yrs; 2Centermost region (nucleus + part of bulge) of the galaxy; 3northeast region of the stellar bar; 4southwest region of the stellar bar; 5position of peak which includes supernova 2003AA type Ia Swift, Pugh & Li (2003); Filippenko & Chornock (2003); – 36 – Table 9. UBVI Colors of foreground stars and a background galaxy in the field of NGC 3367. ID U − B B − V V − I type1 822 -0.03 +0.57 1.16 F7V 83 +0.62 +0.91 1.09 K3V 84 +0.26 +0.75 0.83 G8V 853 +0.39 +0.63 0.65 G2III 86 -0.88 +1.44 2.36 K5III 87 +0.65 +0.90 1.64 G7III 88 +0.29 +0.71 2.31 K3III 894 +0.29 +1.21 1.57 spiral 905 +0.13 +0.30 0.56 F1III 1approximate spectral type Mihalas & Binney (1981); 2dim star within NGC 3367; 3bright star south west in the field of NGC 3367; this star was taken as a test star in order to estimate the uncertainities when obtaining its magnitudes with different aperture radii; 4galaxy with weak surface brightness south south west in the field of NGC 3367; 5bright star south south west in the field of NGC 3367; this star was taken as a test star in order to estimate the uncertainities when ob- taining its magnitudes with different aperture radii. Introduction Observations and Data Reduction Uncertainities in the Photometry As a result of the Method Standard Stars as calibrators Results Total Magnitudes and Colors U B V R I Photometry of Star Associations and structures U B V R I Images Discussion Spatial distribution of structures with different SWB type Estimated ages of the structures throughout NGC 3367 Conclusions

0704.1451

Proton Decay Constraints on Low Scale AdS/CFT Unification

Proton Decay Constraints on Low Scale AdS/CFT Unification James B. Dent∗ and Thomas W. Kephart† Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 (Dated: October 30, 2018) Dark matter candidates and proton decay in a class of models based on the AdS/CFT correspondence are discussed. We show that the present bound on the proton decay lifetime is inconsistent with N = 1 SUSY, and strongly constrains N = 0 non-SUSY, low scale trinification type unification of orbifolded AdS⊗S5 models. ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.1451v1 mailto:[email protected] mailto:[email protected] I. INTRODUCTION Models with low scale (∼ TeV) unification are of great potential interest for LHC physics since the unification scale, and thus any new particles that exist at that scale, will be well within the reach of the LHC. Many models based on the AdS/CFT correspondence are of this type. When orbifolded they can lead to products of SU(N) gauge groups[1], (Actually they are products of U(N) gauge groups, and the extra U(1)s can be anomalous. However, counter terms can be added the Lagrangian to cancel these anomalies. See [2] and references therein. In what follows, we will suppress the U(1) factors, and assume they decouple from the analysis.) with particle representations that can be identified with the Standard Model (SM) when the gauge group G is broken to SU(3)× SU(2)× U(1). The simplest examples are AdS ⊗ S5/Zn models with N = 3 so G = SU(3) n where the matter fields are all in bifundamental representations. ( For n = 3 this leads to a three family N = 1 trinification model [3]. For a general classification of both N = 1 and N = 0 Zn orbifolded models see [4, 5].) While some progress has been made [6, 7, 8], the full phenomenological implications of these models are not known as detailed model building has been lacking. Recently, however, some attempts have been made to accommodate a dark matter candidate [9], the lightest conformal particle (LCP), and a stable proton [10] by the (well motivated yet ad hoc) assignment of discrete charges to particles, which thereby forbids any harmful interactions. Here we will investigate whether such assignments can arise naturally via discrete subgroups upon spontaneous symmetry breaking, and determine their consequences. We will carry out our analysis within a single representative model, and then show how the results generalize. We begin with a review of the simplest non-SUSY low scale example based on a Z7 orbifold [11]. We then follow the quantum numbers through the symmetry breaking from G = SU(3)7 to the trinification [12] group SUC(3)× SUL(3)× SUR(3), then to the group that contains B −L symmetry, SUC(3)× SUL(2)×UY (1)×UB−L(1)×UX(1), and finally to the standard model. We show that a dark matter candidate LCP can only arise if an appropriate R-parity assignments [9] is realized phenomenologically by adding extra scalars in this model. We also show that dimension six proton decay operators are present and therefore proton decay occurs much too rapidly in this model. This is always the case for any low scale N = 1 SUSY trinification model of this type. There is a possible escape in the non-SUSY case, but it will take clever model building to achieve an acceptable result. The set of charge assignments that avoids proton decay given in [10] is such a case, but it is not realized in the Z7 orbifold model studied here. II. NON-SUSY Z7 We begin with a short review of the non-SUSY Z7 orbifold model [11], but before doing so we will first pause to recall how models are constructed by orbifolding AdS ⊗ S5 [1]. When we desire an N = 1 supersymmetric theory, we embed the orbifolding group Γ in the 4 of the SU(4) R-symmetry of the underlying N = 4 theory via 4 = (a1, a2, a3, a4), where a1 = 1 and the other aj are nontrivial elements of Γ . We will only be concerned with abelian Zn choices for Γ which generate chiral supermultiplet matter fields in the bifundamental representations of the gauge group SUn(N) = SU1(N) × SU2(N) × ....SUn(N). When Γ is Zn the aj are of the form aj = e 2πinj n and the matter bifundamentals are (N, N̄) representations of SUi(N) × SUi+nj(N) for all i = 1, 2, ...n and nj . (Note, n1 = 0 in the N = 1 SUSY case.) If we wish to break all the supersymmetry, a1 must also be nontrivial, and therefore n1 6= 0. The fermions are still generated by the embedding of Zn in the 4 of the initial SU(4) R-symmetry, but now the scalars arise from the embedding of Zn in the 6 = (4 × 4)A, generated from the embedding of the 4. The scalar bifundementals are in (N, N̄) representations of SUi(N) × SUi+pk(N), where the pk are obtained from the six antisymmetric combinations e 2πipk n = e 2πi(nj+n n . We now have sufficient background to write down the non-SUSY Z7 model on which we will focus most of our attention. A. Initial Particle Content and sin2 θW The fermions are given by the embedding of the Z7 orbifolding group in the 4 of the SU(4) R symmetry: 4 = (α, α, α2, α3) [11] where α = e 7 , and the scalars can be found from 6 = (α2, α3, α3, α4, α4, α5), which leads to the particle content Fermions 2(3,3̄,1,1,1,1,1) (3,1,3̄,1,1,1,1) (3,1,1,3̄,1,1,1) 2(1,3,3̄,1,1,1,1) (1,3,1,3̄,1,1,1) (1,3,1,1,3̄,1,1) 2(1,1,3,3̄,1,1,1) (1,1,3,1,3̄,1,1) (1,1,3,1,1,3̄,1) 2(1,1,1,3,3̄,1,1) (1,1,1,3,1,3̄,1) (1,1,1,3,1,1,3̄) 2(1,1,1,1,3,3̄,3) (1,1,1,1,3,1,3̄) (3̄,1,1,1,3,1,1) 2(1,1,1,1,1,3,3̄) (3̄,1,1,1,1,3,1) (1,3̄,1,1,1,3,1) 2(3̄,1,1,1,1,1,3) (1,3̄,1,1,1,1,3) (1,1,3̄,1,1,1,3) Scalars (3,1,3̄,1,1,1,1) 2(3,1,1,3̄,1,1,1) 2(3,1,1,1,3̄,1,1) (3,1,1,1,1,3̄,1) (1,3,1,3̄ 1,1,1) 2(1,3,1,1,3̄,1,1) 2(1,3,1,1,1,3̄,1) (1,3,1,1,1,1,3̄) (1,1,3,1,3̄ 1,1) 2(1,1,3,1,1,3̄,1) 2(1,1,3,1,1,1,3̄) (3̄,1,3,1,1,1,1) (1,1,1,3,1,3̄,1) 2(1,1,1,3,1,1,3̄) 2(3̄,1,1,3,1,1,1) (1,3̄,1,3,1,1,1) (1,1,1,1,3,1,3̄) 2(3̄,1,1,1,3,1,1) 2(1,3̄,1,1,3,1,1) (1,1,3̄,1,3,1,1) (3̄,1,1,1,1,3,1) 2(1,3̄,1,1,1,3,1) 2(1,1,3̄,1,1,3,1) (1,1,1,3̄,1,3,1) (1,3̄,1,1,1,1,3) 2(1,1,3̄,1,1,1,3) 2(1,1,1,3̄,1,1,3) (1,1,1,1,3̄,1,3) At the unification scale sin2(θW ) = where SUL(2) ⊂ SU p(3), UY (1) ⊂ SU p(3), and p and q are from the seven initial SU(3)s [4]. We find this by starting with the gauge group SUn(3) and breaking to SUC(3) × SUL(3)× SUR(3) where SUC(3) ⊂ SU r(3), SUL(3) ⊂ SU p(3), and SUR(3) ⊂ SU q(3) where n = p + q + r, then writing Y as the sum of diag(1 ,−1) of SUL(3) and diag(1,1,-2) of SU(3)R we get sin2θW = 3 + 5 (α2/αY ) 3 + 5 at the unification scale. For the model at hand, p = 2 and q = 1, hence the result 2/7. B. Symmetry Breaking We proceed as in [13] by giving a VEV successively to (1,3,1,3̄,1,1,1), (1,1,3,3̄,1,1), (1,1,3,3̄,1), and (1,1,3,3̄). Each VEV breaks an SU(3) × SU(3) pair to its diagonal SU(3) subgroup and thus, after all four of these scalars have obtained VEVs, the gauge group is SUC(3) × SUL(3) × SUR(3) with three massless chiral families of fermions in complex representations 3[(3, 3̄, 1) + (1, 3, 3̄) + 3(3̄, 1, 3)]F plus fermions in real representations that acquire heavy (unification scale) masses, as well as various scalars given by [19(1, 1, 1)+(1, 8, 1)+9(1, 1, 8)+3(3, 3̄, 1)+2(3̄, 3, 1)+8[(1, 3, 3̄)+(1, 3̄, 3)]+4(3̄, 1, 3)+3(3, 1, 3̄)]S We can then break SUL(3) to SUL(2)×UL8(1) (the subscript “8” refers to the λ8 generator of SUL(3)) by giving a VEV to the neutral singlet in the adjoint (1,8,1) which decomposes as 8 = 10 + 23 + 2−3 + 30 under SUL(2) × UL8(1) (the 2s are the fundamental SUL(2) representation and the subscript refers to the UL8(1) charge). We can also break SUR(3) to UR3(1) × UR8(1) by first breaking to SUR(2) × UR8(1) as just described (except that the 8 is now chosen from one of the nine (1,1,8) multiplets) and then giving a VEV to a neutral UR3(1) × UR8(1) singlet (which arises from the decomposition of an SUR(2) adjoint, which is 30 = 12 + 1−2 + 10 under UR3(1)× UR8(1)). The quantum numbers for hypercharge (Y ), baryon-minus-lepton (B−L), and X can be written in terms of the U(1) charges T L,R (i.e., the Cartan subalgebra charges) as follows Y = − T 8L + T 8R − T 3R (1) B − L = − T 8L − T 3R (2) X = −T 8L + 2T R (3) C. Low Energy Particle Content The gauge group is now SUC(3)× SUL(2)× UY (1)× UB−L(1)× UX(1) and the particle content is given by (listing only light fermions, but all the scalars): Family Fermions Q: 3(3, 2) 1 ,1 L: 3(1, 2)− 1 ,−1,3 ū: 3(3̄, 1)− 2 d̄: 3(3̄, 1) 1 ,2 ē: 3(1, 1)1,1,0 N : 3(1, 1)0,1,0 Exotic Fermions h̄: 3(3̄, 1) 1 ,−4 h: 3(3, 1)− 1 Ē: 3(1, 2) 1 ,0,−3 E: 3(1, 2)− 1 ,0,−3 S: 3(1, 1)0,0,6 Scalars 8(1,1)1,0,0 8(1,1)0,1,0 8(1,1)0,1,−6 8(1,1)1,1,−6 8(1,1)−1,0,0 8(1,1)0,−1,0 8(1,1)0,−1,6 8(1,1)−1,−1,6 8(1,1)0,0,6 8(1,1)1,1,0 8(1,2) 1 ,0,3 8(1,2) 1 ,0,−3 8(1,1)0,0,−6 8(1,1)−1,−1,0 8(1,2)− 1 ,0,−3 8(1,2)− 1 8(1,2) 1 ,1,−3 3(3,1)− 1 ,−2 3(3,1) 2 ,−2 3(3,1)− 1 8(1,2)− 1 ,−1,3 2(3̄,1)− 1 ,−2 4(3̄,1)− 2 ,2 4(3̄,1) 1 3(3,1)− 1 ,−2 3(3,2) 1 ,1 26(1,1)0,0,0 4(3̄,1) 1 ,2 2(3̄,2)− 1 As in a usual trinified model, one can see that the fermion content corresponds to the Standard Model (SM) particles plus a right handed neutrino, N , plus eleven additional (exotic) particles. III. GAUGED R-PARITY AND DARK MATTER CANDIDATES Before discussing R-symmetry, we first dispense with U(1)X which we break with a VEV for a (1, 1)0,0,6. In [9] it was shown that if a Z2 R-symmetry is imposed, then one has a natural candidate for dark matter in the lightest conformal particle (LCP). Here we will argue that this Z2 R-symmetry does not naturally arise in the present model and will not generically be present in any trinified model which originates from orbifolded AdS5 × S Our reasoning is similar to that found in [14] where criteria are given for a gauged R-parity surviving in E6 (as well as Pati-Salam and SO(10)). The E6 case is similar to ours in that the E6 sub-group examined is SUC(3)×SUL(2)×UY (1)×UB−L(1)×UX(1), and the fermions are in a 27 of E6 with the same transformations as the fermion content in the model we are examining. The current model contains a continuous U(1)B−L symmetry which, when broken by a scalar VEV carrying an even integer amount of 3(B−L), will result in a discrete (−1)3(B−L) symmetry. This remaining discrete symmetry will then become the Z2 required in [9] and one will be assured of an LCP dark matter candidate. One sees that only the scalars (3, 1)−1/3,−2/3,−2, (3̄, 1)1/3,2/3,2, (3̄, 1)1/3,2/3,−4, and (3, 1)−1/3,−2/3,4 satisfy the criteria that if one were to obtain a VEV then there would remain the desired discrete R-symmetry. Since any such VEV would break SU(3)C , we conclude that the Z2 R-symmetry can not be arranged in this model, since any other color singlet VEV breaks B − L completely. IV. PROTON DECAY In typical grand unified trinification models, unification occurs at MG ≃ 10 14GeV [15], and rapid proton decay is avoided. Due to the low scale of unification that can arise in orb- ifolded AdS/CFT models proton decay must be strictly forbidden by choices of field content and interaction terms. A mechanism for accomplishing this via baryon charge assignment was put forth in [10] where one can assign various baryon numbers to the scalar sector and therefore exclude the unwanted interactions. We will examine if such a mechanism exists in the present Z7 model. Once scalar VEVs are obtained, quark and lepton masses are generated from the Yukawa couplings terms λq(QH1ū+QH2d̄) + λl(LH1ē+ LH2N) (4) where we have defined the scalars H1: (1, 2)1/2,0,−3 and H2: (1, 2)−1/2,0,−3. Additional interactions are seen to exist due to the presence of colored scalars in the model. In particular there are the Yukawa terms λc(QQβ + ēs̄β) (5) involving the colored scalar that we label β which has charges (3,1)− 1 These are seen to be B and L nonconserving operators and will give rise to the interaction shown in Figure 1. Once the scalars are integrated out, one will be left with a dimension six effective four-fermion operator. This is disastrous for the model due to incredibly rapid FIG. 1: Interaction leading to proton decay proton decay via p → K+ + ν. The rate of proton decay due to this interaction is given by Γ = A where mβ is the mass of the scalar β. The dimensionful proportionality constant, A (of total dimension five), contains relatively well known aspects, given by lattice QCD calculations, as well as model dependent Yukawa couplings. In a typical trinified model [12], proton decay (with reasonable couplings) was suppressed since colored triplet scalars acquire masses on the order of the unification scale, 1014GeV. Here one would expect their masses to be on the order of a few TeV and therefore one can immediately conclude that the low scale of unification will produce a proton decay rate in gross disagreement with current experimental bounds. This has severe consequences for any AdS/CFT orbifold SUSY model with low-scale unification due to the fact that these dangerous colored scalars will always be present in the form of the superpartner of the fermion we have labeled h. This leads us to believe that if a viable conformal model with low scale unification exists, it will not be supersymmetric. V. CONCLUSIONS We have examined the Z7 orbifold of AdS5 ⊗ S 5 which is the minimal low scale model that can produce the SM fermion content with all three generations present. It was shown that the scalar content is sufficient to produce the breaking pattern given by SU(3)7 → SUC(3)×SUL(3)×SUR(3) → SUC(3)×SUL(2)×UY (1)×UB−L(1)×UX(1). After breaking UB−L(1) and UX(1) it was shown that there exist two Higgs bosons that will provide the usual Yukawa couplings to give the lepton and quark masses and break SUL(2)× UY (1) → UEM . We also shown that there are no scalars in the spectrum to break the continuous UB−L(1) to leave a Z2 R-symmetry which would have provided an LCP dark matter candidate. This is in contrast to previous work where the Z2 symmetry was implemented by hand. Proton decay was also examined in the model and found to be disastrous due to the presence of colored scalars which mediates rapid proton decay due to their masses naturally occurring at the TeV unification scale (as opposed to the usual high scale of unification of trinified models which will provide agreement can the current bounds on proton decay). It was then argued that this is an impasse for any supersymmetric models for this types of orbifolding, since the rapid proton decay will be a feature of any generic low scale model that contains SUSY. For the non-SUSY case, it still remains possible but challenging to find a model without the problematic scalar fields. As the Z7 is the minimal model from which one can obtain a low scale three generation non-supersymmetric realization, evading the proton decay problem due to the existence of colored scalars will be a steep challenge. As one increases the order of the Abelian orbifold group, one will also increase the particle content and thus the dangerous colored scalars seem to be ubiquitous in models that result in trinification. One possible avenue to explore would be to examine models of type SU(N)n with minimal n, but with N not equal to 3 and determine if the problem persists. Acknowledgments We thank Paul Frampton for a useful discussion. This work was supported by U.S. DoE grant number DE-FG05-85ER40226. [1] A. E. Lawrence, N. Nekrasov and C. Vafa, Nucl. Phys. B 533, 199 (1998) [arXiv:hep-th/9803015]. [2] E. Di Napoli and P. H. Frampton, Phys. Lett. B 638, 374 (2006) [arXiv:hep-th/0603065]. [3] S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998) [arXiv:hep-th/9802183]. [4] T. W. Kephart and H. Pas, Phys. Lett. B 522, 315 (2001) [arXiv:hep-ph/0109111]. [5] T. W. Kephart and H. Pas, Phys. Rev. D 70, 086009 (2004) [arXiv:hep-ph/0402228]. [6] P. H. Frampton, R. M. Rohm and T. Takahashi, Phys. Lett. B 570, 67 (2003) [arXiv:hep-ph/0302074]. http://arxiv.org/abs/hep-th/9803015 http://arxiv.org/abs/hep-th/0603065 http://arxiv.org/abs/hep-th/9802183 http://arxiv.org/abs/hep-ph/0109111 http://arxiv.org/abs/hep-ph/0402228 http://arxiv.org/abs/hep-ph/0302074 [7] P. H. Frampton and T. W. Kephart, Phys. Lett. B 585, 24 (2004) [arXiv:hep-ph/0306053]. [8] P. H. Frampton and T. Takahashi, Phys. Rev. D 70, 083530 (2004) [arXiv:hep-ph/0402119]. [9] P. H. Frampton, Mod. Phys. Lett. A22 (2007, in press), [astro-ph/0607391]. [10] P. H. Frampton, Mod. Phys. Lett. A 22, 347 (2007) [arXiv:hep-ph/0610116]. [11] P. H. Frampton, Phys. Rev. D 60, 085004 (1999) [arXiv:hep-th/9905042]. [12] For a recent detailed phenomological study of trinification see, J. Sayre, S. Wiesenfeldt, and S. Willenbrock, Phys. Rev. D 73, 035013 (2006) [arXiv:hep-ph/0601040]. [13] P. H. Frampton, Phys. Rev. D 60, 121901 (1999) [arXiv:hep-th/9907051]. [14] S. Martin, Phys. Rev. D 46, 2769 (1992) [arXiv:hep-ph/9207218]. [15] S. Willenbrock, Phys. Lett. B 561, 130 (2003) [arXiv:hep-ph/0302168]. http://arxiv.org/abs/hep-ph/0306053 http://arxiv.org/abs/hep-ph/0402119 http://arxiv.org/abs/astro-ph/0607391 http://arxiv.org/abs/hep-ph/0610116 http://arxiv.org/abs/hep-th/9905042 http://arxiv.org/abs/hep-ph/0601040 http://arxiv.org/abs/hep-th/9907051 http://arxiv.org/abs/hep-ph/9207218 http://arxiv.org/abs/hep-ph/0302168 Introduction non-SUSY Z7 Initial Particle Content and sin2W Symmetry Breaking Low Energy Particle Content Gauged R-parity and Dark Matter Candidates Proton Decay Conclusions Acknowledgments References

0704.1452

BPS Black Holes

7 BPS Black Holes Bernard de Wita aInstitute for Theoretical Physics & Spinoza Institute, Utrecht University, Utrecht, The Netherlands The entropy of BPS black holes in four space-time dimensions is discussed from both macroscopic and micro- scopic points of view. 1. INTRODUCTION Classical black holes are solutions of Einstein’s equations of general relativity that exhibit an event horizon. From inside this horizon, noth- ing (and in particular, no light) can escape. The region inside the horizon is therefore not in the backward lightcone of future timelike infinity. However, since the discovery of Hawking radia- tion [1], it has become clear that many of the classical features of black holes will be subject to change. In these lecture we consider static, spherically symmetric black holes in four space-time dimen- sions that carry electric and/or magnetic charges with a flat space-time geometry at spatial in- finity. Such solutions exist in Einstein-Maxwell theory, the classical field theory of gravity and electromagnetism. The most general static black holes of this type correspond to the Reissner- Nordstrom solutions and are characterized by a charge Q and a mass M . In the presence of mag- netic charges, Q is replaced by q2 + p2 in most formulae, where q and p denote the electric and the magnetic charge, respectively. Hence there is no need to distinguish between the two types of charges. For zero charges one is dealing with Schwarzschild black holes. Two quantities associated with the black hole horizon are its area A and the surface gravity κs. The area is simply the area of the two-sphere de- fined by the horizon. The surface gravity, which is constant on the horizon, is related to the force (measured at spatial infinity) that holds a unit test mass in place. On the other hand, the mass M and charge Q of the black hole are not primar- ily associated with the horizon and are defined in terms of appropriate surface integrals at spatial infinity. As is well known, there exists a striking cor- respondence between the laws of thermodynam- ics and the laws of black hole mechanics [2]. Of particular importance is the first law, which, for thermodynamics, states that the variation of the total energy is equal to the temperature times the variation of the entropy, modulo work terms, for instance proportional to a change of the vol- ume. The corresponding formula for black holes expresses how the variation of the black hole mass is related to the variation of the horizon area, up to work terms proportional to the variation of the angular momentum. In addition there can also be a term proportional to a variation of the charge, multiplied by the electric/magnetic potential µ at the horizon. Specifically, the first law of thermo- dynamics, δE = T δS − p δV , translates into + µ δQ+Ω δJ . (1) The reason for factorizing the first term on the right-hand side in this particular form, is that κs/2π equals the Hawking temperature [1]. This then leads to the identification of the black hole entropy in terms of the horizon area, Smacro = 14A , (2) a result that is known as the area law [3]. In these equations the various quantities have been defined in Planck units, meaning that they have http://arxiv.org/abs/0704.1452v1 2 B. de Wit been made dimensionless by multiplication with an appropriate power of Newton’s constant (we will set ~ = c = 1). This constant appears in the Einstein-Hilbert Lagrangian according to LEH = −(16πGN)−1 |g|R. With this normal- ization the quantities appearing in the first law are independent of the scale of the metric. Einstein-Maxwell theory can be naturally em- bedded into N = 2 supergravity. This super- gravity theory has possible extensions with sev- eral abelian gauge fields and a related number of massless scalar fields (often called ‘moduli’ fields, for reasons that will become clear later on). At spatial infinity these moduli fields will tend to a constant, and the black hole mass will depend on these constants, thus introducing additional terms on the right-hand side of (1). For Schwarzschild black holes the only relevant parameter is the mass M and we note the follow- ing relations, A = 16πM2 , κs = , (3) consistent with (1). For the Reissner-Nordstrom black hole, the situation is more subtle. Here one distinguishes three different cases. For M > Q one has non-extremal solutions, which exhibit two horizons, an exterior event horizon and an interior horizon. When M = Q one is dealing with an extremal black hole, for which the two horizons coalesce and the surface gravity vanishes. In that case one has A = 4πM2 , κs = 0 , µ = Q . (4) It is straightforward to verify that this result is consistent with (1) for variations δM = δQ and κs = 0. Because the surface gravity vanishes, one might expect the entropy to vanish as well, as is suggested by the third law of thermodynamics. However, this is not the case because the hori- zon area remains finite for zero surface gravity. Finally, solutions with M < Q are not regarded as physically acceptable. Their total energy is less than the electromagnetic energy alone and they no longer have an event horizon but exhibit a naked singularity. Hence extremal black holes saturate the bound M ≥ Q for physically accept- able black hole solutions. When embedding Einstein-Maxwell theory into a complete supergravity theory, the above classifi- cation has an interpretation in terms of the super- symmetry algebra. This algebra has a central ex- tension proportional to the black hole charge(s). Unitary representations of the supersymmetry algebra must necessarily have masses that are larger than or equal to the charge. When this bound is saturated, one is dealing with so-called BPS supermultiplets. Such supermultiplets are smaller than the generic massiveN = 2 supermul- tiplets and have a different spin content. Because of this, BPS states are stable under (adiabatic) changes of the coupling constants and the rela- tion between charge and mass remains preserved. This important feature of BPS states will be rele- vant for what follows. In these lectures BPS black hole solutions are defined by the fact that they have some residual supersymmetry, so that they saturate a bound implied by the supersymmetry algebra. So far we did not refer to the explicit Schwarzschild or Reissner-Nordstrom black hole solutions, which can be found in many places in the literature. One feature that should be stressed, concerns the near-horizon geometry. For extremal, static and spherically symmetric black holes, this geometry is restricted to the product of the sphere S2 and an anti-de Sitter space AdS2, corresponding to the line element, ds2 = −r2dt2 + dr 2 + r2(dθ2 + sin2 θ dϕ2) . (5) In these coordinates the horizon is located at r = 0, where the timelike Killing vector K = ∂t turns lightlike. Such a horizon is called a Killing horizon. In these notes we discuss various aspects of the relation between black hole solutions and corre- sponding microscopic descriptions. Section 2 gen- erally describes the macroscopic (field theoretic) and microscopic (statistical) approach to black hole entropy and indicates why they are related. Section 3 summarizes the calculation of the black hole entropy based on a fivebrane wrapping on a Calabi-Yau four-cycle in a compactification of BPS Black Holes 3 M-Theory on the product space of a Calabi-Yau threefold and a circle. In section 4 the attractor equations are discussed for extremal black holes on the basis of a variational principle defined for a generic gravitational theory. Section 5 contains a brief review of N = 2 supergravity and the su- perconformal multiplet calculus. This material is used in the description of the BPS attractor equa- tions in N = 2 supergravity presented in section 6. In this case there also exists is a corresponding formulation in terms of a BPS entropy function. Finally section 7 discusses the relation between this entropy function and the black hole partition function. We try to concentrate as much as possible on the conceptual basis of the underlying ideas, but these notes cannot do justice to all aspects rele- vant for the study of black holes in string theory, such as use of the AdS3/CFT2 correspondence, black holes or related objects in other than four space-time dimensions, the relation with indices for BPS states, and the like. We refer to some recent reviews where some of these topics have been discussed [4,5,6,7]. 2. DUAL PERSPECTIVES A central question in black hole thermodynam- cis concerns the statistical interpretation of the black hole entropy. String theory has provided new insights here [8], which enable the identi- fication of the black hole entropy as the loga- rithm of the degeneracy of states d(Q) of charge Q belonging to a certain system of microstates. In string theory these microstates are provided by the states of wrapped brane configurations of given momentum and winding. When calculating the black hole solutions in the orresponding ef- fective field theory with the charges specified by the brane configuration, one discovers that the black hole area is equal to the logarithm of the brane state degeneracy, at least in the limit of large charges. We will be reviewing some aspects of this remarkable correspondence here. The horizon area, which is expected to be pro- portional to the macroscopic entropy according to the Bekenstein-Hawking area law, turns out to grow quadratically with the charges Q. After converting to string units the radius of a black hole is therefore proportional to Rhorizon lstring ∼ gs Q , (6) where lstring and gs are the string length and cou- pling constant, respectively. Since we will be as- suming that the charges are large, the black holes are generically much larger than the string scale. Consequently these black holes are called large and can be identified with the macroscopic black holes we have been discussing earlier. However, there are also situations where the leading contri- bution to the area is only linear in the charges Q. In that case (6) is replaced by Rhorizon lstring Q , (7) Moreover, in that case the string coupling (in- versely proportional to the dilaton field) cannot be taken constant, but tends to zero for large charges according to gs ∼ Q−1/2, so that the ra- dius of the black hole remains comparable to the string scale. These black holes are called small. Their corresponding classical supergravity solu- tions exhibit a vanishing horizon area and a dila- ton field that diverges at the horizon. To reliably compute the right-hand side of (7) therefore re- quires to include appropriate terms in the effec- tive action of higher order in space-time deriva- tives. We return to this issue in subsection 6.2. To further understand the relation between a field-theoretic description and a microscopic de- scription it is relevant that strings live in more that four space-time dimensions. In most situ- ations the extra dimensions are compactified on some internal manifold X and one is dealing with the standard Kaluza-Klein scenario leading to ef- fective field theories in four dimensions, describ- ing low-mass modes of the fields associated with appropriate eigenfunctions on the internal mani- fold. Locally, the original space-time is a product M4×X , where M4 denotes the four-dimensional space-time that we experience in daily life. In this situation there exists a corresponding space X at every point xµ ofM4, whose size is such that it will not be directly observable. However, this 4 B. de Wit space X does not have to be the same at every point in M4, and moving through M4 one may encounter various spaces X that are not necessar- ily equivalent. In principle they belong to some well-defined class of fixed topology parametrized by certain moduli. These moduli will appear as fields in the four-dimensional effective field the- ory. For instance, suppose that the spaces X are n-dimensional tori T n. The metric of T n will ap- pear as a field in the four-dimensional theory and is related to the torus moduli. Hence, when deal- ing with a solution of the four-dimensional theory that is not constant in M4, each patch in M4 has a non-trivial image in the space of moduli that parametrize the internal spaces X . Let us now return to a black hole solution, viewed in this higher-dimensional perspective. The fields, and in particular the four-dimensional space-time metric, will vary nontrivially overM4, and so will the internal spaces X . When moving to the center of the black hole the gravitational fields will become strong and the local product structure intoM4×X could break down. Conven- tional Kaluza-Klein theory does not have much to say about what happens, beyond the fact that the four-dimensional solution can be lifted to the higher-dimensional one, at least in principle. However, there is a feature of string theory that is absent in a purely field-theoretic approach. In the effective field-theoretic context only the lo- cal degrees of freedom of strings and branes are captured. But extended objects may also carry global degrees of freedom, as they can also wrap themselves around non-trivial cycles of the inter- nal space X . This wrapping tends to take place at a particular position inM4, so in the context of the four-dimensional effective field theory this will reflect itself as a pointlike object. The wrapped object is the string theory representation of the black hole! We are thus dealing with two complementary pictures of the black hole. One is based on general relativity where a point mass generates a global solution of space-time with strongly varying grav- itational fields, which we shall refer to as the macroscopic description. The other one, based on the internal space where an extended object is entangled in one of its cycles, does not immedi- ately involve gravitational fields and can easily be described in flat space-time. This description will be refered to as microscopic. To understand how these two descriptions are related is far from easy, but a connection must exist in view of the fact that gravitons are closed string states which inter- act with the wrapped branes. These interactions are governed by the string coupling constant gs and we are thus confronted with an interpolation in that coupling constant. In principle, such an interpolation is very difficult to carry out, so that a realistic comparison between microscopic and macroscopic results is usually impossible. How- ever, reliable predictions are possible for extremal black holes that are BPS! As we indicated earlier, in that situation there are reasons to trust such interpolations. Indeed, it has been shown that the predictions based on these two alternative de- scriptions can be successfully compared and new insights about black holes can be obtained. But how do the wrapped strings and branes represent themselves in the effective action de- scription and what governs their interactions with the low-mass fields? Here it is important to real- ize that the massless four-dimensional fields are associated with harmonic forms on X . Harmonic forms are in one-to-one correspondence with so- called cohomology groups consisting of equiva- lence classes of forms that are closed but not exact. The number of independent harmonic forms of a given degree is given by the so-called Betti numbers, which are fixed by the topology of the spaces X . When expanding fields in a Kaluza-Klein scenario, the number of correspond- ing massless fields can be deduced from an expan- sion in terms of tensors onX corresponding to the various harmonic forms. The higher-dimensional fields Φ(x, y) thus decompose into the massless fields φA(x) according to (schematically), Φ(x, y) = φA(x) ωA(y) , (8) where ωA(y) denotes the independent harmonic forms on X . The above expression, when sub- stituted into the action of the higher-dimensional theory, leads to interactions of the fields φA pro- portional to the ‘coupling constants’, CABC··· ∝ ωA ∧ ωB ∧ ωC · · · . (9) BPS Black Holes 5 These constants are known as intersection num- bers, for reasons that will become clear shortly. We already mentioned that the Betti numbers depend on the topology of X . This is related to Poincaré duality, according to which cohomology classes are related to homology classes. The latter consist of submanifolds of X without boundary that are themselves not a boundary of some other submanifold of X . This is precisely relevant for wrapped branes which indeed cover submanifolds of X , but are not themselves the boundary of a submanifold because otherwise the brane could collapse to a point. Without going into detail, this implies that there exists a dual relationship between harmonic p-forms ω and (dX −p)-cycles, where dX denotes the dimension of X . We can therefore choose a homology basis for the (dX−p)- cycles dual to the basis adopted for the p-forms. Denoting this basis by ΩA the wrapping of an ex- tended object can now be characterized by spec- ifying its corresponding cycle P in terms of the homology basis, P = pAΩA . (10) The integers pA count how many times the ex- tended object is wrapped around the correspond- ing cycle, so we are actually dealing with integer- valued cohomology and homology. The wrap- ping numbers pA reflect themselves as magnetic charges in the effective action. The electric charges are already an integer part of the effective action, because they are associated with gauge transformations that usually originate from the higher-dimensional theory. Owing to Poincaré duality it is thus very nat- ural that the winding numbers interact with the massless modes in the form of magnetic charges, so that they can be incorporated in the effective action. Before closing this section, we note that, by Poincaré duality, we can express the number of intersections by P · P · P · · · = CABC··· pApBpC · · · . (11) This is a topological characterization of the wrap- ping, which will appear in later formulae. 3. BLACK HOLES IN M/STRING THE- ORY – AN EXAMPLE As an example we now discuss the black hole entropy derived from microscopic arguments in a special case. In a later section we will con- sider the corresponding expression for the macro- scopic entropy. We start from M-theory, which is the strong coupling limit of type-IIA string the- ory. Its massless states are described by eleven- dimensional supergravity. The latter is invariant under 32 supersymmetries. Seven of the eleven space-time dimensions are compactified on an in- ternal space which is the product of a Calabi-Yau threefold (a Ricci flat three-dimensional complex manifold, henceforth denoted by CY3) times a cir- cle S1. Such a space induces a partial breaking of supersymmetry which leaves 8 supersymmetries unaffected. In the context of the four-dimensional space-time M4, these 8 supersymmetries are en- coded into two independent Lorentz spinors and for that reason this symmetry is referred to as N = 2 supersymmetry. Hence the effective four- dimensional field theory will be some version of N = 2 supergravity. M-theory contains a five-brane and this is the microscopic object that is responsible for the black holes that we consider: the five-brane has wrapped itself on a 4-cycleP of the CY 3 space [9]. Alternatively one may study this class of black holes in type-IIA string theory, with a 4-brane wrapping the 4-cycle [10]. The 4-cycle is subject to certain requirements which will be mentioned in due course. The massless modes captured by the effective field theory correspond to harmonic forms on the CY3 space; they do not depend on the S 1 co- ordinate. The 2-forms are of particular interest. In the effective theory they give rise to vector gauge fields Aµ A, which originate from the rank- three tensor gauge field in eleven dimensions. In addition there is an extra vector field Aµ 0 cor- responding to a 0-form which is related to the graviphoton associated with S1. This field will couple to the electric charge q0 associated with momentum modes on S1 in the standard Kaluza- Klein fashion. The 2-forms are dual to 4-cyles and the wrapping of the five-brane is encoded in 6 B. de Wit terms of the wrapping numbers pA, which appear in the effective field theory as magnetic charges coupling to the gauge fields Aµ A. Here we see Poincaré duality at work, as the magnetic charges couple nicely to the corresponding gauge fields. In view of the fact that the product of three 2- forms defines a 6-form that can be integrated over the CY3 space, there exist non-trivial triple inter- section numbers CABC . These numbers will ap- pear in the three-point couplings of the effective field theory. There is a subtle topological fea- ture that we have not explained before, which is typical for complex manifolds containing 4-cycles, namely the existence of another quantity of topo- logical interest known as the second Chern class. The second Chern class is a 4-form whose inte- gral over a four-dimensional Euclidean space de- fines the instanton number. The 4-form can be integrated over the 4-cycle P and yields c2A pA, where the c2A are integers. Let us now turn to the microscopic counting of degrees of freedom [9]. These degrees of freedom are associated with the massless excitations of the wrapped five-brane characterized by the wrap- ping numbers pA on the 4-cycle. The 4-cycle P must correspond to a holomorphically embedded complex submanifold in order to preserve 4 super- symmetries. The massless excitations of the five- brane are then described by a (1+1)-dimensional superconformal field theory (the reader may also consult [11]). Because we have compactified the spatial dimension on S1, we are dealing with a closed string with left- and right-moving states. The 4 supersymmetries of the conformal field the- ory reside in one of these two sectors, say the right-handed one. Conformal theories in 1+1 di- mensions are characterized by a central charge, and in this case there is a central charge for the right- and for the left-moving sector separately. The two central charges are expressible in terms of the wrapping numbers pA and depend on the intersection numbers and the second Chern class, according to cL = CABC p ApBpC + c2A p cR = CABC p ApBpC + 1 c2A p Here we should stress that the above result is far from obvious and holds only under the condition that the pA are large. In that case every generic deformation of P will be smooth. Under these cir- cumstances it is possible to relate the topological properties of the 4-cycle to the topological data of the Calabi-Yau space. We can now choose a state of given momentum q0 which is supersymmetric in the right-moving sector. From rather general arguments it fol- lows that such states exist. The corresponding states in the left-moving sector have no bearing on the supersymmetry and these states have a certain degeneracy depending on the value of q0. In this way we have a tower of degenerate BPS states invariant under 4 supersymmetries, built on a supersymmetric state in the right-moving sector. We can then use Cardy’s formula, which states that the degeneracy of states for fixed but large momentum (large as compared to cL) equals exp[2π |q0| cL/6]. This leads to the following ex- pression for the entropy, Smicro(p, q) = |q̂0|(CABC pApBpC + c2A pA) , where q0 has been shifted according to q̂0 = q0 + CABqAqB . (14) Here CAB is the inverse of CAB = CABCp C . This modification is related to the fact that the electric charges associated with the gauge fields Aµ A will interact with the M-theory two-brane [9]. The existence of this interaction can be inferred from the fact that the two-brane interacts with the rank-three tensor field in eleven dimensions, from which the vector gauge fields Aµ A originate. We stress that the above results apply in the limit of large charges. The first term propor- tional to the triple intersection number is obvi- ously the leading contribution whereas the terms proportional to the second Chern class are sub- leading. The importance of the subleading terms will become more clear in later sections. Having obtained a microscopic representation of a BPS black hole, it now remains to make contact with it by deriving the corresponding black hole solu- tion directly in the N = 2 supergravity theory. This is discussed in some detail in section 6. BPS Black Holes 7 4. ATTRACTOR EQUATIONS The microscopic expression for the black hole entropy depends only on the charges, whereas a field-theoretic calculation can in principle depend on other quantities, such as the values of the mod- uli fields at the horizon. To establish any agree- ment between these two approaches, the moduli (as well as any other relevant fields that enter the calculation) must take fixed values at the horizon which may depend on the charges. As it turns out this is indeed the case for extremal black hole so- lutions, as was first demonstrated in the context of N = 2 supersymmetric black holes [12,13,14]. The values taken by the fields at the horizon are independent of their asymptotic values at spatial infinity. This fixed point behaviour is encoded in so-called attractor equations. In the presence of higher-derivative interac- tions it is very difficult to explicitly construct black hole solutions and to exhibit the attractor phenomenon. However, by concentrating on the near-horizon region one can usually determine the fixed-point values directly without considering the interpolation between the horizon and spa- tial infinity. Provided the symmetry of the near- horizon region is restrictive enough, the attrac- tor phenomenon can be described conveniently in terms of a variational principle for a so-called en- tropy function. The stationarity of this entropy function then yields the attractor equations and its value at the attractor point equals the macro- scopic entropy. The purpose of the present sec- tion is to explain this phenomenon. For N = 2 BPS black holes with higher- derivative interactions the attractor equations fol- low from classifying possible solutions with full supersymmetry [15]. As it turns out supersym- metry determines the near-horizon geometry (and thus the horizon area), the values of the moduli fields in terms of the charges and the value of the entropy as defined by the Noether charge defini- tion of Wald [16]. This result can also be de- scribed in terms of a variational principle [17,18], as we shall explain in section 6. Let us now turn to the more generic case of ex- tremal, static, spherically symmetric black holes that are not necessarily BPS, following the ap- proach of [19] which is based on an action prin- ciple. Similar approaches can be found elsewhere in the literature, for instance, in [20,21,22]. When dealing with spherically symmetric solutions, one integrates out the spherical degrees of freedom and obtains a reduced action for a 1 + 1 dimen- sional field theory which fully describes the black hole solutions. Here we consider a general system of abelian vector gauge fields, scalar and other matter fields coupled to gravity. The geometry is thus restricted to the product of the sphere S2 and a 1 + 1 dimensional space-time, and the de- pendence of the fields on the S2 coordinates θ and ϕ is fixed by symmetry arguments. For the moment we will not make any assumption regard- ing the dependence on the remaining two cooor- dinates r and t. Consequently we write the gen- eral field configuration consistent with the various isometries as ds2(4) = gµνdx = ds2(2) + v2 dθ2 + sin2 θ dϕ2 I = eI , Fθϕ sin θ . The Fµν I denote the field strengths associated with a number of abelian gauge fields. The θ- dependence of Fθϕ I is fixed by rotational invari- ance and the pI denote the magnetic charges. The latter are constant by virtue of the Bianchi iden- tity, but all other fields are still functions of r and t. As we shall see in a moment the fields eI are dual to the electric charges. The radius of S2 is defined by the field v2. The line element of the 1+1 dimensional space-time will be expressed in terms of the two-dimensional metric ḡij , whose determinant will be related to a field v1 according |ḡ| . (16) Eventually ḡij will be taken proportional to an AdS2 metric, ds2(2) = ḡij dx idxj = v1 − r2 dt2 + dr . (17) In addition to the fields eI , v1 and v2, there may be a number of other fields which for the moment we denote collectively by uα. 8 B. de Wit As is well known theories based on abelian vec- tor fields are subject to electric/magnetic dual- ity, because their equations of motion expressed in terms of the dual field strengths,1 GµνI = |g| εµνρσ ∂FρσI , (18) take the same form as the Bianchi identities for the field strengths Fµν I . Adopting the conven- tions where xµ = (t, r, θ, ϕ) and εtrθϕ = 1, and the signature of the space-time metric equals (−,+,+,+), which is consistent with (17), it fol- lows that, in the background (15), Gθϕ I = −v1v2 sin θ Grt I = −4π v1v2 These two tensors can be written as qI sin θ/(4π) and fI . The quantities qI and fI are conjugate to pI and eI , respectively, and can be written as qI(e, p, v, u) = −4π v1v2 fI(e, p, v, u) = −4π v1v2 They depend on the constants pI and on the fields eI , v1,2 and uα, and possibly their t and r deriva- tives, but no longer on the S2 coordinates θ and ϕ. Upon imposing the field equations it follows that the qI are constant and correspond to the electric charges. Our aim is to obtain a descrip- tion in terms of the charges pI and qI , rather than in terms of the pI and eI . Electric/magnetic duality transformation are induced by rotating the tensors Fµν I andGµν I by a constant transformation, so that the new linear combinations are all subject to Bianchi identities. Half of them are then selected as the new field strengths defined in terms of new gauge fields, while the Bianchi identities on the remaining lin- ear combinations are regarded as field equations belonging to a new Lagrangian defined in terms 1We assume that the Lagrangian is a function of the abelian field strengths and does not depend explicitly on the gauge fields. of the new field strengths. In order that this du- alization can be effected the rotation of the ten- sors must belong to Sp(2n + 2;R), where n + 1 denotes the number of independent gauge fields [23]. Naturally the duality leads to new quan- tities (p̃I , q̃I) and (ẽ I , f̃I), related to the original ones by the same Sp(2n+2;R) rotation. Since the charges are not continuous but will take values in an integer-valued lattice, this group should even- tually be restricted to an appropriate arithmetic subgroup. Subsequently we define the reduced Lagrangian by the integral of the full Lagrangian over S2, F(e, p, v, u) = dθ dϕ |g| L , (21) We note that the definition of the conjugate quan- tities qI and fI takes the form, qI = − , fI = − . (22) It is known that a Lagrangian does not trans- form as a function under electric/magnetic dual- ities (see, e.g. [24]), but one can generally show that the following combination does [25], E(q, p, v, u) = −F(e, p, v, u)− eIqI . (23) More precisely, this quantity transforms un- der electric/magnetic duality according to Ẽ(q̃, p̃, v, u) = E(q, p, v, u). In view of the first equation (22), the definition (23) takes the form of a Legendre transform. Furthermore the field equations imply that the qI are constant and that the action, dtdr E , is stationary under varia- tions of the fields v and u, while keeping the pI and qI fixed. This is to be expected as E is in fact minus the Hamiltonian density associated with the reduced Lagrangian density (21), at least as far as the vector fields are concerned. In the near-horizon background (17), assuming fields that are invariant under the AdS2 isome- tries, the generally covariant derivatives of the fields vanish and the function E depends only on the fields which no longer depend on the coordi- nates. The equations of motion then imply that the values of the fields v1,2 and uα are determined BPS Black Holes 9 by demanding E to be stationary under variations of v and u, = 0 , qI = constant . (24) The function 2π E(q, p, v, u) then coincides with the entropy function proposed by Sen [19]. The first two equations of (24) are interpreted as the attractor equations. At the attractor point one may prove attractor dθ dϕ |g|Rrtrt ∂Rrtrt , (25) where the right-hand side is evaluated in the near- horizon geometry. This leads to the expression attractor ∂Rµνρσ εµνερσ , (26) where Σhor denotes a spacelike cross section of the Killing horizon, and εµν the normal bivector which acts in the space normal to Σhor. This is precisely the expression for the Wald entropy [16] (applied to this particular case). For the Einstein- Hilbert action, (26) equals a quarter of the hori- zon area in Planck units. For more general La- grangians (26) may lead to deviations from the area law, as we will see in due course. Note that the entropy function does not necessarily depend on all fields at the horizon. The values of some of the fields will then be left unconstrained, but those will not appear in the expression for the Wald entropy either. The above derivation of the entropy function applies to any gauge and general coordinate in- variant Lagrangian, including Lagrangians with higher-derivative interactions. In the absence of higher-derivative terms, the reduced Lagrangian F is at most quadratic in eI and pI and the Leg- endre transform (23) can easily be carried out. The results coincide with corresponding terms in the so-called black hole potential discussed in e.g. [20,21,22]. 5. N=2 SUPERGRAVITY In the previous section the symmetry of the near-horizon geometry played a crucial role. For BPS black holes the supersymmetry enhancement at the horizon is the crucial input that constrains both some of the fields at the horizon as well as the near-horizon geometry itself. The black hole solutions that we will be considering have a resid- ual N = 1 supersymmetry (so that they are BPS) and are solitonic interpolations between N = 2 configurations at the horizon and at spatial infin- ity. Obviously to describe BPS black holes one needs to consider extended supergravity theories. In view of the application described in section 3, N = 2 supergravity is relevant. Moreover, N = 2 supergravity has off-shell formulations (meaning that supersymmetry transformations realize the supersymmetry algebra without the need for im- posing field equations associated with a specific Lagrangian) and this facilitates the calculations in an essential way. This aspect is especially im- portant because we will be considering supersym- metric Lagrangians with interactions containing more than two derivatives. In the following subsections we present a brief introduction to N = 2 supergravity. Supermul- tiplets are introduced in subsection 5.1 and su- persymmetric actions in subsection 5.2. Finally, in subsection 5.3, we elucidate the use of compen- sating fields and corresponding supermultiplets to familiarize the reader with the principles under- lying the superconformal multiplet calculus. Fur- ther details can be found in the literature [26,27]. 5.1. Supermultiplets In this subsection we briefly introduce the su- permultiplets that play a role in the following. Of particular interest are the vector and the Weyl su- permultiplet. Other multiplets are the tensor su- permultiplets and the hypermultiplets, but those will not be discussed as they only play an ancil- lary role. The covariant fields and field strengths of the various gauge fields belonging to the vector or to the Weyl supermultiplet comprise a chiral mul- tiplet. Such multiplets are described in super- space by chiral superfields.At this point it is con- venient to systematize our discussion by using su- perspace notions, although we do not intend to make an essential use of superfields. Scalar chiral fields in N = 2 superspace have 16 + 16 bosonic + fermionic field components, but there exists a 10 B. de Wit constraint which reduces the superfield to only 8+8 field components. This constraint expresses higher-θ components of the superfield in terms of lower-θ components or space-time derivatives thereof. The vector supermultiplet and the Weyl supermultiplet are both related to reduced chiral multiplets. Note, however, that products of re- duced chiral superfields constitute general chiral fields. The vector supermultiplet is described by a scalar reduced chiral superfields, whose lowest- θ component is a complex field which we de- note by X . Then there is a doublet of chiral fermions Ωi, where i is an SU(2) R-symmetry doublet index. The position of the index i in- dicates the chirality of the spinor field: Ωi car- ries positive, and Ωi negative chirality. The fields X̄ and Ωi appear as lowest-θ components in the anti-chiral superfield that one obtains by complex conjugation of the chiral superfield. We recall that the so-called R-symmetry group is defined as the maximal group that rotates the supercharges in a way that commutes with Lorentz symmetry and is compatible with the supersymmetry alge- bra. For N = 2 supersymmetry this group equals SU(2)×U(1), which acts chirally on the spinors. In spite of its name, R-symmetry does not nec- essarily consitute an invariance of supersymmet- ric Lagrangians. Finally, at the θ2-level, we en- counter the field strength Fµν of the gauge field and an auxiliary field which we write as a sym- metric real tensor, Yij = Yji = εikεjlY kl. Here we note that complex conjugation will often be indicated by rasing and/or lowering of SU(2) in- dices. One can easily verify that in this way we have precisely 8+8 independent field components (which we will refer to as off-shell degrees of free- dom, as we did not impose any field equations). Note the difference with on-shell degrees of free- dom. The conventional Lagrangian for the vector supermultiplet describes 4 + 4 physical massless bosonic and fermionic states: 2 scalars associated with X and X̄, the 2 helicities associated with the vector gauge field, and 2 Majorana fermions, each carrying 2 helicities. The Weyl supermultiplet has a rather similar decomposition, but in this case the reduced chi- ral superfield is not a scalar but an anti-selfdual Lorentz tensor. For extended supersymmetry the Weyl superfield is also assigned to the antisym- metric representation of the R-symmetry group, so that its lowest-order θ-component is denoted by Tab ij . Its complex conjugate belongs to the corresponding anti-chiral superfield and its corre- sponding tensor is selfdual and denoted by Tabij . Here the indices a, b, . . . denote the components of space-time tangent space tensors. In view of its tensorial character the Weyl supermultiplet com- prises 24 + 24 off-shell degrees of freedom. The covariant components of the Weyl multiplet are as follows. Linear in θ one has the fermions decom- posing into the field strength of the gravitini and a doublet spinor χi. The gravitino field strengths comprise 16 degrees of freedom so that together with the spinors χi we count 24 off-shell degrees of freedom. At the θ2-level we have the Weyl tensor, the field strengths belonging to the gauge fields associated with R-symmetry, and a real scalar field denoted by D, comprising 5, 4× 3 and 1 off- shell degrees of freedom, respectively. Together with the 6 degrees of freedom belonging to Tab we count 24 bosonic degrees of freedom. The Weyl multiplet contains the fields ofN = 2 conformal supergravity and an invariant action can be written down that is quadratic in its components. Although Tab ij is not subject to a Bianchi identity, it is often called the “gravipho- ton field strength”. The reason for this misnomer is that the gravitini transform into Tab ij and in lo- cally supersymmetric Lagrangians of vector mul- tiplets that are at most quadratic in space-time derivatives, Tab ij acts as an auxiliary field and couples to a field-dependent linear combination of the vector multiplet field strengths. For this class of Lagrangians, all the fields of the Weyl multiplet with the exception of the graviton and the gravitini fields, act as auxiliary fields. In this subsection we mainly describe linearized results. Ultimately we are interested in con- structing a theory of local supersymmetry. This means that the vector multiplets must first be formulated in a supergravity background. This leads to additional terms in the supersymmetry transformation rules and in the superfield com- ponents which depend on the supergravity back- ground. Some of these terms correspond to re- BPS Black Holes 11 placing ordinary space-time derivatives by covari- ant ones. However, we consider only vector mul- tiplets and the Weyl multiplet here and the latter describes the supermultiplet of conformal super- gravity. Consistency therefore requires that we formulate the vector supermultiplet in a super- conformal background, and, indeed, the vector supermultiplet is a representation of the full su- perconformal algebra. Therefore all the super- conformal symmetries can be realized as local symmetries. Naturally the Weyl multiplet itself will also acquire non-linear corrections but those do not involve the vector multiplet fields. The reader may wonder why we are interested in con- formal supergravity here, but this will be clarified shortly. Other than that, the situation is concep- tually the same as when considering multiplets of matter fields coupled to a nonabelian gauge the- ory, or to gravity. 5.2. Supersymmetric actions In view of the fact that both the vector and Weyl supermultiplets are described by chiral su- perfields, even beyond their linearized form, it is clear how to construct supersymmetric ac- tions. Namely one takes some function of the vector superfield (actually we will need several of these superfields, which we label by indices I = 0, 1, 2, . . . , n) and the square of the Weyl superfield. Taking the square implies no loss of generality because we are interested in Lorentz invariant couplings. When expanding the super- fields in θ components, one generates multiple derivatives of this function which depend on the lowest-θ components, X and A = (Tab ijεij) 2 . (27) Because the function is holomorphic, i.e., it de- pends on X and A, but not on their complex conjugates, we take the imaginary part of the re- sulting expression. However, in order that the action is superconformally invariant, the function F (X,A) must be holomorphic and homogeneous, F (λX, λ2A) = λ2F (X,A) . (28) We refrain from giving full results. In principle they are derived straightforwardly, but the for- mulae are often lengthy and require extra defini- tions. Therefore we discuss only a few character- istic terms. First of all, let us consider the scalar kinetic terms. They are accompanied by a coupling to the Ricci scalar and the scalar field D of the Weyl multiplet in the following way, L ∝ i(∂µFI ∂µX̄I − ∂µF̄I ∂µXI) − i(1 R−D)(FI X̄I − F̄I XI) , where FI denotes the derivative of F with respect to XI . Observe that when the function F de- pends on Tab ij this will generate interactions be- tween the kinetic term for the vector multiplet scalars and the tensor field of the Weyl multiplet. Of course, this pattern continues for other terms. A second term concerns the kinetic term of the vector fields, which is proportional to the second derivative of the function F , L ∝ 1 iFIJ (F I − 1 X̄ITab ijεij) × (F−abJ − 1 X̄JT abklεkl) + h.c. . A third term involves the square of the Weyl tensor, contained in the tensor R(M), L ∝ 16 iFA[2R(M)cdab R(M)−cd − 16T abij DaDcTcbij ] + h.c. . Here Da denotes a superconformally covariant derivative (which also contains terms propor- tional to the Ricci tensor). We refrain from giving further details at this point and refer to the liter- ature. 5.3. Compensator multiplets The theories discussed so far are invariant un- der the local symmetries of the superconformal algebra. This high degree of symmetry seems un- necessary for, or even an obstacle to, practical applications. The purpose of this section is to explain that this is not the case. Let us start with a simple example, namely massive SU(N) Yang-Mills theory, with La- grangian, L = 1 Tr[Fµν(W )F µν(W )] M2Tr[Wµ W 12 B. de Wit where we use a Lie-algebra valued notation and the definition Fµν = 2 ∂[µWν] − [Wµ,Wν ]. In- troduce now a matrix-valued field Φ ∈ SU(N) transforming under SU(N) gauge transforma- tions from the left and substitute Wµ = Φ −1DµΦ into the Lagrangian, where the covariant deriva- tive reads DµΦ = (∂µ − Aµ)Φ. The first term is not affected by this transformation which takes the form of a field-dependent gauge transforma- tion. But the mass term changes, and we find the following Lagrangian, L = 1 Tr[Fµν(A)F µν(A)] M2 Tr[DµΦD µΦ−1] , which is manifestly gauge invariant. Clearly, this is a massless gauge theory coupled to N2 − 1 scalars. However, this formulation is gauge equiv- alent to (32), as one readily verifies by imposing the gauge condition Φ = . What do we achieve by rewriting (32) into the form (33)? Both Lagrangians describe the same number of physical states and are based on the same number of off-shell degrees of freedom. In (32) the degrees of freedom are contained in a sin- gle field, Wµ, carrying 4 components per genera- tor. In (33), however, the degrees of freedom are distributed over two fields in a local and Lorentz invariant way, namely 3 components per gener- ator in Aµ (we subtracted the gauge degrees of freedom) and 1 component per generator in Φ. Hence the second formulation can be regarded as reducible, and this reducibility has been achieved by introducing extra gauge invariance. A similar situation exists for gravity, as is shown by the Lagrangian, |g|L ∝ gµν∂µφ∂νφ− 16Rφ , (34) which is invariant under local scale transforma- tion with parameter Λ(x): δφ = Λφ, δgµν = −2Λ gµν. This Lagrangian is gauge equivalent to the Einstein-Hilbert Lagrangian. To see this one either rewrites it in terms of a scale invariant metric φ2gµν , or one simply imposes the gauge condition and sets φ equal to a constant (which will then be related to Newton’s constant). Again the Einstein-Hilbert Lagrangian and (34) contain the same number of off-shell degrees of freedom, but the latter field configuration is reducible: gµν describes only 5 degrees of freedom (in view of the local scale invariance) and the sixth one can be assigned to the scalar field φ. Fields such as Φ and φ are called compensat- ing fields, because they can be used to convert any quantity that transforms under the gauge symme- try into a gauge invariant one. Often the gauge equivalent formulation, based on the introduction of compensator fields and gauge symmetries at the same time, is exploited for reasons of renor- malizability as one can use the gauge freedom to choose a different gauge that leads to better short- distance behaviour. This is not the issue here but the crucial point is that the compensating degrees of freedom must be contained in full supermulti- plets. By keeping the gauge invariance manifest one realizes a higher degree of symmetry which facilitates the construction of Lagrangians and clarifies the geometrical features of the resulting supergravity theories. In this way, pureN = 2 su- pergravity is, for instance, constructed from two compensating supermultiplets, one of which is a vector multiplet and the other one can be a ten- sor multiplet or a hypermultipet. Of these two supermultiplet only the vector field will describe physical degrees of freedom (namely, those corre- sponding to the graviphoton). The other compo- nents play the role of either compensating fields (associated with local scale, R-symmetry and spe- cial conformal supersymmetry transformations), or are constrained by the field equations, either by Lagrange multipliers, or because they are aux- iliary. 6. BPS ATTRACTORS As we already discussed in section 4 the BPS attractor equations follow from the requirement of full N = 2 supersymmetry at the horizon. In the context of an off-shell representation of su- perymmetry, the corresponding equations can be derived in a way that is rather independent of the action. As explained in the previous section, N = 2 vector multiplets contain complex physical scalar fields which we denoted by XI . In the con- text of the superconformal framework these fields are defined projectively, in view of the invariance BPS Black Holes 13 under local scale and U(1) transformations. The action for vector multiplets with additional inter- actions involving the square of the Weyl tensor is encoded in a holomorphic function F (X,A), which is homogenous of second degree (c.f. (28)). Here A is quadratic in the anti-selfdual field Tab as shown in (27). The normalizations of the La- grangian and of the charges adopted below differ from the normalizations used in section 4. Another issue that we should explain concerns electric/magnetic duality, which in principle per- tains to the gauge fields. Straightforward appli- cation of this duality to an N = 2 supersymmet- ric Lagrangian with vector multiplets, leads to a new Lagrangian that no longer takes the canoni- cal form in terms of a function F (X,A). In order to bring it into that form one must simultane- ously apply a field redefinition to the scalar and spinor fields. On the scalar fields, this redefinition follows from the observation that (XI , FI(X,A)) transforms as a sympletic vector analogous to the tensors (Fµν I , GµνI) discussed previously. Hence the (2n + 2)-dimensional vector (XI , FI(X,A)) is rotated into a new vector (X̃I , F̃I). When this rotation belongs to Sp(2n + 2;R) then F̃I can be written as the derivative of a new func- tion F̃ (X̃, A) with respect to X̃I . The new func- tion then encodes the dual action. The need for this additional field redefinition follows from the fact that the gauge fields, and thus their field strengths Fµν I , and the fields XI have a well- defined relation encoded in the supersymmetry transformation rules. Therefore, the transforma- tions of (XI , FI(X,A)) must be taken into ac- count when considering electric/magnetic dual- ity. We refer to [24,28] for further details and a convenient list of formulae. To determine the BPS attractor equations one classifies all possible N = 2 supersymmetric so- lutions. This is done by studying the supersym- metry variations of the fermions in an arbitrary bosonic background. Requiring that these varia- tions vanish then imposes strong restrictions on this background. The analysis was performed in [15] for N = 2 supergravity with an arbitrary number of vector multiplets and hypermultiplets, including higher-order derivative couplings pro- portional to the square of the Weyl tensor. Also the interpolating BPS solutions were studied in considerable detail. It was found that N = 2 su- persymmetric solutions are unique and depend on a harmonic function with a single center. Hence the horizon geometry and the values of the rel- evant fields are fully determined in terms of the charges. The hypermultiplet scalar fields are co- variantly constant but otherwise arbitrary. How- ever, the horizon and the entropy do not depend on these fields, so that they can be ignored. In the absence of charges one is left with flat Minkowski space-time with arbitrary constant moduli and ij = 0. As it turns out the attractor equations have a universal form. Before commenting further on their derivation, let us present the equa- tions, which are manifestly covariant under elec- tric/magnetic duality, PI = 0 , QI = 0 , Υ = −64 , (35) where PI ≡ pI + i(Y I − Ȳ I) , QI ≡ qI + i(FI − F̄I) . Here the Y I and Υ are related to the XI and A, respectively, by a uniform rescaling and FI and FΥ will denote the derivatives of F (Y,Υ) with respect to Y I and Υ. To explain the details of the rescaling, we introduce the complex quantity Z, sometimes refered to as the ’holomorphic BPS mass’, which equals the central charge associated with the vector supermultiplets. In terms of the original variables XI it is defined as Z = exp[K/2] (pIFI(X,A)− qIXI) , (37) where e−K = i (X̄IFI(X,A)− F̄I(X̄, Ā)XI) . (38) At the horizon the variables Y I and Υ are de- fined by Y I = exp[K/2] Z̄ XI , Υ = exp[K] Z̄2 A . Note that Y I and Υ are invariant under arbitrary complex rescalings of the underlying variables XI 14 B. de Wit and A. The reader may easily verify that for fields satisfying the attractor equations (35), one estab- lishes that |Z|2 ≡ pIFI − qIY I , (40) which is obviously real and positive, is equal to i(Ȳ IFI − Y I F̄I). Finally we wish to draw attention to just one aspect of the derivation of the attractor equations (35). Consider the spinor fields belonging to the vector supermultiplets, and concentrate on their supersymmetry variation in terms of a two-rank tensor, I = 1 I − 1 εklTab kl X̄I . (41) This particular linear combination of the field strength F−ab I and the field X̄I arises because the symmetry transformations are evaluated in a su- perconformal background. Full supersymmetry therefore implies that the right-hand side of (41) vanishes, so that I = 1 εklTab kl X̄I . (42) An extension of this argument gives a similar re- sult for the conjugate field strengths, G−abI = εklTab kl F̄I . (43) Note that these two equations are consistent with respect to electric/magnetic duality. Given the fact that the field strengths Fab I and GabI satisfy the Maxwell equations and therefore become pro- portional to the magnetic and electric charges, pI and qI , it is not surprising that one finds the at- tractor equations (35). For further details of the analysis we refer to [15]. 6.1. The BPS entropy function The BPS attractor equations follow also from a variational principle based on the entropy func- tion [17,18], Σ(Y, Ȳ , p, q) = F(Y, Ȳ ,Υ, Ῡ) − qI(Y I + Ȳ I) + pI(FI + F̄I) , where pI and qI couple to the correspond- ing magneto- and electrostatic potentials at the horizon (c.f. [15]) in a way that is consistent with electric/magnetic duality. The quantity F(Y, Ȳ ,Υ, Ῡ), which will be regarded as a ‘free energy’ in what follows, is defined by F(Y, Ȳ ,Υ, Ῡ) = −i Ȳ IFI − Y I F̄I ΥFΥ − ῩF̄Υ where FΥ = ∂F/∂Υ. Just as the entropy function discussed in section 4, the entropy function (44) transforms as a function under electric/magnetic duality [28]. Varying this entropy function with respect to the Y I , while keeping the charges and Υ fixed, yields the result, δΣ = PI δ(FI + F̄I)−QI δ(Y I + Ȳ I) . (46) Here we made use of the homogeneity of the func- tion F (Y,Υ). Under the mild assumption that the matrix NIJ = i(F̄IJ − FIJ), (47) is non-degenerate, it thus follows that station- ary points of Σ satisfy the attractor equations. The macroscopic entropy is equal to the entropy function taken at the attractor point. This im- plies that the macroscopic entropy is the Legen- dre transform of the free energy (45). An explicit calculation yields the entropy formula [29,30,31], Smacro(p, q) = πΣ attractor |Z|2 − 256 ImFΥ Υ=−64 Here the first term represents a quarter of the horizon area (in Planck units) so that the second term defines the deviation from the Bekenstein- Hawking area law. The entropy coincides pre- cisely with the Wald entropy [16] as given by the right-hand side of (26). In fact, the original derivation of (48) was not based on an entropy function and made direct use of the expression (26). In the absence of Υ-dependent terms, the ho- mogeneity of the function F (Y ) implies that the area scales quadratically with the charges, as was discussed already at the beginning of section 2. However, in view of the fact that Υ takes a fixed BPS Black Holes 15 value, the second term will be subleading in the limit of large charges.2 Note, however, that also the area will contain subleading terms, as it de- pends on Υ. The entropy equation (48) has been confronted with the result of microstate counting, for in- stance, in the situation described in section 3. In that case the effective supergravity action is known and based on the function F (Y,Υ) = −1 CABC Y AY BY C − c2A 24 · 64 Substituting this result into (48) and imposing the attractor equations (35) with p0 = 0, one in- deed derives the result (13) for the macroscopic entropy [29]. The entropy formula (48) has also been put to a test in other cases. Some of them will be discussed in due course. The relation between (44) and the entropy function introduced in section 4 was discussed in [25], where it was established that both en- tropy functions lead to identical results for BPS black holes. When the black holes are not BPS (i.e. have no residual supersymmetry), then the entropy function (44) is simply not applicable. In this connection the question arises whether other, independent, higher-derivative interactions associated with matter multiplets will not con- tribute to the entropy either. For instance, La- grangians for tensor supermultiplets that con- tain interactions of fourth order in space-time derivatives, lead to terms quadratic in the Ricci scalar that will in principle contribute to the Wald entropy [32]. Indeed, for non-BPS black holes these terms yield finite contributions to the entropy, but for BPS black holes these cor- rections vanish. A comprehensive treatment of higher-derivative interactions is yet to be given for N = 2 supergravity, but it seems that this result is generic. At any rate, this observation seems in line with more recent findings [33,34] based on heterotic string α′-corrections encoded in a higher-derivative effective action in higher 2 Here one usually assumes that F (Y,Υ) can be expanded in positive powers of Υ. dimensions. In four dimensions this action leads to additional matter-coupled higher-derivative in- teractions. When these are taken into account, the matching of the macroscopic entropy with the microscopic result is established [34]. Another modification concerns possible non- holomorphic corrections to the function F (X,A). This holomorphic function leads to a supersym- metric action that corresponds to the so-called effective Wilsonian action, based on integrating out the massive degrees of freedom. The Wilso- nian action describes the correct physics for en- ergies between appropriately chosen infrared and ultraviolet cutoffs. However, this action does not reflect all the physical symmetries. To preserve those symmetries non-holomorphic contributions should be included associated with integrating out massless degrees of freedom. In the special case of heterotic black holes in N = 4 supersym- metric compactifications, the requirement of ex- plicit S-duality invariance of the entropy and the attractor equations allows one to determine the contribution from these non-holomorphic correc- tions, as was first demonstrated in [31] for BPS black holes. In [35,18] it was established that non-holomorphic corrections to the BPS entropy function (44) can be encoded into a real function Ω(Y, Ȳ ,Υ, Ῡ) which is homogeneous of second degree. The modifications to the entropy func- tion are then effected by substituting F (Y,Υ) → F (Y,Υ) + 2iΩ(Y, Ȳ ,Υ, Ῡ). There are good rea- sons to expect that this same substitution should be applied to the more general entropy function based on (23) [25]. Note that when Ω is harmonic, i.e., when it satisfies ∂2Ω/∂Y I ∂Ȳ J = 0, it can simply be absorbed into the original holomorphic funtion F (Y,Υ). Finally we point out the existence of a formu- lation in terms of real, rather than the complex fields (Y I ,Υ), that we used before. This for- mulation is manifestly covariant with respect to electric/magnetic duality. We first decompose Y I and FI into their real and imaginary parts, Y I = xI + iuI , FI = yI + ivI , (50) where FI = FI(Y,Υ). The real parametriza- tion is obtained by taking (xI , yI ,Υ, Ῡ) instead of (Y I , Ȳ I ,Υ, Ῡ) as the independent variables. 16 B. de Wit This reparametrization is only well defined pro- vided det(NIJ) 6= 0. Subsequently one defines the Hesse potential, the real analogue of the Kähler potential, which equals twice the Legendre trans- form of the imaginary part of the prepotential with respect to uI = ImY I , H(x, y,Υ, Ῡ) = 2 ImF (x+ iu,Υ, Ῡ)− 2 yI uI , Owing to the homogeneity of the function F (Y,Υ) one can show that the free energy (45) equals twice the Hesse potential. The entropy function (44) is now replaced by Σ(x, y, p, q) = 2H(x, y,Υ, Ῡ)− 2 qIxI + 2 pIyI , and it is straightforward to show that the ex- tremization equations are just the attractor equa- tions (35), expressed in terms of the new variables (xI , yI). The value of Σ(x, y, p, q) at the attrator point coincides again with the macroscopic en- tropy. 6.2. Partial Legendre transforms It is, of course, possible to define the macro- scopic entropy as a Legendre transform with re- spect to only a subset of the fields, by substitut- ing part of the attractor equations such that the variational principle remains valid. These par- tial Legendre transforms constitute a hierarchy of Legendre transforms for the black hole entropy. Here we discuss two relevant examples, namely, the one proposed in [36] where all the magnetic attractor equations are imposed, and the dila- tonic one for heterotic black holes, where only two real potentials are left which together constitute the complex dilaton field [31]. A possible dis- advantage of considering partial Legendre trans- forms is that certain invariances may no longer be manifest. As it turns out, the dilatonic for- mulation does not suffer from this shortcoming. Apart from that, there is no reason to prefer one version over the other. This will change in sec- tion 7 when we discuss corresponding partition functions and inverse Laplace transforms for the microscopic degeneracies in semiclassical approx- imation. Let us first impose the magnetic attractor equa- tions so that only the real parts of the Y I will re- main as independent variables. Hence one makes the substitution, Y I = 1 (φI + ipI) . (53) The entropy function (44) then takes the form, Σ(φ, p, q) = FE(p, φ,Υ, Ῡ)− qI φI , (54) where the corresponding free energy FE(p, φ) equals FE(p, φ) = ImF (Y,Υ) + Ω(Y, Ȳ ,Υ, Ῡ) Y I=(φI+ipI)/2 To derive this result one makes use of the homogeneity of the functions F (Y,Υ) and Ω(Y, Ȳ ,Υ, Ῡ). The latter function may contain possible non-holomorphic terms. When extrem- izing (55) with respect to φI we obtain the at- tractor equations qI = ∂FE/∂φI . This shows that the macroscopic entropy is a Legendre trans- form of FE(p, φ) subject to Υ = −64, as was first noted in [36] in the absence of Ω. In the latter case this Legendre transform led to the conjecture that there is a relation with topological strings, in view of the fact that exp[FE] equals the modulus square of the topological string partition function [37]. We return to this in subsection 7.1. Along the same line one can now proceed and eliminate some of the φI as well. A specific exam- ple of this is the dilatonic formulation heterotic black holes, where we eliminate all the φI with the exception of two of them which parametrize the complex dilaton field. This leads to an en- tropy function that depends only on the charges and on the dilaton field [31,35]. Here it is conve- nient to include all the Υ-dependent terms into Ω, which also contains the non-holomorphic cor- rections. The heterotic classical function F (Y ) is given by F (Y ) = −Y 1 Y aηabY , a = 2, . . . , n, (56) with real constants ηab. In the application that we will be considering the function Ω depends only BPS Black Holes 17 linearly on Υ and Ῡ, as well as on the dilaton field S = −iY 1/Y 0 and its complex conjugate S̄. The result for the BPS entropy function then takes the form, Σ(S, S̄, p, q) = −q 2 − ip · q (S − S̄) + p2 |S|2 S + S̄ + 4Ω(S, S̄,Υ, Ῡ) , where q2, p2 and p · q are T-duality invariant bi- linears of the various charges, defined by q2 = 2q0p 1 − 1 abqb , p2 = −2p0q1 − 2paηabpb , q · p = q0p0 − q1p1 + qapa . Note that these bilinears are not positive defi- nite. Furthermore, Ω captures the Υ-dependent corrections to the classical result (56). Its form was derived for N = 4 heterotic string compact- ifications by requiring S-duality of the attractor equations and of the entropy [31,35], Ω(Y, Ȳ ,Υ, Ῡ) = Υ log η24(S) + Ῡ log η24(S̄) (Υ + Ῡ) log(S + S̄)12 where η(S) denotes the Dedekind function. Note the presence of the last term which is non- holomorphic. This term is in accord with the result for the effective action obtained from five-brane instantons [38]. The attractor equa- tions associated with the dilaton take the form ∂SΣ(S, S̄, p, q) = 0. It is interesting to consider the consequences of (57) in the classical case (Ω = 0). Then the attractor equations yield the following values for the real part of the dilaton field and the macro- scopic entropy, S + S̄ = − 2 p2q2 − (p · q)2 , Smacro = π p2q2 − (p · q)2 . Obviously there are two types of black holes, de- pending on whether p2q2 − (p · q)2 is positive or zero. In the context of N = 4 heterotic string compactifications these correspond to 1/4- and 1/2-BPS black holes, respectively. The 1/4-BPS states are dyonic, so that they necessarily carry both electric and magnetic charges. The 1/2-BPS states can be purely electric. We derived these results from the BPS entropy function, but they have also been obtained directly from the full su- pergravity solutions [39,40,41]. Clearly the 1/4-BPS black holes are large black holes as their area (entropy) is nonzero and scales quadratically with the charges. Note that, de- pending on the choice of the charges, the com- plex dilaton field can remain finite in the limit of large charges. This is relevant when studying the asymptotic growth of the dyonic degeneracy of 1/4-BPS dyons in heterotic string theory com- pactified on a six-torus and in the related class of heterotic CHL models [42]. These degeneracies are encoded in automorphic forms Φk(ρ, σ, υ) of weight k under Sp(2,Z), or an appropriate sub- group thereof [43,44]. The torus compactifica- tion corresponds to k = 10 and the CHL mod- els to k = 1, 2, 4, 6. The three modular param- eters, ρ, σ, υ, parametrize the period matrix of an auxiliary genus-two Riemann surface, which takes the form of a complex, symmetric, two-by- two matrix. The microscopic degeneracy of 1/4- BPS dyons is expressed as an integral over an appropriate 3-cycle, dk(p, q) = dρ dσ dυ eiπ[ρ p 2+σ q2+(2υ−1) p·q] Φk(ρ, σ, υ) Since Φk has zeros in the interior of the Siegel half-space in addition to the zeros at the cusps, the value of the integral (61) depends sensitively on the choice of the integration 3-cycles. The charges are in general integer, with the exception of q1 which equals a multiple of N , and p 1 which is fractional and quantized in units of 1/N . Here N and k are related by (k+2)(N +1) = 24. The inverse of the modular form Φk takes the form of a Fourier sum with integer powers of exp[2πiρ] and exp[2πiυ] and fractional powers of exp[2πiσ] which are multiples of 1/N . The 3-cycle is then defined by choosing integration contours where the real parts of ρ and υ take values in the interval (0, 1) and the real part of σ takes values in the 18 B. de Wit interval (0, N). The formula (61) is manifestly invariant under T-duality (the integrand depends on the three T-duality invariant bilinears (58)), as well as under S-duality, which is a subgroup of the full modular group. The integral (61) can be evaluated in saddle- point approximation which yields the leading and subleading contribtions to dk(p, q) [35,44]. As it turns out these contributions are precisely en- coded in (57) and (59), including the Dedekind eta-functions and the non-holomorphic terms. The presence of the non-holomorphic terms is not surprising in view of the S-duality invariance. Note that the expression (59) refers to k = 10 and that there exist similar formulae for k = 1, 2, 4, 6. The 1/2-BPS black holes are small black holes as their area scales linearly with the charges. Ac- cording to (60) their entropy (and horizon area) vanishes while the dilaton field diverges at the horizon, because we have p2 = p · q = 0. To de- scribe the situation more accurately one retains the leading term of (59). In that case one ob- tains the following result (we restrict ourselves to k = 10) [31], S + S̄ ≈ |q2|/2 , Smacro ≈ 4 π |q2|/2− 6 log |q2| , where the logarithmic term is due to the non- holomorphic contribution. Because the dilaton is large in this case, all the exponentials in the Dedekind eta-function are suppressed and we are at weak string coupling gs ∝ (S + S̄)−1/2. We already stressed that small black holes have a size of the order of the string scale and, in- deed, these states are precisely generated by per- turbative heterotic string states arising in N = 4 supersymmetric compactifications to four space- time dimensions. In the supersymmetric right- moving sector these states carry only momen- tum and winding and contain no oscillations, whereas in the left-moving sector oscillations are allowed that satisfy the string matching condi- tion. The oscillator number is then linearly re- lated to q2. These perturbative states received quite some attention in the past [45]. Because the higher-mass string states are expected to be within their Schwarzschild radius, it was conjec- tured that they should have an interpretation as black holes.3 Their calculable level density, pro- portional to the exponent of 4π |q2|/2, implies a nonzero microscopic entropy for these black holes [46]. This result was confronted with explicit black hole solutions [47,48,49] based on standard super- gravity Lagrangians that are at most quadratic in derivatives, which have a vanishing horizon area. Based on the area law one thus obtains a vanishing macroscopic entropy. The fact that (48), which takes into account higher-derivative interactions, can nicely account for the discrep- ancies encountered in the classical description of the 1/2-BPS black holes, was first emphasized in [50,51]. Note also that, since the electric states correspond to perturbative heterotic string states, their degeneracy is known from string theory and given by d(q) = eiπσq η24(σ) ≈ exp |q2|/2− 27 log |q2| where the integration contour encircles the point exp(2πiσ) = 0. This large-|q2| approximation is based on a standard saddle-point approximation. Obviously the leading term of (63) is in agreement with (62, but beyond that there is a disagreement as the logarithmic corrections carry different co- efficients. This discrepancy may be regarded as a first indication that small black holes are not well understood (for a disucssion, see, for instance, [18]). Therefore we will mainly concentrate on large black holes in the next chapter. 7. PARTITION FUNCTIONS AND IN- VERSE LAPLACE TRANSFORMS To again make the connection with microstate degeneracies, we conjecture, in the spirit of [36], that the Legendre transforms of the entropy are indicative of a thermodynamic origin of the vari- ous entropy functions. It is then natural to as- sume that the corresponding free energies are 3 The idea that elementary particles, or string states, are behaving like black holes, has been around for quite some time. BPS Black Holes 19 related to black hole partition functions corre- sponding to suitable ensembles of black hole mi- crostates. Following [18], we define Z(φ, χ) = {p,q} d(p, q) eπ[qIφ I−pIχI ] , (64) where d(p, q) denotes the microscopic degenera- cies of the black hole microstates with black hole charges pI and qI . This is the partition sum over a canonical ensemble, which is invariant un- der the various duality symmetries, provided that the electro- and magnetostatic potentials (φI , χI) transform as a symplectic vector. Identifying a free energy with the logarithm of Z(φ, χ), it is clear that it should, perhaps in an appropriate limit, be related to the macroscopic free energy introduced earlier. On the other hand, viewing Z(φ, χ) as an analytic function in φI and χI , the degeneracies d(p, q) can be retrieved by an inverse Laplace transform, d(p, q) ∝ dχI dφ I Z(φ, χ) eπ[−qIφ I+pIχI ] , where the integration contours run, for instance, over the intervals (φ − i, φ + i) and (χ − i, χ + i) (we are assuming an integer-valued charge lat- tice). Obviously, this makes sense as Z(φ, χ) is formally periodic under shifts of φ and χ by mul- tiples of 2i. These arguments suggest to identify Z(φ, χ) with the Hesse potential (51), {p,q} d(p, q) eπ[qIφ I−pIχI ] ∼ shifts e2πH(φ/2,χ/2,Υ,Ῡ) , where Υ is equal to its attractor value and where we suppressed possible non-holomorphic contri- butions for simplicity. However, the Hesse po- tential is a macroscopic quantity which does not in general exhibit the periodicity that is charac- teristic for the partition function. Therefore, the right-hand side of (66) requires an explicit peri- odicity sum over discrete imaginary shifts of the φ and χ.4 When substituting 2πH into the inverse Laplace transform, we expect that the periodic- ity sum can be incorporated into the integration contour. It is in general difficult to find an explicit repre- sentation for the Hesse potential, as the relation (50) between the complex variables Y I and the real variables xI and yI is complicated. Therefore we rewrite (66) in terms of the original variables Y I and Ȳ I , where explicit results are known, {p,q} d(p, q) eπ[qI (Y+Ȳ ) I−pI(F̂+ ˆ̄F )I ] ∼ shifts eπF(Y,Ȳ ,Υ,Ῡ) . Here F equals the free energy (45) suitably mod- ified with possible non-holomorphic corrections. The latter requires that FI is changed into F̂I = FI + 2iΩI , as was demonstrated in [18]. It is im- portant to note that both sides of (67) (as well as of (66)) are manifestly consistent with duality. Again, it is possible to formally invert (67) by means of an inverse Laplace transform, d(p, q) ∝ d(Y + Ȳ )I d(F̂ + ˆ̄F )I e πΣ(Y,Ȳ ,p,q) dY dȲ ∆−(Y, Ȳ ) eπΣ(Y,Ȳ ,p,q) , where ∆−(Y, Ȳ ) is an integration measure whose form depends on F̂I+ ˆ̄F I . The expression for ∆ as well as for a related determinant ∆+, reads as follows, ∆±(Y, Ȳ ) = ImFKL + 2Re(ΩKL ± ΩKL̄) As before, FIJ and FI refer to Y -derivatives of the holomorphic function F (Y,Υ) whereas ΩIJ and ΩIJ̄ denote the holomorphic and mixed holomorphic-antiholomorphic second derivatives 4In case that the Hesse potential exhibits a periodicity with a different periodicity interval, then the sum over the imaginary shifts will have to be modded out appropriately such as to avoid overcounting. 20 B. de Wit of Ω, respectively. In the absence of non- holomorphic corrections ∆+ = ∆−. A priori it is not clear whether the integral (68) is well-defined and we refer to [18] for a discus- sion. Note that the periodicity sum in (68) is defined in terms of the variables φ and χ, which should have some bearing on the integration con- tour in (68). Leaving aside these subtle points one may consider a saddle-point approximation of the intergral representation (68). In view of the previous results it is clear that the saddle point coincides with the attractor point. Subsequently one evaluates the semiclassical Gaussian integral that emerges when expanding the exponent in the integrand to second order in δY I and δȲ I about the attractor point. When Y I − Ȳ I = ipI , the resulting determinant factorizes into the square roots of two subdeterminants, ∆+ and Here the plus (minus) sign refers to the contribu- tion of integrating over the real (imaginary) part of δY I . Consequently, the result of a saddle-point approximation applied to (68) yields, d(p, q) = ∆−(Y, Ȳ ) ∆+(Y, Ȳ ) attractor eSmacro(p,q) . (70) In the absence of non-holomorphic corrections the ratio of the two determinants is equal to unity and one recovers precisely the macroscopic en- tropy. In the presence of non-holomorphic terms, the deviations from unity are usually suppressed in the limit of large charges, and one recovers the leading and subleading corrections to the entropy [18]. Of course, this is only the case when the saddle-point approximation is appropriate. Alternatively one may choose to integrate only over the imaginary values of the fluctuations δY I in saddle-point approximation. The saddle point then occurs in the subspace defined by the mag- netic attractor equations, so that one obtains a modified version of the OSV integral [36], d(p, q) ∝ ∆−(p, φ) eπ[FE(p,φ)−qIφ I ] , where FE(p, φ) was defined in (55) and ∆−(p, φ) is defined in (69) with the Y I given by (53). Hence this integral contains a non-trivial integra- tion measure factor ∆− in order to remain con- sistent with electric/magnetic duality. Without the integral measure this is the integral conjec- tured in [36]. In view of the original setting in terms of the Hesse potential, we expect that the integration contours in (71) should be taken along the imaginary axes. Inverting (71) to a partition sum over a mixed ensemble, one finds, Z(p, φ) = d(p, q) eπ qIφ shifts ∆−(p, φ) eπFE(p,φ) . It should be noted that this expression and the preceding one is less general than (68) because it involves a saddle-point approximation. More- over the function FE is not duality invariant and the invariance is only recaptured when complet- ing the saddle-point approximation with respect to the fields φI . Therefore an evaluation of (71) beyond the saddle-point approximation will most likely give rise to a violation of (some of) the du- ality symmetries again. 7.1. The integration measure and the mixed partition function As mentioned earlier, the partition function Z(p, φ) was originally conjectured to be equal to the modulus square of the partition function of the topological string [36]. Soon thereafter, how- ever, it was realized that this relationship must be more subtle. The arguments in [18], based on electric/magnetic duality clearly indicate some of the missing ingredients, resulting in the mea- sure factor ∆− and in the presence of the non- holomorphic corrections. In fact, it was already clear from an early analysis of small heterotic black holes that T-duality was not conserved when straightforwardly applying these ideas [52]. Although the presence of the measure factor cor- rects for the lack of duality invariance, the semi- classical results for small black holes seem to re- main inconsistent with the analysis of microstate counting, a fact we already alluded to at the end of subsection 6.2. BPS Black Holes 21 It is possible to test the result (72) in the context of the 1/4-BPS states of the heterotic N = 4 supersymmetric string compactifications, by making use of the degeneracy formula (61). Such a calculation was first performed in [53] and it was subsequently generalized in [18] for more general charge configurations and for CHL mod- els. Using the degeneracies (61) one calculates the mixed partition sum on the left-hand side of (72). As it turns out, the resulting expression is indeed proportional to the square of the partition function of the topological string, |Ztop(p, φ)|2 = e−2πi[F (Y,Υ)−F̄(Ȳ ,Ῡ] , (73) where F (Y,Υ) is now the holomorphic part of (56) and (59), F (Y,Υ) = −Y 1 Y aηabY log η24(−iY 1/Y 0) . Here Υ = −64 and the Y I are given by (53).5 However, there is a non-trivial proportionality factor, which, up to subleading contributions, we expect to coincide with ∆−(p, φ) exp[4πΩnonholo] . (75) The expression for Ωnonholo follows from the last term in (59), and is thus equal to exp[4π Ωnonholo] = (S + S̄) −12 , (76) where S + S̄ = 2(p1φ0 − p0φ1) (φ0)2 + (p0)2 . (77) The factor (S + S̄)−12 cancels against a similar factor in ∆− and, up to subleading terms, (75) becomes ∆−(p, φ) exp[4πΩnonholo] ≈ i(Ȳ I F̂I − Y I F̂I) 2 |Y 0|2 e−K(Y,Ȳ ,Υ,Ῡ) 2 |Y 0|2 Indeed this result coincides with the result found in [53,18]. The expression for e−K was already de- fined in (38) up to non-holomorphic corrections. 5For convenience, we only refer to the case k = 10. The latter can be dropped as they are subleading. Note that e−K has been rescaled by replacing the XI and A by Y I and Υ, respectively. This is to be expected in view of the fact that the right- hand side of (78) must be invariant under such rescalings. However, we should discuss a subtlety related to the fact that we derived the expression for√ ∆− in the context of N = 2 supergravity, whereas the evaluation based on (61) is based on N = 4 supersymmetric compactifications. This means that the number of scalar moduli (related to the number of N = 2 vector multiplets) is not obviously the same. In the case ofN = 4 the rank of the gauge group is equal to 28 (for simplicity we restrict ourselves to the case k = 10). Of the 28 abelian vector gauge fields, 6 will correspond to the graviphotons of pure N = 4 supergrav- ity, and 22 will each belong to an N = 4 vector supermultiplet. In the reduction to N = 2 super- gravity, one of the graviphotons will be contained in an additional N = 2 vector supermultiplet and another one will play the role of the gravipho- ton of N = 2 supergravity. There are thus 24 abelian vector gauge fields, of which 23 are asso- ciated with N = 2 vector multiplets and one is associated with the N = 2 graviphoton. The re- maining 4 graviphotons are associated with two N = 2 gravitino supermultipets, and these vector fields seem to have no place in the N = 2 de- scription. Therefore it is often assumed that the effective rank of the gauge group in the N = 2 description should be taken equal to 24, rather than to 28. Nevertheless, in the calculation of ∆− leading to (78) we assumed that the N = 2 description is based on 28 vector fields, corresponding to 27 vector supermultiplets and one graviphoton field. Only in that case, the factor (S + S̄)−12 cancels so that one obtains the proportionality constant noted in (78) based on the N = 2 expression for√ ∆−. This somewhat confusing issue seems en- tirely due to the fact that the N = 2 description of an N = 4 theory is not fully understood, as the corresponding description of N = 4 supergravity is indeed based on 28 electrostatic potentials φ which are contained in 28, rather than 24, ana- logues of the N = 2 quantities Y I . 22 B. de Wit In a recent series of papers [54,55,56] progress was made towards a further understanding of the relation between the mixed black hole partition function and the partition function of the topo- logical string. Unfortunately no evidence for the presence of the integration measure factor in (71) and (72) was presented. However, a more de- tailed analysis for compact Calabi-Yau [57] mod- els subsequently revealed the presence of the mea- sure factor. 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Moore, hep-th/0702146. http://arxiv.org/abs/hep-th/9602060 http://arxiv.org/abs/hep-th/9603191 http://arxiv.org/abs/hep-th/9812082 http://arxiv.org/abs/hep-th/9904005 http://arxiv.org/abs/hep-th/9906094 http://arxiv.org/abs/hep-th/0606148 http://arxiv.org/abs/hep-th/0607094 http://arxiv.org/abs/hep-th/0608182 http://arxiv.org/abs/hep-th/0412287 http://arxiv.org/abs/hep-th/0405146 http://arxiv.org/abs/hep-th/9309140 http://arxiv.org/abs/hep-th/9610237 http://arxiv.org/abs/hep-th/9507090 http://arxiv.org/abs/hep-th/9512031 http://arxiv.org/abs/hep-th/9605059 http://arxiv.org/abs/hep-th/9505054 http://arxiv.org/abs/hep-th/9607026 http://arxiv.org/abs/hep-th/0510147 http://arxiv.org/abs/hep-th/9405117 http://arxiv.org/abs/hep-th/9411187 http://arxiv.org/abs/hep-th/9504147 http://arxiv.org/abs/hep-th/9506200 http://arxiv.org/abs/hep-th/0409148 http://arxiv.org/abs/hep-th/0410076 http://arxiv.org/abs/hep-th/0507014 http://arxiv.org/abs/hep-th/0508174 http://arxiv.org/abs/hep-th/0602046 http://arxiv.org/abs/hep-th/0608021 http://arxiv.org/abs/hep-th/0608059 http://arxiv.org/abs/hep-th/0702146 INTRODUCTION DUAL PERSPECTIVES BLACK HOLES IN M/STRING THEORY – AN EXAMPLE ATTRACTOR EQUATIONS N=2 SUPERGRAVITY Supermultiplets Supersymmetric actions Compensator multiplets BPS ATTRACTORS The BPS entropy function Partial Legendre transforms PARTITION FUNCTIONS AND INVERSE LAPLACE TRANSFORMS The integration measure and the mixed partition function

0704.1453

Structure factors of harmonic and anharmonic Fibonacci chains by molecular dynamics simulations

Structure factors of harmonic and anharmonic Fibonacci chains by molecular dynamics simulations Michael Engel,∗ Steffen Sonntag, Hansjörg Lipp, and Hans-Rainer Trebin Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany The dynamics of quasicrystals is characterized by the existence of phason excitations in addition to the usual phonon modes. In order to investigate their interplay on an elementary level we resort to various one-dimensional model systems. The main observables are the static, the incoherent, and the coherent structure factor, which are extracted from molecular dynamics simulations. For the validation of the algorithms, results for the harmonic periodic chain are presented. We then study the Fibonacci chain with harmonic and anharmonic interaction potentials. In the dynamic Fibonacci chain neighboring atoms interact by double-well potentials allowing for phason flips. The difference between the structure factors of the dynamic and the harmonic Fibonacci chain lies in the temperature dependence of the phonon line width. If a bias is introduced in the well depth, dispersionless optic phonon bands split off. PACS numbers: 63.20.Ry, 61.44.Br, 02.70.Ns Keywords: Phonons; Phason flip; Quasicrystal. I. INTRODUCTION A. Motivation Quasicrystals are long-range ordered materials lacking translational symmetry1. Their diffraction patterns ex- hibit a dense set of sharp Bragg reflections, that can be indexed by an integer linear combination of a finite number of basis vectors which is larger than the dimen- sion of space. As a consequence the atomic configuration of quasicrystals is described with reference to a higher- dimensional analog of a periodic lattice. Elementary dy- namic excitations within this ‘hyperspace’ description are phonons2 and phasons3. Phasons involve rearrangements of the structure by atomic jumps over short distances, denoted ‘phason flips’. They are connected with many physical proper- ties of quasicrystals as for example elastic deformations4, dislocations5,6, diffusion7,8, and phase transformations9. Recently, indications for phason flips10,11 have been ob- served by in situ transmission electron microscopy. A coherent set of phason flips may form a static phason field, e.g. during a phase transformation or in the neigh- borhood of a dislocation12. By investigating the dynamics of quasicrystals one can find out the influence of the quasiperiodicity on the phonon spectrum2 and one may gain a deeper under- standing of phason flips13. Both points can be studied in x-ray or neutron diffraction experiments by measur- ing the response of the system in frequency (ω) and momentum (q) space. Depending on the experimental setup, different functions can be obtained from the scat- tering experiments: (1) The static structure factor S(q) is the usual – not energy resolved – diffraction image, measured with either x-rays or neutrons. It is used for the determination of the atomic structure. (2) The co- herent structure factor14 S(q, ω) is studied via coherent inelastic neutron scattering15 or alternatively via inelas- tic x-ray scattering16. It allows the determination of the phonon dispersion relations. The experiments on icosa- hedral quasicrystals show well defined acoustic phonon modes at small wave-vectors17 and dispersionless broad optic bands at larger wave vectors18. The cross-over be- tween the two regions is very sharp. (3) The incoherent structure factor14 Si(q, ω) can be measured in quasielas- tic neutron scattering. Neutrons are exclusively used here, due to the necessity of a high energy resolution. The technique also allows the investigation of phason flips. In a series of experiments Coddens et al.13,19,20,21 have found an anomalous q-dependence of the quasielas- tic signal in icosahedral quasicrystals. They interpreted it as correlated simultaneous jumps of several atoms.21 Up to now various calculations of the coherent struc- ture factor of quasiperiodic model systems have been published, see Ref. 2. Amongst them are the perfect one- dimensional Fibonacci chain22, by static phason fields disordered Fibonacci chains23, and three-dimensional tilings24. In these studies the dynamical matrix is diago- nalized, which is a purely analytic method and yields the phonon dispersion relations only. The results are highly structured excitation spectra with a hierarchical system of gaps22. The influence of anharmonicities, however, es- pecially the dynamics of phason flips has not been taken into account. This ‘missing link’ markes the starting point of our study. Here we present calculations of the structure fac- tors of special one-dimensional quasiperiodic model sys- tems by use of molecular dynamics (MD) simulations with either harmonic potentials or potentials that allow for phason flips. Although structure factors play such a central role in the dynamics of solids, not much seems to be known about their exact forms for one-dimensional chains. Even for the simple harmonic chain only few ar- ticles exist25,26,27,28. FIG. 1: Double-well potential V (x) = x4−2x2 of the dynamic Fibonacci chain. The equilibrium distances are S and L. B. Model systems As a simple one-dimensional model for a quasiperiodic system we consider the Fibonacci chain. It consists of particles arranged at two different distances: large ones (L) and small ones (S). The length ratio L/S equals the number of the golden mean τ = 1 5 + 1). The sequence of the distances is created recursively by the mapping {L, S} 7→ {LS,L} with starting condition L. For example, after four iterations the resulting sequence is LSLLSLSL. We want to study chains consisting of identical parti- cles with nearest-neighbor interactions. The Hamiltonian has the form + V (xj − xj+1 − aj), (1) where xj and pj are position and momentum of the jth particle. The dynamic Fibonacci chain (DFC) is defined by the choices aj ≡ a0 = τ3 for the equilibrium distances and V (x) = x4 − 2x2 for the interaction potential, re- spectively. The latter forms a double-well potential with minima at ±1 and a potential hill of height ∆E = 1 as shown in Fig. 1. Because neighboring particles sit in either of the potential minima, two nearest-neighbor dis- tances L = a0 + 1 and S = a0 − 1 are possible. They fulfil the constraint L/S = τ of the Fibonacci chain. The DFC shows two types of elementary excitations: Phonon vibrations in the local minima and phason flips that interchange the particle distances L and S. At low temperatures only phonons are excited, phason flips have to be activated thermally. With its neighbors at rest the activation energy of a particle for a phason flip is 2∆E. This value is a result of the perfect superposition of the potential hills of both neigbors. The value is lowered when the neighbors assist by stepping simultaneously to the inside or outside during the phason flip thus creating a non-perfect superposition of the potential hills. Since the particle distances L and S are energetically degen- erate, the total equilibrium potential energy is invariant under a phason flip. The occurrence of phason flips makes nonlinearity an intrinsic feature of the DFC and an analytical treatment of the dynamics impossible. To understand the influence of the nonlinearity, we study four model systems with increasing complexity concerning their dynamical behav- • Harmonic periodic chain (HPC): VHPC(x) = 4x2 and aj = a = 2 • Harmonic Fibonacci chain (HFC): VHFC(x) = 4x2 and aj = L or S according to the Fibonacci sequence. • Dynamic Fibonacci chain (DFC): VDFC(x) = x4 − 2x2 and aj = τ3. • Asymmetric Fibonacci chain (AFC): VAFC(x) = VDFC +χ(x2−1)2(�x+x2/2−1/2) and aj = τ3 with χ ∈ [0, 1] and � = ±1. The potentials of the HPC, HFC, and DFC are chosen to be identical in the harmonic approximation around the equilibrium separation. The average particle distance a is the same for all four systems. In the case of the Fibonacci chain the occurrence probabilities for L and S are given by τ−1 and τ−2, hence a = 2 For the solution of the equations of motions we use a special molecular dynamics (MD) code. The code is introduced in Sec. II together with a short theoretical background. The simplest system is, of course, the HPC. Exact solutions for the equation of motion exist as a su- perposition of plane waves. We study the dynamics of the HPC in Sec. III as a reference system. The HFC con- sists of particles arranged on the Fibonacci chain with distances L and S interacting with the same harmonic potentials as the HPC, see Sec. IV. The DFC will then be studied in Sec. V. In the case of the AFC the particles in the two potential minima have different eigenfrequen- cies. The parameters χ and � determine the degree of asymmetry. For more details we refer to Sec. VI. We finish with a discussion and conclusion in Sec. VII. II. STRUCTURE FACTORS FROM MOLECULAR DYNAMICS A. Definition of the structure factors We write the particle number density of the chain with N particles as a sum of delta functions positioned along the particle trajectories xl(t), n(x, t) = l=1 δ(x−xl(t)). The time dependent density-density correlation function and the density-density autocorrelation function are de- fined as G(x, t) = 〈n(x′, t)n(x+ x′, 0)〉 dx′ 〈δ(x− xj(t) + xl(0))〉 , (2a) Ga(x, t) = 〈δ(x− xl(t) + xl(0))〉 , (2b) where the brackets denote the thermal average.29 The coherent and incoherent structure factor are the space- time Fourier transforms30, S(q, ω) = e−iωt e−iqxj(t)eiqxl(0) Si(q, ω) = e−iωt e−iqxl(t)eiqxl(0) Both functions are symmetric about q = 0 and ω = 0. The static structure factor is the integral of the coher- ent structure factor, S(q) = S(q, ω) dω, i.e. the Fourier transform of G(x, 0), S(q) = e−iqxj(0)eiqxl(0) . (4) B. Molecular dynamics simulations For further calculations the particle trajectories are re- quired as solutions of the equations of motion. Since in the case of the anharmonic chains only numerical solu- tions exist, we use a simple MD code. Initially the par- ticles are placed on the equilibrium positions of a finite chain of length L with periodic boundary conditions. The velocities are initialized according to a Gaussian distribu- tion. Its width determines the total energy and thus the temperature of the system. The equations of motion are integrated by a Verlet algorithm running for a simulation time Tsim. After starting the simulation, the dynamics is not controlled by a thermostat or in any other way. For the direct numerical calculation of the Eqs. (3) we must compute a fourfold sum: two sums over the par- ticle number N and two over the time Tsim, one sum for the Fourier transform and one for the time average. Note, that by assuming ergodicity the thermal average 〈 〉 can be replaced by a time average 1 dt and ad- ditionally by an average over several independent MD runs. For the sake of clarity the averaging over the MD runs is suppressed in the following notation. We intro- duce a more compact notation by defining the functions fl(q, t) = eiqxl(t). Let us assume tentatively that these functions are periodic in time with period Tsim and in space with period L. Then the Eqs. (3) and (4) are greatly simplified to S(q, ω) = 2πNTsim ∥∥∥∥∥ e−iωt fl(q, t) dt ∥∥∥∥∥ , (5a) Si(q, ω) = 2πNTsim e−iωtfl(q, t) dt ∥∥∥∥2 , (5b) S(q) = NTsim ∫ ∥∥∥∥∥∑ fl(q, t) ∥∥∥∥∥ dt. (6) The equations differ in the order of the absolute square and the particle sum. Since only two sums are left, an efficient numerical computation of the structure factors is possible. Furthermore a fast Fourier transform is used for the time integrals in Eqs. (5). It is left to discuss the periodicity conditions. The spa- tial periodicity follows from the periodic boundaries used in the simulation. Therefore the chain acts like a ring and the excitations can go round during the simulation. To avoid such a behavior that would lead to unwanted cor- relations, we limit the maximum simulation time by the quotient of the length of the chain and the sound velocity cs to Tmax = L/cs. For all the model systems HPC, HFC, and DFC the sound velocity is the same, cs = a 8, and Tmax = N/ 8. We use Tsim = Tmax. The functions fl(q, t) are in general not periodic in time. There is no reason, why the particles should be at the same positions at the end of the simulation as at the beginning. To avoid this problem, we multiply fl(q, t) with a window function w(t) to enforce an arti- ficial periodicity. The function w(t) has to decrease fast enough – both in direct as in Fourier space – towards the boundaries of its domains. We use a normalized broad Gaussian function. Its width is chosen as large as pos- sible with the constraint that the Gaussian has decayed to a small enough value at the interval boundaries. The effect of the Gaussian is a smoothing of the structure fac- tors by convolution with a narrow Gaussian. The exact value of the width has no influence on the results. III. HARMONIC PERIODIC CHAIN AS REFERENCE SYSTEM A. Analytic calculations The harmonic periodic chain (HPC) is used as a ref- erence system to test our algorithms since its equa- tions of motions can be solved analytically. If we put xl(t) = ul(t)+ la, then the ul(t) are expressed by a linear combination of normal modes. The wave vector q and the frequency ω are related according to the dispersion relation ω(q) = 2ω0| sin(qa/2)|. Here, w0 is the eigen- frequency of a single particle. In the case of the model systems HPC, HFC, and DFC we have ω0 = For the HPC the thermal averages in the structure factors, Eqs. (3) can be calculated to be S(q, ω) = e−iωt e−iqal exp q2σ2l (t) Si(q, ω) = e−iωt exp q2σ20(t) dt. (7b) where we used from the literature25 σ2l (t) = [ul(t)− u0(0)]2 ∫ 2ω0t J2l(s)(2ω0t− s) ds . (8) Jn(s) is the Bessel function of the first kind of order n. The only external parameter in these functions is the temperature T . The particle sum and the Fourier transform have to be evaluated numerically. Due to the translation invariance of the HPC the double sum of the Eqs. (3) is reduced to a single sum. The static structure factor of the HPC can also be calculated analytically26 S(q) = sinh(q2σ2/2) cosh(q2σ2/2)− cos(qa) , (9) where σ2 = kBT/ω20 . Let us take a closer look at the incoherent struc- ture factor Si(q, ω). In the limit of small T the term exp(− 1 q2σ20(t)) decays slowly with t and we substi- tute σ0(t) with its approximation for large t: σ0(t) = |t|kBT/ω0 for |t| → ∞. This leads to a Lorentzian peak Si(q, ω) = Γ2 + ω2 , Γ = q2kBT In the limit of large T the term σ0(t) is approximated for small t: σ0(t) = t2kBT for |t| � ω−10 . Hence there is a Gaussian peak Si(q, ω) = , γ = q2kBT . (11) The transition temperature between these two limiting cases is kBT = 4ω20/q The Fourier transform of the Eqs. (7) yields the corre- lation functions G(x, t) = σl(t) (x+ la)2 2σ2l (t) (12a) Ga(x, t) = σ0(t) 2σ20(t) . (12b) The function G(x, t) is shown in Fig. 2. It consists of a sum of Gaussians centered at the equilibrium positions of the particles. The width of the Gaussians increases with temperature, as well as with time and in space: limx,t→∞G(x, t) = ρ and ρ = 1/a. This means that there is no longe-range order. Indeed, for the particle number density we have 〈n(x)〉 = ρ, which is uniform as in liquids25. The autocorrelation function Ga(x, t) corre- sponds to the center peak at x = 0. B. Simulation results In the case of the incoherent structure factor the vari- ables T and q only appear as combination q2kBT in FIG. 2: Density-density correlation function G(x, t) of the HPC for kBT = 0.5. The Gaussians are centered at integer multiples of x = a ≈ 4.47. FIG. 3: Incoherent structure factor Si(q, ω) of the HPC for q = π/a, and different temperatures. The symbols mark the data from MD simulations with N = 1000 particles, the lines the result from the analytical fomula, Eq. (7b). The peaks and edges are at integer multiples of ω = 2ω0 = 2 Eqs. (7) and (8). Therefore it suffices to examine Si at a fixed wave vector for different temperatures. We choose q = π/a arbitrarily. The results from MD simulation and the numerical integration of the analytical formula, Eq. (7b) are shown in Fig. 3 for temperatures ranging from 0.01 to 100.0. There is a maximum at ω = 0, called the quasielastic peak. In the low temperature regime the maximum has a Lorentzian shape. Furthermore a one- phonon peak, a two-phonon edge, a three-phonon edge, etc. are found at 2ω0 ≈ 5.7, 4ω0 ≈ 11.3, 6ω0 ≈ 17.0.31 Note that the multi-phonon contributions rapidly decay for larger ω (logarithmic scale). At higher temperatures the curve smoothes and approaches a Gaussian profile. The coherent structure factor S(q, ω) is shown in Fig. 4. For a low temperature of kBT = 0.01 a one-phonon branch, a two-phonon branch, and very weakly a three- phonon branch are observed. The one-phonon branch has a Lorentzian line shape and follows the phonon- dispersion relation. At higher q-values the branch broad- ens with a width proportional to kBTq2. This is similar to the temperature behavior for the incoherent structure factor. The multi-phonon branches follow the modified relations ω(q) = 2nω0| sin(qa/2n)| with n = 2, 3. In Fig. 4(b) the comparison of the MD simulation (left side) and the analytical formula Eq. (7a) (right side) is shown. The temperature in this figure is kBT = 0.1, which is higher than in Fig. 4(a). As a consequence the one-phonon branch is broader. For both methods of calculating S(q, ω) a high accuracy over 12 orders of magnitude is possible. The accuracy is only limited by the internal floating point precision. The static structure factor from Eq. (9) is compared to the results from the MD simulation in Fig. 5. S(q) consists of a sequence of Lorentzian peaks at the recipro- cal lattice points. The increasing width for larger wave vectors shows again that no long-range order is present in the one-dimensional model system. The MD simulations and analytical formulas show a perfect agreement. Therefore we conclude that MD sim- ulations are a well-suited numerical tool for calculating the structure factors of the one-dimensional model sys- tems. Although we integrate the equations of motions with a good precision only on a short time scale using the simple Verlet-algorithm, the statistics extracted from the trajectories are correct. This confirms our approach and encourages us to proceed studying the phason dynamics of Fibonacci chains in the next section. IV. HARMONIC FIBONACCI CHAIN: INFLUENCE OF THE QUASIPERIODICITY By changing the interparticle equilibrium distances of the HPC to those of a Fibonacci sequence with separa- tions L and S we obtain the harmonic Fibonacci chain (HFC). The interaction potential is left unchanged. Since the incoherent structure factor is a function of the single particle motion only, it does not depend on the equilib- rium distances of the particles but only the interaction potential. Hence, the incoherent structure factor of the HPC and of the HFC are identical. For the coherent structure factor the interparticle distances become im- portant. Instead of Eq. (7a) we now have S(q, ω) = e−iωt e−iqx l exp −q2σ2l (t)/2 Here x0l = j=1 aj for l > 0, x j=l aj for l < 0 and x00 = 0 denote the equilibrium positions of the particles. For kBT = 0 this gives the Fourier transform of the static Fibonacci chain S(q, ω) = δ(ω) e−iqx l (14) which is well known32. It consists of a dense set of delta peaks with varying intensity, positioned at the reciprocal lattice points q = q0 (h+ τh ′) , h, h′ ∈ N (15) with q0 = 2πa ≈ 1.40. As shown in Fig. 6 for the temperature kBT = 0.02, the coherent structure factor S(q, ω) of the HFC consists of many different branches all following the one-phonon dispersion relation. Relative to each other the branches are displaced in the q direction. They start at the recip- rocal lattice points with the intensity of the respective delta peak. Two-phonon branches are also found. The broadening of the branches proportional to kBTq2 has already been discussed for the HPC. From these findings and Fig. 5 one can guess the form of the static structure factor S(q) for the HFC: Lorentzians are positioned at the reciprocal lattice points with varying intensity. In Fig. 7 S(q) is shown for q ∈ [0, 3π]. For small q a large number of Lorentzians occur. With increasing q-value their width increases and the stronger ones hide the weaker ones. Comparing the position of the strong peaks and ignoring the change of their widths and heights, there is a self-similarity in S(q). The deflation factor is ∆q1/∆q2 = τ3 as indicated in Fig. 7. It is interesting to note that there are regions with very few peaks. They are positioned around q = π, q = 2π, etc. . The same regions are also special for the coherent structure factor. As seen in Fig. 6 all the weak one- phonon branches vanish towards q = π. Only the strong one-phonon branch starting from the Bragg peak at q = τq0 ≈ 2.27 remains. V. DYNAMIC FIBONACCI CHAIN: OCCURRENCE OF PHASON FLIPS A. Phason flips Let us now proceed to the anharmonic chains with pha- son flips by investigating the dynamic Fibonacci chain (DFC). It is built from identical particles that interact with a symmetric double-well potential VDFC = x4−2x2. First the notion of a phason flip has to be specified. To do so we identify the position of the changes from L to S and from S to L of the interparticle distances along the particle trajectories. This is done in Fig. 8. Often a L → S change and a S → L change lie next to each other (particle distance 1) and the sequences LS and SL are interchanged. But there are also many cases where the positions of the two changes are separated by 0, 2, 3, or even more particle distances as marked by lines in Fig. 8. Sometimes it is not possible to find a partner lo- cally. Only in the long time average every L→ S change will eventually cancel with a S → L change. In the literature on the Fibonacci chain a phason flip is understood as the exchange of a L and a neighboring FIG. 4: (Color online) Coherent structure factor S(q, ω) of the HPC with N = 6500 particles from MD simulation. The temperatures are for kBT = 0.01 (a) and for kBT = 0.1 (b). One-, two- and three-phonon branches are observed. They start at the reciprocal lattice points 2πn/a. In (b) the output from the MD simulation (left side) is compared to the output of the analytical formula Eq. (7a) (right side). FIG. 5: Static structure factor S(q) of the HPC for kBT = 0.02. The symbols mark the data from a MD simulation with N = 1000 particles, the line the result from the analytical fomula, Eq. (9). S particle distance. As we learned above there are also other types of exchanges in the DFC. In the following we denote by phason flip every pair of flip partners as those connected by lines in Fig. 8. Note that the times and positions of the phason flips are not well defined. Only their number can be estimated by counting the changes in the particle distances as we will do now. The temperature dependence of the average phason flip frequency ωflip is shown in Fig. 9. Phason flips start to appear at about kBT = 0.1. At low kBT the average pha- son flip frequency increases rapidly by thermal excitation and ωflip � ω0. At higher temperatures kBT > 0.4 the average phason flip frequency slowly saturates. In this region the internal energy is comparable to the potential hill. B. Results for the structure factors For the anharmonic chains no analytic results are avail- able, in particular not at elevated temperatures when phason flips occur. The incoherent structure factors of the DFC and of the HFC/HPC differ remarkably and are shown in Fig. 10 for q = π/a. The comparison leads to the following conclusions: 1. At a fixed temperature, there are ω-ranges where the curve for the DFC lies below the curve for the HFC/HPC and vice versa. Since we have∫ Si(q, ω) dω = 1 from Eq. (3b), the integral area between the two curves has to vanish. 2. At very low kBT and ω < 2ω0 ≈ 5.7 the curves of the DFC and the HFC/HPC cannot be distin- guished in logarithmic scale except for two small bumps. They are a consequence of the anharmonic- ity of the interaction potential of the DFC and not related to the phason flips. At larger ω values the multi-phonon edges have different heights. 3. Above kBT = 0.1 the one-phonon peak and the multi-phonon edges in the curves for the DFC broaden and weaken considerably faster than in the curves for the HFC/HPC. They finally disappear at kBT = 1.0. 4. No additional peaks or edges occur at any temper- ature. Further MD runs show that different q-values change the temperature dependence, but the general features re- main unchanged: The phonon peaks and edges broaden and weaken much faster with increasing temperature for the DFC than for the HFC/HPC. Similar conclusions follow for the coherent structure factor of the DFC. S(q, ω) for the DFC looks qualitatively FIG. 6: (Color online) Coherent structure factor S(q, ω) of the HFC with N = 13000 particles for kBT = 0.02. It consists of a dense set of phonon branches starting from the reciprocal lattice points. FIG. 7: Static structure factor S(q) of the HFC from a MD simulation with 2000 particles at the temperature kBT = 0.02. similar to S(q, ω) for the HFC except that the branches broaden more quickly with temperature. To compare the broadening let us look at plane cuts through S(q, ω) for ω = 1.0 fixed. Three cuts for the temperature kBT = 0.05, 0.2, and 0.3 are shown in Fig. 11. A one-phonon branch is located inside the cut resulting in a peak with approximate Lorentzian line shape as indicated by the fits in the figure. Only for lower temperatures the line shape deviates from a Lorentzian, which is seen at the base of the peak for kBT = 0.05. The width of the Lorentzian in Fig. 11 is shown as a function of temperature in Fig. 12 for both, the HFC and the DFC. In the case of the HFC the width increases linearly with temperature, as discussed in Sec. IV. There is a deviation from linearity at low temperatures. This is an artefact from the method of calculation from MD simulation data. As explained in Sec. II B the structure factor is convoluted with a Gaussian due to the finite FIG. 8: Snapshot of the particle trajectories of the DFC at the temperature kBT = 0.6. Changes in the particle distances from L to S and S to L are marked with a cross (×) and a plus (+). A L → S change and a S → L change combined form a phason flip. FIG. 9: Average flip frequency as a function of the temper- ature kBT . In the temperature range of the figure: ωflip < ω0 ≈ 2.83. simulation time. The (narrow) Gaussian generates the observed offset. The convolution with the Gaussian is also responsible for the shape of the cuts in Fig. 11 at low temperature, deviating from the Lorentzian shape. There is one aspect of the DFC that has been ignored up to now. Due to the phason flips the original per- fect quasiperiodic long-range order is slowly decaying. With progressing simulation time the chain becomes ran- domized which, however, has no effect on the incoherent structure factor. To test the influence of the randomiza- tion on the coherent structure factor, a MD simulation was started with a totally randomized Fibonacci chain. The interaction potentials and the occurrence ratio of L and S were not adapted. The result of the simulation is shown in Fig. 13. Most of the details in S(q, ω) are lost by the randomization process and the branches are strongly FIG. 10: Incoherent structure factor Si(q, ω) of the DFC (solid) and the HFC/HPC (dashed) for q = π/a at different temperatures. The data was calculated using MD simulations with N = 1000 particles. FIG. 11: Cuts through S(q, ω) of the DFC for a fixed ω = 1.0 including a one-phonon peak. The solid curves are fits with Lorentzians. broadened, although the simulation has been carried out at a low temperature of kBT = 0.02. We summarize the results of this section: By the in- troduction of the anharmonic double-well potential of the DFC the phonon peaks, edges, and branches are strongly broadened and weakened with increasing tem- perature. There are two effects responsible for the broad- ening: (1) The anharmonicity and single phason flips affect the propagation of phonons. This is seen in the incoherent structure factor. (2) The destruction of the long-range order by a large number of phason flips. This is the main effect of broadening for the coherent structure factor. The phason flips appear more or less uniformly distributed over the simulation time and the particles. FIG. 12: Width of the Lorentzian peak of Fig. 11 as a function of temperature for the HFC (inset) and the DFC. FIG. 13: (Color online) Coherent structure factor S(q, ω) of a randomized Fibonacci chain with 2000 particles at the tem- perature kBT = 0.02. The same interaction potentials as in Fig. 6 have been used. VI. ASYMMETRIC FIBONACCI CHAIN: COMPETING EIGENFREQUENCIES A. Band gaps In the next step we modify the double well potential of the DFC. Like all the previous model systems the asym- metric Fibonacci chain (AFC) is built of identical par- ticles, but they interact with the more complicated po- tential VAFC = VDFC + ∆V , see Fig. 14. The additional term is given by ∆V (x) = χ(x2 − 1)2(�x+ x2/2− 1/2) (16) with χ ∈ [0, 1] and � = ±1. This term has been cho- sen in such a way that the positions of the minima at x = ±1 are left invariant, but the curvatures of the poten- tial at those points is changed to V ′′AFC(±1) = 8(1± �χ). FIG. 14: Interaction potential VAFC(x) of the AFC for differ- ent values of the parameter χ and � = 1. VDFC is shown with dashed lines. In harmonic approximation a particle will feel different eigenfrequencies, depending on the nearest-neighbor con- figuration SS, SL, LS, or LL. In the case of � = 1 it is ωLL > ωLS = ωSL > ωSS and for � = −1 it is ωLL < ωLS = ωSL < ωSS. The sign change of � corre- sponds to a mirror operation of VAFC about the y-axis. The coherent structure factor of the AFC with χ = 0.1, 0.3, 0.5 and � = ±1 is shown in Fig. 15. Band gaps of different widths appear and broaden with increasing val- ues for χ. They are positioned at the frequencies where one-phonon branches intersect each other. Their posi- tions and widths are different for � = 1 and � = −1. For � = 1 three large gaps and several smaller gaps ap- pear, whereas for � = −1 only one very large gap, one medium gap and several small gaps appear. The band gaps are a consequence of the competing eigenfrequen- cies due to the asymmetric potential in the same way as band gaps appear in periodic systems with several parti- cles and eigenfrequencies per unit cell. B. Density of states The gaps are seen more clearly in the density of states (DOS) D(ω). The DOS per particle is calculated from the velocity autocorrelation function by a Fourier trans- D(ω) = e−iωt 〈vl(t)vl(0)〉dt, ω ≥ 0. (17) It is obtained from MD simulation data similar to the incoherent structure factor: fl(q, t) has to be substited by vl(t) in Eq. (5b). By interchanging the Fourier transform and two time derivatives we can alternatively write D(ω) = e−iωt 〈xl(t)xl(0)〉dt. (18) FIG. 15: (Color online) Coherent structure factor S(q, ω) of the AFC for different values of χ and � from MD simulations with 6500 particles at kBT = 0.01. Band gaps are observed. The band gaps also appear in the DOS D(ω). After a taylor expansion of the exponentials in Eq. (3b) a connection to the incoherent structure factor is found, D(ω) = 2ω2 lim Si(q, ω) , ω 6= 0. (19) For comparison we present the DOS of a harmonic chain (HPC or HFC) D(ω) = 4ω20 − ω2 for 0 < ω < 2ω0 (20) and 0 elsewhere. The total number of states is normalized D(ω) dω = kBT . It can be checked, that the DOS of the harmonic chain fits well to the DOS of the AFC for � = 0.0, except small bumps that originate from the anharmonicity of the potentials. In Fig. 15 the DOS of the AFC for different values of χ and � are drawn. The band gaps appear at the same frequencies and the same widths as in the coherent structure factor. VII. DISCUSSION AND CONCLUSION At the end we would like to make some general re- marks concerning phason flips and phason modes: In the context of a hydrodynamic theory, phason flips can be as- sociated with phason modes. It was noted quite early33, that phason modes are diffusive, in contrary to the prop- agating phonons. This means that phason flips are only weakly coherent in space and in time, which of course we have also observed here. As a result their influence on the structure factor is small making it difficult though still interesting to study them by scattering experiments. An- other point concerns the connection of phason flips and quasiperiodicity. There is no reason why phason flips should only occur in quasicrystals. Since interaction po- tentials are not sensitive on the long-range order, phason flips in the form of atomic jumps can also occur in pe- riodic complex intermetallic phases, which is supported by recent experimental results34,35. In the case of our model systems, it is equally possible to compare simula- tions of a periodic LSLSLS . . . chain with harmonic and double-well potentials respectively. In conclusion, we have investigated the dynamics of phonons and phason flips in one-dimensional model sys- tems with molecular dynamics simulations. An efficient algorithm made it possible to calculate the structure fac- tors with high precision and in great detail. As a result multi-phonon contributions, – although weak in compar- ison to the one-phonon peaks and branches – have been identified. By introducing phasons in the model systems we were able to study their influence on the structure factors, which is mainly a broadening of the character- istic peaks, edges, and branches with temperature. The broadening can be further split into a broadening due to the disorder as a result of collective phason flips, i.e. a static phason field, and a broadening due to the anhar- monicity of the interaction potentials and single phason flips. The work presented here is a first step. Further studies on two-dimensional and three-dimensional model systems with phason flips are under way. Acknowledgments We wish to thank T. Odagaki and M. Umezaki for stimulating discussions. ∗ Electronic address: [email protected] 1 An introduction on quasicrystals can be found in: C. Janot, Quasicrystals: A Primer (Oxford Science Publica- tions, Oxford, 1992); M. de Boissieu, P, Guyot, and M. Audier, in Lectures on Quasicrystals, edited by D. G. F. 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B 61, 339 (1985). 29 The correlation functions have the following interpretation: G(x, t)∆x is the probability to find a particle in [x, x+∆x] at time t, if any particle has been at the origin at t = 0. Ga(x, t)∆x is the probability to find a particle in [x, x+∆x] at time t, if the same particle has been at the origin at t = 0. 30 M. Bée, Quasielastic neutron scattering, IOP Publishing, Bristol (1988). 31 The notation ‘x-phonon’ is adopted freely from the lan- guage of neutron scattering. However in the case of a one- dimensional system there is no regular phonon-expansion possible, as it is done for three-dimensional crystals. 32 D. Levine and P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984). 33 T. C. Lubensky, S. Ramaswamy, and J. Toner, Phys. Rev. B 32, 7444 (1985). 34 U. Dahlborg, W. S. Howells, M. Calvo-Dahlborg, and J. M. Dubois, J. Phys.: Condens. Matter 12, 4021 (2000). 35 J. Dolinšek, T. Apih, P. Jeglič, M. Klanǰsek, J. Non-Cryst. Solids 334&335, 280 (2004). Introduction Motivation Model systems Structure factors from molecular dynamics Definition of the structure factors Molecular dynamics simulations Harmonic periodic chain as reference system Analytic calculations Simulation results Harmonic Fibonacci chain: Influence of the quasiperiodicity Dynamic Fibonacci chain: Occurrence of phason flips Phason flips Results for the structure factors Asymmetric Fibonacci chain: Competing eigenfrequencies Band gaps Density of states Discussion and conclusion Acknowledgments References

0704.1454

Ground-Based Direct Detection of Exoplanets with the Gemini Planet Imager (GPI)

Microsoft Word - gpi_wp_astroph.doc GROUND-BASED DIRECT DETECTION OF EXOPLANETS WITH THE GEMINI PLANET IMAGER (GPI) James R. Graham1, Bruce Macintosh2, Rene Doyon3, Don Gavel4, James Larkin5, Marty Levine6, Ben Oppenheimer7, David Palmer2, Les Saddlemyer8, Anand Sivaramakrishnan7, Jean-Pierre Veran8, & Kent Wallace6 Abstract The Gemini Planet (GPI) imager is an “extreme” adaptive optics system being designed and built for the Gemini Observatory. GPI combines precise and accurate wavefront control, diffraction suppression, and a speckle-suppressing science camera with integral field and polarimetry capabilities. GPI’s primary science goal is the direct detection and characterization of young, Jovian-mass exoplanets. For systems younger than 2 Gyr exoplanets more massive than 6 MJ and semimajor axes beyond 10 AU are detected with completeness greater than 50%. GPI will also discover faint debris disks, explore icy moons and minor planets in the solar system, reveal high dynamic range main- sequence binaries, and study mass loss from evolved stars. This white paper explains the role of GPI in exoplanet discovery and characterization and summarizes our recommendations to the NSF-NASA-DOE Astronomy and Astrophysics Advisory Committee ExoPlanet Task Force. 1 UC Berkeley 2 Lawrence Livermore National Lab 3 Université de Montréal 4 UC Santa Cruz 5 UCLA 6 JPL 7 American Museum of Natural History 8 NRC/Herzberg Institute Introduction Direct detection affords access to exoplanet atmospheres, which yields fundamental information including effective temperature, gravity, and composition. Mid-IR exoplanetary light has been detected in secondary eclipses by SPITZER (Grillmair et al. 2007; Richardson et al. 2007). However, such information is necessarily limited by the overwhelming photon shot noise of the primary. Direct detection separates the exoplanet light from that of its primary so that this noise source is suppressed. When the fundamental atmospheric properties are known for a population of planets spanning a range of mass and age it will be possible to conduct critical tests of formation scenarios and evolutionary models. The goals of direct detection include measurement of abundances and reconstruction of the thermal history. Such information will discriminate between idealized adiabatic contraction (Burrows et al. 2004), more realistic accretion models that occurs in core- accretion (Marley et al. 2007), and formation by gravitational instability (Boss 2006). Ultimately, the impact of composition and equation of state on evolution can also be investigated. Figure 1: Planet-to-star contrast for a 1 MJ exoplanet orbiting a G2 V star at 4 AU for ages 0.1–5 Gyr (Burrows et al. 2004). Planets shine by reflected light in the visible. Young exoplanets (< 1 Gyr for 1 MJ) are detectable by their intrinsic luminosity. Note strong J (1.25 µm) and H (1.65 µm) emission where ground- based facilities have sensitive operation. Old exoplanets (> 1 Gyr) shine only by reflected light at short wavelengths (< 2 µm). Direct detection has several practical advantages. Planets at large semimajor axes can be found without waiting for an orbit to complete—a condition that renders indirect detection of Neptune like planets (a = 30 AU, P = 160 yr) impractical. The Fourier decomposition that underlies Doppler and astrometric detection are also subject to aliasing and beat phenomena, and suffer confusion when multiple planets are present. Direct imaging gives an immediate portrait of a planetary system, including sensitivity to the analogs of Zodiacal and Kuiper dust belts. Figure 2: Contrast for a 5 Gyr system orbiting a G2 V star at 4 AU. Masses are 0.5–8 MJ. Massive, solar-system age exoplanets shine only by reflected light at visible wavelengths, but remain sufficiently self-luminous to be detectable at contrast ratios > 3 x 10-8 in the near-IR. Surveys at large semimajor axes are necessary for several fundamental reasons. The structure of our own solar system implies that a full picture of planet formation cannot be constructed without reaching out to 30 or 40 AU. Moreover, if planets form as a consequence of disk instabilities, they are most likely to occur at large semimajor axis separation (> 20 AU for solar type stars) where the cooling time exceeds the Keplerian shear time scale. Indirect searches continue to hint that the semimajor axis distribution is at least flat or even rising in dN/dlog(a) beyond 5 AU. With only 200 Doppler planets known there are too few to make a statistically significant study of trends with planetary and stellar mass, semimajor axis, and eccentricity. Because Doppler methods have reached the domain of diminishing returns (a = P2/3), improving the statistics will be challenging. A direct search that can efficiently tap into the dominant population of giant exoplanets beyond 5 AU can bring statistical significance to these studies. Figure 3: A schematic of the Gemini Planet Imager showing the major subsystems. Detecting Self-Luminous Exoplanets Direct detection of Jovian-mass planets via their reflected sunlight requires a contrast ratio of order 2 ×10−9(a /5AU)−2 relative to their parent star. Because of the inverse square law, reflected light searches are an impractical way to explore the outer regions of solar systems. GPI seeks to detect the energy radiated by the planet itself, which is independent of a, except for very small semimajor axis separations. Old planets are cool and dim, but young planets are hot and therefore bright in the infrared relative to their parent star (Figure 1 & Figure 2). For example, at 1.6 µm it is possible to detect a 10 Myr-old 3 MJ planet, or a 100 Myr-old 7 MJ planet, orbiting a G2V star at a contrast ratio of only 4 × 10-6. With improved contrast, an increasingly large phase space of planets becomes accessible. Better contrast is obviously preferable, but it comes at a penalty. For example, Angel (1994) described an AO system designed to achieve very high contrast ratios using bright guide stars. This system would be suitable for exploring the planetary systems of 13 bright stars (R < 3.8 mag.) in the solar neighborhood (< 8 pc). However, the detection of a few planets, although dramatic, would be insufficient scientific impetus given the success of Doppler searches. It is necessary to show that any proposed instrument can search a scientifically interesting range of semimajor axes and accumulates a statistically significant sample of exoplanets in reasonable observing time. Figure 4: The 1-σ, 1-hour speckle noise at H for GPI with 18-cm subapertures and a maximum update rate of 2.5 kHz. The speckle noise is measured in units of the guide star brightness. Therefore, these curves represent the achievable contrast when speckle noise is dominant. The system performance is a function of guide star magnitude (thick vs. thin lines). The Fried parameter (at 500 nm) for Mauna Kea is 18.7 cm and 14.5 cm for Cerro Pachon. The coronagraph occulter is opaque within 0.09 arc seconds. Overview of GPI The Gemini Planet Imager (GPI) is a high contrast or “extreme” AO (ExAO) system (Macintosh et al. 2006). Basic R&D enabling the construction of a practical system was supported by the NSF’s Center for Adaptive Optics. In 2004, Gemini supported two ExAO conceptual design studies as part of its “Aspen Process” next generation instrumentation program. At the conclusion of those studies, the Gemini Board recently selected GPI to proceed. Work began in June 2006 and delivery of the instrument is expected in late 2010. Imaging a Jovian exoplanet requires a contrast significantly better than that delivered by existing astronomical adaptive optics (AO) systems. Currently achievable contrast, about 10-5, is completely limited by quasi-static wavefront errors, so that contrast does not improve with integration times longer than about 1 minute. Moreover, there are enough slow drifts in these errors that PSF subtraction does not increase contrast by more than a factor of a few, except in the most ideal circumstances. GPI will surpass the performance of existing systems by two orders of magnitude. Table 1: Principal properties of GPI Adaptive Optics Calibration WFS Deformable mirror 4096-actuator Boston Micro- machines Type 1–2.4 µm interferometer Subaperture 18 cm (N = 44 subaps.) Static Accuracy 1 nm RMS Limiting I = 8 mag. (goal: I = 9 mag.) Science Instrument Optics quality < 5 nm RMS WFE per optic Type Lenslet-based integral field Coronagraph Spatial sampling 0.014 arc seconds per lenslet Type Apodized-pupil coronagraph Field of 2.8 arc second square Inner Working Dist. ~ 3 λ/D Spectral coverage Y, J, H or K Throughput 60% Spectral resolution λ/Δλ ≈ 45 To achieve our science goals, GPI integrates four key subsystems (Figure 3): 1) An AO system that makes fast measurement of the instantaneous wavefront, and provides wavefront control via deformable mirrors; 2) A calibration unit that provides precise and accurate measurements of the time-averaged wavefront at the science wavelength, so that the final image is not dominated by persistent speckles caused by quasi-static wavefront errors; 3) A coronagraph that controls diffraction and pinned speckles; and 4) An integral field spectrograph unit (IFU) that records the scientific data, providing low- resolution spectroscopy and suppression of residual speckle noise. The IFU incorporates a dual-channel polarimeter for studies of circumstellar dust. An example of GPI contrast performance based on detailed simulation of this system is shown in Figure 4. Technical aspects of GPI have been presented in the recent literature, including: apodized pupil Lyot coronagraphs (Soummer 2005, Sivaramakrishnan & Lloyd 2005); optimal wavefront control for ExAO (Poyneer et al. 2006); MEMS deformable mirrors (Morzinski et al. 2006); simultaneous spectral differential imaging and the effects of out-of- pupil-plane optical aberrations (Marois et al. 2006); multiwavelength speckle noise suppression (Marois et al. 2004); and the wavefront calibration system (Wallace et al. 2006). GPI’s ExoPlanet Parameter Space The performance of GPI is characterized by the detectable brightness ratio. The achievable contrast will be a function of the brightness of the wavefront reference, the angular separation, and observing and wavefront-sensing wavelengths. The detectability of planets in a given sample of target stars can then be estimated by comparing the distribution of relative exoplanet brightness versus angular separation with the expected performance. This comparison also quantifies selection effects that vary with properties of the planet (age, mass, and orbital elements) and of the primary star (spectral type and distance). Our knowledge of the distribution of planetary properties is incomplete, but a basic premise is that sufficient information exists to make a preliminary estimate of this distribution. Given these predictions it is possible to estimate the scientific impact of different design choices, e.g., precision and accuracy of adaptive optics correction and observing wavelength, and therefore make an informed trade-off between cost and performance. Figure 5: Detectable companion contrast versus angular separation for GPI, showing the direct detection of young luminous planets in a hypothetical survey of field (< 50 pc) stars. The small dots represent the planet population: those detected by GPI are drawn with a box, those detectable in current Doppler surveys are shown with a circle. The dashed line shows the GPI contrast threshold (5 σ) for a 1-hour exposure at 1.65 µm. Within 100 λ/D speckle noise dominates. In this example, which has no speckle noise suppression and a fixed frame rate and simple AO model, a universal contrast curve applies to all targets. The approach we adopt involves making a Monte Carlo model for the population of planets in the solar neighborhood (Graham et al. 2002). This description includes the mass, age and orbital elements of each accompanying planet. When combined with cooling curves, model atmospheres and the distance to the primary star we can compute the brightness ratio and angular separation at any epoch. Our adopted planet properties are based on the precision Doppler monitoring of about 2000 F, G and K stars by eight exoplanets searches at the AAT, Lick, Keck, ESO/Coralie, Provence/Elodie, Whipple Observatory, ESO/CES and McDonald. Approximately 200 exoplanets have been found with periods between a few days and a few years and with M sin i spanning 0.1–10 MJ. Since there is no accepted theoretical model for the planet mass, M, or semimajor axis, a, distributions, we adopt a simple power law distribution: dN/dM ∝ Mα - and dN/da ∝ aβ. Figure 6: Exoplanets from the simulated GPI field star survey. Heavy filled circles are GPI detected planets from. Light dots are planets detected by a hypothetical 8- year astrometric interferometer survey, with a limit of R < 10 mag., and a precision of 30 µas. Exoplanets detected in the Keck/Lick Doppler survey are shown as stars. This illustrates how GPI explores a complementary phase space to indirect searches. Figure 5 shows the results of a Monte Carlo experiment using GPI with ideal apodization, Fried parameter, r0 = 100 cm at the observing wavelength of 1.6 µm, operating at 2.5 kHz. In this example the planet population is parameterized by a planetary mass spectrum with α = -1, a semi major axis distribution with β = -1/2 between 0.1 < a/AU < 50, and a star formation history such that the age of the disk is 10 Gyr and the mean age of stars in solar neighborhood is 5 Gyr. For this simple plot, we assume no suppression of residual speckles, and therefore speckle noise always dominates. When this is true, it is a good approximation to assume a universal contrast curve for all targets, which means that the results can be easily visualized in a contrast versus angular separation plot. Figure 7: Exoplanets from the simulated GPI field star survey. Solid lines show the evolution of 1–20 MJ exoplanets. Dotted lines are isochrones labeled in log10(Gyr). Detected planets are filled circles. The population straddles the H2O cloud condensation line at about 400 K (dashed), and a few objects lie below the NH3 condensation curve (dashed). The only confirmed astronomical object that lies on this plot is Jupiter, with Teff = 120 and log10g = 3.4. This simple system can detect about 5% of the planets in a survey of field stars. Preliminary calculations of this type were first used to argue for the scientific utility of GPI because the exoplanet detection rate is comparable to that delivered by Doppler searches. The GPI IFU will be used to suppress speckles, but the results are not so easily visualized—in subsequent simulations (e.g., Figure 7) all significant noise sources, e.g., photon shot noise, background, flat field noise, detector read noise and dark current, are fully treated. High fidelity calculations using the full sensitivity of the system predict that in a survey of approximately 3800 field stars with I < 8 mag. we would detect over 200 planets. Figure 7 shows two ways to visualize how GPI- detected planets explore the potential discovery phase space. Figure 6 compares the catalog of Doppler planets, a hypothetical astrometric survey and the GPI exoplanets. Figure 8: Exoplanet detection completeness contours for an age-selected (< 2 Gyr) GPI survey of stars in the solar neighborhood as a function of semimajor axis and planetary mass. GPI completeness is good (>50%) for 5 MJ planets beyond 10 AU. Massive planets (12 MJ) are found with similar completeness as close as 5 AU. Approximately 500 such stars are accessible to GPI (I < 8 mag.) in the solar neighborhood and over 100 exoplanets should be detectable. This comparison confirms that GPI is complementary to indirect methods. GPI can probe the outer regions of solar systems and will answer our key science questions related to planet formation and migration. Figure 7 plots the GPI exoplanets atmospheric properties on an effective temperature, log (g) diagram. The youngest detected exoplanets have ages of a few hundred Myr, representing the youngest systems in the field-star sample. The youngest objects sample masses as low as 1 MJ. With increasing age, the mean detected planet mass increases. The median exoplanet age is 1.3 Gyr, the oldest have ages of ~ 5 Gyr. About 70% of the planets are cooler than that water cloud condensation line, and three lie to the left of the ammonia cloud condensation curve. The properties of young planets (< 100 Myr) depend on initial conditions, and have been excluded from these Monte Carlo simulations. It is worth emphasizing that at GPI’s contrast levels we do not require super-luminous “hot start” planets but can detect planets formed through low-entropy core-accretion. Important complements to broad field- star surveys are targeted surveys of young clusters and associations or age-selected field stars, where the mean GPI exoplanet detection rate can approach 50%. Figure 7 (right) shows that the median detected exoplanet age in the field survey is 1.3 Gyr. Old stars can be winnowed using a chromospheric activity indicator such as Ca H & K. Employing a selection criterion that eliminates stars older than 2 Gyr boosts the planet detection rate by a factor of four. The completeness of such a survey is represented in Figure 8. About 500 stars in the solar neighborhood would be selected by such a survey, which should yield over 100 exoplanets. This suggests that a preliminary estimate of the semimajor axis and planetary mass distributions could be made in 15–30 nights of telescope time. The Gemini Observatory plans to devote 100-200 nights to GPI campaigns, therefore a number of comprehensive exoplanet surveys will be possible to accumulate statistics relevant to planet formation and to study exoplanet atmospheres. GPI also enables a broad range of adjunct science. Most relevant to the ExoPTF is sensitive detection and characterization of planetary debris disks. Unseen planets below GPI’s detection threshold may gravitationally sculpt debris disks, and therefore the ability to detect disks with optical depths as small as 8 × 10-5, or approximately one fiftieth of well- known systems such as beta Pic or AU Mic, is a key feature. Other science missions include high contrast imaging of minor planets and icy moons in the solar system, studies of main sequence binaries (GPI can see an M dwarf next to an O star), and investigation of mass loss from evolved stars. Conclusions & Recommendations to the ExoPTF Outer planets (a > 10 AU) take more than thirty years to complete one revolution. These planets will remain undetected in the first-generation Doppler surveys until about 2030. In contrast, direct detection of exoplanets is feasible now using advanced AO techniques. Within three years the GPI instrument will begin to survey for exoplanets beyond 10 AU. GPI will give our first glimpse of the outer regions of exoplanetary systems, where novel planet formation pathways may operate, the counterparts of inward planetary migration may reside, and Kuiper belt analogs may be discovered. Around selected younger targets, GPI will have significant sensitivity in the 5-10 AU range. From low-resolution spectra of the atmospheres of exoplanets GPI will also yield effective temperatures, gravity and composition. As static wavefront errors set the fundamental noise floor for direct detection, whether on the ground or in space, GPI is a unique pathfinder for future NASA missions. The GPI program therefore represents the next major step in direct planet detection and should remain a priority of the Gemini Observatory. The Gemini Observatory has recognized that the full scientific potential of GPI can only be realized if observing with this instrument is coordinated into major campaigns. Substantial resources are needed immediately if the US astronomical community is to compete effectively to conduct these hundred-night observing programs. Foundation science includes the assembly of catalogs of adolescent (0.1–1 Gyr) stars, which are the most fertile hunting grounds for self-luminous exoplanets. Efficient use of observing time will require careful experimental design, and resources will be needed to reduce and analyze GPI data for reliable exoplanet discovery and characterization. Support will also be necessary to prepare for follow up observations of candidates, including building a strong theoretical basis for understanding exoplanets atmospheres in the temperature range of 200–600 K. References Angel, J. R. P. 1994, Nature 368, 203. Boss, A. P. 2006, ApJ, 644L, 79. Burrows, A., Sudarsky, D., & Hubeny, I. 2004, ApJ, 609, 407. Graham, J. R. 2002, BAAS, 201 2102 Grillmair, C. J., et al. 2007, ApJ, 658L, 115. Macintosh B. et al. 2006, SPIE, 6272, 18 Marois, C., Doyon, R., Racine, R., & Nadeau, D. 2004, PASP, 112, 91 Marois, C. et al. 2006, SPIE, 6269, 114 Marley, M. et al. 2007, ApJ, 655, 541. Morzinski, K., M., et al. 2006, SPIE, 6272, 62 Poyneer et al. 2006, 6272, 50 Richardson, L. J., et al. 2007, Nature, 445, 892. Sivaramakrishnan, A., & Lloyd, J. P. 2005, ApJ, 633, 528 Soummer, R. 2005, ApJ 618L, 161 Wallace, J. K., et al. 2006, SPIE, 6273, 22

0704.1455

A Better Good-Turing Estimator for Sequence Probabilities

A Better Good-Turing Estimator for Sequence Probabilities Aaron B. Wagner School of ECE Cornell University [email protected] Pramod Viswanath ECE Department University of Illinois at Urbana-Champaign [email protected] Sanjeev R. Kulkarni EE Department Princeton University [email protected] Abstract— We consider the problem of estimating the prob- ability of an observed string drawn i.i.d. from an unknown distribution. The key feature of our study is that the length of the observed string is assumed to be of the same order as the size of the underlying alphabet. In this setting, many letters are unseen and the empirical distribution tends to overestimate the probability of the observed letters. To overcome this problem, the traditional approach to probability estimation is to use the classical Good-Turing estimator. We introduce a natural scaling model and use it to show that the Good-Turing sequence probability estimator is not consistent. We then introduce a novel sequence probability estimator that is indeed consistent under the natural scaling model. I. INTRODUCTION Suppose we are given a string drawn i.i.d. from an un- known distribution. Our goal is to estimate the probability of the observed string. One approach to this problem is to use the type, or empirical distribution, of the string as an approximation of the true underlying distribution and then to calculate the resulting probability of the observed string. It is well known that this estimator assigns to each string its largest possible probability under an i.i.d. distribution. For large enough observation sizes, this estimator works well; indeed, for large n and a fixed underlying distribution, it is a consistent sequence probability estimator. Motivated by applications in natural language, we focus on a nonstandard regime in which the size of the underlying alphabet is of the same order as the length of the observed string. In this regime, the type of the observation is a poor representation of the true probability distribution. Indeed, many letters with nonzero probability will not be observed at all and the type will obviously assign these letters zero probability. This would not make for a consistent probability estimator. Since probability estimation and compression are closely related, we can turn to the compression literature for suc- cor. The results in this literature are negative, however. For instance, Orlitsky and Santhanam [1] shows that universal compression of i.i.d. strings drawn from an alphabet that grows linearly with the observation size is impossible. As such, the compression literature is unhelpful and even suggests that seeking a consistent universal sequence probability estimator might be futile. Nevertheless, sequence probability estimation is of such im- portance in applications that several heuristic approaches have been developed. The foremost among them is based on the classical Good-Turing probability estimator (see Section IV). The idea is to use the Good-Turing estimator instead of the type to estimate the underlying probability distribution. The probability of the sequence can then be calculated accordingly. Orlitsky et al. [3] have studied the performance of a similar scheme in the context of probability estimation for patterns. No theoretical results regarding the performance of this approach for sequence probability estimation are available, however. To analyze the performance of this scheme, we introduce a natural scaling model in which the number of observations, n, and the underlying alphabet size grow at the same rate. Further, the underlying probabilities vary with n. The only restriction we make is that no letter should be either too rare or too frequent. That is, the probability that any given symbol occurs somewhere in the string should be bounded away from 0 and 1 as the length of the string tends to infinity. In particular, this condition requires that the probabilities of the letters be Θ(1/n). We call this the rare events regime. This scaling model is formally described in the next section. Our model is similar to the one used by Klaassen and Mnatsakanov [4] and Khmaladze and Chitashvili [5] to study related problems. We used this model previously [6] to show consistency of the Good-Turing estimate of the total probabil- ity of letters that occur a given number of times in the observed string. In the present paper, we use this model to first show that the Good-Turing estimator for sequence probabilities performs poorly; in fact, a simple example illustrates that it is not consistent. Drawing from this example, we then provide a novel sequence probability estimator that improves upon the Good-Turing estimator—in fact, we show that it is consistent in the context of the natural scaling model. This is done in Section V. Finally, we discuss the application of our results to universal hypothesis testing problems in the rare event regime in Section VI. http://arxiv.org/abs/0704.1455v2 II. THE RARE EVENTS REGIME Let Ωn be a sequence of finite alphabets. For each n, let pn and qn be probability measures on Ωn satisfying ≤ min(pn(ω), qn(ω)) ≤ max(pn(ω), qn(ω)) ≤ for all ω in Ωn, where č and ĉ are fixed constants that are independent of n. Observe that this requires the cardinality of the alphabet size to grow linearly in n ≤ |Ωn| ≤ We observe two strings of length n. The first, denoted by x, is a sequence of symbols drawn i.i.d. from Ωn according to pn. The second, denoted by y, is a sequence of symbols drawn i.i.d. from Ωn according to qn. We assume that x and y are statistically independent. Note that both the alphabet and the underlying probability measures are permitted to vary with n. Note also that by assumption (1), each element of Ωn has probability Θ(1/n) under both measures and thus will appear Θ(1) times on average in both strings. In fact, the probability of a given symbol appearing a fixed number of times in either string is bounded away from 0 and 1 as n → ∞. In other words, every letter is rare. The number of distinct symbols in either string will grow linearly with n as a result. Our focus shall be on the quantities pn(x) and pn(y). An important initial observation to make is that the distributions of these two random variables are invariant under a relabeling of the elements of Ωn. It is therefore convenient to consider the probabilities assigned by the measures pn and qn without reference to the labeling of the symbols. It is also convenient to normalize these probabilities so that they are Θ(1). Let Pn denote the distribution of (npn(xn), nqn(xn)), where xn is drawn according to pn. Likewise, let Qn denote the distribution of (npn(yn), nqn(yn)), where yn is drawn according to qn. Note that both Pn and Qn are probability measures on C := [č, ĉ] × [č, ĉ]. It follows from the definitions that Pn and Qn are absolutely continuous with respect to each other and the Radon-Nikodym derivative is given by (x, y) = . (2) Note that many quantities of interest involving pn and qn can be computed using Pn (or Qn). For example, the entropy of pn can be expressed as dPn(x, y) and the relative entropy between pn and qn is given by D(pn||qn) = dPn(x, y). We shall assume that Pn converges in distribution to a probability measure P on C. Since Pn and Qn are related by (2), this implies that Qn converges to a distribution Q satisfying (x, y) = III. PROBLEM FORMULATION Recall that the classical (finite-alphabet, fixed-distribution) asymptotic equipartition property (AEP) asserts that logµ(w) = −H(µ) a.s., (3) where w is an i.i.d. sequence drawn according to µ and H(·) denotes discrete entropy. Loosely speaking, (3) says that the probability of the observed sequence, µ(w), is approximately exp(−nH(µ)). In the rare events regime, on the other hand, one expects the probability of an observed sequence to be approximately for some constant h. Indeed, in the rare events regime the following AEP holds true (all proofs are contained in Sec- tion VII). Theorem 1: log(npn(xi)) = log(x) dP (x, y) a.s. Our goal is to estimate the limit in Theorem 1 universally, that is, using only the observed sequence x without reference to the probability measures pn. Of course, in the classical setup, the analogous problem of universally estimating the limit in (3) is straightforward. The distribution µ can be determined from the observed sequence by the law of large numbers, from which the entropy H(µ) can be calculated. In the rare events regime, on the other hand, this approach fails and the problem is more challenging. We shall also study the following variation on this problem. Consider the related quantity pn(y). That is, the sequence is generated i.i.d. according to qn, but we evaluate its probability under pn. This quantity arises in detection problems, where one must determine the likelihood of a given realization under multiple probability distributions. As in the single-sequence setup, it turns out that this probability converges if it is suitably normalized. Theorem 2: log(npn(yi)) = log(x) dQ(x, y) a.s. Our goal is then to estimate the limit in Theorem 2 using only the observed sequences x and y. Again, in a fixed- distribution setup, this problem is straightforward because the two distributions can be determined exactly from the observed sequences in the limit as n tends to infinity. In the rare events regime, however, the problem is more challenging. IV. THE GOOD-TURING ESTIMATOR The Good-Turing estimator can be viewed as an estimator for the probabilities of the individual symbols. Let Ak be the set of symbols that appear k times in the sequence x, and let ϕk = |Ak| denote the number of such symbols. The basic form of the Good-Turing estimator assigns probability (k + 1)ϕk+1 to each symbol that appears k ≤ n − 1 times [2]. The case k = n must be handled separately, but this case is unimportant to us because in the rare events regime the chance that only one symbol appears in x is asymptotically negligible. The Good-Turing formula can also be viewed as an estima- tor for the total probability of all symbols that appear k times in x, i.e., pn(Ak). In particular, the ϕk in the denominator can be viewed as simply dividing the total probability (k + 1)ϕk+1 equally among the ϕk symbols that appear k times. In previ- ous work, we showed that the Good-Turing total probability estimator is strongly consistent in that for any k ≥ 0, (k + 1)ϕk+1 = lim pn(Ak) xke−x dP (x, y) =: λk a.s. (5) (see [6], where the notation is slightly different, for a proof of a stronger version of this statement). The Good-Turing probability estimator in (4) gives rise to a natural estimator for the probability of the observed sequence x (k + 1)ϕk+1 This in turn suggests the following estimator for the limit in Theorem 1 (k + 1)ϕk+1 . (6) This estimator is problematic, however, because for the largest k for which ϕk > 0, (k + 1)ϕk+1 which means that the kth term in (6) equals −∞. Various “smoothing” techniques have been introduced to address re- lated problems with the estimator [2]. Our approach will be to truncate the summation at a large but fixed threshold, K (k + 1)ϕk+1 In the rare events regime, with probability one it will eventu- ally happen that ϕk > 0 for all k = 1, . . . ,K , thus obviating the problem. By the result in (5), this estimator will converge λk−1 log . (7) We next show that this quantity need not tend to the limit in Theorem 1 as K tends to infinity. Let Ωn be the set {1, 2, . . . , 3n}. Suppose that pn assigns probability 1/(4n) to the first 2n elements and probability 1/(2n) to the remaining n. The distribution qn is obviously not relevant here so we shall simply set it equal to pn. The resulting distribution P will place mass 1/2 on each of the points (1/4, 1/4) and (1/2, 1/2). From Theorem 1, the limiting normalized probability of x is −(1/2) log 8. By (7), the Good-Turing estimate converges to e−1/4(1/4)k−1 (k − 1)! 1 + e−1/42k−1 8(1 + e−1/42k) 4(1 + e−1/42k−1) e−1/4(1/4)k−1 (k − 1)! 1 + e−1/42k−1 Now as K tends to infinity, the second sum converges to the correct answer, −(1/2) log 8. But one can verify that every term in the first sum is strictly positive. Thus the Good-Turing estimator is not consistent in this example. The problem is that the Good-Turing estimator is estimating the sum, or equivalently the arithmetic mean, of the probabil- ities of the symbols appearing k times in x. Estimating the sequence probability, on the other hand, amounts to estimating the geometric mean of these probabilities. If pn assigns the same probability to every symbol, then the arithmetic and geo- metric means coincide, and one can show that the Good-Turing sequence probability estimator is asymptotically correct. In the above example, however, pn is not uniform, and the Good- Turing formula converges to the wrong value. In the next section, we describe an estimator that targets the geometric mean of the probabilities instead of the arithmetic mean, and thereby correctly estimates the sequence probability. V. A BETTER GOOD-TURING ESTIMATOR Write ĉ+ č and then let γMk = − (−c)−ℓ (k + ℓ)! m · k! (k + ℓ+ 1)ϕk+ℓ+1 + log(c) (k + 1)ϕk+1 Note that γMk is only a function of x and in particular, it does not depend on pn. The next theorem shows that for large K and M , is a consistent estimator for the limit in Theorem 1. Theorem 3: For any ǫ > 0, log(npn(xi))− ≤ ǫ a.s. (8) provided exp(ĉ)c ĉ− č ĉ+ č ĉK+1c (K + 1)! where c = max(| log č|, | log ĉ|). The idea behind Theorem 3 is this. Recall from (5) that (k + 1)ϕk+1 xke−x dP (x, y) a.s. If one could find a sequence of constants ak such that xke−x = log(x) on [č, ĉ], then one might expect that (k + 1)ϕk+1 log(x) dP (x, y) a.s. This is indeed the approach we took to find the formula for γMk . The estimator can be naturally extended to the two-sequence setup, namely to the problem of universally estimating pn(y). Let ϕk,ℓ be the number of symbols in Ωn that appear k times in x and ℓ times in y. Then let γ̃Mk = − (−c)−ℓ (k + ℓ)! m · k! jϕk+ℓ,j + log(c) jϕk,j Note that γ̃Mk is a function of x and y. Theorem 4: For any ǫ > 0, log(npn(yi))− ≤ ǫ a.s. provided exp(ĉ)c ĉ− č ĉ+ č ĉK+1c (K + 1)! This result shows that although we are unable to determine pn from x, we are able to glean enough information about pn to determine the limit in Theorem 2. VI. UNIVERSAL HYPOTHESIS TESTING The γ̃Mk estimator leads to a natural scheme for the problem of universal hypothesis testing. Suppose that we again observe the sequences x and y, which we now view as training data. In addition, we observe a test sequence, say z, which is generated i.i.d. from the distribution rn. We assume that either rn = pn for all n or rn = qn for all n. The problem is to determine which of these two possibilities is in effect using only the sequences x, y, and z. Using Theorem 4, one can estimate pn(z) and qn(z), and by comparing the two, determine which of the two distribu- tions generated z. This will make for a consistent universal classifier, without recourse to actually estimating the true underlying distributions pn and qn. As a scheme for universal hypothesis testing, however, this approach is quite complicated and there is no reason to believe it would be optimal in an error-exponent sense. We are currently investigating other, more direct approaches to the universal hypothesis testing problem in the rare events regime. For a discussion of universal hypothesis testing in the traditional, fixed-distribution regime, see Gutman [7] and Ziv [8]. VII. PROOFS Due to space limitations, we will only prove Theorem 1 and sketch the proof of Theorem 3. The proofs of Theorems 2 and 4 are similar. Lemma 1: log(npn(xi)) log(x) dP (x, y). Proof: Note that for any i E[log(npn(xi))] = pn(ω) log(npn(ω)) log(x) dPn(x, y). Since log(x) is bounded and continuous over C and Pn converges in distribution to P , the result follows. Lemma 2: log(npn(xi)) log(npn(xi)) = 0 a.s. Proof: Consider the sum log(npn(xi)). If one symbol in the sequence x is altered, then this sum can change by at most It follows from the Azuma-Hoeffding-Bennett concentration inequality [9, Corollary 2.4.14] that log(npn(xi)) log(npn(xi)) ≤ 2 exp 2(log(ĉ/č))2 The result then follows by the Borel-Cantelli lemma. Note that Theorem 1 follows immediately from Lemmas 1 and 2. The key step in the proof of Theorem 3 is showing that γMk converges to the proper limit. This is shown in the next and final lemma. Lemma 3: For any ǫ > 0 and any k ≥ 0, γMk − log(x) exp(−x)xk dP (x, y) ≤ ǫĉ a.s., provided ĉ− č ĉ+ č Proof (sketch): Note that the limit exists by (5). By the triangle inequality, γMk − log(x) exp(−x)xk dP (x, y) ∣γMk − γMk γMk − log(x) exp(−x)xk dP (x, y) , (9) where γMk = − (−c)−ℓ (k + ℓ)! m · k! λk+ℓ + log(c)λk. The first term on the right-hand side of (9) tends to zero by (5). (−c)−ℓ (k + ℓ)! m · k! exp(−x)xk m · k! (−c)−m (−c)m−ℓ xℓ dP (x, y). By the Binomial Theorem, (−c)m−ℓ xℓ = (x− c)m. Substituting these last two equations into (10) yields γMk = − )m exp(−x)xk m · k! dP (x, y) + log(c)λk. Using the well-known power series log(1 + x) = (−1)m+1 valid for −1 < x ≤ 1, one can show that č≤x≤ĉ ĉ− č ĉ+ č by hypothesis. Thus γMk − log(x) exp(−x)xk dP (x, y) )m exp(−x)xk m · k! dP (x, y) ) exp(−x)xk dP (x, y) exp(−x)xk dP (x, y) ≤ Since exp(−x)xk one would expect from Lemma 3 that for large K and M , would be close to log(x) dP (x, y). Indeed, one can prove Theorem 3 using this approach. The details are omitted. REFERENCES [1] A. Orlitsky and N. P. Santhanam, “Performance of universal codes over infinite alphabets,” in Proc. IEEE Data Compression Conference, Mar. 2003, pp. 402–10. [2] I. J. Good, “The population frequencies of species and the estimation of population parameters,” Biometrika, vol. 40, no. 3/4, pp. 237–64, 1953. [3] A. Orlitsky, N. P. Santhanam, and J. Zhang, “Always Good Turing: Asymptotically optimal probability estimation,” Science, vol. 302, pp. 427–31, Oct. 2003. [4] C. A. J. Klaassen and R. M. Mnatsakanov, “Consistent estimation of the structural distribution function,” Scand. J. Statist., vol. 27, pp. 733–46, 2000. [5] E. V. Khmaladze and R. Ya Chitashvili, “Statistical analysis of a large number of rare events and related problems,” Proc. A. Razmadze Math. Inst., vol. 92, pp. 196–245, 1989, in Russian. [6] A. B. Wagner, P. Viswanath, and S. R. Kulkarni, “Strong consistency of the Good-Turing estimator,” in IEEE Int. Symp. Inf. Theor. Proc., July 2006, pp. 2526–30. [7] M. Gutman, “Asymptotically optimal classification for multiple tests with empirically observed statistics,” IEEE Trans. Inf. Theory, vol. 35, no. 2, pp. 401–8, Mar. 1989. [8] J. Ziv, “On classification with empirically observed statistics and universal data compression,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 278–86, Mar. 1988. [9] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applica- tions, 2nd ed. New York: Springer-Verlag, 1998. Introduction The Rare Events Regime Problem Formulation The Good-Turing Estimator A Better Good-Turing Estimator Universal Hypothesis Testing Proofs References

0704.1456

Effective temperature vs line-depth ratio for ELODIE spectra. Gravity and rotational velocity effects

Astron. Nachr. / AN , No. , 1 – 7 () / DOI please set DOI! Effective temperature vs line-depth ratio for ELODIE spectra. Gravity and rotational velocity effects ⋆ K. Biazzo1,2,⋆⋆, A. Frasca1, S. Catalano1, and E. Marilli1 1 INAF - Catania Astrophysical Observatory, via S. Sofia 78, I-95123 Catania, Italy 2 Department of Physics and Astronomy, University of Catania, via S. Sofia 78, I-95123 Catania, Italy The dates of receipt and acceptance should be inserted later Key words stars: fundamental parameters – stars: late-type – techniques: spectroscopy The dependence on the temperature of photospheric line-depth ratios (LDRs) in the spectral range 6190–6280 Å is in- vestigated by using a sample of 174 ELODIE Archive stellar spectra of luminosity class from V to III. The rotational broadening effect on LDRs is also studied. We provide useful calibrations of effective temperature versus LDRs for giant and main sequence stars with 3800 <∼ Teff ∼ 6000 K and v sin i in the range 0–30 km s −1. We found that, with the ex- ception of very few line pairs, LDRs, measured at a spectral resolution as high as 42 000, depend on v sin i and that, by neglecting the rotational broadening effect, one can mistake the Teff determination of ∼100 K in the worst cases. c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The stellar effective temperature is one of the most impor- tant astrophysical parameters. It is not easy to measure it with high accuracy and this situation worsens for stars that have already left the main sequence and that are expanding their envelopes and redistributing their angular momentum. In addition, surface inhomogeneities, such as spots and fac- ulae can significantly affect the average temperature of ac- tive stars. Several diagnostics, based on color indices and spectral- type classification, are available for temperature measure- ments, but errors greater than 100 Kelvin degrees are often encountered. On the other hand, the analysis of line profiles and their dependence upon the stellar effective temperature is a very powerful tool for the study of the photospheric temperature, as well as for the investigation of stellar sur- face structures. As a matter of fact, it has been demonstrated that the ra- tio of the depths of two lines having different sensitivity to temperature is an excellent diagnostics for measuring small temperature differences between stars or small temperature variations of a given star. Although the effective temperature scale can be set to within a few tens of degrees, temperature differences can be measured with a precision down to a few Kelvin degrees in the most favorable cases (e.g., Gray & Jo- hanson 1991, Strassmeier & Schordan 2000, Gray & Brown 2001, Catalano et al. 2002a, 2002b). This allows putting a star sample in a “relative” temperature scale or to detect tiny temperature variations in individual stars. Starspot tempera- ⋆ Based on spectral data retrieved from the ELODIE Archive at Obser- vatoire de Haute-Provence (OHP). ⋆⋆ Corresponding author: e-mail: [email protected]. tures from line-depth ratios (LDRs) variations, indeed, have been determined in the slowly rotating dwarf star σ Draco- nis (Gray et al. 1992), in the very young and rapidly rotating star LQ Hydrae (Strassmeier et al. 1993), in some young solar-type stars (Biazzo et al. 2007), and in three single- lined RS CVn binaries (Frasca et al. 2005). However, all these works do not explicitly take into account the effects of stellar rotation on spectral line depths and consequently on LDRs. In the present work, we primarily explore the influence of the rotational velocity on LDRs and provide calibration relations between individual LDRs and effective tempera- ture at different rotation velocities for high-resolution spec- tra (R = 42 000) taken from the ELODIE Archive (Moul- taka et al. 2004), taking also into account the effect of grav- The effect of slightly different metallicity for the stars in our sample is also examined. Its influence on LDRs is found to be negligible for stars around solar metallicity (within ±0.3). 2 The star sample 2.1 Sample selection We selected 174 stellar spectra of giant and main sequence stars (MS) from the on-line ELODIE Archive (Moultaka et al. 2004). Their spectral types range from F5 to M0 (Hoffleit & Warren 1991). The criteria for the choice of the stellar sample are the following: – low rotation rate, v sin i <∼ 5 km s −1 (values mainly taken from Bernacca & Perinotto 1970, de Medeiros & Mayor 1999, Glebocki et al. 2000); c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://arxiv.org/abs/0704.1456v1 2 K. Biazzo et al.: Effective temperature vs line-depth ratio for ELODIE spectra – good Hipparcos parallaxes (π) with errors less than 15% (ESA 1997); – accurate B−V color indices (Mermilliod & Mermilliod 1998) ranging from 0.49 to 1.37 for the 102 MS stars and from 0.64 to 1.54 for the 72 giant stars; – nearly solar metallicity, [Fe/H]=0.0±0.3 (Soubiran et al. 1998). 2.2 Effective temperature and luminosity In order to determine the Teff and luminosity of the stars in our sample and to put them onto the HR diagram, it is neces- sary to evaluate the interstellar extinction (AV ) and redden- ing (EB−V ). We have evaluated AV from the star distance, assuming a mean extinction of 1.7 mag/kpc for stars on the galactic plane (|b| < 5◦) and of 0.7 mag/kpc for stars out of the plane (|b| > 5◦). The reddening was estimated accord- ing to the standard law AV = 3.1EB−V (Savage & Mathis 1979), with EB−V color excess. However, the color excess of these nearby stars is always less than 0.m03 for MS stars and less than 0.m09 for giant stars. The only two exceptions are HD 176737 and HD 54489 where EB−V ∼ 0.20 since they have |b| < 5◦ and π < 3 mas. Fig. 1 shows the distri- butions of the de-reddened color (B − V )0 for the MS and giant stars. The de-reddened V magnitude was converted into absolute magnitude MV through the parallax and sub- sequently into bolometric magnitude by means of the bolo- metric corrections tabulated by Flower (1996) as a function of the effective temperature. A solar bolometric magnitude of Mbol = 4.64 (Cox 2000) was used to express the stellar luminosity in solar units. From (B − V )0 values we have deduced the effective temperature by means of the follow- ing empirical relation given by Gray (2005) and valid for 0.00≤ (B − V )0 <∼ 1.5: logTeff = 3.981− 0.4728(B − V )0 + 0.2434(B − V )2 − 0.062(B − V )3 We have verified the consistency of Teff values derived in this way with those listed by Prugniel & Soubiran (2001) and compiled from the literature (Fig. 2). The latter temper- atures appear to be systematically lower than Gray’s temper- atures by about 120 K, on average. The root mean square (rms) of data compared to the linear fits is around 130 K both for giant and MS stars. This comparison allows us to estimate an average temperature uncertainty of about 100– 150 K for the FGK stars of our sample, as already found in previous studies (see, e.g., Gray 2005, and references therein). We find analogous results by plotting the Teff de- rived from Gray’s relation with those obtained by means of the Teff–(V − I) calibration by Alonso et al. (1996, 1999). After several tests made on the LDR–Teff calibrations, we have chosen to use Gray’s effective temperatures for the following analysis, because they give the smallest scatter in the calibrations (rms≃0.13, 0.14, and 0.06 for Prugniel & Soubiran, Alonso et al., and Gray temperature sets, re- spectively). Anyway, the use of Alonso et al.’s temperature Fig. 1 B − V distribution of the giant (III) and MS (V) calibration stars. Fig. 2 Comparison between the effective temperatures of the giant (triangles) and MS (squares) stars obtained using the Gray’s calibration (Gray 2005) and those listed by Prug- niel & Soubiran (2001). The two continuous lines represent the linear fits to the two star groups, while the dashed line is the bisector. scale, instead of Gray’s scale, does not affect significantly the LDR–Teff calibrations. The position onto the Hertzsprung-Russel (HR) diagram of the ELODIE stars used in this work is shown in Fig. 3 together with the evolutionary tracks and isochrones calcu- lated by Girardi et al. (2000). 3 LDR analysis and results 3.1 Line identification Within the visible region of cool star spectra, there are sev- eral pairs of lines suitable for temperature determination. Most works are based on line pairs in the spectral domain around 6200 Å (Gray & Johanson 1991, Gray & Brown 2001, Catalano et al. 2002a, 2002b, Biazzo 2006) and 6400 Å (Strassmeier & Fekel 1990, Strassmeier & Schordan 2000). c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN () 3 Fig. 3 HR diagram of the stars in our sample. The evolu- tionary tracks for different masses are taken from Girardi et al. (2000) and are shown by continuous and dotted lines. The isochrones at an age of 6.31×107 yrs (ZAMS) with Z = 0.019 ([Fe/H]=0) and Z = 0.01 ([Fe/H]∼ −0.3) are also displayed with dashed and dash-dotted lines, respec- tively. For this work, several spectral lines have been chosen in the 6190–6280 Å range. A sample of ELODIE spectra of different spectral types is shown in Fig. 4. The lines were identified through the solar (Moore et al. 1966) and Arcturus (Griffin 1968) atlases, choosing those appearing unblended at the very high resolution of the aforementioned atlases. The chosen lines, their excitation potential χ (Moore et al. 1966) and ionization energy I (Allen 1973) in electron volts are listed in Table 1 together with a code number in the fifth column. The lines for each ratio are chosen to be close together in order to minimize the errors in the continuum setting. Fifteen line ratios are used altogether in this work. Each spectral line has an intensity depending specifically on temperature and gravity (or electronic pressure) through the line and continuum absorption coefficients (lν and κν). These parameters are related to the excitation and ionization potentials, χ and I , by means of Boltzmann and Saha equa- tions. For these reasons, for example, the lines λ6243.11 V I and λ6247.56 Fe II change in opposite directions with temperature, as displayed in Fig. 4, where the effect of tem- perature and gravity on line intensities is apparent. The measure of line depths (d), LDRs (r), and the eval- uation of errors has been carried out according to the guide- lines of Catalano et al. (2002a). 3.2 Metallicity effect Before proceeding with the calibration, it is important to evaluate the dependence of line-depth ratios on metallicity. Gray (1994) has investigated the influence of metallicity on color indices, finding an empirical relation between B − V and the logarithm of the iron abundance normalized to the Sun ([Fe/H]). A calibration relation between B−V and Teff depending on [Fe/H] has been also proposed by Alonso et Table 1 Spectral lines. Each line printed in italics is fully blended with the preceding one. λ Element χ I Code (Å) (eV) (eV) 6199.19 V I 0.29 6.74 1 6200.32 Fe I 2.61 7.87 2 6210.67 Sc I 0.00 6.54 3 6213.44 Fe I 2.22 7.87 4 6213.83 V I 0.30 6.74 5 6215.15 Fe I 4.19 7.87 6 6215.22 Ti I 2.69 6.82 6216.36 V I 0.28 6.74 7 6223.99 Ni I 4.10 7.63 8 6224.51 V I 0.29 6.74 9 6232.65 Fe I 3.65 7.87 10 6233.20 V I 0.28 6.74 11 6242.84 V I 0.26 6.74 12 6243.11 V I 0.30 6.74 13 6243.82 Si I 5.61 8.15 14 6246.33 Fe I 3.60 7.87 15 6247.56 Fe II 3.89 16.18 16 6251.83 V I 0.29 6.74 17 6252.57 Fe I 2.40 7.87 18 6255.95 Fe I ? 7.87 19 6256.35 Ni I 1.68 7.63 20 6256.36 Fe I 2.45 7.87 6256.89 V I 0.28 6.74 21 6265.14 Fe I 2.18 7.87 22 6266.33 V I 0.28 6.74 23 6268.87 V I 0.30 6.74 24 6270.23 Fe I 2.86 7.87 25 6274.66 V I 0.27 6.74 26 al. (1999). However, the B − V color index is only very slightly dependent on [Fe/H] for stars of nearly solar metal- licity. Furthermore, Gray (1994) did not find any depen- dence on [Fe/H] for LDRs of weak lines and only a very weak dependence for the line pair λ6251.83 V I–λ6252.57 Fe I, amounting to about ±20 K for [Fe/H] in the range ± 0.3. Since all the stars analysed in this work have metal- licities close to the solar value (within ± 0.3) and taking also into account the large uncertainty on many [Fe/H] litera- ture data, no correction for this parameter has been applied. Anyway, given the high number of stars in our sample, pos- sible residual effects due to small metallicity differences not properly accounted for are not expected to significantly af- fect our calibration. 3.3 Gravity effect Line depth ratios are sensitive to gravity due to the depen- dence on the electron pressure of the H− bound-free contin- uum absorption coefficient. As a consequence, to determine the temperature of stars with different gravity, such as giants and MS, it is necessary to evaluate this dependence and/or to set appropriate temperature scales (Biazzo 2001). For this reason, in the present work different r − Teff calibrations have been performed for MS and giant stars. We www.an-journal.org c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 4 K. Biazzo et al.: Effective temperature vs line-depth ratio for ELODIE spectra Fig. 4 Three representative normalized spectra for giant (upper panel) and MS (bottom panel) stars. Temperature de- creases from top to bottom. The spectral lines used for the LDR computation are labeled with their wavelength (in Å), chemical element and ionization level. The lines, with low excitation potential, such as λ6210.67 Sc I or λ6266.33 V I, increase their depth with decreasing temperature, while the λ6247.56 Fe II line, with high excitation potential, shows an opposite behaviour. The lines with intermediate χ, such as λ6256.35 Ni I, don’t display any relevant variation in this temperature range. c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN () 5 have taken into account the gravity spread in each sample in the following way: i. for MS stars we have considered the difference ∆L be- tween their luminosity and that of ZAMS stars of the same color (Fig. 3) as a gravity index; ii. for giant stars we have computed their gravity, log g, according to the evolutionary tracks of Girardi et al. (2000). Then, the residuals ∆rMS,GIA of LDRs compared to the fits in the (B − V )0 − r plane have been plotted against ∆L and log g for MS and giant stars, respectively. A linear fit to the data provides very small positive slopes, indicating slightly increasing LDRs with gravity (Fig. 5). The linear fit was obtained by means of POLY FIT, an IDL routine using matrix inversion, which performs a weighted least- square polynomial fit with error estimates. If a and b are the intercept and the slope of these linear fits, respectively, the gravity-corrected line-depth ratio, rc, is rc = r − (a+ b∆L), (2) for MS stars and rc = r − (a+ b log g), (3) for giant stars. In Eq. 2 and 3, r is the observed LDR. The final polynomial fits to the corrected LDRs are of the type Teff = c0 + c1rc + ...+ cnr both for MS and giant stars, with c0, ..., cn coefficients of the polynomial fits. An example of calibration is illustrated in Fig. 6 and the polynomial fit coefficients are listed in Table 2 for the 15 different line pairs. With the aim of verifying the internal consistency of our method, we have computed the effective temperature of each star coming from each rc − Teff calibration and we have carried out the weighted average on all the 15 LDR. The resulting < Teff > are plotted as a function of (B−V )0 in Fig. 7, together with the following polynomial fit on the points (continuous line): log < Teff > = 3.85478− 0.0582469(B− V )0 − 0.1883(B − V )2 + 0.0823(B − V )3 In the range 0.5 <∼ (B−V )0 ∼ 1.5, our< Teff > −(B−V )0 calibration is perfectly consistent with the one expressed in Eq. 1 and obtained by Gray (2005). The data scatter is ob- viously larger than the individual errors in < Teff > (origi- nating from the LDR errors), due to the errors in the B − V measurements and to the residual dependence of B − V on the stellar parameters other than the effective temperature, like metallicity and microturbulence. The values of < Teff >, L and rc for all the standard stars are listed in Table 51. 1 Table 5 is only available in electronic form at the Web site http://cdsads.u-strasbg.fr/. Fig. 5 Examples of the residuals of the LDR λ6252 V I- λ6253 Fe I compared to the polynomial fit as a function of the gravity index (dots). The continuous lines represent lin- ear fits to the data of giants and MS stars, respectively. 3.4 Rotational broadening effect In a moderately rotating star, the depth at the line center decreases with increasing v sin i at a rate depending on the line characteristics. Following Stift & Strassmeier (1995), in a weak line for which the saturation effects are very small, the residual intensity directly reflects the run of the opacity profile with frequency. A strong line, in contrast, behaves differently, because saturation leads to a contribution to the intensity at the line center that remains fairly high over a considerable part of the stellar disk, resulting in a slow de- crease in the central line intensity with v sin i. This implies that if two lines are of comparable strength and do not dif- fer radically in the broadening parameters, the LDR will not depend on rotation. Stift & Strassmeier (1995) investi- gated the dependence of the LDR on rotation by synthesiz- ing the spectral region containing the λ6252 V I and λ6253 Fe I lines in a grid of atmospheric models (Kurucz 1993) for main-sequence stars with v sin i values ranging from 0.0 to 6.0 km s−1. They estimated that, for a fixed microturbu- lence, the rotation dependence is always present at all tem- www.an-journal.org c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://cdsads.u-strasbg.fr/ 6 K. Biazzo et al.: Effective temperature vs line-depth ratio for ELODIE spectra Fig. 6 rc − Teff calibration for the line pair λ6211 Sc I-λ6215 Fe I+Ti I at different v sin i’s (open circles: v sin i=0 km s−1; triangles: v sin i=15 km s−1; crosses: v sin i=30 km s−1) for giant stars (left panel) and MS stars (right panel). Fig. 7 < Teff > of the standards as a function of the de- reddened B − V . The continuous and dashed lines are re- ferred to our and Gray’s calibrations, respectively. peratures (from 3500 to 6000 K) for a v sin i ≥ 4 − 5 km We find similar results computing the LDRs obtained from high-resolution synthetic spectra. In particular, we have considered the Synthetic Stellar Library described and made available by Coelho et al. (2005), which contains spectra synthesized by adopting the model atmospheres of Castelli & Kurucz (2003). These spectra are sampled at 0.02 Å, range from the near-ultraviolet (300 nm) to the near- infrared (1.8 µm), and cover the following grid of param- eters: 3500 ≤ Teff ≤7000 K, 0.0 ≤ log g ≤ 5.0, −2.5≤[Fe/H]≤+0.5, α-enhanced [α/Fe]=0.0, 0.4 and mi- croturbulent velocity vt=1.0, 1.8, 2.5 km s −1. In Fig. 8 we show, as an example, the behaviour of synthetic LDRs with v sin i at three different Teff for the line pair λ6211 Sc I- λ6215 Fe I+Ti I. The synthetic-LDR for this couple de- creases with the increase in v sin i from 0 to about 20 km s−1 and then remains nearly constant. This behaviour is ap- parent in all the three temperatures displayed in Fig. 8, and is more evident in the lowest one. Neglecting this effect, a systematic error in the effective temperature can arise. For instance, if we use the rc − Teff calibration obtained by us at v sin i=0 km s−1 for a giant star with Teff=4750 K and v sin i= 20 km s−1, we overestimate its effective tempera- ture by about 80 K. As a consequence, this LDR appears to be quite sensitive to the rotation velocity in the range 0–20 km s−1. Thus, the rotational broadening must be taken into account to perform a reliable calibration for this as well as for several other LDRs. To study in detail the influence of rotational broadening on LDRs, we have computed the LDRs of our star sample broadening the corresponding spectra at different rotation velocities from 0 to 30 km s−1 in steps of 5 km s−1. The LDR calculation has been carried out with a simple code written in IDL2. The code first convolves the spectrum with the required rotational profile and then, automatically, com- putes the 15 LDRs in the selected wavelength range. An example of an rc − Teff calibration (λ6211 Sc I- λ6215 Fe I+Ti I) both for giant and MS stars is displayed in Fig. 6, where only the calibrations for three v sin i values (0, 15, and 30 km s−1) are shown. We find that for some LDRs the effects of the broadening are already visible at 5 km s−1, while some other LDRs are practically not affected by the rotational broadening. The main characteristics of these cal- ibrations can be summarized as follows. – Because of blending problems, for some pairs it is not possible to broaden the lines at any rotational veloc- ity (e.g., λ6214 V I-λ6213 Fe I, λ6233 V I-λ6232 Fe I, λ6242V I-λ6244 Si I, λ6252V I-λ6253 Fe I, λ6257 V I- λ6255 Fe I, λ6257 V I-λ6256 Fe I+Ni I). – Some LDRs seem not to suffer from rotational broaden- ing effects in almost all the effective temperature ranges 2 IDL (Interactive Data Language) is a trademark of Research Systems Incorporated (RSI). c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN () 7 Fig. 8 Synthetic LDRs at 3 differentTeff as a function of v sin i for the line pair λ6211 Sc I-λ6215 Fe I+Ti I. The synthetic spectra are referred to a star with log g=2.5, [Fe/H]=0.0, [α/Fe]=0.0, vt=1.8, and have been degraded to the same resolution of ELODIE. (for example λ6199 V I-λ6200 Fe I, λ6214 V I-λ6213 Fe I, λ6275 V I-λ6270 Fe I), as expected for weak lines. – Some line pairs show variation due to rotational broad- ening for a temperature lower than about 4500 K and 5000 K for giant and MS stars, respectively (e.g., λ6243 V I-λ6247 Fe II, λ6269 V I-λ6270 Fe I). – Other couples display changes at any temperature (e.g., λ6211 Sc I-λ6215 Fe I+Ti I, λ6216 V I-λ6215 Fe I+Ti I, λ6252 V I-λ6253 Fe I, λ6266 V I-λ6265 Fe I). – The slope of rc − Teff calibration in some ratios de- creases with the increase in the rotation (e.g., λ6243V I- λ6246 Fe I), while in some other ratios increases with the rotation (e.g., λ6216 V I-λ6215 Fe I+Ti I). – In any case, the variation of the slope with v sin i seems to be more evident for giant stars compared to MS stars (e.g., λ6243 V I-λ6246 Fe I and λ6269 V I-λ6270 Fe I). This behaviour can be due to the fact that the giants have narrower lines compared to MS stars, due to their lower atmospheric density. As a consequence, they ap- pear to be sensitive to the rotational broadening, while the gravity broadening in MS stars is comparable to the rotational broadening at low rotational velocity and the effect on LDR comes out to be less prominent. 3.4.1 Temperature sensitivity on LDR In order to measure the temperature sensitivity of each LDR, the slopes dT of the polynomial fits have been calculated. The dT absolute values at a temperature of 5000 K are listed in Table 2 for the calibrations of the giant and MS stars not rotationally broadened. Values of about 10–30 K and, in some case, even smaller, have been found for a 0.01 variation of rc, which represents the typical uncertainty for the LDR determination in well-exposed spectra. Stars below 4000 K and above 6200 K have the most uncertain temper- atures because of the influence of molecular bands in the coolest stars and the very small depths of the low-excitation lines in the hottest stars of our sample, respectively. Thus, the temperature sensitivity changes from an LDR to another one, but it is also a function of the rotational ve- locity. Fig. 9 displays an example of the variation of the tem- perature sensitivity with the rotational velocity of the cou- ple λ6211 Sc I-λ6215 Fe I+Ti I obtained for the giant cali- bration. For this ratio, the temperature sensitivity decreases with the increasing rotational broadening. As we mentioned before, other LDRs display different behaviours of the tem- perature sensitivity as a function of v sin i. The coefficients (c0, ..., cn) of the fits and the values of and rms obtained at v sin i=0, 15, and 30 km s−1 are listed in Tables 2, 3 4, respectively. 4 Conclusion In this work we have performed LDR–Teff calibrations from high-resolution spectra taking into account the corrections for the gravity effect and the rotational broadening. To our knowledge, this is the first work in which the dependence of LDRs on rotational velocity has been considered on real spectra. In previous studies, this dependence had not been taken into account, also following Gray & Johanson’s (1991) early suggestions that considered the effect of v sin i to be negligible. Here we demonstrate that the rotational broadening effect is already evident at 5 km s−1 both in synthetic and real spectra of cool stars. Its effect can be ne- glected only in particular temperature domains and for a few line pairs. In other situations, the dependency of LDRs on v sin i must be properly taken into account, at least at a spec- tral resolution as high as 42 000. We show that neglecting this effect can lead to temperature overestimates as high as ≈100 K. www.an-journal.org c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 8 K. Biazzo et al.: Effective temperature vs line-depth ratio for ELODIE spectra Table 2 Coefficients of the fits Teff = c0 + c1rc + ...+ cnrnc for the ELODIE spectra not rotationally broadened. GIANT STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 5658.52 5630.37 5357.34 8808.49 4976.68 5328.31 5375.97 5435.02 c1 −1565.89 −2563.00 −846.737 −10484.1 697.119 −749.330 −292.110 −8.57992 c2 1384.63 2766.46 −176.810 10403.9 −849.779 −150.672 −25.7998 −654.941 c3 −807.719 −1384.76 −80.6700 −3804.58 184.978 −3.71716 0.715242 0.01 | dT |5000 K 10.4 11.0 12.4 14.9 5.2 10.9 4.0 12.8 rms 0.080 0.060 0.070 0.071 0.081 0.078 0.085 0.068 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 5479.94 5630.09 5204.23 5407.48 5324.25 5346.80 5335.75 c1 −325.954 −1935.08 −53.5851 −2091.73 −1625.05 −1054.34 −327.402 c2 4.71410 2496.75 −325.674 2640.38 1431.70 619.856 −313.672 c3 1.43373 −1976.03 74.5653 −1956.14 −1193.13 −343.396 −42.0121 0.01 | dT |5000 K 4.1 13.2 6.3 12.9 13.9 9.2 10.1 rms 0.070 0.054 0.068 0.062 0.063 0.081 0.081 DWARF STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 6592.19 6298.64 6303.41 8075.38 6181.66 6204.32 6122.76 6583.91 c1 −4053.01 −4813.87 −5018.31 −6053.31 −1077.50 −3855.13 −895.008 −3240.32 c2 4020.50 5670.56 6604.68 4843.37 197.649 3379.47 158.882 2763.93 c3 −1663.63 −2457.28 −3562.99 −1688.39 −7.73603 −1175.00 −8.65057 −1211.49 0.01 | dT |5000 K 11.0 10.1 13.4 15.7 4.5 12.4 2.8 13.2 rms 0.065 0.060 0.040 0.057 0.085 0.047 0.087 0.049 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 6250.15 6486.68 6178.37 6210.10 6228.96 6120.07 6271.19 c1 −732.154 −5065.09 −1203.98 −5244.93 −6031.24 −2734.59 −1994.09 c2 143.714 7059.62 261.964 6959.06 9450.38 1909.74 716.031 c3 −9.97940 −4105.26 −19.2823 −3796.17 −5981.44 −559.915 −138.664 0.01 | dT |5000 K 1.4 15.2 5.0 16.5 18.0 8.5 9.8 rms 0.074 0.043 0.067 0.047 0.053 0.067 0.076 This paper provides calibrations that can be used for temperature determinations of stars with 0≤ v sin i <∼ 30 km s−1 and observed with ELODIE as well as with other spectrographs at a similar resolution. We have shown that all the LDRs examined in the present paper display a good sensitivity to the effective temperature which allows us to reach an uncertainty lower than 10–15 K at R = 42 000 for stars rotating up to v sin i=30 km s−1. The simultaneous use of several LDRs allows us to improve the temperature sensitivity and to fully detect and analyse effective tempera- ture modulations produced by cool starspots in active stars, as we have already shown in previous works (Frasca et al. 2005, Biazzo et al. 2007). Acknowledgements. This work has been supported by the Italian National Institute for Astrophysics (INAF), by the Italian Minis- tero dell’Istruzione, Università e Ricerca (MIUR) and by the Re- gione Sicilia which are gratefully acknowledged. We thank the ref- eree for useful suggestions. We are also grateful to Mrs. Luigia Santagati for the English revision of the text. This research has made use of SIMBAD and VIZIER databases, operated at CDS, Strasbourg, France. References Allen, C.W.: 1973, Astrophysical Quantities, London: Athlone Fig. 9 Example of variation of the temperature sensitivity with the rotational velocity obtained for the giant calibration λ6211 Sc I-λ6215 Fe I+Ti I. Alonso, A., Arribas, S., Martı́nez-Roger, C.: 1996, A&A 313, 873 Alonso, A., Arribas, S., Martı́nez-Roger, C.: 1999, A&A 140, 261 Bernacca, P.L., Perinotto, M.: 1970, Contr. Oss. Astrof. Padova in Asiago, 239 Biazzo, K.: 2001, Master Thesis, Catania University c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org Astron. Nachr. / AN () 9 Table 3 Coefficients of the fits Teff = c0 + c1rc + ... + cnrnc for the ELODIE spectra rotationally broadened to 15 km s−1. The empty columns relate to the line pairs completely blended at this rotational velocity. GIANT STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 5249.84 5559.50 8090.67 5453.71 5355.55 5662.56 c1 245.810 −2741.36 −9105.81 −489.599 −879.618 −670.393 c2 −1064.84 3273.61 9705.52 −34.6856 −138.541 −13.6821 c3 306.601 −1888.48 −3948.66 33.7288 35.1345 0.01 | dT |5000 K 9.8 13.5 15.6 5.1 12.6 8.0 rms 0.090 0.057 0.065 0.087 0.068 0.070 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 5400.03 5051.44 5605.92 5297.24 5237.87 5202.02 c1 −231.533 −1505.89 −3632.92 −1585.45 −526.586 203.540 c2 13.1768 4285.89 5882.55 940.850 −111.045 −1183.49 c3 −0.368023 −1829.64 −4511.96 −817.658 47.8455 391.671 0.01 | dT |5000 K 2.4 15.5 15.9 15.8 7.9 10.0 rms 0.072 0.055 0.050 0.056 0.087 0.088 DWARF STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 6537.09 6205.81 7824.72 6216.99 6451.17 6522.90 c1 −3721.86 −5454.44 −5415.18 −1442.34 −5424.86 −2833.56 c2 3167.41 7964.98 3678.15 532.435 5770.74 1687.54 c3 −1119.43 −4361.33 −1198.04 −70.4289 −2439.22 −485.220 0.01 | dT |5000 K 10.7 11.8 17.9 3.2 16.0 11.0 rms 0.061 0.054 0.048 0.089 0.036 0.053 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 6194.16 6312.44 6347.22 5964.93 5988.72 6264.03 c1 −549.734 −4404.66 −7231.99 −4180.21 −2177.96 −2135.85 c2 83.9286 5255.34 12331.5 4047.15 1527.77 1057.32 c3 −4.27056 −3379.88 −8240.08 −1716.93 −483.352 −322.180 0.01 | dT |5000 K 1.0 18.6 16.4 18.7 6.9 10.1 rms 0.075 0.033 0.053 0.049 0.082 0.087 Biazzo, K.: 2006, Ph.D. Thesis, Catania University Biazzo, K., Frasca, A., Henry, G.W. et al.: 2007, ApJ 656, 474 Castelli, F., Kurucz, R.L.: 2003, in Modelling of Stellar Atmo- spheres, IAU Symp., No. 210, Poster Proc., ed. Piskunov N., Weiss W.W., Gray, D.F., A20 Catalano, S., Biazzo, K., Frasca, A., Marilli, E.: 2002a, A&A 394, Catalano, S., Biazzo, K., Frasca, A. et al.: 2002b, AN 323, 260 Coelho, P., Barcuy, B., Meléndez, J. et al.: 2005, A&A 443, 735 Cox, A.N.: 2000, Allen’s Astrophysical Quantities, 4th ed., New York: AIP Press and Springer-Verlag de Medeiros, J.R., Mayor, M.: 1999, A&AS 139, 433 ESA: 1997, The Hipparcos and Tycho Catalog, ESA SP-1200 Flower, P.J.: 1996, ApJ 469, 355 Frasca, A., Biazzo, K., Catalano, S. et al.: 2005, A&A 432, 647 Girardi, L., Bressan, A., Bertelli, G., Chiosi, C.: 2000, A&AS 141, Glebocki, R., Gnacinski, P., Stawikowski, A.: 2000, Acta Astron. 50, 509 Gray, D.F.: 1994, PASP 106, 1248 Gray, D.F.: 2005, The Observation and Analysis of Stellar Photo- spheres, 3rd ed., Cambridge University Press Gray, D.F., Johanson, H.L.: 1991, ApJ 103, 439 Gray, D.F., Baliunas, S.L., Lockwood, G.W. et al.: 1992, ApJ 400, Gray, D.F., Brown, K.: 2001, PASP 113, 723 Griffin, R.F.: 1968, A Photometric Atlas of the Spectrum of Arc- turus: λ 3600–8825 Å, Cambridge Philosophsical Society Hoffleit, D., Warren, W.H.Jr.: 1991, The Bright Star Catalogue, 5th ed. riv., http://cdsweb.u-strasbg.fr/cats/Cats.htx Kurucz, R.L.: 1993, ATLAS9 Stellar Atmosphere Programs and 2 km s−1 grid, Kurucz CD-ROM No. 13 Mermilliod, J.-C., Mermilliod, M.: 1998, http://obswww.unige.ch/gcpd/cgi-bin/photoSys.cgi Moore, C.E., Minnaert, M.G.J., Houtgast, J.: 1966, The Solar Spectrum 2935 Å to 8770 Å, National Bureau of Standards: Washington, Monograph 61 Moultaka, J., Ilovaisky, S.A., Prugniel, P., Soubiran, C.: 2004, PASP 116, 693 Prugniel, P., Soubiran, C.: 2001, A&A 369, 1048 Savage, B.D., Mathis, J.S. 1979, ARA&A 17, 73 Soubiran, C., Katz, D., Cayrel, R.: 1998, A&AS 133, 221 Stift, M.J., Strassmeier, K.G.: 1995, in Stellar Surface Structure, IAU Symp., No. 176, Poster Proc., ed. Strassmeier, K.G., 29 Strassmeier, K.G., Fekel, F.C.: 1990, A&A 230, 389 Strassmeier, K.G., Rice, J.B., Wehlau, W.H. et al.: 1993, A&A 268, 671 Strassmeier, K.G., Schordan, P.: 2000, AN 321, 277 www.an-journal.org c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://cdsweb.u-strasbg.fr/cats/Cats.htx http://obswww.unige.ch/gcpd/cgi-bin/photoSys.cgi 10 K. Biazzo et al.: Effective temperature vs line-depth ratio for ELODIE spectra Table 4 Coefficients of the fits Teff = c0 + c1rc + ... + cnrnc for the ELODIE spectra rotationally broadened to 30 km s−1. The empty columns relate to the line pairs completely blended at this rotational velocity. GIANT STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 5481.25 5496.39 7025.27 5695.10 c1 −596.797 −2961.51 −5597.99 −800.163 c2 −43.7139 4228.71 5957.99 75.3450 c3 −48.3037 −3087.51 −2710.79 0.01 | dT |5000 K 9.3 16.1 16.5 8.4 rms 0.095 0.058 0.061 0.075 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 5337.08 5150.51 5190.59 5357.17 c1 −166.813 −825.095 −500.186 −591.566 c2 8.35369 70.5176 28.5439 −202.882 c3 −0.153085 −989.250 6.34586 92.0652 0.01 | dT |5000 K 1.7 15.1 7.5 9.5 rms 0.075 0.071 0.089 0.085 DWARF STARS LDR 1/2 3/6 5/4 7/6 9/8 11/10 12/14 13/15 c0 6432.25 5999.97 7479.52 6874.46 c1 −3039.71 −4545.30 −3706.65 −3697.12 c2 2161.86 6247.09 1147.46 2312.46 c3 −692.710 −3573.28 −116.692 −595.265 0.01 | dT |5000 K 10.3 11.6 18.9 11.1 rms 0.066 0.046 0.051 0.079 LDR 13/16 17/18 21/19 21/20 23/22 24/25 26/25 c0 6082.76 5895.59 5712.75 6039.12 c1 −329.412 −4065.13 −1362.68 −885.351 c2 30.3591 5641.78 931.915 −863.622 c3 −0.945513 −5297.51 −312.388 527.239 0.01 | dT |5000 K 1.0 20.0 5.0 10.1 rms 0.080 0.033 0.085 0.090 c© WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org Introduction The star sample Sample selection Effective temperature and luminosity LDR analysis and results Line identification Metallicity effect Gravity effect Rotational broadening effect Temperature sensitivity on LDR Conclusion

0704.1457

Chaos and Symmetry in String Cosmology

arXiv:0704.1457v1 [gr-qc] 11 Apr 2007 Chaos and Symmetry in String Cosmology1 Thibault Damour Institut des Hautes Etudes Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Abstract: We review the recently discovered interplay between chaos and symmetry in the general inhomogeneous solution of many string-related Einstein-matter systems in the vicinity of a cosmo- logical singularity. The Belinsky-Khalatnikov-Lifshitz-type chaotic behaviour is found, for many Einstein-matter models (notably those related to the low-energy limit of superstring theory andM-theory), to be connected with certain (infinite-dimensional) hyperbolic Kac- Moody algebras. In particular, the billiard chambers describing the asymptotic cosmological behaviour of pure Einstein gravity in spacetime dimension d + 1, or the metric-three-form system of 11- dimensional supergravity, are found to be identical to the Weyl chambers of the Lorentzian Kac-Moody algebras AEd, or E10, re- spectively. This suggests that these Kac-Moody algebras are hidden symmetries of the corresponding models. There even exists some evidence of a hidden equivalence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10. 1Invited talk at the 11th Marcel Grossmann Meeting on Recent Developments in Gen- eral Relativity, Berlin, Germany, 23-29 July 2006. http://arxiv.org/abs/0704.1457v1 1 Introduction We wish to review a recently discovered intriguing connection between two, seemingly antagonistic, structures present in pure Einstein gravity, and in some, string-theory motivated, Einstein-matter systems. On the one hand, Belinsky, Khalatnikov and Lifshitz (BKL) [1] discov- ered that the asymptotic behaviour of the general solution of the (3 + 1)- dimensional Einstein’s equations, in the vicinity of a cosmological singularity, exhibited a chaotic structure: they showed that, because of non-linearities in Einstein equations, the generic, inhomogeneous solution behaves as a chaotic [2] sequence of “generalized Kasner solutions”. They then showed that pure gravity in 4+1 dimensions exhibits a similar chaotic structure [3]. The exten- sion of the BKL analysis to pure gravity in higher dimensions was addressed in [4, 5]. A surprising result was found: while the general behaviour of the vacuum Einstein solutions remains “chaotic” (i.e. “oscillatory”) for space- time dimensions D ≤ 10, it ceases to be so for spacetime dimensions D ≥ 11, where it becomes monotonic and Kasner-like. On the other hand, a symmetry structure was found to be present (often in a hidden form) in the “dimensional reductions” of Einstein’s gravity. The paradigm of this hidden symmetry structure is the continuous SL(2,R)E symmetry group found long ago by Ehlers [6] for D = 4 Einstein gravity in the presence of one Killing vector. When two commuting Killing vectors are present, the finite-dimensional Ehlers group SL(2,R) was found, through, notably, the work of Matzner and Misner, of Geroch, of Julia, of Breiten- lohner and Maison, and of Belinsky and Zakharov, to be promoted to an infinite-dimensional symmetry group, which can be identified to the affine Kac-Moody extension of SL(2,R) (see [7] for references and a review). A similar pattern was found to take place in supergravity theories, and notably in the “maximal” supergravity theory which lives in D = 11. In that case, it was remarkably found [8] that the successive toroidal dimensional reductions of D = 11 supergravity to lower dimensions (in the presence of an increasing number of commuting Killing vectors) admitted a correspondingly larger and larger hidden symmetry group En(R), where n denotes the number of Killing vectors, and En the (extended) sequence of the groups belonging to the excep- tional series in the Cartan-Killing classification of finite-dimensional simple Lie groups. The latter series culminates into the last finite-dimensional ex- ceptional group E8 in the case where one has 8 Killing vectors, i.e. in the case of toroidal compactification (on T 8) down to a D′ = 3 dimensionally reduced theory. However, as first conceived by Julia [9], this symmetry- increasing pattern is expected to continue to the (infinite-dimensional) affine Kac-Moody group E9 for the reduction of supergravity down to D ′ = 2 (9 Killing vectors). This was indeed explicitly proven later [10]. It was also mentionned by Julia [9] that the still larger symmetry group E10 (which is now an infinite-dimensional hyperbolic Kac-Moody group2) might arise when trying to further reduce maximal supergravity. However, this poses a serious challenge because the naive dimensional reduction to D′ = 1, i.e. the set of D = 11 supergravity solutions which depend only on one (time) variable is much too small to (faithfully) carry such a huge symmetry. At this point, the two separate threads (chaos and symmetry) in our history unexpectedly merged in a sequence of works which, while extend- ing the BKL analysis to D = 11 supergravity, and to the D = 10 models describing the low-energy limit of the various superstring theories [12, 13], found that behind their seeming entirely chaotic3 BKL-type behaviour there were hints of a hidden symmetry structure [14, 15, 16] linked to E10. More precisely, it was first found [14] that the “billiard chamber” (see [17] and below) describing the BKL-type behaviour of D = 11 supergravity (as well as of D = 10 IIA and IIB superstring theories) could be identified with the “Weyl chamber” of E10. [The cosmological billiard chambers of type I and heterotic string theories coincides with the Weyl chamber of BE10. We fo- cus here on the more fundamental E10 case.] Similarly, an examination of 2See [11] for an introduction to the theory of Kac-Moody algebras. 3Though D = 11 pure gravity is monotonically Kasner-like instead of chaotic, the presence of the three-form A in D = 11 supergravity was found to reintroduce a generic chaotic, oscillatory behaviour. the case of pure gravity in spacetime dimension D ≡ d + 1 revealed the presence of the Weyl chamber of the Lorentzian Kac-Moody algebras AEd (which are hyperbolic only when d ≤ 9) [15]. Then it was shown that, up to height 30 in a “height expansion” related to the BKL “gradient expan- sion”, the dynamics of the 11-dimensional supergravity variables could be identified with the dynamics of a null geodesic on the infinite-dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10 [16]. This led to the conjecture that there exists a hidden equivalence between D = 11 supergravity and null geodesic motion on E10/K(E10). This conjecture can be generalized to other gravity models and other coset spaces [17]. In particular, usual (3 + 1)-dimensional gravity might be equivalent to null geodesic motion on AE3/K(AE3). If these conjectures were true, they would mean that the infinite-dimensional numerator Kac-Moody groups, E10 or AE3, are hidden continuous “symmetries” of the respective field equations (supergravity11 or gravity4) which transform solutions into (new) solutions. In the case of D = 4 gravity, this would represent a huge generalization4 of the Ehlers SL(2,R) symmetry group. The present paper is organized as follows. In Section 2 we summa- rize the Hamiltonian “cosmological billiard” approach to BKL behaviour (mainly drawing from [17]). See [18] and the contribution of C. Uggla to these proceedings for a comparison between this Hamiltonian cosmological billiard approach and a “Hubble-normalized” “dynamical systems picture”. In Section 3 we sketch the “correspondence” between a gravity model (e.g. D = 11 supergravity) and a coset geodesic dynamics (e.g. geodesic mo- tion on E10/K(E10)). For more details on the construction and structure of those Kac-Moody coset models see the contribution of A. Kleinschmidt and H. Nicolai to these proceedings. Finally, Section 4 offers some (speculative) conclusions. 4This is a generalization in two separate senses: (i) the conjectured AE3 symmetry would apply in absence of any Killing field, and (ii) AE3 is the hyperbolic Kac-Moody group canonically associated with A1 = SL(2) (while the “Geroch group” is the affine Kac-Moody group associated with A1 = SL(2)). 2 Cosmological billiards Let us start by summarizing the BKL-type analysis of the “near spacelike singularity limit”, that is, of the asymptotic behaviour of the metric gµν(t,x), together with the other fields (such as the 3-form Aµνλ(t,x) in supergravity), near a singular hypersurface. The basic idea is that, near a spacelike singu- larity, the time derivatives are expected to dominate over spatial derivatives. More precisely, BKL found that spatial derivatives introduce terms in the equations of motion for the metric which are similars to the “walls” of a billiard table [1]. To see this, it is convenient [17] to decompose the D- dimensional metric gµν into non-dynamical (lapse N , and shift N i, here set to zero) and dynamical (e−2β , θai ) components. They are defined so that the line element reads ds2 = −N2dt2 + θai θ idxj . (1) Here d ≡ D − 1 denotes the spatial dimension (d = 10 for SUGRA11, and d = 9 for string theory), e−2β represent (in an Iwasawa decomposition) the “diagonal” components of the spatial metric gij, while the “off diagonal” components are represented by the θai , defined to be upper triangular matrices with 1’s on the diagonal (so that, in particular, det θ = 1). The Hamiltonian constraint, at a given spatial point, reads (with Ñ ≡ det gij denoting the “rescaled lapse”) H(βa, πa, P, Q) Gabπaπb + cA(Q,P, ∂β, ∂ 2β, ∂Q) exp − 2wA(β) . (2) Here πa (with a = 1, ..., d) denote the canonical momenta conjugate to the “logarithmic scale factors” βa, while Q denote the remaining configuration variables (θai , 3-form components Aijk(t,x) in supergravity), and P their canonically conjugate momenta (P ia, π ijk). The symbol ∂ denotes spatial derivatives. The (inverse) metric Gab in Eq. (2) is the DeWitt “superspace” metric induced on the β’s by the Einstein-Hilbert action. It endows the d-dimensional5 β space with a Lorentzian structure Gab β̇ aβ̇b. One of the crucial features of Eq. (2) is the appearance of Toda-like exponential potential terms ∝ exp(−2wA(β)), where the wA(β) are linear forms in the logarithmic scale factors: wA(β) ≡ wAa βa. The range of labels A and the specific “wall forms” wA(β) that appear depend on the considered model. For instance, in SUGRA11 there appear: “symmetry wall forms” wSab(β) ≡ βb − βa (with a < b), “gravitational wall forms” w abc(β) ≡ 2βa + e 6=a,b,c βe (a 6= b, b 6= c, c 6= a), “electric 3-form wall forms”, eabc(β) ≡ βa + βb + βc (a 6= b, b 6= c, c 6= a), and “magnetic 3-form wall forms”, ma1....a6 ≡ βa1 + βa2 + ...+ βa6 (with indices all different). One then finds that the near-spacelike-singularity limit amounts to con- sidering the large β limit in Eq.(2). In this limit a crucial role is played by the linear forms wA(β) appearing in the “exponential walls”. Actually, these walls enter in successive “layers”. A first layer consists of a sub- set of all the walls called the dominant walls wi(β). The effect of these dynamically dominant walls is to confine the motion in β-space to a fun- damental billiard chamber defined by the inequalities wi(β) > 0. In the case of SUGRA11, one finds that there are 10 dominant walls: 9 of them are the symmetry walls wS12(β), w 23(β), ..., w 910(β), and the 10th is an elec- tric 3-form wall e123(β) = β 1 + β2 + β3. As noticed in [14] a remarkable fact is that the fundamental cosmological billiard chamber of SUGRA11 (as well as type-II string theories) is the Weyl chamber of the hyperbolic Kac-Moody algebra E10. More precisely, the 10 dynamically dominant wall forms wS12(β), w 23(β), ..., w 910(β), e123(β) can be identified with the 10 sim- ple roots {α1(h), α2(h), ..., α10(h)} of E10. Here h parametrizes a generic el- ement of a Cartan subalgebra (CSA) of E10 . [Let us also note that for Heterotic and type-I string theories the cosmological billiard is the Weyl 510 dimensional for SUGRA11; but the various superstring theories also lead to a 10 dimensional Lorentz space because one must add the (positive) kinetic term of the dilaton ϕ ≡ β10 to the 9-dimensional DeWitt metric corresponding to the 9 spatial dimensions. chamber of another rank-10 hyperbolic Kac-Moody algebra, namely BE10]. In the Dynkin diagram of E10, Fig. 1, the 9 “horizontal” nodes correspond to the 9 symmetry walls, while the characteristic “exceptional” node sticking out “vertically” corresponds to the electric 3-form wall e123 = β 1 + β2 + β3. [The fact that this node stems from the 3rd horizontal node is then seen to be directly related to the presence of the 3-form Aµνλ, with electric kinetic energy ∝ giℓgjmgknȦijkȦℓmn]. α1 α2 α3 α4 α5 α6 α7 α8 α9 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Figure 1: Dynkin diagram of E10. The appearance of E10 in the BKL behaviour of SUGRA11 revived the old suggestion of [9] about the possible role of E10 in a one-dimensional re- duction of SUGRA11. A posteriori, one can view the BKL behaviour as a kind of spontaneous reduction to one dimension (time) of a multidimensional theory. Note, however, that we are always discussing generic inhomogeneous 11-dimensional solutions, but that we examine them in the near-spacelike- singularity limit where the spatial derivatives are sub-dominant: ∂x ≪ ∂t. Note also that the discrete E10(Z) was proposed as a U -duality group of the full (T 10) spatial toroidal compactification of M-theory by Hull and Townsend [19]. 3 Gravity/Coset correspondence Refs [16, 20] went beyond the leading-order BKL analysis just recalled by in- cluding the first three “layers” of spatial-gradient-related sub-dominant walls ∝ exp(−2wA(β)) in Eq.(2). The relative importance of these sub-dominant walls, which modify the leading billiard dynamics defined by the 10 dom- inant walls wi(β), can be ordered by means of an expansion which counts how many dominant wall forms wi(β) are contained in the exponents of the sub-dominant wall forms wA(β), associated to higher spatial gradients. By mapping the dominant gravity wall forms wi(β) onto the corresponding E10 simple roots αi(h), i = 1, ..., 10, the just described BKL-type gradient ex- pansion becomes mapped onto a Lie-algebraic height expansion in the roots of E10. It was remarkably found that, up to height 30 (i.e. up to small corrections to the billiard dynamics associated to the product of 30 leading walls e−2wi(β)), the SUGRA11 dynamics for gµν(t,x), Aµνλ(t,x), considered at some given spatial point x0, could be identified to the geodesic dynam- ics of a massless particle moving on the (infinite-dimensional) coset space E10/K(E10). Note the “holographic” nature of this correspondence between an 11-dimensional dynamics on one side, and a 1-dimensional one on the other side. A point on the coset space E10(R)/K(E10(R)) is coordinatized by a time- dependent (but spatially independent) element of the E10(R) group of the (Iwasawa) form: g(t) = exp h(t) exp ν(t). Here, h(t) = βacoset(t)Ha belongs to the 10-dimensional CSA of E10, while ν(t) = α>0 ν α(t)Eα belongs to a Borel subalgebra of E10 and has an infinite number of components labelled by a positive root α of E10. The (null) geodesic action over the coset space E10/K(E10) takes the simple form SE10/K(E10) = (vsym|vsym) (3) where vsym ≡ 1 (v + vT ) is the “symmetric”6 part of the “velocity” v ≡ (dg/dt)g−1 of a group element g(t) running over E10(R). The correspondence between the gravity, Eq. (2), and coset, Eq. (3), dy- namics is best exhibited by decomposing (the Lie algebra of) E10 with respect 6Here the transpose operation T denotes the negative of the Chevalley involution ω defining the real form E10(10) of E10. It is such that the elements k of the Lie sub-algebra ofK(E10) are “T -antisymmetric”: k T = −k, which is equivalent to them being fixed under ω : ω(k) = +ω(k). to (the Lie algebra of) the GL(10) subgroup defined by the horizontal line in the Dynkin diagram of E10. This allows one to grade the various components of g(t) by their GL(10) level ℓ. One finds that, at the ℓ = 0 level, g(t) is parametrized by the Cartan coordinates βacoset(t) together with a unimodu- lar upper triangular zehnbein θacoset i(t). At level ℓ = 1, one finds a 3-form Acosetijk (t); at level ℓ = 2, a 6-form A coset i1i2...i6 (t), and at level ℓ = 3 a 9-index object Acoseti1|i2...i9(t) with Young-tableau symmetry {8, 1}. The coset action (3) then defines a coupled set of equations of motion for βacoset(t), θ coset i(t), Acosetijk (t), A coset i1...i6 (t), Acoseti1|i2...i9(t). By explicit calculations, it was found that these coupled equations of motion could be identified (modulo terms corre- sponding to potential walls of height at least 30) to the SUGRA11 equations of motion, considered at some given spatial point x0. The dictionary between the two dynamics says essentially that: (0) βagravity(t,x0) ↔ βacoset(t) , θai (t,x0) ↔ θacoset i(t), (1) ∂t Acosetijk (t) corre- sponds to the electric components of the 11-dimensional field strength Fgravity = dAgravity in a certain frame e i, (2) the conjugate momentum of Acoseti1...i6(t) corresponds to the dual (using εi1i2...i10) of the “magnetic” frame compo- nents of the 4-form Fgravity = dAgravity, and (3) the conjugate momentum of Ai1|i2...i9(t) corresponds to the ε 10 dual (on jk) of the structure constants C ijk of the coframe ei (d ei = 1 C ijk e j ∧ ek). The fact that at levels ℓ = 2 and ℓ = 3 the dictionary between supergrav- ity and coset variables maps the first spatial gradients of the SUGRA variables Aijk(t,x) and gij(t,x) onto (time derivatives of) coset variables suggested the conjecture [16] of a hidden equivalence between the two models, i.e. the existence of a dynamics-preserving map between the infinite tower of (spa- tially independent) coset variables (βacoset, ν α), together with their conjugate momenta (πcoseta , pα), and the infinite sequence of spatial Taylor coefficients (β(x0), π(x0), Q(x0), P (x0), ∂Q(x0), ∂ 2β(x0), ∂ 2Q(x0), . . . , ∂ nQ(x0), . . .) formally describing the dynamics of the gravity variables (β(x), π(x), Q(x), P (x)) around some given spatial point x0. 7One, however, expects the map between the two models to become spatially non-local It has been possible to extend the correspondence between the two models to the inclusion of fermionic terms on both sides [21, 22, 23]. Moreover, Ref. [24] found evidence for a nice compatibility between some high-level contributions (height −115!) in the coset action, corresponding to imaginary roots8, and M-theory one-loop corrections to SUGRA11, notably the terms quartic in the curvature tensor. (See also [25] for a study of the compatibility of an underlying Kac-Moody symmetry with quantum corrections in various models). 4 Conclusions and outlook At this stage, we are far from having a proof of the full equivalence between SUGRA11 and the E10/K(E10) model, or of any other gravity/coset con- jectured pair. The partial evidence summarized above is suggestive of the existence of some kind of hidden symmetry structure in General Relativity and Supergravity. However, it is quite possible that this hidden symmetry is present only in a way which cannot be explicitly realized at the level of the classical field equations. Indeed, the situation might be similar to that discussed in the plenary talks of Sasha Polyakov, Igor Klebanov and Eva Silverstein. In the much better understood gravity/gauge correspondence a quasi-classical, weakly curved spacetime corresponds to a strongly quan- tum, strongly self-interacting gauge theory state. Reciprocally, the perturba- tive, weakly-interacting gauge theory states correspond to non-perturbative, strongly curved gravity states. By analogy, we might expect that the sim- ple geodesic coset motions correspond to strongly curved spacetimes, with curvatures larger than the Planck scale. [This would intuitively explain why the coset picture becomes prominent in the formal limit where one tends towards an infinite-curvature singularity.] In addition, it is possible that the for heights ≥ 30. 8i.e. such that (α, α) < 0, by contrast to the “real” roots, (α, α) = +2, which enter the checks mentionned above. equivalence between the gravity and coset models hold only at the level of the quantized models. In this respect, note that the quantum version of the null geodesic dynamics (3) would be, if we neglect polarization effects9 a Klein-Gordon equation10, �Ψ(βa, να) = 0 , (4) where � denotes the (formal) Laplace-Beltrami operator on the infinite- dimensional Lorentz-signature curved coset manifold E10(R)/K(E10(R)). As recently emphasized [28], the gravity/coset correspondence sketched above suggests a new physical picture of the fate of space at a cosmological singularity. It suggests that, upon approaching a spacelike singularity, the description in terms of a spatial continuum breaks down and should be re- placed by a purely abstract Lie algebraic description. In other words, one is led to the conclusion that space actually “disappears” (or “de-emerges”) as the singularity is approached11. The gravity/coset duality suggests that there is no (quantum) “bounce” from an incoming collapsing universe to some out- going expanding one [24]. Rather, it is suggested that “life continues” for an infinite “affine time” at a singularity, with the double understanding that: (i) life continues only in a totally new form (as in a kind of “transmigration”), and (ii) an infinite affine time interval (measured, say, in the coordinate t of Eq. (3) with a coset lapse function n(t) = 1) corresponds to a sub-Planckian interval of geometrical proper time12. 9Actually, Refs [21, 22, 23] indicate the need to consider a spinning massless particle, i.e. some kind of Dirac equation on E10/K(E10). 10This equation, submitted to a condition of periodicity over a discrete group E10(Z), has been considered in Refs [26, 27] in the context of quantum theories with all spatial dimensions being toroidally compactified. 11We have in mind here a “big crunch”, i.e. we conventionally consider that we are tending toward the singularity. Mutatis mutandis, we would say that space “appears” or “emerges” at a big bang. 12Indeed, it is found that the coset time t, with n(t) = 1, corresponds to a “Zeno-like” gravity coordinate time (with rescaled lapse Ñ = N/ g = 1) which tends to +∞ as the proper time tends to zero. References [1] V. A. Belinsky, I. M. Khalatnikov and E. M. Lifshitz, “Oscillatory ap- proach to a singular point in the relativistic cosmology,” Adv. Phys. 19, 525 (1970). [2] I. M. Khalatnikov, E. M. Lifshitz, K. M. Khanin, L. N. Shchur and Y. G. Sinai, “On The Stochasticity In Relativistic Cosmology,” J. Stat. Phys. 38, 97 (1985). [3] V.A. Belinskii and I.M. Khalatnikov, “Effect of scalar and vector fields on the nature of the cosmological singularity”, Sov. Phys. JETP 36, 591 (1973). [4] J. Demaret, M. Henneaux and P. Spindel, Phys. Lett. B 164, 27 (1985). [5] J. Demaret, J. L. Hanquin, M. Henneaux, P. Spindel and A. Taormina, Phys. Lett. B 175, 129 (1986). [6] J. Ehlers, “Transformations of static exterior solutions of Einstein’s grav- itational field equations into different solutions by means of conformal mappings”, in: Les théories relativistes de la gravitation, Colloques in- ternationaux du CNRS 91, 275 (1962). [7] H. Nicolai, “Two-dimensional gravities and supergravities as integrable system,” Lectures presented at 30th Schladming Winter School, Schlad- ming, Austria, Feb 27 - Mar 5, 1991. In: Schladming 1991, Proceedings, Recent aspects of quantum fields 231-273 and Hamburg DESY - DESY 91-038 (91/05,rec.May) 43 p. [8] E. Cremmer and B. Julia, Nucl. Phys. B 159, 141 (1979). [9] B. Julia, “Kac-Moody Symmetry of Gravitation and Supergravity Theo- ries,” in: Lectures in Applied Mathematics, Vol. 21 (1985), AMS-SIAM, p. 335; preprint LPTENS 80/16. [10] H. Nicolai, Phys. Lett. B 194, 402 (1987). [11] V. G. Kac, “Infinite dimensional Lie algebras,” Cambridge, UK: Univ. Pr. (1990) 400 p. [12] T. Damour and M. Henneaux, Phys. Rev. Lett. 85, 920 (2000) [arXiv:hep-th/0003139]. [13] T. Damour and M. Henneaux, Phys. Lett. B 488, 108 (2000) [Erratum- ibid. B 491, 377 (2000)] [arXiv:hep-th/0006171]. [14] T. Damour and M. Henneaux, Phys. Rev. Lett. 86, 4749 (2001) [arXiv:hep-th/0012172]. [15] T. Damour, M. Henneaux, B. Julia and H. Nicolai, Phys. Lett. B 509, 323 (2001) [arXiv:hep-th/0103094]. [16] T. Damour, M. Henneaux and H. Nicolai, Phys. Rev. Lett. 89, 221601 (2002) [arXiv:hep-th/0207267]. [17] T. Damour, M. Henneaux and H. Nicolai, Class. Quant. Grav. 20, R145 (2003) [arXiv:hep-th/0212256]. [18] J. M. Heinzle, C. Uggla and N. Rohr, arXiv:gr-qc/0702141. [19] C. M. Hull and P. K. Townsend, Nucl. Phys. B 438, 109 (1995) [arXiv:hep-th/9410167]. [20] T. Damour and H. Nicolai, “Eleven dimensional supergravity and the E(10)/K(E(10)) sigma-model at low A(9) levels,” arXiv:hep- th/0410245. [21] T. Damour, A. Kleinschmidt and H. Nicolai, Phys. Lett. B 634, 319 (2006) [arXiv:hep-th/0512163]. [22] S. de Buyl, M. Henneaux and L. Paulot, JHEP 0602, 056 (2006) [arXiv:hep-th/0512292]. [23] T. Damour, A. Kleinschmidt and H. Nicolai, JHEP 0608, 046 (2006) [arXiv:hep-th/0606105]. [24] T. Damour and H. Nicolai, Class. Quant. Grav. 22, 2849 (2005) [arXiv:hep-th/0504153]. [25] T. Damour, A. Hanany, M. Henneaux, A. Kleinschmidt and H. Nicolai, Gen. Rel. Grav. 38, 1507 (2006) [arXiv:hep-th/0604143]. [26] O. J. Ganor, arXiv:hep-th/9903110. [27] J. Brown, O. J. Ganor and C. Helfgott, JHEP 0408, 063 (2004) [arXiv:hep-th/0401053]. [28] T. Damour and H. Nicolai, “Symmetries, Singularities and the De- emergence of Space”, essay submitted to the Gravity Research Foun- dation, March 2007.

0704.1458

Radio Emission from the Intermediate-mass Black Hole in the Globular Cluster G1

Radio Emission from the Intermediate-mass Black Hole in the Globular Cluster G1 James S. Ulvestad National Radio Astronomy Observatory P.O. Box O, Socorro, NM 87801 [email protected] Jenny E. Greene1 Department of Astronomy, Princeton University Princeton, NJ [email protected] Luis C. Ho The Observatories of the Carnegie Institute of Washington 813 Santa Barbara St., Pasadena, CA 91101 [email protected] ABSTRACT We have used the Very Large Array (VLA) to search for radio emission from the globular cluster G1 (Mayall-II) in M31. G1 has been reported by Geb- hardt et al. to contain an intermediate-mass black hole (IMBH) with a mass of ∼ 2 × 104 M⊙. Radio emission was detected within an arcsecond of the cluster center with an 8.4 GHz power of 2× 1015 W Hz−1. The radio/X-ray ratio of G1 is a few hundred times higher than that expected for a high-mass X-ray binary in the cluster center, but is consistent with the expected value for accretion onto an IMBH with the reported mass. A pulsar wind nebula is also a possible candidate 1Hubble Fellow http://arxiv.org/abs/0704.1458v1 – 2 – for the radio and X-ray emission from G1; future high-sensitivity VLBI observa- tions might distinguish between this possibility and an IMBH. If the radio source is an IMBH, and similar accretion and outflow processes occur for hypothesized ∼ 1000 M⊙ black holes in Milky Way globular clusters, they are within reach of the current VLA and should be detectable easily by the Expanded VLA when it comes on line in 2010. Subject headings: accretion, accretion disks — galaxies: individual (M31) — globular clusters: individual (G1) — radio continuum: galaxies 1. Introduction Most galaxies with massive spheroidal components appear to harbor central black holes (BHs), with masses ranging from a few ×106 M⊙ to over 10 9 M⊙. These BH masses are well correlated with both the luminosity and the velocity dispersion of the galaxy spheroid (Gebhardt et al. 2000; Ferrarese & Merritt 2000), implying that the formation of the central BHs is connected intimately with the development of the galaxy bulges. However, a bulge is not necessarily a prerequisite for a massive BH. On the one hand, neither the late-type spiral galaxy M33 (Gebhardt et al. 2001) nor the dwarf spheroidal galaxy NGC 205 (Valluri et al. 2005) shows dynamical evidence for a massive BH. On the other hand, both the Sd spiral galaxy NGC 4395 and the dwarf spheroidal galaxy POX 52 contain active BHs with masses ≈ 105 M⊙, although neither contains a classic bulge (Fillipenko & Ho 2003; Barth et al. 2004). Greene & Ho (2004, 2007) find that optically active intermediate-mass BHs (IMBHs, MBH < 10 6 M⊙), while rare, do exist in dwarf galaxies, but optical searches are heavily biased toward sources accreting at high Eddington rates. Alternate search techniques are needed to probe the full demographics of IMBHs. Direct dynamical detection of IMBHs is currently impossible outside of the Local Group. However, Gebhardt, Rich, & Ho (2002, 2005) have found dynamical evidence for an excess dark mass of (1.8±0.5)×104 M⊙ at the center of the globular cluster G1 in M31; the evidence for this IMBH was questioned by Baumgardt et al. (2003), but supported by the improved data and analysis of Gebhardt et al. (2005). The physical nature of the central dark object is difficult to prove: it could be either an IMBH or a cluster of stellar remnants. While the putative presence of a BH in a globular cluster center may appear unrelated to galaxy bulges, the properties of G1, including its large mass, high degree of rotational support, and multi-aged stellar populations, all suggest that G1 is actually the nucleus of a stripped dwarf galaxy (Meylan et al. 2001). Most intriguingly, the inferred BH mass for G1 is about 0.1% of the total mass, consistent with the relation seen for higher mass BHs, and consistent with – 3 – predictions based on mergers of BHs (Miller & Hamilton 2002) or stellar mergers in dense clusters (Portegies Zwart & McMillan 2002). Recently, Pooley & Rappaport (2006) reported an X-ray detection of G1, with a 0.2– 10 keV luminosity of LX ≈ 2 × 10 36 ergs s−1. Although this may represent accretion onto a central BH, it is within the range expected for either accretion onto an IMBH or for a massive X-ray binary. Unfortunately, the most accurate X-ray position determined recently by Kong (2007) does not have sufficient accuracy to determine whether the X-ray source is located within the central core of G1, which would help distinguish between these two possibilities. The radio/X-ray ratio of G1 provides an additional test of the nature of the G1 X-ray source. As pointed out by Maccarone (2004) and Maccarone, Fender, & Tzioumis (2005), deep radio searches may be a very effective way to detect IMBHs in globular clusters and related objects, since, for a given X-ray luminosity, stellar mass BHs produce far less radio luminosity than supermassive BHs. The relation between BH mass, and X-ray and radio luminosity empirically appears to follow a “fundamental plane,” in which the ratio of radio to X-ray luminosity increases as the ∼ 0.8 power of the BH mass (Merloni, Heinz, & di Matteo 2003; Falcke, Körding, & Markoff 2004). For an IMBH mass of 1.8× 104 M⊙ in G1, one thus would expect a radio/X-ray ratio about 400 times higher than for a 10 M⊙ stellar BH. In this paper, we report a deep Very Large Array (VLA) integration on G1 and a radio detection that apparently confirms the presence of an IMBH whose mass is consistent with that found by Gebhardt et al. (2002, 2005). 2. Observations and Imaging We obtained a 20-hr observation of G1 using the VLA in its C configuration (maximum baseline length of 3.5 km) at 8.46 GHz. The observation was split into two 10-hr sessions, one each on 2006 November 24/25 and 2006 November 25/26. Each day’s observation consisted of repeated cycles of 1.4 minutes observation on the local phase calibrator J0038+4137 and 6 minutes observation on the target source G1. In addition, each day contained two short observations of 3C 48 (J0137+3309) that were used to calibrate the flux density scale to that of Baars et al. (1977). Thus, the total integration time on G1 was 14.1 hr. We also obtained a total of 9.5 hr of observing in C configuration at 4.86 GHz on 2007 January 13/14 and 2007 January 14/15, using a similar observing strategy, and achieving a total of 7.3 hr of integration on source. All data calibration was carried out in NRAO’s Astronomical Image Processing System (Greisen 2003). Absolute antenna gains were determined by the 3C 48 observations, then transferred to J0038+4137, which was found to have respective flux densities of 0.52 mJy and – 4 – 0.53 mJy at 8.4 and 4.9 GHz. In turn, J0038+4137 was used to calibrate the interferometer amplitudes and phases for the target source, G1. Erroneous data were flagged by using consistency of the gain solutions as a guide and by discarding outlying amplitude points. The VLA presently is being replaced gradually by the Expanded VLA (EVLA), which includes complete replacement of virtually all the electronic systems on the telescopes. Since antennas are refurbished one at a time, the VLA at the time of our observations consisted of 18–20 “old” VLA antennas and 6 “new” (actually, refurbished) EVLA antennas, having completely different electronics systems. Although all antennas were cross-correlated for our observations, we found subtle errors in some of the EVLA data. Thus, to be conservative, we discarded the data from all EVLA antennas except for 3 antennas that were confirmed to work very well on 2006 November 24/25. The radio data were Fourier transformed and total-intensity images were produced in each band, covering areas of 17′×17′ at each frequency.2 These images were CLEANed in order to produce the final images. At 8.4 GHz, the rms noise was 6.2 µJy beam−1 for a beam size of 2.′′94×2.′′72; at 4.9 GHz, the noise was 15.0 µJy beam−1 for a beam size of 5.′′09×4.′′43. A few radio sources with strengths of hundreds of microjansky to a few millijansky were found in the images, but we discuss only G1 in this Letter.3 At 8.4 GHz, an apparent source with a flux density of 28 ± 6 µJy [corresponding to 2 × 1015 W Hz−1 for distance modulus of (m − M) = 24.42 mag (Meylan et al. 2001)] was found approximately one arcsecond from the G1 optical position reported by Meylan et al. (2001); this radio source has J2000 coordinates of α = 00h32m46.54s, δ = 39◦34′39.2′′. Figure 1 shows our 8.4 GHz image of the 20′′ by 20′′ region centered on G1; this image includes a 1 sigma error circle of 1.′′5 radius for the X-ray position found by Kong (2007). The radio position has an estimated error of 0.′′6 in each dimension (not shown in the figure), derived by dividing the beam size by the signal-to-noise ratio.4 The a priori probability of finding a 4.5 σ noise spike or background source so close to G1 is quite small, as indicated by the lack of any other contours of similar strength in 2The image areas covered were much larger than the primary beams of the individual VLA antennas, in order to provide the best possible subtraction of confusing sources. 3The strongest nearby radio source is a 1 mJy (at 8.4 GHz) object located approximately 100′′ from G1; this object and the other weak sources might be X-ray binaries or supernova remnants if located at the distance of M31, but their numbers also are consistent with the possibility that some could be background extragalactic sources. 4Transfer of the phase from the local phase calibrator makes an insignificant contribution to the radio source position error, relative to the error imposed by the limited signal strength. – 5 – Figure 1. If we hypothesize that there are 9 independent beams (roughly 8′′ by 8′′) within which a source would be considered to be associated with G1, then the probability of a 4.5 σ noise point close to G1 is less than 10−4. Similarly, the expected density of extragalactic radio sources at 28 µJy or above is 0.25 arcmin−2 (Windhorst et al. 1993), or 4 × 10−3 in a box 8′′ on a side, making it unlikely that we have found an unrelated background source. In order to search for possible data errors that might cause a spurious source, we have subjected our data set to additional tests, imaging data from the two days separately, and also imaging the two different intermediate frequency channels separately. The G1 radio source remains in the images made from each data subset, with approximately the same flux density and position. The overall significance is reduced by 21/2 to approximately 3 σ in each image made with about half the data, as expected for a real source with uncontaminated data. Other 2.5 σ–3 σ sources appear in the central 20′′ box in some subsets of half the data, consistent with noise statistics, but none is above the 3.5 σ level in the full data set. Thus, all tests indicate that the detection of G1 is real, and we will proceed on that basis for the remainder of this paper. At 4.9 GHz, we find no detection at the G1 position, but the much higher noise level provides us only with very loose constraints on the source spectrum (see below). 3. Origin of the G1 Radio Emission Merloni et al. (2003) and Falcke et al. (2004) have quantified an empirical relation (or “fundamental plane”) among X-ray and 5 GHz radio luminosity and BH mass; we use the Merloni et al. (2003) relation LR ∝ L M0.78 . Merloni et al. (2003) analyzed this relation in the context of accretion flows and jets associated with massive BHs. One might expect some general relationship among these three quantities, if an X-ray-emitting accretion flow onto a massive BH leads to creation of a synchrotron-emitting radio jet, with the detailed correlation providing some insight into the nature of that flow. By comparing the empirically determined relation with expectations from theoretical models, Merloni et al. (2003) deduced that the data for BHs emitting at only a few percent of the Eddington rate are consistent with radiatively inefficient accretion flows and a synchrotron jet, but inconsistent with standard disk accretion models. Maccarone (2004) scaled the fundamental-plane relation to values appropriate for an IMBH in a Galactic globular cluster; we rescale their equation here to find a predicted radio flux density of S5 GHz = 52 1036 ergs s−1 )0.6 ( MBH 104 M⊙ )0.78 ( d 600 kpc µJy . (1) – 6 – Using the previously cited X-ray luminosity and BH mass for G1 and our adopted distance modulus, this predicts a 5 GHz flux density of 77 µJy for G1. However, taking into account the 30% uncertainty in the IMBH mass, the unknown spectral index of the radio emission, and the dispersion of 0.88 in logLR (Merloni et al. 2003), the predicted 8.4 GHz flux density for G1 is in the range of tens to a few hundred microjansky. Thus, our radio detection of 28 µJy at 8.4 GHz is consistent with the predictions for a 1.8× 104 M⊙ IMBH, but strongly inconsistent with a 10 M⊙ BH. Since neutron star X-ray binaries in a variety of states have radio/X-ray ratios much lower than BH X-ray binaries (Migliari & Fender 2006), and thus another 2 orders of magnitude below the observed value, stellar-mass X-ray binaries of any type are ruled out as the possible origin of the radio emission in G1. We can use the radio/X-ray ratio to assess other possible origins for the radio emission. Here, we use the ratio RX = νLν(8.4 GHz)/LX(2− 10 keV) as a fiducial marker. For G1, RX ≈ 5× 10 −5, which is considerably lower than RX ≈ 10 −2 that is common to the Galactic supernova remnant Cas A, low-luminosity active galactic nuclei (supposing G1 might be a stripped dwarf elliptical galaxy), and most ultraluminous X-ray sources (cf. Table 2 of Neff, Ulvestad, & Campion [2003], and references therein). It is of interest to compare the G1 source to various relatives of pulsars as well. For instance, G1 is within the wide range of both luminosity and radio/X-ray ratio observed for pulsar wind nebulae (PWNs) (Frail & Scharringhausen 1997), less luminous than the putative PWN in M81 (Bietenholz, Bartel, & Rupen 2004), but considerably more luminous than standard pulsars or anomalous X-ray pulsars (Halpern et al. 2005). The 8.4 GHz luminosity of G1 is similar to that of the magnetar SGR 1806 − 20 about 10 days after its outburst in late 2004, and the lack of a 4.9 GHz detection would be consistent with the fading of SGR 1806−20 two months after the outburst (Gaensler et al. 2005). However, there is no published evidence for a gamma-ray outburst from G1, and the relatively steady apparent X-ray flux (Pooley & Rappaport 2006) also argues against a transient source. Thus, the only stellar-mass object that might account for the radio and X-ray emission would be a PWN; using the scaling law given by Frail & Scharringhausen (1997), we find a likely size of ∼ 10 milliarcseconds for a PWN radio source in G1, implying that high-sensitivity VLBI observations could distinguish between a PWN and IMBH origin for the radio emission from Knowledge of the radio spectrum of G1 could provide more clues to the character of the radio emission, although either a PWN or an IMBH accretion flow might have a flat spectrum. In any case, our 5 GHz observation simply is not deep enough. If we choose a 2 σ upper limit of 30.0 µJy at 4.9 GHz (2 σ chosen since we know the position of the 8.4 GHz source with high accuracy), we derive a spectral index limit of α > −0.12 ± 0.99 – 7 – (for Sν ∝ ν +α, 1 σ error in spectral index), which has little power to discriminate among models. The X-ray emission from G1 may be due to Bondi accretion on the IMBH, either from ambient cluster gas or from stellar winds (Pooley & Rappaport 2006). Ho, Terashima, & Okajima (2003) and Pooley & Rappaport (2006) give approximate relations for the Bondi accretion on an IMBH in a globular cluster; for an ambient density of 0.1 cm−3, an ambient speed of 15 km s−1 for the gas particles relative to the IMBH, and a radiative efficiency of 10%, the Bondi accretion luminosity for the G1 IMBH would be ∼ 3× 1038 ergs s−1. The X-ray lumi- nosity of 2 × 1036 ergs s−1 measured by Pooley & Rappaport (2006) thus implies accretion at just under 1% of the Bondi rate. Given that LX/LEdd ≈ 10 −6, a more likely scenario is that G1 accretes at closer to 10% of the Bondi rate but with a radiative efficiency under 1%. In this context, we note that the radio/X-ray ratio for G1 is logRX > −4.3, which is above the value of −4.5 used to divide radio-quiet from radio-loud objects (Terashima & Wilson 2003).5 G1 therefore should be considered radio-loud, as inferred for BHs in galactic nuclei that radiate well below their Eddington luminosities (Ho 2002). If the globular clusters in our own Galaxy also have central BHs that are 0.1% of their total masses, and they accrete and radiate in the same way as G1, many would have expected 5 GHz radio flux densities in the 20–100 µJy range; flux densities often would be in the 1– 10 µJy range even for less efficient accretion and radiation (Maccarone et al. 2005). As Maccarone et al. (2005) summarize, there are few radio images of globular clusters that go deep enough to test this possibility. Fender (2004) points out that the Square Kilometer Array (SKA) will be able to test for the existence of IMBHs in many globular clusters. However, based on our results for G1, we suggest that it is not necessary to wait for the SKA; the current VLA can reach the hypothesized flux densities with some effort. The EVLA (Ulvestad et al. 2006), scheduled to be on line in about 2010, will have 40 times the bandwidth and 6.3 times the sensitivity of the current VLA in the frequency range near 8 GHz. This will enable the EVLA to reach the 1 µJy noise level in approximately 12 hours of integration, thus probing the range of radio emission predicted by Maccarone et al. (2005) for many globular clusters. 5A lower limit is given for RX because this quantity traditionally is given in terms of the 2–10 keV luminosity, whereas logRX = −4.3 would correspond to the value computed for the 0.2–10 keV luminosity given by Pooley & Rappaport (2006). – 8 – 4. Summary We have detected faint radio emission from the object G1, a globular cluster or stripped dwarf elliptical galaxy in M31. The emission has an 8.4 GHz power of 2× 1015 W Hz−1. As- suming that the radio source is associated with the X-ray source in G1 (Pooley & Rappaport 2006), the radio/X-ray ratio is consistent with the value expected for an accreting ∼ 2 × 104 M⊙ BH. Thus, the radio detection lends support to the presence of such an IMBH within G1. The other possible explanation, a pulsar wind nebula, could be tested by making very high-sensitivity VLBI observations of G1. The National Radio Astronomy Observatory is a facility of the National Science Foun- dation operated under cooperative agreement by Associated Universities, Inc. We thank the staff of the VLA that made these observations possible. Support for JEG was provided by NASA through Hubble Fellowship grant HF-01196, and LCH acknowledges support from NASA grant HST-GO-09767.02. Both were awarded by the Space Telescope Science Insti- tute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. We also thank Dale Frail for useful discussions about pulsar wind nebulae, and an anonymous referee for useful suggestions. Facilities: VLA. REFERENCES Baars, J. W. M., Genzel, R., Pauliny-Toth, I. I. K., & Witzel, A. 1977, A&A, 61, 99 Barth, A. J., Ho, L. C., Rutledge, R. E., & Sargent, W. L. W. 2004, ApJ, 607, 90 Baumgardt, H., Makiro, J., Hut, P., McMillan, S., & Portegies Zwart, S. 2003, ApJ, 589, Bietenholz, M. 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D. 1993, ApJ, 405, This preprint was prepared with the AAS LATEX macros v5.2. – 11 – arcsec 10 5 0 -5 -10 Fig. 1.— VLA C configuration 8.4 GHz image of the vicinity of G1 in M31, with contours at intervals of 0.5 times the r.m.s noise. The lowest contour is at 3 times the noise of 6.2 µJy beam−1, and negative contours are shown dashed; the synthesized beam is shown in the box in the lower-left corner. The (0,0) point is at a J2000 position of α = 00h32m46.60s, δ = 39◦34′40.0′′. The 1.′′5-radius circle about this central point represents the 1 σ error in the X-ray position of G1 (Kong 2007). Introduction Observations and Imaging Origin of the G1 Radio Emission Summary

0704.1459

Even infinite dimensional real Banach spaces

7 Even infinite dimensional real Banach spaces Valentin Ferenczi∗ and Elói Medina Galego Abstract This article is a continuation of a paper of the first author [5] about com- plex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, in- cluding HI or unconditional examples from [5] and C(K) examples due to Plebanek [13]. We extend results of [5] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential opera- tors on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [5] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manous- sakis [2] provide examples of essentially incomparable complex structures which are not totally incomparable. 1 2 1 Introduction Any complex Banach space is also a real Banach space. Conversely, the linear structure on a real Banach space X may be induced by a C-linear structure; the corresponding complex Banach space is said to be a complex structure on X . The existing theory of complex structure is up to isomorphism. In this setting, a complex structure on a real Banach space X is a complex space which is R- linearly isomorphic to X . Any complex structure up to isomorphism is associated ∗This article was written during a visit of the first author at the University of São Paulo financed by the Centre National de la Recherche Scientifique. 1MSC numbers: 46B03, 47A53. 2Keywords: complex structures, inessential operators, even Banach spaces, spectral theory on real spaces. http://arxiv.org/abs/0704.1459v1 to an R-linear isomorphism I on X such that I2 = −Id. Conversely, for any such isomorphism I , an associated complex structure may be defined by the law ∀λ, µ ∈ R, (λ+ iµ).x = (λId+ µI)(x), and the equivalent norm |||x||| = sup ‖cos θx+ sin θIx‖ . Isomorphic theory of complex structure addresses questions of existence, uni- queness, and when there is more than one complex structure, the possible structure of the set of complex structures up to isomorphism (for example in terms of car- dinality). It is well-known that complex structures do not always exist (up to isomor- phism) on a Banach space. The HI space of Gowers and Maurey [9] is a good example of this, or more generally any space with the λId+S property (i.e. every operators is a strictly singular perturbation of a multiple of the identity); and note that since this property passes to hyperplanes, complex structures neither exist on finite codimensional subspaces of such a space (relate this observation to the forthcoming Question 17). By the examples of [3] and [10] of complex spaces not isomorphic to their conjugates, there exists real spaces with at least two complex structures up to iso- morphism, and the examples of [3] and [1] (which are separable) actually admit a continuum of complex structures. In [5] the first author showed that for each n ≥ 2 there exists a space with exactly n complex structures. He also gave var- ious examples of spaces with unique complex structure up to isomorphism and different from the classical example of ℓ2, including a HI example and a space with an unconditional basis. A fundamental tool in [5] is an identification of isomorphism classes of com- plex structures on a space X with conjugation classes in some group associated to strictly singular operators on X . It remained open whether the associated map was bijective. In this paper, we show that it is not bijective in general, but that there actually exists a natural bijection between isomorphism classes of complex struc- ture on a space X and on its hyperplanes with conjugation classes in the group associated to strictly singular operators in X , Theorem 11. We also note that this holds as well when one replaces strictly singular operators by any Fredholm ideal in L(X), such as In(X) the ideal of inessential operators. More precisely, it turns out to be fundamental to determine when a given oper- ator of square −Id modulo inessential operators lifts to an operator of square −Id on the space. While the answer is always positive in the complex case, Proposi- tion 6, it turns out that on a real space, such an operator lifts either to an operator on X of square −Id or, in a sense made precise in Lemma 9, to an operator on an hyperplane of X of square −Id, Proposition 10; furthermore the two cases are exclusive, Proposition 8. This implies that operators of square −Id modulo inessential operators characterize complex structures on X and on its hyperplanes. This leads us to define a notion of even and odd real Banach space extending the classical notion for finite dimensional spaces. Even spaces are those spaces which admit complex structure but whose hyperplanes do not. Odd spaces are the hyperplanes of even spaces. We provide characterizations of even and odd spaces in terms of the previously mentioned groups associated to inessential operators on X and on its hyperplanes, Corollary 12 or more precisely, in terms of lifting properties of operators of square −Id modulo inessential operators, Proposition 13. We prove that there exist even infinite dimensional Banach spaces, using various examples from [5], including a HI and an unconditional example, Theorem 14. Moreover we use spaces con- structed in [13] to give examples of even and odd spaces of the form C(K), The- orem 18. We also show that the direct sum of essentially incomparable infinite dimensional spaces is even whenever both spaces are even, Proposition 16. Finally we extend and simplify the proof of some results of [5] about totally incomparable complex structures by showing that essentially incomparable com- plex structures are necessarily conjugate, Theorem 21 and Corollary 22. We also show that the complex version of a space built by S. Argyros and A. Manoussakis [2] provides examples of complex structures on a space which are essentially in- comparable yet not totally incomparable, Proposition 25. 2 Parity of infinite dimensional spaces It may be natural to think that a real infinite dimensional space of the form X⊕X should be considered to be even. This seems to be restrictive however, as we should consider as candidates for spaces with even dimension the spaces which admit a complex structure, and there are spaces with complex structure which are not isomorphic to a cartesian square (actually, not even decomposable, by the examples of [5]). Another problem is that we would wish the hyperplanes of a space with even infinite dimension not to share the same property. In other words, parity should imply a structural difference between the whole space and its hyperplanes. This suggests the following definition, which obviously generalizes the case of finite dimensional spaces, and will be our guideline for this section. Definition 1 A real Banach space is even if it admits a complex structure but its hyperplanes do not admit a complex structure. It is odd if its hyperplanes are even. Equivalently a Banach space is odd if it does not have a complex structure but its hyperplanes do, and clearly 2-codimensional subspaces of an even (resp. odd) space are even (resp. odd). The following crucial fact will be used repeatedly without explicit reference: two complex structures XI and XJ are isomorphic if and only if I and J are conjugate, i.e. there exists an isomorphism P on X such that J = PIP−1 (P is then C-linear from XI onto XJ ). There is therefore a natural correspondance between isomorphism classes of complex structure on X and conjugacy classes of elements of square −Id in GL(X), and we shall sometimes identify the two sets. Our first results are improvements of results from [5]. We recall that an op- erator T : Y → Z is Fredholm if its kernel is finite dimensional and its image is finite codimensional, in which case the Fredholm index of T is defined by i(T ) = dim(Ker(T ))− dim(Z/TY ). We shall use the easy facts that an operator T is Fredholm if and only if T 2 is Fredholm (with i(T 2) = 2i(T )), and that a C-linear operator is Fredholm if and only if it is Fredholm as an R-linear operator (and the corresponding indices are related by iR(T ) = 2iC(T )). A closed two-sided ideal U(X) in L(X) is a Fredholm ideal when an operator T ∈ L(X) is Fredholm if and only if the corresponding class is invertible in L(X)/U(X). It follows from well-known results in Fredholm theory that U(X) is contained in the ideal In(X) of inessential operators, i.e. operators S such that IdX−V S is a Fredholm operator for all operators V ∈ L(X), see for example [6]. Note that by continuity of the Fredholm index, IdX−V S is necessarily Fredholm with index 0 when S is inessential, and V ∈ L(X). Lemma 2 Let X be an infinite dimensional real Banach space. Let I ∈ L(X) satisfy I2 = −Id, and let S be inessential such that (I + S)2 = −Id. Then I and I + S are conjugate, or equivalently, the complex structures XI and XI+S associated to I and I + S respectively are isomorphic. Proof : The map 2I + S is immediately seen to be C-linear from XI into XI+S . Furthermore, it is an inessential perturbation of an isomorphism, and therefore Fredholm with index 0 as an R-linear operator on X . So it is also Fredholm with index 0 as a C-linear map, which implies that XI and XI+S are C-linearly isomorphic. � Let X be a Banach space. The set I(X) denotes the set of operators on X of square −Id. Let U(X) be a Fredholm ideal in L(X) and πU (or π) denote the quotient map from L(X) onto L(X)/U(X). Let (L(X)/U(X))0 denote the group πU(GL(X)), and Ĩ(X) denote the set of elements of (L(X)/U(X))0 whose square is equal to −πU (Id). Lemma 3 Let X be an infinite dimensional real Banach space and U(X) be a Fredholm ideal in L(X). Then the quotient map πU induces an injective map π̃U from the set of GL(X)-conjugation classes on I(X) (and therefore from the set of isomorphism classes of complex structures on X) into the set of (L(X)/U(X))0- conjugation classes on Ĩ(X). Proof : For any operator T on X , we write T̃ = π(T ). Let I and T be operators in I(X). If α is a C-linear isomorphism from XI onto XT , then the C-linearity means that αI = Tα. Therefore α̃Ĩ = T̃ α̃, and Ĩ and T̃ satisfy a conjugation relation. Conversely, if Ĩ = α̃−1T̃ α̃ for some α ∈ GL(X), then α−1Tα = I + S, where S belongs to U(X) and is therefore inessential . Note that (I+S)2 = −Id, and since Tα = α(I + S), α is a C-linear isomorphism from XI+S onto XT . By Lemma 2, it follows that XI and XT are isomorphic. This proves that π̃ is well-defined and injective. � We shall now discuss when the above induced map is actually a bijection. This is equivalent to saying that U(X) has the following lifting property. Definition 4 Let X be an infinite dimensional Banach space. The Fredholm ideal U(X) is said to have the lifting property if any α in (L(X)/U(X))0 satisfying α2 = −1 is the image under πU of an operator T such that T 2 = −Id. The following was essentially observed in [5]. Lemma 5 Let X be an infinite dimensional Banach space and let U(X) be a Fredholm ideal in L(X). If U(X) admits a supplement in L(X) which is a sub- algebra of L(X), then U(X) has the lifting property. Proof : If H(X) is a subalgebra of L(X) which supplements S(X), then let T ∈ L(X) be such that T̃ 2 = −Ĩd; we may assume that T (and therefore T 2) belongs to H(X). Then since T 2 + Id is in U(X) ∩ H(X), T 2 must be equal to −Id. Any class T̃ ∈ Ĩ(X) may therefore be lifted to an element of I(X). � We shall now prove that although any Fredholm ideal in a complex space has the lifting property, this is not necessarily true in the real case. The proof of the complex case is essentially the same as the similar classical result concerning projections (see e.g. [6]), and could be deduced directly from it using the fact that an operator A satisfies A2 = −Id if and only if 1 (Id − iA) is a projection. We shall however give a direct proof of this result for the sake of completeness. Proposition 6 Let X be an infinite dimensional complex Banach space and let U(X) be a Fredholm ideal in L(X). Then every element a ∈ L(X)/U(X) with a2 = −1 is image under the quotient map of some A ∈ L(X) with A2 = −Id. Proof : Recall that π : L(X) → L(X)/U(X) denote the quotient map. We choose B ∈ L(X) such that π(B) = a. So π(B2) = −1 and therefore there exists S ∈ U(X) ⊂ In(X) such that B2 = −Id+ S. Since the spectrum σ(−Id+S) of −Id+S is countable and its possible limit point is -1, it follows by the spectral mapping theorem ([4], Theorem VII.3.11) that the spectrum σ(B) is also countable and its possible limit points are −i and i. Take a simple closed curve Γ in C \ σ(B) such that i is enclosed by Γ and −i is not enclosed by Γ. Define the operator (λI − B)−1dλ. By [4], Theorem VII.3.10, P is a projection. Moreover, according to the continu- ity of π, π(P ) = (λI − a)−1dλ. (1) On the other hand, it is easy to check that (λI − a)−1 = λI + a λ2 + 1 Thus (1) implies that π(P ) = and hence putting A = 2iP − iI , we see that π(A) = a and A2 = −I . � Such a proof is not possible in the real case essentially because there is no formula with real coefficients linking projections and operators of square −Id. Actually it is known that if X is real every element p belonging to the quotient by a Fredholm ideal and satisfying p2 = p may be lifted to a projection (see [6]), and the proof uses the complexification of X and a curve with well-chosen symmetry so that the complex projection produced by the proof in the complex case is induced by a real projection which will answer the question by the positive. However there is no choice of curve such that the complex map A of square −Id obtained in the above proof applied in the complexification of X is induced by a real operator. Actually the result is simply false in the real case: Theorem 7 Let X be a real infinite dimensional Banach space whose hyper- planes admit a complex structure. Then no Fredholm ideal in L(X) has the lifting property. This theorem is a consequence of Proposition 8. We need to recall that the complexification X̂ of a real Banach space X (see, for example, [12], page 81) is defined as the space X̂ = {x + iy : x, y ∈ X}, which is the space X ⊕ X with the canonical complex structure associated to J defined on X ⊕X by J(x, y) = (−y, x). Let A,B ∈ L(X). Then (A+ iB)(x+ iy) := Ax− By + (Ay +Bx) defines an operator A+ iB ∈ L(X̂) that satisfies max{‖A‖, ‖B‖} ≤ ‖A+ iB‖ ≤ 21/2(‖A‖+‖B‖). Conversely, given T ∈ L(X̂), if we put T (x+i0) := Ax+iBx, then we obtain A,B ∈ L(X) such that T = A + iB. We write T̂ = T + i0 for T ∈ L(X), and say that such the operator T̂ is induced by the real operator T . Proposition 8 Let Y be an infinite dimensional real Banach space and J ∈ L(Y ) such that J2 = −Id. Let A be defined on X = R ⊕ Y by the matrix Then A2 is the sum of −Id and of a rank 1 operator, but there is no inessential operator S on X such that (A+ S)2 = −Id. Proof : Assume (A + S)2 = −Id for some inessential operator S. Passing to the complexification X̂ of X , we obtain ( + Ŝ)2 = −Îd. The map from [0, 1] into L(X̂) defined by Tµ =  + µŜ is polynomial, moreover by spectral properties of inessential operators and the spectral theorem, the spectrum Sp(Tµ) is, with the possible exception of i and −i, a countable set of isolated points, which are eigenvalues with associated spectral projections, denoted E(λ, Tµ) for each λ ∈ Sp(Tµ), of finite rank. Furthermore the complex operator Tµ is induced by the real operator A+ µS, therefore Sp(Tµ) is symmetric with respect to the real line. Let n(µ) = λ∈R∩Sp(Tµ) rk(E(λ, Tµ)) and let I1 = {µ ∈ [0, 1] : n(µ) is odd}, I0 = {µ ∈ [0, 1] : n(µ) is even}. Observe that 0 ∈ I1, since  is defined on X̂ = R̂⊕ Ŷ by the matrix and therefore has unique real eigenvalue 1, with associated spectral projection of dimension 1. On the other hand, since T (1)2 = ( + Ŝ)2 = −Îd, T (1) does not admit real eigenvalues and therefore 1 ∈ I0. We now pick some µ1 ∈ I1. Let U be an open set containing the real line, symmetric with respect to it, and such that U ∩ Sp(Tµ1) ⊂ R. Then the spectral projection E(µ) := E(Tµ, U∩Sp(Tµ)) is an analytic ([4] Lemma VII.6.6) projec- tion valued function defined for all µ such that |µ−µ1| < γ for some small enough γ > 0, and for which E(µ1) has rank n(µ1). Therefore by [4] Lemma VII.6.8, the dimension λ∈U∩Sp(Tµ) rk(E(λ, Tµ)) of the image of E(µ) is also n(µ1), for µ in a small enough open set V around µ1. By symmetry of U and of Sp(Tµ) with respect to the real line (with preservation of the ranks of the associated spectral projections), n(µ1) = λ∈R∩Sp(Tµ) rk(E(λ, Tµ)) + 2 λ∈Sp(Tµ),Im(λ)>0 rk(E(λ, Tµ)), when µ ∈ V . So n(µ) = λ∈R∩Sp(Tµ) rk(E(λ, Tµ)) is odd whenever µ is in the neighborhood V of µ1. We have therefore proved that I1 is open. In the same way, I0 is also open (in the special case when n(µ0) = 0, then E(µ0) = 0 and so, E(µ) = 0 and therefore n(µ) = 0 in a neighborhood of µ0). In conclusion, the sets I0 and I1 are open, non-empty, and partition [0, 1], a contradiction. � The obstruction for the lifting property is therefore that a complex structure on a hyperplane of a space X does not correspond to a complex structure on X , although it does induce elements of square −1 in (L(X)/S(X))0, as explicited by the following result. Lemma 9 Let Y be an infinite dimensional Banach space and X = Y ⊕ R. Let U(X) be a Fredholm ideal. Then the map π′U (or π ′) from GL(Y ) into (L(X)/U(X))0 defined by π U(A) = πU( ) maps I(Y ) into Ĩ(X) and induces an injection from the set of conjugation classes on I(Y ) (and therefore from the set of isomorphism classes of complex structures on Y ) into the set of conjugation classes on Ĩ(X). Proof : First note that if the conjugation relation J = PKP−1 is satisfied in GL(Y ) then 0 P−1 is satisfied in GL(X), which provides a conjugation relation in (L(X)/U(X))0. Conversely if ˜(1 0 ˜(1 0 P̃−1, write Then since P is Fredholm with index 0, P|Y is Fredholm with index −1 as an op- erator of L(Y,X), and D = P|Y − b ∗ is also Fredholm with index −1 in L(Y,X), and with index 0 in L(Y ). Therefore some finite rank perturbation D′ of D is an isomorphism. Furthermore it is easy to deduce from the conjugation relation that J = D′KD′ + S, where S ∈ U(Y ). Therefore J − S = D′KD′−1, i.e. J − S and K are conjugate, and by Lemma 3, it follows that Y J is isomorphic to Y K . � The previous lemma does not mean that the sets of isomorphism classes of complex structures on X and on hyperplanes of X are disjoint (only the corre- spoding images by π̃, and π̃′ respectively, are). Actually the sets of isomorphism classes on X and on its hyperplanes can either be equal, when X is isomorphic to its hyperplanes, or disjoint, when it is not. And we have: Proposition 10 Let Y be an infinite dimensional Banach space and let X = R⊕ Y . Let U(X) be a Fredholm ideal in L(X). Let A ∈ GL(X) and assume A2 = −Id+S, S ∈ U(X) (i.e. à ∈ Ĩ(X)). Then there exists s ∈ U(X) such that either (A+ s)2 = −Id or (A+ s)2 = where J ∈ L(Y ) satisfies J2 = −Id. Before the proof, let us observe that by Proposition 8, only one of the two alternatives of the conclusion can hold for a given A such that à ∈ Ĩ(X). Proof : Passing to the complexification X̂ of X , we have that Â2 = −Îd+Ŝ, and Ŝ is inessential. Now let Γ be a rectangular curve with horizontal and vertical edges, symmetric with respect to the horizontal axis, included in the open unit disk, and such that Γ∩Sp(Ŝ) = ∅ and let U be the interior of the domain delimited by Γ, V be the interior of the complement of this domain. Let P̂ be the spectral projection associated to Sp(Ŝ)∩U ; since Γ is rectangular and symmetric with respect to the real axis, it is classical and easy to see that P̂ is indeed induced by a real operator P on X , see e.g. [7] where this principle is used. Let also Q̂ be the spectral projection associated to Sp(Ŝ) ∩ V . Then Ŝ = ŜP̂ + ŜQ̂. The operator ŜP̂ has spectral radius strictly smaller than 1, therefore the series n≥1 bn(ŜP̂ ) n converges to an operator ŝ, where n≥1 bnz n = −1 + (1 − z)−1/2 for all |z| < 1, and since the bn’s are reals, it is indeed induced by a real operator s = bn(SP ) n in U(X). We observe (P̂ + ŝ)2 = P̂ (Îd+ ŝ)2 = P̂ (Îd− Ŝ)−1, therefore (ÂP̂ + Âŝ)2 = −P̂ . Assume now that Q has even rank, then there exists a finite rank operator F on QX such that F 2 = −Id|QX . Let then s ′ = FQ− AQ, therefore ÂQ̂+ ŝ′ = F̂ Q̂ and (ÂQ̂+ ŝ′)2 = −Q̂. If we then let v = s+ As′, we deduce that (Â+ v̂)2 = (ÂP̂ + Âŝ+ ÂQ̂ + ŝ′)2 = −P̂ − Q̂ = −Îd, and therefore (A+ v)2 = −Id, with v ∈ U(X). If Q has odd rank then there exists a finite rank operator F on QX such that , with j2 = −Id, in an appropriate decomposition of QX . Defining v ∈ U(X) in the same way as above, we obtain that A+ v may be written A + v = corresponding to some decomposition R′⊕Y ′ of X where R′ is 1-dimensional. If Y ′ = Y then we may clearly find some rank 1 perturbation f such that A+ v + f may be written A+ v + f = corresponding to the original decomposition R ⊕ Y of X . If Y ′ 6= Y then we consider the space Z = Y ′ ∩ Y ∩ JY , which is stable by J . If Z has codimension 3 then we may decompose Y = G⊕Z, where G has dimension 2, and by using a operator k of square −IdG on G, we may find a rank 3 perturbation f such that A + v + f = 0 J ′ in the original decompositionR⊕Y of X , and with J ′|Z = J|Z and J |G = k, so that J ′2 = −Id|Y . If finally Z has codimension 2 then easily some rank 2 perturbation of A + v on X has square −Id, but because of the decomposition of A + v on R′ ⊕ Y ′, this would contradict Proposition 8, so this case is not possible. � We sum up the results of Proposition 8, Lemma 9 and Proposition 10 in the next theorem: Theorem 11 Let Y be an infinite dimensional real Banach space and let X = Y ⊕ R. Let U(X) be a Fredholm ideal in L(X). Then there exists a partition {Ĩ0(X), Ĩ1(X)} of Ĩ(X) such that πU induces a bijection from the set of con- jugation classes on I(X) (and therefore from the set of isomorphism classes of complex structures on X) onto the set of conjugation classes on Ĩ0(X) and such that π′U induces a bijection from the set of conjugation classes on I(Y ) (and there- fore from the set of isomorphism classes of complex structures on Y ) onto the set of conjugation classes on Ĩ1(X). Corollary 12 An infinite dimensional real Banach space is even if and only if Ĩ(X) = Ĩ0(X) 6= ∅, and odd if and only if Ĩ(X) = Ĩ1(X) 6= ∅. When Ĩ(X) is a singleton then X is either even or odd. The next proposition sums up when a real Banach space has the lifting prop- erty. Proposition 13 Let X be an infinite dimensional real Banach space. Then the following are equivalent: • i) any Fredholm ideal in L(X) has the lifting property. • ii) some Fredholm ideal in L(X) has the lifting property. • iii) the hyperplanes of X do not admit complex structure. • iv) for any Fredholm ideal U(X), the map πU induces a bijection from the set of isomorphism classes of complex structures on X onto the set of con- jugation classes on ĨU(X). • v) for some Fredholm ideal U(X), the map πU induces a bijection from the set of isomorphism classes of complex structures on X onto the set of conjugation classes on ĨU(X). Proof : It is clear by definition that iv) ⇔ i) and v) ⇔ ii). Then i) ⇒ ii) is obvious, ii) ⇒ iii) due to Proposition 8, and iii) ⇒ i) by Proposition 10. � We use this proposition to solve an open question from the first author, which in our formulation asked whether there existed infinite dimensional even Banach spaces. Recall that the space XGM is the real version of the HI space of Gowers and Maurey [9], on which every operator is of the form λId+S, and therefore does not admit complex structure. The space X(C) is a HI space constructed in [5] and such that the algebra L(X(C)) may be decomposed as C⊕S(X(C)), and X(H), also HI, is a quaternionic version of X(C). It is proved in [5] that X(H) admits a unique complex structure, while X(C) admits exactly two complex structures. Finally X(D2) is a space with an unconditional basis on which every operator is a strictly singular perturbation of a 2-block diagonal operator and which also admits a unique complex structure. We refer to [5] for details. Theorem 14 The spaces X(C), X(H), X2nGM for n ∈ N, and X(D2) are even. Proof : For X(C), X(H) and X2nGM this is due to the fact that the ideal of strictly singular operators has the lifting property, because there is a natural subalgebra supplementing the ideal of strictly singular operators in each case, see [5], so Lemma 5 applies. For X(D2) note that it is proved in [5] that X(D2) admits a complex structure and that there is a unique conjugation class in ĨS(X), therefore the induced injection π̃S is necessarily surjective, i.e. v) is verified. � Before giving some more examples, let us note two open problems about even spaces. The first problem is quite simple to formulate. Question 15 Is the direct sum of two even Banach spaces necessarily even? We obtain a positive answer when the spaces are assumed to be essentially incomparable. Recall that two infinite dimensional spaces Y and Z are essentially incomparable when every operator from Y to Z is inessential [8]. More details about this notion may be found in the last section of this article. Proposition 16 The direct sum of two infinite dimensional even Banach spaces which are essentially incomparable is even. Proof : Let X and Y be infinite dimensional even, and essentially incomparable. Clearly X⊕Y admits a complex structure. Assume X⊕(Y ⊕R) admits a complex structure and look for a contradiction. Let T = ∈ L(X ⊕ (Y ⊕ R)) be such that T 2 = −Id. Since X and Y ⊕ R are essentially incomparable, S and S ′ are inessential. Furthermore T 21 + S ′S = −IdX , i.e. T̃1 ∈ Ĩ(X). Since X is even, there exists an inessential operator s1 on X such that T1+s1 = J1 with J −IdX (Theorem 11). Likewise T 2 +SS ′ = −IdY⊕R therefore since Y ⊕R is odd, there exists an inessential operator s2 on Y ⊕R such that T2+s2 = , J2 ∈ L(Y ) with J22 = −IdY . Therefore there exists S0 = s1 −S −S s2 inessential on X ⊕ Y ⊕ R such that T + S0 = J1 0 0 0 J2 0 0 0 1 Since T 2 = −Id this contradicts Proposition 8. � For the second problem, we note that Banach spaces which are not isomorphic to their hyperplanes may be classified in four categories: even spaces, odd spaces, spaces such that neither the whole space neither hyperplanes admit a complex structure, and spaces such that both the whole space and any hyperplane admit complex structure. While we have just produced examples of the first and the second category, and the space XGM belongs to the third, no examples are known which belong to the fourth. Question 17 Does there exist a real Banach space X which is not isomorphic to its hyperplanes and such that both X and its hyperplanes admit complex struc- ture? We now use some spaces constructed by Plebanek [13] to giveC(K) examples of even and odd Banach spaces. Similar C(K) spaces were first constructed by P. Koszmider [11] under the Continuum Hypothesis. Let K be one of the two infinite, separable, compact Hausdorff spaces defined in [13]. Every operator on C(K) is of the form g.Id + S where g ∈ C(K) (therefore g.Id denotes the multiplication by g) and S is strictly singular (or equivalently weakly compact). The first space is connected, and we shall indicate where our proofs simplify due to this additional property. The space K ∪K will denote the space which is the topological union of two copies of K (i.e. open sets are unions of open sets of each copy), while K ∪0 K denotes the amalgamation of two copies of K in some point 0 (open sets are unions of open sets of each copy either both containing 0 or neither containing 0). Both are separable compact Hausdorff spaces. Theorem 18 The space C(K∪K) is even and admits a unique complex structure, and the space C(K ∪0 K) is odd. Proof : The space C(K ∪K) identifies isomorphically with C(K) ⊕ C(K) and C(K ∪0 K) identifies with the quotient C(K) 2/Y , where Y = {(f, g) : f(x0) = g(x0)} for some fixed x0 ∈ K; therefore C(K∪0K) is isomorphic to a hyperplane of C(K ∪K) and it is enough to prove that C(K) ⊕ C(K) is even with unique complex structure. Write X = C(K) and let Is(K) be the set of isolated points of K. Let N ⊂ C(K) be the closed ideal of almost null functions, i.e. g ∈ N iff g vanishes on K \ Is(K) and converges to 0 on Is(K) (i.e. for any ǫ > 0, |g(x)| ≤ ǫ for all x ∈ Is(K) except a finite number of points). We observe the following fact: if an operator on X of the form g.Id is strictly singular, then g belongs to N . Indeed if g(x) 6= 0 for some non-isolated point x then |g| ≥ α > 0 on some infinite subset L of K containing x, and the restriction of gId to the space of functions with support included in L is an isomorphism. Likewise if |g| ≥ α > 0 on some infinite subset of Isol(K) then the corresponding restriction of gId is an isomorphism. In the case where K is connected, we have simply that g.Id is never strictly singular unless g = 0. Since X ⊕ X has a canonical complex structure, to prove that it is even with unique complex structure, it is enough by Theorem 11 to prove that the group G0 := (L(X ⊕ X)/S(X ⊕ X))0 has a unique conjugation class of elements of square −Id. We observe that the group GL2(C(K)) admits a unique class of conjugation of elements of square −I , where I = , i.e. that whenever M = f1 f2 f3 f4 satisfies M2 = −I , then it is conjugate to the canonical element J = Indeed from M2 = −I we deduce easily f1 = −f4 and f 1 + f2f3 = −1. Note that f2 never takes the value 0. Let then P = f1 f2 and let Q = −f1/f2 1/f2 . It is routine to check that Q = P−1 and that QJP = M . When K is connected then N = {0}, therefore L(X) ≃ C(K) ⊕ S(X) and L(X ⊕ X) = M2(C(K)) ⊕ S(X ⊕ X). So S(X ⊕ X) has the lifting property by Lemma 5, and G0 easily identifies with GL2(C(K)), and we therefore deduce that there is a unique G0-conjugacy class of elements of square −Ĩd. The general case is more complicated. Assume T̃ ∈ G0 satisfies T̃ 2 = −Ĩd. Then T 2+ Id is strictly singular, and up to a strictly singular perturbation we may assume that T = M.Id with M ∈ M2(C(K)). Therefore (M 2 + I).Id is strictly singular and M2 + I ∈ N . It follows that M2 = −I + n with ‖|n(x)‖| ≤ 1/2 except for x ∈ F , F a finite subset of Is(K) (here ‖|.‖| denotes the operator norm on M2(R) ≃ L(R⊕2 R)). Note that M(x) and n(x) commute for all x ∈ K. For x ∈ K \ F , let n′(x) = M(x) k≥1 bkn(x) k where (1 − z)−1/2 = 1 + k≥1 bkz k for |z| < 1, and let M ′(x) = (M + n′)(x). Then it is easy to check that (M ′(x))2 = −IdR2 and therefore there exist P (x), Q(x) in M2(R) given by formulas from the coefficients of M ′(x) which are explicited above in the connected case, such that P (x)Q(x) = IdR2 and Q(x)jP (x) = M ′(x), where For x ∈ F we let M ′(x) = (M + n′)(x) = j and P (x) = Q(x) = IdR2 . Note that since the points of F are isolated and by uniform convergence of n′ and the formulas giving P and Q, the matrices P , Q and n′ define elements of M2(C(K)). Actually, since M ′(x) is invertible for all x and by the explicit formulas for the inverses, we deduce that M ′ ∈ GL2(C(K)) and M ′.Id ∈ GL(X ⊕X). Likewise P and Q belong to GL2(C(K)) and P.Id, Q.Id belong to GL(X ⊕X). It is now enough to prove that S = n′.Id is a strictly singular operator on X⊕X . Then the relation QJP = M ′ will imply a G0-conjugacy relation between ˜J.Id and ˜M ′.Id = ˜M.Id+ S̃ = T̃ , as desired. For L ⊂ K, CL(K) denotes the space of functions of support included in L. Let v denote the rank |F | projection onto CF (K) associated to the decomposition C(K) = CF (K) ⊕ CK\F (K), and w = Id − v. Let V on X ⊕X be defined by and let W = It is easy to check that S = k≥1 bk(n k.Id)W + V (n′.Id), (just compute S(f1, f2)(x) for all (f1, f2) ∈ X ⊕ X and x ∈ K). Since n k.Id = (n.Id)k is strictly singular for all k and V has finite rank, it follows that S is strictly singular. To conclude this section we note an open question. Recall that there exist spaces with exactly n complex structures, for any n ∈ N∗ [5], and spaces with exactly 2ω complex structures [1]. Question 19 Does there exist a Banach space with exactly ω complex structures? 3 Essentially incomparable complex structures We recall that two infinite dimensional spaces Y and Z are said to be essentially incomparable if any bounded operator S ∈ L(Y, Z) is inessential, i.e., if IdY−V S is a Fredholm operator (necessarily of index 0) for all operators V ∈ L(Z, Y ). Es- sentially incomparable spaces were studied by M. González in [8]; it is clear that the notion of essential incomparability generalizes the notion of total incompara- bility. We also recall that Y and Z are projection totally incomparable if no infinite dimensional complemented subspace of Y is isomorphic to a complemented sub- space of Y . Essentially incomparable spaces are in particular projection totally incomparable. In this section we show how some results of [5] about totally incomparable complex structures extend to essentially incomparable structures. Interestingly, our more general proof turns out to be much simpler than the original one. As was noted in [5], whenever T 2 = U2 = −Id, it follows that (T + U)T = U(T + U), which means that T + U is C-linear from XT into XU . The similar result holds for T − U between XT and X−U . Lemma 20 Let X be a real Banach space, T, U ∈ I(X). If T + U is inessential from XT into XU , then XT is isomorphic to X−U . Proof : Since T+U is inessential as an operator from XT into XU , and as T+U is also linear from XU into XT , it follows by definition of inessential operators that Id + λ(T + U)2 ∈ L(XT ) is Fredholm with index 0 whenever λ is real. Taking λ = 1/4, we obtain that 4Id+ (T + U)2 = 2Id+ TU + UT = −(T − U)2. Therefore (T −U)2 is Fredholm with index 0 as an operator on XT , and therefore as an operator on X . It follows that T −U is Fredholm with index 0 as an operator of L(X) and therefore as an operator of L(XT , X−U), hence XT and X−U are C-linearly isomorphic. � It was proved in [5] that two totally incomparable complex structures on a real space must be conjugate and both saturated with HI subspaces. We show: Theorem 21 Let X be a real Banach space with two essentially incomparable complex structures. Then these complex structures are conjugate up to isomor- phism and do not contain a complemented subspace with an unconditional basis. Proof : Assume XT is essentially incomparable with XU . Then T +U is inessen- tial from XT into XU and by Proposition 20, XU is isomorphic to X−T . If Y is a C-linear complemented subspace of XT with an unconditional basis, then Y is complemented in X−T and the coordinatewise conjugation map α as- sociated to the unconditional basis is an isomorphism from Y onto Y . Therefore XT and X−T are not projection totally incomparable, contradicting the essential incomparability of XT with XU ≃ X−T . � Note that Proposition 25 will prove that one cannot hope to improve Theorem 21 to obtain HI-saturated in its conclusion, as in the case of totally incomparable complex structures. Corollary 22 There cannot exist more than two mutually essentially incompara- ble complex structures on a Banach space. Recall that two Banach spaces are said to be nearly isomorphic (or sometimes essentially isomorphic) if one is isomorphic to a finite-codimensional subspace of the other. Equivalently this means that there exists a Fredholm operator acting between them. In the next proposition, we consider properties of complex structures which are generalization of the λId + S-property. Note that each of these properties implies that there do not exist non-trivial complemented subspaces. We first state a lemma whose proof was given to us by M. González. Lemma 23 Let X be an infinite dimensional real or complex Banach space such that every operator is either Fredholm or inessential. Then every Fredholm oper- ator on X has index 0 and L(X)/In(X) is a division algebra. Proof : If in the above conditions, T were Fredholm with nonzero index, then by the continuity of the index, K := sT + (1 − s)Id would be inessential for some s with 0 < s < 1. Thus T := s−1 Id + 1 K; hence T has index 0, a contradiction. Moreover, the only noninvertible element in L(X)/In(X) is 0; hence L(X)/In(X) is a division algebra. � Proposition 24 Let X be a real Banach space, and T ∈ I(X). • i) If every operator on XT is either inessential or Fredholm, then either XT is the only complex structure on X , or XT and X−T are the only two complex structures on X and they are not nearly isomorphic. • ii) If every operator on XT is either strictly singular or Fredholm, then either XT is the only complex structure on X , or XT and X−T are the only two complex structures on X and neither one embeds into the other. Proof : i) Let U generate a complex structure on X . We use the relation (T + U)2 + (T − U)2 = −4Id. If (T −U)2, which is an operator on XT , is inessential, then (T +U)2 is Fredholm with index 0 as a perturbation of −4Id, therefore T + U is Fredholm with index 0 and there exists an isomorphism from XT onto XU . Therefore if there exists some U generating a complex structure non isomor- phic to XT , then (T − U)2 is not inessential, and by the property of operators on XT , (T − U)2 is Fredholm, with index 0 by Lemma 23, and therefore T − U as well, so XU is isomorphic to X−T . We deduce that XT and X−T are the only complex structures on X . To see that they are not nearly isomorphic, note that if a map α is Fredholm from XT into X−T , then α2 is Fredholm on XT , hence it is Fredholm with index 0, and α is Fredholm with index 0. Therefore XT is isomorphic to X−T , contradicting our initial assumption. ii) Applying i) we see that any complex structure on X is either isomorphic to XT or X−T , and if α embeds XT into X−T , then α2 embeds XT into itself, hence it is not strictly singular and so it is Fredholm, with index 0, and α is Fredholm with index 0, which implies that XT is isomorphic to X−T . � In [2] S. Argyros and A. Manoussakis constructed a real space which is un- conditionnally saturated yet has the λId + S property. Although their result is stated in the real case, no specific property of the reals is used in their definition and proofs, and so the complex version XAM of their space satisfies the complex version of the properties mentioned above. We observe: Proposition 25 Every R-linear operator on the complex XAM is of the form λId + S, λ ∈ C, S strictly singular. It follows that the complex XAM seen as real admits exactly two complex structures, which are essentially incomparable but not totally incomparable. Proof : Denote XAM the complex version of the space of Argyros and Manous- sakis, and let X be XAM seen as real. Lemma 4.17 from [2] in its complex ver- sion states that every C-linear operator T on XAM satisfies lim d(Ten,Cen) = 0, where (en)n is the canonical complex basis of XAM . A look at their proof shows that actually only the R-linearity of T is required. Then if T is R-linear on X we deduce easily that there exists λ ∈ C such that for any ǫ > 0, there exists M an infinite subset of N such that (T − λId)|[en,n∈M ] is of norm at most ǫ. Here [en, n ∈ M ] denotes the real subspace R-linearly generated by (en)n∈M . Proposition 4.16 in [2] states that for any infinite subset M of N, any (yk) a normalized block-sequence of (en), the distance d(S[en,n∈M ], S[yk,k∈N]) between the respective unit spheres of the complex (i.e. C-linearly generated) block- subspaces [en, n ∈ M ] and [yk, k ∈ N] is 0. A look at the proof shows that one can obtain this using only the real block-subspaces which are R-linearly gen- erated by (en)n∈M and (yk)k∈N (in particular note that [2] Lemma 4.12 used in the proof only uses R-linear combinations of the (en)’s and the (yk)’s). Combining the facts of the first and the second paragraph, we deduce that when T is R-linear on X , there exists λ ∈ C such that for all ǫ > 0, for any complex block-subspace (yk)k of (en)n, there exists a unit vector x in the R-linear span of (yk)k such that (T − λId)x is of norm less than 2ǫ. We deduce easily that every R-linear operator on X is of the form λId+ S, λ ∈ C, S strictly singular. The complex structure properties now follow easily. By Proposition 24 ii) we already know that either X has unique complex structure, or exactly two which are XJ and X−J (where J ∈ L(X) is defined by Jx = ix). Note that whenever α is C-linear from XJ into X−J then α(ix) = −iα(x) for all x ∈ X . Since α = λ.Id + S, λ ∈ C and S R-strictly singular, we deduce that λ = 0 and α = S. The operator S is in particular C-strictly singular; so L(XJ , X−J) = S(XJ , X−J) and XJ and X−J are essentially incomparable. On the other hand XJ is the complex version of XAM and so is unconditionally saturated, therefore it is not HI saturated, and so not totally incomparable with its conjugate X−J , by [5] Corollary 23. � We end the paper with two open questions in the direction of further general- izing the above results. Recall that two spaces are projection totally incomparable if no infinite dimensional complemented subspace of one is isomorphic to a com- plemented subspace of the other, and that essentially incomparable spaces are in particular projection totally incomparable [8]. Question 26 If two complex structures on a real space X are projection totally incomparable, must they be conjugate? Question 27 Assume a complex space is projection totally incomparable with its conjugate, is it necessarily essentially incomparable with it? Acknowledgements The first author thanks B. Maurey for a useful discussion which led the authors to Proposition 8. Both authors thank M. González for his comments on a first version of this paper, and P. Koszmider for information about the C(K) examples used in Theorem 18. References [1] R. Anisca, Subspaces of Lp with more than one complex structure, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2819–2829. [2] S. Argyros and A. Manoussakis, An indecomposable and uncondition- ally saturated Banach space, Studia Math. 159 (2003), no. 1, 1–32. [3] J. Bourgain, Real isomorphic complex Banach spaces need not be com- plex isomorphic, Proc. Amer. Math. Soc. 96 (1986), no. 2, 221–226. [4] N. Dunford and J. T. Schwarz. Linear Operators, Part I: General the- ory, New York 1958. [5] V. Ferenczi, Uniqueness of complex structure and real hereditarily in- decomposable Banach spaces, Advances in Math., to appear. [6] M. González, Banach spaces with small Calkin algebras, Banach Cen- ter Publ. to appear. [7] M. González and J. M. Herrera. Decompositions for real Banach spaces with small spaces of operators, preprint. [8] M. González, On essentially incomparable Banach spaces, Math. Z. 215 (1994), no. 4, 621–629. [9] W.T. Gowers and B. Maurey, The unconditional basic sequence prob- lem, J. Amer. Math. Soc. 6 (1993), 4, 851–874. [10] N. J. Kalton, An elementary example of a Banach space not isomorphic to its complex conjugate, Canad. Math. Bull. 38 (1995), no. 2, 218–222. [11] P. Koszmider, Banach spaces of continuous functions with few opera- tors, Math. Ann. 330 (2004), 151–183. [12] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Springer- Verlag, New York, Heidelberg, Berlin (1979). [13] G. Plebanek, A construction of a Banach space C(K) with few opera- tors, Topology and its applications 143 (2004), 217–239. Valentin Ferenczi, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, Projet Analyse Fonctionnelle, Boı̂te 186, 4, place Jussieu, 75252 Paris Cedex 05, France. E-mail: [email protected], [email protected] Elói Medina Galego, Departamento de Matemática, Instituto de Matemática e Estatı́stica, Universidade de São Paulo. 05311-970 São Paulo, SP, Brasil. E-mail: [email protected]. Introduction Parity of infinite dimensional spaces Essentially incomparable complex structures

0704.1460

Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence

arXiv:0704.1460v1 [hep-th] 11 Apr 2007 hep-th/yymmnnn MIT-CTP-nnnn Imperial/TP/2-07/nn Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence B. Stefański, jr.1,2 1 Center for Theoretical Physics Laboratory for Nuclear Science, Massachusetts Institute of Technology Cambridge, MA 02139, USA 2 Theoretical Physics Group, Blackett Laboratory, Imperial College, London SW7 2BZ, U.K. Abstract We define Landau-Lifshitz sigma models on general coset space G/H, with H a maximal stability sub-group of G. These are non-relativistic models that have G-valued Nöther charges, local H invariance and are classically integrable. Using this definition, we con- struct the PSU(2, 2|4)/PS(U(2|2)2) Landau-Lifshitz sigma-model. This sigma model describes the thermodynamic limit of the spin-chain Hamiltonian obtained from the com- plete one-loop dilatation operator of the N = 4 super Yang-Mills (SYM) theory. In the second part of the paper, we identify a number of consistent truncations of the Type IIB Green-Schwarz action on AdS5×S5 whose field content consists of two real bosons and 4,8 or 16 real fermions. We show that κ-symmetry acts trivially in these sub-sectors. In the context of the large spin limit of the AdS/CFT correspondence, we map the Lagrangians of these sub-sectors to corresponding truncations of the PSU(2, 2|4)/PS(U(2|2)2) Landau- Lifshitz sigma-model. http://arxiv.org/abs/0704.1460v1 1 Introduction The gauge/string correspondence adscft [1] provides an amazing connection between quantum gauge and gravity theories. The correspondence is best understood in the case of the maximally supersymmetric dual pair of N = 4 SU(N) super-Yang-Mills (SYM) gauge theory and Type IIB string theory on AdS5 × S5. Recent progress in understanding this duality has come from investigations of states in the dual theories with large charges bmn,gkp2,ft [3, 4, 5]. In these large-charge limits (LCLs) it is possible to test the duality in sectors where quantities are not protected by supersymmetry. Typically, one compares the energy of some semi-classical string state with large charges (labelled schematically J) to the anomalous dimensions of the corresponding op- erator in the dual gauge theory, using 1/J as an expansion parameter which supresses quantum corrections. A crucial ingredient, which made such comparisons possible, was the observation that computing anomalous dimensions in the N = 4 SYM gauge theory is equivalent to find- ing the energy eigenvalues of certain integrable spin-chains andim [6] (following the earlier work on more generic gauge theories oandim [7]). At the same time the classical Green-Schwarz (GS) action for the Type IIB string theory on AdS5 × S5 was shown to be integrable [21]. The presence of integrable structures has led to an extensive use of Bethe ansatz-type techniques to investigate the gauge/string duality [8]. In particular, impressive results for matching the world-sheet S- matrix of the GS string sigma-model with the corresponding S-matrix of the spin-chain have been obtained smatrix [10]. The matching of anomalous dimensions of gauge theory operators with the energies of semi- classical string states was shown to work up to and including two loops in the ’t Hooft coupling λ. At three loops it was shown that the string and gauge theory results differ. As has been noted many times in the literature, this result should not be interpretted as a falsification of the gauge/string correspondence conjecture. Indeed, while the (perturbative) gauge theory computatons are done at small values in λ, they are compared to dual string theory energies which are computed at large values of λ and as such are not necessarily comparable. It has then been a fortunate coincidence that the one- and two-loop results do match. This match was first established in a number of particular semi-classical string solutions and corresponding single-trace operators [5]. Later it was shown that, to leading order in the LCL, for some bosonic sub-sectors the string action reduced to a generalised Landau-Lifshitz (LL) sigma model, which also could be obtained as a thermodynamic limit of the corresponding spin-chain k1,k2,hl1,st1,k3 [13, 14, 16, 18, 15] (see also [20]). In this way, by matching Lagrangians on both sides one can establish that energies of a wide class of string solutions do indeed match with the corresponding anomalous dimensions of gauge theory operators without having to compute these on a case-by-case basis. A natural extension of this programme is to match, to leading order, the LCL of the full GS action of Type IIB string theory on AdS5 × S5 to the thermodynamic limit of the spin- chain corresponding to the dilatation operator for the full N = 4 SYM gauge theory; including fermions on both sides of the map is interesting given the different way in which they enter the respective actions. On the spin-chain side fermions are on equal footing to bosons st2,hl2 [19, 17] - the LL equation, which describes the thermodynamic limit of the system, relates to a super-coset manifold when fermions are included, as opposed to a coset manifold when there are no fermions. In particular, both fermions and bosons satisfy equations which are first order in τ and second order in σ. On the other hand, fermions in the GS action possess κ-symmetry ws,gs,hm,mt [22, 23, 24, 25] and their equations of motion are first order both in τ and σ. Previous progress on this question was able to match string and spin chain actions in a LCL up to quadratic level in fermions mikh,hl2,st2 [20, 17, 19]. Roughly speaking, on the string side, κ-gauge fixed equations of motion for fermions typically come as 2n first order equations. From these one obtains n second-order equations for n by ’integrating out’ half of the fermions. Taking a non-relativistic limit on the worldsheet one ends up with equations which are first order in τ and second order in σ which can be matched with the corresponding LL equations obtained from the spin chain side. Matching the terms quartic and higher in the fermions had so far not been achieved, though it is expected that this should be possible given the results of fullalgcurve [9]. However, finding a suitable κ-gauge in which this matching could be done in a natural way remained an obstacle. Below we propose a κ gauge which appears to be natural from the point of view of the dual spin-chain and allows for a matching of higher order fermionic terms in the dual Lagrangians. In this paper we first present a compact way of writing LL sigma models for quite general (super-)cosets G/H ; in particular we write down the full PSU(2, 2|4)/PS(U(2|2)2) LL sigma model which arrises as the thermodynamic limit of the one-loop dilatation operator for the full N = 4 SYM theory. This generalises earlier work by [11], and allows one to write down LL-type actions without having to go through the coherent-state [12] thermodynamic limit of the spin chain. We then identify a number of sub-sectors of the classical GS action 1 all of which have two real bosonic degrees of freedom and a larger number of fermionic degrees of freedom (specifically 4,8 and 16 real fermionic d.o.f.s 2). Finally, we define a LCL in which 1By a sub-sector we mean that the classical equations of motion for the full GS superstring on AdS5 × S5 admit a truncation in which all other fields are set to zero in a manner which is consistent with their equations of motion. This is quite familiar in two cases: (i) when one sets all fermions in the GS action to zero and, (ii) when one further restricts the bosons to lie on some AdSp × Sq sub-space (1 ≥ p , , q ≥ 5). 2The 4 fermion model was previously postulated to be a sub-sector of the classical GS action in [28] and represents a starting point for our analysis. the GS actions for these fermionic sub-sectors reduce to corresponding LL actions. In this way we match the complete Lagrangians for these sub-sectors and not just the terms quadratic in fermions. Since the largest of these sectors contains the maximal number of fermions (sixteen) for a κ-fixed GS action the LCL matching to a LL model gives a clear indication of what the natural κ-gauge is from the point of view of the dual spin-chain. The fermionic sub-sectors of the GS action that we find are quite interesting in themselves because on-shell κ-symmetry acts trivially on them - in particular the sub-sector containing 16 fermionic degrees of freedom contains the same number of fermions as the κ-fixed GS superstring on AdS5×S5. Since κ-symmetry acts trivially in this case one cannot use it to eliminate half of the fermions as one does in more conventional GS actions. Further, these fermionic sub-sectors naturally inherit the classical integrability of the full GS superstring on AdS5×S5 found in [21]. Integrating out the metric and the two bosonic degrees of freedom one then arrives at a new class of integrable differential equations for fermions only. This paper is organised as follows. In section 2 we give a prescription for constructing a LL sigma model on a general coset G/H . We also present a number of explicit examples of LL sigma models most relevant to the gauge/string correspondence there and in Appendix A. In section 3 we identify the fermionic sub-sectors of the GS superstring on AdS5×S5. In section we define a LCL in which the GS action of the fermionic sub-sectors reduces, to leading order in J , to the LL sigma models for the corresponding gauge-theory fermionic sub-sectors. Since the GS action for the four fermion subsector is quadratic in the remaining appendices to this paper we present a more detailed discussion of it including a light-cone quantisation in Appendix a discussion of its conformal invariance in Appendix C and a T-dual form of the action in Appendix 2 Landau-Lifshitz sigma models In this section we construct the Lagrangian for a Landau-Lifshitz (LL) sigma model on a coset G/H . 3 The Lagrangian will typically be first (second) order in the worldsheet time (space) coordinate, and so is non-relativistic on the worldsheet. We refer to such models as LL sigma models because in the case of G/H = SU(2)/U(1) the equations of motion reduce to the usual LL equation ∂τni = εijknj∂ σnk , where nini = 1 . (2.1) 3For earlier work on this see [11]. The construction of LL Lagrangians is closely related to coherent states |ω,Λ〉. Recall 4 that to construct a coherent state |ω ,Λ〉 we need to specify a unitary irreducible representation Λ of G acting on a Hilbert space VΛ and a vacuum state |0〉 on which H is a maximal stability sub-group, in other words for any h ∈ H Λ(h) |0〉 = eiφ(h) |0〉 , (2.2) vacuumphase with φ(h) ∈ R. Given such a representation Λ and state |0〉 we define the operator Ω as Ω ≡ |0〉 〈0| . (2.3) Omega The LL sigma model Lagrangian on G/H is defined as LLL G/H = LWZLL G/H + LkinLL G/H (2.4) LLsigmamodel where LWZLL G/H = −iTr Ωg†∂τg , (2.5) LLLWZ LkinLL G/H = Tr g†Dσgg . (2.6) Above, g†Dσg ≡ g†∂σg− g†∂σg|H is just the standard H-covariant current. It is then clear that LkinLL G/H is invariant under gauge transformations g → gh , (2.7) for any h = h(τ , σ) ∈ H . We may also show that the same is true of LWZLL G/H. To see this note that the gauge variation of LWZ LL G/H , using equation ( vacuumphase 2.2), is given by δHLWZLL G/H = e−iφ(h) 〈0| ∂τh |0〉 = e−iφ(h)∂τ (〈0|h |0〉) = i∂τφ(h) . (2.8) This in turn is a total derivative; and so the full action is invariant under local right H action. The Lagrangian also has a global G symmetry g → g0g , (2.9) for any g0 ∈ G with ∂τg0 = ∂σg0 = 0, and the corresponding Nöther current is given by (jτ , jσ) = (gΩg †, 2iDσgg †) . (2.10) 4For a detailed exposition of coherent states see [12]; a brief summary, using the same notation as in this paper, is also presented in Appendix A of [18]. st1,st2 [18, 19] LL actions were written down in terms of Lie algebra matrices denoted typically by N . To make contact with the present notation we note that 5 N ≡ gΩg† − 1 In , (2.11) where the second term on the right hand side is included since N is traceless. Finally, let us note that these LL sigma models admit a Lax pair representation and as a result are integrable. This is most easily seen in terms of the matrix N for which the equations of motion are the LL matrix equation ∂τN = N , ∂2σN . (2.12) This is equivalent to the zero-curvature condition on the following Lax pair L −→ ∂σ − , (2.13) M −→ ∂τ − 4π2x2 − [N, ∂σN} , (2.14) where [· , ·} is the (super)-commutator. In the remainder of this section we construct a number of explicit examples of LL sigma models. Further examples of interest in the gauge/string correspondence are relagated to Appendix A. The reader who is not interested in the details of these examples should skip the remainder of this section. 2.1 The U(1|1)/U(1)2 model This is one of the simplest LL sigma models, 6 in that the Lagrangian is quadratic LLL U(1|1)/U(1)2 = iψ̄∂τψ + ∂σψ̄∂σψ , (2.15) LLu11 with ψ a complex Grassmann-odd field and ψ̄ its complex conjugate. Notice that this result can be obtained using the explicit 2×2 supermatrix representation of U(1|1), with the vacuum state |0〉 being the super-vector (0, 1). 2.2 The SU(3)/S(U(2)× U(1) model Before proceeding to our main example - the PSU(2, 2|4) model - in this subsection we show how the above formal prescription applies to the well known SU(3) Landau-Lifshitz model st1,hl1 [18, 16]. 5The following equation is due to Charles Young. 6There is also the equally simple bosonic U(1) LL sigma model. Recall that the Lagrangian for this is LSU(3)/S(U(2) × U(1)) = −iU i∂τUi − |DσUi|2 + Λ(UiU i − 1) , (2.16) gtsu3 where DµUi ≡ ∂µ − iCµ , Cµ = −iU i∂µUi , (2.17) for µ = τ, σ and U i ≡ U∗i . To show that we can obtain this from our general expression ( LLsigmamodel we write elements of the group SU(3) as 3 × 3 matrix g, split into a 3 × 2 matrix X and a vector Y g = (X, Y ) , (2.18) and because g is in SU(3) (i.e. g†g = 1) we have X†X = 12 , Y †Y = 11 , X †Y = 0 , Y †X = 0 , (2.19) XX† + Y Y † = 13 . (2.20) The kinetic part of the Lagrangian ( LLsigmamodel 2.4) is then given by Lkin SU(3)/S(U(2) × U(1)) = (g−1D1g)(g −1D1g) X†D1X X Y †D1X Y 0 X†∂1Y Y †∂1X 0 X†∂1Y Y †Y Y †∂1X †XX†∂1Y †(1−XX†)∂1X i − YiY j)∂1Yj iD1Yi . (2.21) The final expression is the same as the kinetic term of the usual SU(3) Landau-Lifshitz La- grangian ( gtsu3 2.16) upon identifying Yi with Ui (above Y i ≡ Y †). Above, we have defined D1Yi = ∂1Yi − YiY j∂1Yj , D̄1Y i ≡ (D1Yi)† (2.22) D1X = ∂1X −XX†∂1X , D̄1X ≡ (D1X)† . (2.23) The WZ term of the Lagrangian is given by equation ( LLLWZ 2.5) and can be written as LWZ SU(3)/S(U(2) × U(1)) = iTr(X†∂0X) = −iY i∂0Yi . (2.24) This follows from the fact that g−1∂0g is traceless and so Tr(X†∂0X) = −Y i∂0Yi . (2.25) Upon identifying Yi with Ui, the WZ term above is the same as the usual SU(3) Landau- Lifshitz one ( gtsu3 2.16). Notice that we have also given an alternate parametrisation of the SU(3) Landau-Lifshitz model in terms of X LSU(3)/S(U(2) × U(1)) = iTr(X†∂0X)− + Λ(X†X − 12) , (2.26) which has an explicit SU(2) gauge invariance. Finally, out of X and Y we may define a matrix which takes values in the SU(3) Lie algebra N ij = 3Y iYj − δij = −3XjaXai + 2δij , (2.27) where a = 1, 2. This matrix is however, not a general SU(3) matrix but rather satisfies the identity N2 = N + 2 . (2.28) In terms of N the equations of motion take the form of the matrix Landau-Lifshitz equation ∂0N = − [N , ∂21N ] . (2.29) These are equivalent to the consistency of the following linear problem ψ = 0 , (2.30) 4π2x2 N − b [N, ∂1N ] ψ = 0 . (2.31) 2.3 The SU(2, 2|4)/S(U(2|2)× U(2|2)) model In this sub-section we present an explicit Lagrangian for the complete PSU(2, 2|4) Landau- Lifshitz sigma model Lagrangian following the general discussion at the start of the present section. The action we are interested in is the Landau Lifshitz model as defined in equation ( LLsigmamodel on the coset PSU(2, 2|4) PS(U(2|2)× U(2|2)) , (2.32) or on the coset SU(2, 2|4) S(U(2|2)× U(2|2)) , (2.33) both of which have 32 real components. The derivation is very similar to the SU(3) Lagrangian derived in the previous sub-section, and so we will simply state our results. A general group element g can be written as (X, Y ) where now X (Y ) is a 8× 4 supermatrix, with the diagonal 4 × 4 blocks bosonic (fermionic) and the off-diagonal 4 × 4 blocks fermionic (bosonic). The Lagrangian is then given by LLL PSU(2, 2|4)/PS(U(2|2) × U(2|2)) = iSTr(X†∂0X)− STr(D̄1X †D1X) + Λ(X †X − 1) . (2.34) Note that there are 32 complex degrees of freedom in X , which the constraints reduce to 48 real degrees of freedom. The action also has a local U(2|2) gauge invariance, so in total the above Lagrangian has 32 degrees of freedom - the same as the coset. In fact we may write X as X = (Ũa, Ṽa, Ua, Va) , X † ≡ (Ũa, Ṽ a, Ua, V a) , (2.35) where a = 1, . . . , 8, and ŨaŨa = −1 , Ṽ aṼa = −1 , Ṽ aŨa = 0 , ŨaṼa = 0 , (2.36) UaUa = 1 , V aVa = 1 , V aUa = 0 , U aVa = 0 , (2.37) UaŨa = 0 , U aṼa = 0 , V aŨa = 0 , V aṼa = 0 , (2.38) ŨaUa = 0 , Ũ aVa = 0 , Ṽ aUa = 0 , Ṽ aVa = 0 . (2.39) Above we have defined Ua = U∗bC ba , V a = V ∗b C ba , Ũa = −Ũ∗bCba , Ṽ a = −Ṽ ∗b Cba , (2.40) where Cab = diag(−1,−1, 1, 1, 1, 1, 1, 1). The Lagrangian ( psu224 2.41) written in terms of Ũa, Ṽa, Ua, Va is LLL PSU(2, 2|4)/PS(U(2|2)2) = −iŨa∂0Ũa − iṼ a∂0Ṽa − iUa∂0Ua − iV a∂0Va a∂1Ũa + ∂1Ṽ a∂1Ṽa + ∂1U a∂1Ua + ∂1V a∂1Va −Ũa∂1ŨaŨ b∂1Ũb − Ṽ a∂1ṼaṼ b∂1Ṽb +V a∂1VaV b∂1Vb + U a∂1UaU b∂1Ub +2V a∂1UaU b∂1Vb − 2Ṽ a∂1ŨaŨ b∂1Ṽb + 2Ũa∂1UaU b∂1Ũb +2Ũa∂1VaV b∂1Ũb + 2Ṽ a∂1UaU b∂1Ṽb + 2Ṽ a∂1VaV b∂1Ṽb . (2.41) psu224 One can check explicitly that this action has local U(2|2) invariance (Ũa, Ṽa, Ua, Va)→ (Ũa, Ṽa, Ua, Va)U(τ, σ) , (2.42) for U a U(2|2) matrix. 2.3.1 Subsectors of the the SU(2, 2|4)/S(U(2|2)× U(2|2)) model In the above Lagrangian we may set Ũa = (1, 0 7) , Ṽa = (0, 1, 0 6) , Ua = (0 2, U3, . . . , U8) , Va = (0 2, V3, . . . , V8) , (2.43) where UaUa = 1 , V aVa = 1 , V aUa = 0 , U aVa = 0 . (2.44) The resulting Lagrangian is that of the SU(2|4) sector. If we further set 0 = U3 = U4 = V3 = V4 , (2.45) we can recover the SO(6) Lagrangian ( [18]). Details of this are presented in Appendix B. We may further consistently set 0 = U8 = V3 = V4 = V5 = V6 = V7 , V8 = 1 , (2.46) in which case we obtain the SU(2|3) Lagrangian ( [19]), with the identification (U3, U4) ≡ (ψ1, ψ2). We may instead set Ua = (0 7, 1) , Va = (0 6, 1, 0) , Ũa = (U1, . . . , U6, 0 2) , Ṽa = (V1, . . . , V6, 0 2) , (2.47) where ŨaŨa = −1 , Ṽ aṼa = −1 , Ṽ aŨa = 0 , ŨaṼa = 0 . (2.48) The resulting Lagrangian is that of the SU(2,2|2) sector. If we further set 0 = U3 = U4 = V3 = V4 , (2.49) we recover the SO(2,4) Lagrangian, which is the Wick rotated version of the SO(6) La- grangian ( [18]). In Appendix B we write out this Lagrangian explicitly. A final interesting choice is to set Ua = (0 7, 1) , Va = (0, V2, . . . , V7, 0) , Ũa = (0, U2, . . . , U7, 0) , Ṽa = (1, 0 7) , (2.50) where ŨaŨa = −1 , V aVa = 1 , V aŨa = 0 , ŨaVa = 0 . (2.51) The resulting Lagrangian is that of the SU(1,2|3) sector. If we further set 0 = V2 = V7 = Ũ2 = Ũ7 , (2.52) we get the SU(2|2) Lagrangian. In Appendix B we write out this Lagrangian explicitly. 3 Green-Schwarz actions and fake κ-symmetry in this section we construct GS sigma model actions whose field content are two real bosons and 4,8 or 16 real fermions. These models all come from consistent truncations of the equations of motion for the full Type IIB GS action on AdS5 × S5. Just as any GS sigma model these fermionic actions have a κ-symmetry. However, we show that for these models κ-symmetry is trivial on-shell. As a result one cannot use it to reduce the fermionic degrees of freedom of these models by fixing a κ-gauge as one does in more conventional GS actions. Let us briefly recall the construction of the GS action on a super-coset G/H . We require that: (i) H be bosonic and, (ii) G admit a ZZ4 automorphism that leaves H invariant, acts by −1 on the remaining bosonic part of G/H , and by ±i on the fermionic part of G/H . The currents jµ = g †∂µg can then be decomposed as jµ = j µ + j µ + j µ + j µ , (3.1) currz4dec where j(k) has eigenvalue ik under the ZZ4 automorphism. In terms of these the GS action can be written as LGS G/H = −ggµνStr(j(2)µ j(2)ν ) + ǫµνStr(j(1)µ j(3)ν ) , (3.2) z4gs from which the equations of motion are 0 = ∂α( −ggαβj(2)β )− −ggαβ j(0)α , j j(1)α , j j(3)α , j , (3.3) eom1 −ggαβ + ǫαβ j(3)α , j , (3.4) eom2 −ggαβ − ǫαβ j(1)α , j . (3.5) eom3 3.1 Fermionic GS actions Having briefly reviewed the general construction of GS actions on G/H super-cosets, we now turn to the main focus of this section which is identifying GS actions with a large number of fermionic degrees of freedom, which are consistent truncations of the full AdS5×S5 GS action. To do this consider the following sequence of super-cosets U(1|1)× U(1|1) U(1)× U(1) ⊂ U(2|2) SU(2)× SU(2) ⊂ PS(U(1, 1|2)× U(2|2)) SU(1, 1)× SU(2)3 ⊂ PSU(2, 2|4) SO(1, 4)× SO(5) . (3.6) The ⊂ symbols are valid both for the numerators and denominators and hence for the cosets as written above. Notice that the right-most of these cosets is just the usual Type IIB on AdS5 × S5 super-coset. Further, it is easy to convince onself that each of the cosets above admits a ZZ4 automorphism which is compatible with the ZZ4 automorphism of the Type IIB on AdS5 × S5 super-coset. The ZZ4 automorphisms may be used to write down GS actions for each of these cosets. The fact that the cosets embed into each other as shown above in a manner compatible with the ZZ4 automorphism implies that their GS actions can be thought of as coming from a consistent truncation of the GS action of any coset to the right of it in the above sequence. In particular this reasoning shows that the GS actions for U(1|1)2/U(1)2, U(2|2)/SU(2)2 and U(1, 1|2)×U(2|2)/(SU(1, 1)×SU(2)3 can all be thought of as coming from consistent truncations of the Type IIB GS action on AdS5 × S5. Counting the number of bosonic and fermionic components of the three cosets U(1|1)2/U(1)2, U(2|2)/SU(2)2 and U(1, 1|2)× U(2|2)/(SU(1, 1)× SU(2)3 we see immediately that they each have 2 real bosonic components and, respectively, 4,8 and 16 real fermionic components - which is why we refer to these actions as fermionic GS actions. We might expect that some of the femrionic degrees of freedom could be eliminated from the GS actions by fixing κ-symmetry. In fact, it turns out that for these models κ-symmetry acts trivially on-shell and so cannot be used to eliminate some of the fermionic degrees of freedom. Indeed, the GS actions on the above-mentioned cosets do have 4,8 and 16 real fermionic degrees of freedom, respectively. In the remainder of this sub-section we write down explicitly the GS actions for U(2|2)/SU(2)2 and U(1|1)2/U(1)2 and discuss their κ and gauge transformations; the GS action for U(1, 1|2)× U(2|2)/SU(1, 1)× SU(2)3 may also be written down in an analogous fashion but since we will not need its explicit form later we refrain from writing it out in full. 3.2 The GS action on U(2|2)/SU(2)2 The GS action on action on U(2|2)/SU(2)2 can be written down in terms of the parametrisation of the U(2|2) supergroup-valued matrix written as g = (X, Y ; X̃, Ỹ ) , (3.7) ads2s2param whereX , Y (X̃ , Ỹ ) are four-component super-vectors with the first (last) two entries Grassmann even and the last (first) two entries Grassmann odd. Since the matrix g is unitary we must 1 = X†X = Y †Y = X̃†X̃ = Ỹ †Ỹ , 0 = X†Y = Y †X = X†X̃ = X̃†X = X†Ỹ = Ỹ †X = Y †X̃ = X̃†Y = Y †Ỹ = Ỹ †Y = X̃†Ỹ = Ỹ †X̃ , 1(2|2) = XX † + Y Y † + X̃X̃† + Ỹ Ỹ † , (3.8) ads2s2const where the matrix 1(2|2) is just the 4× 4 identity matrix. The ZZ4 automorphism is given by Ω : M = −AT CT −BT −DT , (3.9) Z4autu22 which acts on the current as Ω(jµ) = −Y †∂µY X†∂µY −Ỹ †∂µY X̃†∂µY Y †∂µX X †∂µX Ỹ †∂µX −X̃†∂µX Y †∂µỸ −X†∂µỸ −Ỹ †∂µỸ X̃†∂µỸ −Y †∂µX̃ X†∂µX̃ Ỹ †∂µX̃ −X̃†∂µX̃ . (3.10) The Green-Schwarz action then is LGS U(2|2)/(SU(2)×SU(2)) = (X†∂µX + Y †∂µY )(X †∂νX + Y †∂νY ) −(X̃†∂µX̃ + Ỹ †∂µỸ )(X̃†∂νX̃ + Ỹ †∂ν Ỹ ) +2iǫµν X†∂µX̃Y †∂νỸ + Ỹ †∂µY X̃ −X†∂µỸ Y †∂νX̃ − X̃†∂µY Ỹ †∂νX . (3.11) u22gs One can easily check that this action has a local SU(2)× SU(2) invariance which acts on the doublets (X, Y ) and (X̃, Ỹ ). The action also has κ-symmetry which acts on the fields as 7 δκX = −X̃(ǭ1 + ¯̃ǫ1)− Ỹ (ǭ2 + ¯̃ǫ2) δκY = iX̃(ǫ2 − ǫ̃2)− iỸ (ǫ1 − ǫ̃1) δκX̃ = X(ǫ1 + ǫ̃1) + iY (ǭ2 − ¯̃ǫ2) δκỸ = X(ǫ2 + ǫ̃2)− iY (ǭ1 − ¯̃ǫ1) , (3.12) ku22coord where ǫi = Π †∂αX + Y †∂αY + X̃ †∂αX̃ + Ỹ †∂αỸ )κi , β ǫ̃i = Π †∂αX + Y †∂αY + X̃ †∂αX̃ + Ỹ †∂αỸ )κ̃i , β , (3.13) ku22eps for i = 1 , 2 with κi , β and κ̃i , β local Grassmann-odd parameters. The world-sheet metric also varies as −ggαβ) = Παγ+ 1 ,+(X̃ †∂γX − iY †∂γ Ỹ ) + κβ2 ,+(Ỹ †∂γX + iY †∂γX̃) + c.c. + α↔ β 1 ,−(X̃ †∂γX + iY †∂γỸ ) + κ̃ 2 ,−(Ỹ †∂γX − iY †∂γX̃) + c.c. + α↔ β . (3.14) ku22metric 7The κ-action below has the nice feature of acting as a local fermionic group action by multiplication from the right. Such a representation was originally suggested in mcarthur [26] and was developed more fully for the AdS5×S5 GS action in glebnotes [27]; the formulas below are a simple extension of this latter construction to the coset at hand. Notice that the above variation is consistent with the symmetries and the unimodularity of √−ggαβ as long as καi = Π + κi , β , κ̃ i = Π − κ̃i , β . (3.15) In the above formulas we have decomposed two-component vectors vα as vα± ≡ Π ± vβ ≡ −ggαβ ± ǫαβ vβ . (3.16) 3.3 The GS action on U(1|1)2/U(1)2 To obtain the GS action on U(1|1)2/U(1)2 we may simply set 0 = X3 = Y4 = X̃1 = Ỹ2 . (3.17) in the action ( u22gs 3.11). This is because now the group element g given in equation ( ads2s2param 3.7) belongs to U(1|1)2 ⊂ U(2|2); this truncation is also consistent with the ZZ4 automorphism ( Z4autu22 3.9). As was argued at the start of this sub-section these facts imply that setting the above components to zero is a consistent truncation of the equations of motion for the action ( u22gs 3.11). The GS action for the truncated theory then is LGS U(1|1)2/U(1)2 = (X†∂µX + Y †∂µY )(X †∂νX + Y †∂νY ) −(X̃†∂µX̃ + Ỹ †∂µỸ )(X̃†∂νX̃ + Ỹ †∂ν Ỹ ) −2iǫµν X†∂µỸ Y †∂νX̃ + X̃ †∂µY Ỹ . (3.18) gsu112 It has two U(1) gauge invariances X → eiθ1X , Y → eiθ1Y , (3.19) X̃ → eiθ2X̃ , Ỹ → eiθ2 Ỹ , (3.20) as well as κ-symmetry which is simply the restriction of equations ( ku22coord 3.12) and ( ku22metric 3.14). If we parametrise the group element g = (X, Y, X̃, Ỹ ) ∈ U(1|1)2 by X = (eit/2(1 + ψ2) , 0 , 0 , −e−iα/2ψ̄) , Y = (0 , eit/2(1 + 1 η2) , −e−iα/2η̄ , 0) ,(3.21) X̃ = (0 , eit/2η , e−iα/2(1− 1 η2) , 0) , Ỹ = (eit/2ψ , 0 , 0 , e−iα/2(1− 1 ψ2)) , (3.22) where ψ2 ≡ ψ̄ψ and η2 ≡ η̄η, the action ( gsu112 3.18) becomes LGS U(1|1)2/U(1)2 = −∂µφ+∂νφ− + i∂µφ+ηi i − ∂µφ+∂νφ+ηiηi −ǫµν∂µφ+(η1 ∂ν η2 − η1 2) , (3.23) gsu112comp This action was postulated in [28] to be a consistent truncation of the full Type IIB GS action on AdS5 × S5, by checking the absense of certain cubic terms in the latter action, using an ex- plicit non-unitary representation for PSU(2, 2|4). Here we have shown that on group-theoretic grounds this action is indeed such a consistent truncation, and have obtained its form using a unitary representation of the group. On the local coordinates defined above κ-symmetry acts as δηi = ǫi , δt = −δα = i ηiǫi + ηiǫ . (3.24) In particular notice that δφ+ = 0. The parameters ǫi are not however free, instead they are given by i + i∂αφ− + iη iηi∂αφ+ καj . (3.25) epskappa Above καi are complex-valued Grassmann functions of the world-sheet; their complex conjugates are denoted by κα i. We will also require that the metric vary under κ-symmetry as −ggαβ) = − i + (−η1∂γφ+ − iη2∂γφ+ + 2i∂γη1 + 2∂γη2) +κ̄(αP + (−η1∂γφ+ + iη2∂γφ+ − 2i∂γη1 + 2∂γη2) +κ̃(αP − (η1∂γφ+ − iη2∂γφ+ − 2i∂γη1 + 2∂γη2) 1∂γφ+ + iη2∂γφ+ + 2i∂γη 1 + 2∂γη2) = − i −ggαγ κβ i∂γηi + κ i ∂γη ∂γφ+(κ β iηi − κβi ηi) +iǫαγ 1∂γη2 + κ 2∂γη1 − κβ 1∂γη2 − κβ 2∂γη1 ǫαγ∂γφ+ 1η2 + κ 2η1 + κ β 1η2 + κβ 2η1 . (3.26) epskappa where a(αbβ) = aαbβ + aβbα and κα1 = α − κ̄α) , κα2 = (κ̃α + κα) , (3.27) with the complex conjugates defined as κ† ≡ κ̄ and κ̃† ≡ ¯̃κ. The above variation of the metric is symmetric and since −ggαβ has unit determinant (is uni-modular) we require that κα = P + κβ , κ̃ α = P − κ̃β . (3.28) Using the above formulas one can check that the action ( gsu112comp 3.23) is indeed invariant under this symmetry. However, as we show below this local symmetry is trivial on-shell. 3.4 Fake κ-symmetry partlim In this sub-section we show that κ-symmetry acts trivially on-shell on the fermionic GS actions studied in this section. To see this most easily we will first consider the particle limit (in other words we remove all σ dependence of fields) for the action LGS U(1|1)2/U(1)2 . This gives Lparticle = − dτe−1φ̇+ φ̇− + φ̇+η iηi − iηiη̇i − iηiη̇i dτe−1φ̇+a , (3.29) supart where for convenience we have defined 8 φ̇− + φ̇+η iηi − iηiη̇i − iηiη̇i . (3.30) Setting e = constant, we may solve the the φ+, φ− and ηi equations of motion to get φ+ = 2κτ , φ− = λτ , ηi = e −iκτη0 i , (3.31) where κ, λ (respectively, η0 i) are complex constant Grassmann-even (odd) numbers. 9 Finally, we turn to the equation for the einbein e which reduces to κλ = 0 . (3.32) or in other words forces us to set either κ or λ to zero. As a result the theory consists of two sectors, one with κ = 0 and the other with λ = 0. The former sector is trivial and uninteresting as all fields apart from φ− are constant and the energy is zero. The physically more relevant sector has λ = 0 and κ 6= 0. Let us now turn to the κ invariance of the action ( supart 3.29). It is easy to see that this action is invariant under δφ+ = 0 , δηi = aκi , δφ− = ia(η iκi + ηiκ δ(e−1) = 2i(η̇iκi + η̇iκ i) + φ̇+(η iκi + ηiκ i) , (3.33) where κi are arbitrary Grassmann-odd functions of τ . Since we are free to pick the parameters κi one might think that we could simply gauge away the femrionic degrees of freedom using this symmetry; had the κ variations been of the form δηi = κi , 8As an aside note that the fermion index i can now run over any number and is not restricted to i = 1, 2 as is the case for the super-string. This is quite typical of κ-invariant particle actions. 9In the above solution we have, without loss of generality, set the constant parts of φ+ and φ− to zero. we would have been able to gauge away the fermions. In fact this is not the case: the κ variation of the fermions instead reads δηi = aκi , (3.34) From the equation for the einbein e we see that in fact a = 0 (in the physically important sector for which κ 6= 0 as discussed above) and so on-shell the above κ symmetry acts trivially on all fields except the einbein itself. But any κ variation of the einbein e can be compensated for by a diffeomorphism. We conclude that while the actions ( supart 3.29) and ( gsu112 3.18) formally have a κ-symmetry, this has a trivial action on-shell and so cannot be used to eliminate any fermions. The argument in the above paragraph relies on the fact that on fermions κ-symmetry was acting as δηi = aκi and on-shell a = 0. Returing to the fermionic GS superstring actions discussed in this section we see from equation ( ku22eps 3.13) that here too κ-symmetry acts as δηi = astringκi, where astring = (X †∂αX + Y †∂αY + X̃ †∂αX̃ + Ỹ †∂αỸ ) . (3.35) It is easy to check that because of the Virasoro constrains astring is also zero on-shell. We conclude that the κ-symmetry of the action ( gsu112comp 3.23) is trivial on-shell and so cannot be used to eliminate any fermions. 4 Large charge limits of fermionic GS actions Given a ZZ4 automorphism on some coset G/H we may construct a Green-Schwarz Lagrangian for it ( 3.2). On general grounds the large charge limit of this Lagrangian should be a generalised Landau Lifshitz sigma model. Further, since we expect the global charges of the two actions to map onto one another, this LL sigma model should be constructed on a coset G/H̃. In this section we will attempt to identify H̃. One step in this direction is to count the number of degrees of freedom that the GS action has and compare it with that of the LL model. For example in the case of the Type IIB superstirng on AdS5×S5 there are 10 real bosonic degrees of freedom, and there are 32/2 = 16 fermionic degrees of freedom (where the factor of 1/2 comes from κ symmetry). In the large charge limit two of the bosonic degrees of freedom are eliminated; the remaining eight are ’doubled’ since the LL Lagrangian should be thought of as a Lagrangian on phase space. The 16 fermions are described by coupled first order equations. When taking the LCL we integrate out half of the fermions, in order to arrive at second order equations [19], leaving us with 8 real fermionic degrees freedom; as in the case of the bosons this should also be ’doubled’, leaving us with 16 fermionic degrees of freedom. At this point we may simply guess what H̃ is in the case of G = PSU(2, 2|4), since the only coset of the form G/H̃ with 16 bosonic and fermionic degrees of freedom each is H̃ = PS(U(1, 1|2)× U(2|2)) , (4.1) though of course in this case H̃ is well known from gauge theory. Let us persue this counting argument further and consider the GS action on U(1|1)2 U(1)2 . (4.2) This is a sub-sector of the classical GS string action on AdS5 × S5. It has 2 real bosonic degrees of freedom and 4 real fermionic degrees of freedom. As was shown in section partlim 3.4, κ- symmetry in this case is trivial on-shell, and so, following the counting argument in the previous paragraph, 10 we expect the LL sigma model corresponding to the LCL of this GS action to have 4 real fermionic degrees of freedom and no bosonic degrees of freedom. The only such coset is U(1|1)2 U(1)4 , (4.3) in other words H̃ = U(1)4. Similarily, we may consider the bigger sub-sector of the full classical superstring on AdS5×S5 U(2|2) SU(2)2 , (4.4) for which κ-symmetry is also trivial on-shell. This sub-sector has 2 bosonic and 8 fermionic d.o.f. As a result we expect the LL sigma-model to have no bosonic d.o.f. and 8 fermionic d.o.f. Again this is enough for us to identify U(2|2) U(2)2 , (4.5) as the coset on which the LL sigma model is constructed. Finally, the largest classical sub- sector of the GS string action on AdS5 × S5 for which κ-symmetry is trivial is the GS action PS(U(1, 1|2)× U(2|2)) SU(1, 1)× SU(2)3 . (4.6) 10For the bosons we subtract two real degrees of freedom in the LCL and double the remaining ones. In the present case this gives 2× (2−2) = 0 d.o.f. For the fermions, the number of d.o.f. in the LL sigma model should be the same as that of the GS string once κ-symmetry is fixed. This is because, once κ-symmetry is fixed, we halve the number of d.o.f. since the GS action gives first order differential equations, and the LL action gives second order differential equations; we then double it because the LL action is an action on phase space. In the present case, since κ-symmetry is trivial on-shell we end up with 2× 4/2 = 4 fermionic d.o.f. By our counting argument the corresponding LCL coset should have 16 fermionic and no bosonic d.o.f. As a result, the LL sigma model which corresponds to the LCL limit of the GS action on (U(1, 1|2)× U(2|2))/SU(1, 1)× SU(2)3 is constructed over the coset PS(U(1, 1|2)× U(2|2)) U(1, 1)× U(2)3 . (4.7) While this counting argument shows how to identify H̃, it is not very clear how the LCL should be taken in practice and in particular how starting from a GS action one arrives at a LL action. The rest of this section will address these issues in the three cases ofG = U(1|1)2 , U(2|2) and U(1, 1|2)×U(2|2). We will restrict our discusion to the leading order term in the LCL and leave the matching of sub-leading terms to a future publication. 4.1 Matching the U(1|1)2 sub-sectors In this subsection we will argue that the large charge limit of the Lagrangian given in equa- tions ( gsu112 3.18) and ( gsu112comp 3.23) which describes the Green-Schwarz string on the coset U(1|1)2 U(1)2 , (4.8) is given by the Landau-Lifshitz Lagrangian on the coset 11 U(1|1)2 U(1)4 . (4.9) We will first arrive at this result in a very pedestrian way. Since general solutions to both the LL and GS cosets can be given explicitly in full generality we will write them down using unconstrained coordinates. On the GS side, φ+ = κτ , (4.10) the general solution takes the form eiωnτψ+n + e −iωnτψ−n , (4.11) where ψ±n are constant Grassmann-odd numbers, and n2 + κ2/4 . (4.12) 11This is somewhat different to the comparison between gauge and string theory done in ( [28]) where it was argued that on the gauge theory side the coset should be U(1|1)/U(1)2. η2 is completely determined via the equation of motion ∂ση2 = i∂τη 1 − κ η1 . (4.13) In the LCL we take κ→∞ in which case we have ei(κ/2+n 2/κ)τψ+n + e −i(κ/2+n2/κ)τψ−n = eiκτ/2 ψ+0 + ψ+n e inσ + ψ+−ne +e−iκτ/2 ψ−0 + ψ−n e inσ + ψ−−ne ≡ eiκτ/2ψ1 LL + e−iκτ/2ψ̄2 LL , (4.14) where ψ1 LL and ψ2 LL are the 2 complex fermionic d.o.f. for the LL sigma model on (see equation ( LLu11 2.15)) U(1|1)2 U(1)4 . (4.15) In particular, after rescaling τ → κτ , they satisfy the equations of motion ∂2σ − i∂τ ψ1,2 LL . (4.16) In this way we match, to leading order in the LCL, the classical string Lagrangian with the corresponding coherent state continuum limit of the gauge theory dilatation operator in the U(1|1)2 sub-sector. Notice that physical string solutions have to satisfy the level-matching condition ∂1φ− = 2πm , for m ∈ ZZ . (4.17) levmatch The winding parameter m does not, however, enter the LCL Lagrangian Rather, it gives a constraint on its solutions. This matches the spin-chain side where m enters as a constraint on the Bethe roots, but does not enter the algebraic Bethe equations or the LL sigma-model action. This feature is very similar to the SL(2) sector discussed in [29]. 4.2 Large Charge Limit of fermionic GS actions In this section we re-phrase the above discussion in terms of the embedding coordinates X, Y . . . , and the currents j µ . This allows for a straightforward generalisation from the U(1|1)2 sub-sector to the U(2|2) and U(1, 1|2) × U(2|2) sub-sectors. We present the explicit discussion only for the case of U(2|2), but the other case follows almost trivially. The first thing to note is that the equation of motion for one of the two bosonic fields, φ+, is particularily simple in the GS models presently considered. This can be obtained as the super-trace of equation ( 3.3). As a result we may set X†∂µX + Y †∂µY − X̃†∂µX̃ − Ỹ †∂µỸ = iκδµ , ,0 . (4.18) phipansatz Using this, in conformal gauge the equation of motion for the off-diagonal component of the worldsheet metric implies that X†∂σX + Y †∂σY + X̃ †∂σX̃ + Ỹ †∂σỸ = 0 , (4.19) offdiagVir while the fermionic equations of motion ( 3.4), ( 3.5) reduce to 12 0 = κ(j(3)τ − j(3)σ ) + . . . , 0 = κ(j(1)τ + j(1)σ ) + . . . . (4.20) fermrel As a result of these relations the WZ term does not contribute to the bosonic equation of motion ( 3.3). 13 This fact allows us to check explicitly that the bosonic equations of motion, together with the ansatz ( phipansatz 4.18), are consistent with the equations of motion for the metric gµν in conformal gauge. In fact these Virasoro constraints then imply that Dµt = δµ ,0 , D̃µα = −δµ ,0 . (4.21) As in the discussion around equation ( levmatch 4.17) above, the level matching condition that follows from the Virasoro constraints does not enter the LCL action. Using equations ( phipansatz 4.18), ( offdiagVir 4.19) and ( fermrel 4.20) together with a rescaling τ → κτ we may re-write 12In terms of X, Y, X̃, Ỹ this implies that we have relations of the form X†∂τ Ỹ = iX̃ †∂σY , X̃ †∂τY = −iX†∂σỸ , etc . 13This is easy to see since the WZ term’s contribution to these equations is proportional to τ , j τ , j . However, since j τ = −j(1)σ and j(3)τ = j(3)σ each of these commutators vanishes seperately. the GS Lagrangian in conformal gauge as follows LGS U(2|2)/SU(2)2 = ηµνStr(j(2)µ j(2)ν ) + ǫµνStr(j(1)µ j(3)ν ) = ηµν X†∂µX + Y †∂µY − X̃†∂µX̃ − Ỹ †∂µỸ X†∂µX + Y †∂µY + X̃ †∂µX̃ + Ỹ †∂µỸ −2Str j(1)σ j X†∂τX + Y †∂τY + X̃ †∂τ X̃ + Ỹ †∂τ Ỹ − STr (j(1)σ + j σ )(j σ + j = LLL U(2|2)/U(2)2 (4.22) The right-hand side of the above equation is nothing but the LL sigma model Lagrangian defined on G/H̃, where H̃ is fixed under the ZZ2 automorphism which is the square of the ZZ4 automorphism used in the construction of the GS action. We have thus shown that to leading order in the LCL the fermionic GS actions constructed in section 3 above reduce to LL sigma model actions in the manner anticipated by the general argument presented at the start of the present section. It would be interesting to consider sub-leading corrections to this LCL for example in a manner similar to [15]. 4.3 A gauge-theory inspired κ gauge The GS sigma model on AdS5 × S5 has κ-symmetry. This, as well as other symmetries of the string action, such as world-sheet diffeomorphisms, are not manifest in the corresponding spin- chain simply because this latter system keeps track only of the physical degrees of freedom. One of the challenges of defining a LCL is to identify suitable gauges for these stringy symmetries in which the physical degrees of freedom are written in the most natural coordinates for the spin-chain: while all gauges should be in principle equivalent it may be much more difficult to define a LCL between the two theories if we pick an unnatural gauge. In the previous sub-section we have defined an LCL which matches all 16 fermionic degrees of freedom from the GS action to the corresponding LL model in a very natural way. This strongly suggests what κ-gauge should be used in the full AdS5 × S5 string action when comparing to gauge theory. Specifically it should be the gauge which keeps non-zero the 16 fermions of the coset PS(U(1, 1|2)×U(2|2))/(SU(1, 1)×SU(2)3). In fact this is the gauge used recently in [30] and the above argument can be interpreted as one motivation for their κ-gauge choice. Acknowledgements I am grateful to Arkady Tseytlin for many stimulating discussions throughout this project and to Chris Hull for a number of detailed conversations on κ-symmetry. I would also like to thanks Charles Young for discussions and Gleb Arutyunov for providing a copy of his notes glebnotes [27]. This research is funded by EPSRC and MCOIF. A Some examples of Landau-Lifshitz sigma models In this appendix we collect some expressions for a number of relevant Landau-Lifshitz sigma models. A.1 The SU(2|3)/S(U(2|2)× U(1)) model The SU(2|3) sub-sector sigma model Lagrangian is LLL SU(2|3) = −iU i∂τUi − iψα∂τψa − |DσUi|2 − aDσψa + Λ(UiU i + ψaψ a − 1) , (A.1) gtsu23 where Dµ ≡ ∂µ − iCµ , D̄µ ≡ ∂µ + iCµ , Cµ = −iU i∂µUi − iψa∂µψa , (A.2) and ψa = ψ∗a and a = 1, 2. A.2 The SU(4)/S(U(2)× U(2)) model The SO(6) ∼ SU(4) sub-sector sigma model Lagrangian is LLL SU(4) = LSU(4) WZ − Tr(∂1m) 2 − 1 Tr(m∂1m) 2 + Λ(m−m3) = −iV i∂τVi − |DσVi|2 + Λ1(V iVi − 1) + Λ2(ViVi − 1) + Λ∗2(V iV i − 1) ,(A.3) gtso6 where mij is a 6× 6 matrix, related to Vi by mij = ViV j − VjV i , (A.4) Dµ ≡ ∂µ − iCµ , Cµ = −iV i∂µVi . (A.5) Let us define MAB = B , mij = tr(Mρij) , (A.6) where ρ are the usual SU(4) ρ-matrices. Notice that TrM = 0 , M † =M , M2 =M . (A.7) and so we can write it as M = 2XX† − 1 = −2Y Y † + 1 , (A.8) where now X and Y are 4× 2 matrices which satisfy X†X = 12 , Y †Y = 12 , X †Y = 0 , Y †X = 0 , (A.9) XX† + Y Y † = 14 . (A.10) Further we can write the 4× 2 matrix X as two four-component vectors uA and vA X = (uA, vA) , (A.11) collvect in terms of which MAB can be written as MAB = 2u AuB + 2v AvB − δAB , (A.12) uAuA = 1 , v AvA = 1 , u AvA = 0 . (A.13) uvconds We can relate uA and vA to Vi by uAρiABv B , V i = iABuB . (A.14) natgencoords It is an easy check to see that these are consistent with i = 1 , ViVi = 0 , M B = ViV jρijAB . (A.15) In terms of these, the Lagrangian is LLL SU(4) = −iuA∂0uA − ivA∂0vA − A∂1uA + ∂1v A∂1vA +uA∂1uAu B∂1uB + v A∂1vAv B∂1vB + 2u A∂1vAv B∂1uB = −iTr(X†∂0X)− Tr(D̄1X †D1X) , (A.16) goodllso6 As before X = (uA, vA) , X , (A.17) DµX = ∂µX −XX†∂µX . (A.18) The action ( goodllso6 A.16) has a local U(2) invariance X → XU(τ, σ) , (A.19) for U(τ, σ) a general U(2) matrix U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.20) In terms of the uA and vA the action ( goodllso6 A.16) is invariant with respect to the following local transformations (uA, vA) → (cos θ(τ, σ) uA + sin θ(τ, σ) vA,− sin θ(τ, σ) uA + cos θ(τ, σ) vA) , (uA, vA) → (eiφ1(τ,σ)uA, eiφ1(τ,σ)vA) , (uA, vA) → (eiφ2(τ,σ)uA, e−iφ2(τ,σ)vA) , (uA, vA) → (eiφ3(τ,σ)vA,−eiφ3(τ,σ)uA) , (A.21) A.2.1 Subsectors of the SU(4)/S(U(2)× U(2)) model so6sub When written in terms of the Vi, the Lagrangian LSU(4) can be reduced to the SU(3) sub-sector by requiring V 2a = −iV 2a−1 ≡ 1√ Ua , a = 1, 2, 3 , (A.22) which can further be restriced to the SU(2) subsector for V 5 = 0 = V 6. In terms of the uA and vA this restriction is easily enforced by setting for example uA = (U1, U2, U3, 0) , vA = (0, 0, 0, 1) . (A.23) Since UaU a = 1, this choice satisfies the constraints ( uvconds A.13). Restricting to the SU(2) sector is achieved by setting u3 = U3 = 0. Upon inserting these ansatze, the Lagrangian ( goodllso6 A.16) reduces to the Lagrangian ( gtsu3 2.16). Another interesting sub-sector is obtained by setting uA = (U1, U2, 0, 0) , vA = (0, 0, V3, V4) , (A.24) su2su2sub together with the conditions U1U1 + U 2U2 = 1 , V 1V1 + V 2V2 = 1 . (A.25) This results in SU(2)×SU(2) subsector consisting of two decoupled SU(2) Landau Lifshitz Lagrangians. A.3 The SU(2, 2)/S(U(2)× U(2)) model For later convenience we present here the SO(2, 4)/S(O(2)×O(4)) ∼ SU(2, 2)/S(U(2)×U(2)) Landau-Lifshitz Lagrangian LLL SU(2,2) = −iṼ i∂0Ṽi − |DσṼi|2 = −iũA∂0ũA − iṽA∂0ṽA − A∂1ũA + ∂1ṽ A∂1ṽA −ũA∂1ũAũB∂1ũB − ṽA∂1ṽAṽB∂1tvB − 2ũA∂1ṽAṽB∂1ũB = iTr(X̃†∂0X̃) + Tr(D̄1X̃ †D1X̃) , (A.26) goodllso24 where Ṽ i ≡ Ṽ ∗j ηji , where ηij = diag(−1,−1, 1, 1, 1, 1) , (A.27) ũA ≡ ũ∗BCBA , ṽA ≡ ṽ∗BCBA , where CAB = (1, 1,−1,−1) . (A.28) The 4× 2 matrix X̃ has two columns X̃ = (ũA, ṽA) , (A.29) and the covariant derivatives are DµṼi = ∂µṼi + Ṽ j∂µṼjṼi , (A.30) DµX̃ = ∂µX̃ − X̃X̃†∂µX̃ . (A.31) We define X̃† ≡ − . (A.32) This is done for convenience, so that the form of the action in terms of X is independent of the signature. The fields in the Lagrangian ( goodllso24 A.26) now satisfy the constraints X̃†X̃ = 12 , (A.33) Ṽ iṼi = −1 , ṼiṼi = 0 , (A.34) ũAũA = −1 , ṽAṽA = −1 , ũAṽA = 0 . (A.35) uvlorconst The action ( goodllso24 A.26) has a local non-compact U(2) invariance X̃ → X̃U(τ, σ) , (A.36) for U(τ, σ) a general U(2) matrix U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.37) In terms of the ũA and ṽA the action ( goodllso24 A.26) is invariant with respect to the following local transformations (ũA, ṽA) → (cos θ(τ, σ) ũA + sin θ(τ, σ) ṽA,− sin θ(τ, σ) ũA + cos θ(τ, σ) ṽA) , (ũA, ṽA) → (eiφ1(τ,σ)ũA, eiφ1(τ,σ)ṽA) , (ũA, ṽA) → (eiφ2(τ,σ)ũA, e−iφ2(τ,σ)ṽA) , (ũA, ṽA) → (eiφ3(τ,σ)ṽA,−eiφ3(τ,σ)ũA) , (A.38) To relate the Ṽi coordinates to the ũA, ṽA coordinates recall that the SU(4) ρ matrices could be combined into 8× 8 γ matrices of SO(6) as follows 0 ρiAB ρiAB 0 , i = 1, . . . , 6 , (A.39) with the γi satisfying the SO(6) anti-commutation relations γi, γj = 2δij . (A.40) dirso6 The SO(2, 4) γ-matrix algebra is instead γ̃i, γ̃j = −2ηij . (A.41) dirso24 Given a set of SO(6) γ matrices we can define γ̃i = γi , i = 1, 2 , iγi , i = 3, . . . , 6 , (A.42) which satisfy ( dirso24 A.41). Similarily we will define ρ̃iAB = ρiAB , i = 1, 2 , iρiAB , i = 3, . . . , 6 , and ρ̃iAB = ρiAB , i = 1, 2 , iρiAB , i = 3, . . . , 6 , (A.43) which now satisfy ρ̃iAB ρ̃ jBC + ρ̃iAB ρ̃ jBC = −2δCAηij , (A.44) as well as ηij ρ̃ jCD = 2(δCAδ B − δDA δCB) . (A.45) Note also that for SU(4) ρ matrices we had (ρiAB) ∗ = −ρiAB , (A.46) while for the SU(2,2) ρ̃ matrices we have (ρ̃iAB) ∗ = ηijρjAB . (A.47) The relationship between the ṽA, ũA and the Ṽi is Ṽi = ũAρ̃iAB ṽ B , Ṽ i = ṽAρ̃ iABũB . (A.48) This can be used to derive the equality between the first and second lines in equation ( goodllso24 A.26). A.3.1 Subsectors of the SU(2, 2)/S(U(2)× U(2)) model so24sub When written in terms of the Vi, the Lagrangian LLL SU(2,2) can be reduced to the SU(1,2) sub-sector by requiring Ṽ 2a = −iṼ 2a−1 ≡ 1√ Ũa , a = 1, 2, 3 , (A.49) which can further be restriced to the SU(2) subsector for Ṽ 5 = 0 = Ṽ 6. In terms of the ũA and ṽA this restriction is easily enforced by setting for example ũA = (Ũ1, Ũ2, Ũ3, 0) , ṽA = (0, 0, 0, 1) . (A.50) We require ηabŨ∗a Ũb = −1 , (A.51) so as to satisfy the constraints ( uvlorconst A.35). Restricting to the SU(1,1) sector is achieved by setting ũ3 = Ũ3 = 0. Upon inserting these ansatze, the Lagrangian ( goodllso24 A.26) reduces to the standard SU(1,2) Landau-Lifshitz Lagrangian LSU(1,2) = −iŨa∂0Ũa − |DσŨa|2 + Λ(ŨaŨa + 1) , (A.52) with a = 1, 2, 3 and Ũa ≡ ηabŨ∗b . A.4 The SU(2|2)/S(U(1|1)× U(1|1)) model Lets construct the LL model on SU(2|2)/S(U(1|1) × U(1|1)). Starting from equation ( LLsigmamodel 2.4), with Tr now replaced by STr we may define g = (X, Y ) , (A.53) with X and Y super-matrices which satisfy X†X = 12 , Y †Y = 12 , XX † + Y Y † = . (A.54) The LL Lagrangian for this model is then LLL SU(2|2) = g−1∂0g (g−1D1g)(g −1D1g) = iSTr(X†∂0X)− , (A.55) where D1X ≡ ∂1X −XX†∂1X . (A.56) The bosonic base of SU(2|2) is SU(2)×SU(2), where in the case of interest to us we write X = (ũA, vA) , A = 1 . . . , 4 , (A.57) ũAũA = −1 , vAvA = 1 , (A.58) ũA ≡ ũ∗BCBA , vA ≡ v∗BCBA , X† ≡ − , (A.59) where CBA ≡ diag(−1,−1, 1, 1). Note that the first (last) two components of uA (vA) are bosonic and the last (first) two components of uA (vA) are fermionic. LLL SU(2|2) = −iũA∂0ũA − ivA∂0vA − A∂1ũA + ∂1v A∂1vA −ũA∂1ũAũB∂1ũB + vA∂1vAvB∂1vB + 2ũA∂1vAvB∂1ũB . (A.60) goodllsu22 The action ( goodllsu22 A.60) has a local non-compact U(1|1) invariance X̃ → X̃U(τ, σ) , (A.61) for U(τ, σ) a general U(1|1) matrix U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.62) In terms of the ũA and vA the action ( goodllsu22 A.60) is invariant with respect to the following local transformations (ũA, vA) → (ũA + vAθ1(τ, σ), vA + ũAθ1(τ, σ)) , (ũA, vA) → (ũA − ivAθ2(τ, σ), vA + iũAθ2(τ, σ)) , (ũA, vA) → (eiφ1(τ,σ)ũA, eiφ1(τ,σ)vA) , (ũA, vA) → (eiφ2(τ,σ)ũA, e−iφ2(τ,σ)vA) , (A.63) where ψ1, ψ2 (θ1, θ2) are real Grassmann-even (-odd) valued function. B Quantising the action ( gsu112 3.18) in the t + α = κτ gauge Given the simple form of the action ( gsu112 3.18), ( gsu112comp 3.23) we present a brief light-cone quantisation of it here. The main point is that, as expected, the Hamiltonian has a non-zero normal ordering constant ( normordconst B.18). Since the equation of motion for φ+ is 0 = ∂µ ( ggµν∂νφ+) , (B.1) we may impose conformal gauge (gµν = ηµν) and set φ+ = 2κτ . (B.2) The fermionic equations of motion then reduce to 0 = (i∂0 + κ)η i + ∂1ηj , (B.3) fermeom1 where i 6= j. The fermionic fields have the following periodicity conditions η1(τ , 2π) = e iαη1(τ , 0) , η2(τ , 2π) = e −iαη1(τ , 0) , (B.4) The fermionic equations of motion then are solved by i(nσ+ωnτ) + θ̃ne i(nσ−ωnτ) , (B.5) −i(nσ+ωnτ) + ξ̃ne −i(nσ−ωnτ) , (B.6) where , (B.7) and for n 6= 0 ωn + κ ωn − κ ξ̃n , (B.8) while for n = 0 0 = θ0 = ξ̃0 . (B.9) The Virasoro constraints can be used to find φ− in terms of the other fields 0 = ∂0φ− + i − κ ηiηi , (B.10) 0 = ∂1φ− + i . (B.11) The Nöther current for time translations t→ t+ ǫ is jtµ = −∂µφ+ − ∂µφ− + iηi i − 2∂µφ+ηiηi − ǫµν ∂ν η2 − η1 . (B.12) We can use the equations of motion to write the Hamiltonian of the system as Hκ ≡ − dσjt0 = 2κ+ i . (B.13) Hkappa The canonical momentum conjugate to ηi is 4iκη i and so upon quantisation we must have ηi(τ, σ), ηj(τ, σ = − 1 δijδ(σ − σ′) . (B.14) As a consequence the mode oscillators have the following non-zero anti-commutators ξ̄n, ξm = −δnm ωn + κ 16πκωn ξn, ξ̃m = −δnm ωn − κ 16πκωn , (B.15) together with ξ̄0, ξ0 = − 1 θ0, θ̃0 = − 1 , (B.16) with all other anti-commutators equal to zero. With the convention that ξn, ξn, ξ0 and θ̃0 are the annihilaiton operators the normal ordered expression for Hκ in the quantum theory is Hκ = 2κ(ξ̄0ξ0 + θ0θ̃0) + 4 n 6=0 ξ̄nξn ωn + κ ωn − κ + aκ . (B.17) The normal ordering constant aκ is ωn . (B.18) normordconst The remaining non-trivial bosonic Nöther current for the rotations η1 → eiǫη1 , η2 → e−iǫη2 , (B.19) jcµ = ∂µφ+(η1η 1 − η2η2) + 2iηµν∂νφ+(η2η1 + η2η1) . (B.20) The corresponding normal-ordered conserved current is J = − 1 dσjc0 = dσ(η2η 2 − η1η1) θ0θ̃0 − 2κξ̄0ξ0 + n 6=0 ωn − κ − ξ̄nξn ωn + κ . (B.21) In this case the normal ordering constant is zero. Since φ− is periodic in σ we require that dσ∂1φ− = i i , (B.22) In the quantum theory this is equivalent to the level matching requirement n 6=0 ωn − κ − ξ̄nξn ωn + κ | physical 〉 . (B.23) Finally, we may compute the four non-zero supercharges Q1 ≡ i Q10 4 = κe −iκτ dσ η1 = κ θ0 , (B.24) Q2 ≡ i Q20 3 = κe −iκτ dσ η2 = κξ̄0 , (B.25) Q̄1 ≡ i Q40 1 = κe iκτ dσ η1 = κθ̃0 , (B.26) Q̄2 ≡ i Q30 2 = κe iκτ dσ η2 = κξ0 , (B.27) The above (super-)charges form a U(1|1)2 algebra and in particular we find [Hκ , J ] = 0 , Qi, Q̄j = − κ δij . (B.28) C Comments on conformal invariance of the action ( gsu112 3.18) In this appendix we entertain the possibility of using the action ( gsu112 3.18), ( gsu112comp 3.23) as a Polyakov string action. As a warm-up let us integrate out φ− in the action ( gsu112comp 3.23). We arrive at an effective action for the fermions in which we may set φ+ = 2κτ . (C.1) Explicitly we then have Leff = (−κ) d2σ iηi i − 2κηiηi + η1 ∂1 η2 − η1 2 . (C.2) We may represent the worldsheet gamma matrices as , ρ1 = , (C.3) and define a world-sheet Dirac spinor as , α = 1, 2 . (C.4) The conjugate spinor is then ψ̄α = (ψ †ρ0)α = −η1, η2 , (C.5) and the effective action may be written as Leff = (−κ) d2σ iδµa ψ̄ρ a←→∂µψ + 2κψ̄ψ , (C.6) where ∂µψ ≡ ψ̄ρa∂µψ − ∂µψ̄γaψ . (C.7) This is simply the Lagrangian for a worldsheet Dirac fermion of mass 2κ. Since such fermions are not conformal, we get the first indication that the Lagrangian ( gsu112 3.18) is also not conformal. With the above definitions for ρa, ψα and ψ̄α we can re-write the action ( gsu112 3.18) as gµν∂µφ+∂νφ− + ie a ψ̄ρ a←→∂µψ + 2mψ̄ψ , (C.8) firstcurved where eµa = gµν∂νφ+ , µν∂νφ+ . (C.9) We have written the above expression in the form of an inverse zwei-bein; we will see shortly that this is indeed justified. The corresponding zwei-bein is eaµ = (∂µφ+ , ǫµν∂ νφ+) , (C.10) and the metric is Gµν ≡ eaµebνηab = − gµν . (C.11) Above m ≡ gµν∂µφ+∂νφ+ = −Gµν∂µφ+∂νφ+ , (C.12) is the norm of φ+, which needs to be non-zero. For completness note that the determinant of the metric and the zwei-bein are G ≡ detGµν = , e ≡ det(eaµ) = − . (C.13) Rescaling fermions in the action ( firstcurved C.8) by ψ → m−1/2ψ , (C.14) gives gµν∂µφ+∂νφ− + im −1eµa ψ̄ρ a←→∂µψ + 2ψ̄ψ . (C.15) Integrating by parts this can be written as gµν∂µφ+∂νφ− + 2im −1eµa ψ̄ρ a∂µψ + −gm−1)√ −g ψ̄ρ aψ + 2ψ̄ψ gµν∂µφ+∂νφ− + 2im −1eµa ψ̄ρ a∂µψ +m −1∂µ(e ψ̄ρaψ + 2ψ̄ψ gµν∂µφ+∂νφ− + 2im −1eµa ψ̄ρ a∂µψ +m −1ω01a ψ̄ρ aρ01ψ + 2ψ̄ψ −Gµν∂µφ+∂νφ− + 2ψ̄(iρµDµ +m)ψ . (C.16) curved The final form of the action is that of a world-sheet Dirac fermion of mass m together with the fields φ± moving in a curved metric Gµν . Above we have used the fact that in two dimensions for any zwei-bein êaµ and corresponding metric ĝµν , the spin connection ω̂ µ can be written as ω̂abµ = −ǫab êcµǫc , (C.17) where ǫab (ǫc d) is the flat Minkowski space ǫ-tensor with non-zero components ǫ01 = −ǫ10 = 1 1 = ǫ1 0 = −1). This formula can be derived from the xpressions presentd in Appendix We may now want to define a string theory path integral for this Lagrangian. To do so we consider the Polyakov path integral for the Lagrangian ( curved C.16). Since the path integral integrates over metrics gµν , and the Lagrangian is a function of the metric Gµν = −m−1gµν we first rescale gµν → −m1/2gµν , (C.18) in order to eliminate the metric Gµν . We arrive at a Polyakov-type path-integral with action gµν∂µφ+∂νφ− + 2ψ̄(iρ µDµ + . (C.19) altu11 This Lagrangian is conformally invariant. One way to see this is to generalise the argument presented in [31] which considered sigma-models on plane-wave backgrounds. Let us integrate out the fermions to obtain an effective Lagrangian for φ± Leff ∼ ηµν∂µφ+∂µφ− + log det = ηµν∂µφ+∂µφ− + log det ∂µ1φ+Π + ∂µ2∂ν1φ+Π − ∂ν2 + = ηµν∂µφ+∂µφ− + log(m2) ∼ ηµν∂µφ+∂µφ− + ηµν∂µφ+∂µφ+ ln Λ , (C.20) altu11 where Λ is the cut-off. We can re-absorb this divergent piece by re-defining φ− φ− → φ− − φ+ ln Λ . (C.21) This shows that the Lagrangian ( altu11 C.19) is conformal. As it stands however, this Lagrangian is not Weyl invariant and, just as in [31], we need to turn on a dilaton Φ = φ2+ . (C.22) D Two dimensional spin connection Let us consider a geenral Lorenzian two dimensional metric gµν which we will parametrise for convenience as gµν = , (D.1) 14I am grateful to A. Tseytlin for a number of discussions and explanations of these issues. where a, b and d are complex functions of τ and σ the coordinates on the manifold. The zwei-bein from which this follows is given by e1µ = (a sinh ρ ,−d sinh ρ) , e2µ = (a cosh ρ , d cosh ρ) , (D.2) where . (D.3) The Christoffel symbols gµκ (gκν ,λ + gκλ ,ν − gνλ ,κ) (D.4) are given by Γ111 = g aba,1 + ad 2a,0 − bb,0 , (D.5) Γ112 = Γ 21 = g d2aa,1 − bdd,0 , (D.6) Γ122 = g d2b,1 − bdd,1 − d3d,0 , (D.7) Γ211 = g −a3a,1 − aba,0 + a2b,0 , (D.8) Γ212 = Γ 21 = g −aba,1 + a2dd,0 , (D.9) Γ222 = g a2dd,1 − bb,1 + bdd,0 . (D.10) where g = det gµν . It is easy to check that these satisfy the defining equation gµν ,λ − gκνΓκµλ − gκµΓκνλ = 0 . (D.11) The spin connection ωmnµ can be determined from the following equation ν = ∂µe ν + ω ν − Γκµνemκ = 0 . (D.12) Since ωmnµ is anti-symmetric in (m,n) the non-zero components are given by ω010 = −ω100 = −2a2da,1 − bda,0 + adb,0 + abd,0 −g , (D.13) ω011 = −ω101 = −bda,1 − adb,1 + abd,1 + 2ad2d,0 √−g . (D.14) E T-dual version of the action ( gsu112 3.18) Performing T-duality for the action ( gsu112 3.18) along α leads to a very simple form for an equivalent action. In this appendix we breifly present these results. To T-dualise along α we replace ∂µα by Aα and adding the Lagrange multiplier term ǫ µνAµ∂ν α̃. The Aµ are then integrated out and we obtain the action Ldκ = 1− ηiηi −(∂µt− i)(∂νt− +(∂µα̃− (η1 ∂µ η2 − η1 2))(∂να̃− (η1 ∂ν η2 − η1 1− ηiηi (∂µt− i)(∂να̃− (η1 ∂ν η2 − η1 (E.1) tdual where we have used the fact that up to total derivatives 1− ηiηi ∂µt∂ν α̃η iηi = 1− ηiηi ∂µt∂ν α̃ . (E.2) At the level of classical equations of motion we may integrate out the metric to get a Nambu- Goto type action LdNGκ = 1− ηiηi (∂µt− iηi i)(∂να̃− (η1 ∂ν η2 − η1 , (E.3) where we have rescaled t→ t/2 and multiplied the whole action by a factor of 2. The Nambu- Goto form of the action is particularily simple due to the ’two-dimensional’ target space form of the action ( tdual E.1). The equations of motion for α̃ and t imply that 1− ηiηi (∂µt− i) = ∂µχ1 , (E.4) 1− ηiηi (∂να̃− (η1 ∂ν η2 − η1 2) = ∂µχ2 , (E.5) where χi are arbitrary Grassmann-even functions of τ and σ. 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0704.1461

Trajectory of neutron$-$neutron$-^{18}C$ excited three-body state

arXiv:0704.1461v4 [nucl-th] 16 Nov 2007 Trajectory of neutron−neutron−18C excited three-body state M. T. Yamashita a, T. Frederico b and Lauro Tomio c 1 aUniversidade Estadual Paulista, 18409-010, Itapeva, SP, Brazil. bDepartamento de F́ısica, ITA, CTA, 12228-900 São José dos Campos, Brazil. cInstituto de F́ısica Teórica, UNESP, 01405-900 São Paulo, Brazil. Abstract The trajectory of the first excited Efimov state is investigated by using a renor- malized zero-range three-body model for a system with two bound and one virtual two-body subsystems. The approach is applied to n− n−18C, where the n− n vir- tual energy and the three-body ground state are kept fixed. It is shown that such three-body excited state goes from a bound to a virtual state when the n−18C binding energy is increased. Results obtained for the n−19C elastic cross-section at low energies also show dominance of an S−matrix pole corresponding to a bound or virtual Efimov state. It is also presented a brief discussion of these findings in the context of ultracold atom physics with tunable scattering lengths. PACS 03.65.Ge, 21.45.+v, 11.80.Jy, 21.10.Dr Key words: Bound states, scattering theory, Faddeev equation, Few-body The interest on three-body phenomena occurring for large two-body scattering lengths have increased in the last years in view of the experimental possibilities presented in ultracold atomic systems, where the two-body interaction can be tunned by using Feshbach resonance techniques. Theoretical predictions, well investigated for three particle systems, such as the increasing number of three-body bound states when the two-body scattering length goes to infinity - known as Efimov effect [1] - can actually be checked experimentally in ultracold atomic laboratories. Indeed, the first indirect evidence of Efimov states came from recent experiments with ultracold trapped Caesium atoms made by the Innsbruck group [2]. 1 Email: [email protected] Preprint submitted to Elsevier Science 24 October 2018 http://arxiv.org/abs/0704.1461v4 In the nuclear context, the investigations of Efimov states are being of renewed interest with the studies on the properties of exotic nuclei systems with two halo neutrons (n−n) and a core (c). One of the most promising candidates to present these states is the 20C [3–5] (c ≡18 C). 20C has a ground state energy of 3.5 MeV with a sizable error in n−18C two-body energy, 160±110 keV [6]. The proximity of an Efimov state (bound or virtual) to the neutron-core elastic scattering cut makes the cross-section extremely sensitive to the corresponding S-matrix pole. However, one should be aware that the analytic properties of the S-matrix for Borromean systems (where all the two-body subsystems are unbound, like the Caesium atoms of the Innsbruck experiment [2]) are expected to be quite different from systems where at least one of the two- body subsystems are bound, as the present case of n− n−18C. The trajectory of Efimov states for three particles with equal masses has been studied in [7,8] using the Amado model [9]. By studying the S−matrix in the complex plane, varying the two-body binding, it was confirmed previous analysis [8,10], that Efimov bound states disappear into the unphysical energy sheet associated to the unitarity cut, becoming virtual states. It was also verified that, by further increasing the two-body binding, the corresponding pole trajectories remain in the imaginary axis and never become resonant. Considering general halo-nuclei systems (n − n − c), in [3] it was mapped a parametric region defined by the s−wave two-body (bound or virtual) energies, where the Efimov bound states can exist. By increasing the binding energy of a two-body subsystem, it was noted that the three-body bound state turns out to a virtual state, remaining as virtual with further increasing of the two- body binding, as already verified in [8,11]. In the other side, starting with zero two-body binding, by increasing the two-body virtual energy we have the three-body energy going from a bound to a resonant state [12]. Actually, the analysis of the trajectory of Efimov states in the complex plane can be relevant to study properties of the 20C. In order to study the behavior of Efimov states, the authors of [4] have recently pointed out the importance of the analysis of low energy n−19C elastic scattering observables. Their results, leading to a 20C resonance prediction near the scattering threshold [4], when the separation energy of the bound halo neutron of 19C is changed, suggest a different behavior from the one found for three equal-mass particles, where a bound Efimov state turns into a virtual one as the two-body binding increases. In view of the importance of the results, not only in the nuclear context, it is interesting to consider a new independent analysis of three-particle systems where two subsystems are bound and one is unbound, considering particularly the case with different masses. As we show in the present letter, the results of our treatment for three-body system with unequal masses are consistent with the expectation derived from equal mass systems. Moreover, they are model independent due to the universal character of the three-body physics at very low energies. See Ref. [13] for a comment on the results obtained in Ref. [4]. In order to clarify the behavior (in the complex energy plane) of a given Efimov state for the n− n−18C system, we consider the n− c subsystem bound with varying energies; the virtual energy of the n − n subsystem and the ground state energy of 20C are fixed, respectively, at −143 keV [14,15] and −3.5 keV (these constraints are convenient to follow the trajetory of the excited Efimov states since their appearance/disappearance depends only on the ratio of the n− c and n−n− c energies, see pages 327-329 of Ref. [16]). Our present study of n−n−18C can be easily extended to similar systems such as 12Be, 15B, 23N and 27F. In the present case, the bound state equations are extended to the second Riemann energy sheet through the n−19C elastic scattering cut (see Fig. 1) using a well-known technique [17]. Consistent with previous results [3,8,11] for three equal masses particles, by using the present approach that will be explained in the following, we conclude that (also in this case) the three- body bound state turns into a virtual (and not a resonance) one when we increase the n−18C two-body binding energy. The full behavior of the Efimov virtual states in the unphysical sheet of the complex energy plane, by further increasing the two-body bound state energy, is still open for investigation. In our present investigation, we also show that, when the three-body S−matrix pole of a virtual or excited Efimov state is near the scattering threshold, the n−19C elastic scattering cross section is dominated by such pole, peaking at zero relative energy and decreasing monotonically with energy. Elastic scattering Dimer breakup Fig. 1. Analytic structure of the S−matrix. The energies of the neutron-core (bound system), the three-body bound and the three-body virtual states are, respectively, given by εnc, E3b, and E3v. The arrow passing through the elastic scattering cut shows the trajectory of an S−matrix pole to the second Riemann sheet. Next, we introduce the basic formalism by starting with the coupled spectator functions for a bound three-body system n − n − c (c ≡ 18C in our specific case). Our units are such that h̄ = mn = 1, where mn is the mass of the neutron, with the n− n, n− c and three-body energies respectively given by Enn = h̄ 2εnn/mn, Enc = h̄ 2εnc/mn, and E3b = h̄ 2E3b/mn, where n − n and n − c refer to virtual and bound subsystems. After partial wave projection, the ℓ-wave spectator functions χℓn and χ c (where the subindex n or c indicates the spectator particle), for a bound three-body system, are given by [3] n(q; E3b) = τnc(q; E3b) 2(q, k; E3b)χ n(k; E3b) +K 1(q, k; E3b)χ c(k; E3b) c(q; E3b) = τnn(q; E3b) 1(k, q; E3b)χ n(k; E3b), (2) where Kℓi=1,2(q, k; E3b)≡G i(q, k; E3b)− δℓ0 G i(q, k;−µ 2) (3) Gℓi(q, k; E3b) = Pℓ(y) E3b − A+Ai−1 q2 − A+1 k2 − kqy , (4) τnn(q; E3b)≡ |εnn|+ q2 − E3b , (5) τnc(q; E3b)≡ |εnc| − (A+ 2) 2(A+ 1) q2 − E3b . (6) In the above, E3b ≡ εnc − 2(A+1) κ2b and A is the mass-number of particle c. The absolute value of the momentum of the spectator particle with respect to the center-of-mass (CM) of the other two particles is given by q ≡ |~q|; with k ≡ |~k| being the absolute value of the relative momentum of these two particles. The n − n virtual state energy is fixed at Enn = −143 keV. In Eqs. (1) and (2), we have the Kronecker delta δℓ0 (= 1 for ℓ = 0 and =0 for ℓ 6= 0) in order to renormalize the equations (using a subtraction procedure) only for the partial wave where such renormalization is necessary, ℓ = 0. In the cases of ℓ > 0, due to the centrifugal barrier, the Thomas collapse is absent and such renormalization is not necessary. In this way, with Eqs. (1) and (2) renormalized, the three-body observables are completely defined by the two-body energy scales, εnc and εnn. Later on, the definitions (3)-(6) will be extended also to unbound systems. The regularization scale µ2, used in the s−wave [see Eq. (3)], is chosen to reproduce the three-body ground-state energy of 20C, E0 = −3.5 MeV [6]. A limit cycle [16,18] for the scaling function of s−wave observables is evidenced when µ is let to be infinity. We note that, a good description of this limit is already reached in the first cycle [11]. The analytic continuation of the bound three-body system given by (1) and (2) to the second Riemann sheet is performed through the n− (n− c) elastic scattering cut, following Refs. [11,17,19]. As we are considering n−n−c nuclei where only the subsystem n − c is bound (“samba-type” nuclei [14,15]), we have only the n − c cut in the complex energy plane. In order to see how to perform the analytical continuation from the first to the second sheet of the complex energy, let us first consider a complex momentum variable ki. As the energies of the two-body sub-systems are fixed and only the n− c subsystem is bound, it is convenient to define this momentum as ki = 2(A+1) (Ei − εnc). In this case, the bound-state energy (Ei = E3b) is given by ki = iκb, with the virtual state energy (Ei = E3v) given by ki = −iκv. Next, by removing E3b in favor of ki in the bound-state equations (1)-(6), with τ̄nc(q; E) ≡ |εnc|+ (A+ 2)q2 2(A+ 1) , (7) and with χc and χn redefined as χℓc(q; E) ≡ h c(q; E), χ n(q; E) ≡ h n(q; E)/(q 2 − k2i − iǫ), (8) we observe that the relevant integrals to be considered have the structure I(ki) = q2dq [F (q2)/(k2i − q 2 + iǫ)] in the first sheet of the complex energy plane. To obtain I(ki) in the second sheet we need to change the contour of integration, as shown in detail in Ref. [7], such that the value of I(ki) in the second sheet of the complex energy plane is given by I ′(ki) = I(ki)−iπkiF (k Following this procedure, from Eqs. (1) and (2) we obtain the corresponding equations in the second sheet of the complex energy plane, where for the virtual state energy we have ki = −iκv = 2(A+1) (E3v − εnc): n(q; E3v) = 2(A+ 1) τ̄nc(q; E3v) 2(q,−iκv ; E3v)h n(−iκv ; E3v)+ Kℓ1(q, k; E3v)h c(k; E3v) + Kℓ2(q, k; E3v) h n(k; E3v) k2 + κ2v  , (9) hℓc(q; E3v) =τnn(q; E3v) 1(−iκv , q; E3v)h n(−iκv ; E3v)+ 1(k, q; E3v) h n(k; E3v) k2 + κ2v  . (10) The above coupled equations, as well as the corresponding coupled equations for bound state, can be written as single-channel equations for hℓn, by defining an effective interaction V and considering I = b, v: hℓn(q; E3I) = 2κvh n(−iκv; E3v)V ℓ(q,−iκv ; E3v)δI,v + dkk2Vℓ(q, k; E3I) hℓn(k; E3I) k2 + κ2I Vℓ(q, k; E3I )≡ π (A+ 1) τ̄nc(q; E3I)× (12) 2(q, k; E3I) + 1(q, k ′; E3I)τnn(k ′; E3I)K 1(k, k ′; E3I) The first term in the right-hand-side (rhs) of (11), which is non-zero only for the virtual state, corresponds to the contribution of the residue at the pole. The virtual states are limited by the cut of the elastic scattering amplitude in the complex plane, corresponding to the second term in the rhs of Eq. (9). In this case, the cut is given by the zero of the denominator of Gℓ2(q, k; E3v) [See Eq. (4)], where −1 < y < 1 and q = k = −iκcut. With |E3v| = |εnc|+ 2(A+1) κ2cut, we obtain the branch points, with the cut given by 2(A+ 1) A + 2 |εnc| < |E3v| < 2(A+ 1) |εnc|. (13) For n − n−18C (A = 18), a virtual state energy can be found in the energy interval between the threshold of the elastic scattering and the starting of the above cut (13): |εnc| < |E3v| < 1.9 |εnc| . (14) 0 100 200 300 400 500 600 90 120 150 180 210 | E19C| (keV) Fig. 2. Three-body n − n−18C results for the first excited state, with respect to the threshold (|E20C − E19C|) for varying 19C binding energies. Three-body bound (virtual) states occur when |E19C| is approximately smaller (larger) than 170 keV. s−wave results (solid line) are also presented in the inset (with dots). Results for the p−and d−waves, divided by a factor 10, are shown with dashed and dotted lines, respectively. Figure 2 shows how the absolute value of the first excited three-body state energy with respect to the two-body bound state, |E − Enc|, varies when increasing the n − c bound-state energy. In this case, with 18C being the core, we have 19C as the two-body bound subsystem. The solid line of Fig. 2 presents the s−wave results, with a close focus in the region of the threshold (E = Enc) given by the inset figure. Although the Efimov states do not appear in higher partial waves due to the existence of the centrifugal barrier, we have also presented results for the virtual states of p− and d− waves in view of their possible relevance for the elastic n−19C low-energy cross-section. Such situation can happen when the s−wave virtual state is far from the elastic scattering region. The results shown in Fig. 2, valid for n−n−c system with n−c bound, together with previous analysis [3,11], are clarifying that the behavior of Efimov states, when one of the particles have a mass different from the other two, follows the same pattern as found in the case of three equal-mass particles [8,10]. No resonances were found for Eqs. (9) and (10) in the complex energy plane. Next, for consistency, we present results for the s−wave elastic cross-sections, which are in agreement with the above. The formalism for the partial-wave elastic n− 19C scattering equations can be obtained from Eqs. (1) and (2), by first introducing the following boundary condition in the full-wave spectator function χn(~q): χn(~q) ≡ (2π) 3δ(~q − ~ki) + 4π hn(~q; E(ki)) q2 − k2i − iǫ , (15) where hn(~q; E(ki)) is the scattering amplitude, and the on-energy-shell in- coming and final relative momentum are related to the three-body energy Ei ≡ E(ki) by ki ≡ |~ki| = |~kf | = [2(A+ 1)/(A+ 2)] (Ei − εnc). With the same formal expressions (7) and (6) for τ̄nc and τnn, by using the definition (13), the partial-wave scattering equation can be cast in the following single channel Lippmann-Schwinger-type equation for hℓn: n(q; Ei) = V ℓ(q, ki; Ei) + ℓ(q, k; Ei) h n(k; Ei) k2 − k2i − iǫ . (16) Virtual states and resonances correspond to poles of the scattering matrix on unphysical sheets. So, the most natural method to look for them is to perform an analytically continuation of the scattering matrix to the unphysical sheet. For that one needs prior knowledge about the analytic properties of the kernel of the integral equation, as well as the scattering matrix properties on the unphysical sheet of the complex energy plane. Although such properties are easy to derive in simple cases, they can be difficult to obtain in more complex situations, which can put some restriction on the approach. For the numerical treatment of Eq. (16), we consider the approach developed in [8] to find virtual states and resonances on the second energy sheet associ- ated with the lowest scattering threshold. Such approach does not require prior knowledge of the analytic properties of the scattering matrix on the unphys- ical sheet. The solution of the scattering equation is written in a form where the analytic structure in energy is clearly exhibited. Then, it is analytically continued to the unphysical sheet. The method relies on the calculation of an auxiliary (resolvent) function [20], which has an integral structure similar to the original scattering equation, but with a weaker kernel due to a subtrac- tion procedure at an arbitrary fixed point k̄i. For real positive energies, this subtraction point k̄i is identified with ki, such that the corresponding integral equation does not have the two-body unitarity cut. For non-real or negative en- ergies, k̄i can be any arbitrary positive real number. For convenience, k̄i = |ki|. The final solution is obtained by evaluating certain integrals over the auxil- iary function. In case of scattering solutions, the two-body unitarity cut is introduced through these integrals. In the present case, we have the following integral equation for the auxiliary function Γ, and the corresponding solution for hℓn(q; Ei): Γℓn(q, ki; Ei) =V ℓ(q, ki; Ei) + k2Vℓ(q, k; Ei)− k̄ ℓ(q, ki; Ei) ] Γℓn(k, ki; Ei) k2 − k2i hℓn(q; Ei) =Γ n(q, ki; Ei) + Γ n(q, k̄i; Ei) Γℓn(k, ki; Ei) k2 − k2i − iǫ Γℓn(k, k̄i; Ei) k2 − k2i − iǫ . (17) For the on-shell scattering amplitude, we have hℓn(ki; Ei) = [1/Γ n(ki, ki; Ei)−J ] −1 (18) Γℓn(k, ki; Ei)/Γ n(ki, ki; Ei)− 1 k2 − k2i + iki In order to obtain the bound and virtual energy states, two independent pro- cedures have been used. The first one, by solving directly the homogeneous coupled equations (1), (2), (9) and (10), looking for zeros of the corresponding determinants. The other one, by verifying the position of the poles in the com- plex energy plane of the scattering amplitude, hℓn(ki; Ei), given by Eq. (18) and corresponding analytic extension to the second Riemann sheet. In this second approach, we solve the corresponding inhomogeneous equation with on-energy- shell momentum ki = +iκb (bound state, Ei = −|E3b|) and ki = −iκv (virtual state, Ei = −|E3v|)[8]. By comparing the results of both approaches, we checked that they give con- sistent results. However, for numerical stability and accuracy of the results, particularly in the case of the search for Efimov states, when the absolute values of the energies are very close to zero, the second approach is by far much better. Results for the total n−19C elastic cross-sections, obtained from dσ/dΩ = |hn(~kf ; Ei)| 2, refering to bound or virtual states E∗, are presented in Fig. 3 as functions of the CM kinetic energy, K(ki) ≡ [(A+ 2) k i ]/[2(A+ 1)] = E(ki)− Enc. (19) 0 25 50 75 100 125 150 10000 0 30 60 90 120150180 10000 0 100 200 300 400 500 ] (keV) ] (keV) Fig. 3. n−19C elastic cross sections (in barns) versus the CM kinetic energies [Eq. (19) with A=18], for different 19C bound energies. In the left-hand-side frame we show results for two cases that generate three-body energies close to the threshold: E19C = −150 keV (main figure) and −180 keV (inset), producing respectively three- -body bound and virtual states. Solid-line are obtained from (16), with dashed-line from (20). In the rhs we have results for E19C = −500 keV. Although each higher partial ℓ−wave have a virtual state, below the breakup the cross-section is completely dominated by the s−wave [19]. In the frame shown in the left-hand-side, we have two cases of energies close to the scatter- ing threshold: n − 18C bound at −150 keV, giving an excited Efimov bound state with E∗ = −150.12 keV; and n − 18C bound at −180 keV, producing a virtual state with E∗ = −180.12 keV. In both two cases the cross-section has a huge peak at zero energy due to the presence of the nearby pole. For comparison, we also show (with dashed-line) the results obtained from the following effective range expansion (approximately valid for small ki near the elastic scattering threshold): σ(ki) = 4πh̄2 1.9mn (Enc − E ∗) + h̄2k2i 2741 keV E(ki)−E∗ barn, (20) where we have used A =18, h̄2/mn =414.42 keV barn and Eq. (19). The case not so close to the threshold (where Eq.(20) fails) is shown in rhs for Enc = −500 keV, with 20C virtual energy E∗ = −568.73 keV. The proximity of an Efimov state (bound or virtual) to the neutron and neutron-core elastic scattering makes the cross-section extremely sensitive to the corresponding S-matrix pole. We remark that if it will be possible to dissociate a “samba-type” halo nuclei, like 20C, and measure the correlation function in the two-body channel corresponding to a neutron and a bound n−c system for small relative momentum, the information on the final state inter- action as well as the halo structure will be clearly probed, as the counterpart seen in the n− n correlation in the breakup of Borromean nuclei [21]. In conclusion, we analyzed the three-body halo system n−n−18C, where two pairs (n−18C) are bound, and the remaining pair n − n has a virtual-state. We study the trajectory of three-body Efimov states in the complex energy plane. As shown, the energy of an excited Efimov state varies from a bound to a virtual state as the binding energy of the subsystem n−18C is increased, while keeping fixed the 20C ground-state energy and the virtual energy of the remaining pair (n − n). In our approach we applied a renormalized zero- range model, valid in the limit of large scattering lengths. Considering that low-energy correlations, as the one represented by the Phillips line [22] (cor- relation between triton and doublet neutron-deuteron scattering length), are well reproduced by zero-range potentials [23], the present conclusions should remain valid also for finite two-body interactions when the scattering length is much larger than the potential range. On the numerical analysis, we should remark that we have considered two approaches that give consistent results for bound and virtual state energies. In the case of Efimov physics, where the poles are very close to zero, the method considered in Ref. [8] was found to give solutions with much better stability and accuracy in the scattering region than by using a contour deformation technique. The present results are extending to n−n−core systems the long ago conclu- sion reached for three equal-mass particles [8,10]: by increasing the binding energy of the n−core subsystem, an excited weakly-bound three-body Efimov state moves to a virtual one and will not become a resonance. From Fig. 3, one can also observe that the n−19C elastic cross-sections at low energies present a smooth behavior dominated by the S−matrix pole corresponding to a bound or virtual three-body state. In contrast with the above conclusion applied to system where n−c is bound, we should observe that it was also verified that an excited Efimov state can go from a bound to a resonant state (instead of vir- tual state) in case of Borromean systems (with all the subsystems unbound), when the absolute value of a virtual-state energy for the n − c system is in- creased [12]. Actually, it should be of interest to extend the present analysis of the trajectory of Efimov states to other possible two-body configurations, with different mass relations of three-particle systems. In view of the exciting possibilities of varying the two-body interaction, it can be of high interest the results of the present study to analyze properties of three-body systems in ultracold atomic experiments. For negative scattering lengths the Efimov state goes to a continumm resonance when |a| is decreased, as observed by the change of resonance peak in the three-body recombination to deeply bound states towards smaller values of |a| by raising the tempera- ture [24]. Alternatively, for positive a the recombination rate has a peak when the Efimov state crosses the threshold and turns into a virtual state when decreasing a. A dramatic effect will appear in the atom-dimer scattering rate when the cross-section is dominated by the S-matrix pole near the scattering threshold. We foreseen that the coupling between atom and molecular con- densate will respond strongly to the crossing of the triatomic bound state to a virtual one by changing a. Determined by the dominance of the cou- pled channel interaction, new condensate phases of the atom-molecule gas are expected. The proximity of the virtual trimer state to the physical region, im- plying in a large negative atom-dimer scattering length, will warrant stability to both condensates, while the positive atom-dimer scattering length, due to a trimer bound state near threshold, make possible the collapse of the condensed phases. Indeed, the occurrence of some interesting effects in the condensate due to Efimov states near the scattering threshold have already been discussed in Refs. [25]. We hope these exciting new consequences of Efimov physics can be explored experimentally in the near future. LT thanks Prof. S.K. Adhikari for helpful suggestions. We also thank Fundação de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico for partial support. References [1] V. Efimov, Phys. Lett. B 33 (1970) 563. [2] T. Kraemer et al, Nature 440 (2006) 315. [3] A. E. A. Amorim, T. Frederico and L. Tomio Phys. Rev. C 56 (1997) R2378. [4] V. Arora, I. Mazumdar, V. S. Bhasin, Phys. Rev. C 69 (2004) 061301(R); I. Mazumdar, A. R. P. Rau, V. S. Bhasin, Phys. Rev. Lett. 97 (2006) 062503. [5] A. S. Jensen, K. Riisager, D. V. Fedorov, E. Garrido, Rev. Mod. Phys. 76 (2004) [6] G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A 729 (2003) 337. [7] S. K. Adhikari, A. C. Fonseca, and L. Tomio, Phys. Rev. C 26 (1982) 77. [8] S. K. Adhikari and L. Tomio, Phys. Rev. C 26 (1982) 83. [9] R. D. Amado, Phys. Rev. 132 (1963) 485. [10] R. D. Amado and J. V. Noble, Phys. Rev. D 5 (1972) 1992; S. K. Adhikari and R. D. Amado, Phys. Rev. C 6 (1972) 1484. [11] M. T. Yamashita, T. Frederico, A. Delfino, L. Tomio, Phys. Rev. A 66 (2002) 052702. [12] F. Bringas, M. T. Yamashita, T. Frederico, Phys. Rev. A 69 (2004) 040702(R). [13] M. T. Yamashita, T. Frederico and L. Tomio, submitted to Phys. Rev. Lett. [14] M. T. Yamashita, T. Frederico, L. Tomio, Nucl. Phys. A 735 (2004) 40. [15] M. T. Yamashita, T. Frederico, M. S. Hussein, Mod. Phys. Lett. A 21 (2006) 1749. [16] E. Braaten and H.-W. Hammer, Phys. Rep. 428 (2006) 259. [17] W. Glöckle, Phys. Rev. C 18 (1978) 564; Walter Glöckle, The Quantum Mechanical Few-Body Problem (Springer-Verlag, Berlin, 1983). [18] S. D. Glazek, K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401; R. Mohr, R. Furnstahl, H. Hammer, R. Perry, and K. Wilson, Ann. of Phys. 321 (2005) 225. [19] A. Delfino, T. Frederico, M. S. Hussein and L. Tomio, Phys. Rev. C 61 (2000) 051301(R). [20] L. Tomio and S. K. Adhikari, Phys. Rev. C 22 (1980) 28; 22 (1980) 2359; 24 (1981) 43. See refs. therein for other similar approaches. For bound-states, see S. K. Adhikari and L. Tomio, Phys. Rev. C 24 (1981) 1186. [21] M. Petrascu, et al., Phys. Rev. C 73 (2006) 057601; M. T. Yamashita, T. Frederico, L. Tomio, Phys. Rev. C 72 (2005) 011601(R). [22] A. C. Phillips, Nucl. Phys. A 107 (1968) 209. [23] S. K. Adhikari and J.R.A. Torreão, Phys. Lett. B 132 (1983) 257; D.V. Fedorov and A.S. Jensen, Nucl. Phys. A 697 (2002) 783; P. F. Bedaque, U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52 (2002) 339. [24] M. T. Yamashita, T. Frederico, L. Tomio, Phys. Lett. A 363 (2007) 468. [25] A. Bulgac, Phys. Rev. Lett. 89 (2002) 050402; B.J. Cusack, T.J. Alexander, E.A. Ostrovskaya, and Y.S. Kivshar, Phys. Rev. A 65 (2002) 013609; E. Braaten, H.-W. Hammer, and T. Mehen, Phys. Rev. Lett. 88 (2002) 040401; E. Braaten, H.-W. Hammer, and M. Kusunoki, Phys. Rev. Lett. 90 (2003) 170402.

0704.1462

Physical parameters of evolved stars in clusters and in the field from line-depth ratios

arXiv:0704.1462v1 [astro-ph] 11 Apr 2007 Physical parameters of evolved stars in clusters and in the field from line-depth ratios K. Biazzo1,2, L. Pasquini2, A. Frasca1, L. da Silva3, L. Girardi4, A. P. Hatzes5, J. Setiawan6, S. Catalano1, and E. Marilli1 1 Osservatorio Astrofisico di Catania-INAF, Catania, Italy [email protected] 2 European Southern Observatory, Garching bei München, Germany 3 Observatório Nacional-MCT, Rio de Janeiro, Brazil 4 Osservatorio Astronomico di Padova-INAF, Padova, Italy 5 Thüringer Landessternwarte, Tauterburg, Germany 6 Max-Planck-Institute für Astronomie, Heidelberg, Germany Summary. We present a high-resolution spectroscopic analysis of two samples of evolved stars selected in the field and in the intermediate-age open cluster IC 4651, for which detailed measurements of chemical composition were made in the last few years. Applying the Gray’s method based on ratios of line depths, we determine the effective temperature and compare our results with previous ones obtained by means of the curves of growth of iron lines. The knowledge of the temperature enables us to estimate other fundamental stellar parameters, such as color excess, age, and mass. 1 Introduction The study of stellar populations in our Galaxy and in its neighborhoods has received in the last years a big impulse, especially thanks to the use of large telescopes and to the detailed spectroscopic analysis performed on high-resolution spectra. In this context, open clusters, that are homogeneous samples of stars having the same age and chemical composition, are very suitable to investigate the stellar and Galactic formation and evolution. In spite of this, the data on stars belonging to open clusters are often insuffi- cient to adequately constrain age, distance, metallicity, mass, color excess, and temperature. This is due to the fact that the main classical tool to study cluster properties is the color-magnitude diagram, which suffers of several un- certainties and intrinsic biases due to, for example, the uncertain knowledge of the chemical composition and the reddening of the stars. As a consequence, spectroscopic methods, being independent of the reddening, are very efficient to evaluate temperatures of stars in clusters. Spectroscopic effective temperatures are usually determined imposing that the abundance of one chemical element with many lines in the spectrum (typically iron) does not depend on the excitation potential of the lines. An- other method for determining effective temperature is based on line-depth ratios (LDRs). It has been widely demonstrated that the ratio of the depths http://arxiv.org/abs/0704.1462v1 2 K. Biazzo et al. of two lines having different sensitivity to temperature is an excellent measure of stellar temperatures with a sensitivity as small as a few Kelvin degrees in the most favorable cases ([14, 13, 12]). In the present paper, we apply the LDR method to high-resolution UVES and FEROS spectra for deriving effective temperatures in nearby evolved field stars with very good Hipparcos distances and in giants of the intermediate-age open cluster IC 4651. For both the star samples, the temperature was already derived spectroscopically, together with the element abundances, with the curves of growth of absorption lines spread throughout the optical spectrum ([17, 18]). In addition, for the stars belonging to the open cluster IC 4651, we make the first robust determination of the average color excess, based on spectroscopic measurements. 2 Star samples We have analysed seventy-one evolved field stars and six giant stars belonging to the open cluster IC 4651. The sample of field stars was already analysed by [18] for the determination of radii, temperatures, masses and chemical composition. The stars in the intermediate-age cluster IC 4651 have been selected from the sample studied by [17] for abundance estimates. The field stars data were acquired with the FEROS spectrograph (R = 48 000) at the ESO 1.5m-telescope in La Silla (Chile), while the IC 4651 spectra were acquired with UVES (R = 100 000) at the ESO VLT Kueyen 8.2m-telescope in Cerro Paranal (Chile). In both cases, the signal-to-noise ratio (S/N) was greater than 150 for all the spectra, which make them very suitable for the temperature determination described in Sec. 3. 3 Effective temperature determination The wavelength range covered by FEROS and UVES spectrographs contain a series of weak metal lines which can be used for temperature determination with the LDR method. Lines from similar elements such as iron, vanadium, titanium, but with different excitation potentials (χ) have indeed different sensitivity to temperature. This is due to the fact that the line strength, de- pending on excitation and ionization processes, is a function of temperature and, to a lesser extent, of the electron pressure. For this reason, the better line couples are those with the largest χ-difference. In the range 6150 Å <∼ λ 6300Å there are several lines of this type whose ratios of their depths have been exploited for temperature calibrations ([14, 13, 7, 5]), and for studies of the rotational modulation of the average effective temperature of mag- netically active stars ([10, 4]) or for investigating the pulsational variations during the phases of a Cepheid star ([15, 3]). In particular, we choose 15 line pairs for which [5] already made suitable calibrations. Physical parameters in evolved stars 3 Field giant stars The comparison between the temperatures obtained by us (T LDR ) and those obtained by [18] (T SPEC ) is plotted in Fig. 1. We find a very good agreement between T SPEC and T LDR in all the temperature range 4000–6000 K. Fig. 1. Comparison between the temperatures obtained from LDR and curve-of- growth analyses for the field stars (left) and the giants belonging to IC 4651 (right). Giant stars in IC 4651 T LDR is systematically lower than T SPEC by an amount typically between 70 and 90 K. The only exception is the star E95 for which the difference amounts to about 320 K. [17] already found for this star the largest difference between photometric and spectroscopic temperature among their giant star sample. However, the position on the HR diagram of this star corresponds to a subgiant and this could be the reason for the disagreement. 4 Color excess of IC 4651 We can evaluate for each star the intrinsic color index (B−V )0 by inverting for example the (B − V ) − Teff calibrations of [12] and [1, 2] with the aim to compute the color excess E(B − V ) of the cluster IC 4651. Thus, for the two temperature sets, T LDR and T SPEC , we obtain E(B − V ) ≈ 0.12 and 0.16 for the Gray’s calibration, and E(B − V ) = 0.13 and 0.17 for the Alonso’s calibration. It is worth noticing that there is not a large difference between color excesses obtained with the two calibrations. From a preliminary analysis, we find an improving of the agreement if we properly take into account the metallicity effects ([6]). Moreover, our color excess values are in good agreement with the results of E(B − V ) = 0.13 and E(b − y) = 0.091 obtained by [9] and [17], respectively (E(b − y) = 0.72E(B − V ), [8]). The 4 K. Biazzo et al. present determination of E(B − V ) is a strong argument in favor of such a low reddening notwithstanding the distance of ≈ 900pc estimated by [16] and the low galactic latitude of IC 4651 (≃ 9◦). 5 Conclusion In this paper we have derived accurate atmospheric parameters for field evolved stars and giant stars in the open cluster IC 4651 by means of high- resolution spectra acquired with the ESO spectrographs FEROS and UVES. For the field giant stars, we find a good agreement between temperatures computed by [18] with the curves-of-growthmethod and by ourselves with the LDR technique. For the giants in the intermediate-age open cluster IC 4651, we have determined the effective temperatures by means of the LDR method, that allowed us to compute the reddening. We find a rather low reddening towards the cluster, E(B − V ) ≃ 0.13, that needs to be explained, given the high distance (≃ 900pc) and the low galactic latitude of IC 4651. We conclude that our technique is well suited to derive accurate effective temperatures and reddening of clusters with a nearly-solar metallicity. The determination of very precise temperatures is of great importance to derive stellar age and mass distributions ([11]), representing a powerful tool for stellar population studies in addition to those based on photometric data. References 1. A. Alonso, S. Arribas & C. Mart́ınez-Roger: A&A, 313, 873 (1996) 2. A. Alonso, S. Arribas & C. Mart́ınez-Roger: A&A, 140, 261 (1999) 3. K. Biazzo, S. Catalano, A. Frasca & E. Marilli: MSAIS, 5, 109 (2004) 4. K. Biazzo, A. Frasca, G.W. Henry et al.: ApJ, in press (2006) 5. K. Biazzo, A. Frasca, S. Catalano & E. Marilli: AN, submitted (2006) 6. K. Biazzo, L. Pasquini, L. Girardi et al.: A&A, in preparation (2006) 7. S. Catalano, K. Biazzo, A. Frasca, E. Marilli: A&A, 394, 1009 (2002) 8. J.A. Cardelli, G.C. Clayton & J.S. Mathis: ApJ, 345, 245 (1989) 9. O.J. Eggen: ApJ, 166, 87 (1971) 10. A. Frasca, K. Biazzo, S. Catalano et al.: A&A, 432, 647 (2005) 11. L. Girardi, L. da Silva, L. Pasquini et al.: in Resolved Stellar Populations, ed by D. Valls-Gaband, M. Chavez (ASP Conf. Ser.: San Francisco) in press 12. D.F. Gray: The Observation and Analysis of Stellar Photospheres, 3rd edn (Cambridge University Press 2005) 13. G.F. Gray & K. Brown: PASP, 113, 723 (2001) 14. D.F. Gray & H.L. Johanson: 1991, ApJ, 103, 439 (1991) 15. V.V. Kovtyukh & N.I. Gorlova: A&A, 358, 587 (2000) 16. S. Meibom, J. Andersen, & B. Nordström: A&A 386, 187 (2002) 17. L. Pasquini, S. Randich, M. Zoccali et al.: A&A 951, 963 (2004) 18. L. da Silva, L. Girardi, L. Pasquini et al.: A&A, 458, 609 (2006)

0704.1463

Large deviations of Poisson cluster processes

Large deviations of Poisson cluster processes Charles Bordenave and Giovanni Luca Torrisi Abstract In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes. Keywords: Hawkes processes, Large deviations, Poisson cluster processes, Poisson processes. 1 Introduction Poisson cluster processes are one of the most important classes of point process models (see Daley and Vere-Jones (2003) and Møller and Waagepetersen (2004)). They are natural models for the location of objects in the space, and are widely used in point process studies whether theoretical or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or Hawkes processes (Hawkes (1971a), (1971b); Hawkes and Oakes (1974)). From a theoretical point of view Hawkes processes combine both a Poisson cluster process representation and a simple stochastic intensity representation. Poisson cluster processes found applications in cosmology, ecology and epidemiology; see, respec- tively, Neyman and Scott (1958), Brix and Chadoeuf (2002) and Møller (2003). Hawkes processes are particularly appealing for seismological applications. Indeed, they are widely used as statistical models for the standard activity of earthquake series; see the papers by Ogata and Akaike (1982), Vere-Jones and Ozaki (1982), Ogata (1988) and Ogata (1998). Hawkes processes have also aspects appealing to neuroscience applications; see the paper by Johnson (1996). More recently, Hawkes processes found applications to finance, see Chavez-Demoulin, Davison and Mc Neil (2005), and to DNA modeling, see Gusto and Schbath (2005). In this paper we derive scalar and sample path large deviation principles for Poisson cluster pro- cesses. The paper is organized as follows. In Section 2 we give some preliminaries on Poisson cluster processes, Hawkes processes and large deviations. In Section 3 we provide scalar large deviation principles for Poisson cluster processes, under a light-tailed assumption on the number of points per cluster. As consequence, we provide scalar large deviations for ergodic Hawkes processes. Section 4 is devoted to sample path large deviations of Poisson cluster processes. First, we prove a sample path large deviation principle on D[0, 1] equipped with the topology of point-wise convergence, under a light-tailed assumption on the number of points per cluster. Second, we give a sample path large deviation principle on D[0, 1] equipped with the topology of uniform convergence, under a super-exponential assumption on the number of points per cluster. In Section 5 we prove large deviations for spatial Poisson cluster processes, and we provide the asymptotic behavior of the void probability function and the empty space function. We conclude the paper with a short discussion. ∗INRIA/ENS, Départment d’Informatique, 45 rue d’Ulm, F-75230 Paris Cedex 05, France. e-mail: [email protected] †Istituto per le Applicazioni del Calcolo ”Mauro Picone” (IAC), Consiglio Nazionale delle Ricerche (CNR), Viale del Policlinico 137, I-00161 Roma, Italia. e-mail: [email protected] http://arXiv.org/abs/0704.1463v1 2 Preliminaries In this section we recall the definition of Poisson cluster process, Hawkes process, and the notion of large deviation principle. 2.1 Poisson cluster processes A Poisson cluster process X ⊂ R is a point process. The clusters centers of X are given by particular points called immigrants; the other points of the process are called offspring. The formal definition of the process is the following: (a) The immigrants are distributed according to a homogeneous Poisson process I with points Xi ∈ R and intensity ν > 0. (b) Each immigrant Xi generates a cluster Ci = CXi , which is a finite point process containing (c) Given the immigrants, the centered clusters Ci − Xi = {Y − Xi : Y ∈ Ci}, Xi ∈ I are independent, identically distributed (iid for short), and independent of I. (d) X consists of the union of all clusters. The number of points in a cluster is denoted by S. We will assume that E[S] < ∞. Let Y be a point process on R and NY(0, t] the number of points of Y in the interval (0, t]. Y is said stationary if its law is translations invariant, is said ergodic if it is stationary, with a finite intensity E[NY(0, 1]], NY(0, t] = E[NY(0, 1]], a.s.. By the above definition of Poisson cluster process it is clear that X is ergodic with finite intensity νE[S]. In particular, NX(0, t] = νE[S], a.s.. (1) 2.2 Hawkes processes We say that X ⊂ R is a Hawkes process if it is a Poisson cluster process with (b) in the definition above replaced by: (b)’ Each immigrant Xi generates a cluster Ci = CXi , which is the random set formed by the points of generations n = 0, 1, . . . with the following branching structure: the immigrant Xi is said to be of generation 0. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of generation n generates a Poisson process on (Y,∞), say Φ, of offspring of generation n + 1 with intensity function h(· − Y ). Here h : (0,∞) → [0,∞) is a non-negative Borel function called fertility rate. We refer the reader to Section 2 in Møller and Rasmussen (2005) for more insight into the branch- ing structure and self-similarity property of clusters. Consider the mean number of points in any offspring process Φ: h(t) dt. As usual in the literature on Hawkes processes, throughout this paper we assume 0 < µ < 1. (2) Condition µ > 0 excludes the trivial case in which there are almost surely no offspring. Recalling that the total number of points in a cluster is equivalent to the total progeny of the Galton-Watson process with one ancestor and number of offspring per individual following a Poisson distribution with mean µ (see p. 496 of Hawkes and Oakes (1974)), the other condition µ < 1 is equivalent to assuming that E[S] = 1/(1 − µ) < ∞. For our purposes it is important to recall that for Hawkes processes the distribution of S is given by P (S = k) = e−kµ(kµ)k−1 , k = 1, 2, . . . (3) This follows by Theorem 2.11.2 in the book by Jagers (1975). Finally, since X is ergodic with a finite and positive intensity equal to ν/(1 − µ) it holds: NX(0, t] 1 − µ, a.s.. (4) 2.3 Large deviation principles We recall here some basic definitions in large deviations theory (see, for instance, the book by Dembo and Zeitouni (1998)). A family of probability measures {µα}α∈(0,∞) on a topological space (M,TM ) satisfies the large deviations principle (LDP for short) with rate function J(·) and speed v(·) if J : M → [0,∞] is a lower semi-continuous function, v : [0,∞) → [0,∞) is a measurable function which increases to infinity, and the following inequalities hold for every Borel set B: − inf J(x) ≤ lim inf log µα(B) ≤ lim sup log µα(B) ≤ − inf J(x), where B◦ denotes the interior of B and B denotes the closure of B. Similarly, we say that a family of M -valued random variables {Vα}α∈(0,∞) satisfies the LDP if {µα}α∈(0,∞) satisfies the LDP and µα(·) = P (Vα ∈ ·). We point out that the lower semi-continuity of J(·) means that its level sets: {x ∈ M : J(x) ≤ a}, a ≥ 0, are closed; when the level sets are compact the rate function J(·) is said to be good. 3 Scalar large deviations 3.1 Scalar large deviations of Poisson cluster processes Consider the ergodic Poisson cluster process X described above. In this section we prove that the process {NX(0, t]/t} satisfies a LDP on R. Define the set DS = {θ ∈ R : E[eθS ] < ∞}. With a little abuse of notation, denote by C0 the cluster generated by an immigrant at 0 and let L = supY ∈C0 |Y | be the radius of C0. We shall consider the following conditions: the function θ 7→ E[eθS ] is essentially smooth and 0 ∈ D◦S (5) E[LeθS ] < ∞ for all θ ∈ D◦S . (6) For the definition of essentially smooth function, we refer the reader to Definition 2.3.5. in Dembo and Zeitouni (1998). Remark 3.1 Since S ≥ 1 we have that the function ϕ(θ) = E[eθS ] is increasing. It follows that S = (−∞, θ0) with θ0 ∈ [0,∞]. By the dominated convergence theorem we have that ϕ′(θ) = E[SeθS ] and ϕ′′(θ) = E[S2eθS ], for all θ ∈ D◦S . Hence, if θ0 < ∞, to prove that ϕ is essentially smooth it suffices to show that E[Seθ0S] = ∞. On the other hand, if θ0 = +∞, the function ϕ is always essentially smooth. It holds: Theorem 3.2 Assume (5) and (6). Then {NX(0, t]/t} satisfies a LDP on R with speed t and good rate function Λ∗(x) = sup (θx − Λ(θ)). (7) where Λ(θ) = ν(E[eθS ] − 1). It is easily verified that Λ∗(νE[S]) = 0. Moreover, this is the unique zero of Λ∗(·). Therefore the probability law of NX(0, t]/t concentrates in arbitrarily small neighborhoods of νE[S] as t → ∞, as stated by the law of large numbers (1). The LDP is a refinement of the law of large numbers in that it gives us the probability of fluctuations away the most probable value. Before proving Theorem 3.2 we show that the same LDP holds for the non-stationary Poisson cluster process Xt,T with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is a fixed constant. Furthermore, the LDP for Xt,T holds under a weaker condition. Theorem 3.3 Assume (5). Then {NXt,T (0, t]/t} satisfies a LDP on R with speed t and good rate function (7). Proof . The proof is based on the Gärtner-Ellis theorem (see, for instance, Theorem 2.3.6 in Dembo and Zeitouni (1998)). We start proving that log E[e θNXt,T (0,t]] = ν(E[eθS ] − 1) if θ ∈ DS +∞ if θ /∈ DS For a Borel set A ⊂ R, let I|A = I∩A be the point process of immigrants in A. Clearly I|(0,t], I|[−T,0] and I|(t,t+T ] are independent Poisson processes with intensity ν, respectively on (0, t], [−T, 0] and (t, T + t]. Since I|(0,t], I|[−T,0] , and I|(t,t+T ] are independent, by the definition of Poisson cluster process it follows that the random sets {Ci : Xi ∈ I|(0,t]}, {Ci : Xi ∈ I|[−T,0]} and {Ci : Xi ∈ I|(t,t+T ]} are independent. Therefore, for all θ ∈ R, θNXt,T (0,t] Xi∈I|(0,t] NCi (0,t]+ Xi∈I|[−T,0] NCi (0,t]+ Xi∈I|(t,t+T ] NCi(0,t] Xi∈I|(0,t] NCi (0,t] Xi∈I|[−T,0] NCi (0,t] Xi∈I|(t,t+T ] NCi(0,t] We shall show log E Xi∈I|(0,t] NCi(0,t] ν(E[eθS ] − 1) if θ ∈ DS +∞ if θ /∈ DS log E Xi∈I|[−T,0] NCi(0,t] = lim log E Xi∈I|(t,t+T ] NCi (0,t] = 0, for θ ∈ DS . (10) Note that (8) is a consequence of (9) and (10). We first prove (9). With a little abuse of notation, denote by C0 the cluster generated by an immigrant at 0. Since {(Xi, Ci) : Xi ∈ I|(0,t]} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have Xi∈I|(0,t] NCi(0,t] Xi∈I|(0,t] NCi−Xi (−Xi,t−Xi] = exp eθNC0 (−x,t−x] − 1 = exp eθNC0 (−tz,(1−z)t] − 1 . (11) Therefore if θ ∈ DS , the expectation in (11) goes to E[eθS − 1] as t → ∞ by the monotone convergence theorem. Hence, for θ ∈ DS the limit (9) follows from the dominated convergence theorem. For θ /∈ DS the expectation in (11) goes to +∞ as t → ∞ by the monotone convergence theorem, and the limit (9) follows by Fatou’s lemma. We now show (10). Here again, since {(Xi, Ci) : Xi ∈ I|[−T,0]} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere- Jones (2003) we have Xi∈I|[−T,0] NCi (0,t] Xi∈I|[−T,0] NCi−Xi(−Xi,t−Xi] = exp eθNC0 (x,x+t] − 1 . (12) Now note that, for θ ∈ DS ∩ [0,∞), we have 0 ≤ 1 log E Xi∈I|[−T,0] NCi(0,t] eθS − 1 dx < ∞ and, for each θ ≤ 0, eθS − 1 eθNC0 (x,x+t] − 1 dx ≤ 0. By passing to the limit as t → ∞ we get that the first limit in (10) is equal to 0. The proof for the second limit in (10) is rigorously the same. Hence we proved (8). Using assumption (5), the conclusion is a consequence of the Gärtner-Ellis theorem. Proof of Theorem 3.2. The proof is similar to that one of Theorem 3.3 and is again based on the Gärtner-Ellis theorem. We start showing that log E[eθNX(0,t]] = ν(E[eθS ] − 1) if θ ∈ D◦S +∞ if θ /∈ DS By similar arguments as in the proof of Theorem 3.3, using the definition of X, we have eθNX(0,t] θNXt,T (0,t] θ(NX(0,t]−NXt,T (0,t]) , for all θ ∈ R, t > 0. By the computations in the proof of Theorem 3.3, in order to prove (13) we only need to check log E θ(NX(0,t]−NXt,T (0,t]) = 0, for all θ ∈ D◦S . (14) It is easily verified for θ ≤ 0 (the argument of the expectation is bounded below by eθS and above by 1). We only check (14) for θ ∈ D◦S ∩ (0,∞). Here again, for a Borel set A ⊂ R, let I|A = I ∩ A denote the point process of immigrants in A. Note that NX(0, t] − NXt,T (0, t] = Xi∈I|(−∞,−T ) NCi(0, t] + Xi∈I|(t+T,∞) NCi(0, t], t > 0. Clearly I|(−∞,−T ) and I|(t+T,∞) are independent Poisson processes with intensity ν, respectively on (−∞,−T ) and (t + T,∞). Thus, by the definition of Poisson cluster process it follows that the random sets {Ci : Xi ∈ I|(−∞,−T )} and {Ci : Xi ∈ I|(t+T,∞)} are independent. Therefore, for all θ ∈ D◦S ∩ (0,∞), θ(NX(0,t]−NXt,T (0,t]) Xi∈I|(−∞,−T ) NCi(0,t] Xi∈I|(t+T,∞) NCi(0,t] Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} and {(Xi, Ci) : Xi ∈ I|(t+T,∞)} are independently marked Poisson processes, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have Xi∈I|(−∞,−T ) NCi(0,t] Xi∈I|(−∞,−T ) NCi−Xi(−Xi,t−Xi] = exp eθNC0 (x,t+x] − 1 Xi∈I|(t+T,∞) NCi (0,t] = exp eθNC0 (−x,t−x] − 1 = exp eθNC0 (−t−z,−z] − 1 Now notice that since θ > 0 we have eθNC0 (x,x+t] − 1 ≤ (eθNC0 (R) − 1)1{x ≤ L}, for all x ≥ T eθNC0 (−t−z,−z] − 1 ≤ (eθNC0 (R) − 1)1{z ≤ L}, for all z ≥ T . Relation (14) follows by assumption (6) noticing that the above relations yield Xi∈I|(−∞,−T ) NCi(0,t] ≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0. Xi∈I|(t+T,∞) NCi (0,t] ≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0. Therefore, (13) is proved. Now, if DS = D S then the claim is a consequence of the Gärtner-Ellis theorem and assumption (5). It remains to deal with the case DS 6= D◦S . We shall show the large deviations upper and lower bounds proving that for any sequence {tn}n≥1 ⊂ (0,∞) diverging to +∞, as n → ∞, there exists a subsequence {sn} ⊆ {tn} such that lim sup log P (NX(0, sn]/sn ∈ F ) ≤ − inf Λ∗(x), for all closed sets F (15) lim inf log P (NX(0, sn]/sn ∈ G) ≥ − inf Λ∗(x), for all open sets G, (16) where Λ∗ is defined by (7) (then the large deviations upper and lower bounds hold for any sequence {tn} and the claim follows). By assumption (5), there exists θ0 > 0 such that DS = (∞, θ0]. Let {tn}n≥1 ⊂ (0,∞) be a sequence diverging to +∞, as n → ∞, and define the extended non-negative real number l ∈ [0,∞] by l ≡ lim sup log E[eθ0NX(0,tn]]. Clearly, there exists a subsequence {sn} ⊆ {tn} which realizes this lim sup, i.e. log E[eθ0NX(0,sn]] = l. By (13) it follows that log E[eθNX(0,sn]] = Λ̃(θ), θ ∈ R where Λ̃(θ) = Λ(θ) if θ < θ0 l if θ = θ0 +∞ if θ > θ0. Note that, irrespective to the value of l, Λ̃ is essentially smooth (however it may be not lower semi-continuous). We now show that the Legendre transform of Λ and Λ̃ coincide, i.e. Λ̃∗(x) = Λ∗(x), x ∈ R. (17) A straightforward computation gives Λ̃∗(x) = Λ∗(x) = +∞, for x < 0, and Λ̃∗(0) = Λ∗(0) = ν. Now, note that since θ0 < ∞, Λ̃∗(x) and Λ∗(x) are both finite, for x > 0. Moreover, since Λ and Λ̃ are essentially smooth, if x > 0 we have that Λ∗(x) = θxx − Λ(θx) and Λ̃∗(x) = θ̃xx − Λ̃(θ̃x), where θx (respectively θ̃x) is the unique solution of Λ ′(θ) = x (respectively Λ̃(θ) = x) on (−∞, θ0). The claim (17) follows recalling that Λ̃(θ) = Λ(θ) = ν(E[eθS ] − 1) on D◦S . Now, applying part (a) of Theorem 2.3.6 in Dembo and Zeitouni (1998) we have (15). Applying part (b) of Theorem 2.3.6 in Dembo and Zeitouni (1998) we get lim inf log P (NX(0, sn]/sn ∈ G) ≥ − inf x∈G∩F Λ∗(x), for any open set G, (18) where F is the set of exposed points of Λ∗ whose exposing hyperplane belongs to (−∞, θ0), i.e. F = {y ∈ R : ∃ θ ∈ D◦S such that for all x 6= y, θy − Λ∗(y) > θx − Λ∗(x)}. We now prove that F = (0,+∞). For y < 0, Λ∗(y) = ∞, therefore an exposing hyperplane satisfying the corresponding inequality does not exist. For y > 0 consider the exposing hyperplane θ = θy, where θy is the unique positive solution on (−∞, θ0) of E[SeθS ] = y/ν. Note that Λ′(θ) = E[SeθS ] and Λ′′(θ) = E[S2eθS ] for all θ < θ0. In particular, since S ≥ 1, we have that Λ is strictly convex on (−∞, θ0). Therefore, for all x 6= y, it follows θyy − Λ∗(y) = Λ(θy) > Λ(θx) + Λ′(θx)(θy − θx) = θyx − Λ∗(x). It remains to check that 0 /∈ F. Notice that since E[SeθxS ] = x/ν, limx↓0 θx = −∞. Also, by the implicit function theorem, x 7→ θx is a continuous mapping on (0,∞). Now assume that 0 ∈ F, then there would exist θ < θ0, such that for all x > 0, −Λ∗(0) > θx−Λ∗(x). However, by the intermediate values theorem, there exists y > 0 such that θ = θy, and we obtain a contradiction. This implies F = (0,+∞) as claimed. Now recall that Λ∗(x) = +∞ for x < 0; moreover, limx↓0 Λ∗(x) = Λ∗(0) = ν (indeed, limx↓0 θx = −∞). Therefore x∈G∩F Λ∗(x) ≤ inf Λ∗(x), for any open set G. Finally, by (18) and the above inequality we obtain (16). 3.2 Scalar large deviations of Hawkes processes Consider the ergodic Hawkes process X described before. In this section we prove that the process {NX(0, t]/t} satisfies a LDP, and we give the explicit expression of the rate function. Our result is a refinement of the law of large numbers (4). The following theorem holds: Theorem 3.4 Assume (2) and th(t) dt < ∞. (19) Then {NX(0, t]/t} satisfies a LDP on R with speed t and good rate function Λ∗(x) = xθx + ν − νxν+µx if x ∈ (0,∞) ν if x = 0 +∞ if x ∈ (−∞, 0) , (20) where θ = θx is the unique solution in (−∞, µ − 1 − log µ) of = x/ν, x > 0, (21) or equivalently of E[eθS ] = ν + xµ , x > 0. Proof . The proof is a consequence of Theorem 3.2. We start noticing that by (3) we have E[eθS ] = (eθ−µ)k(kµ)k−1 and this sum is infinity for θ > µ − 1 − log µ and finite for θ < µ − 1 − log µ (apply, for instance, the ratio criterion). If θ = µ − 1 − log µ the sum above is finite. Indeed, in this case E[eθS ] = (1/µ) e−kkk−1 = 1/µ. Therefore DS = (−∞, µ − 1 − log µ]. The origin belongs to D◦S in that by (2) and the inequality ex > x + 1, x 6= 0, we have eµ−1 > 1. The function θ 7→ E[eθS ] is essentially smooth. Indeed, it is differentiable in the interior of DS and E[Se(µ−1−log µ)S ] = ∞ because E[Se(µ−1−log µ)S ] = (1/µ) e−kkk and this sum is infinity since by Stirling’s formula e 2πk. We now check assumption (6). By the structure of the clusters, it follows that there exists a sequence of independent non- negative random variables {Vn}n≥1, independent of S, such that V1 has probability density h(·)/µ and the following stochastic domination holds: Vn, a.s. (see Reynaud-Bouret and Roy (2007)). Therefore, for all θ < µ − 1 − log µ, we have E[LeθS ] ≤ E = E[V1]E[Se θS ]. Since θ < µ − 1 − log µ, we have E[SeθS ] < ∞; moreover, assumption (19) yields E[V1] = th(t) dt < ∞. Hence, condition (6) holds, and by Theorem 3.2, {NX(0, t]/t} satisfies a LDP on R with speed t and good rate function Λ∗(x) = sup (θx − Λ(θ)) = sup θ≤µ−1−log µ (θx − Λ(θ)). Now Λ∗(x) = ∞ if x < 0, in that in such a case limθ→−∞(θx − Λ(θ)) = ∞. If x > 0, letting θx ∈ (−∞, µ − 1 − log µ) denote the unique solution of the equation (21) easily follows that Λ∗(x) = xθx − Λ(θx). (22) It is well-known (see, for instance, p. 39 in Jagers (1975)) that, for all θ ∈ (−∞, µ − 1 − log µ), E[eθS ] satisfies E[eθS ] = eθ exp{µ(E[eθS ] − 1)}, therefore differentiating with respect to θ we get E[SeθS ] = eθ exp{µ(E[eθS ] − 1)} 1 − µeθ exp{µ(E[eθS ] − 1)} = E[eθS ] 1 − µE[eθS ] . (23) Setting θ = θx in the above equality and using (21) we have E[eθxS ] 1 − µE[eθxS ] , which yields E[eθxS ] = ν + xµ Thus, by (22) we have for x > 0 Λ∗(x) = xθx + ν − ν + µx The conclusion follows noticing that a direct computation gives Λ∗(0) = ν. 4 Sample path large deviations Let X be the ergodic Poisson cluster process described at the beginning. The results proved in this section are sample path LDP for X. 4.1 Sample path large deviations in the topology of point-wise convergence Let D[0, 1] be the space of càdlàg functions on the interval [0, 1]. Here we prove that {NX(0,α·] satisfies a LDP on D[0, 1] equipped with the topology of point-wise convergence on D[0, 1]. The LDP we give is a refinement of the following functional law of large numbers: NX(0, α·] = χ(·) a.s., (24) where χ(t) = νE[S]t. As this is a corollary of the LDP we establish, we do not include a separate proof of this result. Letting Λ∗(·) denote the rate function of the scalar LDP, we have: Theorem 4.1 Assume (5) and (6). If moreover DS is open, then {NX(0,α·]α } satisfies a LDP on D[0, 1], equipped with the topology of point-wise convergence, with speed α and good rate function J(f) = { ∫ 1 Λ∗(ḟ(t))dt iff ∈ AC0[0, 1] ∞ otherwise , (25) where AC0[0, 1] is the family of absolutely continuous functions f(·) defined on [0, 1], with f(0) = 0. While it is tempting to conjecture that the result above holds even if the effective domain of S is not open, we do not have a proof of this claim. If we take χ(t) = νE[S]t, then J(χ) = 0. Moreover this is the unique zero of J(·). Thus the law of NX(0, α·]/α concentrates in arbitrarily small neighborhoods of χ(·) as α → ∞, as ensured by the functional law of large numbers (24). The sample path LDP is a refinement of the functional law of large numbers in that it gives the probability of fluctuations away the most likely path. As in Section 3.1, denote by Xt,T the non-stationary Poisson cluster process with immigrant process empty on (−∞,−T )∪ (t+T,∞), where T > 0 is a fixed constant. Before proving Theorem 4.1 we show that the same LDP holds for Xt,T . Furthermore, the LDP for Xt,T holds under a weaker condition. Theorem 4.2 Assume (5). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the topology of point-wise convergence, with speed α and good rate function (25). To prove this theorem we need Lemma 4.3 below, whose proof can be found in Ganesh, Macci and Torrisi (2005) (see Lemma 2.3 therein). Lemma 4.3 Let (θ1, . . . , θn) ∈ Rn and let w1, . . . , wn ≥ 0 be such that w1 ≤ . . . ≤ wn. Then i=k θiwi ≤ θ∗w∗ for all k ∈ {1, . . . , n}, for any θ∗ ≥ max{max{ i=k θi : k ∈ {1, . . . , n}}, 0} and any w∗ ≥ wn. Proof of Theorem 4.2. With a little abuse of notation denote by C0 the cluster generated by an immigrant at 0. We first show the theorem under the additional condition NC0((−∞, 0)) = 0, a.s.. (26) The idea in proving Theorem 4.2 is to apply the Dawson-Gärtner theorem to ”lift” a LDP for the finite-dimensional distributions of {NXαt,T (0, αt]/α} to a LDP for the process. Therefore, we first show the following claim: (C) For all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1, NXαt1,T (0, αt1]/α, . . . ,NXαtn,T (0, αtn]/α satisfies the LDP in Rn with speed α and good rate function Jt1,...,tn(x1, . . . , xn) = (tj − tj−1)Λ∗ xj − xj−1 tj − tj−1 , (27) where x0 = 0 and t0 = 0. Claim (C) is a consequence of the Gärtner-Ellis theorem in Rn, and will be shown in three steps: (a) For each (θ1, . . . , θn) ∈ Rn, we prove that Λt1,...,tn(θ1, . . . , θn) ≡ lim log E θiNXαti,T (0, αti] (tj−tj−1)Λ where the existence of the limit (as an extended real number) is part of the claim, and Λ(·) is defined in the statement of Theorem 3.2. (b) The function Λt1,...,tn(·) satisfies the hypotheses of the Gärtner-Ellis theorem. (c) The rate function Jt1,...,tn(x1, . . . , xn) ≡ sup (θ1,...,θn)∈Rn θixi − Λt1,...,tn(θ1, . . . , θn) coincides with the rate function defined in (27). Proof of (a). For a Borel set A ⊂ R, denote by I|A = I ∩ A the Poisson process of immigrants in A. Since, for each t, I|(0,t] and I|[−T,0] are independent, it follows from the definition of Poisson cluster process that, for each i, the random sets {Ck : Xk ∈ I|(0,αti]} and {Ck : Xk ∈ I|[−T,0]} are independent. Therefore, θiNXαti,T (0, αti] Xk∈I|(0,αti] NCk(0, αti] Xk∈I|[−T,0] NCk(0, αti] where we used the independence and the assumption that NC0(−∞, 0) = 0 a.s.. In order to prove (28), we treat successively the two terms in (29). Viewing I|(0,αti] as the superposition of the i independent Poisson processes: I|(αtj−1,αtj ] on (αtj−1, αtj ] (j = 1, . . . , i) with intensity ν we get Xk∈I|(0,αti] NCk(0, αti]  = E Xk∈I|(αtj−1,αtj ] θiNCk(0, αti] Xk∈I|(αtj−1,αtj ] θiNCk(0, αti] Xk∈I|(αtj−1,αtj ] θiNCk(0, αti] , (30) where in the latter equality we used the independence of {Ck : Xk ∈ I|(αtj−1,αtj ]} (j = 1, . . . , n). Since, for each j, {(Xk, Ck) : Xk ∈ I|(αtj−1,αtj ]} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have Xk∈I|(αtj−1,αtj ] θiNCk(0, αti] ∫ α(tj−tj−1) θiNC0(−αtj−1 − s, α(ti − tj−1) − s]  . (31) We now show log E Xk∈I|(0,αti] NCk(0, αti] (tj − tj−1)Λ  (32) for each (θ1, . . . , θn) ∈ Rn. We first notice that by (30) and (31) we have log E Xk∈I|(0,αti] NCk(0, αti] (tj − tj−1)Jj(α), where Jj(α) = α(tj − tj−1) ∫ α(tj−tj−1) θiNC0(−αtj−1 − s, α(ti − tj−1) − s]  ds. Now suppose that (θ1, . . . , θn) ∈ Rn is such that i=j θi ∈ DS for each j ∈ {1, . . . , n}. Then by Lemma 4.3 it follows that there exists θ∗ ∈ DS such that θ∗ ≥ 0, i=j θi ≤ θ∗ for all j ∈ {1, . . . , n}, θiNC0(−αtj−1 − s, α(ti − tj−1) − s] ≤ θ∗NC0(R), a.s.. By (33) and the dominated convergence theorem, we have Jj(α) = ν i=j θiS Hence we proved (32) whenever (θ1, . . . , θn) ∈ Rn satisfies i=j θi ∈ DS for every j ∈ {1, . . . , n}. Now suppose that (θ1, . . . , θn) ∈ Rn is such that i=j θi /∈ DS for some j ∈ {1, . . . , n}. We have that Jj(α) is bigger than or equal to α(tj − tj−1) ∫ α(tj−tj−1) 1{θi < 0}θiS + 1{θi > 0}θiNC0 [0, α(tj − tj−1) − s] 1{θi < 0}θiS + 1{θi > 0}θiNC0 [0, α(tj − tj−1)(1 − z)]  − 1  dz. The expectation in the latter formula goes to E[exp( i=j θiS) − 1] as α → ∞ by the monotone convergence theorem. Therefore, by Fatou’s lemma we have Jj(α) ≥ νE  = ∞. Thus, since the quantities J1(α), . . . , Jn(α) are bounded below by −ν, we get (32) also in this case. We now show log E Xk∈I|[−T,0] NCk(0, αti]  = 0 (34) for all (θ1, . . . , θn) ∈ Rn such that i=j θi ∈ DS for each j ∈ {1, . . . , n}. By Lemma 4.3 we have that there exists θ∗ ∈ DS such that θ∗ ≥ 0, i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and Xk∈I|[−T,0] NCk(R) ≤ Xk∈I|[−T,0] NCk(0, αti] ≤ θ∗ Xk∈I|[−T,0] NCk(R), a.s., where θ− ≡ i:θi<0 θi and θ− ≡ 0 if {i : θi < 0} = ∅. Therefore, using again Lemma 6.4 VI in Daley and Vere-Jones (2003), we have νT (E[eθ−S ] − 1) Xk∈I|[−T,0] NCk(0, αti]  ≤ exp νT (E[eθ ∗S ] − 1) Equation (34) follows taking the logarithms in the above inequalities and passing to the limit. The conclusion follows putting together (29), (32) and (34). Proof of (b) and Proof of (c). Part (b) can be shown using assumption (5) and following the lines of the proof of part (b) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005). The proof of part (c) is identical to the proof of part (c) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005). End of the proof under condition (26). By claim (C) and the Dawson-Gärtner theorem, {NXα·,T (0, α·]/α} satisfies the LDP on D[0, 1], equipped with the topology of point-wise convergence, with speed α and good rate function J̃(f) = sup (tk − tk−1)Λ∗ (f(tk) − f(tk−1) tk − tk−1 : n ≥ 1, 0 ≤ t1 < . . . < tn ≤ 1 The conclusion follows noticing that J̃(·) coincides with J(·) in (25), as can be checked following the same lines as in the proof of Lemma 5.1.6 in Dembo and Zeitouni (1998). Removing the additional condition (26). The general case is solved as follows. Since Ck is almost surely finite, there exists a left-most extremal point Yk ∈ Ck such that NCk(−∞, Yk) = 0 a.s.. Note that, given the immigrants, Yk − Xk is an iid sequence. Therefore, by a classical result on Poisson processes we have that {Yk} is a Poisson process with intensity ν. Viewing Xt,T as a Poisson cluster process with cluster centers Yk and clusters Ck, the conclusion follows by the first part of the proof. Proof of Theorem 4.1. The proof uses similar steps as in the proof of Theorem 4.2. Here we sketch the main difference. Assume the additional condition NC0((−∞, 0)) = 0 a.s. (the general case can be treated as in the proof of Theorem 4.2). Define the following subsets of Rn: (θ1, . . . , θn) ∈ Rn : θi ∈ DS for all j ∈ {1, . . . , n} (θ1, . . . , θn) ∈ Rn : θi /∈ DS for some j ∈ {1, . . . , n} We start showing that for all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1 Λt1,...,tn(θ1, . . . , θn) = j=1(tj − tj−1)Λ i=j θi for (θ1, . . . , θn) ∈ A1 +∞ for (θ1, . . . , θn) ∈ A2, where Λt1,...,tn(θ1, . . . , θn) ≡ lim log E θiNX(0, αti] and Λ(·) is defined in the statement of Theorem 3.2. Using the definition of X and the assumption NC0((−∞, 0)) = 0 a.s., we have θiNX(0, αti] Xk∈I|(0,αti] NCk(0, αti] Xk∈I|[−T,0] NCk(0, αti] Xk∈I|(−∞,−T ) NCk(0, αti] As noticed in the proof of Theorem 4.2 we have θiNXαti,T (0, αti] Xk∈I|(0,αti] NCk(0, αti] Xk∈I|[−T,0] NCk(0, αti] Therefore, by the computations in the proof of Theorem 4.2, to prove (35) we only need to check log E Xk∈I|(−∞,−T ) NCk(0, αti]  = 0, for all (θ1, . . . , θn) ∈ A1. (36) Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have Xk∈I|(−∞,−T ) NCk(0, αti]  = E Xk∈I|(−∞,−T ) θiNCk−Xk(−Xk, αti − Xk] = exp i=1 θiNC0 (x,αti+x] − 1 Take (θ1, . . . , θn) ∈ A1. By Lemma 4.3 we have that there exists θ∗ ∈ DS such that θ∗ ≥ 0, i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and θ−NC0(R) ≤ θiNC0(x, αti + x] ≤ θ∗NC0(R), a.s. where θ− ≡ i:θi<0 θi and θ− ≡ 0 if {i : θi < 0} = ∅. Thus, i=1 θiNC0 (x,αti+x] − 1 ≤ (eθ∗NC0 (R) − 1)1{x ≤ L}, for all x ≥ T i=1 θiNC0 (x,αti+x] − 1 ≥ (eθ−NC0(R) − 1)1{x ≤ L}, for all x ≥ T The limit (36) follows by assumption (6) noticing that the above relations yield, for all (θ1, . . . , θn) ∈ Xk∈I|(−∞,−T ) NCk(0, αti]  ≤ exp(νE[L(eθ∗S − 1)]) Xk∈I|(−∞,−T ) NCk(0, αti]  ≥ exp(νE[(eθ−S − 1)(L − T )1{L ≥ T}]). Now since DS is open, the claim follows by applying first the Gärtner-Ellis theorem in R n to get the LDP for the finite-dimensional distributions, and then the Dawson-Gärtner theorem to have the LDP for the process (argue as in the proof of Theorem 4.2 for the remaining steps). 4.2 Sample path large deviations in the topology of uniform convergence In the applications, one usually derives LDPs for continuous functions of sample paths of stochastic processes by using the contraction principle. Since the topology of uniform convergence is finer than the topology of point-wise convergence, it has a larger class of continuous functions. Thus, it is of interest to understand if {NX(0, α·]/α} satisfies a LDP on D[0, 1] equipped with the topology of uniform convergence. In this section we give an answer to this question assuming that the tails of S decay super-exponentially. Theorem 4.4 Assume E[eθS ] < ∞ for each θ ∈ R (37) E[LeθS ] < ∞ for each θ ∈ R. (38) Then {NX(0,α·] } satisfies a LDP on D[0, 1], equipped with the topology of uniform convergence, with speed α and good rate function (25). In this section, without loss of generality we assume that the points of I are {Xi}i∈Z∗ , where ∗ = Z\{0}, Xi < Xi+1, and we set X0 = 0. As usual, we denote by Xt,T the non-stationary Poisson cluster process with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is a fixed constant, and by C0 the cluster generated by an immigrant at 0. Before proving Theorem 4.4 we show that the same LDP holds for Xt,T , under a weaker con- dition. Theorem 4.5 Assume (37). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the topology of uniform convergence, with speed α and good rate function (25). To prove Theorem 4.5 above we use the following Lemma 4.6, whose proof is omitted since it is similar to the proof of Lemma 3.3 in Ganesh, Macci and Torrisi (2005). Let {Sk}k∈Z be the iid sequence of random variables (distributed as S) defined by Sk = NCk(R). Lemma 4.6 Assume (37), NC0(−∞, 0) = 0 a.s., and define (Sk − NCk−Xk(0,Xk]), n ≥ 1. It holds log P (An ≥ nδ) = −∞ for each δ > 0. Proof of Theorem 4.5. We prove the theorem assuming that NC0(−∞, 0) = 0 a.s.. The general case is solved as in the proof of Theorem 4.2. As usual denote by I|A the restriction of I on the Borel set A ⊂ R. Define C(t) = Xk∈I|(0,t] Sk, t > 0. We prove that {NXα·,T (0, α·]/α} and {C(α·)/α} are exponentially equivalent (see, for instance, Definition 4.2.10 in the book of Dembo and Zeitouni, (1998)) with respect to the topology of uniform convergence. Therefore the conclusion follows by a well-known result on sample path large deviations, with respect to the uniform topology, of compound Poisson processes (see, for instance, Borovkov (1967); see also de Acosta (1994) and the references cited therein) and Theorem 4.2.13 in Dembo and Zeitouni (1998). Define CT (t) = Xk∈I|[−T,t] Sk, t > 0. Using Chernoff bound and condition (37) can be easily realized that the processes {C(α·)/α} and {CT (α·)/α} are exponentially equivalent with respect to the topology of uniform conver- gence. Therefore, it suffices to show that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv- alent with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) = 0 a.s. gives NXt,T (0, t] = Xk∈I|(0,t] NCk(0, t] + Xk∈I|[−T,0] NCk(0, t] t > 0, a.s.. Therefore, we need to show that log P (Mα > δ) = −∞, for any δ > 0, (39) where t∈[0,1] CT (αt) − Xk∈I|(0,αt] NCk(0, αt] − Xk∈I|[−T,0] NCk(0, αt] Since Mα ≤ M (1)α + M (2)α , a.s., where M (1)α = Xk∈I|[−T,0] Sk and M t∈[0,1] Xk∈I|(0,αt] (Sk − NCk(0, αt]), the limit (39) follows if we prove log P (M (1)α > δ/2) = −∞, for any δ > 0 (40) log P (M (2)α > δ/2) = −∞, for any δ > 0. (41) The limit (40) easily follows by the Chernoff bound and condition (37). It remains to show (41). Since the random function t 7→ NCk(0, αt] is non-decreasing, it is clear that the supremum over t is attained at one of the points Xn, n ≥ 1. Thus M (2)α = n≥1:Xn≤α (Sk − NCk(0,Xn]). Note that M (2)α ≤ M̃α where M̃α = n≥1:Xn≤α (Sk − NCk−Xk(0,Xn − Xk]) a.s.. Therefore (41) follows if we show log P (M̃α > δ/2) = −∞, for any δ > 0. (42) Since Xn, n ≥ 1, is the sum of n exponential random variables with mean 1/ν, using Chernoff bound and taking the logarithm, we have that, for all η > 0 and all integers K > ν, log P (XK[α] < α) ≤ η + ν + η . (43) Here the symbol [α] denotes the integer part of α. Next, observe that using the union bound we P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max 1≤n≤K[α] (Sk − NCk−Xk(0,Xn − Xk]) ≥ αδ/2 Now we remark that, for n ≥ 1, (Xn − X1, . . . ,Xn − Xn−1) and (Xn−1, . . . ,X1) have the same joint distribution. Moreover, given I, the centered processes Ck − Xk are iid and independent of the {Xk}. Hence, letting An denote the random variable defined in the statement of Lemma 4.6, we have P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max 1≤n≤K[α] P (An ≥ αδ/2). The random variables An are increasing in n, therefore P (M̃α > δ,XK[α] ≥ α) ≤ K[α]P (AK[α] ≥ αδ/2), and by Lemma 4.6 we have log P (M̃α > δ/2,XK[α] ≥ α) = −∞. (44) Now note that P (M̃α > δ/2) ≤ P (M̃α > δ/2,XK[α] ≥ α) + P (XK[α] < α), for arbitrary K > ν. Hence by (43) and(44) we have lim sup log P (M̃α > δ/2) ≤ inf η + K log ν + η = K − ν − K log K Then we obtain (42) by letting K tend to ∞. Proof of Theorem 4.4. Throughout the proof we assume NC0((−∞, 0)) = 0 a.s.. The general case is solved as in the proof of Theorem 4.2. Let {CT (t)} be the process defined in the proof of Theorem 4.5. The claim follows if we show that {CT (α·)/α} and {NX(0, α·]/α} are exponentially equivalent with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) = 0 a.s. implies NX(0, t] = NXt,T (0, t] + Xk∈I|(−∞,−T ) NCk(0, t], t > 0 a.s.. Therefore, since we already proved that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv- alent with respect to the uniform topology (see the proof of Theorem 4.5), the claim follows if we prove that log P Xk∈I|(−∞,−T ) NCk(0, α] > αδ  = −∞, for any δ > 0. (45) Using the Chernoff bound we have, for all θ > 0, Xk∈I|(−∞,−T ) NCk(0, α] > αδ  ≤ e−αθδE Xk∈I|(−∞,−T ) θNCk(0, α] = e−αθδ exp E[eθNC0 (x,α+x] − 1] dx ≤ e−αθδ exp νE[(eθS − 1)L] Taking the logarithm, dividing by α, letting α tend to ∞ and using assumption (38) we get lim sup log P Xk∈I|(−∞,−T ) NCk(0, α] > αδ  ≤ −θδ, for all θ > 0. Relation (45) follows letting θ tend to infinity in the above inequality. 5 Large deviations of spatial Poisson cluster processes 5.1 The large deviations principle A spatial Poisson cluster process X is a Poisson cluster process in Rd, where d ≥ 1 is an integer. The clusters centers are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈ (0,∞). Each immigrant Xi ∈ I generates a cluster Ci = CXi , which is a finite point process. Given I, the centered clusters {CXi −Xi} are iid and independent of I. X is the union of all clusters. As in dimension 1, we denote by S the number of points in a cluster, with a little abuse of notation by C0 the cluster generated by a point at 0, and by L the radius of C0. Moreover, we denote by NX(b(0, r)) the number of points of X in the ball b(0, r), and by ωd(r) = rdπd/2 Γ(1 + d/2) the volume of b(0, r). The following LDP holds: Theorem 5.1 Assume (5) and E[LdeθS ] < ∞, for all θ ∈ D◦S. (46) Then {NX(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r) and good rate function (7). Before proving Theorem 5.1, we show that the same LDP holds for the non-stationary Poisson cluster process Xr,R with immigrant process empty in R d \ b(0, R + r). As usual this LDP holds under a weaker condition. Theorem 5.2 Assume (5). Then {NXr,R(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r) and good rate function (7). Proof of Theorem 5.2 The proof is similar to that one for the non-stationary Poisson cluster process on the line. Here we just sketch the main differences. As in the proof of Theorem 3.3, the claim follows by the Gärtner-Ellis theorem. Indeed, letting I|b(0,r) denote the point process of immigrants in b(0, r), and I|b(0,R+r)\b(0,r) the point process of immigrants in b(0, R + r) \ b(0, r) we have, for each θ ∈ R, θNXr,R (b(0,r)) Xi∈I|b(0,r) NCi (b(0,r)) Xi∈I|b(0,R+r)\b(0,r) NCi(b(0,r)) As usual, with a little abuse of notation denote by C0 the cluster generated by an immigrant at 0. It holds: Xi∈I|b(0,r) NCi (b(0,r)) = exp b(0,r) eθNC0 (b(−x,r)) − 1 for each θ ∈ R; Xi∈I|b(0,R+r)\b(0,r) NCi(b(0,r)) ≤ exp ν(ωd(R + r) − ωd(r))E eθS − 1 for θ ∈ [0,∞) ∩ DS ; 1 ≥ E Xi∈I|b(0,R+r)\b(0,r) NCi (b(0,r)) ≥ exp ν(ωd(R + r) − ωd(r))E eθS − 1 for θ ≤ 0. Therefore, ωd(r) log E Xi∈I|b(0,r) NCi(b(0,r)) = lim ωd(r) b(0,r) E[eθNC0 (b(−x,r)) − 1] dx ωd(1) b(0,1) E[eθNC0 (b(−ry,r)) − 1] dy = νE[eθS − 1], for each θ ∈ R, and, since limr→∞ ωd(R + r)/ωd(r) = 1, for each θ ∈ DS , ωd(r) log E Xi∈I|b(0,R+r)\b(0,r) NCi (b(0,r)) The rest of the proof is exactly as in the one-dimensional case. Proof of Theorem 5.1 The proof is similar to that one of Theorem 5.2 and is again based on the Gärtner-Ellis theorem. We start showing that ωd(r) log E θ(NX(b(0,r))−NXr,R (b(0,r))) = 0, for all θ ∈ D◦S . (47) This relation is easily verified for θ ≤ 0. Thus we only check (47) for θ ∈ D◦S ∩ (0,∞). We have, for all θ ∈ D◦S ∩ (0,∞), θ(NX(b(0,r))−NXr,R (b(0,r))) Xi∈I|Rd\b(0,r+R) NCi (b(0,r)) = exp Rd\b(0,r+R) eθNC0 (b(−x,r)) − 1 Now notice that since θ > 0 we have eθNC0 (b(−x,r)) − 1 ≤ (eθNC0 (Rd) − 1)1{‖x‖ ≤ L + r}, for all x ∈ Rd. The limit (47) follows by assumption (46) noticing that the above relations yield, for all θ ∈ S ∩ (0,∞), r > 0, Xi∈I|Rd\b(0,r+R) NCi(b(0,r)) ≤ exp((νπd/2/Γ(1 + d/2))E[(L + r)d(eθS − 1)]). Now notice that eθNX(b(0,r)) θNXr,R (b(0,r)) θ(NX(b(0,r))−NXr,R (b(0,r))) , for all θ ∈ R, r > 0. Therefore, if DS = D S then the claim is a consequence of the computation of the log-Laplace limit of {NXr,R(b(0, r))} in the proof of Theorem 5.2, the Gärtner-Ellis theorem and assumption (5). It remains to deal with the case DS 6= D◦S . Arguing exactly as in the proof of Theorem 3.2 it can be proved that for any sequence {rn}n≥1 ⊂ (0,∞) diverging to +∞, as n → ∞, there exists a subsequence {qn} ⊆ {rn} such that lim sup log P (NX(b(0, qn))/ωd(qn) ∈ F ) ≤ − inf Λ∗(x), for all closed sets F lim inf log P (NX(b(0, qn))/ωd(qn) ∈ G) ≥ − inf Λ∗(x), for all open sets G, where Λ∗ is defined by (7). Then the large deviations upper and lower bounds hold for any sequence {rn} and the claim follows. 5.2 The asymptotic behavior of the void probability function and the empty space function Apart some specific cases, the void probability function v(r) = P (NX(b(0, r)) = 0), r > 0, of a spatial Poisson cluster process is not known in closed form. Comparing X with the immigrant process I we easily obtain v(r) ≤ P (NI(b(0, r)) = 0) = e−νωd(r), r > 0. (48) A more precise information on the asymptotic behavior of v(·), as r → ∞, is provided by the following proposition: Proposition 5.3 Assume E[Ld] < ∞. Then ωd(r) log v(r) = −ν. Proof Note that v(r) = P (NCi(b(0, r)) = 0, for all Xi ∈ I) 1{NI(b(0, r)) = 0} Xi∈I|Rd\b(0,r) 1{NCi(b(0, r)) = 0} = e−νωd(r)E Xi∈I|Rd\b(0,r) 1{NCi(b(0, r)) = 0} = e−νωd(r) exp Rd\b(0,r) P (NC0(b(−x, r)) > 0) dx where in (49) we used Lemma 6.4.VI in Daley and Vere-Jones (2003). Thus the claim follows if we prove ωd(r) Rd\b(0,r) P (NC0(b(−x, r)) > 0) dx = 0. For this note that 1{NC0(b(0, r)) > 0} ≤ 1{‖x‖ − r ≤ L}, for all x ∈ Rd, r > 0. Therefore, ωd(r) Rd\b(0,r) P (NC0(b(−x, r)) > 0) dx ≤ E[(1 + L/r)d − 1], and the right-hand side in the above inequality goes to 0 as r → ∞ by the dominated convergence theorem (note that E[Ld] < ∞ by assumption). In spatial statistics, a widely used summary statistic is the so-called empty space function, which is the distribution function of the distance from the origin to the nearest point in X (see, for instance, Møller and Waagepetersen (2004)), that is e(r) = 1 − v(r), r > 0. Apart some specific cases, the empty space function of Poisson cluster processes seems to be in- tractable. Next Corollary 5.4 concerns the asymptotic behavior of e(r), as r → ∞. Corollary 5.4 Under the assumption of Proposition 5.3 it holds ωd(r) log log e(r)−1 = −ν. Proof The proof is an easy consequence of Proposition 5.3. By the upper bound (48) we obtain lim sup ωd(r) log log e(r)−1 ≤ lim ωd(r) log log(1 − e−νωd(r))−1 = −ν. To get the matching lower bound we note that the inequality log(1 − x) ≤ −x, x ∈ [0, 1), gives log e(r)−1 ≥ v(r), r > 0, and therefore by Proposition 5.3 we get lim inf ωd(r) log log e(r)−1 ≥ lim inf ωd(r) log v(r) = −ν. 5.3 Spatial Hawkes processes Spatial Hawkes processes have been introduced in Daley and Vere-Jones (2003). Brémaud, Mas- soulié and Ridolfi (2005) considered spatial Hawkes processes with random fertility rate and not necessarily Poisson immigrants, and computed the Bartlett spectrum; the reader is directed to Daley and Vere-Jones (2003) for the notion of Bartlett spectrum. Møller and Torrisi (2005) de- rived the pair correlation function of spatial Hawkes processes; we refer the reader to Møller and Waagepetersen (2004) for the notion of pair correlation function. For the sake of completeness, we briefly recall the definition of spatial Hawkes process. A spatial Hawkes process is a Poisson cluster process X ⊂ Rd where d ≥ 1 is an integer. The clusters centers are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈ (0,∞). Each immigrant Xi ∈ I generates a cluster Ci = CXi which is formed by the points of generations n = 0, 1, . . . with the following branching structure: the immigrant Xi ∈ I is said to be of zero- th generation. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of generation n generates a Poisson process on Rd of offspring of generation n + 1 with intensity function h(· − Y ). Here h : Rd → [0,∞) is a non-negative Borel function. In the model it is assumed that, given the immigrants, the centered clusters {Ci −Xi} are iid, and independent of I. By definition the spatial Hawkes process is X ≡ i Ci. As in the one-dimensional case, it is assumed 0 < µ ≡ h(ξ) dξ < 1. (50) This assumption guarantees that the number of points in a cluster has a finite mean equal to 1/(1−µ), excludes the trivial case where there are no offspring, and ensures that X is ergodic, with a finite and positive intensity given by ν/(1−µ). Due to the branching structure, the number S of offspring in a cluster follows the distribution (3). Finally, we note that the classical Hawkes process considered in the previous sections corresponds to the special case where d = 1 and h(t) = 0 for t ≤ 0. A LDP for spatial Hawkes processes can be obtained by Theorem 5.1. The precise statement is as Theorem 5.1 with (50) and ‖ξ‖h(ξ) dξ < ∞, in place of (5) and (46), moreover the rate function is Λ∗(·) defined by (20). Here the symbol ‖ · ‖ denotes the Euclidean norm. Similarly, the asymptotic behavior of the void probability function and the empty space function of spatial Hawkes processes can be obtained as immediate consequences of Proposition 5.3 and Corollary 5.4, respectively. The precise statements are as Proposition 5.3 and Corollary 5.4, with conditions (50) and ‖ξ‖dh(ξ) dξ < ∞ in place of E[Ld] < ∞. 6 Extensions and open problems In this paper we studied large deviations of Poisson cluster processes. Applications of these results to insurance and queueing models are presently under investigation by the authors. The definition of Hawkes process extends immediately to the case of random fertility rate h(·, Z), where Zk’s are iid unpredictable marks associated to the points Xk (see Daley and Vere- Jones (2003) for the definition of unpredictable marks, and Brémaud, Massoulié and Ridolfi (2005) for the construction of Hawkes processes with random fertility rate specified by an unpredictable mark). Due to the form of the distribution of S in this case (see formula (6) in Møller and Rasmussen (2005)) it is not clear if the LDPs for Hawkes processes proved in this paper are still valid for Hawkes processes with random fertility rate. The generalization of our results to non-linear Hawkes processes (Kerstan (1964); Brémaud and Massoulié (1996); Massoulié (1998); Brémaud, Nappo and Torrisi (2002); Torrisi (2002)) would be interesting. However, since a non-linear Hawkes process is not even a Poisson cluster process, a different approach is needed. Acknowledgements We thank Kamil Szczegot for reporting two mistakes and for a careful reading of a first draft of the paper. References Borovkov, A.A. (1967), Boundary values problems for random walks and large deviations for func- tion spaces, Theory Probab. Appl. 12, 575–595. Brémaud, P. (1981), Point Processes and Queues, Springer, New York. Brémaud, P. and Massoulié, L. (1996), Stability of nonlinear Hawkes processes, Ann. Prob. 24, 1563–1588. Brémaud, P., Massoulié, L. and Ridolfi, A. (2005), Power spectra of random spike fields and related processes, Adv. Appl. Prob. 37, 1116–1146. Brémaud, P., Nappo, G. and Torrisi, G.L. (2002), Rate of convergence to equilibrium of marked Hawkes processes, J. Appl. Prob. 39, 123–136. Brix, A. and Chadoeuf, J. (2002), Spatio-temporal modeling of weeds by shot-noise G Cox processes, Biometrical J. 44, 83–99. Chavez-Demoulin, V., Davison, A.C. and Mc Neil, A.J. (2005), Estimating value-at-risk: a point process approach, Quantitative Finance 5, 227–234. Daley, D.J. and Vere-Jones, D. (2003), An Introduction to the Theory of Point Processes (2nd edition), Springer, New York. de Acosta, A. (1994), Large deviations for vector valued Lévy processes, Stochastic Process. Appl. 51, 75–115. Dembo, A. and Zeitouni, O. (1998), Large Deviations Techniques and Applications (2nd edition), Springer, New York. Ganesh, A., Macci, C. and Torrisi, G.L. (2005), Sample path large deviations principles for Poisson shot noise processes, and applications, Electron. J. Probab. 10, 1026–1043. Gusto, G. and Schbath, S. (2005), F.A.D.O.: a statistical method to detect favored or avoided distances between occurrences of motifs using the Hawkes model, submitted. Hawkes, A.G. (1971a), Spectra of some self-exciting and mutually exciting point processes, Biometrika 58, 83–90. Hawkes, A.G. (1971b), Point spectra of some mutually exciting point processes, J. Roy. Statist. Soc. Ser. B 33, 438–443. Hawkes, A.G. and Oakes, D. (1974), A cluster representation of a self-exciting process, J. Appl. Prob. 11, 493–503. Jagers, P. (1975), Branching Processes with Biological Applications, John Wiley, London. Jonnson, D.H. (1996), Point Process Models of Single-Neuron Discharges, J. Computational Neu- roscience 3, 275–299. Kerstan, J. (1964), Teilprozesse Poissonscher Prozesse. Transactions of the Third Prague Confer- ence on Information Theory, Statistical Decision Functions, Random Processes, 377–403. Massoulié, L. (1998), Stability results for a general class of interacting point processes dynamics, and applications, Stoch. Proc. Appl. 75, 1–30. Møller, J. (2003), A comparison of spatial point process models in epidemiological applications. In Highly Structured Stochastic systems, eds P.J. Green, N.L. Hjort and S. Richardson, Oxford Uni- versity Press, 264–268. Møller, J. and Waagepetersen, R.S. (2004), Statistical Inference and Simulation for Spatial Point Processes, Chapman and Hall, Boca Raton. Møller, J. and Rasmussen, J.G. (2005), Perfect simulation of Hawkes processes, Adv. Appl. Prob. 37, 629–646. Møller, J. and Torrisi, G.L. (2007), The pair correlation function of spatial Hawkes processes, Statistics and Probability Letters, to appear Neyman, J. and Scott, E.L. (1958), Statistical approach to problems of cosmology, J.R. Statist. Soc. B 20, 1–43. Ogata, Y. and Akaike, H. (1982), On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes, J. R. Statist. Soc. B 44, 102–107. Ogata, Y. (1988), Statistical models for earthquake occurrences and residual analysis for point processes, J. Amer. Statist. Assoc. 83, 9–27. Ogata, Y. (1998), Space-time point process model for earthquake occurrences, Ann. Inst. Statist. Math. 50, 379–402. Reynaud-Bouret, P. and Roy, E. (2007), Some non asymptotic tail estimates for Hawkes processes, Bull. Belg. Math. Soc. Simon Stevin 13, 883–896. Torrisi, G.L. (2002), A class of interacting marked point processes: rate of convergence to equilib- rium, J. Appl. Prob. 39, 137–161. Vere-Jones, D. and Ozaki, T. (1982), Some Examples of Statistical Estimation Applied to Earth- quake Data, Ann. Inst. Statist. Math. 34, 189–207. Introduction Preliminaries Poisson cluster processes Hawkes processes Large deviation principles Scalar large deviations Scalar large deviations of Poisson cluster processes Scalar large deviations of Hawkes processes Sample path large deviations Sample path large deviations in the topology of point-wise convergence Sample path large deviations in the topology of uniform convergence Large deviations of spatial Poisson cluster processes The large deviations principle The asymptotic behavior of the void probability function and the empty space function Spatial Hawkes processes Extensions and open problems

0704.1464

Distributed quantum information processing with minimal local resources

Distributed quantum information processing with minimal local resources Earl T. Campbell∗ Department of Materials, Oxford University, Oxford, UK (Dated: Feb, 2006) We present a protocol for growing graph states, the resource for one-way quantum computing, when the available entanglement mechanism is highly imperfect. The distillation protocol is frugal in its use of ancilla qubits, requiring only a single ancilla qubit when the noise is dominated by one Pauli error, and two for a general noise model. The protocol works with such scarce local resources by never post-selecting on the measurement outcomes of purification rounds. We find that such a strategy causes fidelity to follow a biased random walk, and that a target fidelity is likely to be reached more rapidly than for a comparable post-selecting protocol. An analysis is presented of how imperfect local operations limit the attainable fidelity. For example, a single Pauli error rate of 20% can be distilled down to ∼ 10 times the imperfection in local operations. PACS numbers: 03.67.Mn, 03.67.Lx, 03.67.Pp The paradigm of distributed quantum computing (QC) in- volves a number of simple, optically active structures, each capable of representing at least one qubit. Relevant exam- ples include trapped atoms [1, 2, 3], and elementary nanos- tructures such as NV centres within diamond [4, 5, 6]. En- tanglement between structures is to be accomplished through an optical channel, for example by measuring photons af- ter a beam splitter has erased their ‘which path’ information [1, 7, 8, 9, 10, 11, 12], as in figure (1a). Remarkably, recent experimental results [3] demonstrated such an optical channel between ions in separate traps. However, results to-date show that the ‘raw’ entanglement generated in this way is liable to have significant noise, well above fault tolerance thresholds [13, 14]. Thus it is important to ask, can we exploit the mod- est complexity within each local structure in order to distill the entanglement to a higher fidelity? Originally entanglement distillation was intended for se- cure quantum communication [15, 16, 17, 18], but the same protocols naturally carry over to distributed QC [19, 20]. First a noisy entanglement operation produces many noisy Bell pairs between two locations, which these protocols then con- vert into fewer high-fidelity Bell pairs. At each local site there must be a certain number of qubits available, one logical qubit that is directly involved in the computation, and some num- ber of ancilla qubits. Computation is performed by distilling a high-fidelity Bell pair between two ancilla qubits, and then us- ing it to implement a gate between two logical qubits. In addi- tion to allowing purification, ancilla qubits protect the logical qubits against damage from probabilistic gates [1, 4, 19, 20]. Since these protocols emphasize implementing a good fidelity gate, we refer to them as gate-based protocols. For significant purification of noise from a depolarizing source, these propos- als require 3 or 4 ancillary qubits [19, 20]; whereas for phase noise, the number of ancillas can be reduced by one [20]. Another family of distillation protocols emerged after the one-way model of quantum computing showed that all the en- tanglement necessary for computation is present in a class of states called graph, or cluster, states [21, 22, 23, 24]. The dis- ∗Electronic address: [email protected] FIG. 1: An outline of a suitable architecture for a distributed quantum computer. (a) The numbers label local sites, each housing a matter system within a optical cavity. The optical cavities emit photons into an input port of a multiplexer, which can route any input port to any output port. Beam-splitters erase which-path information, so that entanglement is generated conditional on the detector signatures. (a: inset) for our primary protocol, each matter system is assumed to have enough level structure to provide two good qubits, a ancilla and a logical qubit. (b) two example level structures that would be suitable for the ancilla qubit. The L-level has only one logical state that optically couples to an excited state. The Λ-level structure has both levels coupling to a common excited state, however the two transitions are distinguishable by either frequency or polarization. tillation of graph states is akin to error correction, as it con- sists in repeated measurement of the stabilizers that describe the graph state. In virtue of this feature, we refer to these as stabilizer-based protocols. The first such protocol uses noisy copies of a graph state and post-selects upon detection of a single error [25, 26, 27]. Further proposals cast aside the need for post-selection at the cost of a stricter error threshold [28], above which, distillation is possible. These proposals use a combination of noisy copies of the graph state and highly pu- rified GHZ states. Because of the size of the entangled states in the ancilla space, iterating these distillation protocols may take longer than for gate-based protocols. A significant tem- poral overhead will occur when the entangling operation has a high failure rate. Building large entangled states in the an- http://arxiv.org/abs/0704.1464v3 mailto:[email protected] cilla space also restricts the class of employable entangling operations, excluding entangling protocols that only produce Bell pairs [1, 9]. Of course, provided we have enough local qubits to provide ancillas for our ancillas, these disadvantages are easily nullified. However, many systems which are poten- tially well suited for distributed QC may be very limited in the number of qubits they can embody. By blending ideas from the gate-based and stabilizer proto- cols, this paper proposes an entanglement distillation protocol which performs rapidly whilst requiring fewer ancillas than previous protocols. The bulk of this paper shows that one an- cilla is sufficient to distil errors from dephasing noise. We then extend the protocol to cover depolarizing noise; as with other schemes this requires an additional ancilla, which we use to reduce the depolarizing noise to a dephasing noise. Like gate-based protocols, we build up a graph state edge-by-edge, with ancillas never building entangled states larger than Bell pairs. However, as with stabilizer-based protocols, our pro- posal repeatedly makes stabilizer measurements directly onto the qubits constituting the graph state. Ancillas must typically be optically active, such as in an L or Λ level configuration (see Fig. 1b). Qubits are laBelled Ax and Lx for ancilla and logical qubit, respectively, at local site x. First our analysis will focus on the case when the noisy entanglement channel is dominated by one type of Pauli error, which may be very severe. Without loss of generality, we describe the channel as being affected by phase noise, such that two ancillas A1 and A2, can be put in the mixed state: ρA1,A2 = (1− ε)|Ψ+〉〈Ψ+|+ εZA|Ψ+〉〈Ψ+|ZA, (1) where ZA is the Pauli phase-flip operator acting on either A1 or A2, and |Ψ+〉 = |0〉A1|1〉A2 + |1〉A1|0〉A2. If the dominant noise is a different Pauli error, or different Bell pairs are pro- duced, then local rotations can always bring the state into the form of equation 1. Furthermore, only a single Z error is pos- sible as this Bell state is invariant under the bilateral ZA1ZA2 rotation. Scenarios where such a noise model may arise in- clude parity based entangling operations [9, 10] that possess a degree of robustness against bit-flip errors. After producing noisy entanglement between two ancillas, the entanglement is pumped down to the logical qubits, result- ing in a quantum operation on the logical qubits. The target (perfect) entangling operation we aim to eventually achieve is either of the parity projections: P− = 2(|01〉〈01|+ |10〉〈10|), (2) P+ = 2(|00〉〈00|+ |11〉〈11|), which act on the logical qubits L1 and L2, and have an ad- ditional normalization factor of 2 that simplifies later expres- sions. The only assumption we make about the initial state of the logical qubits is that they are part of a graph state (in the constructive definition), such that they have equal magni- tude in both parity subspaces, 〈G |P+|G 〉 = 〈G |P−|G 〉; where |G 〉 denotes the graph state of all the logical qubits. Both P− and P+ allow arbitary graph growth, and which projection we eventually obtain is unimportant as P±|G 〉 differ only by lo- cal rotations [30]. We will see that entanglement distillation results from repetition of an entanglement transfer procedure. FIG. 2: The sequence of operations required to pump entanglement down to two logical qubits L1 and L2, shown in the graph state no- tation. (a) Ancillas are prepared in the (|0〉+ |1〉)/ 2 state, and the logical qubits are part of some larger graph state |G 〉; (b) An entan- gling operation is performed between the ancillas, with the possibil- ity of Z noise; (c) A H ·X is applied to ancilla A1; (d) At both local sites control-Z operations are performed between ancilla and logical qubit; (e) Ancilla A1 is measured in the Y basis; (f ) Ancilla A2 is measured in the X-basis. The possibility of a Z error is tracked by using ZE , where E = 1 tracks an error, and E = 0 tracks the er- rorless state. The measurement outcome is represented by M , where M = +1 for a |0〉 measurement and M = −1 otherwise. The dotted line between the logical qubits represents a projection operator P± between the logical qubits, where the sign is equal to M(−1)E . Each round of our protocol is an entanglement pumping procedure, described graphically in Fig. (2). Every round of purification begins with performing a noisy entangling opera- tion between two ancillas, A1 and A2. This operation may be probabilistic provided that success is heralded, in which case it is repeated until successful. Next, a series of local opera- tions must be performed. First, apply a bit-flip then Hadamard to one ancilla, say A1, and then two control–Z operations be- tween each ancilla and its logical qubit. The resulting state is a graph state, with the possibility of a Z error on A2. Then, measure A1 in the Y -basis (correcting for any by-product), giving the state described by Fig. 2e. Finally, A2 is measured in the X-basis. When no error was present, this measurement performs a parity check on the logical qubits, L1 and L2; that is, we measure the observable ZL1ZL2. On the first round of pumping, the odd and even parity outcomes will occur with 50/50 probability. Accounting for the possibility of a Z error causes a noisy parity measurement, or quantum operation: P∆ (|G 〉〈G |) = α∆P+|G 〉〈G |P+ + α −∆P−|G 〉〈G |P− α∆ + α−∆ where α = (ε−1 − 1) 2 , and ∆ = M1 is +1 for a |0〉 mea- surement outcome, and −1 for |1〉. If we repeat the entanglement pumping procedure n times, then we will get a series of measurements results M1,M2,. . .Mn. Concatenating the mapping, for each mea- surement result, we get back an operation of the same form but with ∆ = Mi. The core of our proposal is that we continue to purify the qubits until |∆| reaches some value FIG. 3: The evolution of ∆ against T , the number of rounds of entan- glement pumping. The state evolves as a biased random walk in ∆. The weighting of probabilities at a some point ∆ is such that there is a probability P|∆| for increasing the magnitude of ∆, where P|∆| is defined in equation 4. The red dashed lines represent the halting lines for ∆, and in this example ∆H = 3. Note that, the paths to the halting line, occur at T = ∆H + 2n, for non-negative integer n. ∆H at which point we halt the procedure. ∆H is chosen such that it corresponds to a target fidelity FT , where the fidelity is simply F (∆) = (1 + α−2|∆|)−1. In contrast to previous gate-based protocols, our protocol is not post-selective (NPS). Analogous post-selecting protocols (PS) using an equivalent entangling pumping procedure to eliminate phase errors have already been proposed [20]. The essential difference for PS is that it resets upon any measure- ment outcome, Mx, that differs from the first measurement outcome, M1; a reset consists in measuring out the qubits be- ing distilled, bringing them back to ∆ = 0. Benefits of NPS are two-fold: (i) since purification is never restarted it is safe to operate directly on the logical qubits, hence we eliminate the need for an additional ancilla that exists in PS protocols; (ii) the probability of success within T rounds is never less than for PS, indeed, we shall show that NPS significantly out- performs PS in this regard. A point in favour of PS is that, if can freely use multiple ancillas, then PS may achieve a higher asymptotic limit of fidelity (due to the effect of errors in local operations). However, we shall see that NPS can still attain fidelities within fault tolerance thresholds. Returning to our consideration of the evolution of ∆ in our protocol, it is clear that at each purification step, T , ∆ can either increase or decrease by 1. Hence, the evolution bears similarities to a random walk, illustrated by Fig.3. It differs from a random walk in two regards: (i) it halts when it reaches ∆ = ±|∆H |; (ii) the probabilities are the biased when ∆ 6= 0. The bias increases the chance of walking in the direction of larger |∆|, which occurs with probability: (1− ε)αD + εα−D αD + α−D , (4) where D = |∆|. On the face of it, it seems that the probability of walking to a state ∆ in T steps is dependent upon which FIG. 4: A comparison of the rapidity of our proposal versus a post- selection protocol, with a target fidelity FT = 1 − 10−4. The three plots represent different values for the error rate ε, and hence require a different value of ∆H ; shown in key. On each plot we show the probability of success against the number of rounds T , for both the protocol NPS (blue) and PS (orange). The yield for NPS and PS (Ynps and Yps) is given on each plot. Notice that for NPS, the prob- ability of success increases in steps. This is explained by figure 3, which shows that successful paths are separated by 2 time steps. path is taken. However, the probability of a kink in the path — D increasing and subsequently decreasing — is independent of D, and is k = PD(1 − PD+1) = ε(1 − ε). Hence, each path occurs with probability: Ppath(D,T ) = ). (5) The total probability of walking to (D,T ) is the product of Ppath(D,T ) with the number of paths to that position. We have calculated the total probability of success, after T rounds, by summing over all the different ways of reaching the halting line. For comparison, we performed the analogous calculation for an otherwise equivalent PS protocol. Figure 4 shows PS and NPS protocols for a target fidelity of FT = 1 − 10−4, with each plot being for a different error rate ε. Note that, for higher error rate or higher target fidelity, the width of the random walk is wider (larger∆H ). In this regime, the superiority of NPS increases, as more entanglement can be lost upon post-selection. Conversely, when ∆H = 2 the protocols are effectively identical, as stepping back will take the walk to the origin. A protocol’s yield is the expected ratio of distilled Bell pairs to used noisy Bell pairs. Since each time step uses a Bell pair, the yield of a protocol is 1/〈T 〉, with some values given in Fig. 4. Since NPS is a faster protocol than PS, it also has a superior yield. For this idealized error model we can asymptotically ap- proach unit fidelity. However, it is important to consider how other errors limit the maximum attainable fidelity. For sim- plicity, we take an aggressive error model where if a single error occurs once, then the overall entangling gate has fidelity zero. We use η to denote the probability per time step that an error occurs, where these errors can result from either faulty FIG. 5: A logarithmic plot of the expected infidelity, 1 − E(F ), when additional error sources affect our distillation protocol. The plot is a function of ε, the probability of a phase error occurring in the long-range entangling operation. Each curve is a different value of η, the probability per distillation round that some other error oc- curs. The blue cross marks an example discussed in the text, where the dephased Bell pairs are obtained by prior distillation on raw de- polarized Bell pairs. local operations or noise in the entanglement channel that is orthogonal to the distilled dominant noise. Furthermore, we approximate the chance of an error after T rounds by its up- per bound, ηT . Once we reach ∆H the fidelity will depend on the number of time steps taken. Therefore, we calculate the expectation of the fidelity, E(F ). Figure 5 shows how the expected infidelity, 1 − E(F ), varies with ε and η. Since the optimal choice of ∆H changes with ε this produces inverted humps along the curves, which are more pronounced for small ε. On all curves the behaviour is roughly the same; we can characterize the performance by noting that when the domi- nant error rate is 0.2 (i.e. 20%), and the probability of other error sources is η, then the protocol brings all error probabili- ties to order 10η. Given that relevant fault tolerance strategies can handle noise of order 1% [13, 14], the single-ancilla dis- tillation may suffice when η is of order 0.1%. If the orthogonal errors are too large, then an additional an- cilla is required. Using the two ancillas, bit-errors are first distilled away by a post-selective protocol, such as in [20]. Orthogonal errors are at their most extreme when the noise is depolarizing, producing Werner states of fidelity F0. As an example, we consider the distillation of Werner states of F0 = 0.85; a rigorous analysis is provided in appendix A. Five rounds of distillation reduces the orthogonal errors to ∼ 10−5, after which the phase noise has accumulated to ε = 0.22; we have not gained fidelity, but we have mapped all noise into phase noise. These dephased Bell pairs are used in our primary distillation protocol, and are distilled to E(F ) ∼ 10−4. This result is marked on Fig. 5 , where η is taken to equal the remaining non-phase errors. As a final re- mark, note that if we create GHZ states among ancillas (rather than Bell-pairs) then our strategy can be combined with the band-aid protocol of [28], increasing tolerance of imperfect local operations. In conclusion, using fewer ancillas than previous pro- posals, an otherwise intolerably-large error is rapidly re- duced below error-correction thresholds [13, 14]. For de- phasing/depolarizing noise models, the protocol needs only one/two ancillas, respectively. The author thanks Simon Ben- jamin, Joseph Fitzsimons, Pieter Kok, and Dan Browne for useful discussions. This research is part of the QIP IRC (GR/S82176/01). [1] L. M. Duan et al., Quantum Information and Computation 4, 165 (2004). [2] D. K. L. Oi, S. J. Devitt, and L. C. L. Hollenberg, Physical Review A 74, 052313 (2006). [3] D. L. Moerhing et al., Nature 449, 68 (2007). [4] S. C. Benjamin et al., New Journal of Physics 8, 141 (2006). [5] J. Wrachtrup, S. Y. Kilin, and A. P. Nizovtsev, Optics and Spec- troscopy 91, 429 (2001). [6] T. Gaebel et al., Nature Physics 2, 408 (2006). [7] C. Cabrillo et al., Phys. Rev. A 59, 1025 (1999). [8] S. Bose et al., Phys. Rev. Lett 83, 5158 (1999). [9] L. M. Duan and H. J. Kimble, Physical Review Letters 90, 253601 (2003). [10] S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310(R) (2005). [11] S. C. Benjamin, Phys. Rev. A 72, 056302 (2005). [12] Y. L. Lim, A. Beige, and L. C. Kwek, Phys. Rev. Lett 95, 030505 (2005). [13] R. Raussendorf, J. Harrington, and K. Goyal, New. Jour. Phys, 9, 199, (2007). [14] C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Physical Review Letters 96, 020501 (2006). [15] C. H. Bennett et al., Phys. Rev. A 54 (1996). [16] C. H. Bennett et al., Phys. Rev. Lett 76 (1996). [17] W. Dür et al., Phys. Rev. A 59, 169 (1999). [18] M. Murao et al., Phys. Rev. A 57, R4075 (1998). [19] W. Dür and H. J. Briegel, Phys. Rev. Lett. 90, 067901 (2003). [20] L. Jiang et al., (2007), quant-ph/0703029. [21] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003). [22] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [23] M. Hein et al., (2005), quant-ph/0602096. [24] M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A 69, 062311 (2004). [25] W. Dür, H. Aschauer, and H. J. Briegel, Phys. Rev. Lett. 91, 107903 (2003). [26] H. Aschauer, W. Dür, and H. J. Briegel, Physical Review A 71, 012319 (2005). [27] C. Kruszynska et al., Physical Review A 74, 052316 (2006). [28] K. Goyal, A. McCauley, and R. Raussendorf, Physical Review A 74, 032318 (2006). [29] D. E. Browne and T. Rudolph, Phys. Rev. Lett 95, 010501 (2005). [30] P+|G 〉 = XL1( j∈N(L1) Zj)P−|G 〉, where N(L1) is the set of neighbours of L1. APPENDIX A: CONVERTING WERNER STATES INTO DEPHASED BELL STATES. In the main article we propose a protocol that uses a sin- gle extra ancilla to distil a graph state in an edge-by-edge fashion, provided an entanglement channel that generates de- phased Bell states. However, if the entanglement channel suf- fers depolarizing noise then this generates a rank-4 Bell diag- onal mixed state, known as a Werner state. Here we show that with an additional ancilla a well-known post-selective proto- col can be used to convert these Werner states into dephased Bell states. As an example, we calculate the accumulated phase noise when the entanglement channel produces Werner states with a fidelity of F0 = 0.85, and non-phase errors must be reduced to order 10−5. Given an entanglement channel that generates Werner states of the form: ρ = F0|Φ +〉〈Φ+|+ 1− F0 Z|Φ+〉〈Φ+|Z (A1) +X |Φ+〉〈Φ+|X + Y |Φ+〉〈Φ+|Y where |Φ+〉 = |00〉 + |11〉. The X and Y noise can be dis- tilled by performing repeated noisy measurements of the ZZ observable. This is implemented by first performing bilateral control-phase rotations, with both controls from one EPR pair, and both targets on another EPR pair. The control qubits are measured in the X-basis, and we post-select on the even parity measurement outcome. After n successful rounds the target qubits are in some Bell diagonal state: ρn = an|Φ +〉〈Φ+|+ bnX |Φ +〉〈Φ+|X (A2) +cnZ|Φ +〉〈Φ+|X + dnY |Φ +〉〈Φ+|Y After another successful round of distillation, the state is transformed such that: ρn+1 ∝ F0P+ρnP+ + 1− F0 P−ρnP− (A3) +ZP+ρnP+Z + ZP−ρnP−Z Where each term comes from considering a different Bell state contribution to the mixed state of the control qubits. Contribu- tions with Z or Y noise generate projections into the opposite parity space, as these errors anti-commute with one of the X- basis measurements. Contributions with X or Y noise will result in a phase error on a target qubit. These errors propa- gate down to the target qubits because X and Y rotations do not commute with control phase gates, rather these rotations change the gate’s control from the |1〉 state to |0〉. Before nor- malization, the density matrix coefficients obey the recursive relations: an+1 = F0an + 1− F0 cn (A4) cn+1 = an + Fcn, (A5) bn+1 = dn+1 = 1− F0 (bn + dn) (A6) Fixing the n = 1 coefficients to those of our undistilled Werner state, we can derive: an + cn = 1 + 2F0 bn = dn = (1− F0) Hence, after n rounds of post-selective distillation, the re- maining X and Y noise has a magnitude: NXY (F0, n) = bn + dn an + bn + cn + dn 1 + 2F0 2− 2F0 )n)−1 To give a numerical example, consider a source of Werner states of fidelity 0.85, and we wish to reduce NXY to order 10−5. It is easy to calculate that five rounds of distillation is sufficient since NXY (0.85, 5) = 1.69 · 10 Finally, we need to calculate how the undistilled phase noise has changed through these five rounds of distillation. From the recursive relations, an unnormalized form of cn can be derived: 1 + 2F 4F − 1 (A10) After normalization, we have the remaining phase noise: NZ(F0, n) = an + bn + cn + dn , (A11) NZ(0.85, 5) = 0.22 (A12) It is an interesting feature of distillation that although this state has a lower fidelity than the Werner state, it is a more useful resource for the subsequent level of distillation. Hence, fi- delity alone is a poor indicator of the distillable entanglement of a mixed state.

0704.1465

X-Raying the MOJAVE Sample of Compact Extragalactic Radio Jets

X-Raying the MOJAVE Sample of Compact Extragalactic Radio Jets M. Kadler∗,†, G. Sato∗, J. Tueller∗, R. M. Sambruna∗, C. B. Markwardt∗, P. Giommi∗∗ and N. Gehrels∗ ∗Astrophysics Science Division, NASA’s Goddard Space Flight Center, Greenbelt, MD 20771, USA †NASA Postdoctoral Research Associate ∗∗ASI Science Data Center, ESRIN, I-00044 Frascati, Italy Abstract. The MOJAVE sample is the first large radio-selected, VLBI-monitored AGN sample for which complete X-ray spectral information is being gathered. We report on the status of Swift survey observations which complement the available archival X-ray data at 0.3-10 keV and in the UV with its XRT and UVOT instruments. Many of these 133 radio-brightest AGN in the northern sky are now being observed for the first time at these energies. These and complementary other multi- wavelength observations provide a large statistical sample of radio-selected AGN whose spectral energy distributions are measured from radio to gamma-ray wavelengths, available at the beginning of GLAST operations in 2008. Here, we report the X-ray spectral characteristics of 36 of these previously unobserved MOJAVE sources. In addition, the number of MOJAVE sources detected by the BAT instrument in the hard X-ray band is growing: we report the detection of five new blazars with Keywords: galaxies: active — galaxies: jets — galaxies: individual (NRAO 140, PKS B 1222+216, PKS B 1510−089, 3C 345, PKS B 2126−158) PACS: 95.85.Bh, 95.85.Nv, 95.85.Pw, 98.54.Cm X-RAY OBSERVATIONS OF MOJAVE AND RELEVANCE FOR GLAST Radio-loud core-dominated (RLCDs) active galactic nuclei (AGN) are an important class of extragalactic supermassive black-hole systems whose bright compact radio cores can be imaged at milliarcsecond resolution with Very-Long- Baseline Interferometry (VLBI). The vast majority of all RLCDs are blazars, BL Lac objects and flat-spectrum radio quasars, but also some Seyfert and broad-line radio galaxies exhibit bright compact radio cores with powerful relativistic jets. Since 1994, the VLBA 2 cm Survey [1] and its follow-up MOJAVE ([4], and Lister 2007, these proceedings) have been monitoring the structural changes in the jets of more than 200 AGN with bright compact radio cores. At present, the MOJAVE monitoring regularly observes 192 objects, including a flux-density limited, radio-selected, statistically complete sub sample of the 133 radio-brightest compact extragalactic jets in the northern sky: the MOJAVE sample1. Compact radio jets of blazars, typically show ejections of relativistic plasma every few months to years related to the formation of new jet component that travel at superluminal speeds of up to ∼ 30 c down the jet. On the other hand, blazars are known to be bright and rapidly variable gamma-ray sources. The connection of jet-formation events to the broadband-SED variability of blazars is an important aspect of both the MOJAVE and GLAST project and calls for multi-wavelength coordination with other large surveys at intermediate observing wavelengths. In order to be able to react swiftly to gamma-ray flares detected by GLAST, to plan and coordinate multiwavelength observations, it is important to gather broadband SED information about the objects of interest in advance to the arrival of first GLAST data. In our case of a radio-selected AGN sample that is continuously monitored with VLBI and single-dish telescopes (see Fuhrmann et al., these proceedings), Swift with its optical/UV (UVOT) and X-ray (XRT) capabilities, is ideally suited to provide these SED data to connect the radio and gamma-ray bands. 1 See the program website for a detailed description of the selection criteria and a regularly updated list of the full sample: http://www.physics.purdue.edu/astro/MOJAVE/ http://arxiv.org/abs/0704.1465v1 http://www.physics.purdue.edu/astro/MOJAVE/ Swift X-Ray Survey of the MOJAVE Sample An archival X-ray spectral survey of all publicly available data for MOJAVE sources has been conducted by [2]. This so-called 2cm-X-Sample currently contains 50 out of 133 MOJAVE sources. 17 additional sources have been observed by Swift before the beginning of our project. This still leaves us with 66 of the radio brightest blazars in the northern sky having never been observed with an X-ray spectroscopic mission above 2 keV. We have started a program to observe all the so-far unobserved MOJAVE sources with the Swift XRT as part of the Swift blazar key project. This will produce the first statistically complete large radio-selected sample of blazars and other RLCDs. The observations will complete the data base of the 2 cm-X-Sample of photon indices, source-intrinsic absorbing column densities and X-ray luminosities. It will allow for the first time statistically robust radio/X-ray correlation analyses of these quantities with relativistic jet parameters from VLBI observations. In addition, because of the ongoing MOJAVE monitoring observations, the Swift data will naturally yield to an unprecedented set of quasi- simultaneous broadband SED data from the radio, optical/UV and X-ray regime. New X-ray spectra for 36 MOJAVE sources:. Our program started in November 2006. At the time of writing (March 2007), 94 objects have been observed. We concentrate here on completed observations (i.e., >∼ 10 ksec) of sources from the statistically complete MOJAVE sample of the 133 radio-brightest, compact AGN in the northern sky. The XRT data were acquired mainly in Photon Counting (PC) mode. We used the program XSELECT to extract source and background counts from the cleaned event lists processed at Swift Science Data Center (SDC). The source spectrum is calculated from a circular region with the radius of 47 arcsec, while the background region is selected as an annulus of the outer radius of 150 arcsec and the inner radius of 70 arcsec. A pile-up correction is applied when the count rate exceeds 0.6 counts/s by excluding the central area of 7 arcsec radius. RLCDs are found to have comparably simple X-ray spectra, typically well-approximated by an absorbed power law. Therefore, we performed a spectral fit to the time-averaged spectra with an absorbed power-law model. As of March 2007, we have completed the observations of 36 out of 66 previously unobserved MOJAVE sources. In Table 1, we report for the first time the basic X-ray spectral characteristics of these objects in the (0.3− 10)keV band. Errors quoted are at the 90% confidence level. All 36 objects have been detected with the XRT in 10 ksec exposures. In some cases of either very low X-ray flux or peculiar spectral or temporal source behavior, we have obtained follow-up observations to improve the photon statistics or to trace source variability over a longer time range. We find an average value of Γave = 1.8, with a standard deviation of 0.2 similar to values of ∼ 1.6–1.7, which are typically found in the old 2 cm-X-Sample ([2] and in prep.). The absorbing column densities determined have been compared to the expected Galactic absorption values from the LAB survey [3]. Significant excess absorption was found in 9 sources (< 1021 cm−2 in all cases; compare Table 1). New hard X-ray detections of five blazars with BAT:. The Burst Alert Telescope (BAT) is the hard X-ray instru- ment on-board Swift and is a continuously operating all-sky hard X-ray monitor and survey instrument [5]. Prior to the beginning of our program, 17 blazars were detected by BAT (Tueller et al., in prep.). These sources were found via “blind search” of the whole sky, using a sliding-cell detection method (BATCELLDETECT), requiring at least ∼ 5σ sig- nificance of each individual source. We have conducted a search of the BAT data base, based on the first nine months of observations, aiming for those MOJAVE sources for which our Swift XRT fill-in observations did yield an extrapolated (14− 195)keV flux above or close to ∼ 10−11 erg s−1 cm−2. This criterion was met for 19 objects which we assign a ranking number, according to the value of the extrapolated flux (rank 1 for the highest to rank 19 for the lowest extrap- olated flux). We found significant excess flux at the positions of five blazars: NRAO 140 (Rank 1, Fextrapol. = 4.73× 10−11 erg s−1 cm−2, 5.1σ ), PKS B 1510−089 (Rank 2, Fextrapol.= 3.68×10−11 erg s−1 cm−2, 4.0σ ), PKS B 2126−158 (Rank 3, Fextrapol. = 3.40× 10 −11 erg s−1 cm−2, 3.4σ ), 3C 345 (Rank 6, Fextrapol. = 2.61× 10−11 erg s−1 cm−2, 3.7σ ), and PKS B 1222+216 (Rank 19, Fextrapol. = 9.36 × 10 −12 erg s−1 cm−2, 3.8σ ). A deeper analysis of the hard X- ray properties of these blazars is currently being performed. Tentative signals (. 3σ ) are found at the position of the 5th-ranked source PKS B 1127−145, the 10th-ranked source PKS B 0403−132, and the 16th-ranked source PKS B 2008−159. The rank distribution demonstrates that our method preferentially picks up those blazars that are just below the BAT detection threshold of a few times 10−11 erg s−1 cm−2. The fact that a significant signal is registered at the position of the relatively faint source PKS B 1222+216 but not from comparably bright objects like the 4th-ranked source PKS B 0723−08 suggests that X-ray variability and spectral breaks above 10 keV may play an important role. We expect to detect more MOJAVE sources with BAT in the future, in particular with increased sensitivity in later stages of the BAT mission. TABLE 1. X-ray spectral fitting parameter of MOJAVE sources Source IAU B 1950 Exposure [ksec] Photon Index [1022 cm−2] NH,Gal [1022 cm−2] χ2red/dof F(0.3−10)keV [erg s−1 cm−2] 0016+731 8950 1.75+0.23 −0.21 0.25 +0.08 −0.07 0.19 0.59/22 3.30×10 0202+149 8983 1.92+1.43 −1.02 < 0.65 0.05 0.74/3 3.04×10 0224+671 4C +67.05 12112 1.89+0.39 −0.34 0.58 +0.25 −0.19 0.40 0.44/9 1.23×10 0336−019 CTA 26 10653 1.94+0.31 −0.27 0.08 +0.07 −0.06 0.06 1.27/11 1.17×10 0403−132 9503 1.56+0.14 −0.13 0.06 +0.04 −0.03 0.04 0.78/37 5.02×10 0529+075 19537 1.86+0.19 −0.18 0.30 +0.19 −0.18 0.17 1.36/31 1.92×10 0529+483 9124 1.28+0.79 −0.63 < 0.37 0.25 0.87/2 9.21×10 1124−186 9515 1.99+0.45 −0.36 0.07 +0.08 −0.06 0.04 1.26/8 9.71×10 1150+812 11244 1.74+0.45 −0.39 < 0.07 0.05 1.24/6 8.48×10 1324+224 13330 1.78+0.19 −0.17 0.07 +0.05 −0.04 0.02 0.86/24 2.01×10 1417+385 11730 2.11+0.52 −0.41 0.08 +0.10 −0.07 0.01 0.46/5 5.56×10 1504−167 12571 1.80+0.70 −0.54 < 0.22 0.07 0.64/7 3.34×10 1538+149 4C +14.60 9527 2.28+0.49 −0.40 0.12 +0.09 −0.07 0.03 0.60/11 1.17×10 1546+027 9657 1.76+0.17 −0.16 0.07 +0.04 −0.04 0.07 1.03/31 3.45×10 1606+106 4C +10.45 49342 1.43+0.10 −0.09 0.04 +0.02 −0.02 0.04 0.93/73 1.92×10 1611+343 DA 406 9789 1.77+0.34 −0.27 < 0.11 0.01 0.74/9 1.18×10 1637+574 10140 1.91+0.14 −0.13 0.05 +0.03 −0.03 0.01 0.90/40 3.95×10 1638+398 NRAO 512 12267 1.49+0.29 −0.23 < 0.04 0.01 1.17/5 7.58×10 1726+455 10722 2.04+0.36 −0.31 0.15 +0.08 −0.07 0.02 1.23/12 1.20×10 1730−130 NRAO 530 9912 1.46+0.32 −0.29 0.29 +0.14 −0.11 0.18 0.98/12 2.18×10 1739+522 4C +51.37 10304 1.55+0.23 −0.20 0.07 +0.06 −0.05 0.03 0.72/18 2.30×10 1741−038 9927 1.72+0.34 −0.31 0.41 +0.18 −0.14 0.18 0.72/12 2.01×10 1751+288 9772 1.77+0.45 −0.39 0.12 +0.11 −0.09 0.05 0.83/7 1.10×10 1758+388 9711 2.14+1.30 −0.90 < 0.38 0.03 0.70/4 2.84×10 1800+440 10098 1.75+0.17 −0.16 0.04 +0.03 −0.03 0.03 0.98/31 3.36×10 1849+670 9952 2.07+0.31 −0.28 0.17 +0.08 −0.07 0.05 1.65/16 1.71×10 1936−155 9929 2.25+0.60 −0.49 0.24 +0.17 −0.12 0.08 0.50/5 7.15×10 1958−179 8997 1.84+0.29 −0.27 0.16 +0.09 −0.07 0.07 0.90/14 1.95×10 2005+403 11986 1.69+0.29 −0.27 0.50 +0.19 −0.15 0.48 0.88/15 1.98×10 2008−159 9657 1.75+0.13 −0.12 0.14 +0.04 −0.03 0.06 1.08/48 5.71×10 2021+317 4C +31.56 28861 1.96+1.04 −0.65 1.21 +0.95 −0.76 0.52 1.98/4 3.24×10 2021+614 17562 0.87+1.26 −0.88 < 2.01 0.14 0.51/3 2.14×10 2037+511 3C 418 10189 1.70+0.45 −0.38 0.66 +0.37 −0.25 0.54 0.91/11 2.02×10 2136+141 OX 161 10140 1.43+0.28 −0.25 < 0.15 0.06 0.48/11 1.67×10 2201+171 9585 1.85+0.34 −0.30 0.10 +0.09 −0.07 0.05 0.89/7 9.80×10 2216−038 9650 1.82+0.33 −0.29 0.07 +0.08 −0.06 0.06 1.16/12 1.53×10 ACKNOWLEDGMENTS The authors wish to acknowledge the efforts and contributions of the Swift BAT and MOJAVE teams. REFERENCES 1. Kellermann, K. I., et al. 2004, ApJ, 609, 539 2. Kadler, M. 2005, PhD thesis, Universität Bonn, Germany 3. Kalberla, P. M. W., Burton, W. B., Hartmann, D., Arnal, E. M., Bajaja, E., Morras, R., Pöppel, W. G. L. 2005, A&A, 440, 775 4. Lister, M. L., & Homan, D. C. 2005, AJ, 130, 1389 5. Markwardt, C. B., Tueller, J., Skinner, G. K., Gehrels, N., Barthelmy, S. D., & Mushotzky, R. F. 2005, ApJ, 633, L77 X-Ray Observations of MOJAVE and Relevance for GLAST Swift X-Ray Survey of the MOJAVE Sample

0704.1466

Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator

arXiv:0704.1466v1 [math.ST] 11 Apr 2007 Sparse Estimators and the Oracle Property, or the Return of Hodges’ Estimator Hannes Leeb Department of Statistics, Yale University Benedikt M. Pötscher Department of Statistics, University of Vienna First version: November 2004 This version: March 2007 Abstract We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity. AMS 2000 Subject Classifications: Primary 62J07, 62C99; secondary 62E20, 62F10, 62F12 Key words and phrases: oracle property, sparsity, penalized maximum likelihood, penal- ized least squares, Hodges’ estimator, SCAD, Lasso, Bridge estimator, hard-thresholding, maximal risk, maximal absolute bias, non-uniform limits http://arxiv.org/abs/0704.1466v1 1 Introduction Recent years have seen an increased interest in penalized least squares and penalized maximum likelihood estimation. Examples are the class of Bridge estimators introduced by Frank and Friedman (1993), which includes Lasso-type estimators as a special case (Knight and Fu (2000)), or the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001) and further discussed in Fan and Li (2002, 2004), Fan and Peng (2004), and Cai, Fan, Li, and Zhou (2005). As shown in Fan and Li (2001), the SCAD estimator, with appropriate choice of the regularization (tuning) parameter, possesses a sparsity property, i.e., it estimates zero components of the true parameter vector exactly as zero with probability approaching one as sample size increases while still being consistent for the non-zero components. An immediate consequence of this sparsity property of the SCAD estimator is that the asymptotic distribution of this estimator remains the same whether or not the correct zero restrictions are imposed in the course of the SCAD estimation procedure. [This simple phenomenon is true more generally as pointed out, e.g., in Pötscher (1991, Lemma 1).] In other words, with appropriate choice of the regularization parameter, the asymptotic distribution of the SCAD estimator based on the overall model and that of the SCAD estimator derived from the most parsimonious correct model coincide. Fan and Li (2001) have dubbed this property the “oracle property” and have advertised this property of their estimator.1 For appropriate choices of the regularization parameter, the sparsity and the oracle property are also possessed by several – but not all – members of the class of Bridge estimators (Knight and Fu (2000), p. 1361, Zou (2006)). Similarly, suitably tuned thresholding procedures give rise to sparse estimators.2 Finally, we note that traditional post-model-selection estimators (e.g., maximum likelihood estimators following model selection) based on a consistent model selection procedure (for example, BIC or test procedures with suitably chosen critical values) are another class of estimators that exhibit the sparsity and oracle property; see Pötscher (1991) and Leeb and Pötscher (2005) for further discussion. In a recent paper, Bunea (2004) uses such procedures in a semiparametric framework and emphasizes the oracle property of the resulting estimator; see also Bunea and McKeague (2005). At first sight, the oracle property appears to be a desirable property of an estimator as it seems to guarantee that, without knowing which components of the true parameter are zero, we can do (asymptotically) as well as if we knew the correct zero restrictions; that is, we can “adapt” to the unknown zero restrictions without paying a price. This is too good to be true, and 1The oracle property in the sense of Fan and Li should not be confused with the notion of an oracle inequality as frequently used elsewhere in the literature. 2These estimators do not satisfy the oracle property in case of non-orthogonal design. it is reminiscent of the “superefficiency” property of the Hodges’ estimator; and justly so, since Hodges’ estimator in its simplest form is a hard-thresholding estimator exhibiting the sparsity and oracle property. [Recall that in its simplest form Hodges’ estimator for the mean of an N(µ, 1)-distribution is given by the arithmetic mean ȳ of the random sample of size n if |ȳ| exceeds the threshold n−1/4, and is given by zero otherwise.] Now, as is well-known, e.g., from Hodges’ example, the oracle property is an asymptotic feature that holds only pointwise in the parameter space and gives a misleading picture of the actual finite-sample performance of the estimator. In fact, the finite sample properties of an estimator enjoying the oracle property are often markedly different from what the pointwise asymptotic theory predicts; e.g., the finite sample distribution can be bimodal regardless of sample size, although the pointwise asymptotic distribution is normal. This is again well-known for Hodges’ estimator. For a more general class of post-model-selection estimators possessing the sparsity and the oracle property this is discussed in detail in Leeb and Pötscher (2005), where it is, e.g., also shown that the finite sample distribution can “escape to infinity” along appropriate local alternatives although the pointwise asymptotic distribution is perfectly normal.3 See also Knight and Fu (2000, Section 3) for related results for Bridge estimators. Furthermore, estimators possessing the oracle property are certainly not exempt from the Hajek-LeCam local asymptotic minimax theorem, further eroding support for the belief that these estimators are as good as the “oracle” itself (i.e., the infeasible “estimator” that uses the information which components of the parameter are zero). The above discussion shows that the reasoning underlying the oracle property is misguided. Even worse, estimators possessing the sparsity property (which often entails the oracle property) necessarily have dismal finite sample performance: It is well-known for Hodges’ estimator that the maximal (scaled) mean squared error grows without bound as sample size increases (e.g., Lehmann and Casella (1998), p.442), whereas the standard maximum likelihood estimator has constant finite quadratic risk. In this note we show that a similar unbounded risk result is in fact true for any estimator possessing the sparsity property. This means that there is a substantial price to be paid for sparsity even though the oracle property (misleadingly) seems to suggest otherwise. As discussed in more detail below, the bad risk behavior is a “local” phenomenon and furthermore occurs at points in the parameter space that are “sparse” in the sense that some of their coordinates are equal to zero. For simplicity of presentation and for reasons of comparability with the literature cited earlier, the result will be set in the framework of a linear regression model, but inspection of the proof shows that it easily extends far beyond that 3That pointwise asymptotics can be misleading in the context of model selection has been noted earlier in Hosoya (1984), Shibata (1986a), Pötscher (1991), and Kabaila (1995, 2002). framework. For related results in the context of traditional post-model-selection estimators see Yang (2005) and Leeb and Pötscher (2005, Appendix C);4 cf. also the discussion on “partially” sparse estimators towards the end of Section 2 below. The theoretical results in Section 2 are rounded out by a Monte Carlo study in Section 3 that demonstrates the extent of the problem in finite samples for the SCAD estimator of Fan and Li (2001). The reasons for concentrating on the SCAD estimator in the Monte Carlo study are (i) that the finite-sample risk behavior of traditional post-model-selection estimators is well-understood (Judge and Bock (1978), Leeb and Pötscher (2005)) and (ii) that the SCAD estimator – especially when tuned to sparsity – has been highly advertised as a superior procedure in Fan and Li (2001) and subsequent papers mentioned above. 2 Bad Risk Behavior of Sparse Estimators Consider the linear regression model yt = x tθ + ǫt (1 ≤ t ≤ n) (1) where the k × 1 nonstochastic regressors xt satisfy n−1 t=1 xtx t → Q > 0 as n → ∞ and the prime denotes transposition. The errors ǫt are assumed to be independent identically distributed with mean zero and finite variance σ2. Without loss of generality we freeze the variance at σ2 = 1.5 Furthermore, we assume that ǫt has a density f that possesses an absolutely continuous derivative df/dx satisfying ((df(x)/dx)/f(x)) f(x)dx < ∞. Note that the conditions on f guarantee that the information of f is finite and positive. These conditions are obviously satisfied in the special case of normally distributed errors. Let Pn,θ denote the distribution of the sample (y1, . . . , yn) ′ and let En,θ denote the corresponding expec- tation operator. For θ ∈ Rk, let r(θ) denote a k×1 vector with components ri(θ) where ri(θ) = 0 if θi = 0 and ri(θ) = 1 if θi 6= 0. An estimator θ̂ for θ based on the sample (y1, . . . , yn)′ is said to satisfy the sparsity-type condition if for every θ ∈ Rk r(θ̂) ≤ r(θ) → 1 (2) 4The unboundedness of the maximal (scaled) mean squared error of estimators following BIC-type model selection has also been noted in Hosoya (1984), Shibata (1986b), and Foster and George (1994). 5If the variance is not frozen at σ2 = 1, the results below obviously continue to hold for each fixed value of σ2, and hence hold a fortiori if the supremum in (3)–(4) below is also taken over σ2. holds for n → ∞, where the inequality sign is to be interpreted componentwise. That is, the estimator is guaranteed to find the zero components of θ with probability approaching one as n → ∞. Clearly, any sparse estimator satisfies (2). In particular, the SCAD estimator as well as certain members of the class of Bridge estimators satisfy (2) for suitable choices of the regularization parameter as mentioned earlier. Also, any post-model-selection estimator based on a consistent model selection procedure clearly satisfies (2). All these estimators are additionally also consistent for θ, and hence in fact satisfy the stronger condition Pn,θ(r(θ̂) = r(θ)) → 1 for all θ ∈ Rk. [Condition (2) by itself is of course also satisfied by nonsensical estimators like θ̂ ≡ 0, but is all that is needed to establish the subsequent result.] We now show that any estimator satisfying the sparsity-type condition (2) has quite bad finite sample risk properties. For purposes of comparison we note that the (scaled) mean squared error of the least squares estimator θ̂LS satisfies n(θ̂LS − θ)′(θ̂LS − θ) = trace which converges to trace(Q−1), and thus remains bounded as sample size increases. Theorem 2.1 6Let θ̂ be an arbitrary estimator for θ that satisfies the sparsity-type condition (2). Then the maximal (scaled) mean squared error of θ̂ diverges to infinity as n → ∞, i.e., n(θ̂ − θ)′(θ̂ − θ) → ∞ (3) for n → ∞. More generally, let l : Rk → R be a nonnegative loss function. Then En,θl(n 1/2(θ̂ − θ)) → sup l(s) (4) for n → ∞. In particular, if the loss function l is unbounded then the maximal risk associated with l diverges to infinity as n → ∞. The theorem says that, whatever the loss function, the maximal risk of a sparse estimator is – in large samples – as bad as it possibly can be. 6Theorem 2.1 and the ensuing discussion continue to apply if the regressors xt as well as the errors ǫt are allowed to depend on sample size n, at least if the errors are normally distributed. The proof is analogous, except that one uses direct computation and LeCam’s first lemma (cf., e.g., Lemma A.1 in Leeb and Pötscher (2006)) instead of Koul and Wang (1984) to verify contiguity. Also, the results continue to hold if the design matrix satisfies δ−1n t=1 xtx t → Q > 0 for some positive sequence δn other than n, provided that the scaling factor n1/2 is replaced by δ n throughout. Upon choosing l(s) = |si|, where si denotes the i-th coordinate of s, relation (4) shows that also the maximal (scaled) absolute bias of each component θ̂i diverges to infinity. Applying relation (4) to the loss function l∗(s) = l(c′s) shows that (4) holds mutatis mutandis also for estimators c′θ̂ of arbitrary linear contrasts c′θ. In particular, using quadratic loss l∗(s) = (c′s)2, it follows that also the maximal (scaled) mean squared error of the linear contrast c′θ̂ goes to infinity as sample size increases, provided c 6= 0. Proof of Theorem 2.1: It suffices to prove (4).7 Now, with θn = −n−1/2s, s ∈ Rk arbitrary, we have l(u) ≥ sup En,θl(n 1/2(θ̂ − θ)) ≥ En,θn l(n1/2(θ̂ − θn)) ≥ En,θn [l(n1/2(θ̂ − θn))1(θ̂ = 0)] = l(−n1/2θn)Pn,θn(r(θ̂) = 0) = l(s)Pn,θn(r(θ̂) = 0). (5) By the sparsity-type condition we have that Pn,0(r(θ̂) = 0) → 1 as n → ∞. Since the model is locally asymptotically normal under our assumptions (Koul and Wang (1984), Theorem 2.1 and Remark 1; Hajek and Sidak (1967), p.213), the sequence of probability measures Pn,θn is contiguous w.r.t. the sequence Pn,0. Consequently, the far r.h.s. of (5) converges to l(s). Since s ∈ Rk was arbitrary, the proof is complete. � Inspection of the proof shows that Theorem 2.1 remains true if the supremum of the risk in (4) is taken only over open balls of radius ρn centered at the origin as long as n 1/2ρn → ∞. Hence, the bad risk behavior is a local phenomenon that occurs in a part of the parameter space where one perhaps would have expected the largest gain over the least squares estimator due to the sparsity property. [If the supremum of the risk in (4) is taken over the open balls of radius n−1/2ρ centered at the origin where ρ > 0 is now fixed, then the proof still shows that the limit inferior of this supremum is not less than sup‖s‖<ρ l(s).] Furthermore, for quadratic loss l(s) = s′s, a small variation of the proof shows that these “local” results continue to hold if the open balls over which the supremum is taken are not centered at the origin, but at an arbitrary θ, as long as θ possesses at least one zero component. [It is easy to see that this is more generally true for any nonnegative loss function l satisfying, e.g., l(s) ≥ l(πi(s)) for every s ∈ Rk and an index i with θi = 0, where πi represents the projection on the i-th coordinate axis.] Inspection of the proof also shows that – at least in the case of quadratic loss – the element s can be chosen to point in the direction of a standard basis vector. This then shows that the bad risk behavior occurs at parameter values that themselves are “sparse” in the sense of having 7Note that the expectations in (3) and (4) are always well-defined. many zero coordinates. If the quadratic loss n(θ̂ − θ)′(θ̂ − θ) in (3) is replaced by the weighted quadratic loss (θ̂ − θ)′ t=1 xtx t(θ̂ − θ), then the corresponding maximal risk again diverges to infinity. More generally, let ln be a nonnegative loss function that may depend on sample size. Inspection of the proof of Theorem 2.1 shows that lim sup ln(u) ≥ lim sup ‖θ‖<n−1/2ρ En,θln(n 1/2(θ̂ − θ)) ≥ sup ‖u‖<ρ lim sup ln(u), (6) lim inf ln(u) ≥ lim inf ‖θ‖<n−1/2ρ En,θln(n 1/2(θ̂ − θ)) ≥ sup ‖u‖<ρ lim inf ln(u) (7) hold for any 0 < ρ ≤ ∞. [In case 0 < ρ < ∞, the lower bounds in (6)-(7) can even be improved to lim supn→∞ sup‖u‖<ρ ln(u) and lim infn→∞ sup‖u‖<ρ ln(u), respectively. 8 It then follows that in case ρ = ∞ the lower bounds in (6)-(7) can be improved to sup 0<τ<∞ lim supn→∞ sup‖u‖<τ ln(u) and sup 0<τ<∞ lim infn→∞ sup‖u‖<τ ln(u), respectively.] Next we briefly discuss the case where an estimator θ̂ only has a “partial” sparsity property (and consequently a commensurable oracle property) in the following sense: Suppose the param- eter vector θ is partitioned as θ = (α′, β′)′ and the estimator θ̂ = (α̂′, β̂ )′ only finds the true zero components in the subvector β with probability converging to one. E.g., θ̂ is a traditional post-model-selection estimator based on a consistent model selection procedure that is designed to only identify the zero components in β. A minor variation of the proof of Theorem 2.1 im- mediately shows again that the maximal (scaled) mean squared error of β̂, and hence also of θ̂, diverges to infinity, and the same is true for linear combinations d′β̂ as long as d 6= 0. [This immediately extends to linear combinations c′θ̂, as long as c charges at least one coordinate of β̂ with a nonzero coefficient.]9 However, if the parameter of interest is α rather than β, Theorem 2.1 and its proof (or simple variations thereof) do not apply to the mean squared error of α̂ (or its linear contrasts). Nevertheless, the maximal (scaled) mean squared error of α̂ can again be shown to diverge to infinity, at least for traditional post-model-selection estimators θ̂ based on a consistent model selection procedure; see Leeb and Pötscher (2005, Appendix C). While the above results are set in the framework of a linear regression model with non- stochastic regressors, it is obvious from the proof that they extend to much more general models such as regression models with stochastic regressors, semiparametric models, nonlinear models, 8Note that the local asymptotic normality condition in Koul and Wang (1984) as well as the result in Lemma A.1 in Leeb and Pötscher (2006) imply contiguity of Pn,θn and Pn,0 not only for θn = γ/n 1/2 but more generally for θn = γn/n 1/2 with γn a bounded sequence. 9In fact, this variation of the proof of Theorem 2.1 shows that the supremum of En,θ l(n 1/2(β̂ − β)), where l is an arbitrary nonegative loss function, again converges to the supremum of the loss function. time series models, etc., as long as the contiguity property used in the proof is satisfied. This is in particular the case whenever the model is locally asymptotically normal, which in turn is typically the case under standard regularity conditions for maximum likelihood estimation. 3 Numerical Results on the Finite Sample Performance of the SCAD Estimator We replicate and extend Monte Carlo simulations of the performance of the SCAD estimator given in Example 4.1 of Fan and Li (2001); we demonstrate that this estimator, when tuned to enjoy a sparsity property and an oracle property, can perform quite unfavorably in finite samples. Even when not tuned to sparsity, we show that the SCAD estimator can perform worse than the least squares estimator in parts of the parameter space, something that is not brought out in the simulation study in Fan and Li (2001) as they conducted their simulation only at a single point in the parameter space (which happens to be favorable to their estimator). Consider n independent observations from the linear model (1) with k = 8 regressors, where the errors ǫt are standard normal and are distributed independently of the regressors. The regressors xt are assumed to be multivariate normal with mean zero. The variance of each component of xt is equal to 1 and the correlation between the i-th and the j-th component of xt, i.e., xt,i and xt,j , is ρ |i−j| with ρ = 0.5. Fan and Li (2001) consider this model with n = 40, n = 60, and with the true parameter equal to θ0 = (3, 1.5, 0, 0, 2, 0, 0, 0) ′; cf. also Tibshirani (1996, Section 7.2). We consider a whole range of true values for θ at various sample sizes, namely θn = θ0 + (γ/ n) × η for some vector η and for a range of γ’s as described below. We do this because (i) considering only one choice for the true parameter in a simulation may give a wrong impression of the actual performance of the estimators considered, and (ii) because our results in Section 2 suggest that the risk of sparse estimators can be large for parameter vectors which have some of its components close to, but different from, zero. The SCAD estimator is defined as a solution to the problem of minimizing the penalized least squares objective function (yt − x′tθ)2 + n pλ(|θi|) where the penalty function pλ is defined in Fan and Li (2001) and λ ≥ 0 is a tuning parameter. The penalty function pλ contains also another tuning parameter a, which is set equal to 3.7 here, resulting in a particular instance of the SCAD estimator which is denoted by SCAD2 in Example 4.1 of Fan and Li (2001). According to Theorem 2 in Fan and Li (2001) the SCAD estimator is guaranteed to satisfy the sparsity property if λ → 0 and nλ → ∞ as samples size n goes to infinity. Using the MATLAB code provided to us by Runze Li, we have implemented the SCAD2 estimator in R. [The code is available from the first author on request.] Two types of performance measures are considered: The ‘median relative model error’ studied by Fan and Li (2001), and the relative mean squared error. The median relative model error is defined as follows: For an estimator θ̂ for θ, define the model errorME(θ̂) byME(θ̂) = (θ̂−θ)′Σ(θ̂−θ), where Σ denotes the variance/covariance matrix of the regressors. Now define the relative model error of θ̂ (relative to least squares) by ME(θ̂)/ME(θ̂LS), with θ̂LS denoting the least squares estimator based on the overall model. The median relative model error is then given by the median of the relative model error. The relative mean squared error of θ̂ is given by E[(θ̂−θ)′(θ̂−θ)]/E[(θ̂LS−θ)′(θ̂LS−θ)].10 Note that we have scaled the performance measures such that both of them are identical to unity for θ̂ = θ̂LS . Setup I: For SCAD2 the tuning parameter λ is chosen by generalized cross-validation (cf. Section 4.2 of Fan and Li (2001)). In the original study in Fan and Li (2001), the range of λ’s considered for generalized cross-validation at sample sizes n = 40 and n = 60 is {δ(σ̂/ δ = 0.9, 1.1, 1.3, . . . , 2}; here, σ̂2 denotes the usual unbiased variance estimator obtained from a least-squares fit of the overall model. For the simulations under Setup I, we re-scale this range of λ’s by logn/ log 60. With this, our results for γ = 0 replicate those in Fan and Li (2001) for n = 60; for the other (larger) sample sizes that we consider, the re-scaling guarantees that λ → 0 and nλ → ∞ and hence, in view of Theorem 2 in Fan and Li (2001), guarantees that the resulting estimator enjoys the sparsity condition. [For another choice of λ see Setup VI.] We compute Monte Carlo estimates for both the median relative model error and the relative mean squared error of the SCAD2 estimator for a range of true parameter values, namely θn = θ0 + (γ/ n) × (0, 0, 1, 1, 0, 1, 1, 1)′ for 101 equidistant values of γ between 0 and 8, and for sample sizes n = 60, 120, 240, 480, and 960, each based on 500 Monte Carlo replications (for comparison, Fan and Li (2001) use 100 replications). Note that the performance measures are symmetric about γ = 0, and hence are only reported for nonnegative values of γ. The results are summarized in Figure 1 below. [For better readability, points in Figure 1 are joined by lines.] 10The mean squared error of θ̂LS is given by E trace((X ′X)−1) which equals trace(Σ−1)/(n−9) = 38/(3n−27) by von Rosen (1988, Theorem 3.1). 0 2 4 6 8 Median Relative Model Error of SCAD2 gamma 0 2 4 6 8 Relative Mean Squared Error of SCAD2 gamma Figure 1: Monte Carlo performance estimates under the true parameter θn = θ0+(γ/ (0, 0, 1, 1, 0, 1, 1, 1)′ , as a function of γ. The left panel gives the estimated median relative model error of SCAD2 for sample sizes n = 60, 120, 240, 480, 960. The right panel gives the corresponding results for the estimated relative mean squared error of SCAD2. Larger sample sizes correspond to larger maximal errors. For comparison, the gray line at one indicates the performance of the ordinary least squares estimator. In the Monte Carlo study of Fan and Li (2001), only the parameter value θ0 is considered. This corresponds to the point γ = 0 in the panels of Figure 1. At that particular point in the parameter space, SCAD2 compares quite favorably with least squares. However, Figure 1 shows that there is a large range of parameters where the situation is reversed. In particular, we see that SCAD2 can perform quite unfavorably when compared to least squares if the true parameter, i.e., θn, is such that some of its components are close to, but different from, zero. In line with Theorem 2.1, we also see that the worst-case performance of SCAD2 deteriorates with increasing sample size: For n = 60, ordinary least squares beats SCAD2 in terms of worst-case performance by a factor of about 2 in both panels of Figure 1; for n = 960, this factor has increased to about 3; and increasing n further makes this phenomenon even more pronounced. We also see that, for increasing n, the location of the peak moves to the right in Figure 1, suggesting that the worst- case performance of SCAD2 (among parameters of the form θn = (γ/ n)× (0, 0, 1, 1, 0, 1, 1, 1)′) is attained at a value γn, which is such that γn → ∞ with n. In view of the proof of Theorem 2.1, this is no surprise.11 [Of course, there may be other parameters at any given sample size for 11See Section 2.1 and Footnote 14 in Leeb and Pötscher (2005) for related discussion. which SCAD2 performs even worse.] Our simulations thus demonstrate: If each component of the true parameter is either very close to zero or quite large (where the components’ size has to be measured relative to sample size), then the SCAD estimator performs well. However, if some component is in-between these two extremes, then the SCAD estimator performs poorly. In particular, the estimator can perform poorly precisely in the important situation where it is statistically difficult to decide whether some component of the true parameter is zero or not. Poor performance is obtained in the worst case over a neighborhood of one of the lower-dimensional models, where the ‘diameter’ of the neighborhood goes to zero slower than 1/ We have also re-run our simulations for other experimental setups; the details are given below. Since our findings for these other setups are essentially similar to those summarized in Figure 1, we first give a brief overview of the other setups and summarize the results before proceeding to the details. In Setups II and III we consider slices of the 8-dimensional performance measure surfaces corresponding to directions other than the one used in Setup I: In Setup II the true parameter is of the form θ0+(γ/ n)× (0, 0, 1, 1, 0, 0, 0, 0)′, i.e., we consider the case where some components are exactly zero, some are large, and others are in-between. In Setup III, we consider a scenario in-between Setup I and Setup II, namely the case where the true parameter is of the form θ0+(γ/ n)×(0, 0, 1, 1, 0, 1/10, 1/10, 1/10)′. The method for choosing λ in these two setups is the same as in Setup I. The results in these additional setups are qualitatively similar to those shown in Figure 1 but slightly less pronounced. In further setups we also consider various other rates for the SCAD tuning parameter λ. By Theorem 2 of Fan and Li (2001), the SCAD estimator is sparse if λ → 0 and nλ → ∞; as noted before, for Figure 1, λ is chosen by generalized cross- validation from the set Λn = {δ(σ̂/ n)(log(n)/ log(60)) : δ = 0.9, 1.1, 1.3, . . . , 2}; i.e., we have nλ = Op(log(n)). The magnitude of λ has a strong impact on the performance of the estimator. Smaller values result in ‘less sparse’ estimates, leading to less favorable performance relative to least squares at γ = 0, but at the same time leading to less unfavorable worst-case performance; the resulting performance curves are ‘flatter’ than those in Figure 1. Larger values of λ result in ‘more sparse’ estimates, improved performance at γ = 0, and more unfavorable worst-case performance; this leads to performance curves that are ‘more spiked’ than those in Figure 1. In Setups IV and V we have re-run our simulations with γ chosen from a set Λn as above, but with log(n)/ log(60) replaced by (n/60)1/10 as well as by (n/60)1/4, resulting in nλ = Op(n 1/10) and nλ = Op(n 1/4), respectively. In Setup IV, where nλ = Op(n 1/10), we get results similar to, but less pronounced than, Figure 1; this is because Setup IV leads to λ’s smaller than in Setup I. In Setup V, where nλ = Op(n 1/4), we get similar but more pronounced results when compared to Figure 1; again, this is so because Setup V leads to larger λ’s than Setup I. A final setup (Setup VI) in which we do not enforce the conditions for sparsity is discussed below after the details for Setups II-V are presented. Setups II and III: In Setup II, we perform the same Monte Carlo study as in Setup I, the only difference being that the range of θ’s is now θn = θ0 +(γ/ n)× (0, 0, 1, 1, 0, 0, 0, 0)′ for 101 equidistant values of γ between 0 and 8. The worst-case behavior in this setup is qualitatively similar to the one in Setup I but slightly less pronounced; we do not report the results here for brevity. In Setup III, we again perform the same Monte Carlo study as in Setup I, but now with θn = θ0 + (γ/ n)× (0, 0, 1, 1, 0, 1/10, 1/10, 1/10)′ for 101 equidistant values of γ between 0 and 80. Note that here we consider a range for γ wider than that in Scenario I and II, where we had 0 ≤ γ ≤ 8. Figure 2 gives the results for Setup III. 0 20 40 60 80 Median Relative Model Error of SCAD2 gamma 8 0 20 40 60 80 Relative Mean Squared Error of SCAD2 gamma Figure 2: Monte Carlo performance estimates under the true parameter θn = θ0+(γ/ (0, 0, 1, 1, 0, 1/10, 1/10, 1/10)′, as a function of γ. See the legend of Figure 1 for a description of the graphics. The same considerations as given for Figure 1 also apply to Figure 2. The new feature in Figure 2 is that the curves are bimodal. Apparently, this is because now there are two regions, in the range of γ’s under consideration, for which some components of the underlying regression parameter θn are neither very close to zero nor quite large (relative to sample size): Components 3 and 4 for γ around 5 (first peak), and components 6, 7, and 8 for γ around 40 (second peak). Setups IV and V: Here we perform the same simulations as in Setup I, but now with the range of λ’s considered for generalized cross-validation given by {δ(σ̂/ n)(n/60)1/10 : δ = 0.9, 1.1, 1.3, . . . , 2} for Setup IV, and by {δ(σ̂/ n)(n/60)1/4 : δ = 0.9, 1.1, 1.3, . . . , 2} for Setup V. Setup IV gives ‘less sparse’ estimates while Setup V gives ‘more sparse’ estimates relative to Setup I. The results are summarized in Figures 3 and 4 below. Choosing the SCAD tuning- parameter λ so that the estimator is ‘more sparse’ clearly has a detrimental effect on the esti- mator’s worst-case performance. 0 2 4 6 8 Median Relative Model Error of SCAD2 gamma 0 2 4 6 8 Relative Mean Squared Error of SCAD2 gamma Figure 3: Monte Carlo performance estimates under the true parameter θn = θ0+(γ/ (0, 0, 1, 1, 0, 1, 1, 1)′ as a function of γ; the SCAD tuning parameter λ is chosen as described in Setup IV. 0 2 4 6 8 Median Relative Model Error of SCAD2 gamma 0 2 4 6 8 Relative Mean Squared Error of SCAD2 gamma Figure 4: Monte Carlo performance estimates under the true parameter θn = θ0+(γ/ (0, 0, 1, 1, 0, 1, 1, 1)′, as a function of γ; the SCAD tuning parameter λ is chosen as described in Setup V. In all setups considered so far we have enforced the conditions λ → 0 and nλ → ∞ to guarantee sparsity of the resulting SCAD estimator as risk properties of sparse estimators are the topic of the paper. In response to a referee we further consider Setup VI which is identical to Setup I, except that the range of λ’s over which generalized cross-validation is effected is given by {δ(σ̂/ n) : δ = 0.9, 1.1, 1.3, . . . , 2}, which is precisely the range considered in Fan and Li (2001). Note that the resulting λ does now not satisfy the conditions for sparsity given in Theorem 2 of Fan and Li (2001). The results are shown in Figure 5 below. The findings are similar to the results from Setup I, in that SCAD2 gains over the least squares estimator in a neighborhood of θ0, but is worse by approximately a factor of 2 over considerable portions of the range of γ, showing once more that the simulation study in Fan and Li (2001) does not tell the entire truth. What is, however, different here from the results obtained under Setup I is that – not surprisingly at all – the worst case behavior now does not get worse with increasing sample size. [This is akin to the boundedness of the worst case risk of a post-model-selection estimator based on a conservative model selection procedure like AIC or pre-testing with a sample-size independent critical value.] 0 2 4 6 8 Median Relative Model Error of SCAD2 gamma 0 2 4 6 8 Relative Mean Squared Error of SCAD2 gamma Figure 5: Monte Carlo performance estimates under the true parameter θn = θ0+(γ/ (0, 0, 1, 1, 0, 1, 1, 1)′, as a function of γ; the SCAD tuning parameter λ is chosen as described in Setup VI. 4 Conclusion We have shown that sparsity of an estimator leads to undesirable risk properties of that estimator. The result is set in a linear model framework, but easily extends to much more general parametric and semiparametric models, including time series models. Sparsity is often connected to a so-called “oracle property”. We point out that this latter property is highly misleading and should not be relied on when judging performance of an estimator. Both observations are not really new, but worth recalling: Hodges’ construction of an estimator exhibiting a deceiving pointwise asymptotic behavior (i.e., the oracle property in today’s parlance) has led mathematical statisticians to realize the importance uniformity has to play in asymptotic statistical results. It is thus remarkable that today – more than 50 years later – we observe a return of Hodges’ estimator in the guise of newly proposed estimators (i.e., sparse estimators). What is even more surprising is that the deceiving pointwise asymptotic properties of these estimators (i.e., the oracle property) are now advertised as virtues of these methods. It is therefore perhaps fitting to repeat Hajek’s (1971, p.153) warning: “Especially misinformative can be those limit results that are not uniform. Then the limit may exhibit some features that are not even approximately true for any finite n.” The discussion in the present paper as well as in Leeb and Pötscher (2005) shows in particular that distributional or risk behavior of consistent post-model-selection estimators is not as sometimes believed, but is much worse. The results of this paper should not be construed as a criticism of shrinkage-type estimators including penalized least squares (maximum likelihood) estimators per se. Especially if the dimension of the model is large relative to sample size, some sort of shrinkage will typically be beneficial. However, achieving this shrinkage through sparsity is perhaps not such a good idea (at least when estimator risk is of concern). It certainly cannot simply be justified through an appeal to the oracle property.12 12In this context we note that “superefficiency” per se is not necessarily detrimental in higher dimensions as witnessed by the Stein phenomenon. However, not all forms of “superefficiency” are created equal, and “superefficiency” generated through sparsity of an estimator typically belongs to the undesirable variety as shown in the paper. Acknowledgements A version of this paper was previously circulated in 2004. We are grateful to the editor Ron Gallant and the referees as well as to Hemant Ishwaran, Paul Kabaila, Richard Nickl, and Yuhong Yang for helpful comments. 5 References Bunea, F. (2004): Consistent covariate selection and post model selection inference in semi- parametric regression. Annals of Statistics 32, 898-927. Bunea, F. & I. W. McKeague (2005): Covariate selection for semiparametric hazard function regression models. Journal of Multivariate Analysis 92, 186-204. Cai, J., Fan, J., Li, R., & H. Zhou (2005): Variable selection for multivariate failure time data, Biometrika 92, 303-316. Fan, J. & R. Li (2001): Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348-1360. Fan, J. & R. Li (2002): Variable selection for Cox’s proportional hazards model and frailty model. Annals of Statistics 30, 74-99. Fan, J. & R. Li (2004): New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. Journal of the American Statistical Association 99, 710- Fan, J. & H. Peng (2004): Nonconcave penalized likelihood with a diverging number of parameters. Annals of Statistics 32, 928-961. Foster D. P. & E. I. George (1994): The risk inflation criterion for multiple regression. Annals of Statistics 22, 1947-1975. Frank, I. E. & J. H. Friedman (1993): A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109-148. Hajek, J. (1971): Limiting properties of likelihoods and inference. In: V. P. Godambe and D. A. Sprott (eds.), Foundations of Statistical Inference: Proceedings of the Symposium on the Foundations of Statistical Inference, University of Waterloo, Ontario, March 31 – April 9, 1970, 142-159. Toronto: Holt, Rinehart & Winston. Hajek, J. & Z. Sidak (1967): Theory of Rank Tests. New York: Academic Press. Hosoya, Y. (1984): Information criteria and tests for time series models. In: O. D. Anderson (ed.), Time Series Analysis: Theory and Practice 5, 39-52. Amsterdam: North-Holland. Judge, G. G. & M. E. Bock (1978): The Statistical Implications of Pre-test and Stein-rule Estimators in Econometrics. Amsterdam: North-Holland. Kabaila, P. (1995): The effect of model selection on confidence regions and prediction regions. Econometric Theory 11, 537-549. Kabaila, P. (2002): On variable selection in linear regression. Econometric Theory 18, 913-915. Knight, K. & W. Fu (2000): Asymptotics of lasso-type estimators. Annals of Statistics 28, 1356-1378. Koul, H. L. & W. Wang (1984): Local asymptotic normality of randomly censored linear regression model. Statistics & Decisions, Supplement Issue No. 1, 17-30. Lehmann, E. L. & G. Casella (1998): Theory of Point Estimation. Springer Texts in Statistics. New York: Springer-Verlag. Leeb, H. & B. M. Pötscher (2005): Model selection and inference: facts and fiction. Econo- metric Theory 21, 21-59. Leeb, H. & B. M. Pötscher (2006): Performance limits for estimators of the risk or distribution of shrinkage-type estimators, and some general lower risk-bound results. Econometric Theory 22, 69-97. (Correction, ibid., forthcoming.) Pötscher, B. M. (1991): Effects of model selection on inference. Econometric Theory 7, 163-185. Shibata R. (1986a): Consistency of model selection and parameter estimation. Journal of Applied Probability, Special Volume 23A, 127-141. Shibata R. (1986b): Selection of the number of regression variables; a minimax choice of generalized FPE. Annals of the Institute of Statistical Mathematics 38, 459-474. Tibshirani, R. J. (1996): Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society, Ser. B 58, 267-288. Von Rosen, D. (1988): Moments for the inverted Wishart distribution. Scandinavian Journal of Statistics 15, 97-109. Yang, Y. (2005): Can the strengths of AIC and BIC be shared? A conflict between model identification and regression estimation. Biometrika 92, 937-950. Zou, H. (2006): The adaptive lasso and its orcale properties. Journal of the American Statistical Association 101, 1418-1429.

0704.1467

Capillary ordering and layering transitions in two-dimensional hard-rod fluids

Capillary ordering and layering transitions in two-dimensional hard-rod �uids Yuri Martínez-Ratón Grupo Interdis iplinar de Sistemas Complejos, Departamento de Matemáti as, Es uela Polité ni a Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E�28911, Leganés, Madrid, SPAIN (Dated: November 16, 2018) In this arti le we al ulate the surfa e phase diagram of a two-dimensional hard-rod �uid on�ned between two hard lines. In a �rst stage we study the semi-in�nite system onsisting of an isotropi �uid in onta t with a single hard line. We have found omplete wetting by the olumnar phase at the wall-isotropi �uid interfa e. When the �uid is on�ned between two hard walls, apillary olumnar ordering o urs via a �rst-order phase transition. For higher hemi al potentials the system exhibits layering transitions even for very narrow slits (near the one-dimensional limit). The theoreti al model used was a density-fun tional theory based on the Fundamental-Measure Fun tional applied to a �uid of hard re tangles in the restri ted-orientation approximation (Zwanzig model). The results presented here an be he ked experimentally in two-dimensional granular media made of rods, where verti al motions indu ed by an external sour e and ex luded volume intera tions between the grains allow the system to explore those stationary states whi h entropi ally maximize pa king on�gurations. We laim that some of the surfa e phenomena found here an be present in two-dimensional granular-media �uids. PACS numbers: 64.70.Md,61.30.Hn,61.20.Gy I. INTRODUCTION The e�e t of �uid on�nement on phase transitions is nowadays an a tive line of s ienti� resear h due to the dire t appli ation of the theoreti ally predi ted surfa e phase diagrams in the nanote hnology industry. Con�n- ing simple �uids, su h as hard [1, 2℄ or Lennard-Jones [3℄ spheres, in a narrow slit geometry, results in a ri h phase behavior, whi h has re ently been studied in de- tail. Liquid rystals on�ned in nanopores is another typi al example of systems with important appli ations in the industry of ele troni devi es. For this reason they have been extensively studied using theoreti al models based on density-fun tional theory. In parti ular, apil- lary phase transitions exhibited by a nemati �uid on- �ned between hard walls[4, 5℄, or walls favoring a par- ti ular an horing [6℄, have been predi ted. When non- uniform liquid- rystal phases, su h as the sme ti phase, are in luded in the study of on�ned systems, the result- ing surfa e phase diagrams display a ri h phenomenol- ogy, whi h in ludes wetting transitions, the appearan e of sme ti defe ts [7℄, and layering transitions [8℄. The e�e t of on�nement on two-dimensional �uids is also an interesting topi of resear h. Langmuir monolay- ers of lipids on the surfa e of water have been extensively studied in the last hundred years [9℄, and the dis ov- ery of stru tures and phase transitions in these systems has experien ed a dramati evolution driven by the new experimental te hniques. Now it is possible to on�ne these two-dimensional systems by external potentials and study the in�uen e of the on�nement on the mole ular Ele troni address: yuri�math.u 3m.es pa king of surfa e monolayers. Another paradigm of two-dimensional systems where the on�nement plays an important role is the pa k- ing stru tures formed by parti les in granular media [10℄. The rystallization of a quasi-two-dimensional one- omponent granular-disk �uid has re ently been studied experimentally [11℄. It was found that the properties of the rystal stru ture obtained (su h as pa king fra - tion, latti e stru ture, and Lindenman parameter) o- in ide with their ounterparts obtained from MC sim- ulations of a hard disk �uid. Re ent experiments have found non-equilibrium steady states in a vibrated granu- lar rod monolayer with tetrati , nemati and sme ti or- relations [12℄. Some of these textures are similar to the equilibrium thermodynami states of two-dimensional anisotropi �uids resulting from density-fun tional al- ulations [13℄ and MC simulations [14℄. It was shown by several authors that the inherent states of some frozen granular systems an be des ribed by equilibrium sta- tisti al me hani s [15℄. Also, an experimental test of the thermodynami approa h to granular media has re- ently been arried out [16℄. Con�ning two-dimensional granular rods in di�erent geometries ( ir ular, re tangu- lar, et .) results in spontaneous formation of patterns, with di�erent orientationally ordered textures and de- fe ts next to the ontainer [17℄. The authors of Ref. [18℄ have arried out MC simulations of a on�ned hard disk �uid. They have found that the rystal phase fails to nu- leate due to formation of sme ti bands when the system is on�ned [18℄. It would be interesting to devi e an ex- periment with on�ned granular disks with the aim of omparing the properties of the non-uniform stationary states with those obtained from the statisti al me hani s applied to the hard disk �uid. The main purpose of this arti le is the study of a on- �ned two-dimensional hard-rod �uid. We are interested http://arxiv.org/abs/0704.1467v1 mailto:[email protected] in the al ulation of the surfa e phase diagram of a hard re tangle (HR) �uid on�ned by a single or two hard lines. We an think on a HR �uid as an experimental re- alization of a system of hard ylinders on�ned between two plates at a distan e less than twi e the ylinder di- ameter. We suggest that some of the surfa e phase tran- sitions obtained here by applying the density fun tional formalism to a on�ned two-dimensional HR �uid should be similar to the steady states of on�ned granular rods. Some experiments are required to verify this hypothesis. The paper is organized as follows. In Se . II we present the theoreti al model: the fundamental-measure density fun tional applied to a HR �uid in the restri ted- orientation approximation. This se tion is divided into two subse tions. In the �rst the model is parti ularized to the study of the bulk phases, while in the se ond part the theoreti al expressions used in the al ulations of the thermodynami and stru tural properties of the inter- fa es are presented. The results are presented in Se . III. First we study the bulk phase diagram of a HR �uid with aspe t ratio equal to 3, and then the resulting sur- fa e phase diagrams of a single wall-HR �uid interfa e and of the �uid on�ned between two hard lines are pre- sented. Some on lusions are drawn in Se . IV. II. THEORETICAL MODEL In this se tion we introdu e the theoreti al model used in the al ulations of the bulk and interfa e equilibrium phases. To study highly inhomogeneous phases su h as those resulting from the on�nement of a �uid in a nar- row slit geometry or the solid phase with a high pa king fra tion, we have used the Fundamental-Measure Theory (FMT) applied to an anisotropi �uid of hard re tangles. It is well known that this formalism presents a great ad- vantage over other te hniques when dealing with highly inhomogeneous phases, and that this is mainly due to the fa t that a basi requirement to onstru t the FMT den- sity fun tional is that it onform with the dimensional ross-over riterium [19, 20℄. To implement the al u- lations we have used the restri ted-orientation approxi- mation, where the axes of the re tangles are restri ted to align only along the oordinate axes x or y. Thus, the whole system is des ribed in terms of density pro�les ρν(r) (ν = x, y). While the ideal part of the free energy density in re- du ed thermal units has the exa t form Φid(r) = ρν(r) [ln ρν(r)− 1] , (1) the FMT intera tion part of the 2D HR �uid is approxi- mated [20℄ by Φexc(r) = −n0(r) ln [1− n2(r)] + n1x(r)n1y(r) 1− n2(r) , (2) where the weighted densities nα's are al ulated as nα(r) = ν=x,y ρν ∗ ω (r), (3) and where the symbol ∗ stands for onvolution, i.e., ρν ∗ ω dr′ρν(r ν (r − r ′). The weights ω are the hara teristi fun tions whose volume integrals onstitute the fundamental measures of a single parti le (the edge lengths and surfa e area). They are de�ned as ω(0)ν (r) = − |x|)δ( − |y|), (4) ω(1x)ν (r) = − |x|)δ( − |y|), (5) ω(1y)ν (r) = − |x|)Θ( − |y|), (6) ω(2)ν (r) = Θ( − |x|)Θ( − |y|), (7) where σνµ = σ+(L− σ)δµν , with L and σ the length and width of the re tangle and δµν the Krone ker fun tion, while δ(x) and Θ(x) are the Dira delta and Heaviside fun tions, respe tively. A. The bulk phases To al ulate the bulk phase diagram we need to mini- mize the Helmholtz free energy fun tional βF [{ρν(r)}] = dr [Φid(r) + Φexc(r)] with respe t to the density pro- �les ρν(r). These density pro�les have the symmetries orresponding to the equilibrium phases, whi h an be the isotropi or nemati �uids, the sme ti phase (with parti les arranged in layers with their long axes pointing perpendi ular to the layers), the olumnar phase (with long axes parallel to the layers), plasti solid (parti les lo ated at the nodes of the square grid with averaged ori- entational order parameter over the ell equal to zero), and oriented solid (with both translational and orienta- tional order). To take proper a ount of all these possible symmetries, we have used a Fourier-series expansion of the density pro�les: ρν(r) = ρ0xν k=(0,0) k1,k2 cos(q1x) cos(q2y), (8) where we de�ned k ≡ (k1, k2) [with N = N(1, 1)℄, q1 = 2πk1/dx, and q2 = 2πk2/dy are the wave ve tor omponents parallel to x and y axes respe tively, and dx, dy are the periods of the re tangular grid along these di- re tions. α k1,k2 are the Fourier amplitudes of the density pro�le of the spe ies ν with the onstraint α 0,0 = 1. ρ0 is the average of the lo al density over the ell, while xν is the ell-averaged o upan y probability of spe ies ν. The Fourier series is trun ated at that value N whi h guar- antees that α N,N < 10 . With this parametrization the weighted density an be al ulated expli ity as nα(r) = ρ0 k1,k2 ω̂(α)ν (k) cos(q1x) cos(q2y), (9) where ω̂ ν (k) are the Fourier transforms of the orre- sponding weights, whi h have the form ω̂(0)ν (k) = χ0(q1σ ν /2)χ0(q2σ ν/2), (10) ω̂(1x)ν (k) = σ νχ1(q1σ ν/2)χ0(q2σ ν/2), (11) ω̂(1y)ν (k) = σ νχ0(q1σ ν/2)χ1(q2σ ν/2), (12) ω̂(2)ν (k) = aχ1(q1σ ν/2)χ1(q2σ ν/2), (13) Here a = Lσ is the surfa e area of the parti le, and χ0(x) = cosx, χ1(x) = sin(x)/x. We have sele ted the orientational dire tor parallel to y. Thus, the equilib- rium sme ti ( olumnar) phase should be found by min- imizing the free energy with respe t to the Fourier am- plitudes α 0,k (α k,0), the sme ti ( olumnar) period dy (dx) and the order parameter QN ∈ [−1, 1] [related to the xν 's through the relations x‖,⊥ = (1 ±QN)/2 where the symbols ‖,⊥ stand for parti le alignment along y and x respe tively℄. For uniform phases [α k1,k2 ∀(k1, k2) 6= (0, 0)℄ QN oin ides with the nemati order parameter. The solid phase is to be found by minimiz- ing the free energy with respe t to all the Fourier ampli- tudes α k1,k2 , the rystal periods dx and dy, and the order parameter QN in the ase of an orientationally ordered solid. To measure the pa king stru ture and the orienta- tional order of the bulk phases we use the lo al density and the order parameter pro�les, ρ(r) = ν ρν(r), and Q(r) = [ρy(r)− ρx(r)] /ρ(r) respe tively. B. The interfa ial phases As we want to study the hard wall-�uid interfa e or the HR �uid on�ned in a slit geometry, we have introdu ed the following external potential: Vν(x) = ∞, x < σxν/2 0, x ≥ σxν/2, for the semi-in�nite system, and Vν(x) = ∞, x < σxν/2 and x > H − σ 0, σxν/2 ≤ x ≤ H − σ ν /2, for the slit geometry, where H is the slit width, and the normal to the wall was sele ted in the x dire tion. Note that this external potential represents a hard line whi h ex ludes the enter of mass of parti les at distan es less than their onta t distan es with the wall. In this sense we an say that the external potential favors par- allel alignment at the wall. This is in ontrast with the favored homeotropi alignment usually onsidered in sev- eral studies of three-dimensional liquid rystals on�ned by a single or two walls (in parti ular that of Ref. [8℄). The one-dimensional equilibrium density pro�les ρν(x) were found by minimizing the ex ess surfa e free energy per unit length + P − ρν(x) [µν − Vν(x)] , (16) where β = (kBT ) , Φ(x) = Φid(x) + Φexc(x), and µν are the hemi al potentials of spe ies ν �xed at the bulk �uid-phase value at in�nite distan e from the wall, while P is the �uid pressure. The hemi al potential of the bulk �uid phase is al ulated, as usual, as µ = xνµν , with xν the molar fra tions of spe ies ν. If the bulk phase is an isotropi �uid then xν = 1/2, and µν = µ, ∀ν. To measure the degree of interfa ial order, we will use the adsorption of the density pro�le, de�ned as dx [ρ(x)− ρ(∞)], and the order parameter pro�le Q(x). The expression (16) oin ides with the de�nition of the surfa e tension of the wall-�uid interfa e for the semi- in�nite ase, whi h is approximately equal to half the ex ess surfa e free-energy for the slit geometry when the wall distan e H is large enough to a ommodate both interfa es. To minimize the fun tional given by (16), we have dis- retized spa e in the x dire tion and minimize γ with respe t to ρν(xi) (xi ∈ [x0, xN ]) using the onjugate- gradient algorithm. III. RESULTS In this se tion we present the main results obtained from the appli ation of the theoreti al model just de- s ribed to the study of surfa e properties of a 2D HR �uid. Parti les were hosen to have aspe t ratio κ ≡ L/σ = 3. This aspe t ratio is hosen be ause one of the aims of the present work is the study of layered phases on�ned by one or two walls. As we will show bellow for κ = 3 the stable phase is the olumnar layered phase. In the �rst subse tion we will summarize the results obtained in the al ulation of the bulk phase diagram of this system, while in the se ond subse tion we will fo us on the study of the surfa e phase diagram. A. Bulk phase diagram We have minimized the free energy density of the HR �uid, de�ned as Φ ≡ V −1 dr [Φid(r) + Φexc(r)], with respe t to the Fourier amplitudes, periods, and mean o - upan y probability, as des ribed in detail in Se . II A. The results are plotted in Fig. 1, where the free-energy densities of all the stable and metastable phases found are plotted as a fun tion of the pa king fra tion η = ρ0a. We have found, apart from the usual isotropi (I) and ne- mati (N) phases, two di�erent sme ti phases (Sm1, and Sm2), a plasti solid (PS), perfe tly oriented solid (OS), and �nally the olumnar phase (C), whi h is the stable one in the whole range of pa king fra tions explored. 0.55 0.6 0.65 0.7 0.75 Figure 1: The res aled free-energy density Φ∗ = Φ+2.9875− 5.8501η is plotted against the mean pa king fra tion for all the stable and metastable phases found. These are: isotropi (dashed line), nemati (dotted line), sme ti -1 and sme ti - 2 (dotted and dashed lines), plasti solid (dashed line la- belled as PS), while the perfe tly oriented solid and the olumnar phases (labelled in the �gure as OS and C respe - tively) are plotted with solid lines. The open ir le indi- ates the isotropi -nemati bifur ation point; the open square, the isotropi -plasti solid bifur ation point; and the solid ir- les represent the oexisting pa king fra tions at isotropi - olumnar phase oexisten e. The oupling between the spatial and orientational de- grees of freedom of the parti les results in the presen e of phases (stable or metastable) with di�erent symmetries. In Fig. (2) we have sket hed some of the parti le on- �gurations orresponding to phases with olumnar (a), sme ti -1 (b), sme ti -2 ( ), and plasti solid (d) sym- metries found from the numeri al minimization of the density fun tional. The dire tions of spatial periodi ities of ea h phase have been depi ted in the �gure. In Fig. 3 (a) we have plotted the density and order- parameter pro�les of the oexisting olumnar phase. The olumnar phase is orientationally ordered in the y dire - tion with the long re tangle axis pointing along the y axis, while the periodi ity of both density and order parame- ter pro�les (whi h are in phase) is along the x dire tion [see Fig. 3 (a)℄. The mean oexisten e pa king fra tions of the I and C phases are ηI = 0.57058 and ηC = 0.60310, respe tively while the period of the C phase, in units of the HR width, was found to be dx/σ = 1.20102. In Fig. 3 (b) we have plotted the order parameter QN, and the period of the olumnar phase as a fun tion of the mean pa king fra tion. (c) (d) Figure 2: Sket h of parti le on�gurations orresponding to di�erent phases: olumnar (a), sme ti -1 (b), sme ti -2 ( ), and plasti solid (d) phases. The dire tion of spatial period- i ities are labeled in the �gure. To ompare the di�erent pa kings of HR parti les in the metastable phases (found as the lo al minima of the free energy density) for a �xed mean pa king fra tion η = 0.7, we have plotted the density and order-parameter pro�les of the Sm1,2 [Fig. 4 (a) and (b)℄, and PS and OS [Fig. 5 (a)�( )℄ phases. As an be seen from Fig. 4 (a), the density pro�le of the Sm1 phase has two max- ima per period. The less pronoun ed maxima, lo ated at the interstitials, re�e t the high population of parti- les with long axes oriented parallel to the sme ti layers [see the sket hed parti le on�gurations in Fig. 2 (b)℄. This alignment is also shown in the order-parameter pro- �le, whi h rea hes high negative values at the intersti- tial positions. This phase bears a strong resemblan e to the �ndings of Refs. [21℄ and [22℄ where the parti- le equilibrium on�gurations in the 3D sme ti phases show the same pattern. As a onsequen e of this (al- ternating population of parti les aligned perpendi ular �sharpest peak in the density pro�le� and parallel to the layers), the sme ti period in units of the parti le length is dy/L = 1.53025, higher than the sme ti period of the Sm2 phase (dy/L = 1.17935). The density and order- parameter pro�les of the Sm2 are shown in Fig. 4 (b). As an be seen from the �gure, these pro�les re�e t the usual pa king in sme ti s, hara terized by a single den- sity peak with vanishingly small population of parti les in the interstitials, while the order parameter rea hes its maximum value at the position of the sme ti layers [see Fig. 2 ( ) for the sket hed parti le on�gurations℄. The density and order parameter pro�les of the PS phase with mean pa king fra tion equal to 0.7 are plot- -0.5 -0.25 0 0.25 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Figure 3: (a): density ρ(x) (solid line) and order parameter Q(x) (dashed line) pro�les of the olumnar phase at oex- isten e with the isotropi phase. (b): order parameter QN and period of the olumnar phase against the mean pa king fra tion. ted in Fig. 5 (a) and (b). The plasti solid has the same periodi ity in the x and y dire tion, i.e. dx = dy = d, and the order parameter averaged over the unit ell is stri tly equal to zero. As we an see from Fig. 5 (b), while the order parameter at the nodes of the square latti e is equal to zero, it rea hes positive (negative) val- ues at the (±0.5, 0) [(0,±0.5)℄ positions along the sides of the ell (the same solution with the x and y dire - tions inter hanged was found in the minimization of the free energy). Finally, the density pro�le of the perfe tly aligned two-dimensional solid is plotted in Fig. 5 ( ). Although the phases des ribed above are metastable with respe t to the olumnar phase, they an be stabi- lized for di�erent values of the parti le aspe t ratio. A detailed study of the omplete phase diagram, ne essary to elu idate this point, is a work in progress. We now pro eed to make a omparison between the results for the 2D Zwanzig model with κ = 3 obtained -0.5 -0.25 0 0.25 0.5 -0.5 -0.25 0 0.25 0.5 Figure 4: Density (solid line) and order parameter (dashed line) pro�les of the sme ti -1 (a) and sme ti -2 (b) metastable phases for a value of mean pa king fra tion �xed at 0.7. above and those for hard parallelepipeds with restri ted orientations and the same value of κ [23℄. This ompari- son will show the di�eren es in phase behaviour between three and two dimensions as predi ted by Fundamental- Measure Theory (whi h, as already pointed out, on- forms with the dimensional rossover riterion). As shown in Ref. [23℄, hard parallelepipeds exhibit a se ond- order phase transition between isotropi and plasti solid phases. As density in reases the system goes to a dis oti sme ti phase ( on�rmed by simulations) via a �rst-order phase transition, whi h in turn dis ontinuously hanges to a olumnar phase and then to an oriented solid. By ontrast, the present model shows that, in two dimen- sions, the isotropi phase exhibits a �rst-order transition to a olumnar phase that is stable until very high pa k- ing fra tions (more stable that plasti , oriented solid and di�erent sme ti phases). As a onsequen e, one expe ts that the orresponding surfa e phase diagrams be also Figure 5: Density (a) and order-parameter (b) pro�les of the plasti solid phase. ( ): Density pro�le of the perfe tly ori- ented solid. di�erent. B. Surfa e phase diagram In this se tion we deal with surfa e phenomena. In the �rst part we will on entrate on the semi-in�nite wall- isotropi interfa e of a HR �uid, while in the se ond part we will fo us on the slit geometry. We will demostrate the presen e of omplete wetting, apillary ordering and layering transitions in the on�ned two dimensional hard rod �uid. For a detailed dis ussion on general grounds of the phase behavior and riti al phenomena of a on�ned by a single wall �uid see Ref. [24℄. 1. The wall-�uid interfa e The intera tion between the isotropi �uid phase and a hard wall was studied by al ulating the one-dimensional equilibrium density ρ(x) and order-parameter Q(x) pro- �les through the ex ess surfa e free-energy minimization [see Eq. (16)℄. The hemi al potential µ of the �uid phase at in�nite distan e from the wall was varied within the range of isotropi -phase stability, i.e. µ ∈ [−∞, µ0] (µ0 being the value at whi h the I-C phase transition o urs). It is well known that the presen e of a hard wall in a sys- tem of elongated parti les indu es parallel alignment of the parti le axes with respe t to the wall [25, 26℄. This preferential alignment is a result of the entropi depletion e�e t. In the parallel on�guration, the enters of mass of the parti les are mu h loser to the wall, so the gain in volume per parti le is larger and, as a onsequen e, the on�gurational entropy of the system is maximized. This e�e t is responsible for the o urren e of a biaxial nemati phase whi h breaks the orientational symmetry in a three-dimensional nemati �uid [4℄. The same deple- tion me hanism is at work in 2D, as we will show below. The results from the minimization are shown in Figs. 6 (a) and (b) for an undersaturation of β∆µ = −1.1×10−4. As we an see from the �gure, the density and order- parameter pro�les indi ate olumnar order near the wall, whi h propagates several olumnar periods into the �uid phase. The wall-�uid intera tion enhan es the orienta- tional order near the surfa e and the adsorption of par- ti les, reating a stru tured layer with olumnar-phase symmetry whi h grows in width with in reasing hem- i al potential and diverges at µ = µ0. Thus, omplete wetting by a olumnar phase o urs at the wall-isotropi interfa e. This result is shown in Fig. 7 (a) where the ex- ess surfa e free-energy γ and the adsorption oe� ient Γ are plotted against ∆µ = µ−µ0. As we an see, Γ grows ontinuously, ultimately diverging logarithmi ally with ∆µ (see inset of �gure). The ex ess surfa e free energy γ has a maximum, and at this point the adsorption passes through zero. This result is dire tly related to the interfa- ial Gibbs-Duhem equation, Γ = −dγ/dµ, whi h relates the adsorption oe� ient with the �rst derivative of the ex ess surfa e free-energy with respe t to bulk hemi al potential. At µ0 the ex ess surfa e energy is equal to the wall-isotropi surfa e tension γWI, whi h is in turn equal to the sum of wall- olumnar and olumnar-isotropi sur- fa e tensions, γ(µ0) = γWI = γWC + γCI (the Young's equation for omplete wetting). 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Figure 6: Density (a) and order parameter (b) pro�les of the wall-isotropi �uid interfa e. The undersaturation is �xed to β∆µ = −1.1× 10−4. We have arried out a logarithmi �t of the adsorp- tion oe� ient with respe t to undersaturation β∆µ = β (µ− µ0), and we �nd that Γσ = τ1+τ2 ln [β|∆µ|], with τ1 = 0.02387 and τ2 = −0.03396. Then, integrating the interfa ial Gibbs-Duhem relation Γ = −dγ/dµ, we �nd the expression βγσ ≈ βγWIσ − [τ1 + τ2 (ln (β|∆µ|) − 1)]β∆µ, (17) whi h approximates the ex ess surfa e free energy near omplete wetting. The above expression is plotted against β∆µ in Fig. 7 (b), where the results from dire t al ulation of βγσ, using the equilibrium density pro�les obtained, are also plotted. As we an see the agreement is ex ellent even for relatively high values of undersatu- ration. 0 2 4 6 8 σ 10-4 10-2 100 0.4(a) 0 0.01 0.02 0.03 0.04 0.05 0.136 0.138 0.142 0.144 Figure 7: (a): ex ess surfa e free energy (solid line) and ad- sorption oe� ient (dashed line), in redu ed units, against β∆µ. The inset shows Γσ vs. β∆µ in logarithmi s ale. (b): ex ess surfa e free energy vs. βµ in the neighborhood of zero undersaturation. The open ir les show the values obtained from the numeri al minimization, while the solid line repre- sents the analyti urve obtained by integrating the interfa ial Gibbs-Duhem relation with the �tted logarithmi dependen e of the adsorption oe� ient (see text). The solid ir le shows the value of the W-I surfa e tension βγWIσ = 0.13822 To al ulate the stru tural and thermodynami prop- erties of the olumnar-isotropi interfa e, we have imple- mented a numeri al s heme already used in Ref. [27℄, onsisting of minimizing the surfa e ex ess free energy γ in a box of width h ontaining a stripe of a few olum- nar layers surrounded by isotropi material with periodi boundary onditions. h is hosen su h that the density pro�les an easily a ommodate the two interfa es and go to the oexisten e �uid density at the periodi bound- ary. A typi al result from this al ulation is plotted in Fig. 8 (a) and (b) for the density and order-parameter pro�les, respe tively. Thus, the I-C interfa ial tension an be al ulated as half the ex ess surfa e free energy resulting from the minimization. We have found a value of βγICσ = 0.00672. -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 Figure 8: Density (a) and order-parameter (b) pro�les of a numeri al box ontaining two isotropi - olumnar interfa es. Finally, to verify that Young's law for omplete wet- ting holds, we need to al ulate the surfa e tension of the wall- olumnar interfa e. To onstru t density pro- �les ompatible with this semi-in�nite interfa e, one has to establish a boundary, at the side of the omputational box opposite to the wall, and pla e, beyond the bound- ary and into the bulk, a periodi ally stru tured pro�le, hoosing the phase (i.e. the value of the pro�le at the boundary) arbitrarily within the bulk period. Although this re ipe an in prin iple be implemented, we have ho- sen to �x bulk I-C oexisten e onditions in a on�ned olumnar phase and al ulate the density pro�le of the system on�ned between two walls; the separation be- tween the walls was hosen large enough so that the ef- fe ts of having a �nite interfa e penetration length aused by the presen e of the on�ned external potential an be negle ted. Also, in order to ensure that ommensurabil- ity e�e ts an be ignored, the distan e between the walls was set to a (large) integer number of equilibrium periods of the olumnar phase. The results from these al ula- tions are plotted in Fig. 9 (a) and (b). The W-C surfa e tension al ulated as half the value of the ex ess surfa e free energy results in βγWCσ = 0.13150, ompatible with Young's law in onditions of omplete wetting of the W-I interfa e by the olumnar phase. 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Figure 9: The density (a) and order parameter (b) pro�les of two wall- olumnar interfa es 2. Capillary ordering This se tion is devoted to a study of the e�e t of on- �nement of a 2D HR �uid on the thermodynami and stru tural properties of the �uid. In parti ular, we are interested in the enhan ement of the orientational and layering ordering due to on�nement, and the ommen- surability e�e ts exhibited by a layered phase sandwi hed between two hard walls at a distan e that may or may not ommensurate with the period of the bulk olumnar phase. It is well known that, under ertain ir umstan es (related to the nature of the �uid-�uid and surfa e-�uid intera tions), a �uid inside a pore an exhibit apillary �rst-order phase transitions between two di�erent phases at a hemi al potential below the bulk oexisten e value. An example of this phenomenon is the re ently studied apillary nematization [4℄ and sme tization [8℄ of a liquid rystal �uid inside a pore. The bulk ondensed phase may have uniform (nemati ) or nonuniform density pro- �les. For the latter ase, apillary layering transitions be- tween interfa ial phases with di�erent number of sme ti layers [8℄ an also be found. Here we will show that these apillary and layering phase transitions are not unique to 3D system. They are also present in 2D anisotropi �uids whi h an stabilize layered phases with di�erent spatial symmetries, su h as the olumnar phase. With a view to �nding the e�e ts of on�nement on olumnar ordering in a HR �uid, we have minimized the ex ess surfa e free energy with respe t to the density pro- �le for the parti ular ase of HR's with κ = 3. The �uid is on�ned by two hard walls at a distan e H/σ = 30 (in units of the parti le width). As already pointed out, hard walls favor alignment parallel to the wall, as well as ad- sorption of parti les at both surfa es (density and order parameters at onta t are mu h higher than their bulk values). This oupled translational-orientational order- ing near the surfa es propagates into the �uid, reating olumnar ordering. We have found that for low values of the hemi al potential of the bath the density pro�le is stru tureless (ex ept just at the wall onta t), similar to the bulk isotropi phase. In reasing the hemi al po- tential several damped olumnar peaks appear near the wall in a ontinuous fashion, i.e. with their heights in- reasing ontinuously. At some value of the hemi al po- tential, the system exhibits a �rst-order phase transition between a phase with highly damped olumnar peaks to a new phase with mu h stronger olumnar ordering even at the enter of the pore. The typi al density and order- parameter pro�les of both interfa ial phases are shown in Fig. 10 (a)-(d). Although the less-ordered phase exhibits strong os illations in both density and order-parameter pro�les, the peak amplitudes are damped into the pore faster than those of the higher ordered phase. We will take the onvention to all the �rst `isotropi ', and the se ond ` olumnar' surfa e phases. This onvention is jus- ti�ed by the fa t that, just before the transition des ribed above, olumnar ordering in reases ontinuously, start- ing from an isotropi -like density pro�le, as the hemi- al potential is in reased. Thus we annot tra e out a de�nite boundary (a value for µ below that orrespond- ing to �rst order phase transition) below or above whi h the pro�le inside the pore an be onsidered isotropi or olumnar. Only the �rst order phase transition des ribed above an really distinguish two di�erent surfa e phases, one of them less ordered (following our onvention, the isotropi phase) than the other (the olumnar phase). As we an see in the �gure, the latter has 25 olumnar peaks. The transition point is al ulated from the dis ontinu- ity in the �rst derivative of the ex ess surfa e free en- 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Figure 10: Isotropi (a)-(b) and olumnar ( )-(d) phases that oexist at the same hemi al potential bellow µ0. (a), ( ): Density pro�les. (b), (d): Order-parameter pro�les. ergy with respe t to the bulk pa king fra tion η. The orresponding plot is shown in Fig 11 (a). At this point the adsorption oe� ient jumps dis ontinuously from the less- (the damped olumnar) to the higher-ordered phase [see Fig. 11 (b)℄. This surfa e transition point is lo ated below the bulk isotropi - olumnar phase transition [see Fig. 11 (a)℄, 0.54 0.55 0.56 0.57 0.58 0.54 0.55 0.56 0.57 0.58 Figure 11: Ex ess surfa e free energy (a) and adsorption oef- � ient (b) against pa king fra tion of the bulk isotropi �uid. In the �gure at top, the solid ir le represents the transition point between both interfa ial phases, while open square in- di ates the point orresponding to the bulk oexisten e value for isotropi and olumnar phases. showing the presen e of olumnar-order enhan ement in the pore. On further in reasing the hemi al potential up to a su� iently high value (above µ0), we �nd a �rst- order layering transition between two interfa ial olum- nar phases whi h di�er by just a single olumnar layer. The behavior of the ex ess surfa e free energy and the adsorption oe� ient is similar to that shown in Fig. 11 (a) and (b). Alternatively we an �nd the transition from n−1 to n olumnar layers by �xing the hemi al potential and in reasing the pore width H . The two surfa e phase transitions des ribed above, namely �rst-order apillary I-C ordering and (n − 1)� n layering transition, are onne ted in the µ−H surfa e phase diagram through the pe uliar stru ture shown in Fig. 12. The parabola below the bulk transition line orre- sponds to �rst-order transition lines separating regions of stability of the isotropi and the olumnar interfa ial phases, while the straight lines indi ate layering transi- tions. In reasing the hemi al potential from low values to those orresponding to the parabola, the density pro- �les always hange ontinously from a stru tureless to damped olumnar density pro�le. Both types of transi- 28 28.5 29 29.5 30 24 C25 Figure 12: µ − H surfa e phase diagram showing �rst-order apillary olumnar ordering and layering transitions. The pore width overs a range whi h goes from 23 to 25 olum- nar layers, as labeled in the �gure. The open ir les indi ate the ases hosen for al ulations, while the solid line is a u- bi spline interpolation. The horizontal dashed line shows the value of the bulk hemi al potential at the isotropi - olumnar phase oexisten e. tions (the isotropi - olumnar and n−1-n layering transi- tions) oales e in triple points, two of whi h are shown in Fig. 12. At the triple points an isotropi and two olum- nar interfa ial phases with n− 1 and n layers oexist in equilibrium. The set of onne ted of Fig. 12 are similar to those found in MC simulations of the on�ned hard- sphere �uid [2℄. In this work the authors have shown the existen e of apillary freezing of the HS �uid, on�ned in the slit geometry, for hemi al potential values below the bulk freezing transition. The transitions lines in the µ−H surfa e phase diagram follow the same topology of onne ted set of parabolas as found in our system. Some of the topologi al features of this surfa e phase diagram an be elu idated from the Clausius-Clapeyron equation as applied to the interfa ial oexisten e lines. The ex ess surfa e free energy γ(µ,H) along oexisten e is a fun tion of two variables, the hemi al potential µ, and the pore width H . Thus, in�nitesimal hanges in these variables along the oexisten e urve are related through the equation dγα − dγβ = ∆ dH = 0, (18) where the oexisting ondition γα = γβ (for α, β =I,Cn−1,Cn) was used, and ∆u = uα − uβ for any fun tion u. Using the interfa ial Gibbs-Duhem equation ∂γ/∂µ = −Γ and the de�nition of the solvation for e f = −∂γ/∂H , we arrive at , (19) whi h relates the �rst derivative of the hemi al potential with respe t to the pore width with hanges in the sol- vation for e and in the adsorption oe� ient at the tran- sition point. The negative slope of the layering urves is a dire t result of Eq. (19), as the in rement in the ad- sorption is always positive for the (n − 1) → n layering transition, while the hange in the solvation for e is also positive (the latter an be interpreted as an in rement with respe t to the bulk of the ex ess surfa e pressure, whi h is obviously larger for the phase with n layers). For values of the pore width that ommensurate with an integer number of olumnar periods of the bulk olumnar phase, the solvation for e be omes zero and we get a min- imum in the I-C apillary transition urve (see Fig. 12). At ea h side of the minimum the solvation for e hange the sign to positive (left side) or negative (right side) as we ompress or expand the �lm, respe tively, while the hange in adsorption remains positive. The Kelvin equation for apillary ondensation in a slit geometry relates the undersaturation in hemi al poten- tial with pore width H as ∆µ = µ(H)− µ0 = − (ρα − ρβ)H , (20) where ρα and ρβ are the bulk oexisting densities of phases α and β (α being the ondensed phase), while γαβ is the surfa e tension of the orresponding interfa e. It was assumed that omplete wetting by the α phase o urs at the W-β interfa e. For a detailed dis ussion of the Kelvin equation in the ontext of liquid rystal phase transitions see Ref. [28℄. Applying this equation using H/σ = 28.88 (the lo ation of the minimum in the µ−H phase diagram of Fig. 12), we obtain an undersaturation β∆µ = −0.0429, while its real value is β∆µ = −0.1255. In the derivation of the Kelvin equation, deviations from the bulk stru ture of the density pro�le arising from the on�nement by the external potential are negle ted. Also, the elasti energy resulting from the ompression or expansion of a layered phase on�ned between two walls is not taken into a ount. These e�e ts might be respon- sible for the di�eren es found between our al ulations and the estimation based on the Kelvin equation. We have he ked that the sequen e of minima in the µ−H phase diagram tends to µ0 as H → ∞, a result predi ted by Eq. (20). Refs. [4℄ and [8℄ showed that the apillary nemati- zation line of the on�ned liquid- rystal �uid ends in a riti al point for small values of the pore width. In or- der to study how the topology of the surfa e phase dia- gram hanges in the regime of small pore widths, we have arried out the orresponding al ulations of interfa ial stru ture. We have found that the I-C apillary ordering transition hanges at some parti ular value of H (near its maximum undersaturation represented by the mini- mum in the I-C interfa e oexisting urve) from �rst to se ond order. For lower values of H two riti al points emerge from this single point, the distan e between them in reasing. In Fig. 13 one of these s enarios is shown. As we an see, there is a range of values of H (near the triple points) where the �rst-order apillary order- ing transitions are still present but, between the riti al points belonging to di�erent layering bran hes, olumnar ordering grows ontinously from the isotropi (damped olumnar interfa ial phase) to a highly-ordered olum- nar phase. Layering transitions are always present even for very small H , as will be shown below. An interest- ing feature of this phase diagram is that the lo ation of the triple points moves above the bulk oexisten e value µ0. This indi ates that the interfa ial olumnar phase just below the triple points an be unstable for values of hemi al potentials orresponding to those of olumnar- phase stability at bulk (similar to the apillary evapora- tion of the on�ned �uid). For wide enough slits (those for whi h the parabolas are onne ted) the triple points are pra ti ally lo ated at µ0, as an be observed from Fig. 12. 18 18.5 19 19.5 20 20.5 Figure 13: µ−H surfa e phase diagram showing riti al points (�lled ir les). Number of olumnar layers are indi ated as subs ripts. For even smaller values of H , only layering transitions remain; these end in riti al points lo ated above µ0, as Fig. 14 shows. When the width H is su h that the pore an only a - ommodate one parti le with its long axis perpendi u- lar to the wall (or not more than four or three parti les aligned parallel to the wall) the system is near the one- dimensional limit. It is known that hard- ore systems in this limit do not exhibit �rst-order phase transitions, but even for very narrow slits we an still �nd �rst-order tran- sitions at whi h the density pro�le experien es an abrupt hange inside the pore. In Fig. 15 (a) and (b) we show two oexisting density pro�les orresponding to oversat- urations, β∆µ = 0.51760 and β∆µ = 0.72836, and pore widths H/σ = 4.32 and H/σ = 3.14 in (a) and (b), re- spe tively. The �uid inside the pore undergoes a phase transition, whi h dramati ally hanges the stru ture of the interfa ial density pro�les by in reasing the heigh of 11.4 11.45 11.5 11.55 11.6 7.8 7.85 7.9 7.95 8 Figure 14: µ −H surfa e phase diagrams for small values of four [Fig. 15(a)℄ or three [(b)℄ density peaks inside the pore. IV. CONCLUSIONS In this arti le we have shown that 2D �uids omposed of anisotropi parti les intera ting via hard- ore repul- sion and on�ned in a slit geometry exhibit a omplex and ri h interfa ial phase behavior. Apart from �rst- order apillary olumnar ordering, we have also found layering transitions in this system. These results are sim- ilar to those found in 3D liquid- rystal �uids on�ned in a pore, where apillary sme tization and layering phe- nomena were also found [8℄. In view of these similarities, we an extra t the on lusion that, independent of the system dimensionality and the pe uliarities of the layered phases, either sme ti or olumnar, if the �uid-wall inter- a tion enhan es layered interfa e ordering (homeotropi in ase of sme ti phases, and the entropi ally favored parallel alignment for the olumnar phase), ompatible with the equilibrium bulk phase, then the on�ned �uid exhibits the interfa ial phase transitions des ribed above. In this study we have used as a model a hard-re tangle �uid, and the density and the order-parameter pro- �les were al ulated by minimizing the ex ess surfa e free-energy fun tional resulting from the Fundamental- Measure Theory applied to the two-dimensional Zwanzig model. The orientational degrees of freedom were dis retized, in order to take advantage of having a free-energy fun tional whi h redu es to the exa t one- dimensional fun tional when the density pro�le is on- strained to lie along a line. This property is ru ial to study strongly on�ned �uids (as is the ase in this study), in parti ular when the pore width has only a few parti le diameters in width. As already pointed out in Se . I, some experiments had shown profound similarities between parti le on- �gurations obtained as stationary states of systems of anisotropi grains and those orresponding to the equi- 0 1 2 3 4 5 0 0.8 1.6 2.4 3.2 Figure 15: (a): density pro�les of two oexisting phases (shown with solid and dashed lines) at β∆µ = 0.5176. The pore width is H/σ = 4.32. (b): same as in (a) but for a pore with H/σ = 3.14 and for β∆µ = 0.72836. librium states obtained by density fun tional minimiza- tion [12℄. These similarities an be explained by applying a maximum-entropy prin iple on granular olle tions of parti les, i.e. for a �xed pa king fra tion, externally- indu ed vibrational motion for es the system to explore those stationary states whi h maximize the on�gura- tional entropy (sin e the grains annot overlap). Of ourse, equilibrium statisti al me hani s is unable to pro- pose an equation of state for granular matter, but it ould be possible to predi t that granular matter omposed of anisotropi parti les and on�ned between parallel walls may support a stationary texture onsisting of layers of parti les oriented parallel to the wall. The manner in whi h the grain orientations propagate into the ontainer would depend on the average pa king fra tion and on the frequen y of the external for e. Only at this qualita- tive level an we give some insight into possible omplete wetting phenomena and apillary ordering in granular rod �uids on�ned between two horizontal plates at a distan e slightly larger than the parti le dimensions in the verti al dire tion (thus simulating a two-dimensional system), and also on�ned by one or two verti al planes (these playing the role of hard walls). Some al ulations (not shown here) on the 2D HR �uid show that, for di�erent aspe t ratios, 2D sme ti and rystal phases an be stable over some range of pa k- ing fra tions. It would be interesting to explore whether on�nement suppresses or enhan es bulk ordering, and to study the hanges in the surfa e phase diagram when phases of di�erent symmetries are in luded. Work along this dire tion is urrently in progress. A knowledgments I thank D. de las Heras, E. Velas o, and L. Mederos for useful dis ussions, and E. Velas o for a riti al reading of the manus ript. The author gratefully a knowledges �nan ial support from Ministerio de Edu a ión y Cien ia under grants No. BFM2003-0180 and from Comunidad Autónoma de Madrid (S-0505/ESP-0299) and (UC3M- FI-05-007). The author was supported by a Ramón y Cajal resear h ontra t from the Ministerio de Edu a ión y Cien ia. [1℄ M. S hmidt, and H. Löwen, Phys. Rev. Lett. 76, 4552 (1996); Phys. Rev. E 55, 7228 (1997). [2℄ M. Dijkstra, Phys. Rev. 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0704.1468

Generation of Entanglement Outside of the Light Cone

Superluminal Generation of Entanglement Generation of Entanglement Outside of the Light Cone J.D. Franson University of Maryland, Baltimore County, Baltimore, MD 21250 The Feynman propagator has nonzero values outside of the forward light cone. That does not allow messages to be transmitted faster than the speed of light, but it is shown here that it does allow entanglement and mutual information to be generated at space-like separated points. These effects can be interpreted as being due to the propagation of virtual photons outside of the light cone or as a transfer of pre-existing entanglement from the quantum vacuum. The differences between these two interpretations are discussed. 1. INTRODUCTION Entanglement is one of the most fundamental and nonclassical aspects of quantum mechanics. It is shown here that entanglement, mutual information, and optical coherence can all be generated at two distant locations in less time than it would take for light to travel between them. These counterintuitive effects are possible because the probability amplitude to emit a photon at one location and annihilate it at another location is proportional to the Feynman propagator FD [1]. As noted by Feynman himself [2, 3], FD has nonzero values outside of the forward light cone, as illustrated in Fig. 1. It should be emphasized that this property of the Feynman propagator does not allow messages to be transmitted between space-like separated points, but it will be shown here that it does allow entanglement to be generated outside of the light cone. The fact that the Feynman propagator is not confined to the light cone raises some issues with regard to causality that have previously been discussed by Feynman [2, 3], Hegerfeldt [4], and others [5-13]. Much of this discussion has been concerned with the probability that an excited atom at one location will emit a photon that is absorbed by a second, distant atom after some time interval , as illustrated in Fig. 2. The probability that the second atom will be excited was first considered by Fermi [14], who made several unwarranted assumptions [15] and concluded that there were no effects at all outside of the forward light cone. Other authors [4-13, 15-20] considered the problem in more detail, some of whom [4, 20] concluded that there should be an increase in in apparent contradiction with causality. It was subsequently shown that cannot change outside the light cone when all possible effects are included [5-13], but that correlations between the states of the two atoms could be produced [5, 7, 11]. tΔ 2P 2P 2P This paper generalizes the earlier results to show that entanglement can be generated outside of the light cone as well. It should be emphasized that the generation of correlations or entanglement in this way does not contradict the results of an earlier paper by Milonni et al [9], who considered the expectation values associated with the state of a distant atom and showed that causality is maintained. That paper did not consider the generation of correlations or entanglement between two distant atoms, which do not violate causality. This paper and the references quoted above are all based on the usual perturbative approach to quantum optics or quantum electrodynamics, where these effects can be interpreted as being due to the propagation of virtual photons outside of the light cone. In a separate series of papers, Summers and Werner [21-25] used the more abstract techniques of algebraic quantum field theory [21-28] to show that Bell’s inequality can be violated in the vacuum state of any quantum field theory that satisfies certain assumptions. In a similar manner, Reznik and his collaborators [29-33] have discussed the generation of entanglement outside of the light cone by scalar or Dirac fields, which they interpreted as being due to the transfer of pre-existing entanglement in the quantum vacuum to the atomic states. Thus the usual perturbative approach and the algebraic quantum field approach lead to very different interpretations of these effects. FIG. 1. A plot of the Feynman propagator FD in the xy plane, which is proportional to the probability amplitude to emit a photon at one location and annihilate it at another location. The plot shown here corresponds to a 1 ns delay after a photon was emitted at the origin. The colors of the contour levels were arbitrarily chosen to reflect the wave-like nature of a photon. The value of FD was truncated on the light cone (the yellow ring), where it diverges, but it can be seen that it has a nonzero value arbitrarily far outside the forward light cone as well. Despite the long history of papers on this subject, there is still no consensus of opinion regarding the significance or interpretation of effects of this kind. The goal of this paper is to consider these issues in more detail and within the context of entanglement, quantum information theory, and the Feynman propagator. An attempt will be made to address all of the significant objections that have been raised in the past, such as the difficulty in localizing relativistic free particles, the use of the dipole approximation, and the virtual photon "cloud" associated with dressed atomic states. It is hoped that this will provide some additional insight into what is entailed by Fig. 1 as well as the difference between these two interpretations. The remainder of the paper is organized as follows. In Section 2, perturbation theory and the Feynman propagator are used to calculate the probability amplitude to excite the second atom in the limit where is much smaller than the time required for light to travel between the two atoms. Those results are used in Section 3 to show that the two systems become entangled outside of the forward light cone, and that the entanglement of the two atoms can be increased using post-selection and entanglement concentration. These effects are compared with quantum teleportation and entanglement swapping in Section 4. FIG. 2. Two distant atoms, one initially in its excited state and the other in its ground state. A photon emitted by atom 1 can be absorbed by atom 2 to produce a transition to its excited state. The probability amplitude for this process is determined by the Feynman propagator of Fig. 1 and is nonzero outside the forward light cone. Section 5 considers the impact of these effects on special relativity, including a proof based on the commutation relations of the field operators that superluminal messages cannot be transmitted; it will be shown that the generation of entanglement and mutual information is not limited by that argument. The generation of optical coherence (as conventionally defined) outside the forward light cone is discussed in Section 6, and it is suggested that the definition of optical coherence may need to be reconsidered. The possibility of new types of quantum information protocols, including a quantum time capsule, is considered in section 7. The two interpretations mentioned above are contrasted and compared in Section 8, while a summary and conclusions are given in Section 9. For simplicity, all of the calculations in the main text will be based on the dipole approximation. The effects of higher-order multipoles are included in Appendix A, where it is shown that they are negligible for the situation of interest here. The calculations in the main text consider only bare atomic states, which provide the most straightforward way to derive the effects of interest. Appendix B includes the effects of the virtual photons associated with the use of dressed atomic states. It is found that similar results are obtained in either the bare-state or dressed-state basis, which justifies the use of bare atomic states in the main text. These results do not depend on any relativistic properties of the atoms, but a covariant calculation using second- quantized Dirac operators is discussed in Appendix C nevertheless. 2. PERTURBATION THEORY CALCULATION The effects of interest can be understood by considering two distant atoms labeled 1 and 2 and located at positions and , as illustrated in Fig. 2. At the initial time , atom 1 is assumed to be in its first excited state 1x 2x 0t = 1E while atom 2 is in its ground state 2G , with the field in its vacuum state (no photons present). The system is then allowed to evolve for a time interval , during which there is some probability amplitude that atom 1 may emit a photon and make a transition back to its ground state 1G while atom 2 absorbs the photon and makes a transition to its excited state 2E . Since the Feynman propagator is nonzero outside the forward light cone, the question is whether or not this process could produce a state of the form 1 2 1 2a E G b G Eψ γ φ⊥= + + (1) even if 2 1 c t− > Δx x . Here the state φ⊥ is orthogonal to the other two states and it includes the possibility that atom 1 may have emitted a photon that was not absorbed by atom 2, for example. Another scenario would be to assume that the excited states 1E and 2E correspond to metastable states with zero dipole moment, such as the 2S state of hydrogen. In that case, atom 1 can be assumed to have been in state 1E for times 0t < with negligible interaction with the electromagnetic field, and similarly for atom 2 in state 2G . A dipole moment could then be produced by applying an external electric field to both atoms over the time interval , as in the Lamb shift experiments. To the extent that the interaction with the field is negligible outside of the time interval , the analysis of this situation is the same as that of the original example of Fig. 2. This example has several advantages when dressed atomic states are considered and it will be discussed in more detail in Appendix B. The probability amplitude to find the system in the state b 1 2G E with no photons present [34] at time can be calculated in a straightforward way using commutators and the Feynman propagator. From second-order perturbation theory, which is equivalent [35] to the use of Wick’s theorem in scattering (S-matrix) calculations, the change in the state of the system at time is given by 1 ˆ ˆ( ) ' '' '( ') '( '') . t dt dt H t H t Δ = ∫ ∫ (2) The interaction Hamiltonian ˆ '( )H t is given [36] by 3 ˆ ˆˆ '( ) ( , ) ( , ) H t d t = ⋅∫ r j r A r t (3) where ( , )tj r is the current operator, is the vector potential operator, and is the charge of the electron. The minimum coupling Hamiltonian of Eq. (3) has the advantage of being manifestly covariant in the Lorentz gauge (see appendix C) and is used in quantum electrodynamics. Similar results for the correlations between the atoms were also obtained by Power and Thirunamachandran [11] using the ˆ ( , )tA r e− ⋅r E form of the Hamiltonian. In the limit of small atomic dimensions (dipole approximation), the vector potential can be evaluated at the centers of the atoms and the atomic matrix elements of j reduce to , where /AiE− d G E=d x is the atomic dipole moment, AE is the energy of the excited atomic states, and is Planck’s constant divided by 2π [36]. (The contribution from higher-order multipoles is negligible, as shown in Appendix A.) The projection of equation (2) onto the state 1 2 0G E then gives ( '' ') / 1 ˆ ˆ' '' 0 ( , ') ( , '') 0 iE t tA b dt dt e A t A − −⎛ ⎞= ⎜ ⎟ ∫ ∫ 2 1x x t (4) where 0 is the vacuum state of the field with no photons and it has been assumed for simplicity that the dipole moment is perpendicular to 2 1−x x and along the x-axis. As usual [37], we can write the vector potential as the sum of its positive and negative frequency components ( ) ( )ˆ ˆ ˆ( , ) ( , ) ( , )x x xA t A t A t + −= +x x x (5) where ( )ˆ ( , )xA t − x creates a photon and ( )ˆ ( , )xA t + x annihilates a photon. Only the product ( ) ( )ˆ ˆ( , ') ( , '')x xA t A t 2 1x x contributes to the matrix element M in Eq. (4) and ( ) ( )ˆ ˆ ˆ ˆ0 ( , ') ( , '') 0 0 ( , ') ( , '') 0 .x x x xM A t A t A t A t + −≡ =2 1 2 1x x x x (6) Equation (6) can be simplified using the commutator of the field operators to give ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ0 ( , '') ( , ') [ ( , '), ( , '')] 0 ˆ ˆ0 [ ( , '), ( , '')] 0 . x x x x M A t A t A t A t M A t A t − + + − 1 2 2 1 x x x x (7) The commutator has a simple form in the Lorentz gauge [37], where ( ) ( )ˆ ˆ[ ( , '), ( , '')] ( , ' '')x x FA t A t ic D t t + − = − −2 1 2 1x x x x − (8) for ' [where ' 't t> ( ' '') 1t tθ − = ]. Combining Eqs. (4), (7), and (8) gives ( )2 ' ( '' ') / ' '' ( , ' '')A A iE t t b dt dt e D − −= − − −∫ ∫ 2 1x x t t (9) For a massless particle such as a photon, FD has the explicit value [38, 39] 2 1 2 2 22 2 2 4 ( ) 4 ( ' '') FD x x i x x i i c t t i − = − − − − −x x (10) Here ε is an infinitesimally small quantity that determines the value of any integrals and we have used the 4-vector notation ( , )x ict= x , which demonstrates the covariant nature of the propagator. We will consider a space-like separation with 2 1r c t= − >> Δx x , which gives 2 21/ 4FD i rπ= − . This approximation is valid far outside of the light cone and it is equivalent to neglecting terms that are of order smaller than the remaining terms, as can be seen from a power series expansion of Eq. (10). 2( / )t rΔ The two integrals over time in Eq. (9) can then be evaluated to give ( 2 2 14 b i t e Δ= − Δ + − ) (11) Here is the fine structure constant and is the resonant frequency of the atomic transition. 2 /e cα = /A AEω = Eq. (11) corresponds to the probability amplitude for atom 1 to emit a photon that is absorbed by atom 2. Although it may seem counterintuitive, there is also a probability amplitude for atom 2 to emit a photon and make a transition to its excited state while atom 1 absorbs the photon and makes a transition to its ground state, since a virtual process of that kind need not conserve energy in the intermediate state. This probability amplitude can be calculated in the same way as that leading to Eq. (11) and the total probability amplitude from both processes is given by [ 02 2 1 cos( ) .2 = − − Δ ]t (12) Eq. (12) shows that there is a probability amplitude for the two atoms to exchange a photon even though they are space-like separated, and this entangles the two systems as discussed in the next section. It can be seen that the effects are only appreciable for atomic separations that are at most a few orders of magnitude larger than . As a result, the predicted correlations are experimentally observable in principle but an actual experimental test would be difficult. It should be emphasized once again that these results do not contradict an earlier paper by Milonni et al. [9], which only considered the expectation value of the state of atom 2 and not the correlations between the two atomic states. Commutator techniques are used in section 5 to give a more general proof that the expectation values associated with atom 2 cannot change outside of the light cone, as would be expected from causality. 3. ENTANGLEMENT GENERATION AND CONCENTRATION By definition, an entangled state is any quantum state that cannot be written as the product of two or more single-particle states. Eq. (1) does not appear to be factorable, but the situation is complicated by the presence of the φ⊥ term and a more careful analysis is required in order to demonstrate that the state ψ actually is entangled. If we could simply ignore the φ⊥ term we would have the entangled state 1 2 1 2' a E G b G Eψ = + , where the coefficients and have been normalized to give unit probability. That term cannot be ignored, however, and one way to include its effects is to ask whether or not the state ψ of Eq. (1) can be written in the form 1 2ψ = Ψ Ψ (13) where 1Ψ represents the most general form of the system consisting of atom 1 and any localized field associated with it: ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 ˆ ˆ( ) 0 ˆ ˆ( ) 0 G c n a a E d n a a Ψ = ⋅⋅ ⋅ + ⋅ ⋅ ⋅ (14) Here creates a photon with wave vector and †ˆia ik { }1( ic n ) and { }1( id n ) are arbitrary complex coefficients. A similar expression exists for the most general form of the system consisting of atom 2 and any field associated with it. Taking the projection onto the vacuum state gives 1 1 1 1 2 2 2 2 0 0 0 (0) (0) (0) (0) ' . c G d E c G d E = ⎡ + ⎤⎣ ⎦ ×⎡ + ⎤ =⎣ ⎦ (15) But 'ψ is entangled and cannot be written in this form [40], which implies that Eq. (13) cannot hold. Thus the two systems and must be entangled as well, even without the post- selection process described below. This result corresponds to the use of bare atomic states, but a more general result is derived in Appendix B for dressed atomic states. 1S 2S Most applications of quantum information would use only the atoms independent of the state of the field, and an average (trace) over the field states would then have to be taken. Although ψ itself is a pure state, the trace would produce a mixed state with a density matrix ρ̂ that would no longer be entangled. It can be shown [11], however, that the atoms are still correlated if a trace is taken over the field states. Entangled states of the atoms alone could be produced, at least in principle, by using an array of single-photon detectors to determine whether or not any photons were present in the final state. It will be assumed that suitable detectors are located throughout the region surrounding the two atoms and that they are all turned on shortly after time tΔ , with a response time that is comparable to . If no photons are detected, the system would be projected into the entangled state 'ψ at a time , which is outside the forward light cone. ' ~tΔ Δt The use of an array of detectors would allow the post-selection process illustrated in Fig. 3a, where the event is rejected if a photon is found after time tΔ , as indicated by the red X in the figure. In principle, this post-selection process could be performed using a large number of pairs of atoms to generate a smaller number of pairs of atoms in entangled states. In order to actually use this entanglement in a quantum information protocol, it would be necessary to wait for a much longer time 2 1 /TΔ ≥ −x x c in order to distribute the results from the detectors to all of the relevant locations. Nevertheless, the entangled states were created at time as recorded by the detectors. FIG. 3. Generation of maximally entangled atoms using post-selection. (a) If a photon is detected after time , the event is rejected. (b) If atom 2 is found in state F, the event is also rejected as described in more detail in Fig. 4. (c) If no photon is found and atom 2 is not in state F, the two atoms are known to have been maximally entangled outside of the light cone. An entangled state of the atoms and their associated fields will be generated outside the light cone without any need for post-selection. Most quantum information protocols would also require that the entanglement of the two atoms be nearly perfect (maximal entanglement). This could be achieved (in principle) using entanglement concentration [41], such as the protocol illustrated in Fig. 4. Here the ground state 2G of atom 2 is coupled with a laser pulse to a third atomic level 2F . By adjusting the intensity of the laser pulse, it is possible to transfer any desired amount of probability amplitude from 2G to 2F and to convert the post-selected state 'ψ into the state ''ψ given by 1 2 1 2 1 2'' 'b E G b G E E Fψ = + + γ (16) where 'γ is a complex constant of no interest. A measurement is then performed to determine whether or not atom 2 is in state 2F . If it is, then the event is discarded as illustrated by the red X in Fig. 3b. If not, then the system will be projected into the maximally entangled state ( )1 2 1 2 / 2E G G E+ . A protocol of this kind can be used to convert pairs of partially entangled atoms into pairs of maximally entangled Bell states. 'N N< It should be noted that the laser pulse and measurements can be completed outside the light cone in a small time interval after time tΔ . Similar techniques have been proposed [42] for use in Zeno quantum logic gates [43], and Reznik et al. [29-33] have considered the use of entanglement distillation with scalar fields. laser FIG. 4. Concentration of entanglement in which partially-entangled pairs of atoms are converted into a smaller number of maximally entangled atoms. A laser beam is used to partially excite atom 2 from its ground state to state F. A pair of atoms is subsequently rejected if a measurement shows that atom 2 is in state F. This process creates a balanced superposition of states 1 2E G and 1 2G E . To summarize this section, it has been shown that an entangled state of the atoms and their associated fields will be generated outside the light cone without any need for post-selection. A maximally entangled state of the two atoms alone can be prepared using post-selection and entanglement concentration, where the entangled states are known to have been produced within time 2 1'tΔ < −x x / c . The atoms are still correlated but not entangled if an average over the field states is taken instead. 4. COMPARISON WITH QUANTUM TELEPORTATION AND ENTANGLEMENT SWAPPING The post-selection processes described above could be used to prepare pairs of maximally entangled atoms. Entanglement between two distant systems can also be prepared using quantum teleportation [44] or entanglement swapping [45], and it may be useful to discuss the fundamental differences between these techniques. In quantum teleportation, two experimenters A (Alice) and B (Bob) would like to transport an unknown quantum state from Alice to Bob. It is assumed that they already share an entangled pair of particles, such as a pair of photons that were generated in an entangled state and then propagated through optical fibers to Alice and Bob. Alice performs a Bell state measurement between her unknown state and her member of the entangled pair, which produces two bits of classical information. Once the classical information is transmitted to Bob, it can be used to regenerate the unknown quantum state by applying a suitable transformation to his member of the entangled pair of particles. Although quantum teleportation is a remarkable and useful process, it is apparent that the information needed to recreate Alice’s unknown state is transmitted in the form of classical information from Alice to Bob at ordinary velocities. This is very different from the situation described above, where the entanglement was created in a time interval 'tΔ outside of the forward light cone and the classical information is used only to determine which events were successful; no corrections are applied to the atoms based on that information. In entanglement swapping [45], a pair of entangled particles is created locally at Alice and Bob. One member of each pair is then transmitted at ordinary velocities to a central location, where a Bell state measurement is made. If we post-select on the results of the measurement, the two distant particles will be left in an entangled state. But in this case the entangled state is not generated until sufficient time has elapsed for two of the particles to have traveled to the same location. In addition, there is no physical interaction of the distant particles via virtual photons as there is in the situation of interest here. 5. SPECIAL RELATIVITY AND CAUSALITY The generation of entanglement outside of the forward light cone may seem counterintuitive, but it does not allow messages to be transmitted faster than the speed of light. These effects are thus consistent with special relativity to that extent, although they raise some questions regarding the postulates of relativity theory. Commutator techniques can be used to provide a simple proof that there can be no net change in the probability of finding atom 2 in its excited state. In analogy with the usual perturbation theory expression for the state vector in Eq. (2), the expectation value of any observable quantity corresponding to an operator at time t is given in the interaction picture [46] by 0 0 0 1ˆ ˆ ˆ ˆ( ) ( ) ' , '( ') 1 ˆ ˆ ˆ' '' , '( ') , '( '') ... t Q t Q dt Q H t dt dt Q H t H t 0ψ ψ ψ ψ ψ ψ ⎡ ⎤= + ⎣ ⎦ ⎡ ⎤⎡ ⎤+ +⎣ ⎦⎣ ⎦ (17) Here the interaction Hamiltonian ˆ '( )H t involves the free-field (Heisenberg picture) operators and , which satisfy the usual commutation relations. The probability of finding atom 2 in its excited state is given by the expectation value of the projection operator ˆ ( , )tA x 2ˆ ( , )tA x 2 2 2p̂ E E= . Since this operator and the corresponding field operator commute with the Hamiltonian of atom 1 and the field at that location, it follows immediately that cannot depend on the state of atom 1 outside the light cone. On the other hand, the correlated probability described by the operator 12 2 1 1 2p̂ E G G E= does not commute with atom 1 and need not be zero outside of the forward light cone. The fact that there is no net change in can be understood from the fact that there are other processes that must be included as well. For example, energy conservation does not strictly apply for a finite time interval tΔ , where the uncertainty relation / 2E tΔ Δ ≥ holds. As a result, atom 2 could emit a photon and make a transition to its excited state even if atom 1 were not present. The probability to find atom 2 in its excited state with a photon present is reduced if atom 1 can absorb the photon, and this tends to cancel the increase in the probability 2PΔ calculated above. A complete calculation of must include all fourth-order terms and gives no dependence on the initial state of atom 1 [5-13]. Thus the exchange of a photon can produce correlations and entanglement between the two atoms, but those correlations cannot be controlled by an experimenter to send superluminal messages between the two locations. The postulates of special relativity are that (i) the laws of nature are the same in all inertial reference frames and (ii) the speed of light is a constant independent of its source. In order to investigate the impact on postulate (i), consider another reference frame R’ that is moving at a velocity with respect to the original reference frame R in which the two atoms are allowed to interact with the field over the same time interval tΔ . (Here we assume metastable excited states with the dipole interaction controlled by an external electric field.) Since the events of interest are space-like separated, it is possible to choose in such a way that atom 1 is allowed to interact over a time interval that does not overlap with the time interval 1tΔ 2tΔ over which atom 2 is allowed to interact. Moreover, the interaction at atom 1 can occur before or after the interaction at atom 2 depending on the choice of . It can be shown that the same entanglement is produced in any inertial reference frame, in agreement with postulate (i). This result is ensured by the fact that perturbation theory can be put into an invariant form [47]. Although the same entanglement is produced in any reference frame, the photon can only be emitted by atom 1 and travel to atom 2 in some reference frames, whereas it can only travel from atom 2 to atom 1 in other reference frames. As a result, any causal interpretation of this process would depend on the choice of reference frame. In that sense, there is some similarity to the collapse or reduction of the wave function in experiments based on Bell’s inequality [48]. The second postulate assumes that the speed of light is a constant in all reference frames regardless of its source. Special relativity was proposed in the context of classical electromagnetism where the speed of light has a well-defined value, and quantum electrodynamics did not exist at the time. The effects described above correspond to the generation of entanglement outside of the forward light cone by the exchange of virtual photons, which raises some possible issues regarding the definition of the speed of light. For example, do individual photons travel faster than the classical speed of light during the time that they are being exchanged? Are the photons really being exchanged if we are not allowed to detect them in the process? And what is the nature of the photons if their direction of travel depends on the choice of reference frame? Similar questions arise in the interpretation of Feynman diagrams, which consist of space-time points (vertices) where two or more particles interact. The particles propagate between the vertices as described by the Feynman propagator, which connects points that are space-like separated with a small but nonzero probability amplitude. One might ask once again whether or not the particles travel faster than the speed of light over such trajectories. As Feynman put it [3], “possible trajectories are not limited to regions within the light cone” but “in reality, not much of the t − x space outside the light cone is accessible”. It seems apparent that these effects are due to the properties of the Feynman propagator and that the atoms can be treated nonrelativistically. Nevertheless, a covariant calculation using second-quantized Dirac operators is outlined in Appendix C. 6. OPTICAL COHERENCE In the conventional theory of single-photon detection developed by R.J. Glauber [49], the probability of detecting a single photon at position and time is given by dP x t ( ) ( )ˆ ˆ( , ) ( , )dP E t E tη − += x x (18) Here η is a constant related to the detector efficiency and time window. This expression can be evaluated in the same way as using the Feynman propagator, with the result that would be nonzero outside of the forward light cone, in apparent violation of causality. However, Eq. (18) is based on an approximation [8, 9, 50] that neglects the possibility that the detector may emit a photon as well as absorb one (the rotating wave approximation). As described above, that process will cancel the more intuitive one and the actual detection probability is unchanged outside of the forward light cone, as can be shown using Eq. (17) or other methods. 2PΔ dP The concept of higher-order optical coherence plays an important role in quantum optics. If we assume that atom 1 was initially in its ground state and then excited with a short laser pulse at time , we can ask whether or not there is any coherence between the remaining laser pulse and the field in the vicinity of atom 2. In analogy with Eq. (18), the second-order optical coherence at two locations is defined [49] as ( ) ( ) ( ) ( ) 1 2 2 1(2) 1 1 2 2 ( ) ( ) ( ) ( ) 1 1 2 ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , ; , ) ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) E t E t E t E t g t t E t E t E t E t − − + + − + − + 1 2 2 1 1 1 2 2 x x x x x x x x 2 (19) If we accept this definition, then it can be shown that second-order coherence would be generated between the laser pulse used to excite atom 1 and the field at a space-like separated point. Once again, this is a non-physical result due to the use of the rotating-wave approximation, and it suggests that the definition of optical coherence may need to be reconsidered [9,50]. 7. QUANTUM INFORMATION PROTOCOLS The nonclassical nature of quantum information allows a number of potentially useful practical applications, such as quantum computing and quantum key distribution. The ability to generate entanglement outside of the forward light cone would, in principle, allow the possibility of additional protocols that would not be possible otherwise. As a practical matter, the rate at which the entanglement can be generated would be too small to be of any real use, but it is interesting to consider protocols of this kind nonetheless. One example of a quantum protocol of this kind is the quantum time capsule. In a conventional time capsule, information and artifacts describing the current environment are sealed into a capsule that is to be left unopened for some length of time, say 100 years. Classically, there is no way to ensure that the capsule might not be opened sooner than intended. This would be a potential problem, for example, if a well-known politician sealed his or her notes and memoirs in a time capsule with the understanding that the material would remain unavailable until long after his or her death. A quantum-mechanical version of a time capsule that cannot be prematurely opened could (in principle) be implemented if entanglement were generated between two distant locations, say 100 light years apart, as illustrated in Fig. 5. Information could be encoded in the qubits at one location by taking the XOR (exclusive OR operation) between the classical information and the entangled qubits. The classical information could then be destroyed, and it could not be recreated from the qubits at the first location since they have random values [51]. The information could only be retrieved by bringing together the entangled pairs of atoms from the two distant locations and comparing the qubits (a second XOR operation), which could only be done after 100 years in this example. Generating the entanglement at a distance eliminates the possibility of cheating that would occur if the entangled pairs were generated at one location and then allegedly separated by a large distance. Although protocols of this kind are not practical, they do illustrate the fact that the entanglement generated in this way can have unique properties. FIG. 5. Implementation of a quantum time capsule whose contents cannot be accessed for a specified period of time. (a) Two distant quantum memories contain a large number of atoms that are pair-wise entangled. Classical information is stored in one of the quantum memories by taking the XOR of the classical bits with the qubits in the memory and destroying the classical information. (b) The information can only be retrieved when the two quantum memories are brought together at less than or equal to the speed of light and a second XOR is performed. It may also be useful to note that these effects allow mutual information as conventionally defined to be established outside of the light cone. Suppose that pairs of maximally entangled qubits are generated at two space-like separated locations as described above. If the values of the qubits are simply measured, then the classical mutual information is defined by 2 ( , ) ( ; ) ( , ) log ( ) ( )y Y x X p x y I X Y p x y p x p y∈ ∈ = ∑∑ (20) where ( , )p x y is the joint probability distribution of the variables and and X Y ( )p x and ( )p y are their marginal probability distributions. Since the resulting measurements are totally correlated, the process yields bits of mutual information that were generated outside of the forward light cone. 8. INTERPRETATION OF RESULTS Effects similar to those described here were independently predicted based on two different approaches. Starting from the work of Fermi [14] and Hegerfeldt [4], the usual perturbative approach to quantum optics or quantum field theory led to the realization that correlations could be generated between two distant atoms. Here that approach was generalized to show that entanglement and mutual information could be generated outside the light cone as well. This approach involves the exchange of virtual photons and the Feynman propagator (or equivalent calculations). Similar effects were also predicted using algebraic quantum field theory, which leads to a different interpretation that does not involve virtual photons [21-33]. Electromagnetic interactions are generally viewed as being produced by the exchange of real or virtual photons, which is consistent with the usual perturbative treatment of quantum electrodynamics. The photons are assumed to carry momentum and energy and that is responsible for the force between two charges, for example. In that case, the most natural interpretation of the results obtained here would be to assume that the entanglement is generated by the propagation of virtual photons outside the forward light, as suggested by Fig. 1 and substantiated by the calculations of Section 2. This interpretation is complicated by the fact that the photons cannot be directly observed without destroying the effects of interest, while their direction of travel depends on the choice of reference frame, as discussed in Section 5. Summers and Werner independently used the more abstract techniques of algebraic quantum field theory to show that Bell’s inequality is violated by the field operators in the vacuum state of a quantum field [21-25]. Later work in algebraic field theory showed that the vacuum state of the field is entangled provided that the theory satisfies certain assumptions [26-33]. This has prompted an interpretation by Reznik and his collaborators [29-33] in which effects similar to those of interest here (but for a scalar field) were assumed to be due to a transfer of pre-existing entanglement from the quantum vacuum to the atoms, with no requirement for any transfer of information outside of the light cone. The assertion that the quantum vacuum is entangled may seem surprising within the context of quantum optics. Whether a system is entangled or not depends on the choice of basis vectors. In the usual plane-wave number-state basis used in quantum optics, the quantum vacuum corresponds to the product of a large number of ground states of independent harmonic oscillators, which is not an entangled state. But the plane waves do not correspond to spatially separated systems and they do not form a suitable basis for a discussion of entanglement. In algebraic quantum field theory, the states of interest are localized to two or more separated regions and the quantum vacuum corresponds to an entangled state in that basis. Regardless of whether the quantum vacuum is entangled or not, the vacuum fluctuations at two different locations are correlated. For example, it follows from Eq. (8) that ˆ ˆ0 ( , ') ( , '') 0 ( , ' '')x x FA t A t ic D t t= − −2 1 2 1x x x x − (22) which shows that the field operators at two different locations are correlated, as qualitatively indicated in Fig. 6a. The algebraic quantum field approach suggests that these local fluctuations may be responsible for the change in the state of the two atoms without any requirement for a transfer of information outside of the light cone. position vacuum state: virtual photon position vacuum state: FIG. 6. (a) Correlations between the electric fields E due to vacuum fluctuations at two distant locations, which could produce correlated changes in the states of two atoms. (b) The state of an atom cannot change without a change in the state of the field, which corresponds to the emission or absorption of virtual photons. But it seems to me that there is something missing here, namely the virtual photons illustrated in Fig. 6b [53]. If a test particle is placed in the electromagnetic field in its vacuum state, then conservation of momentum does not allow a change in the momentum of the test particle unless the state of the field changes as well. Such a change in the field corresponds to the emission or absorption of a virtual photon. The form of the field operators also shows that there can be no change in the state of the atoms unless there is a change in the state of the field corresponding to the emission or absorption of a virtual photon. This suggests that the quantum vacuum alone is not sufficient to produce the effects of interest here, and that the emission and absorption of virtual photons is required. The correlations between the field operators can be further understood if we insert a complete set of basis states in the left-hand side of Eq. (22). Since the vector potential only couples the vacuum to single-photon states, this can be written as ˆ ˆ ˆ ˆ0 ( , ') ( , '') 0 0 ( , ') ( , '') 0x x x x A t A t A t k k A t=∑2 1 2 1x x x x (23) where k is a single-photon state with wave vector . In the presence of two test charges, each of the matrix elements on the right-hand side of the equation correspond to the emission or absorption of a photon, whose propagation is described by the Feynman propagator as before. Eq. (23) suggests that the correlations between the vacuum fluctuations at two locations are maintained by the exchange of virtual photons, which is consistent with the fact that the states of the two atoms cannot change in a correlated way without such an exchange. In contrast, the algebraic quantum field theory approach considers the correlations between the field operators without considering their origin. As a result, the propagation of virtual photons does not enter into those discussions. Unless we are willing to abandon the role of photons in electromagnetic interactions, the discussion above shows that the effects of interest here are due to the propagation of virtual photons outside of the forward light cone. 9. SUMMARY AND CONCLUSIONS It has been shown that entanglement, optical coherence, and mutual information can all be generated between two space-like separated points. An entangled state of the atoms and the field can be generated outside the light cone without any need for post-selection, while the latter can be used to identify pairs of atoms that were maximally entangled after an arbitrarily short amount of time. The analysis in the main text was based on the use of bare atomic states, but similar results are obtained in the dressed-state basis in Appendix B, where it is shown that the entanglement generated during the time interval tΔ is independent of the “cloud” of virtual photons that may have existed before that time. This justifies the use of bare states in the main text. It has been suggested that these results do not demonstrate any “real” entanglement because useful results can only be obtained using post-selection. But Eq. (15) clearly shows that the two systems (including the field) become entangled without any post-selection or entanglement concentration; a generalization of this result to dressed atomic states is given in Appendix B. Even in the case where post-selection is used, no corrections need be applied to the atoms and they must have been entangled during a time interval 2 1'tΔ < −x x / c , as recorded by the photon detectors. These counterintuitive effects are due to the fact that the Feynman propagator has nonzero values outside of the forward light cone. Hegerfeldt [52] has noted that this is an unavoidable feature of any quantum field theory where the energies of the particles are bounded from below. If electromagnetic interactions are viewed as being produced by the exchange of virtual photons that carry energy and momentum, then the results obtained here can be interpreted as being due to the propagation of virtual photons outside of the light cone. Whether such effects should be considered to be superluminal is a semantic issue, since the photons cannot be directly observed and only the resulting correlations between the atoms can be measured. An alternative interpretation has been suggested based on algebraic quantum field theory, where the quantum vacuum is considered to be an entangled state [21-33]. In that case, the effects of interest here could be interpreted instead as being due to the transfer of entanglement from the quantum vacuum to the atoms without the need for an exchange of particles or information outside of the forward light cone. Although that is perhaps a valid interpretation, it does not explain how the correlations in the vacuum fluctuations are produced or maintained. My personal preference would be to retain the usual role of virtual photons in quantum electrodynamics. Regardless of the interpretation, a number of questions remain [53]: Are there any feasible experiments to test effects of this kind outside of the light cone? If an experiment were to be performed, would these effects be observed? And are there any alternative theories that do not have this property? Further research of this kind appears to be a natural extension of the earlier work on Bell’s inequality. ACKNOWLEDGEMENTS I am grateful to B.C. Jacobs, T.B. Pittman, and M.H. Rubin for their comments on the manuscript. APPENDIX A: DIPOLE APPROXIMATION Eq. (4) in the text is based on the dipole approximation, which is valid when the characteristic dimensions of the atom are much smaller than the wavelength of the photons. Since the wavelength does not explicitly appear in the commutator approach used here, it may be useful to verify that the dipole approximation is valid for the situation of interest. It will be shown that the contribution from the higher-order multipoles is negligible for r c tΔ and the only significant contribution is from the dipole term. The basic approach will be to evaluate the commutator of the field operators before the atomic matrix elements are calculated. Combining Eqs. (2) and (3) in the text gives 1 ˆ ˆ0 ( ) ' '' 0 ' ( ', ') ( ', ') ˆ ˆ'' ( '', '') ( '', '') 0 . t dt dt d t d t t E G t⎡ ⎤Δ = ⋅⎣ ⎦ ⎡ ⎤× ⋅⎣ ⎦ ∫ ∫ ∫ r j r A r r j r A r (A1) From Eq. (7) in the text, the matrix element M of the field operators is given by 22 2 2 ˆ ˆ0 ( ', ') ( '', '') 0 ( ' '', ' '') 4 ' '' ( ' '') x x FM A t A t ic D t t c t tπ ≡ = − − − − r r r r (A2) We can expand this in a Taylor series about the point p given by 2' =r x , 1'' =r x , , and ' 0t = '' 0t = ( ) ( )2 22 2 ' ' ... 4 ' ' c f f M x x y r x y ⎡ ⎤∂ ∂⎡ ⎤ = + − + −⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ ∂ ∂⎡ ⎤ ⎡ ⎤ + +⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ (A3) Here we have defined 2 1r = −x x as before and the function f is defined as 2 2 ' '' ( ' '') c t t − − −r r 2 (A4) The partial derivatives are given by 2 12 42 2 2 22 2 2 2 ' '' 2 ' ' '' ( ' '') 2 ' '' ' ' '' ( ' '') x rc t t c t tf t c t t −∂⎡ ⎤ ⎢ ⎥= − = − −⎢ ⎥ ⎢ ⎥∂⎣ ⎦ − − −⎢ ⎥⎣ ⎦ −∂⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥∂⎣ ⎦ − − −⎢ ⎥⎣ ⎦ (A5) for example, with similar expressions for the other terms. For simplicity, we can take along the z axis, so that only the partial derivatives with respect to and are nonzero. 2 1−x x 'z ''z Inserting this into Eq. (A1) and taking the projection onto 1 2G E gives the probability amplitude : b ( ) 2 2 2 ( ) 4 2ˆ' ( ', ') 1 ' ... 2ˆ'' ( '', '') 1 '' ... . b dt dt i c r E d j t z z G G d j t z z E × − − +⎜⎢ ⎥ ⎝ ⎠⎣ ⎦ ⎡ ⎤⎛ ⎞ × + −⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ (A6) It can be seen that the second term in the expansion is smaller than the first by a factor of , where is the characteristic dimensions of the atom. Thus in the limit of we get /Ad r Ad /Ad r 1 2 22 2 2 1 1 ˆ' '' ' ( ', ') ( ) 4 ˆ'' ( '', '') . b dt dt E d j i c r G d j t E ⎡ ⎤= ⎣ ⎡ ⎤× ⎣ ⎦ ∫ ∫ ∫ t G⎦ (A7) The identity /x A xj iE d= − does not depend on the dipole approximation and can be derived from the fact that [36]. Eq. (A7) is thus equivalent to Eqs. (9) and (10) in the text. ˆˆ/ / , /xdx dt p m x H i⎡ ⎤= = ⎣ ⎦ These results show that the contributions from the higher-order multipoles are negligible when the separation between the atoms is much larger than their dimensions. This situation is very different from the usual case in which electromagnetic energy is emitted by atom 1 and then travels at the speed of light to atom 2, where it can be absorbed; there the ratio of the dipole to quadrupole contributions is independent of the distance, for example. In that case the largest contribution is on the light cone where FD diverges and the Taylor’s series expansion is not valid. Here we are considering the opposite limit, where the multipole contributions can be completely neglected. APPENDIX B: DRESSED ATOMIC STATES AND VIRTUAL PHOTONS The analysis in the text assumed an initial state in which atom 1 was in its bare excited state 1E and atom 2 was in its bare ground state 2G with no photons present initially. It is valid to consider the time evolution of an initial state of this kind, at least as a Gedanken experiment, and most of the previous discussion has been based on this example. It is shown here that similar results are obtained in the more realistic case in which the initial state corresponds to dressed atomic states that have some probability amplitude to include one or more virtual photons. The entanglement generated during the time interval tΔ will be shown to be independent of any virtual photons that existed before that time to lowest order in perturbation theory. The presence of divergent diagrams, such as the self-corrections to the mass and charge of the electron, require that the theory be renormalized. The calculations can then be performed using the physical mass and charge, as is implicitly done in most quantum optics calculations. It will be assumed that the theory has been renormalized in this way, so that the remaining interaction is small ( 1α << ) and perturbation theory can be used. After the renormalization, the remaining interaction of the atoms with the field will produce energy eigenstates with some probability amplitude for the presence of a virtual photon, which we will refer to as dressed states. The virtual photon “cloud” associated with a dressed atomic state is illustrated in a very schematic way by the black arrows in Fig. 7a. We will refer to an eigenstate of the Hamiltonian that does not include the interaction with the field (but with the theory renormalized to use the physical mass and charge of the electron) as a bare atomic state. Including dressed atomic states in the analysis raises a number of questions: Does the presence of the virtual photons in the initial dressed states affect the matrix elements and the transition amplitudes? Is the entanglement between dressed or bare states, and how would that affect the results of any measurements? And would the virtual photons in the final dressed states have any effect on the outcome of the detectors used in the post-selection process? In order to investigate these issues, we will make two basic assumptions: (i) The interaction between the field and the atoms is weak ( 1α << ) after renormalization, so that we can consider only the lowest-order effects. (ii) The interaction Hamiltonian that couples the atomic states of interest ( E and G ) can be effectively turned off before time 0t = and after time , or at least reduced to a negligible value outside the time interval t = Δt tΔ . FIG.7. Comparison of the dressed state basis with the bare state basis. (a) After renormalization, the dressed atomic states will include virtual photons (represented by black arrows) produced by atomic transitions other than the one of interest for times . (b) During the time interval 0t < tΔ , the interaction between the ground state and first excited state is turned on by an external field, which produces additional virtual photons represented by red arrows. (c) The effects of the additional photons are independent of the virtual photons that existed before to lowest order, so that the situation is equivalent to using the bare initial state with no virtual photons. (d) During the time interval , the analysis in the bare state basis includes only the virtual photons generated by the atomic transition of interest. To lowest order, this gives the same results as the dressed state analysis of (a) and (b). As mentioned in the text, an example of a situation in which the interaction Hamiltonian can be modulated is the case in which the excited dressed states 1 'E and 2 'E correspond to metastable states with zero dipole moment. Atom 1 can then be assumed to have been in dressed state 1 'E for times and similarly for atom 2 in dressed state 0t < 2 'G . The application of an external electric field can be used to produce a dipole moment for both atoms over the time interval , as in the Lamb shift experiments. We will work in the interaction picture where the unperturbed Hamiltonian ˆ ( )H t is chosen to include the effects of and the coupling to the field that exists for , while the interaction Hamiltonian 0t < ˆ '( )H t will include only the coupling term that is turned on during time tΔ . Modulating the interaction in this way provides a clear definition of the initial conditions that apply to the problem. Moreover, the modulation could be applied using electromagnetic pulses traveling in free space, so that there is no change in the boundary conditions associated with the modulation. This avoids the ambiguities that have been associated with some of the earlier discussions. It should also be noted that the virtual photons that exist prior to all 0t = correspond to atomic transitions other than the G E↔ transition of interest, which has zero matrix elements for . 0t < We begin with a generalization of Eqs. (13-15) in the text, which showed that 1 2ψ ≠ Ψ Ψ in the bare state basis. Here the dressed atomic states may contain correlations between the two atoms even before , and the bare atomic states used in the main text need to be replaced with more general eigenstates, such as 1 2 1 20 ', ',0E G E G→ ' . (B1) Here 1 2', ',0 'E G corresponds to the perturbed eigenstate produced by the interaction with the field in 0Ĥ , which will no longer factor into two single-atom states as before. The 0 ' in Eq. (B1) reflects the fact that the field will no longer be in the bare vacuum state and it may contain virtual photons. Similar notation will be used for the other dressed states of interest. The dressed atomic states that exist outside of the time interval tΔ are eigenstates of 0Ĥ . To lowest order in perturbation theory, they can be written in the form 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ', ',0' 0 ( , ) ( , ) ( , , , ) , ( , ) 0 0 E G E G c i F G d i E F e i j F F f i j F F g G G h E E k k k p k p (B2) for example, with similar expressions for the other dressed states. Here ε is a constant of the same order as ˆ 'H , β is a normalization constant on the order of unity, c through represent arbitrary complex coefficients, and 1iF and 2iF represent atomic states other than the ground state or first excited state. The form of this equation follows from the fact that 0Ĥ does not directly couple the ground and first excited states. The second and third terms in Eq. (B2) correspond to the emission of a single virtual photon and an atomic transition to 1iF or 2iF . The fourth term corresponds to the emission of two virtual photons and two atomic transitions, while the fifth term corresponds to the emission of a virtual photon by one atom followed by its absorption by the other atom. The final two terms correspond to the emission and absorption of a virtual photon by the same atom, which can produce a transition from 1E to 1G , for example, via a virtual state involving 1iF . (Similar transitions to 1 2 0iF G and 1 2 0iE F are of no interest and have not been included.) It should be noted that the term 1 2 0G E can only be produced via two virtual transitions of that kind and is therefore fourth order and negligible. We will consider the subspace S of Hilbert space corresponding to the projection ÊGP onto the ground and first excited atomic states: 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 EGP G G G G G E E G E G G E E E E E (B3) The projection of the initial dressed state onto S gives 1 2 1 2 1 2 ˆ ', ',0 ' 0 0 EGP E G E G g G G h E E (B4) This state can be prepared by local operations on the two atoms and it can be written as the product of two single-atom states to order 2ε . This shows that the initial state is not entangled in the subspace S before the interaction at time 0t = to lowest order. We will now show that the system is entangled in subspace S after the interaction over time interval . The Hamiltonian tΔ 0Ĥ cannot produce any change ψΔ in the state of the system and ψΔ must involve at least one factor of ˆ 'H . To second order in ε , we can therefore drop the 2ε terms in Eq. (B2) and only the first three terms in the initial state can contribute to ψΔ . Since ˆ 'H does not produce any transitions involving the states 1iF , the projection onto the subspace S also has no contribution from the second and third terms of Eq. (B2), so that 1ˆ ˆ ˆ' '' '( ') '( '') 0 . EGP dt dt H t H t Ei Δ = ∫ ∫ G (B5) where we have taken 1β → . This is identical to Eq. (2) in the main text, and the same techniques used there can be used to show that 1 2ÊGP ψ ≠ Ψ Ψ (B6) for . t t> Δ These results show that the system was not entangled in subspace S before the interaction (to second order) while it is entangled in subspace S after the interaction. It is also apparent that the entanglement that is generated during time interval tΔ is independent of any entanglement or virtual photons that may have existed prior to the interaction to lowest order. The situation here is analogous to entanglement in quantum optics, where a photon can be independently entangled in polarization or in energy-time variables, for example. The entanglement in polarization is routinely measured experimentally while simply ignoring any energy-time entanglement. Here the entanglement generated during the interaction over time interval is orthogonal to any entanglement that previously existed, and the latter can be ignored in the same way. We will now consider the effects of using dressed atomic states for post-selection and entanglement concentration, as in Fig. 3. The analysis can be repeated using the same perturbation theory techniques that were used in the main text or by using Eq. (17), where the matrix elements will now be taken in the dressed-state basis. A transition from the dressed state 1 2', ',0 'E G to 1 2', ',0 'G E will require two factors of ˆ 'H as described above, so that a typical term in the integrand is T 1 2 1 2 1 2 1 2ˆ ˆ' '0 ' ' ' ' ' ' ' ' '0 'T G E H G G G G H E G= k k (B7) for example. Since the factors of ˆ 'H already introduce a factor of 2ε , to second order in perturbation theory this reduces to ( ) ( ) ( ) ( ) 2 1 1 2 2 1 1 2 ˆ ' 0 T E G H G G G G H E G . ⎡ ⎤= ⎣ ⎦ ⎡ ⎤×⎣ ⎦ (B8) These matrix elements are the same as in the bare-state basis, and the perturbation calculation will therefore give the same results as were obtained in the main text to lowest order. Measuring the joint state of the two systems is a nonlocal operation [54, 55] that will require a longer time interval as before, but that does not alter the basic conclusion that entanglement can be generated outside of the forward light cone. We can model the detectors as additional two-level atoms initially prepared in their dressed ground states, where a detection event will correspond to finding the atom in its excited dressed state. We do not require a fast response from the detectors, so that they can be turned on slowly after time and the system allowed to evolve over a relatively long time period during the post-selection process (but still shorter than ). As a result, energy conservation will apply. For times , the system is in an eigenstate of the Hamiltonian where no transitions can occur and the detectors could not have registered a count due to any virtual photons in the initial dressed atomic states. The same situation will hold after time tΔ , where atoms 1 and 2 are either in their ground state or a metastable state and cannot supply any energy to excite the detector atoms. As a result, the detector atoms cannot respond to any virtual photons associated with the dressed atomic eigenstates and they can only register photons that were emitted by atom 1 or atom 2 during the time interval tΔ . Thus the post-selection process can be performed as described in the main text. Since we are working in the dressed state basis, Eq. (B8) combined with the post-selection and entanglement concentration described in the main text will produce a final state of the form 1 2 1 2 1 2 1 2 '' ', ',0' ', ',0' / 2 0 0 / 2 E G G E E G G E ψ = ⎡ + ⎤⎣ = ⎡ + ⎤⎣ ⎦ . ⎦ (B9) The second line of the equation holds to second order in the interaction from Eq. (B2), since the dominant terms are already second order. Thus the dressed-state analysis gives the same results as the bare-state analysis in the text to lowest order. A comparison of the dressed-state and bare-state analyses is summarized in Fig. 7. In the dressed-state basis, there will be virtual photons associated with other atomic transitions for , as illustrated by the black arrows in Fig. 7a. Turning on the coupling between the ground and first excited states over time interval tΔ will produce additional virtual photons, as illustrated by the red arrows in Fig. 7b. The effects of these additional virtual photons are independent of the virtual photons that were present in the initial state to lowest order, so that the net result is the same as in the bare-state basis illustrated in Figs. 7c and 7d. This justifies the use of the bare-state basis in the main text and in the discussion of Appendix C. APPENDIX C: COVARIANT CALCULATION It seems apparent that these results are due to the nature of the Feynman propagator for the photons and that the atoms can just as well be treated nonrelativistically. It has been suggested, however, that the calculations should be performed in a covariant way nevertheless to ensure that the results are not an artifact of the nonrelativistic treatment of the atoms. It will be shown in this appendix that the same results are obtained using the second-quantized Dirac theory for the electrons, aside from a small relativistic correction to the atomic matrix elements. In the Lorentz gauge, the interaction Hamiltonian can be written in the form 3 ˆˆ ˆ'H d j Aμμ= −∫ r (C1) Here ĵμ is the current 4-vector in the second-quantized Dirac theory with components (C2) ˆ ˆ ˆ( ) ( ) ( ) ˆ ˆ ˆ( ) ( ) ( ) ρ ψ ψ r r r j r r α r where is the field operator for the electron-positron field and corresponds to the Dirac matrices. The use of perturbation theory will give rise to integrals ˆ ( )ψ r α I of the form 3 ˆˆ ˆ'( ') ' ' I H t dt dt d j Aμμ −∞ −∞ = = −∫ ∫ ∫ r (C3) for example. If the interaction goes to zero after the time of interest, which it does in S-matrix theory as well as in the metastable state example discussed above, then the integral over time can be extended to infinity to give 4 ˆˆI d x j Aμμ = − ∫ (C4) Eq. (C4) is an invariant under Lorentz transformations, and it can be shown that the results of perturbation theory are the same in any reference frame and that perturbation theory is equivalent to the use of Wick’s theorem in scattering (S-matrix) calculations [35]. In the Lorentz gauge, the interaction Hamiltonian includes both the vector and scalar potential operators: 3 ˆ ˆˆ ˆˆ' (H d )ρ= − ⋅ + Φ∫ r j A (C5) Here is the scalar potential operator and Φ̂ ρ̂ is the charge density operator. The contribution from the scalar potential term can be evaluated using the commutator techniques of Appendix A. That contribution vanishes in the limit of r c tΔ , since the only nonzero matrix elements of ρ̂ depend on the partial derivatives of Eq. (A5). Thus the only significant contribution is from the vector potential term that was used in Eq. (3) of the main text. Although the nonrelativistic Schrodinger equation was tacitly assumed in the main text, the actual form of the current operator was never used. Instead, the dipole approximation was derived from the identity (C6) 0ˆˆ/ / , /xdx dt p m x H i⎡ ⎤= = ⎣ ⎦ This relationship must hold at least approximately for the Dirac theory, since the Schrodinger equation corresponds to its nonrelativistic limit. As a result, the matrix elements of in the Dirac theory are related at least approximately to the dipole moment by , just as in the nonrelativistic Schrodinger equation. The dipole moment is to be evaluated between the relativistic eigenstates of the atom [56], which will give a small correction to their value. /A iiE d− Since the matrix elements of the interaction Hamiltonian are the same in the Dirac theory as in the nonrelativistic Schrodinger equation, aside from a small relativistic correction to the dipole moments, it follows from perturbation theory or Eq. (17) that the Dirac theory will give all the same results that were previously described in the main text. This discussion also shows that the same results would be observed in any reference frame, as can be explicitly demonstrated. Hegerfeldt [52] showed that, in relativistic quantum field theory, a free particle that is completely localized inside a finite volume at time 0t = will subsequently have some probability to be found arbitrarily far away after a short time interval. It has been argued [13, 54, 55] that this difficulty in localizing free particles invalidates the usual assumption that the initial state is localized. But Hegerfeldt’s theorem applies to free particles, whereas the electrons in the atomic states of interest here are bound. In addition, they are not strictly localized in the initial state, since the relativistic atomic state corresponds to exponentially decaying probability amplitudes [56], and Hegerfeldt’s theorem does not apply. 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0704.1469

Monte Carlo Simulations of Quantum Spin Systems in the Valence Bond Basis

arXiv:0704.1469v1 [cond-mat.str-el] 11 Apr 2007 Monte Carlo Simulations of Quantum Spin Systems in the Valence Bond Basis Anders W. Sandvik1 and K. S. D. Beach1,2 1 Department of Physics, Boston University, Boston, Massachusetts, USA 2 Theoretische Physik I, Universität Würzburg, Würzburg, Germany Abstract. We discuss a projector Monte Carlo method for quantum spin models formulated in the valence bond basis, using the S = 1/2 Heisenberg antiferromagnet as an example. Its singlet ground state can be projected out of an arbitrary basis state as the trial state, but a more rapid convergence can be obtained using a good variational state. As an alternative to first carrying out a time consuming variational Monte Carlo calculation, we show that a very good trial state can be generated in an iterative fashion in the course of the simulation itself. We also show how the properties of the valence bond basis enable calculations of quantities that are difficult to obtain with the standard basis of Sz eigenstates. In particular, we discuss quantities involving finite-momentum states in the triplet sector, such as the dispersion relation and the spectral weight of the lowest triplet. 1 Introduction Quantum Monte Carlo (QMC) simulations of spin systems have traditionally been carried out in the basis of eigenstates of the spin-z operators Szi , i = 1, . . . , N , i.e., the basis of “up” and “down” spins in the case of S = 1/2 (which is the case we consider here). For the prototypical model of interacting quantum spins, the antiferromagnetic (J > 0) Heisenberg hamiltonian, H = J 〈i,j〉 Si · Sj = J 〈i,j〉 [Szi S (S+i S j + S j )], (1) this basis is clearly natural and convenient, as an off-diagonal operator acting on a basis state just flips two spins or destroys the state. Starting with the work of Suzuki [1], finite-temperature simulation methods employing the spin- z basis were developed in which a quantum mechanical expectation value for a system in D dimensions is mapped onto an anisotropic classical statistical- mechanics problem in D+1 dimensions—the discretized [2–4] or continuous [5,6] imaginary-time path integral. There are now very efficient methods uti- lizing loop-cluster [7,8,5] or “worm” [6] updates of the world-line spin config- urations. These methods have enabled studies of systems with ≈ 104 − 105 spins in the low-temperature (ground-state) limit and much more at elevated temperatures. Loop updates have been developed [9,10] also for the alterna- tive and now frequently used power-series expansion representation [11–14] of the partition function (stochastic series expansion; SSE), where the spin-z http://arxiv.org/abs/0704.1469v1 2 A. W. Sandvik and K. S. D. Beach |Vk> |Vl> <Vl|Vk> Fig. 1. Two valence bond states |Vk〉, |Vl〉 in two dimensions and their overlap graph corresponding to 〈Vl|Vk〉 = 2 N◦−N/2, where N◦ is the number of loops formed (in this caseN◦ = 3 and the number of sitesN = 16). Filled and open circles correspond to sublattices A and B. The sign convention in Eq (3) for a singlet valence bond (i, j) dictates that spins i and j belong to A and B, respectively basis is also normally used. It is in principle possible to adapt these ap- proaches to other local bases, e.g., that of singlet and triplet states of spin pairs on a dimerized lattice. This basis is often used in diagrammatic and series-expansion calculations [15], but its implementation in QMC simula- tions is typically rather cumbersome. Zero-temperature (T = 0) simulations, in which the ground state is pro- jected out of a trial wave function, are also normally carried out in the spin-z basis [16,17]. Here we will discuss an alternative approach to ground- state calculations which turns out to have some unique features enabling access to quantities that are normally difficult to obtain with standard finite- temperature or projector methods. We make use of the valence bond basis, i.e., states in which the spins are paired up into singlets; |V 〉 = |(i1, j1)(i2, j2) · · · (iN/2, jN/2)〉. (2) Here (i, j) denotes a singlet formed by the spins at sites i and j; (i, j) = (| ↑i↓j〉 − | ↓i↑j〉)/ 2, (3) and the total number of sites N is assumed to be even. While in principle one can include all possible pairings of the spins, it is in most cases better to consider a smaller basis in which the sites are first divided into two groups, A and B, of N/2 spins each, and to only consider singlets (i, j) in which the first index i ∈ A and the second j ∈ B [18–20]. In the case of a bipartite lattice, these groups are naturally the two sublattices, as shown in Fig. 1. This restricted VB basis has (N/2)! states and is still massively overcomplete—the singlet space has N !/[(N/2)!]2(N/2+1) dimensions [18]. The VB basis states are all non-orthogonal, overlapping with each other according to the simple loop rule illustrated in Fig. 1. The VB basis was introduced already in the early 1930s [18,21,22] and has played an important role in exactly solvable models [18,23–25]. Later, it became a tool for describing spin liquids—the resonating valence bond (RVB) mechanism introduced by Fezekas and Anderson [26–28], in which the ground state is dominated by short valence bonds. In exact diagonalization Monte Carlo Simulations of Quantum Spin Systems 3 studies, the VB basis is useful in cases where it is a good approximation to only consider a restricted (and incomplete) space of short bonds (spin liquids and other states with no magnetic long-range order) [29–32]. Variational cal- culations in the VB basis have been carried out for the 2D Heisenberg model [20,33]. Furthermore, Liang realized that a variational VB state could be con- siderably improved by stochastically projecting it with an operator (−H)m for large m [34]. Later, Santoro et al. devised a Green’s function method for calculating energies in the VB basis [35]. Despite the promising results ob- tained in these studies, there was, to our knowledge, no further developments of QMC methods in the VB basis until one of us recently introduced two related projector algorithms [36], improving on the schemes of Liang [34] and Santoro et al. [35]. These algorithms have already been applied in studies of quantum phase transitions [37,38] and entanglement entropy [39]. Some previously unnoticed advantages of the VB basis in QMC algorithms were pointed out in Ref. [36]. Here we summarize our recent work on VB projector methods and highlight some of their unique features. We discuss in particular a scheme for “self-optimizing” the trial state out of which the ground state is projected, and also show how to study properties of triplet excitations at finite momentum. 2 Ground state projection Consider a state |Ψ〉 and its expansion in terms of eigenstates |n〉, n = 0, 1, . . ., of some hamiltonian H ; |Ψ〉 = cn|n〉. (4) With C a constant chosen such that the lowest eigenvalue E0 − C is the largest in magnitude, a large number m of repeated operations with C −H projects out the ground state, (C −H)m|Ψ〉 → c0(C − E0)m |0〉+ c1 C − E1 C − E0 |1〉+ . . . , (5) provided that the overlap c0 6= 0. Here we will first consider singlet eigenstates of the Heisenberg model (1), which can be expanded in VB states; |Ψ〉 = fi|Vi〉. (6) Because of the overcompleteness of the VB basis, this expansion is not unique. That, however, does not prohibit the ground state |0〉 to be projected out according to Eq. (5). The Heisenberg hamiltonian can be written in terms of singlet projection operators Hb ≡ Hi(b),j(b) on the interacting spin pairs, {i(b), j(b)}, b = 1, . . . , Nb = DN (for a periodic cubic D-dimensional lattice); H = −J Hb = −J Hi(b),j(b), Hij = −(14 − Si · Sj). (7) 4 A. W. Sandvik and K. S. D. Beach When a singlet projector Hij acts on a VB basis state, one of two things can happen; 1) if i, j belong to the same bond the state is unchanged with a matrix element 1, or 2) if they belong to different bonds these two bonds are reconfigured (“flipped”) with matrix element 1/2; Hij | · · · (i, j) · · ·〉 = | · · · (i, j) · · ·〉, (8) Hij | · · · (i, k) · · · (l, j) · · ·〉 = 12 | · · · (i, j) · · · (l, k) · · ·〉. (9) Here positive-definitness of (9) is directly related to the two sites i and j being in different sublattices. For a frustrated system, where there are operators with both sites i, j in the same sublattice, positive-definitness does not hold [19,20]. For a non-frustrated system the simple bond flip (9) makes for a convenient stochastic implementation of the ground state projection (5). We write the projection operator as (with J = 1 henceforth) (C −H)m = Pr, Pr = Hbr · · ·Hbr , (10) where we have introduced a compact notation Pr, r = 1, . . . , N b , for the different strings of singlet projectors. When a string Pr acts on a VB basis state |V 〉 the result is another basis state, which we denote |V (r)〉, with a prefactor (weight) Wr which is simply given by the number moff of off- diagonal operations in the course of evolving |V 〉 to |V (r)〉; Pr|V 〉 = Wr|V (r)〉, Wr = 2−moff . (11) We here first consider projecting the ground state out of a single VB basis state; later we will consider the use of a more complicated trial state. We consider two ways to calculate expectation values: 〈H〉 = r〈Ψ |HPr|V 〉 r〈Ψ |Pr |V 〉 r Wr〈Ψ |H |V (r)〉 r Wr〈Ψ |V (r)〉 , (12) 〈A〉 = rl〈V |P ∗l APr|V 〉 rl〈V |P ∗l Pr|V 〉 rl WrWl〈V (l)|A|V (r)〉 rl WrWl〈V (l)|V (r)〉 . (13) We will discuss how to estimate these using importance sampling; terms (configurations) of the numerators are illustrated in Fig. 2. We will refer to (12) and (13) as the single and double projection, respectively. In (12), which is an exact (when m → ∞) expectation value only of the hamiltonian (or other operators for which the ground state is an eigenstate) the state |Ψ〉 is in principle arbitrary. It is very convenient to use a state which has equal overlaps with all the VB basis states, e.g., the Néel state |ΨN 〉 (all spins up on sublattice A and down on B). It is easy to see that 〈ΨN |V 〉 = 2−N/2 for any basis state |V 〉. Since H |V (r)〉 = − b Hb|V (r)〉 is a sum of basis states multiplied by factors −1 or −1/2, the overlaps with 〈ΨN | drop out altogether and do not have to be considered further. If the projector Monte Carlo Simulations of Quantum Spin Systems 5 |V ><V | A (a) (b) |V >H Fig. 2. Propagation of a VB state on a 6-site chain. The horizontal bars represent nearest-neighbor Heisenberg interactions (singlet projectors). In the single projec- tion (a) the state is propagated from right to left, and an estimator for the ground state energy is obtained by acting once more with all terms of the hamiltonian. In the double projection (b) the state is projected from the right and the left, and any operator expectation value can be estimated by calculating the corresponding matrix elements between the propagated states strings Pr in (12) are importance-sampled according to their weights Wr, the estimator for the ground state energy is thus E0 = 〈H〉 = −〈md + 12mo〉, (14) where md and mo are, respectively, the number of diagonal and off-diagonal operations Hb|V (r)〉 (and md +mo = Nb). It should be noted that although this estimator is exact in the limit m → ∞, it is not variational. The correct energy may thus be approached with increasing m from above or below. Eq. (13) is valid for any expectation value and in the case of A = H gives a variational estimate of the energy. Using WrWl〈V (l)|V (r)〉 as the sampling weight, the estimator for any 〈A〉 is of the form 〈A〉 = 〈V (l)|A|V (r)〉 〈V (l)|V (r)〉 . (15) In the case of a spin correlation function 〈Si·Sj〉, the matrix element is related to the loop structure of the overlap graph [19,20] (illustrated in Fig. 1): 〈Vl|Si · Sj |Vr〉 〈Vl|Vr〉 +3/4, if i, j ∈ same loop, same sublattice, −3/4, if i, j ∈ same loop, different sublattices, 0, if i, j ∈ different loops. Measuring the spin correlations is hence straight-forward once the overlap- loops have been constructed. Higher-order functions, e.g., dimer-dimer cor- relations 〈(Si · Sj)(Sk · Sl)〉, are also related to the loop structure [40]. Note again that no bond operator Hb can destroy a VB state and that all the states have non-zero overlap with each other. Thus all terms in (13) contribute to the expectation value. This turns out to be an advantage in constructing a Monte Carlo algorithm, as any change made in the operator strings can be accepted with some probability. With an orthogonal basis, such 6 A. W. Sandvik and K. S. D. Beach as the Szi eigenstates, there would be considerable constraints, both in terms of individual operators in the projection [the spin flip operator in (1) can act, without destroying the state, only on anti-parallel spins], and in ensuring a non-zero overlap between the propagated states (the two propagated states have to be identical). Note also that the singlet projectors are non-hermitian in the VB basis. As indicated in (13), and illustrated in Fig. 2, we here propagate two states, |V (l)〉 ∝ Pl|V 〉 and |V (r)〉 ∝ Pr|V 〉, and subsequently compute their overlap and various matrix elements. Propagating |V 〉 with P ∗l Pr and then taking the overlap with |V 〉 is not equivalent term-by-term. To carry out the projection stochastically, the operator strings are stored in arrays [Pα] = [b 1 ][b 2 ] · · · [bαm], where bαi ∈ {1, . . . , Nb} with α = 1, 2 cor- responding to Pr and Pl, respectively, in the double projection; in the single projection α is redundant. A table holds the site pairs i(b), j(b). The state |V 〉 is stored in a list [V ] = [v1][v2] · · · [vN ] where vi = j and vj = i if there is a valence bond at (i, j). Propagation with the bond flips (9) is easily carried out in this representation. The state list [V ] is then first copied into two lists, [V1] and [V2], in which |V (r)〉 and |V (l)〉 are constructed. The simulation can be started with a randomly generated operator string. The strings can can be updated in an trivial way, by changing a number R of operators at random. In either [P1] or [P2], R positions pj , j = 1, . . . , R, pj ∈ {1, . . . ,m} (all different) are generated. Their contents bαpj are picked randomly from the set {1, . . . , Nb} (excluding the old value for each pj). To calculate the Metropolis acceptance probability, the state is propagated with the updated operator string and the number of off-diagonal operationsmoff in (11) is counted. In the single expansion, the acceptance probability is simply Paccept = min W newr W oldr = min[2m −mnew off , 1], (17) whereas in the double projection an overlap ratio appears as well. In the double projection, we change operators only in one of the operator strings at a time, so that only one state has to be propagated. It is customary to define a size-normalized Monte Carlo step (or “sweep”). For projector length m we do m replacement attempts, and so our sweep is independent of the number of operators R replaced in each update. Normally, in Monte Carlo simulations one does not compute the full weight, because it is possible to more speedily calculate just the change in the weight [the weight ratio in (17)]. In the present formulation of the VB projector algorithm the weight is, however, recomputed from scratch each time, because a better scheme is not yet known. Each update hence requires on the order of m operations. In the double expansion, construction of the loops needed to compute the overlap scales as N , but typically m > N and the propagation of the state dominates the simulation. In spite of the need to recalculate the weight, the scheme is sufficiently efficient to compete with other ground state QMC methods. More importantly, as we will discuss in Sec. 4, the VB basis offers access to quantities out of reach for other methods. Monte Carlo Simulations of Quantum Spin Systems 7 1 2 3 4 5 1 2 3 4 5 (a) (b) Fig. 3. (a) Acceptance rate in a double projection and (b) the number of bonds changed in the propagated state in accepted updates for a 16× 16 lattice (symbols with lines; the bare lines are for L = 8) versus the projection length (normalized by the number of sites N). The trial state |V 〉 was a columnar dimer state The optimum number R of operators to replace depends on the accep- tance rate. In Fig. 3(a) we show the acceptance rate for a double-projected 2D system versus the length m of the projection for R = 1− 6. As expected, the acceptance rate decreases with increasing R, but it does not change ap- preciably with m. It also depends only weakly on the lattice size. Multiplying the acceptance rate by R gives the average number of operators changed; it initially increases with R but has a maximum for R = 7 (for L = 16). The optimum R is clearly model/lattice dependent. Another characteristic of the update is the number of bonds changed in the projected state |V (r)〉 as a consequence of the modifications of Pr. This number is shown in Fig. 3(b). It is seen to increase with R, as expected. As a function of m the num- ber of changed bonds decreases. This behavior reflects a loop structure of the singlet-projection operators [41,8], which implies that some changes “up- stream” in the operator string may be healed further downstream in the propagation. For a finite lattice in the limit m → ∞ one would expect a substitution of an operator far upstream in Pr to have no effect on the final propagated state |V (r)〉. This does not imply that this update is inconsequen- tial, as the sampling is over paths, not just the final states in the propagations. In principle, it should be possible to take advantage of the underlying loop structure of the singlet projectors [41,8] to devise a loop update in the VB basis, analogous to such updates in world-line [7,8] and SSE [9] simulations. However, we have not yet been able to construct a scheme that is in practice faster than the trivial random substitution with full state propagation. 3 Self-optimized trial state So far, we have projected the ground state out of a single VB basis state |V 〉. This works well [36], but the rate of convergence with m of course depends 8 A. W. Sandvik and K. S. D. Beach on the state chosen; ideally one would like to maximize the overlap 〈V |0〉. One way to obtain a typically good single-configuration trial state is to first start with an arbitrary one; a regular bond pattern or a randomly generated configuration. After carrying out some projection steps with this state, the current propagated state |V (r)〉 is chosen as a new trial state. Since this state has been generated in the projection it should contribute substantially to the ground state and hence typically will be better than a completely arbitrary one. However, as we will discuss next, we can do much better than this. Liang’s original motivation for introducing a projector technique in the VB basis was to improve on a variational calculation [34]. Previously, Liang, Doucot, and Anderson had studied a variational amplitude-product state for the 2D Heisenberg model of the form [20] |Ψ〉 = fk|Vk〉, fk = h(xbk, ybk), (18) where xbk and ybk are the x- and y-lengths of bond b in VB state k, as il- lustrated in Fig. 4(a). Liang et al. tried power-law and exponential forms depending only on the total length l of the bonds [the ”Manhattan” length l = x + y was used, but defining l = (x2 + y2)1/2 should not change things qualitatively], in addition to keeping several short-bond amplitudes as pa- rameters to optimize [20]. More recently, all the amplitudes were optimized without any assumed form on lattices with up to 32×32 sites, with the result that h(l) ∼ l−3 for long bonds [33]. One of us has also recently showed the more general result h(l) ∼ l−(D+1) within a mean-field approach for a D- dimensional cubic lattice [42]. In 2D the fully-optimized amplitude-product state turns out to be extremely good, with an energy deviating by only ≈ 0.06% from the exact ground state energy and with the long-distance spin-spin correlations reproduced to within 2% [33]. With the state (18) an expectation value is given by 〈A〉 = kp fkfp〈Vp|Vk〉 〈Vp|A|Vk〉 〈Vp|Vk〉 rl fkfp〈Vp|Vk〉 , (19) which can be evaluated using importance sampling of the VB configurations with weight fkfp〈Vp|Vk〉. Liang et al. introduced a very simple scheme for updating the dimer configurations [20], which we here illustrate in Fig. 4(b). Choosing two next-nearest-neighbor sites, i.e., ones on a diagonal of a 4-site plaquette (or, in principle, any two sites on the same sublattice), the two bonds connected to them are reconfigured in the only possible way which maintains only bonds between the A and B sublattices, as shown in the figure. Labeling the two initially chosen sites 1 and 2, and the bonds connected to them b = 1, 2, the Metropolis acceptance probability is, assuming that the bond update was made in |Vk〉, resulting in |Vk′ 〉, Paccept = min h(x1k′ , y1k′)h(x2k′ , y2k′) h(x1k, y1k)h(x2k, y2k) 〈Vp|Vk′ 〉 〈Vp|Vk〉 . (20) Monte Carlo Simulations of Quantum Spin Systems 9 (a) (b) Fig. 4. (a) Definition of the size of a bond b. (b) Reconfiguration of two bonds in an update of the trial state. The two sites marked 1, 2 are chosen at random among all pairs of next-nearest neighbor sites. We can use an amplitude-product state as the trial state in the projector QMC method, using some set of amplitudes not necessarily originating from a variational calculation. In updating the bond configurations, we then also must compute the new weight of the propagation; Pr|Vk′ 〉 = Wk′r|Vk′ (r)〉, and of course 〈Vp|Vk〉 in (20) is replaced by 〈Vp(l)|Vk(r)〉 (in the case of the double projection; for the single projection there is no overlap). The acceptance rate of a state update is similar to that of an operator string update with a smallR, and often we find it advantageous to combine state and operator updates. To save some time, one can tentatively accept/reject a bond update based solely on an amplitude ratio—Eq. (20) without the overlaps—and then calculate the overlap and the propagation weight for a final accept/reject probability only for tentatively accepted bond updates. In the variational calculation 〈H〉 is minimized with respect to all h(x, y). With a recently developed stochastic optimization method [33], all the ∝ N amplitudes can be minimized for moderate-size lattices (up to 32 × 32 sites were considered in Ref. [33]). In principle we could follow Liang [34] and use the best possible variational state as our trial state in the projector QMC method—indeed this can be expected to be the optimum starting point. However, we will now describe a scheme which delivers a trial state nearly as good as the best variational state, at a smaller computational cost. Consider the probability distribution P (x, y) of valence bonds. In an amplitude-product state (18) we would have P (x, y) ∼ h(x, y), were it not for the “hard-core” constraint of only one bond per site. Even with this con- straint, it is clear that the probabilities and amplitudes are related in a mono- tonic way; increasing h(x, y) for some given (x, y), while keeping the other amplitudes fixed, will lead to a larger P (x, y). This fact can be exploited in constructing a good trial state. We define two different probability dis- tributions, P0(x, y) and Pm(x, y), the former being the just discussed bond probability in a trial state of the form (18) and the latter the probability distribution in the projected state. For sufficiently large m, Pm is an exact property of the ground state, whereas P0 is a property of the trial state and is in general different from Pm. However, for given m, we can adjust the ampli- tudes h(x, y) of the trial state such that Pm(x, y) = P0(x, y) for all x, y. If this is done for m sufficiently large, then our trial state has a bond distribution identical to that of the exact ground state. Such a state is often almost as good as the best variational state. The reason that this is useful in practice 10 A. W. Sandvik and K. S. D. Beach 0.0 0.5 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 0.650 0.655 0.660 0.665 0.670 self-optimized projected random columnar dimer 0.0 0.5 1.0 1.5 2.0 0.6694 0.6695 0.6696 0.6697 0.6698 0.6699 0.6700 Projected state Self-optimized trial state Fig. 5. Left: Energy versus projection length for a 16 × 16 system, using three different trial states (labeled in the inset). Right: The energy of the self-optimized trial state compared with the projected energy using that trial state. The dashed line shows the energy evaluated independently using the SSE method [43] is that it is very easy to adjust the amplitudes to achieve self-consistency. Because of the monotonous relationship between h(x, y) and P0(x, y), we can simply increase h(x, y) by some amount if P0(x, y) < Pm(x, y) and decrease it if P0(x, y) > Pm(x, y), and repeat this until self-consistency is achieved. We use the following scheme to update the amplitudes after the kth iteration; ln[h(x, y)] → ln[h(x, y)] + RAN · β(k) · sign[Pm(x, y) − P0(x, y)], (21) where RAN is a random number in the range [0, 1) and β(k) decreases with the iteration step k = 1, 2, . . ., according to β(k) ∝ k−α. For the exponent, we typically use α = 3/4. To evaluate the probabilities P0 and Pm (in two independent simulations; with and without projection of the trial state), the number of Monte Carlo sweeps does not have to be very large, because we only need the sign of the difference of the two probabilities. We normally use on the order of 100-1000 sweeps per iteration. Even if the stochastically evaluated sign in (21) occasionally may be wrong, it is correct on average and the amplitudes typically converge to a self-consistent solution after a few hundred iterations. Due to the stochastic nature of the procedure, self- consistency of course obtains only to within some statistical error, which can be reduced by increasing the number iterations and/or sweeps per iteration. Fig. 5 shows results for the energy of a 16× 16 system obtained in double projections with three different trial states; a columnar dimer state, a ran- domly picked state generated while projecting the columnar state, as well as the self-optimized state. Already for the shortest projection, m = N/8 = 32, the self-optimized state gives a projected energy which deviates by only 0.06% from the exact ground state energy, and for largerm the energy is exact within statistical errors. The other two trial states also lead to the correct energy but only for much larger m. The energy of the self-optimized trial state itself is shown in the right panel—its error is as small as that of the best variational amplitude-product state [33]. Monte Carlo Simulations of Quantum Spin Systems 11 0.0 0.5 1.0 1.5 2.0 self-optimized random projected columnar dimer 0.0 0.5 1.0 1.5 2.0 0.116 0.118 0.120 0.122 0.124 0.126 Projected state Self-optimized trial state Fig. 6. Long-distance spin correlations in the same runs as in Fig. 5. A different “random projected” state is picked for each m, which leads to an un-smooth curve. Fluctuations beyond the small statistical errors in the right panel reflect differences in how closely the rather short self-optimization runs have approached P0 = Pm Fig. 6 shows the long-distance spin correlation calculated in the same runs. Again, the self-optimized state delivers superior results, although here the convergence is not as fast as for the energy. The error of the spin correla- tion in the trial state (right panel) is about 4% for large m, which is twice the error in the best variational state [33]. Thus, the self-optimized state is not identical to the best variational state, and an even faster convergence could be achieved by using the fully optimized variational state. However, the vari- ational calculation is much more time consuming than the self-optimization. We can go beyond the amplitude-product state by taking into account bond correlations. We are currently exploring this with both variational and self-optimized states. 4 Triplet excitations A unique advantage of the VB basis is that an mz = 0 triplet state can be projected simultaneously with the singlet, with essentially no additional overhead. Any triplet can be expanded in VB states where one of the bonds corresponds to a triplet [18]; (i, j) → [i, j], where [i, j] = (| ↑i↓j〉+ | ↓i↑j〉)/ 2. (22) Formally, such a triplet can be created by acting on a singlet with Szi − Szj ; (Szi − Szj )(i, j) = [i, j]. (23) To create a triplet |τ(q)〉 with some momentum q, we can apply Szq = eiq·rjSzj (24) 12 A. W. Sandvik and K. S. D. Beach to a singlet state |σ(0)〉 with zero momentum; Szq|σ(0)〉 ∝ |τ(q)〉. (25) The amplitude-product state (18) for a periodic-boundary system has q = 0 if the number of bonds, N/2, is even, whereas for odd N/2 it has q = (π, π). This simply follows from the fact that sublattice A → B and B → A when translating by one lattice constant, whence each singlet (3) acquires a minus sign. We typically work with systems with even N/2 (e.g., L×L lattices with even L) and so we will here consider q = 0 singlets. At the antiferromagnetic wave-vector, Q = (π, π), SzQ acting on an ar- bitrary VB basis state |V 〉 can be written as a sum of N/2 terms of the form (Szi − Szj )|V 〉, with i, j corresponding to the sites connected by bonds. Operating on a q = 0 singlet thus gives SzQ|σ(0)〉 = SzQ fk|(i1, j1)k(i2, j2)k · · · (iN/2, jN/2)k〉 fk|(i1, j1)k · · · [ib, jb]k · · · (iN/2, jN/2)k〉, (26) where the unspecified coefficients fk, e.g., the amplitude products in (18), have translational invariance built in. Thus, for a triplet with q = Q the wave function phases are buried in the definition of the singlets. Often, the lowest excitation of a Heisenberg system is a q = Q triplet, which we thus can sample without any difficulties with signs or phases. We consider this case first, before turning to triplets with arbitrary momentum. There are two possible actions of a singlet projectorHij on a triplet bond; Hij | · · · [i, j] · · ·〉 = 0, (27) Hij | · · · [i, k] · · · (l, j) · · ·〉 = 12 | · · · (i, j) · · · [l, k] · · ·〉, (28) i.e., a diagonal operation on a triplet bond destroys the state whereas an off-diagonal operation on one triplet and one singlet bond creates a singlet at the sites on which the operator acts and moves the triplet to the other two sites involved. Importantly, the matrix element remains the same as in the off-diagonal operation on two singlet bonds. The weight of a triplet path is therefore the same as the corresponding singlet path (11), except that the triplet dies (giving zero weight) if an operator in Pr acts diagonally on the triplet bond. We can thus measure triplet properties using paths generated in a singlet simulation, by considering only those triplet paths that do survive the propagation. In the trial state we have N/2 possible locations of the triplet. All of them can be attempted collectively in a single propagation, by keeping counters t(i) for the number of surviving states in which the triplet is connected to site i (with i on sublattice A). Initially t(i) = 1 for all i. During the propagation, for each diagonal operation (27) t(i) → t(i) − 1, and for each off-diagonal operation (28) t(i) → t(i) − 1, t(l) → t(l) + 1. Eventually, Monte Carlo Simulations of Quantum Spin Systems 13 as m → ∞, all triplets die; t(i) = 0 for all i, but typically there are enough survivors left at large enough m to compute converged triplet properties. An added advantage of calculating singlet and triplet properties in the same run is that there are error cancellations which in some cases can increase the statistical precision of differences, e.g., the singlet-triplet gap [36], ∆ = ET (π, π)− ES(0, 0), (29) by up to orders of magnitude relative to two independent calculations. The triplet energy ET (π, π) can be estimated using (15), taking into account that a diagonal operation on a triplet bond gives zero, i.e., nd → nd − nt, where, for surviving triplet configurations, nt = 0, 1 is the number of triplet bonds of length 1. Other triplet properties have been discussed in Ref. [40]. We now discuss calculations with triplets of arbitrary momentum q. The energy can be evaluated according to E(q) = r〈σ(0)|Sz−qHPrSzq|σ(0)〉 r〈σ(0)|Sz−qPrSzq|σ(0)〉 . (30) We want to evaluate this expression using the sampled q = 0 singlet bond configurations, and so we rewrite it as E(q) = r〈σ(0)|Sz−qHPrSzq|σ(0)〉 r〈σ(0)|Pr |σ(0)〉 r〈σ(0)|Sz−qPrSzq|σ(0)〉 r〈σ(0)|Pr |σ(0)〉 . (31) The two factors can be evaluated based on sampling the propagations Pr and the amplitude-product state |σ(0)〉 [which we have not explicitly written as a sum of bond configurations in (30) and (31)]. In the singlet energy (12), we could pick the Néel state for 〈Ψ | and then obtained the very simple expression (15). Now we must consider the overlap with a momentum q triplet state. We use 〈σ(0)|Sz−q, but in (31) have rewritten E(q) so that only the overlap with 〈σ(0)| has to be considered for the sampling weight. Phases arising from Szq only appear in the measurements, but in the end we have to evaluate the ratio of the two quantities in (31), which can be challenging in practice. However, close to q = (π, π) and (0, 0) [q 6= (0, 0)] we find that it can be done; in some cases the method works even far away from these momenta. We define the dispersion relative to the gap (29) at (π, π); ω(q) = ET (q)−∆. (32) Fig. 7 shows results for∆ and ω(q1), where q1 is the momentum closest to but not equal to (π, π); q1 = (π−2π/L, π). We show the convergence as a function of N/m for 4× 4 and 16× 16 lattices, comparing with exact diagonalization results in the former case. From (5) one would expect the convergence to be asymptotically exponential, which is seen clearly for L = 4. For L = 16, the three largest-m points are equal within statistical errors, suggesting that these results are also close to converged. Using ω(q1) = 0.62 for the L = 16 14 A. W. Sandvik and K. S. D. Beach 0.578 0.580 0.582 0.584 0.586 0 0.5 1 1.5 2 1.760 1.765 1.770 1.775 1.780 0 0.5 1 1.5 2 L=4 L=16 Fig. 7. Projection-length convergence of the singlet-triplet gap and the excitation energy at momentum q1 = (π − 2π/L, π) for L = 4 (left) and 16 (right) lattices. The horizontal lines show the exact results for L = 4 system gives the spin-wave velocity c = 1.58, in very close agreement with the known value [43]. Another useful quantity accessible with the VB projector is the matrix element 〈τ(q)|Szq|σ(0)〉, the square of which gives the single-magnon weight in the dynamic structure factor (which is experimentally measurable using neutron scattering). It can be calculated in a way similar to E(q), and we have done so successfully for q close to (π, π). Results will be presented elsewhere. Finally, we also note that the triplet bond-length distribution gives a di- rect, albeit basis dependent, window into the “spinon” aspects [44] of the excitations. Spinon deconfinement should be manifested as a delocalized dis- tribution function, whereas two spinons bound into a magnon or “triplon” should be reflected in a well-defined peak in the distribution function. We are currently exploring this. Acknowledgments We would like to thank H. G. Evertz for useful discussions. This work was sup- ported by the National Science Foundation under grant No. DMR-0513930. References 1. M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976); M. Suzuki, S. Miyashita, and A. Kuroda, Prog. Theor. Phys. 58, 1377 (1977) Monte Carlo Simulations of Quantum Spin Systems 15 2. M. Barma and B. S. Shastry, Phys. Rev. B 18, 3351 (1977) 3. J. E. Hirsch, R. L. Sugar, D. J. 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0704.1470

N-dimensional sl(2)-coalgebra spaces with non-constant curvature

N -dimensional sl(2)-coalgebra spaces with non-constant curvature A. Ballesterosa∗ A. Encisob† F.J. Herranza‡ O. Ragniscoc§ a Depto. de F́ısica, Universidad de Burgos, 09001 Burgos, Spain b Depto. de F́ısica Teórica II, Universidad Complutense, 28040 Madrid, Spain c Dip. di Fisica, Università di Roma 3, and Istituto Nazionale di Fisica Nucleare, Via Vasca Navale 84, 00146 Rome, Italy Abstract An infinite family of ND spaces endowed with sl(2)-coalgebra symmetry is introduced. For all these spaces the geodesic flow is superintegrable, and the explicit form of their common set of integrals is obtained from the underlying sl(2)-coalgebra structure. In particular, ND spherically symmetric spaces with Euclidean signature are shown to be sl(2)-coalgebra spaces. As a byproduct of this construction we presentND generalizations of the classical Darboux surfaces, thus obtaining remarkable superintegrable ND spaces with non-constant curvature. PACS: 02.30.Ik 02.40.Ky KEYWORDS: Integrable systems, geodesic flow, coalgebras, curvature, Darboux spaces. 1 Introduction An N -dimensional (ND) Hamiltonian H(N) is called completely integrable if there exists a set of (N −1) globally defined, functionally independent constants of the motion that Poisson-commute with H(N). Whereas completely integrable systems are quite unusual [1], they have long played a central role in our understanding of dynamical systems and the analysis of physical models. Moreover, in case that some additional independent integrals do exist, the system H(N) is called superintegrable [2] (there are different degrees of superintegrability, as we shall point out later). It is well known that superintegrability is strongly related to the separability of the corresponding Hamilton–Jacobi and Schrödinger equations [3] in more than one coordinate systems, and gives a fighting chance (which can be made precise in several contexts) of finding the general solution of the equations of motion by quadratures [4, 5]. In this paper we consider a specific class of (classical) ND Hamiltonian systems: the geodesic flows on ND Riemannian manifolds defined by the corresponding metrics. Contrary to what happens in the constant curvature cases, these kinetic-energy Hamiltonians can exhibit extremely complicated dynam- ics in arbitrary manifolds, the prime example being the chaotic geodesic flow on Anosov spaces. The ∗[email protected] †[email protected] ‡[email protected] §[email protected] http://arxiv.org/abs/0704.1470v2 complete integrability of a free Hamiltonian on a curved space and the separability of its Hamilton– Jacobi equation are rather nontrivial properties, and the analysis of such systems is being actively pursued because of its significant connections with the geometry and topology of the underlying man- ifold [6, 7]. In physics, curved (pseudo-)Riemannian manifolds (generally, of dimension higher than four) arise as the natural arena for general relativity, supergravity and superstring theories, and integrable geodesic flows in arbitrary dimensions are thus becoming increasingly popular in these areas [8]. Particularly, the case of Kerr–AdS spaces has attracted much attention due to its wealth of applications [9, 10, 11, 12]. In the studies performed so far, the explicit knowledge of the Stäckel–Killing integrals of motion in Kerr–AdS spaces has already proven to be an essential ingredient in these contexts (see e.g. [13, 14, 15] and references therein), which suggests that an explicit analysis of integrable geodesic flows on curved manifolds would certainly meet with interest from this viewpoint. Very recently, an in-depth analysis of the integrability properties and separability of the Hamilton–Jacobi equation on Kerr–NUT–AdS spacetimes have been achieved in [16, 17, 18], thus showing the relevance of developing the required machinery to deal with superintegrable spaces of non-constant curvature. From a quite different perspective, quantum groups (in an slz(2) Poisson coalgebra version) have been recently used to generate a family of distinguished ND hyperbolic spaces whose curvature is governed by the deformation parameter z [19]. In these slz(2)-coalgebra spaces the geodesic flow is completely integrable and the corresponding (N − 1) quadratic first integrals (which give rise to generalized Killing tensors) are explicitly known. Moreover, these flows turn out to be superintegrable, since the quantum slz(2)-coalgebra symmetry provides an additional set of (N−2) integrals. We stress that in several interesting situations (such as in the N = 2 case), Lorentzian analogs of these spaces can be obtained through an analytic continuation method; this procedure has actually been used to construct a new type of (1 + 1)D integrable deformations of the (Anti-)de Sitter spaces [20]. In this letter we present a class of ND spaces with Euclidean signature whose geodesic flow is, by construction, superintegrable. This is achieved by making use of an undeformed Poisson sl(2)- coalgebra symmetry. Furthermore, their (2N − 3) constants of the motion, which turn out to be quadratic in the momenta, are given in closed form. In fact, these invariants have the same form for all the spaces under consideration as a direct consequence of the underlying Poisson coalgebra structure, so we can talk about “universal” first integrals. As it has been pointed out in [21], spaces of constant curvature belong to this class of sl(2)-coalgebra spaces, but the former are only a small subset of the superintegrable spaces that can be obtained through this construction. Here we shall present four new significant ND examples with non-constant scalar curvature: the ND generalizations of the so- called (2D) Darboux spaces, which are the only surfaces with non-constant curvature admitting two functionally independent, quadratic integrals [22, 23, 24]. The paper is organized as follows. In the next section we briefly sketch the construction of generic sl(2)-coalgebra spaces and discuss their superintegrability properties; we also show how spherically symmetric spaces (with non-constant curvature) arise in this approach. In Section 3 we exploit the sl(2)-coalgebra symmetry of the 2D Darboux spaces to constructND counterparts. Some brief remarks of global nature are made. Finally, the closing section includes some comments and open problems. 2 sl(2)-coalgebra spaces and superintegrability An ND completely integrable Hamiltonian H(N) is called maximally superintegrable (MS) if there exists a set of 2N − 2 functionally independent global first integrals that Poisson-commute with H(N). As is well known, at least two different subsets of N − 1 constants in involution can be found among them. In the same way, a system will be called quasi-maximally superintegrable (QMS) if there are 2N − 3 independent integrals with the aforementioned properties, i.e. if the system is “one integral away” from being MS. Let us now consider the sl(2) Poisson coalgebra generated by the following Lie–Poisson brackets and comultiplication map: {J3, J+} = 2J+, {J3, J−} = −2J−, {J−, J+} = 4J3, ∆(Jl) = Jl ⊗ 1 + 1⊗ Jl, l = +,−, 3. The Casimir function is C = J−J+ − J 3 . Then, the following result holds [21]: Let {q,p} = {(q1, . . . , qN ), (p1, . . . , pN)} be N pairs of canonical variables. The ND Hamiltonian H(N) = H (J−, J+, J3) , (2) with H any smooth function and i ≡ q , J+ = ≡ p2 + , J3 = qipi ≡ q · p, (3) where bi are arbitrary real parameters, is a QMS system. The (2N − 3) functionally independent “universal” integrals of motion for H(N) read C(m) = 1≤i<j (qipj − qjpi) C(m) = N−m+1≤i<j (qipj − qjpi) i=N−m+1 bi, (4) where m = 2, . . . , N and C(N) = C(N). Moreover, the sets of N functions {H (N), C(m)} and {H(N), C(m)} (m = 2, . . . , N) are in involution. The proof of this result is based on the fact that, for any choice of the function H, the Hamiltonian H(N) has an sl(2) Poisson coalgebra symmetry [19, 21]; the generators (3) fulfil the Lie–Poisson brackets of sl(2) and the integrals (4) are obtained through the m-th coproducts of the Casimir C within an m-particle symplectic realization of type (3). With the previous general result in mind, we shall say that an ND Riemannian manifold is an sl(2)-coalgebra space if the kinetic energy Hamiltonian H T corresponding to geodesic motion on such a space has sl(2)-coalgebra symmetry, i.e., if H T can be written as T = HT (J−, J+, J3) = HT 2,p2 + ,q · p , (5) where HT is some smooth function on the sl(2)-coalgebra generators (3). Since H T has to be homogeneous quadratic in the momenta, we are forced to restrict ourselves to the specific sl(2) sym- plectic realizations (3) with all bi = 0, so that the most general Hamiltonian corresponding to an sl(2)-coalgebra space reads HT = A(J−)J+ + B(J−)J 3 = A(q 2)p2 + B(q2) (q · p)2, (6) where A and B are arbitrary functions. At this point, we stress that for any choice of both functions, T is a QMS Hamiltonian system with integrals given by (4). Thus, an infinite family of ND spaces with QMS geodesic flow is defined by (6) or, equivalently, by the pair of functions (A,B) that will characterize the ND metric. 2.1 Spaces of constant curvature In the previous discussion it is implicit that the pair (q,p) is an arbitrary set of canonically conjugated positions and momenta, for which no a priori geometric interpretation is given. This becomes apparent by considering the (simply connected) ND Riemannian spaces with constant sectional curvature κ (the sphere SN and the hyperbolic HN space), which are distinguished examples of sl(2)-coalgebra spaces. It can be shown [21] that the corresponding Hamiltonians can be written in the following ways (among others), both of them compatible with (6): HPT = (1 + κJ−) 1 + κq2 HBT = (1 + κJ−) J+ + κJ (1 + κq2) 2 + κ(q · p)2 The associated coordinate systems are classical: in the first case, q denotes the Poincaré coordinates in SN or HN , coming from the stereographic projection in RN+1 [25], whereas in the second one q are the Beltrami coordinates, which are associated with the central projection. We recall that the image of both the stereographic projection (Poincaré coordinates) and central projection (Beltrami coordinates) is the subset of RN determined by 1+ κq2 > 0, which means that for HN with κ = −1 such an image is the open subset q2 < 1. In both cases, we recover the standard Cartesian coordinates in RN when we set κ = 0. Note that in this language both Hamiltonians can immediately be interpreted as deformations (in terms of the curvature parameter κ) of the motion on Euclidean space EN , to which they reduce when κ = 0. This is analogous to the analysis in terms of the quantum parameter z carried out in [20]. By construction, the above Hamiltonians admit the universal integrals (4), whose concrete geometric realization depends on the interpretation of (q,p) as either Poincaré or Beltrami coordinates. The generalized Killing vectors of these spaces are the Hamiltonian vector fields (in phase space) associated with the latter first integrals. It should be noted that these spaces admit in fact an additional first integral which makes them MS: while they are not strictly the only ones having this property (see [26] and references therein), this final symmetry is not of coalgebraic nature and, when it exists, must be found by ad hoc methods. 2.2 Spherically symmetric spaces Any ND spherically symmetric metric of the type ds2 = f(|q|)2 dq2, (8) where |q| = q2, dq2 = i and f is an arbitrary smooth function, leads to a geodesic motion described by the Hamiltonian f(|q|)2 , (9) which is clearly of the form (6). Therefore, the metric (8) corresponds to an sl(2)-coalgebra space with A(J−) = , B(J−) = 0, (10) so that for any choice of f its geodesic flow defines a QMS system whose generalized Killing symmetries are the Hamiltonian vector fields associated with (4). It is apparent that these spaces are conformally flat, so that its Weyl tensor vanishes. The scalar curvature R of (8), which is generally non-constant, can be computed to be R = −(N − 1) (N − 4)f ′(|q|)2 + f(|q|) 2f ′′(|q|) + 2(N − 1)|q|−1f ′(|q|) f(|q|)4 . (11) If we define ND spherical coordinates (r, θ1, . . . , θN−1) as qj = r cos θj sin θk, qN = r sin θk, where j = 1, . . . , N − 1, r = |q| and hereafter a product k=1 is assumed to be equal to 1, the metric (8) can be alternatively written as ds2 = f(r)2(dr2 + r2dΩ2N−1) . dΩ2N−1 = sin2 θk, denotes the metric of the unit (N − 1)-sphere SN−1, with dΩ21 = dθ 1 . In these coordinates the free Hamiltonian (9) can be equivalently expressed as p2r + r f(r)2 where (sin θk) , (12) is the squared angular momentum and (pr, pθ1 , . . . , pθN−1) are the conjugate momenta of (r, θ1, . . . , θN−1). Finally, it is also convenient for our purposes to consider the modified spherical system given by (ρ, θ1, . . . , θN−1), where ρ = ln r. If pρ stands for the conjugate momentum of ρ, this yields ds2 = F (ρ)2 (dρ2 + dΩ2N−1) , (13) p2ρ + L F (ρ)2 , (14) where the arbitrary function F is defined as F (ρ) = r f(r). Hence any metric of the form (13) defines an sl(2)-coalgebra space given by A(J−) = F (ln , B(J−) = 0, (15) and the geodesic flows on (13) are QMS for any choice of F . The scalar curvature (11) now reads R = −(N − 1) (N − 4)F ′(ρ)2 − (N − 2)F (ρ)2 + 2F (ρ)F ′′(ρ) F (ρ)4 . (16) 3 Darboux spaces The (2D) Darboux surfaces are the 2-manifolds with non-constant curvature admitting two quadratic first integrals, so that its geodesic motion is quadratically MS. There are only four types of such spaces [22], which we will represent by D i (i = I, II, III, IV) following the notation in [23, 24]. In this section, the spaces D i will be initially described in terms of isothermal coordinates [27] (u, v) with canonically conjugate momenta (pu, pv). In [23, 24] it has been explicitly shown that their natural free Hamiltonians can be expressed in these variables as H(2) = p2u + p F (u)2 , (17) which implies that ds2 = F (u)2 (du2 + dv2). (18) Occasionally we shall need to consider other different isothermal charts of D i with isothermal coor- dinates (ξ, η) and conjugate momenta (pξ, pη). Immediately, from (18) we realize that the four 2D Darboux spaces D i are sl(2)-coalgebra spaces of the type (A, 0) with A(J−) given by (15). As a consequence, we can use the underlying sl(2)- coalgebra symmetry to define ND, spherically symmetric, conformally flat generalizations D i of the Darboux surfaces that will be thoroughly described in the following subsections. By construction, the four ND spaces so constructed will have QMS geodesic motions and their (2N − 3) independent integrals will be given by (4). It should be highlighted that, at least for the space D III the additional integral giving rise to an ND MS system can be explicitly constructed [26]. 3.1 Type I The Hamiltonian for geodesic motion on D I is given by p2u + p Therefore the corresponding metric reads [23] ds2 = u(du2 + dv2) . (19) The construction of the ND space D I can be conveniently performed via the substitution u → ρ = ln r , dv2 → dΩ2N−1 . (20) Note that we have used the same letter for the functions F (u) and F (ρ) appearing in the coordinate expressions of the metric (Eqs. (18) and (13)) because upon this substitution they define, in fact, the same function F : R → R+. Therefore, we find that the generic ND metric (13) is in this case characterized by the function F (ρ) = ρ1/2, yielding the following metric for D ds2 = ρ (dρ2 + dΩ2N−1) = ln |q| dq2 . (21) In other words, D I is the sl(2)-coalgebra space (A, 0) given by A(J−) = . (22) This space is certainly not flat; its scalar curvature can be readily computed to be (N − 1) 4(N − 2)ρ2 −N + 6 Some remarks on the global properties of the type I Darboux N -manifold are in order. We define I to be the exterior of the closed unit ball q : |q| > 1 covered with the coordinates q and endowed with the metric (21). It is not difficult to see that this space is incomplete by integrating its radial geodesics. In fact, the radial motion on D I is obtained from the Lagrangian L = r−2 ln r ṙ2 ≡ G(r)2 ṙ2, so that a straightforward calculation shows that the radial geodesics are complete at infinity and incomplete at the hypersphere |q| = 1 since the integral G(r) dr diverges at infinity but converges at 1. It should be remarked that one can also replace the conformal factor ln r by its absolute value and define D I to be the interior of the unit ball q : |q| < 1 together with the metric (21). The latter manifold is complete at 0 and incomplete at 1. 3.2 Type II In this case the free Hamiltonian reads [24] p2u + p 1 + u−2 The QMSND extension is performed again using the substitution (20). In this case, F (ρ) = (1+ρ−2)1/2 and we have p2ρ + L 1 + ρ−2 1 + (ln |q|)−2 Thus, D II is the sl(2)-coalgebra space determined by A(J−) = 1 + (ln J−)−2 . (23) The metric of D II is then given by ds2 = (1 + ρ−2)(dρ2 + dΩ2N−1) = 1 + (ln |q|)−2 dq2 , with non-constant scalar curvature (N − 1) N [(ρ3 + ρ)2 − 1]− 2ρ2(ρ4 + 2ρ2 + 4) (ρ2 + 1)3 If we set G(r)2 = r−2(1 + ln−2 r), the same arguments discussed in the previous subsection show that the radial geodesics of D II are complete at 0, 1 and at infinity. Hence both manifolds (M±, ds are complete. 3.3 Type III The 2D free Hamiltonian reads now [24] III = 1 + eu (p2u + p In order to obtain the ND spherically symmetric generalization, it suffices to take F (ρ) = e−ρ(1+eρ)1/2 and apply the map (20), so that the metric of D III becomes ds2 = e−2ρ(1 + eρ)(dρ2 + dΩ2N−1) = 1 + |q| dq2 , and the space is characterized by A(J−) = . (24) The scalar curvature reads (recall that r = eρ): r3(N − 1)[N(3r + 4)− 6(r + 2)] 4(r + 1)3 In this case the radial geodesics are obtained from the Lagrangian L = r−4(1+r) ṙ2 ≡ G(r)2 ṙ2. The integral G(r) dr diverges at 0 but is finite at ∞, and therefore D III = (R (N)\{0}, ds2) is complete at 0 but incomplete at ∞ (i.e., free particles in this space escape to infinity in finite time). Again, the whole manifold is conveniently described in terms of the coordinates q. It should be pointed out that the Hamiltonian H III can be written in a different coordinate system (ξ, η) as III = p2ξ + p 1 + ξ2 + η2 which admits the ND coalgebraic generalization III = 1 + q2 1 + J− The complete manifold (RN\{0}, (1 + q2) dq2) was thoroughly studied in [26], showing that it is in fact MS. 3.4 Type IV The Darboux Hamiltonian of type IV is given by [24] sin2 u a+ cosu p2u + p , (25) where a is a constant. This Hamiltonian admits a QMS ND generalization via the substitution u → ρ = ln r , p2v → L with F (ρ) = sin−1 ρ (a+ cos ρ)1/2. More precisely, the system has the form sin2 ρ a+ cos ρ p2ρ + L q2 sin2(ln |q|) a+ cos(ln |q|) so that the sl(2)-coalgebra space corresponds to setting A(J−) = J− sin ln J−) a+ cos(1 ln J−) , (26) and the metric in DNIV is given by ds2 = a+ cos ρ sin2 ρ (dρ2 + dΩ2N−1) = a+ cos(ln |q|) q2 sin2(ln |q|) dq2 . Its scalar curvature is found to be R = − N − 1 32(a+ cos(ρ))3 64a2 + 40(N + 1) cos(ρ)a+ 8(3N − 5) cos(3ρ)a+ 15N + 4[8(N − 2)a2 + 3(N + 2)] cos(2ρ) + 5(N − 2) cos(4ρ)− 14 If we take a > 1, it is obvious from inspection that the metric becomes singular at r = 1 and r = eπ. The radial geodesics are obtained from the Lagrangian L = sin−2 ρ (a + cos ρ) ρ̇2 ≡ G(ρ)2 ρ̇2. As the integral G(ρ) dρ diverges both at 0 and π, it immediately follows that the Riemannian manifold IV = (M, ds 2) is complete, M being the annulus q ∈ RN : 1 < |q| < eπ 4 Concluding remarks In the framework here discussed, the notion of sl(2)-coalgebra spaces arises naturally when analyzing (generalized) symmetries in Riemannian manifolds, and can be rephrased in terms of an sl(2)⊗sl(2)⊗ · · ·(N)⊗sl(2) dynamical symmetry of the free Hamiltonian on these spaces. As a matter of fact, we have shown that spherically symmetric spaces are sl(2)-coalgebra ones. We stress that once a non-constant curvature space is identified within the family (6), the underlying coalgebra symmetry ensures that this is, by construction, QMS. It should be explicitly mentioned that not every integrable geodesic flow is amenable to the sl(2)-coalgebraic approach developed in this paper by means of an appropriate change of variables. For instance, the completely integrable Kerr–NUT–AdS spacetime studied in [16] does not fit within this framework, even after euclideanization. Moreover, any potential with sl(2)-coalgebra symmetry, i.e. given by a function V (J−, J+, J3), can be added to the kinetic energy HT of an sl(2)-coalgebra background space without breaking the superintegrability of the motion. In this respect, we stress that the symplectic realization (3) with arbitrary parameters bi’s would give rise to potential terms of “centrifugal” type. It is well known that the latter terms can be often added to some “basic” potentials (such as the Kepler–Coulomb and the harmonic oscillator potentials) without breaking their superintegrability. Among the infinite family of sl(2)-coalgebra spaces, the four ND Darboux spaces here introduced are from an algebraic viewpoint the closest ones to constant curvature spaces, since they are the only spaces other than EN , HN and SN whose geodesic motion can be expected to be (quadratically) MS for all N . In the case N = 2, this statement is the cornerstone of Koenigs classification [22], whereas in the case of D III such maximal superintegrability has been recently proven in [26]. The search for the additional independent integral of motion in the three remaining Darboux spaces is currently under investigation, as is the exhaustive analysis of ND versions of the 2D potentials given in [23, 24]. Another interesting problem is the construction of the Lorentzian counterparts of the Riemannian sl(2)-coalgebra spaces presented in this letter. We expect that such an extension should be feasible by resorting to an analytic continuation procedure similar to the one used in [20]. In this direction, we believe that an appropriate shift to the Lorentzian signature should not affect the superintegrability properties of the geodesic flows, in the same way that separability is not altered by the standard analytic continuation tecniques [15]. Acknowledgements This work was partially supported by the Spanish MEC and by the Junta de Castilla y León under grants no. FIS2004-07913 and VA013C05 (A.B. and F.J.H.), by the Spanish DGI under grant no. FIS2005-00752 (A.E.) and by the INFN–CICyT (O.R.). Furthermore, A.E. acknowledges the financial support of the Spanish MEC through an FPU scholarship, as well as the hospitality and the partial support of the Physics Department of Roma Tre University. 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Winternitz, J. Math. Phys. 44 (2003) 5811. [25] B. Doubrovine, S. Novikov, A. Fomenko, Géométrie Contemporaine, Méthodes et Applications First Part MIR, Moscow, 1982. [26] A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, math-ph/0612080. [27] M.P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, 1976. http://arxiv.org/abs/math-ph/0611040 http://arxiv.org/abs/math-ph/0612080 Introduction sl(2)-coalgebra spaces and superintegrability Spaces of constant curvature Spherically symmetric spaces Darboux spaces Type I Type II Type III Type IV Concluding remarks

0704.1471

The Generalized PT-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton-Jacobi Formalism

The Generalized PT-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton-Jacobi Formalism Özlem Yeşiltaşa, S. Bilge Ocakb,* a Gazi University, Faculty of Arts and Sciences, Department of Physics, 06500, Teknikokullar, Ankara, Turkiye b Sarayköy Nuclear Research and Training Center Kazan, Ankara, Turkiye. Abstract The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh- Gordon potential are found for 0, 1n  states. Keywords: Quantum Hamilton Jacobi, Sinh-Gordon Potential. * Corresponding Author: [email protected] 1. Introduction The discovery of new class of physically significant spectral problems, called quasi-exactly solvable (QES) models, has attracted much attention [1-3]. Several methods [4-5] for the generation of QES model have been worked out. These are the models for which a part of the bound state energy spectrum and corresponding wave functions can be obtained exactly. These models have been constructed and studied extensively by means of Lie algebraic approach. In order that a part of the spectrum can be obtained exactly, the parameters appearing in the relation must satisfy a condition known as the condition for quasi-exact solvability. Within the Quantum Hamiltonian Jacobi approach (QHJ) [6-7], it has been found to be an elegant and the simple method to determine the energy spectrum of exactly solvable models in quantum mechanics. The advantage of this method is that it is possible to determine the energy eigen-values without having to solve for the eigen-functions. In this formalism, a quantum analog of classical action angle variables is introduced [6-7] . The quantization condition represents well known results on the number nodes of the wave function, translated in terms of logarithmic derivative, also is called quantum momentum function (QMF) [3, 6-7]. The equation satisfied by the QMF is a non-linear differential equation, called quantum Hamilton-jacobi equation leads to two solutions. The application of QHJ to eigen-values has been explored in great detail in ref. [3, 6-9]. In recent years, PT -symmetric Hamiltonians have generated much interest in quantum mechanics [10-11]. Physical motivation for the PT - symmetric but non-Hermitian Hamiltonians have been emphasized by many authors [12-16]. Recently, Mostafazadeh [16, 17] has introduced a different concept which is known as the class of pseudo-Hermitian Hamiltonians, and argued that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo- Hermiticity [17]. Following these detailed works, non-Hermitian Hamiltonians with real or complex spectra have been analyzed by using various numerical and analytical techniques [18-22]. In the applications, Ranjani and her collaborators applied the QHJ formalism, to Hamiltonians with Khare- Mandal potential and Scarf potential, characterized by discrete parity and time reversal PT symmetries [3]. The aim of the present work is to calculate the energy eigenvalues and the corresponding eigenfunctions of the Sinh-Gordon potential field which is used in coherent spin states [23]. Originally the form of the potential is a hyperbolic one given as ( ) sinh ( ) cosh V x x B S x    (1) We study general form of this hyperbolic effective potential which is called as Sinh-Gordon potential and is also in PT-symmetric form where the potential parameters 1V and 2V are real [23], xCoshVxSinhVxV  2 1)(  (2) in a Hamiltonian system generating quasi-exactly-solvable problems. In this letter, we first give a brief description of the QHJ formalism. The Sinh-Gordon potential is studied in one dimension. In each of these cases the condition of quasi-exactly solvability is derived within the QHJ approach. The energy and eigenfunctions of the generalized hyperbolic effective potential are obtained by using the QES condition. 2. Quantum Hamiltonian-Jacobi Formalism In the quantum Hamilton-Jacobi (QHJ) formalism, the logarithmic derivative of the wave function )(x is given by [3, 6-7] : )(ln ip  (3) which is called as quantum momentum function QMF. QMF plays an important role since it is stated analogous to the classical momentum function as, p  where S is the Hamilton’s characteristic function and it is related to the wave function by  x exp)( . Substituting )(x in terms of S in the Schrödinger equation  EH  and using the potential relation in Eq.(2), the following expression is obtained:  )(2'2 xVEmpip   . (4) Leacock and Padgett [6-7] proposed the quantization condition for the bound states in order to find the eigenvalues. Quantum action variable is defined as QMF and exact quantization condition for the bound states of a real potential is given as [3, 6-7] npdx (5) where, C is the contour enclosing the n moving poles in the complex domain [3, 6-7]. 3. The Sinh-Gordon Potential The Sinh-Gordon potential which is given by Eq.(2) is PT symmetric if 1V and 2V are real. It is worth noting that, parity operation is given by pp  , xx  in the Hamiltonian whereas the time reversal remains same as the conventional pp  , xx  and ii  . The QHJ equation with 12  m , can be introduced as: 0]coshsinh[ 2 2  xVxVE p  (6) In order to transform Eq. (6) into a rational form, one can change the variable xy cosh in Eq.(6) and obtain the following relation: 0])1([)(1)( 2 '22  yVyVEypyiyp  (7) The coefficient of the 1  in Eq.(7). Thus, in order to bring it to the Riccati type equation form, p can be defined as;  12  yip (8) And and another transformation is given as below,  12 2  y  . (9) Using Eqs.(8) and (9) in Eq.(7), one can obtain a Riccati type equation as       0 12222 2'   VyVyVE  (10) Assume that  has a finite number of moving poles in the complex y plane. In addition to,  has not only moving poles, but also fixed poles at 1y and it is bounded at y . Then,  can be written in the form of, 11 (11) where 1b and 1b are the residues at 1y . )(yPn is a polynomial of degree n and C is a constant due to the Liouville’s theorem (for more details, see [3] ) . At 1y ,  is expanded in a Laurent series as: ...)1( 1   yaa  (12) Substituting Eq.(12) in Eq. (10), one obtains the values of residue 1b as: 1 b (13) and also a  is found. For large y ,  which are the values of C . If the same procedure is applied to  for 1y 1   yaa  (14) One can obtain the residue at 1y as 1 b (15) In order to the discuss the behaviour of  at infinity, one expands  as: 0   (16) Substitution of Eq.(16) in (10) gives  (17) Behaviour of  approaches to nbb  '11 for large y . Hence one can obtain [3],  nbb '11 (18) It can be seen that Eq.(18) is positive. Therefore, one can choose positive value of  which is and it leads to choose Ca  because of the physical solutions for the wavefunction. In order to obtain a QES condition, one must take all possible combinations of the residues '11 , bb . These combinations with a constraint are given in table 3.1 as: Table3.1. The QES condition and the n for each '11 , bb . 1b 2b 11 bbn   Condition on QES condition 1M M : odd, 1M 12  nM 3M M : odd, 3M 32  nM 2M M : even, 2M 22  nM 2M M : even, 2M 22  nM As it is seen from the table3.1 that sets 1 and 2 are valid if and only if M is odd, sets 3 and 4 are valid if and only if M is even. When it comes to the form of the wavefunction, using   dxpix exp)( and writing p in terms of  , it is obtained as    exp)(  (19) Substitute Eq.(19) in (10) to obtain,          .011 yVyVE (20) Using Eq.(20), one can find the energy eigenvalues and corresponding wavefunctions for 3M and 2M cases. For 3M state, it can be seen that 1n and the )(yPn is a first order polynomial given as   yP1 . The results are illustrated in table3.2 and table3.3 respectively. Table3.2. When 3M and for set 1-2, the energy eigenvalues and wave-functions. 3M '11 , bb Energy, nE Wave- function n Set 1 11  bb , 1n   cosh)( Set 2 11  bb , 0n 20 E  xex  sinh)( where 1 and  are found for 0)sgn( 1 Vc and  is found for 0)sgn( 1 Vc . Similarly for 2M case, the table3.3 is given as Table3.3. When 2M and for set 3-4, the energy eigenvalues and wave-functions. 2M '11 , bb Energy, nE Wave- function n Set 3 11  bb , 0n  1cosh)( Set 4 11  bb , 0n  1cosh)( From table3.2 and table 3.3, the energy eigenvalues are obtained for the odd and even values of for Sinh-Gordon potential. If the complex forms of the potential given in Eq.(2) are written for 2 as below, xiVxVxV 2cosh2sinh)( 2 1  (21) xVxiVxV 2cosh2sinh)( 2 1  (22) the solutions of QHJ equation for these complex potentials can be discussed. Eq.(21) is PT symmetric under parity reflections as x , ii  and Eq.(22) is non- PT symmetric. If we take Eq.(21) into consideration, values of  can be found by following the same procedure as,  (23) which also means the values of M in the table3.1 are complex. The same procedure is valid for the potential relation in Eq.(22), too. Hence, these conditions don’t lead to physical solutions (for the wavefunctions). But, in ref.[3] the potential has the form given below  22cosh)( iMxxV   (24) which is PT symmetric under parity reflections x , ii  and leads to real energy spectrum [3]. Conclusions In this letter quantum Hamilton-Jacobi formalism is briefly given and the generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. We have obtained the quasi exact solutions of eigenvalues and eigenfunctions of the potential for 0, 1, 2n  states. The general Sinh-Gordon potential is applied to the coherent spin states. The eigenvalues and eigenfunctions comply with the results of ref.[23], for 2/1,0S ( 1 ) spin states if the potential parameters are chosen appropriately. Interesting features of quantum expectation theory for PT -violating potentials may be affected by changing from complex to real systems. Finally, we have pointed out that our exact results of the Sinh-Gordon potential may increase the number of applications in the study of various nonlinear potential applications in quantum systems. Acknowledgements This research was partially supported by the Scientific and Technological Research Council of Turkiye. References [1] A. Khare , U. Sukhatme, J.Math.Phys. 40 (1999) 5473-5494. [2] F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267. [3] K. G. Geojo, S. S. Ranjani and A. K. Kapoor, J. of Phys. A: Math and Gen.,36 (2003) 309 ; S.S. Ranjani et al, Mod. Phys. Lett. A 19 (2004) 1457-1468; S.S. Ranjani et al, Int. J. of Modern Physics A, 20 (2005) 4067-4077. [4] Carl M. Bender, Stefan Boettcher, J.Phys. A 31 (1998) L273-L277. [5] B. Bagchi, P. Gorain et al, Czech. J. Phys. 54 (2004) 1019-1026. [6] R. A. Leacock and M. J. Padgett, Phys. Rev. Lett., 50 (1983) 3. [7] R. A. Leacock and M. J. Padgett, Phys. Rev. D 28 (1983) 2491. [8] R. S. Bhalla, A.K. Kapoor and P.K. Panigrahi, Am. J. Phys. 65 (1997) 1187. [9] R. S. Bhalla, A.K. Kapoor and P.K. Panigrahi, Mod. Phys. Lett. A, 12 (1997) 295. [10] C. M. Bender, J. of Math. Phys., 37 (1996) 6-11 . [11] C. M. Bender and S. Boettcher, Phys. Rev. Lett., 80 (1998) 5243; C. M. Bender and S. Boettcher and P. N. Meisenger, J. Math. Phys., 40 (1999), 2201; C. M. Bender, G. V. Dunne and P. N. Meisenger, Phys. Lett. A 252 (1999) 272. [12] Ö.Yeşiltaş, M. Şimşek, R. Sever, C. Tezcan, Physica Scripta, 67 (2003) 472. [13] M. Aktaş, R. Sever, Mod.Phys.Lett. A 19 (2004) 2871-2877. [14] M. Znojil, J.Phys., A 32 (1999) 7419-7428. [15] M. Znojil, J. Phys. A:Math. and Gen., 36 (2003) 7639-7648. [16] A. Mostafazadeh, J.of Math. Phys., 43 (2002) 205-214 . [17] A. Mostafazadeh, A. Batal, J.Phys. A 37 (2004) 11645-11680; A. Mostafazadeh, J. Phys. A: Math. Gen., 38 (2005) 3213-3234; A. Mostafazadeh, J. Math. Phys. 46, (2005) 1-15. [18] F. Cannata et all, Phys. Lett. A246 (1998) 219. [19] A. Khare, B.P. Mandal, Phys. Lett.A 272 (2000) 53. [20] B. Bagchi, C. Quesne, Phys. Lett. A 273 (2000) 285. [21] Z. Ahmed, Phys. Lett. A 282 (2000) 343. [22] C. M. Bender et all, Phys. Lett. A 291 (2000) 197. [23] O. B. Zaslavskii and V. V. Ulyanov, Sov. Pyhs. JETP 60 (1984) 991.

0704.1472

Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy

Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy Jie Lou,1 Anders W. Sandvik,1 and Leon Balents2 Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215 Department of Physics, University of California, Santa Barbara, CA 93106-4030 (Dated: October 21, 2018) We study the three-dimensional XY model with a Zq anisotropic term. At temperatures T < Tc this dangerously irrelevant perturbation is relevant only above a length scale Λ, which diverges as a power of the correlation length; Λ ∼ ξaq . Below Λ the order parameter is U(1) symmetric. We derive the full scaling function controlling the emergence of U(1) symmetry and use Monte Carlo results to extract the exponent aq for q = 4, . . . , 8. We find that aq ≈ a4(q/4) 2, with a4 only marginally larger than 1. We discuss these results in the context of U(1) symmetry at “deconfined” quantum critical points separating antiferromagnetic and valence-bond-solid states in quantum spin systems. PACS numbers: 75.10.Hk, 75.10.Jm, 75.40.Mg, 05.70.Fh A salient feature of the recently proposed theory of ”deconfined” quantum critical points, which separate Néel and valence-bond-solid (VBS) ground states of an- tiferromagnets on the square lattice, is the emergence of U(1) symmetry [1]. The VBS is either dimerized on columns or forms a square pattern with plaquettes of four strongly entangled spins [2, 3]. In both cases there are four degenerate patterns and, thus, Z4 symmetry is broken. However, as the critical point is approached the theory predicts a length scale Λ, diverging faster than the correlation length, Λ ∼ ξa, a > 1, below which the dis- tinction between columnar and plaquette VBS states dis- appears. The nature of the VBS state is manifested only when coarse-graining on length-scales l > Λ, whereas for l < Λ the Z4 symmetry is unbroken and is replaced by an emergent U(1) symmetry characterizing the fluctuations between columnar and plaquette order. Quantum Monte Carlo simulations [4] of an S = 1/2 Heisenberg model with four-spin couplings have recently provided concrete evidence for a continuous Néel–VBS transition, and also detected U(1) symmetry in the VBS order-parameter distribution P (Dx, Dy), where Dx and Dy are VBS order parameters for horizontal and vertical dimers. There is no trace of the expected Z4 anisotropy in the VBS phase—the distribution is ring shaped— although the finite-size scaling of the squared order pa- rameter shows that the system is long-range ordered. This can be interpreted as the largest studied lattice size L = 32 < Λ. A ring-shaped distribution was also found in simulations of an SU(N) generalization of the S = 1/2 Heisenberg model [5]—possibly a consequence of proxim- ity of this system to a deconfined quantum-critical point. In order to better understand the U(1) features of these VBS states, and to guide future studies of them, we here exploit a classical analogy. In the three-dimensional XY model including a Zq-anisotropic term, H = −J (i,j) cos(θi − θj)− h cos(qθi), (1) the anisotropy is dangerously irrelevant for q ≥ 4 [6, 7, 8, 9, 10], i.e., the universality class is that of the isotropic XY model but the perturbation is relevant for T < Tc above a length-scale Λ. In the closely related q-state clock model, the anisotropy is dangerously irrelevant for q ≥ 5. While numerical studies [11, 12, 13] have confirmed the irrelevance of the anisotropy at Tc, the associated Λ has, to our knowledge, not been extracted numerically, except for an analysis of the 3-state antiferromagnetic Potts model, which corresponds to Z6 [9, 14]. Here we report results of Monte Carlo simulations for 4 ≤ q ≤ 8 on periodic-boundary lattices with N = L3 sites and L up to 32. In addition to Metroplis single-spin updates, we also use Wolff cluster updates [15] to reduce critical slowing down. We sample the order-parameter distribution P (mx,my), where cos(θi), my = sin(θi). (2) The standard order parameter can be defined as 〈m〉 = dmyP (mx,my) m2x +m dθr2P (r, θ). (3) We will compare this with an order parameter 〈mq〉 which is sensitive to the angular distribution; 〈mq〉 = dθr2P (r, θ) cos(qθ). (4) While the finite-size scaling of 〈m〉 is governed by the correlation length ξ, 〈mq〉 should instead be controlled by the U(1) length scale Λ [9], becoming large for a system of size L only when L > Λ. Fig. 1 shows magnetization histograms at h/J = 1 for Z4 and Z8 systems with L = 4 and 32. The angular distribution P (θ) = drrP (r, θ) is also shown. The average radius of the distribution is the magnetization 〈m〉, which decreases with increasing L. The anisotropy, on the other hand, increases with http://arxiv.org/abs/0704.1472v2 FIG. 1: (Color online) P (mx,my) at h/J = 1 for q = 4, 8, L = 4, 32. The temperature T/J = 2.17 for Z4 and 1.15 for Z8; both less than Tc/J ≈ 2.20. The size of the histograms corresponds to mx,y ∈ [−1, 1]. Angular distributions P (θ) with θ ∈ [0, 2π] are shown above each histogram. L. This is particularly striking for Z8, where the L = 4 histogram shows essentially no angular dependence, even though T is very significantly below Tc, whereas there are 8 prominent peaks for L = 32. Thus, in this case the U(1) length scale 4 < Λ < 32. For the Z4 system T is much closer to Tc but still some anisotropy is seen for L = 4; it becomes much more pronounced for L = 32. It is instructive to examine a spin configuration with mx ≈ my, i.e., θ ≈ π/4. Fig. 2 shows one layer of a Z4 system with L = 10 below Tc. The spins align predomi- nantly along θ = 0 and θ = π/2, with only a few spins in the other two directions. Clearly there is some cluster- ing of spins pointing in the same direction—the system consists of two interpenetrating clusters. Essentially, the configuration corresponds to a size-limited domain wall between θ = 0 and θ = π/4 magnetized states. Hove and Sudbø studied the q-state clock model and performed a course graining at criticality [13]. They found that the structure in the angular distribution di- minished with the size of the block spins for q ≥ 5, as would be expected if the anisotropy is irrelevant. Here we want to quantify the length scale Λ at which the anisotropy becomes relevant for T < Tc. Consider first what would happen in a course graining procedure for a single spin configuration of an infinite system in the ordered state very close to Tc. The individual spins will of course exhibit q preferred directions, as is seen clearly in Fig. 2, i.e., there would be q peaks in the probability distribution of angles θi. Constructing block spins of l spins, we would expect the angular dependence to first FIG. 2: (Color online) Spins in one layer of the Z4 model with L = 10 at h/J = 1, T/J = 1.9 < Tc. Here mx ≈ my, corresponding to θ ≈ π/4 in P (r, θ). Arrows are color-coded according to the closest Z4 angle; nπ/2, n = 0, 1, 2, 3. become less pronounced because of the averaging over spins pointing in different directions (again, as is seen in Fig. 2). Sufficiently close to Tc we would expect the dis- tribution to approach flatness. However, since we are in an ordered state, one of the q preferred angles eventually has to become predominant, and thus one peak in the histogram will start to grow. This happens at l ≈ Λ. We cannot simulate the infinite system and instead we carry out an analogous procedure as a function of the lattice size L, sampling a large number of configurations. We calculate the order parameters 〈m〉 and 〈mq〉, defined in Eqs. (3,4), and analyze them using 〈m〉 = L−σf(tL1/ν), (5) 〈mq〉 = L −σg(tL1/νq). (6) Here (5) is the standard finite-size ansatz with σ = β/ν, and the XY exponents are β ≈ 0.348 and ν ≈ 0.672 [17]. Eq. (6) is an intuitive generalization of (5), which was proposed and used also in Ref. [9], but we can actually also derive the scaling function g(X) exactly. Let us consider the scaling behavior of the order- parameter distribution P (~m). It depends upon the sys- tem size L and the size of scaling operators that perturb the critical theory. Specifically, we consider the temper- ature deviation t = Tc − T and the presumed irrelevant q-fold anisotropy strength h. By conventional scaling ar- guments, we expect P (~m;L, t, h) = Lσ/2P̂ (Lσ ~m, tL1/ν , H = hL3−∆q), (7) where ∆q > 3 is the scaling dimension of the irrelevant anisotropy. The prefactor above is determined from nor- malization of the probability distribution. In the scaling regime, |t| ≪ 1, L ≫ 1, so H is small. When the first two arguments are O(1), P̂ can be well-approximated by taking H = 0 [with “corrections to scaling” of O(H), i.e. suppressed by L3−∆q for a large system]. At H = 0, the distribution is fully XY symmetric, and the integral in Eq. (4) vanishes. Thus, in this regime 〈mq〉 is small, O(H), and should be considered as arising from correc- tions to scaling. This simply reflects the irrelevance of the anisotropy at the critical point. Because the anisotropy is dangerously irrelevant, a larger contribution, however, emerges when tL1/ν ≫ 1, i.e. L ≫ ξ ∼ t−ν . In this limit, the system can be re- garded as possessing long-range XY order, and the only significant fluctuations are the global fluctuations of the XY phase θ. This is biased by the anisotropy. The scale κ of the total anisotropy (free) energy can be estimated by its typical magnitude within an XY correlation vol- ume, hξ3−∆q , multiplied by the number of correlation volumes, (L/ξ)3, i.e. κ = hL3ξ−∆q . Note that although the energy per correlation volume is small (due to the ir- relevance of anisotropy at the critical point), the number of correlation volumes becomes very large and more than compensates for this smallness for L/ξ sufficiently large. From this argument, we see that for L/ξ ≫ 1, the dis- tribution of angles θ is just determined from a Boltzmann factor for a single XY spin with the q-fold anisotropy energy ∼ −κ cos qθ. Furthermore, for L/ξ ≫ 1, the magnitude |~m| ≈ 〈m〉 is approximately non-fluctuating. Thus the distribution factors into the form P (|m|, θ) = 〈m〉−2δ(|m| − 〈m〉)P (θ), with P (θ) = eκ cos(qθ). (8) Here Z = dθ eκ cos(qθ) is the single-spin partition func- tion. It is then straightforward to obtain from Eq. (4) 〈mq〉 = 〈m〉 I1(κ) I0(κ) , (9) where In is the modified Bessel function of order n. Os- hokawa obtained a similar expression in a different way, but we disagree with his scaling variable. Comparing this with the scaling form in Eq. (6), we see that νq = ∆qν/3 (aq = ∆q/3), κ = h(tL 1/νq )3νq , and g(X) ∝ I1(h̃X I0(h̃X . (10) Here h̃ should be viewed as a non-universal scale factor. From the above discussion, one sees that this form is valid for L/ξ ≫ 1 but L/ξq arbitrary. For L/ξ of O(1) or smaller, L/ξq ≪ 1 (implying κ,X ≪ 1), and the scaling form for 〈mq〉 becomes small and of order the expected correction to scaling in the critical regime. In Fig. 3 we show results for the two order parame- ters for systems with q = 4, 5, 6. We have studied several values of h/J and here show results for a different value for each q. We have extracted Tc using finite-size scaling of 〈m〉 with Eq. (5) and the XY exponents. This works very well for all q, confirming the irrelevance of h. The magnetization for T < Tc is seen to decrease marginally with increasing q in Fig. 3. The Zq order-parameter 〈mq〉 changes more drastically, being strongly suppressed close to Tc for large q. This is expected, as 〈mq〉 should van- ish for all T in the XY limit q → ∞. For Z4, the 〈mq〉 2.1 2.15 2.2 2.25 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 q=4, h/J=1 q=5, h/J=10 q=6, h/J=5 FIG. 3: (Color online) The XY order parameter 〈m〉 (solid curves) and the Zq order parameter 〈mq〉 (dashed curves) vs temperature for q = 4, 5, 6. The system sizes are L = 8, 10, 12, 14, 16, 24, and 32. The curves become sharper (in- creasing slope) around Tc (indicated by vertical lines). The ratios h/J used are indicated on the graphs. curves for different L cross each other, with the crossing points moving closer to Tc as L increases. This is consis- tent with the above discussion of course-graining: In the ordered state close to Tc, 〈mq〉 should first, for small L, decrease with increasing L as the q-peaked structure in P (θ) diminishes due to averaging over more spins. For larger L, 〈mq〉 starts to grow with L as the length-scale Λ is exceeded. This behavior is more difficult to observe directly for q = 5, 6 because 〈mq〉 is small and dominated by statistical noise close to Tc where the curves cross. Fig. 4 shows finite-size scaling of the Zq order- parameter 〈mq〉, using the hypothesis (6) and the XY value for σ. Adjusting νq = aqν for each q we find sat- isfactory data collapse using a4 = 1.07(3), a5 = 1.6(1), a6 = 2.4(1), and, not shown in the figure, a8 = 4.2(3). These results are consistent with the form aq = a4(q/4) in qualitative agreement with the ǫ-expansion by Os- hikawa, which gave aq → q 2/30 for large q [9]. However, in the ǫ-expansion there are significant deviations from the q2 form in the range of q values considered here. Our a6 is also smaller than the value ≈ 3.5 obtained on the basis of the 3-state Potts antiferromagnet [9]. In Fig. 4 we also show the scaling function (10). It does not match exactly the collapsed data, but the agreement -4 -2 0 2 4 -6 -3 0 -4 -3 -2 -1 0 q=4, h/J=1 q=5, h/J=10 q=6, h/J=5 =0.72 =1.05 =1.60 FIG. 4: (Color online) Scaling of the Zq magnetization. We use σ = 0.52 for all q, and νq as indicated in the plots. The colors of the curves correspond to L as in Fig. 3. The dashed curves are the predicted scaling functions with h̃ and prefac- tors adjusted to fit the data approximately. improves as q increases. As we have discussed above, the scaling function represents the dominant behavior for T < Tc, but exactly at Tc this contribution vanishes and the critical-point scaling form 〈mq(Tc)〉 ∼ L 3−∆q be- comes dominant. For q = 4, ∆4−3 is small; our estimate is ∆4 − 3 = 0.21(9), in good agreement with previous es- timates of the scaling dimension [8, 10]. Thus it is clear that very large systems would be required for this contri- bution to become invisible on the scale used in our graph. It is also clear that for t < 0, close to t = 0, there will be a similarly significant correction to the asymptotically dominant scaling function. As q increases, we have seen that ∆q = 3aq increases rapidly, and we thus expect sig- nificantly smaller correction to scaling. For q = 6 the agreement is already seen to be quite good, considering that our lattices are not very large. To conclude, we relate our results to the quantum VBS states discussed in the introduction. Returning to Fig. 2, associating θi ≈ 0 arrows with two adjacent horizontal dimers on even-numbered columns and θi ≈ π/2 with vertical adjacent dimers on even rows, 〈θ〉 = 0, π/2 cor- respond to columnar VBS states. A plaquette is a super- position of horizontal and vertical dimer pairs, whence a plaquette VBS corresponds to 〈θ〉 = π/4 [3]. Rotating the arrows by 90◦ corresponds to translating or rotating a VBS. Either a columnar or plaquette VBS should ob- tain in the infinite-size limit, but close to a deconfined quantum-critical point, for L < Λ, the system fluctuates among all mixtures of plaquette and columnar states. This corresponds to a ring-shaped VBS order-parameter histogram. In numerical studies of quantum antiferro- magnets [4, 5] no 4-peak structure was observed in the angular distribution, and hence it is not clear what type of VBS finally will emerge (although a method using open boundaries favors a columnar state in [4]). It seems un- likely that the U(1) symmetry should persist as L → ∞. In the classical Z4 model we never observe a perfectly U(1)-symmetric histograms far inside the ordered phase, in contrast to Refs. [4, 5]. On the other hand, aq is larger for q > 4, and in Fig. 1 we have shown a prominently U(1)-symmetric histogram for the Z8 model deep inside the ordered phase. Thus, the exponent a may be larger for the Z4 quantum VBS than a4 ≈ 1 obtained here for the classical Z4 model. There is of course no reason to expect them to be the same, as the universality class of deconfined quantum-criticality is not that of the classical Z4 model [1, 4]. Future numerical studies of VBS states and deconfined quantum-criticality can hopefully reach sufficiently large lattices to extract the U(1) exponent using the scaling method employed here. We would like to thank Kevin Beach, Masaki Os- hikawa, Andrea Pelissetto, and Ettore Vicari for useful discussions and comments. This research is supported by NSF Grant No. DMR-0513930. [1] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004). [2] S. Sachdev, Rev. Mod. Phys. 75, 913 (2003). [3] M. Levin and T. Senthil, Phys. Rev. B 70, 220403 (2004). [4] A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007). [5] N. Kawashima and Y. Tanabe, Phys. Rev. Lett. 98, 057202 (2007). [6] J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977). [7] D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 29, 5250 (1984). [8] M. Caselle and M. Hasenbusch, J. Phys. A 31, 4603 (1998). [9] M. Oshikawa, Phys. Rev. B 61, 3430 (2000). [10] J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61, 15136 (2000). [11] P. D. Scholten and L. J. Irakliotis, Phys. Rev. B 48, 1291 (1993). [12] S. Miyashita, J. Phys. Soc. Jpn. 66, 3411 (1997). [13] J. Hove and A. Sudbø, Phys. Rev. E 68, 046107 (2003). [14] A related Z6 problem has been studied by A. Aharony, R. J. Birgeneau, J. D. Brock, and J. D. Litster, Phys. Rev. Lett. 57, 1012 (1986). [15] U. Wolff, Phys. Rev. Lett. 62, 361 (1989). [16] A. P. Gottlob and M. Hasenbusch, Physica A 201, 593 (1993). [17] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 63, 214503 (2001).

0704.1473

Existence of Universal Entangler

Existence of Universal Entangler Jianxin Chen,1, ∗ Runyao Duan,1, † Zhengfeng Ji,1, ‡ Mingsheng Ying,1, § and Jun Yu2, ¶ 1 State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China 2 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (Dated: November 18, 2018) A gate is called an entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This Letter demonstrates that a universal entangler always exists except for a degen- erated case. Nevertheless, the problem how to find a universal entangler remains open. PACS numbers: 02.20.-a, 02.20.Uw, 03.65.Ud, 03.67.Lx Introduction. It is a common sense in the quantum computation and quantum information community that entanglement is an extremely important kind of phys- ical resources. How to generate this kind of resources therefore becomes an important problem. One possibil- ity one may naturally conceive is to generate entangle- ment from product states which can be prepared spatially separately. If a gate can transform some product states to entangled states, then we call it an entangler. For some given product states, it is not difficult to find an entangler that maps them to entangled states. Here we consider a more challenging problem: Does there exist some entangler that can transform all product states to entangled states. We call such an entangler a universal entangler [9, 17]. One may even wonder at the existence of universal entanglers. Formally, the problem of existence of universal entan- glers in bipartite systems can be stated as follows: Problem 1. Suppose Alice and Bob have two quan- tum systems with state spaces HA and HB respectively. Whether there exists a unitary operator U acting on HA ⊗HB such that U(|φ〉 ⊗ |ψ〉) is always entangled for any |φ〉 ∈ HA and |ψ〉 ∈ HB? The purpose of this letter is to present a complete so- lution to the above problem. We have our main theorem as follows: Theorem 1. Given a bipartite quantum system Hm ⊗H\, where Hm and H\ are Hilbert space with di- mension m and n respectively, then there exists a uni- tary operator U which maps every product state of this system to an entangled bipartite state if and only if min(m,n) ≥ 3 and (m,n) 6= (3, 3). There are many literatures and different approaches which are potentially relevant to the topic of entanglers. Besides universal entangler, Zhang et al. discussed per- fect entanglers which are defined as the unitary opera- tions that can generate maximal entangled states from some initially separable states [16, 19, 20]. Bužek et al. [9] considered the entangler that entangles a qubit in unknown state with a qubit in a reference, also they proved the nonexistence of universal entangler for qubits, which answers a special case of our problem. Further- more, nonexistence of universal entangler in 2 ⊗ n bi- partite system can be derived straightforwardly from Parthasarathy’s recent work on the maximal dimension of completely entangled subspace [14], which is tightly connected to the unextendible product bases introduced by Bennett et al. [6, 7, 10]. But the proof of the above theorem requires some mathematical results from basic algebraic geometry. We will first review some basic notions in algebraic geometry in the next section. To make a clear presentation, Theo- rem 1 is divided into two parts, namely Theorems 2 and 3 below, and detailed proofs of them are given. Finally, a brief conclusion is drawn and some open problems are proposed. Preliminaries. For the convenience of the reader, we recall some basic definitions in algebraic geometry [3, 11]. The n× n general linear group and the n× n unitary group are denoted by GL(n,C) and U(n), respectively. An affine n-space, denoted by An, is the set of all n- tuples of complex numbers. An element of An is called a point, and if point P = (a1, a2, · · · , an) with ai ∈ C, then the ai’s are called the coordinates of P . The polynomial ring in n variables, denoted by C[x1, x2, · · · , xn], is the set of polynomials in n variables with coefficients in a ring. A subset Y of An is an algebraic set if it is the common zeros of a finite set of polynomials f1, f2, · · · , fr with fi ∈ C[x1, x2, · · · , xn] for 1 ≤ i ≤ r, which is also denoted by Z(f1, f2, · · · , fr). It is not hard to check that the union of a finite number of algebraic sets is an algebraic set, and the intersection of any family of algebraic sets is again an algebraic set. Thus by taking the open subsets to be the complements of algebraic sets, we can define a topology, called the Zariski topology on An. http://arxiv.org/abs/0704.1473v2 A nonempty subset Y of a topological spaceX is called irreducible if it cannot be expressed as the union Y = Y1 ∪ Y2 of two proper closed subsets Y1, Y2. The empty set is not considered to be irreducible. Let X , Y be two topological spaces, then we have two useful facts: Fact 1. If X is irreducible and F : X → Y be a contin- uous function, then F (X) with induced topology is also irreducible. Fact 2. Let Y ⊂ X be a subset, if X is irreducible and Y is open, then Y is irreducible; if Y is irreducible and dense in X, then X is irreducible. An affine algebraic variety is an irreducible closed sub- set of some An, with respect to the induced topology. We define projective n-space, denoted by Pn, to be the set of equivalence classes of (n + 1)−tuples (a0, · · · , an) of complex numbers, not all zero, under the equivalence relation given by (a0, · · · , an) ∼ (λa0, · · · , λan) for all λ ∈ C, λ 6= 0. A notion of algebraic variety may also be introduced in projective spaces, called projective algebraic variety: a subset Y of Pn is an algebraic set if it is the com- mon zeros of a finite set of homogeneous polynomials f1, f2, · · · , fr with fi ∈ C[x0, x1, · · · , xn] for 1 ≤ i ≤ r. We call open subsets of irreducible projective varieties as quasi-projective varieties. We will mainly use the following two varieties: Example 1. A1 is irreducible, because its only proper closed subsets are finite, yet it is itself infinite. This fact is somewhat trivial. If T is an algebraic subset of A1, then we should say, there is a finite set F of polynomi- als over C, such that T = Z(F ). For all f ∈ C[x], if degree(f) is 0 or f is a constant function, then T should be the degenerate subset of C, i.e.,empty set ∅ or C. Oth- erwise, we can choose a f ∈ C[x], such that f is not a constant function, and degree(f) is no less than 1. From the fundamental theorem of algebra, we know the num- ber of its roots is at most its degree, so its solution set should be a finite set. Thus, a subset of the solution set must be finite too, and we proved that T is a finite subset of C. Example 2. The Segre embedding is defined as the map: σ : Pm−1 × Pn−1 → Pmn−1 taking a pair of points ([x], [y]) ∈ Pm−1 × Pn−1 to their product σ : ([x0 : x1 : · · · : xm−1], [y0 : y1 : · · · : yn−1]) 7−→ [x0y0 : x0y1 : · · · : xm−1yn−1] Here, Pm−1 and Pn−1 are projective vector spaces over some arbitrary field, [x0 : x1 : · · · : xn−1] is the homogeneous coordinates of x, and similarly for y. The image of the map is a variety, called Segre variety, written as Σm−1,n−1. What concerns us is that Segre variety represents the set of product states [8, 12, 13]. If X is a topological space, we define the dimension of X, denoted by dim(X), to be the supremum of all integers n such that there exists a chain Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of n+1 distinct irreducible closed subsets of X. The dimension of a quasi-projective variety is then defined according to the Zariski topology. Main results. With the notations introduced above, Problem 1 can be restated as follows: Problem 2. Let Z denote the Segre variety Σm−1,n−1. Whether there exists a gate Φ ∈ U(mn) s.t. Φ(Z)∩Z = Note that Z is the set of product states, and thus Φ(Z) is the set of states generated by gate Φ from product states. So, Φ(Z)∩Z = ∅means that all states in Φ(Z) are entangled, and the above problem coincides with Prob- lem 1. We claim that the answer to the above question is af- firmative except for some degenerated cases. First, we consider the degenerated cases that min(m,n) ≤ 2 or (m,n) = (3, 3). The following theorem gives a negative answer for this case: Theorem 2. Given a bipartite quantum system Hm ⊗H\, where Hm and H\ are Hilbert spaces with di- mension m and n respectively, if min (m,n) ≤ 2 or (m,n) = (3, 3), then ∀Φ ∈ U(mn), we have Φ(Z) ∩ Z 6= ∅ In other words, universal entangler does not exist. To prove the above theorem, we need the following: Lemma 1. [Projective Dimension Theorem] Let Y , Z be varieties of dimensions r, s in Pn. Then every irre- ducible component of Y ∩ Z has dimension ≥ r + s− n. Furthermore, if r + s− n ≥ 0, then Y ∩ Z is nonempty. For the proof of Lemma 1, see [11], chapter I, theorem 7.2. Then we can easily prove Theorem 2: Proof. For any Φ ∈ U(mn), dim(Z) = m − 1 + n − 1 = m+n−2. Then we have dim(Φ(Z))+dim(Z)−dim(Y ) = 2 dim(Z)−dim(Y ) = 2(m+n−2)− (mn−1) = 1− (m− 2)(n − 2) ≥ 0. So we have Φ(Z) ∩ Z 6= ∅ according to Lemma 1. We now turn to consider the general case where min(m,n) ≥ 3 and (m,n) 6= (3, 3). The following lemmas are needed in the proof of our main theorem. Lemma 2. U(k) is Zariski dense in GL(k,C) Proof. For the case k = 1, the only Zariski closed subsets of G = GL(1,C) = C\{0} ∼= {(z1, z2) ∈ C 2 : z1z2 = 1} are ∅, G and non-empty finite subsets of G. This follows immediately from Example 1. Since GL(1,C) is an open subset of affine line C, its closed subsets are intersections of closed subsets of C and C\{0}. Now we see that U(1) is infinite, thus U(1) = G. In general, H(k) = {diag(z1, z2, · · · , zk) : z1, z2, · · · , zk ∈ C\{0}} ∼= (C\{0}) k is the Fi- bre product of k copies of C\{0} [11]. Then H(k) ∩ U(k) = {diag(z1, z2, · · · , zk) : |z1| = |z2| = · · · = |zk| = 1} ∼= U(1) By the results of the case k = 1, we also have H(k) ∩ U(k) = H(k) ⊇ A(k) = {diag(z1, z2, · · · , zk) : z1 > 0, z2 > 0, · · · , zk > 0}. Now for any B ∈ U(k), LB : X → B · X and RB : X → X ·B are two isomorphisms of G, and LB(U(k)) = U(k) = RB(U(k)). Thus LB(U(k)) = U(k) = RB(U(k)). Since A(k) ⊆ H(k) ∩ U(k) ⊆ U(k), we have U(k)A(k)U(k) ⊆ U(k). By singular value decomposition for GL(k,C), we get U(k)A(k)U(k) = GL(k,C) ⊆ U(k). Thus U(k) = GL(k,C). The following lemma [18] establishes a connection be- tween the dimensions of domain and codomain of a va- riety morphism. A morphism Φ : Z1 → Z2 is called a dominant morphism if Φ(Z1) is dense in Z2. Lemma 3. 1 Z1 and Z2 are both irreducible varieties over C, and φ : Z1 → Z2 is a dominant morphism, then dim(Z2) ≤ dim(Z1). 2 Z1 and Z2 are both varieties over C, and φ : Z1 → Z2 is a morphism, r = max dim(φ−1(z)), then dim(Z1) ≤ r + dim(Z2). 3 If V = ∪si=1Vi is a finite open covering, and ∀i, Vi is irreducible, ∀i, j, Vi ∩ Vj 6= ∅, then V is irre- ducible. Let X = {Φ|Φ ∈ GL(mn,C),Φ(Z) ∩ Z 6= ∅}. We are able to give an upper bound on the dimension of the closure of X with respect to the Zariski topology. Lemma 4. dim(X) ≤ m2n2 − (m− 2)(n− 2)+ 1, where X is the Zariski closure of X. Proof. We have a morphism F : G×Y → Y which is just the left action of G on Y , defined by F (g, [w]) = [g · w] where G = GL(k,C), Y = Pmn−1, k = mn. Let y0 = (1, 0, · · · , 0) T be a column vector with k entries. For any given y1, y2 ∈ Y , we choose g1, g2 ∈ GL(k,C), such that [g1 ·y0] = [y1] and [g2 ·y0] = [y2]. Then we have [g · y1] = [y2] ⇐⇒ [gg1 · y0] = [g2 · y0] ⇐⇒ [g−1 gg1 · y0] = [y0] From above observations, F has the following prop- erty: for any y1, y2 ∈ Y , F −1(y2) ∩ {G × {y1}} ∼= : z1 ∈ C\{0}, g ′ ∈ GL(k − 1,C), α ∈ k−1 is a row vector.}. Hence dim(F−1(y2) ∩ G× {y1}) = m 2n2 − (mn− 1). Let P1, P2 be projections of G×Y to G, Y respectively. Now we only look atG×Z ⊆ G×Y , to get F : G×Z → Y . Then we have a characterization of X: X = P1F −1(Z). In fact, g ∈ X ⇐⇒ g(Z) ∩ Z 6= ∅ ⇐⇒ ∃z1, z2 ∈ Z, s.t.g(z1) = z2 ⇐⇒ ∃z1, z2 ∈ Z, s.t.(g, z1) ∈ F −1(z2) ⇐⇒ ∃z2 ∈ Z, s.t.g ∈ P1F −1(z) ⇐⇒ g ∈ P1F −1(Z) Let X ⊆ G be the Zariski closure of X in G, then P1 : F−1(Z) → X is a dominant morphism. Furthermore, consider Ψ : F−1(Z) → Z × Z given by Ψ(g, [z]) = ([z], [g · z]) For all z1, z2 ∈ Z, we have Ψ −1(z1, z2) = (g2Tg 1 , z1), where T = { : z0 ∈ C\{0}, g ′ ∈ GL(k − 1,C), α ∈ Ck−1 is a row vector}, and g1, g2 ∈ GL(k,C), s.t. g1(y0) = z1, g2(y0) = z2. Ψ is a dominant morphism since G acts transitively on Pk−1. Then we obtain dim(F−1(Z)) ≤dim(T ) + dim(Z × Z) =m2n2 −mn+ 1 + 2 · dim(Z) It is required in Lemma 3.1 that varieties Z1 and Z2 are irreducible, but we haven’t proved F−1(Z) is irreducible. Actually, this condition can be weakened: Lemma 3.1 is still true for the more general case that Z1 and Z2 are closed subsets of irreducible varieties [11]. Thus, we can fill out the gap and apply this lemma. Indeed, the irreducibility of F−1(Z) andX really holds, but the proof is not easy (see [1] for a brief proof ). From Lemma 3.1, we have dim(X) ≤ dim(F−1(Z)) ≤(m2n2 − (mn− 1)) + 2 · dim(Z) =m2n2 − (mn− 1) + 2(m+ n− 2) =m2n2 − (m− 2)(n− 2) + 1 With the above lemmas, we can now derive the main result: Theorem 3. Given a bipartite quantum system Hm ⊗H\, where Hm and H\ are Hilbert spaces with di- mensions m and n respectively, if min (m,n) > 2 and (m,n) 6= (3, 3), then there exists an unitary operator Φ ∈ U(mn), s.t. Φ(Z) ∩ Z = ∅ That is, Φ is a universal entangler. Proof. If U(mn) ⊂ X , then it follows that GL(mn,C) = U(mn) ⊂ X from Lemma 2. And in this assumption, we also have dim(X) ≤ m2n2− (m−2)(n−2)+1< m2n2 = dim(GL(mn,C)) from Lemma 4 [2]. It’s a contradiction. So U(mn) 6⊂ X , i.e. a unitary operator Φ ∈ U(mn) with Φ(Z) ∩ Z = ∅ exists. Conclusion. In summary, it is shown that a universal entangler of bipartite system exists except for the cases of 1⊗n, 2⊗n, m⊗ 1, m⊗ 2 and 3⊗ 3. So we have com- pletely determined when a universal entangler exists. It seems that the method employed in this Letter can be extended to the multipartite case. This extension will lead us to explore the geometric structure of entangled states and other related objects. Furthermore, we can consider some more practical questions: (1) How to con- struct such a universal entangler for 3⊗4 bipartite system explicitly? (2) Given a universal entangler, what is the minimum entanglement it guarantees to output with re- spect to the definition of entanglement measure for pure states [4, 5, 15]? (3) Furthermore, what is the optimal universal entangler which maximized the minimally pos- sible entanglement the entangler outputs. Intuitively, for a bipartite system of sufficiently large dimensions,a ran- domly chosen unitary operator seems to be a universal entangler with high probability. Nevertheless, we failed to give a proof of this conjecture. Acknowledgements. We are thankful to the colleagues in the Quantum Computation and Information Research Group of Tsinghua University for helpful discussions. And one of us (J.-X. Chen) enjoyed delightful discus- sions with W. Huang, L. Xiao, Z. Q. Zhang and Q. Lin. Jun Yu thank Professor X.-J. Tan and Q.-C. Tian who ask him to study Hartshorne ’s book. This work was partly supported by the Natural Science Foundation of China(Grant Nos.60621062 and 60503001) and the Hi- Tech Research and Development Program of China(863 project)(Grant No.2006AA01Z102). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] [1] Here, we will present a short proof of the irreduciblity of F−1(Z). ∀i, j: 0 ≤ i ≤ m− 1, 0 ≤ j ≤ n− 1, let Pm,i = {[x0, x1, · · · , xi, · · · , xm−1]|xi 6= 0}, and Pn,j = {[y0, y1, · · · , yj , · · · , yn−1]|yi 6= 0}. Let Zi,j = σ(Pm,i × Pn,j), we will have Z = ∪i,jZi,j , and for ∀i, j, Zi,j ∼= Pm,i × Pn,j ∼= A m−1 × An−1 ∼= A m+n−2, and each Zi,j is open in Z. ∀i, j, we can construct a morphism Ti,j : Zi,j → GL(mn,C) s.t. ∀y ∈ Zi,j , (Ti,j(y))(y0) = y. Then ∀i, j, î, ĵ, (g, y) ∈ Ψ î,ĵ Zi,j) ⇐⇒ g·y ∈ Zi,j and y ∈ Zî,ĵ ⇐⇒ (g·Tî,ĵ(y1))y0 = Ti,j(y2)(y0) for some y2 ∈ Zi,j , y1 ∈ Zî,ĵ ⇐⇒ g ∈ Ti,j(Zi,j)·H ·(Tî,ĵ(Zî,ĵ)) −1. Thus we have F−1(Z) i,j,̂i,ĵ {(g, y) ∈ Ψ−1(Z î,ĵ × Zi,j)} is irreducible since it’s the image of a morphism Zi,j × H × Zî,ĵ → F −1(Z). Each −1(Z) i,j,̂i,ĵ is also open in F−1(Z), then the irreducib- lity follows from lemma 3.3. [2] In fact, we proved Ψ : F−1(Z) → Z × Z is an alge- braic fibre bundle thus dim(F−1(Z)) = m2n2 − mn + 1 + 2 · dim(Z). We also conjecture that dim(X) = dim(F−1(Z)) and X = X. [3] M. Artin. Alegbra. Prentice Hall; United States Ed edi- tion, 1991. [4] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu- macher. Concentrating partial entanglement by local op- erations. Phys. Rev. A., 53(2046), 1996. [5] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters. Purification of noisy entanglement and faithful teleportation via noisy chan- nels. Phys. Rev. Lett., 76(5):722–725, Jan 1996. [6] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal. Unextendible product bases and bound entanglement. Physical Review Letters, 82:5385, 1999. [7] B.V.R. Bhat. A completely entangled subspace of maxi- mal dimension. International Journal of Quantum Infor- mation, to appear. [8] D.C. Brody and L.P. Hughston. Geometric quantum me- chanics. Journal of Geometry and Physics, 2001. [9] Vladimı́r Bužek and Mark Hillery. Optimal manipula- tions with qubits: Universal quantum entanglers. Phys. Rev. A, 62(2):022303, Jul 2000. [10] David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, and Barbara M. Terhal. Unextendible prod- uct bases, uncompletable product bases and bound en- tanglement. Communications in Mathematical Physics, 238:379, 2003. [11] R. Hartshorne. Alegbraic Geometry. Springer-Verlag New York, 1977. [12] H. Heydari and G. Bjork. Complex multi-projective va- riety and entanglement. J.Phys.A:Math.Gen., 38:3203, 2005. [13] A. Miyake. Classification of multipartite entangled states by multidimensional determinants. Phys.Rev.A, 67(1):012108, Jan 2003. [14] Parthasarathy. On the maximal dimension of a com- pletely entangled subspace for finite level quantum sys- tems. http://www.arxiv.org/abs/quant-ph/0405077. [15] S. Popescu and D. Rohrlich. Thermodynamics and the measure of entanglement. Phys. Rev. A, 56(5):R3319– mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] R3321, Nov 1997. [16] A. T. Rezakhani. Characterization of two-qubit perfect entanglers. Phys. Rev. A., 70, 2004. [17] Fabio Sciarrino, Francesco De Martini, and Vladimir Buzek. Realization of the optimal universal quantum entangler. Physical Review A (Atomic, Molecular, and Optical Physics), 70(6):062313, 2004. [18] P. Tauvel and R.W.T. Yu. Lie Algebras and Algebraic Groups. Springer Monographs in Mathematics. 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0704.1474

Fan-shaped and toric textures of mesomorphic oxadiazoles

Fan-shaped and toric textures of mesomorphic oxadiazoles A. Sparavigna , A. Mello , and B. Montrucchio Dipartimento di Fisica, Politecnico di Torino Dipartimento di Automatica ed Informatica, Politecnico di Torino C.so Duca degli Abruzzi 24, Torino, Italy Keywords: Liquid crystals, phase transitions Author for correspondence: A. Sparavigna, [email protected] Abstract When a family of non symmetrical heterocycled compounds is investigated, a variety of mesophases can be observed with rather different features. Here we report the behaviour of seven different members among a family of such materials, that consists of mesomorphic oxadiazole compounds. In two of these compounds, the optical microscope investigation shows very interesting behaviours. In their smectic phases, fan-shaped and toric textures, sometimes with periodic instability, are observed. Moreover, the nematic phase displays a texture transition. Texture transitions have been previously observed only inside the nematic phase of some compounds belonging to the families of the oxybenzoic and cyclohexane acids. In these two oxadiazole compounds we can observe what we define as a "toric nematic phase", heating the samples from the smectic phase. The toric nematic texture disappears as the sample is further heated, changing into a smooth texture. Introduction The correlation between the existence of mesophases and their features and the chemical structure of molecules is the core of the research in liquid crystals. As we have seen studying some alkyloxybenzoic acids, an increased length of the alkyl chain produces a strong increase of the nematic mesophase range [1,2]. These acids are materials where the mesogenic unit structure is a symmetrical hydrogen-bonded dimer. A simple binary mixture of two members of the same group of acids is able to enlarge the smectic range [3]. When a family of non symmetrical structures is under investigation, the variety of results is strongly enhanced. Here we discuss the mesomorphic behaviour of some members among a family of non symmetrical compounds, consisting of mesomorphic oxadiazole compounds [4-6]. These materials have been developed because their molecular structure with heterocycles as substituents gives smectic and nematic mesophases. It was observed in particular, that not only the chemical structure of the substituents, but also their position with respect to the oxadiazolic ring of the molecule is relevant for the mesophases. The optical microscope investigations, which we are discussing in this paper, show very interesting behaviour of the smectic and the nematic phase of some of these oxadiazole compounds. In the smectic phases, fan-shaped and toric textures, with periodic instability in the toric structure, are observed. Moreover, it is also remarkable the behaviour of the nematic phase, displaying a texture transition. These transitions have been previously observed inside the nematic phase of some mesomorphic thermotropic materials belonging to the families of the alkyloxybenzoic and cyclohexane acids [7-13]. It is a more or less abrupt change, driven by the temperature, in the nematic texture seen by means of the polarised light microscope. In some cases, it is also possible to observe the transition in the calorimetric investigations and dielectric spectroscopy [8,14]. In other cases, it is necessary an image processing aided procedure to enhance the observation of the transition [1,9]. Alkyloxybenzoic and cyclohexane acids are mesogenic because the mesogenic units are dimers. The more common explanation for the presence of a texture transition in the nematic range is based on the existence of cybotactic clusters of dimers, favouring a local smectic order in the nematic range [15,16]. Our observation of a texture transition inside the nematic phase of two oxadiazole compounds suggests a possible existence of cybotactic clusters in these materials too. We start with a short description of the oxadiazole family used for investigations and then the more deep discussion of the smectic phases and of the texture transitions. The oxadiazole compounds. The members of the oxadiazole family we are now discussing are those whose structures are reported in Fig.1. These compounds have a non symmetrical heterocycled structure. They have been previously studied with the DSC differential scanning calorimetry [5]. As it is possible to see in Table 1, which reports the transition temperatures, the molecular structure is strongly influencing the mesophase ranges. Not only the presence of benzene or cyclohexane rings, but also their position, with respect to the thiadiazolic or imidazolic part of the molecule, is important for the existence of mesophases. All the samples were inserted in the cell when the material was in the isotropic phase. The walls of liquid crystal cells are untreated clean glass surfaces. No treaments were done to favor planar or homeotropic alignments. The liquid crystal cells are heated and cooled in a thermostage and textures observed with a polarized light microscope. Let us start the discussion from compound A of Figure 1: it has only the smectic phase. Compound A It is an interesting material, that displays at room temperature the texture shown in the upper part of Figure 2. In Table I, we guess the phase at room temperature to be a crystal phase, or a rigid smectic phase. On heating, the sample changes into a smectic phase with a fan-shaped texture at the temperature of 140 °C. The material becomes an isotropic liquid at the temperature of 210°C. The material does not show a nematic phase. On cooling the sample has a more complex behaviour. First of all, beautiful bâtonnets appears cooling from the isotropic liquid, with periodic instabilities insides. Then larger domains grow, and the texture changes very fast in the image frame as in a scene played by moving fans. The smectic phase has a strong hysteresis on cooling, because the fan-shaped texture is observed till down 110°C. Then the sample probably becomes a crystal or a rigid smectic F. The transition is shown in Fig.3. A further reduction of the temperature gives the texture in Figure 2. In this rigid phase observed with crossed polarisers, the crystal shows thick dark lines among the lamellar structure. These lines are cracks as it is possible to see when they arise as the sample is cooled down from the smectic phase. Compound B On heating, we observed the following temperature transitions: 156°C from crystal to smectic, 172°C from smectic to nematic and 200°C from the nematic to the isotropic phase. On cooling, we have the transition from the isotropic liquid to the nematic phase at 198°C and the transition into the smectic phase at 168°C. The crystal phase appears at 145°C. This compound possesses a nematic phase too. The smectic phase has not only the fan-shaped texture but also regions with macroscopic toric arrangement of the smectic planes. The toric domains are focal conic domains where the ellipse is degenerated in a circle and the hyperbola is degenerated in a straight line [17]. If we look at the sequence in Figure 4, the nematic, the smectic and the crystal phases have textures where the same overall arrangement of defects is unchanged on cooling. It could be a memory effect of the surface layers of the material, close to the cell walls. Compound C The sample has a crystal phase till 94°C. Then a smectic phase till 172°C. We observed the clearing point at 233°C. On cooling, the nematic phase appears at 232°C, the smectic phase at 163°C, and the crystal at 65°C. What is remarkable in this compound is a texture transition inside the nematic phase. We can observe what we define as a "toric nematic phase" on heating till 195-200°C, and then a smooth nematic texture till the clearing point. On cooling, the toric phase appears around 200°C. This behaviour of the nematic phase is observed also in the compound D, that we will describe in the next section. The smectic phase of sample C has a fan shaped and toric smectic phase. The Figure 5 shows in its upper part, the arrangement of the toric defects in the smectic phase. Sometimes on cooling the smectic phase, it happens that the toric texture displays inside an undulation instability (lower part of Fig.5). A recent theory of the lamellar phases could be invoked to explain the appearance of undulations in the planes of the lamellar phases [18,19]. Let us discuss more deeply the nematic phase. On heating, the nematic phase substitutes the smectic phase, the transition is shown in Fig.6. The nematic phase in the figure has a texture that shows a "toric" arrangement of the defects, and this is why we call this nematic phase a "toric nematic phase" . The texture remembers the shape of domains in the smectic phase. On heating, this nematic texture disappears between 195-200°C, depending on the position in the cell observed at microscope, and a smooth texture is displayed by the nematic. This is the texture that is observed on cooling from the isotropic phase. On cooling, around 200°C, the toric texture starts to grow in the cell as shown in the Figure 7. It is remarkable that a compound, rather different from the dimeric acids [1,2], has a texture transition in the nematic phase. If we assume as a reasonable explanation for the existence of a texture transition, the presence in the nematic melt of cybotactic clusters with local smectic order, we can invoke the presence of such clusters also in this material. The local smectic order is responsible for the toric nematic texture. Compound D The sample has a crystal phase that changes in another crystal phase at 82°C. The crystal becomes to be smectic at 103°C. The smectic phase persists till 110°C. We observed the clearing point at 232°C. On cooling, the nematic phase appears at 230°C, the smectic phase at 97°C, and the crystal at 71°C. This compound, as the previous one, has a texture transition inside the nematic phase. We can observe the toric nematic phase on heating till 150°C, and then a nematic texture till the clearing point. On cooling, the toric phase appears around 150°C. The Figure 8 shows a beautiful toric nematic phase, with the bent structure of the defects. The bent contours between the nematic domains are clearly visible in this figure. In fact, samples C and D are similar in the behaviour of the smectic and nematic phases. They have a very slight change in the structure of the molecule, as we can see in the Figure 1, with respect of the oxadiazolic group. The different position changes the transition temperatures. Compounds E,F and G These three compounds have only the nematic mesophase. In the nematic range there is no evidence of a texture transition. Sample E has a transition from a crystal phase in the nematic phase at 46°C, on heating. The clearing point is at 109°C. On cooling, the nematic phase appears at 107°C. The nematic phase remains till the room temperature and just after a very long relaxation time, the crystal is formed. The nematic phase has always a visible thermal flickering of the texture. The phase sequence for sample F is: a crystal phase till 72°C that changes into a nematic phase till 99°C. On cooling, the nematic starts from 98°C with a very fast transition. The sample remains nematic till 50°C. For sample G, we observed a transition from crystal to nematic at 80°C, and a short nematic range till 87°C. On cooling, the nematic range extends till 51°C. This behaviour, a wide nematic range on cooling, is common to these three materials and confirmed by previous DSC measurements. Discussion. As already observed in some members of the alkyloxybenzoic acid family (6-,7-,8-,9OBAC), in the binary mixtures of 6OBAC with other members of this family and in a cyclohexane acid (C6), the nematic phase exhibits a different order at low and at high temperatures. This different order is observed with the optical microscopy because the cell has two different textures in the nematic range. For the previously mentioned compounds, the low temperature nematic texture is composed of very small domains and looks like a granular structure. If the sample has a smectic phase, the texture of the smectic phase is also showing a granular texture. In the case of the oxadiazole compounds C and D, we observe a texture transition in the nematic phase driven by the temperature, with a low temperature subphase textured with bent-shaped contours between domains. We defined this texture as a toric texture, because a further cooling of the sample gives a smectic phase with toric domains. This is consistent with the hypothesis that a concentration of cybotactic clusters is at the origin of the texture transition in the nematic phase. Let us remember that the cybotactic clusters are groups of molecules with a local smectic phase; they can be imagined as smectic seeds in the nematic fluid. In a previous paper, we proposed a model to explain the origin of different textures in the nematic phase [1]. Let us imagine a thin liquid crystal cell in the smectic phase, with defects in the smectic structure. Rising the temperature, the material phase changes into a nematic phase, the smectic planes disappear but, in some places of the cell, these planes remain as little clusters, anchored at the cell walls. We have then these cybotactic clusters preserving a local smectic order. Moreover, defects on the walls can contribute to the persistence of these clusters. When the temperature further increases, the smectic order in cybotactic clusters is suppressed and the texture changes, more or less abruptly, in a smooth texture. The persistence of a local smectic order at the walls explains the memory effect observed in the cells of compounds C and D, if we suppose that the mechanism in the formation of the textures in these compounds is the same as in the alkyloxybenzoic materials. The memory effect is also displayed in the cells prepared with compounds A and B, the materials with a more "smectic" behaviour. Samples E,F and G have only the nematic phase that possesses, on cooling, a wider range compared with that observed on heating. No memory effect is seen. The presence of cybotactic clusters can be excluded in E,F and G compounds. Or, it is better to tell in other words, cybotactic clusters do not grow with the decrease in the temperature, consistent with the fact that the medium does not support a smectic phase. This can be due to a kind of frustration developed in the medium, wherein a phase transition into layered structure is not possible due to the molecular structure. As it is shown in the molecular scheme of Fig.1, the responsible can be the position of the Cl atom. All the results reported in this paper, on the smectic and nematic phases, confirm that the family of oxadiazoles is very promising for further researches in the investigation of texture transitions, and the relationship between the molecular structure and the features of the mesophases. Let us remember once more that two of these compounds shows a texture transition in the nematic range. References [1] A. Sparavigna, A. Mello, B. Montrucchio, Phase Trans. 79 293 (2006). [2] R.F. Bryan, P. Hartley, R.W. Miller and Sheng Ming-Shing, Mol. Cryst. Liq. Cryst. 62 281 (1980). [3] A. Sparavigna, A. Mello, B. Montrucchio, Phase Trans.(2007) in print. [4] O.Francescangeli, L.A.Karamysheva, S.I.Torgova, A.Sparavigna and A.Strigazzi, Proc. SPIE Liquid Crystals: Chemistry and Structure 3319 139 (1998) [5] L.A. Karamysheva, S.I.Torgova, I.F. Agafonova, A.Sparavigna and A.Strigazzi, Mol. Cryst. Liq. Cryst. 352 335 (2000) [6] S. Torgova, L. Karamysheva and A. Strigazzi, Brazilian J. of Physics, 32 593 (2002) [7] M. Petrov, A. Braslau, A.M. Levelut and G. Durand, J. Phys. II (France) 2 1159 (1992). [8] L. Frunza, S. Frunza, M. Petrov, A. Sparavigna and S.I. Torgova, Mol. Mater. 6 215 (1996). [9] B. Montrucchio, A. Sparavigna and A. Strigazzi, Liq. Cryst. 24 841 (1998). [10] B. Montrucchio, A. Sparavigna, S.I. Torgova and A. Strigazzi, Liq. Cryst. 25 613 (1998). [11] P. Simova and M. Petrov, Phys. Stat. Sol. A 80 K153 (1983). [12] M. Petrov and P.Simova, J. Phys. D: Appl. Phys. 18 239 (1985). [13] L.S. Chou, Chinese J. Phys. 18 114 (1980). [14] L. S. Chou and E. F. Carr, Phys. Rev. A 7, 1639–1641 (1973) [15] A. DeVries, Mol. Cryst. Liq. Cryst. 10 31 (1970). [16] G.W. Gray and B. Jones, J. Chem. Soc. 41 79 (1953). [17] M. Kleman, C. Meyer and Yu. A. Nastishin, Phil. Mag. 86 4439 (2006) [18] G.K. Auernhammer, H.R. Brand and H. Pleiner, Rheologica Acta 39 215 (2000). [19] G.K. Auernhammer, H. Pleiner, and H.R. Brand, Phys. Rev. E 66 061707 (2002). Compound Transition temperatures on heating A K-140°C-Sm-210°C-I B K-156°C-Sm-172°C-N-200°C-I C K-94°C-Sm-172°C-N-233°C-I D K-103°C-Sm-108°C-N-232°C-I E K-46°C-N-109°C-I F K-72°C-N-99°C-I G K-80°C-N-87°C-I TABLE I: Transition temperatures of the compounds on heating. The first compound (A) has only the smectic mesophase. E,F and G possess only a nematic mesophase. Figure captions Figure 1: Molecular structures of the seven oxadiazole compounds under investigation. Figure 2: The crystal texture of compound A (upper part). Note the thick dark lines among the lamellar structure. These are cracks appearing on cooling from the smectic phase. In its lower part, the figure shows the bâtonnets growing in the isotropic melt on cooling (image dimensions 0.38mm x 0.5mm). Figure 3: The sequence shows the cooling of compound A from the smectic fan-shaped texture in the lamellar texture of the crystal phase (image dimensions 0.38mm x 0.5 mm). Figure 4: The sequence shows the cooling of compound B. In the upper part the nematic, in the middle the smectic and, in the lower part of the figure, the crystal phase . Figure 5: The smectic phase of compound C. Note the toric focal conic domains. In the lower part of the figure, domains with undulations inside. Figure 6: The transition from the smectic into the nematic phase of compound C: note the toric texture of the nematic phase. Figure 7: The texture transition for sample C, on cooling. Note the growth of the low temperature nematic texture with the bent-shaped lines. Figure 8: The toric nematic phase as it appears when heating the smectic phase of compound D. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8

0704.1475

Williams' decomposition of the L\'evy continuous random tree and simultaneous extinction probability for populations with neutral mutations

WILLIAMS’ DECOMPOSITION OF THE LÉVY CONTINUUM RANDOM TREE AND SIMULTANEOUS EXTINCTION PROBABILITY FOR POPULATIONS WITH NEUTRAL MUTATIONS ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS Abstract. We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity propor- tional to the size of the population. First we establish a Williams’ decomposition of the genealogy of the total population given by a continuum random tree, according to the an- cestral lineage of the last individual alive. This allows us to give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population. 1. Introduction We consider an initial Eve-population whose size evolves as a continuous state branching process (CB), Y 0 = (Y 0t , t ≥ 0), with branching mechanism ψEve. We assume this population gives birth to a population of irreversible mutants. The new mutants population can be seen as an immigration process with rate proportional to the size of the Eve-population. We assume the mutations are neutral, so that this second population evolves according to the same branching mechanism as the Eve-population. This population of mutants gives birth also to a population of other irreversible mutants, with rate proportional to its size, and so on. In [2], we proved that the distribution of the total population size Y = (Yt, t ≥ 0), which is a CB with immigration (CBI) proportional to its own size, is in fact a CB, whose branching mechanism ψ depends on the immigration intensity. The joint law of (Y 0, Y ) is characterized by its Laplace transform, see Section 4.1.4. This model can also be viewed as a special case of multitype CB, with two types 0 and 1, the individuals of type 0 giving birth to offsprings of type 0 or 1, whereas individuals of type 1 only have type 1 offsprings, see [13, 6] for recent related works. In the particular case of Y being a sub-critical or critical CB with quadratic branching mechanism (ψ(u) = α0u + βu 2, β > 0, α0 ≥ 0), the probability for the Eve-population to disappear at the same time as the whole population is known, see [17] for the critical case, α0 = 0, or Section 5 in [2] for the sub-critical case, α0 > 0. Our aim is to extend those results for the large class of CB with unbounded total variation and a.s. extinction. Formulas given in [2] could certainly be extended to a general branching mechanism, but first computations seem to be rather involved. In fact, to compute those quantities, we choose here to rely on the description of the genealogy of sub-critical or critical CB introduced by Le Gall and Le Jan [12] and developed later by Duquesne and Le Gall [7], see also Lambert [10] for the genealogy of CBI with Date: September 4, 2021. 2000 Mathematics Subject Classification. 60G55, 60J70, 60J80, 92D25. Key words and phrases. Continuous state branching process, immigration, continuum random tree, Williams’ decomposition, probability of extinction, neutral mutation. http://arxiv.org/abs/0704.1475v2 2 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS constant immigration rate. Le Gall and Le Jan defined via a Lévy process X the so-called height process H = (Ht, t ≥ 0) which codes a continuum random tree (CRT) that describes the genealogy of the CB (see the next section for the definition of H and the coding of the CRT). Initially, the CRT was introduced by Aldous [4] in the quadratic case: ψ(λ) = λ2. Except in this quadratic case, the height process H is not Markov and so is difficult to handle. That is why they also introduce a measure-valued Markov process (ρt, t ≥ 0) called the exploration process and such that the closed support of the measure ρt is [0,Ht] (see also the next section for the definition of the exploration process). We shall be interested in the case where a.s. the extinction of the whole population holds in finite time. The branching mechanism of the total population, Y , is given by: for λ ≥ 0, (1) ψ(λ) = α0λ+ βλ (0,∞) π(dℓ) e−λℓ−1 + λℓ where α0 ≥ 0, β ≥ 0 and π is a Radon measure on (0,∞) such that (0,∞) (ℓ∧ ℓ2) π(dℓ) <∞. We shall assume that Y is of infinite variation, that is β > 0 or (0,1) ℓπ(dℓ) = ∞. We shall assume that a.s. the extinction of Y in finite time holds, that is, see Corollary 1.4.2 in [7], we assume that ∫ +∞ dv We suppose that the process Y is the canonical process on the Skorokhod space D(R+,R+) of càdlàg paths and that the pair (Y, Y 0) is the canonical process on the space D(R+,R+) 2. Let Px denote the law of the pair (Y, Y 0) (see [2]) started at (Y0, Y 0 ) = (x, x). The probability measure Px is infinitely divisible and hence admits a canonical measure N: it is a σ-finite measure on D(R+,R+) 2 such that (Y, Y 0) (Y i, Y 0,i) where ((Y i, Y 0,i), i ∈ I) are the atoms of a Poisson measure on D(R+,R+) 2 with intensity xN(dY, dY 0). In particular, we have (3) Ex[e −λYt ] = exp(−xN[1− e−λYt ]) and u(λ, t) = N[1 − e−λYt ] is the unique non-negative solution of u(λ,t) = t, for t ≥ 0 and λ ≥ 0. Let τY = inf{t > 0;Yt = 0} be the extinction time of Y . Letting λ go to ∞ in the previous equalities leads to Px(τY < t) = exp−xN[τY ≥ t], where the positive function c(t) = N[τY ≥ t] solves = t, for t > 0. Let us consider the exploration process (ρt, t ≥ 0) associated with this CB. We denote by N its excursion measure. Recall that the closed support of the measure ρt is [0,Ht], where H is the height process. Let La be the total local time at level a of the height process H (well-defined under N). Then, the process (La, a ≥ 0) under N has the same distribution as the CB Y under N. WILLIAMS’ DECOMPOSITION 3 We decompose the exploration process, under the excursion measure, according to the maximum of the height process. In terms of the CRT, this means that we consider the longest rooted branch of the CRT and describe how the different subtrees are grafted along that branch, see Theorem 3.3. When the branching mechanism is quadratic, the height process H is a Brownian excursion and the exploration process ρt is, up to a constant, the Lebesgue measure on [0,Ht]. In that case, this decomposition corresponds to Williams’ original decomposition of the Brownian excursion (see [18]). This kind of tree decomposition with respect to a particular branch (or a particular subtree) is not new, let us cite [9, 14] for instance, or [16, 15, 8] for related works on superprocesses. We present in the introduction a Poisson decomposition for the CB only, and we refer to Theorem 3.3 for the decomposition of the exploration process. Conditionally on the extinction time τY equal to m, we can represent the process Y as the sum of the descendants of the ancestors of the last individual alive. More precisely, let N ′(dℓ, dt) = δ(ℓi,ti)(dℓ, dt) be a Poisson point measure with intensity 1[0,m)(t) e −ℓc(m−t) ℓπ(dℓ)dt, (6) κmax(dt) = ℓiδti(dt) + 2β1[0,m)(t)dt. Let Nt(dY ) denote the law of (Y (s − t), s ≥ t) under N and δ(tj ,Y j) be, conditionally on N ′, a Poisson point measure with intensity κmax(dt)Nt[dY,1{τY ≤m}] where Nt[dY,1{τY ≤m}] denotes the restriction of the measure Nt to the event {τY ≤ m}. The next result is a direct consequence of Theorem 3.3. Proposition 1.1. The process j∈J Y j is distributed as Y under N, conditionally on {τY = Let τY 0 = inf{t > 0;Y t = 0} be the extinction time of the Eve-population. In the particular case where the branching mechanism of the Eve-population is given by a shift of (7) ψEve(·) = ψ(θ + ·)− ψ(θ), for some θ > 0 and β = 0, the pruning procedure developed in [1] gives that the nodes of width ℓi correspond to a mutation with probability 1− e −θℓi . As β = 0 there is no mutation on the skeleton of the CRT outside the nodes. In particular, we see simultaneous extinction of the whole population and the Eve-population if there is no mutation on the nodes in the ancestral lineage of the last individual alive. This happens, conditionally on κmax, with probability i∈I ℓi . 4 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS Integrating w.r.t. the law of N gives that the probability of simultaneous extinction, condi- tionally on {τY = m}, is under N, given by N[τY 0 = m|τY = m] = exp− 1[0,m)(t) e −ℓc(m−t) ℓπ(dℓ)dt 1− e−θℓ = exp− [ψ′(c(m− t) + θ)− ψ′(c(m− t))] dt = exp− φ′(c(t)) dt, where φ = ψEve − ψ. Now, using that the distribution of (Y 0, Y ) is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that Px(τY 0 = m|τY = m) = N[τY 0 = m|τY = m] that is Px(τY 0 = m|τY = m) = exp− φ′(c(t)) dt. Notice that this formula is also valid for the quadratic branching mechanism (ψ(u) = α0u+ βu2, β > 0, α0 ≥ 0), see Remark 5.3 in [2]. In fact this formula is true in a general framework. Following [2], we consider the branching mechanisms of the total population and Eve-population are given by ψ(λ) = α0λ+ βλ (0,∞) π(dℓ)[e−λℓ−1 + λℓ], ψEve(λ) = αEveλ+ βλ (0,∞) πEve(dℓ)[e −λℓ−1 + λℓ], and the immigration function φ(λ) = ψEve(λ)− ψ(λ) = αImmλ+ (0,∞) ν(dℓ)(1 − e−λℓ), where αImm = αEve − α0 − (0,∞) ℓν(dℓ) ≥ 0 and π = πEve + ν, where πEve and ν are Radon measures on (0,∞) with (0,∞) ℓν(dℓ) <∞. Notice the condition (0,∞) ℓν(dℓ) <∞ is stronger than the usual condition on the immigration measure, (0,∞) (1 ∧ ℓ) ν(dℓ) < ∞, but is implied by the requirement that (1,∞) ℓν(dℓ) < (1,∞) ℓπ(dℓ) <∞. Inspired by Theorem 3.3, we consider N (dℓ, dt, dz) = δ(ℓi,ti,zi)(dℓ, dt, dz) a Poisson point measure with intensity (8) 1[0,m)(t) e −ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt. Intuitively, the mark zi indicates if the ancestor (of the last individual alive) alive at time ti had a new mutation (zi = 1) or not (zi = 0). Note however that if β > 0 we have to take into account mutation on the skeleton. More precisely, let T1 = min{ti, zi = 1} be the first mutation on the nodes in the ancestral lineage of the last individual alive and let T2 be an exponential random time with parameter αImm independent of N . The time T2 corresponds to the first mutation on the skeleton for the ancestral lineage of the last individual alive. We T0 = min(T1, T2) if min(T1, T2) < m, T0 = +∞ otherwise. In particular there is simultaneous extinction if and only if T0 = +∞. WILLIAMS’ DECOMPOSITION 5 For t ≥ 0, let us denote by Nt(dY 0, dY ) the joint law of ((Y 0(s− t), Y (s− t)), s ≥ t) under N. Recall κmax given by (6). Conditionally on N and T2, let δ(tj ,Y 0,j ,Y j) be a Poisson point measure, with intensity κmax(dt)Nt[(dY 0, dY ), 1{τY ≤m}]. We set (10) (Y ′ , Y ′) = tj<T0 (Y 0,j, Y j) + tj≥T0 (0, Y j). We write Qm for the law of (Y ′0, Y ′) computed for a given value of m. Theorem 1.2. Under Qm, (Y ′0, Y ′) is distributed as (Y 0, Y ) under N[·|τY = m] , or equiv- alently, under |c′(m)|Qm(·)dm, (Y ′0, Y ′) is distributed as (Y 0, Y ) under N. Let us remark that this Theorem is very close to Theorem 3.3 but only deals with CB and does not specify the underlying genealogical structure. This is the purpose of a forthcoming paper [3] where the genealogy of multi-type CB is described. Intuitively, conditionally on the last individual alive being at time m, until the first mu- tation in the ancestral lineage (that is for tj < T0) , its ancestors give birth to a population with initial Eve type which has to die before time m, and after the first mutation on the ancestral lineage (that is for tj ≥ T0), there is no Eve-population in the descendants which still have to die before time m. Now, using that the distribution of (Y 0, Y ) is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that Px(τY 0 = m|τY = m) = N[τY 0 = m|τY = m]. As N[τY 0 = m|τY = m] = Qm(T0 = +∞) = Qm(T1 = +∞)Qm(T2 ≥ m) (0,∞) e−ℓc(m−t) ℓν(dℓ) e−αImmm dt φ′(c(t)), we deduce the following Corollary. Corollary 1.3 (Probability of simultaneous extinction). We have for almost every m > 0 Px(τY 0 = m|τY = m) = exp− φ′(c(t)) dt, where c is the unique (non-negative) solution of (5). The paper is organized as follows. In Section 2, we recall some facts on the genealogy of the CRT associated with a Lévy process. We prove a Williams’ decomposition for the exploration process associated with the CRT in Section 3. We prove Theorem 1.2 in Section 4. Notice that Proposition 1.1 is a direct consequence of Theorem 1.2. 2. Notations We recall here the construction of the Lévy continuum random tree (CRT) introduced in [12, 11] and developed later in [7]. We will emphasize on the height process and the exploration process which are the key tools to handle this tree. The results of this section are mainly extracted from [7]. 6 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS 2.1. The underlying Lévy process. We consider a R-valued Lévy process (Xt, t ≥ 0) with Laplace exponent ψ (for λ ≥ 0 E e−λXt = etψ(λ)) satisfying (1) and (2). Let I = (It, t ≥ 0) be the infimum process of X, It = inf0≤s≤tXs, and let S = (St, t ≥ 0) be the supremum process, St = sup0≤s≤tXs. We will also consider for every 0 ≤ s ≤ t the infimum of X over [s, t]: Ist = inf s≤r≤t The point 0 is regular for the Markov process X − I, and −I is the local time of X − I at 0 (see [5], chap. VII). Let N be the associated excursion measure of the process X − I away from 0, and σ = inf{t > 0;Xt − It = 0} the length of the excursion of X − I under N. We will assume that under N, X0 = I0 = 0. Since X is of infinite variation, 0 is also regular for the Markov process S −X. The local time, L = (Lt, t ≥ 0), of S −X at 0 will be normalized so that t ] = e−tψ(β)/β , where L−1t = inf{s ≥ 0;Ls ≥ t} (see also [5] Theorem VII.4 (ii)). 2.2. The height process and the Lévy CRT. For each t ≥ 0, we consider the reversed process at time t, X̂(t) = (X̂ s , 0 ≤ s ≤ t) by: X̂(t)s = Xt −X(t−s)− if 0 ≤ s < t, and X̂ t = Xt. The two processes (X̂ s , 0 ≤ s ≤ t) and (Xs, 0 ≤ s ≤ t) have the same law. Let Ŝ(t) be the supremum process of X̂(t) and L̂(t) be the local time at 0 of Ŝ(t) − X̂(t) with the same normalization as L. Definition 2.1 ([7], Definition 1.2.1 and Theorem 1.4.3). There exists a process H = (Ht, t ≥ 0), called the height process, such that for all t ≥ 0, a.s. Ht = L̂ t , and H0 = 0. Because of hypothesis (2), the height process H is continuous. The height process (Ht, t ∈ [0, σ]) under N codes a continuous genealogical structure, the Lévy CRT, via the following procedure. (i) To each t ∈ [0, σ] corresponds a vertex at generation Ht. (ii) Vertex t is an ancestor of vertex t′ if Ht = H[t,t′], where (11) H[t,t′] = inf{Hu, u ∈ [t ∧ t ′, t ∨ t′]}. In general H[t,t′] is the generation of the last common ancestor to t and t (iii) We put d(t, t′) = Ht +Ht′ − 2H[t,t′] and identify t and t ′ (t ∼ t′) if d(t, t′) = 0. The Lévy CRT coded by H is then the quotient set [0, σ]/ ∼, equipped with the distance d and the genealogical relation specified in (ii). Let (τs, s ≥ 0) be the right continuous inverse of −I: τs = inf{t > 0;−It > s}. Recall that −I is the local time of X − I at 0. Let Lat denote the local time at level a of H until time t, see Section 1.3 in [7]. Theorem 2.2 ([7], Theorem 1.4.1). The process (Laτx , a ≥ 0) is under P (resp. N) defined as Y under Px (resp. N). In what follows, we will use the notation N instead of N for the excursion measure to stress that we consider the genealogical structure of the branching process. WILLIAMS’ DECOMPOSITION 7 2.3. The exploration process. The height process is not Markov. But it is a simple function of a measure-valued Markov process, the so-called exploration process. If E is a Polish space, let B(E) (resp. B+(E)) be the set of real-valued measurable (resp. and non-negative) functions defined on E endowed with its Borel σ-field, and let M(E) (resp. Mf (E)) be the set of σ-finite (resp. finite) measures on E, endowed with the topology of vague (resp. weak) convergence. For any measure µ ∈ M(E) and f ∈ B+(E), we write 〈µ, f〉 = f(x)µ(dx). The exploration process ρ = (ρt, t ≥ 0) is a Mf (R+)-valued process defined as follows: for every f ∈ B+(R+), 〈ρt, f〉 = [0,t] t f(Hs), or equivalently (12) ρt(dr) = 0<s≤t Xs−<I (Ist −Xs−)δHs(dr) + β1[0,Ht](r)dr. In particular, the total mass of ρt is 〈ρt, 1〉 = Xt − It. For µ ∈ M(R+), we set (13) H(µ) = sup Supp µ, where Supp µ is the closed support of µ, with the convention H(0) = 0. We have Proposition 2.3 ([7], Lemma 1.2.2). Almost surely, for every t > 0, • H(ρt) = Ht, • ρt = 0 if and only if Ht = 0, • if ρt 6= 0, then Supp ρt = [0,Ht]. In the definition of the exploration process, as X starts from 0, we have ρ0 = 0 a.s. To state the Markov property of ρ, we must first define the process ρ started at any initial measure µ ∈ Mf (R+). For a ∈ [0, 〈µ, 1〉], we define the erased measure kaµ by kaµ([0, r]) = µ([0, r]) ∧ (〈µ, 1〉 − a), for r ≥ 0. If a > 〈µ, 1〉, we set kaµ = 0. In other words, the measure kaµ is the measure µ erased by a mass a from the top of [0,H(µ)]. For ν, µ ∈ Mf (R+), and µ with compact support, we define the concatenation [µ, ν] ∈ Mf (R+) of the two measures by: [µ, ν], f ν, f(H(µ) + ·) , f ∈ B+(R+). Finally, we set for every µ ∈ Mf (R+) and every t > 0 ρ k−Itµ, ρt]. We say that t , t ≥ 0) is the process ρ started at ρ 0 = µ, and write Pµ for its law. Unless there is an ambiguity, we shall write ρt for ρ Proposition 2.4 ([7], Proposition 1.2.3). The process (ρt, t ≥ 0) is a càd-làg strong Markov process in Mf (R+). Notice that N is also the excursion measure of the process ρ away from 0, and that σ, the length of the excursion, is N-a.e. equal to inf{t > 0; ρt = 0}. 8 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS 2.4. The dual process and representation formula. We shall need the Mf (R+)-valued process η = (ηt, t ≥ 0) defined by ηt(dr) = 0<s≤t Xs−<I (Xs − I t )δHs(dr) + β1[0,Ht](r)dr. The process η is the dual process of ρ under N thanks to the following time reversal property: recall σ denotes the length of the excursion under N. Proposition 2.5 ([7], Corollary 3.1.6). The processes ((ρs, ηs); s ≥ 0) and ((η(σ−s)−, ρ(σ−s)−); s ≥ 0) have the same distribution under N. It also enjoys the snake property: for all t ≥ 0, s ≥ 0 (ρt, ηt)[0,H[t,s]) = (ρs, ηs)[0,H[t,s]), that is the measures ρ and η between two instants coincide up to the minimum of the height process between those two instants. We recall the Poisson representation of (ρ, η) under N. Let N∗(dx dℓ du) be a Poisson point measure on [0,+∞)3 with intensity dx ℓπ(dℓ)1[0,1](u)du. For every a > 0, let us denote by Ma the law of the pair (µa, νa) of finite measures on R+ defined by: for f ∈ B+(R+) 〈µa, f〉 = N∗(dx dℓ du)1[0,a](x)uℓf(x), 〈νa, f〉 = N∗(dx dℓ du)1[0,a](x)ℓ(1 − u)f(x). We finally set M = da e−α0aMa. Proposition 2.6 ([7], Proposition 3.1.3). For every non-negative measurable function F on Mf (R+) F (ρt, ηt) dt M(dµ dν)F (µ, ν), where σ = inf{s > 0; ρs = 0} denotes the length of the excursion. We can then deduce the following Proposition. Proposition 2.7. For every non-negative measurable function F on Mf (R+) F (ρt, ηt) dL = e−α0a Ma(dµ dν)F (µ, ν), where σ = inf{s > 0; ρs = 0} denotes the length of the excursion. 3. Williams’ decomposition We work under the excursion measure. As the height process is continuous, its supremum Hmax = sup{Hr; r ∈ [0, σ]} is attained. Let Tmax = inf{s ≥ 0;Hs = Hmax}. For every m > 0, we set Tm(ρ) = inf{s > 0,Hs(ρ) = m} the first hitting time of m for the height process. When there is no need to stress the dependence in ρ, we shall write Tm for Tm(ρ). Recall the function c defined by (5) is equal to (14) c(m) = N[Tm <∞] = N[Hmax ≤ m]. WILLIAMS’ DECOMPOSITION 9 We set ρd = (ρTmax+s, s ≥ 0) and ρg = (ρ(Tmax−s)+, s ≥ 0), where x+ = max(x, 0). For every finite measure with compact support µ, we write P∗µ for the law of the exploration process ρ starting at µ and killed when it first reaches 0. We also set µ := lim µ( · |H(µ) ≤ Hmax ≤ H(µ) + ε). We now describe the probability measure P̂∗µ via a Poisson decomposition. Let (αi, βi), i ∈ I be the excursion intervals of the process X − I away from 0 (well defined under P∗µ or under P̂∗µ). For every i ∈ I, we define hi = Hαi and the measure-valued process ρ i by the formula 〈ρit, f〉 = (hi,+∞) f(x− hi)ρ(αi+t)∧βi(dx). We then have the following result. Lemma 3.1. Under the probability P̂∗µ, the point measure δ(hi,ρi) is a Poisson point measure with intensity µ(dr)N[·, H ≤ m− r]. Proof. We know (cf Lemma 4.2.4 of [7]) that the point measure δ(hi,ρi) is under P Poisson point measure with intensity µ(dr)N(dρ). The result follows then easily from standard results on Poisson point measures. � Remark 3.2. Lemma 3.1 gives also that, for every finite measure with compact support µ, if we write µa = µ(· ∩ [0, a]), µ = lim a→H(µ) ( · |Hmax ≤ H(µ)). Theorem 3.3 (Williams’ Decomposition). (i) The law of H is characterized by N[H ≤ m] = c(m), where c is the unique non-negative solution of (5). (ii) Conditionally on H = m, the law of (ρTmax , ηTmax) is under N the law of viriδti + β1[0,m](t)dt, (1− vi)riδti + β1[0,m](t)dt where δ(vi,ri,ti) is a Poisson measure with intensity 1[0,1](v)1[0,m](t) e −rc(m−t) dv rπ(dr) dt. (iii) Under N, conditionally on H = m, and (ρTmax , ηTmax), (ρd, ρg) are independent and ρd (resp. ρg) is distributed as ρ (resp. η) under P̂ ρTmax (resp. P̂∗ηTmax Notice (i) is a consequence of (14). Point (ii) is reminiscent of Theorem 4.6.2 in [7] which gives the description of the exploration process at a first hitting time of the Lévy snake. The end of this section is devoted to the proof of (ii) and (iii) of this Theorem. Let m > a > 0 be fixed. Let ε > 0. Recall Tm = inf{t > 0;Ht = m} is the first hitting time of m for the height process, and set Lm = sup{t < σ;Ht = m} for the last hitting time of m, with the convention that inf ∅ = +∞ and sup ∅ = +∞. We consider the minimum of H between Tm and Lm: H[Tm,Lm] = min{Ht; t ∈ [Tm, Lm]}. 10 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS We set ρ(d) = (ρTmax,a+t, t ≥ 0), with Tmax,a = inf{t > Tmax,Hs = a}, the path of the exploration process on the right of Tmax after the hitting time of a, and ρ(g) = (ρ(Lmax,a−t)−, t ≥ 0), with Lmax,a = sup{t < Tmax;Ht = a}, the returned path of the exploration process on the left of Tmax before its last hitting time of a. Let us note that, by time reversal (see Proposition 2.5), the process ρ(g) is of the same type as η. This remark will be used later. To prove the Theorem, we shall compute A0 = N (g))F2(ρ (d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{m≤Hmax<m+ε} and let ε go down to 0. We shall see in Lemma 3.4, that adding 1{H[Tm,Lm]>a} in the integrand does not change the asymptotic behavior as ε goes down to 0. Intuitively, if the maximum of the height process is between m and m+ε, outside a set of small measure, the height process does not reach level a between the first and last hitting time of m. So that we shall compute first (15) A = N (g))F2(ρ (d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε} Notice that on {H[Tm,Lm] > a}, we have Tmax,a = Tm,a := inf{s > Tm,Hs(ρ) = a} and, from the snake property, ρTmax |[0,a] = ρTm |[0,a] and ηTmax |[0,a] = ηTm |[0,a], so that A = N (g))F2((ρTm,a+t, t ≥ 0))F3(ρTm |[0,a])F4(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε} Let us remark that, we have 1{H[Tm,Lm]>a,m≤Hmax<m+ε} = 1{m≤sup{Hu,0≤u≤Tm,a}<m+ε}1{sup{Hu,u≥Tm,a}<m}. By using the strong Markov property of the exploration process at time Tm,a, we get (g))F4(ηTm |[0,a])1{m≤sup{Hu,0≤u≤Tm,a}<m+ε}F3(ρTm |[0,a])E ρTm |[0,a] F2(ρ)1{Hmax<m} and so, by conditioning, we get A = N (g))F4(ηTm |[0,a])G2(ρTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε} where G2(µ) = F3(µ)E µ[F2(ρ)|Hmax < m]. Using time reversibility (see Proposition 2.5) and the strong Markov property at time Tm,a again, we have A = N (d))F4(ρTm |[0,a])G2(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε} G1(ρTm |[0,a])G2(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε} where G1(µ) = F4(µ)E µ[F1(ρ)|Hmax < m]. Now, we use ideas from the proof of Theorem 4.6.2 of [7]. Let us recall the excursion decom- position of the exploration process above level a. We set τas = inf du1{Hu≤a} > s Let Ea be the σ-field generated by the process (ρ̃s, s ≥ 0) := (ρτas , s ≥ 0). We also set η̃s = ητas . WILLIAMS’ DECOMPOSITION 11 Let (αi, βi), i ∈ I be the excursion intervals of H above level a. For every i ∈ I we define the measure-valued process ρi by setting 〈ρis, ϕ〉 = (a,+∞) ραi+s(dr)ϕ(r − a) if 0 < s < βi − αi, ρs = 0 if s = 0 or s ≥ βi − αi, and the process ηi similarly. We also define the local time at the beginning of excursion ρi by ℓi = L . Then, under N, conditionally on Ea, the point measure δ(ℓi,ρi,ηi) is a Poisson measure with intensity 1[0,Laσ](ℓ)dℓN[dρ dη]. In particular, we have A = N j 6=i 1{Tm(ρj)=+∞}G1(ραi)G2(ηαi)1{m≤Hmax(ρi)<m+ε} Let us denote by (τaℓ , ℓ ≥ 0) the right-continuous inverse of (L s , s ≥ 0). Palm formula for Poisson point measures yields A = N j 6=i 1{Tm(ρj)=+∞}G1(ραi)G2(ηαi)1{m≤Hmax(ρi)<m+ε} ∫ Laσ dℓG1(ρτa )G2(ητa )N[m ≤ Hmax < m+ ε]N 1{Tm(ρj)=+∞} A time-change then gives (16) A = v(m− a, ε)N dLasG1(ρs)G2(ηs) e −c(m−a)Laσ where v(x, ε) = c(x) − c(x+ ε) = N[x ≤ Hmax < x+ ε]. We have A = v(m− a, ε)N dLasG1(ρs)G2(ηs) e −c(m−a)Las e−c(m−a)(L = v(m− a, ε)N dLasG1(ρs)G2(ηs) e −c(m−a)Las e−〈ρs,N[1−e −c(m−a)L where we used for the last equality that the predictable projection of e−λ(L s ) is given by e−〈ρs ,N[1−e σ ]〉. Notice that by using the excursion decomposition above level 0 < r < m, we have c(m) = N[Tm <∞] = N[1− e −c(m−r)Lrσ ]. In particular, we get A = v(m− a, ε)N dLasG1(ρs)G2(ηs) e −c(m−a)Las e−〈ρs,c(m−·)〉 Using time reversibility, we have A = v(m− a, ε)N dLasG1(ηs)G2(ρs) e −c(m−a)(Laσ−L s ) e−〈ηs ,c(m−·)〉 12 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS Similar computations as those previously done give A = v(m− a, ε)N dLasG1(ηs)G2(ρs) e −〈ηs+ρs,c(m−·)〉 = v(m− a, ε)N dLasG1(ρs)G2(ηs) e −〈ρs+ηs,c(m−·)〉 Using Proposition 2.7, we get A = v(m− a, ε) e−α0a Ma(dµ dν)G1(µ)G2(ν) e −〈µ+ν,c(m−·)〉 . We can give a first consequence of the previous computation. Lemma 3.4. We have N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = c c(m− a)− c(m− a+ ε) c′(m− a) Proof. Taking F1 = F2 = F3 = F4 = 1 in (16), we deduce that N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = v(m− a, ε)N Laσ e −c(m−a)Laσ Let a0 > 0 and let us compute B(a0, a) = N Laσ e −c(a0)L . Thanks to Theorem 2.2, notice B(a0, a) = N −c(a0)Ya ∂a0N[1− e −c(a0)Ya ] c′(a0) On the other hand, we have c(a+ a0) = N[Ya+a0 > 0] = N[1− EYa[Ya0 = 0]] = N 1− e−Yac(a0) where we used the Markov property of Y at time a under N for the second equality and (3) with λ going to infinity for the last. Thus, we get B(a0, a) = c′(a0 + a) c′(a0) . We deduce that N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = v(m− a, ε)B(a−m,a) = c′(m) c(m− a)− c(m− a+ ε) c′(m− a) Since F1, F2, F3 and F4 are bounded, say by C, we have |A−A0| ≤ C 4N[H[Tm,Lm] < a,m ≤ Hmax < m+ ε]. From Lemma 3.4, we deduce that |A−A0| N[m ≤ Hmax < m+ ε] 1− lim N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] N[m ≤ Hmax < m+ ε] WILLIAMS’ DECOMPOSITION 13 We deduce that (g))F2(ρ (d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{m≤Hmax<m+ε} N[m ≤ Hmax < m+ ε] c′(m− a) c′(m) e−α0a Ma(dµ dν)G1(µ)G2(ν) e −〈µ+ν,c(m−·)〉 Ma(dµ dν)G1(µ)G2(ν) e −〈µ+ν,c(m−·)〉 Ma(dµ dν) e −〈µ+ν,c(m−·)〉 M̃a(dµ dν)G1(µ)G2(ν) M̃a(dµ dν)F4(ν)E ν [F1(ρ (d))|Hmax < m]F3(µ)E µ[F2(ρ (d))|Hmax < m], where µ(dt) = uiℓiδti + β1[0,a](t)dt ν(dt) = (1− ui)ℓiδti + β1[0,a](t)dt, i∈I δ(xi,ℓi,ti) is under M̃a a Poisson point measure on [0,+∞) 3 with intensity 1[0,a](t)dt ℓ e −ℓc(m−t) π(dℓ)1[0,1](u)du. Standard results on measure decomposition imply there exists a regular version of the probability measure N[ · |Hmax = m] and that, for almost every non-negative m, N[ · |Hmax = m] = lim N[ · |m ≤ Hmax < m+ ε]. This gives (ii) and (iii) of Theorem 3.3 since F1, F2, F3, F4 are arbitrary continuous functionals and by Remark 3.2. 4. Proof of Theorem 1.2 The proof of this Theorem relies on the computation of the Laplace transform for (Y ′ , Y ′) and is given in the next three paragraphs. The next paragraph gives some preliminary computations. 4.1. Preliminary computations. 4.1.1. Law of T0. Recall the definition of Qm as the law of (Y ′0, Y ′) defined by (10) and T0 defined by (9) as the first mutation undergone by the last individual alive. For r < m, we have Qm(T0 ∈ [r, r + dr], T0 = T2) = Qm(T2 ∈ [r, r + dr])Qm(T1 > r) = dr αImm e −αImmr exp− (0,∞) e−ℓc(m−t) ℓν(dℓ) = dr αImm e φ′(c(m−t)) dt, 14 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS and, with the notation φ0(λ) = φ(λ)− αImmλ, Qm(T0 ∈ [r, r + dr], T0 = T1) = Qm(T2 > r)Qm(T1 ∈ [r, r + dr]) = dr φ′0(c(m− r)) e −αImmr exp− (0,∞) e−ℓc(m−t) ℓν(dℓ) = dr φ′0(c(m− r)) e φ′(c(m−t)) dt . In particular, we have for r < m Qm(T0 ∈ [r, r + dr]) = dr φ ′(c(m− r)) e− φ′(c(m−t)) dt . (17) Qm(T0 > r) = e φ′(c(m−t)) dt . Notice we have Qm(T0 = ∞) = exp− φ′(c(t)) dt. 4.1.2. Conditional law of N given T0. Recall N is under Qm a Poisson point measure with intensity given by (8). Conditionally on {T0 = r, T0 = T2}, with m > r > 0, N is under Qm a point Poisson measure with intensity 1[0,r)(t) e −ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt+ 1(r,m)(t) e −ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt. Conditionally on {T0 = r, T0 = T1}, with r < m, N is distributed under Qm as Ñ + δ(L,r,1) where Ñ is a point Poisson measure with intensity 1[0,r)(t) e −ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt + 1(r,m)(t) e −ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt, and L is a random variable independent of Ñ with distribution e−ℓc(m−r) ℓν(dℓ) (0,∞) ′c(m−r) ℓ′ν(dℓ′) Conditionally on {T0 = ∞}, N is under Qm a point Poisson measure with intensity 1[0,m)(t) e −ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt. 4.1.3. Formulas. The following two formulas are straightforward: for all x, γ ≥ 0, ψ′Eve(x+ γ)− ψ Eve(γ) = 2βx+ (0,∞) e−ℓγ ℓπEve(dℓ)[1 − e −ℓx],(18) ψ′(x+ γ)− ψ′(γ) = 2βx+ (0,∞) e−ℓγ ℓπ(dℓ)[1− e−ℓx],(19) Finally we deduce from (5) that ψ(c) = −c′, ψ′(c)c′ = −c′′ and ψ′(c) = − log(c′). WILLIAMS’ DECOMPOSITION 15 4.1.4. Laplace transform. Recall τY = inf{t > 0;Yt = 0} is the extinction time of Y . Let µEve and µTotal be two finite measures with support a subset of a finite set A = {a1, . . . , an} with 0 = a0 < a1 < · · · < an < an+1 = ∞. For m ∈ (0,+∞) \A, we consider wm(t) = N[1− e Y 0r−t µEve(dr)− Yr−t µTotal(dr) 1{τY <m−t}], w∗m(t) = N[1− e Yr−t µTotal(dr) 1{τY <m−t}]. By noticing that N-a.e. 1{τY <m−t} = limλ→∞ exp− Yr−t µ λ(dr), where µλ(dr) = λδm(dr), we deduce from Lemma 3.1 in [2] that (wm, w m) are right continuous and are the unique non-negative solutions of : for k ∈ {0, . . . , n}, m ∈ (ak, ak+1), t ∈ (−∞,m), (21) w∗m(t) + [t,ak] ψ(w∗m(r))dr = [t,ak] µTotal(dr) + c(m− ak), (22) wm(t) + [t,ak] ψEve(wm(r))dr [t,ak] µEve(dr) + [t,ak] µTotal(dr) + c(m− ak) + [t,ak] φ(w∗m(r))dr. We define (23) ām = max{ak; ak < m, k ∈ {0, . . . , n}}. Notice that wm(t) = w m(t) = c(m− t) for t ∈ (ām,m). 4.2. Proof of Theorem 1.2. 4.2.1. Aim. Theorem 1.2 will be proved as soon as we check that the following equality w(0) = −c′(m)Qm[1− e r µEve(dr)− Y ′r µTotal(dr)]dm holds for all the possible choices of measures µEve and µTotal satisfying the assumptions of Section 4.1.4, with w = w∞ defined by (22). Notice the integrand of the right-hand side is null for m < a1. Let ∆ denote the right-hand side. We have for 0 < ε ≤ a1: dm (−c′(m))Qm[1− e r µEve(dr)− Y ′r µTotal(dr)] = c(ε) + dm 1Ac(m)c ′(m)Qm[Z], with, thanks to the definition (6) of κmax, Z = exp− ∫ ām κmax(dt) [nt1{t<T0} + n t1{t≥T0}] nt = N[(1 − e r−t µEve(dr)− Yr−t µTotal(dr))1{τY ≤m−t}] = wm(t)− c(m− t) n∗t = N[(1 − e Yr−t µTotal(dr))1{τY ≤m−t}] = w m(t)− c(m− t), with (wm, w m) the non-negative solutions of (21) and (22). Notice that wm(t) = w m(t) = c(m− t) for t ∈ (ām,m) and thus nt = n t = 0 when t ∈ (ām,m). 16 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS We set ∆ = c(ε) + 1Ac(m)(∆1 +∆2 +∆3) dm with ∆1 = c ′(m)Qm[Z|T0 > ām]Qm(T0 > ām), ∆2 = c ∫ ām Qm[Z|T0 = r, T0 = T1]Qm(T0 ∈ [r, r + dr], T0 = T1), ∆3 = c ∫ ām Qm[Z|T0 = r, T0 = T2]Qm(T0 ∈ [r, r + dr], T0 = T2). We shall assume m 6∈ A. 4.2.2. Computation of ∆1. We have, using formula (6), ∆1 = c ′(m)Qm(T0 > ām)Qm[e R ām κmax(dt)nt |T0 > ām] = c′(m) e− R ām φ′(c(m−t))dt ∫ ām (wm(t)− c(m− t)) dt− ∫ ām dt e−ℓc(m−t) ℓπEve(dℓ)[1− e −ℓ(wm(t)−c(m−t))] = c′(m) e− φ′(c(m−t))dt exp ∫ ām dt[ψ′Eve(wm(t))− ψ Eve(c(m− t))] = c′(m) e m−ām ψ′(c(t))dt dt ψ′Eve(wm(t)) = c′(m− ām) e dt ψ′Eve(wm(t)), where we used (20) for the last equality to get (24) e m−ām ψ′(c(t))dt = e−[log(c ′(t))]m m−ām = c′(m− ām) c′(m) 4.2.3. Computation of ∆2. Using Section 4.1.2, we get Qm[Z|T0 = r, T0 = T1] = e (wm(t)−c(m−t)) dt−2β R ām (w∗m(t)−c(m−t)) dt exp(− dt e−ℓc(m−t) ℓπEve(dℓ)[1 − e −ℓnt ]) exp(− ∫ ām dt e−ℓc(m−t) ℓπ(dℓ)[1 − e−ℓn (0,∞) ν(dℓ′) ′c(m−r) ℓ′ e−ℓ φ′0(c(m− r)) = exp(− dt[ψ′Eve(wm(t))− ψ Eve(c(m− t))]) exp(− ∫ ām dt[ψ′(w∗m(t))− ψ ′(c(m− t))]) φ′0(w m(r)) φ′0(c(m− r)) WILLIAMS’ DECOMPOSITION 17 We deduce from Section 4.1.1 ∆2 = c ∫ ām Qm[Z|T0 = r, T0 = T1]Qm(T0 ∈ [r, r + dr], T0 = T1), = c′(m) ∫ ām dr φ′0(w m(r)) e φ′(c(m−t)) dt exp(− dt[ψ′Eve(wm(t)) − ψ Eve(c(m − t))]) exp(− ∫ ām dt[ψ′(w∗m(t))− ψ ′(c(m− t))]) = c′(m) e ψ′(c(m−t)) dt ∫ ām dr φ′0(w m(r)) e dt ψ′Eve(wm(t))− dt ψ′(w∗m(t)) = c′(m− ām) ∫ ām dr φ′0(w m(r)) e dt ψ′ (wm(t))− R ām dt ψ′(w∗m(t)), where we used (24) for the last equality. 4.2.4. Computation of ∆3. Using Section 4.1.2, we get Qm[Z|T0 = r, T0 = T2] = e (wm(t)−c(m−t)) dt−2β (w∗m(t)−c(m−t)) dt dt e−ℓc(m−t) ℓπEve(dℓ)[1 − e −ℓnt ] ∫ ām dt e−ℓc(m−t) ℓπ(dℓ)[1 − e−ℓn = exp(− dt[ψ′Eve(wm(t)) − ψ Eve(c(m − t))]) exp(− ∫ ām dt[ψ′(w∗m(t))− ψ ′(c(m− t))]). We deduce from Section 4.1.1 ∆3 = c ∫ ām Qm[Z|T0 = r, T0 = T2]Qm(T0 ∈ [r, r + dr], T0 = T2), = c′(m) ∫ ām dr αImm e φ′(c(m−t)) dt exp(− dt[ψ′Eve(wm(t))− ψ Eve(c(m− t))]) exp(− ∫ ām dt[ψ′(w∗m(t))− ψ ′(c(m− t))]) = c′(m) e ψ′(c(m−t)) dt ∫ ām dr αImm e dt ψ′Eve(wm(t))− dt ψ′(w∗m(t)) = c′(m− ām) ∫ ām dr αImm e dt ψ′ (wm(t))− R ām dt ψ′(w∗m(t)), where we used (24) for the last equality. 4.2.5. Computation of ∆2 +∆3. We have ∆2 +∆3 = c ′(m− ām) ∫ ām dr φ′(w∗m(r)) e dt ψ′Eve(wm(t))− dt ψ′(w∗m(t)) . Differentiating (21) w.r.t. time and m, we get for t < m ′(t)− ∂mw m(t)ψ ′(w∗m(t)) = 0. 18 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS Notice also that for m > t ≥ ām, we have ∂mw ∗(t) = c′(m− t) and thus ∗(ām) = c ′(m− ām). We get exp(− ∫ ām dtψ′(w∗m(t))) = ∂mw∗m(ām) c′(m− ām) Differentiating (22) w.r.t. time and m, we get for t < m m(t)− ∂mwm(t)ψ Eve(wm(t)) = −∂mw m(t)φ ′(w∗m(t)). We deduce that ∆2 +∆3 = ∫ ām dr∂mw m(t)φ ′(w∗m(r)) e dt ψ′ (wm(t)) ∫ ām dr[∂mw m(r)− ∂mwm(r)ψ Eve(wm(r))] e dt ψ′Eve(wm(t)) ∂mwm(r) e dt ψ′Eve(wm(t)) = ∂mwm(0) − ∂mwm(ām) e dt ψ′Eve(wm(t)) . Notice also that for m > t ≥ ām one has ∂mw(t) = c ′(m − t), in particular ∂mw(ām) = c′(m− ām). This implies that ∆2 +∆3 = ∂mwm(0) − c ′(m− ām) e dt ψ′Eve(wm(t)) . 4.3. Conclusion. Thus, for m 6∈ A, we have ∆1 +∆2 +∆3 = ∂mwm(0), ∆ = c(ε) + ∂mwm(0) = c(ε) + w∞(0)− wε(0) = w(0). This ends the proof of the Theorem. Acknowledgments. The authors wish to thank an anonymous referee for his numerous and useful comments which improved the presentation of the paper. References [1] R. ABRAHAM and J.-F. DELMAS. Fragmentation associated with Lévy processes using snake. Probab. Th. Rel. Fields, 141 (1–2): 113-154, 2008. [2] R. ABRAHAM and J.-F. DELMAS. Changing the branching mechanism of a continuous state branching process using immigration. Annales de l’IHP, To appear, 2008. [3] R. ABRAHAM, J.-F. DELMAS and G. VOISIN. Pruning a continuum random tree. Preprint arXiv:0804.1027. [4] D. ALDOUS. The continuum random tree I. Ann. Probab., 19 (1):1–28, 1991. [5] J. BERTOIN. Lévy processes. Cambridge University Press, Cambridge, 1996. [6] J. BERTOIN. The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations. Preprint. [7] T. DUQUESNE and J.-F. LE GALL. Random trees, Lévy processes and spatial branching processes, volume 281. Astérisque, 2002. [8] A.M. ETHERIDGE and D.R.E. WILLIAMS. A decomposition of the (1+ β)-superprocess conditioned on survival. Proc. Roy. Soc. Edinburgh Sect. A, 133 (4):829–847, 2003. [9] J. GEIGER and L. KAUFFMANN. The shape of large Galton-Watson trees with possibly infinite variance. Rand. Struct. Alg., 25 (3):311-335, 2004. http://arxiv.org/abs/0804.1027 WILLIAMS’ DECOMPOSITION 19 [10] A. LAMBERT. The genealogy of continuous-state branching processes with immigration. Probab. Th. Rel. Fields, 122 (1):42–70, 2002. [11] J.-F. LE GALL and Y. LE JAN. Branching processes in Lévy processes: Laplace functionals of snake and superprocesses. Ann. Probab., 26:1407–1432, 1998. [12] J.-F. LE GALL and Y. LE JAN. Branching processes in Lévy processes: The exploration process. Ann. Probab., 26:213–252, 1998. [13] G. MIERMONT. Invariance principles for spatial multitype Galton-Watson trees. Preprint. [14] J. PITMAN and M. WINKEL. Growth of the Brownian forest. Ann. Probab., 33 (6):2188–2211, 2005. [15] T. SALISBURY and J. VERZANI. On the conditioned exit measures of super Brownian motion. Probab. Th. Rel. Fields, 115:237-285, 1999. [16] L. SERLET. The occupation measure of super-Brownian motion conditioned on non-extinction. J. The- oretical Probab., 9:561-578, 1996. [17] J. WARREN. Branching processes, the Ray-Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI, volume 1655 of Lecture Notes in Math., pages 1–15. Springer, Berlin, 1997. [18] D. WILLIAMS. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc., 28 (3):738–768, 1974. Romain Abraham, MAPMO, Fédération Denis Poisson, Université d’Orléans, B.P. 6759, 45067 Orléans cedex 2, France. E-mail address: [email protected] Jean-François Delmas, CERMICS, Univ. Paris-Est, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France. E-mail address: [email protected] 1. Introduction 2. Notations 2.1. The underlying Lévy process 2.2. The height process and the Lévy CRT 2.3. The exploration process 2.4. The dual process and representation formula 3. Williams' decomposition 4. Proof of Theorem ?? 4.1. Preliminary computations 4.2. Proof of Theorem ?? 4.3. Conclusion References

0704.1476

TeV-scale gravity in Horava-Witten theory on a compact complex hyperbolic threefold

TeV-scale gravity in Hořava-Witten theory on a compact complex hyperbolic threefold Chris Austin1 33 Collins Terrace, Maryport, Cumbria CA15 8DL, England Abstract The field equations and boundary conditions of Hořava-Witten theory, compactified on a smooth compact spin quotient of CH3, where CH3 denotes the hyperbolic cousin of 3, are studied in the presence of Casimir energy density terms. If the Casimir energy densities near one boundary result in a certain constant of integration taking a value greater than around 105 in units of the d = 11 gravitational length, a form of thick pipe geometry is found that realizes TeV-scale gravity by the ADD mechanism, with that boundary becoming the inner surface of the thick pipe, where we live. Three alternative ways in which the outer surface of the thick pipe might be stabilized consistent with the observed value of the effective d = 4 cosmological constant are considered. In the first alternative, the outer surface is stabilized in the classical region and the constant of integration is fixed at around 1013 in units of the d = 11 gravitational length for consistency with the observed cosmological constant. In the second alternative, the four observed dimensions have reduced in size down to the d = 11 gravitational length at the outer surface, and there are Casimir effects near the outer surface. In the third alternative, the outer surface is stabilized in the classical region by extra fluxes of the three-form gauge field, whose four-form field strength wraps three-cycles of the compact six-manifold times the radial dimension of the thick pipe. Some problems related to fitting the strong/electroweak Standard Model are considered. 1Email: [email protected] http://arxiv.org/abs/0704.1476v2 Contents 1 Introduction 4 2 Thick pipe geometries 11 2.1 Hořava-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The complex hyperbolic space CH3 . . . . . . . . . . . . . . . . . . . . 25 2.3 The field equations and boundary conditions . . . . . . . . . . . . . . . 32 2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor . . . 34 2.3.2 The Yang-Mills coupling constants in four dimensions . . . . . . 35 2.3.3 The problem of the higher order corrections to Hořava-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.4 The Casimir energy density corrections to the energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.3.5 The orders of perturbation theory that the terms in the Casimir energy densities occur at . . . . . . . . . . . . . . . . . . . . . . 64 2.3.6 The expansion parameter . . . . . . . . . . . . . . . . . . . . . 66 2.3.7 Witten’s topological constraint . . . . . . . . . . . . . . . . . . 71 2.3.8 The field equations and boundary conditions for the three-form gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.3.9 The field equations and boundary conditions for the metric . . . 80 2.4 Analysis of the Einstein equations and the boundary conditions for the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.4.1 Beyond the proximity force approximation . . . . . . . . . . . . 90 2.4.2 The region near the inner surface of the thick pipe . . . . . . . . 93 2.4.3 The boundary conditions at the inner surface of the thick pipe . 104 2.4.4 The classical solutions in the bulk . . . . . . . . . . . . . . . . . 107 2.5 Solutions with both a and b large compared to κ2/9, at the outer surface of the thick pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.5.1 Newton’s constant and the cosmological constant for solutions with the outer surface in the classical region . . . . . . . . . . . 115 2.6 Solutions with a as small as κ2/9, at the outer surface of the thick pipe 120 2.6.1 Newton’s constant and the cosmological constant . . . . . . . . 126 2.6.2 Comparison with sub-millimetre tests of Newton’s law . . . . . 132 2.6.3 Comparison with precision solar system tests of General Relativity135 2.6.4 Further consequences of the warp factor decreasing to a small value, at the outer surface of the thick pipe . . . . . . . . . . . . 136 2.7 Stiffening by fluxes wrapping three-cycles of the compact six-manifold times the radial dimension . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.7.1 The region near the outer surface . . . . . . . . . . . . . . . . . 142 2.7.2 Newton’s constant and the cosmological constant in the presence of the extra fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3 Smooth compact quotients of CH , H6, H3 and S3 160 3.1 Smooth compact arithmetic quotients of CHn and Hn . . . . . . . . . 167 3.1.1 Compactness of G/Γ for the examples of arithmetic lattices . . . 174 3.1.2 Obtaining finite index torsion-free subgroups of Γ by Selberg’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.2 Smooth compact quotients of S3 . . . . . . . . . . . . . . . . . . . . . . 191 4 The Casimir energy densities 191 4.1 The Salam-Strathee harmonic expansion method . . . . . . . . . . . . . 196 4.2 AdS5 ×CP3 compactification of d = 11 supergravity . . . . . . . . . . 214 5 E8 vacuum gauge fields and the Standard Model 222 5.1 The lightest massive modes of the supergravity multiplet . . . . . . . . 232 5.2 An SU (9) basis for E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.3 Dirac quantization condition for E8 vacuum gauge fields . . . . . . . . 241 5.4 Nonexistence of models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× (SU (2))3 × (U (1))3 . . . . . . . . . . . . . . . . . 254 5.5 Models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× (SU (2))2 × (U (1))4 . . . . . . . . . . . . . . . . . . . . . . . . 263 5.6 Models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× SU (2)× (U (1))5 . . . . . . . . . . . . . . . . . . . . . . . . . 270 5.7 Generalized seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . 277 1 Introduction The observed physical universe is a very stiff structure, approximately flat up to dis- tances larger, by a factor of 1061, than the radius of curvature that would be expected on the basis of the Standard Model, plus General Relativity, in 3 + 1 dimensions. Large two-dimensional structures, such as the hull of an oil tanker, are often stiffened by structures that extend a short distance into the third dimension. So it is natural to wonder whether compact additional spatial dimensions, not yet observed, could play an active role in stiffening the universe. To study the possibility of such a mechanism, I shall consider, in this paper, the compactification of Hořava-Witten theory [1, 2] on a smooth compact spin Kähler manifold, that is obtained from CH3, the hyperbolic cousin of CP3, by quotienting out the free, holomorphic action of a cocompact, torsionless, discrete subgroup of the isometry group of CH3, which is SU(3,1). I shall look for solutions that realize TeV- scale gravity by the ADD mechanism [3, 4, 5] in a form of thick pipe geometry [6, 7, 8], such that the two boundaries of the Hořava-Witten universe become the inner and outer surfaces of the thick pipe, the eleventh dimension becomes the radial direction of the thick pipe, and the diameter of the compact six-manifold increases with increasing distance from the inner surface of the thick pipe, where we live. The choice of a smooth compact spin quotient of CH3, rather than a Calabi-Yau threefold [9], as the compact six-manifold M6, means that all supersymmetries are bro- ken by the compactification. By a fundamental theorem of Mostow, known as Mostow rigidity [10], the geometry of M6 is now completely determined by its fundamental group, up to an overall scale factor, so that M6 has no shape moduli. There are an infinite number of topologically distinct smooth compact quotients of CH3, but only a finite number with |χ (M6)| up to a given value, where χ (M6) denotes the Euler number of M6, and only a small fraction of these are likely to be spin manifolds. The possible values of |χ (M6)| are constrained by the fact that the squares of the Yang-Mills coupling constants, at unification, are inversely proportional to |χ (M6)|, and by combining an estimate of the Yang-Mills coupling constants, at unification, with an estimate by Giudice, Rattazzi, and Wells [11] of the effective expansion parameter for quantum gravity in eleven dimensions, the upper limit on |χ (M6)| is provisionally estimated in subsection 2.3.6, on page 66, to be around 7×104. This upper limit might possibly be slightly increased by an effect considered by Robinson and Wilczek [12]. This limit on the value of |χ (M6)| means that TeV-scale gravity cannot be realized simply by choosing an extremely large value of |χ (M6)|. Instead, it is necessary that the boundary conditions at the inner surface of the thick pipe, and the Casimir energy density corrections to the energy-momentum tensor on and near the inner surface, result in a certain constant of integration taking a value greater than around 105 in units of the gravitational length in eleven dimensions. Specifically, if y denotes the geodesic distance from the inner surface of the thick pipe, up to an additive constant, and the d = 11 metric on M6 is b2hABdxAdxB, where b depends only on y, and hAB is the standard metric on CH 3 introduced in subsection 2.2, on page 25, then in the main part of the bulk, where there are no significant source terms in the Einstein equations, we find )1.8990 , (1) where B is a constant of integration, that for TeV-scale gravity has to have a value greater than around 105κ2/9, where κ is the gravitational coupling constant in eleven dimensions. The value of B is completely determined by the region close to the inner surface of the thick pipe, because the only other physically significant constant of integration, which is an overall constant multiplying the warp factor that multiplies the metric in the four extended dimensions, does not occur in any significant terms in the field equations or boundary conditions in this region. Thus the two boundary conditions at the inner surface fix B and b1, the value of b at the inner surface. A perturbative mechanism by which a large value of B could occur is identified in subsection 2.4.2, on page 93. In essence, the bulk power law (1) holds only for )0.6551 , which is greater than around 103, while for 1 ∼ b1 )0.6551 , we find self-consistently that , (2) when the Casimir energy density corrections are taken into account, as discussed in subsection 2.3.4, on page 57, and subsection 2.4.1, on page 90. Thus there is a quantum region of thickness greater than around 8κ2/9 adjacent to the inner surface, in which b increases exponentially with y. The linear relation (2) starts to round off to a broad peak at b ∼ 103κ 29 , followed smoothly by the classical power law (1). The only requirement for obtaining the linear relation (2) is that a certain sign is positive rather than negative, so it seems possible that a value of B significantly larger than κ2/9 could be found for as many as fifty percent of the smooth compact quotients of CH3 that are spin manifolds. The actual value of b at which the quantum relation (2) transforms into the classical relation (1), and the corresponding value of B, will be determined by how close to the self-consistent linear relation (2) the system is set by the boundary conditions at b1 ∼ κ2/9. This mechanism is completely perturbative, and could be tested by one-loop calcu- lations, for smooth compact quotients of CH3 that are spin manifolds. The numerical coefficient in the linear relation (2) is found to be ∼ 1 if b1 is at least a factor of 2 or so larger than the minimum value allowed by the Giudice, Rattazzi, and Wells estimate of the expansion parameter, which is b1 ≃ 0.2κ2/9. Thus it seems likely that b1 will be somewhere in the range from 0.4κ2/9, which corresponds to |χ (M6)| ≃ 103, to 0.8κ2/9, which corresponds to |χ (M6)| ≃ 20. There are inevitably significant Casimir energy density terms in the energy-momen- tum tensor on and near the inner surface of the thick pipe, due to the Hořava-Witten relation λ ≃ 5.8κ 23 between the d = 10 Yang-Mills coupling constant λ and κ [2], and the fact that the d = 4 Yang-Mills coupling constants at unification are not much smaller than 1, which implies that b1 is comparable to κ Although the mechanism for realizing TeV-scale gravity considered in this paper is completely perturbative, it would be desirable to be able to calculate corrections beyond one loop, and the problem of the higher order corrections to Hořava-Witten theory is considered in subsection 2.3.3, on page 38. The derivation of type IIA superstring theory [13] from the Cremmer-Julia-Scherk (CJS) theory of d = 11 supergravity [14] compactified on a small S1 [1] is reviewed, and M-theory on a smooth background is observed to be the same as the CJS theory. The superspace constructions of higher-derivative counterterms for the CJS theory [15, 16] are considered, and I suggest that an obstruction might exist that prevents the geometrical transformations in superspace [17, 18] from matching the CJS super- symmetry variations for a general solution of the CJS field equations beyond a certain power of θ. This would mean that with the exception of the possibly unique counter- term constructible by the superform or ectoplasm method [19, 20, 21, 22, 16, 23], the superspace counterterms do not result in locally supersymmetric deformations of the CJS theory, so that since the coefficient of the unique dimension 8 counterterm [24] is fixed by cancellation of the tangent bundle anomaly on five-branes [25, 26, 27, 28, 29], it might be possible to calculate the predictions of the CJS theory and Hořava-Witten theory in the framework of effective field theory, without the occurrence of undeter- mined parameters connected with the short distance completion of the theory. If the d = 11 metric on the four observed dimensions is a2gµνdx µdxν , where a depends only on y, and gµν is a metric on de Sitter space with de Sitter radius equal to 1, then in the classical region corresponding to (1), we find a = A )0.7753 , (3) where A is a constant of integration whose value is determined by the region close to the outer surface of the thick pipe. And in the quantum region corrresponding to (2), we find a = A1 , (4) where the constant A1 is determined by continuity with (3) at the transition between the classical and quantum regions, and the exponent τ is determined by the Casimir terms in the energy-momentum tensor for the self-consistent linear relation (2). For B ≫ κ2/9, the existence of a solution of the boundary conditions at the outer surface with b2 = 2a2 ≫ κ2/9, where b2 and a2 are the values of b and a at the outer surface, is demonstrated in subsection 2.5, on page 111, and this type of solution is found in subsection 2.5.1, on page 115, to fit the observed values of Newton’s constant and the cosmological constant for TeV-scale gravity if τ ≃ −3 and B ∼ 1013κ2/9 ∼ 10−5 metres. This type of solution does not fully satisfy the condition for a valid reduction to a four dimensional effective action, due to the fact that a (y) decreases from the observed de Sitter radius of around 1026 metres at the inner surface of the thick pipe, to around 10−5 metres at the outer surface. The fact that Newton’s law is recovered for the gravitational force between point particles on the Planck brane [30] of the first Randall-Sundrum model [31] suggests there is a possibility that Newton’s law might be obtained between point particles on the inner surface of the thick pipe, but this question is not resolved in this paper. Solutions in which a (y) has decreased to around κ2/9 at the outer surface, and there are Casimir effects near the outer surface, are considered in subsection 2.6, on page 120. The three observed spatial dimensions are in this case assumed to be compactified to a smooth compact quotient of H3, whose topology is significant for the Casimir effects near the outer surface. There is now an additional large constant of integration, Ã, which is the analogue of B for the quantum region near the outer surface, and by increasing τ from around −3 towards the exponent −0.7753 in the classical relation (3), the value of B can be reduced from around 1013 towards a limiting value of around 105, at a cost of rapidly increasing the value of à For the case when τ = −0.7753, this type of solution is demonstrated in subsection 2.6.2, on page 132, to be consistent with the precision sub-millimetre tests of Newton’s law [32], because most of the decrease of a (y) takes place in a very narrow region near the outer surface, so that only a fraction ∼ 10−6 of the integral that determines Newton’s constant comes from values of y for which a (y) is smaller than around 1018 metres. Solutions with extra fluxes of the four-form field strength of the three-form gauge field of d = 11 supergravity [14] wrapping three-cycles of the compact six-manifold M6 times the radial dimension are considered in subsection 2.7, on page 137. The outer surface is in the classical region b2 ≫ κ2/9, a2 ≫ κ2/9, and there is an additional large constant of integration, G̃, whose square corresponds to an average value of the energy-momentum tensor of the extra fluxes. The value of B can again be reduced from around 1013 towards a limiting value of around 105, by increasing τ from around −3 towards −0.7753, at a cost now of rapidly increasing the value of G̃ . This results in greatly increasing the value of a2, so that a2 is around 10 22 metres for τ = −0.7753, while b2 remains ∼ B. These solutions are therefore also consistent with the precision sub-millimetre tests of Newton’s law, for τ in a range including −0.7753. The value of G̃ in this type of solution does not appear to be quantized, which suggests that cosmological models involving G̃ might resemble quintessence models [33]. Most of the results of this paper are also valid, with minor modifications, for smooth compact spin quotients of H6, and the construction of an infinite family of smooth compact quotients of CH3 and H6, called arithmetic quotients, which is due to Borel and Harish-Chandra [34], is reviewed in subsection 3.1, on page 167. Non-arithmetic smooth compact quotients of H6 have been constructed by Gromov and Piatetski- Shapiro [35]. Non-arithmetic smooth compact quotients of CH2 have been constructed by Mostow [36], and non-arithmetic smooth finite-volume, but non-compact, quotients of CH3 have been constructed by Deligne and Mostow [37], but it does not at present seem to be known whether there exist non-arithmetic smooth compact quotients of The compact six-manifold M6 is required to be a spin manifold, because the three- form gauge field [38] only enters the generalized spin connection through its four-form field strength, which is well-defined globally, so there is no possibility of defining an analogue of a spinc structure [39] in the bulk. I do not know whether any of the arithmetic smooth compact quotients ofCH3 orH6 are spin manifolds, but the simplest known smooth compact quotient of H4, which is called the Davis manifold [40], is both an arithmetic quotient and a spin manifold [41, 42]. A counting argument considered in section 3, on page 160, suggests that for sufficiently large |χ (M6)|, non-arithmetic smooth compact quotients of H6 will exist that are spin manifolds. The value of the integration constant B is likely to depend on the choice of the spin structure on M6. The value of B is also affected by the presence of topologically stabilized vacuum Yang-Mills fields tangential to M6 on the inner surface of the thick pipe, and the further Casimir energy density terms in the energy-momentum tensor, to which they in turn give rise. Such vacuum Yang-Mills fields also affect the four-form field strength of the three-form gauge field of d = 11 supergravity [38, 14], due to the boundary condition derived by Hořava and Witten [2], and this also results in terms in the energy-momentum tensor that are significant near the inner surface of the thick pipe, and thus affect the value of B. By considering certain Wilson lines formed from trees of hairpins, I demonstrate in subsection 5.3, on page 241, that integrals over closed orientable two-dimensional surfaces in M6, of the field strengths of Yang-Mills fields in the Cartan subalgebra of E8, whose field strengths are proportional to Hodge - de Rham harmonic two-forms, are restricted by a form of Dirac quantization condition to lie on a certain discrete lattice in the Cartan subalgebra of E8, and more generally, that Abelian configurations of the E8 Yang-Mills fields, with field strengths proportional to Hodge - de Rham harmonic two-forms, can be topologically stabilized in magnitude, and partly also in orientation within E8, by a form of Dirac quantization condition. Such topologically stabilized Abelian vacuum Yang-Mills fields are restricted only by the requirements that they break E8 to the Standard Model [43, 44] in the correct way, as studied in subsection 5.5, on page 263, and subsection 5.6, on page 270, and that a topological constraint derived by Witten [45] is satisfied, and that the correct spectrum of chiral fermions, namely three Standard Model generations, plus possible singlet neutrinos, is obtained. Witten’s topological constraint ensures that the effective field theory, in the four extended dimensions, is free of chiral anomalies. The first of these requirements leaves a substantial amount of flexibility in the choice of the topologically stabilized Abelian vacuum Yang-Mills fields, and Witten’s topological constraint also leaves a substantial amount of flexibility, unless it should happen that the symmetric trilinear form which defines the topologically invariant cup product H2 × H2 → H4 of M6 is either positive definite or negative definite as a bilinear form when one of its indices takes some fixed values, thus preventing Witten’s topological constraint from being satisfied by cancellations between contributions from different elements of the Cartan subalgebra when the free index takes one of those fixed values. It seems reasonable to expect that this is increasingly unlikely to occur, the larger the second Betti number of M6 is. Now Mostow rigidity does not imply that M6 has no Kähler shape moduli, so that h1,1, the dimension of the Dolbeault cohomology group H1,1, is equal to 1, or that the second Betti number of M6 is small. Rather, just as with any Kähler-Einstein metric with a nonvanishing Ricci scalar, each Kähler modulus is equal to a fixed multiple of the corresponding element of the first Chern class. However, by a theorem of Gro- mov [46], all the Betti numbers of M6 are bounded by a constant times |χ (M6)|. It seems reasonable to expect that the second Betti number of M6 will be comparable to |χ (M6)|, and thus around 104. If the embedding of the Standard Model in E8 is such that only a small number of types of exotic fermion could occur, then the requirements of anomaly cancellation, which are automatically satisfied when Witten’s topological constraint is satisfied, may already be sufficient to prevent the occurrence of exotic chiral fermions. This happens for the embeddings of the Standard Model in E8 studied in subsection 5.6, on page 270, where there is only one type of exotic fermion, and the only solutions of the anomaly cancellation constraints are an integer number of Standard Model generations. In this case there would still be a substantial amount of flexibility in the choice of the topologically stabilized Abelian vacuum Yang-Mills fields, when all three requirements are satisfied. It might also be possible to introduce partially topologically stabilized Yang-Mills instantons in SU (2) subgroups of E8, associated with non-contractible closed four- dimensional surfaces in M6 [47], and this might be necessary for the more complicated types of embedding of the Standard Model in E8 studied in subsection 5.5, on page 263. However, it is not certain that this is possible, because it does not seem likely that non-contractible closed four-dimensional surfaces in M6 will be simply connected, and it is also unclear to what extent the orientation of such SU (2) subgroups in E8 could be topologically stabilized [48]. The introduction of topologically stabilized Abelian vacuum Yang-Mills fields of Hosotani type [49, 50, 51], with vanishing field strength, is usually associated with a torsion element of the fundamental group of the compact six-manifold, or in other words, a nontrivial element a such that an = 1 for some finite integer n [9]. A smooth compact quotient of CH3 necessarily has torsionless fundamental group, due to the fact that CH3 is the quotient of the isometry group, SU (3, 1), by its maximal compact subgroup, SU (3) × U (1), but examples in three dimensions suggest that it might be possible for H1 (M6,Z) to have torsion even though the fundamental group of M6 has no torsion, and I show in subsection 5.6, on page 270, that this would be sufficient to enable Abelian vacuum Yang-Mills fields of Hosotani type to be topologically stabilized. The breakings of E8 to the Standard Model considered in subsections 5.5 and 5.6 partly suppress proton decay by a mechanism related to the Aranda-Carone mechanism [52], but I do not know whether the suppression is sufficient for consistency with current experimental limits [53, 54, 55]. The breakings also produce natural candidates for light sterile neutrinos [56, 57] that might be relevant if the forthcoming results of the MiniBooNE experiment [58, 59] confirm the evidence for light sterile neutrinos from the LSND experiment [60]. The possibility that the existence of multiple oscillation channels involving light sterile neutrinos could improve the compatibility between the KARMEN [61] and LSND experiments was recently demonstrated in [62]. CH3 has previously been considered in the context of M theory by Kehagias and Russo [63]. Compact hyperbolic spaces have been considered in the context of large extra dimensions by Kaloper, March-Russell, Starkman, and Trodden [64], and by Tabbash [65]. 2 Thick pipe geometries I shall now briefly review Hořava-Witten theory, in Subsection 2.1, on page 17, then summarize the relevant facts about CH3, in Subsection 2.2, on page 25. The metric ansatz is introduced, and the field equations and boundary conditions derived, in the presence of assumed Casimir energy densities, in Subsection 2.3, on page 32, and the equations are studied in Subsection 2.4, on page 86. I use units such that h̄ = c = 1. The metric signature is (−,+,+, . . . ,+). The definitions of the Riemann and Ricci tensors are chosen to agree with the conventions of Weinberg [66]. The Riemann tensor is defined by: [Dµ, Dν ]Vσ = −RµνστV τ = −R τµνσ Vτ (5) Hence: R τµνσ = ∂µΓ νσ − ∂νΓτµσ + ΓτµρΓρνσ − ΓτνρΓρµσ (6) where Γτµν , the Christoffel symbol of the second kind, is defined by: Γτµν = gτσ (∂µgνσ + ∂νgµσ − ∂σgµν) (7) The Ricci tensor is defined by: Rµν = R µτν = ∂µΓ τν − ∂τΓτµν + ΓτµρΓρτν − ΓττρΓρµν = ∂µ∂ν ln |g| − ∂τΓτµν + ΓτµρΓρτν − Γρµν∂ρ ln |g| (8) where g is the determinant of the metric, gµν . These conventions are consistent with references [2, 67, 68, 69, 70, 71] on Hořava-Witten theory, but the Riemann and Ricci tensors, as defined here, have the opposite signs to those used in Chapters 15 and 16 of [72], and the Ricci tensor also has the opposite sign, to that defined in Chapter 18 of [43]. Laboratory and astrophysical observations, excluding the hypothesized period of inflation, in the very early universe, are consistent with an action Stot = SEin + Svac + SSM + SDM (9) where SEin = − 16πGN −ggµνRµν (10) is the Einstein action, Svac = −ρvac −g = − Λ −g (11) is the vacuum energy, SSM is the Standard Model matter action, and SDM is the action for the unknown dark matter, provided that the metric, gµν , is treated classically, rather than quantum mechanically, and all contributions to the vacuum energy, other than ρvac, are discarded. This means, in particular, that the contributions to the vacuum energy from the VEV of the Standard Model Higgs field, the chiral symmetry breaking condensate and possible other condensates of QCD, and vacuum Feynman diagrams of the Standard Model fields and the dark matter fields, in the metric gµν , are all to be discarded. GN is Newton’s constant, with the value [43] GN = 6.7087× 10−39 GeV−2 (12) so that GN = 8.1907× 10−20 GeV−1 = 1.6160× 10−35 metres. Variation of Stot, with respect to the metric, gives Einstein’s field equations: Rµν − Rgµν − Λgµν + 8πGNTµν = 0 (13) where the energy-momentum tensor, Tµν , is defined by: T µν = The observed large-scale structure of the universe is consistent with a Friedmann- Robertson-Walker metric ds2 = −dt2 +R2 (t) g̃ij (x) dxidxj (15) where the spatial metric g̃ij (x) is maximally symmetric, and satisfies R̃ij = −2kg̃ij , where k = +1 for spherical spatial sections, k = 0 for flat spatial sections, and k = −1 for hyperbolic spatial sections. The large-scale structure of Tµν is consistent with a perfect fluid form: Tµν = pgµν + (p+ ρ) uµuν (16) with pressure p and energy density ρ, where uµ = (1, 0, 0, 0) is the velocity vector of the fluid in co-moving coordinates. Einstein’s equations then lead to the Friedmann- Lemâıtre equation 8πGN ρ −H2 = H2 where H (t) = Ṙ is the Hubble parameter, Ṙ = dR , ρc = is the critical value of ρtot = ρ + ρvac for which k vanishes, and Λ = 8πGNρvac is the cosmological constant. The Hubble Space Telescope Key Project [73] has given the value H−10 = 13.6± 1.4 Gyr = (1.29± 0.13)× 1026 metres (18) for the present value of the Hubble parameter. By combining WMAP data with other astronomical data, Spergel et al [74] give the value = 1.02± 0.02 (19) However, there is no theoretical restriction on the magnitude of HR = Ṙ, so this value is consistent with any of the three possibilities k = +1, 0, or −1, although k = −1 is disfavoured by a standard deviation. In fact, visual inspection of the lower two panels, of Fig. 13 of [74], does not suggest any strong preference for k = +1, as opposed to k = −1. It seems likely that the class of models considered in the present paper will prefer k = −1 to k = +1, due to the infinitely greater variety of the smooth compact quotients ofH3, in comparison to the smooth compact quotients of S3, and the correspondingly improved chances of finding a quotient whose Casimir energy densities are such that, in combination with a suitable quotient of CH3, the observed values of GN and Λ can be fitted. There are, however, only 18 distinct topologies with k = 0, of which ten are compact, and the remaining eight have one or more uncompactified dimensions, [75]. This is far too small a number of distinct topologies, for there to be any likelihood of any of them satisfying the requirements on the Casimir energy densities, that will make it possible to fit the observed values of GN and Λ, so I do not expect any model, of the type studied in this paper, to have k = 0. Furthermore, most of the flat topologies have one or more shape moduli, unlike the hyperbolic topologies, and possibly also unlike the spherical topologies. I shall therefore, for simplicity, assume k 6= 0. The individual values of ρ , and ρvac , are not so precisely measured as their sum. Chapter 2 of [43] quotes the values of [74]: ρ = 0.27 ± 0.04, and ρvac = 0.73 ± 0.04. The baryonic and dark matter contributions to ρ are quoted as ρb = 0.044 ± 0.004, and ρdm = 0.22 ± 0.04. Chapter 19 of [43] quotes the values of Tonry et al [76]: = 0.72± 0.05, and ρ = 0.28± 0.05, if k = 0 is assumed. And Chapter 21 of [43] quotes best-fit values from SNe Ia and CMB data of ρ ≈ 0.3 and ρvac ≈ 0.7. Using the middle value ρvac = 0.72, and the above value of H−10 , we have: Λ = 3H20 = 0.012 Gyr−2 = 1.3× 10−52 metres−2 = = 3.4× 10−122G−1N = 5.1× 10−84 GeV2 (20) Hence: ρvac = Λ/(8πGN) = 3.0× 10−47 GeV4 = 2.3× 10−3 eV If ρ is set to zero, in the Friedmann-Lemâıtre equation (17), then for Λ > 0, the equation has the solutions R = (t− t0), for k = +1, R = R0e t, for k = 0, and R = (t− t0), for k = −1, t > t0. All three of these solutions satisfy Rµν = −Λgµν , and all three are in fact pieces of the maximally symmetric de Sitter space dS4 [77]. The k = +1 solution covers the full de Sitter hyperboloid, in the global coordinates of [77], the k = 0 solution covers the future of a single point in the t→ −∞ “boundary” of the hyperboloid, in the planar coordinates of [77], which cover precisely half the hyperboloid, and the k = −1 solution, for t > t0, covers the future of an ordinary point of the hyperboloid, in the hyperbolic coordinates of [77]. known as the de Sitter radius. For the measured value of Λ, the de Sitter radius is: = 16.0 Gyr = 1.51× 1026 metres = 0.94× 1061 GN (22) For each of the cases k = +1, 0, and −1, we can quotient the spatial sections of the solutions by discrete subgroups of the isometry groups of the spatial sections, that act freely, or in other words, without fixed points, on the spatial sections, in order to obtain locally de Sitter solutions, with Rµν = −Λgµν , and non-trivial spatial topology. For each of the cases k = +1 and k = −1, there are an infinite number of distinct such topologies, so it seems plausible, especially for k = −1, that there will exist topologies for which Bose - Fermi cancellations occur in the Casimir energy densities, for the compactifications of supergravity in eleven dimensions, and supersymmetric Yang-Mills theory in ten dimensions, on quotients with those topologies, with just the relative precisions I will show are needed, in order for solutions involving those quotients, together with a suitable quotient of CH3, to fit the observed values of GN and Λ. The action (9) is not applicable to the hypothesized period of inflation, since accord- ing to section 19.3.5 of [43], most current models of inflation are based on an unknown symmetry breaking involving a new scalar field, the “inflaton”. Models of the type considered in the present paper are expected to give very different behaviour from the standard hot big bang model, at times earlier than the time, t, at which the hot big bang model predicts the temperature, T , in units in which Boltzmann’s constant is equal to 1, to be comparable to the Planck mass in eleven dimensions, in the models of the present paper. According to [78], for temperatures higher than all particle masses, the standard hot big bang model gives T 2 = where Nb denotes the number of bosonic degrees of freedom that are effectively massless at temperature T , for example the photon contributes two units to Nb, and Nf is the corresponding number for fermions, for example electrons and positrons together contribute four units to Nf . If we set T = 1 TeV, and count only the observed Standard Model particles plus the graviton, so that Nb = 26, and Nf = 90, (if the neutrinos are assumed left-handed, so that their masses are Majorana), then 2.78× 10−3 eV = 0.709× 10−4 metres = 2.36× 10−13 seconds (24) which is comparable to the inverse one quarter power of the observed vacuum energy density ρvac, equation (21). I do not yet know whether models of the type studied here have problems with initial conditions, analogous to the horizon and flatness problems, that led to the hypothesis of inflation [78, 79, 80, 81, 82, 83]. To answer this question it will be necessary to study cosmological versions of these models, which will involve partial differential equations, with the time, and the radial coordinate of the thick pipe, as independent variables. In the present paper I shall only seek solutions such that the metric in the four observed dimensions is locally maximally symmetric, with the correct values of Newton’s constant and the cosmological constant. Thus the metric in the four extended dimensions will be locally de Sitter, although I will also consider whether or not flat and AdS solutions are possible. The aim of this section is to determine the circumstances under which the observed values of GN and Λ can be fitted, in a certain class of compactifications of Hořava- Witten theory. Rather than seeking supersymmetric solutions, I shall seek solutions in which the universe is stiff and strong, in the sense that the forces, that make it big and flat, are much stronger, than the forces that occur in any other physical process. In addition to fitting GN and Λ, I shall also require that the gauge coupling constants have approximately the correct values at unification, which typically means that the E8 fine structure constant, resulting from the compactification to 3 + 1 dimensions, is about 1 2.1 Hořava-Witten theory Hořava-Witten theory [1, 2] is supergravity in eleven dimensions, on a manifold with two boundaries, or, more precisely, on the orbifold M10 × S1/Z2, where M10 is a ten-dimensional manifold. At one-loop order in the Feynman diagram expansion, it is necessary to introduce a supersymmetric Yang-Mills theory, with gauge group E8, on each of the ten-dimensional boundaries, in order to cancel anomalies. The Hořava-Witten action in the bulk is the standard Cremmer-Julia-Scherk (CJS) action [14]. In the “upstairs” picture, working on the orbifold M10 × S1/Z2, and omitting terms quartic in the gravitino, this is: SCJS = IJKDJψK − GIJKLG IJKLMNψN + 12ψ̄ JΓKLψM GJKLM gI1I2...I11CI1I2I3GI4...I7GI8...I11 where gI1I2...I11 is the tensor gI1I2...I11 = 1√−gǫ I1I2...I11 , ǫ0 1 2 ...9 10 = 1, and GIJKL = 24∂[ICJKL]. Coordinate indices I, J,K, . . . run over all directions on M11. The Dirac matrices ΓI are 32 × 32 real matrices satisfying {ΓI ,ΓJ} = 2gIJ . A suitable representation of the Γa, where a, b, c, . . . are local Lorentz indices, is given, for example, in section 2.5 of [84]. The matrices ΓI1I2...In are defined by ΓI1I2...In = Γ[I1ΓI2 . . .Γ In], so that when the indices are all different, ΓI1I2...In = ΓI1ΓI2 . . .ΓIn . Spinor indices are written α, β, γ, . . .. The matrices Γ0ΓI1I2...In , where the index of Γ0 is a local Lorentz index, are symmetric for n = 1, 2, 5, 6, 9, 10, and antisymmetric for n = 0, 3, 4, 7, 8, 11. The charge conjugation matrix is the antisymmetric matrix C = −Γ0, where the index of Γ0 is a local Lorentz index. We note that the right-hand spinor index of a Dirac matrix transforms under lo- cal Lorentz transformations by matrix multiplication on the right by where λab are the local Lorentz transformation parameters, and the left-hand spinor index of CΓI transforms by matrix multiplication on the left by −C 1 + 1 , which is equivalent to acting on the left-hand spinor index of CΓI by matrix multiplication on the right by . Thus the left-hand spinor index of CΓI is an index with the Lorentz transformation properties of the right-hand index of a Dirac matrix, so if spinor indices with the Lorentz transformation properties of the left-hand and right-hand spinor indices of a Dirac matrix are distinguished by writing them as upper and lower spinor indices respectively, then C acts as a “metric”, that lowers a spinor index. This is consistent with CΓa being an invariant tensor under SO (10, 1) local Lorentz transformations, since (δab + λ δγα − δδβ − = (CΓa)αβ , (26) up to terms quadratic in λab, where the identity Γ aΓcd = Γacd + ηacΓd − ηadΓc was used. The inverse “metric” Cαβ is defined in terms of Cαβ by CαβCβγ = δαγ . The invariant tensor (CΓa)αβ can be written as Γaαβ , since the position of the first spinor index distinguishes it from (Γa) β = Γ β. All spinors in ten or eleven dimensions will be Majorana, which for a real representation of the Dirac matrices, means real [85]. The conjugate Majorana spinor is ψ̄α = −ψβCβα = Cαβψβ. The manifold M11 is assumed to have the topology M10 × S1. Coordinate indices U, V,W, . . . will run over all directions on M10, and I will use the lower-case letter y for the coordinate in the S1 direction, and also for the coordinate index in the S1 direction. There is assumed to be an orbifold fixed point at y = y1, and another one at y = y2 > y1. All fields are periodic in the y direction, with period 2 (y2 − y1). The bosonic fields gUV , gyy, and CUV y are even under the reflections y → (2y1 − y) and y → (2y2 − y), and gUy and CUVW are odd. The gravitino satisfies ΨU (y) = ΓyΨU (2y1 − y) Ψy (y) = −ΓyΨy (2y1 − y) (27) together with the corresponding conditions, with y1 replaced by y2. The integral over M11, in (25), includes two copies of the physical region y1 ≤ y ≤ y2, namely the original region, and its reflection in one of the two fixed point sets. I shall adopt the viewpoint of Hořava and Witten, that it should be possible to switch, as convenient, between the “upstairs” viewpoint, of working on the full M11, with these reflection symmetries imposed on the fields, and the “downstairs” viewpoint, of working on a manifold with boundary, with the topology M10 × I1, where I1 denotes the interval y1 ≤ y ≤ y2. For this to work, it is essential, as noted in footnote 3 of [2], that when working on the manifold with boundary, the factor 1 , in (25), should be replaced by 2 The conditions (27) imply that the gravitino is chiral on the ten-dimensional orb- ifold fixed point sets, which results in a gravitational anomaly, localized on the ten- dimensional fixed point sets. Hořava and Witten argued, in [1], that this gravitational anomaly could be cancelled by introducing an E8 supersymmetric Yang-Mills multi- plet, on each of the ten-dimensional fixed point sets, and they studied the required couplings in [2]. The supersymmetric Yang-Mills action, on the orbifold fixed point set at y1, is SYM = − −g tr FUV F χ̄ΓUDUχ . (28) and the action at y2 is obtained from this by the substitution M101 → M102 . The action (28) is written in Hořava and Witten’s notation, in which “tr”, for E8, denotes 1 the trace in the adjoint representation, which they denote by “Tr”. I will also use this notation. Hořava and Witten do not explicitly specify the normalization of the E8 generators they use, or, equivalently, their choice of normalization of the E8 structure constants. This needs to be determined for the present study, because in Section 5 I shall use an SU(9) basis for E8, rather than an SO(16) basis, and the correct normalization of the generators, in the SU(9) basis, has to be determined. It is clear from (28) that Hořava and Witten use hermitian E8 generators, and I shall assume that their hermitian E8 generators are given by i, or alternatively −i, times antihermitian generators, normal- ized so that, in the SO(16) basis, the E8 Lie algebra is as given in Appendix 6.A of [72]. Specifically, let γi, 1 ≤ i ≤ 16, be a Majorana-Weyl representation of the SO(16) gamma matrices, so the γi are real and off block diagonal, and let σij = [γi, γj]. Then the σij are real, antisymmetric, and block diagonal, with two 128× 128 blocks, which are the two irreducible spinor representations of SO(16). Choose one of the two spinor representations, say the first, and let σ̄ij denote the restriction of σij to the corre- sponding block. Then the generators of E8 are the 120 generators Jij of SO(16), where Jji = −Jij , together with 128 generators Qα, whose label, α, runs over the chosen spin representation of SO(16). The commutation relations are: [Jij , Jkl] = Jilδjk − Jjlδik − Jikδjl + Jjkδil (29) [Jij, Qα] = (σ̄ij)αβ Qβ (30) [Qα, Qβ] = (σ̄ij)αβ Jij (31) We therefore find that the matrix elements of the generators, in the adjoint represen- tation of E8, which is also the fundamental, are given by: Jij = −fij,pq,rs 0 0 − (σ̄ij)βγ 0 (σ̄pq)αγ (σ̄rs)βα 0 where fij,pq,rs = (δiqδpsδrj − δjqδpsδri − δipδqsδrj + δjpδqsδri − δiqδprδsj + δjqδprδsi +δipδqrδsj − δjpδqrδsi) (34) are the SO(16) structure constants, from (29). These generators are correctly normal- ized so that, in doing matrix multiplications with the generators (32) and (33), the vector index pairs (p, q), and (r, s), are to be summed over the full ranges of all the vector indices, without restrictions, so there is no restriction, for example, to p < q. In fact, (Jij)pq,rs = −fij,pq,rs are the correctly normalized generators of SO(16), in the adjoint representation, with Young tableau shape (1, 1), and can be obtained, al- ternatively, by Young tableaux methods, starting from the generators for the vector representation of SO(16), which are: (Jij)ef = δieδjf − δjeδif (35) Using (32) and (33), we find that: Tr (JijJkl) = (δilδjk − δikδjl) = −60 (δikδjl − δilδjk) = −120δij,kl (36) Tr (QαJij) = 0 (37) Tr (QαQβ) = (σ̄pq)αγ (σ̄pq)γβ + (σ̄rs)δα (σ̄rs)βδ = −60δαβ − 60δβα = −120δαβ (38) where in obtaining (36) I used that, for SO(d), we have: fij,pq,rsfkl,rs,pq = (2d− 4) (δilδjk − δikδjl) (39) and δij,kl = (δikδjl − δilδjk) is the unit matrix, in the space of matrices whose rows and columns are labelled by antisymmetrized pairs of vector indices. Hence, denoting the 248 generators (Jij , Qα) collectively by ΛA, we have: Tr (ΛAΛB) = −120δAB (40) On the other hand, for the vector representation (35) of SO(16), we have: (Jij)ef (Jkl)fe = −4δij,kl (41) Thus the trace of the square of a generator of SO(16), in the adjoint of E8, is 30 times the trace of the square of the corresponding generator, in the vector representation of SO(16). Seeking to extend (28) to a locally supersymmetric action, coupled in a locally supersymmetric manner to the bulk supergravity multiplet, Hořava and Witten found it necessary to modify the Bianchi identity of the four-form gauge field, so that it reads: dGyUVWX = −3 δ (y − y1) trF (1)[UV F WX] + δ (y − y2) trF [UV F where dGIJKLM = 5∂[IGJKLM ], and F UV denotes the E8 gauge fields at y = yi. This, in turn, implies that the three-form, CIJK , is not invariant under Yang-Mills gauge trans- formations. It also implies, in the “upstairs” picture, that GUVWX has a discontinuity, at y = y1, given by GUVWX = − ǫ (y − y1) trF (1)[UV F WX] + . . . (43) where ǫ (x) is 1 for x > 0, and −1 for x < 0, and . . . denotes terms that are regular near y = y1, and thus vanish at y = y1. While in the “downstairs” picture, on the interval y1 ≤ y ≤ y2, (43) becomes a boundary condition: GUVWX |y=y1+ = − [UV F WX] (44) Corresponding results also hold in the region of y = y2. The non-vanishing variation of the three-form, CIJK , under Yang-Mills gauge trans- formations, now implies that the Chern-Simons term, CGG, in the Cremmer-Julia- Scherk action (25), has a non-vanishing variation, under Yang-Mills gauge transforma- tions. Hořava and Witten found that this non-vanishing variation, under Yang-Mills gauge transformations, of the Cremmer-Julia-Scherk Chern-Simons term, precisely can- cels the one-loop quantum gauge anomaly, of the Majorana-Weyl fermions in the su- persymmetric Yang-Mills multiplets on the orbifold fixed points, provided that λ2 = 2π A slightly different result was found by Conrad [86], who found λ2 = 2 32π (4πκ2) 4π (2πκ2) 3 . This difference will not have a major impact on the results of the present paper, so I shall use the Hořava-Witten result (45), and not attempt to resolve the issue here. The relation (45) implies that so the Yang-Mills action is of relative order κ Having cancelled the Yang-Mills gauge anomalies, by relating the Yang-Mills cou- pling constant to the gravitational coupling constant as just discussed, Hořava and Witten returned to the original purpose of introducing the Yang-Mills multiplets on the orbifold fixed points, which was to cancel the gravitational anomalies of the graviti- nos, on the orbifold fixed points. As explained in Section 2 (i) of [1], the “irreducible” part of the formal twelve-form, from which the gravitino anomaly in ten dimensions is constructed, can only be cancelled by the introduction of 248 vector multiplets on each of the orbifold fixed point hyperplanes. This requirement is fulfilled by the E8 supersymmetric Yang-Mills multiplets. Hořava and Witten then argued that, in conse- quence of factorization properties of the remaining terms in the full gravitational and mixed gravitational - gauge anomalies, in ten dimensions, the remaining terms in the gravitational and mixed anomalies can all be cancelled, provided that, in the equations (42), (43), and (44), above, the substitutions [UV F WX] → trF [UV F WX] − trR[UV RWX] (47) are made uniformly, where RUV is the curvature two-form, and trR[UV RWX] must be defined, by analogy with Section 16.1 of [72], as R Y Z[UV RWX]Y Z , and provided that, in the quantum effective action, or in other words, the generating functional of the proper vertices [87, 88], which is, in general, a non-local functional of the fields, a certain local term, called the bulk Green-Schwarz term, appears in the bulk, with an appropriate finite coefficient. The required form of the bulk Green-Schwarz term was in agreement with the form already found from a one-loop calculation for Type IIA superstrings [89], and from anomaly cancellation for five-branes in eleven dimensions [25, 90], and its coefficient was studied by de Alwis [91, 92] and Conrad [86]. Hořava and Witten then completed the calculation of the action at relative order 3 , and found a problem with a term in a supersymmetry variation, proportional to δ (0). This led to a further problem, with a term in the action at relative order κ with a coefficient proportional to δ (0). They suggested this implies that the full theory must have a built-in cutoff, that would replace δ (0) by a finite constant times κ− 9 , for example, by having the gauge fields propagate in a boundary layer, of thickness about κ2/9, rather than precisely on the orbifold fixed point hyperplanes. However Moss has presented an improved form of Hořava-Witten theory [69, 70, 71], in which the δ (0) terms are absent. The modifications introduced by Moss include the introduction of a supersymmetrized Gibbons-Hawking boundary term [93, 94, 95], additional terms bilinear in the gauginos in (42), (43), and (44) above, and a modification to the chirality conditions (27) on the gravitino, in the neighbourhood of an orbifold fixed point, which for the components ψU , in the upstairs picture, amounts to introducing a step function term in the behaviour of (1− Γy)ψU , near the fixed point, analogous to (43) above. The existence of Moss’s improved form of the theory suggests it is reasonable to assume that the Yang-Mills multiplets do not, after all, spread into a boundary layer of nonzero thickness in the bulk, and do, indeed, stay in the orbifold fixed point hyper- planes, of zero thickness. The study of the boundary conditions, and of the field equa- tions in the bulk, near the boundaries, in the present paper, depend on this assumption for their validity, so the conclusions about the existence of thick pipe geometries, and the possibility of fitting both Newton’s constant and the cosmological constant, for topologies such that the Casimir energy densities cancel to the required relative preci- sions, depend on the existence of Moss’s improved form of the theory. However, these studies do not involve the fermi fields, so I will not need to use the explicit form of the modifications introduced by Moss. The assumption that the Yang-Mills multiplets do, indeed, stay in the orbifold fixed-point hyperplanes, of zero thickness, means that, for the further development of Hořava-Witten theory, it is essential to treat the step functions, such as in (43) above, and their derivatives, by a consistent limiting procedure, from properly regularized versions, as discussed by Bilal and Metzger [96, 28]. However this is not necessary in the present paper. Lukas, Ovrut, and Waldram, [67], have pointed out that, corresponding to the replacement (47), supersymmetry is likely to require that, in the Yang-Mills action (28), the corresponding replacement (i)UV → trF (i)UV F (i)UV − RUVWXR UVWX − 4RUVRUV +R2 = trF (i)UV − 3R [UV[UV R WX] (48) is made, where RUV now denotes the Ricci tensor. This would be analogous to the situation for the E8 × E8 heterotic superstring [97, 98], whose effective low-energy field theory action contains the expression to the right of the arrow in (48), summed over both the E8 groups. In this case, the Lovelock-Gauss-Bonnet term [99, 100, 101] [UV R WX] is stated, in Section 16.1 of [72], to be related, by supersymme- try, to the Lorentz Chern-Simons term that is included in the field strength of the d = 10, N = 1 supergravity two-form, by the original Green-Schwarz anomaly can- cellation mechanism [102]. The Lovelock-Gauss-Bonnet term, for the E8 × E8 het- erotic superstring, was found by Gross and Witten [103], by means of a low energy expansion of tree-level superstring scattering amplitudes. The relative coefficients of RUVWXR UVWX, RUVR UV , and R2 were fixed to the Lovelock-Gauss-Bonnet form by Zwiebach [104], who pointed out this linear combination contains no terms quadratic in the graviton, and thus does not lead to the occurrence of ghosts, in the free graviton propagator. An analogue of the positive energy theorem [105, 106] for the Einstein action, together with the Lovelock-Gauss-Bonnet term, as it occurs in the effective low-energy field theory action for the E8 × E8 heterotic superstring, was proved by Kowalski-Glikman [107], and the Lovelock-Gauss-Bonnet term was found by Candelas, Horowitz, Strominger, and Witten [9], to make it possible to circumvent the no-go theorem [108], for compactifications of supersymmetric Yang-Mills theory coupled to N = 1 supergravity in ten dimensions [109, 110]. To the best of my knowledge, the corresponding Lovelock-Gauss-Bonnet terms for Hořava-Witten theory, given by making the substitutions (48) in the Yang-Mills ac- tion (28), have not yet been directly derived, nor explicitly related by supersymmetry to the modified Bianchi identity (42), with the substitutions (47). This would pre- sumably require the systematic study of Slavnov-Taylor identities [111, 112, 113] for BRST quantized [114, 115] Hořava-Witten theory, perhaps in the Batalin-Vilkovisky framework [116, 117, 118, 119, 120]. The Lovelock-Gauss-Bonnet terms would, then, presumably be found as local terms, in the generating functional of proper vertices, on the orbifold fixed-point hyperplanes, with the expected finite coefficients, in a similar manner to the bulk Green-Schwarz term. I shall simply follow Lukas, Ovrut, and Wal- dram [67], and assume these terms to be present, with the coefficients implied by the substitutions (48). 2.2 The complex hyperbolic space CH I shall assume that six of the nine spatial dimensions, of M10, are compactified on a smooth compact spin quotient of CH3, the complex hyperbolic space with three complex dimensions. A detailed account of the geometry of complex hyperbolic space has been given by Goldman [121], but for the present study of the field equations and boundary conditions, I shall only need the very simplest properties of CH3, which I shall now summarize. The study of CH3 is facilitated by the use of complex coordinates. I shall consider the transformation from Cartesian coordinates to complex coordinates to be a special case of a general coordinate transformation, and use the corresponding notation. For 2n real dimensions, we define a 2n by 2n complex matrix Uµ ν by: U r2s−1 = δrs, U 2s−1 = δrs, U δrs, U 2s = − i√2 δ s (49) for 1 ≤ r ≤ n and 1 ≤ s ≤ n. Then we define complex coordinates zµ by a complex general linear transformation: zµ = Uµ νx ν (50) zr = U r νx ν = U r2s−1x 2s−1 + U r2sx x2r−1 + ix2r zr̄ = U r̄ νx ν = U r̄2s−1x 2s−1 + U r̄2sx x2r−1 − ix2r for 1 ≤ r ≤ n. Thus zr̄ = (zr)∗, where ∗ denotes complex conjugation. We define the inverse, V , of U , by: V 2r−1 s = δr s, V δr s, V s = − i√2δ δr s (53) xµ = V µ νz ν (54) In general, on the change to complex coordinates, a contravariant index is transformed by U , thus xµ → Uµ νxν , and a covariant index is transformed by V , thus ∂µ → V ν µ∂ν . The metric in flat Cartesian coordinates, namely the Kronecker delta, δµν , is not preserved by the transformation to complex coordinates. Its components in the complex coordinate basis, which I will denote by δ̃µν , are given by: δ̃µν = V νδστ (55) Explicitly: δ̃rs = δ̃r̄s̄ = 0, δ̃rs̄ = δ̃r̄s = δrs (56) Thus δ̃µνz µzν = 2zrzr̄ = δστx σxτ , where I have introduced a summation convention specific to complex coordinates, namely that if a holomorpic contravariant index, i.e. an unbarred contravariant index that runs from 1 to n, has the same letter as an antiholomorphic contravariant index, i.e. a barred contravariant index that runs from 1 to n, then the formula is to be summed over all values of that letter, from 1 to n. I shall also use the corresponding convention when a holomorphic covariant index, and an antiholomorphic covariant index, have the same letter, and summation from 1 to n also applies, when a holomorphic contravariant index, and a holomorphic covariant index, have the same letter, and it also applies, when an antiholomorphic contravariant index, and an antiholomorphic covariant index, have the same letter. Similarly: δ̃µν = Uµ σU στ (57) The components are: δ̃rs = δ̃r̄s̄ = 0, δ̃rs̄ = δ̃r̄s = δ rs (58) We then find, for example, that δ̃rµδ̃µs = δ̃ r̄µδ̃µs̄ = δ s, δ̃ rµδ̃µs̄ = δ̃ r̄µδ̃µs = 0. It is convenient also to define δrs̄, δr̄s, δ rs̄, and δr̄s, (without tildes), by: δrs̄ = δr̄s = δ rs̄ = δr̄s = δrs = δ rs (59) I shall adopt the convention that, when using complex coordinates, the indices of the coordinates are lowered and raised by the flat space complex metric, δ̃µν and δ̃ µν , not by whatever curved metric is under consideration. We thus have: zr = z r̄ = (zr) , zr̄ = z r = (zr̄) zrzr̄ = zrz r = zr̄zr̄ = z r̄zr (61) regardless of the curved metric under consideration. This is convenient for working with CHn and CPn, because it makes the SU(n) properties of formulae manifest, and facilitates the study of the transformation properties under SU(n, 1) and SU(n + 1), respectively. The metric on CHn is now defined by: gµν = 0 grs̄ gr̄s 0  (62) where: grs̄ = gs̄r = (1− zt̄zt) zrzs̄ (1− zt̄zt)2 (1− zt̄zt) δrs̄ + zrzs̄ (1− zūzu) so that: gµνdz µdzν = grs̄dz rdzs̄ + gr̄sdz r̄dzs = (1− zt̄zt) δrs̄ + zrzs̄ (1− zūzu) dzrdzs̄ (64) The complex hyperbolic space CHn corresponds to the region zrzr̄ < 1. We note that gr̄s is the complex conjugate of grs̄, or in other words, gr̄s = (grs̄) In general, when working with complex coordinates, I shall choose definitions in ac- cordance with a convention such that if every index of a vector, tensor, or matrix is of definite holonomic type, i.e. either holonomic or antiholonomic, but not an index, such as µ in this section, which can be either, then replacing every unbarred index by the corresponding barred index, and every barred index by the corresponding unbarred index, is equivalent to complex conjugation. From (62) and (63) we find: (1− zt̄zt)2n+2 (1− zt̄zt)d+2 where g denotes, as usual, the determinant of gµν , and d = 2n. Also: gµν = 0 grs̄ gr̄s 0  (66) where grs̄ = gs̄r = (1− zv̄zv) (δrs̄ − zrzs̄) (67) We observe that grs̄ = −∂r∂s̄ ln 1− zt̄zt so the metric is Kähler, with Kähler potential − ln 1− zt̄zt . The Kähler form is ωrs̄ = −igrs̄, ωs̄r = igrs̄ (69) and has real components in a real coordinate system. The nonvanishing Christoffel symbols of the second kind are: Γrst = δr tzs + δ (1− zv̄zv) , Γr̄s̄t̄ = δr̄ t̄zs̄ + δ s̄zt̄ (1− zv̄zv) Hence, recalling the sign convention (6) for the Riemann tensor, we have: R urs̄t = −∂s̄Γurt = −δutgrs̄ − δurgts̄, R ūrs̄t̄ = ∂rΓūs̄t̄ = δūt̄grs̄ + δūs̄grt̄ (71) Rrs̄tū = −grs̄gtū − grūgts̄ (72) and the Ricci tensor: Rrs̄ = (n+ 1) grs̄ (73) so the metric is Kähler-Einstein. To calculate the quantity ǫν1...ν2nǫ µ1...µ2nR ν1ν2µ1µ2 . . . R ν2n−1ν2n µ2n−1µ2n , which occurs in the generalized Gauss-Bonnet formula [122], we note that, if we define the ten- sor, gν1...ν2n , by gν1...ν2n ≡ gǫν1...ν2n , then on transforming to complex coordinates, as described after (54), gν1...ν2n becomes det V gν1...ν2n . I shall assume that the complex coordinates are taken in the order 1, 1̄, 2, 2̄, 3, 3̄, . . ., so that V is block diagonal. Then det V = in. Thus g11̄22̄33̄...nn̄ = i n√gǫ123...nǫ1̄2̄3̄...n̄, so ǫ11̄22̄33̄...nn̄ = inǫ123...nǫ1̄2̄3̄...n̄. We then find, from (72), that: ǫν1...ν2nǫ µ1...µ2nR ν1ν2µ1µ2 . . . R ν2n−1ν2n µ2n−1µ2n = 22nn! (n+ 1) ! (74) If we now formally introduce an (n+ 1)th coordinate, zn+1, that is actually set equal to 1, so that dzn+1 = 0, let indices R, S, . . ., run from 1 to n + 1, and define ηRS̄ = (δrs̄,−1), then the above formula (64), for the squared line element of CHn, can be written: gµνdz µdzν = −ηT̄ Y zT̄ zY ηRS̄dz RdzS̄ + zX̄ηRX̄dz zV ηS̄V dz −ηŪWzŪzW = − 2 ηT̄ Y z T̄ zY dzR − zR ηZX̄z X̄dzZ ηB̄Cz dzS̄ − zS̄ ηĀV z V dzĀ ηŪW z which, apart from the overall minus sign, and the factor of 2, which results from my choice of normalization in (63), would become equal to the Fubini-Study metric on CPn, as in equation (15.3.14) of [72], if each ηDĒ was now replaced by δDĒ. Conversely, if we now relax the condition zn+1 = 1, (75) can be regarded as the metric on CHn in homogeneous coordinates, since, just as with the Fubini-Study metric in homogeneous coordinates, (75) is invariant under rescaling of the zR, and also vanishes if dzR is a multiple of zR, or dzS̄ is a multiple of zS̄. Indeed, just as in the case of the Fubini- Study metric, (75) is invariant under holomorphic position-dependent rescalings of the coordinates: zR → f (z) zR. For (75) is the square of the distance between z, and a nearby point z′ = z + dz. And zR → f (z) zR implies: R → f (z′) (z′)R = f (z) zR + f (z) dzR + (∂Df (z)) zR (76) Hence dzR → f (z) dzR+(∂Df (z)) zR, and the second term here cancels because it is a multiple of zR. CHn corresponds to the region, in the space of the homogeneous coordinates, such that ηRS̄z RzS̄ < 0 (77) Thus zn+1 does not vanish anywhere on CHn, so, unlike the case of CPn, CHn is naturally covered by a single coordinate patch. The metric (75) is manifestly invariant under linear SU(n, 1) transformations of the homogeneous coordinates: zR → LRSzS = LRszs + LRn+1zn+1 (78) where the matrix LRS satisfies: ηRS̄ = ηT ŪL S̄ (79) so that ηRS̄z RzS̄ is invariant. The matrix LŪ in (79) is by definition equal to i.e. the complex conjugate of LUS, in accordance with the convention stated above. Now the hypersurface zn+1 = 1, in the space of the homogeneous coordinates, is equivalent to the whole of CHn, in the original coordinates. The action of (78), on points in this hypersurface, is: zr → Lr szs + Lr n+1 zn+1 → Ln+1 szs + Ln+1n+1 (80) Thus this hypersurface is not, in general, left invariant by (78). However, as noted above, (75) is also invariant under holomorphic position-dependent rescalings of the coordinates: zR → f (z) zR. Hence (75) is invariant under (78), followed by division by the new value of zn+1. This compound transformation leaves the hypersurface zn+1 = 1 invariant, and transforms the points in this hypersurface, which correspond to the points of CHn, by the projective transformations: Lr sz s + Lr n+1 Ln+1 sz s + Ln+1n+1 Thus, since the original squared line element, (64), is the restriction of (75) to the hypersurface zn+1 = 1, the original squared line element, (64), is also invariant under the projective SU(n, 1) transformations, (81). This can also be verified directly. Returning now to the original coordinate system, or in other words, setting zn+1 = 1, and restricting z to represent the n-vector, (z1, . . . , zn), we can send the origin of CHn to an arbitrary point, z, of CHn, by means of an SU(n, 1) projective transforma- tion, (81), by choosing the matrix LRS to be the SU(n, 1) “boost”: LRS = Lr s L Ln+1 s L δrs + ztzt̄ zrzs γz γzs γ  (82) where 1− ztzt̄ For n ≥ 2, CHn is not maximally symmetric, and the sectional curvature is not constant. In general, for linearly independent vectors Aµ, Bν , the sectional curvature, at a point of a Riemannian manifold, is defined, bearing in mind the sign convention (5), by: K (S) = − RµνστA µBνAσBτ (gµσgντ − gµτgνσ)AµBνAσBτ where S denotes the linear space spanned by Aµ and Bν . To apply this to CHn, in the complex coordinates, we note that, if a vector, Aµ, is real, in real coordinates, then after transforming to complex coordinates, as described after (54), its components satisfy Ar̄ = (Ar) , just as for the complex coordinates themselves. For the CHn metric, (62), (63), with the Riemann tensor components (72), we find: RµνστA µBνAσBτ = gtr̄A (gsūB sBū)+2 gtūA (gsr̄B sAr̄)−2 (grs̄ArB s̄) 2−2 (gsr̄BsAr̄) (gµσgντ − gµτgνσ)AµBνAσBτ = = 4 (grs̄A rAs̄) gtūB − 2 (grs̄ArB s̄) gtūB − (gsr̄BsAr̄) 2 − (grs̄ArB s̄) If we now work at the origin of the complex coordinates, so that grs̄ = δrs̄, we can define real magnitudes |A|, |B|, and real angles, θ, ϕ, by: |A| = grs̄ArAs̄, |B| = grs̄BrB s̄, θ = arccos (grs̄ArB s̄) (gtūBtAū) |A| |B| arcsin grs̄A rB s̄ gtūBtAū − gtūB grs̄ArB s̄ Thus: RµνστA µBνAσBτ (gµσgντ − gµτgνσ)AµBνAσBτ (1 + cos2 θ (1− 2 cos 2ϕ)) (2− cos2 θ (1 + cos 2ϕ)) Now, if n = 1, the angle θ is 0, so the sectional curvature is −2. For n ≥ 2, and for all values of ϕ, such that cos 2ϕ 6= 1, the right-hand side of (88) varies between a minimum of −2, when cos2 θ = 1, and a maximum of −1 , when cos2 θ = 0, since if we replace cos2 θ by x, the function has no maximum or minimum between x = 0 and x = 1. For cos 2ϕ = 1, the right-hand side of (88) is equal to −1 , except for cos2 θ = 1. And when cos 2ϕ and cos2 θ are both equal to 1, Aµ and Bν are no longer linearly independent. Thus for n ≥ 2, the sectional curvature of CHn, with the metric (62), (63), lies in the range −2 to −1 The equation for a geodesic is: + Γrst (1− zv̄zv) = 0 (89) For the geodesics through the origin, we have zr = Zr tanh (αs), where Zr is a fixed complex n-vector such that ZsZ s = 1, and α is a real constant. If we choose α = 1√ then s is the geodesic distance from the origin to z, in accordance with (64). Hence the geodesic distance from the origin to z, is 1 + |z| 1− |z| where |z| ≡ zrzr. To find the geodesic distance from a point, z, to a point, w, we can use the invariance of the geometry under the projective SU(n, 1) transformations (81). Sending z to the origin, by the inverse of (82), sends w to a point w̃, such that: w̃rw̃ (wr − zr) (wr − zr)− zrzrwsws + wrzrzsws (1− ztwt) (1− wuzu) Hence: cosh2 1− w̃rw̃r (1− zrwr) (1− wszs) (1− ztzt) (1− wuwu) where s now denotes the geodesic distance from z to w. Thus in the homogeneous coordinates, we have: ∣ηRS̄z ∣ηT Ūz T zŪ ∣ηV X̄w where |·| denotes the absolute value of a complex number. 2.3 The field equations and boundary conditions I shall now assume that Hořava-Witten theory has been quantized in accordance with standard procedures for quantizing supergravity in eleven dimensions [123, 124], to- gether with an appropriate treatment of the orbifold fixed-point hyperplanes, and seek solutions of the field equations, and boundary conditions, that follow from varying the quantum effective action, or in other words, the generating functional of the proper vertices, Γ, [125, 126, 87, 88], with respect to the fields. The quantum effective action is expanded in terms of the number of loops in Feynman diagrams, which in the bulk, is an expansion in powers of κ2, and on the orbifold fixed-point hyperplanes, is an expansion in powers of λ2, where λ and κ are related by (45). Since κ and λ are di- mensional constants, the actual expansion parameters have the form κ , and λ , where L will be, in general, the smallest physically relevant distance, in a particular region of the geometry. I shall seek solutions with a “thick pipe” form of geometry, so that, in particular, if L denotes the geodesic distance between the two orbifold fixed-point hyperplanes, then κ ≪ 1. Furthermore, if L denotes a radius of curvature of the compact six-manifold, which will in general be either smaller than, or comparable to, its diameter, on account of the hyperbolic nature of the manifold, then we will again have κ ≪ 1, throughout the main part of the bulk. Thus, throughout the main part of the bulk, it will be a good approximation to neglect all quantum corrections to Γ, and approximate Γ as the gauge-fixed classi- cal action, together with the Fadeev-Popov terms. We then seek a solution, of the field equations that follow from varying Γ with respect to the fields, in which all the Fadeev-Popov fields, and also any other fields introduced in the course of the gauge- fixing, vanish. The field equations then reduce to the classical field equations for the supergravity multiplet, which are the Euler-Lagrange equations for the Cremmer-Julia- Scherk action (25), together with gauge-fixing conditions. We can always solve such equations by solving the Cremmer-Julia-Scherk field equations in any convenient gauge we choose, then applying gauge transformations to the solution, in order to satisfy the required gauge conditions. I shall now denote the full eleven-dimensional metric by GIJ . It will be distin- guished from GIJKL by context, and the number of indices. Other conventions are as in Subsection 2.1, on page 17. Furthermore, coordinate indices A,B,C, . . . will be tangent to the compact six-manifold, and coordinate indices µ, ν, σ, . . . will be tangent to the four observed space-time dimensions, which at the inner surface of the thick pipe, where we live, in this type of model, are the extended dimensions. I shall use the gauge freedom of general coordinate invariance, in order to choose Gaussian normal coordinates, such that Gyy = 1, and GUy = 0, and thus seek a solution where the metric has the form: ds211 = GIJdx IdxJ = a (y) gµνdx µdxν + b (y) hABdx AdxB + dy2 (94) where gµν is the metric on a four-dimensional locally de Sitter space, whose de Sitter radius I shall set equal to 1, and whose spatial sections may have been compactified, as discussed after equation (22), and hAB is the metric on a smooth compact quotient of CH3, and is locally equal to the metric specified in (62) and (63). Thus Rµν (g) = −3gµν , and RAB (h) = 4hAB. I shall also consider the possibility of flat and AdS spacetimes, which would have Rµν (g) equal to zero, and a positive multiple of gµν , respectively. I shall seek solutions such that the inner and outer surfaces of the thick pipe, or, in other words, the orbifold fixed point hyperplanes, are at y = y1 and y = y2, where y1 and y2 are determined by the boundary conditions, and are independent of position in the four observed dimensions, and on the compact six-manifold. We are free to shift y by a constant, and I shall use this freedom to obtain the simplest formulae for the solution in the bulk, rather than to set y1 or y2 to any particular value. The inner surface of the thick pipe, where we live, will be at y = y1, so in the de Sit- ter case, it follows from (22), that we require a (y1) = 16.0 Gyr = 1.51×1026 metres = 0.94× 1061 2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor The non-vanishing Christoffel symbols of the second kind, for the metric ansatz (94), Γµνσ = gµτ (∂νgστ + ∂σgντ − ∂τgνσ) Γµνy = Γ Γyµν = −aȧgµν = − ΓABC = hAD (∂BhCD + ∂ChBD − ∂DhBC) ΓABy = Γ AB = −bḃhAB = − GAB (95) where a dot denotes differentiation with respect to y. From this, and the formula (6), for the components of the Riemann tensor, it follows that the only non-vanishing components of the form R JUV I , of the Riemann tensor, in eleven dimensions, are of the forms R τµνσ , R ABC , R µAν , R Aµν , R µAB , and R AµB . In particular, neither I, nor J , can be y. The non-vanishing components of the Riemann tensor, when one or more of the indices is y, are R yµyν , R yµν , R µyy , R yµy , R AyB , R yAB , R Ayy , and R yAy . We find: R τµνσ = R µνσ (g) + ν − Gνσδ τµ R DABC = R ABC (h) + B −GBCδ DA R BµAν = A , R AµB = R yµyν = Gµν , R yµy = AyB = GAB, R yAy = δ BA (96) where R τµνσ (g) denotes the Riemann tensor calculated from the four-dimensional met- ric gµν , and R ABC (h) denotes the Riemann tensor calculated from the six-dimensional metric hAB. From (96), we find that the non-vanishing Ricci tensor components, in eleven dimensions, are: Rµν = Rµν (g) + Gµν = RAB = RAB (h) + GAB = Ryy = 4 where I used the relations Rµν (g) = −3gµν , and RAB (h) = 4hAB, from above. For smooth compact quotients ofH6, we choose the metric hAB forH 6 to have radius of curvature equal to 1, so that RABCD (h) = hAChBD−hADhBC , and RAB (h) = 5hAB. Thus for a smooth compact quotient of H6, the term 4 GAB in RAB is replaced by We also need the Riemann tensor components, on the orbifold fixed-point hyper- planes, calculated from the ten-dimensional metric, on the orbifold fixed-point hyper- planes. The ten-dimensional metric is obtained from (94), by setting dy = 0, and either y = y1, or y = y2, as appropriate. Then M10 is simply the Cartesian product, of a four dimensional locally de Sitter space, with de Sitter radius a (y1), or a (y2), as appropriate, and a smooth compact quotient of CH3, with the metric (63) multiplied by a factor b2 (y1), or b 2 (y2), as appropriate. All the Christoffel symbols and Rie- mann tensor components with mixed indices now vanish, and the only non-vanishing Riemann tensor components are now R τµνσ (g) and R ABC (h). 2.3.2 The Yang-Mills coupling constants in four dimensions There are inevitably significant Casimir energy density terms in the energy-momen- tum tensor on and near the inner surface of the thick pipe, due to the Hořava-Witten relation λ ≃ 5.8κ 23 between the d = 10 Yang-Mills coupling constant λ and κ [2], and the fact that the d = 4 Yang-Mills coupling constants at unification are not much smaller than 1, which implies that b1 = b (y1), the value of b at the inner surface of the thick pipe, is comparable to κ2/9. The value of b1 is fixed by the value of the Yang-Mills fine structure constants in four dimensions at unification, αU = , which will be equal to the value of the QCD fine structure constant at unification, and the magnitude |χ (M6)| of the Euler number of the compact six-manifold M6. For by the generalized Gauss-Bonnet theorem [122], the Euler characteristic, or Euler number, χ (M2n), of an arbitrary smooth 2n-manifold M2n, is given by: gǫν1...ν2nǫ µ1...µ2nR ν1ν2µ1µ2 . . . R ν2n−1ν2n µ2n−1µ2n . (98) Thus defining V (M6) ≡ h, we find from (74), on page 28, that for a smooth compact quotient of CH3, with the standard metric (62), (63), as used in the metric ansatz (94): = −10.3354χ And for a smooth compact quotient of H6, with the metric normalized such that RABCD (h) = hAChBD − hADhBC , as stated after (97), we have: = −8π = −16.5367χ (100) Then on using (45), and reducing (28) to four dimensions, we find [127] that when M6 is a smooth compact quotient of CH3: (4πκ2) 2b61V (M6) 0.2615 |χ (M6)| (101) And for a smooth compact quotient of H6, the same relation is obtained, but with the coefficient 0.2615 replaced by 5 × 0.2615 = 0.1634. The result (101) depends on the factor 1 , in the definition of tr in (28), cancelling with a factor 30, in the ratio of the trace of the square of a generator of SU (3), naturally embedded in E8, in the adjoint of E8, to the trace of the square of the corresponding generator, in the fundamental representation of SU(3). For standard Grand Unification, this follows from the corresponding ratio for generators of SO(16), already derived in subsection 2.1, via the natural embedding SU(3) ⊂ SU(5) ⊂ SO(10) ⊂ SO(16). I will be using a different chain of natural embeddings in Section 5, namely SU (3) ⊂ SU (9) ⊂ E8, but the embedding of SU(3) in E8, by this chain, is equivalent to the embedding of SU(3) in E8, by the above SO(16) chain, as follows from, firstly, the equivalence of the embedding of SU(3) in E8 by the above SO(16) chain, and the embedding of SU(3) in E8 by the chain SU(3) ⊂ SU(5) ⊂ SO(10) ⊂ E6, secondly, the equivalence of the embeddings in E8, of all four SU(3)’s, in the chain SU (3)×SU (3)×SU (3)×SU (3) ⊂ E6×SU (3) ⊂ E8, and thirdly, the equivalence of the embeddings in E8, of any three of the four SU(3)’s in the preceding chain, and the embeddings in E8, of the three SU(3)’s in the chain SU (3)× SU (3)× SU (3) ⊂ SU(9) ⊂ E8. Thus the required relation also holds for the embedding of SU(3) in E8, via the subgroup chain I will be using in Section 5. This result will also be verified directly in Section 5. The subgroup chains just listed all follow simply by identifying appropriate subsets of the roots of E8, in the weight diagram of E8, without the need to project the roots to a subspace, and take linear combinations of roots that coincide after the projection, as required, for example, for embedding SO(n) into SU(n). In Section 5, I shall consider E8 vacuum gauge fields, that break E8 to the Standard Model SU (3)×SU (2)×U (1), in such a way, that the values of the coupling constants, at unification, are approximately equal to the observed values of the Standard Model coupling constants, as evolved in the Standard Model, to around 142 to 166 TeV. How- ever, if κ− 9 is around a TeV, it seems possible that the higher dimensional accelerated unification mechanism of Dienes, Dudas, and Gherghetta (DDG) [128, 129], might perhaps reduce the unification energy to not much larger than a TeV, since, provided |χ (M6)| is not too large, b−11 would also then be not much larger than a TeV. The supersymmetry in the higher dimensions, required for the DDG mechanism to work, would of course automatically be present, since it is only the compactification that breaks the supersymmetry, in the models studied in the present paper. The embed- ding of SU (3)QCD, in E8, will be equivalent to the usual embedding of SU (3)QCD, in conventional Grand Unification, as discussed in the preceding paragraph, so, assuming that the DDG mechanism reduces the unification energy, without altering the unifi- cation value of the coupling constants, I shall provisionally estimate αU as the value of the QCD fine structure constant, α3, as evolved to around 142 to 166 TeV, in the Standard Model, which gives the value: αU ≃ 0.0602 ≃ (102) Robinson and Wilczek [12] calculated the one loop gravitational correction to the renormalization group running of the Standard Model gauge coupling constants in a four-dimensional framework, and found that, within the region of validity of their one loop result, the gravitational correction reduces the magnitudes of the Yang-Mills gauge coupling constants as energies are reached where quantum gravitational effects become significant. However this result is not directly applicable in the present context, where higher dimensional effects and quantum gravitational effects become significant together, so I shall not adjust the provisional estimate (102) for this effect. Pietrykowski [130] found that the Robinson-Wilczek effect is gauge-dependent, and vanishes in a class of gauges different from the gauge choice made by Robinson and Wilczek. Thus when the compact six-manifold is a smooth compact quotient of CH3, we find: ≃ 1.2772 |χ (M6)| (103) And when the compact six-manifold is a smooth compact quotient ofH6, the coefficient 1.2772 is replaced by 1.1809. 2.3.3 The problem of the higher order corrections to Hořava-Witten the- At the inner surface of the thick pipe, where we live, we must necessarily have λ2 ∼ |χ (M6)| b61, up to factors of order 1, where b1 = b (y1), and χ (M6) is the Euler number of the compact six-manifold M6, which is an integer ≤ −1. Equivalently, from the Hořava-Witten relation (45), we must have κ 3 ∼ |χ (M6)| b61, up to factors of order 1. The relation including all factors of order 1 is given in (103), on page 37. This follows from the fact that the Yang-Mills coupling constant, g, in four dimensions, at unification, is given by g2 ∼ λ2|χ(M6)|b6 , up to factors of order 1, and g, which is equal to the QCD coupling constant, at unification, is of order 1. Thus quantum effects must necessarily be relevant, at the inner surface of the thick pipe, and, from (45), also in the bulk, near the inner surface of the thick pipe. One type of quantum effect has already been taken into account, namely the very existence of the supersymmetric Yang-Mills multiplets, which, as discussed above, are required to cancel the one-loop chiral anomalies of the gravitinos, which are chiral on the orbifold fixed-point hyperplanes. In order to consider what other relevant quantum effects may occur, it is necessary to consider how Hořava-Witten theory is defined, beyond the long distance limit. Hořava-Witten theory was formally defined [1, 2] as M -theory on M10 × S1/Z2, where M -theory is an unknown theory in eleven dimensions, whose defining properties are that it is the strong coupling limit of type IIA superstring theory [72], and its low energy limit is supergravity in eleven dimensions. Hořava and Witten suggested that the theory would have a built in short-distance cutoff, but left open the question of whether the supermembrane in eleven dimensions [131, 132] would play a role in the physics of that short-distance cutoff, because at the time, it appeared that, although the supermembrane contained the states of the Cremmer-Julia-Scherk supergravity multiplet in eleven dimensions [133], it could not be consistently quantized. The prob- lem was that, due to supersymmetry, there was no energy cost to deforming the shape of a membrane by drawing “infinitely thin” tubes out from it, even when the zero point oscillations of the thickness of the tubes were taken into account, and the spectrum was therefore continuous [134]. However, the supermembrane has more recently been reinterpreted as a second- quantized theory [135], the idea being that little bubbles of supermembrane, connected to one another by infinitely thin tubes, are like independently moving single particles, with the sums over paths, of the infinitely thin tubes connecting the bubble “parti- cles”, presumably building up the eleven-dimensional analogue of the static Newtonian gravitational forces between them. Moreover, from section 12 of [135], it is possible that the supermembrane mass spectrum (in flat space) corresponds simply to the single particle and multi-particle states of supergravity. It is thus possible that the supermembrane is, in fact, a kind of second quantized version of supergravity in eleven dimensions. If this is correct, there is then no known physical effect to provide the basis for any difference, at the quantum level, between M -theory and supergravity, on a smooth background, in eleven uncompactified dimen- sions, because the classical membrane [136] and five-brane [137] solutions are infinitely massive, on a smooth background, in eleven uncompactified dimensions, and thus do not take part in quantum processes. Now the classical membrane solution of d = 11 supergravity was reinterpreted in [138] as a sourceless solitonic solution, with the singularity at the origin found in [136] being reinterpreted as a coordinate singularity at an event horizon, through which the solution can be continued, although there is a curvature singularity hidden inside the event horizon. And by an argument of Hull and Townsend [139], involving U -duality after toroidal compactification to four dimensions, it is known that M-theory cannot contain a separate fundamental supermembrane, distinct from the solitonic membrane of d = 11 supergravity. This is consistent with the fact that the full dynamics of type IIA superstring theory [13] arises from the solitonic membrane of the CJS theory, on toroidal compactification to ten dimensions. Let us recall how this works [1]. We first recall that, by a generalization of Dirac’s argument [140] for the quantiza- tion of the product of electric charge and the magnetic charge of a magnetic monopole, the tensions T2 and T5, of a solitonic membrane and a solitonic fivebrane, are con- strained quantum mechanically to satisfy [141, 142, 143]: 2κ2T2T5 = 2πn, n ∈ Z (104) Thus there must be a fundamental membrane tension, that is a numerical multiple of 3 , and a fundamental fivebrane tension, that is a numerical multiple of κ− 3 , such that the tensions of all solitonic membranes and solitonic fivebranes are constrained quantum mechanically to be integer multiples of these fundamental tensions. I will confirm below, without reference to fivebranes, that the fundamental membrane tension , when the CJS action is 1 M11 d R + . . . , as in (25) for Hořava- Witten theory in the “upstairs” picture [25, 91]. Now since a solitonic membrane in eleven uncompactified dimensions is infinitely extended and has a nonzero minimum tension, it is infinitely massive, and cannot be produced in any physical process. Solitonic membranes of finite extent do not exist quantum mechanically in eleven uncompactified dimensions, because a membrane of finite extent and tension T2 would contract into a region of size T 2 , the smallest size allowed by the uncertainty principle, which for T2 not smaller than the fundamental membrane tension is comparable to or smaller than the thickness ∼ κ 13T 2 of the membrane [136], so it would look like a lump rather than a membrane. And while such a lump could exist classically as a black hole, it has the wrong geometry to be a source of CIJK , so it cannot carry any charge to stabilize it, as an extreme charged state of nonzero mass, against decay by Hawking radiation [144], so it will not lead to the existence of any massive single particle states in the spectrum of the uncompactified CJS theory. On the other hand, if a solitonic membrane of infinite extent, and tension T2, already exists in the vacuum, then its effective dynamics, at distances ≫ both κ2/9 and the thickness ∼ κ 13T 2 of the membrane, can be studied in terms of collective coordinates, by deriving a worldvolume effective action for the membrane, by the method of Callan, Harvey, and Strominger [145]. The first step is the same as in studying a Kaluza-Klein compactification, treating the dimensions parallel to the worldvolume of the membrane as the “extended” dimensions, and the dimensions perpendicular to the worldvolume of the membrane as the “compact” dimensions. The CJS fields are decomposed into blocks according to which of their tensor indices are parallel to or perpendicular to the membrane, and the spinor index of the gravitino is written as a pair of a two-valued SO (2, 1) spinor index and a sixteen-valued SO (8) spinor index. Then all the fields are expanded in terms of a complete set of states on the “compact” dimensions, with coefficients that depend on position in the “extended” dimensions, or in other words, on position on the membrane worldvolume. The membrane thickness ∼ κ 13T 2 now plays the role of the size of the compact dimensions, and at distances large compared to both κ2/9 and κ 2 , only the massless modes are dynamically significant. The massless modes correspond to the zero modes of the solitonic membrane, which have been studied by Kaplan and Michelson [146]. The solitonic membrane is a BPS solution of the CJS theory, so in accordance with the general analysis of Callan, Harvey, and Strominger, 16 of the 32 supersymmetries of the CJS theory are realized linearly, as supersymmetries of the world-sheet action, and the remaining 16 supersymmetries are realized nonlinearly, as massless fermionic Goldstone modes. Half of the fermionic Goldstone modes vanish on the mass shell, so there are 8 bosonic Goldstone modes, which are the 8 translational zero modes, corresponding to translations of the membrane in the 8 directions perpendicular to the world sheet. Choosing coordinates such that the membrane is in the (1, 2) plane, let xµ, 0 ≤ µ ≤ 2 denote the coordinates on the worldvolume, and ym, 3 ≤ m ≤ 10 denote the coordinates perpendicular to the worldvolume. Then an arbitrary diffeomorphism ym → ym − ξm (y), with infinitesimal parameters ξm (y), generates a zero mode, by: δgIJ = ξ K∂KgIJ + gKJ + gIK (105) δCIJK = ξ L∂LCIJK + CLJK + CILK + CIJL (106) In the right-hand sides, here, gIJ and CIJK are as given by the classical membrane solution: ds2 = Λ− 3ηµνdx µdxν + Λ 3 δmndy mdyn (107) Cµνρ = ± εµνρΛ −1 (108) where Λ = 1 + , (109) ymym, and the membrane thickness, rh, is related to the membrane tension T2 by r6h = , where Ω7 is the volume of the unit seven-sphere S 7. The horizon is located at ρ = 0 in these coordinates. We note that the metric (107) tends to Minkowski space as ρ → ∞. Let ǫK(i) (y) be a set of eight linearly independent vector fields in the eight dimensions perpen- dicular to the membrane, (so ǫ (i) = 0), such that limρ→∞ ǫ (i) = δ i, and such that when any of the ǫK(i) is used as the diffeomorphism parameter, ξ K , in (105) and (106), the corresponding zero modes δgIJ and δCIJK are normalizable, in the sense that −ggIKgJMδgIJδgKM and −ggILgJMgKNδCIJKδCLMN are finite, where the integral extends over the region ρ ≥ 0 outside the horizon. Then when the small fluctuations of gIJ and CIJK are expanded as δgIJ = λ(i) (x) δgIJ ǫ(i), y , δCIJK = λ(i) (x) δCIJK ǫ(i), y (110) where δgIJ ǫ(i), y and δCIJK ǫ(i), y are given by (105) and (106), respectively, with ξK taken as ǫK(i) (y), the corresponding change of the CJS Lagrangian, defined as the integrand of the Hořava-Witten bulk action in the “upstairs” picture, (25), including −g factor, has been calculated by Kaplan and Michelson [146] as: δ2L = (111) (i)∂nǫ (j) + Λ∂mǫ (i)∂nǫ (j) − (i)∂mǫ (j) − (i)∂nǫ ηµν∂µλ(i)∂νλ(j) We see that δ2L vanishes for constant ǫm(i), as expected, due to the global translation invariance of the classical membrane solution, in the directions perpendicular to the membrane. But constant ǫm(i) do not lead to normalizable modes δgIJ ǫ(i), y δCIJK ǫ(i), y . Let us try, instead, ǫm(i) = f (ρ) δ i, where f (ρ) → 0 as ρ → 0, and f (ρ) → 1 as ρ→ ∞. Then (111) becomes: δ2L = 1 Λ,ρf∂ρf + Λ∂ρf∂ρf Λδij∂ρf∂ρf ηµν∂µλ(i)∂νλ(j) (112) Thus after doing the angular integral over the yi, we find: δ2SCJS = − ∂ρf∂ρf d3x ηµν∂µλ(i)∂νλ(i) (113) Thus with this ansatz for the ǫm(i) (y), the coefficient of η µν∂µλ(i)∂νλ(i) in δ 2SCJS is nonzero and has the correct sign. Kaplan and Michelson [146] suggest that the uncer- tainty in the magnitude of the coefficient should be absorbed into the definition of the λ(i). Considering, now, the restrictions on the choice of f (ρ) that result from the re- quirement that the zero modes are normalizable, so that −ggIKgJMδgIJδgKM −ggILgJMgKNδCIJKδCLMN are finite, we note that the ρ integrals will cer- tainly converge at large ρ if f (ρ) tends to 1 rapidly enough as ρ → ∞. While for ρ→ 0, we find from (105) and (106) that −ggµσgντδgµνδgστ and −ggµρgντgσλδCµνσδCρτλ lead to integrals of the form 0 dρρ 3f 2, while −ggjlgkmδgjkδglm leads to integrals of the forms 0 dρρ 3f 2, 0 dρρ 4f∂ρf , and 0 dρρ 5 (∂ρf) Thus we can choose f = for 0 ≤ ρ ≤ L, and f = 1 for ρ ≥ L, where L > 0 and n ≥ 1. Then for L→ 0 and n→ ∞, f (ρ) increases very rapidly from 0 to 1 in a small interval near ρ = 0, and then stays equal to 1 for all larger ρ. In this limit, we see from (105) and (106) that λ(i) can be interpreted as X i, the x-dependent transverse displacement of the membrane. And with this choice of f (ρ), the integral in (113) tends to 1 as L→ 0, so for L→ 0 and n→ ∞, we find from (113) that: δ2SCJS = − d3x ηµν∂µX jδij (114) We now use the fact that the worldbrane effective action is completely determined by its supersymmetries, up to an overall normalization factor. From the general prin- ciples discussed in [147], and the fact that the classical solitonic membrane is a BPS solution that preserves half of the 32 supersymmetries, with the broken supersymme- tries being realized nonlinearly as Goldstone modes, it follows that the worldbrane effective action must be the d = 11 supermembrane action found by Bergshoeff, Sez- gin, and Townsend [131, 132]. However the supermembrane action will be obtained in “static” gauge, as discussed in section 4 of [132], such that Siegel’s κ symmetry [148, 149, 150] has been fixed by the gauge choice Xµ = xµ, where XI are the bosonic coordinates of the supermembrane, and xµ are the coordinates on the worldvolume, as above. Reversing the gauge fixing of the κ symmetry, and allowing a general background satisfying the CJS field equations, the worldvolume effective action of the infinitely extended classical solitonic membrane solution, that describes its dynamics at distances large compared to both κ2/9 and the thickness ∼ κ 13T 2 of the solitonic membrane, is thus the d = 11 supermembrane action of [131]. The bosonic part of the worldvolume action is then [136, 25]: −γγµν∂µXI∂νXJGIJ (X) + 2ǫµνσ∂µX KCIJK (X) , (115) where the sign choice in the third term in (115), called the Wess-Zumino term, is the same as in (108) [136]. Here γµν is the metric on the worldvolume, and XI (x) are the bosonic coordinates of the supermembrane, so that the Xm (x), in (115), should correspond to the X i (x) in (114). We note, however, that there is a factor of 1 discrepancy between (114) and (115), which I shall here leave unresolved. We now note, following [25] and [91], that since CIJK (X) is not gauge-invariant, the worldvolume action with the bosonic part (115) will not lead to a well-defined quantum theory, unless changing the gauge of CIJK (X) can only change the worldvolume action by an integer multiple of 2π. Let us consider a configuration such that XI (x) sweeps out some closed three-dimensional surface S3, as x sweeps out some region v of the worldvolume. Then the requirement that the worldvolume action leads to a well-defined quantum theory implies that if C ′IJK is any gauge transformation of CIJK , the integral 2ǫµνσ∂µX K (CIJK (X)− C ′IJK (X)) = 3! (C − C ′) (116) must be an integer multiple of 2π. Following [91], we now apply this requirement with S3 being the equator of a topological 4-sphere S4, such that C and C ′ are the three- form gauge field on coordinate patches that cover the north and south hemispheres of S4 respectively. Then from Stokes’s theorem, and recalling that the four-form G was defined in section 2.1 by GIJKL = 24∂[ICJKL], so that G = 6dC, it follows that√ S4 G must be an integer multiple of 2π. On the other hand, by an argument of Witten [151], which I review in subsection 2.7, on page 137, the vanishing of the Pontryagin number of S4 implies that is an integer. Thus the smallest possible nonzero value of S4 G is . The requirement that S4 G is an integer multiple of 2π must be satisfied, in particular, for this value of S4 G. Hence we find that , n ∈ Z (117) which is in agreement with [25], as amended by [91]. Now the above argument for identifying the worldvolume effective action of the infinitely extended classical solitonic membrane solution, that describes its dynamics at distances large compared to both κ2/9 and the thickness ∼ κ 13T 2 of the solitonic membrane, as the d = 11 supermembrane action, also applies for compactification on S1, when one dimension of the solitonic membrane wraps the S1 and its other dimension extends infinitely, and for compactification on S1×S1, when the membrane wraps both S1’s, since these are also BPS solutions of the CJS field equations, that preserve 16 of the 32 supersymmetries. Furthermore, because these solutions are BPS, it is expected that the semiclassical quantization of the worldvolume effective action will be exact [152], and will thus be valid even when the circumference of one or both of the S1’s is small compared to κ2/9, which is a strong coupling limit for the CJS theory. The semiclassical quantization of the supermembrane wrapping a two-torus was studied in [153]. Let us now consider compactification of the CJS theory on an S1 of circumference L ≪ κ2/9, such that a solitonic membrane of the minimum tension 1 wraps once around the S1. If we use the same unit of length in ten dimensions as in eleven dimensions, the solitonic membrane now looks like a string of tension 1 while its thickness is ∼ κ2/9, as in eleven dimensions. The argument above for the nonexistence quantum mechanically of solitonic membranes of finite extent in eleven uncompactified dimensions no longer applies, because a string-like solitonic membrane of finite extent would contract into a region of size ∼ α′ ∼ κ , the smallest size allowed by the uncertainty principle, which can be made arbitrarily large compared to the string thickness ∼ κ2/9, by choosing L sufficiently small compared to κ2/9. Furthermore, the analogue of the Planck length, in ten dimensions, is , and α′ can also be made arbitrarily large compared to this, by choosing L sufficiently small compared to κ2/9. Thus the worldvolume effective action of the classical solitonic membrane solution, namely the d = 11 supermembrane action, can be made arbitrarily accurate at distances comparable to or larger than α′, by choosing L sufficiently small compared to κ2/9. Now the compactification of the d = 11 supermembrane action on S1, when the membrane also wraps the S1, was calculated by Duff, Howe, Inami, and Stelle [154], and found to equal the covariant Green-Schwarz action for the superstring [149]. Once the covariant Green-Schwarz action is obtained, the standard spectrum of the type IIA superstrings can be obtained in the light cone gauge [13]. However, we still have to take account of the fact that the worldvolume effective action of the classical solitonic membrane solution has only been related to the d = 11 supermembrane action in the BPS configurations, so each of the two dimensions of the solitonic membrane is either infinitely extended or wraps a compactification S1. So, following Hořava and Witten [1], we consider compactification of the CJS theory on S1 × S1, such that a solitonic membrane of the minimum tension 1 wraps once around each S1. This is a BPS solution of the CJS theory, so we can choose the circumference L of one S1 to be ≪ κ2/9. Then the solitonic membrane now looks like a closed string of tension 1 L, wrapping once around the second S1, whose radius R we choose to be comparable to or larger than . Thus we now obtain the Green-Schwarz action for the closed superstring wrapping once around the second S1. Going to light-cone gauge in the limit L≪ κ2/9, the oscillator degrees of freedom of the closed superstring completely decouple from the wrapping degrees of freedom. If the large S1 is in the X9 direction, then the bosonic coordinate X9 of the superstring has the standard expansion: X9 = x9 + α′p9τ +N9Rσ + i n 6=0 −in(τ−σ) + α̃9ne −in(τ+σ) , (118) where τ and σ, 0 ≤ σ ≤ 2π, are the timelike and spacelike worldsheet coordinates of the closed superstring, p9 = M9 , for some integer M9, and N9 is the number of times the closed superstring wraps the large S1, which is 1 for the configuration in which we have obtained the closed superstring. The oscillators αnµ and α̃ µ, n 6= 0, satisfy αnµ, α = nδm,−nηµν and α̃nµ, α̃ = nδm,−nηµν , because the semiclassical quantization of the BPS solution is exact [152]. And although we have only obtained (118) in the case when N9 = 1, the fact that the oscillators in (118) are completely decoupled from the wrapping degrees of freedom shows that these same oscillators also create the massive single particle states of freely moving superstrings. Applying the same treatment to the fermionic collective coordinates of the solitonic membrane solution, we thus obtain all the massive single-particle superstring states of the type IIA supertring, while the massless single particle states arise from the dimensional reduction of the d = 11 supergravity multiplet. Finally, since α′ ∼ κ when we measure distances in ten dimensions in the same units as in eleven dimensions, we should instead use a unit of length in ten dimensions that is longer than the unit of length used in eleven dimensions by a factor κ , if we want to keep α′, as measured in the new unit of length introduced for ten dimensions, fixed as κ → ∞ with κ fixed. This can be implemented by writing the Kaluza-Klein ansatz for the d = 11 metric as ds2 = gUV dx UdxV + dy2 (119) and interpreting the case where gUV = ηUV as ten-dimensional Minkowski space. Here U and V run from 0 to 9 as in subsection 2.1, and y, the coordinate along the small S1, runs from 0 to L. The metric gUV in (119) is called the string metric [155], because it corresponds to choosing a unit of length, in ten dimensions, with respect to which = (2π2) 9 is independent of L. Now since the Green-Schwarz action, which describes free superstrings, becomes exact in the limit L → 0 with κ fixed, we expect that the string coupling constant λ = eφ, where φ is the dilaton, should tend to zero in this limit. This was demonstrated by Witten [155], by showing that the string metric gUV in (119) is the correct metric to use for comparison with the low energy effective action of the type IIA superstring, written in a standard form such that the kinetic terms for the massless fields from the NS-NS sector include a factor 1 = e−2φ, and the kinetic terms for the massless fields from the RR sector are independent of the dilaton. A dynamical dilaton corresponds to the possibility that L, the circumference of the small S1, can depend on position in the ten large dimensions. To allow for this possibility, we define L = κ2/9eγ , where γ can depend on xU , and y = L ỹ, so that ỹ runs from 0 to κ2/9. The RR vector field AU is the Kaluza-Klein vector field, and allowing also for a possible nonvanishing AU , the d = 11 metric ansatz (119) becomes: ds2 = e−γgUV dx UdxV + e2γ dỹ −AUdxU (120) The massless NS-NS fields are the graviton, the dilaton, and BUV = CUV y, and the other massless RR field, besides AU , is CUVW . Substituting (120) into the CJS action (25), the bosonic kinetic terms in the CJS action then become schematically: R + (∂γ) + |dB|2 + |dA|2 + |dC|2 (121) where this expression shows only the dependence on κ and γ, not the correct numerical coefficients of the terms. Comparison with the low energy effective action of the type IIA superstring, in the standard form described above, then shows that the string coupling constant is given by λ = eφ ∼ e 32γ = L , and thus does, indeed, tend to 0 as L→ 0 with κ fixed. Thus the full dynamics of type IIA superstring theory is already contained in the CJS theory of supergravity in eleven dimensions. But since the defining properties of M-theory are that it is the strong coupling limit of type IIA superstring theory, and its low energy limit is supergravity in eleven dimensions, there is then no detectable differ- ence, on a smooth background, between M-theory, and the CJS theory of supergravity in eleven dimensions. In section 1.2 of [156], Green, Russo, and Vanhove noted that on compactification of d = 11 supergravity on a 2-torus of radii rA and rB, terms of the form e −crB that arise in the string theory 4-graviton amplitude are not reproduced by Feynman diagrams at any number of loops. However for the case rB ≪ κ2/9 that they consider, the d = 11 solitonic membrane wrapping rB can form finite mass solitonic closed strings with mass proportional to rB, that would give terms of this form by propagating as internal lines of the Feynman diagrams. The contribution of these solitonic closed strings to the 4-graviton amplitude could presumably be calculated, for example, by the collective coordinate techniques developed by Gervais, Jevicki, and Sakita [157, 158, 159, 160]. Type IIA superstring theory is thought to be UV complete [161, 162, 163, 164, 165, 166, 167, 168], so apart from the factor of 1 discrepancy between (114) and (115) that I left unresolved, the CJS theory of d = 11 supergravity, with the non- perturbative effects of the classical membrane and 5-brane solutions properly included where appropriate, appears to contain the full dynamics of the UV complete type IIA superstring theory. However the CJS theory has been argued to be UV incomplete [169, 170, 171, 172], on the basis of the existence of the linearized 4-field counterterms of dimensions 8, 12, 14, 16, . . . , constructed by Deser and Seminara [173, 174, 175], which have been proved by Metsaev [176] to be the complete set of linearized 4-field countert- erms, and the existence of an infinite set of counterterms [15] constructed as integrals over the full d = 11 superspace [17, 18], together with a 2-loop dimensional regulariza- tion calculation [169], using the methods developed earlier in [177, 178], which found that the dimension 20 Deser-Seminara linearized 4-field counterterm would occur with an infinite coefficient. Green, Vanhove, Kwon, and Russo have found that the coefficients of some local counterterms of dimensions ≥ 12 in the d = 11 theory are fixed by calculations in the type II superstring theories [179, 180, 181, 156], so the paradox of the UV-incomplete CJS theory containing the full dynamics of the UV-complete type IIA superstring theory cannot be resolved by ambiguities in the UV completion of the CJS theory, which would arise as undetermined coefficients of the Deser-Seminara and superspace counterterms of dimension ≥ 12 in the quantum effective action of the CJS theory, somehow disappearing during the compactification of the CJS theory on a small circle to obtain the type IIA theory. A possible resolution of half of the paradox, that appears to be consistent with all known results, follows from noting that the Noether completion of the Deser-Seminara linearized 4-field invariants, to fully non-linear counterterms, invariant under the full non-linear CJS supersymmetry variations, up to terms which vanish when the CJS field equations are satisfied, and can thus be cancelled by the addition of higher dimension terms to the CJS supersymmetry variations [182, 183], has never been carried out, and with the exception of the unique dimension 8 invariant [176, 24], whose Noether completion must exist, if M-theory is consistent, because it occurs in the quantum effective action of d = 11 supergravity with a non-zero coefficient that is fixed by the tangent bundle anomaly cancellation on five-branes [25, 26, 27, 28, 29], and confirmed by anomaly cancellation in Hořava-Witten theory [92, 86, 184, 185, 186, 187, 28, 188, 71], and by comparison with types IIA and IIB superstring theory [189, 190], it is possible that their Noether completions do not exist. In the case of d = 4, N = 1 supergravity [191, 192], Noether completions were always found to exist, but this follows from the existence of the auxiliary field formu- lations [193, 194, 195, 196, 197, 198]. However for the CJS theory in 11 dimensions, Rivelles and Taylor showed that no similar auxiliary field formulation can exist [199]. An example of an obstruction to Noether completion in 11 dimensions was found by Nicolai, Townsend, and van Nieuwenhuizen, when they tried to construct an analogue of the CJS theory using a 6-form gauge field instead of a 3-form gauge field [200]. A possible resolution to the other half of the paradox would be obtained if the can- didate counterterms constructed as integrals over the full d = 11 superspace [15, 16] all vanished identically, or alternatively, if an obstruction existed that prevented the geometrical transformations in superspace [17, 18] from matching the CJS supersym- metry variations, for a general solution of the CJS field equations, beyond a certain power of θ. The mapping of the component fields and supersymmetry variations of a supersymmetric theory into superspace, such that the geometrical transformations in superspace match the supersymmetry variations of the component fields, is called gauge completion [201, 202], and for the CJS theory, this was initially carried out only to leading order in θ [17]. The first terms beyond leading order in the gauge completion mapping of the CJS theory into superspace were studied by de Wit, Peeters, and Plefka [203], and to consider whether an obstruction to gauge completion appears in their results, I shall temporarily adopt their notation. Thus for the following discussion of [203], coordinate indices µ, ν, ρ, . . . will temporarily run over all eleven bosonic coordinate directions, r and s are bosonic tangent space indices, and α, β, γ, . . . are fermionic coordinate indices. The relations between the normalizations of the fields are ψHWµ = 2ψ CHWµνρ = CdWPPµνρ , and Gµνρσ = Fµνρσ. The first place to look for an obstruction is equation (4.5) of [203], which must be satisfied by the terms proportional to the supercovariant field strength F̂µνρσ = 4∂[µCνρσ] + 12ψ̄[µΓνρψσ], in the conventions of [203], at order θ 2 in the θ expansion of the superspace diffeomorphism parameter. Denoting these terms by ǫβNβ α, where ǫβ is the parameter of a CJS local supersymmetry variation, that is to be matched by a combination of superspace diffeomorphisms, local Lorentz transformations, and possibly also gauge transformations of a superspace three-form superfield, if one is included, equation (4.5) of [203] reads: 2∂βNγ θ̄Γµǫ2 νρσλǫ1 F̂νρσλ − (1 ↔ 2) = (Γrsθ) ǭ2 (Γ νρσλ + 24eν sΓσλ) ǫ1F̂ νρσλ (122) Here ∂β = , and Tµ νρσλ = 1 νρσλ − 8δ[νµ Γρσλ] . This equation is to be satisfied for arbitrary local supersymmetry variation parameters ǫ1 and ǫ2, and is thus a three- index equation for a two-index quantity. Thus it will have no solution, unless the “source” terms satisfy an appropriate integrability condition. In fact, from the identity: ∂ε (∂βNγ α + ∂γNβ α) + ∂β (∂γNε α + ∂εNγ α) + ∂γ (∂εNβ α + ∂βNε α) = 0, (123) which follows from the fact that the spinor derivatives anticommute, we find that integrability of (122) requires that the following expression vanish for arbitrary F̂νρσλ: (Tµνρσλ) νρσλ − 2 (Tµνρσλ) (Tµνρσλ) νρσλ + (Γµκ) Γ0 (Γµκνρσλ + 24gµνgκρΓσλ) F̂ νρσλ (Γµκ) Γ0 (Γµκνρσλ + 24gµνgκρΓσλ) F̂ νρσλ (Γµκ) Γ0 (Γµκνρσλ + 24gµνgκρΓσλ) F̂ νρσλ (124) Thus integrability of (122) requires the following expression, which is antisymmetric in ν, ρ, σ, and λ, and symmetric in ε, β, and γ, to vanish identically: (Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α γ (Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α β (Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α ε + (Γµκ) Γ0 (Γµκνρσλ + 4gµνgκρΓσλ + 4gµσgκλΓνρ + 4gµρgκσΓνλ + 4gµνgκλΓρσ +4gµσgκνΓρλ + 4gµρgκλΓσν))βγ + (Γ Γ0 (Γµκνρσλ + 4gµνgκρΓσλ + 4gµσgκλΓνρ +4gµρgκσΓνλ + 4gµνgκλΓρσ + 4gµσgκνΓρλ + 4gµρgκλΓσν))γε + (Γ Γ0 (Γµκνρσλ +4gµνgκρΓσλ + 4gµσgκλΓνρ + 4gµρgκσΓνλ + 4gµνgκλΓρσ + 4gµσgκνΓρλ + 4gµρgκλΓσν))εβ (125) This expression (125) is the type of expression that might vanish by a Fierz identity. To find out whether or not it vanished, I used the fact, reviewed for example in [84], that for a real representation of the d = 11 Dirac matrices, as assumed here, the 1024 matrices (Γτ1...τn) α, 0 ≤ n ≤ 5, form a complete basis for real 32 × 32 matrices. We can therefore find out whether or not (125) vanishes, by contracting it with a general matrix Xγα, which turns it into an ordinary sum of matrices, with indices εβ or βε, multiplied, in the case of the first five terms and the last seven terms, by a trace, and then taking Xγα to be each of these 1024 matrices in turn. However, due to Lorentz invariance, it is not necessary to take Xγα to be all 1024 of these matrices. Instead, we first note that (125) vanishes by antisymmetry, unless ν, ρ, σ, and λ are all different. Thus it is sufficient to evaluate (125) for a fixed choice of ν, ρ, σ, and λ, all different from each other. I chose ν = 0, ρ = 8, σ = 9, and λ = y, where, as throughout this section, y denotes the tenth spatial direction. We then find, when we choose Xγα to be a matrix (Γτ1...τn) α, for any specific value of n, and any specific values for the indices τ1, τ2, . . . , τn, that each term in (125) is equal to a coefficient, times either the matrix (Γ0Γκ1...κm)εβ or the matrix (Γ 0Γκ1...κm)βε, where κ1, . . . , κm are the indices in {ν, ρ, σ, λ}, that are not in {τ1, τ2, . . . , τn}, and the indices in {τ1, τ2, . . . , τn} that are not in {ν, ρ, σ, λ}, and may be taken in ascending order. Furthermore, due to the symmetry of (125) in β and ε, the result vanishes automat- ically, unless m is one of the numbers for which the matrix (Γ0Γκ1...κm)βε is symmetric, namely 1, 2, 5, 6, 9, and 10. Furthermore, for each value of n, 0 ≤ n ≤ 5, it is sufficient to consider just one choice of the indices {τ1, τ2, . . . , τn} that gives each of these values of m, since if the result vanishes for one choice, it will also vanish for any other choice that gives the same value of m. I chose X to be the ten matrices Γ1, Γ08, Γ12, Γ89y, Γ12y, Γ189y , Γ123y, Γ1089y , Γ1239y , and Γ12345. For each of these ten choices of X , the contraction of (125) with X α was found to vanish. The expression (125) is therefore identically zero, so this potential obstruction to the completion of the gauge completion procedure, at order θ2, in fact vanishes. However, this does not yet imply that there is no obstruction to completion of the gauge completion procedure at order θ2, because the spin-spin components of the supervielbein also contain terms of order F̂ θ2, and these terms are required to satisfy equation (4.9) of [203], which is again a three-index equation for a two-index quantity, and thus will have a nontrivial integrability condition, since it is required to be satisfied for arbitrary supersymmetry variation parameter ǫ. The “source” terms of equation (4.9) of [203] include terms similar in structure, although different in detail, from the source terms in equation (4.5) of [203], reproduced as equation (122) above, and also a term involving the solution Nβ α of equation (4.5) of [203], which cannot be eliminated by use of equation (4.5) of [203], because it does not occur in the combination (∂γNβ α + ∂βNγ α). However Nβ α does occur in the combination (∂γNβ α + ∂βNγ α) in the integrability condition for equation (4.9) of [203], so that integrability condition could be checked by substituting for (∂γNβ α + ∂βNγ α) from equation (4.5) of [203], without actually solving equation (4.5) of [203], but I will not do that in this paper. The evaluation of (125), contracted with each of the ten choices of Xγα listed above, was speeded up by use of the well-known identities [84]: ΓµΓν1...νnΓµ = (−1) (d− 2n) Γν1...νn (126) ΓµσΓν1...νnΓµσ = − (d− 2n)2 − d Γν1...νn (127) valid in d dimensions. For example, to evaluate the term (Γ0Γµκ089yΓ1239yΓ , which arises for the choice X = Γ1239y, we note that we can treat µ and κ here as summed only over the seven dimensions different from 0, 8, 9, and y. So we split Γ1239y as Γ123Γ9y and commute the Γ9y to the right, and, with the understanding that µ and κ are summed only over the range 1 to 7, we also split Γµκ089y as Γ089yΓµκ. We then use the identity (127) above, with d = 7 and n = 3, to obtain: Γ0Γµκ089yΓ1239yΓ (7− 6)2 − 7 Γ0Γ089yΓ123Γ9y Γ0Γ01238 (128) We see from above that already at order θ2, the possibility of mapping the CJS theory into superspace such that the geometrical transformations in superspace match the CJS supersymmetry variations, for a general solution of the CJS field equations, requires that nontrivial integrability conditions be satisfied. Thus it is not possible to conclude, from the construction of a counterterm in standard d = 11 superspace, that there exists a corresponding higher derivative term, local in the CJS component fields, whose variation under the CJS supersymmetry transformations is a total derivative when the CJS field equations are satisfied, without explicitly checking that there are no nonvanishing obstructions to the gauge completion mapping of the CJS theory into superspace, up to the highest power of θ that occurs in the superspace counterterm. For the Duff-Toms superspace counterterms [15] that would be θ32. In the pure spinor framework of Berkovits [204, 205], there are superspace invariants involving an inte- gration over only nine components of θ, but it would be necessary to check that there are no nonvanishing obstructions when the pure spinor constraint is satisfied, at least through order θ9. Turning now to the occurrence of fractional powers of κ, in the expansion of the quantum effective action Γ, Γ is formally given by an expansion in powers of κ2, starting with the classical action, of order κ−2, followed by the one-loop term, which is formally independent of κ. However, it is inevitable that other powers of κ will occur, especially if Γ, which is a non-local functional of the fields, is developed in a low energy expansion, as a series of local terms, with increasing numbers of derivatives on the fields. Indeed, the bulk Green-Schwarz term, mentioned at the end of subsection 2.1, which occurs in such an expansion, is a sum of terms formed from a three-form gauge field, and four Riemann tensors, with their indices contracted in various ways, using the metric, and one antisymmetric eleven-tensor, and is of order κ− 3 . As already noted, if there had been a built-in short distance cutoff, of order κ2/9, then such fractional powers of κ would have been interpreted as arising from powers of the short distance cutoff. But we now need to understand where they come from, when there is no short distance cutoff. Figure 1 shows a typical Feynman diagram, in the loop expansion of Γ, that can contribute to the bulk Green-Schwarz term. It has a three-form gauge field prop- agating in the loop, and the CI1J1K1 vertex comes from the Cremmer-Julia-Scherk Chern-Simons term in (25), while the ϕIiJi vertices come from the three-form gauge field kinetic term, with the metric expanded as GIJ = ηIJ +ϕIJ . Each propagator has two derivatives acting on it, one from the vertex at each end of it, so that, for purposes of power counting, the line between two neighbouring vertices, say xi and xj , behaves as |xi − xj |−11. On the basis of power counting, there is a logarithmic divergence when- ever any n consecutive vertices, such that 2 ≤ n ≤ 4, cluster together, but the position space integral is in fact conditionally convergent in these regions, and these apparent divergences, associated with tree subdiagrams, can be dealt with by the method used ϕI2J2 ϕI3J3ϕI4J4 ϕI5J5 CI1J1K1 Figure 1: A pentagon contributing to the bulk Green-Schwarz term to prove Theorem 2 of [206]. However, the diagram as a whole has degree of diver- gence 11, so that, if we choose the three-form gauge field vertex, x1, as the contraction point of the diagram, then, in the BPHZ framework [207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218], we have to subtract a counterterm, which, in this instance, has the form of the “internal function” of the diagram, namely the propagators, with the derivatives acting on them out of the vertices, times the terms, of degree up to and including degree 11, of the Taylor expansion of the “external function” of the di- agram, namely the function CI1J1K1 (x1)ϕI2J2 (x2)ϕI3J3 (x3)ϕI4J4 (x4)ϕI5J5 (x5), about the point (x1, x2, x3, x4, x5) = (x1, x1, x1, x1, x1). After integrating over x2, x3, x4, and x5, this counterterm includes terms with the structure of the bulk Green-Schwarz term, although with a divergent coefficient, as well as many other terms. Let us now consider a term of degree 11 in this counterterm, which has a total of eleven derivatives acting on ϕI2J2 (x1)ϕI3J3 (x1)ϕI4J4 (x1)ϕI5J5 (x1), and a total of eleven factors, in the counterterm integrand, of the form (xr − x1)Ir , where the index r runs from 1 to 11, and xr is one of x2, x3, x4, or x5. Suppose we now integrate over the vertex positions, in the sequence x2, x3, x4, then x5. We see that, when we come to integrate over the position of x5, the counterterm has an uncancelled logarithmic divergence at large distances, in consequence of the masslessness of the propagators. There was no such large distance divergence at all, in the original diagram, if the classical fields, CIJK , and ϕIJ , are assumed to fall off sufficiently rapidly, at large distances. Such large distance divergences, occurring in BPHZ counterterms, but not in the original diagrams, are a well-known problem of BPHZ renormalization, when there are massless particles. Traditionally, the problem has beeen dealt with by the BPHZL method [219, 220, 221, 222], which involves the introduction of regulator masses for the massless particles, performing additional infra-red subtractions, in addition to the short-distance subtractions, then letting the regulator masses tend to zero, at the end of the calculation. An alternative method was presented in [206], where a generalized BPHZ convergence proof was presented, in Euclidean position space, that allowed the propagators in the counterterms to differ, at large distances, from the propagators in the original diagram, without altering the propagators in the uncontracted diagram. This enables massless propagators, in the counterterms, to be cut off smoothly, at large distances, so that the large distance divergences are eliminated from the counterterms, without spoiling the convergence proof, and without altering the propagators in the uncontracted diagram. The proof in [206] applies only in Euclidean signature position space, but it seems plausible that Hepp’s convergence proof [214], which can be ap- plied in Minkowski signature, could be generalized in an analogous way, allowing the parameter integrals, of the exponentiated propagators, to be cut off at large values of the exponentiation parameters, in the counterterms, without altering them in the terms coming from the uncontracted diagram. When this method is used for a theory such as massless QCD, with no dimensional parameters in the classical action, the distance at which the smooth long distance cutoffs of the propagators in the counterterms begin, becomes the distance that provides the basis for dimensional transmutation [223]. In the case of supergravity in eleven dimensions, the classical action has precisely one parameter with the dimension of length, namely κ 9 , so the distance, at which the smooth long distance cutoffs of the propagators in the counterterms begin, will be a numerical multiple of κ Now the convergence proof in [206] assumed that the same modified propagators, differing at long distances from the propagators in the terms coming from the uncon- tracted diagram, are used in all the terms of the Taylor expansions that occur in the counterterms, so we will also be using these same modified propagators, with a long distance cutoff commencing at some fixed numerical multiple of κ 9 , in those terms in the Taylor expansions in the counterterms, where this is not actually needed, to ensure convergence at large distances. However, there is not, a priori, any reason to choose any particular numerical multiple of κ 9 , as the distance at which the smooth long distance cutoffs of the propagators in the counterterms begin, and if we choose a different numerical multiple of κ 9 , the result will change by the addition of local finite counterterms, whose coefficients will involve powers of κ, as determined by dimensional analysis. In particular, the term with the structure of the bulk Green-Schwarz term, which contains eight derivatives, will include a factor of κ− 3 . Then, when we require that the Slavnov-Taylor identities [111, 112, 113], which follow from local supersym- metry, in the BRST-BV framework [114, 115, 116, 117, 118, 119, 120], are satisfied, and impose appropriate gauge-fixing conditions, and use the freedom to redefine the fields, in order to set to zero the coefficients of terms that vanish, when the classical field equations are satisified, the coefficients of the possible finite counterterms will be fixed, up to the addition of linear combinations of terms, that correspond to nontrivial locally supersymmetric higher-derivative deformations, of the CJS theory. And as I discussed above, it is possible, and consistent with all known results, that the only non-trivial higher-derivative deformation of the CJS theory, that is locally supersymmetric at the full non-linear level, might be the deformation whose lowest- dimension term is the unique dimension-8 CJS on-shell invariant [176, 24] that contains the bulk Green-Schwarz term. The numerical coefficient of the unique dimension-8 CJS on-shell invariant [176, 24], in the quantum effective action of d = 11 supergravity, is fixed by the tangent bundle anomaly cancellation on five-branes [25, 26, 27, 28, 29], and confirmed by anomaly cancellation in Hořava-Witten theory [92, 86, 184, 185, 186, 187, 28, 188, 71], and by comparison with types IIA and IIB superstring theory [189, 190]. This in turn depends on the Dirac quantization of the two-brane and five-brane tensions [140, 141, 142, 143, 224, 25, 225, 226, 91, 185]. Thus if the unique dimension-8 CJS on-shell invariant is the only non-trivial higher- derivative CJS on-shell invariant that is locally supersymmetric at the full non-linear level, it might be possible to calculate the predictions of the CJS theory and Hořava- Witten theory in the framework of effective field theory, without the occurrence of undetermined parameters connected with the short distance completion of the theory. 2.3.4 The Casimir energy density corrections to the energy-momentum tensor Having now considered some of the problems involved in the definition of Hořava- Witten theory, or more specifically, the bulk M -theory aspect of it, beyond the long- wavelength limit, I shall now consider the Casimir-type effects resulting from the com- pactification on the compact six-manifold. The Casimir corrections to the energy-momentum tensor, in Einstein’s equations, arise from the variation of the one loop, and higher loop terms, in the quantum ef- fective action, Γ, with respect to the classical metric, GIJ . In general, these terms give corrections to the classical Einstein equations, that are non-local functionals of the classical metric, GIJ . However, for a given classical metric GIJ , the Casimir terms in the energy-momentum tensor will be specific functions of position. We can there- fore adopt an iterative approach to solving the quantum-corrected Einstein equations, calculating the Casimir terms in a trial classical metric GIJ , then solving the Einstein equations with these Casimir terms, and if the resulting “output” metric differs from the “input” metric, repeating the process with an improved “input” metric, until agree- ment is reached. This method will be used, at the level of rough order of magnitude estimates, in subsection 2.4.2, on page 93. The classical metric GIJ will not, in general, be a solution of the classical field equations, in regions where the Casimir corrections to the energy-momentum tensor are significant. Nevertheless the gauge-fixed quantum effective action, Γ, is still well defined, up to possible ultraviolet divergences, as the generating functional of proper vertices [125, 126, 87, 88]. Moreover it can be calculated, for a classical action A (ϕ), and for an arbitrary classical field configuration Φ, as the sum of all the one-line- irreducible vacuum diagrams, calculated from the action A (Φ + ϕ), with the term linear in ϕ deleted, where ϕ denotes the quantum fields. In other words, using DeWitt’s compact index notation [227], where a single index, i, runs over all combinations of type of field, space-time position, and coordinate and other indices, the quantum effective action, as a function of the classical fields, Φ, is given by the sum of all the one line irreducible vacuum diagrams, calculated with the action: A (Φ + ϕ)− ϕi δA (Φ) (129) where the summation convention is applied to the index i. The derivation of this result is reviewed in section 4, on page 191. I shall look for solutions such that all physical quantities are covariantly constant in directions tangential to the four observed dimensions, which is consistent with the choice of the de Sitter metric for the four observed dimensions, in the metric ansatz (94). The compactification of CH3 or H6 to the compact six-manifold M6 usually breaks the homogeneity of the hyperbolic space, so the Casimir terms in the energy- momentum tensor will not, in general, be covariantly constant in directions tangential toM6. Furthermore, in the models considered in section 5, on page 222, there are topo- logically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe, with non-vanishing field strengths, whose contributions to the energy-momentum tensor ex- plicitly break covariant constancy in directions tangential to M6. However, following Lukas, Ovrut, and Waldram [67], we can introduce a harmonic expansion on the com- pact six-manifold. I shall work throughout this section at the level of the leading term in such a harmonic expansion of the energy-momentum tensor, which I shall assume has the form: Tµν = t (1)(y)Gµν , TAB = t (2)(y)GAB, Tyy = t (3)(y) (130) Using the expressions (95), on page 34, for the non-vanishing Christoffel symbols of the second kind, the conservation equation, DIT IJ = 0, for the energy-momentum tensor, now reduces to: 0 = DIT Iy = ∂yT Γµµy + Γ T yy + ΓyµνT µν + Γ = ∂yt (3) + t(3) − 4 ȧ t(1) − 6 ḃ t(2) (131) We will find that for thick pipe geometries that realize TeV-scale gravity by the ADD mechanism [3, 5], the energy-momentum tensor, including the contributions of the four-form field strength GIJKL of the three-form gauge field, is negligible in the main part of the bulk, well away from the boundaries. Thus the Einstein equations in the main part of the bulk will, indeed, be consistent with all physical quantities being covariantly constant on M6. We note that when M6 is a smooth compact quotient of 3, there will be h1,1− 1 additional harmonic (1, 1)-forms on M6 besides the Kähler form, but only the Kähler form will be covariantly constant. The Kähler moduli do not correspond to massless modes because, just as for any Kähler-Einstein metric with a nonvanishing Ricci scalar, each Kähler modulus is equal to a fixed multiple of the corresponding element of the first Chern class. It seems reasonable to expect that the effects of the higher harmonics in the Lukas-Ovrut-Waldram harmonic expansion will decrease rapidly relative to the effects of the leading harmonic, as the distance from the nearest boundary increases, so that the effects of the higher harmonics will not be significant, in the main part of the bulk. The functions t(i) (y) in (130) will be significant near the inner surface of the thick pipe, where b (y) is ∼ κ2/9. I shall consider three alternative ways in which the outer surface of the thick pipe might be stabilized, consistent with the observed value (20) of the effective d = 4 cosmological constant, and in one of the three alternatives, a (y) is ∼ κ2/9 near the outer surface, so in that case, which is studied in subsection 2.6, on page 120, the t(i) (y) will also be significant near the outer surface. To calculate the quantum effective action Γ, and the functions t(i) (y), for a par- ticular classical metric (94), the propagators and heat kernels for the d = 11 super- gravity fields, and also for the Fadeev-Popov ghosts [228, 229, 230, 231], the ghosts for ghosts for the three-form gauge field [232, 233, 123], and possible Nielsen-Kallosh ghosts [234, 235], are needed for that metric. These can be obtained from the corre- sponding propagators and heat kernels on an uncompactified CH3 or H6 background, as appropriate, with the same a (y) and b (y), by the sum over images method of Müller, Fagundes, and Opher [236, 237, 238], provided the sum over images converges. For the case of a massless scalar, the sum over images marginally converges when the action of the massless scalar is as simple as possible, with no “conformal improve- ment” term, but diverges exponentially, due to the exponential growth of volume with distance, when a “conformal improvement term” is added to the action, to make the classical energy-momentum tensor traceless. If the sum over images diverges for any of the required propagators or heat kernels, it might be possible to obtain the result by a resummation method [239, 240], or a theta function method [241, 242, 243]. The propagators and heat kernels on a flat R5 times uncompactified CH3 or H6 background can be obtained from the corresponding propagators and heat kernels on a flat R5 times CP3 or S6 background, which can be calculated by using the Salam- Strathdee harmonic expansion method [244], and summing the expansions by means of a generating function. This calculation is currently in progress for CH3, and the scalar heat kernel on CH3, obtained by this method, is presented in subsection 4.1, on page 196. The leading terms at short distances in the propagators and heat kernels have been calculated for all the relevant fields on general smooth backgrounds by Burgess and Hoover [245, 246], using the heat kernel expansion [247, 227]. Casimir effects for compactification on hyperbolic quotients have also been studied in [248, 249]. Considering, now, the form of the functions t(i) (y) near the inner surface of the thick pipe, where b ∼ κ2/9, we note that the low energy expansion of the M-theory quantum effective action, Γ, is known to contain local terms formed from the Rie- mann tensor and its covariant derivatives. The first such term is formed from four Riemann tensors, and usually referred to as the t8t8R 4 term [189, 190, 250, 179, 251, 182, 180, 183, 181, 252], where tI1I2J1J2K1K2L1L28 denotes the tensor obtained from −6gI1J2gJ1I2gK1L2gL1K2 + 24gI1J2gJ1K2gK1L2gL1I2 by antisymmetrizing under I1 ⇀↽ I2, and symmetrizing under all permutations of (I, J,K, L), with total weight one. Recalling the definition (14), on page 13, of the energy-momentum tensor, and looking at the Riemann tensor components (96), on page 34, for the metric ansatz (94), we see that near the inner surface of the thick pipe, the t8t8R 4 term will result in terms in the t(i) functions that are numerical multiples of κ , where the origin of the non-integer power of κ, in the framework of effective field theory, was explained in the preceding subsection, and there will also be terms where κ is multiplied by up to four powers of ḃ2 or bb̈. We will find in subsection 2.3.8, on page 77, that the vacuum configurations of the three-form gauge field CIJK , that result, due to the Hořava-Witten modified Bianchi identity (42), on page 21, from the presence of topologically stabilized vac- uum Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, with non-vanishing field strengths tangential to the compact six-manifoldM6, also produce terms in the t(i) functions that are numerical multiples of κ , and in this case, there are no additional terms involving derivatives of b with respect to y. Calculations of Casimir energy effects often make use of the proximity force ap- proximation [253, 254], which in the present case would correspond to treating b as independent of y, so that all terms with factors of ḃ, b̈, or higher derivatives of b with respect to y, could be neglected. Thus in this approximation the y direction would effectively be uncompactified, so that t(3) would be equal to t(1), and t(1) and t(2) would correspond to d = 11 supergravity on flat R5 times M6. In this case, the first terms dependent on the topology of M6 would be the one-loop contributions from the terms in the sum over images other than the identity term. None of these terms contain short-distance divergences, so their contributions to Γ are independent of κ. The cor- responding terms in the t(i) functions are thus numerical multiples of 1 Several different indications have been found [189, 190, 250, 179, 180, 181, 255, 256], that suggest that the canonical dimensions of non-vanishing terms, in the low- energy expansion of the M -theory quantum effective action Γ, in eleven uncompactified dimensions, will have the form 2 (3k + 1), for integer k, or in other words, 2, 8, 14, . . . . In that case, the next powers of b, whose coefficients, in the t(i) functions, can get contributions from local terms in the low-energy expansion of the quantum effective action, in the context of the proximity force approximation, will be b−14 and b−20. Neither of these terms would be expected to get contributions at one loop, but both could get contributions at two loops. It thus seems reasonable to assume that, within the context of the proximity force approximation, the functions t(i), near the inner surface of the thick pipe, have expan- sions of the form: t(i) = C + . . . (132) where the C(i)n are numerical constants, that depend only on the topology and spin structure of the compact six-manifold M6, and on the topologically stabilized configu- rations of the Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, and of the three-form gauge field CIJK in the bulk. The conservation equation (131) takes a particularly simple form, near the inner surface of the thick pipe, when t(3) = t(1), and the t(i) depend only on b, as in the context of the proximity force approximation. Specifically, when t(3) = t(1): dt(1) t(1) − 6 t(2) = 0 (133) Hence, in this case: C(2)n = − (2 + 3n) C(1)n , C n = C n n ≥ 0 (134) For n = 1, this implies that, in the context of the proximity force approximation, the topology dependent part of the one-loop Casimir energy-momentum tensor, which is the b−11 terms in (132), is traceless [257]. This is presumably connected with the formal relation between the trace of the energy-momentum tensor, and the divergence of the “dilation current”, and the fact that the b−11 terms in (132) are independent of κ. The limitations of the proximity force approximation are discussed, for example, in subsection 4.3 of [258]. In the present case, the proximity force approximation would not be valid unless ḃ, bb̈, and similar dimensionless quantities formed from b and its higher derivatives with respect to y, all had magnitude small compared to 1, and we will find in subsection 2.4, on page 86, that this is not the case. It will therefore be necessary to go beyond the proximity force approximation, as I will discuss in subsection 2.4.1, on page 90. However, for a given trial classical metric GIJ , and in the approximation of dropping all but the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of TIJ on M6, we can still assume that the t(i) functions have an expansion of the form (132) near the inner surface of the thick pipe, provided that b depends monotonically on y in this region, except that other powers of b, not included in (132), may occur, and we have to check that when the boundary conditions are satisfied, the “input” t(i) functions lead self-consistently to a metric that results in “output” t(i) functions equal to the “input” t(i) functions. Considering, now, the energy-momentum tensor on the Hořava-Witten orbifold fixed-point hyperplanes, let T̃ [i]UV , i = 1, 2, be defined by (14), on page 13, with (SSM + SDM) replaced by the boundary action at y = yi, and the metric gµν replaced by the induced metric, GUV , on the boundary at y = yi. This is a change of notation from earlier sections, where the fixed-point hyperplanes were distinguished by a superscript in round parentheses. Then in the approximation of dropping all but the leading terms in the Lukas-Ovrut-Waldram harmonic expansions of the T̃ [i]UV on M6, I shall assume that the T̃ UV have the block diagonal structure: T̃ [i]µν = t̃ [i](1)Gµν , T̃ AB = t̃ [i](2)GAB (135) The coefficients t̃[1](j) will receive contributions that are numerical multiples of κ from the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of the energy- momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe. It would seem reasonable to expect these contributions to be roughly a positive numerical multiple of the energy-momentum tensor that results from embedding the spin connection in the gauge group for CH3, which will be calculated in subsection 2.3.9, on page 80. The Lovelock-Gauss-Bonnet terms in the quantum effective action on the Hořava-Witten orbifold hyperplanes, discussed in connection with (48), on page 23, also result in terms in the t̃[1](j) coefficients that are numerical multiples of κ , which will also be calculated in subsection 2.3.9. Thus by analogy with (132), I shall assume that within the context of the proximity force approximation, the coefficients t̃[1](i) can be expanded as: t̃[1](i) = D + . . . (136) where the D(i)n are numerical constants, that depend only on the topology and spin structure of M6, and on the topologically stabilized vacuum configurations of the Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, and at higher orders, on the vacuum configuration of the three-form gauge field in the bulk. We note that in consequence of the Hořava-Witten relation (45), on page 21, between κ, and the Yang- Mills coupling constant λ on the orbifold hyperplanes, the expansion (136) is equivalent to an expansion in integer powers of λ. For the solutions I shall consider in subsection 2.6, on page 120, where a (y) becomes as small as κ2/9 at the outer surface of the thick pipe, and the three observed spatial dimensions are assumed to be compactified to a smooth compact quotient M3 of H3, the expansion analogous to (132) is t(i) = C̃ + . . . , (137) and the expansion analogous to (136) is t̃[2](i) = D̃ + . . . . (138) The situation where a (y) becomes as small as κ 9 at the outer surface differs from the situation near the inner surface, in that one of the four dimensions scaled by a (y) is the time dimension, and only the three spatial dimensions scaled by a (y) are assumed to be compactified. The compactification of the three observed spatial dimensions to M3 breaks d = 4 Lorentz invariance globally, although not locally, so the Casimir effects near the outer surface will not, in general, be Lorentz invariant. Thus for the solutions where a (y) becomes as small as κ2/9 at the outer surface, the Gµν form of Tµν , in (130), would in general have to be replaced, near the outer surface, by a more general Robertson-Walker form, and the Gµν form of T̃ µν , in (135), would also have to be replaced by a Robertson-Walker form. However in subsection 2.6 of this paper, I shall consider the case where the Casimir effects near the outer surface are, to sufficient accuracy, consistent with (130) and (135). The coefficients C̃(i)n in (137) and D̃(i)n in (138) are then numerical constants that depend only on the topology and spin structure of M3. 2.3.5 The orders of perturbation theory that the terms in the Casimir energy densities occur at We recall that in subsection 2.3.3, on page 38, we defined the homogeneity number, of a local monomial in the CJS fields and their derivatives, to be the number of derivatives, plus half the number of gravitinos. Let us now extend this definition to an arbitrary product of the CJS fields and their derivatives, not necessarily all at the same point, and denote the homogeneity number by h. Then the overall degree of divergence of an L loop Feynman diagram contributing to a term in the quantum effective action, or in other words, the generating function of proper vertices, in eleven dimensions, corresponding to a product of the CJS fields and their derivatives, with homogeneity number h, is 9b+ 10f + (2− 11) v0 + (1− 11) v1 − 11v2 + 11−N = 9L+ 2− h, (139) where in the left-hand side of (139), b denotes the number of boson propagators, f denotes the number of fermion propagators, vq denotes the number of vertices with 2q fermion legs, and N denotes the number of derivatives acting on the CJS fields, which are here the “background” fields, and we noted that the number of fermion propagator ends is 2f = 2v1+4v2−F , where F is the number of gravitinos among the “background” fields, and L = b+ f + 1− v0 − v1 − v2. The maximum power of κ that can occur for an L loop Feynman diagram contribut- ing to the quantum effective action, in eleven dimensions, is 2 for each propagator, minus 2 for each vertex, hence 2 (L− 1). However, as discussed in the second part of subsection 2.3.3, starting around page 54, when we use BPHZ renormalization, with propagators in the counterterms that differ from the propagators in the direct terms, by being cut off at large distances, as allowed by the convergence proofs in [206], so as to avoid the occurrence of divergences at large distances in the BPHZ counterterms, due to the presence of massless particles, terms involving lower powers of κ also arise naturally at L loops. Specifically, according to the prescription in [206], the same modification of the propagator, at long distances, is used in all the internal lines of a counterterm part. Since a unit of distance, namely κ2/9, occurs in the CJS action (25), it is natural to cut off the propagators, in the counterterms, at distances greater than κ2/9, where the numerical multiple of κ2/9, at which the cutoff occurs, is likely to get modified later, in effect, when finite counterterms are added so as to satisfy Slavnov-Taylor identities. The position-space integral for the BPHZ counterterm that has p extra derivatives acting on the CJS “background” fields, and contributes to cancelling the short-distance divergence of a direct term of overall ultraviolet degree of divergence D, where 0 ≤ p ≤ D, then has the schematic form ∫ κ2/9 xpdx , where the divergence at small x cancels against a corresponding ultraviolet divergence in the direct terms. Thus this integral gives ∼ κ 29 (p−D), which is a power ≤ 0 of κ. The CJS fields, mostly at separated points in the direct term, are now collected into a local monomial, in the CJS fields and their derivatives, at a single point, in the counterterm, whose homogeneity number is hf ≡ h+ p. The total power of κ, including the overall factor κ2(L−1), is κ2(L−1)κ (p−D) = κ 9(hf−11), (140) by (139). This is the correct power of κ to multiply a local monomial, in the CJS fields and their derivatives, of homogeneity number hf , in order for the quantum effective action to be dimensionless. Thus we see that the terms of index n in the expansions (132), namely C(i)n (n−1) b8+3n 9 C(i)n )8+3n , first occur at a number of loops L, where L is the smallest integer ≥ n+2 , and we also find the corresponding conclusion, for the terms in the expansions (137). Now for the local terms in the low energy expansion of the quantum effective action, such as terms formed from products of Riemann tensors, possibly with covariant derivatives acting on them, and their indices contracted in various ways, a number of indications have been found that, for at least some terms, their coefficients, which will be independent of the topology of the background field configuration, do not receive any further modifications, beyond certain finite orders of perturbation theory [25, 89, 189, 190, 250, 179, 180, 181, 252]. However, for smooth compact quotients of CH3 or H6, the coefficients in (132) and (137) also receive nonlocal contributions, for example via the sums over images in the propagators, if these converge, so we would expect the coefficients C(i)n and C̃ n to receive contributions from all loop orders L such that L ≥ n+2 Considering, now, the terms of index m in the expansions (136), namely D(i)m b7+3m an analogous argument, using power counting as appropriate for Feynman diagrams in ten dimensions, indicates that D(i)m first receives contributions at a number of loops L, where L is the smallest integer ≥ m+1 , with a corresponding conclusion, for the coefficients D̃(i)m in (138). However, Hořava-Witten theory is fundamentally defined in eleven dimensions, and from the Hořava-Witten relation (45), we see that λ(m−1) is a numerical multiple of κ (m−1), so by analogy with the bulk case, it seems likely that the coefficients D(i)m and D̃ m will, in fact, receive contributions from all loop orders L such that L ≥ m+2 . For m = −1, this is in agreement with the fact that, in Hořava-Witten theory, the Yang-Mills actions, on the orbifold ten-manifolds, first arise as a one-loop effect, while for m ≥ 0, it gives an onset value of L that is less than or equal to that given by the “d = 10” estimate. 2.3.6 The expansion parameter Now we found in subsection 2.3.2, on page 35, that for a reasonable estimate, (102), on page 37, of the d = 4 Yang-Mills fine structure constant at unification, the value b1 = b (y1) of b (y), at the inner surface of the thick pipe, is related to |χ (M6)|, the magnitude of the Euler number of the compact six-manifold, by (103), on page 37, which states that b1 ≃ 1.28 |χ(M6)| , when M6 is a smooth compact quotient of CH3, and we also find that b1 ≃ 1.18 |χ(M6)| , when M6 is a smooth compact quotient of H6. Thus to find out whether a particular value of |χ (M6)| is possible, and indeed, whether |χ (M6)| ≥ 1 is possible, we need to know whether the expansions (132), for the bulk Casimir energy density coefficients near the inner surface of the thick pipe, and the expansions (136), for the Casimir energy density coefficients on the inner surface of the thick pipe, allow b1 to be as small as the value given by (103), for that value of |χ (M6)|, or whether the expansions (132) and (136) already become infinite, for a value of b larger than that value of b1. We assume that the expansion coefficients in (132), from C 1 onwards, and the expansion coefficients in (136), from D 1 onwards, depend on the topology of M6, and in particular, that their signs depend on the topology of M6. Kenneth and Klich [259] and Bachas [260] have recently discovered that Casimir forces are always attractive in certain circumstances, but their result does not apply in the present context because M6 has no shape moduli, so that regions of M6 cannot be moved closer together without also being squeezed at the same time. Now we know that the Casimir energy densities have local contributions, indepen- dent of the topology of M6, such as the terms quartic in the Riemann tensor, discussed in [251, 182, 183], that would contribute terms C in (132), and the terms on the boundaries, quadratic in the Riemann tensor, discussed in [67], and mentioned in sub- section 2.1 above, in connection with equation (48), on page 23, that would contribute terms D in (136), which will be calculated in (188), on page 85, when M6 is a smooth compact quotient ofCH3, and in (192), on page 85, when M6 is a smooth com- pact quotient ofH6. There will also be local terms built from more covariant derivatives and powers of the Riemann tensor [189, 190, 250, 179, 180, 181, 252, 255, 256], that will contribute to the C(i)n and D n with larger n. Thus for the phenomenological estimates in this paper, I shall assume that the signs of the C(i)n and D n , n ≥ 1, depend on the topology of M6, and that their magnitudes can depend on the topology of M6 though a factor of order 1, but that, apart from this factor of order 1, the magnitudes of the C(i)n and D n , n ≥ 1, are determined by their typical values, for a geometry of roughly constant curvature. We therefore need to know what those typical values are. According to Giudice, Rattazzi, and Wells (GRW) [11], the expansion parameter for graviton loop corrections in D dimensions, in the sense that perturbation theory is reliable when the expansion parameter is less than 1, is 2(2π)D , where SD−1 = Γ(D2 ) is the (D − 1)-volume of a unit radius SD−1, E is the relevant energy of the process, and MD is defined such that the Einstein equation, in D dimensions, is RAB − 12gABR = − (2π)(D−4) TAB. Thus from (25), with replaced by 2 , so as to work in the downstairs picture, we find that for Hořava-Witten theory, 2 9M11 = 2π and the GRW estimate of the expansion parameter for graviton loop corrections is 1890 (2π) 0.0304κ2/9E (141) Considering, now, the value of E that would apply for the expansions (132) and (136), we note, from the discussion after (88), on page 31, that with the metric (62), (63), the sectional curvature of CH3, at each point of CH3, lies in the range −2 to −1 with the actual value depending on the choice of the two-dimensional section through the point, so that the magnitude of the corresponding “radius of curvature” lies in the range 1√ 2. Thus when M6 is a smooth compact quotient of CH3, its “radius of curvature”, at the inner surface of the thick pipe, lies in the range 1√ b1 to 2b1. And if M6 is a smooth compact quotient of H6, and hAB is in that case normalized so that RABCD (h) = hAChBD − hADhBC , as assumed after (188), on page 85, then its “radius of curvature”, at the inner surface of the thick pipe, has the fixed value b1. Thus for both cases, it is reasonable to take b1 as the typical “radius of curvature” of M6, at the inner surface of the thick pipe. Now for the related cases of CP3 and S6, b1 would be the actual radius of curvature, so the corresponding “wavelength” would be λ = 2πb1, and the corresponding energy would be E = 2π . Thus if we also use this estimate of E for the cases of smooth compact quotients of CH3 and H6, the minimum value of b1 , allowed by the requirement that the GRW estimate of the expansion parameter be ≤ 1, would be: ≃ 0.03 (142) which by (103), implies that |χ (M6)| could not be larger than around 6× 109. On the other hand, since E occurs in the combination E in (141), it seems possible that the appropriate value of E should, in fact, be 2π , in which case the minimum value of b1 , allowed by the requirement that the GRW estimate of the expansion parameter be ≤ 1, would be: ≃ 0.2 (143) which by (103), implies that |χ (M6)| could not be larger than around 7× 104. As a first check of the GRW estimate of the expansion parameter, we note that, for D = 4, their estimate of the expansion parameter becomes GN E2, where GN is Newton’s constant, (12). Looking now at Donoghue and Torma’s formula for the one- loop graviton-graviton scattering cross section in D = 4, equation (29) in their paper [261], and noting their convention for the coupling constant, from their equation (2), or just after their equations (1) or (2), we see that the expansion parameter is 2GN where E is the square of the centre of mass energy, times a sum of terms, the first of which is ln −t ln −u , where s, t, and u are the Mandelstam invariants of the scattering process. Thus in a kinematic region where this sum of terms is ∼ 1, the GRW estimate of the expansion parameter is, in this case, smaller than the actual parameter, by a factor ∼ 1 And looking at equation (15) of Donoghue’s calculation of one-loop corrections to the gravitational scattering of two heavy masses, for D = 4 [262], and noting that his convention for the coupling constant is the same as Donoghue and Torma’s, we see that the expansion parameter is GN |q2|, where q is the momentum transfer, times a sum of two terms, one of which is −3 ln (−q2), and the other of which, with a heavy mass in the numerator, is identified, by considering the non-relativistic limit, as a post-Newtonian correction of classical general relativity, rather than a quantum correction. Thus, in this case, the expansion parameter for quantum gravitational corrections is 3GN |q2| ln (−q2), where −q2 would be multiplied, in the argument of the logarithm, by an undetermined multiple of GN , that would have to have to be fixed by an experimental measurement, due to the non-renormalizability of quantum gravity for D = 4, although it might be determined in a resummation of quantum gravity, for D = 4, recently developed by Ward [263]. So if we identify |q2| as the GRW E2, we see that, in the kinematic region where the argument of the logarithm is ∼ 1, the GRW estimate of the expansion parameter is, in this case, smaller than the actual parameter, by a factor ∼ 2 . So it appears that, for D = 4, the GRW estimate of the expansion parameter is reasonable, in kinematic regions where the logarithmic factors it omits are not too large. Considering, now, how the GRW estimate of the expansion parameter might be understood in D dimensions, let us choose the Hořava-Witten downstairs conven- tion for the gravitational action in D dimensions, so that the Einstein term in the action is 1 −gR. The GRW estimate of the expansion parameter is then π2SD−1 (2π)D . Working, now, in Euclidean signature momentum space, there is a factor κ D for each loop, a kinematic factor for each propagator, where kµ is the momentum in the propagator, and two numerator momentum factors for each vertex. Considering, now, a ladder diagram formed from graviton propagators, with an ex- ternal momentum pµ, with p 2 = E2, running along the ladder, the momentum integral for each loop of the ladder will be ∼ ∫ dDk|k|4 3 , which we would expect to be cut off for |k| larger than around E, by BPHZ counterterms, and thus to give around D−2 E D−2, for D > 2. Thus, without considering sums over diagrams with a given number of loops, and the Lorentz index structure of the graviton propagator and vertices, the GRW estimate of the expansion parameter is obtained for D > 2, up to a factor (D−2)π2 D−2 . This factor is 1 for D = 4, so, in view of the two examples above, the estimate so far is as good as the GRW estimate, for D = 4. It is not clear, without further investigation, why the magnitude of the Euclidean loop momentum would tend to be cut off, by BPHZ counterterms, at around E , rather than at around E, as suggested by the GRW estimate of the expansion parameter, and it is also not clear where the extra factor of π2, in the GRW estimate, comes from. This seems to suggest that, in applying the GRW estimate to the expansions (132) and (136), E should have been taken as 2π , resulting in the estimate (143), above, for the minimum possible value of b1 , rather than the estimate (142), above, except that the estimate (143), above, could possibly be reduced by a factor of π− 9 , to around 0.15, with a corresponding increase in the maximum possible value of |χ (M6)|, to around 4× 105. Considering, now, the effects of sums over diagrams with a given number of loops, and the Lorentz index structure of propagators and vertices, the fact that the esti- mate so far includes a factor 1 D−2 , which is absent from the GRW estimate, and is thus presumably cancelled by Lorentz index contractions, for some diagrams, suggests considering the limit D → ∞. The D → ∞ limit of the Feynman diagram expansion of quantum gravity was considered by Strominger [264], and recently reconsidered by Bjerrum-Bohr [265], and the D → ∞ limit of quantum gravity was also considered, in the context of a lattice regularization, by Hamber and Williams [266]. The graviton propagator has twoD-vector indices at each end, and includes terms in which two index-contraction lines run along it, so is in this respect similar to the gluon propagator at large Nc, when the SU (Nc) adjoint indices, of the gluon propagator, are written as pairs of an SU (Nc) fundamental index and an SU (Nc) antifundamen- tal index [267], but that is as far as the similarity with large-Nc Yang-Mills theory goes. One difference is that the graviton interaction vertices all include two factors of momentum, and in terms where the D-vector indices of these two momentum factors are not contracted with each other, a D-vector index line ends on each of them. But the main difference is that the three-graviton vertex, Vµ1ν1,µ2ν2,µ3ν3, in an expansion about flat space, includes, in Euclidean signature momentum space, terms of structure δµ1µ2δν1ν2pµ3qν3 , which allow both index lines from one propagator ending at the vertex, to pass through the vertex “in parallel”, like a railway track, and leave the vertex to- gether along another propagator, without getting separated. There are no such terms in the vertices of SU (Nc) Yang-Mills theory, with its usual action, since they could only arise from Lagrangian terms with at least two traces, such as tr (FµνFστ ) tr (FµνFστ ). The presence of such “railway track” terms, in the three-graviton vertex, means that for some diagrams, there are two factors of D per loop, at large D, and these are therefore the leading diagrams at large D, so far as index contractions go. For diagrams built from propagators and three-graviton vertices only, the loops have to be separated from each other, as one-loop propagator insertions, or one-loop vertex insertions, in order to be able to have two factors ofD per loop, so in this respect, the large-D limit of quantum gravity is much simpler than the large-Nc limit of SU (Nc) Yang-Mills theory. When (2n+m)-graviton vertices, containing terms with n “railway tracks” through them, with n ≥ 2, m ≥ 1, are included, loops with two factors of D per loop can now touch one another, and the leading terms at large D, in the quantum effective action, so far as index contractions go, are “trees” built from one-loop bubbles, that meet one another at (2n+m)-graviton vertices, n ≥ 2, m ≥ 1, that have n “railway tracks” through them. Thus from considering the index contractions from the diagrams that are leading at large D, so far as index contractions go, the estimate of the expansion parameter now gets an additional factor D2, so for large D, our estimate of the expansion parameter is now larger than the GRW estimate, by a factor D(2π)D−2 . However, for D = 11, the factor D is approximately 1, and the factor (2π) is accomodated by taking E as 2π in the GRW estimate (141), rather than 1 , as we would initially have expected. Thus it appears that the second estimate, (143), is at present the best rough estimate of the minimum value of b1 , and the best rough estimate of the upper bound on |χ (M6)| is therefore around 7 × 104. To check this estimate further, it would be necessary to consider diagrams involving the gravitino and the three-form gauge field, but that will not be done in this paper. I shall seek solutions of the Casimir energy density corrected field equations and boundary conditions, such that all the fermionic fields vanish. Thus, in the bulk, the only non-vanishing fields will be the metric, and the three-form gauge field. I shall now consider the implications of a topological constraint, that was discussed by Witten in the context of superstrings, then consider the field equation, and boundary conditions, for the three-form gauge field. 2.3.7 Witten’s topological constraint By analogy with a constraint on the compactification of superstrings, discussed by Witten [45], the fact that the gauge-invariant field strength, GIJKL, is globally well- defined, implies that for any closed five-dimensional surface, S, we must have S dG = 0. If we now work in the “upstairs” picture, so that M11 is M10 × S1, and the fields transform under reflection in the orbifold fixed-point hyperplanes y = y1, and y = y2, as discussed in Subsection 2.1, on page 17, and choose S to be the Cartesian product of the circle S1, and a closed four-dimensional surface, Q, in M10, then this relation, together with (42), after making the substitutions (47), implies that the sum, over the two orbifold fixed-point hyperplanes, of the integral: trF (i) ∧ F (i) − trR ∧ R (144) must be equal to zero. We recall, from the discussion after (28), that for E8, “tr” denotes 1 of the trace in the adjoint representation, and from the discussion after (47), that trR[UV RWX] is defined as R [UV RWX]Y Z , so that the trace is effectively in the vector representation, of the SO(10) tangent space group of M10. I will show that, for the metric ansatz (94), the implications of the topological constraint are the same, regardless of whether the Riemann tensors, in trR[UV RWX], are calculated entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold fixed-point hyperplane, or, alternatively, treated as the restriction to the orbifold fixed-point hyperplane, of the Riemann tensors calculated from the metric in eleven dimensions. Now in the problem studied here, M10 is the Cartesian product, of a four dimen- sional locally de Sitter space, whose three spatial dimensions may have been compact- ified, and a smooth compact quotient of CH3. Suppose, first, that Q is the Cartesian product of a topologically non-trivial closed four-dimensional surface, in CH3, and a point of the locally de Sitter space. And suppose, first, that the Riemann tensors, in trR[UV RWX], are calculated entirely in ten dimensions, from the restriction of the met- ric to the appropriate orbifold fixed-point hyperplane. Then Q trR ∧ R is generically non-zero, and, moreover, is a topological invariant, specifically a Pontrjagin number, so both the orbifold fixed points give the same contribution, to the quantity that is required to vanish. Thus it is necessary to choose nonvanishing E8 vacuum gauge fields, on one or both of the orbifold fixed-point hyperplanes, in order to cancel the contributions, to the sum, from Q trR ∧ R. We recall, from the discussion after (36), that the trace of the square of a generator of SO(16), in the adjoint representation of E8, is 30 times the trace of the square of the corresponding generator, in the vector representation of SO(16). Thus we can satisfy the topological constraint, for all Q of this type, by choosing one of the two orbifold fixed-point hyperplanes, and choosing an SO(6) subgroup of the E8 gauge group on that orbifold fixed-point hyperplane, em- bedded in that E8 gauge group by the natural embedding SO (6) ⊂ SO (16) ⊂ E8, and setting the E8 Yang-Mills gauge fields, in that SO(6) subgroup of that E8, equal to the spin connection, while the E8 gauge fields, in the E8 on the other orbifold fixed-point hyperplane, are zero. Furthermore, the classical Yang-Mills field equation is automat- ically satisfied for such a configuration, in consequence of the fact that the compact six-manifold is locally symmetric, so that the covariant derivative of the Riemann ten- sor, DURVWXY , vanishes identically, which implies that DUR V W vanishes identically, where x and y are local Lorentz indices. More generally, the topological constraint, for this type of Q, will also be satisfied, for arbitrary Yang-Mills field configurations, in the same topological class, as the configuration just described. This is known as the standard embedding of the spin connection of the compact six-manifold, in one of the two E8 gauge groups. In the present case, CH 3 is a Kähler manifold, so the spin connection will lie in a U(3) subgroup of the SO(6). Now suppose, instead, that the Riemann tensors, in trR[UV RWX], are treated as the restriction to the orbifold fixed-point hyperplane, of the Riemann tensors calculated from the metric in eleven dimensions. In that case, we find, from (96), that: R JABI R CDJ = R ABE R CDF = = R FABE (h)R CDF (h)+4ḃ 2RABDC (h)+2ḃ 4 (hADhBC−hAChBD)(145) Thus R IJ[AB RCD]IJ = R [AB RCD]EF = R [AB (h)RCD]EF (h), so for the metric ansatz (94), the topological constraint, for a closed four-surface Q, that has the form of the Cartesian product of a topologically non-trivial four-dimensional closed surface in the compact six-manifold, and a point in the locally de Sitter space, has exactly the same form, regardless of whether the Riemann tensors, in trR ∧ R, are calculated entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold fixed-point hyperplane, or are the components, in the orbifold fixed-point hyperplane, of the eleven-dimensional Riemann tensor, and, indeed, it still has the same form, even if “tr” sums the contracted indices, over all eleven dimensions. Suppose, now, that Q is the Cartesian product, of a topologically non-trivial n- dimensional closed surface, in CH3, such that 1 ≤ n ≤ 3, and a topologically non- trivial (4− n)-dimensional closed surface, in the locally de Sitter space. And as before, suppose, first, that the Riemann tensors, in trR[UV RWX], are calculated entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold fixed-point hyperplane. Then all the Riemann tensor components, with mixed indices, vanish identically, so Q trR ∧ R vanishes identically, for any such Q. Furthermore, the E8 vacuum gauge fields already introduced, have no components tangential to the locally de Sitter space, so Q trF (i) ∧ F (i) also vanishes identically, for both E8 gauge groups, for all such Q. So no nontrivial topological constraint arises from any such Q. Now suppose, instead, that the Riemann tensors, in trR[UV RWX], are treated as the restriction to the orbifold fixed-point hyperplane, of the Riemann tensors calculated from the metric in eleven dimensions. Then it follows from the list, in subsection 2.3.1, of the components of the Riemann tensor, of the form R JUV I , that do not vanish automatically, for the metric ansatz (94), that Q trR ∧ R vanishes identically, unless n = 2. We then find, from (96), that: R IJµA RνBIJ = R µA RνBUV = 2R µA RνBσC = 2 GµνGAB (146) Hence R IJ[µA RνB]IJ = R [µA RνB]UV = 0, hence Q trR∧R vanishes identically, for the metric ansatz (94), regardless of whether the Riemann tensors, in trR∧R, are calculated entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold fixed-point hyperplane, or are the components, in the orbifold fixed-point hyperplane, of the eleven-dimensional Riemann tensor, and, moreover, this is still true, even if “tr” sums the contracted indices, over all eleven dimensions. Finally, there are no topologically non-trivial 4-dimensional closed surfaces, in the locally de Sitter space, since the time dimension has not been compactified. Thus Witten’s topological constraint is completely satisfied, by the standard em- bedding of the spin connection of the compact six-manifold, in the E8 gauge group, on one of the two orbifold fixed-point hyperplanes, as just described, and this is true, for the metric ansatz (94), regardless of whether the Riemann tensors, in trR ∧R, are calculated entirely in ten dimensions, from the restriction of the metric to the appropri- ate orbifold fixed-point hyperplane, or are the components, in the orbifold fixed-point hyperplane, of the eleven-dimensional Riemann tensor, and, furthermore, this is still true, even if “tr” sums the contracted indices, over all eleven dimensions. The fact that the spin connection is embedded in the E8 gauge group, on just one of the two orbifold fixed-point hyperplanes, breaks the symmetry between the two orbifold fixed-point hyperplanes, and it is known from calculations by Witten [127], and by Lukas, Ovrut, Stelle, and Waldram [68], that when the compact six-manifold is a Calabi-Yau manifold, the volume of the compact six-manifold is greater, on the orbifold hyperplane that has the spin connection embedded in its E8 gauge group, than it is on the other orbifold hyperplane. I will show that this is also true, when the compact six-manifold is a smooth compact quotient of CH3, so the spin connection will be embedded in the E8 gauge group, on the outer surface of the thick pipe. This is fortunate, because we must expect that, in order to find smooth compact quotients of CH3, such that Fermi-Bose cancellations occur in the Casimir energy densities, to the precisions required for thick pipe geometries to exist, the Euler number of the compact quotient will have to be of larger order of magnitude than 1. Thus, if the spin connection was embedded in the E8 gauge group on the inner surface of the thick pipe, where we live, the number of generations of chiral fermions would be of larger order of magnitude than 1, in contradiction with experiment. In fact, as studied by Pilch and Schellekens [268], the fact that the holonomy group of the compact six-manifold, in the present case, is SU (3)× U (1), rather than SO(6) or SU(3), implies that there exist additional ways in which the spin connection could be embedded in E8×E8, such that the topological constraint is satisfied, and in some of these ways, part of the U(1) part of the spin connection, is embedded in the E8 on the inner surface of the thick pipe. However, it would seem likely that, for any uniform embedding of part of the U(1) part of the spin connection, in the E8 on the inner surface of the thick pipe, in a manner that is independent of position on the compact six-manifold, the number of chiral fermion modes, on the inner surface of the thick pipe, would still be comparable, in order of magnitude, to the Euler number of the compact six-manifold. Thus I shall assume that the entire spin connection is embedded in the E8 gauge group, on the outer surface of the thick pipe. In Section 5, on page 222, I shall introduce some E8 vacuum gauge fields, on the inner surface of the thick pipe, localized on Hodge - de Rham harmonic two-forms, and partly topologically stabilized by a form of Dirac quantization condition, in order to break E8 to the Standard Model at around 140 TeV, if the couplings are evolved in the Standard Model up to unification, and produce a small number of chiral fermions, on the inner surface of the thick pipe, where we live. This has to be done without spoiling the satisfaction of Witten’s topological constraint, and I shall also require that, in the context of Lukas, Ovrut, and Waldram’s harmonic expansion, as discussed above, the modification to the leading term in each harmonic expansion, resulting from the introduction of these localized E8 vacuum gauge fields, on the inner surface of the thick pipe, is a small perturbation of the value which the leading term had, in the absence of these localized E8 vacuum gauge fields. Thus the analysis of the Casimir energy density corrected field equations and boundary conditions, in the present Section, should still be a good first approximation, when the localized E8 vacuum gauge fields are introduced, in Section 5. The idea is that the compactifications studied in the present Section, should provide a strong, stiff, robust “platform”, that will only be slightly perturbed, by the interesting physics of the Standard Model, taking place on the “platform”. However, it is necessary to note that, since Fermi-Bose cancellations will be required to take place, to a certain precision, in the leading term in the harmonic expansions of the Casimir energy density contributions to the energy-momentum tensor, on the inner surface of the thick pipe, even the small changes to this leading term, resulting from the introduction of the localized E8 vacuum gauge fields, on the inner surface of the thick pipe, might imply that a smooth compact quotient of CH3, for which the cancellations occur to the required precision, in the absence of the localized E8 gauge fields, on the inner surface of the thick pipe, might have to be replaced by a different smooth compact quotient of CH3, when those localized E8 gauge fields, are introduced. I will not be able to determine, in the present paper, whether such a substitution would be likely to be necessary, and I will simply assume that, if such a substitution is necessary, then it is made. If I had chosen the compact six-manifold to be a smooth compact quotient of the real hyperbolic space H6, rather than of CH3, then RABCD (h) would have been a constant multiple of (hAChBD − hAChBC), and trR[ABRCD] would have vanished identically, so that Witten’s topological constraint would not have given any nontrivial constraints, and there would not have been any need to embed the spin connection in the gauge group. The symmetry between y1 and y2 would, in that case, have remained unbroken, at this stage. Nevertheless, we will see that, in this case, thick pipe solutions, very similar to those obtained for suitable quotients of CH3, will still exist, provided that the Casimir energy-momentum tensor coefficients satisfy relations similar to those required for quotients of CH3. The reason for this is that the terms in the energy- momentum tensor, quadratic in GABCD, the field strength of the three-form gauge field, as determined by the Hořava-Witten boundary conditions, are only significant, for the CH3 thick pipe solutions, in at most a very small fraction of the bulk, and, in fact, at most, only for a small fraction of the region y1 < y < κ 2/9, while for the case of TeV-scale gravity, we will find that (y2 − y1) ∼ 1015κ2/9. Typical solutions of the Einstein equations break the symmetry between y1 and y2, even when it is unbroken to start with, because either b (y) increases monotonically with increasing y, while a (y) decreases, or vice versa. I always choose the solutions for which b (y) increases with increasing y, while the warp factor, a (y), decreases, since, by assumption, we live at y1, with y1 < y2. Although I mainly consider compactification on quotients of CH3, in this paper, there are two reasons why compactification on quotients of H6 might turn out to be preferable. Firstly, on the basis of existing knowledge, the number of smooth compact quotients of H6, up to a given value of the modulus of the Euler number, might be very much larger than the number of smooth compact quotients of CH3, up to the same value of the modulus of the Euler number, as I shall discuss in Section 3, on page 160. And secondly, if the large number of chiral fermion modes, on the outer surface of the thick pipe, for smooth compact quotients of CH3, should turn out to be a phenomenological problem, it might be preferable to look for suitable smooth compact quotients of H6, since there is no need to embed the spin connection of the compact six-manifold in the gauge group, for smooth compact quotients of H6. However, it seems possible that the most important criterion, that might favour either CH3 or H6, is that the local contributions to the coefficients D 1 , in (136), should vanish, if this is necessary, in order to have an infinite number of smooth compact quotients, with arbitrarily small, but nonvanishing, values of the D 2.3.8 The field equations and boundary conditions for the three-form gauge field When the compact six-manifold is a smooth compact quotient of CH3, we can use the ansatz of Lukas, Ovrut, Stelle, and Waldram [68], (LOSW), for the four-form field strength in the bulk, namely that GIJKL vanishes unless all four indices are on the compact six-manifold, and: GABCD = αhABCDEFh EGhFHωGH (147) for y1 < y < y2, where α is a fixed number, to be determined by the boundary conditions, hABCDEF is the tensor hǫABCDEF , where h is given by (65), and ǫ123456 = 1, and ωGH is the Kähler form, given by (69). This satisfies the Bianchi identities, and field equations, in the bulk, due to its independence from y, and from position in the four-dimensional locally de Sitter space, the covariant constancy of hABCDEF , h and ωFH , and the fact that there are not enough non-vanishing components of GIJKL, for the GI1...I11G I4...I7GI8...I11 term in the field equations, to be nonzero. Here GI1...I11 denotes the tensor −GǫI1...I11 . To confirm that the vanishing of GIJKL, unless all four indices are on the compact six-manifold, is consistent with the boundary conditions (43) or (44), after making the substitutions (47), we recall, from the preceding subsection, that for the metric ansatz (94), and for all cases of trR[UV RWX], other than trR[µνRστ ], which was not considered there, the value of trR[UV RWX] = R [UV RWX]Y Z is the same, regardless of whether the Riemann tensors are calculated from the restriction of the metric to the ten-dimensional orbifold hyperplanes, or are taken to be the components on the orbifold hyperplanes, of the Riemann tensor in eleven dimensions. Furthermore, all cases of trR[UV RWX] with mixed components vanish identically, and trR[ABRCD] = R EF[AB (h)RCD]EF (h). For the case of trR[µνRστ ], we find, by a calculation pre- cisely analogous to the case of trR[ABRCD], that R [µν Rστ ]IJ = R [µν Rστ ]ρη = [µν (g)Rστ ]ρη (g), so, as with the other cases, the result is the same, regardless of whether the Riemann tensors are calculated from the restriction of the metric to the ten-dimensional orbifold hyperplanes, or are taken to be the components on the orbifold hyperplanes, of the Riemann tensor in eleven dimensions, and this remains true, even if “tr” sums the contracted indices, over all eleven dimensions. However, the metric gµν is locally de Sitter, specifically dS4, with de Sitter radius equal to 1, so we have Rµνρη (g) = gµηgνρ − gµρgνη. Hence R ρηµν (g)Rστρη (g) = 2 (gµσgντ − gµτgνσ), hence R [µν (g)Rστ ]ρη (g) = 0. Thus the boundary conditions are, indeed, consistent with the vanishing of GIJKL, unless all four indices are on the compact six-manifold. To determine α, we note that, in consequence of the decision to embed the spin connection, of the compact six-manifold, in the E8 at the outer surface of the thick pipe, it follows from (43), after making the substitutions (47), that near y = y1, we have: GABCD = ǫ (y − y1) trR[ABRCD] + . . . (148) while near y = y2, we have: GABCD = − ǫ (y − y2) trR[ABRCD] + . . . (149) Thus, setting y = y1+, in (148), and y = y2−, in (149), we see that the boundary conditions are consistent with GABCD taking the constant value: GABCD = trR[ABRCD] = R EF[AB (h)RCD]EF (h) (150) for y1 < y < y2. Now in the complex coordinate system of subsection 2.2, for the compact six- manifold, we have, from (71), and (72), that: R EFrs̄ (h)RtūEF (h) = −10hrs̄htū − 2hrūhts̄ (151) Hence: R EF[rs̄ (h)R tū]EF (h) = −4 (hrs̄htū − hrūhts̄) (152) On the other hand, in the complex coordinate system, we have: hrstūv̄w̄ = i hǫrstǫūv̄w̄ = 6i hArstδrūδsv̄δtw̄ (153) where the factor of i is present because hABCDEF is a tensor, which is real in a real coordinate system, and on transforming to complex coordinates, for example, by the matrix U , in (49), h11̄22̄33̄ acquires a factor (detU) = (−i)−3 = −i, h is given by (65), with n = 3, and g rewritten as h, and the symbol A, with a list of indices underneath it, denotes the antisymmetrization of the expression that follows it, under permutations of those indices. Thus, from (147), we have: Grst̄ū = αhrst̄ūEFh EGhFHωGH = iαhrst̄ūvw̄h vx̄hw̄zhx̄z = − hǫrsvǫt̄ūw̄ hw̄v = α (hrx̄hsz̄hvq̄ǫxzqǫt̄ūw̄)h w̄v = −1 α (hrt̄hsū − hrūhst̄) (154) Comparing with (150), and (152), we see that the ansatz (147) is, indeed, consistent with the boundary conditions, and that α = −9 = − 9√ (155) where I used (46), at the last step. However, it is also interesting to consider compact- ification on smooth compact quotients of H6, for which there is no need to embed the spin connection in the gauge group, so that we can set GIJKL = 0. I shall therefore often leave the above coefficient, α, in the Einstein equations, so that the results for CH3 can be obtained by setting α = − 9√ , and the results for H6 obtained by setting α = 0. Making use of the Kähler geometry identity ωABh BCωCDh DE = −δ EA , and the relation GAB = 1 hAB, we find, from (147), that: GBFGCGGDHGABCDGEFGH = GAE (156) GAEGBFGCGGDHGABCDGEFGH = (157) Now, in the upstairs framework, the contribution of the three-form gauge field, to the energy-momentum tensor, (14), for the bulk action (25), in eleven dimensions, is: GKNGLOGMPGIKLMGJNOP − QRGKNGLOGMPGQKLMGRNOP (158) Hence the non-vanishing components of T IJ are: T (3f)µν = − 6κ2b8 Gµν , T 18κ2b8 GAB, T yy = − 6κ2b8 (159) These contributions to the energy-momentum tensor are of the form (134), on page 61, for n = 0, with C 0 negative. They have been calculated here, for nonzero α, only for the special case of the standard embedding of the spin connection in the gauge group on the outer surface of the thick pipe, when the compact six-manifold M6 is a smooth compact quotient of CH3. However it seems reasonable to expect that in the approximation of restricting the energy-momentum tensor to the leading term in the Lukas-Ovrut-Waldram harmonic expansion on M6 [67], as done throughout this section, the same result would be obtained, but with a different value of α, for the contributions to the energy-momentum tensor from the vacuum configurations of the three-form gauge field that result, due to the Hořava-Witten modified Bianchi identity (42), from the presence of general topologically stabilized vacuum Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, with non-vanishing field strengths tangential to M6. It seems unlikely that the bulk Green-Schwarz term [89, 25] would have a significant effect, near the inner surface of the thick pipe, because the extended dimensions can to a good approximation be treated as flat, in this region, and the bulk Green-Schwarz term includes an antisymmetric tensor, with eleven indices, and the expression con- tracted with this antisymmetric tensor would, in the approximation that the extended dimensions are treated as flat, not have any nonvanishing components with enough different indices, to give a nonvanishing result. I shall assume that the bulk Green- Schwarz term does not have any significant effect on the field equations of either the three-form gauge field or the metric, for the geometries considered in the present paper. 2.3.9 The field equations and boundary conditions for the metric By analogy with (13), the field equations for the gravitational field GIJ , in the upstairs picture, in eleven dimensions, are: RIJ − RGIJ + κ 2TIJ = 0 (160) where TIJ is now defined by (14), with (SSM + SDM) replaced by the sum of all terms in the quantum effective action Γ, in the upstairs picture in eleven dimensions, except for the Ricci scalar term in (25). We note that, due to the incompatibility of a cosmological constant in eleven dimensions with local supersymmetry in eleven dimensions [200, 269, 124], there is not expected to be any d = 11 cosmological constant term, in the low energy expansion of Γ. The Einstein equations (160) can alternatively be written: RIJ + κ TIJ − KLTKL = 0 (161) Now subject to the assumptions and approximations discussed in subsections 2.3.4 and 2.3.8, TIJ will have the block diagonal structure (130), on page 58. Thus, using the Ricci tensor components (97), on page 34, the Einstein equations (161) become: 5t(1) (y)− 6t(2) (y)− t(3) (y) = 0 (162) −4t(1) (y) + 3t(2) (y)− t(3) (y) = 0 (163) −4t(1) (y)− 6t(2) (y) + 8t(3) (y) = 0 (164) where the t(i) (y) satisfy the conservation equation (131), on page 58. We next need the boundary conditions for the metric, at y1 and y2. Because of the simple structure of the metric ansatz (94), we can obtain these either directly from the above Einstein equations, with appropriate delta function terms in the t(i) (y), located on the orbifold fixed point ten-manifolds, or alternatively, from the Israel matching conditions [270, 271], which are obtained by including a Gibbons-Hawking term [93, 94] in the action on the boundary. We recall from subsection 2.1, that Moss’s improved form of Hořava-Witten theory, which for the purposes of the present paper I assume to be valid, includes a supersymmetrized Gibbons-Hawking boundary term. Considering first the direct approach, the energy-momentum tensor T̃ UV , i = 1, 2, on the Hořava-Witten orbifold hyperplane at y = yi, has the block-diagonal struc- ture (135), on page 62, by assumption. Hence GKLT̃ KL = 4t̃ [i](1) + 6t̃[i](2). Thus, by (161) and (130), the first Einstein equation (162) will include delta function terms 5t̃[i](1) − 6t̃[i](2) δ (y − yi), the second Einstein equation (163) will include delta function terms κ −4t̃[i](1) + 3t̃[i](2) δ (y − yi), and the third Einstein equation (164) will include delta function terms 4t̃[i](1) + 6t̃[i](2) δ (y − yi). To match these delta function terms, the slopes of a (y), and b (y), must be discontinuous, at y1, and y2. Furthermore, by the orbifold conditions, a (y), and b (y), are to be symmetric, under reflection about y1, and under reflection about y2. Thus, near y = y1, we must have, for example: a (y) = a1 + σ |y − y1|+O (y − y1)2 (165) If we now consider the Einstein equations, (162), (163), and (164), in the vicinity of y1 and y2, and drop all terms except the delta function terms, we find: 5t̃[1](1) − 6t̃[1](2) δ (y − y1) + 5t̃[2](1) − 6t̃[2](2) δ (y − y2) = 0 (166) −4t̃[1](1) + 3t̃[1](2) δ (y − y1) + −4t̃[2](1) + 3t̃[2](2) δ (y − y2) = 0 (167) 4t̃[1](1) + 6t̃[1](2) δ (y − y1)− 4t̃[2](1) + 6t̃[2](2) δ (y − y2) = 0 (168) The third of these three equations follows from the first two, so we only need to consider the first two. Considering the first equation, near y = y1, we find that σ, in (165), is given by σ = −κ2 5t̃[1](1) − 6t̃[1](2) a (y1). Thus we find y=y1+ 5t̃[1](1) − 6t̃[1](2) . The other boundary conditions follow similarly, and we find: y=y1+ −5t̃[1](1) + 6t̃[1](2) y=y1+ 4t̃[1](1) − 3t̃[1](2) (169) y=y2− 5t̃[2](1) − 6t̃[2](2) y=y2− −4t̃[2](1) + 3t̃[2](2) (170) Alternatively, we can obtain the boundary conditions from the Israel matching condi- tions [270, 271], which read: {KUV −KHUV } = −κ2T̃UV (171) Here HUV is defined to be the components tangential to the orbifold fixed-point hyper- plane, of the projection tensor HIJ = GIJ −nInJ , where nI is the unit normal pointing out of the fixed-point hyperplane, on one side. The curly braces denote summation over both sides of the fixed-point hyperplane. KUV is the extrinsic curvature of the fixed-point hyperplane, defined by KUV = H V DInJ , which is symmetric under swapping U and V , because nJ will be the gradient of a scalar function, that takes a fixed value on the fixed-point hyperplane, and whose gradient is normalized, at each point on the fixed-point hyperplane, so that GIJnInJ = 1 there. K = H UVKUV . And T̃UV is the energy-momentum tensor on the fixed-point hyperplane, as above. In the present case, if we first consider the y = y1+ side of the fixed point hyperplane at y = y1, we have ny = 1, and all other components of nI vanish, and HUV is simply the components GUV of GIJ . Furthermore, KUV = −ΓyUV , hence, from (95), Kµν |y=y1+ = y=y1+ Gµν , KAB|y=y1+ = y=y1+ GAB (172) K|y=y1+ = 4 y=y1+ y=y1+ (173) At y = y1−, ny = −1, and ȧ and ḃ have also been multiplied by −1, so we recover the boundary conditions (169), from the Israel matching conditions (171). And we also recover the boundary conditions (170), in a similar manner. The energy-momentum tensors T̃ UV , corresponding to the bosonic part of the Yang- Mills action (28), are given by: [i]YM GCDtrF CDGEF trF (174) T̃ [i]YMµν = − CDGEF trF DF (175) Now for compact quotients of CH3, the spin connection has been embedded in the E8 at the outer surface of the thick pipe, while F AB, and consequently T̃ UV , is zero. And for compact quotients of H6, the Yang-Mills fields are zero on both surfaces of the thick pipe, and consequently T̃ UV = 0, for both i = 1 and i = 2. For the case of CH3, and i = 2, we recall, from subsections 2.1 and 2.3.7, that for E8, “tr” means of the trace in the adjoint representation, and that the trace of the square of a generator of SO(16), in the adjoint representation of E8, is 30 times the trace of the square of the corresponding generator, in the vector representation of SO(16). Furthermore, the E8 generators being used, are hermitian. Thus we have: BD = R AC RBDEF (176) I shall now assume that the Riemann tensor R FACE , that is embedded in the E8 on the outer surface of the thick pipe, is the Riemann tensor R FACE (h), calculated from the induced metric GUV , on the outer surface of the thick pipe, and not the restriction to the outer surface of the thick pipe, of the eleven-dimensional Riemann tensor. We then have R EFAC RBDEF = R AC (h)RBDEF (h), hence, from (151), we have, in the complex coordinate system, that: rs̄ F tū = −10hrs̄htū − 2hrūhts̄ (177) GCDtrF ūD = 16 hrū (178) We also have GCDtrF uD = G CDtrF ūD = 0. Hence: GCDtrF BD = 16 hAB = 16 GAB (179) T̃ [2]YMµν = − [2]YM AB = − (180) t̃[2](1)YM = − , t̃[2](2)YM = − (181) When the functions b (y), or a (y), are shown without arguments, they are evaluated at the appropriate value of y, which for T̃ UV , F UV , and t̃ [2](i), is at y2. Now, as discussed in subsection 2.1, the low energy expansion of the quantum effective action, Γ, on the orbifold fixed point hyperplanes, is believed to contain terms quadratic in the Riemann tensor, of the Lovelock-Gauss-Bonnet form, obtained from the Yang-Mills actions (28), by the substitutions (48). The corresponding term in Γ, at yi, is: LGB = −GR [UV[UV R WX] (182) I shall now assume, as in the calculation above, of the Yang-Mills energy-momentum tensor, when the spin connection is embedded in the gauge group, that the Riemann tensor R XUVW , in (182), is the Riemann tensor calculated from the induced metric GUV , on the orbifold fixed point ten-manifold M10i , and not the restriction to the outer surface of the thick pipe, of the eleven-dimensional Riemann tensor. Then, since M10i is the Cartesian product of the four observed dimensions, and the compact six- manifold, all the Riemann tensor components, with mixed indices, vanish identically. The energy-momentum tensors T̃ UV , corresponding to (182), are: [i]LGB UV = − R̃UWXY R̃ V − 2R̃UWVXR̃WX − 2R̃UW R̃ WV + R̃UV R̃ GUV R̃WXY ZR̃ WXY Z +GUV R̃WXR̃ WX − 1 GUV R̃ (183) where I have now denoted curvatures calculated from the induced metric GUV , on the orbifold fixed point ten-manifold M10i , by a tilde. To evaluate (183), we note that when the compact six-manifold is a quotient of CH3, we have, from (72), that for the metric induced on a fixed point ten-manifold, by the metric ansatz (94): R̃ACDER̃ B = 16 GAB, R̃ACBDR̃ CD = 16 GAB (184) Hence, recalling that gµν , in the metric ansatz (94), is normalized such that Rµν (g) = −3gµν , and when the compact six-manifold is a quotient of CH3, hAB is normalized such that RAB (h) = 4hAB, we find, when the compact six-manifold is a quotient of CH3, that: R̃WXY ZR̃ WXY Z = , R̃WXR̃ , R̃ = (185) [i]LGB GAB (186) T̃ [i]LGBµν = Gµν (187) t̃[i](1)LGB = , t̃[i](2)LGB = (188) To evaluate (183) when the compact six-manifold is a quotient of H6, we recall that in this case we have chosen hAB, in the metric ansatz (94), to be normalized such that RABCD (h) = hAChBD−hADhBC , so that RAB (h) = 5hAB, as stated after (97), on page 34. We then find, when the compact six-manifold is a quotient of H6, that: R̃WXY ZR̃ WXY Z = , R̃WXR̃ , R̃ = (189) [i]LGB GAB (190) T̃ [i]LGBµν = Gµν (191) t̃[i](1)LGB = , t̃[i](2)LGB = (192) Now at the inner surface of the thick pipe, we will have a (y1) ∼ 1026 metres, while b (y1) will be less than about 10 −19 metres, so for i = 1, we can neglect the terms with negative powers of a, in (188) and (192). On the other hand, we will find solutions where a is comparable to b, at the outer surface of the thick pipe, but these solutions will not be able to fit the observed values of Newton’s constant and the cosmological constant, and other solutions where a is small compared to b, at the outer surface of the thick pipe, some of which will be able to fit the observed values of Newton’s constant and the cosmological constant. 2.4 Analysis of the Einstein equations and the boundary con- ditions for the metric The Einstein equations (162), (163), and (164), with the range of y restricted to y1 < y < y2, together with the boundary conditions (169) and (170), now constitute a system of coupled ordinary differential equations, and boundary conditions, for the functions a (y) and b (y). The functions t(i) (y), defined by (130), on page 58, receive contributions from the energy-momentum tensor of the three-form gauge field, given by (159) for quotients of CH3, and 0 for quotients of H6, and from Casimir effects in the bulk, near the inner surface of the thick pipe, and, for solutions such that a (y) becomes sufficiently small near the outer surface of the thick pipe, also from Casimir effects in the bulk, near the outer surface of the thick pipe. The coefficients t̃[i](j), defined by (135), receive contributions from the energy- momentum tensor of the Yang-Mills fields on the outer surface of the thick pipe, given by (181) for quotients of CH3, and 0 for quotients of H6; from the leading terms in the Lukas-Ovrut-Waldram harmonic expansion, on the compact six-manifold M6, of the energy-momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe; from the Lovelock-Gauss-Bonnet energy-momentum tensor on the surfaces of the thick pipe, given by (188) for quotients of CH3, and by (192) for quotients of H6; and from Casimir effects on the inner surface of the thick pipe, and, for solutions such that a (y) becomes sufficiently small at the outer surface of the thick pipe, also from Casimir effects on the outer surface of the thick pipe. The functions t(i) (y), and the coefficients t̃[i](j), are required to be recovered self- consistently, when they are recalculated for the solution of the Einstein equations and the boundary conditions. The equations are invariant under a uniform shift of y, y1, and y2, but as already noted, in the discussion following (94), I shall use this freedom to obtain the simplest form of the solution in the bulk, near the inner surface of the thick pipe, rather than to set y1 or y2 to any particular value. Eliminating the double derivatives between the three Einstein equations, we find: κ2t(3) = 0 (193) When the functions t(i) (y) are shown without arguments, they are evaluated at y. From (193), we find: = −2 ḃ κ2t(3) (194) The second Einstein equation, (163), now becomes: − 3 ḃ ± 2 ḃ κ2t(3) + −4t(1) + 3t(2) − t(3) = 0 (195) Now differentiating (194) with respect to y, we find: = −2 b̈ − 12 ḃ − 8 ȧ κ2ṫ(3) (196) where I defined κ2t(3) (197) Now, using the formula (196) for ä, the left-hand side of the first Einstein equation, (162), becomes: −2± 3ḃ − 3 ḃ ± 2 ḃ −4t(1) + 3t(2) − t(3) κ2t(3) t(3) − t(1) −2 ḃ ṫ(3) + t(3) − 4 t(1) − 6 (198) and thus vanishes when (194) and (195) and the conservation equation (131) are sat- isfied, provided that the square root, (197), is nonvanishing. Now the third Einstein equation, (164), is equivalent to (194), provided that the first two Einstein equations are satisfied. Thus (194) and (195), taken together, imply that all three Einstein equations are satisfied, provided that the conservation equation (131) is satisfied, and the square root, (197), is nonvanishing. This is true whichever choice of sign we take in (194) and (195), provided that we choose either the upper sign in both equations, or the lower sign in both equations. Now we are seeking solutions in the region y1 ≤ y ≤ y2, such that for y close to y1, a (y) is very large, and b (y) is very small. Thus we may neglect the term 4 , in the square root, for y close to y1, in this region. In that case, (195) becomes an ordinary differential equation for b (y), since, in the approximations discussed above, the t(i) (y) only depend on y, through b (y), in this region. Moreover, we are looking for solutions that realize the ADD mechanism [3, 5], by a form of thick pipe geometry, so we require ḃ > 0, for y greater than y1, and close to y1. It is convenient to define c (y) ≡ ḃ, so that b̈ = c dc . Then (195) reduces to a first order differential equation, for c as a function of b: κ2t(3) + −4t(1) + 3t(2) − t(3) = 0 (199) We can now carry out a qualitative analysis of the differential equation (199), in the (b, c) plane. We are interested in the quadrant b > 0, c > 0. For a fixed choice of the sign of the square root, (199) defines a unique curve through each point in the quadrant b > 0, c > 0, such that the argument of the square root is non-negative. We can follow such a curve from the inner surface of the thick pipe, where b is very small. Suppose, first, we choose the lower sign of the square root, so (199) becomes: κ2t(3) + −4t(1) + 3t(2) − t(3) = 0 (200) Now the functions t(i) all decrease rapidly in magnitude, with increasing b, and become negligible as soon as b is large compared to κ2/9. In that case, (200) reduces to: 3c2 − 4 2 (3c2 − 4) (201) We require c ≥ , in order for the square root to be real. Then dc ≥ 0, and will typically be ∼ κ− 29 , or larger, once b is ∼ κ2/9. Then once b has increased by a few multiples of κ2/9, c will be large enough that we can to a reasonable approximation replace 3c2 − 4 by 3c2, and this becomes a better approximation as c increases further. Then (201) becomes: 3 + 2 ≃ 7.8990 (202) Thus as soon as b is as large as a few multiples of κ2/9, we have c = db )7.8990 , for some constant B, that cannot be much larger than κ2/9, but could be smaller, because we could be on a trajectory which starts out with a large value of c, near the inner surface of the thick pipe. Then ≃ 0.7559 (y3−y) )0.1449 , where y3 is some constant greater than y1, but such that y3 − y1 cannot be large compared to κ2/9, unless b somehow remains smaller than B, all the way from y = y1 to y ∼ (y3 −B), which would require c to be smaller than around κ (y3−y1) for most of this range. However, even if the functions t(i) were such that this was possible, and the boundary conditions could be satisfied, such a solution has no classical bulk, because as soon as a value of y is reached, such that b is larger than B, b starts increasing very rapidly, and would reach infinity, if y increased further by more than B. Thus it is not possible to find solutions with a thick pipe form of geometry, that can realize the ADD mechanism in a simple way, without considering the upper choice of sign, in (194) and (195). We now, therefore, choose the upper sign of the square root, so (199) becomes: κ2t(3) + −4t(1) + 3t(2) − t(3) = 0 (203) We start again, at the inner surface of the thick pipe, where b is very small, and follow a curve in the (b, c) plane as before, but defined, this time, by (203). The functions t(i) all become negligible, as before, as soon as b is large compared to κ2/9. Then (203) reduces to: 3c2 − 4  (204) We again require c ≥ , in order for the square root to be real. The right hand side of (204) is < 0 for all c > . The simple dependence on b, of the right hand side of (204), means that the general solution of (204) has the form c = f , for some function f , where B is the constant of integration. Thus all trajectories, in this region, are related to one another, by rescaling b. I shall call the solutions of (204) the bulk-type trajectories. Now for c large compared to , (204) reduces to ≃ −1.8990c (205) Thus when b is large compared to κ2/9, and c large compared to , we have )1.8990 , (206) for some constant B. There is now no upper limit to how large B can be, but it cannot be much smaller than κ2/9. And for large B, this approximate solution will be valid, throughout the range from b somewhat larger than κ2/9, to b somewhat smaller than B, and this range of b can be made arbitrarily large, by choosing a sufficiently large value of B. The above approximate form (206) of c, as a function of b, corresponds to ≃ 1.4436 y − y0 )0.3449 (207) for some y0, which we could choose to set to 0, by using the invariance of the equations and boundary conditions, under a uniform shift of y, y1, and y2. It is convenient to regard a as a function of b, in the same way as c = db is being treated as a function of b. Then in the region where all the t(i) are negligible, the equation (194) for ȧ , with the upper choice of sign, and dropping the term 4 in the square root, becomes: (208) When c is sufficiently large, that we are on a bulk type power law trajectory, this becomes: = −0.7753a (209) Hence: a = A )0.7753 (210) where A is a constant of integration. Now we will find, in subsection 2.6.1, on page 126, and subsection 2.7, on page 137, that for TeV-scale gravity, B , whose value is determined by the boundary conditions at the inner surface of the thick pipe, is required to have a value around 1.5×10 |χ(M6)|0.1715 . Thus with our best rough estimate, (143), on page 68, of the minimum value of b1 and the corresponding best rough estimate of the upper bound on |χ (M6)| as around 7×104, we see that B will be around 104. Thus if the bulk power law, (206), was valid down to the inner surface of the thick pipe, the value of c = db , at the inner surface of the thick pipe, would be around 108. Thus the proximity force approximation will certainly not be an adequate approximation for the Casimir energy densities near the inner surface of the thick pipe, and it is necessary to consider the effects of going beyond the proximity force approximation. 2.4.1 Beyond the proximity force approximation I shall now consider the effects of including, in the expansions (132), of the c(i) near the inner surface of the thick pipe, and the expansions (137), of the c̃(i) near the outer surface of the thick pipe, terms depending on c = ḃ, b̈, and higher derivatives of b, with respect to y, in the case of (132), and terms depending on ȧ, ä, and higher derivatives of a, with respect to y, in the case of (137). For definiteness, I shall consider (132), near the inner surface of the thick pipe, with similar considerations applying to (137), near the outer surface of the thick pipe. Now the terms proportional to b−8, in (132), get contributions (159), from the classical energy-momentum tensor, (158), of the three-form gauge field, CIJK , for the case of smooth compact quotients of CH3, and contributions from the t8t8R 4 term, in the low energy expansion of the quantum effective action of supergravity in eleven dimensions. The proximity force approximation is in fact exact, for the three-form gauge field configuration (150), but we expect there to be contributions involving c, b̈, , and d , coming from the metric variation of the t8t8R 4 term. As a guide to the derivatives of b with respect to y that might be expected, and the powers to which they might occur, at higher orders in the expansion in κ 3 , in (132), we note that for even n ≥ 0, the term C(i)n κ (n−1) b8+3n , in (132), could come from terms built from 4 + 3 Riemann tensors, in the low energy expansion of the quantum effective action. Considering, first, just the powers of c that might occur, we see, from the Riemann tensor components, (96), that each power of 1 , can bring in up to one power of c. If we extend this to odd n ≥ 1 as well, and bear in mind that for the bulk power-law solution, (206), c will be very large compared to 1, near the inner surface of the thick pipe, the strongest dependence on c, that we expect at order κ (n−1), is (n−1) c8+3n b8+3n We now need to determine the range of values of b, and of y, where such a term could significantly alter the results of the study of the Einstein equations, and the boundary conditions for the metric, in the preceding subsections. If we consider the second Ein- stein equation, in the form (203), the ratio of c , to the new term, will be )6+3n And for b small compared to B, in (206), we have c ≃ )1.8990 , almost right up to the inner surface of the thick pipe, according to subsection 2.4.3. Thus the ratio of c the new term, will be )2.8990 ( )1.8990 )6+3n )0.6551 )2.8990×(6+3n) This is larger than 1, for )0.6551 . And, for B ≫ 1, this will be for most of the range κ2/9 < b < B. And by (207), ignoring factors of order 1, y > κ2/9 implies b > B )0.3449 = κ2/9 )0.6551 . Thus the new terms, involving c, will be signifi- cant for y < κ2/9, and will be likely to alter the conclusions of subsection 2.4.3, about this region, but they will be negligible for y ≫ κ2/9, which for B ≫ 1, will be most of the bulk. We note that the point where b )0.6551 , and y ∼ κ2/9, is the point where b Considering, now, terms involving higher derivatives of b, with respect to y, we see, from (96), that in addition to terms proportional to c , one Riemann tensor can also bring in terms proportional to b̈ , which, by (205), is ∼ c2 in the first bulk power law region, to the extent that (204), and (205), are not significantly altered by the new terms. In general, from terms in the low energy expansion of the effective action, built from polynomials in the Riemann tensor and its covariant derivatives, we expect terms involving products of expressions c , . . ., and non-negative powers of . But by repeated use of (205), we find that 1 , where ∼ means up to constant factors of order 1. Thus, at each mass dimension (8 + 3n), the largest terms, in the first bulk power law region, where (205) and (206) are approximately valid, that we can build by use of factors involving higher derivatives of b with respect to y, are no larger than the terms κ (n−1) c8+3n b8+3n , whose effect has already been considered. Thus the effect of going beyond the proximity force approximation, is that the bulk power law solutions (206), (207), and (210), are no longer expected to be approximately valid throughout the whole range from b somewhat larger than κ 9 , to b somewhat smaller than B, but rather, only over the slightly smaller range, from where y ∼ κ2/9, and b ∼ κ2/9 )0.6551 , to b somewhat smaller than B. Now as I mentioned just before the start of this subsection, we will find, in sub- sections 2.6.1 and 2.7, that for TeV-scale gravity, we require B ∼ 1.5×104|χ(M6)|0.1715 . And from subsection 2.3.6, on page 66, the minimum value of b1 allowed by the Giudice- Rattazzi-Wells estimate of the effective expansion parameter in quantum gravity in eleven dimensions [11] is b1 ≃ 0.2κ2/9, which means that |χ (M6)| cannot be larger than around 7× 104. Thus B ∼ 104, so if the bulk power law (206) continued to be valid until very close to the inner surface of the thick pipe, we would find c = db ∼ 108 near the inner surface of the thick pipe. Thus it is clear that the proximity force approximation, in which the Casimir co- efficients in (132) are approximated by their values as calculated on flat R5 times the compact six-manifold, will not, in fact, be an adequate approximation. One way to take account of this would be to generalize the expansions (132), so as to include ex- plicit dependence also on c = db , and on higher derivatives of b with respect to y, as in the order of magnitude estimates above. However this is not an appropriate way to study the detailed form of the quantum corrections, just as it is not appropriate to study the relativistic corrections to the Schrödinger equation for atoms, by expanding m2 + ~p2 ≃ m + ~p2 + . . ., with ~p interpreted as −i~∂, to higher orders in ~p, because this results in differential equations of higher and higher order, and correspondingly, larger and larger numbers of constants of integration, making it difficult to single out the particular solution of physical interest. Instead, the appropriate way to study the quantum corrections is to use an iterative approach, calculating the Casimir corrections for a trial form of b (y) in the quantum region b1 ∼ κ2/9 ≤ b ≤ κ2/9 )0.6551 , and expressing the results, in the approximation of neglecting dependence on position in the compact six-manifold, or in other words, of neglecting all but the leading term, in the harmonic expansions of Lukas, Ovrut, and Waldram [67], as expansions of the form (132), depending only on b, and not on c, or any higher derivatives of b with respect to y, but with coefficients that now depend on the trial form of b (y) in the quantum region, and possibly, also, intermediate powers of b, not present in (132), then solving the field equations and boundary conditions with these Casimir coefficients, and if the resulting b (y) differs from the trial b (y), repeating the process with a new trial b (y), until a self-consistent solution is found for b (y) in the quantum region, that joins smoothly onto the bulk power law (206), with the required value of B ∼ 104 for TeV-scale gravity, at b≫ κ 29 )0.6551 . Of course, the possibility of finding such a self-consistent b (y), in the quantum region, is likely to depend on the choice of the compact six-manifold M6. 2.4.2 The region near the inner surface of the thick pipe We now consider the region near the inner surface of the thick pipe, to find out whether a value of B greater than around 105κ2/9 could occur, as required for TeV-scale gravity. From the discussion above, we know that the proximity force approximation will not be adequate. However, we can start by assuming that the t(i) functions have expansions of the form (132), on page 61, near the inner surface, and see whether the solution of the Einstein equations can self-consistently reproduce the t(i) functions that we started with, and also produce the required large value of B . I shall consider first the case where t(3) = t(1), near the inner surface of the thick pipe, so the C(2)n and C n will be given by (134), in terms of the C(1)n . We recall from subsection 2.3.2, on page 35, that the value b1 of b at the inner surface of the thick pipe cannot be larger than around 1.2κ2/9, which corresponds to |χ (M6)| ≃ 1, and from subsection 2.3.6, on page 66, that it cannot be smaller than around 0.2κ2/9, which corresponds to |χ (M6)| ≃ 7× 104. Let us consider, first, the case where all the C(1)n are zero, except for a single value of n. Then (203) becomes: (n+2) b8+3n 4 + n C(1)n (n+2) b8+3n = 0 (211) for some fixed value of n ≥ 0. Let us now consider b in the region ∣C(1)n κ2/9 (212) so that we can neglect the term − 8 in the square root, and the term 4 , in (211). Then (211) becomes: (n+2) b8+3n 4 + n C(1)n (n+2) b8+3n = 0 (213) Let us try for a power law trajectory, c = σ , (214) for some numerical constant σ, and exponent ρ. We then find that C(1)n must be negative, which is satisfied for the energy-momentum tensor of the three-form gauge field, (159), and corresponds to a positive contribution to the energy density, T00, and that: ρ = − (6 + 3n) , σ = n (4 + n) 3 (16 + 24n+ 3n2) (215) Thus we see that, in contrast to the situation for the bulk-type trajectories, where every trajectory is approximately a power law trajectory, for a certain range of b, that depends on the trajectory, there is now just a single power law trajectory. If we now try for a solution of the form c = c0 (1 + s), where c0 is the power law trajectory, and s (b) is a small perturbation, we find that s = Sbη, η = 16 + 24n+ 3n2 2 (4 + n) (216) where S is a constant of integration. Now η ≥ 2 for n ≥ 0, so as we follow trajectories near the power law trajectory, in the direction of decreasing b, they tend to converge towards the power law trajectory, in the sense that s decreases in magnitude, so in this sense, the power law trajectory is an attractor, in the direction of decreasing b. Now in the proximity force case, using t(3) = t(1) and (133), the equation (203) can be written, in the region where −κ2t(1) ≫ 1 , as: κ2t(1) + −2t(1) + b dt(1) = 0 (217) The small b power law solutions, (214) and (215), can all be written as: c2 = −b κ2t(1) (b)− 4 κ2t(1) (x) (218) The integral is convergent at x → ∞, because t(1) (x) decreases at least as rapidly as x−8, as x → ∞. Now (218) does not give an exact solution of (217), except when t(1) (b) is a pure power of b. In fact, on substituting (218) into (217), the left-hand side of (217) reduces to: κ2t(1) +  κ2t(1) (x) (219) When t(1) (b) is not a pure power of b, there are cross terms between different powers of b, that do not cancel out of the first term in (219), but the second term in (219) is a linear combination of the contributions from different powers of b. However, if t(1) (b) is a linear combination of two different pure powers, say b−(8+3n) and b−(8+3m), with n and m large, then the remainder term, (219), is ∼ 1√ , while the leading terms, in (217), are ∼ n or m. Furthermore, for a pure power b−(8+3n), with large n, the integral term, in (218), is of order 1 compared to the leading term. Thus it seems likely that, when the coefficients in t(1) (b) are all ≤ 0, a reasonable approximation to the small b attractor trajectory, generalizing the small b power laws (214), (215), valid when t(1) (b) is a pure power, will, in the limit of large −t(1) (b), be: κ2t(1) (b) (220) Now for any sufficiently large value of c, the trajectory passing through the point (b, c) will still be of the bulk power law type (206), even for b in the range (212). However, when all the coefficients in t(1) (b) are ≤ 0, any power law trajectory, of the type (206), will intersect the small b attractor trajectory, (220), for sufficiently small b. From the perturbative analysis carried out in connection with (216), it is clear that what actually happens, when all the coefficients in t(1) (b) are ≤ 0, is that each bulk power law trajectory, (206), curves upwards as it approaches the small b attractor trajectory, (220), and then approaches the small b attractor trajectory gradually, without ever actually crossing it. Now regarding a as a function of b again, and considering the case where C(1)n is only nonzero, for the same n as in (211), the equation (194) for ȧ , with the upper choice of sign, and dropping the term 4 in the square root, becomes: = −2c (n+2) b8+3n (221) On the unique small b power law trajectory, defined by (214) and (215), we can neglect the − 8 term in the square root in this equation, which then becomes: (4 + 2n) (4 + n) (222) Hence: a = A1 (223) where A1 is another constant of integration. For n = 0, the solution defined by (214), (215), and (223), which corresponds to b = βκ2/9 , a = A1 , where β is a constant, has the functional form of the supersymmetric solution found by Lukas, Ovrut, Stelle, and Waldram [68], for the case when the compact six-manifold is a Calabi-Yau threefold with h11 = 1, transformed to the coordinate system where the metric has the form (94). We now have to consider whether these solutions can be self-consistent, when we recalculate the expansion coefficients C(i)n in (132) for b (y) corresponding to these solu- tions, in accordance with the discussion in the preceding subsection. We can no longer assume that t(3) = t(1), but since we are now just considering orders of magnitude, it will be adequate to consider the case where t(3) = t(1). Let us suppose that in the quantum region, where b )0.6551 , we have a power law, c = , with γ ≥ 0, which joins continuously onto the bulk power law (206), at b )0.6551 . Then )0.6551(1+γ) ≃ 4001+γ, (224) where I used that B ≃ 104 for TeV-scale gravity. Now by the preceding subsection, we expect a term κ− 9 C(i)n )8+3n , in (132), to be accompanied by an additional term ∼ κ− 229 C(i)n )8+3n c8+3n. This now becomes: 9 C(i)n )8+3n )γ(8+3n) 9 C(i)n )(1+γ)(8+3n) (225) and thus contributes expansion coefficients C ñ ≃ 4008+3ñC(i)n to the recalculated t(i), where ñ = (1 + γ)n + 8 γ. If we now consider the case where C ñ is significant only for one value of ñ, and assume that the significant C ñ is negative, then by (214) and (215), the recalculated c, calculated from the recalculated t(i), is ∼ 4004+ )3+ 3 , (226) where I dropped all factors of order 1. This is in agreement with the c we started with at the upper limit of the quantum region, where b ≃ 400, but for all γ ≥ 0, and all n ≥ 0, increases much more rapidly with decreasing b than the c we started with, and for b ∼ κ2/9, is very large compared to the c we started with. We would not expect the discrepancy to be any smaller if more than one C ñ is significant, provided all the significant C ñ are negative. Thus we cannot obtain a self-consistent solution if all the significant C ñ are negative. Now since the proximity force approximation is not valid in the quantum region b1 ∼ κ2/9 ≤ b ≤ κ2/9 )0.6551 , we cannot assume that t(3) = t(1), but for the purpose of illustration, I shall continue to consider the case where t(3) = t(1). Then by the result above, if a self-consistent solution with B ≃ 104 exists, the self-consistent t(1), in (130) and (132), must contain at least one significant C ñ that is positive, which corresponds to a negative contribution to the energy density T00. This is expected to be possible for Casimir energy densities, whose sign often depends on the detailed geometry of a physical situation [258], although recent results of Kenneth and Klich [259] and Bachas [260] have shown that Casimir forces are always attractive in certain circumstances. If t(1) is dominated by a single term κ− 9 C(1)n )8+3n in (132), where C(1)n is positive, then as noted after (214), there is no small b power law solution of (213), for that C(1)n . Instead, the generic solution of (213), with C n > 0, with c viewed as a function of b, in the quadrant b > 0, c > 0 of the (b, c) plane, has a peak at a point (bp, cp) where 6n2 + 36n+ 64 + 3n + 4 )3+ 3n . (227) Every point in the quadrant b > 0, c > 0 of the (b, c) plane must now lie on a trajectory that has such a peak, for if dc is positive at the point (b, c), and we follow the trajectory in the direction of increasing b, the terms in (213) proportional to 1 b8+3n will eventually become negligible, and the trajectory will then take the form (206), for some B > 0, so that dc is now negative. Now suppose that dc is negative at the point (b, c), and follow the trajectory in the direction of decreasing b. If C(1)n (3n+6) b8+3n small compared to c , then the trajectory has the form (206), so C(1)n (3n+6) b8+3n increases more rapidly than c with decreasing b, and a value of b > 0 will be reached where the two terms are comparable in magnitude. Then either the two terms continue to be comparable in magnitude as b decreases further, or C(1)n (3n+6) b8+3n becomes large compared to c , as b decreases further. But if the two terms continue to be comparable in magnitude as b decreases further, then we have c ≃ αC(1)n κ (3n+6) b8+3n , for some constant α > 0, for all b from the value > 0 where the two terms first become comparable in magnitude, down to b = 0. But this is the characteristic property of the small b power law trajectory (214), (215), and the trajectories that asymptotically approach it, in the direction of decreasing b, in the sense described after (216), and, as noted after (214), there is no small b power law trajectory for C(1)n > 0. Thus C (3n+6) b8+3n must become large compared to c , as b decreases further, beyond the value > 0 where the two terms are comparable. The trajectory then tends to the form (4 + n) 9 (2 + n) )6+3n )6+3n , (228) where bs > 0 is a constant of integration, so that is positive. Now if such a peak occurs, then for self-consistency, when we include the Casimir energy density corrections beyond the proximity force approximation, as discussed in the preceding subsection, the peak must occur at the upper limit of the quantum re- gion, so bp ∼ 400κ2/9. This is because we have the bulk power law (206) to the right of the peak, and from the discussion above, we cannot self-consistently have any power law c ≃ , with γ ≥ 0, in the quantum region. The peak will be broad, with width ∼ bp, so in the region of the peak, we can treat c as a constant ∼ 400. The additional term ∼ κ− 229 C(1)n )8+3n c8+3n, which by the preceding subsection, we expect to ac- company the term κ− 9 C(1)n )8+3n in (132), now becomes ∼ κ− 229 C(1)n )8+3n Substituting this into the right-hand side of (227), and dropping all factors of order 1, we see that we have self-consistency in the region of the peak. However, from compar- ison of (227) and (228), we see that cannot be large compared to 1, because if was much larger than 1, (228) would allow c to become substantially larger than the maximum value given by (227), in the region where (228) is still valid. Thus since b1 cannot be smaller than bs, we cannot obtain a self-consistent result with b1 ∼ κ2/9, and bp ∼ 400κ2/9, in this way. A similar result is also expected when no single term is dominant in t(1) in (132), because if the C(1)n term in t (1) is multiplied by c8+3n for all n ≥ 0, with c a constant > 1, the effect is to multiply the minimum possible value of b, as derived in subsection 2.3.6, on page 66, from the Giudice-Rattazzi-Wells estimate [11] of the effective expansion parameter for quantum gravity in eleven dimensions, by c. Since there is no difficulty obtaining self-consistency at the upper limit b ∼ 400κ2/9 of the quantum region, but we cannot obtain consistency inside the quantum region for any power law c ≃ , with γ ≥ 0, we now try for a power law of this form with γ < 0. We see that if γ = −1, corresponding to a linear dependence of c on b, then ñ = (1 + γ)n + 8 γ is independent of n, and equal to −8 . We can now simply have Bq ≃ κ2/9, in which case, if the magnitude of C(1)n in (132) is ∼ 0.23nC(1), for some constant C(1) of order 1, as suggested by the minimum value of b estimated in subsection 2.3.6, the additional terms ∼ κ− 229 C(1)n )8+3n c8+3n sum up to no more than around κ− 9 (1− 0.23)−1C(1) ≃ κ− 229 C(1). And considering the equation (213) for n = −8 , we see from (214) and (215) that we do indeed have a unique linear solution, , (229) provided that the effective C is found to be positive. We see that C will be self-consistently determined as a fixed number of order 1, provided that this number is positive. Thus it seems reasonable to expect that for around fifty percent of all possible choices of a smooth compact quotient M6 of CH3 or H6 that is a spin manifold, a spin structure on M6, and a topologically stabilized configuration of vacuum Yang- Mills fields on the inner surface of the thick pipe, consistent with Witten’s topological constraint [45], a value of B larger than κ 9 will be found by this mechanism. The actual value of b at which the self-consistent quantum linear relation (229) transforms into the classical relation (206), and the corresponding value of B, will be determined by how close to the self-consistent quantum linear relation (229) the system is set by the boundary conditions at b = b1 ∼ κ2/9. We note that η, in (216), is equal to −10 when n = −8 , so the linear solution (229), of (213) with n = −8 , is a very strong attractor in the direction of increasing b. However this has not taken into account the fact that in the presence of deviations from the self-consistent linear relation (229), the equation to be solved will no longer be precisely (213), with n = −8 . We also note, from the discussion above, that it is consistent for c to be approximately constant in the region of the peak at b ∼ κ2/9 )0.6551 , although not for b≪ κ2/9 )0.6551 we expect the transition from (229) to (206) to occur smoothly across a broad peak of width ∼ κ2/9 )0.6551 The linear relation (229) means that b depends exponentially on y in the quantum region: b = bq exp (y − yq) , (230) where bq ≡ κ2/9 )0.6551 and yq ≡ 0.3449κ2/9, for agreement with (207) at b = bq, for y0 = 0. The thickness in y of the quantum region is ∼ κ2/9 ln bqκ2/9 , which for TeV-scale gravity, with bq greater than around 2000κ 9 , is ∼ 8κ2/9. We note that if the percentage of possible choices of M6, its spin structure, and the vacuum Yang-Mills fields, for which the thickness in y of the quantum region is greater than a certain value, decreases roughly exponentially with that value, then the percentage of possible choices, for which B is greater than a certain value, will be roughly given by a fixed negative power of that value. From (222) and (223), with n = −8 , we see that in the quantum region, where the linear relation (229) applies, a also depends linearly on b: a = A1 , (231) where A1 is a constant of integration. However this linear dependence of a on b in the quantum region, for n = −8 , is a consequence of the proximity force relation t(3) = t(1), which would apply for compactification on flat R5 times the compact six-manifold M6, and as noted above, there is no reason to expect this relation to hold when a and b depend nontrivially on y. Consideration of the special case where this relation holds was adequate for the order of magnitude studies above, where only the dependence of b on y was considered, but to determine the possible dependences of a on b in the quantum region, I shall now assume that the t(i), in (130), on page 58, are constrained only by the conservation equation (131). Considering the region κ2/9 ≪ b≪ bq, only the terms κ− , in the self-consistent versions of the expansions (132), on page 61, will be significant. The relevant equations are now (194) and (195), on page 87, with the upper choice of sign, and t(i) ≃ κ− , (232) where the C are numerical constants, to be determined self-consistently, as discussed above. The only possible power-law dependence of c on b, with this form of the t(i), is again c = σ b , where σ is a numerical constant, and this linear dependence of c on b leads self-consistently to the form (232) of the t(i) in this region, as before. However a no longer has to depend linearly on b in this region, so we try an ansatz a = A1 . (233) The conservation equation (131) then reduces to: (4τ + 6)C − 4τC(1)− 8 − 6C(2)− 8 = 0. (234) Choosing C and C as independent, equations (194) and (195) reduce in this region (2τ + 4)σ = 6σ2 + (235) − σ2 + σ 6σ2 + − (τ + 2)C(1)− 8 + (τ + 1)C = 0, (236) from which we find: (2τ + 4) 2 − 6 (237) (2τ + 3) σ2 = (τ + 2)C − (τ + 1)C(3)− 8 . (238) Thus almost any value of τ can be obtained, if there exists a suitable smooth compact quotient M6 of CH3 or H6 that is a spin manifold, and a choice of a spin structure on M6 and a topologically stabilized configuration of the Yang-Mills gauge fields on the inner surface of the thick pipe, that results self-consistently in the appropriate values and C . In particular, the bulk power law value τ = −0.7753 is one of the two solutions if C > 0 and C = 0. However τ = −2 would imply that the square root vanished, so that we could not conclude that all three Einstein equations would be satisfied. The calculation of C and C for a particular example requires, in particular, the calculation of the propagators and heat kernels for all the CJS fields on a flat R5 times uncompactified CH3 or H6 background, as appropriate. These can be obtained from the corresponding propagators and heat kernels on a flat R5 times CP3 or S6 background, which can be calculated by using the Salam-Strathdee harmonic expan- sion method [244], and summing the expansions by means of a generating function. This calculation is currently in progress for CH3, and the scalar heat kernel on CH3, obtained by this method, is presented in subsection 4.1, on page 196. For t(3) 6= t(1), we can no longer study trajectories near the self-consistent linear trajectory c = σ b by perturbing only the dependence of c on b as c = σ b (1 + Sbη), where S is a small constant of integration. The dependence (233) of a on b also has to be perturbed as a = A1 (1 + Ubη), where U is a small constant of integration, and the t(i) functions (232) in the region κ2/9 ≪ b ≪ bq have to be perturbed as t(i) ≃ κ− 229 C(i)− 8 1 + V (i)bη , where the V (i) are small constants. The Einstein equations (194) and (195) and the conservation equation (131) impose three relations among the six constants describing the perturbation, and we would now expect the exponent η to depend on ratios of the small constants S, U , and the V (i), rather than having the unique value −10 as for the case when t(3) = t(1). The possibility of having both a self-consistent quantum region, in which c increases linearly with b as σ b , with σ a numerical coefficient of order 1, and a self-consistent classical region where c satisfies the classical bulk power law (206), on page 89, is due to the presence, beyond the proximity force approximation, of additional terms 9 C(i)n,n κ2/9c )8+3n , (239) with n ≥ 0, in the expansions (132), on page 61, of the t(i) functions in (130), on page 58. These terms sum to finite constant terms κ− at low orders of perturbation theory in the quantum region, provided σ is not too large, and thus result self-consistently in the linear dependence of c on b in the quantum region, provided the C are consistent with σ2 > 0, as determined by (237) and (238). While if c is related to b by the classical bulk power law (206), and b is larger than bq = κ )0.6551 , so the value of c given by (206) is smaller than the value that would be given by extrapolating the linear relation from the quantum region, then the terms (239) rapidly decrease in magnitude with further increase in b, and quickly become negligible, so that the classical bulk power law (206) becomes self-consistent. Thus it is consistent for the quantum region to transform into the classical region at any point bq > b1, and the value of bq at which the transition occurs in a particular ex- ample, and consequently the value of B, will depend on how close to the self-consistent linear trajectory c = σ b , with σ determined by (237) and (238), the system is set by the boundary conditions at b = b1, and on whether the self-consistent linear trajectory attracts or repels neighbouring trajectories, in the direction of increasing b, and how strongly it does so. From the discussion above, we see that for t(3) 6= t(1), the space of relevant neighbouring trajectories is three-dimensional, and parametrized, for example, by small quantities S, V (1), and V (3). The actual transition from the quantum region to the classical region will take place gradually, over a broad peak of width around bq, as discussed just before (230). The presence of the additional terms (239) in the t(i) functions, beyond the prox- imity force approximation, follows from their presence in the local terms formed from powers of the Riemann tensor, and the components (96), on page 34, of the Rie- mann tensor for the metric ansatz (94). In particular, RABC D contains both a term D (h), which for a local term in the quantum effective action Γ formed from 4+3m powers of the Riemann tensor leads both for CH3, on using the CHn Riemann tensor components (71), on page 28, and also for H6, to terms in the t(i) functions of the form κ− )8+6m , in agreement with the even order terms in (132), and a term GACδB D, which for the same term in Γ leads to even order terms of the form (239). We note, furthermore, that since, on a power law trajectory, ȧ is equal to ḃ times a fixed number of order 1, the RµAν B and RAµB ν components, and the ȧ terms in τ , will lead both in the quantum region and the classical region to terms similar in magnitude to the terms (239). And since ä and b̈ are equal, on a power law trajectory, to ḃ times fixed numbers of order 1, except that b̈ vanishes on the self-consistent linear trajectory in the quantum region, the Rµyν y and Ryµy ν components will also lead both in the quantum region and the classical region to terms similar in magnitude to the terms (239), and the RAyB y and RyAy B components will lead in the classical region to terms similar in magnitude to the terms (239). Now by definition, the metric gµν , in the metric ansatz (94), has de Sitter radius equal to 1. Hence the value a1 of a, at the inner surface of the thick pipe, is equal to the observed de Sitter radius (22). We recall that in subsection 2.3.2, on page 35, we found, by combining an estimate of the d = 4 Yang-Mills coupling constants at unification, with the Hořava-Witten relation (45), that b1 ≃ 1.2772 |χ(M6)| , when the compact six-manifold M6 is a smooth compact quotient of CH3, and b1 ≃ 1.1809 |χ(M6)| when M6 is a smooth compact quotient of H6. Thus from (233), we find: |χ (M6)| 1.2772 × de Sitter radius, (240) when M6 is a smooth compact quotient of CH3, and the same relation, with 1.2772 replaced by 1.1809, when M6 is a smooth compact quotient of H6. And from matching the bulk power law (210) for a in terms of b to (233), at b = bq, we find: A = A1 )τ+0.7753 )0.6551τ+0.5079 . (241) Thus: |χ (M6)| 1.2772 )0.6551τ+0.5079 × de Sitter radius (242) for a smooth compact quotient of CH3, and the same relation, with 1.2772 replaced by 1.1809, holds for a smooth compact quotient of H6. 2.4.3 The boundary conditions at the inner surface of the thick pipe Now, treating b as the independent variable, the boundary conditions (169), on page 82, become: b=b1+ −5t̃[1](1) + 6t̃[1](2) b=b1+ 4t̃[1](1) − 3t̃[1](2) (243) where b1 ≡ b (y1). The coefficients t[1](i) receive contributions from the Lovelock-Gauss- Bonnet terms, given by (188), on page 85, for quotients of CH3, and by (192) for quotients of H6; from the leading terms in the Lukas-Ovrut-Waldram harmonic expan- sion, on the compact six-manifold M6, of the energy-momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe; and from Casimir effects on the inner surface of the thick pipe. The terms in (188) and (192) that involve negative powers of a are negligible at the inner surface of the thick pipe, so the Lovelock-Gauss-Bonnet terms, for quotients of CH3, are: t̃[1](1)LGB = λ2b41 , t̃[1](2)LGB = λ2b41 (244) and for quotients of H6, they are: t̃[1](1)LGB = λ2b41 , t̃[1](2)LGB = λ2b41 (245) It would seem reasonable to expect that the contributions to the coefficients t̃[1](i), from the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of the energy- momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe, will be roughly a positive numerical multiple of the contribu- tions that result from embedding the spin connection in the gauge group for CH3, as given in (181), on page 84, for the outer surface of the thick pipe. Thus we estimate the vacuum Yang-Mills field contribution to the coefficients t̃[1](i) as: t̃[1](1)YM ≃ − 24 λ2b41 N, t̃[1](2)YM ≃ − 8 λ2b41 N, (246) where the numerical constant N ≥ 0 is given by 96V (M6) hhCDhEF trF DF , (247) in terms of the topologically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe, with V (M6) given by (99), on page 36, for a smooth compact quotient of CH3, and by (100), for a smooth compact quotient of H6. Now the results (188), (192), (244), and (245), for the Lovelock-Gauss-Bonnet con- tributions, have been calculated assuming that the Riemann tensor in the Lovelock- Gauss-Bonnet term in (48), on page 23, is the d = 10 Riemann tensor calculated from the induced metric GUV on the Hořava-Witten orbifold hyperplanes, and not the re- striction to the orbifold hyperplanes of the d = 11 Riemann tensor. This would seem to be a reasonable assumption, because it implies that for a Calabi-Yau compactification [9], with the standard embedding of the spin connection in the gauge group, the Rie- mann tensor term in (48) is automatically equal to −1 times the Yang-Mills term, at an arbitrary point of the Calabi-Yau moduli space. For the compactifications consid- ered here, it implies that when we go beyond the proximity force approximation, there are no related terms with factors of c = db that can cancel the 1 factor in the region κ2/9 < b < bq = κ )0.6551 , where, from the previous subsection, c is ∼ b If we assume, by analogy with this, that when we go beyond the proximity force approximation, there are also no terms related to the higher order terms in the expan- sions (136), on page 62, with enough powers of c to cancel all the powers of 1 in those terms in the region κ2/9 < b < bq, then the boundary conditions (243), at the inner surface of the thick pipe, cannot be solved for any value of b1 much larger than κ This is in agreement with the result from subsection 2.3.2, on page 35, that to fit a reasonable estimate of the unification value of the observed d = 4 Yang-Mills coupling constants, the value of b1 cannot be larger than around 1.2κ 2/9, which corresponds to |χ (M6)| ≃ 1. On the other hand, it would seem reasonable to expect that for perhaps around three percent of choices of a smooth compact quotient M6 of CH3 or H6 that is a spin manifold, a spin structure on M6, and a topologically stabilized configuration of vacuum Yang-Mills fields tangential to M6, a solution of the boundary conditions (243) will exist with B larger than around 5κ2/9, and b1 no smaller than around twice the minimum value ≃ 0.2κ2/9 derived in subsection 2.3.6, on page 66, from the GRW estimate [11] of the expansion parameter of quantum gravity in eleven dimensions, so that the boundary conditions can be solved perturbatively. For let us suppose that we have done the one-loop calculation for a trial classical metric GIJ , and have an approximation to the expansions (132), on page 61, of the t that contains terms with at least two different powers of b. Then from the preceding subsection, we expect there to be roughly a fifty percent chance of having at least a small region b1 < b < bq in which c increases roughly linearly with b, so that the lowest power of 1 in the self-consistent t(i) will be zero, as in (232). The approximation to the t(i) contains only a few terms, so for b somewhat smaller than κ2/9, it will be dominated by the terms with the largest power of 1 , and there is around a fifty percent chance that these will lead to a small b power law trajectory. And similarly, the perturbative approximation to the expansions (136), on page 62, will contain only a few terms, so in this approximation, the ratio of the right-hand sides of the boundary conditions (243) will have an approximately fixed value for b somewhat larger than κ2/9, and generically some other approximately fixed value for b somewhat smaller than κ2/9. Now on any power law trajectory in the bulk, da is a fixed multiple of a , where the fixed multiple is characteristic of the trajectory, so c is a fixed multiple of c on the trajectory. Thus in this approximation the boundary conditions (136) generically have no simultaneous solution for any value of b1 either much larger or much smaller than κ2/9, while there is perhaps a fifty percent chance there will be a solution in the region with b1 ∼ κ2/9 where each term is ∼ 1 in magnitude, and the ratios of the left-hand sides and right-hand sides are moving between their limiting values. Finally, if there is such a solution of the boundary conditions in this approximation, we would expect there to be roughly a fifty percent chance that it will have b1 greater than the minimum value of around 0.2κ2/9 estimated in subsection 2.3.6, on page 66, and perhaps another fifty percent chance that it will have b1 greater than around twice this value, so that the one-loop calculation would give a reasonable approximation to the correct result. 2.4.4 The classical solutions in the bulk I shall now consider solutions of the Einstein equations in the classical part b > bq = )0.6551 of the bulk, that start out in the classical region on a trajectory of the form (206), on page 89, with B large compared to κ2/9, and follow such solutions further into the bulk, towards the region where c is no longer large compared to so that (204) no longer reduces to (205). From (209) and (210), we know that a is decreasing in magnitude, as b increases, in this region, and, depending on the values of the integration constant B, in (206), which is determined by the boundary conditions at the inner surface of the thick pipe, and the integration constant A, in (210), which is not determined by the boundary conditions at the inner surface of the thick pipe, and is at present a free parameter, that will eventually be determined by the boundary conditions at the outer surface of the thick pipe, we may or may not have to stop neglecting the term 4 , in the square root (197), as it occurs in the classical Einstein equation (204), before we reach the region where c is no longer large compared to I shall first consider the case where the term 4 , in the square root (197), continues to be negligible, into the region where c is no longer large compared to , so the equation to study is (204). We first note that (204) has the solution c = . However, when the terms 4 and 2 κ2t(3) are negligible in the square root R, defined in (197), as presently assumed, R vanishes identically for this solution, hence we cannot conclude, from (198), that (194), (195), and (131) imply that all three Einstein equations are satisfied. And indeed, this special solution, of (204), does not correspond to a solution of all three Einstein equations. Now the second term in the parentheses, in (204), is smaller in magnitude that times the first term in the parentheses, for all c ≥ , and tends to 0 relative to the first term as c→ from above. Hence for c ≥ , but near , (204) reduces to: 2 (3c2 − 4) (248) The solution of this is:  (249) where B1 > 0 is a constant of integration, different from the constant of integration, B, in (206). Now (249) gives:  (250) This is negative for b < B1, vanishes for b = B1, at which point c = , and positive for b > B1. Thus we see that the solutions (249), with different values of B1, all osculate with the line c = , at different points along this line, and that, moreover, as each solution (249) passes the point b = B1, in the direction of increasing b, it moves from the positive sign to the negative sign of the square root, in (248), or in other words, from the upper sign, to the lower sign, of the square root, in (199). dc now becomes positive, so, if the term 4 , in the square root, R, defined in (197), remains negligible, c now starts increasing without limit, and, when c is large compared to , we reach another power law region, where, instead of (205), (206), and (207) we have (202), and )7.8990 (251) for some B2, which in general will be different from both B and B1. (251) corresponds 6 + 2 y3 − y ≃ 0.7558 y3 − y )0.1449 (252) for some y3 > y1. Now we already found, in the first region of the classical part of the bulk, where we choose the upper sign in (194), (195), and (199), and c is sufficiently large, that we are on a bulk power law trajectory (205), (206), and (207), that a depends on b through a power law, (209) and (210), so that a decreases as b increases. In the second bulk power law region, where we have the lower sign in (194), (195), and (199), and c is sufficiently large, that we are on a bulk power law trajectory (202), (251), and (252), we have: = −3.2247 (253) so that: a = A2 )3.2247 (254) where A2 is constant, which will in general be different from A, in (210), so that a now decreases much more rapidly, with increasing b, than it did in the first bulk power law region, (209) and (210). A convenient interpolating function, that agrees with (207) in the first bulk power law region, when y0 is set to 0, and agrees in form with (252), in the second bulk power law region, is: (1− αy) = 1.4436 )0.3449 (1− αy)0.1449 (255) To fix α, we note that, from (201) and (204), cdc = b̈ should vanish, when c = ḃ = The zero of b̈, at y < 1 , is at: y = − 6− 387 6− 241 0.4747 (256) Imposing the requirement that ḃ = at this value of y, we find that: 0.9094 (257) Thus in terms of B, the zero of b̈, at y < 1 , is at: y ≃ 0.4747 = 0.5220B (258) Thus B1, in (249), which is the value of b, at the zero of b̈, is given by: B1 ≃ 1.2664B (259) In the second bulk power law region, the interpolating function, (255), approxi- mately reduces to: 6 + 2 6 + 1 (260) Comparing with the bulk power law in the second bulk power law region, (252), we see that: = 1.0996B (261) 6 + 1 B = 1.8327B (262) A convenient interpolating function for the dependence of a on b, that agrees with (209) in the first bulk power law region, and agrees in form with (253), in the second bulk power law region, is: )(2− 12 )0.7753 1 + 4.9154 )2.4495 ) (263) The coefficient of b 6, in the denominator of (263), has been chosen so that da = −2a when c = , so that b = B1, as follows from (221), on neglecting the Casimir energy term in the square root. Comparing with (254), we see that: A2 = 0.2034A )2.4495 (264) From (194), with the lower choice of sign, together with (251) and (254), we find that, in the second bulk power law region, ȧ is related to a, by: − ȧ ≃ )1.1394 (265) where: ≃ 2.7944 )0.8777 )6.0550 A2 ≃ 0.002109 )0.8777 )1.4556 A (266) The power law (265) is analogous to (206), on page 89, so by analogy with the region near the inner surface of the thick pipe studied in subsection 2.4.2, on page 93, we would expect it to be possible to realize a large value of à by the occurrence of a quantum region ãq = κ )0.5326 > a > a2 ∼ κ2/9 adjacent to the outer surface, in which −ȧ self-consistently grows linearly with a, and −ȧ and a increase exponentially with the geodesic distance (y2 − y) from the outer surface. The boundary conditions at the outer surface will determine both the value a2 of a at the outer surface, and the integration constant Ã, in (265). The integration constant A is then determined in terms of Ã, B, and κ2/9, by (266), which then determines the relation between a and b, by (263), and hence the value of b, at the outer surface of the thick pipe. This then determines y2, or in other words, the value of y, at the outer surface of the thick pipe, by (255). We note that for this type of solution, in which a is comparable to κ2/9 at the outer surface of the thick pipe, the term 4 , in the square root in the Einstein equations (194) and (195), which is the only term in (194) and (195) that depends on the existence and sign of the effective cosmological constant in the four observed dimensions, is not very important, since it is negligible except near the outer surface, where there will be Casimir terms of comparable magnitude. Before considering this type of solution in more detail, I shall now look for solutions such that both a and b are classical, or in other words, large compared to κ2/9, at the outer surface of the thick pipe. We will see that in contrast to the solutions where a is comparable to κ2/9 at the outer surface, the 4 term, in the square root in (194) and (195), is essential for obtaining this type of solution, which thus will exist only when the effective cosmological constant, in the four observed dimensions, is greater than zero. 2.5 Solutions with both a and b large compared to κ2/9, at the outer surface of the thick pipe I shall now look for solutions of the Einstein equations, and the boundary conditions (170) at the outer surface of the thick pipe, such that both a and b are large compared to κ2/9, at the outer surface, assuming that the boundary conditions at the inner surface have already been solved, such that in the first bulk power law region, we are on a trajectory (206), (207), and (210), with a large value of B , but A not yet determined. We see from (170), (181), (188), (192), the expansions (138), and the relations (45) and (46), that when both a and b are large compared to κ2/9, at the outer surface of the thick pipe, the largest terms, in the right-hand sides of the boundary conditions, (170), are the Yang-Mills terms, (181), forCH3, and the Lovelock-Gauss-Bonnet terms, (188), for CH3, and (192), for H6, and that these terms are of order 1 , times κ , or κ , and thus negligible. Thus we are now looking for solutions such that , and ḃ , are zero, at the outer surface of the thick pipe. From (194), we therefore require that 8 , at the outer surface of the thick pipe, for this type of solution, or in other words, b = 2a, at the outer surface of the thick pipe. Furthermore, the term 4 , in the square root, in (194), arose from the Ricci tensor of four-dimensional de Sitter space, Rµν (g), in (97), and would have been absent, if the effective cosmological constant, in the four observed dimensions, had been zero, and would have had the opposite sign, if the effective cosmological constant, in the four observed dimensions, had been zero. Thus there will be no solutions, such that both a and b are large compared to κ2/9, at the outer surface of the thick pipe, unless the effective cosmological constant, in the four observed dimensions, is greater than zero. We now have to study the coupled equations (194) and (195), when the t(i) are negligible, but the term 4 , in the square root, is not negligible. We can still express the two equations as first order differential equations for a, and c = ḃ, as functions of b, but the two equations are now coupled. Equation (194), with the upper choice of sign, now becomes: 6c2 − 8 + 4 (267) And (195), with the upper choice of sign, now becomes: 3c2 − 4 6c2 − 8 + 4 b (268) Qualitatively, when the b term starts to become significant, in the square root in the right-hand side of (268) the trajectory, in the (b, c) plane, starts to peel off below the = 0 trajectory. We are looking for a solution where ȧ , or in other words, c , and , or in other words, c , both tend to zero, at the boundary, while b tends to 2, at the boundary, and b tends to a finite nonzero limit. Thus c→ 0 at the boundary, while da must remain finite. Then 4 b needs to increase rapidly enough, to compensate for the decrease in 6c2, so as to keep 6c2 − 8 + 4 b2 > 0, as c tends towards 0. And as c → 0, will be determined by the − 4 term, which → ∞. Now the equation dc = − 4 has the solution c2 = 8 ln , where b2 is a constant of integration, that we would like to identify as b (y2), the value of b at the outer surface of the thick pipe. This solution applies in the region c → 0, b ≃ b2, b ≤ b2, so we can expand the logarithm, to find: (269) If we now define u ≡ b , the equations (267), and (268) become: 6c2 − 8 + 4u2 (270) 3c2 − 4 6c2 − 8 + 4u2 (271) And the boundary conditions, at b = b2, become: 2, c = 0 (272) We now see that there are two different possible behaviours of u near the boundary, consistent with (269), (270), and (271). Specifically, expanding u in the small quantity , as u = 1 + α , we find, from (269), and (270), that − α = 3− 6 + 2α (273) which has the solutions: α = −5 , α = −3 (274) We note that the first of these is only a solution, for the particular sign of the square root in (273), while the second is a solution for both signs of the square root, since the square root vanishes for it. Both the solutions (274) are consistent with (271), and substituting one of them into (271), fixes the term in c2, that is quadratic in Then substitution into (270) fixes the quadratic term in u, and so on. Now (270) and (271) imply that: 6c2 − 8 + 4u2 6c2 − 8 + 4u2 6c2 + u2 (6c2 − 8 + 4u2) (275) Hence 6c2 − 8 + 4u2 = 0 is a solution of (270) and (271). However, the square root, R, defined in (197), vanishes identically for this solution, when the t(i) are negligible, so we cannot infer, from (198), that 6c2 − 8 + 4u2 = 0 is a solution of all three Einstein equations, and, in fact, it does not correspond to a solution of all three Einstein equations. It is, in fact, the generalization, to the case where u 6= 0, of the line c = that the trajectories in the (b, c) plane, that corresponded to actual solutions of the Einstein equations, in the limit u = 0, osculated with, as they switched from the first to the second branch of the square root, in (194) and (195). We see, furthermore, that the case α = −3, in (274), satisfies 6c2 − 8 + 4u2 = 0, to the order given, and thus is the c → 0 limit of this special solution of (270) and (271), that does not correspond to a solution of all three Einstein equations. We note, furthermore, that this special solution, of (270) and (271), never rises above the line , in the (b, c) plane. It in fact approaches this line from below, as b → 0, since u → 0, as b → 0. Furthermore, when 6c2 − 8 + 4u2 = 0, (270) reduces to du hence u = , where, by (272), b2 is the integration constant in (269). Hence , which does, indeed, solve (271). Considering, now, the case α = −5 , in (274), we see that 6c2−8+4u2 ≃ 8 c2 near the boundary, hence the square root, R, is nonvanishing, as soon as we move away from the boundary, so, by (198), this solution will correspond to a solution of all three Einstein equations. Furthermore, du starts positive, specifically du at b = b2, hence u decreases, as b decreases downwards, away from b = b2, hence, provided du never becomes negative, and the square root stays real, the square root is bounded above, by 6c ≃ 2.45c, hence, by (270), we have 3u 1.78u , and, by (275), we have d (6c2 − 8 + 4u2) ≤ − 24− 6 6c2 − 8 + 4u2 ≃ −9.30 c 6c2 − 8 + 4u2, hence du never does become negative, and the square root does stay real. We can also confirm directly from (271), by considering separately the cases and c ≤ , that dc is negative irrespective of the value of u, provided the square root is real. Furthermore, ≤ u ≤ )1.77 , hence , hence d (6c2 − 8 + 4u2)√ 6c2 − 8 + 4u2 ≤ −9.30c db ≤ −10.73 db (276) Hence 6c2 − 8 + 4u2 ≥ 2.68 − arcsin (277) With the bound u ≤ )1.77 , this implies that 6c2 − 8 is positive for b < 0.61, and is greater than 9.86 for b = 0.2, by which point u2 < 0.006. Thus this solution merges into a solution of (204), as b continues to decrease, and for c large compared , will follow a trajectory of the form (206), in the (b, c) plane, with B a fixed number of order 1, that will be the same for all solutions, of this type. Thus we do, indeed, have a solution of the boundary conditions, such that both a and b are large compared to κ2/9, at the outer surface of the thick pipe. And moreover, for solutions of this type, namely with α = −5 in (274), the constant of integration, b2, in (269), can be identified as b2 = b (y2), the value of b at the outer surface of the thick pipe. From the behaviour (269), of c near the outer boundary, we see that near the outer boundary, y2 − y ≃ b2 c, so y tends to a finite value, y2, at the outer boundary, even though dy goes to ∞, right at the boundary. y2 will be equal to a number of order 1, times the value of y at which b̈ vanishes for the interpolating function (255), on page 109, which by (258), is at y = 0.5220B. Thus the geodesic distance from the inner surface to the outer surface of the thick pipe is around B. An alternative method of studying solutions of this type, is to take the ratio of (270) and (271). Then b cancels out, and we get a single first order differential equation, that expresses du , as a function of u and c. 2.5.1 Newton’s constant and the cosmological constant for solutions with the outer surface in the classical region We now need to consider whether a solution of this type can fit the observed values of Newton’s constant, (12), and the cosmological constant, (20). Considering first the value of Newton’s constant, the current observational limits on extra dimensions in high energy physics experiments [272], and in measurements of the gravitational force at short distances [32], imply that the maximum values of y, and b, namely y2, and b2 = b (y2), are required to be sufficiently small, that a four-dimensional effective field theory description can be used, for all observations up to the present time. Assuming, provisionally, that y2 and b2 are, indeed, sufficiently small, the four-dimensional effec- tive field theory description can be obtained by following the method of Randall and Sundrum [31]. The first step is to identify the massless gravitational fluctuations about the classical solution found above. These provide the gravitational fields of the effective theory. They are the zero-modes of the classical solution, and correspond to replacing the locally de Sitter metric, gµν , in (94), on page 33, by g̃µν = gµν + hµν , where hµν is a small perturbation, that, like gµν , depends on position in the four extended dimensions, but not on y, nor on the coordinates of the compact six-manifold. We note that since the de Sitter radius of gµν has been set equal to 1, hµν is allowed to very rapidly, with “Fourier modes” of wavelength down to ∼ 10−29 of the de Sitter radius, corresponding to the current short distance limit of about a millimetre, on short-distance tests of Newton’s law, in units of the observed de Sitter radius (22). The four-dimensional effective theory follows by substituting the zero modes of the classical solution into the original Hořava-Witten action, (25) plus (28), plus the analogue of (28) for y2. To determine the value of Newton’s constant, we focus on the term, in the Einstein action term in (25), that produces the Einstein action, (10), in four dimensions. The Riemann tensor for the perturbed metric is still given by (96), on page 34, with gµν replaced by g̃µν , since the derivation of (96) did not make use of the locally de Sitter property of gµν . I shall denote the metric in eleven dimensions, with the locally de Sitter metric, gµν , replaced by the perturbed metric, g̃µν , by G̃IJ . The relevant term, in (25), is then: G̃µνR τµτν (g̃) −g̃g̃µνR τµτν (g̃) dya2b6 (278) where I replaced 1 , because we are here working in the downstairs picture, and integrating over only one copy of the bulk, rather than over two copies, one of which is reflected, as in (25), and I have denoted the coordinates on the compact six-manifold by zA. The factor V (M6) = h is given by (99), on page 36, for a smooth compact quotient of CH3, and by (100), for a smooth compact quotient of H6. To evaluate the factor dya2b6, we note that b is a monotonically increasing function of y, for the solutions considered in this subsection, so this integral is equal to a2b6, where c = db For the region b1 ≤ b ≤ bq = κ2/9 )0.6551 , we have c ∼ b , up to a factor of order 1, and a = A1 , where A1 is related to the integration constant A in the classical bulk power law (210), on page 90, by (241), on page 104. Thus: a2b6 ∼ )1.3102τ+3.9306 (τ > −3) |6+2τ |A 1 (τ < −3) (279) up to a factor of order 1. While for the region bq ≤ b ≤ b2, we note that the equations (270) and (271), on page 113, are invariant under rescaling of b by a constant factor, so for the general solution, c = f , where f (1) = 0. Thus the integration constant B, in the classical bulk power law (206), on page 89, is a fixed number times b2, and we see from the discussion around (276) and (277) that this fixed number is of order 1. We also note that the classical bulk power law (210), on page 90, for a in terms of b, will be approximately valid, up to a factor of order 1, right up to the outer surface, for the solutions considered in this subsection. Thus we find: a2b6 ∼ )5.4494 )1.3102τ+6.4652 , (280) up to a factor of order 1, where I used (241). Comparing (279) and (280), we see that for B ≫ κ2/9, the contribution from the classical region is large compared to the contribution from the quantum region for all τ > −4.9345, while for τ ≤ −4.9345, there is no enhancement of the integral for large We now note that, by definition, the de Sitter radius of the unperturbed metric, gµν , is equal to 1, so since g̃µν differs from gµν only by a small perturbation, the use of the metric g̃µν corresponds to measuring distances in units of the de Sitter radius. We therefore define a rescaled metric ḡµν by ḡµν = (de Sitter radius) g̃µν , which corresponds to measuring distances in ordinary units rather than in units of the de Sitter radius. Then from (278) and (280), together with (99) or (100), on page 36, and (240), on page 104, we find that for τ > −4.9345, the Einstein action term, in the four-dimensional effective action, is for the solutions considered in the present subsection, equal to: )1.3102τ+6.4652 −ḡḡµνRµτντ (ḡ) (281) up to a factor of order 1. Comparing with (10), on page 12, we find that for the solutions considered in the present subsection, with τ > −4.9345: )1.3102τ+6.4652 , (282) up to a factor of order 1. This is the form taken by the ADD mechanism [3, 5], for the solutions considered in the present subsection. And for τ ≤ −4.9345, we find the same result, but without the B-dependent factor. Thus for these solutions, there is no ADD mechanism, unless τ > −4.9345. Considering, now, the case of TeV-scale gravity, I shall take κ− 9 ≃ 0.2 TeV, so that κ2/9 ≃ 10−18 metres, as a representative example, which according to Mirabelli, Perelstein, and Peskin [273] will for six flat extra dimensions be just out of reach at the Tevatron, but comfortably accessible at the LHC, as I shall review further in subsection 2.6.1, on page 126, and section 5, on page 222. Then from (282) and (12) we find that )1.3102τ+6.4652 3 ∼ 1032 (283) for TeV-scale gravity, up to a factor of order 1. Now as noted above, the bulk power law (210), on page 90, is valid up to a factor of order 1, for the solutions considered in this subsection, right up to the outer surface, where b = 2a for these solutions. We also noted that b2 ∼ B, up to a factor of order 1. Thus for these solutions: )1.7753 , (284) up to a factor of order 1. Thus from (242), on page 104, we find that for TeV-scale gravity, the condition for the solutions considered in this subsection to fit the observed de Sitter radius (22), on page 15, is: )1.2674−0.6551τ 6 ∼ 1044, (285) up to a factor of order 1. The solution of (283) and (285) for the minimum value |χ (M6)| = 1 is τ = −3.103, B ∼ 1013κ2/9 ∼ 10−5 metres, (286) and the solution for the maximum value |χ (M6)| ≃ 7× 104 is τ = −3.023, B ∼ 1013κ2/9 ∼ 10−5 metres. (287) The value of bq corresponding to (286) and (287) is bq ∼ 108κ2/9 ∼ 10−10 metres, so the thickness in y of the quantum region is ∼ 20κ2/9 ∼ 10−17 metres. We note that the value of B in (286) and (287) is about a factor of 10 smaller than the shortest distance so far studied in precision sub-millimetre tests of Newton’s law [32]. Nevertheless, we cannot directly conclude that the solutions just obtained correctly reproduce the d = 4 Newton’s law over any distance even up to the de Sitter radius ∼ 1026 metres, because we have not fully satisfied the requirement for a valid reduction to a four-dimensional effective theory, due to the fact that a (y) decreases from ∼ 1026 metres at the inner surface of the thick pipe, to ∼ 10−5 metres at the outer surface. Thus any perturbation of gµν , of wavelength less than the de Sitter radius, will have wavelength less than around B, at the outer surface. We note that a2 corresponds to the warp factor of the first Randall-Sundrum model (RS1) [31], and that we live on the “wrong” boundary, from the point of view of the RS1 model, because the reverse RS1 effect is outweighed by the ADD effect [3, 5], which is absent from the RS1 model. Arnowitt and Dent [30] have studied Newtonian forces in the RS1 model, and found that Newton’s law is obtained correctly between point sources on the RS1 “Planck brane”, which corresponds to the inner surface of the thick pipe, even though there are problems with Newton’s law between point sources on the RS1 “TeV brane”. This suggests there is a possibility that Newton’s law might be obtained correctly for the solutions found in this subsection, even though the requirement for a valid reduction to a four-dimensional effective theory is not completely satisfied. However to check this would require repeating the analysis of Arnowitt and Dent for the solutions found in this subsection, and that will not be done in this paper. I shall now consider two alternative ways in which the outer surface of the thick pipe might be stabilized, consistent with the observed values of Newton’s constant and the effective d = 4 cosmological constant, for which the value of τ is not fixed uniquely, and the problem noted above can be avoided. In the first alternative, considered in the next subsection, a (y) has decreased to around κ2/9 at the outer surface, and there are Casimir effects near the outer surface. However for τ around the bulk power law value −0.7753, the main part of the decrease of a (y) takes place in a very narrow part of the classical region near the outer surface, corresponding to y near 1 = 1.0996B in the interpolating function (255), on page 109, and in the quantum region near the outer surface, so that only a fraction ∼ 10−6 of the integral that determines Newton’s constant comes from values of y for which a (y) is smaller than around 1018 metres. And in the second alternative, considered in subsection 2.7, on page 137, the outer surface is stabilized in the classical region by extra fluxes of the three-form gauge field, whose four-form field strength wraps three-cycles of M6 times the radial dimension of the thick pipe, and for τ around −0.7753, the value of a (y) at the outer surface is around 1022 metres. 2.6 Solutions with a as small as κ2/9, at the outer surface of the thick pipe I shall now look for solutions of the Einstein equations (162), (163), and (164), on page 81, and the boundary conditions (170) at the outer surface of the thick pipe, such that the term 4 , in the square root, R, defined in (197), is still extremely small, compared to the term − 8 , when c = db is no longer large compared to , assuming, as in the preceding two subsections, that the boundary conditions at the inner surface of the thick pipe have already been solved, such that in the first bulk power law region, we are on a trajectory (206), (207), and (210), on page 89, with a large value of B , and A not yet determined, but such that A )0.7753 ≫ B. We are therefore, in the main part of the bulk, where both a and b are large compared to κ2/9, and both (y − y1) and (y2 − y) are large compared to κ2/9, on a solution of the form studied in subsection 2.4.4, on page 107, so that the interpolating function (255), for b as a function of y, with α given by (257), and the interpolating function (263), for a as a function of b, are approximately valid, throughout the main part of the bulk. There is now no possibility of satisfying the boundary conditions at the outer surface until a has become as small as κ2/9, so that there are Casimir effects on and near the outer surface. So we are now looking for a solution in which the constant of integration Ã, in (265), whose value is determined by the boundary conditions at the outer surface, obtains a very large value in units of κ2/9, by a mechanism analogous to the way in which the constant of integration B, whose value is determined by the boundary conditions at the inner surface, can obtain a large value in units of κ 9 , as studied in subsection 2.4.2, on page 93, but with the roles of b and a now reversed. We therefore now assume that the three observed spatial dimensions, whose curva- ture has become very large at the outer surface, due to the small size of the warp factor, a, there, are compact hyperbolic, so as to maximize the available range of dependences of the Casimir energy densities, at the outer surface, on a. This violates rotational invariance and Lorentz invariance globally, but not locally. The violation of Lorentz invariance globally means that the Casimir energy-momentum tensors on and near the outer surface will not necessarily have the forms (130) and (135), but I shall consider the case where they do have these forms. We will find that a large value of à can be obtained self-consistently in the same way as a large value of B . By analogy with the region near the inner surface, I shall first consider the case where t(3) = t(2), as would be appropriate for a dS4 times flat R7 background. In that case, by analogy with (133), on page 61, the conservation equation (131), on page 58, reduces to: dt(2) t(2) − 4 t(1) = 0 (288) Hence, in this case, the expansion coefficients C̃(i)n , in the expansions (137), on page 63, are related by: C̃(1)n = − (4 + 3n) C̃(2)n , C̃ n = C̃ n n ≥ 0 (289) By analogy with (211), on page 94, I shall first consider the case where all the C̃(2)n are zero, except for a single value of n, and consider the region ∣C̃(2)n κ2/9 (290) The Einstein equations (194) and (195), with the lower choice of sign, as appropriate for this region, and dropping the − 8 and 4 terms in the square root, and the 4 in (195), now become: = −2c (n+2) a8+3n (291) (n+2) a8+3n 2 + n C̃(2)n (n+2) a8+3n = 0 (292) The natural independent variable in this region would be a, and we would expect a trajectory analogous to (214), with ȧ = cda being given by a power law as a function of a, with a fixed coefficient, and b being given by a power law as a function of a, with an undetermined coefficient, analogous to (223). So we try an ansatz: = −σ̃ , b = B̃ (293) This implies da , and c = − σ̃τ̃ . We then find, from (291), and (292), that ρ̃ = −6+3n , which is the same as ρ, at the inner surface of the thick pipe, and κ (n+2) a8+3n σ̃2τ̃2 . Thus (291) and (292) reduce to = −2 − 1 σ̃2τ̃ 2 (294) 6 + 3n − 2− 1 σ̃2τ̃ 2 2 + n C̃(2)n σ̃2τ̃ 2 = 0 (295) which imply 2 C̃(2)n σ̃2τ̃2 = 10 + 16 , and (32 + 10n) τ̃ 2 + (32 + 13n) τ̃ + (8 + 4n) = 0 (296) which has the solutions τ̃ = −1 , and τ̃ = − 4n+8 5n+16 . However, τ̃ = −1 implies, by (294), that the square root, R, defined in (197), vanishes identically, so we cannot conclude, from (198), that all three Einstein equations are satisfied, and this also applies to the solution τ̃ = − 4n+8 5n+16 , when n = 0. Thus I now assume τ̃ = − 4n+8 5n+16 , with n ≥ 1. We then find: √− C̃ n (5n+ 16) 6 (15n2 + 96n+ 96) (297) We note, from (289), that this requires C̃(1)n to be positive, which corresponds to a negative contribution to the energy density, T00, which is opposite to the situation at the inner surface of the thick pipe. These results can be checked by solving (193) for ḃ , which gives: −4 ȧ κ2t(3)  (298) and then using this result, to eliminate ḃ from the first Einstein equation, (162), which gives: κ2t(3) − 5t(1) − 6t(2) − t(3) = 0 (299) Comparing with (253), we see that the upper sign in (298) and (299) corresponds to the lower sign in (194) and (195), and is thus the appropriate sign for the region nearer the outer surface, in the case under consideration in the present subsection, where a ∼ κ2/9 at the outer surface. Considering, again, the case where t(3) = t(2), and all the C̃(2)n are zero, except for a single value of n, and trying for a solution of the form c̃ ≡ ȧ = c̃0 (1 + s), where c̃0 is the small a power law trajectory found above, and s is a small perturbation, we find, similarly to the region near the inner surface, that s = Saη̃, η̃ = 15n2 + 96n+ 96 10n+ 32 (300) where S is a constant of integration. And since η̃ ≥ 3 for n ≥ 0, trajectories near the small a power law trajectory tend to converge towards it in the direction of decreasing a, or in other words, of increasing y, in the sense that s decreases in magnitude in this direction, so in this sense, the small a power law trajectory is an attractor in the direction of decreasing a, for n ≥ 0. Now, by analogy with (218), the small a power law trajectories, found above, can all be written as: ȧ2 = − κ2t(2) (a) − κ2t(2) (x) (301) And as near the inner surface of the thick pipe, the integral is convergent, as x → ∞, because t(2) (x) decreases at least as rapidly as x−8, as x → ∞, although of course t(2) (x), near the outer surface of the thick pipe, is not the same function as t(2) (x), near the inner surface of the thick pipe, and (301) does not give an exact solution of (299), except when t(2) (a) is a pure power of a. However, if t(2) (a) is a linear combination of two different pure powers, say a−(8+3n) and a−(8+3m), with n and m large, then the remainder term, in (299), will be ∼ 1√ , while the leading terms will be ∼ n or m. And for a pure power a−(8+3n), the integral, in (301), is of order 1 compared to the leading term. Thus, by analogy with the situation near the inner surface of the thick pipe, we expect that for large −t(2) (a), there will be an attractor trajectory in the (a, ȧ) plane, such that nearby trajectories approach it, in the direction of decreasing a, or in other words, in the direction towards the outer surface of the thick pipe, in the sense that the relative separation of the two trajectories decreases, in the direction of decreasing a, and this attractor trajectory will approximately be given by: κ2t(2) (a) (302) This trajectory will eventually intersect every second bulk power law region bulk power law trajectory (265), and, as near the inner surface of the thick pipe, we expect each bulk power law trajectory to curve upwards as it approaches the attractor trajectory, and then approach the attractor trajectory gradually. The square roots in (298) and (299) approximately vanish on the approximate small a attractor trajectory (302), but for n ≥ 1, the small a power law trajectories, which are approximately described by (302), are already known to be approximate solutions of all three Einstein equations near the outer surface of the thick pipe, when all the C̃(2)n are zero, except for a single value of n, and that C̃(2)n is negative, so it seems reasonable to expect that (302) will also give an approximate solution of all three Einstein equations in the more general case, when C̃(2)n ≤ 0 for all n ≥ 1. Now in the same way as in the discussion beginning just before (224), on page 96, for the region near the inner surface of the thick pipe, we have to consider whether these solutions can be self-consistent, when we recalculate the expansion coefficients C̃(i)n in (137), on page 63, for a (y) corresponding to these solutions, in accordance with the discussion in subsection 2.4.1, on page 90. We first recall that the bulk power law in the second classical power law region, that corresponds to the bulk power law (206), on page 89, in the first classical power law region, is (265), on page 110, as we can confirm from (299) above, with the upper choice of sign. And from the discussion in subsection 2.4.1, a term κ− 9 C̃(i)n )8+3n in (137) will be associated with additional terms 9 C̃(i)n,m )8+3n c̃m, 1 ≤ m ≤ n, as well as terms with factors of higher derivatives of a with respect to y, which can, however, be bounded by constant multiples of the terms without factors of higher derivatives, when the dependence of a on y is by a power law. The constant of integration Ã, in (265), will have to have a very large value, in units of κ2/9, in order to fit the observed value (22) of the de Sitter radius, so the largest additional terms will be those with m = n. Thus the Casimir terms in (298) and (299), with the upper choice of sign, will be significant near the outer surface for κ 2/9c̃ ≥ 1, which from (265) corresponds to )0.5326 . Defining ãq to be the value of a where this is an equality, we then find, from (266), that: )0.5326 = 0.03757 )1.2427 (303) And defining b̃q to be the corresponding value of b, we find from (254), on page 109, and (264), on page 110, that = 1.6884 )1.1450 (304) Then in the same way as in subsection 2.4.2, on page 93, for the region near the inner surface, we find that the only self-consistent way to obtain a large value of the integration constant Ã, is for c̃ to depend linearly on a in the quantum region a ≤ ãq, which results in (291) and (292) for n = −8 , and an effective coefficient C̃ , for ãq ≥ a≫ κ2/9. This results self-consistently in (293), with ρ̃ = τ̃ = 1, and σ̃ = 13 so that the effective coefficient C̃ has to be > 0 in order to obtain the linear relation. We note that η̃, in (300), takes the value −10 when n = −8 , so the linear trajectory is a very strong attractor in the direction of increasing a. However, in the same way as for the corresponding result for the region near the inner surface, this has not taken account of the fact that in the presence of deviations from the linear trajectory, the equations to be solved will no longer be precisely (291) and (292), with n = −8 Now a continues to decrease with increasing y in the quantum region near the outer surface, since c̃ = da is still negative in the quantum region. Hence since b has the fixed value B̃ in the quantum region, b stops increasing with increasing y at the upper limit b̃q of the classical region, and decreases with increasing y in the quantum region. Thus a necessary condition for the existence of a solution to the boundary conditions at the outer surface is that b must be comparable to or larger than a at the start of the quantum region, or in other words, ãq ≫ b̃q must not hold, for if ãq was ≫ b̃q, the boundary conditions at the outer surface would not depend significantly on the integration constant Ã, so that à would be undetermined, and B would be overdetermined. In the next subsection, I shall determine the values of B and à required to fit the observed values of Newton’s constant and the cosmological constant, for given values of τ and κ− 9 , assuming that this consistency condition is satisfied. We will then find that for τ = 1, which follows from assuming that t(3) = t(1) in the quantum region near the inner surface of the thick pipe, the consistency condition cannot be satisfied unless κ− 9 is much smaller than the minimum value ∼ 0.1 TeV allowed by current observations. However the linear relation between b and a in the quantum region near the outer surface, which follows from (293) and (296) for n = −8 , on rejecting the solution τ̃ = −1 , is a consequence of the assumption that t(3) = t(2) near the outer surface, and there is no reason to expect this relation to be valid when a and b depend exponentially on y. Thus in a similar way to the discussion following (231), on page 100, we should discard the assumption that t(3) = t(2) near the outer surface, and assume that the t(i), in (130), on page 58, are constrained only by the conservation equation (131). We would then expect, by analogy with the situation near the inner surface, that almost any value of τ̃ could be obtained, provided there exists a suitable smooth compact quotient M3 of H3, and a choice of spin structure on M3, that results self-consistently in the appropriate values of the independent coefficients C̃ and C̃ 2.6.1 Newton’s constant and the cosmological constant I shall now determine the values of the integration constants B, in (255), on page 109, and A, in (263), or equivalently, Ã, in (265), and the constants τ , in (233), on page 101, and κ2/9, for which the solutions found above can fit the observed values of Newton’s constant, (12), and the cosmological constant, (20), and check that this type of solution is consistent with observational limits on the existence of large extra dimensions, and can avoid the possible problem noted in the discussion following (287), on page 118, for the solutions studied in subsection 2.5, on page 111. I shall then check that this type of solution is consistent with experimental limits on deviations from Newton’s law at sub-millimetre distances, in subsection 2.6.2, on page 132, and with precision solar system tests of General Relativity, in subsection 2.6.3, on page 135. Some further consequences of the warp factor decreasing to a small value, at the outer surface of the thick pipe, in this type of solution, are considered briefly in subsection 2.6.4, on page We can follow the same method as used in subsection 2.5.1, on page 115. The term, in the Einstein action term in (25), that produces the Einstein action, (10), in four dimensions, is again given by (278), where b (y) is now given by (255), and a, as a function of b, is given by (263). Thus we now have: a2b6 ≃ A2B6 )1.5506 (305) where f (Y ) is defined by: f (Y ) ≡ 5.1220 (1− 0.9094Y ) 0.0651 Y 1.5346 (1− 0.9094Y )0.3549 + 12.0816Y 0.8448 (306) The function f (Y ) is illustrated in Figure 2. The peak is at Y = 0.5777, at which point the value of the function is 0.03002. The function is 0 at Y = 0, and at Y = 1.0996, and by use of PARI/GP [274], we find: ∫ 1.0996 dY f (Y ) = 0.02967 (307) 0.0 0.5 1.0 Figure 2: The function f (Y ) defined in (306) The contribution to this integral, from the regions 0 ≤ Y ≤ y1 , and y2 ≤ Y ≤ 1.0996, will be negligible, to the accuracy to which we are working, so we now find: dya2b6 ≃ 0.02967κ )5.4494 = 0.02967κ 9 A21 )1.3102τ+6.4652 (308) instead of (280). The numerical coefficient in (308) should now be approximately correct, for the solutions found in subsection 2.6, on page 120, to the extent that the interpolating functions (255), and (263), are approximately valid, whereas the numerical coefficient, in (280), was only valid up to a factor of order 1. The result (308) is for a smooth compact quotient of CH3. To obtain the cor- responding result for a smooth compact quotient of H6, we note that throughout the range where they give significant contributions to the integral, a and b are solu- tions of the vacuum Einstein equations, and a is so large that the curvature of the four-dimensional de Sitter space can be neglected. We recall that we have chosen the metric hAB for H 6 to have radius of curvature equal to 1, so that RABCD (h) = hAChBD−hADhBC , and RAB (h) = 5hAB, as stated after (97), on page 34. Then looking at the Ricci tensor components (97), and noting that RAB (h) = 4hAB for the stan- dard metric on CH3 introduced in subsection 2.2, on page 25, we see that the vacuum Einstein equations for CH3, when a is so large that the curvature of the dS4 can be neglected, can be transformed into the corresponding equations for H6, by rescaling y by a factor . Furthermore, derivatives with respect to y are larger for H6 than for CH3 by a factor , so the range of y is smaller for H6 than for CH3, by a factor . Thus the integral dya2b6 for H6 is obtained from the corresponding integral for CH3 by multiplying by a factor , or in other words, replacing the coefficient 0.02967, in (308), by 0.02654. The integral over the compact six-manifold, in terms of the Euler number of the compact six-manifold, will be the same as before, so we find that when the compact six-manifold is a smooth compact quotient of CH3, the Einstein action term, in the four-dimensional effective action, for the solutions considered in subsection 2.6, will be equal to: 0.3067 )5.4494 −g̃g̃µνRµτντ (g̃) (309) And when the compact six-manifold is a smooth compact quotient of H6, we get the same result as in (309), but with the numerical coefficient replaced by ×0.3067 ≃ 0.4389. Thus from the relation (242), on page 104, between A, and the observed de Sitter radius (22), and the discussion following (281), on page 117, we see that when we define the rescaled metric ḡµν by ḡµν = (de Sitter radius) g̃µν as before, so as to measure distances in ordinary units, rather than in units of the de Sitter radius, the Einstein action term, in the four-dimensional effective action, for the solutions considered in subsection 2.6, will for smooth compact quotients of CH3 be equal to: − 0.3067 1.27722τ )1.3102τ+6.4652 −ḡḡµνRµτντ (ḡ) (310) And for smooth compact quotients of H6, we get the same result as in (310), but with the numerical coefficient replaced by 0.4389 1.18092τ Thus, comparing with (10), we find that for smooth compact quotients of CH3: ≃ 15.416 1.27722τ )1.3102τ+6.4652 (311) And for smooth compact quotients of H6, the numerical coefficient is replaced by 22.062 1.18092τ . This is the form taken by the ADD mechanism [3, 5], for the solutions consid- ered in subsection 2.6, on page 120. We see that in the same way as for the solutions considered in subsection 2.5, on page 111, there is no ADD effect unless τ > −4.9345. This is due to the fact that for the classical region in the bulk, and for τ < 0, also for the quantum region near the inner surface of the thick pipe, we live on the wrong boundary, from the point of view of the first Randall-Sundrum model [31], and for τ < −4.9345, the reverse RS1 effect outweighs the ADD effect. For τ = 1, we find from (311) that for smooth compact quotients of CH3: ≃ 0.7491 |χ (M6)|0.1715 0.1286 (312) Considering, now, the case of TeV-scale gravity, we will find in section 5, on page 222, that κ is related to the gravitational masses M , Mp, and MD, with D = 11, defined respectively by Mirabelli, Perelstein, and Peskin [273], Giddings and Thomas [275], and Giudice, Rattazzi, and Wells [11], by M =Mp = 2 9MD = 2π . I shall use the results of Mirabelli, Perelstein, and Peskin, for six flat extra dimensions, as an indication of the current experimental limits on κ− 9 . Thus from their Table 1, we see that in 1998, the LEP 2 lower bound on κ− 9 was around 107 GeV, and the Tevatron lower bound was around 125 GeV. And the final lower bound on κ− 9 attainable at the Tevatron is expected to be around 166 GeV, and the final lower bound on κ− attainable at the LHC is expected to be around 677 GeV. As a representative example of TeV-scale gravity, I shall consider the case where the Giudice, Rattazzi, and Wells gravitational massMD, for D = 11, is equal to 1 TeV, which corresponds to κ− 9 = 0.2217 TeV, so that κ2/9 = 8.899×10−19 metres. We then find from (12), on page 13, that for τ = 1, and smooth compact quotients of CH3: B ≃ 1.515× 10 |χ (M6)|0.1715 κ2/9 ≃ 1.348× 10 −14 metres |χ (M6)|0.1715 (313) Thus from (242), on page 104, and (22), on page 15, we have for τ = 1, and smooth compact quotients of CH3: A ≃ 8.6× 10 30 metres |χ (M6)|0.0328 ≃ 5.4× 10 |χ (M6)|0.0328 (314) Thus from (266), on page 110, the integration constant Ã, in (265), is given for τ = 1, and smooth compact quotients of CH3, by: à ≃ 3.15× 1061 0.3386 metres ≃ 3.54× 1079 0.3386 κ2/9 (315) Thus from (303) and (304), on page 124, we find that for τ = 1, and smooth compact quotients of CH3: ≃ 2.33× 1042 0.1803 1.033× 105 |χ (M6)|0.1964 (316) Thus the consistency requirement that when the exponent τ̃ in (293), on page 121, is equal to 1, ãq must not be large compared to b̃q, is violated for τ = 1. Thus the relation t(3) = t(1) near the inner surface of the thick pipe and the relation t(3) = t(2) near the outer surface cannot both be satisfied, but as noted in subsections 2.4.2, on page 93, and 2.6, on page 120, there is no reason for either of these relations to be satisfied, since a and b depend exponentially on y in the quantum regions. If τ̃ is < 0, there is no consistency condition on ãq and b̃q, since b (y) continues to increase with increasing y in the quantum region near the outer surface. When τ = −0.7753, as for the classical power law (210), on page 90, in the first classical region, we find from (311) that for smooth compact quotients of CH3: 0.5646 |χ (M6)|0.1361 0.1835 (317) And for smooth compact quotients of H6, the coefficient 0.5646 is replaced by 0.5406. Considering, again, the case of TeV-scale gravity, with κ− 9 = 0.2217 TeV, we find that for τ = −0.7753, and smooth compact quotients of CH3: B ≃ 7.864× 10 |χ (M6)|0.1361 κ2/9 ≃ 6.998× 10 −13 metres |χ (M6)|0.1361 , (318) 1.83× 1026 metres |χ (M6)|0.1292 1.14× 1061 |χ (M6)|0.1292 , (319) à ≃ 5.29× 1048 0.0750 metres = 5.95× 1066 0.0750 κ2/9. (320) Comparing (315) and (320), we see that the cost of decreasing B by a factor of around 50, by increasing τ from −0.7753 to 1, is to increase à by a factor of around 6 × 1012. Thus it does not seem likely that B will be much smaller than the value (318) corresponding to τ = −0.7753. Thus from the upper bound of around 7 × 104 on |χ (M6)| found in subsection 2.3.6, on page 66, it does not seem likely that B be much smaller than 105, for TeV-scale gravity. By decreasing τ below −0.7753, it will be possible to decrease à at a cost of increasing B , until as τ approaches the values near −3 in (286) and (287), on page 118, the assumption made in subsection 2.6, on page 120, that the term 4 in the square root R defined in (197), on page 87, is still extremely small compared to the term − 8 , when c = db is no longer large compared to , will no longer be valid, and the type of solution considered in subsection 2.6 will resemble the solutions studied in subsection 2.5, on page 111, except in the region close to the outer surface. However a will still decrease to around κ2/9 at the outer surface for the solutions considered in subsection 2.6, because there are no solutions of the Einstein equations where c goes to zero at a finite value of b on the second branch of the square root. For if such a solution existed, then expanding u = b near the boundary as in subsection 2.5, we would find an equation that is obtained from (273), on page 113, by reversing the sign of the square root. This leads to the same quadratic equation as before, with the same solutions, (274), as before. But the solution α = −3 corresponds to the osculating line, which is not a solution of all three Einstein equations, and the solution α = −5 longer solves the original equation. The large values of B and à are the large numbers built into the structure of the universe, that make the universe into the stiff, strong structure that we observe. We note that due to the unique properties of smooth compact quotients of H3, it might be easier to obtain large values of à than of B . The three-volume V (M3) of a compact hyperbolic three-manifold M3 is a topological invariant when the Ricci scalar has a fixed value, which is usually chosen to be 6, corresponding to sectional curvature equal to −1. And uniquely to three dimensions, for any given three-volume V1, there is a finite, larger three-volume V2, such that there are an infinite number of topologically distinct compact hyperbolic three-manifolds M3 with Ricci scalar equal to 6, such that V1 ≤ V (M3) ≤ V2. The existence of this property follows from a construction of Thurston [276], and its uniqueness to three dimensions follows from a theorem of Wang [277], as I shall briefly discuss in section 3, on page 160. There is no observational upper limit to the topological invariant V (M3). For approximately homogeneous M3 the Casimir terms in the energy-momentum tensor near the outer surface of the thick pipe may tend to become independent of the topology of M3 for large V (M3), but all but a finite number of the M3 with volumes in a finite range V1 to V2 produced by the Thurston construction are significantly inhomogeneous. The inhomogeneity takes the form of a finite number of finite length “spikes” with smooth rounded ends, that approximate the infinite length “cusps” of the finite volume non-compact quotients of H3 to which the smooth compact quotients of H3 produced by the Thurston construction are related. The value of à depends only on the average over M3 of the functions t(i), and it would seem reasonable to expect that for the majority of the smooth compact M3 produced by the Thurston construction, these averages will continue to depend on the topology of M3 for arbitrarily large V (M3), and perhaps might tend to populate some ranges of values densely. Comparing (318) with (320), we see that when τ has the value −0.7753, correspond- ing to the classical power law (210) in the first classical region, the value of B required for TeV-scale gravity is relatively small in comparison to the very large value required for à . Moreover, from (242), (22), (317), and (266), we find that for τ = −0.7753 and smooth compact quotients of CH3, with a general value of κ2/9: ≃ 3.53× 10112 )1.3670 0.0750 (321) Thus for τ = −0.7753 the required value of à is minimized by choosing κ2/9 as large as possible, which means TeV-scale gravity, provided this is consistent with the precision tests of Newton’s law down to sub-millimetre distances [32], which I will check in the next subsection. 2.6.2 Comparison with sub-millimetre tests of Newton’s law We now need to check that TeV-scale gravity, in the type of model considered here, is consistent with the precision tests of Newton’s law, down to sub-millimetre distances. The shortest distance over which Newton’s law has been tested precisely is currently about 0.2 millimetres, so to be sure of the validity of the four dimensional effective action description, we require that for all y that make a significant contribution to the integral (308), a(y1) times 0.2 millimetres is large compared to both y and b (y). From subsection 2.5.1, on page 115, we know that if τ > −4.9345, so that there is an ADD effect, then the dominant contribution to the integral on the first branch of the square root comes from the classical region. I shall consider the case where τ = −0.7753, as in the classical power law (210), on page 90, in the first part of the classical region, and τ̃ = −0.3101, corresponding to the classical power law (254), on page 109, in the second part of the classical region. Then the interpolating function (263), on page 110, will be approximately valid throughout the whole range from the inner surface to the outer surface of the thick pipe. I shall make the approximation of treating the interpolating function (255), on page 109, as if it was also valid throughout the whole range from the inner surface to the outer surface. The condition to be sure of the validity of the four dimensional effective action description will be strictest as y approaches the outer surface of the thick pipe, at y ≃ 1.0996B, since a (y) decreases monotonically with increasing y, and b (y) increases monotonically with increasing y. Moreover, we see, from (255), that b (y) is comparable to y in the mid-region of the thick pipe, but becomes large compared to y, as either boundary of the thick pipe is approached. Thus it is sufficient to check the requirement for b (y), in the region where b is approaching the outer surface of the thick pipe. We then have, from (255), (263), and (242), that: b ≃ 1.5123B 1.0996B − y )0.1449 (322) a ≃ 0.2459 |χ (M6)|0.1292 )2.4495 )3.2247 × de Sitter radius (323) Thus, by (323), the requirement is that for all y that make a significant contribution to the integral (308): 0.2459 |χ (M6)|0.1292 )2.4495 × 0.2 millimetre )4.2247 (324) And for TeV-scale gravity, this becomes, by (313): 1.89× 103 0.1124 )4.2247 (325) And by (322), this becomes: 1.0996− 7.72× 10−5 |χ (M6)|0.1836 (326) Now |χ (M6)| ≥ 1, hence (326) will be satisfied, provided: 1.0996− ≫ 7.72× 10−5 (327) Now the contribution to the integral (307), from the region where Y is within 7.72×10−5 of the upper limit, is 1.35 × 10−6, which is a fraction 4.54 × 10−5 of the full integral (307). Thus for tests of Newton’s law at distances around 0.2 millimetres, we anticipate deviations from Newton’s law, in the shape of a small change in the effective value of Newton’s constant, at the level of about 50 parts in a million, or 5× 10−3 percent. To compare this result with the measurements of Hoyle et al [32], we note that one of the ways they expressed their results, was by giving 95% confidence level limits on the magnitude of the parameter α, as a function of λ, in a modified Newtonian potential of the form: V (r) = −Gm1m2 1 + αe−r/λ (328) The 95% confidence level limits on |α|, as a function of λ, are given in their Table XIII, from which we see that for λ = 0.10 millimetre, |α| ≤ 1.8 × 101. For λ = 0.25 millimetre, |α| ≤ 4.3×10−1. And for λ in the range 1.00 millimetres to 10.0 millimetres, the upper bound on |α| is around 10−2. The form of equation (328) is such that for r large compared to λ, the correction to Newton’s law is negligible, but for r comparable with λ, or smaller than λ, there is effectively a modification of Newton’s constant, by a factor ∼ (1 + α). Thus for the form of TeV-scale gravity considered in the present paper, the expected deviations from Newton’s law, at distances around a millimetre, are around 5×10−3 times smaller than the current best experimental limits of Hoyle et al. It is interesting to note that the upper bound, (326), on y, for the four dimensional reduction to be valid, for submillimetre tests of Newton’s law, from the inner surface of the thick pipe, up to y, corresponds, by (313), to: 1.0996B − y ≫ 5.40× 10−17 metres |χ (M6)|0.3197 (329) or in other words, since κ2/9 = 8.899× 10−19 metres, for TeV-scale gravity, to: 1.0996B − y ≫ 60.7 |χ (M6)|0.3197 κ2/9 (330) On the other hand, from (303), on page 124, the value of that corresponds to the value (320) of à is 3.67× 1035 |χ (M6)|0.0399, so the thickness in y of the quantum region near the outer surface is around κ2/9 ln ≃ 81.9κ2/9. Thus if the precision of the submillimetre tests of Newton’s law could be increased by another three decimal places, they would be probing the quantum region near the outer surface of the thick pipe, for the solutions considered in subsection 2.6, when τ = −0.7753. From (322) and (318), the value of b, at the value of y where the inequalities in (329) and (330) become equality, is: b ≃ 5.96 0.0266 B ≃ 4.17× 10 −12 metres |χ (M6)|0.1095 ≃ 4.69× 10 |χ (M6)|0.1095 κ2/9 (331) And from (323) and (22), the corresponding value of a is: a ≃ 2.09× 10 |χ (M6)|0.1095 ×de Sitter radius = 3.16× 10 18 metres |χ (M6)|0.1095 3.55× 1036 |χ (M6)|0.1095 κ2/9 (332) 2.6.3 Comparison with precision solar system tests of General Relativity There are also very precise tests of General Relativity, via lunar laser ranging mea- surements of the lunar orbit, using reflectors left on the surface of the moon by Apollo astronauts, and by unmanned Soviet lunar missions [278, 279]. In particular, a test of the equivalence principle, obtained from a fit of lunar laser ranging data, gives a value for the difference in the ratio of gravitational mass to inertial mass, MG/MI , between the Earth (e) and the Moon (m). The value quoted in [278], which has been corrected for solar radiation pressure, is: = (−1.0± 1.4)× 10−13 (333) To check the consistency with this measurement, of the models studied here, we need to decide, if this measurement is interpreted as giving a bound on the variation of Newton’s constant with distance, what the shortest relevant distance is. The lunar orbit is determined by the gravitational interaction between the Moon and the Earth, while both move in the gravitational field of the Sun (s). From equation (2) of [278], the effective acceleration of the Moon with respect to the Earth, ~a = ~am − ~ae, for the three-body Earth-Moon-Sun system, is: ~a = −GN −GNMs +GNMs (334) The last two terms in (334) represent the solar effect on the motion of the Moon with respect to the Earth. A violation of the equivalence principle would produce a lunar orbit perturbation proportional to the difference in the two MG/MI ratios. From the form of equation (334), it appears that a small percentage difference inGN , between the first term, and the last two terms, corresponding to a small percentage difference in GN , for the Earth-Moon distance, and the Earth-Sun distance, might result in an orbital perturbation different in form, but of the same order of magnitude, as the perturbation resulting from a similar percentage difference in the two MG/MI ratios. Thus I shall provisionally interpret the measurement (333), as also giving an order of magnitude bound on the percentage difference of Newton’s constant for the Earth-Moon distance, and for the Earth-Sun distance. Thus we have to repeat the calculation performed above, for tests of Newton’s law over distances of around 0.2 millimetres, for distances around the Earth-Moon distance, which is around 4 × 108 metres. Instead of (327), we now find that a sufficient condition on y, for the four- dimensional reduction to be valid from the inner surface of the thick pipe up to y, is that: 1.0996− ≫ 6.20× 10−25 (335) It follows immediately from the flat topped shape of the function f (Y ), Figure 2, together with the fact that the peak of the function is outside the range excluded by (335), that the contribution to the integral (307), from the region excluded by (335), is not more than a fraction ∼ 10−23 of the value of the integral, if ≫ is interpreted as meaning larger by a factor of at least 10. Thus for the form of TeV-scale gravity considered in the present paper, the fractional difference of GN for the Earth-Moon distance, from GN for the Earth-Sun distance, will not be more than around 10 at the most, which is smaller than the bound given by (333), interpreted as discussed above, by a factor of around 10−10. Thus, notwithstanding the remarkable precision of the lunar laser ranging measure- ments, the submillimetre tests of Newton’s law are currently closer to testing the form of TeV-scale gravity considered in subsection 2.6, with τ = −0.7753. 2.6.4 Further consequences of the warp factor decreasing to a small value, at the outer surface of the thick pipe The fact that the warp factor, a2 (y), decreases to a small value as y approaches y2, in the solutions considered in subsection 2.6, implies that there are short spacelike paths through the bulk between points that are separated by large distances in the observed universe. However, for the solutions considered in the present paper, it is not possible, even in principle, to send signals through the bulk to distant parts of the observed universe, at what would appear to be superluminal speeds, from the point of view of observers on the inner surface of the thick pipe, because the time dimension scales with exactly the same scale factor, a (y), as the three observed spatial dimensions. It would be interesting to find out whether or not this conclusion could be modified in cosmological-type solutions, which would require the analysis of some coupled partial differential equations, with the time, and y, as independent variables. In particular, it would be interesting to find out whether or not an effect of this type could provide an alternative to inflation, for solving the horizon problem of the early universe [78]. Alternative solutions to the horizon problem, of this type, have been discussed in [280, 281, 282, 283, 284, 285, 286, 287, 288]. It would also be interesting to find out whether or not an effect of this type would be consistent with the type of causality constraints recently discussed by Arkani-Hamed et al [289]. However these questions will not be addressed in the present paper. 2.7 Stiffening by fluxes wrapping three-cycles of the compact six-manifold times the radial dimension The occurrence of non-vanishing fluxes of form fields in the de Sitter backgrounds for type IIB superstrings constructed by Kachru, Kallosh, Linde, and Trivedi [290] suggests that it might also be interesting to consider solutions with extra fluxes of the four-form field strength of the three-form gauge field in the present context, so I shall now consider the possible effects of fluxes wrapping three cycles of the compact six-manifold, times the Hořava-Witten one-cycle along the eleventh dimension, in the upstairs picture. I shall assume, to start with, that there will not be enough non-vanishing components of the three-form gauge field, for the non-linear term in the classical field equation for the three-form gauge field to be non-vanishing, so that we can treat the classical field equation for the three-form gauge field as a linear equation, and add solutions. We then seek a classical solution, such that only the components GABCy are non-zero, and GABCy (z, y), where z denotes the coordinates on the compact six-manifold, has the factorized form GABCy (z, y) = GABC (z) f (y) (336) Now the Bianchi identity reads: ∂IGJKLM + ∂JGKLMI + ∂KGLMIJ + ∂LGMIJK + ∂MGIJKL = 0 (337) With the ansatz (336), one component of this reads: (∂AGBCD (z)− ∂BGCDA (z) + ∂CGDAB (z)− ∂DGABC (z)) f (y) = 0 (338) which, since f (y) 6= 0 by assumption, is the Bianchi identity for the three-form factor GABC (z). Now when the gravitino field vanishes, the classical field equation for the three-form gauge field CIJK , from the action (25), is: −GGIMGJNGKOGLPGMNOP −GGJKLI4...I7I8...I11GI4...I7GI8...I11 = 0 (339) where the metric in eleven dimensions is denoted GIJ , as in (94), so that G I1I2...I11 de- notes the tensor 1√−Gǫ I1I2...I11. Let us now assume that GIJKL is zero, if any component is along the four observed dimensions. Then there are at most seven possible values for each index, such that GIJKL is non-zero, so the term in (339) bilinear in GIJKL vanishes, and the field equation reduces to: −GGIMGJNGKOGLPGMNOP = 0 (340) Now, bearing in mind the metric ansatz (94), one set of components of this equation, for the factorized ansatz (336), reads: hhADhBEhCFGDEF (z) a (y) f (y) = 0 (341) Now, since −g, a (y), and f (y) are assumed to be non-vanishing, this equation, together with (338), implies that GABC (z) is a Hodge - de Rham harmonic three-form on the compact six-manifold. So by standard Hodge - de Rham theory, there are B3 linearly independent solutions GABC (z) of (338) and (341), where B3 is the third Betti number of the compact six-manifold. I shall now assume that GABC (z) is a Hodge - de Rham harmonic three-form on the compact six-manifold. The remaining set of components of (340), that are not satisfied identically for the factorized ansatz (336), are: a (y) f (y) hhADhBEhCFGDEF (z) = 0 (342) Thus f (y) is equal to a fixed number, times a (y) , so, absorbing the fixed number into GABC (z), we find that: GABCy (z, y) = GABC (z) a (y) (343) where GABC (z) is a Hodge - de Rham harmonic three-form on the compact six- manifold. We note that (343) applies for all y, in the upstairs picture, since under reflec- tion in the orbifold hyperplane at y = y1, we have GUVWy (x, 2y1 − y) = GUVWy (x, y), and also a (2y1 − y) = a (y), where x here denotes the coordinates on M10. Now, following an argument given by Witten, in section 2 of [151], we consider a four-cycle X in M10, on the y1+ side of the orbifold hyperplane at y = y1, and apply the relation (44), with the substitution (47). If the Pontryagin number of X is zero, then the RR term in (47) will not contribute to the integral of the right-hand side of (47) over X , so we find that G|y=y1+ = dxUdxV dxWdxX GUVWX |y=y1+ (344) is equal to 1 X trF (1) ∧ F (1), which Witten indicates is a four-dimensional charac- teristic class of the E8 bundle at y1, and is equal to an integer. However, (344) is a topological invariant for smoothly varying G, and thus has the same value no matter what value of y it is evaluated at, and, indeed, has the same value for any four-cycle in M11 = M10 × S1/Z2 that is topologically equivalent to X . Following Witten’s argument, if we now consider Hořava-Witten theory with a large value of (y2 − y1), specifically, much larger than the diameter of X , and the integral (344) at some value of y a long distance away from both y1 and y2, then it would seem unlikely that the value of the integral would depend on whether or not there exist orbifold hyperplanes a very large distance away, at y1 and y2. Thus we expect that (344) should be equal to an integer for an arbitrary four-cycle X with zero Pontryagin number, for smoothly varying G, in supergravity in eleven dimensions. In other words, (344) gives a form of Dirac quantization condition on the integral of the Cremmer-Julia-Scherk four-form field strength G, over a four-cycle with zero Pontryagin number. Witten gives further arguments supporting this interpretation, and also, a generalization of the quantization condition, to four-cycles with non-zero Pontryagin number. Witten’s arguments do not cover the case of a four-cycle, in the upstairs formulation of Hořava-Witten theory, that has the form of a three-cycle in M10, times a one-cycle that wraps the S1 in the y direction. However, since the Hořava-Witten boundary conditions, at the orbifold fixed-point hyperplanes, imply that GUVWy is continuous across the orbifold fixed-point hyperplanes, and such a four-cycle automatically has zero Pontryagin number, I shall assume that (344) also has an integer value, for such a four-cycle, and that this applies, in particular, for the factorized ansatz (336). Thus, from (343), we find that, for any three-cycle, Z, of the compact six-manifold: dzAdzBdzCGABC (z) dya (y) (345) must be equal to an integer. We now have to calculate the modified value of the contribution (158), on page 79, of the three-form gauge field to the energy-momentum tensor, (14), in the presence of fluxes wrapping three-cycles of the compact six-manifold, with the ansatz (336). Since the three-form field configurations considered in subsection 2.3.8 are only significant, in the energy-momentum tensor, near the inner surface of the thick pipe, while, from (343), the three-form field configurations considered in the present subsection are sup- pressed by the very small factor a (y) , near the inner surface of the thick pipe, I shall provisionally assume that cross terms in the energy-momentum tensor, between the three-form field configurations considered in subsection 2.3.8, and those considered in the present subsection, can be neglected, for compactifications on smooth compact quotients of CH3, while for compactifications on smooth compact quotients of H6, the three-form field configurations of the type considered in subsection 2.3.8 are absent, since Witten’s topological constraint is satisfied with zero G, as noted at the end of subsection 2.3.7. From the metric ansatz (94), we find: GKNGLOGMPGAKLMGBNOP = hCEhDFGACD (z)GBEF (z) (346) I shall now assume, as in the study of the Casimir contributions to the energy- momentum tensor in subsection 2.3.4, that the Einstein equations are expanded in harmonics on the compact six-manifold, following the procedure of Lukas, Ovrut, and Waldram [67], and I shall consider the Einstein equations in the approximation of drop- ping all but the lowest harmonic. I shall also assume that GABC (z), which is a sum of constant multiples of B3 linearly independent Hodge - de Rham harmonic three-forms, where B3 is the third Betti number of the compact six-manifold, has been chosen such hhCEhDFGACD (z)GBEF (z) = CFhDGhEHGCDE (z)GFGH (z) (347) hhABh CFhDGhEHGCDE (z)GFGH (z) = G hhAB (348) for a suitable real constant G > 0. These conditions (347) and (348) constitute at most 20 + 20 linearly independent constraints on the B3 independent coefficients in GABC (z), and thus can presumably always be satisfied, for sufficiently large B3, unless this somehow conflicts with the requirement that (345) be an integer for all three-cycles Z, which I shall assume does not occur. Then in the approximation of dropping all but the lowest harmonic, (346) becomes: GKNGLOGMPGAKLMGBNOP = 2b6a8 2 (349) Similarly, we find: GKNGLOGMPGyKLMGyNOP = hADhBEhCFGABC (z)GDEF (z) (350) And in the approximation of dropping all but the lowest harmonic, this becomes: GKNGLOGMPGyKLMGyNOP = G̃2 (351) where the real constant G̃ > 0 is defined by hhADhBEhCFGABC (z)GDEF (z) = G̃ h (352) We also find: GQRGKNGLOGMPGQKLMGRNOP = hADhBEhCFGABC (z)GDEF (z) (353) Thus, in the approximation of dropping all but the leading harmonic, we have: QRGKNGLOGMPGQKLMGRNOP = G2GAB (354) QRGKNGLOGMPGQKLMGRNOP = G̃2Gµν (355) QRGKNGLOGMPGQKLMGRNOP = G̃2 (356) Hence, from (158), we find the following additional contributions to the energy- momentum tensor of the three-form gauge field, to be added to (159), in the approx- imation of neglecting cross terms between the three-form gauge field configurations considered in subsection 2.3.8, and those considered in the present subsection: T (3f)µν = − 12κ2b6a8 Gµν , T AB = 0, T 12κ2b6a8 (357) These components satisfy the conservation equation (131), with t(1), t(2), and t(3), defined by (130), now interpreted as unrestricted functions of y. 2.7.1 The region near the outer surface Now comparing with (130) on page 58, and with (194) and (195), on page 87, with the upper choice of sign for the square root, we see that the new term in the square root, resulting from (357), has the correct sign, namely the same sign as the 4 term, to make possible a solution of the boundary conditions, at the outer surface of the thick pipe, with both a and b large compared to κ2/9 at the outer surface, by the same mechanism as in subsection 2.5, on page 111. Furthermore, from (210), on page 90, we see that 1 behaves as b0.2024 in the first bulk type power law region, governed by (205), on page 89, and (209), on page 90. Thus the correction terms from (357) do indeed grow in importance with increasing b, or equivalently, with increasing y, and thus the most important of the correction terms, which is the one in the square root, has the correct qualitative behaviour, for a suitable value of G̃, to make possible a solution of the boundary conditions at the outer surface in the classical region, along the lines of subsection 2.5. Furthermore, for a sufficiently large value of G̃, it might be possible to arrange for a≫ b at the outer surface, and thus avoid the problem that prevented the solutions of subsection 2.5 from being in agreement with observation. To study this possibility in detail, I shall now retrace the steps in subsection 2.5, but assuming, now, that G̃ is sufficiently large, and the integration constant A, in (210), is also sufficiently large, that as y increases, in the first bulk power law region, (205), (206), and (207), with the integration constant, B, large compared to κ2/9, the correction terms in (194) and (195), due to (357), first become significant long before the 4 term, in the square root, becomes significant. This assumption will be satisfied, if we find a solution in the classical region, with a≫ b at the outer surface. With these assumptions, we find from (130), (194), (195), and (357), that the relevant equations, away from the inner surface of the thick pipe, but still on the first branch of the square root, where we take the upper sign in (194) and (195), are: 6c2 − 8 + 18b4a8 (358) 3c2 − 4 6c2 − 8 + G̃ 18b4a8 36b5a8c (359) Here c = ḃ = db , as defined just before (199), on page 88. These two equations replace the equations (267) and (268) of subsection 2.5. Moreover, we are again seeking solutions such that both a and b are large compared to κ2/9, at the outer surface of the thick pipe, so, as explained at the start of subsection 2.5, on page 111, the boundary conditions are now that both ȧ , and ḃ , are zero, at the outer surface of the thick pipe. Thus c = 0 at the outer surface, and from (358), rewritten in its original form, like (194), on page 87, we see that G̃ 18b4a8 = 8, at the outer surface. The qualitative difference from subsection 2.5, is that there we had b = 2a, b ≫ κ2/9, at the outer surface, and here we have a new adjustable parameter, namely G̃ , related to the extra fluxes, and we are going to try to choose a sufficiently large value of G̃ , that we find a solution with a≫ b≫ κ2/9, at the outer surface. Now the equations (358) and (359) have the family of solutions (206), (210), for large values of the integration constants A and B, in the region c≫ , which means b ≪ B, by (206), provided also that G̃ 12b2a4 ≪ 1, or in other words, by (210), provided also that b )0.9081 since, as noted above, for this class of solutions, 1 grows as b0.2024 with increasing b, hence 1 grows as b2.2024 with increasing b. Let us now follow a solution of (358) and (359) in the class (206), (210), from small b, in the direction of increasing b, and suppose that the G̃2 terms start to become significant while the solution of (359), which is decoupled from (358) while these terms are negligible, is still on the first branch of the square root. Thus the integration constants A and B, in (206) and (210), must be such that κ2/9 )0.9081 is not large compared to B. Then, in a similar manner to the situation in subsection 2.5, when the G̃ terms first start to become significant, the solution of (359), in the (b, c) plane, starts to peel off below the G̃ = 0 trajectory. Equation (359) then starts to become coupled to equation (358), and we are looking for a solution such that the trajectory, in the (b, c) plane, curves downwards and meets the line c = 0, at a finite value of b, which will be b2 = b (y2), the value of b at the outer surface of the thick pipe. At this point, a will take the value In the limit c → 0, the other boundary condition, G̃2 18b4a8 = 8, implies that (359) reduces to dc = − 8 . This, in turn, reduces to dc = − 8 , in the region of the boundary, so that, in the region of the boundary, we have: (360) This replaces equation (269) of subsection 2.5, in the present context. Following the method of subsection 2.5, we now define v ≡ 1 . The above equa- tions then become: 6− 2 6c2 − 8 + G̃  (361) 3c2 − 4 6c2 − 8 + v2 (362) Now, as noted shortly after (277), one way of studying a pair of equations of this type, would be to take the ratio of (361) and (362). Then b cancels out, and we get a single first order differential equation, that expresses dv , as a function of v and c. However, I shall follow the method of subsection 2.5. The boundary conditions, at b = b2, are now that: , c = 0 (363) Near the boundary, we expand v in the small quantity v = 12 1 + α . Then from (360) and (361), we find that: − α = 6− 2 6 + α (364) which has the solutions: α = −2, α = −6 (365) We note that, analogously to the situation in subsection 2.5, the first of these is only a solution, for the particular sign of the square root in (364), while the second is a solution for both signs of the square root, since the square root vanishes for it. The solutions can be developed to higher order in , by substituting next into (362), to fix the next term in c2, then after that into (361) again, to fix the next term in v, and so on, in the same way as in subsection 2.5. Furthermore, in a similar manner to subsection 2.5, (361) and (362) imply that: 6c2 − 8 + G̃ 6c2 − 8 + G̃ 6c2 − 8 + G̃ v2 − 1 4c2 + (366) Thus 6c2 − 8 + G̃2 v2 = 0 is a solution of (361) and (362). However the square root, R, defined in (197), vanishes identically for this solution, so we cannot infer, from (198), that 6c2 − 8 + G̃2 v2 = 0 is a solution of all three Einstein equations, and by analogy with subsection 2.5, we would expect that it does not correspond to a solution of all three Einstein equations, but is, rather, the generalization to the case where v 6= 0 of the line c = , which is the line in the (b, c) plane that actual solutions of the Einstein equations, in the limit v = 0, osculate with as they switch from the first to the second branch of the square root. Moreover, the case α = −6, in (365), satisfies 6c2 − 8 + G̃2 v2 = 0, to the order given, and is thus the c→ 0 limit of this particular solution of (361) and (362), and is thus not expected to correspond to a solution of all three Einstein equations. We note that this particular solution of (361) and (362) satisfies c ≤ , and thus can never rise above the line c = , in the (b, c) plane. Furthermore, when 6c2 − 8 + G̃2 v2 = 0, (361) reduces to dv , hence v = 12 , where, by (363), b2 is the integration constant in (360). Hence c = , which does, indeed, also solve (362). Considering now, the case α = −2, in (365), and still following the method of subsection 2.5, we see that 6c2−8+ G̃2 v2 ≃ 64 ≃ 4c2 near the boundary, hence the square root, R, is nonvanishing, as soon as we move away from the boundary, so, by (198), this solution will correspond to a solution of all three Einstein equations. Furthermore, for c ≥ , the right-hand side of (362) is ≤ the right-hand side of (204), and as noted before (248), the right-hand side of (204) is ≤ 0 for all c ≥ , so the right-hand side of (362) is ≤ 0 for all c ≥ , and the right-hand side of (362) is certainly ≤ 0 for c ≤ such that the square root is real, so dc is ≤ 0 for all c ≥ 0 and b ≥ 0 such that the square root is real. Furthermore, dv starts positive, specifically at b = b2, hence v decreases, as b decreases downwards, away from b = b2, hence, provided dv never becomes negative, and the square root stays real, the square root is bounded above by 6c ≃ 2.45c, hence, by (361), we have dv 1.10v , and, by (366), we have d 6c2 − 8 + G̃2 24− 6 6c2 − 8 + G̃2 −9.30 c 6c2 − 8 + G̃2 v2, hence dv never does become negative, and the square root does stay real. Furthermore v ≤ 12 )1.10 , hence c ≥ , hence 6c2 − 8 + G̃2 6c2 − 8 + G̃2 ≤ −9.30c db ≤ −10.73 db (367) Hence 6c2 − 8 + G̃ v2 ≥ 2.68 − arcsin  (368) With the bound v ≤ 12 )1.10 , this implies that 6c2−8 is positive for b ≤ 0.54, and is greater than 13.45 for b = 0.1, by which point G̃ v2 < 0.05. Thus this solution merges into a solution of (204), as b continues to decrease, and for c large compared to will follow a trajectory of the form (206), in the (b, c) plane, with B a fixed number of order 1, that will be the same for all solutions of this type. And for solutions of this type, namely with α = −2 in (365), the constant of integration, b2, in (360), can be identified as b2 = b (y2), the value of b at the outer surface of the thick pipe. To estimate the integration constants B, in (206), and A, in (210), in terms of b2 and G̃, it is convenient to define ṽ ≡ G̃ v. The equations (361) and (362) then become: 6c2 − 8 + 8ṽ2 (369) 3c2 − 4− 4ṽ2 6c2 − 8 + 8ṽ2 (370) The boundary conditions, at b = b2, are now that: ṽ = 1, c = 0 (371) Near the boundary, we have (360) and ṽ ≃ 1−2 ≃ 1− c2 . Moreover, c increases monotonically, and ṽ decreases monotonically, as b decreases downwards from b2. And as b tends to zero, and c becomes large compared to 1, c tends to the form (206), where B will be a fixed multiple of b2, that we now wish to estimate, and from (369), ṽ tends to the form ṽ ≃ V )1.1010 )1.1010 c−0.5798 (372) where the second form follows from (206), and the constant V is given, from (210), and the relation ṽ = G̃ 12b2a4 (373) where A is the constant of integration that occurs in (210). A simple estimate of the dependence of ṽ on c, with the required behaviour ṽ ≃ 1− c2 as c→ 0, and the power law behaviour (372) as c→ ∞, is 24− 8 1 + 0.4312c2 )−0.2899 (374) which gives: )1.1010 ≃ 0.4312−0.2899 ≃ 1.2762 (375) As a check on (374) and (375), we note that, on dividing (369) by (370), we have: 6c− 2 6c2 − 8 + 8ṽ2 3c2 − 4− 4ṽ2 − 2c 6c2 − 8 + 8ṽ2 (376) And with the help of Maxima [291], we find that the solution of (376), that behaves as ṽ ≃ 1− c2 near c = 0, has the Taylor expansion: 46080 − 3559c 10321920 15167c10 3715891200 + · · · (377) where . . . denotes terms of order c12. And using the graphical facility of Maxima, we see that (377) is accurately approximated by its first four terms up to around c = 1.3, at which point (377) is 0.8256, while (374) is 0.8533, and (377) is accurately approximated by its first five terms up to around c = 2.0, at which point (377) is 0.6562, while (374) is 0.7478. And (377) starts to curve rapidly downwards above around c = 2.0, and would thus appear likely to depart from the true dependence of ṽ on c, starting at around c = 2.0. Thus it seems likely that for large c, the estimate (374) of ṽ will be around 15 to 20 percent too large, and the estimate (375), of V )1.1010 , will also be around 15 to 20 percent too large. So a better estimate of V )1.1010 would perhaps be around 1.1. A simple estimate of the dependence of c on b, with the required behaviour (360) as c→ 0, and b→ b2 from below, and the power law behaviour (206) for c≫ , is: )1.8990 (378) which also has the required property that c only depends on b, and the integration constant b2, through the ratio , as follows from (370), after substituting for ṽ as a function of c, with a Taylor expansion that begins as in (377). The estimate (378) leads to the estimate: B ≃ 4 1.8990 b2 ≃ 2.0751b2 (379) To check (378) and (379), we note that, from (370), we have: 4 + 4ṽ2 + 2x 6x2 − 8 + 8ṽ2 − 3x2 ) (380) where, in the integrand, ṽ is given as a function of x, by rewriting c as x, in the dependence of ṽ on c as above, whose Taylor expansion begins as in (377). To calculate 0 fdx by (380) c for this by (378) (378) c for this by (381) (381) 0.5 0.01561 0.9845 0.5130 1.0260 0.5032 1.0064 1.0 0.06170 0.9402 1.0997 1.0997 1.0209 1.0209 1.5 0.1329 0.8756 1.8156 1.2104 1.5577 1.0385 2.0 0.2177 0.8044 2.6746 1.3373 2.1081 1.0541 3.0 0.3821 0.6824 4.6576 1.5525 3.1991 1.0664 5.0 0.6286 0.5333 9.0170 1.8034 5.3261 1.0652 10.0 0.9866 0.3728 20.631 2.0631 10.621 1.0621 20.0 1.3504 0.2591 44.747 2.2374 21.220 1.0610 80.0 2.0801 0.1249 194.41 2.4301 84.842 1.0605 Table 1: The numerical dependence of c on b. the integral in (380), I used the numerical integration facility of PARI/GP [274], with the dependence of ṽ on c given by (377) for c ≤ 2, and by (374), multiplied by 0.6562 0.7478 0.8775, so as to obtain continuity at c = 2, for c ≥ 2. In this way, writing the integral in the right hand side of (380) as 0 fdx, we find the entries in the second column of Table 1. The entries in the third column are the values of b which correspond by (380) to the entries in the second column, and the entries in the fourth column are the values of c which the estimate (378) gives, for the values of b in the third column. The fifth column gives the ratio of the estimated value of c calculated in the fourth column by the estimate (378), to the original value of c in the first column. From the form of the discrepancy factor in the fifth column, we would expect that the estimate (378) could be improved by replacing the factor 4 , in (378), by a factor of the form 4√ , which has the same limiting behaviour as b → b2 from below, and where n ∼ 2.43 ∼ 6. Thus, taking n = 6, we try an estimate: c ≃ 1.633 )1.8990 (381) The values of c given by the estimate (381) are listed in the sixth column of the table, and from the discrepancy factor, in the seventh column of the table, we see that the error now stays below 7 percent, and is actually slowly decreasing, as b continues to decrease below 0.5. I shall therefore use (381) as a reasonable estimate of the ṽ from Taylor series (377) ṽ from estimate (374) (374) 0.5 0.9696 0.9696 0.9708 1.0012 1.0 0.8882 0.8882 0.9013 1.0147 1.5 0.7830 0.7818 0.8215 1.0492 2.0 0.6830 0.6562 0.7478 1.0949 3.0 0.5383 - 0.6315 1.1731 5.0 0.3919 - 0.4892 1.2483 10.0 0.2581 - 0.3336 1.2925 20.0 0.1719 - 0.2243 1.3048 80.0 0.07680 - 0.1006 1.3099 Table 2: The numerical dependence of ṽ on c. dependence of c on b, in the presence of the extra fluxes. The corresponding estimate of the integration constant B, in (206), is: B ≃ 1.633 1.8990 b2 ≃ 1.2947b2 (382) which now replaces the estimate (379). Returning, now, to the dependence of ṽ on c, I used a standard fourth-order Runge- Kutta method [292] to integrate (376) from c = 0.3, where ṽ is reliably given by (377) as ṽ ≃ 0.9889, into the power law region, where c≫ . The same result was obtained with a Runge-Kutta interval h = 0.01 as with h = 0.00001, even for c = 80. In fact, to four significant digits, the same result was also obtained with h = 0.1, even for c = 80. The results are shown in Table 2. Thus the error of the estimate (374) stabilizes at about 31 percent in the power law region, and the estimate (375) should be replaced by: )1.1010 ≃ 0.4312 −0.2899 ≃ 0.9742 (383) We next consider the dependence of b on y, and note, following the discussion shortly after (277), in subsection 2.5, that the behaviour (360), for c near the boundary, implies that near the boundary, y2− y ≃ b22 c, thus y does, indeed, tend to a finite value, y2, at the boundary, even though goes to ∞, right at the boundary. And using the approximate relation (382), we find that near the outer boundary: ≃ 0.7724 1− 6.7050 y2 − y (384) For a first estimate of y2, we could simply use the form (207), with y0 = 0, all the way from y1 to y2, and determine y2 as the point where this gives b = b2 ≃ 0.7724B, which gives y2 ≃ 0.7724 1.4436 0.3449 B ≃ 0.1631B. This underestimates y2 by a factor of order 1, because, by (384), the curve of b (y), in the (y, b) plane, curves to the right as the outer boundary is approached, so that b2 is not reached until a larger value of y than would be indicated by (207) with y0 = 0. For a better estimate of y2, a convenient interpolating function would be b = 1.4436 )0.3449 (1 + αyβ) γ (385) with β > 0, and either α > 0 and βγ > 0.3449, or α < 0 and γ < 0. This agrees with (207) for y ≪ 1 , and has a peak at y = 0.3449 α(βγ−0.3449) , which we attempt to identify with y2. Requiring agreement with (384) for b (y2) leads to the requirement that: 1.4436 0.3449 0.3449 (βγ − 0.3449) βγ−0.3449 β = 0.7724 (βγ) (386) This is written for the α > 0 case, and for the α < 0 case should be rewritten in the equivalent form with the contents of each of the three pairs of parentheses multiplied by −1. And requiring agreement with (384) for 1 , evaluated at y = y2, leads to the requirement that: βγ − 0.3449 0.3449 = 112.73γ (387) This is also written for the α > 0 case, and for the α < 0 case should be rewritten in the equivalent form with the contents of each of the two pairs of parentheses multiplied by −1, and the right hand side also multiplied by −1. Eliminating αBβ between (386) and (387), we find: βγ − 0.3449 )γ+0.1725 1.0059 β0.1725 (388) Trying first β = 1, there is no solution in the α > 0 case, but there is a solution with γ ≃ −0.1672 in the α < 0 case. We note that the improved estimate, (381), of the dependence of c on b, shows that the power law behaviour remains a good approximation until b is quite close to b2. For small , the correction to the power β γ αBβ y2 1 −0.1672 −2.3996 0.2807 2 −0.6666 −3.1744 0.2545 2.5 2.4194 2.0098 0.2462 4 0.1771 340.47 0.2298 Table 3: Parameters for the interpolating function (385) for b as a function of y. law behaviour, in (381), is by a term of relative size , which is ∼ )2.0694 in the power law region, which suggests that β = 2 might be a good choice in (385). However with β = 2, there still appears to be no solution in the α > 0 case, while there is a solution with γ ≃ −0.6666 in the α < 0 case. Trying β = 2.5, there is a solution with γ ≃ 2.4194 in the α > 0 case, but apparently no solution in the α < 0 case. And trying β = 4, there is a solution with γ ≃ 0.1771 in the α > 0 case, and apparently no solution, again, in the α < 0 case. These example solutions, and the corresponding values of αBβ and y2 , are listed in Table 3. We see that, notwithstanding the substantial differences between the parameters of the interpolating function, for the different choices of β, the corresponding values of only differ by around 20 percent, and are around 1.4 to 1.7 times larger than the value 0.1631 given by the uncorrected power law (207). They show a trend towards the uncorrected power law value with increasing β, corresponding to a later and more rapid onset of the corrrections to the power law. We can also obtain an approximate value of y2 by integrating the approximate formula (381). From (381) and (382) we obtain: ∫ 0.7724 x1.8990dx 0.7724 ≃ y2 − y1 (389) But b1 ∼ κ2/9, which by assumption is small compared to B, so we can extend the lower limit of the integral in the left hand side of (389) to zero, and choosing, as usual, the integration constant y0 in (207) to be zero, we have y1 ≪ κ2/9 ≪ B, so we can drop the y1 term in the right hand side of (389). Then by use of the numerical integration facility of PARI-GP [274], plus an analytic approximation for the contribution from the region close to the upper limit, the integral in the left hand side of (389) is found to be ≃ 0.2536. And comparing with the estimates of y2 as given in Table 3, for the different choices of β in the interpolating function (385), we see that the best agreement is obtained for the choice β = 2, as expected from the discussion following (388). 2.7.2 Newton’s constant and the cosmological constant in the presence of the extra fluxes Turning now to fitting the observed values of Newton’s constant and the cosmological constant, we again follow the method used in subsection 2.5.1, on page 115. The term, in the Einstein action term in (25), that produces the Einstein action, (10), in four dimensions, is again given by (278), where b (y) is now given approximately by (385), with β preferably chosen as 2, and γ and α as given by the row corresponding to β = 2 in the above table, and a, as a function of b, is given, as a first approximation, in terms of the approximate dependence of ṽ on c, in (374), the approximate dependence of c on b, in (381), and the relation ṽ = G̃ 12b2a4 . The worst approximation here is the estimate (374) of the dependence of ṽ on c, which has a percentage error that stabilizes at around 31 percent in the power law region, as found above. We then have: dya2b6 = a2b6 = ∫ 0.7724B (390) We note that since ṽ occurs in (390) only through its square root, the contribution to the error percentage resulting from the use of (374) will be roughly halved, to not more than around 16 percent. And in the power law region, where by (372), ṽ ∼ c−0.5798, the denominator, in the last integral in (390), is ∼ c0.7101, so since (381) overestimates c by not more than about 7 percent, and (374) will overestimate ṽ, as a function of c, by not more than about 16 percent, the use of (381) and (374), in (390), is expected to give a result that will be smaller than the correct result, but by not more than about 16+0.71× 7 ≃ 21 percent. Furthermore, since c ∼ b−1.8990 in the power law region, by (206), the integrand, in the last integral in (390), is ∼ b6.3485 in the power law region, and goes to infinity as (0.7724B − b)− 2 at the upper limit, due to the behaviour (360) of c, so the last integral in (390) is substantially dominated by the contribution from the region near the upper limit, where (381) and (374) are accurate, so the error is in fact expected to be substantially smaller than 21 percent. Inserting the approximate expressions (381) and (374), we find: dya2b6 ≃ B6 ∫ 0.7724 x6.8990dx 1 + 0.4312 x3.7980 0.7724 ))0.1450 0.7724 (391) Now as before, b1 ∼ κ2/9 , so we can set the lower limit of the integral in the right hand side of (391) to zero. Then using again the numerical integration facility of PARI-GP, plus an analytic approximation for the contribution from the region near the upper limit of the integration domain in the integral in the right hand side of (391), we find: dya2b6 ≃ 0.03896B6 (392) And, as explained above, the coefficient 0.03896 is expected to be smaller than the correct value, but the percentage error is expected to be substantially smaller than 21 percent. Now from (373) and (383), we have: ≃ 0.9742κ )1.1010 (393) Hence from (392), and (241), on page 104, we have: dya2b6 ≃ 0.03845κ )5.4495 ≃ 0.03845κ 9 A21 )1.3102τ+6.4653 (394) which replaces (280), on page 117, and (308), on page 127, for the present situation, where the outer boundary is controlled by the extra fluxes. We see that, as found in subsections 2.5.1 and 2.6.1, there is no ADD effect unless τ > −4.9346. Continuing to follow subsections 2.5.1 and 2.6.1, we now find that when the compact six-manifold is a smooth compact quotient of CH3, the Einstein action term, in the four-dimensional effective action, for the solutions considered in the present subsection, will be equal to: 0.3974 )5.4495 −g̃g̃µνRµτντ (g̃) (395) I shall now consider the case where τ = −0.7753, which corresponds to the clas- sical power law (210), on page 90. Then defining the rescaled metric ḡµν by ḡµν = (de Sitter radius) g̃µν , as in subsections 2.5.1 and 2.6.1, so as to measure distances in ordinary units rather than in units of the de Sitter radius, we find from (242), on page 104, that the Einstein action term in the four-dimensional effective action, for the solutions considered in the present subsection, will for τ = −0.7753 and smooth compact quotients of CH3 be equal to − 0.5808 )5.4495 0.7416 −ḡḡµνRµτντ (ḡ) (396) Thus, comparing with (10), we find that for τ = −0.7753 and smooth compact quotients of CH3: ≃ 29.19 1 )5.4495 0.7416 (397) This is the form taken by the ADD mechanism [3, 5], for the solutions considered in the present subsection with τ = −0.7753. Thus for τ = −0.7753 and smooth compact quotients of CH3: ≃ 0.5384 |χ (M6)|0.1361 0.1835 (398) Considering, now, the case of TeV-scale gravity, I shall again consider the case where 9 = 0.2217 TeV, so that κ2/9 = 8.899 × 10−19 metres, and the Giudice, Rattazzi, and Wells [11] gravitational mass MD, for D = 11, is equal to 1 TeV. We then find, from (12), that for τ = −0.7753 and smooth compact quotients of CH3: 7.499× 105 |χ (M6)|0.1361 κ2/9 ≃ 6.673× 10−13 metres |χ (M6)|0.1361 (399) On the other hand, for τ = −0.7753, the integration constant A in (210), on page 90, is from (242), on page 104, fixed directly in terms of |χ (M6)| and the observed de Sitter radius (22), on page 15, and given for smooth compact quotients of CH3 by (319), on page 130. Hence from (393) we find, for τ = −0.7753 and smooth compact quotients of CH3, that: G̃ ≃ 3.53× 10 63metres6 |χ (M6)|0.3670 ≃ 7.11× 10 |χ (M6)|0.3670 4.39× 1028κ2/9 |χ (M6)|0.3670 (400) where G̃ is defined in (352). This is the large constant of integration, not constrained by the field equations or boundary conditions, that is built into the structure of the universe, to make it into the stiff, strong structure that we observe, for the solutions considered in the present subsection with τ = −0.7753, in the case of TeV-scale gravity. In a similar way to the situation with τ , à , and B in subsection 2.6.1, it will be possible, by decreasing τ below −0.7753, to decrease G̃ at a cost of increasing B until as τ approaches the values near −3 in (286) and (287), on page 118, it will no longer be a good approximation to neglect the term 4 in the square root in comparison to the term G̃ 18b6a8 , and the solutions considered in this subsection will tend as G̃ → 0 to those studied in subsection 2.5.1, on page 115. More generally, from (242), (22), (398), and (393), we find that for τ = −0.7753 and smooth compact quotients of CH3, with a general value of κ2/9: 3.90× 10245 |χ (M6)|0.3670 )2.2020 (401) Thus the required value of G̃ is minimized by choosing κ2/9 to be as large as possible, which means TeV-scale gravity, provided this is consistent with the precision tests of Newton’s law down to sub-millimetre distances [32]. To check that this requirement is satisfied, we now determine the values of b, y, and a, at the outer surface of the thick pipe, for the solutions considered in the present subsection with τ = −0.7753, in the case of TeV-scale gravity, with κ− 9 = 0.2217 TeV. From (399) and (382), we find that b2, the value of b at the outer surface of the thick pipe, is given, for τ = −0.7753 and smooth compact quotients of CH3, by: 5.15× 10−13 metres |χ (M6)|0.1361 (402) And since, from above, y2, the value of y at the outer surface of the thick pipe, which is also the “radius” of the thick pipe, is approximately given by y2 ≃ 0.254B, we find, from (399), that for τ = −0.7753 and smooth compact quotients of CH3, y2 is approximately given, for TeV-scale gravity, by: y2 ≃ 0.254B ≃ 1.70× 10−13 metres |χ (M6)|0.1361 (403) Furthermore, from (400), (402), the relation ṽ = G̃ 12b2a4 , and the boundary condition (371) on ṽ at the outer boundary, where b = b2, we find that for τ = −0.7753 and smooth compact quotients ofCH3, a2 = a (y2), the value of a (y) at the outer boundary, is given by: 12b22 5.77× 1021 metres |χ (M6)|0.0237 (404) We note that, since |χ (M6)| ≥ 1, this is large compared to b2, as assumed near the beginning of this subsection. Furthermore, since a1 = a (y1) is the de Sitter radius (22), and |χ (M6)| is bounded above by around 7 × 104, the ratio a1 is bounded above by around 3.4×104. Thus 0.2 millimetres on the inner surface of the thick pipe corresponds on the outer surface to a distance no shorter than around 6 nanometres, so the four dimensional effective field theory description is certainly valid for distances down to 0.2 millimetres, and the realization of TeV-scale gravity considered in this subsection is for τ = −0.7753 consistent with the precision tests of Newton’s law at sub-millimetre distances. Turning now to the flux quantization condition, (345), we find, from the relation ṽ = G̃ 12b2a4 , and the approximate relation (382) between b2 = b (y2) and B, that dya (y) −4 ≃ 12 ∫ 0.7724B b2ṽ (405) Inserting the approximate dependence of ṽ on c, in (374), and the approximate depen- dence of c on b, in (381), we find: dya (y) ∫ 0.7724 x3.8990dx 0.7724 1 + 0.4312 x3.7980 0.7724 ))0.2899 (406) For small b, or equivalently, for large c, the integrand in the right-hand side of (405) behaves as b c1.5798 , and thus as b5, so the integral is dominated by the contribution from the region near the upper limit. The estimate (374) of the dependence of ṽ on c is accurate near the upper limit, and becomes too large by about 31 percent at large c, and the estimate (381) of the dependence of c on b is accurate near the upper limit, and becomes around 6 percent too large at small b, so the integrand in the right-hand side of (406) is accurate near the upper limit, and too large by around 22 percent near the lower limit. Thus we expect (406) to give a result that is too large, but by a lot less than 22 percent. We can again set the lower limit to zero, since b1 ∼ κ2/9 , and using again the numerical integration facility of PARI-GP, plus an analytic approximation near the upper limit, we find: dya (y) −4 ≃ 1.0907B (407) Thus from the flux quantization condition (345), the quantity that is required to be an integer, for each three-cycle, Z, of the compact six-manifold, is approximately: 0.2212 dzAdzBdzCGABC (z) (408) Now comparing with the definition (352) of the constant G̃, we see that G̃ cancels out of (408), which is thus independent of the overall normalization ofGABC (z). Thus the flux quantization condition (345) does not constrain the integration constant A in (210), the de Sitter radius (22), or the effective cosmological constant in four dimensions, (20). Furthermore, we recall that GABC (z) is a linear combination, with position- independent coefficients, of the B3 linearly independent Hodge - de Rham harmonic three-forms on the compact six-manifold M6, where B3 is the third Betti number of M6, that has been assumed to satisfy the conditions (347) and (348), which constitute at most 20 + 20 linearly independent constraints on the B3 independent coefficients in GABC (z). We can always choose a linearly independent set of B3 Hodge - de Rham harmonic three-forms g ABC (z), 1 ≤ i ≤ B3, and a set of B3 three-cycles Z(j) of M6, 1 ≤ j ≤ B3, linearly independent in the sense of homology, such that dzAdzBdzCg ABC (z) = (j). Choosing a basis of harmonic three-forms and a set of B3 three-cycles that satisfy this condition, the requirement that (408) be an integer, for each three-cycle Z(j), 1 ≤ j ≤ B3, implies that the (B3 − 1) independent ratios of the coefficients in GABC (z) are rational numbers. The overall normalization of the coefficients, which cancels out of (408), is fixed by (400) for TeV-scale gravity, and by (401) in general, together with the definition (352) of G̃. If we now define ρj ≡ 1G̃ dzAdzBdzCGABC (z), 1 ≤ j ≤ B3, the requirement that (408) be an integer for all three-cycles Z of M6 reduces to the requirement that 0.2212B ρj be an integer for all 1 ≤ j ≤ B3. Now the value of B has been assumed to be fixed by the boundary condition at the inner surface of the thick pipe, with its actual value determined by the Casimir energy densities on and near the inner surface of the thick pipe, so B would be overconstrained if the flux quantization conditions significantly restricted its value. However for TeV-scale gravity, (399) implies that the value of B is around 1016, pro- vided |χ (M6)| is not too large, so provided none of the nonvanishing ρj are too small in magnitude, and the nonvanishing are expressible as ratios of sufficiently small integers, an alteration of a (y) by a tiny percentage in the region near the outer sur- face, where the alteration would have the greatest effect on the integral (405), would be sufficient to satisfy all the flux quantization conditions. Furthermore we are free to choose the independent ratios of the ρj , and thus to set them equal to ratios of small nonvanishing integers, in which case it seems plausible that the magnitudes of the ρj would generally lie more or less within the range . Thus provided |χ (M6)| and B3 are not too large, it seems plausible, at least for the case of TeV-scale gravity, that the flux quantization conditions, (345), will not significantly restrict the solutions considered in the present subsection. We note that, notwithstanding the large value (400) of G̃ in the case of TeV-scale gravity, and its large value (401) in general, the extra fluxes of the four-form field strength of the three form gauge field considered in the present subsection, which wrap three-cycles of the compact six-manifold times the radial dimension of the thick pipe, never have a large enough field strength that we would expect them to produce quantum effects. To estimate whether we would expect the extra fluxes to produce quantum effects, we note that we expect quantum gravitational effects when the Ricci scalar has magnitude ∼ κ− 49 or larger. Hence from the supergravity action (25), we would expect the four-form field strength GIJKL to produce quantum effects when GIMGJNGKOGLPGIJKLGMNOP has magnitude ∼ κ− 9 or larger. And, noting that there are no cross terms in GIMGJNGKOGLPGIJKLGMNOP between the extra fluxes and the standard Witten fluxes that follow for smooth compact quotients of CH3 from Witten’s topological constraint, as studied in subsections 2.3.7 and 2.3.8, we find, from (351), that in the approxiation of dropping all but the leading harmonic, the contribution to GIMGJNGKOGLPGIJKLGMNOP from the extra fluxes is given by GQRGKNGLOGMPGQKLMGRNOP = G̃2 = 576 (409) where the relation ṽ = G̃ 12b2a4 was used. Thus since ṽ = 1 at the outer surface of the thick pipe, by (371), and b ∼ B at the outer surface of the thick pipe, which for TeV- scale gravity is ≫ κ2/9 by (399), unless |χ (M6)| is extremely large, which seems very unlikely since it would require a correspondingly small value of b1 , by (103), we see that for TeV-scale gravity the extra fluxes are not large enough at the outer surface of the thick pipe that we would expect them to cause quantum effects there. Furthermore, from (210), on page 90, 1 behaves as b0.2024 in the bulk power law region, and thus decreases with decreasing b, and from the form of the solutions studied above, neither ṽ nor b changes significantly in order of magnitude between the power law region and the outer surface of the thick pipe, hence for TeV-scale gravity (409) is small in magnitude compared to κ− 9 throughout the whole thick pipe. Thus for TeV- scale gravity, we do not expect the extra fluxes considered in the present subsection to produce any significant quantum effects at all, and away from the inner surface of the thick pipe, the solutions studied in the present subsection are entirely classical in character. We note, furthermore, that even though G̃, and the integration constant, A, in (210), cancel out of the reduced Einstein equations (369) and (370) and boundary conditions (371), and also cancel out of the contribution of the extra fluxes to GIMGJNGKOGLPGIJKLGMNOP , they are nevertheless physically significant. For first of all, if G̃ and A had not been sufficiently large, it would not have been possible to neglect the term 4 in the square root in (194) and (195), in comparison with the new term in 2 κ2t(3) coming from (357), as explained between (357) and (358), and it would not then have been possible to eliminate G̃ from the Einstein equations by defining ṽ = G̃ 12b2a4 . And secondly, from the metric ansatz (94), and (210), the observed de Sitter radius (22), as estimated from observations of type Ia supernovae [293, 294, 295], and significantly bounded below by a great variety of astronomical observations, as well as by the approximate flatness of the everyday world, is equal to A )0.7753 , where b1 = b (y1) is expected, by (103), to be ∼ κ2/9. It is the large value of A, which for the solutions considered in the present subsection results from the large value of G̃, that results in the existence of a large and approximately flat platform at the inner surface of the thick pipe, on which the interesting processes of intermediate range astronomy, and the everyday world, can take place. The value of G̃ is not constrained by the field equations or the boundary conditions, since the relevant field equation, (342), is satisfied both in the bulk and on the orbifold fixed-point hyperplanes, in the upstairs picture, in consequence of the Hořava-Witten orbifold conditions, as summarized after (25), which imply that the components GyUVW are even under reflections in the orbifold hyperplanes. It seems that the existence of the arbitrary constant G̃, defined in (352), in compactifications of Hořava-Witten theory of the type studied in the present paper, is implicit in the field content of supergravity in eleven dimensions [38, 14], and the boundary conditions or orbifold conditions of Hořava-Witten theory, as summarized after (25). The question of how G̃ came to have the large value required to fit the observed small value of the cosmological constant, and the related question of whether G̃ has any effects on the dynamics of the early universe, other than preventing the occurrence of a large cosmological constant, in models of this type, will not be considered in the present paper. Finally we should check whether the solutions considered in the present subsection are consistent with the precision tests of Newton’s law down to submillimetre distances, and the very high precision tests of Newton’s law over solar system distances, as carried out for the solutions of subsection 2.6 in subsection 2.6.1, starting shortly after equation (321), on page 132. However, it does not seem very likely that the constraints from these tests will be more stringent for the solutions considered in the present subsection than for the solutions of subsection 2.6, and this will not be considered in detail in the present paper. The fact that the value of G̃ does seem to be quantized seems to suggest that if the metric ansatz (94) is generalized to a cosmological ansatz of the form ds211 = −a (y)2 dt2 + ã (y, t)2 gijdxidxj + b̃ (y, t)2 hABdxAdxB + c̃ (y, t)2 dy2, which is consistent with (94) if the metric on the four-dimensional de Sitter space, in (94), is taken in Friedmann-Robertson-Walker form, a parameter related to G̃ might evolve with time, as in quintessence models [33]. This could perhaps be investigated by studying cosmo- logical solutions that are small perturbations of the de Sitter solutions studied in this section. 3 Smooth compact quotients of CH , H6, H3 and S3 For the compactifications of Hořava-Witten theory considered in the present paper, the compact six-manifold, M6, is a smooth compact quotient of either the symmetric space CH3, or the symmetric space H6, by a discrete subgroup of the isometry group of the symmetric space, and for the solutions considered in subsection 2.6, on page 120, the three observed spatial dimensions are also compactified to a smooth compact quotient of either the symmetric space H3, or the symmetric space S3, by a discrete subgroup of the symmetric space. I shall first consider smooth compact quotients of the non-compact symmetric spaces CH3, H6, and H3, then briefly consider smooth compact quotients of S3, at the end of this section. Let G be the identity component, or in other words, the connected component that contains the identity, of either SU(n,1), for n ≥ 1, or SO(n,1), for n ≥ 2, and let K be the maximal compact subgroup of G, which is SU (n)×U (1) for SU(n,1), and SO(n), for SO(n,1). Then the non-compact symmetric space, S ≡ G/K, is CHn for SU(n,1), and Hn for SO(n,1). I shall assume that the metric of CHn is normalized such that the Riemann tensor is given by (72), in complex coordinates, and that the metric of Hn is normalized such that the Riemann tensor is given by Rµνστ = gµσgντ − gµτgνσ, consistent with the choice made between (188) and (189), and between (99) and (100), so that the sectional curvature (84) of Hn is equal to −1, which is the conventional value. We choose a point of S, called O, to be the origin of S. For example, for CHn, we could choose O to be the origin of the coordinates used for CHn, in subsection 2.2. For any subgroup, H , of G, let C (H) denote the set of all the set of all the images of O, by the action of elements of H . A discrete subgroup, Γ, of G, is a subgroup of G such that there is a real number ρ > 0, such that for all members x of C (Γ) different from O, the geodesic distance from O to x is ≥ ρ. For any discrete subgroup, Γ, of G, and any member, x, of C (Γ), the Wigner-Seitz cell, or Voronoi cell, W (Γ, x), is the set of all points of S, that are closer to x, than to any other member of C (Γ). The fundamental domain of the quotient S/Γ is W (Γ, O). Γ is called a lattice in G, if W (Γ, O) has finite volume. When Γ is a lattice in G, the set C (Γ), of all the images of O in S, looks like a hyperbolic analogue of a crystal lattice. G always has an infinite family of lattices called “arithmetic lattices”, whose exis- tence was demonstrated by Borel and Harish-Chandra [34]. A very helpful introduction to arithmetic lattices has been provided by Morris [296]. Some examples of arithmetic lattices in G are considered in subsection 3.1, on page 167. For SO(n,1), SU(2,1), and SU(3,1), there also exist additional lattices called “non-arithmetic lattices”. Non- arithmetic lattices in SO(n,1), for n ≤ 5, were constructed by Makarov and Vinberg [297, 298], and non-arithmetic lattices in SO(n,1), for all n, were constructed by Gro- mov and Piatetski-Shapiro [35]. The construction of Gromov and Piatetski-Shapiro involves cutting two different quotients of S into two parts along totally geodesic n- dimensional submanifolds, and smoothly joining together one part from each of the two different quotients. There is no analogous construction for SU(n,1) for n ≥ 2, because for n ≥ 2, CHn has no totally geodesic (2n− 1)-dimensional submanifolds [121]. Non-arithmetic lattices in SU(2,1) were constructed by Mostow [36], and non- arithmetic lattices in SU(3,1) were constructed by Deligne and Mostow [37]. To the best of my knowledge, it is not yet known whether or not any non-arithmetic lattices exist for SU(n,1), n ≥ 4. For the models considered in the present paper, I assume that the quotient S/Γ is a smooth manifold, not an orbifold, so Γ is required to act on S without fixed points, or in other words, no non-trivial element of Γ is allowed to leave any point of S invariant. The fact that S is the quotient of G, by its maximal compact subgroup, K, implies that a necessary condition for Γ to act without fixed points, is that Γ must have no torsion, in the sense of discrete group theory, or in other words, Γ must not contain any element g 6= 1, such that gn = 1, for some finite n. For the finite group generated by such a g is a compact subgroup of G, and every compact subgroup of G is contained in a maximal compact subgroup, and all maximal compact subgroups of G are conjugate. Furthermore, K is the subgroup of G that leaves O invariant, and the conjugate hKh−1 of K, where h is a fixed element of G, leaves the point hO invariant. Conversely, the requirement that Γ have no torsion is also sufficient to ensure that Γ acts on S without fixed points. For suppose an element g 6= 1 of Γ leaves a point x of S invariant. Then since Γ is an isometry, and maps members of C (Γ) to members of C (Γ), g must permute the members of C (Γ), at any given fixed distance from x, among themselves. In particular, g must permute the nearest neighbours of x, in C (Γ), amongst themselves. But the discreteness of Γ implies that the number of nearest neighbours of x, in C (Γ), is finite. Hence g is an element of a finite group, hence gn = 1, for some finite n. Now let Γ be a lattice in G, that acts without fixed points on S, so that the quotient S/Γ is a smooth manifold, of finite volume. Let d denote the real dimension of S, which is 2n for SU(n,1), and n for SO(n,1). Then for d ≥ 3, Mostow’s rigidity theorem [299, 300, 10, 301, 302] implies that the locally symmetric space S/Γ is completely determined, up to isometry, by its fundamental group, which is Γ. This result is not true for d = 2, since smooth compact quotients of CH1, which differs from H2 only in the normalization of its metric, in the conventions adopted here, have shape moduli, as is well known in superstring theory. An orientable smooth compact quotient of CH1 of genus g ≥ 2, which is topologically equivalent to a sphere with g handles, has a moduli space of dimension 6g − 6. The moduli are called Teichm”uller parameters, and are the minimum number of parameters needed to characterize conformally inequivalent closed Riemann surfaces. They correspond to the positions and radii of six circles in the complex plane, which are identified in pairs to produce the closed surface, less six parameters that relate conformally equivalent surfaces. Mostow’s rigidity theorem implies, in particular, that for d ≥ 3, the volume, V (S/Γ), is a topological invariant. For d even, V (S/Γ) is a fixed multiple of the Euler number, given for d = 6 by (99) for smooth compact quotients of CH3, and by (100) for smooth compact quotients of H6, but for d odd, the Euler number is zero, and, at least for d = 3, there is no corresponding restriction on the possible values of V (S/Γ). If there is a finite upper bound, on the geodesic distance between pairs of points in the fundamental domain W (Γ, O), then the quotient S/Γ is compact, while if, for any given finite distance, there exist pairs of points, in the fundamental domain, such that the geodesic distance between them is greater than that given distance, then the quotient is non-compact. In the coordinate system used for CHn, in subsection 2.2, a quotient of CHn is compact, if all points in the closure of its fundamental domain have zrzr̄ < 1, and non-compact if the closure of its fundamental domain has one or more vertices on the sphere zrzr̄ = 1. If a quotient is non-compact, then the non- compactness is associated with a finite number of tubular regions, called cusps, which extend out to infinite distances, but become narrow so rapidly, that their contribution to the total volume is finite. Inspection of the list in section (14.4) of [37] shows that the non-arithmetic quotients of CH3 found by Deligne and Mostow, which correspond to the lattice in SU (3, 1) denoted 753333 in their notation, are not compact. This same lattice is also the only non-arithmetic lattice in SU (3, 1) listed in the Appendix in [303], where it occurs as No. 66 in the list. Thus to the best of my knowledge, it is at present not known whether or not any compact non-arithmetic quotients of CH3 exist. However, the non-arithmetic quotients of CH2 found by Mostow in [36] are compact. For the models considered in the present paper, I assume that the quotient is compact. However, smooth non-compact finite volume quotients of H3 are important, because by a construction of Thurston [301, 304, 305, 276], there exist infinite sequences of smooth compact quotients of H3, all with distinct topology, whose volumes converge to the volumes of smooth non-compact finite volume quotients of H3. This cannot happen in any dimension larger than 3, because by a theorem ofWang [277], the number of topologically distinct smooth finite volume quotients of a non-compact symmetric space of dimension ≥ 4, whose volume V (S/Γ) is less than a given volume, is finite. Moreover, Borel [306] demonstrated that the number of topologically distinct smooth compact arithmetic quotients of H3, whose volume is less than a given volume, is finite, so all but a finite number of the smooth compact quotients of H3 resulting from Thurston’s construction, whose volume is less than a given volume, are non-arithmetic. Each cusp of a non-compact finite volume quotient of H3 is topologically equivalent to the Cartesian product of a two-torus and the infinite half-line. Thurston’s construction makes use of a method of modifying the topology of a three-manifold M3, called Dehn surgery [307, 308, 309]. A Dehn surgery consists of removing a tubular neighbourhood N of an S1 embedded in the manifold, then putting it back, in a twisted fashion. The surface of N is a two-torus, and twists can be introduced in two independent senses. We choose two oriented simple closed curves m and l, called the meridian and the longitude, embedded in the common boundary torus of N and its complement, that generate the fundamental group of that torus. When M3 is S3, l is chosen such that it bounds a surface in the complement of N , and m is chosen such that it crosses l exactly once. This gives any oriented simple closed curve c on that torus two coordinates p and q, which correspond to the net number of times c crosses m and l respectively. These coordinates depend only on the homotopy class of c. The Dehn surgery with slope u = p , where p and q are coprime integers, then corresponds to gluing back N by means of a homeomorphism of its two-torus boundary to the two-torus boundary of its complement, such that the meridian curve of the boundary of N maps to a (p, q) curve in the boundary of its complement. By a theorem of Lickorish [310] and Wallace [311], every closed, orientable, connected three- manifold can be obtained from S3 by Dehn surgery around finitely many copies of S1 embedded disjointly in S3. Now let M3 a smooth non-compact finite volume quotient of H3 with n cusps, where n ≥ 1. Then because each cusp of M3 is topologically equivalent to the Cartesian product of a two-torus and the infinite half-line, M3 is topologically equiv- alent to the interior of a compact three-manifold with n connected boundary compo- nents, each of which is topologically equivalent to a two-torus. We choose a merid- ian and longitude for each boundary torus, as in the case of Dehn surgery. Let M3(u1, u2, . . . , un) denote the manifold obtained from M3 by filling in each bound- ary two-torus with a solid torus using the slopes ui = pi/qi, where each pair pi and qi are coprime integers. This is called Dehn filling. Thurston’s hyperbolic Dehn surgery theorem then states that M3(u1, u2, . . . , un) is topologically equivalent to a smooth compact quotient H3/Γ (u1, u2, . . . , un) of H 3, provided a finite set Ei of slopes is avoided for each i. Furthermore, V (H3/Γ (u1, u2, . . . , un)) < V (M3), and the volumes V (H3/Γ (u1, u2, . . . , un)) converge to V (M3) as all p2i + q2i → ∞, pi 6= 0, qi 6= 0. It is also known that only a finite number of topologically distinct smooth compact quotients of H3 with any given volume exist [304]. Many smooth non-compact finite volume quotients of H3 have been discovered by studying the complement of disjoint tubular neighbourhoods of finitely many copies of S1 embedded disjointly in S3, to see if it can be constructed by gluing together a small number of hyperbolic tetrahedra, some of whose vertices stretch out to infinity as parts of cusps. A hyperbolic polyhedron, one or more of whose vertices stretches out to infinity as part of a cusp, is called an ideal hyperbolic polyhedron. This method was originally applied by Thurston to show that the complement of the figure of eight knot was hyperbolic, by constructing it by gluing together two ideal hyperbolic tetrahedra. It had earlier been shown to be hyperbolic by Riley, and by Jorgensen, using other methods. Weeks’s computer program SnapPea [312], which can perform Dehn surgeries au- tomatically, includes a census of smooth non-compact finite volume quotients of H3 constructed by gluing together up to seven hyperbolic tetrahedra. The smooth non- compact finite volume quotient ofH3 that is topologically equivalent to the complement of the figure of eight knot is designated m004 in SnapPea, and has volume 2.02988 . . .. This has been shown by Cao and Meyerhoff [313] to be the smallest possible volume of an orientable cusped hyperbolic three-manifold. There is one other known smooth non-compact finite volume quotient of H3 with this volume, which is designated m003 in SnapPea, and can be obtained by a (5, 1) Dehn filling on the complement of the Whitehead link. The Whitehead link is a disjoint embedding of two copies of S1 in S3, such that neither S1 is knotted by itself, but the two copies of S1 are linked such that one S1 has a loose twist to resemble a figure of eight, and the other S1 links both loops of the figure of eight. The smooth compact quotient of H3 of smallest known volume is called the Weeks manifold or the Fomenko-Matveev-Weeks manifold [314, 315], and can be obtained by a (5,2) Dehn filling on m003 or by a (3,−1) Dehn filling on m003, and has volume 0.9427 . . .. The smooth compact quotient of H3 of second smallest known volume is called the Thurston manifold, and can be obtained by a (−2, 3) Dehn filling on m003, and has volume 0.9814 . . .. By Thurston’s hyperbolic Dehn surgery theorem, there are already an infinite num- ber of topologically distinct smooth compact quotients of H3 with volume less than the volume 2.02988 . . . of m003 and m004, while as noted above, when the dimension d of S is ≥ 4, the number of topologically distinct smooth finite volume quotients of S, whose volume V (S/Γ) is less than a given volume, is finite. Let ρS (v) denote the number of topologically distinct smooth finite volume quotients of S, whose volume V (S/Γ) is less than v. Then Gelander [316] has proved that when the dimension d of S is ≥ 4, there is a constant c, depending on S, such that log ρS (v) ≤ cv log v (410) for all v > 0. And for Hn, n ≥ 4, Burger, Gelander, Lubotzky, and Mozes (BGLM) [317] have proved that there exist constants cn > bn > 0 and vn > 0, such that bnv log v ≤ log ρHn (v) ≤ cnv log v (411) whenever v > vn. Thus the number of topologically distinct smooth finite volume quotients of H6 with |χ (M6)| < n grows as ncn for sufficiently large n, where c is a constant > 0. Furthermore, for both the smooth finite volume quotients ofH3 obtained by Thurston’s construction, and for the arithmetic quotients considered in the following subsection, the vast majority of the smooth finite volume quotients are in fact compact, so it seems likely that the number of topologically distinct smooth compact quotients of H6 with |χ (M6)| < n also grows as ncn for sufficiently large n, with the same constant c > 0. However, since supergravity in eleven dimensions does not contain any Yang-Mills fields, and the three-form gauge field only enters the supercovariant derivative on the gravitino through its three-form field strength, which is well defined globally, there is no possibility of introducing an analogue of a spinc structure to compensate for the compact six-manifold M6 not being a spin manifold, so M6 is required to be a spin manifold. To the best of my knowledge, none of the known examples of smooth compact quotients of CH3 or H6 have yet been shown to be spin manifolds. For smooth compact quotients of CH3, the fraction of smooth compact quotients that are spin would be expected to be ∼ 2−B2 , where B2 is the second Betti number of the quotient, since the second Steifel-Whitney class is the mod 2 reduction of the first Chern class, hence the vanishing of the second Steifel-Whitney class requires that h11 ∼ B2 integers be even. Now by a theorem of Gromov [46], there is a number β such that for all smooth compact quotients M6 of CH3 or H6 all the Betti numbers of M6 are bounded by β |χ (M6)|. Thus if we suppose that the fraction of the smooth compact quotients of H6 that are spin is also ∼ 2−B2 , then the result of Burger, Gelander, Lubotzky, and Mozes stated above implies that for sufficiently large |χ (M6)|, there will be smooth compact quotients M6 of H6 that are spin. It is known that all smooth compact orientable manifolds of dimension ≤ 3 are spin [318], and the Davis manifold [40], which is the simplest known smooth compact quotient of H4, has been shown to be a spin manifold [41]. We note that most of the smooth compact hyperbolic threefolds associated by Thurston’s construction with a given smooth non-compact finite volume quotient of H3 will be highly inhomogeneous. In fact, by a theorem of Cheeger [319], summarized recently in [320], if a sequence of Riemannian manifolds is such that there is a fixed upper bound on the magnitudes of all sectional curvatures of all manifolds in the se- quence, a fixed lower bound > 0 on the volumes of all manifolds in the sequence, and a fixed upper bound on the diameters of all manifolds in the sequence, then the sequence contains only a finite number of diffeomorphism types. Thus since the members of a Thurston infinite sequence of topologically distinct smooth compact quotients of H3, whose volumes tend to the volume of a finite volume cusped hyperbolic threefold, satisfy the first two requirements of Cheeger’s theorem, they must violate the third require- ment, which means that the diameters of the members of the sequence must increase without limit. This suggests that the members of such a Thurston sequence develop longer and longer spikes, that approximate more and more closely to the infinite cusps of the finite volume cusped hyperbolic threefold, and that, moreover, the differences in topology between the members of the sequence might become localized further and further out along these spikes, where the spikes become narrower and narrower. 3.1 Smooth compact arithmetic quotients of CHn and Hn I shall now outline the construction of some smooth compact arithmetic quotients of CHn and Hn. The first step is to construct some cocompact arithmetic lattices in SU(n,1) and SO(n,1), whose existence follows from section 12 of [34]. I shall then briefly review Selberg’s lemma [321] for the case of these arithmetic lattices, which states that certain finite index subgroups of these discrete groups are torsion-free, or in other words, have no nontrivial finite subgroups. A subgroup Γ1 of a discrete group Γ is said to have finite index in Γ, if Γ1 divides Γ into a finite number of left cosets. We recall that an algebraic number field [322] is a finite-dimensional (and therefore algebraic) field extension of the field Q of rational numbers. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. To form an algebraic number field, we recall that for any field F , the ring of polynomials with coefficients in F is denoted by F [x]. A polynomial p(x) in F [x] is called irreducible over F [323] if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from F [x]. Then if α is a root of some irreducible polynomial f (x) in F [x], the extension field F (α) is the set of all polynomials g (α), with two polynomials g (α) and h (α) being defined to be equal, if f (α) = 0 implies g (α) = h (α). In practice this means that, if f (x) is of degree m, every polynomial g (α), of degree ≥ m, is equal to some polynomial of degree < m. The field extension F (α) will then be of degree m, as a vector space over the field F , and a possible basis for F (α) is the set of monomials 1, α, α2, . . . , αm−1. If an element a 6= 0 of F (α) is expressed in this basis as a = a1 + a2α+ a3α2 + . . .+ amαm−1, where the coefficients ai are elements of F , then the reciprocal of a, expressed in this basis as a−1 = b1 + b2α+ b3α 2 + . . .+ bmα m−1, where the coefficients bi are elements of F , can be found by solving the 2m− 1 linear equations, that result from equating coefficients of all powers of α, up to and including α2m−2, in the equation a1 + a2α + a3α 2 + . . .+ amα b1 + b2α+ b3α 2 + . . .+ bmα = 1 + c1 + c2α + c3α 2 + . . .+ cm−1α f (α) (412) for the m coefficients bi, and the m − 1 coefficients ci, which are also elements of F . The extension field F (α) is sometimes written F ({α}), to allow for the possibility of adjoining more than one new element to F . In general, if S is a set of elements not in F , the extension field F (S) is the smallest field that contains F and S. We recall, also, that an algebraic number is a root of a polynomial with integer coefficients. For any algebraic number, α, there is a unique polynomial f (x) in Q [x], such that f (x) is irreducible over Q, the coefficient of the highest power of x in f (x) is equal to 1, and α is a root of f (x). This is called the minimal polynomial of α, and the degree of this polynomial is called the degree of α. Every polynomial g (x) in Q [x], such that g (α) = 0, is a multiple of the minimal polynomial of α. The roots of the minimal polynomial of α, including α itself, are called the conjugates of α, and are all distinct. By the primitive element theorem [324], every algebraic number field F is of the formQ (α), where α is a root of a polynomial f (x) inQ [x], such that f (x) is irreducible over Q. An element α of F , such that F is generated by adjoining α to F , is called a primitive element of F . A primitive element of F can also be characterized by the fact that it does not belong to any proper subfield of F , and it can also be characterized by the fact that the degree of its minimal polynomial is equal to the degree of F . An algebraic number field F has only a finite number of subfields K such that Q ⊆ K ⊆ F , and since these correspond to subspaces of F as a vector space over Q, most elements of F are in fact primitive elements. Considering the field F = Q , for example, neither 2 nor 3 is a primitive element of F , since they are respectively elements of the subfields Q . We cannot form F by adjoining to Q a root of the polynomial x4−5x2+6, whose roots are ± 2 and ± 3, because this polynomial factors as (x2 − 2) (x2 − 3), and is thus not irreducible overQ. The nfinit function of PARI/GP [274], for example, simply rejects an attempt to form an algebraic number field with this polynomial. However α = 3, whose minimal polynomial is x4 − 10x2 + 1, is a primitive element of F , and, indeed, we have α 2, and −α3 + 11α 3. The conjugates of α are ± 3 with all four sign choices allowed. If F is an algebraic number field of degree m over Q, and vi, 1 ≤ i ≤ m, is a basis for F , as a vector space over Q, then we may associate to each element x of F , a square matrix xij with rational elements, defined by xvi = vjxji, where the summation convention is used. We then find, for any elements x and y of F , that xyvi = xvjyji = vjxjkyki. Thus the matrices xij form a matrix representation of the elements of F , called the regular representation for the basis given by the vi. For example, for the field Q , we find that 1 is represented by , and is represented by . Invariants of the matrix xij representing an element x of F , such as its trace, determinant, and characteristic polynomial, are properties of x, and do not depend on the basis. In particular, the characteristic polynomial of xij is a polynomial of degree m, with the coefficient of λ m equal to 1, and by the Cayley-Hamilton theorem has x as a root. If x is a primitive element of F , then the characteristic polynomial if its matrix representation, xij , is irreducible over Q, and is the minimal polynomial of x. We recall, also, that an algebraic integer [325] is a root of a polynomial with integer coefficients, such that the coefficient of the highest power of x is equal to 1. The sum, difference and product of two algebraic integers is an algebraic integer. If F is an algebraic number field of degree m over Q, then the elements x of F , whose regular representation matrices xij have a characteristic polynomial with integer coefficients, are the algebraic integers in F . The set of all the algebraic integers in F is a ring, called the ring of algebraic integers of F , that is often denoted OF . It is always possible to use a basis for F consisting of algebraic integers, called an integral basis, in which every algebraic integer x is represented by a matrix xij with integer matrix elements. When we use an integral basis for F , the algebraic integers of F are then precisely those elements x of F which, when expressed as a linear combination x = x1v1 + . . .+ xmvm of elements of the basis, are such that the xi are all integers. Most of the algebraic integers of F are primitive elements of F , since those which are not primitive are one of the finite number of proper subfields K such that Q ⊆ K ⊂ F , and thus lie in one of a finite number of linear spaces over Q, each of dimension < m over Q. Thus F can always be expressed in the form Q (α), where α is an algebraic integer in F , of degree equal to m. If F is an algebraic number field of degree m over Q, and α is an algebraic integer of F , of degree equal to m, then F always has an integral basis whose first element is 1, whose second element has the form a21+α , whose third element has the form a31+a32α+α and so on, where the aij and the di are ordinary integers in Z. This is called a canonical basis of F [326]. If t is an integer not divisible by the square of an integer > 1, then for t = 1 (mod4), an integral basis for the quadratic number field Q is given by 1, 1 , while for t = 2 or 3 (mod 4), an integral basis for Q is given by 1, t [327]. If F is an algebraic number field, then an embedding of F into the field C of complex numbers, sometimes called an isomorphism of F into C, means a one to one map of F into C, that preserves all the structure of F . In particular, the subfield Q of F is mapped by the identity map to the subfieldQ ofC. The number of distinct embeddings of F into C is finite. In particular, if F is defined by adjoining to Q a root, α, of a polynomial f (x) in Q [x], such that f (x) is irreducible over Q, then an embedding of F = Q (α) into C is specified by saying which root of f (x), in C, α corresponds to. An algebraic number field F is called totally real , if every embedding of F into C, is equivalent to its complex conjugate. If F is defined by adjoining to Q a root, α, of a polynomial f (x) in Q [x], such that f (x) is irreducible over Q, then F = Q (α) is totally real, if all the roots of f (x) are real. If an object X , such as a number, matrix, or group, is defined for a specific embed- ding, I, of an algebraic number field, F , into C, then the object corresponding to X , for an embedding σ of F into C, is denoted by Xσ, and called the Galois conjugate of X by σ. For example, if F is Q , then there is only one embedding of F into C different from I, and taking the Galois conjugate of an object by that embedding, corresponds to replacing all occurrences of 2 by − The discreteness of the arithmetic lattices, to be constructed below, will follow from the fact, noted in section 12 of [34], that if F is an algebraic number field, OF is the ring of algebraic integers of F , and ΦF is the set of all distinct embeddings σ of F into C, then for any positive number r, there are only a finite number of elements b of OF , such that all the Galois conjugates bσ of b, σ ∈ ΦF , have magnitude less than r. For example, in the case when F = Q and OF = Z , there are an infinite number of elements a + b 2, a, b ∈ Z, of OF , whose magnitude is less than 1, but no elements at all of OF , such that both a + b 2 and a − b 2 have magnitude less than 1. To check the result for general F , we note, first, that for any member, b, of F , the set of all the Galois conjugates bσ of b, σ ∈ ΦF , is the same as the set of all the conjugates of b, as an algebraic number. This set will have the same number of members as the degree of F , if b is a primitive element of F , and a smaller number, if b is not a primitive element of F . Furthermore, the product of all the conjugates of an algebraic number of degree s is equal to (−1)s times the constant term in its minimal polynomial, which must be non-zero, for otherwise the polynomial would be reducible over Q. Thus the product of all the conjugates of an algebraic integer has magnitude ≥ 1, hence the result is certainly true when r ≤ 1, since there are then no elements of OF , all of whose Galois conjugates have magnitude less than r. For general r, we note that if all the conjugates of an algebraic integer of degree s have magnitude less than r, then, denoting this algebraic integer and its conjugates by b1, b2, . . . , bs, we have |b1 + b2 + . . .+ bs| < sr, |b1b2 + b1b3 + . . .+ bs−1bs| < s(s−1)2 r 2, . . ., |b1b2 . . . bs| < rs, hence every coefficient of the minimal polynomial of that algebraic integer is bounded by a binomial coefficient times a power of r, hence since these coefficients are integers, there are, in fact, only a finite number of distinct algebraic integers of degree s, all of whose conjugates have magnitude less than r. Hence, since all elements of OF are algebraic integers of degree ≤ the degree of F , there are only a finite number of elements of OF , all of whose Galois conjugates, or equivalently, all of whose conjugates, have magnitude less than r. I will call this result the bounded conjugates lemma. We can now construct examples of cocompact lattices in SU(n,1) and SO(n,1), by choosing: 1. A totally real algebraic number field F 6= Q; 2. A specific embedding I of F into R; and 3. A diagonal (n+ 1) × (n+ 1) matrix B, with signature (+,+, . . . ,+,−), and diagonal matrix elements in the ring of algebraic integers OF of F , such that for all embeddings σ 6= I of F into R, the Galois conjugate Bσ is either positive definite or negative definite. For example, we could choose F to be Q (α), where α is a root of the polynomial x2 − 2, we could specify I by choosing α to be 2 rather than − 2, and we could choose B to be the diagonal (n + 1) × (n + 1) matrix with diagonal matrix elements 1, 1, . . . , 1,− , so that for the one embedding σ different from I, the Galois conju- gate Bσ is the positive definite diagonal matrix with diagonal matrix elements 1, 1, . . . , 1, We note that SU (B), the group of all complex (n + 1)× (n+ 1) matrices with unit determinant, that preserve the quadratic form BRS̄z RzS̄, is isomorphic to SU(n,1), and that the matrix ηRS̄, defined between (73) and (75), could be transformed to equal BRS̄ , by a suitable rescaling of the coordinates, and similarly, SO (B), the group of all real (n + 1)× (n+ 1) matrices with unit determinant, that preserve the quadratic form BRSx RxS, is isomorphic to SO(n,1), and that the standard Minkowski metric ηRS could be transformed to equal BRS, by a suitable rescaling of the coordinates. Here zS̄ = , in accordance with the conventions of subsection 2.2. If we now identify G with the identity component, or in other words, the connected component containing the identity, of either SU (B) or SO (B), then the required co- compact lattice, Γ, is in the unitary case, the subgroup GOF [i] of G, consisting of the elements of G, all of whose matrix elements are in OF [i], the extension of the ring of algebraic integers OF of F , by the square root of −1, and in the orthogonal case, the subgroup GOF of G, consisting of the elements of G, all of whose matrix elements are in OF [34]. To check the discreteness of Γ I will consider the case of SU (n, 1) and SU(B), since the corresponding discussion for SO(n,1) and SO(B) will follow by dropping the extension of OF to OF [i]. We first note that, for all embeddings σ 6= I of F into R, the Galois conjugate group Gσ = SU (Bσ) is isomorphic to SU (n+ 1), and thus compact. Furthermore, for any element γ of Γ, the Galois conjugate γσ will be a member of Γσ, and thus of Gσ. Thus γσ could be transformed to a unitary matrix by a rescaling of the coordinates, so there is a number rσ, depending only B σ, such that every matrix element of γσ has magnitude less than rσ. Let r be any number ≥ the maximum of the numbers rσ, for all the embeddings σ 6= I of F into R. Then since OF is the ring of algebraic integers of the algebraic number field F , the bounded conjugates lemma implies that there are only a finite number of elements b of OF , such that all the Galois conjugates bσ of b, including b itself, have magnitude less than r. Thus if γ is an element of Γ, such that every matrix element (a+ ib), a, b ∈ OF , of γ, has magnitude less than r, then there are only a finite number of elements c of OF [i], that can be matrix elements of γ. Thus there are only a finite number of elements γ of Γ, such that every matrix element of γ has magnitude less than r. In particular, there is a number r > 0 such that there are only a finite number of elements γ of Γ, such that every matrix element of γ − 1, where 1 denotes the unit matrix, has magnitude less than r. Hence Γ is discrete [34]. We note that if r is the maximum of the numbers rσ, for all the embeddings σ 6= I of F into R, and γ is any element of Γ, then every matrix element (a+ ib) of γ has the property that each of a and b is an algebraic integer in OF , such that all its conjugates, other than itself, have magnitude less than r. An algebraic integer, such that all its conjugates, other than itself, have magnitude less than r, is called an r-Pisot number. Pisot numbers are sometimes called Pisot-Vijayaraghavan numbers, or PV numbers. Fan and Schmeling [328] have demonstrated that for any real algebraic number field, and any r > 0, there exists a number L such that for all x ∈ R, there is at least one r-Pisot number η in that algebraic number field, such that x ≤ η ≤ x + L. This result gives an indication of the distribution of algebraic integers that can be matrix elements of an element of Γ. For example, if F is the field Q , G is isomorphic to SU(1,1), and B is the diagonal matrix with diagonal entries , then every matrix element of every element of Γ is a 2 4 -Pisot number. Now for every integer, b, there is an integer, a, such that ∣a− b ∣ ≤ 1 4 , so that a + b 2 is a 2 4 -Pisot number. And for that integer, a, we have 2b 2 − 1 ≤ a + b 2 ≤ 2b 2 + 1 . Thus in this case we can take L = 2 2 + 1. Some examples of elements of Γ, in this case, are: 3 + 2 2 4 + 2 2 + 2 2 3 + 2 33 + 24 2 40 + 28 28 + 20 2 33 + 24  (413) Let PF,r denote the set of all the r-Pisot numbers in OF . We note that, for all real numbers s > 0, there are only a finite number of elements of PF,r with magnitude < s. For it is sufficient to prove this for s ≥ r. And for s ≥ r, every element of PF,r is an s-Pisot number in F . And by definition, all the conjugates σ 6= I, of an s-Pisot number, have magnitude < s. Hence by the bounded conjugates lemma, there are only a finite number of s-Pisot numbers in F , whose magnitude is < s, hence there are only a finite number of elements of PF,r, whose magnitude is < s. Hence, in particular, PF,r is discrete. 3.1.1 Compactness of G/Γ for the examples of arithmetic lattices The compactness of G/Γ, for groups such as those in the examples given above, was originally proved by Borel and Harish-Chandra [34], making use of their proof that Γ is a lattice in G, and their proof of a compactness criterion that had been conjectured by Godement. The following direct proof of compactness is adapted from sections (6.36) and (6.45) of [296], and the proof of Mahler’s compactness theorem [329] in section (5.34) of [296]. To check the compactness of the quotient G/Γ, it is sufficient to check that, given any infinite sequence {gk} of elements of G, there exists a sequence {γk} of elements of Γ, such that the sequence {gkγk} has a convergent infinite subsequence. We first note that if G = SU (B) ∼= SU (n, 1), the subgroup GF (i) of G, consisting of the elements of G, all of whose matrix elements are in F (i), the extension of F , by the square root of −1, is dense in G, while if G = SO (B) ∼= SO(n, 1), the subgroup GF of G, consisting of the elements of G, all of whose matrix elements are in F , is dense in G. For an arbitrary element V of SU(B) satisfies V †BV = B (414) where V † denotes the hermitian conjugate of V , and (414) is also satisfied by an arbitrary element V of SO (B), since in that case V † = V T , where V T denotes the transpose of V . And in general, if B is a nonsingular hermitian matrix, and V is a complex matrix that satisfies (414), then the matrix A ≡ B (V − 1) (V + 1)−1 (415) is antihermitian, and V is expressed rationally in terms of the antihermitian matrix A V = (B − A)−1 (B + A) (416) Furthermore, the matrix (B − A) = 2B (V + 1)−1 is nonsingular. Moreover, if A is an arbitrary antihermitian matrix, such that (B − A) is nonsingular, and V is defined in terms of A by (416), then V satisfies V †BV = B. Thus, for an arbitrary element V of SU (B) or SO (B), such that (V + 1) is non-singular, we can define the antihermitian matrix A by (415), and then, by choosing an antihermitian matrix Ã, with matrix elements in F (i) or F , as appropriate, that approximates A sufficiently closely, and is such that B − à is nonsingular, we can obtain an element Ṽ of SU (B)F (i) or SO (B)F , as appropriate, such that every matrix element of Ṽ − V has magnitude less than any given number > 0. And if (V + 1) is singular, we can follow the same procedure, for an element V1 of SU (B) or SO (B), as appropriate, such that that matrix elements of (V1 − V ) are sufficiently small in magnitude, and (V1 + 1) is nonsingular, so as to obtain, again, an element Ṽ of SU (B)F (i) or SO (B)F , as appropriate, such that every matrix element of Ṽ − V has magnitude less than any given number > 0. Thus it is sufficient to check that, given any infinite sequence {gk} of elements of GF (i) or GF , as appropriate, there exists a sequence {γk} of elements of Γ, such that the sequence {gkγk} has a infinite Cauchy subsequence, or in other words, an infinite subsequence {gpγp}, such that for any given ε > 0, there exists an integer t, such that for all p > t and all q > t, every matrix element of (gpγp − gqγq) has magnitude less than ε. The matrices γk will be constructed as one block of a block diagonal matrix that includes all the Galois conjugates of Γ along its block diagonal, because we can then transform these block diagonal matrices to matrices with integer matrix elements, by a similarity transformation that consists of multiple copies of the inverse of the similarity transformation that diagonalizes the matrix representations of the elements of F in an integral basis, discussed above. Once we are working with matrices with integer matrix elements, we can construct the sequence {γk} by a method due to Mahler [329]. It is convenient, first, if G = SU (B) ∼= SU (n, 1), to embed G in a group of 2 (n+ 1) × 2 (n + 1) matrices with real matrix elements. For each element g of G, let ḡ denote the 2 (n+ 1)× 2 (n + 1) matrix with real matrix elements, obtained from g by replacing each complex matrix element (a+ ib) by the real matrix We note that, by this rule, the hermitian conjugate g† of g corresponds to the transpose ḡT of ḡ. And let B̄ be obtained from B by the same rule. Thus B̄ is a diagonal matrix whose first and second diagonal matrix elements are equal to one another, whose third and fourth diagonal matrix elements are equal to one another, and so on. B̄ has sig- nature (+,+, . . . ,+,−,−), so we are embedding G in a group isomorphic to SO (n, 2). However the following discussion will not depend on the detailed signature of B̄ or B, beyond the fact that B̄ or B is indefinite, while its Galois conjugates for σ 6= I are either positive definite or negative definite. When an element g of SU (B) acts on a complex (n + 1)-vector, each complex matrix element (c+ id) of that (n+ 1)-vector is replaced by the real column vector , so ad− bc bc+ ad , and the complex (n + 1)-vector becomes a real 2 (n+ 1)-vector. These two ways of representing a complex number, as a real matrix, or as a real column vector, are an example of the relation between the representation of an element x, of an algebraic number field, by the matrix xij , and by its components xi, in a particular basis, as discussed above. We note that if ḡ corresponds to an element g of SU (B) by the transformation described above, then each 2 × 2 block in ḡ can be diagonalized by the similarity transformation 1√ a+ ib 0 0 a− ib , hence all the 2 × 2 block matrices in ḡ can be diagonalized by applying a block diagonal similarity transformation with n + 1 copies of  along the block diagonal. Then by permuting rows and columns, ḡ can be brought to the form of a block diagonal matrix with two (n+ 1) × (n+ 1) blocks along the block diagonal, one of which is g, and the other of which is g∗, the complex conjugate of g. Hence det ḡ = |det g|2. But det g = 1, hence det ḡ = 1. Let J be the block diagonal matix with n copies of  along the block diagonal. Then the subgroup SU (B) of SO is the group of all elements g of SO such that J g = gJ . Hence by renaming B̄ as B, we can now assume that either B is a diagonal matrix with n positive diagonal matrix elements and one negative diagonal matrix element, and G = SO(B), or B is a diagonal matrix with 2n positive diagonal matrix elements and 2 negative diagonal matrix elements, and G is the subgroup of SO (B) that commutes with J . And for every Galois conjugate σ of F different from the identity I, Bσ is in both cases either a positive definite matrix or a negative definite matrix. We define p = (n + 1) in the orthogonal case, and p = 2 (n + 1) in the unitary case. Let H be the set of all p × p matrices with matrix elements in F , and m be the degree of F . We now choose a fixed sequence of the m Galois conjugates σ of F , starting with the identity σ = I, and for an arbitrary element h of H , we define ĥ to be the pm× pm block diagonal matrix which has h as its top left p× p matrix elements, then the first Galois conjugate σ 6= I of h as its second block of p× p matrix elements on the block diagonal, then the second Galois conjugate σ 6= I of h as its third block of p× p matrix elements on the block diagonal, and so on, and all other matrix elements equal to zero. We also define Ĥ to be the set of all the matrices ĥ, for h ∈ H , Ĝ to be the set of all the matrices ĝ, for g ∈ GF , and Γ̂ to be the set of all the matrices ĝ, for g ∈ Γ. We note that if g is any element of GF , then since ĝ is block diagonal, and every block on the block diagonal of ĝ has determinant equal to 1, det ĝ = 1. For an arbitrary p-vector x in F p, we define x̂ to be the pm-vector whose first p components are x, whose next p components are the first Galois conjugate σ 6= I of x, whose third set of p consecutive components are the second Galois conjugate σ 6= I of x, and so on. We also define F̂ p to be the set of all the pm-vectors x̂, for x ∈ F p. Thus for all ĥ ∈ Ĥ , and all x̂ ∈ F̂ p, the pm-vector ĥx̂ is an element of F̂ p. We also define ÔpF to be the set of all the pm-vectors x̂, for x ∈ O We note that, if x is any nonzero p-vector in F p, then the quadratic form xTBx is nonzero. For by assumption F 6= Q, hence F has at least one Galois conjugate σ different from I, and by assumption (xσ) Bσxσ = is either positive definite or negative definite. Furthermore, no nonzero element of F can have any Galois conjugate equal to 0, for 0 is of degree 1, hence has no conjugates other than itself. Thus if x̂ is any nonzero element of F̂ p, then the quadratic form x̂T B̂x̂ is nonzero. Furthermore, if x is any nonzero p-vector in OpF , then the value of the quadratic form x̂T B̂x̂ is an ordinary integer in Z, and its magnitude is ≥ 1. For all the matrix elements of B are in OF , hence xTBx is an algebraic integer in OF . Furthermore, x̂T B̂x̂ is the sum of all the Galois conjugates of xTBx, which if the degree of the algebraic integer xTBx is equal to m, is −1 times the coefficient of xm−1 in the minimal polynomial of xTBx, and thus an integer in Z, while if the degree k of xTBx is less than m, it must divide m, and x̂T B̂x̂ is equal to the integer m , times −1 times the coefficient of xk−1 in the minimal polynomial of xTBx, and thus again an integer in Z. And furthermore, by the preceding paragraph, the ordinary integer x̂T B̂x̂ cannot be equal to zero, hence it has magnitude ≥ 1. We now carry out a similarity transformation ĥ → S−1ĥS on the elements ĥ of Ĥ , such that S consists of p copies of the inverse, s, of a similarity transformation that diagonalizes the matrix representations of the elements of F in an integral basis, discussed above. Each of the p copies of s is “spread out”, so that, for example, the first copy acts from the right only on the first column of each of the m Galois conjugates of elements of GF , the second copy acts from the right only on the second column of each of the m Galois conjugates of elements of GF , and so on. For example, if F = Q , we can choose the similarity transformation S−1ĥS to consist of p copies of the similarity transformation: 0 a− b  (417) And if, in addition, G ∼= SO (1, 1), so that p = 2, the similarity transformation S−1ĥS would have the form: 1 0 1 0 0 − 1√ 0 1 0 1 0 − 1√ 2 c+ d 2 0 0 e + f 2 g + h 2 0 0 0 0 a− b 2 c− d 0 0 e− f 2 g − h 2 0 0 0 0 1 2 0 0 0 0 1 − a 2b c 2d b a d c e 2f g 2h f e h g (418) We see that for each element h of H , the similarity transformation S−1ĥS transforms ĥ into a p × p block matrix, each block of which is the m ×m matrix representation of the corresponding matrix element of h, in the chosen integral basis. Thus the matrix elements of S−1ĥS are rational numbers. For each element h of H , we define h̃ ≡ S−1ĥS, where ĥ is the element of Ĥ that corresponds to h as above. We see that the elements of the subgroup Γ̂ of Ĝ are precisely those elements ĝ of Ĝ for which g̃ = S−1ĝS has integer-valued matrix elements. We define H̃ to be the set of all the matrices h̃, for h ∈ H , G̃ to be the set of all the matrices g̃, for g ∈ GF , and Γ̃ to be the set of all the matrices g̃, for g ∈ Γ. We note that if g is any element of GF , then since g̃ is related to ĝ by a similarity transformation, and det ĝ = 1, we have det g̃ = 1. Now if α is a primitive element of F , or in other words, an algebraic number in F , whose degree is equal to m, and Qm×m denotes the set of all m×m matrices with rational matrix elements, then the elements of Qm×m that are matrix representations of elements of F , in the chosen integral basis, are precisely those that commute with the matrix representation of α in the chosen integral basis. For every element of F commutes with α, and if an element χ of Qm×m commutes with the matrix repre- sentation of α, then since (1, α, α2, . . . , αm−1) is a possible basis for F , so the matrix representations of 1, α, α2, . . . , αm−1 are linearly independent elements of Qm×m, and are thus a complete basis for the elements of Qm×m that commute with α, χ is a linear combination, with coefficients in Q, of the matrix representations of 1, α, α2, . . . , αm−1, and is thus the matrix representation of an element of F . We now choose an algebraic integer ϕ of F , such that ϕ is primitive in F . Such an algebraic integer ϕ of F always exists, because, as noted above, most algebraic integers in F are primitive in F . We define F to be the element of H that is the p× p diagonal matrix, with each matrix element on the diagonal equal to ϕ, so that, in other words, F is equal to ϕ times the p×p unit matrix. The elements F̂ of Ĥ, and F̃ of H̃ , are then defined in the standard way, as above. Thus F̃ is the pm× pm block diagonal matrix, such that each m×m block on the block diagonal is equal to the matrix representation of ϕ, in the chosen integral basis. Then if Qpm×pm denotes the set of all pm × pm matrices with matrix elements in Q, the elements of H̃ are precisely the elements of Qpm×pm that commute with F̃ , since an element ξ of Qpm×pm commutes with F̃ if and only if every m ×m block of ξ commutes with the matrix representation of ϕ, in the chosen integral basis. We note that B̃ will have integer-valued matrix elements, and be block diagonal, with each block on the block diagonal being an m × m matrix which, for at least one block, in the example when F is Q , will not be symmetric. F̃ will also have integer-valued matrix elements, since ϕ is an algebraic integer. And in the unitary case, J̃ will have integer-valued matrix elements, which will in fact be +1 or −1, and be block diagonal, with each block on the block diagonal being a 2m× 2m antisymmetric matrix. We note that in the above example, (417), s has been chosen such that every matrix element in its first column is equal to 1. This has the consequence that for an arbitrary element h of H , the first column of the matrix ĥS is the element x̂ of F̂ p, where x denotes the first column of h, and the element x̂ of F̂ p is related to the p-vector x in F p, as described above. Furthermore, the first m matrix elements of the first column of S−1ĥS, or in other words, the first m matrix elements of S−1x̂, are the components of the first matrix element of x, with respect to the integral basis and the next m matrix elements of the first column of S−1ĥS, or in other words, the next m matrix elements of S−1x̂, are the components of the second matrix element of x, with respect to the integral basis . Therefore, since for an arbitrary p-vector x in F p, we can write down an element h of H , such that the first column of h is x, it follows, for this example, that for an arbitrary p-vector x in F p, the components of the pm-vector S−1x̂, where the element x̂ of F̂ p is related to the p-vector x in F p as described above, are the m components of the first matrix element of x, with respect to the integral basis , followed by the m components of the second matrix element of x, with respect to the integral basis Thus, for this example, for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has integer components in Z, and conversely, for an arbitrary pm-vector x̃ in Zpm, the pm-vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O F in the manner described above. Specifically, we have: 1 0 1 0 0 − 1√ 0 1 0 1 0 − 1√ a + b (419) This corresponds to the fact that, with s chosen as in the example (417), s−1 acts on a column m-vector, that consists of all the Galois conjugates of an element of F , in the standard order, as: a + b  (420) I shall now show that for an arbitrary algebraic number field F , and thus, in particular, for an arbitrary totally real algebraic number field F , we can always choose an integral basis {vi} for F , such that the elements of the first column of the matrix representation of each element of F , in the basis {vi}, are the expansion coefficients of that element of F in the basis {vi}, 1 ≤ i ≤ m. And furthermore, in such a basis for F , the inverse, s, of the similarity transformation that diagonalizes the matrix representations of elements of F in the integral basis {vi}, will be such that all the matrix elements in its first column are nonzero, and moreover, all the matrix elements in its first column can be chosen equal to 1. When we use such a basis for F , and choose all the matrix elements in the first column of s to be equal to 1, then it immediately follows, as in the above example, that for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has integer components in Z, which are in fact the expansion coefficients of the successive matrix elements of x, with respect to the integral basis {vi}, and conversely, for an arbitrary pm-vector x̃ in Zpm, the pm-vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O in the manner described above. We choose an integral basis vi, 1 ≤ i ≤ m, for F , such that v1 = 1. For example, we could choose a canonical integral basis, associated with the algebraic integer ϕ, in terms of which we defined the matrices F , F̂ , and F̃ above. We recall, from the beginning of this subsection, that the matrix elements xij of the representation of an element x of F , for the basis given by the vi, are defined by xvi = vjxji, where the summation convention is used. Thus for the basis element vk we find vkv1 = vk = vj (vk)j1, hence by the linear independence of the basis elements, we must have (vk)j1 = δkj . On the other hand, we can also express a general element x of F , in the integral basis vi, as x = xkvk. Hence xi1 = xk (vk)i1 = xi. Thus the elements of the first column of xij are the expansion coefficients of x in the integral basis vi. Let s be the inverse of the similarity transformation that diagonalizes the matrix representations of elements of F in the integral basis vi, so that S consists of p copies of s, each “spread out”, as described above. Thus if x is a general element of F , and x̂ is the diagonal m×m matrix, whose matrix elements on the diagonal are the m Galois conjugates of x, taken in the same order as we chose above, then (s−1x̂s)ij = xij , where xij are the matrix elements of x in the basis vi, which by assumption has v1 = 1. Now this equation remains true, with the same x̂ and xij , if we pre-multiply s by an arbitrary diagonal matrix, so that s−1 gets post-multiplied by the inverse of that diagonal matrix. Thus by pre-multiplying s by a suitable diagonal matrix, we can assume that every nonzero matrix element, in the first column of s, is equal to 1. Furthermore, no matrix element in the first column of s can be zero. For by assumption, each matrix element of the first column of x̂s is either the appropriate Galois conjugate of x, or zero. And by the preceding paragraph, the set of the first columns of (s−1v̂is), for the m basis elements vi, is a set of m linearly independent column vectors of real numbers, namely , . . . , . But this would be impossible, if any matrix element of the first column of s was zero, because the set of the first columns of (v̂is) would not then be a set of m linearly independent column vectors of real numbers. Thus we can assume that every matrix element, in the first column of s, is equal to 1, as in the example above. We now choose all the matrix elements in the first column of s to be equal to 1, so that for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has integer components in Z, which are the expansion coefficients of the successive matrix elements of x, with respect to the integral basis {vi}, and for an arbitrary pm-vector x̃ in Zpm, the pm- vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O F in the manner described above. We next note that by choosing each matrix element in the first column of s to be equal to 1, we have guaranteed that the matrix V ≡ sT s has rational matrix elements. For by the definition of s, we have, for an arbitrary element x of F , that x̃s−1 = s−1λ, where λ is a diagonal matrix. Hence x̃ = s−1λs, and x̃T = sTλ (s−1) = V x̃V −1, hence (V x̃) = V x̃, or in other words, V x̃ is a symmetric matrix. If we now regard V as an independent symmetric matrix, and impose this condition on V , for an arbitrary primitive element x of F , then since all the eigenvalues of x̃ are distinct for x primitive, we find 1 m (m− 1) linearly independent equations among the matrix elements of V , with coefficients linear in the matrix elements of x̃, and thus rational numbers. For if we regard V as an independent symmetric matrix, and express the equation x̃TV = V x̃ in a basis in which x̃ is diagonal, or in other words, if we write the equation as λ (s−1) V s−1 = (s−1) V s−1λ, and treat this as an equation for the symmetric matrix V , without making use of the relation between s and V , then the fact that λ is a diagonal matrix, all of whose eigenvalues are different, implies that the equation x̃TV = V x̃ is equivalent to 1 m (m− 1) linearly independent relations among the matrix elements of the symmetric matrix V , of the form (λi − λj) (s−1) V s−1 = 0, 1 ≤ i < j ≤ m, where the summation convention is not applied to i and j. But the number of linearly independent relations in the matrix equation x̃TV = V x̃ is independent of what basis we express it in, hence for x primitive this matrix equation gives 1 m (m− 1) linearly independent linear relations, with rational coefficients, among the 1 m (m+ 1) independent matrix elements of the symmetric matrix V , which we can use to express m (m− 1) matrix elements of V as linear combinations, with rational coefficients, of the remaining m independent matrix elements. Furthermore, the form of the equation, in the basis in which all the x̃ are diagonal, which simply states that the symmetric matrix (s−1) V s−1 is diagonal, shows that no further information can be obtained, by imposing the relation x̃TV = V x̃, for any further x̃. We choose to use the relation x̃TV = V x̃, for one primitive x, to express all the m (m− 1) independent Vij , 2 ≤ i ≤ m, 2 ≤ j ≤ m, as linear combinations, with rational coefficients, of the Vi1, 1 ≤ i ≤ m. And making use, now, of the definition V = sT s, and the fact that we have set si1 = 1, for all 1 ≤ i ≤ m, we find that: m s12 + s22 + . . .+ sm2 . . . s1m + s2m + . . .+ smm s12 + s22 + . . .+ sm2 s 12 + s 22 + . . .+ s 32 . . . s12s1m + . . .+ sm2smm . . . s1m + s2m + . . .+ smm s12s1m + . . .+ sm2smm . . . s 1m + s 2m + . . .+ s (421) Thus it remains to check that the linear combinations s12 + s22 + . . .+ sm2, s13 + s23 + . . .+ sm3, ..., s1m + s2m + . . .+ smm, are rational numbers. To do this, we use the fact that sT is the matrix of eigenvectors of the x̃, or in other words, for all x in F , we have x̃T sT = sTλ, where λ is the diagonal matrix of eigenvalues of x̃T . We have chosen the top matrix element of each column of sT to be equal to 1. The sums whose rationality we want to determine, are the sums of the matrix elements across the rows 2 to m of sT . In other words, for each row of sT after the first, we need to check that the sum of all the matrix elements in that row of sT is rational. We choose a primitive element x of F , so that the eigenvalues λi of x̃ T are all distinct, and for each column i of sT , use rows 2 to m of the equation x̃T sT = sTλ, to express each matrix element after the first of that column of sT , or in other words, the sij, 2 ≤ j ≤ m, as a ratio of multinomials formed from the matrix elements of x̃T , the eigenvalue λi for that column of s T , and the top matrix element of that column of sT , which is 1. When we do this for all the columns i of sT , 1 ≤ i ≤ m, we find that for all the sij in each row j of sT , we have a formula of the form sij = fj (λi) gj (λi) (422) where fj (λi) and gj (λi) are multinomials in the matrix elements of x̃ T and λi, such that the dependence of fj (λi) and gj (λi) on the matrix elements of x̃ T is the same for all the sij in the row j of s T . For example, for m = 3 we find, for row 2 of sT , that: s12 = − x̃12 (λ1 − x̃33) + x̃13x̃32 λ1x̃33 + x̃22 (λ1 − x̃33) + x̃23x̃32 − λ21 s22 = − x̃12 (λ2 − x̃33) + x̃13x̃32 λ2x̃33 + x̃22 (λ2 − x̃33) + x̃23x̃32 − λ22 s32 = − x̃12 (λ3 − x̃33) + x̃13x̃32 λ3x̃33 + x̃22 (λ3 − x̃33) + x̃23x̃32 − λ23 (423) From the structure of the formula (422), we see that i=1 sij is a symmetric function of the eigenvalues λi, and can in fact be expressed as the ratio of two symmetric multinomials in the λi. Thus it is equal to the ratio of two multinomials in the matrix elements of x̃, so it is a rational number. Thus the matrix elements of the matrix V = sT s are rational numbers. Now the quadratic form xTBx is preserved by all elements g of GF , for by definition of G, we have gTBg = B, for all elements g of GF . And similarly, we have ĝ T B̂ĝ = B̂, for all elements ĝ of Ĝ, because this equation is block diagonal, with each of the m blocks on the block diagonal being one of the m Galois conjugates of the first p × p block on the block diagonal, which is an equation of the form gTBg = B, with g ∈ GF . We now define the symmetric matrix B̆ = ST B̂S = STSB̃, where B̃ = S−1B̂S as above, and the S is constructed from p “spread out” copies of s, as described above. Then since the matrix elements of B̃ are ordinary integers, and the matrix elements of STS are rational numbers, the matrix elements of B̆ are rational numbers. We next note that, for all elements g̃ of G̃, we have: g̃T B̆g̃ = S−1ĝS ST B̂S S−1ĝS = ST ĝT B̂ĝS = ST B̂S = B̆ (424) Now let x̃ be an arbitrary nonzero pm-vector with integer components, or in other words, an arbitrary nonzero element of Zpm. Then x̂ = Sx̃ is a nonzero element of ÔpF , that corresponds to a nonzero element x of OpF in the manner described above. Hence, from above, the value of x̃T B̆x̃ = (S−1x̂) ST B̂S S−1x̂ = x̂T B̂x̂ is an ordinary integer in Z, and its magnitude is ≥ 1. The fact that the set of possible values of x̃T B̆x̃ is discrete, and that there is a minimum distance > 0 between adjacent possible values of x̃T B̆x̃, also follows directly from the fact that the matrix elements of B̆ are rational numbers. We next note that there is a real number λ > 0, such that for all g ∈ GF , and all nonzero pm-vectors x̃ ∈ Zpm, |g̃x̃| ≥ λ. Here, and throughout the following, |g̃x̃| has its usual meaning of |g̃x̃| = x̃T g̃T g̃x̃. For if |g̃x̃| could be arbitrarily small, then the value of x̃T g̃T B̆g̃x̃ = x̃T B̆x̃ could be arbitrarily close to 0. But by the preceding paragraph, the magnitude of x̃T B̆x̃ is ≥ 1, for arbitrary nonzero x̃ ∈ Zpm. Hence for nonzero x̃ ∈ Zpm, x̃T B̆x̃ cannot be arbitrarily close to 0, hence |g̃x̃| cannot be arbitrarily small. Let λ > 0 be the largest number such that for all g ∈ GF , and all nonzero pm-vectors x̃ ∈ Zpm, |g̃x̃| ≥ λ. Given an infinite sequence {gk} of elements of GF , the plan now is to find, first, a sequence {βk} of elements of SL (pm,Z), such that the sequence {g̃kβk} has an infinite Cauchy subsequence {g̃jβj}, and then show that this infinite Cauchy subsequence itself has an infinite subsequence {g̃iβi}, such that βiβ−11 is an element γ̃i of Γ̃, for all i in this infinite subsequence. The sequence {ĝiγ̂i} = {Sg̃iγ̃iS−1} is then an infinite Cauchy sequence of block diagonal matrices in Ĝ, such that all the γ̂i are in Γ̂, and the sequence {giγi} of the first p× p blocks, on the block diagonal, is an infinite Cauchy sequence in GF , such that the sequence {gi} is an infinite subsequence of the given sequence {gk}, and all the γi are in Γ. To construct the required sequence βk of elements of SL (pm,Z), we first use a method of Mahler [329] to construct, for an arbitrary element g of GF , an element β of SL (pm,Z), such that all matrix elements of g̃β are bounded above in magnitude in terms of λ, where λ > 0 was defined above to be the largest number such that for all g ∈ GF , and all nonzero pm-vectors x̃ ∈ Zpm, |g̃x̃| ≥ λ. The following form of Mahler’s construction is adapted from section (5.34) of [296]. We define l ≡ pm. Given an element g of GF , the required element β of SL (l,Z) will be constructed column by column, as a sequence of nonzero column vectors in Zl, that I shall call v1, v2, v3, . . . , vl. We first choose v1 ∈ Zl\{0}, where \ means “outside”, such that |g̃v1| has its mini- mum possible value, for v ∈ Zl\{0}. This is always possible, because g̃ is nonsingular, hence vT g̃T g̃v is a positive definite quadratic form, with no flat directions. Let π1 denote the projection to the line Rg̃v1, and π 1 denote the projection to the subspace orthogonal to this line. We next choose v2 ∈ Zl\Rv1, such that ∣π⊥1 g̃v2 ∣ has its minimum possible value, for v ∈ Zl\Rv1. This is always possible, for the same reason as before. Moreover, π⊥1 g̃v2 is unaltered by adding a multiple of v1 to v2. For k ∈ Z, the values of |π1g̃ (v2 + kv1)| = g̃v1(vT1 g̃ T g̃(v2+kv1)) vT1 g̃ T g̃v1 = |g̃v1| vT1 g̃ T g̃v2 vT1 g̃ T g̃v1 are spaced by |g̃v1|, so by replacing v2 by v2 + kv1, with a suitable value of k, we can assume that |π1g̃v2| ≤ 12 |g̃v1|. Then from the minimality of |g̃v1|, we have that: |g̃v1| ≤ |g̃v2| ≤ ∣π⊥1 g̃v2 ∣+ |π1g̃v2| ≤ ∣π⊥1 g̃v2 |g̃v1| (425) Hence: ∣π⊥1 g̃v2 |g̃v1| (426) Let π2 denote the projection to the plane Rg̃v1 +Rg̃v2, and π 2 denote the projection to the subspace orthogonal to this plane. We next choose v3 ∈ Zl\(Rv1 +Rv2), such that ∣π⊥2 g̃v3 ∣ has its minimum possible value, for v ∈ Zl\(Rv1 +Rv2). This is always possible, for the same reason as before. Moreover, π⊥2 g̃v3 is unaltered by adding multiples of v1 and v2 to v3. We first add a suitable integer multiple of v2, to arrange that 1 g̃v3 ∣ ≤ 1 ∣π⊥1 g̃v2 ∣. Then, without affecting this bound, we add a suitable integer multiple of v1, to arrange that |π1g̃v3| ≤ |g̃v1|. The minimality of ∣π⊥1 g̃v2 ∣ now implies: ∣π⊥1 g̃v2 ∣π⊥1 g̃v3 ∣π⊥2 g̃v3 1 g̃v3 ∣π⊥2 g̃v3 ∣π⊥1 g̃v2 ∣ (427) Hence: ∣π⊥2 g̃v3 ∣ ≥ 1 ∣π⊥1 g̃v2 ∣ (428) Then we carry on after this pattern, until we eventually choose vl ∈ Zl\(Rv1 + . . .+Rvl−1), such that ∣π⊥l−1g̃vl ∣ has its minimum possible value, for v ∈ Zl\(Rv1 + . . .+Rvl−1). Then by successively adding suitable integer multiples of vl−1, vl−2, ..., v2, and v1, we arrange that ∣πl−1π l−2g̃vl ∣ ≤ 1 ∣π⊥l−2g̃vl−1 ∣πl−2π l−3g̃vl ∣π⊥l−3g̃vl−2 ∣, ..., |π1g̃vl| ≤ 12 |g̃v1|. Then from the minimality of ∣π⊥l−2g̃vl−1 ∣, we find, in the same way as before, that: ∣π⊥l−1g̃vl ∣ ≥ 1 ∣π⊥l−2g̃vl−1 ∣ (429) We next note that the successive minimality of |g̃v1|, ∣π⊥1 g̃v2 ∣π⊥2 g̃v3 ∣, ..., ∣π⊥l−1g̃vl implies in turn that the convex hull of {0, v1} contains no points of Zl other than 0 and v1, the convex hull of {0, v1, v2} contains no points of Zl other than 0, v1, and v2, ..., and finally that the convex hull of {0, v1, v2, . . . , vl} contains no points of Zl other than 0, v1, v2, ..., vl. Hence the parallelepiped generated by the vectors v1, v2, ..., vl, whose vertices are the expressions of the form k1v1 + k2v2 + . . . + klvl, where each ki can independently take the values 0 or 1, contains no points of Zl in its convex hull, other than its 2l vertices. Now by considering tesselations of Rl by lattice parallelepipeds, the volume of a lattice parallelepiped is given, in terms of the points of Zl in its convex hull, by: f2 + . . .+ fl−1 + fl, (430) where f0 is the number of points of Z l that are vertices of the parallelpiped, f1 is the number of points of Zl that lie within the “interiors” of edges of the parallepiped, ..., and fl is the number of points of Z l that lie within the interior of the l-volume of the parallepiped. Hence in the present instance, the volume of the parallelepiped generated by the vectors v1, v2, ..., vl, is 1, hence the determinant of β, which is defined to be the l× l matrix whose columns are v1, v2, ..., vl, is ±1. And if det β = −1, we note that we can replace vl by −vl, which is also in Zl\(Rv1 + . . .+Rvl−1), without affecting the minimality of ∣π⊥l−1g̃vl ∣, and we therefore replace vl by −vl, to obtain det β = 1. Furthermore: ∣π⊥l−1g̃vl ∣π⊥l−2g̃vl−1 ∣π⊥l−3g̃vl−2 ∣ . . . ∣π⊥1 g̃v2 ∣ |g̃v1| = |det (g̃β)| = 1 (431) Hence from (426), (428), ..., and (429), we find that: |g̃v1| ≤ 2 2 (432) Furthermore, since v1 6= 0, we have |g̃v1| ≥ λ > 0. Therefore, returning to (431), and the bounds (426), (428), ..., we find that ∣π⊥1 g̃v2 ∣ ≤ 2 ∣π⊥2 g̃v3 ∣ ≤ 2 . . . ∣π⊥k−1g̃vk ∣ ≤ 2 (k−2)(k−1) 2(l+1−k) 2 l+1−k . . . ∣π⊥l−1g̃vl ∣ ≤ 2 (l−2)(l−1) (433) Furthermore, since |π1g̃v2| ≤ 12 |g̃v1|, we find: |g̃v2| ≤ ∣π⊥1 g̃v2 ∣+ |π1g̃v2| ≤ 2 2 (434) And similarly: |g̃v3| ≤ ∣π⊥2 g̃v3 1 g̃v3 ∣+ |π1g̃v3| ≤ 2 (435) And so on. Thus, since λ > 0, all matrix elements of g̃β are, indeed, bounded, inde- pendently of g̃. Given an infinite sequence {gk} of elements of GF , we now take, for each k, βk to be the matrix β, as constructed above, with g̃ taken as g̃k. The elements of the sequence {g̃kβk} are then bounded in terms of λ as above, independently of k. Hence this sequence has a Cauchy subsequence. We can find a Cauchy subsequence by subdi- viding the bounded l2-dimensional domain of the matrix elements into a finite number of subsectors, choosing a subsector in which the sequence has an infinite number of elements, subdividing that subsector into an finite number of subsectors, choosing one of them in which the sequence has an infinite number of elements, and so on. Let {g̃jβj} be an infinite Cauchy subsequence of the sequence {g̃kβk}. Now we found above that the matrix elements of B̆ are rational numbers. Hence there is an integer a ∈ Z such that all the matrix elements of aB̆ are ordinary integers in Z. On the other hand, the fact that {g̃jβj} is a Cauchy sequence implies that the sequence βTj g̃ j B̆g̃jβj βTj B̆βj (436) is a Cauchy sequence. Hence since all the matrix elements of aβTj B̆βj are ordinary integers in Z, there must be a value q1 of j such that for all s ≥ q1 and all t ≥ q1, aβTs B̆βs = aβ t B̆βt, hence B̆βsβ t = B̆. This result can also be obtained without directly using the fact that the matrix elements of B̆ are rational numbers, by using the fact that for an arbitrary element x̃ of Zl, the value of x̃T B̆x̃ is an ordinary integer in Z, and considering the Cauchy sequence x̃TβTj g̃ j B̆g̃jβj x̃ x̃TβTj B̆βj x̃ for 1 l (l + 1) suitable choices of x̃, such as the l unit vectors in the positive coordinate directions, and the 1 l (l − 1) distinct sums of two such unit vectors. This procedure can also be used to give an alternative proof that the matrix elements of B̆ are rational numbers, without using the fact that the matrix elements of sT s are rational numbers. Furthermore, the fact that {g̃jβj} is a Cauchy sequence implies that the sequence β−1j g̃ j F̃ g̃jβj β−1j F̃βj (437) is a Cauchy sequence. Hence since the matrix elements of F̃ are ordinary integers in Z, hence the matrix elements of β−1j F̃βj are ordinary integers in Z, there must be a value q2 of j such that for all s ≥ q2 and all t ≥ q2, β−1s F̃βs = β−1t F̃βt, hence F̃βsβ−1t = βsβ−1t F̃ . And finally, in the unitary case, the preceding paragraph is also valid with F̃ re- placed by J̃ , hence there must be a value q3 of j such that for all s ≥ q3 and all t ≥ q3, J̃ βsβ−1t = βsβ−1t J̃ . Hence there is a value r of j, namely the maximum of q1 and q2 in the orthogonal case, and the maximum of q1, q2, and q3 in the unitary case, such that for all s ≥ r and all t ≥ r, βsβ−1t is an element of G̃. Let γ̃j ≡ βjβ−1r for all j ≥ r, and let {g̃iβi} be the infinite Cauchy sequence obtained from {g̃jβj} by dropping all terms with j < r. Then {g̃iβiβ−1r } = {g̃iγ̃i} is an infinite Cauchy sequence in G̃, such that {g̃i} is an infinite subsequence of {g̃k}, and γ̃i ∈ G̃, for all i. Then as anticipated above, the sequence {ĝiγ̂i} = {Sg̃iγ̃iS−1} is an infinite Cauchy sequence of block diagonal matrices in Ĝ, such that all the γ̂i are in Γ̂, and the sequence {giγi} of the first p × p blocks, on the block diagonal, is the required infinite Cauchy sequence in GF , such that the sequence {gi} is an infinite subsequence of the given sequence {gk}, and all the γi are in Γ. 3.1.2 Obtaining finite index torsion-free subgroups of Γ by Selberg’s lemma We recall from above that for compact quotients of SU (n, 1) or SO (n, 1), the re- quirement that the quotient be smooth, rather than an orbifold, or in other words, that all elements 6= 1 of the discrete subgroup Γ act on the symmetric space CHn = SU (n, 1) /(SU (n)× U (1)) or Hn = SO(n, 1) /SO (n) without fixed points, is equiva- lent to the requirement that Γ have no torsion, or in other words, no nontrivial finite subgroups. Selberg’s lemma [321], for the case of arithmetic lattices such as those in the examples above, states that certain finite index subgroups of these discrete groups have no torsion. We recall that a subgroup Γ1 of a discrete group Γ is said to have finite index in Γ, if Γ1 divides Γ into a finite number of left cosets. Thus if G/Γ is compact, and Γ1 has finite index in Γ, then G/Γ1 is also compact, so for all the examples above, we can obtain smooth compact quotients of CHn or Hn by using any of the subgroups of Γ specified by Selberg’s lemma for this case. I shall briefly review Selberg’s lemma for the case of these arithmetic lattices, following section (5.60) of [296]. As in the previous subsection, we define p = 2 (n + 1) if G = SU (n, 1), and p = n+1 if G = SO(n, 1), and l ≡ pm, where m is the degree of F . We choose an integral basis for F , and represent each element γ of Γ as a p× p block matrix, each block of which is the m × m matrix representation of the corresponding matrix element of γ, in the chosen integral basis. Thus each element γ of Γ is represented by an l× l matrix, with matrix elements in Z. In the preceding subsection, such l × l matrices representing elements γ of Γ were denoted γ̃, and the group of all of them was denoted Γ̃, but since Γ̃ is isomorphic to Γ, and this representation of Γ, as a group of l × l matrices, with matrix elements in Z, is the only representation of Γ that will be used in the present subsection, I shall not use the tildes in this subsection. Thus we now regard Γ as a subgroup of SL (l,Z). The following construction is valid for all subgroups Γ of SL (l,Z). For k ∈ Z, such that k ≥ 2, let Γk denote the set of all elements of Γ of the form (1 + kT ), where 1 denotes the l× l unit matrix, and the matrix elements of T are in Z. Then Γk is a group, and is moreover a normal subgroup of Γ, since if γ is an element of Γ, then γ is a matrix with matrix elements in Z, and determinant equal to 1, so γ−1 (1 + kT ) γ has the form 1 + kT1, where the matrix elements of T1 are in Z. We next note that elements γ1 and γ2 of Γ are in the same left left coset of Γk in Γ, if and only if corresponding matrix elements of γ1 and γ2 are equal, mod k. For if corresponding matrix elements of γ1 and γ2 are equal, mod k, then γ2 = γ1 + kT , for some matrix T with matrix elements in Z, hence γ−11 γ2 = 1 + kγ 1 T , which is in Γk, so γ1 and γ2 are in the same left coset of Γk in Γ, while if γ1 and γ2 are in the same left coset of Γk in Γ, then γ2 = γ1 (1 + kT ) for some matrix T with matrix elements in Z, hence corresponding matrix elements of γ1 and γ2 are equal, mod k. Thus the quotient group Γ/Γk, of Γ by its normal subgroup Γk, is the group obtained from Γ, by considering its matrix elements mod k. Thus each matrix element of Γ/Γk takes values in the finite set {0, 1, 2, . . . , k − 1}, hence Γ/Γk cannot have more than 2) elements, and is thus a finite group, and Γk has finite index in Γ. We now demonstrate that for k ≥ 3, Γk has no torsion. It is sufficient to demonstrate that for an arbitrary element γ of Γk, such that γ 6= 1, no integer power s ≥ 1 of γ is equal to 1, for if γ is an element of a finite group, the sequence {1, γ, γ2, γ3, . . .} cannot contain more distinct elements than the number of elements of that finite group. We assume now that k ≥ 3. Then k is divisible by either 22 or an odd prime. Furthermore, Γj is a subgroup of Γk whenever k is a divisor of j, so it is sufficient to prove that Γk has no torsion when k is either 2 2 or an odd prime. Thus we now assume k = pr, where p is prime, and r = 2 for p = 2, and r = 1 for p ≥ 3. Furthermore, it is sufficient to prove that for an arbitrary element γ of Γk, such that γ 6= 1, no power γs is equal to 1 for s prime, since if s factorizes as s = tq, where q is prime, we can write γs = (γt) . Thus we now assume s is prime, so either p does not divide s, or p = s. We can write a general element of Γk as (1 + p uT ), where u ≥ r ≥ 1, and not every matrix element of T is divisible by p. If p does not divide s, we note that (puT ) is equal to 0, mod pu+1, hence (1 + puT ) is equal to 1 + spuT , mod pu+1, which is 6= 1, mod pu+1. And if p = s, we note that (puT )3 is equal to 0, mod pu+2, hence (1 + puT ) = 1+pu+1T + p2u+1(p−1) T 2, mod pu+2. Furthermore, if p ≥ 3, then (p− 1) is even, hence p2u+1(p−1) is an integer that is equal to 0, mod pu+2, hence (1 + puT ) p 6= 1, mod pu+2, while if p = 2, then u ≥ 2, hence again p 2u+1(p−1) is an integer that is equal to 0, mod pu+2, hence (1 + puT ) p 6= 1, mod pu+2. An alternative method of constructing a torsion-free subgroup of Γ has been consid- ered by Everitt and Maclachlan [42], who applied their method to obtain a construction of the Davis manifold [40], which is the smallest known smooth compact quotient of 3.2 Smooth compact quotients of S3 For even n, every real antisymmetric (n+ 1) × (n + 1) matrix has a zero eigenvalue, hence no element of SO (n + 1) acts without fixed points on Sn, and the only smooth compact quotient of Sn is the non-orientable n-dimensional real projective space, ob- tained from Sn by identifying every point with its antipode. On the other hand, S3 is well known to have smooth compact quotients, which fall into a small number of families, that were first classified by Seifert and Threlfall [330, 331]. Smooth com- pact quotients of S3 have been considered recently as possible topologies for the three observed spatial dimensions, in consequence of the current slight preference of astro- physical data for k = +1 rather than k = −1, as discussed in section 2, and have recently been reclassified by Gausmann, Lehoucq, Luminet, Uzan, and Weeks [332]. 4 The Casimir energy densities The validity of the realization of TeV-scale gravity by the thick pipe geometries studied in section 2, for the compactification of Hořava-Witten theory on a particular smooth compact quotient of CH3 or H6 that is a spin manifold, and a particular choice of spin structure on that spin manifold, depends on the Casimir energy densities on and near the inner surface of the thick pipe resulting in the integration constant B, in (206), taking the value (313), or (399), and in the case where the outer surface of the thick pipe is stabilized by Casimir effects, also on the Casimir energy densities on and near the outer surface resulting in the integration constant Ã, in (265), taking the value (315). The Casimir energy densities are, by definition, the correction terms in the field equations and boundary conditions for the graviton, when they are derived by varying the full quantum effective action, or in other words, the generating functional of the proper vertices, with respect to the graviton field, rather than by varying the classical Cremmer-Julia-Scherk action, augmented by supersymmetrized Gibbons-Hawking [93, 94, 95, 69, 70, 71] terms and the semiclassical Hořava-Witten supersymmetric Yang- Mills actions on the orbifold fixed-point hyperplanes. Now, as noted in connection with (129), on page 57, the quantum effective action Γ (Φ), for a properly gauge-fixed classical action A (ϕ), where ϕ denotes all the fields occurring in the gauge invariant classical action, together with all the Faddeev-Popov ghosts [228, 227, 230, 231], and also the Nielsen-Kallosh ghosts [234, 235] if appropriate, can be calculated, for an arbitrary classical field configuration Φ, as the sum of all the one line irreducible vacuum diagrams, calculated from the action A (Φ + ϕ), with the term linear in ϕ deleted, where ϕ denotes the quantum fields. In other words, using DeWitt’s compact index notation [227], where a single index, i, runs over all combinations of type of field, space-time position, and coordinate and other indices, the quantum effective action, as a function of the classical fields, Φ, is given by the sum of all the one line irreducible vacuum diagrams, calculated with the action A (Φ + ϕ)− δA(Φ) , as in (129), where the summation convention is applied to the index i. To check this, we note that, with the functional integral defined as Z (J) = e−iW (J) = [dϕ] exp i (A (ϕ) + Jiϕi), the classical field Φi defined as Φi ≡ − δWδJi , and Γ (Φ) defined by a Legendre transformation by the relation Γ (Φ) + JiΦi = −W (J), we have Ji = −ιii δΓ , where ιij is −1 if both ϕi and ϕj are fermionic, and 1 otherwise, indices on ιij are ignored in applying the summation convention, and all derivatives act from the left [125, 126, 333]. We then have [334]: eiΓ(Φ) = [dϕ] ei(A(ϕ)+Ji(ϕi−Φi)) = [dϕ] e A(ϕ)−(ϕi−Φi) δΓδΦi (438) which can be regarded as an alternative definition of Γ (Φ). Shifting the integration variables ϕi by Φi, we have: eiΓ(Φ) = [dϕ] e A(Φ+ϕ)−ϕi δΓδΦi [dϕ] e A(Φ+ϕ)−ϕi δAδΦi (439) where the loop expansion Γ (Φ) = A (Φ)+Γ1 (Φ)+Γ2 (Φ)+ . . . was introduced. Now if the term −i + . . . in the exponent in the right-hand side of (439) was neglected, (439) would express Γ (Φ) as the sum of all connected, but not necessarily one line irreducible, vacuum bubbles, calculated with the action A (Φ + ϕ)− ϕi δA(Φ)δΦi An arbitrary such vacuum bubble can be regarded as a tree diagram, such that a vertex of the tree diagram on which n propagators end corresponds to i δnΓ(Φ) δΦi1 ...δΦin , and a propagator of the tree diagram corresponds to iG (Φ)i1i2 , where G (Φ)ij is the inverse of the matrix δ2A(Φ) δΦiδΦj And when the effects of the term −i + . . . in the exponent are included, the only change is that each vertex, of a tree diagram, on which precisely one propagator ends, can now come from either of two alternative sources, namely either as a one line irreducible diagram built from the propagators G (Φ)ij and vertices δnA(Φ) δΦi1 ...δΦin , n ≥ 3, of the action A (Φ + ϕ)− ϕi δA(Φ)δΦi , as before, or alternatively from the term of appropriate loop order ≥ 1 in the extra term −i + . . . . The result of this is that a tree diagram that contains m vertices i δΓ , on which precisely one propagator ends, gets a factor (1− 1)m. Hence since every tree diagram with more than one vertex contains at least one such vertex, all the tree diagrams cancel out except for those with precisely one vertex, and these are the one line irreducible vacuum bubbles calculated with the action A (Φ + ϕ)− ϕi δA(Φ)δΦi , as stated. Continuing to use DeWitt’s abstract index notation, the one-loop effective action, Γ1 (Φ), is given, by (439), by: eiΓ1(Φ) = K1 [dϕ] e δ2A(Φ) δΦjδΦi superdet δ2A(Φ) δΦjδΦi = K2e supertr ln δ2A(Φ) δΦjδΦi (440) where K1 and K2 are constants independent of the fields Φi, and the matrix δ2A(Φ) δΦjδΦi has been assumed to have a bose-bose part δ2A(Φ) δΦjδΦi , for which the indices j and i denote bosonic fields, and a fermi-fermi part δ2A(Φ) δΦjδΦi , for which the indices j and i denote fermionic fields, but no non-vanishing matrix elements such that one of the indices i and j is bosonic, and the other fermionic, in which case the superdeterminant [335, 336, 337] is defined by: superdet δ2A (Φ) δΦjδΦi δ2A(Φ) δΦjδΦi δ2A(Φ) δΦjδΦi (441) and the bose-bose part δ2A(Φ) δΦjδΦi has been assumed to have an infinitesimal positive- definite imaginary part. We assume now that A (Φ) has an expansion: A (Φ) = AijΦjΦi + AijkΦkΦjΦi + AijklΦlΦkΦjΦi + . . . (442) and define G (Φ)ij to be the inverse of the matrix δ2A(Φ) δΦjδΦi , and Gij to be the inverse of the matrix Aij. Then since A (Φ) is bosonic, and the assumed vanishing of all bose-fermi matrix elements of δ2A(Φ) δΦjδΦi thus implies that all non-vanishing matrix elements of δ2A(Φ) δΦjδΦi are bosonic, we have: G (Φ)ij = Gij −Gik δ2A (Φ) δΦkδΦm − Akm δ2A (Φ) δΦkδΦm − Akm δ2A (Φ) δΦnδΦp − Anp Gpj − . . . (443) And from (440) we have, up to an additive constant, independent of all the fields Φi: ιjjGjk δ2A (Φ) δΦkδΦj ιjjGjk δ2A (Φ) δΦkδΦl δ2A (Φ) δΦmδΦj + . . . (444) where the effect of the factors ιjj is to introduce a − sign when the field circulating in the loop is fermionic. The expression (444) is well-known to be real, in Minkowski signature. Checking it is real is simplest for the field equations. We have: δ3A (Φ) δΦiδΦkδΦj Gjk −Gjm δ2A (Φ) δΦmδΦn − Amn Gnk + . . . δ3A (Φ) δΦiδΦkδΦj ιjjG (Φ)jk (445) Thus, from the definition (14) of the energy-momentum tensor, the one-loop Casimir energy density contributions to the energy-momentum tensor are obtained from this equation, by choosing the field Φi to be the metric, gµν , and multiplying by Considering, now, the contribution to the one-loop Casimir energy densities from a real scalar boson, with the classical action: Ascalar = − gµν (Dµϕ) (Dνϕ) +m 2ϕ2 − ξRϕ2 (446) where R is the Ricci scalar, and ξ is a real constant, sometimes called the conformal coupling, when d = 4, we find, by use of the Palatini identity δRµν = DµδΓ τν−DτδΓτµν , and the identity δΓτµν = gτσ (Dµδgσν +Dνδgσµ −Dσδgµν), that: δAscalar δΦgxµν (1− 2ξ) gµσgντ (Dσϕ) (Dτϕ)− gµνgστ (Dσϕ) (Dτϕ) −2ξgµσgντϕDσDτϕ+ 2ξgµνgστϕDσDτϕ− ξ Rµν − 1 ϕ2 − 1 gµνm2ϕ2 (447) For the particular case ξ = 1 , this is in agreement with the formulae of Muller, Fagun- des, and Opher [236, 238], after for allowing for their sign convention for the Riemann tensor, which results in the opposite sign of the Ricci tensor to (8). From (14), (445), and (447), we find: 1,scalar = δΓ1,scalar δΦgxµν (1− 2ξ) gµσgντ (DyσDxτ +DxσDyτ ) gµνgστ (DyσDxτ +DxσDyτ )− 2ξgµσgντ (DxσDxτ +DyσDyτ ) +2ξgµνgστ (DxσDxτ +DyσDyτ )− 2ξ Rµν − − gµνm2 G (Φ)ϕxϕy (448) In the particular case of 3+ 1 dimensional Minkowski space, the scalar contribution to the one-loop energy density is given by (448) as: T 001,scalar = (2− 4ξ) ∂y0∂x0 + (1− 4ξ) −∂y0∂x0 + ~∂y~∂x − 2ξ (∂x0∂x0 + ∂y0∂y0) −∂2x0 + ~∂2x − ∂2y0 + ~∂2y G (Φ)ϕxϕy (449) Furthermore, for the scalar propagator, requiring that the Fresnel integral in (440) be well-defined uniquely selects the Feynman iε prescription for the propagator. Thus the scalar propagator is: Gϕxϕy = − ei(−p0(x0−y0)+~p.(~x−~y)) −p20 + ~p2 +m2 − iε (450) Substituting (450) into (449), and taking the limit y → x from either y0 > x0 or y0 < x0, we find: T 001,scalar = ~p2 +m2 (451) which is real, as required, and is the standard divergent expression for the one-loop vacuum energy density of a real scalar field. For models that have unbroken super- symmetry in 3 + 1 spacetime dimensions, and do not involve gravitons, the one-loop vacuum energy densities cancel between fermions and bosons, and, moreover, the vac- uum energy density is exactly zero to all orders in the coupling constants [338], and the one-loop vacuum energy densities have also been found to vanish in some models with broken supergravity [339, 340, 245, 246], whilst for d = 11 supergravity [14], it appears that a cosmological constant is not consistent with supersymmetry [200, 269, 124], so that divergences corresponding to a cosmological constant term would be cancelled unambiguously within the framework of BPHZ renormalization, to all orders in the semiclassical expansion in the number of loops in the Feynman diagrams. Considering the gravitino propagator, for the compactification of Hořava-Witten theory on M6, as a sum over images: ψxµiψyνj Φ|CH3 ψxµiψγ(y)νj , (452) where M6 is the quotient of CH3 by the cocompact, torsionless, discrete subgroup Γ of SU (3, 1), we see that if the sign of the gravitino field, at the image γ (y) of y by an element γ of Γ, depends on the route taken from y to γ (y), then the sum of G (Φ|CH3)ψxµiψγ(y)νj , for y close to x, over all elements γ of Γ different from the identity, will not be well defined, even if it converges. But this sum is directly physically significant, because it determines the finite part of the gravitino contribution to the one-loop Casimir contribution to the energy-momentum tensor, by a formula analogous to (448). Furthermore, the three-form gauge field, CIJK , only enters the gravitino field equation through its four-form field strength GIJKL, which is globally well defined, so a background configuration of the three-form gauge field cannot make any difference to whether or not the sum over γ ∈ Γ in G (Φ|M6)ψxµiψyνj well defined. Thus it does, indeed, seem that models of this type are not physically well defined, unless M6 is a spin manifold. Of course, a rotation through 2π changes the sign of a spinor field, so it is natural to wonder whether introducing twists or rotations in the local Lorentz part of the vielbein between different coordinate patches, which will cancel out of the relations between the metric on the different coordinate patches, can cancel the ambiguity, but this is presumably taken into consideration in determining whether or not a manifold is a spin manifold. A direct explanation of why CP2 is not a spin manifold has been given by Hawking and Pope [39], and recently reviewed in Appendix B of [341]. 4.1 The Salam-Strathee harmonic expansion method For the explicit calculation of the Casimir energy densities for compactifications on smooth compact quotients of CH3 or H6, by means of the sum over images method of Muller, Fagundes, and Opher [236, 237, 238], for obtaining the propagators on the quotients, or some extension of their method if the sums diverge at large distances due to the masslessness of the fields, the propagators and heat kernels for the d = 11 supergravity multiplet are needed for flat R5, times CH3 or H6, and the propagators and heat kernels for the d = 10 supersymmetric Yang-Mills multiplet are needed for flat R4, timesCH3 orH6. The propagators and heat kernels forCH3 orH6 can be obtained from the corresponding propagators and heat kernels for CP3 or S6, which can in turn be obtained by a straightforward but lengthy application of the harmonic expansion method of Salam and Strathdee [244, 342, 343], which is currently in progress. The harmonic expansions can be summed by means of a generating function, and for the heat kernel of a massive scalar, on CH3, we find the integral representation: H (χ, s) = 4e−sm d cosh (2χ) + (cosh (2χ) + 1) d cosh (2χ) (dy) e− 4s sinh (2y) (cosh (2y)− cosh (2χ))− 2 (453) Here χ is proportional to the geodesic distance between the two position arguments of the heat kernel. The same integral, but with different differential operators acting on it, occurs in the heat kernel of a massive scalar, on real hyperbolic spaces of all even dimensions ≥ 2, while for real hyperbolic spaces of odd dimension ≥ 3, the heat kernel can be written in closed form, as found by Muller, Fagundes, and Opher, for d = 3. The application of the Salam-Strathdee method to CP3, which is a spin manifold, was begun by Strathdee [343], and developed by Sobczyk [344, 345]. A special effect in a related background was discovered by Gibbons and Nicolai [340], who calculated the one-loop vacuum energy density of the Freund-Rubin AdS4 × S7 compactification of d = 11 supergravity [346], and found that it vanished “floor by floor”, or in other words, separately for each N = 8 supersymmetric Kaluza-Klein level, whereas to preserve the supersymmetry of the vacuum, it would have been sufficient for the sum over all the Kaluza-Klein levels to vanish. The contribution of the lowest Kaluza-Klein level, namely the N = 8 supergravity multiplet, had earlier been found to vanish by Allen and Davis [339]. There is also a AdS4 × CP3 compactification of type IIA d = 10 supergravity, discovered by Watamura [347], that was shown by Nilsson and Pope [348] to have N = 6 supersymmetry, and also to be related to the Freund-Rubin compactification of d = 11 supergravity, via the fact that S7 is a non-trivial U (1) fiber bundle over CP3, called a Hopf fibration. What this means is that the Watamura AdS4 × CP3 compactification of type IIA d = 10 supergravity can be identified with a particular AdS4×CP3×S1 compactification of d = 11 supergravity, such that the metric ansatz (94) has been modified by the replacement dy2 → dy −AAdzA , (454) where AA is proportional to a potential for the Kähler form of the CP 3, and y is now the coordinate around the S1. The Watamura N = 6 compactification is then obtained in an appropriate limit, where the radius of the S1 tends to 0, while for another special case, where the radius of the S1 is appropriately related to the diameter of the CP3, the supersymmetry is presumably extended to N = 8, and the Freund-Rubin compactification is obtained. Nilsson and Pope showed that the complete spectrum of small fluctuations of the Watamura N = 6 compactification can be directly obtained from the known spectrum of small fluctuations of the Freund-Rubin solution [349, 350, 351, 352, 353]. I shall now obtain the complete list of the modes by the Salam-Strathdee method, and check it against the list given by Nilsson and Pope, and then repeat the Gibbons-Nicolai calculation, for all but the lowest two Kaluza-Klein levels, for the Watamura N = 6 compactification. The isometry group of CP3, with the standard Fubini-Study metric [72], is SU (4), and the subgroup of the isometry group, that leaves a chosen point fixed, which I shall call the tangent space isometry group, is SU (3) × U (1). The tangent space group of CP3 is SO (6), because CP3 has six real dimensions, so by accident, the tangent space group is locally isomorphic to the isometry group, although the tangent space group and the isometry group are completely distinct, and the tangent space isometry group, SU (3) × U (1), is found to be embedded in the tangent space group, SO (6), and the isometry group, SU (4), in different ways. I shall put a tilde above the irreducible representations of the tangent space group, SO (6), to distinguish them from the irreducible representations of the isometry group, SU (4). The first step of the Salam-Strathdee method is to decompose all the fields in- volved, which are here the metric, the three-form gauge field, and the gravitino, of d = 11 supergravity, into irreducible representations of the product of the tangent space isometry groups SO (3, 1), of the four extended dimensions, and SU (3)× U (1), of CP3, and possible components along the S1, that does not have a nontrivial con- tinuous tangent space isometry group. The next step is then to determine, for each irreducible represention of the tangent space isometry group SU (3)×U (1) of CP3 that arises, the list of all the irreducible representations of SU (4), the isometry group of CP3, that contain that irreducible representation of SU (3)×U (1), under the subgroup decomposition SU (4) → SU (3)×U (1). This is then the list of all the harmonics that occur, in the harmonic expansion, on CP3, of that particular irreducible representation of the tangent space isometry group of CP3. According to Salam and Strathdee’s general prescription, [244], for harmonic ex- pansions on the quotient space, G/H , of a Lie group, G, and a Lie subgroup, H , of G, the quotient space CP3 = SU (4) /(SU (3)× U (1)) is coordinatized by the “boosts” generated by the six generators of SU (4), that are not generators of its subgroup SU (3)×U (1). Now SU (3)×U (1) is contained in SU (4) such that the 4 of SU (4) has the SU (3)× U (1) content: 4 = 3 1 + 1− 3 (455) where the relative value of the U (1) charges, which are shown as subscripts, is deter- mined by the tracelessness of the SU (4) generators, and the overall normalization of the U (1) charges is a convention, that I have chosen to agree with Strathdee, [343], and Sobczyk, [344, 345]. From (455), we find that the SU (3) × U (1) content of the adjoint of SU (4) is determined by: 15 + 1 = 4× 4̄ = + 1− 3 3̄− 1 + 1 3 = 80 + 10 + 31 + 3̄−1 + 10 (456) Thus the generators of SU (4), that are not generators of SU (3) × U (1), have the SU (3)×U (1) content 31+3̄−1, so the tangent space isometry group, SU (3)×U (1), of CP3, is embedded in the tangent space group, SO (6), of CP3, such that the tangent space six-vector, the 6̃ of SO (6), has the SU (3)× U (1) content [343]: 6̃ = 31 + 3̄−1 (457) The decomposition (457) now determines the decompositions of all the other irreducible representations of SO (6). In particular, if we consider the 4̃ of SO (6) that contains the 1 and the 3̄ of SU (3), and write its decomposition as 4̃ = 1a + 3̄b, where the U (1) charges a and b are to be determined, we find that: 4̃× 4̃ = 12a + 3̄a+b + 3̄a+b + 6̄2b + 32b (458) However, we know that the antisymmetric part of 4̃× 4̃ is the 6̃, so for consistency with (457), we must have 2b = 1, and a+ b = −1, so that we find 4̃ = 1− 3 + 3̄ 1 . The other 4̃ of SO (6) then decomposes as 1 3 + 3− 1 , so, comparing with (455), we see that the tangent space isometry group, SU (3)×U (1), of CP3, is, indeed, embedded differently in the tangent space group, SO (6), and the isometry group, SU (4), as stated above [343]. To determine which irreducible representations of SU (4) contain a given irreducible representation of SU (3)× U (1), it will be convenient to use Young tableau notations for the irreducible representations of SU (3) and SU (4). I shall denote the irreducible representation of SU (3), that corresponds to a Young tableau with rows of lengths a, b, and c, such that a ≥ b ≥ c ≥ 0, by [a, b, c], with a corresponding notation for SU (4). Then [a+ n, b+ n, c + n], for all integer n such that c + n ≥ 0, all correspond to the same irreducible representation of SU (3), whose Dynkin label is (a− b, b− c). It will be very convenient also to allow Young tableaux with negative length rows, which means that a, b, and c, in the Young tableau [a, b, c], are restricted only by a ≥ b ≥ c, without the restriction to c ≥ 0. Negative length rows are represented by rows of blocks extending out to the left of what would normally be the left-hand side of the Young tableau. The corresponding irreducible representations of SU (3) are constructed from appropriately symmetrized Kronecker products of the fundamental and the antifundamental representations, one fundamental representation factor for each block in a positive length row, and one antifundamental representation factor for each block in a negative length row, with all traces that can be formed by contracting the SU (3) invariant tensor δrs̄ with an antifundamental index and a fundamental index, both from among the left-hand indices of the representation matrix, or both from among the right-hand indices of the representation matrix, removed. Analogous constructions also apply for all the other special unitary groups. For example, the [p, 0, 0,−p] representation of SU (4), whose Dynkin label is (p, 0, p), with the three components of the Dynkin label corresponding to the three vertices of the SU (4) Dynkin diagram, written in sequence from end to end along the line, has the representation matrices: UI1I2...IpJ̄1J̄2...J̄p,K̄1K̄2...K̄pL1L2...Lp = (p! ) SISJSKSL (−1)r (2p+ 2− r) ! ((p− r) ! )2 r! (2p+ 2) ! ×δL1K̄1δI1J̄1 . . . δLrK̄rδIrJ̄rUIr+1K̄r+1UJ̄r+1Lr+1 . . . UIpK̄pUJ̄pLp (459) where, in the last line of this expression, δL1K̄1δI1J̄1 . . . δLrK̄rδIrJ̄rUIr+1K̄r+1UJ̄r+1Lr+1 . . . UIpK̄pUJ̄pLp is interpreted as having no δs, if r < 1, and as no Us, if r + 1 > p. In other words, the subscripts, on the subscripts, are to increase from 1 to p, going from left to right, along this expression. UIJ̄ here denotes the fundamental representation of an element of SU (4), and UĪJ ≡ (UIJ̄) denotes the antifundamental representation of that same element of SU (4), in accordance with the conventions of subsection 2.2, on page 25, for barred and unbarred indices, and denotes the sum over all permutations of the “I” subscripts, and thus contains p! terms, since there are p such subscripts, and so on. The formula (459) is used in the derivation of the scalar heat kernel on CP3, (453), by the Salam-Strathdee method. We then find, using indices µ, ν, σ, . . ., for the four extended dimensions, r, s, t, . . ., and r̄, s̄, t̄, . . ., in the complex coordinate notation of subsection 2.2, for CP3, and y for the S1, that the d = 11 gravition, hIJ , contains the d = 4 graviton, hµν , in the Young tableau representation [0, 0, 0]0 of SU (3) × U (1), where the subscript denotes the U (1) charge, and also a d = 4 vector, hµy, and a d = 4 scalar, hyy, in the [0, 0, 0]0 of SU (3) × U (1), a d = 4 vector, hµr, and a d = 4 scalar, hyr, in the [1, 0, 0]1 of SU (3) × U (1), a d = 4 vector, hµr̄, and a d = 4 scalar, hyr̄, in the [0, 0,−1]−1 of SU (3) × U (1), and d = 4 scalars, hrs̄, in the [1, 0,−1]0, hrs, in the [2, 0, 0]2, and hr̄s̄, in the [0, 0,−2]−2, of SU (3)×U (1). The decomposition of the three-form gauge field, CIJK , is worked out similarly, bearing in mind that for d = 4, a two-form gauge field is equivalent to a scalar [354, 355, 356, 233], and a three-form gauge field has no degrees of freedom. To work out the decomposition of the gravitino, we first determine the decomposi- tion of a d = 11 spinor. We decompose the 32-valued spinor index into the Cartesian product of an 8-valued spinor index, for the CP3, and a four-valued spinor index, for the four extended dimensions and the S1, considered together as a five-dimensional space, and consider the decomposition of the 8-valued spinor index, which is the sum of the two opposite chirality 4̃’s of SO (6). Thus, from above, the 8-valued spinor index decomposes into the 1− 3 +3̄ 1 +3− 1 = [0, 0, 0]− 3 +[0, 0,−1]1 +[0, 0, 0]3 +[1, 0, 0]− 1 of SU (3) × U (1). The d = 11 gravitino, ψI , thus contains d = 4 gravitinos, ψµ, and d = 4 spinors, ψy, in these four representations of SU (3)× U (1), together with d = 4 spinors, ψr, in the SU (3)×U (1) representations that result from forming the Cartesian product of the 31 with these four representations, namely 3 +3̄ 1 [1, 0, 0]− 1 + [1, 0,−1] 3 + [0, 0, 0]3 + [1, 0, 0]5 + [2, 0, 0]1 + [0, 0,−1] 1 , and d = 4 spinors, ψr̄, in the complex conjugates of these six representations. To determine which irreducible representations of SU (4) contain these represen- tations of SU (3) × U (1), we first recall the general rule for the irreducible represen- tations of SU (q − 1) contained in an irreducible representation of SU (q), which, for the present case, states that the irreducible representations of SU (3), contained in the irreducible representation of SU (4) that corresponds to a Young tableau [n1, n2, n3, n4], n1 ≥ n2 ≥ n3 ≥ n4, are the irreducible representations of SU (3) that correspond to all the Young tableaux [m1, m2, m3], such that n1 ≥ m1 ≥ n2 ≥ m2 ≥ n3 ≥ m3 ≥ n4. This general rule is the basis for the Gelfand-Tsetlin patterns that can be used to label the basis vectors of the irreducible representations of SU (q), via the subgroup chain U (1) ⊂ SU (2) ⊂ SU (3) ⊂ . . . ⊂ SU (q − 1) ⊂ SU (q), as reviewed, for example, in [357]. We next note that if the SU (3) representation, corresponding to a Young tableau [m1, m2, m3], is contained in an SU (4) representation, corresponding to a Young tableau [n1, n2, n3, n4], with n4 ≥ 0, then m1 +m2 +m3 of the n1 + n2 + n3 + n4 copies of the SU (4) fundamental, from which the SU (4) representation is constructed, branch to the 3 of SU (3), and the remaining n1 + n2 + n3 + n4 − (m1 +m2 +m3) copies of the SU (4) fundamental branch to the 1 of SU (3), so from (455), the U (1) charge of the SU (3) representation is m1 +m2 +m3 − (n1 + n2 + n3 + n4) (460) Furthermore, this relation, like the rule for the SU (3) irreducible representations con- tained within a given SU (4) irreducible representation, is unaltered by adding a con- stant to all the ni and all the mi, so it remains true when the assumption that n4 ≥ 0 is no longer satisfied. We now find that the rule (460), in combination with the rule that n1 ≥ m1 ≥ n2 ≥ m2 ≥ n3 ≥ m3 ≥ n4, is very restrictive. For example, the SU (3) representation, corresponding to the Young tableau [0, 0, 0], is contained in all the SU (4) representa- tions [p, 0, 0,−p′], with p ≥ 0 and p′ ≥ 0, but the SU (3)×U (1) representation [0, 0, 0]0 is contained only in the representations [p, 0, 0,−p], with p ≥ 0, whose representation matrices are given in (459). The Dynkin labels of these representations of SU (4) are (p, 0, p), as already noted, and in general, the Dynkin label of an SU (4) irreducible representation, that corresponds to a Young tableau [a, b, c, d], is (a− b, b− c, c− d), with the three components of the Dynkin label corresponding to the three vertices of the SU (4) Dynkin diagram, written in sequence from end to end along the line. We next note that each massive d = 4 graviton mode hµν , corresponding to har- monics on CP3 with Dynkin labels (p, 0, p), p > 0, will absorb a d = 4 vector with the same Dynkin label, for which hµy is available, and a d = 4 scalar with the same Dynkin label, for which hyy is available. Many of the other d = 4 massive modes, with a subscript y, also get absorbed by higher spin d = 4 massive modes, with matching Dynkin labels, in a similar way, although for the modes Cµνr, Cµyr, Cµνr̄, and Cµyr̄, the situation is reversed, with the d = 4 vector fields Cµyr and Cµyr̄ absorbing the fields Cµνr and Cµνr̄, which are equivalent to d = 4 scalar fields. In this way, we find that the unabsorbed d = 4 massive modes, and the SU (4) irreducible representations that occur in their harmonic expansions on CP3, are as shown in Table 4 for the bosons, and in Table 5 for the fermions. Most of the SU (4) irreducible representations that occur in the harmonic expansions of the metric, and of a vector field, were listed by Sobczyk, [344], in the context of a CP3 compactification of a d = 10 Einstein-Yang-Mills theory. We see that for each d = 4 spin, the SU (4) multiplets in Tables 4 and 5 are in one to one correspondence with the SU (4) multiplets listed by Nilsson and Pope [348] for that d = 4 spin, except that in some cases, p has to be shifted by a small number. This was to be expected, because simply listing the harmonics corresponding to each d = 4 state does not determine the corresponding masses, nor does it determine how the states are organized into N = 6 supermultiplets. I have listed the SU (4) multiplets in Tables 4 and 5 so that the complete set of harmonics entering the expansion, on CP3, of the corresponding d = 4 state, is given by the SU (4) multiplets shown, for all p ≥ 0, whereas p, in Nilsson and Pope’s Table 1, identifies the distinct N = 6 supermultiplets, with p = 0 corresponding to the standard N = 6 supergravity multiplet. Thus in Nilsson and Pope’s Table 1, many of the SU (4) Dynkin labels have a negative component for small values of p, and in particular, for p = 0, indicating the absence of an SU (4) representation in that p-series, in the corresponding low-lying N = 6 supermultiplet. Nilsson and Pope also listed the parities of the d = 4 boson states. I have not calculated the parities of the d = 4 boson states by the Salam-Strathdee method, but we note that if we assume that the parity of a d = 4 boson state is the product of a factor of −1 if the state arises from the d = 11 three-form gauge field, a factor of d = 11 compo- nents SU(3)×U(1) tableau SU (4) Dynkin labels p ≥ 0 2 hµν [0, 0, 0]0 (p, 0, p) 1 hµr [1, 0, 0]1 (p+ 1, 0, p+ 1) + (p, 1, p+ 2) 1 hµr̄ [0, 0,−1]−1 (p+ 1, 0, p+ 1) + (p+ 2, 1, p) 0 hrs̄ [1, 0,−1]0 + [0, 0, 0]0 (p+ 1, 0, p+ 1) + (p, 1, p+ 2) + (p+ 2, 1, p) + (p, 2, p) + (p, 0, p) 0 hrs [2, 0, 0]2 (p+2, 0, p+2)+ (p+1, 1, p+3) + (p, 2, p+4) 0 hr̄s̄ [0, 0,−2]−2 (p+2, 0, p+2)+ (p+3, 1, p+1) + (p+4, 2, p) 0 Cµνy [0, 0, 0]0 (p, 0, p) 1 Cµyr [1, 0, 0]1 (p+ 1, 0, p+ 1) + (p, 1, p+ 2) 1 Cµyr̄ [0, 0,−1]−1 (p+ 1, 0, p+ 1) + (p+ 2, 1, p) 1 Cµrs̄ [1, 0,−1]0 + [0, 0, 0]0 (p+ 1, 0, p+ 1) + (p, 1, p+ 2) + (p+ 2, 1, p) + (p, 2, p) + (p, 0, p) 1 Cµrs [0, 0,−1]2 (p, 1, p+ 2) + (p, 0, p+ 4) 1 Cµr̄s̄ [1, 0, 0]−2 (p+ 2, 1, p) + (p+ 4, 0, p) 0 Crst [0, 0, 0]3 (p, 0, p+ 4) 0 Crst̄ [0, 0,−2]1 + [1, 0, 0]1 (p, 0, p+ 4) + (p, 1, p+ 2) + (p, 2, p) + (p+ 1, 0, p+ 1) + (p, 1, p+ 2) 0 Crs̄t̄ [2, 0, 0]−1 + [0, 0,−1]−1 (p+ 4, 0, p) + (p+ 2, 1, p) + (p, 2, p) + (p+ 1, 0, p+ 1) + (p+ 2, 1, p) 0 Cr̄s̄t̄ [0, 0, 0, ]−3 (p+ 4, 0, p) Table 4: Boson harmonics for type IIA d = 10 supergravity compactified on CP3. d = 4 d = 11 compo- nents SU(3)×U(1) tableau SU (4) Dynkin labels p ≥ 0 ψµ [0, 0, 0]− 3 (p+ 2, 0, p) ψµ [0, 0,−1] 1 (p, 0, p+ 2) + (p, 1, p) ψµ [0, 0, 0]3 (p, 0, p+ 2) ψµ [1, 0, 0]− 1 (p+ 2, 0, p) + (p, 1, p) ψr [1, 0, 0]− 1 (p+ 2, 0, p) + (p, 1, p) ψr [1, 0,−1] 3 (p+1, 0, p+3)+ (p, 1, p+4) + (p+1, 1, p+1) + (p, 2, p+ 2) ψr [0, 0, 0]3 (p, 0, p+ 2) ψr [1, 0, 0]5 (p+ 1, 0, p+ 3) + (p, 1, p+ 4) ψr [2, 0, 0]1 (p+ 2, 0, p) + (p+ 1, 1, p+ 1) + (p, 2, p+ 2) ψr [0, 0,−1] 1 (p, 0, p+ 2) + (p, 1, p) ψr̄ [0, 0,−1]− 5 (p+ 3, 0, p+ 1) + (p+ 4, 1, p) ψr̄ [0, 0,−2]− 1 (p, 0, p+ 2) + (p+ 1, 1, p+ 1) + (p+ 2, 2, p) ψr̄ [1, 0, 0]− 1 (p+ 2, 0, p) + (p, 1, p) ψr̄ [0, 0,−1] 1 (p, 0, p+ 2) + (p, 1, p) ψr̄ [1, 0,−1]− 3 (p+3, 0, p+1)+ (p+4, 1, p) + (p+1, 1, p+1) + (p+ 2, 2, p) ψr̄ [0, 0, 0]− 3 (p+ 2, 0, p) Table 5: Fermion harmonics for type IIA d = 10 supergravity compactified on CP3. −1 if the d = 4 boson state has spin 1, a factor of −1 if the second component of the SU (4) Dynkin label is an odd number, and a factor of −1 for each index y of the d = 11 components that the state arises from, then the SU (4) boson multiplets listed in Table 4, for each d = 4 spin and parity, can be paired one to one with the SU (4) boson multiplets listed by Nilsson and Pope, of the same d = 4 spin and parity, up to small shifts of p in some cases, as before. We can now calculate the one-loop vacuum energy of the Watamura N = 6 com- pactification of type IIA d = 10 supergravity, by the method of Gibbons and Nicolai [340], which uses the zeta function regularization method of Hawking [358]. We can directly use Gibbons and Nicolai’s formula (9) for the contribution to the vacuum en- ergy from a spin s state, s > 0, and their formula (11) for the contribution to the vacuum energy from spin 0 state, except that the last term in their formula (9), in the scanned version of the preprint from KEK [340], which seems to be a misprint, has to be replaced by − 20s+9 . I have verified, using Maxima [291], that the one- loop vacuum energy of the Freund-Rubin compactification does, indeed, vanish floor by floor, for all Kaluza-Klein levels above the lowest, when this replacement is made in their formula (9). The lowest Kaluza-Klein level, namely the ordinary supergravity multiplet, requires a separate calculation, which was carried out by Allen and Davis [339], because the formula for the dimension of an SO (8) irreducible representation does not vanish for some of the SO (8) Dynkin labels with a negative component that arise in this case, such as (−2, 1, 0, 0). The dimensions of the irreducible representations of SU (4) with Dynkin labels (a, b, c), where the integers a ≥ 0, b ≥ 0, and c ≥ 0 are associated with the three verteices of the SU (4) Dynkin diagram, taken in sequence from end to end along the line, can be calculated from Weyl’s dimension formula [359, 360], or from the combinatorial result, summarized in section 3.I.(b) of [361], that the dimension of the irreducible representation of GL (n,C) associated with an ordinary Young tableau with n rows, and no negative length rows, is the product over all the boxes x of the tableau, n+c(x) , where c (x) is the horizontal position of x minus its vertical position, counting from left to right and downwards, starting from the box at the top left-hand corner of the tableau, and hx, the length of the hook whose top left-hand corner is x, is the number of boxes directly under x, plus the number of boxes directly to the right of x, plus 1. The result is: D (a, b, c) = (1 + a) (1 + b) (1 + c) a+ b+ c (461) Then using Maxima [291], we find, by a formula analogous to Gibbons and Nicolai’s formula (14), with z = −1, but using the entries in Nilsson and Pope’s Table 1, instead of from Gibbons and Nicolai’s Table 1, that in units of , where Λ is the cosmological constant of the AdS4, the contributions to the vacuum energy, from the states in the N = 6 supersymmetry multiplet at Kaluza-Klein level p, for p ≥ 2, where p = 0 corresponds to the N = 6 supergravity multiplet, are as follows. Spin 2: −20p9 − 270p8 − 1580p7 − 5250p6 − 10888p5 − 14565p4 − 12506p3 −6597p2 − 1916p− 228 (462) Spin 3 320p9 + 4320p8 + 24800p7 + 78960p6 + 152188p5 + 181290p4 + 129622p3 +50049p2 + 7275p− 414 (463) Spin 1: −180p9 − 2430p8 − 13740p7 − 42210p6 − 76152p5 − 80685p4 − 45930p3 −9045p2 + 2892p+ 1260 (464) Spin 1 960p9 + 12960p8 + 72480p7 + 216720p6 + 370644p5 + 353070p4 +153770p3 − 7065p2 − 28859p− 6690 (465) Spin 0: −420p9 − 5670p8 − 31500p7 − 92610p6 − 152928p5 − 134865p4 − 45442p3 +14211p2 + 13360p+ 2004 (466) The fact that the contribution of the spin 2 states is negative is presumably an artifact of the zeta function regularization used. The sum of these contributions is zero, so for all the Kaluza-Klein levels with p ≥ 2, the one-loop vacuum energy vanishes “floor by floor” for the Watamura-Nilsson-Pope N = 6 CP3 compactification of type IIA d = 10 supergravity, just as it does for the N = 8 Freund-Rubin compactification of d = 11 supergravity. The cases of p = 0 and p = 1 require separate calculations, because some SU (4) multiplets occur that should be omitted in these cases, and some Dynkin labels with a negative component occur, for which the formula (461) for the dimension of an SU (4) irreducible representation does not give zero. For the N = 6 supergravity multiplet, which is the case with p = 0, the one-loop result vacuum energy was found to vanish by Allen and Davis [339]. The case of p = 1 requires further study, and will not be considered in this paper. But it does not seem very likely that the one-loop vacuum energy would fail to vanish for this one Kaluza-Klein level, when it does for all the others. We note that this calculation has not included any Kaluza-Klein excitations asso- ciated with the S1, so in the context of Nilsson and Pope’s compactifications of d = 11 supergravity, interpolating between the Freund-Rubin compactification of d = 11 su- pergravity, and Watamura’s N = 6 compactification of type IIA d = 10 supergravity, the d = 4 states listed here are appropriate for the limit in which the radius of the S1 tends to zero. To consider the opposite limit, in which the radius of the S1 tends to infinity, which is presumably related to the N = 6 supergravity in five dimensions listed by Cremmer [362], it would be necessary to repeat the calculation with the ex- tra modes included. However, at a certain value of the radius of the S1, the N = 6 supersymmetry would be extended to the N = 8 supersymmetry of the Freund-Rubin compactification, for which it is known from the Gibbons-Nicolai calculation that the one-loop vacuum energy vanishes floor by floor. So it is perhaps plausible that the one-loop vacuum energy might also vanish floor by floor for all values of the radius of the S1, from 0 to ∞. If the numbers of fermion and boson helicity states are equal for all the N = 6 massive multiplets, which I have not explicitly checked, then we would presumably find that the one-loop vacuum energy would still vanish when the background is flat four-dimensional Minkowski space times CP3, even though this background is not a solution of the classical field equations, and is not supersymmetric, and, on the basis of relations between the propagators and heat kernels on Minkowski space times CP3, and on Minkowski space times CH3, it might then also vanish when the background is flat four-dimensional Minkowski space times CH3. And for similar reasons, it seems possible that the one-loop vacuum energy might also vanish when the background is flat five-dimensional Minkowski space times CH3. However, in consequence of the rule, discussed at the beginning of this section, that the quantum effective action of the BRST-BV gauge-fixed theory, in a background that is not a solution of the classical field equations, is the sum of all the one-line-irreducible vacuum bubbles, calculated with an action given by the BRST-BV gauge-fixed classical action in the presence of the background field, but with the terms linear in the quantum fields deleted, the action used in the calculation of the quantum effective action of the BRST-BV gauge-fixed theory, on a background that is flat four-dimensional or five-dimensional Minkowski space, times CP3 or CH3, would presumably not satisfy identities needed to use Zumino’s arguments [338] for the vanishing of the higher loop vacuum energies. Nevertheless, if the massive N = 6 multiplets satisfied Curtright’s spin sum rules [363] for a theory with N = 6 supersymmetry in d = 4, which I have not explicitly checked, then some of the ingredients for a possible cancellation of higher loop vacuum energies, on a flat four-dimensional Minkowski space times CP3 or CH3 background, would be in place, so the possibility that such cancellations might occur is not yet excluded. However, the grounds for expecting such higher loop cancellations to occur are not very strong, and it does not seem very likely that the higher loop vacuum energies of type IIA d = 10 supergravity on a four-dimensional Minkowski space times uncompactified CH3 background, and of d = 11 supergravity on a five-dimensional Minkowski space times uncompactified CH3 background, will vanish, notwithstanding the special properties of the Watamura N = 6 compactification of type IIA d = 10 supergravity, and its oxidation to d = 11 by Nilsson and Pope, for three reasons. Firstly, the lowest Kaluza-Klein energies of the states in the N = 6 supermultiplet at Kaluza-Klein level p, are not all the same. Instead, they differ by up to four units within the same multiplet, so the energy differences, between the lowest energies of states within one multiplet, are up to four times greater than the energy difference between corresponding states within successive multiplets, and these energy differences, between the states within a multiplet, are likely to be essential for the cancellation of the vacuum energy of a multiplet, at least when Λ is nonzero. The contribution to the vacuum energy, from a state of lowest energy E0, is a quartic polynomial in E0, by Gibbons and Nicolai’s equations (9) and (11). E0 is linear in the Kaluza-Klein level number p. When the relation between the sectional curvature of the AdS4, and the minimum sectional curvature of the CP 3, is broken, there are two independent units of energy, namely , where ΛCP3 is defined in terms of the minimum sectional curvature of the CP 3. We would now expect the lowest energy E0 of a state to contain a term p , associated with its Kaluza-Klein level number p, and a term q , where q is the integer or half integer, such that 1 ≤ q ≤ 5, that determines the offset of E0 from p listed by Nilsson and Pope, for the case when ΛCP3 = Λ. Vanishing of the one-loop vacuum energy floor by floor, for independent Λ and ΛCP3 , would then require that the coefficients of the different powers of in the vacuum energy, which are polynomials in p of degree up to 9, all vanish separately, and, although this has not been excluded, there is no reason to expect it to happen, to the best of my knowledge, except that, if the numbers of fermion and boson states in each N = 6 multiplet are equal, we would expect the coefficients of the terms independent of Λ, which are polynomials in p of degree 9, to vanish, since the one-loop vacuum energy of each N = 6 multiplet would then vanish in four-dimensional Minkowski space. Thus, although the one-loop vacuum energy of each N = 6 multiplet would vanish for Λ = 0, if the numbers of fermion and boson states in each N = 6 multiplet are equal, it does not seem very likely that the one-loop vacuum energy of each N = 6 multiplet would vanish for values of the ratio Λ strictly between 0 and 1, so the vanishing of the one-loop vacuum energy of each N = 6 multiplet, for Λ = 0, would be an isolated phenomenon, not continuously connected to the supersymmetric system with Λ = ΛCP3 , so it does not seem very likely that the supersymmetric system could result in the vanishing of the higher loop vacuum energies of type IIA d = 10 supergravity on a four-dimensional Minkowski space times CP3 background, or that its d = 11 oxidation could result in the vanishing of the higher loop vacuum energies of d = 11 supergravity on a five-dimensional Minkowski space times CP3 background, when there are no other reasons to expect this to happen. Secondly, there is a second Watamura-Nilsson-Pope CP3 compactification of type IIA d = 10 supergravity, that has no supersymmetry, but differs from the N = 6 com- pactification only by the relative orientation of a four-form field strength Fµνστ , which is proportional to the d = 4 tensor density ǫµνστ , and a two-form field strength FAB, which is proportional to the Kähler form of the CP3. The relative orientation of Fµνστ and FAB is detected by the supersymmetry variations of the fermions. Now, by the Salam-Strathdee construction, the small fluctuation modes, about this N = 0 compact- ification, will consist of exactly the same collection of series of SU (4) representations as listed above for the N = 6 compactification, but the lowest energies, of the smallest representations of some of the series, will be shifted up or down, by a small number of units of , so that the vacuum energy will presumably no longer vanish. And to distinguish the two cases, both Fµνστ and FAB would still have to be nonzero, when the AdS4 is replaced by Minkowski space, and the two cases would still have to be distinguished, when the CP3 is replaced by CH3, so it seems unlikely that the higher loop vacuum energies will vanish for a four-dimensional Minkowski space times CH3 background, without nonvanishing background fields corresponding to Fµνστ and FAB. And for the corresponding compactifications of d = 11 supergravity, Nilsson and Pope showed that these twoCP3 compactifications of type IIA d = 10 supergravity are “Hopf fibrations” of the Freund-Rubin AdS4 × S7 compactification of d = 11 supergravity, which means that the metric ansatz (94) would have to be modified by the replacement (454), where AA is proportional to a potential for the Kähler form of the CP 3 or CH3. And thirdly, there is an AdS5 × CP3 compactification of d = 11 supergravity, such that the only nonvanishing form field, in the background, has the form of the Lukas-Ovrut-Stelle-Waldram [68] ansatz (147). This compactification is investigated in the following subsection 4.2, and found to have no supersymmetry. Its one-loop vacuum energy will thus presumably be nonvanishing, and d = 11 supergravity, on a five-dimensional Minkowski space times CP3 or uncompactified CH3 background, is as closely related to this compactification, as it is to Nilsson and Pope’s d = 11 oxidation of Watamura’s N = 6 compactification of type IIA d = 10 supergravity. This suggests, again, that if the higher loop vacuum energies of d = 11 supergravity were to vanish on any five-dimensional Minkowski space times CH3 background, there would have to be a nonvanishing field strength Fµνστ in the background, and the metric ansatz (94) would have to be modified by the replacement (454), with AA proportional to a potential for the Kähler form of the CH3, in order to relate the background to the d = 11 oxidation of the Watamura N = 6 compactification, and distinguish it from a background related to the AdS5 ×CP3 compactification. These arguments do not exclude the possibility that the higher loop vacuum energies of a four-dimensional Minkowski space times uncompactifiedCH3 timesR1 background for d = 11 supergravity, with suitable dependences of a (y) and b (y), in the metric ansatz (94), on the position y along the R1, and a suitable y-dependent value of the field strength Fµνστ , proportional to ǫµνστ , and the replacement (454) in the metric ansatz (94), with AA a suitable y-dependent multiple of a potential for the Kähler form of the CH3, might vanish. However, the reasons for expecting such a background to exist, for which the higher loop vacuum energies vanish, are not very strong, so for the phenomenological estimates in this paper, I assume that the higher-loop vacuum energies are nonvanishing on an uncompactified CH3 background, and, moreover, that they have their typical order of magnitude, in terms of the magnitude of the curvature of the background, which means that b1 cannot be smaller than the value ∼ 0.03 to 0.2 estimated in subsection 2.3.6, on page 66, on the basis of Giudice, Rattazzi, and Wells’s estimate [11] of the expansion parameter for quantum gravitational corrections in d dimensions, so that, in consequence of the relation (103), on page 37, between and |χ (M6)|, which follows from the estimate (102), of the d = 4 Yang-Mills fine structure constant at unification, values of |χ (M6)| larger than ∼ 7 × 104 to 6 × 109 are excluded. If it turned out that cancellations of higher loop vacuum energies of Hořava-Witten theory, on a suitable uncompactified CH3 background, actually did occur, allowing to be smaller than ∼ 0.03 to 0.2, and |χ (M6)| to be larger than ∼ 7 × 104 to 6 × 109, when M6 is a smooth compact quotient of CH3 that is a spin manifold, then the phenomenological estimates in this paper would presumably still be valid, with minor modifications, for smooth compact quotients of H6 that are spin manifolds, since, to the best of my knowledge, there is no reason to expect the vacuum energy of Hořava-Witten theory to vanish on an uncompactified H6 background. It would be interesting to find out whether Nilsson and Pope’s d = 11 “oxidation” of the Watamura N = 6 CP3 compactification of type IIA d = 10 supergravity, as discussed in this subsection, can be extended by the addition of non-vanishing components GABCD of the four-form field strength of the three-form gauge field, given by the ansatz (147) of Lukas, Ovrut, Stelle, and Waldram (LOSW) [68], so as to obtain a supersymmetric AdS4 × CP3 compactification of Hořava-Witten theory, consistent with Witten’s topological constraint [45], when the SU (3) part of the spin connection of the CP3 is embedded in the E8 on one of the two orbifold hyperplanes, and the U (1) part of the spin connection of the CP3 is embedded in the E8×E6 left unbroken by the SU (3) embedding, in one of the four ways listed by Pilch and Schellekens, in subsection 4.3 of [268]. However the components Gµνστ of the four-form field strength of the three-form gauge field, like the components GABCD, are odd under reflection in the Hořava-Witten orbifold hyperplanes, so if they do not vanish as one or both of the orbifold hyperplanes are approached, they would have to have discontinuities at the orbifold hyperplanes in the upstairs picture, which would then, by (43), require the existence of non-vanishing components Fµν of the E8 Yang-Mills field strength on the corresponding orbifold hyperplane, which would break invariance under the SO (3, 2) Anti de Sitter group. Thus to preserve invariance under the Anti de Sitter group, Gµνστ would have to vanish on both orbifold hyperplanes. This is not necessarily inconsistent with the existence of a compactification, since there also exists an AdS5×CP3 compactification of d = 11 supergravity, whose only non-vanishing components of GIJKL are given by the LOSW ansatz (147), but I shall show in the next subsection that this AdS5×CP3 compactification has no supersymmetry, so to have a chance of having a supersymmetric AdS4×CP3 compactification of Hořava-Witten theory, Gµνστ would have to be nonzero in the bulk, away from the orbifold hyperplanes. The boundary conditions, on Gµνστ , would then be that these components vanish on both orbifold hyperplanes. A new feature, in the bulk, would be that GIJKL now has enough nonvanishing components, in the bulk, to turn on the nonlinear GIJKL1...L8G L1...L4GL5...L8 term in the field equation for CIJK, where GIJKL1...L8 denotes the tensor −GǫIJKL1...L8. We would thus expect also to find some nonvanishing components of GIJKL that have an index y, and some nonvanishing components of CIJK , with an index y, were in fact found in Witten’s original investigation of supersymmetric compactifications of Hořava-Witten theory [127]. Furthermore, the metric components GAy, which are nonzero in the Nilsson-Pope d = 11 oxidation of the Watamura N = 6 compactification of type IIA d = 10 su- pergravity, due to the replacement (454) in the metric ansatz (94), are also odd under reflection in the Hořava-Witten orbifold hyperplanes, and must thus presumably vanish on the orbifold hyperplanes, since, to the best of my knowledge, there is no analogue, for the metric components GUy, of the discontinuity equation (43), for the components GUVWX of the four-form field strength of the three-form gauge field. This is, again, not necessarily inconsistent with the existence of a compactification, due to the existence of the AdS5×CP3 compactification studied in the next subsection, and would give the boundary conditions on GAy. In the presence of a boundary, half of the bulk supersymmetry is always broken [364]. However the N = 3 d = 4 supergravity supermultiplet contains three vector bosons, which naturally transform as the adjoint of SO (3), and do not fit naturally into aCP3 compactification. However, as noted by Nilsson and Pope [348], the SU (4)× U (1) gauge bosons, found in the Watamura CP3 compactification of type IIA d = 10 supergravity, could be consistent with N = 2 or N = 1 supersymmetry, as well as with N = 6 supersymmetry. N = 2, d = 4 supersymmetry is not consistent with the existence of chiral fermions, and three of the four embeddings of the SU (3) × U (1) spin connection of CP3, in E8×E8, found by Pilch and Schellekens [268], have chiral fermions, so could have at most N = 1 supersymmetry, whereas the fourth embedding found by Pilch and Schellekens, their case 4.3.(a), has no chiral fermions for CP3, and thus might possibly be consistent with N = 2 supersymmetry. We note that for gauged N -extended d = 4 supergravity, with Nleq4, and not cou- pled to any matter multiplets, Allen and Davis [339] found that the one-loop vacuum energy, in the AdS4 background, is nonvanishing, so that there would be no possibility of an analogue of the Gibbons-Nicolai floor by floor vanishing of the one-loop vacuum energy when the contributions of the Kaluza-Klein multiplets above the supergravity multiplet are included. However, if the Nilsson-Pope d = 11 oxidation of the Watamura N = 6 solution could be modified to obtain a supersymmetric CP3 compactification of Hořava-Witten theory, in the manner just discussed, there would be additional su- persymmetric Yang-Mills multiplets, together with the Kaluza-Klein multiplets above them, so there would be a possibility that the floor by floor vanishing of the one-loop vacuum energy might be restored. The question of whether or not there exists, in the bulk, a supersymmetric deforma- tion of the Nilsson-Pope d = 11 oxidation of the Watamura N = 6 compactification, whose nonvanishing components of GIJKL include components GABCD given by the LOSW ansatz (147), where α might now depend on y, could perhaps be investigated, in the first instance, by Witten’s method [127], in which the new components of GIJKL would be treated as a perturbation. 4.2 AdS5 ×CP3 compactification of d = 11 supergravity The value of the integration constant B, in (206), that is required for TeV-scale gravity, is given by (313), when the outer surface of the thick pipe is stabilized in the quantum region by Casimir effects, and by (399), when the outer surface is stabilized in the classical region by extra fluxes. From these equations, we see that the value of B required for TeV-scale gravity is reduced if the Euler number χ (M6) of the compact six-manifold, which is a negative integer for the compact six-manifolds considered in the present paper, is large in magnitude. However, |χ (M6)| is also related to b1 the relation (103), which follows from the value (102) of the Yang-Mills fine structure constant assumed at unification, which is the value of the QCD fine structure constant, α3, as evolved in the Standard Model to around 150 TeV. Here b1 = b (y1) is the value of b (y) at the inner surface of the thick pipe, where b (y) was introduced in the metric ansatz (94) as the scale factor that determines the diameter of the compact six-manifold, once its topology is fixed by selecting a specific smooth compact quotient of CH3 or H6. And b1 is determined by Casimir effects near the inner surface of the thick pipe, and thus, as discussed in subsection 2.4.3, cannot be small compared to 1, unless, for some reason, not only are the one-loop coefficients in the Casimir energy densities (132) and (136) small compared to 1, but also the multi-loop coefficients are all suppressed by the appropriate powers of the small number b1 , either to all loop orders, or at least up to some high loop order. Thus we cannot have a very large value of |χ (M6)|, and also obtain a reasonable value value of the Yang-Mills fine structure constant at unification, unless the coefficients in the Casimir energy densities (132) and (136), either to all loop orders, or at least to some high loop order, all tend to zero as the appropriate power of b1 , where b1 is given by (103), as |χ (M6)| becomes very large. However there is no reason for this to happen unless some special effect occurs, because the limit |χ (M6)| → ∞ does not correspond to any restoration of supersymmetry. A special effect of the required type was, however, discovered by Gibbons and Nicolai [340], who calculated the one-loop Casimir energy density of the Freund-Rubin compactification of d = 11 supergravity on the round seven-sphere [346], including the effects of all the Kaluza-Klein states, and found that not only did the Casimir energy density vanish, as required to preserve the supersymmetry of the solution at one loop, but also the contributions to the Casimir energy density vanished “floor by floor”, or in other words, at each separate Kaluza-Klein level or Osp(8|4) multiplet, which is not required to preserve the supersymmetry. This appears to suggest that the one-loop Casimir energy density of this compactification would still vanish “floor by floor” even if the Freund-Rubin relation between the AdS4 radius and the S 7 radius was broken, in which case the background would no longer satisfy the classical Cremmer-Julia-Scherk field equations, but the Casimir energy density would nevertheless still be defined by the general formula for the quantum effective action, Γ, as a function of arbitrary background fields, as described before (129). Thus the Gibbons-Nicolai result would seem to imply that the one-loop Casimir energy density of d = 11 supergravity, defined in this way, would vanish “floor by floor” even when the background is flat R4, times S7. And furthermore, since there will be relations between the propagators and heat kernels on a flat R4, times S7, background, and the propagators and heat kernels on a flatR4, timesH7, background, analogous to those discussed above for the flatR5, times CP3, and the flat R5, times CH3, backgrounds, the Gibbons-Nicolai result would seem to suggest that the one-loop Casimir energy density of d = 11 supergravity, defined by the quantum effective action, Γ, as a function of arbitrary background fields, will also vanish when the background is flat R4, times H7, for arbitrary radius of curvature of the H7. In that case, the one-loop Casimir energy density of d = 11 supergravity, on a flat R4, times M7, background, where M7 is a smooth compact quotient of H7, would presumably tend to zero, in the limit as the volume of the M7 at fixed Ricci scalar, which is a topological invariant by Mostow’s rigidity theorem even though the Euler number vanishes for a smooth compact manifold of odd dimension, tends to infinity. Thus it is appropriate to ask if there exist supersymmetric compactifications of d = 11 supergravity on AdS5 ×CP3 or AdS5 × S6, which might lead, by an analogue of the Gibbons-Nicolai effect, to the vanishing of the one-loop Casimir energy density of d = 11 supergravity, as defined by the quantum effective action, Γ, on a flat R5, times CH3, background, or a flat R5, times H6, background. To the best of my knowledge, there is no classical solution of the Cremmer-Julia-Scherk field equations on an AdS5×S6 background, that has a maximally symmetric metric on both factors, because there is no natural ansatz for the four-form field strength of the three-form gauge field. However, there is, indeed, a classical solution of the Cremmer-Julia-Scherk field equations on an AdS5 ×CP3 background, with the Lukas-Ovrut-Stelle-Waldram (LOSW) ansatz (147) for the four-form field strength of the three-form gauge field. I shall seek a solution with the metric ansatz (94), such that AdS5 is realized as flat four- dimensional Minkowski space times the y direction, with a (y) depending exponentially on y, as in the Randall-Sundrum model [31], and b (y) independent of y. Comparing the Ricci tensor components (97), the energy-momentum tensor components contributed by the three-form gauge field with the LOSW ansatz (159), the definition of the t(i) (y) energy-momentum tensor coefficients (130), and the Einstein equations (162), (163), and (164), we see that on replacing CH3 by CP3, so that the relation RAB (h) = 4hAB is replaced by RAB (h) = −4hAB, and replacing four-dimensional de Sitter space by four dimensional Minkowski space, so that the relation Rµν (g) = −3gµν is replaced by Rµν (g) = 0, and setting the t (i) (y) energy-momentum tensor coefficients to the values given by the LOSW ansatz, the Einstein equations become: = 0 (467) = 0 (468) = 0 (469) Requiring that ḃ = 0, b̈ = 0, the second of these equations reduces to α2 = 36b6 (470) The first and third equations then reduce to: (471) (472) which have the solutions a = Ae b and a = Ae− b . And from the formulae (96) for the Riemann tensor components, we see that Rµνστ = (GµσGντ −GνσGµτ ) , Rµyνy = Gµν (473) hence since Gµy = 0 and Gyy = 1, we have: Rµ̄ν̄σ̄τ̄ = (Gµ̄σ̄Gν̄τ̄ −Gν̄σ̄Gµ̄τ̄ ) (474) where the barred Greek indices run over four-dimensional Minkowski space and y. Thus the five-dimensional space formed from four-dimensional Minkowski space and the y direction is maximally symmetric, and in consequence of its (−++++) signature and the relation Rµ̄σ̄ = Gµ̄σ̄, is AdS5. We now need to determine whether this solution has any supersymmetries. There are no Majorana spinors in five dimensions, but a symplectic-Majorana condition can be imposed on a pair of spinors [362], in consequence of which the possible numbers of supersymmetries in five dimensions are even, and there do, indeed, exist supergrav- ities with 2, 4, 6, and 8 supersymmetries in five dimensions [362]. We know from the Figueroa-O’Farrill - Papadopoulos theorem [365] that the solution cannot have 8 super- symmetries. The SU(4) isometry group of CP3 with its standard metric implies there will be 15 Yang-Mills vector bosons in the adjoint of SU(4), and looking at the table of states of the extended supergravities in five dimensions given by Cremmer [362], we see that N = 6 supergravity in five dimensions has precisely 15 vector fields, which on toroidal compactification to four dimensions join the extra vector field coming from the metric, to produce the standard 15+ 1 = 16 vector fields of N = 6 supergravity in four dimensions. Furthermore, Nilsson and Pope [348] found that a known compact- ification [347] of Type IIA supergravity in ten dimensions on AdS4 × CP3 has either N = 6 supersymmetry or no supersymmetry, depending on the relative sign of form field fluxes on the AdS4 and CP 3 factors. However, notwithstanding these positive indications, the AdS5×CP3 compactification of d = 11 supergravity considered above has no supersymmetry. To check this, I shall use the notations of subsection 2.1 for supergravity in eleven dimensions, so coordinate indices I, J,K, . . . run over all directions on M11. The Dirac matrices ΓI satisfy ΓI ,ΓJ = 2GIJ , and ΓI1I2...In ≡ Γ[I1ΓI2 . . .Γ In]. Coordinate indices µ, ν, σ, . . . will now run over all directions on AdS5, which is a change from the meaning of the Greek indices used above and in section 2, and coordinate indices A,B,C will run over all directions on the compact six-manifold, which is in agreement with section 2, although the compact six-manifold is now CP3. Local Lorentz indices will be indicated by putting a bar over the corresponding coordinate indices, so the meaning of barred Greek indices is also now changed from their meaning in equation (474) above. A real representation of the Γ matrices for eleven dimensions does not decompose neatly into Dirac matrices for the five extended dimensions with signature (−++++) and Dirac matrices for the six compact dimensions with signature (+ + + +++), so I shall instead use a representation of the form used by Lukas, Ovrut, Stelle, and Waldram [68], with ΓI = 1 γµ × λ, 1× λA , where γµ and λA are the five- and six-dimensional Dirac matrices, respectively. Here, λ is the chiral projection matrix in six dimensions with λ2 = 1. For a specific representation of the λĀ we can choose λ5 = σ1 × 1 × 1, λ6 = σ2×1×1, λ7 = σ3×σ1×1, λ8 = σ3×σ2×1, λ9 = σ3×σ3×σ1, λ10 = σ3×σ3×σ2. We define λ = iλ5λ6λ7λ8λ9λ10 = σ3 × σ3 × σ3. For a specific representation of the γµ̄ we can choose γ1 = σ1 × 1, γ2 = σ2 × 1, γ3 = σ3 × σ1, γ4 = σ3 × σ2, γ0 = iσ3 × σ3. Then for the charge conjugation matrix C in eleven dimensions, which satisfies as usual C = −CΓI , CT = −C, we can take C = C5×C6, where C5 = σ1×σ2 is the charge conjugation matrix in five dimensions, and satisfies (γµ) C5 = C5γ µ, CT5 = −C5, in agreement with [362], and C6 = σ2 × σ1 × σ2 is the charge conjugation matrix in six dimensions, and satisfies C6 = −C6λA, λTC6 = −C6λ, CT6 = C6. Now the gravitino field is zero in the above classical solution, so if it has any super- symmetries, there must exist supersymmetry variation parameters η (x, z), where xµ are coordinates on AdS5, and z A are coordinates on CP3, such that the supersymme- try variation of the gravitino vanishes. The supersymmetry variation of the gravitino, about a configuration in which the gravitino field is zero, is [14, 2, 127]: δψI = DIη + (ΓIJKLM − 8GIJΓKLM)GJKLMη (475) To study the condition on η that results from setting this variation equal to zero, when GJKLM is given by the LOSW ansatz (147), I shall follow the method of Nilsson and Pope [348]. It is convenient, first, to note the identities: ΓIJKLMG JKLM = (ΓIΓJKLM − 4GIJΓKLM)GJKLM (476) 8GIJΓKLMG JKLM = [ΓI ,ΓJKLM ]G JKLM (477) Thus the supersymmetry variation of the gravitino, (475), can be written: δψI = DIη + (−ΓIΓJKLM + 3ΓJKLMΓI)GJKLMη (478) Now from the definition (69) of the Kähler form, we have: C = −hAC (479) Furthermore, for an arbitrary 2n × 2n antisymmetric matrix M , with real matrix elements, we have the identity: εi1i2i3i4...i2n−1jMi1i2Mi3i4 . . .Mi2n−1k = 2 n−1 (n− 1) ! detMδjk (480) This is proved by applying an orthogonal similarity transformation to transform M to a block diagonal matrix M̃ , such that each block in the block diagonal of M̃ is an antisymmetric 2× 2 matrix with real matrix elements, then replacing each index i by an index pair aI, where a runs from 1 to 2, and I runs from 1 to n, so that M̃ can be expressed as a Kronecker product M̃aI,bJ = εabM̄IJ , where εab = , and M̄ is an n× n diagonal matrix with real matrix elements. Applying this to the Kähler form, we have the identity: ωABωCDh ABCDEF = 8ωEF (481) Following Nilsson and Pope, it is convenient to define: Q ≡ −iωABλABλ (482) We note that [Q, λ] = 0. And from the definition of the ΓA in terms of the λA, as above, {λA, λB} = 2hAB. Thus from the identities λABλCD = λABCD−hACλBD+hADλBC+hBCλAD−hBDλAC−hAChBD+hADhBC (483) λABCD = hABCDEFh EGhFHλGHλ (484) and (481), we find that: Q2 = 4Q+ 12 (485) Hence the eigenvalues of Q are −2 and 6, hence since Q is traceless, there are six eigenvalues −2 and two eigenvalues 6. We now assume that η (x, z) factorizes in the form η (x, z) = ε (x) η̃ (z), where ε (x) is a four component spinor acted on by the first factor in the Kronecker product expressions for the ΓI , and η̃ (z) is an eight component spinor acted on by the second factor in the Kronecker product expressions for the ΓI . Substituting in the LOSW ansatz (147), and requiring that δψI = 0, we find from the components of (478) with I along CP3 that: DAη̃ + 3456b3 (−λAλBCDE + 3λBCDEλA)hBCDEFGωFGη̃ = 0 (486) Now from (482) and (484), we find: λBCDEh BCDEFGωFG = −24Q (487) Hence (486) reduces to: DAη̃ ± (λAQ− 3QλA) η̃ = 0 (488) where I also used (470), and the sign choice corresponds to α = ±6b3. And from (482), we also have: {Q, λA} = −4iω BA λBλ (489) Hence (488) is equivalent to: DAη̃ ± λAQ+ 3iω A λBλ η̃ = 0 (490) A necessary condition for the existence of solutions of (490) is the integrability condi- tion: λAQ+ 3iω A λCλ , DB ± λBQ+ 3iω B λDλ = 0 (491) To evaluate the left-hand side of (491), we note first that with the convention (5) for the Riemann tensor, we have [DA, DB] = −14RABCDΓ CD = −1 R̃ABCDλ CD, where RABCD is the Riemann curvature of CP3 with the metric GAB = b 2hAB, and R̃ABCD = RABCD is the Riemann curvature of CP3 with the metric hAB. And secondly, there are no cross terms between DA or DB, and the extra terms that came from the G JKLM term in (475), because the extra terms are built from the Kähler form and the vielbein, which are covariantly constant, and the Dirac matrices with local Lorentz indices, and λ, which are position-independent invariant tensors with respectively a vector index and two spinor indices, and two spinor indices, and thus also covariantly constant. To evaluate the commutator of the extra terms, we note that: [λAQ, λBQ] = −4iω DB λAλDλQ + 4iω DA λBλDλQ+ 2λBA (4Q+ 12) (492) [λAQ, λDλ] = −4iω CD λAλC + 2λDAλQ (493) where (485) was used to obtain (492). Thus we find: λAQ + 3iω A λCλ λBQ+ 3iω B λDλ ω DB λDA − ω DA λDB ωABλQ+ λBA (Q + 6)+ω B λCD (494) Terms of the same structure, but with different coefficients, occurred in Nilsson and Pope’s calculation of the corresponding commutator for the AdS4 × CP3 compactifi- cation of Type IIA supergravity in ten dimensions [347], and in that case, for one of two alternative choices of a relative sign, the result was that after adding the Riemann tensor term, each nonvanishing term had a factor of (Q+ 2) at its right-hand side, so that acting on any linear combination of the six linearly independent eigenvectors of Q with eigenvalue −2, the commutator vanished. That does not happen in the present case, so we have to check whether there is any further relation between the terms in the left-hand side of (491) that might result in (491) being satisfied when acting on an appropriate eigenvector of Q. It is convenient now to switch to complex coordinates, as in subsection 2.2, on page 25. Barred Latin indices will now denote antiholomorphic indices, as in subsection 2.2. Then corresponding to the Riemann tensor (72) for CHn, the Riemann tensor for CP3 R̃rs̄tū = hrs̄htū + hrūhts̄ (495) Evaluating the left-hand side of (491) for A = r, B = s, the Riemann tensor term does not contribute, and the result is: λsr (10λQ+ 4Q+ 33) (496) which is nonvanishing for any combination of λ = +1 or −1, and Q = −2 or +6, and thus proves the absence of supersymmetry. And similarly, for A = r̄, B = s̄, the left- hand side of (491) is 1 λs̄r̄ (−10λQ+ 4Q+ 33). And for A = r, B = s̄, the Riemann tensor term −1 R̃ABCDλ CD contributes 1 (iωrs̄Qλ + 2λrs̄), and the left-hand side of (491) is 1 (25iωrs̄Qλ + λs̄r (16Q+ 42)). In fact, if the numerical coefficients of the terms in the parentheses in (490) had had the values ±3 i, ∓3 i, instead of their actual values 1 and 3, the left-hand side of (494) would have been equal to 1 λAB (Q+ 2), and would thus have been consistent with N = 6 supersymmetry. About 18 months after version 1 of this article was published on arXiv, I learned from [366] that the AdS5 ×CP3 solution was studied by Pope and van Nieuwenhuizen in 1989, who showed that it is not supersymmetric [367]. The lack of supersymmetry could also have been deduced from a general study of supersymmetric AdS5 solutions of M-theory by Gauntlett, Martelli, Sparks, and Waldram [368]. 5 E8 vacuum gauge fields and the Standard Model In the present paper, we have considered the compactification of Hořava-Witten the- ory on a smooth compact quotient of either CH3 or H6, which breaks supersymmetry completely. The fact that the observed gauge coupling constants are ∼ 1 in magnitude implies that the six-volume of the inner surface of the thick pipe is ∼ κ 43 , as discussed in subsection 2.6.1, following (309), on page 128. Thus the energy at which supersym- metry is broken at the inner surface of the thick pipe will be ∼ κ− 29 . Thus if κ− 29 was large compared to the energy ∼ 174 GeV at which the electroweak symmetry is broken, we would have a hierarchy problem of the original type [369], without supersymmetry just above the electroweak breaking energy, to stabilize the parameters of the effective electroweak Higgs sector. Thus in models of the present type we would expect to find the simplest physical picture if κ− 9 is as close above the electroweak breaking energy as allowed by present experimental constraints, which in practice means TeV-scale gravity [3, 5]. In the present section I shall consider how the Standard Model [44] might be realized in the framework considered in the preceding sections, if κ− 9 is of order a TeV. No positive experimental evidence for the existence of large extra dimensions and TeV-scale gravity has yet been reported. However, in the approximation that the seven extra dimension are flat, the branching ratio for emitting a graviton, in any process, is ∼ κ2/9E , where E is the energy available to the graviton [3]. Thus if quantum gravitational effects are observed at the LHC, the effects will start very suddenly, as the energy of the beams is gradually increased, with no detectable effects at all up to a certain energy, and very large effects, with large amounts of missing energy, at slightly higher beam energies, as gravitons start radiating into the bulk of the thick pipe. This is in agreement with the general expectation that, although new physics is not yet observed at colliders, it cannot be far away [370]. The perturbative contributions of virtual graviton exchange to scattering amplitudes and cross sections, not yet observed, also increase very rapidly with increasing beam energies [11, 273], and once they become observable above the background, are expected to saturate rapidly at the nonperturbative rate for production of short-lived microscopic black holes, whose production cross section increases much more slowly with increasing energy, specifically κ2/9E [275, 371]. To estimate the current experimental limits on κ− 9 , I shall use the results of Mirabelli, Perelstein, and Peskin [273], who consider the case of flat extra dimensions. From the discussion around their equations (3) and (4), we see that their fundamental gravitational mass M is defined such that for seven flat extra dimensions, compactified to volume V7, Newton’s constant GN is given by π 16πGN . On the other hand, comparing (10) and (25), and remembering that for working in the “downstairs” picture, on the manifold with boundary, the coefficient 1 in (25) is to be replaced by , we see that V7 16πGN . Hence κ is related to Mirabelli, Perelstein, and Peskin’s M by 1 . Hence κ− 9 = π ≃ 0.2053M . The nearest case to the models of the present paper, for which they give results, is for six flat extra dimensions. Thus from the limits on M in their Table 1, we see that in 1998, the LEP 2 lower bound on 9 was around 107 GeV, and the Tevatron lower bound was around 125 GeV. And the final lower bound on κ− 9 attainable at the Tevatron is expected to be around 166 GeV, and the final lower bound on κ− 9 attainable at the LHC is expected to be around 677 GeV. The relations between Mirabelli, Perelstein, and Peskin’s M , and Mp, the Planck mass in D dimensions, as defined by Giddings and Thomas [275], and MD, the Planck mass in D dimensions, as defined by Giudice, Rattazzi, and Wells [11], for the case D = 11, and κ, are M =Mp = 2 9MD = 2π . (497) Considering, now, the massless vector bosons in the effective theory in four dimen- sions, we note that a smooth compact Einstein space of negative curvature cannot have any continuous symmetries. For a vector field V A that generates a continuous symmetry on a differentiable manifold M must satisfy the Killing vector equation DAVB + DBVA = 0. Hence 0 = D A (DAVB +DBVA). But from (5), on page 12, we have DADBVA = DBD AVA − RBDV D, and from the Killing vector equation, we have DAVA = 0. And if M is an Einstein space of negative curvature, then RBD = αgBD, where α > 0 is independent of position by the contracted Bianchi identity. Thus we find DADAVB = αgBDV D, hence V BDADAVB = αV BgBDV D. Thus if M is compact, we find on integrating by parts that: DAV B (DAVB) = −α ddxV BgBDV D (498) The left-hand side of this equation is ≥ 0, but for nonzero V A, the right-hand side is < 0, so there can be no such nonzero V A. Thus since there is certainly no contin- uous symmetry under translation in the radial direction of the thick pipe, the only massless vector bosons in four dimensions, in the models considered in this paper, are those which originate from the E8 Yang-Mills multiplets on the orbifold fixed-point hyperplanes. In standard compactifications of the weak coupling E8 heterotic superstring [97, 98], the E8 containing the Standard Model [43, 44] is first broken to E6 by embedding the spin connection in the gauge group [9, 72], and the E6 is then further broken by the Hosotani mechanism [49, 50, 51]. However, in the models considered in the present paper, the Standard Model is contained in the E8 on the inner surface of the thick pipe, whereas if the compact six-manifold, M6, is a smooth compact quotient of CH3, the spin connection is embedded in the E8 on the outer surface of the thick pipe, and if M6 is a smooth compact quotient of H6, the spin connection is not embedded in either of the two E8’s. The fundamental group of M6 necessarily has no torsion in the sense of group theory, or in other words, has no non-trivial finite subgroup, so if the vacuum contains Hosotani configurations of the Yang-Mills fields, or in other words, topologically non- trivial configurations of the Yang-Mills fields, with identically vanishing Yang-Mills field strengths, they might have to be stabilized dynamically, by radiative corrections, or partly dynamically and partly topologically, rather than purely topologically, as in [9]. The dynamical Hosotani fields in the Cartan subalgebra of E8, analogous to the Hosotani modes on a torus [49, 50, 51], would be proportional to harmonic 1-forms on M6, which are associated with the non-torsion part of the first homology group H1 (M6;Z), while Hosotani fields in the Cartan subalgebra of E8 that are associated with the torsion part of H1 (M6;Z) would be partly topologically stabilized, and might modify the potential for the dynamical Hosotani fields. I shall assume that the first stage of breaking the E8 on the inner surface of the thick pipe is by topologically non-trivial E8 vacuum gauge fields, localized on Hodge - de Rham harmonic two-forms of M6, whose field strengths are topologically stabilized in magnitude, and also partly in orientation within E8, by a form of Dirac quantization condition, studied in subsection 5.3. When these Hodge - de Rham “monopoles” are all in the Cartan subalgebra of E8, they break E8 either to SU (3)× (SU (2))3 × (U (1))3, or to SU (3) × (SU (2))2 × (U (1))4, or to SU (3) × SU (2) × (U (1))5, and the U (1)’s, other than U (1)Y , are also broken by a form of Higgs mechanism involving the CABy components of the three-form gauge field, that was discussed by Witten [45], and by Green, Schwarz, and West [372]. This arises, in the case of Hořava-Witten theory, from the redefinition of GyUVW to include a term δ (y − y1)ω(1)UVW , and an analogous term involving δ (y − y2), in order to solve the modified Bianchi identity (42). Here UVW is the Chern-Simons form constructed from the E8 gauge fields at y1: UVW =tr W −∂WA V , A + cyclic permutations of U, V,W (499) This redefinition of GyUVW corresponds to the redefinition of the three-form field strength of the two-form gauge field of N = 1 supergravity in ten dimensions, in the Bergshoeff-de Roo-de Wit-van Nieuwenhuizen [109] and Chapline-Manton [110] couplings of N = 1, d = 10 supergravity to Abelian gauge fields, and Yang-Mills fields, respectively. ω µAB contains a term 2tr A(1)µ F , and when F AB has a vacuum ex- pectation value in the Cartan subalgebra of E8, this leads, through the kinetic term GIJKLG IJKL of the three-form gauge field, to a mass term for the corresponding gauge field in the Cartan subalgebra. However, when GyUVW is redefined as above, the re- sulting ωUVWω UVW term in the action is formally infinite, being proportional to δ (0), so it would presumably be preferable to use Moss’s improved form of Hořava-Witten theory [69, 70, 71], mentioned shortly after (47), on page 22, in which the δ (0) terms are absent. It was noted by Witten, and by Green, Schwarz, and West, in the papers cited above, that if the gauge field of a U (1) subgroup of E8 develops a vacuum expec- tation value, but commutes with the gauge fields in the vacuum, it can be anomalous, so consistency would require any such field that is anomalous to be massive also in Moss’s form of the theory, so the δ (0) term would have to be replaced by a finite term, rather than zero. The Hodge - de Rham “monopoles” have non-vanishing Yang-Mills field strength, and thus contribute to the vacuum energy on the inner surface of the thick pipe. However, in the models considered in the present paper, the universe is stiffened by effects largely determined by the region near the outer surface of the thick pipe, and in particular, in the case studied in subsection 2.7, the universe is stiffened by the large value of the integration constant G̃, defined in (352). Thus the presence of the Hodge - de Rham monopoles, on the inner surface of the thick pipe, does not lead to a large value of the effective cosmological constant in four dimensions. When the Hodge - de Rham “monopoles” in the Cartan subalgebra break E8 directly to SU (3) × SU (2) × (U (1))5, there is no need for any Hodge - de Rham monopoles outside the Cartan subalgebra, but unification of the Yang-Mills coupling constants then depends entirely on the accelerated unification mechanism studied by Dienes, Dudas, and Gherghetta [128, 129], and by Arkani-Hamed, Cohen, and Georgi [373]. In this case, the Hodge - de Rham monopoles automatically satisfy the classical Yang- Mills field equations. When the Hodge - de Rham monopoles in the Cartan subalgebra break E8 to SU (3)×(SU (2))2×(U (1))4, the (SU (2))2 must then be broken to the diagonal subgroup SU (2)diag by monopoles outside the Cartan subalgebra, so that, at unification, the Yang-Mills coupling constant of SU (2)diag is smaller than the Yang-Mills coupling constant of SU (3), by a factor of 1√ , and the Yang-Mills coupling constants, as evolved in the Standard Model, approximately unify at around 150 TeV, so there is still a need for an accelerated unification effect, to achieve unification at around a TeV. The study of the Dirac quantization condition, in subsection 5.3, only covers the case where all the Hodge - de Rham monopoles are in the Cartan subalgebra, and I do not know whether it is possible, by topological means, to prevent the Hodge - de Rham monopoles outside the Cartan subalgebra, that break (SU (2)) to SU (2)diag, from “rotating”, or “sliding”, back into the Cartan subalgebra. In the study of this case, I shall assume, without proof, that this is possible. And finally, when the Hodge - de Rham monopoles in the Cartan subalgebra break E8 to SU (3)× (SU (2))3 × (U (1))3, the (SU (2))3 must also be broken to the diagonal subgroup SU (2)diag by monopoles outside the Cartan subalgebra, so that, at unifi- cation, the Yang-Mills coupling constant of SU (2)diag is smaller than the Yang-Mills coupling constant of SU (3), by a factor of 1√ , and the SU (3) and SU (2)diag coupling constants, as evolved in the Standard Model, now unify at around 413 GeV. However, it is not possible to do this without breaking SU (2)diag×U (1)Y , and at the same time, obtain an acceptable value of sin2 θW , which would have to be close to the value ≃ 0.23 observed at mZ , so this case appears to be excluded. It is not possible to stabilize the absolute orientation of the Cartan subalgebra within E8 topologically, and there will therefore, by Goldstone’s theorem [374], be 248 − 12 = 236 potentially massless Goldstone boson fields, corresponding to extra- dimensional Lorentz components of the Yang-Mills fields, proportional to generators of E8 outside the Standard Model SU (3))×SU (2)×U (1)Y , that can rotate different pos- sible choices of the Standard Model SU (3))× SU (2)×U (1)Y into one another. These modes, which are independent of position on M6, do not correspond to physical mass- less Lorentz scalar multiplets, but rather become the longitudinal degrees of freedom of the massive E8 gauge bosons outside the Standard Model SU (3))× SU (2)×U (1)Y [375, 376, 377, 378]. I shall assume that M6 has first Betti number B1 > 0. There are then B1 linearly independent harmonic 1-forms on M6, so that before the Dirac-quantized harmonic 2-form Hodge - de Rham monopoles in the E8 Cartan subalgebra are introduced, there are at tree level B1 physical massless Lorentz scalar multiplets in the E8 fundamen- tal/adjoint, one for each linearly independent harmonic 1-form. When E8 is broken by the Hodge - de Rham monopoles, some of the resulting scalar multiplets have the quantum numbers of the Standard Model Higgs field. The Hodge - de Rham monopoles can also produce a potential for some or all of the scalar multiplets at tree level, which can result in some of the scalars becoming tachyonic and developing vacuum expecta- tion values, so that the Standard Model SU (3) × SU (2) × U (1)Y is broken as in the ordinary Higgs effect [379, 380]. After the inclusion of radiative corrections, the potential is expected to depend on all the scalar multiplets originating from harmonic 1-forms on M6, including any that are not affected by the Hodge - de Rham monopoles, by the Coleman-Weinberg mech- anism [381, 382], or equivalently, the Hosotani mechanism [49, 50, 51]. I shall assume that this potential has a minimum in which a scalar multiplet with the quantum num- bers of the Standard Model Higgs field has a vacuum expectation value, which after integration over position on M6, produces masses for the Standard Model W± and Z bosons, equivalent to the masses produced by the Standard Model Higgs boson with a vacuum expectation value of 246 GeV ≃ 2× 174 GeV, and breaks the electroweak SU (2)×U (1)Y to U (1)e.m., as in the Standard Model. The original Coleman-Weinberg mechanism resulted in a Higgs mass that was much smaller than the current exper- imental lower bound of around 95 to 120 GeV, but more recent studies, taking into account the large Yukawa coupling of the top quark, have found consistent solutions, with a Higgs mass consistent with the current experimental constraints [383, 384, 385]. The vacuum expectation value of the scalar multiplet that serves as the Standard Model Higgs field is proportional to a harmonic 1-form on M6, and is thus expected to depend on position on M6. I shall assume that this enables the effective Yukawa couplings of this scalar multiplet, identified as the Standard Model Higgs field, to different pairs of chiral fermion zero modes to have different values, so as to realize the fermion mass hierarchy, and the CKM [386, 387] and PMNS [388, 389] mixing matrices, by a version of the Arkani-Hamed - Schmaltz mechanism [390]. I shall also assume that all the other scalar multiplets that originate from harmonic 1-forms on M6 are sufficiently massive at the minimum of the potential to be consistent with experimental limits, even though they do not develop vacuum expectation values. The Hodge - de Rham monopoles are required to satisfy Witten’s topological con- straint [45], that was discussed in subsection 2.3.7. But since the vacuum field con- figuration already satisfies this constraint in the absence of the Hodge - de Rham monopoles, this means that the configuration of the Hodge - de Rham monopoles is required to satisfy the requirement that for each closed four-dimensional submanifold Q of the compact six-manifold M6, the integral Q tr (F ∧ F ) is equal to zero. For a given configuration F , of the E8 gauge fields on the inner surface of the thick pipe, this integral only depends on the cohomology class of Q, and thus gives B4 constraints, where B4 is the fourth Betti number of M6. But by Poincaré duality, B4 = B2, where B2 is the second Betti number of M6. Hence there is one constraint per harmonic two-form. However, the embedding of each harmonic two-form, in the Cartan subal- gebra of E8, is determined by eight independent numbers, which, as I will show in subsection 5.3, are constrained only to lie on a certain lattice in the Cartan subalgebra of E8. Thus it seems likely that there will be non-trivial solutions of Witten’s topo- logical constraint, even when the Hodge - de Rham monopoles are required to leave SU (3)×(SU (2))n×(U (1))6−n unbroken, for the required value 3, 2, or 1, of n, and also to be perpendicular to U (1)Y , so that the U (1)Y does not become massive by Witten’s Higgs mechanism. However, when Witten’s topological constraint is imposed in addi- tion to these requirements, there only remain 4− n degrees of freedom per monopole, for the embedding in the E8 Cartan subalgebra, so the greatest flexibility is obtained for n = 1. The Hodge - de Rham monopoles result in the existence of chiral fermion zero modes, for chiral fermions in various irreducible representations of the subgroup of E8 left unbroken by the monopoles and Witten’s Higgs mechanism involving the three- form gauge field, and the number of chiral fermion zero modes, in each such irreducible representation, is determined by the Atiyah-Singer index theorem [391]. Many of these irreducible representations have the quantum numbers of a fermion representation in the Standard Model, subject to the necessary accelerated unification of the Yang-Mills coupling constants. And, as shown by Witten [45], and Green, Schwarz, and West [372], Witten’s topological constraint ensures that there will be no gauge anomalies involving only the gauge bosons left massless by the Hodge - de Rham monopoles and Witten’s Higgs mechanism. Green, Schwarz, and West also state that the anomalies involving the U (1) gauge bosons that commute with the vacuum Yang-Mills fields, but become massive by Witten’s Higgs mechanism, due to having nonvanishing vacuum expectation values themselves, are harmless. For all the breakings of E8 considered in the present paper, there exists a U (1) gauge boson Bµ that becomes massive by Witten’s Higgs mechanism, and one or more irreducible representations with the quantum numbers of each left-handed fermion representation in the Standard Model, such that the coupling of Bµ to each of those fermion representations is a fixed multiple of the baryon number of that fermion repre- sentation in the Standard Model. Sums of triangle diagrams with one or more external Bµ’s are expected to be anomalous, but as explained by Witten [45], this does not matter, due to the fact that Bµ has become massive by the Higgs mechanism involving the CABy components of the three-form gauge field. Thus there might be a possibility of stabilizing the proton in a manner similar to the Aranda-Carone mechanism [52], although Aranda and Carone required the massive gauge boson, whose couplings to the observed fermions are proportional to baryon number, to be non-anomalous. In the case where the Hodge - de Rham monopoles in the Cartan subalgebra break E8 directly to SU (3)×SU (2)× (U (1))5, and there are no Hodge - de Rham monopoles outside the Cartan subalgebra, realizing the Standard Model requires: 1. finding a linear combination of the U (1)’s to serve as U (1)Y , such that there exist SU (3)×SU (2) irreducible representations in the E8 fundamental, with the correct SU (3)× SU (2) quantum numbers and U (1)Y charges to be identified as the left-handed fermions of one or more generations, and the Higgs boson of the Standard Model; 2. finding another linear combination of the U (1)’s to serve as U (1)B, such that for each of the five types of SU (3)× SU (2) multiplet with non-vanishing U (1)Y charge in the Standard Model, and also for the left-handed antineutrino, if these are required, there exists at least one SU (3)× SU (2) irreducible representation in the E8 fundamental, with those SU (3)× SU (2) quantum numbers and U (1)Y charge, such that the U (1)B charge of that irreducible representation is a fixed multiple of the baryon number of the corresponding fermion; and 3. finding, for each of the B2 linearly independent Hodge - de Rham harmonic two forms of M6, where B2 is the second Betti number of M6, a point perpendicular to U (1)Y , in the eight-dimensional lattice of points in the E8 Cartan subalgebra that is allowed by the Dirac quantization condition, such that: (a) Witten’s topological constraint is satisfied, for all B4 = B2 linearly indepen- dent harmonic four-forms ofM6, or equivalently, for a set of B4 topologically non-trivial closed four-dimensional surfaces in M6, linearly independent in the sense of homology; and (b) for each of the five or six types of SU (3)× SU (2) left-handed fermion mul- tiplet in the Standard Model, depending on whether or not left-handed antineutrinos are required: i. every occurrence of that multiplet in the E8 fundamental, that has the correct U (1)Y charge, and U (1)B charge equal to the correct multiple of baryon number, has a net number of chiral fermion zero modes, as given by the Atiyah-Singer index theorem, ≥ 0; and ii. the sum, over all occurrences of that multiplet in the E8 fundamental, that have the correct U (1)Y charge, and U (1)B charge equal to the correct multiple of baryon number, of the net number of chiral fermion zero modes, as given by the Atiyah-Singer index theorem, is equal to 3; iii. every occurrence of that multiplet in the E8 fundamental, that either has the wrong U (1)Y charge, or has U (1)B charge equal to the wrong multiple of baryon number, has a net number of chiral fermion zero modes, as given by the Atiyah-Singer index theorem, equal to 0; (c) for each SU (3) × SU (2) multiplet in the E8 fundamental, that does not correspond to a fermion multiplet in the Standard Model, or the complex conjugate of a fermion multiplet in the Standard Model, the net number of chiral fermion zero modes, as given by the Atiyah-Singer index theorem, is equal to 0; and (d) if there are sufficiently many left-handed antineutrinos, a Majorana mass matrix, with one or more very light eigenstates by a generalized seesaw mechanism, as discussed in subsection 5.7 below, is generated for them by the Hodge - de Rham monopoles; and (e) a potential is generated for all the “Higgs” bosons, by the Coleman-Weinberg mechanism, that has a minimum at which all the “Higgs” bosons are mas- sive, and the electrically neutral component of a “Higgs” boson, with the quantum numbers of the Standard Model Higgs boson, has a vacuum expec- tation value, possibly dependent on position on M6, whose value, averaged over position on M6, produces masses for the Standard Model W± and Z bosons, equivalent to the masses produced by the Standard Model Higgs boson, with a vacuum expectation value of 246 GeV; and (f) the mass matrices with entries given by the overlap integrals of pairs of chiral fermion zero modes, with the vacuum expectation of the “Higgs” boson, which may depend on position on M6, produce the observed mass spectra of the quarks and the electrically charged leptons, and the CKM mixing matrix of the quarks, by a version of the Arkani-Hamed - Schmalz mechanism; and (g) the masses of the Standard Model neutrinos, and the PMNS mixing matrix of the Standard Model leptons, arise in some way. In the present paper, I will present some solutions to the requirements 1. and 2. above, both for the case when the Hodge - de Rhammonopoles in the Cartan subalgebra of E8 break E8 directly to SU (3)×SU (2)× (U (1))5, and for the case when they break E8 directly to SU (3)×(SU (2))2×(U (1))4. In the solutions where the Hodge - de Rham monopoles in the Cartan subalgebra break E8 directly to SU (3)× (SU (2))2× (U (1))4, there exist components of the E8 fundamental, outside the Cartan subalgebra, that could break (SU (2)) to SU (2)diag, without breaking SU (3) × SU (2)diag × U (1)Y , if they could be given topologically stabilized vacuum expectation values, as Hodge - de Rham monopoles, but, as mentioned above, I do not know whether or not there is any topological obstruction to prevent the orientation in E8, of such Hodge - de Rham monopoles, from “rotating”, or “sliding”, back into the Cartan subalgebra. The necessary first step for studying the requirements 3. (a) - (g) is to find explicit examples of smooth compact quotients of CH3 or H6 that are spin manifolds. This is unavoidable, because Witten’s topological constraint depends on the cohomology cup product of the manifold [392, 393], that expresses the wedge product of pairs of har- monic two-forms as linear combinations of harmonic four-forms, and this cohomology cup product is a topological invariant of the manifold. I shall now consider the lightest massive modes of the supergravity multiplet, in the following subsection 5.1. The SU (9) basis for E8 is studied in subsection 5.2, on page 235. The Dirac quantization condition on the field strengths of Hodge - de Rham harmonic two-forms, in the Cartan subalgebra of E8, is studied in subsection 5.3, on page 241. I show that there are no models with an acceptable value of sin2 θW , such that the Hodge - de Rham monopoles, in the Cartan subalgebra of E8, break E8 to SU (3)×(SU (2))3×(U (1))3, in subsection 5.4, on page 254. Models where the Hodge - de Rham monopoles, in the Cartan subalgebra of E8, break E8 to SU (3)× (SU (2))2× (U (1)) , are studied in subsection 5.5, on page 263, and models where they break E8 to SU (3)× SU (2)× (U (1))5, are studied in subsection 5.6, on page 270. 5.1 The lightest massive modes of the supergravity multiplet From the point of view of the effective theory in four dimensions, supersymmetry is broken explicitly in the models considered in the present paper, even though, from the point of view of Hořava-Witten theory in eleven and ten dimensions, the supersymmetry is broken spontaneously, by the compactification. Thus the gravitinos, four of which are allowed, by the Hořava-Witten boundary conditions, to couple directly to the matter on the inner surface of the thick pipe, and the associated spin-1 fermions, and also the vectors and scalars which correspond, in four dimensions, to the three-form gauge field, couple to ordinary matter with at most gravitational strength, and there is no enhancement of the coupling of the gravitino to ordinary matter, as can happen in models where N = 1 supersymmetry is broken spontaneously in four dimensions, through the absorption of the goldstino by the gravitino [394, 395]. To study the Kaluza-Klein modes of the supergravity multiplet we have to expand the quantum effective action to quadratic order in small fluctuations about the relevant background solution, which is here one of the solutions found in subsections 2.5, 2.6, and 2.7. For a first estimate I shall instead consider a massless scalar field Φ in the bulk, which is intended to represent a small fluctuation of a component of any of the supergravity fields, and retain only its classical action. Dropping also RΦ and H2Φ terms, the equation for the small fluctuation Φ is then: −GGIJ∂JΦ = 0 (500) Trying an ansatz Φ xµ, xA, y = ϕ (xµ)ψ xA, y , where coordinate indices µ, ν, σ, . . . are tangent to the four observed space-time dimensions, and coordinate indices A,B, C, . . . are tangent to M6, as in subsection 2.3, we find from (500) that: (501) b2ψ (xC , y) hhAB∂Bψ xC , y a4b6ψ (xC , y) a4b6∂yψ xC , y ϕ (xσ) −ggµν∂νϕ (xσ) The left-hand side of (501) is independent of xµ and the right-hand side is independent of xA and y, hence each side must be a constant. The left-hand side is a positive operator on a compact manifold so must be a non-negative constant m2 ≥ 0. From the metric ansatz 94, on page 33, the metric Gµν at the inner surface of the thick pipe, where we live, in the models considered here, is Gµν = A dSgµν , where AdS is the observed de Sitter radius 22, since by definition the de Sitter radius of gµν is 1. Thus in terms of the metric Gµν at the inner surface of the thick pipe, the wave equation along the 4 extended dimensions, for a Kaluza-Klein mode ψ xC , y for which each side of 501 is equal to m2, is: −GGµν∂νϕ (xσ) ϕ (xσ) = 0. (502) For the solution found in subsection 2.5, starting on page 111, a and b are roughly constant ∼ B over the main part of the classical region around y ∼ B, so there are modes spread in this region for which −∂2yψ ∼ n ψ, so that m ∼ n, for all integers n > 0. Thus there are very light Kaluza-Klein modes of the bulk whose mass, as seen at the inner surface of the thick pipe, is ∼ n , for all integers n > 0. ψ xC , y suppressed in the region of the inner surface of the thick pipe for these modes, so the situation is qualitatively similar to the situation considered by Randall and Sundrum in [396], where the modifications to Einstein gravity in the 4 extended dimensions, on the brane we live on, from modes of this form, were found to be negligibly small. However the model considered in [396] did not include the ADD effect, so further study would be needed to determine whether these very light Kaluza-Klein modes, localized in the classical region of the bulk, prevent the solution found in subsection 2.5 from being consistent with the precision Solar System tests of Einstein gravity [278, 279], and with the sub-millimetre tests of Newton’s law [32]. For the solution found in subsection 2.6, starting on page 120, there are modes in the second quantum region, adjacent to the outer surface of the thick pipe, that oscillate sufficiently rapidly as y increases, that a4b6 is approximately constant over ∼ 10 or more cycles, and wavepackets localized in this region can be formed from these modes. For such a wavepacket localized at a ≃ acent and independent of position on M6, the left-hand side of (501) is approximately −a ∂2yψ (y) ≃ m2, so a representative mode is ψ (y) = cos my acent times a wavepacket profile. In this region a decreases exponentially with increasing y , with a coefficient ∼ 1 in the exponent, and b is a constant times aτ̃ , where τ̃ is a constant of magnitude ∼ 1. Thus the requirement that a4b6 changes over one wavelength by at most a factor close to 1 is that (4+6τ̃)2πacent κ2/9m ≪ 1. For example m ∼ 103 would be adequate, for acent roughly at the outer boundary and hence ∼ κ2/9. Thus from (502), there are very light Kaluza-Klein modes of the bulk whose mass, as seen at the inner surface of the thick pipe, is ∼ 103 n , for all integers n > 0. ψ xC , y is again suppressed in the region of the inner surface of the thick pipe for these modes, and further study would be needed to determine whether these modes prevent the solution found in subsection 2.6 from being consistent with the precision Solar System tests of Einstein gravity [278, 279], and with the sub-millimetre tests of Newton’s law [32]. For the solution found in subsection 2.7, starting on page 137, where the outer surface is stabilized in the classical region by fluxes, a and b are roughly constant, with a ∼ 1022 metres, from (404), on page 155, and b ∼ B, over the main part of the classical region around y ∼ B, so there are modes spread in this region for which −∂2yψ ∼ n so that from (501), m ∼ n × 1022 metres, for all integers n > 0. Thus from (502), and (22), on page 15, the mass of these modes, as seen from the inner surface of the thick pipe, is ∼ 10−4 n , which from (399), on page 154, is ∼ n 10−8 metres ∼ 10n eV. The wavefunctions of these modes are again suppressed in the region of the inner surface of the thick pipe. 5.2 An SU (9) basis for E8 Throughout this section, I shall use an SU(9) basis for E8, as in [8]. On breaking E8 to SU(9), the 248 of E8 splits to the 80, 84, and 84 of SU(9). Here the 80 is the adjoint of SU(9), the 84 has three totally antisymmetrized SU(9) fundamental subscripts, and the 84 has three totally antisymmetrized SU(9) antifundamental subscripts. The fundamental representation generators (tα)ij̄ of SU (9) are normalized to satisfy [44] tr (tαtβ) = (503) The generators of the required representations are as follows: Antifundamental (Tα)īj = − (tα)jī (504) Adjoint (Tα)ij̄,k̄m = (tα)ik̄ δmj̄ − δik̄ (tα)mj̄ (505) 84 (Tα)ijk,m̄p̄q̄ = ((tα)im̄ δjp̄δkq̄ ± seventeen terms) (506) 84 (Tα)̄ij̄k̄,mpq = − (tα)mī δpj̄δqk̄ ± seventeen terms (507) where the additional terms in (506) and (507) antisymmetrize with respect to permu- tations of (i, j, k), and with respect to permutations of (m, p, q). We can check directly that these generators satisfy the same commutation relations as (tα)ij̄ , with the same structure constants. It is convenient to define: δijk,r̄s̄t̄ ≡ (δir̄δjs̄δkt̄ + δis̄δjt̄δkr̄ + δit̄δjr̄δks̄ − δir̄δjt̄δks̄ − δis̄δjr̄δkt̄ − δit̄δjs̄δkr̄) (508) which is the unit matrix in the space of matrices whose rows and columns are labelled by antisymmetrized triples of indices, and projects expressions with three indices to their antisymmetric part. Then we have: (Tα)ijk,m̄p̄q̄ = δijk,r̄s̄t̄ (tα)rm̄ δsp̄δtq̄ + δrm̄ (tα)sp̄ δtq̄ + δrm̄δsp̄ (tα)tq̄ (tα)ir̄ δjs̄δkt̄ + δir̄ (tα)js̄ δkt̄ + δir̄δjs̄ (tα)kt̄ δrst,m̄p̄q̄ (509) (Tα)̄ij̄k̄,mpq = −δrst,̄ij̄k̄ (tα)mr̄ δps̄δqt̄ + δmr̄ (tα)ps̄ δqt̄ + δmr̄δps̄ (tα)qt̄ (tα)rī δsj̄δtk̄ + δrī (tα)sj̄ δtk̄ + δrīδsj̄ (tα)tk̄ δmpq,r̄s̄t̄ (510) We define the totally antisymmetric SU (9) structure constants fαβγ by [tα, tβ ] = ifαβγtγ , noting, from (503), that the SU (9) generators tα, in the SU (9) fundamen- tal representation, have been chosen to be hermitian. The generators of E8 are now the 80 generators Tα of SU (9), together with 84 generators Trst, antisymmetric in rst, whose label, rst, runs over the 84 of SU (9), and 84 generators Tr̄s̄t̄, antisymmetric in r̄s̄t̄, whose label, r̄s̄t̄, runs over the 84 of SU (9). Indices A,B, C, . . . will run over all 248 generators of E8, as in the discussion of the SO (16) basis, in subsection 2.1. The E8 structure constants will be written FABC , and defined such that [TA, TB] = iFABCTC , and in a similar way to the discussion of the SO (16) basis in subsection 2.1, I shall use a summation convention such that each index in a multi-index, such as rst, gets summed over its full range, without restrictions, and no compensating factor, such as , is included. Thus when C, in iFABCTC, refers to r̄s̄t̄ on FABC , and to rst on TC , the contribution is iFAB,r̄s̄t̄Trst, with the normal summation convention, so that each of the distinct generators Trst, 1 ≤ r < s < t ≤ 9, actually gets counted 6 times in the sum. This is analogous to the convention used in the discussion of the SO (16) basis in subsection 2.1, where the definition (34), of the orthogonal group structure constants, means that the orthogonal group commutation relation (29) takes the form [Jij , Jkl] = fij,kl,rsJrs, so that each of the distinct generators Jij , 1 ≤ i < j ≤ 16, actually gets counted twice in the sum. The structure constants FABC are totally antisymmetric under permutations of A,B, C, and the non-vanishing matrix elements of the E8 generators are: (Tα)βγ = ifβαγ = iFβαγ (511) (Tα)ijk,m̄p̄q̄ = − (Tα)m̄p̄q̄,ijk = − (Tijk)α,m̄p̄q̄ = (Tijk)m̄p̄q̄,α = = (Tm̄p̄q̄)α,ijk = − (Tm̄p̄q̄)ijk,α = iFijk,α,m̄p̄q̄ (512) (Trst)ijk,mpq = ǫrstijkmpq = iFijk,rst,mpq (513) (Tr̄s̄t̄)̄ij̄k̄,m̄p̄q̄ = ǫr̄s̄t̄̄ij̄k̄m̄p̄q̄ = iFīj̄k̄,r̄s̄t̄,m̄p̄q̄ (514) The matrix representations of the generators are not antisymmetric in this basis, even though the structure constants are totally antisymmetric, because it is necessary to take the three types of index group in a different order for rows and columns, to ensure that SU (9) anti-fundamental indices contract with SU (9) fundamental indices. The matrix representations of the generators can be written as: m̄p̄q̄ γ mpq īj̄k̄ iFijk,α,m̄p̄q̄ 0 0 0 iFβαγ 0 0 0 iFīj̄k̄,α,mpq (515) Trst = m̄p̄q̄ γ mpq īj̄k̄ 0 0 iFijk,rst,mpq iFβ,rst,m̄p̄q̄ 0 0 0 iFīj̄k̄,rst,γ 0 (516) Tr̄s̄t̄ = m̄p̄q̄ γ mpq īj̄k̄ 0 iFijk,r̄s̄t̄,γ 0 0 0 iFβ,r̄s̄t̄,mpq iFīj̄k̄,r̄s̄t̄,m̄p̄q̄ 0 0 (517) To check the Jacobi identities, we first note that from the SU (9) commutation relation for (Tα)ijk,m̄p̄q̄, we have: Frst,α,m̄p̄q̄Fmpq,β,̄ij̄k̄ + Fīj̄k̄,α,mpqFm̄p̄q̄,rst,β + FβαγFγ,̄ij̄k̄,rst = 0 (518) We next note that: Fijk,α,ūv̄w̄Fuvw,rst,mpq + Fmpq,α,ūv̄w̄Fuvw,ijk,rst + Frst,α,ūv̄w̄Fuvw,mpq,ijk = = − i (tα)iū ǫrstujkmpq + (tα)jū ǫrstukimpq + (tα)kū ǫrstuijmpq + (tα)mū ǫrstijkupq + (tα)pū ǫrstijkuqm + (tα)qū ǫrstijkump + (tα)rū ǫustijkmpq + (tα)sū ǫutrijkmpq + (tα)tū ǫursijkmpq (519) The right-hand side is totally antisymmetric in rstijkmpq, for each value of α, and is thus equal to an α-dependent multiple of ǫrstijkmpq. The α-dependent coefficient of ǫrstijkmpq is found by contracting with ǫr̄s̄t̄̄ij̄k̄m̄p̄q̄, which gives zero, due to the traceless- ness of tα. Similarly, we find: Fīj̄k̄,α,uvwFūv̄w̄,r̄s̄t̄,m̄p̄q̄ + Fm̄p̄q̄,α,uvwFūv̄w̄,̄ij̄k̄,r̄s̄t̄ + Fr̄s̄t̄,α,uvwFūv̄w̄,m̄p̄q̄,̄ij̄k̄ = 0 (520) We next note that, due to the tracelessness and the normalization (503) of the SU (9) generators, we have: (tα)rī (tα)sj̄ = δrj̄δs̄i − δrīδsj̄ (521) We now consider the expression: Fuvw,̄ij̄k̄,αFα,m̄p̄q̄,rst + Frst,̄ij̄k̄,αFα,uvw,m̄p̄q̄ = (tα)uī δvj̄δwk̄ ± seventeen terms ((tα)rm̄ δsp̄δtq̄ ± seventeen terms) (tα)rī δsj̄δtk̄ ± seventeen terms ((tα)um̄ δvp̄δwq̄ ± seventeen terms) (522) I will show that this is equal to: δrīδsj̄δtk̄δum̄δvp̄δwq̄ ± 719 terms ǭij̄k̄m̄p̄q̄x̄ȳz̄ǫrstuvwxyz = −Fm̄p̄q̄,̄ij̄k̄,x̄ȳz̄Fxyz,rst,uvw (523) where the additional terms in the first line of (523) antisymmetrize with respect to permutations of i,j̄, k̄, m̄, p,q̄ . We first note that the terms in the first line of (523) can be classified by the number n of elements of {u, v, w} that are joined by Kronecker deltas to elements of {m̄, p,q̄}. We see that when we use (521) in the right-hand side of (522), terms with n = 0 can only come from the second term in the right-hand side of (521), used in the first term in the right-hand side of (522), terms with n = 1 can only come from the first term in the right-hand side of (521), used in the first term in the right-hand side of (522), terms with n = 2 can only come from the first term in the right-hand side of (521), used in the second term in the right-hand side of (522), and terms with n = 3 can only come from the second term in the right-hand side of (521), used in the second term in the right-hand side of (522). Thus the first term in the first line of (523), which has n = 3, comes from the second term in the right-hand side of (521), used in the second term in the right-hand side of (522). Considering, now, the coefficient of the first term in the first line of (523), we see that it gets contributions from three terms in the first factor of the second term in the right-hand side of (522), namely (tα)rī δsj̄δtk̄ + δrī (tα)sj̄ δtk̄ + δrīδsj̄ (tα)tk̄, times three terms in the second factor of the second term in the right-hand side of (522), namely (tα)um̄ δvp̄δwq̄+δum̄ (tα)vp̄ δwq̄+δum̄δvp̄ (tα)wq̄. Hence its coefficient is − , as required. Considering, next, the coefficient of a term with n = 2, namely δrm̄δsj̄δtk̄δuīδvp̄δwq̄, in the first line of (523), we see that the locations of the tα’s are now fixed, and this term only gets a contribution from the first term in the first factor of the second term in the right-hand side of (522), times the first term in the second factor of the second term in the right-hand side of (522). Hence its coefficient is − 1 = − 1 , as required. And in a similar manner, we confirm the coefficient of a term with n = 1, namely δum̄δvj̄δwk̄δrīδsp̄δtq̄, as , and the coefficient of a term with n = 0, namely δuīδvj̄δwk̄δrm̄δsp̄δtq̄, as − 172 . The coefficients of the remaining 716 terms in the first line of (523), of which 62− 1 have n = 3, (34 × 22)− 1 have n = 2, (34 × 22)− 1 have n = 1 and 62 − 1 have n = 0, are then determined by the separate antisymmetries of the left-hand side of (522) in (u, v, w), (r, s, t), i,j̄, k̄ , and (m̄, p,q̄). And furthermore, all 182 + 182 terms in the right-hand side of (522) have now been accounted for. Thus we find the final Jacobi identity: Fuvw,̄ij̄k̄,αFα,m̄p̄q̄,rst + Frst,̄ij̄k̄,αFα,uvw,m̄p̄q̄ + Fm̄p̄q̄,̄ij̄k̄,x̄ȳz̄Fxyz,rst,uvw = 0 (524) We next calculate Tr (TATB), where we recall, from just after (28), on page 19, that we are using Hořava and Witten’s notation for traces in E8, so that for E8, “tr” denotes of the trace in the adjoint representation, which is denoted by “Tr”. We also recall, from above, our summation convention, that each index in a multi-index, such as rst, gets summed over its full range, without restrictions, and no compensating factor, such , is included. We first note that, from (503), and the definition, [tα, tβ ] = ifαβγtγ , the SU (9) structure constants fαβγ are given by fαβγ = −2itr ([tα, tβ] tγ). Hence we find: fδαγfδβγ = −2tr ([tδ, tα] , [tδ, tβ]) = 9δαβ (525) We next note that: (Tα)ijk,m̄p̄q̄ (Tβ)mpq,̄ij̄k̄ = = 3 (tα)rx̄ δsȳδtz̄ (tβ)xū δyv̄δzw̄ + δxū (tβ)yv̄ δzw̄ + δxūδyv̄ (tβ)zw̄ δuvw,r̄s̄t̄ = δαβ (526) Thus: Tr (TαTβ) = (Tα)γδ (Tβ)δγ + (Tα)ijk,m̄p̄q̄ (Tβ)mpq,̄ij̄k̄ + (Tα)īj̄k̄,mpq (Tβ)m̄p̄q̄,ijk = = 9δαβ + δαβ + δαβ = 30δαβ (527) We next note that: − Fijk,rst,mpqFm̄p̄q̄,ūv̄w̄,̄ij̄k̄ = − ǫrstijkmpqǫūv̄w̄m̄p̄q̄īj̄k̄ = 10δrst,ūv̄w̄ (528) And from (521), we find that: −Fα,rst,m̄p̄q̄Fmpq,ūv̄w̄,α = 3δrst,f̄ ḡh̄ (tα)fx̄ δgȳδhz̄ (Tα)xyz,ūv̄w̄ = − 1− 1 − 1− 1 δrst,ūv̄w̄ = 10δrst,ūv̄w̄ (529) Thus: Tr (TrstTūv̄w̄) = −Fijk,rst,mpqFm̄p̄q̄,ūv̄w̄,̄ij̄k̄ − Fα,rst,m̄p̄q̄Fmpq,ūv̄w̄,α − Fīj̄k̄,rst,βFβ,ūv̄w̄,ijk = = (10 + 10 + 10) δrst,ūv̄w̄ = 30δrst,ūv̄w̄ (530) And from the block matrix structure of the generators (515), (516), and (517), we see Tr (TαTrst) = Tr (TαTr̄s̄t̄) = Tr (TrstTuvw) = Tr (Tr̄s̄t̄Tūv̄w̄) = 0 (531) We note that we can choose a set of generators for the SU (9) Cartan subalgebra, such that in the SU (9) fundamental, the generators of the Cartan subalgebra are diag- onal matrices, and their nonzero matrix elements are equal to integers, times an overall normalization factor that depends on the generator, and that there is an infinite variety of such choices of the generators of the SU (9) Cartan subalgebra, consistent with (503). And from (505), (506), (507), and (515), we see that for any such set of generators of the SU (9) Cartan subalgebra, each generator of the E8 Cartan subalgebra, in the E8 fundamental / adjoint, will be a 248 × 248 diagonal matrix, whose nonzero matrix elements are equal to integers, times an overall normalization factor that depends on the generator. The occurrence of the 84 and 84 of SU (9), in the fundamental of E8, is connected to the presence of the three-form gauge field in d = 11 supergravity [38, 14], through the embedding of SO (9), the little group of the d = 11 Poincare group, in E8, by the subgroup chain SO (9) ⊂ SU (9) ⊂ E8. For, as reviewed in subsection 2.2 of [397], half of the 32 supercharges of d = 11 supergravity vanish on the mass shell, and the representation space of the 16 nonvanishing supercharges decomposes into the two chiral spinor representations of SO (16), one of which corresponds to the bosonic states, and the other to the fermionic states. The 16 nonvanishing spinor charges transform according to a single spinor representation of the little group, SO (9), and the helicity content of the bosonic and fermion states is determined by the branching of the two different 128’s of SO (16), when SO (9) is embedded into SO (16) such that the spinor of SO (9) becomes the vector of SO (16). This results in one of the 128’s of SO(16) branching into the 44+84 of SO (9), corresponding to the graviton and the three-form gauge field, while the other 128 of SO (16) becomes the 128 vector-spinor of SO (9), corresponding to the gravitino, as can be checked by studying weight diagrams. On the other hand, the adjoint of SO (16) branches into the antisymmetrized square of the spinor of SO (9), which contains the 36 of SO (9), which is the adjoint, and the 84 of SO (9), which is the three-form. And under the embedding SO (9) ⊂ SU (9), the adjoint of SU (9) branches to the adjoint and the 44 of SO (9), and the 84 and 84 of SU (9) both become the 84 of SO (9). Thus the decomposition of the adjoint of E8 into irreducible representations of SO (9) is the same, when SO (9) is embedded into E8 according to the subgroup chains SO (9) ⊂ SU (9) ⊂ E8 and SO (9) ⊂ SO (16) ⊂ E8, provided SO (9) is embedded into SO (16) in the manner that determines the helicity content of the d = 11 supergravity states on the mass shell, and the spinor of SO (16), in the fundamental of E8, is the one which branches to the 44+ 84 of SO (9). 5.3 Dirac quantization condition for E8 vacuum gauge fields In this subsection, I will show that the field strengths of the Hodge - de Rham monopoles are restricted in their possible magnitudes, and partly also in their pos- sible orientations within E8, by a form of Dirac quantization condition. In particular, if the configuration of the Yang-Mills fields is gauge equivalent to a configuration where they lie everywhere within the Cartan subalgebra of E8, then for an arbitrary closed smooth orientable two-dimensional surface S in the compact six-manifoldM6, the inte- gral of the field strengths, over S, in a gauge where the Yang-Mills fields lie everywhere within the Cartan subalgebra of E8, is a 248 × 248 diagonal matrix, that must be a lattice point of a certain discrete lattice in the eight dimensional Cartan subalgebra of We recall that for an arbitrary Yang-Mills gauge group, and for matter fields ψ transforming in an arbitrary representation of the gauge group, with hermitian gener- ators Tα satisfying [Tα, Tβ] = ifαβγTγ , with totally antisymmetric structure constants fαβγ , the covariant derivative is Dµψ = (∂µ − igAµαTα)ψ, where Aµα are the Yang- Mills fields and g is the coupling constant, and the Wilson line, or gauge covariant path ordered phase factor, for a continuous path x (s), smin ≤ s ≤ smax, differentiable except at a finite number of values of s, is: W ({A},{x (s)})ij̄ = (−ig)n ds1 . . . dsnθ (s1 − smin) θ (s2 − s1) . . . θ (sn − sn−1)× ×θ (smax − sn) dxµ1 (s1) . . . dxµn (sn) Aµ1α1 (x (s1)) . . . Aµnαn (x (sn)) (Tα1 . . . Tαn)ij̄ , (532) where θ (s) is the step function, θ (s) = 1 for s ≥ 0, and θ (s) = 0 for s < 0. For paths x1 (s), smin ≤ s ≤ smid, and x2 (s), smid ≤ s ≤ smax, such that x1 (smid) = x2 (smid), W ({A} , {x (s)})ij̄ satisfies the product formula: W ({A} , {x1 (s)})ik̄W ({A} , {x2 (s)})kj̄ =W ({A} , {x1 (s)} ∪ {x2 (s)})ij̄ (533) where {x1 (s)} ∪ {x2 (s)} denotes the union of the two paths, which is a map with domain smin ≤ s ≤ smax. We now consider the transformations of Aµα, ψ, and W ({A} , {x (s)}), under fi- nite gauge transformations, that might be topologically non-trivial, and might not be connected to the identity. I shall assume that the gauge transformation parameters Λα (x) are continuous and differentiable on each coordinate patch, and that the gauge transformation acts on ψ by ψ (x) → U (x)ψ (x), where U (x) = eiΛ(x) = eiΛα(x)Tα . Then the gauge-transformed Yang-Mills fields A′µα are required to satisfy Dµ (A ′)Uψ = UDµ (A)ψ. Thus we require: A′µαTα = AµαUTαU (∂µU)U † (534) Using the Baker-Campbell-Hausdorff formula [398] eABe−A =B+[A,B]+ 1 [A, [A,B]]+ [A, [A, [A,B]]] + . . ., and also, for expanding eiΛ(x+δx)e−iΛ(x) to first order in δx, the relation es(A+B)e−sA = 1 + tABe−tAdt + O (B2), we find that (534) is satisfied for an arbitrary representation with generators Tα, if: A′µα = Aµβ e−iΛ̆(x) (∂µΛβ (x)) e−iΛ̆(x) − 1 −iΛ̆ (x) (535) where the matrix Λ̆βγ (x) is defined in terms of the generators = −ifαβγ of the adjoint representation, by Λ̆βγ (x) = Λα (x) . The Wilson line (532) then transforms as: W ({A},{x (s)})ij̄ →W ({A ′},{x (s)})ij̄ = Uik̄ (x (smin))W ({A},{x (s)})kl̄ U (x (smax)) (536) Now on a topologically non-trivial manifold, such as the compact six-manifolds M6 considered in the present paper, the Yang-Mills fields can also be topologically non- trivial. This means that Aµα (x) is not well-defined globally as a continuous and differ- entiable function of the coordinates, which are themselves not defined globally. Instead Aµα (x) is a continuous and differentiable function of the coordinates on each coordinate patch, and where two patches i and j overlap, A(i)µα is related to A by both a general coordinate transformation, and a finite gauge transformation. This is the case, for example, when the Yang-Mills fields are in the Cartan subalgebra of the gauge group, and their field strengths are nonzero and proportional to Hodge - de Rham harmonic two-forms. The simplest example of this is a two-sphere centred on a Dirac magnetic monopole [140] in the Wu-Yang gauge [399, 400]. The vector potential is tangential to the two- sphere, and is well defined on two coordinate patches, one of which covers the northern hemisphere, and a strip of the southern hemisphere along the equator, and the other of which covers the southern hemisphere, and a strip of the northern hemisphere along the equator. More generally, there will be three or more coordinate patches, and at any point where three coordinate patches i, j, and k overlap, the gauge transformations U i→j , U j→k, and Uk→i are required to satisfy U i→jU j→kUk→i = 1. When a Wilson line crosses from a patch i to a patch j, we choose a point x on the line in the overlap region between the two patches, at which to make the transition from patch i to patch j, and the Wilson line is then defined to be the matrix product of the segment of the line in patch i, and the segment of the line in patch j, as in (533), but with the gauge transformation matrix U i→j (x) inserted between the two segments. If we consider two different choices of the point x on the line in the overlap region, at which to make the transition between the two patches, we find, from the gauge transformation (536) of the segment of the Wilson line between the two different choices of the transition point, that the Wilson line is independent of the choice of the transition point. Now if a Wilson line doubles back on itself like a hairpin, and exactly retraces its path back to its starting point, then it is identically equal to the unit matrix, even if the hairpin path crosses between several coordinate patches. And this is also true for a hairpin path that has “branches”, that are themselves hairpins. Furthermore, by the result just noted, this is also true if, for a segment of the hairpin path that lies in the overlap region of two coordinate patches, we make the transition between the two coordinate patches, at different points on the “outward” and “return” sections of the hairpin path. Let us now consider a configuration of the Yang-Mills fields that is gauge equivalent to a configuration where the gauge fields are everywhere in the Cartan subalgebra of the Lie algebra, and choose a gauge where the gauge fields are everywhere in the Cartan subalgebra. Let us also assume that the manifold has non-vanishing second Betti number, and that the gauge field configuration is topologically non-trivial, due for example to including Hodge - de Rham harmonic two forms. We now consider an arbitrary closed orientable two-dimensional surface in the man- ifold, that is embedded in the manifold in a topologically non-trivial manner, in the sense that it cannot be contracted to a point. Such surfaces exist due to the assumption that the manifold has non-vanishing second Betti number. We do not know what the intrinsic topology of the surface is, but it was shown by Seifert and Threlfall that the most general closed orientable two-dimensional manifold is topologically equivalent to a sphere with n handles, n ≥ 0. I shall consider a particular Wilson line that has the form of a branched hairpin, and is thus equal to the identity matrix. However, the hairpin branches will loop round and meet at their tips, in such a way that, due to the assumption that the field configuration is Abelian, we can also express the Wilson line as a diagonal matrix, such that each matrix element on the diagonal has the form F , where F denotes the integral over the closed two-dimensional surface, of the two-form field strength of the corresponding diagonal matrix element of AµαTα. This will be non-zero, if the field configuration includes a Hodge - de Rham harmonic two- form, with non-zero coefficient in that matrix element, that has non-zero integral over that surface. Thus g F must be an integer multiple of 2π. Considering, first, the case when the intrinsic topology of the two-dimensional sur- face is an ordinary two-sphere, the intersections of the coordinate patches of the mani- fold will define coordinate patches on the two-dimensional surface. Let us suppose, first, that the coordinate patches on the two-dimensional surface are topologically equivalent to the northern hemisphere, plus a strip of the southern hemisphere, and the southern hemisphere, plus a strip of the northern hemisphere, as in the case of the Wu-Yang gauge for the Dirac monopole. Then we choose a simple hairpin that starts at a point on the equator, and wraps once round the equator, so that the point where the hairpin doubles back on itself is the same as the point where it started. We choose the hairpin to start on the northern hemisphere patch, and remain on the northern hemisphere patch all the way around the equator to the point where it doubles back on itself, and it makes the transition to the southern hemisphere patch at the point where it doubles back on itself, and it remains on the southern hemisphere patch for the entire “return” section of the hairpin, until it reaches the starting point, where it finally makes the transition back to the northern hemisphere patch again. Then due to the Abelian na- ture of the gauge field, each of the two transitions from one patch to the other simply introduces a phase factor, and the two phase factors cancel one another because the two transitions occurred at the same point. Furthermore, for an Abelian field configuration, whose only non-vanishing matrix elements are on the diagonal, each non-vanishing ma- trix element of the Wilson line has the form exp dxµ(s) Aµ (x (s)) , where Aµ denotes the corresponding diagonal matrix element of AµαTα. We then uses Stokes’s theorem to equate the line integral in the exponent, for the “outward” section of the hairpin path, to the integral of F over the northern hemisphere, and the line integral in the exponent, for the “return” section of the hairpin path, to the integral of F over the southern hemisphere. And if the coordinate patches on the two-dimensional surface, topologically equiva- lent to a two-sphere, are not topologically equivalent to the northern hemisphere, plus a strip of the southern hemisphere, and the southern hemisphere, plus a strip of the northern hemisphere, we can introduce two new coordinate patches in the manifold, whose intersections with the two-dimensional surface do have this form, and choose suitable gauges on these two coordinate patches, such that we can use the intersections of these two coordinate patches with the two-dimensional surface, as the coordinate patches on the two-dimensional surface, and then use the argument as above. Considering, now, the case where the intrinsic topology of the two-dimensional surface is a sphere with n handles, n ≥ 1, it will be sufficient to show that we can always find a suitable branched hairpin, that divides the surface into suitable sectors, so that we can use the same arguments as above. We note, first, that we can always cut a sphere with n handles, n ≥ 1, in such as way as to transform it into a polygon with 4n sides, such that opposite sides are identified. Figure 3 (i) shows a way of doing this for n = 3, that extends directly to all n ≥ 1. In this diagram, paired circles AA, BB, and CC are identified by reflection in the vertical midline of the diagram, to form ✭✭✭❳❳❳ ✟✟✟✘✘✘✘ ❅❤❤❤❤✭✭✭✭✟ ❤❤❤❤❤ ✭✭✭❤❤❤ ❳❳❳❳❍❍❍❍ ✑✑✭✭✭✭✭✭❛❛ S S′Q ✏✏✏✏✏✏✏PP ✟✟✟✟✟✟✟✟ ✻ ❄ ❏ ✟✟✟✙✟ PPPPPPP ✏✏✏✏✏✏✏ ▲▲ ✁✁ ✏✏✏✏✮ ❈❈❈❈❲ ❅❅❘❅❅❅ PPPPPq ✏✏✶✏✏ ✔✔ ❆❆✔✔ ✔✔ ✔✔ ❍❍❍❍❍❍❍ �� ✦✦ ❜❜ ✦✦ ✟✟✟✟✟✟✟✟ Figure 3: (i) A sphere with three handles, cut so as to transform it into a twelve-sided polygon with opposite sides identified. (ii) A multi-hairpin Wilson line for a sphere with three handles. Opposite sides of the polygon, for example P and P′, are identified. handles, and the remaining lines are the cuts. Figure 3 (ii) shows a branched hairpin dividing the sphere with three handles into twelve triangular regions, which we can assume correspond to the main parts of the coordinate patches on the two-dimensional surface in this case. We make the transitions between the coordinate patches, such that the three sections of the Wilson line directly surrounding each triangle, are on the coordinate patch corresponding to that triangle. The individual branches of the hairpin all branch out of the Wilson line at a single point, which is the central point of Figure 3 (i), and corresponds to all twelve vertices of the polygon in Figure 3 (ii). Six of the 6 + 12− 1 = 17 hairpins that branch out of this point loop round and meet this point again at their tips. These are the hairpins PP ′, QQ′, RR′, SS ′, TT ′, and UU ′, along the edges of the polygon in Figure 3 (ii). The Wilson line starts and ends at a different point, corresponding to the centre of the polygon in Figure 3 (ii), which could be any other point of the sphere with three handles shown in Figure 3 (i), and the remaining 12− 1 = 11 hairpins, which are the hairpins running from vertices b to l of the polygon in Figure 3 (ii), to the centre of that polygon, also loop round to meet that point at their tips. These hairpins reach that point in Figure 3 (i), by passing along the handles, as necessary. For example, if the Wilson line starts and ends at a point somewhere in the external region of Figure 3 (i), the hairpin that runs from vertex b of the polygon in Figure 3 (ii), to the centre of that polygon, reaches that point from b in Figure 3 (i), by first passing along handle A, then along handle B, and finally along handle C. If we label a hairpin that runs from a vertex of the polygon in Figure 3 (ii) to the centre of that polygon, by the letter of the corresponding vertex, then after the initial section from the centre of the polygon to vertex a, the Wilson line runs along the hairpins in the sequence PP ′, h, c, RR′, j, e, TT ′, l, g, b, QQ′, i, d, SS ′, k, f, UU ′, then finally along the final section from vertex a back to the centre of the polygon. We see that each transition, from one coordinate patch to another, that occurs across a side of the polygon in Figure 3 (ii), is matched by a reverse transition through the same point, so that all the phase factors associated with these transitions cancel out. While for the transitions at the centre of the polygon in Figure 3 (ii), we see that, since the Wilson line must end with a transition back to the coordinate patch it started on, we have transitions corresponding to diagonal matrices U1→2, U2→3, . . . , U12→1, all at the same point, where the patches are labelled 1 to 12 anticlockwise around the polygon, and the product of all these is equal to 1. Furthermore, each of the twelve triangular regions is circled anticlockwise by the Wilson line sections around its edge, which are the sections of the Wilson line on the coordinate patch corresponding to that triangle, so we can use Stokes’s theorem for each triangle. Considering, now, how this works for general n ≥ 1, we draw the corresponding 4n-sided polygon with an opposite pair of its vertices pointing east and west. We draw a T , consisting of the initial and final sections of the Wilson line, and two half hairpins, with the centre of its top at the easternmost vertex, as in Figure 3 (ii). And for each of the remaining 2n−1 sides of the upper half of the polygon, we draw an L, consisting of one and a half hairpins, with the foot of the L pointing anticlockwise as in Figure 3 (ii). And for each of the remaining 2n−1 sides of the lower half of the polygon, we draw an L, consisting of one and a half hairpins, with the foot of the L pointing clockwise, as in Figure 3 (ii). And finally we draw an I, consisting of a single hairpin, with its foot at the westernmost vertex, as in Figure 3 (ii). We draw an arrow pointing anticlockwise on every Wilson line section running along an edge of the polygon, as in Figure 3 (ii), and add arrows to the Wilson line sections directly joined to these sections, consistent with these arrows, so that every triangular section is circled anticlockwise by the three Wilson line sections around its edge. The cancellation of the phase factors associated with the transitions between co- ordinate patches, and the use of Stokes’s theorem, will now work exactly as for the n = 3 case, so it remains to check that, starting at the start of the Wilson line, we pass along each Wilson line section exactly once, and in the correct direction. To check this, we number the Wilson line sections running along the perimeter of the top half of the polygon 0, 1, . . . , (2n− 1) in sequence anticlockwise, starting at the easternmost section, which is half the top of the T , and labelled P in Figure 3 (ii). And we number the Wilson line sections running along the perimeter of the lower half of the poly- gon 0′, 1′, . . . , (2n− 1)′ in sequence anticlockwise, starting at the westernmost section, which is labelled P ′ in Figure 3 (ii). Thus the L’s in the top half of the polygon are numbered 1, 2, . . . (2n− 1), and the L’s in the lower half of the polygon are numbered 0′, 1′, . . . , (2n− 2)′. We observe that, due to the directions of the arrows on the Wilson line sections, each pair of opposite L’s of the form mm′, 1 ≤ m ≤ (2n− 2), is traversed in the sequence: first m, then m′. Furthermore, the upper half of the top of the T , labelled P in Figure 3 (ii), and 0 in the general numbering scheme, is traversed immediately after the initial section of the Wilson line, and immediately before the L labelled 0′, which is labelled P ′ in Figure 3 (ii), and the lower half of the top of the T , labelled U ′ in Figure 3 (ii), and (2n− 1)′ in the general numbering scheme, is traversed immediately after the L labelled (2n− 1), which is labelled U in Figure 3 (ii), and immediately before the final section of the Wilson line. Furthermore, the hairpin based at the westernmost vertex of the polygon, labelled g in Figure 3 (ii), is traversed immediately after the L labelled (2n− 2)′, which is labelled T ′ in Figure 3 (ii), and immediately before the L labelled 1, which is labelled Q in Figure 3 (ii). And finally, for 0 ≤ m ≤ (2n− 3), L number m′, in the lower half of the polygon, is immediately followed by L number (m+ 2), in the upper half of the polygon. Thus the 4n Wilson line sections running along the perimeter of the polygon, and the Wilson line sections directly connected to them in the diagram, and the hairpin based at the westernmost vertex of the polygon, which together comprise the 4n + 1 pieces of Wilson line that are directly connected in the diagram, are traversed in the sequence: 0, 0′, 2, 2′, 4, 4′, . . . , (2n− 2) , (2n− 2)′, then the hairpin based at the westernmost vertex of the polygon, then 1, 1′, 3, 3′, 5, 5′, . . . , (2n− 1) , (2n− 1)′. If there is just one coordinate patch, as is natural when a compact hyperbolic manifold is specified by giving a Dirichlet domain for it in uncompactified hyperbolic space, together with the face-pairing maps for the Dirichlet domain, a simpler tree of hairpins can be obtained from the one shown in Figure 3 (ii), by moving the start and end point to just inside the 12-sided polygon at a, and shrinking the eleven hairpins that meet at the centre of the polygon, back to the perimeter of the polygon, so that all that remains are the hairpin halves around the perimeter of the polygon, which are traversed in the sequence PP ′RR′TT ′QQ′SS ′UU ′. Thus we have shown that if the configuration of the Yang-Mills fields lies entirely within the Cartan subalgebra of the gauge group, then for an arbitrary representation of the gauge group, with generators Tα, such that matter fields exist that transform under that representation of the gauge group, and for each matrix element on the leading diagonal of that representation of the gauge group, and for an arbitrary closed orientable two-dimensional surface embedded smoothly in the manifold, the integral F must be an integer multiple of 2π, where F denotes the integral over the closed two-dimensional surface, of the two-form field strength of the corresponding diagonal matrix element of AµαTα. And we noted that this integral will be non-zero, if the field configuration includes a Hodge - de Rham harmonic two-form, with non-zero coefficient in that matrix element, that has non-zero integral over that surface. Now for the fundamental / adjoint representation of E8, in the SU (9) basis used in this section, each of the eight generators of the Cartan subalgebra of E8, which are the eight generators of the Cartan subalgebra of SU (9), in a reducible representation of SU (9) that comprises the 80, 84, and 84 of SU (9), is such that its nonzero ma- trix elements are integer multiples of an overall normalization factor, specific to that generator. Let us now consider Aµα such that α denotes a fixed one of the eight gen- erators of the Cartan subalgebra of E8. Let B2 denote the second Betti number of the manifold, which by assumption is > 0. Then there are B2 linearly independent Hodge - de Rham harmonic two-forms, and there are also just B2 non-contractible closed two-dimensional surfaces in the manifold, that are linearly independent in the sense of homology. Thus we can choose a basis of B2 non-contractible closed two-dimensional surfaces in the manifold, such that for an arbitrary closed two-form F , or in other words, for an arbitrary two-form F that satisfies the Bianchi identity dF = 0, or in components, ∂[µFνσ] = 0, and an arbitrary closed two-dimensional surface in the man- ifold, the integral F , of F over the surface, is equal to a linear combination of the corresponding integrals for the B2 surfaces in the basis. Thus if we consider one particular matrix element in the diagonal of Tµα, for the particular α in the Cartan subalgebra we are considering, and restrict Aµα to be a linear combination of the one-form vector potentials of the B2 harmonic two-forms, so that the Yang-Mills field equations will automatically be satisfied, for this Abelian field configuration, there are just B2 linearly independent quantization conditions, to be satisfied by B2 independent coefficients. And if we now extend the consideration to all 248 matrix elements on the leading diagonal of Tµα, but still for the fixed value of α in the Cartan subalgebra, we see that, because the ratios of the matrix elements are fixed rational numbers, there will be a finite integer p, such that if Aµα satisfies the quantization condition for one particular matrix element on the diagonal, such that that matrix element of Tµα is nonzero, then pAµα will satisfy the quantization condition for all the nonzero matrix elements of the diagonal matrix Tµα. Thus, still considering Aµα for just one fixed value of α in the Cartan subalgebra, the quantization condition can be satisfied by an infinite number of non-trivial field configurations, and for field configurations that satisfy the Yang-Mills field equations, and are thus a linear combination of the B2 Hodge - de Rham harmonic two-forms, the allowed values of the coefficients of the B2 Hodge - de Rham harmonic two-forms will lie on a discrete B2-dimensional lattice, because B2 linearly independent linear combinations of the coefficients have quantized values, so that after a suitable change of basis, each coefficient would be quantized independently. And when we choose such a basis for the B2 Hodge - de Rham harmonic two-forms, so that we can consider each Hodge - de Rham harmonic two-form in the basis independently, the allowed values of the Aµα, associated with any one Hodge - de Rham harmonic two form, will be integer multiples of a basic “monopole”. Let us now choose such a basis for the B2 Hodge - de Rham harmonic two-forms, and consider one Hodge - de Rham harmonic two-form in the basis, so that the allowed values of Aµα will be integer multiples of a basic “monopole”. We now allow Aµα to be nonzero for all the eight values of α in the Cartan subalgebra. Then the solutions of the quantization condition will include, in particular, a discrete eight-dimensional lattice, in the Cartan subalgebra of E8, whose lattice points correspond to Yang-Mills fields of the form of the sum over the Cartan subalgebra of qαAµαTα, where the eight qα are the integers that define the lattice point, and Aµα (x) is the Hodge - de Rham harmonic two- form under consideration, times a normalization factor, dependent on α, that makes it into the correspnonding basic “monopole”, for the element α of the Cartan subalgebra. There may now be additional solutions of the quantization conditions, such that some or all of the qα are non-integer rational numbers, but the number of such additional solutions, in each unit cell of the lattice defined by integer qα, will be finite, since 248 linear combinations of the eight qα, not necessarily all distinct, are required to satisfy quantization conditions, which are, however, mutually consistent, and among these 248 linear combinations, there are eight that are linearly independent. Furthermore, given a point in the eight-dimensional space of the qα, that satisfies all the quantization conditions, and such that not all eight of the qα are integers, other non-integer solutions of the quantization conditions can be obtained by adding arbitrary integers to the qα. Thus the solutions of the quantization conditions form an infinite discrete lattice in the space of the qα, which is, however, not necessarily hypercubic. Thus, for each separate Hodge - de Rham harmonic two-form, in a basis in which we can apply the quantization conditions to each separate Hodge - de Rham harmonic two- form independently, we can have Abelian configurations of the E8 Yang-Mills fields, that solve the classical Yang-Mills field equations, and, within the Cartan subalgebra, are topologically stabilized, and whose field strengths have the spatial dependence of the Hodge - de Rham harmonic two form, and an embedding within E8, that lies on any lattice point of an infinite eight-dimensional lattice in the eight-dimensional Cartan subalgebra of E8. Thus, provided the different lattice points are not connected to one another by orbits within E8 that go outside the Cartan subalgebra, we can break E8 to a wide variety of subgroups, in a topologically stabilized manner, by introducing such Hodge - de Rham harmonic two-forms in the vacuum, embedded in E8 on suitable lattice points of this infinite eight-dimensional lattice in the eight-dimensional Cartan subalgebra of E8. Furthermore, for breakings to smaller subgroups of E8, such as the subgroups SU (3)×(SU (2))3×(U (1))3, SU (3)×(SU (2))2×(U (1))4, and SU (3)×SU (2)×(U (1))5, considered in this paper, there is a multi-dimensional space of embeddings in the Car- tan subalgebra of E8, that achieve the required breaking. Thus we can choose a differ- ent embedding, consistent with the required breaking, for each independent Hodge - de Rham harmonic two-form, subject to the requirement of satisfying Witten’s topological constraint [45], as discussed in subsection 2.3.7, and thus seek to satisfy the conditions 3. (a) to (g), in the list above. Specifically, for breaking to SU (3)×(SU (2))3×(U (1))3, the space of embeddings that achieve the required breaking is three-dimensional, while for breaking to SU (3)× (SU (2))2× (U (1))4, it is four-dimensional, and for breaking to SU (3)×SU (2)×(U (1))5, it is five-dimensional. However, in each case, we also need to ensure that the embeddings of all the monopoles are perpendicular to U (1)Y , so that U (1)Y is not broken by Witten’s Higgs mechanism involving the components CABy of the three-form gauge field [45], which reduces the dimensionalities of the spaces of available embeddings to two, three, and four, respectively. And if we want to make the unwanted U (1)’s massive by Witten’s Higgs mechanism, rather than by monopoles outside the Cartan subalgebra, we also have to ensure that the embeddings of at least some of the monopoles are not perpendicular to the unwanted U (1)’s. We can ensure that we really do get the expected multi-dimensional lattice of em- beddings within the E8 Cartan subalgebra, consistent with the required breaking, by choosing a basis for the Cartan subalgebra such that a certain subset of the gen- erators automatically preserve the required subgroup. For example, the subgroup SU (3) × (SU (2))3 × (U (1))3 is preserved by an arbitrary element of the Cartan sub- algebra of E8, which in the basis used in the present section, is also the Cartan sub- algebra of SU (9), whose diagonal matrix elements, in the SU (9) fundamental, are (σ1, σ1, σ1, σ2, σ2, σ3, σ3, σ4, σ4), with 3σ1 + 2σ2 + 2σ3 + 2σ4 = 0. Of course, for cer- tain values of the σi, a larger subgroup is preserved. For example, (σ1, σ2, σ3, σ4) = (2,−1,−1,−1) preserves E7, (0, 0, 1,−1) preserves SO (14), (4, 4,−5,−5) preserves SU (5) × SU (4), (2, 2, 2,−7) preserves SU (7) × SU (2), (2, 2,−1,−4) preserves E6 × SU (2), (4,−5, 1,−2) preserves SO (10) × SU (3), and (0, 1,−2, 1) preserves SU (3) × SO (10). However, for most of the points of the lattice, which in this case is three- dimensional, the required breaking is obtained. This partial topological stabilization no longer applies for continuous variations of the gauge field configuration that are allowed to go outside the Cartan subalgebra. For a generic path that has the same start and end point as the tree of hairpins, and is homotopic to the tree of hairpins, the logarithm F ≡ lnW , of the Wilson phase factor W for the path, can generically be defined by continuity under continuous variations of the path and of the gauge field configuration. For an assumed small variation δ of lnW , we have: W ′ = ei(F+δ) = eiF + dseiF siδeiF (1−s) + . . . = = eiF + i eiF̆ − 1 iF + . . . , (537) where F̆βγ = Fα(T̆α)βγ = −iFαfαβγ is in the adjoint representation, and the Baker- Campbell-Hausdorff formula has been used as in the derivation of (535). Thus δ could fail to be determined by W ′ if F̆ has one or more eigenvalues equal to non-zero multiples of 2π, and this must inevitably happen for variations of a path that transform it into the tree of hairpins, for we can transform the tree of hairpins continuously to the trivial path by continuously retracting the hairpins. Thus lnW is undefined for the tree of hairpins, for general variations of the gauge field configuration that go outside the Cartan subalgebra. The above discussion has involved only the components of the gauge field tangential to a particular closed smooth orientable two-dimensional surface S embedded in M6, and for this restricted system, the question of the existence of any possible absolute topological stabilization of a non-trivial configuration of the gauge field reduces to the corresponding question for S. From subsection 4.1 of [401], if the gauge group G is connected and simply connected, then a G bundle over a two-dimensional surface is trivial. From [402], the compact Lie group E8 is simply connected and appears also to be connected. Furthermore, it has trivial centre, so it is not a covering group of any other connected Lie group [403]. Thus the Dirac-quantized harmonic 2-formmonopoles considered in this subsection are apparently not absolutely stabilized topologically, although they might be separated by potential energy barriers from other solutions of the classical Yang-Mills equations, including pure gauge configurations. 5.4 Nonexistence of models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× (SU (2))3 × (U (1))3 In models of TeV-scale gravity based on Hořava-Witten theory, such as those considered in the present paper, the gauge couplings have to unify at around a TeV, rather than at 1016 GeV. One way this could work is if the running of the coupling constants somehow accelerates, so that the couplings run to their conventional unification values at the TeV scale, rather than at 1016 GeV. This possibility was studied by Dienes, Dudas, and Gherghetta [128, 129], and by Arkani-Hamed, Cohen, and Georgi [373]. An alternative possibility, considered in [8], is to embed SU(3)× SU(2)×U(1) into E8 in an unusual way, so that the values of the coupling constants, at unification, are equal to their observed values, as evolved conventionally to the TeV scale. Usually the coupling constant of a simple non-Abelian subgroup of a Grand Unification group, at unification, is equal to the coupling constant of the Grand Unification group, irrespec- tive of how the subgroup is embedded in the Grand Unification group. An exception occurs [6, 373] if the initial breaking of the Grand Unification group produces N copies of the simple subgroup, and the N copies of the simple subgroup then break into their “diagonal” subgroup. In this case, after the second stage of the breaking, the coupling constant of the “diagonal” subgroup is equal to 1√ times the coupling constant of the Grand Unification group. Effectively, the gauge field, in each of the N copies of the simple non-Abelian subgroup, becomes equal to 1√ times the “diagonal” gauge field, plus massive vector terms that can be ignored at low energies. The sum of the N copies of the Yang-Mills action, of the simple non-Abelian subgroup, then becomes equal to the Yang-Mills action of the “diagonal” subgroup, whose coupling constant is times the coupling constant of the Grand Unification group. Looking at the observed values of the reciprocals of the SU(3)× SU(2)×U(1) fine structure constants, at MZ , normalized so as to meet at unification in SU(5) Grand Unification, [404], (Mohapatra [405], page 22): α−13 (MZ) = 8.47± .22 α−12 (MZ) = 29.61± .05 α−11 (MZ) = 58.97± .05 (538) we see that they are quite close to being in the ratios 1, 3, 6. If we evolve them in the Standard Model, [44], then α−13 and α 2 reach an exact ratio of 1, 3, at 413 GeV, at which point α−13 is equal to 10.12. At this point, α First generation LH states Multiplet Y SU(3)× SU(2) content SU(5) coefficient required coefficient uR uG uB dR dG dB (3,2) 1√ ūR ūG ūB (3̄, 1) −4√ d̄R d̄G d̄B (3̄, 1) 2√  −1 (1,2) −3√ 2 (1,1) 6√ 0 (1,1) absent 0 Table 6: Weak hypercharge, SU(3)× SU(2) assignments, coefficient of the coupling to the U(1)Y vector boson in SU(5), and the required coefficient of the coupling to the U(1)Y vector boson, for the left-handed fermions of the first generation. equal to 58.00, which is 4% off being 6 times α−13 , and sin 2 θW ≃ 0.239. Thus it is natural to consider the breaking of E8 to SU(3) × (SU(2))3 × U(1)Y , followed by the breaking of (SU(2))3 to SU(2)diag, and seek an embedding of U(1)Y that gives the correct hypercharges at unification. I have summarized the required left-handed fermions of the first generation, together with their hypercharges, Y , [44], the coefficients of their U(1)Y couplings in SU(5) Grand Unification, and the required coefficients of their U(1)Y couplings, in Table 6. Here I have assumed that α 3 and α−11 are in the ratio 1, 6, at unification, but it would be useful to study models that achieve this within a few percent, since the correct form of running to unification is not yet known. Since the running of the coupling constants is always by small amounts, the additional states in these models, not yet observed experimentally, will not alter the unification mass, or the value of the SU(3) coupling constant at unification, which is equal to the E8 coupling constant at unification, by a large amount. Thus this class of models generically predicts that the unification mass is about a TeV, and the E8 fine structure constant, at unification, is about 1 The breaking of E8 to SU(3)× (SU(2))3 × U(1)Y can be studied, following [8], by analyzing the breaking of SU(9) to SU(3)× (SU(2))3 × U(1)Y . It is convenient to use block matrix notation. Each SU(9) fundamental index is replaced by a pair of indexes, an upper-case letter and a lower-case letter. The upper-case letter runs from 1 to 4, and indicates which subgroup in the sequence SU(3)×SU(2)×SU(2)×SU(2) the block belongs to. Thus B = 1 denotes the SU(3), B = 2 denotes the first SU(2), B = 3 denotes the second SU(2), and B = 4 denotes the third SU(2). The lower-case index is a fundamental index for the subgroup identified by the upper-case index it belongs to. It is important to note that the range of a lower-case index depends on the value of the upper-case index it belongs to, so we have to keep track of which lower-case indexes belong to which upper-case indexes. Each SU(9) antifundamental index is treated in the same way, except that the lower-case index is now an antifundamental index for the appropriate subgroup. The summation convention is applied to both upper-case letters and lower-case letters that derive from an SU(9) fundamental or antifundamental index, but we have to remember that lower-case indexes are to be summed over first, because their ranges of summation depend on the values of the upper-case indexes they belong to. Each SU(9) adjoint representation index, which in the notation above, is a lower-case Greek letter, is replaced by a pair of indexes, an upper-case letter and a lower-case letter, where the upper-case letter runs from 1 to 5, and identifies which subgroup in the sequence SU(3)×SU(2)×SU(2)×SU(2)×U(1)Y a generator belongs to, and the lower-case letter runs over all the generators of the subgroup identified by the upper-case letter it belongs to. When an upper-case adjoint representation index takes the value 5, the associated lower-case index takes a single value, 1. The summation convention is applied to a lower-case letter that derives from an SU(9) adjoint representation index, but not to an upper-case letter that derives from an SU(9) adjoint representation index. We can now list all the blocks in the 80, the 84, and the 84, and display their SU(3)× SU(2) content. This is displayed in Table 7 for the 80, and in Table 8 for the 84, with all the lower-case indexes suppressed. The SU(9) generators, in the SU(3) × SU(2) × SU(2) × SU(2) × U(1)Y subgroup, may be taken as follows, in the block matrix notation. BiC̄j̄ = δABδAC̄ (tAa)ij̄ (1 ≤ A ≤ 4) (539) BiC̄j̄ σ1δ1Bδ1C̄δij̄ + σ2δ2Bδ2C̄δij̄ + σ3δ3Bδ3C̄δij̄ + σ4δ4Bδ4C̄δij̄ σAδABδAC̄δij̄ (540) Here (tAa)ij̄ denotes the fundamental representation generator number a, of non- States in the 80 Blocks Number of distinct blocks SU(3)× SU(2) content Number of states coefficient of coupling to U(1) ψ11̄ 1 (8,1) 8 0 ψ22̄ 1 (1,3) 3 0 ψ33̄ 1 (1,3) 3 0 ψ44̄ 1 (1,3) 3 0 ψdiag ψdiag ψdiag applicable (1, 1) + (1, 1)+ +(1, 1) ψ12̄ 1 (3,2) 6 σ1−σ2 ψ13̄ 1 (3,2) 6 σ1−σ3 ψ14̄ 1 (3,2) 6 σ1−σ4 ψ21̄ 1 (3̄, 2) 6 −σ1+σ2 ψ31̄ 1 (3̄, 2) 6 −σ1+σ3 ψ41̄ 1 (3̄, 2) 6 −σ1+σ4 ψ23̄ 1 (1, 3) + (1, 1) 4 σ2−σ3 ψ24̄ 1 (1, 3) + (1, 1) 4 σ2−σ4 ψ34̄ 1 (1, 3) + (1, 1) 4 σ3−σ4 ψ32̄ 1 (1, 3) + (1, 1) 4 −σ2+σ3 ψ42̄ 1 (1, 3) + (1, 1) 4 −σ2+σ4 ψ43̄ 1 (1, 3) + (1, 1) 4 −σ3+σ4 Table 7: The states in the 80, organized by their SU(3) × SU(2) × SU(2) × SU(2) assignments, showing their SU(3) × SU(2)diag content, and the coefficients of their couplings to a U(1) gauge field, parametrized as in equation (540). States in the 84 Blocks Number of distinct blocks SU(3)× SU(2) content Number of states coefficient of coupling to U(1) ψ1̄1̄1̄ 1 (1,1) 1 ψ2̄1̄1̄ ψ1̄2̄1̄ ψ1̄1̄2̄ 1 (3, 2) 6 −2σ1−σ2 ψ3̄1̄1̄ ψ1̄3̄1̄ ψ1̄1̄3̄ 1 (3, 2) 6 −2σ1−σ3 ψ4̄1̄1̄ ψ1̄4̄1̄ ψ1̄1̄4̄ 1 (3, 2) 6 −2σ1−σ4 ψ2̄2̄1̄ ψ2̄1̄2̄ ψ1̄2̄2̄ 1 (3̄, 1) 3 −σ1−2σ2 ψ3̄3̄1̄ ψ3̄1̄3̄ ψ1̄3̄3̄ 1 (3̄, 1) 3 −σ1−2σ3 ψ4̄4̄1̄ ψ4̄1̄4̄ ψ1̄4̄4̄ 1 (3̄, 1) 3 −σ1−2σ4 ψ1̄2̄3̄ ψ2̄1̄3̄ ψ2̄3̄1̄ ψ1̄3̄2̄ ψ3̄1̄2̄ ψ3̄2̄1̄ 1 (3̄, 3) + (3̄, 1) 12 −σ1−σ2−σ3 ψ1̄2̄4̄ ψ2̄1̄4̄ ψ2̄4̄1̄ ψ1̄4̄2̄ ψ4̄1̄2̄ ψ4̄2̄1̄ 1 (3̄, 3) + (3̄, 1) 12 −σ1−σ2−σ4 ψ1̄3̄4̄ ψ3̄1̄4̄ ψ3̄4̄1̄ ψ1̄4̄3̄ ψ4̄1̄3̄ ψ4̄3̄1̄ 1 (3̄, 3) + (3̄, 1) 12 −σ1−σ3−σ4 ψ2̄2̄2̄ ψ3̄3̄3̄ ψ4̄4̄4̄ these three blocks are empty ψ2̄2̄3̄ ψ2̄3̄2̄ ψ3̄2̄2̄ 1 (1,2) 2 −2σ2−σ3 ψ2̄2̄4̄ ψ2̄4̄2̄ ψ4̄2̄2̄ 1 (1,2) 2 −2σ2−σ4 ψ3̄3̄2̄ ψ3̄2̄3̄ ψ2̄3̄3̄ 1 (1,2) 2 −σ2−2σ3 ψ3̄3̄4̄ ψ3̄4̄3̄ ψ4̄3̄3̄ 1 (1,2) 2 −2σ3−σ4 ψ4̄4̄2̄ ψ4̄2̄4̄ ψ2̄4̄4̄ 1 (1,2) 2 −σ2−2σ4 ψ4̄4̄3̄ ψ4̄3̄4̄ ψ3̄4̄4̄ 1 (1,2) 2 −σ3−2σ4 ψ2̄3̄4̄ ψ3̄2̄4̄ ψ3̄4̄2̄ ψ4̄3̄2̄ ψ4̄2̄3̄ ψ2̄4̄3̄ (1, 4) + (1, 2)+ +(1, 2) 8 −σ2−σ3−σ4 Table 8: The states in the 84, organized by their SU(3) × SU(2) × SU(2) × SU(2) assignments, showing their SU(3) × SU(2)diag content, and the coefficients of their couplings to a U(1) gauge field, parametrized as in equation (540). Abelian subgroup number A, in the list above. Thus for A = 1, the subgroup is SU(3), a runs from 1 to 8, and i and j each run from 1 to 3, while for A = 2, 3, or 4, the subgroup is SU(2), a runs from 1 to 3, and i and j each run from 1 to 2. σ1, σ2, σ3, and σ4 are real numbers parametrizing the embedding of the U(1)Y subgroup in SU(9), and thus in E8, and θ is a normalization factor. In using this notation, we have to take sensible precautions, such as grouping within brackets, to keep track of which lower-case indexes belong to which upper-case indexes. In equation (540), it would be wrong to “factor out” the δij̄ , because it represents a 3 by 3 matrix in one term, and a 2 by 2 matrix in the other three terms. The tracelessness of BiC̄j̄ implies: 0 = 3σ1 + 2 (σ2 + σ3 + σ4) (541) and the normalization condition (503) implies: θ2 = 6σ21 + 4 σ22 + σ 3 + σ (542) As an example, I consider the states in the left-handed 84. The covariant derivative is [44] Dµ = ∂µ − igAµαTα (543) so, for unbroken SU(9), and with (+,+,+, 1) metric, and {γµ, γν} = 2gµν , the massless Dirac action in this case is [44]: ψ̄γµDµψ = ψ̄γµ (∂µ − igAµαTα)ψ = = ψ̄ijkγ δmīδpj̄δqk̄ ± five terms −igAµα − (tα)mī δpj̄δqk̄ ± seventeen terms ψm̄p̄q̄ = = ψ̄ijkγ µ∂µψīj̄k̄ + 3igAµαψ̄ijkγ µ (tα)mī ψm̄j̄k̄ (544) where I used (507), the antisymmetry of ψ̄ijk and ψm̄p̄q̄ in their indexes, and the re- labelling of dummy indexes. ψ̄ijk are the right-handed 84 states, and ψmpq are the left-handed 84 states. Breaking SU(9) to SU(3)× (SU(2))3×U(1)Y , and using the block matrix notation, this becomes: ψ̄BiCjDkγ µ∂µψB̄īC̄j̄D̄k̄ + 3ig AµAaψ̄BiCjDkγ EmB̄ī ψĒm̄C̄j̄D̄k̄ = ψ̄BiCjDkγ µ∂µψB̄īC̄j̄D̄k̄ + 3ig AµAaψ̄AiCjDkγ µ (tAa)mī ψĀm̄C̄j̄D̄k̄ +3igAµY σAψ̄AiCjDkγ µψĀīC̄j̄D̄k̄ (545) where I used (539) and (540). We can now extract the covariant derivative Dirac action terms for the various entries in Table 8, and thus determine the coefficients of their couplings to AµY . For example, a block in ψB̄īC̄j̄D̄k̄, where two upper-case indexes take the value 1, and the remaining upper-case index takes the value 2, 3, or 4, is a candidate to be a (3,2) quark multiplet. The sum of all terms in (545), where two upper-case indexes take the value 1, and the remaining upper-case index takes the value 2, is: 3ψ̄1i1j2kγ µ∂µψ1̄ī1̄j̄2̄k̄+6igAµ1aψ̄1i1j2kγ µ(t1a)mī ψ1̄m̄1̄j̄2̄k̄+3igAµ2aψ̄2i1j1kγ µ(t2a)mī ψ2̄m̄1̄j̄1̄k̄ +6igAµY σ1ψ̄1i1j2kγ µψ1̄ī1̄j̄2̄k̄ + 3igAµY σ2ψ̄2i1j1kγ µψ2̄ī1̄j̄1̄k̄ (546) Now ψ1̄ī1̄j̄2̄k̄ is antisymmetric under swapping the two SU(3) antifundamental in- dexes i and j, so that we may write: ψ1̄ī1̄j̄2̄k̄ = εīj̄m̄φmk̄ (547) and analogously: ψ̄1i1j2k = εijmφ̄m̄k (548) We can then use relations such as ψ̄1i1j2kγ µψ1̄ī1̄j̄2̄k̄ = εijpφ̄p̄kγ µεīj̄m̄φmk̄ = 2δpm̄φ̄p̄kγ µφmk̄ = 2φ̄m̄kγ µφmk̄ (549) ψ̄1i1j2kγ µ (t1a)mī ψ1̄m̄1̄j̄2̄k̄ = εijpφ̄p̄kγ µ (t1a)mī εm̄j̄q̄φqk̄ = = (δim̄δpq̄ − δiq̄δpm̄) φ̄p̄kγµ (t1a)mī φqk̄ = φ̄p̄kγ µ (t1a)īi φpk̄ − φ̄m̄kγ µ (t1a)mī φik̄ = = −φ̄m̄kγµ (t1a)mī φik̄ (550) to express (546) as: φ̄ījγ µ∂µφij̄ − igAµ1aφ̄ījγµ (t1a)ik̄ φkj̄ + igAµ2aφ̄ījγ µ(t2a)mj̄ φim̄ −igAµY (−2σ1 − σ2) φ̄ījγµφij̄ (551) Thus we see that the index i of φij̄ is an SU(3) fundamental index. The SU(2) antifundamental is equivalent to the fundamental, the relation being given by matrix multiplication by εjk, and we could, if we wished, make a further transformation to replace the SU(2) antifundamental index j̄ of φij̄, by an index that is manifestly in the SU(2) fundamental. When (SU(2))3 is broken to SU(2)diag, the Aµ2a, in the third term in (551), will be replaced, at low energy, by 1√ Bµa, where Bµa is the gauge field of SU(2)diag. The overall factor of 6 can be absorbed into the normalizations of φij̄ and φ̄īj , so from the fourth term in (551), we can read off what the coefficient of gAµY φ̄ījγ µφij̄ would be, if the φ̄īijiγ µ∂µφij̄ term had standard normalization, and thus complete the entries in the second row of Table 8. The entries in the third column of Tables 7 and 8 can be completed by similar methods. The entries in the fifth column of Table 8 can be completed by a simple mnemonic: for each upper-case index, of the untransformed ψB̄īC̄j̄D̄k̄, that takes the value A, 1 ≤ A ≤ 4, include a term −1 σA. For Table 7, the mnemonic is that when the index B of ψBiC̄j̄ takes the value A, 1 ≤ A ≤ 4, so that i is in the fundamental of non-Abelian subgroup number A, include a term 1 σA, and when the index C̄ of ψBiC̄j̄ takes the value A, 1 ≤ A ≤ 4, so that j is in the antifundamental of non-Abelian subgroup number A, include a term −1 Indeed, suppose we extract all terms from (545) such that ψB̄īC̄j̄D̄k̄ has nA upper- case indexes with the value A, 1 ≤ A ≤ 4, so that n1 + n2 + n3 + n4 = 3. We get n1!n2!n3!n4! contributions from the ψ̄BiCjDkγ µ∂µψB̄īC̄j̄D̄k̄ term. The number of times we get σA, from the third term in (545), is ñ1!ñ2!ñ3!ñ4! , where ñB = nB if B 6= A, and ñA = nA − 1. But this is equal to 2!nAn1!n2!n3!n4! . The factor n1!n2!n3!n4! combines with the explicit factor of 3, in the third term in (545), to produce the same overall factor of n1!n2!n3!n4! as found for the first term, so the coefficient of the contributions from the third term, if the contributions from the first term had standard normalization, would be −1 nAσA. The mnemonic for Table 7 can be justified in a similar manner. We know that we have to find couplings of the observed fermions, to the U(1)Y gauge field, that are smaller than those found in the SU(5) model [404, 405], by an overall factor that is within a few percent of 1√ , so it is useful to apply the same techniques to calculate the corresponding coefficients in the SU(5) model. In this case, the relations (503) and (541) completely determine the U(1) generator, up to sign, and we find the entries in the fourth column of Table 6. The entries in the fifth column have been filled in, assuming the overall factor is exactly 1√ If we now choose (σ1, σ2, σ3, σ4) = (4, 3,−3,−6), so that θ2 = 312, we can identify ψ12̄ as a q, the (3̄, 1) state in ψ1̄2̄3̄ as a ū, ψ3̄3̄1̄ as a d̄, ψ2̄2̄3̄ as an l, and the (1, 1) state in ψ23̄ as an e +. We note that another U (1), defined by (σB1, σB2, σB3, σB4) = (2, 0, 0,−3), couples to these states in proportion to their baryon number. Furthermore, ψ3̄3̄2̄, which has Y = 1, can be identified as the Standard Model Higgs field. To determine the value of sin2 θW at unification, for this choice of the σi, let us denote the Higgs field, ψ3̄3̄2̄, by φi. Then by the methods above, we find that at low energies, its covariant derivative, times i, becomes: i∂µφi + g (σa)ik φk + g AµY φi, (552) where Bµa is the gauge field of SU (2)diag. While from equation (117) on page 33 of Rosner’s review [44], the standard covariant derivative, times i, on the Standard Model Higgs field is: i∂µφi + gRosnerBµa (σa)ik φk + g Rosner AµY φi (553) Thus since the Standard Model Higgs field has Y = 1, we see that: gRosner = g , g′Rosner = g (554) Now by definition, tan θW = Rosner gRosner . Thus we find that, for this choice of the σi, sin2 θW = ≃ 0.257 at unification. This is the closest I have found to the required value of sin2 θW ≃ 0.239, when the Hodge - de Rham harmonic two-forms, in the Cartan subalgebra, break E8 to SU (3)× (SU (2))3 × (U (1))3. However, even this value of sin2 θW cannot actually be realized. For to break (SU (2)) to SU (2)diag, without breaking SU (3)× SU (2)diag × U (1)Y , we need to find states in the E8 fundamental / adjoint, that transform nontrivially under (SU (2)) but are singlets of SU (3) × SU (2)diag, and have Y = 0. The only states which transform nontrivially under (SU (2)) , but are singlets of SU (3)× SU (2)diag, are the (1, 1) states in ψ23̄, ψ24̄, ψ34̄, and their complex conjugates. Looking at the U (1) couplings of these states, in Table 7, we see that none of them have Y = 0, for (σ1, σ2, σ3, σ4) = (4, 3,−3,−6). Furthermore, to ensure that SU (2)diag is the diagonal subgroup of all three SU (2)’s, and not just two of them, we need at least two of ψ23̄, ψ24̄, and ψ34̄, to have Y = 0. That means we require σ1 = σ2 = σ3, which means it is impossible to have ū and d̄ states with different values of Y . Thus we cannot real- ize the Standard Model, if the Hodge - de Rham harmonic two-forms, in the Cartan subalgebra of E8, break E8 to SU (3)× (SU (2))3 × (U (1))3. 5.5 Models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× (SU (2))2 × (U (1))4 In this subsection, I shall consider some models where the Hodge - de Rham harmonic two-forms break E8 to SU (3)×(SU (2))2×(U (1))4. We will find that there are states of E8 that transform nontrivially under (SU (2)) , and can break (SU (2)) to SU (2)diag, without breaking SU (3) × SU (2)diag × U (1)Y , for a reasonable value of the U (1)Y coupling constant at unification. The SU (2)diag coupling constant is now at uni- fication, so the SU (3) and SU (2)diag coupling constants, as evolved in the Standard Model, now meet at around 145 TeV, although this could presumably be reduced to around a TeV by the accelerated unification mechanism [128, 129, 373]. We will find two distinct types of solution for U (1)Y , both of which give sin 2 θW = = 0.300 at unification, roughly halfway between the observed value ≃ 0.23 at mZ , and the value = 0.375 found in conventional grand unification [404]. The observed value of sin2 θW evolves to ≃ 0.270 at around 145 TeV, in the Standard Model. An element of the E8 Cartan subalgebra, and hence of the SU (9) Cartan subalgebra, that can have a vacuum expectation value without breaking this subgroup of E8, is, in the SU (9) fundamental, a diagonal matrix, with diagonal matrix elements (σ1, σ1, σ1, σ2, σ2, σ3, σ3, σ4, σ5) (555) such that: 3σ1 + 2σ2 + 2σ3 + σ4 + σ5 = 0 (556) The normalization condition is now: θ2 = 6σ21 + 4 σ22 + σ σ24 + σ (557) The states in the 80 are shown in Table 9, and the states in the 84 are shown in Table 10. To break (SU (2)) to SU (2)diag, without breaking SU (3)× SU (2)diag ×U (1)Y , we need an SU (3) singlet state that transforms non-trivially under (SU (2)) , but contains a singlet of SU (2)diag, to have Y = 0, so that the singlet of SU (2)diag can have a non-vanishing vacuum expectation value, without breaking U (1)Y . Thus at least one of ψ23̄, ψ2̄3̄4̄, and ψ2̄3̄5̄ is required to have vanishing U (1)Y charge, so at least one of (σ2 − σ3), (−σ2 − σ3 − σ4), and (−σ2 − σ3 − σ5) is required to be zero. There are nine SU (3) × SU (2)diag singlets, plus their complex conjugates, whose U (1) charges do not vanish identically. However, only three of the U (1) charges of these nine SU (3)× SU (2)diag singlets are linearly independent, so it is not possible to raise the masses of more than two of the three unwanted U (1)’s as much as required, without breaking SU (3)× SU (2)diag × U (1)Y , and without relying on Witten’s Higgs mechanism. I did a computer search to determine whether the number of distinct choices of U (1)Y , such that there is at least one set of q, u, d, l, and e states with the correct relative Y values, and such that two (1, 1) states of SU (3)×SU (2)diag, with independent U (1)Y charges, have Y = 0, is finite or infinite. Specifically, I generated all sets of integer-valued (σ1, σ2, σ3, σ4), in order of increasing |σ1|+ |σ2|+ |σ3|+ |σ4|, up to |σ1|+ |σ2|+ |σ3|+ |σ4| = 300, with σ5 fixed by (556), and tested for the existence of at least one set of (3, 2), (3, 1), (3, 1), (1, 2), and (1, 1) states of SU (3)×SU (2)diag, with U (1)Y charges in the ratios 1, 4,−2,±3,±6, respectively. The result was that, excluding (σ1, σ2, σ3, σ4) with greatest common divisor > 1, and solutions related to solutions already found, by multiplying by −1, or by swapping σ2 and σ3, or by swapping σ4 and σ5, thirteen distinct solutions were found with |σ1| + |σ2| + |σ3| + |σ4| ≤ 10, and no new solutions were found with 11 ≤ |σ1| + |σ2| + |σ3| + |σ4| ≤ 300. Thus it looks likely that the thirteen distinct solutions, found with |σ1|+ |σ2|+ |σ3|+ |σ4| ≤ 10, are the only distinct solutions. All thirteen solutions were found to satisfy the requirement that at least one of (σ2 − σ3), (−σ2 − σ3 − σ4), and (−σ2 − σ3 − σ5) is zero, so that (SU (2))2 can be broken to SU (2)diag, without breaking SU (3)× SU (2)diag × U (1)Y . Furthermore, all thirteen solutions were found to admit a choice of a set of q, u, d, l, and e states with the correct relative Y values, such that there exists a U (1)B, defined by a different set of σi, whose couplings to that set of q, u, d, l, and e states are proportional to their baryon number, so that there is a chance of stabilizing the proton by a version of the Aranda-Carone mechanism. For a given set of q, u, d, l, and e states, the requirement for such a U (1)B to exist is four homogeneous linear equations for (σ1, σ2, σ3, σ4), and is thus equivalent to the vanishing of the determinant of the matrix of the coefficients of these equations. To try to find out if any of the thirteen solutions might be physically equivalent to Block SU (3)× SU (2)diag Coupling 3Y1 3Y2 3Y3 3B3 ψ11̄ (8, 1) 8 0 0 0 0 0 ψ22̄ (1, 3) 3 0 0 0 0 0 ψ33̄ (1, 3) 3 0 0 0 0 0 ψS (1, 1) 1 0 0 0 0 0 ψT (1, 1) 1 0 0 0 0 0 ψX (1, 1) 1 0 0 0 0 0 ψY (1, 1) 1 0 0 0 0 0 ψ12̄ (3, 2) 6 σ1−σ2 1 q 1 q 1 q 1 ψ13̄ (3, 2) 6 σ1−σ3 1 q 1 q −5 1 ψ14̄ (3, 1) 3 σ1−σ4 1 4 u −2 d 1 ψ15̄ (3, 1) 3 σ1−σ5 −5 10 −8 4 ψ21̄ (3̄, 2) 6 −σ1+σ2 −1 q̄ −1 q̄ −1 q̄ −1 ψ31̄ (3̄, 2) 6 −σ1+σ3 −1 q̄ −1 q̄ 5 −1 ψ41̄ (3̄, 1) 3 −σ1+σ4 −1 −4 ū 2 d̄ −1 ψ51̄ (3̄, 1) 3 −σ1+σ5 5 −10 8 −4 ψ23̄ (1, 3) + (1, 1) 4 σ2−σ3 0 ν̄ 0 ν̄ −6 e− 0 ψ24̄ (1, 2) 2 σ2−σ4 0 3 l̄ −3 l 0 ψ25̄ (1, 2) 2 σ2−σ5 −6 9 −9 3 ψ34̄ (1, 2) 2 σ3−σ4 0 3 l̄ 3 l̄ 0 ψ35̄ (1, 2) 2 σ3−σ5 −6 9 −3 l 3 ψ45̄ (1, 1) 1 σ4−σ5 −6 e− 6 e+ −6 e− 3 ψ32̄ (1, 3) + (1, 1) 4 −σ2+σ3 0 ν̄ 0 ν̄ 6 e+ 0 ψ42̄ (1, 2) 2 −σ2+σ4 0 −3 l 3 l̄ 0 ψ52̄ (1, 2) 2 −σ2+σ5 6 −9 9 −3 ψ43̄ (1, 2) 2 −σ3+σ4 0 −3 l −3 l 0 ψ53̄ (1, 2) 2 −σ3+σ5 6 −9 3 l̄ −3 ψ54̄ (1, 1) 1 −σ4+σ5 6 e+ −6 e− 6 e+ −3 Table 9: The states in the 80 for the SU (3)× (SU (2))2 × (U (1))4 case. Block SU (3)× SU (2)diag Coupling 3Y1 3Y2 3Y3 3B3 ψ1̄1̄1̄ (1, 1) 1 0 ν̄ −6 e− 6 e+ −3 ψ1̄1̄2̄ (3, 2) 6 −2σ1−σ2 1 q −5 7 −2 ψ1̄1̄3̄ (3, 2) 6 −2σ1−σ3 1 q −5 1 q −2 ψ1̄1̄4̄ (3, 1) 3 −2σ1−σ4 1 −2 d 4 u −2 ψ1̄1̄5̄ (3, 1) 3 −2σ1−σ5 −5 4 u −2 d 1 ψ1̄2̄2̄ (3̄, 1) 3 −σ1−2σ2 2 d̄ −4 ū 8 −1 ψ1̄3̄3̄ (3̄, 1) 3 −σ1−2σ3 2 d̄ −4 ū −4 ū −1 ψ1̄2̄3̄ (3̄, 3) + (3̄, 1) 12 −σ1−σ2−σ3 2 d̄ −4 ū 2 d̄ −1 ψ1̄2̄4̄ (3̄, 2) 6 −σ1−σ2−σ4 2 −1 q̄ 5 −1 ψ1̄2̄5̄ (3̄, 2) 6 −σ1−σ2−σ5 −4 5 −1 q̄ 2 ψ1̄3̄4̄ (3̄, 2) 6 −σ1−σ3−σ4 2 −1 q̄ −1 q̄ −1 ψ1̄3̄5̄ (3̄, 2) 6 −σ1−σ3−σ5 −4 5 −7 2 ψ1̄4̄5̄ (3̄, 1) 3 −σ1−σ4−σ5 −4 ū 8 −4 ū 2 ψ2̄2̄3̄ (1, 2) 2 −2σ2−σ3 3 l̄ −3 l 3 l̄ 0 ψ2̄3̄3̄ (1, 2) 2 −σ2−2σ3 3 l̄ −3 l −3 l 0 ψ2̄2̄4̄ (1, 1) 1 −2σ2−σ4 3 0 ν̄ 6 e+ 0 ψ2̄2̄5̄ (1, 1) 1 −2σ2−σ5 −3 6 e+ 0 ν̄ 3 ψ3̄3̄4̄ (1, 1) 1 −2σ3−σ4 3 0 ν̄ −6 e− 0 ψ3̄3̄5̄ (1, 1) 1 −2σ3−σ5 −3 6 e+ −12 3 ψ2̄3̄4̄ (1, 3) + (1, 1) 4 −σ2−σ3−σ4 3 0 ν̄ 0 ν̄ 0 ψ2̄3̄5̄ (1, 3) + (1, 1) 4 −σ2−σ3−σ5 −3 6 e+ −6 e− 3 ψ2̄4̄5̄ (1, 2) 2 −σ2−σ4−σ5 −3 l 9 −3 l 3 ψ3̄4̄5̄ (1, 2) 2 −σ3−σ4−σ5 −3 l 9 −9 3 Table 10: The states in the 84 for the SU (3)× (SU (2))2 × (U (1))4 case. one another, I calculated several numerical properties of each solution, to see if they distinguished between the solutions. Specifically, I made an arbitrary, but fixed, choice of one of each charge conjugate pair of (1, 2) states, to include in the tests, and an arbitrary, but fixed, choice of one of each charge conjugate pair of (1, 1) states, not in the SU (3)× (SU (2))2× (U (1))4 subgroup, to include in the tests, and then calculated N , the number of distinct possible choices of a set of q, u, d, l, and e states with the correct relative Y values, and NB, the number of distinct possible choices of a set of q, u, d, l, and e states with the correct relative Y values, that admit the existence of a U (1)B coupling to their baryon number. And, defining integer-valued charges, for these integer-valued (σ1, σ2, σ3, σ4, σ5), by the numerators in the fourth columns of Tables 9 and 10, I calculated nq, the number of (3, 2) states with Y = 1; xq, the number of (3, 2) states with Y = −1; nu, the number of (3, 1) states with Y = 4; xu, the number of (3, 1) states with Y = −4; nd, the number of (3, 1) states with Y = −2; xd, the number of (3, 1) states with Y = 2; nl, the number of (1, 2) states tested with Y = ±3; ne, the number of (1, 1) states tested with Y = ±6; nν , the number of (1, 1) states tested with Y = 0; and ndiag, the number of (σ2 − σ3), (−σ2 − σ3 − σ4), and (−σ2 − σ3 − σ5) that are zero. The result was that the thirteen solutions fell into three groups, with all these numerical quantities, and also θ2, having the same values, for all the members of each group. Thus it seems possible that there might be just three physically distinct solutions, one from each group. I have tabulated the Y values for one representative solution from each group, in Tables 9 and 10. The solutions in the first group are (σ1, σ2, σ3, σ4, σ5) = (0,−1,−1,−1, 5), (1, 0,−3, 0, 3), (−2, 0, 0, 3, 3), (−1, 1,−2, 1, 4), and (2,−2,−2, 1, 1). They all have N = 48, NB = 20, θ 2 = 60, nq = 4, xq = 0, nu = 1, xu = 0, nd = 3, xd = 0, nl = 4, ne = 1, nν = 2, and ndiag = 1. The Y values for (0,−1,−1,−1, 5) are tabulated in Tables 9 and 10 as Y1. The solutions in the second group are (σ1, σ2, σ3, σ4, σ5) = (2, 1, 1,−2,−8), (−2, 3, 3, 0,−6), and (0,−1, 5,−4,−4). They all have N = 400, NB = 208, θ2 = 168, nq = 4, xq = 0, nu = 5, xu = 0, nd = 1, xd = 0, nl = 4, ne = 5, nν = 4, and ndiag = 2. The Y values for (2, 1, 1,−2,−8) are tabulated in Tables 9 and 10 as Y2. The solutions in the third group are (σ1, σ2, σ3, σ4, σ5) = (−2,−3, 3, 0, 6), (0,−1,−1,−4, 8), (−4, 1, 1, 4, 4), (2, 1,−5,−2, 4), and (4,−3,−3, 0, 0). They all have N = 1296, NB = 592, θ 2 = 168, nq = 4, xq = 0, nu = 3, xu = 0, nd = 3, xd = 0, nl = 6, ne = 6, nν = 2, and ndiag = 1. The Y values for (−2,−3, 3, 0, 6) are tabulated in Tables 9 and 10 as Y3. For this example, we can choose 3B = (1, 0, 0, 0,−3), which gives the correct baryon number, except for the first four states of the 84, and states involving σ5, other than the fifth state of the 84. The B values for this choice of B are tabulated in Tables 9 and 10 as B3. To determine sin2 θW at unification, for the three groups of models, we recall that the Standard Model Higgs field is a (1, 2) state of SU (3) × SU (2)diag, with Y = 1. In the examples in Tables 9 and 10, this could, for example, be an extra-dimensional component AA245 of AU245, for the Y1 case, AA24̄ of AU24̄, for the Y2 case, and AA34̄ of AU34̄, for the Y3 case. Denoting this field by φi, we find, by the methods of the preceding subsection, that for the Y2 and Y3 cases, its covariant derivative, times i, becomes, at low energies: i∂µφi + g (σa)ik φk + g Aµ4φi (558) where Aµ4 is the gauge field of U (1)Y , in a notation similar to the previous subsection, and Bµa = (Aµ2a + Aµ3a) is the gauge field of SU (2)diag. While from equation (117) of Rosner’s review of the Standard Model [44], the standard covariant derivative, times i, on the Standard Model Higgs field is: i∂µφi + gRosnerBµa (σa)ik φk + g Rosner Aµ4φi (559) Now, as noted above, the Standard Model Higgs field has Y = 1. Thus we see that, for the Y2 and Y3 cases: gRosner = g , g′Rosner = g (560) And by definition, tan θW = Rosner gRosner . Hence we find that, for the Y2 and Y3 cases, sin2 θW = = 0.300 at unification, which is roughly halfway between the value ≃ 0.23 observed at mZ , and the value = 0.375 found in conventional grand unification [404], and reasonably consistent with the unification of the SU (3) and SU (2)diag coupling constants at around 145 TeV, in the absence of accelerated unification. On the other hand, for the Y1 case, the 168, in (558) and (560), gets replaced by 60, which gives sin2 θW = ≃ 0.545 at unification, so the Y1 case does not seem very likely. Let us now suppose that we have found a smooth compact quotient of CH3 or H6, and a set of Hodge - de Rham harmonic two-forms embedded in the E8 Cartan subalgebra as above, such that the net number of chiral zero modes of each of the left-handed states of one generation of the Standard Model, as in Table 6, is three, and the net number of chiral zero modes of each fermion state not in the Standard Model, is zero, and that SU (2) can be broken to SU (2)diag, in a topologically stabilized manner, by a “monopole”, embedded in E8 in one or more of whichever of the states ψ23̄, ψ2̄3̄4̄, and ψ2̄3̄5̄ have vanishing U (1)Y charge, in the example under consideration, without spoiling this. Then it seems reasonable to expect that the Hodge - de Rham harmonic two-forms will lead to masses ∼ a TeV for all chiral zero modes that can be matched in left-handed and right-handed pairs, so that the only light fermions will be the three generations of Standard Model fermions, except possibly for one or more light singlet neutrino states, which could obtain very small masses by the generalized seesaw mechanism to be discussed in subsection 5.7. Let us now consider an arbitrary proton decay process, proceeding via a dimension 6 term in the Standard Model effective action, such as qqql cdcucec cucqq , or d cucql [406], with the SU (3) and SU (2)diag indices contracted in an appropriate manner, where Λ is an effective cutoff, that determines the size of the term. Then for any four specific states from Tables 9 and 10, that have nonvanishing amplitudes in those four types of Standard Model state, the condition for the existence of a U (1)B, that couples as a nonzero multiple of baryon number, just on those four states, is three homogeneous linear equations on the four linearly independent σi, so is always satisfied. Thus those parts of the arguments of Aranda and Carone [52], that depend only on the existence of such a U (1)B, would seem to suggest that the contribution of those four states, to the corresponding term in the Standard Model effective action, will be suppressed. And since this argument applies to all sets of states from Tables 9 and 10, that have nonvanishing amplitudes in the Standard Model fermion fields in the effective action term concerned, we expect the same suppression to apply to the overall coefficient of that term in the effective action, leading to a large value of the effective cutoff Λ, even though the relevant U (1)B may be different, for different relevant sets of states from Tables 9 and 10. Of course, it was not necessary to require that two of the SU (3)× SU (2) singlets, with independent U (1) charges, have Y = 0, since the unwanted U (1)’s will become massive by Witten’s Higgs mechanism, provided that none of them is orthogonal to all the Hodge - de Rham monopoles in the E8 Cartan subalgebra. So additional solutions might exist, such that the largest number of SU (3) × SU (2) singlets, with linearly independent U (1) charges, that have Y = 0, is less than two. 5.6 Models where the Abelian Hodge - de Rham monopoles break E8 to SU (3)× SU (2)× (U (1))5 I shall now consider some models where the Hodge - de Rham harmonic two-forms break E8 to SU (3) × SU (2) × (U (1))5. An element of the E8 Cartan subalgebra, and hence of the SU (9) Cartan subalgebra, that can have a vacuum expectation value without breaking this subgroup of E8, is, in the SU (9) fundamental, a diagonal matrix, with diagonal matrix elements (σ1, σ1, σ1, σ2, σ2, σ3, σ4, σ5, σ6) , (561) such that: 3σ1 + 2σ2 + σ3 + σ4 + σ5 + σ6 = 0 (562) The states in the 80, omitting the states in the unbroken SU (3)×SU (2)×(U (1))5, whose U (1) charges vanish identically, are shown in Table 11, and the states in the 84 are shown in Table 12. There are now fifteen SU (3)×SU (2) singlets, plus their complex conjugates, whose U (1) charges do not vanish identically, and the U (1) charges of five of these fifteen SU (3) × SU (2) singlets are linearly independent, so there is now a possibility of raising the masses of all four unwanted U (1)’s as much as required, without breaking SU (3) × SU (2) × U (1)Y , and without relying on Witten’s Higgs mechanism, by choosing the σi such that four SU (3)× SU (2) singlets outside the E8 Cartan subalgebra, with linearly independent U (1) charges, all have vanishing U (1)Y charge, and could thus have vacuum expectation values without breaking SU (3) × SU (2)× U (1)Y . I did a computer search through all 15! 4!11! = 1365 choices of which four of the fifteen SU (3) × SU (2) singlets should be set to have Y = 0, to determine which choices led to the existence of at least one set of q, u, d, l, and e states with the correct Y values, such that there exists a U (1)B, defined by a different set of σi, whose couplings to at least one set of these states are proportional to their baryon number, so that there is a chance of stabilizing the proton by a version of the Aranda-Carone mechanism. There were only six distinct solutions, four of which are related by permutations of σ3, σ4, σ5, and σ6. Taking only one of these four, the three solutions are: 3Y1 = (−2, 3, 0, 0, 0, 0) (563) Block Multiplet Coupling 3Y1 3B1 3Y2 3Y3 ψ12̄ (3,2) 6 σ1−σ2 −5 1 1 q 1 q ψ13̄ (3,1) 3 σ1−σ3 −2 d 1 4 u 4 u ψ14̄ (3,1) 3 σ1−σ4 −2 d 1 4 u −2 d ψ15̄ (3,1) 3 σ1−σ5 −2 d 1 4 u −2 d ψ16̄ (3,1) 3 σ1−σ6 −2 d 4 4 u −2 d ψ21̄ (3̄, 2) 6 −σ1+σ2 5 −1 −1 q̄ −1 q̄ ψ31̄ (3̄, 1) 3 −σ1+σ3 2 d̄ −1 −4 ū −4 ū ψ41̄ (3̄, 1) 3 −σ1+σ4 2 d̄ −1 −4 ū 2 d̄ ψ51̄ (3̄, 1) 3 −σ1+σ5 2 d̄ −1 −4 ū 2 d̄ ψ61̄ (3̄, 1) 3 −σ1+σ6 2 d̄ −4 −4 ū 2 d̄ ψ23̄ (1, 2) 2 σ2−σ3 3 l̄ 0 3 l̄ 3 l̄ ψ24̄ (1, 2) 2 σ2−σ4 3 l̄ 0 3 l̄ −3 l ψ25̄ (1, 2) 2 σ2−σ5 3 l̄ 0 3 l̄ −3 l ψ26̄ (1, 2) 2 σ2−σ6 3 l̄ 3 3 l̄ −3 l ψ34̄ (1, 1) 1 σ3−σ4 0 0 0 −6 e− ψ35̄ (1, 1) 1 σ3−σ5 0 0 0 −6 e− ψ36̄ (1, 1) 1 σ3−σ6 0 3 0 −6 e− ψ45̄ (1, 1) 1 σ4−σ5 0 0 0 0 ψ46̄ (1, 1) 1 σ4−σ6 0 3 0 0 ψ56̄ (1, 1) 1 σ5−σ6 0 3 0 0 ψ32̄ (1, 2) 2 −σ2+σ3 −3 l 0 −3 l −3 l ψ42̄ (1, 2) 2 −σ2+σ4 −3 l 0 −3 l 3 l̄ ψ52̄ (1, 2) 2 −σ2+σ5 −3 l 0 −3 l 3 l̄ ψ62̄ (1, 2) 2 −σ2+σ6 −3 l −3 −3 l 3 l̄ ψ43̄ (1, 1) 1 −σ3+σ4 0 0 0 6 e+ ψ53̄ (1, 1) 1 −σ3+σ5 0 0 0 6 e+ ψ63̄ (1, 1) 1 −σ3+σ6 0 −3 0 6 e+ ψ54̄ (1, 1) 1 −σ4+σ5 0 0 0 0 ψ64̄ (1, 1) 1 −σ4+σ6 0 −3 0 0 ψ65̄ (1, 1) 1 −σ5+σ6 0 −3 0 0 Table 11: The states in the 80 for the SU (3) × SU (2) × (U (1))5 case, omitting the states in the SU (3)× SU (2)× (U (1))5 subgroup. Block Multiplet Coupling 3Y1 3B1 3Y2 3Y3 ψ1̄1̄1̄ (1, 1) 1 6 e+ −3 −6 e− 0 ψ1̄1̄2̄ (3, 2) 6 −2σ1−σ2 1 q −2 −5 1 q ψ1̄1̄3̄ (3, 1) 3 −2σ1−σ3 4 u −2 −2 d 4 u ψ1̄1̄4̄ (3, 1) 3 −2σ1−σ4 4 u −2 −2 d −2 d ψ1̄1̄5̄ (3, 1) 3 −2σ1−σ5 4 u −2 −2 d −2 d ψ1̄1̄6̄ (3, 1) 3 −2σ1−σ6 4 u 1 −2 d −2 d ψ1̄2̄2̄ (3̄, 1) 3 −σ1−2σ2 −4 ū −1 −4 ū 2 d̄ ψ1̄3̄4̄ (3̄, 1) 3 −σ1−σ3−σ4 2 d̄ −1 2 d̄ 2 d̄ ψ1̄3̄5̄ (3̄, 1) 3 −σ1−σ3−σ5 2 d̄ −1 2 d̄ 2 d̄ ψ1̄3̄6̄ (3̄, 1) 3 −σ1−σ3−σ6 2 d̄ 2 2 d̄ 2 d̄ ψ1̄4̄5̄ (3̄, 1) 3 −σ1−σ4−σ5 2 d̄ −1 2 d̄ −4 ū ψ1̄4̄6̄ (3̄, 1) 3 −σ1−σ4−σ6 2 d̄ 2 2 d̄ −4 ū ψ1̄5̄6̄ (3̄, 1) 3 −σ1−σ5−σ6 2 d̄ 2 2 d̄ −4 ū ψ1̄2̄3̄ (3̄, 2) 6 −σ1−σ2−σ3 −1 q̄ −1 −1 q̄ 5 ψ1̄2̄4̄ (3̄, 2) 6 −σ1−σ2−σ4 −1 q̄ −1 −1 q̄ −1 q̄ ψ1̄2̄5̄ (3̄, 2) 6 −σ1−σ2−σ5 −1 q̄ −1 −1 q̄ −1 q̄ ψ1̄2̄6̄ (3̄, 2) 6 −σ1−σ2−σ6 −1 q̄ 2 −1 q̄ −1 q̄ ψ2̄2̄3̄ (1, 1) 1 −2σ2−σ3 −6 e− 0 0 6 e+ ψ2̄2̄4̄ (1, 1) 1 −2σ2−σ4 −6 e− 0 0 0 ψ2̄2̄5̄ (1, 1) 1 −2σ2−σ5 −6 e− 0 0 0 ψ2̄2̄6̄ (1, 1) 1 −2σ2−σ6 −6 e− 3 0 0 ψ2̄3̄4̄ (1, 2) 2 −σ2−σ3−σ4 −3 l 0 3 l̄ 3 l̄ ψ2̄3̄5̄ (1, 2) 2 −σ2−σ3−σ5 −3 l 0 3 l̄ 3 l̄ ψ2̄3̄6̄ (1, 2) 2 −σ2−σ3−σ6 −3 l 3 3 l̄ 3 l̄ ψ2̄4̄5̄ (1, 2) 2 −σ2−σ4−σ5 −3 l 0 3 l̄ −3 l ψ2̄4̄6̄ (1, 2) 2 −σ2−σ4−σ6 −3 l 3 3 l̄ −3 l ψ2̄5̄6̄ (1, 2) 2 −σ2−σ5−σ6 −3 l 3 3 l̄ −3 l ψ3̄4̄5̄ (1, 1) 1 −σ3−σ4−σ5 0 0 6 e+ 0 ψ3̄4̄6̄ (1, 1) 1 −σ3−σ4−σ6 0 3 6 e+ 0 ψ3̄5̄6̄ (1, 1) 1 −σ3−σ5−σ6 0 3 6 e+ 0 ψ4̄5̄6̄ (1, 1) 1 −σ4−σ5−σ6 0 3 6 e+ −6 e− Table 12: The states in the 84 for the SU (3)× SU (2)× (U (1))5 case. q q̄ u ū d d̄ l l̄ e+ e− ν q5 q̄5 8 3 Y total Y1 in 80 0 0 0 0 4 4 4 4 0 0 12 1 1 1 1 5 80 Y1 in 84 1 4 4 1 0 6 6 0 1 4 4 0 0 0 0 0 84 Y1 in 84 4 1 1 4 6 0 0 6 4 1 4 0 0 0 0 0 84 Y1 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248 Y2 in 80 1 1 4 4 0 0 4 4 0 0 12 0 0 1 1 5 80 Y2 in 84 0 4 0 1 4 6 0 6 4 1 4 0 1 0 0 0 84 Y2 in 84 4 0 1 0 6 4 6 0 1 4 4 1 0 0 0 0 84 Y2 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248 Y3 in 80 1 1 1 1 3 3 4 4 3 3 6 0 0 1 1 5 80 Y3 in 84 1 3 1 3 3 4 3 3 1 1 7 0 1 0 0 0 84 Y3 in 84 3 1 3 1 4 3 3 3 1 1 7 1 0 0 0 0 84 Y3 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248 Table 13: The numbers of each type of state, for the three choices of Y in the SU (3)× SU (2)× (U (1))5 case. 3Y2 = (2, 1,−2,−2,−2,−2) (564) 3Y3 = (0,−1,−4, 2, 2, 2) (565) All three of these have θ2 = 60, so by the same method as in the previous two subsec- tions, we find sin2 θW = at unification, as for SU(5) grand unification, so unification depends entirely on the accelerated unification mechanism [129, 129, 373]. For Y1, we could choose 3B = (1, 0, 0, 0, 0,−3), and the resulting values of B are tabulated as B1 in Tables 11 and 12. The number of states of each type, for each of the three choices of Y , are given in Table 13. The total number of states of each type is the same for all three choices, so it seems possible that the three different choices of Y might be physically equivalent. We note that all the fermion states in the E8 fundamental, that are not in the SU (3) × SU (2) × (U (1))5 subgroup, and can thus be given a nonzero net number of chiral zero modes by the Hodge - de Rham harmonic two-forms in the E8 Cartan subalgebra, are now either Standard Model fermions, as in Table 6, or singlet neutrinos, apart from the single q5 state with Y = −5 , and the single q̄5 state with Y = 5 is well known that the possible sets of left-handed chiral fermions, in four dimensions, are very strongly constrained by the requirement of the absence of anomalies [407, 408, 409, 410, 411, 412, 413], and we will now find that an arbitrary set of Hodge - de Rham harmonic two-forms, of a smooth compact quotient of CH3 or H6 that is a spin manifold, embedded in the E8 Cartan subalgebra as above, such that Witten’s topological constraint is satisfied, will result in a set of chiral zero modes that is simply a number of Standard Model generations. If the net numbers of left-handed chiral zero modes are nq q’s, nu u’s, nd d’s, nl l’s, ne e’s, and n5 q 5’s, then the conditions for the absence of gauge anomalies [407, 408], and mixed gauge-gravitational anomalies [414, 415, 416], in four dimensions, are as follows. From a triangle diagram with three external SU (3) gauge bosons: 2nq + nu + nd + 2n5 = 0 (566) From a triangle diagram with two external SU (3) gauge bosons, and one external U (1)Y gauge boson: nq + 2nu − nd − 5n5 = 0 (567) From a triangle diagram with two external SU (2) gauge bosons, and one external U (1)Y gauge boson: nq − nl − 5n5 = 0 (568) From a triangle diagram with three external U (1)Y gauge bosons: nq + 32nu − 4nd − 9nl + 36ne − 125n5 = 0 (569) And from a triangle diagram with two external gravitons, and one external U (1)Y gauge boson [414, 415, 416]: nq + 2nu − nd − nl + ne − 5n5 = 0 (570) The five equations (566), (567), (568), (569), and (570), are linearly independent, and the general solution, with integer values for the ni, is an integer multiple of one Standard Model generation, which has (nq, nu, nd, nl, ne, n5) = (1,−1,−1, 1, 1, 0). Thus for an arbitrary set of Hodge - de Rham harmonic two-forms, in the Cartan subalgebra of E8, that break E8 to SU (3)× SU (2)×U (1)Y as considered in this subsection, and satisfy Witten’s topological constraint, the chiral fermions will consist of an integer number of Standard Model generations. Let us now consider the case where U (1)Y is (−2,−2,−2, 3, 3, 0, 0, 0, 0). To ensure that U (1)Y does not get a mass by Witten’s Higgs mechanism [45], the Abelian vacuum gauge fields (σ1, σ1, σ1, σ2, σ2, σ3, σ4, σ5, σ6) with non-vanishing field strength must be perpendicular to U (1)Y . But this implies that σ1 = σ2, so that the Abelian vacuum gauge fields with non-vanishing field strength must actually leave SU (5) unbroken. Nevertheless, we would still be able to break E8 to the Standard Model by topologically stabilized vacuum gauge fields in the E8 Cartan subalgebra, if we could topologically stabilize a Hosotani vacuum gauge field with vanishing field strength [49, 50, 51] that is in the Cartan subalgebra but not perpendicular to U (1)Y . This could be achieved if the fundamental group of the compact six-manifold M6 included a non-trivial element a such that an = 1 for some finite integer n, because an Abelian Wilson line looping once round the closed path corresponding to a will then be a phase factor f satisfying fn = 1. But as noted in section 3, on page 160, the smooth compact quotients M6 considered in this paper have no such non-trivial elements a, called torsion elements. Nevertheless, examples in three dimensions show that it is possible for the first homology groupH1 (M,Z) of a hyperbolic manifoldM to have torsion even though the fundamental group of M has no torsion. For example, using Weeks’s program SnapPea [312], the Weeks manifold, which is the compact hyperbolic three-manifold of smallest known volume, and designated m003(-3,1) by SnapPea, is found to have first homology group Z/5+Z/5. This can be checked using the presentation of the fundamental group given by SnapPea, which has generators a, b, and relations a2b2a2b−1ab−1 = 1 and a2b2a−1ba−1b2 = 1. We obtain the first homology group from the fundamental group by treating the generators as commuting in the relations, which then collapse to a5 = 1 and b5 = 1. SnapPea also confirms that the Weeks manifold is oriented. Thus it seems reasonable to expect that there may exist smooth compact quotients M6 of CH3 or H6 such that H1 (M6,Z) has torsion. This would be sufficient to obtain a topologically stabilized Hosotani Abelian vacuum gauge field with vanishing field strength, even though the fundamental group of M6 has no torsion. For suppose there exists a one-cycle l that is not a boundary, such that nl, for some finite integer n, is a boundary. We consider an E8 Wilson line w that loops once around l. Suppose there is a Hosotani U (1) vacuum field that is locally pure gauge, but for which w is a non-trivial phase factor. Then wn is a phase factor along a one-cycle that is a boundary. Thus since the Hosotani field is locally pure gauge, we find wn = 1 by Stokes’s theorem. As in the preceding section, it seems reasonable to expect that the Hodge - de Rham harmonic two-forms will lead to masses ∼ a TeV for all chiral zero modes that can be matched in left-handed and right-handed pairs, so that the only light fermions will be the three generations of Standard Model fermions, except possibly for one or more light singlet neutrino states, which could obtain very small masses by the generalized seesaw mechanism to be discussed in the following subsection. And as in the preceding subsection, let us now consider an arbitrary proton decay process, proceeding via a dimension 6 term in the Standard Model effective action, such as qqql cdcucec cucqq , or d cucql [406], with the SU (3) and SU (2)diag indices contracted in an appropriate manner, where Λ is an effective cutoff, that determines the size of the term. Then for any four specific states from Tables 11 and 12, that have nonvanishing amplitudes in those four types of Standard Model state, the condition for the existence of a U (1)B, that couples as a nonzero multiple of baryon number, just on those four states, is three homogeneous linear equations on the five linearly independent σi, so is always satisfied. Thus those parts of the arguments of Aranda and Carone [52], that depend only on the existence of such a U (1)B, would seem to suggest that the contribution of those four states, to the corresponding term in the Standard Model effective action, will be suppressed. And since this argument applies to all sets of states from Tables 11 and 12, that have nonvanishing amplitudes in the Standard Model fermion fields in the effective action term concerned, we expect the same suppression to apply to the overall coefficient of that term in the effective action, leading to a large value of the effective cutoff Λ, even though the relevant U (1)B may be different, for different relevant sets of states from Tables 11 and 12. To find out whether the mass hierarchy of the observed quarks and charged leptons could occur by a version of the Arkani-Hamed - Schmaltz mechanism [390], in the type of model considered here, it would be necessary to find the explicit form of the Hodge - de Rham harmonic two-forms, for examples of smooth compact quotients of CH3 or H6 that are spin manifolds, and the corresponding chiral fermion zero modes, to find out how spread out or localized they are. However, we note that in the examples considered by Arkani-Hamed and Schmaltz [390], and by Acharya and Witten [417], the chiral fermion modes have a Gaussian shape, even though the fermion “mass terms” only depend linearly on position. The explicit forms of the chiral fermion zero modes in monopole backgrounds on the two-sphere have been given by Deguchi and Kitsukawa [418]. Of course, it was not necessary to require that four of the SU (3)× SU (2) singlets, with linearly independent U (1) charges, have Y = 0, since the unwanted U (1)’s will become massive by Witten’s Higgs mechanism, provided that none of them is orthogo- nal to all the Hodge - de Rham monopoles in the E8 Cartan subalgebra. So additional solutions might exist, such that the largest number of SU (3) × SU (2) singlets, with linearly independent U (1) charges, that have Y = 0, is three or less. 5.7 Generalized seesaw mechanism With regard to how small neutrino masses, ∼ 1 eV or smaller, might arise in models of the type considered in this paper, it seems to be possible to produce a very small, but nonzero, eigenvalue, from a matrix whose matrix elements are integers in the range 0 to 10, if all the matrix elements in the lower right triangle, below the lower left to upper right diagonal, are zero, the matrix elements on the lower left to upper right diagonal are 1, and the matrix elements in the upper left triangle, above the lower left to upper right diagonal, are ∼ 10. For example the matrix: 10 10 10 1 10 10 1 0 10 1 0 0 1 0 0 0 (571) has eigenvalues 22.891, −7.6024, 4.7127, and −0.001219. If this effect occurs because all but one of the eigenvalues tend to be comparable to the large matrix elements in the upper left triangle, but the determinant, and hence the product of the eigenvalues, is equal to 1, it would presumably be possible to obtain an eigenvalue as small as required, by considering larger matrices with this structure. We note that in the models considered in the preceding subsection, it might be natural to find a number ∼ 10 or more of singlet neutrinos, which could perhaps sometimes have a mass matrix of this type. To obtain the required small eigenvalue, the matrix elements in the lower right triangle would presumably have to be exactly zero. This would presumably be possible, if the matrix elements were integer multiples of an overall factor, but I do not know of a reason why this should be so. Acknowledgements I would like to thank Savas Dimopoulos, David E. Kaplan, and Karin Slinger for or- ganizing a very enjoyable and helpful visit to Stanford University ITP, where part of the work that led to this paper was carried out, Nima Arkani-Hamed, Savas Dimopou- los, Michal Fabinger, Simeon Hellerman, Veronika Hubeny, Shamit Kachru, Nemanja Kaloper, Renata Kallosh, David E. Kaplan, Matt Kleban, Albion Lawrence, Andrei Linde, John McGreevy, Michael Peskin, Steve Shenker, Eva Silverstein, Matt Strassler, and Lenny Susskind for helpful discussions or comments, and Fyodor Tkachov and Kasper Peeters for helpful emails. At an early stage of this work, some of the calculations were carried out using TeXaide [419] and TeXnic Center [420], rather than by using pen and paper. Subse- quently, after migrating to Debian GNU/Linux [421], the work was done almost entirely by means of GNU TeXmacs [422], without using pen and paper at all. The paper was largely written using GNU TeXmacs, and ported to KTeXmaker2, now renamed to Kile [423], for completion. 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Salam, Ordercite, a program to establish whether your bibliography is in the same order as the citations to it. http://www.lpthe.jussieu.fr/~salam/ordercite/ 1 Introduction 2 Thick pipe geometries 2.1 Horava-Witten theory 2.2 The complex hyperbolic space CH^3 2.3 The field equations and boundary conditions 2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor 2.3.2 The Yang-Mills coupling constants in four dimensions 2.3.3 The problem of the higher order corrections to Horava-Witten theory 2.3.4 The Casimir energy density corrections to the energy-momentum tensor 2.3.5 The orders of perturbation theory that the terms in the Casimir energy densities occur at 2.3.6 The expansion parameter 2.3.7 Witten's topological constraint 2.3.8 The field equations and boundary conditions for the three-form gauge field 2.3.9 The field equations and boundary conditions for the metric 2.4 Analysis of the Einstein equations and the boundary conditions for the metric 2.4.1 Beyond the proximity force approximation 2.4.2 The region near the inner surface of the thick pipe 2.4.3 The boundary conditions at the inner surface of the thick pipe 2.4.4 The classical solutions in the bulk 2.5 Solutions with both a and b large compared to ^2/9, at the outer surface of the thick pipe 2.5.1 Newton's constant and the cosmological constant for solutions with the outer surface in the classical region 2.6 Solutions with a as small as ^2/9, at the outer surface of the thick pipe 2.6.1 Newton's constant and the cosmological constant 2.6.2 Comparison with sub-millimetre tests of Newton's law 2.6.3 Comparison with precision solar system tests of General Relativity 2.6.4 Further consequences of the warp factor decreasing to a small value, at the outer surface of the thick pipe 2.7 Stiffening by fluxes wrapping three-cycles of the compact six-manifold times the radial dimension 2.7.1 The region near the outer surface 2.7.2 Newton's constant and the cosmological constant in the presence of the extra fluxes 3 Smooth compact quotients of CH^3 , H^6 , H^3 and S^3 3.1 Smooth compact arithmetic quotients of C H^n and H^n 3.1.1 Compactness of G / for the examples of arithmetic lattices 3.1.2 Obtaining finite index torsion-free subgroups of by Selberg's lemma 3.2 Smooth compact quotients of S^3 4 The Casimir energy densities 4.1 The Salam-Strathee harmonic expansion method 4.2 AdS_5 C P^3 compactification of d = 11 supergravity 5 E8 vacuum gauge fields and the Standard Model 5.1 The lightest massive modes of the supergravity multiplet 5.2 An SU ( 9 ) basis for E_8 5.3 Dirac quantization condition for E8 vacuum gauge fields 5.4 Nonexistence of models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) ( SU ( 2 ) )^3 ( U ( 1 ) )^3 5.5 Models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) ( SU ( 2 ) )^2 ( U ( 1 ) )^4 5.6 Models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) SU ( 2 ) ( U ( 1 ) )^5 5.7 Generalized seesaw mechanism

0704.1477

Balance of forces in simulated bilayers

Untitled Balance of forces in simulated bilayers. J. Stecki Department III, Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warszawa, Poland November 21, 2018 Dedicated to Professor Keith Gubbins on the occasion of his 70th birthday Abstract Two kinds of simulated bilayers are described and the results are reported for lateral tension and for partial contributions of intermolecular forces to it. Data for a widest possible range of areas per surfactant head, from tunnel formation through tensionless state, transition to floppy bilayer, and its disintegration, are reported and discussed. The significance of the tensionless state, is discussed. Keywords: bilayer, lateral tension, buckling transition, tensionless. http://arxiv.org/abs/0704.1477v3 I. Introduction: two-dimensional sheets in three dimensions. Great interest in the properties of membranes and bilayers is reflected in the very large volume of work, reporting simulations of a variety of models and simulation meth- ods. However, simulations including a range of areas are relativery rare and those including a widest possible range of areas - rarer still1−6. Most papers concentrate on the ”tensionless state” for which the lateral tension vanishes. The latter is the direct counterpart of the surface (interfacial) tension between two liquids and in fact it is computed according to the same Kirkwood-Buff formula7. At this stage we must mention that the physico-chemical properties of a sheet of surfactant molecules, forming a bilayer, are positively exotic. Compare with a planar inerface between e.g. a liquid and its vapor or two immiscible liquids. Then the surface tension γ is defined by (∂F/∂A)V = γ (1.1) where F is the free energy, A the area, V the volume and the constancy of particle numbers and temperature T is understood. But γ itself is independent of the area; it is a material property, a function of state. In a shocking contrast to that, the lateral tension of a bilayer (again defined after (1.1)) is area-dependent; moreover in the same system as to composition, density, and temperature, γ(A) can be positive or negative. Its zero defines the ”tensionless state” which is of particular interest and some physical significance. Other properties, such as internal energy U or correlation functions including the structure factor S(q), also display the area dependence, if a sufficiently wide interval of areas is investigated. The peculiar shape of the bilayer isotherm i.e. of the function and plot of γ(A), as shown and discussed in Section II, raises the question as to how it originates. A partial answer to that is provided by the split of γ into its components. Not all simulations are ”atomistic” i.e. not all simulations construct the inter- molecular total energy E({r}) from model intermolecular potentials depending only on the spatial coordinates of the constituent particles (atoms or molecules), which energy is then used in a Monte-Carlo or Molecular Dynamics simulation scheme. One of the advantages of an atomistic simulation is the possibillity of examining the role of the con- stituent components such as surfactant heads, solvent molecules, and surfactant tails. This is put to use in this paper, in which we report the split of lateral tension. Our results are reported for two kinds of simulated bilayers; these are defined in Section II. In Section III we show the split of lateral tension into components and we discuss the physical meaning of the ”tensionless state”. In Section IV we return to the discussion of bilayer properties and deeper dis- tinctions between interfaces and membranes or bilayers. The discussion introduces the concept of constraint. It also appears to be necessary to point out that a very important category of objects, called ”vesicles”, is entirely outside the realm of simulated bilayers. Merging ”membranes and vesicles” in one sentence greatly oversimplifies matters, because vesi- cles, like spherical micelles floating in a solvent, enclose a finite volume. The bilayers do not. The ”widest possible range of areas” of simulated bilayer, mentioned above, refers to the limits of existence of a stable bilayer. When the imposed area is extended too much, the bilayer recedes creating a spherical hole or rather a tunnel filled with solvent joining the two sides of the solvent volume. When compressed, the bilayer flips into a ”floppy” state (see Section III) which then if compressed further, disintegrates into structureless object(s) which cannot be called a bilayer any more. Section II. Two kinds of bilayers. A planar bilayer is formed by amphiphilic surfactant molecules made each of a hydrophilic polar ”head” and hydrophobic non-polar hydrocarbon chains as ”tails”. The tails form the center of the bilayer and the two outer layers of heads separate the tails from the polar solvent (water). Therefore the cohesive energy density (CED)8 is high (solvent), high (first layer of heads), low (tails), low (second layer of tails), high (second layer of heads), high (solvent on the other side of the bilayer). The same forces and the same preferences operate in the formation of micelles of various shapes. Normally the solvent is polar, of high CED, most commonly water, and the micelles shield their hydrocarbon tails by an outer layer of heads in contact with water. However, there exist rare examples of reverse micelles which are formed by amphiphilic molecules in a non-polar solvent of low CED. In reverse micelles, the (hydrocarbon) tails form the outward shell and the (polar) heads are in the center of the micelle. By analogy with these I have2,4 constructed in simulations reverse bilayers which are formed in a non-polar solvent of low CED. On crossing the reverse bilayer along the z-direction perpendicular to the x-y plane of the bilayer, the sequence is now: solvent, tails, heads, heads, tails, solvent or, in terms of CED : low,low,high,high,low,low. These cases can be and have been modelled with the use of Lennard-Jones (6-12) potential (LJ) with the minimum of 3 kinds of particles: s(solvent), h (head), t (tail). With 6 energies, 6 collision diameters, and 6 cut-off radii, various simplifications have been used in bilayer simulations with the LJ potentials1−4,5,9−13. In our work the solvent was made of structureless spherical particles; freely jointed chains of such particles of the same size have modelled the surfactant molecules making up the bilayer. We found certain regularities in the stability of the model bilayers. We found that the chain lengths of the tails can be shortened even down to unity, the molecule becomes then a dimer (”h-t”) made of two particles. We also found that it is worthwhile to keep the presence of the solvent; in some simulations very unusual, in fact unphysical, intermolecular forces were required in order to ensure existence (in simulation) of stable bilayers14 made of trimers. The reverse bilayers made of trimers were successfully simulated without a solvent 15, although it appears this was not followed with further work. We also found that longer chains stabilize the bilayer; of the tail lengths l = 1, 4, 8 the dimers were difficult to stabilize without an extra repulsion (replacing the hydrophobic effect) from the solvent whereas, for longer chains, this extra repulsion was not necessary for stability. Chains with l = 8 produced stable bilayers with great ease. The modelling of solvent as made of spherical structureless particles interacting with a central potential creates a certain conceptual difficulty because of the hydrophobic effect. It has been partially resolved by an introduction9 of an additional repulsive force between the solvent and the tails, e.g. of the form c/rn with c > 0 and n=10 or more9−13. A temperature dependence would be needed too, to take into account the entropic effects of the hydrophobicity. In the case of a reverse bilayer, the reasons for its formation are mostly energetic: a pair of heads (now inside the bilayer) attract each other more strongly than any other pair and there is no hydrophobic effect since the solvent is non-polar with a weak CED. Section III. The Lateral Tension and its Components. A bilayer originally planar, lies in the x−y plane in the middle of a simulation box which is a parallelepiped of volume V = Lx×Ly×Lz; its area is A = LxLy. It is made of Nm molecules which contain Nm heads, originally Nm/2 in each of the two monolayers, so that the ”area per head” a = A/(Nm/2). The bilayer is surrounded ”from above” and ”from below” by Ns molecules of the solvent. The overall density is ρ = N/V with N = Ns + (l + 1)Nm. The lateral tension is defined 7 as the response to the process of increasing the area at constant volume. With Lx → Lx + δLx, Ly → Ly + δLy, Lz → Lz + δLz, the constraint of constant volume requires δLx/Lx + δLy/Ly + δLz/Lz = 0 and the Kirkwood-Buff equation follows γA = Lz(2pzz − pxx − pyy)/2 (3.1) with the known definitions pαα =< > (3.2) valid for a rectangular box. The average is a canonical average at given temperature T , all N ′s, A, and V . In the simulation, for each state point of T, V,Nm, Ns, A we obtain one value of γ(A). In our work, Lx = Ly; changing Lx to any new value, we calculate the new value of Lz needed to keep the volume V = LxLxLz at exactly the same value. Each time γ is computed, it is a sum of binary terms and these are grouped into partial sums so as to produce the split γ = γss + γsb + γbs + γbb. (3.3) Here the indices ss refer to the solvent-solvent part, the indices sb and bs to the solvent- bilayer part, and bb refers to the bilayer-bilayer part. There is no approximation in- volved. Fig.1 shows an example of a bilayer made of chain molecules of tail length l = 4 plus one more particle as the head, immersed in a solvent at high liquid-like density. Each γαβ is plotted against the area per head a; the lines joining the data points are there to guide the eye, only. The contribution of solvent-solvent pairs is about 0.2-0.3 and almost constant i.e. independent of a. All pairs made of solvent particle and any particle of any bilayer molecule make up the sb or bs contribution; it is also positive, varying smoothly with a and taking values between 0.5 and 1.05 . The bb contribution shows the characteristic break into two curves, nearly linear with a, but one with positive slope and the other with a negative one. This pattern is transferred to the sum, eq.(3.3), total γ(a). Fig.2 shows the effect of size for the same chain-length l = 4. The solvent-solvent contribution γss(a) is practically independent of size, as is nearly so the solvent-bilayer part. The bb part and, consequently, the total γ display the break into two regions; the floppy part strongly affected by size and the extended region negligibly so. The bigger size shows flatter γ(a), still negative, but closer to zero. Lowest values of γ(a), which may be taken as the border between the floppy region and the extended region, are negative but lower value corresponds to smaller size. These remarkable patterns are repeated for chain length l = 8, with breaks much more pronounced. Fig.3 shows γαβ(a) for two sizes; the intermediate size is omitted, just as was in Fig.2, in order to make the plot clearer. Again the ss part is constant and size-independent, the sb part nearly so, and the total γ along with the sb part, show these remarkable features: breaks and abrupt changes of slope. For large a the slope is positive and the size-dependence negligible. The transition to the floppy region occurs near a = 1.8 for the smaller system and the slope dγ/da changes sign; the break is very pronounced. The transition is shifted to a = 1.9 in the bigger system, the break is more pronounced in the bb part than in the total γ. Most characteristically in the bigger system γ(a) is flat and near zero (though still always negative) in the floppy region. The tensionless point appears to lie on the r.h.s. curve ( extended bilayer region) in this case. All bilayer isotherms i.e. all plots of γ(a) we have obtained1−4 show the same pattern of two curves joined at some negative value of the lateral tension γ. We interpret the region of lower a as the region of the floppy state of the bilayer which escapes into the third dimension, buckles, and takes a fuzzy and irregular form. A gently fluctuating planar bilayer is found in the other region at higher a’s. Generally, the region of extended bilayer includes all states with positive γ. The crossing of the isotherm with the a-axis marks the tensionless state . In the plots of Fig.1-3 the region of the extended bilayer has a large positive slope of γ(a). Such positive slopes have been found independently5,6 and also earlier9−13. The entire range including the floppy region is found only in few references2−6. The corresponding split of the lateral tension is found in this paper only. The status of the tensionless state appears now in a new light, as an accidental event resulting from the algebraic sum of positive and negative contributions after (3.3). The partial contributions suggest that the break point dividing the two branches of γ(a) (or γbb(a) ) is a truly characteristic point which ought to have a physical significance, rather than γ(a0) = 0. The shortest possible tail of one particle is present in a dimer. We have attempted and partially succeeded1 in simulating bilayers made of dimers, but these appeared to be only imperfectly stabilized by strong head-head and head-solvent interactions or/and the extra-repulsion introduced artificially to mimick the hydrophobic effect, as invented earlier9. Possibly at other densities and temperatures the simulations would have been more encouraging. We do not show these results. As mentioned in Section II, we turned to the new case of reverse bilayers formed in a weakly interacting solvent and succeeded in producing (in simulation) stable reverse bilayers of amphiphilic dimers2,4. Fig.4 and 5 show a selection of several characteristic cases, each for two sizes. These cases are: (a) extra repulsion added, (b) no extra repulsion, (c) no extra repulsion and longer range of attractive forces. In cases (a) and (b) the cut-off was the generally used r = 2.5, in the case (c) it was r = 3.2. For clarity, we split the data into two Figures: Fig.4 shows γ total only, for all 5 systems whereas Fig.5 shows the components ss, sb, and bb. All curves for all 5 systems in Fig.4 show a pattern similar to γ(a) in Fig.3 (for l = 8). This characteristic pattern is : almost linear rise with a for larger a and almost constant negative γ for small a. Such a description fits bigger systems better. The biggest system of an area about 100× 100 of reverse dimers case (a) is shown in Reference 4. As can be seen from Fig.4, the plot again strongly suggests a transition between two forms of the bilayer and certainly the existence of two regions1−5. In the floppy region γ < 0 and the bigger the system the closer to zero γ becomes. Fig.5 shows the components of γ but without γ itself. The pattern is similar to those shown in Fig.1,2,3 except that the ss contribution to γ is now negative, between -0.6 and -0.2, the sb contribution much larger, between 2. and 3.6. The bb contribution is, for all sizes, practically linear in a with again negative slope and negative value in the region of ”floppy” state. The transition appears softer than for chain molecules, especially those with longer tails where the breaks were sharp (see Fig.3) in small systems. The status of the tensionless state appears to be relegated to the category of acci- dental coincidences. An analogy may be drawn with the equation of state of liquid in equilibrium with its saturated vapor; certainly P = 1 is a particular point on the vapor pressure curve, but it has no particular physical meaning - in contrast with e.g. the triple point. The only physical consequence of zero in lateral tension is the special form of the structure factor10,1−4,5. However, even this has been questioned recently12,4. In a detailed investigation of the structure factor S(q) of a simulated bilayer, which should, for small q, asymptotically vary as 1/S ∼ kq4 + gq2 where k is the rigidity constant and g - the surface tension, we found g ≥ γ systematically4. The explanation advanced elsewhere12 is that γδA is not the correct work of deforming a bilayer initially planar, but gδAtrue is. Here Atrue is the true area of the interface or bilayer, obtained by following its surface, whereas A ≡ LxLy is the projected area. Since S(q) measures the spontaneous fluctuations, gq2 is correct and γq2 is not. On the other hand, when measuring the response of the system to an imposed change in the projected area, γδA is the correct free energy increment. The increase in the projected area is under control of the experimentalist; the change of the true area induced by the twists and bends of the bilayer, is not. By taking these arguments into account, the correct definition of the tensionless state was assumed12 to be g = 0 and interpolation procedures were used12 to determine the new correct tensionless state. Whether the details of the advanced explanation12 are correct or not, it is obvious that g parametrizes the spectrum S(q) of spontaneous shape fluctuations. The difference between g and γ was also discussed elsewhere13,6,15. The zero of γ(a) is unambiguous in small systems; in large system of reverse bilayers4 it falls within the narrow region of fastest change in slope, signalling the transition of the bilayer to the floppy state. Then the precision of its determination is doubtful. Some of this ambiguity is seen in Fig.4 and in Figures of Reference 4. Section IV. Discussion. We have noted in the Introduction that, to a physical chemist familiar with the interfaces, some properties of a bilayer are truly exotic. Let us list these properties: (1) the derivative given by eq.(1.1) depends on the specific area; (2) it can be negative; (3) extension of the area leads to a hole or tunnel as the bilayer resists further extension; (4) there is distinction between actual area and the projected area but the bilayer keeps its true area nearly constant; (5) as there are two areas, there are two interfacial tensions. Definitely this is not what we learn from classical textbooks on thermodynamics. The feature (1) has been seen in all simulations, including these over a wide range of (projected) areas1−6,9−15. The feature (2) has also been seen2−6,11,12,13,15. The feature (3) has not only been seen2,4,15 but also investigated15. Distinction (4) has been the subject of papers13,5 specifically devoted to it. Finally, the existence of an(other) surface tension(s), besides the lateral tension, has been repeatedly surmised5,6,11,13; equations have even been derived14. The explanation of such specific behaviour ultimately lies with intermolecular forces and steric effects which any future theory will have to take into account. The bilayers are made of amphiphilic molecules, each containing a hydrophilic part and a hydrophobic part (or more generally, ”solvophilic” and ”solvophobic”). These molecules use their very special setup of intermolecular forces to create a stable sheet of constant mass (and approximately constant area). This creates a new situation with a constraint of given total area almost conserved when the projected area is varied by varying the simulation The constraint of constant mass (or anything related to it) is entirely absent in fluctuating interfaces. The molecular mechanism of fluctuations of shape of e.g. a liquid- vapor interface, involves diffusion from either phase; a local protrusion or excursion of the interface takes place not only by deforming the existing surface but also by absorbing or releasing molecules from/to either phase. The surface of a perfect crystal in equilibrium with its vapor is a good example; all shape fluctuations are due to evaporation or condensation. Such processes clearly take part in shape fluctuations of a liquid surface as well. Hence an interface is a system open with respect to particle number. It is not so with the bilayer. Not only the solvent is virtually absent from the bilayer, but also the surfactant molecules making the bilayer never leave it. In principle the surfactant must be present in the solvent but the bilayer changes shape by changing the position of its molecules, not by exchanging them with the solvent solution. The shape fluctuations take place under the constraint of constant number of particles, i.e. constant mass. A constant mass is implicit in the elastic theory of solids and the links to that field have already been explored5,10. Conversely, application of the elastic theory of solids to liquid interfaces, (the so called ”drumhead model”), is clearly not quite correct by not allowing for diffusion described above. We discuss now the new evidence for the transition. All plots of γbb(a) and γ(a) show some kind of break into two branches, one corresponding to the ”floppy” bilayer. Although the word ”floppy” or ”buckling” appeared occasionally4,5, in fact the status of the floppy region and of the transition into that region are not very clearly estab- lished. There are suggestive analogies with the crumpling transition of solid (tethered) membranes or with coiling transitions of polymers. This would suggest it is a first-order transition. The breaks in γbb(a) (see Section III) strongly support such a hypothesis also suggesting it is an internall reorganization of the bilayer structure, in which the solvent plays a secondary role. Further support is found with the radius of gyration of the bilayer, shown in Fig.6; the break in slope is very clear. Finally, the issue of the other surface tension is not fully resolved. As we discussed above in Section III, one may define two surface tensions, one coupled to the projected area and the other coupled to something else, perhaps the true area. The first one is the lateral tension and the other may be that given by the virial expression derived or rather proposed recently14 or g parametrizing the structure factor S(q). (See the end of Section III). These last two may be equal or may be not. I hypothesize that g may be related to that given by the Zwanzig-Triezenberg (Z- T) equation16,7 - because the latter is derived by considering a spontaneous fluctuation. Now, in simulations of liquid-vapor interface, I have determined17 the surface tension from both expressions and these two numbers were equal within 0.01 percent. Thus I have confirmed the validity of the Schofield proof7 of the equivalence. Since the (Z-T) equation is derived in a grand-canonical ensemble, g can only be ”related to it” as stated above; the constraint would have to be introduced. We finally remark that the constraint appears as a most natural thing in description of polymers when the polymer string has a constant number of segments. Indeed a model one-dimensional membrane/bilayer embedded in two dimensions, is identical to a model of a polymer string on a plane. Appendix. Details of the Model Used . The simulations themselves were in the past and also recently based on the Verlet leap-frog algorithm with Nose-Hoover thermostat and were never shorter than half a million steps, about 0.9 million in almost all runs. The intermolecular potential energy was a sum over pairs of particles; all pairs were interacting Lennard-Jones 6-12 potentials u(r) = 4ǫ (σ/r)12 − (σ/r)6 with different parameters ǫ and σ and with different cutoffs rc. A constant u0 may be added at will and this makes no difference to forces. u(r) has a minimum at r = r∗ of depth u(r∗) = −ǫ. There are two kinds of spherical particles ”a” and ”b”. The solvent was made of particles ”a” and the amphiphilic molecules of chains ”abb..b”. The collision diameter σ was common to all pairs. The cutoff parameter was rc = 2.5σ for all like pairs and rc = r ∗ for unlike-pair intermolecular force. All potentials were cut-and-shifted to assure continuity of force. Molecular Dynamics algorithms use forces. The bilayer-forming amphiphilic molecules were freely-jointed chains and each in- tramolecular bond was made of two LJ halves thus: ubond = u(r) r < r ubond = u(2r ∗ − r) (r∗ < r < 2r∗) ubond = +∞ r > 2r for either ”a-b” or ”b-b” bond. As explained in the text, for the special case of ”reverse bilayers” made of dimers, a− b, the ”bb” intermolecular attractions were enhanced by making ǫbb >> ǫaa. Thus ”b”’s became ”heads” located in the center of the bilayer. Acknowledgements. This paper is dedicated to Professor Keith Gubbins on the occasion of his 70th birthday. The author gratefully acknowledges most useful interactions with Professor Keith Gubbins while visiting him at Cornell University. REFERENCES . 1 J. Stecki, Intl. J. Thermophysics 22,175 (2001). 2 J. Stecki, J. Chem. Phys. 120, 3508 (2004). 3 J. Stecki, J. Chem. Phys. Comm.122, 111102 (2005). 4 J. Stecki, J. Chem. Phys. 125, 154902 (2006). 5 W. K. den Otter, J. Chem. Phys.123, 214906 (2005). 6 Hiroshi Noguchi and Gerhard Gompper, Phys. Rev. E 73, 021903 (2006). 7 J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Clarendon, Ox- ford, 1982), page 89-90. 8 J. S. Rowlinson, Cohesion (Cambridge U.P., 2002). 9 R. Goetz and R. Lipowsky, J. Chem. Phys. 108, 7397 (1998). 10 G. Gompper, R. Goetz, and R. Lipowsky, Phys. Rev. Lett. 82, 221 (1999). 11 A. Imparato, J. C. Shilcock, and R. Lipowsky, Eur. Phys. J. E. 11, 21 (2003); see also Reference 13. 12 A. Imparato, J. C. Shilcock, and R. Lipowsky, Europhys. Lett. 69, 650 (2005). 13 A. Imparato, J. Chem. Phys. 124, 154714 (2006). 14 O. Farago, ibid. 119, 596 (2003); for further work on this model see O. Farago and P.Pincus, ibid. 120, 2934 (2004). 15 Ira R. Cooke and Marcus Deserno, cond-mat/0509218 (8 Sept.2006); Europhysics Letters, 16 D. G. Triezenberg and R. Zwanzig, Phys. Rev. Lett. 28, 1183 (1972). 17 see J. Stecki, J. Chem. Phys. 114, 7574 (2001); ibid. 108, 3788 (1998). http://arxiv.org/abs/cond-mat/0509218 Captions to Figures Caption to Figure 1 The general pattern of splits of lateral tension γ according to (3.3) is shown as plots against the area-per-head a; here for an intermediate size of simulated bilayer with Nm = 1800 molecules with tails l=4 segments long, with all collision diameters σ and all energies ǫ equal , at T = 1., Ns+(l+1)Nm = 49000, liquid density ρ = 0.89204. The ss contribution is marked with squares, the sb contribution - with stars, the bb contribution - with crosses , and the sum - with black circles . All data are quoted everywhere in LJ units reduced by the energy depth ǫ and collision diameter σ. The lines are there to guide the eye. Caption to Figure 2 The split of lateral tension γ plotted against the area-per-head a for two sizes (SM) and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules with tails l=4 segments long, with all collision diameters σ and all energies ǫ equal, at T = 1., Ns + (l + 1)Nm = 12250 (SM) and 105134 (B), liquid-like density ρ = 0.89204. With the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and squares (B) near 0.2-0.3; sb - plus signs (SM) and stars (B) near 0.5-0.7; bb - open circles (SM) and crosses (B) with negative slope for small a and positive slope for large a. The sum after (3.3), γ, is shown with thick lines, black squares (SM), and black circles (B). See text and Caption to Fig.1. Compare with Fig.3. Caption to Figure 3 The split of lateral tension γ plotted against the area-per-head a for two sizes (SM) and (B) of simulated bilayer, made of Nm = 450 (SM) and Nm = 3973 (B) molecules with tails l=8 segments long, with all collision diameters σ and all energies ǫ equal , at T = 1., N = Ns + (l+ 1)Nm = 14050 (SM) and 121026 (B), density ρ = 0.89204. With the notation of eq.(3.3), the contribution ss is shown with triangles (SM) and squares (big) near 0.2-0.3; sb - with plus signs (SM) and stars (big) near 0.5-1.1; bb - with open circles (SM) and crosses (big). Note the negative slope for small a and positive slope for large a. The sum after (3.3) is shown with thick lines, black squares (SM), and black circles (big). See text. Compare with Fig.2 and note: the unique breaks in γsb, almost constant slope in the region of floppy bilayer, translated into zero slope in total γ. Caption to Figure 4 Total lateral tension γ of reverse bilayers made of dimer molecules plotted against area-per-head a for biggest (B) or smallest (SM) size, and 3 cases: (a) extra repulsion added and head-head attraction enhanced (H-H); (b) only (H-H); (c) the cutoff of at- traction is a long-range rc = 3.2, not rc = 2.5; otherwise (a). The other parameters are: T = 1.9, ρ = 0.89024, Nm = 2238 dimers, N = Ns+2Nm = 40000(SM) and Nm = 5760 dimers, N = Ns + 2Nm = 160000 (B). These had areas near 36× 36 and near 55× 55. The points are: open squares - Ba; plus signs - SMa; black squares - Bb; crosses -SMb; stars -SMc. The lines are there to guide the eye only. The deviation of data-point at a ∼ 1.13 (black square) is due to the tunnel (hole). see Fig.5 and text. Caption to Figure 5 The split of lateral tension after (3.3) of systems shown in Fig.4. See caption to Fig.4 for parameters. Here the points are: case ”Ba” contribution ss - open squares, sb - black triangles, bb - pentagons; case ”SMa” and ss - plus signs, sb - open circles, bb -black triangles; case ”Bb” and ss - black squares, sb - open triangles, bb -black pentagons; case ”SMb” and ss - crosses, sb - black circles, bb -diamonds; case ”SMc” and ss - stars, sb - open triangles, bb - black diamonds. All ss contributions cluster near -0.5, all sb are > 2., and all bb show two regions interpreted as floppy bilayer and extended bilayer. See text. Caption to Figure 6 One example of the radius of gyration for the same bilayer whose γ is plotted in Fig.1, also plotted against area-per-head a. It appears to support, along with other correlations3,4, the hypothesis of a discontinuous transition between the extended and floppy bilayer. 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 1.4 1.6 1.8 2 2.2 2.4 2.6 0.9 0.95 1 1.05 1.1 1.15 1.2 0.9 0.95 1 1.05 1.1 1.15 1.2 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

0704.1478

Search for the radiative leptonic decay B+ --> gamma l+ nu

BABAR-PUB-06/067 SLAC-PUB-12442 Search for the Radiative Leptonic Decay B+ → γℓ+νℓ B. Aubert,1 M. Bona,1 D. Boutigny,1 Y. Karyotakis,1 J. P. Lees,1 V. Poireau,1 X. Prudent,1 V. Tisserand,1 A. Zghiche,1 E. Grauges,2 A. Palano,3 J. C. Chen,4 N. D. Qi,4 G. Rong,4 P. Wang,4 Y. S. Zhu,4 G. Eigen,5 I. Ofte,5 B. Stugu,5 G. S. Abrams,6 M. Battaglia,6 D. N. Brown,6 J. Button-Shafer,6 R. N. Cahn,6 Y. Groysman,6 R. G. Jacobsen,6 J. A. Kadyk,6 L. T. Kerth,6 Yu. G. Kolomensky,6 G. Kukartsev,6 D. Lopes Pegna,6 G. Lynch,6 L. M. Mir,6 T. J. Orimoto,6 M. Pripstein,6 N. A. Roe,6 M. T. Ronan,6, ∗ K. Tackmann,6 W. A. Wenzel,6 P. del Amo Sanchez,7 M. Barrett,7 T. J. Harrison,7 A. J. Hart,7 C. M. Hawkes,7 A. T. Watson,7 T. Held,8 H. Koch,8 B. Lewandowski,8 M. Pelizaeus,8 K. Peters,8 T. Schroeder,8 M. Steinke,8 J. T. Boyd,9 J. P. Burke,9 W. N. Cottingham,9 D. Walker,9 D. J. Asgeirsson,10 T. Cuhadar-Donszelmann,10 B. G. Fulsom,10 C. Hearty,10 N. S. Knecht,10 T. S. Mattison,10 J. A. McKenna,10 A. Khan,11 P. Kyberd,11 M. Saleem,11 D. J. Sherwood,11 L. Teodorescu,11 V. E. Blinov,12 A. D. Bukin,12 V. P. Druzhinin,12 V. B. Golubev,12 A. P. Onuchin,12 S. I. Serednyakov,12 Yu. I. Skovpen,12 E. P. Solodov,12 K. Yu Todyshev,12 M. Bondioli,13 M. Bruinsma,13 M. Chao,13 S. Curry,13 I. Eschrich,13 D. Kirkby,13 A. J. Lankford,13 P. Lund,13 M. Mandelkern,13 E. C. Martin,13 W. Roethel,13 D. P. Stoker,13 S. Abachi,14 C. Buchanan,14 S. D. Foulkes,15 J. W. Gary,15 O. Long,15 B. C. Shen,15 L. Zhang,15 E. J. Hill,16 H. P. Paar,16 S. Rahatlou,16 V. Sharma,16 J. W. Berryhill,17 C. Campagnari,17 A. Cunha,17 B. Dahmes,17 T. M. Hong,17 D. Kovalskyi,17 J. D. Richman,17 T. W. Beck,18 A. M. Eisner,18 C. J. Flacco,18 C. A. Heusch,18 J. Kroseberg,18 W. S. Lockman,18 G. Nesom,18 T. Schalk,18 B. A. Schumm,18 A. Seiden,18 D. C. Williams,18 M. G. Wilson,18 L. O. Winstrom,18 J. Albert,19 E. Chen,19 C. H. Cheng,19 A. Dvoretskii,19 F. Fang,19 D. G. Hitlin,19 I. Narsky,19 T. Piatenko,19 F. C. Porter,19 G. Mancinelli,20 B. T. Meadows,20 K. Mishra,20 M. D. Sokoloff,20 F. Blanc,21 P. C. Bloom,21 S. Chen,21 W. T. Ford,21 J. F. Hirschauer,21 A. Kreisel,21 M. Nagel,21 U. Nauenberg,21 A. Olivas,21 J. G. Smith,21 K. A. Ulmer,21 S. R. Wagner,21 J. Zhang,21 A. Chen,22 E. A. Eckhart,22 A. Soffer,22 W. H. Toki,22 R. J. Wilson,22 F. Winklmeier,22 Q. Zeng,22 D. D. Altenburg,23 E. Feltresi,23 A. Hauke,23 H. Jasper,23 J. Merkel,23 A. Petzold,23 B. Spaan,23 T. Brandt,24 V. Klose,24 H. M. Lacker,24 W. F. Mader,24 R. Nogowski,24 J. Schubert,24 K. R. Schubert,24 R. Schwierz,24 J. E. Sundermann,24 A. Volk,24 D. Bernard,25 G. R. Bonneaud,25 E. Latour,25 Ch. Thiebaux,25 M. Verderi,25 P. J. Clark,26 W. Gradl,26 F. Muheim,26 S. Playfer,26 A. I. Robertson,26 Y. Xie,26 M. Andreotti,27 D. Bettoni,27 C. Bozzi,27 R. Calabrese,27 G. Cibinetto,27 E. Luppi,27 M. Negrini,27 A. Petrella,27 L. Piemontese,27 E. Prencipe,27 F. Anulli,28 R. Baldini-Ferroli,28 A. Calcaterra,28 R. de Sangro,28 G. Finocchiaro,28 S. Pacetti,28 P. Patteri,28 I. M. Peruzzi,28, † M. Piccolo,28 M. Rama,28 A. Zallo,28 A. Buzzo,29 R. Contri,29 M. Lo Vetere,29 M. M. Macri,29 M. R. Monge,29 S. Passaggio,29 C. Patrignani,29 E. Robutti,29 A. Santroni,29 S. Tosi,29 K. S. Chaisanguanthum,30 M. Morii,30 J. Wu,30 R. S. Dubitzky,31 J. Marks,31 S. Schenk,31 U. Uwer,31 D. J. Bard,32 P. D. Dauncey,32 R. L. Flack,32 J. A. Nash,32 M. B. Nikolich,32 W. Panduro Vazquez,32 P. K. Behera,33 X. Chai,33 M. J. Charles,33 U. Mallik,33 N. T. Meyer,33 V. Ziegler,33 J. Cochran,34 H. B. Crawley,34 L. Dong,34 V. Eyges,34 W. T. Meyer,34 S. Prell,34 E. I. Rosenberg,34 A. E. Rubin,34 A. V. Gritsan,35 A. G. Denig,36 M. Fritsch,36 G. Schott,36 N. Arnaud,37 M. Davier,37 G. Grosdidier,37 A. Höcker,37 V. Lepeltier,37 F. Le Diberder,37 A. M. Lutz,37 S. Pruvot,37 S. Rodier,37 P. Roudeau,37 M. H. Schune,37 J. Serrano,37 A. Stocchi,37 W. F. Wang,37 G. Wormser,37 D. J. Lange,38 D. M. Wright,38 C. A. Chavez,39 I. J. Forster,39 J. R. Fry,39 E. Gabathuler,39 R. Gamet,39 K. A. George,39 D. E. Hutchcroft,39 D. J. Payne,39 K. C. Schofield,39 C. Touramanis,39 A. J. Bevan,40 F. Di Lodovico,40 W. Menges,40 R. Sacco,40 G. Cowan,41 H. U. Flaecher,41 D. A. Hopkins,41 P. S. Jackson,41 T. R. McMahon,41 F. Salvatore,41 A. C. Wren,41 D. N. Brown,42 C. L. Davis,42 J. Allison,43 N. R. Barlow,43 R. J. Barlow,43 Y. M. Chia,43 C. L. Edgar,43 G. D. Lafferty,43 T. J. West,43 J. C. Williams,43 J. I. Yi,43 C. Chen,44 W. D. Hulsbergen,44 A. Jawahery,44 C. K. Lae,44 D. A. Roberts,44 G. Simi,44 G. Blaylock,45 C. Dallapiccola,45 S. S. Hertzbach,45 X. Li,45 T. B. Moore,45 E. Salvati,45 S. Saremi,45 R. Cowan,46 G. Sciolla,46 S. J. Sekula,46 M. Spitznagel,46 F. Taylor,46 R. K. Yamamoto,46 H. Kim,47 S. E. Mclachlin,47 P. M. Patel,47 S. H. Robertson,47 A. Lazzaro,48 V. Lombardo,48 F. Palombo,48 J. M. Bauer,49 L. Cremaldi,49 V. Eschenburg,49 R. Godang,49 R. Kroeger,49 D. A. Sanders,49 D. J. Summers,49 H. W. Zhao,49 S. Brunet,50 D. Côté,50 M. Simard,50 P. Taras,50 F. B. Viaud,50 H. Nicholson,51 N. Cavallo,52, ‡ G. De Nardo,52 F. Fabozzi,52, ‡ C. Gatto,52 L. Lista,52 D. Monorchio,52 P. Paolucci,52 D. Piccolo,52 C. Sciacca,52 M. A. Baak,53 G. Raven,53 H. L. Snoek,53 C. P. Jessop,54 http://arxiv.org/abs/0704.1478v1 J. M. LoSecco,54 G. Benelli,55 L. A. Corwin,55 K. K. Gan,55 K. Honscheid,55 D. Hufnagel,55 P. D. Jackson,55 H. Kagan,55 R. Kass,55 J. P. Morris,55 A. M. Rahimi,55 J. J. Regensburger,55 R. Ter-Antonyan,55 Q. K. Wong,55 N. L. Blount,56 J. Brau,56 R. Frey,56 O. Igonkina,56 J. A. Kolb,56 M. Lu,56 C. T. Potter,56 R. Rahmat,56 N. B. Sinev,56 D. Strom,56 J. Strube,56 E. Torrence,56 A. Gaz,57 M. Margoni,57 M. Morandin,57 A. Pompili,57 M. Posocco,57 M. Rotondo,57 F. Simonetto,57 R. Stroili,57 C. Voci,57 E. Ben-Haim,58 H. Briand,58 J. Chauveau,58 P. David,58 L. Del Buono,58 Ch. de la Vaissière,58 O. Hamon,58 B. L. Hartfiel,58 Ph. Leruste,58 J. Malclès,58 J. Ocariz,58 L. Gladney,59 M. Biasini,60 R. Covarelli,60 C. Angelini,61 G. Batignani,61 S. Bettarini,61 G. Calderini,61 M. Carpinelli,61 R. Cenci,61 F. Forti,61 M. A. Giorgi,61 A. Lusiani,61 G. Marchiori,61 M. A. Mazur,61 M. Morganti,61 N. Neri,61 E. Paoloni,61 G. Rizzo,61 J. J. Walsh,61 M. Haire,62 D. Judd,62 D. E. Wagoner,62 J. Biesiada,63 P. Elmer,63 Y. P. Lau,63 C. Lu,63 J. Olsen,63 A. J. S. Smith,63 A. V. Telnov,63 F. Bellini,64 G. Cavoto,64 A. D’Orazio,64 D. del Re,64 E. Di Marco,64 R. Faccini,64 F. Ferrarotto,64 F. Ferroni,64 M. Gaspero,64 L. Li Gioi,64 M. A. Mazzoni,64 S. Morganti,64 G. Piredda,64 F. Polci,64 F. Safai Tehrani,64 C. Voena,64 M. Ebert,65 H. Schröder,65 R. Waldi,65 T. Adye,66 B. Franek,66 E. O. Olaiya,66 S. Ricciardi,66 F. F. Wilson,66 R. Aleksan,67 S. Emery,67 A. Gaidot,67 S. F. Ganzhur,67 G. Hamel de Monchenault,67 W. Kozanecki,67 M. Legendre,67 G. Vasseur,67 Ch. Yèche,67 M. Zito,67 X. R. Chen,68 H. Liu,68 W. Park,68 M. V. Purohit,68 J. R. Wilson,68 M. T. Allen,69 D. Aston,69 R. Bartoldus,69 P. Bechtle,69 N. Berger,69 R. Claus,69 J. P. Coleman,69 M. R. Convery,69 J. C. Dingfelder,69 J. Dorfan,69 G. P. Dubois-Felsmann,69 D. Dujmic,69 W. Dunwoodie,69 R. C. Field,69 T. Glanzman,69 S. J. Gowdy,69 M. T. Graham,69 P. Grenier,69 V. Halyo,69 C. Hast,69 T. Hryn’ova,69 W. R. Innes,69 M. H. Kelsey,69 P. Kim,69 D. W. G. S. Leith,69 S. Li,69 S. Luitz,69 V. Luth,69 H. L. Lynch,69 D. B. MacFarlane,69 H. Marsiske,69 R. Messner,69 D. R. Muller,69 C. P. O’Grady,69 V. E. Ozcan,69 A. Perazzo,69 M. Perl,69 T. Pulliam,69 B. N. Ratcliff,69 A. Roodman,69 A. A. Salnikov,69 R. H. Schindler,69 J. Schwiening,69 A. Snyder,69 J. Stelzer,69 D. Su,69 M. K. Sullivan,69 K. Suzuki,69 S. K. Swain,69 J. M. Thompson,69 J. Va’vra,69 N. van Bakel,69 A. P. Wagner,69 M. Weaver,69 W. J. Wisniewski,69 M. Wittgen,69 D. H. Wright,69 H. W. Wulsin,69 A. K. Yarritu,69 K. Yi,69 C. C. Young,69 P. R. Burchat,70 A. J. Edwards,70 S. A. Majewski,70 B. A. Petersen,70 L. Wilden,70 S. Ahmed,71 M. S. Alam,71 R. Bula,71 J. A. Ernst,71 V. Jain,71 B. Pan,71 M. A. Saeed,71 F. R. Wappler,71 S. B. Zain,71 W. Bugg,72 M. Krishnamurthy,72 S. M. Spanier,72 R. Eckmann,73 J. L. Ritchie,73 C. J. Schilling,73 R. F. Schwitters,73 J. M. Izen,74 X. C. Lou,74 S. Ye,74 F. Bianchi,75 F. Gallo,75 D. Gamba,75 M. Pelliccioni,75 M. Bomben,76 L. Bosisio,76 C. Cartaro,76 F. Cossutti,76 G. Della Ricca,76 L. Lanceri,76 L. Vitale,76 V. Azzolini,77 N. Lopez-March,77 F. Martinez-Vidal,77 A. Oyanguren,77 Sw. Banerjee,78 B. Bhuyan,78 K. Hamano,78 R. Kowalewski,78 I. M. Nugent,78 J. M. Roney,78 R. J. Sobie,78 J. J. Back,79 P. F. Harrison,79 T. E. Latham,79 G. B. Mohanty,79 M. Pappagallo,79, § H. R. Band,80 X. Chen,80 S. Dasu,80 K. T. Flood,80 J. J. Hollar,80 P. E. Kutter,80 B. Mellado,80 Y. Pan,80 M. Pierini,80 R. Prepost,80 S. L. Wu,80 Z. Yu,80 and H. Neal81 (The BABAR Collaboration) 1Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France 2Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain 3Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy 4Institute of High Energy Physics, Beijing 100039, China 5University of Bergen, Institute of Physics, N-5007 Bergen, Norway 6Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 7University of Birmingham, Birmingham, B15 2TT, United Kingdom 8Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany 9University of Bristol, Bristol BS8 1TL, United Kingdom 10University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 11Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom 12Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 13University of California at Irvine, Irvine, California 92697, USA 14University of California at Los Angeles, Los Angeles, California 90024, USA 15University of California at Riverside, Riverside, California 92521, USA 16University of California at San Diego, La Jolla, California 92093, USA 17University of California at Santa Barbara, Santa Barbara, California 93106, USA 18University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 19California Institute of Technology, Pasadena, California 91125, USA 20University of Cincinnati, Cincinnati, Ohio 45221, USA 21University of Colorado, Boulder, Colorado 80309, USA 22Colorado State University, Fort Collins, Colorado 80523, USA 23Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany 24Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany 25Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France 26University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 27Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy 28Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy 29Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy 30Harvard University, Cambridge, Massachusetts 02138, USA 31Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 32Imperial College London, London, SW7 2AZ, United Kingdom 33University of Iowa, Iowa City, Iowa 52242, USA 34Iowa State University, Ames, Iowa 50011-3160, USA 35Johns Hopkins University, Baltimore, Maryland 21218, USA 36Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany 37Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France 38Lawrence Livermore National Laboratory, Livermore, California 94550, USA 39University of Liverpool, Liverpool L69 7ZE, United Kingdom 40Queen Mary, University of London, E1 4NS, United Kingdom 41University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom 42University of Louisville, Louisville, Kentucky 40292, USA 43University of Manchester, Manchester M13 9PL, United Kingdom 44University of Maryland, College Park, Maryland 20742, USA 45University of Massachusetts, Amherst, Massachusetts 01003, USA 46Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA 47McGill University, Montréal, Québec, Canada H3A 2T8 48Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy 49University of Mississippi, University, Mississippi 38677, USA 50Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 51Mount Holyoke College, South Hadley, Massachusetts 01075, USA 52Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy 53NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands 54University of Notre Dame, Notre Dame, Indiana 46556, USA 55Ohio State University, Columbus, Ohio 43210, USA 56University of Oregon, Eugene, Oregon 97403, USA 57Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy 58Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France 59University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 60Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy 61Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy 62Prairie View A&M University, Prairie View, Texas 77446, USA 63Princeton University, Princeton, New Jersey 08544, USA 64Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy 65Universität Rostock, D-18051 Rostock, Germany 66Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 67DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France 68University of South Carolina, Columbia, South Carolina 29208, USA 69Stanford Linear Accelerator Center, Stanford, California 94309, USA 70Stanford University, Stanford, California 94305-4060, USA 71State University of New York, Albany, New York 12222, USA 72University of Tennessee, Knoxville, Tennessee 37996, USA 73University of Texas at Austin, Austin, Texas 78712, USA 74University of Texas at Dallas, Richardson, Texas 75083, USA 75Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy 76Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy 77IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 78University of Victoria, Victoria, British Columbia, Canada V8W 3P6 79Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 80University of Wisconsin, Madison, Wisconsin 53706, USA 81Yale University, New Haven, Connecticut 06511, USA (Dated: November 5, 2018) We present the results of a search for B+ → γℓ+νℓ, where ℓ = e, µ. We use a sample of 232 million BB̄ pairs recorded at the Υ (4S) with the BABAR detector at the PEP-II B Factory. We measure a partial branching fraction ∆B in a restricted region of phase space that reduces the effect of theoretical uncertainties, requiring the lepton energy to be between 1.875 and 2.850GeV, the photon energy to be between 0.45 and 2.35GeV, and the cosine of the angle between the lepton and photon momenta to be less than −0.36, with all quantities computed in the Υ (4S) center-of-mass frame. We find ∆B(B+ → γℓ+νℓ) = (−0.3 −1.5(stat)± 0.6(syst)± 0.1(th))× 10 −6, assuming lepton universality. Interpreted as a 90% C.L. Bayesian upper limit, the result corresponds to 1.7 × 10−6 for a prior flat in amplitude, and 2.3× 10−6 for a prior flat in branching fraction. PACS numbers: 13.20.He, 13.30.Ce, 12.38.Qk, 14.40.Nd At tree level, the branching fraction (BF) for radiative leptonic B decays is given by: B(B+ → γℓ+νℓ) = α G2F |Vub| 288π2 f2BτBm where mB is the B + meson mass, mb is the MS b quark mass, τB is the B + meson lifetime, fB is the B meson decay constant, Qi is the charge of quark flavor i and λB is the first inverse moment of the B light-cone distribu- tion amplitude [1, 2], a quantity that enters into theo- retical calculations [3] of the BF of hadronic B decays such as B → ππ, and is typically taken to be of the order of ΛQCD. Thus, a measurement of B(B + → γℓ+νℓ) can provide a determination of λB free of hadronic final-state uncertainties. The best current 90% C.L. upper limit on the full BF is 5.2× 10−5[4], for B+ → γµ+νµ. However, Eq.(1) is based on the assumption that the factorization relation for the vector and axial-vector form factors is valid over the entire phase space. Instead, one can relate, at tree-level, λB to a partial BF, ∆B, over a restricted region of phase space [5]: ∆B = α G2F |Vub| f2BτBm a+ bL+ cL2 , (2) where L = (mB/3)(1/λB + 1/(2mb)), the first term de- scribes the effects of photon radiation from the lepton, the third term the internal photon emission, and second their interference. The constants a, b, and c can be pre- dicted model-independently using factorization at large photon energy, the kinematic region for our analysis. We present herein the results of a search for charged B meson decays B+ → γℓ+νℓ, where ℓ = e, µ (“elec- tron channel”,“muon channel”)[6]. Our measurements are based on a sample of 232 million BB pairs recorded with the BABAR detector [7] at the PEP-II asymmetric- energy e+e− storage rings, comprising an integrated lu- minosity of 210.5 fb−1 collected at the Υ (4S) resonance (“on-peak”). We also use 21.6 fb−1 recorded approxi- mately 40 MeV below the Υ (4S) (“off-peak”). The analysis procedure consists of selecting a lepton and photon recoiling against a reconstructedB, and iden- tifying signal candidates by reconstructing the neutrino using missing energy and momentum. We use a variety of selection criteria, optimized using Monte Carlo (MC) samples, to discriminate signal from background. We then extract the number of signal events in data using a binned maximum-likelihood (ML) fit. The backgrounds are divided into three categories: continuum (non-BB), specific exclusive b → uℓνℓ decays, and “generic B” decays, defined as a combination of all B hadronic decays, b → cℓνℓ decays, and the remain- ing inclusive b → uℓνℓ decays. In particular, we study the seven exclusive b → uℓνℓ modes: B + → h0ℓ+νℓ (h0 = π0, ρ0, η, η′, ω) and B0 → h−ℓ+νℓ (h − = π−, ρ−), referred to below, for each h, as the “h mode”. Our signal MC samples were generated using the tree- level model of Ref. [1]. The π0 and π± mode samples were generated using the form factor parameterization of Ref. [8], with the value of the shape parameter based on lattice QCD results [9]. Light cone sum rule-based form factor models were used to generate samples for the ρ0, ρ±, and ω modes [10], and η and η′ modes [11]. We find an excess of events in the off-peak data com- pared to continuum MC (e+e− → qq, τ+τ−, and in the muon channel, µ+µ−γ), with the excess more pronounced in the electron channel. This is likely to result from unmodeled higher-order QED and hadronic two-photon events. We thus use off-peak data instead of continuum MC to represent continuum background in our analysis. We take as the signal lepton and photon the high- est center-of-mass (CM) energy electron (muon) and the highest CM energy photon candidate in each event. The remaining charged tracks, each assigned a pion mass, and neutral clusters, treated as photons, are assigned to the “recoil B” candidate. We reconstruct the recoil B in two ways: we construct an “unscaled” recoil momentum as the sum of the CM 3-momenta of its constituents, and we define a “scaled” recoil momentum in the direction of the unscaled recoil, with its magnitude determined from the CM energy of the Υ (4S) and the B± mass. Using either the scaled or unscaled momentum, we reconstruct the 3-momentum of a corresponding scaled or unscaled signal neutrino candidate. The reconstructed neutrino CM energy is calculated as the difference between the CM beam energy and the sum of the lepton and photon candidates’ CM energies. We optimize a set of selection criteria for the best sig- nal sensitivity at a significance of 3σ using MC samples, splitting each in half, with one sample used for the opti- mization and the other used to evaluate its performance. On the signal side, we require that the electron (muon) have a CM energy between 2.00 and 2.85 (1.875 and 2.775)GeV. We require that the photon have a CM en- ergy between 0.65 and 2.35 (0.45 and 2.35) GeV. We define cos θℓγ to be the cosine of the angle between the lepton and photon in the CM frame, and require its value to be less than −0.42 (−0.36). We require −1.10(−1.05) < cos θBY < 1.10(1.00), where cos θBY is the cosine of the angle between the signal B and the lepton-photon combination Y in the CM frame [12], com- puted from the known B mass, the beam energy, and the 3-momenta of the signal lepton and photon. In order to reduce background from neutral hadrons, we require the lateral moment [13] of the electromagnetic calorimeter energy distribution of the signal photon can- didate to be less than 0.55 for both channels. The polar angle of the photon candidate in the laboratory frame is required to be between 0.326 and 2.443 rad for both channels. We pair the candidate with every other neutral cluster in the event and reject events with a pair invariant mass in the π0 mass range 123–147 (116–148) MeV. We require the difference between the total CM energy of the recoil B constituents and the CM beam energy to be between −5.0 and 0.9 (−2.5 and 0.7) GeV. For the neutrino reconstruction, we require that both the scaled and unscaled neutrino polar angle in the laboratory frame be between 0.300 and 2.443 rad for both channels. To reduce continuum background, we require the ratio of the second to zeroth Fox-Wolfram moment [14] of all charged tracks and neutral clusters to be less than 0.5, and the absolute value of the cosine of the angle between the CM thrust axes of the recoil B and the lepton-photon system be less than 0.98 (0.86). We use a Fisher discrim- inant, F ≡ a0L0+a2L2, calculated from the momentum- weighted zeroth and second Legendre moments, L0 and L2, of the recoil B about the lepton-photon CM thrust axis, with coefficients a0 and a2 equal to 0.43 and −1.86 (0.008 and −1.590), respectively. F is required to be greater than 1.50 (0.310). In the electron channel, we veto two-photon events via the charge-angle correlation of the signal lepton aris- ing from the initial state. For a positively (negatively)- charged signal electron, we require the cosine of its CM polar angle to be between −0.74 and 0.78 (−0.94 and 0.70). In the muon channel, we require this variable to be between −1.00 and 0.78 for both charges. These crite- ria were optimized on a loosely-selected sample of events, where the off-peak data are used for the continuum, and the MC for the signal and other backgrounds. We also reject two-photon events using a parameter- ized combination of the missing CM momentum in the beam direction and the invariant mass of the hypotheti- cal two-photon system. For the muon channel, the entire observed event is taken as the two-photon system, while (GeV)ESm 4.9 4.95 5 5.05 5.1 5.15 5.2 5.25 5.3 FIG. 1: Electron-channel ∆EP vs. mES signal MC, using a color scale to represent relative contents of each bin. for the electron channel, the signal electron is assumed to be from the initial state, and so is excluded from the two-photon system. The selection criterion was adjusted to preserve a 94% efficiency for signal for both channels. After applying our selection criteria, we use the two- dimensional distribution of ∆EP , the difference between the scaled neutrino candidate’s CM energy and the mag- nitude of its 3-momentum, and mES, the invariant mass of the recoil B, calculated from its unscaled CM 3- momentum and the CM beam energy, as inputs to the ML fit. These distributions provide distinct signatures for signal, B background, and continuum, with the sig- nal distribution shown in Fig. 1. The signal (S) and three sideband (B1, B2, B3) regions were selected to maximize separation of signal fromBB and continuum background. We extract signal events by fitting on-peak data for the contributions of signal and background, while allowing the predicted shapes of signal and background to vary within statistical uncertainties. The scale of signal and generic B contributions are allowed to vary, while the scale of off-peak data is fixed using the on-peak/off-peak luminosity ratio. For the seven semileptonic (SL) modes, we fit for three of the BFs and relate the other four to them as follows: The π± and ρ± mode BFs are obtained from BABAR measurements [12], and the η mode BF is obtained from CLEO [15]. The charged and neutral π and ρ modes are related by the lifetime ratio, τB±/τB0 = 1.071 ± 0.009 [16], and an isospin factor of 2. The ω mode BF is taken as equal to the ρ0 mode BF. We take the ratio of the η to η′ mode BFs to be 2.057± 0.020[17]. We maximize a likelihood function consisting of the product of four Poisson probability distribution functions (PDFs), modeling the total counts in each of the four regions, three Gaussian PDFs for the BFs of the three SL modes, and 40 Poisson PDFs for the 4-region shapes of the various samples. All of the shapes are obtained from MC, except for continuum, where off-peak data are used, introducing a larger statistical uncertainty. Each Poisson PDF that models the total count in one of the four fit regions has a measured value obtained from the on-peak data count, and an expected value based on the fitted contributions of signal and background, in- cluding fitted variations of the shapes. For the seven SL modes (where three of the fitted BFs are indepen- dent), the variances of the three Gaussian likelihoods are obtained from the published statistical and experimental systematic uncertainties, combined in quadrature. In all, there are 47 PDFs, and 45 free parameters. We fit for the partial BF ∆B for the kinematic region with lepton CM energy between 1.875 and 2.850GeV, photon CM energy between 0.45 and 2.35GeV, and cos θℓγ less than −0.36 — the union of the electron and muon channel regions. We perform three fits: separate electron and muon channel fits, and a joint fit in which the signal and three SL BFs are constrained to be equal for the two channels. For each fit, errors on the fitted sig- nal BF are obtained by finding the two values at which the signal BF likelihood decreased by a factor of e−1/2. Table I shows the results from the joint fit. TABLE I: Comparison of fit results and experimental obser- vations for the joint fit to the muon and electron channels. For each of the four fit regions, the individual fitted contri- butions from continuum (cont.), BB background, and signal are shown, along with their total. The on-peak and off-peak (scaled to the integrated on-peak luminosity) observations are shown for comparison, in indented rows, and are not included in the “Total fit” value shown. Muon channel S B1 B2 B3 Fit cont. 20.0±11.8 116.3±14.7 42.6±12.8 213.2±42.1 Off-peak 23.0±16.2 158.1±40.8 17.4±12.3 219.7±45.8 Fit BB 59.1± 8.5 61.0± 9.9 61.7± 9.8 286.6±46.6 Fit signal −5.2±13.8 −1.3± 3.4 −0.4± 1.0 −0.2± 0.5 Total fit 74.0± 8.1 176.0±12.4 103.9± 9.8 500.0±22.1 On-peak 73.0± 8.5 170.0±13.0 111.0±10.5 498.0±22.3 Electron channel S B1 B2 B3 Fit cont. 55.4±20.5 181.1±16.2 48.9±14.1 356.7±54.4 Off-peak 41.4±20.7 239.7±48.9 79.0±27.9 294.5±52.9 Fit BB 69.2± 8.5 59.2± 8.5 140.1±15.5 393.8±57.2 Fit signal −8.4±22.3 −1.5± 3.9 −1.2± 3.3 −0.4± 1.0 Total fit 116.2±10.3 238.7±14.5 187.7±12.5 750.2±26.5 On-peak 119.0±10.9 231.0±15.2 176.0±13.3 764.0±27.6 Table II shows all systematic uncertainties on ∆B ex- cept for theoretical uncertainties on the signal model, which are shown in Table III. The experimental systematic errors result from un- certainties on the data/MC consistency with respect to tracking efficiency of the signal lepton, particle identifica- tion efficiency of the signal lepton, reconstruction of the signal photon energy, selection criteria efficiency uncer- tainties, and uncertainties on the data/MC consistency in the shape of ∆EP andmES. All of these were evaluated using a number of control samples, including µ+µ−γ, e+e−γ, and B+ → π+D0(→ K+π−). The uncertainty on the number of produced BB pairs is 1.1%. In our fits, we further assumed a charged-to- neutral B production ratio of 1.0, and determine the systematic uncertainty by varying within the measured interval 1.020± 0.034 [16]. For systematic errors due to the theoretical uncertain- ties on the π and ρ mode BFs and form factor models, we refit, applying correlated variations in the BFs and form factor models, and take the magnitude of the largest change in signal BF as the associated systematic. For systematic errors arising from the theoretical un- certainties on the η and η′ mode BFs, we vary the as- sumed BFs by ±10%[15] and refit to obtain a systematic error. We find a negligible systematic from the uncer- tainty on the ratio of the η to η′ mode BFs. In the generic B sample, a significant fraction of events contributing to our fit are non-resonant B → Xuℓ events. We obtain the systematic error due to uncer- tainty on the B → Xuℓ +νℓ BF by fixing the total contri- bution of generic B decays, as predicted by MC. TABLE II: Systematic uncertainties on ∆B. All additive sys- tematic values have been multiplied by 106. Multiplicative Muon Electron Joint Tracking efficiency 1.3% 1.3% 1.3% Particle ID 3.5% 2.2% 2.1% Neutral reconstruction 1.6% 1.6% 1.6% Selection efficiency 6.0% 5.0% 6.0% B counting 1.1% 1.1% 1.1% Charged to neutral B ratio 9.4% 9.4% 9.4% Additive Shape of ∆EP vs. mES 0.3 0.2 0.3 η mode BF 0.3 0.1 0.2 π,ρ mode BF, ff 0.3 0.4 0.4 B → Xuℓ νℓ BF 0.4 0.2 0.3 The theoretical uncertainty within the kinematic re- gion of ∆B is conservatively estimated by evaluating the change in efficiency when the model of Ref. [1] is modified by setting the axial vector form factor equal to zero. The results for ∆B are given in Tables III and IV. We determine 90% C.L. Bayesian upper limits by integrating the signal BF likelihood with two different priors, both of which take values of 0 for negative values of the signal BF: a prior flat in the BF (“flat BF prior”), and a prior flat in the square root of the BF (“flat amplitude prior”), equivalent to assuming a flat prior for |Vub| or fB. For our kinematic region, the constants a, b, and c of Eq.(2) are 0.88, −3.24, and 3.25, respectively[5]. Us- ing input values of fB = 216MeV [18], |Vub| = 4.31 × 10−3 [16], τB = 1.638 ps [16], and mb = 4.20GeV [16], TABLE III: Comparison of ∆B two-sided results for all three fits. All values have been multiplied by 106. All Central Statistical Systematic Theoretical values ×106 value uncertainty uncertainty uncertainty Muon −1.33 +1.74 +0.80 −2.20 −0.87 Electron 0.11 +1.73 +0.61 −2.13 −0.59 Joint −0.25 +1.33 +0.60 −1.53 −0.64 TABLE IV: The 90% Bayesian upper-limits for all three fits, for the two different choices of prior, in terms of ∆B. Prior flat in amplitude Prior flat in BF Muon < 1.5× 10−6 < 2.1× 10−6 Electron < 2.2× 10−6 < 2.8× 10−6 Joint < 1.7× 10−6 < 2.3× 10−6 our 90% C.L. Bayesian limits on ∆B correspond to val- ues of λB of > 669MeV and > 591MeV, for the choice of the flat amplitude and flat BF priors, respectively. Given a theoretical model, a measurement of ∆B may be converted into an estimate of the total BF. In the model of Ref. [1], the result of the joint fit corresponds to a BF of (−0.6+3.0−3.4(stat) −1.4(syst)) × 10 −6, and 90% C.L. Bayesian upper limits of 3.8× 10−6 and 5.0× 10−6 for the flat amplitude and flat BF priors, respectively. We thank D. Pirjol for help on the signal model, and C. S. Kim and Y. Yang for advice on the η mode. We are grateful for the excellent luminosity and machine con- ditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organi- zations that support BABAR. The collaborating institu- tions wish to thank SLAC for its support and kind hospi- tality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS- IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), MEC (Spain), and PPARC (United Kingdom). Individuals have received support from the Marie Curie EIF (European Union) and the A. P. Sloan Foundation. ∗ Deceased † Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ‡ Also with Università della Basilicata, Potenza, Italy § Also with IPPP, Physics Department, Durham Univer- sity, Durham DH1 3LE, United Kingdom [1] G. P. Korchemsky, D. Pirjol, and T. M. Yan, Phys. Rev. D 61, 114510 (2000). [2] S. Descotes-Genon and C. T. Sachrajda, Nucl. Phys. B 650, 356 (2003). [3] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachra- jda, Phys. Rev. Lett. 83, 1914 (1999). [4] T. E. Browder et al. [CLEO Collaboration], Phys. Rev. D 56, 11 (1997). [5] The calculation of the partial BF and its dependence on λB , based on Ref. [1], was performed by D. Pirjol. [6] Charge-conjugate modes are included implicitly. [7] B. Aubert et al. [BABAR Collaboration], Nucl. Instrum. Meth. A 479, 1 (2002). [8] D. Becirevic and A. B. Kaidalov, Phys. Lett. B 478, 417 (2000). [9] M. Okamoto et al., Nucl. Phys. Proc. Suppl. 140, 461 (2005). [10] P. Ball and R. Zwicky, Phys. Rev. D 71, 014029 (2005). [11] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005). [12] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 72, 051102 (2005). [13] A. Drescher et al., Nucl. Instrum. Meth. A 237, 464 (1985). [14] G. C. Fox and S. Wolfram, Phys. Rev. Lett. 41, 1581 (1978). [15] S. B. Athar et al. [CLEO Collaboration], Phys. Rev. D 68, 072003 (2003). [16] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006). [17] C. S. Kim and Y. D. Yang, Phys. Rev. D 65, 017501 (2002). The uncertainty on this factor was determined in consultation with the authors. [18] A. Gray et al. [HPQCD Collaboration], Phys. Rev. Lett. 95, 212001 (2005). References

0704.1479

Possible experimental manifestations of the many-body localization

Possible experimental manifestations of the many-body localization D. M. Basko,1, ∗ I. L. Aleiner,1 and B. L. Altshuler1, 2 Physics Department, Columbia University, New York, NY 10027, USA NEC-Laboratories America, Inc., 4 Independence Way, Princeton, NJ 085540 USA Recently, it was predicted that if all one-electron states in a non-interacting disordered system are localized, the interaction between electrons in the absence of coupling to phonons leads to a finite- temperature metal-insulator transition. Here, we show that even in the presence of a weak coupling to phonons the transition manifests itself (i) in the nonlinear conduction, leading to a bistable I-V curve, (ii) by a dramatic enhancement of the nonequilibrium current noise near the transition. Introduction.— Low-temperature charge transport in disordered conductors is governed by the interplay be- tween elastic scattering of electrons off static disorder (impurities) and inelastic scattering (electron-electron, electron-phonon, etc.). At low dimensions an arbitrarily weak disorder localizes [1] all single-electron states [2, 3], and there would be no transport without inelastic pro- cesses. For the electron-phonon scattering, the dc con- ductivity σ(T ) at low temperatures T is known since long ago [4]: in d dimensions lnσ(T ) ∝ −1/T γ, γ = 1/(d+ 1) . (1) What happens if the only possible inelastic process is electron-electron scattering? The answer to this ques- tion was found only recently [5]: σ(T ) = 0 identically for T < Tc, the temperature of a metal-insulator transition. Here we discuss experimental manifestations of this tran- sition in real systems, where both electron-electron and electron-phonon interactions are present. We show that (i) the I-V characteristic exhibits a bistable region, and (ii) non-equilibrium current noise is enhanced near Tc. The notion of localization was originally introduced for a single quantum particle in a random potential [1]. Subsequently, the concept of Anderson localization was shown to manifest itself in a broad variety of phenomena in quantum physics. This concept can also be extended to many-particle systems. Statistical physics of many-body systems is based on the microcanonical distribution, i. e., all states with a given energy are assumed to be realized with equal probabilities. This assumption means delocal- ization in the space of possible states of the system. It does not hold for non-interacting particles; however, it is commonly believed that an arbitrarily weak interaction between the particles eventually equilibrates the system and establishes the microcanonical distribution. Many-body dynamics of interacting systems and its re- lation to Anderson localization has been discussed in the context of nuclear [6] and molecular [7] physics. For in- teracting electrons in a chaotic quantum dot this issue was raised in Ref. [8], where it was shown that electron- electron interaction may not be able to equilibrate the system. This corresponds to Anderson localization in the many-body space. Recently it was demonstrated that in an infinite low-dimensional system of (weakly) inter- Tc δζ/λ δζ/λ σ(T ) Insulator Metal FIG. 1: Schematic temperature dependence of the dc con- ductivity σ(T ) for electrons subject to a disorder potential localizing all single particle eigenstates, in the presence of weak short-range electron-electron λδζ , λ ≪ 1, established in Ref. [5]. Below the point of the many-body metal-insulator transition, T < Tc ∼ δζ/|λ lnλ|, no inelastic relaxation occurs and σ(T ) = 0. At T ≫ δζ/λ 2 the high-temperature metallic perturbation theory[9] is valid, and corrections to the Drude conductivity σ∞ are small. In the interval δζ/λ ≪ T ≪ δζ/λ electron-electron interaction leads to electron transitions be- tween localized states, and the conductivity depends on tem- perature as a power-law. acting electrons, subject to a static disorder, Anderson transition in the many-body space manifests itself as a finite-temperature metal-insulator transition [5]. Let single-particle eigenstates be localized on a spatial scale ζloc (localization length). The characteristic energy scale of the problem is the level spacing within the lo- calization volume: δζ = 1/(νζ loc), ν being the density of states per unit volume. Below we neglect energy depen- dences of ν, ζloc, and δζ . According to Ref. [5], as long as electrons are not coupled to any external bath (such as phonons), a weak short-range electron-electron inter- action (of typical magnitude λδζ , with the dimensionless coupling constant λ <∼ 1) does not cause inelastic relax- ation unless the temperature T exceeds a critical value: |λ lnλ| . (2) http://arxiv.org/abs/0704.1479v2 The small denominator |λ ln λ| represents the character- istic matrix element of the creation of an electron-hole pair. The ratio Tc/δζ is the number of states available for such a pair (in other words, the phase volume) at T = Tc. Only provided that this number is large enough to compensate the smallness of the matrix element, the interaction delocalizes the many-body states and thus leads to an irreversible dynamics. As a result, the finite- temperature dc conductivity σ(T ) vanishes identically if T < Tc, while σ(T > Tc) is finite, i. e. at T = Tc a metal- insulator transition occurs. The overall dependence σ(T ) is summarized in Fig. 1. In any real system the electron-phonon interaction is always finite. This makes σ(T ) finite even at T < Tc: σ(T ) is either exponentially small [Eq. (1)] at T ≪ δζ , or follows a power-law [10] at δζ ≪ T ≪ Tc. At the transition point the phonon-induced conductivity is not exponentially small, i. e. phonons smear the transition into a crossover. Are there any experimental signatures of the many-body localization? In what follows we show that if the electron-phonon coupling is weak enough, a qualitative signature of the metal-insulator transition can be identified in the nonlinear conduction. Namely, in a certain interval of applied electric fields E and phonon temperatures Tph both metallic and insulating states of the system turn out to be stable! As a result, the I- V curve exhibits an S-shaped bistable region (Fig. 3). Moreover, we show that the many-body character of the electron conduction dramatically modifies the non- equilibrium noise near the transition [Eq. (12)]. Bistable I-V curve.— Our arguments are based on two observations. First, in the absence of phonons a weak but finite electric field cannot destroy the insulating state – it rather shifts the transition temperature. Let us neglect the effect of the field on the single-particle wave functions, representing it as a tilt of the local chemical potential of electrons. Then at T = 0 the role of the field in the insu- lating regime is increase the energy of the electron-hole (e-h) excitation of a size L by eEL. This provides in addi- tional phase volume of the order of eEL/δζ. However, for L > ζloc the matrix element for creation of such an ex- citation quickly vanishes. In the diagrammatic language for the effective model of Ref. [5] this means that each electron-electron interaction vertex must be accompanied by tunneling vertices which describe coupling between lo- calization volumes and whose number is (i) at least one in order to gain phase volume (in contrast to the finite-T case when tunneling had to be included only to overcome the finiteness of the phase space in a single grain [5]), and (ii) not much greater than one, otherwise the diagram is exponentially small. As a result, at T = 0 the insulator state is stable provided that E < Ec ∼ Tc/(eζloc). In the same way one can analyze the finite-temperature correction to the critical field, and the finite-field correc- tion to the critical temperature can be found by taking into account the extra phase volume in the calculation of TelTph Tc(E) (Tel − Tph)/eE , Lph(Tel) FIG. 2: Sketch of the dependences (Tel − Tph)/(eE) (dashed line) and Lph(Tel) (solid lines, electron-phonon coupling strength being weaker for higher curves). Actual value of Tel is determined by their crossing. Ref. [5]. One obtains σ(T ) = 0 for T < Tc(E), where Tc(E) = Tc − c1eEζloc . (3) with a model-dependent factor c1 ∼ 1, weakly dependent on E [here and below Tc without the argument E is the zero-field value given by Eq. (2)]. As a consequence, at T < Tc(E) the nonlinear transport, as well as the linear one, has to be phonon-assisted. The second observation is that when both σ and E are finite, there is Joule heating. The thermal balance is qualitatively different in the insulating and the metallic phases. Deep in the insulating phase (T ≪ Tc) each electron transition is accompanied by a phonon emis- sion/absorption, i. e. electrons are always in equilibrium with phonons whose temperature Tph ≪ Tc we assume to be fixed. On the contrary, in the metallic phase elec- trons gain energy when drifting in the electric field, i. e. they are heated. Due to this Joule heating the effective electron temperature Tel deviates from the bath temper- ature. The role of phonons is then to stabilize Tel. For weak electron-phonon coupling Tel and Tph can differ sig- nificantly. A self-consistent estimate for Tel follows from Tel − Tph ∼ eELph(Tel) , (4) Lph(Tel) = D(Tel) τph(Tel) . (5) Here τph(Tel) is the time it takes an electron to emit or absorb a phonon, Lph(Tel) is the typical electron dis- placement during this time, and D(Tel) = σ(Tel)/(e is the electron diffusion coefficient. We sketch in Fig. 2 (Tel − Tph)/(eE) and Lph(Tel) for different electron-phonon coupling strengths as func- tions of Tel. It is taken into account that (i) Lph coin- cides with variable range hopping length at Tel ≪ δζ , (ii) Lph ∼ ζloc at δζ ≪ Tel ≪ Tc(E), (iii) D(Tel) quickly rises to its large metallic value D∞ ∼ δζζ2loc near eEζloc ∼ Tc∼ (ζloc/L ph)Tc eEζloc FIG. 3: (a) sketch of the bistable I-V curve for a fixed value of Tph; (b) (E , Tph)-plane with the bistable region schemat- ically shown by shading. The dashed line represents the crossover between the metallic state at high electric field E or high phonon temperature Tph, and the insulating state at low E and low Tph. Tc(E), (iv) τph(Tel) decreases as a power law with in- creasing Tel. The peak of the curve rises with decreas- ing electron-phonon coupling strength, and eventually the curve crosses the straight line. After that, in ad- dition to Tel = Tph, Eq. (4) acquires two more solutions, both with Tel > Tc(E), of which only the rightmost so- lution is stable. The maximum of Lph can be estimated as L∗ph ∼ D∞τph(Tc), so three solutions appear when Tc(E) − Tph ≪ eEL∗ph. At the same time, as seen from Eq. (3), electric field is unable to break down the insu- lator as long as eEζloc ≪ Tc − Tph. Thus, the interval of electric fields where both regimes are stable, is deter- mined by Tc(E)− Tph eL∗ph ≪ E ≪ Tc − Tph eζloc . (6) The two conditions are compatible provided that L∗ph ≫ ζloc , (7) which is realistic when electron-phonon coupling is weak. In the bistable region (6), for a given value of E one finds two stable solutions for Tel, giving two possible values of the conductivity and the current, which corre- sponds to an S-shape current-voltage characteristic [11], the third (unstable) solution corresponding to the neg- ative differential conductivity branch. The macroscopic consequences of such behavior depend on the dimension- ality. In a 2d sample the two phases of different electronic temperature and current density can coexist, separated by a boundary of the width ∼ L∗ph, parallel to the di- rection of the electric field. The particular state of the system determined by the boundary conditions (proper- ties of the external circuit), as well as by the history. Noise enhancement.— In the vicinity of the criti- cal point conduction is dominated by correlated many- electron transitions (electronic cascades). Each cascade is triggered by a single phonon. As Tel → Tc(E), the typ- ical value n̄ of the number n of electrons in the cascade diverges together with the time duration of a cascade. The results of Ref. [5], adapted for a finite electric field, give the following probability for an n-electron transition to go with the rate Γ: Pn(Γ) = e−Γ̄n/(4Γ) , Γ̄n ∝ Tel + c1eEζloc which gives 1/n̄ ∼ ln[Tc/(Tel + c1eEζloc)]. (9) The divergence in n̄ is cut off when electron-phonon cou- pling is finite. The largest n̄ is such that the phonon broadening of the single-electron levels, 1/τph(Tc), is comparable to n̄-particle level spacing (in other words, time duration of a cascade cannot exceed τph): τph(Tc) δζ/n̄ ⇒ n̄max ∼ ln(δζτph) ln(Tc/δζ) with logarithmic precision; n̄α represents the divergent spatial extent of the cascade (correlation length) [12]. Each many-electron transition can be characterized, besides its rate Γ, by the total dipole moment ~d it produces. The corresponding backward transition pro- duces the dipole moment −~d and goes with the rate ~E·~d)/Tel [13]. The average current 〈I(t)〉 is deter- mined by the difference between forward and backward rates; obviously, it vanishes for E = 0. At the same time, the noise power (second cumulant) S2 ≡ [〈I(t)I(t′)〉 − 〈I(t)〉〈I(t′)〉] dt′ is determined by the sum of the forward and backward rates; at E = 0 it is given by the equilib- rium Nyquist-Johnson expression. Equilibrium noise carries no information about the na- ture of conduction. To see a signature of many-electron transitions it would be natural to analyze the shot noise, whose power is proportional to the charge transferred in a single event. Many-electron cascades would then cor- respond to “bunching” of electrons, thus increasing the shot noise. However, shot noise is observed in the limit when transitions transferring charge only in one direc- tion (namely, ~d · ~E > 0) are allowed, i. e. Tel ≪ eEζloc, which is impossible to satisfy in the insulating state, as Tel ∼ max{Tph, eEζloc} [14]. Thus, S2 inevitably has both equilibrium and non-equilibrium contributions, which are difficult to separate. To see the “bunching” effect unmasked by a large ther- mal noise at low fields one should study the third Fano factor S3 of the current fluctuations [15]. Indeed, being proportional to an odd power of the current, it vanishes in equilibrium, so it is not subject to the problems de- scribed above for S2. In a wire of length L the ratio S3/〈I〉 is given by L−1 〈〈Γd〉〉 . (11) The double angular brackets on the right-hand side mean the sum over all allowed transitions. For nearest-neighbor single-electron transitions with d = ±eζloc Eq. (11) gives S3/〈I〉 ∼ e2(ζloc/L)2, which is analogous to the Schot- tky expression reduced by the effective number of tunnel junctions in series, L/ζloc, for S2 [16]. Since d3 diverges stronger than d as n̄ → ∞, we ex- pect a divergence in Eq. (11). The critical index of d depends on the order of limits: d ∼ neζloc if the lin- ear response limit E → 0 is taken prior to n̄ → ∞, while d ∼ neζloc(eEζloc/Tc) for a small but finite E . As a result, eζloc n̄eEζloc , n̄ <∼ n̄max , where n̄ is given by Eq. (9), and the saturation of the divergence is determined by Eq. (10). Upon further in- crease of the temperature, the system crosses over to the metallic state, and n̄ starts to decrease. This decrease is governed by the same Eq. (10) with the phonon inelas- tic rate substituted by the typical value of the electron- electron inelastic rate, which grows with temperature. As the critical behavior of Γ on the metallic side of the transition is unknown, we cannot give any quantitative estimate of S3 above Tc. Conclusions.— In conclusion, we have shown that the finite-temperature metal-insulator transition, predicted theoretically in Ref. [5], can manifest itself on the macro- scopic level as an S-shape current-voltage characteristic with a bistable region. In fact, the hysteretic behaviour of the current in YxSi1−x [17] is a possible candidate for the effect discussed in the present paper. Besides, we have shown that the many-body nature of the conduction near the transition manifests itself in the dramatic increase of the non-equilibrium current noise: the noise depends on the total charge transferred in each random event, while the number of electrons, involved in such an event, increases as one approaches the transition. We acknowledge discussions with M. E. Gershenson, C. M. Marcus, A. K. Savchenko, M. Sanquer, and thank H. Bouchiat for drawing our attention to Ref. [17]. ∗ Electronic address: [email protected] [1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [2] M. E. Gertsenshtein and V. B. Vasil’ev, Teoriya Veroy- atnostei i ee Primeneniya 4, 424 (1958) [Theory of Prob- ability and its Applications 4, 391 (1958)]. [3] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). [4] N. F. Mott, J. Non-Cryst. Solids 1, 1 (1968); Mott for- mula does not into account Coulomb interaction, which changes the value of γ in Eq. (1) [A. L. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975)]. [5] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. 321, 1126 (2006). See also cond-mat/0602510. [6] D. Agassi, H. A. Weidenmüller, and G. Mantzouranis, Phys. Rep. 22, 145 (1975). [7] D. E. Logan and P. G. Wolynes, J. Chem. Phys. 93, 4994 (1990). [8] B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. 78, 2803 (1997). [9] B. L. Alshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Systems, ed. by A. L. Efros and M. Pollak (Elsevier, Amsterdam, 1985). [10] A. A. Gogolin, V. I. Melnikov, and É. I. Rashba, Zh. Eksp. Teor. Fiz. 69, 327 (1975) [Sov. Phys. JETP 42, 168 (1976)]. [11] E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors (Cambridge University Press, Cambridge, 2001). [12] The index α depends on the dimensionality only. Argu- ments given in Sec. 6.5 of Ref. [5] fix α = 1 in 1d, and restrict 1/2 ≤ α ≤ 3/4 in 2d, 1/3 ≤ α < 3/5 in 3d, 1/d ≤ α ≤ 1/2 in higher dimensions. [13] Strictly speaking, the electric field appearing in Eq. (8) is the local electric field, resulting from the spatial re- distribution of charge over the random resistor network. Here we neglect fluctuations of the local field (more pre- cisely, their correlations with the spatial fluctuations in Γ and ~d), and understand E as the external field. This is a good approximation for the distribution (8), as the cur- rent pattern is determined by the typical resitors, Γ ∼ Γ̄n. [14] B. I. Shklovskii, Fiz. Tekh. Poluprovodn. 6, 2335 (1972) [Sov. Phys. Semicond. 6, 1964 (1973)]; B. I. Shklovskii, E. I. Levin, H. Fritzsche, and S. D. Baranovskii, in Trans- port, Correlation and Structural Defects, edited by H. Fritzsche (World Scientific, Singapore, 1990). [15] L. S. Levitov and M. Reznikov, cond-mat/0111057; Phys. Rev. B 70, 115305 (2004). [16] R. Landauer, Physica B 227, 156 (1996); A. N. Korotkov and K. K. Likharev, Phys. Rev. B 61, 15975 (2000). [17] F. Ladieu, M. Sanquer, and J. P. Bouchaud, Phys. Rev. B 53, 973 (1996). mailto:[email protected] http://arxiv.org/abs/cond-mat/0602510 http://arxiv.org/abs/cond-mat/0111057

0704.1480

Magnetic structure of Sm2IrIn8

Magnetic structure of Sm2IrIn8 C. Adriano,1, ∗ R. Lora-Serrano,1 C. Giles,1 F. de Bergevin,2 J. C. Lang,3 G. Srajer,3 C. Mazzoli,2 L. Paolasini,2 and P. G. Pagliuso1 Instituto de F́ısica ”Gleb Wataghin”,UNICAMP,13083-970, Campinas-São Paulo, Brazil. European Synchrotron Radiation Facility, Grenoble 38043, France. Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439. (Dated: December 28, 2018) The magnetic structure of the intermetallic antiferromagnet Sm2IrIn8 was determined using x-ray resonant magnetic scattering (XRMS). Below TN = 14.2, Sm2IrIn8 has a commensurate antiferro- magnetic structure with a propagation vector ~η = (1/2, 0, 0). The Sm magnetic moments lie in the ab plane and are rotated roughly 18o away from the a axis. The magnetic structure of this compound was obtained by measuring the strong dipolar resonant peak whose enhancement was of over two orders of magnitude at the L2 edge. At the L3 edge both quadrupolar and dipolar features were observed in the energy line shape. The magnetic structure and properties of Sm2IrIn8 are found to be consistent with the general trend already seen for the Nd-, Tb- and the Ce-based compounds from the RmMnIn3m+2n family (R = rare earth; M=Rh or Ir, m = 1, 2; n = 0, 1), where the crystalline electrical field (CEF) effects determine the direction of magnetic moments and the TN evolution in the series. The measured Néel temperature for Sm2IrIn8 is slightly suppressed when compared to the TN of the parent cubic compound SmIn3. PACS numbers: 75.25.+z, 75.50.Ee, 75.30.-m, 75.30.Kz I. INTRODUCTION The microscopic details of 4f -electron magnetism play a fundamental role in the physical properties of vari- ous classes of rare-earth based materials such as heavy fermions, magnetically ordered alloys and permanent magnets. The existence of structurally related families of rare-earth based compounds provides a great opportunity to explore how the details of the 4f -electrons magnetism evolve as a function of changes in the dimensionality, lo- cal symmetry and electronic structure along each related family. The recently discovered1,2,3,4,5,6,7,8 family of in- termetallic compounds RmMnIn3m+2n (M = Co, Rh or Ir, m = 1, 2; R = La, Ce, Pr, Nd, Sm, Gd) have proved to be very promising in this regard, since it possesses many members of structurally related heavy-fermions su- perconductors (HFS), for R = Ce, antiferromagnets (R = Nd, Sm, Gd and Tb) and paramagnetic metals (R = La, Pr). Within this family, the physical properties of a par- ticular R-member can also be compared to compounds based on the same R with three different related struc- tures [the cubic RIn3 and the tetragonal RMIn5(1-1-5) and R2MIn8 (2-1-8)] 9,10,11 and/or to the same R formed with three distinct transition metals (M = Rh, Ir and Co - not for all R -) in the same structure. For the Ce-based HFS in this family, extensive investigation has revealed fascinating physical prop- erties such as quantum criticality, non-fermi-liquid- behavior and an intriguing interplay between mag- netism and superconductivity, reflected in very rich phase diagrams.12,13,14,15,16,17,18,19,20 Because the HFS mem- bers of this family are structurally related, its investiga- tion has been used to provide some insights on the ques- tion why some structure types are favorable to host many superconductors. A possible relationship between the su- perconducting critical temperature Tc and the crystalline anisotropy13,21,22, the role of the 4f -electron hybridiza- tion with the conduction electrons in the occurrence of superconductivity23,24,25 and the effects of quasi-2D elec- tronics structures26,27,28are some of the physical phenom- ena that have been brought to the scenario to answer the question above. Further, motivated by this experimental trend, new materials search based on the 1-1-5 structures has led to the discovery of the Pu-based HFS PuMGa5 (M = Rh and Co).29,30 On the other hand, as these HFS are presumably mag- netically mediated, others studies5,6,7,21,31,32,33,34,35,36,37 have been focused in understanding the evolution of the 4f local magnetism, not only for the magnetically or- dered Ce-based members of this family such as CeRhIn5 and Ce2RhIn8, but also for their antiferromagnetic coun- terparts RmMnIn3m+2n (M = Rh or Ir, m = 1, 2;) for R = Nd, Sm, Gd and Tb. From these studies, it was es- tablished the role of tetragonal crystalline electrical field (CEF) in determining the spatial direction of the ordered R-moments with respect to the lattice and the evolution of the Néel temperature, TN , in the series. 5,6,7,32,37 A key set of experiments allowing the above conclu- sions was the experimental determination of the magnetic structures of various members of the RmMnIn3m+2n (M = Rh or Ir, m = 1, 2;) family.7,31,36,38,39,40,41 Up to date, however, none of the Sm-based compounds from this fam- ily have had their magnetic structures determined. In fact, the compounds of this series containing Sm ions may be particularly important in testing the extension of the CEF trends in this family because the presence of excited J-multiplet states in Sm3+ and quadrupolar interactions have to be taken into account in order to understand their magnetic phase diagrams.42,43,44,45 Es- pecially interesting is Sm2IrIn8 which presents a first or- http://arxiv.org/abs/0704.1480v2 der antiferromagnetic transition at TN = 14.2 K. 5 This value is slightly smaller than the TN ∼ 16 K of the cubic SmIn3 11 which according to the CEF trends observed in other members of this family7,37 suggest that the ordered Sm-moments should lie the ab-plane. To further explore the magnetic properties of Sm2IrIn8 and to check the extension of the CEF trends observed for R = Nd, Tb, and Ce,5,6,7,32,37 to the Sm-based compounds, we report in this work the solution of the magnetic structure of the intermetallic antiferromagnet Sm2IrIn8 by means of the x-ray resonant magnetic scat- tering (XRMS) technique. The XRMS technique has proved to be a very important tool for the investigation of microscopic magnetism in condensed matter, specially for highly neutrons absorber ions such as Sm. Sm2IrIn8 presents, below TN = 14.2 K, a commensu- rate antiferromagnetic structure with a propagation vec- tor ~η = (1 , 0, 0). The Sm magnetic moments lie in the ab plane. In terms of relative orientation, the propa- gation vector ~η indicates that the Sm-spins are ordered antiferromagnetically along the a axis and ferromagnet- ically along the b axis and, because of the presence of two Sm ions per unit cell along c axis, some calculations have to be performed in order to determine the type of ordering along this direction. Furthermore, as it could be expected for such spin arrangement in a tetragonal com- pound, antiferromagnetic domains were observed in the ordered state of Sm2IrIn8. These domains were removed by field-cooling the sample at a field of H = 10 T. II. EXPERIMENT Single crystalline samples of Sm2IrIn8 were grown from Indium flux as described previously.5,46 The crystal struc- ture, unit cell dimensions and macroscopic properties of the Sm2IrIn8 single crystals used in this work were in agreement with the data in Ref. 5. For the XMRS exper- iments of this work, selected crystals were extracted and prepared with polished (0,0,l) flat surfaces, and sizes of approximately 4 mm x 3.4 mm x 1.5 mm. The preferred crystal growth direction of this tetragonal compound is columnar along the [00l ] direction and the (001) facet is relatively large. The mosaic spread of the sample was found to be < 0.08◦ by a rocking curve (θ scan) on a Phillips four circle diffractometer. XRMS studies were performed at the 4-ID-D beam- line at the Advanced Photon Source (APS) and at the ID-20 beamline at the European Synchrotron Radiation Facility (ESRF). The 4-ID-D x-ray source is a 33 mm period planar undulator and the energy is selected with a double crystal Si(111) monochromator. A toroidal mir- ror focuses the beam to a 220 µm (horizontal) x 110 µm (vertical) spot, yielding an incident flux of ∼3.5 x 1013 photons/s with an energy resolution of δE/E = 1.4 x 10−4. The sample was cooled in a closed-cycle He refrig- erator (with a base temperature of 4 K) with a dome Be window. Our experiments were performed in the copla- nar geometry with σ-polarized incident photons, i.e., in the vertical scattering plane, using a four-circle diffrac- tometer. Except for azimuthal scans, the sample was mounted with the b axis perpendicular to the scattering plane. In most measurements, we have performed a polar- ization analysis, whith Cu(220), Graphite (006) and Au(111) crystal analysers, appropriate for the energies of Sm L2 and L3 edges. The diffractometer configuration at the APS allowed measurements at different azimuthal angles (φ) by rotating the sample around the scatter- ing vector Q. This was particularly useful to probe the magnetic moment components at the dipolar resonant condition with σ incident polarization. The x-ray source on the ID-20 beamline was a linear undulator with a 32 mm period. The main optical com- ponents are a double Si(111) crystal monochromator with sagital focusing and two meridional focusing mirrors on either side of the monochromator. At 7.13 keV using the first harmonic of the undulator u32, the standard inci- dent flux at the sample position was approximately 1 x 1013 ph/s at 200 mA with a beam size of 500 µm (hori- zontal) x 400 µm (vertical). The sample was mounted on a cryomagnet (with a base temperature of 2 K), installed on a horizontal six-circle diffractometer, with the b axis parallel to the cryomagnet axis and perpendicular to the scattering plane. This configuration allowed π-polarized incident photons in the sample and the application of an external magnetic field up to 10 T perpendicular to the scattering plane. III. RESULTS A. Temperature dependence and resonance analysis Magnetic peaks were observed in the dipolar resonant condition at temperatures below TN = 14.2 K at recip- rocal lattice points forbidden for charge scattering and consistent with an antiferromagnetic structure with prop- agation vector (1 , 0, 0). Their temperature dependence was studied for increasing and decreasing temperature sweeps. Figure 1 shows the temperature dependence of (0, 1 , 9) magnetic reflection at an incident photon energy of 7.313 keV (L2 edge) and measured at π incident polar- ization without polarization analysis. The squared root of the integrated intensity, which is proportional to a Sm sub-lattice magnetization, is displayed. A pseudo-voigt peak shape was used to fit transversal θ scans through the reciprocal lattice points in order to obtain the integrated intensities of the reflection peak. This peak intensity de- creases abruptly to zero for T > 13 K and its critical be- havior can not be described by a power-law function with a critical exponent β. This result is in agrement with the first order character of the magnetic transition at 14.2 K, revealed by heat capacity data, from which a latent heat of ∼ 10 J/mol was extracted.5 Consistently, we found 4 6 8 10 12 14 1.2 (0,1/2,9) Peak Heating Cooling T (K) FIG. 1: Temperature dependence of one Sm2IrIn8 sub-lattice magnetization measured with transverse (θ) scans at the (0, 1 , 9) peak. evidence of small hysteresis for T . 14.2 when changing from warming to the cooling temperature sweep. The energy line shape curves for the polarization chan- nels σ - π’ and σ - σ’ of the (1 ,0,9) diffraction peak at (a) the L2 and (b) the L3 absorption edges of Sm 3+ ion at T = 5.9 K are shown in Figure 2. The solid lines in both panels represent the absorption spectrum, µ(E), extracted from fluorescence yield. The data of Figure 2 were collected at the 4-ID-D beamline of APS by count- ing the photons reaching the detector at a fixed Q while changing the incident energy. The strong resonant en- hancement of the x-ray scattering at this reciprocal space position provide clear evidence of the magnetic origin of the observed peaks. The energy scan curve in Figure 2(a) has a maximum at 7.312 keV which is only ∼2.5 eV larger than the L2 absorption edge (defined by the inflection point of the absorption spectrum), revealing the electric dipolar char- acter (E1) of this transition (from 2p to 5d states). Fig- ure 2 also shows the polarization analysis performed to unambiguously confirm the magnetic origin of the super- structure peaks. Polarization analysis was also used to verify whether the anomaly at approximately 8 eV be- low the dipolar peak in Figure 2(a) could be associated with a quadrupolar transition47 or it simply represents an enhanced interference between the non-resonant and the resonant part of the scattering amplitude. For the ex- perimental configuration used (incident σ-polarization), the electric dipole transitions E1 rotate the plane of po- larization into the scattering plane (π-polarization). Our data in Figure 2(a) reveals a strong enhancement of the scattered intensities at the σ - π’ channel (closed cir- cles) and no enhancement at the σ − σ’ channel for the same energy range. These results confirm the magnetic origin of the (h, 0, l)±(1 , 0, 0) reflections due to the ex- 6.68 6.70 6.72 6.74 6.76 7.28 7.30 7.32 7.34 Energy (keV) Sm L3 edge 6.716 keV Sm L2 edge 7.312 keV FIG. 2: Energy scan of the ( 1 ,0,9) magnetic peak at T = 5.9 K for σ - π’ (closed circles) and σ - σ’ (open circles) polarization channels at the L2 (top) and L3 (bottom) absorption edges. The data have been corrected for absorption, µ(E), using the measured fluorescence yield. Arrows indicate the scales for the fluorescence yield (right) and the observed data (left). istence of an antiferromagnetic structure doubled along the crystallographic â direction, with a propagation vec- tor ~η = (1 , 0, 0). The energy profile around the Sm L3 edge is pre- sented in Figure 2(b). Firstly, the observed intensities are roughly one order of magnitude weaker than those obtained at the L2 resonance, in agreement with previ- ous measurements on pure Sm.45 Secondly, there are two peaks in the σ - π’ channel signal, as also observed for other light rare-earth48,49 and Sm-based compounds.45,50 A high energy peak appears at 6.716 keV, while a low en- ergy and more intense enhancement can be observed at 6.708 keV. Interestingly, Stunault et al.45 have demon- strated that for pure Sm the quadrupolar E2 resonance is more intense than the dipolar E1 at the L3 edge and they found that the energy difference between the E2 and the E1 resonances is of the order of 8 eV, the same as the one found in this work. Furthermore, in the σ - σ’ channel only an enhancement at 6.708 keV could be ob- served which is consistent with the quadrupolar character of this resonance, since scattering signal in σ - σ’ chan- nel for dipolar transitions is strictly forbidden.47,51 Thus, the presence of this pre-edge enhancement in the energy curves of Figure 2 confirms an expected quadrupole (E2) 2p to 4f contribution to the resonant x-ray scattering in Sm2IrIn8. B. The magnetic structure The magnetic structure of the Sm2IrIn8 was experi- mentally investigated using dipolar resonant x-ray mag- netic scattering with polarization analisys. In general, the magnetic scattering intensities are given by:47,50 µ∗sin(2θ) ~k, ǫ̂, ~k′, ǫ̂′)ei ~Q·~Rn , (1) where µ∗ is the absorption correction for asymmetric re- flections, 2θ is the scattering angle, ~Q = ~k′ − ~k is the wave-vector transfer, ~k and ~k′ (ǫ̂ and ǫ̂′) are the incident and scattered wave (polarization) vectors, respectively. ~Rn is the position of the nth resonant atom in the lattice, and ẑn is the moment direction of this atom. The reso- nant scattering amplitude contains both dipole (E1) and quadrupole (E2) contributions. For the determination of the magnetic structure of this work we have used the second term of the electric dipole transition (E1) form factor which produces magnetic peaks. In this case we have: nE1 ∝ 0 k̂′ · ẑn −k̂′ · ẑn (k̂ × k̂′) · ẑn 0 z1cosθ + z3sinθ −z1cosθ + z3sinθ −z2sin(2θ) , (2) where θ is the Bragg angle, z1, z2 and z3 are the compo- nents of the magnetic moment at the nth site, according to the commonly used geometry convention of Ref. 52; σ, π, σ’ and π’ describe the incident (non-primed terms) and scattered (primed) photon polarizations. As described previously, two experimental setups have been used in this work, in the vertical (4-ID-D beamline) and horizontal (ID-20) scattering configurations. This permitted us to access all four polarization channels of the 2x2 matrix in (2) and to determine the magnetic mo- ment orientations through their polarization dependence at the E1 resonance by comparing the relative intensities of experimental (1 , 0, l) magnetic peaks with the calcu- lated ones using the appropriate terms of matrix (2).50 In the case of Sm2IrIn8 the magnetic propagation vec- tor ~η = (1 , 0, 0) does not unequivocally determine the magnetic structure due to the presence of two magnetic Sm atoms per chemical unit cell along the ĉ direction. Therefore, as stated above, we have an antiparallel or- dering of the Sm moments along the â direction and a parallel ordering along b̂. Along ĉ there are, however, two possibilities of coupling that can take place: a parallel arrangement (Model I), in which the moments of neigh- boring Sm ions along c axis are parallel to each other (sequence ++ ++ ++ . . . ), or the antiparallel coupling (Model II), with the sequence (+− +− +− . . . ). These two possibilities have been considered into the calculated magnetic structure factor while orienting the magnetic moment along the three crystallographic directions for five different (1 , 0, l) magnetic Bragg peaks, with l = 6, 7, 8, 9, 10. The calculated intensities are strongly de- pendent on the projections of magnetic moments along the crystallographic axis through the product k̂′ · ẑn of equation (2). Therefore, they were compared to the rel- ative observed intensities for each case. This evaluation was performed at the vertical geometry of the 4-ID-D beamline at 9 K by performing rocking scans with the crystal analyzer and numerically integrating the data.50 We show this analysis in Table I, where “Model I” stands for the ++ ++ ++ . . . sequence and “Model II” for the +−+−+−. . . one. This comparison shows that the model which best fits the experimental data is the one assum- ing antiparallel coupling along c axis (Model II) with the magnetic moments approximately oriented along the a axis (according to matrix (2), for a σ polarized incident beam and peaks at reciprocal space positions with the (001) normal surface contained in the scattering plane, contributions from an oriented moment along b̂ direction cannot be detected). In addition, we have also measured the π − σ′ and π − π′ polarization channels at the horizontal geometry of the ID-20 beamline. Measuring these two channels we gained access to the z1 and z3 components (in equation 2) of magnetic moment vector in one case [π − σ′, Fig- ure 3(a)] and to z2 in the other [π − π ′, Figure 3(b)]. There is a clear indication that for the π − σ′ channel the observed data are well fit when considering the mo- ments along the â direction [dotted curve in Figure 3(a)] instead of ĉ direction [short dashed curve]. Also in this case the E1 terms are not sensitive to the component of the ordered moment perpendicular to the scattering plane, i.e. along b axis. Further, when measuring the channel (π − π′) we are only allowed to measure the b component, which is confirmed by the good fit of exper- imental data when assuming magnetic moments along such direction [dash-dotted curve in Figure 3(b)]. These two last results indicate that the Sm moments actually have components along both a and b real space axis and not perfectly aligned along any of these two directions. To determine the exact orientation of the magnetic mo- ments within the ab plane, we have performed azimuthal scans (φ scan) through the (1 , 0, 9) reflection (Figure 4) TABLE I: Comparison between observed and calculated intensities of magnetic Bragg reflections, assuming either parallel (model I) or antiparallel (model II) alignment between the moments of two Sm ions along the c axis in the same chemical unit cell. MODEL I MODEL II (h, k, l) Exp. Data m//c m//a m//c m//a (1/2,0,6) 66 13 29 24 55 (1/2,0,7) 78 17 29 39 68 (1/2,0,8) 5 77 100 3.4 4.5 (1/2,0,9) 100 3 3 100 100 (1/2,0,10) 12 100 68 32 23 ( ')channel z //a-axis z //c-axis ( ')channel z //b-axis Q (r.l.u) FIG. 3: Analysis of the possible magnetic moment directions for Sm2IrIn8 at the L2 resonance. Q-dependence of the inte- grated intensities of: (a) six satellite peaks signal in the π−σ′ channel with the moments along â and ĉ, and (b) in the π−π′ with moments parallel to b̂. at the E1 resonance. At the σ − π′ polarization channel this procedure warrants the determination of moments directions with no ambiguity because the magnetic cross section is strongly dependent of the magnetic moment direction and the polarization of the incoming and scat- tered radiation, the maximum (minimum) intensity in the curve will occur with the magnetic moment being parallel (perpendicular) to the diffraction plane. With the experimental setup of 4-ID-D beamline we had ac- cess to record points at azimuthal angles φ between -50o -120 -80 -40 0 40 80 120 //18o Azimuth angle (degree) , (1/2,0,9), T = 6.5 K Exp. Data ( ') FIG. 4: Azimuth scan analysis. Normalized integrated in- tensities of the ( 1 , 0, 9) magnetic peak at T = 6.5 K (open circles). The other curves represent the integrated intensi- ties behavior considering the magnetic moments along the â (dotted line), b̂ (dashed) and 18o away from â (solid line) direction. and 60o. In order to compare with the observed data, one can calculate the intensities for the σ − π′ channel using the expressions (1) and (2) and a reasonably simple geometry analysis considering the projections of both k̂′ and ẑn on the coordinate system of Ref. 52 when the azimuth angle is changed. Then, the calculated inten- sity is proportional to Iσπ ∝ |-cosθ cosφ cosα + sinθ sinα|2, where α represents the assymetry angle between the scattering and the normal surface vector.50 Figure 4 shows the experimental and the calculated relative in- tensities considering the moment along the a and b axis, as well as 18o tilted from the a axis, which is the value that nicely adjust the experimental data. Considering the experimental errors we can then conclude that the magnetic moment is in the ab plane making (18o ± 3o) with the â direction of the sample. Using all the above results, a model of the magnetic unit cell of Sm2IrIn8 can be constructed and is shown in Figure 5. FIG. 5: Magnetic structure of Sm2IrIn8 below TN = 14.2 K (left) and a Sm-In plane top view (right) showing the in-plane arrangement of Sm moments. As it was observed in the magnetic structure of other members of the RmMIn3m+2 series such as NdRhIn5 TbRhIn5, 7 GdRhIn5, 41 and Gd2IrIn8 36 the magnetic structure of Sm2IrIn8 presents a lower symmetry than the crystallographic structure, as the Sm spins present different relative orientations along the â and b̂ direc- tions even though a and b are indistinguishable. This spin arrangement was explained by considering the first (J1) and second (J2) R-neighbors exchange interactions in the case of a small J1/J2 ratio. Considering the observation of this kind of magnetic structure in tetragonal compounds, it may be expected that at zero magnetic field the antiferromagnetic ordering takes place with the formation of antiferromagnetic do- mains where the relative orientation of the magnetic mo- ments along a given direction (â or b̂) changes from paral- lel to antiparallel between the domains. The presence of a twinned magnetic structure with symmetry-related do- mains was evidenced by the observation of both (1 , 0, l) and (0, 1 , l) reflection-types in this work. To further in- vestigate the presence of antiferromagnetic domains in the ordering state of Sm2IrIn8 we follow the behavior of the magnetic (1 , 0, l) and (0, 1 , l) reflections under an applied magnetic field. Figure 6 presents the behavior of the (1 , 0, 9) and (0, 1 , 9) intensities as a function of the applied magnetic field of 10 T along one of the tetragonal axis in the plane (defined as b̂ direction). At zero field and T = 6 K, both (1 , 0, 9) [open circles] and (0, 1 , 9) [closed squares] intensities can be observed with comparable magnitude [Figure 6(a)]. The (1 , 0, 9) intensity is roughly 66% that of the (0, 1 , 9) peak. The sample was then field cooled (H = 10 T) from the paramagnetic (16 K) to the ordered state (6 K) with the field applied along the b̂ direction. As can be seen in Figure 6(b) the (0, 1 , 9) diffraction peak disappears as the magnetic field favors the parallel spin orientation along the b axis. The same effect was also ob- served for the other five (0, 1 , l) reflections (not shown). The results under applied magnetic field shown in Fig- 38.7 38.8 38.9 39.0 39.1 39.2 0.000 0.075 0.150 45.9 46.0 46.1 46.2 46.3 46.4 (1/2,0,9) H = 0 T = 6 K Degrees (0,1/2,9) H = 10 T Field Cooled uo to T = 6 K (0,1/2,9) Degrees alized sity x 10 -3 (a.u (1/2,0,9) FIG. 6: Field-dependence of the integrated intensities of the , 0, 9) and (0, 1 , 9) magnetic peaks taken with transverse (θ) scans around each reciprocal space lattice points. (a) For H = 0 applied field at T = 6 K, (b) for H = 10 T and (c) field cooled from 16 K to 6 K at H = 10 T. ure 6 confirm the existence of a twinned magnetic struc- ture for Sm2IrIn8 which allows the observation of both (0, 1 , l) and (1 , 0, l) magnetic reflections at zero field. IV. DISCUSSION Early studies on the antiferromagnetic cubic com- pound SmIn3 have shown multiple magnetic transitions associated with quadrupolar ordering, magnetoelastic and magnetocrystalline competitive effects at 14.7, 15.2 and 15.9 K (the former two temperatures being associ- ated with successive magnetic dipolar, antiferromagnetic, orders and the last one due to quadrupolar ordering).42,43 For the tetragonal Sm2IrIn8, the insertion of two addi- tional SmIn3 atomic layers into the crystalline structure slightly decreases TN compared to that of SmIn3 (14.2 and 15.2 K for the Sm2-1-8 and Sm1-0-3 TN ’s, respec- tively) and an additional anomaly at 11.5 K has been ob- served in the specific heat and resistivity measurements,5 probably related to the successive transitions seen in the ordered phase of the SmIn3. Following the investigation of the isostructural mag- netic non-Kondo compounds from the RmMIn3m+2 fam- ily, where the details the 4f magnetism along the series may be important to understand the possible magnetic- mediated superconductivity in the compounds with R = Ce, we have studied the magnetic structure of Sm2IrIn8, which is the only compound from this family with a clear first order antiferromagnetic transition and now it is the first Sm-member from this family with a solved magnetic structure, which is the main result of this work. The de- termination of the Sm2-1-8 magnetic structure allows for the investigation of the CEF driven trends of magnetic properties within the RmMIn3m+2 family to be extended to the Sm-based members. Our results confirm the complex resonance profile of Sm-based compounds (at one satellite reciprocal point, Figure 2), as seen in previous studies of pure Sm.45 It has been argued that the larger intensity of E2 resonance at Sm L3 edge compared to its intensity at the L2 edge may be explained qualitatively by the spin-orbit splitting of the intermediate 4f levels involved.45 The L3 transitions connect the j = 7 state while L2 involves transitions to the j = 5 level, which lie lower in energy and therefore can be preferentially populated by the five 4f Sm elec- trons. This reduces the number of vacant j = 5 states from 6 to 1, in contrast to the 8 states available for the j = 7 level, which increases the transition probability of the E2 resonance at Sm L3 in Sm2IrIn8. Considering the additional magnetic transitions ob- served for SmIn3, 42,43 and the additional anomaly at T = 11.5 K in heat capacity and electrical resistivity mea- surements for Sm2IrIn8, 37 we did not observe any dis- continuities, within the resolution of our experiment, in the integrated intensities of the (0, 1 , 9) magnetic peak from roughly 4 K up to 16 K (Figure 1). Therefore we conclude that there are no changes of the magnetic prop- agation vector ~η = (1 , 0, 0) below TN . For completeness, on going field-dependent heat capacity and thermal ex- pansion measurements (not shown and will be published elsewhere) have revealed no field-induced transitions up to H=9 and 18 T, respectively, similarly to SmIn3 where no additional transition was found with applied field up to H=32 T.44 On the other hand, recent works have shown that the low temperature CEF configuration plays a fundamen- tal role on the behavior of TN and the magnetic mo- ment directions within the RmMIn3m+2 family. 7,32,37,40 Further, Kubo et al.53 has also proposed an orbital con- trolled mechanism for superconductivity in the Ce-based compounds from this family. For the Sm members, CEF effects confine the magnetic moments to the ab plane, consistent with the experimental CEF trends observed for R = Ce, Nd and Tb5,6,7,32 and also by the predictions of a recently developed mean field theoretical model.7,37 If the magnetic ordered moments lie in the ab-plane but they are more magnetically susceptible along the c axis the magnetic order can be frustrated to lower TN val- ues than for their cubic relatives. The mean-field model of Ref. 37, however, only includes the contributions of tetragonal CEF and first neighbor isotropic dipolar ex- change interaction. Therefore, it may not be expected to work for Sm containing compounds, because for the Sm3+ ion the first excited J -multiplet lying just above the ground state is closer in energy. Thus, the tetragonal CEF splitting can mix both the excited and ground state CEF scheme and this particular effect should be consid- ered into the calculations. Indeed, this is the responsible for the non-linear response of the inverse of magnetic susceptibility at high temperatures on SmIn3 and other Sm-based compounds,11,54 as well as in Sm2IrIn8. 5 Fur- thermore, as it was found for SmIn3, 42,43 quadrupolar magnetic interactions also have to be considered in order to achieve a complete description of the magnetic prop- erties of the Sm-based compounds in the RmMIn3m+2 family. Apart from the higher complexity of the magnetic properties of the Sm-compounds, it was found exper- imentally that TN is decreased (roughly ∼ 10%) for the tetragonal compounds when compared to the cu- bic SmIn3. In addition, we have found that the mag- netic structure of Sm2IrIn8 shows the ordered Sm mo- ments in the ab plane, as expected in the case of TN suppression.7,37 Although the changes in TN for the Sm compounds are much smaller (perhaps due to the partic- ularities of the Sm3+ ion discussed above) than that ob- served for R = Ce, Nd and Tb in the RmMIn3m+2 family, we can conclude with the solution of the magnetic struc- ture reported here, that the general CEF trend of the RmMIn3m+2 is also qualitatively present in Sm2IrIn8. V. CONCLUSION In summary, we have presented the results of the mag- netic structure determination of the intermetallic anti- ferromagnet Sm2IrIn8. The magnetic order is commen- surate with propagation vector ~η = (1 , 0, 0) and the Sm moments oriented in the ab plane. We used differ- ent scattering geometries (exploring the polarization de- pendences of magnetic intensities) and azimuthal scans around a magnetic reciprocal space point to determine without ambiguity that the moments are aligned approx- imately 18o away from the a axis. The temperature be- havior of the magnetic satellites have been probed at the (0, 1 , 9) reciprocal node and show no evidence of changes in the magnetic structure within the studied temperature range. Besides, an abrupt (non-power law) decrease of magnetic intensities at TN was found, consistent with the first order character of the antiferromagnetic transition of Sm2IrIn8. The resonance properties at the Samarium L2 and L3 absorption edges revealed both resonant E1 and E2 process with roughly one order of magnitude more in- tense resonance peaks at the L2 edge and a much stronger quadrupole resonance in the L3 edge. The orientation of Sm moments in the ab plane and the small decrease of TN compared to its value for SmIn3 agrees with a general CEF trend found in the RmMIn3m+2 family. Acknowledgments This work was supported by FAPESP (SP-Brazil) Grants No. 05/55272-9, 05/00962-0, 04/08798-2 and 03/09861-7, CNPq (Brazil) Grants No. 307668/03, 04/08798-2, 304466/20003-4 and 140613/2002-1, and FAEPEX (SP-Brazil) Grant No. 633/05. Use of the Advanced Photon Source was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE- AC02-06CH11357. The staff at the 4-ID-D and ID-20 beam lines are gratefully acknowledged for providing an outstanding scientific environment during these experi- ments. ∗ Electronic address: [email protected] 1 H. Hegger, C. Petrovic, E. G. Moshopoulou, M. F. Hund- ley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Phys. Rev. Lett. 84, 4986 (2000). 2 C. Petrovic, R. Movshovich, M. Jaime, P. G. Pagliuso, M. F. Hundley, J. L. 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0704.1481

TEXES Observations of Pure Rotational H2 Emission From AB Aurigae

arXiv:0704.1481v1 [astro-ph] 11 Apr 2007 Draft version November 20, 2021 Preprint typeset using LATEX style emulateapj v. 2/19/04 TEXES OBSERVATIONS OF PURE ROTATIONAL H2 EMISSION FROM AB AURIGAE Martin A. Bitner , Matthew J. Richter , John H. Lacy , Thomas K. Greathouse , Daniel T. Jaffe Geoffrey A. Blake Draft version November 20, 2021 ABSTRACT We present observations of pure rotational molecular hydrogen emission from the Herbig Ae star, AB Aurigae. Our observations were made using the Texas Echelon Cross Echelle Spectrograph (TEXES) at the NASA Infrared Telescope Facility and the Gemini North Observatory. We searched for H2 emission in the S(1), S(2), and S(4) lines at high spectral resolution and detected all three. By fitting a simple model for the emission in the three transitions, we derive T = 670 ± 40 K and M = 0.52±0.15 M⊕ for the emitting gas. Based on the 8.5 km s −1 FWHM of the S(2) line, assuming the emission comes from the circumstellar disk, and with an inclination estimate of the AB Aur system taken from the literature, we place the location for the emission near 18 AU. Comparison of our derived temperature to a disk structure model suggests that UV and X-ray heating are important in heating the disk atmosphere. Subject headings: circumstellar matter, infrared:stars, stars:planetary systems, protoplanetary disks, stars:individual(AB Aur), stars:pre-main sequence 1. INTRODUCTION Disks around young stars are a natural part of the star formation process and, as the likely site of planet formation, have generated much interest. A consider- able amount has been learned about the structure of circumstellar disks through the study of dust emission and detailed modeling of the spectral energy distribu- tions (SED) of these stars. However, if we assume the standard gas-to-dust ratio for the interstellar medium, dust makes up only 1% of the mass of the disk. Thus there is an interest in developing direct tracers of the gas component. Studies of gas in disks have so far focused mainly on gas at either large radii (> 50 AU), using ob- servations at submillimeter wavelengths (Semenov et al. 2005), or small radii, using observations of near-infrared CO lines (Blake & Boogert 2004; Najita et al. 2003). Mid-infrared spectral diagnostics may probe disks at in- termediate radii (1-10 AU) (Najita et al. 2006). Molecular hydrogen diagnostics are promising because they trace the dominant constituent of the disk and so do not rely on conversion factors to determine the mass of the emitting gas. H2 has been observed in circumstellar environments at ultraviolet (Johns-Krull et al. 2000) and near-infrared (Bary et al. 2003) wavelengths. These ob- servations trace hot circumstellar gas, or gas excited by fluorescent processes, and are therefore difficult to trans- 1 Department of Astronomy, University of Texas at Austin, Austin, TX 78712; [email protected], [email protected], [email protected] 2 Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement no. NCC 5-538 with the National Aeronautics and Space Administration, Science Mission Directorate, Planetary Astronomy Program. 3 Physics Department, University of California at Davis, Davis, CA 95616; [email protected] 4 Lunar and Planetary Institute, Houston, TX 77058; [email protected] 5 Division of Geological & Planetary Sciences, California Institute of Technology, MS 150-21, Pasadena, CA 91125; [email protected] late into gas masses. The pure rotational mid-infrared H2 lines are useful probes because the level populations should be in LTE at the local gas temperature, and so line ratios allow determination of the excitation temper- ature and mass of the warm gas. The line ratios of the three low-J lines accessible from the ground are sensitive to gas at temperatures of 200-800 K. Other lines in the mid-infrared such as [NeII] at 12.8 µm (Glassgold et al. 2007), [SI] at 25.2 µm and [FeII] at 26 µm (Pascucci et al. 2006) should be useful probes of gas in disks but their interpretation requires detailed modeling. Molecular hy- drogen emission does not trace the cold, optically thick regions of disks, and therefore estimates of the amount of warm gas based on H2 emission represent a lower limit to the total disk mass. In order to see H2 emission aris- ing from a disk, the warm gas must be either physically separated from optically thick dust or at a different tem- perature. These conditions can be met in the disk sur- face layer or in gaps or holes in the disk. In either case, gas can be physically separated (due to dust settling or clearing) as well as thermally decoupled (due to lower densities) from the dust and heated to a higher temper- ature by X-ray and UV radiation (Glassgold et al. 2004; Gorti & Hollenbach 2004; Nomura & Millar 2005). From the ground, three H2 mid-infrared rotational lines are accessible: S(1) (λ = 17.035 µm), S(2) (λ = 12.279 µm), and S(4) (λ = 8.025 µm). The high spec- tral resolution achievable with the Texas Echelon Cross Echelle Spectrograph (TEXES; Lacy et al. 2002) is cru- cial for these observations in order to separate the lines from nearby telluric features and to maximize the line contrast against the dust continuum. Observations at high spectral resolution also have the advantage that, when coupled with information about the disk inclina- tion, they allow an estimate for the location of the emit- ting gas. For emission lines with small equivalent width, ground-based observations with TEXES can be more sensitive than Spitzer IRS observations. AB Aur is one of the brightest and thus one of the most well-studied of the Herbig Ae stars. It is located http://arxiv.org/abs/0704.1481v1 2 BITNER ET AL. at a distance of 144 pc based on Hipparcos measure- ments (van den Ancker et al. 1998), has a spectral type of A0-A1 (Hernández et al. 2004) and is surrounded by a disk/envelope structure that extends to at least r ∼ 450 AU (Mannings & Sargent 1997). Observations at 11.7 and 18.7 µm by Chen & Jura (2003) show that AB Aur varies in the mid-infrared. The mass of the star is 2.4 M⊙ with an age of 2-4 Myr (van den Ancker et al. 1998). AB Aur has a SED which is well-fit by the passive irradi- ated disk with puffed-up inner rim model of Dullemond et al. (2001). There have been a wide range of inclina- tion estimates for AB Aur (see Brittain et al. 2003 for an extended discussion of this issue). Recent results by Se- menov et al. (2005) derive an inclination of 17+6 −3 deg for the AB Aur system by modeling millimeter observations. We will assume this value for analysis of our data. Several attempts have been made to observe molecular hydrogen emission from the disk of AB Aur. Thi et al. (2001) claimed the detection of H2 S(0) and S(1) emis- sion associated with the AB Aur system using the In- frared Space Observatory (ISO), but subsequent obser- vations from the ground with improved spatial resolu- tion were unable to confirm the emission (Richter et al. 2002). Richter et al. (2002) and Sheret et al. (2003) saw some evidence of emission in the S(2) line but the data quality was such that a definitive statement could not be made. They saw no evidence of the S(1) line. In this paper, we present definite detections of the S(1), S(2), and S(4) lines. 2. OBSERVATIONS AND DATA REDUCTION All of the data were acquired using TEXES in its high- resolution mode on the NASA Infrared Telescope Facil- ity (IRTF) between December 2002 and October 2004 and on Gemini North in November 2006 under program ID GN-2006B-Q-42. Table 1 lists details of the obser- vations. We nodded the source along the slit to remove background sky emission. We observed flux and telluric standards at each setting. Other calibration files taken include blank sky and an ambient temperature black- body used to calibrate the wavelength and to flatfield the data. The data were reduced using the standard TEXES pipeline (Lacy et al. 2002) which produces wavelength- calibrated one dimensional spectra. The TEXES pipeline data reduction gives a first order flux calibration. How- ever, because of the different slit illumination of the am- bient temperature blackbody and point sources, we ob- served flux standards to determine a correction factor. We assumed that the slit illumination was the same for the target as for the flux standard. Guiding and seeing variations can introduce uncertainty into our flux cali- bration. 3. RESULTS Figure 1 shows flux-calibrated spectra for all three set- tings taken at the Gemini North Observatory. Overplot- ted on each spectrum is a Gaussian with integrated line flux equal to that derived from the best-fit model as- suming the emission arises from an isothermal mass of optically thin H2 gas. The three lines were fit simulta- neously while varying the temperature and mass of the emitting gas in the model. For the simultaneous fitting, we adopted the FWHM and centroid of the S(2) line for all three lines. The quoted errors for the temperature and mass are 1-σ based on the contour plot of the χ2 values. Our best-fit model based on Gemini data has T = 670 ± 40 K and M = 0.52 ± 0.15 M⊕. A similar analysis of our IRTF data, which consisted of a detec- tion at S(2) and upper limits for S(1) and S(4), gives T = 630± 60 K and M = 0.7± 0.2 M⊕. Though within 2-σ uncertainties, the lower measured S(2) flux observed at Gemini than from the IRTF combined with the fact that our slit is smaller on the sky at Gemini suggests we may be resolving out some of the flux we see at the IRTF. To investigate this possibility, we examined spatial plots of our 2-D echellograms but found no clear evidence for the existence of spatially resolved line emission. Table 2 contains a summary of our results. We fit each line individually with a Gaussian to determine centroid, FWHM, and flux values. The lines are all centered near the systemic velocity of AB Aur and have FWHM ∼10 km s−1. Roberge et al. (2001) find A = 0.25 for AB Aur, so we assume no extinction at our wavelengths. We quote equivalent widths in addition to line fluxes due to uncertainty in the determination of the continuum level. In an attempt to test how sensitive our temperature and mass estimates are to errors in the flux calibration, we fit our data after normalizing to the continuum flux values from ISO SWS observations of AB Aur (Meeus et al. 2001). The resulting temperature and mass do not differ significantly from the values based on our internal flux calibration. We derive T = 630 K and M = 0.71 M⊕ when normalizing to the ISO continuum values of 13.1 Jy at 8 µm, 21.0 Jy at 12 µm, and 30.8 Jy at 17 µm before fitting. The upper limits for S(1) and S(4) from the IRTF quoted in Table 2 are based on a Gaussian fit at the expected position of each line plus a 1-σ error. The 1-σ line flux errors were computed by summing over the number of pixels corresponding to the FWHM of the lines in regions of the spectrum with similar atmospheric transmission. Figure 2 shows a population diagram based on our observations. The points marked are based on the line fluxes derived from Gemini observations listed in Table 2, and the error bars are 1-σ. The overplotted solid line is not a fit to the population diagram; rather it is based on the best-fit model temperature and mass. Even though we only have three points, the deviation from a single temperature (a straight line on this plot) is significant and may indicate that the emission is coming from a mix of temperatures, as expected for a disk with a radial tem- perature gradient. We note that curvature in the popu- lation diagram can also occur if the ortho-to-para (OTP) ratio is different from our assumed value of 3. Fuente et al. (1999) derived a an OTP ratio between 1.5-2 for H2 in a photodissociation region with gas temperatures of 300-700 K. However, the sense of the curvature in our population diagram would require an OTP ratio greater than 3 to explain the increased S(1) flux, which seems unlikely. We discuss the derived temperature of the gas and its location in the next section. 4. DISCUSSION We have detected all three mid-IR H2 lines with good signal to noise. The results are consistent with the pre- vious upper limits and the 2-σ detection of the S(2) line (Richter et al. 2002; Sheret et al. 2003). Our observations OBSERVATIONS OF H2 EMISSION FROM AB AURIGAE 3 of all three lines allow us to put tighter constraints on the temperature and mass of the emitting gas. The analysis of our data has three components. First, we determine a characteristic temperature of the emitting gas by fitting a simple model to the three lines. Second, we locate the po- sition in the disk where the emission arises by comparing our observed line profiles to line profiles calculated from a Keplerian disk model. Finally, in order to comment on the structure of the disk, we compare our results to the predictions of a well-established disk structure model which assumes well-mixed gas and dust at the same tem- perature and find that, under such assumptions, we do not reproduce our observations. This argues for the need for additional gas heating mechanisms which may come in the form of X-ray and UV heating. The importance of UV and X-ray heating in the surface layers of disks has been demonstrated observationally by Bary et al. (2003) and Qi et al. (2006). The procedure used to derive a temperature of T = 670 ± 40 K and mass of M = 0.52 ± 0.15 M⊕ from a simulaneous fit to the H2 S(1), S(2), and S(4) lines as- sumes the gas is optically thin and in LTE at a single temperature. If the emission arises in the atmosphere of a disk where radial and vertical temperature gradients exist, the derived temperature should be considered a characteristic or average temperature. In fact, Figure 2 shows evidence that the emission arises in gas at a mix of temperatures. It should be noted that radial tem- perature gradients in a disk also affect the detectability of various lines. Since we are most sensitive to narrow emission lines, the S(1) line, originating in cooler gas farther out in the disk, is easier to detect than emission in the S(4) line. Emission arising very close to the star from the inner rim of the disk would be more difficult for us to detect at our high spectral resolution due to line broadening. Our derived temperature, 670 K, is in agreement with the temperature found by Richter et al. (2002) who de- rived T > 380 K based on S(2) and the upper limit on S(1). Brittain et al. (2003) made observations toward AB Aur at 4.7 µm of 12CO v = 1-0 emission. The low-J lines in the resulting population diagram have a steeper slope, interpreted as emission coming from cooler gas than the high-J lines. Fits to the population diagrams assuming LTE gas give T = 70 K for the cool gas and T = 1540 K for the hot gas. The authors explain the hot CO emission as coming from the inner rim of the disk, while the cool emission originates in the outer flared part of the disk after it emerges from the inner rim’s shadow. Blake & Boogert (2004) also observed 4.7 µm CO emission from AB Aur. Isothermal fits with T ∼ 800 K are able to reproduce their line fluxes. The line widths for the CO lines are generally broader than what we observed, sug- gesting that the CO emission comes from smaller radii or outflowing gas (Blake & Boogert 2004). Our derived mass, 0.52 M⊕, is four orders of magni- tude less than the total disk mass of AB Aur (Semenov et al. 2005). This is partly due to the fact that AB Aur’s disk is optically thick at mid-infrared wavelengths (Sheret et al. 2003). Most of the gas in the disk is hid- den by optically thick dust and is also too cool to emit at our wavelengths. The mass we derive should therefore be interpreted as a lower limit to the amount of warm gas on the surface of the disk facing us. By comparing the FWHM of our observed S(2) line profile with computed line profiles from a simple Keple- rian disk model, we derive an estimate of where the line emission originates. We generate line profiles under the assumption of emission from a Keplerian disk inclined at 17◦ (Semenov et al. 2005) orbiting a 2.4 M⊙ star (van den Ancker et al. 1998) with equal emissivity at all points within an annulus 2 AU in extent. We convolve the line profiles with our instrumental broadening function plus a thermal broadening function based on our derived tem- perature, T = 670 K, and compare the resulting FWHM of the line profiles to our observed S(2) line. Under these assumptions, the line profiles imply that our emission arises near 18 AU in the disk. We used parameters for the disk based on the fit to AB Aurigae’s SED from Dullemond et al. (2001) as input to the vertical disk structure model of Dullemond et al. (2002) in order to derive a grid of temperature and den- sity values as a function of disk radius and height above the midplane. The Dullemond et al. (2001) model does an excellent job fitting the SED, although we note that their derived inclination angle, 65◦, is significantly larger than our assumed value of 17◦. As noted in Dullemond et al. (2001), the large discrepancy between their derived inclination and what most observations suggest is proba- bly due to the inclusion in their SED model of a perfectly vertical inner rim which may not be physically accurate. We input this radial and vertical temperature and den- sity profile into an LTE radiative transfer program. We assume the gas is in LTE with the dust and compute the emergent spectrum. The emission lines produced by our model under these conditions are much weaker than our observations. This is not especially surprising given our assumption of well-mixed gas and dust at the same tem- perature and suggests the need for additional gas heating mechanisms. The vertical structure model of Dullemond et al. (2002) is mainly concerned with modeling the dust emission from disks. By assuming equal gas and dust temper- ature, the gas only reaches a temperature of ∼ 260 K at the top of the disk at 18 AU, much cooler than our derived temperature of 670 K. To explain the observed emission, an additional mechanism is needed to heat the gas. X-ray and UV heating are likely possibilities which can heat the gas to temperatures significantly hotter than the dust. Glassgold et al. (2007) computed the gas tem- perature in X-ray irradiated disks around T Tauri stars. At 20 AU, the temperature can reach over 3000 K at the top of the disk before dropping to 500-2000 K in a transi- tion zone and then to much cooler temperatures deep in the disk. Nomura & Millar (2005) considered UV heat- ing of protoplanetary disks and the resulting molecular hydrogen emission. The temperature in their model at 10 AU reaches over 1000 K and predicts an H2 S(2) line flux of 2×10−15 ergs s−1 cm−2, somewhat lower than our observed flux, but their model was for a less massive star and disk. H2 emission could also arise in regions where there is spatial separation of the gas from the dust due to dust settling or coagulation of dust into larger parti- cles. The SED of AB Aur shows no evidence for gaps in the AB Aur disk and we are unable to make definitive statements based on our observations. A UV-heated H2 layer in a protostellar disk ought to exhibit at least some similarities to the comparable hot 4 BITNER ET AL. layer at the surface of photodissociation regions (PDRs) with high densities and strong radiation fields. We there- fore compare our observations of AB Aur with earlier mid-IR H2 spectroscopy of the Orion Bar PDR. The UV field expressed in terms of the mean interstellar radiation field that was adopted by Allers et al. (2005) (3× 104) is somewhat smaller than that expected at 18 AU around an A0 star (∼ 105). Jonkheid et al. (2007) have com- puted the strength of the UV field throughout a disk around a Herbig Ae star finding values between 105 - 106 near the disk surface at 20 AU. Our derived tempera- ture, 670 K, is in the range of temperatures (400-700 K) derived for the Orion bar PDR based on the H2 S(1), S(2), and S(4) lines (Allers et al. 2005). As shown in Allers et al. (2005), the line ratios are determined by a complex mix of the temperature gradient, H2 abundance and dust properties. As another check of the plausibility that the surface of AB Aur’s disk is similar to a PDR, we computed the surface brightnesses of our three lines. Assuming our observed emission arises from an annulus 2 AU in extent centered at 18 AU, we derive surface brightnesses of ∼ 10−5 ergs cm−2 s−1 sr−1, roughly an order of magnitude lower than those seen by (Allers et al. 2005). However, Allers et al. (2005) estimated that their line intensities were brightened by a factor of 10 since the Orion Bar PDR is seen nearly edge-on making our surface brightnesses consistent with each other. We thank the Gemini staff, and John White in par- ticular, for their support in getting TEXES to work on Gemini North. We thank Rob Robinson for useful dis- cussions on the analysis of our data. The development of TEXES was supported by grants from the NSF and the NASA/USRA SOFIA project. Modification of TEXES for use on Gemini was supported by Gemini Observa- tory. Observations with TEXES were supported by NSF grant AST-0607312. This work is based on observa- tions obtained at the Gemini Observatory, which is op- erated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the Na- tional Science Foundation (United States), the Particle Physics and Astronomy Research Council (United King- dom), the National Research Council (Canada), CON- ICYT (Chile), the Australian Research Council (Aus- tralia), CNPq (Brazil) and CONICET (Argentina). 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H. 2002, ApJ, 572, L161 Roberge, A. et al. 2001, ApJ, 551, L97 Semenov, D., Pavlyuchenkov, Y., Schreyer, K., Henning, T., Dullemond, C., & Bacmann, A. 2005, ApJ, 621, 853 Sheret, I., Ramsey Howat, S.K., & Dent, W.R.F. 2003, MNRAS, 343, L65 Thi, W.F. et al. 2001, ApJ, 561, 1074 van den Ancker, M.E., de Winter, D., & Tjin A Djie, H.R.E. 1998, A&A, 330, 145 OBSERVATIONS OF H2 EMISSION FROM AB AURIGAE 5 TABLE 1 Observing Parameters Date Telescope Linea Resolving Powerb Slit Widthc Integration Time (R ≡ λ/δλ) (arcsec, AU) (s) Dec 2002 IRTF S(1) 60,000 2.0, 288 2591 Dec 2002 IRTF S(2) 85,000 1.4, 202 4792 Dec 2003 IRTF S(2) 81,000 1.4, 202 2072 Oct 2004 IRTF S(4) 88,000 1.4, 202 8549 Nov 2006 Gemini S(1) 80,000 0.81, 117 2007 Nov 2006 Gemini S(2) 100,000 0.54, 78 2072 Nov 2006 Gemini S(4) 100,000 0.54, 78 1295 aS(1) is at 587.0324 cm−1. S(2) is at 814.4246 cm−1. S(4) is at 1246.098 cm−1. bThe resolving power is deduced from observations of stratospheric emission from Titan or Saturn at similar frequencies or unresolved emission lines in a low pressure gas cell. cThe slit width in AU is calculated at AB Aur’s distance of 144 pc. 6 BITNER ET AL. TABLE 2 Summary of Results Telescope λ Fν Line Flux Equivalent Width FWHM (µm) (Jy) (10−14 ergs s−1 cm−2) (km s−1) (km s−1) 8 12.9 (1.1) < 1.2 ... ... IRTF 12 13.3 (0.4) 0.93 (0.25) 0.86 (0.23) 7.0 17 25.2 (1.8) < 1.1 ... ... 8 12.7 (0.4) 1.47 (0.34) 0.93 (0.21) 10.4 Gemini 12 14.7 (0.3) 0.53 (0.07) 0.44 (0.06) 8.5 17 24.6 (0.4) 1.1 (0.30) 0.76 (0.21) 9.0 OBSERVATIONS OF H2 EMISSION FROM AB AURIGAE 7 Fig. 1.— Regions of the three H2 pure rotational lines available from the ground. The data shown were obtained at Gemini North in November 2006 under Gemini program GN-2006B-Q-42. Each spectrum has been flux-calibrated. The lines were individually fit with a Gaussian to determine their centroid, flux, and FWHM. The lines are all centered near the systemic velocity of AB Aur and have FWHM ∼10 km s−1. Line fluxes are listed in Table 2. The overplotted Gaussians are based on the simultaneous fit and are not the individual fits to each line. The three spectral regions were simultaneously fit with a model assuming the observed emission arises from a mass of H2 gas at a single temperature. The best-fit model with T = 670 K and H2 mass = 0.52 M⊕ is overplotted. 8 BITNER ET AL. Fig. 2.— Population diagram based on Gemini North observations. The points marked are based on fluxes derived by fitting a Gaussian to each of the lines and shown with 1-σ error bars. The overplotted solid line is not a fit to the population diagram; rather it is based on the best-fit model temperature and mass found by simultaneously fitting all three lines. The deviation of the points from the single-temperature line is significant and suggests that the emission may arise from gas at a range of temperatures.

0704.1482

Deciphering top flavor violation at the LHC with B factories

arXiv:0704.1482v2 [hep-ph] 7 Sep 2008 UCB–PTH–07/06 YITP–SB–07–11 Deciphering top flavor violation at the LHC with B factories Patrick J. Fox,1 Zoltan Ligeti,1 Michele Papucci,1, 2 Gilad Perez,3 and Matthew D. Schwartz4 Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720 Department of Physics, University of California, Berkeley, CA 94720 C.N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840 Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles St., Baltimore, MD 21218 The LHC will have unprecedented sensitivity to flavor-changing neutral current (FCNC) top quark decays, whose observation would be a clear sign of physics beyond the standard model. Although many details of top flavor violation are model dependent, the standard model gauge symmetries relate top FCNCs to other processes, which are strongly constrained by existing data. We study these constraints in a model independent way, using a low energy effective theory from which the new physics is integrated out. We consider the most important operators which contribute to top FCNCs and analyze the current constraints on them. We find that the data rule out top FCNCs at a level observable at the LHC due to most of the operators comprising left-handed first or second generation quark fields, while there remains a substantial window for top decays mediated by operators with right-handed charm or up quarks. If FCNC top decays are observed at the LHC, such an analysis may help decipher the underlying physics. I. INTRODUCTION The Large Hadron Collider (LHC) will have unprecedented sensitivity to flavor changing neutral currents (FCNCs) involving the top quark, such as t → cZ. With a tt pair production cross section of about 800 pb and after 100 fb−1 of integrated luminosity, the LHC will explore branching ratios down to the 10−5 level [1, 2]. Flavor changing neutral currents are highly suppressed in the standard model (SM), but are expected to be enhanced in many models of new physics (NP). Because top FCNCs are clean signals, they are a good place to explore new physics. There are important constraints from B physics on what top decays are allowed, and understanding these constraints may help decipher such an FCNC signal. In this paper, we calculate the dominant constraints on top FCNCs from low energy physics and relate them to the expected LHC reach using a model-independent effective field theory description. Flavor physics involving only the first two generations is already highly constrained, but the third generation could still be significantly affected. Of course, the new flavor physics could be so suppressed that it will not be observable at all at the LHC. However, since the stabilization of the Higgs mass is expected to involve new physics to cancel the top loop, it is natural to expect some new flavor structure which may show up in the top quark couplings to other standard model fields. Thus, one may expect flavor physics to be related to the electroweak scale, and then flavor changing effects involving the top quark are a natural consequence. Although there are many models which produce top FCNCs, the low energy constraints are independent of the details of these models. The new physics can be integrated out, leaving a handful of operators relevant at the weak scale involving only standard model fields. These operators mediate both FCNC top decays and flavor-changing transitions involving lighter quarks. Thus, the two can be related without reference to a particular model of new physics, provided there is no additional NP contributing to the B sector. The low energy constraints can be applied to any model in which top FCNCs are generated and the constraints on the operators may give information on the scale at which the physics that generates them should appear. Analyses of FCNC top decays have been carried out both in the context of specific models [3] and using model independent approaches [4]. However, in most cases the effective Lagrangian analyzed involved the SM fields after electroweak symmetry breaking. As we shall see, the scale Λ at which the operators responsible for top FCNC are generated has to be above the scale v of electroweak symmetry breaking. Thus, integrating out the new physics should be done before electroweak symmetry breaking, leading to an operator product expansion in v/Λ. The requirement of SU(2)L invariance provides additional structure on the effective operators [5], which helps constrain the expectations for top FCNCs. For example, an operator involving the left-handed (t, b) doublet, the SU(2) gauge field, and the right handed charm quark, can lead to b → sγ at one loop, but also directly to a b → c transition. If we ignored SU(2)L invariance, we would only have the b → sγ constraint, and the resulting bound would be different. An important feature of our analysis is that, after electroweak symmetry breaking, the resulting operators can modify even SM parameters which contribute at tree level to B physics observables, such as |Vcb|. The organization of this paper is as follows. In Sec. II we introduce the effective Lagrangian relevant for top FCNCs. We also explain why some operators can be neglected and introduce conventions used throughout the paper. In Sec. III we calculate how these operators affect top quark decays and integrate out theW and Z bosons and the top quark to match onto the relevant effective theory at the weak scale. In Sec. IV we relate the experimental constraints http://arxiv.org/abs/0704.1482v2 to the Wilson coefficients calculated in Section III, focusing mostly on observables related to B physics. This leads directly to predictions for the top branching ratio. Sec. V contains a summary of the results and our conclusions. We include an Appendix with details of the calculations. II. EFFECTIVE LAGRANGIAN FOR TOP FCNC We consider an effective Lagrangian Leff = (CiOi + C i) . (1) where the Oi operators involve third and second generation quarks and the O i involve the third and first generations. Since we are interested in top quark decays, we define Oi and O i in the mass basis for the up-type quarks. A complete set of dimension-six operators which give a tcZ or tcγ vertex are OuLL = i H̃†Q2 + h.c. , OhLL = i + h.c. , OwRL = g2 µνσaH̃ µν + h.c. , ObRL = g1 tRBµν + h.c. , OwLR = g2 µνσaH̃ µν + h.c. , ObLR = g1 cRBµν + h.c. , OuRR = i tRγ + h.c. . (2) The brackets mean contraction of SU(2) indices, Q3 and Q2 are the left-handed SU(2) doublets for the third and second generations, tR and cR are the right-handed SU(2) singlets for the top and charm quarks, H is the SM Higgs doublet, H̃ = iσ2H ∗, and the index a runs over the SU(2) generators. The first lower L or R index on the operators denotes the SU(2) representation of the third generation quark field, while the second lower index refers to the representation of the first or second generation field. In this basis all of the derivatives act on the Higgs fields. We could also consider operators directly involving gluons, but since the indirect constraints on gluonic currents are very weak (see, e.g., [6]), we restrict our focus to the electroweak operators in Eq. (2). The form of the operators in Eq. (2) after electroweak symmetry breaking are given in the Appendix. Throughout the paper we focus on those new operators that contribute to t → cZ, cγ. In any particular model there may be additional contributions to Eq. (1) that contribute to ∆F = 1 and ∆F = 2 processes in the down sector (e.g., four-fermion operators). These operators have suppressed contributions to top FCNCs. When we bound the coefficients of the operators in Eq. (2) from B physics, we neglect these other contributions. In any particular model these two sets of operators may have related coefficients. Unless there are cancellations between the different operators, the bounds will not get significantly weaker. There are other dimension-six operators that can mediate FCNC top decays (for example tRγ µDνcRBµν). But these can always be reduced to a linear combination of the operators included in Eq. (2) plus additional four-fermion operators and operators involving QLqRHHH fields. For instance, operators involving two quark fields and three covariant derivatives can be written in terms of operators involving fewer derivatives using the equations of motion. Operators involving two quark fields and two covariant derivatives (e.g., Q3DµcRD µH̃) can be written in terms of operators involving the commutator of derivatives included in Eq. (2) plus operators with one derivative and four- fermion operators. Finally, operators involving two quark fields and one covariant derivative can be written in a way that the derivative acts on the H field, as in Eq. (2), plus four-fermion operators. Of the four-fermion operators which appear after the reduction of the operator basis, some are suppressed by small Yukawa couplings and can simply be neglected. However, some are not suppressed, and of those, the biggest concern would be semileptonic four-fermion operators, like (tc)(ℓℓ). These contribute to the same final state as t → cZ → cℓ+ℓ−. (We emphasize Z → ℓ+ℓ−, because the LHC is expected to have the best sensitivity in this channel [1, 2].) However, the invariant mass of the ℓ+ℓ− pair coming from a four-fermion operator will have a smooth distribution and not peak around mZ , so the Z-mediated contribution can be disentangled experimentally. Operators with (tc)(qq) flavor structure also contribute to t → cℓ+ℓ− or t→ cγ at one loop, but their contributions are suppressed by α/(4π). Finally, operators with the QLqRHHH structure either renormalize Yukawa couplings, or contribute to FCNCs involving the Higgs (e.g., t→ ch), but we do not consider such processes, as explained later. Throughout most of this paper we consider each of the operators one at a time and constrain its coefficient. This is reasonable as the operators do not mix under renormalization. One exception is that OuLL and O LL mix with one another between the scales Λ and v, so it would be unnatural to treat them independently. Their mixing is given by d lnµ CuLL(µ) ChLL(µ) CuLL(µ) ChLL(µ) , (3) where α2 = α/ sin 2 θW is the SU(2) coupling. (The zero in the anomalous dimension matrix is due to the fact that custodial SU(2) preserving operator OhLL cannot mix into the custodial SU(2) violating O LL.) So, we will also carry out a combined analysis for these two operators. We have written the operators in Eq. (2) in terms of a single SM Higgs doublet. In principle there may be many new Higgs scalars, but only those that acquire a vev will contribute to t → cZ and cγ. Since a triplet Higgs vev is tightly constrained by electroweak precision tests, we concentrate on the possibility of multiple Higgs doublets. With the introduction of extra Higgs doublets, there are more operators of each particular type (OuLL, O LL, etc.), one linear combination of which gives rise to t → cZ and cγ. There are also several physical Higgs states that can contribute in loops in low energy processes. For each type of operator, a different linear combination of couplings enter in low energy measurements. However, without cancellations this will only differ from the one Higgs case by a number of order one. This allows our results to be applied to the general case of multiple Higgs doublets.1 Of course, the Higgs sector is also relevant to FCNCs involving the Higgs, such as t → ch, but we do not consider such processes as they are more model dependent. Once we go beyond models with minimal flavor violation (MFV) [7], the possibility of new CP violating phases in the NP should be considered. In MFV models, top FCNC is not observable at the LHC. In models such as next-to- minimal flavor violation (NMFV) [8] top FCNCs could be observable and the Wilson coefficients can be complex. It is not always the case that the constraints are weaker when the NP Wilson coefficients are real (in the basis where the up type Yukawa matrix is real and diagonal). Rather, interference patterns realized in some of the observables mean the constraints are weakest when some of the new phases are different from 0 or π. We shall point out the places where phases associated with the new operators can play an important role and how we treat them. In addition to the B physics related constraints we will derive in this paper, one can also use constraints from electroweak precision observables. However, these bound flavor-diagonal operators strongly, and the flavor non- diagonal operators in Eq. (2) which contribute to top FCNCs are far less constrained. For instance, the OuLL operator corrects the W propagator at one loop and so contributes to the T parameter. The loops involve a t or c quark, and have one insertion of OuLL and one insertion of Vts or Vcb. Thus, the contribution is suppressed by |Vts| ∼ |Vcb| ∼ 0.04 relative to an insertion of the flavor diagonal equivalent of OuLL, Q3H̃D/H̃ †Q3. In contrast, when considering low energy FCNC processes, OuLL will be more strongly constrained then its flavor diagonal version. That is, flavor diagonal operators are more tightly constrained by electroweak observables than by low energy FCNCs, while the off diagonal operators are more tightly constrained by low energy FCNCs. Moreover, the mixing between these two classes of operators is small. It occurs at one loop proportional to y2b |Vcb|, where the factor of yb, the bottom Yukawa coupling, is due to a GIM mechanism. Thus, we can think of the flavor diagonal and off diagonal operators as independent. And so for the purpose of studying top FCNCs, we are justified in neglecting flavor diagonal operators and the relatively weak constraints from electroweak precision tests. III. WEAK SCALE MATCHING In this section we derive how the NP operators modify flavor changing interactions at the electroweak scale and derive the effective Hamiltonian in which the t,W , and Z are integrated out. For numerical calculations we use besides the Higgs vev, v = 174.1GeV, and other standard PDG values [9], |Vts| = 41.0× 10−3 [10] and mt = 171GeV [11]. 1 One possible exception is if an extended Higgs sector allows Yukawa couplings larger than in the SM, for example, in a two Higgs doublet model at large tan β. Then a Higgs loop may give additional unsuppressed contributions when we match to the Wilson coefficients at the electroweak scale. A. Top quark decays After electroweak symmetry is broken, the operators in Eq. (2) give rise to t→ cZ and t→ cγ FCNC decays. The analytic expressions for the partial widths of these decays are given in Eq. (A2) in the Appendix. Numerically, the t→ cZ branching ratio in terms of the Wilson coefficients is B(t→ cZ) = × 10−4 × |CbLR|2 + |CbRL|2 − 9.6Re CbLRC + CbRLC |CwLR|2 + |CwRL|2 − 8.3Re (ChLL + C ∗ − CbLRCuRR + 28Re ChLL + C ∗ − CwLRCuRR ∣ChLL + C + |CuRR|2 . (4) The tcγ vertex, which has a magnetic dipole structure as required by gauge invariance, is induced only by the left-right operators. The branching ratio for t→ cγ is B(t→ cγ) = × 10−4 × 8.2 ∣CbLR + C ∣CbRL + C . (5) The analogous expressions for t→ u decays are obtained by replacing Ci by C′i in Eqs. (4) and (5). The LHC will have unparalleled sensitivity to such decays. With 100 fb−1 data, the LHC will be sensitive (at 95% CL) to branching ratios of 5.5 × 10−5 in the t → cZ channel and 1.2 × 10−5 in the t → cγ channel [1]. In the SM, B(t → cZ, cγ) are of order α(Vcbαm2b/m2W )2 ∼ 10−13, so an experimental observation would be a clear sign of new physics. Equations (4) and (5) will allow one to translate the measurements or upper bounds on these branching ratios to the scale of the individual operators. B. B decays Many of the operators in Eq. (2) modify SM interactions at tree level (this possibility was discussed in [5]). After electroweak symmetry breaking, OuLL gives rise to a bWc vertex with the same Dirac structure as the SM, so the measured value of Vcb (which we denote V cb ) will be the sum of the two. This allows us to absorb the new physics contribution of CuLL into the known value of V cb — in processes where Vcb and C LL enter the same way, the dependence on CuLL cannot be disentangled. For example, the SM unitarity condition, V tbVtd + V cbVcd + V ubVud = 0, would be violated if one simply shifted the SM values by the NP contributions. However, the CKM fits have unitarity built in, so the NP contribution to Vcb causes a shift in the values of Vts and Vtd extracted from the CKM fit, V and V fittd . Since we cannot measure all CKM elements independently, we have to replace Vts and Vtd by V ts and V plus modified NP contributions. (Recall that Vts and Vtd are only constrained from loop processes where they enter together with new physics contributions.) With these redefinitions we can use V cb , V ts and V td in the CKM fit, and the NP will only have distinguishable effects in SM loop processes. An analogous procedure applies to the t → u contribution to V ub , V td and V ts . Some other operators such as C LR do not generate a bWc vertex with the same Dirac structure as the SM. Thus, their contributions to observables from which Vcb is extracted may be disentangled as discussed in the following. At leading order in the Wolfenstein parameter (Cabibbo angle), λ, these relations are: Vcb = V cb + (v 2/Λ2)CuLLVtb , Vub = V ub + (v 2/Λ2)C′uLLVtb , V ∗ts = V ts − (v2/Λ2) (CuLLV ∗cs + C′uLLV ∗us) , V ∗td = V td − (v2/Λ2) (CuLLV ∗cd + C′uLLV ∗ud) . (6) The OwLR (O LR) also modifies the bWc (bWu) vertex, but with different Dirac structure from the SM, so its effects can be separated from the SM contribution. Finally, OhLL (O LL) gives tree-level FCNC, since it contains a bZs (bZd) interaction. At the one-loop level, the operators in Eq. (2) contribute to b→ s transitions. The constraints from B physics are easiest to analyze by matching these operators onto operators containing only the light SM fields at a scale µ ∼ mW . We use the standard basis as defined in [12]. Integrating out the top, W , and Z, the most important operators for B → Xsγ and B → Xsℓ+ℓ− which are affected by NP are O7γ = [mbs σ µν(1 + γ5)b]Fµν , O9V = [sγ µ(1− γ5)b] [ℓγµℓ] , O10A = [sγ µ(1− γ5)b] [ℓγµγ5ℓ] . (7) For example, the diagram in Fig. 1 gives a contribution from OwRL (denoted by ⊗) to O7γ . The coefficients of the QCD and electroweak penguin operators, O3,...,10, are also modified, but their effect on the processes we consider are suppressed. Summing the relevant diagrams, the contributions of all operators can be expressed in terms of generalized Inami- Lim functions, presented in the Appendix. Setting Λ = 1TeV, the numerical results are2 C7γ(mW ) = −0.193 + 0.810CuLL + 0.179C LL + 0.310C RL − 0.236CbRL + 0.004CwLR − 0.003CbLR C9V (mW ) = 1.56 + −0.562CuLL + 44.95ChLL − 0.885CwRL − 1.127CbRL + 0.046CwLR + 0.004CbLR C10A(mW ) = −4.41 + −7.157CuLL − 598ChLL + 3.50CwRL − 0.004CuRR . (8) The first term in each expression is the SM contribution. Note that the OhLL contribution is large because it is at tree level, while ObLR, O LR, and O RR are tiny because they are suppressed by mc/mW and so the constraints on these will be weaker. In the case of b → d transitions the NP contribution has to be rescaled by the O(1/λ) factor, |V ∗tsVud/V ∗tdVcs| ≈ 5.6, and Ci should be replaced with C′i. C. ∆F = 2 transitions The operators OuLL, C LL, and O RL also contribute to ∆F = 2 transitions, i.e., neutral meson mixings. Again, the contribution from OhLL is present at tree level, while the other two contribute starting at one-loop order. The relevant functions are again listed in the Appendix. The modifications relative to the SM Inami-Lim function can be parameterized as S0 → S0(1 + hMe2i σM ) for each neutral meson system. Numerically (setting Λ = 1TeV), for B0sB0s mixing, the effect of the t→ c operators is given by 2iσBs = 800(ChLL) 2 + 0.92ChLLC LL − 6.84(CuLL)2 + 1.55ChLL − 2.64CuLL − 0.32(CwRL)2 − 1.03CwRL . (9) The contributions of the O′i operators to B s mixing is given by replacing Ci with C i in Eq. (9) and multiplying its right-hand side by λ. The contribution of the Oi operators to B d mixing is obtained by multiplying the right-hand side of Eq. (9) by eiβ , where β is the CKM phase, β = arg(−VcdV ∗cb/VtdV ∗tb). Whereas the contribution of the O′i operators to B0dB0d mixing is obtained again by replacing Ci with C i in Eq. (9) and multiplying its right-hand side by −eiβ/λ. Finally, the O′i contribution to K 0K0 mixing is the same as that to B0dB d mixing, up to corrections suppressed by powers of λ. For the Oi contribution to K 0K0 mixing, one has to replace in Eq. (9) each Wilson coefficient Ci by Ci + C iβ (see Eq. (A20) in the Appendix), and add to it the additional contribution ∆(hKe 2iσK ) = 2.26Re(ChLLC LL) e iβ − 5.17 |CuLL|2 eiβ − 8.35 |CwRL|2 eiβ . (10) These expressions are valid up to corrections suppressed by λ2 or more. b t s FIG. 1: A one-loop contribution from OwRL (denoted by ⊗) to O7γ . 2 Throughout this paper we will bound Ci(1TeV/Λ) 2 and quote numerical results setting Λ = 1TeV. IV. EXPERIMENTAL CONSTRAINTS In this section we use low energy measurements to constrain the Wilson coefficients of the operators in Eq. (2). Throughout we assume that there are no cancellations between the contributions from different operators. A. Direct bounds The best direct bounds on the operators in Eq. (2), as summarized in [9], come at present from searches for FCNCs at the Tevatron, LEP, and HERA. The strongest direct constraints on t→ cZ and t→ uZ come from an OPAL search for e+e− → tc in LEP II [13]. The upper limit on the branching ratio B(t→ cZ, uZ) < 0.137 bounds the LL and RR operators. For neutral currents involving a photon, there is a constraint from ZEUS that looked for e±p→ e±tX [14]. This bounds B(t→ uγ) < 0.0059, and is the strongest constraint on the RL and LR operators with an up quark. The other bounds come from a CDF search in Tevatron Run I, which bounds B(t → cγ, uγ) < 0.032 [15] and constrains the LR and RL involving a charm. We translate these branching ratios into bounds on the Wilson coefficients and list them in the first rows of Tables I and II. The LHC reach with 100 fb−1 data, as estimated in the ATLAS study [1] is B(t→ cZ, uZ) < 5.5× 10−5 and B(t→ cγ, uγ) < 1.2× 10−5. These will improve the current direct constraints on the Wilson coefficients by one and a half orders of magnitude, as summarized in the second rows of the tables. B. B → Xsγ and B → Xsℓ We first consider the constraints from B → Xsγ. At the scale mb, O7γ gives the leading contribution. Using the NLO SM formulae from Ref. [16], we obtain B(B → Xsγ) = 10−4 × 0.07 + ∣1.807 + 0.081 i+ 1.81∆C7γ(mW ) , (11) where ∆C7γ(mW ) is the NP contribution to C7γ at the µ = mW matching scale. The current experimental average [17], B(B → Xsγ) = (3.55± 0.26)× 10−4, implies at 95% CL3 (setting Λ = 1TeV) − 0.07 < CuLL < 0.04 or 1.2 < CuLL < 1.3 , −0.3 < ChLL < 0.16 or 5.3 < ChLL < 5.8 , −0.2 < CwRL < 0.1 or 3.1 < CwRL < 3.4 , −0.1 < CbRL < 0.24 or −4.5 < CbRL < −4.1 , (12) The first (left) intervals are consistent with the SM, while the second (right) ones require new physics at the O(1) level. The non-SM region away from zero is disfavored by b → sℓ+ℓ− discussed below, but we include it here for completeness. For the operators whose contributions are suppressed by mc, we find − 14 < CwLR < 7 , −10 < CbLR < 19 , (13) and no meaningful bound for CuRR. To obtain the results in Eq. (12) and (13), we assumed that the NP contributions are real relative to the SM, i.e., that there are no new CP violating phases. Had we not made this assumption, the allowed regions would be annuli in the complex Ci planes. Next we consider B → Xsℓ+ℓ−. The theoretically cleanest bound at present comes from the inclusive B → Xsℓ+ℓ− rate measured for 1GeV2 < q2 < 6GeV2 [18] B(B → Xsℓ+ℓ−)1GeV2<q2<6GeV2 = (1.61± 0.51)× 10−6 . (14) Due to the unusual power counting in B → Xsℓ+ℓ−, the full set of O(αs) corrections are only included in what is called NNLL order, achieving an accuracy around 10%. For the SM prediction we use the NNLL calculation as implemented in Ref. [19]. This calculation does not normalize the rate to the B → Xℓν̄ rate; doing so would not 3 Hereafter all constraints are quoted at 95% CL, unless otherwise specified. -0.5 0 0.5 1 1.5 -0.03 -0.02 -0.01 FIG. 2: Constraints from B → Xsγ and B → Xsℓ − in the CuLL – C LL plane. The red, green, and blue regions denote 68%, 95%, and 99% CL, respectively. The region between the dashed lines is beyond the LHC sensitivity. improve the prediction significantly and would unnecessarily couple different operators’ contributions. We include the modifications of C7γ , C9V , and C10A due to the new operators at lowest order. With our input parameters, we obtain B(B → Xsℓ+ℓ−)1<q2<6GeV2 = 10−6 × 1.55 + 35100 |∆C9V (mW )|2 + |∆C10A(mW )|2 + 0.45 |∆C7γ(mW )|2 (180 + 5i)∆C9V (mW ) − 360Re ∆C10A(mW ) (0.17 + 0.04i)∆C7γ(mW ) − 200Re ∆C9V (mW ) ∗∆C7γ(mW ) The simplest way to proceed would be to bound C7γ , C9V , and C10A separately at µ = mW , assuming that the others have their SM values, and use this to constrain new physics. This procedure would not be consistent, since the NP necessarily affects these Wilson coefficients in a correlated way. Instead, we directly constrain the coefficients of OuLL, OhLL, O LR, and O LR, which also yields stronger constraints. With Λ = 1TeV, we obtain − 1.1 < CuLL < 0.3 , −1.8× 10−2 < ChLL < −1× 10−2 or − 5× 10−3 < ChLL < 3× 10−3 , −0.5 < CwRL < 0.7 or 1.7 < CwRL < 3 , −2.0 < CbRL < 3.5 . (16) The combined constraints from B → Xsγ and B → Xsℓ+ℓ− on these four Wilson coefficients are shown in Table I in the Conclusions. We plot in Fig. 2 the bound on the LL operators in the CuLL – C LL plane. The SM corresponds to the point (0, 0). A measurement or a bound on the t→ cZ branching ratio corresponds to a nearly vertical band. The LHC is sensitive to this whole plane, except for the band between the dashed lines. The above bounds were derived assuming that the NP contribution is real relative to the SM. It is conceivable that improved measurements of B → Xsℓ+ℓ− will lead to constraints on the CP violating phases before the LHC is be able to probe top FCNCs. Thus we postpone a full analysis with complex NP Wilson coefficients until more data become available. C. Exclusive and inclusive b → cℓν̄ decays In this subsection we investigate the constraints on the operators in Eq. (2) due to measurements of semileptonic b→ c decays. They will allow us to bound the coefficient of the operator OwLR, which contains a right handed charm field and is weakly constrained otherwise. We focus on three types of constraints coming from the ratio of exclusive D and D∗ rates, the polarization in the D∗ mode, and moments in inclusive spectra. We begin with the exclusive case where the B → Dℓν and B → D∗ℓν rates can be calculated in an expansion in ΛQCD/mb,c using heavy quark effective theory. The form factors at zero recoil, where w = v · v′ = 1 (v and v′ are the four-velocities of the B and D(∗) mesons, respectively), have been determined from lattice QCD [20]. In the SM the ratio of rates is independent of Vcb, and therefore it provides a good test for non-SM contributions. The presence of the new operator, OwLR, affects the two rates differently. The rates are given by [21] dΓ(B → Dℓν) r3(w2 − 1)3/2(1 + r)2|Vcb|2(FD)2 , dΓ(B → D∗ℓν) w2 − 1 (1 + w)2 (1 − r∗)2 + 4w 1 + w (1− 2wr∗ + r∗2) |Vcb|2(FD∗)2 , (17) where r = mD/mB and r ∗ = mD∗/mB. The form factors FD and FD∗ can be decomposed in terms of 6 form factors, h+,−,V,A1,A2,A3 [21]. At leading order in the heavy quark limit F(w) = F∗(w) = h+,V,A1,A3 = ξ(w), where ξ(w) is the Isgur-Wise function [22], while h−,A2 = 0. Therefore, it is useful to define the following ratios of form factors R1(w) = , R2(w) = hA3 + r , (18) which are equal to unity in the heavy quark limit and have been measured experimentally. Following the analysis of [23], we can absorb the new physics contributions in the form factors. We obtain ∆h+ = k(1 + r)(1 − w)ξ(w) , ∆h− = −k(1− r)(1 + w)ξ(w) , ∆hA1 = −2k(1− r∗)ξ(w) , ∆hA2 = −2kξ(w) , ∆hA3 = −2kξ(w) , ∆hV = 2k(1 + r∗)ξ(w) , where 2 vmB CwLRVtb . (20) For the new physics contribution we include only the leading term, so we set ξ(1) = 1. Setting Λ = 1TeV, we obtain FD(1) ≈ FSMD (1)− 1.01× 10−3 × Re CwLRVtb FD∗(1) ≈ FSMD∗ (1)− 2.02× 10−3 × Re CwLRVtb R1(1) ≈ RSM1 (1) + 6.52× 10−3 × Re CwLRVtb R2(1) ≈ RSM2 (1)− 2.48× 10−3 × Re CwLRVtb . (21) Recent lattice QCD calculations [20] give FSMD (1) = 1.074± 0.024 and FSMD∗ (1) = 0.91± 0.04. For RSM1 and RSM2 we use the results of [24], scanning over the hadronic parameters that enter. The experimental results are |Vcb|FD(1) = (42.4± 4.4)× 10−3, |Vcb|FD∗(1) = (36.2± 0.6)× 10−3 [17], R1(1) = (1.417± 0.075), and R2(1) = (0.836± 0.043) [25]. We set |Vtb| = 1 and do a combined fit for CwLR and |Vcb|. We find − 0.2 < Re(V LRVtb) |Vcb| < 1.6 . (22) We next turn to inclusive B → Xcℓν̄ decays, which is also sensitive to the presence of the additional operators. We concentrate on the partial branching ratio and moments constructed from the charged lepton energy spectrum (see, e.g., [26]), M0(E0) = τB dEℓ , M1(E0) = , Mk(E0) = [Eℓ −M1(E0)]k dΓdEℓ dEℓ . (23) -1 -0.5 0 0.5 1 1.5 2 FIG. 3: Constraints on OwLR in the Re(C LR) – |Vcb| plane from semileptonic B → Xcℓν̄ (solid curves) and B → D ℓν̄ decays (dashed curves) and their combination (shaded areas). For each constraint the 68%, 95% and 99% CL regions are shown. These are well measured and can be reliably calculated. We use the SM prediction including 1/m2b and αs corrections and compare it in a combined fit with the 20 Babar [27] and a subset [28] of the 45 Belle [29] measurements, including their correlations. The modification of dΓ/dEℓ due to the C LR coupling is given by dΓNP(B → Xcℓν̄) 2 Re(CwLRVcb) 2π3Λ2 ρ y2(3 − y)(1− y − ρ)3 (1− y)3 2G3Fm 4 |CwLR|2 3π3Λ4 y2(3− y)(1− y − ρ)4 (1− y)3 , (24) where y = 2Eℓ/mb and ρ = m b . It is known that the data cannot be fitted well with the OPE truncated at 1/m2b. Including the 1/m b corrections in a more complicated fit would make the agreement with the SM better, and therefore our bounds stronger. The combined constraints on CwLR and |Vcb| from exclusive and inclusive decays is shown in Fig. 3. The solid curves show the constraints from inclusive decays, the dashed curves show the bounds from exclusive semileptonic decays to D and D∗, and the shaded regions show the combined constraints (the confidence levels are as in Fig. 2). D. Exclusive and inclusive b → uℓν̄ decays We now turn to some 3rd → 1st generation transitions. While the experimental constraints are less precise for these than for 3rd → 2nd generation transitions, the SM also predicts smaller rates, and therefore NP could more effectively compete with the SM processes. These constraints are particularly important as they bound the O′i contributions relevant for t→ u decays, which might not be distinguishable at the LHC from t→ c. As is the case for 3rd → 2nd generation transitions, exclusive and inclusive semileptonic b→ u decays can constrain the operator O′wLR in t → u transition. Similarly to b → cℓν̄, this operator distorts the lepton energy spectrum, so information on the lepton energy moments could constrain it. However, such measurements are not yet available for B → Xuℓν̄. Therefore, to distinguish between the SM Vub contribution and CwLR, we use B → πℓν̄ in addition to the inclusive data. For exclusive B → πℓν̄ decay, we use for the SM prediction the parameterization of Ref. [30], which relies on -2 -1 0 1 2 FIG. 4: Constraints on O′wLR in the Re(C LR) – |Vub| plane from B → Xuℓν̄ (solid curves) and B → πℓν̄ (dashed curves) and their combination (shaded areas). For each constraint the 68%, 95% and 99% CL regions are shown. analyticity constraints and lattice QCD calculations of the form factors at large q2 [31, 32]. The NP contribution is dΓNP(B → πℓν̄) G2F |pπ|3 4m2Bv 2|C′wLR|2 (1 + q̂2)f− + (1− q̂2)f+ − 4mBvRe(VtbC (1− q̂2)f2+ + (1 + q̂2)f−f+ . (25) where the f± form factors are functions of the dilepton invariant mass, q 2, q̂2 = q2/m2B, and we neglected terms suppressed by m2π/m For inclusive B → Xuℓν̄ decay, we focus on the measurement utilizing combined cuts [33] on q2 and the hadronic invariant mass, mX , and compare it with the Belle and Babar measurements [34]. Using this determination of Vub is particularly simple for our purposes, because in the large q2 region the mild cut on mX used in the analysis only modifies the rate at a subleading level. Working to leading order in the NP contribution, we can neglect the effect of the mX cut on the NP and include the NP contribution to the rate via dΓNP(B → Xuℓν̄) 192π3 32m2Bv |C′wLR|2 q̂2(q̂2 − 1)2(q̂2 + 2) . (26) Since the interference between the SM and NP is suppressed by mu/mb (see the ρ factor in Eq. (24) in the first term), there is no dependence on the weak phase of C′wLR in the inclusive decay. Using other determinations of Vub would be harder to implement and would not change our results significantly. The combined constraint on C′wLR and |Vub| from inclusive and exclusive decays is shown in Fig. 4. (This uses the lattice QCD input from Fermilab [31], and the one using the HPQCD calculation [32] would also be similar.) E. B → ργ and B → µ The inclusive B → Xdγ decay has not been measured yet, and there is only limited data on B → ργ. Averaging the measurements [35] using the isospin-inspired4 relation B(B → ργ) = B(B± → ρ±γ) = 2(τB±/τB0)B(B0 → ρ0γ), 4 Isospin is not a symmetry of the electromagnetic interaction. This average relies on the heavy quark limit to argue that the dominant isospin violation is ΛQCD/mb suppressed. With more precise data, using onlyB 0 decays will be theoretically cleaner, because annihilation is suppressed in the B0 compared to the B± modes. At present, this would double the experimental error, so we include the B± data. and the PDG value τB±/τB0 = 1.07, we obtain B(B → ργ) = (1.26± 0.23)× 10−6 . (27) To reduce the sensitivity to form factor models, we normalize this rate to B(B → K∗γ) = B(B± → K∗±γ) + (τB±/τB0)B(B0 → K∗0γ) /2 = (41.4± 1.7)× 10−6, B(B → ργ) B(B → K∗γ) 2 ( m2B −m2ρ m2B −m2K∗ |C7γ |2 |CSM7γ |2 . (28) We use ξγ = 1.2 ± 0.15, where this error estimate accounts for the fact that we consider the rates to be determined by O7γ(mb) alone. The contributions of other operators have larger hadronic uncertainties and are expected to partially cancel [36]. If first principles lattice QCD calculations of the form factor become available then one can avoid taking the ratio in Eq. (28), and directly compare the calculation of B(B → ργ) with data. We obtain the following constraints − 0.26 < C′uLL < −0.21 or −0.026 < C′uLL < 0.03 , −1.2 < C′hLL < −0.9 or −0.11 < C′hLL < 0.13 , −0.7 < C′wRL < −0.5 or −0.07 < C′wRL < 0.08 , −0.1 < C′bRL < 0.09 or 0.7 < C′bRL < 0.9 . (29) Note that there are no constraints on O′wLR or O LR, because of their mu/mW suppression. As for B → Xsγ, the two solutions in Eq. (29) correspond to the sign ambiguity in interpreting the constraint on |C7γ |2 when we assume that the NP contributions are real relative to the SM. Had we not made this assumption, the allowed regions would be annuli in the complex C′i planes. The NP operators we consider also contribute to the rare decays Bd,s → µ+µ−. This is most interesting for Bd → µ+µ−, since one expects that the NP contribution is enhanced compared to the SM by [(v2/Λ2)(1/|Vtd|)]2, which is around 20 for Λ = 1TeV. Moreover, O′hLL contributes at tree level, so its contribution is enhanced by an additional factor of (4π/α)2. Although this decay mode has not yet been observed and the present upper bound B(B → µ+µ−) < 3× 10−8 [37] is two orders of magnitude above the SM expectation, it still gives a useful constraint on O′hLL. In particular, for Λ = 1TeV, we obtain − 0.023 < C′hLL < 0.026 . (30) The combined constraints fromB → ργ and B → µ+µ− on O′uLL and O′hLL are shown in Figure 5. The region between the dashed lines is beyond the LHC reach, but the LHC will be able to exclude (though perhaps not completely) the non-SM region in Fig. 5. In the case of O′uLL and O RL the present data are not strong enough to exclude the non-SM region allowed by B → ργ. F. ∆F = 2 transitions In this section we present the results of the analysis of the NP effects in ∆F = 2 processes. Since their contribution appear at the same time in BdBd, BsBs and K 0K0, we performed a full fit using the CKMFitter code [10], after having suitably modified it to include the results of Sec. III C. The code simultaneously fits experimental data for the Wolfenstein parameters ρ and η and for NP (extending earlier studies in ∆F = 2 processes [38, 39]). The observables used here include the Bd and Bs mass differences, the time dependent CP asymmetries in B → J/ψK, the CP asymmetries in B → ππ, ρρ, ρπ, the ratio of |Vub| and |Vcb| measured in semileptonic B decays, the CP asymmetries in B → DK, the width difference in the BsBs system, ∆Γs, the semileptonic CP asymmetry in B decays, ASL, and the indirect CP violation in K decays, ǫK . We allowed the NP Wilson coefficients to be complex and performed a scan over their phases. Thus, the results in this section are quoted in terms of the absolute values of the Ci and C Keeping only one operator at a time, we get |CuLL| < 0.07 , |ChLL| < 0.014 , |CwRL| < 0.14 , |C′uLL| < 0.11 , |C′hLL| < 0.018 , |C′wRL| < 0.26 . (31) As before, we also performed a combined analysis for the LL operators. This is particularly interesting for O′uLL and O′hLL, since until B → Xdℓ+ℓ− data becomes available, only ∆F = 2 processes are sensitive to the complex phases. In -0.4 -0.2 0 0.2 0.4 -0.04 -0.02 FIG. 5: Constraints from B → ργ and B → µ+µ− in the C′uLL – C LL plane. The red, green, and blue regions denote 68%, 95%, and 99% CL, respectively. The region between the dashed lines is beyond the LHC sensitivity. general, allowing for a variation of the phases of C′uLL and C LL, a cancellation can occur between the two contributions and the above bounds are relaxed. If their absolute values satisfy |C′hLL| ∼ 0.1 |C′uLL| then arbitrarily large values of the Wilson coefficients is allowed for some values of the phases. This possibility is ruled out when the B → ργ and B → µ+µ− constraints are included. Indeed, combining ∆F = 2 with these measurements, we obtain |C′uLL| < 0.26 , |C′hLL| < 0.026 . (32) V. COMBINED CONSTRAINTS AND CONCLUSION In this paper, we studied constraints on flavor-changing neutral current top quark decays, t → cZ, uZ, cγ, uγ. We used an effective field theory in which beyond the SM physics is integrated out. In the theory with unbroken electroweak symmetry the leading contributions to such FCNC top decays come from seven dimension-6 operators of Eq. (2). We assumed that the new physics scale, Λ, is sufficiently above the electroweak scale, v, to expand in v/Λ and neglect higher dimension operators. We find different and sometimes stronger constraints than starting with an effective theory which ignores SU(2)L invariance. The 95% CL constraints on the Wilson coefficients of the operators involving 3rd and 2nd generation fields are summarized in Table I. We consider one operator at a time, i.e., that there are no cancellations. The top two rows show the present direct constraints and the expected LHC bounds. The next three rows show the bounds from B physics. In the B → Xsγ, Xsℓ+ℓ− row the combined bounds from these processes are shown. The two allowed regions are obtained neglecting the complex phases of the operators (see Fig. 2 and the discussion in Sec. IVB). This assumption can be relaxed in the future with more detailed data on B → Xsℓ+ℓ−. In the ∆F = 2 row the numbers refer to upper bounds on the magnitudes of the Wilson coefficients and are derived allowing the phase to vary. The best bound for each operator is listed and then translated to a lower bound on the scale Λ (in TeV, assuming that the C’s are unity), and to the maximal t → cZ and t→ cγ branching ratios still allowed by each operator. The last row indicates whether a positive LHC signal could be explained by each of the operators alone. In this row, the star in “Closed∗” for CuLL and C LL refers to the fact that small values of these Wilson coefficients cannot give an observable top FCNC signal, however, there is an allowed region with cancellations between the SM and the NP, which may still give a signal. In the same row “Ajar” means that CwRL and C RL cannot yield an LHC signal in t → cZ but may manifest themselves in t→ cγ. It is remarkable that the coefficients of several operators are better constrained by B physics than by FCNC top decays at the LHC. direct bound 9.0 9.0 6.3 6.3 6.3 6.3 9.0 LHC sensitivity 0.20 0.20 0.15 0.15 0.15 0.15 0.20 B → Xsγ, Xsℓ − [−0.07, 0.036] [−0.017, −0.01] [−0.005, 0.003] [−0.09, 0.18] [−0.12, 0.24] [−14, 7] [−10, 19] — ∆F = 2 0.07 0.014 0.14 — — — — semileptonic — — — — [0.3, 1.7] — — best bound 0.07 0.014 0.15 0.24 1.7 6.3 9.0 Λ for Ci = 1 (min) 3.9 TeV 8.3 TeV 2.6 TeV 2.0 TeV 0.8 TeV 0.4 TeV 0.3 TeV B(t → cZ) (max) 7.1×10−6 3.5×10−7 3.4×10−5 8.4×10−6 4.5×10−3 5.6×10−3 0.14 B(t → cγ) (max) — — 1.8×10−5 4.8×10−5 2.3×10−3 3.2×10−2 — LHC Window Closed∗ Closed∗ Ajar Ajar Open Open Open TABLE I: 95% CL constraints on the Wilson coefficients of the operators involving 3rd and 2nd generation fields for Λ = 1TeV. The top two rows show the present direct constraints and the expected LHC bounds. The second part shows the bounds from B physics, which is then translated to a lower bound on the NP scale, Λ, and to the maximal t → cZ and t → cγ branching ratios each operator could still give rise to (the ATLAS sensitivity with 100 fb−1 is 5.5 × 10−5 and 1.2 × 10−5, respectively). The last line concludes whether a positive LHC signal could be explained by each of the operators. direct bound 9.0 9.0 2.7 2.7 2.7 2.7 9.0 LHC sensitivity 0.20 0.20 0.15 0.15 0.15 0.15 0.20 B → ργ, µ+µ− [−0.26, −0.21] [−0.026, 0.03] [−0.023, 0.026] [−0.7, −0.5] [−0.07, 0.08] [−0.1, 0.09] [0.7, 0.9] — — — ∆F = 2 0.11 0.02 0.26 — — — — semileptonic — — — — [−0.9, 0.1] [0.8, 1.4] combined bound 0.10 0.02 0.16 0.9 1.4 2.7 9.0 Λ for Ci = 1 (min) 3.2 TeV 7.2 TeV 2.5 TeV 1.1 TeV 0.8 TeV 0.6 TeV 0.3 TeV B(t → uZ) (max) 1.6×10−5 6.4×10−7 4.1×10−5 1.2×10−4 3.2×10−3 1.0×10−3 0.14 B(t → uγ) (max) — — 2.1×10−5 6.7×10−4 1.6×10−3 5.9×10−3 — LHC Window Closed Closed Ajar Open Open Open Open TABLE II: Constraints on the Wilson coefficients of the operators involving 3rd and 1st generation fields. The entries in the table are analogous to Table I. Table II shows the constraints on the operators involving the 3rd and 1st generation quarks. We studied this because the LHC may not be able to distinguish between t→ c and t→ u FCNC decays, and these processes are also interesting in their own rights. In this case there are two allowed regions of C′wLR from semileptonic decays, as can be seen in Fig. 4. The entries in the “combined bound” row show the result of the fit to all the B decay data above it, as discussed in Sec. IVF. We see from the last row that the LHC window remains open for all of the RR, LR, and RL operators, except O′wRL. We conclude from Tables I and II that if the LHC sees FCNC t decays then they must have come from LR or RR operators, unless there are cancellations. Moreover, if t→ cZ is seen but t→ cγ is not, then only OuRR could account for the data. Our analysis used the currently available data and compared it to an estimate of the LHC reach with 100 fb−1. However, by that time many of the constraints discussed above will improve, and new measurements will become available. The direct bounds will be improved by measurements from Run II of the Tevatron in the near future. All the B decay data considered in this paper will improve, and the calculations for many of them may become more precise. Important ones (in no particular order) are: (i) improved measurements of B → Xsℓ+ℓ− to better constrain the magnitudes and especially the phases of CuLL, C LL, C RL, and C RL; (ii) measurements of the lepton energy and the hadronic mass moments in B → Xuℓν̄ to constrain C′wLR; (iii) improvements in B → ργ and measurement of B → Xdγ to reduce the uncertainties of C′uLL, C′hLL, C′wRL, and C′bRL; (iv) measurement of and B → Xdℓ+ℓ− to reduce the errors and constrain the weak phases of these last four coefficients. Additional information will also come from LHCb. For example, the measurement of the CP violating parameter SBs→ψφ, the direct measurements of the CKM angle γ, and some of the above rare decays will help improve the constraints. With several of these measurements available, one can try to relax the no-cancellation assumption employed throughout our analysis. Note that not all NP-sensitive B factory measurements can be connected to FCNC top decays; e.g., the CP asymmetry SK∗γ is sensitive to right-handed currents in the down sector and cannot receive a sizable enhancement from the operators in Eq. (2). Thus, there are many ways in which there can be interesting interplays between measurements of or bounds on FCNC t and b quark decays. If an FCNC top decay signal is observed at the LHC, the next question will be how to learn more about the underlying physics responsible for it. With a few tens of events one can start to do an angular analysis or study an integrated polarization asymmetry [40]. These could discriminate left-handed or right-handed operators (say OuRR or OuLL). Such interactions could arise in models in which the top sector has a large coupling to a new physics sector, predominantly through right-handed couplings [41]. However, a full angular analysis that could also distinguish OuRR from OwLR requires large statistics, which is probably beyond the reach of the LHC. The observation of FCNC top decays at the LHC would be a clear discovery of new physics, and therefore it would be extremely exciting. Our analysis shows that an LHC signal requires Λ to be less than a few TeV. This generically implies the presence of new particles with significant coupling to the top sector. If the new particles are colored, we expect that they will be discovered at the LHC. It would be gratifying to decipher the underlying structure of new physics from simultaneous information from top and bottom quark decays and direct observations of new heavy particles at the LHC. Acknowledgments We acknowledge helpful discussions with Chris Arnesen and Iain Stewart on B → πℓν̄ [30], Frank Tackmann on B → Xsℓ+ℓ− [19], and Phillip Urquijo about the Belle measurement of B → Xcℓν̄ [29]. P.F., M.P., and G.P. thank the Aspen Center for Physics for hospitality while part of this work was completed. The work of P.F., Z.L., and M.P. was supported in part by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under contract DE-AC02-05CH11231. M.S. was supported in part by the National Science Foundation under grant NSF-PHY-0401513. APPENDIX A: ANALYTIC EXPRESSIONS We give the form of the operators of Eq. (2) after electroweak symmetry breaking, keeping only trilinear vertices which do not involve the Higgs: OuLL = −sL + bLW/ 2mZmW tLZ/ cL + . . . , OhLL = 2mZmW tLZ/ cL + bLZ/ sL + . . . , OwRL = mW sLσ µνtRW 2mW cLσ µνtR (cwZµν + swAµν) + . . . , ObRL = 2mW cLσ swAµν − + . . . , OwLR = mW bLσ µνcRW 2mW tLσ µνcR (cwZµν + swAµν) + . . . , ObLR = 2mW tLσ swAµν − + . . . , OuRR = 2mZmW tRZ/ cR + . . . . (A1) Here sw = sin θw, cw = cos θw, and the dots denote hermitian conjugate and the neglected vertices involving Higgs and higher number of fields. Throughout this paper the covariant derivative is defined as Dµ = ∂µ + igA a + ig′Bµ. The analytic expressions for the contributions of the operators in Eq. (2) to the top FCNC partial widths are Γ(t→ cZ) = mt v2m2t (1− y)2 |ChLL + CuLL|2 + |CuRR|2 (1 + 2y) + 2g22 cos 2 θW (2 + y) |CbLR tan2 θW − CwLR|2 + |CbRL tan2 θW − CwRL|2 2g2 sin θW tan θW (CbRL) ∗(ChLL + C LL)− (CbLR)∗CuRR 2g2 cos θW (CwRL) ∗(ChLL + C LL)− (CwLR)∗CuRR , (A2) Γ(t→ cγ) = αmt v2m2t |CbLR + CwLR|2 + |CbRL + CwRL|2 , (A3) where y = m2Z/m t . The analogous expressions for t → u decays are obtained by replacing Ci by C′i above. This expression makes it straightforward to relate the Wilson coefficients used in this paper with different notation present in the literature, which defines the couplings in the effective Lagrangian after electroweak symmetry breaking. Next we present the analytic expression for the Wilson coefficients originating from the operators in Eq. (2). We use the MS scheme and match at the scale µ = mW . It is easiest to express the results as modifying the Inami- Lim functions B0, C0, and D0/D 0, coming from box diagrams, Z-penguins, and γ-penguins, respectively. Using the standard normalization of the effective Hamiltonian Heff = − CiOi , (A4) the Wilson coefficients at the matching scale can be written as C7γ = − D′0(x) , C9V = sin2 θW B0(x) + sin2 θW C0(x) −D0(x) C10A = sin2 θW B0(x) − C0(x) , (A5) where x = m2t/m W . In the SM, we have the well-known expressions [12] B0(x) = (x− 1)2 C0(x) = 3x+ 2 (x − 1)2 D0(x) = − lnx− 19x 3 − 25x2 36(x− 1)3 x2(5x2 − 2x− 6) 18(x− 1)4 lnx , D′0(x) = 8x3 + 5x2 − 7x 12(x− 1)3 3 − 2x2 2(x− 1)4 lnx . (A6) The contributions of the OuLL, O LR, andO LR operators introduced in Eq. (2) can be included by adding the following terms to Eq. (A6) ∆B0(x) = − x ln x (x− 1)2 , (A7) ∆C0 = 20(x− 1) sin2 θW + 23x+ 7 − 6x(x 2 + x+ 3) (x− 1)2 − 2πκ ChLL + − x lnx (x− 1)2 − x− 4 , (A8) ∆D0 = − 47x3 − 237x2 + 312x− 104 6(x− 1)3 4 − 30x3 + 54x2 − 32x+ 8 (x− 1)4 49x2 − 89x+ 34 6(x− 1)3 3 − 9x2 + x+ 1 (x− 1)4 2κg CbRL x lnx 59x− 68 9(x− 1) 3x− 2 (x− 1)2 , (A9) ∆D′0 = 68x3 − 291x2 + 297x− 92 18(x− 1)3 + x2(3x− 2) (x− 1)4 lnx ChLL(sin 2 θW + 3) 3x3 + 33x2 − 25x+ 1 2(x− 1)3 4 − 6x3 + 33x2 − 32x+ 8 (x− 1)4 − 2x(x − 4) (x − 1)2 CwLR − CbLR , (A10) where xc = m W and V ∗cs V ∗ts . (A11) Note that the contribution of OhLL to ∆C0 occurs at tree level, as indicated by its 1/α2 enhancement in Eq. (A8), so OhLL gives tree-level contributions to C9V and C10A. Nevertheless, we shall not include the matrix element of O LL to one higher order in α2, in analogy with the conventional approach in which the NNLL calculation of B → Xsℓ+ℓ− does not include the O(α2s) matrix element of O9V . Finally we calcuate the ∆F = 2 contributions due to CuLL and C LR. The shift in the SM contributions read SSM0 → SSM0 + κi∆Si(x) + κi κj∆S′i,j(x) + κij ∆S′′ij(x) , (A12) where i = u, h, w labels the contributions from the operators OuLL, O LL and O RL, respectively. The expressions for ∆S and ∆S′ are ∆Su = − x(4x2 − 11x+ 1) (x− 1)2 2x(x3 − 6x+ 2) (x− 1)3 lnx , (A13) ∆Sh = − x[(1 + x) sin2 θW + 2x− 6] 2x[x(x + 2) sin2 θW − 6] 3(x− 1)2 lnx , (A14) ∆Sw = 3 g x(x+ 1) (x− 1)2 − (x− 1)3 lnx , (A15) ∆S′u,u = 7x3 − 15x2 + 6x− 4 (x − 1)2 − 2x(2x 3 + 3x2 − 12x+ 4) (x − 1)3 ln x , (A16) ∆S′h,h = , (A17) ∆S′u,h = 2[(x+ 1)(x+ 2) sin2 θW + 3(x 2 − 9x+ 4)] 3(x− 1) 2x[x(x − 3− 2 sin2 θW ) + 6] (x− 1)2 lnx , (A18) ∆S′w,w = g −6x(x+ 1) (x− 1)2 (x− 1)3 , (A19) and κi depends on the flavor transition, Ci Vcs/Vts for t→ c contribution in b→ s , Ci Vcd/Vtd for t→ c contribution in b→ d , C′i Vus/Vts for t→ u contribution in b→ s , C′i Vud/Vtd for t→ u contribution in b→ d , (Ci V tsVcd + C csVtd)/(V tsVtd) for t→ c contribution in s→ d , (C′i V tsVud + C usVtd)/(V tsVtd) for t→ u contribution in s→ d . 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0704.1483

Exploring Infrared Properties of Giant Low Surface Brightness Galaxies

Exploring Infrared Properties of Giant Low Surface Brightness Galaxies Nurur Rahman1, Justin H. Howell, George Helou, Joseph M. Mazzarella, and Brent Buckalew Infrared Processing and Analysis Center (IPAC)/Caltech, Mail Code 100-22, 770 S. Wilson Avenue, Pasadena, CA 91125, USA [email protected] ABSTRACT We present analysis of Spitzer Space Telescope observations of the three low surface brightness (LSB) optical giant galaxies Malin 1, UGC 6614 and UGC 9024. Mid- and far-infrared morphology, spectral energy distributions, and in- tegrated colors are used to derive the dust mass, dust-to-gas mass ratio, total infrared luminosity, and star formation rate (SFR). We also investigate UGC 6879, which is intermediate between high surface brightness (HSB) and LSB galaxies. The 8 µm images indicate that polycyclic aromatic hydrocarbon (PAH) molecules are present in the central regions of all three metal-poor LSB galax- ies. The diffuse optical disks of Malin 1 and UGC 9024 remain undetected at mid- and far-infrared wavelengths. The dustiest of the three LSB galaxies, UGC 6614, has infrared morphology that varies significantly with wavelength; 160 µm (cool) dust emission is concentrated in two clumps on the NE and NW sides of a distinct ring seen in the 24 and 8 µm images (and a broken ring at 70 µm) at a radius of ∼40′′ (18 kpc) from the galaxy center. The 8 and 24 µm emission is co-spatial with Hα emission previously observed in the outer ring of UGC 6614. The estimated dust-to-gas ratios, from less than 10−3 to 10−2, support previous indications that the LSB galaxies are relatively dust poor compared to the HSB galaxies. The total infrared luminosities are approximately 1/3 to 1/2 the blue band luminosities, suggesting that old stellar populations are the primary source of dust heating in these LSB objects. The SFR estimated from the infrared data ranges ∼0.01− 0.88 M⊙ yr−1, consistent with results from optical studies. Subject headings: galaxies: spirals - galaxies: dust - galaxies: ISM - galaxies: structure - galaxies: individual (Malin 1, UGC 6614, UGC 6879, UGC 9024) 1National Research Council (NRC) Postdoc Fellow http://arxiv.org/abs/0704.1483v1 – 2 – 1. Introduction The unprecedented imaging and spectroscopic sensitivity and higher spatial resolution of the Spitzer Space Telescope (SST; Werner et al. 2004) compared to past IR missions such as the Infrared Astronomical Satellite (IRAS; Neugebauer et al. 1984) and the Infrared Space Observatory (ISO; Kessler et al. 1996) provide a unique opportunity to probe the inter-stellar medium (ISM) of optically faint sources such as low surface brightness (LSB) galaxies. The goal of this study is to use the Spitzer data to analyze the IR properties of three LSB optical giants: Malin 1, UGC 6614, and UGC 9024. This is the first time we are able to view these galaxies in ∼3-160 µm wavelength range. The LSB galaxies are usually defined as diffuse spiral disks with low B-band central surface brightness (e.g. µB,0 ≥ 23 mag arcsec−2; Bothun et al. 1997; Impey & Bothun 1997). These galaxies are either blue (B-V . 0.5) or red (B-V & 0.8) in color (O’Neil et al. 1997), metal poor ([O/H] . 1/3Z⊙; McGaugh 1994; de Blok & van der Hulst 1998b; de Naray et al. 2004), rich in neutral hydrogen (H I) (Schombert et al. 1992; O’Neil et al. 2004), deficient in H II emission (McGaugh et al. 1995; de Naray et al. 2004), and have low star formation rate (SFR) . 0.1 M⊙ yr −1 (van den Hoek et al. 2000; Burkholder et al. 2001). The majority of these galaxies lack molecular (CO) gas (Schombert et al. 1990; de Blok & van der Hulst 1998a); only a handful have been reported to have molecular emission (O’Neil & Schinnerer 2004; Matthews et al. 2005; Das et al. 2006).The observed properties suggest that LSB disks are relatively unevolved systems and may have a different evolutionary history compared to their high surface brightness (HSB) counterparts (McGaugh 1994; van den Hoek et al. 2000; Vallenari et al. 2005; Zackrisson et al. 2005). Most of our knowledge regarding the composition and structure of the ISM of LSB spirals comes from optical (Impey et al. 1996) and H I surveys (O’Neil et al. 2004). These surveys have demonstrated that the neutral hydrogen is by far the dominant component of the ISM in these galaxies (∼95% by mass; Matthews 2005). While we have improved understanding of the gaseous component of the ISM from decade long H I surveys, our knowledge of the inter-stellar dust, the component of the ISM radiating from mid-IR (∼8 µm) through sub- millimeter (∼850 µm) wavelengths, is still very limited. Because of the scarcity of information in this wavelength range, complementary observational facts such as low metal abundance (McGaugh 1994), strong similarities in optical and near-IR morphology (Bergvall et al. 1999; Bell et al. 2000), transparency of the stellar disks (O’Neil et al. 1998; Matthews et al. 1999), and deficiency in molecular emission (Schombert et al. 1990; de Blok & van der Hulst 1998a) have been used to probe the ISM of LSB galaxies. All these observations lead to a general consensus that the LSB disks are deficient in dust and molecular gas. Given that LSB spirals comprise ∼50% of the local disk galaxy population (McGaugh – 3 – et al. 1995b), they deserve equal attention as their HSB cousins. To develop a consistent picture of the local galaxy populations it is therefore necessary to probe each population at all wavelength regimes as have been done in most cases for HSB galaxies. Previous long wavelength studies on LSB galaxies involved a few cases in sub-millimeter and millimeter wavelengths (de Blok & van der Hulst 1998b; Pickering & van der Hulst 1999; O’Neil & Schinnerer 2004; Matthews et al. 2005; Das et al. 2006). Hoeppe et al. (1994) made the first attempt to investigate long wavelength (60 µm, 100 µm, and 20 cm) properties of LSB dwarfs; however, no study has been made in the mid-IR and far-IR for LSB disks. In this study we have made the first attempt to explore ∼3-160 µm properties of three LSB giants: Malin 1, UGC 6614, and UGC 9024, recently observed by the SST. We focus on the IR morphology to probe extent of dust in the ISM, on the SEDs for the IR energy budget and dust content, and on the IR colors to establish the dust temperature of the ISM. The organization of this paper is as follows: we describe observation and data reduction in section §2 and present our results in section §3. Discussions and conclusions are given in section §4. 2. Observation and Data Reduction We describe the Infrared Array Camera (IRAC; Fazio et al. 2004) and Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004) imaging data for Malin 1, UGC 6614, and UGC 9024. These extended disk galaxies with radial scale length hr,R > 5 kpc are observed as part of a larger guaranteed time observing program (Spitzer PID #62). The program also includes two LSB galaxies (UGC 5675 and UGC 6151) with 2 kpc < hr,R < 3 kpc, an edge-on disk (UGC 6879) with hr,R ∼2.5 kpc, and a HSB dwarf (UGC 10445) with hr,R ∼1 kpc. The central brightness of UGC 6879 is µB,0 ∼ 20.4 mag arcsec−2. A simple correction for inclination, using µ0,face−on = µ0,observed+2.5 log(a/b) brings its central brightness to ∼ 21.71 mag arcsec−2. It is lower than the conventional choice (µ0 ≈ 23 mag arcsec−2) but close to the Freeman value (µ0 ∼ 21.65 mag arcsec−2; Freeman 1970). The central brightness of this galaxy falls in between the range of LSB and HSB galaxies and hence, we include it as a representative of the intermediate class. The properties of UGC 5675 and UGC 6151 will be explored in a forthcoming paper. The readers are refereed to Hinz et al. (2006) for an analysis of UGC 10445. The IRAC 3.6, 4.5, 5.8, and 8 µm images and the MIPS 24, 70, and 160 µm images were acquired, respectively, in the mapping and photometry modes. The IRAC images were reduced with the standard Spitzer Science Center data pipeline, and aligned, re-sampled, and – 4 – combined into a mosaic image using the Mopex1 software. The MIPS 24 µm data required the use of the self-calibration procedure described in the MIPS Data Handbook2 to remove latent image artifacts. The corrected images were then combined into a mosaic using Mopex. Time filtering and column filtering was applied to the 70 µm images using IDL routines created by D. Fadda. The filtered images were then combined using Mopex. The 160 µm images were combined into a mosaic using Mopex. The IRAC spatial resolution is ∼2′′ for all bands. The MIPS spatial resolutions are 6 , and 40 for the respective bands. Sky subtraction was carried out through the use of multiple sky apertures placed near the source which do not overlap with the faintest isophotes visible from the galaxy. For each galaxy we measured flux densities from the sky subtracted images within the aperture covering the entire galaxy. The flux density contributed by foreground stars within a galaxy aperture was removed by measuring each such star in a small aperture and subtracting the result from the total flux within the galaxy aperture. The calibration uncertainty in the IRAC flux densities is at the level of ∼10% (Reach et al. 2005; Dale et al. 2005). Aperture corrections have been applied to all IRAC flux densities (T. H. Jarrett 2006; private communication). The MIPS flux density calibration uncertainties are 10% at 24 µm and 20% at 70 and 160 µm. Near-IR (1.3, 1.7, and 2.2 µm) flux densities from the Two Micron All Sky Survey (2MASS; Jarrett et al. 2000), upper limits on IRAS flux densities, and those derived from the IRAC and MIPS bands are given in Table 1. Basic properties of the galaxies obtained from the literature, the NASA/IPAC Extragalactic Database (NED), the Lyon-Meudon Ex- tragalactic Database (LEDA), and derived in this study are summarized in Table 2. The IRAS flux densities for UGC 6614 and UGC 6879 were computed using the SCANPI tool available from IRSA as linked via NED where IRAS flux density limits represent SCANPI’s inband total measurement, fν(t). No IRAS detections were available for Malin 1 and UGC 9024. Distances for the galaxies are estimated from heliocentric radial velocity after correct- ing for the local group, infall into the Virgo cluster, the Great Attractor, and the Shapley supercluster following the Mould et al. (2000) flow model. 1http://ssc.spitzer.caltech.edu/postbcd/ 2http://ssc.spitzer.caltech.edu/mips/dh/ – 5 – 2.1. Contamination from Galactic Cirrus A basic concern about faint extragalactic sources with highly diffuse disk structure is confusion by foreground Galactic “cirrus” emission. This is especially critical in the case of far-IR cool sources (defined below) such as LSB galaxies. The far-IR ratio of typical local cirrus is S60µm/S100µm ≤ 0.2 (Gautier 1986). UGC 6614 and UGC 6879 are well above this limit (Fig. 8c). From its Galactic latitude b ∼ +22o and the observed S70µm/S160µm ratio, it is reasonably safe to assume that UGC 9024 has not been affected by cirrus emission. In the absence of far-IR information it is uncertain for Malin 1. However, its high galactic latitude b ∼ +14◦ can be used to argue against any cirrus contamination. We should also stress the fact that whether there is cirrus in the foreground is less significant as compared to whether the cirrus in the foreground varies on scales of the IRAC and MIPS field of view. If its spatial variation is negligible across the IRAC and MIPS mosaic, it gets subtracted out as “sky” in the data reduction process. 3. Results In this section we present IR morphology, SEDs, and IR colors of LSB galaxies. Using this information we estimate dust mass (Md), dust-to-(atomic) gas ratio (D), total infrared luminosity (LTIR), and star formation rate (SFR). To obtain a qualitative assessment of the observed properties of LSB disks compared to their HSB counterparts, we take the Spitzer Infrared Nearby Galaxy Survey (SINGS; Kennicutt et al. 2003) sample as a representative sample of local HSB galaxies. This sample contains 75 galaxies of various Hubble types as well as dwarfs and irregular galaxies, and thus making it a suitable reference for comparative analysis with LSB spirals. 3.1. Infrared Morphology The IR emission beyond ∼25 µm is dominated by the inter-stellar dust under various heating conditions. On the other hand, mid-IR (∼5-25 µm) emission marks the transition from stellar photospheres to inter-stellar dust dominating the emission. Whereas the mor- phology of a galaxy at 3.6 and 4.5 µm represents the stellar disk, at ∼5 µm and beyond it shows the structure of the ISM. The IRAC 3.6 and 4.5 µm bands are sensitive to the under- lying stellar populations typically consisting of red giants and old stars. In some galaxies 3.6 µm band is also known to contain emission from a hot dust component (Bernard et al. 1994; Hunt et al. 2002; Lu et al. 2003; Helou et al. 2004). The hot dust is also visible in the other – 6 – IRAC bands. While 3.6 µm is only sensitive to hot dust near the sublimation temperatures (∼1000 K), the longer wavelength IRAC bands can detect dust at lower temperatures down to several hundred Kelvin. The IRAC 5.8 and 8 µm bands are primarily sensitive to the PAH emission at 6.2, 7.7, and 8.6 µm (Puget & Leger 1989; Helou et al. 2000; Lu et al. 2003). The PAH is the hot component of the inter-stellar dust with effective temperature Td > 100 K stochastically ex- cited to high energy levels by stellar photons. Stellar photospheric emission also contributes to these two IRAC wavebands. The fraction of stellar emission at 5.8 and 8 µm, respectively, are ∼40% and ∼20% (Helou et al. 2004; Pahre et al. 2004). The emission detected by the MIPS bands are coming from dust grains with different size distributions. Very small grains (∼1-10 nm) emit in the mid-IR region (&15 µm), intermediate between thermal equilibrium and single-photon heating. Large, classical grains (∼100-200 nm) are in thermal equilibrium with the radiation field and responsible for far-IR emission (Desert et al. 1990). To begin with we should bear in mind that the demarcation lines among various heating environments are ad-hoc and we assume the following effective color temperature ranges for large grains in thermal equilibrium: warm (40 K . Td . 100 K), cool (20 K . Td . 30 K), cold (10 K . Td . 20 K), and very cold (10 K . Td). The Solan Digital Sky Survey (SDSS) images of target galaxies are shown in Fig. 1. The Spitzer images of the LSB galaxies are shown in Fig. 2 (Malin 1), Fig. 3 (UGC 6614), and Fig. 4 (UGC 9024). UGC 6879 is shown in Fig. 5. Galaxy images are shown using Gaussian equalization (Ishida 2004). The images are oriented such that north is up and east is to the left. Malin 1 is too faint to be detected by the MIPS 70 and 160 µm channels. The other three galaxies were detected by all IRAC and MIPS bands. We do not show 5.8 µm images in these figures since the morphological appearances of each of these galaxies at 5.8 µm closely follow that at 8 µm. For all galaxies, the IRAC 4.5 and 8 µm images are shown without subtracting stellar photospheric emission. The contours represent surface brightness with intervals of 10 where the lowest level is 4σ above the background. The lowest level of contours at different bands are 0.04 (3.6 µm), 0.05 (4.5 µm), 0.30 (8 µm), 0.08 (24 µm), 1.6 (70 µm), and 2.1 (160 µm) expressed in MJy/Sr. The giant optical disks in Malin 1 and UGC 9024 appear as point sources in the IRAC images. While we detect the stellar bulges of these galaxies, their optically diffuse disks remain undetected long-ward of the IRAC bands. This suggests that the low surface bright- ness structures at larger radii might be photometrically distinct components rather than smoothed extensions of the normal inner disks (see Barth 2007 for a discussion on Malin 1 based on Hubble data). That the disks appear in the B-band but are undetected at 3.6 µm suggest that these disks have a small population of young stars rather than a large popu- – 7 – lation of old stars. The bulge spectrum of Malin 1 is consistent with a predominantly old stellar population (Impey & Bothun 1989). For both of these galaxies, the mid-IR emissions at 8 and 24 µm are concentrated in the central few kpc, within a region of 12′′ (20 kpc) radius for Malin 1 and 24′′ (5 kpc) radius for UGC 9024. Undetected far-IR emission from the disk of Malin 1 implies that it contains either very cold (Td < 10 K) cirrus-like dust emitting in the submillimeter and millimeter wavelengths or it lacks cold dust altogether and contains only neutral gas. For UGC 9024, 70 µm emission comes from the central region but 160 µm emission is very hard to measure because of large scale diffuse emission in the field. This result in an upper limit of S160 µm < 268 mJy. The optical morphology of UGC 6614 shows a massive bulge and spiral structure. A thin ring ∼40′′ from the core is prominent in Hα (McGaugh et al. 1995). The 3.6 and 4.5 µm emission is spread over the entire disk of this galaxy. The 3.6 µm image shows a discernible spiral arm pattern closely resembling the optical morphology. At 4.5 µm this feature disappears and the disk shrinks in radius showing only its inner region. This galaxy appears markedly different at 5.8 and 8 µm compared to the other IRAC bands. The 8 µm morphology suggests that the PAH emission is coming from two distinct regions: the central bulge and an outer ring surrounding the bulge. The 8 µm morphology closely traces the Hα image. The MIPS 24 µm morphology is similar to the 8 µm PAH emission although the outer ring appears a bit more disjointed at this band. The dust emission at 24 µm is coming mostly (∼70%) from the central disk. The lower resolution image at 70 µm indicates that only ∼25% of the dust emission is coming from the central part of the galaxy, with the remaining ∼75% emission co-spatial with the ring of radius ∼40′′. Surprisingly a dumbbell shaped region is the dominant source (∼90%) of 160 µm emission; these two peaks are located on the NE and NW sides of the ring. The far-IR images of this galaxy show a small, localized region within the ring, SW from the center. Whether or not this region coincide with the spatial location showing the CO emission in this galaxy disk (Das et al. 2006) is not entirely clear. We will investigate this in the forthcoming paper. UGC 6879 is an edge on spiral with a red central part and a blue (optical) disk. This radial color gradient is perhaps related to greater concentrations of dust in the nucleus than in the overall disk. Being a transitional disk, with a central surface brightness in between HSB and LSB spirals, it is not unexpected to find that UGC 6879 is a strong IR emitting source compared to the LSB galaxies. Both PAH emission and warm dust (24 µm ) emission show spatial variation along the disk. These emissions peak at the central region and diminish toward the edge. – 8 – 3.2. Infrared Diagnostics 3.2.1. IR Spectral Energy Distributions The flux densities obtained from the 2MASS and IRAS archives and those estimated from the IRAC and MIPS images are given in Table 1. These flux densities are used to construct the observed infrared SEDs of LSB galaxies as shown in Fig. 6. The open and solid circles represent, respectively, 2MASS and IRAC data. The open triangles represent the MIPS data whereas the IRAS upper limits for UGC 6614 and UGC 6879 are shown by filled triangles. Since Malin 1 was undetected by the MIPS far-IR channels we show the detection limits at 70 and 160 µm for the total integration time (∼252 sec. and ∼42 sec. respectively). We include four SINGS galaxies with different ISM properties for comparison. These are NGC 0337 (a normal star forming galaxy), NGC 2798 (a star burst galaxy), NGC 2976 (a normal galaxy with nuclear H II region), and NGC 3627 (a Seyfert galaxy). They are shown, respectively, by dotted, dashed, dashed-dotted, and long dashed lines. The motive is to make a visual comparison between SEDs of LSB and HSB galaxies. All flux densities are normalized by the 3.6 µm flux density. There are several noticeable features in these SEDs. First, the amplitude of the mid-IR and far-IR dust emissions of LSB galaxies are lower compared to those of HSB galaxies. An obvious and expected result is that the LSB galaxies have less dust content and hence have lower IR emission. Second, Malin 1 is deficient in the integrated 8 µm emission compared to its 4.5 µm emission. This is quite opposite for other LSB galaxies and more like the SED of an elliptical galaxy (see Fig. 4 in Pahre et al. 2004). Third, the 24 µm emission is suppressed in both Malin 1 and UGC 9024. For UGC 6614 it is slightly above the IRAC bands. This feature suggests that the ISM of UGC 9024 lacks warm dust emission, and is made mostly of cool dust. Fourth, the SEDs show a tendency to turnover at relatively longer wavelength, a signature that the low density and low surface brightness ISM have low radiation intensity. On the other hand, the shape of the SED of UGC 9024 is quite similar to those of the representative HSB galaxies. We discuss this in detail in section §3.3 3.2.2. Dust Mass Dust mass is frequently estimated by fitting the far-IR peak by a single temperature modified blackbody function (Hildebrand 1983). However, the inability of a single tempera- ture or even two temperature blackbody function to fit the observed flux densities suggests that a more sophisticated model of the IR SED is needed. The global SED models of Dale et al. (2001) and Dale & Helou (2002; hereafter DH02) provide a robust treatment of the – 9 – multiple grain populations that contribute to the IR emission in a galaxy. This model allows a realistic derivation of dust mass since it combines information from the full range of heating environments (∼10K-1000K). Previous studies have shown that dust mass is underestimated by a factor of ∼5-10 for quiescent galaxies (i.e. IR cool) if one simply fits the far-IR and sub-millimeter continuum data points with a simple single temperature black body instead of exploiting information from the full range of the SED. Figure 7 shows the fits to the observed SEDs obtained from the DH02 model (solid line). The dashed and dotted lines represent, respectively, empirical stellar SED (Pahre et al. 2004) and stellar synthesis model prediction fitted only to the 2MASS fluxes from Vazquez & Leitherer (2005). We use the model fit (solid line) to estimate Md and D noting that DH02 model does not provide ideal fits to very extended, low density, diffuse disks like Malin 1 and UGC 9024. However, within the measurement and observational uncertainties the model fits provide useful insight. A rigorous and detailed treatment of infrared SEDs of LSB disks will be presented in a future study. The estimated mass is given in Table 2. We find that the ISM of UGC 6614 has the highest amount of dust with dust-to-gas ratio, D ∼0.01. Both Malin 1 and UGC 9024 are ∼3 times less dusty than UGC 6614. Given that IR emission coming only from the central regions of the latter two galaxies, it is not surprising that they show low dust content. In a recent study Das et al. (2006) detected CO(1-0) emission localized in a specific region on the disk of UGC 6614. They estimated molecular gas mass (MH2 ∼ 2.8 × 108M⊙) which is almost equal to the total dust content (Md ∼ 2.6 × 108 M⊙) that we measure distributed over the bulge and disk. The difference between D and dust-to-(total) gas mass is negligible. The (systematic) calibration uncertainty in the observed flux densities and the uncer- tainty in the distance estimates result in a ∼30% uncertainty in Md. Additional uncertainty comes from the mass absorption coefficient. The different sizes used to measure IR and H I fluxes will also attribute additional uncertainty in D. Besides, considering that the long-wavelength end of the SED is poorly constrained the estimate of overall dust mass and gas-to-dust ratio is a bit uncertain. All of these errors compound to make Md and D uncertain by a factor of ∼2 or more. 3.2.3. Infrared Luminosity DH02 proposed a simple relation to compute total IR luminosity using the MIPS bands (see Eq. 4 in DH02). Due to uncertainties in the MIPS flux densities for Malin 1 and UGC 9024 we use the empirical relation given by Calzetti et al. (2005) to estimate LTIR. Calzetti – 10 – et al. relate flux densities at 8 µm and 24 µm to derive LTIR for M 51, a normal star forming galaxy. Estimated total IR luminosities are given in Table 2 with estimated uncertainty of ∼35%. We find comparable IR luminosity for both Malin 1 and UGC 6614. UGC 9024 is the least luminous because of its suppressed 24 µm warm dust emission. In spite of its border line HSB nature, the IR output of UGC 6879 resembles a normal quiescent galaxy. We also estimate LTIR using DH02 model fits. We follow Sanders & Mirabel (1996) to define LTIR where flux densities at 12, 25, 60, and 100 µm are obtained from the model SEDs by interpolation. The result is presented in Table 2. Interestingly, these two estimates are within a factor of ∼2.5 where the model estimates are always higher. The largest difference is noted for UGC 9024 which is a result of relatively poor fit to the data. Within the uncertainty, the IR energy output of LSB galaxies are smaller by a factor of a few compared to their B-band luminosities LB. The infrared-to-blue ratio, LTIR/LB, compares the luminosity processed by dust to that of escaping starlight (see Table 2). The ratio ranges from <0.01 (in quiescent galaxies) to ∼100 (in ultra-luminous galaxies). It can be used to characterize optical depth of a system composed of dust and stars as well as recent (∼100 Myr) SFR to the long term (∼1 Gyr) average rate. The ratio is in the range 0.3-0.5 (see Table 2) indicating that the current level of star formation is low, a consistent result with previous studies on star formation in the LSB ISM (van den Hoek et al. 2000; Burkholder 2001). However, there is a potential degeneracy in this parameter and one can only make an indirect assessment of the ISM of a galaxy if this degeneracy can be lifted. The fact that LTIR/LB is less than unity can arise from two different physical conditions. In one hand, a galaxy may be undergoing intense heating by young stars (large LB) but have very little neutral ISM left (less IR emission) resulting in low LTIR/LB. On the other hand, a quiescent galaxy may generate most of its IR emission in H I clouds heated by older stellar populations and will display a similarly low LTIR/LB (Helou 2000). 3.2.4. Star Formation Rate A widely used recipe for estimating SFR from IR luminosity is given by Kennicutt (1998). However, far-IR luminosity in Kennicutt model is based on IRAS data. Without a proper calibration between IRAS and MIPS far-IR flux densities the uncertainty will loom large in the SFR estimate. We use a new SFR estimator, derived recently by Alonso-Herrero et al. (2006; hereafter AH06) using 24 µm flux density. Our estimates are given in Table 2. The error associated with this SFR is ∼10%. van den Hoek et al. (2000) derived a current SFR ranging ∼0.02-0.2 M⊙ yr−1 from – 11 – I-band photometry for a sample of LSB galaxies generally found in the field. Their estimate agree within a factor of two with our results based on infrared data. For Malin 1 and UGC 6614 the infrared SFR are ∼0.38 M⊙ yr−1 and ∼0.88 M⊙ yr−1 , respectively, whereas it is ∼0.01 M⊙ yr−1 for UGC 9024. The fact that these galaxies have low dust content indicates that extinction is less likely to cause the difference between SFR derived from IR and optical data. The higher SFR of UGC 6614 compared to the other two galaxies is consistent with the fact that this galaxy has a prominent Hα morphology which indicate modest level of current star formation. A very low SFR for UGC 9024 implies that more light is scattered off from the central disk than being absorbed by the ISM. The SFR of these three LSB galaxies, as derived from IR data, are thus significantly below the rate ∼5-10 M⊙ yr−1 derived for the HSB galaxies (Kennicutt 1998) but consider- ably larger than the rate ∼0.001-0.01 M⊙ yr−1 observed typically in dwarf irregular galaxies (Hunter & Elmegreen 2004). The star formation efficiency (SFE), quantified by LTIR/MHI, is also shown in Table 2. This measure represents the amount of unprocessed gas available to be consumed in subsequent star formation. As expected the LSB galaxies have a SFE that is ≤ 1/5 of the SFE of HSB galaxies. 3.3. Infrared Colors Panels 8a and 8b show, respectively, mid-IR colors and the well known PAH-metallicity connection. In these diagrams we highlight two extreme classes (in terms of their IR SEDs) of HSB galaxies: the dwarf systems represented by the decimal numerals and the massive elliptical galaxies represented by the roman numerals. All of these galaxies are obtained from the SINGS sample. Panel 8a compares PAH emission in various classes of galaxies. Metal-rich HSB galaxies preferentially show high ratio in S8µm/S24µm and low ratios of non-stellar 4.5-to-8 µm dust emission. Metal poor (e.g. dwarfs and irregulars) and extremely metal poor HSB galaxies such as blue compact dwarfs (BCDs), show the opposite trend and fall within the PAH- deficient box (Engelbracht et al. 2005; hereafter E05). Note that E05 only looked at star- forming galaxies. “Red and dead” ellipticals should have little or no PAH emission, yet are nowhere near the PAH-deficient region. So this region will not necessarily contain all PAH deficient galaxies. Also note that the result of E05 could also arise from selection bias since recent studies have shown that dwarf galaxies are not necessarily PAH defficient systems (Rosenberg et al. 2006). – 12 – Panel 8b, on the other hand, illustrates the PAH-Z connection in HSB galaxies. E05 showed that galaxies with low PAH emission have relatively unpolluted ISMs. They noted a sharp boundary between galaxy metallicity with and without PAH emission, although the trend may have been affected by selection bias. We show this trend in panel 8b, where the PAH deficient galaxies reside inside the dashed region and galaxies with higher metallicites avoid two regions in the diagram which are shown by the dotted lines. We are interested to see where the LSB galaxies fit in these diagrams. Panel 8a shows that the LSB galaxies are different than the PAH-deficient dwarf galaxies in terms of their mid-IR colors. In the color space, LSB galaxies occupy a region similar to HSB galaxies and reside significantly farther (> 3σ, along the horizontal axis) from PAH- deficient galaxies. While UGC 6614 stays right in the middle of the locus, both Malin 1 and UGC 9024 falls in the edge because of the shapes of their SEDs at 8 and 24 µm. The mid-IR colors of these galaxies closely resemble those of elliptical galaxies which is surprising given that their apparent different star formation histories. The LSB galaxies are metal poor with Z ≤ 1/3Z⊙ (McGaugh 1994; de Bloke & van der Hulst 1998b; de Naray et a. 2004). McGaugh (1994) provided an estimate of oxygen abundance for UGC 6614 but it is highly uncertain. Following the general trend shown by the LSB galaxies we assign one-third solar value to Malin 1 and a solar value to UGC 6879. The full range and the median values of published oxygen abundance are shown for UGC 6614 and UGC 9024.Having very limited information it is, therefore, extremely difficult to explore the PAH-metallicity connection for these galaxies. We are interested in the question whether LSB galaxies being low Z systems, will appear close to the PAH-deficient box or will they fall in the region shunned both by HSB dwarfs and HSB spirals with extended disks (dotted regions; Fig 8b)? While it is tempting to give more weight to the latter case, only three data points with large errors are insufficient to derive any trend. A larger sample of LSB galaxies with 8 µm detections is needed to shed more light on this topic. Panel 8c shows the connection between mid-IR and far-IR colors which basically describe the nature of dust emission at these wavelengths. In this panel, the far-IR color of Malin 1 is shown with respect to a far-IR flat SED, i.e. same flux density at 70 and 160 µm. For the other two galaxies the diagram shows rather low S70µm/S160µm and high S8µm/S24µm ratios. While low far-IR color suggest that they are IR cool sources, the mid-IR color may be linked with the destruction of very small grains (but not the PAH molecules) and thus can be used as a parameter which can signal evolutionary stages of an ISM. A higher value of S8µm/S24µm can mean that S8µm is high (large amount of PAH) or that S24µm is low (little or no warm dust). Ellipticals are in the latter category (see Pahre et al. 2004) whereas LSB systems are in the first category. It should be noted that the 24 µm emission is very closely associated – 13 – with H II regions (Helou et al. 2004). Therefore, the lack of emission at this wavelength is more likely a deficiency in H II region which is quite consistent with the Hα images of LSB galaxies (McGaugh et al. 1995; de Naray et al. 2004). The vertical arrow in panel 8d represents a probable range of IRAS far-IR color for UGC 9024 assuming that Spitzer far-IR ratio is the same as the IRAS far-IR ratio. The sequence of IR colors can be associated with a progression toward greater dust heating intensity and thus with a sequence of star formation activity. The cool end of the color sequence corresponds to cool diffuse H I medium and quiescent molecular clouds, whereas the warm end corresponds to the colors of H II regions, starbursts, and galaxies with higher LTIR/LB ratios. Although the IR nature of these three LSB galaxies are explicit in this diagram, the interesting feature is that they are not the extreme cases in terms of IR SEDs as shown by some of the SINGS galaxies. Two primary sources have been proposed to explain the heating of the dust which produces IR luminosity in spiral galaxies - massive young (OB) stars and associated H II regions (Helou et al. 1985; Devereux & Young 1990, 1993), and non-ionizing A and later stars (Lonsdale & Helou 1986; Walterbos & Schwering 1987; Bothun et al. 1989). Some authors, however, suggest contribution from both sources (Smith 1982; Habing et al. 1984; Rice et al. 1990; Sauvage & Thuan 1992; Smith et al. 1994; Devereux et al. 1994). The dominance of the heating source is, therefore, governed by the availability of discrete and dense star forming regions in the ISM. While observational evidence suggests that the global IR emission from luminous IR spiral galaxies provides a measure of high mass SFR (Kennicutt 1998), the diffuse IR emission in quiescent galaxies is caused mainly by the thermal radiation of the interstellar dust heated by the interstellar radiation field (ISRF) (Jura 1982; Mezner et al. 1982; Cox et al. 1986; Jura et al. 1987). The dust color temperature (Td) deduced for galaxies from the IRAS 60 and 100 µm flux densities are typically 25-40 K assuming emissivity index β = 2 (Rahman et al. 2006). This range is similar to the temperature of dust in Galactic star-forming regions (Wynn- Williams & Becklin 1974; Scoville & Good 1989), and considerably greater than the 15-20K temperatures expected for dust heated by the ambient ISRF (Jura 1982; Mezner et al. 1982; Cox et al. 1986; Jura et al. 1987). The color temperature derived from the MIPS 70 and 160 µm of LSB galaxies ranges ∼17-21 K. From a statistical study of a large sample of optically selected galaxies, Bothun et al. (1989) demonstrated that in the absence of UV radiation, far-IR color ratio of S60µm/S100µm ≤ 0.3 can result from dust which is heated by old stars. Galaxies with S60µm/S100µm ≈ 0.3 − 0.5 requires a steadily increasing proportion of UV heated dust, while galaxies with S60µm/S100µm ≥ 0.5 are entirely dominated by the UV heated dust. From panel 8d we – 14 – find that the LSB galaxies have cool effective dust temperatures and therefore lack intense heating from massive stars. It should be mentioned here that in the IR color analysis we did not subtract the stellar contributions from the 8 and 24 µm measurements to exactly reproduce the E05 diagram. Assuming conservatively that the 3.6 µm emission is coming from the stellar photosphere, the stellar contributions obtained from the empirical stellar SED of Pahre et al. (2004) derived for early type galaxies are ∼59%, ∼38%, and ∼19% at 4.5, 5.8, and 8 µm, respectively. The stellar contribution at 24 µm emission is ∼5% or less. Applying corrections to the corresponding flux densities do not change our results since these corrections systematically shift the parameters in the color space. Note that we use the same set of numbers for each galaxy ignoring the variation in amount of stellar contributions from galaxy to galaxy. However, error due to this is negligible. To find stellar contributions, preference is given to the empirical SED than any stellar synthesis model because of unknown SFHs and poor metallicity constraints for LSB galaxies. In summary, from the four panels of Fig. 8 we conclude that: (a) Malin 1, UGC 6614, and the intermediate HSB/LSB object UGC 6879 have mid-IR colors similar to quiescent HSB disk galaxies; UGC 9024 falls in the region of HSB elliptical galaxies in this color plane. (b) There is insufficient data to conclude whether LSB galaxies have PAH emission properties significantly different from HSB galaxies with comparably low metallicities. Observations of many more LSB galaxies are needed to settle this. (c) Available far-IR detections and upper limits indicate that LSB galaxies are far-IR cool sources. The dust temperatures derived from the MIPS 70 and 160 µm of LSB galaxies ranges Td ∼17-21 K, similar to many quiescent HSB spiral and elliptical galaxies. 3.4. Molecular ISM The LSB galaxies are rich in neutral hydrogen (H I). Molecular hydrogen (H2) gas inferred from CO emission has been detected in only a handful of such galaxies (Matthews & Gao 2001; O’Neil & Schinnerer 2004; Matthews et al. 2005; Das et al. 2006). The low rate of CO detection has been attributed to an ISM with low dust content and a low surface density of neutral gas. Dust opacity is crucial for the formation and survival of molecules since it provides them necessary shielding from the ISRF. A larger column density is needed to self-shield the H2 molecule. A low density and less dusty environment exposes H2 to UV photons which can easily dissociate these molecules. The low star formation and far-IR cool nature of LSB galaxies implies lower energy density of the radiation field and consequently lower dissociation of CO and H2 (de Blok & van der Hulst 1998b). – 15 – The deficiency of molecular gas detections in LSB galaxies may point to a dynamical condition such as the absence of large-scale instability in the disk preventing formation of giant molecular clouds (Mihos et al. 1997). Local instabilities may lead to cloud condensation resulting in localized star formation which may escape detection by current observations (de Blok & van der Hulst 1998b). Local instability in the disk invokes energetic phenomena such as SN explosions and the frequency of such occurrence in LSB galaxies is low (Hoeppe et al. 1994; McGaugh 1994). The enhanced cooling by molecules is crucial in the onset of instability of molecular clouds. Therefore, the effect of less efficient cooling of the ISM can also prevent local instabilities. Long cooling time leads to higher cloud temperatures and thus makes it difficult for a cold molecular phase to exist (Gerritson & de Blok 1999). If the low density ISM truly lacks H2 and CO molecules, what other types of molecule can exist in this environment? Are they very cold or very warm molecules of known types? To which dust components do they belong? Along with millimeter wavelength observations, can we use currently available IR data to shed light into these questions? One of the three LSB galaxies, UGC 6614, has been reported to have CO(1-0) emission at a certain localized region in the disk (Das et al. 2006), while the other two galaxies have not been observed in this wavelength. On the other hand, the Spitzer observation of UGC 6614 shows the presence of enhanced PAH emission on the bulge and almost entirely along the outer disk. This 8 µm emission, however, is concentrated in the central regions of the other two galaxies. A possible source of the origin of PAH molecules is the dense, high temperature, carbon- rich ([C]/[O] > 1) environment of circumstellar envelopes surrounding mass-losing AGB car- bon stars. PAH formation by stars with more normal, oxygen-rich, photospheric abundances ([C]/[O] < 1) will be negligible because nearly all the available carbon is bound up in the CO molecules (Latter 1991). Therefore, if the stellar population responsible for the enrichment of the ISM are dominantly old carbon-rich stars, the warm PAH molecules will be ubiquitous resulting in lower abundance of CO molecules. While a detailed investigation is beyond the scope of this study, we believe the observed PAH emission and lack of CO emission holds the potential clue to probe not only the ISM but also to better understand the SFHs in LSB galaxies. 3.5. Mid-IR Photometry of UGC 6614 The optical spectra of large LSB disks show an unexpected high occurrence of low-level active galactic nuclei (AGN) type activity (Spraybarry et al. 1995; Schombert 1998). UGC 6614, a optical giant LSB galaxy, is suspected to harbor a weak AGN from the optical spectrum (Schombert 1998) and from excess emissions at millimeter (Das et al. 2006) and – 16 – radio wavelengths (Condon et al. 2000). Integrated mid-IR photometry provides a robust technique to identify AGN in HSB galaxies where AGN tend to be redder than normal star forming galaxies in the mid-IR (Lacy et al. 2004; Stern et al. 2005). To determine whether one can use a similar technique to detect AGN signatures in a LSB bulge, we analyze the IRAC colors ([3.6]-[4.5] vs. [5.8]-[8.0]) of UGC 6614. The colors are measured for two regions of radius ∼1 and ∼5.5 kpc, respectively, encircling the galaxy center (Fig. 9; left panel). We find that in the color space, along the vertical [3.6]-[4.5] axis, the galaxy resides well outside the Stern et al “AGN box”. Although the lower end of the AGN box is within the error bar of this galaxy, its overall mid-IR color put it in a region occupied mostly by star forming galaxies suggesting that broad band colors may not be an efficient tracer for weak AGNs. The contribution from an AGN to the measured IRAC fluxes can be estimated by combining image subtraction with assumptions about the nature of the SED coming from starlight and AGN emission. We use Pahre et al. (2004) for stellar flux ratios, and a ν−1 power law SED for AGN (Clavel et al. 2000). Following the procedure of Howell et al. (2007), a 4.5 µm image of the non-stellar emission was constructed (Fig. 9; right panel). Unlike the procedure of E05 which simply measures S4.5 − αS3.6, this procedure includes an additional factor to account for the contribution of non-stellar emission to the 3.6 µm image. The non-stellar flux density measured this way indicates that, within a 12′′ aperture, the AGN contributes ∼12% of the light at 4.5 µm and ∼6% at 3.6 µm. At 8 µm, starlight and the AGN each contribute ∼35% of the light, with PAHs contributing the remaining. Note that the selected aperture includes only the bulge of the galaxy, excluding the spiral arm/ring structure. UGC 6614 illustrates that although [3.6]-[4.5] color can identify strong AGN, weaker AGN will not be clearly separated from pure stellar sources. The procedure of E05, measuring S4.5µm−αS3.6µm, will identify regions of non-stellar emission but will not provide quantitative picture of stellar emission. Given reasonable assumptions the procedure of Howell et al. allows a quantitative decomposition of the stellar and non-stellar flux densities. 4. Summary and Conclusions The Spitzer observations of the three optical giant low surface brightness galaxies Malin 1, UGC 6614, and UGC 9024 have been examined to study the mid and far-IR morphology, spectral energy distributions, and IR color to estimate dust mass, dust-to-(atomic) gas mass ratio, total IR luminosity, and star formation rate (SFR). We also investigate UGC 6879, – 17 – which is intermediate between HSB and LSB galaxies. The 8 µm images indicate that polycyclic aromatic hydrocarbon (PAH) molecules are present in the central regions of all three metal-poor LSB galaxies. The diffuse optical disks of Malin 1 and UGC 9024 remain undetected at mid- and far-infrared wavelengths. The dustiest of the three LSB galaxies, UGC 6614, has infrared morphology that varies significantly with wavelength; 160 µm (cool) dust emission is concentrated in two clumps on the NE and NW sides of a distinct ring seen in the 24 and 8 µm images (and a broken ring at 70 µm) at a radius of ∼40′′ (18 kpc) from the galaxy center. The 8 and 24 µm emission is co-spatial with Hα emission previously observed in the outer ring of UGC 6614. The estimated dust-to-gas ratios, from less than 10−3 to 10−2, support previous indications that LSB galaxies are relatively dust poor compared to HSB galaxies. The total infrared luminosities are approximately 1/3 to 1/2 the blue band luminosities, suggesting that old stellar populations are the primary source of dust heating in these LSB objects. The SFR estimated from the infrared data ranges ∼0.01− 0.88 M⊙ yr−1, consistent with results from optical studies. The mid-IR colors of UGC 6614 shows the presence of a weak AGN at the central bulge. Questions can be raised such as what is the most viable reason for these LSB galaxies to have LTIR/LB < 1? To answer this question we first note that observables such as stellar populations and SED shapes can be used to break the degeneracy in infrared-to-blue ratio (Helou 2000). That LSB galaxies have low infrared-to-blue luminosities, stellar populations spanning a wide range of mean ages, are not dominated by OB stars (McGaugh 1994), have less dust than the HSB galaxies (see Fig. 6), and are IR cool sources (see Fig. 8) suggest a composite scenario. The LSB disks are less dusty and the older stellar populations are the primary source of the IR emission from their ISMs. The presence of PAH emission in these three galaxies indicates that the ISM of the region contributing to the emission in these galaxies have significant amount of carbon enrichment over cosmic time and the ISRF in the ISM must have been significantly weak and thus it is unable to reduce the strength of PAH emission. In other words, the small grains are more exposed to the ISRF so that their destruction rate is larger than for PAH molecules. The detection of mid and far-IR emission from a larger sample will be crucial to un- derstand the properties of ISM in LSB galaxies and probing their star formation histories. This will have a significant effect on analytical modeling of galaxy formation and evolution, the role of different galaxy populations in observed number counts and possibly metallicity effects in the observed number counts. Whether star formation in LSB disks occurred in a continuous fashion but with a low rate, or in an exponentially decreasing rate, or as sporadic bursts with quiescent periods in between is still a matter of debate. Since each type of for- – 18 – mation history will lead to a stellar population that could be traced by optical photometry (i.e. blue or red), the formation scenario must follow the route where the ISM would be in a state having significant carbon enrichment with substantial amount of dust. The constraints coming from SST such as mid-IR 8 µm emission and moderate dust mass could be used as probes to understand the nature of LSB spirals. Metal poor HSB objects such as blue compact dwarf galaxies are PAH deficient systems (E05; Wu et al. 2006). Their SEDs are markedly different in the ∼5-15 µm wavelength range compared to metal rich HSB galaxies. The LSB galaxies are metal poor but have substantial PAH emission. Although these galaxies are not the extreme cases in metal deficiency such as BCDs, they fill an interesting niche among local populations, distinct from HSB dwarfs and from HSB regular galaxies. They may also represent a significant fraction of the galaxy population at earlier epochs (Zackrisson et al. 2005), and therefore, may have important implication in the interpretation of galaxy number counts in the infrared/submillimeter as well as in the visible and near-IR wavelengths. Previous analytical studies suggest that metal abundances have profound implications on galaxy number counts observed at 24, 70, and 160 µm (Lagache et al. 2003; Dale et al. 2005). To date, in analytical models metallicity effects have been incorporated in an ad hoc manner by artificially manipulating SEDs of HSB galaxies in the wavelength range mentioned above (Lagache et al. 2003). When template SEDs of many nearby metal-poor LSB galaxies become available, one can incorporate these in galaxy evolution models as an independent class along with various other classes such as normal star forming, starburst, luminous and ultra-luminous, and AGNs to understand the observed galaxy number counts and the origin of the IR background. The anonymous referee is thanked for constructive comments and suggestions. We happily thank D. Dale for his model fits. We also thank Y. Wu, B. R. Brandl, J. R. Houck for helpful communications. We acknowledge useful discussions from A. Blain, G. D. Bothun, and S. S. McGaugh on LSB galaxy population. One of us (NR) gratefully acknowledges the support of a Research Associateship administered by Oak Ridge Associated Universities (ORAU) during this research. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, USA under contract with the National Aeronautics and Space Administration, and the LEDA database in France. This study is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407. This study has made use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and IPAC/Caltech, funded by NASA and the National Science Foundation. This study also – 19 – acknowledges use of data products from Solan Digital Sky Survey. REFERENCES Alfonso-Herrero, A., Reike, G. H., Reike, M. J., Colina, L., Perez-Gonzalez, P. G., & Ryder, S. D. 2006, ApJ, 650, 835 Asplund, M., Grevesse, N., Sauval, A. J., Allende Prieto, C., & Kiselman, D. 2004, A&A, 417, 751 Barth, A. 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J. G., & van der Hulst, J. M. 1998b, A&A, 336, 49 de Naray R. K., McGaugh S. S., & de Blok, W. J. G. 2004, MNRAS, 355, 887 – 20 – Fig. 1.— The Solan Digitial Sky Survey (SDSS) composite images of target galaxies. In each image north is up and east is to the left. The field of view is 2.5′ × 2.5′ with 0.4′′ resolution. Note that the diffuse disks of Malin 1 and UGC 9024 are barely visible in these images. To help visualize these extended disks readers are referred to Barth (2007) for I-band Hubble image of Malin 1 and the following website for deeper B-band images of UGC 6614 and UGC 9024 (http://zebu.uoregon.edu/sb2.html). http://zebu.uoregon.edu/sb2.html – 21 – Malin 1 Fig. 2.— The Spitzer view of Malin 1. The IRAC 3.6, 4.5, and 8 µm images are at left and the MIPS 24, 70, and 160 µm images are at right. The IRAC 4.5 and 8 µm images are shown without subtracting stellar photospheric emission. In each image north is up and east is to the left. The field of view is 2.5′ × 2.5′ in all bands. Pixel sizes are 0.61′′ for the IRAC bands and 1.8′′, 4.0′′, and 8.0′′ for the MIPS 24, 70, and 160 µm bands, respectively. Galaxies from Figs. 3-5 are presented in a similar manner. There is no detection of Malin 1 at 70 and 160 µm and hence the position of 24 µm peak emission is shown by the “+” sign at these bands. The contours represent surface brightness (MJy/Sr) with intervals of 10 where the lowest level is 4σ above the background. See text for values of the lowest contour levels at different bands. – 22 – UGC 6614 Fig. 3.— The Spitzer view of UGC 6614. The position of 24 µm peak emission is shown by the “+” sign at 160 µm image. – 23 – UGC 6879 Fig. 4.— The Spitzer view of UGC 9024. The position of 24 µm peak emission is shown by the “+” sign at 160 µm image. – 24 – UGC 9024 Fig. 5.— The Spitzer view of UGC 6879. The B-band central surface brightness µB,0 of this galaxy is intermediate between LSB and HSB galaxies. – 25 – Fig. 6.— The observed SEDs of LSB galaxies and UGC 6879 at near, mid, and far-IR wavelengths. The 2MASS, IRAC, and MIPS points are shown by the open circles, filled circles, and open triangles, respectively. The IRAS upper limits are shown by the filled triangles. For all galaxies the flux densities are normalized at 3.6 µm. Dotted, dashed, dashed-dotted, and long dashed lines are used, respectively, to show the SEDs of: NGC 0337 (normal star forming galaxy), NGC 2798 (starburst galaxy), NGC 2976 (galaxy with nuclear H II region), and NGC 3627 (seyfert II galaxy). Malin 1 was undetected by the MIPS far-IR channels and hence the detection limits are shown for the total integration time (∼252 sec. at 70 µm and ∼42 sec. at 160 µm). – 26 – Fig. 7.— The observed SEDs of LSB galaxies and UGC 6879 using the DH02 model. The symbol styles are similar to Fig. 6. The dashed represents the empirical stellar SED of Pahre et al. (2004). The dotted lines represents the stellar synthesis model prediction from Vazquez & Leitherer (2005) fitted only to the 2MASS fluxes. The dust mass is estimated using the fitted SEDs (solid line). – 27 – Fig. 8.— The mid and far-IR color-color diagrams highlighting the LSB spirals (shown by open stars) with respect to different classes of HSB galaxies (shown by black dots). The SINGS (Kenni- cutt et al. 2003 ) sample galaxies are taken as the representative of local HSB galaxies. This sample contains various types of galaxies such as dwarfs, normal starforming, starburst, Syfert I, Seyfert II and ellipticals. In all panels galaxies represented by the decimal numerals are dwarfs systems whereas those shown by the roman numerals are ellipticals. The rest of the points (black dots) represent other population types where one galaxy from each population are shown by the open circle: NGC 0337 (normal star forming galaxy), NGC 2798 (starburst galaxy), NGC 2976 (galaxy with nuclear H II region), and NGC 3627 (Seyfert II galaxy). In panel (a) α ≈ 0.58 is the stellar contribution at 4.5 µm estimated from the empirically derived stellar SED of Pahre et al. (2004). The regions covered by the dotted boxes (panel b) are from a similar diagram of Eangelbracht et al. (2005). No HSB galaxies (dwarfs or extended disks) occupy these regions. Far-IR color of Malin 1 (panel c) is shown with respect to a flat far-infrared SED. The vertical arrow in panel (d) represents a probable range of IRAS color for UGC 9024. A few galaxies shown by numerals do not appear in panel (d) because of a lack of IRAS data. Representative error bars based on calibration uncertainty are shown. – 28 – Fig. 9.— Left panel: The IRAC colors for UGC 6614 as shown by solid symbols. The triangle (square) represents the color measured within a central region of radius of 2′′ (12′′); this corresponds to a physical radius of ∼1 kpc (∼5.5 kpc). The circle represents integrated flux from the entire galaxy (Table 1 and 2). In this diagram, stars occupy a locus around (0,0) with various exceptions (Stern et al. 2005). Star forming galaxies mostly lie along the the horizontal axis depending on the amount of 8 µm PAH emission. UGC 6879 is shown by open symbols to highlight the behavior of non-AGN type galaxies. A typical error bar is shown at the bottom. Right panel: an image (radius 12′′) of non-stellar emission at 4.5 µm after taking into account the contribution of similar emission at 3.6 µm image. See text for detail. Desert, F. X., Boulanger, F., & Pugetet, J. 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Flux densities from 2MASS, IRAS, and Spitzer Source λ(µm) Malin 1 UGC 6614 UGC 6879 UGC 9024 Sν r Sν r Sν r Sν r (mJy) (arcsec) (mJy) (arcsec) (mJy) (arcsec) (mJy) (arcsec) 2MASS 1.3 3.93 10 26.80 26 21.10 63 2.67 13 2MASS 1.7 3.41 10 35.50 26 40.90 63 3.62 13 2MASS 2.2 2.00 10 28.00 26 28.70 63 2.65 13 IRAC 3.6 1.61 12 22.31 60 16.03 75 1.94 24 IRAC 4.5 1.10 12 13.74 60 9.93 75 1.18 24 IRAC 5.8 0.73 12 14.11 60 14.73 75 1.57 24 IRAC 8 0.74 12 16.33 60 25.78 75 2.38 24 IRAS 12 – – <60 – <60 – – – MIPS 24 0.57 30 22 60 28.35 75 1.00 30 IRAS 25 – – <170 – <330 – – – IRAS 60 – – <150 – <360 – – – MIPS 70 <5 – 147 60 447 75 98 60 IRAS 100 – – <430 – <1080 – – – MIPS 160 <25 – 677 60 1529 75 <268 60 Note. — The IRAC flux densities are aperture corrected. Malin 1 was un- detected by the MIPS far-IR channels and hence the detection limits at 70 and 160 µm are shown for the total integration time: ∼252 sec. at 70 µm and ∼42 sec. at 160 µm. In general, r represents the radius of a circu- lar aperture which is used to estimate the flux density. However, the 2MASS flux densities are given for “total” elliptical aperture radius in the archive (http://irsa.ipac.caltech.edu/applications/2MASS/PubGalPS) where elliptical ra- dius represents the semi-major axis. UGC 6879 is edge-on and for this galaxy we use “effective radius”, defined by r = ab, where a and b are semi-major and minor axes. http://irsa.ipac.caltech.edu/applications/2MASS/PubGalPS – 34 – Table 2. Basic Properties of LSB galaxies Property Malin 1 UGC 6614 UGC 6879∗ UGC 9024 Reference Type S? SAa SABc SAab 1,2 redshift, z 0.0834 0.0237 0.009 0.0088 1,2 DA (Mpc) 320.47 93.51 35.87 34.86 1,2 inclination, i 16.1o 34o 90o 32.2o 1,2 D25 (arcsec) 16.15 75.54 87.75 104.26 1,2 D (kpc) 25.09 34.24 15.25 17.62 1,2 mB (mag) 17.92 14.12 13.40 17.11 1,2 MB (mag) -19.83 -20.70 -19.52 -15.64 1,2 µB,0 (mag arcsec −2) 26.50 24.50 20.40 24.71 4,5,6 hr,R (kpc) 73.3 16.00 2.50 7.47 4,5 Yo <8.66 7.35-9.16 – 7.98-8.35 3,9 log(LB/L⊙) 9.94 10.29 9.83 8.21 1,2,5 log(MHI/M⊙) 10.66 10.42 9.08 9.40 5,7,8 log(MH2/M⊙) – 8.45 – – 10 log(Md/M⊙) <8.14 8.42 7.81 <6.94 11 D <0.003 0.01 0.053 <0.004 11 log(LTIR/L⊙) 9.49 9.72 9.05 7.90 11 log(L /L⊙) <9.50 9.80 9.28 <8.30 11 LTIR/LB 0.35 0.27 0.17 0.49 11 SFR (M⊙ yr −1) 0.38 0.88 0.20 0.01 11 LTIR/MHI (L⊙/M⊙) 0.07 0.20 0.93 0.03 11 Note. — ∗UGC 6879 is not strictly a LSB galaxy. Its central surface brightness µB,0 is intermediate between LSB and HSB galaxies. We assume ΩM = 0.3, ΩΛ = 0.7, and H0 = 75 km sec−1 Mpc−1 to be consistent with the literature. Notation: DA is the angular diameter distance; D25 is the B-band 25 mag arcsec −2 isophotal diameter; D is the physical diameter; mB and MB , respectively, are B-band apparent and absolute magnitude corrected for Galactic extinction, internal extinction, and K-correction; µB,0 is the B-band central surface brightness; disk scale length hr,R is in kpc; the solar value of the oxygen abundance Yo = 12 + log[O/H] ∼ 8.66 from Asplund et al. (2004); MHI is neutral hydrogen mass; MH2 is molecular hydrogen mass; Md is dust mass; the dust-to-gas ratio is D = Md/MHI; total IR luminosity LTIR is estimated from – 35 – Calzetti et al. (2005); total IR luminosity L is estimated from the DH02 model fit; star formation rate SFR is calculated from Alfonso-Herrero et al. (2006). References: 1) LEDA; 2) NED; 3) McGaugh 1994; 4) Sprayberry et al. 1995; 5) Impey et al. 1996; 6) McGaugh & de Blok 1998; 7) Matthews et al. 2001; 8) Sauty et al. 2003; 9) de Naray et al. 2004; 10) Das et al. 2006; 11) This study. Introduction Observation and Data Reduction Contamination from Galactic Cirrus Results Infrared Morphology Infrared Diagnostics IR Spectral Energy Distributions Dust Mass Infrared Luminosity Star Formation Rate Infrared Colors Molecular ISM Mid-IR Photometry of UGC 6614 Summary and Conclusions

0704.1484

One-time pad booster for Internet

One-time pad booster for Internet Geraldo A. Barbosa∗ QuantaSec – Research in Quantum Cryptography Ltd., 1558 Portugal Ave., Belo Horizonte MG 31550-000 Brazil. (Dated: 11 April 2007) One-time pad encrypted files can be sent through Internet channels using current Internet proto- cols. However, the need for renewing shared secret keys make this method unpractical. This work shows how users can use a fast physical random generator based on fluctuations of a light field and the Internet channel to directly boost key renewals. The transmitted signals are deterministic but carries imprinted noise that cannot be eliminated by the attacker. Thus, a one-time pad for Internet can be made practical. Security is achieved without third parties and not relying on the difficulty of factoring numbers in primes. An informational fragility to be avoided is discussed. Information-theoretic analysis is presented and bounds for secure operation are determined. PACS 89.70.+c,05.40.Ca,42.50.Ar,03.67.Dd Unconditionally secure one-time pad encryption [1] has not find wide applicability in modern communications. The difficult for users to share long streams of secret keys beforehand has been an unsurmountable barrier pre- venting widespread use of one-time pad systems. Even beginning with a start sequence of shared secret keys, no amplification method to obtain new key sequences or key “refreshing” is available. This work proposes a practical solution for this problem and discusses its own limita- tions. Assume that (statistical) physical noise n = n1, n2, ... has been added to a message bit sequence X = x1, x2, ... according to some rule fj(xj , nj) giving Y = f1(x1, n1), f2(x2, n2), ... (Whenever binary physical sig- nals are implied, use fj(xj , nj) will represent fj = ⊕ (=addition mod2)). When analog physical signals are made discrete by analog-to-digital converters, a sum of a binary signal onto a discrete set will be assumed). The addition process is performed at the emitter station and Y becomes a binary file carrying the recorded noise. Y is sent from user A to user B (or from B to A) through an insecure channel. The amount of noise is assumed high and such that without any knowledge beyond Y, neither B (or A) or an attacker E could extract the sequence X with a probability P better than the guessing level of P = (1/2)N , where N is the number of bits. Assuming that A and B share some knowledge before- hand, the amount of information between A (or B) and E differs. Can this information asymmetry be used by A and B to share secure information over the Internet? It will be shown that if A and B start sharing a secret key sequence K0 they may end up with a practical new key sequence K ≫ K0. The security of this new sequence is discussed including an avoidable fragility for a-posteriori attack with a known-plaintext attack. Within bounds to be demonstrated, this makes one-time pad encryption practical for fast Internet communications (data, image or sound). It should be emphasized that being practical does not imply that K0 or the new keys have to be open to the attacker after transmission. These keys have to be kept secret as long as encrypted messages have to be protected, as in a strict one-time pad. The system gives users A and B direct control to guarantee secure commu- nication without use of third parties or certificates. Some may think of the method as an extra protective layer to the current Internet encryption protocols. The system operates on top of all IP layers and does not disturb current protocols in use by Internet providers. Anyway, one should emphasize that the proposed method relies on security created by physical noise and not just on math- ematical complexities such as the difficulty of factoring numbers in primes. This way, its security level does not depend on advances in algorithms or computation. Random events of physical origin cannot be determin- istically predicted and sometimes are classified in clas- sical or quantum events. Some take the point of view that a recorded classical random event is just the record of a single realization among all the possible quantum trajectories possible [2]. These classifications belong to a philosophical nature, and are not relevant to the practical aspects to be discussed here. However, what should be emphasized is that physical noise is completely different from pseudo noise generated in a deterministic process (e.g. hardware stream ciphers) because despite any com- plexity introduced, the deterministic generation mecha- nism can be searched, eventually discovered and used by the attacker. Before introducing the communication protocol to be used, one should discuss the superposition of physical sig- nals to deterministic binary signals. Any signal transmit- ted over Internet is physically prepared to be compatible with the channel being used. This way, e.g., voltage lev- els V0 and V1 in a computer may represent bits. These values may be understood as the simple encoding V (0) ⇒ V0 → bit 0 V1 → bit 1 Technical noise, e.g. electrical noise, in bit levels V0 and V1 are assumed low. Also, channel noise are as- sumed with a modest level. Errors caused by these noises http://arxiv.org/abs/0704.1484v1 are assumed to be possibly corrected by classical error- correction codes. Anyway, the end user is supposed to receive the bit sequence X (prepared by a sequence of V0 and V1) as determined by the sender. If one of these deterministic binary signals xj is repeated over the chan- nel, e.g. x1 = x and x2 = x, one has the known prop- erty x1 ⊕ x2 = 0. This property has to be compared to cases where a non-negligible amount of physical noise nj (in analog or a discrete form) has been added to each emission. Writing y1 = f1(x1, n1) = f1(x, n1) and y2 = f2(x2, n2) = f2(x, n2) one has f(y1, y2) = neither 0 or 1 in general. This difference from the former case where x1 ⊕ x2 = 0 emphasizes the uncontrollable effect of the noise. The V (0) encoding shown above allows binary values V0 and V1 to represent bits 0 and 1, respectively. These values are assumed to be determined without ambiguity. Instead of this unique encoding consider that two distinct encodings can be used to represent bits 0 and 1: Either V (0) over which x 0 and x 1 represent the two bits 0 and 1 respectively, or V (1), over which x 1 = x 0 +ǫ and 0 = x 1 + ǫ (ǫ ≪ 1) represent the two bits 1 or 0 (in a different order from the former assignment). These en- codings represent physical signals as, for example, phase signals. Assume noiseless transmission signals but where noise nj has been introduced or added to each j th bit sent (This is equivalent to noiseless signals in a noisy channel). Consider that the user does not know which encoding V (0) or V (1) was used. With a noise level nj superposed to signals in V (0) or V (1) and if |x00 − x10| ≫ nj ≫ ǫ, one cannot distinguish between signals 0 and 1 in V (0) and V (1) = V (0)+ǫ but one knows easily that a signal belongs either to the set (0 in V (0) or 1 in V (1)) or to the set (1 in V (0) or 0 in V (1)). Also note that once the encoding used is known, there is no question to identify between xj and xj+ǫ. In this case, it is straightforward to determine a bit 0 or 1 because values in a single encoding are widely separated and, therefore, distinguishable. One may say that without information on the encoding used, the bit values cannot be determined. Physical noise processes will be detailed ahead but this indistinguishability of the signals without basis informa- tion is the clue for A and B to share random bits over the Internet in a secure way. Physical noise has been used before in fiber-optics based systems using M -ry levels [3] to protect information (αη systems). However, the sys- tem proposed here is completely distinct from those αη systems and it is related to the key distribution system presented in [4]. A brief description of protocol steps will be made, be- fore a theoretic-security analysis is shown and the sys- tem’s limitations discussed. It was said that if A and B start sharing a secret key sequence K0 beforehand they may end up with a secure fresh key sequence K much longer than K0 (K ≫ K0). Assume that K0 gives encoding information, that is to say, which encod- ing (V (0) or V (1)) is being used at the jth emission. As- sume that K0 = k 1 , k 2 , ... has a length K0 and that the user A has a physical random generator PhRG able to generate random bits and noise in continuous levels. A generates a random sequence K1 = k 1 , k 2 , ...k (say, binary voltage levels) and a sequence of K0 noisy- signals n (e.g., voltage levels in a continuum). The de- terministic signal (carrying recorded noise) Y1 = k 1 , n 1 ), k 2 ⊕f2(k 2 , n 2 ), ... is then sent to B. Is B able to extract the fresh sequence K1 from Y1? B applies Y1 ⊕ K0 = f1(k(1)1 , n 1 ), f2(k 2 , n 2 ), ...fN (k N , n As B knows the encoding used and the signals rep- resenting bits 0 or 1 in a given encoding are eas- ily identifiable: f1(k 1 , n 1 ) → k 1 , f2(k 2 , n 2 ) → 2 , ...fN(k N , n N ) → k N . B then obtains the new ran- dom sequence K1 generated by A. Is the attacker also able to extract the same sequence K1? Actually, this was a one-time pad with K0 with added noise and, therefore, it is known that the attacker cannot obtainK1. The security problem arises for further exchanges of random bits, e.g. if B wants to share further secret bits with A. Assume that B also has a physical random genera- tor PhRG able to generate random bits and noise in continuous levels. B wants to send in a secure way a freshly generated key sequence K2 = k 1 , k 1 , ...k from his PhRG to A. B record the signals Y2 = k 1 , n 2 ), k 2 ⊕ f2(k 2 , n 2 ), ... and sends it to A. As A knows K1 he(or she) applies Y2⊕K1 and extracts K2. A and B now share the two new sequences K1 and K2. For speeding communication, even a simple round- ing process to the nearest integer would produce a simple binary output for the operation fj(kj , nj). The security of this process will be shown ahead. The simple description presented show a key distribu- tion from A to B and from B to A, with the net re- sult that A and B share the fresh sequences K1 and K2. These steps can be seen as a first distribution cycle. A could again send another fresh sequence K3 to B and so on. This repeated procedure provides A and B with sequences K1,K2,K3,K4, .... This is the basic key dis- tribution protocol for the system. A last caveat should be made. Although the key sharing seems adequate to go without bounds, physical properties impose some constraints and length limita- tions. Besides these limitations, the key sequences shared should pass key reconciliation and privacy amplification steps [5] to establish security bounds to all possible E attacks. The length limitation arises from the physical constraints discussed as follows. A and B use PhRGs to generate physical signals creat- ing the random bits that define the key sequences K and the continuous noise n necessary for the protocol. Be- ing physical signals, precise variables have to discussed and the noise source well characterized. Interfaces will transform the physical signals onto binary sequences ad- equate for Internet transmission protocols. Optical noise sources can be chosen for fast speeds. PhRGs have been discussed in the literature and even commercial ones are now starting to be available. Without going into details one could divide the PhRG in two parts, one generating random binary signals and another providing noise in a continuous physical variable (e.g., phase of a light field). These two signals are detected, adequately formatted and can be added. Taking the phase of a light field as the physical vari- able of interest, one could assume laser light in a coherent state with average number of photons 〈n〉 within one co- herence time (〈n〉 = |α|2 ≫ 1) and phase φ. Phases φ = 0 could define the bit 0 while φ = π could de- fine the bit 1. It can be shown [4] (see also ahead) that two non-orthogonal states with phases φ1 and φ2 (∆φ12 = |φ1 − φ2| → 0 and 〈n〉 ≫ 1) overlap with (un- normalized) probability pu ≃ e−(∆φ12) 2/2σ2 φ , (2) where σφ = 2/〈n〉 is the standard deviation measure for the phase fluctuations ∆φ. For distinguishable states, pu → 0 (no overlap) and for maximum indistinguishabil- ity pu = 1 (maximum overlap). With adequate format- ting φ1−φ2 gives the spacing ǫ (∆φ12 = ǫ) already intro- duced. Eq. (2) with ∆φ12 replaced by ∆φ describes the probability for generic phase fluctuations ∆φ in a coher- ent state of constant amplitude (|α| = 〈n〉 =constant) but with phase fluctuations. The laser light intensity is adjusted by A (or B) such that σφ ≫ ∆φ. This guarantees that the recorded infor- mation in the files to be sent over the open channel is in a condition such that the recorded light noise makes the two close levels φ1 and φ2 indistinguishable to the attacker. In order to avoid the legitimate user to confuse 0s and 1s in a single basis, the light fluctuation should obey σφ ≪ π/2. These conditions can be summarized as 2/〈n〉 ≫ ∆φ . (3) This shows that this key distribution system depends fun- damentally on physical aspects for security and not just on mathematical complexity. The separation between bits in the same encoding is easily carried under condition π/2 ≫ 2/〈n〉. The con- dition 2/〈n〉 ≫ ∆φ implies that that set of bits 0–in encoding 0, and 1–in encoding 1 (set 1) cannot be easily identifiable and the same happens with sets of bit 1–in encoding 0, and bit 0–in encoding 1 (set 2). Therefore, for A, B and E, there are no difficulty to identify that a sent signal is in set 1 or 2. However, E does not know the encoding provided to A or B by their shared knowledge on the basis used. The question “What is the attacker’s probability of error in bit identification without repeating a sent signal?” has a general answer using information theory applied to a binary identification of two states [6]: The average probability of error in identifying two states |ψ0〉 and |ψ1〉 is given by the Helstrom bound [6] 1− |〈ψ0|ψ1〉|2 . (4) Here |ψ0〉 and |ψ1〉 are coherent states of light [7] with same amplitude but distinct phases |ψ〉 = |α〉 = ||α|e−iφ〉 = e− 12 |α| |n〉 , (5) defined at the PhRG. |ψ0〉 define states in encoding 0, where bits 0 and 1 are given by |ψ0〉 = |α〉, for bit 0, and | − α〉, for bit 1 , (6) |ψ1〉 define states in encoding 1, where bits 1 and 0 are given by |ψ1〉 = ||α|e−i 2 〉, for bit 1, and ||α|e−i( +π)〉, for bit 0 , where |φ0 − φ1| = ∆φ. |〈ψ0|ψ1〉|2 is calculated in a straightforward way and gives |〈ψ0|ψ1〉|2 = e−2〈n〉[1−cos ] . (8) For 〈n〉 ≫ 1 and ∆φ≪ 1, |〈ψ0|ψ1〉|2 ≃ e− ∆φ2 ≡ e−∆φ 2/(2σ2φ) , (9) where σφ = 2/〈n〉 is the irreducible standard deviation for the phase fluctuation associated with the laser field. One should remind that in the proposed system the measuring procedure is defined by the users A and B and no attack launched by E can improve the deterministic signals that were already available to him(her). Thus, the noise frustrating the attacker’s success, cannot be eliminated or diminished by measurement techniques. One should observe that each random bit defining the key sequence is once sent as a message by A (or B) and then resent as a key (encoding information) from B (or A) to A (or B). In both emissions, noise is superposed to the signals. In general, coherent signal repetitions implies that a better resolution may be achieved that is propor- tional to the number of repetitions r. This improvement in resolution is equivalent to a single measurement with a signal r × more intense. To correct for this single rep- etition 〈n〉 is replaced by 2〈n〉 in |〈ψ0|ψ1〉|2. The final probability of error results 1− e− . (10) 0.5001 0.5002 0.5003 basesH∆ FIG. 1: ∆Hbases as a function of 〈n〉 and ∆φ. This error probability can be used to derive some of the proposed system’s limitations. The attacker’s probability of success Ps (= 1 − Pe) to obtain the basis used in a single emission may be used to compare with the a-priori starting entropy Hbases of the two bases that carry one bit of the message to be sent (a random bit). If the attacker knows the basis, the bit will also be known, with the same probability → 1 as the legitimate user. Hbases,bit = −p0 log p0 − p1 log p1 = 1 , (11) where p0 and p1 are the a-priori probabilities for each basis, p0 = p1 = 1/2, as defined by the PhRG. The en- tropy defined by success events is Hs = −Ps logPs. The entropy variation ∆H = Hbases,bit − Hs, statistically obtained or leaked from bit measurements show the sta- tistical information acquired by the attacker with respect to the a-priori starting entropy: ∆Hbases = Hbases,bit −Hs . (12) Fig. 1 shows ∆Hbases for some values of 〈n〉 and ∆φ. Value ∆Hbases = 1/2 is the limiting case where the two bases cannot be distinguished. ∆Hbases deviations from this limiting value of 1/2 indicates that some amount of information on the basis used may potentially be leaking to the attacker. It is clear that the attacker cannot obtain the basis in a bit-by-bit process. In order to be possible to obtain statistically a good amount of information on a single one encoding used, L should be given by ∆Hbases − ≫ 1 . (13) Fig. 2 shows estimates for L for a range of values 〈n〉 and ∆φ satisfying L× ∆Hbases − 12 = 1 (∆φ is given in powers of 2, indicating bit resolution for analog-to-digital converters). It is assumed that error correction codes can correct for technical errors in the transmission/reception steps for 10000 15000 20000 FIG. 2: Estimates for the minimum length of bits L ex- changed between A and B that could give one bit of infor- mation about the bases used to the attacker. the legitimate users. The leak estimate given by Eq. (13) do not imply that the information actually has leaked to the attacker. However, for security reasons, one takes for granted that this deviation indicate a statistical fraction of bits acquired by the attacker. Privacy amplification procedures can be applied to the shared bits in order to reduce this hypothetical informa- tion gained by the attacker to negligible levels [5]. These procedures are beyond the purposes of the present dis- cussion but one can easily accept that A and B may dis- card a similar fraction of bits to statistically reduce the amount of information potentially leaked. Reducing this fraction of bits after a succession of bits are exchanged between A and B implies, e.g., that the number of bits to be exchanged will decrease at every emission. Eventually, a new shared key K0 has to start the process again to make the system secure. Nevertheless, the starting key length K0 was boosted in a secure way. Without further procedures, the physical noise allowed K ≫ 103K0, a substantial improvement over the classical one-time pad factor of 1. One may still argue that the ultimate secu- rity relies onK0’s length because ifK0 is known no secret will exist for the attacker. This is also true but does not invalidate the practical aspect of the system, because the K0 length can be made sufficiently long to frustrate any brute-force attack at any stage of technology. Therefore, the combination of physical noise and complexity makes this noisy-one-time pad practical for Internet uses. Although the security of the process has been demon- strated, one should also point to a fragility of the system (without a privacy amplification stage) that has to be avoided when A and B are encrypting messages X be- tween them. As it was shown, knowledge of one sequence of random bits lead to the knowledge of the following sequence. This makes the system vulnerable to know- plaintext attacks in the following way: E has a perfect record of both sequences Y1 and Y2 and tries to recover any bit sequence from them, K2, K1 or K0. E will wait until A and B uses these sequences for encryption before trying to brake the system. A and B will encrypt a mes- sage using a new shared sequence, K1 or K2. This mes- sage could be a plain-text, say X = x1, x2, ...xK0 known to the attacker. Encrypting this message with sayK1 in a noiseless way, gives Y = x1⊕k(1)1 , x2⊕k 2 , ...xK0 ⊕k Performing the operation Y ⊕ X, E obtains K1. The chain dependence of Kj on Kj−1 creates this fragility. Even addition of noise to the encrypted file does not elim- inate this fragility, because the attacker can use his/her knowledge of X –as the key– to obtain K–as a message. The situation is symmetric between B or the attacker: one that knows the key (X for E, and K for B) obtains the desired message (K for E, and X for B) [8] . In general, random generation processes are attractive to attackers and have to be carefully controlled. Well identifiable physical components (e.g. PHRG) are usu- ally a target for attackers that may try to substitute a true random sequence by pseudo-random bits generated by a seed key under his/her control. Electronic compo- nents can also be inserted to perform this task replacing the original generator; electric or electromagnetic signal may induce sequences for the attacker and so on. In the same way, known-plaintext attacks also have to be care- fully avoided by the legitimate users. The possibility of further privacy amplification procedures to eliminate the known-plaintext attack presented is beyond the purposes of this work. Many protocols that use secret key sharing may profit from this one-time pad booster system. For example, besides data encryption, authentication procedures can be done by hashing of message files with sequences of shared secret random bits. Challenge hand-shaking may allow an user to prove its identity to a second user across an insecure network. As a conclusion, it has been shown that Internet users will succeed in generating and sharing, in a fast way, a large number of secret keys to be used in one-time-pad encryption as described. They have to start from a shared secret sequence of random bits obtained from a physical random generator hooked to their computers. The physical noise in the signals openly transmitted is set to hide the random bits. No intrusion detection method is necessary. Privacy amplification protocols eliminate any fraction of information that may have eventually obtained by the attacker. As the security is not only based on mathematical complexities but depend on physical noise, technological advances will not harm this system. This is then very different from systems that would rely entirely, say, on the difficulty of factoring large numbers in their primes. It was then shown that by sharing secure secret key sequences, one- time pad encryption over the Internet can be practically implemented. ∗E-mail: [email protected] [1] G. S. Vernam, J. Amer. Inst. Elec. Eng. 55, 109 (1926). C. E. Shannon, Bell Syst. Tech. J. 28, 656 (1949). [2] V. P. Belavkin, Int. J. of Theoretical Physics 42, 461 (2003). [3] G. A. Barbosa, E. Corndorf, P. Kumar, H. P. Yuen, Phys. Rev. Lett. 90, 227901 (2003). E. Corndorf, G. A. Bar- bosa, C. Liang, H. P. Yuen, P. Kumar, Opt. Lett. 28, 2040 (2003). G. A. Barbosa, E. Corndorf, and P. Ku- mar, Quantum Electronics and Laser Science Conference, OSA Technical Digest 74, 189 (2002). [4] G. A. Barbosa, Phys. Rev. A 68, 052307 (2003); Phys. Review A 71, 062333 (2005); quant-ph/0607093 v2 16 Aug 2006. [5] S. Wolf, Information-Theoretically and Computationally Secure Key Agreement in Cryptography, PhD thesis, ETH Zurich 1999. [6] C. W. Helstrom, Quantum Detection and Estimation Theory, ed. R. Bellman (Academic Press, 1976), pg. 113 Eq. (2.34). [7] R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963); Quantum Optics and Electronics, eds. C. DeWitt, A. Blandin, C. Cohen-Tannoudji (Dunod, Paris 1964), Proc. École d’Été de Physique Théorique de Les Houches, 1964. [8] The author thanks one of the reviewers for a demonstra- tion of this fragility. http://arxiv.org/abs/quant-ph/0607093

0704.1485

A Local Concept of Wave Velocities

arXiv:0704.1485v1 [math-ph] 11 Apr 2007 On a Local Concept of Wave Velocities I. V. Drozdov A. A. Stahlhofen † University of Koblenz Department of Physics Universitätsstr.1, D-56070 Koblenz, Germany June 4, 2018 Abstract The classical characterization of wave propagation , as a typical con- cept for far field phenomena, has been successfully applied to many wave phenomena in past decades. The recent reports of superluminal tunnelling times and negative group velocities challenged this concept. A new lo- cal approach for the definition of wave velocities avoiding these difficulties while including the classical definitions as particular cases is proposed here. This generalisation of the conventional non-local approach can be applied to arbitrary wave forms and propagation media. Some applications of the formalism are presented and basic properties of the concept are summa- rized. 1 Introduction The velocities conventionally used for the characterization of wave propagation are the phase velocity vph := ω/k, the group velocity vgr := dω/dk, the front velocity vf := limω→∞ vph, the signal velocity vs with vf ≥ vs ≥ vgr, the energy velocity v̄E := P̄ /ρ and the phase time velocity vϕ := x/(dϕ/dω), where ω denotes the angular frequency, k - the wave number, φ- a phase shift accumulated in course of propagation, x- the distance of propagation, P̄ - the energy current density and ρ - the energy density. ∗e-mail: [email protected] †e-mail: [email protected] http://arxiv.org/abs/0704.1485v1 This classical concept [1, 2] and its improvements [3, 4] have been successfully applied to many wave phenomena in past decades. The recent reports of super- luminal and even negative velocities [5, 6] the advent of near field optics and the generation of ultra-short pulses [7, 8], for instance, revealed intrinsic difficulties and limitations of the classical concept. It is the purpose of the present note to resolve these difficulties by a new local approach for the characterization of wave velocities containing the classical Ansatz as special case and being applicable to arbitrary wave forms and media of propagation. The material is organized as follows: the next section contains a formal discus- sion of the classical approach pinpointing its intrinsic difficulties and summarizing the goals of the new approach. For this purpose it is completely sufficient to focus on the notions of phase and group velocity . This prepares the ground for the new approach outlined in the section 2 and an elucidation of its properties by application to known examples in section 3. The behavior of the wave velocities defined here under relativistic transforma- tions is outlined in the section 4. In order to pave the way for applications of the formalism to experimental results like those mentioned above, the discussion is concluded by a practical interpretation of the definitions and the extension from a local to a global evaluation of signal velocities . 2 Some remarks on wave velocities The classical concept of wave velocities analyses wave propagation in the far field, where the wave is far away from the source. The wave is described via propagating harmonic functions containing a periodic factor of the form eiϕ. The argument of this periodic functions is interpreted as a phase and identified with (t− x/v) via the free wave equation , where v is the phase velocity [1]. A second fundamental concept of characterizing wave propagation is that of the ”group velocity”. This concept is - mathematically speaking - a comprehen- sive definition for a specified (linear) superposition of solutions of the free wave equation with the same periodicity properties usually expressed by the frequency distribution of the constituent periodic waves. This definition is also an inher- ent far field concept considering ”source-free” waves propagating in a strongly homogeneous and isotropic medium. This medium is characterized only by its ”dispersion” (supposed to satisfy additionally the Kramers-Kronig relations), i.e. by a dependence of the complex index of refraction n(ω) on the periodicity pa- rameter of the wave - the frequency ω. The different notions of velocities (and the accompanying notion of a signal) developed in this context were a major step forward allowing to accomodate many experimental data and to characterize many wave phenomena. (cf. e.g. [2]) It is, however, not obvious to what extent these far field concepts can be applied to near field problems or any of the other challenges mentioned above. A possible answer to this question simultaneously preserving the classical concepts whenever applicable is the focus of the present work. The basic idea is to employ a local definition of a propagation velocity . Such local approaches can be found in many areas physics and its applications in concerned natural sciences. A need for an alternative local approach to the analysis of wave propagation was justified already since years in different fields of application such as geophysics [21], physical chemistry [20, 25], biophysics [22], acoustics [23], dealing with a relative ”slow” waves, compared to the optics and electronics, thus being long ago in the near-field region, where the actual problems require a fine resolution of wave behavior. There is thus a justified reason for a local approach to wave velocities since this allows to resolve the intrinsic problems accompanying the classical global definitions. To elucidate this remark we now list some of these problems. The conventional analysis of wave propagation is based on a representation via periodic functions by means of Fourier analysis. A well defined phase velocity is assigned to each Fourier component equipped with its frequency; a corresponding group velocity is subsequently defined for the complete Fourier superposition, i.e. for a wave packet. The classical wave theory is based on such definitions relying canonically of the phase of periodic functions and on group of such functions . This Ansatz is supposed to be the inherent kernel of any characterization of wave phenomena and has also been used as a guideline for characterizing general - not necessary periodic - electromagnetic pulses [1]. A close inspection, however, reveals several shortcomings of this Ansatz with respect to mathematics and physics. Let us start with the local features of wave propagation . A propagating wave is per definition a local space-time distribution which is itself a solution of a local differential equation . A definition of a wave velocity as a space-time relation should respect this frame being local as well. This is not the case in the conventional definitions of wave velocities : these involve the frequency and the wave vector, which are (temporally and spatially) nonlocal parameters in following the independent space- and time-periodicity. Since these parameters do not enter the wave equation , they are usually plugged in from the outside. The source free wave equation for instance, admits solutions of arbitrary shapes without an a priori assumption of periodicity. Simultaneously, the classical approach always takes recourse to a representa- tion of an arbitrary solution by a set of periodic harmonic functions , where each element of the set does in general not obey the wave equation . Any attempt to apply this strategy to waves resulting from non-linear equations such as solitons, for instance, elucidates this problem in a clear way since in this case even a linear superposition of solutions is not anymore a solution. Second, the subject of transmission and velocity of a signal is based on suf- ficiently local procedures of measurement [7]. Roughly expressed, the intervals between several space-time points are measured. Definitions of velocities based on periodicity parameters can possess generically neither time- nor space-locality. Moreover, any Fourier transformation is basically a global object, and all manip- ulations concerned are in general mathematically exact only with integration over the whole space and whole time, as well as over an infinite frequency band. The conventional definitions of the phase and group velocity mentioned above [1] are based on the special case of a propagation of periodic harmonic waves in homogeneous media. Thus there is already an essential contradiction between the local character of wave propagation and the inherent globality contained in the definitions of wave velocities. Any attempt to construct a local measurable object using Fourier sums or integrals contain an essential contradiction as outlined above and provides indeed no real locality. This is the source of several problems arising when replacing originally local features by ”microglobal” ones [3, 12, 13]. Even if the approach is supported by mathematical consistency (like the as- sumption of an infinite frequency band [1]), it does not lend itself easily to a transparent physical interpretation. A nice example thereof is the phenomenon of signal transmission in (optically active) media, which triggered a lot of discus- sions [3, 11, 10] The classical papers [1] and later improvements thereof [13, 4] still contain an essential mismatch between local and global wave features, based on several mathematically correct but physically misleading definitions (like the notion of front and signal velocities [9, 26]). This Ansatz is bound to lead to controversial results in applications. For example, it seems not more to be surprising, that the subject of ”group velocity” fails to describe a propagation of ultra-short pulses [8]. These shortcomings of the classical concept of wave velocities motivated the Ansatz presented here for the following reasons: - The classical harmonic approach represents local wave features in terms of generically non-local attributes [15, 16]. Hence: - The mathematical equipment is not exact for finite physical values (like a limited frequency band); - It is therefore not obvious how to apply it to near-field effects, if the size of space-time regions are of an order of magnitude smaller than the wave periodicity parameter; - There is no canonical application thereof for inhomogeneous and anisotropic media, as well as for ultra-short pulses of arbitrary form and for nonlinear waves [12, 17]; - As a consequence, the applicability of this approach for several fields of modern quantum optics, nano-optics and photonics is difficult, since these topics deal with parameter areas outside its range of validity [5]. Further motivations outside of optics have been mentioned above. The classical approach is also to some extent problematic from a pure technical point of view: it is an essential restriction of generality and universality of the theory enforcing via the Fourier analysis - a number of problematical artifacts. A prominent example thereof being the assumption of infinite frequency bands for signals. These additional problems are then resolved via the analysis of dispersion relations. A first step towards a more general concept should be based on an indepen- dent alternative approach without doubting the canonical harmonic criteria, i.e. leading to the same (verified) results in known cases. The aim of the present paper is to establish a transparent criterion for evalu- ation of the propagation velocity for a quite arbitrary signal, that does not need an explicit representation in terms of periodic or exponential functions at all, and with minimal loss of generality in other respects. First of all, we have to recall, what is being measured in experiments and what is meant when speaking about a ”wave velocity”, thereby providing the spectrum of wave velocities. 3 N-th order phase velocity (PV) The following discussion is restricted to a 1+1 dimensional space-time for sim- plicity. To evaluate the speed of a moving matter point one has to check the change of the space coordinate ∆x during the time interval ∆t. How can this Ansatz be applied to waves ? An arbitrary wave is a function defined on the 2-dimensional space-time (x, t) and one has no a priori defined fixed points to follow up as in the former case. Let us therefore analyze a continuous smooth function ψ(x, t) of arbitrary shape. In a first step towards a local definition of velocity we consider a vicinity U(x) of a certain point x at the certain time t. Further we assume the local information about the function ψ(x, t) to be measurable, i.e. we should be able (at least in principle) to evaluate the values of the function ψ(x, t) itself and all its derivatives in the point x as well, and in the vicinity U(x). A description of propagation is based on monitoring the fate of a certain attribute (”labelled point”) of this shape, i.e. one finds the same attribute at the next moment t1 in the next point x1. Let this attribute be labelled by some one-point fixed value ψ0 of the func- tion ψ(x, t). Suppose, one follows up this local ”attribute” and manages a local observation of the condition: ψ(x, t)− ψ0 = 0, (3.1) which fixes the space-time points {x, t} where this condition holds (Fig.1). Thus we have a function x(t) given in an implicit form in eq.(3.1). The first order implicit derivation provides the velocity of this one-point local (x,t) t t0 1 Figure 1: The propagation of a signal ψ(x, t) without a change of shape with the value ψ0 of the amplitude as an attribute to be traced in course of propagation attribute: v(0)(x, t) := , (3.2) called from here on the ”zero-order” or ”one-point” phase velocity (0-PV). (Here the subject ”phase” is treated in sence of local charactreristics in the point of shape including all N-order derivatives, N = 0, 1, 2, ... i.e. all terms of the Taylor expansion in this point). It describes in the simplest case the speed of translation of some arbitrary pulse, that can be treated as the signal propagation velocity, provided that the measured value ψ0 of the amplitude is the signal considered. To proceed with the local description of the shape propagation we now con- sider as the measurable attribute of ψ(x, t) at some time t0 not the single value ψ0 = ψ(x0; t0) at the point x0, but a set of values ψ in a local neighborhood (like U(x)) close to this point. Then another local attribute can be constructed to trace their propagation. If, for instance, one looks at a certain value of the first derivative ∂ψ(x; t0)/∂x := χ0 of the shape ψ, (like ∂ψ(x; t0)/∂x = 0, i.e. at a maximum or minimum point), one should consider the propagation of the condition: ∂ψ(x, t) = χ0; (3.3) this provides its propagation velocity via v(I)(x, t) := − , (3.4) called in view of (3.2) the ”first order” or ”two-point” phase velocity (1-PV) respectively. When tracking the propagation of a maximum (minimum), the conditions ∂ψ(x,t) = 0 and ∂ < 0 have to hold simultaneously. This Ansatz is easily iterated to phase velocities of order N interpreted as the propagation velocities of higher order local shape attributes via v(N)(x, t) := − ∂N+1ψ ∂t∂xN ∂N+1ψ ∂xN+1 , (3.5) leading to N-th order (”N+1-point”) PV, describing the propagation of higher order local shape attributes. The phase velocities so defined are obviously local features depending on space-time coordinates. It has to be noted, that any given problem at hand might require a particular choice of a PV allowed for by the definitions given above. The following examples elucidate these requirements and demonstrate the flexibility of these definitions. Before proceeding to it should be recalled, that the PV-spectrum has been obtained wanting to be able to describe the propagation of an arbitrary pulse in terms of local attributes in a medium whose properties depend on several vari- ables, especially time and space coordinates being the most prominent and natu- ral examples thereof. Any initial shape ψ(x0, t0) thus should be deformed during the propagation (or evolution, as typicaly encountered for dispersive media). The ordinary phase velocity v0 therefore is not a relevant criterion to characterize the shape propagation and one has to choose an appropriate PV v(N). For example, let the pulse ψ(x, t0) be subject to damping (Fig.2). As a con- cequence, the zeroth order PV measured in the point x1 gives a magnitude much smaller as the same magnitude measured in the point x2. For an amplified signal (Fig.3, like a signal propagating in a laser excited medium), by comparison, the zero order PV from the point x1 provides alto- gether even a backward propagation. In both cases an appropriate approach would be to apply the first order PV v(I) which describes the propagation of the maximum up from the point x0 properly. For a propagation of a kink front that experiences a deformation, one can check the translation of the second derivative of the shape, keeping track of the turning-point of the kink shape (Fig.4). In this case the second order (”three- point”) PV turns out to be the relevant velocity of propagation. (x,t) 2x1 1x x2 Figure 2: A damping deformation of a signal ψ(x, t). A natural attribute allowing to describe propagation is the peak location (maximum) Summarized, the local concept is based on local attributes of the wave shape being traced in course of propagation. The phase of an arbitrary wave in some point is characterized by the local Taylor expansion in this point. The local attribute of N-th order is therefore the N-th spatial derivative of the wave shape. The phase velocity of N-th order (N-PV) is the velocity of propagation of the corresponding N-th order attribute. Finally, it should be noted, that the definiton of phase velocities of order zero and one, the v(0) and v(I) respectively, admits a straightforward generalization to two-, three-, and higher-dimensional propagation, while the phase velocities of second ( v(II)) and higher orders inherently contain a certain element of ambiguity in their definition, since a possibility to choose a second-order attribute to be traced is not unique is not unique [19]. 4 Examples Let us consider the conventional 1+1-dimensional wave equation ψ(x, t) = 0 (4.1) possessing translational solutions of the form ψ(x, t) = ψ(t± x ). (4.2) ∆xmax x Figure 3: The propagation of an amplified signal ψ(x, t). The location of the maximum describes the wave propagation It is easy to check that in this case the PV’s of all orders defined above are identical and read v(N)(x, t) = a, N = 0, 1, 2, ... (4.3) which is nothing else but the classical phase velocity ±a, thereby satisfying the a priori definition of ”phase velocity” itself as a medium constant in (4.1). Let a propagating shape Ψ now be subject to a temporal damping similar to (Fig.2) with Ψ(x, t) = ψ(t− )e−λt ≡ ψ(φ)e−λt (4.4) that obeys the wave equation (a typical field equation with a dynamical dissipa- tion and a mass term) Ψ(x, t) = 0 (4.5) The ordinary 0-PV velocity reads v(0) = a , (4.6) where the prime denotes the derivative of ψ with respect to its argument φ ≡ t− x/a. Figure 4: The propagation of a growing kink (”tsunami model”). The turning point is chosen to trace the propagation This result provides a velocity with bad physical features: the velocity grows for a descending shape, for an ascending shape it decreases, can even be neg- ative, and it diverges exactly for the peak point (under the condition that the (measured) amplitude Ψ as well as parameter a, λ have positive values). A relevant physical velocity in this case is for instance the 1-PV v(I) = a 1− λ ψ (4.7) providing for a peak being traced exactly the canonical phase velocity entering in the wave equation (4.5). For this shape further PV’s of higher orders are given by (3.5): v(N) = a 1− λ ψ ψ(n+1) 1− λ(log′ ψ(n))−1 (4.8) where ψ(n) denotes the n-th derivative of ψ with respect to its argument as mentioned above. For an amplified signal as in Fig.3 we can e.g. change the sign of λ. Especially, for the case of a kink (Fig. 4) ψ(t− x ) ≡ ψ(φ) = arctanφ, (4.9) the spectrum of phase velocities reads by comparison : v(0) = a 1 + λ(1 + φ2) arctanφ v(I) = a 1− λ1 + φ , (4.10) v(II) = a φ3 + φ 3φ2 − 1 The ordinary 0-PV has an oscillating sign at λ and is multiple defined because of the arctan function . Therefore it cannot be interpreted as a well-defined physical velocity. If the point being traced should be labelled by a derivative attribute, it turns out to be a physically inconvenient choice since the shape possesses no real peaks, whose propagation could be traced. Moreover the velocity v(I) diverges at φ = 0. The possible labelled attribute is also the turning-point traced by the 2-PV. The velocity v(II) also possesses two singularities at φ = ±1/ 3 which do not coincide with the labelled point φ = 0, so it can be successfully followed up at the measurement. Historical remark The definition of a wave velocity proceeded with an extension of a translational solution of (4.1): ψ(x, t) = ψ(t− x ) ≡ ψ[ 1 (ωt− ω x)],≡ φ(ωt− kx), k ≡ ω , (4.11) where the parameter ω has been interpreted as a frequency of a necessary perodic harmonic function φ. usually e±ix as mentioned above in the introduction. It is not surprising that the 0-order PV provides in this case the value ω/k, canonically interpreted as a phase velocity of periodic wave In case of any interrelations between k and ω that are not encountered in the wave equation, in particular any dispersion relation between frequency and wave-number, the PV v(0) is in fact a proportionality factor dω = v(0)dk, (4.12) which is identical with the classical definiton of a group velocity [1]. At this point it should be recalled, that the original idea of the group velocity U , as summarized e.g. in [28], = 0 (4.13) was of the similar form as the recent definition (3.2) of 0-PV. It should be noted here that equation (4.13) involves local derivatives of the non-local parameter,i.e. the wavelength λ. For a variable wavelength much smaller than a vicinity of the point, this group velocity provides therefore a natural approximation to the zero-order phase velocity v(0). In the present approach this feature appears as a physical phase velocity following in a straightforward way from the interpretation of phase propagation and does not require an interpretation of ω and k as a frequency and wave number, as well as a constancy of some group respectively. It is noted in passing, that in the case of a kink there are no suitable definitions of a wave group and of a concerned group velocity for this solution since a Fourier decomposition does not work on a non-compact support. 5 Lorentz covariance The zero order PV (3.2 ) possesses an interesting property, i.e. a local covariance in the sense of special relativity, as shown now. This is not the case for PV’s of higher orders. The PV v(0), measured in some stationary system X , takes in some other system X ′, moving with a constant speed V , via the Lorentz transformations x− V t 1− V 2 , t′ = t− V x 1− V 2 (5.1) the form v′(0) = v(0) + V v(0)V . (5.2) This means that the zero order PV respects the relativistic velocity addition. Especially, a subluminal zero order PV remains also subluminal in any other moving system X ′. This result should not be a surprise, since the definition of the v(0) is implied by the condition: ψ(x, t) = const, (5.3) which remains to be of the same form under arbitrary transormations x = x(x′, t′); t = t(x′, t′), implying dψ(x, t) ≡ ∂ψ dx = 0 (5.4) for the first order differential form (or simply first differential), which possesses a form-invariance property under transformations. Since v(0)(x, t) = (5.5) per definition, the equation (5.4) inplies the definition (3.2) of 0-PV, so it should behave under space-time transformations as a usual velocity of a matter point. The first order PV (3.4), by comparison, evaluated in some system X trans- formes in the moving system X ′ to: v′(I) = − 1 + V − 2 V . (5.6) If the pulse ψ(x, t) obeys the free wave equation (4.1), the transformation becomes v′(I) = − 1 + V v(I) + 2V 1 + V 2V v(I) , (5.7) i.e. a relation that should be called the ”first order velocity addition”. It differs obviously from the corresponding transformation of v(0), since the definition of the velocity v(I) results from the condition ∂ψ(x, t) = 0 (5.8) whose form is explicitely non-invariant under space-time transformations. It can be shown that a subluminal first order PV is still preserved by this transformation as well. Note, that for a signal which does not obey the free equation (4.1), this restriction is in general not guaranteed anymore. 6 A global velocity of signal transmission and ”dynamic separation” The discussion of local velocities was aimed towards an evaluation of global fea- tures of signal propagation, namely the propagation through a finite spatial inter- val during a finite temporal interval. In other words, a global velocity, practically measured, means roughly the length of the distance ∆x traveled by a traced attribute divided by the time interval ∆t. The local PV’s of N-th order analyzed above can be interpreted as a first order differential equations of the form v(N)(x, t) = , (6.1) that can be illustrated graphically as a field of isoclines (Fig.5). Here the local PV is the tangent function of the tangent vector of the isocline, and the averaged (total) velocity between (t0, x0) and (t1, x1) is represented by the tangent of the hypotenuse of the triangle {(t0, x0), (t1, x1), (t0, x1)}, (cf. Fig.5). This procedure is elucidated best by some clear and well known examples. Consider the propagation of a translation mode of the form ψ(x, t) = ψ(ξt− k(x)) := ψ(φ) (6.2) describing, for instance, an electromagnetic field propagation in an inhomoge- neous dielectric medium. In this case, if the transversality of the wave not as- sured, the differential equation for any components of the field can possess e.g. a ordinary phase velocity Figure 5: Phase velocities as a family of isoclines v(N)(x, t) and averaged global veloc- ities between two events (measurements) form like this: n2(x) ψ + ln′ n(x) ψ = 0 (6.3) where n(x) is the spatially dependent index of refraction. From the Ansatz (6.2) it results for the k(x) from the optical inhomogeneity over the spatial coordinate n(x) = k′(x), k(x) = n(x)dx, (6.4) Then the local zero order PV is provided by eq.(3.2) and results in the first order equation v(0)(x, t) ≡ , (6.5) that is a quite trivial result. We proceed now with phase velocities of higher orders. For the 1-PV velocity v(I) defined by eq.(3.4) the same procedure leads to v(I)(x, t) ≡ n(x)− c ln′ n(x) (6.6) where x and t denote the space and time location of the first oder attribute being traced; ψ′andψ′′ are the first and the second derivative of the wave function ψ with respect to its argument φ (cf.6.2). Similarly, for the 2-PV we obtain: v(II)(x, t) = n(x)− 3 c ln′ n(x) + c . (6.7) Suppose, we measured at the moment t0 some value of the fisrt order attribute ψ′(x0) in the point x0, and we are interested, where is this attribute to recover at the next moment t1. For a maximum or a minimum point the result does not distinguish from the zero order velocity, as in the case considered above. Now, let we measured at t0 some values: ψ′(0, x0) := ψ 0 and ψ ′′(0, x0) := ψ 0 ; Λ := ψ ′(0, x0)/ψ 0 (6.8) An explicit integration of eq.(6.6) leads to c(t1 − t0) = n(x)dx− Λ n(x1) n(x0) (6.9) for the interval of two events, where the signal attribute choosen for tracing has been checked at the time t0 in the point x0 and afterwards in x1 at t1. Let us assume that the point (t0, x0) was not a critical point (i.e. of the knot type) of the equation (6.1). Then the global transition velocity between two points x0, x1 is evaluated unique in such a way: divided by the spatial interval x1 − x0 the eq.(6.9 ) provides the factor γ: γ := c t1 − t0 x1 − x0 x1 − x0 n(x)dx− Λ c n(x1) n(x0)  (6.10) It follows, that a value of γ less than 1 is not prohibited in principle even for real n > 1, that means a superluminal averaged global velocity along the finite distance. A similar result appears for 2-PV and higher orders. A numerical examples for dielectric media of special modulation confirms this possibility [27]. The discussion of physical interpretation of this result is still open. It is also worth to notice, that for the first order PV (as well as the second and higher orders) in media with a variable refraction index n(x) an essential dependence on the parameter ξ enters. The meaning of ξ can be derived from the given form of the solution ψ. It can be interpreted, for instance, as a frequency factor for a periodic mode or a damping factor for an evanescent mode. This parameter simply means an enumeration of solutions of a solution family (space of solutions). The merit gained above is the separation of these solutions with respect to the parameter ξ by the different first (and higher)-order PV’s through the inhomogeneity of the medium. In the case of a periodic wave for instance, it can be interpreted as a dispersion although an explicit frequency-dependence of n(x) was not assumed. Moreover, the parameter ξ has not necessarily to be interpreted as a frequency of some temporally periodic oscillation, but rather as a time component of the space- time wave vector {ω,k} for a translation mode (4.2) in a general form. The phenomenon considered thus has another origin and is much more general as a conventional non-localized frequency dispersion n(ω) Thus we established for the first (and higher) order PV’s in optical inhomoge- neous media the essential dependence on the t-component ω of the wave transla- tion vector, especially frequency, even for local non dispersive media, which could be called ”dynamic separation”. The global averaged dynamic separation of eq. (6.9) survives as a corollary of the local separation of (6.6). This phenomena does not occur for the ordinary zero order PV. 7 Concluding remarks The inherent inconsistency of the classical subjects of phase velocity and group velocity, (as well as signal velocity based therein) has been discussed. The inap- plicability of these concepts for actual studies in photonics, near-field and nano- optics has been shown to result from the essential non-locality of these definitions. An alternative approach for description of a propagation velocity has been proposed. It is strictly local and is based on the natural assumption of an ordinary measurement procedure. It does not need any a priori condition of periodicity, frequency, groups and packets or other canonical attributes. The definitions presented above are applicable in a natural way for arbitrary waves and pulses. In a mathematical sense they describe a propagation of a perturbation in any field in space-time. Examples could be: an acoustic wave as a pressure perturbation, a gravitation wave on a fluid surface, a spin wave in a solid state a.s.f. In particular, the formalism is very suited for the description of particle propagation in field theory, where particles are considered as field perturbations. This approach results in the set of measurable propagation velocities. In zeroth order the propagation velocity coincides with the ordinary phase velocity and appears to be ordinary Lorentz covariant; further application gives rise to generate ”Lorentz covariance of higher orders”. For propagation in inhomogeneous media, for instance for light in a medium with a space-dependent index of refraction, phase velocities of higher orders show some remarkable properties: a) they are not restricted in general by the speed of light in vacuum; b) they exhibit an essential dependence on the time component of the wave vector solely as a result of inhomogeneity, treated canonically as a ”dispersion”. This appears to be a more general phenomena, and does not presuppose any periodic frequency and dispersive properties of media. It should be called for this reason ”dynamic separation” [27]. Further applications, like photonics and optoelectronics are dealing with sig- nals i.e. non-localized objects, represented as a cut of wave shape between la- belled points possessing a certain spatial length (or temporal duration). These two points are marked by a value of zeroth order attribute (e.g. a voltage am- plitude), or by a first order one (a block between an ascent and a descent of amplitude). In this sense the signal is considered as not a local, but at least a bi-local object. Its propagation can be traced by an averaged bilocal velocity, constructed from local ones in a natural way. The same concerns a propagation of single photons, especially of low energy, which are non-localized objects. For the case of evanescent modes we are dealing yet with a sufficient delocalization. Another approach to the signal transmission is based on a pure local signals, which can be, for instance, a point of discontinuity, propagating away from the perturbation [18]. For these problems the velocities of higher orders, especially 1-PV and 2-PV provide a suitable tool, as shown above. In this context are the velocities 1-PV and 2-PV of special interest, being treated in pure physical sense as a true speed of interaction. References [1] L. Brillouin Wave Propagation and Group Velocity NEW YORK (U. A.): ACADEMIC PR., 1960. - XI,154 S. , 1960 . [2] M. Born, E. Wolf Principles of Optics, Cambridge University Press, 1999 [3] G. C. Sherman, K. E. Oughstun, Phys.Rev.Lett., 47 (20), 1451-1454 (1981) [4] K. E. Oughstun, N. A. Cartwright J.Mod.Opt., 52 (8), 1089-1104 (2005) [5] M. Centini, M. Bloemer, K. Myneni, M. Scalora, C. Sibilia, M. Bertolotti, G. D’Aguanno, Phys.Rev.E, 68 016602 (2003) [6] G. Nimtz and W. Heitman, Prog.Quant.Electronics 21, (1997) [7] A. Gürtler, C. Winnewisser, H. Helm, P. U. Jepsen J.Opt.Soc.Am.A, 17 (1), 74-83 (2000) [8] T. Kohmoto, H. Tanaka, S. Furue, K. Nakayama, M. Kunimoto, Y. Fukuda, Phys.Rev. A, 72, 025802 (2005) [9] A. Bers, Am.J.Phys., 68(5), 482-484 (2000) [10] N. S. Shiren, Phys.Rev.A, 128 (5), 2103-2112 (1962) [11] R. Loudon, J.Phys.A , 3 233-245(1970) [12] T. Kohmoto, S. Furue, Y. Fukui, K. Nakayama, M. Kunimoto, Y. Fukuda, Technical Digest. Summaries of papers presented at the Quantum Electronics and Laser Science Conference. Conference Edition (IEEE Cat. No.02CH37338) (2002) Vol.1 p.241 Vol.1 of (271+40 suppl.)pp., [13] S. C. Bloch, Am.J.Phys., 45 (6), 538-549 (1976) [14] H. M. Nussenzveig, Causality and dispersion relations New York Acad. Press 1972 [15] E. Sonnenschein, I. Rutkevich, D. Censor, J.Electromagneic Waves and Appl., 14 (4), 563-565 (2000) [16] H. P. Schriemer, J. Wheeldon, T. Hall Proc.SPIE, Int.Soc.Opt.Eng.2005, 5971, 597104.1-597104.9 (2005) [17] A. Gomez, A. Vegas, M. A. Solano Microwave and Opt.Technol.Lett., 44 (3), 302-303 (2005) [18] M. D. Stenner et al., Phys.Rev.Lett.94 053902 (2005); M. D. Stenner et al., Nature 425, (2003)625 [19] I. Drozdov, A. Stahlhofen, arXiv: math-ph/0701022, to be published [20] D. K. Hoffmann, M. Arnold, D. J. Kouri J.Phys.Chem.(1993), 97(6) ,1110- [21] K. Aric, R. Gutdeutsch, A. Sailer J.Pure and Applied Geophys.(1980), 118(3), 796-806 [22] Z. Xiaoming, J. M Greenleaf Ultrasound in Med.and Biol. (Nov.2006) 32(11) 1655-60 [23] J. A. Simmons J.Acoust.Soc.Am. (1992) 92(2) 1061-90 [24] A. Gonzalez-Serrano, J. F. Claerbout Geophysics(Sept 1984) 49(9) 1432-56 [25] S. Al Rabadi, L. Friedel, A. Al Salaymeh Chem.Eng.Tec.(2007) 30(1) 71-83 [26] C. Pacher, W. Boxleitner, E. Gornik Phys.Rew.B, 71 (12), 125317-1-11 (2005) [27] I. Drozdov, A. Stahlhofen, in preparation [28] H. Bateman, Partial Differential Equations of Mathematical Physics Cam- bridge University Press, 1969 (orig.1931) p.231

0704.1486

Nodal/Antinodal Dichotomy and the Two Gaps of a Superconducting Doped Mott Insulator

Nodal/Antinodal Dichotomy and the Two Gaps of a Superconducting Doped Mott Insulator M. Civelli1, M. Capone2, A. Georges3, K. Haule4, O. Parcollet5, T. D. Stanescu6 and G. Kotliar4 1 Theory Group, Institut Laue Langevin, Grenoble, France 2 SMC, CNR-INFM, and Physics Department, University of Rome “La Sapienza”, Piazzale A. Moro 5, I-00185, Rome, Italy and ISC-CNR, Via dei Taurini 19, I-00185, Rome, Italy 3 Centre de Physique Théorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France 4 Physics Department and Center for Materials Theory, Rutgers University, Piscataway NJ USA 5 Service de Physique Théorique, CEA/DSM/SPhT-CNRS/SPM/URA 2306 CEA-Saclay, F-91191 Gif-sur-Yvette, France and 6 Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA (Dated: November 8, 2018) We study the superconducting state of the hole-doped two-dimensional Hubbard model using Cellular Dynamical Mean Field Theory, with the Lanczos method as impurity solver. In the under- doped regime, we find a natural decomposition of the one-particle (photoemission) energy-gap into two components. The gap in the nodal regions, stemming from the anomalous self-energy, decreases with decreasing doping. The antinodal gap has an additional contribution from the normal com- ponent of the self-energy, inherited from the normal-state pseudogap, and it increases as the Mott insulating phase is approached. PACS numbers: 71.10.-w,71.10.Fd,74.20.-z,74.72.-h Superconductivity in strongly correlated materials such as the high-Tc cuprates has been the subject of in- tensive research for more than twenty years (for a re- view see, e.g.,[1]). From the theoretical side, low-energy descriptions in terms of quasiparticles interacting with bosonic modes have been widely studied starting from the weak correlation limit. A different approach views the essence of the high-Tc phenomenon as deriving from doping with holes a Mott insulator [2]. The strong corre- lation viewpoint has not been yet developed into a fully quantitative theory and whether the weak- and strong- coupling pictures are qualitatively or only quantitatively different is an important open issue. The development of Dynamical Mean Field Theory (DMFT) and its cluster extensions [3] provides a new path to investigate strongly correlated systems. These methods construct a mean-field theory for Hubbard- like models using a cluster of sites embedded in a self- consistent bath. Extensive investigations have been car- ried out for intermediate interaction-strength using the Dynamical Cluster Approximation on large clusters [4]. The strong coupling limit is more difficult, as only small clusters can be employed. Many groups however have identified interesting phenomena, such as the competi- tion between superconductivity and antiferromagnetism [5], the presence of a pseudogap (PG) [6], the forma- tion of Fermi arcs [7, 8, 9, 10] and the existence of an avoided critical point [11]. In this work we use Cellu- lar DMFT (CDMFT) to explore the energy gap in the one-particle spectra of the superconducting state when correlations are strong. The goal is to identify qualita- tive aspects of the approach to the Mott transition in the light of recent experimental studies on superconducting under-doped cuprates [12, 13, 14, 15, 16], which report the presence of two distinct energy scales. We consider the two-dimensional Hubbard Model: H = − i,j,σ tij c cj,σ + U ni↑ni↓ (1) ci,σ destroys a σ-spin electron on site i, niσ = c iσciσ is the number operator and tii ≡ µ is the chemical po- tential. Only next-neighbor t and nearest-next-neighbor t′ = −0.3t hoppings are considered. The on-site repulsion is set U = 12t. We implement CDMFT on a 2×2 pla- quette. Though this is the minimal configuration allow- ing to study a d-wave superconducting state, it already presents a rich physics and we think that its deep under- standing is an essential step to be accomplished before challenging bigger clusters (hardly accessible by the com- putational methods presently available). H is mapped onto a 2×2-cluster Anderson impurity model which is solved using the Lanczos method [17]. The CDMFT self- consistency condition [18] is then enforced via the Dyson relations Σ̂(iωn) = Ĝ−1(iωn) − Ĝ−1[Σ̂](iωn), which de- termines the cluster self-energy Σ̂. The hat denotes 8×8 matrices with cluster-site indices containing both normal and anomalous components (Nambu notation). Ĝ is the ”Weiss field” describing the bath, Ĝ[Σ̂] is the one-particle cluster Green’s function [18] and ωn = (2n + 1)π/β the Matsubara frequencies, with βt = 300. The bath is de- scribed by 8 energy levels determined through a fit on the Matsubara axis (0 < ωn < 2U), which weights more the low frequencies [8]. Our main result is the presence of two energy-scales on the under-doped side of the phase diagram. We first show that this can be established directly from an anal- ysis of quantities inside the 2 × 2 cluster, which are the output of the CDMFT procedure. In the left panel of http://arxiv.org/abs/0704.1486v3 0 0.08 0.16 -0.15 0 0.15 0 0.5 1 0 0.1 12/t -Im G11/π FIG. 1: (Color online). Left: ReΣano12 vs. ωn. In the inset, the ωn → 0 value as a function of doping δ. Right: The distance from the Fermi level (ω = 0) of the left (circle) and right (square) edge-peaks in the local DOS − 1 ImG11 (see inset) are displayed as a function of δ. The dashed line is the average of the left and right values. In the inset G11 is shown for δ = 0.06 using a broadening η ∼ 7×10−3t to display poles. Fig. 1 we display the real part of the anomalous clus- ter self-energy Σano on the Matsubara axis. Only the nearest-neighbor component ReΣano (iω) is appreciably non-zero. The main observation is that at low energy (0) presents a non-monotonic behavior with dop- ing δ, as emphasized in the inset. A first characteris- tic energy-scale, measuring the superconducting contri- bution to the one-particle energy-gap, can be defined as ZnodΣ (0), where Znod is the quasiparticle spec- tral weight at the nodal k-points, where quasiparticles are well defined. As shown below, and as physically expected, Znod decreases as the doping is reduced to- wards the Mott insulator. Hence, ZnodΣ (0) decreases too due to the behavior of both Znod and Σ (0). We stress the sharp contrast of this result with resonating valence bond mean-field (RVB-MF) theories [19], where ZnodΣ (0) corresponds to the spinon pairing amplitude which is largest close to half-filling. In the right panel of Fig. 1 we show that there is actually another energy- scale, which increases when the doping level is reduced. This is revealed by looking at the local density of state (DOS) − 1 ImG11 in Ĝ[Σ̂]. In the Lanczos-CDMFT the spectral function on the real axis is obtained as a discrete set of poles (shown in the inset), which are displayed by adding a small imaginary broadening iη. We extract rel- evant energy scales by measuring the distance from the Fermi level of the gap edge-peaks. While for δ > 0.08 the spectrum is symmetric, an asymmetry appears for δ < 0.08. The total energy gap (dashed line in Fig. 1) grows with decreasing doping δ, as in RVB-MF theories. In order to make contact with experimental observ- ables it is useful to obtain momentum-resolved quantities from the local cluster quantities. For this we need a pe- riodization procedure restoring the translational invari- ance of the lattice. Several schemes have been proposed [3]. Building on previous normal-state studies [7, 9] we use a mixed scheme which is able to reconstruct the local cluster Green’s function (upon integrating over k the lat- tice Green’s function) in the nodal and antinodal points better than uniform periodization schemes. Our method is based on the idea that, when the self-energies are reg- ular the, most suitable choice is to periodize the cluster self-energy via the formula Σσ(k, ω) = e−ikµ Σµν,σ(ω) e ikν (2) (µ, ν label cluster sites). The anomalous self-energy Σano and the normal self-energy Σnor in the nodal re- gions, where we expect to find quasiparticles, are well behaved quantities, therefore we extract them through formula (2). In particular, the anomalous self-energy acquires a standard dx2−y2 -wave form: Σ ano(k, ω) = (ω) (cos kx − cos ky). On the other hand, when the self-energies develop singularities, the cluster self-energy is not a good quantity to be periodized. In Ref. [9], it has been shown that this takes place in the normal self- energy Σnor in the antinodal regions, when the system approaches the Mott insulator. In this case, a more suit- able quantity to be periodized is the the irreducible two- point cluster cumulant Mnorσ (ω) = (ω + µ)1̂− Σ̂norσ which is a more local and regular quantity. In the antin- odal region, therefore, we can apply formula (2) to Mnor, to obtain Mnor(k, ω) and finally extract the normal lat- tice self-energy Σnor(k, ω) = ω + µ− 1/Mnor(k, ω). The k-dependent Green’s function can be written as a matrix in Nambu’s space. (ω) = ω − tk − Σnorσ (k, ω) −Σano(k, ω) −Σano(k, ω) ω + tk +Σnorσ (k,−ω)∗ The imaginary part of the diagonal entry yields the spec- tral function A(k, ω) measured in photoemission. In order to compare our results with experiments, it is useful to disentangle the normal and superconducting contributions to the spectral gap. To this end, we first set Σano = 0 in Eq. (3). The results are shown in Fig. 2. The k-points along the nodal and antinodal directions are chosen as those where the highest peak is observed in A(k, ω), as done, e.g., in Ref. [13]. Their actual values are shown in the inset of panel C of Fig. 2. Near the nodal point (panel A) a quasiparticle peak is well defined at the Fermi level (ω = 0) and decreases by decreasing doping. In the antinodal region (panel B), a quasiparticle peak is also found at the Fermi level for δ > 0.08. For δ < 0.08, however, the spectral weight shifts to negative energies signaling the opening of a PG, whose size increases as δ → 0. The behaviour of the PG in the superconducting solution smoothly connects to the PG previously found in the normal state CDMFT study [9]. The approach to the Mott transition is characterized by a strong reduction in the area of the nodal spectral peak Znod, which is plot- ted in panel C (green circles). We also plot the area of -0.2 0 0.2 -0.2 0 0.2 δ=0.16 δ=0.13 δ=0.08 δ=0.05 δ=0.02 -0.2 0 0.2 0 0.08 0.16 antinodal antinodal nodal nodal (π,π)(0,π) (0,0) antinodal 0 0.08 0.16δ FIG. 2: (Color online) Spectral function A(k, ω) for different δ. Broadening η = 0.03t. Panel A: nodal quasiparticle peak; Panel B, normal component (set Σano = 0 in Eq. (3)) of the antinodal quasiparticle peaks; Panel C, nodal and antinodal quasiparticle weights. The inset shows the k-positions of the nodal and antinodal points; Panel D, spectra at the antinodes. the antinodal peak Zanod, which shows a constant value upon the opening of the PG (δ > 0.08). In panel D, we restore Σano, and examine the actual superconducting so- lution. The superconducting gap opens in the antinodal region (the nodal region is practically unaffected). For δ > 0.08 the spectra are almost symmetric around the Fermi level, as in a standard BCS d-wave superconduc- tor. In contrast, close to the Mott transition the PG, which originates from the normal component, is super- imposed to the superconducting gap, resulting in asym- metric spectra. This reveals the origin of the left/right asymmetry in the cluster DOS discussed in Fig. 1. In the nodal region the quasiparticle peaks are well defined at all dopings and we can expand the self-energies at low frequencies. The quasiparti- cle residue (1− ∂ωReΣk(ω))−1 (blue crosses in panel C of Fig. 2) numerically coincides with the area of the quasiparticle peak Znod. From Eq. (3), we get A(k, ω) ≃ Znod δ k2⊥ + v where vnod = Znod|∇k (tk − Σnor(k, 0)) | and v∆ =√ 2ZnodΣ (0)| sin knod| are the normal and anomalous velocities respectively perpendicular and parallel to the Fermi surface. v∆ physically expresses the superconduct- ing energy-scale discussed in the left panel of Fig. 1. We display them as a function of doping δ on the left side of Fig. 3. vnod does not show a special trend for δ → 0 and it stays finite, consistently with experiments [20]. The anomalous velocity, v∆ ≪ vnod presents a 0 0.05 0.1 0.15 v∆ /(aot) 0 0.05 0.1 0.15 γ/γδ=0.16 α/αδ=0.16 FIG. 3: (Color online) Left: vnod and v∆ as a function of doping δ (ao is the lattice spacing). Right: low-frequency co- efficients of local DOS γ and of the Raman B2g and superfluid density response α, renormalized by the value at δ = 0.16. dome-like shape. This behavior (confirmed by continuous time quantum Monte Carlo (CTQMC) calculations [21]) is in agreement with recent experiments on under-doped cuprates showing that, contrary to the antinodal gap, the nodal gap decreases by reducing doping [12, 13, 14]. The low-energy behaviour of several physical observ- ables in the superconducting state is controlled by nodal- quasiparticle properties and hence can be related to vnod, v∆ and Znod. Two specific ratios are particu- larly significant, namely: γ = Znod/(vnodv∆) and α = /(vnodv∆). The first one is associated with the low- energy behaviour of the local DOS measured in tun- neling experiments: N(ω) = A(k, ω) ∼ γ ω (for ω → 0). Neglecting vertex corrections [12], the sec- ond one determines the low-energy B2g Raman response function χ′′(ω) ∝ αω and the low-temperature (T → 0) behaviour of the penetration depth (superfluid density) ρs(T ) − ρs(0) ∝ αT . We display α and γ in the right panel of Fig. 3 as a function of δ. α is monotonically decreasing (see also CTQMC results [21]) and, on the under-doped side δ < 0.08, it saturates to a constant value, in agreement with Raman spectroscopy [12] and penetration depth measurements [22]. Also γ neatly de- creases in going from the over-doped to the under-doped side, but it presents a weak upturn for low doping. The low-frequency linear behavior of N(ω) is well established in scanning tunneling experiments on the cuprates [23]. However, it is not currently possible to determine the ab- solute values of the tunneling slope α from experiments, hence the behavior we find is a theoretical prediction. We finally turn to the one-electron spectra in the antin- odal region, shown in Fig. 4, physically interpreting the cluster energy-scales observed in Fig. 1. We evaluate the antinodal gap in the superconducting state ∆tot by measuring the distance from the Fermi level (ω = 0) at which spectral peaks are located (panel D of Fig. 2). ∆tot monotonically increases by reducing doping, as observed in experiments. The data of panel B in Fig. 2, where Σano = 0, allow us to extract the normal contribution ∆nor. We notice that the peaks found there at negative 0.05 0.1 0.15 /t= ( ∆ /t= Z FIG. 4: (Color online) Antinodal energy gap ∆tot (circles), obtained from the spectra of panel D in Fig. 2, as a func- tion of doping δ, and decomposed in a normal contribution ∆nor (squares), obtained from panel B in Fig. 2, and in a superconducting contribution ∆sc (diamonds). frequency ωpg do not represent Landau quasiparticles in a strict sense, but we can estimate the PG as |ωpg|. We also display the anomalous contribution to the antinodal gap ∆sc = ∆2tot −∆2nor, and find that, within numerical precision, ∆sc ≃ Zanod|Σano(kanod, ωpg)|. The appear- ance of ∆nor signs a downturn in ∆sc. We interpret ∆tot as the monotonically increasing antinodal gap observed in cuprate superconductors, while the superconducting gap ∆sc, detectable as the nodal-slope v∆ (Fig. 3), is decreasing in approaching the Mott transition. The concept of two energy gaps with distinct doping dependence in the cuprates has recently been brought into focus from an analysis of Raman spectroscopy [12], and photoemission experiments [13, 14], which have re- vived experimental and theoretical debate [16]. Our the- oretical dynamical mean-field study of superconductivity near the Mott transition establishes the remarkable co- existence of a superconducting gap, stemming from the anomalous self-energy, with a PG stemming from the normal self-energy. This is reminiscent of slave-boson RVB-MF of the t − J model [19, 24], which uses or- der parameters defined on a link and includes the pos- sibility of pairing in both the particle-particle and the particle-hole channels. Compared to the self-energy of the RVB-MF, the CDMFT lattice-self-energy has con- siderably stronger variations on the Fermi surface [9] and additional frequency dependence, which makes the elec- tron states near the antinodes very incoherent even in the superconducting state. Furthermore, in the RVB-MF theory the anomalous self-energy monotonically increases by decreasing doping, in contrast to our CDMFT results which reveal a second energy scale associated with su- perconductivity, distinct from the PG, which decreases with decreasing doping. Whether this feature survives in larger clusters, representing a property of the real ground-state, or it requires some further ingredient to be stabilized against competing instabilities (above all antiferromagnetism at low doping [5]) remains an impor- tant open question addressed to future developments. We think however that the assumption of a d-wave super- conducting ground-state is a reasonable starting point, and the importance of our 2×2-plaquette-CDMFT result stands in the natural explanation it provides of the prop- erties of under-doped cuprates. We thank E.Kats, P.Nozières, P.Phillips, C.Castellani, A.-M. Tremblay, B.Kyung, S.S. Kancharla, A. Sacuto and M. Le Tacon for useful discussions. M.Ca. was sup- ported by MIUR PRIN05 Prot. 200522492, G.K. by the NSF under Grant No. DMR 0528969. [1] ”Physics of Superconductors II”, K.H. Bennemann and J.B. Ketterson, Springer-Verlag Berlin (2004). [2] P.W. Anderson, Science 235, 1196 (1987). [3] For reviews see: A. Georges et al., Rev. of Mod. Phys. 68, 13 (1996); A.-M.S. Tremblay et al., Low Temp. Phys., 32, 424 (2006); Th. Maier et al., Rev. of Mod. Phys. 77, 1027-1080 (2005); G. Kotliar et al., Rev. of Mod. Phys. 78, 865 (2006). [4] T.A. Maier et al., Phys. Rev. Lett. 95, 237001 (2005); T.A. Maier, M. Jarrell and D.J. 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Phys. 2, 537 (2006); [13] K. Tanaka et al., Science 314, 1910 (2006). [14] T. Kondo et al., Phys. Rev. Lett 98, 267004 (2007). [15] K.K. Gomes et al., Nature 447, 569 (2007) [16] For earlier discussions of related ideas see: G. Deutscher, Nature 397, 410 (1999); B. Kyung and A.-M.S. Trem- blay, cond-mat/0204500; For more recent discussions, see: Kai-Yu Yang, T.M. Rice and F.C. Zhang, Phys. Rev. B 73, 174501 (2006); A.J. Millis, Science 314, 1888 (2006); A. Cho, Science 314, 1072 (2006); S. Huefner et al., arXiv:0706.4282, B. Valenzuela and E. Bascones, Phys. Rev. Lett. 98, 227002 (2007) and M. Aichhorn et al., Phys. Rev. Lett. 99, 257002 (2007). [17] M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545 (1994). http://arxiv.org/abs/cond-mat/0204500 http://arxiv.org/abs/0706.4282 [18] S.S. Kancharla et al., cond-mat/0508205. [19] See e.g. P.A. Lee et al., Rev. Mod. Phys. 78, 17 (2006). [20] X.J. Zhou et al., Nature 423, 398 (2003). [21] K. Haule and G. Kotliar, Phys. Rev. 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0704.1487

Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions

Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions Lúıs Daniel Abreu Abstract. The Fourier transforms of Laguerre functions play the same canon- ical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by Gröchenig and Lyubarskii in Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007). Building on Seip´s work Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993), concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames by translations and dila- tions of the Fourier transform of the nth Laguerre function. As in Gröchenig- Lyubarskii theorem, the density increases with n, and in the special case of the hyperbolic lattice in the upper half plane it is given by b log a < 2n+ α where α is the parameter of the Laguerre function. 1. Introduction The ”frame density problem” is one of the fundamental questions in applied harmonic analysis: how coarse we need to sample a continuous object so that the resulting discrete object is a frame in a certain Hilbert space? This question was sharply solved for frames in the Bargmann-Fock [8], [24], [25] and in the Bergmann space [28]. However, in the cases of general wavelet and Gabor transforms, very little is known and only a few very special windows and analysing wavelets are understood. The short time Fourier (Gabor) transform with respect to a gaussian window can be written in terms of the Bargmann transform, mapping isometrically the space L2(R) onto the Bargmann-Fock space of entire functions. This is the reason why everything is known about the geometry of sequences that generate frames by sampling the Gabor transform with gaussian windows g(t) = e−πt . Apart from this example, the only cases where a description is known of the lattice sequences that generate frames are the hyperbolic secant g(t) = (coshat)−1 [18] and the characteristic function of an interval [19], which turned out to be a nontrivial problem. There is also a necessary density condition for Gabor frames due to Ramanathan and Steger [23] and it was recently observed that the frame property is stable under small perturbations of the lattice, given the window belongs to the Feichtinger algebra [10]. http://arxiv.org/abs/0704.1487v1 2 LUÍS DANIEL ABREU If we consider the wavelet case, things appear to be even more mysterious. Al- though the density and irregular grid problems have attracted some attention lately (see [1], [31], [22], [15] and references there in), there is no known counterpart of Ramanathan and Steeger theorem (there is, however, a version of the HAP property for wavelet systems [16]) and the only information available so far, on particular systems, concerns a special family of analyzing wavelets that maps the problem into Bergmann spaces: The wavelet transform, with positive dilation parameter, with respect to a wavelet of the form (Paul´s Wavelet in some literature) (1.1) ψα(t) = can be rescaled as an isometrical integral transform between spaces of analytical functions, namely, between the Hardy space on the upper half plane H2(U) and the Bergman space in the upper half plane A2α+1(U). It is natural to refer to this isomorphism as the Bergman transform (the designation analytic wavelet is also frequently used). Since the upper half plane can be mapped isomorphically onto the unit disc by using linear fractional transformations, we can construct a transform mapping the Hardy space onto the Bergman space in the unit disc. The approaches used to deal with the special situations mentioned in the above paragraphs are based in techniques which differ from case to case. It seems highly desirable to follow a more structured approach. The natural place to look for this structure is within the context of Hilbert spaces of analytic functions, where the powerful methods from complex analysis may answer questions that seem hopeless otherwise. Following this line of reasoning implies using windows that allow to carry the problem to such spaces. In this direction a major step was taken recently by Göchenig and Lyubarskii [12], by considering Gabor systems with Hermite func- tions of order n as windows. They have proved that if the size of the lattice Λ is < (n + 1)−1 then the referred Gabor system is a frame and provided an example supporting their conjecture that the result is sharp. In the Bargmann-Fock setting the Hermite functions play a very special role [11, pag. 57]. They are, up to normalization constants, pre-images, under the Bargmann transform, of the monomials {zn} and since the latter constitute an orthogonal basis of the Bargmann-Fock space, their pre-images also constitute an orthogonal basis of L2(R). That is, to the isomorphism (1.2) L2(R) B→ F 2(C) corresponds (1.3) hn B→ cnzn where B is the Bargmann transform, F 2(C) is the Bargmann-Fock space, hn are the Hermite functions and cn some constants dependent on n. They are canonical to time-frequency analysis in an additional sense, since they constitute the eigenfunc- tions of the time-frequency-localization operator with Gaussian window [3]. The Hermite functions are eigenfunctions of the Fourier transform and they can be used [20] to describe Feichtinger´s algebra So. Wavelet and Gabor analysis share many similarities and many of their struc- tural aspects can be bound together in a more general theory using representations of locally compact abelian groups [13], [9]. In looking for a Wavelet-analogue of Gröchenig-Lyubarskii structured approach to the density problem, we must first WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 3 clarify what functions should be used instead of the Hermite functions. In analogy to the above paragraph it is natural to consider the pre-images, under the Bergman transform, of the monomials {zn} in the unit disc (up to an isomorphism with the upper half plane). Since the {zn} form a basis of the weighted Bergman spaces in the unit disc (independently of the weight), their pre-images must be an orthogonal basis of the Hardy space H2(U) and it is reasonable to expect that such functions will play a similar role in Wavelet analysis as do the Hermite functions in Gabor analysis. Such functions are the Fourier transforms of the Laguerre functions and they will be ad-hoc denoted by Sαn . Additional structural evidence that the functions Sαn are canonical in Wavelet analysis comes from the work of Daubechies and Paul [4], where it is shown that they are the eigenfunctions of a differential operator that commutes with a time– scale localization operator, once windows of the form (1.1) are chosen. This com- pletely parallels the situation with the time-frequency-localization operators with gaussian windows, which commute with the harmonic oscillator and therefore have as eigenfunctions the Hermite functions [3]. It was observed by Seip [26] that these problems have a more natural formulation when mapped into the convenient spaces of analytical functions. 2. Description of the results There exists an analogue of the correspondence between (1.2) and (1.3), but involving four different functional spaces and the corresponding bases: To the se- quence of isomorphisms between the Hilbert spaces (2.1) L2(0,∞) F←→ H2(U) Ber → A2α+1(U) Tα→ A2α+1(D), where A2α+1(U) and A2α+1(D) are the weighted Bergmann spaces in the upper- half plane and in the unit disc, respectively, corresponds the relations between the basis of the respective spaces: (2.2) lαn F←→ Sαn Berα→ cαnΨαn Tα→ cαnzn. for some constants cαn, where l n is the nth Laguerre function of order n and pa- rameter α, Sαn is its Fourier transform and Ψ n is a basis of A2α+1(U) to be defined in section 4. This correspondence was implicit in the conection between papers [4] and [26] but since it was not stated explicitly, we devote section 4 to clarify how exactly does it work. The functions Sαn were computed recently in closed form [29], but the connec- tion to the Wavelet transform seems to have been unnoticed. It is also shown in [29] that Sαn are, up to a fractional transformation, defined in terms of a certain system of orthogonal polynomials on the unit circle. We will show that this family of polynomials is nothing more but the circular Jacobi orthogonal polynomials for which there is, for computational purposes, a very convenient three term recurrence formula. This connection will make our case to call the functions Sαn the rational Jacobi orthogonal functions. With a computational method available for evaluating the functions Sαn and graphic evidence of their good localization properties (see for example the plots of their real versions in page 44 of [2]), it is natural to investigate how to use such func- tions as analysing wavelets and to obtain frames from the resulting discretization. 4 LUÍS DANIEL ABREU We will obtain the following sufficient condition on the density of the parameters of the hyperbolic lattice {(ajbk, aj)}j,k∈Z: b log a < 2n+ α that is, as in Gröchenig and Lyubarskii result [12], this density increases with n. This will follow as a special case of the following more general theorem, which uses Seip´s notion of lower Beurling density in the unit disk and constitutes the main result in this paper: Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform. If D−(Γ) > n+ α then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H 2(U). Case α = 0 gives the sufficient part of theorem 7.1 in [28]. The proof of theorem 1 will be dramatically simplified with the observation of the following remarkable ocurrence: There exist constants {C} and sequences of numbers {ak}, both de- pending on α and n, such that (2.3) Sαn ( ) = C akψk+α This will allow to obtain a formula which expresses the wavelet transform with window Sαn as a linear combination of simple but non-analytic functions. As a result we will have to deal with a situation that is reminiscent of the non-analiticity problem that Gröchenig and Lyubarskii faced with formula (15) of [12] and which they solved by using the so called Wexler-Raz identities [11]. Since there is no known analogue of the Wexler-Raz identities in wavelet analysis, we have to deal with this in a more direct way. The key idea is to map the problem into the unit circle and use a deep sampling formula proved by Seip in [28]. Althought this approach lacks the simplicity of [12] it gives the result for more general sequences, while the method in [12] works only for the lattice case due to the restritions imposed by the use of the Wexler-Raz identities. We organize our ideas as follows. The next section contains the main definitions and facts concerning wavelet transforms, Bergman spaces and Laguerre functions. Section four contains the evaluation of the pre-images required to build up the correspondences (2.1) and (2.2). The fifth section contains our main results on wavelet frames with Fourier transforms of Laguerre functions. We conclude the paper collecting some further properties of the functions Sαn and clarifying their classification within known families of orthogonal polynomials. 3. Tools 3.1. The Bergman transform. Now we present a sinthesis of ideas that appeared in the section 3.2 of [13] and in [4] (see also [2, pag. 31]) Here they will be exposed in such a way that the role of the Bergman spaces is emphasized. Consider the dilation and translation operators Dsf(x) = |s|− 2 f(s−1x) Txf(t) = f(t− x) and define ψx,s(t) = TxDsψ(t) = |s|− WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 5 The wavelet transform of a function f , with respect to the wavelet ψ is Wψf(x, s) = 〈f, TxDsψ(t)〉L2(R) = f(t)ψx,s(t)dt. A function ψ ∈ L2(R) is said to be admissible if |Fψ(s)|2 ds where K is a constant. If ψ is admissible, then for all f ∈ L2(R) we have (3.1) s−2 |Wψf(x, s)|2 dxds = K ‖f‖2 We will restrict ourselves to parameters s > 0 and functions f ∈ H2(U), where H2(U) is the Hardy space in the upper half plane U = {z = x+ is : s > 0} H2(U) = {f : f is analytic in U and sup 0<s<∞ |f(x+ is)|2 dx <∞}. Let F denote the Fourier transform (Ff)(t) = 1√ e−itxf(x)dx By the Paley-Wiener theorem, H2(U) is constituted by the functions whose Fourier transform is supported in (0,+∞) and belongs to L2(0,∞). Now take the special window ψα(t) defined in (1.1). Since (3.2) Fψ−x,sα (t) = 1[0,∞]sα+ 2 tαei(x+is)t (3.3) Wψαf(−x, s) = s tαeizt(Ff)(t)dt where the function defined by the integral is analytic in z = x + si. The identity (3.1) gives (3.4) ∣Wψαf(−x, s) s−2dxds = ‖f‖H2(R) This motivates the definition of the Bergman transform, or the analytic wavelet transform: (3.5) Berα f(z) = tαeizt(Ff)(t)dt = s−α− 12Wψαf(−x, s), where z = x+ is (see for instance [15], where the authors use this Bergman trans- form with the same normalization in the case α = 1). Introducing the scale of weighted Bergman spaces Aα(U) = {f analytic in U such that |f(z)|2 sα−2dxds <∞}, it is clear from (3.4) and (3.5) that Berα f(z) ∈ A2α+1(U). We have therefore an isometric transformation Berα : H2(R)→ A2α+1(U) 6 LUÍS DANIEL ABREU The weighted Bergman spaces in the unit disc are denoted by Aα(D) and defined Aα(D) = f analytic in D such that |f(z)|2 (1− |z|)α−2dxdy <∞ For detailed expositions of the theory of Bergman spaces we point out the mono- graphs [32], [6], [14] 3.2. Frames and sampling sequences. A sequence of functions {ej} is said to be a frame in a Hilbert space H if there exist constants A and B such that (3.6) A ‖f‖2 ≤ |〈f, ej〉|2 ≤ B ‖f‖2 . In spaces of analytic functions a related concept to frames is the one of a sampling sequence. A set Γ = {zj} is said to be a sampling sequence for the Bergman space Aα(U) if there exist positive constants A and B such that (3.7) A |f(z)|2 yα−2dxds ≤ |f(zj)|2 yαj ≤ B |f(z)|2 yα−2dxds. The corresponding definition of a sampling sequence for the Bergman space on the unit disc is |f(z)|2 (1− |z|2)α−2dz ≤ |f(zj)|2 (1− |z|2)α |f(z)|2 (1− |z|2)α−2dz. Define the pseudohyperbolic metric on the unit disk by ̺(z, ζ) = z − ζ 1− ζz Following [28], a sequence Γ = {zj} ⊂ D is separated if infn6=j zj−zn zj−zn > 0 and its lower density D− is given by D−(Γ) = lim inf inf ̺(zj ,z)<r (1− ̺(zj , z)) log 1 The next Theorem is from [28]: Theorem A A separated sequence Γ ⊂ D is a sampling sequence for Aα(D) iff D−(Γ) > α It is a generalization of an earlier lattice result [27]: Theorem B Let Γ(a, b) = {zjk}j,k∈Z, where zjk = aj(bk + i). Γ(a, b) is a sampling sequence for Aα(U) if and only if b log a < 3.3. Fourier transforms of Laguerre functions. The Laguerre polynomi- als will play a central role in our discussion. One way to define them is by means of the Rodrigues formula (3.8) Lαn(x) = exx−α e−xxα+n WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 7 and this gives, in power series form Lαn(x) = (α + 1)n (−n)k (α+ 1)k For information on specific systems of orthogonal polynomials, we suggest [17]. The Laguerre functions are defined as lαn(x) = 1[0,∞](x)e −x/2xα/2Lαn(x) and they are known to constitute an orthogonal basis for the space L2(0,∞). By the Paley-Wiener theorem, the Fourier transform is an isomorphism between H2(U) and L2(0,∞). Therefore, the Fourier transform of the functions lαn form an orthog- onal basis for the space H2(U). From now on we will set (3.9) FSαn (t) = lαn(t) The functions Sαn (t) can be evaluated explictly using (3.8). This was already done by Shen in [29], where the author was interested in describing the orthogonal poly- nomials which arise from an application of the Fourier transform on the Laguerre polynomials. A description of this method of generating new families of orthogonal polynomials with an interesting historical account is in the paper [21] where a link is provided between Jacobi and Meixner-Polaczek polynomials. Following [29], we (3.10) Sαn (t) = (1 + α)n (−n)k(α2 + 1)k k!(α+ 1)k 4. Bergman transform of Sαn (t) In this section the correspondences (2.1) and (2.2) will be established via a direct calculation. The complex linear fractional transformations will play an im- portant role, in a style that is reminiscent of the way they are used in the discrete series representation of SL(2,R) over the Bergman space [7, chapter IX]. We first define a set of functions that, as we shall see later, constitute a basis of Aα(U). For every n ≥ 0 and α > −1 let Ψαn(z) = iz + 1 iz − 1 )−α−1 Now define a map Tα such that for every function f ∈ Aα(U) the action of Tα (4.1) Tαf(w) = f w + 1 w − 1 The range space of Tα is a weighted Bergman space in the unit disc. Lemma 4.1 The map Tα : Aα(U)→ Aα(D) is an unitary isometry between Hilbert spaces. Proof. Argue as in the proof of lemma 1 in [7, pag.185]. � Proposition 4.1 For n = 0, 1, ..., the following relations hold: (4.2) Berα(S2αn ) = c n , with c n = (−1)α+1(2α+ n)!/n! 8 LUÍS DANIEL ABREU and, for |z| < 1, (4.3) Tα(Ψ n) = z In other words, the function S2αn is the pre-image, under Tα ◦ , of zn ∈ Aα(D). Proof. Using the definition of the Bergman transform (3.5) and (3.9) we have Berα S2αn (z) = tαeizt(FS2αn (t))(t)dt e(iz− )tt2αL2αn (t)dt e−xx2α+n (−iz + 1 = cαnΨ n (z), where in the last two identities we are applying Rodrigues formula (3.8) for the Laguerre polynomials and writing the integral in terms of the Laplace transform L, whose well known properties establish the last identity. Now, the linear fractional transformation iz + 1 iz − 1 is an analytic isomorphism between the upper half plane and the unit circle. Since the inverse of this transformation is given by w + 1 w − 1 a short calculation with the definition of Tα gives(4.3). � Remark 1. {Ψαn(z)} is a basis of Aα(U) and the map Tα is an unitary isomorphism between Aα(U) and Aα(D). Indeed, since {zn} is an orthogonal basis of the space Aα(D) [7, pag. 186] and it is contained in the range of Tα, Tα is onto and therefore an unitary isomorphism.The functions {Ψαn(z)} form a basis for the space Aα(U) since they are the pre-images of the basis {zn}. As a consequence we obtain a new proof of the (known) isomorphic property of the Bergman transform. Corollary 1. The transform Berα : H2(R)→ A2α+1(U) is an isometric isomor- phism. Proof. The isometry is a consequence of the isometric property of the wavelet transform, so we need only to prove that the Ber 2 is onto. But in view of the preceding remark and Theorem 5.1, the range of Ber 2 contains a basis of Aα(U). Therefore, Ber 2 is onto. � Remark 2. Similar calculations as we have seen here also play a role in [5], in the context of Laplace transformations and group representations and in [4], to obtain an explicit formula for the eigenvalues of the time-scale localization operator. WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 9 5. Wavelet frames with Fourier transforms of Laguerre functions 5.1. Preparation. We wish to construct wavelet frames with analysing wavelets Sαn , by using the common discretization of the continuous wavelet transform via the hyperbolic lattice. Discretizing the scale parameter s in the Wavelet transform by a sequence aj and the parameter x by ajbm gives (5.1) Wψf(a jbm, aj) = 〈f, TajbmDajψ〉 . We want to know conditions in a and b under which {TajbkDajψ} is a frame, for a given wavelet ψ. More generally we can consider the non lattice problem replacing (ajbm, aj) by more sequences (ajm, bjm). Remark 3. Observe that it follows immeadiately from Theorem A and Theorem B that if Γ ⊂ D is the sequence obtained from mapping the sequence {zk,j = aj(bk + i)} ⊂ U into the unit disk via a Cayley transform, then D−(Γ) = 2π b log a (to move between the notation of [28] and [27] set α = 2n+ 1). Remark 4. Observe that {TajbmDajψα}j,k∈Z is a frame of H2(U) iff Γ(a, b) = {z = ajbm+ aji}j,k∈Z is a set of sampling for A2α+1(U). Indeed, since functions in A2α+1(U) can be identified with Ber α transforms of H2(U) functions, it follows from (3.7) and (??) that Γ(a, b) is a set of sampling for A2α+1(U) if and only if ∣Wψαf(x, s) 2 dxds ∣Wψαf(a jbm, aj) 2 ≤ B ∣Wψαf(x, s) 2 dxds Using (3.4) this is equivalent to A ‖f‖2H2(U) ≤ |〈f, TajbmDajψα〉| 2 ≤ B ‖f‖2H2(U) which says that {TajbmDajψα}j,k is a frame in H2(U). We now collect the required preliminary lemmas for the proof of the main result. The first was quoted in the introduction and is just a simple modification of the representation (3.10). The second appeared in [27] (see [6, pag. 160] for a proof) and the third one is a quite deep sampling theorem which is formula (30) in [28] with s = 0 and ǫ = 0 (see also [6, pag. 216] for a direct derivation with this choice of parameters). Lemma 5.2 The functions Sαn can written as the linear combination (2.3) of analyzing wavelets ψk+α (t) defined by Sαn ( ) = C akψk+α where the constants C and the coeficients ak given as C = Cα,n = (1 + α)n ak = a k = (2i) (−n)k(α2 + 1)k k!(α+ 1)k Lemma 5.3 For 1 < s < t we have (5.2) (1− |zj |2)s ∣1− ζzj ≤ C(1− |ζ|2)s−t 10 LUÍS DANIEL ABREU for any separated sequence {zj}. Lemma 5.4 Let D({wj}) > β. Then every f ∈ A2β+1(D) satisfies (5.3) (1− |w|2)βf(w) = (1 − |wj |2)βf(wj)hj(w), with the estimate for hj(w) (5.4) |hj(w)| ≤ C (1− |w|2)(1 − |wj |2) |1− wjw|2 5.2. Main result. To prove our main result we will adhere to the following plan: we begin by deriving from Lemma 5.2 a formula expressing the wavelet trans- form with analysing wavelet Sαn in terms of the wavelet transform with analysing wavelet ψk+α . The proof of the right frame inequality is then straightforward (as usual in frame inequalities). To prove the left hand inequality we first rewrite the formula in terms of complex functions defined on the upper half plane and then map it into the unit disk by means of a Cayley transform. Manipulating the resulting formula and using lemma 5.3 and lemma 5.4 gives an estimate in the unit disk. We proceed backwards to the half plane to end up with the required estimate in terms of wavelet frames. Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform. If D−(Γ) > n+ α then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H 2(U). Proof. The definition of the wavelet transform gives, using (2.3), )f(−x, s) = f, T−xDsS f, T−xDsψk+α akWψk+α f(−x, s). The right frame inequality is now easy: observe that, since {Tam,jDbm,jψk+α is a frame, then for every k = 1, ..., n there exists a Bk such that f, Tam,jDbm,jψk+α ≤ Bk ‖f‖2H2(U) . Let Bn = max1≤k≤nBk. Then, f, Tam,jDbm,jS f, Tam,jDbm,jψk+α |ak|2 f, Tam,jDbm,jψk+α ≤ B ‖f‖2 , where B = BnC k=0 |ak| and we are done. WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 11 Now the left frame inequality. Define the function F (z) as (5.5) F (z) =WSα )f(−x, s) = C 2Berk+ 2 f(z) where z = x+is belongs to the upper half plane U. We construct a related function in the unit disc, by setting w = z−i G(w) = C ak(1− |w|2)k+ 2Gk+α where G(w) = F (i 1 + w (w) = 2k+α+1 Berk+ 2 f(i 1 + w ) ∈ A2k+α+1(D). Therefore, if D({wj}) > n+ α2 we can use (5.3) to write, for every k ≤ n, (1 − |w|2)k+ 2 Gk+α (w) = (1 − |wj |2)k+ 2 Gk+α (wj)hj(w). As a result, (1 − |w|2)− 12G(w) = Cα,n ak(1− |w|2)k+ 2 Gk+α (1− |wj |2)k+ 2 Gk+α (wj)hj(w) hj(w)C ak(1− |wj |2)k+ 2 Gk+α hj(w)(1 − |wj |2)− 2G(wj). Now use estimate (5.4), then Cauchy-Schwarz and finally (5.2) in the following way. (1− |w|2)−1 |G(w)|2 ≤ (1 − |w|2) |1− wjw|2 (1− |wj |2) 2G(wj) (1 − |w|2)2 |G(wj)|2 (1− |wj |2)1 |1− wjw|4 (1 − |w|2)−1 |G(wj)|2 . Thus the factor (1− |w|2)−1 cancels and we have |G(w)|2 ≤ C |G(wj)|2 . 12 LUÍS DANIEL ABREU This last inequality gives at once |G(w)|2 (1− |w|2)−2dw ≤ C |G(wj)|2 or, in the upper half plane, |F (z)|2 s−2dxds ≤ |F (zj)|2 . In the wavelet notation this is )f(−x, s) s−2dxds ≤ jbm, aj) Taking into account that Sαn ( ) is admissible, we can apply (3.1) and (5.1) to write the above inequality as (5.6) A ‖f‖2 ≤ f, Tam,jDbm,jS This is the left frame inequality. The sequence {Tam,jDbm,jSαn ( t2 )}j,m is thus a frame. � In view of remark 5.1, the lattice result is a special case of our main theorem. Corollary 2. If b log a < 4π , then {TajbmDajSαn ( t2 )}j,m is a frame of H 2(U). Remark 5. Theorem 5.1 parallels theorem 3.1 in [12], which states that, in the lattice case of the time-frequency plane, if the density of the lattice Λ is > n + 1 (or if the size of Λ is < (n + 1)−1) then the Gabor system {e2πiλ2tHn(t − λ1) : λ = (λ1, λ2) ∈ Λ}, where Hn stands for the Hermite function of order n, is a frame for L2(R). In particular, if one is dealing with the Von Neumann lattice with parameters a and b, the condition is ab < (n+1)−1. Corollary 5.2 makes this analogy even more explicit. Remark 6. It is interesting to notice that, since Berα f(z) = tαeizt(Ff)(t)dt, we clearly have ikBerk+ 2 f(z) = 2 f(z). This allows to rewrite (5.5) as )f(−x, s) = C 2 f(z) which is, in some sense, reminiscent of Proposition 3.2 in [12]. Remark 7. Observe that combining the Paley-Wiener with the Plancherel theorem, we have ‖f‖H2(U) = ‖f‖L2(0,∞) and the results of Theorem 1 and 2 say also that we have a frame for L2(0,∞). WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 13 6. Further properties of the functions Sαn (t) The notation F (a, b; c;x) = (a)k(b)k k!(c)k for the hypergeometric function is used in this section. Rewriting Sαn in this notation gives Sαn (t) = C F (−n, α + 1;α+ 1; (observe that the infinite sum becomes a polynomial of order n, since (−n)k = 0 if k > 0). Composing the functions Sαn (t) with the fractional linear transformation (6.1) z = 2t− i 2t+ i the result is (6.2) Sαn (t) = Γ (1 − z)α2 +1gαn(z) where gαn(z) = (α/2)n F (−n, α + 1;−n− α/2 + 1; z) is a polynomial in z of degree n. This was pointed out in [29]. It was also shown that these polynomials satisfy the orthogonality (6.3) |z|=1 gαn(z)g n(z) |1− z| = 0 if m 6= n and therefore are orthogonal on the unit circle with respect to the weight (6.4) w(z) = sinα This fact implies many properties, since there exists a very rich theory for orthogo- nal polynomials on the unit circle (see [30] and references therein and also chapter 8 of [17]). For example, the general theory assures that all the zeros of gαn(z) lay within the unit disc. Remark 8. Setting a = α in Example 8.2.5 at [17], and using the identity 1−eiθ = 4 sin2 θ to write the measure (8.2.21) as (6.4) we recognize that the polynomials gαn(z) are, up to a normalization, a family of orthogonal polynomials on the unit circle known as the circular Jacobi orthogonal polynomials. Remark 9. From (6.3) the functions z− 2 (1−z)α2 gαn(z) are orthogonal on the circle and form a basis of the Hardy space on the unit disc H2(D) = {f : f is analytic in D and sup ∣f(reit) dt <∞} it is also clear that Sαn (t) are orthogonal on the real line (the boundary of the upper half place). Since gαn(z) are the circular Jacobi orthogonal polynomials, the basis functions z− 2 (1 − z)α2 gαn(z) are the circular Jacobi orthogonal functions and it is therefore natural to call Sαn (t) the rational Jacobi orthogonal functions. 14 LUÍS DANIEL ABREU Remark 10. From the general theory [17, (8.2.10)] follows that, if κn is the leading coefficient of the polynomial, then the sequence of polynomials {gαn(z)} satisfies a three term recurrence relation n(0)g n+1(z) + κn−1g n+1(0)zg n−1(z) = [κng n+1(0) + κn+1g n(0)z]g where κn is the leading coefficient of the polynomial. From the explicit representa- tion of the polynomials gαn(z) it is easily seen that + 1)n ; φn(0) = + 1)n−1 This three term recurrence relation provides a effective method for computational purposes: To evaluate the functions Sαn (t) it is sufficient to combine this recurrence relation with formulas (6.1) and (6.2). References [1] A. Aldroubi, C. Cabrelli, U. M., Molter Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L2(Rd), Appl. Comput. Harmon. Anal. 17, no. 2, 119–140 (2004). [2] I. Daubechies, ”Ten lectures on wavelets”, CBMS-NSF Regional conference series in applied mathematics (1992). [3] I. Daubechies, Time-frequency localization operators: A geometric phase space approach IEEE Trans. Inform. Theory, 34, pp. 605-612. (1988). [4] I. Daubechies, T. Paul, Time-frequency localization operators: A geometric phase space ap- proach II. The use of dilations and translations. Inverse Prob., pp 661-680. (1988). [5] M. Davidson, G. Ólafsson, G. Zhang, Laguerre polynomials, restriction principle, and holo- morphic representations of SL(2;R). Acta Appl. Math. 71 , no. 3, 261–277 (2002). [6] P. Duren, A. Schuster, ”Bergman spaces”, Mathematical Surveys and Monographs, 100. American Mathematical Society, Providence, RI, (2004). [7] S. Lang, ”SL(2,R)”, Springer (1985). [8] Y. Lyubarskii, Frames in the Bargmann space of entire functions, Entire and subharmonic functions, 167-180, Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI (1992) [9] H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (2), 307-340 (1989). [10] H. G. Feichtinger, N. Kaiblinger, Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc. 356 (2004), no. 5, 2001–2023 [11] K. Gröchenig, ”Foundations of time-frequency analysis”, Birkhäuser, Boston, (2001) [12] K. Gröchenig, Y. Lyubarskii, Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007). [13] A. Grossman, J. Morlet, T. Paul, Transforms associated to square integrable group represen- tations II: examples, Ann. Inst. H. Poincaré Phys. Théor., 45, 293-309, (1986). [14] H. Hedenmalm, B. Korenblum, K. Zhu, ”Theory of Bergman spaces.” Graduate Texts in Mathematics, 199. Springer-Verlag, New York, (2000). [15] C. Heil, G. Kutyniok, Density of wavelet frames, J. Geom. Anal. 13, no. 3, 480-493. (2003). [16] C. Heil, G. Kutyniok, The Homogeneous Approximation Property for wavelet frames , J. Approx. Theory., online first. [17] M. E. H. Ismail, ”Classical and quantum orthogonal polynomials in one variable”, Encyclo- pedia of Mathematics and its Applications No. 98. (2005). [18] A. J. E. M. Janssen; T. Strohmer, Hyperbolic secants yield Gabor frames. Appl. Comput. Harmon. Anal. 12 , no. 2, 259–267 (2002). [19] A. J. E. M. Janssen, Zak transforms with few zeros and the tie, in ”Advances in Gabor Analysis” (H.G. Feichtinger, T.Strohmer, eds.), Boston, 2003, pp. 31-70. [20] A. J. E. M. Janssen, Hermite function description of Feichtinger´s Space S0, J. Fourier Anal. Appl., 11 (5) 577-588 (2005). [21] H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124, no. 3, 887-898 (1996). WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 15 [22] G. Kutyniok, Affine density, frame bounds, and the admissibility condition for wavelet frames. Constr. Approx., 25, no. 3, 239-253 , (2006) [23] J. Ramanathan, T. Steger, Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2 , no. 2, 148–153 (1995). [24] K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math. 429, 91-106 (1992). [25] K. Seip, R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math. 429 (1992), 107-113. [26] K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces, SIAM J. Math. Anal. 22, No. 3 pp. 856-876 (1991). [27] K. Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. 117, no. 1, 213-220 (1993). [28] K. Seip, Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993). [29] L. C. Shen, Orthogonal polynomials on the unit circle associated with the Laguerre polyno- mials, Proc. Amer. Math. Soc. 129, no. 3, 873-879 (2001). [30] B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42 (2005), 431-460. [31] W. Sun, X. Zhou, Density and stability of wavelet frames, Appl. Comput. Harmon. Anal. 15, no. 2, 117–133 (2003). [32] K. H. Zhu, ”Operator theory in function spaces”. Monographs and Textbooks in Pure and Applied Mathematics, 139. Marcel Dekker, Inc., New York, 1990. 1. Introduction 2. Description of the results 3. Tools 4. Bergman transform of Sn0=x"010B(t) 5. Wavelet frames with Fourier transforms of Laguerre functions 6. Further properties of the functions Sn0=x"010B(t) References

0704.1488

On Beltrami fields with nonconstant proportionality factor on the plane

On Beltrami fields with nonconstant proportionality factor on the plane Vladislav V. Kravchenko1, Héctor Oviedo2 Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 MEXICO e-mail: [email protected]∗ SEPI, ESIME Zacatenco, Instituto Politécnico Nacional, Av. IPN S/N, C.P. 07738, D.F. MEXICO† November 2, 2018 Abstract We consider the equation B + α B = 0 (1) on the plane with α being a real-valued function and show that it can be reduced to a Vekua equation of a special form. In the case when α depends on one Cartesian variable a complete system of exact solutions of the Vekua equation and hence of equation (1) is constructed based on L. Bers’ theory of pseudoanalytic formal powers. 1 Introduction Solutions of the equation B + α B = 0 (2) ∗Research was supported by CONACYT, Mexico via the research project 50424. †During the preparation of this work the second-named author was supported by CONACYT on a postdoctoral stay at the Department of Mathematics, CINVESTAV del IPN, Unidad Querétaro http://arxiv.org/abs/0704.1488v2 where α is a scalar function of space coordinates are known as Beltrami fields and are of fundamental importance in different branches of modern physics (see, e.g., [22], [18], [7], [21], [1], [9], [8], [11]). For simplicity, in this work we consider the real-valued proportionality factor α and real-valued solutions of (2), though the presented approach is applicable in a complex- valued situation as well with a considerable complication of mathematical techniques involved (instead of complex Vekua equations their bicomplex generalizations should be considered [5], [14]). We consider equation (2) on a plane of the variables x and y, that is α and B are functions of two Cartesian variables only. In this case as we show in section 3 equation (2) reduces to the equation + αu = 0. (3) This second-order equation can be reduced (see [14]) to a corresponding Vekua equation (describing generalized analytic functions) of a special form. This reduction under quite general conditions allows us to construct a com- plete system of exact solutions of (3) explicitly (see [13] and [15]). For the reduction of (3) to a Vekua equation it is sufficient to find a particular solu- tion of (3). In the present work (section 4) we show that in a very important for applications case of α being a function of one Cartesian variable a particu- lar solution of (3) is always available in a simple explicit form. This situation corresponds to models describing waves propagating in stratified media (see, e.g., [16]). As a result in this case we are able to construct a complete system of solutions explicitly which for many purposes means a general solution. We give an example of such construction. We show in this work that when α = α(y) (of course in a similar way the case α = α(x) can be considered) equation (3) and hence equation (2) reduce to the Vekua equation of the following form ∂zW (x, y) = if ′(y) 2f(y) W (x, y) (4) where sinA+ c2√ cosA; A is an antiderivative of α with respect to y, c1 and c2 are arbitrary real con- stants, z = x + iy and ∂z = (∂x + i∂y). A complete (in a compact uniform convergence topology) system of exact solutions to (4) can be constructed explicitly. The system represents a set of formal powers [3], [6] which gen- eralize the usual analytic complex powers (z − z0)n, n = 0, 1, 2, . . . and in a sense give us a general solution of (4). Thus, in the case when α is a function of one Cartesian variable the Vekua equation equivalent to (2) in a two-dimensional situation can be solved and a complete system of solutions of (2) is obtained. 2 Preliminaries We need the following definition. Consider the equation ∂zϕ = Φ (5) on a whole complex plane or on a convex domain, where Φ = Φ1 + iΦ2 is a given complex valued function such that its real part Φ1 and imaginary part Φ2 satisfy the equation ∂yΦ1 − ∂xΦ2 = 0 (6) then there exist real valued solutions of (5) which can be easily constructed in the following way ϕ(x, y) = 2 Φ1(η, y)dη + Φ2(x0, ξ)dξ + c (7) where (x0, y0) is an arbitrary fixed point in the domain of interest and c is an arbitrary real constant. By A we denote the integral operator in (7): A[Φ](x, y) = 2 Φ1(η, y)dη + Φ2(x0, ξ)dξ Note that formula (7) can be extended to any simply connected domain by considering the integral along an arbitrary rectifiable curve Γ leading from (x0, y0) to (x, y) ϕ(x, y) = 2 Φ1dx+ Φ2dy Thus if Φ satisfies (6), there exists a family of real valued functions ϕ such that ∂zϕ = Φ, given by the formula ϕ = A[Φ]. Let f denote a given positive twice continuously differentiable function defined on a domain Ω ⊂ C. Consider the following Vekua equation W in Ω (8) where the subindex z means the application of the operator ∂z , W is a complex-valued function and W is its complex conjugate function. As was shown in [12], [13], [14], [15], [17] equation (8) is closely related to the second- order equation of the form (div p grad+q)u = 0 in Ω (9) where p and q are real-valued functions. In particular the following state- ments are valid. Theorem 1 [14] Let u0 be a positive solution of (9). Assume that f = and W is any solution of (8). Then u = 1√ ReW is a solution of (9) and p ImW is a solution of grad+q1)v = 0 in Ω (10) where q1 = − . (11) Theorem 2 [14] Let Ω be a simply connected domain, u0 be a positive so- lution of (9) and f = p1/2u0. Assume that u is a solution of (9). Then a solution v of (10) with q1 defined by (11) such that W = p 1/2u+ ip−1/2v is a solution of (8) is constructed according to the formula v = u−10 A(ipu 0∂z(u 0 u)) (12) and vice versa, let v be a solution of (10), then the corresponding solution u of (9) such that W = p1/2u + ip−1/2v is a solution of (8), is constructed according to the formula u = −u0A(ip−1u−20 ∂z(u0v)). (13) Thus the relation between (8) and (9) is very similar to that between the Cauchy-Riemann system and the Laplace equation. Moreover, choosing p ≡ 1, q ≡ 0 and u0 ≡ 1 we obtain that (12) and (13) become the well known formulas from the classical complex analysis for constructing conjugate har- monic functions. For a Vekua equation of the form Wz = aW + bW where a and b are arbitrary complex-valued functions from an appropriate function space [20] a well developed theory of Taylor and Laurent series in formal powers was created (see [3], [4]) containing among others the expan- sion and the Runge theorems as well as more precise convergence results (see, e.g., [19]) and a general simple algorithm [15] for explicit construction of formal powers for the Vekua equation of the form (8). 3 Reduction of (2) to a Vekua equation We consider equation (2) where both α and B are supposed to be dependent on two Cartesian variables x and y. Then equation (2) can be written as the following system ∂yB3 + αB1 = 0 (14) − ∂xB3 + αB2 = 0 (15) ∂xB2 − ∂yB1 + αB3 = 0. Solving this system for B3 leads to the equation ∆B3 − + α2B3 = 0 (16) where 〈·, ·〉 denotes the usual scalar product of two vectors. Note that α div = ∆B3 − and hence (16) can be rewritten as follows + αB3 = 0. (17) Thus equation (2) reduces to an equation of the form (9) with p = 1/α and q = α. Let us notice that (see, e.g., [14]) ∇+ α = 1√ (∆− r) 1√ where r = −1 − α2. (18) That is B3 is a solution of (17) iff the function f = B3/ α is a solution of the stationary Schrödinger equation (−∆+ r) f = 0 (19) with r defined by (18). As was explained in section 2, given its particular solution this equation reduces to the Vekua equation (8). Unfortunately, in general we are not able to propose a particular solution of (17). Nevertheless in an important special case when α depends on one Cartesian variable, a particular solution of (17) is always available in explicit form. We give this result in the next section. 4 Solution in the case when α is a function of one Cartesian variable Let us consider equation (19) where α = α(y). We assume that α is a nonvanishing function and look for a solution of the corresponding ordinary differential equation f0 = 0. Its general solution is known (see [10, 2.162 (14)]) and is given by the ex- pression f0(y) = sinA(y) + cosA(y) (20) where A is an antiderivative of α and c1, c2 are arbitrary real constants. Choosing, e.g., c1 = 1, c2 = 0 and calculating the coefficient (∂zf0) /f0 we arrive at the following Vekua equation which is equivalent to (2) in the case under consideration (and which is considered in any simply connected domain where sinA(y) does not vanish), ∂zW (x, y) = α(y) cotA(y)− α 2α(y) W (x, y). Note that F = f0 = sinA(y)√ and G = i sinA(y) represent a generating pair for this Vekua equation (see [13], [15]) and hence if W is its solution, the corresponding pseudoanalytic function of the second kind ω = 1 ReW + if0 ImW satisfies the equation 1− f 20 1 + f 20 ωz (21) which can be written in the form of the following system ψy, φy = − where φ = Reω and ψ = Imω. For f0 being representable in a separable form f0(x, y) = X(x)Y (y) the formulas for constructing corresponding formal powers explicitly were pre- sented already by L. Bers and A. Gelbart (see [3] and [6]). Using them we obtain the following representation for the formal powers corresponding to (n)(a, z0; z) = a1 (x− x0)(n−k)ikY k + ia2 (x− x0)(n−k)ikỸ k (we preserve the notations from [3]) where z0 = x0+ iy0 is an arbitrary point of the domain of interest, a is an arbitrary complex number: a = a1 + ia2, Y k and Ỹ k are constructed as follows Y (0)(y0, y) = Ỹ (0)(y0, y) = 1 and for n = 1, 2, . . . Y (n)(y0, y) = n Y (n−1)(y0, η)f 0 (η)dη n odd Y (n)(y0, y) = n Y (n−1)(y0, η) f 20 (η) n even Ỹ (n)(x0, x) = n Ỹ (n−1)(y0, η) f 20 (η) n odd Ỹ (n)(x0, x) = n Ỹ (n−1)(y0, η)f 0 (η)dη n even. The system (n)(1, z0; z), ∗Z (n)(i, z0; z) represents a complete (in a compact uniform convergence topology [2]) system of solutions of (21) that means that any solution ω of (21) in a simply connected domain Ω can be represented as a series ω(z) = (n)(an, z0; z) = a′n ∗Z (n)(1, z0; z) + a (n)(i, z0; z) where a′n = Re an, a n = Im an and the series converges normally (uniformly on any compact subset of Ω). Consequently the system of functions f0(y) Re(∗Z (n)(1, z0; z)), f0(y) Re(∗Z (n)(i, z0; z)) represents in the same sense a complete system of solutions of (19) with r defined by (18), and α(y)f0(y) Re(∗Z (n)(1, z0; z)), α(y)f0(y) Re(∗Z (n)(i, z0; z)) is a complete system of solutions of (17). Thus in the case under consideration any solution B3 of (17) can be represented in the form B3(x, y) = (an sinA(y) Re(∗Z(n)(1, z0; z)) + bn sinA(y) Re(∗Z(n)(i, z0; z))) where an and bn are real constants. The other two components of the vector B are obtained from (14) and (15): B1 = − ∂yB3 and B2 = ∂xB3 (23) that gives us a complete system of solutions of (2) in the case under consid- eration. On the following example we explain how this procedure works. Example 3 Let us consider the following relatively simple situation in which the corresponding integrals are not difficult to evaluate. Let α(y) = 1− y2 and Ω be an open unitary disk with a center in the origin. We take in (20) c1 = 0 and c2 = 1. Then it is easy to verify that f0(y) = (1− y2) The first three formal powers with a centre in the origin can be calculated as follows (1)(1, 0; z) = x+ i[ y(1− y2) 32 3y(1− y2) 12 arcsin y], (1)(i, 0; z) = − y (1 − y2) 12 + ix, (2)(1, 0; z) = x2 − 1 y2 − 3 y arcsin y (1− y2) 12 + 2ix y(1− y2) 32 3y(1− y2) 12 arcsin y (2)(i, 0; z) = − 2xy (1− y2) 12 x2 − y2 − 1 (3)(1, 0; z) = x3 − 3x y arcsin y (1− y2) 12 + 3ix2 y(1− y2) 32 3y(1− y2) 12 arcsin y − i(− 3 y(1− y2) y(1− y2) 2 + y(1− y2) (1− y2)2 arcsin y + 33 arcsin y), (3)(i, 0; z) = − (1− y2) 12 y(1 + y2) (1− y2) 12 arcsin y+ix x2 − 3 Now taking the real parts of these formal powers and multiplying them by the factor αf0 (see (22)) we obtain the first elements of the complete system of solutions of (17), that is any solution B3 of (17) in a simply connected domain can be represented as an infinite linear combination of the functions {(1− y2) 2 , x(1 − y2) 2 , −y, (1− y2) x2 − 1 y2 − 3 y arcsin y (1− y2) 12 − 2xy, (1− y2) x3 − 3x y arcsin y (1− y2) 12 − 3x2y + y(1 + y2)− (1− y2) 2 arcsin y, ...} and the corresponding series converges normally. From (23) it is easy to calculate the corresponding components B1 and B2 respectively, {y, xy, (1− y2) (1− y2) 2 arcsin y + y(x2 − 3 2x(1− y2) x(1 − y2) 2 arcsin y + y x3 − 9 xy2 + y2 − 3x2)(1− y2) 2 − 3 y arcsin y, ...} {0, (1− y2), 0, 2x(1− y2), −2y(1− y2) (3x2 − y2)(1− y2)− y(1− y2) 2 arcsin y, −6xy(1− y2) 2 , ...}. Thus, we obtain the following complete system of solutions of (2) with the proportionality factor α defined by (24), B 0 = (1− y2) 12 B 1 = (1− y2) x(1 − y2) 12 B 2 = (1− y2) 12 B 3 = (1− y2) 12 arcsin y + y(x2 − 3 y2 + 5 2x(1− y2) (1− y2) 12 x2 − 1 y2 − 3 y arcsin y (1−y2)  , B 4 = 2x(1− y2) 12 −2y(1− y2) 12 B 5 = x(1 − y2) 12 arcsin y + y x3 − 9 xy2 + 15 (3x2 − 3 y2)(1− y2)− 9 y(1− y2) 12 arcsin y (1− y2) 12 x3 − 3x y2 + 3 y arcsin y (1−y2)  , B 6 = y2 − 3x2)(1− y2) 12 − 3 y arcsin y −6xy(1− y2) 12 −3x2y + 3 y(1 + y2)− 3 (1− y2) 12 arcsin y . . . . 5 Concluding remarks We presented two new results. 1. We showed that the system of equations describing Beltrami fields on the plane can be reduced to a Vekua equation of the form (8) whenever any particular solution of the corresponding second-order equation of the form (17) or (19) is known. 2. In the case when the proportionality factor α depends on one Carte- sian variable we obtain a particular solution explicitly, construct the corresponding Vekua equation (section 4), and solve it in the sense that a complete system of solutions is obtained which gives us a complete system of solutions of the original vector equation describing Beltrami fields. Of course not always the integrals involved in the construction of the complete system of solutions are sufficiently easy to evaluate explicitly as in the example 3. Nevertheless our numerical experiments confirm that in general the formal powers and hence the solutions of (2) can be calculated with a remarkable accuracy. For example, the vector B 40 (see notations in the example 3) in the Matlab 7 package on a usual PC can be calculated with a precision of the order 10−4. Thus, the use of formal powers for numerical solution of boundary value problems corresponding to (2) and more generally to equations of the form (9) is really promising. The work in this direction will be reported elsewhere. References [1] Athanasiadis C, Costakis G and Stratis I G 2000 On some properties of Beltrami fields in chiral media. Reports on Mathematical Physics 45 257-271. [2] Bers L 1950 The expansion theorem for sigma-monogenic functions. American Journal of Mathematics 72 705-712. [3] Bers L 1952 Theory of pseudo-analytic functions. New York University. [4] Bers L 1956 Formal powers and power series. Communications on Pure and Applied Mathematics 9 693-711. [5] Castañeda A and Kravchenko V V 2005 New applications of pseudoana- lytic function theory to the Dirac equation. J. of Physics A: Mathemat- ical and General 38, No. 42 9207-9219. [6] Courant R and Hilbert D 1989 Methods of Mathematical Physics, v. 2. Wiley-Interscience. [7] Feng Qingzeng 1997 On force-free magnetic fields and Beltrami flows. Applied Mathematics and Mechanics (English Edition) 18 997-1003. [8] Gonzalez-Gascon F and Peralta-Salas D 2001 Ordered behaviour in force-free magnetic fields. Physics Letters A 292 75-84. [9] Kaiser R, Neudert M and vonWahl W 2000 On the existence of force-free magnetic fields with small nonconstant α in exterior domains. Commu- nications in Mathematical Physics 211 111-136. [10] Kamke E 1976 Handbook of ordinary differential equations. Moscow: Nauka (Russian translation from the German original: 1959 Differen- tialgleichungen. Lösungsmethoden und Lösungen. Leipzig). [11] Kravchenko V V 2003 On Beltrami fields with nonconstant proportion- ality factor. J. of Phys. A 36, 1515-1522. [12] Kravchenko V V 2005 On the reduction of the multidimensional station- ary Schrödinger equation to a first order equation and its relation to the pseudoanalytic function theory. J. of Phys. A 38, No. 4, 851-868. [13] Kravchenko V V 2005 On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrödinger equation and Taylor series in formal powers for its solutions. J. of Phys. A , 38, No. 18, 3947-3964. [14] Kravchenko V V 2006 On a factorization of second order elliptic opera- tors and applications. Journal of Physics A: Mathematical and General 39, No. 40, 12407-12425. [15] Kravchenko V V Recent developments in applied pseudoanalytic func- tion theory. To appear, available from www.arxiv.org. [16] Kravchenko V V and Oviedo H 2003 On a quaternionic reformulation of Maxwell’s equations for chiral media and its applications. Zeitschrift für Analysis und ihre Anwendungen 22, No. 3, 569 - 589. [17] Kravchenko V V and Oviedo H 2007 On explicitly solvable Vekua equa- tions and explicit solution of the stationary Schrödinger equation and of the equation div(σ∇u) = 0. Complex Variables and Elliptic Equations 52, No. 5, 353 - 366. [18] Lakhtakia A 1994 Beltrami fields in chiral media. Singapore: World Scientific. [19] Menke K 1974 Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome. Manuscripta Math. 11 111-125. [20] Vekua I N 1959 Generalized analytic functions. Moscow: Nauka (in Russian); English translation Oxford: Pergamon Press 1962. [21] Yoshida Z 1997 Applications of Beltrami functions in plasma physics. Nonlinear Analysis, Theory & Applications 30 3617-3627. [22] Zaghloul H and Barajas O 1990 Force-free magnetic fields. American Journal of Physics 58 (8) 783-788. Introduction Preliminaries Reduction of (??) to a Vekua equation Solution in the case when is a function of one Cartesian variable Concluding remarks

0704.1489

Where the monotone pattern (mostly) rules

7 Where the monotone pattern (mostly) rules Miklós Bóna Department of Mathematics University of Florida Gainesville FL 32611-8105 [email protected] ∗ Abstract We consider pattern containment and avoidance with a very tight definition that was used first by Riordan more than 60 years ago. Us- ing this definition, we prove the monotone pattern is easier to avoid than almost any other pattern of the same length. We also show that with this definition, almost all patterns of length k are avoided by the same number of permutations of length n. The corresponding state- ments are not known to be true for more relaxed definitions of pattern containment. This is the first time we know of that expectations are used to compare numbers of permutations avoiding certain patterns. 1 Introduction The classic definition of pattern avoidance on permutations is as follows. Let p = p1p2 · · · pn be a permutation, let k < n, and let q = q1q2 · · · qk be another permutation. We say that p contains q as a pattern if there exists a subsequence 1 ≤ i1 < i2 < · · · < ik ≤ n so that for all indices j and r, the inequality qj < qr holds if and only if the inequality pij < pir holds. If p does not contain q, then we say that p avoids q. In other words, p contains q if p has a subsequence of entries, not necessarily in consecutive positions, which relate to each other the same way as the entries of q do. Classic pattern avoidance has been a rapidly developing field for the last decade. One of the most fascinating subjects in this field was the enumera- tion of permutations avoiding a given pattern. Let Sn(q) denote the number of permutations of length n (or in what follows, n-permutations) that avoid ∗Partially supported by an NSA Young Investigator Award. http://arxiv.org/abs/0704.1489v1 the pattern q, and let us consider the numbers Sn(q) for each pattern q of length k. A very interesting and counter-intuitive phenomenon is that in this multiset of k! numbers, the number Sn(q) will, in general, not be the largest or the smallest number. There are several results on this fact (see [1], [2] or [3]), but the phenomenon is still not perfectly well understood. In 2001, Elizalde and Noy [4] proposed another definition of pattern con- tainment. We will say that the permutation p = p1p2 · · · pn tightly contains the permutation q = q1q2 · · · qk if there exists an index 0 ≤ i ≤ n− k so that qj < qr if and only if pi+j < pi+r. In other words, for p to contain q, we require that p has a consecutive string of entries that relate to each other the same way the entries of q do. For instance, 246351 contains 132 (take the second, third, and fifth entries, for instance), but it does not tightly con- tain 132 since there are no three entries in consecutive positions in 246351 that would form a 132-pattern. If p does not tightly contain q, then we say that p tightly avoids q. Let Tn(q) denote the number of n-permutations that tightly avoid q. Elizalde and Noy conjectured in [4] that no pattern of length k is tightly avoided by more n-permutations than the monotone pattern. In other words, if q is a pattern of length k, then Tn(q) ≤ Tn(12 · · · k). (1) This conjecture is still open. (In the special case of k = 3, it was proved in [4].) Still, it is worth pointing out that changing the definition of pattern avoidance changed the status of the monotone pattern among all patterns of the same length. With this definition, it is believed that the monotone pattern is the easiest pattern to avoid. This perceived change in the status of the monotone pattern led us to the following direction of research. Let us take the idea of Elizalde and Noy one step further, by restricting the notion of pattern containment further as follows. Let p = p1p2 · · · pn be a permutation, let k < n, and let q = q1q2 · · · qk be another permutation. We say that p very tightly contains q if there is an index 0 ≤ i ≤ n− k and an integer 0 ≤ a ≤ n− k so that qj < qr if and only if pi+j < pi+r, and, {pi+1, pi+2, · · · , pi+k} = {a+ 1, a+ 2, · · · , a+ k}. That is, p very tightly contains q if p tightly contains q and the entries of p that form a copy of q are not just in consecutive positions, but they are also consecutive as integers (in the sense that their set is an interval). For example, 15324 tightly contains 132 (consider the first three entries), but does not very tightly contain 132. On the other hand, 15324 very tightly contains 213, as can be seen by considering the last three entries. If p does not very tightly contain q, then we will say that p very tightly avoids q. Note that in the special case when q is the monotone pattern, this notion was studied before pattern avoidance became widely known. The literature of permutations very tightly avoiding monotone patterns goes back at least to [7]. More recent examples include [6] and [5]. However, we did not find any examples where the notion was used in connection with any other pattern. Let Vn(q) denote the number of n-permutations that very tightly avoid q. While we cannot prove that Vn(q) ≤ Vn(12 · · · k) for all patterns q of length k, we will be able to prove that this inequality holds for most patterns q of length k. As a byproduct, we will prove that for all k, there exists a set Wk of patterns of length k so that limk→∞ = 1, and Vn(q) is identical for all patterns q ∈ Wk(q). In other words, almost all patterns of length k are equally difficult to very tightly avoid. There are no comparable statements known for the other two discussed notions of pattern avoidance. Our argument will be a probabilistic one. Once the framework is set up, the computation will be elementary. However, this is the first time we know of that expectations are successfully used to compare the number of permutations avoiding a given pattern (admittedly, with a very restrictive definition of pattern avoidance). We wonder whether more sophisticated methods of enumeration could extend the reach of this technique to less restrictive definitions of pattern avoidance. 2 A Probabilistic Argument 2.1 The outline of the argument For the rest of this section, let k ≥ 3 be a fixed positive integer. Let α = 12 · · · k, the monotone pattern of length k. Recall that Vn(α) is the number of n-permutations very tightly avoiding α. Our goal is true prove Vn(q) ≤ Vn(α) for any pattern q of length k. Let q be any pattern of length k. For a fixed positive integer n, let Xn,q be the random variable counting the occurrences of q in a randomly selected n-permutation. As the following straightforward proposition shows, the expectation of Xn,q does not depend on q; it only depends on n, and the length k of q. Proposition 1 For any fixed n, and q ∈ Sk, we have E(Xn,q) = (n− k + 1)2 Proof: Let Xi be the indicator random variable of the event that the string pi+1 · · · pi+k is a q-pattern in the very tight sense. Then E(Xi) = P (pi+1 · · · pi+k ≃ q) = n−k+1 , since there are n− k+1 favorable choices for the set of the entries pi+1, · · · , pi+k, and there is 1/k! chance that their pattern is q. Now note that E(Xn,q) = i=0 Xi, and the statement is proved by the linearity of expectation. ✸ Let pn,i,q be the probability that a randomly selected n-permutation contains exactly i copies of q, and let Pn,i,q be the probability that a randomly selected n-permutation contains at least i copies of q. Set m = n− k + 1, and observe that no n-permutation can very tightly contain more than m copies of any given pattern q of length k. Now note that by the definition of expectation E(Xn, q) = ipn,i,q pn,m−i,q = pn,m,q + (pn,m,q + pn,m−1,q) + · · ·+ (pn,m,q + · · · + pn,1,q) P (n, i, q). By Proposition 1, we know that E(Xn,q) = E(Xn,α), and then previous displayed equation implies that P (n, i, q) = P (n, i, α). (2) So if we could show that for i ≥ 2, the inequality P (n, i, q) ≤ P (n, i, α) (3) holds, then (2) would imply that P (n, 1, q) ≥ P (n, 1, α), which is just what we set out to prove. The simple counting argument that we present in this paper will not prove (2) for every pattern q. However, it will prove (2) for most patterns q. We describe these patterns in the next subsection. 2.2 Condensible and Non-condensible Patterns Let us assume that the permutation p = p1p2 · · · pn very tightly contains two non-disjoint copies of the pattern q = q1q2 · · · qk. Let these two copies be q and q(2), so that q(1) = pi+1pi+2 · · · pi+k and q (2) = pi+j+1pi+j+2 · · · pi+j+k for some j ∈ [1, k − 1]. Then |q(1) ∩ q(2)| = k − j + 1 = s. Furthermore, since the set of entries of q(1) is an interval, and the set of entries of q(2) is an interval, it follows that the set of entries of q(1) ∩ q(2) is also an interval. So the rightmost s entries of q, and the leftmost s entries of q must form identical patterns, and the respective sets of these entries must both be intervals. For obvious symmetry reasons, we can assume that q1 < qk. We claim that then the rightmost s entries of q must also be the largest s entries of q. This can be seen by considering q(1). Indeed, the set of these entries of q(1) is the intersection of two intervals of the same length, and therefore, must be an ending segment of the interval that starts on the left of the other. An analogous argument, applied for q(2), shows that the leftmost s entries of q must also be the smallest s entries of q. The following Proposition collects the observations made in this subsec- tion. Proposition 2 Let p be a permutation that very tightly contains copies q(1) and q(2) of the pattern q = q1q2 · · · qk. Let us assume that q1 < qk. Then q(1) and q(2) are disjoint unless all of the following hold. There exists a positive integer s ≤ k − 1 so that 1. the rightmost s entries of q are also the largest s entries of q, and the leftmost s entries of q are also the smallest s entries of q, and 2. the pattern of the leftmost s entries of q is identical to the pattern of the rightmost s entries of q. It is easy to see that if q satisfies both of these criteria, then two very tightly contained copies of q in p may indeed intersect. For example, the pattern q = 2143 satisfies both of the above criteria with s = 2, and indeed, 214365 very tightly contains two intersecting copies of q, namely 2143 and 4365. Definition 1 Let q be a pattern that satisfies both conditions of Proposition 2. Then we say that q is condensible. It is not difficult to prove that almost all patterns of length k are non- condensible. We do not want to break the course of our proof with this, so we postpone this computation until the Appendix. 2.3 The Computational Part of the Proof The following Lemma is the heart of our main result. Lemma 1 Let q be a non-condensible pattern, and let i > 1. Then P (n, i, q) ≤ P (n, i, α). Proof: We point out that if n < ik, then the statement is clearly true. Indeed, P (n, i, q) = 0 since any two copies of q in an n-permutation p would have to be disjoint, and n is too small for that. Therefore, in the rest of the proof, we can assume that n ≥ ik. For k ≤ 2, the statement is trivial, so we assume k ≥ 3 as well. First, we prove a lower bound on P (n, i, α). The number of n-permutations very tightly containing i copies of α is at least as large as the number of n-permutations very tightly containing the pattern 12 · · · (i + k − 1). The latter is at least as large as the number of n-permutations that very tightly contain a 12 · · · (i + k − 1)-pattern in their first i+ k − 1 positions, that is, (n− k − i+ 2) · (n− k − i+ 1)! = (n− k − i+ 2)!. Therefore, (n− k − i+ 2)! ≤ P (n, i, α). (4) We are now going to find an upper bound for P (n, i, q). Let S be an i- element subset of [n] so that the elements of S can be the starting positions of i (necessarily disjoint) very tight copies of q in an n-permutation. If S = {s1, s2, · · · , si}, then this is equivalent to saying that 1 ≤ s1 < s2 − k+ 1 ≤ s3 − 2k + 2 ≤ · · · ≤ si − (i− 1)(k − 1) ≤ n− i(k − 1). Therefore, there are n−i(k−1) possibilities for S. Now let AS be the event that in a random permutation p = p1 · · · pn, the subsequence pjpj+1 · · · pj+k−1 is a very tight q-subsequence for all j ∈ S. Let Ai,q be the event that p contains at least i very tight copies of q. Then P (Ai,q) = P (n, i, q). Fur- thermore, Ai,q = ∪SAS , where the union is taken over all n−i(k−1) possible subsets for S. Therefore, P (n, i, q) = P (Ai,q) ≤ P (AS). (5) Let us now compute P (AS). We will see that this probability does not de- pend on the choice of S. Indeed, just as there are n−i(k−1) possibilities for S, there are n−i(k−1) possibilities for the entries in the positions belonging to S. Once those entries are known, the rest of the q-patterns starting in those entries are determined, and there are (n− ik)! possibilities for the rest of the permutation. This shows that P (AS) = n−i(k−1) (n − ik)! 1 for all S. Therefore, (5) implies P (n, i, q) ≤ n− i(k − 1) (n− ik)! . (6) Comparing (6) and (4), we see that our lemma will be proved if we show that for i > 1, the inequality n− i(k − 1) (n− ik)! ≤ (n− i− k + 2)!, or, equivalently, (n− i(k − 1))i ≤ i! 2(n− k− i+ 2)(n− k− i+ 1) · · · (n− i(k − 1) + 1)) (7) holds. Where (z)j = z(z−1) · · · (z− j+1). Note that the left-hand side has i factors, while the right-hand side, not counting i!2, has (k − 1)(i − 1) > i factors, each of which are larger than the factors of the left-hand side. Therefore, (7) holds, and the Lemma is proved. ✸ The proof of our main result is now immediate. Theorem 1 Let q be any pattern of length k. Then Vn(q) ≤ Vn(α). Proof: Lemma 1 and formula (2) together imply that P (n, 1, q) ≥ P (n, 1, α), which means that there are at least as many n-permutations that very tightly contain q as n-permutations that very tightly contain α. ✸ 2.4 A Result on Non-condensible Patterns We have seen in Proposition 2 that if q is non-condensible and q1 < qk, then any two copies of q contained in a given permutation p are disjoint. Therefore, the number of n-permutations that very tightly avoid q can be computed by the Principle of Inclusion-Exclusion. Indeed, in this case, the following holds. Proposition 3 Let q be a non-condensible pattern. Then Vn(q) = n!− ⌊n/k⌋ n− i(k − 1) (n− ik)!. In particular, Vn(q) does not depend on the choice of q. Proof: In the proof of Lemma 1, more precisely, in our argument showing that (6) holds, we showed that there are n−i(k−1) ways to choose an i- element set of positions that can be the starting positions of i disjoint very tight copies of q, and there are n−i(k−1) ways to choose the sets of entries forming these same copies. Once these choices are made, the rest of the permutation can be chosen in (n− ik)! ways. The statement now follows by the Principle of Inclusion-Exclusion. ✸ As we said, we will prove in Proposition 4 that almost all patterns are non-condensible. Note that nothing comparable is known for the other two notions of pattern avoidance. Numerical evidence suggests that similarly strong results will probably not hold if the traditional definition or the tight definition is used. 3 Further Directions The novelty of this paper, beside the notion of very tight containment, was the application of expectations to compare the numbers of permutations avoiding two patterns. The computations themselves were elementary. This leads to the following question. Question 1 Is it possible to apply our method to compare the numbers Tn(q) and Tn(α), or the numbers Sn(q) and Sn(α), for at least some patterns q? As we used a very simple estimate in our proof of Lemma 1, there may be room for improvement at that point. There are several natural questions that can be raised about the enumer- ation of permutations that very tightly avoid a pattern. Let us recall that we proved a formula for Vn(q) for the overwhelming majority of patterns q, namely for non-condensible patterns. A formula for Vn(α), where α is the monotone pattern, can be found in [6] and [5]. This raises the following question. Question 2 Are there other patterns q for which Vn(q) can be explicitly determined? Let us call patterns q and q′ very tightly equivalent if Vn(q) = Vn(q ′) for all n. We have seen that almost all patterns of length k are very tightly equivalent. This raises the following questions. Question 3 How many equivalence classes are there for very tight patterns of length k? Question 4 Can we say anything about the same topic for tight pattern containment, or traditional pattern containment? If equivalence is defined for them in an analogous way, how many equivalence classes will be formed, (for patterns of length k) and how large will the largest one be? 4 Appendix In this section we prove the following simple fact. Proposition 4 Let hn be the number of condensible permutations of length n. Then Proof: We prove that another class of permutations, one that contains all condensible permutations, is also very small. Let an be the number of decomposable permutations, that is, permutations p1p2 · · · pn for which is there is an index i so that pj < pm if j ≤ i < m. In other words, p can be cut into two parts so that everything before the cut is less than everything after the cut. (Note that if a permutation is not decomposable, then it is called indecomposable, and Exercise 1.32 of [8] contains more information about these permutations.) Counting according to the index i of the above definition, we see that i!(n − i)! where in the last step we used the well-known unimodal property of binomial coefficient, in particular the inequality that if 2 ≤ i ≤ n− 2. Therefore, an/n! converges to 0 as n goes to infinity. Clearly, all con- densible patterns are decomposable since their first i entries are also their i smallest entries, so our claim follows. ✸ References [1] M. Bóna, Combinatorics of Permutations, CRC Press, 2004. [2] M. Bóna, The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns, J. Combin. Theory Ser. A 110 (2005) no. 2, 223–235. [3] New Records on Stanley-Wilf Limits. European Journal of Combina- torics, 28 (2007), vol. 1, 75-85. [4] S. Elizalde; M. Noy, Consecutive patterns in permutations. Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001). Adv. in Appl. Math. 30 (2003), no. 1-2, 110–125. [5] D. M. Jackson; R. C. Reid, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343. [6] D. M. Jackson; J. W. Reilly, Permutations with a prescribed number of p-runs. Ars Combinatoria 1 (1976), no. 1, 297-305. [7] J. Riordan, Permutations without 3-sequences, Bull. Amer. Math. Soc., 51 (1945), 745-748. [8] R. Stanley, Enumerative Combinatorics, Volume 1. Cambridge Uni- versity Press, Cambridge UK, second edition 1997. Introduction A Probabilistic Argument The outline of the argument Condensible and Non-condensible Patterns The Computational Part of the Proof A Result on Non-condensible Patterns Further Directions Appendix

0704.1490

Staggered Heavy Baryon Chiral Perturbation Theory

Staggered Heavy Baryon Chiral Perturbation Theory Jon A. Bailey∗ Washington University in St. Louis (Dated: August 7, 2021) Abstract Although taste violations significantly affect the results of staggered calculations of pseudoscalar and heavy-light mesonic quantities, those entering staggered calculations of baryonic quantities have not been quantified. Here I develop staggered chiral perturbation theory in the light-quark baryon sector by mapping the Symanzik action into heavy baryon chiral perturbation theory. For 2+1 dynamical quark flavors, the masses of flavor-symmetric nucleons are calculated to third order in partially quenched and fully dynamical staggered chiral perturbation theory. To this order the expansion includes the leading chiral logarithms, which come from loops with virtual decuplet- like states, as well as terms of O(m3π), which come from loops with virtual octet-like states. Taste violations enter through the meson propagators in loops and tree-level terms of O(a2). The pattern of taste symmetry breaking and the resulting degeneracies and mixings are discussed in detail. The resulting chiral forms are appropriate to lattice results obtained with operators already in use and could be used to study the restoration of taste symmetry in the continuum limit. I assume that the fourth root of the fermion determinant can be incorporated in staggered chiral perturbation theory using the replica method. ∗[email protected] http://arxiv.org/abs/0704.1490v2 mailto:[email protected] I. INTRODUCTION The staggered formulation [1] of lattice QCD incorporates the rooting conjecture of [2] to eliminate remnant sea quark doublers. The past year has seen good progress in our understanding of this conjecture [3, 4] and the corresponding replica or quark flow method [5, 6, 7, 8] of staggered chiral perturbation theory [9]. However, the results of [9] have yet to be extended to the baryon sector, and many of the assumptions of [4, 9] have not yet been tested. Lacking proof of these assumptions, successful calculations of experimentally well-known hadronic quantities serve as essential tests of the rooting conjecture and the replica method. The advantages [10] of staggered QCD have allowed successful calculations of many hadron masses, decay constants, form factors, and other quantities of phenomenological importance [11, 12, 13, 14, 15, 16]. However, to date this success has been most pronounced in the pseudoscalar and heavy-light meson sectors, where the development of staggered chi- ral perturbation theory (SχPT) [7, 8, 17, 18] facilitated calculations of quark masses, meson masses, and meson decay constants [13, 14, 16, 19]. Progress calculating baryonic quantities has been impeded by comparatively large statistical errors and two sources of systematic errors: questions of interpretation specific to the staggered valence sector, and a lack of control over the chiral and continuum extrapolations. In contemporary studies employing staggered fermions, three physical quark flavors are explicitly included in the action [15]; the four pseudoflavors per physical quark flavor are called “tastes.” The presence of taste quantum numbers implies that the valence sector of staggered QCD contains baryons that have no counterpart in nature. However, assuming taste symmetry is restored in the continuum limit, the staggered baryons become degenerate within irreps of the taste symmetry group, SU(4)T , and baryons composed of valence quarks of the same taste are seen to correspond to physical states [20]. At nonzero lattice spacing, taste violations lift the SU(4)T degeneracies and mix baryons with different SU(4)T quantum numbers. The resulting spectrum contains many sets of nearly degenerate, mixed states. Since states with the same conserved (lattice) quantum numbers mix, interpolating fields typically overlap multiple nearly degenerate, mixed members of these sets. This situation complicates the task of extracting the masses of physical baryons. To date, staggered calculations of the masses of light-quark baryons have been performed by first taking the continuum limit of lattice data and then using continuum baryon chiral perturbation theory to take the chiral limit [15]. In principle this approach is perfectly rigorous, but using it to quantify the effects of taste violations, which are O(α2sa2), is difficult and has not been done. However, using SχPT to study the splittings and mixings due to taste violations is comparatively straightforward. Reference [20] considered the questions of interpretation specific to the mass spectrum of light-quark staggered baryons. It was found that, in calculations of the masses of the nucleon, the lightest decuplet, and certain excited states, one can avoid the complications due to the splittings and mixings introduced by taste violations provided one appropriately chooses the interpolating operators and quark masses. The work reported here enhances our control over the chiral and continuum limits by developing SχPT in the baryon sector. As an example, the masses of certain staggered nucleons are calculated in staggered heavy baryon chiral perturbation theory (SHBχPT) through leading chiral logarithms; the resulting chiral forms could be used to study the restoration of taste symmetry in the continuum limit and to improve calculations of the nucleon mass. Calculations of the chiral forms for the masses of other states highlighted in [20] are in progress and will be reported elsewhere [21]. In SHBχPT, the calculation of baryon masses through leading chiral logarithms requires the inclusion of leading loops, which enter at third order in the staggered chiral expansion. After discussing the power counting in Sec. II, I write down the Symanzik action and use its symmetries to map the quark-gluon operators into the corresponding interactions of SHBχPT. As anticipated in [17], the O(a2) Lagrangian breaks the continuum spin-taste symmetry down to the hypercubic group of the lattice. However, in the rest frame of the heavy baryons, a new remnant spin-taste symmetry eliminates O(a2) mixing between spin-1 and spin-3 baryons. For calculations to third order, this symmetry allows one to consider the baryon mass matrix in a smaller spin-flavor-taste subspace. The nucleon operators currently in use interpolate to baryons that are completely sym- metric in flavor [15]. In Sec. IIIA I recall the symmetry considerations used in [20, 22] to identify staggered baryons that are degenerate with the nucleon in the continuum limit and to identify the irreducible representations (irreps) of interpolating fields for the flavor- symmetric nucleons. Sec. III B contains a discussion of the symmetries of the various terms in the staggered chiral expansion of the masses of the flavor-symmetric nucleons and the im- plied form of the mass matrix. Sec. IIIC contains the staggered chiral forms for the masses of the flavor-symmetric nucleons in partially quenched and fully dynamical SHBχPT. In Sec. IIID, I show that the pattern of symmetry breaking furnishes the connection between the staggered chiral forms and the irreps of the corresponding interpolating fields; the rea- soning is the same as that used to establish the connection between continuum states and lattice irreps [20, 23]. Sec. IV summarizes and discusses future work. II. THE STAGGERED HEAVY BARYON LAGRANGIAN Counting a2 as O(mq), where mq is any of the light quark masses, the staggered chiral power counting is a straightforward generalization of the power counting of the continuum heavy baryon chiral perturbation theory [24, pp. 360-362, 476-477]. The staggered heavy baryon Lagrangian is expanded in increasing powers of the quark masses, the squared lattice spacing, and derivatives of the heavy baryons and pseudoscalar mesons. The baryon masses are calculated in increasing powers of the square roots of the quark masses, the residual off-shell momentum of the heavy baryon, the momenta of the pseudoscalar mesons, the squared lattice spacing, and the average mass splitting between the lightest spin-1 spin-3 baryons. For convenience O(ε) ≡ O(m1/2q ) = O(k) = O(a) = O(∆), where k is the residual heavy baryon momentum or pseudoscalar meson momentum and ∆ is the octet-decuplet mass splitting in the continuum and chiral limits. In the continuum theory, the leading order (LO) heavy baryon Lagrangian is of O(ε), while the LO meson Lagrangian is of O(ε2); the next-to-leading order (NLO) heavy baryon Lagrangian is of O(ε2), while the NLO meson Lagrangian is of O(ε4), and so on. First order contributions to the baryon self-energies do not arise, while tree-level contributions of O(ε3) vanish. One-loop diagrams first enter the expansion at O(ε3) and can only include vertices from the LO Lagrangian; two-loop diagrams first enter at O(ε5). Tree-level contributions at O(ε3) vanish. Contributions at O(ε2) can contain at most one vertex from the NLO heavy baryon Lagrangian, together with vertices from the LO heavy baryon and meson Lagrangians; in fact, no diagrams with vertices from the LO Lagrangians arise. Nontrivially, the situation in SHBχPT is the same. The staggered symmetries forbid corrections to the Symanzik action ofO(a) and O(a3) [25, 26], so the power counting remains the same through NLO. Calculating the baryon masses to O(ε3) requires at most the LO heavy baryon and meson Lagrangians for one-loop diagrams, which are O(ε3), and the NLO heavy baryon Lagrangian for O(ε2) tree-level diagrams. Tree-level contributions to the baryon masses do not arise at O(ε) and O(ε3). To derive the LO heavy baryon and meson Lagrangians and the NLO heavy baryon Lagrangian needed for O(ε2) tree-level corrections, we require the Symanzik action through O(a2). In partially quenched staggered QCD [7, 17], the Symanzik action to O(a2) is Seff = S4 + a d4x(L4 + a2L6), where L4 = 12Tr(FµνFµν) +Q( /D +m)Q and L6 = Lglue6 + Lbilin6 + L FF (A) 6 + L FF (B) Lglue6 contains terms constructed from gluon fields only, Lbilin6 contains quark bilinears, and LFF (A,B)6 contain four-fermion operators constructed from products of quark bilinears. The terms in LFF (A)6 respect the remnant taste symmetry Γ4 ⋊ SO(4)T [17], while the terms in LFF (B)6 break the continuum rotation-taste symmetry down to the hypercubic group of the staggered lattice. Because the staggered formulation introduces four quark tastes for each physical quark flavor, the chiral symmetry of L4 is SU(24|12)L × SU(24|12)R. There are twelve valence quarks, twelve sea quarks, and twelve ghost quarks; Q is a 36-element column vector in flavor-taste space. The quark mass matrix is m = diag(mx, my, mz;mu, md, ms;mx, my, mz)⊗ I4, where mx,y,z are the masses of the valence quarks, mu,d,s, the masses of the sea quarks, and I4 is the identity in taste space. The masses of the ghost quarks are set equal to the masses of the valence quarks so that valence quarks do not contribute to the fermion determinant. In the fully dynamical limit, one can drop the graded formalism and consider a Symanzik theory with the chiral symmetry SU(12)L × SU(12)R. For the fully dynamical case with N physical flavors, the symmetries of the various terms in S4 and S6 are given in detail in the Appendix of [7]. The fourth root of the fermion determinant is implemented in SχPT calculations by the method of quark flows [7, 27] or, equivalently, the replica method [5, 9]. In the continuum limit, the resulting chiral forms are equivalent to the forms that result by considering a partially quenched theory with only three flavors of sea quarks. The development of SχPT in the baryon sector proceeds as in the meson sector [7, 8]. In addition to the pseudoscalar mesons, the Lagrangian now includes the lightest spin-1 and spin-3 baryons. In the fully dynamical continuum chiral theory [28], the lightest spin-1 baryons transform in the adjoint of SU(3)F , so they can be collected in a 3 × 3 matrix in flavor space: Σ0 + 1√ Λ Σ+ p Σ− − 1√ Σ0 + 1√ Ξ− Ξ0 − 2√ In the quenched and partially quenched continuum chiral theories [27, 29], the lightest spin- baryons transform within mixed-symmetry irreps of the appropriate graded symmetry groups; the presence of more than three light quark flavors implies that the baryons no longer transform in the adjoint. The above matrix parameterization remains valid only in restricted subsectors of the flavor space. The generalization to an arbitrary number of flavors involves embedding the independent fields in a rank-3 mixed-symmetry tensor of the graded group, Bijk. The embedding respects the defining indicial symmetries of this tensor but is otherwise arbitrary; any convenient basis of the graded irrep suffices. In the fully dynamical case (alternatively, restricting the indices to fermionic quarks), the standard defining relations of B are Bikj = Bijk and Bijk +Bjki +Bkij = 0. (1) Under the chiral symmetry group, the spin-1 baryon tensor transforms as Bijk → U li U k Blmn, where U is a local, nonlinear realization of the chiral symmetry defined by the transformation law of the meson fields [18, 24, 28, 30] Σ → σ′ = LσU † = UσR†. The tastes of SχPT are effectively additional quark flavors, so the situation in SχPT is similar to that in the quenched and partially quenched continuum theories: Even in the fully dynamical case, one makes use of the rank-3 mixed-symmetry tensor in Eq. (1). If the indices range from 1 to N , then the number of independent components is 1 N(N2 − 1). For three quark flavors, N = 12, so the lightest spin-1 baryons transform within a 572M of SU(12)f , the diagonal (vector) subgroup of SU(12)L × SU(12)R. The lightest spin-3 baryons are incorporated in SχPT similarly. In the fully dynamical case, the independent fields are embedded in a completely symmetric rank-3 tensor, Tijk: Tijk = Tjki = Tkij = Tjik = Tikj = Tkji, Tijk → U li U k Tlmn. The dimension of the symmetric irrep is 1 N(N +1)(N +2), so for three quarks, the lightest spin-3 baryons transform within a 364S of SU(12)f . So far the spacetime indices of the heavy baryon fields have been suppressed. B is a Dirac spinor, while T is a spin-3 Rarita-Schwinger field and therefore carries a spinor index and a vector index. The anti-particle components of both fields are projected out in an arbitrary but specific Lorentz frame, and the relativistic chiral Lagrangian is expanded in increasing powers of 1/mB, where mB is the average octet baryon mass in the chiral limit [28, 31]. The resulting heavy baryon Lagrangian contains no explicit reference to the Dirac matrices; they are replaced by the classical baryon four-velocity vµ and the covariant spin vector Sµ ≡ i2γ5σµνv ν [24, 31]. For convenience in making contact with the literature on the continuum theory, I use Minkowski space notation throughout; accordingly, the Lorentz indices here and below take values from {0, 1, 2, 3}. The Green’s functions of SHBχPT are defined in Euclidean space; a Wick rotation is implicit throughout. A. O(a0) Lagrangian In the continuum chiral theory [29, 31], the heavy baryon Lagrangian to O(ε2) is con- structed of all Hermitian operators containing one derivative or quark mass matrix that respect chiral symmetry, Lorentz invariance, parity, and, before non-relativistic reduction, charge conjugation. The building blocks are the heavy baryon fields, the covariant deriva- tives Dµ of the heavy baryon fields, the mesonic axial vector current Aµ ≡ i2(σ∂µσ †−σ†∂µσ), the mass matrices M± ≡ 12(σ †mσ† ± σmσ), the four-vectors vµ and Sµ, and the Lorentz- invariant tensors, gµν and ǫ µναβ . In the heavy baryon Lagrangian, the vector current Vµ ≡ 12(σ∂µσ † + σ†∂µσ) and derivatives of the heavy baryon fields only appear together in the covariant derivative Dµ; the explicit definition of Dµ is given in [29] but will not be required here. For calculating the baryon masses to O(ε3), the covariant derivative may be replaced with the partial derivative ∂µ, M− does not contribute, and M+ may be replaced with the quark mass matrix, m. The meson fields therefore enter the required operators in the heavy baryon Lagrangian only through Aµ. In SχPT, operators mapped from S4 obey all the symmetries of the continuum theory (after Wick rotation) except for the chiral symmetries, which are simply enlarged to the appropriate flavor-taste group: SU(6|3)L × SU(6|3)R → SU(24|12)L × SU(24|12)R SU(3)L × SU(3)R → SU(12)L × SU(12)R in the partially quenched or fully dynamical cases, respectively. It follows that the stag- gered heavy baryon Lagrangian through O(ε2) mapped from L4 has the same form as the continuum Lagrangian through O(ε2) [29], where the flavor indices in the latter are simply replaced by flavor-taste indices: L4 → L(1)φB + L φB + . . . L(1)φB ≡ Biv µDµB + 2αBS µBAµ + 2βBS − T ν(ivµDµ −∆)Tν + AµB +BAµT + 2HT SµAµTν L(2)φB ≡ 2αMBBM+ + 2βMBM+B + 2σ MBBstr(M+) + 2γMT M+Tµ − 2σ′MT Tµstr(M+) + . . . where the notation is closely based on that of [17, 27, 29]; α, β, αM , βM , C, H , γM , σ and σ′M are low-energy couplings (LECs), ∆ is the average 572M-364S mass splitting in the continuum and chiral limits, and “str” denotes the supertrace. The couplings σ′M and σ M are normalized so that the supertrace terms reproduce the continuum tree-level contributions to the baryon masses in the continuum limit; σ′M = σM and σ σM . The neglected terms in L(2)φB are higher order in 1/mB, contain two derivatives, or are O(a2); through O(ε3), only the terms of O(a2) contribute to the masses of the lightest spin-1 baryons. Of the operators listed above, those that contribute are L(1)′φB = Biv µ∂µB + BSµB∂µφ+ BSµ(∂µφ)B (2) − T ν(ivµ∂µ −∆)Tν + (∂µφ)B +B(∂µφ)T L(2)′φB,m = 2αMBBm+ 2βMBmB + 2σ MBBstr(m) where the definitions of M+, Aµ, and σ = e iφ/(2f) = 1 + iφ + O(φ2) [7] were used; f is the pion decay constant in the continuum and chiral limits. Terms with explicit factors of φ contribute vertices for the leading loops, and terms with explicit factors of m contribute analytically at O(ε2). B. O(a2) baryon operators for analytic corrections In addition to the chiral operators in L(1)φB and L φB mapped from L4, one must find the chiral operators in L(2)φB corresponding to L6. Here I consider only the subset of these operators needed to calculate irrep-dependent corrections to the masses of the lightest spin- baryons through O(ε3); through this order, such operators contribute analytic tree-level corrections of O(a2). I neglect operators contributing only generic O(a2) corrections to the LECs in the Lagrangian mapped from L4 and operators that contribute only to the masses of spin-3 baryons. Operators containing the quark mass matrix, the axial vector current, and derivatives of the heavy baryon fields can be ignored because such operators first contribute at O(a2mq) = O(ε4). Tree-level corrections come only from operators that are bilinear in the heavy baryon fields and have terms that are free of meson fields. The previous set of building blocks is thus reduced to the heavy baryon fields, the four-vectors vµ and Sµ, and the Lorentz-invariant tensors, gµν and ǫ µναβ . In addition to these building blocks, one may promote the taste matrices to spurion fields and use the meson field σ to construct objects that transform in the adjoint under vector (chiral diagonal) flavor-taste transformations. Expanding σ = 1+O(φ) at the end of the enumeration, one recovers the Lagrangian that contributes at tree-level. Finally, the fact that the heavy baryon Lagrangian may be obtained as the non-relativistic limit of the relativistic chiral theory implies that Sµ occurs at most once in each baryon bilinear, while it suffices for the present purposes to consider operators in which vµ occurs at most once for each heavy baryon field. Three cases arise. Chiral operators corresponding to Lglue6 and Lbilin6 respect all the symmetries of the LO Lagrangian except for spacetime rotation invariance, which is broken to the hypercubic rotations [7, 17]. Chiral operators corresponding to LFF (A)6 break chiral flavor-taste to the remnant flavor-taste group; in the fully dynamical case, we have U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ], where U(3)l ×U(3)r is the analog of the U(1)vec ×U(1)A of the single-flavor case [7]. These operators are invariant under all other symmetries of the LO Lagrangian, including spacetime rotations. Allowing the taste matrices to transform as spurions, these operators also respect chiral flavor-taste transformations. Finally, chiral operators corresponding to LFF (B)6 break chiral flavor-taste and spacetime rotations to the symmetry group of the lattice theory [17, 26]. Allowing the taste matrices to transform appropriately, one can work with all the symmetries except spacetime rotations, which are broken to the hypercubic rotations. We will consider each of these cases in turn. 1. Lglue6 and Lbilin6 The chiral operators corresponding to Lglue6 and Lbilin6 contribute nothing but a generic O(a2) correction to the average spin-1 baryon mass in the chiral limit. The Symanzik operators contain no taste matrices, so there are no spurions in the corresponding chiral operators. Displaying flavor-taste indices explicitly, the only possibilities that respect parity are, up to additional factors of vµ, Bijkv µvµgµµ Tµ,ijk + T µ Bijk)v µ Tµ,ijkv ijkgµµ where the repeated index µ is summed from 0 to 3. Chiral operators containing Sµ do not arise because it is an axial vector; bilinears containing it are odd under parity. The second operator is constructed so that it is invariant under charge conjugation. One Lorentz index is raised in each operator because I have Wick rotated to Minkowski space. The first term is a velocity-dependent correction to the average spin-1 baryon mass in the chiral and continuum limits. The underlying lattice theory is not Lorentz invariant, so there is no prohibition against such corrections. The second term mixes spin-1 and spin-3 baryons; the continuum spin and taste symmetries are broken by L6. The last two terms are generic corrections to the average difference ∆ between the masses of the spin-1 and spin-3 baryons. Specializing to the rest frame of the heavy baryon, v = (1, 0), and only the first and last terms survive. The other two vanish because [31] µ = 0, ∀ v =⇒ T 0 = 0 for v = (1, 0). For v = 0, the surviving terms respect all the symmetries of the LO heavy baryon La- grangian, including invariance under spatial rotations. The spacetime symmetry is restored for this special case. Operators with additional factors of vµ yield nothing new; such opera- tors either vanish or reduce to the first type. The fact that spin-1 and spin-3 baryons do not mix in the rest frame considerably simpli- fies, in this frame, the analysis of Sec. III and calculations of the baryon masses. Moreover, simulations performed with zero momentum sources must be analyzed in this frame. Ac- cordingly, the baryon masses given in Sec. IIIC are specific to this case. 2. LFF (A)6 There are twelve types of operators in LFF (A)6 ; following [17, 26], I label these with the spacetime-taste irreps in which the quark bilinears transform: LFF (A)6 ∼ [V × S] + [V × P ] + [V × T ] + (V → A) [S × V ] + [S ×A] + (S → P ) + [T × V ] + [T × A]. (3c) In these operators, the indices of the spin and taste matrices are contracted separately; e.g., [A× T ] ≡ Q(γρ5 ⊗ ξµν)QQ(γ5ρ ⊗ ξνµ)Q. Here ξµν is shorthand for I9 ⊗ ξµν , where I9 is the identity matrix in flavor space. We recall the definitions of the taste matrices: ξµν ≡ iξµξν and ξρ5 ≡ iξρξ5, where ξ5 ≡ ξ1ξ2ξ3ξ4, and similarly for the spin matrices. In Sec. III B we will specialize to the Weyl basis for the taste matrices, in which  0 −iσ  and ξ4 =  0 I2 . (4) For now, the taste basis is arbitrary; different choices correspond to different interpreta- tions of the underlying staggered degrees of freedom in terms of continuum taste degrees of freedom. Promoting the taste matrices to spurion fields and displaying the chiral structures of these operators explicitly, one can write each of the operators in LFF (A)6 in one of three distinct forms [17, 26]. The six operators with vector and axial spin structure (3a) can each be written in the form OF = ± QR(γµ ⊗ FR)QR ±QL(γµ ⊗ FL)QL , (5) where FL,R are spurions transforming so that OF is invariant under chiral transformations, Euclidean rotations, parity, and charge conjugation: SU(N)L × SU(N)R : FL → LFLL† (6) FR → RFRR† SO(4)E : FL → FL FR → FR P : FL ↔ FR C : FL ↔ F TR . The four operators with scalar and pseudoscalar spin structure (3b) can be written in the O′F = QL(I4 ⊗ F̃L)QR ±QR(I4 ⊗ F̃R)QL , (7) where the spurions F̃L,R transform as follows: SU(N)L × SU(N)R : F̃L → LF̃LR† (8) F̃R → RF̃RL† SO(4)E : F̃L → F̃L F̃R → F̃R P : F̃L ↔ F̃R C : F̃L → F̃ TL F̃R → F̃ TR . The two tensor operators (3c) can be written O′′F = QL(γµν ⊗ F̃L)QR QR(γνµ ⊗ F̃R)QL , (9) where the spurions are the same as in O′F . For convenience in mapping to the heavy baryon theory, I use the meson fields σ to construct objects with the spurions that transform in the adjoint of the vector subgroup of the chiral group. Let l ≡ σ†FLσ and l̃ ≡ σ†F̃Lσ† r ≡ σFRσ† r̃ ≡ σF̃Rσ. SU(N)L × SU(N)R : l → UlU † l̃ → Ul̃U † (10) r → UrU † r̃ → Ur̃U † SO(4)E : l → l l̃ → l̃ r → r r̃ → r̃ P : l ↔ r l̃ ↔ r̃ C : l ↔ rT l̃ → l̃T r̃ → r̃T . The spurion-meson objects l, r, l̃, and r̃ can be further combined to construct definite-parity objects that are Hermitian: l ⊗ l ± r ⊗ r and l ⊗ r ± r ⊗ l (11) l̃ ⊗ l̃ + r̃ ⊗ r̃ l̃ ⊗ r̃ + r̃ ⊗ l̃ −i(l̃ ⊗ l̃ − r̃ ⊗ r̃) −i(l̃ ⊗ r̃ − r̃ ⊗ l̃) The objects involving l̃ and r̃ are effectively Hermitian because one sets the left- and right- handed spurion fields equal to one another at the end of the enumeration, which implies l̃† = r̃. This relation also implies that the above list is complete. Moreover, in constructing the chiral operators, the only axial vector that is allowed is Sµ, which can occur once. However, LFF (A)6 is Lorentz invariant, and v·S = 0 is the only Lorentz invariant containing Sµ that can be constructed. Therefore, Sµ does not appear in the chiral operators corresponding to LFF (A)6 , and only the parity-even spurion-meson combinations in (11) are allowed. Lorentz invariance also implies that operators mixing the spin-3 fields Tµ and the spin- fields do not appear; to be Lorentz invariant, such operators at this order must contain either v · T or S · T , but both vanish. A priori, chirally invariant baryon bilinears may contain zero, one, two, or three pairs of contractions between the flavor-taste indices of the baryon fields and the flavor-taste indices of the meson fields. Explicitly, WBijk B Xni Bnjk B ij Bnqk B ijkBnql, (12) where W , X , Y , and Z are constructed out of the non-baryonic building blocks and are quadratic in the spurions. However, it turns out that only operators with two pairs of con- tractions are relevant here. The W - and X-type operators contribute only irrep-independent corrections to the masses, while Z-type operators only modify the LECs of the twice- contracted (Y -type) operators. First consider the W -type operators. Since there are no contractions between the flavor- taste indices of the meson fields inW and the baryon fields, W can be any operator appearing in the remnant-taste symmetric potential V of the meson sector [7, 17]. Setting Σ = 1+O(φ) in these terms, V collapses to a single constant. Therefore, W -type operators contribute an O(a2), irrep-independent (generic) correction to the masses of the baryons. The X- and Z-type operators reduce similarly. For example, consider Xni ≡ (lr + rl) ni and Z ijk ≡ (l j + r j )(Σ + Σ †) lk . The taste matrices all square to the identity, so setting the spurions to their final values and σ = 1 gives Xni ∝ δni , and the X-type operators reduce to W -type operators. As for the Z-type operators, the fact that all operators must be quadratic in the spurions means that only Σ can be used to provide a third set of indices in Z. But Σ is set to unity at the end of the enumeration, so these operators reduce to the twice-contracted form. Therefore, all chiral operators leading to distinct spin-taste breaking (irrep-dependent) corrections to the baryon masses have two pairs of contractions between the flavor-taste indices on the baryon fields and those on the meson fields. Assuming the indicial symmetries in Eqs. (1), there are only two linearly independent, Y -type contraction structures: ij Bnqk and B ij Bknq. (13) Although the symmetries in Eqs. (1) are not generally valid for the partially quenched case, for which various minus signs must be inserted, only required are those O(a2) operators needed for the calculation of tree-level corrections to masses of baryons composed of valence quarks. In this case, the graded indicial symmetries reduce to those given in Eqs. (1), so the latter are correct for the operators considered here, and the contraction structures (13) exhaust the possibilities for the relevant operators. For operators with vector or axial spin structure (3a, 5), setting ij = l j + r j , l j + r inserting the scalar, pseudoscalar, and tensor taste matrices in the spurions, and introducing constants of proportionality gives [V × S] and [A× S] → (−(±b1 + b2) + 2(±c1 + c2))B [V × P ] and [A× P ] → ±b1B (σ†ξ5σ) †ξ5σ) j + (σξ5σ †) ni (σξ5σ + b2B (σ†ξ5σ) i (σξ5σ j + (σξ5σ †) ni (σ †ξ5σ) ± c1B (σ†ξ5σ) †ξ5σ) j + (σξ5σ †) ni (σξ5σ + c2B (σ†ξ5σ) i (σξ5σ j + (σξ5σ †) ni (σ †ξ5σ) [V × T ] and [A× T ] → ±b1B (σ†ξµνσ) †ξνµσ) j + (σξµνσ †) ni (σξνµσ + b2B (σ†ξµνσ) i (σξνµσ j + (σξµνσ †) ni (σ †ξνµσ) ± c1B (σ†ξµνσ) †ξνµσ) j + (σξµνσ †) ni (σξνµσ + c2B (σ†ξµνσ) i (σξνµσ j + (σξµνσ †) ni (σ †ξνµσ) Here the taste matrix ξτ is being used as shorthand for I3 ⊗ ξτ , where I3 is the identity matrix in valence quark flavor space. For operators with scalar or pseudoscalar spin structure (3b, 7), setting ij = l̃ j + r̃ j , l̃ j + r̃ inserting the vector and axial taste matrices in the spurions, and introducing constants of proportionality gives [S × V ] and [P × V ] → e1B (σ†ξνσ †) ni (σ j + (σξνσ) i (σξνσ) ± e2B (σ†ξνσ †) ni (σξνσ) j + (σξνσ) + f1B (σ†ξνσ †) ni (σ j + (σξνσ) i (σξνσ) ± f2B (σ†ξνσ †) ni (σξνσ) j + (σξνσ) [S × A] and [P ×A] → e1B (σ†ξν5σ †) ni (σ †ξ5νσ j + (σξν5σ) i (σξ5νσ) ± e2B (σ†ξν5σ †) ni (σξ5νσ) j + (σξν5σ) †ξ5νσ + f1B (σ†ξν5σ †) ni (σ †ξ5νσ j + (σξν5σ) i (σξ5νσ) ± f2B (σ†ξν5σ †) ni (σξ5νσ) j + (σξν5σ) †ξ5νσ Finally, for operators with tensor spin structure (3c, 9), one proceeds as before except that, noting the form of O′′F (9), one omits operators corresponding to cross-terms of left- and right-handed spurions. Setting ij = l̃ j + r̃ inserting vector and axial taste matrices, and introducing constants of proportionality gives [T × V ] → e′1B (σ†ξνσ †) ni (σ j + (σξνσ) i (σξνσ) + f ′1B (σ†ξνσ †) ni (σ j + (σξνσ) i (σξνσ) [T × A] → e′1B (σ†ξν5σ †) ni (σ †ξ5νσ j + (σξν5σ) i (σξ5νσ) + f ′1B (σ†ξν5σ †) ni (σ †ξ5νσ j + (σξν5σ) i (σξ5νσ) Although these operators have been deduced within the heavy baryon framework, all the operators listed here do in fact arise when deriving the heavy baryon Lagrangian from the relativistic baryon chiral Lagrangian. To confirm this assertion, one may map LFF (A)6 to the relativistic chiral theory in Euclidean space, Wick rotate, and then execute the non- relativistic reduction. For the operators listed above, this analysis has been carried out. 3. LFF (B)6 The development parallels that for LFF (A)6 . There are four types of operators in LFF (B)6 [17, 26]: LFF (B)6 ∼ [Vµ × Tµ] + [Aµ × Tµ] (14a) + [Tµ × Vµ] + [Tµ × Aµ]. (14b) In these operators, the indices of the spin and taste matrices are contracted together; e.g., [Vµ × Tµ] ≡ ν 6=µ Q(γµ ⊗ ξµν)QQ(γµ ⊗ ξνµ)Q−Q(γµ ⊗ ξµν5)QQ(γµ ⊗ ξ5νµ)Q Up to a taste singlet component that has no influence on the map to the chiral theory [26], these four operators may each be written in one of two forms. The two operators with vector and axial spin structure (14a) can be written in the form OFF (B)F = ± QR(γµ ⊗ FRµ)QR ±QL(γµ ⊗ FLµ)QL , (15) where (FL,R)µ are spurions transforming so that OFF (B)F is invariant under chiral transfor- mations, Euclidean hypercubic rotations, parity, and charge conjugation. Except for the Euclidean spacetime index on the spurions, OFF (B)F is the same as OF (5). Therefore, ex- cepting Euclidean rotations, the spurions transform as in (6); under Euclidean rotations, the spurions transform in the fundamental representation (rep): SO(4)E : FLµ → ΛµνFLν FRµ → ΛµνFRν . The two operators with tensor spin structure (14b) can be written in the form O′′FF (B)F = ν 6=µ QL(γµν ⊗ F̃Lµ)QRQR(γνµ ⊗ F̃Rµ)QL, (16) where, once again excepting Euclidean rotations, the spurions (F̃L,R)µ transform as in (8). Under Euclidean rotations, the spurions again transform as vectors: SO(4)E : F̃Lµ → ΛµνF̃Lν F̃Rµ → ΛµνF̃Rν . For purposes of mapping to the heavy baryon theory, I again use the meson fields σ to construct spurion-meson objects that transform in the adjoint of the vector subgroup of the chiral group. Let lµ ≡ σ†FLµσ and l̃µ ≡ σ†F̃Lµσ† rµ ≡ σFRµσ† r̃µ ≡ σF̃Rµσ. Then these objects transform as in (10) but are to be treated as vectors under Euclidean rotations. The Hermitian, definite-parity objects are constructed as before (cf. (11)). How- ever, unlike the operators in LFF (A)6 , the operators in L FF (B) 6 break Lorentz invariance. The argument given in the paragraph following (11) no longer applies, and the operators can contain both Sµ and parity-odd spurion-meson objects. A priori, the chiral operators can have any of the contraction structures given in (12). However, the previous arguments against X- and Z-type operators go through unchanged, while W -type operators cannot contribute because they require derivatives to break Lorentz invariance and are therefore necessarily O(a2k2)=O(ε4) or higher, in accord with the central result of [17]. Therefore only the two contraction structures given in (13) enter. For operators with vector or axial spin structure (14a, 15), setting ij = vµv µ((lµ) i (lµ) j + (rµ) i (rµ) j ), vµv µ((lµ) i (rµ) j + (rµ) i (lµ) vµSµ((lµ) i (rµ) j − (rµ) ni (lµ) inserting the tensor taste matrices in the spurions, introducing constants of proportionality, and summing the remaining tensor index (cf. [26]) gives [Vµ × Tµ] and [Aµ × Tµ] → ν 6=µ (σ†ξµνσ) †ξνµσ) j + (σξµνσ †) ni (σξνµσ ν 6=µ (σ†ξµνσ) i (σξνµσ j + (σξµνσ †) ni (σ †ξνµσ) + h′2 ν 6=µ SµBknq (σ†ξµνσ) i (σξνµσ j − (σξµνσ†) ni (σ†ξνµσ) ν 6=µ (σ†ξµνσ) †ξνµσ) j + (σξµνσ †) ni (σξνµσ ν 6=µ (σ†ξµνσ) i (σξνµσ j + (σξµνσ †) ni (σ †ξνµσ) + p′2 ν 6=µ SµBnqk (σ†ξµνσ) i (σξνµσ j − (σξµνσ†) ni (σ†ξνµσ) Two points deserve additional comment. First, the operator multiplying h′2 is not Hermitian. The culprit is the coincidence of the second contraction structure in (13) with a spurion- meson object that is anti-symmetric under simultaneous interchange of the upper and lower flavor-taste indices. This consideration did not arise in mapping LFF (A)6 because parity and Lorentz invariance conspired to allow only symmetric spurion-meson objects. The remedy is to find a Hermitian linear combination of the contraction structures given in (13) and replace the operator multiplying h′2 with its Hermitian counterpart. For anti-symmetric spurion-meson objects, the sum of the operators multiplying h′2 and p 2 is anti-Hermitian. Using the cyclic property in (1), one may take ij Bqkn as the second linearly independent contraction structure for anti-symmetric spurion-meson objects. Then the operator multiplying h′2 above is replaced with the Hermitian operator ν 6=µ SµBqkn (σ†ξµνσ) i (σξνµσ j − (σξµνσ†) ni (σ†ξνµσ) Second, at the level of the relativistic baryon chiral Lagrangian, charge conjugation invari- ance and Hermiticity conspire to forbid operators containing the spurion-meson object ij = v µSµ((lµ) i (lµ) j − (rµ) ni (rµ) and such operators have been eliminated from the above list (h′1 = p 1 ≡ 0). For tree-level corrections (σ = σ† = 1) or quantities calculated in the rest frame (v = 0), this detail does not affect the results. For operators with tensor spin structure (14b, 16), one proceeds as before except that, noting the form of O′′FF (B)F (16), one omits operators corresponding to direct-terms of left- and right-handed spurions. Setting ij = vµv µ((l̃µ) i (r̃µ) j + (r̃µ) i (l̃µ) j ), −ivµSµ((l̃µ) ni (r̃µ) j − (r̃µ) ni (l̃µ) inserting vector and axial taste matrices, and introducing constants of proportionality gives [Tµ × Vµ] → s (σ†ξµσ †) ni (σξµσ) j + (σξµσ) SµBqkn (σ†ξµσ †) ni (σξµσ) j − (σξµσ) ni (σ†ξµσ†) (σ†ξµσ †) ni (σξµσ) j + (σξµσ) −ivµBijkSµBnqk (σ†ξµσ †) ni (σξµσ) j − (σξµσ) ni (σ†ξµσ†) [Tµ ×Aµ] → s (σ†ξµ5σ †) ni (σξ5µσ) j + (σξµ5σ) †ξ5µσ SµBqkn (σ†ξµ5σ †) ni (σξ5µσ) j − (σξµ5σ) ni (σ†ξ5µσ†) (σ†ξµ5σ †) ni (σξ5µσ) j + (σξµ5σ) †ξ5µσ −ivµBijkSµBnqk (σ†ξµ5σ †) ni (σξ5µσ) j − (σξµ5σ) ni (σ†ξ5µσ†) where the procedure followed for the h′2 operator above has already been performed; opera- tors multiplying s′ have been replaced with their Hermitian counterparts. As before, all the operators given here do arise when deriving the heavy baryon Lagrangian from the relativistic baryon chiral Lagrangian; I have mapped LFF (B)6 to the relativistic chiral theory in Euclidean space, performed the Wick rotation, and performed the non-relativistic reduction. In addition to the operators given here, there are operators that mix spin-1 spin-3 fields and operators that contain more powers of the baryon four-velocity. In the rest frame, the former vanish, while the latter contribute nothing new. I therefore ignore such operators. C. Staggered heavy baryon Lagrangian for O(ε3) baryon octet masses In the rest frame of the heavy baryon, v = 0, so the only non-vanishing component of the four-velocity is v0 = 1. But v = 0 implies that S0 = 0, and it follows from the form of the enumerated operators that O(a2) operators containing Sµ do not contribute to the rest frame baryon masses. Setting σ = σ† = 1 in the remaining operators and introducing an LEC for each taste-violating term in the Lagrangian that is distinct at tree level, L(2)A′ φB,a2 = A1B Bknq(ξ5) i (ξ5) j + A2B Bnqk(ξ5) i (ξ5) j (17) + A3B (ξµν) i (ξµν) j + A4B (ξµν) i (ξµν) + A5B Bknq(ξν) i (ξν) j + A6B Bnqk(ξν) i (ξν) + A7B Bknq(ξν5) i (ξν5) j + A8B Bnqk(ξν5) i (ξν5) L(2)B′ φB,a2 = B1B ν 6=4 (ξ4ν) i (ξ4ν) j +B2B ν 6=4 (ξ4ν) i (ξ4ν) j (18) + B3B Bknq(ξ4) i (ξ4) j +B4B Bnqk(ξ4) i (ξ4) + B5B Bknq(ξ45) i (ξ45) j +B6B Bnqk(ξ45) i (ξ45) In the fully dynamical case, the flavor-taste indices ijknq take the values 1 to 12. Recalling that ξτ for τ ∈ {I, µ, µν(µ < ν), µ5, 5} is shorthand for I3⊗ξτ , we see that the O(a2) heavy baryon Lagrangian is invariant under arbitrary flavor transformations, SU(3)F ⊂ SU(12)f . However, the taste matrices in L(2)A′ φB,a2 break taste down to the remnant taste group of [17], Γ4 ⋊ SO(4)T ⊂ SU(4)T , while in an arbitrary reference frame, the chiral operators arising from LFF (B)6 (Wick rotated to Euclidean space) break taste to the lattice symmetry group, Γ4 ⋊ SW4,diag ⊂ [Γ4 ⋊ SO(4)T ]× SO(4)E. SW4,diag is the hypercubic group in the diagonal of the taste and spacetime SO(4)’s; Γ4 is the Clifford group generated by shifts by one lattice site [17]. In contrast, the operators appearing in L(2)B′ φB,a2 are manifestly invariant under a larger group: Under the taste subgroup that corresponds to the spatial rotations, which excludes (taste) boosts, the taste matrices ξν for ν = 1, 2, 3 transform as components of a three- vector, and the taste matrices ξ4 and ξ5 are invariant. Moreover, in the rest frame we Term in Lχ Case Flavor-taste-spacetime symmetry L(1)′φB + L φB,m + L φ isospin limit, m 6= 0, any v, a SU(8)x,y × SU(4)z × SO(4)E V any m, v, a U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ]× SO(4)E L(2)A′ φB,a2 any m, v = 0, any a U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ]× SU(2)E L(2)B′ φB,a2 any m, v = 0, any a U(3)l × U(3)r × [Γ4 ⋊ SU(2)T ]× SU(2)E TABLE I: The valence quark symmetries of terms in the Euclidean staggered heavy baryon La- grangian that are needed for computing the octet baryon masses to O(ε3). The taste symmetry in the rest frame of the heavy baryons is not completely broken to the lattice symmetry group. are free to make arbitrary spatial rotations. Recalling the taste subgroup corresponding to spatial rotations of spinors, SU(2)T ⊂ SO(4)T , one concludes that the O(a2) Lagrangian is invariant under U(3)l × U(3)r × [Γ4 ⋊ SU(2)T ]× SU(2)E , (19) where SU(2)E is the group of spatial rotations. Table I lists the symmetries of the various terms of the Lagrangian used in the calculation, Lχ = L(1)′φB + L φB,m + L φ − a 2V + a2L(2)A′ φB,a2 + a2L(2)B′ φB,a2 where L(1)′φ denotes the leading staggered chiral Lagrangian in the meson sector at zero lattice spacing, and V is the generalized Lee-Sharpe potential [7, 17]. III. THE FLAVOR-SYMMETRIC NUCLEONS A. Identifying staggered nucleons and irreps of interpolating fields As shown in [20], the 572M and 364S contain not only baryons that are degenerate in the continuum limit with the octet and decuplet states of nature, but also baryons that have unphysical masses; the latter states are degenerate with certain partially quenched octet and decuplet baryons. To calculate the masses of baryons degenerate with a given member of the octet or decuplet, one must choose a basis in the flavor-taste space in which degeneracies with the desired states are evident. Consider the valence sector of the partially quenched theory. Decomposing the 572M and 364S of SU(12)f into irreps of the flavor-taste subgroup gives SU(12)f ⊃ SU(3)F × SU(4)T 572M → (10S, 20M)⊕ (8M, 20S)⊕ (8M, 20M)⊕ (8M, 4̄A)⊕ (1A, 20M) (20a) 364S → (10S, 20S)⊕ (8M, 20M)⊕ (1A, 4̄A) (20b) Assume that taste is restored in the continuum limit. Taking the valence quark masses equal so that SU(3)F is exact in the valence sector, SU(12)f is a good symmetry as well, and all the baryons of the 572M are degenerate, as are all the baryons of the 364S. The symmetry of the spin-1 baryons is the same as the symmetry of the octet, and the symmetry of the spin-3 baryons, the same as that of the decuplet. Therefore, setting the masses of all three valence quarks and the masses of two sea quarks equal to the average up-down quark mass while setting the mass of the remaining sea quark equal to that of the strange quark mass, the baryons of the 572M are degenerate with the nucleon (in the limit of exact isospin), and those of the 364S, with the ∆. Increasing the mass of the strange valence quark to its physical value does not change the masses of baryons that do not contain a strange valence quark; such baryons remain de- generate with the nucleon or the ∆. In particular, the isospin-3 members of the (10S, 20M) remain degenerate with the nucleon. These states are flavor-symmetric; interpolating fields for the single-flavor members of the (10S, 20M) were constructed in [23]. In the contin- uum limit, the SU(8)x,y ×SU(4)z symmetry of the valence sector allows one to rotate these flavor-symmetric nucleons into those with physical flavor and simple taste structure, i.e., baryons that are manifestly degenerate with the nucleon [20]. The interpolating fields for the flavor-symmetric nucleons fall into irreps of the geometrical time-slice group (GTS) [23]. Decomposing the continuum irrep of the flavor-symmetric nucleons into irreps of GTS gives [23] , 20M) → 3(8)⊕ 16, where a direct product with continuum parity is suppressed on both sides of the decom- position. At nonzero lattice spacing, discretization effects lift the degeneracy of the GTS irreps and introduce mixing among corresponding members of the 8’s. Interpolating fields transforming in the 8 of GTS overlap corresponding members of each of the three 8’s, while interpolating fields transforming in the 16 overlap corresponding members of the 16. B. Taste symmetry and the mass matrix Consider one of the single-flavor, isospin-3 members of the 10S of the (10S, 20M). At nonzero lattice spacing, taste violations partially lift the degeneracy of the members of the 20M. The baryons are degenerate within irreps of the remnant taste symmetries respected by the relevant propagators and vertices at a given order in the staggered chiral expansion. To O(ε3) in the staggered chiral expansion, the masses receive analytic contributions of O(mq) and O(a2) at tree level, non-analytic contributions of O(m3/2q ) from loops with virtual spin- baryons, and non-analytic contributions of O(∆mq lnmq) from loops with virtual spin-32 baryons. The analytic contributions proportional to mq come from L(2)′φB,m; referring to Table I, we see that they respect SU(8)x,y × SU(4)z × SO(4)E. The vertices and heavy baryon propagators in the loops come from L(1)′φB , while the meson propagators come from L φ and V. Therefore the loops respect SU(2)x,y ×U(1)z × [Γ4⋊SO(4)T ]×SO(4)E; the loops break taste from SU(4)T to the remnant taste symmetry Γ4 ⋊ SO(4)T . Under Γ4 ⋊ SO(4)T , the 20M of SU(4)T decomposes into one 12-dimensional irrep and two 4-dimensional irreps: SU(4)T ⊃ Γ4 ⋊ SO(4)T 20M → 12⊕ 2(4). Finally, analytic contributions proportional to a2 come from L(2)A′ φB,a2 and L(2)B′ φB,a2 . The former do not break taste further than the loops; the latter break taste to Γ4 ⋊ SU(2)T . Under Γ4 ⋊ SU(2)T , the 12 of Γ4 ⋊ SO(4)T decomposes further into an 8 and a 4, while the two 4’s of Γ4 ⋊ SO(4)T are irreducible under Γ4 ⋊ SU(2)T : Γ4 ⋊ SO(4)T ⊃ Γ4 ⋊ SU(2)T 12 → 8⊕ 4 4 → 4. Taste violations in the loops and in the O(a2) terms from L(2)A′ φB,a2 lift the continuum degener- acy between baryons in the 12 and those in the two 4’s of Γ4⋊SO(4)T and introduce mixing between corresponding states in the two 4’s; such states have the same Γ4 ⋊ SO(4)T quan- tum numbers. Taste violations in the O(a2) terms from L(2)B′ φB,a2 lift the remaining degeneracy between baryons in the 8 and those in the 4 of Γ4⋊SU(2)T and mix corresponding states in the three 4’s of Γ4 ⋊ SU(2)T ; in this case, corresponding states have the same Γ4 ⋊ SU(2)T quantum numbers. The quantum numbers of these remnant taste symmetries are eigenvalues of maximal sets of commuting observables (MSCO) corresponding to each symmetry. To construct such sets, first consider the decomposition of the Γ4⋊SO(4)T irreps under the SO(4)T subgroup: Γ4 ⋊ SO(4)T ⊃ SO(4)T (21a) (21b) where the isomorphism SO(4) ≃ SU(2) × SU(2)/Z2 has been used to label the SO(4)T irreps; the conserved observables are simply the spins corresponding to the SU(2)’s. The decomposition (21b) is explicit in the Weyl representation of the taste matrices, so it is convenient to work in this representation. In the Weyl representation, the only diagonal member of Γ4 that commutes with but is not redundant with the spins is ξ5. Therefore, one MSCO of Γ4 ⋊ SO(4)T is {ξ5, J2L, J2R, JLz, JRz}, where JL and JR denote the spins. Tree-level corrections of O(a2) from L(2)B′ φB,a2 break Γ4⋊SO(4)T to Γ4⋊SU(2)T by breaking SO(4)T to SU(2)T , where SU(2)T is the subgroup of SO(4)T obtained by rotating left- handed and right-handed Weyl spinors together; decomposing the SO(4)T irreps under SU(2)T is an elementary exercise in addition of angular momentum: SO(4)T ⊃ SU(2)T (22a) (22b) The sum of left- and right-handed spins is respected by Γ4⋊SU(2)T , and the corresponding MSCO is {ξ5, (JL + JR)2, JLz + JRz}. Table II lists the MSCO’s of the remnant taste symmetries and notation for the corresponding quantum numbers. It is not difficult to see that Γ4 contains a taste parity operation: ξ4 interchanges the left- and right-handed spin quantum numbers in the product irreps of (21); moreover, under Remnant taste symmetry MSCO Quantum numbers Γ4 ⋊ SO(4)T ξ5, J R, JLz, JRz ξ5, jL, jR, mL, mR Γ4 ⋊ SU(2)T ξ5, (JL + JR) 2, JLz + JRz ξ5, j, m TABLE II: Maximal sets of commuting observables (MSCO’s) for each remnant taste symmetry. The quantum numbers of each MSCO are used to distinguish generically nonzero mixing elements in the mass matrix from off-diagonal elements that must vanish. SU(2)T in Γ4 ⋊ SU(2)T , Γ4 ⋊ SU(2)T ⊃ SU(2)T 8 → 3 (23a) 4 → 1 (23b) where taste parity interchanges the two SU(2)T irreps in each of the decompositions (23). Taste parity allows one to identify irreps of the remnant taste symmetries from the spin irreps appearing in the decompositions (21) and (23). The explicit form of the mass matrix depends on the basis chosen for the 20M. Taking tensor products of fundamental representations of SU(4)T , projecting onto the 20M, and demanding that the states also be eigenvectors of the MSCO of Γ4 ⋊ SO(4)T leads to |Naab〉 ≡ 1√6 (|aab〉 + |aba〉 − 2|baa〉) , a 6= b, a, b ∈ {1, 2, 3, 4} (24a) |Σabc〉 ≡ 1√12 (|abc〉 + |bac〉 − 2|cab〉+ |acb〉+ |bca〉 − 2|cba〉) (24b) |Λabc〉 ≡ 12 (|abc〉 + |acb〉 − |bac〉 − |bca〉) (24c) where in (24b) and (24c), abc ∈ {123, 124, 341, 342}. The names for the states are motivated by analogy with the states of the mixed irrep of SU(3)F ; of the 20 vectors spanning the 20M, 12 have a taste structure analogous to the flavor structure of the nucleon, 4 have a taste structure analogous to the flavor structure of the neutral Σ, and 4, taste structure analogous to the flavor structure of the Λ. The convenience of this basis is evident when one considers taste violations from loops and tree-level O(a2) contributions respecting Γ4 ⋊ SO(4)T . Although they are both linear combinations of states with the same taste labels, Σabc and Λabc do not mix because they are State of 20M ξ5 jL, jR mL mR N112 +1 N221 +1 N334 −1 N443 −1 Λ341 +1 Λ342 +1 Λ123 −1 Λ124 −1 N331 +1 +1 +1 N332 +1 +1 −1 N441 +1 −1 +1 N442 +1 −1 −1 Σ341 +1 Σ342 +1 N113 −1 N114 −1 N223 −1 N224 −1 Σ123 −1 Σ124 −1 TABLE III: Quantum numbers of Γ4⋊SO(4)T for each of the states in the 20M. States having the same set of Γ4⋊SO(4)T eigenvalues are mixed by taste violations. States with distinct eigenvalues are not mixed by interactions that respect Γ4 ⋊ SO(4)T . eigenstates of J2L and J R with different eigenvalues; the Σ’s are symmetric under interchange of the first two indices, and so are triplets, jL(R) = 1, while the Λ’s are antisymmetric under interchange of the first two indices, and so are singlets, jL(R) = 0. The Γ4 ⋊ SO(4)T quantum numbers of each of the states in the basis (24) are listed in Table III. As a set, the Γ4⋊SO(4)T quantum numbers of members of the 12 differ from the Γ4⋊SO(4)T quantum numbers of the two 4’s; interactions respecting Γ4 ⋊ SO(4)T do not mix members of the 12 with members of the two 4’s. In the same way, the Γ4 ⋊SO(4)T quantum numbers serve to completely distinguish members within each of the Γ4 ⋊ SO(4)T irreps; within each irrep, the mass submatrix of contributions that respect Γ4⋊SO(4)T is diagonal. This observation is consistent with the fact that members of a given irrep are not mixed by interactions respecting the corresponding symmetry group. However, each member of each 4 has the same quantum numbers as one member of the other 4. The taste violations mix N112 with Λ341, N221 with Λ342, and so on. An 8-dimensional submatrix corresponding to the two 4’s contains generically nonzero off-diagonal elements due to contributions from the loops and tree-level O(a2) corrections that respect Γ4 ⋊ SO(4)T . Note that states having the same Γ4 ⋊ SO(4)T quantum numbers are nonetheless distinct states; they have different SU(4)T quantum numbers. The form of the contributions to the mass matrix that respect Γ4 ⋊ SO(4)T follows from the symmetry of the quark flows and the Γ4 ⋊ SO(4)T symmetry; in the basis given in Table III, the 8-dimensional submatrix corresponding to the two 4’s (for any given member of the 10S) has the form  c1 0 0 0 c3 0 0 0 0 c1 0 0 0 −c3 0 0 0 0 c1 0 0 0 c3 0 0 0 0 c1 0 0 0 −c3 c3 0 0 0 c2 0 0 0 0 −c3 0 0 0 c2 0 0 0 0 c3 0 0 0 c2 0 0 0 0 −c3 0 0 0 c2  To O(ε3) in the staggered chiral expansion, the contributions parameterized here are given in Sec. IIIC. The results have been checked to verify that they conform to the pattern of degeneracies and mixings given here. The minus signs in the off-diagonal elements arise because some of the states in Table III are unconventionally normalized with respect to the ladder operators. Specifically, JR−|N112〉 = −|N221〉 JL−|N334〉 = −|N443〉 while JR−|Λ341〉 = |Λ342〉 JL−|Λ123〉 = |Λ124〉. In the continuum limit, the restoration of SU(4)T implies that c3 must vanish and c1 must equal c2. The tree-level taste violations vanish trivially; a straightforward exercise with the loop contributions completes the consistency check (cf. Appendix A). Tree-level O(a2) contributions from L(2)B′ φB,a2 may be parameterized in much the same way. The taste violations in such contributions split the baryons in the Γ4 ⋊ SO(4)T 12 between the 8 and a 4 of Γ4 ⋊ SU(2)T and introduce mixing between corresponding states in the resulting three 4’s of Γ4 ⋊ SU(2)T . The two 4’s of Table III are also irreducible under Γ4 ⋊ SU(2)T ; however, because the MSCO of Γ4 ⋊ SU(2)T includes the total angular mo- mentum (JL + JR) 2, the components JLz and JRz are no longer separately conserved, and contributions from L(2)B′ φB,a2 are not diagonal in the basis given in Table III for the 12. Con- structing eigenstates of the new MSCO gives the bases for the 8 and 4 of Γ4 ⋊ SU(2)T shown in Table IV. These states are related to those in Table III by the Clebsch-Gordan coefficients for adding spin 1 and spin 1 From the eigenvalues listed in Table IV, we see that the submatrix containing generically nonzero off-diagonal elements is 12-dimensional and corresponds to mixing among corre- sponding members of the three 4’s of Γ4⋊SU(2)T . In the basis of Table IV, the form of the mixing matrix implied by the symmetry of the quark flows and Γ4 ⋊ SU(2)T is  c4 0 0 0 c7 0 0 0 c8 0 0 0 0 c4 0 0 0 −c7 0 0 0 −c8 0 0 0 0 c4 0 0 0 c7 0 0 0 c8 0 0 0 0 c4 0 0 0 −c7 0 0 0 −c8 c7 0 0 0 c5 0 0 0 c9 0 0 0 0 −c7 0 0 0 c5 0 0 0 c9 0 0 0 0 c7 0 0 0 c5 0 0 0 c9 0 0 0 0 −c7 0 0 0 c5 0 0 0 c9 c8 0 0 0 c9 0 0 0 c6 0 0 0 0 −c8 0 0 0 c9 0 0 0 c6 0 0 0 0 c8 0 0 0 c9 0 0 0 c6 0 0 0 0 −c8 0 0 0 c9 0 0 0 c6  The contributions parameterized here are presented in Sec. IIIC; a straightforward exercise shows that they are consistent with this parameterization (cf. A). The minus signs again arise from the phase convention for |N221〉 and |N443〉. C. Masses of the flavor-symmetric nucleons We now consider the staggered chiral forms for the masses of the flavor-symmetric nu- cleons having degenerate valence quarks of mass mx. The forms are given through O(ε3) in fully dynamical and partially quenched SHBχPT. Let the tree-level contributions pro- portional to mq be denoted by Tree(mq), the tree-level contributions proportional to a by Tree(a2), the loops with virtual spin-1 baryons, by Loop(1 ), and the loops with vir- tual spin-3 baryons, by Loop(3 ). These contributions are of O(mq), O(a2), O(m3/2q ), and O(∆mq lnmq), respectively. The mass matrix is given by M = M0 + Σ(0), (27) Σ(0) = Tree(mq) + Tree(a 2) + Loop(1 ) + Loop(3 ) (28) where M0 is the average mass of the 572M in the continuum and chiral limits, and Σ(0) is the heavy baryon self-energy evaluated at v · r = 0; here r is the residual 4-momentum of the heavy baryon. State of 20M ξ5 j m N112 +1 N221 +1 N334 −1 12 + N443 −1 12 − Λ341 +1 Λ342 +1 Λ123 −1 12 + Λ124 −1 12 − N332 − 1√ Σ341 +1 N441 + Σ342 +1 N114 − 1√ Σ123 −1 12 + N223 + Σ124 −1 12 − N331 +1 N332 + Σ341 +1 N441 + Σ342 +1 N442 +1 N113 −1 32 + N114 + Σ123 −1 32 + N223 + Σ124 −1 32 − N224 −1 32 − TABLE IV: Eigenvalues of the MSCO of Γ4 ⋊ SU(2)T for each of the eigenstates in the 20M. States with the same eigenvalues are mixed by contributions respecting Γ4 ⋊ SU(2)T . In accord with the discussion of Sec. III B, M0 and Tree(mq) are diagonal matrices in the baryon-taste subspace corresponding to the 20M of SU(4)T ; explicitly, M0 + Tree(mq) = [M0 − 2(αM + βM)mx − 2σM(mu +md +ms)] I20, where I20 is the identity matrix. Loop( ), Loop(3 ), and the contributions of L(2)A′ φB,a2 Tree(a2) are diagonal in the 12 of Γ4 ⋊ SO(4)T but have the form given in (25) in the subspace corresponding to the two 4’s of Γ4 ⋊ SO(4)T . The contributions of L(2)B′φB,a2 to Tree(a2) are diagonal in the 8 of Γ4 ⋊ SU(2)T but have the form given in (26) in the three 4’s of Γ4 ⋊ SU(2)T . For convenience in writing the results, let Loop(1 σconn1/2 + σ Loop(3 (2π)2 σconn3/2 + σ Tree(a2) = −a2 σFF (A) + σFF (B) where σconn is the sum of loops with connected meson propagators, σdisc is the sum of loops with disconnected (hairpin) propagators, and σFF (A,B) are tree-level contributions from L(2)A,B′ φB,a2 . At tree-level, the pseudoscalar meson masses are degenerate within irreps of Γ4 ⋊ SO(4)T , so we label them with an index t ∈ {I, V, T, A, P} denoting the SO(4)T irrep: (mtij) 2 = λ(mi +mj) + a where i, j ∈ {u, d, s, x, y, z}, λ is proportional to the chiral condensate, and ∆t is the meson taste-splitting [7]. Let τ ∈ {I, µ, µν(µ < ν), µ5, 5} be the meson taste index, and st, the set of meson tastes in irrep t: sI = {I}, sV = {µ}, sT = {µν(µ < ν)}, sA = {µ5}, and sP = {5}. Finally, let nt denote the number of meson tastes corresponding to irrep t. 1, and f(τ) = for any function f(τ). If f is a function of t only, then f(τ) = f(t) = ntf(t). For the contributions to Loop(1 ), we have σconn1/2 = csea1/2,t (mtxq) 3 + cval1/2,t(m σdisc1/2 = 6(2(α + β)) 2iDIXX + (4c 1/2,V + c 1/2,V )iD XX + (V → A) where q ∈ {u, d, s}, the DtXX are hairpin loop integrals, and csea1/2,t and cval1/2,t are symmetric State(s) 〈cval 1/2,I 〉 〈cval 1/2,V 〉 〈cval 1/2,T 〉 〈cval 1/2,A 〉 〈cval 1/2,P N112 ( , 2, 10) (0, 0, 0) (1, −20, −28) (0, 0, 0) (7 , 2, 10) N331 ( , 2, 10) (1 , −20 , −28 ) (11 , −20 , −28 , −10 , −14 Λ123 ( , 2, 10) (5, −4, 4) (−3, −12, −24) (5, −4, 4) (−5 , 2, −2) N112|Λ341 (0, 0, 0) ( 6) (0, 0, 0) (− 6, −4 6, −8 6) (0, 0, 0) TABLE V: Coefficients of (α2, β2, αβ) for the distinct, nontrivial matrix elements of cval 1/2,t matrices depending on the irrep t and containing the LECs. In dimensional regularization, DIXX ≡ iµ4−n (2π)n (S · k)2 v · k + iǫ (k2 − (mIxx)2 + iǫ)2 XX ≡ a 2δ′V,Aµ (2π)n (S · k)2 v · k + iǫ (k2 − (mV,Axx )2 + iǫ)2 (k2 −m2η′ + iǫ) where (k2 − (mtuu)2 + iǫ) (k2 − (mtdd)2 + iǫ) (k2 − (mtss)2 + iǫ) (k2 −m2πt + iǫ) (k2 −m2ηt + iǫ) Contributions from quark flows that contain a sea quark circulating in the loop are rep- resented by csea 1/2,t ; an overall factor of 1 included in csea 1/2,t accounts for taking the fourth root of the fermion determinant in simulations. Contributions from quark flows containing only valence quarks in the meson propagator are represented by cval 1/2,t . In the basis (24) of Table III, the diagonal elements of csea 1/2,t 〈csea1/2,t〉 = nt(58α 2 + β2 + 1 and the off-diagonal elements of csea 1/2,t all vanish. Unlike csea 1/2,t , cval 1/2,t depends on the rep- resentation used for the taste matrices. Employing the Weyl representation, the distinct, nontrivial matrix elements of cval 1/2,t are listed in Table V. The results for c sea,val 1/2,t are consis- tent with the degeneracies and mixings parameterized in (25). Appendix A gives the general forms of the matrix elements of cval1/2,t in terms of the elements of the taste matrices. These forms are independent of the representation used for the taste matrices and can therefore be used to show that the chiral forms of continuum HBχPT are recovered in any representation. For the contributions to Loop(3 ), we have σconn3/2 = csea3/2,t F(mtxq) + cval3/2,tF(mtxx) σdisc3/2 = (4c 3/2,V + c 3/2,V )iE XX + (V → A) where F(m) ≡ ∆ (3m2 − 2∆2) ln(m2/µ2)− 4m2 + 10 m3g(∆/m), g(x) = (1− x2)3/2 arccosx 0 ≤ x ≤ 1 (x2 − 1)3/2 ln(x+ x2 − 1) x > 1 XX ≡ a 2δ′V,Aµ (2π)n kνkλP v · k −∆+ iǫ (k2 − (mV,Axx )2 + iǫ)2 (k2 −m2η′ + iǫ) the projection operator P νλ, in n spacetime dimensions, is P νλ = vνvλ − gνλ − 4 SνSλ. As before, the csea 3/2,t and cval 3/2,t are symmetric matrices in the baryon taste space. However, the coefficients for spin-3 contributions are simpler than for the spin-1 contributions. The diagonal elements of csea3/2,t are 〈csea3/2,t〉 = 12nt, while the off-diagonal elements vanish. In the Weyl representation, the distinct, nontrivial matrix elements of cval3/2,t are listed in Table VI. As for the spin- contributions, the results for sea,val 3/2,t are consistent with the degeneracies and mixings parameterized in (25). The general forms of the matrix elements of cval 3/2,t , in terms of the elements of the taste matrices, are given in Appendix A. Like the loops, σFF (A) is a symmetric matrix. In the Weyl representation, the distinct, nontrivial matrix elements of σFF (A) are listed in Table VII; they are consistent with the degeneracies and mixings parameterized in (25). The LECs Ai, i = 1, . . . , 8 are defined in Eq. (17). Appendix B contains the general forms of the matrix elements of σFF (A) in terms of the taste matrices appearing in (17). For the contributions to the symmetric matrix σFF (B), we consider the basis of Table IV. In the Weyl representation, the distinct, nontrivial matrix elements are listed in Table VIII; State(s) 〈cval 3/2,I 〉 〈cval 3/2,V 〉 〈cval 3/2,T 〉 〈cval 3/2,A 〉 〈cval 3/2,P N112 −2 0 20 0 −2 N331 −2 203 Λ123 −2 4 12 4 −2 N112|Λ341 0 −4 6 0 4 TABLE VI: The distinct, nontrivial matrix elements of cval 3/2,t State(s) A1 A3 A5 A7 A2 A4 A6 A8 N112 −12 −1 0 0 1 −4 0 0 N331 −16 − 0 −1 −1 0 −3 0 0 N112|Λ341 0 0 0 0 0 0 TABLE VII: Coefficients of Ai, i = 1, . . . , 8, for the matrix elements of σ FF (A). they are consistent with the degeneracies and mixings parameterized in (26). The LECs Bi, i = 1, . . . , 6 are defined in Eq. (18). Appendix B contains the general forms of the matrix elements of σFF (B) in terms of the taste matrices appearing in (18). Taking the continuum and isospin limits, the staggered chiral forms reduce to that of State(s) B1 B3 B5 B2 B4 B6 N112 −12 0 0 −2 0 0 N331 −16 − Λ123 0 −14 − N332 − 1√ Σ341 −23 N112|Λ341 0 0 0 0 N112| N332 − 1√3 Σ341 0 − Λ341| N332 − 1√ TABLE VIII: Coefficients of Bi, i = 1, . . . , 6, for the matrix elements of σ FF (B). continuum HBχPT [29, 32] for the partially quenched nucleon. For the fully dynamical 2+1 flavor case, the hairpin loop integrals simplify significantly; we have DIUU = iµ (2π)n (S · k)2 v · k + iǫ UU = a 2δ′V,Aµ (2π)n (S · k)2 v · k + iǫ X ′V,A (k2 −m2 + iǫ) UU = a 2δ′V,Aµ (2π)n kνkλP v · k −∆+ iǫ X ′V,A (k2 −m2 + iǫ) where X ′t ≡ (k2 − (mtuu)2 + iǫ)2 (k2 − (mtss)2 + iǫ) (k2 −m2πt + iǫ)(k2 −m2ηt + iǫ) Recalling the residue relations of [8], evaluating the integrals, and renormalizing the tree- level LECs gives DIUU → (m3ηI − 3m UU → − ia2δ′V,A R[3,1]πV, Am πV, A +R[3,1]ηV, Am ηV, A [3,1] ia2δ′V,A (2π)2 R[3,1]πV, AF(mπV,A) +R [3,1] ηV, A F(mηV, A) +R [3,1] F(mη′ where R [3,1] lV, A is shorthand for the residue R [3,1] lV, A ({mπV, A, mηV, A , mη′V, A}; {m ss }). Combining these results for the hairpin integrals with the coefficients csea and those listed in Tables V, VI, VII, and VIII, the parameters of (25) and (26) are, in the fully dynamical 2+1 flavor case, c1 = M0 − 2(αM + βM + 2σM )mu − 2σMms (31) 192πf 2 α2 + β2 + 1 t nt(2m +m3Kt) α2 + 2β2 + 10αβ)(m3πI +m ) + (α2 − 20β2 − 28αβ)m3πT + 4(α + β)2(m3ηI − 3m )− 16a2(5 α2 + β2 + 1 δ′V (R [3,1] m3πV +R [3,1] m3ηV +R [3,1] ) + (V → A) t nt[2F(mπt) + F(mKt)] − 2[F(mπI) + F(mπP )] + 20F(mπT ) − 8a2 δ′V (R [3,1] F(mπV ) +R[3,1]ηV F(mηV ) +R [3,1] F(mη′ )) + (V → A) A1 −A3 + A2 − 4A4 c2 = M0 − 2(αM + βM + 2σM)mu − 2σMms (32) 192πf 2 α2 + β2 + 1 nt(2m +m3Kt) α2 + 2β2 + 10αβ)m3πI + (5α 2 − 4β2 + 4αβ)(m3πV +m + (−3α2 − 12β2 − 24αβ)m3πT + (− α2 + 2β2 − 2αβ)m3πP + 4(α+ β)2(m3ηI − 3m α2 + β2 + 1 αβ) + (5α2 − 4β2 + 4αβ) δ′V (R [3,1] m3πV +R [3,1] m3ηV +R [3,1] ) + (V → A) t nt[2F(mπt) + F(mKt)] − 2[F(mπI ) + F(mπP )] + 4[F(mπV ) + F(mπA)] + 12F(mπT ) − 12a2 δ′V (R [3,1] F(mπV ) +R[3,1]ηV F(mηV ) +R [3,1] F(mη′ )) + (V → A) A1 − A5 − A7 − 3A4 c3 = − 192πf 2 6(α2 + 4β2 + 8αβ)(m3πV −m )− a2 6(α2 + 4β2 + 8αβ)× (33) δ′V (R [3,1] m3πV +R [3,1] m3ηV +R [3,1] )− (V → A) 6[F(mπV )− F(mπA)] + 4a2 δ′V (R [3,1] F(mπV ) +R[3,1]ηV F(mηV ) +R [3,1] F(mη′ ))− (V → A) 6 [A6 −A8] , c4 = −a2 B1 − 2B2 , (34) c5 = −a2 B3 − 14B5 − , (35) c6 = −a2 , (36) c7 = −a2 , (37) c8 = −a2 B4 − 12√6B6 , (38) c9 = −a2 B1 − 16B3 − . (39) In addition to the parameters ci, i = 1, . . . , 9, which correspond to members of Γ4⋊SU(2)T 4’s, define the parameters d1 and d2 to respectively correspond to the Γ4 ⋊ SO(4)T 12 and Γ4 ⋊ SU(2)T 8. Then d1 = M0 − 2(αM + βM + 2σM)mu − 2σMms (40) 192πf 2 α2 + β2 + 1 t nt(2m +m3Kt) α2 + 2β2 + 10αβ)m3πI + (α2 − 20β2 − 28αβ)(m3πV +m (11α2 − 4β2 + 16αβ)m3πT + α2 − 10β2 − 14αβ)m3πP + 4(α+ β)2(m3ηI − 3m α2 + β2 + 1 αβ) + 1 (α2 − 20β2 − 28αβ) δ′V (R [3,1] m3πV +R [3,1] m3ηV +R [3,1] ) + (V → A) nt[2F(mπt) + F(mKt)] − 2F(mπI) + 203 [F(mπV ) + F(mπA)] + F(mπT ) + 103 F(mπP ) δ′V (R [3,1] πV F(mπV ) +R [3,1] ηV F(mηV ) +R [3,1] F(mη′ )) + (V → A) A1 + 2A3 + A5 + A7 + 2A2 −A4 + 4A6 + 4A8], d2 = a B5 + 2B2 + 2B4 + 2B6 . (41) The mass submatrices equal to d1 I12 and d2 I8 were suppressed in (25) and (26), respectively. In the 2+1 flavor partially quenched case, double poles in the integrands of the hairpin loop integrals complicate the results for c1, c2, c3, and d1. Evaluating the integrals and renormalizing the tree-level LECs gives DIXX → − [2,2] · (mIxx) +D [2,2] XI ,XI (mIxx) [2,2] ηI , XI (mηI ) ia2δ′V,A [3,2] XV, A · (mV,Axx ) +D [3,2] XV, A, XV, A (mV,Axx ) [3,2] ηV, A,XV, A (mηV, A) [3,2] ,XV, A XX → − ia2δ′V,A (2π)2 [3,2] XV, A F ′(mV,Axx ) [3,2] XV, A, XV, A F(mV,Axx ) [3,2] ηV, A, XV, A F(mηV, A) +D [3,2] , XV, A F(mη′ where XI, V, A denote the states with masses m I, V, A xx , and, adapting the notation of [8], R [3,2] lV, A is shorthand for R [3,2] lV, A ({mV,Axx , mηV, A, mη′V, A}; {m uu , m ss }), [3,2] lV, A, XV, A xx )2] [3,2] lV, A [2,2] is shorthand for R [2,2] ({mIxx, mηI}; {mIuu, mIss})|m2→−m2 , and [2,2] lI , XI ≡ + d d[(mIxx) [2,2] Working in Minkowski space leads to the plus signs before the derivatives and the necessity of evaluating R [2,2] with all masses m replaced, relative to the conventions of [8], by im, where i2 = −1; R[3,2]lV, A is invariant under this substitution. The prime on F ′(m) represents the derivative with respect to m. Combining the results for the hairpin integrals with the coefficients csea, the coefficients listed in Tables V and VI, and the results (29) and (30), the parameters c1, c2, c3, and d1 are, in the 2+1 flavor partially quenched case, c1 = M0 − 2(αM + βM)mx − 2σM(2mu +ms) (42) 192πf 2 α2 + β2 + 1 t nt (2(m 3 + (mtxs) α2 + 2β2 + 10αβ) (mIxx) 3 + (mPxx) + (α2 − 20β2 − 28αβ)(mTxx)3 − 8(α+ β)2 [2,2] · (mIxx) +D [2,2] XI , XI (mIxx) [2,2] ηI , XI (mηI ) + 16a2(5 α2 + β2 + 1 [3,2] · (mVxx) +D [3,2] XV , XV (mVxx) [3,2] ηV , XV (mηV ) [3,2] + (V → A) t nt[2F(mtxu) + F(mtxs)] − 2[F(mIxx) + F(mPxx)] + 20F(mTxx) + 8a2 [3,2] F ′(mVxx) 2(mVxx) [3,2] XV ,XV F(mVxx) [3,2] ηV ,XV F(mηV ) +D [3,2] F(mη′ + (V → A) A1 −A3 + A2 − 4A4 c2 = M0 − 2(αM + βM)mx − 2σM(2mu +ms) (43) 192πf 2 α2 + β2 + 1 t nt (2(m 3 + (mtxs) α2 + 2β2 + 10αβ)(mIxx) 3 + (5α2 − 4β2 + 4αβ)((mVxx)3 + (mAxx)3) + (−3α2 − 12β2 − 24αβ)(mTxx)3 + (−52α 2 + 2β2 − 2αβ)(mPxx)3 − 8(α+ β)2 [2,2] · (mIxx) +D [2,2] XI , XI (mIxx) [2,2] ηI ,XI (mηI ) α2 + β2 + 1 αβ) + (5α2 − 4β2 + 4αβ) [3,2] · (mVxx) +D [3,2] XV , XV (mVxx) [3,2] ηV ,XV (mηV ) [3,2] + (V → A) t nt[2F(mtxu) + F(mtxs)] − 2[F(mIxx) + F(mPxx)] + 4[F(mVxx) + F(mAxx)] + 12F(mTxx) + 12a2 [3,2] F ′(mVxx) 2(mVxx) [3,2] XV ,XV F(mVxx) [3,2] ηV , XV F(mηV ) +D [3,2] F(mη′ + (V → A) A1 − A5 − A7 − 3A4 c3 = − 192πf 2 6(α2 + 4β2 + 8αβ)((mVxx) 3 − (mAxx)3) + a2 6(α2 + 4β2 + 8αβ)× (44) [3,2] · (mVxx) +D [3,2] XV ,XV (mVxx) [3,2] ηV , XV (mηV ) [3,2] + (V → A) 6[F(mVxx)− F(mAxx)] − 4a2 [3,2] F ′(mVxx) 2(mVxx) [3,2] XV ,XV F(mVxx) [3,2] ηV ,XV F(mηV ) +D [3,2] F(mη′ + (V → A) 6 [A6 −A8] , d1 = M0 − 2(αM + βM)mx − 2σM (2mu +ms) (45) 192πf 2 α2 + β2 + 1 t nt (2(m 3 + (mtxs) α2 + 2β2 + 10αβ)(mIxx) 3 + 1 (α2 − 20β2 − 28αβ)((mVxx)3 + (mAxx)3) (11α2 − 4β2 + 16αβ)(mTxx)3 + 13( α2 − 10β2 − 14αβ)(mPxx)3 − 8(α + β)2 [2,2] · (mIxx) +D [2,2] XI ,XI (mIxx) [2,2] ηI ,XI (mηI ) α2 + β2 + 1 αβ) + 1 (α2 − 20β2 − 28αβ) [3,2] · (mVxx) +D [3,2] XV ,XV (mVxx) [3,2] ηV , XV (mηV ) [3,2] + (V → A) t nt[2F(mtxu) + F(mtxs)] − 2F(mIxx) + 203 [F(m xx) + F(mAxx)] + 43F(m xx) + F(mPxx) [3,2] F ′(mVxx) 2(mVxx) [3,2] XV , XV F(mVxx) [3,2] ηV ,XV F(mηV ) +D [3,2] F(mη′ + (V → A) A1 + 2A3 + A5 + A7 + 2A2 − A4 + 4A6 + 4A8]. The distinct, nontrivial elements of the mass matrix of the single-flavor nucleons are simply related to the parameters ci and dj , j = 1, 2: 〈N112|M |N112〉 = c1 + c4 (46a) 〈Λ341|M |Λ341〉 = c2 + c5 (46b) N332 − 1√3 Σ341|M | N332 − 1√3 Σ341〉 = d1 + c6 (46c) 〈N331|M |N331〉 = d1 + d2, (46d) 〈N112|M |Λ341〉 = c3 + c7 (46e) 〈N112|M | N332 − 1√3 Σ341〉 = c8 (46f) 〈Λ341|M | N332 − 1√3 Σ341〉 = c9. (46g) The O(a2) tree-level terms can be roughly estimated by noting that Tree(mq) − a2σFF (A) absorbs the renormalization-scale dependence of the loop diagrams with virtual spin-3 baryons [33]. At a given scale, one expects the counterterms to be at least as large as the change in the loop diagrams under a change in the renormalization scale by an amount of order unity. If at some scale the counterterms were much smaller, then changing the scale would make the magnitude of the counterterms comparable to the change in the loops. For quark masses and lattice spacings for which the staggered chiral power counting is meaning- ful, the tree-level corrections of O(mq) should be roughly equal to the tree-level corrections of O(a2); one expects the expansion in the tree-level masses of the staggered mesons to be meaningful at tree-level. Varying the renormalization scale in the loops and associating the resulting O(a2) terms with valence and sea contributions of O(mq) gives Tree(a2) & −2(αM + βM) nt + c 3/2,t)− 2σM where the term proportional to (αM + βM) comes from the valence sector, and the term proportional to σM , from the sea. Given an estimate of the LECs αM , βM , σM , and λ from the continuum and the measured values of the meson mass splittings ∆t [14], one estimates the splittings to be roughly 10-40 MeV for lattice spacings of current interest. This splitting in the flavor-symmetric nucleon masses may have been observed already with unimproved staggered quarks [34]. Different nucleon operators transforming in the 8 of GTS would generically have differing overlaps with the nondegenerate states in the 20M, which could generate the observed operator dependence of the central mass values. D. Correspondence between chiral forms and interpolating fields The connection between the interpolating operators and the staggered chiral forms may be obtained by decomposing the irreps of the forms into irreps of the operators. To O(ε3), the staggered chiral forms respect the parity–spin–remnant-taste symmetry P × SU(2)E × [Γ4⋊SU(2)T ], while the operators transform within irreps of GTS. For the flavor-symmetric nucleons, the relevant decompositions are SU(2)E × [Γ4 ⋊ SU(2)T ] ⊃ GTS , 8) → 16 , 4) → 8, where the direct product with continuum parity has been suppressed on both sides of the decompositions. Operators transforming in the 8 of GTS overlap members of SU(2)E × [Γ4 ⋊ SU(2)T ] ( , 4)’s; operators transforming in the 16 overlap members of (1 , 8)’s. To O(ε3) in SχPT, members of a given SU(2)E × [Γ4 ⋊ SU(2)T ] irrep are degenerate, so the exact linear combination of states within the SU(2)E × [Γ4 ⋊ SU(2)T ] multiplet created by a given operator is unimportant; the staggered chiral forms are degenerate. To O(ε3), the flavor-symmetric nucleons transform in one (1 , 8) and three (1 , 4)’s. Operators transforming in the GTS 8 overlap states in the three 4’s of Γ4 ⋊ SU(2)T : Rep- resentative members of these irreps are the N112, Λ341, and N332 − 1√3 Σ341. Operators transforming in the 16 overlap states in the 8 of Γ4⋊SU(2)T : for example, the N331. For the case of 2+1 fully dynamical flavors, the staggered chiral forms for the masses of these states are given in Eq. (46) with Eqs. (31) through (41); for the 2+1 flavor partially quenched case, the corresponding results for c1,2,3 and d1 are given in Eqs. (42), (43), (44), and (45). IV. SUMMARY AND FURTHER DIRECTIONS Staggered, partially quenched chiral perturbation theory has been formulated in the light-quark baryon sector by introducing taste degrees of freedom in heavy baryon chi- ral perturbation theory and breaking the taste symmetry by mapping the operators of the O(a2) Symanzik action to the heavy baryon Lagrangian. Including operators of O(a2) in the Symanzik action allows one to calculate octet and decuplet baryon masses toO(ε3)=O(m3/2q ) in the staggered chiral expansion. As an example, the masses of the single-flavor nucleons have been calculated; the result for the mass matrix is consistent with the pattern of degen- eracies and mixings implied by the remnant taste symmetries, Γ4⋊SO(4)T and Γ4⋊SU(2)T . In the rest frame of the heavy baryon, the symmetry Γ4⋊SU(2)T emerges as the taste sym- metry of the chiral Lagrangian mapped from type B four-fermion operators in the Symanzik action [17]; the resulting spin-taste SU(2)E× [Γ4⋊SU(2)T ] protects against mixing between states of different spin. In the continuum limit, taste restoration forces all off-diagonal ele- ments of the mass matrix to vanish, while the diagonal elements reduce to the result obtained for the partially quenched nucleon in continuum HBχPT [29, 32]. The splittings in the nucleon mass must vanish as taste is restored, so they could be used to test taste restoration. The staggered chiral forms given here are those needed to quantify taste violations in simulation results obtained from the local corner-wall operators of [15]. At small quark masses, the lattice spacing and quark mass dependences of the data qualitatively agree with the staggered chiral forms given here. At larger quark masses, the O(ε3) chiral forms decrease as −m3φ, while the data continues to increase. Fits to continuum χPT that include analytic terms of O(m2q) seem to accurately describe the general trend of the data at large quark mass. At sufficiently large quark mass and fixed lattice spacing, continuum O(ε4)=O(m2q) terms will dominate over terms that are formally O(ε4)=O(mqa2)=O(a4), so it might be possible to describe the data by supplementing the results given here with con- tinuum terms of O(m2q). Alternatively, some authors argue that dimensional regularization in BχPT incorporates spurious high-energy physics [35]. If true, then it is possible that the data at larger quark masses would be better described by using a cut-off regulator. Such an approach amounts to resumming the perturbative expansion and might not require terms of O(ε4). Minimally, departing from dimensional regularization would require recalculating the loop integrals at O(ε3). Operators transforming in the 8 of GTS interpolate to three nucleon states and two ∆ states. Operators transforming in the 16 of GTS interpolate to only one nucleon, but to three ∆’s [20]. For extracting the mass spectrum, one would prefer operators that do not interpolate to states in SU(4)T -degenerate multiplets, which are split and mixed by discretization effects. By these criteria, operators that would be ideal for extracting the masses of the nucleon and the lightest decuplet were identified and constructed in [20]. The associated chiral forms for the nucleon, the ∆, and the Ω− have been calculated to O(ε3); these and the chiral forms for the Σ∗ and Ξ∗ will be reported in a future publication. There are plans to use these operators and chiral forms in the near future [36]. Acknowledgments The guidance and assistance of C. Bernard were essential at nearly every stage of this project. Funding was supplied in part by the U.S. Department of Energy under grant DE- FG02-91ER40628. APPENDIX A: QUARK-FLOW COEFFICIENTS AND TASTE MATRICES Here I write down the matrix elements of the quark-flow coefficients c sea,val 1/2,t and c sea,val 3/2,t in terms of the elements of the taste matrices ξτ , τ ∈ {I, µ, µν(µ < ν), µ5, 5}. These explicit forms can be used to verify that, in the Weyl representation of the taste matrices, the loop contributions are consistent with the parameterization (25). Because they are independent of the representation used for the taste matrices, they can also be used to verify that the continuum limits of the O(ε3) staggered chiral forms equal, in any representation, the corresponding O(ε3) chiral forms of continuum HBχPT. The result is immediate if one uses the completeness relation of the taste matrices: ξτabξ cd = 4δadδbc, where ξτ ≡ ξτ and ξτab ≡ 〈a|ξτ |b〉, the (a, b)-element of ξτ in the fundamental representation of SU(4)T . In the basis (24) of Table III, the diagonal elements of csea1/2,t and c 1/2,t are 〈Naab|csea1/2,t|Naab〉 = nt(58α 2 + β2 + 1 〈Naab|cval1/2,t|Naab〉 = (ξτabξ ba + 10ξ bb + 11(ξ (−(ξτaa)2 − 5ξτabξτba + 4ξτaaξτbb) + 2αβ (−7ξτabξτba + 11ξτaaξτbb + 4(ξτaa)2) 〈Σabc|csea1/2,t|Σabc〉 = nt(58α 2 + β2 + 1 〈Σabc|cval1/2,t|Σabc〉 = (11(ξτaaξ bb + ξ ba) + (ξτacξ ca + ξ cb) + 5(ξ cc + ξ (2(ξτaaξ bb + ξ ba) + 5(ξ ca + ξ cb)− 4(ξτaaξτcc + ξτbbξτcc)) + 2αβ (4(ξτaaξ bb + ξ ba)− 72(ξ ca + ξ cb) + (ξτaaξ cc + ξ 〈Λabc|csea1/2,t|Λabc〉 = nt(58α 2 + β2 + 1 〈Λabc|cval1/2,t|Λabc〉 = (ξτaaξ bb − ξτabξτba + 52(ξ ca + ξ cb) + 3(ξ cc + ξ + β2 [2(ξτaaξ bb − ξτabξτba)− (ξτacξτca + ξτbcξτcb)] + 2αβ (4(ξτaaξ bb − ξτabξτba) + (ξτacξτca + ξτbcξτcb) + 3(ξτaaξτcc + ξτbbξτcc)) The remnant taste symmetry Γ4⋊SO(4)T forces many of the off-diagonal elements to vanish. The distinct off-diagonal elements that do not vanish by this symmetry are 〈Naab|csea1/2,t|Λdfg〉 = 0 〈Naab|cval1/2,t|Λdfg〉 = (7ξτag(δadξ bf − δafξτbd) + 4ξτbg(δadξτaf − δafξτad) + 3δag(ξ bf − ξτafξτbd) + 11ξτag(δbfξτad − δbdξτaf)) (3δag(ξ bd − ξτadξτbf) + δbfξτag(ξτad − ξτaf ) + ξτag(δafξ bd − δadξτbf) + 2ξτbg(δadξτaf − δafξτad)) + 2αβ (ξτbg(δafξ ad − δadξτaf ) + 5ξτag(δadξτbf − δafξτbd) + 6δag(ξ bf − ξτafξτbd) + 4ξτag(δbfξτad − δbdξτaf)) The matrix element 〈Naab|csea1/2,t|Λdfg〉 vanishes accidentally; calculation shows it is propor- tional to δag(δbfδad − δbdδaf ), and no two of the indices dfg are ever equal for the states Λdfg. In the Weyl representation, these matrix elements of c sea,val 1/2,t are consistent with the degeneracies and mixings parameterized in (25). In the Weyl representation, the distinct, nontrivial matrix elements of cval1/2,t are listed in Table V. The diagonal elements of csea 3/2,t and cval 3/2,t 〈Naab|csea3/2,t|Naab〉 = 12nt 〈Naab|cval3/2,t|Naab〉 = [(ξτaa) 2 + 5ξτabξ ba − 4ξτaaξτbb] 〈Σabc|csea3/2,t|Σabc〉 = 12nt 〈Σabc|cval3/2,t|Σabc〉 = [2(ξτaaξ bb + ξ ba) + 5(ξ ca + ξ cb)− 4(ξτaaξτcc + ξτbbξτcc)] 〈Λabc|csea3/2,t|Λabc〉 = 12nt 〈Λabc|cval3/2,t|Λabc〉 = [−2(ξτaaξτbb − ξτabξτba) + ξτacξτca + ξτbcξτcb] , while the off-diagonal elements that are not required to vanish by symmetry are 〈Naab|csea3/2,t|Λdfg〉 = 0 〈Naab|cval3/2,t|Λdfg〉 = (3δag(ξ bd − ξτadξτbf) + δbfξτag(ξτad − ξτaf ) + ξτag(δafξ bd − δadξτbf) + 2ξτbg(δadξτaf − δafξτad)) In the Weyl representation, these matrix elements of c sea,val 3/2,t are consistent with the degen- eracies and mixings parameterized in (25). In this representation, the distinct, nontrivial matrix elements of cval 3/2,t are listed in Table VI. APPENDIX B: FOUR-FERMION CONTRIBUTIONS AND TASTE MATRICES Here I write down the matrix elements of σFF (A) and σFF (B) in terms of the elements of the taste matrices appearing in (17) and (18). These results can be used to show that, in the Weyl representation, σFF (A) and σFF (B) are consistent with the forms (25) and (26), respectively. For the diagonal σFF (A) contributions, we have 〈Naab|σFF (A)|Naab〉 = A1 (2(ξ5aa) 2 + ξ5abξ ba + ξ (2ξµνaa ξ aa + ξ ba + ξ (2ξνaaξ aa + ξ ba + ξ (2ξν5aaξ aa + ξ ba + ξ ((ξ5aa) 2 − 4ξ5abξ5ba + 5ξ5aaξ5bb) (ξµνaa ξ aa − 4ξ ba + 5ξ (ξνaaξ aa − 4ξνabξνba + 5ξνaaξνbb) (ξν5aaξ aa − 4ξν5ab ξν5ba + 5ξν5aaξν5bb ) 〈Σabc|σFF (A)|Σabc〉 = A1 (4(ξ5abξ ba + ξ bb) + ξ ca + ξ cc + ξ cb + ξ ba + ξ bb ) + ξ ca + ξ cc + ξ cb + ξ (4(ξνabξ ba + ξ bb) + ξ ca + ξ cc + ξ cb + ξ (4(ξν5ab ξ ba + ξ bb ) + ξ ca + ξ cc + ξ cb + ξ (2(ξ5abξ ba + ξ bb)− 4ξ5acξ5ca + 5ξ5aaξ5cc − 4ξ5bcξ5cb + 5ξ5bbξ5cc) ba + ξ bb )− 4ξµνac ξµνca + 5ξµνaa ξµνcc − 4ξ cb + 5ξ (2(ξνabξ ba + ξ bb)− 4ξνacξνca + 5ξνaaξνcc − 4ξνbcξνcb + 5ξνbbξνcc) (2(ξν5ab ξ ba + ξ bb )− 4ξν5ac ξν5ca + 5ξν5aaξν5cc − 4ξν5bc ξν5cb + 5ξν5bb ξν5cc ) 〈Λabc|σFF (A)|Λabc〉 = A1 (ξ5acξ ca + ξ cc + ξ cb + ξ (ξµνac ξ ca + ξ cc + ξ cb + ξ (ξνacξ ca + ξ cc + ξ cb + ξ (ξν5ac ξ ca + ξ cc + ξ cb + ξ (2(ξ5aaξ bb − ξ5abξ5ba) + ξ5aaξ5cc + ξ5bbξ5cc) (2(ξµνaa ξ bb − ξ ba ) + ξ cc + ξ (2(ξνaaξ bb − ξνabξνba) + ξνaaξνcc + ξνbbξνcc) (2(ξν5aaξ bb − ξν5ab ξν5ba ) + ξν5aaξν5cc + ξν5bb ξν5cc ) while the nontrivial off-diagonal terms are 〈Naab|σFF (A)|Λdfg〉 = A1 (δaf (ξ bg + ξ bd) + 2δbdξ − δad(ξ5afξ5bg + ξ5agξ5bf )− 2δbfξ5adξ5ag) (δaf (ξ bg + ξ bd ) + 2δbdξ − δad(ξµνaf ξ bg + ξ bf )− 2δbfξ (δaf (ξ bg + ξ bd) + 2δbdξ − δad(ξνafξνbg + ξνagξνbf )− 2δbfξνadξνag) (δaf (ξ bg + ξ bd ) + 2δbdξ − δad(ξν5afξν5bg + ξν5agξν5bf )− 2δbfξν5adξν5ag ) (δaf (ξ bg − 2ξ5agξ5bd) + δad(2ξ5bfξ5ag − ξ5afξ5bg) + δbfξ ag − δbdξ5afξ5ag + 3δag(ξ5adξ5bf − ξ5afξ5bd)) (δaf (ξ bg − 2ξµνag ξ bd ) + δad(2ξ ag − ξ + δbfξ ag − δbdξ ag + 3δag(ξ bf − ξ bd )) (δaf (ξ bg − 2ξνagξνbd) + δad(2ξνbfξνag − ξνafξνbg) + δbfξ ag − δbdξνafξνag + 3δag(ξνadξνbf − ξνafξνbd)) (δaf (ξ bg − 2ξν5agξν5bd ) + δad(2ξν5bf ξν5ag − ξν5afξν5bg ) + δbfξ ag − δbdξν5afξν5ag + 3δag(ξν5adξν5bf − ξν5afξν5bd )) Repeated meson taste indices µν, ν, and ν5 are summed as in (17); as for the quark- flow coefficients, the matrix is symmetric: 〈Λdfg|σFF (A)|Naab〉 = 〈Naab|σFF (A)|Λdfg〉. In the Weyl representation, the matrix elements of σFF (A) are consistent with the degeneracies and mixings parameterized in (25). For this representation the distinct, nontrivial matrix elements are listed in Table VII. For the diagonal σFF (B) contributions, first consider 〈Naab|σFF (B)|Naab〉 = B1 (2ξ4νaaξ aa + ξ ba + ξ (2ξ4aaξ aa + ξ ba + ξ (2ξ45aaξ aa + ξ ba + ξ (ξ4νaaξ aa − 4ξ4νab ξ4νba + 5ξ4νaaξ4νbb ) (ξ4aaξ aa − 4ξ4abξ4ba + 5ξ4aaξ4bb) (ξ45aaξ aa − 4ξ45abξ45ba + 5ξ45aaξ45bb ) 〈Σabc|σFF (B)|Σabc〉 = B1 (4(ξ4νab ξ ba + ξ bb ) + ξ ca + ξ cc + ξ cb + ξ (4(ξ4abξ ba + ξ bb) + ξ ca + ξ cc + ξ cb + ξ (4(ξ45abξ ba + ξ bb ) + ξ ca + ξ cc + ξ cb + ξ (2(ξ4νab ξ ba + ξ bb )− 4ξ4νac ξ4νca + 5ξ4νaaξ4νcc − 4ξ4νbc ξ4νcb + 5ξ4νbb ξ4νcc ) (2(ξ4abξ ba + ξ bb)− 4ξ4acξ4ca + 5ξ4aaξ4cc − 4ξ4bcξ4cb + 5ξ4bbξ4cc) (2(ξ45abξ ba + ξ bb )− 4ξ45acξ45ca + 5ξ45aaξ45cc − 4ξ45bc ξ45cb + 5ξ45bb ξ45cc ) 〈Λabc|σFF (B)|Λabc〉 = B1 (ξ4νac ξ ca + ξ cc + ξ cb + ξ (ξ4acξ ca + ξ cc + ξ cb + ξ (ξ45acξ ca + ξ cc + ξ cb + ξ (2(ξ4νaaξ bb − ξ4νab ξ4νba ) + ξ4νaaξ4νcc + ξ4νbb ξ4νcc ) (2(ξ4aaξ bb − ξ4abξ4ba) + ξ4aaξ4cc + ξ4bbξ4cc) (2(ξ45aaξ bb − ξ45abξ45ba ) + ξ45aaξ45cc + ξ45bb ξ45cc ) Repeated meson taste indices are summed as in (18). The Naab and Λabc terms above include the diagonal σFF (B) contributions. However, the basis of Γ4 ⋊ SU(2)T eigenstates does not include the Σabc. Instead, we consider the states of Table IV, Naab − 1√3Σacd and 1√ Naab + Σacd. In terms of the Γ4 ⋊ SO(4)T basis, the corresponding diagonal contributions to σFF (B) are Naab − 1√3Σacd|σ FF (B)| Naab − 1√3Σacd〉 = 2〈Naab|σ FF (B)|Naab〉 − 〈Σacd|σFF (B)|Σacd〉 Naab + Σacd|σFF (B)| 1√3Naab + Σacd〉 = 2〈Σacd|σFF (B)|Σacd〉 − 〈Naab|σFF (B)|Naab〉, where abcd ∈ {1423, 2314, 4132, 3241}, and I used |Σabc〉 = |Σbac〉 (cf. (24b)). The off- diagonal terms of σFF (B) are 〈Naab|σFF (B)|Λdfg〉 = B1 (δaf (ξ bg + ξ bd ) + 2δbdξ − δad(ξ4νafξ4νbg + ξ4νagξ4νbf )− 2δbfξ4νadξ4νag ) (δaf (ξ bg + ξ bd) + 2δbdξ − δad(ξ4afξ4bg + ξ4agξ4bf )− 2δbfξ4adξ4ag) (δaf (ξ bg + ξ bd ) + 2δbdξ − δad(ξ45afξ45bg + ξ45agξ45bf )− 2δbfξ45adξ45ag) (δaf (ξ bg − 2ξ4νagξ4νbd ) + δad(2ξ4νbf ξ4νag − ξ4νafξ4νbg ) + δbfξ ag − δbdξ4νafξ4νag + 3δag(ξ4νadξ4νbf − ξ4νafξ4νbd )) (δaf (ξ bg − 2ξ4agξ4bd) + δad(2ξ4bfξ4ag − ξ4afξ4bg) + δbfξ ag − δbdξ4afξ4ag + 3δag(ξ4adξ4bf − ξ4afξ4bd)) (δaf (ξ bg − 2ξ45agξ45bd ) + δad(2ξ45bf ξ45ag − ξ45afξ45bg ) + δbfξ ag − δbdξ45afξ45ag + 3δag(ξ45adξ45bf − ξ45afξ45bd )) 〈Naab|σFF (B)| Nccd − 1√3Σcfg〉 = B1 (2δbc(2ξ ad − ξ4νafξ4νag ) + 4(δadξ4νac ξ4νbc − δbdξ4νac ξ4νac ) + δac(ξ bg + ξ bf − 2ξ4νac ξ4νbd − 2ξ4νadξ4νbc ) + 4δbgξ af − 2(δbfξ4νagξ4νac + δag(ξ4νac ξ4νbf + ξ4νafξ4νbc )) + δaf (ξ bc + ξ bg )) (2δbc(2ξ ad − ξ4afξ4ag) + 4(δadξ4acξ4bc − δbdξ4acξ4ac) + δac(ξ bg + ξ bf − 2ξ4acξ4bd − 2ξ4adξ4bc) + 4δbgξ af − 2(δbfξ4agξ4ac + δag(ξ4acξ4bf + ξ4afξ4bc)) + δaf (ξ bc + ξ (2δbc(2ξ ad − ξ45afξ45ag) + 4(δadξ45acξ45bc − δbdξ45acξ45ac ) + δac(ξ bg + ξ bf − 2ξ45acξ45bd − 2ξ45adξ45bc ) + 4δbgξ af − 2(δbfξ45agξ45ac + δag(ξ45acξ45bf + ξ45afξ45bc )) + δaf (ξ bc + ξ bg )) (2(δbdξ ac − δadξ4νac ξ4νbc ) + δbc(ξ4νafξ4νag − 2ξ4νac ξ4νad) + δac(4ξ bf + 10ξ bd − 8ξ4νbc ξ4νad − 5ξ4νafξ4νbg ) − 2δbgξ4νac ξ4νaf + δbfξ4νac ξ4νag + δag(ξ4νac ξ4νbf + ξ4νafξ4νbc ) + δaf (4ξ bc − 5ξ4νac ξ4νbg )) (2(δbdξ ac − δadξ4acξ4bc) + δbc(ξ4afξ4ag − 2ξ4acξ4ad) + δac(4ξ bf + 10ξ bd − 8ξ4bcξ4ad − 5ξ4afξ4bg) − 2δbgξ4acξ4af + δbfξ4acξ4ag + δag(ξ4acξ4bf + ξ4afξ4bc) + δaf (4ξ bc − 5ξ4acξ4bg)) (2(δbdξ ac − δadξ45acξ45bc ) + δbc(ξ45afξ45ag − 2ξ45acξ45ad) + δac(4ξ bf + 10ξ bd − 8ξ45bc ξ45ad − 5ξ45afξ45bg ) − 2δbgξ45acξ45af + δbfξ45acξ45ag + δag(ξ45acξ45bf + ξ45afξ45bc ) + δaf (4ξ bc − 5ξ45acξ45bg )) 〈Λabh|σFF (B)| Nccd − 1√3Σcfg〉 = 〈Nccd|σFF (B)|Λabh〉. In the Weyl representation, the matrix elements of σFF (B) are consistent with the degenera- cies and mixings parameterized in (26). The distinct, nontrivial matrix elements are listed in Table VIII. [1] J. B. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975). [2] E. Marinari, G. Parisi, and C. Rebbi, Nucl. Phys. B190, 734 (1981). [3] C. Bernard, M. Golterman, and Y. Shamir, Phys. Rev. D73, 114511 (2006), hep-lat/0604017. [4] Y. Shamir (2006), hep-lat/0607007. [5] P. H. Damgaard and K. Splittorff, Phys. Rev. D62, 054509 (2000), hep-lat/0003017. [6] C. Aubin and C. Bernard, Nucl. Phys. Proc. Suppl. 129, 182 (2004), hep-lat/0308036. [7] C. Aubin and C. Bernard, Phys. Rev. D68, 034014 (2003), hep-lat/0304014. [8] C. Aubin and C. Bernard, Phys. Rev. D68, 074011 (2003), hep-lat/0306026. [9] C. Bernard, Phys. Rev. D73, 114503 (2006), hep-lat/0603011. [10] D. Toussaint, Nucl. Phys. Proc. Suppl. 106, 111 (2002), hep-lat/0110010. [11] C. T. H. Davies et al. (HPQCD, UKQCD, MILC, and Fermilab Lattice), Phys. Rev. Lett. 92, 022001 (2004), hep-lat/0304004. [12] C. Aubin et al. (HPQCD, MILC, and UKQCD), Phys. Rev. D70, 031504 (2004), hep- lat/0405022. [13] C. Aubin et al. (Fermilab Lattice, MILC, and HPQCD), Phys. Rev. Lett. 94, 011601 (2005), hep-ph/0408306. [14] C. Aubin et al. (MILC), Phys. Rev. D70, 114501 (2004), hep-lat/0407028. [15] C. Aubin et al. (MILC), Phys. Rev. D70, 094505 (2004), hep-lat/0402030. [16] C. Aubin et al. (Fermilab Lattice, MILC, and HPQCD), Phys. Rev. Lett. 95, 122002 (2005), hep-lat/0506030. [17] W.-J. Lee and S. R. Sharpe, Phys. Rev. D60, 114503 (1999), hep-lat/9905023. [18] C. Aubin and C. Bernard, Phys. Rev. D73, 014515 (2006), hep-lat/0510088. [19] C. Bernard (MILC), Phys. Rev. D65, 054031 (2002), hep-lat/0111051. [20] J. A. Bailey (2006), hep-lat/0611023. [21] J. A. Bailey, work in progress. [22] J. A. Bailey and C. Bernard, PoS LAT2005, 047 (2006), hep-lat/0510006. [23] M. F. L. Golterman and J. Smit, Nucl. Phys. B255, 328 (1985). [24] S. Scherer, Adv. Nucl. Phys. 27, 277 (2003), hep-ph/0210398. [25] S. R. Sharpe, Nucl. Phys. Proc. Suppl. 34, 403 (1994), hep-lat/9312009. [26] S. R. Sharpe and R. S. Van de Water, Phys. Rev. D71, 114505 (2005), hep-lat/0409018. [27] J. N. Labrenz and S. R. Sharpe, Phys. Rev. D54, 4595 (1996), hep-lat/9605034. [28] E. Jenkins and A. V. Manohar, Phys. Lett. B255, 558 (1991). [29] J.-W. Chen and M. J. Savage, Phys. Rev. D65, 094001 (2002), hep-lat/0111050. [30] H. Georgi, Weak Interactions and Modern Particle Theory (Menlo Park, Usa: Ben- jamin/Cummings, 1984), pp. 92-95. [31] E. Jenkins and A. V. Manohar (1991), talk presented at the Workshop on Effective Field Theories of the Standard Model, Dobogoko, Hungary. [32] A. Walker-Loud, Nucl. Phys. A747, 476 (2005), hep-lat/0405007. [33] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984). [34] M. Fukugita, N. Ishizuka, H. Mino, M. Okawa, and A. Ukawa, Phys. Rev. D47, 4739 (1993). [35] J. F. Donoghue, B. R. Holstein, and B. Borasoy, Phys. Rev. D59, 036002 (1999), hep- ph/9804281. [36] J. A. Bailey et al. (MILC), work in progress. Introduction The Staggered Heavy Baryon Lagrangian O(a0) Lagrangian O(a2) baryon operators for analytic corrections L6glue and L6bilin L6FF(A) L6FF(B) Staggered heavy baryon Lagrangian for O(3) baryon octet masses The Flavor-Symmetric Nucleons Identifying staggered nucleons and irreps of interpolating fields Taste symmetry and the mass matrix Masses of the flavor-symmetric nucleons Correspondence between chiral forms and interpolating fields Summary and Further Directions Acknowledgments Quark-flow coefficients and taste matrices Four-fermion contributions and taste matrices References

0704.1491

Imaging Magnetic Focusing of Coherent Electron Waves

Microsoft Word - Aidala_focusing_arXiv.doc Imaging Magnetic Focusing of Coherent Electron Waves Katherine E. Aidala,1* Robert E. Parrott,2 Tobias Kramer,2 E.J. Heller,2,3 R.M. Westervelt,1,2 M.P. Hanson4 and A.C. Gossard4 1Div. of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 2Dept. of Physics, Harvard University, Cambridge, MA 02138 3Dept. of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 4Dept. of Materials Science, University of California, Santa Barbara, CA 93106 * Present Address: Dept. of Physics, Mount Holyoke College, South Hadley, MA 01075 The magnetic focusing of electrons has proven its utility in fundamental studies of electron transport.1-4 Here we report the direct imaging of magnetic focusing of electron waves, specifically in a two-dimensional electron gas (2DEG). We see the semicircular trajectories of electrons as they bounce along a boundary in the 2DEG, as well as fringes showing the coherent nature of the electron waves. Imaging flow in open systems is made possible by a cooled scanning probe microscope.5-17 Remarkable agreement between experiment and theory demonstrates our ability to see these trajectories and to use this system as an interferometer. We image branched electron flow11 as well as the interference of electron waves.10,11,18 This technique can visualize the motion of electron waves between two points in an open system, providing a straightforward way to study systems that may be useful for quantum information processing and spintronics.19-21 For nanoscale devices at low temperatures, both the particle and wave aspects of electron motion are important. Electrons can travel ballistically through an unconfined two- dimensional electron gas (2DEG) while their quantum phase remains coherent. The development of unconfined or open devices in this regime1,22-26 clears the way for new applications in electron interferometry, spintronics, and quantum information processing.18- 21,27 The most basic question for open systems is: "What is the pattern of flow for electron waves?" Imaging flow is difficult, because the electrons are often buried inside a heterostructure, and because low temperatures and strong magnetic fields are needed to observe quantum phenomena. A cooled scanning probe microscope (SPM) provides a way to image flow: a conducting SPM tip above the sample capacitively couples to the 2DEG below.5-17 Pathways for electron waves emerging from a quantum point contact (QPC) were revealed10 and found to have a dramatic branched form.11 The branching was shown to be the real-space consequence of small-angle scattering by charged donor atoms - the well known mechanism that limits the electron mobility. A charged tip was able to image this flow, because it backscattered electrons; some of them follow a time-reversed path back through the QPC, measurably reducing its conductance. Images of flow were obtained by displaying the QPC conductance as the tip was scanned in a plane above the sample.10,11 However, this backscattering imaging technique cannot be used in a perpendicular magnetic field, because the electrons follow curved paths and no longer reverse. Magnetic fields are important for spintronics and quantum information processing,19-21 so it is important to understand electron flow with a field present. In this letter, we demonstrate a new way to image the flow of electron waves between two points in an open system using a cooled SPM, and we use this approach to image magnetic focusing of electron waves in a 2DEG. Magnetic focusing plays an important role in the study of ballistic transport.1-4 Images were obtained by using an electron lens formed in the 2DEG beneath a charged SPM tip to redirect flow, by throwing a shadow downstream. We obtain clear pictures of semicircular bouncing-ball patterns of particle flow that cause magnetic focusing. Branching of the flow is visible. In addition, we see fringes created by the interference of electron waves bouncing off the tip with those travelling directly between the two QPCs. In this way, the cooled SPM acts as a new type of electron interferometer for solids. The observed bouncing-ball patterns of flow and interference fringes are in excellent agreement with full quantum simulations that include the tip-induced electron lens and small- angle scattering, demonstrating that this new imaging technique accurately views the flow of electron waves through the device. Figure 1 illustrates the device geometry and the imaging technique. Two QPCs are formed in a 2DEG by surface gates (Fig. 1a); a scanning electron micrograph of the device is shown in Fig. 1b. The separation between the QPC centers is L = 2.7 μm. A 2DEG with density 3.8x1011 cm−2 and mobility 500,000 cm2/Vs is located 47 nm beneath the surface of a GaAs/AlGaAs heterostructure with the following layers: 5 nm GaAs cap, 20 nm Al0.3Ga0.7As barrier, Si delta-doping, then a 22 nm Al0.3Ga0.7As barrier next to the 2DEG in GaAs. Metal gates that define the QPCs were fabricated using e-beam lithography. The device is mounted in the SPM, inside a superconducting solenoid, and cooled to 4.2 K. A computer controls the SPM and records the images. Magnetic focusing (Fig. 1d) occurs because electrons leaving one QPC over a range of angles circle around and rejoin at the second QPC, when the spacing L is close to the diameter of a cyclotron orbit. For GaAs the electron cyclotron orbit is circular with radius: rc = kF eB , (1) where e is the electron charge, B is the perpendicular magnetic field, and kF is the Fermi wavevector. As B is increased, the first focusing peak occurs when L = 2rc . Additional peaks occur at higher fields when L = 2nrc is an integer multiple of the cyclotron diameter at fields: Bn = 2n kF eL . (2) The shape, clarity and spacing of magnetic focusing peaks provide information about ballistic flow in the sample material. The effects of small-angle scattering are shown in the simulation of Fig. 1e (and in supplementary material): magnetic focusing still occurs, but the flow now contains branches11 similar to flow in the SPM image in Fig. 1b for B = 0. To image electron flow from one point to another, a small movable electron lens is created by creating a dip in electron density immediately below the charged SPM tip. The lens deflects electrons (Fig. 1c), throwing a V-shaped shadow downstream. An image of electron flow is obtained by displaying the transmission T between the QPCs as the tip is scanned across a plane 10 nm above the surface. T is measured by recording the voltage across the second QPC as a known current is passed through the first. When electrons are ballistically injected into the second QPC from the first, a voltage develops that drives a current in the opposite direction to cancel the influx of electrons; this voltage is proportional to T. Simulations of electron flow that show how the imaging technique works are presented in Figs. 1c-f. A Gaussian ϕ tip = Vo exp − r − rtip( ) 2a2( ) is used to model the tip potential in the 2DEG, where rtip is the tip position, a is the width, and Vo is the height. For this paper Vo > 0, and the tip creates a dip in electron density. The relative strength of the tip potential is η = , (3) where EF is the Fermi energy. For η <1, the electron gas is partially depleted, and an imperfect diverging lens is created with a focal length determined by η. When η ≥1, the 2DEG is fully depleted, and electrons can backscatter. Simulations (Fig. 1c) at B = 0 show how a weak lens (η = 0.2) creates a V-shaped shadow downstream by forcing electrons to the sides, where they form two caustics in flow along the legs of the V. Figure 1f shows how we image magnetic focusing: when the shadow cast by the lens beneath the tip hits the second QPC, T is reduced. The amount of reduction ΔT is proportional to the original flux, before the tip was introduced. By displaying ΔT vs. tip position rtip as the tip is raster scanned in a plane above, an image of the original flow is created.28 Experimental images of magnetic focusing near the first, second, and third peaks are presented in Figs. 2a-c for a weak tip (η ≈ 0.5), and in Figs. 2d-e for a strong tip (η ≈1.0). The bouncing semicircular cyclotron orbits imaged here are an experimental visualization of the origins of magnetic focusing. Figure 2a shows a single, semicircular crescent characteristic of the first peak. Branches in flow from small-angle scattering are also visible. For Fig. 2b the bounce characteristic of the second peak is clearly seen. For Fig. 2c, recorded near the third focusing peak, it becomes difficult to see distinct bounces, although circular features are evident, with radii comparable to rc. The bouncing-ball orbits begin to form the semi-classical equivalent of an edge state. With a strongly scattering tip (η ~ 1.0) the SPM images (Figs. 2d-f) have new and distinctly finer features, resulting from the interference of electron waves, some traveling along new pathways created by scattering at the tip. In the absence of the tip, electrons move from one QPC to the other by simultaneously traveling along multiple paths, which add up with a particular overall phase. As quantified below, a strongly scattering tip introduces new trajectories (and removes some of the old). These new trajectories interfere with the original ones, and create interference fringes as the tip is scanned that can be seen in the experimental images. The images in Figs. 2d-f also show bouncing ball orbits similar to their counterparts in Figs. 2a-c. The striking difference is the appearance of narrow fringe-like features. In some locations, a noticeably periodic structure of fringes exists, shown in the blowups Figs. 2g and 2h. We can understand both the classical and quantum behavior by using full, thermally averaged quantum simulations of an SPM image including tip scattering. Figures 3a-c show simulations of an image on the first magnetic focusing peak for weak (η = 0.2), moderate (η = 0.6) and strong (η =1.2) tips. These were obtained at 1.7 K using a thermal wavepacket calculation (26) with λF = 40 nm, and they include a random background potential from donor ions. For a weak tip in Fig. 3a, the dark area of reduced transmission (ΔT < 0) corresponds to a classical cyclotron orbit. For moderate and strong tips in Figs. 3b and 3c, quantum interference fringes created by tip scattering become visible. The increase in fringe- like structure as the tip strength is increased is in excellent qualitative agreement with the behavior of the experimental SPM images in Figs. 2a-c and Figs. 2d-f. To understand the source of contrast in the experimental images, it is very useful to compare quantum simulations with classical trajectories.10-11 Figure 3d joins quantum simulations from Fig. 3a (beige surface) with classical trajectories (red lines) computed without the tip present using ray tracing. The background potential is shown in blue. Areas with ΔT < 0 are eliminated for ease of comparison. We see that the original trajectories line up with paths of decreased transmission, because the flow is blocked by the tip. These simulations reveal the unusual, and yet very informative way that electron flow is encoded in the experimental images. The origin of fringing can be understood by a simple semiclassical argument pictured in Fig. 4a. When a new trajectory with phase φ deflects from the tip and reaches the target QPC, it contributes an amplitude φiae which interferes coherently with the background amplitude Aoe iφo from the nascent trajectories, such that the change in transmission depends on the phase difference ΔT ∝ cos(φ − φo) . This is illustrated in Fig. 4a by the interference of a direct path between the QPCs along a single cyclotron orbit, with a path deflected by the tip, composed of two cyclotron orbit segments. The phase of the deflected trajectory φ = S is proportional to the classical action S accumulated along the trajectory. When the tip is moved, S changes, and so does φ . This leads to a simple equation for the fringe spacing d: d = , (4) where θ is the angle by which the trajectory was deflected by the tip, shown in Fig. 4a. When the tip backscatters by θ = π , the fringe spacing is λF /2 as has been seen in previous experiments.10,11 When using a weak tip withη <1, the maximum possible scattering angle is θ < π , because it cannot backscatter. This implies that the minimum fringe spacing is dmin > λF /2 . For sufficiently weak tips η ~ 0.1, the fringe spacings are so large that they are indistinguishable from classical structures such as branches. A direct comparison of fringing between theory (Figs. 4b-d) and experiment (Figs. 4e- g) shows remarkable agreement. A series of quantum simulations in a small region (outlined in black in Fig. 3c) is compared with SPM images of a comparable area. The simulations in Figs. 4b-d are for η = 0.2, 0.6 and 1.0 respectively, with the corresponding experimental images for these tip strengths displayed below in Figs. 4e-g. The agreement is excellent. The fringe spacing for a strong tip is comparable to λF , to be expected when the electrons are scattered by θ ~ π 3. In this geometry the SPM acts as an interferometer that could be used to extract information from the fringes about the momentum and energy of the electrons. We have successfully visualized coherent electron transport patterns in a 2DEG magnetic focusing experiment and provided theoretical explanations of phenomena not previously predicted nor measured. The complex interaction of focusing, branching, and tip scattering has been unraveled, revealing the true nature of electron pathways in a real device. References 1. van Houten, H. et al. Coherent electron focusing with quantum point contacts in a two- dimensional electron gas. Phys. Rev. B 39, 8556 (1989). 2. 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Quantized conductance of point contacts in a two-dimensional electron gas. Phys. Rev. Lett. 60, 848 (1988). 24. Shepard, K.L., Roukes, M.L., & Van der Gaag, B.P. Direct measurement of the transmission matrix of a mesoscopic conductor. Phys. Rev. Lett. 68, 2660 (1992). 25. Crommie, M.F., Lutz, & C.P., Eigler, D.M. Confinement of Electrons to Quantum Corrals on a Metal Surface. Science 262, 218 (1993). 26. Kikkawa, J.M., & Awschalom, D.D. Lateral drag of spin coherence in gallium arsenide. Nature 397, 139 (1999). 27. Katine, J.A., et al., Point Contact Conductance of an Open Resonator. Phys. Rev. Lett. 79, 4806 (1997). 28. Aidala, K.E., Parrott, R.E., Heller, E.J., & Westervelt, R.M. Imaging electrons in a magnetic field. Physica E 34, 409 (2006). 29. Heller, E.J. et al. Thermal averages in a quantum point contact with a single coherent wave packet. Nano Letters 5, 1285 (2005). Author Contributions: Katherine E. Aidala conducted the experiments with R.M. Westervelt; Robert E. Parrot and Tobias Kramer did classical and quantum simulations of electron flow with E.J. Heller, and M.P. Hanson grew the semiconductor heterostructure with A.C. Gossard. Acknowledgements: This work has been performed with support at Harvard University from the ARO, the NSF-funded Nanoscale Science and Engineering Center (NSEC), and the DFG (Emmy-Noether program). Work at Santa Barbara has been supported in part by the Institute for Quantum Engineering, Science and Technology (iQUEST). We would also like to thank the National Nanotechnology Infrastructure Network (NNIN) and the Harvard CrimsonGrid for computing resources. Correspondence and requests for materials should be addressed to R.M.W. (e-mail: [email protected]). Figure 1 (a) Schematic showing a conducting scanning probe microscope (SPM) tip scanned above the surface of a device. (b) Scanning electron micrograph of the magnetic focusing device; the inset shows an SPM image of the electron flow at zero magnetic field, displaying branches in electron flow. (c) Quantum simulation showing the scattering of an incoming plane wave by an SPM tip, with 2DEG Fermi energy EF = 13 meV, and a Gaussian tip potential with height 0.2 EF and width 50 nm. (d-f) Classical simulations of magnetic focusing in a magnetic field: (d) in a flat potential, showing circular cyclotron orbits, (e) with small-angle scattering, showing added branches, and (f) in a flat potential with an SPM tip. Figure 2 Experimental SPM images of magnetic focusing in a 2DEG at 4.2 K recorded near the first three magnetic focusing peaks: (a-c) Weakly focusing (η ≈ 0.5) tip 90 nm above the 2DEG recorded at B = 100 mT, B = 174 mT, and B = 262 mT with cyclotron radii rc = 970 nm, rc = 560 nm and rc = 370 nm, respectively. The left QPC is on the first conductance plateau, the right is on the third. (d-f) Strongly focusing (η ≈1.0) tip 60 nm above the 2DEG recorded at B = 74 mT, B = 169 mT, and B = 254 mT with cyclotron radii rc = 1310 nm, rc = 580 nm and rc = 380 nm, respectively. Both QPCs are on the second conductance plateau. The color scale shows the change in transmission between the QPCs induced by the tip. (g,h) Closeup surface plots in the yellow rectangles of (d,f) that show the regularity and consistency of the quantum fringe structure. Figure 3 Quantum simulations of SPM images of the first magnetic focusing peak (B = 77 mT) for λF = 40 nm at 1.7 K, showing the change in transmission ΔT between QPCs as the tip is scanned above, including small-angle scattering. (a) For a weak tip (η = 0.2) that scatters into small angles, the dark area of reduced transmission (ΔT < 0) shows a classical cyclotron orbit. For (b) moderate (η = 0.6) and (c) strong (η = 1.2) tips, quantum interference fringes become visible. (d) Correspondence between the simulated SPM image (beige surface) from (a) and ray tracing calculations of the originally transmitted electron trajectories (red lines) before the tip was present; regions with ΔT < 0 are omitted for ease of comparison. The blue surface is the smoothly disordered background potential. The dark areas with reduced transmission (ΔT < 0) line up very well with the original classical trajectories. For stronger tips, imaging of the original classical trajectories becomes more difficult, but fringing reveals regions of high coherent flux of electron waves. Figure 4 Direct comparison of interference fringes between experiment and theory for different tip strengths. (a) How interference occurs between a path deflected by the SPM tip with a direct path between the two QPCs. (b-d) Quantum simulations of interference fringes at 1.7 K and B = 77 mT for weak (η = 0.2), moderate (η = 0.6) and strong (η = 1.0) tip strengths; the Fermi wavelength is λF = 40 nm. The panels are located in the black box in Fig. 3c. (e-g) SPM images at B = 173 T showing fringes that appear and move closer as the tip strength η increases, in good agreement with the simulations. The images have dimensions 600 x 450 nm2 and are located 750 nm above and 500 nm to the right of the midpoint between the two QPCs.

0704.1492

On the solution of the static Maxwell system in axially symmetric inhomogeneous media

On the solution of the static Maxwell system in axially symmetric inhomogeneous media Kira V. Khmelnytskaya1, Vladislav V. Kravchenko2, Héctor Oviedo3 CINVESTAV-Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 MEXICO Department of Mathematics, CINVESTAV-Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 MEXICO e-mail: [email protected] SEPI, ESIME Zacatenco, Instituto Politécnico Nacional, Av. IPN S/N, C.P. 07738, D.F. MEXICO∗ December 13, 2018 Abstract We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions which can be used for analytic and numerical solution of corresponding boundary value problems. 1 Introduction Consider the static Maxwell system div(εE) = 0, rotE = 0 (1) where we suppose that ε is a function of the cylindrical radial variable r =√ x21 + x 2: ε = ε(r). Two important situations are usually studied: the meridional field and the transverse field. ∗Research was supported by CONACYT, Mexico http://arxiv.org/abs/0704.1492v1 The first case is characterized by the condition that the vector E is inde- pendent of the angular coordinate θ and the component Eθ of the vector E in cylindrical coordinates vanishes identically. The vector of such field belongs to a plane containing the axis x3 and depends only on the distance r to this axis as well as on the coordinate x3. The field then is completely described by a two-component vector-function in the plane (r, x3). The second case is characterized by the condition that the vector E is in- dependent of x3 and the component E3 is identically zero. The vector of such field belongs to a plane perpendicular to the axis x3 and the corresponding model reduces to a two-component vector-function in the plane (x1, x2). In the present work in both cases we construct a complete system of solutions of the corresponding model. We use the fact that in both cases the system (1) reduces to a system describing so-called p-analytic functions [1], [4], [8], [10], [11], [13], [17], [18] vy, uy = − vx. (2) In the first case the function p is a function of one Cartesian variable x meanwhile in the second it is a function of r = x2 + y2. In both cases we construct an infinite system of so-called formal powers [2], [5]. This is a complete system of exact solutions of equations (2) generalizing the system of usual complex powers (z− z0)n, n = 0, 1, 2, . . .. Locally, near the center z0 the formal powers behave asymptotically like powers. Nevertheless in general their behaviour can be arbitrarily different from that of powers but with a guarantee of their completeness in the sense that any solution of the consid- ered equations can be represented as a uniformly convergent series of formal powers. The general theory of formal powers was developed by L. Bers [2] as a part of his pseudoanalytic function theory. Its application was restricted by the fact that only for a quite limited class of pseudoanalytic functions the explicit construction of formal powers was possible. L. Bers’ results allow us to construct a complete system of formal powers in the meridional case. Nevertheless they are not applicable to the model arising from the trans- verse case. In the recent works [12] and [15] the class of solvable in this sense systems (2) was substantially extended. In the present work we use these results for solving the static Maxwell system in the transverse case. This combination of the relation between the static Maxwell system (1) and the system (2) together with the classical results of L. Bers on pseudoanalytic formal powers and new developments in [12] and [15] allow us to obtain a general solution of the static Maxwell system in the axially symmetric case in the sense that we construct a complete system of its solutions for both the meridional and the transverse fields. 2 Reduction of the static Maxwell system to p-analytic functions 2.1 The meridional case Introducing the cylindrical coordinates and making the assumptions that E is independent of the angular variable θ and that the component Eθ is identically zero we obtain that (1) can be written as follows ∂(rεEr) ∂(εE3) Denote x = r, y = x3, u = E3 and v = rεEr. Then the system takes the xε(x) vy, uy = − xε(x) where the subindices denote the derivatives with respect to the corresponding variables. Thus, in the case of a meridional field the vector E is completely described by an xε(x)-analytic function ω = u+ iv. 2.2 The transverse case We assume that E is independent of the longitudinal variable x3 and E3 ≡ 0. Then from (1) we have that the vector (E1, E2) T is the gradient of a function u = u(x1, x2) which satisfies the two-dimensional equation div(ε∇u) = 0. (3) Denote x = x1, y = x2, z = x+ iy and consider the system vy, uy = − vx. (4) It is easy to see that if the function ω = u + iv is its solution then u is a solution of (3), and vice versa [14], if u is a solution of (3) in a simply connected domain Ω then choosing v = A(iεuz), (5) where A [Φ] (x, y) = 2 ReΦdx+ ImΦdy + c, (6) c is an arbitrary real constant, Γ is an arbitrary rectifiable curve in Ω leading from (x0, y0) to (x, y) we obtain that ω = u + iv is a solution of (4). Here the subindex z means the application of the operator ∂z = (∂x + i∂y). Note that due to the fact that u is a solution of (3) the function Φ = iεuz satisfies the condition ∂y ReΦ − ∂x ImΦ = 0 and hence the integral in (6) is path- independent. For a convex domain then expression (6) can be written as follows A [Φ] (x, y) = 2 ReΦ(η, y)dη + ImΦ(x0, ξ)dξ Note that v is a solution of the equation ∇v) = 0. Thus, equation (3) (and hence the system (1) in the case under consideration) is equivalent to the system (4) in the sense that if ω = u+ iv is a solution of (4) then its real part u is a solution of (3) and vice versa, if u is a solution of (3) then ω = u+ iv, where v is constructed according to (5) is a solution of (4). We reduced both considered cases the meridional and the transverse to the system describing p-analytic functions. In the first case p = xε(x) is a function of one Cartesian variable and in the second p = ε(r), r = x2 + y2. As we show below in both cases we are able to construct explicitly a complete system of formal powers and hence a complete system of exact solutions of the corresponding Maxwell system. Let us notice that equation (3) with ε being a function of the variable r was considered in the recent work [6] whith applications to electrical impedance tomography. The algorithm proposed in that work implies numerical solution of a number of ordinary differential equations arising after a standard separation of variables. Our construction of a complete system of solutions of (3) is based on essentially different ideas and does not require solving numerically any differential equation. 3 p-analytic functions and formal powers 3.1 The main Vekua equation Consider the system describing p-analytic functions vy, uy = − vx, (7) where we suppose that p is a positive and continuously differentiable function of x and y. Together with tis system we consider the following Vekua equation which due to its importance in relation to second-order elliptic equations of mathematical physics is called [14], [15] the main Vekua equation W, (8) where f = p. The function ω = u + iv is a solution of (7) iff [15] W = uf + iv/f is a solution of (8). In [15] there was proposed a method for explicit construction of the system of formal powers corresponding to the main Vekua equation under a quite general condition on f . Here we briefly describe the method for which we need first to recall some basic definitions from L. Bers’ theory of formal powers [2]. Let F and G be a couple of solutions of a Vekua equation Wz = a(F,G)W + b(F,G)W in Ω (9) such that Im(FG) > 0. Then (F,G) is said to be a generating pair corre- sponding to (9). The complex functions a(F,G) and b(F,G) are called charac- teristic coefficients of the pair (F,G) and it can be seen that a(F,G) = − FGz − FzG FG− FG , b(F,G) = FGz − FzG FG− FG Together with these characteristic coefficients another pair of characteristic coefficients is introduced in relation to the notion of the (F,G)-derivative: A(F,G) = − FGz − FzG FG− FG , B(F,G) = FGz − FzG FG− FG where the z means the application of the operator ∂z = − i ∂ . As in the present work we do not use explicitly the notion of the (F,G)-derivative, we refer the interested reader to [2] for its definition and properties. However we do need the concept of characteristic coefficients for defining the following important object. Definition 1 Let (F,G) and (F1, G1) - be two generating pairs in Ω corre- sponding to the Vekua equations with coefficients a(F,G), b(F,G) and a(F1,G1) and b(F1,G1) respectively. Then (F1, G1) is called successor of (F,G) and (F,G) is called predecessor of (F1, G1) if a(F1,G1) = a(F,G) and b(F1,G1) = −B(F,G). Definition 2 A sequence of generating pairs {(Fm, Gm)}, m = 0,±1,±2, . . . , is called a generating sequence if (Fm+1, Gm+1) is a successor of (Fm, Gm). If (F0, G0) = (F,G), we say that (F,G) is embedded in {(Fm, Gm)}. For any generating pair (F,G) the corresponding (F,G)-integral is defined as follows wd(F,G)ζ = F (z) Re G∗wdζ +G(z) Re F ∗wdζ where Γ is a rectifiable curve leading from z0 to z and (F ∗, G∗) is an adjoint generating pair defined by the equations F ∗ = − FG− FG , G∗ = FG− FG If w is an (F1, G1) - pseudoanalytic function (i.e., it is a solution of the Vekua equation with the coefficients a(F1,G1) and b(F1,G1)) then its (F,G)-integral is path-independent. Now we are ready to introduce the definition of formal powers. Definition 3 The formal power Z m (a, z0; z) with center at z0 ∈ Ω, coeffi- cient a and exponent 0 is defined as the linear combination of the genera- tors Fm, Gm with real constant coefficients λ, µ chosen so that λFm(z0) + µGm(z0) = a. The formal powers with exponents n = 1, 2, . . . are defined by the recursion formula Z(n)m (a, z0; z) = n (n−1) m+1 (a, z0; ζ)d(Fm,Gm)ζ. (10) This definition implies the following properties. m (a, z0; z) is an (Fm, Gm)-pseudoanalytic function of z. 2. If a′ and a′′ are real constants, then Z ′+ia′′, z0; z) = a m (1, z0; z)+ m (i, z0; z). 3. The asymptotic formulas Z(n)m (a, z0; z) ∼ a(z − z0)n, z → z0 (11) hold. Writing Z(n)(a, z0; z) we indicate that the formal power corresponds to the generating pair (F,G). The definition of formal powers shows us that in order to obtain Z(n)(a, z0; z) we need to have first the formal power Z (n−1) 1 (a, z0; z) for which it is necessary to calculate Z (n−2) 2 (a, z0; z) and so on. Thus, the problem of construction of formal powers of any order for a given generating pair (F,G) reduces to the construction of a corresponding generating sequence. Then definition 3 gives us a simple algorithm for constructing the formal powers. In other words, one needs a pair of exact solutions for each of the infinite number of Vekua equations corresponding to a generating sequence. In the next subsection we show how this seemingly difficult task can be accomplished in a quite general situation. Meanwhile here we recall some well known results in order to explain that the system of formal powers in fact represents a complete system of solutions of a corresponding Vekua equation. First of all, let us notice that due to the property 2 of formal powers for every n (and for a fixed z0) it is sufficient to construct only two formal powers: Z(n)(1, z0; z) and Z (n)(i, z0; z), then for any coefficient a the corresponding formal power Z(n)(a, z0; z) is a linear combination of the former two. An expression of the form n=0 Z (n)(an, z0; z) is called a formal polyno- mial. Under the conditions imposed in this work on the function ε and on the domain of interest Ω (see section 4 and for more details [15]) the fol- lowing Runge-type theorem is valid where following [2] we say that a series converges normally in a domain Ω if it converges uniformly on every bounded closed subdomain of Ω. Theorem 4 [3] A pseudoanalytic function defined in a simply connected do- main can be expanded into a normally convergent series of formal polynomi- In other words a pseudoanalytic function can be represented as an infinite linear combination of the functions Z(n)(1, z0; z), Z (n)(i, z0; z) Moreover, if we know that a pseudoanalytic function W satisfies the Hölder condition on the boundary of a domain of interest Ω (a common require- ment when a boundary value problem is considered) then, e.g., the following estimate in the C(Ω)-norm is available. Theorem 5 [16] Let W be a pseudoanalytic function in a domain Ω bounded by a Jordan curve and satisfy the Hölder condition on ∂Ω with the exponent α (0 < α ≤ 1). Then for any ǫ > 0 and any natural n there exists a pseudopolynomial of order n satisfying the inequality |W (z)− Pn(z)| ≤ Const for any z ∈ Ω where the constant does not depend on n, but only on ǫ. These and other results on interpolation and on the degree of approxima- tion by pseudopolynomials which can be found in the vast bibliography ded- icated to pseudoanalytic function theory (see, e.g., [7], [9]) show us that the system of formal powers is as good for solving corresponding boundary value problems as is the system of usual complex powers (z − z0)n, n = 0, 1, 2, . . .. The real (or imaginary) parts of {(z − z0)n}∞n=0 are harmonic polynomials successfully applied to the numerical solution of boundary value problems for the Laplace equation. In a similar way the real parts of formal powers{ Z(n)(1, z0; z), Z (n)(i, z0; z) corresponding to the main Vekua equation (8) (where f = ε) can be used for the numerical solution of boundary value problems for the conductivity equation (3) because as was shown in [14] the system of functions ReZ(n)(1, z0; z), ReZ(n)(i, z0; z) is complete in the space of solutions of (3) in the sense of theorems 4 and 5. A formal power Z(n)(a, z0; z) related to the Vekua equation (8) corre- sponds to a formal power ∗Z (n)(a, z0; z) (we use the notation of L. Bers) related to the system (7) in the following way (n)(a, z0; z) = ReZ(n)(a, z0; z) + if ImZ (n)(a, z0; z). As we will see in the meridional case it is convenient to work directly with formal powers ∗Z (n)(a, z0; z). Any solution ω = u+ iv of the system (7) can be expanded into a normally convergent series of real linear combinations of the complex functions (n)(1, z0; z), ∗Z (n)(i, z0; z) 3.2 Construction of generating sequences In [15] the following result was obtained. Theorem 6 Let F = U(u)V (v) and G = i U(u)V (v) where U and V are ar- bitrary differentiable nonvanishing real valued functions, Φ = u + iv is an analytic function of the variable z = x + iy in Ω such that Φz is bounded and has no zeros in Ω. Then the generating pair (F,G) is embedded in the generating sequence (Fm, Gm), m = 0,±1,±2, . . .in Ω defined as follows Fm = (Φz) F and Gm = (Φz) G for even m F and Gm = (Φz) U2G for odd m. This theorem opens the way for construction of generating sequences and consequently of formal powers in a quite general situation (see [15]) and in particular in both cases considered in the present work. In the meridional case the theorem reduces to the result of L. Bers [2] which we use in the next subsection while in the transverse case this and other classical results are insufficient for constructing formal powers explicitly and theorem 6 is indispensable. 4 Construction of formal powers 4.1 Formal powers in the meridional case As was shown in subsection 2.1 in the meridional case the Maxwell system reduces to the following couple of equations xε(x) vy, uy = − xε(x) which is equivalent to the system considered in [2, N18.1] σ(x)φx = τ(y)ψy, σ(x)φy = −τ(y)ψx. Taking σ(x) = xε(x) and τ ≡ 1 we can use the elegant formulas for the generating powers obtained by L. Bers. Let X(0)(x0, x) = X̃ (0)(x0, x) = 1 and for n = 1, 2, ...denote X(n)(x0, x) = n X(n−1)(x0, t) tε(t) dt for odd n X(n)(x0, x) = n X(n−1)(x0, t)tε(t)dt for even n X̃(n)(x0, x) = n X̃(n−1)(x0, t)tε(t)dt for odd n X̃(n)(x0, x) = n X̃(n−1)(x0, t) tε(t) dt for even n Then the formal powers in the meridional case are given by the expressions (n)(a′ + ia′′, z0; z) = a X(n−k)ik(y − y0)k + ia′′ X̃(n−k)ik(y − y0)k for odd n (n)(a′ + ia′′, z0; z) = a X̃(n−k)ik(y − y0)k + ia′′ X(n−k)ik(y − y0)k for even n. 4.2 Formal powers in the transverse case As was shown in subsection 2.2 the Maxwell system (1) in the transverse case reduces to the system vy, uy = − where ε is a positive differentiable function of r = x2 + y2. This system describing ε-analytic functions is equivalent to the main Vekua equation (8) where f = ε. In order to apply theorem 6 we denote u = ln r and U(u) =√ ε(eu). Then taking V ≡ 1 we obtain the generating pair (F,G) for equation (8) in the desirable form F = U(u), G = . (12) The analytic function Φ (from theorem 6) corresponding to the polar coor- dinate system has the form Φ(z) = ln z and consequently Φz(z) = 1/z. We note that Φz has a pole in the origin and a zero at infinity. Thus, theorem 6 is applicable in any domain Ω which does not include these two points. Moreover, as for constructing formal powers we need to use the recursive integration defined by (10) in what follows we require Ω to be any bounded simply connected domain not containing the origin. From theorem 6 we have that a generating sequence corresponding to the generating pair (12) can be defined as follows and Gm = for even m and Gm = for odd m. As was explained in subsection 3.1 in order to have a complete system of formal powers for each n we need to construct Z(n)(1, z0; z) and Z (n)(i, z0; z). For n = 0 we have Z(0)(1, z0; z) = λ 1 F (z) + µ 1 G(z) Z(0)(i, z0; z) = λ i F (z) + µ i G(z) where λ 1 , µ 1 are real constants chosen so that 1 F (z0) + µ 1 G(z0) = 1 and λ i , µ i are real constants such that i F (z0) + µ i G(z0) = i. Taking into account that F is real and G is imaginary we obtain that F (z0) 1 = 0, i = 0, µ i = F (z0). Thus, Z(0)(1, z0; z) = F (z) F (z0) ε(r0) Z(0)(i, z0; z) = iF (z0) F (z) ε(r0) where r0 = |z0|. For constructing Z(1)(1, z0; z) and Z (1)(i, z0; z) we need first the formal powers Z 1 (1, z0; z) and Z 1 (i, z0; z). According to definition 3 they have the form 1 (1, z0; z) = λ 1 F1(z) + µ 1 G1(z) 1 (i, z0; z) = λ i F1(z) + µ i G1(z) where λ 1 , µ 1 are real numbers such that 1 F1(z0) + µ 1 G1(z0) = 1 and λ i , µ i are real numbers such that i F1(z0) + µ i G1(z0) = i. Thus in order to determine λ 1 , µ 1 and λ i , µ i we should solve two systems of linear algebraic equations: z0ε1/2(r0) iε1/2(r0) z0ε1/2(r0) iε1/2(r0) which can be rewritten as follows 1 + µ 1 iε(r0) = ε 1/2(r0)z0 i + µ i iε(r0) = iε 1/2(r0)z0. From here we obtain 1 = ε 1/2(r0)x0, µ 1 = ε −1/2(r0)y0, λ i = −ε1/2(r0)y0, µ i = ε −1/2(r0)x0. Let us notice that in general for odd m we have Z(0)m (1, z0; z) = zmε1/2(r) 1/2(r) Z(0)m (i, z0; z) = zmε1/2(r) 1/2(r) where 1 = ε 1/2(r0) Re z 0 = ε 1/2(r0)r 0 cosmθ0, 1 = ε −1/2(r0) Im z 0 = ε −1/2(r0)r 0 sinmθ0, i = −ε1/2(r0) Im zm0 = −ε1/2(r0)rm0 sinmθ0, i = ε −1/2(r0) Re z 0 = ε −1/2(r0)r 0 cosmθ0, θ0 is the argument of the complex number z0. Thus, for odd m: Z(0)m (1, z0; z) = cosmθ0 ε(r0) + i sinmθ0 ε(r0) Z(0)m (i, z0; z) = − sinmθ0 ε(r0) + i cosmθ0 ε(r0) In a similar way we obtain the corresponding formulas for even m: Z(0)m (1, z0; z) = cosmθ0 ε(r0) + i sinmθ0 ε(r0) Z(0)m (i, z0; z) = − sinmθ0 ε(r0) + i cosmθ0 ε(r0) In order to apply formula (10) for constructing formal powers of higher orders we need to calculate the adjoint generating pairs (F ∗m, G m). For odd m we F ∗m = − ε1/2(r) , G∗m = ε 1/2(r)zm. For even m we obtain F ∗m = −izmε1/2(r), G∗m = ε1/2(r) Now the whole procedure of construction of formal powers can be easily algorithmized. The obtained system of formal powers Z(n)(1, z0; z), Z (n)(i, z0; z) is complete in the space of all solutions of the main Vekua equation (8) with f = ε1/2(r), i.e., any regular solution W of (8) in Ω can be represented in the form of a normally convergent series W (z) = Z(n)(an, z0; z) = (n)(1, z0; z) + a (n)(i, z0; z) where a′n = Re an, a n = Im an and z0 is an arbitrary fixed point in Ω. References [1] Aleksandrov A Ya and Solovyev Yu I 1978 Spatial problems of elasticity theory: application of methods of the theory of functions of complex variable Moscow: Nauka (in Russian). [2] Bers L 1952 Theory of pseudo-analytic functions. New York University. [3] Bers L 1956 Formal powers and power series. Communications on Pure and Applied Mathematics 9 693-711. [4] Chemeris V S 1995 Construction of Cauchy integrals for one class of nonanalytic functions of complex variables. J. Math. Sci. 75, no. 4, 1857– 1865. [5] Courant R and Hilbert D 1989 Methods of Mathematical Physics, v. 2. 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Math. 66, No.4, 2369-2376. [12] Kravchenko V V 2005 On a relation of pseudoanalytic function theory to the two-dimensional stationary Schrödinger equation and Taylor series in formal powers for its solutions. J. of Phys. A , 38, No. 18, 3947-3964. [13] Kravchenko V V 2005 On the relationship between p-analytic functions and the Schrödinger equation. Zeitschrift für Analysis und ihre An- wendungen, 24, No. 3, 487-496. [14] Kravchenko V V 2006 On a factorization of second order elliptic opera- tors and applications. Journal of Physics A: Mathematical and General, 2006, 39, No. 40, 12407-12425. [15] Kravchenko V V Recent developments in applied pseudoanalytic func- tion theory. To appear, available from www.arxiv.org. [16] Menke K 1974 Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome. Manuscripta Math. 11 111-125. [17] Polozhy G N 1965 Generalization of the theory of analytic functions of complex variables: p-analytic and (p, q)-analytic functions and some applications. Kiev University Publishers (in Russian). [18] Zabarankin M and Ulitko A F 2006 Hilbert formulas for r-analytic func- tions in the domain exterior to spindle. SIAM Journal of Applied Math- ematics 66, No. 4, 1270-1300. Introduction Reduction of the static Maxwell system to p-analytic functions The meridional case The transverse case p-analytic functions and formal powers The main Vekua equation Construction of generating sequences Construction of formal powers Formal powers in the meridional case Formal powers in the transverse case

0704.1493

On a {K_4,K_{2,2,2}}-ultrahomogeneous graph

On a {K4, K2,2,2}-ultrahomogeneous graph Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00931-3355 [email protected] Abstract The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 induced copies of K2,2,2, with no more copies of K4 or K2,2,2 existing in G. Moreover, each edge of G is shared by exactly one copy of K4 and one of K2,2,2. While the line graphs of d-cubes, (3 ≤ d ∈ ZZ), are {Kd,K2,2}-ultrahomogeneous, G is not even line-graphical. In addition, the chordless 6-cycles of G are seen to play an interesting role and some self-dual configurations associated to G with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered. 1 Introduction Let H be a connected regular graph and let m,n ∈ ZZ with 1 < m < n. An {H}mn -graph is a connected graph that: (a) is representable as an edge-disjoint union of n induced copies of H ; (b) has exactly m copies of H incident to each vertex, with no two such copies sharing more than one vertex; and (c) has exactly n copies of H as induced subgraphs isomorphic to H . We remark that an {H}mn -graph G is {H}-ultrahomogeneous (as in [7]) if every isomor- phism between two copies of H in G extends to an automorphism of G. Graph ultrahomo- geneity is a concept that can be traced back to [9, 6, 8]. Notice that a connected graph G is m-regular if and only if it is a {K2} |E(G)|-graph. In this case, G is arc-transitive if and only if G is {K2}-ultrahomogeneous. Thus, {H}- ultrahomogeneity is a notion of graph symmetry stronger than arc-transitivity. If G is an {Hi} -graph, where i = 1, 2, and H1 6= H2, then G is said to be an -graph. If, in addition, G is {Hi}-ultrahomogeneous, for both i = 1, 2, then G is {H1, H2}-ultrahomogeneous, again as in [7]. If each edge of G is in exactly one copy of Hi, for both i = 1, 2, then G is said to be fastened. If min(m1, m2) = m1 = 2 and http://arxiv.org/abs/0704.1493v3 H1 is a complete graph, then G is said to be line-graphical. For example, the line graph of the d-cube, where 3 ≤ d ∈ ZZ, is a line-graphical fastened {Kd, K2,2}-ultrahomogeneous {K2,2} d(d−1)2d−3 -graph. The first case here, known as the cuboctahedron, is a fastened {K3, K2,2, C6}-ultrahomogeneous {K3} 8{K2,2} 6{C6} 4-graph, where C6 is 6-cycle. In Sections 3-5, a 12-regular fastened {K4, K2,2,2}-ultrahomogeneous {K4} 42{K2,2,2} graph G of order 42 and diameter 3 is presented. The role that d-cliques Kd and squares K2,2 play in the line graph of the d-cube is performed in G by tetrahedra K4 and octahedra K2,2,2, but in this case with min(m1, m2) =min(4, 3) > 2, so G is non-line-graphical. The graph G has automorphism-group order |A(G)| = 1008 = 4|E(G)|. In Section 5, the 252 edges of G can be seen as the left cosets of a subgroup Γ ⊂ A(G) of order 4, and its vertices as the left cosets of a subgroup of A(G) of order 24. These two equivalence classes of subgraphs of G, i.e. tetrahedra and octahedra, allow in Section 6 to define several combinatorial configurations ([3]) related to G, 3 of which are self-dual, with their Levi graphs as: (1) a 4-regular 2-arc-transitive graph ([2]) on 84 vertices and 1008 automorphisms, with diameter = girth = 6, reflecting a natural duality property of G; (2) an 8-regular arc-transitive graph on 42 vertices and 2016 automorphisms, with diameter = 3 and girth = 4; and (3) a 6-regular semisymmetric graph ([5]) on 336 vertices and 1008 automorphisms, with diameter = girth = 6 and just two slightly differing distance distributions. The Menger graph and dual Menger graph associated to this Levi graph have common degree 24 and diameter = girth = 3, with 1008 and 2016 automorphisms, respectively. Section 7 distinguishes the k-holes (or chordless k-cycles) of G with the least k > 4, namely k = 6, and studies their participation in some toroidal subgraphs of G that together with the octahedra of G can be filled up to form a closed piecewise linear 3-manifold. After some considerations on the Fano plane, we pass to define G and study its properties. 2 Ordered Fano pencils The Fano plane F is the (73)-configuration with points 1,2,3,4,5,6,7 and Fano lines 123, 145, 167, 246, 257, 347, 356. The map Φ that sends the points 1, 2, 3, 4, 5, 6, 7 respectively onto the lines 123, 145, 167, 246, 257, 347, 356 has the following duality properties: (1) each point p of F pertains to the lines Φ(q), where q ∈ Φ(p); (2) each Fano line ℓ contains the points Φ(k), where k runs over the lines passing through Φ(ℓ). Given a point p of F , the collection of lines through p is a pencil of F . A linearly ordered presentation of these lines is an ordered pencil through p. An ordered pencil v through p is denoted v = (p, qara, qbrb, qcrc), orderly composed, in reality, by the lines pqara, pqbrb, pqcrc. Note that there are 3! = 6 ordered pencils through any point p of F . 3 The {K4, K2,2,2}-ultrahomogeneous graph G Ordered pencils constitute the vertex set of our claimed graph G, with any two vertices v = (p, qara, qbrb, qcrc) and v ′ = (p′, q′ar c) adjacent whenever the following two conditions hold: (1) p 6= p′; (2) |piri ∩ p i| = 1, for i = a, b, c. The 3 points of intersection resulting from item (2) form a Fano line, which we consider as an ordered Fano line by taking into account the subindex order a < b < c, and as such, set it as the strong color of the edge vv′. This provides G with an edge-coloring. An alternate definition of G can be given via Φ−1, in which the vertices of G can be seen as the ordered Fano lines xaxbxc, with any two such vertices adjacent if their associated Fano lines share the entry in F of exactly one of its 3 positions, either a or b or c. We keep throughout, however, the ordered-pencil presentation of G, but the first self-dual configura- tion of Subsection 6.1 and accompanying example show that the suggested dual presentation of G is valid as well. Notice that the vertices of G with initial entry p = 1 appear in lexicographic order as: (1, 23, 45, 67), (1, 23, 67, 45), (1, 45, 23, 67), (1, 45, 67, 23), (1, 67, 23, 45), (1, 67, 45, 23), which may be simplified in notation by using super-indices a through f to denote the shown order, that is: 1a, 1b, 1c, 1d, 1e, 1f , respectively. A similar lexicographic presentation may be given to the vertices of G having p = 2, . . . , 7. This treatment covers the 42 vertices of G. As an example of the adjacency of G, the neighbors of 1a = (1, 23, 45, 67) in G are: (2, 13, 46, 57), (2, 13, 57, 46), (3, 12, 47, 56), (3, 12, 56, 47), (4, 26, 15, 37), (4, 37, 15, 26), (5, 27, 14, 36), (5, 36, 14, 27), (6, 24, 35, 17), (6, 35, 24, 17), (7, 25, 34, 16), (7, 34, 25, 16), or in the continuation of the simplified notation above: 2a, 2b, 3a, 3b, 4c, 4e, 5c, 5e, 6d, 6f , 7d, 7f . The strong colors of the resulting edges are: 167, 154, 176, 154, 356, 246, 347, 451, 321, 231, 321, respectively. Given vertices v = (p, qara, qbrb, qcrc) and w = (p ′, q′ar c) adjacent in G, there exists a well-defined j ∈ {a, b, c} such that (1) p ∈ q′jr j; (2) p ′ ∈ qjrj ; (3) the lines pqjrj and p′q′jr j intersect at either qj or rj , which coincides with either q j or r j. Say that these lines pqjrj and p ′q′jr j intersect at qj . Then qj (including the subindex j) is taken as the weak color for the edge vw. This provides G with another edge-coloring, with symbols qj, where q ∈ F and j ∈ {a, b, c}. For example, the weak colors qj corresponding to the 12 edges incident to 1a, as cited above, are: 3a, 3a, 2a, 2a, 5b, 5b, 4b, 4b, 7c, 7c, 6c, 6c, respectively. 3.1 The automorphism group A(G) of G The 12 neighbors of 1a displayed above induce a subgraph NG(1 a) of G, called the open neighborhood of 1a in G, which is isomorphic to the graph Λ of the hemi-rhombicubocta- hedron (obtained from the rhombicuboctahedron by identification of antipodal vertices and edges). This is a 4-regular vertex-transitive graph on 12 vertices embedded in the projective plane with 13 faces realized by 4 disjoint triangles and 9 additional 4-holes. The 4-holes are of two types: (1) 6 have two opposite sides adjacent each to a triangle; (2) the other 3 have only their vertices in common with the 4 triangles. We also have the graph homomorphism f : Λ → K4 of Figure 1, where f(ji) = i for i ∈ {0, 1, 2, 3}, j ∈ {a, b, c} and Λ is depicted in two different ways inside (dotted) fundamental polygons of the real projective plane. ❜ ❜❜ ❜ b0❜ ❜ b0 b2 2 3❜ ❜ Figure 1: The homomorphism f : Λ → K4 with f(ji) = i for i ∈ {0, 1, 2, 3}, j ∈ {a, b, c} Moreover, we may identify Λ with NG(1 a) via a graph isomorphism g : Λ → NG(1 a) given g(a0) = 5 c, g(a1) = 4 c, g(a2) = 5 e, g(a3) = 4 g(b0) = 6 d, g(b1) = 7 d, g(b2) = 7 f , g(b3) = 6 g(c0) = 2 b, g(c1) = 2 a, g(c2) = 3 b, g(c3) = 3 Moreover, the graph homomorphism f induces, at the level of automorphism groups of graphs, a group isomorphism f ∗ : A(Λ) → A(K4) = S4. In fact, f ∗ is given by sending the following generators of A(Λ) into corresponding generators of S4 (that can be better visualized from the leftmost Λ to the rightmost K4 depicted in Figure 1): (a1b2c3)(b1c2a3)(c1a2b3)(a0b0c0) → (123), (a0a1)(a2a3)(b0b1)(b2b3)(c0c1)(c2c3) → (01)(23), (a0a1a2a3)(c0b1c2b3)(b0c1b2c3) → (0123). Thus, A(NG(1 a)) = A(Λ) = S4 has 24 elements, which is consistent with the size of a vertex stabilizer of G. Furthermore, since G has 42 vertices that behave exactly in the same geometric way as ordered pencils in F , we conclude that |A(G)| = 42× 24 = 1008. 4 Copies of K2,2,2 and K4 in G Notice that f maps bijectively the 3 4-cycles and 4 triangles of K4 respectively onto the 3 4-holes of Λ of type (2) above and the 4 triangles of Λ. Notice also that these 7 holes form a cycle-decomposition of Λ. Inside the closed neighborhood NG[w] of each vertex w of G (induced in G by w and the open neighborhood NG(w)), we obtain 3 copies of K2,2,2 and 4 copies of K4, which are induced by w together respectively with the mentioned 3 4-holes and 4 triangles. Observe that these 7 induced subgraphs of G have intersection formed solely by w. The rest of this section is dedicated to the study of these polyhedral subgraphs. First, notice that the inverse image f−1 of each edge of K4 is one of the 6 4-holes of Λ of type (1) above. This yields another cycle-decomposition of Λ, which in turn makes explicit the remaining 4-holes of G, apart from the 4-holes contained in the copies of K2,2,2 of G. However, these new 4-holes are not contained in any copy of K2,2,2 in G. 4.1 Copies of K2,2,2 in G Each vertex of G belongs to 3 induced copies of K2,2,2 in G. For example, the sets of weak colors qj of the edges of such copies for the vertex 1 a, that contain the 4-holes g(c0, c2, c1, c3), g(a0, a1, a2, a3) and g(b0, b1, b3, b2) arising in Subsection 3.1, are respectively: {1a, 2a, 3a}, {1b, 4b, 5b}, {1c, 6c, 7c}. Each qj colors the edges of a specific 4-hole in its corresponding copy of K2,2,2. The 3 weak colors appearing in each copy of K2,2,2 correspond bijectively with its 3 4-holes, the edges of each 4-hole bearing a common weak color of its own. A similar situation holds for any other vertex of G. There is a copy of K2,2,2 in G whose set of weak colors of edges is {xj , yj, zj}, for each line xyz of F and index j ∈ {a, b, c}. We denote this copy of K2,2,2 by [xyz]j . As a result, there is a total of 21 = 7 × 3 copies of K2,2,2 in G. In fact, triangles with weak colors qj sharing a common j (but q varying) are only present in the said copies of K2,2,2 in G. Each 4-hole in a copy of K2,2,2 in G have: (1) edges sharing a common weak color qj and (2) opposite vertices representing ordered pencils through a common point of F , which yields a total of two such points per 4-hole. For example, the strong colors of the triangles [xyz]j composing the copies of K2,2,2 incident to 1a conform triples of strong colors having: for [123]a, a-entries covering line 123, and another fixed entry equal to each one of 4,5,6,7: (145, 246, 347), (154, 257, 356), (167, 257, 347), (176, 275, 374), (154, 264, 374), (145, 275, 365), (176, 275, 374), (167, 246, 365); for [145]b, b-entries covering line 145, and another fixed entry equal to each one of 2,3,6,7: (213, 246, 257), (312, 347, 356), (617, 642, 653), (716, 743, 752), (312, 642, 752), (213, 743, 653), (716, 246, 356), (617, 347, 257); for [167]c, c-entries covering line 167, and another fixed entry equal to each one of 2,3,4,5: (231, 246, 257), (321, 356, 347), (451, 426, 437), (541, 536, 527), (321, 426, 527), (231, 536, 437), (541, 246, 347), (451, 356, 257). In fact, these triangles are respectively: (2a, 3a, 1a), (2b, 3b, 1a), (2a, 3b, 1a), (2b, 3a, 1a), (2b, 3b, 1b), (2a, 3a, 1b), (2b, 3a, 1b), (2a, 3b, 1b); (4c, 5c, 1a), (4e, 5e, 1a), (4c, 5e, 1f), (4e, 5c, 1f), (4e, 5e, 1f), (4c, 5c, 1f), (4e, 5c, 1a), (4c, 5e, 1a); (6d, 7d,1a), (6f , 7f ,1a), (6d, 7f , 1c), (6f , 7d,1c), (6f , 7f ,1c), (6d, 7d, 1c), (6f , 7d, 1a), (6d, 7f , 1a). The sets of strong colors for the respective composing 4-holes are: (145, 167, 154, 176), (246, 257, 264, 275), (347, 356, 374, 365); (213, 617, 312, 716), (246, 347, 624, 743), (257, 356, 752, 653); (231, 451, 321, 541), (246, 356, 426, 536), (257, 347, 527, 437). In fact, these 4-holes are respectively: (2a, 3a, 2b, 3b), (3a, 1a, 3b, 1b), (2a, 1a, 2b, 1b); (4c, 5e, 4e, 5c), (5c, 1a, 5e, 1f), (4c, 1a, 4e, 1f); (7d, 6d, 7f ,6f), (7d, 1a, 7f ,1c), (6d, 1a, 6f , 1c). 4.2 Copies of K4 in G There is one copy of K4 in G for each ordered Fano line xyz. Such a copy, denoted 〈xyz〉, is formed by 3 pairs of equally weakly-colored opposite edges, with weak colors xa, yb and zc. For each p ∈ F \ {x, y, z}, there is exactly one vertex (p, qara, qbrb, qcrc) of 〈xyz〉, with x ∈ qara, y ∈ qbrb, z ∈ qcrc. The strong colors of the edges of 〈xyz〉 are precisely xyz. For example, the triangles g(c1, a2, b2), g(c3, a0, b1), g(c2, a1, b0) and g(c0, a3, b2) from Subsection 3.1 are contained respectively in 〈347〉, 〈246〉, 〈257〉 and 〈356〉. Since there are 42 such copies of K4 in G, we arrive at the following result. Theorem 4.1 The graph G is a 12-regular {K4} 42{K2,2,2} 21-graph of order 42 and diameter 3. Each vertex of G is incident to exactly 3 copies of K2,2,2 and 4 copies of K4. Proof: Let G′ be the graph defined by the same rules that define G on the unordered Fano lines. Then it is not hard to prove that G′ is isomorphic to the graph 2K7, the complete graph on 7 vertices with each edge doubled. The graph G is then a 6-fold covering graph over G′. Also, the lexicographically smallest path realizing the diameter of G is the 3-path (1a, 2a, 4a, 1d). The statement follows. 4.3 Disposition of copies of K2,2,2 and K4 in G ❝ ❝ ❝ ❝ 4f7a 7e 6c 6c 1c 1c 6c 6c 5a 2b 5b 4a Figure 2: Disposition of copies of K2,2,2 and K4 at vertex 7 f in G Each point p of F determines a Pasch configuration PC(p), formed by the 4 lines of F that do not contain p. This PC(p) may be denoted also pc(qara, qbrb, qcrc), where pqara, pqbrb, pqcrc are the lines of F containing p. None of the lines of PC(p) contains either qara or qbrb or qcrc. The 7 possible Pasch configurations here are: PC(1) = pc(23, 45, 67) = {246, 257, 347, 356}, PC(2) = pc(13, 46, 57) = {145, 167, 347, 356}, PC(3) = pc(12, 47, 56) = {145, 167, 246, 257}, PC(4) = pc(15, 26, 37) = {123, 167, 257, 357}, PC(5) = pc(14, 27, 36) = {123, 167, 246, 347}, PC(6) = pc(17, 24, 35) = {123, 145, 257, 347}, PC(7) = pc(16, 25, 34) = {123, 145, 246, 356}. Figure 2 shows the disposition of the induced copies of K2,2,2 and K4 incident to the vertex 7f in G, represented by 3 octahedra and 4 tetrahedra, respectively, with vertices and edges accompanied by their respective simplified notations and weak colors. The 4 tetrahedra in the figure are also shown as separate entities, for better distinction, while the 3 octahedra are integrated in the central drawing as an upper-left, an upper-right and a lower-central oc- tahedron, radiated from the central vertex, 7f . This 7 polyhedra can be blown up to 3-space without more intersections than those of the vertices and edges shown in the figure. Starting from the right upper corner in the figure and shown counterclockwise, the 3 octahedra have respective composing 4-holes, each sub-indexed with its common weak color, as follows: [347] : (7f , 4e, 7e, 4f)3a , (7 f , 3d, 7e, 3e)4a , (3 d, 4e, 3e, 4f)7a ; [257] : (7f , 5a, 7a, 5f)2b , (7 f , 2b, 7a, 2d)5b , (2 b, 5a, 2d, 5f)7b ; [167] : (7f , 6d, 7d, 6f)1c , (7 f , 1a, 7d, 1c)6c , (1 a, 6d, 1c, 6f)7c . The triangles in each octahedron here differ from those in the copies of K4 in G in the way their edges are weakly colored. For example, the copies of K4 in Figure 2, namely those denoted 〈321〉, 〈426〉, 〈356〉, 〈451〉, have their corresponding sets of constituent triangles with the clockwise sequences of simplified notations and weak colors of respective alternate incident vertices and edges, as follows: {(7f , 2b, 5 f , 1c, 4 f , 3a), (6 f , 3a, 5 f , 1c, 4 f , 2b), (7 f , 1c, 6 f , 3a, 5 f , 2b), (7 f , 3a, 4 f , 2b, 6 f , 6c)}; {(7f , 6c, 1 c, 4a, 5 a, 2b), (3 e, 2b, 1 c, 4a, 5 a, 2b), (7 f , 4a, 3 e, 2b, 1 c, 4a), (7 f , 2b, 5 a, 6c, 3 e, 4a)}; {(7f , 3a, 4 e, 5b, 1 a, 6c), (2 b, 6c, 4 e, 5b, 1 a, 3a), (7 f , 6c, 1 a, 3a, 2 b, 5b), (7 f , 5b, 2 b, 6c, 4 e, 3a)}; {(2d, 1c, 3 d, 5b, 6 d, 4a), (7 f , 5b, 2 d, 1c, 3 d, 4a), (7 f , 6c, 6 d, 4a, 2 d, 5b), (7 f , 4a, 3 d, 5b, 6 d, 6c)}. This reflects the fact that the vertex 7f = (7, 34, 25, 16) = (p, qa, ra, qbrb, qcrc) is associated with the Pasch configuration pc(34, 25, 16) = pc(qara, qbrb, qcrc) given with its triples ordered according to the presence of the different symbols 6= 7 at the 3 pair positions a, b, c, which is shown in the ordered Fano lines 321 = qaqbqc, 426 = raqbrc, 356 = qarbrc, 451 = rarbqc, or in their respectively associated tetrahedra: 〈321〉, 〈426〉, 〈356〉, 〈451〉. These ordered lines form the ordered Pasch configuration pc(7f) = {321, 426, 356, 451}. Similarly, an ordered Pasch configuration is associated to the set of copies of K4 incident to any other vertex of G. Moreover, the following two results are readily checked. Theorem 4.2 Any vertex v = (p, qara, qbrb, qcrc) of G can be expressed in such a way that 〈qaqbqc〉, 〈qarbrc〉, 〈raqbrc〉, 〈rarbqc〉 are its 4 incident copies of K4, reflecting their notation and that of its 3 incident octahedra. Proof: The ordered Pasch configuration pc(v) associated to v determines the ordered lines qaqbqc, qarbrc, raqbrc, rarbqc associated to the copies of K4, while the 3 remaining triples of F provide the data for the octahedra incident to v: [pqara]a, [pqbrb]b, [pqcrc]c. Theorem 4.3 For any edge e of G, there exists exactly one copy of K2,2,2 and one of K4 in G that intersect at e. Moreover, e is the only edge at which those copies intersect. Thus, G is fastened. Proof: Let e = vv′ have weak color qj , where v = (p, qara, qbrb, qcrc) and v ′ = (p′, q′ar c). Then, the octahedron [pp ′p′′]j and the tetrahedron 〈xyz〉 are the copies of K2,2,2 and K4 in the statement, where: (a) pp ′p′′ is the Fano line containing p and p′, (b) j ∈ {a, b, c} is such that pp′′ = q′jr j and pp ′ = qjrj and (c) xyz, one of the 4 ordered lines cited in Theorem 4.2 with respect to v, is the strong color of e. For example, the edge 7f5a has weak color 2b and strong color 426. This is the only edge shared by the octahedron [257]b and the tetrahedron 〈426〉. 5 Symmetric properties of G Each automorphism τ ∈ A(G), is the composition of a permutation φτ of F with a per- mutation ψτ of {a, b, c}. A set of 16 generators τi of A(G), (i = 1 . . . 16), is given by τi = ψi ◦ φi = φi ◦ ψi, where we denote φi = φ τi, ψi = ψ τi, and φ1 = (23)(67), φ2 = (45)(67), φ3 = (13)(57), φ4 = (46)(57), φ5 = (12)(56), φ6 = (47)(56), φ7 = (15)(37), φ8 = (26)(37), φ9 = (14)(27), φ10 = (27)(36), φ11 = (17)(35), φ12 = (24)(35), φ13 = (16)(34), φ14 = (25)(34), ψ15 = (ac), ψ16 = (bc), with ψi and φj taken as the identity maps of F and {a, b, c}, respectively, for 1 ≤ i ≤ 14 and j = 15, 16. The subgroup Γ ⊂ A(G) that sends the lexicographically smallest arc (1a, 2a) onto it- self, either directly or inversely oriented, includes exchanging, or not, its incident triangles (2a, 3a, 1a) and (2a, 3b, 1a) in [123]a, or (1 a, 2a, 6f) and (1a, 2a, 5e) in 〈347〉. Thus, Γ contains 4 elements and has generating set {τ6 ◦ τ16, τ5}. Moreover, Γ is a subgroup of A([1]a), which has generating set {τ1, τ2, τ5, τ6, τ16}. Furthermore, {τ1, τ2, τ5, τ6, τ15, τ16} is a generating set for A(∪cj=a[123]j). The remaining automorphisms τi map A(∪ j=a[123]j) onto its nontrivial cosets in A(G) by left multiplication. The subgroup of A(G) that fixes 1a has order 24 and generating set {φ1, φ2, φ4 ◦φ16, φ6 ◦φ16, φ8 ◦φ15, φ10 ◦φ15, φ12 ◦φ15 ◦φ16, φ14 ◦φ15 ◦φ16}. Theorem 5.1 G is a fastened {K4, K2,2,2}-ultrahomogeneous {K4} 42{K2,2,2} 21-graph which is non-line-graphical, with |A(G)| = 1008 = 4|E(G)|. The edges of G can be seen as the left cosets of a subgroup Γ ⊂ A(G) of order 4, and its vertices as the left cosets of a subgroup of A(G) of order 24. Proof: Recall from Subsection 3.1 that |A(G)| = 1008. Notice that A(〈347〉) = S4 is formed by 24 automorphisms. Since |A(G)| |A(〈347〉)| = 1008 = 42, then 〈347〉 is sent by an automorphism of G onto any other copy of K4 in G, from which it is not difficult to see that G is K4-ultrahomogeneous. Similarly, A([123]a) is formed by 48 automorphisms. Since |A(G)| |A([123]a)| = 1008 = 21, then [123]a is sent by an automorphism of G onto any other copy of K2,2,2 in G, from which it is not difficult to see that G is K2,2,2-ultrahomogeneous. On the other hand, |A(K2,2,2)| = 48 and |E(K2,2,2)| = 12 agree with the fact that |Γ| = |A(K2,2,2)| |E(K2,2,2)| = 4. Since G is the edge-disjoint union of 21 copies of K2,2,2, it contains a total of 21|E(K2,2,2)| = 21×12 = 252 edges. Now, |A(G)| = 1008 = 21×48 = 21|A(K2,2,2)|. This is 4 times the number 252 of edges of G. These edges correspond to the left cosets of Γ in A(G) and its vertices to the left cosets of the stabilizer of 1a = (1, 23, 45, 67) in A(G), whose order is 24. 6 Configurations associated with G The symmetrical disposition of objects in G gives place to several combinatorial point-line configurations and to their associated Levi, Menger, and dual Menger graphs. We present the points and lines of 3 self-dual configurations obtained from G, and their incidence relations: 1. the 42 vertices and 42 tetrahedra of G, and incidence given by inclusion of a vertex in a tetrahedron; this is a self-dual (424)-configuration with 2-arc-transitive Levi graph of diameter = girth = 6, automorphism-group order 2016, stabilizer order 24, distance distribution vector (1, 4, 12, 24, 27, 14, 2) and isomorphic arc-transitive Menger graphs of diameter = girth = 3, degree 12 and automorphism-group order 1008; 2. the 168 tetrahedral triangles and 168 octahedral triangles in G and their sharing of an edge; this is a self-dual (1686)-configuration with semisymmetric Levi graph of diame- ter = girth = 6, automorphism-group order 1008, common stabilizer order 6, distance distribution vectors (1, 6, 24, 60, 111, 102, 32) and (1, 6, 24, 60, 108, 102, 35), (just differ- ing at distances 4 and 6 by 3 vertices) and vertex-transitive Menger graphs of common degree 24, diameter = girth = 3 and automorphism-group orders 1008 and 2016, re- spectively. 3. the 168 tetrahedral triangles and 168 octahedral triangles in G and their sharing of an edge; this is a self-dual (1686)-configuration with semisymmetric Levi graph of diame- ter = girth = 6, automorphism-group order 1008, common stabilizer order 6, distance distribution vectors (1, 6, 24, 60, 111, 102, 32) and (1, 6, 24, 60, 108, 102, 35), (just differ- ing at distances 4 and 6 by 3 vertices) and vertex-transitive Menger graphs of common degree 24, diameter = girth = 3 and automorphism-group orders 1008 and 2016, re- spectively. Another interesting configuration associated to G is formed by the 42 tetrahedra and 21 octahedra of G, and their sharing of an edge; this is a flag-transitive (426, 2112)-configuration. Example. Let L be the Levi graph of the (424)-configuration in item 1 above. Then ((1, 23, 45, 67), 〈246〉, (3, 12, 47, 56), 〈145〉, (6, 17, 24, 35), 〈725〉, (1, 67, 23, 45)) and (〈123〉, (4, 15, 26, 37), 〈167〉, (2, 13, 46, 57), 〈347〉, (5, 36, 15, 27), 〈312〉) are the lexicographically smallest paths realizing the diameter of L and departing from each one of the two vertex parts of L. The second lexicographically smallest paths are ((1, 23, 45, 67), 〈246〉, (3, 12, 47, 56), 〈176〉, (4, 15, 37, 26), 〈572〉, (1, 45, 67, 23)) and (〈123〉, (4, 15, 26, 37), 〈167〉, (3, 12, 56, 47), 〈264〉, (5, 27, 36, 14), 〈231〉). We reach this way to the only two vertices realizing the diameter of L starting from (1, 23, 45, 67), namely (1, 67, 23, 45) and (1, 45, 67, 23); respectively: starting from 〈123〉, namely 〈312〉 and 〈231〉. Those two pairs of paths reflect the correspondence between both parts of L induced by the map Φ in Section 2. 7 On 6-holes and other subgraphs of G 1 2 3 5b6 4 ✓✏ 5 2 7 1b6 4 ✓✏ 7 2 5 3b6 4 ✓✏ 3 2 1 7b6 4 ❝6d 4c2a 6a 4a 6a 4a 4c 6d 4d 6a 4a 6a 4a 1 2 3 5c6 4 ✓✏ 5 2 7 1c6 4 ✓✏ 7 2 5 3c6 4 ✓✏ 3 2 1 7c6 4 ❝6c 4d2a 6a 4a 6a 4a 4d 6c 4c 6a 4a 6a 4a Figure 3: Octahedral triangles of [246]a Let us depict the Fano plane in an octahedral triangle, as shown for example in Figure 3 for the 8 triangles of [246]a, as follows. For each such triangle t, there is a unique i ∈ {a, b, c} and a unique point p ∈ F such that the 3 vertices of t (looked upon as ordered pencils) have p present in position i. The central point of the depiction of F inside each such t is set to be p, subindexed by i. For example in Figure 3, this i appears as subindices c and b, respectively, in the 4 top and 4 bottom triangles. Also, for each edge e of t with weak color qj and strong color ℓ, the point in the depiction of F in t at the middle of e is set to be q, and the ’external’ line of F containing q is set to be ℓ, with the points of ℓ \ {q} set near the endvertices of e. G contains 84 6-holes obtainable from the octahedral triangles of G based on these depictions of F . This is exemplified on the left side of Figure 4, where the upper-left triangle in Figure 3 appears as the bottom triangle, sharing its edge of weak color 2a with the central 6-hole. The 6-cycle of weak colors associated to this 6-hole is (2a3b1a2b3a1b). 1 2 3 5c6 4 ❝ ❝ ❝ 1 7 6 5c3 4 4 7 3 5c6 1 6 7 1 3 2 1 3 7 4 7b 7e 7a 7a 7b 7b 6a 4a 6b 4b 2465c 2463c 2465c 2465a 2461a 2467a 2467c 2463a 2461c 2467c 1235c 2465c 2465c 3475c 167 1675c 347 1a 3b 3a1b 7a 5a7f5f 3d 1f3f 1d 5c 7c5d7d 1e 3c1c 3e 7e 5e7b5b Figure 4: A 6-hole and a star subgraph of G Since the central color of each octahedral triangle having an edge in common with this 6-hole is 5c, and the line 123 has its points composing the weak colors of its edges, we denote this 6-hole by 1235c , as indicated in the figure. Similar denominations are given to the 6-holes neighboring this 1235c in the figure. We observe that the 6-holes neighboring 123 c on left and right coincide with 2465c ; on upper-left and lower-right with 347 c; on lower-left and upper- right with 1675c. These 4 6-holes form, together with the 8 octahedral triangles that appear in their generation (i.e. the 6 in the figure plus (1b, 2a, 3a) and (1e, 2c, 3c)), a non-induced toroidal subgraph of G that we may denote [5]c. In the same way, a non-induced subgraph [w]d is obtained, for each w ∈ F and d ∈ {a, b, c}. The subgraph [[w]]d induced by [w]d in G is formed by its union with the copies of K4 in G of the form 〈xaxbxc〉, with xd = w, a total of 6 copies of K4 sharing each a 4-cycle with [w]d. Each [x1xbxc] here is the union of such a 4-cycle plus two additional edges of weak color wd. We get 21 subgraphs [[w]]d of G. The 6-holes of the form xyzwd , where xyz is a fixed line of F , d varies in {a, b, c} and w in F , are 12 in number and conform a subgraph [xyz] of G isomorphic to the star Cayley graph ST4, that can be defined as the graph with vertex set S4 and each vertex (a0, a1, a2, a3) ∈ S4 adjacent solely to (a1, a0, a2, a3), (a2, a1, a0, a3) and (a3, a1, a2, a0); see [1, 4]. For example, the right side of Figure 4 depicts a (dotted) fundamental polygon of the torus whose convex hull contains a representation of the subgraph [246] of G. This subgraphs [xyz] are not induced in G. However, the graph [[xyz]] induced in G by each [xyz] is the edge-disjoint union of [xyz] with the edge-disjoint union of six copies of K4 in G, namely: 〈xyz〉, 〈xzy〉, 〈yxz〉, 〈yzx〉, 〈zxy〉, 〈zyx〉. Figure 4 has 4 vertices painted black, which span a copy of K4 in G but not in [246]. We get 7 subgraphs [xyz] of G. Now, two new flag-transitive configurations associated to G (apart from those cited in Section 6) are given by: (a) the 42 tetrahedra and 21 subgraphs [wd] in G, and inclusion of a tetrahedron in a copy of T ; this is a (423, 216)-configuration; (b) the 21 subgraphs [wd] and 7 copies of ST4 in G, and their sharing of a 6-hole; this is a (214, 712)-configuration. When considered immersed in 3-space, the 21 octahedra and 28 toroidal subgraphs of G presented above have faces that appear in canceling pairs, allowing the visualization of a closed piecewise-linear 3-manifold. We ask: which are the properties of this manifold? 8 Open problems It remains to see whether G is a Cayley graph or not. On the other hand, the definition of G may be extended by means of projective planes, like the Fano plane, but over larger fields than GF (2), starting with GF (3). Moreover, the two conditions of the definition of G in Section 3 may be taken to 3 conditions, replacing F by a binary projective space P (r− 1, 2) of dimension r−1, and the Fano lines by subspaces of dimension σ < r−1, where 2 < r ∈ ZZ and σ ∈ (0, r−1)∩ZZ, and requiring, as a third condition, that the points of intersection of a modified condition (b) form a projective hyperplane in P (r− 1, 2), (which was not required for G, since it was a ready conclusion). The resulting graph, that appears in place of G, may not be even connected, but the study of the component containing the lexicographically smallest vertex could still be interesting. Another step would be taking the study over other fields, starting with the ternary one. Acknowledgement: The author is grateful to Josep Rifà and Jaume Pujol, for their friend- ship and support, and to the referee, for his helpful observations, among which was suggesting the hemi-rhombicuboctahedron as the open neighborhood of each vertex of G. References [1] S. B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnec- tion networks, IEEE Trans. Comput., 38(1989) 555–565. [2] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1993. [3] H.S.M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc., 56(1950) 413–455. [4] I. J. Dejter and O. Serra, Efficient dominating sets in Cayley graphs, Discrete Applied Mathematics, 119 (2003) 319–328. [5] J. Folkman, Regular line-symmetric graphs, J. Combin. Theory, 3(1967), 215–232. [6] A. Gardiner, Homogeneous graphs, J. Combinatorial Theory (B), 20 (1976), 94-102. [7] D. C. Isaksen, C. Jankowski and S. Proctor, On K∗-ultrahomogeneous graphs, Ars Com- binatoria, Volume LXXXII, (2007), 83–96. [8] C. Ronse, On homogeneous graphs, J. London Math. Soc. (2) 17 (1978), 375–379. [9] J. Sheehan, Smoothly embeddable subgraphs, J. London Math. Soc. (2) 9 (1974), 212–218. Introduction Ordered Fano pencils The {K4,K2,2,2}-ultrahomogeneous graph G The automorphism group A(G) of G Copies of K2,2,2 and K4 in G Copies of K2,2,2 in G Copies of K4 in G Disposition of copies of K2,2,2 and K4 in G Symmetric properties of G Configurations associated with G On 6-holes and other subgraphs of G Open problems

0704.1494

Unconventional approaches to combine optical transparency with electrical conductivity

Unconventional approaches to combine optical transparency with electrical conductivity J. E. Medvedeva∗ Department of Physics, University of Missouri–Rolla, Rolla, MO 65409 Combination of electrical conductivity and optical transparency in the same material – known to be a prerogative of only a few oxides of post-transition metals, such as In, Sn, Zn and Cd – manifests itself in a distinctive band structure of the transparent conductor host. While the oxides of other elements with s2 electronic configuration, for example, Mg, Ca, Sc and Al, also exhibit the desired optical and electronic features, they have not been considered as candidates for achieving good electrical conductivity because of the challenges of efficient carrier generation in these wide- bandgap materials. Here we demonstrate that alternative approaches to the problem not only allow attaining the transport and optical properties which compete with those in currently utilized transparent conducting oxides (TCO), but also significantly broaden the range of materials with a potential of being developed into novel functional transparent conductors. The key attribute of any conventional n-type TCO host is a highly dispersed single free-electron-like conduction band [1, 2, 3, 4, 5, 6, 7, 8]. Upon proper doping, it provides both (i) high mobility of extra carriers (elec- trons) due to their small effective mass, and (ii) low op- tical absorption in the visible part of the spectrum due to high-energy inter-band transitions, e.g., Fig. 1. For the complete transparency in the visible range, the tran- sitions from the valence band, Ev, and from the partially filled conduction band, Ec, should be larger than 3.1 eV, while the intra-band transitions as well as the plasma fre- quency should be smaller than 1.8 eV. The high energy dispersion also ensures a pronounced Fermi energy dis- placement, so-called Burstein-Moss (BM) shift, so that the optical transparency can be achieved in a material with a relatively small bandgap, for example, in CdO where the optical (direct) band gap is 2.3 eV. Figures 1(a) and 1(b) illustrate the typical conduction band of a conventional n-type transparent conductor and how doping alters the electronic band structure of the TCO host affecting the optical transitions. It is seen that upon introduction of extra carriers into the host, a large BM shift which facilitates higher-energy transitions from the valence band (Ev), leads to a reduced energy of the transitions from the Fermi level up into the conduction band (Ec), i.e., Ev and Ec are interconnected [9]. In other words, large carrier concentrations desired for a good conductivity, may result in an increase of the optical absorption because the Ec transitions become smaller in energy. In addition, the transitions within the partially filled band as well as plasma frequency may lead to the absorption in the long-wavelength range. The mutual exclusiveness of the optical transmittance and electrical conductivity (see Refs. [5, 10, 11]) makes it challenging to achieve the optimal performance in a transparent conductor. Below we outline novel, uncon- ventional ways to balance the optical and transport prop- erties and to improve one without making a sacrifice of the other. (a) Γ N P Γ H N (b) N H Γ P N Γ (c) N H Γ P N Γ N H Γ P N Γ (d) FIG. 1: Electronic band structure of pure (a), 6.25% Sn-doped (b), and 6.25% Mo-doped In2O3 for the majority (c) and the minority (d) spin channels. Magnetically mediated transparent conductors One of the possible routes to avoid compromising the optical transparency is to enhance conductivity via mo- bility of the carriers rather than their concentration [11]. Recently, the mobility with more than twice the value of the commercial Sn-doped indium oxide (ITO) was ob- served in Mo-doped In2O3 (IMO), and it was shown that the conductivity can be significantly increased with no changes in the spectral transmittance upon doping with Mo [12, 13, 14, 15]. Surprisingly, introduction of the http://arxiv.org/abs/0704.1494v1 Complex M Eg(0) k [110] [111] [010] Mo•••In(1) 1.85 1.38 0.152 0.148 0.157 1.63 Mo•••In(2) 1.32 1.18 0.194 0.187 0.201 2.05 [Mo•••In O • 0.50 1.26 0.125 0.123 0.132 1.27 Sn•In(1) — 0.98 0.201 0.203 0.205 2.29 In2O3+e — 1.16 0.206 0.204 0.213 2.38 TABLE I: Calculated magnetic moments on the Mo atoms, M, in µB ; the fundamental band gap values Eg(0), in eV; the Fermi wave vectors kF , in atomic units; and the plasma fre- quency ωp, in eV, for the different substitutional complexes with 6.25% Mo doping level. Calculated values for pure (rigid- band model) and 6.25% Sn-doped In2O3 are given for com- parison. transition metal Mo which donates two more carriers per substitution compared to Sn, does not lead to the ex- pected increase of the optical absorption or a decrease of the mobility due to the scattering on the localized Mo d-states. Our electronic band structure investigations of IMO revealed [8] that the magnetic interactions which have never been considered to play a role in combining opti- cal transparency with electrical conductivity, ensure both high carrier mobility and low optical absorption in the visible range. As one can see from Figs. 1(c) and 1(d), strong exchange interactions split the Mo d-states located in the vicinity of the Fermi level. These d-states are res- onant states, while the conductivity is due to the delo- calized In s-states which form the highly dispersed free- electron-like conduction band. In other words, the free carriers in the system flow in a background of the Mo defects which serve as strong scattering centers. Because of the exchange splitting of the Mo d-states, the carriers of one spin is affected by only a half of the scattering centers, i.e., only by the Mo d-states of the same spin. Therefore, the concentration of the Mo scattering cen- ters is effectively lowered by half compared to the Mo doping level. Figs. 1(b) and 1(c,d) show that the BM shift is less pronounced in the IMO case – despite the fact that Mo6+ donates two extra carriers as compared to Sn4+ at the same doping level. Such a low sensitivity to doping ap- pears from the resonant Mo d-states located at the Fermi level that facilitates the d-band filling (pinning) and thus hinders further displacement of the Fermi level deep into the conduction band. Smaller BM shift in IMO leads to the following advantageous features to be compared to those of ITO: (i) Smaller increase in the effective mass is expected upon Mo doping. In addition, the resonant Mo d-states do not hybridize with the s-states of indium and so do not affect the dispersion of the conduction band. There- fore, the effective mass remains similar to the one of pure indium oxide. This is borne out in experimental observa- tions [14] showing that the effective mass does not vary with doping (up to 12 % of Mo) and/or carrier concen- tration. (ii) Larger (in energy) optical transitions from the par- tially occupied band (cf., Figs. 1(b) and 1(c,d)) along with the fact that transitions from d- to s-states are for- bidden ensure lower short-wavelength optical absorption. (iii) The calculated plasma frequency, ωp, in IMO is be- low the visible range and significantly smaller than that of ITO (Table I). This finding suggests a possibility to in- troduce larger carrier concentrations without sacrificing the optical transmittance in the long wavelength range. (iv) Smaller BM shift does not lead to the appear- ance of the intense inter-band transitions from the va- lence band, Ev, in the visible range due to the large op- tical band gap in pure indium oxide, namely, 3.6 eV [16]. Furthermore, in contrast to ITO where the bandgap nar- rowing has been demonstrated both experimentally [16] and theoretically [2], doping with Mo shows an opposite (beneficial) trend: the fundamental band gap increases upon introduction of Mo, Table I, because the asymmet- ric d-orbitals of Mo rotate the p-orbitals of the neighbor- ing oxygen atoms leading to an increase of the overlap between the latter and the In s-states. It is important to note that the optical and transport properties in IMO are sensitive to specific growth condi- tions, namely, the ambient oxygen pressure. It is found [8] that an increased oxygen content facilitates the forma- tion of the oxygen compensated complexes which reduces the number of free carriers – from 3 to 1 per Mo substitu- tion – but, at the same time, improves the carrier mobil- ity due to smaller ionized impurity scattering and hence longer relaxation times. On the other hand, the inter- stitial oxygen significantly supresses the magnetic inter- actions, Table I, which should be strong enough to split the transition metal d-states in order to provide good conductivity in one (or both) spin channels. Thus, the transition metal dopants can be highly ben- eficial in providing the transport and optical properties which compete with those of commercially utilized ITO. Similar behavior is expected upon doping with other transition metal elements and other hosts – provided that the magnetic interactions are small enough to keep the d↑-d↓ transitions out of the visible range. Multicomponent TCO with layered structure Complex transparent conductors consisting of struc- turally and/or chemically distinct layers, such as InGaO3(ZnO)m, m=integer, offer a way to increase con- ductivity by spatially separating the carrier donors (tra- ditionally, oxygen vacancies or aliovalent substitutional dopants) and conducting layers which transfer the carri- ers effectively, i.e., without charge scattering on the im- purities [17, 18, 19]. 1.2 1.6 2 2.4 2.8 Energy, eV 0.005 0.01 0 0.005 0 0.005 InGaZnO4 In s−states Ga s−states Zn s−states O1 s, p−states O2 s, p−states 0.4 0.8 1.2 1.6 2 Energy, eV 0 0.005 0.01 0.005 0.01 InAlMgO4 In s−states Al s−states Mg s−states O1 s, p−states O2 s, p−states FIG. 2: Partial density of states for InGaZnO4 and InAlMgO4. The homologous series InGaO3(ZnO)m and In2O3(ZnO)m, also known for their promising ther- moelectric properties [20], have been extensively studied experimentally [1, 18, 19, 21, 22, 23, 24, 25]. In these materials, octahedrally coordinated In layers alternate with (m + 1) layers of oxygen tetrahedrons around Zn (and Ga) [26, 27, 28]. Because the octahedral oxygen co- ordination of cations was long believed to be essential for a good transparent conductor [1, 4, 17, 29, 30, 31, 32], it has been suggested that the charge is transfered within the InO1.5 layers while Ga and Zn atoms were proposed as candidates for efficient substitutional doping [1, 18, 19]. However, accurate electronic band structure investiga- tions for InGaZnO4, m=1, showed [33] that the atoms from both InO1.5 and GaZnO2.5 layers give compara- ble contributions to the conduction band, Fig. 2(a). This resulted in a three-dimensional distribution of the charge density: the interatomic (or “background”) elec- tron density is similar in and across the [0001] layers. The isotropy of the electronic properties in this layered compound manifests itself in the electron effective masses being nearly the same in all crystallographic directions (Table II). Most strikingly, we found that the effective mass re- mains isotropic when the cation(s) in InGaZnO4 are re- placed by other elements with s2 electronic configuration, for example, Sc, Al and/or Mg. This finding may seem to be counterintuitive, since the s-states of Sc, Al and Mg are expected to be located deeper in the conduc- tion band due to significantly larger band gaps in Sc2O3, Al2O3 and MgO as compared to those in In2O3, Ga2O3 and ZnO. Analysis of the partial density of states shows that although the contributions from the Sc, Al and Mg atoms to the bottom of the conduction band are notably reduced, cf., Fig. 2 and Table II, the states of these Compound N1 N2 Eg(0) m[100] m[010] m[001] mab mz InGaZnO4 48% 52% 1.30 0.23 0.22 0.20 0.23 0.23 InGaMgO4 58% 42% 2.15 0.27 0.27 0.24 0.28 0.29 ScGaZnO4 26% 74% 2.48 0.33 0.33 0.34 0.33 0.53 InAlMgO4 72% 28% 2.78 0.32 0.31 0.35 0.31 0.34 TABLE II: Net contributions to the conduction band at the Γ point from the states of the atoms that belong to the In(Sc)O1.5, N1, or Ga(Al)Zn(Mg)O2.5, N2, layers, in per cent; the LDA fundamental bandgap values Eg(0), in eV; the electron effective masses m, in me, along the specified crys- tallographic directions; and the components of the electron effective-mass tensor, ma,b and mz, calculated via simple av- eraging of those of the corresponding single-cation oxides. atoms are still available for the electron transport. Con- sequently, the interatomic charge density distribution is three-dimensional for all these layered multi-cation ox- ides – in accord with the isotropic electron effective mass. Moreover, we found [33] that the electron effective mass in these complex materials can be predicted via simple averaging over those of the corresponding single-cation oxides (Table II). It is important to stress that the isotropic character of the intrinsic transport properties in the TCO hosts with layered structure may not be maintained when extra car- riers are introduced. Different valence states (In3+ and Ga3+ vs Zn2+) and oxygen coordination (octahedral for In vs tetrahedral for Ga and Zn) are likely to result in preferential (non-uniform) arrangement of aliovalent sub- stitutional dopants or oxygen vacancies. We believe that the observed anisotropic conductivity [20, 21] as well as its dependence on the octahedral site density [1, 32] in the layered TCO’s is a manifestation of a specific carrier generation mechanism. While proper doping can help make either or both structurally distinct layers conduct- ing, leading to a highly anisotropic or three-dimensional electron mobility, respectively, amorphous complex ox- ides [22, 24, 25] readily offer a way to maintain isotropic transport properties. Thus, we believe that other cations with s2 electronic configuration, beyond the traditional In, Sn, Zn and Cd, can be effectively incorporated into novel complex mul- ticomponent TCO hosts – such as the layered materi- als decribed above, ordered ternary oxides [32, 34, 35], solid solutions [1] as well as their amorphous counterparts [22, 35], important for flexible electronics technologies [24]. Significantly, the sensitivity of the bandgap value to the composition of a multicomponent oxide (Table II) offers a possibility to manipulate the optical properties as well as the band offsets (work functions) via proper composition of an application-specific TCO. Finally, it should be mentioned, that the efficient dop- ing of the wide-bandgap oxides is known to be a chal- lenge [36, 37, 38]. Alternative carrier generation mech- anisms, for example, magnetic dopants discussed above, FIG. 3: Crystal structure of H-doped 12CaO·7Al2O3. On the left, only three of the 12 cages in the unit cell are shown. The cube (right) represent the unit cell. Only the atoms that par- ticipate in hopping transport are shown: Ca (yellow), OH− (blue) and H− (green spheres). Red line represent the elec- tron hopping path. introduction of hydrogen [39] or ultraviolet irradiation in nanoporous calcium aluminate [5, 40, 41], are being ac- tively sought and have already yielded promising results – as outlined in the following section. Novel UV-activated transparent conductors Cage-structured insulating calcium-aluminum oxide, 12CaO·7Al2O3, or mayenite, differs essentially from the conventional TCO’s not only by its chemical and struc- tural properties but also by the carrier generation mecha- nism: a persistent conductivity (with a ten-order of mag- nitude change) has been achieved upon doping with hy- drogen followed by UV irradiation [40, 41]. Mayenite belongs to the CaO-Al2O3 family of Portland cements which are known for their superior refractory properties. The unique structural features of mayenite, Fig. 3, namely, the encaged “excess” oxygen ions, al- low incorporation of hydrogen according to the chem- ical reaction: O2−(cage) + H2(atm.) → OH −(cage) + H−(another cage). While the H-doped mayenite re- mains insulating (Fig. 4(a)), the conductivity results from the electrons excited by UV irradiation off the H− ions into the conduction band formed from Ca d-states. The charge transport occurs by electron hopping through the encaged “defects” – the H0 and OH− located inside the large (more than 5.6 Å in diameter) structural cavi- ties. Understanding of the conduction mechanism on the microscopic level [41, 42] resulted in prediction of ways to control the conductivity by targeting the particular atoms that participate in the hopping. These predictions have been confirmed experimentally [41, 43]. The low conductivity in H-doped UV-irradiated mayenite (∼1 S/cm, Ref. [40, 41]) was attributed to the strong Coulomb interactions between the UV released electrons which migrate along a narrow conducting chan- (a) (b) N H Γ P N Γ FIG. 4: Electronic band structure of (a) H-doped and (b) H-doped UV-irradiated Ca12Al14O33 and (c) [Ca12Al14O32] 2+(2e−). nel – the hopping path, Fig. 3. Alleviation of the elec- tronic repulsion [42] resulted in the observed [44] 100-fold enhancement of the conductivity in the mayenite-based oxide, [Ca12Al14O32] 2+(2e−), although the carrier con- centration was only two times larger than that in the H- doped UV-irradiated Ca12Al14O33. The improved con- ductivity, however, came at the cost of greatly increased absorption [42, 44] due to an increased density of states at the Fermi level, making this oxide unsuitable for prac- tical use as a transparent conducting material. Despite the failure to combine effectively the opti- cal transparency and with useful electrical conductiv- ity in Ca12Al14O33, the band structure analysis of the mayenite-based oxides suggests that these materials be- long to a conceptually new class of transparent conduc- tors where a significant correlation between their struc- tural peculiarities and electronic and optical properties allows achieving good conductivity without compromis- ing their optical properties. In striking contrast to the conventional TCO’s, where there is a trade-off between optical absorption and conductivity, as discussed above, nanoporous materials allow a possibility to combine 100% optical transparency with high electrical conductivity. The schematic band structure of such an “ideal” TCO is shown in Fig. 5. Introduction of a deep impurity band in the bandgap of an insulating material would help to keep intense interband transitions (from the valence band to the impurity band and from the impurity band to the conduction band) above the visible range. This requires the band gap of a host material to be more than 6.2 eV. Furthermore, the impurity band should be narrow enough (less than 1.8 eV) to keep intraband transitions (as well as the plasma frequency) below the visible range. In order to achieve high conductivity, the concentra- tion of impurities should be large enough so that their electronic wavefunctions overlap and form an impurity band. The formation of the band would lead to a high carrier mobility due to the extended nature of these states resulting in a relatively low scattering. For this, a ma- terial with a close-packed structure should not be used, because large concentration of impurities would result in (i) an increase of ionized impurity scattering which Conduction band Valence band FIG. 5: Schematic band structure of conventional (left) and “ideal” (right) transparent conductor. limits electron transport; and (ii) a large relaxation of the host material, affecting its electronic structure and, most likely, decreasing the desired optical transparency. Therefore, an introduction of a deep impurity band into a wide-band insulator with a close-packed structure would make the material neither conducting nor transparent. Alternatively, materials with a nanoporous structure may offer a way to incorporate a large concentration of impuri- ties without any significant changes in the band structure of the host material. Zeolites have been proposed [5] as potential candidates for the “ideal” TCO’s because they possess the desired structural and optical features, i.e., spacious interconnected pores and large bandgaps, and also exhibit the ability to trap functional “guest” atoms inside the nanometersized cavities which would govern the transport properties of the material. Thus, understanding the principles of the conventional transparent conductors provide a solid base for further search of novel TCO host materials as well as efficient car- rier generation mechanisms. Ab-initio density-functional band structure investigations [45] are valuable not only in providing a thorough insight into the TCO basics but also in predicting hidden capabilities of the materials be- yond the traditionally employed. ∗ E-mail:[email protected] [1] A.J. Freeman, K.R. Poeppelmeier, T.O. Mason, R.P.H. Chang, T.J. Marks, MRS Bull. 25, 45 (2000) [2] O.N. Mryasov, A.J. Freeman, Phys. Rev. B 64, 233111 (2001) [3] R. Asahi, A. 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0704.1495

Exponential Decay of Correlations for Randomly Chosen Hyperbolic Toral Automorphisms

EXPONENTIAL DECAY OF CORRELATIONS FOR RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS ARVIND AYYER1 AND MIKKO STENLUND2 Abstract. We consider pairs of toral automorphisms (A,B) sat- isfying an invariant cone property. At each iteration, A acts with probability p ∈ (0, 1) and B with probability 1 − p. We prove exponential decay of correlations for a class of Hölder continuous observables. 1. Introduction 1.1. Background. Toral automorphisms are the simplest examples of Anosov maps. For deterministic Anosov maps, many ergodic and sta- tistical properties such as ergodicity, existence of SRB measures and exponential decay of correlations for Hölder continuous observables are known. Dynamical systems with randomness have been studied extensively in recent years. A typical model has been an Anosov map with noise. There are several good books on the subject [Ki, Ar]. Products of random matrices are used in physics to model mag- netic systems with random interactions and localization of electronic wave functions in random potentials. They also play a central role in chaotic dynamical systems. In such applications, Lyapunov exponents provide information on the thermodynamic properties, electronic trans- port, and sensitivity for initial conditions. For more on the applications of products of random matrices, see [CrPaVu]. We consider the action of two separate toral automorphisms A and B, satisfying a cone condition, but which cannot be considered pertur- bations of one another. At each iteration, the matrix A is picked with a certain probability and the matrix B is picked if A is not picked and applied on the torus. Date: August 3, 2021. 2000 Mathematics Subject Classification. 37A25; 37H15, 37D20. M. S. would like to thank the Finnish Cultural Foundation for funding. The work was supported in part by NSF DMR-01-279-26 and AFOSR AF 49620-01-1-0154. http://arxiv.org/abs/0704.1495v1 2 ARVIND AYYER AND MIKKO STENLUND We are interested in the ergodic and statistical properties of the model for fixed realizations of the sequence of A’s and B’s obtained in this way. One might consider the “environment” (given by the sequence of matrices) to be fixed and the randomness to be associated with choosing the initial point on the torus. That is, our point of view is quenched randomness; one rolls the dice and lives with the outcome. Acknowledgements. We are grateful to Giovanni Gallavotti, Shel- don Goldstein, Joel Lebowitz, Carlangelo Liverani, David Ruelle, and Lai-Sang Young for useful discussions. 1.2. Toral automorphisms. Let T2 be the 2-torus R2/2πZ2. Definition 1. A map A : T2 by the matrix action x 7→ Ax (mod 2π) is called a toral automorphism if the matrix A has integer entries and detA = ±1. It is a hyperbolic toral automorphism if, further, the eigenvalues of the matrix A have modulus different from 1. Since the eigenvalues of a 2× 2 matrix A are given by the formula (trA)2 − 4 detA , (1) we see that a toral automorphism is hyperbolic precisely when the eigenvalues are in R \ {1}. The hyperbolicity condition reduces to |trA| > 2 if detA = +1, trA 6= 0 if detA = −1. Under the hyperbolicity assumption, the matrix has an eigenvalue whose absolute value is greater than 1, which we call the unstable eigenvalue and denote by λAu . Similarly, it has a stable eigenvalue, λAs , with absolute value less than 1. The corresponding eigenvectors eAu and e s span linear subspaces E u and E s , respectively. These we refer to as the unstable (eigen)direction and the stable (eigen)direction, respectively. From now on, we will always assume that our toral automorphisms are hyperbolic and have determinant +1. This is necessary for the cone property formulated below. It is important to notice that the eigenvalues in (1) are irrational. Consequently, the corresponding eigendirections EAs,u have irrational slopes; if A = , the eigenvectors are given by the formula eAs,u = (λAs,u − a)/b . (3) RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 3 Here b 6= 0 by hyperbolicity; see (2). Irrationality is equivalent to EAu,s ∩ Z 2 = {0}. (4) An alternative, “dynamical”, way of proving (4) starts with assuming the opposite; suppose that 0 6= q ∈ EAs ∩Z 2. Then Anq → 0 as n → ∞, which is a contradiction because A is an invertible integer matrix. The case of EAu is similar using A In fact, the slopes of EAs,u—call them α s,u—are not only irrational but satisfy a stronger arithmetic property called the Diophantine condition: there exist ǫ > 0 and Kǫ > 0 such that (q1, q2) ∈ Z 2, q1 6= 0 =⇒ ∣∣∣αAs,u − ∣∣∣ ≥ |q1|2+ǫ . (5) This tells us that in order for EAu,s to come close to a point on the integer lattice Z2 \ {0}, that point has to reside far away from the origin. Finally, we point out that a toral automorphism is symplectic. That is, setting , (6) any 2× 2 matrix A with determinant one satisfies ATJA = J. (7) For future reference, we define à := (AT )−1 = JAJ−1. (8) 1.3. Invariant cones. A matrix on R2 maps lines running through the origin into lines running through the origin. Therefore, it is natural to consider cones, i.e., sets in R2 spanned by two lines intersecting at the origin [Al, Wo]. Cones have also been used in the study of the spectrum of the transfer operator. See, for example, [BlKeLi, GoLi, Ba]. Definition 2 (Cone property). A pair (A0, A1) of hyperbolic toral au- tomorphisms has the cone property if the following cones exist (Fig. 1): An expansion cone, E , is a cone such that (1) AiE ⊂ E , (2) there exists λE > 1 such that |Aix| ≥ λE |x| for x ∈ E , (3) The EAiu do not lie along the boundary ∂E : E u ∩ ∂E = {0}. A contraction cone, C, is a cone such that E ∩ C = {0} and (1) A−1i C ⊂ C, (2) there exists λC < 1 such that |A i x| ≥ λ C |x| for x ∈ C, (3) The EAis do not lie along the boundary ∂C: E s ∩ ∂C = {0}. 4 ARVIND AYYER AND MIKKO STENLUND ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� 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��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� PSfrag replacements eAu eBu eAseBs Figure 1. An example of expansion and contraction cones. Remark 1. A cone is a contraction cone with rate λC (respectively expansion cone with rate λE) for (A,B) if and only if it is an expansion cone with rate λ−1C (respectively contraction cone with rate λ E ) for (A−1, B−1). With the aid of (8), one checks that if one of the pairs (A,B), (AT , BT ), (A−1, B−1), (Ã, B̃) has the cone property then all of them do (with different cones). Moreover, the rates coincide for the corresponding cones of (A,B) and (Ã, B̃). The name “expansion cone” is obvious, whereas “contraction cone” deserves some caution: |Aix| ≤ λC|x| holds in general only under the assumption Aix ∈ C, as opposed to the weaker x ∈ C. Given a line L passing through the origin transversely to EAis , it is a consequence of hyperbolicity that the image line Ani L tends to E as n → ∞. Therefore, an expansion cone has to contain the unstable eigendirection, EAiu . Similarly, considering the backward iterates, a contraction cone has to contain the stable eigendirection, EAis . In brief, EAiu ⊂ E and E s ⊂ C. (9) RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 5 The expansion and contraction rates in the cones are naturally bounded by the eigenvalues of the matrices: 1 < λE < |λ u | and 1 > λC > |λ s |. (10) In particular, our results will hold for any two toral automorphisms with positive entries, as well as for any two toral automorphisms with negative entries. This is because the union of the first and third quad- rant is automatically an expansion cone and the complement a con- traction cone, as can be easily checked. By inverting the matrices, our results apply just as well to any two toral automorphisms whose diago- nal elements are positive (respectively negative) and off-diagonal ones negative (respectively positive). 1.4. Random toral automorphisms. For definiteness, let A be cho- sen with probability p and B be chosen whenever A is not chosen. In order to model randomness, we first set A0 := A and A1 := B. (11) On the space of sequences, Ω := {0, 1}N, we define the shift operator τ : (ω(0), ω(1), . . . ) 7→ (ω(1), ω(2), . . . ), (12) and independently for each index n ∈ N prescribe the probability p to “ω(n) = 0” and the probability 1 − p to “ω(n) = 1”. The resulting product measure, P, is a τ -invariant ergodic probability measure on Ω. The map Φ : Ω× T2 : (ω, x) 7→ (τω, Aω(0)x) (13) is called a skew product and defines a random dynamical system. If m stands for the normalized Lebesgue measure on T2, i.e., (2π)2 , (14) then P×m is a Φ-invariant probability measure, because m is Ai- invariant for i = 0, 1. As usual, we will write µ(f) := dµf for the integral of a function f over a measure space with some measure µ. A basis of the space L2(Ω,P) of square integrable functions on Ω can be constructed as follows. Set σi(ω) := p/(1− p) if ω(i) = 1, and σi(ω) := − (1− p)/p if ω(i) = 0. Then define σA := i∈A σi for any finite subset A of N. The set {σA |A ⊂ N finite} is a countable orthonormal basis of L2(Ω,P). Let us denote Anω := Aω(n−1) · · ·Aω(0), such that A ω = A Akω for 1 ≤ k < n. Using the notation à := (AT )−1 for any matrix A, we get Ãnω := Ãω(n−1) · · · Ãω(0) = ((A T )−1. (15) 6 ARVIND AYYER AND MIKKO STENLUND Moreover, Ãnω = à Ãkω for k = 1, . . . , n− 1. (16) 1.5. Sensitive dependence on initial conditions. As an ingredient of chaos, we discuss how the iterates of a point on the torus depend sensitively on the point chosen. By this we roughly mean that the distance, |Anωx − A ωy|, between iterates of two close by points, x and y, typically diverges at an exponential rate as n grows. To this end, we use the Multiplicative Ergodic Theorem (MET) that is originally due to Oseledets in the context of differential equations and smooth flows; see [Os] and also [Ru]. Theorem 2 (2 × 2 MET). Suppose that ω 7→ Aω(0) is a measurable mapping from Ω to the space of real 2 × 2 matrices and the mapping ω 7→ ln+‖Aω(0)‖ is in L 1(Ω,P). Here ln+ t ≡ max(ln t, 0) and ‖ · ‖ is any matrix norm. Then there exists a set Γ ⊂ Ω with τΓ ⊂ Γ and P(Γ) = 1 such that the following holds if ω ∈ Γ: (1) The limit (Anω) )1/2n = Λω (17) exists. (2) Let expχ ω < · · · < expχ ω be the eigenvalues of Λω with ω , · · · , U ω the corresponding eigenspaces. Further, denote V (0)ω = {0} (18) V (r)ω = U ω ⊕ · · · ⊕ U ω for r = 1, · · · , s (19) Then s is either 1 or 2 and for r = 1, . . . , s we have ln|Anωx| = χ ω for x ∈ V ω \ V (r−1) ω . (20) The numbers χ ω are called Lyapunov exponents. Both s and ω generically depend on ω but are τ -invariant. Notice that det Λω = 1, or r=1 χ ω = 0, yields two possibilities: (1) s = 1 and χ ω = 0, (2) s = 2 and χ ω = −χ ω > 0. The MET guarantees that the Lyapunov exponents are invariant under the flow ω 7→ τω. By ergodicity of the τ -invariant measure P, they are constant almost surely (P = 1). RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 7 Corollary 3. Replacing Anω by à ω (see (15)) does not change the Lya- punov exponents. In fact, setting J := (Ãnω) T Ãnω )1/2n = JΛωJ −1 = Λ̃ω = (Λω) −1. (21) Proof. By (8), Ãnω = JA −1. Moreover, JT = J−1 and the symmet- ric matrix (Anω) TAnω is diagonalizable, such that (Ãnω) T Ãnω )1/2n reads −1 = J (Anω) )1/2n J−1 for some matrix On and di- agonal matrix Λn. � With respect to the Lebesgue measure on R2, almost all (a.a.) points x belong to the set V (s−1) ω corresponding to the largest Lyapunov exponent χ ω . P-almost surely the latter equals a constant χ (s). There- fore, for a.a. ω, for a.a. x, ln|Anωx| = χ (s). (22) A priori, there might not exist a positive Lyapunov exponent (s = 1). This would rule out sensitive dependence on initial conditions in the meaning of the notion described in the beginning of the subsection. In particular, it does not follow from the classical works of Fursten- berg [Fu], Kesten [FuKe], and Virtser [Vi] that the largest Lyapunov exponent is positive, because the Bernoulli measure used to choose a matrix at each step is concentrated at two points, A and B, on SL(2,R). Theorem 4 (The largest Lyapunov exponent is positive). Suppose that (A,B) has the cone property. Then there are two distinct Lyapunov ex- ponents, and χ(2) = −χ(1). In fact, 0 < lnλE ≤ χ (2) ≤ maxi ln|λ u | and mini ln|λ s | ≤ χ (1) ≤ lnλC < 0. In particular, χ (2) ≥ lnmax(λ−1C , λE). Proof. The largest Lyapunov exponent is positive, because the expan- sion cone has nonzero measure. We conclude that s = 2 in the MET. Consider the intersection Eω := k≥0(A −1C of preimages of the contraction cone C. It is a line inside C whose forward iterates forever remain in C. That is, if x ∈ Eω, then A ωx ∈ C such that |A ωx| ≤ λ C |x| for all n ≥ 0. We must have V ω = Eω. The construction of Eω is similar to that of random stable manifolds [Yo]. The bounds are now obvious. � 1.6. Observables and a Hölder continuity condition. 8 ARVIND AYYER AND MIKKO STENLUND Definition 3. We say that a function (“observable”) f : Ω× T2 → C satisfies the strong Hölder condition with exponent β ∈ [0, 1], if ‖f‖β := sup |f̂(ω, q)||q|β < ∞. (23) Here f̂ is the Fourier transform of f . Observe that if f satisfies (23), then |f̂(ω, q)| ≤ ‖f‖β |q| −β, q ∈ Z2 \ {0}. (24) Because 0 ≤ β ≤ 1, |eit − 1|/|t|β is uniformly bounded in t ∈ R, and we see that (23) implies Hölder continuity of f(ω, ·) with exponent β: |f(ω, x+ y)− f(ω, x)| ≤ q∈Z2\{0} |f̂(ω, q)||q · y|β |eiq·y − 1| |q · y|β ≤ C|y|β (25) for all x, y ∈ T2. The opposite is not true; hence the adjective “strong”. 1.7. Decay of correlations and mixing. We define the nth (time) correlation function of two observables f and g as Cf,g(ω, n) := dm(x) f(ω,Anωx)g(ω, x)−m(f(ω, ·))m(g(ω, ·)). (26) We also need the related CΦf,g(ω, n) := dm(x) (f ◦Φn ·g)(ω, x)−m(f(τnω, ·))m(g(ω, ·)) (27) CΦf,g(n) := d(P×m) (f ◦ Φn · g)− (P×m)(f) (P×m)(g), (28) where Φ refers to the skew product (13). Our result is the following: Theorem 5 (Decay of Correlations). Let the pair (A,B) satisfy the cone property (2); see Remark 1. There exist c > 0 and ρ > 0 such that, if f and g are two observables satisfying the strong Hölder condition with exponent β ∈ (0, 1], then for all ω ∈ Ω, |Cf,g(ω, n)| ≤ c ‖f‖β‖g‖β e −ρβn (29) |CΦf,g(ω, n)| ≤ c ‖f‖β‖g‖β e −ρβn (30) hold for all n ∈ N. In fact, we can take ρ = lnmin(λ−1C , λE). RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 9 For all ǫ > 0, there exists a constant C(ǫ) such that, for almost all ω, the upper bounds above can be replaced by c ‖f‖β‖g‖β C(ǫ) −β e−(χ (2)−ǫ)βn, (31) where χ(2) ≥ lnmax(λ−1C , λE) is the a.e. constant, positive, Lyapunov exponent. Remark 6. Without additional information concerning the conver- gence rates of 1 ln|Ãnωx| to the corresponding Lyapunov exponents, we have no control over C(ǫ) beyond the fact that it is an increasing func- tion of ǫ. The proof of Theorem 5 follows in Section 2. If f, g are trigonometric polynomials (finite linear combination of exponentials of the form eiq·x), they satisfy (23) trivially and thus, also (29). Trigonometric polynomials form a countable basis of L2(T2,m). This implies that limn→∞ dm(x)F (Anωx)G(x) = m(F )m(G) for any functions F,G ∈ L2(T2,m). We say that every fixed realization of the random sequence (Aω(0), Aω(1), . . . ) of maps is mixing on T Similarly, one should interpret (30) as a mixing result for the skew product, keeping ω fixed. Remark 7. It is true that the positivity of the Lyapunov exponent χ(2) is enough for mixing, even if the cone condition is not satisfied. However, we need the cone condition to (1) check that χ(2) actually is positive, and (2) obtain estimates on correlation decay, i.e., on the mixing rate. Corollary 8. If (A,B) satisfies the cone property, then the skew prod- uct Φ, or the random dynamical system, is mixing. If, moreover, f and g satisfy the strong Hölder condition and m(f(ω, ·)) ≡ m(g(ω, ·)) ≡ 0, then |CΦf,g(n)| ≤ c ‖f‖β‖g‖β min e−ρβn, C(ǫ)−βe−(χ (2)−ǫ)βn Proof. For f, g ∈ L2(Ω×T2,P×m), the difference CΦf,g(n)−P(C f,g(·, n)) has the expression dP(ω)m(f(τnω, ·))m(g(ω, ·))− (P×m)(f) (P×m)(g), (32) and tends to zero, because τ is mixing. Since {σA |A ⊂ N finite} is a countable basis of L2(Ω,P), we get L2(Ω × T2,P×m) ∼= L2(Ω,P) ⊗ L2(T2,m). The functions σA(ω)e iq·x form a countable basis of the latter and trivially satisfy (23) and an estimate corresponding to (30). Hence, for any f, g ∈ L2(Ω × T2,P×m), limn→∞C f,g(n) = 0, such that Φ is 10 ARVIND AYYER AND MIKKO STENLUND mixing. The second claim follows from (30) and (31) because CΦf,g(n) = P(CΦf,g(·, n)) when m(f(ω, ·)) ≡ m(g(ω, ·)) ≡ 0. � The correlation function in (26) has the Fourier representation Cf,g(ω, n) = q∈Z2\{0} f̂(ω,−Ãnωq) ĝ(ω, q). (33) The proof of Theorem 5 is based on the decay (24) of f̂(q), ĝ(q) with increasing |q| and on controlling |Ãnωq| with lower bounds that are in- creasing in n but not too heavily decreasing in |q|, such that summa- bility persists. Here we gain by working with the sequence (Ãnω)n∈N instead of ((Anω) T )n∈N, because the former has a Markov property—not shared by the latter—due to the order in which the matrix factors are multiplied. More precisely, if qn := Ãnωq, then q n+1 is completely de- termined by qn and ω(n) as opposed to the entire history (ω(k))0≤k≤n. 1.8. Comments. Notice that Theorem 5 is not needed for Theorem 4; ergodicity of the shift τ alone is relevant for Theorem 4. Without affecting the proofs much, the cone property can be weak- ened by relaxing, for instance in the case of the expansion cone, the assumption that every iteration results in expansion. Assuming instead the existence of a number N such that |ANω x| ≥ λE |x| for all ω if x ∈ E is equally sufficient. In fact, it is true that the system is mixing even when the cones are merely invariant without any contraction/expansion assumption. More precisely, if Ai are hyperbolic toral automorphisms and there exist non- overlapping cones E and C with the properties AiE ⊂ E and A i C ⊂ C, then mixing occurs. This is so, because all possible products Ãnω turn out to be hyperbolic. However, we have no control over the mixing rate (the speed at which correlations decay) in this case. Our results remain valid for any number of automorphisms, if all of the matrices have a mutual contraction cone and a mutual expansion cone with contraction and expansion rates bounded away from 1. One can pursue a line of analysis different from ours by emphasizing randomness in ω. For instance, fixing x ∈ T2, the sequence (Anωx)n∈N of random variables on Ω is a Markov chain. Furthermore, with a more probabilistic approach, it should be possible to do without the cone property (excluding extreme cases such as B = A−1 with p = 1 ) and still obtain results for almost all ω. RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 11 2. Proof of Theorem 5 Before entering the actual proof, we explain our method for the sake of motivation. Starting with (33), we have |Cf,g(ω, n)| ≤ q∈Z2\{0} |f̂(ω,−Ãnωq) ĝ(ω, q)|. (34) The idea is to split Z2 \ {0} into suitable pieces using cones. By the cone property and Remark 1, we have the following at our disposal: Contraction cone, C. With some positive λC < 1, |Ãiq| ≤ λC|q| if Ãiq ∈ C, i ∈ {0, 1}. (35) Moreover, E s ⊂ C. The complement C c is invariant in the sense that q ∈ Cc =⇒ Ãiq ∈ C c. (36) Expansion cone, E . With some λE > 1, |Ãiq| ≥ λE |q| for q ∈ E , i ∈ {0, 1}. (37) Moreover, E u ⊂ E . The cone E itself is invariant in the sense that q ∈ E =⇒ Ãiq ∈ E . (38) The complement (C ∪ E)c and the number M . There exists a positive integer M depending on only the choice of cones C, E such that, for any “random” sequence ω, q ∈ (C ∪ E)c =⇒ ÃMω q ∈ E . (39) Let us define λ := max (λC, λ E ) < 1. (40) It follows from the bound (35) in the definition of C that, if q, · · · , Ãnωq ∈ C and q 6= 0, then 1 ≤ |Ãnωq| ≤ λ n|q|, (41) where the first inequality is due to the fact that Ãnωq is a nonzero integer vector. But the right-hand side of (41) tends to zero as n → ∞, which is a contradiction unless eventually Ãnωq /∈ C. We conclude that there exists a unique integer Nω(q), called the contraction time, satisfying q ∈ C \ {0} =⇒ ÃNω(q)ω q ∈ C and à Nω(q)+1 ω q ∈ C c. (42) Let us list some bounds used in proving Theorem 5. The proof of this lemma is given at the end of the section. 12 ARVIND AYYER AND MIKKO STENLUND Lemma 9. The contraction time, Nω(q), obeys the ω-independent bound Nω(q) ≤ ln|q| lnλ−1 . (43) The following complementary bounds hold: 1 ≤|Ãnωq| ≤ λ n|q| for q ∈ C \ {0}, n ≤ Nω(q), (44) |Ãnωq| ≥ Cλ −n|q|−1 for q ∈ C \ {0}, n > Nω(q), (45) |Ãnωq| ≥ Cλ −n|q| for q ∈ Cc, n ∈ N. (46) In fact, (43) is never used but we state it for the sake of completeness. We now have a natural way of decomposing the sum in (34) for each fixed value of n, by means of the disjoint partition 2 \ {0} = {q ∈ C |n ≤ Nω(q)} ∪ {q ∈ C |n > Nω(q)} ∪ C c \ {0}. (47) Namely, we can rearrange the series (of nonnegative terms) as q∈Z2\{0} q∈C\{0}: n≤Nω(q) q∈C\{0}: n>Nω(q) . (48) In the first series on the right-hand side we use (44), i.e., the fact that |q| is large for large n. In the second and the third series it is |Ãnωq| that is large for large n, by (45) and (46), respectively. Similarly, the last part of Theorem 5 is based on the following re- finement of Lemma 9: Lemma 10. For all ǫ > 0, there exists a constant C(ǫ) such that, for almost all ω, the bounds 1 ≤|Ãnωq| ≤ C(ǫ) −1e−n(χ (2)−ǫ)|q| for q ∈ C \ {0}, n ≤ Nω(q), (49) |Ãnωq| ≥ C(ǫ)e n(χ(2)−ǫ)|q|−1 for q ∈ C \ {0}, n > Nω(q), (50) |Ãnωq| ≥ C(ǫ)e n(χ(2)−ǫ)|q| for q ∈ Cc, n ∈ N, (51) hold true. Proof of Theorem 5. We first prove (29) and leave it to the reader to check that the upper bounds below apply just as well to the case of (30). By the same token, we keep the first argument ω of the observables implicit and just write ĝ(q) instead of ĝ(ω, q) etc. Let us proceed case by case in the decomposition (48) of (34): Case q ∈ C \ {0}, n ≤ Nω(q). By (24) and (44), |ĝ(q)| ≤ ‖g‖β|q| −β ≤ ‖g‖βλ βn, (52) RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 13 such that q∈C\{0}: n≤Nω(q) |f̂(−Ãnωq) ĝ(q)| ≤ ‖g‖βλ q∈C\{0}: n≤Nω(q) |f̂(−Ãnωq)| ≤ ‖f‖β‖g‖βλ Case q ∈ C \ {0}, n > Nω(q). By (24) and (45), |f̂(−Ãnωq)| ≤ ‖f‖β|à −β ≤ ‖f‖βC −βλβn|q|β, (54) and we have∑ q∈C\{0}: n>Nω(q) |f̂(−Ãnωq) ĝ(q)| ≤ ‖f‖βC −βλβn q∈C\{0}: n>Nω(q) |ĝ(q)||q|β ≤ ‖f‖β‖g‖βC −βλβn. Case q ∈ Cc. By (24) and (46), |f̂(−Ãnωq)| ≤ ‖f‖β|à −β ≤ ‖f‖βC −βλβn|q|−β, (56) which implies |f̂(−Ãnωq) ĝ(q)| ≤ ‖f‖βC −βλβn |ĝ(q)||q|−β ≤ ‖f‖β‖g‖βC −βλβn. Choosing c ≥ 1 + 2C−β and ρ := − lnλ, (58) the bound (29) follows. By the same argument, the fact that the upper bound (31) applies is an immediate consequence of Lemma 10. � Proof of Lemma 9. Equation (44) repeats (41) and the related dis- cussion, whereas (43) is just another way of writing it in the case n = Nω(q). By the definition of E , |Ãnωq| ≥ λ −n|q| when q ∈ E , n ∈ N. (59) For each index i and any q ∈ Z2 \ {0}, we also have the a priori bound |Ãiq| ≥ µi|q| with µi := ‖à −1 = |λAis |. (60) Therefore, if q ∈ (C ∪ E)c and m ≥ 1, (16) implies |ÃM+mω q| ≥ λ −m|ÃMω q| ≥ Cλ −M−m|q|, (61) where C := minω(λ Mµω(0) . . . µω(M−1)). Observing C < 1 and sacrific- ing (59), the latter bound extends to the whole of Cc, yielding (46). 14 ARVIND AYYER AND MIKKO STENLUND We also need a lower bound on |Ãnωq|—one that grows exponentially with n but does not decrease too much with |q|—assuming that q ∈ C and n > Nω(q). To this end, we notice à Nω(q) ω q ∈ C c and compute |Ãnωq| = |à n−Nω(q) τNω(q)ω ÃNω(q)ω q| ≥ Cλ Nω(q)−n|ÃNω(q)ω q|. (62) From (44) wee see |à Nω(q) ω q| ≥ λ −Nω(q)|q|−1, so that (45) follows. � Proof of Lemma 10. Modifying the constant, (50) follows from (49) and (51), just like (45) follows from (44) and (46) in the proof of Lemma 9. ÃnωC c becomes a thinner and thinner cone inside E as n increases. If x ∈ ∂C (the worst case), then 1 ln(|Ãnωx|/|x|) tends to χ (2). Thus, for each ǫ > 0 and n ∈ N, we have |Ãnωx|/|x| > C(ǫ)e n(χ(2)−ǫ) for some choice of C(ǫ), and (51) follows. Next, we prove (49). Let es(ω) be a unit vector spanning the random stable line k≥0(à −1C (see the proof of Theorem 4). Then Ãnωes(ω) ∈ C for all n. Let us also pick an arbitrary unit vector v in E . By an elementary geometric argument, there exists a constant K such that if x ∈ C, y ∈ E , and x + y ∈ C, then |y| ≤ K|x| and |x| ≤ K|x + y| (the worst case: x, x + y ∈ ∂C and y ∈ ∂E). Recall that we are assuming Ãnωq ∈ C for each n = 0, . . . , Nω(q). If we split q = q1es(ω) + q2v, then à ωq = q1à ωes(ω) + q2à ωv, where the first term belongs to C and the second to E . We gather that |q2à ωv| ≤ K|q1à ωes(ω)|. Because 1 ln(|Ãnωes(ω)|) tends to −χ (2), for all ǫ > 0 there exists a Cǫ such that, for all n, |à ωes(ω)| ≤ Cǫe −n(χ(2)−ǫ). In conclusion, |Ãnωq| ≤ Cǫ(1+K)|q1|e −n(χ(2)−ǫ), and (49) follows from |q1| ≤ K|q|. � References [Al] Alekseev, V. Quasirandom dynamical systems. I. Quasirandom diffeomor- phisms, Math. USSR-Sb 5 (1968), no. 1, 73-128. II. One-dimensional nonlin- ear vibrations in a periodically perturbed field, Math. USSR-Sb 6 (1968), no. 4, 505-560. III. Quasirandom vibrations of one-dimensional oscillators, Math. USSR-Sb 7 (1969), no. 1, 1-43. [Ar] Arnold, L. Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [Ba] Baladi V. Positive Transfer Operators and Decay of Correlations Advanced Series in Nonlinear Dynamics, vol 16, Singapore, World Scientific. [BlKeLi] Blank, M., Keller, G., Liverani, C. Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), no. 6, 1905-1973. RANDOMLY CHOSEN HYPERBOLIC TORAL AUTOMORPHISMS 15 [CrPaVu] Crisanti, A., Paladin, G., Vulpiani, A. Products of random matrices in statistical physics, Springer Series in Solid-State Sciences, 104, Springer- Verlag, Berlin, 1993. [Fu] Furstenberg, H. Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428. [FuKe] Furstenberg, H., Kesten, H. Products of random matrices, Ann. Math. Statist, 31 (1960), 457-469. [GoLi] Gouëzel, S., Liverani, C. 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Systems, 6 (1986), no. 2, 311-319. 1 Department of Physics, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA 2 Department of Mathematics, Rutgers University, 110 Frelinghuy- sen Road, Piscataway, NJ 08854, USA E-mail address : {ayyer,mstenlun}@math.rutgers.edu 1. Introduction 1.1. Background Acknowledgements 1.2. Toral automorphisms 1.3. Invariant cones 1.4. Random toral automorphisms 1.5. Sensitive dependence on initial conditions 1.6. Observables and a Hölder continuity condition 1.7. Decay of correlations and mixing 1.8. Comments 2. Proof of Theorem ?? References

0704.1496

Follow-up observations of pulsating subdwarf B stars: Multisite campaigns on PG 1618+563B and PG 0048+091

Follow-up observations of pulsating subdwarf B stars: Multisite campaigns on PG 1618+563B and PG 0048+091 M. D. Reed 1, S. J. O’Toole2,3, D.M. Terndrup4, J. R. Eggen1, A.-Y. Zhou1, D. An4, C.-W. Chen5, W. P. Chen5, H.-C. Lin5, C. Akan6, O. Cakirli6, H. Worters7,8, D. Kilkenny7, M. Siwak9, S. Zola9,10, Seung-Lee Kim11, G. A. Gelven1, S. L. Harms1, and G. W. Wolf1 1 Department of Physics, Astronomy, & Materials Science, Missouri State University, 901 S. National, Springfield, MO 65897 U.S.A. [email protected] 2 Anglo-Australian Observatory, PO Box 296, Epping NSW 1710, Australia 3 Dr Remeis-Sternwarte, Astronomisches Institut der Universität Erlangen-Nürnberg, Sternwartstr. 7, Bamberg 96049, Germany 4 The Ohio State University, 140 W. 18th Avenue, Columbus, OH 43210, U.S.A. 5 Graduate Institute of Astronomy, National Central University, Chung-Li, Taiwan 6 Ege University Observatory, 35100 Bornova-Izmir, Turkey 7 South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa 8 Centre for Astrophysics, University of Central Lancashire, Preston, PR1 2HE, UK 9 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Cracow, Poland 10 Mt. Suhora Observatory of the Pedagogical University, ul. Podchora̧zych 2, PL-30-084 Cracow, Poland 11 Korea Astronomy and Space Science Institute, Daejeon, 305-348, South Korea ABSTRACT We present follow-up observations of pulsating subdwarf B (sdB) stars as part of our efforts to resolve the pulsation spectra for use in asteroseismologi- cal analyses. This paper reports on multisite campaigns of the pulsating sdB stars PG 1618+563B and PG 0048+091. Data were obtained from observatories placed around the globe for coverage from all longitudes. For PG 1618+563B, our five-site campaign uncovered a dichotomy of pulsation states: Early during the campaign the amplitudes and phases (and perhaps frequencies) were quite http://arxiv.org/abs/0704.1496v1 – 2 – variable while data obtained late in the campaign were able to fully resolve five stable pulsation frequencies. For PG 0048+091, our five-site campaign uncovered a plethora of frequencies with short pulsation lifetimes. We find them to have observed properties consistent with stochastically excited oscillations, an unex- pected result for subdwarf B stars. We discuss our findings and their impact on subdwarf B asteroseismology. Subject headings: variable stars: general — asteroseismology, close binaries 1. Introduction Subdwarf B (sdB) stars are thought to have masses about 0.5M⊙, with thin (<10 −2M⊙) hydrogen shells and temperatures from 22 000 to 40 000 K (Saffer et al. 1994). They are horizontal branch stars that have shed nearly all of their H-rich outer envelopes near the tip of the red giant branch and as He-flash survivors, it is hoped that asteroseismology can place constraints on several interesting phenomena. Subdwarf B star pulsations come in two varieties: short period (90 to 600 seconds; EC 14026 stars after that prototype, officially V361 Hya stars, or sdBV stars) with amplitudes typically near 1%, and long period (45 minutes to 2 hours; PG 1716 stars after that prototype or LPsdBV stars) with amplitudes typically <0.1%. For more on pulsating sdB stars, see Kilkenny (2001) and Green et al. (2003) for observational reviews and Charpinet, Fontaine, & Brassard (2001) for a review of the proposed pulsation mechanism. For this work, our interest is the sdBV (EC 14026) class of pulsators. In order for asteroseismology to discern the internal conditions of variable stars, the pulsation “mode” must be identified from the temporal spectrum (also called pulsation spectrum or Fourier transform; FT). The mode is represented mathematically by spherical harmonics with quantum numbers n (or k), ℓ, and m. For nonradial, multimode pulsators the periods, frequencies, and/or the spacings between them are most often used to discern the spherical harmonics (see for example Winget 1991). These known modes are then matched to models that are additionally constrained by non-asteroseismic observations, typically Teff and log g from spectroscopy. Within such constraints, the model that most closely reproduces the observed pulsation periods (or period spacing) for the constrained modes is inferred to be the correct one. Occasionally such models can be confirmed by independent measurements (Reed et al. 2004, Reed, Kawaler, & O’Brien 2000, Kawaler 1999), but usually it is impossible to uniquely identify the spherical harmonics and asteroseismology cannot be applied to obtain a unique conclusion. Such has been the case for sdBV stars, which seldom show multiplet structure (i.e., even frequency spacings) that may be used to observationally constrain the – 3 – pulsation modes. However, relatively few sdBV stars have been observed sufficiently to know the details of their pulsation spectra. The goal of our work is to fully resolve the pulsation spectrum, search for multiplet structure, and examine the characteristics of the pulsation frequencies over the course of our observations. In this paper we report on multi-site follow-up observations of the pulsating sdB stars PG 1618+563B (hereafter PG 1618B) and PG 0048+091 (hereafter PG 0048) obtained during 2005. PG 1618B was discovered to be a variable star by Silvotti et al. (2000; hereafter S00) who detected frequencies of 6.95 and 7.18 mHz (P ≈ 144 and 139 s respectively) from short data runs (< 2.3 hrs) obtained during seven nights, three of which were separated by three months. PG 1618 is an optical double consisting of a main sequence F-type star (component A) with an sdB star (component B) at a separation of 3.7 arcseconds. The combined brightness is V = 11.8 while the sdB component has V ≈ 13.5. The discovery data used a combination of photoelectric photometry, which did not resolve the double, and CCD data which did. The combined flux of the double in the former would have reduced the pulsation amplitudes. From spectra obtained at Calar Alto, S00 determined that Teff = 33 900± 1 500K and log g = 5.80 ± 0.2. PG 0048 was discovered to be a variable star during 10 observing runs varying in length from 1 to 4 hours obtained in 1997 and 1998 (Koen et al. 2004; hereafter K04). As they used a variety of instruments spread over the span of a year, K04 were only able to combine two consecutive runs, from which they detected seven frequencies. However, it is obvious from their temporal spectrum that pulsation amplitudes and possibly frequencies were changing (their Fig. 3). K04 contributed this to unresolved frequencies caused by their short duration data runs. K04 also obtained an optical spectrum and examined 2MASS colors to determine that PG 0048 has a G0V-G2V companion; though the orbital parameters are unknown and no Teff or log g estimates were given. Here we report the results of a new program to resolve the pulsation spectra of these two stars. Section 2 describes the observations, reductions, and analysis for PG 1618B, and §3 the same for PG 0048. Section 4 compares the results for both stars and discusses the implications for asteroseismology. 2. PG 1618B+563B 2.1. Observations PG 1618B was observed from 5 observatories (Baker, MDM, McDonald, Lulin, and Suhora) over a 45 day period during spring 2005. Data obtained at MDM (2.4 m) and – 4 – McDonald (2.1 m) observatories used the same Apogee Alta U47+ CCD camera. This camera is connected via USB2.0 for high-speed readout, and our binned (2×2) images had an average dead-time of one second. Observations at Baker (0.4 m) and Lulin (1.0 m) observatories were obtained with Princeton Instruments RS1340 CCD cameras. Data obtained at Baker Observatory were binned 2× 2 with an average dead-time of one second, while observations from Lulin Observatory used a 392× 436 subframe at 1 × 1 binning with an average dead- time of six seconds. The Mt. Suhora Astronomical Observatory (0.6 m) data were obtained with a photomultiplier tube photometer which has microsecond dead-times. Observations obtained at McDonald, Baker, and MDM observatories used a red cut-off (BG40) filter, so the transmission is virtually the same as the blue photoelectric observations from Suhora observatory. Observations from Lulin Observatory used a Johnson B filter, which slightly reduced the amount of light collected compared to other observations, but does not impose any significant phase and/or amplitude changes compared to other observations (Koen 1998; Zhou et al. 2006). Accurate time was kept using NTP (Baker, McDonald, and MDM observatories) or GPS receivers (Lulin and Suhora observatories) and corrected to barycentric time during data reductions. Standard procedures of image reduction, including bias subtraction, dark current and flat field correction, were followed using iraf1 packages. Differential magnitudes were ex- tracted from the calibrated images using momf (Kjeldsen & Frandsen 1992) or occasionally they were extracted using iraf aperture photometry with extinction and cloud corrections using the normalized intensities of several field stars, depending on conditions. Photoelec- tric data reductions proceeded using standard Whole Earth Telescope reduction packages (Nather et al. 1990). As sdB stars are substantially hotter, and thus bluer, than typical field stars, differential light curves using an ensemble of comparison stars are not flat due to differential atmospheric and color extinctions. A low-order polynomial was fit to remove these trends from the data on a night-by-night basis. Finally, the lightcurves are normalized by their average flux and centered around zero so the reported differential intensities are ∆I = (I/〈I〉) − 1. Amplitudes are given as milli-modulation amplitudes (mma) with an amplitude of 10 mma corresponding to an intensity change of 1.0% or 9.2 millimagnitudes. The companion of PG 1618B adds a complication to the reductions in that data obtained at McDonald and MDM observatories resolved the optical double, but those from other observatories did not. Using our data for which the stars are resolved, we determined that component A contributes 67.3% of the total flux. To correct the unresolved data, we created iraf is distributed by the National Optical Astronomy Observatories, which are operated by the As- sociation of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. – 5 – a fitting function by smoothing the data over many points (around 50 points per box), multiplying it by 0.673 and subtracting it from the unresolved data. While this process effectively removes the flux from PG 1618A, it cannot correct for the noise of this component, which remains behind. As such, the corrected data are noisier, limiting their usefulness. Multiple-longitude coverage was only obtained during the first week of the campaign. A total of 73.5 hours of data were collected from three observatories (McDonald, Lulin, and Suhora) which provided a 47% duty cycle. Subsequent data were obtained only in Missouri (Baker Observatory) and Arizona (MDM Observatory). These data serve to extend the timebase of observations (increasing the temporal resolution) and to decrease the noise in the temporal spectrum. Lightcurves showing the coverage of the first six nights of observations, as well as a portion of a typical MDM run, are provided in Fig. 1. 2.2. Analysis Our campaign was quite long (about 45 days) with a concentration of data at the beginning, but the best data (highest S/N and best conditions) were obtained at the end. We therefore grouped combinations of nightly runs into the subsets given in Table 2 for analysis. Table 2 also provides the temporal resolution (calculated as 1/trun where trun is the length of the observing run) and the 4 σ detection limit (calculated using areas adjacent to the pulsation but outside of their window functions). The temporal spectra and window functions of these subsets are plotted in Fig. 2. A window function is a single sine wave of arbitrary, but constant amplitude sampled at the same times as the data. The central peak of the window is the input frequency, with other peaks indicating the aliasing pattern of the data. Each peak of the data spectrum intrinsic to the star will create such an aliasing pattern. As is evident from Fig. 2, the MDM data was significantly better than the rest, so we began our analysis with that subset. Analysis of the MDM data was relatively easy and straightforward. In Fig. 3, the top panel shows the original FT, while the bottom panel shows the residuals after prewhitening by the frequencies indicated by arrows. The insets show the window function (top right) and an expanded view of a 65µHz region around the close doublet. Frequencies, amplitudes and phases were determined by simultaneously fitting a nonlinear least-squares solution to the data. Since during the MDM observations the amplitudes were relatively constant, the solution proceeded as expected and prewhitening effectively removed the peaks and their aliases. The formal solution and errors for the MDM subset are given in Table 3. Examination of the other data sets indicates that while PG 1618B was well-behaved – 6 – during the MDM observations, it was not at other times. This was most noticeable in our examination the data collected during the first week. While the temporal spectrum has the cleanest window function, no peaks are detected above the 4σ detection limit (1.53 mma) even though peaks are detected in individual Lulin and McDonald runs (see Fig. 4 which will be discussed in §2.3). Combining the well-behaved MDM data with any other data set results in a decrease of amplitudes, indicating that outside of the MDM data, the amplitudes, phases, or even frequencies are not constant. If the pulsation properties were consistent throughout the campaign, data collected at smaller telescopes, with low S/N would still have been useful for reducing the overall noise. Unfortunately, such was not the case so all we can really conclude is that the MDM data detected all the pulsations that were occurring (above the detection limit) at that time while the pulsations intrinsic to PG 1618B must have been more complex at other times. It would be interesting to study the longer-term variability of PG 1618B, but using only 2 m-class telescopes. Outside of the combined data sets, there are two frequencies that are detected above the 4σ detection limit during individual runs. The least-squares solutions for these frequencies are provided at the end of Table 3. The frequency at ≈ 9199µHz was above the noise only in the March 22 McDonald data, though a peak at the same frequency also appears in the Suhora data during March 16 and 21. The frequency at ≈ 8179µHz was above the noise only in the April 30 MDM data though corresponding peaks appear in the McDonald March 18, Lulin March 18, and Suhora March 21 runs. Since they are detected above the 4σ detection criteria for those runs, we include them in our discussion that follows. 2.3. Discussion Silvotti et al. (2000) detected two frequencies in their discovery data while we clearly resolve four frequencies from our MDM dataset, and two more from individual data runs, bringing the total to six independent frequencies. We calculate the S00 resolution to be 5.5 µHz and estimate their noise to be about 1 mma though this is misleading in that because of their short data runs, their window function effectively covers all of the remaining pulsations. However, for a strict comparison, we can say that our MDM data alone are 3× better in resolution and have a detection limit twice as good, though in a practical sense our MDM data are far superior solely based on the duration of our individual data runs. Had S00 observed for longer durations (particularly with their CCD setup), their data would likely have been similar to the same number of runs from our MDM set. However, it is clearly safe to say that our MDM data alone are insufficient to describe the complexity of pulsations occurring within PG 1618B. As such, the remainder of our discussion which is based on the – 7 – MDM data, can only be a minimum of what is really occurring. 2.3.1. Constraints on the pulsation modes One of our goals is to observationally identify or constrain the pulsation modes of indi- vidual frequencies. Differing m components of the same degree ℓ have degenerate frequencies unless perturbed, typically by rotation. If a star is rotating, then each degree will separate into a multiplet of 2ℓ+ 1 components with spacings nearly that of the rotation frequency of the star. As such, observations of multiplet structure can constrain the pulsation degree (for examples, see Winget et al. 1991 for pulsating white dwarfs and Reed et al. 2004 for sdB stars). For PG 1618B, there are no two frequency spacings that are similar, though there are not many frequencies to work with. The lack of observable multiplet structure is typical of sdBV stars but is likely limited to four possibilities: i) Rotation is sufficiently slow that all m values remain degenerate within the frequency resolution of our data; ii) our line of sight is along the pulsation axis, with sin i ≈ 0, leaving only the m = 0 mode observable because of geometric cancellation (Pesnell 1985; Reed, Brondel, & Kawaler 2005); iii) rapid internal rotation is such that m multiplets are widely spaced and uneven (Kawaler & Hostler 2005); or iv) at most one pair is part of a multiplet with an unobserved component of the multiplet. Spectroscopy can only rule out large splittings for possibility (i) as spectroscopic limits are typically ≈ 10 km/s and from Fig. 1 of S00, PG 1618B appears as a “normal” sdB devoid of rapid rotation. Possibility (ii) can only be determined for cases in which the sdB star is part of a close binary such that the rotation and orbital axes can be inferred to be aligned. Since PG 1618 is only an optical double at wide separation, it does not constraint the alignment of the surface spherical harmonics. Similarly, possibility (iii) is virtually impossible to decipher unless the star pulsates in many (tens of) frequencies, and would still require some interrelation of spacings for modes of the same degree (Kawaler & Hostler 2005). Possibility (iv) also remains an option, though a difficult one to constrain. Higher resolution (and perhaps longer duration) spectroscopy would help to answer this question, and multicolor photometry or time-series spectroscopy might also be able to discern the spherical harmonics (see Koen 1998 and O’Toole et al. 2002 for examples of each). Another quantity that can be used to constrain the pulsation modes is the frequency density. Using the assumptions that no two frequencies share the same n and ℓ values (except possibly the close pair at 6946µHz), and that high-degree ℓ ≥ 3 modes are not observationally favored because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel, & Kawaler 2005), we can ascertain whether the frequencies are too dense to be accounted for using only ℓ ≤ 2 modes. From stellar models, a general rule of thumb is to allow three frequencies – 8 – per 1000µHz. We will ignore f1 which is too distant in frequency space and count f5 and f6 as a single degree ℓ. This leaves four frequencies within 1235 µHz; which can easily be accounted for using only ℓ ≤ 2 modes. Indeed, even if f5 and f6 do not share their n and ℓ values, the frequency spectrum can still accommodate all of the detected frequencies without invoking higher degree modes. Of course this does not mean that they are not ℓ ≥ 3 modes, only that the pulsation spectrum is not sufficiently dense to require their postulation. 2.3.2. Amplitude and phase stability If pulsating sdB stars are observed over an extended time period, it is common to detect amplitude variability in many, if not all, of the pulsation frequencies (eg. O’Toole et al. 2002; Reed et al. 2004; Zhou et al. 2006). Such variability can occasionally be ascribed to beating between pulsations too closely spaced to be resolved in any subset of the data. However, variations often appear in clearly resolved pulsation spectra where mode beating cannot be the cause. For PG 1618B, frequencies f1 and f2 are only detected during a single run each and frequencies f5 and f6 are too closely spaced to be resolved during individual runs, leaving only frequencies f3 and f4 available for analysis of amplitude variations. Figure 4 shows the amplitude and phases of these two frequencies for individual data runs from McDonald and MDM observatories as well as a single Lulin run (marked by a triangle); these frequencies were not detected elsewhere. During the MDM observations, the amplitudes and phases for both frequencies are nearly constant (to within the errors) except for one low amplitude, but they have a significant variation in the McDonald and Lulin data. Of particular interest are the phases and amplitudes of f4, especially those during day three, in which we have both a McDonald and Lulin run that do not overlap in time. Between these two runs, the amplitude, which had been decreasing during the previous three days, suddenly increases to begin the same declining pattern again. The phases also show a bimodal structure early in the campaign with phases near −0.20 and +0.25 with the first phase jump occurring coincident with the amplitude increase. Except for the lack of sinusoidal amplitude variation, this has the appearance of unresolved pulsations. However, if the two unresolved frequencies had intrinsic amplitude variability, then it could reproduce the observations. However the MDM observations, which are not only steady, but have f4 phases intermediate to the McDonald and Lulin data, do not support this. Clearly, during the week of MDM observations, PG 1618B had neither amplitude nor phase variations and since the MDM phases do not coincide with phases from earlier in the campaign, unresolved pulsations are unlikely. Since the data obtained at MDM and McDonald observatories used the same acquisition system and time server (NTP), errors in timing also seem unlikely. – 9 – 3. PG 0048+091 3.1. Observations We originally observed PG 0048 as a secondary target during a campaign on KPD 2109+4401 (Zhou et al. 2006). Those data revealed a complex pulsation spectrum which we could not resolve with such limited sampling and a short time base. As such, PG 0048 was re-observed as a multisite campaign during Fall 2005. Five observatories participated in the campaign with the specifics of each run provided in Table 4. Though we were a bit unlucky with weather, over the course of our 16 night campaign we obtained 167.4 hours of data for a duty cycle of 44%. Details of the observing instruments and configurations are the same as for PG 1618B, except for the following: SAAO (1.9 m) used a frame transfer CCD with millisecond dead-times but only an ≈ 30 × 40 arcsecond field of view, which resulted in no comparison stars within the CCD field. As such no transparency variations could be cor- rected and only photometric nights were used. Tubitak Observatory used a Fairchild CCD447 detector; during the first run the images had 1× 1 binning with a dead-time of 102 seconds, while subsequent runs used 2×2 binning with a dead-time of 29 seconds. Bohyunsan Optical Astronomy Observatory (BOAO 1.9 m) data were obtained with a SiTe-424 CCD windowed to 580×445 pixels, binned 2×2 with an average dead time of 14 seconds. MDM and SAAO used red cut-off filters, making their responses very similar to blue-sensitive photoelectric observations, while Lulin, BOAO, and Tubitak used no filter making their sampling more to the red. As pulsations from sdB stars have little amplitude dependence in the visual and no phase dependence (Koen 1998; Zhou et al. 2006), mixing these data is not seen as a problem. The standard procedures of image reduction, including bias subtraction, dark current and flat field correction, were followed using iraf. Differential magnitudes were extracted from the calibrated images using momf (Kjeldsen & Frandsen 1992), except for the SAAO data for which we used aperture photometry because there were no comparison stars. As described for PG 1618B, we again used low-order polynomials to remove airmass trends between our blue target star and the redder comparison stars. The lightcurves are normalized by their average flux and centered around zero, so the reported differential intensities are δI = (I/〈I〉) − 1. Figure 5 shows the lightcurve of PG 0048 with each panel covering two days. – 10 – 3.2. Analysis During the campaign, we completed a “quick-look” analysis of data runs as early as possible to ascertain the data quality and the pulsation characteristics of the star. We noticed early on that the temporal spectra of PG 0048 changed on a nightly basis with pulsation frequencies appearing and then disappearing on subsequent nights. Likewise, we knew that our analysis would be complicated by severe amplitude variations which would limit the usefulness of prewhitening techniques and could create aliasing. Figure 6 shows the effects of amplitude variations. The full panels are pulsation spectra for three groups of data: All of the data; data obtained from September 30 through October 3; and from October 7 through October 11. The right insets are the corresponding window functions plotted on the same horizontal scale. At such large scales, the windows appear as single peaks and show that the changes in the FTs are not caused by aliasing. The central insets are individual data runs within the larger set and show the variability between runs. When sets of data are combined in which the peak amplitudes are not constant, an FT will show the average amplitude. For the frequencies that appear in only a few runs, the amplitudes are effectively quashed in the combined FT. As PG 0048 is the most pulsation variable sdB star currently known, our immediate goal is to glean as many observables from these data as possible. While we do provide some interpretation, our aim is to provide sufficient information for theorists to test their models. The complexity of the data meant it was necessary to analyze it using multiple tech- niques: We performed standard Fourier analyses on combined sets of observations to increase temporal resolution and lower the overall FT noise and analyzed individual runs of the best quality data. The analysis of individual runs represents a time-modified Fourier analysis, which is essentially a Gabór transform, except that we replace a Gaussian time discrimina- tor with the natural beginnings and endings of the individual runs. As the best individual runs are not continuous with time (and nearly all are from MDM Observatory) the use of a Gaussian-damped traveling temporal wave discriminator (a standard Gabór transform) would not enhance the results. The temporal spectra of these runs are shown in Fig. 7. Runs mdm1005 and mdm1009, though long in duration, have gaps in them because of inclement weather, whereas the other 12 runs are gap-free. For these 12 runs aliasing in the FT is not a problem and the only constraints are the width of the peaks, which are determined by run length, and the noise of the FT, which is a combination of the signal-to-noise of each point and the number of data points within the run. Frequencies, amplitudes and phases were determined using two different software pack- ages, Period04 (Lenz & Breger 2004) and a custom (Whole Earth Telescope) set of non-linear least squares fitting and prewhitening routines. Each of the three data combinations in Fig. 6, – 11 – the 12 gap-free data runs plotted in Fig. 7, the three data runs obtained during 2004, and the 10 runs from the discovery data (kindly provided by Chris Koen) were reduced using both software packages. Overall, more than 35 frequencies were fit during at least one data run. Table 5 provides information for 28 frequencies which have been detected above the 4σ detection limit. Column 1 lists a frequency designation; column 2 the frequency as fit to the highest temporal resolution data set in which each frequency is detected with the formal least-squares errors in column 3. Column 4 provides the standard deviation of the corresponding frequencies detected in individual runs and column 5 gives the number N of individual runs in which that frequency was detected (from twelve 2005 runs and three 2004 runs). Tables 6 and 7 provide the corresponding amplitudes as fit for individual runs and var- ious combinations of data acquired during the 2005 campaign, a re-analysis of the discovery data, and the 2004 MDM data. The last two rows of these tables provide the 4σ detection limits and temporal resolutions for the runs. Our determination that these frequencies are real and intrinsic to the star is based on i) detection by both fitting software packages, and amplitude(s) higher than the 4σ detection limit with ii) detection during several data runs, and/or iii) detection at amplitudes too large to be associated with aliasing. 3.3. Discussion 3.3.1. Frequency content During our 2005 multisite campaign, we detected 24 pulsation frequencies from indi- vidual data runs plus an additional frequency from the combined data set (which was also detected in 2004). We recover all seven of the frequencies detected in the discovery data (K04), but only 14 of the 16 frequencies detected from our 2004 data. As can be seen from Table 5, PG 0048 shows an atypically large range of frequencies for sdBV-type pulsators, especially when considering that only one frequency, f23 (11103.3µHz), can be identified as a linear combination (of f6 and f7). As noted in Table 5, only one frequency is detected in all of our data (f2: ∼5244.9µHz) while the next most common frequency (f13: ∼7237.0µHz) is detected in only 11 of the 15 runs. Several frequencies are only detected once or twice (e.g. f12: ∼7154.3µHz), and so we should test if their amplitudes are sufficient to consider them to be real. If a particular frequency has amplitudes that are 1σ above the detection limit, then we might only expect to detect it 68% of the time2. Figure 8 shows the amplitudes and 1σ errors for 12 different frequencies (four frequencies per panel) and the detection limit (solid 2This is a lower limit since statistically, the pulsation amplitude is equally likely to be higher than 1σ from the detected level rather then below it. – 12 – line) for individual runs. The (black) circles in the top panel are for f2, which is detected in every run. However, the two frequencies indicated by (magenta) squares in the middle and bottom panels are only detected once, even though they are > 1σ above the detection limit. If their amplitudes were nearly constant (to within their errors), they would be detected at least 68% of the time. Another way to show this is in panel a of Fig. 9 where the detections are plotted against their significance. We detect a total of 24 frequencies from 12 individual runs from our 2005 data. If we detected all 24 frequencies from every run, we would have made 288 detections, while we only actually made 75. The solid line shows the number of individual detections cumulative with significance (the number of standard deviations the detection was above the detection limit). In other words, 49 of our 75 detections were 2σ or less above the detection limit. The dashed line is the standard Gaussian probability distribution which shows that at 1σ significance, we should have made at least 196 (68%) detections. Since 64% of our detections are ≥ 1σ, our 75 detections is well short of what we should have detected, indicating that the pulsation amplitudes are really falling below the detection limit. Panel b compares the number of actual detections to the maximum pulsation amplitude. As expected, there is some correlation as the higher the amplitude, the easier it is to detect that frequency. Additionally, the highest amplitude (and therefore most easily detected) frequency is the same during 1997, 1998, and 2004 (f8: ∼5612.2µHz) but is only detected in ∼ 1/3 of our 2005 data runs during which f2 (∼5244.9µHz) had the highest amplitudes. This change in pulsation amplitudes will be further discussed in §3.3.3. 3.3.2. Constraints on mode identifications As in §2.3.1, one of the best ways to relate pulsation frequencies to pulsation modes is via multiplet structure. With such a rich pulsation spectrum, it seems likely that some of the frequencies should be related by common frequency splittings. If not, then the pulsation spectrum is too dense (discussed below) for the frequencies to consist of only low-order ℓ ≤ 2 modes. There are two different spacings that occur many times with splittings near 972 and 41.1 µHz. Table 8 lists these frequencies and the deviation from the average spacing between them, while Fig. 10 shows them graphically. While the spacings may be important, it is difficult to attach any physical meaning to them. A spacing of 972 µHz is far too large to be associated with stellar rotation, as it corresponds to a rotation period of only 17 minutes. There is currently no high-resolution spectrum of PG 0048, and the star’s main-sequence companion would complicate any attempts to measure its rotation velocity. However, the typical v sin i of sdBs is less than ∼ 5 km s−1 (Heber, Reid, & Werner 2000). If we try to explain this large splitting using asymptotic theory (consecutive n overtones rather than m multiplets), we would expect successive overtones of the radial index to be roughly evenly – 13 – spaced for large n, with one series for each ℓ degree. But the spacings observed in PG 0048 are irregular, requiring eight degrees (one for every line in Table 8), and 3 ≤ ℓ ≤ 7 modes have reduced visibility because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel, & Kawaler 2005). Theory suggests that sdBV stars should pulsate in low overtone modes and the spacings between successive low overtone modes should differ by several hundreds of µHz (Charpinet et al. 2002), which again does not match what is observed. Another asymptotic-like relation is the “Kawaler scheme” which has been recently pre- sented by Kawaler et al. (2006) and Vuc̆ković et al. (2006). Though it is not compatible with the low overtone (n) pulsation theory associated with sdB pulsators, an improved frequency fit can sometimes be obtained using an asymptotic-like formula; f(i, j) = fo + i× δ + j ×∆ where i has integer values, j is limited to values of −1, 0, and 1, δ is a large frequency spacing and ∆ a small one. However, for the case of PG 0048, the small spacings are unrelated to the larger spacings and the larger spacings themselves do not interrelate, but rather appear in sets with differing spacings between the sets. So this scheme is not applicable for PG 0048. The smaller, 41.1 µHz spacings are more akin to what we would expect for rotationally split multiplets, though still large compared to typical measured rotation rates. If the low- frequency set are all components of a single multiplet, it would require very high degree (ℓ ≥ 5) pulsations, which are not observationally favored (Charpinet et al. 2005; Reed, Brondel, Kawaler 2005). The same is true for the high-frequency set if f13 and f16 belong to the same set. Since they are separated by 7×41.0µHz, it is certainly possible that f12-f13 and f16-f18 are just two pairs, but if they are all combined into a single multiplet, it would require ℓ ≥ 5 again. Yet these could also be just chance superpositions. Since PG 0048 pulsates in so many frequencies, it is important to test the significance of the spacings discussed above. We did this by producing Monte Carlo simulations, randomly placing 28 frequencies within 6000 microHz of each other, and counting how often we could detect 14 frequency splittings the same to within about 5%. This criterion would find all but one of the splittings we actually observe. After analysing over one million simulations, we detected at least 14 splittings in nearly every case. In other words, the splittings we observe are not statstically significant. Another tool we can use to place constraints on mode identifications is the mode density. As mentioned in §2.3.1, models predict roughly one overtone (n) per degree (ℓ) per 1000 µHz and though we have detected 28 frequencies, they are spread across nearly 6000 µHz. Since we do not detect any multiplets that can be unambiguously associated with rotational splitting, it is likely the pulsations are degenerate in m. The average mode density is 4.8 frequencies – 14 – per 1000 µHz, which is too high to accommodate only ℓ ≤ 2, with degenerate m modes and it gets worse as the frequencies are not quite distributed equally, but rather fall into loose groups, enhancing the density locally. The regions between 5200 and 6200 and 6600 to 7600 µHz each contains 9 frequencies, and the region from 8800 to 9800 µHz has 5 frequencies. If all 2ℓ+1 multiplets were filled (9 frequencies per 1000 µHz), the frequency density would not require any ℓ ≥ 3 modes. However, as we do not detect appropriate multiplets within these frequency regions, the most likely result is that PG 0048 has too high a frequency density to exclude ℓ ≥ 3 modes using current models. 3.3.3. Amplitude and phase variability Nearly all sdBV stars show some amount of amplitude variability. However, no other sdBV star has shown variability like that detected in PG 0048. An example of this is shown in Fig. 11, where the same 350 µHz region is shown for the three data sets in Fig. 6. The bottom two panels are subsets of the top panel, which includes all of the data, and indicate how strikingly the pulsation spectrum changes with time. We can apply some constraints to the timescale of amplitude variability. Since pulsa- tion frequencies can change amplitudes (even to the point of being undetectable) between individual runs, and in particular between runs from different observatories but for the same date, the next step was to divide up our longer data runs into halves. The FTs for six such runs are shown in Fig. 12 with each half containing about four hours of data. Particularly for runs mdm1007 (which can also be compared to saao1007) and mdm1010, the amplitudes change by factors of two over times as short as four hours. Low amplitude frequencies can easily become undetectable within that time. Figure 13 compares the maximum and aver- age amplitudes detected in individual runs (the black circles and blue squares, respectively) to detections in groups of data (the lines) from Table 6. If the amplitudes were simply wandering around between values detected in individual runs, then the amplitude of the combined data would be an average of these values (the blue squares). However, since the combined amplitudes are significantly lower then the average amplitudes from individual runs, something else must be occurring to reduce the amplitudes. Since changes in phase can impact pulsation amplitudes, we investigated that next. Figure 14 shows phases for eight of the most-often detected frequencies. f2 is the only frequency detected in every run, and we include the half-night analysis for it. For the other seven frequencies, we only determined phases for those runs in which they were detected above the 4σ limit. While Fig.14 indicates that most phases do not appear constant with time, most are within 20-30% of a central value. – 15 – To isolate and test the impact phase variation creates on data like ours, we analyzed simulated data with the following properties: The data are represented by a noise-free, single- frequency sine wave sampled corresponding to our 12 best individual runs with frequency f2. The amplitudes of each individual run are fixed at the measured values for f2 from Table 6 and we assume that no phase changes occur during an individual run. While the properties of f2 may not be the most representative of the variations detected, it is the only frequency detected every time, and so by using it, we sample the full range of amplitudes. If we used a different frequency, we could not know the actual pulsation amplitude during those runs without detections, and so would only be sampling the higher amplitude data points. We created simulations with the following phase properties: No change in phase, a fixed change of ±10% from the previous phase, a fixed change of ±20% from the previous phase, phases that are randomly set at the beginning of each run, and with the actual phase values for f2. The results of the simulations are given in Table 9 and the ratios are used in Fig. 15 where they are compared with our observations. The last line of Table 9 presents results with unique answers. As expected, there is a correlation between the amplitude and the amount of phase variation in that increasing changes in phase between individual runs decreases the measured amplitude of the data set as a whole. More useful are the ratios of the average amplitude to the average and maximum of the individual amplitudes (〈A〉/〈A〉ind and 〈A〉/Amax). These ratios can be compared with ratios from all frequencies, as has been done in Fig. 15. The shaded regions are the 1σ ratios produced from the phase simulations and the circles represent ratios with G4 data and the squares are for ratios using G1 data. The frequency ordering is the same as for Fig. 12, and like Fig. 12, the results indicate that for all but f2 and possibly f6, the amplitudes detected in groups of data are too low compared to individual amplitudes. Additionally, except for f2, the phases of Fig. 14 (and their standard deviations given in Table 10) are in discord with the amplitude ratios in that the amplitudes are too low. There does not seem to be sufficient phase variation to produce the low amplitudes of the group data. As such, it seems that more extreme circumstances are required. However, that leads into a more speculative area which we save for §3.3.4. We conclude our observational portion of this paper with a summary provided in Ta- ble 10. In this table we have included all measurables (not in previous tables) discussed in this section and some that will be useful for the next section. Columns 2, 3, and 4 consider the number of expected detections based on the average significance of the actual detections; Columns 5 through 8 detail amplitudes detected for individual 2005 observing runs; Columns 9 through 12 provide ratios of individual to group amplitudes; and Column 13 lists the 1σ deviations of pulsation phases. – 16 – 3.3.4. A possible cause of the amplitude/phase variability Can we determine the cause of the erratic behaviour of PG 0048’s frequencies and amplitudes? We suggest that, with the possible exception of f2, the oscillations may be stochastically excited. While this is counter to current theory, supporting observations that we have in hand are 1) amplitudes that vary significantly between every individual run, and in less than ∼4 hrs; 2) the combined amplitudes are significantly lower than the average value indicating that phases are not coherent on these timescales; 3) the peaks in the FTs appear similar to those of known stochastic pulsators (compare Fig. 11 to Fig. 1 of Bedding et al. (2005) or Fig. 2 of Stello et al. (2006) for stochastic oscillators to those in, for example, Reed et al (2004) for “normal” sdB stars); and 4) the number of actual frequency detections compared to the expected number based on significance (the ratio of the two lines in the left panel of Fig. 9). We note that this is not conclusive evidence, but is suggestive and so we will pursue a stochastic nature for PG 0048’s pulsations in the remainder of this section. Recently, Stello et al. (2006) found that short mode lifetimes in red giants can severely limit the possibility of measuring reliable frequencies. The difficulty arises because the frequencies can disappear entirely and when they are re-excited (even if this occurs prior to complete damping), they do not maintain the same phase. The parallel with our analysis of PG 0048 are clear and this kind of variability has been seen before in sdBVs. In a study of KPD2109+4401, Zhou et al. (2006) found substantial variation in the amplitudes of two modes during their 32 night campaign. A brief analysis found that at least one of these modes, and possibly both of them, satisfied the criterion outlined by Christensen-Dalsgaard et al. (2001; hereafter JCD01) for stochastically excited pulsations, rather than overstable driving. The criterion compares the ratio of amplitude scatter to the mean amplitude; for stochastic pulsations, this ratio should be ≈ 0.52. Stochastic processes in pulsating sdB stars have also been discussed by Pereira & Lopes (2005) in the context of the complex sdB pulsator PG1605+072, which is known to have variable amplitudes (O’Toole et al. 2002; Reed et al. 2007, in press). Using the JCD01 criterion, Pereira & Lopes deduced that none of the modes of that star were consistent with stochastic excitation. However, O’Toole et al. (2002) noted amplitude changes between years, while Pereira & Lopes (2005) only studied 7 nights of data, and as such their analysis was likely affected by the short length of their time series. A limitation to the JCD01 test is that the damping times of the oscillations should be longer than the timescale used for determining the amplitudes. Our analysis of ∼ 4 hour segments of PG 0048 data indicate that amplitude variations are on very short timescales that are shorter than the observing time for individual runs (see Figs. 7 and 12). We provide the JCD01 parameter values σA/〈A〉 in Column 7 of Table 10, but we suggest that the JCD01 – 17 – test is not appropriate for PG 0048. Aside from the JCD01 test, we can attempt to reproduce some of the observational properties using simple simulations with damped and randomly re-excited frequencies. The complexity of the actual data is such that we cannot hope to reproduce it directly, but instead will strive to fit the observations listed at the beginning of this section. Our simulations follow the simple prescription (equations 2 and 3) of Chaplin et al. (1997) summarized as follows: The pulsations themselves are described by sine waves of the form A(t) = A cos((2π · f)(t − φ)), with the amplitude modified in two ways; it is damped exponentially as A = Ao exp(−te/td) where Ao is the maximum amplitude, te is the time since the last excitation and td is the damping timescale. The pulsation is re-excited by setting te = 0 when time t exceeds an excitation timescale (texc). The time before the first re-excitation is randomly set to some fraction of the excitation timescale and every time the pulsation is re-excited, the phase is randomly set and the excitation timescale and pulsation amplitude can vary randomly by up to 20% or their original values. The free parameters of the simulation are the input amplitude, which is the maximum amplitude attainable and the excitation and damping timescales (Ao , texc and td, respectively). The simulation includes frequencies, amplitudes, and phases for up to 100 pulsations with an unlimited number of data runs (input as run start time, the number of data points, and cycle time). We will concentrate on matching the MDM runs in Table 6. This is the G1 data set with an average run length of 9.35 hours (∼ 0.4d), containing ∼ 2800 data points each, an average detection limit of 0.88 mma, and average ratios AG1/〈A〉 = 0.44± 0.14 and AG1/Amax = 0.37± 0.09. To match the observational constraint that the pulsation amplitude can reduce by half in a four hour span, the damping timescale is necessarily less than 5.8 hours. With this constraint, we produced a grid of simulations with 1 ≤ td ≤ 7 hours in 1 hour increments and 1 ≤ texc ≤ 33 hours in 2 hour increments. Qualitatively, if no re-excitations occur during an individual run, the FT is single-peaked whereas multiple re-excitations create a variety of complex, multi-peaked FTs, depending on how similar the randomized phases were (the less alike the phases for each re-excitation, the lower the overall amplitude and more and similar-amplitude peaks appear in the FT). For small texc, the FT becomes increasingly complex with most simulations resulting in many low-amplitude peaks distributed across a couple hundred µHz. However, such complex patterns are not consistent with observations and so we discount small values for texc. Amplitudes in the FT are reduced with large values of texc and small values of td while the scatter increases with increasing values for both. An increase in amplitude scatter is necessary to produce the low rate of detections. As we now have all the pieces in place, we can ask how well the simulations reproduce the observational constraints we set at the beginning of this section. A selection of the results are shown in Figs. 16 and 17. Panels a, b, and c of Fig. 16 have td fixed at 5 hours and vary texc whereas panels d, e, and f fix texc at 19 hours and vary td. Panels a and d – 18 – are the results for individual runs and panels b and e are for the combined nine-run data set. The points represent the average (〈A〉) with 1σ deviations while the lines indicate maximum (Amax) and minimum amplitudes. Panels c and f show the ratios 〈AG1〉/Amax and 〈AG1〉/〈Aind〉; the average amplitude from the combined data divided by the maximum or average amplitude from the individual runs. Figure 17 shows the expected rate of detections calculated in the following manner: The detection limit was calculated using the observed ratios 〈Amax〉/0.88 = 1.56 and 〈A〉/0.88 = 1.32 for the MDM data with an average detection limit of 0.88 mma and solving for the new detection limit. The dotted line is the observed detection rate of 26% and it is interesting that none of simulations are that low if using the average detected amplitude. While the overall amplitudes can become quite small, the average follows that, which is used to calculate our detection limit in the top panel. However, most values of td matched the observed rate near texc = 19-21 hours using Amax. The simulations that best fit the observational constraints are those which have 4 ≤ td ≤ 6 hrs and 13 ≤ texc ≤ 21 hrs. The lower values of texc better fit the 〈AG1〉/A ratios while the larger values are a better match for the detection rate. These relatively simple simulations are able to fit all of the observed constraints, thus explaining the amplitude variations, their lower detection limits in groups of data, the relatively low detection rate and the appearance of the peaks in the FT. What they cannot explain however, is the relative lack of phase variability in some frequencies (though relatively few were measurable) and why stochastic processes should occur in the first place. Stochastic oscillations are usually presumed to be driven by random excitations caused by convection (see Christensen-Dalsgaard 2004 for a review concerning the Sun). The He II/He III convection zone in sdB stars was investigated by Charpinet et al. (1996), who determined that it could not drive pulsations. However, it appears that they did not investigate this zone for convective motions but rather as a driver for the κ mechanism. So some ambiguity remains here. There is also convection or semi-convection in the cores of sdB stars, and it is possible that the eigenfunctions could be sampling this region. Whatever the case, the extreme amplitude and phase variability of PG 0048 poses a significant challenge to the iron driving mechanism found by Charpinet et al. (2001) to excite pulsations in sdB stars. Though it is beyond the scope of this paper to ascertain the cause of the random amplitude variations, we find that the observed properties are consistent with our simplified randomly excited simulations and that the amplitude spectrum resembles those of pulsators that are stochastically driven. – 19 – 4. Conclusions and Future Work We have carried out multisite campaigns for two sdB pulsators, PG 1618+563B and PG 0048+091 and in both cases, our observations were superior to the published discovery data, yet questions concerning these two stars still remain. Our MDM observations of PG 1618B (obtained under good conditions) show characteristics typical of about half the objects in the sdBV (V361Hya) class: A small number of stable (in amplitude) frequencies with a closely spaced pair. In contrast, the data obtained at McDonald observatory – under non-photometric conditions – show PG 1618B to be a complex pulsator with four “regions” of power showing amplitude and phase variability. An ensemble analysis of any combinations of data other than the MDM set are hindered by poor least-squares fitting and amplitudes reduced below detectability. Such poor fitting can be caused by unresolved frequencies with intrinsic amplitude variability (O’Toole et al. 2002) or randomly excited pulsations (Christensen-Dalsgaard 2004). So despite having expended considerable effort to obtain not only multisite, but extended time-base observations, we were only reasonably successful at detecting pulsations from our 2 m telescope data; and these show the star to be two-faced. With this dichotomy of observational results, PG 1618B remains an interesting target for more follow-up observations; particularly to examine its long-term frequency stability. PG 0048 is much more complex than PG 1618B, yet it too has shown somewhat stable pulsation amplitudes at one epoch (the discovery and 2004 data) and wildly variable ampli- tudes at another (2005). Though an extremely rich pulsator with at least 28 independent frequencies, many modes are only excited to amplitudes above the noise occasionally, often for very short lengths of time. These behaviors are consistent with stochastic pulsations and we have performed several tests along these lines. We simulated damped and re-excited pul- sations and found that the observations were best matched with damping timescales between 4 and 6 hours and excitation timescales between 13 and 19 hours. We detected common fre- quency splittings of 972 and 41 µHz which may be related to multiplet structure, but could reproduce these using Monte Carlo splittings of random spacings. So while they may be intrinsic to the pulsation of PG 0048, we cannot be sure. We can be sure that PG 0048’s rich pulsation spectrum is too dense to be accounted for using only ℓ ≤ 2 modes regardless of how many m components are present. The observations presented in this paper provide some very interesting and confusing results. Pulsations that appear stable during some times and variable at others; attributes that have also been observed in other sdBV stars as well (KPD 2109+2752, PG 1605+072, and HS 1824+5745, just to name a few). The pulsations in PG 0048 present observables that seem best described by randomly excited oscillations which would be in contrast to the proposed driving mechanism (Charpinet et al. 2001). If validated, it would represent a new – 20 – direction in sdB pulsations (and modeling too!). However, a longer time series may be the only way to clarify the nature of the oscillations in this star. Ideally this would take place on at least 2m-class telescopes and cover several weeks. We would like to thank the time allocation committees for generous time allocations, without which this work would not have been possible; Dave Mills for his time and help with the Linux camera drivers; Chris Koen for contributing the discovery data and helpful discussions; and the anonymous Referee for a helpful re-organization of the paper. This material is based in part upon work supported by the National Science Foundation under Grant Numbers AST007480. 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(The Whole Earth Telescope Collaboration) 1991, ApJ, 378, 326 Zhou A.-Y., et al. 2006, MNRAS, 367, 179 This preprint was prepared with the AAS LATEX macros v5.2. – 22 – Table 1: Observation record for PG 1618B. The first two runs were obtained in 2003, while the rest were obtained in 2004. Run Date Start Length Int. Observatory UT hr:min:sec (Hrs) (s) suh16mar 17 Mar 00:07:58 3.3 10 Suhora 0.6 m lul031705 17 Mar 16:46:33 5.6 10 Lulin 1.0 m McD031805 18 Mar 04:34:00 1.1 5 McDonald 2.1 m lul031805 18 Mar 15:14:24 6.1 10 Lulin 1.0 m McD031905 19 Mar 08:40:40 3.1 5 McDonald 2.1 m lul031905 19 Mar 19:58:41 1.3 15 Lulin 1.0 m baker032005 20 Mar 04:29:30 5.6 25 Baker 0.4 m McD032005 20 Mar 06:35:00 5.8 5 McDonald 2.1 m lul032005 20 Mar 15:17:19 6.0 10 Lulin 1.0 m suh20mar 20 Mar 18:59:00 8.3 10 Suhora 0.6 m lul032105 21 Mar 16:44:59 0.7 15 Lulin 1.0 m suh21mar 21 Mar 18:20:20 8.6 20 Suhora 0.6 m McD032205 22 Mar 04:33:00 7.8 5 McDonald 2.1 m suh22mar 22 Mar 18:40:40 2.2 20 Suhora 0.6 m McD032305 23 Mar 04:12:10 8.0 5 McDonald 2.1 m mdr299 29 Mar 04:11:15 4.7 15 Baker 0.4 m mdr301 31 Mar 04:28:10 7.0 15 Baker 0.4 m mdr302 02 Apr 02:53:10 8.1 10 Baker 0.4 m bak040305 03 Apr 03:20:46 7.9 15 Baker 0.4 m bak040405 04 Apr 04:23:10 5.6 15 Baker 0.4 m bak040505 05 Apr 03:17:10 6.6 10 Baker 0.4 m bak041405 14 Apr 03:45:00 7.1 10 Baker 0.4 m bak041505 15 Apr 02:26:30 8.4 10 Baker 0.4 m bak041605 16 Apr 02:50:30 7.9 10 Baker 0.4 m bak041705 17 Apr 03:18:50 5.3 15 Baker 0.4 m bak041805 18 Apr 02:56:00 7.7 15 Baker 0.4 m mdm042605 26 Apr 04:25:30 7.3 5 MDM 2.4 m mdm042705 27 Apr 04:16:50 7.6 5 MDM 2.4 m mdm042805 28 Apr 04:14:00 7.8 3 MDM 2.4 m mdm042905 29 Apr 04:18:00 1.6 5 MDM 2.4 m mdm043005 30 Apr 04:04:30 7.8 3 MDM 2.4 m mdm050105 01 May 04:05:30 7.8 5 MDM 2.4 m mdm050205 02 May 03:36:40 5.3 5 MDM 2.4 m – 23 – Table 2: Subsets of data for PG 1618B. For column 2, observatories are 1) McDonald Obser- vatory; 2) Suhora Observatory; 3) Lulin Observatory; 4) Baker Observatory; and 5) MDM Observatory. Set Observatory(ies) Inclusive Dates Resolution 4σ detection limit µHz mma McD 1 18 - 23 March 2.2 1.64 MDM 5 26 Apr - 02 May 1.9 0.55 McD+MDM 1, 5 18 - 02 May 0.3 0.59 Week 1 1, 2, 3 17 - 23 March 1.5 1.53 All 1, 2, 3, 4, 5 17 - 02 May 0.2 0.77 – 24 – Table 3: Our least-squares fit solution for the pulsation periods, frequencies, and amplitudes detected in PG 1618B. Formal least-squares errors are given in parentheses. Frequencies marked with a dagger (†) were only detected during individual runs (one each) while the remaining frequencies are from the MDM data set. Des. Period Frequency Amplitude (s) (µHz) (mma) f1† 108.7092 (0.0518) 9198.85 (4.38) 1.79 (0.39) f2† 122.2574 (0.0597) 8179.46 (4.00) 1.04 (0.21) f3 128.9549 (0.0008) 7754.64 (0.05) 1.71 (0.09) f4 139.0571 (0.0008) 7191.28 (0.04) 2.04 (0.09) f5 143.9290 (0.0011) 6947.87 (0.05) 2.22 (0.10) f6 143.9759 (0.0014) 6945.60 (0.07) 1.64 (0.10) – 25 – Table 4: Observations of PG 0048 Run Date Start Length Int. Observatory UT hr:min:sec (Hrs) (s) mdr285 10 Oct 04:21:00 6.2 15 MDM 1.3 m mdr290 12 Oct 03:39:00 7.3 15 MDM 1.3 m mdr295 14 Oct 03:34:00 7.4 15 MDM 1.3 m boao 26 Sept. 10:30:20 5.0 10 BOAO 1.9 m mdm092805 28 Sep 07:32:30 4.5 15 MDM 1.3 m mdm092905 29 Sep 02:53:00 9.5 15 MDM 1.3 m mdm093005 30 Sep 02:44:00 9.5 15 MDM 1.3 m turkSep3sdb 30 Sep 17:37:54 9.3 10 Tubitak 1.5 m mdm100105 01 Oct 02:46:00 9.4 10 MDM 1.3 m turk1Octsdb 01 Oct 21:26:58 4.1 10 Tubitak 1.5 m mdm100205 02 Oct 09:57:30 2.3 15 MDM 1.3 m turk2Octsdb 02 Oct 20:51:36 4.8 10 Tubitak 1.5 m mdm100305 03 Oct 02:32:00 9.1 10 MDM 1.3 m turk3Octsdb 03 Oct 17:41:49 8.5 10 Tubitak 1.5 m mdm100405 04 Oct 09:24:00 2.2 15 MDM 1.3 m mdm100505 05 Oct 02:33:00 9.4 12 MDM 1.3 m mdm100605 06 Oct 02:23:00 9.4 10 MDM 1.3 m mdm100705 07 Oct 02:15:00 9.6 12 MDM 1.3 m lul7Oct 07 Oct 18:32:20 1.6 20 Lulin 1.0 m a024 07 Oct 20:28:14 6.2 10 SAAO 1.9 m mdm100805 08 Oct 03:17:00 8.5 15 MDM 1.3 m a039 08 Oct 21:31:48 5.4 10 SAAO 1.9 m mdm100905 09 Oct 02:17:00 9.3 10 MDM 1.3 m a057 09 Oct 20:11:53 4.1 10 SAAO 1.9 m mdm101005 10 Oct 02:06:00 9.6 10 MDM 1.3 m a077 10 Oct 19:58:49 6.5 10 SAAO 1.9 m mdm101105 11 Oct 02:00:00 9.6 10 MDM 1.3 m – 26 – Table 5. Frequencies detected for differing subsets of PG 0048 data. Column 1 is the frequency designation; column 2 is the frequency determined from a combined data set; column 3 is the formal least-squares error from the combined data set; column 4 is the standard deviation of frequencies detected from individual runs; and column 5 is the number of individual 2005 and 2004 runs in which that frequency is detected. All frequencies are given in µHz. († indicates frequencies that were not detected in 2005.) Des. Freq. σfit σ N f1 5203.1 - 5.9 2 f2 5244.9 0.16 4.6 15 f3 5287.6 0.03 3.6 8 f4 5356.9 0.05 9.8 8 f5 5407.0 0.07 3.2 1 f6 5465.1 - - 1 f7 5487.2 0.05 8.5 7 f8 5612.2 0.05 4.5 8 f9 5652.9 - - 1† f10 6609.2 - 9.0 2 f11 6834.3 0.15 9.0 3† f12 7154.3 0.08 1.4 1 f13 7237.0 0.04 7.0 10 f14 7430.1 - - 1 f15 7501.3 0.07 - 1 f16 7523.9 0.06 15.0 8 f17 7560.0 - - 1 f18 7610.1 0.07 5.6 5 f19 8055.5 0.06 10.1 8 f20 8651.4 0.08 - 1 f21 8820.6 0.08 6.4 4 f22 9352.8 0.07 16.3 1 f23 9385.3 0.28 8.9 1† f24 9694.6 - - 1 f25 9795.1 0.08 18.8 2 f26 10366.8 0.08 1.4 2 f27 11103.3 - - 1† f28 11159.8 0.07 - 1 – 27 – Table 6. Amplitudes detected for differing subsets of PG 0048 data during the 2005 multisite campaign. The last two rows give the 4σ detection limit and the temporal resolution for each subset. Note that all frequencies are given in µHz. Runs 1 through 12 are boao, mdm0929, mdm9030, mdm1001, mdm1003, mdm1006, mdm1007, saao1007, mdm1008, mdm1010, saao1010, and mdm1011 and runs G1 through G4 are MDM only, Oct. 1 - 3, Oct. 7 - 11, and all the 2005 data, respectively. Des. 1 2 3 4 5 6 7 8 9 10 11 12 G1 G2 G3 G4 f1 1.11 1.69 f2 2.44 1.03 1.48 1.25 2.26 1.12 0.93 2.34 1.70 2.19 2.42 1.77 1.38 1.07 1.77 1.40 f3 1.21 1.39 1.22 1.01 1.26 1.77 0.60 0.62 0.78 0.83 f4 1.13 1.30 1.24 1.07 1.01 0.59 0.94 0.53 0.54 f5 1.19 0.41 0.52 0.42 f6 0.86 0.59 f7 1.06 1.05 1.02 0.87 1.59 0.56 0.66 f8 0.98 1.12 1.39 1.06 1.02 0.92 0.53 f10 0.94 f12 0.81 0.36 f13 1.36 1.04 0.95 1.68 1.50 1.46 2.13 1.07 0.79 1.08 0.78 f14 0.83 f15 1.52 0.37 0.64 0.79 0.37 f16 0.97 1.36 2.26 1.39 1.78 1.09 0.53 0.86 0.48 f17 1.10 0.50 0.48 f18 1.00 1.09 1.10 0.99 0.48 0.63 0.59 0.36 f19 1.12 0.89 1.08 1.13 1.74 0.43 0.58 0.69 0.47 f20 0.90 0.36 0.44 f21 1.41 0.81 0.94 0.39 0.63 f22 1.10 0.41 0.46 f23 0.44 f24 0.85 f25 1.22 0.88 0.36 0.48 0.35 f26 0.91 1.01 0.36 0.56 f28 1.18 0.40 0.54 0.35 4σ 0.96 0.75 0.84 0.85 0.90 1.02 0.76 1.60 0.84 1.07 1.00 0.91 0.31 0.47 0.43 0.28 1/T 53 29 29 31 29 29 29 46 31 29 39 36 0.9 2.2 2.2 0.8 – 29 – Table 7. The same as Table 6 for the discovery data (1997 and 1998; courtesy of C. Koen) and 2004 MDM data. Runs dd1 through dd10 are tex151, tex157, tex177, tex182, tex186, dmk157, dmk163, ck261, ck263, and ck265, runs 1 through 3 are mdr285, mdr290, and mdr295, and runs G1 through G4 are all 1997, October 1998, November 1998, and 2004 data, respectively. Des. dd1 dd2 dd3 dd4 dd5 dd6 dd7 dd8 dd9 dd10 1 2 3 G1 G2 G3 G4 f1 2.63 f2 3.18 1.11 1.95 2.01 2.78 1.65 f3 5.86 3.52 2.77 1.76 4.29 2.10 1.54 f4 3.39 2.33 3.40 1.96 1.14 1.90 1.93 1.34 1.57 f5 1.96 0.55 f7 1.96 1.20 2.12 0.93 0.84 1.94 2.16 0.75 f8 4.94* 5.80 3.97 4.39 3.10 3.43 3.57 3.95 4.96 2.85 3.10 3.14 4.05 3.34 2.01 2.97 f9 1.28 0.65 f10 1.18 0.68 f11 1.03 1.16 0.98 1.56 1.00 f13 2.76 1.55 1.56 1.56 f16 4.31* 2.28 1.22 1.26 f17 1.55 2.17 1.69 1.40 1.56 f18 1.25 0.77 f19 2.36 3.29 1.21 1.08 2.03 1.17 1.99 1.06 f20 1.75 f21 0.93 0.74 f22 2.43 1.26 f23 1.04 0.65 f24 0.53 f27 1.00 4σ 5.38 3.54 1.50 1.20 1.70 2.06 1.63 3.80 3.17 2.44 0.75 0.82 0.97 1.14 1.21 1.97 0.50 1/T 194 132 69 96 84 96 84 174 185 111 41 38 37 0.8 5.4 0.9 – 31 – Table 8. PG 0048 pulsation frequencies split by integer multiples near 972 and 41.1 µHz. The first column gives the minimum degree ℓ for the multiplet and the order for each row is from the top of Table 5, or in descending frequencies and the number in parentheses indicates the integer multiple between itself and the previously listed frequency and the percent deviation from 972 or 41.1 µHz. ℓmin Des. Designations for related frequencies Spacings of 972 µHz 3 f1 f12 (2, 0.2%); f27 (4, 1.4%) 3 f7 f14 (2, 0.2%); f23 (2, 0.4%); f26 (1, 0.8%) 1 f8 f10 (1, 2.4%); f17 (1, 2.4%) 2 f9 f18(2, 0.5%); f20 (1, 6.9%) 2 f11 f21 (2, 2.0%); f25 (1, 0.07%) 1 f3 f13 (2, 0.4%) 2 f6 f22 (4, 0.2%) Spacings of 41.1 µHz 6 f1 f2 (1, 1.7%); f3 (1, 3.9%); f5 (3, 3.2%); f7 (2, 2.4%); f8 (3, 1.5%); f9 (1, 0.1%) 6 (1, 1) f12 f13 (2, 0.05%); f16† (7, 0.02%); f18 (2, 5.9%) – 32 – Table 9. Results of simulations for phase changes using f2 as the template and a comparison with observations. Column 1 indicates how the phase was changed in the simulation and columns 2 and 3 provide the average amplitude and standard deviation of > 100, 000 simulations. Columns 4 and 5 provide ratios comparing amplitudes between individual runs and that of the combined simulated data set. The bottom row provides the average amplitude from the individual runs and results from the simulations with constant phase and fixed amplitudes and phases with those observed for f2. Sim 〈A〉 σ 〈A〉/〈A〉ind 〈A〉/Amax 10% 1.50 0.03 0.97 0.66 20% 1.35 0.10 0.88 0.60 Random 1.06 0.11 0.69 0.47 〈A〉ind constant phase Aobs & φobs 1.54 1.55 1.49 Table 10. Observed properties of individual frequencies. Des. #det 〈σd〉 #exp 〈A〉 σA σA/〈A〉 Amax AG4/〈A〉 AG4/Amax AG1/〈A〉 AG1/Amax σφ (%) f1 2 0.38 3 1.30 0.41 0.32 1.69 - - - - - f2 12 3.69 12 1.69 0.53 0.31 2.44 0.83 0.57 0.82 0.57 9.6 f3 5 2.63 11 1.32 0.22 0.17 1.77 0.63 0.47 0.45 0.34 - f4 5 1.59 10 1.14 0.15 0.13 1.30 0.47 0.44 0.52 0.48 8.4 f5 2 0.92 7 1.06 0.19 0.18 1.19 0.40 0.35 0.39 0.34 - f6 1 0.11 1 0.86 - - 0.86 0.67 0.67 - - - f7 5 1.16 8 1.09 0.19 0.17 1.59 - - 0.51 0.35 7.9 f8 5 1.50 10 1.11 0.09 0.08 1.39 0.48 0.38 - - 6.3 f9 1 - - - - - - - - - - - f10 1 1.06 8 0.94 - - 0.94 - - - - - f11 1 - - - - - - - - - - - f12 1 0.33 3 0.81 - - 0.81 - - 0.44 0.44 - f13 7 2.32 11 1.43 0.36 0.25 2.13 0.55 0.37 0.75 0.50 15.8 f14 1 0.41 3 0.83 - - 0.83 - - - - - f15 1 2.75 11 1.52 - - 1.52 0.24 0.24 0.24 0.24 - f16 6 2.60 11 1.38 0.44 0.32 2.26 0.35 0.21 0.38 0.23 15.0 f17 1 0.50 4 1.10 - - 1.10 0.44 0.44 0.45 0.45 - f18 4 0.94 7 1.05 0.18 0.17 1.10 0.43 0.33 0.46 0.44 25.9 f19 5 1.79 10 1.19 0.26 0.22 1.74 0.39 0.27 0.36 0.25 7.3 f20 1 0.32 2 0.90 - - 0.90 - - 0.40 0.40 - f21 3 1.16 8 1.03 0.26 0.25 1.41 - - 0.38 0.28 - f22 1 1.32 9 1.10 - - 1.10 - - 0.37 0.37 - f23 1 - - - - - - - - - - - f24 1 0.00 1 - - - - - - - - - f25 2 2.25 11 1.03 0.24 0.23 1.22 0.34 0.29 0.35 0.30 - f26 3 0.76 6 0.94 0.17 0.18 1.01 - - 0.41 0.36 - f27 1 - - - - - - - - - - - f28 1 1.42 9 1.18 - - 1.18 0.30 0.30 0.34 0.34 - – 34 – Fig. 1.— Lightcurves for PG 1618B data. The top four panels show the coverage during the multisite portion of the campaign from March 17 – 24, 2005, while the bottom panel shows an enlarged section of a run obtained at MDM. Note that the scales are different for the bottom panel. The line indicates our fit to the data. – 35 – Fig. 2.— Temporal spectra of various subsets of PG 1618B data. Insets are the window functions. Solid (blue) horizontal line is the 4σ detection limit while the dashed (blue) lines are at 2 mma in all panels. – 36 – Fig. 3.— Temporal spectrum of PG 1618B data. Top panel shows the original FT of the combined MDM data while the bottom panel shows the residuals after prewhitening. Prewhitened frequencies are indicated by arrows. Insets show an enlarged view of the fre- quency doublet and the window function. – 37 – Fig. 4.— Amplitudes and phases of the two frequencies in PG 1618B that are resolvable nightly. The data are only those from McDonald and MDM observatories except for a single Lulin run (marked by a triangle). The dashed lines indicate our fits to the MDM data. Note that the time axis is discontinuous. – 38 – Fig. 5.— Lightcurves for the PG 0048 data. Each panel is two days. – 39 – Fig. 6.— Comparison of temporal spectra for varying groups of PG 0048 data. The large panels show FTs for combinations of data (inclusive dates are labeled). The insets on the right are the window functions for the combinations of data to scale. The central insets are slightly enlarged FTs of individual runs obtained congruent with the larger panels and are labeled. They go from early to late within the campaign. – 40 – Fig. 7.— Temporal spectra for individual PG 0048 observing runs. – 41 – Fig. 8.— Amplitudes and errors of 12 frequencies detected in PG 0048. Each panel contains amplitudes for four frequencies (differentiated by point type and color) which may be shifted by ±0.1 day if they overlap. The solid line is the 4σ detection limit. – 42 – Fig. 9.— a) The number of detections of frequencies from individual runs for PG 0048 compared to Gaussian probability. The number of possible detections (dashed line) compared to the amount of actual detections (solid line) depending on the detection significance. b) A comparison between the number of detections for individual frequencies and the maximum amplitude detected. – 43 – Fig. 10.— A schematic of the pulsation content of PG 0048 to indicate the nearly evenly spaced frequencies (provided in Table 8). The top panel shows the spacings that are integer multiples near 972 µHz and the bottom panel shows those near 41.1 µHz. For clarity, each set of multiplets are connected via horizontal bars at differing heights and have different colors (electronic version only, and black is used twice in the top panel). – 44 – Fig. 11.— A 350µHz region of PG 0048’s FT covering three modes at ∼7497.4µHz, ∼7524.1µHz and ∼7605.9µHz. The panels correspond to those in Fig. 6. The spectral windows at the top of each panel cover the same frequency range as the observations, to give the reader an appreciation of how dense these regions are. – 45 – Fig. 12.— Temporal spectra for halves of individual PG 0048 observing runs. – 46 – Fig. 13.— Amplitudes of frequencies detected in both individual runs and grouped data during 2005 for PG 0048. Dots (black) indicate the maximum amplitude, squares (blue) the average amplitude from individual runs. The lines indicate amplitudes from the following group datasets (from Table 6): dotted (blue) is G1, short-dashed (green) is G2, long-dashed (magenta) is G3, solid (black) is G4, and the dot-dashed (black) line is the G4/Amax ratio (or G1/Amax if no G4 value). – 47 – Fig. 14.— Phases of pulsation frequencies. Frequency designations are provided in each panel with phases determined only for individual runs in which the frequency was detected. – 48 – Fig. 15.— Comparison of amplitude ratios from simulations to those observed for PG 0048. Black circles are for G4 and blue squares are for G1 data. The verticle-lined (top and green) area is for simulations with 10% phase variations, the +45o-lined (middle and red) area is for simulations with 20% phase variations, and the −45o-lined (bottom and blue) area is for simulations with random phases. – 49 – Fig. 16.— Simulations of randomly excited and damped pulsations. Panels a, b, and c show the effect of changing the excitation timescale for a fixed damping timescale of td = 5 hours. Panel a and b show the results of simulating a single 9.35 hour run and the nine MDM runs in Table 2, respectively. The points indicate the average amplitude with 1σ error bars, the solid line is the maximum amplitude from an individual simulation and the dotted line indicated the minimum amplitude detected from an individual simulation. Panel c shows the ratios 〈AG1〉/Amax (blue circles) and 〈AG1〉/〈Aind〉 (black triangles). The dashed blue (dotted black) lines indicate the observed range in PG 0048 for 〈AG1〉/Amax (〈AG1〉/〈Aind〉). Panels d, e, and f correspond to panels a, b, and c except that the excitation timescale is fixed at texc = 19 hours and the damping timescale is varied. – 50 – Fig. 17.— Expected fraction of detected frequencies based on simulations. Top panel used average amplitudes while the bottom panel used maximum amplitudes from the simulations. Differing lines represent different values of td given in the legend. Dotted line is the observed 26% detection rate. See the electronic edition of the Journal for a color version of this figure. Introduction PG 1618B+563B Observations Analysis Discussion Constraints on the pulsation modes Amplitude and phase stability PG 0048+091 Observations Analysis Discussion Frequency content Constraints on mode identifications Amplitude and phase variability A possible cause of the amplitude/phase variability Conclusions and Future Work

0704.1497

Connecting microscopic simulations with kinetically constrained models of glasses

Connecting microscopic simulations with kinetically constrained models of glasses Matthew T. Downton and Malcolm P. Kennett Physics Department, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada (Dated: August 23, 2021) Kinetically constrained spin models are known to exhibit dynamical behavior mimicking that of glass forming systems. They are often understood as coarse-grained models of glass formers, in terms of some “mobility” field. The identity of this “mobility” field has remained elusive due to the lack of coarse-graining procedures to obtain these models from a more microscopic point of view. Here we exhibit a scheme to map the dynamics of a two-dimensional soft disc glass former onto a kinetically constrained spin model, providing an attempt at bridging these two approaches. PACS numbers: 05.10.-a, 61.20.Gy, 61.43.Fs, 64.70.Pf I. INTRODUCTION The origin of the onset of ultra-slow dynamics in glassy systems, and in particular, glass-forming liquids, remains a murky subject, with many competing ideas and tanta- lizing clues as to underlying causes, despite years of ef- fort by a large community of researchers [1]. Recently it has become increasingly clear that dynamical hetero- geneities, regions of atypically fast dynamics that are lo- calized in space and time, are intimately connected to the phenomenon of glassiness [2, 3, 4, 5, 6], becoming increas- ingly important at lower temperature scales towards and below the glass transition temperature Tg [7, 8, 9, 10]. Early ideas about heterogeneous dynamics focused on the idea of co-operatively rearranging regions which grow with decreasing temperature [11]. Currently, molecular dynamics (MD) simulations of supercooled liquids allow much greater access to the microscopic details of this het- erogeneity [12, 13, 14]. This has included the observation of “caging” of particles, and string-like excitations that allow particles to escape these cages [9, 15, 16, 17], which has been confirmed in experiments on colloidal glasses [18]. However, MD simulations have the drawback that it is difficult to reach the low temperatures and long times characteristic of the glassy phase. An alternative approach to reach low temperatures and long times, but without a microscopic foundation, is to study simple models of glassiness, kinetically con- strained models (KCMs) [19, 20, 21, 22, 23, 24], such as the Fredrickson-Andersen (FA) model [20] or the East model [25], or variations such as the North-East model [23], which mimic the constrained dynamics of real glassy systems but have trivial thermodynamics. These may be viewed as effective models for glasses, in terms of some coarse-grained degree of freedom often labelled “spins”, also termed a “mobility field” by some authors [23]. In Fredrickson and Andersen’s original work, they posited that the degrees of freedom may be high and low den- sity regions, related to earlier suggestions by Angell and Rao [26]. Despite these appealing physical pictures, it has not been particularly clear to what physical quan- tity this “mobility field” corresponds. If KCMs are truly effective models of glassy behavior then it should be pos- sible to make a connection between some set of degrees of freedom, in a MD simulation, for instance, and a KCM. In this paper we propose a specific coarse-graining pro- cedure to explore whether a link can be made between MD simulations and KCMs of glassiness. Previous work in this direction found evidence of dynamic facilitation in MD simulations [10, 27], however, there was no attempt to map the dynamic facilitation onto a KCM. We use an approach directly related to the idea of a “mobility” field, using the local mean-square displacement (MSD) in a suitably defined box to define a spin variable. Re- gions with large average MSD correspond to “up” spins and those with low average MSD correspond to “down” spins. We give specific details of our procedure below. We investigate the time and length-scale dependence of this coarse-graining. The two characteristic time scales are the beta relaxation time scale, tβ (corresponding physically to the time for relaxation within a cage) and the longer alpha relaxation time scale, tα, which corre- sponds to the time-scale on which structural relaxation of cages occurs. We find that evidence of dynamic facil- itation becomes much stronger at longer times of order tα, than at earlier times of order tβ . We also study the effect of changing the size of the coarse-graining box, l, in space and consider values 0.02 ≤ l/L . 0.25, where L is the system size. With appropriate choices of time and lengthscales, we find a clear mapping from our MD simulations onto a KCM similar to the 1-spin facilitated FA model. This is not what one might naively expect. Since we study a fragile glass-former, we expect to find a KCMwhich exhibits super-Arrhenius relaxation, whereas the 1-spin facilitated FA model has Arrhenius relaxation – the super-Arrhenius growth of timescales is absorbed into the coarse graining time, which is most effective at capturing kinetically constrained behaviour when it is of order tα contrary to expectations based on the idea of a mobility field [20, 23]. The demonstration of a coarse-graining procedure to translate from a microscopic model to a coarse-grained KCM for glasses can shed light on the following: it pro- vides a physical interpretation to the “mobility field”; it can give a stronger theoretical justification for the use of KCMs to study glassy dynamics; and may open the door to further exploration of the link between micro- scopic models and long-time features of dynamics, i.e. http://arxiv.org/abs/0704.1497v3 answering the question: for a given interparticle interac- tion potential, how will the dynamics of the glass behave? This paper is structured as follows: in Sec. II we give details of our molecular dynamics simulations, in Sec. III we describe our coarse-graining procedure and present the results of coarse-graining MD simulations, and then in Sec. IV we discuss our results. II. MOLECULAR DYNAMICS SIMULATIONS We study the dynamics of a well-characterized glass former: the binary soft disc model with a potential of the form ǫ in two dimensions [13, 15, 28]. This mixture of discs with size ratio 1:1.4 inhibits crystalliza- tion upon cooling. We choose a 75:25 ratio of small to large particles and cool the system with the density fixed at ρ = 0.85σ−2 . All temperatures and lengths are quoted in the standard reduced units of the Lennard-Jones po- tential using the small disc diameter σ11, as a length scale. This mixture has the same glassy characteristics as the model previously documented in Ref. [28]. Post- equilibration calculations are performed using the re- cently introduced iso-configurational (IC) ensemble [29]. Our results are qualitatively similar if we follow a single trajectory with no IC averaging, but IC averaging gives smoother trends as a function of temperature. FIG. 1: Averaged mean square displacement from IC simula- tions of length τe at T = 0.360. On this time scale there is movement throughout the cell.Not all of the the larger groups grow into regions of large displacements at longer timescales. We show the corresponding spin configuration, with up spins represented as black squares. In Fig. 1 we show the MSD for each particle averaged over 500 independent trajectories started from the same particle configuration in a N = 1600 particle system. Each trajectory is evolved for a time τe, where τe is the time that it takes for the self-intermediate scattering function for the small particles Fs(k, t) = i 〈sin (k |ri(t)− ri(0)|) / (k |ri(t)− ri(0)|)〉 (note that the form we use has already been averaged over angle) to decay by a factor of 1/e – this is roughly tα. The k value chosen is that of the first peak of the static structure factor. It is clear that there are regions of much higher MSD than the average, and that these regions are reasonably widely spaced. The important question for mapping the dynamics to a KCM is how such regions influence the behavior of their neighbors. The MSD in the IC ensemble simulations can be seen as a measurement of the propensity for a particle to move based on the initial configuration. As noted by Widmer- Cooper and Harrowell, each trajectory within the ensem- ble does not reproduce the same dynamics [29]. The final propensity is therefore the composite of a set of trajec- tories that is determined solely by the initial particle po- sitions. We can follow the change of the propensity in time by following a single trajectory and performing IC simulations separated by a time τs (which we mostly take to be ∼ τe). The KCM that we determine is one that is obtained from an IC average over 500 trajectories. III. COARSE GRAINING PROCEDURE There are two parts to our coarse-graining procedure. First, we identify the spins that enter in the KCM (using the results of the MD simulations). Second we infer the dynamics of these spins. Specificly, we construct a model which has a Hamiltonian si, (1) where si is a “spin” variable on a site i ∈ [1, NS] (where NS is the number of sites in the spin model) for which up (si = 1) corresponds to an active region and down (si = −1) corresponds to an inactive region, and J is some (yet to be determined) energy scale. The first part of the coarse-graining procedure is to find a way to deter- mine the separation of regions into up and down spins. In general one might also consider terms in the Hamil- tonian related to spin-spin interactions [24], but in their simplest forms, KCMs are usually taken to have the form in Eq. (1). This implies that at high temperatures there are no static correlations, as one expects in a liquid. The model Eq. (1) has no interactions and any glassy phe- nomenology must come from the dynamical rules that govern how spins flip. These dynamical rules are usually stated in the form that the probability of a spin flipping is dependent on the state of its neighbors [20]. To be more precise, we can note that Glauber rates for flipping a spin i are given by [19, 24]: wi(s) = fi(s) n↓, si = 1 n↑, si = −1 which respects detailed balance, and the concentration of up spins (with n↑ + n↓ = 1) is 1 + eJ/T , (3) and s = (s1, . . . , sNS ). We determine the function fi(s) assuming that it has the form f(m), where m is the num- ber of up spins on sites neighboring site i, similarly to the formulation of the FA model [20]. We analyze the data from the MD simulations to determine f(m). It is desir- able that the results be relatively insensitive to the pa- rameters entering the coarse-graining procedure, which is what we find. Our coarse-graining procedure to determine spins and sites is as follows: we perform a set of simulations to give n ≃ 100 timesteps in the IC ensemble. The timesteps are chosen to be tβ , 0.6τe, and τe to check the coarse-graining in time. We find that the fitting form τe = works well over the entire temperature range we consider (T = 0.36 to T = 0.96), although for T ≤ 0.48 the form τe = 0.025e (1.1T ) works equally well. We take each of the snapshots of IC averaged particle configurations and coarse-grain in space, by dividing the sample into (l/L)2 boxes lying on a square lattice [30], so that each particle is assigned to a box. We take l to be small and fixed to l = 2 most of the time (this appears to be roughly the length-scale of the cage-breaking process, and also of the order of the dynamic correlation length, ξ [28]) and assume l to be temperature and time independent. Since we have at most 1600 particles in our system, when we go to large coarse graining lengths (l ≥ 8), we start to get close to the system size and finite size effects are important, i.e. l/L & 0.2. We associate a spin with each box, either up or down depending on whether the MSD per particle in the box is larger or smaller than some cutoff. We adjust the cut- off so that n↑ takes its equilibrium value, Eq. (3). This leaves the freedom to choose the energy scale J . A seem- ingly natural energy/temperature scale associated with glassy dynamics appears to be that where the relaxation time for the small particles starts to stretch more quickly and there is a marked onset of dynamic heterogeneity; roughly T ∼ 0.5. This is also the temperature where the scaling between diffusion and relaxation times changes [28], motivating us to choose J = 0.5 (in the same units as T ). However, we check that our results are robust un- der varying this choice (see Fig. 2). In general we find that if J & 0.3 the probabilities of spin flips that we identify are identical. Now, one of the assumptions that underlies writing down Eq. (1) is that there are no static correlations. Given that the spins we consider are themselves defined through dynamics, albeit within a single coarse-graining time, it is difficult to be certain that one can extract truly static correlations. Nevertheless, we checked for static correlations in the spin model defined above and find that at high temperatures there are no static corre- lations, whereas for temperatures below about T = 0.45, there are “static” correlations on a lengthscale of up to two lattice spacings (at T = 0.36), that appears to be growing with decreasing temperature. This is in accord with the expectation that as the spins diffuse, they lead to local relaxations that appear as a static correlation in a single snapshot, but can be resolved, for example, by the dynamic four-point susceptibility that has been discussed extensively in the literature [31, 32, 33, 34, 35, 36, 37, 38]. It is also possible that extra terms involving spin-spin in- teractions should perhaps be included in the model as these can also lead to static correlations, but the rela- tively short correlation length, and the existence of dy- namic correlations as discussed above suggests that we can ignore these interactions as a first approximation. We shall proceed under this assumption of non-interacting spins and discuss some consequences of interactions that appear to give small corrections to our non-interacting results. 0 0.5 1 1.5 2 FIG. 2: P (m) at m = 0, 1, 2, 3, and 4, for spin flips down to up as J is varied, with a coarse-graining timescale of τe at T = 0.36 with l = 2. In Figs. 2 and 3 we show some of our checks on the coarse-graining procedure. All of these are at T = 0.36. In particular, in Fig. 2 we show how P (m), the prob- ability of a spin flip between consecutive time steps changes with variations in J at fixed l = 2 and T for spin flips from down to up. [At a fixed temperature, P (m) ∝ f(m).] In Fig. 3 we show how P (m) changes with variations in l at fixed J and T for spin flips with up to down. Comparable results are found for the spin flips not shown. Statistical error bars are comparable to the size of the symbols. We have thus defined our spins. Now we must under- stand their dynamics. To do this, we ask the question, for an up or down spin with m nearest neighbors that are up spins, what is the probability that it will flip in a given time-step? This is the way that the classic FA model is posed. We display the function f(m) as a func- tion of temperature in Figs. 4 and 5 for coarse graining 0 2 4 6 8 10 FIG. 3: P (m) as a function of m for spin flips up to down as l is varied, with a coarse-graining timescale of τe at T = 0.36 with J = 0.5. times of tβ and τe respectively. In Figs. 4a) and 5a) we consider m = 0, 1, and 2, whilst for clarity, in Figs. 4b) and 5b) we show m = 3 and m = 4. The behaviour between these two coarse- graining times is quite distinct. For a coarse-graining time of tβ there is no strong tendency towards kineti- cally constrained dynamics, whereas for a coarse graining time of τe there are quite strong indications. For a coarse graining time of τe, at low temperatures (T . 0.5), f(0), f(1) and f(2) are distinctly different, and there is qual- itative agreement between f(m) determined from either up to down spin flips or down to up spin flips. Similar results are seen for f(3) and f(4). Detailed balance implies that f(m) determined from either type of spin flip should be the same if the system is described precisely by a model of the type in Eq. (1). In Figs. 4 and 5 there are small but clear differences be- tween f(m) determined from the two types of spin flips. We believe that there are two sources for this discrepancy. Probably most important is the presence of spin-spin in- teractions which are not accounted for in Eq. (1). Such interactions mean that f(m) does not depend solely on m, although at low temperatures it is clear that m is the most important variable controlling the behaviour of f(m). Secondly, the magnitude of the discrepancy between rates varies from temperature to temperature point. This is likely to be from biases that are forced on us by computational restrictions. In principle, one would like to equilibrate a large number of independent parti- cle configurations, then perform an IC average for each initial condition to determine f(m). In practice, it is computationally expensive to equilibrate at low temper- atures, so we equilibrate one particle configuration and then use this to generate subsequent particle configura- tions from one member of the IC ensemble. This intro- duces a sampling bias that is likely to contribute to the discrepancy between the rates. We note that even for a single trajectory, there are signatures of kinetically con- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f(0) ud f(0) du f(1) ud f(1) du f(2) ud f(2) du 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f(3) ud f(3) du f(4) ud f(4) du FIG. 4: f(m) as a function of temperature from both up to down and down to up spin flips a) m = 0, 1, 2; b) m = 3, 4 with coarse graining time tβ strained dynamics, but cleaner results are obtained from our IC averaging, and we expect averages over more in- dependent particle configurations to yield more precise results. Despite these caveats, Fig. 5 clearly illustrates that the coarse-graining procedure we have devised gives strong evidence of kinetically constrained dynamics, and appears to give a mapping of a microscopic simulation to a KCM. The signatures of kinetically constrained behavior only start to show up in the coarse grained spin model for T . 0.5. which is around the temperature at which the relaxation time for small particles increases quickly with decreasing temperature, and is also around where the Stokes-Einstein relation breaks down. An important point to note about KCMs is that they are expected to give the best account of glassy dynamics when n↑ ≪ 1. In the temperature range we consider, this is only approx- imately true, for instance, at T = 0.36, n↑ ≃ 0.2 (with J = 0.5, although for larger J , n↑ is considerably smaller, and the f(m) we have determined are unchanged). The requirement of a vanishing number of up spins suggests that to extract a specific KCM from our data, one should consider the limit that T → 0. Figure 5 certainly suggests that as T → 0, f(0) → 0, but it is harder to determine 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f(0) ud f(0) du f(1) ud f(1) du f(2) ud f(2) du 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f(3) ud f(3) du f(4) ud f(4) du FIG. 5: f(m) as a function of temperature from both up to down and down to up spin flips a) m = 0, 1, 2; b) m = 3, 4 with coarse graining time τe the fate of f(m) for m 6= 0. However, our results suggest a KCM that might apply in the limit T → 0 defined by (with α constant) f(m) = 0, m = 0 α, m ≥ 1. The most important feature of the model that is realised, whether it is exactly as in Eq. (4) or not, is that it is of the one-spin facilitated type [19]. It appears unlikely from Fig. 5 that f(1) → 0 as T → 0 as would be required for a two-spin facilitated KCM. We have verified with Monte Carlo simulations that this model gives timescales that diverge in an Arrhenius fashion at low temperatures. However, the fact that the timestep in the KCM is also strongly temperature dependent (through τe) leads to a fragile behaviour of timescales when expressed in terms of the time units of the MD simulations, i.e. τ ∼ eA/T for some constant A. IV. DISCUSSION There are several goals in trying to map MD simula- tions of a glass former onto a KCM. The first is making a connection between microscopic particle motions and some effective theory of glassy dynamics. A second is to determine the long-time dynamics of a given glass for- mer at very low temperatures where MD simulations are ineffective, say through Monte Carlo simulations of the KCM, or in some cases, analytic calculations. The mapping of MD simulations to a KCM that we have achieved is not what one might naively expect. From the discussions in the literature [20, 23], the expec- tation would be that for coarse-graining on some small lengthscale (as we do), and on a timescale much less than tα or τe, one obtains a KCM which has within it the physics of the alpha relaxation time. Our attempts along these lines are shown in Fig. 4, which clearly in- dicates that this expected behaviour does not hold. It is only when one coarse-grains on timescales of order tα that we start to see effective kinetic constraints emerg- ing. In order to get fragile glass like behaviour, as seen in the MD simulations, from the constant coarse-graining time scenario, one would require the KCM obtained from the coarse-graining procedure to be multi-spin facilitated, since single-spin facilitated models of the type found in Eq. (4) are known to have activated dynamics [19]. This suggests that there may be alternative coarse graining schemes that capture a connection between MD simula- tions and KCM. Nevertheless, to the best of our knowl- edge, we have demonstrated the first effective mapping of MD simulations onto a KCM. The “spins” of our model correspond to regions of high or low MSD per particle and hence are in the spirit of the “mobility field”[23]. Given that the mapping we exhibit does not allow us to obtain the fragile glass behaviour of the original model from our KCM, it is interesting to ask what physics the spins that we map to are sensitive to. We believe they are sensitive to slow structural relaxation that proceeds on time scales even longer than tα. This appears to be consistent with recent work by Szamel and Flenner [39] that suggests that continuing relaxations beyond tα lead to the timescale for the onset of Fickian diffusion to be an order of magnitude longer than tα and grow faster than tα at low temperatures. The same authors also proposed a non-gaussian parameter which differs from that con- ventionally used in studies of glass formers [40]. This non-gaussian parameter has a peak at later times than the conventional one, at timescales of order tα, where it appears that heterogeneity in the distribution of parti- cle displacements is maximal. This might be related to why a “spin” definition based on mean square displace- ment, as we use here, is most sensitive to physics on the timescale of tα. This suggests that in order to make a mapping to a KCM more in line with naive expecta- tions, one should use quantities that have maximum con- trast on timescales which are considerably shorter than tα. Such an approach may be a viable way to construct effective theories of glassy dynamics in this and in other systems. We have demonstrated a particular numerical coarse-graining procedure, and we hope that our results may help point the way to analytic approaches to con- nect microscopic models of glasses to KCMs, and further insight into the glass problem. V. ACKNOWLEDGEMENTS Calculations were performed on Westgrid. 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0704.1498

DIRBE Minus 2MASS: Confirming the CIRB in 40 New Regions at 2.2 and 3.5 Microns

DIRBE Minus 2MASS: Confirming the CIRB in 40 New Regions at 2.2 and 3.5 Microns L.R. Levenson & E.L. Wright Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1562 [email protected] B.D. Johnson Dept. of Astronomy, Columbia University, 550 W. 120th St., New York, NY 10027 ABSTRACT With the release of the 2MASS All-Sky Point Source Catalog, stellar fluxes from 2MASS are used to remove the contribution due to Galactic stars from the intensity measured by DIRBE in 40 new regions in the North and South Galactic polar caps. After subtracting the interplanetary and Galactic foregrounds, a consistent residual intensity of 14.69 ± 4.49 kJy sr−1 at 2.2 µm is found. Allowing for a constant calibration factor between the DIRBE 3.5 µm and the 2MASS 2.2 µm fluxes, a similar analysis leaves a residual intensity of 15.62 ± 3.34 kJy sr−1 at 3.5 µm. The intercepts of the DIRBE minus 2MASS correlation at 1.25 µm show more scatter and are a smaller fraction of the foreground, leading to a still weak limit on the CIRB of 8.88 ± 6.26 kJy sr−1 (1 σ). Subject headings: cosmology: observations — diffuse radiation —infrared: gen- 1. Introduction The Cosmic InfraRed Background (CIRB) is the aggregate of the short wavelength radiation from the era of structure formation following the decoupling of matter and radiation in the early universe, redshifted to near infrared wavelengths by cosmological redshifting and far infrared wavelengths by dust-reprocessing, i.e. absorption and re-emission of starlight by intervening dust. Particle decay models also allow for a contribution to the CIRB from photons produced in the decay of weakly interacting massive particles such as big bang http://arxiv.org/abs/0704.1498v3 – 2 – relic neutrinos (Bond et al. 1986). The Diffuse InfraRed Background Experiment (DIRBE) on the COsmic Background Explorer (COBE, see Boggess et al. (1992)), which observed the entire sky in 10 infrared wavelengths from 1.25 to 240 µm, was primarily intended to measure the CIRB. A determination of the CIRB turned out to be exceedingly difficult due to bright foreground contamination including Galactic starlight and the strong thermal emission and reflection of sunlight by dust in the solar neighborhood in many of the DIRBE bands, especially in the near infrared (NIR). The DIRBE team estimated this zodiacal contribution by fitting a computed brightness, based on a physical model of the InterPlanetary Dust (IPD) cloud, to the weekly variations in the DIRBE brightness due to COBE’s changing position in the solar system as the Earth orbited the Sun (Kelsall et al. 1998). The modeled brightness was computed by integrating a source function, multiplied by a three dimensional density distribution function, along the line of sight. Using this model for zodiacal light subtraction, the DIRBE team succeeded in measuring an isotropic CIRB of 25 ± 7 nW m−2 sr−1 (1170 ± 328 kJy sr−1) and 14 ± 3 nW m−2 sr−1 (1123 ± 241 kJy sr−1) at 140 and 240 µm respectively (Hauser et al. 1998). In the NIR bands, they reported only upper limits on the CIRB. Using the “very strong no-zodi principle” of Wright (1997), another model of the zodiacal light was determined and is described in Wright (1998) and Gorjian et al. (2000). After creating DIRBE all sky maps with this new model subtracted, the problem of subtracting Galactic starlight in the NIR was addressed by Gorjian et al. (2000) at 2.2 and 3.5 µm by directly observing Galactic stars in a 2◦ × 2◦ region of the sky, smoothing the observed intensities to the DIRBE pixel size and subtracting the resulting intensities from DIRBE maps with the Wright (1998) zodiacal light model subtracted. Wright (2001) used this same procedure of subtracting Galactic starlight from the DIRBE maps using stellar fluxes from the 2MASS 2nd incremental release Point Source Catalog (PSC) in four regions in the North and South Galactic poles. Dwek & Arendt (1998) showed that the DIRBE 3.5 µm intensity was strongly correlated with the DIRBE 2.2 µm intensity at high Galactic latitudes and used this correlation to convert a lower limit on the 2.2 µm CIRB, based on galaxy counts, into a lower limit on the 3.5 µm CIRB. Wright & Johnson (2001) extended the Wright (2001) analysis to 13 regions at 2.2 µm and combined the Wright (2001) and Dwek & Arendt (1998) techniques to obtain estimates of the CIRB for those 13 regions at 3.5 µm. In this work we extend the Wright & Johnson (2001) analysis to 40 new regions to further constrain the CIRB at 1.25, 2.2 and 3.5 µm. – 3 – 2. Data Sets The two main datasets in this paper are the zodi-subtacted DIRBE maps and the 2MASS All-Sky Point Source Catalog (PSC). DIRBE has a large 0.7◦ × 0.7◦ square beam with a diagonal of 1◦. Pixel intensities in the DIRBE maps are averages of all observations made while the beam was centered in a given pixel in the COBE Quadrilateralized Spherical Cube (CSC) projection. Due to the large beam size, bright stars outside of a particular pixel will, depending on the exact center position and position angle of the beam, occasionally affect the observed brightness in that pixel. Thus a thick buffer ring is needed around any studied field to keep bright stars out- side the field from influencing the measured DIRBE intensity. The resulting inefficiency was minimized by using the darkest possible 2◦ circular regions which have the largest possible area:perimeter ratio. A list of dark spots was created by smoothing the 3.5 µm DIRBE Zo- diacal Subtracted Mission Average (ZSMA) map to 1/64th it’s original resolution, reducing the 393,216 0.32◦ × 0.32◦ pixels to 6144 pixels approximately 2.5◦ × 2.5◦ using a straight average of nearest neighbor pixels and sorting the resulting low resolution map. The 40 darkest regions not used in previous analyses were used. However, the DIRBE ZSMA project data set only uses a fraction of the DIRBE data since extreme solar elongations were dropped. Therefore, a set of mission averaged zodiacal subtracted DIRBE maps were created by Wright (2001) using the zodiacal light model of Wright (1998). The physical model is similar to the Kelsall et al. (1998) model, with an added constraint. The “very strong no-zodi principle,” described in Wright (1997) and Gorjian et al. (2000), requires that at high Galactic latitudes, at 25 µm, the background emission should be isotropic and adds a single pseudo-observation of zero emission to the nearly 105 observations used to fit the time variation in the weekly maps to the model brightness. At 1.25 and 2.2 µm, no correction for interstellar dust emission is needed, while at 3.5 µm, there is a very small correction (Arendt et al. 1998). The ith pixel in these maps provides the DIRBE data DZi. Using the NASA/IPAC InfraRed Science Archive (IRSA), fluxes from all stars in each of the 40 regions brighter than K = 14 in the all-sky release of the 2MASS PSC were obtained. Again due to the large beam size and effectively random distribution of the beam center and position angle, these fluxes were converted into intensities by smearing with a 0.7◦ × 0.7◦ square beam with a center uniformly distributed in the DIRBE pixel and orientation uniformly distributed in position angle, to obtain the cataloged star contribution, Bi, to the DIRBE intensity. This smearing process is described fully in Section 3. For the J-band contribution, only stars with J and K less than 14 were used. This dual wavelength magnitude selection is essentially equivalent to a simple J < 14 selection (Wright 2001). – 4 – 3. Analysis The zodi-subtracted DIRBE intensity in the ith pixel, DZi, should be the sum of the cataloged stars, Bi; the faint stars, Fq, assumed constant for all pixels in the q th region; and the CIRB, C ,which is isotropic, i.e. DZi = Bi + Fq + C . (1) The cataloged star contribution was computed using the “smearing” described in Gorjian et al. (2000) and Wright (2001), pijSj , (2) where Ωb is the solid angle of the DIRBE beam, Sj is the flux of the j th star and pij is the probability of the jth star affecting the ith pixel under the assumptions that over the many observations in which the DIRBE beam was centered in the ith pixel, the beam center was uniformly distributed within the pixel and the beam orientation is uniformly distributed in position angle. Due to the random distribution of the orientation of the square 0.7◦ × 0.7◦ beam, the probablility as a function of the angular distance, r[(α, δ), (α⋆, δ⋆)], between the beam center (α, δ) and a particular star at (α⋆, δ⋆) is P (r) = 0, for r > l√ arccos l , for l ≤ r ≤ l√ 1, for r < l where l is the width of the beam (0.7 deg) and, r = arccos[cos(α− α⋆) cos(δ) cos(δ⋆) + sin(δ) sin(δ⋆)] . (4) Then, to account for the random position of the beam center within the pixel, this probability must be averaged over the area of the pixel by integrating P(r) over the solid angle of the pixel and dividing by the pixel solid angle, Ωi, so that pij = P (r)dΩi . (5) Uncertainties in Bi were calculated as in Wright (2001): σ2(Bi) = [pij(1− pij) + pij 2(0.001 + (0.4 ln 10)2σ2(mj))]Sj 2 . (6) The first term is noise due to stars near the edge of the DIRBE beam which will“flicker” in and out of the beam when observed with various centers and position angles. The second – 5 – term is the flux error with an added allowance for variation in the fluxes between the DIRBE and 2MASS observations as in Wright (2001), with the modification that the allowance for variable stars was reduced from 0.1 to 0.001, reducing this allowance from σ = 0.34 to σ = 0.03 magnitudes. Statistically, the σ’s computed with the Wright (2001) value of 0.1 were too large to be justifiable. Upon dividing the residuals from the linear fit to the DIRBE vs. 2MASS intensities (described below) by the computed σ’s, we noticed that the residual/σ at all pixels was less than one. The new value of 0.001 gives a statistically more reasonable distribution of residual/σ values which is described at the end of this Section. The error estimates for all stars brighter than K = 5.5 or J = 6.5 were set to ± 1 mag to effectively remove pixels affected by confusion due to saturation from the final analysis. Stars with reported null uncertainties were assigned an uncertainty of ± 0.5 mag. Since the error in the DIRBE data is negligible (Hauser et al. 1998), all of the error comes from the calculation of Bi and is ascribed to DZi for the fits. Figures 1, 2 and 3 show plots of DZi vs. Bi for all pixels in each of the 40 regions in K, J, and L respectively where the point sizes are inversely proportional to the above σ’s. The fits in K, J and L have slopes of κK = 0.88, κJ = 0.97 and κL = 0.43 respectively with 40 independent intercepts in each band, derived using a weighted median procedure, i.e. finding the values of κ and DZ(0) that minimize the sum: |(DZi − κBi −DZ(0))/σi| . (7) Derived intercepts for each field are given in Tables 1, 2 and 3 for K, J, and L respectively. The contributions from stars fainter than the 14th magnitude were evaluated statistically by fitting a power series of the form nq(m) = n◦,q10 αm to counts of 2MASS stars in each of the 40 regions, binned into 3 one-magnitude bins centered on m= 11.5, 12.5 and 13.5. The fits resulted in 40 individual n◦,q and αK = 0.288 and αJ = 0.276 where any α < 0.4 results in a converging flux contribution. The intensity contribution from faint stars in the qth region with solid angle Ωq is then F◦(λ)n◦,q m=14.5 10(α−0.4)m (8) or, in the limit of infinitely fine bins, F◦(λ)n◦,q 10(α−0.4)mdm , (9) which was computed analytically. At L, the faint source contribution in each region is the 2.2 µm value multiplied by the calibration ratio of 0.491. These Fq are listed in Tables 1, – 6 – 2 and 3 for K, J and L. An uncertainty of 20% of the total prediction is assigned to this correction, and is listed in Table 5 under “Faint Source.” The CIRB in each region is then the derived intercept, DZ(0), minus the faint star contribution Fq, C = DZ(0) - F. These values are also listed in Tables 1, 2 and 3. The mean of these CIRB estimates are 14.59 ± 0.14, 8.83 ± 0.51 and 15.57 ± 0.20 kJy sr−1 for K, J and L. These standard deviations of the means are listed in Table 5 as “Scatter.” 2MASS magnitudes at K and J were converted into fluxes using F◦(K) = 614 Jy and F◦(J) = 1512 Jy which were derived by Wright (2001) and Gorjian et al. (2000). However, the derived slopes at K and J indicate DIRBE fluxes for a zero magnitude 2MASS star of 540 and 1467 Jy respectively. The ratio of the calibration factors at 3.5 and 2.2 µm is 0.491, consistent with those found by Dwek & Arendt (1998), Wright & Reese (2000) and Wright & Johnson (2001). Uncertainties in the CIRB due to calibration error were estimated using the change in the median DZ(0) when the slopes, κ, were forced to change by ±5% at J and L, or ±10% at K due to the large difference (Wright & Johnson 2001) between the fitted value of 0.88 and the expected value of 1. The change in the medians are ∓ 2.24, 1.77, and 0.60 kJy sr−1 at K, J and L respectively and are listed in Table 5 under “Calibration.” There is a small correction for faint galaxies that appear in the 2MASS PSC. These have been subtracted along with the Galactic stars, but should be included in the CIRB. Wright (2001) estimates this correction is 0.1 and 0.05 kJy sr−1 at 2.2 and 1.25 µm. The 0.1 kJy sr−1 correction at 2.2 µm implies a 0.05 kJy sr−1 correction at 3.5 µm due to the relative calibration factor of 0.491. These corrections have been added back after taking the mean of C = DZ(0) - F in the 40 regions. Thus, the final reported values of the CIRB are the mean of the values C in Tables 1, 2 and 3 plus this correction. An uncertainty of 100% of this correction is included in Table 5 under “Galaxies.” Gorjian et al. (2000) adopt an uncertainty of 5% of the zodiacal intensity at the ecliptic poles. These uncertainties are listed in Table 5 under “Zodiacal.” After adding errors in quadrature, we obtain a CIRB of 14.69 ± 4.49 kJy sr−1 at 2.2 µm, a weak limit of 8.88 ± 6.26 kJy sr−1 at 1.25 µm and a CIRB of 15.62 ± 3.34 kJy sr−1 at 3.5 µm. Figures 4, 5, and 6 show histograms of the residuals DZi − κBi − DZ(0) for all 2971 pixels in K, J and L with interquartile ranges of 2.85, 3.57 and 2.11 respectively. Dividing these residuals by σ(Bi) at each pixel gives a non-Gaussian distribution which is tightly packed near zero with a few pixels extending out to large values. The number of pixels with |(DZi − κBi − DZ(0))/σ(Bi)| less than {0.5, 1.0, 2.0, 3.0} at K, J and L are {2424, 2832, 2930, 2951}, {2588, 2873, 2955, 2955} and {2590, 2867, 2947, 2955}. For comparison, with – 7 – a Gaussian distribution, one would expect these numbers to be, {1138, 2028, 2835, 2963}; more spread out in the center, with fewer pixels in the extended tail. We have used a robust, least sum of absolute values fitting method, which, by it’s nature, is insensitive to the few pixels with large residual/σ. Thus the fit for the slopes, κ, and intercepts, DZ(0), is to the pixels in the narrow core, and the outliers have little effect on their derived values and the final CIRB values. Figures 7, 8, and 9 show the derived intercepts, DZ(0), vs. ecliptic latitude in the three bands. We see here the same trends with ecliptic latitude as were apparent in Wright & Johnson (2001). While the K-band intercepts appear reasonably independent of ecliptic latitude, there is a strong trend in J and a slight negative trend in the L-band. The zodiacal light is fainter at 3.5 µm than at 2.2 µm, and so the stronger dependence on β at L than at K may seem surprising. However, at 3.5 µm, we begin to see thermal emis- sion from the interplanetary dust, in addition to the scattered sunlight. From a modeling standpoint, this gives another free parameter which should provide a better fit, but from a physical standpoint, we are likely seeing a more complicated emission/scattering pattern on the sky which is more difficult to model correctly. There is an overall scaling factor at each wavelength in the zodiacal light model, but the parameters that determine the physical shape of the dust cloud were fit simultaneously to observations at 8 DIRBE bands. The trend with ecliptic latitude at both J and L indicates a problem with the modeled shape of the cloud. The better fit to the scattered sunlight at K could be a coincidence, rather than better modeling in that band. This remaining solar system dependence in two of the three analyzed bands is evidence of a problem with the zodiacal light model that still prevents us from claiming a detection at 1.25 µm. The model may be improved by requiring that these DIRBE minus 2MASS intercepts be ecliptic independent simultaneously in all three bands. Improvements to the zodiacal light model will be addressed in future work. 4. Discussion The results of this DIRBE minus 2MASS subtraction in these 40 regions of the sky give a statistically significant isotropic background of 14.69 ± 4.49 kJy sr−1 at 2.2 µm and 15.62 ± 3.34 kJy sr−1 at 3.5 µm where the uncertainty has not been significantly reduced since the dominant sources of error are systematic. These results are consistent with earlier results, summarized in Table 4 , including Gorjian et al. (2000), Wright & Reese (2000), Wright (2001) and with the 13 similarly analyzed regions from Wright & Johnson (2001) all of which used the same zodiacal light model considered here (Wright 1998). In the J-band, we have weak limit on the 1.25 µm CIRB of 8.88 ± 6.26 kJy sr−1. This – 8 – is also consistent with the 1.25 µm values in Table 4, which were reported in Wright (2001) and Wright & Johnson (2001), also using the same zodiacal light model used here. The Kelsall et al. (1998) zodiacal light model gives a zodiacal intensity at the ecliptic pole 3.9, 9.2 and 4.0 kJy sr−1 lower at K, J and L. Allowing for this difference in the models, these results are also consistent with those reported in Cambresy et al. (2001) and Matsumoto (2000) at 1.25 and 2.2 µm and Matsumoto et al. (2005) at 2.2 and 3.5 µm, also shown in Table 4. The cumulative light from galaxies is a strict lower limit on the CIRB. A determination of the total contribution of resolved galaxies to the CIRB was determined via galaxy number counts by Fazio et al. (2004). Using the InfraRed Array Camera (IRAC) on the Spitzer Space Telescope (Eisenhardt et al. 2004), surveys were done of the Boötes region, the Extended Groth Strip and a deep image surrounding the QSO HS 1700+6416. After integrating the light from galaxies from the 10th to the 21st magnitudes, a total integrated intensity of 5.4 nW m−2 sr−1 (6.5 kJy sr−1) at 3.6 µm is reported, which is less than half of the CIRB determined at that wavelength by this and other similar studies. This 3.6 µm value is the final entry in Table 4. This discrepancy may be partially resolved by improving the photometry of these and other Spitzer surveys. But, the dominant source of error in these DIRBE minus 2MASS measurements comes from the estimation of the zodiacal light, and based on the strong ecliptic dependence of our J-band results, it will likely take an improvement of the zodiacal light models to make significant progress in determining whether we can resolve this discrepancy with improvements in data analysis, or if we truly require some exotic diffuse source as suggested by Cambresy et al. (2001). It is also interesting to note another indirect constraint on the CIRB from the attenua- tion of TeV γ-rays by e+e− pair production through collisions with CIRB photons. A recent attempt by Mapelli et al. (2004) to fit the TeV spectrum of the blazar H1426+428 found that the spectrum can not be fit using only the integrated light from galaxies. Their best fit uses the Wright (2001) determination of the CIRB favoring the higher values determined from direct observations of the total sky brightness. However, H.E.S.S. observations of the blazars H2356-309 and 1ES 1101-232 by Aharonian et al. (2006) suggest that the CIRB can not be much higher than the lower limits from galaxy counts. The Wright (2001) 1.25 and 2.2 µm values and the Dwek & Arendt (1998) 3.5 µm value were used by Aharonian et al. (2006) in fitting a model of the Extragalactic Background Light (EBL) needed for estimating γ-ray attenuation, resulting in their P1.0 EBL model which is the top curve in Figure 10. This model gives (11,18,16) kJy sr−1 at 1.25, 2.2 and 3.5 µm and assuming a power law blazar spectrum (dN/dE [photons cm−2 s−1 TeV−1] ∝ E−Γ) results in a power law index for the spectrum of the blazar 1ES 1101-232 of Γ = -0.1. The model must be scaled down by a factor of 0.45 to the P0.45 model (lower curve in Figure 10) to give a power law index of at – 9 – least 1.5, which is considered by Aharonian et al. (2006) to be the lowest acceptable value. Mapelli et al. (2006) show however that an EBL model, also based on the Wright (2001) values at 1.25 and 2.2 µm, but with a steeper decline from 4 to 10 µm results in a power law index of Γ = +0.5. They consider Γ = 0.6 to be the lowest acceptable index based on physical considerations and suggest then that while the DIRBE minus 2MASS CIRB values at 1.25 and 2.2 µm from Wright (2001) do require a hard spectrum, and the lower limits from galaxy counts are favored, they, and the slightly lower values reported here ((9,15,16) kJy sr−1), can not be ruled out based on the current H.E.S.S. data. While the implications of γ-ray attenuation for CIRB measurement are still limited by the small number of observed sources, this independent limit on the CIRB will only improve as more blazars are observed by H.E.S.S. and eventually, VERITAS. There is still a substantial difference between the Fazio et al. (2004) lower limits from galaxy counts and the intensities determined here. Progress can be made in resolving this discrepancy with improvements in the photometry of survey data from Spitzer, as well as improvements in the zodiacal light models and further data on gamma-ray attenuation. However, the dominant source of error in directly measuring the CIRB, the model based subtraction of the zodiacal light, will not be significantly reduced with currently available data. A directly measured map of the zodiacal light, which would have to be observed from outside the bulk of the IPD cloud, beyond about 3AU, would allow accurate removal of the zodiacal light from the DIRBE maps and thus an accurate, direct measurement of the Cosmic Infrared Background. We have shown here that the subtraction of catalogued stars from low resolution maps works well, thus an instrument with a field of view of a few square degrees and a resolution of a few arcminutes would suffice. The main requirements would be sensitivity to extremely low surface brightnesses, down to less than 1 nW m−2 sr−1 for good signal to noise, and an accurate, absolute flux calibration. Such an instrument could be a camera on a probe to one of the outer planets. It would also be useful to have observations from different positions with respect to the IPD cloud. This could be accomplished either by observing the same fields at widely different solar elongations during a long lived mission as the craft orbits the sun, or by observing during the cruise from 1 to 3 AU as the dust density decreases. While we will continue to improve our understanding of the CIRB in the mean time, a space mission of this type will ultimately be required. The COBE datasets were developed by the NASA Goddard Space Flight Center under the direction of the COBE Science Working Group and were provided by the NSSDC. This publication makes use of the data product from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, funded by the National Aeronautics and Space Administration and the National – 10 – Science Foundation. This research has also made use of the NASA/ IPAC Infrared Sci- ence Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration REFERENCES Aharonian, A. G. et al. 2006, Nature 440 1018-1021 Arendt, R. et al. 1998, ApJ, 507, 74 Boggess, N. W. et al. 1992, ApJ, 397, 420 Bond, J., Carr, B. & Hogan, C. 1986, ApJ, 306, 428 Cambresy, L., Reach, W., Beichman, C. & Jarrett, T. 2001, ApJ, 555, 563 Dwek, E. & Arendt, R. 1998, ApJ, 508, L9 Eisenhardt, P., et al. 2004, ApJS, 154, 48E Fazio, G., et al. 2004, ApJS, 154, 39F Gorjian, V., Wright, E., & Chary, R. 2000, ApJ, 536, 550 Hauser, M. et al. 1998 ApJ, 508, 25 Kelsall, T., et al. 1998, ApJ, 508, 44 Mapelli, M., Salvaterra, R. & Ferrara, A. 2004, PoS BDMH2004 012 astro-ph/0411134 Mapelli, M., Salvaterra, R. & Ferrara, A. 2006, New Astronomy 11 420-430 Matsumoto, T. 2000, The Institute of Space and Astronautical Science Report SP No. 14, p. 179 Matsumoto, T. et al. 2005, ApJ, 626, 31 Wright, E. 1997 BAAS, 29, 1354 Wright, E. 1998 ApJ, 496, 1 Wright, E. & Reese, E. 2000, ApJ, 545, 43 Wright, E. 2001, ApJ, 553, 538 http://arxiv.org/abs/astro-ph/0411134 – 11 – Wright, E. & Johnson, B. 2001, astro-ph/0107205 v2 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0107205 – 12 – Table 1. K-band Results l b β Npix DZ(0) [kJy sr −1] F [kJy sr−1] C [kJy sr−1] 204.4 67.8 20.9 79 18.83 4.11 14.72 293.9 -72.5 -46.2 74 19.71 4.55 15.16 191.5 72.1 26.6 75 19.43 3.83 15.60 227.8 75.1 19.6 75 19.37 3.70 15.67 150.5 67.6 41.0 73 19.11 3.86 15.25 193.5 76.9 27.5 75 18.23 3.68 14.55 208.7 78.3 24.7 77 20.89 3.72 17.17 146.7 69.3 41.6 75 19.09 3.88 15.21 150.6 72.6 38.9 72 19.18 3.90 15.28 164.0 79.0 33.2 77 17.72 3.73 13.99 158.3 81.1 38.2 74 20.53 3.77 16.76 127.6 53.8 57.0 76 19.03 4.93 14.10 143.2 79.4 36.5 73 18.41 3.62 14.79 166.0 86.3 31.2 71 16.98 3.67 13.31 133.5 80.8 36.7 73 18.00 3.63 14.37 317.5 76.8 19.7 74 18.24 3.96 14.28 115.4 65.7 52.4 74 19.40 4.18 15.22 355.1 81.7 27.7 75 18.29 4.07 14.22 43.5 82.9 34.1 73 18.05 3.88 14.17 115.8 56.4 60.2 76 18.99 4.74 14.25 72.2 81.9 36.9 73 18.55 3.89 14.66 15.3 81.9 30.9 77 18.15 3.85 14.30 79.3 79.1 40.4 72 17.78 3.90 13.88 347.9 76.1 24.5 74 18.85 4.10 14.75 49.7 79.4 36.5 74 18.32 4.67 13.65 111.6 57.0 60.6 75 18.97 4.61 14.36 93.3 70.0 49.5 72 16.84 4.02 12.82 103.0 57.8 61.5 73 19.89 4.80 15.09 42.2 72.6 39.0 72 18.98 4.23 14.75 56.4 68.5 44.7 75 18.28 4.25 14.03 83.8 63.3 55.3 73 18.65 4.44 14.21 109.1 46.2 70.8 73 19.66 5.75 13.91 84.7 60.4 58.2 75 18.48 4.73 13.75 74.3 62.1 54.5 76 18.76 4.52 14.24 44.6 64.0 43.2 76 19.31 4.61 14.70 69.8 61.2 53.8 76 17.79 4.82 12.97 45.1 59.8 44.4 71 20.29 5.07 15.22 252.6 -71.8 -45.9 74 18.93 4.42 14.51 255.8 -59.2 -57.6 77 20.02 5.13 14.89 254.1 -53.0 -61.4 74 20.27 5.60 14.67 Note. — Mean of the CIRB over all 40 regions (2971 pixels) at 2.2 microns is 14.59 ± 0.14 kJy sr−1 . This uncertainty is listed in Table 5 under “Scatter.” aIntercepts derived using a weighted median procedure resulting in a global slope of 0.88 and 40 separate intercepts, DZ(0). – 13 – Table 2. J-band Results l b β Npix DZ(0) b [kJy sr−1] F [kJy sr−1] C [kJy sr−1] 204.4 67.8 20.9 79 8.77 5.60 3.17 293.9 -72.5 -46.2 74 16.21 6.20 10.01 191.5 72.1 26.6 75 12.35 5.38 6.97 227.8 75.1 19.6 75 7.94 5.12 2.82 150.5 67.6 41.0 73 15.55 5.32 10.23 193.5 76.9 27.5 75 10.14 4.92 5.22 208.7 78.3 24.7 77 11.95 5.11 6.84 146.7 69.3 41.6 75 15.50 5.40 10.10 150.6 72.6 38.9 72 14.26 5.29 8.97 164.0 79.0 33.2 77 11.65 5.14 6.51 158.3 81.1 38.2 74 16.27 5.17 11.10 127.6 53.8 57.0 76 18.06 6.74 11.32 143.2 79.4 36.5 73 12.90 4.95 7.95 166.0 86.3 31.2 71 10.38 5.01 5.37 133.5 80.8 36.7 73 11.44 4.89 6.55 317.5 76.8 19.7 74 7.20 5.45 1.75 115.4 65.7 52.4 74 18.38 5.81 12.57 355.1 81.7 27.7 75 10.38 5.51 4.87 43.5 82.9 34.1 73 12.10 5.41 6.69 115.8 56.4 60.2 76 18.19 6.44 11.75 72.2 81.9 36.9 73 12.22 5.30 6.92 15.3 81.9 30.9 77 11.05 5.14 5.91 79.3 79.1 40.4 72 13.80 5.11 8.69 347.9 76.1 24.5 74 8.69 5.54 3.15 49.7 79.4 36.5 74 12.37 6.50 5.87 111.6 57.0 60.6 75 18.00 6.23 11.77 93.3 70.0 49.5 72 15.01 5.56 9.45 103.0 57.8 61.5 73 18.95 6.57 12.38 42.2 72.6 39.0 72 14.35 5.97 8.38 56.4 68.5 44.7 75 16.42 5.82 10.60 83.8 63.3 55.3 73 18.89 5.98 12.91 109.1 46.2 70.8 73 20.99 8.11 12.88 84.7 60.4 58.2 75 19.38 6.43 12.95 74.3 62.1 54.5 76 17.71 6.28 11.43 44.6 64.0 43.2 76 17.78 6.42 11.36 69.8 61.2 53.8 76 17.37 6.64 10.73 45.1 59.8 44.4 71 18.61 6.91 11.70 252.6 -71.8 -45.9 74 16.19 6.12 10.07 255.8 -59.2 -57.6 77 19.50 7.05 12.45 254.1 -53.0 -61.4 74 20.69 7.76 12.93 Note. — Mean of the CIRB over all 40 regions (2971 pixels) at 1.25 microns is 8.83 ± 0.51 kJy sr−1 . This uncertainty is listed in Table 5 under “Scatter.” bIntercepts derived using a weighted median procedure resulting in a global slope of 0.97 and 40 separate intercepts, DZ(0). – 14 – Table 3. L-band Results l b β Npix DZ(0) c [kJy sr−1] F [kJy sr−1] C [kJy sr−1] 204.4 67.8 20.9 79 19.54 2.00 17.54 293.9 -72.5 -46.2 74 18.35 2.21 16.14 191.5 72.1 26.6 75 19.50 1.86 17.64 227.8 75.1 19.6 75 20.18 1.80 18.38 150.5 67.6 41.0 73 18.07 1.88 16.19 193.5 76.9 27.5 75 18.78 1.79 16.99 208.7 78.3 24.7 77 20.18 1.81 18.37 146.7 69.3 41.6 75 17.44 1.89 15.55 150.6 72.6 38.9 72 17.77 1.90 15.87 164.0 79.0 33.2 77 18.02 1.81 16.21 158.3 81.1 38.2 74 18.37 1.84 16.53 127.6 53.8 57.0 76 16.46 2.40 14.06 143.2 79.4 36.5 73 17.70 1.76 15.94 166.0 86.3 31.2 71 17.17 1.79 15.38 133.5 80.8 36.7 73 17.60 1.77 15.83 317.5 76.8 19.7 74 18.76 1.93 16.83 115.4 65.7 52.4 74 17.30 2.03 15.27 355.1 81.7 27.7 75 19.11 1.98 17.13 43.5 82.9 34.1 73 17.74 1.89 15.85 115.8 56.4 60.2 76 16.68 2.31 14.37 72.2 81.9 36.9 73 17.46 1.90 15.56 15.3 81.9 30.9 77 17.65 1.87 15.78 79.3 79.1 40.4 72 16.60 1.90 14.70 347.9 76.1 24.5 74 18.71 1.99 16.72 49.7 79.4 36.5 74 16.85 2.27 14.58 111.6 57.0 60.6 75 16.02 2.24 13.78 93.3 70.0 49.5 72 15.86 1.96 13.90 103.0 57.8 61.5 73 16.52 2.34 14.18 42.2 72.6 39.0 72 17.72 2.06 15.66 56.4 68.5 44.7 75 16.26 2.07 14.19 83.8 63.3 55.3 73 16.54 2.16 14.38 109.1 46.2 70.8 73 16.30 2.80 13.50 84.7 60.4 58.2 75 17.07 2.30 14.77 74.3 62.1 54.5 76 16.22 2.20 14.02 44.6 64.0 43.2 76 17.36 2.24 15.12 69.8 61.2 53.8 76 16.09 2.34 13.75 45.1 59.8 44.4 71 17.57 2.47 15.10 252.6 -71.8 -45.9 74 18.90 2.15 16.75 255.8 -59.2 -57.6 77 17.91 2.50 15.41 254.1 -53.0 -61.4 74 17.76 2.73 15.03 Note. — Mean of the CIRB over all 40 regions (2971 pixels) at 3.5 microns is 15.57 ± 0.20 kJy sr−1 . This standard deviation of the mean is listed in Table 5 under “Scatter.” cIntercepts derived using a weighted median procedure resulting in a global slope of 0.43 and 40 separate intercepts, DZ(0). – 15 – Table 4. Previous determinations of the CIRB [kJy sr−1] Authors 1.25 µm 2.2 µm 3.5 µm Model Gorjian et al. (2000) · · · 16.4 ± 4.4 12.8 ± 3.8 Wright (1998) Wright & Reese (2000) · · · 16.9 ±4.4 14.4 ±3.7 Wright (1998) Wright (2001) 12 ±7 14.8 ± 4.6 · · · Wright (1998) Wright & Johnson (2001) 10.1 ± 7.4 17.6 ± 4.4 16.1 ± 4 Wright (1998) This Work 8.9 ±6.3 14.7 ± 4.5 15.6 ± 3.3 Wright (1998) Matsumoto (2000) 25 ± 6.3 20.5 ± 3.7 · · · Kelsall et al. (1998) Cambresy et al. (2001) 22.9 ± 7.0 20.4 ± 4.9 · · · Kelsall et al. (1998) Matsumoto et al. (2005) · · · 22.3 ± 4 16.9 ± 3.5 Kelsall et al. (1998) Fazio et al. (2004) · · · · · · > 6.5 N.A. (galaxy counts) Note. — The Kelsall et al. (1998) model gives a CIRB 9.2, 3.9 & 4.0 kJy sr−1 higher at 1.25, 2.2 and 3.5 µm. – 16 – Table 5. Error Budget for the CIRB Component 1.25 µm 2.2 µm 3.5 µm Scatter 0.51 0.14 0.20 Faint Source 1.17 0.85 0.42 Galaxies 0.05 0.10 0.05 Calibration 1.77 2.24 0.60 Zodiacal 5.87 3.79 3.25 Quadrature Sum 6.26 4.49 3.34 – 17 – 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 Intensity at K from 2MASS stars [kJy/sr] Fig. 1.— K-band: DIRBE 2.2 µm intensities vs. 2MASS 2.2 µm stellar intensities for the forty new regions. Fitted lines show a weighted median fit, resulting in a common slope of 0.88 and 40 different intercepts, to the data points. Intercepts along with region l,b and β are listed in Table 1. Reading left to right and top to bottom, the panels are in the same order as Table 1. Point sizes are inversely proportional to σi. – 18 – 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 Intensity at J from 2MASS stars [kJy/sr] Fig. 2.— J-band: DIRBE 1.25 µm Intensities vs. 2MASS 1.25 µm stellar intensities for the forty new regions. Fitted lines show a weighted median fit, resulting in a common slope of 0.97 and 40 different intercepts, to the data points. Intercepts along with region l,b and β are listed in Table 2. Reading left to right and top to bottom, the panels are in the same order as Table 2. Point sizes are inversely proportional to σi. – 19 – 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 50 Intensity at K from 2MASS stars [kJy/sr] Fig. 3.— DIRBE L vs. 2MASS K: DIRBE 3.5 µm Intensities vs. 2MASS 2.2 µm stellar intensities for the forty new regions. Fitted lines show a weighted median fit, resulting in a common slope of 0.43 and 40 different intercepts, to the data points. Intercepts along with region l,b and β are listed in Table 3. Reading left to right and top to bottom, the panels are in the same order as Table 3. Point sizes are inversely proportional to σi. – 20 – Fig. 4.— Histogram of DZi − κBi −DZ(0) at 2.2 µm for all regions combined where κ = 0.88. – 21 – Fig. 5.— Histogram of DZi − κBi −DZ(0) at 1.25 µm for all regions combined where κ = 0.97. – 22 – Fig. 6.— Histogram of DZi − κBi −DZ(0) at 3.5 µm for all regions combined where κ = 0.43. – 23 – Fig. 7.— Derived intercepts, DZ(0), versus Ecliptic Latitude at 2.2 µm. – 24 – Fig. 8.— Derived intercepts, DZ(0), versus Ecliptic Latitude at 1.25 µm. – 25 – Fig. 9.— Derived intercepts, DZ(0), versus Ecliptic Latitude at 3.5 µm. – 26 – Fig. 10.— Filled black squares are the CIRB values reported here. Gray stars are Matsumoto et al. (2005) values from IRTS observations. The open gray triangle is the Fazio et al. (2004) lower limit at 3.6µm. For comparison, the upper and lower black curves are the P1.0 and P0.45 models used by Aharonian et al. (2006) to estimate the attenuation of TeV γ-rays by the CIRB. P1.0 was normalized by Aharonian et al. (2006) to fit the 1.25-3.5 µm values from Dwek & Arendt (1998) and Wright (2001). P0.45 is the P1.0 scaled down by a factor of 0.45, which was required, using this shape for the CIRB, to give blazar spectra with power law spectral indices of at least 1.5. Introduction Data Sets Analysis Discussion

0704.1499

Averaging of the electron effective mass in multicomponent transparent conducting oxides

Averaging of the electron effective mass in multicomponent transparent conducting oxides Julia E. Medvedeva∗ Department of Physics, University of Missouri–Rolla, Rolla, MO 65409 (USA) We find that layered materials composed of various oxides of cations with s2 electronic config- uration, XY2O4, X=In or Sc, Y=Ga, Zn, Al, Cd and/or Mg, exhibit isotropic electron effective mass which can be obtained via averaging over those of the corresponding single-cation oxide con- stituents. This effect is due to a hybrid nature of the conduction band formed from the s-states of all cations and the oxygen p-states. Moreover, the observed insensitivity of the electron effective mass to the oxygen coordination and to the distortions in the cation-oxygen chains suggests that similar behavior can be expected in technologically important amorphous state. These findings significantly broaden the range of materials as efficient transparent conductor hosts. Transparent conducting oxides (TCO) – the vital part of optoelectronic devices – have been known for a century and employed technologically for decades [1, 2, 3, 4, 5]. Yet, the current TCO market is dominated by only three materials, In2O3, SnO2 and ZnO, and the research efforts are primarily focused on the oxides of post-transition metals with (n − 1)d10ns2 electronic configuration. De- spite excellent optical and thermal properties as well as low cost, oxides of the main group metals, such as Al2O3, SiO2, MgO and CaO, have never been considered as can- didates to achieve useful electrical conductivity due to the challenges of efficient carrier generation in these wide- bandgap materials [6, 7, 8]. Multicomponent TCO with layered structure, e.g., InGaZnO4 [9, 10, 11, 12, 13, 14], drew attention due to a possibility to separate carrier donors (traditionally, oxy- gen vacancies or aliovalent substitutional dopants) and the conducting layers where carriers are transfered with- out charge scattering on the impurities. In InGaZnO4, octahedrally coordinated In layers alternate with double layers of oxygen tetrahedrons around Ga and Zn, Fig. 1. Because octahedral oxygen coordination of cations was long believed to be essential for a good transparent con- ductor [9, 15, 16, 17, 18], it has been suggested that in InGaZnO4 the charge is transfered within the InO1.5 lay- ers while the atoms in GaZnO2.5 layers were proposed as candidates for efficient doping [9, 10, 11]. Conversely, it has been argued that InGaO3(ZnO)m is a Zn 4s conduc- tor [12]. To understand the role of local symmetry in the intrin- sic transport properties of TCO’s and to determine the functionality of structurally and chemically distinct lay- ers in InGaZnO4, we employ ab-initio density functional approach to study the electronic properties of various sin- gle and multi-cation oxides. Further, using InGaZnO4 as a test model which examplifies not only the structural but also combinatorial peculiarities of complex TCO’s, we survey other ns2 cations – beyond the traditional In, Zn and Ga – for a possibility of being effectively incor- porated into novel multicomponent TCO hosts. Isotropy of the electronic properties in InGaZnO4. FIG. 1: The unit cell of InGaZnO4 where three similar blocks consisting of one InO1.5 and two GaZnO2.5 alternate along the [0001] direction. Ga and Zn atoms are distributed randomly. Three-dimensional interatomic (background) charge density distribution is evident from the contour plot calculated in the (011) plane. The plotted charge density corresponds to the carrier concentration of ∼1×1018cm−3. The electronic band structure calculations for InGaZnO4 show that the atoms from both InO1.5 and GaZnO2.5 layers give comparable contributions to the conduction band, fig. 2, leading to a three-dimensional distribution of the charge density, fig. 1. Moreover, the isotropy of the electronic properties in this layered material mani- fests itself in the electron effective masses being nearly the same in all crystallographic directions (table I). The conduction band in InGaZnO4 consists of a set of highly dispersed parabolic bands, fig. 3(a). Since the band gap values in the corresponding single metal oxides are different, one may expect that each band is attributed to a certain cation in this multicomponent compound. However, we find that each band cannot be assigned to a state of a particular atom since all atoms in the cell, including the oxygen atoms, give non-negligible contri- butions to the conduction band wavefunction, fig. 2 and http://arxiv.org/abs/0704.1499v2 1.2 1.6 2 2.4 2.8 Energy, eV 0.005 0.01 0 0.005 0 0.005 InGaZnO4 In s−states Ga s−states Zn s−states O1 s, p−states O2 s, p−states 0.4 0.8 1.2 1.6 2 Energy, eV 0 0.005 0.01 0.005 0.01 InAlMgO4 In s−states Al s−states Mg s−states O1 s, p−states O2 s, p−states FIG. 2: Partial density of states at the bottom of the conduc- tion band for InGaZnO4 and InAlMgO4. Atoms from both In-O1 and Ga-Zn-O2 (or Al-Mg-O2) layers give non-negligible contributions. table I. The conduction band dispersion calculated along the [0001] crystallographic direction for a single unit cell, fig. 3(b), reveals that the multiple bands can be at- tributed to a “folding” of one parent band. Triple unfold- ing of the conduction band corresponds to the three-time reduction (expansion) of the conventional unit cell (Bril- louin zone) in the z direction. Since the block of three layers (one InO1.5 and two GaZnO2.5 layers, fig. 1) is repeated in the unit cell via translation, the splitting be- tween the resulting bands [for the k-vector equal to π , ..., fig. 3(c)] is negligible. Although the subsequent unfolding into three individual layers is not justified be- cause the three layers are structurally and chemically dis- similar, fig. 1, we find that the band can be unfolded again, fig. 3(c). The resulting highly dispersed band is in accord with the stepless increase of the density of states, fig. 3(d). Thus, the conduction band can be “unfolded” nine times that corresponds to the total number of layers in the unit cell. Therefore, the electronic properties of the individual layers are similar. Unconventional s2-cations at work. A two-dimensional electronic structure could be expected in InAlMgO4 and ScGaZnO4 since the band gap values in Sc2O3, Al2O3 and MgO are at least twice larger than those in In2O3, Ga2O3, CdO and ZnO and, hence, the unoccupied s- states of Sc, Al and Mg should be located deeper in the conduction band. From the analysis of the partial den- sity of states for InGaMgO4, InAlCdO4, ScGaZnO4 and InAlMgO4, we find that although the contributions to the bottom of the conduction band from Sc, Al and Mg atoms are notably reduced, these states are available for electron transport, fig. 2. Thus, similar to InGaZnO4 where the cations s-states are energetically compatible, in all multicomponent oxides considered, the conduction band wavefunction is a combination of the s-states of all cations and the p-states of the oxygen atoms. The con- tributions from the chemically distinct layers are compa- rable, table I, and, consequently, these complex oxides exhibit three-dimensional network for the electron trans- port and isotropic electron effective mass, table I. Comparison to single-cation TCO’s. The unfolded conduction band in the layered multicomponent mate- rials resembles those of single-cation TCO’s, e.g., In2O3, cf. Figs. 3 (c) and (e). Such a highly-dispersed single conduction band is the key attribute of any conventional [19] n-type TCO host [9, 17, 20, 21, 22, 23]. Upon proper doping, it provides both high mobility of extra carriers (electrons) due to their small effective mass, and low optical absorption due to high-energy inter-band tran- sitions from the valence band, Ev, and from the par- tially filled conduction band, Ec, fig. 4. Even in rela- tively small bandgap oxides, e.g., CdO where the optical band gap, Eg, is 2.3 eV, the high energy dispersion en- sures a pronounced Fermi energy displacement with dop- ing (so called Burstein-Moss shift) which helps to keep the intense transitions from the valence band out of the visible range. However, large carrier concentrations re- quired for good electrical conductivity may result in an increase of the optical absorption due to low-energy tran- sitions from the Fermi level up into the conduction band as well as plasma frequency. Application-specific optical properties and desired band offsets (work functions) can be attained in a multicomponent transparent conductor with a proper composition. In both single and multi-cation oxides, the conduc- tion band is formed from the empty s-states of the metal atoms and the oxygen antibonding p-states, e.g., fig. 2. For multicomponent oxides we find that even at the bot- tom of the conduction band, i.e., at Γ point, the contri- butions from the oxygen p-states are significant, table I. Thus, the key feature of the conduction band in a conven- tional TCO – its high energy dispersion – originates in a strong interaction between the cation s-states and the anion antibonding p-states [17]. The direct s-s overlap is insignificant, fig. 1, and therefore, the s-s interactions which were commonly assumed to play to key role in the electronic properties of TCO [12, 14, 16], do not gov- ern the transport properties in these oxides. Indeed, the small electron effective mass in s2-cation oxides is deter- mined by the strong s-p interactions [26]. From orbital symmetry considerations, one can see that the oxygen coordination of cations does not affect the s-p overlap. Instead, the largest overlap should be at- tained in materials where the oxygen atom is coordinated octahedrally by cations with the extended s-orbitals [17]. Our systematic comparison of the calculated electron ef- fective mass in the oxides of metals with s2 electronic con- figuration, fig. 4, shows that the mass slightly decreases as the ionic radii of the cations from the same group or the symmetry of the same-cation oxide increases. How- [010] Γ [100] (a) Γ [001] (b) 9k 6k 3k 0 (c) 0 5 10 H Γ P (e) −2k −k 0 k 2k β1 β ββ 1 2 22 ε ε ε01 2 n FIG. 3: Electronic band structure of single and multi-cation oxides. (a) Conduction band dispersion calculated in the ab-plane and (b) along the [0001] direction in InGaZnO4. (c) The conduction band unfolded nine times that corresponds to the number of [0001] layers in the unit cell, k=π , c is the lattice parameter. The resulting single free-electron-like band is in accord with the stepless total density of states (d), in states/eV. (e) Conduction band of In2O3 is given for comparison. (f) Tight-binding conduction band (solid line) calculated for one-dimensional atomic chain depicted above the plot. Two types of metal atoms (red and purple spheres) alternate with oxygen atoms (blue spheres) and only the nearest-neighbor hopping β is assumed. To illustrate the effective mass averaging, cf. eq. (1), the conduction bands for the corresponding single-metal oxide chains (dashed lines) are aligned with (ε1 + ε2)/2. FIG. 4: The key electronic features of n-type TCO hosts. High energy dispersion of the conduction band (left) is essential for both high mobility of extra carriers due to their small effective mass and low optical absorption. For complete transparency in the visible range, the inter-band transitions Ev and Ec should be >3.1 eV, while the intra-band transitions as well as plasma frequency should be <1.8 eV. Such a dispersed conduction band along with the above-visible optical transitions at the Γ point are found in a variety of oxides with s2-cation(s). The electron effective mass calculated for different oxide phases shows little dependence on the oxygen coordination and is isotropic (δ=(ma +mb)/2mc − 1). The highlighted areas are to guide the eye; red circles represent currently known TCO hosts. TABLE I: Net contributions from the states of the atoms that belong to the X-O1 or Y2-O2 layers (X=In or Sc, Y=Ga, Zn, Al, Cd and/or Mg) to the conduction band wavefunction at Γ point, in per cent; and the electron effective masses m, in me, calculated from the band structure of the layered oxides and the components of the electron effective-mass tensor, ma,b and mz, calculated from eqs. (1) and (2) using the effective masses of the corresponding single-cation oxides. XY2O4 NX NO1 NY2 NO2 m[100] m[010] m[001] mab mz InGaZnO4 23 25 29 23 0.23 0.22 0.20 0.23 0.23 InAlCdO4 27 27 18 28 0.26 0.25 0.20 0.27 0.27 InGaMgO4 27 31 21 21 0.27 0.27 0.24 0.28 0.29 InAlMgO4 33 40 12 15 0.32 0.31 0.35 0.31 0.34 ScGaZnO4 8 19 40 33 0.33 0.33 0.34 0.33 0.53 ever, variations in the oxygen coordination as well as strong distortions in the metal-oxygen chains in different oxide phases lead to insignificant changes in the effec- tive mass. For example, for cubic (octahedral coordina- tion) and hexagonal (tetrahedral) ZnO or for corundum (distorted tetrahedral) and monoclinic β-phase (both dis- torted tetrahedral and trigonal) Ga2O3 the correspond- ing electron effective masses vary by 15% or 9%, respec- tively. The largest deviation in the effective mass values for various SiO2 phases is 26%. Furthermore, the effec- tive mass remains isotropic for all phases of the s2-cation oxides – including silica ITQ-4 zeolite with large pore channels [24]. These observations explain the success of amorphous transparent conducting oxides – in marked contrast to the amorphous Si where the directional inter- actions between the conduction p-orbitals lead to strong anisotropy of the transport properties which are sensitive to the orbital overlap and hence to the distortions in the atomic chains [14]. Finally, we note that the fact that the calculated as well as the observed [25] isotropic effective mass in rutile SnO2 where the direct overlap between Sn s-orbitals is possible (only) along the [001] direction, cor- roborate our conclusion that the s-s interactions do not govern the transport properties as discussed above. Effective mass averaging. Because the local symme- try (nearest neighbors), the cation-oxygen bond lengths and, hence, the s-p overlap are similar in the single and multi-cation oxides, the intrinsic transport properties in the layered materials should be related to those in the single-cation oxides. Moreover, due to the hybrid nature of the conduction states in the multicomponent oxides, the states of all cations should give the same order of magnitude contributions to the effective mass. Thus, we expect the later to be an “effective” average over the ef- fective masses of the corresponding single-cation oxides. Indeed, we find that in case of one-dimensional atomic chain where two types of metal atoms alternate with oxy- gen atoms, the effective mass averaging can be derived analytically, fig. 3(f). We formulate a simple approach which allows one to estimate the effective mass of the multicomponent ox- ides as follows. With proper doping, the Fermi energy is shifted up into the conduction band (Burstein-Moss shift). When the extra electrons propagate along the z direction, i.e., across the layers, the resulting resistivity is a sum of the resistivities of each layer. Therefore, the z component of the average effective-mass tensor can be found as: mz = (m1 +m2 +m3)/3, (1) wherem1,2,3 are the effective masses of the corresponding single metal oxides – In2O3, Ga2O3 and ZnO in the case of InGaZnO4. For the in-plane charge transport in the layered mate- rials, the effective-mass tensor components can be found in a parallel fashion. Note, that one needs to average the effective mass for the mixed GaZnO2.5 layers: (m2 +m3) . (2) The resulting ma,b and mz are presented in table I. We find that the increase of the electron effective masses in the order InGaZnO4 < InAlCdO4 < InGaMgO4 < InAlMgO4 < ScGaZnO4 is well reproduced by the above averaging. Moreover, thema,b andmz values nearly coin- cide with the corresponding effective masses of the multi- cation oxides with the exception of the Sc case. In fact, for the Sc-containing compounds the effective mass aver- aging is not legitimate due to the presence of the empty d-states of Sc near the conduction band edge. In Sc2O3, the Sc d-states are located at 0.5 eV above the conduction band edge, and therefore give significant contributions to the effective mass which is about the mass of an electron. In the multi-cation oxide, the Sc d-states are found to be at ∼2 eV above the conduction band edge, so that the resulting small effective mass, table I, is determined pri- marily by the s − p interactions – similar to the rest of the s2-cation oxides. The effective mass averaging procedure, eqs. (1) and (2), can be generalized for materials consisting of any number of layers, e.g., for InGaO3(ZnO)m, m=integer. Furthermore, since the intrinsic transport properties are determined entirely by the local symmetry, i.e., the near- est neighbors, the effective mass averaging should apply to TCO’s in amorphous state. In this case, one needs to average the components of the effective-mass tensor, mamorph=(ma+mb+mz)/3. Importance of carrier generation. Upon doping of a TCO host material, the resulting conductivity depends not only on the effective mass but also on the carrier generation mechanism, carrier concentration and carrier relaxation time. Doping of a structurally anisotropic material may lead to non-uniform distribution of the carrier donors and, therefore, the isotropic behavior of the host may not be maintained as, for example, in oxygen-deficient β-Ga2O3 [27]. In multicomponent InGaO3(ZnO)m, different valence states (In 3+ and Ga3+ vs Zn2+) and oxygen coordination (octahedral for In vs tetrahedral for Ga and Zn) are likely to result in pref- erential arrangement of aliovalent substitutional dopants or oxygen vacancies. Consequently, an anisotropic mo- bility should be expected in the layered materials due to the spatial separation of the carrier donors and the layers where the extra carriers are transfered efficiently, i.e., without charge scattering on the impurities. While targeted doping can help make either or both struc- turally distinct layers conducting, the amorphous com- plex oxides readily offer a way to maintain isotropic transport properties. Indeed, experimental observa- tions that the mobility and conductivity are indepen- dent of the large variations in the composition in amor- phous InGaO3(ZnO)n with n≤4 [12] and that the effec- tive masses of amorphous and crystalline InGaZnO4 are nearly the same [28] support our conclusions. For efficient doping of wide-bandgap oxides such as MgO, CaO, SiO2 and Al2O3, novel carrier generation mechanisms should be sought. A non-traditional ap- proach has already yielded promising results in calcium aluminates [29, 30, 31] – a conceptually new class of transparent conductors [19]. Multicomponent oxides – such as those considered in this work, ordered ternary oxides [32] as well as their solid solutions and amorphous counterparts – represent an alternative way to utilize the abundant main-group elements such as Ca, Mg, Si and Al towards novel TCO hosts with a predictable effective mass and optical and transport properties controllable via the composition. Acknowledgement. The work is supported by Uni- versity of Missouri Research Board. ∗ Electronic address: [email protected] [1] K.L. Chopra, S. Major, D.K. Pandya, Thin Solid Films 1983, 102, 1. [2] G. Thomas, Nature 1997, 389, 907. [3] Special issue on Transparent Conducting Oxides, MRS Bull. 2000, 25. [4] H. Ohta, H. Hosono, Mater. Today 2004, 7, 42. [5] P.P. Edwards, A. Porch, M.O. Jones, D.V. Morgan, R.M. Perks, Dalton Trans. 2004, 19, 2995. [6] G.F. 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Solid State Chem. 1989, 78, APPENDIX Theoretical methods First-principles full-potential linearized augmented plane wave method [33, 34] with the local density ap- proximation is employed for electronic band structure investiga- tions of the XY2O4 compounds, X = In or Sc and Y = Ga, Zn, Al, Cd and/or Mg, and single-cation oxides. Cut-offs of the plane- wave basis (16.0 Ry) and potential representation (81.0 Ry), and expansion in terms of spherical harmonics with ℓ ≤ 8 inside the muffin-tin spheres were used. Summations over the Brillouin zone were carried out using 14 special k points in the irreducible wedge. Structure optimization. XY2O4 compounds have rhombohedral R3̄m crystal structure of YbFe2O4 type [35, 36]. Indium (or scan- dium) atoms substitute Yb in 3(a) position, while both Y 3+ and Y 2+ atoms replace Fe in 6(c) position and are distributed ran- domly [37]. Our total energy calculations for several structures in the (a,2a,c) supercell with various arrangements of the Y 2+ and Y 3+ atoms suggest that their arrangement is not ordered but ran- dom – in agreement with the experiment. We note here that the electronic band structure features are similar for different spatial distributions of the Y atoms for the reasons discussed in the paper. Since the valence state and ionic radii of Y 2+ and Y 3+ are dif- ferent, the site positions of these atoms as well as their oxygen sur- rounding should be different. Because the exact internal positions of atoms are unknown, we used those of the YbFe2O4 [35] as the starting values and then optimized each structure via the total en- ergy and atomic forces minimization. During the optimization, the lattice parameters were fixed at the experimental values [36, 38]. We find that the optimized cation-anion distances correlate with ionic radii of the cations. Single-cation oxides. For the single-cation oxides, the following phases have been calculated: Fm3̄m for MgO, CaO, SrO, BaO and YbO; Ia3̄ for Sc2O3 and Y2O3; Fm3̄m and P63mc for ZnO; Fm3̄m for CdO; R3̄c for Al2O3; R3̄c and C2/m for Ga2O3; Ia3̄, R3̄c and I213 for In2O3; P3221, P21/c, P63/mmc, P41212 and I2/m for SiO2; P41212 and P42/mnm for GeO2; P42/mnm and Pbcn for SnO2. For each structure, the internal positions of all atoms have been optimized via the total energy and atomic forces minimization, while the lattice parameters were fixed at the exper- imental values. One-dimensional model of complex oxides. The effective mass averaging, cf. eq. (1), can be shown analytically using a one- dimensional model in the tight-binding approximation. To capture the key features of complex oxides, we consider a chain consisting of two types of metal atoms which alternate with oxygen atoms, fig. 3(f), and assume only the nearest-neighbor interactions given by the hopping integrals β1 and β2. The Hamiltonian of this model system is: |n, l〉εl〈n, l|+ n,n′,l,l′ |n′, l′〉βl〈n, l|. (3) Here, l is the atom index in the unit cell, n enumerates the cells and n′, l′ in the second sum run over the nearest neighbors. For the bottom of the conduction band, the dispersion relation can be simplified to ε(k) = ε1 + ε2 if |ε1 − ε2| < 2 . Here ε0, ε1 and ε2 are the atomic level energies of the oxygen and two types of metal atoms, respectively, and it is assumed that ε0 < ε1,2 and ε1 ∼ ε2; ∆ = (ε1 + ε2)− ε0 and a is a half of the lattice parameter. Similar considerations for the chain consisting of only one type of metal atoms alternat- ing with oxygen atoms show that the quantity ∆ represents the effective mass of the system. Therefore, eq. (4) represents the ef- fective mass averaging over those of the corresponding single-metal “oxide” chains, cf., fig. 3(f) – in agreement with the results of our first-principles calculations. The following parameters were used to plot fig. 3(f): ε0 = 1.00, ε1 = 2.00, ε2 = 2.05, β1 = 0.4 and β2 =

0704.1500

A Search for Electron Neutrino Appearance at the Delta m**2 ~ 1 eV**2 Scale

A Search for Electron Neutrino Appearance at the ∆m2 ∼ 1 eV2 Scale A. A. Aguilar-Arevalo5, A. O. Bazarko12, S. J. Brice7, B. C. Brown7, L. Bugel5, J. Cao11, L. Coney5, J. M. Conrad5, D. C. Cox8, A. Curioni16, Z. Djurcic5, D. A. Finley7, B. T. Fleming16, R. Ford7, F. G. Garcia7, G. T. Garvey9, C. Green7,9, J. A. Green8,9, T. L. Hart4, E. Hawker15, R. Imlay10, R. A. Johnson3, P. Kasper7, T. Katori8, T. Kobilarcik7, I. Kourbanis7, S. Koutsoliotas2, E. M. Laird12, J. M. Link14, Y. Liu11, Y. Liu1, W. C. Louis9, K. B. M. Mahn5, W. Marsh7, P. S. Martin7, G. McGregor9, W. Metcalf10, P. D. Meyers12, F. Mills7, G. B. Mills9, J. Monroe5, C. D. Moore7, R. H. Nelson4, P. Nienaber13, S. Ouedraogo10, R. B. Patterson12, D. Perevalov1, C. C. Polly8, E. Prebys7, J. L. Raaf3, H. Ray9, B. P. Roe11, A. D. Russell7, V. Sandberg9, R. Schirato9, D. Schmitz5, M. H. Shaevitz5, F. C. Shoemaker12, D. Smith6, M. Sorel5, P. Spentzouris7, I. Stancu1, R. J. Stefanski7, M. Sung10, H. A. Tanaka12, R. Tayloe8, M. Tzanov4, R. Van de Water9, M. O. Wascko10, D. H. White9, M. J. Wilking4, H. J. Yang11, G. P. Zeller5, E. D. Zimmerman4 (The MiniBooNE Collaboration) 1University of Alabama; Tuscaloosa, AL 35487 2Bucknell University; Lewisburg, PA 17837 3University of Cincinnati; Cincinnati, OH 45221 4University of Colorado; Boulder, CO 80309 5Columbia University; New York, NY 10027 6Embry Riddle Aeronautical University; Prescott, AZ 86301 7Fermi National Accelerator Laboratory; Batavia, IL 60510 8Indiana University; Bloomington, IN 47405 9Los Alamos National Laboratory; Los Alamos, NM 87545 10Louisiana State University; Baton Rouge, LA 70803 11University of Michigan; Ann Arbor, MI 48109 12Princeton University; Princeton, NJ 08544 13Saint Mary’s University of Minnesota; Winona, MN 55987 14Virginia Polytechnic Institute & State University; Blacksburg, VA 24061 15Western Illinois University; Macomb, IL 61455 16Yale University; New Haven, CT 06520 (Dated: February 1, 2008) The MiniBooNE Collaboration reports first results of a search for νe appearance in a νµ beam. With two largely independent analyses, we observe no significant excess of events above background for reconstructed neutrino energies above 475 MeV. The data are consistent with no oscillations within a two-neutrino appearance-only oscillation model. This Letter reports the initial results from a search for νµ → νe oscillations by the MiniBooNE Collaboration. MiniBooNE was motivated by the result from the Liquid Scintillator Neutrino Detector (LSND) experiment [1], which has presented evidence for ν̄µ → ν̄e oscillations at the ∆m2 ∼ 1 eV2 scale. Although the Karlsruhe Ruther- ford Medium Energy Neutrino Experiment (KARMEN) observed no evidence for neutrino oscillations [2], a joint analysis [3] showed compatibility at 64% CL. Evidence for neutrino oscillations also comes from solar-neutrino [4, 5, 6, 7, 8] and reactor-antineutrino experiments [9], which have observed νe disappearance at ∆m ∼ 8×10−5 eV2, and atmospheric-neutrino [10, 11, 12, 13] and long- baseline accelerator-neutrino experiments [14, 15], which have observed νµ disappearance at ∆m ∼ 3×10−3 eV2. If all three phenomena are caused by neutrino oscilla- tions, these three ∆m2 scales cannot be accommodated in an extension of the Standard Model that allows only three neutrino mass eigenstates. An explanation of all three mass scales with neutrino oscillations requires the addition of one or more sterile neutrinos [16] or further extensions of the Standard Model (e.g., [17]). The analysis of the MiniBooNE neutrino data pre- sented here is performed within a two neutrino appearance-only νµ → νe oscillation model which uses νµ events to constrain the predicted νe rate. Other than oscillations between these two species, we assume no ef- fects beyond the Standard Model. The experiment uses the Fermilab Booster neutrino beam, which is produced from 8 GeV protons incident on a 71-cm-long by 1-cm-diameter beryllium target. The proton beam typically has 4 × 1012 protons per ∼ 1.6 µs beam spill at a rate of 4 Hz. The number of pro- tons on target per spill is measured by two toroids in the beamline. The target is located inside a focusing horn, which produces a toroidal magnetic field that is pulsed in time with the beam at a peak current of 174 kA. Posi- tively charged pions and kaons, focused by the horn, pass through a 60-cm-diameter collimator and can decay in a 50-m-long tunnel, which is 91 cm in radius and filled with http://arXiv.org/abs/0704.1500v3 air at atmospheric pressure. The center of the detector is 541 m from the front of the beryllium target and 1.9 m above the center of the neutrino beam. There is about 3 m of dirt over- burden above the detector, which is a spherical tank of inner radius 610 cm filled with 800 tons of pure mineral oil (CH2) with a density of 0.86 g/cm 3 and an index of refraction of 1.47. The light attenuation length in the mineral oil increases with wavelength from a few cm at 280 nm to over 20 m at 400 nm. Charged particles pass- ing through the oil can emit both directional Cherenkov light and isotropic scintillation light. An optical bar- rier separates the detector into two regions, an inner vol- ume with a radius of 575 cm and an outer volume 35 cm thick. The optical barrier supports 1280 equally-spaced inward-facing 8-inch photomultiplier tubes (PMTs), pro- viding 10% photocathode coverage. An additional 240 tubes are mounted in the outer volume, which acts as a veto shield, detecting particles entering or leaving the detector. Two types of PMT are used: 1198 Hamamatsu model R1408 with 9 stages and 322 Hamamatsu model R5912 with 10 stages. Approximately 98% of the PMTs have worked well throughout the data taking period. The experiment triggers on every beam spill, with all PMT hits recorded for a 19.2 µs window beginning 4.4 µs before the spill. Other triggers include a random trig- ger for beam-unrelated measurements, a laser-calibration trigger, cosmic-muon triggers, and a trigger to record neutrino-induced events from the nearby Neutrinos at the Main Injector (NuMI) beamline [18]. The detector elec- tronics, refurbished from LSND [19], digitize the times and integrated charges of PMT hits. PMT hit thresholds are ∼ 0.1 photoelectrons (PE); the single-PE time reso- lutions achieved by this system are ∼ 1.7 ns and ∼ 1.2 ns for the two types of PMTs. One PE corresponds to ∼0.2 MeV of electron energy. Laser calibration, consist- ing of optical fibers that run from the laser to dispersion flasks inside the tank, is run continuously at 3.33 Hz to determine PMT gains and time offsets. Averaged over the entire run, the beam-on livetime of the experiment is greater than 98%. The νµ energy spectrum peaks at 700 MeV and extends to approximately 3000 MeV. Integrated over the neutrino flux, interactions in MiniBooNE are mostly charged- current quasi-elastic (CCQE) scattering (39%), neutral- current (NC) elastic scattering (16%), charged-current (CC) single pion production (29%), and NC single pion production (12%). Multi-pion and deep-inelastic scatter- ing contributions are < 5%. NC elastic scattering, with only a recoil nucleon and a neutrino in the final state, typically produces relatively little light in the detector and contributes only 3 events to the final background estimate. Table I shows the estimated number of events with reconstructed neutrino energy, EQEν , between 475 MeV and 1250 MeV after the complete event selection from TABLE I: The estimated number of events with systematic error in the 475 < EQEν < 1250 MeV energy range from all of the significant backgrounds, together with the estimated num- ber of signal events for 0.26% νµ → νe transmutation, after the complete event selection. Process Number of Events νµ CCQE 10 ± 2 νµe → νµe 7 ± 2 Miscellaneous νµ Events 13 ± 5 NC π0 62 ± 10 NC ∆ → Nγ 20 ± 4 NC Coherent & Radiative γ < 1 Dirt Events 17 ± 3 νe from µ Decay 132 ± 10 νe from K + Decay 71 ± 26 νe from K L Decay 23 ± 7 νe from π Decay 3 ± 1 Total Background 358 ± 35 0.26% νµ → νe 163 ± 21 all of the significant backgrounds, where EQEν is deter- mined from the reconstructed lepton energy and angle with respect to the known neutrino direction. The back- ground estimate includes antineutrino events, which rep- resent < 2% of the total. Also shown is the estimated number of νe CCQE signal events for the LSND central expectation of 0.26% νµ → νe transmutation. Studies of random triggers have established that no significant backgrounds survive the analysis cuts other than those due to beam related neutrinos, which can be divided into either νµ-induced or νe-induced backgrounds. The small fraction of νe from µ, K, and π decay in the beamline gives a background that is indistinguishable from oscil- lations except for the energy spectrum. CC νµ events are distinguished from νe events by the distinct patterns of Cherenkov and scintillation light for muons and elec- trons, as well as by the observation of a delayed electron from the muon decay, which is observed > 80% of the time from νµ CCQE events. NC π 0 events with only a single electromagnetic shower reconstructed are the main νµ-induced background, followed by radiative ∆ decays giving a single photon, and then neutrino interactions in the dirt surrounding the detector, which can mimic a sig- nal if a single photon, mostly from π0 decay, penetrates the veto and converts in the fiducial volume. We use PMT charge and time information in the 19.2 µs window to reconstruct neutrino interactions and iden- tify the product particles. This time window is defined as an “event” and is divided into “subevents”, collec- tions of PMT hits clustered in time within ∼ 100 ns. A νµ CCQE event with a muon stopping within the tank may have two subevents: the first subevent from par- ticles produced at the neutrino interaction, the second from the muon decay to an electron. A νe CCQE event has a single subevent. To ensure stable, well-targeted beam at full horn cur- rent, it is required that the two monitoring toroids agree to within 5%, the estimated transverse containment of the beam in the target be greater than 95%, and the measured horn current be within 3% of its nominal value. The event time at the detector must be consistent with the beam delivery time (both determined by GPS), and the event must pass a number of data integrity checks. The beam quality requirements reject 0.7% of the events, while the detector time and quality requirements remove a further 1.8%, with the remaining data corresponding to (5.58 ± 0.12)× 1020 protons on target. Next, events with exactly one subevent (as expected for νe CCQE events) are selected. By requiring that the subevent have fewer than 6 hits in the veto and more than 200 hits in the main tank (above the muon-decay electron endpoint), entering cosmic-ray muons and their associated decay electrons are eliminated. The average time of hits in the subevent is required to be within the beam time window of 4-7 µs. These cuts yield a cosmic ray rejection of greater than 1000:1. After these initial cuts, the surviving events are re- constructed under four hypotheses: a single electron-like Cherenkov ring, a single muon-like ring, two photon-like rings with unconstrained kinematics, and two photon- like rings with Mγγ = mπ0 (see Fig. 1). Photon-like rings are assumed to be identical to electrons, but al- lowed to be independently displaced from the neutrino interaction vertex. The reconstruction uses a detailed model of extended-track light production and propaga- tion in the tank to predict the charge and time of hits on each PMT. Event parameters are varied to maximize the likelihood of the observed hits, yielding the vertex posi- tion and time of the event and the direction, energy, and, for photons, the conversion distance of the ring(s). For νe events, the event vertex, direction, and energy are recon- structed on average with resolutions of 22 cm, 2.8◦, and 11%, respectively, while NC π0 events are reconstructed with a π0 mass resolution of 20 MeV/c2. The final analysis cuts were designed to isolate a sam- ple of νe-induced events that were primarily CCQE. The only data that were used in developing the analysis were samples that Monte Carlo (MC) simulation had indicated could not contain a significant number of νµ → νe oscil- lation events. We require that the electron-hypothesis event vertex and muon-hypothesis track endpoint occur at radii < 500 cm and < 488 cm, respectively, to en- sure good event reconstruction and efficiency for pos- sible muon decay electrons. We require visible energy Evis > 140 MeV. We then apply particle identification (PID) cuts to reject muon and π0 events. These are Evis- dependent cuts on log(Le/Lµ), log(Le/Lπ0), and Mγγ, where Le, Lµ, and Lπ0 are the likelihoods for each event maximized under the electron 1-ring, muon 1-ring, and fixed-mass 2-ring fits, and Mγγ is from the unconstrained two-ring fit. These also enhance the fraction of CCQE FIG. 1: Events in MiniBooNE are reconstructed as either a muon event, an electron event, or a π0 event. TABLE II: The observed number of νe CCQE candidate events and the efficiency for νµ → νe CCQE oscillation events after each cut is applied sequentially. Selection #Events νe CCQE Efficiency Cosmic Ray Cuts 109,590 100% Fiducial Volume Cuts 68,143 55.2 ± 1.9% PID Cuts 2037 30.6 ± 1.4% 475 < EQEν < 1250 MeV 380 20.3 ± 0.9% events among the surviving electron candidates. Table II shows the observed number of νe CCQE candidate events and the efficiency for νµ → νe CCQE oscillation events after each cut is applied sequentially. A total of 380 data events remains after the complete selection. Detailed Monte Carlo simulations of the beam and de- tector were used to make initial estimates of the flux and detector efficiencies. The Booster neutrino beam flux at the detector is modeled using a GEANT4-based simu- lation [20] of the beamline geometry. Pion and kaon production in the target is parametrized [21] based on a global fit to proton-beryllium particle production data [22]. The kaon flux has been cross-checked with high- energy events above 1250 MeV and with an off-axis muon spectrometer that viewed the secondary beamline from an angle of 7◦. This detector determined the flux of muons with high transverse momentum, which originate mostly from kaon decays, to be consistent with the MC predictions. The v3 NUANCE [23] event generator simulates neu- trino interactions in mineral oil. Modifications are made to NUANCE which include adjustment of the axial form factor of the nucleon for quasi-elastic scattering, the Pauli blocking model, and coherent pion production cross sec- tions based on fits to MiniBooNE νµ data [24]. In addi- tion, the final state interaction model has been tuned to reproduce external pion-carbon rescattering data [25], an explicit model of nuclear de-excitation photon emission for carbon has been added [26], and the angular correla- tions for ∆ decay are modified to be in accord with the model of Rein and Sehgal [27]. Particles from NUANCE-generated final states are propagated through a GEANT3-based simulation [28] of the detector, with the subsequent decays, strong, and electroweak interactions in the detector medium in- cluded. Most particular to MiniBooNE, the emission of optical and near-UV photons via Cherenkov radiation and scintillation is simulated, with each photon individ- ually tracked, undergoing scattering, fluorescence, and reflection, until it is absorbed [29]. Small-sample mea- surements of transmission, fluorescence, and scattering are used in the model. Muon decay electrons are used to calibrate both the light propagation in the detector and the energy scale. The amount of scintillation light is constrained from NC elastic scattering events. The charge and time response of the electronics is simulated, and from this point onward, data and MC calculations are treated identically by the analysis programs. All of the major νµ-induced backgrounds are con- strained by our measurements outside the signal region. The inclusive CC background is verified by comparing data to MC calculations for events with two subevents, where the second subevent has < 200 tank hits and is consistent with a muon-decay electron. As the probabil- ity for µ− capture in the oil is 8%, there are an order of magnitude more CC inclusive scattering events with two subevents than with only one subevent, so that this background is well checked. These data events are also modified by moving the hits of the second subevent ear- lier in time to model early, inseparable decays which can look more like an electron. To determine the NC π0 background, π0 rates are mea- sured in bins of momentum by counting events in the γγ mass peak. The MC simulation is used to correct the production rate for inefficiency, background and resolu- tion (corrections are ∼ 10%). To match the data angular distribution, the π0 candidates are fit to MC templates (in mass and angle) for resonant and coherent production (generated using the model of Rein and Sehgal [27]) as well as a template for non-π0 background events. The fitted parameters are used to reweight π0 from the MC calculations and to constrain the ∆ → Nγ rate, which has a branching ratio at the peak of the ∆ resonance of 0.56%. NC coherent γ background [30] and NC radiative γ background [31] are both estimated to be negligible. The background from interactions in the dirt surround- ing the detector is measured from a sample of inward- pointing events inside the tank at high radius. A sample of ∼105 candidate νµ CCQE events is ob- tained by requiring a µ-decay electron with a recon- structed vertex consistent with the estimated endpoint of the parent muon’s track (60% efficiency). The observed rate of these νµ CCQE events is used to correct the MC predictions for νe signal events, νµ CC backgrounds, and νe from µ backgrounds (which share their π parentage with the νµ CCQE events). These constraints increase the event normalization by 32% and greatly reduce the rate uncertainties on these three components of the final analysis sample. Systematic errors are associated with neutrino fluxes, the detector model, and neutrino cross sections. The neutrino flux systematic errors are determined from the uncertainties of particle production measurements, the detector model systematic errors are mostly determined from fits to MiniBooNE data, and the neutrino cross sec- tion systematic errors are determined from MiniBooNE data as well as from external sources, both experimen- tal and theoretical. These groups of errors are taken to be independent, and, for each, an individual error ma- trix is formed that includes the full correlation among the systematic parameters. This is mapped to a matrix describing the correlated errors in predicted background plus possible signal in eight νe E ν bins. The final co- variance matrix for all sources of uncertainty (statistical and systematic) is the sum of the individual error ma- trices. The signal extraction is performed by computing the χ2 comparing data to predicted background plus a (sin2(2θ), ∆m2)-determined contribution from νµ → νe two-neutrino oscillations in the eight EQEν bins and min- imizing with respect to these two oscillation parameters across their physical range. With the analysis cuts set, a signal-blind test of data- MC agreement in the signal region was performed. The full two-neutrino oscillation fit was done in the range 300 < EQEν < 3000 MeV and, with no information on the fit parameters revealed, the sum of predicted back- ground and simulated best-fit signal was compared to data in several variables, returning only the χ2. While agreement was good in most of the comparisons, the Evis spectrum had a χ2 probability of only 1%. This triggered further investigation of the backgrounds, focusing on the lowest energies where νµ-induced backgrounds, some of which are difficult to model, are large. As part of this study, one more piece of information from the signal re- gion was released: unsigned bin-by-bin fractional discrep- ancies in the Evis spectrum. While ambiguous, these re- inforced suspicions about the low-energy region. Though we found no specific problems with the background es- timates, it was found that raising the minimum EQEν of the fit region to 475 MeV greatly reduced a number of backgrounds with little impact on the fit’s sensitivity to oscillations. We thus performed our oscillation fits in the energy range 475 < EQEν < 3000 MeV and opened the full data set. The top plot of Fig. 2 shows candidate νe events as a function of EQEν . The vertical dashed line indicates the minimum EQEν used in the two-neutrino oscillation anal- ysis. There is no significant excess of events (22±19±35 events) for 475 < EQEν < 1250 MeV; however, an ex- cess of events (96±17±20 events) is observed below 475 MeV. This low-energy excess cannot be explained by a two-neutrino oscillation model, and its source is under investigation. The dashed histogram in Fig. 2 shows the predicted spectrum when the best-fit two-neutrino oscil- lation signal is added to the predicted background. The bottom panel of the figure shows background-subtracted data with the best-fit two-neutrino oscillation and two oscillation points from the favored LSND region. The oscillation fit in the 475 < EQEν < 3000 MeV energy range yields a χ2 probability of 93% for the null hypoth- esis, and a probability of 99% for the (sin2 2θ = 10−3, ∆m2 = 4 eV2) best-fit point. analysis threshold oscillationν2 y MiniBooNE data µνg expected background µνeν→µν BG + best-fit backgroundµν backgroundeν 300 600 900 1200 1500 (MeV)νreconstructed E data - expected background eν→µν best-fit 2=1.0 eV2m∆)=0.004, θ(22 sin 2=0.1 eV2m∆)=0.2, θ(22 sin FIG. 2: The top plot shows the number of candidate νe events as a function of EQEν . The points represent the data with sta- tistical error, while the histogram is the expected background with systematic errors from all sources. The vertical dashed line indicates the threshold used in the two-neutrino oscilla- tion analysis. Also shown are the best-fit oscillation spec- trum (dashed histogram) and the background contributions from νµ and νe events. The bottom plot shows the number of events with the predicted background subtracted as a func- tion of EQEν , where the points represent the data with total errors and the two histograms correspond to LSND solutions at high and low ∆m2. A single-sided raster scan to a two neutrino appearance-only oscillation model is used in the energy range 475 < EQEν < 3000 MeV to find the 90% CL limit corresponding to ∆χ2 = χ2limit − χ bestfit = 1.64. As shown by the top plot in Fig. 3, the LSND 90% CL al- lowed region is excluded at the 90% CL. A joint analysis as a function of ∆m2, using a combined χ2 of the best fit values and errors for LSND and MiniBooNE, excludes at 98% CL two-neutrino appearance oscillations as an explanation of the LSND anomaly. The bottom plot of Fig. 3 shows limits from the KARMEN [2] and Bugey [32] experiments. A second analysis developed simultaneously and with the same blindness criteria used a different set of recon- struction programs, PID algorithms, and fitting and nor- malization processes. The reconstruction used a simpler model of light emission and propagation. The PID used 172 quantities such as charge and time likelihoods in an- gular bins, Mγγ, and likelihood ratios (electron/pion and electron/muon) as inputs to boosted decision tree algo- rithms [33] that are trained on sets of simulated signal events and background events with a cascade-training technique [34]. In order to achieve the maximum sen- sitivity to oscillations, the νµ-CCQE data sample with two subevents were fit simultaneously with the νe-CCQE candidate sample with one subevent. By forming a χ2 using both data sets and using the corresponding covari- ance matrix to relate the contents of the bins of the two distributions, the errors in the oscillation parameters that best describe the νe-CCQE candidate data set were well constrained by the observed νµ-CCQE data. This pro- cedure is partially equivalent to doing a νe to νµ ratio analysis where many of the systematic uncertainties can- The two analyses are very complementary, with the second having a better signal-to-background ratio, but the first having less sensitivity to systematic errors from detector properties. These different strengths resulted in very similar oscillation sensitivities and, when unblinded, yielded the expected overlap of events and very similar oscillation fit results. The second analysis also sees more events than expected at low energy, but with less signif- icance. Based on the predicted sensitivities before un- blinding, we decided to present the first analysis as our oscillation result, with the second as a powerful cross- check. In summary, while there is a presently unexplained discrepancy with data lying above background at low energy, there is excellent agreement between data and prediction in the oscillation analysis region. If the oscil- lations of neutrinos and antineutrinos are the same, this result excludes two neutrino appearance-only oscillations as an explanation of the LSND anomaly at 98% CL. We acknowledge the support of Fermilab, the Depart- ment of Energy, and the National Science Foundation. We thank Los Alamos National Laboratory for LDRD funding. We acknowledge Bartoszek Engineering for the design of the focusing horn. We acknowledge Dmitri Top- tygin, Anna Pla, and Hans-Otto Meyer for optical mea- surements of mineral oil. This research was done using resources provided by the Open Science Grid, which is supported by the NSF and DOE-SC. We also acknowl- edge the use of the LANL Pink cluster and Condor soft- LSND 90% C.L. LSND 99% C.L. -310 -210 -110 1 ) upper limitθ(22sin yMiniBooNE 90% C.L. MiniBooNE 90% C.L. sensitivity BDT analysis 90% C.L. )θ(22sin -310 -210 -110 1 LSND 90% C.L. LSND 99% C.L. yMiniBooNE 90% C.L. KARMEN2 90% C.L. Bugey 90% C.L. FIG. 3: The top plot shows the MiniBooNE 90% CL limit (thick solid curve) and sensitivity (dashed curve) for events with 475 < EQEν < 3000 MeV within a two neutrino oscilla- tion model. Also shown is the limit from the boosted decision tree analysis (thin solid curve) for events with 300 < EQEν < 3000 MeV. The bottom plot shows the limits from the KAR- MEN [2] and Bugey [32] experiments. The MiniBooNE and Bugey curves are 1-sided upper limits on sin2 2θ correspond- ing to ∆χ2 = 1.64, while the KARMEN curve is a “unified approach” 2D contour. 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0704.1501

Surface Structure Analysis of Atomically Smooth BaBiO$_3$ Films

Surface Structure Analysis of Atomically Smooth BaBiO3 Films A. Gozar∗, G. Logvenov, V. Y. Butko, and I. Bozovic Brookhaven National Laboratory, Upton, New York 11973-5000, USA Using low energy Time-of-Flight Scattering and Recoil Spectroscopy (TOF-SARS) and Mass Spectroscopy of Recoiled Ions (MSRI) we analyze the surface structure of an atomically smooth BaBiO3 film grown by molecular beam epitaxy. We demonstrate high sensitivity of the TOF-SARS and MSRI spectra to slight changes in the orientation of the ion scattering plane with respect to the crystallographic axes. The observed angle dependence allows us to clearly identify the termination layer as BiO2. Our data also indicate that angle-resolved MSRI data can be used for high resolution studies of surface structure of complex oxide thin films. PACS numbers: 68.03.Hj, 68.47.Gh, 68.49.Sf, 81.15.Hi Introduction - The plethora of remarkable electrical and magnetic properties of transition metal oxides made them both a focus of basic research and very appeal- ing candidates for integration in electronic devices. For the latter, it is important to reproducibly synthesize and characterize atomically perfect surfaces and interfaces. However, this goal is difficult to attain in complex ox- ides because of the complicated phase diagrams and high sensitivity to growth conditions.1 This explains the large disparity between what is known about the structure of the very top atomic layers in these materials compared to widely used semiconductor or metal surfaces. In this work we provide a new route to circumvent these problems. This is done by combining the power of atomic layer-by-layer molecular beam epitaxy (ALL- MBE)2, enabling production of films with perfect sur- faces, with the extreme surface sensitivity of the low en- ergy Time-of-Flight Scattering and Recoil Spectroscopy (TOF-SARS)3,4 and the Mass Spectroscopy of Recoiled Ions (MSRI)5 techniques. Angle-Resolved (AR) TOF- SARS could determine in principle inter-atomic spac- ings with a resolution approaching 0.01 Å, comparable to lateral values obtained in surface X-ray crystallogra- phy and even better in the direction perpendicular to the surface.4 This comparison is to be judged also from the perspective of having a table-top experimental setup as opposed to the requirement of very intense synchrotron light. However, in order to reach such accuracy one needs samples with very smooth surfaces. For this rea- son TOF-SARS and MSRI data in oxides have been pri- marily used for monitoring surface composition rather than the structure.6 Exceptions are very few materials like Al2O3 or SrTiO3 which are commercially available and commonly used as single crystal substrates for film growth.7,8 Ba1−xKxBiO3 is a family of superconductors with the maximum Tc ≃ 32 K (at x = 0.4) being the highest in an oxide material without copper.9,10 BaBiO3 , the parent compound, is insulating and non-magnetic. These inter- esting properties are believed to arise due to a charge- density-wave instability11 which leads to the lowering of the crystal symmetry from the simple cubic perovkite structure, see Fig. 1. The driving force of this transi- tion and the persistence of the charge order in supercon- ducting Ba1−xKxBiO3 are still a matter of debate. 11,12,13 This is largely due to the difficulty in obtaining high quality single crystals from these materials. A recent study brought to the forefront the important problem of dimensionality in Ba1−xKxBiO3 suggesting that this compound has in fact a layered structure, in analogy to the high Tc superconducting cuprates. 14 In the con- text of superconducting electronics, the interest in un- doped and K doped BaBiO3 stems from succesful fabrica- tion of superconductor-insulator-superconductor tunnel junctions, a task that has bee quite ellusive with either cuprates or MgB2 superconductors. For these reasons it is critically important to understand and control the surface properties in this family of compounds. Real space crystallography using TOF-SARS is based on the concepts of shadowing and blocking cones.3,15 TOF-SARS is sensitive to both ions and neutrals so it is not dependent on charge exchange processes at the sur- face. The drawback is that these spectra display broad ’tails’ at high times (see Fig. 2) associated with multi- ple scattering events that are difficult to analyze quan- titatively. This drawback is eliminated by MSRI which achieves ’time focussing’ according to t = t0+k(M/e) i.e. the flight time of the ions is only a function of their mass to charge ratio. The broad continua seen in TOF- SARS are turned into very sharp features allowing iso- topic mass resolution5, see Fig. 3. MSRI is thus easier to interpret and one is tempted to use AR MSRI for surface structure analysis. However, because MSRI detects only ions, one should worry about possible anisotropic neu- tralization effects which makes a quantitative interpreta- tion problematic.16 In fact this aspect is a long standing problem in ion based mass spectrometry affecting both compositional and structural studies. Here we report on succesful and reproducible synthe- sis of large area single crystal thin films of BaBiO3 using ALL-MBE. This opens the path of improved basic exper- iments including high resolution surface crystallography in oxides based on ion scattering. Next we present results of surface analysis of a BaBiO3 film using AR TOF-SARS and MSRI. We show that atomically smooth surfaces lead to high sensitivity of both type of spectra with respect http://arxiv.org/abs/0704.1501v1 0 25 50 m [100] [210] [110] Roughness Rms < 0.4 nm 0 20 40 60 80 100 2 (degrees) (001) (002) (003) (004) 1.2 10 0.6 0.8 1 1.2 1.4 Incident Angle (deg.) (a) (b) FIG. 1: (a) The cubic perovskite structure of undistorted BaBiO3 . (1) and (2) correspond to K trajectories as described in the text. (b) A 50 × 50 µm AFM image of the BaBiO3 film. (c) ω − 2θ scan of the BaBiO3 film; STO labels Bragg peaks from the SrTiO3 substrate. The inset shows X-ray reflectance intensity oscillations at grazing incidence. to as small as one degree variation in the azimut angle. Comparison between the AR TOF-SARS and MSRI data shows that the latter can be a powerful tool for quan- titative surface structure analysis. To the best of our knowledge this result has not been reported before. The angular dependence of the spectra along with numerical simulations allows us to unambiguously determine that the BaBiO3 film terminates by a BiO2 layer. Experimental - A BaBiO3 thin film was grown on a SrTiO3 substrate. The thickness of 96 nm was deter- mined from low angle X-ray reflectometry, a value in very good agreement with the prediction from the pro- grammed growth rate. Atomic force microscopy data show absence of any secondary phase precipitates and a roughness Rms that is below 4 Å in a typical 25µm2 area, see Fig. 1b. This was expected based on our observation of strong specular reflection and undamped RHEED os- cillations during growth. The pronounced finite thickness reflectance oscillations (Fig. 1c - inset) clearly demon- strate atomically flat film interfaces. The high quality of the film is furthermore reflected in the sharp Bragg diffraction peaks seen in the ω−2θ scans shown in Fig. 1c. The lattice constants of the bulk (96 nm thick) film at T = 300 K were determined to be c = 4.334(2) Å while in-plane a ≈ b ≃ 4.353(4) Å. Note also that the unit cell of our BaBiO3 film appears to be very close to the sim- ple perovskite structure shown in Fig. 1b, which is not the case for the body-centered monoclinic structure char- acteristic of single crystals at this temperature.11 The experimental observation of excellent epitaxy is intrigu- ing taking into account the relatively large mismatch, of about 11%, between the in-plane lattice constants of SrTiO3 and bulk single crystals of BaBiO3 . The ion scattering data were taken using a recently built Ionwerks TOF system, based on principles de- scribed in Refs.5,17 We used a K+ ion source tuned to provide a monochromatic beam of 10 keV. Microchannel plate detectors were mounted at θ = 270 and 370 total scattering angles. The incident angle was α = 150. Time resolved spectra were achieved by pulsing the source beam at a 20 kHz over a 0.5×2 mm aperture. The typical pulse width was 12 ns. This, together with the I ≈ 0.1µA value of the ion current on the aperture, allows us to esti- mate an ion dose of about 3×1011 ions/cm2 per spectrum. The surface density is about 1015 atoms/cm2 indicating that the technique was not invasive for the duration of the experiment. More important, measurements from pristine regions of the 1 cm2 sample during and after the experiment ensured that the surface was neither dam- aged nor charged. The TOF-SARS and MSRI spectra were collected at T ≈ 6000 C in ozone atmosphere at a pressure p ≃ 5×10−6 Torr. In the following KS(X) de- notes K+ ions undergoing single scattering events from the element X on the surface while the notation XR stands for particles (X = Ba,Bi or O) recoiled from the BaBiO3 surface. The numerical calculations were per- formed using the Scattering and Recoil Imaging Code (SARIC) which is a classical trajectory simulation pro- gram based on the binary collision approximation.18 Surface structure analysis of BaBiO3 - Fig. 2a illus- trates the dependence of the TOF-SARS spectra on the azimuth angle Φ, defined as the angle between the scat- tering plane and the [100] direction of the cubic structure. One minute long scans taken with Φ varied in 10 incre- ments between 3.50 and 10.50 are shown. The continuum above t ≥ 9.9 µs is due to K multiple scattering. Based on the predictions of elastic binary collision the two sharp features in Fig. 2a can be immediately assigned to single scattering events of K ions. The peak at t = 9.75 ns cor- responds toKS(Bi) and the one at t ≃ 9.8 ns to KS(Ba). Dramatic changes are seen in the behavior of the higher time KS(Ba) peak as Φ is varied. Surface roughness typically smears out the structure in AR scattering or recoil features. The strong sensitivity of the spectra in Fig. 1 to small variations of the azimuthal angle is a consequence (and a direct proof) of high surface quality. The angle independent intensity of the KS(Bi) peak along with the strong variation in the KS(Ba) fea- ture suggest that the film terminates with BiO2 planes which shadow the subsurface BaO layers. We show be- low that this is fully supported by more detailed analysis including numerical simulations. Information about the surface arrangement and dy- namics can be obtained by studying the details of the spectral shapes. For example the inset of Fig. 2a, which shows a zoomed in region around the KS(Bi) peak at Φ = 450, reveals a shoulder on the low time side. Indeed, 9.7 9.8 9.9 10 time ( Sec) 9.72 9.75 9.78 = 45 time ( Sec) 17 18 19 time ( Sec) recoil FIG. 2: (Color online) Time-of-Flight K scattering data from BaBiO3 recorded at θ = 27 0 total scattering angle. (a) The spectra are taken for azimuthal angles Φ from 3.50 to 10.50. The arrow indicates the direction of increasing angle. The peaks at t = 9.74 µs and t ≈ 9.8 µs correspond to KS(Bi) and KS(Ba) respectively. Inset: zoomed in area around the KS(Bi) peak for Φ = 45 0. The line through the data points is a two Gaussian peaks fit. These two peaks, denoted by (1) and (2), correspond to the K trajectories shown in Fig. 1a. (b) The BiR peak at higher times for Φ = 0 0, 150 and 210. the data can be well fitted by two Gaussian peaks which correspond to trajectories involving single and double scattering events denoted by (1) and (2) in Fig. 1a. This assignment to events involving only K and Bi atoms along the [110] direction is based on the elastic binary collision model which predicts a difference of 33 ns between these two trajectories, in good agreement with the experimen- tal value δt = 30 ns. We do not observe the low time feature for Φ = 00; this is understood as the effect of the intervening O atom along [100] direction. Since the intensity of peak (2) depends strongly on the atomic ar- rangement at the surface, angular dependencies of the relative intensity of these two peaks could be used to get information about the symmetry and vibration ampli- tudes of the top layer atoms. Shown in Fig. 2b is the BiR peak which is found around t ≃ 18µs. We turn now to the question whether AR MSRI can be a quantitative probe for high resolution surface study. It is natural to ask (a) if one can see structure in the AR MSRI data and (b) whether such dependence, if present, provides quantitative information about the surface. The latter problem is related to the fact that it is hard to quantify the yield of ionic fraction which, moreover, could be itself an anisotropic function with respect to the ori- entation of the crystallographic axes.16 The answer to the first question is given in Fig. 3. The main panel shows a MSRI spectrum taken at Φ = 00 and the inset shows the Ba isotopes region for three azimuths. Clearly, the intensities of the corresponding peaks vary substantially when this angle is changed. Note also the sharpness of the peaks which allows for easy separation of atomic isotopes. The mass resolution, m/∆m ≈ 380, is about one order of magnitude better than that obtained 20 40 60 80 100 120 140 160 180 200 Mass (AMU) = 0 134 135 136 137 138 139 140 Mass (AMU) FIG. 3: (Color online) The main panel displays a MSRI spectrum taken at Φ = 00 azimuth. The inset shows a zoomed in area of the Ba isotopes region from the main panel, for three values of Φ: 00, 50 and 200. in typical TOF-SARS spectra. The second question is addressed in the top panel of Fig. 4 where two data sets are compared. One data set corresponds to the Φ dependence of the intensities of MSRI BaR peaks derived from the spectra shown in the inset of Fig. 3. The second data set refers to BaR peak from the TOF-SARS spectra taken with the detector at the same total scattering angle θ = 370. The latter data set was taken with the time focussing analyser not biased, making it sensitive to both ion and neutral particles. The two dependencies are essentially identical. Equally good agreement is also observed if the BiR feature is consid- ered instead of BaR. We conclude that anisotropic neu- tralization effects are not important which proves that AR MSRI can be used as a quantitative probe for sur- face analysis. The insulating nature of the film and the use of alkali ion source beam could be responsible for this effect.16 Note also that the steep decrease in the BaR sig- nal between Φ = 50 and 100 seen in Fig. 3 is consistent with the disappearance of the KS(Ba) peak at t ≃ 9.8µs in the TOF-SARS data at θ = 270 from Fig. 2a. We address now the problem of film surface termina- tion. The spectra shown in the bottom panel of Fig. 4 provide the answer to this question. The experimental points correspond to the Φ dependence of the integrated intensity of the BiR peak shown in Fig. 2b. For every data point we subtracted the background due to the high time tail associated with K multiple scattering (as seen in Fig. 2a). The results of two SARIC simulations are also shown in Fig. 4. The solid and dashed lines correspond to assumed BiO2 and BaO terminations respectively. The simulation based on BiO2 termination reproduces well the main features of the experiment, i.e. the two peaks around 200 and 330. In contrast, the lower angle fea- ture is absent if BaO is assumed to be the topmost layer. These numerical simulations clearly show that the film terminates in a BiO2 surface. One advantage of real space structure analysis is the intuitive picture it immediately provides: the dips in the 300 TOF-SARS 37 MSRI 37 Recoil 0 5 10 15 20 25 30 35 40 45 azimuth (degrees) simulations TOF-SARS 27 experiment Recoil FIG. 4: (Color online) Top: azimuth dependence of the inte- grated intensity of the Ba recoil peak from two sets of mea- surements taken at θ = 370 scattering angle: MSRI (squares) and TOF-SARS spectra (circles). The curves are matched at Φ = 50. Bottom: Φ dependence of the experimental in- tegrated intensity of the BiR peak from TOF-SARS data of Fig. 2b. The solid and dashed lines correspond to SARIC simulations assuming BiO and a BaO film terminations re- spectively. The data in this panel are matched at Φ = 450. azimuth scans generally correspond to low index crys- tallographic directions in the material. In Fig. 4b the minima in the experimental scan found at Φ = 00, 270 and 450 are associated with the [100], [210] and [110] di- rections of the cubic structure respectively, see Fig. 1. Because these effects strongly depend on the inter- atomic distances as well as the type of atoms on the surface, the magnitude of these minima can also be ex- plained qualitatively. The severe shadowing and bloking due to both Bi and O atoms lying along the [100] azimuth are the cause of the absence of BiR signal for Φ = 0 Although Oxygen atoms contribute to this effect, due to their lower mass they cannot completely shadow or block the incoming K ions or the recoiled Bi along [210] direc- tion. As a result only a shallow minimum is seen around Φ = 270. This is not the case for Bi atoms which are responsible for the more pronounced dip at Φ = 450 in spite of the larger interatomic separation along the [110] direction. Similarly, the absence of a peak for Φ ≃ 200 in the simulated BiR intensity for assumed BaO termina- tion (dashed line in Fig. 4b) can be easily understood in terms of Ba shadowing effects on Bi atoms along the [141] direction and the value α = 150 for the incidence angle. A detailed analysis of the lattice constants and possible surface relaxation based on numerical calculations of the shadowing and blocking cones are the purpose of future work. Conclusions - Achieving atomically smooth surfaces is proven to have a great impact on the possibility to use AR TOF-SARS and MSRI for structure and interface anal- ysis in complex oxides. This is a stepping stone for fu- ture characterization of artificially layered superconduct- ing compounds like Ba1−xKxBiO3 or the cuprates. The quantitative agreement between TOF-SARS and MSRI spectra of a BaBiO3 film show that AR MSRI can be a powerful tool for high resolution surface analysis. Data and simulations allowed us to clearly identify that the BaBiO3 film terminates in a BiO2 layer. Acknowledgements - This work was supported by DOE grant DE-AC-02-98CH1886. We thank W. J. Rabalais for providing us with the SARIC simulation code and J. A. Schultz for useful discussions. * E-mail: [email protected] 1 I. Bozovic, IEEE Trans. Appl. Supercond. 11, 2686 (2001) 2 I. Bozovic, G. Logvenov, I. Belca, B. Narimbetov, and I. Sveklo, Phys. Rev. Lett. 89, 107001 (2002). 3 J. W. Rabalais, Principles and Applications of Ion Scat- tering Spectrometry, Wiley-Interscience, 2003. 4 V. Bykov, L. Houssiau, and J. W. Rabalais, J. Phys. Chem. 104, 6340 (2000). 5 V. S. Smentkowski, A. R. Krauss, D. M. Gruen, J. C. Hole- cek, and J. A. Schultz, J. Vac. Sci. Technol. A 17, 2634 (1999). 6 O. Auciello, J. Appl. Phys. 100, 051614 (2006) and refer- ences therein. 7 J. Ahn, and J. W. Rabalais, Surf. Sci. 388, 121 (1997). 8 P. A. W. van der Heide, Q. D. Jiang, Y. S. Kim, and J. W. Rabalais, Surf. Sci. 473, 59 (2001). 9 L. F. Maththeiss, E. M. Gyorgy, and D. W. Johnson, Jr., Phys. Rev. B 37, R3745 (1988). 10 R. J. Cava, B. Batlogg, J. J. Krajewski, R. Farrow, L. W. Rupp, A. E. White, K. Short, W. F. Peck, and T. Kometani, Nature 332, 814 (1988). 11 S. Pei, J. D. Jorgensen, B. Dabrovski, D. G. Hinks, D. R. Richards, A. W. Mitchell, J. M. Newsam, S. K. Sinha, D. Vaknin, and A. J. Jacobson, Phys. Rev. B 41, 4126 (1990). 12 T. Thonhauser, and K. M. Rabe, Phys. Rev. B 73, 212106 (2006). 13 A. V. Puchkov, T. Timusk, M. A. Karlow, S. L. Cooper, P. D. Han, and D. A. Payne, Phys. Rev. B 54, 6686 (1996). 14 L. A. Klinkova, M. Uchida, Y. Matsui, V. I. Nikolaichik, and N. V. Barkovskii, Phys. Rev. B 67, R140501 (2003). 15 M. Aono, Y. Hou, C. Oshima, and Y. Ishizawa, Phys. Rev. Lett. 49, 567 (1982). 16 H. Niehus, W. Heiland, and E. Taglauer, Surf. Sci. Rep. 17, 213 (1993). 17 A. R. Krauss, Y. Lin, O. Auciello, G. J. Lamich, D. M. Gruen, J. A. Schultz, and R. P. H. Chang, J. Vac. Sci. Technol. A 12, 1943 (1994); J. A. Schultz, S. Ulrich, L. Woolverton, W. Burton, K. Waters, C. Kleina, and H. Wollnik, Nucl. Intrum. Methods Phys. Res. B 118, 758 (1996). 18 V. Bykov, C. Kim, M. M. Sung, K. J. Boyd, S. S. Todorov, and J. W. Rabalais, Nucl. Intrum. Methods Phys. Res. B 114, 371 (1996).

0704.1503

A Diagrammatic Category for the Representation Theory of U_q(sl_n)

A Diagrammatic Category for the Representation Theory of Uq(sln) Scott Edward Morrison B.Sc. (Hons) (University of New South Wales) 2001 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy Mathematics in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Vaughan Jones, Chair Professor Richard Borcherds Professor Steve Evans Spring 2007 http://arxiv.org/abs/0704.1503v1 A Diagrammatic Category for the Representation Theory of Uq(sln) Copyright 2007 Scott Edward Morrison Abstract A Diagrammatic Category for the Representation Theory of Uq(sln) Scott Edward Morrison Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vaughan Jones, Chair This thesis provides a partial answer to a question posed by Greg Kuperberg in [24] and again by Justin Roberts as problem 12.18 in Problems on invariants of knots and 3-manifolds [28], essentially: Can one describe the category of representations of the quantum group Uq(sln) (thought of as a spherical category) via generators and relations? For each n ≥ 0, I define a certain tensor category of trivalent graphs, modulo iso- topy, and construct a functor from this category onto (a full subcategory of) the category of representations of the quantum group Uq(sln). One would like to describe completely the kernel of this functor, by providing generators for the tensor category ideal. The re- sulting quotient of the diagrammatic category would then be a category equivalent to the representation category of Uq(sln). I make significant progress towards this, describing certain elements of the ker- nel, and some obstructions to further elements. It remains a conjecture that these elements really generate the kernel. The argument is essentially the following. Take some trivalent graph in the diagrammatic category for some value of n, and consider the morphism of Uq(sln) representations it is sent too. Forgetting the full action of Uq(sln), keeping only a Uq(sln−1) action, the source and target representations branch into direct sums, and the morphism becomes a matrix of maps of Uq(sln−1) representations. Arguing inductively now, we attempt to write each such matrix entry as a linear combination of diagrams for n − 1. This gives a functor dGT between diagrammatic categories, realising the forgetful functor at the representation theory level. Now, if a certain linear combination of diagrams for n is to be in the kernel of the representation functor, each matrix entry of dGT applied to that linear combination must already be in the kernel of the representation functor one level down. This allows us to perform inductive calculations, both establishing families of elements of the kernel, and finding obstructions to other linear combinations being in the kernel. This thesis is available electronically from the arXiv, at arXiv:0704.1503, and at http://tqft.net/thesis. http://arxiv.org/abs/0704.1503 http://tqft.net/thesis Contents 1 Introduction 1 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Temperley-Lieb algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Kuperberg’s spiders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 The ‘diagrammatic’ category Symn 5 2.1 Pivotal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quotients of a free tensor category . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Flow vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Polygonal webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Just enough representation theory 16 3.1 The Lie algebra sln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 The quantum groups Uq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Strictifying RepUq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Generators for FundRepUq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 The representation functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 The diagrammatic Gel‘fand-Tsetlin functor 31 4.1 Definition on generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Descent to the quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Calculations on small webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 A path model, and polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Describing the kernel 41 5.1 The I = H relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 The square-switch relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.3 The Kekulé relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.4 More about squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.6 Examples: Uq(sln), for n = 2, 3, 4 and 5. . . . . . . . . . . . . . . . . . . . . . 46 5.7 Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2 . . . . . . . . . . . . . . . . . . . . . 52 6 Relationships with previous work 66 6.1 The Temperley-Lieb category . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.2 Kuperberg’s spider for Uq(sl3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.3 Kim’s proposed spider for Uq(sl4) . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Tags and orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.5 Murakami, Ohtsuki and Yamada’s trivalent graph invariant . . . . . . . . . 68 6.6 Jeong and Kim on Uq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7 Other work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Future directions 71 Bibliography 74 A Appendices 77 A.1 Boring q-binomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.2 The Uq(sln) spider cheat sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Acknowledgements First I’d like to thank Vaughan for being such a great advisor. I’m glad he took a chance on me, and I hope he’s not too disappointed by the complete absence of subfactors in what fol- lows! I’m grateful for so many things; good advice, lots of mathematics, constant interest and encouragement, gentle reminders to keep climbing and work in balance, windsurfing lessons, and a great friendship. Thanks also to Dror Bar-Natan and Kevin Walker. I’ve learnt a ton from each of them, thoroughly enjoyed working with them, and look forward to more! And further thanks to Greg Kuperberg, for introducing me to the subject this thesis treats, and taking interest in my work. Conversations with Joel Kamnitzer, Mikhail Khovanov, Ari Nieh, Ben Webster, Noah Snyder and Justin Roberts helped me along the way. Thanks to my family, Pam, Graham and Adele Morrison, for endless love and support. And finally, thanks to my friends Nina White, Yossi Farjoun, Rahel Wachs, Erica Mikesh, and Carl Mautner. Chapter 1 Introduction 1.1 Summary The eventual goal is to provide a diagrammatic presentation of the representation theory of Uq(sln). The present work describes a category of diagrams, along with certain relations amongst these diagrams, and a functor from these diagrams to the representation theory. Further, I conjecture that the relations given are in fact all of them—that this functor is an equivalence of categories. We begin by defining a ‘freely generated category of diagrams’, and show that there’s a well-defined functor from this category to the category of representations of Uq(sln). Essentially, this is a matter of realising that the representation category is a piv- otal category, and, as a pivotal category, it is finitely generated. It’s then a matter of trying to find the kernel of this functor; if we could do this, the quotient by the kernel would give the desired diagrammatic category equivalent to the representation category. This work extends Kuperberg’s work [24] on RepUq(sl3), and agrees with a pre- viously conjectured [21] description of RepUq(sl4). Some, but not all, of the relations have been presented previously in the context of the quantum link invariants [11, 27]. There’s a detailed discussion of connections with previous work in §6. For each n ≥ 0 we have a category of (linear combinations of) diagrams Pivn =  , , , a+ b+ c = n 1 ≤ k ≤ n− 1 pivotal with edges labelled by integers 1 through n−1, generated by two types of trivalent vertices, and orientation reversing ‘tags’, as shown above. We allow arbitrary planar isotopies of the diagrams. This category has no relations; it is a free pivotal category. We can construct a functor from this category into the representation theory Repn : Pivn → RepUq(sln) . This functor is well-defined, in that isotopic diagrams give the same maps between rep- resentations. Our primary goal is thus to understand this functor, and to answer two questions: 1. Is Repn full? That is, do we obtain all morphisms between representations? 2. What is the kernel of Repn? When do different diagrams give the same maps of rep- resentations? Can we describe a diagrammatic quotient category which is equivalent to the representation theory? The first question has a relatively straightforward answer. We do not get all of RepUq(sln), but if we lower our expectations to the subcategory containing only the fun- damental representations, and their tensor products, then the functor is in fact full. Ku- perberg gave a proof of this fact for n = 3, by recognising the image of the functor using a Tannaka-Krein type theorem. This argument continues to work with only slight modifica- tions for all n. I’ll also give a direct proof using quantum Schur-Weyl duality, in §3.5. The second question has proved more difficult. Partial answers have been known for some time. I will describe a new method for discovering elements of the kernel, based on branching. This method also gives us a limited ability to find obstructions for further relations. The core of the idea is that there is a forgetful functor GT : RepUq(sln) → RepUq(sln−1) , which forgets the full Uq(sln) action but does not change the underlying linear maps, and that this should be reflected somehow in the diagrams. A diagram in Pivn ‘represents’ some morphism in RepUq(sln); thinking of this as a morphism in RepUq(sln−1) via GT , we can hope to represent it by diagrams in Pivn−1. This hope is borne out—in §4 I con- struct a functor dGT : Pivn → Mat(Pivn−1) (and explain what a ‘matrix category’ is), in such a way that the following diagram commutes: // RepUq(sln) Mat(Pivn−1) Mat(Repn−1) // RepUq(sln−1) With this functor on hand, we can begin determining the kernel of Repn. In particular, given a morphism in Pivn (that is, some linear combination of diagrams) we can consider the image under dGT . This is a matrix of (linear combinations of) diagrams in Pivn−1. Then the original diagrammatic morphism becomes zero in the Uq(sln) representation the- ory exactly if each entry in this matrix of diagrams is zero in the Uq(sln−1) representa- tion theory. Thus, if we understand the kernel of Repn−1, we can obtain quite strong restrictions on the kernel of Repn. Of course, the kernels of Rep2 and Rep3 are well known, given by the relations in the Temperley-Lieb category and Kuperberg’s spider for sl3. Moreover, the kernel of Rep1 is really easy to describe. The method described allows us to work up from these, to obtain relations for all Uq(sln). In §5 I use this approach to find three families of relations, and to show that these relations are the only ones of certain types. It remains a conjecture that the proposed rela- tions are in fact complete. The first family are the I = H relations, essentially correspond to 6− j symbols: = (−1)(n+1)a . For a given boundary, there are two types of squares, and each can be written as a linear combination of the others. I call these relations the ‘square-switch’ relations. When n + Σa− Σb ≥ 0, for max b ≤ l ≤ min a+ n we have min b m=max a n+Σa− Σb m+ l − Σb and when n+Σa− Σb ≤ 0, for max a ≤ l ≤ min b we have n+mina m=max b Σb− n− Σa m+ l − Σa− n Finally, there are relations amongst polygons of arbitrarily large size (but with a cutoff for each n), called the ‘Kekulé ’ relations. For each Σb ≤ j ≤ Σa+ n− 1, (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j 1.2 The Temperley-Lieb algebras The n = 2 part of this story has, unsurprisingly, been understood for a long time. The Temperley-Lieb category gives a diagrammatic presentation of the morphisms between tensor powers of the standard representation of Uq(sl2). The objects of this category are natural numbers, and the morphisms from n to m are Z[q, q−1]-linear combinations of dia- grams drawn in a horizontal strip consisting of non-intersecting arcs, with n arc endpoints on the bottom edge of the strip, and m on the top edge. (Notice, in particular, that I’m an optimist, not a pessimist; time goes up the page.) Composition of morphisms is achieved by gluing diagrams, removing each closed circle in exchange for a factor of [2]q. 1.3 Kuperberg’s spiders The n = 3 story, dates back to around Kuperberg’s paper [24]. There he defines the notion of a ‘spider’ (in this work, we use the parallel notion of a pivotal category), and constructs the spiders for each of the rank 2 Lie algebras A2 = su(3), B2 = sp(4) and G2 and their quantum analogues. Translated into a category, his A2 spider has objects words in (+,−), and morphisms (linear combinations of) oriented trivalent graphs drawn in a horizontal strip, with orientations of boundary points along the top and bottom edges coinciding with the target and source word objects, and each trivalent vertex either ‘oriented inwards’ or ‘oriented outwards’, subject to the relations = [3]q = q 2 + 1 + q−2 (1.3.1) = − [2]q (1.3.2) = + . (1.3.3) (Note that there’s a ‘typo’ in Equation (2) of [24], corresponding to Equation (1.3.3) above; the term has been replaced by another copy of the term.) Kuperberg proves that this category is equivalent to a full subcategory of the category of representations of the quantum group Uq(sl3); the subcategory with objects arbitrary tensor products of the two 3-dimensional representations. It is essentially an equinumeration proof, showing that the number of diagrams (modulo the above relations) with a given boundary agrees with the dimension of the appropriate Uq(sl3) invariant space. I’m unable to give an analogous equinumeration argument in what follows. Note that the n = 3 special case of my construction will not quite reproduce Kuperberg’s relations above; the bigon relation will involve a + [2]q, not a − [2]q. This is just a normalisation issue, resolved by multiplying each vertex by Chapter 2 The ‘diagrammatic’ category Symn Just as permutations form groups, planar diagrams up to planar isotopy form pivotal categories. In what follows, we’ll define a certain ‘free (strict) pivotal category’, Pivn along with a slight modification called Symn obtained by adding some symmetries and some relations for degenerate cases. Essentially, Pivn will be the category of trivalent graphs, with edges carrying both orientations and labels 1 through n − 1, up to planar isotopy. For lack of a better place, I’ll introduce the notion of a matrix category here; given any category C in which the Hom spaces are guaranteed to be abelian groups, we can form a new category Mat (C), whose objects are formal finite direct sums of objects in C, and whose morphisms are matrices of appropriate morphisms in C. Composition of mor- phisms is just matrix multiplication (here’s where we need the abelian group structure on Hom spaces). If the category already had direct sums, then there’s a natural isomorphism C ∼= Mat (C). 2.1 Pivotal categories I’ll use the formalism of pivotal categories in the following. This formalism is essentially interchangeable with that of spiders, due to Kuperberg [24], or of planar algebras, due to Jones [13].1 A pivotal category is a monoidal category2 C equipped with 1. a cofunctor ∗ : Cop → C, called the dual, 2. a natural isomorphism τ : 1C → ∗∗, 3. a natural isomorphism γ : ⊗ ◦ (∗ × ∗) → ∗ ◦ ⊗op, 1These alternatives are perhaps more desirable, as the pivotal category view forces us to make a artificial distinction between the domain and codomain of a morphism. I’ll have to keep reminding you this distinction doesn’t matter, in what follows. On the other hand, the categorical setup allows us to more easily incorporate the notion of direct sum. 2For our purposes, we need only consider strict monoidal categories, where, amongst other things, the tensor product is associative on the nose, not just up to an isomorphism. The definitions given here must be modified for non-strict monoidal categories. 4. an isomorphism e → e∗, where e is the neutral object for tensor product, and 5. for each object c ∈ C, a ‘pairing’ morphism pc : c∗ ⊗ c → e. with the natural isomorphisms satisfying certain coherence conditions, and the pairing morphisms certain axioms, all given in [2].3 The primary example of a pivotal category is the category of representations of an involutory or ribbon Hopf algebra [2]. A pivotal functor between two pivotal categories should intertwine the duality cofunctors, commute with the isomorphisms e → e∗, and take pairing morphisms to pairing morphisms. It should also intertwine the natural isomorphisms τ ; that is, for a functor F : C → D, we require τD = F(τCa ). There’s a similar condition for γ. In the case that τ and γ are simply the identity on each object (and e = e∗), we say the pivotal category is strict. Unfortunately, representations of a Hopf algebra do not generally form a strict pivotal category; τ cannot be the identity. However, every pivotal category is equivalent to a strict pivotal category [2]. We’ll take advantage of this later! In a pivotal category with e = e∗ and γ the identity (but not necessarily also τ ), the axioms satisfied by the natural isomorphism τ simplify to the requirements that it’s a tensor natural transformation: τa⊗b = τa ⊗ τb, and that τ∗a : a∗∗∗ → a∗ and τa∗ : a∗ → a∗∗∗ are inverse morphisms.4 In this same situation, the axioms for the pairing morphisms simplify to: 1. For each object a, = . (2.1.1) 2. For each morphism f : a → b, = . (2.1.2) 3. For all objects a and b, = . (2.1.3) 3Actually, [6] is an earlier reference for the strict case (see below), and they cite a preprint of [15] for the full version; however, the published version of that paper ended up introducing a slightly different notion, of ‘autonomy’ for duals. 4This condition might also be stated as τ commuting with the cofunctor ∗. From these, we can derive Lemma 2.1.1. The dual of a morphism can be written in terms of the original morphism, compo- nents of τ , and the pairing morphisms as Proof. using by the naturality of τ , which becomes by Equation (2.1.2), and finally by Equation (2.1.1). Note that the category of matrices over a pivotal category is still pivotal, in an essentially obvious way. 2.2 Quotients of a free tensor category To begin with, let’s just define a free (strict) monoidal category on some generating mor- phisms, which we’ll call Tn. We’ll then add some relations implementing planar isotopy to obtain Pivn, and some more relations to obtain Symn. The objects of Tn form a monoid under tensor product, with neutral object 0 (sometimes also called n), generated by the set {1, . . . , n− 1, 1∗, . . . , (n− 1)∗}. The ‘generating morphisms’ are diagrams , , , for each a = 1, . . . , n− 1 (but not for the ‘dual integers’ 1∗, . . . , (n − 1)∗), along with , , , and (2.2.1) again for each a = 1, . . . , n − 1, and finally and (2.2.2) for each a, b, c = 0, 1, . . . , n − 1 such that a + b + c = n. We’ll sometimes speak of the ‘type’ of a vertex; the first, outgoing, vertex here is of +-type, the second, incoming, vertex is of −-type. We say a vertex is ‘degenerate’ if one of its edges is labelled with 0. Notice there are no nondegenerate trivalent vertices for n = 2, and exactly one of each type for n = 3. To read off the source of such a morphism, you read across the lower boundary of the diagram; each endpoint of an arc labelled a gives a tensor factor of the source object, either a if the arc is oriented upwards, or a∗ if the arc is oriented downwards. To read the target, simply read across the upper boundary. Thus the source of is 0, and the target is a ⊗ b ⊗ c. All morphisms are then generated from these, by formal tensor product and composition, subject only to the usual identities of a tensor category. Next Pivn. This category has exactly the same objects. The morphisms, however, are arbitrary trivalent graphs drawn in a strip, which look locally like one of the pictures above, up to planar isotopy fixing the boundary of the strip. (Any boundary points of the graph must lie on the boundary of the strip.) Thus the graphs are oriented, with each edge carrying a label 1 through n, and edges only ever meet bivalently as in Equation (2.2.1) or trivalently as in Equation (2.2.2). Being a little more careful, we should ask that the diagrams have product structure near the boundary, that this is preserved throughout the isotopies, and that small discs around the trivalent vertices are carried around rigidly by the isotopies, so we can always see the ordering (not just the cyclic ordering) of the three edges incident at a vertex. (Note, though, that we’re not excited about being able to see this ordering; we’re going to quotient it out in a moment.) The source and target of such a graph can be read off from the graph exactly as described for the generators of Tn above. Next, we make this category into a strict pivotal category. For this, we need to define a duality functor, specify the evaluation morphisms, and then check the axioms of §2.1. The duality functor on objects is defined by 0∗ = 0 and otherwise (k)∗ = k∗, (k∗)∗ = k. On morphisms, it’s a π rotation of the strip the graph is drawn in. Clearly the double dual functor ∗∗ is the identity on the nose. The evaluation morphisms for a = 1, . . . , n − 1 are ‘leftwards-oriented’ cap diagrams; the evaluation morphisms for a = 1∗, . . . , n − 1∗ are ‘rightwards-oriented’. The axioms in Equation (2.1.1) and (2.1.2) are then satisfied auto- matically, because we allow isotopy of diagrams. The evaluation morphisms for iterated tensor products are just the nested cap diagrams, with the unique orientations and labels matching the required source object. This definition ensures the axiom of Equation (2.1.3) is satisfied. There’s a (tensor) functor from Tn to Pivn, which I’ll call Draw. Simply take a morphism in Tn, which can be written as a composition of tensor products of generating morphisms, and draw the corresponding diagram, using the usual rules of stacking boxes to represent composition, and juxtaposing boxes side by side to represent tensor product. The resulting diagram can then be interpreted as a morphism in Pivn. This is well-defined by the usual nonsense of [14], that the identities relating tensor product and composition in a tensor category correspond to ‘rigid’ isotopies (that is, isotopies which do not rotate boxes). The functor is obviously full; or at least, obviously modulo some Morse theory. The kernel of this functor is generated by the extra isotopies we allow in Pivn. Thus, as a tensor ideal of Tn, kerDraw is generated by = = , (2.2.3) = = , (2.2.4) = = , (2.2.5) = = . (2.2.6) There are other obvious variations of Equations (2.2.3) (rotating the vertex other way) and (2.2.5) (tags pointing the other way), but these follow easily from the ones given here. Whenever we want to define a functor on Pivn by defining it on generators, we need to check these morphisms are in the kernel. Finally, we can define the category we’re really interested in, which I’ll call Symn. We’ll add just a few more relations to Pivn; these will be motivated shortly when we define a functor from Tn to the representations category of Uq(sln). This functor will descend to the quotient Pivn, and then to the quotient Symn, and the relations we add from Pivn to Symn will be precisely the parts of the kernel of this functor which only involve a single generator. The real work of this thesis is, of course, understanding the rest of that kernel! We add relations insisting that the trivalent vertices are rotationally symmetric = , = , (2.2.7) that opposite tags cancel = , (2.2.8) that dual of a tag is a ±1 multiple of a tag = (−1)(n+1)a , = (−1)(n+1)a , (2.2.9) and that trivalent vertices ‘degenerate’ to tags = = . (2.2.10) Notice here we’re implicitly using the canonical identifications between the objects 0 ⊗ a, a, and a⊗ 0, available because our tensor categories are strict. Clearly the element of kerDraw in Equation (2.2.3) can be constructed by tensor product and composition out of the briefer rotations in Equation (2.2.7), and so in checking the well-definedness of a functor on Symn, we only need to worry about the latter. 2.3 Flow vertices We’ll now introduce two new types of vertices. You could add them as diagrammatic gen- erators, then impose as relations the formulas below, but it’s less cumbersome to just think of them as a convenient notation. In each of these vertices, there will be some ‘incoming’ and some ’outgoing’ edges, and the sum of the incoming edges will be the same as the sum of the outgoing edges. Definition 2.3.1. The ‘flow vertices’ are The convention here is that the ‘hidden tag’ lies on the ‘thick’ edge, and points counterclockwise. These extra vertices will be convenient in what follows, hiding a profu- sion of tags. ‘Splitting’ vertices are of +-type, ‘merging’ vertices are of −-type. 2.4 Polygonal webs To specify the kernel of the representation functor, in §5, we’ll need to introduce some notations for ‘polygonal webs’. These webs will come in two families, the ‘P’ family and the ‘Q’ family. In each family, the vertices around the polygon will alternate in type. A boundary edge which is connected to a +-vertex in a P-polygon will be connected to a −-vertex in a Q-polygon. For a, b ∈ Zk define5 the boundary label pattern L(a, b) = (bk − a1,−)⊗ (bk − ak,+)⊗ · · · ⊗ (b2 − a2,+)⊗ (b1 − a2,−)⊗ (b1 − a1,+). We’ll now define some elements of HomSymn(∅,L(a, b)), Pna,b;l for max b ≤ l ≤ min a + n and Qna,b;l for max a ≤ l ≤ min b by Pna,b;l = = (2.4.1) 5I realise this definition is ‘backwards’, or at least easier to read from right to left than from left to right. Sorry—I only realised too late. Qna,b;l = = . (2.4.2) (The diagrams are for k = 3, but you should understand the obvious generalisation for any k ∈ N.) You should consider the first of each pair of diagrams simply as notation for the second. Each edge label is a signed sum of the ‘flows labels’ on either side, determining signs by relative orientations. It’s trivial6 to see that for web diagrams with only ‘2 in, 1 out’ and ‘1 in, 2 out’ vertices, it’s always possible to pick a set of flow labels corresponding to an allowable set of edge labels. Not every set of flow labels, however, gives admissible edge labels, because the edge labels must be between 0 and n. The allowable flow labels for the P- and Q-polygons are exactly those for which ai, ai+1 ≤ bi ≤ n + ai, n + ai+1. Further, there’s a Z redundancy in flow labels; adding a constant to every flow label in a diagram doesn’t actually change anything. Taking this into account, there’s a finite set of pairs a, b for each n and k. The inequalities on the ‘internal flow label’ l for both Pna,b;l and Qna,b;l simply demand that all the internal edges have labels between 0 and n, inclusive. We denote the subspace of HomSymn(∅,L(a, b)) spanned by all the P-type poly- gons by APna,b, and the subspace spanned by the Q-type polygons by AQna,b. The space APna,b is min a −max b+ n − 1 dimensional (or 0 dimensional when this quantity is nega- tive), and the space AQna,b is max a−min b+ 1 dimensional. A word of warning; a and b each having k elements does not necessarily mean that Pna,b;l or Qna,b;l are honest 2k-gons. This can fail in two ways. First of all, if some ai = bi, ai + n = bi, ai+1 = bi or ai+1 + n = bi, then one of the external edges carries a trivial label. Further, when l takes on one of its extremal allowed values, at least one of the internal edges of the polygon becomes trivial, and the web becomes a tree, or a disjoint union of trees and arcs. For example (ignoring the distinction between source and target of morphisms; strictly speaking these should all be drawn with all boundary points at the 6Actually, perhaps only trivial after acknowledging that the disk in which the diagrams are drawn is simply connected. top of the diagram), P4(0,0,0),(1,1,1);1 = P (0,0,0),(1,1,1);2 = P4(0,0,0),(1,1,1);3 = P (0,0,0),(1,1,1);4 = P5(0,1,1),(2,2,2);2 = P (0,1,1),(2,2,2);3 = P5(0,1,1),(2,2,2);4 = P (0,1,1),(2,2,2);5 = Finally, it’s actually possible for a P-type polygon and a Q-type polygon to be equal in Symn. This can only happen in the case that a and b each have length 2, or at any length, when either a or b is constant. This only involves polygons with extreme values of the internal flow label l. Specifically Lemma 2.4.1. If a and b are each of length 2, Pna,b;max b = Qna,b;min b Pna,b;min a+n = Qna,b;max a. Further, even if a and b have length greater than 2, when a is a constant vector a = −→a Pn−→a ,b;a+n = Q −→a ,b;a and when b is a constant vector b = in Sym. Otherwise, the P- and Q-polygons are linearly independent in Symn. In particular, dim(APna,b ∩ AQna,b) is 0, 1 or 2 dimensional, depending on whether neither a nor b are constant, one is, or either both are or a and b have length 2. For example P3(0,0),(1,1);1 = = Q (0,0),(1,1);1, P3(0,0),(1,1);2 = , P3(0,0),(1,1);3 = = Q (0,0),(1,1);0. 2.4.1 Rotations It’s important to point out that the two types of polygons described above are actually closely related. In fact, by a ‘rotation’, and adding some tags, we can get from one to the other. This will be important in our later descriptions of the kernel of the representation functor. Unsurprisingly, this symmetry between the types of polygons is reflected in the kernel, and we’ll save half the effort in each proof, essentially only having to deal with one of the two types. First, by rotl : Zk → Zk we just mean ‘rotate left’: rotl(a1, a2, . . . , ak) = (a2, . . . , ak, a1), and by rotr : Zk → Zk ‘rotate right’. Lemma 2.4.2. Writing the identity out by explicit tensor products and compositions would be tedious; much easier is to write it diagrammatically first off: = , (2.4.3) although keep in mind that we intend the obvious generalisation to arbitrary size polygons, not just squares. Proof. It’s just a matter of using isotopy, the definition of the ‘flow vertices’ in §2.3 and cancelling tags by Equation (2.2.8). = = . Notice that it’s actually not important which way the tags point in Equation (2.4.3); we could reverse them, since by Equation (2.2.9) we’d just pick up an overall sign of (−1)(n+1)(2Σb−2Σa) = 1. We’ll sometimes write Equation (2.4.3) as Pnb−n,rotl(a);l = drotl Qna,b;l or equivalently, re-indexing Pna,b;l = drotl Qnrotr(b),a+n;l where the operation drotl is implicitly defined by Equation (2.4.3). (You should read drotl as ‘diagrammatic rotation’, and perhaps have the ‘d’ prefix remind you that we’re not just rotating, but also adding a tag, called d in the representation theory, to each external edge.) The corresponding identity writing a Q-polygon in terms of a P-polygon is Qnrotr(b),a+n;l = drotl Pna,b;l Qna,b;l = drotl Pnb−n,rotl(a);l Chapter 3 Just enough representation theory In this chapter, we’ll describe quite a bit of representation theory, but hopefully only what’s necessary for later! There are no new results in this section, although possi- bly Proposition 3.5.8, explaining roughly that ‘the fundamental representation theory of Uq(sln) is generated by triple invariants’ has never really been written down. 3.1 The Lie algebra sln The Lie algebra sln of traceless, n-by-n complex matrices contains the commutative Cartan subalgebra h of diagonal matrices, spanned by Hi = Ei,i − Ei+1,i+1 for i = 1, . . . , n − 1. (Here Ei,j is simply the matrix with a single nonzero entry, a 1 in the (i, j) position.) When sln acts on a vector space V , the action of h splits V into eigenspaces called weight spaces, with eigenvalues in h∗. The points Λ = {λ ∈ h∗ | λ(Hi) ∈ Z} ∼= Zn−1 form the weight lattice. We call h∗+ = {λ ∈ h∗ | λ(Hi) ≥ 0} the positive Weyl chamber, and the lattice points in it, Λ+ = Λ ∩ h∗+, the dominant weights. The positive Weyl chamber looks like Rn−1≥0 , and the dominant weights like N n−1, generated by certain fundamental weights, λ1, . . . , λn−1, the dual basis to {Hi} ⊂ h. Under the adjoint action of sln on itself, sln splits into h, as the 0-weight space, and one dimensional root spaces, each spanned by a root vector Ei 6=j with weight [Hk, Ei,j] = (δi,k + δj,k+1 − δi,k+1 − δj,k)Ei,j . In particular, the simple roots are defined E+i = Ei,i+1, for i = 1, . . . , n− 1. Similarly define E−i = Ei+1,i. Together, E i , E i and Hi for i = 1, . . . , n− 1 generate sln as a Lie algebra. The universal enveloping algebra U(sln) is a Hopf algebra generated as an asso- ciative algebra by symbols E+i , E i and Hi for i = 1, . . . , n− 1, subject only to the relations that the commutator of two symbols agrees with the Lie bracket in sln. (We won’t bother specify the other Hopf algebra structure, the comultiplication, counit or antipode. See the next section for all the details for quantum sln.) Trivially, sln and U(sln) have the same rep- resentation theory. We denote the subalgebra generated by Hi and E i for i = 1, . . . , n − 1 by U± (sln). 3.2 The quantum groups Uq(sln) We now recall the q-deformation of the Hopf algebra U(sln). Unfortunately we can’t straightforwardly recover U(sln) by setting q = 1, but we will see in §3.3 that this is the case at the level of representation theory. Let A = Q(q) be the field of rational functions in an indeterminate q, with co- efficients in Q. The quantum group Uq(sln) is a Hopf algebra over A, generated as an associative algebra by symbols X+i ,X i ,Ki and K i , for i = 1, . . . , n − 1, subject to the relations i = 1 = K KiKj = KjKi j = q ±(i,j)X±j Ki where (i, j) = 2 if i = j, −1 if |i− j| = 1, or 0 if |i− j| ≥ 2 X+i X i = δij Ki −K−1i q − q−1 X±i X j = X i when |i− j| ≥ 2 and the quantum Serre relations X±j − [2]q X = 0 when |i− j| = 1. The Hopf algebra structure is given by the following formulas for the comultiplication ∆, counit ε and antipode S. ∆(Ki) = Ki ⊗Ki, ∆(X+i ) = X i ⊗Ki + 1⊗X ∆(X−i ) = X i ⊗ 1 +K ε(Ki) = 1 ε(X i ) = 0, S(Ki) = K i S(X i ) = −X i S(X i ) = −KiX−i . (See, for example, [4, §9.1] for verification that these do indeed fit together to form a Hopf algebra.) There are Hopf subalgebras Uq(sln) , leaving out the generators X∓i . 3.3 Representations The finite-dimensional irreducible representations of both U(sln) and Uq(sln) can be suc- cinctly catalogued. A representation V of U(sln) is said to be a highest weight representation if there is a weight vector v ∈ V such that V = U(sln)− (v). The representation is said to have weight λ if v has weight λ. As with sln, representations of Uq(sln) split into the eigen- spaces of the action of the commutative subalgebra generated by the Ki and K i . These eigenspaces are again called weight spaces. A representation V of Uq(sln) is called a high weight representation if it contains some weight vector v so V = Uq(sln) (v). The finite- dimensional irreducible representations (henceforth called irreps) are then classified by: Proposition 3.3.1. Every irrep of U(sln) or of Uq(sln) is a highest weight representation, and there is precisely one irrep with weight λ for each λ ∈ Λ+, the set of dominant weights. Proof. See [7, §23] for the classical case, [4, §10.1] for the quantum case. Further, the finite dimensional representation theories are tensor categories and the tensor product of two irreps decomposes uniquely as a direct sum of other irreps. The combinatorics of the the tensor product structures in RepU(sln) and RepUq(sln) agree. A nice way to say this is that the Grothendieck rings of RepU(sln) and RepUq(sln) are isomorphic, identifying irreps with the same highest weight [4, §10.1]. 3.3.1 Fundamental representations Amongst the finite dimensional irreducible representations, there are some particularly simple ones, whose highest weights are the fundamental weights. These are called the fundamental representations, and there are n− 1 of them for U(sln) or Uq(sln). We’ll write, in either case, Va<n for the fundamental representation with weight λa. For U(sln) this is just the representation ∧aCn. It will be quite convenient to agree that V0<n and Vn<n (poor notation, I admit!) both denote the trivial representation. We can now define FundRepUq(sln); it’s the full subcategory of RepUq(sln), whose objects are generated by tensor product and duality from the trivial representa- tion A, and the fundamental representations Va<n. Note that there are no direct sums in FundRepUq(sln). All dominant weights are additively generated by fundamental weights, and this is reflected in the representation theory; the irrep with high weight λ = amaλa is con- tained, with multiplicity one, as a direct summand in the tensor product ⊗aV ⊗maa<n . This observation explains, to some extent, why it’s satisfactory to simply study FundRepUq(sln), instead of the full representation theory. Every irrep, while not neces- sarily allowed as an object of FundRepUq(sln), reappears in the Karoubi envelope [17, 37], since any irrep appears as a subrepresentation of some tensor product of fundamental rep- resentations. In fact, there’s a canonical equivalence of categories Kar (FundRepUq(sln)) ∼= RepUq(sln) . 3.3.2 The Gel‘fand-Tsetlin basis We’ll now define the Gel‘fand-Tsetlin basis [9, 10], a canonical basis which arises from the nice multiplicity free splitting rules for Uq(sln) →֒ Uq(sln+1). (Here, and hereafter, this inclusion is just Ki 7→ Ki,X±i 7→ X Lemma 3.3.2. If V is an irrep of Uq(sln+1), then as a representation of Uq(sln) each irrep appearing in V appears exactly once. More specifically, if V is the Uq(sln+1) irrep with highest weight (λ1, λ2, . . . , λn+1), then an Uq(sln) irrep W of weight (µ1, µ2, . . . , µn) appears (with multiplicity one) if and only if ‘µ fits inside λ’, that is λ1 ≤ µ1 ≤ λ2 ≤ µ2 ≤ · · · ≤ λn ≤ µn ≤ λn+1. In particular, the fundamental irreps Va<n break up as Va<n ∼= Va−1<n−1 ⊕ Va<n−1. Proof. See [4, §14.1.A], [23]. This allows us to inductively define ordered bases for Uq(sln) irreps, at least pro- jectively. This was first done in the quantum case in [12]. Choose, without much effort, a basis for the only irreducible representation of Uq(sl1), the trivial representation C(q). Now for any representation V of Uq(sln+1), decompose V over Uq(sln), as V ∼= α Wα, ordered lexicographically by highest weight, and define the Gel‘fand-Tsetlin basis of V to be the concatenation of the inclusions of the bases for each Wα into V . These inclusions are unique up to complex multiples, and we thus obtain a canonical projective basis. We’ll call this forgetful functor the ‘Gel‘fand-Tsetlin’ functor, GT : RepUq(sln) → RepUq(sln−1). Restricted to the fundamental part of the representation theory (see §3.3.1), it becomes a functor GT : FundRepUq(sln) → Mat (FundRepUq(sln)). Next, I’ll describe in gory detail the action of Uq(sln) on each of its fundamental representations Va<n, using the Gel‘fand-Tsetlin decomposition. We introduce maps p−1 and p0, the Uq(sln−1)-linear projections of Va<n ։ Va−1<n−1 and Va<n ։ Va<n−1. We also introduce the inclusions i−1 : Va−1<n−1 →֒ Va<n and i0 : Va<n−1 →֒ Va<n. Proposition 3.3.3. We can describe the action of Uq(sln) on Va<n recursively as follows. On V0<n and Vn<n, Uq(sln) acts trivially: X i by 0, and Ki by 1. On the non-trivial representations, we X+n−1|Va<n = i−1i0p−1p0, (3.3.1) X−n−1|Va<n = i0i−1p0p−1, Kn−1|Va<n = i−1(i−1p−1 + qi0p0)p−1 + i0(q i−1p−1 + i0p0)p0, (3.3.2) and for Z ∈ Uq(sln−1) Z |Va<n = i−1Z |Va−1<n−1p−1 + i0Z |Va<n−1p0. (3.3.3) Remark. Relative to the direct sum decomposition Va<n ∼= (Va−2<n−2 ⊕ Va−1<n−2) ⊕ (Va−1<n−2 ⊕ Va<n−2) under Uq(sln−2), we can write these as matrices, as X+n−1 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 , X−n−1 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Kn−1 = 1 0 0 0 0 q 0 0 0 0 q−1 0 0 0 0 1 , Z = Z 0 0 0 0 Z 0 0 0 0 Z 0 0 0 0 Z Proof. We need to check the relations in Uq(sln) involving X n−1 and Kn−1. We’re induc- tively assuming the others hold. To begin, the Ki’s automatically commute, as they’re diagonal with respect to the direct sum decomposition. Further, it’s clear from the defini- tions that X±n−1 and Kn−1 commute with the subalgebra Uq(sln−2). We then need to check the relation KiX j = q ±(i,j)X±j Ki, for |i− j| ≤ 1. Now Kn−1X n−1 = (i−1(i−1p−1 + qi0p0)p−1 + i0(q −1i−1p−1 + i0p0)p0)i−1i0p−1p0 = qi−1i0p−1p0, while X+n−1Kn−1 = i−1i0p−1p0(i−1(i−1p−1 + qi0p0)p−1 + i0(q −1i−1p−1 + i0p0)p0) = q−1i−1i0p−1p0, so Kn−1X n−1 = q 2X+n−1Kn−1 and similarly Kn−1X n−1 = q −2X−n−1Kn−1. Further n−1 = (i−1(i−1(i−1p−1 + qi0p0)p−1 + i0(q i−1p−1 + i0p0)p0)p−1+ + i0(i−1(i−1p−1 + qi0p0)p−1 + i0(q −1i−1p−1 + i0p0)p0)p0)i−1i0p−1p0 = i−1i0(q i−1p−1 + i0p0)p−1p0, while X+n−1Kn−2 = i−1i0p−1p0(i−1(i−1(i−1p−1 + qi0p0)p−1 + i0(q i−1p−1 + i0p0)p0)p−1+ + i0(i−1(i−1p−1 + qi0p0)p−1 + i0(q −1i−1p−1 + i0p0)p0)p0) = i−1i0(i−1p−1 + qi0p0)p−1p0, so Kn−2X n−1 = q −1X+n−1Kn−2, and by similar calculations Kn−2X n−1 = qX n−1Kn−2, Kn−1X n−2 = q −1X+n−2Kn−1 and Kn−1X n−2 = qX n−2Kn−1. Next, we check X+i X j − X i = δij q−q−1 , for i = j = n − 1, and for i = n− 1, j = n− 2. X+n−1X n−1 = i−1i0p−1p0i0i−1p0p−1 = i−1i0p0p−1 X−n−1X n−1 = i0i−1p0p−1i−1i0p−1p0, = i0i−1p−1p0 while Kn−1 −K−1n−1 = i−1(i−1p−1 + qi0p0)p−1 + i0(q−1i−1p−1 + i0p0)p0− − i−1(i−1p−1 + q−1i0p0)p−1 − i0(qi−1p−1 + i0p0)p0 = (q − q−1)i−1i0p0p−1 + (q−1 − q)i0i−1p−1p0, so X+n−1X n−1 −X n−1 = Kn−1−K q−q−1 , as desired, and X+n−1X n−2 = i−1i0p−1p0(i−1i0i−1p0p−1p−1 + i0i0i−1p0p−1p0) = i−1i0p−1i0i−1p0p−1p0 = 0 X−n−2X n−1 = (i−1i0i−1p0p−1p−1 + i0i0i−1p0p−1p0)i−1i0p−1p0 = i−1i0i−1p0p−1i0p−1p0 = 0 so X+n−1X n−2 −X n−2 = 0. Finally, we don’t need to check the Serre relations. By a result of [16], if you defined a quantum group without the Serre relations, you’d see them reappear in the ideal of elements acting by zero on all finite dimensional representations. Corollary 3.3.4. We’ll collect here some formulas for the generators acting on the dual representa- tions. Abusing notation somewhat, we write i0 : V a<n−1 →֒ V ∗a<n, i−1 : V ∗a−1<n−1 →֒ V ∗a<n, p0 : V ∗a<n ։ V a<n−1 and p−1 : V a<n ։ V a−1<n−1. With these conventions, (i0|Va<n−1) ∗ = p0|V ∗a<n , and so on. Then X+n−1|V ∗a<n = −qi0i−1p0p−1, X−n−1|V ∗a<n = −q i−1i0p−1p0, Kn−1|V ∗a<n = i−1(i−1p−1 + q i0p0)p−1 + i0(q −1p−1 + i0p0)p0, and for Z ∈ Uq(sln−1) Z |V ∗a<n = i−1Z |V a−1<n−1 p−1 + i0Z |V ∗ a<n−1 3.4 Strictifying RepUq(sln) We’ll now make something of a digression into abstract nonsense. (But it’s good abstract nonsense!) It’s worth understanding at this point that while RepUq(sln) isn’t a strict piv- otal category, it can be ‘strictified’. That is, the natural isomorphism τ : 1 → ∗∗ isn’t the identity, but RepUq(sln) is equivalent to another pivotal category in which that ‘pivotal isomorphism’ τ is the identity. This strictification will be a full subcategory of RepUq(sln) (as a tensor category), but with a new, modified duality functor. While this discussion may seem a little esoteric, it actually pays off! Our eventual goal is a diagrammatic category equivalent to RepUq(sln), with diagrams which are only defined up to planar isotopy. Such a pivotal category will automatically be strict. The ‘strictification’ we perform in this section bridges part of the inevitable gap between such a diagrammatic category, and the conventionally defined representation category. Of course, it would be possible to have defined RepUq(sln) in the first place in a way which made it strict as a pivotal category, but this would have required a strange and unmotivated definition of the dual of a morphism. Hopefully in the strictification we describe here, you’ll see the exact origin of this unfortunate dual. First of all, let’s explicitly describe the pivotal category structure on RepUq(sln), in order to justify the statement that it is not strict1. We need to describe the duality functor ∗, pairing maps, and the ‘pivotal’ natural isomorphism τ : 1 → ∗∗. The contravariant2 duality functor ∗ on RepUq(sln) is given by the usual dual- ity functor for linear maps. We need to dress up duals of representations of Uq(sln) as representations again, which we do via the the antipode S, just as we use the comultipli- cation to turn tensor products of representations into representations. Thus for V some representation of Uq(sln), Z ∈ Uq(sln), f ∈ V ∗ and v ∈ V , we say (Zf)(v) = f(S(Z)v). The pairing maps pV : V ∗ ⊗ V → A are just the usual duality pairing maps for duals of vector spaces. That these are maps of representations follows easily from the Hopf algebra axioms. Be careful, however, to remember that the other vector space pairing map V ⊗ V ∗ → A is not generally a map of representations of the Hopf algebra. Similarly, the copairing cV : A → V ⊗ V ∗ is equivariant, while A → V ∗ ⊗ V is not. The pivotal isomorphism is where things get interesting, and we make use of the particular structure of the quantum group Uq(sln). Definition 3.4.1. We’ll define the element τn = K j=1 K j(n−j) j of Uq(sln). Lemma 3.4.2. Abusing notation, this gives an isomorphism of Uq(sln) representations, τn : V V ∗∗, providing the components of the pivotal natural isomorphism. Proof. This relies on the formulas for the antipode acting on Uq(sln); we need to check i = S 2(X±i )τn. Since S2(X±i ) = q ±2X±i , this becomes the condition n = q ±2X±i , 1Not strict as a pivotal category, that is. We’ve defined it in such a way that it’s strict as a tensor category, meaning we don’t bother explicitly reassociating tensor products. 2In fact, it’s doubly contravariant; both with respect to composition, and tensor product. which follows immediately from the definition of τn and the commutation relations in the quantum group. We also need to check that τ satisfies the axioms for a pivotal natural isomor- phism. Since the Ki are group-like in Uq(sln) (i.e. ∆(Ki) = Ki ⊗Ki), τ is a tensor natural transformation. Further τ∗V |V ∗∗∗ ◦ τV ∗ |V ∗ = (Kρ)∗|V ∗∗∗ ◦Kρ|V ∗ = (Kρ)S(Kρ) so τ∗V and τV ∗ are inverses, as required. To construct the strictification of RepUq(sln), written as SRepUq(sln), we be- gin by defining it as a tensor category, taking the full tensor subcategory of RepUq(sln) whose objects are generated by tensor product (but not duality) from the trivial represen- tation A, Va<n and V ∗a<n, for each a = 1, . . . , n − 1. Before describing the pivotal struc- ture on SRepUq(sln), we might as well specify the equivalence; already there’s only one sensible choice. We can include SRepUq(sln) into RepUq(sln) in one direction. For the other, we simply send V a<n (that is, the n-th iterated dual) to Va<n or V a<n, depending on whether n is even or odd, and modify each morphism by pre- and post-composing with the unique3 appropriate tensor product of compositions of τ and τ−1. Clearly the compo- sition RepUq(sln) → SRepUq(sln) → RepUq(sln) is not the identity functor, but equally easily it’s naturally isomorphic to the identity, again via appropriate tensor products of compositions of τ . Here, we’re doing nothing more than identifying objects in a category related by a certain family of isomorphisms, and pointing out that the result is equivalent to what we started with! The interesting aspect comes in the pivotal structure on SRepUq(sln). Of course, we define the duality cofunctor on objects of SRepUq(sln) so it exchanges Va<n and V ∗a<n. It turns out (inevitably, according to the strictification result of [2], but explicitly here) that if we define pairing maps on SRepUq(sln) by taking the image of the pairing maps for RepUq(sln), we make SRepUq(sln) a strict pivotal category (and of course, we thus make the equivalence of categories from the previous paragraph an equivalence of pivotal categories). Explicitly, then, the pairing morphisms on SRepUq(sln) are pSa : V ∗a<n ⊗ Va<n → A = pa and pSa∗ : Va<n ⊗ V ∗a<n → A = pa∗ ◦ (τa ⊗ 1a∗). We’ve yet to define the duality cofunctor at the level of morphisms in SRepUq(sln); there’s in fact a unique definition forced on us by Lemma 2.1.1. Thus f∗ = (1a∗ ⊗ pb∗) ◦ (1a∗ ⊗ (τb ◦ f ◦ τ−1a )⊗ 1b∗)) ◦ (p∗a ⊗ 1b∗). The processes of strictifying, and of limiting our attention to the fundamental part of the representation theory, are independent; the discussion above applies exactly to FundRepUq(sln). In that case, we obtain the full subcategory SFundRepUq(sln). 3It’s unique, given the axioms for τ described in §2.1, and established for our particular τ in Lemma 3.4.2. 3.5 Generators for FundRepUq(sln) In this section we’ll define certain morphisms in FundRepUq(sln), which we’ll term ‘el- ementary’ morphisms. Later, in Proposition 3.5.8 we’ll show that they generate all of FundRepUq(sln) as a tensor category. We already have the duality pairing and copairing maps, discussed above, pV : V ∗ ⊗ V → A, and cV : A → V ⊗ V ∗, for each fundamental representation V = Va<n, and for arbitrary iterated duals V = V a<n . We also have the non-identity isomorphisms τ identifying objects with their double duals. Beyond those, we’ll introduce some more interesting maps: • a map which identifies a fundamental representations with the dual of another fun- damental representation, da,n : Va<n =→ V ∗n−a<n, along with the inverses of these maps, • a ‘triple invariant’, living in the tensor product of three fundamental representations vna,b,c : A → Va<n ⊗ Vb<n ⊗ Vc<n with a+ b+ c = n, and • a ‘triple coinvariant’, wna,b,c : Va<n ⊗ Vb<n ⊗ Vc<n → A, again with a+ b+ c = n. Definition 3.5.1. The map da,n : Va<n → V ∗n−a<n is specified recursively by d0,n : 1 7→ 1∗, dn,n : 1 7→ 1∗, da,n = i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0. Lemma 3.5.2. The map da,n is a map of Uq(sln) representations. Proof. Since da,n is defined in terms of Uq(sln−1) equivariant maps, we need only check da,n commutes with X n−1 and Kn−1. We’ll do one calculation explicitly: da,nX n−1 = i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0 i−1i0p−1p0 = i0da−1,n−1i0p−1p0 i0da−2,n−2p−1 + (−q)1−ai−1da−1,n−2p0 i0p−1p0 = (−q)1−ai0i−1da−1,n−2p−1p0, while X+n−1da,n = −qi0i−1p0p−1 i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0 = (−q)1−ai0i−1p0da,n−1p0) = (−q)1−ai0i−1p0 i0da−1,n−2p−1 + (−q)−ai−1da,n−2p0 = (−q)1−ai0i−1da−1,n−2p−1p0). The other two cases (commuting with X−n−1 and Kn−1) are pretty much the same. Proposition 3.5.3. The duals of the maps da,n satisfy: d∗a,nτn = (−1)(n+1)adn−a,n. (3.5.1) We need two lemmas before proving this. Lemma 3.5.4. j |Va<n = q n−ai−1p−1 + q −ai0p0. Proof. Arguing inductively, and using the formula for Kn−1|Va<n from Equation (3.3.2), we j |Va<n = j |Va−1<n−1  p−1 + i0 j |Va<n−1 Kn−1n−1 |Va<n i−1(q n−ai−1p−1 + q −a+1i0p0)p−1 + i0(q n−a−1i−1p−1 + q −ai0p0)p0 i−1(i−1p−1 + q i0p0)p−1 + i0(q i−1p−1 + i0p0)p0 = qn−ai−1i−1p−1p−1 + q n−ai−1i0p0p−1 + q −ai0i−1p−1p0 + q −ai0i0p0p0 = qn−ai−1p−1 + q −ai0p0. Lemma 3.5.5. τn|Va<n = q i−1τn−1|Va−1<n−1p−1 + q i0τn−1|Va<n−1p0 Proof. Making use of Equation (3.3.3) and Lemma 3.5.4: τn|Va<n = τn−1|Va<n × j |Va<n i−1τn−1|Va−1<n−1p−1 + i0τn−1|Va<n−1p0 qn−ai−1p−1 + q −ai0p0 = qn−ai−1τn−1|Va−1<n−1p−1 + q i0τn−1|Va<n−1p0. Proof of Proposition 3.5.3. The proposition certainly holds for a = 0 or a = n, where all three of the maps d∗a,n, τn and dn−a,n are just the identity, and the sign is +1. Otherwise, we proceed inductively. First, write d∗a,n = i−1d a−1,n−1p0 + (−q)−ai0d∗a,n−1p−1. Using Lemma 3.5.5 we then have d∗a,nτn = a−1,n−1p0 + (−q)−ai0d∗a,n−1p−1 qai−1τn−1p−1 + q i0τn−1p0 = (−q)−aqai0d∗a,n−1τn−1p−1 + qa−ni−1d∗a−1,n−1τn−1p0 = (−1)ai0(−1)nadn−1−a,n−1p−1 + qa−ni−1(−1)n(a−1)dn−a,n−1p0 = (−1)(n+1)ai0dn−1−a,n−1p−1 + qa−n(−1)n(a−1)i−1dn−a,n−1p0, while (−1)(n+1)adn−a,n = (−1)(n+1)a(i0dn−a−1,n−1p−1 + (−q)a−ni−1dn−a,n−1p0) = (−1)(n+1)ai0dn−a−1,n−1p−1 + qa−n(−1)n(a−1)i−1dn−a,n−1p0). Finally, we need to define the triple invariants vna,b,c and w a,b,c. Definition 3.5.6. When a + b + c = m, we define vna,b,c ∈ Va<n ⊗ Vb<n ⊗ Vc<n and wna,b,c ∈ V ∗a<n ⊗ V ∗b<n ⊗ V ∗c<n by the formulas v00,0,0 = 1⊗ 1⊗ 1, w00,0,0 = 1 ∗ ⊗ 1∗ ⊗ 1∗, vna,b,c = (−1)cqb+c(i−1 ⊗ i0 ⊗ i0)(vn−1a−1,b,c) + (3.5.2) (−1)aqc (i0 ⊗ i−1 ⊗ i0)(vn−1a,b−1,c) + (−1)b (i0 ⊗ i0 ⊗ i−1)(vn−1a,b,c−1), wna,b,c = (−1)c (wn−1a−1,b,c)(p−1 ⊗ p0 ⊗ p0) + (3.5.3) (−1)aq−a (wn−1a,b−1,c)(p0 ⊗ p−1 ⊗ p0) + (−1)bq−a−b(wn−1a,b,c−1)(p0 ⊗ p0 ⊗ p−1). Lemma 3.5.7. The maps vna,b,c and w a,b,c are maps of Uq(sln) representations. Proof. As in Lemma 3.5.2, we just need to check that these maps commute with X±n−1 and Kn−1. We’ll do the explicit calculation for X n−1; here we need to check that X+n−1|Va<b⊗Vb<n⊗Vc<bv a,b,c = 0 wna,b,cX n−1|Va<b⊗Vb<n⊗Vc<b First, we need to know how X+n−1 acts on the tensor product of three representations, via ∆(2)(X+n−1) = X n−1 ⊗Kn−1 ⊗Kn−1 + 1⊗X n−1 ⊗Kn−1 + 1⊗ 1⊗X Next, let’s use two steps of the inductive definition of vna,b,c to write vna,b,c = q 2b+2c(i−1i−1 ⊗ i0i0 ⊗ i0i0)(vn−2a−2,b,c) + q2c (i0i0 ⊗ i−1i−1 ⊗ i0i0)(vn−2a,b−2,c) + (i0i0 ⊗ i0i0 ⊗ i−1i−1)(vn−2a,b,c−2) + (−1)a+bqc(−i0i0 ⊗ i−1i0 ⊗ i0i−1 + q−1i0i0 ⊗ i0i−1 ⊗ i−1i0)(vn−2a,b−1,c−1) + (−1)b+cqb+c(i−1i0 ⊗ i0i0 ⊗ i0i−1 − q−1i0i−1 ⊗ i0i0 ⊗ i−1i0)(vn−2a−1,b,c−1) + (−1)a+cqb+2c(−i−1i0 ⊗ i0i−1 ⊗ i0i0 + q−1i0i−1 ⊗ i−1i0 ⊗ i0i0)(vn−2a−1,b−1,c). We’ll now show X+n−1 kills each term (meaning each line, as displayed above) separately. In each case, it follows immediately from the formulas for X+n−1 and Kn−1 in Equations (3.3.1) and (3.3.2). In the first three terms, we use X+n−1i−1i−1 = X i0i0 = 0. In the fourth term, we see ∆(2)(X+n−1)(−i0i0 ⊗ i−1i0 ⊗ i0i−1 + q−1i0i0 ⊗ i0i−1 ⊗ i−1i0) = −i0i0 ⊗ i−1i0 ⊗ i−1i0 + q−1qi0i0 ⊗ i−1i0 ⊗ i−1i0 Here only one of the three terms of ∆(2)(X+n−1) acts nontrivially on each of the two terms. The fifth and sixth terms are exactly analogous. For wna,b,c we use the same trick; write it in terms of w a−2,b,c, w a,b−2,c, w a,b,c−2, wn−2a,b−1,c−1, w a−1,b,c−1 and w a−1,b−1,c, and show that each of these terms multiplied by X gives zero separately. Remark. Going through this proof carefully, you’ll see that it would still work with some variation allowed in constants the definitions of vna,b,c and of w a,b,c in Equations (3.5.2) and (3.5.3). However, given the normalisation for da,n that we’ve chosen in Definition 3.5.1, these normalisation constants are pinned down by Lemmas 4.2.4 and 4.2.6 below. Later, we’ll discuss relationships between these elementary morphisms, but for now we want to justify our interest in them. Proposition 3.5.8. The elementary morphisms generate, via tensor product, composition and lin- ear combination, all the morphisms in FundRepUq(sln). Remark. Certainly, they can’t generate all of RepUq(sln), simply because the sources and targets of elementary morphisms are all in FundRepUq(sln)! Remark. A fairly abstract proof of this fact for n = 3 has been given by Kuperberg [24].4 Briefly, he specialises to q = 1, and considers the subcategory of FundRepU(sl3) ∼= FundRepSU(3) generated by the elementary morphisms. He extends this by formally adding kernels and cokernels of morphisms (that is, by taking the Karoubi envelope). This extension being (equivalent to) all of RepSU(3) is enough to obtain the result. To see this, he notes that the extension is the sort of category to which an appropriate Tannaka-Krein theorem applies, allowing him to say that it is the representation category of some com- pact Lie group. Some arguments about the symmetries of the ‘triple invariant’ morphisms allow him to conclude that this group must in fact be SU(3). While I presume this proof can be extended to cover all n, I prefer to give a more direct proof, based on the quantum version of Frobenius-Schur duality. Proof. It’s a sort of bootstrap argument. First, we notice that the action of the braid group Bm on tensor powers of the standard representation V1<n can be written in terms of ele- mentary morphisms. Next, we recall that the braid group action generates all the endo- morphisms of V ⊗m1<n , and finally, we show how to map arbitrary tensor products of fun- damental representations into a tensor power of the standard representation, using only elementary morphisms. 4His other results additionally give a direct proof for n = 3, although quite different from the one here. These ideas are encapsulated in the following four lemmas. Lemma 3.5.9. The action of the braid group Bm on V 1<n , given by R-matrices, can be written in terms of elementary morphisms. Proof. We only need to prove the result for V ⊗21<n. There, we can be particularly lazy, taking advantage of the fact that dimHomUq(sln) V ⊗21<n, V = 2. (There are many paths to see- ing this, the path of least effort perhaps being that HomUq(sln) V ⊗21<n, V is isomorphic to HomUq(sln) V1<n ⊗ V ∗1<n, V ∗1<n ⊗ V1<n , and that both the source and target representations there decompose into the direct sum of the adjoint representation and the trivial represen- tation.) The map p = (wn1,1,n−2 ⊗ 1V1<n ⊗ 1V1<n) ◦ (1V1<n ⊗ 1V1<n ⊗ vnn−2,1,1) is not a multiple of the identity, so every endomorphism of V ⊗21<n is a linear combination of compositions of elementary morphisms, in particular the braiding. Remark. In fact, the map associated to a positive crossing is qn−11 − qnp. It’s easy to prove that this, along with the negative crossing (obtained by replacing q with q−1), is the only linear combination of 1 and p which satisfies the braid relation. One could presumably also check that this agrees with the explicit formulas given in [4, §8.3 and §10.1]5. Lemma 3.5.10. The image of Bm linearly spans HomUq(sln) V ⊗m1<n , V Proof. This is the quantum version of Schur-Weyl duality. See [4, §10.2B] and [12]. Lemma 3.5.11. For any tensor product of fundamental representations i Vαi , there’s some nat- ural number m and a pair of morphisms constructed out of elementary ones ι : i Vαi → V and π : V ⊗m1 → i Vαi such that π ◦ ι = 1Ni Vαi . Proof. We just need to do this for a single fundamental representation, then tensor together those morphisms. For a single fundamental representation, first define ι′ : Va → V1 ⊗ Va−1 = (pVn−a ⊗ 1V1 ⊗ 1Va−1) ◦ (da ⊗ vnn−a,1,a−1) and π′ : V1 ⊗ Va−1 → Va = x(wn1,a−1,n−a ⊗ da) ◦ (1V1 ⊗ 1Va−1 ⊗ cVn−a). The composition π′ ◦ ι′ is an endomorphism of the irreducible Va, which I claim is nonzero, and so for some choice of the coefficient x is the identity. Now build the maps ι : Va → V ⊗a1 and π : V → Va as iterated compositions of these maps. Lemma 3.5.12. In fact, given two such tensor products of fundamental representations i Vαi j Vβj , such that there is some nonzero Uq(sln) map between them, it’s possible to choose ια, πα, ιβ and πβ as in the previous lemma, with the same value of m. 5Although be careful there — the formula for the universal R-matrix in §8.3.C is incorrect, although it shouldn’t matter for such a small representation. The order of the product is backwards [5]. Proof. (This argument is due to Ben Webster. Thanks!) First just pick ια, πα, ι β and π as in the previous lemma, with possibly different values of m, say mα and mβ . Now, if there’s some Uq(sln) map between i Vαi and j Vβj , then there’s also a Uq(sln) map between V ⊗mα1 and V 1 . This can happen if and only if there’s some sln map, and in fact some SL(n) map, between the corresponding classical representations,Cn⊗mα and Cn⊗mβ . This SL(n) intertwiner guarantees that the center of SL(n) acts in the same way on each representation; the element 1I acts by 1mα/n and 1mβ/n on the two representations, so mα and mβ must be congruent mod n. Assuming, without loss of generality, that mβ ≤ mα we now define ιβ = ι β ⊗ ι⊗(mα−mβ)/n, and πβ = π′β ⊗ π⊗(mα−mβ)/n, where ι : A → V and π : V ⊗n1 → A are diagrammatic morphisms satisfying π ◦ ι = 1A. (These certainly exist!) Putting all this together, we take an arbitrary morphism φ : i Vαi → j Vβj in FundRepUq(sln), and pick ια, πα, ιβ and πβ as in the last lemma. Then φ = πβ ◦ ιβ ◦ φ ◦ πα ◦ ια. The middle of this composition, ιβ ◦ φ ◦πα, is an endomorphism of V ⊗m1 , so can be written as a linear combination of braids, and hence as a linear combination of morphisms built from elementary morphisms. 3.6 The representation functor Finally, everything is in place for a definition of the representation functor Rep : Symn → SFundRepUq(sln). (Notice that we’re defining the functor with target the strictification SFundRepUq(sln), as discussed in §3.4. To obtain the functor to the more usual represen- tation theory FundRepUq(sln), you need to compose with the equivalences of categories described in that section; of course, this is just the inclusion of a full subcategory!) First define Rep′ on the free tensor category Tn, by ′(a ∈ {1, . . . , n− 1}) = Va<n, Rep′(a∗) = V ∗a<n, on objects, and = pa, Rep = pa∗ ◦ (τn ⊗ 1), = ca, Rep = (Id⊗ τ−1n ) ◦ ca∗ , = da,n, Rep = d−1a,n, = (−1)(n+1)ada,n, Rep′ = (−1)(n+1)ad−1a,n,  = vna,b,c Rep  = wna,b,c on morphisms. Proposition 3.6.1. The functor Rep′ descends to a functor defined on the quotient Symn → SFundRepUq(sln). This is proved in the next chapter; we could do it now, but the proof will read more nicely once we have better diagrams available. Chapter 4 The diagrammatic Gel‘fand-Tsetlin functor It’s now time to define the diagrammatic Gel‘fand-Tsetlin functor, dGT : Symn → Mat Symn−1 The first incarnation of the diagrammatic functor will be a functor defined (in §4.1) on the free version of the diagrammatic category, dGT ′ : Tn → Mat Symn−1 . We’ll need to show that it descends to the quotient Symn (in §4.2). This definition is made so that the perimeter of the diagram dGT ′ // SFundRepUq(sln) Symn−1 ) Mat(Rep) // Mat (SFundRepUq(sln−1)) (4.0.1) commutes. Happily, as an easy consequence of this, we’ll see that the representation functor Rep : Tn → SFundRepUq(sln) also descends to the quotient Symn. At the end of the chapter in §4.4 we describe how to compute the diagrammatic Gel‘fand-Tsetlin functor, and perform a few small calculations which we’ll need later. 4.1 Definition on generators The target category for the diagrammatic Gel‘fand-Tsetlin functor is the matrix category over Symn−1. We thus need to send each object of Tn to a direct sum of objects of Symn−1, and for each generating morphism of Tn, we need to pick an appropriate matrix of mor- phisms in Symn−1. On objects, we use the obvious dGT ′(a) = (a− 1)⊕ a (4.1.1) dGT ′(a∗) = (a− 1)∗ ⊕ a∗ (omitting the nonsensical direct summand in the case that a = 0, 0∗, n or n∗), extending to tensor products by distributing over direct sum. For morphisms we’ll take advantage of the lack of ‘multiplicities’ in Equation 4.1.1 to save on some notation. An arbitrary morphism in Mat Symn−1 has rows and columns indexed by tensor products of the fun- damental (and trivial) objects and their duals, 0, 1, . . . , n − 1, 0∗, 1∗, . . . , (n − 1)∗. Notice, however, that morphisms in the image of dGT ′ have distinct labels on each row (and also on each column). Moreover, a (diagrammatic) morphism in Symn−1 explicitly encodes its own source and target (reading across the top and bottom boundary points). We’ll thus abuse notation, and write a sum of matrix entries, instead of the actual matrix, safe in the knowledge that we can unambiguously reconstruct the matrix, working out which term should sit in each matrix entry. For example, if f : a → b is a morphism in Tn, then dGT ′(f) is a matrix (a − 1) ⊕ a → (b − 1) ⊕ b, which we ought to write as f11 f12 f21 f22 , but will simply write as f11 + f12 + f21 + f22. This notational abuse is simply for the sake of brevity when writing down matrices with many zero entries. When composing matrices written this way, you simply distribute the composition over summation, ignoring any non-composable terms. That said, we now define dGT ′ on trivalent vertices by dGT ′  = (−1)cqb+c + (−1)aqc + (−1)b , dGT ′  = (−1)c + (−1)aq−a + (−1)bq−a−b (the right hand sides of these equations are secretly 8× 1 and 1× 8 matrices, respectively), on the cups and caps by dGT ′ = + dGT ′ = qn−a + q−a dGT ′ = + dGT ′ = qa−n + qa , (4.1.2) and on the tags by dGT ′ = + (−1)aq−a dGT ′ = (−1)n+a + q−a dGT ′ = qa + (−1)n+a dGT ′ = (−1)aqa + . (4.1.3) The functor dGT ′ then extends to all of Tn as a tensor functor. Proposition 4.1.1. The outer square of Equation (4.0.1) commutes. Proof. Straight from the definitions; compare the definition above with the definition of Rep′ from §3.6, and use the inductive Definitions 3.5.1 and 3.5.6 of the generating mor- phisms over in the representation theory. 4.2 Descent to the quotient Proposition 4.2.1. The functor dGT ′ descends from Tn to the quotient Symn, where we call it simply dGT . Proof. Read the following Lemmas 4.2.2, 4.2.3, 4.2.4, 4.2.5 and 4.2.6. Lemma 4.2.2. We first check the relation in Equation (2.2.4). dGT ′  = dGT ′ = dGT ′ dGT ′  = dGT ′ = dGT ′ Proof. This is direct from the definitions in Equation (4.1.2), and using the fact that we can straighten strands in the target category Symn−1. Lemma 4.2.3. Next, the relations in Equation (2.2.5). dGT ′ = dGT ′ dGT ′ = dGT ′ Proof. dGT ′ = qn−a + (−1)a = qn−a + (−1)a = dGT ′ making use of the definitions in Equations (4.1.2) and (4.1.3), and the identity appearing in Equation (2.2.5) for Symn−1, and dGT ′ = + (−1)aq−a = + (−1)aq−a = dGT ′ Lemma 4.2.4. We check the ‘rotation relation’ from Equation (2.2.7) is in the kernel of dGT ′, which of course implies the ‘2π rotation relation from Equations (2.2.3) is also in the kernel. dGT ′ (4.2.1) = (−1)c + (−1)aq−a + (−1)bq−a−b = dGT ′ dGT ′ (4.2.2) = (−1)cqb+c + (−1)aqc + (−1)b = dGT ′ Proof. The statement of the lemma essentially contains the proof; we’ll spell out Equation (4.2.1) in gory detail: dGT ′ = dGT ′ 1c∗ ⊗ dGT ′ ⊗ 1c dGT ′ ⊗ 1a⊗b⊗c 1c∗ ⊗ (−1)b + (−1)cq−c + + (−1)aq−c−a qc−n + qc ⊗ 1a⊗b⊗c = (−1)c + (−1)aq−a + (−1)bq−a−b = dGT ′ Going from the third last to the second last line, we simply throw out all non-composable cross terms. The next two lemmas are similarly direct. Lemma 4.2.5. dGT ′ = + (−1)aq−a = (−1)(n+1)adGT ′ dGT ′ = qa + (−1)n+a = (−1)(n+1)adGT ′ Lemma 4.2.6. dGT ′  = dGT ′ dGT ′  = dGT ′ We can now give the Proof of Proposition 3.6.1. Somewhat surprisingly, the fact that dGT ′ descends from Tn to Symn implies that Rep′ : Tn → SFundRepUq(sln) also descends to Symn, simply because the outer square of Equation (4.0.1) commutes, as pointed out in Proposition 4.1.1. 4.3 Calculations on small webs We’ll begin with some calculations of dGT on the flow vertices introduced in §2.3.  = (−1)a  + (−1)n+b + q−a  = (−1)a + (−1)n+bqb + = (−1)a + (−1)n+bqb + = (−1)a qa + + (−1)n+bqa−n We’ll also need the following formulas for two vertex webs:  = + + q−a + + + q−a + (−1)b+cqb (4.3.1) = qb + + + + + qb + (−1)a+cqb+c−n (4.3.2) 4.4 A path model, and polygons. Call a diagram in Symn in which all vertices are flow vertices and there are no tags (except those implicitly hidden inside the vertices) a flow diagram. In particular, the flow vertices themselves, and the P- and Q-polygons described in §2.4 are flow diagrams. A reduction path on a flow diagram is some disjoint union of cycles and paths beginning and ending at the boundary of the diagram, always matching the orientations on the edges of the diagram. We then notice that if v is a flow vertex, the three terms of dGT (v) correspond to the three reduction paths on the vertex. Specifically, for each reduction path, in the corresponding term of dGT (v) the label on each traversed edge has been reduced by one. Each term additionally has a coefficient ±qk for some integer k. It’s easy to see that this condition must also hold for larger flow diagrams. For a reduction path π on a diagram D ∈ Symn, write π(D) for the diagram in Symn−1 obtained from D by reducing the label on each edge traversed by π. Further, write t(π) for the subset of the tensor factors in the target of D which π reaches, and s(π) for the subset of the tensor factors in the source. Proposition 4.4.1. The result of applying dGT to a flow diagram D is a sum over the reduction paths on D: dGT (D) = reduction paths π ±q•π(D). (4.4.1) Further, two terms in this sum lie in same matrix entry (remember, dGT is a functor Symn → Symn−1 ) iff the boundaries of the two reduction paths agree. Proof. Certainly this is true for the generators which can appear in a flow diagram; caps, cups, and flow vertices. The condition is also preserved under tensor product and compo- sition. To see this, suppose we have two flow diagrams D1 and D2 in Symn, with fami- lies of reduction paths P1 and P2, so that dGT (Di) = ±q•π(Di). Now the reduction paths P for D1 ⊗ D2 are exactly those of the form π1 ∪ π2 for some π1 ∈ P1, π2 ∈ P2 (remember tensor product of diagrams is side-by-side disjoint union). Thus dGT (D1 ⊗D2) = dGT (D1)⊗ dGT (D2) π1∈P1 (−1)•q•π1(D1) π2∈P2 (−1)•q•π2(D2) π1∈P1 π2∈P2 (−1)•q•(π1 ∪ π2)(D1 ⊗D2) (−1)•q•π(D1 ⊗D2). The argument for compositions is a tiny bit more complicated. Supposing D1 and D2 are composable, the reduction paths P for D1 ◦ D2 are those of the form π1 ∪ π2, for some π1 ∈ P1, π2 ∈ P2 such that s(π1) = t(π2). Thus dGT (D1 ◦D2) = dGT (D1) ◦ dGT (D2) π1∈P1 ±q•π1(D1) π2∈P2 ±q•π2(D2) π1∈P1 π2∈P2 ±q•δs(π1),t(π2)(π1 ∪ π2)(D1 ◦D2) ±q•π(D1 ⊗D2). We’ll now calculate dGT on the P- and Q-polygons. On a P-polygon, there are two reduction paths which do not traverse any boundary points; the empty reduction path, and the path around the perimeter of the polygon. We’ll call these πP;∅ and πP;◦. Otherwise, there is a unique reduction path for each (cyclic) subsequence of boundary points which alternately includes incoming and outgoing edges. We’ll write πP;s for this, where s is the subset of {1, . . . , 2k} corresponding to the points on boundary traversed by the path components. The reduction path has one component for each such pair of incoming and outgoing edges, and traverses the perimeter of the polygon counterclock- wise between them. On a Q-polygon, there are two reductions paths which traverse every boundary point; in one, each component enters at an incoming edge, and traverse coun- terclockwise, departing at the nearest outgoing edge, while in the other each component traverses clockwise. We’ll call these πQ; and πQ;�. Otherwise, there is a unique reduction path for each disjoint collection of adjacent pairs of boundary edges; the path connects each pair, traversing a single edge of the polygon between them, and again, we’ll write πQ;s, where s is the subset of {1, . . . , 2k} corresponding to the subset of the boundary con- sisting of the union of all the pairs. All these statements about reduction paths should be immediately obvious, looking at the definition of the P- and Q-polygons in Equations (2.4.1) and (2.4.2). πP;∅(Pna,b;l) = Pn−1a,b;l πP;◦(Pna,b;l) = Pn−1a+−→1 ,b+−→1 ;l = P a,b;l−1. Further πP;s(Pna,b;l) = Pn−1a+a′,b+b′;l, where a′, b′ ∈ {0, 1}k is determined from s by 2i − 1 ∈ s iff a′i = 1 and b′i = 0, and 2i ∈ s iff a′i+1 = 1 and b i = 0. The condition that s contains alternately incoming and outgoing edges translates to b′i ≤ a′i, a′i+1; we’ll call such pairs a′, b′ P-admissible. Note that πP;∅ and πP;◦ correspond to two exceptional pairs, a ′ = b′ = 0 and a′ = b′ = 1 respectively. On the Q-polygons, we have πQ; (Qna,b;l) = Qna,b+−→−1;l = Q 1 ,b;l+1 , πQ;�(Qna,b;l) = 1 ,b;l and πQ;s(Qna,b;l) = Q a+a′,b+b′;l , where a′ ∈ {0, 1, }k and b′ ∈ {−1, 0}k are deter- mined by 2i − 1 ∈ s iff ai = 1 or bi = −1, and 2i ∈ s iff ai+1 = 1 or bi = −1. The condition that the points in s come in disjoint adjacent pairs translates to the condition i = 0 = a i; such pairs a ′, b′ are called Q-admissible. Note that πQ; and πQ;� cor- respond to two exceptional pairs, a′ = 0 , b′ = −→−1 and a′ = −→1 , b′ = −→0 respectively. Also, note that a pair a′, b′ is both P- and Q-admissible exactly if b′ = −→0 ; any a′ ∈ {0, 1}k is allowed. Thus we have Pna,b;l a′,b′ P-admissible dGT a′,b′ Pna,b;l Qna,b;l a′,b′ Q-admissible dGT a′,b′ Qna,b;l where dGT a′,b′ Pna,b;l = (−1)b′·a+rotl(a)ql(Σb′−Σa′+1)qrotl(a′)·b−b′·a−a1−na′1Pn−1 a+a′,b+b′;l , (4.4.2) dGT a′,b′ Qna,b;l = (−1)b′·a+rotl(a)ql(Σa′−Σb′− +1)qΣb−nΣb ′+b′·rotl(a)−a′·b−a1−na 1Qn−1a+a′,b+b′;l. (4.4.3) The coefficients here are products of the coefficients appearing in the formulas for dGT on the two vertex webs in Equations (4.3.1) and (4.3.2). We’ll often also use the notation dGT ∅ for the terms corresponding to reduction paths not traversing any boundary edges, and dGT ∂ for the terms corresponding to reduction paths traversing all boundary edges. Thus dGT ∅ Pna,b;l = dGT −→ Pna,b;l + dGT −→ Pna,b;l = ql−a1Pn−1a,b;l + q l−a1+Σb−Σa−nPn−1a,b;l−1, (4.4.4) dGT ∅ Qna,b;l = dGT −→ Qna,b;l = ql−a1qΣb− 2 Qn−1a,b;l , (4.4.5) dGT ∂ Pna,b;l = dGT −→ Pna,b;l = ql−a1−nqΣb− 2 Pn−1 1 ,b;l , (4.4.6) dGT ∂ Qna,b;l = dGT −→ Qna,b;l + dGT −→ Qna,b;l = ql−a1+Σb−Σa− 2 Qn−1 1 ,b;l+1 + ql−a1−nQn−1 1 ,b;l . (4.4.7) Chapter 5 Describing the kernel Finally, in this chapter we’ll use the methods developed to this point to try to pin down the kernel of the representation functor. In particular, Theorems 5.1.1, 5.2.1 and 5.3.2, stated in the three subsequent sections, describe particular classes of relations amongst diagrams. Theorems 5.2.1 and 5.3.2 additionally rule out any other similar relations. 5.1 The I = H relations Theorem 5.1.1. For each a, b, c ∈ {0, . . . , n} with a+ b+ c ≤ n, there’s an identity = (−1)(n+1)a , (5.1.1) and another = (−1)(n+1)a . (5.1.2) The proof appears in §5.7. Remark. It’s somewhat unfortunate that there’s a sign here in the first place, and that it depends unsymmetrically on the parameters. (See also §6.1 and §6.3, for a discussion of sign differences between my setup and previous work.) Replacing the vertices in Equation (5.1.1) with standard ones, we can equivalently write this identity in the following four ways (obtaining each by flipping tags, per Equation (2.2.9)): = (−1)(n+1)a = (−1)(n+1)b (5.1.3) = (−1)(n+1)c = (−1)(n+1)d , where here a+ b+ c+ d = n. These signs are special cases of the 6− j symbols for Uq(sln). 5.2 The square-switch relations We can now state the ‘square switch’ relations, which essentially say that every P-type square can be written as a linear combination of Q-type squares, as vice versa. (Refer back to §2.4 for the definitions of P- and Q-polygons.) Theorem 5.2.1. The subspace (SS for ‘square switch’) SSna,b = spanA Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb Qna,b;m }mina+n l=max b if n+Σa− Σb ≥ 0 spanA Qna,b;l − n+mina m=max b Σb− n− Σa m+ l − Σa− n Pna,b;m }min b l=maxa if n+Σa− Σb ≤ 0 of APna,b +AQna,b ⊂ HomSymn(∅,L(a, b)) lies in the kernel of the representation functor. The proof appears in §5.7. Notice that we’re stating this result for all tuples a and b, not just those for which SS is non-trivial. This is a minor point, except that it makes the base cases of our inductive proof easy. Remark. In the case n+Σa− Σb = 0, this becomes simply SSna,b = spanA Pna,b;l −Qna,b;Σb−l }min a+n l=max b Remark. For n + Σa − Σb ≤ 0, at the maximal or minimal values of l, we simply get Pna,b;max b − Qna,b;min b (since the only terms for which the q-binomial is nonzero are m ≥ min b) and Pna,b;mina+n − Qna,b;max a (again, the q-binomials vanish unless m ≤ max a). We already knew these were identically zero in Symn, by the discussion at the end of §2.4 on relations between P- and Q-polygons. A similar statement applies when n+Σa−Σb ≤ 0. Loops, and let bigons be bygones Over on the representation theory side, any loops or bigon must become a multiple of the identity, by Schur’s lemma. We can see this as a special case of the SS relations. (Alterna- tively, we’d be able to derive the same formulas from Theorem 5.3.2 in the next section.) Consider the boundary flow labels a = (0, 0), b = (k, 0), so L(a, b) = ((0,−), (0,+), (k,−), (k,+)). Then APna,b = spanA Pna,b;l , while AQna,b = spanA Qna,b;0 . By 5.2.1, we have SSna,b = spanA Pna,b;l − l − k Qna,b;0 Diagrammatically, this says that in the quotient by kerRep, l − k or, removing trivial edges, and cancelling tags l − k The special case k = 0 evaluates loops: 5.3 The Kekulé relations Friedrich August Kekulé is generally thought to have been the first to suggest the cyclic structure of the benzene molecule [38]. A simplistic description of the benzene molecule is as the quantum superposition of two mesomers, each with alternating double bonds The similarity to the relation between the two hexagonal diagrams in Sym4/ ker, + = + (5.3.1) discovered by Kim [21], prompted Kuperberg to suggest the name ‘Kekulé relation’, even though in my classification all of the relations for Uq(sl2) and Uq(sl3) are also of this type! In the following, we’ll use the convenient notation x = Σx − max x, Σx−minx. (With the convention that ∅ = 0.) Definition 5.3.1. We’ll say a set of flow labels (a, b) is n-hexagonal if there’s a sequence of six elements of L(a, b), each non-trivial for n, i.e. between 1 and n− 1 inclusive, which are alternately incoming and outgoing edges. Remark. It’s not enough to simply say there are 6 non-trivial edges. For example, the boundary labels ((5,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−)) seem to allow hexagonal webs, but modulo the I = H relations, the P-polygons are all actually bigons in disguise. Theorem 5.3.2. The subspaces APRna,b ⊂ APna,b ⊂ HomSymn(∅,L(a, b)) and AQRna,b ⊂ AQna,b ⊂ HomSymn(∅,L(a, b)) defined by APRna,b = spanA (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j Pna,b;j+k Σa+n−1 (5.3.2) AQRna,b = spanA (−1)j+k j + k −max a j − Σa min b− j − k Σb− n( |∂| − 1)− 1− j Qna,b;j+k Σb−n( −1)−1 (5.3.3) are in the kernel of the representation functor. Moreover, when the representation functor is re- stricted to APna,b, its kernel is exactly APRna,b, and similarly for AQna,b and AQRna,b. Further, in the case that (a, b) is n-hexagonal, when the representation functor is restricted to APna,b +AQna,b, its kernel is exactly APRna,b +AQRna,b. The proof appears in §5.7. Remark. It’s worth being a little careful here; remember from Lemma 2.4.1 that sometimes APna,b∩AQna,b 6= 0; when a or b is constant, the intersection is 1 dimensional, and when both are constant, but not equal, it’s 2 dimensional. Besides this, however, you can understand the second half of the above Theorem as saying that there are no relations between P-type polygons, and Q-type polygons, when those polygons are at least hexagons. In particular, this contradicts the conjecture of [11], which expected to find relations looking like the ‘square-switch’ relations described in Theorem 5.2.1 amongst larger polygons. The elements of APRna,b described above have ‘breadth’ a + 2; that is, they have that many terms. We can generalise the definition of n-hexagonal to say that a pair (a, b) has n-circumference at least 2k if there’s a sequence of 2k edges in L(a, b), which are nontrivial for n and alternately incoming and outgoing. Then Lemma 5.3.3. If (a, b) has n-circumference at least 2k, the breadth of the relations in APRna,b is at least k + 1. In particular, the only place that relations of breadth 2 appear is when (a, b) has n-circumference no more than 2. The proof appears in §5.7, after Lemma 5.7.7. 5.4 More about squares Theorem 5.4.1. Further, when n+ Σa− Σb ≥ 0, the space AQRna,b of relations amongst the Q- squares becomes 0, and in fact APRna,b ⊂ SSna,b. (Conversely, when n+Σa−Σb ≤ 0, APRna,b = 0 and AQRna,b ⊂ SSna,b.) When n+Σa− Σb ≥ 0, defining SS ′na,b = spanA Qna,b;m − n+mina l=n+Σa−m (−1)m+l+n+Σa m+ l − 1− Σb m+ l − n− Σa Pna,b;l }min b m=max a we find that SSna,b = APR a,b ⊕ SS′na,b. When n+Σa− Σb ≤ 0, we define instead SS ′na,b = spanA Pna,b;m − min b l=Σb−m (−1)m+l+Σb m+ l − n− 1− Σa m+ l − Σb Qna,b;l }min a+n m=max b and find that SSna,b = AQRna,b ⊕ SS′na,b. Corollary 5.4.2. The kernel of Rep on APna,b +AQna,b, in the case of squares, is exactly SSna,b. Proof. This follows from Theorems 5.3.2 and 5.4.1. Suppose x ∈ kerRepn ∩(APna,b + AQna,b), we’ll show that it’s zero as an element of (APna,b +AQna,b)/SSna,b. In fact, (AP a,b + AQna,b)/SSna,b = AP a,b/SS a,b = AP a,b/APRna,b, by Theorem 5.4.1, and by Theorem 5.3.2, (kerRepn ∩APna,b)/APRna,b = 0. 5.5 A conjecture I wish I could prove the following Conjecture. The kernel of the representation functor is generated, as a pivotal category ideal, by the elements described in Theorems 5.1.1, 5.2.1 and 5.3.2. Evidence. First of all, we have Corollary 5.4.2, which confirms that the conjecture holds for diagrams with at most four boundary points. Second, Theorem 5.3.2 tells us more than simply that certain elements are in the kernel; it tells us exactly the kernel of the representation functor when restricted to dia- grams with a single (hexagonal or bigger) polygonal face. Any relations not generated by those discovered here must then be ‘more nonlocal’, in the sense that they involve dia- grams with multiple polygons. Finally, there’s some amount of computer evidence. I have a Mathematica pack- age, sadly unreleased at this point, which can explicitly produce intertwiners between rep- resentations of arbitrary quantum groups Uqg. A program which can translate diagrams into instructions for taking tensor products and compositions of elementary morphisms (i.e. an implementation of the representation functor described in §3.6) can then auto- matically look for linear dependencies amongst diagrams. For small values of n (up to 8 or 9), I used such a program to look for relations involving two adjacent hexagons, but found nothing besides relations associated, via Theorem 5.3.2, to one of the individual hexagons. 5.6 Examples: Uq(sln), for n = 2, 3, 4 and 5. For each n, we can enumerate the finite list of examples of each of the three families of re- lations. To enumerate APRn relations, we find each set of flow labels (a, b) such that every element of L(a, b) is between 1 and n − 1 inclusive, and dimAPRna,b = n + Σa − Σb > 0. When sets of flow labels (a, b) and (rotlk(a), rotlk(b)) differ by a cyclic permutation, we’ll only discuss the lexicographically smaller one. There’s no need to separately enumerate the AQRn relations; they’re just rotations of the APRn relations. To enumerate SSn re- lations, we’ll do this same, but weakening the inequality to n + Σa − Σb ≥ 0. Recall that we don’t need to look at the extreme values of l in the SS relations in Theorem 5.2.1; those relations automatically hold already in Symn. For n = 2 and n = 3, there are no APRna,b relations with a, b of length 3 or more; the relations of length 2 or less follow from the SSn relations. We recover exactly the Temperley-Lieb loop relation for n = 2, and Kuperberg’s loop, bigon and square relations for n = 3, from SS3(0,0),(0,0),SS (0,0),(0,1) and SS (0,0),(1,1) respectively. For n = 4, we have non-trivial APR4a,b spaces for (a, b) = (∅, ∅) when k = 0 ((0), (1)), ((0), (2)), or ((0), (3)) when k = 1 ((0, 0), (1, 1)), ((0, 0), (1, 2)), or ((0, 0), (2, 2)) when k = 2 and for k = 3 (a, b) = ((0, 0, 0), (1, 1, 1)). Thus we have relations involving loops: APR4∅,∅ = spanA [4]q P ∅,∅;0 − P ∅,∅;1,− [3]q P ∅,∅;1 + [2]q P ∅,∅;2, [2]q P ∅,∅;2 − [3]q P ∅,∅;3,−P ∅,∅;3 + [4]q P ∅,∅;4 = spanA [4]q − ,− [3]q + [2]q , [2]q − [3]q ,− + [4]q , (5.6.1) and bigons: APR4(0),(1) = spanA − [3]q P (0),(1);1 + P (0),(1);2, [2]q P (0),(1);2 − [2]q P (0),(1);3, −P4(0),(1);3 + [3]q P (0),(1);4 = spanA − [3]q + , [2]q − [2]q , (5.6.2) APR4(0),(2) = spanA [2]q P (0),(2);2 − P (0),(2);3,−P (0),(2);3 + [2]q P (0),(2);4 = spanA [2]q − , (5.6.3) APR4(0),(3) = spanA −P4(0),(3);3 + P (0),(3);4 = 0, (5.6.4) (there’s a redundancy in each APR4 (0),(k) space, since P4 (0),(k);k (0),(k);4 ) and then rela- tions involving squares: APR4(0,0),(1,1) = spanA − [3]q P (0,0),(1,1);1 + [2]q P (0,0),(1,1);2 − P (0,0),(1,1);3, P4(0,0),(1,1);2 − [2]q P (0,0),(1,1);3 + [3]q P (0,0),(1,1);4 = spanA − [3]q + [2]q − , − [2]q + [3]q , (5.6.5) APR4(0,0),(1,2) = spanA P4(0,0),(1,2);2 − P (0,0),(1,2);3 + P (0,0),(1,2);4 = spanA , (5.6.6) APR4(0,1),(2,2) = spanA P4(0,1),(2,2);2 − P (0,1),(2,2);3 + P (0,1),(2,2);4 = spanA , (5.6.7) and finally the eponymous Kekulé relation, involving hexagons: APR4(0,0,0),(1,1,1) = spanA −P4(0,0,0),(1,1,1);1 + P (0,0,0),(1,1,1);2− −P4(0,0,0),(1,1,1);3 + P (0,0,0),(1,1,1);4 = spanA − + − + (5.6.8) Here we’re telling some small lies; APR4a,b is always a subspace of HomSym4(∅,L(a, b)), but I’ve drawn some of these diagrams as elements of other Hom spaces, via rotations. The interesting SS4 spaces are SS4(0,0),(1,1) = spanA P4(0,0),(1,1);2 −Q (0,0),(1,1);0 − [2]q Q (0,0),(1,1);1, P4(0,0),(1,1);3 − [2]q Q (0,0),(1,1);0 −Q (0,0),(1,1);1 = spanA − − [2]q , − [2]q − SS4(0,0),(1,2) = spanA P4(0,0),(1,2);3 −Q (0,0),(1,2);0 −Q (0,0),(1,2);1 = spanA SS4(0,0),(2,2) = spanA P4(0,0),(2,2);3 −Q (0,0),(2,2);1 = spanA SS4(0,1),(2,2) = spanA P4(0,1),(2,2);3 −Q (0,1),(2,2);1 −Q (0,1),(2,2);2 = spanA Notice the redundancies here: SS4(0,0),(1,1) = APR (0,0),(1,1), SS (0,0),(1,2) = APR (0,0),(1,2) and SS4(0,1),(2,2) = APR (0,1),(2,2), but SS (0,0),(2,2) is independent of any of the APR rela- tions. For n = 5, we’ll just look at non-trivial APR5a,b spaces where a and b are each of length at least 3; we know what the length 0 and 1 relations look like (loops and bigons), and the length 2 relations all follow from SS relations. Thus we have (a, b) = ((0, 0, 0), (1, 1, 1)), ((0, 0, 0), (1, 1, 2)), ((0, 0, 1), (1, 2, 2)), ((0, 1, 1), (2, 2, 2)) = ((0, 0, 0, 0), (1, 1, 1, 1)). APR5(0,0,0),(1,1,1) = = spanA − [4]q P + [3]q P − [2]q P + P5−→ − [2]q P + [3]q P − [4]q P = spanA − [4]q + [3]q − [2]q + , − [2]q + [3]q − [4]q (5.6.9) APR5(0,0,0),(1,1,2) = spanA P5(0,0,0),(1,1,2);2 − P (0,0,0),(1,1,2);3+ +P5(0,0,0),(1,1,2);4 − P (0,0,0),(1,1,2);5 = spanA − + − (5.6.10) APR5(0,0,1),(1,2,2) = spanA P5(0,0,1),(1,2,2);2 − P (0,0,1),(1,2,2);3+ +P5(0,0,1),(1,2,2);4 − P (0,0,1),(1,2,2);5 = spanA − + − (5.6.11) APR5(0,1,1),(2,2,2) = spanA P5(0,1,1),(2,2,2);2 − P (0,1,1),(2,2,2);3+ +P5(0,1,1),(2,2,2);4 − P (0,1,1),(2,2,2);5 = spanA − + − (5.6.12) APR5(0,0,0,0),(1,1,1,1) = spanA −P5(0,0,0,0),(1,1,1,1);1 + P (0,0,0,0),(1,1,1,1);2 − P (0,0,0,0),(1,1,1,1);3+ +P5(0,0,0,0),(1,1,1,1);4 − P (0,0,0,0),(1,1,1,1);5 = spanA − + − + (5.6.13) The interesting SS5 relations are SS5(0,0),(1,1) = spanA P5(0,0),(1,1);2 −Q (0,0),(1,1);0 − [3]q Q (0,0),(1,1);1, P5(0,0),(1,1);3 − [3]q Q (0,0),(1,1);0 − [3]q Q (0,0),(1,1);1, P5(0,0),(1,1);4 − [3]q Q (0,0),(1,1);0 −Q (0,0),(1,1);1 = spanA − − [3]q , − [3]q − [3]q , − [3]q − SS5(0,0),(1,2) = spanA P5(0,0),(1,2);3 −Q (0,0),(1,2);0 − [2]q Q (0,0),(1,2);1, P5(0,0),(1,2);4 − [2]q Q (0,0),(1,2);0 −Q (0,0),(1,2);1 = spanA − − [2]q , − [2]q − SS5(0,0),(1,3) = spanA P5(0,0),(1,3);4 −Q (0,0),(1,3);0 −Q (0,0),(1,3);1 = spanA SS5(0,0),(2,2) = spanA P5(0,0),(2,2);3 −Q (0,0),(2,2);1 −Q (0,0),(2,2);2, P5(0,0),(2,2);4 −Q (0,0),(2,2);0 −Q (0,0),(2,2);1 = spanA − − , SS5(0,1),(2,2) = spanA P5(0,1),(2,2);3 −Q (0,1),(2,2);1 − [2]q Q (0,1),(2,2);2, P5(0,1),(2,2);4 − [2]q Q (0,1),(2,2);1 −Q (0,1),(2,2);2 = spanA − − [2]q , − [2]q − SS5(0,1),(2,3) = spanA P5(0,1),(2,3);4 −Q (0,1),(2,3);1 −Q (0,1),(2,3);2 = spanA SS5(0,1),(3,2) = spanA P5(0,1),(3,2);4 −Q (0,1),(3,2);1 −Q (0,1),(3,2);2 = spanA SS5(0,2),(3,3) = spanA P5(0,2),(3,3);4 −Q (0,2),(3,3);2 −Q (0,2),(3,3);3 = spanA 5.7 Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2 5.7.1 The I = H relations Proof of Theorem 5.1.1. We’ll just show the calculation for Equation (5.1.1), with ‘merging’ vertices; the calculation for Equation (5.1.2) is identical. It’s an easy induction, beginning by calculating dGT of both sides, using the calculations from §4.3. = (−1)c + (−1)n+b+cq−a + + (−1)bq−a−b + (−1)b , while = (−1)n+a+c + (−1)n+a+b+cq−a + + (−1)a+bq−a−b + (−1)a+b = (−1)n(a−1)(−1)n+a+c + (−1)na(−1)n+a+b+cq−a + + (−1)na(−1)a+bq−a−b + (−1)na(−1)a+b , which differs by exactly an overall factor of (−1)(n+1)a. For the base cases of the induction, we need consider the possibility that a, b or c is 0 or a + b + c = n. It’s actually more convenient to consider one of the equivalent relations in Equation (5.1.3), where the corresponding base cases are a, b, c or d = 0. Then I claim = (−1)(n+1)a (5.7.1) in Symn (and similarly in the other 3 cases when instead b, c or d is zero), and so the difference is automatically in the kernel of Rep. Each of the four variations of Equation (5.7.1) follows from the ‘degeneration’ relations in Symn, given in Equation (2.2.10). If a = 0, we have = = = (−1)(n+1)0 . (5.7.2) The middle equality here uses ‘tag cancellation’ from Equation (2.2.8). The other cases are similar, but use ‘tag flipping’ from Equation (2.2.9), producing the desired signs. 5.7.2 The square-switch relations Proof of Theorem 5.2.1. This is quite straightforward, by induction. We’ll just do the n + Σa− Σb ≥ 0 case; the other follows immediately by rotation, by §2.4.1. The base cases are easy; since either each bi ≥ ai, or APna,b = AQna,b = 0, when we’re at n = 0 the only interesting case is a = b = (0, 0). Then SS0(0,0),(0,0) is simply the span of P0 (0,0),(0,0);0 (0,0),(0,0);0 ; this element is actually zero in Sym0, by Lemma 2.4.1, and so automatically in the kernel of Rep. For the inductive step, we simply show that for l = max b, . . . ,min a + n the component of Pna,b;l − min b m=max a n+Σa−Σb m+ l − Σb Qna,b;m (5.7.3) in HomSymn−1(∅,L(a+ a′, b+ b′)) lies in SS a+a′,b+b′ for each a ′, b′. The expression in Equa- tion (5.7.3) has one matrix for each subset of the four boundary points. The only nonzero matrix entries correspond to those subsets for which there is a reduction path with end- points coinciding with the subset. Thus, numbering the four boundary points 1, 2, 3 and 4, we have dGT = dGT ∅+dGT {1,2}+dGT {1,4}+dGT {3,4}+dGT {3,2}+dGT ∂ . In the notation of §4.4, dGT {1,2}|AP = dGT (1,1),(0,1), dGT {1,2}|AQ = dGT (0,0),(−1,0), dGT {1,4}|AP = dGT (1,0),(0,0), dGT {1,4}|AQ = dGT (1,0),(0,0), dGT {3,4}|AP = dGT (1,1),(1,0), dGT {3,4}|AQ = dGT (0,0),(0,−1), dGT {3,2}|AP = dGT (0,1),(0,0), dGT {3,2}|AQ = dGT (0,1),(0,0). We now compute all of the components of Equation (5.7.3). First, use Equations (4.4.2) and (4.4.3) to write down dGT {1,2} Pna,b;l = (−1)Σaq−n−Σa+ΣbPn−1 a+(1,1),b+(0,1);l dGT {1,2} Qna,b;l = (−1)Σaq−n−Σa+ΣbQn−1 a+(0,0),b+(−1,0);l = (−1)Σaq−n−Σa+ΣbQn−1 a+(1,1),b+(0,1);l+1 dGT {1,4} Pna,b;l = q−n−a1+b2Pn−1 a+(1,0),b+(0,0);l dGT {1,4} Qna,b;l = q−n−a1+b2Qn−1 a+(1,0),b+(0,0);l dGT {3,4} Pna,b;l = (−1)Σaq−n−Σa+ΣbPn−1 a+(1,1),b+(1,0);l dGT {3,4} Qna,b;l = (−1)Σaq−n−Σa+ΣbQn−1 a+(0,0),b+(0,−1);l = (−1)Σaq−n−Σa+ΣbQn−1 a+(1,1),b+(1,0);l+1 dGT {3,2} Pna,b;l = q−a1+b1Pn−1 a+(0,1),b+(0,0);l dGT {3,2} Qna,b;l = q−a1+b1Qn−1 a+(0,1),b+(0,0);l and notice that the coefficients appearing are in each case independent of whether we’re acting on AP or AQ, and moreover that the coefficients are independent of the internal flow labels l. We can use this to show that dGT {i,j} Pna,b;l − ∑min b m=max a n+Σa−Σb m+l−Σb Qna,b;m lies in SSn−1 for each of the four pairs {i, j}. For example, dGT {1,2} Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb Qna,b;m = (−1)Σaq−n−Σa+Σb× a+(1,1),b+(0,1);l min b m=max a n+Σa− Σb m+ l − Σb a+(1,1),b+(0,1);m+1 = (−1)Σaq−n−Σa+Σb× a+(1,1),b+(0,1);l min b m=max a n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1)) m+ 1 + l − Σ(b+ (0, 1)) a+(1,1),b+(0,1);m+1 which becomes, upon reindexing the summation, (−1)Σaq−n−Σa+Σb× a+(1,1),b+(0,1);l min b+1 m=max a+1 n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1)) m+ l − Σ(b+ (0, 1)) a+(1,1),b+(0,1);m Now, making use of the fact that if min b+1 > min (b+ (0, 1)), the last term vanishes since a+(1,1),b+(0,1);min b+1 = 0, we can rewrite the summation limits and obtain dGT {1,2} Pna,b;l − min b m=max a n+Σa− Σb m+ l −Σb Qna,b;m = (−1)Σaq−n−Σa+Σb × a+(1,1),b+(0,1);l min (b+(0,1)) m=max (a+(1,1))+1 n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1)) m+ l − Σ(b+ (0, 1)) a+(1,1),b+(0,1);m The parenthesised expression is exactly an element of the spanning set for SSn−1 a+(1,1),b+(0,1) It remains to show that dGT ∅ and dGT ∂ carry SSn into SSn−1. Both are similar; here’s the calculation establishing the first. dGT ∅ Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb Qna,b;m = q−a1 qlPn−1a,b;l + q l−n−Σa+ΣbPn−1a,b;l−1 − min b m=max a q−mqΣb n+Σa− Σb m+ l −Σb Qn−1a,b;m which, using the q-binomial identity + qt−s , we can rewrite as = q−a1 qlPn−1a,b;l − min b m=max a n− 1 + Σa− Σb m+ l − Σb Qn−1a,b;m+ ql−n−Σa+ΣbPn−1 a,b;l−1 − min b m=max a ql−n−Σa+Σb n− 1 + Σa− Σb m+ l − 1− Σb a,b;m which is in SSn−1a,b . 5.7.3 The Kekulé relations We now prove Theorem 5.3.2, describing the relations amongst polygonal diagrams. We begin with a slight reformulation of the goal; it turns out that the orthogonal complement APR⊥ of the relations in the dual space AP∗ is easier to describe. Lemma 5.7.1. The orthogonal complement APRna,b⊥ in APna,b∗ is spanned by the elements a,b;j∗ k∗=Σb n+Σa− Σb k∗ − Σb (Pna,b;k∗+j∗)∗ for j∗ = − b, . . . ,− Proof. First note that APRna,b ⊂ APna,b is an (n+Σa−Σb)-dimensional subspace of APna,b; the spanning set we’ve described for it really is linearly independent in Symn. It’s a sub- space of a (min a−max b+ n + 1)-dimensional vector space, so we expect the orthogonal complement to be (1 + a)-dimensional, as claimed in the lemma. We thus need only check that each e a,b;j∗ annihilates each d a,b;j , for j = Σb, . . . ,Σa + n − 1. Thus we calculate j∗ (d j ) = min(− a+1,j∗−j+n+Σa) k=max(− b,Σb+j∗−j) (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j n+Σa−Σb j − j∗ + k − Σb and observe that the limits of the summation are actually irrelevant; outside the limits at least one of the three quantum binomial coefficients is zero. Lemma A.1.1 in the appendix §A.1 shows that this q-binomial sum is zero. Corollary 5.7.2. The orthogonal complement AQRna,b⊥ in AQna,b∗ is spanned by the elements a,b;j∗ = k∗=Σa Σb−Σa− n k∗ − Σa Qna,b;j∗+k∗⊥ for j∗ = − a, . . . ,− Proof. This follows immediately from the previous lemma, by rotation, as in §2.4.1. The proof of Theorem 5.3.2 now proceeds inductively. We’ll assume that kerRep∩APn−1 = APRn−1 for all flows a, b. We’ll look at dGT ∅ first; we know that anything in kerRep∩APna,b must lie inside dGT −1 (APRn−1a,b ), and we’ll discover that in fact dGT (APRn−1a,b ) = APR a,b, in Lemma 5.7.3. Subsequently, we need to check, in Lemma 5.7.4, that dGT a′,b′(APRna,b) ⊂ APRn−1 a+a′,b+b′ for each allowed pair of boundary flow reduction patterns a′, b′. Modulo these lemmas, that proves the first part of Theorem 5.3.2 (at least the statements about APR; Lemma 5.7.5 deals with the statements about AQR). Later, Lemma 5.7.8 deals with the other parts of Theorem 5.3.2. Explicit formulas for dGT ∅ and dGT a′,b′ are easy to come by, from §4.4: dGT ∅ Pna,b;l = ql−a1 Pn−1a,b;l + q Σb−Σa−nPn−1a,b;l−1 (5.7.4) dGT a′,b′(Pna,b;l) = κn,a,b,a′,b′q−l(m(a ′,b′)−1)Pn−1a+a′,b+b′;l (5.7.5) where here κn,a,b,a′,b′ is some constant not depending on l, which for now we don’t need to keep track of, but can be reconstructed from Equation (4.4.2), and m(a′, b′) is the number of components of the reduction path indexed by a′, b′, which is just Σa′ −Σb′. It’s worth noting at this point that the P-webs appearing in the right hand sides of Equations (5.7.4) and (5.7.5) are sometimes zero; the internal flow value might be out of the allowable range. In particular, the allowed values of l for Pna,b;l ∈ APna,b are l = max b, . . . ,min a+n, and so the target of dGT ∅ is one dimension smaller than its source. Moreover, it’s easy to see from Equation (5.7.4) that the kernel of dGT ∅ is exactly one dimensional. Life is a little more complicated for dGT a′,b′ ; if b′i = 1 for any i where bi = max b, then dGT a′,b′ Pna,b;max b and if a′i = 1 for every i where ai = min a, then dGT a′,b′ Pna,b;mina+n and otherwise dGT a′,b′ is actually nonzero on each P-web. Thus the kernel of dGT a′,b′ may be 0, 1 or 2 dimensional. Lemma 5.7.3. The only candidates for the kernel of the representation functor on APna,b are in APRna,b, since dGT −1 (APRn−1a,b ) = APR Proof. We’ve already noticed that the kernel of dGT ∅ is 1-dimensional, so we actually only need to check that dGT ∅ APRna,b ⊂ APRn−1 , or, equivalently, that dGT ∗∅(APR ) ⊂ APRna,b⊥. Easily, we obtain dGT ∗∅(P a,b;l∗ ) = ql Pna,b;l∗∗ + qΣb−Σa−n+1Pna,b;l∗+1∗ simply by taking duals in Equation (4.4.4). Then dGT ∗∅(e P,n−1 a,b;j∗ ) = dGT n−1+Σa k∗=Σb n− 1 + Σa− Σb k∗ − Σb (Pn−1a,b;k∗+j∗) = q−a1 n−1+Σa k∗=Σb n− 1 + Σa− Σb k∗ − Σb Pna,b;k∗+j∗∗ + qΣb−Σa−n+1Pna,b;k∗+j∗+1∗ and by reindexing the summation of the second terms, we can rewrite this as dGT ∗∅(e P,n−1 a,b;j∗ ) = q j∗−a1+Σb k∗=Σb n− 1 + Σa− Σb k∗ − Σb ∗−n−Σa n− 1 + Σa− Σb k∗ − 1− Σb Pna,b;k∗+j∗∗ ∗−a1+Σb k∗=Σb n+Σa− Σb k∗ − Σb Pna,b;k∗+j∗∗ ∗−a1+Σbe a,b;j∗ tidily completing the proof. Remark. By rotation, we can also claim that dGT −1∂ (AQR a,b ) = AQR and that if you’re in the kernel of Rep, and in AQna,b, you must also be in AQRna,b. Lemma 5.7.4. The candidate relations APRna,b really are in the kernel of the representation functor, since dGT a′,b′(APRna,b) ⊂ APRn−1a+a′,b+b′ for each pair a′, b′ (other than the pairs a′ = b′ = 0 and a′ = b′ = 1 ; the previous Lemma dealt with that matrix entry of dGT ). Proof. For this we need the quantum Vandermonde identity [39], = qyz q−(x+y)i z − i Throughout, we’ll write m = Σa′ − Σb′ for the number of components of the reduction path. We want to show dGT ∗a′,b′(e P,n−1 a+a′,b+b′;j∗ ) = κn,a+a′,b+b′,a′,b′× n+Σa+m−1 k∗=Σb ∗+k∗+Σb′)(m−1) n− Σa− Σb+m− 1 k∗ − Σb Pna,b;k∗+j∗+Σb∗ (5.7.6) is equal to a,b,j∗+Σb+i = k∗=Σb n+Σa− Σb k∗ − Σb Pna,b;k∗+j∗+i+Σb′ ∗ ∈ APRna,b⊥ for some coefficients Xi. Looking at the coefficient of Pna,b;k∗+j∗+Σb′ n in Equation (5.7.6) and applying the quantum Vandermonde identity with x = n + Σa− Σb, y = m − 1, and z = k∗ − Σb, we obtain κn,a+a′,b+b′,a′,b′q −(j∗+k∗+Σb′)(m−1)q(m−1)(k ∗−Σb)× q−(n+Σa−Σb+m−1) n+Σa− Σb k∗ − Σb− i Choosing Xi = κn,a+a′,b+b′,a′,b′q (m−1)(−j∗−Σb−Σb′)q−(n+Σa−Σb+m−1)i, (which, note, is inde- pendent of k∗, the index of the term we’re looking at) this is exactly the coefficient of Pna,b;k∗+j∗+Σb′ i=0 Xie a,b,j∗+Σb+i Lemma 5.7.5. The intersection of kerRep and AQ is exactly AQR. Proof. Again, this is just by rotation, see §2.4.1. The next two Lemmas, and the subsequent proof of Lemma 5.3.3, are kinda hairy. Hold on tight! Lemma 5.7.6. For any n-hexagonal flows (a, b) of length k ≥ 3 such that APna,b 6= 0 one of the following must hold: 1. The flows (a, b) are also (n− 1)-hexagonal. 2. The flows (a+ 1 , b) are (n− 1)-hexagonal. 3. There is some (a′, b′) ∈ ({0, 1}k)2, so (a + a′, b + b′) is (n − 1)-hexagonal and moreover dGT a′,b′ maps APna,b faithfully into APn−1a+a′,b+b′ . 4. n ≤ 3. Proof. For (a, b) not to be (n− 1)-hexagonal, at least k − 5 edge labels must be (n− 1), and for (a+ 1 , b) not to be (n− 1)-hexagonal, at least k− 5 edge labels must be 1. Already, this establishes the lemma for k > 10 and n > 2 (and it’s trivially true for n ≤ 2). We thus need to deal with k = 6, 8 or 10. If k = 10, one of the first two alternatives hold, unless L(a, b) is some permutation of (1, 1, 1, 1, 1, n − 1, n− 1, n− 1, n− 1, n− 1). It’s easy enough to see that no permutation is possible, given the condition that the incoming edges and outgoing edges have the same sum. For k = 6 or 8, we can be sure that both 1 and n − 1 ∈ L(a, b). Notice that this implies APRna,b = AQRna,b = 0, and both APna,b and AQna,b are at most 2-dimensional, with APna,b spanned by Pna,b;max b and Pna,b;min a+n and AQna,b spanned by Qna,b;max a and Qna,b;min b. Let’s now assume that we never see (not even ‘cyclically’) as contiguous sequences of boundary edges labels either ((n− 1,−), (∗ ≤ 1,+), (∗,−), (∗ ≤ 1,+), (∗,−), . . . , (∗,−), (∗ ≤ 1,+), (n − 1,−)) (5.7.7a) ((n− 1,+), (∗ ≤ 1,−), (∗,+), (∗ ≤ 1,−), (∗,+), . . . , (∗,+), (∗ ≤ 1,−), (n − 1,+)). (5.7.7b) We can then prescribe a reduction path on the P-polygons, which corresponds to a pair (a′, b′) establishing the third alternative above. Consider the reduction path which starts at each incoming edge labeled n− 1, and ends at the next (heading counterclockwise) out- going edge with a label greater than 1, and which additionally ends at each outgoing edge labeled n−1, having started at the previous incoming edge with a label greater than 1. That this forms a valid reduction path (i.e. there are no overlaps between the components de- scribed) follows immediately from the assumption of the previous paragraph, that certain sequences do not appear. The pair (a′, b′) corresponding (via the correspondence discussed in §4.4) to this reduction path then satisfies the first condition of the third alternative above, that (a+ a′, b+ b′) be (n− 1)-hexagonal. However, it doesn’t obviously satisfy the second condition. The only way that dGT a′,b′ might not be faithful on APna,b is if it kills Pna,b;max b or Pna,b;mina+n (remember max b = min a+ n− 1 or min a+ n, since 0 < dimAP a,b ≤ 2). To kill Pna,b;max b, the reduction path we’ve described would have to traverse an internal edge labeled 0. However, it turns out there’s ‘no room’ for this. Consider some component of the reduction path starting at an incoming edge labeled n− 1. It then looks like: for some 0 ≤ l ≤ n, 1 < κ ≤ n, λi = 0 or 1 and 0 ≤ µi ≤ n. If any internal 0 edge gets reduced, there must be some i so that the internal edge li also gets reduced. However, l1 = l + n − 1 − λ1 ≥ l + n − 2, and generally li = l + n − 1 + j=1(−λj + µi) − λi ≥ l + n+ ♯ (µi ≥ 1)− i− 1. Thus for li to be 0, we must have m ≥ i ≥ n− 1 + ♯ (µi ≥ 1) . (5.7.8) Now, if k = 6, none of the external edges can be zero, since (a, b) is n-hexagonal. Thus ♯ (µi ≥ 1) = m, so as long as n ≥ 2, the inequality of Equation (5.7.8) cannot be satisfied. If, on the other hand k = 8, we can only say ♯ (µi ≥ 1) ≥ m − 1 (since any non-adjacent pair of external edges being zero makes it impossible to satisfy the ‘alternating’ part of the condition for being n-hexagon), so m ≥ n− 1 +m− 1, or n ≤ 2. The argument preventing dGT a′,b′ from killing Pna,b;min a+n is much the same. Finally, we need to show that the appearance of one of the ‘forbidden sequences’ from Equation (5.7.7) forces n ≤ 3, so that the last alternative holds. Suppose that, for some m ≥ 0, ((n− 1,−), (λ1,+), (κ1,−), . . . , (κm,−), (λm+1,+), (n− 1,−) with λi = 0 or 1 appears as a contiguous subsequence of L(a, b). The internal edge labels on either side of this subsequence then differ by 2n − 2 + i=1 κi − i=1 λi. We know this quantity must be no more than n, since APna,b 6= 0, i.e., there is some internal flow value l so Pna,b;l 6= 0. However, n−2+ i=1 κi− i=1 λi ≥ n−2+♯ (κi 6= 0)−♯ (λi = 1) ≥ n+♯ (κi 6= 0)−m−3. Thus we must have n ≤ m + 3 − ♯ (κi 6= 0). If such a subsequence appears with m = 0, this immediately forces n ≤ 3. If such a subsequence appears with m > 0, either n ≤ 3 or ♯ (κi 6= 0) < m, so there’s at least one κi = 0. This certainly isn’t possible for k = 3; (a, b) couldn’t be n-hexagonal. We’re now left with a fairly finite list of cases to check for the k = 4 case; for each one, even in the presence of a ‘forbidden sequence’ we’ll explicitly describe an ad hoc reduction path, so that the corresponding (a′, b′) pair satisfies the third alternative. We need to consider both m = 1 and m = 2; there’s no room for m ≥ 3 if k = 4. For m = 1, the boundary edges (up to a rotation) are ((n − 1,−), (λ1,+), (0,−), (λ2,+), (n − 1,−), (µ1,+), (µ2,−), (µ3,+)) with λi = 0 or 1 and some µi. In order for this to be n-hexagon, we can’t have µ1 = n, µ2 = 0, µ3 = n or both λi = 0. Subject then to the condition that incoming and outgoing labels have the same sum, we must have one of µ2 = 2 λ1 = 1 λ2 = 1 µ1 = n− 1 µ3 = n− 1, µ2 = 1 λ1 = 1 λ2 = 1 µ1 = n− 2 µ3 = n− 1, µ2 = 1 λ1 = 1 λ2 = 1 µ1 = n− 1 µ3 = n− 2, µ2 = 1 λ1 = 0 λ2 = 1 µ1 = n− 1 µ3 = n− 1, µ2 = 1 λ1 = 1 λ2 = 0 µ1 = n− 1 µ3 = n− 1. For the first three, however, there’s an (a′, b′) pair: ((1, 0, 1, 1), (0, 0, 0, 0)), ((1, 1, 1, 0), (0, 1, 0, 0)), ((1, 0, 1, 1), (0, 0, 0, 1)), and for the last two, we find that APna,b = 0. For m = 2, the boundary edges are ((n− 1,−), (λ1,+), (κ1,−), (λ2,+), (κ2,−), (λ3,+), (n − 1,−), (µ,+)) with λi = 0 or 1 and at least one of the κi = 0. If both κi = 0, or if µ = n, we can’t achieve n-hexagon-ness. But then the sum of the incoming labels is 2n−2+κ1+κ2 ≥ 2n−1, while the sum of the outgoing edges is λ1 +λ2 +λ3 +µ ≤ n+2. If 2n− 1 ≤ n+2, n ≤ 3 anyway, and we’re done. The argument for the forbidden sequence with opposite orientations is almost identical. Lemma 5.7.7. The intersection APRna,b ∩ spanA Pna,b;max b,Pna,b;max b+1, . . . ,Pna,b;Σb−Pa zero. Proof. Say x = i=0 xiPna,b;max b+i, write imax for the greatest i so xi 6= 0. Define j∗ = imax − Σb, and note that since max b ≤ imax ≤ Σb− a we must have − b ≤ j∗ ≤ − Then apply e a,b;j∗ , as defined in Lemma 5.7.1 to x, obtaining: a,b;j∗(x) = ximax 6= 0, so x /∈ APRna,b. We now give the proof of an earlier lemma, as we’re about to need it. Proof of Lemma 5.3.3. We can just prove the ∞-circumference case. Moreover, if we’re as- suming (a, b) has ∞-circumference 2k, we can further assume that a and b each actually have length k. (Otherwise, there’s some i so bi = ai, or bi = ai+1; in that case, ‘snip out’ those entries from a and b.) Now, pick some imin so aimin = min a, and then pick as imax the first value of i ≥ imin (in the cyclic sense) so that bimax = max b. If imax = imin, we’re done; i 6=imax bi − ai ≥ k − 1. Otherwise, we’ll show how to modify a and b, preserving the value of a, to make imax − imin (again, measured cyclically) smaller. Specifically, for each imin ≤ i < imax, increase bi by bimax − bimax−1, and for each imax < i ≤ imax, increase ai by the same. The resulting sequence is still valid, in the sense that bi ≥ ai, ai+1 for each i. Clearly, this doesn’t change any of max b, min a, Σb or Σa, but it reduces the minimal value of imax − imin by one. We the previous three monstrosities tamed, we can finish of Theorem 5.3.2. Lemma 5.7.8. If (a, b) is a pair of flow labels of length at least 3, and all the external edge labels in L(a, b) are between 1 and n − 1 inclusive, the intersection of kerRep and APna,b + AQna,b is APRna,b +AQRna,b. Proof. If AQna,b 6= 0 but APna,b = 0, first apply a rotation, as described in §2.4.1. We need to deal with the four alternatives of the previous lemma. Supposen−1 /∈ L(a, b), so we’re in the first alternative. Then after applying dGT ∅, we’re in a situation where 5.3.2 holds in full; the kernel of Rep is just APRn−1+AQRn−1. (The argument below works also if 1 /∈ L(a, b) and we’re in the second alternative, mutatis mutandis, exchanging the roles of dGT ∅ and dGT ∂ .) Thus suppose X ∈ APn +AQn is in kerRep. Then dGT ∅ (X) = X∅;P +X∅;Q, for some X∅;P ∈ APRn−1 and X∅;Q ∈ AQRn−1. Pick some XP ∈ dGT −1∅ (X∅;P), which we know by Lemma 5.7.3 must also be an element of APRn. Now dGT ∅ (X −XP) = X∅;Q ∈ AQRn−1, so X −XP must lie in AQn. We want to show it’s actually in AQRn. We know (via Lemma 5.7.4), that X − XP ∈ kerRep, so dGT ∂(X−XP) ∈ AQRn−1, by the remark following Lemma 5.7.3. Thus X−XP ∈ AQRn. The decomposition X = XP + (X −XP) establishes the desired result. Suppose that there’s some (a′, b′) with dGT a′,b′ |APn faithful and (a + a′, b + b′) (n−1)-hexagonal. Now, if 1, n−1 /∈ L(a, b), as in the last lemma we know APna,b is spanned by Pna,b;max b and Pna,b;mina+n and AQ a,b is spanned by Qna,b;max a and Qna,b;min b (remember that in each case the pairs of elements might actually coincide). Thus dGT a′,b′ Pna,b;max b = (−1)b′·a+rotl(a)qmax b(Σb′−Σa′+1)× × qrotl(a′)·b−b′·a−a1−na′1Pn−1a+a′,b+b′;max b 6= 0, dGT a′,b′ Pna,b;min a+n = (−1)b′·a+rotl(a)q(min a+n)(Σb′−Σa′+1)× × qrotl(a′)·b−b′·a−a1−na′1Pn−1 a+a′,b+b′;mina+n 6= 0, dGT a′,b′ Qna,b;max a = (−1)b′·a+rotl(a)qmax a(Σa′−Σb′− × qΣb−nΣb′+b′·rotl(a)−a′·b−a1−na′1Qn−1 a+a′,b+b′;max a dGT a′,b′ Qna,b;min b = (−1)b′·a+rotl(a)qmin b(Σa′−Σb′− × qΣb−nΣb′+b′·rotl(a)−a′·b−a1−na′1Qn−1 a+a′,b+b′;min b Thus suppose x1Pna,b;max b + x2Pna,b;min a+n + y1Qna,b;max a + y2Qna,b;min b is in kerRepn. Then x′1Pn−1a+a′,b+b′;max b + x 2Pn−1a+a′,b+b′;mina+n + y 1Qn−1a+a′,b+b′;max a + y 2Qn−1a+a′,b+b′;min b is in kerRepn−1, where x 1 is a nonzero multiple of x1, and x 2 is a nonzero multiple of x2, by the formulas above. Since (a+a′, b+b′) is (n−1)-hexagonal, we can inductively assume kerRepn−1 = APRn−1a+a′,b+b′ + AQR a+a′,b+b′ , so x′1Pn−1a+a′,b+b′;max b + x′2P a+a′,b+b′;mina+n APRn−1 a+a′,b+b′ . However, by Lemma 5.7.7, along with the knowledge that the relations in APRn−1 a+a′,b+b′ have ‘breadth’ at least 3, as per Lemma 5.3.3, this implies x′1 = x 2 = 0, and thus x1 = x2 = 0. Now we know y1Qna,b;max a + y2Qna,b;min b is in kerRepn, which when restricted to AQna,b is just AQRna,b, by Lemma 5.7.5. Again by Lemma 5.7.7, y1 = y2 = 0. The final alternative is n ≤ 3, where the lemma is obvious from the known com- plete descriptions of kerRep2 and kerRep3. 5.7.4 More about squares To prove Theorem 5.4.1, that SS = APR ⊕ SS ′, we need to establish the following three lemmas. (Trivially, APR ∩ SS ′ = 0.) We’ll in fact only deal with the n+Σa− Σb > 0 case of Theorem 5.4.1; the n + Σa − Σb = 0 is trivial, and, as usual, the n + Σa − Σb < 0 case follows by rotation. Lemma 5.7.9. APR ⊂ SS . Proof. In the quotient (AP +AQ)/SS , (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j Pna,b;j+k = (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j min b m=max a n+Σa− Σb m+ j + k − Σb Qna,b;m. The coefficient of Qna,b;m in this is min b m=max a (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j n+Σa− Σb m+ j + k − Σb Not only does this vanish, but each term indexed by a particular value of m vanishes separately: replacing m with −j∗, this is precisely the q-binomial identity of Lemma A.1.1. Lemma 5.7.10. SS ′ ⊂ SS. Proof. We need to show that each element of the spanning set of SS ′ presented in Theorem 5.4.1 is in SS . In SS ′/(SS ∩ SS ′), Qna,b;m − n+mina l=n+Σa−m (−1)m+l+n+Σa m+ l − 1− Σb m+ l − n− Σa Pna,b;l = = Qna,b;m − n+mina l=n+Σa−m (−1)m+l+n+Σa m+ l − 1− Σb m+ l − n− Σa min b m′=max a n+Σa− Σb m+ l − Σb Qna,b;m′ . The coefficient of Qna,b;m′ here is δmm′ − (−1)m+n+Σa n+mina l=n+Σa−min b (−1)l m+ l − 1− Σb m+ l − n− Σa n+Σa− Σb m′ + l − Σb That this is zero follows from the q-binomial identity proved in §A.1 as Lemma A.1.2. Lemma 5.7.11. SS ⊂ SS ′ +APR. Proof. We take an element of the spanning set described for SS, considered as an element of (AP +AQ)/SS ′. Using the relations from SS ′, we write this as an element of AP/SS ′, and check that this element is annihilated by each element of the spanning set for APR⊥ described in Lemma 5.7.1. Thus, Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb Qna,b;m = = Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb n+min a l′=n+Σa−m (−1)m+l′+n+Σa m+ l′ − 1−Σb m+ l′ − n− Σa Pna,b;l′ . Applying ∑n+Σa k∗=Σb n+Σa−Σb k∗−Σb (Pna,b;k∗+j∗)∗ to this we get n+Σa− Σb k∗ − Σb δl,k∗+j∗− min b m=max a n+mina l′=n+Σa−m k∗=Σb (−1)m+l′+n+Σa n+Σa− Σb m+ l − Σb m+ l′ − 1− Σb m+ l′ − n− Σa n+Σa− Σb k∗ − Σb δl′,k∗+j∗ . First noticing that the lower limit for l′ is redundant, a slight variation Lemma A.1.2 lets us evaluate the sum over m, obtaining n+Σa− Σb k∗ − Σb δl,k∗+j∗ − n+mina l′=−∞ k∗=Σb δl,l′ n+Σa− Σb k∗ − Σb δl′,k∗+j∗ n+Σa− Σb k∗ − Σb δl,k∗+j∗ − k∗=Σb n+Σa− Σb k∗ − Σb δl,k∗+j∗ n+Σa− Σb k∗ − Σb δl,k∗+j∗ − n+Σa− Σb k∗ − Σb δl,k∗+j∗ Chapter 6 Relationships with previous work 6.1 The Temperley-Lieb category The usual Temperley-Lieb category (equivalently, the Kauffman bracket skein module) has unoriented strands, and a ‘loop value’ of − [2]q. The category for Uq(sl2) described here has oriented strands, and a ‘loop value’ of [2]q, along with orientation reversing ‘tags’, which themselves can be flipped for a sign. Nevertheless, the categories are equivalent, as de- scribed below (in parallel with a description of the relationship between my category for Uq(sl4) and Kim’s previous conjectured one) in §6.4. The modification of Khovanov homol- ogy described by myself and Kevin Walker in [26], producing a fully functorial invariant, is based on a categorification of the quantum group skein module, rather than the Kauffman skein module. 6.2 Kuperberg’s spider for Uq(sl3) Kuperberg’s work on Uq(sl3) is stated in the language of spiders. These are simply (strict) pivotal categories, with an alternative set of operations emphasised; instead of composi- tion and tensor product, ‘stitch’ and ‘join’. Further, in a spider the isomorphisms such as Hom(a, b⊗ c) ∼= Hom(a⊗ c∗, b) are replaced with identifications.In this and the following sections, I’ll somewhat freely mix vocabularies. In his work, the only edges that appear are edges labelled by 1, and there are no tags. As previously pointed out, the only trivalent vertices in my construction at n = 3 involve three edges labelled by 1 (here I’m not counting the flow vertices; remember they’re secretly defined in terms of the original vertices and tags). This means that in any diagram, edges labelled by 2 only appear either at the boundary, connected to a tag, or in the middle of an internal edge, with tags on either sides. Since tags cancel without signs in the n = 3 theory, we can ignore all the internal tags. There’s thus an easy equivalence between Kuperberg’s n = 3 pivotal category (defined implicitly by his spider) and mine. (In fact, this equivalence holds before or after quotienting out by the appropriate relations.) In one direction, to his category, we send (1,±) → (1,±) and (2,±) → (1,∓) at the level of objects, and ‘chop off’ any external 2- edges, and their associated tag, at the level of morphisms.1 The functor the other direction is just the ‘inclusion’. Of the two composition functors, the one on his category is actually the identity; the one on mine is naturally isomorphic to the identity, via the tensor natural transformation defined by φ(1,+) = φ(1,−) = φ(2,+) = φ(2,−) = 6.3 Kim’s proposed spider for Uq(sl4) In the final chapter of his Ph.D. thesis, Kim [21] conjectured relations for the Uq(sl4) spider. These relations agree exactly with mine at n = 4. He discovered his relations by calculating the dimensions of some small Uq(sl4) invariant spaces. As soon as it’s possible to write down more diagrams with a given boundary than the dimension of the corresponding space in the representation theory, there must be linear combinations of these diagrams in the kernel. Consistency conditions coming from composing with fixed diagrams which reduce the size of the boundary enabled him to pin down all the coefficients, although, as with my work, he’s unable to show that he’s found generators for the kernel. 6.4 Tags and orientations In this section I’ll describe an equivalence between my categories and the categories pre- viously described for n = 2, in §6.1 and for n = 4, in §6.3. The description will also encompass the equivalence described for n = 3 in §6.2; I’m doing these separately because further complications arise when n is even. The usual spider for Uq(sl2), the Temperley-Lieb category, has unoriented edges. Similarly, Kim’s proposed spider for Uq(sl4) does not specify orientations on the ‘thick’ edges, that is, those edges labelled by 2. Moreover, as in Kuperberg’s Uq(sl3) spider, only a subset of the edge labels I use appear; he only has edges labelled 1 and 2. Nevertheless, those spiders are equivalent to the spiders described here. First, the issue of the edge label 3 being disallowed is treated exactly as above in §6.2; we notice that at n = 4, there are no vertices involving edges labelled 3, so we can remove any internal 3-edges by cancelling tags, and, at the cost of keeping track of a natural isomorphism, remove external 3-edges too. There’s a second issue, however, caused by the unoriented edges. The category equivalences we define will have to add and remove orientation data, and tags. 1This picturesque description needs a patch for the identity 2-edge; there we create a pair of tags first, so we have something to chop off at either end. Thus we define two functors, ι, which decorates spider diagrams with ‘complete orientation data’, and π, which forgets orientations and tags (entirely for n = 2, and only on the 2 edges for n = 4). The forgetful functor also ‘chops off’ any 3 edges (in the n = 4 case), just like the functor in §6.2. The decorating functor ι simply fixes an up-to-isotopy representative of a diagram, and orients each unoriented edge up the page, placing tags at critical points of unoriented edges as follows: It’s well defined because opposite tags cancel. The composition π ◦ ι is clearly the identity. The other composition ι ◦π isn’t quite, but is natural isomorphic to the identity, via the tag maps. 6.5 Murakami, Ohtsuki and Yamada’s trivalent graph invariant In [27], Murakami, Ohtsuki and Yamada (MOY, hereafter) define an invariant of closed knotted trivalent graphs, which includes as a special case the HOMFLYPT polynomial.2 Their graphs carry oriented edge labels, and the trivalent vertices are exactly as my ‘flow vertices’. (They don’t have ‘tags’.) They don’t make any explicit connection with Uq(sln) representation theory, although this is certainly their motivation. In fact, they say: We also note that our graph invariant may be obtained (not checked yet) by direct computations of the universal R-matrix. But the advantage of our defi- nition is that it does not require any knowledge of quantum groups nor repre- sentation theory. One of their closed graphs can be interpreted in my diagrammatic categorySymn; pushing it over into the representation theory RepUq(sln), it must then evaluate to a num- ber. Presumably, this number must be a multiple of their evaluation (modulo replacing their q with my q2), with the coefficient depending on the vertices appearing in the dia- gram, but not their connectivity. Thus given a suitable closed trivalent graph D, the two evaluations would be related via 〈D〉MOY = λ(D)Repn (D) . (6.5.1) Knowing this evaluation coefficient λ(D) explicitly would be nice. You might approach it by either ‘localising’ the MOY formulation3, or deriving a recursive version of the MOY 2There’s also a ‘cheat sheet’, containing a terse summary of their construction, at http://katlas.math. toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg. 3If you’re interested in trying, perhaps ask Dror Bar-Natan or myself, although we don’t have that much to say; the localising step is easy enough. http://katlas.math.toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg http://katlas.math.toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg evaluation function, writing the evaluation of a graph for n in terms of the evaluation of slightly modified graphs for n − 1. In particular, there’s an evaluation function in my category, obtained by branching all the way down to n = 0, which is of almost the same form as their evaluation function. It’s a sum over multiple reduction paths, such that each edge is traversed by as many reductions paths as its label. Moreover, each reduction path comes with an index, indicating which step of the branching process it is ‘applied’ to the diagram at. This index corresponds exactly to the labels in N in MOY, after a linear change of variable. Writing down the details of this should produce a formula for λ(D). Modulo the translation described in the previous paragraph, their (unproved) Proposition A.10 is presumably equivalent to the n+Σa− Σb ≥ 0 case of Theorem 5.2.1. 6.6 Jeong and Kim on Uq(sln) In [11], Jeong and Kim independently discovered a result analogous to both cases of Theo- rem 5.2.1, using a dimension counting argument to show that there must be such relations, and finding coefficients by gluing on other small webs. (Of course, they published first, and have priority on that theorem.) They never describe an explicit map from trivalent webs to the representation theory of Uq(sln), however; they posit relations for loops and bigons, and I = H relations, which differ from ours up to signs, and use these to show that the relations of Theorem 5.2.1 must hold in order to get the dimensions of spaces of diagrams right. They later make a conjecture which says (in my language) that even for 2k-gons with k ≥ 3, each P-2k-gon can be written in terms of Q-2k-gons and smaller polygons. My results show this conjecture is false. 6.7 Other work Sikora, in [34], defines an invariant of oriented n-valent braided ribbon graphs. His graphs do not have labels on the edges, and allow braidings of edges. The invariant is defined by some local relations, including some normalisations, a ‘traditional skein relation’ for the braiding, and a relation expressing a pair of n-valent vertices as a linear combination of braids Easily, every closed graph evaluates to a number: the second relation above lets you re- move all vertices, resulting in a linear combination of links, which can be evaluated via the first relation. He further explains the connection with Murakami, Ohtsuki and Yamada’s work, giving a formula for their evaluation in terms of his invariant. Although his work does not expressly use the language of a category of diagrams, it’s straightforward to make the translation. He implicitly defines a braided tensor cate- gory, with objects generated by a single object, the unlabeled strand, and morphisms gen- erated by caps, cups, crossings, and the n-valent vertices. This category, with no more relations than he explicitly gives, ought to be equiv- alent to the full subcategory of RepUq(sln) generated by tensor powers of the standard representation. Note that this isn’t the same as the category FundRepUq(sln) used here; it has even fewer objects, although again its Karoubi envelope is the entire representation category. However, I don’t think that this equivalence is obvious, at least with the cur- rently available results. Certainly he proves that there is a functor to this representation category (by explicitly construct Uq(sln) equivariant tensors for the braiding, coming from the R-matrix, and for the n-valent vertices). Even though he additionally proves that there are no nontrivial quotients of his invariant, this does not prove that the functor to the rep- resentation theory is faithful. Essentially, there’s no reason why ‘open’ diagrams, such as appear in the Hom spaces of the category, shouldn’t have further relations amongst them. Any way of closing up such a relation would have to result in a linear combination of closed diagrams which evaluated to zero simply using the initially specified relations. Al- ternatively, we could think of this as a question about‘nondegeneracy’: there’s a pairing on diagrams, giving by gluing together pairs of diagrams with the same, but oppositely oriented, boundaries. It’s C(q)-valued, since every closed diagram can be evaluated, but it may be degenerate. That is, there might be elements of the kernel of this pairing which do not follow from any of the local relations. See [25, §3.3] for a discussion of a similar issue in sl3 Khovanov homology. Perhaps modulo some normalisation issues, one can write down functors be- tween his category and mine, at least before imposing relations. In one direction, send an edge labeled k in my category to k parallel ribbons in Sikora’s category, and send each vertex, with edges labeled a, b and c to either the incoming or outgoing n-valent vertex in Sikora’s category. In the other direction, send a ribbon to an edge labeled 1, an n-valent vertex to a tree of trivalent vertices, with n leaves labeled 1 (up to a sign it doesn’t mat- ter, by the I = H relations, which tree we use), and a crossing to the appropriate linear combination of the two irreducible diagrams with boundary ((1,+), (1,+), (1,−), (1,−)). This suggests an obvious question: are the elements of the kernel of the repre- sentation functor I’ve described generated by Sikora’s relations? They need not be, if the diagrammatic pairing on Sikora’s category is degenerate. Yokota’s work in [40] is along similar lines as Sikora’s, but directed towards giv- ing an explicit construction of quantum SU(n) invariants of 3-manifolds. Chapter 7 Future directions Sad though it is to say, I’ve barely scratched the surface here. There are a number of obvious future directions. • Prove Conjecture 5.5; that the whole kernel of Rep is generated by the relations I’ve described via tensor product and composition. • Try to find a ‘confluent’ set of relations, according to the definition of [35]. There’s no obvious basis of diagrams modulo the relations I’ve given, for n ≥ 4; this is essentially just saying that the relations I’ve presented are not confluent. • Find an evaluation algorithm; that is, describe how to use the relations given here to evaluate any closed web. The authors of [11] claim to do so, using only the I = H and ‘square-switch’ relations, although I have to admit not being able to follow their proof. • Complete the discussion from §6.5, explicitly giving the relationship between the evaluation of a closed diagram in my category, and its evaluation in the MOY theory. • In fact, the first Kekulé relation, appearing in Equation (5.3.1) at n = 4, follows from the ‘square switch’ and ‘bigon’ relations, as follows. First, applying Equation (5.6.6) across the middle of a hexagon, we see and then, using the I = H relation to switch two of the edges in the first term, and Equation (5.6.5) to resolve each of the squares in the second term, q + [2]q + [2]q + Next, we resolve the remaining squares using Equation (5.6.6), obtaining  + + + q + [2]q + [2]q + and then resolve the newly created squares using Equation (5.6.5) again, and resolve the newly created bigons using Equation (5.6.2) 1 + [3]q − [2] q + 1 = + − . Is there any evidence that this continues to happen? Perhaps all the Kekulé relations follow in a similar manner from ‘square-switch’ relations? • Try to do the same thing for the other simple Lie algebras. Each Lie algebra has a set of fundamental representations, corresponding to the nodes of its Dynkin diagram. You should first look for intertwiners amongst these fundamental representations, until you’re sure (probably by a variation of the Schur-Weyl duality proof given here) that they generate the entire representation theory. After that, describe some rela- tions, and perhaps prove that you have all of them. I have a quite useful computer program for discovering relations, which is general enough to attempt this problem for any simple Lie algebra. On the other hand, the methods of proof used here, based on the multiplicity free branching rule for sln, will need considerable modification. For the notion of Gel‘fand-Tsetlin basis in the orthogonal and symplectic groups, see [36]. • Describe the representation theory of the restricted quantum group at a root unity in terms of diagrams; in particular find a diagrammatic expression for the idempotent in ⊗aV maa<n which picks out the irrep of high weight amaλa, for each high weight. Kim solved this problem for Uq(sl3) in his Ph.D. thesis [22]. It would be nice to start simply by describing the ‘lowest root of unity’ quotients, in which only the funda- mental representations survive. After this, one might learn how to use these cate- gories of trivalent graphs to give a discussion, parallel to that in [3], of the modular tensor categories appearing at roots of unity. • Categorify everything in sight! Bar-Natan’s work [1] on local Khovanov homology provides a categorification of the Uq(sl2) theory. 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A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping. In Special functions, q-series and related topics (Toronto, ON, 1995), volume 14 of Fields Inst. Commun., pages 179–210. Amer. Math. Soc., Providence, RI, 1997. MR1448687, (online). [30] Peter Paule and Markus Schorn. A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. J. Symbolic Comput., 20(5-6):673–698, 1995. Symbolic computation in combinatorics ∆1 (Ithaca, NY, 1993). MR1395420, (online). [31] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. A = B. A K Peters Ltd., Wellesley, MA, 1996. Available from http://www.cis.upenn.edu/~wilf/AeqB.html. [32] A. Riese. Contributions to symbolic q-hypergeometric summation, 1997. Ph.D. Thesis, RISC, J. Kepler University, Linz, (online). [33] M. Schorn. Contributions to symbolic summation, December 1995. Diploma Thesis, RISC, J. Kepler University, Linz, (online). 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DOI:10.1007/s002080050025. http://www.ams.org/mathscinet-getitem?mr=MR1448687 http://www.risc.uni-linz.ac.at/research/combinat/risc/software/qZeil/pub/pauleRiese97.pdf http://www.ams.org/mathscinet-getitem?mr=MR1395420 http://www.risc.uni-linz.ac.at/research/combinat/risc/software/PauleSchorn/pub/pauleSchorn95.pdf http://www.cis.upenn.edu/~wilf/AeqB.html http://www.risc.uni-linz.ac.at/research/combinat/software/qZeil/pub/riese97.pdf http://www.risc.uni-linz.ac.at/research/combinat/risc/software/PauleSchorn/pub/schorn95.pdf http://dx.doi.org/10.2140/agt.2005.5.865 http://arxiv.org/abs/math.QA/0407299 http://arxiv.org/abs/math.QA/0609832 http://www.ams.org/mathscinet-getitem?mr=MR1010814 http://en.wikipedia.org/w/index.php?title=Karoubi_envelope&oldid=51453957 http://en.wikipedia.org/w/index.php?title=Friedrich_August_Kekul%C3%A9_von_Stradonitz&oldid=101345265 http://en.wikipedia.org/w/index.php?title=Q-Vandermonde_identity&oldid=105022267 http://dx.doi.org/10.1007/s002080050025 Appendix A Appendices A.1 Boring q-binomial identities In this section we’ll prove some q-binomial identities needed in various proofs in the body of the thesis. The identities are sufficiently complicated that I’ve been unable to find com- binatorial arguments for them. Instead, we’ll make use of the Zeilberger algorithm, de- scribed in [31]. In particular, we’ll use the Mathematica‘ implementation in [30, 33], and the implementation of the q-analogue of the Zeilberger algorithm described in [29, 32]. If you’re unhappy with the prospect of these identities being proved with com- puter help, you should read Donald Knuth’s foreword in [31] Science is what we understand well enough to explain to a computer. Art is ev- erything else we do. During the past several years an important part of math- ematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically. For each use made of the the q-Zeilberger algorithm, I’ve included a Mathematica notebook showing the calculation. These are available in the /code/ subdirectory, after you’ve downloaded the source of this thesis from the arXiv. Unfortunately, the conventions I’ve used for q-binomials don’t quite agree with those used in the implementation of the q-Zeilberger algorithm. They are related via qk(k−n) ]qZeil , although in my uses of their algorithms, I’ve replaced q2 with q throughout; this apparently produces cleaner looking results. Lemma A.1.1. For Σb ≤ j ≤ Σa+ n− 1 and − b ≤ j∗ ≤ − min(− a+1,j∗−j+n+Σa) k=max(− b,Σb+j∗−j) (−1)j+k j + k −max b j −Σb min a+ n− j − k Σa+ n− 1− j n+Σa− Σb j − j∗ + k − Σb = 0. (A.1.1) Proof. Writing Σn for the sum above (suppressing the dependence on Σa,Σb,min a,max b, j and j∗), the q-Zeilberger algorithm reports (see /code/triple-identity.nb/) that a+Σb−j+j∗−1) (1− q(2+2n+2min a−2max b)) 1− q(2Σa+2 b−2j+2j∗+2n) as long as n+Σa− Σb 6= 0 and n+Σa+ b− j + j∗ 6= 0. (A.1.2) In the case that n+Σa−Σb = 0, the top of the third q-binomial in Equation (A.1.1) is zero, so the sum is automatically zero unless k = Σb+ j∗ − j, in which case the sum collapses to (−1)Σb+j∗ Σb+ j∗ j − Σb a− j∗ Σb− 1− j However, since j ≥ Σb, the bottom of the second q-binomial here is less than zero, so the whole expression vanishes. In the case that n+Σa+ b−j+j∗ = 0, we can replace n everywhere in Equation (A.1.1), obtaining (−1)j+k j + k −max b j − Σb b− j∗ − k b− j∗ − 1 b+ j − j∗ j − j∗ − k − Σb Again, the second q-binomial vanishes, since j∗ ≥ − Now that we’re sure the identity holds when either of the inequalities of Equation (A.1.2) are broken, we can easily establish the identity when they’re not; if it holds for some value of n, it must hold for n + 1 (notice that the restriction on j in the hypothesis of the lemma doesn’t depend on n). Amusingly, the easiest starting case for the induction is not n = 0, but n sufficiently large and negative, even though this makes no sense in the original representation theoretic context! In particular, at n ≤ −Σa+ Σb− 1, the third binomial in Equation (A.1.1) is always zero, so the sum vanishes. For the next lemma, we’ll need q-Pochhammer symbols, (a; q)k = j=0(1− aqj) if k > 0, 1 if k = 0, j=1(1− aq−j)−1 if k < 0. In particular, note that (q; q) k<0 = 0, and (q; q)k>0 6= 0. Lemma A.1.2. For any n,m,m′ ∈ Z, and a = (a1, a2), b = (b1, b2) ∈ Z2, with n+Σa−Σb > 0 δmm′ − (−1)m+n+Σa n+mina l=n+Σa−min b (−1)l m+ l − 1− Σb m+ l − n− Σa n+Σa−Σb m′ + l − Σb Proof. First, notice that the expression is invariant under the transformation adding some integer to each of m,m′, l, a1, a2, b1, b2. We’ll take advantage of this to assume Σb + 2m − 1,Σb+m+m′−1, 2Σb+2m−1−n−Σa, min a+m+n, min a+m′+n, and Σb−maxa+m are all positive, and −2Σb−m−m′ + n− Σa and max a− Σb−m′ − 1 are both negative. Again, writing Σn for the left hand side of the above expression, the q-Zeilberger algorithm reports (see /code/SS-identity.nb/) that as long as m 6= m′, Σn = (−1)•q• (q; q)Σb+2m−1 (qm − qm′) (q; q)Σb+m+m′−1 (q; q)2Σb+2m−1−n−Σa (q; q)−2Σb−m−m′+n−Σa + (−1)•q• (q; q)mina+m+n (qm − qm′) (q; q)mina+m′+n (q; q)Σb−max a+m (q; q)max a−Σb−m′−1 This is exactly zero, using the inequalities described in the previous paragraph, and the definition of the q-Pochhammer symbol. When m = m′, the q-Zeilberger algorithm reports that Σn = (−1)•q• (qn+Σa−Σb − 1) (qmin a+m+n − 1) (q; q)Σb−max a+m (q; q)−Σb+max a−m +Σn−1. (A.1.3) The first term is zero, since −Σb+max a−m < 0, so (q; q)−1−Σb+max a−m = 0. Thus we just need to finish off the case m = m′, n + Σa− Σb = 1, where the left hand side of Equation (A.1.2) reduces to δmm′ − (−1)m+Σb+1 (−1)l m+ l − 1− Σb m+ l − 1− Σb m′ + l − Σb There are then two terms in the summation, l = Σb−m′ and l = Σb−m′ +1, so we obtain δmm′ − (−1)m−m m−m′ − 1 m−m′ − 1 − (−1)m−m′ If m = m′, the first q-binomial vanishes, but the second is 1, while if m > m′, both q- binomials are equal to 1, and cancel, and if m < m′, both q-binomials vanish. A.2 The Uq(sln) spider cheat sheet. A.2.1 The diagrammatic category, §2.2. Generators. , , , for each a = 1, . . . , n− 1, , , , and for each a = 1, . . . , n− 1, and for 0 ≤ a, b, c ≤ n, with a+ b+ c = n. Relations: planar isotopy. Relations: (anti-)symmetric duality. = (−1)(n+1)a = (−1)(n+1)a Relations: degeneration. A.2.2 The map from diagrams to representation theory, §3.6.  = Rep = Rep = (−1)(n+1)a Rep = (−1)(n+1)a = Rep = Rep A.2.3 The diagrammatic Gel‘fand-Tsetlin functor, §4. Pairings and copairings. dGT ′ = + dGT ′ = qn−a + q−a dGT ′ = + dGT ′ = qa−n + qa , (4.1.2) Identifications with duals. dGT ′ = + (−1)aq−a dGT ′ = (−1)n+a + q−a dGT ′ = qa + (−1)n+a dGT ′ = (−1)aqa + . (4.1.3) Curls. = qn−a + q−a dGT = qa−n + qa Trivalent vertices. dGT ′  = (−1)cqb+c + (−1)aqc + (−1)b , dGT ′  = (−1)c + (−1)aq−a + (−1)bq−a−b  = (−1)a  + (−1)n+b + q−a  = (−1)a + (−1)n+bqb + = (−1)a + (−1)n+bqb + = (−1)a qa + + (−1)n+bqa−n Larger webs.  = + + q−a + + + q−a + (−1)b+cqb (4.3.1) = qb + + + + + qb + (−1)a+cqb+c−n (4.3.2) P’s and Q’s, §2.4. dGT a′,b′(Pna,b;l) = (−1)b ′·(a+rotl a)ql(Σb ′−Σa′+1)qrotl a ′·b−b′·a−a1−na 1Pn−1 a+a′,b+b′;l dGT a′,b′(Qna,b;l) = (−1)b ′·(a+rotl a)ql(Σa ′−Σb′− +1)qΣb+nΣb ′+b′·rotl a−a′·b−a1−na 1Qn−1a+a′,b+b′;l dGT ∅(Pna,b;l) = dGT −→0 ,−→0 (P a,b;l) + dGT −→1 ,−→1 (P a,b;l) = ql−a1 Pn−1a,b;l + q −n−Σa+ΣbPn−1a,b;l−1 dGT ∅(Qna,b;l) = dGT −→0 ,−→0 (Q a,b;l) = ql−a1qΣb− 2 Qn−1a,b;l dGT ∂(Pna,b;l) = dGT −→1 ,−→0 (P a,b;l) = ql−a1qΣb− −nPn−1 1 ,b;l dGT ∂(Qna,b;l) = dGT −→0 ,−→−1(Q a,b;l) + dGT −→1 ,−→0 (Q a,b;l) = ql−a1 qΣb−Σa− 2 Qn−1 1 ,b;l+1 + q−nQn−1 1 ,b;l A.2.4 Conjectural complete set of relations, §5. The I = H relations = (−1)(n+1)a , = (−1)(n+1)a . The ‘square-switching’ relations SSna,b = spanA Pna,b;l − min b m=max a n+Σa− Σb m+ l − Σb Qna,b;m }mina+n l=max b if n+Σa− Σb ≥ 0 spanA Qna,b;l − n+mina m=max b Σb− n− Σa m+ l − Σa− n Pna,b;m }min b l=maxa if n+Σa− Σb ≤ 0 The Kekulé relations APRna,b = spanA a,b;j = (−1)j+k j + k −max b j − Σb min a+ n− j − k Σa+ n− 1− j Pna,b;j+k Σa+n−1 (A.2.1) AQRna,b = spanA a,b;j = (−1)j+k j + k −max a j − Σa min b− j − k Σb− n( |∂| − 1)− 1− j Qna,b;j+k Σb−n( −1)−1 (A.2.2) with orthogonal complements APRna,b⊥ = spanA a,b;j∗ = k∗=Σb n+Σa− Σb k∗ − Σb Pna,b;j∗+k∗⊥ AQRna,b⊥ = spanA a,b;j∗ k∗=Σa Σb− Σa− n k∗ − Σa ×Qna,b;j∗+k∗⊥ (A.2.3) Introduction Summary The Temperley-Lieb algebras Kuperberg's spiders The `diagrammatic' category Symn Pivotal categories Quotients of a free tensor category Flow vertices Polygonal webs Just enough representation theory The Lie algebra sln The quantum groups Uq(sln) Representations Strictifying Rep Uq(sln) Generators for FundRepUq(sln) The representation functor The diagrammatic Gel`fand-Tsetlin functor Definition on generators Descent to the quotient Calculations on small webs A path model, and polygons. Describing the kernel The I=H relations The square-switch relations The Kekulé relations More about squares A conjecture Examples: Uq(sln), for n=2,3,4 and 5. Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2 Relationships with previous work The Temperley-Lieb category Kuperberg's spider for Uq(sl3) Kim's proposed spider for Uq(sl4) Tags and orientations Murakami, Ohtsuki and Yamada's trivalent graph invariant Jeong and Kim on Uq(sln) Other work Future directions Bibliography Appendices Boring q-binomial identities The Uq(sln) spider cheat sheet.

0704.1504

The Formation of Lake Stars

The Formation of Lake Stars Victor C. Tsai1∗ and J. S. Wettlaufer2† Department of Earth & Planetary Sciences, Harvard University, Cambridge, Massachusetts, 02138 and Departments of Geology & Geophysics and Physics, Yale University New Haven, Connecticut 06520-8109 (Dated: 11 April, 2007) Star patterns, reminiscent of a wide range of diffusively controlled growth forms from snowflakes to Saffman-Taylor fingers, are ubiquitous features of ice covered lakes. Despite the commonality and beauty of these “lake stars” the underlying physical processes that produce them have not been explained in a coherent theoretical framework. Here we describe a simple mathematical model that captures the principal features of lake-star formation; radial fingers of (relatively warm) water-rich regions grow from a central source and evolve through a competition between thermal and porous media flow effects in a saturated snow layer covering the lake. The number of star arms emerges from a stability analysis of this competition and the qualitative features of this meter-scale natural phenomena are captured in laboratory experiments. I. INTRODUCTION The scientific study of the problems of growth and form occupies an anomalously broad set of disciplines. Whether the emergent patterns are physical or biological in origin, their quantitative description presents many challenging and compelling issues in, for example, applied mathematics [1], biophysics [2], condensed matter [3] and geophysics [4] wherein the motion of free boundaries is of central interest. In all such settings a principal goal is to predict the evolution of a boundary that is often under the influence of an instability. Here we study a novel variant of such a situation that occurs naturally on the frozen surfaces of lakes. Lakes commonly freeze during a snowfall. When a hole forms in the ice cover, relatively warm lake water will flow through it and hence through the snow layer. In the process of flowing through and melting the snow this warm water creates dark regions. The pattern so produced looks star-like (see Figure 1) and we refer to it as a “lake star”. These compelling features have been described qualitatively a number of times (e.g. [5, 6, 7]) but work on the formation process itself has been solely heuristic. Knight [5] outlines a number of the physical ideas relevant to the process, but does not translate them into a predictive framework to model field observations. Knight’s main idea is that locations FIG. 1: Typical lake star patterns. The branched arms are approximately 1 m in length. Quonnipaug Lake, Guilford, Connecticut, 8 March, 2006. ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.1504v1 mailto:[email protected] mailto:[email protected] FIG. 2: Schematic of the geometry of the model. The perspective is looking down on a nascent star. The equations (refer to text for numbering) are shown in the domains of the system where they are applicable. with faster flow rates melt preferentially, leading to even faster flow rates and therefore to an instability that results in fingers. This idea has features that resemble those of many other instabilities such as, for example, those observed during the growth of binary alloys [8], in flow of water through a rigid hot porous media [9], or in more complex geomorphological settings [10], and we structure our model accordingly. Katsaros [6] and Woodcock [7] attribute the holes from which the stars emanate and the patterns themselves to thermal convection patterns within the lake, but do not measure or calculate their nature. However, often the holes do not exhibit a characteristic distance between them but rather form from protrusions (e.g. sticks that poke through the ice surface) [5] and stars follow thereby ruling out a convective mechanism as being necessary to explain the phenomena. The paucity of literature on this topic provides little more than speculation regarding the puncturing mechanism but lake stars are observed in all of these circumstances. Therefore, while hole formation is necessary for lake star formation, its origin does not control the mechanism of pattern formation, which is the focus of the present work. II. THEORY The water level in the hole is higher than that in the wet snow–slush–layer [5] and hence we treat this warm water [11] region as having a constant height above the ice or equivalently a constant pressure head, which drives flow of water through the slush layer, which we treat as a Darcy flow of water at 0◦C. We model the temperature field within the liquid region with an advection-diffusion equation and impose an appropriate (Stefan) condition for energy conservation at the water-slush interface. The water is everywhere incompressible. Finally, the model is closed with an outer boundary condition at which the pressure head is assumed known. Although we lack in-situ pressure measurements, circular water-saturated regions (a few meters in radius) are observed around the lake stars. Hence, we assume that the differential pressure head falls to zero somewhere in the vicinity of this circular boundary. The actual boundary at which the differential pressure head is zero is not likely to be completely uniform (as in Figure 4 of Knight [5]) but treating it as uniform is a good approximation in the linear regime of our analysis. Finally, we treat the flow as two-dimensional. Thus, although the water in direct contact with ice must be at 0◦C, we consider the depth-averaged temperature, which is above freezing. Additionally, the decreasing pressure head in the radial direction must be accompanied by a corresponding drop in water level. Therefore, although the driving force is more accurately described as deriving from an axisymmetric gravity current, the front whose stability we assess is controlled by the same essential physical processes that we model herein. Our analysis could be extended to account for these three-dimensional effects. The system is characterized by the temperature T , a Darcy fluid velocity u, pressure p, and an evolving liquid- slush interface a. The liquid properties are κ (thermal diffusivity), CP (specific heat at constant pressure) and µ (dynamic viscosity) and the slush properties are Π (permeability), ξ (solid fraction) and L (latent heat). We non- dimensionalized the equations of motion by scaling the length, temperature, pressure and velocity with r0, T0, p0, and Πp0 , respectively. Thus, our model consists of the following system of dimensionless equations: + u · ∇θ = ǫ∇2θ r0 < r < a(φ, t), (1) θ = 0 a(φ, t) < r < 1, (2) p = 1 ri < r < a(φ, t), (3) ∇2p = 0 a(φ, t) < r < 1, (4) ∇ · u = 0 ri < r < a(φ, t), (5) = u |a+ r = a(φ, t), (6) u = −∇p a(φ, t) < r < 1, (7) with boundary conditions ȧ = − ∇θ r = a(φ, t), (8) 1 r = ri 0 r = a(φ, t) 0 r = 1 , (9) 1 r = ri 1 r = a(φ, t) 0 r = 1 , (10) where (1) describes the temperature evolution in the liquid, (4) and (5) describe mass conservation with a Darcy flow (7) in the slush, (8) is the Stefan condition, and (9) and (10) are the temperature and pressure boundary conditions, respectively (see Figure 2). Note that (3) and (5) can both be satisfied since the liquid region has an effectively infinite permeability. The dimensionless parameters ǫ and S of the system are given by , and S ≡ , (11) which describe an inverse Peclet number and a Stefan number respectively. Because the liquid must be less than or equal to 4◦C, we make the conservative estimates that T0 < 4 ◦C, ξ > 0.3, and use the fact that L/CP ≈ 80 from which we see S > 6 ≫ 1. Using κ ≈ 10−7m2s−1, and the field observations of Knight [5] to constrain u0 (1cm/hr < u0 < 10cm/hr) and r0 (0.3m < r0 < 3m), we find that ǫ < 0.1 ≪ 1. We therefore employ the quasi- stationary (S ≫ 1) and large Peclet number (ǫ ≪ 1) approximations, and hence equations (1) - (10) are easily solved for a purely radial flow with cylindrical symmetry (no φ dependence) and circular liquid-slush interface. This (boundary layer) solution is u = ur̂ = − ln(a0) r̂ ri < r < 1, (12) ln(r) ln(a0) r > a0, (13) θ0 = 1− (−1/ ln(a0)+2ǫ) r < a0, (14) 0 0.5 1 1.5 2 2.5 3 σ /ln(a /ln2(a = 0.06 = 0.5 approx FIG. 3: Stability curve: Non-dimensional growth rate σ versus non-dimensional wavenumber k′. Scales for the axes are given at the upper left (σ axis) and the lower right corners (k′ axis). σ is plotted for the range of plausible a0 (dot-dashed blue and dashed red curves) and for the approximation (18) (solid green curve). Sa0ȧ0 −1/ ln(a0) + 2ǫ = 1, (15) where equation (15) has an approximate implicit solution for a0 given by a20 ln(a0) = . (16) We perform a linear stability analysis around this quasi-steady cylindrically symmetrical flow. Proceeding in the usual way, we allow for scaled perturbations in θ and a with scaled wavenumber k′ = ǫk, non-dimensional growth rate σ, and amplitudes f(r) and g respectively. Keeping only terms linear in ǫ, 1/S and g, we solve (4) subject to (10), substitute into (6) and satisfy (5) and (1). This gives the non-dimensional growth rate (σ) as a function of scaled wave number (k′): 2a0 ln 2(a0)S 1 + 4k′2 ln2(a0)− 1 −k′ ln(a0) . (17) Equation (17) can be approximated in 0 ≤ x . 1 as ln2(a0)S x(1− x), (18) where x ≡ −k′ ln(a0)/a0. The stability curve (17) and the approximation (18) are plotted in Figure 3. The essential features of (17) are a maximum in the range 0 < k′ < a0/ ln(a0), zero growth rate at k ′ = a0/ ln(a0) and a linear increase in stability with k′ for large k′. The long-wavelength cut-off is typical of systems with a Peclet number, here with the added effect of latent heat embodied in the Stefan number. This demonstrates the competition between the advection and diffusion of heat and momentum (in a harmonic pressure field); the former driving the instability and the latter limiting its extent. The maximum growth rate occurs at approximately k′max ≈ −2 ln(a0) , (19) with (non-dimensional) growth rate σmax ≈ 4S ln . (20) FIG. 4: Schematic showing r0, r , rLS and rℓ. Translating (19) and (20) back into dimensional quantities, we find that the most unstable mode has angular size given by φdegrees = 720◦κ , (21) and has growth rate given by σdim = 4Sr0 ln 2(r0/a0) . (22) III. EXTRACTING INFORMATION FROM FIELD OBSERVATIONS Field observations of lake stars cannot be controlled. A reasonable estmate for r0 is the radius of the wetted (snow) region around the lake stars, and observations [5, 6, 7] bound the value as 1.5m . r0 . 4m. This is simply because if there were significant excess pressure at this point then the wetting front would have advanced further. However, it is also possible that the effective value of r0, say r 0 , is less than this either because the wetted radius is smaller earlier in the star formation process or because the ambient pressure level is reached at smaller radii. Here, we take a0 to be the radius of the roughly circular liquid-filled region at the center of the lake star (rℓ) as the best approximation during the initial stages of star formation (see Figure 4). Field observations show that 0.1m . rℓ . 0.5m, [5, 6, 7] and hence 0.07 . rℓ/r0 . 0.15. We note that equations (21) and (22) are more sensitive to a0/r0 than a0 or r0 independently[12]. With this interpretation of r0 we find a reasonable estimate of u0 as 1.4 · 10−5m/s . u0 . 2.8 · 10 −5m/s. Using these parameter values, the most unstable mode should have wavelength between 8◦ and 130◦. Letting the number of branches be N = 360◦/φdeg, then 3 < N < 45 and we clearly encompass the observed values for lake stars (4 < N < 15), but note that values (N > 15) are never seen in the field. Despite the dearth of field observations, many qualitative features embolden our interpretation. For example, the stars with larger values of a0/r0 have a larger number of branches. Moreover, for any value of a0/r0, our analysis predicts an increase in N with r0 and u0. Indeed, u0 increases with p0 (higher water height within the slush layer) and Π (less well-packed snow). Therefore, we ascribe some of the variability among field observations to variations in these quantities (which have not been measured in the field) and the remainder to nonlinear effects. Because the dendritic arms are observed long after onset and are far from small perturbations to a radially symmetric pattern, as one might see in the initial stages of the Saffman-Taylor instability, the process involves non-linear cooperative phenomena. Hence, our model should only approximately agree with observations. Although a rigorous non-linear analysis of the long term star evolution process (e.g. [3]) may more closely mirror field observations, the present state of the latter does not warrant that level of detail. Instead, we examine the model physics through simple proof of concept experimentation described presently. FIG. 5: Typical experimental run where small- scale fingers are present. For scale, the nozzle head has diameter of 5 mm. FIG. 6: Typical run where channels form. This picture is taken from the underside. Note: part of the slush broke off when it was flipped to image it. The ruler scale is in cm. IV. DEMONSTRATING LAKE STARS IN THE LABORATORY A 30 cm diameter circular plate is maintained below freezing (≈ −0.5◦C), and on top of this we place a 0.5 to 1 cm deep layer of slush through which we flow 1◦C water. Given the technical difficulties associated with its production, the grain size, and hence the permeability, of the slush layer, is not a controlled variable. This fact influences our results quantitatively. In fourteen runs we varied the initial size of the water-filled central hole (a0), that of the circular slush layer (r0), and the flow rate (Q), which determines u0. The flow rate is adjusted manually so that the water level (h0) in the central hole remains constant [13]. Fingering is observed in every experimental run and hence we conclude that fingers are a robust feature of the system. Two distinct types of fingering are observed: small-scale fingering (see Figure 5) that forms early in an experimental run, and larger channel-like fingers (see Figure 6) that are ubiquitous at later times and often extend from the central hole to the outer edge of the slush. Since the channel-like fingers provide a direct path for water to flow, effectively shorting Darcy flow within the slush, their subsequent dynamics are not directly analogous to those in natural lake stars. However, in all runs, the initial small-scale fingers have the characteristics of lake stars and hence we focus upon them. We note that because the larger channel-like fingers emerge out of small-scale fingers, they likely represent the non-linear growth of the linear modes of instability, a topic left for future study. Finally, we measure the distance between fingers (df ), so that for each experiment we can calculate u0 = Q/(2πr0h0), φcalc ≡ φdegrees, from equation (21), and φobs = 180 ◦df/(πa0), and we can thereby compare experiment, theory and field observations. In Figure 7 we plot φobs versus φcalc for the various field observations for which we have estimates of parameters, the laboratory experiments described above, and the model [equation (21)]. There is a large amount of scatter in both 0 100 200 300 400 500 (deg) FIG. 7: Comparison of theory, experiment and field observations. Circles are field observations (cyan = best constrained field observation, black = range of plausible field observations), triangles are experimental results (blue upward-pointing triangles were unambiguous; red left-pointing triangles have channels but show no clear small-scale fingers, so channel spacing is taken for df ; green right-pointing triangles were compromised by the quality of the images). Errors are approximately 0.3 cm, 0.5 cm, 2 mm, 5 ml/min and 0.2 cm (respectively) for the five measured quantities. All experimental results thus have error bars of at least a factor of two in the x-coordinate and 30% in the y-coordinate. Typical error bars are shown on one measurement. The solid red line is the theoretical prediction; the dotted green line is the best fit line to the blue triangles. the experimental and observational data and the data does not lie on the one-to-one curve predicted by the model. However, the experiments are meant to demonstrate the features of the model predictions, and the results have the correct qualitative trend (having a best-fit slope of 0.34). We also attempt to find trends in the experimental data not represented by the model by comparing y ≡ φobs/φcalc vs. various combinations of control parameters (≡ x) including r0, a0, r0/a0, r0u0, r0/a0 ln(r0/a0) and ln(r0/a0)/(a0u0). For all plots of y vs. x, our model predicts a zero slope (and y-intercept of 1). A non-random dependence of y on x would point to failure of some part of our model. Thus, to test the validity of our model, we perform significance tests on all non-flagged data with the null hypothesis being a non-zero slope. In all cases, the null hypothesis is accepted (not rejected) at the 95% confidence level. Thus, although the agreement is far from perfect, the simple model captures all of the significant trends in the experimental data. V. CONCLUSIONS By generalizing and quantifying the heuristic ideas of Knight [5], we have constructed a theory that is able to explain the radiating finger-like patterns on lake ice that we call lake stars. The model yields a prediction for the wavelength of the most unstable mode as a function of various physical parameters that agrees with field observations. Proof of concept experiments revealed the robustness of the fingering pattern, and to leading order the results also agree with the model. There is substantial scatter in the data, and the overall comparison between field observations, model and experiment demonstrates the need for a comprehensive measurement program and a fully nonlinear theory which will yield better quantitative comparisons. However, the general predictions of our theory capture the leading order features of the system. VI. ACKNOWLEDGEMENTS We thank K. Bradley and J. A. Whitehead for laboratory and facilities support. This research, which began at the Geophysical Fluid Dynamics summer program at the Woods Hole Oceanographic Institution, was partially funded by National Science Foundation (NSF) grant OCE0325296, NSF Graduate Fellowship (VCT), NSF grant OPP0440841 (JSW), and Department of Energy grant DE-FG02-05ER15741 (JSW). [1] T.Y. Hou, J.S. Lowengrub, and M.J. Shelley, J. Comp. Phys. 169, 302 (2001). [2] M.P. Brenner, L.S. Levitov, and E.O. Budrene, Biophys. J. 74, 1677 (1998); H. Levine and E. Ben-Jacob, Phys. Biol. 1, 14 (2004). [3] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993); I.S. Aranson and L.S. Tsimring, Rev. Mod. Phys. 78, 64 (2006) [4] N. Goldenfeld, P.Y. Chan and J. Veysey, Phys. Rev. Lett. 96, 254501 (2006); M.B. Short, J.C. Baygents, and R.E. Goldstein, Phys. Fluids 18, 083101 (2006). [5] C.A. Knight, in Structure and dynamics of partially solidified systems, ed. D.E. Loper, (Martinus Nijhoff, Dordrecht, 1987), pp. 453-465. [6] K.B. Katsaros, Bull. Amer. Meteor. Soc. 64, 277 (1983). [7] A.H. Woodcock, Limnol. Oceanogr. 10 R290 (1965). [8] M.G. Worster, Annu. Rev. Fl. Mech. 29, 91 (1997). [9] S. Fitzgerald and A.W. Woods, Nature 367, 450 (1994). [10] N. Schorghofer, B. Jensen, A. Kudrolli and D.H. Rothman, J. Fluid Mech. 503, 357 (2004). [11] A finite body of fresh water cooled from above will have a maximum below ice temperature below of 4 ◦C. [12] For the later stages of growth, clearly in the nonlinear regime not treated presently, a0 may also be interpreted as the radius of the lake star (rLS). Field observations show 1m . rLS . 2m [5, 6, 7] and hence 0.3 . rLS/r0 . 0.6. [13] In many of the runs, we begin the experiment without the central hole. In practice, however, the first few drops of warm water create a circular hole with radius one to three times the radius of the water nozzle (0.5cm < a0 < 1.0cm). It is significantly more difficult to prepare a uniform permeability sample with a circular hole initially present; these runs are therefore more difficult to interpret. Introduction Theory Extracting information from field observations Demonstrating Lake Stars in the Laboratory Conclusions Acknowledgements References

0704.1505

A Higgs-Higgs bound state due to New Physics at a TeV

UCSD/PTH 07-03 A Higgs-Higgs bound state due to New Physics at a TeV. Benjamin Grinstein and Michael Trott1 1Department of Physics, University of California at San Diego, La Jolla, CA, 92093 Abstract We examine the effects of new physics on the Higgs sector of the standard model, focusing on the effects on the Higgs self couplings. We demonstrate that a low mass higgs, mh < 2mt, can have a strong effective self coupling due to the effects of a new interaction at a TeV. We investigate the possibility that the first evidence of such an interaction could be a higgs-higgs bound state. To this end, we construct an effective field theory formalism to examine the physics of such a low mass higgs boson. We explore the possibility of a non relativistic bound state of the higgs field (Higgsium) at LHC and construct a non relativistic effective field theory of the higgs sector that is appropriate for such studies (NRHET). http://arxiv.org/abs/0704.1505v3 I. INTRODUCTION Currently, global fits to all precision electro weak give the higgs mass to be 113+56 with an upper bound given by mh < 241GeV at 95% CL (see, e.g., J. Erler and P. Langacker in Sec. 10 of Ref. [1]). LEP has also placed a lower bound limit of mh > 114.4GeV [2]. Assuming the Standard Model (SM) of electroweak interactions, one expects that the higgs will soon be found at LHC. However, there are at least two reasons why the SM with a single higgs doublet is ex- pected to be incomplete. The first is the triviality problem. This asserts that the higgs self interaction, and hence its mass, must vanish unless the theory has a finite cut-off. Triviality has been rigorously established only for simpler models, but it is widely believed to hold for the SM higgs. The other is the hierarchy problem that quadratic divergences need to be finely tuned to keep the scale of electroweak breaking smaller than the natural cut-off of the theory (which in the absence of new physics would be the Planck scale). In this paper, we investigate the effects that new physics, invoked to cure these problems, may have on the higgs sector of the SM. We assume that the scale of the masses of new quanta, M, is sufficiently higher than the scale of electro-weak symmetry breaking (v ∼ 246GeV) so that the quanta of the unknown new physics can be integrated out. As we want this new physics to address the hierarchy and triviality problems, and for phenomenological reasons, we are interested in new physics where M ∼ TeV. The resulting low energy effective theory is the one higgs standard model supplemented with non-renormalizable local operators, of dimension D > 4, which are constructed of standard model fields invariant under the SU(3) × SU(2) × U(1) gauge symmetry. This approach has been applied to precision electroweak observables[3, 4, 5, 6, 7, 8] and has recently been the subject of further investigations[9, 10, 11, 12, 13, 14, 15]. The advantage of this approach is that it is model independent: any new physics scheme that results in a low energy spectrum coinciding with the SM’s can be described in this way. The disadvantage is that the new physics is parametrized in terms of several arbitrary parameters, the coefficients of higher dimension operators, and nothing is known a priori about these coefficients. For a particular extension to the standard model, consistency requires that fits such as [1] be reconsidered with the new operators, severely relaxing the constraint on the higgs mass. In fact, it has been shown that the effect of higher dimension operators [16, 17, 18] can elminate the mass limit on the higgs. While more exotic possibilities are tantalizing, in this paper, we focus on the possibility that the new physics integrated out is strongly interacting and effecting the higgs sector above the scale M while the higgs itself has a relatively low mass mh <∼ 2mt. Various bounds can be placed on M from low energy experiments. In particu- lar, flavor changing neutral current bounds such as those arising from K0 − K̄0 mix- ing impose strong constraints, M ≥ 104TeV. These bounds can be relaxed by re- stricting the higher dimensional operator basis through adopting the MFV hypothesis [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. This allows one to consider M to be a few TeV while naturally suppressing FCNC. However, even utilizing the MFV hypothesis to justify new physics at a TeV, higher dimensional corrections to the standard model could exist that modify the relation mW = mZ cos θW . This relationship is experimentally required to be respected to a fraction of a percent. The PDG quotes ρ0 = 1.0002 +0.0007 −0.0004 for the global fit [29] of precision electro-weak observables. This fact motivates the consideration of new physics being integrated out that preserves ρ0 ≈ 1 naturally, even with possible strong dynamics effecting the higgs at the scale M. This can be accomplished assuming an approximate custodial SU(2)C symmetry [30, 31, 32], where the weak SU(2) gauge vector bosons transform as a triplet and the higgs field transforms as a triplet and a singlet. The higgs vacum expectation value is in the singlet representation of SU(2)C , so the approximate symmetry is explicitly realized, and is ex- plicitly broken only by isospin splitting of fermion yukawa couplings and by hypercharge. We require that the operator extensions to LSM respect this SU(2)C symmetry up to hyper- charge and Yukawa coupling violations, as in the standard model. Operators that break the custodial symmetry are allowed but their coefficients are taken to be naturally suppressed. In the SM, the higgs cubic and quartic couplings are not independent parameters, but given in terms of the higgs vacuum expectation value v and massmh. The obvious immediate effect of D > 4 operators is to shift all these quantities in independent ways, so that effectively the higgs cubic and quartic couplings become independent parameters. Of course, the shift from the SM values is somewhat restricted, of order (v2/M2)C, where C is a dimensionless coefficient, C ∼ 1. The effect on single higgs production rates of modifications to the coupling of the higgs to weak vector bosons or to itself was investigated in Ref. [11]. The modification of higgs decay widths and this general class of models was also examined in [33]. In this paper we address the question of whether a bound state of higgs particles can form. In the SM a higgs bound state forms only if the higgs is very heavy[34, 35]. There is a competition between the repulsive interaction of the quartic coupling, λ1, and the attractive interaction of the higgs exchange (between higgs particles) which is determined by the cubic coupling, λ1v. For large enough coupling the exchange interaction is strong enough to produce binding, but since the mass, λ1v, is also given in terms of the coupling, the higgs mass is large. In the effective theory context the three parameters (mass and cubic and quartic couplings), are independent and a bound state is possible for smaller higgs mass. The question of detail becomes, how is the bound on the higgs mass for a bound state to form relaxed by the coefficients of D = 6 operators? Can one have a bound state of light higgses? We find that the effect of these operators can be significant, allowing for bound states for much lighter higgs particles. Discovery of such bound states would give valuable information on the scale of new physics. There is no know solution to the bound state problem for identical scalar particles inter- acting via cubic and quartic interactions. The higgs bound state problem has been addressed using different approximations, the N/D method is Ref. [34] and a truncated version of the homogeneous Bethe-Salpeter equation in Ref. [35]. Our aim here is to find a necessary con- dition on the coupling for which a non relativistic (NR) bound state may form. To this end we introduce a new method. We propose to study the formation of the bound state in a non-relativistic effective theory for higgs-higgs interactions. We begin by listing the D = 6 operators of the effective theory. We take two approaches. In the first, linear realization, we consider operators that can be built out of the higgs doublet and the fields in the gauge sector of the SM. Our primary interest here is in the higgs sector per se, so we focus on higgs self interactions. The second approach, uses a non-linear realization of the symmetry. Since the higgs field is intimately connected to the symmetry breaking of the SM gauge symmetry, it is natural to expect that below the scale of new physics the effects of symmetry breaking are already apparent. Were the higgs mass as large as the scale of new physics, the SM would be supplemented not with a higgs doublet but with a triplet of would-be goldstone bosons that are eaten by the W and Z vector bosons. The higgs, if somewhat lighter than the scale M, would appear as a singlet under the gauge symmetry. We then proceed to construct the effective theory at low energies. If mh <∼ 2mt, one can incorporate the virtual effects of the top by integrating it out and constructing a top-less effective theory. In order to investigate the minimal coupling for which a NR higgs-higgs bound state may form we then construct a non-relativistic higgs effective theory, and proceed to determine this condition. II. HIGGS EFFECTIVE FIELD THEORY: LINEAR REALIZATION A. The D = 6 Custodial SU(2) Higgs Sector The Lagrangian density of the standard model containing the higgs field1 is given by L4φ = (Dµ φ) (Dµ φ)− V (φ) (1) where φ is the higgs scalar doublet. The covariant derivative of the φ field is given by Dµ = 1 ∂µ − i Bµ − i g2 W Iµ (2) where σI are the pauli matrices, W Iµ , Bµ, are the SU(2) and U(1) SM gauge bosons and the hypercharge of 1/2 has been assigned to the higgs. The higgs potential at tree level is given V (φ) = −m2 φ† φ+ λ1 . (3) No dimension five operator can be constructed out of higgs fields and covariant derivatives that satisfies Lorentz symmetry and the standard model’s gauge symmetry.2 Utilizing the equation of motion of the higgs field and partial integration the number of dimension six operators is reduced. The effective Lagrangian density of the extended standard model is given by Lφ = L4φ + +O( v ), (4) 1 We have omitted Yukawa interactions with fermions here. 2 To satisfy Lorentz invariance an even number of covariant derivatives are required. To be invariant under the SU(2)×U(1) gauge group the operator must be bilinear in φ† and φ. where the dimension six operators that preserve the symmetries of the standard model and custodial SU(2)C in the Higgs sector are given by L6φ = C1φ ∂µ (φ† φ)∂µ (φ† φ) + C2φ (Dµ φ) (Dµ φ)− λ2 . (5) Note that the operators considered here preserve custodial symmetry and can result from tree level topologies in the underlying theory.[36] As such, these operators need not be suppressed by loop factors of 1/16π2 or proportional to a small custodial symmetry breaking parameter. For these reasons these operators are expected to have the dominant effects on the higgs self couplings and we take their coefficients to be O(1). There is only one operator in the Higgs sector that violates custodial symmetry and could come from an underlying tree topology, (φ†Dµ φ)2. The underlying topology in this case determines that the symmetry breaking parameter is given by g21. The coefficient of this operator has been determined [37] to be C < 4× 10−3 where we have used Λ = 1TeV. We neglect this operator. We expand the higgs field about its vacuum expectation value with 〈h(x)〉 = 0 and treat v2/M2 as a small perturbation. We expand the field as usual around a vacuum expectation value v so that φ(x) = U(x)√ v + h(x)  . (6) Here U(x) = ei ξ a(x)σa/v and the would-be goldstone boson fields of the broken symmetry are ξa. In unitary gauge, the gauge transformation can be used to remove the goldstone boson fields. We then redefine the higgs field (h) so that the kinetic term is normalized to 1/2, using the field redefinition (1 + 2CKh ) , (7) where CKh = (v 2/M2)(C1φ + 14C φ). The effective Lagrangian density is given, in terms of the rescaled field, by L4φ + ∂µ h′ ∂µ h ′ − Veff(h′) + C i,jh′ O h′ + CW W OW W + CZ Z OZ Z h′ W W O h′ W W + C h′ Z Z O h′ Z Z , (8) summed over i, j such that i+ j = 2 , where (h′)i vj ∂µ h′ ∂µ h OW W = W µ , OZ Z = Z h′ W W = (h′)i vj W+µ W µ , O h′ Z Z = (h′)i vj Z0µ Z µ. (9) The coefficients are given by h′ = 0, C 4C1φ + C C1φ + , CW W = m 1 + C2φ CZ Z = 1 + C2φ h′ W W = m C2φ − 2C1φ + h′ W W = m C2φ − 2C1φ + 3 ,−1 h′ W W = 2m 4 ,−2 h′ W W = m2W C h′ Z Z = C2φ − 2C1φ + h′ Z Z = C2φ − 2C1φ + 3 ,−1 h′ Z Z = m 4 ,−2 h′ Z Z = C2φ. (10) The effective potential is Veff (h h′3 + h′4 + 30 λ2 5 !M2 v h′5 + 30 λ2 6 !M2 h′6, (11) which is written in terms of the rescaled mass term and the effective couplings, which are given by 1− 2CKh ), (12) 3 = 3 λ1 1− 3CKh ), (13) 4 = 3 λ1 1− 4CKh +O( v ). (14) We will suppress the prime superscript on the higgs field for the remainder of the paper. B. D = 6 SM Field Strength Operators The operators that can be constructed out of the higgs scalar doublet and the field strengths (or duals) of the standard model are as follows. We restrict our attention to those operators listed in [6, 15] that preserve the SU(2)C custodial symmetry: L6φ,V = − cG g GAµ ν G Aµν − cW g W Iµ ν W I µ ν − cB g Bµ ν B − c̃G g G̃Aµ ν G Aµν − c̃W g W̃ Iµ ν W I µ ν − c̃B g B̃Aµ ν B µν . (15) Here GAµν , W µ ν and Bµ ν stand for the field strength tensors of the SU(3) × SU(2) × U(1) gauge bosons, and a tilde denotes the dual field strengths, F̃µν = ǫµ ν λσ F λσ /2. Note that the operator that is proportional to the S parameter given by − cW B g1 g2 φ† σI φ Bµ ν WI µ ν (16) violates custodial symmetry and is naturally suppressed in our approach 3. C. D = 6 Fermion Sector Operators of dimension 5 and higher that couple the higgs to fermions, or purely fermionic operators, can give rise to unacceptably large flavor changing neutral currents (FCNC). If the coefficient of such operators are generically of order 1 the scale of new physics must be taken to be M >∼ 104 TeV in order to suppress FCNC effects. We adopt the Minimal Flavor Violation hypothesis (MFV) to naturally suppress the dangerous operators while maintaining a low scale of new physics, M >∼ 1 TeV. In the absence of quark and lepton masses the SM has a large flavor symmetry group, GF = SU(3) 5. The MFV asserts that there is a unique source of breaking of this symmetry. All operators that break the symmetry must transform precisely the same way under GF . As a result FCNC operators are suppressed by the familiar factors of the Kobayashi-Maskawa (CKM) matrix in the quark sector and by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and small neutrino masses in the lepton sector. Since the effects of fermionic operators are not needed for the rest of this investigation, we do not list the operators. The interested reader can find a complete description of the operators and their effects in [6]. 3 See Appendix A III. HIGGS EFFECTIVE FIELD THEORY: NON LINEAR REALIZATION The construction in the previous section assumes that the field content of the effective theory includes a higgs doublet. This is not necessary. If the electroweak symmetry is spontaneously broken by a strong interaction the spectrum below the scale of this new physics does not have to be described by a higgs doublet field, beyond the SM fields. Only fields describing the would-be goldstone bosons need be introduced. Such higgs-less theories have been discussed in the literature[38]. However, if the higgs particle is somewhat lighter than the scale of new physics it has to be incorporated in the low energy description and symmetry alone does not dictate that it appears as a member of an iso-doublet. It is sufficient to have the goldstone bosons realize the broken symmetry non-linearly, and the higgs field is then a singlet under the symmetry. The situation is entirely analogous to the case of π’s and the σ in QCD. A phenomeno- logical Lagrangian density describing π and σ interactions does not have to be a linear realization of the chiral SU(2) × SU(2) symmetry. Instead, the π-fields have a better de- scription through a non-linear chiral Lagrangian. Then the σ can be included through interactions that satisfy the non-linearly realized symmetry and the usual rules for naive dimensional analysis [39]. In the non-linear realization, the Lagrangian density in Eq. (1) is replaced by LNL = 14v 2TrDµU †DµU + 1 µh− V (h), (17) where the would-be goldstone bosons ξa appear through the matrix U(x) = ei ξ a(x)σa/v that transforms under SU(2)L × SU(2)R linearly, U → LUR†, and h is a singlet field, describing the higgs particle. Custodial symmetry SU(2)C is the diagonal subgroup of SU(2)L×SU(2)R and the higgs field is invariant under it. 4 This Lagrangian is supplemented by higher order terms suppressed by powers of M. In the case of the higgs potential, this can be included simply as V (h) = M4f(h/M), (18) where f(x) is an arbitrary function with a minimum at zero. The mass and couplings of the 4 The custodial symmetry is discussed in more detail in Appendix A. higgs are given in terms of this dimensionless function by m2h = M2f (′′)(0), (19) 3 = Mf (′′′)(0), (20) 4 = f (iv)(0). (21) It is not a surprise that in the non-linear realization of the symmetry the couplings and mass are completely independent, and that they are all naturally of order 1 times the appropriate power of the dimensionfull scale, M. The natural scale for the higgs mass is M, and we are considering here the class of theories for which f (′′)(0) happens to be small, while higher derivatives may remain of order 1. We stress that the natural scale for the cubic coupling is M. Unless the mechanism (or numerical accident) that keeps the higgs mass small compared to M also acts to suppress the cubic coupling, one must naturally expect λeff3 ∼ M/v ≫ 1. We will also need the corrections to the derivative interactions. We write, generally, L = 1 1 + c ∂µ h ∂µ h− m2h h 2 − v λ h3 − λ h4 + · · · (22) In the linear realization the derivative interaction couplings are related, 1 CKh = (v 2/M2)(C1φ + C2φ/4), but in the non-linear realization they are independent. And, as in the case with λ 3 naive dimensional scaling gives an enhancement of c 1 that could arise from the non-perturbative dynamics of the symmetry breaking sector. Naively, c (v/M), which is enhanced over the linear realization value by a power of (M/v). As we mentioned earlier, non-linear realizations have been extensively studied for higgs- less theories, but have been neglected in studies including a light higgs. There are two important consequences of the non-linear realization outside the pure higgs sector that we point out here. It has been noted that significant corrections to the coupling of a higgs to gluons are possible from D > 4 operators. The modifications can be large because there is no SM contribution at tree level. In the linear realization there is a D = 6 operator that contributes at tree level, and therefore competes with the SM one loop, top mediated amplitude: GaµνG µν , (23) Note that the linear realization implies a relation between the one and two higgs couplings to two gluons. However, in the non-linear realization the two couplings are completely independent, GaµνG µν . (24) In Ref. [40] it was noted that a heavy quark with Yukawa coupling λ → ∞ produces a coupling of two gluons to one or more higgs particles that cannot be described by the effective theory operator in (23). Instead a nonpolinomial interaction was introduced to describe this effect, )GaµνG There is no problem accommodating such interactions in the non-linear realization, by ln(1 + h/v)GaµνG [h/v + (h/v)2 + · · · ]GaµνGaµν . (25) In much of what follows we implicitly assume the linear realization. However, results in terms of the arbitrary parameters mh, λ 3 and λ 4 can be interpreted readily as arising from the non-linear realization. IV. A LOW ENERGY EFFECTIVE THEORY FOR THE HIGGS In this section we will construct an effective theory for the light higgs, integrating out momentum modes heavier than the higgs. This is useful in discussing physical effects with a typical energy of order of the higgs mass. In particular, we integrate out the top quark. As the coupling of the top quark to the higgs is fairly large, we would like to estimate the effects of the top quark on the possibility of forming a higgs bound state. If the top quark mass is much heavier than the higgs it is appropriate and convenient to describe the higgs self interactions in a top-less theory. When the top quark has been integrated out, its effects are accounted for through modifications of coupling constants and mass of the higgs. While this is clearly appropriate when the top quark is much larger than the higgs mass, we use this approximation even when the higgs is slightly heavier than the top. For mh < 2mt the approximation is known, ipso facto, to work better than one would expect. This is due, in part, to the fact that there is no non-analytic dependence on the mass since the higgs is the pseudo-goldstone boson of spontaneously broken scale invariance[41, 42, 43]. It is also known that soft gluon effects are large and correctly reproduced by the effective theory [44]. For single and double higgs production, comparisons between the the full theory calcu- lation and the effective top-less theory find that the latter is a good approximation for the total rate for mh <∼ 2mt. For example, with the appropriate K factor, the resulting topless effective field theory calculated to two loops is known to accurately describe the full NLO result for g g → h to better than 5% accuracy in the full range 0 < mh < 2mt [44]. As another concrete example, consider the higgs mass dependence in the higgs IPI self- energy. The first graph of Fig. 1 is the contribution of the top quark to the IPI self-energy which we label −iΠ(p2). At 1-loop we find Π(p2) = 1− x(1− x) 1 + 3 log m2t − x(1− x)p2 , (26) where we have used the MS subtraction scheme. The quantity 1− Π(0)/Π(p2) (at µ = mt) never exceeds 30% when p2 ranges from zero to 2mt. For these reasons we consider it appropriate to integrate out the top quark for mh <∼ 2mt in this initial study. When mh >> mt, these corrections should be taken only as indications of the size of virtual top effects. While the approximation of neglecting higher order terms in the p2/m2t expansion is known to work better than expected for the applications we will consider, there is no guarantee that it will work well for processes not considered here. [45, 46, 47, 48, 49] A. Running to mt The coefficients of the D = 6 operators at the scale M are unknown. We are assuming that the new physics couples to the higgs field and is strongly interacting at the scale M. In this context, it is natural to take C iφ (M) , λ2 (M) ∼ 1. (27) Similarly it is natural to assume that the coefficients of the D > 4 operators that couple the higgs to other fields, like those in Eq. (15) or those that couple the higgs to quarks while satisfying the MFV hypothesis, are all order unity. The anomalous dimensions of the extended operator basis can be determined system- atically. This is beyond the scope of this paper. But the effect of the running is easy to understand. With minimal subtraction the calculation of the running of coefficients of higher dimension operators can be done in the symmetric, massless phase. There is operator mixing among the D = 6 operators with common quantum numbers. The anomalous dimension matrix is a function of the relevant couplings (λ1, g1, g2 and the top quark Yukawa, λt). The running is always proportional to these coefficients so the effect is roughly of the form C iφ (mt) ∼ C iφ (M) , (28) where mixing is implicit, and c1α stands for a linear combination of λ1 and the squares of g1, g2 and λt. Since log (mt/M) ∼ 1 and the coefficients c1 ∼ 1 the running produces a small, calculable shift in the unknown coefficients. Hence, we continue to take the unknown Wilson coefficients at the scale mt to be ∼ 1 . At mt the top quark is integrated out and this produces a different effect, a shift in the C iφ(mt) by a C φ-independent amount. This can be numerically significant, and we estimate this next. Note that once the top is integrated out we continue to run down to the mass of the higgs scalar mh. The effect of the running of these coefficients from mt to mh is again small, so we take C iφ (mh) , λ2 (mh) ∼ 1. (29) B. Integrating out the top quark Integrating out the top leads to further corrections to the higgs sector of the standard model. The top mass is a result of symmetry breaking, so the resulting effective theory is better presented in unitary gauge, as in (8) and (11). In unitary gauge, the top mass term and coupling to the higgs is given by LY = −mt q̄t qt . (30) We begin by considering effects on the higgs self-couplings. Figure 1 show the Feynman graphs that contribute to modifications of the higgs self-couplings. The solid line denotes a top quark, the dashed external lines denote the higgs. We perform the calculation to lowest order in p2/m2t . Some details of the computation are given in the appendix. The effect of these corrections is to further modify the effective potential of the higgs scalar field h. The effective couplings and mass term of Eqs. (12)–(14) FIG. 1: Integrating out the top quark. are shifted by these corrections, and are now are given by 1− 2CKh m2t m ), (31) 3 = 3 λ1 1− 3CKh m2t m ), (32) 4 = 3 λ1 1− 4CKh − 4Nc +O( v m2t m ). (33) As emphasized above, these corrections are not multiplicative, that is, they are present even for λ2 = C h = 0. Whether they are important depends on the scale and strength of the new physics. The condition . (34) is satisfied for λ2 ≈ 1 when M ≈ 2πv = 1.6 TeV. So the corrections are numerically comparable to these new physics terms. Similarly, for λ1 ∼ 1 the condition CKh λ1 = (C1φ + C2φ)λ1 ∼ . (35) still requires M ≈ 1.6 TeV for C1φ + 14C φ ∼ 1. C. Corrections to Field Strength operators Integrating out the top quark also results in effective operators of the higgs field and the SM field strengths. The dominant SM production mechanisms for the higgs at LHC is the gluon fusion process g g → h. We restrict our attention to such operators that effect the production processes of the higgs through gluon fusion. Figure 2 shows the 1-loop Feynman diagram for the top contribution to g g → h. For a higgs with mh < 2mt, the expected production cross section of the g g → h process has been determined up to NNLO [50, 51, 52]. For SM gluon fusion, the single higgs production mechanism is given by the mt → ∞ effective Lagrangian density comprised of a dimension five operator Lmt = C1GGh (αs) Gaµ ν G a , (36) where the coefficient is given in the MS scheme, in terms of αs for five active flavors, by [53, 54, 55, 56, 57, 58] C1GGh (αs) = 11α2s 48 π2 +O(α3s). (37) FIG. 2: The gluon fusion g g → h production process. The production process through the effective local operators in shown in the second column. The effective local operators come from integrating out the top quark and new physics at M. The interactions in the effective Lagrangian of Eq. (15) also contribute to single higgs pro- duction through gluon fusion. Combining results, at the scale mh, the effective Lagrangian density for single higgs production is given by Leff = CeffGGh GAµν G Aµν + C̃ G̃Aµ ν G Aµν , (38) where GGh = C GGh − 2 π αs cG , (39) GGh = −2 π αs c̃G . (40) Assuming that the new physics degrees of freedom carry the SU(3) gauge charge, the Wilson Coefficients cG, c̃G will be approximately the same size as the coefficients C φ, λ2 we are interested in. If the new physics degrees of freedom are charged under SU(2)×U(1) but not SU(3), below the scales M, mt effective local operators of this form will still be induced. However, the corresponding Wilson Coefficients will be suppressed by factors of 16 π2. The effect of these interactions on higgs production rates was examined in [14]. Note that in the standard model, contributions to the operator G̃Aµν G Aµν are highly suppressed[59] and therefore neglected. FIG. 3: The gluon fusion g g → hh production process and the effective local operators. The production process of two higgs in the standard model is shown in Fig. 3. In analogy with the single higgs production case we characterize the process in the effective theory by Leff = CeffGGhh GAµν G Aµν + C̃ G̃Aµν G Aµν , (41) where the coefficients are given by GGhh = C GGhh − cG π αs v , (42) GGhh = − c̃G π αs v . (43) Here the top quark contribution is[56] C1GGhh(µ 2) = − 11α2s 48 π2 +O(α3s). (44) The Wilson coefficients for two higgs production in the effective theory is not suppressed relative to the corresponding Wilson coefficient for single higgs production. Note that unlike the case of single higgs production the expansion in p2/m2t does not, in general, have kine- matics such that p2/m2t ∼ m2h/m2t . In two higgs production, higher order terms in p2/m2t have p2 = s, t, u and in general (s, t, u)/m2t is not small. We calculate the next order in the expansion of p2/m2t in Appendix B. These terms are neglected, and our application of the expansion is valid for finite values of mt due to our interest in establishing a necessary condition for a NR bound state to form. The kinematics for the production of a NR bound state at threshold dictate (s, t, u)/m2t ∼ m2h/m2t . V. PHENOMENOLOGY OF HIGGS EFFECTIVE THEORY A. The Magnitude of Self Couplings The effect of the D = 6 operators in the effective potential cause corrections to the three and four point contact interactions and mh. To illustrate that the induced effects on the higgs sector are under control, consider extending the effective potential with a single dimension eight term. We find the following while neglecting the effects of integrating out the top quark 1− 2CKh + 3× 10−2 λ2 1− 2CKh + 4.5× 10−4 λ3, (45) 3 = 3 λ1 1− 3CKh + 7.5 (CKh )2 + 1.5× 10−1 λ2 1− 3CKh + 3.1× 10−3 λ3, (46) 4 = 3 λ1 1− 4CKh + 12 (CKh )2 + 4.5× 10−1 λ2 1− 4CKh + 1.6× 10−2 λ3. (47) From which one sees we are examining the potential of the theory in a controlled expansion, even for M ∼ 1TeV. Eliminating the self-coupling λ1 in favor of the higgs mass, we can write for the effective cubic and quartic higgs-self couplings, 3 = 3 1− CKh ) m2h − 7Nc , (48) 4 = 3 1− 2CKh ) m2h + 6λ2 − 19Nc . (49) With v = 246GeV, mt = 174GeV and M = 1 TeV, and taking mh = v/2, these are 3 = 0.62− 0.05(C1φ + 14C φ) + 0.06λ2, (50) 4 = 0.39− 0.09(C1φ + 14C φ) + 0.36λ2. (51) For negative λ2 of order one one can greatly reduce the repulsive contact interaction, λ in a putative higgs-higgs bound state. Of course, this comes at the price of reducing the attractive interaction, governed by λ B. g g → hh Production From our results in Section IVC, the production of two higgs in our effective theory framework is straightforward to write down. The contributions to the amplitude are shown in Fig. 4. h h h h FIG. 4: The two higgs production process in the effective theory. The amplitude for two higgs production, to O(αs), is given by 〈h h|i A|Aα(P1)Aβ(P2)〉 = 〈h h|i A1|Aα(P1)Aβ(P2)〉+ 〈h h|i A2|Aα(P1)Aβ(P2)〉 (52) where we have 〈i A1〉αβ = 2 i CF (CeffGGh) fαβ(P1, P2) (P1 + P2)2 −m2h + i ǫ P 23 + P 4 + (P1 + P2) 〈i A2〉αβ = 4 i CF (CeffGGhh) f αβ(P1, P2), (53) where fαβ(P1, P2) ≡ P α1 P 2 + P 2 − 2 gαβ P1 · P2. (54) Using two higgs production as a test of the cubic self coupling of the higgs has been examined in [60] where testing for the MSSM with this signal was investigated. Ref. [40] advocated the examination of g g → h h to compare the one and two higgs production coefficients in Eq. (24) since the naive relation between the two coefficients could be upset by the presence of novel operators like that in (25). As we have discussed, in the non linear realization of broken electro-weak symmetry, the relationship between g g → h h production and g g → h production is not fixed as in the linear realization. Any deviation from the SM value for g g → h h must be interpreted with care. The g g → h h production rate in our effective theory construction (in the linear realization) depends on at least six unknowns, namely, M, λ2, C1φ, C2φ, cG, c̃G. The effects of the operator advocated in [40] increase the number of unknown parameters still further. Clearly two higgs production is an important signal to test the higgs mechanism in the standard model. The cross section of g g → h h is suppressed compared to the cross section of g g → h by a factor of 1000, due to the effects of parton distribution functions and phase space suppression[60]. The cross section falls from 50 fb to 10 fb as the higgs ranges in mass from 100GeV to 200GeV. Thus once LHC enters its high luminosity running of 100 fb−1/Year one can expect roughly 1000 events per year. A significant excess or deficit of this signal should be observable. However, the reconstruction of exactly what form of new physics is present requires more information. One could obtain more information on the unknown parameters involved by further probes of the physics of the self interaction of the higgs. In the remainder of the paper we examine the sensitivity of a higgs bound state (Higgsium) in the appropriate low energy effective field theory on TeV scale physics to these parameters. If a bound state forms, one can use the properties of the bound state such as its binding energy, as an a probe of the physics above the scale M. C. Higgsium: Production and Decay time To get some rough understanding of the conditions under which a higgs-higgs bound state may form, consider the non-relativistic Schrodinger equation −∇2r + V (r)− E ψ(r) = 0, (55) with the potential from a yukawa exchange and a contact interaction, V (r) = − e−mh r + κ δ3(r). (56) We are interested in the case g ∼ λeff3 and κ ∼ λ 4 as a non relativistic approximation of the higgs self interactions. Neglect for now the contact interaction. It is well known that the Yukawa potential produces bound states provided >∼ 1.7, (57) Neglecting new physics effects, 3 ≈ 3 , (58) non-relativistic bound states could be expected for mh >∼ 1.2v. (59) The effect of TeV scale new physics changes the relationship between the mass and the coupling. The above Yukawa bound state condition is modified to 1.54 + 0.09(C1φ + C2φ)− 0.02 λ2 2 v. (60) This demonstrates the point that if the higgs self coupling is significantly stronger due to strong TeV scale new physics that contributes large Wilson coefficients, then a low energy signal of this higher scale physics might be a NR bound state formed by two higgs. However, one can see that it is difficult to realize the NR bound state condition when we identify the couplings in this Schrodinger equation with our effective couplings. This identification is in fact incorrect. We will demonstrate in Section VI that the correct NR limit of the higgs sector is described by a Lagrangian containing only contact interactions and higher derivative operators. The formation time of the bound state can be approximated by the ratio of 4R0/u where R0 is the characteristic radius of the NR bound state and u is the relative velocity of the two higgs. This is roughly the period of oscillation for S wave states [61]. For a NR bound state we can approximate the relative momenta of the two higgs by p ∼ mh u so that . (61) The SM higgs decays predominantly via h → b b̄ pairs through Yukawa interactions if 114.4 < mh ≪ 2MZ . We take these decays as dictating the decay width of Higgsium. Neglecting the effects of our new operators, this decay has the decay width . (62) This gives an approximate decay time . (63) The condition that the bound state has time to form is that τf < τb which can be satisfied . (64) Thus a non relativistic bound state has time to form before it decays. Above 135GeV and below the threshold of W+W− production, the dominant decay of the higgs is through a virtual W pair, h → W W ⋆. Above mh > 2mW the decay into W+W− predominates and the decay width is given by 1− aW 4− 4 aW + 3 a2W , (65) where aW = 4m h using the notation of [29]. Comparing the formation and decay time for a higgs whose mass is above the threshold of W+W− production we find η(mh, mw) , (66) where η(mh, mw) ∼ 1. The lower bounds on u in either case are compatible with the NR approximation for the full range of higgs masses we consider. A relativistic bound state is also possible in either case, although an approximation scheme that can estimate its formation time is lacking. In the remainder of the paper we focus on the possibility of a NR bound state being formed by a relatively light higgs, mh < 2mt, due to our treatment of the top quark. We also note that a NR bound state may also have observable effects on the spectrum of two higgs production even if a bound state does not fully form as in the case of top quark pair production near threshold in e+e− collisions [61]. VI. NON RELATIVISTIC HIGGS EFFECTIVE THEORY If the two higgs are created with small relative velocity and form a non relativistic bound state it is appropriate to describe the physics of this state with a non relativistic effective field theory of the higgs sector. We refer to our effective theory derived in Section I through Section IVC as higgs Effective Theory (HET) and now match onto a non-relativistic version of this theory (NRHET) where we take the c → ∞ limit of the scalar field Lagrangian density of HET. Recall the Lagrangian density is of the form 1 + c ∂µ h ∂µ h− m2h h h4 +O . (67) We wish to construct the non relativistic limit of this Lagrangian density systematically, retaining h̄ = 1 and making factors of c explicit with [c] ∼ [x]/[t]. The dimensionful quantities can be expressed in units of length [x] and time [t]. As h̄ = 1, we still have [E] ∼ 1/[t] and [p] ∼ 1/[x]. As the action S = dt d3xL is dimensionless, we have [L] ∼ [x]−3 [t]−1. For the time and spatial derivatives to have the same units in L we take , (68) and so ∂µ ∼ 1/[x]. This gives [h] ∼ 1/ [x] [t]. We require [mh c 2] ∼ [E] ∼ 1/[t], so that we have [mh] ∼ [t]/[x]2, and choose the electroweak symmetry breaking expectation value to have the same dimensions as the field h, [v] ∼ 1/ [x] [t]. The Lagrangian density with these unit conventions is given by L = 1 1 + c ∂µh ∂µh− m2h c 2 h2 − v λ 3 ! c h3 − λ 4 ! c h4 +O Now consider the non-relativistic limit of this theory. The interaction terms will be determined below by matching. Consider first the theory of a free real scalar field of mass mh given by L = 1 ∂µϕ∂µϕ− m2h c 2 ϕ2. (70) The field ϕ must also be expanded in the c → ∞ limit. We remove a large energy scale 2 from this field with a field redefinition ϕ(x) = e−imh c 2 r·x ϕ+(x) + e imh c 2 r·x ϕ−(x), (71) where r = (1, 0) and ϕ+(x), ϕ−(x) correspond to the creation and annihilation components of the scalar field ϕ(x). Expanding the Lagrangian density in terms of ϕ±(x) we neglect terms multiplied by factors of en imh c 2t, (n 6= 0, n ∈ I). (72) These are terms in the Lagrangian density where some of the fields are far off shell. Their effect is only to modify coefficients of local operators in the effective action. With this substitution we find L = ∂0 ϕ− ∂0 ϕ+ − ∂i ϕ− ∂i ϕ+ + imh c 0 ϕ+ − ϕ+ ∂0 ϕ− . (73) The first term, with two time derivatives, is suppressed by 1/c2 and is suppressed in the c→ ∞ limit. Integrating by parts the remaining kinetic terms and rescaling h± = 2mh ϕ± gives L0NR = h− h+. (74) One can extend this effective Lagrangian by adding interactions, including higher order terms suppressed by |u|/c and v2/M2 where u is the relative velocity of the two higgs in a non-relativistic bound state. Scattering is described by a contact interaction which can be parameterized by a coupling CNR, LNR = h− +. (75) There is no cubic interaction because this necessarily involves at least one far off shell particle. The effect of the cubic interaction in the HET is incorporated in the coupling CNR, and we will compute this in terms of the parameters of the HET below, in Sec. VIA. In this effective theory, the energy and the momenta of the system are given by mh |u|2, (76) q = mh u, (77) where |u| ≪ c is the relative velocity of the two higgs. It is advantageous to have power counting rules in |u| that are as manifest as possible in the Lagrangian density as demon- strated in [62]. We rescale so that the natural sizes of the coordinates are given by the above energy and momentum and define a new field H±(x) and new, dimensionless coordinates X and T by x = λx X, t = λt T, h±(x) = λhH±(x). (78) To ensure the rescaled energy and momenta are of order unity we have λt = mhλ x and mh |u| , (79) mh |u|2 , (80) λh = m h |u| 3/2, (81) mh |u|2 , (82) mh |u| . (83) The form of the Lagrangian density when we implement these re-scalings and introduce an appropriately rescaled contact coupling, ĈNR = 4m hCNR is given by LNRH = H− i ∂0 + . (84) This form of the Lagrangian makes power counting explicit in the small parameter u/c. Physical quantities, such as the energy of bound states, can be equally calculated from the theories in Eqs. (75) or (84). Which is used is a matter of convenience: the former has familiar dimensions while the latter has explicit power counting.5 A. Matching onto NRHET To determine the matching coefficient CNR we take the non relativistic limit of the h h → h h scattering determined in HET. We neglect the running from m2t down to our matching scale µ2 = m2h in this initial study, and perform the matching at tree level only. 1. Linear Realization The HET contact interaction is given by AL0 = −3 λ1 + 20 λ1CKh − . (85) 5 The c → ∞ limit of NR effective field theories was studied in [63]. The reader interested in bound states at threshold in NRHET would also profit from an examination of the treatment of bound states at threshold in NN effective field theory, reviewed in [64]. The Yukawa exchange feynman diagrams, shown in Fig. 5, give the amplitude iALy (s, t, u) = i(AL1 (t) + AL1 (u) + AL1 (s)), (86) where s, t, u are the usual Mandelstam variables, and AL1 (x) = −3v2 λ1 x−m2h + i ǫ 3 λ1 + 5 λ2 − 4 (x+ 2m2h) − 2Nc − 18 λ1CKh .(87) A1(s) A1(u) A1(t) A0 FIG. 5: Tree level hh → hh scattering in the extended higgs theory. Time flows left to right. The total amplitude for h h→ h h scattering is given by ALhh→hh(s, t, u) = AL0 +ALy (s, t, u). (88) To perform the matching we take the momenta of the higgs particles to be off-shell by a small residual momenta p̃ with energy and momenta that scale as p̃0 ∼ mhu2 and p̃ ∼ mhu. The momenta of the higgs are decomposed as (recall r = (1, 0)) p = mh r + p̃, k = mh r + k̃, p′ = mh r + p̃′, k ′ = mh r + k̃′. (89) This gives, in the center of mass frame s = 4m2h + 4 |q|2, t = − |q|2 (1− cos(θ)) , (90) u = − |q|2 (1 + cos(θ)) , with q ∼ mhu. In the non relativistic limit we retain the lowest order in |u| and we have ALNR = AL0 +AL1 (4m2h) + 2AL1 (0) = 12 λ1 + 10 λ2 − 64 λ1CKh − . (91) To determine the coupling C4NR in the NRHET Lagrangian, Eq. (75), we compute the four point amplitude and insist that it equals ANR. Inserting four factors of 2mh to account for relativistic normalization of states, we finally arrive at (2mh) 2CLNR = Ĉ NR = 12 λ1 + 10 λ2 − 64 λ1CKh − . (92) 2. Non-Linear Realization For a non-linear realization we find the following for the HET contact interaction ANL0 = −λ 4 + 4 2 . (93) The yukawa exchange diagrams give ANL1 (x) = x−m2h + i ǫ 2m2h + x . (94) The matching is performed as in a linear realization and we find ANLNR = ANL0 +ANL1 (4m2h) + 2ANL1 (0) 2 − λeff4 − 2c 2 − (c ) m2h . (95) This gives the effective HET coupling in the non-linear realization (2mh) 2CNLNR = Ĉ 2 − λeff4 − 2c 2 − (c ) m2h . (96) B. NRHET Bound State Energy To find the approximate bound state energy of the higgs, we calculate the bubble sum in our NRHET theory and interpret the pole in the re-summed bubble chain as the bound state energy of Higgsium. Note that this calculation is formally justified in the large N limit [65] where the higgs sector is equivalent to and O(4) theory [66]. The Feynman rules for the NRHET Lagrangian in (75) are shown in Fig. 6. The bubble sum is straightforward to calculate in NRHET. The leading order term is directly obtained from the Feynman rules, we use the Lagrangian given by Eqs. (75) in the following. The leading bubble graph is given by iA1-loop = (i CNR)2 dk0 ddk (2π)d (E + k0)− k2/2mh + iǫ −k0 − k2/2mh + i ǫ P0 − P2/ 2 FIG. 6: Feynman rules for NRHET. We have chosen to work in the center of mass frame, and E = P 01 +P 2 stands for the center of mass energy. Performing the first integral by residues and the remaining integrations with dimensional regularization, we find iA1−loop = −i mh(CNR) (−mhE)1/2. (98) FIG. 7: The bubble sum of graphs leading to the bound state pole in NRHET. The terms in the bubble sum of diagrams shown in Fig. 7 are given by the geometric series 1− mhCNR (−mhE)1/2 + mhCNR (−mhE)1/2 + · · · i CNR 1 + mhCNR (−mhE)1/2 This result agrees with [62, 67] and indicates a bound state with a bound state for CNR > 0 with binding energy mhCNR . (99) There is an implicit renormalization condition introduced by dimensional regularization. The integral has no pole as d → 3, so it is interesting to ask what subtraction has been made. This is easily understood by performing the d = 3 integration with a momentum cut-off |k| < Λ in terms of the bare coupling C0NR: iAΛ1−loop = imh(C0NR)2 (−mhE)1/2 . (100) The renormalized coupling CNR(µ) can be defined as the amplitude at a fixed energy E = −µ [68]. Then the combination CNR(µ) (mhµ) (101) is renormalization group invariant. This is precisely the coupling that appears in (98). It would appear that for any positive value of CNR we have bound states. However for our NR description to be self consistent we require that the binding energy of the bound state satisfy Eb < mh, that is, ĈNR > 16π. (102) 1. Linear Realization In the case of a linear realization a heavy higgs seems necessary for the bound state to form, but the new physics effects may allow significantly smaller masses for the bound state. If we neglect the effects of new physics (λ2 and C h ) the bound (102) translates into mh > 2.0 v. Retaining the effectsof λ2 and C h one determines a condition for the NRHET calculation of the bound state energy to be self consistent 16π − 4 λ2 v + 51Nc 12− 40CKh . (103) Alternatively, for a given value of the higgs mass, say mh = ξ v, the self-consistency condition implies a constraint on the coefficients of the higher dimension operators: 12 ξ2 + 4 λ1 − 10 ξ2 (C1φ + 14C > 16 π + (104) Using M = 1 TeV and the PDG value for the top quark mass, this condition simplifies to 1.2 ξ2 + 0.024λ1 − 0.24 ξ2 (C1φ + 14C φ) > 5.1 (105) So, for example, for |C1φ + 14C φ| = 1 or 5 the minimal higgs mass for a NR bound state is reduced by 6% or 28%, respectively. Near the limit of validity of our calculation mh ∼ 2mt, for negative values of C1φ + C2φ we find that a bound state is possible for O(1) wilson coefficients as we illustrate in Fig 7. -6.5 -6 -5.5 -5 -4.5 ������� M = 1 TeV, mh = 350 GeV FIG. 8: In the linear realization the allowed parameter space for NR bound state formation is above the line. 2. Non-Linear Realization In the nonlinear realization this condition is easily satisfied even for a light higgs, mh < v. Recall that λ 3 and c 1 are both enhanced by powers of M/v. Taking, for example, mh = 120GeV and M = 1TeV, neglecting the contribution of ceff2 , the NR bound state condition is )2 − 2 c̃1eff λ̃3 eff − eff)2 > 16 π + λ4, (106) where 1 . (107) Note that asM grows larger the region that satisfies the NR bound state condition grows. This is due to the fact that the attractive interaction given by λ 3 ∼ M/v is a relevant operator. We find that as mh grows and as M is larger the allowed parameter space of the NR bound state condition is significant, demonstrating that a bound state is likely to form in the non linear realization. -10 -5 0 5 10 mh = 120 GeV FIG. 9: In the Non-linear realization, holding mh fixed and set λ4 = 0 as it is O(1) and suppressed by 16π. We vary M for the values M = 1TeV (dotted line), M = 3TeV (dashed line) and M = 10TeV (solid line). The region above (the upper) and below (the lower) hyperbolic curves satisfy NR the bound state condition. -10 -5 0 5 10 M = 1 TeV FIG. 10: In the Non-linear realization, holding M fixed and set λ4 = 0 as it is O(1) and suppressed by 16π. We vary mh = 300GeV (solid line), mh = 200GeV (dashed line) and mh = 100GeV (dotted line). The region above (the upper) and below (the lower) hyperbolic curves satisfy the NR bound state condition. VII. SUMMARY AND CONCLUSIONS If a new strong interaction is responsible for electroweak symmetry breaking but a higgs particle, the pseudo-goldstone boson of broken scale invariance, remains unnaturally light, the self-interactions of this higgs particles could be quite strong. If strong enough these self-interactions could bind two higgs particles. To study these questions we formulated two different effective theories of the light, self- interacting higgs below the scale M of the new physics. In the first, the symmetry is realized linearly and the higgs field is described as one component of an SU(2)L doublet, just as in the standard model of electroweak interactions. In the second approach the symmetry is realized non-linearly: the triplet of would-be goldstone bosons and the higgs field are not in a common multiplet. We note that operators of dimension 3 in the effective Lagrangian in the non-linear realization are naturally expected to be enhanced by a power of M/v relative to their linear realization counterparts. In order to study how large these couplings need be, we have studied the case of non- relativistic bound states. To this end we constructed a non-relativistic higgs effective theory (NRHET) describing self-interacting higgs particles in the rest frame of the bound state, in the non-relativistic limit. The effects of the top quark are small but non-negligible. We estimated them by including the virtual top quark effects as a modification to the couplings in the NRHET. Our results show, perhaps not surprisingly, that in the non-linear realization it is quite easy to form light Higgsium, as we call the higgs-higgs bound state. For natural couplings in the linear realization a bound state is only likely to form for mh ∼ v. Relativistic bound states are possible in both the linear and nonlinear realizations. There are many questions that we have not addressed. The most immediate one is how to search for Higgsium. Assuming a light higgs is found, one could imagine strategies involving invariant mass distributions of higgs-pair production. A dedicated study is required to determine if this or other strategies are viable. Another, related question is whether the effects of a short lived bound state could be seen indirectly, much like would-be toponium affecting the line shape in top quark pair production near threshold in e+e− collisions. It would also be interesting to solve the bound state equation in the more general, fully relativistic case. We hope to return to these problems in the future. Acknowledgments Work supported in part by the US Department of Energy under contract DE-FG03- 97ER40546. APPENDIX A: CUSTODIAL SYMMETRY AND THE S PARAMETER There is some confusion in the literature regarding custodial symmetry and the operator − cW B g1 g2 φ† σI φ Bµ ν WI µ ν (A1) which corresponds to the S parameter. Consider the matrix representation of this operator [69] where the Higgs doublet field is given by  . (A2) Then ǫ φ⋆ is also an SUL(2) doublet with components ǫ φ⋆ =  , (A3) where φ− = φ+ ⋆. The Higgs bi-doublet field is given by (ǫ φ⋆, φ) , φ0 ⋆ φ+ −φ− φ0  . (A4) The SUL(2)×UY(1) gauge symmetry acts on the Higgs bi-doublet as SUL(2) : Φ → Lφ (A5) UY (1) : Φ → Φ e−iσ3 θ/2. (A6) In the limit that hyper charge vanishes the Lagrangian also has the following global symmetry SUR(2) : Φ → φR†. (A7) When the Higgs acquires a vacuum expectation value, both SUL(2) and SUR(2) are broken, however the subgroup SUL=R(2) is unbroken, ie L 〈Φ〉L† = 〈Φ〉. (A8) This is explicitly the custodial symmetry, and the corresponding transformation of the Higgs bi-doublet under this symmetry. It is easy to see that − cW B g1 g2 Φ† σI WI µ νΦ Bµ ν (A9) is invariant under this symmetry. The Higgs bi-doublet transforms as above and the field strength σI WI µ ν transforms as σI WI µ ν → LσI WI µ ν L†. (A10) However, it is also easy to see that this representation of the operator vanishes by explicitly performing the trace; one finds Φ† σI WI µ ν Φ = 0. (A11) The non trivial representation of the operator in terms of the bi-doublet is given by Φ† σI Φσ3 . (A12) With this factor of σ3, required for a non trivial representation in terms of the Higgs bi- doublet, one finds that this operator violates custodial symmetry. APPENDIX B: TOP QUARK OPE As an example of the effect of the neglected terms in in the top quark OPE, consider the OPE corrections to the four point function of the higgs. The amplitude is given by i A4(s, t, u) = −6NC (2π)d (k/+mt) k2 −m2t (k/+ a/ +mt) (k + a)2 −m2t (k/+ b/ +mt) (k + b)2 −m2t (k/+ c/+mt) (k + c)2 −m2t We find the leading order in p2/m2t → 0 the amplitude is given by i A04(s, t, u) = −24NC (2π)d (m4t + 6 k 2m2t + k (k2 −m2t ) = − i Nc 16 π2 − 64 + 24 log . (B1) The leading order matching gives a factor of −4NC m4t/v4. Consider performing the top quark OPE to higher orders. We find for the next order in p2/m2t i A14(s, t, u) = − 16 π2 80m2t a2 + b2 + c2 + a · b+ 6 a · c+ b · c . (B2) The invariants of the external momenta a, b, c averaged over the sum of all A4 diagrams can be expressed in the Mandelstam variables. We find that our momenta expressed in terms of these variables are 〈a2〉 = 4 !m2h, 〈b2〉 = 8 (s+ t+ u) , 〈c2〉 = 4 ! 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0704.1506

Measuring two-photon orbital angular momentum entanglement

Measuring Two-Photon Orbital Angular Momentum Entanglement G.F. Calvo, A. Picón and A. Bramon Grup de F́ısica Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain (Dated: July 7, 2021) We put forward an approach to estimate the amount of bipartite spatial entanglement of down- converted photon states correlated in orbital angular momentum and the magnitude of the transverse (radial) wave vectors. Both degrees of freedom are properly considered in our framework which only requires azimuthal local linear optical transformations and mode selection analysis with two fiber detectors. The coincidence distributions predicted by our approach give an excellent fit to the distributions measured in a recent experiment aimed to show the very high-dimensional transverse entanglement of twin-photons from a down-conversion source. Our estimate for the Schmidt number is substantially lower but still confirms the presence of high-dimensional entanglement. PACS numbers: 03.67.Mn, 42.50.Dv, 42.65.Lm I. INTRODUCTION The phenomenon of quantum entanglement whereby distant systems can manifest perfectly random albeit per- fectly correlated behavior is now recognized as the es- sential ingredient to perform tasks which cannot be ac- complished with classically correlated systems [1]. The presence of entanglement has been traditionally revealed in the violation of Bell-type inequalities [2]. However, detecting such a violation does not provide in general a measure of the amount of entanglement. This is par- ticularly significant in systems correlated in multidimen- sional degrees of freedom [3]. Several techniques have been proposed to assess the presence of entanglement for different quantum scenarios. These include state to- mography [4, 5, 6], that yields a complete reconstruction of a quantum state but requires many setting measure- ments, entanglement witnesses [7], which detect some entangled states with considerably less measurements, and the experimental determination of concurrence [8, 9]. In Ref. [9], by using two copies of a down-converted two-photon state entangled in polarization and trans- verse momentum, the measurement of concurrence was achieved with a single, local measurement on one of the photons. Although bipartite entanglement is well understood, finding experimentally feasible procedures to quantify it for systems correlated in multidimensional degrees of freedom turns out to be quite challenging and relevant. Indeed, access to higher dimensional Hilbert spaces in which information can be encoded and manipulated has recently attracted great interest, with proof-of-principle demonstrations using quantum communication protocols in three-level systems (qutrits), such as entanglement concentration [10], quantum bit commitment [11] and quantum coin-tossing [12]. Likewise, a complete char- acterization of states hyperentangled in polarization, or- bital angular momentum and frequency has been exper- imentally implemented [13]. The aim of this paper is to address the problem of how, by performing a certain set of local linear optical operations affecting one of two multidimensional spatial degrees of freedom in which two-photon states can be entangled, it is possible to obtain an explicit measure of the amount of bipartite transverse entanglement. Specif- ically, according to the interplay between both spatial degrees of freedom -orbital angular momentum (OAM) and the magnitude of the radial wave vectors- fundamen- tally different predictions are expected in the subsequent joint photodetection process, via mode-selection analysis with two fiber detectors preceded by azimuthal transfor- mations (acting only on the OAM). The application of our framework is compared with the results of a recent experiment [14]. The paper is organized as follows: Section II presents the general setting of the problem and includes the Schmidt decomposition technique for describing bipar- tite spatial entanglement. In Section III we reveal the generic features that arise in the photodetection coinci- dences according to the transverse structure of the two- photon wave function. Section IV illustrates the results found in Section III with an example of a realistic two- photon wave function that yields a full analytical solution to the problem. In Section V a closed-form expression for the Schmidt number is obtained in terms of easily accessible experimental parameters. We then proceed to exploit this Schmidt number in two interesting examples of optical transformations with azimuthal phase plates. These enable us to estimate the amount of spatial entan- glement of a two-photon source. Conclusions of the paper are drawn in Section VI. Details of our calculations, to- gether with some useful background material, have been included in two Appendices. II. MODAL SCHMIDT DECOMPOSITION FOR TWO-PHOTON STATES Two-photon pure quantum states are described in a Hilbert space by a continuous bilinear superposition of spatiotemporal multimode states. Sources of such non- classical states of light are mostly realised in the process of spontaneous parametric down-conversion [15], where an intense quasimonochromatic laser pump illuminates a crystal endowed with a quadratic nonlinearity produc- ing pairs of photons (idler and signal). Conservation of energy and momentum impose that the state be spec- trally and spatially correlated. Here we explore entan- glement involving spatial degrees of freedom that de- pend on the transverse structure of these states. For simplicity, we assume that the down-converted photons are linearly polarized, monochromatic and frequency de- generated. The two-photon state can then be written as |ψ〉 = dqidqsΦ(qi,qs)â †(qi)â†(qs)|vac〉, where qi,s are the transverse components of the idler and signal wave vectors. Under conditions of paraxial and nearly collinear propagation of the pump, idler and signal photons, the two-photon amplitude Φ is given by [16] Φ(qi,qs) = E(qi + qs)G(qi − qs). The function E represents the transverse profile of the pump, whereas G originates from the phase-matching condition in the longitudinal direc- tion and depends on the specific orientation and cut of the nonlinear crystal. Since the arguments of E and G enforce correlations in different manifolds of the idler and signal wave vector space, it is the global structure of Φ the one dictating the entanglement degree of |ψ〉. In order to extract the amount of entanglement con- tained in Φ, one may resort to the Schmidt decomposi- tion that provides the spatial information modes of the two-photon pair. Suppose that the transverse spatial fre- quency field of the pump beam has a Gaussian profile of the form E(qi + qs) ∝ e−w 0|qi+qs| 2/4, where w0 is the width (at the beam waist). The chosen pump profile peaks when its argument vanishes. This imposes that the idler and signal transverse wave vectors should be mostly anticorrelated (qi ' −qs). Remarkably, it was shown by Law and Eberly [17] that with such a Gaussian profile E the normalised two-photon amplitude can be expressed as Φ(qi,qs) = (−1)` λ`n u`,n(qi)u−`,n(qs) , (1) where u`,n(qi) = ei`φiv`,n(qi)/ 2π and u−`,n(qs) = e−i`φsv−`,n(qs)/ 2π are the normalised polar Schmidt mode functions for the idler and signal photons with topological charge ` and radial index n corresponding to eigenvalues λ`n (they satisfy 1 ≥ λ`n ≥ 0 and∑ `,n λ`n = 1) of the reduced density matrices for each photon. Knowledge of λ`n yields a direct measure of the degree of transverse entanglement given by the Schmidt number [18] K = ( `,n λ −1, which is the reciprocal of the purity of the idler and signal density matrices, it is invariant under free propagation and yields an av- erage of the number of relevant spatial modes involved in the decomposition. The larger the value of K, the higher the transverse entanglement. For instance, prod- uct states correspond to K = 1 (there is only one nonvan- ishing eigenvalue equal to 1), whereas states with K > 1 are entangled. A distinguishing feature of decomposi- tion (1) is that it represents Φ in terms of a perfectly correlated discrete basis of paraxial eigenstates of the FIG. 1: (color online). Two-photon coincidence detection configuration. Down-converted idler and signal beams from a nonlinear crystal traverse two optical systems (h(i,s)) which include azimuthal phase plates and are coupled into single- mode fibers (SMF) gated by a coincidence circuit (CC). OAM operator along the direction of light propagation (with corresponding eigenvalues `~) [19, 20], rather than in a continuous plane-wave modal expansion. The precise form of the radial idler and signal eigenmodes v`,n(qi) and v−`,n(qs) depends on the specific phase-matching function G. In particular, when G is approximated by a constant there are striking consequences: Eq. (1) be- comes a (non-normalisable) superposition in which all OAM eigenstates have an equal weight and the radial de- pendence becomes just a global factor. It is important to emphasize that to attain decomposition (1) the appropri- ate choice of the widths, wi and ws, for the idler and sig- nal radial eigenmodes, has to be made. If the two-photon amplitude is of the form Φ(qi,qs) = E(qi+qs)G(qi−qs), then wi = ws ≡ wS , where wS is the so-called Schmidt width. For widths different from wS an additional sum- mation over the radial indexes occurs, and the perfect correlation between the idler and signal radial modes is absent (see Appendix A). Moreover, the fact that the idler and signal mode functions are anticorrelated with respect to their topological charge numbers is a conse- quence of a more general process: the conservation of OAM, which is transferred from the pump photon (car- rying zero OAM for a Gaussian mode) to the down- converted photon pair [21]. III. AZIMUTHAL TRANSFORMATIONS ON THE TWO-PHOTON STATE The usefulness of the Schmidt decomposition (1) be- comes apparent when analyzing the propagation of two- photon states through optical systems. Each of the in- tervening modes evolves and transforms independently of the others. It is also clear that the correlation proper- ties displayed by the two-photon amplitude (1) are pre- served in the position representation. Therefore, sup- pose that the idler and signal photon beams, described now by the paraxial two-photon wave function in the transverse position representation [20, 22, 23] ψ(ri, rs) = 〈ri, rs|ψ〉 = `,n(−1) λ`nu`,n(ri)u−`,n(rs), are each transmitted through different linear optical systems that include diffractive (or refractive) azimuthal phase plates (see Fig. 1). The role of the plates in each path is to imprint an azimuth-dependent phase factor on the incoming Schmidt modes. Their (separate) action on the two-photon wave function can be represented by the unitary and radially symmetric impulse response functions h(i,s)(φ, φ′). These functions locally trans- form the spiral harmonic mode content of each pho- ton via ei`φ/ (i,s) `′,` e i`′φ/ 2π, with h(i,s)`′,` =∫ 2π dφ dφ′ei`φe−i` ′φ′h(i,s)(φ, φ′)/2π, so that the re- sulting output two-photon wave function is ψout(ri, rs) = `i,`s,`,n (−1)` λ`n h `s,−` × ei`iφiv`,n(ri) ei`sφsv−`,n(rs)/2π , (2) where the initial perfect correlation in OAM is lost, re- maining only that corresponding to the radial modes. Upon traversing their respective linear optical systems the idler and signal photons, having OAM `i = `A, `s = `B and radial indexes nA,B , are detected in coincidence. By placing photodetectors at the output ports of suitable arrays of deterministic mode sorter interferometers [24, 25], or computer-generated holograms [10], it is possible to distinguish modes bearing different OAM. In practice, the most straightforward procedure involves projecting into single-mode fibers, where the propagated mode has a fundamental Gaussian profile (`A = `B = nA = nB = 0). The probability that the idler and signal photons will be projected into modes u`A,nA and u`B ,nB is found to be PnA,nB`A,`B = | dridrsu∗`A,nA(ri)u `B ,nB (rs)ψout(ri, rs)|2, with an additional incoherent (and weighted) sum over the radial indexes nA,B when taking into account multi- mode detection. This photodection probability is ex- pressed in terms of the spatial overlap of the Schmidt and the fiber modes at the planes where the phase plates are located. Notice that their corresponding widths, wS and wG, are not necessarily equal, and thus the orthogonal- ity between these modes when their radial indexes differ does not hold in general. To evaluate PnA,nB`A,`B , we de- fine RA,B` ≡ rirsdridrsv `A,nA (ri)v `B ,nB ×v(wS)`,n (ri)v −`,n(rs), where the two different widths of the radial modes have been specified for clarity. The co- incidence probability can then be written as PnA,nB`A,`B = ∣∣∣∣ ∞∑ (−1)`RA,B` h `B ,−` ∣∣∣∣2 . (3) All the radial dependence is contained in the functions RA,B` that modulate the angular impulse response func- tions h(i,s). We emphasize that although the radial part of the Schmidt modes does not experience any signif- icant transformation when the idler and signal beams traverse their respective linear optical systems, its proper inclusion in the detection process is essential. This is re- flected in the structure of Eq. (3) with the presence of `-dependent radial functions RA,B` , and is a consequence of a non-constant phase-matching function G. Had G been a constant then the input two-photon wave func- tion (1) would have been represented by a common radial function times a (non-normalisable) maximally entangled superposition of spiral harmonic modes. In this limit- ing case one derives a fundamentally different prediction for the coincidences: P`A,`B ∝ | `(−1) `B ,−`| where now all the radial dependence of the detected modes appears only as a global function. IV. AN EXACTLY SOLUBLE MODEL FOR THE TWO-PHOTON AMPLITUDE To illustrate the previous results, suppose that the phase-matching function G is also Gaussian (see Ap- pendix C, where a justification of this model is provided), an approximation implying that photons are generated near the phase-matching region of wave vectors where the down-conversion process occurs most effectively. The normalised two-photon amplitude reads Φ(qi,qs) = 0|qi+qs| 2/4e−b 2|qi−qs|2/4 , (4) where b < w0 plays the role of an effective width of the phase-matching function, and depends on the non- linear crystal thickness, although no specific relation is assumed here. At variance with other models, where G is often approximated by a constant (b → 0), the two-photon amplitude (4) captures the relevant features of the transverse wave-vector correlation between the idler and signal photons, and provides an analytically amenable model that yields explicit formulas for Eq. (3). We show in Appendix A that for the two-photon ampli- tude (4), the Schmidt mode functions belong to the well- known Laguerre-Gaussian basis [17], or, more generally, to a continuum family of spatial modes generated via metaplectic mappings (rotations on the orbital Poincaré sphere) from the Laguerre-Gaussian modes [20, 26, 27]. Furthermore, we also find that the Schmidt eigenvalues are λ`n = (1 − ξ2)2ξ2|`|+4n, with ξ = (w0 − b)/(w0 + b). This allows us to express the Schmidt number in the fol- lowing closed form K = [(1 + ξ2)/(1− ξ2)]2. Large values of K occur when b→ 0. The minimum K = 1 is attained when b → w0 (the two-photon amplitude (4) becomes a separable function in the idler and signal wave vectors). The Schmidt width for any of the above families of eigen- modes is always the same: wS = 2w0b. We stress that the Schmidt width does not represent the actual cross- section widths Wi,s of the idler and signal beams. These widths, that can be measured experimentally, can also be obtained by resorting to the partially reduced density matrix of the idler and signal photons. For amplitude (4) one finds Wi,s = 2w20 + b 2, which is consistent with the widths employed in [28] in the thin crystal approximation (b→ 0). Let us focus on the most usual encountered situation where the measured fiber modes are the fundamental Gaussian modes (of width wG at the phase plates). Since the involved eigenfunctions in the Schmidt decomposi- tion (1) have cylindrical symmetry, we use the Laguerre- Gaussian modes as the convenient computational basis to derive the radial functions R` ≡ R `A,B=nA,B=0 ` (see Ap- pendix B). Remarkably, it turns out that the functions R` can be cast in terms of a sole parameter s R`(s) = Γ2(1 + |`| Γ(1 + |`|) ; 1 + |`|; s2 s|`| , (5) where F (a, b; c; d) denotes the hypergeometric function, 1 + ξ2 + (1− ξ2)(wS/wG)2 w20 − b2 w20 + b 2 + (2w0b/wG)2 . (6) The parameter s corrects ξ by taking into account the spatial overlapping of the Schmidt and the fiber modes. Indeed, if and only if wS = wG, does s = ξ hold. When b → 0 then s → 1, which corresponds to a constant phase-matching function. The functions R`(s) increase monotonically from R`(0) = 0 to R`(1) = 1 for ` 6= 0 (R0(s) = 1). The fact that all R`(s) depend on a single parameter s will be exploited in the next Section to show how one can estimate the Schmidt number in a particular experimental scenario. V. EXPERIMENTAL SCHMIDT NUMBER Let us now examine the problem of measuring the amount of transverse entanglement of two-photon sources which, in our case, is characterised by the Schmidt num- ber K. In principle, a complete quantum tomography of the two-photon state could yield the desired amount of entanglement [11], but generally this would require a very large number of measurements, each for every possi- ble pair of spatial modes, and this is technically demand- ing. Here we propose an alternative approach. In our simple model for the two-photon amplitude (4) we have found that the radial part of the coincidence probabili- ties (3) depends on a single parameter s. This parame- ter s involves the characteristic widths appearing in the two-photon amplitude (4), namely the pump beam width w0 and the phase-matching width b, together with the fiber mode width wG (at the phase plate locations). The phase-matching width b could be measured by scanning in the plane of detection, but in our case it is not neces- sary. If, instead, one rotates the azimuthal phase plates (maintaining the detectors fixed), then, the recording of the coincidence distributions allows one to extract the value of s as a fitting parameter for Ps ≡ P nA=0,nB=0 `A=0,`B=0 via Eqs. (3) and (5). Therefore, if s is conceived as a parameter to be directly measured (rather than b), the amount of transverse entanglement can now be written in the form (1 + 2sµ2)2 (1− s)(1 + s+ 4sµ2) , (7) FIG. 2: (Left column) Normalised coincidence distributions as a function of the relative orientation between the idler and signal angular diaphragms. Solid, dashed and dotted curves are calculated for s = 1, 0.7, and 0.4, respectively. (a) β = π/2 and (c) β = π, both with η = 0.5. (Right column) The corresponding configurations of the angular diaphragms. where µ ≡ w0/wG. This experimental Schmidt number depends on quantities that are easily accessible: the fit- ting parameter s and the widths w0 and wG (the presence of the width ratio µ should be interpreted as a correcting geometrical factor). It increases from K = 1, when s = 0, to infinity as s→ 1. By properly engineering the impulse response func- tions h(i,s) it is possible to enhance the sensitivity of Ps with s, thus improving the accuracy of the estimated K. To this end, we consider two simple types of transpar- ent azimuthal phase plates, and examine the dependence of Eq. (3) when the idler and signal phase plates are mutually rotated a relative angle α ≡ αi − αs. Their dispersionless impulse response functions are of the form h(i,s)(φ, φ′) = eiθi,s(φ)δ(φ− φ′). Owing to the initial per- fect anticorrelation in OAM of the down-converted pho- ton pairs, one expects that only when both photons are subjected to complementary azimuthal transformations the perfect anticorrelation is preserved and the coinci- dences are maximal. As soon as the phase plates are ro- tated in such a way that they are not longer oriented in a complementary arrangement, the photon coincidences decrease. Notice that the axes with respect of which the angles αi and αs are taken do not coincide. For symmet- ric phase plates these reference axes are inverted 180◦. In what follows, we assume that each plate is characterised by a noninteger parameter η = (n0 − 1)d/λ, where d is the relative step height introduced by the plates, n0 their refractive index, and λ the wavelength of the idler and signal beams. The first phase plate type we consider is an an- gular diaphragm [see inset in Fig. 2(a)]. It con- sists of a thin uniform dielectric circular slab with a “cake-slice” indentation that subtends an angle β, with a nonzero θi(φ) = −θs(φ) ≡ 2πη (mod 2π) only if αi,s + β < φ < αi,s + 2π. Similar angu- lar diaphragms have been employed in proof-of-principle demonstrations of the uncertainty relation for angu- lar position and OAM [29]. The probability (3) can be cast as P(AD)s (α) = R0(s)[(β − π)2 + π2 cot2(πη)] FIG. 3: Normalised coincidence distributions as a function of the relative orientation between the idler and signal spiral phase plates. Solid, dashed and dotted curves are calculated for s = 1, 0.7, and 0.4, respectively. (a) η = 0.5; and (b) η = 4.5. (c) Configuration of the spiral phase plates. `=1R`(s) cos(`α) sin 2(`β/2)/`2 . The main fea- tures of the normalized coincidence P(AD)s (α)/P s (0) are: (i) it does not depend on the integer part of η (it suffices to consider 0 < η < 1), the visibility be- ing maximal when η = 1/2; (ii) the coincidence dis- tributions are identical whether the aperture angle is β or 2π − β and symmetrical around α = π; (iii) for β 6= π and η fixed, the visibility diminishes as s decreases; and (iv) the maximum visibility always occurs in the limit s→ 1 (constant phase-matching function and thus very high Schmidt number) where one has P(AD)s→1 (α) = π[cot2(πη)− 1] + |2π − α− β|+ |α− β| . Figures 2(a) and (c) depict the characteristic profiles of the (nor- malised) P(AD)s when the aperture angles of the angular diaphragms are β = π/2 and β = π [Figs. 2(b) and (d) show, respectively, the configurations of the angu- lar diaphragms that yield the coincidences (a) and (c)]. Comparing Figs. 2(a) and (c) one sees that the visibility of the former exhibits a much stronger variation with s than the latter. This suggests that the phase plate con- figuration of Fig. 2(b) is preferred to that of Fig. 2(d) for achieving a more accurate estimation of the Schmidt number via Eq. (7). Finally, we should add that if η is an integer number then the profiles of P(AD)s are constant (independent of α). The second type of azimuthal (refractive) component is a spiral phase plate [see inset in Fig. 3(a)] [19, 30]. Its impulse response function includes the phase de- pendence θi(φ) = −θs(φ) = ηφ, for αi,s < φ < αi,s + 2π. In this case, the joint probability (3) can be written as P(SPP )s (α) = `R`(s)e i`α(`+ η)−2 ∣∣2. At variance with P(AD)s , the coincidence distributions ex- hibit the development of interference ripples as s de- creases and η increases [see Figs. 3(a) and (b)]. When s → 1 a parabolic profile is obtained P(SPP )s→1 (α) = π2 cot2(πη) + (α− π)2 / sin2(πη). In this limit the coincidences become independent of the integer part of η, and, as with angular diaphragms, the maximum coin- cidence visibility is attained (when η = 1/2). FIG. 4: Coincidence distribution dependence on the relative orientation between idler and signal spiral phase plates (SPP with η = 3.5). Circles are experimental values from Ref. [14], solid and dashed curves correspond to P(SPP )s fitted with s = 0.66, and P(SPP )s→1 , respectively. The slight asymmetry in the experimental values was probably caused by the presence of a small anomaly in the center of the spiral phase plates [14]. (Inset) Schmidt number K as a function of the width ratio µ = w0/wG for s = 0.66. We have considered other types of azimuthal phase plates and found the same parabolic dependence of the coincidences as s→ 1. This is consistent with the above- mentioned prediction that for very high transverse en- tanglement the joint probability becomes independent of the radial structure of the Schmidt modes. On the other hand, the absence of a vanishing minimum in the co- incidence distributions is a signature of a finite amount of transverse entanglement. This effect gives rise to a contribution in the photocounts which always exists re- gardless of the detector efficiencies [see Fig. 2(a) and Figs. 3(a),(b)]. It can however be diminished for cer- tain phase plate configurations [see Fig. 2(c)] and/or by employing a large wG (wG � 2w0b). Spiral phase plates were recently used in an elegant ex- periment aimed to show the high-dimensional spatial en- tanglement of a two-photon state from a down-conversion source [14]. The relevant parameters of the experiment are: η = 3.5 for the phase plates, λ = 0.8 µm, pump width w0 = 780 µm, and thickness of the nonlinear crys- tal L = 1 mm [14]. According to the model presented in Ref. [31], which corresponds to our P(SPP )s→1 , it was concluded that K > 3700 ± 100. Figure 4 depicts the predicted distribution for P(SPP )s (dashed line) together with the measured coincidences reported in Ref. [14]. According to our approach, a probability distribution P(SPP )s fitted with s = 0.66 (solid line in Fig. 4) shows excellent agreement with the experimental results. Equa- tion (7) then suggests that the corresponding Schmidt number should be much smaller than the above K. How- ever, we cannot perform a definite estimate for K since the value of the fiber mode width wG at the location of the spiral phase plates, needed to calculate µ = w0/wG appearing in Eq. (7), was not provided in Ref. [14]. The inset in Fig. 4 plots K for a wide range of ratios µ = w0/wG. For instance, using our fitting parameter s = 0.66, together with w0 = 780 µm and assuming wG & 250 µm, leads to µ . 3 and an estimate for K . 20, at least two orders of magnitude smaller than the quoted K in Ref. [14]. A very large value for K could only be ex- pected if, in the experiment, the fiber mode width wG in the plane containing the phase plates was much smaller than the pump width w0. VI. CONCLUSIONS In view of the situation described above, it is rea- sonable to conclude that additional experimental results and further theoretical analyses -a recent example can be found in Ref. [32]- are necessary and of interest in order to clarify the problem of quantifying the amount of transverse entanglement of photon pairs produced in down-conversion sources. If future values for the Schmidt number coming from accurate and solid measurements confirm the estimate of Ref. [14] and cannot be repro- duced within our approach, one should conclude that the conventional Gaussian profile constitutes a poor approx- imation for the phase-matching function G in Eq. (4). However, the ability of our predicted coincidence distri- butions to fit the experimental distributions of Ref. [14] suggests the correctness of our analysis and that the value for the Schmidt number quoted in Ref. [14] could be over- estimated. In this case, our approach not only would pro- vide a simple an accurate procedure to identify the rele- vant spatial modes and extract the degree of transverse entanglement; it would go in fact beyond by character- izing the action of all local bipartite azimuthal optical transformations on a broad family of two-photon states using only two detectors. Our results could also be of in- terest in other fields such as in identifying the intervening spatial modes when preparing well-controlled superposi- tions of photon states carrying OAM (to be coherently transferred onto Bose-Einstein condensates [33, 34] or by entangling photons with ensembles of cold atoms [35]), in high-resolution ghost diffraction experiments with ther- mal light [36], or in decoherence processes such as the en- tanglement sudden death mediated by the simultaneous action of several weak noise sources on bipartite photon systems [37]. Acknowledgments We thank S. Barnett, D. Diego, G. Molina-Terriza, M.J. Padgett and S. Walborn for useful discussions and gratefully acknowledge financial support from the Spanish Ministry of Science and Technology through Project FIS2005-01369, Juan de la Cierva Grant Pro- gram, CONSOLIDER2006-00019 Program and CIRIT Project SGR-00185. APPENDIX A In this Appendix we outline the derivation of the eigen- values and eigenfunctions in the Schmidt decomposi- tion (1) when the two-photon amplitude Φ is given by Eq. (4). We start with some well-known facts. For a general two-photon pure state, |ψ〉, it is possible to express |ψ〉 as a bilinear sum of idler and signal basis states |τi,s〉 belonging to the Hilbert space of the system |ψ〉 = τi,τs Cτi,τs |τi〉 ⊗ |τs〉 , (A1) where τi and τs label the set of quantum numbers for the idler and signal photons, respectively, whereas the coefficients Cτi,τs describe the probability amplitudes for each tensor product of basis states. Our first aim is to evaluate the coefficients Cτi,τs for the down-converted state |ψ〉 = dqidqsΦ(qi,qs)â†(qi)â†(qs)|vac〉. Let uτi,s(qi,s) = 〈vac|â(qi,s)|τi,s〉 denote the basis wave func- tions in the transverse momentum representation. Insert- ing the closure relations |τi,s〉〈τi,s| = 1̂, it can be readily seen that the coefficients Cτi,τs are given by Cτi,τs = dqidqsΦ(qi,qs)u (qi)u (qs). (A2) Since we are interested in elucidating the correlation properties of the spatial degrees of freedom (OAM and the magnitude of the transverse radial wave vectors), we choose as the computational basis the complete set of normalised Laguerre-Gaussian modes [20] u`,n(q, φ) = 2π(|`|+ n)! L|`|n × exp i`φ− i (2n+ |`|) ≡ exp (i`φ) v`,n(q)/ 2π, (A3) where q and φ denote the radial and azimuthal variables in momentum space, w is the mode width (at the beam waist), and L|`|n (x) are the associated Laguerre polynomi- als. The indices ` = 0,±1,±2, . . . and n = 0, 1, 2, . . . rep- resent the winding (or topological charge) and the num- ber of nonaxial radial nodes of the modes. Combining Eqs. (4), (A2) and (A3), it follows that ni,ns `i,`s dqi dqs e −w20|qi+qs| 2/4e−b 2|qi−qs|2/4 × e−i`iφie−i`sφsv`i,ni(qi) v`s,ns(qs) . (A4) By means of the well-known Anger-Jacobi identity e−x cos(φi−φs) = m=−∞(−1) mIm(x)eim(φi−φs), where Im(x) is the modified Bessel function of the first kind, the two angular integrals yield the selection rule m = `i = −`s ≡ `. This shows that the idler and signal photons are perfectly anticorrelated with respect to their topological charge, which is a manifestation of OAM entanglement. Hence, Cni,ns`i,`s = C ni,ns `,−` δ``iδ−``s , where ni,ns `,−` = (−1) `2w0b qidqiqsdqse −(w20+b 2)(q2i+q × v`,ni(qi) v−`,ns(qs) I` (w20 − b2)qiqs . (A5) The first of the radial integrals in (A5) can be performed by resorting to the following formula∫ ∞ x|`|+1L|`|n (x 2)e−γx I`(xy)dx (γ − 1)ny|`| 2|`|+1γ|`|+n+1 4γ L|`|n 4(γ − 1)γ valid for all real y, integers n ≥ 0 and `, and complex γ (Re(γ) > 0). The second radial integral can also be done analyti- cally. However, a dramatic simplification occurs if the widths wi and ws of the idler and signal radial modes, which at this stage have not been specified, are properly selected: wi = ws = 2w0b. In this case, one obtains a second selection rule for the radial indices. Namely, ni = ns ≡ n. That is, with such a choice of the widths, which is unique, the idler and signal radial modes are perfectly correlated. But this shows that it is precisely for wi = ws = 2w0b when one derives the Schmidt de- composition. Thus, the Schmidt width, wS , corresponds to wS = 2w0b. The coefficients reduce then to ni,ns `i,`s = (−1)` (w0 + b)2 w0 − b w0 + b )|`|+2n × δ``iδ−``sδnniδnns . (A6) Let ξ = (w0−b)/(w0+b). It is now clear from Eq. (A6) that the Schmidt eigenvalues are λ`n = (1− ξ2)2ξ2|`|+4n. We therefore conclude that the Schmidt decomposition of the two-photon amplitude (4) is Φ(qi,qs) = (1− ξ2) (−1)`ξ|`|+2nu`,n(qi) × u−`,n(qs), (A7) where u`,n(qi) = ei`φiv`,n(qi)/ 2π and u−`,n(qs) = e−i`φsv−`,n(qs)/ 2π represent the idler and signal Schmidt eigenmodes. These eigenmodes belong to the Laguerre-Gaussian basis. It is worth mentioning that although the Schmidt width is unique, other decom- positions (actually infinitely many) are possible in dif- ferent eigenmode bases. For instance, the Hermite- Gaussian modes (in Cartesian variables and the same 2w0b) constitute another possible basis. In fact, all eigenmode bases connected via the following unitary (metaplectic) transformation Û(θ, ϕ) = exp(−iθ L̂ · uϕ), where uϕ = (− sinϕ, cosϕ, 0) is a unit vector in the equatorial plane of the so-called orbital Poincaré sphere (parameterised by the spherical angles θ and ϕ) and L̂ is an angular momentum operator, form Schmidt bases for (A7). The components of the angular momentum op- erator: L̂x, L̂y, and L̂z are the three SU(2) generators. Of these, only L̂z represents real spatial rotations along the propagation of light, and it is thus the only compo- nent associated with the orbital angular momentum of light that can be measured in experiments [20]. APPENDIX B In this second Appendix we explicitly show how to cal- culate the radial functions R`(s) given in Eq. (5). For the chosen two-photon amplitude (4) and the measure- ment scenario, these radial functions depend on the spa- tial overlap of the Schmidt and the fiber (fundamental Gaussian) modes at the planes where the azimuthal phase plates are located. We need to evaluate the integrals rirsdridrsv `A=0,nA=0 (ri)v `B=0,nB=0 ×v(wS)`,n (ri)v −`,n(rs). We use the Schmidt eigenvalues λ`n = (1 − ξ2)2ξ2|`|+4n found in Appendix A. The idler and signal Schmidt modes v(wS)`,n (ri) and v −`,n(rs) be- long to the Laguerre-Gaussian mode basis. We stress once again the fact that the corresponding fiber wG and Schmidt wS widths are not necessarily equal. This im- plies that the orthogonality relation for these modes,∫ rdr v(wG)`G,nG(r)v `S ,nS (r) = δ`G`SδnGnS , does not hold in general when wS 6= wG. We start by recalling a remarkable identity between Laguerre polynomials L|`|n and the modified Bessel func- tions I` [38] n!L|`|n (x)L n (y) Γ(|`|+ n+ 1) (xyz)−|`|/2 (x+ y)z , (A8) valid for all real x, y, integer ` and complex z (|z| < 1). In detail, the integrals to calculate read R` = (1− ξ2)ξ|`| ridri (|`|+ n)! L|`|n rsdrs (|`|+ n)! L|`|n . (A9) Inserting identity (A8) in (A9) and employing the follow- ing result [38]∫ ∞ r e−γr I`(2νr) dr = Γ(1 + |`| 2Γ(1 + |`|)ν M− 12 , valid for Re(γ) > 0 and all complex ν, with Ma,b repre- senting the Whittaker functions, we obtain Γ(1 + |`| Γ(1 + |`|) 8(1− ξ2)3sw2S ξ3w4G dr exp (2− s2)ξr2 (1− ξ2)sw2S × M− 12 , |`|2 2sξr2 (1− ξ2)w2S , (A10) where we have introduced the parameter s = 2ξ/[1+ξ2 + (1− ξ2)(wS/wG)2]. The remaining integral is carried out through the use of formula e−γ̄t M− 12 , (t) = Γ(1 + |`| γ̄ − 1 ) |`| ; 1 + |`|; valid when γ̄ > 1/2. The function F (a, b; c; d) denotes the hypergeometric function. Equation (A10) is finally given by 1− ξ2 1 + (1−ξ )2 Γ2(1 + |`|2 )Γ(1 + |`|) ; 1 + |`|; s2 s|`|. (A11) This expression reduces to Eq. (5) when one leaves aside all the `-independent factors (they do not play any sig- nificant role in the normalised coincidence profiles). In this way the two-photon detection probabilities (3) can be cast in terms of the impulse response functions h(i)`A,`, `B ,−` and the radial functions R`(s) that solely depend on the parameter s (when `A = `B = 0). APPENDIX C In this appendix we would like to justify the model pro- posed here, the double gaussian profile as phase matching function. 1. Classical Energy for Non-linear Media In this section we derive explicitly the energy of an anisotropic and non-linear medium. This classical de- scription enables us to find a reasonable quantized Hamil- tonian which allow us to describe the PDC process. In anisotropic and non-linear media we must be cautious in the description of physical variables. For a deep analysis, we begin by assuming that the polarization of the mate- rial obeys the usual expansion from non-linear optics Pi = εo[χ ij Ej + χ ijkEjEk + χ ijklEjEkEl + ...] , (A12) where χ(n) (with n > 1) represent the nonlinear suscep- tibility tensors of order n+1 responsible for the coupling of n + 1 fields. The energy of the electromagnetic field can be written as E · dD , (A13) where the displacement vector field is defined as usual D = �oE + P. In linear media, the energy density of the electromagnetic field depends quadratically on the electric field ∝ E2, whereas in non-linear (parametric) media, polarization exhibits a dependence on higher or- der electric field terms (A12). We can expand the energy (A13) in two terms, one related to the linear contribution of the electric field and the other one related to the non- linear part. Taking the lowest order nonlinearity (A12), the energy due to this contribution can be written as [39, 40], d3r χ̃ ijk Ei(r, t)Ej(r, t)Ek(r, t) (A14) where the integration extends over the volume V of the nonlinear medium. Notice that the second order suscep- tibility is not the same as the susceptibility in equation (A12). We will consider that the medium response to the electric field is not instantaneous. Hence, the second order polarization should be written as i (r, t) = �o dt1dt2 χ ijk(t− t1, t− t2) × Ej(r, t1)Ek(r, t2) . (A15) We can see, taking the Fourier transformation of the elec- tric fields in equation (A15), that this specific form of the susceptibility implies that different monochromatic waves (with a well-defined frequency) of the electric field can excite other frequencies. Hence, the energy of the elec- tromagnetic field due to the bilinear susceptibility (A14) can be expressed as [41], ∫ ∫ ∫ dωdω′dω′′ χ̃ ijk(ω, ω ′, ω′′) × Ei(r, ω′′)Ej(r, ω − ω′)Ek(r, ω′) . (A16) This energy gives rise to the quantized Hamiltonian fre- quently found in the literature [15] to describe the pro- cess of spontaneous parametric down conversion, where an incident beam interacts with a non-linear medium and splits into two lower-frequency signal and idler photons. 2. Parametric Down-Conversion If we have a crystal with nonzero χ(2), pumped with a laser beam, there is a small, but non negligible, probabil- ity that a pump photon will decay into an idler and signal photon pair. We take the incident pump light beam to be in the form of an intense quasi-monochromatic plane wave propagating along the z-direction, Ep(r, t) = �p unm(r⊥, z) e iωpt , (A17) where �p is the polarization of the pump beam and unm(r⊥, z) is a slowly-variant-amplitude (for the mode having indices n and m) along z which obeys the paraxial equation in anisotropic media [42]. That is, the paraxial equation for free propagation cannot be used and must be modified. The pump beam is described classically, within the undepleted approximation, which means that one as- sumes that the input state is made up of a large number of photons and only few of them interact with the non- linear medium, producing pairs of correlated photons. In other words, the pump beam, at the output of the crystal, remains almost unaffected. A form for the interaction Hamiltonian operator ĤI(t) can be given motivated by the classical electromagnetic field energy (A16), and reads (we are always in the inter- action picture, though it can also be done in the Heisen- berg picture [40]), ĤI(t) = ki,σi ks,σs ijk(ωp, ωi, ωs) (� ki,σi × (�∗ks,σs) j(�p) k unm(r⊥, z) â ki,σi ks,σs e−i(qi+qs)·r⊥ × e−i[(kiz+ksz)z+(ωp−ωi−ωs)t] +H.c. , (A18) where k(i,s) = q(i,s) + k(i,s)z ûz. In equation (A18) two quantized electric fields are considered; they will be re- sponsible for idler and signal photons, whereas the pump beam is associated with the classical field [39]. As we have mentioned before, we are carrying out the calcula- tions under the interaction picture, where we know that a state evolves as |Φ(t)〉 = T 0 dτĤI (τ) |Φo〉 ' |Φo〉 − dτ : ĤI(τ) : |Φo〉 dτ ′ ĤI(τ)ĤI(τ |Φo〉+ ... , (A19) with T denoting temporal ordering. Since the interaction is assumed to be weak enough, it suffices to expand the exponential (A19) up to first order term. The action of the Hamiltonian operator on the evolution of the initial vacuum state yields the initial state plus a two-photon state (contributions due to higher order processes involve the generation of four, six,... photons, and are exceed- ingly small). Notice that applying the Wick theorem in the above expansion, temporal ordering becomes normal ordering up to first order. Therefore, |Φ(t)〉e = T 0 dτĤI (τ) |Ω〉 ' |Ω〉 − ki,σi ks,σs ijk(ωp, ωi, ωs) (� ki,σi ks,σs) −i(kiz+ksz)z r⊥ unm(r⊥, z) e −i(qi+qs)·r⊥ −i(ωp−ωi−ωs)t/2 sin [(ωp − ωi − ωs)t/2] (ωp − ωi − ωs)/2 |ki, σi; ks, σs〉 , (A20) where we have performed the integration with respect to time τ . Let L denote the length of the crystal along z, whereas S corresponds to the transverse area (the vol- ume of the crystal is V = LS). Equation (A20) is a gen- eral expression for any input paraxial beam. Assuming that the transverse area of the crystal S is much larger than the characteristic width of the input beam, the inte- gral with respect to the transverse position becomes the transverse Fourier transform. Now taking into account that the paraxial wave is propagating in an uniaxially anisotropic crystal [42], this integral becomes r⊥ unm(r⊥, z) e −i(qi+qs)·r⊥ → eikpne(θ,ωp)ze ne(θ,ωp) 2n2e(ωp)kp q2y+q n2e(θ,ωp) n2o(ωp) −iqxsθcθ n2e(θ,ωp) ñ2(ωp) z eunm(qi + qs) , (A21) where, for simplicity, we assume that the optical axis is contained in the plane xz and the pump beam is lin- early polarized along the x-direction (the pump beam has extraordinary polarization). Here, θ refers to the angle between the optical axis and the z-axis, and q = qi + qs ; kp = ; cθ ≡ cos θ ; sθ ≡ sin θ , n2e(θ, ωp) sin2 θ n2e(ωp) cos2 θ n2o(ωp) ñ2(ωp) n2o(ωp) n2e(ωp) Knowledge of the pump spectrum at the input crystal face enables, via equation (A21), to describe the evolu- tion of the pump beam through the crystal. In order to satisfy both the undepleted approximation and the fact that the pump profile is assumed not to vary significantly during the nonlinear propagation, the crystal must be very thin. The z-dependence in equation (A21) will play a crucial role to correctly describe the phase-matching conditions below. Defining ne(θ, ωp) n2e(ωp) q2y + q n2e(θ, ωp) n2o(ωp) −qx sin θ cos θ n2e(θ, ωp) ñ2(ωp) we can perform the integral with respect to z in equation (A20), resulting in the so-called phase-matching function:Z L i[kpne(θ,ωp)−Θ−kiz−ksz ]z = i∆(kp,ki,ks)L/2 sin [∆(kp,ki,ks)L/2] ∆(kp,ki,ks)/2 , (A22) where the argument ∆(kp,ki,ks) ≡ kpne(θ, ωp)−Θ− kiz − ksz , (A23) describes the wave vector mismatching. Therefore, equa- tion (A20) can be cast (excluding the vacuum state) as |Φ(t)〉(2ph)e = − ki,σi ks,σs ijk(ωp, ωi, ωs) (� ki,σi )i(�∗ks,σs) j(�p) k ei∆(kp,ki,ks)L/2 sin [∆(kp,ki,ks)L/2] ∆(kp,ki,ks)/2 × ũnm(qi + qs) e−i(ωp−ωi−ωs)t/2 sin [(ωp − ωi − ωs)t/2] (ωp − ωi − ωs)/2 |ki, σi; ks, σs〉 . (A24) 3. PDC in type II In this section we are going to consider the paramet- ric down conversion under type II phase-matching con- ditions. This implies that the pump beam is extraor- dinarily (linearly) polarized, whereas the idler and idler photons are ordinarily (the polarization vector is perpen- dicular to the optical axis) and extraordinarily polarized, respectively. Our first aim will be to further simplify the down-converted state (A24). If the interaction time is sufficiently large, we can replace the temporal-sinc factor in equation (A24) by a delta function, sin [(ωp − ωi − ωs)t/2] (ωp − ωi − ωs)/2 → 2π δ(ωp − ωi − ωs) , This is a good approximation, and the oscillatory factor e−i(ωp−ωi−ωs)t/2 can then be discarded by a regulariz- ing procedure. In particular, one obtains the frequency matching condition ωp = ωi + ωs , (A25) that is, energy conservation imposes that the annihila- tion of the pump photon gives rise to the creation of the idler and signal photons. In what follows, and effective nonlinear susceptibility χeff will be used to represent the relevant components of χ(2)ijk contracted with the cor- responding components of the pump, idler and signal po- larizations (we also absorb in χeff additional constants). Considering the continuity of the wave vectors of signal and idler photons (assuming that the volume of the crys- tal is large enough), equation (A24) reduces to |Φ〉(2ph)e = − ks χeff (ωp, ωi, ωs) e i∆(kp,ki,ks)L/2 sin [∆(kp,ki,ks)L/2] ∆(kp,ki,ks)L/2 eunm(qi + qs) × δ(ωp − ωi − ωs) |ki, σo,ks, σe〉 . Notice that the idler and signal photons have fixed po- larization, ordinary (σo) and extraordinary (σe) respec- tively. For convenience, we change the integral in the z com- ponent of the wave vector by the frequency. Therefore, for the idler integral, the Jacobian due to the change of variables is qidkiz = ∂(qi, kiz) ∂(qi, ωi) qidωi = qidωi =˛̨̨̨ n2o(ωi) ω2i no(ωi) c2kiz ∂no(ωi) qidωi , while for the signal integral, we have d2qsdksz = ∣∣∣∣∂(qs, ksz)∂(qs, ωs) ∣∣∣∣ d2qsdωs = ∣∣∣∣∂ksz∂ωs ∣∣∣∣ d2qsdωs =∣∣∣∣∣∣∣ n3e(θ,ωs) ∂ne(θ,ωs) − kszqsx sin 2θ ∂∂ωs ñ2(ωs) − q2sx n′2e (θ,ωs) n3e(ωs) ∂ne(ωs) 2 ksz n2e(θ,ωs) + sin 2θ ñ2(ωs) ∣∣∣∣∣∣∣ d2qsdωs . (A26) Proof: For ordinary waves the dispersion relation in an anisotropic medium is ω2i n o(ωi) iz + q Solving the quadratic equation for the variable kiz kiz = ω2i n o(ωi) − q2i . Notice that we only choose the positive solution of kiz. Fi- nally, taking the derivative with respect to ωi n2o(ωi) ω2i no(ωi) c2kiz ∂no(ωi) For extraordinary waves the dispersion relation in an anisotropic uniaxial medium is more complex: n2e(θ, ωs) + (qsx cosϕ+ qsy sinϕ) ksz sin 2θ ñ2(ωs) (qsx cosϕ+ qsy sinϕ) n′2e (θ, ωs) (qsx sinϕ− qsy cosϕ)2 n2e(ωs) ,(A27) where {θ, ϕ} are the polar and the azimuthal angles respec- tively in spherical coordinates, with θ referring to the angle subtended by the optical axis and the z-axis. For simplicity, as we have mentioned before, we assume the optical axis to be contained in the plane x-z (ϕ = 0). We have to take into account the common definitions in anisotropic media n2e(θ, ωs) sin2 θ n2e(ωs) cos2 θ n2o(ωs) n′2e (θ, ωs) cos2 θ n2e(ωs) sin2 θ n2o(ωs) ñ2(ωp) n2o(ωp) n2e(ωp) In this case, performing the implicit derivative with respect to ωs in the dispersion relation (A27), n2e(θ, ωs) 2k2sz n3e(θ, ωs) ∂ne(θ, ωs) sin 2θ ñ2(ωs) + kszqsx sin 2θ ñ2(ωs) n′2e (θ, ωs) n3e(ωs) ∂ne(ωs) . (A28) where ñ2(ωs) n3o(ωs) ∂no(ωs) n3e(ωs) ∂ne(ωs) n′2e (θ, ωs) cos2 θ n3e(ωs) ∂ne(ωs) sin2 θ n3o(ωs) ∂no(ωs) Solving the equation (A28) for ∂ksz we find equation (A26). Substituting equations (A26) into the output state of idler and signal photons (A26), we obtain a general ex- pression for PDC in type II. So far, we have only assumed the undepleted approximation and that the transverse area of the crystal is much larger than the characteristic width of the pump beam. 4. PDC in type II, collinear regime We now focus on the collinear regime in PDC where we only consider those idler and signal photons which are produced nearly in the same direction as that of the pump beam (in our framework along the z-axis). In this particular case, the approximation q (s,i) (s,i) � 1 is justified. Therefore, equations (A26) reduce to d2qidkiz = no(ωi) d2qidωi , for the idler, and d2qsdksz = ne(θ, ωs) d2qsdωs , for the signal, where we have also made the reasonable assumption that the variation of refractive indices with respect to frequency is small. Thus, Jacobians due to the change of variables are just refractive indices, which depend on frequency. In the collinear regime more simplifications can be done. Let us explicitly write the argument of the phase matching function (A23) as ∆(kp,ki,ks) = kpne(θ, ωp)− ksne(θ, ωs)− kino(ωi) + sin 2θ n2e(θ, ωp) ñ2(ωp) n2e(θ, ωs) ñ2(ωs) + qix sin 2θ n2e(θ, ωp) ñ2(ωp) ne(θ, ωs) 2n2e(ωp)kp (qiy + qsy) + (qix + qsx) e(θ, ωp) n2o(ωp) n3e(θ, ωs) 2n2e(ωs)n o(ωs)ks ne(θ, ωs) 2n2e(ωs)ks q2ix + q 2no(ωi)ki (A29) It is clearly an strongly anisotropic function. Notice that the linear terms in the transverse wave vectors are re- sponsible for the walk-off effect (the Poynting and wave vectors of the signal and pump photons are not collinear, i.e., their energy flows in a different direction from its cor- responding phase front). We should mention a previous study by Torres and coworkers [43] where the walk-off ef- fect was taken into account to describe entangled photon states generated in type I PDC. The state for idler and signal photons within type II PDC (A26) is |Φ〉(2ph)e = qidωi qsdωs χeff (ωp, ωi, ωs) × ei∆(kp,ki,ks)L/2 sinc [∆(kp,ki,ks)L/2] eunm(qi + qs) × δ(ωp − ωi − ωs) |ki, σo,ks, σe〉 (A30) where sinc(x) ≡ sin(x)/x. The symbol ∆Ω represents the fact that the transverse idler and signal wave vectors are constrained to be very small (this is the condition for the collinear regime). This can be achieved experimen- tally by using a suitable pin hole placed in the z axis. Now, due to this restriction, only the maximum of the sinc function contributes to the integral (the side lobes being blocked). Furthermore, the action of the pin hole truncating the sinc function can actually be modeled by a simple exponential function. In addition, we can reg- ularize the phase factor ei |qi−qs|2 by just shifting our coordinate system (z = 0) to the middle of the crystal, see equation (A23). We will also focus, for simplicity, on the frequency degenerated case, where the idler and sig- nal photons have the same frequency (ωp = ωi/2 = ωs/2, which can be realized by using appropriate narrow-band filters in front of the photodetectors). Thus, the two- photon state reads |Φ〉(2ph)e = χeff (ωp) d2qs e −∆(kp,ki,ks)L/2 × ũnm(qi + qs) |ki, σo,ks, σe〉 . (A31) The effective susceptibility, due to the energy conserva- tion and the frequency degeneration in idler and signal photons, now depends only on the frequency of the pump beam. The argument of the phase-matching function G = e−∆(kp,ki,ks)L/2 is still too complex. Since we wish to maintain our framework at a sufficiently simple level, and we will pursue a full analytical study, some further (and drastic) approximations are required: we take all refrac- tive indices (ordinary and extraordinary) to be identical and independent of frequency. Hence, the argument of the phase matching function reduces to ∆(kp,ki,ks) = 2kpneff |qi − qs|2 , (A32) which is, in contrast to equation (A29), an isotropic (with cylindrical symmetry) function. The effective refractive index neff can be interpreted as a suitable fitting pa- rameter (in principle, it can be quite different from the ordinary and extraordinary refractive indices). In spite of the strong approximations performed above, it is illustrative to determine their goodness by compar- ing the sinc and the exponential phase-matching func- tions when their arguments are given by equations (A29) and (A32), respectively. Figure 5 depicts both profiles. It can be seen that for sufficiently small transverse wave vectors both profiles are coincident (by suitably choosing neff ); the deviations arise, typically, when the pin hole is already truncating both functions. This qualitatively jus- tifies the validity of the approximations made above. Ac- tually, this approximation was previously used (without any justification) by Law and Eberly [17] to study bipar- tite transverse entanglement from type II PDC. Further- more, in other previous works, the sinc phase matching function was even replaced by a constant [14, 31, 44, 45]. In summary, equation (A31) finally becomes (simpli- fying the notation and absorbing the constant factors) |ψ〉 = qs Φ(qi,qs) |qi,qs〉 , (A33) where Φ = EG is the two-photon amplitude, with E(qi + qs) = ũnm(qi + qs) denoting the transverse (spectrum) profile of the pump beam, and G(qi − qs) = e − L4kpneff |qi−qs| being the phase-matching func- tion. 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[41] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, USA, chap. 10. [42] G.F. Calvo and A. Picón, Opt. Lett. 32, 838 (2007). [43] J. P. Torres, G. Molina-Terriza, and L. Torner, J. Opt. B: Quantum Semiclass. Opt. 7, 235 (2005). [44] S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, Phys. Rev. A 65, 033823 (2002). [45] J. P. Torres, Y. Deyanova, L. Torner, and G. Molina- Terriza, Phys. Rev. A 67, 052313 (2003). INTRODUCTION MODAL SCHMIDT DECOMPOSITION FOR TWO-PHOTON STATES AZIMUTHAL TRANSFORMATIONS ON THE TWO-PHOTON STATE AN EXACTLY SOLUBLE MODEL FOR THE TWO-PHOTON AMPLITUDE Experimental Schmidt Number Conclusions Acknowledgments APPENDIX B APPENDIX C Classical Energy for Non-linear Media Parametric Down-Conversion PDC in type II PDC in type II, collinear regime References

0704.1507

HST/ACS Coronagraphic Observations of the Dust Surrounding HD 100546

Accepted in ApJ HST/ACS Coronagraphic Observations of the Dust Surrounding HD 1005461 D.R. Ardila1,2, D.A. Golimowski2, J.E. Krist3, M. Clampin4, H.C. Ford2, & G.D. Illingworth5 ABSTRACT We present ACS/HST coronagraphic observations of HD 100546, a B9.5 star, 103 pc away from the sun, taken in the F435W, F606W, and F814W bands. Scattered light is detected up to 14” from the star. The observations are consis- tent with the presence of an extended flattened nebula with the same inclination as the inner disk. The well-known “spiral arms” are clearly observed and they trail the rotating disk material. Weaker arms never before reported are also seen. The inter-arm space becomes brighter, but the structures become more neutral in color at longer wavelengths, which is not consistent with models that assume that they are due to the effects of a warped disk. Along the major disk axis, the colors of the scattered-light relative to the star are ∆(F435W −F606W ) ≈ 0.0– 0.2 mags and ∆(F435W − F814W ) ≈ 0.5–1 mags. To explain these colors, we explore the role of asymmetric scattering, reddening, and large minimum sizes on ISM-like grains. We conclude each of these hypotheses by itself cannot explain the colors. The disk colors are similar to those derived for Kuiper Belt objects, suggesting that the same processes responsible for their colors may be at work here. We argue that we are observing only the geometrically thick, optically thin envelope of the disk, while the optically thick disk responsible for the far-IR emission is undetected. The observed spiral arms are then structures on this envelope. The colors indicate that the extended nebulosity is not a remnant of the infalling envelope but reprocessed disk material. 1Spitzer Science Center, California Institute of Technology, MS 220-6, Pasadena, CA 91125; [email protected] 2Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218. 3NASA Jet Propulsion Laboratory, Pasadena, CA 91109. 4NASA Goddard Space Flight Center, Code 681, Greenbelt, MD 20771. 5UCO/Lick Observatory, University of California, Santa Cruz, CA 95064. http://arxiv.org/abs/0704.1507v1 – 2 – Subject headings: stars: pre-main sequence – circumstellar matter – planetary systems: protoplanetary disks – stars: imaging – stars: individual HD100546 1. Introduction Resolved images of circumstellar disks in optical scattered light provide a unique way of establishing the spatial distribution of circumstellar material. They allow for the highest spatial resolution of all techniques available and are sensitive to light scattered by small particles regardless of their temperature. However, at optical wavelengths one has to contend with the brightness of the host star and the use of a coronagraphic mask becomes a necessity. Scattered-light observations complement thermal observations, and constrain models based on spectroscopic data only. In this paper, we present coronagraphic optical images in which the circumstellar mate- rial around the B9.5V Herbig Ae/Be star HD 100546 is resolved in scattered light. HD 100546 is a single star, with d=103+7 −6 pc and reddening of AV=0.28 (van den Ancker et al. 1998), located at the edge of the Lynds dark cloud DC 296.2-7.9. Isochrone fitting using the mod- els developed by Palla & Stahler (1993) suggests its age is &10 Myr (van den Ancker et al. 1998). The age cannot be much larger than this, as the system presents evidence of strong accretion in the Hα lines (Vieira, Pogodin, & Franco 1999). Its fractional excess luminosity is LIR/L∗ = 0.51 (LIR is the excess infrared luminosity and L∗ is the stellar luminosity, see Meeus et al. 2001). The circumstellar material was first resolved from the ground in the J and Ks bands, us- ing the Adaptive Optics Near Infrared System (ADONIS) mounted on the European South- ern Observatory’s 3.6 m telescope at La Silla (Pantin, Waelkens, & Lagage 2000). Those coronagraphic observations revealed nebulosity at angular distances ranging from 0.”4 to 2” from the star. The authors concluded that the observations were consistent with the presence of a disk with an inclination 50◦ ± 5◦ from edge-on. Augereau et al. (2001) performed Near Infrared Camera and Multi-Object Spectrometer (NICMOS) coronagraphic observations at 1.6 µm and detected scattered light from ∼0.”5 to 3.”8 from the star, as well as significant azimuthal asymmetries in the circumstellar disk. The observations had a spatial resolution of ∼0.”2 (Mobasher & Roye 2004). The highest spatial resolution images to date (∼0.”1 per resolution element) have been published by Grady et al. (2001). These are broadband 1For the astro-ph version, the images have been reduced in size and resolution. Please contact D. Ardila for a full resolution PDF. – 3 – coronagraphic observations taken with the Space Telescope Imaging Spectrograph (STIS) that probe angular distances as small as ∼0.”8 from the star (at selected azimuthal angles). The observations reveal a complex circumstellar structure which the authors interpret as an inclined disk surrounded by an envelope of material extending up to ∼10” from the star. In addition, the disk presents bright and dark lanes, reminiscent of spiral structure. Thermal observations at ∼10 µm using nulling interferometry reveal that the disk has a radius of 30 AU at that wavelength (Liu et al. 2003). Those observations also suggest the presence of an inner hole. The spectral energy distribution (SED) of this system has been intensively studied. The star is an example of a class of Herbig Ae/Be objects that show substantial silicate emission at 10 µm and for which there is strong emission from the mid-infrared (mid-IR) relative to the near-infrared (near-IR). These have been called group Ia sources by Meeus et al. (2001). AB Aurigae (4 Myrs, A0V, see van den Ancker et al. 1997; DeWarf et al. 2003) is another group Ia source, often mentioned in the same context as HD 100546. Meeus et al. (2001) suggested that the SED for group Ia sources could be explained by a geometrically thin, op- tically thick disk responsible for the far-infrared (FIR) emission, as well as a flared optically thin component responsible for mid-IR emission. However, they did not develop quantita- tive models to confirm if this configuration could indeed reproduce the SED. More recently, Vinković et al. (2006) concluded from the surface brightness profiles derived from the STIS coronagraphic observations (Grady et al. 2001) that the configuration of the material close to the star is that of a geometrically thin, optically thick disk surrounded by a flat, geomet- rically thick, optically thin envelope. Vinković et al. (2006) used this model to successfully reproduce the near-IR SED (< 6µm). For HD 100546, the 3 µm excess is even smaller and the mid-IR excess larger than in other Herbig Ae/Be stars (Bouwman et al. 2003). This is significant because, histori- cally, the source of the (generally) strong 3 µm “bump” in Herbig Ae/Be stars has been difficult to explain, as it is not obviously produced by an optically thick disk. For example, Hartmann, Kenyon, & Calvet (1993) have suggested that it is due to emission by an infalling envelope. Recently, Dullemond, Dominik, & Natta (2001) showed that the puffed-up inner edge of a Chiang-Goldreich disk (a flared disk with well-mixed dust and gas and a super- heated atmosphere, see Chiang & Goldreich 1997) subjected to direct irradiation from the star will emit in the near-IR. If this inner rim is the source of the 3 µm bump, the low 3 µm emission from HD 100546 suggests a modest rim (outside the bounds of the self-consistent model by Dullemond et al. 2001) and a shallow dust surface density for the inner disk. With a model of these characteristics on a 400 AU disk Dominik et al. (2003) were able to fit the system’s SED, although the fit underpredicts the submillimeter flux by almost an or- der of magnitude. Because of the small 3 µm excess, Harker et al. (2005) were able to fit – 4 – the SED with an unmodified Chiang-Goldreich disk (no puffed-up rim), 150 AU in radius. Harker et al. (2005) did not include any submillimeter measurements in their SED model, which may explain some of the difference in radius with the models from Dominik et al. (2003). A completely different model was developed by Bouwman et al. (2003), who fitted the SED by a set of three optically thin spherical shells. The purpose of their model was not to determine the geometry of the emission but the mass distribution of the dust as a function of temperature. They concluded that the low near-IR emission can be explained by a mostly empty region (an inner hole) within 10 AU and speculated that the hole had been created by a protoplanet. Evidence for a hole close to the star has also emerged from spectroscopic data. Ultravi- olet observations using STIS (Grady et al. 2005) reveal a deficit of molecular hydrogen close to the star, which the authors interpret as a central gas and dust cavity, 13 AU in radius. In addition Acke & van den Ancker (2006) have used the 6300 Å line of [OI] to conclude that there is a 4.3 AU-wide gap in the disk, 6.5 AU away from the star. They argue that this is consistent with the presence of a 20 MJ object at that position. The mid-IR spectrum of HD 100546 is similar to that of Comet C/1995 O1 (Hale- Bopp) Malfait, Bogaert, & Waelkens (1998) in that it shows strong emission in crystalline silicates. In this respect it resembles that of the optically thin debris disk around the K0 dwarf HD 69830 (Beichman et al. 2005). Models of the material around HD 100546 suggest a relatively large fraction of small (0.1 µm) forsterite and large (1.5 µm) enstatite grains (Malfait et al. 1998; van Boekel et al. 2005). The models by Bouwman et al. (2003) indicate that large amorphous silicate grains and small forsterite grains have different temperatures, unlike the case in Hale-Bopp, although they overlap in radial distance from the star. The mineralogical composition of the dust led Bouwman et al. (2001) to suggest that thermal annealing may not have been the origin of the forsterite and that it may result from the destruction of a differentiated larger parent body. All these spectroscopic models indicate a highly evolved disk both in terms of particle size and crystalline fraction. Optical and ultraviolet observations indicate that, in spite of the inner hole, the star is actively accreting and has a stellar wind (Vieira et al. 1999; Deleuil et al. 2004). While hot molecular hydrogen has been detected close to the star (Lecavelier des Etangs et al. 2003), the cold gas component has not been detected (Loup et al. 1990; Nyman et al. 1992; Wilner et al. 2003). From measurements of the 1.3 mm dust continuum Henning et al. (1998) concluded that there are 240 M⊕ of dust within a region 11” × 11” in size, which would imply 100× larger gas mass, if the mixture has ISM proportions. This is certainly an upper limit as the non-detection of the HCO+ line at 89 GHz by Wilner et al. (2003) suggests that – 5 – the gas may be strongly depleted. The lack of detectable gas, the presence of large particles, and the evidence for an inner hole suggest that HD 100546 is an evolved pre-main-sequence star, a “transitional” disk perhaps on the way to becoming a more collisionally evolved object like HD 141569 (Clampin et al. 2003) or β Pictoris (Golimowski et al. 2006). In this, it joins a class of objects with signs of evolution in their inner disk (flux deficits in the near IR relative to the median SED of the class) which have been identified in the last decade (Natta et al. 2006). However, the exceptionally high fraction of crystalline silicates and the characteristics of its SED suggest that it may be in a class by itself. Because of its relevance in this poorly understood transitional phase, we targeted it for observation using the Advanced Camera for Surveys (ACS; Ford et al. 2003, Pavlovski 2006) on board the Hubble Space Telescope (HST). In this paper, we present F435W, F606W, and F814W coronagraphic images of HD 100546’s circumstellar environment. These images reveal the environment between 160 and 1300 AU from the star with unprecedented spatial resolution, suppression of the wings of the point spread function, and photometric precision. In addition, they allow us to measure for the first time the color of the circumstellar dust. The scattered-light observations presented here sample the small dust population, far (&160 AU) from the star. They are then complementary to spectroscopic observations and SED modeling that focus primarily on the inner disk. 2. Observations and Processing In this section we present a summary of the reduction procedure, which involves the subtraction of a point spread function (PSF) from the target star. The reduction follows the standard procedures common to other coronagraphic observations. The resulting images were then deconvolved using a Lucy-Richardson algorithm. Those deconvolved images form the core data used for the analysis. The science-quality images on which most of the conclusions in this paper are based are available from the Journal’s Web site. These observations were conducted as part of the guaranteed observing time awarded to the ACS Investigation Definition Team (proposals 9987 and 9295). They were performed using the coronagraphic mode of the ACS High-Resolution Channel (HRC), with the 0”.9 oc- culting spot, which suppresses the stellar PSF wings by factors of the order of ten (Pavlovski 2006). It is possible to improve the contrast by subtracting a star of the same spectral type as the target, which suppresses the remaining PSF wings. To do this subtraction, we observed HD 129433 as a PSF reference star. The star was chosen because it is bright, has – 6 – similar color (V=5.73, B-V=-0.01, Crawford 1963). The observing log is shown on table 1. The images were taken with the F435W (Johnson B), F606W (broad V) and the F814W (broad I) filters. The HRC has a pixel scale of 25 mas pixel−1, and a coronagraphic field PSF FWHM of 50 mas in the F435W passband, 63 mas in the F606W passband, and 72 mas in then F814W passband. For each band and each star, a short direct exposure was followed by a coronagraphic exposure. In each band, HD 100546 was observed at two different telescope roll angles. The process of PSF subtraction produces image artifacts and the two orientations help distinguish between these and real features. 2.1. Reduction The initial stages of the image reduction (i.e., subtraction of bias and dark frames, and division by a non-coronagraphic flat field) were performed by the ACS image calibration pipeline at the Space Telescope Science Institute (STScI) (Pavlovski 2006). We started our own image reduction procedures on the separate flat-fielded images before combination, cosmic-ray rejection, and correction of the HRC’s geometric distortion by the STScI data pipeline. Coronagraphic images with the same exposure time were averaged and cosmic rays were rejected using conventional IRAF tools2. To account for vignetting around the occulting spot, the coronagraphic images were manually divided by the spot flats available from the ACS reference files page at STScI’s Web site. Each spot flat was shifted to the appropriate position corresponding to the time of observation. This step is done automatically since the CALACS pipeline version 4.5.0. The coronagraphic images obtained in each band and roll position were then sky- subtracted. The sky was measured by taking the median in the number of counts from circles 20 pixels in radius at the four corners of the images. The effect of this step is minimal in the overall photometry. After sky subtraction, the coronagraphic images were divided by the exposure time. In the longest exposures, light diffracted by the coronagraphic mask saturates some pixels in the region within 2” from the star and we replaced those saturated pixels by non-saturated pixels from shorter exposures. 2IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the As- sociation of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. – 7 – 2.2. A Note on HST’s Photometric systems The F435W filter is a good approximation of Johnson B, but the other two filters used in these observations do not correspond to standard filters. To calibrate our observations and compare them with published results, we make use of three different photometric systems: an “instrumental” vegamag system, a standard vegamag system, and the standard Johnson- Cousins system as described by Landolt (1992). The instrumental vegamag system is described by Sirianni et al. (2005). In this system Vega has zero magnitudes and zero colors in all bands. The transformation from e−/sec to magnitudes involves only a zero-point and no color term. We will refer to magnitudes in this system with the filter name. For example, the F435W magnitude, etc. This is the system most used in this paper. If the spectrum of the source is known, it is possible to obtain magnitudes in the standard vegamag system. These magnitudes are given in the Johnson B, Johnson V, and Cousins I bandpasses, but the calibration is still tied to Vega. Transformations between the instrumental and the standard vegamag system are possible using the synthetic photometry package SYNPHOT, which is distributed by STScI. We will refer to the magnitudes in this system with the subscript Vega. For example, BVega, VVega, etc. Ground based observations usually present measurements in a standard photometric system, involving standard filters and a grid of primary calibrators. Sirianni et al. (2005) provide color transformations between the instrumental vegamag system and the standard system described by Landolt (1992). Sirianni et al. (2005) argue that the transformations are strongly dependent on the details of the source spectra and so we use them sparingly. 2.3. Stellar Photometry A measurement of the stellar flux is required for two aspects of the analysis. First, in order to subtract the PSF reference star from the target, the former needs to be scaled by a multiplicative factor. This factor is the ratio of the brightness of the reference star and that of HD 100546 in each band. Very accurate photometry is not crucial for this procedure because visual inspection of the subtracted images is enough to determine the scaling factor within ±3%. Second, in order to determine the intrinsic color of the circumstellar material, we divide the images by the stellar flux. For this, we need to know the stellar photometry as accurately as possible. As one of the main conclusions of this paper is that the circumstellar material is intrinsically red, we now describe in detail the procedure followed to obtain this photometry. – 8 – For F435W we used the direct (non-coronagraphic) images of HD 100546 and HD 129433 to measure the number of e−/sec within 5”. While the cores of the stellar PSFs are saturated, observations performed with the parameter GAIN set to 4 (as are these) preserve the number of electrons over the whole image (Gilliland 2004). In other words, as long as the aperture is large enough to encompass all the saturated pixels, the integrated number of counts can be recovered. Direct non-coronagraphic observations in the other bands were executed with GAIN=2 and so the number of counts in saturated images is not conserved. We used the number of counts measured in F435W, a model spectrum of the target and the PSF reference star, and the measured colors of HD 100546, to predict the number of counts in F606W and F814W. For HD 100546, de Winter et al. (2001) measured B-V=0.012 ± 0.005 mags and V- I=0.042 ± 0.01 mags, from which van den Ancker et al. (1998) derived AV=0.28 mags, as- suming that the intrinsic color was B-V=-0.075 mags. In order to compare with our data, we assumed that the colors in the standard vegamag system are the same as those in the standard Johnson-Cousins system. This is a very good assumption for HD 100546, whose spectral type is B9.5 and so the color-dependent terms in the transformation equations are small (Sirianni et al. 2005). As the model spectrum for the target we used the B9V star HD 189689, from the Bruzual-Persson-Gunn-Stryker (BPGS) Atlas provided by STScI. This star has BVega-VVega=-0.11. We reddened the spectrum using the standard extinction law (Cardelli, Clayton, & Mathis 1989) until BVega-VVega=0.012 mags, the average measured color for HD 100546. The resulting extinction is AV=0.36 mags. With this spectral model and the number of counts measured in the F435W band, we used the calcphot routine within SYNPHOT to derive a BVega magnitude. The BVega magnitude derived in this way (table 2) is 0.03 mags less than the average B magnitude derived by de Winter et al. (2001), consis- tent the difference between the standard system, in which Vega has B=0.03 mags, and the vegamag system, for which Vega has BVega=0.00 mags. From the stellar colors of HD 100546 given by de Winter et al. (2001) we derived VVega and IVega. With these magnitudes, the cal- cphot routine, and the same spectral model, we obtained the number of counts in F606W and F814W. The result of this procedure is that the number of counts in each band corresponds to the measured stellar color. The BPGS Atlas contains four more dwarf stars with spectral types between B9 and A0. For each of them, a different value of AV is required to reproduce the observed color for HD 100546. They range from AV=-0.077 mags in the case of 58 Aql to AV=0.18 mags for θ Vir (both classified as A0V in the BPGS Atlas). However, following the same procedure described above results in changes of the number of counts for each band of <1% with respect to those reported in table 2. The procedure is robust to the assumed spectral model because – 9 – each model is forced to have to observed colors of HD 100546, independently of its intrinsic colors. For HD 129433, the number of counts in F606W and F814Wwas obtained from the num- ber of counts in F435W by using as spectral model the unreddened spectrum of HD 189689. The PSF reference star has a larger BVega-VVega (by 0.1 mags) than HD 189689, and so this procedure will systematically underestimate its flux by ≈ 10%. Accurate photometry of the reference star is not crucial for the conclusions of this paper. Sirianni et al. (2005) indicates that the best aperture photometry using the HRC will have 3% errors. For HD 100546, the color measurements from de Winter et al. (2001) intro- duce an additional 2% error (3% error) in the determination on the F606W (F814W) flux. For the PSF reference star, we assume that the errors are those of the aperture photometry only. These are the errors quoted in table 2. 2.4. Subtraction of the Coronagraphic PSF For each band, the normalized, cleaned, and combined coronagraphic images of the PSF reference star HD 129433 were subtracted from each roll of the combined images of HD 100546. The positions of the occulted stars were determined within one pixel using the central peaks of the coronagraphic PSFs that result from the re-imaging of the incompletely occulted, spherically aberrated starlight (Krist et al. 2003). Further alignment was done visually, by shifting the PSF reference images until subtraction residuals were minimized. The uncertainty in the registration of the images along each axis is 0.125 pixels. For the normalization constants we performed the following visual, iterative adjustment. Starting with the flux ratios implied by the values from table 2 (0.391, 0.443, 0.474 for F435W, F606W, and F814W, respectively), we adjusted the scaling until the subtraction residuals were minimized. This occurred for normalization constants 0.383, 0.382, and 0.387 in F435W, F606W, and F814W. Visually, the uncertainty of these normalization values is 3% and in what follows we will propagate this error linearly (as a systematic error) to estimate uncertainties in calculated quantities (Section 3.4). The values of the normalization constants are consistently lower than implied by the values quoted in table 2, as expected by the mismatch between the colors of the PSF reference star HD 129433 and those of its spectral model. After subtracting the coronagraphic images of HD 129433 from each image of HD 100546, we corrected for the geometric distortion in the HRC image plane, using the coefficients of the biquartic-polynomial distortion map provided by STScI (Meurer et al. 2002) and cubic – 10 – convolution interpolation to conserve the imaged flux. By using standard IRAF routines, we aligned the two rolls in each band, using other stars in the field for reference, and averaged the images from each roll. The resulting surface brightness maps (divided by the stellar fluxes) are shown in figure 1. These maps are available from the Journal Web page. The mid and bottom rows of figure 1 highlight different regions of the circumstellar environment. Differences in the shape of the PSF between HD 100546 and the reference star HD 129433 (due to differences in color, telescope focus, or position behind the mask) result in subtraction residuals. The radial streaks in the top two rows in figure 1 are produced by differences in the way light is scattered by telescope optics for the two stars while the concentric dark and bright rings are due to differences in the diffraction pattern produced by the coronagraphic mask (Krist 2001). The rings are more noticeable along the NE- SW direction, which coincides with the position of the scattering “strip”. This well-known artifact of the ACS coronagraph is a bar-like region of increased number of counts, centered on the star and ≈6” long (e.g. Krist et al. 2005b). The presence of these residuals makes the analysis within ≈ 1.6” of the star impossible. Beyond this limit we can use the images in the two separate telescope rolls to distinguish subtraction residuals from features associated with the material around HD 100546, as the residuals are nearly fixed on the detector’s axes. To trace the contours shown in the bottom row of figure 1 we perform a 10-pixel median azimuthal smoothing on the middle-panel images, followed by a 5-pixel boxcar average smoothing. In figure 1, the difference in the spatial resolution among the three bands is seen as a smoothing of the images with increasing wavelength. In the top row, the imprints of the larger coronagraphic spot and the occulting bar are visible towards the SW direction (one for each roll angle). 2.5. Red Halo and Image Deconvolution For HRC observations in the F814W band, one has to contend with the effect of the red halo (Sirianni et al. 2005). This may appear as a diffuse, smooth halo of light, slightly asymmetric but fixed with respect to the detectors axes. The halo is composed of photons from the core of the stellar PSF which have been scattered to large angles by the CCD substrate. The halo asymmetry can be seen in the direct (non-coronagraphic) images from Krist et al. (2005b) (See their figure 2). In this section we show that deconvolution of the stellar PSF, which partially corrects this effect, changes modestly the slope of the surface brightness profiles. – 11 – To deconvolve the images, we use the PSFs generated by the Tiny Tim software package, distributed by STScI (Krist & Hook 2004). The simulated “off-spot” coronagraphic PSFs can be generated for an array of spectral types. These PSFs include a simplified model of the red halo but do not include the red halo asymmetries mentioned before. We do not believe those asymmetries are important in our images: upon subtraction of the two rolls no residual asymmetric structures are revealed. Unlike the non-coronagraphic images from Krist et al. (2005b), the coronagraphic images presented here do not produce a very strong core to be scattered at large angular distances. The deconvolution is performed on the image after subtraction of the reference PSF. Therefore, only the occulted star subtraction residuals contribute to the deconvolution pro- cess (and then basically as noise) as the effects of vignetting near the occulter are removed with the division by the coronagraphic spot flat field. The only effect of the coronagraphic setup on emission outside of the focal plane mask is to alter the PSF, which is then defined by the Lyot stop rather than the entrance aperture of the telescope. Simulations (White 2007) confirm that beyond 1.5” from the center of the coronagraphic mask, the off-spot PSF does not depend on distance. We generated distorted off-spot PSFs in each band, using BPGS model of the A0 star θ Vir as our model. This is the closest spectrum to our target provided by Tiny Tim. We applied the Lucy-Richardson deconvolution algorithm (Lucy 1974; Richardson 1972) as implemented in IRAF to each (single-roll, distorted, PSF-subtracted) image outside a circular region 1” in radius, centered on HD 100546. We terminated the computations at 50 iterations. By this point, we could not detect changes in the FWHM of the field stars, nor in the surface brightness profiles of the image. After deconvolution, we corrected for camera distortion, rotated the two rolls in each band to a common orientation and averaged them. In this way, we obtained a deconvolved image in each band. A comparison of the “before” and “after” surface brightness profiles in each band is shown in figure 2. The resulting images are shown in figure 3. Figure 2 shows that the deconvolution has its strongest effect in the F814W images, where it results in a sharpening of features within 2.”5 and a modest reduction of the large scale surface brightness from 2.”5 to 10” away of HD 100546. As mentioned before, the deconvolution process is not valid within 1.”5 from the coronagraphic mask. In the case of the direction shown in figure 2 (corresponding to the SE semi-major disk axis), the deconvolved profiles are ∼15%, ∼5%, and ∼3% fainter in average than the non-deconvolved ones, for F814W, F606W, and F435W, respectively. These values are similar to those along other directions. Therefore, the ∆(F435W −F814W ) color of the deconvolved images (defined in section 3.4) is ∼0.12 mags less (i.e. not as red) than the color of the non-deconvolved ones. – 12 – As we will show in section 3.5 this is comparable to the errors in the color. So in terms of the large scale behavior of the color it does not make much difference if the analysis is carried out in the deconvolved or in the non-deconvolved images. Figure 3 shows the deconvolved equivalent of figure 1. After deconvolution, the features in the circumstellar environment become sharper, and the images become noisier. The amplified, correlated noise in the deconvolved images is characteristic of the Lucy-Richardson algorithm and it is a consequence of the algorithm’s requirements of a low-background signal, positive pixel values and flux conservation in local and global scales. The higher noise levels require that the images be further smoothed before calculating the contours. To measure them, we smooth the images as in figure 1 and then convolve them with a gaussian function having FWHM=10 pixels, which acts as a low-pass filter. It is in these heavily smoothed figures that the contours are traced. The deconvolved F435W, F606W, and F814W surface brightness maps are available from the Journal Web page. 2.6. Color-combined images In order to grasp the overall appearance of the system, we created color composites of the images by mapping the bands in three color channels: F435W in Blue, F606W in Green and F814W in Red. Before combining them, each image was divided by the stellar flux in its band. To produce the color composites, the images were scaled by the inverse hyperbolic sine of the average in all three bands, using the softening parameter β = 0.005, as described in Lupton et al. (2004). This scaling makes the apparent color independent of the brightness. In effect, the scaling is linear for very faint regions and logarithmic for brighter ones. The results are shown in figure 4. Regions of extreme colors, like the circular purple features close to the coronagraphic mask, indicate places in the image for which data from at least one of the bands are missing. These occur because the position of the coronagraphic subtraction residuals is wavelength dependent. The deconvolution process (figure 4, right) makes the radial streaks resulting from PSF mismatch very evident. 3. Results Figures 1, 3, 4, show the well known circumstellar structure of HD 100546 (Grady et al. 2001) but with an unprecedented degree of clarity. Within 1.”6 the images are dominated by subtraction residuals. From 1.”6 to 10” from the star, the total scattered-light relative fluxes, – 13 – measured in the deconvolved images, are (LCircum/L∗)λ = 1.3±0.2 ×10 −3, 1.5±0.2 ×10−3, and 2.2± 0.2 × 10−3, in the F435W, F606W, and F814W bands respectively. These values are ≈ 200− 400 times less than the total IR excess. In this section we describe the main observational results associated with the general characteristics of the scattered light distribution around HD 100546. 3.1. Photometry of Field Stars HD 100546 has a galactic latitude of -8.32 degrees, which suggests that the other point sources in the images are unassociated field stars. In table 3 we list the standard photometry for the brightest stars within ∼6” of HD 100546. See also figure 5. We only list those objects that are observed in both rolls (and are outside the occulting bar in both) in at least one band. To compare with known stellar colors, the photometry is given in the standard Johnson-Cousins system, using the color corrections from Sirianni et al. (2005). The largest V-I color is ∼ 2. If the object is unreddened, this color would suggest that it is an early M dwarf. Such a star would have an I magnitude of ≈ 12, much brighter than observed (Leggett 1992). All other objects have bluer colors, indicating earlier spectral types and brighter predicted apparent magnitudes. Therefore, all objects in table 3 are farther away than HD 100546. 3.2. General Morphology Overall, the images show a broad band of light wrapping around the SW side of the star (figure 4, top), reminiscent of the configuration of the scattered light images around AB Aurigae (Grady et al. 1999). A red glow can be seen on the NE side of the target both in the deconvolved and non-deconvolved images, similar to the strong F160W scattered light signal found closer to the star by Augereau et al. (2001). Interpreting the inner circumstellar surface brightness as due to an inclined disk, we use the deconvolved images to fit elliptical isophotes to the images, with semi-major axes ranging from 1.′′6 to ∼ 2′′. At distances larger than 2” the contours become “boxy” because of the bright SW feature (structure 2, see section 3.3) and the scattered light cannot be fit with an ellipse. For all the bands, the position angle (PA) of the semi-major axis is ≈ 145 degrees. From the ellipticity of the isophotes we derive an inclination angle of 42 degrees from face-on. There are no significative differences in these values among the three bands. This is an observational determination of the inclination and PA based on isophotal contours, – 14 – and ignores the effect of anisotropic scattering of the stellar light by the dust grains. These results are summarized in table 4. Table 4 also compares the disk geometry derived in this work with other published results. All disk inclinations are consistent, but the position angles of the major axes are only marginally so. Our results differ from those by Pantin et al. (2000), Grady et al. (2001), and Augereau et al. (2001) at the 2σ level. Pantin et al. (2000) derived the PA by using symmetrized image of the disk. Grady et al. (2001) do not describe the procedure used to determine the PA. Augereau et al. (2001) determined the PA by performing isophot fitting within 3”. These differences may reflect the fact that a measurement of the ratio of the axes of an ellipse is less sensitive to uncertainties than measurements of a single axis. Furthermore, there may be differences between the value of the scattering phase function in the optical and the near-IR. For the F814W images, the contours within 2” (bottom row of figures 1 and 3) are closer together at the SW side than at the NE side: the image brightness changes faster in the former than in the later. The contours in the other bands are not as well defined at these distances. In the sections that follow and for the purposes of the morphological description, we divide the circumstellar environment in three regions, where the distances are measured along the disk major axis: the inner disk (from 1.”6 to 3”), the mid-disk (from 3” to 8”) and the extended envelope (beyond 8”). Each region has its own dynamical and/or radiative characteristics. The inner disk and the material under the coronagraphic mask, comprise the region believed to be responsible for most of the far-IR excess. 3.3. Structures in the Disk The well-known “spiral arms” of HD 100546 (e.g. Grady et al. 2001) are clearly seen beyond 1.”5 to ≈3” from the star, as elongated structures at the NW, SW, and SE (figure 6). The simpler mask shape of the ACS/HRC coronagraph compared to the STIS one, allows for a more complete description of the environment. In the rest of the paper, we refer to these features as structures or spiral arms, interchangeably. Structure 1 is a narrow band of brightness visible between 2.”3 and 2.”8 from the star, and figure 6 suggests that it may continue on the NE side of the disk. Structure 2 describes a broad arc at 2”, and is separated from the inner part of the emission by a dark region ∼0.”5 wide. However, the exact extent of the dark lane is hard to judge because of the subtraction residuals along the minor axis. Structures 1 and 2 are separated by a region – 15 – that is half as bright as the maximum of structure 2. Structure 3 extends from 1.”5 to 2.”5 from the star, and it seems more open than structure 1. Because of this, structure 1 seems to wrap itself around the NE side of the disk, while structure 3 stabs the SW side. Structures 1 and 3 are superficially reminiscent of galactic spiral arms. The morphology of structure 2 is reminiscent of that produced when starlight is scattered from the disk back side towards the observer (see for example, the observations of GM Aur in Schneider et al. 2003). However, both in observations and simulations (Whitney & Hartmann 1992), such structure is generally narrower and it wraps around the disk over a larger angle than seen here. Observations of the [OI] 6300 Å line by Acke & van den Ancker (2006), reveal blueshifted emission from the SE of the disk and redshifted emission from the NW part, suggesting that the inner part of the disk is rotating counter-clockwise. Their observations trace gas at distances <100 AU from the star. Assuming that the disk rotates in the same direction at larger distances from the star and that the SW side is oriented towards the Earth (section 3.4.2), those spectral observations indicate that the structures are trailing the direction of rotation. Before deconvolution, the behavior of the structures with wavelength is tangled with the changing resolution of the telescope at different passbands (figure 1). The deconvolved images (figure 3) reveal no obvious morphological differences in each band, for each of the structures. The contour levels traced on the deconvolved images show that the space between the inner part of the disk and the structures becomes brighter at longer wavelengths, while the value of the peak brightness does not change much between bands. This color behavior is discussed in the section 3.4. The circumstellar disk is illuminated by the starlight, which decays as the inverse dis- tance squared from the star. An appropriate correction for this effect is only possible for face-on disks. We therefore deproject the disk by the inclination, divide by the Henyey- Greenstein scattering phase function with g = 0.15 (derived in section 3.4.2), and multiply every pixel by the square of the projected distance to the star. This procedure will reveal the correct geometry and brightness of the circumstellar material only in the case of a disk in which every dust particle is illuminated by the full stellar radiation field. Such is not the case for HD 100546, as every reasonable model of the SED indicates that some part of the disk is optically thick. However, independently of the optical depth, the procedure helps to reveal weak disk features and clarifies the disk structure, although the brightness of the structures and their distance to the star, particularly along the minor axis, will be incorrect. The result is shown in the top left panel of figure 7 for the non-deconvolved images in the F435W band (images obtained from other bands and from the deconvolved data look – 16 – similar). Because the hemispheric subtraction residuals are circular in the original image, they appear elliptical in the deprojected one. In order to accentuate the sharp structures, we subtract from the deprojected image a smoothed version of itself (figure 7 top right). The smoothed image has been convolved with a gaussian function having a 30-pixel FWHM. The subtraction of this smoothed image from the deprojected one leaves only the sharp features. An unsharp masking technique similar to this has recently been used to clarify the arm structure of our galaxy (Levine, Blitz, & Heiles 2006). This unsharp masking also sharpens the subtraction residuals present in the images, and so identification of real features requires care. In the bottom right panel of figure 7 we have marked those features that are observed in all three bands. The original structures 1, 2, and 3 are traced with solid lines, and marked as 1a, 2a, and 3a, while new structures are traced with thick dashed lines. The brightest new features are structure 1c, an inner arm at ∼ 2” that seems to emanate from 1a, and structure 1d, a feature behind the disk. The deprojected images show that structure 2a curves toward the star, like another spiral arm. The 1a+1b arm is ∼5” long, while 2a+2b and 3a+3b are ∼4” long. On the deprojected, unsharp masked image, arms 1a and 3a have the same opening angle with respect to the disk and appear like symmetric images of each other. Together with arms 2a and 1d, the overall configuration is that of a four-armed spiral. The unsharp masking traces only the sharp ridges, and so the features may extend beyond these limits. Structures analogous to these have also been detected around AB Aur, based on ground- based coronagraphic images in the H-band Fukagawa et al. (2004), although they do not seem as well ordered. In those observations, at least one of the arms appears split, as structure 1 is split between 1a and 1c. The arms around AB Aur are also believed to be trailing the disk. 3.4. Surface Brightness Profiles Figures 8 and 9 show the relative surface brightness profiles (SBPs) and colors, along the major and minor axes of the disk. The SBPs are obtained from the deconvolved images by taking median values along rectangular strips (0.”25 wide) centered on the star for each band. In addition, each profile has been median-smoothed with a boxcar function 10 pixels long. The bottom plots show the colors measured with respect to the star: ∆(F435W−F606W ) = −2.5 log(SBPF435W/SBPF606W ), ∆(F435W − F814W ) = −2.5 log(SBPF435W/SBPF814W ). An aid to the interpretation of the profiles is provided in figure 10. – 17 – 3.4.1. Sources of Error and Comparison with Previous Measurements There are three main sources of error in the SBPs: photometric errors due to the uncertainty in the stellar flux (which varies between 3 and 5%, as given in table 2 and result in errors small compared to those from other sources), photometric errors due to the uncertainty in the flux ratio between HD 100546 and HD 129433, and errors (variability) within the 0.”25 segments. In figures 8 and 9 the total 1σ errors are shown for some representative points. For these profiles, the 3σ detection limits are 2×10−7/arcsec2, corresponding to VVega=23.4 magnitudes/arcsec2. The uncertainty in the flux ratio produces errors which scale with the brightness of the circumstellar material. We propagate them linearly, as they are systematic and not random. For the major axis, they range from 10% for distances <8” to 5% farther out. For the minor axis they range from 20 to 40%. The errors are larger along the minor axis because that axis coincides with the position of the scattering strip (see section 2.4). Finally, errors in the median within the 0.”25 segments are a measure of the variability within the strips used for the calculation of the SBPs, and range from ≈5%, 2” from the star, to ≈10% at 12” for the major axis or ≈40% at 8” for the minor one. In summary, errors in the photometry (introduced by errors in the normalization con- stant) dominate the error budget at small distances and uncertainties due to variability within the median strip dominate at large distances, with a turnover point at 6-8” from the star. In addition, there are regions of the profiles strongly affected by the systematic mis- matches between HD 100546 and the PSF reference star HD 129433. For example, the “dip” in the SE semi-major axis in F606W at 1.”5 is due to over-subtraction of the PSF at that po- sition, as it is obvious from figure 3. Within ≈ 1.′′6 the SBPs in all directions are dominated by subtraction residuals. As mentioned in the introduction, previous coronagraphic observations of the circumstel- lar environment are available. Over the unfiltered STIS passband, Grady et al. (2001) mea- sured a total count rate of 126 cts/sec at 2”, on 9 pixels, which implies 5382 e−/sec/arcsec2. With this measurement, the reddened spectrum of HD 189689 from section 2.3, and calcphot, we predict that the count rate in the F606W passband should be 5× 103 e−/sec/arcsec2, or 1×10−4 per arcsec2 (after dividing by the stellar flux in the F606W passband). This point is shown in figure 8. We repeat the analysis for a point at 8”. Both points are broadly consis- tent with our observations. To compare with the NICMOS coronagraphic images we use the fact that Augereau et al. (2001) measured 16 mag/arcsec2 at 2.”5 from the star, which im- plies 1.7×10−4 /arcsec2, at the F160W band. To obtain this number we used the magnitude – 18 – zero-points reported by Augereau et al. (2001) and the reddened spectrum of HD 189689 as a spectral model, which implies that the F160W flux density of HD 100546 is 2.5 Jy. We also plot the NICMOS measurement at 3.”5 for which Augereau et al. (2001) measured 18 mag/arcsec2. The NICMOS measurements imply ∆(F435W − F160W ) =1.5 at 2.”5 and ∆(F435W − F160W ) =1.4 at 3.”5. As before, uncertainties in the stellar model trans- late into uncertainties of ≈ 4% in the calculated flux. Comparison with the ground-based ADONIS measurements reported by Pantin et al. (2000) yields results that are difficult to understand. They measured 0.3 Jy/arcsec2 at 0.”5 from the star in the J-band. This is one order of magnitude larger than the F160W NICMOS measurement at that point (≈ 11 mag/arcsec2 or 0.04 Jy/arcsec2). Schütz et al. (2004) has shown that variations in the PSF shape due to atmospheric fluctuations may result in false disk detections, when using ADO- NIS paired with coronography. On the other hand, the PA and inclination reported by Pantin et al. (2000) are roughly consistent with those derived using other instruments. 3.4.2. Morphology of the Nebulosity The top panels of figure 8 show that at 8” the F814W SBP drops by a factor of ∼2. The drop is very abrupt along the SE direction. Beyond this distance, the ∆(F435W −F814W ) and ∆(F435W −F606W ) colors become indistinguishable from each other. Consistent with the measured inclination, there is a decrease along the SW minor axis at 6”–7”. This is why we define the mid-disk as the disk between 3” and 8” from the star, and the extended envelope as material beyond 8”. Figure 10 shows that the nebulosity is slightly more extended to the NW side than to the SE. Along the NW side, the major-axis SBPs show scattered light to the edge of the detector, ∼ 14′′. Along the SE side the emission drops at 12”, from 6×10−7/arcsec2 to below the detection limit. Again, this is consistent with the decrease along the NE semi-minor axis at ∼ 9′′. In other words, the extended envelope is a flattened nebular structure ≈12” – 14” in radius, seen with an inclination similar to the inner disk. In figure 11 we compare the profiles along each semi-axis to each other. The semi-major axis profiles along each direction are very similar to each other, while the profiles along the semi-minor axis are very different. Along the major axis, the main differences between the two sides are in the presence of structure 1 at 2.”8, structure 3 at 2.”3, and field star # 8 at 5.5”. Along the minor axis, the amount of scattered light in all bands is larger in the SW side than in the NE side, until the sharp decrease of the SW side at 6”–7”. The ratio of the scattered light in the SW side to that in the NE side, in each band, between 2” and 7”, presents strong local variations, with values oscillating between 1 and 4. The ratio averaged – 19 – over all bands in this range is 2.3. For larger distances, the ratio at all bands is consistent with one. If the brightness asymmetry between the SW and NE sides is interpreted as being the result of forward scattering, it implies that the SW is oriented towards the Earth. Assuming that the scattering phase function can be parameterized as a Henyey-Greenstein function (Henyey & Greenstein 1941) the average value of 2.3 for the SBPs ratios implies a scattering asymmetry factor g ≈ 0.15. A ratio of 4 would imply g ≈ 0.23. For comparison, ISM grains have g≈ 0.4 to 0.6 in the optical (Weingartner & Draine 2001). In other words, the grains around HD 100546 scatter light more isotropically than grains in the ISM. This determination of g is not strictly correct as the procedure is only valid for optically thin dust distributions. 3.4.3. Power-law Fits In any direction, the SBPs are considerably more complex than any previously published. For the purposes of comparison with previous works, we derive the power-law fits (see table 5) for the SE Major axis. The profile can be broken in four parts: 1.”6 to 5”, 5” to 8”, and 8” to 12”. To do the fit, we interpolated over the signature of structure 3. The results of the fit for F435W are shown in figure 11. While it seems clear that the profile to 5” can be described by a power-law, such de- scription is less warranted beyond this limit. In particular, there seems to be a transition region from 5” to 8”, before a flatter edge at longer distances. Within the errors, the fits are the same across wavelengths. Power-law fits to SBPs were derived by Grady et al. (2001) based on STIS observations, using a different PA for the disk, different width for the evaluation of the SBP, and different distance ranges for the power-laws. In table 5 we also derive the power laws using the Grady et al. (2001) ranges and tracks, but based on our data. For the exponents, they derive −3.1± 0.1 between 1” and 2.”7, −3.1± 0.1 between 2.”7 and 5”, and −2.2± 0.2 between 5” and 8”. Within the errors, their results are consistent with ours except in the range between 2.”7 and 5”, were our slopes are significantly steeper. The origin of this discrepancy cannot be ascribed to the differing observation bands, as the power-law indices derived in all our filters are alike within the errors. From NICMOS F160W data, Augereau et al. (2001) derive −2.92 ± 0.04 between 0.”5 and 2.”5 and -5.5±0.2 from 2.”7 to 3.”7. Between 2.”7 to 5.”0, our measured index is between those of STIS and NICMOS. The difference with NICMOS may be due to the differing optical depths of the material at different wavelengths. Beyond 5”, Grady et al. (2001) fitted a power-law index of -2.2 along the major axis – 20 – of the disk. Based on this fit, and the models from Whitney & Hartmann (1993), they concluded that they had detected an infalling envelope in the star. The applicability of the models by Whitney & Hartmann (1993) presumes that the material is optically thick beyond 500 AU from the star. By using the same position angle and strip size to measure the major axis, we can roughly reproduce the power-law dependence from Grady et al. (2001), within the errors (See table 5). However, at large distances from the star, the material will be optically thin along the line of sight. The index of the power-law then reflects the surface density of scatterers, and not the shape of the scattering surface. As such, the use of the models developed by Whitney & Hartmann (1993) to explain the large scale power-law is unwarranted. 3.5. Disk Color The SBPs show that along the principal axes of the disk the circumstellar material is brightest in F814W than in the other bands. Furthermore along the major axis, ∆(F435W− F814W ) increases from 0.3 at 2”, to ≈ 1 at 5”, indicating that the dust is between ≈ 1.3 and 2.5 times brighter in the F814W band than in F435W. After 8” there is a sharp decrease in the value of ∆(F435W − F814W ). The noise precludes an accurate determination of the value, but at least from 8” to 12” it seems positive or neutral. For the other color, ∆(F435W − F606W ) ≈ 0.0–0.2. Typical errors in these colors are 0.1 mags. A succinct description of the colors along the semi-minor axes is more difficult, as the errors are larger. Both colors are strongly affected by the scattering strip which produces a peak of emission at 3” in the NE, and 3” and 3.”7 at the SW. Outside these regions, ∆(F435W − F814W ) ≈ 0.3–0.5 mags, with indications of a decrease beyond 5” along the SW side. The SBPs sample just narrow strips of all the data, and specially in regards to the colors may not give the full picture. Figures 12 and 13 show the ∆(F435W − F606W ) and ∆(F435W −F814W ) colors of the disk relative to the star. We show the colors both before and after deconvolution. In both figures, the data have been convolved with a 4-pixel FWHM gaussian. Formally, the deconvolved image is the correct one to explore color information, but it is marred by streaks resulting from incorrect PSF matching, amplified by the Lucy- Richardson procedure. The effects of the scattering strip are manifested as two clumps both at the NE and SW sides of the star. As we mentioned before, the ∆(F435W − F814W ) color is ∼0.12 mags less beyond 2.”5 in the deconvolved data. These color maps make evident that the star is surrounded by an elongated nebulosity – 21 – 8” in size (measured along the semi-major axes). Beyond this limit there is an abrupt change in the characteristics of the dust, as only a weak color signal is detected (however, nebulosity is detected beyond this limit). Within the mid-disk, ∆(F435W − F814W ) increases from the .0.5 at the SW side, to &0.5 at the NE side, but it is mostly positive. If we assume that the SW side is closest to the Earth, this suggests a decrease in the value of the scattering factor g as a function of increasing wavelength. For ∆(F435W − F606W ) the situation is less clean-cut, although overall it is also true that the color index is larger along the NE than along the SW side. Comparing figures 9 and 12 indicates that the NE semi-minor axis is oriented along a particularly blue direction. In section 3.3 we noted that the arms have more neutral colors than the neighboring material. The effect is clearly seen in figure 13, on which the arms can be roughly traced by the darker color. Along the NW side of the disk, ∆(F435W − F814W ) ∼ 0.1 mags within 3”, on the region occupied by structure 1. In the case of structures 2 and 3, the disk becomes neutral at the center of the features. (Along the smoothed SBPs the effect is only evident along the broad structure 2. See figure 9.) The space within the arms has a non-neutral color in ∆(F435W − F814W ). 4. Discussion As we have shown, these multicolor observations reveal a wealth of features in the cir- cumstellar environment of HD 100546. Any model of this system should be able to reproduce the following characteristics: • The circumstellar environment can be divided in three regions, with different brightness and color characteristics: the inner disk (from 1.”6 to 3”), the mid-disk (from 3” to 8”) and the extended envelope (beyond 8”). For all distances and at all angles, the disk is intrinsically brighter in the F814W band than in the other bands. • The inner disk is the region of the spiral arms. In addition to the well-known spiral structure, there are smaller arms never before reported. If we assume that the disk rotates counterclockwise the main arms trail the direction of disk rotation. Colors of the arms are more neutral than those of the surrounding material. The space between them becomes brighter as one goes to longer wavelengths, even after deconvolving by the coronagraphic PSF. The morphology does not change as a function of wavelength. • The mid-disk is brighter and redder to the SW than to the NE. If the larger brightness is interpreted as the result of scattering asymmetry, it implies that the SW side is oriented – 22 – towards the Earth. A description with a single value scattering asymmetry parameter implies g ∼ 0.15, although there are local variations suggesting values as large as g ∼ 0.2 The mid-disk has ∆(F435W −F814W ) ∼ 0.5 and ∆(F435W −F606W ) ∼ 0.1 although strong local variations are present. Comparison with NICMOS observations indicates that ∆(F435W − F160W ) = 1.5 at 2.”5. The SW-NE color asymmetry suggests that g decreases with increasing wavelength. • An abrupt change of color and brightness marks the end of the mid-disk at 8” along the major axis. Farther out, we detect nebulosity to 12”–14”. Overall, the shape is roughly consistent with what is expected from a flattened envelope with the same inclination as the inner disk. • Power-law fits to the SBPs produce indices steeper than −3 to 8”. This dependence has also been observed in NICMOS and STIS data. A detailed model of this system is beyond the scope of this paper, but in the following sections we explore possible interpretations of these observations that may serve as a basis for such a model. We argue that the behavior of the SBPs with distance suggest that we are not detecting the optically thick disk responsible for the far-IR emission in the SED but only the optically thin envelope. The spiral arms are then structures in this envelope. Overall, the explanation for the colors of the dust remain a mystery, but we show that they are similar to colors of Kuiper Belt objects in the solar system. This suggests the processes like cosmic ray irradiation may be important to understand the dust color in HD 100546. 4.1. The Color of the Circumstellar Material Given the large fractional infrared excess (Ldisk/L∗ = 0.51), we expect some part of the circumstellar material to be optically thick at visible wavelengths. This is not inconsistent with the very low relative scattered fluxes, as the amount of light scattered by an optically thick disk is controlled by the disk flaring angle. Neglecting the wavelength dependence of the scattering phase function, one expects that a situation in which every light path reaching the observer scatters off an optically thick structure to result in gray scattering. This is because interstellar dust albedo is approximately constant with wavelength in the optical (Whitney 1995). Therefore, the colors of the circumstellar material, especially those of material within ≈3”, which has been modeled as a flared, thick disk, are somewhat unexpected. On the other hand, optically thin lines of sight should result in blue scattering, if the dust around HD 100546 has the same optical constants and size distribution (with dust particles ranging in size from tens of Angstroms to a few microns) of the diffuse ISM (Draine 2003). – 23 – By analyzing the extinction as a function of distance for B-stars near the line of sight to HD 100546, Malfait et al. (1998) concluded that the reddening of the latter is consistent with the ISM column density at 100 pc. This means that the surrounding nebulosity does not contribute significantly to the reddening and the star is in front of the observed structures (or that the optical depth of circumstellar material in front of the star is τV . 0.05, from the error of the extinction measurements). In principle, some of the nebulosity could be associated with the nearby Lynds dark cloud DC 296.2-7.9, whose nominal center is only ∼25’ away from the star. The ISM dust associated with the dark cloud should scatter more strongly in the blue than in the red. From figure 8, the brightness in F814W along the major disk axis would have to be 60% smaller (or the brightness in F435W 60% larger) for the scattering to be gray (using 0.5 mags as F435W-F814W color). This is much larger than the uncertainties in the measured stellar colors (Section 2.3). This red color is present even before the Lucy-Richardson deconvolution procedure, the effect of which is to reduce the F435W-F814W color at large angular distances. The disk is also red also in the F435W-F606W color, although not as strongly. Multicolor, resolved, scattered-light observations of disks around Herbig Ae/Be stars, that could be used to compare with our observations of HD 100546 are not very common. There are 14 Herbig Ae/Be stars for which the disk has been resolved3. Of those, six or seven (in addition to HD 100546 these are: AB Aur, Grady et al. 1999; HD 163296, Grady et al. 2000; HD 150193A, Fukagawa et al. 2003; PDS144 N, Perrin et al. 2006; HD 142527, Fukagawa et al. 2006; and perhaps HD 169142, Hales et al. 2006) have been resolved in scattered light. Be- sides HD 100546, HD 142527 (F6, 2 Myr-old) and PDS144 N (∼A2, age uncertain) have observations in multiple bands – in the near-IR – performed by the same group, ensuring that the photometric calibration is consistent. In the case of HD 142527 those observations reveal a gray disk. PDS144 N is an edge-on system, and so the disk appears as a dark gray lane. Observations of T Tauri stars are more common, and for the most part reveal gray disks (e.g. TW Hya, Weinberger et al. 2002). An interesting exception is GG Tau, which has a red, optically thick circumbinary disk (Krist et al. 2005b). The large optical color of the disk relative to the star (∆(V −Ic) = 0.8, Krist, Stapelfeldt, & Watson 2002) has been attributed to a combination of reddened illumination by the central starlight upon passing through in- ner circumstellar disks and the scattering characteristics of the dust (Duchêne et al. 2004). Observations of edge-on T Tauri stars reveal outflow-cavity walls blue in color (relative to their illuminating stars), and whose optical characteristics can be explained by modified ISM extinction laws (Stapelfeldt et al. 2003; Watson & Stapelfeldt 2004). 3Catalog of Resolved Circumstellar Disks, C. McCabe, www.circumstellardisks.org/ – 24 – If one considers the wavelength dependence of the phase function, it is possible to pro- duce non-grey, optically thick disks. McCabe, Duchêne, & Ghez (2002) and Duchêne et al. (2004) argue that at least some fraction of the observed color of the GG Tau disk is due to a combination of the decrease of the scattering asymmetry parameter g with increasing wave- length, and the inclination of the disk. If g is large (g ≈ 0.5 in McCabe et al. 2002), this effect can produce very red disk colors relative to the star, even in optically thick disks. Figures 12 and 13 show evidence that the phase function plays a role in the color of the disk, because the dust becomes redder as the scattering angle becomes larger (assuming that the SW side is closest to Earth). However, if the angular dependence of the scattering function can be modeled by a Henyey-Greenstein function, an explanation based solely on the wavelength dependence of the phase function does not reproduce the colors. The Henyey-Greenstein function is such that if backward scattering results in ∆(F435W − F814W ) = 0.5, forward scattering should result in ∆(F435W − F814W ) = −0.5. In other words, if the back of the disk is very red, the front should be very blue. This is not what is observed in HD 100546. Here we discuss three additional possibilities to explain the color of the observed circum- stellar material: (1) reddening due to dust very close to the star, (2) the dust characteristics of an optically thin envelope, and (3) selective absorption on the surface of the grains. 4.1.1. Obscuring dust By analogy with GG Tau, we consider the possibility of obscuring material near the star which reddens the stellar flux illuminating the disk. Using the standard ISM extinction law with a normalized optical extinction value RV=3.1 (Cardelli et al. 1989), we conclude that the 60% excess in F814W with respect to F435W, requires AV = 0.65 mag to produce a gray disk. In other words, if there is a cloud of ISM material close to the star with AV = 0.65 mag, the disk would appear red although it would really be gray. Structures 1, 2, and 3 would then be slightly blue. However, the SE Major axis F435W-F606W color is at most 0.2 mags, which implies AV = 0.45 mag. The same value of extinction can reproduce both colors if the extinction law changes more slowly in the optical. If RV ≈ 12, the colors will match with AV ≈ 1.3 mag. This would be in addition to the photometrically measured stellar extinction (AV=0.28 mag). Such large value of RV has never been measured (Draine 2003). Some stellar obscuration is expected from the models developed by Dullemond et al. (2001), due to shadowing of the starlight by the inner disk rim. For the particular case of HD 100546, such a rim is quite small, which suggests that this source of optical depth is not important at large distances from what we call the inner disk. Notice, however, that we – 25 – detect red colors at all distances &1.”6, well within the ≈4” disk radius predicted by the models by Dominik et al. (2003) mentioned in the introduction. 4.1.2. A Thin Envelope We may be detecting an optically thin envelope, as suggested by Vinković et al. (2006), made up of a mixture of particles whose total opacity increases with increasing wavelength. This behavior of the opacity seems to be common of most debris disks with resolved circum- stellar material in scattered light (Krist et al. 2005a). For debris disk systems the circumstel- lar material is more evolved (there is less gas, dust grains are larger, the dust is optically thin) than in Herbig Ae/Be stars. Observations of HD 141569, a 10 Myr old A0 star surrounded by an optically thin debris disk show that the disk colors are ∆(F435W − F606W ) = 0.1 and ∆(F435W−F814W ) = 0.25 (Clampin et al. 2003) and recent observations of β Pictoris reveal similar colors along the spine of its disk (Golimowski et al. 2006). No known debris disk has optical colors as large as HD 100546. A scattering opacity increasing to the red will result if the longest wavelength of ob- servation is comparable to the lower limit in the particle distribution, even for a standard astronomical silicate. The scattering efficiency is given by Q = σ/πa2, where σ is the scatter- ing cross section and a is the particle radius. Using the smoothed astronomical silicate from Laor & Draine (1993), and integrating over the bandpass (see Eqn. 8 of Golimowski et al. 2006) we obtain the effective scattering efficiency, Qeff given the left panel of figure 14. The secondary peaks have been smoothed, as they are the result of the assumption of perfectly spherical particles in the scattering opacities calculated by Laor & Draine (1993). Qeff has a broad maximum at a particle size comparable to the wavelength of observa- tion divided by 4(nλ − 1), where nλ is the real part of the complex refraction index of the grain (Augereau & Papaloizou 2004). Notice that the Qeff from figure 14 is not the same quantity derived by Golimowski et al. (2006). In that work, the authors’ “effective scattering efficiency” includes effects due to the dependence of the optical depth on the line of sight. To compare with observations, one should integrate the effective scattering cross section over the range of particle sizes, weighted by the particle size distribution. The resulting observable, the normalized surface brightness, is given by: I/F∗ ∝ ∫ amax a2−pQefffg(θ)da where I is the observed surface brightness, F∗ is the observed stellar flux and fg(θ) – 26 – is the scattering phase function, which we assume to be a Henyey-Greenstein function. It depends on the particle size through g. We have assumed that the particle size distribution is dn/da ∝ a−p. The proportionality constant depends on the surface density of particles as a function of distance from the star. The ratio of I/F∗ between two bands is shown in figure 14 (right), where we plot the ∆(F435W − F606W ), ∆(F435W − F814W ) and the ∆(F435W − F160W ) colors. For this figure we assume that p = 3.5 (Dohnanyi 1969) and that the scattering angle is θ =90 degrees. In other words, these should be the colors along the major disk axis. If the lower limit in particle size is ∼ 0.2 µm, colors like ∆(F435W − F606W ) ≈ 0.2 mags and ∆(F435W − F814W ) ≈ 0.5 mags are possible, but the predicted ∆(F435W − F160W ) color is too small, compared to the observations. If the lower limit in particle size is ∼ 0.7 µm, ∆(F435W − F160W ) ≈ 1.5 mags, ∆(F435W − F606W ) ≈ 0.2 mags, but ∆(F435W − F814W ) is too large. Analogous results are obtained with pure graphite dust and with different grain porosities (as done in Golimowski et al. 2006). We also experimented with different particle size distributions, ranging from uniform distributions (p = 0) to single- sized particles (p = ∞). The three colors cannot be reproduced simultaneously. These arguments seek to explain the color of the scattered light as being a function of particle size, and assume that there is no selective absorption due to surface effects. The fact that the disk is not gray then implies the existence of “small” particles. For the dust model shown in the right panel of figure 14, the disk becomes neutral in the optical if the minimum size particles are &5 µm. If the system does not have any gas, the lower particle size limit is set by radiation pressure. For HD 100546, amin is between ∼5 and ∼50 µm, depending on the exact dust model used (Takeuchi & Artymowicz 2001; Li & Lunine 2003). The presence of particles smaller than the radiation pressure limit is possible in systems with even modest amounts of gas (Ardila et al. 2005). Indeed, any explanation of the richly- featured ISO spectrum near 10 µm (Malfait et al. 1998) requires the presence of particles . 2µm in size close to the star. For example, van Boekel et al. (2005) models the 10 µm feature with dust whose crystalline component is dominated by forsterite grains 0.1 µm in radius. While models of the mid-IR spectrum consider dust much closer to the star than the one described in our observations, Bouwman et al. (2003) argue that small forsterite grains (. 1µm) are present in the disk at all distances. They attribute this result to collisional stirring by an inner (∼ 10 AU) planet which expels small particles off the disk plane. The particles are then pushed out by radiation pressure tempered by gas drag. The lack of information about the gas content of the disk makes it impossible to predict the dynamics in detail. Segregation of dust by size is also possible if dust grains have grown and some have – 27 – settled in the disk mid-plane. Smaller dust particles will remain suspended at larger scale heights (e.g. Tanaka, Himeno, & Ida 2005). 4.1.3. Selective Absorption If there is selective absorption (i.e. if there are strong albedo changes in the observed wavelength range, beyond what is expected from models based on ISM-like dust) the previous analysis will be incorrect. The scattering characteristics of the dust may then be dominated by the optical constants of materials deposited on the surfaces of the grains. In the solar system, Kuiper Belt and Centaur objects (KBOs) present relative colors with respect to the sun ranging from neutral to ∆(B−I) ≈ 2.5 mags (Luu & Jewitt 1996). This trend continues into the near-IR (Delsanti et al. 2004). Optical colors like those observed in HD 100546 are not uncommon (figure 15). In this context, the F160W observation is within the range of colors sampled by the KBOs, having a normalized reflectivity (to V ) of ≈ 3. It has been argued (see Luu & Jewitt 1996; Delsanti et al. 2004 and references therein) that the diversity of colors among KBOs and Centaurs is the result of the competing pro- cesses of formation of organic compounds on the icy surfaces (due to space weathering) and resurfacing (due to mutual collisions and cometary activity). The sources of space weather- ing are UV irradiation from the sun (most important in the case of solar flares), solar cosmic rays, and Galactic cosmic rays (Gil-Hutton 2002). Independently of whether or not this is the correct explanation for the observations presented here, weathering and resurfacing pro- cesses will also be at work around HD 100546, because the water-ice sublimation boundary (T∼120 K) occurs at R ≈ 30 AU for large particles. Delsanti et al. (2004) developed a model of the reddening process in KBOs, in which the reddening timescale (τ ∼ 108±1 yrs, Shul’Man 1972) is controlled by the Galactic cosmic ray flux. In the case of β Pic, Golimowski et al. (2006) used this model (with the timescale scaled by the stellar luminosity) to conclude that the dust grains escape the system before they are significantly altered. However, this scaled model may not be applicable to β Pic or HD 100546. As they are early type stars, their astrosphere will be smaller than the sun’s (reducing the role of stellar cosmic rays) and the effect of galactic cosmic rays will be more important. Furthermore, for HD 100546 the UV flux (λ < 1500 Å) at 300 AU is ∼300 times larger than the solar UV flux at 40 AU. Recent analysis of the colors of the debris disk around HR 4796 (an A0 star, ≈ 10Myrs old) also suggest a scattering opacity similar to that of some KBOs (A. Weinberger, 2006, personal communication). In summary, space weathering is likely to play a role in our understanding of the colors of dust around A stars. However, a model appropriate for the conditions around those stars has not been developed. – 28 – 4.1.4. Summary Our observations reveal two separate, independent facts. First, the circumstellar mate- rial has a non-zero color relative to the star. This is true even within 3”, where the thermal emission has been modeled as being due to an optically thick disk. The color suggests that there is scattering asymmetry in the dust, extinction close to the star, optically thin lines of sight, or selective absorption. Second, the circumstellar material is red, even to large dis- tances of the star, suggesting that all has been processed in a similar way. As we have shown, extinction requires an anomalously large value of RV to explain both sets of colors: values of RV this large have not been observed. An optically thin disk halo composed of astronomical silicate cannot explain the measured scattering asymmetry. The colors are within the range of colors observed in KBOs, which are believed to be produced by selective absorption on the surfaces of dust gains. As in GG Tau, it is possible that more than one process is at work in the system. Beyond 3”, the role on inner disk extinction on the light scattered off the mid-disk should be relevant for our understanding of the system. The intense UV radiation coupled with the effect of Galactic cosmic rays may redden the dust before is dissipated. These ideas await better modeling efforts. Except for an explanation of the colors based only on extinction, the remaining ex- planations imply that the dust opacity increases with wavelength. The explanations based on optically thin scattering of a yet-to-be-determined dust mixture and that of a KBO-like material are not necessarily different, as the polymerized ices believed to be responsible for the KBO colors may provide the required dust characteristics. Formally, dust coated with this kind of ice mixture will be red, no matter what its optical depth. However, as we show below, there are other pieces of evidence that suggest that we are observing an optically thin dust distribution. 4.2. The Envelope Our fit to the SBPs within 2.”7 using the Grady et al. (2001) ranges (see bottom portion of table 5) is consistent with a r−3 dependence, which suggests that the structure responsible for the scattered light is a geometrically thin, optically thick disk (Whitney & Hartmann 1992). This same r−3 dependence has been observed by Grady et al. (2001) (from 1 to 2.7”) and Augereau et al. (2001) (from 0.5 to 2.5”). However, the SBP for a flat disk is only ≈ 10−7/φ2 arcsec−2, where φ is the field distance (in arcseconds) to the star (Whitney & Hartmann 1992). This is about 3 orders of magnitude less than what is observed. A flared, optically – 29 – thick disk would result in a shallower SBP. Notice that for AB Aurigae the H-band scattered- light surface brightness profiles decrease with stellar distance as r−3 (Fukagawa et al. 2004). This near-IR dependence is steeper than the r−2 dependence found by Grady et al. (1999) in the optical. On the other hand, the SBP of an optically thin disk may present any radial power-law, depending on the surface density of the scatterers and/or changes in the scattering cross section as a function of distance. If the material is optically thin along the line of sight, the surface brightness profile then implies a surface density of r−1 or r−1.8 (depending on whether one uses the Grady et al. (2001) tracks or our 0.”25 wide tracks), ignoring variations in the scattering cross section. However, it is difficult to understand how the material around HD 100546 can be optically thin within 3”, because the dust in this region is the bulk source of the IR emission. A similar power law dependence is observed from 0.”5 to 3” suggesting that the disk does not change from optically thick to optically thin within this range. It is unlikely that the transition to optically thin occurs within 0.”5 (50 AU) from the star as in this case, the spectral signature from the disk interior would appear much hotter than the ≈ 50K observed (e.g. Bouwman et al. 2003). An argument analogous to this led Vinković et al. (2006) to posit the existence of a flattened optically thin envelope or halo surrounding a geometrically thin, optically thick disk. The disk surface (the surface where the optical depth from the observer to the disk ≈ 1) has to be shallow enough that it does not intercept significant amounts of the stellar radiation, but otherwise is unconstrained. The objective of Vinković et al. (2006) was to explain the 3 µm bump mentioned in the introduction. They concluded that the envelope has constant optical depth in the visible given by τV = 0.35, although this refers only to relatively hot material responsible for the bump. In the context of this picture, our observations would only be detecting light scattered by the optically thin component. The optically thick disk is undetected in these optical observations. A remarkably similar picture has also been suggested by Meeus et al. (2001) (see their figure 8, top panel) to explain the mid-IR SED. In summary, the distance dependence of the SBPs suggest the presence of an optically thick shallow disk with a geometrically thick, optically thin, flattened envelope, similar to that advanced by Meeus et al. (2001) and Vinković et al. (2006). No thermal or scattering model with these characteristics has been published and so it is not clear if this picture may reproduce the SED and the large fractional infrared excess. A flat, optically thick disk can only produce LIR/L∗ = 0.25 (Adams & Shu 1986; Whitney & Hartmann 1992). A more realistic, flared, optically thick disk, would need to have a total opening angle of 60◦ to explain the excess. If the disk is flatter than this, the envelope would have to cover a larger – 30 – angle (as seen from the star) and/or have a large radial optical depth. Our observations show an envelope extending to 1200 AU from the star, but whether this will be enough to reproduce the SED remains uncertain. 4.3. Spiral Arms or Shadows? Various explanations have been advanced to understand the spiral structure seen in the scattered-light images. On the one hand, it is possible that they are “true” arms, due to spiral density waves or tidal effects. Quillen et al. (2005) conclude that the disk would have to be unreasonably thin for the Toomre Q parameter to be less than one, suggesting that the disk is stable to spiral-density waves. Quillen et al. (2005) also examined the possibility that the spiral structures are driven by a compact object external to the disk, either a bound planet or a passing star. They per- formed hydrodynamical simulations (i.e. a pure gas disk) of the effect of bound planets and unbound stellar encounters on a 300 AU disk and concluded that a 20 MJ mass planet/brown dwarf in an eccentric orbit would be necessary to excite the observed spiral structure, which they argued would be detectable in the NICMOS images. They also concluded that the hypothesis of an encounter with a passing star lacks suitable candidates. Could we detect such a body in our images? Figure 16 shows the 3σ detection limits for point sources around HD 100546. Because the differences in detection along the major and minor axes are small, we average them. To obtain these curves the number of counts has been transformed into instrumental vegamag magnitudes using the quantities from Sirianni et al. (2005). Such transformation ignores color terms. Let us consider a 10 MJ mass planet, 10 Myrs old. According to the models by Baraffe et al. (2003), such an object has MIC = 14.8, which implies mIC = 19.8 at 100 pc, and mF814W ≈ 19.4, in the instrumental vegamag system. The V-I color of such an object would be 5.5 mags (Baraffe et al. 2003), and so it would be undetected in the F606W band. We searched for objects that were present in the F814W images but not in the F606W ones, but did not find any. Clearly, we would have been able to detect such object in any place outside the mask. This planet would have mH ≈ 16.3 mags. Augereau et al. (2001) do not provide detection limits as a function of distance for their observations, and so it is difficult to say if it would be visible in the NICMOS images. The objects predicted by Quillen et al. (2005) to generate the structures would certainly be detectable in the images presented here. A number of researchers (e.g. Bouwman et al. 2003; Grady et al. 2005; Acke & van den Ancker 2006) have indicated that the presence of a hole in the gas and dust distribution at ≈ 10 AU – 31 – suggests the presence of a planet. In this case, the perturber is internal to the disk, unlike the simulation by Quillen et al. (2005). The effects of this configuration on HD 100546 have not been explored. Warps in the cast shadows on the outer disk that may appear as spiral arms. Quillen (2006) modeled the effect of warps on a disk that is optically thin perpendicular to the disk plane (within 80 AU) but partially optically thick along it. In this model, the dark lanes are due to opacity of the mid-plane and the bright lanes represent optically thin material partially oriented along the line of sight. The model predicts that, as the disk is observed at wavelengths in which the opacity is larger, the dark lanes will change little, or they may become darker (because the mid-plane optical depth will increase) and the bright lanes will be brighter (as the optical depth along the line of sight increases). If we assume that the dust opacity increases from F435W to F814W, the brightening of the inter-arm space contradicts the models from Quillen (2006) and is more in tune with what one expects if the structures are “real arms”: changes in the volumetric density of material separated from the inner disk by relatively empty regions, that become brighter at optically thicker wavelengths. However, as mentioned in section 4.2, we believe the spiral arms are observed in the envelope, not in the optically thick disk. They may well reflect structures in the optically thick disk, but the latter is undetected. In summary, the observations presented here are inconsistent with the published simu- lations that seek to explain the origin of the spiral arms. Whether a model can be developed that generates the observed structure with an interior planet, with a very small exterior planet, or via warps in a more realistic, larger disk, remains uncertain. 5. Conclusions We present ACS/HRC coronagraphic images of HD 100546’s circumstellar environment in the F435W, F606W, and F814W bands. The star is a B9.5, &10 Myr old, 103 pc away from the sun. The observations were performed with the 0.”9 occulting spot. For each band, we observed the system in two different telescope rolls. To improve the contrast between the star and the circumstellar material, we subtracted a PSF reference star from HD 100546. We used HD 129433, which has a similar color as the target. To scale the two, we used direct measurements in the F435W band, coupled with color measurements of HD 100546 taken from the literature. We deconvolved the images using a simulated stellar PSF, obtained with the PSF simulator Tiny Tim. Both the deconvolved and non-deconvolved images in each band are available from the Journal Web page. – 32 – Subtraction residuals dominate the dataset within ∼1.”6. Color-combined images re- veal a large scale structure of scattered light to distances ≈14” away from the star. The well-known “spiral arms” (which we call structures 1, 2, and 3) of the disk are clearly seen. Photometric analyses by other groups reveal that the star is in front to the observed nebu- losity. The scattered-light relative fluxes measured from 1.”6 to 10” in the F435W, F606W, and F814W bands are (Lcircum/L∗)λ = 1.3±0.2 ×10 −3, 1.5±0.2 ×10−3, and 2.2±0.2 ×10−3, respectively. If we interpret the circumstellar nebulosity within 2” as due to an inclined disk, the PA of the semi-major axis is ≈ 145 degrees, with an inclination of ≈ 42 degrees. We analyzed the scattered light by taking median cuts along segments 0.”25 wide, oriented along the disk’s principal axes. For these profiles, the 3σ detection limits are 2 × 10−7/arcsec2 (corresponding to VVega=23.4 magnitudes/arcsec 2). Analysis of these profiles reveal that the circumstellar material scatters more strongly towards larger wave- lengths. Along the major axis, the ∆(F435W − F606W ) color ranges from 0 to ≈ 0.2. The ∆(F435W − F814W ) varies between ≈ 0.5 and ≈1, increasing slowly with distance. Com- parison with NICMOS results show that ∆(F435W − F160W ) ≈1.5. The SW side of the nebulosity is brighter (at all distances) than the NE side, by factors between 1 and 3, which we interpret as being due to forward scattering. This suggests that the SW side is the side closest to the Earth. The brightness ratio can be explained invoking a Henyey-Greenstein scattering phase function with g = 0.15 (assuming an optically thin disk). However, there are local variations that suggests values of g as large as 0.23. Evidence of a change in g with wavelength is deduced by the fact the the SW side of the disk is bluer than the NE side. Conceptually, we divide the disk in three regions, described by their size as measured along the major axis. The inner disk (from 1.”6 to 3”) has a size comparable to the optically thick disk used to explain the system’s far-IR SED (Dominik et al. 2003). This is also the region occupied by the spiral arms. The outer edge of the mid-disk (from 3” to 8”) is marked by a factor of two decrease in brightness and by the disk color becoming almost neutral. Beyond that we detect nebulosity to ≈14” along the major axis. The SBPs are consistent with the mid-disk and the extended nebulosity having the same inclination as the inner disk. The well-known spiral arms are clearly detected in the images. They are seen as a narrow bands of nebulosity that wrap around the disk. If the SW side is closest to Earth, spectroscopic observations that spiral structure trails the rotation. There are no significant morphological differences among the images in the different bands. The space between the arms and the part of the inner disk closer to the star becomes brighter at longer wavelengths, but the peak brightness of the structures does not change much, i.e. the structures become – 33 – grayer at longer wavelengths. The brightening of the inter-arm space with wavelengths is inconsistent with models that assume that they are due to the effects of a warped disk. We deproject the disk and subtract a smoothed version of itself to increase the contrast of the sharp structures. This procedure suggest the existence of smaller, fainter arms, not easily seen in the original images. The NW arm along the major axis is split in two and the SW arm is revealed to have a companion along the NE side. The strongest arms seem to trace a four-armed spiral. Dynamical models that seek to explain the structures as due to perturbations from external planets predict the presence of objects that are not seen in the images presented here. Observations of hot gas suggest the presence of a planet at 10 AU, but detailed models of the effect of this object in the disk of HD 100546 have not been published. The cause of these structures and the mechanism for their preservation remains unknown. If one ignores the wavelength dependence of the scattering phase function, the fact that the inner disk has a color is surprising, given that the optical albedo derived for ISM grains is constant with wavelength, and that a large optical depth is predicted for this part of the disk. Most young stellar objects in a similar evolutionary stage are gray. The more evolved debris disks have mostly red colors, although not as red as those of HD 100546. If the colors were due purely to the behavior of the scattering phase function with wavelength, the disk would be bluer in the front side that what is observed. We examine the possibility that the color is due to the presence of dust between the star and the disk, beneath the coronagraphic mask. This would require different amounts of dust to explain different colors among the same direction, if the dust is ISM-like. RV would have to be ≈ 12 before we can reproduce the colors with the same amount of dust. We also explore the effect of the minimum particle size on the colors of a mixture of “astronomical silicate” grains. We conclude that a size distribution dn/da ∝ a−p with p = 3.5 is not capable of reproducing the colors. This general conclusion is also true for pure graphite grains, non-compact silicate grains, and for all values of p. In the context of an explanation for the color based on particle size, the color of the system implies the presence of particles .5µm at distances up to ∼800 AU from the star. We show that the observed colors are within the range of colors observed in KBOs in the solar system. For KBOs the colors are believed to be the result of a competition between space weathering (cosmic ray and UV flux irradiation) and resurfacing due to collisions. This is a promising scenario and deserves to be explored more, by observing the circumstellar material at other near-IR bands. Within 2”.5, the major axis SBPs has a steep r−3.8 dependence. This suggests that the structure responsible for the material is not an optically thick disk. A flat optically thick disk would result in SBPs with a r−3 dependence on distance. A flared disk would result in a – 34 – shallower power law. The distance dependence of SBPs requires invoking the presence of an optically thin halo or envelope in addition to the optically thick disk. Our observations are then consistent with the qualitative pictures from Meeus et al. (2001) and Vinković et al. (2006), in which an optically thick disk is responsible for the far-IR emission but a large scale-height, optically thin disk (with a scale-height larger than predicted by standard disk models) surrounds the optically thick component. Given its colors, the envelope cannot be composed of remnant infalling ISM material but has to have been reprocessed by the star+disk system. The optically thick disk itself is undetected, suggesting that it has a very shallow surface profile. More detailed models are required before the characteristics of this envelope are better known. The crowded field of HD 100546 suggests a way to probe the envelope material, by taking spectra of stars 3, 4, 5, 8, and 9 (Figure 5). However, for optically thin dust, the extinction in V due to the envelope is of the order of SB × 4πφ2, where SB is the normalized surface brightness at F606W, and φ is the angular distance between HD 100546 and the measurement position. This implies that at 5”, AV ≈ 0.001 mags. While this is a lower limit (some light may be blocked by the inner disk) trying to detect a small amount of extinction on the spectrum of a star 5” away from a source forty thousand times brighter (comparing star 4 to HD 100546), is a formidable obstacle for this kind of experiment. Further observations and models are required before we can fully understand this sys- tem. In particular, it is crucial to measure the cool gas content of the circumstellar material, which will determine the dynamics of the dust (Ardila et al. 2005). Additional coronagraphic observations at other NICMOS bands will provide much needed color points at longer wave- lengths, which will help decide whether or not a new kind of dust model is needed to un- derstand old protoplanetary (and/or young debris) disks. The observations presented here show how powerful multicolor coronagraphic data can be in advancing the understanding of protoplanetary disk evolution. The authors wish to thank Mario Van den Ancker and Dejan Vinković for their will- ingness to provide guidance regarding our interpretations of their work. Richard White was kind enough to provide us with the results of his PSF simulations in advance of publication, as well as extensive guidance on the use of the Lucy-Richardson algorithm. Dale Cruikshank was kind enough to illuminate the intricacies of KBO colors for us. Karl Stapefeldt pro- vided advice in frequent discussions on the astrophysics of HD 100546. ACS was developed under NASA contract NAS 5-32865, and this research has been supported by NASA grant NAG5-7697. We are grateful for an equipment grant from Sun Microsystems, Inc. The Space Telescope Science Institute is operated by AURA, Inc., under NASA contract NAS5-26555. We are also grateful to K. Anderson, J. McCann, S. Busching, A. Framarini, and T. Allen – 35 – for their invaluable contributions to the ACS project at JHU. This research has made use of the Catalog of Resolved Circumstellar Disks, the NASA’s Astrophysics Data System Bibli- ographic Services, and of the SIMBAD and Vizier databases, operated at CDS, Strasbourg, France. REFERENCES Acke, B. & van den Ancker, M. E. 2006, A&A, 449, 267 Adams, F. C. & Shu, F. H. 1986, ApJ, 308, 836 Ardila, D. R. et al. 2005, ApJ, 627, 986 Augereau, J.-C. & Beust, H. 2006, A&A, 455, 987 Augereau, J. C., Lagrange, A. M., Mouillet, D., & Ménard, F. 2001, A&A, 365, 78 Augereau, J. C. & Papaloizou, J. C. 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R. 2004, ApJ, 602, 860 Weinberger, A. J. et al. 2002, ApJ, 566, 409 Weingartner, J. C. & Draine, B. T. 2001, ApJ, 548, 296 White, R. et al. 2007, in preparation Whitney, B. A. 1995, in Revista Mexicana de Astronomia y Astrofisica Conference Series, ed. S. Lizano & J. M. Torrelles, 201 Whitney, B. A. & Hartmann, L. 1992, ApJ, 395, 529 —. 1993, ApJ, 402, 605 Wilner, D. J., Bourke, T. L., Wright, C. M., Jørgensen, J. K., van Dishoeck, E. F., & Wong, T. 2003, ApJ, 596, 597 This preprint was prepared with the AAS LATEX macros v5.2. – 40 – Table 1. Log of Coronagraphic Exposures Object Band Obs. Start Total Exp.Time P.A.a Gain (UT) (secs) (deg.) (e−/photon) HD 100546 F435W 2004-04-26 13:51:34 3 -138.31 4 2004-04-26 22:39:00 160 -138.35 2 2004-04-26 22:43:00 2400 -138.35 2 2004-04-26 23:54:00 160 -118.35 2 2004-04-26 23:58:00 2530 -118.35 2 F606W 2003-03-26 19:42:39 3 179.09 2 2003-03-26 03:53:00 130 179.05 2 2003-03-26 03:57:00 2600 179.05 2 2003-03-26 07:05:00 130 -152.95 2 2003-03-26 07:09:00 2600 -152.95 2 F814W 2003-03-26 19:42:39 6 179.09 2 2003-03-26 02:25:00 130 179.05 2 2003-03-26 02:28:00 2350 179.05 2 2003-03-26 05:33:00 160 -152.95 2 2003-03-26 05:37:00 2520 -152.95 2 HD 129433b F435W 2004-04-27 13:56:20 2 151.52 4 2004-04-27 01:32:00 16 151.48 2 2004-04-27 01:34:00 160 151.48 2 2004-04-27 01:40:00 400 151.48 2 2004-04-27 01:48:00 1300 151.48 2 F606W 2004-04-13 19:45:02 1 117.76 2 2003-03-26 09:33:00 16 117.72 2 2003-03-26 09:34:00 1020 117.72 2 F814W 2003-03-26 19:45:11 2 117.72 2 2003-03-26 09:13:00 16 117.72 2 aPosition angle of image y axis (deg. E of N) bPSF Reference star for HD 100546 Note. — Only observations made with the parameter GAIN=4 preserve the number of – 41 – counts in a saturated image. – 42 – Table 2. Photometry HD 100546 (B9.5Vne) HD 129433 (B9.5V) F435W e−/sec (107) 2.5 (3%) 6.4 (3%) Flux Density (Jy) 8.3 (3%) 21.2 (3%) BVega a 6.70 (0.03) 5.69 (0.03) F606W e−/sec (107) 4.9 (4%) 11.1 (3%) Flux Density (Jy) 7.3 (4%) 16.4 (3%) VVega a 6.69 (0.04) 5.79 (0.03) F814W e−/sec (107) 1.9 (5%) 4.1 (3%) Flux Density (Jy) 5.4 (5%) 11.5 (3%) IVega a 6.64 (0.05) 5.82 (0.03) aVegamag system. The instrumental system produces the same magnitudes within the errors. Note. — HD 129433 is the star used a PSF reference. For F435W the count rate (e−/sec) is measured from the image. The F606W and F814W count rates are estimated from the F435W values and synthetic photometry using proxy spectra. The F435W measure- ments have errors of ∼3% (Sirianni et al. 2005). For HD 100546, the errors in F606W and F814W include errors in the measured col- ors of the target star in these bands, while the errors in HD 129433 are only those of the measured photometry in F435W. – 43 – Table 3. Standard Photometry of Field Stars Number B V I 1 23.7± 0.5 22.2± 0.1 20.8± 0.1 2 > 24.5 24.2± 0.3 22.1± 0.3 3 22.1± 0.2 21.2± 0.09 20.2± 0.1 4 19.00± 0.05 18.11± 0.05 17.14± 0.05 5 19.37± 0.06 18.42± 0.03 17.44± 0.05 6 22.1± 0.2 21.8± 0.1 21.1± 0.2 7 > 24.0 23.8± 0.3 22.5± 0.3 8 23.2± 0.4 21.3± 0.1 19.00± 0.06 9 19.5± 0.06 18.60± 0.05 17.64± 0.05 Note. — Aperture photometry for field stars, ob- tained from non-deconvolved images. See figure 5 for a key to the numbers. Color transformations from the instrumental to the standard system given by Sirianni et al. (2005). The error includes uncertain- ties in the measurement of the local sky and Poisson noise. It does not include the (systematic) errors intro- duced by transformation to the standard photometric system. – 44 – Table 4. Geometric Parameters for the Disk Reference Inclination (deg from face-on) PA of Major Axis This Work 42± 5a 145± 5 Pantin et al. (2000) 50± 5 127± 5 Grady et al.(2001) 49± 4 127± 5 Augereau et al. (2001) 51± 3 161± 5 aFrom isophot fitting between 1.”6 and 2” from the star. The numbers quoted are averages for all bands. This determination ignores the angular dependence of dust scattering. – 45 – Table 5. Power-law fits to the SE semi-major axis SE Major Axis Distance F435W F606W F814W 1.”6-5.”0 −3.8± 0.1 −3.8± 0.1 −3.7± 0.1 5.”0-8.”0 −3.4± 0.3 −2.8± 0.3 −2.7± 0.3 8.”0-12.”0 −0.6± 0.4 −0.4± 0.3 −1.2± 0.5 Grady et al. (2001) ranges 1.”6-2.”7 −3.6± 0.4 −3.5± 0.4 −3.5± 0.3 2.”7-5.”0 −4.2± 0.2 −4.2± 0.2 −3.9± 0.2 5.”0-8.”0 −2.5± 0.3 −2.5± 0.3 −3.0± 0.3 Note. — The “SE Major Axis” fits are obtained along a strip 0.”25-wide, centered on the star and with PA=145 deg. For the Grady et al. (2001) ranges, PA=127 deg, each strip is 0.”46 wide and the fits are performed on the average of the SE and NW axes – 46 – Fig. 1.— Surface brightness maps of the circumstellar environment of HD 100546, in a loga- rithmic stretch. All images have been normalized to the stellar brightness in their respective bands. The approximate size of the coronagraphic mask is shown as a black circle 1” in ra- dius. The top row has a different color stretch and spatial scale than the other two, in order to showcase different regions. The top stretch goes from 10−7 arcsec−2 to 10−3 arcsec−2. The middle and bottom rows show a stretch from 10−5 arcsec−2 to 10−3 arcsec−2. The contours in the bottom row are obtained from images that have been heavily smoothed. The contour values are 2, 3, 4, 5, 6, 7, 9, 12, and 15 ×10−5 arcsec−2. – 47 – Fig. 2.— The effect of the deconvolution in the surface brightness profiles. The plot shows the median surface brightness in segments 0.”25 long, taken along the SE extension of the major axis of the disk (145 degrees East of North). Each profile has been smoothed by a median boxcar 10 pixels wide, and offset by an arbitrary amount in the y-axis. The dashed lines show the profiles before, and the solid lines show the profiles after the deconvolution. The dotted line is placed at 1.”6 from the star, the distance within which the images are dominated by subtraction residuals. Note that the effect of the deconvolution is to sharpen the features within ∼2.”5 At larger distances, the deconvolution is most important for the F814W band, resulting in a surface brightness profile reduced by ∼15% with respect to the non-deconvolved one. – 48 – Fig. 3.— PSF-deconvolved images of the circumstellar environment of HD 100546. The color and spatial scales are the same as shown in Figure 1. In order to draw the contours, the images have been smoothed like those in the bottom row of figure 1, and then convolved with a 10-pixel-FWHM gaussian (see Section 2.6). – 49 – Fig. 4.— RGB composite of the individual images. As in previous figures, the black central circle is 1” in radius. North is up and East is left. Before combining them, each image has been normalized by the stellar brightness. In order to map the number of counts to the RGB channels, we have used the algorithm described by Lupton et al. (2004). Left: Color combined image. Right: Color combination of the deconvolved images. – 50 – Fig. 5.— Guide to table 3 (Color-combined image of non-deconvolved bands). – 51 – Fig. 6.— The inner circumstellar region as seen in the F435W band. We have annotated structures 1, 2, and 3, which are observed in all bands. The scaling is the same as in the mid-panels of figure 1. If the SW side is closes to Earth, spectroscopic observations suggest that the disk is rotating counterclockwise. – 52 – Fig. 7.— Unsharp masking of the non-deconvolved image in the F435W band. Upper left: Deprojected image, with each pixel corrected by scattering asymmetry (g = 0.15), and multiplied by φ2, where φ is the angular distance to the star. Notice that the hemispheric subtraction residuals are circular in the original image, so they look elliptical here. Upper right: The same image, smoothed by a gaussian kernel with 30 pixels FWHM. Bottom left: Unsharp masking, the result of subtracting the upper-right image from the upper-left. Bottom right: Same as bottom left, with features identified. We only mark those features that are identified in all the bands. Features 1a, 2a, and 3a correspond to structures 1, 2, and 3 in figure 6. – 53 – Fig. 8.— Top: Surface brightness profiles (SBPs) of the deconvolved images divided by the stellar flux. Bottom: Disk colors relative to the star. The SBPs are calculated by taking the median flux within segments 10 by 1 pixels (0.”25 by 0.”025) along an angle 145o East of North (the PA of the disk), to 13” away from the star. In the plot, the profiles have been also smoothed using a median boxcar 10 pixels long. The dotted lines mark 1.”6, 5”, 8”, and 12”, corresponding to the circles shown in figure 10 and the ranges for the power-law fits (table 5). The errors are shown only at representative points, as filled circles. On the scale of the top panels they are of the order of the size of the circles. Subtraction residuals dominate within 1.”6. In the top right panel, the squares are STIS measurements (Grady et al. 2001) and the triangles are NICMOS measurements in the F160W band (Augereau et al. 2001). – 54 – Fig. 9.— Same as in figure 8 but centered at an angle 55o East of North. Between 2.”5 and 3.”7 in the NE direction, we see the effect of the scattering strip as an increase in the colors. Beyond 11.”75 in the NE, we reach the end of the region sampled by both rolls in F435W. Beyond 8” in the SW the profiles are affected by the large coronagraphic spots. – 55 – Fig. 10.— Annotated, deconvolved image of F435W divided by the stellar flux, to use as a guide interpreting the surface brightness profiles of figures 8 and 9. The circles indicate radii 1.”6, 5”, 8”, and 12”, corresponding to the limits of the power-law fits. The rectangular boxes are 0.”25 wide, and are oriented in the position angles extracted to produce the SBPs. The image is in a logarithmic stretch, with limits 5× 10−6 arcsec−2 to 10−3 arcsec−2. – 56 – Fig. 11.— Comparison of the profiles at both sides of the disk. The profiles of different bands have been shifted by an arbitrary amount. Blue, green, and red colors correspond to F435W, F606W, and F814W respectively. (Left) Solid: SE side; Dashed: NW side. (Right) Solid: NE side; Dashed: SW side. Also shown are the power-law fits for the F435W band in the major axis (starting closest to the star, the indices are -3.8, -3.4, and -0.6) – 57 – Fig. 12.— The ∆(F435W − F606W ) = −2.5 log(SBPF435W/SBPF606W ) color. Left: Color image obtained from non-deconvolved data. Right: Color image obtained from deconvolved data. In the NE the effect of the scattering strip is seen as a brightening of the colors. These features have weaker counterparts on the SW. Fig. 13.— Same as figure 13 but with ∆(F435W − F814W ) = −2.5 log(SBPF435W/SBPF814W ). – 58 – Fig. 14.— The color of the dust. Left: The scattering efficiency (averaged over observation band) as a function of grain size for the standard astronomical silicate. Blue (solid): F435W; green (dotted): F606W; red (dashed): F814W; black (dot-dash): F160W. Right: Expected color of the dust as a function of minimum particle size. We assume a size distribution in the form of a−p, where p=3.5. Blue: ∆(F435W − F606W ). Red: ∆(F435W − F814W ). Black: ∆(F435W − F160W ). – 59 – Fig. 15.— Colors of KBO and Centaurs (black lines), compared with the measured colors of HD 100546. The ordinate axis is the normalized reflectivity, RN = 10 0.4∆(mλ−V ), where V is the star’s magnitude (Luu & Jewitt 1996). For the HD 100546 we use the colors measured 3.5” away from the star along the SE semi-major axis, ∆(F606W − F435W ) = −0.18, ∆(F606W−F814W ) = 0.29, and ∆(F606W−F160W ) = 1.24. The colors were transformed from the instrumental to the standard vegamag system. The data for the solar system objects was taken from table 2 of Delsanti et al. (2004). We include only those objects with BVRIJH photometry. Notice that the colors of the circumstellar material are well within the range of KBO colors. – 60 – Fig. 16.— Detection limits for point sources. The curves show the 3σ detection limits for point sources in F606W (dashed) and F814W (solid) magnitudes. The curves were obtained by planting stellar sources at different distances from the star and calculating how bright would those sources have to be to be detected at the 3σ level. The noise is calculated based on the local background value. These detection limits are averages of detection limits along the same NE, SE, SW, and NW directions used to obtained the profiles from figures 8 and Introduction Observations and Processing Reduction A Note on HST's Photometric systems Stellar Photometry Subtraction of the Coronagraphic PSF Red Halo and Image Deconvolution Color-combined images Results Photometry of Field Stars General Morphology Structures in the Disk Surface Brightness Profiles Sources of Error and Comparison with Previous Measurements Morphology of the Nebulosity Power-law Fits Disk Color Discussion The Color of the Circumstellar Material Obscuring dust A Thin Envelope Selective Absorption Summary The Envelope Spiral Arms or Shadows? Conclusions

0704.1508

Anomalous diffusion and stretched exponentials in heterogeneous glass-forming liquids: Low-temperature behavior

Anomalous diffusion and stretched exponentials in heterogeneous glass-forming liquids: Low-temperature behavior J. S. Langer and Swagatam Mukhopadhyay Dept. of Physics, University of California, Santa Barbara, CA 93106-9530 (Dated: November 5, 2018) We propose a model of a heterogeneous glass forming liquid and compute the low-temperature behavior of a tagged molecule moving within it. This model exhibits stretched-exponential decay of the wavenumber-dependent, self intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities are much larger in extent than the molecular spacing, the time dependence of the scattering function crosses over from stretched- exponential decay with an index b = 1/2 at large wave numbers to normal, diffusive behavior with b = 1 at small wavenumbers. There is a clear separation between early-stage, cage-breaking β relaxation and late-stage α relaxation. The spatial representation of the scattering function exhibits an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times. I. INTRODUCTION A growing body of evidence, both experimental [1, 2, 3] and numerical [4, 5], points to the existence of intrinsic spatial heterogeneities in glass forming liquids at tem- peratures slightly above the glass transition, and the rel- evance of these heterogeneities to stretched exponential decay of correlations and anomalous, non-Gaussian diffu- sion. Here we propose a simple model of a heterogeneous glass former and use it to compute observable proper- ties of a tagged molecule moving within it. In particular, we show that this model naturally predicts stretched- exponential decay when the scale of the heterogeneity is much larger than the molecular spacing; and we compute deviations from Gaussian displacement distributions dur- ing intermediate stages of the relaxation process. Our model is loosely motivated by the excitation-chain (XC) theory of the glass transition proposed recently by one of us (JSL) [7, 8] This theory suggests that a glass- forming liquid at a temperature T not too far above the glass transition temperature T0 consists of fluctuating do- mains of linear size R∗(T ), within which the molecules are frozen in a glassy state where they have little or no mobility. R∗(T ) is a length scale that characterizes dy- namic fluctuations in the equilibrium states of these sys- tems. The theory predicts that R∗(T ) diverges near T0 like (T − T0)−1 and decreases to zero at the upper limit of the super-Arrhenius region, T = TA. Characterizing the more mobile material that lies out- side the hypothetical frozen domains remains one of the deeper challenges in glass physics. In the language intro- duced by Widmer-Cooper and Harrowell [9], the mobile molecules reside in regions of high “propensity.” Those authors perform molecular dynamics simulations of a glasslike, two-dimensional, binary mixture. They average the mean-square displacement of each molecule over an isoconfigurational ensemble in which the initial positions of all the molecules are fixed, but the initial velocities are selected at random from a Boltzmann distribution. The propensity of a molecule is its mean-square displacement during a structural relaxation time. Contour maps of propensity as a function of initial molecular positions do indeed exhibit well defined domains of low and high mo- bility. Widmer-Cooper and Harrowell compare their propen- sity maps with corresponding plots of the potential en- ergy, which is taken to be indicative of the local struc- ture. Interestingly, the propensity appears to be almost entirely uncorrelated with the potential energy. In retro- spect, this lack of correlation between short-range struc- ture and mobility may not be surprising. Such a corre- lation would be expected for a slowly coarsening system consisting of domains of two structurally distinct phases, for example, a slightly supersaturated (mobile) fluid in which (frozen) crystallites are growing. In that case, the domains are very nearly but not quite in thermodynamic coexistence with each other. A true glass-forming liquid, however, is in a state of thermodynamic equilibrium. The temperatures and chemical potentials are uniform every- where, independent of propensity; and the local pair cor- relations, which ought to be determined primarily by the temperature in an equilibrium state, appear to be statis- tically uniform as well. A local quantity that does correlate with the propen- sity is the mean-square vibration amplitude of a molecule averaged over times much longer than its period of oscil- lation but much shorter than the time required for irre- versible rearrangements to occur. A large mean-square vibration amplitude – or, equivalently, a small Debye- Waller factor – implies that a molecule is participat- ing in a soft, low frequency, elastic mode. It is well known that soft modes are abundant in amorphous mate- rials near jamming transitions.[10] Moreover, it appears (for example, in Fig.4 of [10]) that the largest displace- ments in such modes lie predominantly along one dimen- sional, chainlike paths. Thus, regions of high propensity may be elastically soft, and may somehow be correlated with excitation-chain activity. Of course, local variations of elastic stiffness are ultimately structural in nature; but soft modes are collective phenomena involving many molecules, and are not easily detected by measuring near- http://arxiv.org/abs/0704.1508v2 neighbor pair correlations. In the model proposed here, glassy domains of low propensity are separated from each other by regions of higher propensity in which many of the molecules are mobile. This domain structure undergoes persistent fluc- tuations on time scales of the order of τ∗α = R ∗2/Dα, where Dα = ℓ 2/τα is the diffusion constant associ- ated with α relaxation, ℓ is approximately the average molecular spacing, and τα is the strongly temperature- dependent, super-Arrhenius, α relaxation time. In con- trast to the slow fluctuations of domain boundaries, indi- vidual molecules within high-propensity regions are ran- dom walkers with a diffusion constant DM = ℓ 2/τM , where τM is a temperature-dependent time scale – pre- sumably not super-Arrhenius – that characterizes mobile molecular displacements. To a first approximation, a tagged molecule at any given time is either frozen in a glassy domain or is mobile. If the former, then it waits a time of the order of τ∗α before it is encountered by mobile neighbors and itself starts to undergo displacements. If the latter, then it diffuses for some distance until it finds itself frozen again in a glassy domain. That distance is at least a few molecular spacings ℓ; but it seems more likely to be proportional to R∗ if the domains are large and if diffusive hopping takes place more readily within the mobile regions than across their boundaries and into the glassy domains. Therefore, we assume that the time elapsed during mobile motion is of the order of R∗2/DM = (R ∗/ℓ)2 τM . The ratio of these two time scales is τα/τM ≡ ∆. At temperatures well below TA, we expect that ∆ ≫ 1. A similar picture of diffusion in fluctuating regions of varying mobility emerges in kinetically constrained mod- els [11, 12] and in a recent model of a gel [13]. Our analysis is most closely related to the diffusion model of Chaudhuri et al [14], which has been a valuable start- ing point for the present investigation. We go beyond [14] by including a mechanism for producing late-stage stretched-exponential relaxation, and by distinguishing that mechanism from the diffusion of mobile molecules. In contrast to [14], our molecules switch back and forth between frozen and mobile behavior and thus explore the geometry of the domain structure. As a result, our model exhibits a clear separation between the slow α relaxation and the faster mobile motions that we interpret as β re- laxation. It also respects time-translational symmetry, which seems to be violated in [14]. The preceding discussion pertains just to the case of large-scale heterogeneities at temperatures only a little above T0 and well below TA. Some of the most interesting and experimentally accessible physics, however, occurs near the crossover to small-scale heterogeneity near TA. Constructing a theory of diffusion in the crossover region requires a temperature-dependent analysis, which will be described in a following paper. II. BASIC INGREDIENTS OF THE MODEL IN THE LIMIT OF LARGE-SCALE HETEROGENEITY As outlined above, there are two different mechanisms to be described probabilistically in this sytem, each op- erating on its own characteristic time scale. The separa- tion between time scales, characterized by a large value of ∆, suggests that the natural mathematical language in which to discuss this system is that of a continuous time random walk (CTRW) in which a tagged molecule al- ternates between long waits in glassy domains and faster displacements in mobile regions. In this Section, we com- pute the waiting time distributions for molecules in glassy and mobile domains, and the corresponding probability distributions for the mobile displacements. We immediately encounter a problem, however, be- cause there are more than just two characteristic time scales in this system. Analog experiments [3] and numer- ical simulations [4] typically follow the motions of parti- cles starting from their initial positions, and resolve these trajectories on length scales smaller than the interparti- cle spacing, and on time scales of the order of the shortest vibrational periods. This is a continuous range of scales. At its long-time limit, it includes the slow modes that are involved in molecular rearrangements, whose ampli- tudes may exceed the Lindemann melting criterion in the mobile regions. Thus, the cage-breaking, β-relaxation mechanism may be part of the same continuous range of time scales that includes the intra-cage vibrational mo- tions. Indeed, no sharp distinction between vibrational and cage breaking time scales seems to appear in exper- imental data. If these are not clearly distinct kinds of events, then the CTRW approximation is not valid in the short-time limit. In view of this difficulty, we choose here to consider only the β and α time scales, τM and τα respectively. By neglecting the shorter time scales, i.e. those of the order of τ0 ∼ femtoseconds - picoseconds, we imply that we are not resolving length scales much smaller than the molec- ular spacing ℓ. We then assume that the probability that a molecule has left its cage at times of order τM ≫ τ0 is determined by the probability that it has mobile neigh- bors and participates in local molecular rearrangements. At the longest time scales τα ∼ seconds (by definition, the viscous relaxation time at the glass temperature), the probability that a molecule has escaped from a large glassy domain is determined by the probability that it has been encountered by the boundary of the domain. Consider first the slowest motions, i.e. those associated with fluctuations of the domain boundaries on length scales R∗ and time scales τ∗α. Define t ∗ ≡ t/τ∗α, where t is the physical time in seconds; and let ψG(t ∗) be the normalized probability distribution for the time that a molecule spends in a glassy domain before entering a mo- bile region. Write this distribution in the form dρW (ρ) , (2.1) where ρ is the linear size of a domain in units R∗. W (ρ) is a normalized distribution over these sizes, and the re- maining factor inside the integrand is a normalized dis- tribution over t∗. The quantity ρ−2 appearing in the exponential in Eq.(2.1) is the lowest eigenvalue of the diffusion kernel for a molecule moving in a domain of size ρ. Note the similarity to the trapping models dis- cussed, for example, in [15, 16, 17]; but also note that the “trap” here is a two dimensional subspace bounding a three dimensional domain. The quantity that is fluc- tuating in a normal diffusive manner is the mean square of the distance between the tagged molecule and the do- main boundary. Because the boundary surrounds the molecule, the direction of the diffusive drift is irrelevant; any point of contact on the boundary is equivalent to any other. Therefore, it makes little difference whether the molecule is the diffuser and the boundary is the target, or – as in this case – the boundary diffuses and the molecule is the target. Strictly speaking, Eq.(2.1) is a long-time approxima- tion; at shorter times, the higher eigenmodes of the diffu- sion kernel make non-negligible contributions. However, this approximation is qualitatively adequate for our pur- poses at shorter times as well because ψG(t ∗) is well be- haved at small t∗ and, as mentioned previously, we are not trying to include the short-time behavior in this func- tion. If we assume that the distribution over values of ρ is Gaussian, W (ρ) ∝ ρ2 exp(−ρ2) in three dimensions, ∗) = 2 e−2 t∗ . (2.2) That is, we find a rudimentary but nontrivial stretched exponential of the form exp (−const.× t∗b) with b = 1/2. There is little reason to expect that W (ρ) remains Gaussian out to the large values of ρ that determine the long-time behavior of ψG(t ∗). Deviations from the Gaussian would produce different indices. For example, as the temperature increases and the glassy domains be- come smaller, their size distributions might cut off more sharply than a Gaussian. Thus, if W (ρ) ∝ exp(−ρm), with m ≥ 2, then b = m/(m + 2) → 1 as m → ∞. The problem of computing this distribution or, equivalently, values of m from first principles may eventually become solvable as we learn more about the statistical mechan- ics of glass forming liquids; but that problem is beyond the scope of this investigation. For present purposes, the important points are that a plausible distribution W (ρ) produces stretched-exponential behavior, and that the mechanism by which this happens could produce a tem- perature dependent index b. Next, consider the waiting-time distribution ψM (t for a molecule in a mobile region. This situation is qual- itatively different from that of a molecule in a glassy do- main because the distance traveled by the molecule is related to the time during which it remains mobile. As stated earlier, we assume that a mobile molecule diffuses a distance of order R∗ before reentering a glassy region, so that its residence time in the mobile region is of order τM (R ∗/ℓ)2 which, in t∗ units, is simply ∆−1. Then, for simplicity, assume an exponential waiting-time distribu- tion: ψM (t ∗) = ∆ e−∆ t . (2.3) Compared to ψG(t ∗) in Eq.(2.2), ψM (t ∗) is sharply peaked near t∗ = 0 if, as expected, ∆ is large. To com- plete the model of mobile motion, we need the conditional probability pM (r ∗, t∗) for diffusion over a scaled distance r∗ = r/R∗ in time t∗. The natural choice is pM (r ∗, t∗) = (2 π∆ t∗)3/2 2∆ t∗ , (2.4) which is a normalized, three dimensional distribution over r∗. III. CONTINUOUS-TIME RANDOM WALKS The next step is to translate these physical ingredients of the model into the language of continuous-time ran- dom walks.[19, 20] Define two different probability distri- bution functions, nG(r ∗, t∗) and nM (r ∗, t∗), for molecules starting, respectively, in glassy domains or mobile re- gions, and moving distances r∗ in times t∗. Each molec- ular trajectory consists of a sequence of transitions be- tween glassy domains and mobile regions. A single tran- sition in which a molecule starts in a glassy domain at time t∗1 and ends in a mobile region at time t 2 (without having changed its actual position) occurs with probabil- ity ψG(t 2−t∗1). Similarly, a transition in which a molecule starts in a mobile region at time t∗1 and position r 1, and ends in a glassy domain at time t∗2 and position r 2, has probability ψM (t 2 − t∗1) pM (|r∗2 − r∗1|, t∗2 − t∗1). The prob- ability that a molecule starts or arrives in a glassy do- main at time t∗1 and is still there at the final time t ∗ − t∗1), where ∗′) dt∗ = (1 + 2 t∗) e−2 t∗ . (3.1) Finally, the probability that a molecule starts or arrives in a mobile region at time t∗1 and position r 1 and is still in that region at the final time t∗ and at position r∗, is φM (t ∗ − t∗1) pM (|r∗ − r∗1|, t∗ − t∗1), where, in analogy to Eq.(3.1), φM (t ψM (t 1) dt 1 = e −∆ t∗ . (3.2) The multiple convolutions that describe each trajec- tory are, as usual, converted into products by computing Fourier-Laplace transforms of each function: ñj(k, u) = e−ik·r ∗, t∗) dt∗, (3.3) where j = G,M . For jumps starting in glassy regions, we need: ψ̃G(u) = ∗) dt∗, (3.4) φ̃G(u) = 1− ψ̃G(u) . (3.5) For the mobile regions, the spatial Fourier transform of the conditional probability distribution pM (r ∗, t∗) is p̂M (k, t ∗) = exp ; (3.6) therefore, define fM (k, u) ≡ p̂M (k, t ∗)ψM (t ∗) dt∗ u+∆(1 + k2/2) , (3.7) gM (k, u) ≡ p̂M (k, t ∗)φM (t ∗) dt u+∆(1 + k2/2) . (3.8) Putting these pieces together, and summing over in- definitely many individual jumps in each trajectory, we ñG(k, u) = φ̃G(u) + gM (k, u) ψ̃G(u) 1− ψ̃G(u) fM (k, u) NG(k, u) W (k, u) ; (3.9) ñM (k, u) = φ̃G(u) fM (k, u) + gM (k, u) 1− ψ̃G(u) fM (k, u) NM (k, u) W (k, u) ; (3.10) where NG(k, u) = 1− ψ̃G(u) (1 + k2/2) + u/∆; (3.11) NM (k, u) = 1− ψ̃G(u) + u/∆; (3.12) W (k, u) = 1− ψ̃G(u) + k2/2 + u/∆. (3.13) In both Eqs.(3.9) and (3.10), in the numerators of the first expressions, the first term corresponds to trajecto- ries that end in glassy domains, and the second to those that end in mobile regions. -6 -4 -2 0 2 FIG. 1: Intermediate scattering function Fs(k, t ∗) with PG = 0.5 and ∆ = 100, for k = 0.8, 1, 1.2, 1.5, 2, 3, 5, 10, reading from top to bottom. IV. INTERMEDIATE SCATTERING FUNCTION The conventional way to study the diffusion mecha- nisms discussed here is to measure the self intermediate scattering function Fs(k, t), which, in the present no- tation, is the mixed Fourier-time representation of the weighted average of nG and nM . That is, Fs(k, t ∗) = PG n̂G(k, t∗) + (1− PG) n̂M (k, t∗), (4.1) where PG is the probability that a molecule starts its motion in a frozen, glassy environment; and n̂j(k, t ∫ +i∞ ñj(k, u), j = G, M. (4.2) The inversions of Laplace-transforms in Eq.(4.2) are nontrivial because the stretched-exponential function ψ̃G(u), defined in Eq.(3.4), has an essential singularity at u = 0. For t∗ > 0, the integrations over u in Eq.(4.2) must be performed by closing the contour in the negative half u-plane, which requires that ñj(k, u) be analytically continued to points u = |u| exp(iθ). To evaluate ψ̃G(u) at such points, using Eq.(3.4), let t∗ = y2 and then ro- tate the y contour away from the positive real axis to a line y = ξ exp(−iθ/2), 0 < ξ < ∞ so that the inte- grand always decreases rapidly at infinity. For example, for u→ −w ± iǫ, θ = ±π, ψ̃G(−w ± iǫ) = −4 ξ dξ e−wξ 2±2iξ e−1/w ≡ A(w) ∓ i B(w), (4.3) where Erfi is the imaginary error function. The imagi- nary part of ψ̃G(−w ± iǫ) is the discontinuity across a cut along the negative u axis. With this formula, it is straightforward to compute the discontinuity across the cut for the ñj(k, u) in Eqs.(3.9) and (3.10), and in this way to compute the scattering functions by closing the contour around this cut. That is: n̂j(k, t e−w t Nj(k,−w + i0) W (k,−w + i0) , j = G, M. (4.4) The integrands are well behaved as w → 0+, therefore it is possible to let the lower limit of integration be w = 0. The graphs in Fig.1 show Fs(k, t ∗), computed nu- merically from Eq.(4.4), as a function of log10(t ∗), for ∆ = 100, PG = 0.5, and for a sequence of wavenumbers k. These are low-temperature scattering functions, where ∆ ≫ 1, which means that molecules spend much longer times frozen in glassy domains than they do when moving in mobile regions. The choice PG = 0.5 is made primar- ily for clarity; but it seems likely that this value of PG describes a system that is well into its non-Arrhenius, anomalous-diffusion regime. The set of curves in Fig.1 closely corresponds to those shown, for example, in [6]. Howewver, as explained in Sec.II, we do not claim to resolve the small-time behavior accurately. For the larger values of k shown in Fig.1, the scat- tering functions exhibit two-stage relaxation. The first drop, which we identify as the β relaxation, occurs be- cause the initially mobile molecules are diffusing beyond their cages. Then, after much longer times indicated by the plateaus, the scattering curves cross over to typical α behavior in which all molecules, independent of whether or not they were initially mobile, make slow transitions back and forth between mobile and frozen states. This two-step relaxation disappears at small k, where the scat- tering functions are averages over distances larger than R∗ and exhibit normal, diffusive behavior. The most interesting feature of these scattering func- tions is that they exhibit a continuous range of stretched exponential relaxation modes. To see this, we next look at the long-time behavior of n̂G(k, t ∗) for relatively large values of t∗ in the range 1 < t∗ < 1000, and for a sequence of values of k. In Fig.2, we plot − log10 [− log10 n̂G(k, t∗)] as a function of log10(t ∗), so that the slopes of the curves are equal to (minus) the stretched-exponential index b. The results are again for ∆ = 100. Each of the curves in Fig.2 has a constant slope over about two decades in t∗, indicating apparently well de- fined values of b in n̂G ∝ exp (−t∗b). In the limit of large k, n̂G(k, t ∗) is indistinguishable from the glassy waiting time distribution φG(t ∗) defined in Eq.(3.1) and shown by the red curve in the figure; and b is accurately equal to 1/2 for large t∗. b increases toward unity as k decreases toward values of order unity or smaller. For any nonzero k, these curves also exhibit a crossover from b > 1/2 at small t∗ to b = 1/2 at large t∗. -1 0 1 2 3 FIG. 2: (Color online) Intermediate scattering function n̂G(k, t ∗) for k = 0.7, 1, 1.5, 5, 10, reading from top to bot- tom. The red curve is the glassy waiting time distribution ∗), which in this plot is indistinguishable from the large- k limit of n̂G(k, t To see this behavior analytically, we can deduce from Eq.(4.3) that A(w) ≈ 1+ 3w/2 for the very small values of w that are relevant at very large t∗. Therefore, for large ∆, W (k,−w + i0) ≈ k + i B(w); (4.5) and, because B(w) ≪ 1 for w ≪ 1, the integrand in Eq.(4.4) is sharply peaked at w = k2/3 so long as k2 ≪ 1. For t∗ ≫ 1, the integrand in Eq.(4.4) has another sharp peak at the maximum of the function exp (−w t∗)B(w), i.e. at w = w∗ = 1/ t∗. If k2/3 < 1/ t∗ ≪ 1, then the diffusive peak is dominant, and n̂G ∼ exp (−k2 t∗/3). On the other hand, if 1/ t∗ < k2/3 ≪ 1, then the anoma- lous peak at w∗ is dominant, and a saddle-point esti- mate yields n̂G ∼ 2 t∗ exp (−2 t∗) as expected from Eq.(3.1). Thus, at any fixed large time t∗, n̂G becomes purely diffusive in the limit of small k. Conversely, at fixed small k, the diffusion becomes anomalous at large enough t∗. A complementary set of behaviors is illustrated in Fig.3, which is the analog of Fig.2 for the initially mobile molecules. That is, Fig.3 shows − log10 [− log10 n̂M (k, t∗)] as a function of log10(t∗), but for a broader range of times extending in two cases down to t∗ ∼ 10−3. At large times, the preceding analysis remains valid; for large k, n̂M ∼ φG(t∗) ∼ t∗ exp (−2 t∗). As may be expected, however, and consistent with the behavior seen in Fig.1, n̂M decays diffusively at small times (with slope −1 in this graph), -3 -2 -1 0 1 2 3 FIG. 3: (Color online) Intermediate scattering function n̂M (k, t ∗) for k = 0.7, 1, 2, 5, reading from top to bottom. The red curve is the glassy waiting time distribution φG(t and then enters a flat plateau before crossing over to the long-time anomalous behavior characteristic of both of these scattering functions. For large enough k and/or ∆, the early-time diffusive behavior can be deduced analyti- cally. In this case, the relevant values of w are large, and A(w) ≈ −2/w becomes negligibly small. Therefore, W (k,−w + i0) ≈ 1 + + i B(w), (4.6) where, for large w, B(w) ≈ 2 π/w3/2 again becomes small. Now the integrand has a sharp peak at w = ∆(1+ k2/2). This peak dominates the integrand at large w for n̂M , but its amplitude is smaller by a factor ∆ −1 for n̂G. Thus, for small t ∗, n̂M ∼ exp [−(1 + k2/2)∆ t∗] is normally diffusive, while no such behavior occurs in n̂G. V. SPATIAL DISTRIBUTION FUNCTIONS Yet another view of the normal and anomalous dif- fusive behaviors is obtained by looking at the spatial distribution functions themselves, that is, by comput- ing nG(r ∗, t∗) and nM (r ∗, t∗). The Fourier transforms in Eqs.(3.9) and (3.10) can be inverted analytically, yielding ∗, t∗) = φG(t ∗) δ(r∗)− 1 4 π r∗ ∗, t∗), (5.1) where ∗, t∗) = ∫ +i∞ 2π iu ψ̃G(u)κ(u) e −κ(u) r∗ , (5.2) κ(u) = 1− ψ̃G(u) + u/∆ , Reκ ≥ 0. (5.3) 0 2 4 6 8 10 Displacement x* FIG. 4: Weighted spatial distribution function F̄s(x ∗, t∗) as a function of the scaled displacement x∗ for scaled times t∗ = 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, reading from left to right along the bottom of the graph. Similarly, nM (r ∗, t∗) = − 4 π r∗ ΓM (r ∗, t∗), (5.4) where ΓM (r ∗, t∗) = ∫ +i∞ 2π iu κ(u) e−κ(u) r . (5.5) It is convenient to project these distributions onto, say, the x∗ axis, that is, to integrate out the perpendicular directions. The formulas analogous to Eqs.(5.1) and (5.4) for the projected distributions, denoted n̄G(x ∗, t∗) and n̄M (x ∗, t∗), are n̄G(x ∗, t∗) = φG(t ∗) δ(x∗) + ∗, t∗), (5.6) n̄M (x ∗, t∗) = ΓM (x ∗, t∗), (5.7) where the functions ΓG and ΓM are the same as those given in Eqs.(5.2) and (5.5) with r∗ replaced by x∗. The δ functions in nG(r ∗, t∗) and n̄G(x ∗, t∗) are the results of our having neglected the intra-cage motion dis- cussed earlier. In this approximation, with spatial res- olution only of the order of ℓ, molecules in frozen do- mains remain exactly at their initial positions with de- caying probability φG(t ∗). The distributions nM (r ∗, t∗) and n̄M (x ∗, t∗) have no δ-function contributions because diffusion starts immediately in mobile regions. In graphs of experimental or computational data, these δ functions are visible as narrow Gaussian displacement distributions centered at x∗ = 0, whose root-mean-square widths, say ℓ∗C = ℓC/R ∗, are the average intra-cage displacements that we are neglecting here. As a first look at the spatial distribution functions, in Fig.4, we show log10[F̄s(x ∗, t∗)] as a function of the displacement x∗ for a sequence of times t∗. In analogy to Eq.(4.1), we write F̄s(x ∗, t∗) = PG n̄G(x∗, t∗) + (1−PG) n̄M (x∗, t∗), (5.8) For plotting these graphs, and for this purpose only, we replace the δ function in Eq.(5.6) by a normalized, Gaus- sian, intra-cage distribution δ(x∗) → p̄C(x∗) = (2 π ℓ∗C 2)1/2 2 ℓ∗C . (5.9) We make this replacement primarily because the result- ing graphs look – and indeed are – more realistic this way; but we emphasize that this is not a systematic cor- rection of the small-time behavior because the intra-cage fluctuations are not otherwise included in n̄G(x ∗, t∗) or n̄M (x ∗, t∗). Again, we choose ∆ = 100 and PG = 0.5; and, so that the small-x∗, early-time behavior be visible in the figure, a relatively large but possibly realistic value of ℓ∗C = 0.1. Because of the factor φG(t ∗) in Eq.(5.6), this central peak in n̄G(x ∗, t∗) disappears after times of the order of τ∗α. At small times t∗, and for sufficiently large displace- ments x∗, the graphs in Fig.4 exhibit the broad exponen- tial tails reported elsewhere in the literature (see [3, 14]). The exponential tail is a robust mathematical feature of this class of models, closely associated with the decou- pling of the mobile and glassy behaviors, and not depen- dent on any details of the glassy waiting-time distribution ∗). The degree of decoupling is reflected here by the magnitude of the parameter ∆, which must be large for strong decoupling. The only requirement on ψG(t ∗) is that it decays rapidly enough that ψ̃G(u) becomes van- ishingly small at large u. To see how the exponential function emerges, evaluate the u-integrations in Eqs.(5.2) and (5.5) by first integrat- ing around a circle of radius u0 centered at the origin, and then closing the contour around the cut on the neg- ative u-axis from u = −u0 to u → −∞. (The curves in Fig.4 were computed numerically with u0 = 2.) If ∆ ≫ 1, we can choose 1 ≪ u0 ≪ ∆, so that ψ̃G(u) ≈ 0 and κ(u) ≈ 2 everywhere around the circle. Then, for small t∗, the integration around the circle yields ∗, t∗) ≈ ΓM (x∗, t∗) ≈ exp (− 2 x∗). This limiting behavior is most apparent in Fig.4 for the two earliest times, t∗ = 0.03 and 0.1, where the slope has the pre- dicted value of − At larger times, and in the limit of indefinitely large ∆, the preceding argument implies that all of these curves approach the same slope at large x∗. On the other hand, at very early times t∗ ≤ ∆−1, the integration is domi- nated by the discontinuity across the cut near u = −∆, and the displacement distribution is dominated by the early diffusive motion of the initially mobile molecules. Another interesting case is the limit of large t∗ and x∗ ≪ t∗. The integrands in Eqs.(5.2) and (5.5) vanish like exp (−1/w) in the limit−u = w → 0+. Therefore, we can use the long-time, small-w approximation, A(w) ≈ 1 + 3w/2, to estimate ∗, t∗) ≈ ΓM (x∗, t∗) ∝ exp ; (5.10) thus the spatial distribution reverts to a Gaussian. Eq.(5.10) fits the curve in Fig.4 for t∗ = 10 reasonably well for x∗ < 5. VI. MOMENTS OF THE SPATIAL DISTRIBUTIONS Another way of looking at these spatial distributions is to compute their time-dependent moments 〈r∗k(t∗)〉. We use the identities 〈r∗2(t∗)〉 = −3 ∂2Fs(k, t , (6.1) 〈r∗4(t∗)〉 = 15 ∂4Fs(k, t . (6.2) When inverting the Laplace transforms as in Eq.(4.2), we again close the contour of integration in the negative u plane, but in this case it is mathematically essential to include the circle around the origin in order to account for the singularities that occur there. The mathematical situation becomes clear by inspection of the formula for 〈r∗2(t∗)〉M : 〈r∗2(t∗)〉M = 6 ∫ +i∞ κ2(u) ∫ +i∞ 1− ψ̃G(u) + u/∆ . (6.3) Because ψG(u) ≈ 1−3 u/2 for small u, this integrand has a first-order pole at the origin. Higher-order moments have higher-order poles. We can use the analogs of Eq.(6.3) for higher moments and for both the nG and nM distributions to obtain ex- act asymptotic results in the limits of very small and very large t∗. For vanishingly small values of t∗, close the contour around a circle at large values of u where ψG(u) ≈ 2/u. In this limit, the integrand behaves like exp (u t∗)/u2, and 〈r∗2〉M ≈ 3∆ t∗. (6.4) Similar small-t∗ calculations yield: 〈r∗4〉M ≈ 15∆2 t∗2; (6.5) -6 -5 -4 -3 -2 -1 0 1 2 FIG. 5: (Color online) log (t∗)〉F ] as a function of (t∗) for PG = 0, 0.5, 0.75, 0.9, 0.97, 0.999, and 1.0 (red curve), reading from top to bottom. 〈r∗2〉G ≈ 3∆ t∗2; (6.6) 〈r∗4〉G ≈ 10∆2 t∗3. (6.7) For very large t∗, close the contour in Eq.(6.3) on a vanishingly small circle around the origin and use ψG(u) ≈ 1−3 u/2. The result is that, after times so long that the tagged molecule no longer remembers whether it started in a mobile or a glassy region, both 〈r∗2〉M and 〈r∗2〉G converge to the same slowly diffusing Gaussian distribution for which 〈r∗2〉 ≈ 2 t∗ (with corrections of the order of 1/∆). Note several features of these results. As expected, the initially mobile molecules exhibit rapid (large ∆) Gaus- sian diffusion at small times. On the other hand, the displacements of the initially frozen molecules are non- Gaussian and “pseudo-ballistic” with 〈r∗2〉G ∼ ∆ t∗2. This behavior has nothing to do with early-stage bal- listic motion of molecules within their cages but, rather, is an intrinsic feature of intermediate-stage, anomalous diffusion in this model. We turn finally to the full time dependence of the mo- ments. The results for the weighted average, 〈r∗2(t∗)〉F = PG 〈r∗2(t∗)〉G+(1−PG) 〈r∗2(t∗)〉M , (6.8) are shown in Fig.5 for a series of different values of PG, as functions of log10(t ∗). At very short times, these mo- ments rise as functions of t∗ according to the estimates in Eqs.(6.4) - (6.7). The pseudo-ballistic behavior is ap- parent only for PG ≈ 1 because, for smaller PG, these weighted moments are dominated by the displacements of the initially mobile molecules. After the initial rise, at roughly t∗ ≈ ∆−1 = 10−2, the moments cross over from -3 -2 -1 0 1 2 FIG. 6: Non-Gaussian parameter α2 as a function of log10(t for PG = 0. fast β relaxation to slow α decay as the tagged molecules are repeatedly trapped and then escape from glassy do- mains. Finally, in Fig.6, we show the non-Gaussian pa- rameter 3 〈r∗4(t∗)〉M 5 〈r∗2(t∗)〉2M − 1 (6.9) only for PG = 0 because, in this approximation where we have neglected intra-cage vibrational motions, only the mobile molecules exhibit Gaussian displacement dis- tributions at early times. Here we see explicitly that the non-Gaussian behavior occurs during the crossover from β to α relaxation. Both of these last two figures are ar- tificial in the sense that we are independently varying the parameter PG, which in fact ought to be determined uniquely by the temperature along with the parameters R∗ and ∆. VII. CONCLUDING REMARKS The model proposed here contains a relatively simple mechanism for producing stretched exponential decay of molecular correlations, as observed via the self interme- diate scattering function. That mechanism emerges di- rectly from the spatial heterogeneity of a glass-forming liquid. The model illustrates how anomalous diffusion, as exemplified by a broad exponential (non-Gaussian) tail of the molecular displacement distribution, is related to heterogeneity and – indirectly – to the accompanying stretched exponential behavior. Yet another feature of the model is the way in which it illustrates how the tran- sition from relatively fast β relaxation to slow α decay correlates with the onset of anomalous diffusion. Our analysis neglects the very short time, strongly localized, intra-cage fluctuations. The transition between the latter motions and cage-breaking events seems to us to be out- side the range of validity of the continuous-time random- walk theory used here. There are, of course, many missing ingredients, almost all of which pertain to the temperature dependence of the various parameters introduced here. One of us (JSL) plans to address those issues in a subsequent report. Acknowledgments This research was supported by U.S. Department of Energy Grant No. DE-FG03-99ER45762. [1] M.D. Ediger, Ann. Rev. Phys. Chem. 51, 99 (2000) [2] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D. El Masri, D.L’Hote, F. Ladieu, and M. Pierno, Science 310, 1797 (2005). [3] E. R. Weeks et al, Science 287, 627 (2000) [4] W. Kob et al, Phys. Rev. Lett. 79, 2827 (1997) [5] R. Yamamoto and A. Onuki, Phys. Rev. E 58, 3515 (1998) [6] W. Kob and H. C. Andersen, Phys. Rev. Lett. 73, 1376 (1994); Phys. Rev E 51, 4626 (1995); Phys. Rev E 52, 4134 (1995); W. Kob in “Slow relaxation and nonequi- librium dynamics in condensed matter,” Les Houches ses- sion LXXVII (2002) [7] J. S. Langer, Phys. Rev. E 73, 041504 (2006) [8] J. S. Langer, Phys. Rev. Lett. 97, 115704 (2006). [9] A. Widmer-Cooper and P. Harrowell, Phys. Rev. Lett. 96, 185701 (2006); J. Chem. Phys. 126, 154503 (2007). [10] L.E. Silbert, A.J. Liu, and S.R. Nagel, Phys. Rev. Lett. 95, 098301 (2005). [11] Y. Jung, J. Garrahan, and D. Chandler, Phys. Rev. E 69, 061205 (2004). [12] L. Berthier, D. Chandler, and J. Garrahan, Europhysics Lett. 69 (3), 320 (2005). [13] P. Hurtado, L. Berthier, and W. Kob, Phys. Rev. Lett. 98, 135503 (2007). [14] P. Chaudhuri, L. Berthier, and W. Kob, Phys. Rev. Lett. 99, 060604 (2007). [15] P. Grassberger and I. Procaccia, J. Chem. Phys. 77, 6281 (1982). [16] C. Monthus and J.-P. Bouchaud, J. Phys. A: Math. Gen. 29 3847 (1996). [17] J. Bendler, J. Fontanella and M. Shlesinger, Chem. Phys. 284, 311 (2002). [18] D. Kivelson, G. Tarjus, X. Zhao, and S.A. Kivelson, Phys. Rev. E 53(1) 751 (1996). [19] E. W. Montroll and M. F. Shlesinger in Studies in Sta- tistical Mechanics XI, editors J. L. Lebowitz and E. W. Montroll, North-Holland Physics Publishing (1984) [20] J.-P. Bouchaud and A. Georges, Physics Reports 195, 127 (North Holland, 1990).

0704.1509

Modeling the Solar Chromosphere

The Physics of Chromospheric Plasmas ASP Conference Series, Vol. 368, 2007 Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds. Modeling the Solar Chromosphere Mats Carlsson Institute of Theoretical Astrophysics, University of Oslo, Norway Abstract. Spectral diagnostic features formed in the solar chromosphere are few and difficult to interpret — they are neither formed in the optically thin regime nor in local thermodynamic equilibrium (LTE). To probe the state of the chromosphere, both from observations and theory, it is therefore necessary with modeling. I discuss both traditional semi-empirical modeling, numerical exper- iments illustrating important ingredients necessary for a self-consistent theoret- ical modeling of the solar chromosphere and the first results of such models. 1. Introduction My keynote talk was similar in content to a recent talk at a Sacramento Peak workshop celebrating the 70th birthday of Robert F. Stein. This written version builds to a large extent on that writeup (Carlsson 2006), but it is updated and some sections have been expanded. Before discussing models of the solar chromosphere it is worthwhile dis- cussing the very definition of the term “chromosphere”. The name comes from the Greek words “χρωµα” (color) and “σϕαιρα” (ball) alluding to the colored thin rim seen above the lunar limb at a solar eclipse. The color comes mainly from emission in the Balmer Hα line. This is thus one possible definition — the chromosphere is where this radiation originates. At an eclipse this region has a sharp lower edge, the visible limb, but a fuzzy upper end with prominences protruding into the corona. The nature of this region is difficult to deduce from eclipse observations since we see this region edge on during a very short time span and we have no way of telling whether it is homogeneous along the line of sight or very inhomogeneous in space and time. It was early clear that the emis- sion in Hα must mean an atmosphere out of radiative equilibrium — without extra heating the temperature will not be high enough to have enough hydrogen atoms excited to the lower or upper levels of the transition. Early models were constructed to explain observations in Hα and in resonance lines from other abundant elements with opacity high enough to place the formation in these regions even in center-of-disk observations (lines like the H and K resonance lines from singly ionized calcium). These early models were constructed assum- ing one dimensional plane-parallel geometry and they resulted in a temperature falling to a minimum around 4000K about 500 km above the visible surface, a temperature rise to 8000K at a height of about 2000 km and then a very rapid temperature rise to a million degree corona. These plane-parallel models have also at Center of Mathematics for Applications, University of Oslo, Norway http://arxiv.org/abs/0704.1509v1 50 Carlsson led to a common notion that there is a more or less homogeneous, plane-parallel region between these heights that is hotter than the temperature minimum. In such a picture the chromosphere may be defined as a region occupying a given height range (e.g. between 500 and 2000 km height over the visible surface) or a given temperature range. We may also use physical processes for our definition: the chromosphere is the region above the photosphere where radiative equilib- rium breaks down and hydrogen is predominantly neutral (the latter condition giving the transition to the corona). This discussion shows that there is no unique definition of the term “chromosphere”, not even in a one-dimensional, static world. It is even more difficult to agree on a definition of the “chromo- sphere” that also encompasses an inhomogeneous, dynamic atmosphere. As mentioned above, the first models of the chromosphere were constructed with a large number of free parameters to match a set of observational con- straints. Since some equations are used to restrict the number of free parameters (not all hydrodynamical variables at all points in space and time are determined empirically) we call this class of models semi-empirical models. Typically one assumes hydrostatic equilibrium and charge conservation but no energy equa- tion. The temperature as function of height is treated as a free function to be determined from observations. In the other main class of models one tries to minimize the number of free parameters by including an energy equation. Such theoretical models have been very successful in explaining radiation from stellar photospheres with only the effective temperature, acceleration of gravity and abundances as free parameters. In the chromosphere, an additional term is needed in the energy equation — e.g. energy deposition by acoustic shocks or energy input in connection with magnetic fields (e.g. currents or reconnection). It is thus clear from observations that the chromosphere is not in radiative equilibrium — there is a net radiative loss. This loss has to be balanced by an energy deposition, at least averaged over a long enough time span, if the atmo- sphere is to be in equilibrium. This is often called the problem of chromospheric “heating”. It is important to bear in mind, though, that the radiative losses may be balanced by a non-radiative energy input without an increase in the average temperature. The term “chromospheric heating” may thus be misleading since it may be interpreted as implying that the average temperature is higher than what is the case in a radiative equilibrium atmosphere. In the following we will use the term “heating” in a more general sense: a source term in the energy equation, not necessarily leading to an increased temperature. Chromospheric heating is needed not only for the quiet or average Sun but also in active regions, sunspots and in the outer atmospheres of many other stars. I will in the following mainly discuss the quiet Sun case. The outline of this paper is as follows: In Section 2 we discuss semi-empirical models of the chromosphere. In Section 3 we discuss theoretical models; first we elaborate on 1D hydrodynamical models, then we discuss the role of high frequency acoustic waves for the heating of the chromosphere and finally we describe recent attempts to model the chromosphere in 3D including the effects of magnetic fields. Modeling the Solar Chromosphere 51 2. Semi-empirical Models Semi-empirical models can be characterized by the set of observations used to constrain the model, the set of physical approximations employed and the set of free parameters to be determined. Spectral diagnostics used to constrain chromospheric models must have high enough opacity to place the formation above the photosphere. The continuum in the optical part of the spectrum is formed in the photosphere so the only hope for chromospheric diagnostics lies in strong spectral lines in this region of the spectrum. Candidates are resonance lines of dominant ionization states of abundant elements and lines from excited levels of the most abundant elements (hydrogen and helium). Most resonance lines are in the UV but the resonance lines of singly ionized calcium (Ca II), called the H and K lines, fulfill our criteria. These lines originate from the ground state of Ca II, the dominant ionization stage under solar chromospheric conditions, and the opacity is therefore given by the density directly and the optical depth is directly proportional to the column mass (i.e. to the total pressure in hydrostatic equilibrium). Also the source function has some coupling to local conditions even at quite low densities (in contrast to the strongly scattering resonance lines of neutral sodium). Other chromospheric diagnostic lines in the optical region are the hydrogen Balmer lines and the helium 1083 nm line. They all originate from highly excited levels and thus have very temperature sensitive opacity. The population of He 1083 is also set by recombination such that its diagnostic potential is very difficult to exploit. With the advent of space based observatories, the full UV spectral range was opened up. Continua shortward of the opacity edge from the ground state of neutral silicon at 152 nm are formed above the photosphere and can be used to constrain chromospheric models. Together with observations in Ly-α, such UV continuum observations were used by Vernazza et al. (1973, 1976, 1981) in their seminal series of papers on the solar chromosphere. The VAL3 paper (Vernazza et al. 1981) is one of the most cited papers in solar physics (1072 citations in ADS at the time of writing) and the abstract gives a very concise description of the models and the principles behind their construction: “The described investigation is concerned with the solution of the non-LTE optically thick transfer equations for hydrogen, carbon, and other constituents to determine semi-empirical models for six components of the quiet solar chromosphere. For a given temperature-height distribution, the solution is obtained of the equations of statistical equilibrium, radiative transfer for lines and continua, and hydrostatic equilibrium to find the ionization and excitation conditions for each atomic constituent. The emergent spectrum is calculated, and a trial and error approach is used to adjust the temperature distribution so that the emergent spectrum is in best agreement with the observed one. The relationship between semi-empirical models determined in this way and theoretical models based on radiative equilibrium is discussed by Avrett (1977). Harvard Skylab EUV observations are used to determine models for a number of quiet-sun regions.” The VAL3 models are thus characterized by them using Ly-α and UV- continuum observations for observational constraint, hydrostatic equilibrium and non-LTE statistical equilibrium in 1D as physical description and temper- ature as function of height as free function. To get a match with observed line-strengths, a depth-dependent microturbulence was also determined and a 52 Carlsson corresponding turbulent pressure was added. The number of free parameters to be determined by observations is thus large — in principle the number of depth-points per depth-dependent free function (temperature and microturbu- lence). In practice the fitting was made by trial and error and only rather smooth functions of depth were tried thus decreasing the degrees of freedom in the optimization procedure. The models have a minimum temperature around 500 km above the visible surface (optical depth unity at 500 nm), a rapid temperature rise outwards to about 6000K at 1000 km height and thereafter a gradual temperature increase to 7000K at 2000 km height with a very rapid increase from there to coronal temperatures. The Ca II lines were not used in constraining the VAL3 models and the agreement between the model representing the average quiet Sun, VAL3C, and observations of these lines was not good. An updated model with a different structure in the temperature minimum region was published in Maltby et al. (1986) (where the main emphasis was on similarly constructed semi-empirical models for sunspot atmospheres). A peculiar feature with the VAL models was a temperature plateau intro- duced between 20000 and 30000K in order to reproduce the total flux in the Lyman lines. This plateau was no longer necessary in the FAL models where the semi-empirical description of the transition region temperature rise was replaced by the balance between energy flowing down from the corona (conduction and ambipolar diffusion) and radiative losses (Fontenla et al. 1990, 1991, 1993). One goal of semi-empirical models is to obtain clues as to the non-radiative heating process. From the models it is possible to calculate the amount of non- radiative heating that is needed to sustain the model structure. For the VAL3C model this number is 4.2 kWm−2 with the dominant radiative losses in lines from Ca II and Mg II, with Ly-α taking over in the topmost part. The models described so far do not take into account the effect of the very many iron lines. This was done in modeling by Anderson & Athay (1989). Instead of using the temperature as a free parameter and observations as the constraints, they adjusted the non-radiative heating function until they obtained the same temperature structure as in the VAL3C model (arguing that they would then have an equally good fit to the observational constraints as the VAL3C model). The difference in the physical approximations is that they included line blanketing in non-LTE from millions of spectral lines. The radiation losses are dominated by Fe II, with Ca II, Mg II, and H playing important, but secondary, roles. The total non-radiative input needed to balance the radiative losses is three times higher than in the VAL3C model, 14 kWm−2. The VAL3 and FAL models show a good fit to the average (spatial and temporal) UV spectrum but fail to reproduce the strong lines from CO. These lines show very low intensities in the line center when observed close to the solar limb, the radiation temperature is as low as 3700K (Noyes & Hall 1972; Ayres & Testerman 1981; Ayres et al. 1986; Ayres & Wiedemann 1989; Ayres & Brault 1990). If the formation is in LTE this translates directly to a temperature of 3700K in layers where the inner wings of the H and K lines indicate a temperature of 4400K. The obvious solution to the problem is that the CO lines are formed in non-LTE with scattering giving a source function below the Modeling the Solar Chromosphere 53 Planck-function. Several studies have shown that this is not the solution — the CO lines are formed in LTE (e.g. Ayres & Wiedemann 1989; Uitenbroek 2000). The model M CO constructed to fit the CO-lines (Avrett 1995) give too low UV intensities. One way out is to increase the number of free parameters by abandoning the 1D, one-component, framework and construct a two component semi-empirical atmosphere. The COOLC and FLUXT atmospheric models of Ayres et al. (1986) was such an attempt where a filling factor of 7.5% of the hot flux tube atmosphere FLUXT and 92.5% of the COOLC atmosphere repro- duced both the H and K lines and the CO-lines. The UV continua, however, are overestimated by a factor of 20 (Avrett 1995). A combination of 60% of a slightly cooler model than M CO and 40% of a hot F model provides a better fit (Avrett 1995). Another way of providing enough free parameters for a better fit is to introduce an extra force in the hydrostatic equilibrium equation provid- ing additional support making possible a more extended atmosphere. With this extra free parameter it is possible to construct a 1D temperature structure with a low temperature in the right place to reproduce the near-limb observations of the CO lines and a sharp temperature increase to give enough intensity in the UV continua (Fontenla 2007). A word of caution is needed here. Semi-empirical models are often im- pressive in how well they can reproduce observations. This is, however, not a proper test of the realism of the models since the observations have been used to constrain the free parameters. The large number of free parameters (e.g., temperature as function of height, microturbulence as function of height and angle, non-gravitational forces) may hide fundamental shortcomings of the un- derlying assumptions (e.g., ionization equilibrium, lateral homogeneity, static solution). It is not obvious that the energy input required to sustain a model that reproduces time-averaged intensities is the same as the mean energy input needed in a model that reproduces the time-dependent intensities in a dynamic atmosphere. Semi-empirical modeling may give clues as to what processes may be important but we also need to study these underlying physical processes with fewer free parameters. This is the focus of theoretical models. 3. Theoretical Models In contrast to semi-empirical models theoretical models include an energy equa- tion. To model the full 3D system with all physical ingredients we know are important for chromospheric conditions is still computationally prohibitive — various approximations have to be made. In one class of modeling one tries to illustrate basic physical processes without the ambition of being realistic enough to allow detailed comparison with observations. Instead the aim is to fashion a basic physical foundation upon which to build our understanding. The other approach is to start with as much realism as can be afforded. Once the mod- els compare favourably with observations, the system is simplified in order to enable an understanding of the most important processes. I here comment on both types of approaches. 54 Carlsson 3.1. 1D radiation hydrodynamic simulations Acoustic waves were suggested to be the agent of non-radiative energy input al- ready by Biermann (1948) and Schwarzschild (1948). Such waves are inevitably excited by the turbulent motions in the convection zone and propagate outwards, transporting mechanical energy through the photospheric layers into the chromo- sphere and corona. Due to the exponential decrease of density with height, the amplitude of the waves increases and they steepen into shocks. The theory that the dissipation of shocks heats the outer atmosphere was further investigated by various authors, see reviews by Schrijver (1995); Narain & Ulmschneider (1996). In a series of papers, Carlsson & Stein (1992, 1994, 1995, 1997, 2002a) have explored the effect of acoustic waves on chromospheric structure and dynamics. The emphasis of this modeling was on a very detailed description of the radia- tive processes and on the direct comparison with observations. The full non-LTE rate equations for the most important species in the energy balance (hydrogen, helium and calcium) were included thus including the effects of non-equilibrium ionization, excitation, and radiative energy exchange on fluid motions and the effect of motion on the emitted radiation from these species. To make the cal- culations computationally tractable, the simulations were performed in 1D and magnetic fields were neglected. To enable a direct comparison with observations, acoustic waves were sent in through the bottom boundary with amplitudes and phases that matched observations of Doppler shifts in a photospheric iron line. These numerical simulations of the response of the chromosphere to acous- tic waves show that the Ca II profiles can be explained by acoustic waves close to the acoustic cut-off period of the atmosphere. The simulations of the be- haviour of the Ca IIH line reproduce the observed features to remarkable detail. The simulations show that the three minute waves are already present at pho- tospheric heights and the dominant photospheric disturbances of five minute period only play a minor modulating role (Carlsson & Stein 1997). The waves grow to large amplitude already at 0.5 Mm height and have a profound effect on the atmosphere. The simulations show that in such a dynamic situation it is misleading to construct a mean static model (Carlsson & Stein 1994, 1995). It was even questioned whether the Sun has an average temperature rise at chro- mospheric heights in non-magnetic regions (Carlsson & Stein 1995). The simu- lations also confirmed the result of Kneer (1980) that ionization/recombination timescales in hydrogen are longer than typical hydrodynamical timescales under solar chromospheric conditions. The hydrogen ionization balance is therefore out of equilibrium and depends on the previous history of the atmosphere. Since the hydrogen ionization energy is an important part of the internal energy equation, this non-equilibrium ionization balance also has a very important effect on the energetics and temperature profile of the shocks (Carlsson & Stein 1992, 2002a). Kneer (1980) formulated this result as strongly as “Unless confirmed by consis- tent dynamical calculations, chromospheric models based on the assumption of statistical steady state should be taken as rough estimates of chromospheric structure.” Are observations in other chromospheric diagnostics than the Ca II lines consistent with the above mentioned radiation-hydrodynamic simulations of the propagation of acoustic waves? The answer is “No”. The continuum observa- tions around 130 nm are well matched by the simulations (Judge et al. 2003) Modeling the Solar Chromosphere 55 but continua formed higher in the chromosphere have higher intensity in the observations than in the simulations (Carlsson & Stein 2002c). The chromo- spheric lines from neutral elements in the UV range are formed in the mid to upper chromosphere. They are in emission at all times and at all positions in the observations, and they show stronger emission than in the simulations. The failure of the simulations to reproduce diagnostics formed in the mid- dle to upper chromosphere gives us information on the energy balance of these regions. The main candidates for an explanation are the absence of magnetic fields in the simulations and the fact that the acoustic waves fed into the compu- tational domain at the bottom boundary do not include waves with frequencies above 20mHz. The reason for the latter shortcoming is that the bottom boundary is deter- mined by an observed wave-field and high frequency waves are not well deter- mined observationally. I first explore the possibility that high frequency acoustic waves may account for the increased input and address the issue of magnetic fields in the next section. 3.2. High frequency waves Observationally it is difficult to detect high frequency acoustic waves for two reasons: First, the seeing blurs the ground based observations and makes these waves hard to observe. Second, for both ground based and space based observa- tions the signal we get from high frequency waves is weakened by the width of the response function. Wunnenberg et al. (2002) have summarized the various attempts at detecting high frequency waves, and we refer to them for further background. Theoretically it is also non-trivial to determine the spectrum of generated acoustic waves from convective motions. Analytic studies indicate that there is a peak in the acoustic spectrum around periods of 50 s (Musielak et al. 1994; Fawzy et al. 2002) while results from high-resolution numerical simulations of convection indicate decreasing power as a function of frequency (Goldreich et al. 1994; Stein & Nordlund 2001). Recently, Fossum & Carlsson (2005b) & Fossum & Carlsson (2006) ana- lyzed observations from the Transition Region And Coronal Explorer (TRACE) satellite in the 1600 Å passband. Simulations were used to get the width of the response function (Fossum & Carlsson 2005a) and to calibrate the observed in- tensity fluctuations in terms of acoustic energy flux as function of frequency at the response height (about 430 km). It was found that the acoustic energy flux at 430 km is dominated by waves close to the acoustic cut-off frequency and the high frequency waves do not contribute enough to be a significant contributor to the heating of the chromosphere. Waves are detected up to 28mHz frequency, and even assuming that all the signal at higher frequencies is signal rather than noise, still gives an integrated energy flux of less than 500 Wm−2, too small by a factor of ten to account for the losses in the VAL3C model. For the field free internetwork regions used in the TRACE observations it is more appropriate to use the VAL3A model that was constructed to fit the lowest intensities observed with Skylab. It has about 2.2 times lower radiative losses than VAL3C (Avrett 1981) so there is still a major discrepancy. One should also remember that Anderson & Athay (1989) found three times higher energy requirement than in 56 Carlsson the VAL3C model when they included the radiative losses in millions of spec- tral lines, dominated by lines from Fe II. As pointed out by Fossum & Carlsson (2006), the main uncertainty in the results is the limited spatial resolution of the TRACE instrument (0.5′′ pixels corresponding to 1′′ resolution with a possible additional smearing from the little known instrument PDF): “There is possibly undetected wave power because of the limited spatial resolution of the TRACE instrument. The wavelength of a 40 mHz acoustic wave is 180 km and the hor- izontal extent may be smaller than the TRACE resolution of 700 km. Several arguments can be made as to why this effect is probably not drastic. Firstly, 5 minute waves are typically 10–20′′ in coherence, 3 minute waves 5–10′′. In both cases 3–6 times the vertical wavelength. This would correspond to close to the resolution element for a 40mHz wave. Secondly, even a point source excitation will give a spherical wave that will travel faster in the deeper parts (because of the higher temperature) and therefore the spherical wavefront will be refracted to a more planar wave. With a distance of at least 500 km from the excitation level it is hard to imagine waves of much smaller extent than that at a height of 400 km. There is likely hidden power in the subresolution scales, especially at high frequencies. Given the dominance of the low frequencies in the integrated power, the effect on the total power should be small. It is possible to quantify the missing power by making artificial observations of 3D hydrodynamical sim- ulations with different resolution. This is not trivial since the results will be dependent on how well the simuation describes the excitation of high frequency waves and their subsequent propagation. Preliminary tests in a 3D hydrody- namical simulation extending from the convection zone to the corona (Hansteen 2004) indicate that the effect of the limited spatial resolution of TRACE on the total derived acoustic power is below a factor of two. Although it is thus unlikely that there is enough hidden subresolution acoustic power to provide the heating for the chromosphere, the effect of limited spatial resolution is the major uncertainty in the determination of the shape of the acoustic spectrum at high frequencies.” Another effect that goes in the opposite direction is that the analysis as- sumes that all observed power above 5 mHz corresponds to propagating acous- tic waves. Especially at lower frequencies we will also have a signal from the temporal evolotion of small scale structures that in this analysis is mistakenly attributed to wave power. In a restrictive interpretation the result of Fossum & Carlsson (2005b) is that acoustic heating can not sustain a temperature structure like that in static, semi-empirical models of the Sun. Whether a dynamic model of the chromo- sphere can explain the observations with acoustic heating alone has to be an- swered by comparing observables from the hydrodynamic simulation with ob- servations. This was done by Wedemeyer-Böhm et al. (2007). They come to the conclusion that their dynamic model (Wedemeyer et al. 2004) is compatible with the TRACE observations (the limited spatial resolution of the TRACE instru- ment severly affects the synthetic observations) and that acoustic waves could provide enough heating of the chromosphere. The synthetic TRACE images do not take into account non-LTE effects or line opacities. It is also worth noting that the model of Wedemeyer et al. (2004) does not have an average tempera- ture rise in the chromosphere and the dominant wave power is at low frequencies close to the acoustic cut-off and not in the high frequency part of the spectrum. Modeling the Solar Chromosphere 57 Their study, however, is a major step forward — the question will have to be resolved by more realistic modeling and synthesis of observations paired with high quality observations. The results by Fossum & Carlsson also show that the neglect of high- frequency waves in the simulations by Carlsson & Stein is not an important omission. Comparing their simulations with observations shows that the agree- ment is good in the lower chromosphere (Carlsson & Stein 1997; Judge et al. 2003) but lines and continua formed above about 0.8Mm height have much too low mean intensities. This is probably also true for the Wedemeyer et al. (2004) model (since it has similar mean temperature) but this needs to be checked by proper calculations. It thus seems inevitable that the energy balance in the middle and upper chromosphere is dominated by processes related to the mag- netic field. This is consistent with the fact that the concept of a non-magnetic chromosphere is at best valid in the low chromosphere — in the middle to upper chromosphere, the magnetic fields have spread and fill the volume. Even in the photosphere, most of the area may be filled with weak fields or with stronger fields with smaller filling factor (Sanchez Almeida 2005; Trujillo Bueno et al. 2004). 3.3. Comprehensive models in 3D 3D hydrodynamic simulations of solar convection have been very successful in re- producing observations (e.g. Nordlund 1982; Stein & Nordlund 1998; Asplund et al. 2000; Vögler et al. 2005). It would be very natural to extend these simulations to chromospheric layers to study the effect of acoustic waves on the structure, dynamics and energetics of the chromosphere. This approach would then include both the excitation of the waves by the turbulent motions in the convection zone and their subsequent damping and dissipation in chromospheric shocks. For a realistic treatment there are several complications. First, the approximation of LTE that works nicely in the photosphere will overestimate the local coupling in chromospheric layers. The strong lines that dominate the radiative coupling have a source function that is dominated by scattering. Second, shock forma- tion in the chromosphere makes it necessary to have a fine grid or describe sub-grid physics with some shock capturing scheme. Third, it is important to take into account the long timescales for hydrogen ionization/recombination for the proper evaluation of the energy balance in the chromosphere (Kneer 1980; Carlsson & Stein 1992, 2002a). Skartlien (2000) addressed the first issue by extending the multi-group opac- ity scheme of Nordlund to include the effects of coherent scattering. This mod- ification made it possible to make the first consistent 3D hydrodynamic simula- tions extending from the convection zone to the chromosphere (Skartlien et al. 2000). Due to the limited spatial resolution, the emphasis was on the excitation of chromospheric wave transients by collapsing granules and not on the detailed structure and dynamics of the chromosphere. 3D hydrodynamic simulations extending into the chromosphere with higher spatial resolution were performed by Wedemeyer et al. (2004). They employed a much more schematic description of the radiation (gray radiation) and did not include the effect of scattering. This shortcoming will surely affect the amount of radiative damping the waves undergo in the photosphere. The neglect of 58 Carlsson strong lines avoids the problem of too strong coupling with the local conditions induced by the LTE approximation so in a way two shortcomings partly balance out. The chromosphere in their simulations is very dynamic and filamentary. Hot gas coexists with cool gas at all heights and the gas is in the cool state a large fraction of the time. As was the case in Carlsson & Stein (1995) they find that the average gas temperature shows very little increase with height while the radiation temperature does have a chromospheric rise similar to the VAL3C model. The temperature variations are very large, with temperatures as low as 2000K and as high as 7000K at a height of 800 km. It is likely that the approximate treatment of the radiation underestimates the amount of radiative damping thus leading to too large an amplitude. The low temperatures in the simulations allow for a large amount of CO to be present at chromospheric heights, consistent with observations. For a proper calculation of CO concentrations it is important to take into account the detailed chemistry of CO formation, including the timescales of the reactions. This was done in 1D radiation hydrodynamic models by Asensio Ramos et al. (2003) and in 2D models by Wedemeyer-Böhm et al. (2005). The dynamic formation of CO was also included in the 3D models and it was shown that CO-cooling does not play an important role for the dynamic energy balance at chromospheric heights (Wedemeyer-Böhm & Steffen 2007). Including long timescales for hydrogen ionization/recombination is non- trivial. In a 1D simulation it is still computationally feasible to treat the full non-LTE problem in an implicit scheme (avoiding the problem of stiff equations) as was shown by Carlsson & Stein. The same approach is not possible at present in 3D; the non-local coupling is too expensive to calculate. Fortunately, the hy- drogen ionization is dominated by collisional excitation to the first excited level (local process) followed by photo-ionization in the Balmer continuum. Since the radiation field in the Balmer continuum is set in the photosphere, it is possi- ble to describe the photoionization in the chromospheric problem with a fixed radiation field (thus non-local but as a given rate that does not change with the solution). This was shown to work nicely in a 1D setting by Sollum (1999). Leenaarts & Wedemeyer-Böhm (2006) implemented the rate equations in 3D but without the coupling back to the energy equation. The non-equilibrium ion- ization of hydrogen has a dramatic effect on the ionization balance of hydrogen in the chromosphere in their simulation. Magnetic fields start to dominate over the plasma somehere in the chromo- sphere. Chromospheric plasma as seen in the center Hα (e.g., Rutten (2007), De Pontieu et al. (2007)) is very clearly organized along the magnetic struc- tures. It is very likely that acoustic heating alone is not sufficient to account for the radiative losses in the chromosphere. It is thus of paramount importance to include magnetic fields in chromospheric modeling but unfortunetely the in- clusion of magnetic fields increase the level of complexity enormously. As was the case with acoustic waves, it is necessary to perform numerical experiments and modeling in simplified cases in order to fashion a basic physical founda- tion upon which to build our understanding. A number of authors have studied various magnetic wave modes and how they couple, see Bogdan et al. (2003) and Khomenko & Collados (2006) for references. Rosenthal et al. (2002) and Bogdan et al. (2003) reported on 2D simulations in various magnetic field con- Modeling the Solar Chromosphere 59 figurations in a gravitationally stratified isothermal atmosphere, assuming an adiabatic equation of state. Carlsson & Stein (2002b) and Carlsson & Bogdan (2006) reported on similar calculations in the same isothermal atmosphere but this time in 3D and also studying the effect of radiative damping of the shocks. Hasan et al. (2005) studied the dynamics of the solar magnetic network in two dimensions and Khomenko & Collados (2006) studied the propagation of waves in and close to a structure similar to a small sunspot. The picture that emerges from these studies is that waves undergo mode conversion, refraction and reflection at the height where the sound speed equals the Alfvén speed (which is typically some place in the chromosphere). The crit- ical quantity for mode conversion is the angle between the magnetic field and the k-vector: the attack angle. At angles smaller than 30 degrees much of the acoustic, fast mode from the photosphere is transmitted as an acoustic, slow mode propagating along the field lines. At larger angles, most of the energy is refracted/reflected and returns as a fast mode creating an interference pattern between the upward and downward propagating waves. When damping from shock dissipation and radiation is taken into account, the waves in the low-mid chromosphere have mostly the character of upward propagating acoustic waves and it is only close to the reflecting layer we get similar amplitudes for the up- ward propagating and refracted/reflected waves. It is clear that even simple magnetic field geometries and simple incident waves create very intricate inter- ference patterns. In the chromosphere, where the wave amplitude is expected to be large, it is crucial to include the effects of the magnetic fields to understand the structure, dynamics and energetics of the atmosphere. This is true even in areas comparably free of magnetic field (such regions may exist in the lower chromosphere). The fact that wave propagation is much affected by the magnetic field topology in the chromosphere can be used for “seismology” of the chromo- sphere. Observational clues have been obtained by McIntosch and co-workers: McIntosh et al. (2001) & McIntosh & Judge (2001) find a clear correlation be- tween observations of wave power in SOHO/SUMER observations and the mag- netic field topology as extrapolated from SOHO/MDI observations. These re- sults were extended to the finding of a direct correlation between reduced oscilla- tory power in the 2D TRACE UV continuum observations and the height of the magnetic canopy (McIntosh et al. 2003) and the authors suggest using TRACE time-series data as a diagnostic of the plasma topography and conditions in the mid-chromosphere through the signatures of the wave modes present. Such helioseismic mapping of the magnetic canopy in the solar chromosphere was performed by Finsterle et al. (2004) and in a coronal hole by McIntosh et al. (2004). The chromosphere is very inhomogeneous and dynamic. There is no simple way of inverting the above observations to a consistent picture of the chromo- spheric conditions. One will have to rely on comparisons with full 3D Radiation- Magneto-Hydrodynamic forward modeling. Steiner et al. (2007) tracked a plane- parallel, monochromatic wave propagating through a non-stationary, realistic atmosphere, from the convection-zone through the photosphere into the mag- netically dominated chromosphere. They find that a travel time analysis, like the ones mentioned above, indeed is correlated with the magnetic topography and 60 Carlsson Figure 1. Temperature structure in a 3D simulation box. The bottom plane shows the temperature at 1.5 Mm below τ500=1 ranging from 15 700K in down-flowing plumes to 16 500 in the gas flowing into the simulation domain. The next plane is in the photosphere and shows hot granules and cool inter- granular lanes. In the chromosphere the isothermal surfaces show pronounced small scall structures and corrugated shock fronts. The upper 8Mm is filled with plasma at transition region and coronal temperatures up to 1MK. that high frequency waves can be used to extract information on the magnetic canopy. Including not only 3D hydrodynamics but in addition the magnetic field, and extending the computational domain to include the corona is a daunting task. However, the development of modern codes and computational power is such that it is a task that is within reach of fulfillment. Hansteen (2004) reported on the first results from such comprehensive modeling. The 3D computational box is 16 × 8 × 12Mm in size extending 2Mm below and 10Mm above the photosphere. Radiation is treated in detail, using multi-group opacities including the effect of scattering (Skartlien 2000), conduction along field-lines is solved for implicitly and optically thin losses are included in the transition region and corona. For a snapshot of such a simulation, see Fig.1. After a relaxation phase from the initial conditions, coronal temperatures are maintained self-consistently by the injection of Pointing flux from the con- Modeling the Solar Chromosphere 61 vective buffeting of the magnetic field, much as in the seminal simulations by Gudiksen & Nordlund (2002, 2005a, 2005b). It is clear from these simulations that the presence of the magnetic field has fundamental importance for chromospheric dynamics and the propagation of waves through the chromosphere. It is also clear that magnetic fields play a role in the heating of the chromosphere (Hansteen et al. 2007). There are several hotly debated topics in chromospheric modeling today: Is the internetwork chromosphere wholly dynamic in nature or are the dynamic variations only minor perturbations on a semi-static state similar to the state in semi-empirical models (e.g., Kalkofen et al. 1999)? Is there a semi-permanent cold chromosphere (where CO lines originate) or is the CO just formed in the cool phases of a dynamic atmosphere? Is there enough chromospheric heating in high frequency waves of small enough spatial extent that they are not detected by the limited spatial resolution of TRACE? What is the role of the magnetic field (mode conversion of waves, reconection, currents, channeling of waves)? A reason for conflicting results is the incompleteness of the physical description in the modeling and the lack of details (spatial and temporal resolution) in the ob- servations. We are rapidly progressing towards the resolution of this situation. 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0704.1510

A compact star rotating at 1122 Hz and the r-mode instability

A compact star rotating at 1122 Hz and the r-mode instability Alessandro Drago1, Giuseppe Pagliara2 and Irene Parenti1 ABSTRACT We show that r-mode instabilities severely constraint the composition of a compact star rotating at a sub-millisecond period. In particular, the only vi- able astrophysical scenario for such an object, present inside the Low Mass X- ray Binary associated with the x-ray transient XTE J1739-285, is that it has a strangeness content. Since previous analysis indicate that hyperonic stars or stars containing a kaon condensate are not good candidates, the only remaining possibility is that such an object is either a strange quark star or a hybrid quark- hadron star. We also discuss under which conditions sub-millisecond pulsars are rare. Subject headings: star: neutron — star: rotation — r-mode — quark star — submillisecond pulsar — gravitational waves Very recently Kaaret et al. (2007) reported the evidence of a X-ray transient with a pulsed component of the emission having a frequency f = 1122 ± 0.3 Hz. This signal is interpreted as due to the rotation of the central neutron star. As such this object would be the most rapidly rotating compact star discovered up to now. This single observation clearly needs to be confirmed, maybe by the analysis of future X-ray transients of the same object. The implications of this rapid rotation on the Equation of State (EOS) and in particular on the allowed values of the mass and the radius have been discussed in Lavagetto et al. (2006) and in Bejger et al. (2007). Here we discuss the problem of the stability respect to r-modes of such a rapidly rotating object (for a review on r-modes see Andersson & Kokkotas (2001)). It has been widely discussed in the literature the possibility that r-mode instabilities can very efficiently drag angular momentum from a rotating compact star if its temporal evolution in the Ω−T plane (angular velocity and temperature) enters the r-mode instability window, see e.g. Andersson (1998) and Friedman & Morsink (1998). Therefore huge regions of the Ω− T plane are excluded. Moreover, the size and position of that window is strictly 1Dipartimento di Fisica, Università di Ferrara and INFN, Sezione di Ferrara, 44100 Ferrara, Italy 2Inst. Theoretische Physik, Goethe Universität, D-60438, Frankfurt am Main, Germany and INFN, Italy http://arxiv.org/abs/0704.1510v3 – 2 – related to the composition of the star, since it is strongly dependent on the value of bulk and shear viscosity. It is particularly important to recall that for stars containing strangeness, as hyperon stars (Lindblom & Owen 2002), hybrid stars (Drago et al. 2005) and strange quark stars (Madsen 2000) there is also a contribution to the bulk viscosity associated with the formation of strangeness. Due to this, the instability window splits into two parts: one which starts at temperatures larger than (7 ± 3) 109 K (High Temperature Instability Window HTIW) and a lower temperature window at temperatures smaller than (5±4) 108 K (Low Temperature Instability Window, LTIW) 1. Concerning the left border of the LTIW, its position is regulated by the shear viscosity and by the so called viscous boundary layers located at the interface between the crust and the fluid composing the inner part of the star (Bildsten & Ushomirsky 2000). In particular, in bare quark stars, either composed by normal or by superconducting quark matter, due to the absence of a significant crust, the left border of the LTIW extends to much lower temperatures and the LTIW has a minimum corresponding to a significantly lower temperature than in the case of stars containing a crust. In this Letter we discuss in which region of the Ω−T plane the compact star originating XTE J1739-285 is most likely located, due to its composition. Let us start by discussing the simplest possibility, i.e. that the object is a neutron star. In principle a neutron star can rapidly rotate in two cases, either if its temperature is very large, above a few MeV, or if it is recycling, spinning up due to mass accretion (see Fig. 1). Concerning the first possibility, a hot neutron star would be a newly born one, since the time needed to cool below one MeV is of the order of one minute. This is clearly not the case of the stellar object under discussion. Concerning recycling, it should take place on the left side of the instability window, located at lower temperatures (see Fig. 1). The main result of the analysis of Andersson et al. (2000), revisiting previous analysis (Levin 1999; Bildsten & Ushomirsky 2000) is that a neutron star can never spin-up to a rotational period shorter than ∼ 1.5 ms. This result is based on the estimates of the temperature and of the mass accretion rates of Low Mass X-ray Binaries (Brown & Bildsten 1998; Bhattacharyya 2002) indicating that temperatures lower than ∼ 108 K cannot be reached. Therefore a sub-millisecond neutron star cannot be present at the center of a Low Mass X-ray Binary (LMXB). It is important to remark that this conclusion is confirmed by more recent analysis, taking into account the composition stratification of the rigid crust (Glampedakis & Andersson 2006) and discussing the nonlinear development of the r-mode instability (Bondarescu et al. 2007). Let us now discuss the case in which the compact star contains strangeness. It has been 1Here and in the following we are discussing the temperature of the region of the star where r-mode instabilities can develop. Typically it is located at a depth of a few kilometers below the surface. – 3 – shown, in the case of hyperon stars (Wagoner 2002; Reisenegger & Bonacic 2003), and for hybrid stars (Drago et al. 2006) (strange quark stars also have a similar behavior), that due to the large value of the bulk viscosity associated with the non-leptonic weak decays, there are two windows of instability, the LTIW and the HTIW introduced above. The HTIW does not affect significantly the angular velocity of the star because the cooling of a newly born star is so fast that there is not enough time for the r-mode instability to drag a significant fraction of the angular momentum. Therefore, the star exits the HTIW with an angular velocity close to the initial one. When the temperature drops down to a few 108 K the star reaches the LTIW and it starts to lose angular momentum due to r-mode instability. In Fig. 1 we show examples of the instability windows in the case of a pure quark star and of a hybrid star. Notice that the position of the LTIW depends rather strongly on the mass of the strange quark ms. For large values of ms the instability windows shrink considerably. In our analysis we have considered two possibilities concerning the value of ms: a small value ∼ 100 MeV, of the order of the strange quark current mass, and a large value ∼ 300 MeV, similar to what has been obtained in NJL-like models of quark matter in which a density-dependent constituent mass has been computed (Buballa 2005). Concerning the left side of the LTIW, it is easy to see that it cannot be used to accom- modate a submillisecond pulsar. Indeed, in the case of hybrid or hyperonic stars the left side of the LTIW is similar to the one of neutron stars, discussed above, and we can apply here the same analysis done for a neutron star and concerning the minimal temperature of the compact object at the center of a LMXB. Therefore also hybrid or hyperonic stars cannot recycle to frequencies significantly exceeding 700–800 Hz if they are to the left of the LTIW. Moreover, in the case of quark stars the absence of viscous boundary layers implies that they can rotate rapidly only on the right side of the LTIW (a very light crust suspended over the surface of a quark star is irrelevant from the viewpoint of r-mode instability, although it can be important for the production of x-ray bursts, see Page & Cumming (2005)). In conclusion, stars containing strangeness can rotate at submillisecond periods only if they are to the right of the LTIW. Let us now discuss the results of our analysis also considering the constraints posed onto the EOS by the mass shedding limit. The main result of Lavagetto et al. (2006) and of Bejger et al. (2007) is that soft EOSs, and in particular the ones based on hyperonic matter or on matter in which kaon condensation takes place, are rather unlikely. In fact, not only the range of allowed masses is rather small, but moreover the only configurations satisfying the stability constraint for mass shedding turn out to be supra massive (Bejger et al. 2007), making it very difficult to use these stellar structures in a mass accretion scenario. This result confirms previous discussions of other astrophysical objects ruling out soft EOSs (Ozel 2006). On the other hand, stars containing deconfined quark matter are not excluded if the quark – 4 – EOS is stiff enough (Alford et al. 2007). We can therefore conclude that the object under discussion is either a quark or a hybrid star. We can now study the temporal evolutions of the angular velocity of the star Ω, of its temperature T and of the amplitude of the r-modes α which are calculated by solving the set of differential equations given in Andersson et al. (2002) (Eqs. 15-23-24). The only technical difference in our calculation is the inclusion of the reheating associated with the dissipation of r-modes by bulk viscosity, as discussed in various papers (Wagoner 2002; Reisenegger & Bonacic 2003; Drago et al. 2006). Due to the reheating, the time needed for the star to cool down along the right border of the LTIW is rather long. In the equation regulating the thermal evolution (Andersson et al. 2002) we have added a term: Ėbv = 2E/τbv (1) where E ∼ 1051α2M1.4R erg is the energy of the r-modes (the mass, radius and ro- tation period of the star are here expressed in units of 1.4 M⊙, 10 km and in millisecond, respectively) and τbv is the bulk viscosity. Due to the large value of the bulk viscosity for strange matter at temperatures of a few 108 K, the related dissipation of r-modes strongly reheats the star. The trajectory in the Ω−T plane describing the time evolution of the star follows essentially the border of the LTIW and the star keeps rotating as a sub-millisecond pulsar for a very long time, strongly dependent on the value of ms and ranging from ∼ 10 years when ms = 100 MeV to ∼ 10 8 years when ms = 300 MeV, as shown in Fig. 2. In Fig. 3 we show the temporal dependence of the r-mode amplitude. Notice that, in the absence of reheating the time needed to slow-down to frequencies below 1 kHz is at maximum of the order of ∼ 104 years, and can be much shorter than a year if a small value of ms is adopted. In the past this branch of the instability window was only discussed in connection with very young pulsars because reheating due to bulk viscosity was not taken into account. Before discussing the possible astrophysical scenarios for a submillisecond pulsar inside a LMXB, it is important to remark two constraints that a realistic model should fulfil. First, a LMBX is an old object, with a typical age of 108–109 years which is the time needed by the companion (having a low mass) to fill its Roche lobe and to start accreting on the neutron star. Therefore we need to provide a mechanism allowing the central object inside the LMXB to rotate rapidly while being so old. Second, submillisecond pulsars are certainly rather rare, and in particular there is no evidence of a uniform distribution of pulsar rotational frequencies extending from a few hundred Hz up to more than a kHz. Therefore a realistic model should also indicate why most of the compact stars will not be detected rotating at a submillisecond period. As we will show, these two constraints can be satisfied in two different astrophysical scenarios: a first scenario in which a old hybrid or quark star is accelerated up to frequencies exceeding one kHz by mass accretion and a second scenario in which the quark or hybrid – 5 – star is born with a submillisecond period and it is now spinning down by r-mode instability (see Fig. 4). The possible realization of these scenarios depends on two main ingredients: • the value ofms, which, as we have already shown, regulates the magnitude and position of the LTIW; • the cooling rate of the central region of the star which determines the inner temperature T of a star which is accreting material from the companion (Miralda-Escudé et al. 1990). Concerning the cooling rate, if strange quark matter is present in a compact star, direct URCA processes are possible and therefore the cooling is (generally) fast. It turns out from Miralda-Escudé et al. (1990) that in this case the inner temperature is ∼ 5×107 K for a mass accretion rate of Ṁ = 10−10M⊙/year and that the temperature scales as T ∝ Ṁ 1/6. Another possibility recently proposed is that, due to the formation of diquark condensates, URCA processes are strongly suppressed and the cooling turns out to be slow (Blaschke et al. 2000). At the same time, bulk viscosity can still be large enough to suppress r-mode instability (Blaschke & Berdermann 2007). In the following we discuss both possibilities, either of a fast or of a slow cooling. Let us consider first the case in which ms = 300 MeV. The first scenario, i.e. the star span up by accretion, can indeed be realized if T ≥ 108 K (see lower panel of Fig. 1). Such a temperature can be reached via reheating due to mass accretion. In the case of fast cooling a rather large value of mass accretion rate is needed, Ṁ ∼ 10−8M⊙/year, and this stringent request can explain why submillisecond pulsars are rare. A model in which the cooling is slow is instead excluded, because it would be extremely easy to re-accelerate the star to very large frequencies and therefore submillisecond pulsars would not be rare. Also the second scenario in which the star is spinning down due to r-mode instabilities is possible. As shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 108 years and it is therefore compatible with the typical age of LMXBs. In this case submillisecond pulsars are rare because only a (small) fraction of newly born compact stars can rotate with submillisecond periods (see e.g. the discussions in Andersson & Kokkotas (2001) and in Perna et al. (2007)). In this scenario the distribution of millisecond and submillisecond pulsars is determined by the mass of the star at birth: if the mass is large enough, strange quark matter can form in the core of the star, r-modes are efficiently damped by the large bulk viscosity and the star can be a submillisecond pulsar at birth; if the mass of the star is small, strange quark matter can not form, the bulk viscosity of nucleonic matter can not – 6 – damp r-modes and therefore in a short period the star loses almost all its angular momentum (Lindblom et al. 1998). We consider now the case in which ms = 100 MeV. The first scenario in which the star is spinning up by accretion can be realized but T must be at least ∼ 5×108 K for the star to be accelerated up to 1122 (see upper panel of Fig. 4). Such a temperature can be reached but it needs strong reheating due to a large mass accretion rate and slow cooling. The second scenario, i.e. the star is spinning down by r-mode instability, is ruled out since, as shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 100 years and this is incompatible with the typical age of a LMXB. In conclusion, the two astrophysical scenarios proposed to explain a submillisecond pulsar inside a LMXB can both be realized by a quark or a hybrid star if ms ∼ 300 MeV. Instead, if the ms is small, only the first scenario is possible and it requires a slow cooling. Our results are summarized in Table 1. The main uncertainties in our analysis are due to the possible existence of other damping mechanisms, taking place on the left side of the LTIW. For instance magnetic fields can be important to suppress r-mode instabilities (Rezzolla et al. 2000), but their effect is probably negligible for frequencies exceeding ∼ 0.35 ΩK , if the internal magnetic field is not larger than ∼ 1016 G. Obviously, even larger uncertainties exist concerning quark matter. As discussed above, the bulk viscosity of quark matter strongly depends on the strange quark mass and on the possible formation of a diquark condensate (Alford & Schmitt 2007). Clearly, our analysis can provide much needed constraints on the EOS of quark matter. Finally, let us stress that the outcome of our analysis is that a compact star rotating at a submillisecond period inside a LMXB can only be a quark or a hybrid star. Future obser- vations will be important to clarify if the object at the center of XTE J1739-285 constitutes indeed an example. It is a pleasure to thanks Carmine Cuofano for many useful discussions. REFERENCES Alford, M., Blaschke, D., Drago, A., Klahn, T., Pagliara, G. & Schaffner-Bielich, J. 2007, Nature, 445, E7 Alford, M. G. & Schmitt, A. 2007, J.Phys.G, 34, 67 – 7 – Andersson, N. 1998, ApJ, 502, 708 Andersson, N., Jones, D. I., Kokkotas, K. D. & Stergioulas, N. 1998, ApJ, 534, L75 Andersson, N., Jones, D. I., & Kokkotas, K. 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Rev., D65, 063006 Lindblom, L. & Owen, B. J. & Morsink, S. M. (1998) Phys. Rev. Lett.80:4843 Madsen, J. 2000, Phys. Rev. Lett., 85, 10 Miralda-Escudé, & J., Haensel, P., & Paczynski, B. 1990, ApJ362, 572 http://arxiv.org/abs/0704.0799 http://arxiv.org/abs/astro-ph/0612061 – 8 – Ozel, F., 2006, Nature, 441, 1115 Page, D. & Cumming, A. 2005, ApJ, 635, L157 Perna, R. & Soria, R. & Pooley, D. & Stella, L. (2007) e-Print: arXiv:0712.1040 Reisenegger, A. & Bonacic, A. A. 2003, Phys. Rev. Lett., 91, 201103 Rezzolla, L., Lamb, F. K., & Shapiro. S. L. 2000, ApJ, 531, L141 Wagoner, R. V. 2002, ApJ, 578, L63 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/0712.1040 – 9 – Table 1: Microphysics and astrophysical scenarios ms [MeV] cooling scenarios (I or II) conclusions 300 fast I: O.K., Ṁ ≥ 10−8M⊙/y O.K. for II: O.K., needs rapid rotation at birth both scenarios 300 slow I: no, submillisecond pulsars too frequent excluded II: O.K. due to scenario I 100 fast I: no, T is too low not possible II: no, slow down too rapid in both scenarios 100 slow I: O.K., needs large Ṁ O.K. for II: no, slow down too rapid scenario I Table 2: Connection between microphysics (mass of the strange quark and cooling rate) and astrophysical scenarios for a submillisecond pulsar inside a LMXB. The first scenario corresponds to a compact star which is spinning up due to mass accretion; the second scenario to a compact star born as a submillisecond pulsar and still in the process of slowing down. – 10 – 7 8 9 10 11 Log10T@KD f@HzD ms=300 MeV 7 8 9 10 11 Log10T@KD f@HzD 7 8 9 10 11 Log10T@KD f@HzD ms=100 MeV 7 8 9 10 11 Log10T@KD f@HzD Fig. 1.— r-mode instability windows. The star is unstable and loses angular momentum by emitting gravitational waves in the regions above the instability lines displayed in figure. Thin solid lines correspond to neutron stars, for two extreme values of masses and radii allowed by the analysis done by Bejger et al. (2007). The shaded area in between is repre- sentative of intermediate values of masses and radii. Dashed lines delimit the LTIW and the HTIW for strange quark stars. Thick solid lines correspond to hybrid stars. In both cases the mass and radius of the star are M = 2M⊙ and R = 13 km. In the upper panel ms = 100 MeV and in the lower panel ms = 300 MeV. The bulk and shear viscosities are in general from Andersson & Kokkotas (2001). Bulk viscosity of strange quark matter at high temperatures is from Madsen (2000). The arrow indicates the minimal temperature considered possible in LMXBs. The dot indicates the actual position of the compact star associated with XTE J1739-285, as it results from our analysis. – 11 – 4 6 8 10 12 14 16 Log10@t@sDD f@HzD 100 300 300100 Fig. 2.— Time dependence of the rotational frequency of the compact star. The thin lines take into account only the reheating due to shear viscosity, the thick lines take into account also the effect of reheating due to bulk viscosity. We show results for two different values of the strange quark mass, ms =100 MeV and ms =300 MeV. The dashed horizontal line corresponds to a frequency of 1122 Hz. – 12 – 4 6 8 10 12 Log10@t@sDD 4 6 8 10 12 Log10@t@sDD Fig. 3.— Temporal evolution of the amplitude α of the r-modes. Results without reheating in panel (a) and taking into account reheating in panel (b). The amplitude of the r-mode is significantly damped when reheating is taken into account, the maximum amplitude is reduced by roughly 2 orders of magnitude. Therefore non linear effects should be suppressed. – 13 – 7 7.5 8 8.5 9 Log@T@KDD f@HzD HbL 108 yr ms=300 MeV HaL 10 yr 7.5 8 8.5 9 Log@T@KDD f@HzD ms=100 MeV HdL 108yr HbL 109yr Fig. 4.— Upper panel: first scenario (see text). The star is born with a low angular velocity. After a cooling period at a constant angular velocity (line (a)), the star reaches the instability window and loses angular momentum during a long period, ∼ 109 years (line (b)). After this long period the mass accretion process starts and the star is heated (line (c)) and re- accelerated (line (d)) to a frequency of 1122 Hz (indicated by the thick dot). Lower panel: second scenario (see text). The star is born with a large angular velocity. After a short period of cooling at a constant frequency (line (a)) the star enters the instability window and loses angular momentum via r-mode (line (b)). The frequency of the star can be larger than 1122 Hz (indicated by the thick dot) for a period of ∼ 108 years. The mass and radius of the star are M = 1.4M⊙ and R = 10 km, respectively.

0704.1512

Gauge Invariant Factorisation and Canonical Quantisation of Topologically Massive Gauge Theories in Any Dimension

CP3-06-05 ICMPA-MPA/2006/35 April 2007 Gauge Invariant Factorisation and Canonical Quantisation of Topologically Massive Gauge Theories in Any Dimension Bruno Bertrand Center for Particle Physics and Phenomenology (CP3) Institut de Physique Nucléaire Département de Physique, Université catholique de Louvain (U.C.L.) 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium E-mail: [email protected] Jan Govaerts1 Institute of Theoretical Physics Department of Physics, University of Stellenbosch Stellenbosch 7600, Republic of South Africa UNESCO International Chair in Mathematical Physics and Applications (ICMPA) University of Abomey-Calavi 072 B.P. 50, Cotonou, Republic of Benin Abstract Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to gen- erate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as par- ticular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever. 1On sabbatical leave from the Center for Particle Physics and Phenomenology (CP3), Institut de Physique Nucléaire, Université catholique de Louvain (U.C.L.), 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium, E-mail: [email protected]. http://arxiv.org/abs/0704.1512v1 1 Introduction Topological field theories (TFT, see [1] for a review) have played an important role in a wide range of fields in mathematics and physics ever since they were first constructed by A. S. Schwarz [2] and E. Witten [3]. These theories actually possess so large a gauge freedom that their physical, namely their gauge invariant observables solely depend on the topology (more precisely, the diffeomorphism equivalence class) of the underlying manifold. Another related feature of TFT is the absence of propagating physical degrees of freedom. Upon quantisation, these specific properties survive, possibly modulo some global aspects related to quantum anomalies. As a consequence, topological quantum field theories (TQFT) often have a finite dimensional Hilbert space and are quite generally solvable, even though their formulation requires an infinite number of degrees of freedom. There exists a famous classification scheme for TQFT, according to whether they are of the Schwarz or of the Witten type [1]. As a class of great interest, TFT of the Schwarz type have a classical action which is explicitly independent of any metric structure on the underlying manifold and does not reduce to a total divergence or surface term. The present work focuses on all such theories defined by a sequence of abelian B ∧ F theories for manifolds M of any dimension (d+ 1) [2, 4, 5]. Given a real valued p-form field A in Ωp(M) and a real valued (d−p)-form field B in Ωd−p(M), the general TFT action of interest is of the form SB∧F [A,B] = κ (1− ξ)F ∧B − (−1)p ξ A ∧H, (1) κ being some real normalisation parameter of which the properties are specified throughout the discussion hereafter. This action is invariant under two independent classes of finite abelian gauge transformations acting separately in either the A- or B-sector, A′ = A+ α, B′ = B + β, (2) where α and β are closed p- and (d− p)-forms on M, respectively. The derived quantities F = dA and H = dB are the gauge invariant field strengths associated to A and B. The arbitrary real variable ξ introduced in order to parametrise any possible surface term is physically irrelevant for an appropriate choice of boundary conditions on M. Given the definition of the wedge product, ∧, the integrand in (1) is a (d+ 1)-form, the integration of which over M does not require a metric. In the particular situation when the number of spatial dimensions d is even and such that d = 2p with p itself being odd, in addition to the B ∧ F theories defined by (1) there exist TFT of the Schwarz type involving only the single p-form field A with the following action1, SA∧F [A] = κ A ∧ F. (3) These theories are said to be of the A ∧ F type. They include the abelian Chern-Simons theory in 2+1 dimensions [2, 6]. This sequence of TFT of the Schwarz type formulated in any dimension, and related to one another through dimensional reduction [7], possesses some fascinating properties. First, the space of gauge inequivalent classical solutions is isomorphic to Hp(M)×Hd−p(M), Hp(M) being the pth cohomology group of the manifold M. Second, the types of topological terms contributing to these actions define generalisations to arbitrary dimensions of ordinary two-dimensional anyons. Namely, non local holonomy effects give rise to exotic statistics for the extended objects which may be coupled to the higher order tensor fields [8, 9]. Third, these types of quantum field theories display 1If p is even with d = 2p, this action reduces to a surface term. profound connections between mathematics and physics for what concerns topological properties related, say, to the motion group, the Ray-Singer torsion and link theory. These connections appear within the canonical quantisation2 of these systems [10, 11]. Furthermore, within the context of dynamical relativistic (quantum) field theories in any spacetime dimension, which is a general framework of potential relevance to fundamental particle physics as well as mathematical investigations for their own sake, such topological B ∧ F terms may be considered to define couplings between two independent tensor fields whose dynamics is characterised by the following action, provided the spacetime manifold M is endowed now with a Lorentzian metric structure (of mostly negative signature) allowing for the introduction of the associated Hodge ∗ operator, STMGT[A,B] = (−1)p F ∧ ∗F + 1 (−1)d−p H ∧ ∗H (1− ξ)F ∧B − (−1)p ξ A ∧H. (4) The notations are those introduced previously. Given a choice of units such that c = 1, the phys- ical dimensions of A and B are L−p and L−d+p, respectively, whereas the multiplicative constant κ possesses the same physical dimension as the action. The parameters e and g are arbitrary real constants corresponding to coupling constants when matter fields coupled to A and B are intro- duced. Without loss of generality for the present analysis, the parameters e and g are assumed to be strictly positive. In 3+1 dimensions, one recovers the famous Cremmer-Scherk action [12, 13] and in 2+1 dimensions, the doubled Chern-Simons theory [14]. It is well known that the topo- logical terms generate a mass for the dynamical tensor fields without breaking gauge invariance. Introducing an appropriate choice of gauge fixing, it is possible to render one of the tensor fields massive through a combination with the other tensor field [13]. In the particular circumstance that d = 2p with p odd, a topological term of the A∧F type (3) generates also a mass even though the action involves a single p-form field A, STMGT[A] = F ∧ ∗F + A ∧ F. (5) In 2+1 dimensions, this action defines the well-known Maxwell-Chern-Simons theory [15]. The outline of the paper is as follows. Section 2 discusses a new property of the abelian TMGT valid whatever the number of space dimensions and the value of 0 ≤ p ≤ d for the p- and (d − p)- form fields: the “Physical-Topological” factorisation of their degrees of freedom. This result is achieved within the Hamiltonian formulation through a canonical transformation of classical phase space leading to two independent and decoupled sectors3. The first of these sectors, namely the “physical” one, consists of gauge invariant variables which are canonically conjugate and describe free massive propagating physical degrees of freedom. The second sector, namely the “topological” one, consists of canonically conjugate gauge variant variables which are decoupled from the total Hamiltonian and, hence, are non dynamical. This sector is equivalent to a pure TFT of the A∧ F or B ∧ F type. This factorisation enables the identification of a mass generating mechanism for any p-form (or, by dualisation, any (d− p)-form) without introducing any gauge fixing conditions or second-class constraints whatsoever as has heretofore always been the case in the literature. Section 3 addresses the Dirac quantisation of these systems, with the identification of the spectrum of physical states through a likewise factorisation extended to the space of quantum states. Finally, Sect. 4 discusses how the factorisation leads to a most transparent understanding of the projection 2When M = R× Σ, the physical Hilbert space is the set of square integrable functions on Hp(Σ) [4]. 3In other words, the Poisson brackets of variables belonging to the two distinct sectors vanish identically. from TMGT onto TQFT in whatever spacetime dimension in the limit of an infinite topological mass. 2 Gauge Invariant Factorisation of the Classical Theory 2.1 Hamiltonian formulation Because of the built-in gauge invariances of these systems, the analysis of the constraints [16, 17] of topologically massive gauge theories is required in order to identify their Hamiltonian formulation. Given the total action (4) written out in component form the associated Lagrangian density reads, LTMGT = (−1)p (p + 1)! Fµ1···µp+1 F µ1···µp+1 + (−1)d−p (d− p+ 1)! Hν1···νd−p+1 H ν1···νd−p+1 (1− ξ) (1 + p)! (d − p)! ǫµ1···µp+1ν1···νd−p Fµ1···µp+1 Bν1···νd−p − κ ξ (−1) p! (d− p+ 1)! ǫµ1···µpν1···νd−p+1 Aµ1···µp Hν1···νd−p+1 , (6) where Greek indices, µ, ν = 0, 1, . . . , d, denote the coordinate indices of the spacetime manifold M while h is the absolute value of determinant of the metric. According to our conventions, the components of the field strength tensors are given as Fµ1···µp+1 = ∂[µ1 Aµ2···µp+1], Hν1···νd−p+1 = (d− p)! ∂[ν1 Bν2···νd−p+1] , (7) where square brackets on indices denote total antisymmetrisation. The above expression with the single parameter κ multiplying each of the topological B∧F and A∧H terms while ξ parametrises a possible surface term does not entail any loss of generality. Had two independent parameters κ and λ multiplying each of the topological terms been introduced, only their sum, (κ + λ), would have been physically relevant, the other combination corresponding in fact to a pure surface term. In order to proceed with the Hamiltonian analysis, the spacetime manifold M is now taken to have the topology of M = R × Σ where Σ is a compact orientable d-dimensional Riemannian space manifold without boundary. Adopting then synchronous coordinates on M, the spacetime metric takes the form ds2 = dt2 − h̃ij dxi dxj, h̃ij(~x ) being the Riemannian metric on Σ. Here Latin indices, i = 1, . . . , d, label the space directions in Σ. The configuration space variable A(t, ~x ) may then be separated into its temporal component dt∧A0(t, ~x ) with A0(t, ~x ) being a (p−1)-form on Σ, and its remaining components Ã(t, ~x ) restricted to Ωp(Σ), A0(t, ~x ) = (p − 1)! A0i1···ip−1(t, ~x ) dx i1 ∧ . . . ∧ dxip−1 , Ã(t, ~x ) = Ai1···ip(t, ~x ) dx i1 ∧ . . . ∧ dxip . (8) A similar decomposition applies to the (d− p)-form B(t, ~x ). The actual phase space variables are then the spatial components à and B̃ along with their conjugate momenta P̃ and Q̃ defined to be the following differential forms on Σ, h̃i1j1 . . . h̃ipjp P i1···ip dxj1 ∧ . . . ∧ dxjp , (d− p)! h̃i1j1 . . . h̃id−pjd−p Q i1···id−p dxj1 ∧ . . . ∧ dxjd−p , (9) of which the pseudo-tensorial space components are P i1···ip and Qi1···id−p . Expressed in terms of the configuration space variables, these latter quantities are given as P i1···ip = F0j1···jp h̃ i1j1 . . . h̃ipjp + κ (1− ξ) (d− p)! ǫi1···ipj1···jd−p Bj1···jd−p , Qi1···id−p = H0j1···jd−p h̃ i1j1 . . . h̃id−pjd−p − κ (−1)p(d−p) ǫi1···id−pj1···jpAj1···jp, (10) while the symplectic structure of Poisson brackets is characterised by the canonical brackets Ai1···ip(t, ~x ), P j1···jp(t, ~y ) . . . δ δ(d)(~x− ~y ), Bi1···id−p(t, ~x ), Q j1···jd−p(t, ~y ) . . . δ id−p] δ(d)(~x− ~y ). (11) A priori , phase space also includes the canonically conjugate variables A0 and P 0, and B0 and Q The Legendre transform of the Lagrangian (6) leads to the total gauge invariant Hamiltonian, ∗P̃ − κ (1 − ξ) B̃ u, P 0 ∗Q̃+ κ ξ (−1)p(d−p)à + (surface term) (−1)p (u′ +A0) ∧ d ∗P̃ + κ ξ B̃ (−1)d−p(v′ +B0) ∧ d ∗Q̃− κ (1 − ξ) (−1)p(d−p) à . (12) In this expression as well as throughout hereafter, the Hodge ∗ operation is now considered only on the space manifold Σ endowed with the Riemannian metric h̃ij . In (12) the inner product on Ωk(Σ)× Ωk(Σ) is constructed as 2 = (ωk, ωk) with (ωk, ηk) = ωk ∧ ∗ηk. (13) The quantities u′ and v′ are Lagrange multipliers for the two first-class constraints associated to abelian gauge symmetries while u and v are those for the first-class constraints P 0 = 0 and Q0 = 0 arising because the fields A0 and B0 are auxiliary degrees of freedom of which the time derivatives do not contribute to the action. Upon reduction to the basic layer of the Hamiltonian nested structure [16], P 0 and Q0 decouple from the system whereas A0 and B0 play the role of Lagrange multipliers enforcing the two Gauss laws. These constraints generate those gauge transformations in (2) which are continuously connected to the identity transformation, namely the so-called small gauge symmetries, one generated by the fields P̃ and B̃ and the other by à and Q̃, respectively. Note that given Hodge duality, the phase space variables are associated to isomorphic spaces, Ωp(Σ) ≡ Ωd−p(Σ). Hence at any given spacetime point, phase space has dimension 4Cpd . 2.2 The Physical-Topological (PT) factorisation The above results are well-known. However the fields used to construct the theory do not necessarily create physical states since these are not gauge invariant variables. Therefore, let us now introduce the new Physical-Topological factorisation of the classical theory, by also requiring that these field redefinitions are canonical and preserve canonical commutation relations. First consider the quantities A = − (−1)p(d−p) ∗ Q̃+ (1− ξ) Ã, B = ∗ P̃ + ξ B̃, (14) defined on the dual sets Ωp(Σ) and Ωd−p(Σ). This choice is made in such a way that the two Gauss laws are expressed in term of these variables only, as is the case for a topological B ∧ F theory, κ (−1)p(d−p) dA = 0, (−1)p κdB = 0. (15) As a matter of fact, these variables are canonically conjugate, Ai1···ip(t, ~x ),Bj1···jd−p(t, ~y ) ǫi1···ipj1···jd−p δ (d)(~x− ~y ). (16) The two finite gauge transformations in (2) act on these new variables according to the relations, A′ = A+ α, B′ = B + β. (17) At a given spacetime point, these canonically conjugate variables carry 2C degrees of freedom. The remaining 2C degrees of freedom are associated to the following pair of gauge invariant variables, G = Q̃+ κ ξ ∗ Ã, E = P̃ − κ (1 − ξ) (−1)p(d−p) ∗ B̃. (18) Their pseudo-tensor Lorentz components are defined as in (9) while they possess the following non vanishing canonical Poisson brackets, Ei1···ip(t, ~x ), Gj1···jd−p(t, ~y ) = −κ ǫi1···ipj1···jd−p δ(d)(~x− ~y ). (19) When considered in combination with the equations of motion, these variables correspond to the non commutative electric fields associated, respectively, to the field strength tensors of A and B, see (7). Consequently, we have achieved a coherent reparametrisation of phase space which, in fact, factorises the system into two orthogonal sectors, namely sectors of which mutual Poisson brackets vanish identically, Ai1···ip(t, ~x ), Ej1···jp(t, ~y ) Ai1···ip(t, ~x ), Gj1···jd−p(t, ~y ) Bi1···id−p(t, ~x ), E j1···jp(t, ~y ) Bi1···id−p(t, ~x ), G j1···jd−p(t, ~y ) Finally in order to obtain the basic nested Hamiltonian formulation [16] within the factorised parametrisation, the Lagrange multipliers in (12) may be redefined in a convenient way as u = Ȧ0, A0 = A0 + u′ + (−1)(p−1)(d−p) 2 g2 κ2 ∗ d (κB − 2 ∗ E) , v = Ḃ0, B0 = B0 + v′ + (−1)p 2 e2 κ2 κA + (1)p(d−p) 2 ∗G where a dot stands for differentiation with respect to the time coordinate, t ∈ R. Consequently the basic total first-class Hamiltonian of the system reads, H[E,G,A,B] = e 2κ2 g2 2 e2 κ2 (−1)p A0 ∧ dB − (−1)(p+1)(d−p) B0 ∧ dA, (20) where d† = ∗d∗ is the coderivative operator. Obviously, A0 and B0 are Lagrange multipliers enforcing the first-class constraints which generate the small gauge transformations in (17), G(1) = dA G(2) = dB. (21) When restricted to the physical subspace for which these constraints are satisfied, the above gauge invariant Hamiltonian reduces to a functional depending only on the dynamical physical sector, given by the expression in the first line of (20). These redefinitions of the phase space variables have indeed achieved the announced fac- torisation. A first sector is comprised of the variables constructed in (14), which decouple from the physical Hamiltonian and are therefore non propagating degrees of freedom. Furthermore, the canonically conjugate variables A and B actually share the same Poisson brackets, Gauss law con- straints and gauge transformations as the phase space description of a pure B ∧F topological field theory constructed only from the topological terms in the action (6). Hence this “topological field theory (TFT) sector” accounts for the B∧F theory embedded into the topologically massive gauge theory. Physical and non physical degrees of freedom are mixed in the original phase space. Our redefinition of fields deals with the original degrees of freedom in such a way that within the Hamiltonian formalism, non propagating (and gauge variant) degrees of freedom are decoupled from the dynamical sector. This latter sector describes only physical degrees of freedom, namely the gauge invariant canonically conjugate electric fields, which diagonalise the physical Hamiltonian (20) in such a way that they acquire a mass through a mixing of the original variables (18). However the Poisson bracket structure remains unaffected since these field redefinitions define merely a canonical transformation. On account of Hodge duality between Ωp(Σ) and Ωd−p(Σ), one readily identifies in the dynamical sector the Hamiltonian of a massive p-form field of mass m = ~µ, H[C,E,A,B] = +HTFT[A,B]. In comparison with (20) the following identifications have been applied, µ = |κ e g| E → ∗G = e κ (−1)p(d−p) C, where C is a p-form field of which the Lorentz components are covariant in the manner of (8). Physical phase space is then endowed with the elementary Poisson brackets Ci1···ip(t, ~x ), E j1···jp(t, ~y ) . . . δ δ(d)(~x− ~y ) . Alternatively one may also obtain the Hamiltonian of a massive (d− p)-form field of mass m = ~µ, H[C,G,A,B] = µ +HTFT[A,B], in which, in comparison with (20), the following identifications have been applied, ∗E = −g κC. In this case, C is a (d− p)-form field with covariant Lorentz components as in (8). The elementary Poisson brackets for these physical phase space variables are Ci1···id−p(t, ~x ), G j1···jd−p(t, ~y ) . . . δ id−p] δ(d)(~x− ~y ). To conclude this discussion of the factorised Hamiltonian formulation of these TMGT, let us em- phasize once more that no gauge fixing procedure whatsoever was applied, in contradistinction to all discussions available until now in the literature leading to an identification of the physical con- tent of these theories. Through the present approach, the TFT content of TMGT is made manifest in a most transparent and simple manner, with in addition a decoupling of the actual physical and dynamical sector of the system from its purely topological one, the latter carrying only topological information characteristic of the underlying spacetime manifold. 2.3 Hodge decomposition The space manifold Σ having been assumed to be orientable and compact, let us now consider the consequences of its cohomology group structure, especially in the case when the latter could be non trivial. Throughout the discussion it is implicitly assumed that the p- and (d − p)-form fields A and B are globally defined differentiable forms in Ωp(M) and Ωd−p(M). When parametrising the theory in terms of the PT factorised variables, the latter assumption of a topological character concerns only the TFT sector. The variables of the dynamical sector are already globally defined whatever the topological properties of the original variables. By virtue of the Hodge theorem [18], the phase space variables of the TFT sector, thus globally defined on Σ itself endowed with the Riemannian metric h̃ij , may uniquely be decomposed for each time slice into the sum of an exact, a co-exact and a harmonic form with respect to the inner product specified in (13), A = Ae +Ac +Ah, B = Be +Bc +Bh. (22) A likewise decomposition applies to the dynamical sector. Such a decomposition amounts to a split of the fields into a longitudinal part (subscript L), a transverse part (subscript T ) and a “global” part. The transverse and longitudinal parts are associated to idempotent orthogonal projection operators, ΠT(p) = (p+1) d(p), Π (p) = d(p−1)d ΠT(p) : Ω p(Σ) → (Z† )p(Σ), ΠL(p) : Ω p(Σ) → Zp (Σ), (23) where △⊥ is the Laplacian operator acting on the space Ω (Σ) of p-forms from which the kernel ker△(p) of the Laplacian △(p) has been subtracted, while (Z )p (resp. Z ) is the space of co-closed (resp. closed) p-forms non cohomologous to zero. One therefore has the following properties, (−1)p(d−p)ΠT(p) = ∗Π (d−p)∗, Π (p) +Π (p) = Id where ∗ is the Hodge star operator on Σ and Id⊥ the identity operator on the subspace Ω In order that the longitudinal and transverse components possess the same physical dimen- sions as the original fields, the Hodge decomposition of fields may be expressed in terms of a convenient normalisation, △⊥A = dAL + d†AT , △⊥B = dBL + d†BT . (24) Let us then define a new set of variables in the TFT sector, using the projection operators (23), ϕ = ΠT(p−1)AL, ∗Qϑ = Π (p+1)AT , ϑ = ΠT(d−p−1)BL, ∗Pϕ = Π (d−p+1)BT , (25) where the components of ∗Pϕ and ∗Qϑ are pseudo-tensors defined in a manner analogous to the conjugate momenta in (9). In terms of these new variables the non vanishing Poisson brackets are ϕi1···ip−1(t, ~x ), P j1···jp−1 ϕ (t, ~y ) )j1···jp−1 i1···ip−1 δ(d)(~x− ~y ), ϑi1···id−p−1(t, ~x ), Q j1···jd−p−1 (t, ~y ) )j1···jd−p−1 i1···id−p−1 δ(d)(~x− ~y ). In conclusion, in the TFT sector, rather than working in terms of the phase space variables A and B one may parametrise these degrees of freedom in terms of the “longitudinal” fields ϕ and ϑ as well as their conjugate momenta, namely the “transverse” fields Pϕ and Qϑ, to which the harmonic components Ah and Bh of A and B must still be adjoined. The same procedure may be applied to the variables of the dynamical sector. The Hamiltonian (20) then decomposes into a transverse, a longitudinal and a harmonic contribution from these latter variables only. A natural consequence of the Hodge decomposition is the isomorphism between the pth de Rham cohomology group, Hp(Σ,R), and the space of harmonic p-forms, ker△(p). This means that each equivalence class of Hp(Σ,R) has an unique harmonic p-form representative identified through the inner product (13). It is possible to choose a basis for ker△(p) in such a way that the harmonic component of any p-form is expressed in a topological invariant way. This may be achieved by defining a topological invariant isomorphism between the components of an equivalence class of the pth (singular) homology group Hp(Σ,R) and the components of a form in ker△(p) (the p-homology group is the set of equivalence classes of p-cycles differing by a p-boundary). Thus, instead of constructing the basis from the Hodge decomposition inner product (13), one uses the bilinear, non degenerate and topological invariant inner product Λ defined by Λ : Hp(Σ)×Hp(Σ) → R : Λ ([Γ] , [ω]) = ω, (26) making explicit the Poincaré duality between homology and cohomology groups [18]. Given the Hodge theorem, this inner product naturally induces a topological invariant inner product between the equivalent classes of Hp(Σ) and the elements of ker△(p). Therefore, if one introduces generators of the free abelian part of the pth singular homology group of rank Np, , a convenient dual basis {Xγ} of ker△(p) may be chosen such that Σα(p) = δαβ . Using the duality (26), the harmonic component Ah of the p-form variable A is thus decomposed according to These components of Ah in the basis {Xγ} are topological invariants because they express the periods of A over the cycle generators of Hp(Σ). This is thus nothing other than the classical Wilson loop argument over these generators, Ah . (27) In other words, the variables aγ(t) specify the complete set of remaining “global” degrees of freedom in the TFT sector for the field A, Ah(t, ~x ) = aγ(t)Xγ(~x ). In a likewise manner, the harmonic component of the (d− p)-form variable B may be decomposed according to (d−p) Y γ , where {Y γ} is the dual basis of the cycle generators in Hd−p(Σ), (d−p) . Hence, the compo- nents of harmonic (d− p)-forms are expressed as (d−p) Bh, (28) leading to a similar decomposition of the “global” degrees of freedom for the dual field B, Bh(t, ~x ) = bγ(t)Y γ(~x ). The Poisson brackets between the above global variables are topological invariants, aγ , bγ , (29) namely the signed intersection matrix of which each entry is the sum of the signed intersections of the generators of Hp(Σ) and Hd−p(Σ), (d−p) . (30) Within our approach, we recover the results of [19, 11, 10] in 2+1 (on the torus), 3+1 and d+1 dimensions, respectively. 2.4 Large and small gauge transformations Only the TFT sector is not gauge invariant. Its phase space variables transform exactly like in a pure B ∧ F theory, see (17). Let us recall that in (17), α and β are, respectively, closed p- and (d− p)-forms on Σ. In the case of a homologically trivial space Σ any closed form is also exact. In the case of a homologically non trivial space Σ, according to the Hodge theorem any closed form α or β may uniquely be decomposed (for a given metric structure) into the sum of an exact and a harmonic form. The exact parts of α and β define small gauge transformations, generated by the two Gauss law first-class constraints (15). Given the Hodge decompositions in the TFT sector (24) and (25), these constraints, which require that the phase space variables A and B of the TFT sector be closed forms, reduce to G(1) = △⊥ ∗Qϑ, G(2) = △⊥ ∗ Pϕ. (31) Small gauge transformations act only on the exact part of the TFT sector fields by translating them, namely in terms of the longitudinal (p− 1)- and (d− p− 1)-form fields defined in (25), ϕ′ = ϕ+ αL, ϑ ′ = ϑ+ βL, where αL and βL are, respectively, the longitudinal (p−1)- and (d−p−1)-forms defining the exact components of the gauge transformation forms α and β through a construction similar to that in (25). The harmonic components of α and β define the associated large gauge transformations. The physical classical phase space in the TFT sector is the set of all field configurations A and B obeying the first-class constraints setting to zero their transverse degrees of freedom, see (31), and identified modulo the action of all gauge transformations, whether small or large. Since under small transformations the longitudinal modes ϕ and ϑ are gauge equivalent to the trivial configuration of vanishing longitudinal fields, like in any pure B ∧F TFT the physical phase space of the TFT sector, so far for what concerns small gauge symmetries, is thus finite dimensional and isomorphic to the ensemble of harmonic forms defined modulo exact forms, P = Hp(Σ,R)⊕Hd−p(Σ,R), (32) where Hp(Σ,R) is the pth de Rham cohomology group. Let us recall that according to Poincaré duality, Hp(Σ) is isomorphic to Hd−p(Σ). Hence, whether one considers functionals of harmonic p-forms or (d − p)-forms is of no consequence. The finite dimension of this group is given by the corresponding Betti number Np (for example in the case of the torus, Σ = Td, Np = C d ). The physical phase space of the TFT sector is thus spanned by the global degrees of freedom aγ(t) and bγ(t), which are indeed obviously invariant under all small gauge transformations. However, this phase space is subjected to further restrictions still, stemming from large gauge transformations. In a manner similar to the above characterisation of the physical phase space in the TFT sector, the modular group is the quotient of the full gauge group by the subgroup of small gauge transformations generated by the first-class constraints, namely essentially the set of large gauge transformations (LGT) defined modulo small gauge transformations. Large gauge transformations cannot be built from a succession of infinitesimal transformations. They correspond to the co- homologically non trivial, namely the harmonic components of α and β. Rather than requiring strict invariance of the global phase space variables aγ and bγ under large gauge transformations, having in mind compact U(1) abelian gauge symmetries defined in terms of univalued pure imag- inary exponential phase factors, the global physical observables to be considered and thus to be required to remain invariant under large gauge transformations are the holonomy or Wilson loop operators of the TFT sector around compact orientable submanifolds Σp and Σd−p in Σ. The only non trivial Wilson loops are those around homotopically non trivial cycles, namely elements [Γp] of Hp(Σ,Z) which may be decomposed in the basis . Consequently, given the basis of ker△(p) constructed from (26) one has the following set of global Wilson loop observables W [Γ(p)] = exp = exp σγ aγ W [Γ(d−p)] = exp (d−p) = exp σ̃γ bγ where σγ , σ̃γ are arbitrary integers. Large gauge transformations associated to closed forms α and β act on the global variables aγ and bγ according to a′γ = aγ + αγ , b′γ = bγ + βγ , (33) where αγ and βγ are given by α, βγ = (d−p) Although the Wilson loops are constructed on the free abelian homology group Hp(Σ,Z), the cohomology group including the large gauge transformation parameters is dual to the singular homology group Hp(Σ,R). Hence, the only allowed large gauge transformations correspond to components of the harmonic content of the forms α and β which are discrete and quantised, α = 2π ℓ , βγ = (d−p) β = 2π ℓ (d−p) . (34) Here ℓ and ℓ (d−p) are integers which characterise the winding numbers of the large gauge transfor- mations, namely the periods of these transformations around the homology cycle generators. The requirement of gauge invariance of all Wilson loops hence constrains the parameters of large gauge transformations to belong to the dual of the free abelian homology group. As a consequence, finally the physical classical phase space in the TFT sector is the quotient of the de Rham cohomology group Hp(Σ,R)⊕Hd−p(Σ,R) by the additive lattice group defined by the transformations, a′γ = aγ + 2π ℓ , b′γ = bγ + 2π ℓ (d−p) namely a finite dimensional compact space having the topology of a torus of dimension 2Np. 3 Canonical Quantisation and Physical States 3.1 Physical Hilbert space factorisation The BRST formalism offers a powerful and elegant quantisation procedure for TMGT but requires the introduction of ghosts. In some respects, this formalism has also been used for the definition and characterisation of topological quantum field theories [1]. In a related manner, the path integral quantisation of these theories also brings to the fore the characterisation of topological invariants through concepts of quantum field theory. For example, the two-point correlation function of B∧F (and A∧F ) theories provides a quantum field theoretic realisation of the linking number of two sur- faces of dimensions p and (d− p) embedded in M and its path integral representation through the Ray-Singer analytic torsion of the underlying manifold. Notwithstanding these achievements, this paper will not rely on such methods which necessarily require some gauge fixing procedure. Rather, ordinary Dirac canonical quantisation methods will be implemented to unravel the physical content of TMGT. First, this quantisation procedure is best adapted to a condensed matter interpretation. It also enables to deal with large gauge transformations on homologically non trivial manifolds. Sec- ond, the new Physical-Topological (PT) factorisation identified within the Hamiltonian formulation independently of any gauge fixing procedure makes canonical quantisation especially attractive. Canonical quantisation readily proceeds from the correspondence principle, according to which classical Poisson brackets are mapped onto equal time quantum commutation relations for the classical variables which are promoted to linear self-adjoint operators acting on the Hilbert space of quantum states in the Schrödinger picture at the reference time t = t0, Âi1···ip(t0, ~x ), B̂j1···jd−p(t0, ~y ) ǫi1···ipj1···jd−p δ (d)(~x− ~y ), Êi1···ip(t0, ~x ), Ĝ j1···jd−p(t0, ~y ) ǫi1···ipj1···jd−p δ(d)(~x− ~y ). A possible representation of the associated Hilbert space is in terms of functionals Ψ[A, E] with their canonical hermitean inner product defined in terms of the field degrees of freedom A(~x ) and E(~x ). It should be clear that the PT factorisation identified at the classical level extends to the quantum system. The full Hilbert space of the system factorises into the tensor product of two separate and independent Hilbert spaces, each of which is the representation space of the operator algebra of either the gauge invariant dynamical sector or the TFT sector. As a consequence of the complete decoupling of these two sectors, one of which contributes to the physical Hamiltonian only, the other to the first-class constraint operators only, a basis of the space of quantum states may be constructed in terms of a likewise factorisation of wave functionals. Symbolically one has Ψ[A, E] = Φ[E] Ψ[A]. The component Φ[E] associated to the dynamical sector is manifestly gauge invariant and is the only one which contributes to the energy spectrum. The physical Hilbert subspace associated to the TFT sector consists of those states wave functionals ΨP [A] which are invariant both under small gauge transformations, namely which belong to the kernel of the first-class constraint operators generating these transformations, and under the large gauge transformations4 characterised in the previous Section in terms of their lattice action on harmonic p- and (d− p)-forms. When the space manifold Σ is topologically trivial, for instance in the case of the hyperplane, quantisation of TMGT does not offer much interest per se besides the free dynamics of the dynam- ical sector, since the TFT sector then possesses a single gauge invariant quantum state. However in the presence of external sources, or when the space manifold Σ does have non trivial topology, new and interesting features arise. In the latter situation, to be addressed hereafter, the finite though multi-dimensional gauge invariant content of the TFT sector, ΨP [A], does not contribute to the energy spectrum. Hence it induces a degeneracy of the energy eigenstates of the complete system. As demonstrated later, this degeneracy is restricted by gauge invariance under large gauge transformations (LGT). Since the physical wave functional ΨP [A] in the TFT sector coincides with that of a pure topological quantum field theory, one recovers the results of R. J. Szabo [10] who solved in the Schrödinger picture the pure topological B ∧ F theory (as well as in the presence of sources) in any dimension. 3.2 The topological sector: Gauss’ constraints and LGT 3.2.1 Hilbert space and holomorphic polarisation At the classical level, phase space has been separated into two decoupled sectors: the TFT and the dynamical sectors. According to the Hodge decomposition theorem (22), each of the corresponding fields may in turn be decomposed into three further subsectors in terms of their longitudinal, transverse and global components. The Gauss law constraints in conjunction with invariance under small gauge transformations reduce the TFT sector to its global variables only, characterised by the vector space P of the de Rham cohomology group in (32), which is to be restricted further into a compact torus by the lattice action of the appropriate discrete large gauge transformations. Likewise in the dynamical sector, the global degrees of freedom of phase space are also purely topological and are again isomorphic to the 2Np-dimensional symplectic vector space P in (32). In each case, these spaces are spanned by the global variables defined as in (27) and (28), namely (aγ , bγ) and (Eγ , Gγ), respectively. It is quite natural to introduce for these even dimensional vector spaces a complex structure parametrised by a Np ×Np complex symmetric matrix, τ = ℜ(τ) + i ρ, such that (−τ) takes its values in the Siegel upper half-space. Such a complex structure introduced over the phase space of global degrees of freedom enables the definition of a holomorphic phase space polarisation, hence quantisation of these sectors. The same decomposition in terms of longitudinal, transverse and global degrees of freedom applies at the quantum level. Through the correspondence principle, these three subsectors of 4Otherwise, the physical wave functional carries a projective representation of the group of LGT [10]. quantum operators obey the Heisenberg algebra, whether for the TFT or dynamical sector. Let us presently restrict to the TFT sector. For what concerns the local operators, one has ϕ̂i1···ip−1(t0, ~x ), P̂ j1···jp−1 ϕ (t0, ~y ) )j1···jp−1 i1···ip−1 δ(d)(~x− ~y ), ϑ̂i1···id−p−1(t0, ~x ), Q̂ j1···jd−p−1 (t0, ~y ) = − i~ )j1···jd−p−1 i1···id−p−1 δ(d)(~x− ~y ), (35) while for the global operators, âγ(t0), b̂ γ′(t0) Introducing now the holomorphic combinations of the latter operators5, ĉγ = Iγδ â δ + τγδ b̂ , ĉ†γ = Iγδ â δ + τγδ b̂ , (36) where Iγδ is the inverse of the intersection matrix, Iγδ I δγ′ = δγ one finds the Fock type algebra ĉγ , ĉ = ℑ(τ)γγ′ = ργγ′ , (37) all other possible commutators vanishing identically. Note that this result implies that the inner product in this sector of Hilbert space is to be defined in terms of the imaginary part (ρ−1)γγ , in a manner totally independent from the Riemannian metric structure of the compact space submani- fold Σ. A priori , physical observables in pure topological quantum field theories ought nevertheless to be independent from any extraneous ad hoc structure introduced through the quantisation pro- cess such as the present complex structure. Gauss law constraints and large gauge transformations are to be considered in the wave functional representation of Hilbert space. The latter is spanned by the direct product of basis vectors for the representation spaces of the algebras (35) and (37). These consist of functionals Ψ[ϕ, ϑ, c] of the infinite dimensional space of field configurations in the TFT sector. Accordingly, the inner product of such states is defined by 〈Ψ1 |Ψ2〉 = [Dϕ] [Dϑ] Ψ∗1[ϕ, ϑ, c] Ψ2[ϕ, ϑ, c], which requires the specification of a functional integration measure. This measure is taken to be the gaussian measure for fluctuations in the corresponding fields, which is induced by the complex structure τ in the global sector or else by the Riemannian metric on Σ for fluctuations in ϕ and ϑ, δϕ2 = d~xhi1k1(~x ) . . . hip−1kp−1(~x ) δϕi1···ip−1(~x ) δϕk1···kp−1(~x ), δϑ2 = d~xhj1l1(~x ) . . . hjd−p−1ld−p−1(~x ) δϑj1···jd−p−1(~x ) δϑl1···ld−p−1(~x ), δc2 = γ,γ′=1 (ρ−1)γγ δcγ δcγ′ . (38) 5It is implicitly assumed here that the parameter κ is strictly positive. If κ is negative, the roles of the operators γ and b̂γ are simply exchanged in the discussion hereafter. In contradistinction to an ordinary pure topological quantum field theory, such a space metric is readily available within the context of TMGT, being necessary for the specification of the dynamical fields. Independently from the complex structure introduced in the global sector, independence of the physical Hilbert space measure in the (ϕ, ϑ) sector on the metric on Σ will be established hereafter. Consequently the canonical commutation relations (35) and (37) in the TFT sector are represented by the following functional operators acting on the Hilbert space wave functionals, ϕ̂i1···ip−1(~x ) ≡ ϕi1···ip−1(~x ), P̂ i1···ip−1 ϕ (~x ) ≡ − )i1···ip−1 j1···jp−1 δϕj1···jp−1(~x ) , (39) ϑ̂i1···id−p−1(~x ) ≡ ϑi1···id−p−1(~x ), Q̂ i1···id−p−1 ϑ (~x ) ≡ )i1···id−p−1 j1···jd−p−1 δϑj1···jd−p−1(~x ) , (40) ĉγ ≡ cγ , ĉ†γ ≡ − . (41) 3.2.2 Gauss law constraints The physical Hilbert space is invariant under all gauge transformations. A first restriction arises by requiring the physical quantum states to be invariant under small gauge transformations generated by the first class constraints. This set is the kernel of the Gauss law constraint operators (21) which remain defined as in the classical theory since no operator ordering ambiguity is encountered, Ĝ(1) = 0 ⇒ δ δϑi1···id−p−1(~x) ΨP [ϕ, ϑ, c] = 0, Ĝ(2) = 0 ⇒ δϕi1···ip−1(~x) ΨP [ϕ, ϑ, c] = 0. Hence physical quantum states necessarily consist of wave functionals which are totally independent of the longitudinal variables (ϕ, ϑ). When restricted to such states and properly renormalised, the inner product integration measure is constructed from the definition (38) of the gaussian metric on the space of fluctuations in the global coordinates, 〈Ψ1 |Ψ2〉 = dcγ (det ρ) Ψ∗1(c)Ψ2(c). This measure on the physical Hilbert space is thus indeed independent of the Riemannian metric on Σ, and involves only the ad hoc complex structure τ introduced towards the quantisation of the global TFT sector. 3.2.3 LGT and global variables The structure of the physical Hilbert space dramatically depends on the way one deals with LGT. Given the holomorphic parametrisation (36), under the lattice action of LGT of periods (d−p) ) ≡ (ℓ(p), ℓ(d−p)) as defined in (34) the new global operators should transform as, c′γ = cγ + 2π2 κ Iγγ′ ℓ + τγγ′ ℓ (d−p) = c†γ + 2π2 κ Iγγ′ ℓ + τγγ′ ℓ (d−p) . (42) Using the Baker-Campbell-Hausdorff (BCH) formulae for any two operators  and B̂ commuting with their own commutator, e B̂ e− = B̂ + Â, B̂ , eÂ+B̂ = e− 2 [Â,B̂] e eB̂ , (43) it may be seen that the quantum operator generating the LGT of periods (k(p), k(d−p)) is k(p), k(d−p) k(p), k(d−p) γ,γ′,ǫ ×(ρ−1)γγ′ Iγǫ k (p) + τγǫ k (d−p) ĉγ′ − Iγǫ k (p) + τγǫ k (d−p) . (44) The 1-cocycle C k(p), k(d−p) will be determined presently. This operator (44) defines the action of LGT on the Hilbert space in the global TFT sector, k(p), k(d−p) Ψ(cγ) = γ,γ′,δ Iγδ k +τγδ k (d−p) (ρ−1)γγ Iγ′ǫ k +τγ′ǫ k (d−p) k(p), k(d−p) cγ + π Iγγ′ k + τγγ′ k (d−p)  , (45) where the BCH formula (43) has been used. However, a U(1)×U(1) 2-cocycle ω2(k; ℓ) appears in the composition law of this quantum representation, k(p) + ℓ(p), k(d−p) + ℓ(d−p) = e2 i π ω2(k;ℓ) Û k(p), k(d−p) ℓ(p), ℓ(d−p) ω2(k; ℓ) ≡ ω2 k(p), k(d−p); ℓ(p), ℓ(d−p) γ,γ′=1 (d−p) (d−p) The 1-cocycle C k(p), k(d−p) appearing in (44) may be determined by requiring that the abelian group composition law for LGT is recovered. This implies that ω2(k; ℓ) is a coboundary, ω2(k; ℓ) = C1 k(p) + ℓ(p), k(d−p) + ℓ(d−p) k(p), k(d−p) ℓ(p), ℓ(d−p) (mod Z), C(k) ≡ C k(p), k(d−p) 2 i π C1(k(p),k(d−p)). A careful analysis, analogous to the one in [20], finds that the unique solution to this coboundary condition is I k, C k(p), k(d−p) γ,γ′=1 i π k I k (d−p) Iγγ′ k (p) , (46) where k ∈ Z and6 I = det ∈ N. It is noteworthy to recall that although Iγγ′ is a rational valued matrix, I Iγγ′ is integer valued. Note also the quantisation condition arising for the coefficient κ multiplying the topological terms in the original action of TMGT. If k is rational, namely if k = k1/k2 with k1, k2 strictly positive natural numbers, invariance of physical states under LGT cannot be achieved. However in this case the LGT group has a finite 6Recall that κ, hence k is assumed to be strictly positive in the present discussion, while the situation for a negative κ or k is obtained through the exchange of the sectors aγ and bγ . dimensional projective representation which may be constructed by finding a normal subgroup gen- erated by the LGT operators. As demonstrated in [10], the TFT part of the physical wave functions carries a projective representation of the group of LGT while the above discussion establishes that the dimension of Hilbert space is δ=1 k1 k2 IMin(Iδδ′). Any state of a given irreducible repre- sentation gives the same matrix element for a physical observable. The characterisation of Hilbert space changes qualitatively for integer or rational values of k, but the theory remains well-defined. If we take k to be an integer, see (46), wave functions of the physical Hilbert space may be classified in terms of irreducible representations of the group of LGT (45), η1; cγ + π I k Iγγ′ k + τγγ′ k (d−p) γ,γ′,δ=1 Iγδ k (p) + τγδ k (d−p) (ρ−1)γγ cγ′ + Iγ′ǫ k (p) + τγ′ǫ k (d−p) γ,γ′=1 2 i π η1(k(p), k(d−p))− i π k I k (d−p) Iγγ′ k Ψ(η1; cγ) , (47) where the 1-cocycle η1 k(p), k(d−p) characterises the irreducible representation. Since for an abelian group each of its irreducible representations is one-dimensional, physical states correspond- ing to a given irreducible representation are singlet under LGT. As is well-known, functions obeying such a double periodicity condition are nothing other than the generalised Riemann theta functions defined in any dimension on the complex Np-torus [10], with the compact reduced phase space resulting from the requirement of invariance under LGT, (cγ) = γ,γ′=1 cγ (ρ −1)γγ (aδ′ + rδ′) − I k τ , (48) where rδ ∈ [0, k IMin(Iδδ′)− 1] ⊂ N. Each physical subspace, characterised by the 1-cocycle 1 (k(p), k(d−p)) = aγ k + bγ k (d−p) where aγ , bγ ∈ [0, 1[⊂ R, is invariant under a particular irreducible representation of LGT. The TFT component of each physical Hilbert space is of dimension k IMin(Iδδ′). In general, the choice of physical Hilbert space which is invariant under all LGT is the representation space with η1(k(p), k(d−p)) ∈ Z, namely corresponding to aγ , bγ = 0. 3.3 The dynamical sector: Hamiltonian diagonalisation Based on Hodge’s theorem, (22) and (24) define the decomposition of the dynamical sector into three decoupled subsectors of canonically conjugate variables: the global harmonic sector and the local (EL, PE) and (GL, QG) sectors. In turn the classical Hamiltonian (20) decomposes into three separate contributions, one for each subsector. When quantising the system in each subsector, the total quantum Hamiltonian follows from the classical one without any operator ordering ambiguity, Ĥ[Ê, Ĝ] = Ĥh[Êh, Ĝh] + Ĥ1[ÊL, P̂E ] + Ĥ2[ĜL, Q̂E ]. The physical spectrum is thus identified by diagonalising each of these contributions separately. 3.3.1 Global degrees of freedom The choice of normalisation used previously in the harmonic sector relies on the Poincaré duality between the basis elements [Xγ ] and [Y γ ] of the relevant cohomology groups and their associated homology generators Σ and Σ (d−p) , respectively, see (26). This choice is of a purely topological character. However in the dynamical sector, there is a remaining freedom as far as the normalisation of the choice of the harmonic representative of the cohomology group is concerned, depending on the metric structure, and thus fixing the basis elements Xγ of ker△(p) and Y γ of ker△(d−p). This choice involves the inner product in (13) on which the Hodge decomposition relies. Hence one sets Xγ ∧ ∗Xγ′ = Ωγγ′ , Yγ ∧ ∗Yγ′ = Ω̃γγ′ , (49) where Ωγγ′ and Ω̃γγ′ are Np × Np real symmetric matrices. Given this normalisation, the global part of the metric dependent quantum Hamiltonian operator constructed from (20) is expressed as Ĥh[Ê γ , Ĝγ γ,γ′=1 Êγ Êγ Ωγγ′ + Ĝ Ω̃γγ′ , (50) while the non vanishing commutation relations between the global phase space operators read Êγ , Ĝγ = −i~κ Iγγ′ . (51) As in the TFT sector, see (36), the following holomorphic polarisation of the global dynamical sector is used, Iγα Ê α − υγα Ĝα , d†γ = Iγα Ê α − υγα Ĝα where υ = ℜ(υ)+i σ is the Np×Np complex symmetric matrix characterising the complex structure introduced in the global dynamical phase space sector, of which the imaginary part determines the non vanishing commutation relations of the Fock like algebra dγ , d = σγγ′ . (52) In order to readily diagonalise the Hamiltonian in the global sector which is of the harmonic oscillator form, it is convenient to make the following choice for the complex structure matrix v as well as for the normalisation quantities specified in (49), ℜ(υ) = 0, σγγ′ = Ω̃γγ′ = δγγ′ , Ωγγ′ = α,β=1 Iαγ Iβγ′ δ αβ , (53) where δγγ is the Np × Np Kronecker symbol. With these choices, the contribution of the global variables to the Hamiltonian is indeed diagonal, ~µNp + ~µ γ,γ′=1 d†γ dγ′ δ γγ′ , µ = e g κ. One recognizes the Hamiltonian of a collection of Np independent harmonic oscillators of angular frequency7 µ = e g κ, which turns out to be the mass gap of the quantum field theory. The operators 7Recall that under the assumptions of the analysis, this combination of parameters is indeed positive. dγ and d γ are, respectively, annihilation and creation operators obeying the Fock algebra (52) now with σγγ′ = δγγ′ . The energy spectrum in the global dynamical sector of the system is readily identified. The normalised fundamental state is the kernel of all annihilation operators, dα |0〉 = 0, εh(0) = Np ~µ, 〈0|0〉 = 1, where εh is the vacuum energy. Excited states, |nγ〉, are obtained through the action of the Np creation operators d γ on the fundamental state. This leads to the energy eigenvalue for any of these states, |nγ〉 = |0〉, ε(nγ ) = ε (0) + ~µ nγ , (54) γ=1 being the eigenvalues of each of the number operators d γdγ , hence positive integers. 3.3.2 Local degrees of freedom on the torus The canonical treatment of the global degrees of freedom in both the TFT and dynamical sectors does not require the explicit specification of the space manifold Σ with its topology and Riemannian metric, yet allowing the general discussion of the previous Sections. However, in order to identify the full spectrum of dynamical physical states, the space manifold Σ including its geometry has now to be completely specified. The explicit choice to be made for the purpose of the present discussion is that of the d-dimensional Euclidean torus, Σ = Td, enabling straightforward Fourier mode analysis of the then infinite discrete, thus countable set of degrees of freedom, and diagonalisation of the harmonic oscillator structure of the Hamiltonian. This particular choice of the d-torus is motivated by the fact that this manifold is the simplest flat yet homologically non trivial manifold. The notations used are those of [21] where pure quantum electrodynamics is solved on the torus, of which the conventions are extended to any p-form. Accordingly, the variables E and G of the dynamical sector are periodic around the torus p- and (d− p)-cycles, respectively. Their Fourier mode expansions read i1...ip (~x ) = δi1j1 . . . δipjp k 6=0 αp−1=1 α1···αp j1··· jp (k)Eα1···αp(k) e2 i π k(~x) , where Eα1···αp(k) is a complex valued antisymmetric tensor and k are discrete vectors of the torus dual lattice of which the components are measured in units of L−1. Their norm is expressed as ω(k) = ki kj δ ij . Note that the zero modes of the fields are not included in these expressions, as emphasized by the subscript ⊥. In fact, these zero modes are the global degrees of freedom which have already been dealt with in the previous Section. The real valued tensors ε α1···αp i1···ip (k) define a basis of orthonormalised polarisation tensors for each k 6= 0. In our conventions, these tensors are constructed from a orthonormalised basis of polarisation vectors εαi (k) for a vector field such that εαi (k) ε j (k) δ ij = δαβ , (55) where δαβ is the Kronecker symbol in polarisation space. This basis is chosen in such a way that, for each k 6= 0, the dual lattice vector εd(k) is longitudinal whereas the vectors εα(k) are transverse for α = 1, · · · , d− 1. Finally, it is convenient to choose for the longitudinal vector εd(k) = , k 6= 0. Given the recursion relation induced by the Hodge decomposition theorem, the general polarisation tensor of any p-tensor field may be expressed as α1···αp i1···ip (k) = (k) . . . ε which may likewise be decomposed into transverse and longitudinal components, Longitudinal : α1···αp−1 d i1··· ip−1 ip α1,...,αp−1=1 ; Transverse : α1···αp i1···ip α1,...,αp=1 . (56) Given any mode, the C d degrees of freedom of a phase space field then separate into C d−1 longi- tudinal and C d−1 transverse degrees of freedom. These notations having been specified, and using the decompositions defined in (24), the relevant quantum operators are Fourier expanded as i1···ip (~x ) = k 6=0 δi1j1 . . . δipjp p αp−1=1 α1···αp−1 d j1··· jp−1 jp (k) Ê α1···αp−1 L (k) ǫi1···ip j1···jd−p (d− p− 1)! αd−p−1=1 α1···αd−p−1 d j1··· jd−p−1 jd−p (k) Q̂ α1···αd−p−1 G (k) e2 i π k(~x), ip···id−p (~x ) = k 6=0 δi1j1 . . . δid−pjd−p (d− p) αd−p−1=1 α1···αd−p−1 d j1··· jd−p−1 jd−p (k) Ĝ α1···αd−p−1 L (k) (p − 1)! ǫj1···jp i1···id−p αp−1=1 α1···αp−1 d j1··· jp−1 jp (k) P̂ α1···αp−1 E (k) e2 i π k(~x) . The self-adjoint property of the operator Êi1···ip(~x ) translates into the following relations between the associated mode operators and their adjoint, αp−1=1 α1···αp−1 d j1··· jp−1 jp (k) Ê α1···αp−1 L (k) = αp−1=1 α1···αp−1 d j1··· jp−1 jp (−k) ʆ α1···αp−1L (−k), αd−p−1=1 α1···αd−p−1 d i1··· id−p−1 id−p (k) Q̂ α1···αd−p−1 G (k) = αd−p−1=1 α1···αd−p−1 d i1··· id−p−1 id−p (−k) Q̂† α1···αd−p−1G (−k). Similar relations apply for the modes of the self-adjoint operator Ĝip···id−p(~x ). Consequently, this decomposition of the non zero modes of the field operators in the dynamical sector leads to two decoupled subsectors, each of which is comprised of a countable set of mode operators with k 6= 0. In the first subsector one has the operators ÊL(k) and P̂E(k) with the following non vanishing commutation relations, † α1···αp−1 L (k), P̂ β1···βp−1 δα1[β1 . . . δαp−1 βp−1]δkk′ , (57) while in the second subsector the operators ĜL(k) and Q̂G(k) possess the commutator algebra, † α1···αd−p−1 L (k), Q̂ β1···βd−p−1 δα1[β1 . . . δαd−p−1 βd−p−1]δkk′ , (58) V being the volume of the space torus Σ = Td. This Fourier mode decomposition reduces the problem of diagonalising the Hamiltonian to a simple exercise in decoupled quantum oscillators, with Ĥ1[ÊL, P̂E ] = 2 (p − 1)! k 6=0 κ2 g2 α1···αp−1 E (k) ω̃2(k) κ2 g2 α1···αp−1 L (k) , (59) Ĥ2[ĜL, Q̂E ] = V κ2 e2 2 (d − p− 1)! k 6=0 α1···αd−p−1 G (k) ω̃2(k) κ4 e4 α1···αd−p−1 L (k) . (60) In these expressions the following notation is being used, α1···αp−1 L (k) α1,...,αp−1=1 β1,...,βp−1=1 α1···αp−1 L (k) Ê † β1···βp−1 L (k) δ α1β1 . . . δαp−1βp−1 . The operators (59) and (60) are nothing other than the Hamiltonians of a collection of C d−1 and d−1 independent harmonic oscillators, respectively, all of angular frequency ω̃(k) = 4π2 ω2(k) + µ2, µ = e g κ. The physical spectrummay easily be constructed by introducing annihilation and creation operators associated to the algebras (57) and (58). The annihilation operators are defined by α1···αp−1(k) = V ω̃(k) α1···αp−1 L (k) + i g2 κ2 ω̃(k) α1···αp−1 E (k) α1···αd−p−1(k) = V ω̃(k) α1···αd−p−1 L (k) + i κ2 e2 ω̃(k) α1···αd−p−1 G (k) whereas the creation operators a† α1···αp−1(k) and b† α1···αd−p−1(k) are merely the adjoint operators of aα1···αp−1(k) and bα1···αd−p−1(k), respectively. One then establishes the Fock algebras, α1···αp−1(k), a† β1···βp−1(k′) = δα1 [β1 . . . δαp−1 βp−1] δkk′ , α1···αd−p−1(k), b† β1···βd−p−1(k′) = δα1 [β1 . . . δαd−p−1 βd−p−1] δkk′ , (61) whereas (59) and (60) then reduce to the simple expressions, Ĥ1[a, a †] = ~ k 6=0 ω̃(k) d−1 + α1<···<αp−1 † α1···αp−1(k) aα1···αp−1(k) Ĥ2[b, b †] = ~ k 6=0 ω̃(k) d−1 + α1<···<αd−p−1 † α1···αd−p−1(k) bα1···αd−p−1(k) The Fock space representation is based on the normalised Fock vacuum |0〉, 〈0|0〉 = 1, which is the kernel of all annihilation operators α1···αp−1(k) |0〉 = 0, bα1···αd−p−1(k) |0〉 = 0, ε1+2 k 6=0 ω̃(k), where ε1+2 is the divergent total vacuum energy. Excited states are obtained through the action onto the Fock vacuum of all C creation operators, see (61). This leads to states |nγ(k)〉 with energy eigenvalues ε(nγ(k)) = ε k 6=0 nγ(k) ω̃(k), (62) where {nγ(k)} γ=1 are positive integers corresponding to number operator eigenvalues. A shorthand notation is used in (62) with the index γ labelling the C d possible combinations of a set of p distinct integers in the range [1, d], {α1, . . . , αi, . . . , αp}dαi=1, which will be referred to as Γ 4 Spectrum and Projection onto the TFT Sector 4.1 Physical spectrum on the torus Combining all the results of the previous Sections for what concerns the diagonalisation of the physical TMGT Hamiltonian on the spatial d-torus Σ = Td, the complete energy spectrum of states is given as ε(nγ(k)) = ε(0) + ~ nγ(k) ω̃(k) , (63) which is the sum of the contributions (54) and (62). Note that on the d-torus, the pth Betti number, Np, equals C . The components of the vector k of the dual lattice may take any integer values since it is implicit in (63) that {nγ(0)} γ=1 = {nγ} γ=1. However, the index γ has a different meaning whether k 6= 0 or k = 0. In the first case it refers to a value in the set Γpd and denotes one of the possible C d polarisations, while in the second case it is a (co)homology index, γ = 1, · · · , C d . The total vacuum energy ε(0) in (63) is divergent, ε(0) = ω̃(k) , and must be subtracted from the energy spectrum. The positive integer valued functions nγ(k) count, for each k 6= 0, the number of massive quanta of a p- or (d− p)-tensor field of momentum 2π~ k, of polarisation (56), namely Transverse : ε i1···ip (k), γ ∈ Γp ; Longitudinal : ε i1···ip (k), γ ∈ Γp−1 and of rest mass8 M = ~µ = ~κ e g. (64) There are also the contributions of the global quanta of the p- and (d − p)-tensor fields, where {nγ(0)} γ=1 count the numbers of excitations along the homology cycle generators Σ and Σ (d−p) 8A quantity indeed positive under the assumptions made. In the particular case when p = 1, the integers {nγ(k)}dγ=1 count, for each k 6= 0, the number of massive photons of momentum 2π~ k, of rest mass M and of polarisation Transverse : {εγi (k)} , Longitudinal : εdi (k) . Depending on how one deals with large gauge transformations in the TFT sector, each energy state is either infinitely degenerate for a real valued k, see (46), or ( δ=1 k1 k2 IMin(Iδδ′ )) times degenerate if k is a rational number of the form k = k1/k2. If k is an integer, each energy state is k IMin(Iδδ′)) times degenerate and the mass gap is then quantised, I k e g . In the Maxwell-Chern-Simons (MCS) case in 2 + 1 dimensions, we recover in the global sector a quantum mechanical system corresponding to the Landau problem of condensed matter physics on the 2-torus. 4.2 Projection onto the topological field theory sector At least formally, the naive limit of infinite coupling constants, e → ∞ and g → ∞, in the classical Lagrangian of topologically massive gauge theories, see (4) and (5), must lead to a pure topological field theory (TFT) of the B ∧ F or A ∧ F type, see (1) or (3). However, as pointed out by several authors (see for example [19, 22]), a paradox seems to arise at the quantum level (as well as within the classical Hamiltonian formulation) when the pure Chern-Simons (CS) theory is viewed as the limit e → ∞ of the Maxwell-Chern-Simons (MCS) theory. The Hilbert space of the CS theory is constructed from the algebra of the non commuting configuration space operators which are in fact canonically conjugate phase space operators. As far as the MCS theory is concerned, its Hilbert space is constructed from the Heisenberg algebras of twice as many phase space operators. This problem is generic whenever a pure quantum TFT (TQFT) is considered as the limit of its associated TMGT because two distinct Hilbert spaces are being compared. Actually, due to the second-class constraints appearing in the Hamiltonian analysis of a TFT which is already in Hamiltonian form, non vanishing commutation relations apply to the configuration space operators. Furthermore, the Gauss law constraints of pure TFT are not the limit of the Gauss law constraints of TMGT. The former operators tend to restrict too drastically the physical Hilbert space in comparison to the limit of the TMGT physical Hilbert space. This problem of an ill-defined limit is usually handled by projecting from the Hilbert space of the TMGT onto its degenerate ground state. This projection actually acts in a manner similar to second-class constraints which then lead to a reduced phase space and non vanishing configuration space commutation relations determined from the associated Dirac brackets. The global sector of the MCS theory is analogous to the classical Landau problem of a charged point particle of mass m moving in a two dimensional surface in the presence of an uniform external magnetic field B perpendicular to that surface. Within the latter context the mass gap (64) corresponds to the cyclotron frequency ωc [22, 23]. The spectrum of the quantised model is organised into Landau levels (with a degeneracy dependent on the underlying manifold), of which the energy separation ωc is proportional to the ratio B/m. The limit B → ∞ or m → 0 effectively projects onto the lowest Landau level (LLL) in which one obtains a non commuting algebra for the space coordinates. By analogy, projection onto the ground state reduces the phase space of the MCS theory to the canonically conjugate configuration space operators of a pure CS theory. In the global sector, the projection from a general TMGT, (4), to a pure TQFT offers in some sense a generalisation of the LLL projection in any dimension. Considering that the mass gap (64) of the TMGT becomes infinite for coupling constants running to infinity, all excited states decouple from the physical spectrum, leaving over only the degenerate ground states. Projection onto these ground states restricts the Hilbert space to that of a TQFT. Interestingly, the PT factorisation established in this work enables the usual projection from TMGT to TQFT to be defined in a natural way. Already in the classical Hamiltonian formulation phase space is separated into two decoupled sectors, the first being dynamical and manifestly gauge invariant, and the second being equivalent to a pure TFT with identical Gauss law constraints and commutation relations. As a matter of fact in the present approach which does not require any gauge fixing procedure whatsoever, the non commuting sector of a CS theory or, more generally, the reduced phase space of a TQFT appears no longer after the projection onto the ground state at the quantum level (or after the introduction of Dirac brackets) but is manifest already at the classical Hamiltonian level. By letting e or g grow infinite, the mass gap (64) becomes infinite, hence dynamical massive excitations decouple whereas the TFT sector, which is independent of the coupling constants, remains unaffected. In this limit, the system looses any dynamics, the latter being intimately related to the Riemannian metric structure of the spacetime manifold, while all that is then left is a wave function depending on global variables only, namely the quantum states of a TQFT. 5 Conclusion and Outlook The main result of this paper is the identification of a Physical-Topological (PT) factorisation of the classical phase space of abelian topologically massive gauge theories (TMGT) in any dimension, into a manifestly gauge invariant and dynamical sector of non commuting “electric fields” and a gauge variant purely topological sector of the B ∧ F or A ∧ F type. This factorisation is achieved through a canonical transformation in the phase space of TMGT. The discussion considers the most general action for abelian TMGT in any dimension and for any p-form fields, including the two possible types of topological terms related through an integration by parts. The clue to this PT factorisation relies on the identification of a topological field theory embedded in the full TMGT, which is not manifest in the original Hamiltonian formulation. Let us emphasize that the procedure does not require any gauge fixing choice whatsoever, with its cortege of second-class constraints or ghost degrees of freedom. Rather, the PT classical factorisation readily allows for a straightforward quantisation of these systems and the identification of their spectrum of gauge invariant physical states, accounting also for all the topological features inherent to such dynamics. In the early 1990’s, A. P. Balachandran and P. Teotonio-Sobrinho [24] established that it is possible to identify among the phase space variables of a TMGT combinations corresponding to those of a TFT. They noticed that, in a particular case of an underlying manifold with boundary, edge states may be understood in terms of a TFT, already at the classical level. Nevertheless, this paper did not realise the powerful gauge fixing free factorisation of the theory into two decoupled sectors as described in the present work. Incidentally, it should be of interest to analyse how this new approach may shed new light onto this paper, in a manner akin to that in which it makes most transparent and natural the projection onto a topological field theory through the limits e, g → ∞, generalising the concept of projection onto the lowest Landau level of the Landau problem. In the present approach, the TFT sector with its reduced phase space is actually made manifest already at the classical level, independently of any projection onto the quantum ground state, or any classical projection onto physical edge states in the case of a manifold with boundary. The formalism of TMGT defined by the actions in (4) or (5) offers a possible description of some phenomenological phenomena such as effective superconductivity [24, 25], Josephson ar- rays [26], etc. Furthermore, it is certainly of interest to investigate the perspectives offered by the PT factorisation when a B ∧F or A∧F field theory is coupled to (non)relativistic matter fields as an effective description of phenomena related, for example, to QCD confinement [27]. It is also of interest to extend this approach to Yang-Mills-Chern-Simons theories [15, 23] or to the nonabelian generalisation of the Cremmer-Scherk theory which requires the introduction of extra fields or to allow for non renormalisable couplings since the generalisation to a local, power counting renor- malisable action while preserving the same field content and the same number of local symmetries as the abelian theory (4) is not possible (see [28] and references therein). However the long term goal is to gain a deeper understanding of the influence of topological terms, such as the topological mass gap, and of topological sectors in field configuration space on the nonperturbative dynamics of gauge theories, beginning with Yang-Mills theories coupled to whether fermionic or bosonic matter fields. Acknowledgments The work of B. B. is supported by a Ph.D. Fellowship of the “Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture” (FRIA), of the Associated Funds of the National Fund for Scientific Research (F.N.R.S., Belgium). J. G. acknowledges the Institute of Theoretical Physics for an Invited Research Staff position at the University of Stellenbosch (Republic of South Africa). He is most grateful to Profs. Hendrik Geyer and Frederik Scholtz, and the School of Physics for their warm and generous hospitality during his sabbatical leave, and for financial support. His stay in South Africa is also supported in part by the Belgian National Fund for Scientific Research (F.N.R.S.) through a travel grant. J. G. acknowledges the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) Visiting Scholar Programme in support of a Visiting Professorship at the ICMPA. This work is also supported by the Institut Interuniversitaire des Sciences Nucléaires and by the Belgian Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity Attraction Poles (IAP) P5/27 and P6/11. References [1] D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Phys. Rept., 209, 129 (1991). [2] A. S. Schwarz, Lett. Math. Phys. 2, 247 (1978); Commun. Math. Phys. 67, 1 (1979). [3] E. Witten, J. Diff. Geom. 17, 661 (1982). [4] G. T. Horowitz, Commun. Math. Phys. 125, 417 (1989). [5] M. Blau and G. Thompson, Ann. Phys. 205, 130 (1991). [6] E. Witten, Commun. Math. Phys. 121, 351 (1989). [7] J. Barcelos-Neto and E. C. Marino, Europhys. Lett. 57, 473 (2002) [e-print arXiv:hep-th/0107109]. [8] X. Fustero, R. Gambini and A. Trias, Phys. Rev. Lett. 62, 1964 (1989); J.A. Harvey and J. Liu, Phys. Lett. B240, 369 (1990). [9] J. C. Baez, D. K. Wise and A. S. Crans, Exotic Statistics for Loops in 4d BF Theory, e-print arXiv:gr-qc/0603085 (March 2006). http://arxiv.org/abs/hep-th/0107109 http://arxiv.org/abs/gr-qc/0603085 [10] R. J. Szabo, Ann. Phys. 280, 163 (2000) [e-print arXiv:hep-th/9908051]. [11] M. Bergeron, G. W. Semenoff and R. J. Szabo, Nucl. Phys. B437, 695 (1995) [e-print arXiv:hep-th/9407020]. [12] E. Cremmer and J. Scherk, Nucl. Phys. B72, 117 (1974); C. R. Hagen, Phys. Rev. D19, 2367 (1979). [13] T. J. Allen, M. J. Bowick and A. Lahiri, Mod. Phys. Lett. A6, 559 (1991). [14] R. Jackiw, R. and S.-Y. Pi, Phys. Lett. B403, 297 (1997) [e-print arXiv:hep-th/9703226]. [15] R. Jackiw and S. Templeton, Phys. Rev. D23, 2291 (1981); J. F. Schonfeld, Nucl. Phys. B185, 157 (1981). [16] J. Govaerts, Hamiltonian Quantisation and Constrained Dynamics (Leuven University Press, Leuven,1991). [17] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992). [18] M. Nakahara, Geometry, Topology and Physics (Adam Hilger, Bristol, 1990). [19] A. P. Polychronakos, Ann. Phys. 203, 231 (1990). [20] J. Govaerts and B. Deschepper, J. Phys. A33, 1031 (2000) [e-print arXiv:hep-th/9909221]. [21] F. Payen, Théories de Yang-Mills pures et théories des champs quantiques topologiques, Master Degree Thesis, Catholic University of Louvain (Louvain-la-Neuve, September 2004), unpub- lished. [22] G. V. Dunne, R. Jackiw and C. A. Trugenberger, Phys. Rev. D41, 661 (1990). [23] G. V. Dunne, Aspects of Chern-Simons Theory, e-print arXiv:hep-th/9902115 (February 1999). [24] A. P. Balachandran and P. Teotonio-Sobrinho, Int. J. Mod. Phys. A8, 723 (1993). [25] M. C. Diamantini, P. Sodano and C. A. Trugenberger, Nucl. Phys. B474, 641 (1996) [e-print arXiv:hep-th/9511168]. [26] M. C. Diamantini, P. Sodano and C. A. Trugenberger, Eur. Phys. J. B53, 19 (2006) [e-print arXiv:hep-th/0511192]. [27] A. Sugamoto, Phys. Rev. D19, 1820 (1979) ; C. Chatterjee and A. Lahiri, Europhys. Lett. 76, 1068 (2006) [e-print arXiv:hep-ph/0605107]. [28] M. Henneaux, V. E. R. Lemes, C. A. G. Sasaki, S. P. Sorella, O. S. Ventura and L. C. Q. Vilar, Phys. Lett. B410, 195 (1997) [e-print arXiv:hep-th/9707129]. http://arxiv.org/abs/hep-th/9908051 http://arxiv.org/abs/hep-th/9407020 http://arxiv.org/abs/hep-th/9703226 http://arxiv.org/abs/hep-th/9909221 http://arxiv.org/abs/hep-th/9902115 http://arxiv.org/abs/hep-th/9511168 http://arxiv.org/abs/hep-th/0511192 http://arxiv.org/abs/hep-ph/0605107 http://arxiv.org/abs/hep-th/9707129 Introduction Gauge Invariant Factorisation of the Classical Theory Hamiltonian formulation The Physical-Topological (PT) factorisation Hodge decomposition Large and small gauge transformations Canonical Quantisation and Physical States Physical Hilbert space factorisation The topological sector: Gauss' constraints and LGT Hilbert space and holomorphic polarisation Gauss law constraints LGT and global variables The dynamical sector: Hamiltonian diagonalisation Global degrees of freedom Local degrees of freedom on the torus Spectrum and Projection onto the TFT Sector Physical spectrum on the torus Projection onto the topological field theory sector Conclusion and Outlook

0704.1513

Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium

Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium M. A. Noginov1*, V. A. Podolskiy2, G. Zhu1, M. Mayy1, M. Bahoura1, J. A. Adegoke1, B. A. Ritzo1, K. Reynolds1 1 Center for Materials Research, Norfolk State University, Norfolk, VA 23504 * [email protected] 2 Department of Physics, Oregon State University, Corvallis, OR 97331-6507 Abstract: We report the suppression of loss of surface plasmon polariton propagating at the interface between silver film and optically pumped polymer with dye. Large magnitude of the effect enables a variety of applications of ‘active’ nanoplasmonics. The experimental study is accompanied by the development of the analytical description of the phenomenon and the solution of the controversy regarding the direction of the wavevector of a wave with a strong evanescent component in an active medium. Surface plasmon polaritons (SPPs) – special type of electromagnetic waves coupled to electron density oscillations – allow nanoscale confinement of electromagnetic radiation [1]. SPPs are broadly used in photonic and optoelectronic devices [1-7], including waveguides, couplers, splitters, add/drop filters, and quantum cascade lasers. SPP is also the enabling mechanism for a number of negative refractive index materials (NIMs) [8-12]. Many applications of SPPs suffer from damping caused by absorption in metals. Over the years, several proposals to compensate loss by incorporating active (gain) media into plasmonic systems have been made. Theoretically, field-matching approach was employed to calculate the reflectivity at surface plasmon excitation [13]; the authors of [14] proposed that the optical gain in a dielectric medium can elongate the SPP’s propagation length; gain-assisted excitation of resonant SPPs was predicted in [15]; SPP propagation in active waveguides was studied in [16]; and the group velocity modulation of SPPs in nano-waveguides was discussed in [17]. Excitation of localized plasmon fields in active nanosystems using surface plasmon amplification by stimulated emission of radiation (SPASER) was proposed in [18]. Experimentally, the possibility to influence SPPs by optical gain was demonstrated in Ref. [19], where the effect was as small as 0.001%. Here we report conquering the loss of propagating SPPs at the interface between silver film and optically pumped polymer with dye. The achieved value of gain, ≈ 420 cm-1, is sufficient to fully compensate the intrinsic SPP loss in high-quality silver films. This, together with the compensation of loss in localized surface plasmons, predicted in [20] and recently demonstrated in [21], enables practical applications of a broad range of low-loss and no-loss photonic metamaterials. The experimental attenuated total reflection (ATR) setup consisted of a glass prism with the real dielectric permittivity e0=n02, a layer of metal with the complex dielectric constant e1 and thickness d1, and a layer of dielectric medium characterized by the permittivity e2, Fig. 1a. The wave vector of the SPP propagating at the boundary between media 1 and 2, is given by e1 + e2 , (1) where w is the oscillation frequency and c is the speed of light. SPP can be excited by a p polarized light falling on the metallic film at the critical angle q0, such that the projection of the wave vector of the light wave to the axis x, kx q( ) = (w /c)n0 sinq0 , (2) is equal to Re(kx 0) . At this resonant condition, the energy of incident light is transferred to the SPP, yielding a minimum (dip) in the angular dependence of the reflectivity R(q) [1]: R q( ) = r01 + r12 exp 2ikz1d1( ) 1+ r01r12 exp 2ikz1d1( ) , (3) where rik = kziek - kzkei( ) kziek + kzkei( ) and kzi = ± ei - kx q( ) 2 , i = 0,1,2 . (4) The parameter kzc /w defines the field distribution along the z direction. Its real part can be associated with a tilt of phase-fronts of the waves propagating in the media [16], and is often discussed in the content of positive vs. negative refractive index materials (see also Refs. [8-12, 22-24]), while its imaginary part defines the wave attenuation or growth. The sign of the square root in Eq. (4) is selected to enforce the causal energy propagation. For dielectrics excited in total internal reflection geometry, as well as for metals and other media with Re[kz 2] < 0 , which do not support propagating waves, the imaginary part of the square root should be always positive regardless of the sign of e”. For other systems, the selection should enforce the wave decay in systems with loss (e”>0) and the wave growth in materials with gain (e”<0) [23]. This selection of the sign can be achieved by the cut of the complex plane along the negative imaginary axis. Although such cut of the complex plane is different from the commonly accepted cuts along the positive [22] or negative [8-12,23] real axes, our simulations (Fig.1b) show that this is the only solution guaranteeing the continuity of measurable parameters (such as reflectivity) under the transition from a weak loss regime to a weak gain regime. Our sign selection is the only one consistent with previous results on gain-assisted reflection enhancement, predictions of gain- assisted SPP behavior [13-15,25], and the experimental data presented below. The implications of selecting different signs of kz2 are shown in Fig.1b. Note that active media excited above the angle of total internal reflection, as well as the materials with e’<0 and e”<0 formally fall under negative index materials category. However, since | kz"| in this case is greater than | kz ' |, the “left handed” wave experiences very large attenuation (in the presence of gain!!!), which in contrast to claims of Ref. [22], makes the material unsuitable for superlenses and other proposed applications of NIMs [8-12]. In the limit of small plasmonic loss/gain, when the decay length of SPP, L, is much greater than 2p/kx0’, and in the vicinity of q0, Eq. (3) can be simplified, revealing the physics behind the gain-assisted plasmonic loss compensation: R q( ) ª r01 4g ig r + d(q) (kx - kx 0 - Dkx 0)2 + g i + g r( ) , (5) where ' -e1 ' + e1 3 / 2 0 = r01(q0) , and d(q) = 4(kx - kx 0 - Dkx 0)Im(r0 )Im(e 0d1 ) x . The shape of R(q) is dominated by the Lorentzian term in Eq. (5). Its width is determined by the propagation length of SPP, L = 2 g i + g r( )[ ] , (6) which, in turn, is defined by the sum of the internal (or propagation) loss g i = kx ' + e2 3 / 2 ˜ . (7) and the radiation loss caused by SPP leakage into the prism, g r = Im r01e 0d1( ) /x . (8) The radiation loss also leads to the shift of the extremum of the Lorentzian profile from its resonant position kx0 , 0 = Re r01e 0d1( ) /x . (9) The term d in Eq. (5) results in the asymmetry of R(q). The excellent agreement between exact Eq. (3), solutions of Maxwell equations using transfer matrix method [26] and approximate Eqs. (5,6) for the 60 nm silver film are shown in Figs. 1,3. The gain in the medium reduces internal loss gi of SPP, Eq. 7. In reasonably thick metallic films (where gi>gr in the absence of gain) the “dip” in the reflectivity profile Rmin is reduced when gain is first added to the system, reaching Rmin=0 at gi≈gr (Fig. 2a). With further increase of gain, gi becomes smaller than gp, leading to an increase of Rmin. The resonant value of R is equal to unity when internal loss is completely compensated by gain (gi=0) at e2"= - e1"e2' 2 . (10) In the vicinity of gi=0, the reflectivity profile is dominated by the asymmetric term d. When gain is increased to even higher values, gi becomes negative and the dip in the reflectivity profile converts into a peak, consistent with predictions of Refs. [13,14]. The peak has a singularity when the gain compensates total SPP loss (gi+gr=0). Past the singularity point, the system becomes unstable and cannot be described by stationary Eqs. (3-5) [27]. Instead, one should consider the rate equations describing populations of energy states of dye as well as a coupling between excited molecules and the SPP field. In thin metallic films (when gi<gr at e2”=0), the resonant value of R monotonically grows with the increase of gain, Fig. 2b. Experimentally, SPPs were studied in the attenuated total internal reflection setup of Fig. 1a. The 90o degree prism was made of glass with the index of refraction n0=1.784. Metallic films were produced by evaporating 99.99% pure silver. Rhodamine 6G dye (R6G) and polymethyl methacrylate (PMMA) were dissolved in dichloromethane. The solutions were deposited to the surface of silver and dried to a film. In the majority of experiments, the concentration of dye in dry PMMA was equal to 10 g/l (2.1x10-2 M) and the thickness of the polymer film was of the order of 10 mm. The prism was mounted on a motorized goniometer. The reflectivity R was probed with p polarized He-Ne laser beam at l=594 nm. The reflected light was detected by a photodiode or a photomultiplier tube (PMT) connected to the integrating sphere, which was moved during the scan to follow the walk of the beam. The permittivity of metallic film was determined by fitting the experimental reflectivity profile R(q) of not pumped system with Eq.(3), inset of Fig. 4a. As a rule, experimental values e1’ and e1” did not coincide with the commonly used data of Ref. [28]. In the measurements with optical gain, the R6G/PMMA film was pumped from the back side of the prism (Fig. 1a) with Q-switched pulses of the frequency doubled Nd:YAG laser (l=532 nm, tpulse=10 ns, repetition rate 10 Hz). The pumped spot, with the diameter of ~3 mm, completely overlapped the smaller spot of the He-Ne probe beam. Reflected He-Ne laser light was directed to the entrance slit of the monochromator, set at l=594 nm, with PMT attached to the monochromator’s exit slit. Experimentally, we recorded reflectivity kinetics R(q,t) under short-pulsed pumping at different incidence angles (Fig. 4b). In samples with relatively thin (≈ 40 nm) metallic films, strong emission signal from the R6G- PMMA film was observed in the absence of He-Ne probe beam. We therefore performed two measurements of kinetics for each data point: one in the absence of the probe beam, and one in the presence of the beam. We then subtracted “emission background” (measured without He-Ne laser) from the combined reflectivity and emission signal. The kinetics measurements had a relatively large data scatter, which was partially due to the instability of the Nd:YAG laser. The results of the reflectivity measurements in the 39 nm silver film are summarized in Fig. 4a. Two sets of data points correspond to the reflectivity without pumping (measured in flat parts of the kinetics before the laser pulse) and with pumping (measured in the peaks of the kinetics). By dividing the values of R measured in the presence of gain by those without gain, we calculated the relative enhancement of the reflectivity signal to be as high as 280% – a significant improvement in comparison to Ref. [19], where the change of the reflectivity in the presence of gain did not exceed 0.001%. Fitting both reflectivity curves with Eq. (3) and known e'1=-15, e"2=0.85 and e2’= n22=2.25, yields e2”≈-0.006. For l =594 nm, this corresponds to optical gain of g=420 cm-1. In thicker silver films, calculations predict initial reduction of the minimal reflectivity R(q) at small values of gain followed by its increase (after passing the minimum point R=0) at larger gains, Fig. 2a. The predicted reduction of R was experimentally observed in the 90 nm thick film, where instead of a peak in the reflectivity kinetics, we observed a dip, inset of Fig. 4b. For the silver film parameters measured in our experiment, Eq. (5) predicts complete compensation of intrinsic SPP loss at optical gain of 1310 cm-1. For a better quality silver characterized by the dielectric constant of Ref. [28], the critical gain is smaller, equal to 600 cm-1. In addition, if a solution of R6G in methanol (n=1.329) is used instead of the R6G/PMMA film, then the critical value of gain is further reduced to 420 cm-1. This is the value of gain achieved in our experiment. Thus, in principle, at the available gain, one can fully compensate the intrinsic SPP loss in silver. For complete compensation of plasmonic loss in the system, one must also compensate radiation losses. The huge gain equal to 4090 cm-1 is required to completely compensate attenuation of SPP in the 39 nm thick film used in our experiment. This value is dramatically reduced in thicker metallic films, since radiative loss strongly depends on the film thickness. For relatively thick (≥100 nm) metallic films, the total loss is almost identical to the internal loss. In the experiment described above, the concentration of R6G molecules in the PMMA film was equal to 1.26x1019 cm-3 (2.1x10-2M). Using the spectroscopic parameters known for the solution of R6G dye in methanol and neglecting any stimulated emission effects, one can estimate that 18 mJ laser pulses used in the experiment should excite more than 95% of all dye molecules. At the emission cross section equal to 2.7x10-16 cm2 at l=594 nm, this concentration of excited molecules corresponds to the gain of 3220 cm-1. Nearly eight-fold difference between this value and the one obtained in our experiment is probably due to the combined effects of luminescence quenching of R6G due to dimerization of rhodamine 6G molecules occurring at high concentration of dye [29], and amplified spontaneous emission (ASE). While the detailed study of the ASE-induced effects in the R6G/PMMA-silver systems is beyond the scope of this work, we note that at the value of gain equal to 420 cm-1 and the diameter of the pumped spot equal to 3 mm, the optical amplification is enormously large. Obviously, these giant values of the amplification and the gain cannot be maintained in a cw regime, and ASE appears to be a detrimental factor controlling the gain in the pulsed regime. Correspondingly, the choice of a more efficient amplifying medium (as was proposed in Ref. [19]) may not help in compensating the SPP loss by gain. To summarize, in our study of the propagating surface plasmon polariton in the attenuated total reflection setup, we have established the relationship between (i) the gain in the dielectric adjacent to the metallic film, (ii) the internal, radiative and total losses, (iii) the propagation length of the SPP, and (iv) the shape of the experimentally measured reflectivity profile R(q). We have experimentally demonstrated the optical gain in the dielectric (PMMA film with R6G dye) equal to 420 cm-1, yielding nearly threefold increase of the resonant value of the reflectivity. In the case of thick low-loss silver film [28] and low index dielectric, the demonstrated value of gain is sufficient for compensation of the total loss hindering the propagation of surface plasmon polariton. The work was supported by the NSF PREM grant # DMR 0611430, the NSF CREST grant # HRD 0317722, the NSF NCN grant # EEC-0228390, the NASA URC grant # NCC3-1035, and the Petroleum Research Fund. The authors cordially thank Vladimir M. Shalaev for useful discussions. References 1. H. Raether, “Surface plasmons on smooth and rough surfaces and on gratings”, Springer- Verlag, (Berlin, 1988). 2. S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, Phys. Rev. Lett 89, 186801 (2002). 3. A. Boltasseva, S. I. Bozhevolnyi, , T. Søndergaard, T. Nikolajsen, and L. Kristjan, Optics Express 13, 4237-4243 (2005). 4. S. A., Maier, et. al., Nature Materials 2, 229-232 (2003). 5. A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and S. Marin, Phys. Rev. Lett. 95, 063901 (2005). 6. M. Stockman, Phys. Rev. Lett. 93, 137404 (2004). 7. C. Sirtori et.al., Opt. Lett. 23, 1366-1368 (1998). 8. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonl. Opt. Phys. Mat. 11, 65-74 (2002). 9. H. Shin, and S. Fan, Phys. Rev. Lett. 96, 073907 (2006). 10. V. M. Shalaev et. al., Opt. Lett. 30, 3356-3358 (2005). 11. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Science 312, 892-894 (2006). 12. A. Alu, and N. Engheta, J. Opt. Soc. Am. B 23, 571-583 (2006). 13. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, J. Opt. Soc. Am. 69, 419-421 (1979). 14. A. N. Sudarkin and P. A. Demkovich, Sov. Phys. Tech. Phys. 34, 764-766 (1989). 15. I. Avrutsky, Phys. Rev. B 70, 155416 (2004). 16. M. P. Nezhad, K. Tetz and Y. Fainman, Optics Express 12, 4072-4079 (2004). 17. A. A. Govyadinov and V. A. Podolskiy, Phys. Rev. Lett. 97, 223902 (2006). 18. D. Bergman and M. Stockman, Phys. Rev. Lett. 90, 027402 (2003). 19. J. Seidel, S. Grafstroem, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005). 20. N. M. Lawandy, Appl. Phys. Lett. 85, 5040-5042 (2004). 21. M. A. Noginov et. al., Opt. Lett. 31, 3022-3024 (2006). 22. Y. Chen, P. Fisher, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005). 23. T. Mackay, and A. Lakhtakia, Phys. Rev. Lett. 96. 159701 (2006) 24. Y. Chen, P. Fisher and F. W. Wise, Phys. Rev. Lett. 96, 159702 (2006). 25. B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, JETP Lett. 16, 100-105 (1972). 26. I. Avrutsky, J. Opt. Soc. A 20, 548-556 (2003). 27. X. Ma and C. M. Soukoulis, Physica B, 296, 107-111 (2001). 28. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370-4379 (1972). 29. K. Selanger, A. J. Falnes and T. Sikkeland, J. Phys. Chem. 81, 1960-1963 (1977). Figure captions Fig.1 (a) Schematic of SPP excitation in ATR geometry. (b) Reflectivity R as a function of angle q. Traces – solutions of exact Eq. (3). Dots – solution of approximate Eq. (6). For all data sets: e1=-15.584+0.424i, d1=60 nm. Trace 1: dielectric with very small loss, e2=2.25+10-5i. Traces 2-4: dielectric with very small gain, e2=2.25-10-5i. Trace 2 and dots: complex cut along negative imaginary axis (correct; nearly overlaps with trace 1; no discontinuity at the transition from small loss to small gain). Trace 3: complex cut along positive real axis (yields incorrect predictions for incident angles below total internal reflection). Trace 4: complex cut along negative real axis (yields incorrect predictions for incident angles above total internal reflection). Fig.2. Reflectivity R [Eq.(5)] of the three-layer system depicted in Fig. 1a as a function of angle q and pumping (given by imaginary part of e2); panel (a) illustrates the evolution of reflectivity in a relatively thick metallic film (d1=70nm); panel (b) corresponds to a thin film (d2=39 nm). Fig. 3. Inverse propagation length of SPP, L-1, in the system depicted in Fig.1a as a function of gain in dielectric, e2”. Solid line – solution of Eq. (11), dots – exact numerical solution of Maxwell equations. Top inset: intensity distribution across the system. Bottom inset: Exponential decay of the SPP wave intensity E2 (shown in the top inset) along the propagation in the x direction. Fig. 4. (a) Reflectivity R(q) measured without (diamonds) and with (circles) optical pumping in the glass-silver-R6G/PMMA system. Dashed lines – guides for eye. Solid lines – fitting with Eq. (3) at e0’=n02=1.7842=3.183, e0”=0, e1’=-15, e1”=0.85, d1=39 nm, e2’= n22=1.52=2.25, e2”≈0 (trace 1) and e2”≈-0.006 (trace 2). Inset: Angular reflectivity profile R(q) recorded in the same system without pumping (dots) and its fitting with Eq. (3) (solid line). (b) Reflectivity kinetics recorded in the glass-silver-R6G/PMMA structure under pumping. The angle q corresponds to the minimum of the reflectivity; d1=39 nm. Inset: Reflectivity kinetics recorded in a thick film (d1=90 nm) shows a ‘dip’ at small values of gain. Fig. 1 Fig.1 (a) Schematic of SPP excitation in ATR geometry. (b) Reflectivity R as a function of angle q. Traces – solutions of exact Eq. (3). Dots – solution of approximate Eq. (6). For all data sets: e1=-15.584+0.424i, d1=60 nm. Trace 1: dielectric with very small loss, e2=2.25+10 Traces 2-4: dielectric with very small gain, e2=2.25-10 -5i. Trace 2 and dots: complex cut along negative imaginary axis (correct; nearly overlaps with trace 1; no discontinuity at the transition from small loss to small gain). Trace 3: complex cut along positive real axis (yields incorrect predictions for incident angles below total internal reflection). Trace 4: complex cut along negative real axis (yields incorrect predictions for incident angles above total internal reflection). Fig. 2. Fig.2. Reflectivity R [Eq.(5)] of the three-layer system depicted in Fig. 1a as a function of angle q and pumping (given by imaginary part of e2); panel (a) illustrates the evolution of reflectivity in a relatively thick metallic film (d1=70nm); panel (b) corresponds to a thin film (d2=39 nm). Fig. 3. Fig. 3. Inverse propagation length of SPP, L-1, in the system depicted in Fig.1a as a function of gain in dielectric, e2”. Solid line – solution of Eq. (11), dots – exact numerical solution of Maxwell equations. Top inset: intensity distribution across the system. Bottom inset: Exponential decay of the SPP wave intensity E2 (shown in the top inset) along the propagation in the x direction. Fig. 4. 60 62 64 66 68 70 72 Angle (q) 0.0005 0.001 0.0015 0.002 0.0025 0.0E+00 2.0E-07 4.0E-07 6.0E-07 8.0E-07 1.0E-06 Time (s) Fig. 4. (a) Reflectivity R(q) measured without (diamonds) and with (circles) optical pumping in the glass-silver-R6G/PMMA system. Dashed lines – guides for eye. Solid lines – fitting with Eq. (3) at e0’=n02=1.7842=3.183, e0”=0, e1’=-15, e1”=0.85, d1=39 nm, e2’= n22=1.52=2.25, e2”≈0 (trace 1) and e2”≈-0.006 (trace 2). Inset: Angular reflectivity profile R(q) recorded in the same system without pumping (dots) and its fitting with Eq. (3) (solid line). (b) Reflectivity kinetics recorded in the glass-silver-R6G/PMMA structure under pumping. The angle q corresponds to the minimum of the reflectivity; d1=39 nm. Inset: Reflectivity kinetics recorded in a thick film (d1=90 nm) shows a ‘dip’ at small values of gain. 60 62 64 66 68 70 72 Angle (q) 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0E+00 2.5E-07 5.0E-07 7.5E-07 1.0E-06 Time (s)

0704.1514

Towards Functional Flows for Hierarchical Models

arXiv:0704.1514v2 [hep-th] 21 Sep 2007 CERN-TH/2007-037 Towards Functional Flows for Hierarchical Models Daniel F. Litim1 Department of Physcis and Astronomy University of Sussex, Brighton, BN1 9QH, U.K. Theory Group, Physics Division CERN, CH-1211 Geneva 23. Abstract The recursion relations of hierarchical models are studied and contrasted with functional renormalisation group equations in corresponding approximations. The formalisms are compared quantitatively for the Ising universality class, where the spectrum of universal eigenvalues at criticality is studied. A significant correlation amongst scaling exponents is pointed out and analysed in view of an underlying optimisation. Continuous functional flows are provided which match with high accuracy all known scaling exponents from Dyson’s hi- erarchical model for discrete block-spin transformations. Implications of the results are discussed. 1 [email protected], [email protected] http://arxiv.org/abs/0704.1514v2 1. Introduction Renormalisation group methods [1], and in particular Wilson’s renormalisation group [2], play an important role in the study of physical systems at strong coupling and/or large correlations lengths. Differential implementations of Wilson’s idea [3–10] rely on an appropriately-introduced momentum cutoff leading to flow equations for running couplings and N -point functions, which can be studied with a large variety of analytical and numerical methods. Numerical stability and reliability in the results is ensured through powerful control and optimisation techniques [11–14]. A different implementation of Wilson’s idea is realised in hierarchical models of lattice scalar theories [15–19]. Hierarchical renormalisation group transformations are often discrete rather than continuous. Here, sophisticated numerical methods have been developed to extract the relevant physics, most notably for high-accuracy studies of scaling exponents for scalar models at criticality [20–22] and related theories (see [19] and references therein). Given the close similarity of the underlying principles, it is natural to ask whether Wilsonian (functional) flows can be linked explicitly, and on a fundamental level, to hier- archical models. If so, this link would provide a number of benefits. It will make powerful functional and numerical methods available to the study of hierarchical models. Vice versa, the numerical tools for hierarchical models could be employed for functional flows in specific approximations. Furthermore, an explicit link may lead to a path integral representation of hierarchical models, allowing for systematic improvements beyond a standard kinetic term. Finally, well-developed optimisation techniques for functional flows could be taken over for hierarchical models as well. In the limit of continuous hierarchical block-spin transformations, an explicit link between Dyson’s hierarchical model [15, 17] and the Wilson-Polchinski flow [23] in the local potential approximation has been established long ago by Felder [24]. Following a conjecture of [13], this link has been extended [25, 26] to include optimised versions [11, 12] of Wetterich’s flow for the effective average action [27]. These interrelations have recently been backed-up by extensive numerical studies of critical potentials and scaling exponents to high accuracy from either formalism [28]. In this paper, we evaluate the more general case and ask whether hierarchical models for discrete block-spin transformations are linked to functional flows with continuous renormali- sation group transformations. We first contrast the basic setups for functional flows (Sec. 2), background field flows (Sec. 3), and hierarchical models (Sec. 4). At a Wilson-Fisher fixed point, underlying similarities and differences are worked out and compared for the leading scaling exponent (Sec. 5). An extensive numerical study of the eigenvalue spectrum of the Ising universality class from functional flows is performed (Sec. 6). A strong correlation of scaling exponents is established and analysed (Sec. 7). It is shown that specific functional flows match the leading and subleading scaling exponent from Dyson’s hierarchical models for discrete transformation parameter to high accuracy (Sec. 8). We close with a discussion of the results and further implications (Sec. 9). 2. Functional Flows Wilsonian (functional) flows integrate-out quantum fluctuations within a path integral representation of quantum field theory. In their simplest form, they are generated through a cutoff term quadratic in the field added to the Schwinger functional, where the (classical) action is replaced by S → S+∆Sk and ∆Sk ∼ dqφ(q)Rk(q 2)φ(−q). The infrared momentum cutoff Rk(q 2) ensures that the propagation of small momentum modes q2 ≪ k2 is suppressed, while the large momentum modes q2 ≫ k2 remain unaffected. Under an infinitesimal change in the Wilsonian (infrared) cutoff scale k, the effective action Γk changes according to its functional flow, which reads (t = ln k) ∂tΓk = ∂tRk (1) in the form put forward by Wetterich [27]. The trace denotes a momentum integration and a summation over fields. The factor ∂tRk in the integrand is peaked in the vicinity of q 2 ≈ k2. The cutoff function Rk obeys Rk(q 2) → 0 as k2/q2 → 0, Rk(q2) > 0 as q2/k2 → 0, and 2) → ∞ as k → Λ, and can be chosen freely elsewise, e.g. [11]. It ensures that the flow is well-defined, thereby interpolating between an initial action S at k = Λ in the ultraviolet (UV) and the full quantum effective action Γ ≡ Γk=0 in the infrared k → 0. In addition to providing a momentum cutoff, the function Rk(q 2) also controls the stability and convergence of subsequent expansions [11–13, 29]. Therefore, it is possible to identify optimised momentum cutoffs – within given systematic expansions – which improve the physical result [11, 12, 30]. The construction of optimised cutoffs [11–14] is central to extract reliable results also in more complex theories including e.g. QCD [31], quantum gravity [32], thermal physics [5, 6, 33] and critical phenomena [28, 30, 34]. Below, we are interested in 3d scalar theories at criticality, where we can sent the ultraviolet scale Λ → ∞. To leading order in the derivative expansion, the effective action reads Γk = d3x[1 ∂µφ∂µφ+Uk(ρ̄)] and ρ̄ = φ2. Introducing r(y) = Rk(q 2)/q2 with y = q2/k2, we find ∂tu = −3u+ ρu′ + −y3/2 r′(y) y(1 + r) + u′ + 2ρu′′ with u(ρ) = Uk(ρ̄)/k 3 and ρ = ρ̄/k. An irrelevant constant originating from the angular integration has been rescaled into the potential and the fields. For the optimal cutoff Ropt = (k2 − q2)θ(k2 − q2) with ropt = (1/y − 1)θ(1− y), the flow reads [12] ∂tu = −3u+ ρu′ + 1 + u′ + 2ρu′′ after an additional rescaling. This flow is integrated analytically in the limit of a large number of scalar fields [35]. We note that the universal content of the flow (3) is equivalent to the Wilson-Polchinski flow in the local potential approximation [13, 25, 26]. 3. Background Field Flows A different form of the flow (1) is obtained for momentum cutoffs which depend addi- tionally on a background field φ̄. Background fields are most commonly used for the study of gauge theories [36], see [32, 37, 38] for applications. They have also been employed for a path integral derivation of (generalised) proper-time flows [39, 40]. In the presence of background fields, the functional Γk[φ] turns into a functional of both fields, Γk[φ, φ̄]. In order to maintain the one-loop exactness of (1), the momentum cutoff can only depend on the background field, but not on the propagating field. Following [39], we introduce x = Γ(2,0)[φ, φ] and x̄ = x[φ = φ̄], where Γ (n,m) k [φ, φ̄] ≡ δ nδmΓk/δφ nδφ̄m. We chose momentum cutoffs of the form Rk(q 2)→ x̄ r[x̄], which depend now on the background field. Here, the regulator cuts off both large momentum modes q2 ≫ k2 and large field amplitudes with Γ (2,0) ≫ k2. The full advantage of background fields becomes visible once they are identified with the physical mean, leading to the functional Γk[φ, φ̄ = φ] → Γk[φ]. The resulting flow is closed provided Γ k [φ] = Γ (2,0) k [φ, φ]. For scalars, this relation becomes exact in the infrared limit studied below (for gauge fields, see [38]). Using the momentum cutoffs [39] rPT,m[x] = exp 2F1[m,m;m+ 1;−mk − 1 , we are lead to the background field flow ∂tΓk = Tr k2 + x/m rPT,m x(1 + rPT,m) k2 + x/m ∂tx . (4) If the term ∼ ∂tx on the right hand side is dropped – meaning that additional flow terms originating from the implicit scale dependence in the momentum cutoff are neglected over the leading term – the flow (4) reduces to the proper-time flow of Liao [41]. A general proper-time flow is a linear combination of the first term in (4) for various m [40]; see [42, 43] for applications. Next, we specialise to the proper-time approximation to leading order in the derivative expansion. The flow equation for the effective potential takes the very simple form ∂tu = −3u+ ρu′ + (m+ u′ + 2ρu′′)m−3/2 , (5) where m parameterises the momentum cutoff, and an irrelevant constant factor has been rescaled into the potential and the fields. For m ∈ [1, 5 ], the flow (5) is mapped onto the flow (2) [39]. At m = 5 , the flow (5) is equivalent to (3), modulo a trivial rescaling. As a final remark, we note that this proper-time flow is also obtained from linear combinations of higher scale-derivatives of Callan-Szymanzik flows, without relying on background fields [40]. In this representation, the approximation leading to (5) consists in the neglection of higher order flow terms ∼ ∂nt Γ 4. Hierarchical Models Several hierarchical models for an effective potential v(ϕ) of a lattice scalar field have been introduced in the literature [15–17] (see also [19]). The hierarchical transformation laws relate the potential v(ϕ) at momentum scale k/ℓ with an average in field space over v(ϕ) at momentum scale k, where ℓ ≥ 1 is the renormalisation group step parameter. We restrict ourselves to the three-dimensional case; the generalisation to arbitrary dimensions is straightforward. In Dyson’s model [15, 17], the renormalisation group step k → k/ℓ for the potential is expressed as e−vk/ℓ(ϕ) = dξ µℓ(ξ) e −ℓ3 vk(ℓ −1/2ϕ+ξ) (6) The details of the averaging procedure are encoded in the measure factor µℓ(ξ), in the ξ- dependence of the potential on the right-hand side of (6), and in the choice for the decimation parameter ℓ. As is evident from (6), a decimation parameter ℓ = 21/3 – employed for most numerical studies [19–22] – corresponds to a volume decimation of ℓ3 = 2 at each iteration. For Dyson’s model, the measure is chosen as µℓ(ξ) = (π σ(ℓ)) −1/2 exp(−ξ2/σ(ℓ)) [19], where we require σ(ℓ) > 0 for ℓ 6= 1, and σ(1) = 0 with σ′(1) 6= 0. A standard choice is σ(ℓ) = 2(ℓ−1) [24]. By definition, (6) describes a flow towards the infrared for decimation parameters ℓ ≥ 1. For ℓ → 1, the hierarchical transformation (6) becomes continuous and the measure factor turns into a δ-function µℓ→1(ξ) → δ(ξ). Performing −ℓ∂ℓ(6), which is equivalent to k∂k(6), we arrive at a differential flow equation for the effective potential [24] ∂tv = −3v + 12ϕv ′ − v′′ + (v′)2 , (7) where an irrelevant factor is rescaled into the fields and the potential; t = ln k. The interaction terms in (7) originate from the scale-derivative of the measure −ℓ∂ℓ µℓ(ξ), which reads 1 σ′(1) δ′′(ξ) in the limit ℓ → 1. This highlights the relevance of the measure factor in hierarchical models. Our normalisation corresponds to the choice σ′(1) = 4 to match with [28]. The limit (7) is independent of σ, but at ℓ 6= 1, we expect that scaling solutions and exponents from (6) depend on it. Eq. (7) is the well-known Wilson-Polchinski flow [2, 23]. We therefore conclude that the potential in (7) is related to the potential in (3) by a Legendre transformation [26, 28]. A different version of a hierarchical model has been introduced by Wilson [16]. Here, the recursion relation is written as e−vk/ℓ(ϕ) = dξ µℓ(ξ) e ℓ3 [vk(ℓ −1/2ϕ+ξ)+vk(ℓ −1/2ϕ−ξ)] . (8) In Wilson’s original model, the ξ-dependence of the measure is µℓ(ξ) = Nℓ exp(−ξ2), where the normalisation factor Nℓ is ξ-independent [19]. The measure factor is different from the one in Dyson’s model, because the Gaussian width is ℓ-independent. If instead we employ the measure of Dyson’s model, the limit ℓ→ 1 can be performed analytically.2 Up to a trivial rescaling, we find ∂tv = −3v + 12ϕv ′ − v′′ . (9) In contrast to the Wilson-Polchinski flow (7), the non-linear term (v′)2 is absent. This comes about because the integrand of (8) – as opposed to the integrand of (6) – is manifestly symmetric under ξ → −ξ. Numerical evaluations of (8) have been reported in [16, 44]. For other representations of hierarchical models we refer to [19] and references therein. 5. Matching Hierarchical Models In order to match hierarchical models by functional flows, we have to detail the scheme dependences of physical observables in either formalism. In the functional RG framework, the fully integrated flow is independent of the momentum cutoff Rk(q 2) chosen for the integration. Scheme dependences, which enter as a consequence of truncations of Γk[φ], have been discussed extensively in the literature [11–13, 29, 45]. Their origin is easily understood. Since the momentum cutoff R in (1) couples to all operators in the theory, the missing back-coupling of operators neglected in a given truncation can result in a spurious dependence of physical observables evaluated either from Γ0[φ], or from a fixed point solution Γ∗[φ]. The scheme dependence is reduced by identifying those momentum cutoffs, which, in a given truncation, lead to an improved convergence and stability of the flow. In Fig. 1, we discuss the scheme dependence quantitatively for the leading order scaling exponent ν at a fixed point of the 3d Ising universality class [29]. Within exact flows (2), the full Rk-dependence has been studied in [30] by evaluating the fixed points of (2) for general cutoffs (Fig. 1, first column). The main result is that the range of achievable values is bounded from above and from below. The upper bound is attained for Callan-Symanzik type flows with Rk ∼ k2. The lower bound with ν = νopt is attained with the optimal flow (3), and hence equivalent to the Wilson-Polchinski flow. The sharp cutoff result is indicated for comparison. The proper-time flow (5) rests on an intrinsically different truncation, because implicit dependences on the background field have been neglected as well as higher order flow terms proportional to the flow of Γ(2); see Sec. 3. Therefore scheme dependences are quantitatively different. In the approximation (4), the m-dependence of scaling exponents from (5) has been studied in [39, 42] (Fig. 1, second column). The range of values is again bounded from above by a Callan-Symanzik flow. The lower bound is achieved for m → ∞. We note that the range of values exceeds those achievable within (standard) exact flows. The lower bound 2 The variance of the Gaussian measure in (8) can be changed by an explicit rescaling of the fields as ϕ → ϕ/ σ for finite σ, see [19]. Rescaling also the integration variable ξ → ξ/ σ, and denoting the potential in terms of the rescaled fields again as v(ϕ), we obtain (8) with a rescaled measure µℓ(ξ) = −1/2 exp(−ξ2/σ). It agrees with the measure of Dyson’s model for Nℓ = π −1/2 and σ = σ(ℓ). I thank Y. Meurice for e-mail correspondence on this point. Callan-Symanzik sharp cutoff WP/opt mean field exact RG proper-time RG Wilson HM Dyson HM 0.626 0.649562 0.649570 0.650163 0.653 0.6895 Figure 1: (colour online) Comparison of scaling exponent ν from different functional flows (RG) in the local potential approximation, and hierarchical models (HM). The solid lines indicate the range of values obtained in the literature. The dashed lines, if present, indicate that the underlying parameter space has not been exhausted. The horizontal lines from top to bottom indicate the results for Callan-Symanzik flows, the sharp cutoff flow, the Wilson-Polchinski (optimal) flow, and the mean field result. A non-linear rescaling of the ν-axis is introduced for display purposes only (see main text). Colour coding: exact RG (red), exact background field RG in the proper-time approximation (violet), Wilson’s hierarchical model (blue) and Dyson’s hierarchical model (light blue). may be overcome once the additional flow terms, neglected here, are taken into account [39]. This is indicated by the dashed line. Next we consider scheme dependences of hierarchical models. Based on their construction, we expect that physical observables depend on the averaging procedure, on the measure factor µℓ, and on the decimation parameter ℓ. It has proven difficult to systematically include wave function renormalisations and higher order operators in hierarchical models, and it is therefore not known whether the scheme dependence vanishes upon higher order corrections [18, 19]. Still, the scheme dependence should give a reasonable estimate for the underlying error in the model assumptions, in particular in comparison with functional methods. The ℓ-dependence of Wilson’s model (8), originally constructed for ℓ ≈ 2, has been studied in [44] in the range ℓ ∈ [21/3, 2] (Fig. 1, third column). The full line covers the range of values obtained in the literature, while the dashed lines indicate that the underlying parameter space has not been exhausted. Because of (9) being linear in the potential as opposed to (7), we expect a strong ℓ-dependence, possibly a discontinuity, in the limit ℓ→ 1. The slope ℓ∂ℓ ν(ℓ) along the data points with ℓ > 21/3 is negative, meaning that ν(ℓ) increases for smaller ℓ. We stress that Wilson’s HM has an overlap both with exact flows and proper-time flows. Therefore, it is possible to map ν(ℓ) of Wilson’s model for certain decimation parameters ℓ onto ν(R) from functional flows with appropriately chosen R. On the other hand, for some decimation parameter ℓ, Wilson’s model can only be mapped onto proper-time flows but not on exact flows, while for some decimation parameter it cannot be mapped onto either of them. The ℓ-dependence of Dyson’s model (6) is displayed in Fig. 1, fourth column. The full line connects the known results at ℓ = 1 [28, 30], ℓ = 21/3 [20, 22] and ℓ = 2 [20]. The dashed line towards larger values for ν indicates that the parameter space ℓ ≥ 1 has not been exhausted. We note that the ℓ-dependence is very weak, with a tiny slope in the range of ℓ-values covered. The important observation is that the slope ℓ∂ℓ ν(ℓ) is positive in the vicinity of ℓ ≈ 1−2, implying that ν(ℓ) > ν(1) for ℓ > 1. Consequently, it is possible to map the scaling exponent ν(ℓ) at discrete block-spin transformation ℓ > 1 onto ν(R) from functional flows for specific momentum cutoff R, both within the standard exact flows and within proper-time flows. This supports the conjecture that Dyson’s model can be mapped onto functional flows. 6. Spectrum of Eigenvalues Whether the observations of the preceeding section can be promoted to a full map between the formalisms crucially depends on further observables including the subleading scaling exponents. Here and in the following section, we study the spectrum of universal eigenvalues (scaling exponents) form functional flows (2) to high accuracy. A fixed point solution u∗ 6= const. of (2) is characterised by the universal eigenvalues of eigenperturbations in its vicinity. We denote the ordered set of eigenvalues as O(R) = {ωi(R), i = 0, · · · ,∞}, with ωi < ωj for i < j. 3 In addition to the leading exponent ν(R) ≡ −1/ω0, we study the first three subleading scaling exponents ω(R) ≡ ω1(R), ω2(R) and ω3(R) within the exact flow (2) for various cutoffs and coarse graining parameters. For the numerical analysis, we introduce several classes of momentum cutoffs defined through rmexp = b/((b+1) y−1), rexp = 1/(exp cyb−1); rmod = 1/(exp[c(y+(b−1)yb)/b]−1), with c = ln 2; and ropt,n = b(1/y − 1)nθ(1 − y). These cutoffs include the sharp cutoff (b → ∞) and asymptotically smooth Callan-Symanzik type cutoffs Rk ∼ k2 as limiting cases. The larger the parameter b, for each class, the ‘sharper’ the corresponding momentum cutoff. The cutoff ropt,n probes a two-dimensional parameter space in the vicinity of ropt to which it reduces for b = 1 and n = 1. For integer n, ropt,n is a C (n+1) function. In addition, we consider the cutoffs rmix = exp[−b( y − 1/√y)] and rmix,opt = exp[− 1b (yb − y−b)], which obey rmix(1/y) = 1/rmix(y). Note that we have covered a large variety of qualitatively different momentum cutoffs including exponential, algebraic, power-law, sharp cutoffs and cutoffs with compact support. Except for ropt,n, all cutoffs are C (∞)-functions. We employ the numerical techniques developed in [28, 30]. 3 In our conventions, the sole negative eigenvalue at the Wilson-Fisher fixed point is ω0. a) ω vs. −ω0 d) ω vs. ω2 1.46 1.48 1.5 1.52 1.54 2.8 2.9 3 3.1 3.2 b) ω2 vs. −ω0 e) ω2 vs. ω3 1.48 1.5 1.52 1.54 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 c) −ω0 vs. ω3 f) ω vs. ω3 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 ropt,1 rmexp (rPT) rmix,opt rpower Figure 2: (colour online) Six two-dimensional projections of the four leading scaling exponents in the Ising universality class from the functional flow (2) for various cutoffs and coarse graining (approx- imately 103 data sets). Here, ν ≡ −1/ω0. The Wilson-Polchinski result from the optimal flow (3) (large black dot) corresponds to a local extremum for all scaling exponents. Data sets based on rpower, rmix,opt, rmix, rexp, rmexp, ropt,1 and rmod; data from rPT is included in Fig. 4. Our results for the universal eigenvalues at criticality are displayed in Fig. 2 for the six two-dimensional projections of the four-dimensional subspace {−ω0, ω, ω2, ω3} of observables. The plot contains roughly 103 data points, the different classes of cutoffs are colour-coded. We focus on the relevant 10%-vicinity of the Wilson-Polchinski result with scaling exponents Oopt ≡ O(Ropt) from (3), indicated by a large black dot, see [28] for the high-accuracy numerical values. The central result of Fig. 2 is that scaling exponents are very strongly correlated. Despite having probed the space of observables by many qualitatively different momentum cutoffs, we find that only a small subset of values can actually be achieved. The correlations increase the closer the eigenvalues O move towards Oopt. In the immediate vicinity of the Wilson-Polchinski result, we only find a very narrow “throat” connecting observables O(R) with Oopt. For the sub-leading scaling exponents ωi, the throat remains very narrow even further away from Oopt. This is seen most clearly in the correlation of ω2 with ω3 in Fig. 2e), as well as in the correlations of ω with both ω2 and ω3 in Fig. 2d) and f). In turn, for the leading exponents ν, the throat opens up more rapidly once its value is further away from νopt, see Fig. 2a), b) and c). 7. Correlations of Eigenvalues The strong correlation of scaling exponents is a structural fingerprint of Wilsonian flows (2). Since the Wilson-Polchinski result is distinguished in the space of scaling exponents, it is natural to normalise the data of Fig. 2 with respect to it. We introduce the distance of any pair of scaling exponents (x, y) from the optimal result (xopt, yopt) as ρ(x, y) = (xopt − x)2 + (yopt − y)2 ≡ 10−Nρ(x,y) (10) We have chosen a standard metric in the space of observables (other choices can be applied as well). In this representation, full agreement with the (optimal) Wilson-Polchinski result is achieved for ρ→ 0 and Nρ →∞. We also introduce the angles ϕ(x, y) = arctan(x/y) . (11) The critical indices ν and ωi, i ≥ 1, are positive numbers. Therefore, they can cover the range x/y ∈ [0,∞] and ϕ ∈ [0, π ], and ρ ≥ 0 for any pair of observables (x, y). In the subspace (ν, ω), the extremal values (νopt, ωopt) have the polar coordinates (ρopt, ϕopt), where the angle reads ϕopt = 0.78066 · · · which is close to π/4 = 0.785398 · · · , and ρopt = 0. The radial distance from the origin is ρ(0, 0) = 0.923002 · · · . In the representation (10), we can study the close vicinity of the Wilson-Polchinski result. In Fig. 3, we display our data points as functions of the angles ϕ, and their distance from ρopt in a semi-logarithmic basis. It is noteworthy that only a very narrow range of angles ϕ is actually achieved by the data, despite the fact that large fractions of the underlying space of momentum cutoffs is covered. Also, and in contrast to Fig. 2a), many data points are degenerate in the representation (Nρ, ϕ). A priori, the Wilson-Polchinski value (ρopt, ϕopt) could have been approached along many different paths. Instead, we find that only a narrow range of (Nρ, ϕ)-values is achieved for arbitrary momentum cutoff. ↑ Wilson-Polchinski (optimal flow)Nρ(ν, ω) ← hierarchical model (Dyson, ℓ = 21/3) sharp cutoff → Callan-Symanzik → 0 π/8 π/4 3π/8 π/2 ϕ(ν, ω) ropt,1 rmexp (rPT) rmix,opt rpower Figure 3: (colour online) Distance Nρ of the pair of scaling exponents (ν, ω) from the optimal Wilson- Polchinski values (νopt, ωopt) in the representation (10), (11). Only a narrow range of angles ϕ(ν, ω) in the vicinity of ϕ ≈ π/4 is achieved by the data. Many data points are nearly degenerate. Data from functional flows in the local potential approximation (2); same data sets and colour coding as in Fig. 2, plus further high resolution data points from ropt,1 in the close vicinity of (νopt, ωopt). Results from the sharp cutoff limit, the Callan-Symanzik type flow (with Rk ∼ k2) and Dyson’s hierarchical model (with ℓ = 21/3) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result corresponds to ϕ = ϕopt and Nρ →∞. This pattern is further highlighted in Fig. 4, where we have magnified the non-trivial range of data sets from functional flows (2). In addition, we have added data points from the background field flow (5) using the cutoff rPT,m for m < . It is remarkable that these data sets display the same pattern as the data from (2). Our results from this and the preceeding section are summarised as follows: Extremum.— The Wilson-Polchinski (optimal flow) result in Fig. 2 corresponds to an extremum in the space of physical observables with |ωi| ≤ |ωi,opt| for all obervables in the vicinity of Oopt. The extremum is local, because the exponents approach ωi = 2i − 1, i ≥ 0, for very soft (Callan-Symanzik-type) momentum cutoffs [30]. For the eigenvalue products Πni=0(ωi/ωi,opt), the Wilson-Polchinski extremum is a global one. Uniqueness.— Our result indicates that the correlations of eigenvalues at the Wilson- Pochinski result are strongest, in the sense that any flow of the form (2) with the exponent ν(R) = νopt automatically also agrees with the Wilson-Polchinski result in all other observables O(R) = Oopt. In general, for ν(R) > νopt, this is clearly not the case. Nρ(ν, ω) ↑ Wilson-Polchinski (optimal flow) ← hierarchical model (Dyson, ℓ = 21/3) sharp cutoff ← Callan-Symanzik 1 1.05 1.1 1.15 1.2 ropt,1 rmexp rmix,opt rpower Figure 4: (colour online) Magnification of Fig. 3 in the vicinity of ϕ ≈ π/4 where ν/ω ≈ 1. The data points for the distance Nρ(ν, ω) as a function of ν/ω remain highly degenerate. Same data sets and colour coding as in Fig. 3, plus additional high resolution data points from background field flows (5) using rPT,m with m < . Results from the sharp cutoff limit, the Callan-Symanzik type flow (with Rk ∼ k2) and Dyson’s hierarchical model (with ℓ = 21/3) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result corresponds to νopt/ωopt = 0.9905692 · · · and Nρ →∞. Redundancy.— The eigenvalue correlations are so strong that the first two scaling exponents ν(R) and ω(R), for a given R, contain enough information to fix the remaining observables on the percent level or below. These ‘dynamical’ constraints point at a major redundancy of (2) with respect to the underlying momentum cutoffs R. A relevant parameter has been identified previously. The gap miny≥0 y(1 + r) for normalised cutoffs [11], when maximised, leads towards the Wilson-Polchinski result [12, 13, 30]. Optimised observables.— Previous reasonings in favour of an optimisation only invoked properties of the underlying flow (1), e.g. its convergence, locality, stability and boundedness, allowing for improved physical predictions. This has been exemplified quantitatively for the observable ν(R) which obeys 1 ≥ ν(R) ≥ νopt [30], where the lower bound νopt is closest to the physical result [25]. Fig. 2 now shows that this pattern extends to subleading eigenvalues. This equally extends to asymmetric corrections-to-scaling [46]. Therefore, one may turn the original reasoning around and argue that – because of the extremum property of the observables O(R) – an extremisation of the functional flow along the lines discussed in [11, 12, 14], or similar, should naturally lead towards the values Oopt. Stated differently, Figs. 2-4 show that observables derived from (1) admit an optimisation. method cutoff parameter ν ω hierarchical model Dyson (ℓ = 1) 0.649 561 773 880a 0.655 745 939 193a ropt,n (n = 1, b = 1) 0.649 561 773 880 a 0.655 745 939 193a functional RG rcompact (b→ 0) 0.649 561 773 880 0.655 745 939 193 rint (b→ 1) 0.649 561 773 880 0.655 745 939 193 rPT,m (m = 5/2) 0.649 561 773 880 a 0.655 745 939 193a Table 1: Matching scaling exponents ν and ω from continuous hierarchical transformations with functional flows. Results agree at least to the order 10−12. Data from this work, and from a) [28]. Finally, we note that the data point from Dyson’s hierarchical model with ℓ = 21/3 – as plotted in Figs. 3 and 4 – nicely fits into the set of data points covered by functional flows, extending the link observed in Sec. 5 beyond the leading exponent. This observation is addressed quantitatively in the following section. 8. Matching Beyond the Leading Exponent To further substantiate our conjecture that hierarchical models could be mapped onto functional flows, we have to show quantitatively that results from hierarchical model are reproduced by specific functional flows. Here, we study the close vicinity of the Wilson- Polchinski result Oopt, where the correlations are strongest, see Figs. 2 and 3. We have to restrict our search to Dyson’s hierarchical model, where high-accuracy data for the first sub-leading scaling exponent ω is available. No subleading exponents have been computed for Wilson’s model. For the numerical analysis, we introduce additional classes of momentum cutoffs R which contain the optimal flow (3) in some limit. In addition to the two-parameter family of cutoffs ropt,n, we also study the cutoff rcompact = y −1 exp[−e−1/y/(b − y)]θ(b − y) for b > 0 which is C(∞), and the cutoff rint = exp(−y)θ(1 − y)θ(y − b) with b ∈ [0, 1], which is effective for a finite interval of momenta q2 ∈ [bk2, k2]. In the limit b → 0 (b → 1), the corresponding flows are equivalent to (3). Hence, ropt,n, rcompact and rint parametrise substantially different classes of cutoffs. More generally, there are infinitely many cutoffs Rk leading to scaling exponents identical with Oopt, and the examples provided above serve to illustrate this. At ℓ = 1, Dyson’s hierarchical transformation is continuous, and the scaling exponents are equivalent to those from the optimal flow (3) and the Wilson-Polchinski flow (7). In Tab. 1, we compare exponents from different functional flows. We confirm numerically, and with high accuracy, that the cutoffs ropt,n, rcompact and rint lead to the Wilson-Polchinski result for specific parameter values. method cutoff parameter ν ω hierarchical model Dyson (ℓ = 21/3) 0.649 570b 0.655 736b ropt,n (n = 1, b = 1.048) 0.649 570(9) 0.655 736(6) ropt,n (n = 1, b = 0.9545) 0.649 570(9) 0.655 736(9) ropt,n (n = 1.135, b = 1) 0.649 570(6) 0.655 736(8) ropt,n (n = 1.1, b = 1.028) 0.649 570(6) 0.655 736(8) functional RG rcompact (b = 0.04775) 0.649 570(9) 0.655 736(9) rint (b = 0.944) 0.649 570(9) 0.655 736(8) rint (b = 0.9444) 0.649 570(7) 0.655 736(9) rPT,m (m = 2.499785) 0.649 564(9) 0.655 736(1) rPT,m (m = 2.49944) 0.649 570(1) 0.655 720(6) Table 2: Matching scaling exponents ν and ω from discrete hierarchical transformations with functional flows. Results agree to the order 10−6 for all cutoffs except the proper-time flow, which matches up to the order 10−5. Data from this work, and from b) [22]. At ℓ = 21/3, Dyson’s hierarchical transformation is discrete. The reference data reads = 0.649570 and ω = 0.655736 [22].4 These values differ only at the order 10−5 from the optimal (Wilson-Polchinski) result, and are therefore sufficiently close to Oopt to confirm or refute the correlations observed in the previous section. Figs. 3 and 4 indicate that the result from Dyson’s hierarchical model is fully matched by functional flows. Our numerical results are given in Tab. 2; brackets indicate that a digit is possibly affected by numerical errors. We have found several sets of parameter values, such that the scaling exponents agree with Dyson’s model to order 10−6. More importantly, the momentum cutoffs are quite different. Hence, our analysis also confirms the strong correlation of scaling exponents in the immediate vicinity of the Wilson-Polchinski result. Based on the eigenvalue correlations within functional flows, we conjecture that the subleading eigenvalues ωi with i ≥ 2 of Dyson’s model at ℓ = 21/3 also agree to the corresponding accuracy with the values implied through the functional flows in Tab. 2. Our results based on the proper-time flow (5) with rPT has also been given in Tab. 2. Full agreement is achieved either with the exponent ν or the subleading exponent ω , but not with both of them. Once one of them is matched, the deviation in the other observable is of the order 10−5. The relevant parameters are m < 5 , the regime where (5) is mapped onto (2) [39]. Therefore, the values in Tab. 2 reflects well the range covered by standard Wilsonian flows (2). We expect that full agreement is achieved for proper-time flows which are linear combinations of (5) for different m, but we did not attempt to do so here. 4 In [22], high-accuracy results at ℓ = 21/3 have been given for γ = 2 ν and ∆ = ν ω (and η = 0) with 13 significant digits. They imply νDHM = 0.649570365 · · · and ωDHM = 0.655736286 · · · . For the present study, only the first six figures are required. In summary, we have provided numerical maps from several functional flows onto Dyson’s model at a non-trivial ℓ 6= 1, with an accuracy of the order 10−6. The set of achievable values for scaling exponents from functional flows in the close vicinity of the optimal result is just wide enough to accommodate for the data from Dyson’s model. This is a non-trivial result, also showing that the ℓ-dependence of Dyson’s model and the Rk-dependence of functional flows are very intimately related. Based on our results for ν and ω at ℓ 6= 1, and on continuity in ℓ, we expect that this map extends to other universal quantities in the same approximation, analogous to the full map which is known for ℓ = 1. Data for further symmetric and asymmetric corrections-to-scaling exponents, once available, will allow for additional checks of this picture. Full equivalence is guaranteed as soon as an explicit link in the form ℓ = ℓ(Rk) or Rk(q 2) = Rk(q 2, ℓ) is furnished. For the local potential approximation, our results indicate that this map, if it exists, is not unique. 9. Discussion and Conclusions Establishing equivalences between implementations of Wilson’s renormalisation group as different as discrete hierarchical models of lattice scalar fields on one side and continuous functional flows on the other, allows for new views and insights on the respective formalisms and on the underlying physics. Previously, equivalences were known only in the limit where the hierarchical transformation becomes continuous. In this paper, based on similarities in the dependences related to the underlying coarse-graining, we have extended this link towards discrete hierarchical transformations. This correspondence shows that continuous RG flows (1) are sensitive to implicit discretisation effects via the momentum cutoff. Specifically, for the 3d Ising universality class, we have compared the formalisms on the level of scaling exponents. Their dependence on the step-size parameter ℓ within Dyson’s hierarchical model (6) is qualitatively and quantitatively similar to their dependence on the momentum cutoff R within functional flows (2). In either case, scaling exponents are bounded by the Wilson-Polchinski values obtained for ℓ → 1 and R → Ropt. Once the hierarchical transformations are discrete, ℓ 6= 1, slight variations in all known scaling exponents from Dyson’s model are matched by functional flows with non-optimal momentum cutoffs R 6= Ropt. This is quite remarkable, particularily in view of the strong eigenvalue correlations found amongst functional flows. In this light, the optimisation of functional flows with R→ Ropt can now, alternatively, be viewed as the removal of discretisation effects, at least to leading order in a derivative expansion as studied here. It will be interesting to contrast these findings with the construction of improved or perfect actions on the lattice. More generally, it is conceivable that Dyson’s model for arbitrary ℓ is mapped by functional flows on a fundamental level beyond the numerical map provided for ℓ = 21/3. An explicit map would be very welcome, also in view of linking hierarchical models to a path integral representation of the theory. In Wilson’s hierarchical model (8), the range covered by the leading scaling exponent indicates that a partial map onto functional flows exists, though only for a restricted domain of ℓ-values. Interestingly, the overlap with background field flows is even larger. Whether these maps extend beyond the leading exponent cannot be settled presently due to a lack of data for subleading exponents from Wilson’s model. In addition, we found a distinct correlation of scaling exponents from functional flows (2). The eigenvalue spectrum, a fingerprint of the physics in a local potential approxima- tion, is severely constrained and achieves the Wilson-Polchinski values as an extremum. Furthermore, the full space of physical observables is described by very few parameters only, instead of the infinitely many moments of the momentum cutoff. Resolving this redundancy should prove useful for studies of e.g. non-trivial momentum structures and higher orders in the derivative expansion. 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0704.1515

High Precision CTE-Measurement of SiC-100 for Cryogenic Space-Telescopes

High Precision CTE-Measurement of SiC-100 for Cryogenic Space-Telescopes K. Enya1, N. Yamada2, T. Onaka3, T. Nakagawa1, H. Kaneda1, M. Hirabayashi4, Y. Toulemont5, D. Castel5, Y. Kanai6, and N. Fujishiro6 [email protected] Received ; accepted 1 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan 2 National Metrology Institute of Japan, Advanced Industrial Science and Technology, 3 Tsukuba Central, Tsukuba, Ibaraki 305-8563, Japan 3 Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4 Sumitomo Heavy Industries, Ltd., 5-2 Niihama Works, Soubiraki-cho, Niihama, Ehime 792-9599, Japan 5 Astrium Satellites, Earth Observation, Navigation & Science (F), 31 av des Cosmo- nautes, 31042 Toulouse Cedex 4, France 6 Genesia Corporation, Mitaka Sangyo Plaza 601, 3-38-4 Shimorenjyaku, Mitaka, Tokyo 181-0013, Japan http://arxiv.org/abs/0704.1515v1 – 2 – ABSTRACT We present the results of high precision measurements of the thermal expan- sion of the sintered SiC, SiC-100, intended for use in cryogenic space-telescopes, in which minimization of thermal deformation of the mirror is critical and pre- cise information of the thermal expansion is needed for the telescope design. The temperature range of the measurements extends from room temperature down to ∼ 10K. Three samples, #1, #2, and #3 were manufactured from blocks of SiC produced in different lots. The thermal expansion of the samples was measured with a cryogenic dilatometer, consisting of a laser interferometer, a cryostat, and a mechanical cooler. The typical thermal expansion curve is presented using the 8th order polynomial of the temperature. For the three samples, the coefficients of thermal expansion (CTE), α#1, α#2, and α#3 were derived for temperatures between 293K and 10K. The average and the dispersion (1σ rms) of these three CTEs are 0.816 and 0.002 (× 10−6/K), respectively. No significant difference was detected in the CTE of the three samples from the different lots. Neither in- homogeneity nor anisotropy of the CTE was observed. Based on the obtained CTE dispersion, we performed an finite-element-method (FEM) analysis of the thermal deformation of a 3.5m diameter cryogenic mirror made of six SiC-100 segments. It was shown that the present CTE measurement has a sufficient accu- racy well enough for the design of the 3.5m cryogenic infrared telescope mission, the Space Infrared telescope for Cosmology and Astrophysics (SPICA). Subject headings: instrumentation: miscellaneous — methods: laboratory — techniques: miscellaneous – 3 – 1. Introduction Development of cryogenic light-weight mirrors is a key technology for infrared astronomical space-telescope missions, which have large advantages owing to being free from the turbulence, thermal background, and absorption caused by atmosphere. The light-weight mirror technology is essential to bring large mirrors into space that enable a high sensitivity and a high spatial resolution. The infrared sensitivity of the space-telescope is vastly improved by reduction of the thermal background by cooling the telescope to cryogenic temperatures. Thus infrared space telescopes badly need cooled light-weight mirrors with a sufficient optical quality. Silicon-carbide (SiC) is one of the most promising materials for space telescopes because of its high ratio of stiffness to density. Japanese space mission for infrared astronomy, AKARI, carries a 68.5 cm aperture telescope whose mirrors are made of sandwich type SiC (Murakami 2004; Kaneda et al. 2005). The entire AKARI telescope system is cooled down to 6K by helium gas vaporizing from liquid helium. AKARI was launched in February 2006 and the telescope system performance has been confirmed to be as expected from pre-launch ground tests (Kaneda et al. 2007). The Herschel Space Observatory, a submillimeter satellite mission by the European Space Agency (Pilbratt 2004), employs mirrors of sintered SiC (SiC-100) provided by Boostec Industries and EADS-Astrium (Breysse et al. 2004). The 3.5m diameter primary mirror of the Herschel telescope is made of 12 segments brazed together. The Herschel Space Observatory will be launched in 2008 to make observations in 60–670µm, whilst the telescope will be kept to ∼ 80K by passive cooling. SiC-100 is one of the most frequently used SiCs for space optics. It has been used for the telescopes of ALADIN, GAIA, ROCSAT and other missions (Breysse et al. 2004 and the references therein). The Space Infrared telescope for Cosmology and Astrophysics (SPICA) is the next – 4 – generation mission for infrared astronomy, planned by the Japan Aerospace Exploration Agency (Nakagawa 2004; Onaka et al. 2005). The SPICA telescope is required to have a 3.5m diameter aperture and will be cooled down to 4.5K by the combination of radiative cooling and mechanical coolers (Sugita et al. 2006). SPICA is planned to be launched in the middle of the 2010s and execute infrared observations in 5–200µm. SiC-100 is one of the promising candidate materials for the mirrors and structures of the SPICA telescope, whilst carbon-fiber reinforced silicon-carbide is another candidate now being investigated (Ozaki et al. 2004; Enya et al. 2006; Enya et al. 2004). Monolithic primary mirrors can be manufactured by the joint segment technology, similar to the primary mirror for the Herschel telescope. The requirement for the surface figure accuracy of the SPICA primary mirror is, however, better than 0.06µm rms. This requirement is ∼ 20 times more severe than the Herschel Space Observatory because of the difference in the targeted wavelength range. In the development of cryogenic space-telescopes, it is important to suppress the thermal deformation of the mirror caused by cooling to satisfy the requirement for the surface figure accuracy (Kaneda et al. 2003; Kaneda et al. 2005). Therefore, the study of the coefficients of thermal expansion (CTE) of the material used for the mirror is important. The CTE data of the mirror material are indispensable for the design of cryogenic space-telescopes because the actual telescope mirror needs to accommodate complicated support structures consisting of materials different from the mirror. If the CTE measurement of test pieces of the mirror material has a sufficient accuracy, it will enable us to predict the thermal deformation of a segmented mirror caused by the dispersion in the CTE of each segment. The most direct and highly sensitive test of the thermal deformation is an interferometer measurement of the actual mirror at cold temperatures. The CTE data are useful to interpret the result of direct measurements of the mirror and investigate the origin of the observed deformation. – 5 – However, measurements of the CTE of SiC and its dispersion have not so far been performed with a sufficient accuracy and for a wide temperature range. Especially, little data are available at temperatures lower than 77K, which can be realized by only liquid nitrogen cooling. Prior to this work, available CTE data of SiC-100 were limited at temperatures higher than ∼ 77K (Toulemont 2005). Pepi & Altshuler (1995) presented the CTE data of reaction bonded optical grade (RBO) SiC down to 4K based on measurements with samples made from one block of the RBO SiC. The CTE of the new-technology SiC (NT-SiC), developed high-strength reaction-sintered SiC, has been reported down to 20K for one sample by Suyama et al. (2005). In this work, we present the results of high precision CTE measurements of SiC-100 down to cryogenic temperatures for samples from three different lots. We set two major goals for the present work: one is to provide the typical CTE of SiC-100 for the development of the SPICA telescope, and the other is to estimate the thermal deformation of segmented mirrors from the measured CTEs. 2. Experiment 2.1. Sample Figure 1 shows three samples (#1, #2, and #3) of SiC-100 measured in this work. All of the samples were manufactured by Boostec Industries and EADS- Astrium (Breysse et al. 2004). Each of the samples was extracted from blocks of SiC produced in different lots. The locations of each sample in the SiC blocks are arbitrary. The samples are of a rectangular parallelepiped shape with the dimension of 20.00+0.05 −0.00mm×20.00 +0.05 −0.00mm×6.0 −0.0 mm. Flatness, parallelism, and roughness of the 20.00mm×6.00mm surfaces are important for the measurement of this work: The flatness – 6 – of these surfaces is ≤ λ/10 rms, where λ is the wavelength of the He-Ne laser, 632.8 nm. The parallelism of the opposing 20.00mm×6.00mm surfaces is less than 2′′, whilst the parallelism of the opposing 20.00mm×20.00mm surfaces is less than 1.0◦. All of the 20.00mm ×6.00mm surfaces are polished to an optical grade and the surface roughness finally achieved is less than 3 nm rms. Owing to the polished surface, the directly reflected laser light by the sample was used for the measurement. As the result, the measurement was free from the uncertainty due to any additional mirrors or coating, which would be needed if the direct reflected light could not be used. 2.2. Measurement The measurement of the thermal expansion was carried out with the laser interferometric dilatometer system for low temperature developed in the National Metrology Institute of Japan, Advance Industrial Science and Technology (Yamada & Okaji 2000, hereafter the paper–I; Okaji & Yamada 1997; Okaji et al. 1997). The system consists of the cryostat, the cryogenic mechanical refrigerator of the GM cycle (V204SC) by Daikin Industries Ltd., and the interferometer utilizing acousto-optical modulators and the stabilized He-Ne laser system (05STP905) by Melles Griot. The cooling is performed with the refrigerator alone and no cryogen is needed. The minimum temperature achieved in this work is about 10K. The space around the cold stage of the cryostat, containing the installed sample, is filled with 130Pa helium gas and sealed off at room temperature before cooling to ensure thermal uniformity. The configuration of the whole cryostat and the sample installation into the cold stage of the cryostat are shown in figures 1 and 2 of the paper–I, respectively. The change of length of the sample, ∆L, is measured with the double path type laser interferometer of the optical heterodyne method with the digital lock-in amplifier (Model SR-850) from Stanford Research System. Details of this system are given in the paper–I. – 7 – We made totally 6 measurements of the CTEs of SiC-100. In each measurement, the sample was initially cooled down to ∼ 10K and then the proportional integral derivative (PID) temperature control was applied. After the the temperature had been stabilized, the temperature and the change of the sample length ∆L were measured. The temperature stability during the measurement was less than 0.02K per hour. This process was repeated at approximately 16 temperatures of roughly equal intervals up to room temperature. One dataset of the temperature vs. ∆L was obtained for one cooling cycle. To compensate the systematic uncertainty, each measurement was repeated by rotating the sample by 90◦ as described in figure 2 of the paper–I. We measured CTEs of two orthogonal directions (A and B directions as shown in figure 1) of each sample. The directions A and B were arbitrary chosen. 3. Result and discussion 3.1. Typical thermal expansion The results of the measurement of the thermal expansion are presented in figure 2. A fit with the 8th order polynomial is applied to each of the six datasets and the thermal expansion (contraction) ∆L/L is set as 0 at 293K for each of the six curves. The six datasets are plotted in figure 2 (a). We present the curve derived from the fit with all of the six datasets as the typical thermal contraction of SiC-100. It is shown by the solid line in figure 2 (a). The coefficients of the 8th order polynomial, ∆L/L = i, are presented in table 1. Figure 2 (b) shows the residual dispersion of the data after subtracting the fit curve. The shape of the curve in figure 2 (a) is roughly compatible with those of the RBO SiC (Pepi & Altshuler 1995) and NT-SiC (Suyama et al. 2005), though slightly negative CTEs observed in the RBO SiC and NT-SiC observed at temperatures less than 50K are not seen in the present measurements for SiC-100. Because of the uncertainties in the other – 8 – measurements, it is difficult to further investigate the origins of the differences at present. For each of the six curves, the average ∆L/L per temperature between 293K and 10K is derived and summarized in table 2. The average of the six values and their dispersion (1σ) are 0.816 and 0.005 (× 10−6/K), respectively. The dispersion is smaller than the previous upper-limit obtained with high-purity single crystal silicon in paper–I, 0.01 (× 10−6/K), indicating that the present measurements have reached the limit set by the instrument. Pepi & Altshuler (1995) have shown the 1σ dispersion of 0.04 (× 10−6/K) for their measurements. Karlmann et al. (2006) presented the repeatability of the CTE measurement to be 0.004 (× 10−6/K) from 35K to 305K of single crystal silicon by the interferometer based cryogenic dilatometer. Comparing with Karlmann et al. (2006), our measurement reached lower temperatures. Thus the present dispersion is concluded to be well below the measurement uncertainty and we do not detect any significant variations in the CTEs of the present 6 measurements. We do not detect either any differences in the CTEs in different directions of the same sample or any differences in the samples extracted from different lots. Thus SiC-100 we have measured is homogeneous and isotropic within the present measurement uncertainty. Finally, we average two α in directions A and B of each of three samples to obtain α#1 = 0.8145, α#2 = 0.8160, and α#3 = 0.8185 (× 10 −6/K). The average and the dispersion (1σ rms) of these three CTEs are 0.816 and 0.002 (× 10−6/K), respectively. 3.2. Alternative measurements of the dispersion of CTE Differential tests of the CTE dispersion is another strong tool to investigate the variations in the CTEs. The quite small dispersion in the CTEs of SiC-100 has been confirmed by the systematic check made by Astrium in the manufacturing process of the – 9 – mirror segments of the Herschel Space Observatory. All the 12 segments of the Herschel primary mirror came from different SiC powder batches. The measurement of the bending deflection was made for a brazed couple of two thin SiC samples coming from different batches of SiC. The couples were placed in a vacuum chamber and the thermal deformation of the surface figure of the samples was measured by a interferometer between room temperature and 150K. The deformation data is directly linked to the curvature of the bending deflection and to the difference of the CTE between the two samples. In order to validate the sufficient homogeneity of the SiC material, two kinds of verification on the material have been performed: One of the tests was of homogeneity inside one spare segment. For this test, samples were cut out from one spare sintered segment at several locations (along radial, tangent, and thickness directions). The other tests was of homogeneity between samples belonging to different flight segments. For this test, as it is was not possible to take samples from the flight segments after sintering, the samples were cut out from different segments at their green body stage, before the sintering of the segments. Those samples were taken from arbitrary locations of the segments (different orientations and locations were therefore present in the test samples). Those samples were made similar to the segments themselves by taking care of sintering them in the same run as the associated segments. By this way, it was possible to reproduce the differential CTE characteristics. In the telescope manufacturing process, 12 samples of the 12 segments were brazed on the reference samples and tested at 150K. The results indicate the dispersion (1σ rms) to be smaller than 0.0025 (× 10−6/K). The agreement of the dispersion of the CTE derived by two different methods signifies that both of the measurements are reliable and the uniformity of the CTE of the SiC-100 is well confirmed. The direct measurement of the CTE of this work and the differential measurement are complementary: the direct measurement provides absolute CTE data – 10 – down to 10K with high accuracy, which is indispensable to design the space telescope including the surrounding structures, whilst the differential test checks the uniformity of the CTE for a large number of the samples. 3.3. Simulation of the mirror deformation It is fruitful to relate the accuracy of the CTE measurement with corresponding thermal deformation of the mirror. In this section the thermal deformation of the segmented mirror is estimated on the basis of the measured dispersion of the CTE values. To examine this issue, we perform a case study by using a simple model of the finite-element-method (FEM) analysis. All of the simulated thermal deformation is derived for the case of cooling down from 293K to 10K. Figure 3 shows the model used in the FEM analysis. One mirror with a center hole is constructed from six segments with the rib structure as shown in figures 3 (a) and b as a light-weight mirror design. In this model, it was assumed that the mirror surface is flat for simplicity. Figures 3 (c) and d show a three-dimensional view and the geometry of one segment, respectively. The diameter of the whole mirror is 3.5m, equating to the designed diameter of the primary mirror of the SPICA telescope and the Herschel Space Observatory. The thicknesses of the rib structure and mirror surface are 3 mm. The Young’s modulus and Poisson’s ratio of SiC-100 at room temperature are reported to be 420GPa and 0.17, respectively (Breysse et al. 2004; Toulemont 2005). We use these values in the simulation of the thermal deformation since the temperature dependence of these quantities is usually small and not available at present. In the FEM analysis, the rotationally symmetric axis of the mirror is set along the z axis of the Cartesian coordinates. The constraints of the model are shown in figure 3 (b): Three points on the rib of the mirror, indicated by triangles, are constrained on the lines in the x-y plane as shown by the arrows in figure 3 (b). These points are free along the radial direction within the constraint – 11 – lines. Therefore, this constraints cause no inner stress in the mirror in the simulation of the cooling and thermal deformation. The results of the simulation are presented in figure 4. Figure 4 (a) shows the thermal deformation toward the z-direction of the mirror surface obtained from the FEM analysis, in which α#1, α#2, and α#3 are given for the segment s5 and s6, s1 and s2, s3 and s4, respectively (case–I). Figure 4 (b) shows the simulated z-direction deformation, in which α#1, α#2, and α#3 were given for the segment s3 and s6, s2 and s5, s1 and s4, respectively (case–II). The case–I and case–II are the configuration, in which the two segments having the same CTE are allocated to be adjoined and confronted position, respectively. As the result, the configuration of the case–I corresponds to a mirror, which consists of three segments having α#1, α#2, and α#3, whilst the configuration of the case–II corresponds to a mirror having a CTE distribution of 180◦ rotationally symmetric. Both of figures 4 (a) and (b) show the surface after the tilt correction. For the case–I and case–II, 0.032µm and 0.040µm(1σ rms) are obtained as the surface deformation after the tilt correction, respectively. Since the requirement for the surface figure accuracy of SPICA is 0.06µm rms, the FEM analysis indicates that the thermal deformation estimated based on the measured CTE dispersion among the segments is sufficiently small for the SPICA telescope. The temperature expected for the SPICA is 4.5K, whilst the lowest temperature of the CTE measurement in this work is ∼ 10K. However, the thermal contraction between 10K and 4.5K is negligible and does not affect the mirror deformation at all. It is shown that the accuracy of the CTE measurement achieved in the present study is sufficient to investigate the thermal deformation for the wave front error of less than 0.06µm for a segmented SiC mirror of 3.5m size. – 12 – 4. Conclusion In this work, we performed high precision measurement of the thermal expansion of the sintered SiC, SiC-100 for use in cryogenic space-telescopes. Three samples of SiC-100 from different lots are measured. The temperature the measurement ranges from room temperature to ∼ 10K. The following results are obtained. 1. The typical thermal expansion of SiC-100 is given in the form of ∆L/L = The coefficients are shown in table 1. 2. The CTEs were measured for three samples of two orthogonal directions., The average and dispersion (1σ) of these six values between 293K and 10K are 0.816 and 0.005 (10−6/K), respectively. The dispersion is well below the present measurement uncertainty. 3. The homogeneity and the anisotropy of the CTE of SiC-100 has been confirmed within the present measurement accuracy. 4. The small dispersion of the absolute CTE obtained is compatible with the results of the differential CTE measurements using brazed samples made at Astrium. 5. For the three samples, nominal CTEs, α#1, α#2, and α#3 were derived for temperatures between 293K and 10K by averaging two CTE data of tow directions for each sample. 6. The thermal deformation of a segmented mirror is estimated by a FEM analysis, using α#1, α#2, and α#3. The result indicates that the present measured dispersion is sufficiently small and well below the SPICA requirement on thermal deformation of the mirror, 0.06µm rms. – 13 – We are grateful for Dr. Masahiro Okaji in National Metrology Institute of Japan, Advance Industrial Science and Technology for kind and large support for the measurement. This work was supported in part by a grant from the Japan Science and Technology Agency. – 14 – REFERENCES Breysse, J., Castel, D., Laviron, B., Logut, D., & Bougoin, M., 2004, Proc. of the 5th International Conference on Space Optics, 659 Enya, K., Nakagawa, T., Kataza, H., Kaneda, H., Y.Y.Yamashita, T. Onaka, T. Oshima, & T. Ozaki, 2004, Proc. of SPIE, 5487, 1092 Enya, K., Nakagawa, T., Kaneda, H., Onaka, T., Ozaki, T., & Kume, M., 2007, Applied Optics, in press Kaneda, H., Onaka, T., Yamashiro, R., & Nakagawa, T., 2003, Proc. of SPIE, 4850, 230 Kaneda, H., Onaka, T., Nakagawa, T., Enya, K., Murakami, H., Yamashiro, R., Ezaki, T., Numao, Y., & Sugiyama, Y., 2005, Applied Optics, 44, 6823 Kaneda, H., Kim, W., Onaka, T., Wada, T., Ita, Y., Sakon, I., & Takagi, T., 2007, Publ. Astron. Soc. Japan, submitted Karlmann, P. B., Klein, K. J., Halverson, P. G., Peters, R. D., Levine, M. B., van Buren, D., & Dudik, M. J., 2006, Proc. of AIP Conferenc, Advances in Cryogenic Engineering, 824, 35 Murakami, H., 2004, Proc. of SPIE, 5487, 330 Nakagawa, T., & SPICA working group, 2004, Adv. Sp. Res., 34, 645 Okaji, M., & Yamada, N., 1997, High Temp. & High Press., 29, 89 Okaji, M., Yamada, N., Kato, H., & Nara, H., 1997, Cryogenics, 37, 251 Onaka, T., Kaneda, H., Enya, K., Nakawaga, T., Murakami, H., Matsuhara, H., & Kataza, H., 2005, Proc. of SPIE, 5494, 448 – 15 – Ozaki, T., Kume, M., Oshima, T., Nakagawa, T., Matsumoto, T., Kaneda, H., Murakami, H., Kataza, H., Enya, K., T. Onaka, & M. Krodel, 2004, Proc. of SPIE, 5494, 366 Sugita, H., Nakagawa, T., Murakami, H., Okamoto, A., Nagai, H., Murakami, H., Narasaki, K., & Hirabayashi, M., 2006, Cryogenics, 46, 149 Toulemont, Y., private communication, 2005 Pepi, J. W., & Altshuler, T. L., 1995, Proc. of SPIE, 2543, 201 Pilbratt, G. T., 2004, Proc. of SPIE, 5487, 401 Suyama, S., Itoh, Y., Tsuno, K., & Ohno, K., 2005, Proc. of SPIE, 5868, 96 Yamada, N., & Okaji, M., 2000, High Temp. & High Press., 32, 199 This manuscript was prepared with the AAS LATEX macros v5.2. – 16 – Fig. 1.— Samples measured in this work. The geometry of the samples was 20.00mm× 20.00mm× 6.0mm. The surfaces of ×20.00mm× 6.0mm were polished as de- scribed in the text. – 17 – Fig. 2.— Result of the measurements. (a): Thermal expansion data for the direction A and B of the sample #1, #2, and #3. The solid line represents the fitting curve with 8th order polynomial which is expressed by the coefficients presented in table 1. (b): Residual dispersion of the thermal expansion data around the fitting curve. – 18 – Fig. 3.— The model for the FEM analysis. (a) 3D view of the model of the whole mirror seen from the reflective side. (b) The same model but seen from the back side of the mirror. Three triangles on the rib indicate points for constraints. All of the three points were constrained in the three lines shown by the arrows. s1∼ s6 are for ID of the segments (see also figure 4). (c) and (d) shows a 3D view and the geometry of one segment, respectively. – 19 – Fig. 4.— The simulated surface deformation by the FEM analysis. z-direction deforma- tion (i.e., perpendicular to the surface) after tilt correction is shown by color map. (a): The result of the case–I simulation. α#1, α#2, and α#3 were given for the segment s5 and s6, s1 and s2, s3 and s4, respectively. (b): The result of the case–II simulation. α#1, α#2, and α#3 were given for the segment s3 and s6, s2 and s5, s1 and s4, respectively. – 20 – Table 1: Coefficients of the 8th order polynomial ∆L/L = i to represent the typical thermal expansion of SiC-100 below 300 K. coefficient value a0 +2.43165× 10 a1 −9.25541× 10 a2 +7.38688× 10 a3 +2.44225× 10 a4 +5.68470× 10 a5 +5.94436× 10 a6 −4.04320× 10 a7 +8.70017× 10 a8 −6.64445× 10 – 21 – Table 2: Summary of the measured CTE. Samplea Directionb α (10−6/K) c #1 A 0.820 #1 B 0.809 #2 A 0.821 #2 B 0.811 #3 A 0.815 #3 B 0.822 a, b: #1, #2, and #3 correspond to the sample number and A and B correspond to the direction of the sample for measurement shown in figure 1. xc: ∆L/L between 293K and 10K. Introduction Experiment Sample Measurement Result and discussion Typical thermal expansion Alternative measurements of the dispersion of CTE Simulation of the mirror deformation Conclusion

0704.1516

Compton Scattering of Fe K alpha Lines in Magnetic Cataclysmic Variables

Mon. Not. R. Astron. Soc. 000, 1–11 (2006) Printed 22 November 2018 (MN LATEX style file v2.2) Compton Scattering of Fe Kα Lines in Magnetic Cataclysmic Variables A. L. McNamara1⋆, Z. Kuncic1⋆, K. Wu1,2, D. K. Galloway3† and J. G. Cullen4 1School of Physics, University of Sydney, NSW 2006, Australia 2Mullard Space Science Laboratory, University College London, Holmbury St Mary, Surrey, RH5 6NT, UK 3School of Physics, University of Melbourne, Victoria 3010, Australia 4Thales, Garden Island, Cowper Wharf Rd, Potts Point, NSW 2011, Australia Accepted ... . Received ...; in original form ... ABSTRACT Compton scattering of X-rays in the bulk flow of the accretion column in magnetic cat- aclysmic variables (mCVs) can significantly shift photon energies. We present Monte Carlo simulations based on a nonlinear algorithm demonstrating the effects of Comp- ton scattering on the H-like, He-like and neutral Fe Kα lines produced in the post- shock region of the accretion column. The peak line emissivities of the photons in the post-shock flow are taken into consideration and frequency shifts due to Doppler effects are also included. We find that line profiles are most distorted by Compton scattering effects in strongly magnetized mCVs with a low white dwarf mass and high mass accretion rate and which are viewed at an oblique angle with respect to the ac- cretion column. The resulting line profiles are most sensitive to the inclination angle. We have also explored the effects of modifying the accretion column width and using a realistic emissivity profile. We find that these do not have a significant overall effect on the resulting line profiles. A comparison of our simulated line spectra with high resolution Chandra/HETGS observations of the mCV GK Per indicates that a wing feature redward of the 6.4 keV line may result from Compton recoil near the base of the accretion column. Key words: accretion – line: profiles – scattering – binaries: close – white dwarfs – X-rays 1 INTRODUCTION Magnetic cataclysmic variables (mCVs) are close interact- ing binaries consisting of a magnetic white dwarf (WD) and a low mass red dwarf (Warner 1995). Near the white dwarf surface the accretion flow in mCVs is confined by the mag- netic field of the WD and is channelled to the magnetic pole region(s) of the WD, forming an accretion column (see Warner 1995; Cropper 1990; Wu 2000; Wu et al. 2003, for reviews). Near the base of the accretion column, material in su- personic free-fall is brought to rest on the WD surface, form- ing a standing shock which heats and ionizes the accreting plasma. The shock temperature Ts depends mainly on the mass Mwd and radius Rwd of the WD and is given by (e.g. Wu 2000) kTs = GMwdµmH Rwd + xs , (1) ⋆ E-mail:aimee, [email protected] † Centenary Fellow where µ is the mean molecular weight and xs is the shock height. For WD masses Mwd ≈ (0.5 − 1.0)M⊙ and typical mCV parameters, the shock temperature is kTs ≈ (10 − 40) keV ≈ (1 − 4) × 10 8 K. The plasma in the post-shock region of the column cools by emitting bremsstrahlung X-rays and optical/IR cyclotron radiation (Lamb & Masters 1979; King & Lasota 1979). Since cooling occurs along the flow, the post-shock region is stratified in density and temperature. The height of the post-shock re- gion is determined by the cooling length. For a flow with only bremsstrahlung cooling, the shock height is given by (Wu, Chanmugam & Shaviv 1994) xs ≈ 3× 10 1 g cm−2 s−1 0.5M⊙ 109cm )−3/2 cm (2) where ṁ is the specific mass accretion rate. A plasma temperature of kT ≈ 10 keV is sufficient to fully ionize elements such as argon, silicon, sulphur, alu- minium or calcium. Heavier elements such as iron can be c© 2006 RAS http://arxiv.org/abs/0704.1516v1 2 McNamara et al. highly ionized, resulting in H-like Fe XXVI and He-like Fe XXV ions. K-shell transitions in Fe XXVI and Fe XXV ions give rise to Kα lines at 6.97 keV and 6.675 keV, respectively (see e.g. Wu, Cropper & Ramsay 2001). The irradiation of low ionized and neutral iron by X-rays above the Fe K edge produces fluorescent Kα emission at approximately 6.4 keV inside the WD atmosphere, beneath the accretion column and in surrounding areas. The natural widths of the Fe Kα lines are small, but the lines can be Doppler broadened by the bulk and thermal motions of the emitters in the post- shock flow. The bulk velocity immediately downstream of the shock is ≈ 0.25 (GMwd/Rwd) ∼ 1000 kms−1 for typ- ical mCV parameters. Lines can also be broadened by scat- tering processes. Compton (electron) scattering is expected to be more important than resonance (ion) scattering for the Kα transitions (Pozdnyakov, Sobol & Sunyaev 1983). For an mCV with specific accretion rate ṁ ∼ 10 g cm−2 s−1 the electron number density is ne ∼ 10 16 cm−3 for a shock heated region of thickness xs ∼ 10 7 cm, giving a Thompson optical depth of τ ∼ 0.1. Thus, one in every ten photons would encounter an electron before leaving the post-shock region. The relative importance of Doppler shifts, thermal Doppler broadening and Compton scattering depends on the ionization structure in the post-shock flow. X-ray observations by Chandra/HETGS and ASCA/SIS have revealed significant broadening of some Fe Kα lines in mCVs (Hellier, Mukai, & Osborne, 1998; Hellier & Mukai 2004). It was suggested that Compton scattering in the accretion column is largely responsible for the broadening in the observed lines. Doppler broadening should only be significant in lines emitted close to the shock. The absence of Doppler shifts in the observed H-like and He-like lines suggests that these photons may be emitted predominantly from regions of lower velocity near the base of the accretion column (Hellier & Mukai 2004). The observed line profiles have yet to be fully interpreted with a quantitative model that takes into account Compton scattering effects in a complex ionization structure. In this paper, we study the effects of Compton scat- tering in the accretion column of mCVs using a non- linear Monte Carlo algorithm (Cullen 2001a,b) that self- consistently takes into account the density, velocity and tem- perature structure in the column (Wu et al. 2001). The ef- fects of dynamical Compton scattering are also included. In a preliminary investigation (Kuncic, Wu & Cullen 2005), it was found that Fe Kα line photons emitted from the dense base of the accretion column undergo multiple Compton scatterings and as a result, the base of the line profile is substantially broadened. Photons emitted near the shock can also undergo scatterings with hot electrons immediately downstream of the shock, as well as cold electrons in the pre-shock flow before escaping the column. The resulting line profiles display a shoulder-like feature redward of the line centre. More significant broadening is observed when cy- clotron cooling is sufficiently strong to produce a dense, com- pact post-shock region (see Wu 2000, for example). Here, we make three substantial improvements to the previous study: (i) we include Doppler effects; (ii) the photon source regions in the post-shock column are determined from the ioniza- tion structure, rather than specified arbitrarily; and (iii) the effects of different viewing angles are fully explored. The paper is organized as follows: the theoretical outline and ge- ometry of the model are described in Section 2. Numerical results for cases where cyclotron cooling is negligible and when it dominates are presented and discussed in Section 3. A summary and conclusions are presented in Section 4. 2 THEORETICAL MODEL 2.1 Physical Processes Line photons can undergo energy changes when scattering with electrons, resulting in distortions in the line profile. The energy change of a photon per scattering is given by (e.g. see Pozdnyakov et al. 1983), 1− µβ 1− µ′β + E γmec2 (1− cosα) where E is the initial photon energy, γmec 2 is the electron energy, with γ = (1 − β2)−1/2 and where β = v/c includes both thermal and bulk motion, θ = cos−1 µ is the incident photon propagation angle measured relative to the electron’s direction of motion, and the scattering angle is α. The prime superscript denotes quantities after a scattering event. Al- though the energy change per scattering is typically small, the line profile can be broadened considerably as a result of multiple scatterings if the optical depth is large. Photons scattering with hot electrons (i.e. kTe > Ec where Te is the electron temperature and Ec is the line centre energy) will gain energy, while photons scattering with cold electrons (kTe ≪ Ec) will lose energy due to recoils. In the mCV context, Compton recoil can be important for photons scattering with electrons in the cold pre-shock flow and also near the base of the column, where β rapidly decreases and where the optical depth is high. From equation (3), the frac- tional energy change due to Compton recoil E ≫ γmec (1− cosα) . (4) In the post-shock accretion column in mCVs, line pho- tons undergo thermal Doppler broadening as well as Doppler shifts. The bulk velocity of the accreting material immedi- ately downstream of the shock is ∼ 1000 km s−1. For all inclination angles i (see Figure 1), the bulk flow is moving away from our line of sight, so the line centre energy is red- shifted by an amount ∆E/E ∼ β cos i. However, since the bulk velocity in the post-shock flow in mCVs is always less than a few × 1000 kms−1, giving β < 10−2, Doppler shifts are expected to be negligible. Thermal Doppler broaden- ing, on the other hand, is expected to be of order ∆E/E ∼ (2kTs/mic2) ≈ 0.002 for lines emitted in the hottest re- gions of the post-shock flow (i.e. immediately downstream of the shock), where mi is the mass of the ion. In strongly magnetized mCVs, cyclotron emission is the dominant cooling process. The effect of this additional elec- tron cooling process in the post-shock region is to reduce the shock height and modify the density and temperature structure of the region (Wu et al. 1994). This can enhance Compton scattering features in line profiles (Kuncic et al. 2005). c© 2006 RAS, MNRAS 000, 1–11 Compton Scattering of Fe Kα Lines in mCVs 3 Figure 1. Schematic illustration of the geometry of the magne- tized white dwarf accretion column showing approximate loca- tions of Fe Kα source regions. 2.2 Geometry of the Accretion Column The geometry of an mCV accretion column is shown in fig- ure 1. The column is modelled as a cylinder and is divided into a shock heated region and a cool pre-shock region. In the pre-shock region, the density and velocity of the accret- ing material are constant. The velocity of the pre-shock flow is approximated by the free-fall velocity at the shock height, vff(xs) = [GMwd/(Rwd+xs)] 1/2, and the plasma is assumed to be cold (kTe ≪ 1 keV). The electron number density in the pre-shock flow is ne = ṁ/(µmHvff), where ṁ is the spe- cific mass accretion rate. In the post-shock region, the den- sity, velocity and temperature profiles are calculated using the hydrodynamic solution described in Wu et al. (1994), for bremsstrahlung and cyclotron cooling. The parameters that determine the structure of the post-shock region are the WD mass Mwd, the WD radius Rwd, the specific mass accretion rate ṁ and the ratio of the efficiencies of cyclotron cooling to bremsstrahlung cooling ǫs. The accretion column is viewed from an inclination an- gle i, measured relative to the column axis (see Fig. 1). An inclination angle i = 0◦ corresponds to viewing the column along its axis, towards the WD, while viewing the column from an inclination angle i = 90◦ is equivalent to viewing the column from the side. The line photons are injected into the post-shock region at a specific dimensionless height ζ ≡ x/xs at or above the WD surface, where ζ = 0 and ζ = 1 correspond to the WD surface and shock surface, respectively (see Fig. 1). For each set of mCV parameters, the injection point of the line corresponds to the location in the post-shock region where the emissivity of the specific line peaks, according to the ionization structure determined by Wu et al. (2001). The bulk and thermal velocities at the line injection height ζ are used to calculate the Doppler shift and broadening of the lines. The Monte Carlo technique is used to model Comp- ton scattering effects on photon propagation in the column. The distance to a tentative scattering point is determined using an algorithm based on a nonlinear transport tech- nique (Stern et al. 1995) which integrates the mean free path over the spatially varying electron density (Cullen 2001a,b). The scattering cross-section is determined from the Klein- Nishina formula and the momentum vector at the scattering point is drawn from an isotropic Maxwellian distribution at the local temperature. A rejection algorithm is used to de- cide whether the scattering is accepted (Cullen 2001b). For an accepted event the energy and momentum changes of the photon are calculated as described in Pozdnyakov et al. (1983). In each simulation photons are followed until they leave the column and binned to form a spectrum. A full de- scription of the numerical algorithm can be found in Cullen (2001a). In the simulations described below, we make the follow- ing simplifying assumptions: a fixed number of 108 photons are used to simulate each line; a single energy is used for each line (whereas in reality, the neutral and Lymann α transitions are doublets, with energies 6.391/6.404 keV and 6.952/6.973 keV, and the He-like transition has both reso- nant, inter-combination and forbidden components); a fixed accretion column width is used and a single line injection site is used. 3 RESULTS AND DISCUSSION We present simulations of Compton scattering of Fe Kα lines in an mCV accretion column for two different WD mass-radius values: Mwd = 0.5M⊙, Rwd = 9.2 × 10 and Mwd = 1.0M⊙, Rwd = 5.5× 10 8 cm (Nauenberg 1972). For each mass, we consider two different specific mass ac- cretion rates, ṁ = 1 g cm−2 s−1 and ṁ = 10 g cm−2 s−1, and inclination angles i = 0◦, 45◦ and 90◦ with ±5◦ range. The cross-sectional radius of the accretion column is fixed at 0.1Rwd. We investigate the effect of varying the column width in section 3.3. Figure 2 shows the temperature, velocity and density profiles of an accretion column for an mCV with WD mass Mwd = 0.5M⊙ and ṁ = 10 g cm −2 s−1, for cases where the ratio of cyclotron to bremsstrahlung cooling at the shock is ǫs = 0, 10, and 100. The temperature decreases monotoni- cally from a shock temperature kTs ≈ 14 keV (ζ = 1), to a small finite value at the base of the column (ζ = 0).The mass density in the post-shock region is determined by ρ(ζ) = 4ṁ/vb(ζ) and is a minimum at the shock and reaches a maximum at the base of the column. The bulk velocity of the accreting material at the shock is 0.25vff and decreases to zero at the base of the column, where the plasma settles on the WD surface. Figures 3 and 4 show the simulated line spectra for the case where cyclotron cooling is negligible and bremsstrahlung cooling dominates (ǫs = 0). Figure 5 shows the simulated profiles for Mwd = 0.5M⊙ and Mwd = 1.0M⊙ when cyclotron cooling dominates bremsstrahlung (ǫs = 10). In these cases, the neutral Fe Kα line is emitted at the WD surface (ζ = 0), where the bulk velocity is ap- c© 2006 RAS, MNRAS 000, 1–11 4 McNamara et al. 0.0 0.2 0.4 0.6 0.8 1.0 εs = 0 εs = 10 εs = 100 0.0 0.2 0.4 0.6 0.8 1.0 εs = 100 εs = 10 εs = 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 εs = 0 εs = 10 εs = 100 Figure 2. The profiles of (a) electron temperature Te, (b) mass density ρe and (c) bulk velocity vb = βbc in the post-shock accretion column of an mCV with Mwd = 0.5M⊙ and ṁ = 10 g cm −2 s−1. ζ = 0 corresponds to the base of the accretion column and ζ = 1 to the shock. The solid line shows the profile for the case where the ratio of cyclotron to bremsstrahlung cooling at the shock is ǫs = 0, the dotted line for ǫs = 10 and the dashed line for ǫs = 100. In (a) the corresponding mean thermal electron velocity βe = ve/c is also shown as a function of ζ. proximately zero and the thermal velocity of the plasma is small. The 6.675 keV Fe Kα line is emitted from the lowest few percent of the column (ζ ∼ 0.003) where the velocity of the infalling material and the thermal electron velocity are still relatively small. These lines thus show very little Doppler broadening. The 6.97 keV line, however, is emit- ted much closer to the shock (ζ ∼ 0.16), in regions where the temperature of the accreting material is considerably higher and thus suffers more substantial Doppler broadening (see Wu et al. 2001). The simulated line spectra for the case where the accretion column radius is fixed at 4.6 × 107 cm is shown in Figure 6. Figure 7 shows the simulated pro- files for the case where the line photon injection along the flow is specified according to the emissivity profile model of Wu et al. (2001). In figure 8 we compare our simulated line spectra with an observation of the mCV GK Per detected by Chandra/HETGS. Table 1 shows a comparison of the FWHM of the Fe Kα lines for Mwd = 1.0M⊙ and Mwd = 0.5M⊙ with low and high ṁ and for ǫs = 0 and ǫs = 10. 3.1 No Cyclotron Cooling When cyclotron cooling is negligible (ǫs = 0), the 6.4, 6.675 and 6.97 keV lines are mostly emitted at heights above the WD surface of x ≈ 0, 0 and 1.3× 106 cm for Mwd = 1.0M⊙ and x ≈ 0, 1.0 × 105, and 5.1 × 106 cm for Mwd = 0.5M⊙, respectively (Wu et al. 2001). Note that Doppler broaden- ing is most significant for the 6.97 keV line which is emit- ted further up in warmer regions of the post-shock column. Figures 3 and 4 show that in all cases, there is substantial Compton broadening near the base of the line profiles due to scatterings in the post-shock region where photons may lose and gain energy. For example, for the 6.4 keV line emerging at i = 90◦ in Fig. 3a, the Compton broadened wings con- tain ≈ 28% of the total line photons. The blue wing extends up to ∆E ≈ 0.09 keV above the line centroid, while the red wing extends down to ∆E ≈ 0.16 keV (see Appendix). Compton features are generally more prominent at higher accretion rates since this gives higher optical depths in the column and hence, a higher scattering probability. For com- parison, the average number of scatterings per photon in the Table 1. FWHM (eV) values of the (a) 6.4 keV, (b) 6.675 keV and (c) 6.97 keV lines calculated for two different values of ṁ and for Mwd = 1.0M⊙ and Mwd = 0.5M⊙. The ratios of cyclotron to bremsstrahlung cooling at the shock are ǫs = 0 and ǫs = 10. (a) Mwd = 1.0M⊙ Mwd = 0.5M⊙ 6.4 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1 ǫs = 0 5 (7) 4 (5) = 10 4 (5) 3 (4) (b) Mwd = 1.0M⊙ Mwd = 0.5M⊙ 6.675 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1 ǫs = 0 6 (7) 4 (5) = 10 5 (6) 3 (5) (c) Mwd = 1.0M⊙ Mwd = 0.5M⊙ 6.97 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1 ǫs = 0 8 (9) 10 (10) = 10 7 (15) 8 (10) ṁ = 1g cm−2 s−1 case is 0.5 but this increases to 2 in the ṁ = 10 g cm−2 s−1 case. For Mwd = 0.5M⊙ (Fig. 4), a recoil tail redward of the line centroid can be seen in the cases i = 45◦ and 90◦ specif- ically for high ṁ (Fig. 4b) The shock temperature for these cases is kTs ≈ 14 keV, while for the Mwd = 1.0M⊙ case, the shock temperature is much higher (kTs ≈ 34 keV for low ṁ and kTs ≈ 46 keV for high ṁ). The plasma temperature in the 1.0 M⊙ case is too high to produce any downscattering features and a large Mwd (small Rwd) has an accretion col- umn with a lower optical depth, hence the absence of these features in Fig. 3. Again the downscattering features are more enhanced in the high ṁ cases since the optical depth is higher (c.f. Figs. 4a and 4b for i = 90◦). Note also that the 6.97 keV line centre is Doppler shifted slightly redward which is most noticeable for the 0.5 M⊙ case. In addition to the recoil signatures, upscattering fea- tures are also seen in the line spectra for the high ṁ cases (Figs. 3b and 4b) when i = 90◦. For these inclination an- c© 2006 RAS, MNRAS 000, 1–11 Compton Scattering of Fe Kα Lines in mCVs 5 (a) Mwd = 1.0M⊙, ṁ = 1 g cm −2 s−1, kTs = 34 keV, no cyclotron cooling i = 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 45 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 90 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) (b) Mwd = 1.0M⊙, ṁ = 10 g cm −2 s−1, kTs = 46 keV, no cyclotron cooling i = 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 45 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 90 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) Figure 3. Profiles of Fe Kα lines scattered by electrons in the accretion column of an mCV with Mwd = 1.0M⊙ and with (a) ṁ = 1 g cm−2 s−1 and (b) ṁ = 10 g cm−2 s−1. The shock temperature kTs is indicated and the ratio of cyclotron to bremsstrahlung cooling is ǫs = 0. gles, photons emerge from the column with a final scattering angle cosine µ′ ≈ 0. Equation 3 then gives ≈ 1− µβ . (5) Since head on collisions (µ < 1) have a higher probability, E′ ≃ (1+|µ|β)E. The upscattering features are more promi- nent in Fig. 3b than in Fig. 4b because the shock temper- ature, and hence β, is higher. The sharpness of the upscat- tering features seen in the 6.4 keV profiles in particular can be attributed to the maximum possible energy gain when µ ≈ −1 and to very few hot electrons with energies beyond a few standard deviations of the mean thermal (Maxwellian) energy. Scattering features are less prominent in the low ṁ cases (Figs. 3a and 4a), since the overall optical depth is smaller. For Mwd = 1.0M⊙ and ṁ = 1 g cm −2 s−1 for in- stance, the optical depth across the column at the shock is τ ≈ 0.04, which is a factor 10 smaller than that for the ṁ = 10 g cm−2 s−1 case (since τ ∝ ne ∝ ṁ). For higher WD masses (i.e. smaller Rwd), the optical depth across the column is smaller, so fewer photons are scattered, especially for low ṁ cases. This explains the difference in scattering features in the profiles shown in Figs. 3 and 4 (in particular, the high ṁ, i = 90◦ cases). Photons observed at an inclination angle of i = 0◦ prop- agate through the entire length of the column before es- caping. These photons propagate through a thick section of cold pre-shock flow and can thus downscatter, resulting in broadening redward of the line centre. However, the electron number density, and hence, optical depth, in the pre-shock flow is small so recoil effects are correspondingly small. Fur- thermore, fewer photons are detected at i = 0◦ because the solid angle centered around i = 0◦ is considerably smaller than that centered around larger inclination angles. 3.2 Cyclotron Cooling Dominated Flows Figure 5 shows the profiles of Fe Kα lines emitted from mCVs with Mwd = 1.0M⊙ and Mwd = 0.5M⊙ where i = 90◦, ṁ = 10 g cm−2 s−1 and the ratio of cyclotron to bremsstrahlung cooling at the shock is ǫs = 10. Also plotted is the thermal Doppler broadened line profile before scat- tering (dotted curve). The additional cooling of the plasma in the post-shock flow enhances the density and results in larger scattering optical depths. Because the electron tem- perature of the shock-heated region decreases as ǫs increases, the peak emissivities of the 6.675 keV and 6.97 keV Fe lines are found at larger values of ζ, as can be seen in Fig. 2, which shows the temperature, density and velocity profiles for dif- ferent cases of ǫs. The additional cyclotron cooling results in a decrease in the electron temperature and bulk velocity in the post-shock column at a fixed ζ. For ǫs = 10, the 6.4, 6.675 and 6.97 keV lines are emit- ted from heights above the WD surface of x ≈ 0, 3.1 × 105 and 2.4 × 107 cm for Mwd = 0.5M⊙, and x ≈ 0, 0 and 5.3 × 107 cm for Mwd = 1.0M⊙, respectively (Wu et al. 2001). The emission height of the 6.97 keV line is signifi- cantly higher than in the ǫs = 0 case and consequently, c© 2006 RAS, MNRAS 000, 1–11 6 McNamara et al. (a) Mwd = 0.5M⊙, ṁ = 1 g cm −2 s−1, kTs = 14 keV, no cyclotron cooling i = 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 45 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 90 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) (b) Mwd = 0.5M⊙, ṁ = 10 g cm −2 s−1, kTs = 14 keV, no cyclotron cooling i = 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 45 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) i = 90 6.0 6.2 6.4 6.6 6.8 7.0 7.2 Energy (keV) Figure 4. Profiles of Fe Kα lines scattered by electrons in the accretion column of an mCV with Mwd = 0.5M⊙ and with (a) ṁ = 1 g cm−2 s−1 (top panels) and (b) ṁ = 10 g cm−2 s−1 (bottom panels). The shock temperature kTs is indicated and the ratio of cyclotron to bremsstrahlung cooling is ǫs = 0. 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 Energy (keV) 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 Energy (keV) Figure 5. Simulated profiles of Compton scattered Fe Kα lines in the accretion column of an mCV with (a) Mwd = 1.0M⊙ and (b) Mwd = 0.5M⊙. In both cases the inclination angle is i = 90 ◦, the specific mass accretion rate is ṁ = 10 g cm−2 s−1 and the ratio of cyclotron to bremsstrahlung cooling at the shock is ǫs = 10. The dotted curves show the contribution to each line profile by thermal Doppler broadening only. thermal Doppler broadening is more prominent. The addi- tional cooling in the accretion column also results in a lower plasma temperature at the base of the column, ζ = 0 (see Fig. 2), near where irradiation of neutral iron occurs and the fluorescent 6.4 keV line is emitted. This line thus undergoes less thermal Doppler broadening than in the ǫs = 0 case. As a result of Compton scattering, the line profiles show addi- tional broadening and recoil tails redward of the line centre (Fig. 5b). Upscattering features are also evident in the 6.4 and 6.675 keV lines especially for Mwd = 1.0M⊙ (Fig. 5a). Overall, Compton scattering features are more promi- nent for cyclotron cooling dominated accretion columns, especially for high accretion rates. For photons viewed at i = 90◦, upscattering and recoil features that are only marginally seen in the ǫs = 0 case are considerably more pronounced in the ǫ = 10 case. Thus, we expect Compton scattering features to be most conspicuous in Fe Kα lines emitted in strongly magnetized mCVs accreting at a high rate. c© 2006 RAS, MNRAS 000, 1–11 Compton Scattering of Fe Kα Lines in mCVs 7 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 Energy (keV) 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 Energy (keV) Figure 6. Simulated profiles of Compton scattered Fe Kα lines in an accretion column of an mCV with (a) Mwd = 1.0M⊙ and (b) Mwd = 0.5M⊙ both with a specific mass accretion rate of ṁ = 10 g cm −2 s−1, a ratio of cyclotron to bremsstrahlung cooling of ǫs = 10, an inclination angle of i = 90◦ and accretion column radius of 4.6×107 cm . The dotted curves show the profiles when the column radius is fixed at 0.1Rwd (equivalent to the profiles shown in Fig.5). 3.3 Effect of Accretion Column Radius The width of the accretion column in mCVs is poorly known and may vary significantly from system to system. In the results presented so far, we have fixed the accretion col- umn radius to be 0.1Rwd. Here, we investigate the effect of relaxing this assumption. Figure 6 shows the resulting profiles of Fe Kα lines emitted from two mCVs of different masses (Mwd = 1.0M⊙ and Mwd = 0.5M⊙) but with the same absolute accretion column radius of 4.6× 107 cm. This corresponds to 0.08Rwd for the 1.0M⊙ case (Fig. 6a) and 0.05Rwd for the 0.5M⊙ case (Fig. 6b). The other parame- ters used to produce the profiles are the same as those used for Fig. 5, namely ṁ = 10 g cm−2 s−1, ǫs = 10 and i = 90 Fig. 6 shows that the changes in the accretion column width generally result in changes near the base of the line profiles, which is broadened by multiple scatterings. The effect of in- creasing (decreasing) the width of the accretion column is to increase (decrease) the overall scattering optical depth. The Thompson optical depth across the base of the ac- cretion column is ∼ 20 for the 1.0M⊙ case (Fig. 6a) and ∼ 40 for the 0.5M⊙ case (Fig. 6b). In comparison, the Thompson optical depth across the column base for the same lines when the column has a width of 0.1Rwd (dot- ted curves in Fig. 6) are ∼ 25 for Mwd = 1.0M⊙ and ∼ 80 for Mwd = 0.5M⊙. Fig. 6 indicates that due to the nonlin- ear nature of multiple scatterings in the accretion column, particularly near the base, small changes in optical depth associated with the accretion column geometry can result in significant changes in line profiles. These changes mostly effect the base of the line profiles. 3.4 Emissivity Profile Effects Realistically, the photon source regions are determined by an emissivity profile along the post-shock region of the ac- cretion column. In the results presented so far, all the pho- tons were injected at a single height in the accretion column corresponding to the location of the peak in the emissiv- ity profile, as calculated by Wu et al. (2001). Here, we in- vestigate the effect of spreading the photon injection site over a finite range of heights in the post-shock region ac- cording to the calculated emissivity profiles. Only the 6.675 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 Energy (keV) Figure 7. Simulated profiles of 6.675 keV and 6.97 keV Fe lines in an mCV accretion column with Mwd = 1.0M⊙, a specific mass accretion rate of ṁ = 1 g cm−2 s−1, no cyclotron cooling and viewed at an inclination angle of i = 45◦ (solid lines). An emissivity profile is used to disperse the photon injection site over a finite range of heights along the post-shock column. The dotted curves show the corresponding profiles for photons injected at a single height where the emissivity peaks for the same mCV parameters (equivalent to the profiles shown in Fig 3a, middle panel). keV and the 6.97 keV line are studied in this manner since much of the fluorescence 6.4 keV yield derives from beneath the column, where X-ray irradiation is strongest so the in- jection site is always at the base of the column. Figure 7 shows an example of the emissivity profile effect for the 6.675 keV and 6.97 keV Fe Kα lines emitted in an mCV with Mwd = 1.0M⊙, ṁ = 1 g cm −2 s−1, ǫs = 0 and i = 45 The dotted curves show the corresponding profiles for the case where photons are injected at a single height (where the emissivity peaks) for the same mCV parameters. These pro- files are the same as those shown in Fig. 3a for the i = 45◦ case (middle panel). The line profiles calculated using a realistic emissivity profile (solid curves in Fig. 7) show some additional smear- ing, particularly redward of the line centre, compared to the single injection site case (dotted curves in Fig. 7). This can be attributed to a small fraction of photons now being emitted from regions closer to the shock, where the tem- perature is ≈ 34 keV and the flow speed is ≈ 1230 km.s−1. c© 2006 RAS, MNRAS 000, 1–11 8 McNamara et al. Thus, these photons are more affected by Doppler effects than other photons emitted from regions further away from the shock. The small smearing redward of the line centre is therefore due to a combination of Doppler shift and ther- mal broadening. The effect is not prominent because only a small fraction of photons are emitted close to the shock, as predicted by the emissivity profile (Wu et al. 2001). The FWHM for the 6.675 keV and 6.97 keV line is ≈ 8 eV and ≈ 9 eV respectively. This corresponds to an additional broad- ening of approximately 25% for the 6.675 keV line and 11% for the 6.97 keV line. 3.5 Comparison with Observations Chandra/HETGS observations of a number of mCVs were reported by Hellier & Mukai (2004). The highest signal-to- noise spectrum was obtained in two observations of the mCV GK Per during its 2002 outburst. This system is classified as an intermediate polar (IP); the WD accretor is thought to have a magnetic field of ≈ 1MG (Hellier & Mukai 2004), and a mass of Mwd > 0.87±0.24M⊙ (Morales Rueda et al. 2002). Its typical X-ray luminosity in the 0.17−15 keV band- pass is Lx ≈ 7.14 × 10 33 erg s−1 (Vrielmann et al. 2005), which places a lower limit on the specific mass accretion rate, ṁ > 7.3 g cm−2 s−1 (adopting Rwd ≈ 6.7× 10 8 cm, in- ferred using the Nauenberg (1972) mass-radius relation, and assuming that accretion proceeds onto a fraction ≈ 10−3 of the WD surface area; see e.g. Frank, King & Raine 2002). Here we present a pilot study of GK Per and produce a simple comparison between the profiles of the simulated and observed spectra. A more detailed analysis of the data is left for future study. The summed first-order HETGS spectrum from the 2002 Chandra observations of GK Per is shown in Figure 8. The He-like and fluorescence Fe Kα lines are detected, and there is a possible excess at the expected en- ergy for the H-like line at 6.97 keV. Also shown in Fig. 8 are the upper and lower mass simulated HETGS spectra from a 10 ks Chandra observation of an mCV with parameters appropriate for GK Per: an upper mass of Mwd = 1.11M⊙ with Rwd = 5×10 8 cm, a lower mass ofMwd = 0.63M⊙ with Rwd = 8.2× 10 8 cm and for both masses ṁ = 10 g cm−2 s−1 and ǫs = 4 × 10 −5. We adopted inclination angles aver- aged over an orbital period of GK Per (50◦ 6 i 6 90◦) (Hellier, Harmer & Bendmore 2004). We also added a nor- malized bremsstrahlung continuum with an electron tem- perature kTe = 11 keV (Vrielmann et al. 2005). The measured equivalent widths of the 6.4, 6.675 and 6.97 keV lines in the Chandra/HETGS spectra of GK Per are 260, 117 and 80 eV, respectively. The relative strength of the lines in the simulated spectra is fixed by the number of photons used in each simulation and the assumed contin- uum flux level. We used 108 photons for all three lines, so that their relative strengths are comparable. We intend to relax this condition in future work. There is a remarkable similarity between the observed and the simulated spectra for the 6.4 keV line in particular. Realistically, the actual strengths of the 6.675 and 6.97 keV lines depend on the ion- isation structure of the flow (Wu et al. 2001), since this de- termines the emissivity of the lines. The H-like and He-like lines are thus potentially powerful diagnostic tools that can be used to deduce the physical properties of the post-shock accretion column in mCVs. The 6.4 keV line strength, on the other hand, depends on the flux of the illuminating X-rays, which is largest near the WD surface. Fluorescent lines may also be produced in the surface atmospheric region around the accretion column (Hellier & Mukai 2004), and this may contribute to the observed equivalent width of the 6.4 keV line. The observed fluorescent line in GK Per exhibits a red wing extending to 6.33 keV which Hellier & Mukai (2004) attribute to Doppler shifts. In general we find that bulk Doppler shifts have a negligible effect on Fe Kα lines. Our simulated spectra show a weaker red wing arising from downscatterings near the base of the accretion column. The observed spectrum also exhibits a shoulder extending 170 eV redward of the 6.4 keV line centre, which Hellier & Mukai (2004) suggest may be due to Compton downscattering. Our simulations confirm that recoil can indeed affect the fluores- cent line when it is emitted at the base of the accretion column, although in our spectra (Fig. 8) there is no clear evidence that recoil extends down to 170 eV redward of the 6.4 keV line. 4 SUMMARY AND CONCLUSION We have investigated line distortion and broadening effects due to Compton scattering in the accretion column of mCVs using a nonlinear Monte Carlo technique that takes into ac- count the nonuniform temperature, velocity and density pro- files of the post-shock column. Scattered Fe Kα lines were simulated for a range of different physical parameters: white dwarf mass, accretion rate, magnetic field strength and in- clination angle. The photon source regions in the post-shock flow were determined from the ionization structure and the effects due to the bulk velocity Doppler shift and thermal broadening are also considered. We find that line profiles are most affected by Compton scattering in the cases of low white dwarf mass, high specific mass accretion rate, strong cyclotron cooling and oblique in- clination angles. Both a lower white dwarf mass (or equiv- alently a larger white dwarf radius) and a higher accretion rate result in a higher optical depth, hence more pronounced scattering effects. Strong cyclotron cooling associated with a high magnetic field strength results in a higher density in the post-shock flow and hence, a correspondingly larger optical depth. Recoil signatures are evident for oblique view- ing angles when cyclotron cooling becomes important. These are due to scatterings near the base of the flow, where both the dynamical and thermal velocities are small. Sharp up- scattering features are also seen in the line profiles for large viewing angles. These upscattering features are attributed to head-on collisions with hot electrons near the shock. Pho- tons emitted close to the shock in the accretion column dis- play more thermal Doppler broadening. This is generally the case for the Fe XXVI 6.97 keV line. The Fe 6.4 keV and Fe XXV 6.675 keV lines are generally emitted at or near the base of the accretion column respectively, and show a rel- atively small amount of thermal Doppler broadening. Bulk velocity induced Doppler shifts are negligible compared with scattering. We also investigated the effects of dispersing the 6.675 and 6.97 keV line photons using an emissivity calculated self-consistently from the ionization structure of the post- c© 2006 RAS, MNRAS 000, 1–11 Compton Scattering of Fe Kα Lines in mCVs 9 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 Energy (keV) 0.0001 0.0002 0.0003 0.0004 0.0005 Observed Spectrum Low Mass Simulated Spectrum High Mass Simulated Spectrum Figure 8. Fe Kα emission lines for the mCV GK Per detected by Chandra/HETGS (bottom) and simulated by our model for a lower and upper white dwarf mass Mwd = 0.63M⊙ (middle) and Mwd = 1.11M⊙ (top) both with a specific mass accretion rate ṁ = 10 g cm −2 s−1. The simulated spectra have been convolved with the Chandra/HETGS response function. Cyclotron cooling is negligible and we have averaged the simulated lines over all inclination angles in an orbital period. An offset has been added to the simulated plot to allow a comparison between the spectra. See text for further details. shock column. We find that, compared to injecting the line photons at a single height where the emissivity peaks, us- ing an emissivity profile does not significantly change the overall profiles of the 6.675 and 6.97 keV lines. The result- ing line profiles show a small degree of smearing redward of the line center due to additional dynamical and thermal Doppler effects associated with a small fraction of line pho- tons emitted close to the shock. The effect is most important for the 6.675 keV line. We estimate a fractional increase in the FWHM of no more than ≈ 25%. In general, our simulations predict that recoil in scat- tering with cool electrons near the base of the column as well as upscattering by hot electrons near the shock can im- print signatures on the profile of lines emitted near the base of the flow. Such Compton signatures may thus be used to determine the primary source region of the fluorescent line. We predict that when 6.4 keV photons are emitted in the dense, cool plasma at the base of the accretion column, they can suffer strong downscattering when propagating through the electrons streaming downward in the flow. The scatter- ing probability is governed by the effective scattering optical depth, which increases with the mass accretion rate of the system. We convolved simulations with the Chandra response function and compared them to Chandra/HETGS observa- tions of GK Per. Our results indicate that the red wing seen in the 6.4 keV line in GK Per could be attributed to Comp- ton recoil near the base of the flow. Finally we remark that tighter constraints on the dy- namics and flow geometry in magnetized accreting com- pact objects can be obtained by considering the polariza- tion properties of the lines (see Sunyaev & Titarchuk 1985; Matt 2004). There is currently considerable effort to develop X-ray polarimeters which can detect degrees of polarization of the order of one percent (Costa et al. 2001). The spectral resolution of these detectors should be adequate to search for different polarization degrees in emission iron lines. It is possible to include a polarization treatment in our calcula- tions, but the computational algorithm in our Monte Carlo code will need to be revised, which we leave for a future study. ACKNOWLEDGMENTS ALM thanks a University of Sydney Denison Scholarship. KW’s visit to Sydney University was supported by a NSW State Expatriate Researcher Award. We thank an anony- mous referee whose comments helped improve the paper considerably. REFERENCES Costa, E., Soffitta P., Bellazzini R., Brez, A., Lumb, N. & Spandre, G. 2001, Nature, 411, 662 Cropper, M. 1990, Sp. Sci. Rev., 54, 195 Cropper, M., Wu, K. & Ramsay, G. 2000, NewAR, 44, 57 Cullen, J.G. 2001a, PhD Thesis, University of Sydney Cullen, J.G. 2001b, JCoPh, 173, 175 Frank, J., King, A. & Raine, D. 2002, Accretion Power in Astrophysics, Cambridge: Cambridge University Press Hellier, C., Mukai, K. & Osborne, J. P. 1998, MNRAS, 297, Hellier, C. & Mukai, K. 2004, MNRAS, 352, 1037 Hellier, C., Harmer, S. & Bendmore, A.P. 2004, MNRAS, 349, 710 King, A.R., Lasota, J.P., 1979, MNRAS, 188, 653 Kuncic, Z., Wu, K. & Cullen, J. 2005, PASA, 22, 56 Lamb, D.Q. & Masters, A.R. 1979, ApJ, 234, L117 Matt, G. 2004, A&A, 423, 495 Morales Rueda, L., Still, M.D., Roche, P., Wood, J.H. & Lockley, J.J. 2002, MNRAS, 329, 597 c© 2006 RAS, MNRAS 000, 1–11 10 McNamara et al. Nauenberg M. 1972, ApJ, 175, 417 Pozdnyakov, L.A., Sobol, I.M., & Sunyaev, R.A. 1983, AS- PRv, 2, 189 Stern, B., Begelman, M., Sikora, M. & Svensson, R. 1995, MNRAS, 272, 291 Sunyaev, R.A. & Titarchuk, L.G. 1985, A&A, 143, 374 Vrielmann, S., Ness, J.-U. & Schmitt, J.H.M.M. 2005, A&A, 439, 287 Warner, B. 1995, Cataclysmic Variable Stars, Cambridge: Cambridge University Press Wu, K. 2000, SSRv, 93, 611 Wu, K., Chanmugam, G. & Shaviv, G. 1994, ApJ, 426, 664 Wu, K., Cropper, M. & Ramsay, G. 2001, MNRAS, 327, Wu, K., Cropper, M., Ramsay, G., Saxton, C. & Bridge, C. 2003, Chin. J. Astron. Astrophys., 3 (Suppl.), 235 APPENDIX A: THE WIDTH OF THE COMPTON SHOULDER Here, we derive an estimate for the width of the prominent wings near the base of the line profiles. Let the accretion flow be along the z-axis and consider an observer on the x−z plane. Let i be the line-of-sight inclination angle of the accretion column, and let β be the local velocity of the flow, normalized to the speed of light c, which can expressed as ~β = −β ẑ . The normalized vector of a scattered photon of energy E′ propagating in the direction to the observer is = sin i x̂ + cos i ẑ . Suppose that the normalized vector of the incident photon, with energy E, is k̂ = sin θ cosφ x̂ + sin θ sinφ ŷ + cos θ ẑ . Then the change in energy of the photon, after scattering with an electron in the flow, due to bulk motion is given by 1− µβ (1− µ′β) + E γmec2 (1− k̂ · k̂′) where γ = (1−β2)−1/2, µβ = k̂·~β, and µ′β = k̂′ ·~β . In terms of the viewing inclination angle and the photon propagation vectors, 1 + β cos θ (1 + β cos i) + E γmec2 (1− sin i sin θ cosφ− cos i cos θ) Then for β ≪ 1 and E ≪ mec ≈ 1 + β(cos θ − cos i)− λ(1− sin i sin θ cos φ− cos i cos θ) . It follows that ≈ β(cos θ − cos i)− λ(1− sin i sin θ cosφ− cos i cos θ) , where ∆E = E′ − E and λ = E/mec The maximum energy downshift is caused by the recoil process when the photons are scattered by “cold” electrons (with β ≈ 0). This occurs when φ = π, θ = π − i, which yields max down = −2λ . This result is practically independent of the WD mass and the viewing inclination angle. For the 6.4 keV line, ∆E/E|max down ≈ −2.50× 10 The maximum energy upshift is caused by a Doppler shift when the photons are scattered by the fastest available downstream electrons (i.e. β is no longer a negligible factor). The condition for its occurrence can be derived as follows. = 0 , which gives two conditions for extrema with respect to the azimuthal coordinate: φ = 0 and φ = π, corresponding re- spectively to = β(cos θ − cos i) − λ(1∓ sin i sin θ − cos i cos θ) . Differentiating the above expression with respect to θ and setting the resulting expression to zero yields the following condition for the extrema: sin θ cos θ = ±ξ , where λ sin i β + λ cos i The first case (φ = 0) leads to a maximum energy upshift, which requires sin θ = 1 + ξ2 cos θ = 1 + ξ2 At the viewing inclination angle i = π/2, sin θ = β2 + λ2 cos θ = β2 + λ2 Hence, the maximum energy upshift is given by max up β2 + λ2 − λ . If we assume that the maximum β takes the free-fall veloc- ity at the WD surface, then β is simply the reciprocal of the square root of the WD radius in the Schwarzschild unit, i.e. 2GMwd/Rwdc2. For Mwd = 1.0M⊙, β = 2.31×10 and for Mwd = 0.5M⊙, β = 1.23 × 10 −2 (assuming the Nauenberg mass-radius relation (Nauenberg 1972)), this gives ∆E/E|max up ≈ 1.36 × 10 −2 and 5.06 × 10−3 respec- tively for the 6.4 keV line. The Compton shoulder of an Fe line is due to a single scattering event. For the 6.4 keV line, the broadening ex- tends over 6.24 keV 6 E 6 6.49 keV for a 1.0 M⊙ WD and 6.24 keV 6 E 6 6.43 keV for a 0.5 M⊙ WD (omitting ther- mal broadening and flow Doppler broadening), when viewed at i = π/2. c© 2006 RAS, MNRAS 000, 1–11 Compton Scattering of Fe Kα Lines in mCVs 11 This paper has been typeset from a TEX/ L ATEX file prepared by the author. c© 2006 RAS, MNRAS 000, 1–11 Introduction Theoretical Model Physical Processes Results and Discussion No Cyclotron Cooling Effect of Accretion Column Radius Emissivity Profile Effects Comparison with Observations Summary and Conclusion The width of the Compton shoulder

0704.1517

The Limits of Special Relativity

THE LIMITS OF SPECIAL RELATIVITY B.G. Sidharth International Institute for Applicable Mathematics & Information Sciences Hyderabad (India) & Udine (Italy) B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India) Abstract The Special Theory of Relativity and the Theory of the Electron have had an interesting history together. Originally the electron was studied in a non relativistic context and this opened up the interest- ing possibility that lead to the conclusion that the mass of the elec- tron could be thought of entirely in electromagnetic terms without in- troducing inertial considerations. However the application of Special Relativity lead to several problems, both for an extended electron and the point electron. These inconsistencies have, contrary to popular belief not been resolved satisfactorily today, even within the context of Quantum Theory. Nevertheless these and subsequent studies bring out the interesting result that Special Relativity breaks down within the Compton scale or when the Compton scale is not neglected. This again runs contrary to an uncritical notion that Special Relativity is valid for point particles. 1 Introduction When Einstein proposed his Special Theory of Relativity, there were two ruling paradigms, which continue to hold sway even today, though not so universally. The first was that of point elementary particles and the second was that of space time as a differentiable manifold. Little wonder therefore that as the relativistic theory of the electron de- veloped, there were immediate inconsistencies which were finally ostensibly resolved only with the intervention of Quantum Theory. This was because, http://arxiv.org/abs/0704.1517v1 historically the original concept of the electron was that of a spherical charge distribution [1, 2, 3]. It is interesting to note that in the non-relativistic case, it was originally shown that the entire inertial mass of the electron equalled its electromagnetic mass. This motivated much work and thought in this interesting direction. To put it briefly, in non relativistic theory, we get [1], Kinetic energy = (β/2) where R is the radius of the electron and β is a numerical factor of the order of 1. So we could possibly speak of the entire mass of the electron in terms of its electromagnetic properties. It might be mentioned that it was still possible to think of an electron as a charge distribution over a spherical shell within the relativistic context, as long as the electron was at rest or was moving with a uniform velocity. However it was necessary to introduce, in addition to the electromagnetic force, the Poincare stresses - these were required to counter balance the repulsive ”explosion” of the different parts of the electron. When the electron in a field is accelerated, the above picture no longer holds. We have to introduce the concept of the electron self force which is given by, in the simple case of one dimensional motion, ẍ+ γ ẍ+ 0(R2) (1) where dots denote derivatives with respect to time, and R is the radius of the spherical electron. More generally (1) becomes a vector equation. As can be seen from (1), as R the size of the electron → 0 the first term → ∞ and this is a major inconsistency. In contrast the second term which contains the non Newtonian third time derivative remains unaffected while the third and following terms → 0. It may be mentioned that the first term (which → ∞) gives the electromagnetic mass of the electron while the second term gives the well known Schott term (Cf.ref.[1, 2, 4]). Let us now see how it was possible to rescue the relativistic electron theory, though at the expense of introducing some unphysical concepts. 2 The Advanced and Retarded Fields To proceed, from a classical point of view a charge that is accelerating ra- diates energy which dampens its motion. This is given by the well known Maxwell-Lorentz equation, which in units c = 1, and τ being the proper time, while ı = 1, 2, 3, 4, is (Cf.[8]), is = eF ık , (2) The first term on the right side is the usual external field while the second term is the damping field which is added ad hoc by the requirement of the energy loss due to radiation. In 1938 Dirac introduced instead = e{F ık +R where Rık ≡ {F ık(ret) − F k(adv)} (4) In (4), F(ret) denotes the retarded field and F(adv) the advanced field. While the former is the causal field where the influence of a charge at A is felt by a charge at B at a distance r after a time t = r , the latter is the advanced field which acts on A from a future time. In effect what Dirac showed was that the radiation damping term in (2) or (3) is given by (4) in which an antisymmetric difference of the advanced and retarded fields is taken. Let us elaborate a little further. The Maxwell wave equation has two independent solutions, one having sup- port on the future light cone, this is the retarded solution and the other having support on the past light cone which has been called the advanced solution. The retarded solution is selected to describe the physical situa- tion in conventional theory taking into account the usual special relativistic concept of causality. This retarded solution is physically meaningful, as it de- scribes electromagnetic radiation which travels outward from a given charge with the speed of light and reaches another point at a later instant. It has also been called for this reason the causal solution. On the grounds of this causality, the advanced solution has been rejected, except in a few formula- tions like those of Dirac above, or Feynman and Wheeler (F-W) to be seen below. In the F-W formulation, the rest of the charges in the universe react back on the original electron through their advanced waves, which arrive at the given charge at the same time as the given charge radiates its electromagnetic waves. More specifically, when an electron is accelerated at the instant t, it interacts with the other charges at a later time t′ = t + r/c where r is the distance of the other charge–these are the retarded interactions. However the other charges react back on the original electron through their advanced waves, which will arrive at the time t′ − r/c = t. Effectively, there is in- stantaneous action at a distance. It must be mentioned that in the F-W formulation there is no self force or radiation damping. This is provided in- stead by the action of all other charges in the universe on the original charge. There is also no electromagnetic mass, like the first term on the right side of It must also be mentioned that Dirac’s prescription lead to the so called run- away solutions, with the electron acquiring larger and larger velocities in the absence of an external force [7]. This he related to the infinite self energy of the point electron. To elaborate further we use the difference of the advanced and retarded fields (that is (9) in (1), in the following manner: We use successively F(ret) and F(adv) in (1) and take the difference in which case the self force becomes (Cf.[4]) F = − (ẍ) + 0(R) In the above, the troublesome infinity generating term of (1) is absent, while the third derivative term is retained. On the other hand this term is required on grounds of conservation of energy, due to the fact that an accelerated electron radiates energy (Cf.[5]). Except for the introduction of advanced fields, we have infinity free results. However, in this formulation too, there is no electromagnetic mass term, and further, as will be seen below, we have to extend our considerations to a small neighborhood of the electron, and not just the point electron itself. To see this in detail, we observe that the well known Lorentz Dirac equation (Cf.[1]), can be written as maµ(τ) = Kµ(τ + ατ0)e −αdα (5) where aµ is the accelerator and Kµ(τ) = F in + F ext − ∼ 10−23sec (6) τ ′ − τ where τ denotes the time and R is the total radiation rate. Incidentally this is a demonstration of the non locality in Compton time τ0, referred to above. It can be seen that equation (5) differs from the usual equation of Newtonian Mechanics, in that it is non local in time. That is, the acceleration aµ(τ) depends on the force not only at time τ , but at subsequent times also. Let us now try to characterise this non locality. We observe that τ0 given by equation (6) is the Compton time ∼ 10−23secs. So equation (5) can be approximated by maµ(τ) = Kµ(τ + ξτ0) ≈ K µ(τ) (7) Thus as can be seen from (7), the Lorentz-Dirac equation differs from the usual local theory by a term of the order of ȧµ (8) the so called Schott term. It is well known that the time component of the Schott term (8) is given by (Cf.ref.[1]) ≈ R ≈ where E is the energy of the particle. Whence integrating over the period of non locality ∼ τ0 the Compton time, we can immediately deduce that r the scale of spatial non locality is given by r ∼ cτ0, which is of the order of the Compton wavelength. So far as the breakdown of causality is concerned, this takes place within a period ∼ τ , the Compton time as we briefly saw [1, 7]. It was at this stage that Wheeler and Feynman reformulated the above action at a distance formalism in terms of what has been called their Absorber Theory. In their formulation, the field that a charge would experience because of its action at a distance on the other charges of the universe, which in turn would act back on the original charge is given by (ẍ) (9) The interesting point is that instead of considering the above force in (9) at the charge e, if we consider the response at an arbitrary point in its neighborhood as was shown by Feynman and Wheeler (Cf.ref.[19]) and, in fact a neighborhood at the Compton scale, as was argued recently by the author [13], the field would be precisely the Dirac field given in (3) and (4). The net force emanating from the charge is thus given by F ret = F ret + F adv F ret − F adv which is the acceptable causal retarded field. The causal field now consists of the time symmetric field of the charge together with the Dirac field, that is the second term in (10), which represents the response of the rest of the charges. Interestingly in this formulation we have used a time symmetric field, viz., the first term of (10) to recover the retarded field with the correct arrow of time. Feynman and Wheeler stressed that the universe has to be a perfect absorber or to put it simply, every charged particle in the universe should respond back to the action on it by the given charge in our instantaneous action at a distance scenario. In any case, it was realized that the limits of classical physics are reached in the above considerations, at the Compton scale. There are two important inputs which we can see in the above more recent formulation. The first is the action of the rest of the universe at a given charge and the other is minimum spacetime intervals which are of the order of the Compton scale. The minimum spacetime interval removes, firstly the advanced field effects which take place within the Compton time and secondly the infinite self energy of the point electron disappears due to the Compton scale. 3 Quantum Mechanical Considerations The Compton scale comes as a Quantum Mechanical effect, within which we have zitterbewegung effects and a breakdown of Causal Physics [14]. Indeed Dirac had noted this aspect in connection with two difficulties with his elec- tron equation. Firstly the speed of the electron turns out to be the velocity of light. Secondly the position coordinates become complex or non Hermi- tian. His explanation was that in Quantum Theory we cannot go down to arbitrarily small spacetime intervals, for the Heisenberg Uncertainty Princi- ple would then imply arbitrarily large momenta and energies. So Quantum Mechanical measurements are an average over intervals of the order of the Compton scale. Once this is done, we recover meaningful physics. All this has been studied afresh by the author more recently, in the context of a non differentiable spacetime and noncommutative geometry [21]. Weinberg also notices the non physical aspect of the Compton scale [15]. Starting with the usual light cone of Special Relativity and the inversion of the time order of events, he goes on to add, and we quote at a little length and comment upon it, “Although the relativity of temporal order raises no problems for classical physics, it plays a profound role in quantum theories. The uncertainty principle tells us that when we specify that a particle is at position x1 at time t1, we cannot also define its velocity precisely. In conse- quence there is a certain chance of a particle getting from x1 to x2 even if x1 − x2 is space-like, that is, |x1 − x2| > |x 1 − x 2|. To be more precise, the probability of a particle reaching x2 if it starts at x1 is nonnegligible as long 0 ≤ (x1 − x2) 2 − (x01 − x · · · (11) where h̄ is Planck’s constant (divided by 2π) and m is the particle mass. (Such space-time intervals are very small even for elementary particle masses; for instance, if m is the mass of a proton then h̄/m = 2×10−14cm or in time units 6× 10−25sec. Recall that in our units 1sec = 3× 1010cm.) We are thus faced again with our paradox; if one observer sees a particle emitted at x1, and absorbed at x2, and if (x1 − x2) 2 − (x01 − x 2 is positive (but less than or = h̄2/m2), then a second observer may see the particle absorbed at x2 at a time t2 before the time t1 it is emitted at x1. “There is only one known way out of this paradox. The second observer must see a particle emitted at x2 and absorbed at x1. But in general the particle seen by the second observer will then necessarily be different from that seen by the first.” There is another way to view (11). The light cone of special relativity viz., (x1−x2) 2− (x01−x 2 = 0 now gets somewhat distorted because of Quantum Mechanical effects. Let us consider the above in the context of a non zero photon mass. Such a mass ∼ 10−65gms was recently deduced by the author, and it is not only consistent with experimental restrictions, but also predicts a new effect viz., a residual cosmic radiation ∼ 10−33eV , which in fact has been observed [11, 12, 16, 17, 18]. Such a photon would have a Compton length ∼ 1028cms, that is the radius of the universe itself. This would then lead to the following scenario: An observer would see a photon leaving a particle A and then reaching another particle B, while a different observer would see exactly the opposite for the same event - that is a photon leaves B and travels backward in time to A, as in the Weinberg interpretation. This latter gives the advanced potential. The distinction between the advanced and retarded potentials of the old electromagnetic theory thus gets mixed up and we have to consider both the advanced and retarded potentials. We consider this in a little more detail: The advanced and retarded solutions of the wave equation are given by the well known advanced and retarded potentials given by, in the usual notation, the well known expression ret(adv)(x) = jµ(x′) |r − r′| δ (|r − r′| ∓ c(t− t′)) d4x′ (The retarded part of which leads to the Lienard Wiechart potential of earlier theory). It can be seen in the above that we have the situation described within the Compton wavelength, wherein there are two equivalent descriptions of the same event–a photon leaving the charge A and reaching the charge B or the photon leaving the charge B and reaching the charge A. The above expression for the advanced and retarded potentials immediately leads to the advanced and retarded fields (4) and (10) of the F-W description except that we now have a rationale for this formulation in terms of the photon mass and the photon compton wavelength rather than the perfect absorber ad hoc prescription. In fact there is now an immediate explanation for this of the Instantaneous Action At a Distance Theory alluded to. In this case the usual causal electromagnetic field would be given by half the sum of the advanced and retarded fields. We note that as the photon mass is so small, the usual theory is still a good approximation. To sum up [19], the Feynman Wheeler Perfect Absorber Theory required that every charge should interact instantaneously with every other charge in the universe, that is that the universe must be a perfect absorber of all electromagnetic fields emanating from within. If this condition were satisfied, then the net response of all charged particles along the future light cone of the given charge is expressed by an integral that converges. We have argued that this ad hoc prescription of Feynman and Wheeler as embodied by the inclusion of the advanced potential is automatically satisfied if we consider the photon to have a small mass 10−65gms as deduced by the author elsewhere, and which is consistent with the latest experimental limits-this leading to the effect mentioned by Weinberg within the Compton wavelength, which is really the inclusion of the advanced field as well. 4 The Limits of Special Relativity What we have witnessed above is that it is still possible to rescue the clas- sical relativistic theory of the electron, but at the expense of introducing the advanced fields into the physics, fields which have been considered to be unphysical. Another perspective is, as seen above, that there is instantaneous action at a distance, which apparently goes against relativistic causality. But let us now note that in both the Dirac and the Feynman-Wheeler approaches, we are no longer dealing with point particles alone, but rather with a small neighbor- hood of such a point particle, a neighborhood of a Compton length dimension. Furthermore within the Compton scale, relativistic causality breaks down. We can then reformulate the above considerations in the following manner: The limit of applicability or the limit of validity of the Special Theory of Relativity is the Compton scale. The points within the Compton scale no longer obey Special Relativity and see a non relativistic, instantaneous action at a distance universe. 5 Discussion 1. We would like to sum up the foregoing considerations. In Classical Physics the point electron leads to infinite self energy via the term e2/R, where R is the radius which is made to tend to zero. If on the other hand R does not vanish, in other words we have an extended electron, then we have to intro- duce non electromagnetic forces like the Poincare stresses for the stability of this extended object, though on the positive side this allows the radiation damping or self force that is required by conservation laws. Dirac could get rid of the infinity by introducing the difference between the advanced and retarded potentials: This was the content of the Lorentz Dirac equation. The new term represents the radiation damping effect, but we then have to contend with the advanced potential or equivalently a non locality in time. However this non locality takes place within the Compton time, within which the electron attains a luminal velocity. The Lorentz Dirac equation on the other hand had unsatisfactory features like the derivative of the acceleration, the non locality in time and the run away solutions, features confined to the Compton scale. The Feynman-Wheeler approach bypasses the infinity and the extended elec- tron. Moreover the net result is that there is a retarded potential. But an instantaneous interaction with the rest of the charges of the universe is re- quired. It is this interaction with the remaining charges which leads to the point electron’s self energy. Surprisingly however the interaction with the rest of the charges in the immediate vicinity of the given charge in the Feynman- Wheeler formula gives us back the Dirac antisymmetric difference with its non locality within the Compton scale. There is thus a reconciliation of the Dirac and the Feynman Wheeler approaches, once we bring into the picture, the Compton scale. In the Feynman-Wheeler approach on the other hand the self force is dis- pensed with but at the expense of invoking the instantaneous interaction of the electron in question with the rest of the charges in the universe, though even here the Compton scale of the electron comes into question. Outside this scale, the theory is causal that is uses only the retarded potential be- cause effectively the advanced potential gets canceled out as it appears as the sum of the symmetric and antisymmetric differences. The important point however is that all this can be explained consistently in the context of the photon having a non zero mass, consistent with experiment ∼ 10−65gms. The final conclusion was that in a Classical context a totally electromagnetic electron is impossible as also the concept of a point electron. It was believed therefore that the electron was strictly speaking the subject of Quantum Theory. Nevertheless in Dirac’s relativistic Quantum Electron, we again en- counter the electron with the luminal velocity within the Compton scale, precisely what was encountered in Classical Theory as well, as noted above. This again is the feature of a point space time approach. At this stage a new input was given by Dirac - meaningful physics required averages over the Compton scale, in which process, the unphysical zitterbewegung effects were eliminated. Nor has Quantum Field Theory solved the problem - one has to take recourse to renormalization, and as pointed out by Rohrlich, one still has a non electromagnetic electron. In any case, it appears that further progress would come either from giving up point space time or from an elec- tron that is extended (or has a sub structure) in some sense [3, 2, 7, 1]. 2. Nevertheless it is curious to notice that there is some convergence be- tween the Dirac and the Feynman-Wheeler approaches if we consider the fact that special relativity, as seen above, does not hold within the Comp- ton wavelength. This explains the non locality in time. This justifies the use of the advanced potential or non locality in time of the Lorentz-Dirac approach or also the fact that a point inside the Compton wavelength sees a non relativistic instantaneous action at a distance universe around it - this is the instantaneous action at a distance of the Feynman-Wheeler approach. Furthermore, the radiation of photons emitted by the accelerated electrons (in the Dirac self force) are meaningful only if they impinge on other charges as in the Field Theory. We now briefly note the following. 3. As Rohrlich [20] observes, ”Classical Physics ceases to be valid at or below the Compton wavelength and this cannot be valid for a point object.” 4. The self interaction we encountered above gives rise to radiation reaction for an extended object which for a point charge appears as a self acceleration or pre acceleration and an extra inertia, the electromagnetic mass whose be- havior was thought to contradict special relativity [3]. 5. This apart the contradiction of run away accelerations or a divergent elec- tromagnetic mass do not apply for an extended electron. 6. If we take special relativity into account, we get the undesirable factor 4/3 indicating that part of the mass is not electromagnetic - the beauty of getting a unified theory for mass is lost. 7. Even in the non relativistic theory, Poincare stresses are required for the stability of the extended electron. 8. On the other hand Fermi and others showed that relativistically the elec- tromagnetic momentum need not be associated with the Poynting vector, in which case the undesirable 4/3 factor does not arise and there is no need for Poincare stresses. 9. In classical relativistic theory, there appeared an impasse. We could get a special relativistic electron with cohesive forces in an extended model but at the expense of purely electromagnetic electron. On the other hand point electrons were not meaningful as their self energy diverged. Consequently the structure dependent terms had to be taken seriously. 10. We have arrived at the Compton scale from two different approaches. Classically, there was the electron radius and Quantum Mechanically the Compton length, both of the same order except for a factor of the order of the fine structure constant: h̄/mc ∼ β · e2/mc2 We could consider this to a derivation of the value of the Planck constant of Quantum Mechanics, in an order of magnitude sense. 11. In any case the above considerations at the Compton scale lead in recent studies to a noncommutative geometry and the limit to a point particle no longer becomes legitimate. This has been discussed in detail in [21]. References [1] Rohrlich, F., “Classical Charged Particles”, Addison-Wesley, Reading, Mass., 1965, pp.145ff. [2] Barut, A.O., “Electrodynamics and Classical Theory of Fields and Par- ticles”, Dover Publications, Inc., New York, 1964, p.97ff. [3] Jimenez, J.L., and Campos, I., Found. of Phys.Lett., Vol.12, No.2, 1999, p.127-146. [4] Sidharth, B.G., ”The Lorentz Dirac and Dirac Equations” to appear in Electromagnetic Phenomena. [5] Sidharth, B.G., Found. Lett.Phys., Vol. II, 28-7. [6] Wheeler, J.A., and Feynman, R.P., Rev.Mod.Phys., 17, 157, 1945. [7] Hoyle, F., and Narlikar, “Action at a Distance in Physics and Cosmol- ogy”, W.H. Freeman, New York, 1974, pp.12-18. [8] Hoyle, F., and Narlikar, J.V., “Lectures on Cosmology and Action at a Distance Electrodynamics”, World Scientific, Singapore, 1996. [9] Chubykalo, A., and Smirnov-Rueda, R., Phys.Rev.E 53 (5), 5373, 1996. [10] Chubykalo, A., and Smirnov-Rueda, R., Phys.Rev.E 57 (3), 1, 1998. [11] Sidharth, B.G., Found.Phys.Lett. 19 (1), 2006, pp.87ff. [12] Sidharth, B.G., Found.Phys.Lett. 19 (4), 2006. [13] Sidharth, B.G., in Instantaneous Action at a Distance in Modern Physics: “Pro and Contra”, Eds., A.E. Chubykalo et al., Nova Science Publishing, New York, 1999. [14] Dirac, P.A.M., ”The Principles of Quantum Mechanics”, Clarendon Press, Oxford, 1958, pp.4ff, pp.253ff. [15] Weinberg, S., ”Gravitation and Cosmology”, John Wiley & Sons, New York, 1972, p.61ff. [16] B.G. Sidharth, ”A Brief Note on a New Cosmic ”Coincidence” and New Physics”, to appear in Foundation of Physics Letters. [17] Mersini-Houghton, L., Mod.Phys.Lett.A., Vol.21, No.1, (2006), 1-21. [18] Sidharth, B.G., physics/0608222. [19] Sidharth, B.G., ”The Feynman Wheeler Perfect Absorber Theory in a New Light” to appear in Foundation of Physics Letters. [20] Rohrlich, F., Am.J.Phys. 65 (11), November 1997, p.1051-1056. [21] B.G. Sidharth, “The Universe of Fluctuations”, Springer, Dordrecht, 2005. http://arxiv.org/abs/physics/0608222 Introduction The Advanced and Retarded Fields Quantum Mechanical Considerations The Limits of Special Relativity Discussion

0704.1519

Comprehensive simulations of superhumps

Mon. Not. R. Astron. Soc. 000, 1–18 (2005) Printed 18 November 2021 (MN LATEX style file v2.2) Comprehensive simulations of superhumps Amanda J. Smith1,⋆, Carole A. Haswell1, James R. Murray2, Michael R. Truss3 and Stephen B. Foulkes1 1Department of Physics & Astronomy, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK 2Department of Astrophysics & and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia 3Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK ABSTRACT We use 3D SPH calculations with higher resolution, as well as with more realistic viscosity and sound-speed prescriptions than previous work to examine the eccentric instability which underlies the superhump phenomenon in semi-detached binaries. We illustrate the importance of the two-armed spiral mode in the generation of super- humps. Differential motions in the fluid disc cause converging flows which lead to strong spiral shocks once each superhump cycle. The dissipation associated with these shocks powers the superhump. We compare 2D and 3D results, and conclude that 3D simulations are necessary to faithfully simulate the disc dynamics. We ran our sim- ulations for unprecedented durations, so that an eccentric equilibrium is established except at high mass ratios where the growth rate of the instability is very low. We collate the observed data on superhumps. Our improved simulations give a closer match to the observed relationship between superhump period excess and bi- nary mass ratio than previous numerical work. The observed black hole X-ray transient superhumpers appear to have systematically lower disc precession rates than the cat- aclysmic variables. This could be due to higher disc temperatures and thicknesses. No high-resolution 3D disc with mass ratio q > 0.24 developed superhumps, in agreement with analytical expectations. The modulation in total viscous dissipation on the superhump period is over- whelmingly from the region of the disc within the 3 : 1 resonance radius. The pre- cession rates of our high resolution 3D discs match the single particle dynamical pre- cession rate at 0.87R3:1. As the eccentric instability develops, the viscous torques are enhanced, and the disc consequently adjusts to a new equilibrium state, as suggested in the thermal-tidal instability model. We quantify this enhancement in the viscosity, which is ∼ 10 per cent for q = 0.08. The disc motions can be described as superpo- sitions of the S(k,l) modes, and the disc executes complex standing wave dynamics which repeat in the inertial frame on the disc precession period. We characterise the eccentricity distributions in our accretion discs, and show that the entire body of the disc partakes in the eccentricity. Key words: accretion, accretion discs — hydrodynamics — instabilities — methods: numerical — binaries: close — novae, cataclysmic variables 1 INTRODUCTION Cataclysmic variables (CVs) are semi-detached binaries with a Roche-lobe filling low-mass donor star, mass M2, transfer- ring matter onto a white dwarf (WD) primary, mass M1, via an accretion disc. The SUUMa-type dwarf novae (DNe) are short-period cataclysmic variables which display two distinct modes of outburst. The normal outbursts are at- tributed to a thermal–viscous limit–cycle between low and ⋆ E-mail: [email protected] high viscosity states (Osaki 1974; Hōshi 1979; Meyer & Meyer-Hofmeister 1981), and consequently between low and high mass transfer states (see Lasota 2001, for a review). The larger amplitude and longer-lasting superoutbursts are characterised by a periodic photometric modulation known as superhumps. Superhumps are attributed to an eccentric apsidally-precessing accretion disc (Vogt 1982). The super- hump period, Psh, is a few percent longer than the orbital period, Porb: the orientation of the mass donor star relative to the progradely precessing eccentric disc repeats on Psh. Whitehurst (1988) and Lubow (1991a) explained that a disc c© 2005 RAS http://arxiv.org/abs/0704.1519v1 2 A. J. Smith et al. which encounters a 3:1 eccentric inner Lindblad resonance with the tidal potential of the secondary star may become eccentric and precess. A mass ratio q = M2/M1 . 1/4 is required for the tidal truncation radius, Rtides, to lie outside the 3:1 resonance radius, R3:1 (Paczynski 1977). The thermal–tidal instability (TTI) model (Osaki 1989) at- tributes the increased brightness and duration of superout- bursts over normal outbursts to an enhanced viscous torque acting once the disc becomes eccentric. Superhumps arise in several guises, summarised by Pat- terson et al. (2002a). ‘Common’ or ‘normal’ superhumps en- sue after the onset of superoutburst in SUUMa systems. They are powered by the periodically varying tidal interac- tion which modulates dissipation in the disc (e.g. Foulkes et al. 2004). ‘Late’ superhumps sometimes follow common superhumps and are roughly anti-phased to them. The mod- ulation in energy dissipation at accretion stream’s impact on the non-axisymmetric disc powers late superhumps (Rolfe, Haswell & Patterson 2001). ‘Negative’ superhumps, with Psh slightly less than Porb, are sometimes observed simultane- ously with the more usual ‘positive’ superhumps, and may be related to retrograde precession of a warped accretion disc (Patterson et al. 1993b; Murray & Armitage 1998; Foulkes, Haswell & Murray 2006). ‘Permanent’ superhumps are seen in high mass transfer rate systems: the nova-like variables, old novae and some AMCVn systems. Some low mass X-ray binaries (LMXBs, the neutron star and black hole analogues of CVs) also show superhumps (Bailyn 1992; Haswell 1996; O’Donoghue & Charles 1996). In LMXBs optical emission arises overwhelmingly from the reprocessing of X-rays, and superhumps arise from a modulation in reprocessing caused by the changing solid angle subtended by the tidally flexing eccentric disc (Haswell et al. 2001). Recently superhumps were reported in the microquasar GRS1915+105, which has an orbital period exceeding 30 days (Neil, Bailyn & Cobb 2006). In all the above cases, the fractional superhump excess, ǫ = (Psh − Porb)/Porb varies with q, with |ǫ| increasing with higher values of q. The exact relationship has proved difficult to determine. We have performed unprecedentedly compre- hensive numerical simulations of apsidally precessing accre- tion discs, and we present them in the context of previous numerical work, the observational data and salient analyt- ical theory. Section 2 describes our simulations and exam- ines the growth rates of the eccentric instability; the en- hanced viscous torques; the superhump light curves which result from the eccentric instability; and compares 2D and 3D simulation results. In section 3 we use two methods to quantify the eccentricity distributions in our simulated discs. Section 4 focuses on the ǫ − q relationship. In section 5 we discuss our findings. Section 6 gives a summary list of our principle conclusions. 2 SPH SIMULATIONS OF A PRECESSING ACCRETION DISC SPH is a Lagrangian method which models fluid flow as a set of moving particles. A detailed review is given by Monaghan (1992). SPH simulations by Murray (1998) provide consid- erable support for the TTI model, showing that the energy released from a disc that has become tidally unstable is suffi- cient to account for the excess luminosity of a superoutburst. Foulkes et al. (2004) carried out 2D SPH simulations of a binary system with mass ratio 0.1. They show an eccentric, non-axisymmetric precessing disc of changing density, which is continuously flexing and relaxing on the superhump pe- riod. Very clear too in the surface density maps are tightly wrapped spiral density waves which extend from the outer- most regions to small radii. They produce shear and dissi- pation in the outer disc, and propagate angular momentum outwards, allowing disc gas to move inward. 2.1 Simulation details For the calculations presented here, an SPH code is used which has been designed specifically for accretion disc prob- lems. Detailed description of the code can be found in Mur- ray (1996, 1998). The code is normalised in units of a, the binary separation, for distance; Mt = M1+M2 for mass; and Porb/(2π) for time. This code has since been updated to in- clude adaptive spatial resolution, allowing the SPH smooth- ing length λ to vary in both space and time (Murray, de Kool & Li 1999). Here, λ is set to a maximum value of 0.005 a. The code has also been extended to three dimensions (Murray & Armitage 1998). Simulations were run for a range of mass ratios, de- tailed in Table 1. The simulations were built up from zero mass and a single particle was injected at the L1 point each timestep ∆t, so simulating the mass-transfer stream from the secondary. We ran simulations at different mass resolu- tions, where ∆t is between 0.01Ω−1orb and 0.0025 Ω orb. This latter value is at resolution higher than previous calculations (Murray 1996, 1998; Foulkes et al. 2004), and we find our results stable to a further reduction in ∆t. The number of particles in the accretion disc in these high-resolution simu- lations is of order 100 000, and the average number of ‘neigh- bours’, i.e. the average number of particles found within 2λ of each other that are used in the SPH update equations, is between 20 and 30. The stream boundary conditions at L1 are a function of q, calculated by Lubow & Shu (1975) from perturbation analysis of L1. We follow their calculations to determine the direction prograde of the binary axis at which particles are to be injected, θinj. The initial speed of the particles, vinj = 0.1 aΩorb, is determined from cs at L1, where the z-velocity is an arbitrary fraction of this. The actual value is not critical as the gas will rapidly accelerate to become supersonic, and the stream is not well-resolved in our sim- ulations. For some simulations the third dimension is sup- pressed. For the inner boundary condition a hole of radius R1 centred on the primary was used, with R1 set to values between 0.03 a and 0.1 a, and particles entering this were removed from the simulation. The calculations presented in Murray (1998) and Mur- ray, Warner & Wickramasinghe (2000) were of cool isother- mal discs. Here we model a more realistic steady-state disc where cs is a function of disc radius, r, and is given by cs = C r −3/8. C is a constant and in general we have set it equal to 0.05 aΩorb. This means that cs at the resonance radius in each of our simulated discs will be≃ 0.067 aΩorb as detailed in Table 2. The SUUMa systems ZCha and OYCar in outburst have brightness temperatures, TBR, in the outer disc of ∼ 6000 – 7000K (Horne & Cook 1985; Bruch, Beele c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 3 Table 1. Summary of simulations and results. The first column denotes the run number. Columns 2 to 6 describe the simulation parameters, recording respectively whether the simulation is conducted in 2D or 3D, the mass ratio, the injection time step, the constant describing the sound speed (cs = C r −3/8) and the radius of the primary (the central hole). The time at which the simulation was terminated is given in column 7. The remaining columns record outcomes of the simulations: the mean superhump excess as measured from the simulated lightcurve, the final total number of particles in the simulation, the final average number of neighbours, the strengths of the eccentric and 2-armed spiral modes averaged over the final 10 superhump periods in each simulation, and, as a measure of when the disc initially encounters the resonance, the time at which S(1,0) = 0.01. ζ = 1.0 in all cases. Run 2D/ q ∆t C R1 tend mean ǫ np nne S(1,0) S(2,2) tecc 3D (Ω ) (aΩorb) (a) (Ω 1 3D 0.3333 0.0025 0.050 0.03 3798.0575 — 123995 28.28 0.001 0.104 — 2 3D 0.2422 0.0025 0.050 0.03 8616.6650 — 141474 28.50 0.001 0.099 — 3 3D 0.2346 0.0025 0.050 0.03 15496.7600 1 0.062 143274 28.48 0.007 0.099 — 4 3D 0.2270 0.0025 0.050 0.03 14340.9425 1 0.058 143088 28.67 0.065 0.0948 6089.7625 5 3D 0.2195 0.0025 0.050 0.03 15487.8575 0.056 143730 28.18 0.092 0.092 4542.1675 6 3D 0.2121 0.0025 0.050 0.03 8738.4900 0.054 144674 28.62 0.107 0.090 3188.7950 7 3D 0.1765 0.0025 0.050 0.03 4590.9525 0.043 150628 27.41 0.151 0.081 1271.1125 8 3D 0.1429 0.0025 0.050 0.03 4173.9300 0.034 160459 27.63 0.161 0.078 777.2475 9 3D 0.1111 0.0025 0.050 0.03 3134.1350 0.024 171971 26.92 0.172 0.075 770.7275 10 3D 0.0811 0.0025 0.050 0.03 4323.1750 0.017 175989 24.76 0.227 0.062 477.1300 11 3D 0.0526 0.0025 0.050 0.03 3535.2575 0.015 2 227047 26.76 0.001 0.080 426.7250 12 3D 0.0256 0.0025 0.050 0.03 3491.8375 — 257057 24.07 0.001 0.070 — 13 3D 0.2400 0.0025 0.050 0.05 2000.0000 0.05 118947 21.87 0.001 — 14 3D 0.1900 0.0025 0.050 0.05 1204.2430 0.04 128135 21.56 0.003 — 15 3D 0.1500 0.0025 0.050 0.05 2000.0000 0.04 141277 22.09 0.074 1028.375 16 3D 0.1000 0.0025 0.050 0.05 2135.4880 0.03 145893 20.1 0.186 622.850 17 3D 0.0300 0.0025 0.050 0.10 942.0200 — 127310 27.06 0.001 — 18 3D 0.0700 0.0025 0.045 0.05 1014.2300 0.01 431782 98.03 0.008 — 19 2D 0.5385 0.0100 0.050 0.03 3081.36 — 18148 14.95 0.008 0.088 — 20 2D 0.4815 0.0100 0.050 0.03 21798.61 0.093 17973 14.41 0.23 0.057 1858.91 21 2D 0.4286 0.0100 0.050 0.03 18638.44 0.079 18293 14.35 0.30 0.043 974.19 22 2D 0.3333 0.0100 0.050 0.03 18285.39 0.055 18493 13.85 0.43 0.020 764.33 23 2D 0.2500 0.0100 0.050 0.03 18660.42 0.037 18822 14.89 0.53 0.019 591.20 24 2D 0.1765 0.0100 0.050 0.03 17355.41 0.026 20438 13.27 0.56 0.021 400.43 25 2D 0.1111 0.0100 0.050 0.03 14731.33 0.016 24766 13.97 0.54 0.020 308.67 26 2D 0.0526 0.0100 0.050 0.03 8872.88 — 41753 16.00 0.001 0.081 — 27 2D 0.0256 0.0100 0.050 0.03 1608.00 — 46093 15.47 0.001 0.077 — 1 equilibrium not yet reached, simulation continues to run 2 temporary appearance of superhumps & Baptista 1996). For the eight SUUMa eclipsing systems in Table 3 we obtain 0.031 aΩorb . cs . 0.036 aΩorb at the disc midplane at R3:1, assuming a mass transfer rate of Ṁ = 10−9 M⊙ yr −1 and a fully ionised cosmic mixture of gases. Hence our simulations have a sound speed within a factor of two of that prevailing in reality, a much better match than previously possible. We have also improved on the viscosity used in previous calculations. Our code includes an artificial viscosity term which generates a shear viscosity in the disc, ν(r) = κ ζ cs H, (1) where ζ is the dimensionless artificial viscosity parameter. ζ = 1 here, and H is the disc scaleheight. κ may be found analytically, and for a standard cubic spline kernel in three dimensions, κ = 1/10, whilst in two dimensions, κ = 1/8. The bulk viscosity is fixed to be twice the shear viscosity. Using the Shakura–Sunyaev viscosity parametrisation (ν = α cs H) (Shakura & Sunyaev 1973), our simulated 3D and 2D discs have α(R3:1) = 0.1 and 0.125 respectively. This matches estimates of α ∼ 0.1−0.2 derived from observation of systems in the high viscosity state (Smak 1999). 2.2 Simulation Results Table 1 summarises our simulations. In some cases these have run for > 2000 orbits without reaching mass-transfer equilibrium; one has run for over a year of elapsed time. The rate of energy dissipation (i.e. viscously-generated lu- minosity with units M a2 Ω3orb) from different regions in the disc is recorded each timestep and used to produce simula- tion lightcurves. Psh was determined from timings of maxima in dissipation in a smoothed lightcurve for the disc region r > 0.3 a. Column 8 of Table 1 gives the mean value of ǫ so obtained, over the time in which the system has reached equilibrium where applicable. Figure 1 shows the disc evo- lution for the 3D simulations 1 to 12. Here the disc mass is taken to be the number of SPH particles in the disc. The disc eccentricity is estimated from the eccentric mode strength, c© 2005 RAS, MNRAS 000, 1–18 4 A. J. Smith et al. Table 2. Characterisation of the simulated accretions discs at the 3:1 resonance radius. The 3:1 resonance radius is recorded in the fourth column, followed by the sound speed, the scale height, the characteristic value of the ratio of the disc semi-thickness to the radius as given in Goodchild & Ogilvie (2006), the shear viscosity, the bulk viscosity, the Shakura–Sunyaev parameter for the shear viscosity and that for the bulk viscosity. Run 2D/ q R3:1 cs(R3:1) H(R3:1) h(R3:1) νsh(R3:1) νbk(R3:1) αsh(R3:1) αbk(R3:1) 3D (a) (aΩorb) (a) (a) (a 2Ωorb) (a 2Ωorb) 1 3D 0.3333 0.437 0.068 0.023 0.039 1.55× 10−4 3.10× 10−4 0.100 0.200 2 3D 0.2422 0.447 0.068 0.023 0.038 1.52× 10−4 3.05× 10−4 0.100 0.200 3 3D 0.2346 0.448 0.068 0.023 0.037 1.52× 10−4 3.04× 10−4 0.100 0.200 4 3D 0.2270 0.449 0.068 0.023 0.037 1.52× 10−4 3.04× 10−4 0.100 0.200 5 3D 0.2195 0.450 0.067 0.022 0.037 1.52× 10−4 3.03× 10−4 0.100 0.200 6 3D 0.2121 0.451 0.067 0.022 0.037 1.51× 10−4 3.03× 10−4 0.100 0.200 7 3D 0.1765 0.455 0.067 0.022 0.037 1.50× 10−4 3.01× 10−4 0.100 0.200 8 3D 0.1429 0.460 0.067 0.022 0.036 1.49× 10−4 2.98× 10−4 0.100 0.200 9 3D 0.1111 0.464 0.067 0.022 0.036 1.48× 10−4 2.96× 10−4 0.100 0.200 10 3D 0.0811 0.468 0.066 0.022 0.035 1.47× 10−4 2.94× 10−4 0.100 0.200 11 3D 0.0526 0.473 0.066 0.022 0.035 1.46× 10−4 2.92× 10−4 0.100 0.200 12 3D 0.0256 0.477 0.066 0.022 0.034 1.45× 10−4 2.91× 10−4 0.100 0.200 13 3D 0.2400 0.447 0.068 0.023 0.041 1.52× 10−4 3.05× 10−4 0.100 0.200 14 3D 0.1900 0.454 0.067 0.022 0.040 1.51× 10−4 3.02× 10−4 0.100 0.200 15 3D 0.1500 0.459 0.067 0.022 0.039 1.49× 10−4 2.99× 10−4 0.100 0.200 16 3D 0.1000 0.466 0.067 0.022 0.038 1.48× 10−4 2.96× 10−4 0.100 0.200 17 3D 0.0300 0.476 0.066 0.022 0.037 1.45× 10−4 2.91× 10−4 0.100 0.200 18 3D 0.0700 0.470 0.060 0.020 0.034 1.19× 10−4 2.38× 10−4 0.100 0.200 19 2D 0.5385 0.416 0.069 0.023 0.042 2.01× 10−4 4.02× 10−4 0.125 0.250 20 2D 0.4815 0.422 0.069 0.023 0.041 1.99× 10−4 3.98× 10−4 0.125 0.250 21 2D 0.4286 0.427 0.069 0.023 0.040 1.97× 10−4 3.94× 10−4 0.125 0.250 22 2D 0.3333 0.437 0.068 0.023 0.039 1.94× 10−4 3.88× 10−4 0.125 0.250 23 2D 0.2500 0.446 0.068 0.023 0.038 1.91× 10−4 3.82× 10−4 0.125 0.250 24 2D 0.1765 0.455 0.067 0.022 0.037 1.88× 10−4 3.76× 10−4 0.125 0.250 25 2D 0.1111 0.464 0.067 0.022 0.036 1.85× 10−4 3.70× 10−4 0.125 0.250 26 2D 0.0526 0.473 0.066 0.022 0.035 1.83× 10−4 3.66× 10−4 0.125 0.250 27 2D 0.0256 0.477 0.066 0.022 0.034 1.82× 10−4 3.63× 10−4 0.125 0.250 S(1,0), where the strength of the (k θ−lΩorb t) mode, S(k,l), is obtained by Fourier decomposing the simulated disc density distributions in azimuth and time (Lubow 1991b; Murray 1996). 2.2.1 The growth of the eccentricity Figure 1 shows that initially, the eccentricity grows expo- nentially. Lubow (1991a) found an exponential eccentricity growth rate which is proportional to the square of the mass ratio; in contrast we see a very low growth rate for high mass ratios (q & 0.2) (Figure 2). Our results can be prof- itably compared with those of Goodchild & Ogilvie (2006) who formulated a single equation to describe the resonant excitation, propagation and viscous damping of the eccen- tricity in a 2D accretion disc. They showed that the res- onance may have the effect of locally suppressing the ec- centricity which consequently leads to extremely low eccen- tricity growth rates. We find for q = 0.2195, for example, a steady state is not reached until ∼ 1700 orbital periods, and for q = 0.2346 a steady state is still not achieved after ∼ 2500 orbital periods. Figure 2. Eccentricity growth rate (growth rate of the strength of the (1, 0) eccentric mode) as a function of mass ratio for sim- ulations presented in this work, and in previous works (Murray 1998). Simulation parameters are as indicated in the legend. c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 5 Table 3. Parameters of the eight eclipsing SUUMa systems: mass ratio, primary mass, total system mass (M1 + M2), observed disc precession rate (from the measured ǫ value), dynamical precession rate as calculated from Equation 6 at the location of the 3:1 resonance, inferred pressure contribution to precession (ωobs − ωdyn) and pitch angle of the spiral wave (see Section 4 for details). Errors on ωobs and ip correspond to the range of superhump periods observed. In the last but one column the sound speed, in SPH units, is calculated for the midplane of each disc at the 3:1 resonance radius, assuming a mass transfer rate of 10−9 M⊙ yr −1 and a fully ionised ‘cosmic’ mixture. References for M1 and Mt are provided in the final column. Details and references for all other observational data can be found in Table 5. System q M1 Mt ωobs ωdyn(R3:1) ωpr ip cs(R3:1) Ref (M⊙) (M⊙) (rad d −1) (rad d−1) (rad d−1) (◦) aΩorb OY Car 0.102 ± 0.003 0.685 ± 0.011 0.755± 0.011 2.189 +0.447 −0.181 3.649 −1.460 12.69 +2.44 −0.70 0.0334 XZ Eri 0.1098 ± 0.0017 0.767 ± 0.018 0.851± 0.018 2.697 +0.328 −0.0035 4.015 −1.319 13.02 +1.91 −0.16 0.0321 IY UMa 0.125 ± 0.008 0.79± 0.04 0.89± 0.04 2.208 +0.329 −0.120 3.714 −1.506 11.17 +1.42 −0.41 0.0322 Z Cha 0.1495 ± 0.0035 0.84± 0.09 0.965± 0.091 3.161+0.634 −0 4.282 −1.121 14.31 +6.85 −0 0.0360 HT Cas 0.15 ± 0.03 0.61± 0.04 0.70± 0.04 2.725 −0 4.343 −1.618 11.62 −0 0.0347 DV UMa 0.1506 ± 0.0009 1.041 ± 0.024 1.198± 0.024 2.349 +0.080 −0.286 3.738 −1.389 10.54 +0.31 −0.92 0.0314 OU Vir 0.175 ± 0.025 0.90± 0.19 1.06± 0.19 2.732+0.033 −0 4.987 −2.255 8.76 +0.06 −0 0.0305 V2051 Oph 0.19 ± 0.03 0.78± 0.06 0.93± 0.07 2.748 +0.815 −0.319 6.201 −3.453 7.83 +1.11 −0.33 0.0312 1 Wood et al. (1989) 2 Feline et al. (2004c) 3 Steeghs et al. (2003) 4 Wade & Horne (1988) 5 Horne, Wood & Stiening (1991) 6 Feline et al. (2004a,b) 7 Baptista et al. (1998) 2.2.2 The enhanced viscous torques of an eccentric disc For all calculations in which the disc (eventually) becomes sufficiently eccentric we see similar behaviour but on dif- ferent timescales. As Figure 1 shows, initially the disc mass builds, exponentially approaching a steady-state value. Then as eccentricity increases, the disc mass tends to a new lower steady state value. This reveals the non-eccentric accretion disc approaching an equilibrium between the tidal removal of angular momentum and the angular momentum added by material from the L1 point. As the disc becomes eccen- tric it readjusts to a new equilibrium in which tidal removal of angular momentum is more efficient. This is exactly the premise of the TTI model. Table 4 quantitatively examines this. As a measure of the increase in efficiency of tidal removal of angular momen- tum in the eccentric disc, we took the decrease in disc mass between maximum and the value it finally reached in equi- librium (column 4 of Table 4). For a steady state accretion disc the total mass is Mtot = Z Rout 3ν(R) dR (2) (Frank, King & Raine 1985). The mass transfer rate through L1 remains constant, and we are comparing the equilibria with and without an eccentric precessing disc. In both cases, the mass accretion at all disc radii must equal the mass transfer rate at L1. Thus, the change in disc mass can then be related to a change in viscosity by Mb −Ma = ∆M = A f (R) dR νb(R) f (R) dR νa(R) , (3) where A is a constant, and the subscripts b and a respec- tively refer to before and after the disc became fully eccen- tric. Defining f(R)dR νb(R) f (R) dR , (4) where ν̄b is some unknown weighted mean value of νb(R), and similarly for 1 , then we have f (R) dR . (5) This quantity is given in column 5 of Table 4. The radial de- pendence of H is cs(r/µ) 1/2r, where µ = 1/(q+1). Together with the sound speed prescription that we use, then this allows us, via Equation 1, to give explicitly the fractional change in viscous torque necessary to bring about the de- crease in disc mass we see (column 6 of Table 4). We assume Rout = Rtides. Here, we use the formulation Rtides ≃ 0.9R1 (Frank, King & Raine 1985), where R1 is the effective Roche lobe radius (Eggleton 1983), though we note the limitation of this approximation (e.g. Murray, Warner & Wickramas- inghe 2000; Truss 2007). For runs 6, 7, 9 and 10 we measured the average outer disc radius, averaging the position of the outermost particle as a function of azimuth over a super- hump period. We found in each case the difference between this value and Rtides to be < 0.006 a. 2.2.3 The superhump The development of the superhump for two selected simula- tions is shown in Figure 3. For q = 0.0526, superhumps were only temporarily present at a time when the eccentricity was highest. For q = 0.2121 we see that the superhump profile c© 2005 RAS, MNRAS 000, 1–18 6 A. J. Smith et al. Figure 1. Disc mass (red) and disc eccentricity (green) as a function of time for the 3D simulations 1 to 12. Every 5000th and 2000th timestep is plotted respectively. Also plotted is the superhump period (blue) in the form of O-C. This is calculated relative to the mean superhump period from each run, derived from the point at which the superhump signal becomes well-formed. The dissipation (smoothed) from the disc for r > 0.3 a is shown in grey. The mass ratio is shown in the upper left of each panel. evolves with time as the disc is reaching eccentric equilib- rium. In the case of other more extreme mass ratios (smaller values of q) there are higher frequency periodic components present in the early development of the superhumps (Smith, PhD thesis in prep.). Our discs accumulate from zero mass with no switch between viscosity states, so the development of their superhump will differ from that of discs observed in superoutburst. Variations in the superhump period are displayed in the form of O-C (‘observed’ minus ‘calculated’) in Figure 1. In- tervals of non-zero 2nd derivative in the value of O-C indi- cate period changes. Generally Psh decreases as the system approaches eccentric steady-state. For q = 0.0526 a steady- state eccentric disc is not achieved, and the period evolution differs. Each simulated lightcurve was folded on the mean su- perhump period once equilibrium was reached (Figure 4). These are asymmetric with, for most, a steep rise and slower decline. A secondary hump structure is seen, the profile dif- ferent for different mass ratios. In general, these superhump profiles resemble those observed in CVs cf. figure 8 of Patter- son et al. (1995), figure 4 of Imada et al. (2006) and figure 6 of Maehara, Hachisu & Nakajima (2006). In Figure 5 we show the density distributions of the accretion disc for run 9 at selected superhump phases, and compare these with the density profile of a disc which does not show superhumps. The disc is not maximally distorted at the time of maximum energy production (φsh = 0) as might be expected, but at φsh = 0.71. For q = 0.1111, this coincides with a small secondary peak seen in Figure 4. The superhumping disc is most similar to the non-superhumping disc at superhump maximum, the most visible difference be- ing in the strongly enhanced spiral density waves in the su- perhumping disc. Furthermore, the appearance of the spiral density wave changes dramatically over Psh. This illustrates the crucial importance of the spiral density waves to the superhump phenomenon, as suggested by Lubow (1991a) and Osaki (2003) who considered analytic theory and obser- vations respectively. Comparing superhump maximum with superhump minimum, then we see more a more open spiral structure at superhump maximum with outer reaches show- ing enhanced density. After superhump maximum, differen- tial motion in the superhumping disc causes the spiral to become eccentric and more tightly wrapped with lower den- sity contrast (see the bottom two panels in Figure 5). Each radius in the disc has its own characteristic Keplerian angu- lar velocity, and eccentric mode precession rate. Differential precession in the eccentric fluid disc cannot persist, because c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 7 Table 4. Enhanced viscous torque resulting from the accretion disc encounter with the eccentric resonance. The fourth column records the fractional decrease in disc mass before and after the disc becomes fully eccentric. The fifth column gives a measure of the change in viscous torque indicated by this mass decrease as described in the text and given by Equation 5. The final column assumes the radial dependence of viscosity is given by Equation 1, and records, in that case, the fractional change in viscous torque necessary to bring about the decrease in disc mass. Run 2D/ q ∆M/M 1 νa/νb 3D 1/(a2Ωorb) 5 3D 0.2195 0.0223 175 1.019 6 3D 0.2121 0.0270 211 1.023 7 3D 0.1765 0.0464 360 1.040 8 3D 0.1429 0.0360 273 1.031 9 3D 0.1111 0.0465 353 1.040 10 3D 0.0811 0.1063 800 1.098 20 2D 0.4815 0.0735 423 1.060 21 2D 0.4286 0.1145 665 1.097 22 2D 0.3333 0.205 1200 1.190 23 2D 0.2500 0.267 1538 1.258 24 2D 0.1765 0.257 1392 1.231 25 2D 0.1111 0.165 823 1.127 Figure 3. Evolution of the simulated lightcurve for runs 6 and 11. Each panel covers 5 orbits. The lightcurve, which has been smoothed somewhat, is given by the rate of energy dissipation for radii r > 0.3 a. The mass ratio is shown in the uppermost panel of each. this would cause widespread orbit crossings. This converg- ing fluid motion causes the strong spiral shock shown in the upper right panel of Figure 5, and it is the dissipation associ- ated with this shock which powers the observed superhump. Figure 5 illustrates the mechanism by which spiral waves power superhumps. Figure 4. Equilibrium lightcurves (energy dissipation rate for radii r > 0.3 a) folded on the derived mean superhump period for runs 5 to 10. The superhump cycle is repeated for clarity. Figure 5. Density profiles comparing the non-superhumping disc of run 2 (upper left panel) with the superhumping disc of run 9 at 3 selected superhump phases, φsh. The coordinates are centred on the binary system centre of mass. The solid line is the primary Roche lobe and the dashed line is the 3:1 resonance radius. The upper left panel also shows our definition of azimuth used in Sec- tion 3.1. The secondary star is at an azimuth of π radians with respect to the primary, the position of which is marked with a cross. c© 2005 RAS, MNRAS 000, 1–18 8 A. J. Smith et al. Figure 6. Comparison of results for 2D and 3D simulations with q = 0.1111, runs 24 and 9 respectively. The following are shown above and below respectively for each: the disc evolution, details of the simulated lightcurves, folded superhump lightcurves and density profiles. Here symbols are as in Figure 5, and the colour scale used is the same for both the 2D and 3D discs. The simulated lightcurves cover 5 orbital cycles once the disc mass and eccentricity had stabilised. The raw lightcurve is shown in the top panel and increasingly smoothed lightcurves below. 2.2.4 2D versus 3D In Figure 6 we compare the results of q = 0.1111 calcu- lations in 2D and 3D. In the 2D simulations the accretion disc achieves a far higher eccentricity more quickly. This in turn affects the superhump profile. We see from Table 1 that these 2D accretion discs (runs 19 to 27) can become eccentric at much larger values of mass ratio than in the 3D simulations, and well beyond the bounds that theory sug- gests, indicating that three dimensions and the high mass resolution (∆ t = 0.0025) is necessary to accurately simu- late the processes involved. In this case, (3D, ∆ t = 0.0025) no disc develops superhumps where q & 0.24. We also see that for the same value of q, the 2D discs precess at smaller rates than the 3D discs; compare for example run 24 with run 7. 3 ECCENTRICITY DISTRIBUTION IN ACCRETION DISCS An eccentric accretion disc underlies the superhump phe- nomenon, but how eccentric does the disc become? How does the eccentricity vary throughout the disc, and how does it change with superhump phase? We plotted an estimate of the disc eccentricity in our simulations in Figure 1. Calculating the strength of the (1,0) mode takes into consideration the disc as a whole. For those 3D discs which reached equilibrium and showed superhumps, final values of eccentricity e between ∼ 0.09 and ∼ 0.23 were seen, with more extreme mass ratios har- bouring more eccentric discs. This agrees with modelling of line profiles in AMCVn assuming a constant eccentric- ity throughout the disc: Patterson, Halpern & Shambrook (1993c) found e = 0.1 − 0.2. In their analytic study Good- child & Ogilvie (2006) examined the spatial distribution of eccentricity, but explicitly avoided examination of the be- haviour on the orbital timescale. Their eccentricity distri- bution was locally suppressed by the presence of dynamical resonances. Next we examine the eccentricity distributions of our simulated discs, using two complementary methods to characterise their spatial and temporal variation. 3.1 Eccentricity distribution from particle trajectories An instantaneous snapshot of the radial eccentricity distri- bution was found by projecting the elliptical orbit of each particle in the disc using its position and velocity. Particles in the mass transfer stream were discounted. Each parti- cle was assigned a radius given by an average along this elliptical path, weighted by the time spent at each radius. This method calculates the trajectories particles would have if they orbited an isolated primary star. This simplification may introduce certain artefacts into the calculated eccentric- ity distributions, but does give an adequate approximation. Figure 7 shows the results for each of the 3D simula- tions 1 to 12, at superhump maximum where applicable. In this and subsequent similar figures, each point represents the mean eccentricity of particles, obtained in the manner described above, in each of a number of azimuthal and ra- dial bins of size 0.1π rad and 0.1 a respectively. Overplot- ted on these figures are the 3:1, 4:1 and 5:1 resonance radii from right to left respectively. For simulations 11 and 12 the 2:1 resonance is also shown. Each particle is colour coded according to the azimuthal position of the particles it rep- resents, as shown in the key, where θ is the angle defined in Figure 5. Purple and blue particles are approaching the mass donor star, while green and yellow particles are re- ceding from it. The eccentricity distributions at different azimuths are clearly distinct. Caution is required, however, in interpreting the radii assigned in this section. The parti- cles at r > R3:1 are concentrated at azimuths close to the x-axis, where in fact the disc edge is relatively close to the compact object: these particles have larger radii assigned to c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 9 Figure 7. Instantaneous eccentricity distribution plotted as a function of radius for 3D runs 1 to 12, once equilibrium is reached (see text for details). Each point represents the average of all disc particles in each of 20 equal azimuthal sections and binned in radius. These are colour coded by their azimuthal position in the disc, as defined in Figure 5, according to the scale on the right. The time is shown in the upper right of each panel, together with the superhump phase where applicable. them than those found further out at other azimuths be- cause they have relatively circular projected orbits. For all the discs with q > 0.0811 the eccentricity in the outer disc has a wide spread around e ∼ 0.2. This agrees reasonably with the eccentricities found in the outer discs of OYCar: Hessman et al. (1992) found e = 0.38±0.10; and IY UMa: Patterson et al. (2000a) estimate e = 0.29± 0.06. For 0.2121 6 q 6 0.3333 the eccentricity distributions are low for the disc within the 5:1 resonance radius, then increase at larger radii to e up to ∼ 0.3, this apparent eccentric- ity likely due to tidal distortions in the outer disc. For this range of q, if every particle were individually plotted, the ec- centricity distribution outside R5:1 is “double horned”, with a concentration of points near the upper and lower edges of the distribution and a dearth of points midway between the envelope. These two “horns” correspond to the halves of the disc which are approaching and receding from the mass donor star. Figure 7 demonstrates that the body of the disc within the 5:1 resonance appears eccentric for 0.081 6 q 6 0.2195. For these discs the entire mass appears to partake in the ec- centric instability. This roughly flat region of the eccentricity distribution has an eccentricity dependent on q, increasing to ∼ 0.12 as we move to smaller q. Figure 8 shows how the eccentricity distribution devel- ops in run 9 as the disc becomes eccentric. Before super- humps set in (top left panel) the eccentricity distribution resembles those seen in the less extreme mass ratio systems i.e. the top row of panels in Figure 7. The remaining 5 pan- els of Figure 8 span the interval where the disc eccentricity is growing and the disc is emptying as it approaches the new mass transfer equilibrium caused by the enhanced vis- Figure 8. Development of accretion disc radial eccentricity dis- tribution for run 9. Azimuthal colour-coding is as in Figure 7. cous torque (c.f. Figure 1). In each case superhump max- imum is plotted. Within R5:1 the eccentricity of the body of the disc grows, while the range of eccentricities at any given radius remains more or less unchanged. Outside R5:1 the eccentricity distribution becomes more scattered, as the range of eccentricities at any given radius increases. The last panel of Figure 8 shows the distribution when the strength of the (1,0) mode approaches maximum, but mass equilib- rium is not yet reached. It resembles the q = 0.1111 panel c© 2005 RAS, MNRAS 000, 1–18 10 A. J. Smith et al. Figure 9. As Figure 7, showing the changing radial eccentricity distribution with superhump phase, for run 9. in Figure 7, which occurs almost 230 orbits later, after mass equilibrium is established. The eccentricity distribution is shown as a function of superhump phase in Figure 9. At some phases and radii there is little spread in e, while at other phases there is a large spread in e at the same radius. This behaviour may be related to the cusps at the resonance points discussed by Goodchild & Ogilvie (2006), but their treatment explicitly excluded the behaviour of the disc on timescales comparable to or shorter than the orbital period. Numerical simulation facilitates examination of the flexing of the disc during a single orbit, which is important as it is this flexing which gives rise to the modulation in viscously-generated or repro- cessed light which constitutes an observed superhump. The flexing of the disc over the superhump period appears most dramatic at radii between R5:1 and R3:1, in accordance with Pearson’s (2006) point that the dynamic precession rate at the 4:1 resonance actually agrees far better with the obser- vations. Figure 9 neatly illustrates the changes which occur as the binary’s gravitational potential moves relative to the disc. Even as far in as r=0.15a we see the more eccentric particles belonging to the approaching side of the disc at superhump maximum and the receding part of the disc at superhump minimum. 3.2 Eccentricity from the mass distribution How good a representation of the disc eccentricity are the distributions we calculated in section 3.1? If we were to fol- low a single particle as it orbits in the accretion disc it would not move on the elliptical orbit we extrapolated from its in- stantaneous velocity and position. We now use an alterna- tive way of examining the eccentricity of the accretion disc, looking at the mass distribution. The disc in run 9 was split into 100 azimuthal sections. For each, we recorded the number of particles contained within each radial step outwards. In Figure 10, we show the contour maps which result. Every step of 24 particles was recorded and colour-coded, so each colour represents a ‘contour’ of enclosed mass. To each contour we fitted an el- lipse which has one focus at the WD. The fitted parameters are the semi-major axis, asemi, the eccentricity, e, and the Figure 11. As Figure 10 for the non-superhumping disc of run angle the semi-major axis makes with the positive x-axis in an anti-clockwise direction, α0, measured in radians. This is analogous to our definition of θ in Section 3.1. For the out- ermost complete contour, that is the outermost contour for which each azimuthal section is represented, marked in dark blue, these fitted parameters are displayed in the bottom right of each panel in Figure 10 and apply to the overplot- ted magenta ellipse. Figure 10 allows us to look at the disc mass distribution in a more quantitative way than simply looking at the den- sity distribution. The eye is drawn to the pronounced non- axisymmetry outside the 3:1 resonance, though this mass constitutes less than 9 per cent of the mass in the disc, and is the lowest density region. This mass contributes only 4 per cent of the total dissipation, and this contribution to the modulation on Psh is not in phase with the overall su- perhump. The region of the disc within R3:1 is overwhelm- ingly responsible for generating the dissipation-powered su- perhump in the simulations. The four panels in Figure 10 are equally spaced in su- perhump phase. It is noticeable that the outermost disc does not simply precess as the binary frame moves. Instead the flexure of the disc combines with the orbital motion of the binary frame to leave the outer edge of the disc almost fixed in the binary frame between φsh = 0.02 and φsh = 0.27; similarly the outer edge of the disc remains almost the same in the two lower panels at φsh = 0.52 and φsh = 0.77. Fig- ure 11 shows that a non-superhumping disc is extended at similar azimuths. These extended disc edges are analogous to the raised tides in Earth’s oceans. As Figure 11 shows, this effect produces an elliptical shape centred on the pri- mary. This illustrates the distinction that needs to be made between different contributions to the eccentricity distribu- tions found here and in Section 3.1. In the outer disc tidal distortions are important, and are present in discs at all mass ratios. This is to be distinguished from the (1,0) ec- centric mode eccentricity found in the superhumping discs which has the primary at one focus. Truss (2007) finds that for discs which have not yet become tidally unstable, the exact azimuth of the extended ‘wing’ depends on the mass ratio, viscosity parameter and sound speed of the gas (the major axis of the outer streamline moves clockwise with de- creasing q, or increasing α and cs). The interior regions of the disc (e.g. around R4:1) more closely approximate simple relative motion between a slowly apsidally precessing disc orientation and a rapidly moving orbital frame. c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 11 Figure 10. Pictorial representation of the mass distribution in the disc for run 9 at four superhump phases. The disc is split into 100 equal azimuthal sections (dashed lines). Each colour graduation represents ‘contours’ of particle numbers. Moving outwards, each contour represents an increase of 24 particles in each azimuthal section (90 contours in all). Further details are described in the text. The outermost dark red points mark the positions of the outermost particles in each azimuthal section. Also plotted are three circles at the 3:1, 4:1 and 5:1 resonance radii moving inward respectively (black, dark and light grey) which also act as a guide to the eye to show the non-circularity of the contours. Figure 12 shows the fitted ellipse parameters for the four phases shown in Figure 10. The average radius of all points making up the contour was used. The general trends in Figs. 12 and 10 are seen in analogous plots for q = 0.1765, including the peaks and troughs in the radial distribution of the eccentricity parameter. Within R3:1, the orientation of the semi-major axis, α0, changes systematically over the superhump period and in the opposite sense to the motion of the gas in the disc i.e. precession of the slowly-moving disc as viewed from the rapidly rotating binary reference frame. Figure 12 is complementary to Figure 9. Figure 9 shows the eccentricity distributions deduced from instantaneous velocities, while Figure 12 shows the eccentricity distribution deduced from the instantaneous mass distribution. Since the disc is continuously flexing in a complex way, these are not the same. The disc motions can be described as superpo- c© 2005 RAS, MNRAS 000, 1–18 12 A. J. Smith et al. Figure 12. Fitted ellipse parameters for each ‘contour’ in Fig- ure 10, as a function of radius. See Figure 10 and text for details. Results for four superhump phases are presented corresponding to the four panels in Figure 10. Vertical lines indicate the 3:1, 4:1 and 5:1 resonance radii from right to left respectively. Red points indicate contours which are not complete. sitions of the S(k,l) modes, and the resonance radii act as nodes and antinodes in the complex standing wave dynamics the disc executes over a full precession period. To summarise contributions to these disc motions we compare, in table 1, the strengths of the (1, 0) and (2, 2) modes. For comparison, we show the equivalent of Figures 10 and 12 for q = 0.3333 in Figures 11 and 13, a disc which does not show superhumps. Here, as we would expect, the mass distribution remains approximately constant over time, and the eccentricity is much lower. 4 PERIOD EXCESS VERSUS MASS RATIO: DRAWING TOGETHER OBSERVATION, THEORY AND SIMULATION If a reliable relationship between ǫ and q can be deduced, this would be immensely useful. ǫ can be easily measured using relatively modest equipment, while the mass ratio q is more fundamental and less easily determined. Patterson Figure 13. As Figure 12 and corresponding to the panels in Figure 11. et al. (2005) fitted observations of eclipsing systems with ǫ = 0.18 q + 0.29 q2. Figure 14 collates observation, theory and simulations of positive superhumps. All observed sys- tems with q determined by some means independent of ǫ are plotted. Errors in q are formal errors given by the au- thors and do not necessarily reflect the uncertainty in the method. The eclipsing systems (red circles) should therefore be given more weight. We note, however, the scatter of the eclipsing systems seems typical of the scatter of the other points. Error bars for ǫ denote the range of values observed rather than errors in individual values, except in cases where only one measurement has been made. Observational data is tabulated in Table 5. Our high-resolution 3D simulations, which are repre- sented by blue squares in Figure 14, provide a far better match with observed systems than previous studies by Mur- ray (1998, 2000). As in Murray (2000) we see that simple dynamical precession as given by ωdyn = p(r) 1 + q Ωorb, (6) poorly represents observed systems. This is true even if the location of the resonance for a gaseous disc, rather than for isolated particles, is used; the location of the resonance changes only by . 1 per cent. In a real gaseous disc the retrograde effect that pressure forces have on disc preces- sion rates must be taken into account (Lubow 1992; Murray 2000). In a gaseous disc, the excited eccentricity propagates through the disc as a wave and is wrapped into a spiral by the differential precession of the gas. Murray (2000) assumed that the hydrodynamical precession is given by ω = ωdyn + ωpr, (7) where ωpr is the pressure contribution to the precession, and showed that, under the assumption that the eccentricity is tightly wound (that is if it is wound up on a length-scale much smaller that the disc radius) then this pressure con- tribution at the 3:1 resonance radius is ωpr ≃ − Ωorb a tan ip , (8) c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 13 Table 5. Observed superhump systems with independently determined mass ratio. All periods are given in days. Columns 5 and 6 list the minimum and maximum observed superhump periods respectively. Column 7 gives the superhump excess where the errors indicate the range of values as calculated from min Psh and max Psh. In column 8, the object type, (E) denotes a WD eclipsing system, while (e) denotes a system which shows eclipse of the accretion stream/disc impact region only. System q Porb Psh min Psh max Psh ǫ Type Ref OY Car 0.102(3) 0.0631209180(2) 0.06454(2) 0.064245 0.06466 0.0225 +0.0019 −0.0047 SU UMa(E) 1,2,3,4,5 XZ Eri 0.1098(17) 0.061159491(5) 0.062808(17) 0.062603 0.06283 0.0270 +0.0003 −0.0034 SU UMa(E) 6,6,7,7,8 IY UMa 0.125(8) 0.07390897(5) 0.07588(1) 0.07558 0.07599 0.0267 +0.0015 −0.0041 SU UMa(E) 9,9,10,11,11 Z Cha 0.1495(35) 0.074499 0.07740 0.0768 0.0389 −0.0080 SU UMa(E) 12,13,14,14 HT Cas 0.15(3) 0.07364720309(7) 0.076077 0.0330 SU UMa(E) 15,16,17 DV UMa 0.1506(9) 0.0858526521(14) 0.08870(8) 0.0886 0.08906 0.0332 +0.0042 −0.0012 SU UMa(E) 6,6,18,18,18 OU Vir 0.175(25) 0.072706113(5) 0.07508(9) 0.07505 0.0327 −0.0005 SU UMa(E) 19,19,8,20 V2051 Oph 0.19(3) 0.0624278634(3) 0.06418(16) 0.06365 0.06439 0.0281 +0.0033 −0.0085 SU UMa(E) 21,22,23,23,23 WZ Sge 0.060(7) 0.0566878460(3) 0.05726(1) 0.05716 0.05738 0.0101 +0.0021 −0.0018 WZ Sge(e) 24,25,26,5,27 VY Aqr 0.11(2) 0.06309(4) 0.06437(9) 0.0642 0.06489 0.0203 +0.0082 −0.0027 SU UMa 28,29,5,5,5 CU Vel 0.115(5) 0.0785(2) 0.08085(3) 0.0799 0.0299 −0.0121 SU UMa 30,30,20,31 SW UMa 0.14(4) 0.056815(1) 0.058182(7) 0.05790 0.05833 0.0241 +0.0026 −0.0050 SU UMa 32,33,34,35,36 HS 2219+1824 0.19(1) 0.0599 0.06184 0.0324 SU UMa 37,37,37 EK TrA 0.20(3) 0.06288(5) 0.06492(10) 0.0648 0.0324 −0.0019 SU UMa 38,38,38,39 EI Psc 0.21(2) 0.044572(2) 0.04654 0.04579 0.0442 −0.0169 SU UMa 40,41,42,42 VW Hyi 0.21 +0.03 −0.02 0.074271038(14) 0.07714(5) 0.07621 0.07824 0.0386 +0.0148 −0.0125 SU UMa 43,44,44,45,45 YZ Cnc 0.22 0.0868(2) 0.09204 0.0905 0.0604 −0.0178 SU UMa 46,46,47,47 WX Hyi 0.23 +0.07 −0.04 0.0748134(2) 0.07737 0.0783 0.0342 +0.0124 SU UMa 43,48,49,50 T Leo 0.71(15) 0.0588190(5) 0.06022(2) 0.06021 0.06025 0.0238 +0.0005 −0.0002 SU UMa IP? 32,51,52,53,52 U Gem 0.357(7) 0.1769061898(30) 0.20 0.197 0.203 0.131 +0.017 −0.017 U Gem(e) 54,55,56,56,56 V603 Aql 0.22(3) 0.13809(12) 0.14640(6) 0.144854 0.14686 0.0602 +0.0033 −0.0112 Fast nova 57,58, ,59, 60 UU Aqr 0.30(7) 0.163580429(5) 0.17510(18) 0.0704 NL(E) 61,61,8 DW UMa 0.39(12) 0.136606527(3) 0.1454(1) 0.1461 0.0644 +0.0051 NL(E) 62,63,64,63 MV Lyr 0.43 +0.19 −0.13 0.132335 0.1377(4) 0.1487 0.0405 +0.0832 NL 65,66,67,68 AM CVn 0.18(1) 0.011906623(3) 0.012167 0.012164 0.012169 0.0218 +0.0002 −0.0002 AM CVn 69,70,70,70,70 KV UMa 0.037(7) 0.1699339(2) 0.170529(6) 0.17049 0.17073 0.0035 +0.0012 −0.0002 BHXRT 71,72,73,74,8 (XTE J1118+480) QZ Vul 0.042(12) 0.3440915(9) 0.3469(1) 0.3474 0.0082 +0.0014 BHXRT 75,75,76,77 (GS 2000+2) V1487 Aqr 0.058(33) 30.8(2) 31.4 31.2 31.6 0.0195 +0.0065 −0.0065 BHXRT 78,79,79,79,79 (GRS 1915+105) V518 Per 0.111 +0.027 −0.033 0.2121600(2) 0.2157(10) 0.0167 BHXRT 80,80,81 (GRO J0422+32) GU Mus 0.13(2) 0.432602(1) 0.4376(10) 0.0116 BHXRT 82,83,77 (N Mus 1991) 1 Wood et al. (1989) 2 Pratt et al. (1999) 3 Bruch et al. (1996) 4 Schoembs (1986) 5 Patterson et al. (1993a) 6 Feline et al. (2004c) 7 Uemura et al. (2004) 8 Patterson et al. (2005) 9 Steeghs et al. (2003) 10 Uemura et al. (2000) 11 Patterson et al. (2000a) 12 Wood et al. (1986) 13 Baptista et al. (2002) 14 Warner & O’Donoghue (1988) 15 Horne et al. (1991) 16 Feline et al. (2005) 17 Zhang, Robinson & Nather (1986) 18 Patterson et al. (2000b) 19 Feline et al. (2004a) 20 Kato et al. (2003) 21 Baptista et al. (1998) 22 Baptista et al. (2003) 23 Patterson et al. (2003) 24 Skidmore et al. (2002) 25 Patterson et al. (1998) 26 Ishioka et al. (2002) 27 Patterson et al. (2002a) 28 Augusteijn (1994) 29 Thorstensen & Taylor (1997) 30 Mennickent & Diaz (1996) 31 Vogt (1981) 32 Shafter (1983) 33 Howell & Szkody (1988) 34 Nogami et al. (1998) 35 Semeniuk et al. (1997) 36 Robinson et al. (1987) 37 Rodŕıguez-Gil et al. (2005) 38 Mennickent & Arenas (1998) 39 Vogt & Semeniuk (1980) 40 Thorstensen et al. (2002) 41 Uemura et al. (2002b) 42 Skillman et al. (2002) 43 Smith, Haswell & Hynes (2006) 44 van Amerongen et al. (1987) 45 Vogt (1983) 46 Shafter & Hessman (1988) 47 Patterson (1979) 48 Schoembs & Vogt (1981) 49 Bailey (1979) 50 Walker, Marino & Freeth (1976) 51 Shafter & Szkody (1984) 52 Kato (1997) 53 Lemm et al. (1993) 54 Naylor, Allan & Long (2005) 55 Smak (1993) 56 Smak & Waagen (2004) 57 Arenas et al. (2000) 58 Patterson et al. (1993b) 59 Haefner & Metz (1985) 60 Patterson et al. (1997) 61 Baptista, Steiner & Cieslinski (1994) 62 Araujo-Betancor et al. (2003) 63 Stanishev et al. (2004) 64 Patterson et al. (2002b) 65 Hoard et al. (2004) 66 Ritter & Kolb (2003) 67 Skillman, Patterson & Thorstensen (1995) 68 Pavlenko & Shugarov (1999) 69 Roelofs et al. (2006) 70 Skillman et al. (1999) 71 Orosz (2001) 72 Torres et al. (2004) 73 Uemura et al. (2002a) 74 Zurita et al. (2002) 75 Harlaftis, Horne & Filippenko (1996) 76 Charles et al. (1991) 77 O’Donoghue & Charles (1996) 78 Harlaftis & Greiner (2004) 79 Neil et al. (2006) 80 Webb et al. (2000) 81 Kato, Mineshige & Hirata (1995) 82 Orosz et al. (1996) 83 Casares et al. (1997) c© 2005 RAS, MNRAS 000, 1–18 14 A. J. Smith et al. Figure 14. Superhump period excess plotted as a function of binary mass ratio for both observed systems and for SPH simulation. Also plotted are the dynamical and hydrodynamical theoretical predictions. Simulations presented in this paper are displayed as filled squares, as detailed in the top right legend. The second legend down refers to previous works by Murray. The third legend refers to theoretical predictions, and the legend for observational data is in the bottom right. The inset shows data over a large range of q, whilst the main panel focuses on the data at low q where most of the points are clustered, and is plotted on a logarithmic x-axis. where ip is the pitch angle of the spiral wave, and cs is the sound speed. For each eclipsing SUUMa system we have calculated an inferred pressure contribution to the precession rate, ωpr (column 7 in Table 3) in the manner of Murray (2000). A weighted mean gives ωpr = −1.36 rad d−1. This is combined with the mean mass of the WD in systems below the period gap (Smith & Dhillon 1998) to calculate predicted hydro- dynamical precession rates assuming that the precessional pressure contribution is similar for all systems.This is plot- ted as a solid curve on Figure 14. We note that Pearson (2006) considers the inclusion of pressure effects in an al- ternative way. Whilst a reasonable fit to the observations is achieved, the fact that there is a distribution in the eclipsing systems above and below this curve, demonstrates that the situation is not so simple. Possible contributory factors are distributions in primary mass or in disc temperature. We have plotted two further curves on Figure 14, one represent- ing the hydrodynamical prediction for larger WD masses, and the other encompassing both larger M1 and higher ac- cretion disc temperature. These are the mean value of M1 for the eight SUUMa eclipsing systems and the value of ωpr found by Murray (2000) respectively. We tabulate M1, the total system mass, Mt, and the mid-plane sound speed at the resonance radius for each of the eclipsing SUUMa sys- tems (Table 3). These numbers do not seem to provide any clue as to the true cause of the spread in observed systems. In calculating cs, we assumed the same mass transfer rate throughout (10−9 M⊙ yr −1). Undoubtedly this, and conse- quently the temperature, varies from system to system. Figure 14 also shows the best fit that Goodchild & Ogilvie (2006) found to their analytic curve. They found that retrograde pressure terms are in fact only ∼ 1 per cent of the dynamical term, far smaller than the values we infer above. They suggest that the offset of observation from dy- namical precession rate be due instead to averaging over the disc, the eccentricity being distributed throughout the disc rather than being sharply peaked at the resonance itself. This is in accordance with our findings that dissipation- powered superhump overwhelmingly originates in the disc regions within R3:1. Goodchild & Ogilvie (2006) found that their eccentricity distributions peaked close to 0.37 a, a value put forward by Patterson (2001) to match observation. We have plotted the dynamical curve as evaluated at 0.37 a (dot- c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 15 dashed light-blue line). We have also plotted two further lines which show dynamical curves evaluated at some frac- tion of the resonance radius, one to give the best weighted fit to the eclipsing SUUMa systems (red) and the second to give the best fit to our 3D simulations that have reached equilibrium, runs 5 to 10, (blue). We find values of 0.83R3:1 and 0.87R3:1 respectively. 5 DISCUSSION The precession rates for the simulated discs provide a much closer match with observation than has been achieved previ- ously. There are a number of reasons for this. The improved mass resolution changes the stream–disc impact, leading to a different angular momentum distribution in the disc and a different precession rate. The radial sound speed distribution is correct for a steady-state disc (as opposed to an isother- mal one as in the case of Murray (1998)), which means that the variation of density with radius is more realistic, which in turn will determine whether the disc precesses and at what rate. These updated simulations have hotter discs. This means the retrograde effect of pressure on precession will be greater (Lubow 1991a). The viscosity is more in line with what is inferred for an α-disc in the high state. The simulations of Murray (1998) were too viscous, which al- lowed the disc to grow and penetrate the resonance too eas- ily. The propagation of the eccentricity inward through the disc would also have been inhibited. If the precession rate is dictated by a weighted average of the eccentricity as sug- gested by Goodchild & Ogilvie (2006), then the inhibition of inward eccentricity propagation would lead to high preces- sion rates. Finally, the extension to 3D changes the character of the resonance. We can see in Figure 6 that the accretion disc look very different in 2D and 3D. One reason for this is that the character of the stream-disc interaction differs: for 2D mass injection all the particles follow one another exactly and so the stream tends to punch through the outer disc and deposit its angular momentum at r < R3:1. Conversely in the 3D case the particles are more easily captured by the outer disc and so they effectively reduce the specific angu- lar momentum of the outer disc. The growth and decline of the eccentric mode is strongly influenced by the interaction with the mass transfer stream. The picture is further com- plicated by the results of Kunze, Speith & Hessman (2001) who found substantial stream overflow in their simulations. An important extension of this work would be to system- atically isolate the importance of each of the effects which contribute to the disc precession rate. It appears that the offset of both observation and simu- lation from dynamical expectations of precession rates is due to the averaging over the radii interior to R3:1 which partici- pate in the disc precession, and the inclusion of a retrograde effect of pressure forces is insufficient (Goodchild & Ogilvie 2006). In Section 3 and Figure 5 we show that the eccen- tric instability is indeed manifest at radii as small as 0.15a, so the majority of the disc area contributes to the mean precession rate. We note that Goodchild & Ogilvie (2006) required a very low semi-thickness parameter (h = 0.003) to fit observations, whereas in our simulations this is an order of magnitude higher (Table 2) and in better agreement with observational constraints on, and theoretical expectations of, disc thickness. Empirically, systems which show superhumps generally have q < 0.24. There are two classes of exceptions: nova-likes and UGem, and magnetic systems. For the former, Osaki (2005) pointed out that Paczynski’s derivation of Rtides as- sumes the disc to be cold. A sufficiently hot accretion disc may expand beyond this owing to a weakening of shocks by strong pressure effects at the last non-intersecting orbit. Osaki (2005) argues the persistently high-state nova-likes, and the unusually long 1985 outburst of UGem in which su- perhumps were reported, may satisfy this temperature cri- terion. The magnetic systems, TVCol and possibly TLeo, may have discs which are pushed out by magnetic forces (Retter et al. 2003). The high resolution 3D simulations presented here re- produce this distribution well; the upper limit for which su- perhumps are observed is also q ≃ 0.24, albeit at this mass ratio it is a very slow process indeed. This also compares well with the theoretical upper limit of q . 0.25. For q = 0.2422 it appears that the eccentricity starts to grow more than once, but fails each time and the eccentricity falls away again, presumably because the encounter with the resonance was only marginal. The final eccentricity is highest for q ∼ 0.1, and declines gradually as q becomes less extreme. This is expected as the disc has more room to grow beyond the 3:1 resonance before being tidally truncated for extreme mass ratios. For the two most extreme mass ratio simulations, with q = 0.0526 and q = 0.0256 the final eccentricity is neg- ligible. In the q = 0.0256 case there is no eccentricity growth. The q = 0.0526 case is interesting: as the disc is growing, the eccentricity initially increases and superhumps are appar- ent, albeit weak and ill-formed (Figure 3). This eccentricity is, however, damped away again. What is the source of the damping? One possibility could be the action of the 2:1 reso- nance which may be excited in ultra-low mass ratio systems and which can act to damp eccentricity (Lubow 1991b). Per- haps this results in a close competition between the 2:1 and 3:1 resonance where the 2:1 resonance is marginally excited in the case of q = 0.0526, and the 2:1 resonance only comes out on top once the disc has grown and this resonance is sufficiently populated. In reality, the most extreme mass ra- tio system in which superhumps are observed is the black hole X-ray transient KVUMa (XTE J1118 +480), which has q = 0.037± 0.007. In 2D, the range for which the simulated discs become eccentric extends to much higher mass ratios (q 6 0.4815) (Table 1, Figure 14). The confinement to 2D means that the character of the resonance is different and the strength of the resonance is increased. We find that the growth rate of the eccentricity (the growth rate of S(1,0)) is highly dependent on q, with high mass ratio systems taking a very long time indeed to be- come eccentric (Table 1). We see in Figure 2 that our new simulations have slower growth rates than previous 2D sim- ulations by Murray (1998). Observationally, the rise times of superoutbursts is of order a day or so, and superhumps are generally detected within a day or so. The outburst on- set is, however, probably governed by the thermal-viscous instability and hence its timescale is independent of the disc eccentricity. Generally there are no observations suitable for assessing whether or not the disc was eccentric before the superoutburst began, with intensive observations beginning c© 2005 RAS, MNRAS 000, 1–18 16 A. J. Smith et al. after the rise to outburst, so timescales on which SUUMa discs typically develop their eccentricity is ill-constrained. We note further that our simulations do not include the thermal-viscous instability, so their evolution differs from that of SUUMa discs. Lubow (1991a) found an analytical expression for the growth rate of the eccentricity, finding it to be proportional to q2. This is applied to an ideal narrow fluid ring. Clearly the eccentricity growth rate of our simu- lated discs is not proportional to q2. Osaki (2005) uses the dependence of eccentricity growth on q2 to propose, as a re- finement to the TTI model, an explanation for type A/type B superoutbursts, namely those which show a precursor and those which do not. He suggests that it depends on whether the eccentricity growth rate is large enough to excite a signif- icantly eccentric disc within the duration of a normal out- burst, and that this is why most SUUMa systems having relatively low mass ratio show only Type B superoutbursts. However, if it is the case that high q means lower growth rates as we see, then this argument fails, and probably other factors contribute. Goodchild & Ogilvie (2006) also find extremely low growth rates in their analytical work. They use the growth term found by Lubow to describe the rate at which eccentric- ity is created at the resonance, but consider further how this eccentricity propagates through the disc. They explain their low growth rates as due to the eccentricity being strongly suppressed at the resonance itself. They find that the growth rates depend most strongly on mass ratio and on bulk viscos- ity, with further weaker dependence on disc semi-thickness. In column 7 of Table 2 we list the parameter, h, referred to by Goodchild & Ogilvie (2006) as the characteristic disc semi-thickness, and in column 11 the bulk viscosity at R3:1 in our simulations. For the 3D simulations, these are ∼ 0.036 and 0.2 respectively. Comparing our Figure 2 with Good- child & Ogilvie (2006)’s results, we see that the trend in our 3D points matches quite well with the h = 0.01 line (the highest value of h given) in their figure 10. In particular we see in both cases low growth rates at high q, a maximum at q ∼ 0.08 and a steep drop in growth rate at mass ratios below this. Our growth rates, though, are about a factor of 10 higher. Looking to their figure 9, this could be explained by our high semi-thickness parameter. Our bulk viscosity, though, is also rather high. It would be very interesting to make a study of empirical growth rates and their depen- dence on q and on other known parameters to compare with these findings. This would require systematic monitoring of dwarf novae to catch the onset of outburst. There are, however, distinct differences between the work of Goodchild & Ogilvie (2006) and the simulations that we present. Tidal modes, which would presumably act to truncate the eccentric mode, are not included in their work. The outer boundary conditions differ. We see too in Section 3 that the eccentricity distribution as a function of radius appears quite different from their findings. The situa- tion in the simulations is further complicated by the presence and importance of the tidal 2-armed spiral structure which is not included in Goodchild & Ogilvie (2006)’s work. In our simulations we see the superhump period de- creasing (Figure 1) as is often observed over the course of a superoutburst (Patterson et al. 1993a). As the period changes, the disc eccentricity is increasing, and, in most cases, the disc mass has begun to decrease in response to the enhanced tidal torques. The superhump period decrease in the simulations can then be explained by the eccentric wave propagating inward, and additionally by radial shrink- ing of the disc. We are unsure how to explain the exception, q = 0.0526, which shows an opposite behaviour for the ini- tial part of the simulation. Perhaps it is related to the ideas of Uemura et al. (2005), where they suggest an explanation for +ve Ṗsh observed in a few cases. They suggest Ṗsh is related to the amount of matter beyond R3:1, so allowing for an outward propagation of the eccentric wave. For low q, the distance between R3:1 and Rtides is greater. The observational data points to a many-valued ǫ(q) relation (Figure 14). In particular 3 of the BHXRTs show systematically lower precession rates than those of CVs. QZVul (GS 2000+2) might be an exception to this. However the superhump period measurement is uncertain as it has been sparsely observed. The microquasar V1487 Aqr (GRS 1915+105) is a much longer period system and may not be comparable to the other BHXRTs. The accretion discs in BHXRTs are irradiated by the central X-ray source. We would expect these discs to be both hotter and also thicker due to the bloating effect of irradiation. Both of these would act to reduce the precession rate, due to the retrograde ef- fect of pressure and due to the dependence on semi-thickness found by Goodchild & Ogilvie (2006). It would be very inter- esting if ǫ and q for further BHXRTs could be determined. The only ultra-compact helium binary included, AMCVn, also lies below the main cluster of points in Figure 14. As Roelofs et al. (2006) noted, the helium accretion disc in AMCVn could be thicker than its hydrogen-rich counter- parts. 6 SUMMARY The main findings in this work can be summarised as follows: • We present improved accretion disc simulations for a range of q. The main improvements are both numerical and physical: a higher mass resolution, extension to 3D, more realistic disc temperature and viscosity, and a radial depen- dence of sound speed appropriate to a steady-state accretion disc. We ran the simulations until equilibrium was reached. For 0.08 < q < 0.24 the 3D discs reach an eccentric equilib- rium and show a superhump signal in their energy dissipa- tion rate (which we refer to as a simulated lightcurve). • The ǫ(q) dependence for the SPH simulations presented in this work shows a greatly improved match with observa- tion than previous simulations. • No high resolution 3D disc with q > 0.24 developed superhumps. This agrees with theoretical expectations and matches the majority of observations. • The region of the disc within R3:1 is overwhelmingly re- sponsible for generating the dissipation-powered superhump in the simulations. • If the difference between observed precession rates and dynamical precession rates calculated at the 3:1 resonance radius is due to averaging over the disc as Goodchild & Ogilvie (2006) suggest, then we find that the best fit char- acteristic radius of the eccentricity distribution at which the dynamical precession rate is evaluated to be 0.87R3:1 and 0.83R3:1 for the 3D simulated discs and the observed eclips- ing systems respectively. The differences between these two c© 2005 RAS, MNRAS 000, 1–18 Comprehensive simulations of superhumps 17 best-fit radii may be partly due to the differing surface den- sity distributions in the two cases. • Our simulations show the effect of the increased effi- ciency of tidal return of angular momentum to the binary for an accretion disc which has become eccentric. The disc mass approaches a new lower steady-state value as the disc becomes eccentric. This is exactly as asserted by the TTI model. With the assumption of a radial dependence of vis- cosity, we deduce an effective ∼ 4 per cent increase viscous torque between a disc which is circular and one that is ec- centric. The increase depends on q. • As the eccentricity grows and the disc mass falls, the superhump period decreases. • The dependence of eccentricity growth rates on q that we see in the simulations presented here is comparable to the work of Goodchild & Ogilvie (2006). Particularly, we find that for high mass ratios the growth rates are very low indeed, in contrast to the result of Lubow (1991a). This needs to be reconciled with observation. • We show that superhumping discs have noticeable ec- centricity even in their inner regions (r ∼ 0.15a). Conversely, non-superhumping discs are seen to be eccentric only in their outer regions. In this case however, this ‘eccentricity’ is steady-state and has origin in tidal distortions, being there- fore different from that which dominates the main body of the superhumping discs. We characterise the eccentricity dis- tributions using two different methods. • The disc motions can be described as superpositions of the S(k,l) modes, and the resonance radii act as nodes and antinodes in the complex standing wave dynamics the disc executes over a full precession period. We characterise the disc motions on Psh, the key timescale for the powering of the observed superhumps. • The 4:1 and 5:1 resonances may play roles in the dy- namics of eccentric discs for q < 0.24. This may explain why the observed precession rates are closer to the dynamic precession rate at the 4:1 resonance than they are to the dynamic precession rate at the 3:1 resonance. • The observational data shows a multi-valued ǫ(q) rela- tion. In particular, the BHXRTs show systematically lower precession rates than those of the CVs, which may be ex- pected when the higher temperature and thickness of their irradiated discs is considered. 7 ACKNOWLEDGEMENTS Helpful comments from the referee were much appreciated. We acknowledge the use of the supercomputing facilities at the Centre for Astrophysics and Supercomputing, Swin- burne University of Technology. AJS was supported by a PPARC studentship. REFERENCES Araujo-Betancor S., et al., 2003, ApJ , 583, 437 Arenas J., Catalán M. 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0704.1520

Singular Energy Distributions in Granular Media

Singular Energy Distributions in Granular Media E. Ben-Naim1, ∗ and A. Zippelius2, † Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA Institut für Theoretische Physik, Georg-August-Universität, 37077 Göttingen, Germany We study the kinetic theory of driven granular gases, taking into account both translational and rotational degrees of freedom. We obtain the high-energy tail of the stationary bivariate energy distribution, depending on the total energy E and the ratio x = Ew/E of rotational energy Ew to total energy. Extremely energetic particles have a unique and well-defined distribution f(x) which has several remarkable features: x is not uniformly distributed as in molecular gases; f(x) is not smooth but has multiple singularities. The latter behavior is sensitive to material properties such as the collision parameters, the moment of inertia and the collision rate. Interestingly, there are preferred ratios of rotational-to-total energy. In general, f(x) is strongly correlated with energy and the deviations from a uniform distribution grow with energy. We also solve for the energy distribution of freely cooling Maxwell Molecules and find qualitatively similar behavior. PACS numbers: 45.70.Mg, 47.70.Nd, 05.40.-a, 81.05.Rm I. INTRODUCTION Energy dissipation has profound consequences in granular materials, especially in dilute gases, where the dynamics are controlled by collisions [1, 2, 3]. Dissipation is responsible for many interesting collective phenomena including clustering [4, 5, 6, 7, 8], formation of shocks [9, 10, 11, 12, 13], and hydrodynamic instabilities [14, 15]. Another consequence is the anomalous statistical physics that includes the non-Maxwellian velocity distributions [16, 17, 18, 19, 20, 21] and the breakdown of energy equipartition in mixtures [22, 23]. For an elastic gas in equilibrium, the temperature, defined as the average kinetic energy, characterizes the entire distribution function including all of the moments, the bulk of the distribution, as well as the tail of the distribution. Outside of equilibrium, the temperature is not sufficient to characterize the energy distribution. Granular gases are inherently out of equilibrium and a complete characterization must therefore include the behavior of typical particles, the behavior of energetic particles, as well as the moments of the distribution. For example, the energy distribution may have power-law tails with divergent high-order moments [24, 25, 26] and consequently, the moments exhibit multiscaling [27]. Generally, nonequilibrium effects are pronounced in the absence of energy input to balance the dissipation but can be suppressed by injection of energy where the deviation from a Maxwellian distribution affects only extremely energetic particles [17, 28, 29, 30]. While there is substantial understanding of the energy distribution of frictionless granular gases, much less is known theoretically [31, 32, 33, 34, 35, 36, 37, 38] and experimentally [39, 40, 41] when the rotational degrees of freedom are taken into account. It is difficult to measure the rotational motion experimentally, and the few available measurements are restricted to two-dimensions. Surface roughness and friction have important consequences and the hydrodynamic theory [42, 43, 44, 45] must be modified, if the particles have spin [46]. Equipartition does not hold for the average rotational and translational temperature – neither in the free cooling case [33, 34, 35, 36] nor for a driven system [37]. In general, rotational and linear degrees of freedom are correlated in direction [47]. In this paper, we investigate the nature of the full energy distribution, that is, the bivariate distribution of rotational and translational energy. Motivated by the fact that on average the total energy is not partitioned equally between rotational and translational degrees of freedom, we focus on the bivariate distribution P (E, x) of total energy E and the modified ratio x = Ew/E of rotational to total energy. We thereby generalize the understanding of frictionless granular matter in terms of the energy distribution to rough grains. Our starting point is the nonlinear Boltzmann equation with a collision rule that accounts for the coupling of translational and rotational motion due to tangential restitution. We study stationary solutions of the inelastic Boltzmann equation that describe steady states achieved through a balance between energy injections that are powerful but rare and energy dissipation through inelastic collisions. For high-energy particles we derive a linear equation for ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.1520v1 mailto:[email protected] mailto:[email protected] the bivariate energy distribution. The latter can be shown to factorize – P (E, x) = p(E)f(x) – into a product of the distribution of the total energy, p(E), and the distribution of the fraction of energy stored in the rotational degrees of freedom, f(x). The former distribution decays algebraically with energy: p(E) ∼ E−ν . The fraction of energy stored in rotational motion is universal for energetic particles in the sense that f(x) approaches a limiting distribution independent of energy. Furthermore, this quantity has a number of interesting features. First, the distribution is not uniform, as it would be, if equipartition were to hold. Second, the distribution is not analytic but has singularities at special energy ratios. Third, the distribution and in particular its singularities depend sensitively on the moment of inertia and the collision parameters. Only for energetic particles is this distribution well defined. In general, the partition of energy into rotational and translational motion depends on the magnitude of the energy. This paper specifically addresses two-dimensions, although the theoretical approach and the reported qualitative behavior are generic. We also develop a general framework for describing high-energy collisions and we use this framework to study freely cooling Maxwell Molecules where the moments of the energy distribution can be found in a closed form. For example, the two granular temperatures corresponding to the rotational and translational motions are coupled and generally, they are not equal. The high-energy behavior found for driven steady-states extends to freely cooling gases. The rest of this paper is organized as follows. We review the collision rules and introduce the nonlinear kinetic theory in section II. We then derive the linear kinetic theory for high-energy particles in section III. Next, in section IV, we study driven steady states and solve for the stationary energy distribution. Freely cooling Maxwell molecules are discussed in section V and we conclude in section VI. The Appendices detail technical derivations. II. THE NONLINEAR KINETIC THEORY Our system consists of an infinite number of identical particles with mass m = 1, radius R, and moment of inertia I = qR2 where 0 ≤ q ≤ 1 is a dimensionless quantity. Each particle has a linear velocity v and an angular velocity w. Its total energy is shared by the linear and the rotational motion, E = Ev + Ew, or explicitly, v2 + qR2w2 where v ≡ |v| and w ≡ |w|. In a collision between two particles, their velocities (vi,wi) with the labels i = a, b, change according to (va,wa) + (vb,wb) → (v′a,w′a) + (v′b,w′b) (2) where the postcollision velocities are denoted by primes. In a binary collision, rotational and translational energy are exchanged, while the total energy decreases. In this study, we consider tangential restitution in addition to the standard normal restitution. Let ri be the position of particle i, then the directed unit vector connecting the centers of the colliding particles is n̂ = (ra − rb)/|ra − rb|. We term this vector the impact direction. The collision rules are most transparent in terms of ui the particle velocity at the contact point ua = va +R n̂×wa (3a) ub = vb −R n̂×wb. (3b) The inelastic collision laws state that the normal component of the relative velocity U = ua − ub is reversed and reduced by the normal restitution coefficient 0 ≤ rn ≤ 1. The tangential component is either reversed (rough particles) or not (smooth particles) and in any case reduced by the tangential restitution coefficient −1 ≤ rt ≤ 1, according to the following collision rules: ′ · n̂ = −rnU · n̂, (4a) ′ × n̂ = −rt U× n̂. (4b) Inelastic collisions conserve linear and angular momentum. Conservation of linear momentum implies that the total linear velocity does not change, and conservation of angular momentum enforces that the angular momentum of each particle with respect to the point of contact remains the same, because there is no torque acting at the point of contact. The collision laws (4) combined with these conservation laws specify the postcollision velocities as linear combinations of the precollision velocities [33] a = va − ηnV · n̂ n̂− ηt (V −V · n̂ n̂)− ηtR n̂×W w′a = wa + n̂×V + ηt n̂× n̂×W (5a) b = vb + ηnV · n̂ n̂+ ηt (V −V · n̂ n̂) + ηtR n̂×W w′b = wb + n̂×V + ηt n̂× n̂×W (5b) where the shorthand notations V = va − vb and W = wa +wb were introduced. These collision rules involve the normal and tangential collision parameters, defined as 1 + rn , and ηt = 1 + q 1 + rt . (6) Their range of values is bounded by 1/2 ≤ ηn ≤ 1 and 0 ≤ ηt ≤ q/(1 + q). Details of the derivation of the collision rules are given in Appendix A, as they are relevant for our discussion. The energy dissipation, ∆E = Ea+Eb−E′a−E′b, is given by 1− r2n (V · n̂)2 + q 1 + q 1− r2t (V −V · n̂ n̂+R n̂×W)2. (7) The energy dissipation is always positive, except when the collisions are elastic, rn = 1 and rt = −1 (perfectly smooth spheres) or rt = 1 (perfectly rough spheres). The collision rate K(va,vb) is the rate by which the two particles approach each other. For hard spheres, this rate is simply the normal component of the relative velocity, but we study the general case K(va,vb) = |(va − vb) · n̂|γ (8) with 0 ≤ γ ≤ 1. Of course, the collision rate vanishes, K = 0, when the particles are moving away from each other, (va − vb) · n̂ > 0. When particles interact via the central potential r−κ then γ = 1 − 2 d−1κ [48]. The two limiting cases are hard spheres (γ = 1) and Maxwell molecules (γ = 0) where the collision rate is independent of the velocity [49, 50, 51, 52]. The central quantity in kinetic theory is the probability P (v,w, t) that a particle has the velocities (v,w) at time t. We study spatially homogeneous situations where this velocity distribution function is independent of position. Under the strong assumption that the velocities of the two colliding particles are completely uncorrelated, the velocity distribution obeys the Boltzmann equation ∂P (v,w) dvadwadvbdwb |(va − vb) · n̂|γ P (va,wa)P (vb,wb) (9) δ(v − v′a)δ(w −w′a) + δ(v − v′b)δ(w −w′b)− δ(v − va)δ(w −wa)− δ(v − vb)δ(w −wb) We integrate over all impact directions with dn̂ = 1 [53] and over the precollision velocities weighted by the respective probability distributions. There are two gain terms and two loss terms, because the velocities of interest (v,w) can be identified with any one of the four velocities in the collision rule (2) and the kernel is simply the collision rate (8). III. THE LINEAR KINETIC THEORY The focus of this study is the energy distribution that generally depends only on two variables: Ev and Ew. It is our aim to compute the distribution P (Ev, Ew) for asymptotically large energies. This will be done for a system which is driven at very high energies as well as for an undriven system. As a first step to this goal, we simplify the Boltzmann equation in the limit of large energies. Extremely energetic particles are rare and as a result it is unlikely that such particles will encounter each other. Hence, energetic particles typically collide with much slower particles. Since the collision rules are linear, the velocity of the slower particle barely affects the outcome of the collision. We can therefore neglect the slower velocity. Substituting (va,wa) = (v0,w0) and (vb,wb) = (0,0) or (va,wa) = (0,0) and (vb,wb) = (v0,w0) into (5) gives the cascade process [54, 55] (v0,w0) → (v1,w1) + (v2,w2) (10) where (v0,w0) is the precollision velocity of the energetic particle and (vi,wi) with i = 1, 2 are the consequent postcollision velocities. With these definitions, the collision rules for extremely energetic particles are v1 = (1− ηn)v0 · n̂ n̂+ (1− ηt)(v0 − v0 · n̂ n̂)− ηtn̂×w0 w1 = 1− ηt n̂× v0 (11a) v2 = ηnv0 · n̂ n̂+ ηt(v0 − v0 · n̂ n̂) + ηtn̂×w0 w2 = − n̂× v0, (11b) where we have set R = 1, so that the moment of inertia, I = q, is dimensionless. A collision between a high-energy particle and a typical-energy particle produces two energetic particles with an energy total that is smaller than the initial energy. This cascade process transfers energy from large scales to small scales. Since the cascade process (10) involves only one particle, the tail of the probability distribution P (v,w) obeys the linear equation ∂P (v,w) dn̂dv0dw0|v0 · n̂|γP (v0,w0) δ(v−v1)δ(w−w1)+δ(v−v2)δ(w−w2)−δ(v−v0)δ(w−w0) . (12) There are two gain terms and one loss term according to the cascade process (10). Formally, this linear rate equation can be obtained from the full nonlinear equation (9) by treating either one of the precollision velocities as negligible and then integrating over this small velocity. This procedure leads to four gain terms and two loss terms and thus, the factor 1/2 in (9) drops out. We stress that the linear equation (12) is valid only in the high-energy limit. We also comment that the linear equation (12) for the high-energy tail of the velocity distribution may be valid in cases where the full nonlinear equation is not. Whereas the nonlinear equation requires that all possible velocities are uncorrelated, the linear equation merely requires that energetic particles are uncorrelated with typical particles. This is a much weaker condition. In this paper, we restrict ourselves to two space dimensions, i.e. rotating disks. In that case the rotational velocities are always perpendicular to the linear velocities. Thus, we conveniently denote the unit vector in the tangential direction by t̂ and the unit vector coming out of the plane by ẑ, such that n̂ · t̂ = 0 and n̂× t̂ = ẑ. The precollision velocities of the energetic particle v0 = vn n̂ + vt t̂ and w0 = w ẑ are compactly written as [vn, vt, w]. With this notation, the postcollision velocities specified in (12) are (1− ηn)vn, (1− ηt)vt + ηtw, (ηt/q)vt + (1− ηt/q)w , and ηnvn, ηt(vt − w), (ηt/q)(vt − w) , (13) respectively. We now treat the three velocity components, namely the normal component of the velocity vn, the tangential component of the velocity vt, and the scaled angular velocity qw as a three dimensional vector with magnitude V0, polar angle θ0, and azimuthal angle φ0: (vn, vt, qw) = (V0 sin θ0 cosφ0, V0 sin θ0 sinφ0, V0 cos θ0). (14) The magnitude V0 gives the energy E0 = V 20 = (v2n + v t + qw 2) while the polar angle characterizes the fraction of energy stored in the rotational degree of freedom, 1 qw2/E0 = cos 2 θ. In this representation, the postcollision velocities are three-dimensional vectors with magnitude Vi, polar angle 0 ≤ θi ≤ π, and azimuthal angle 0 ≤ φi ≤ 2π. The collision rules (11) allow us to express these quantities in terms of V0, θ0, φ0: (Vi sin θi cosφi, Vi sin θi sinφi, Vi cos θi) = (V0Ai, V0Bi, V0Ci) (15) where i = 1, 2. The magnitudes of the postcollision velocities are proportional to the magnitude of the precollision velocity. The three velocity components are scaled by three dimensionless constants Ai, Bi and Ci, that depend on the angles θ0 and φ0 of the energetic particle, the collision parameters ηn and ηt, and the moment of inertia q, A1 = (1− ηn) sin θ0 cosφ0 (16a) B1 = (1− ηt) sin θ0 sinφ0 + (ηt/ q) cos θ0 (16b) C1 = (ηt/ q) sin θ0 sinφ0 + (1− ηt/q) cos θ0 (16c) A2 = ηn sin θ0 cosφ0 (16d) B2 = ηt sin θ0 sinφ0 − (ηt/ q) cos θ0 (16e) C2 = (ηt/ q) sin θ0 sinφ0 − (ηt/q) cos θ0. (16f) The new energies are proportional to the precollision energies Ei = αiE0, with αi = A i + C i . (17) We term the parameters 0 < αi < 1 the contraction parameters. Since the collisions are dissipative, these parameters satisfy the inequality α1 + α2 ≤ 1. The equality α1 + α2 = 1 holds only for elastic collisions (rn = |rt| = 1). The energy dissipation is ∆E = E0 − E1 − E2 = ΛE with Λ = 1− α1 − α2 or explicitly, 1− r2n sin2 θ0 cos 2 φ0 + 1 + q 1− r2t sin2 θ0 sin 2 φ0 + cos2 θ0 . (18) The polar and azimuthal angles are given by cos θi = A2i +B i + C and tanφi = , (19) respectively. Let us represent solid angles by Ω ≡ cos θ, φ. With this definition, the cascade process (11) is (E0,Ω0) → (E1,Ω1) + (E2,Ω2) (20) with Ei and Ωi given by (17) and (19). Energetic particles have an important property: the solid angle is not coupled to the energy! Indeed, the postcollision angles depend only on the precollision angle. The cascade process has the following geometric interpretation: a three dimensional vector is duplicated into two vectors. Subsequently, these two vectors are scaled down by the contraction parameters (17), and rotated according to the angular transformation (19). We can now write the linear Boltzmann equation for P (E,Ω), the distribution of energy and solid angle, in a closed ∂P (E,Ω) dE0dΩ0 E0sin θ0cosφ0 P (E0,Ω0) δ(E−E1)δ(Ω−Ω1)+δ(E−E2)δ(Ω−Ω2)−δ(E−E0)δ(Ω−Ω0) Time was rescaled, t → 2γ/2t, to absorb the constant which arises from replacing velocity by energy in the collision rate (8). Henceforth, we implicitly assume that the distribution P (E,Ω) is independent of φ because the distribution of linear velocities must be isotropic. The integration over the energy is performed using the collision rule (17), leading to the linear rate equation for the tail of the energy distribution ∂P (E,Ω) =Eγ/2 ∣ sin θ0 cosφ0 δ(Ω− Ω1) 1+γ/2 δ(Ω− Ω2) 1+γ/2 −P (E,Ω0)δ(Ω− Ω0) . (21) This is a non-local equation as the density of particles with energy E is coupled to the density of particles with the higher energies E/α1 and E/α2. We stress that this equation is a straightforward consequence of the cascade process (20) and that it can also be derived from the full nonlinear Boltzmann equation. Yet, there may be situations where the linear equation (21) is valid, while the nonlinear equation (9) is not valid. The bivariate energy distributions P (E,Ω) and P (Ev, Ew) are completely equivalent but we analyze the former because the cascade process (20) is transparent in terms of the total energy and the solid angle. IV. DRIVEN STEADY-STATES The inelastic Boltzmann equation admits stationary solutions for frictionless particles. These stationary solutions describe driven steady-states with rare but powerful injection of energy. The injected energy cascades from high- energies down to small energies, thereby balancing the energy lost in collisions. At energies below the injection scale, Eqs. (9), (12) and (21) are not altered by the energy source and consequently, the stationary solution of the inelastic Boltzmann equation holds up to this large energy scale [54, 55]. Here, we seek a corresponding stationary solution for particles with rotational degrees of freedom in the high energy limit. The stationary solution has to fulfill Eq. (21) with the left hand side set to zero ∣ sin θ0 cosφ0 1+γ/2 δ(Ω− Ω1)+ 1+γ/2 δ(Ω− Ω2)−P (E,Ω0)δ(Ω− Ω0) . (22) At high-energies, the solid angle is not coupled to the energy, as follows from Eq. (19). This fact has a ma- jor consequence: the bivariate energy distribution P (E,Ω) takes the form of a product of the energy distribution p(E) = dΩP (E,Ω) and the distribution of solid angle, g(Ω), P (E,Ω) → p(E) g(Ω) (23) as E → ∞. The angle distribution is normalized, dΩ g(Ω) = 1. It does not depend on the azimuthal angle, because on average the two components of the linear velocity are equivalent. Due to the equi-dimensional (in E) structure of the steady-state equation (22), the product ansatz (23) is a solution when the distribution p(E) decays algebraically p(E) ∼ E−ν , (24) -1 -0.5 0 0.5 1 0 FIG. 1: The exponent ν for hard spheres (γ = 1) as a function of the coefficients of normal, rn, and tangential, rt, restitution coefficients. The numerical procedure for solving (25) is detailed below. as E → ∞ [54, 55]. We obtain a closed equation for the distribution g(Ω) by substituting the product ansatz (23) with the power-law form (24) into the steady-state equation (22) dΩ0 g(Ω0) ∣ sin θ0 cosφ0 ν−1−γ/2 1 δ(Ω− Ω1) + α ν−1−γ/2 2 δ(Ω− Ω2)− δ(Ω− Ω0) . (25) This equation is linear in g(Ω). However, it is nonlinear in ν and moreover, the solid angles Ωi ≡ Ωi(Ω0) in (19) and the contraction parameters αi ≡ αi(Ω0) in (17) are complicated functions of the solid angle Ω0. Equation (25) involves two unknowns quantities, the exponent ν and the distribution function g(Ω). A solution does not exist for arbitrary values of ν. In fact, there is one and only one value of ν for which there is a solution for g(Ω). This is the value selected by the cascade dynamics! In other words, (25) is an eigenvalue equation: ν is the eigenvalue and g(Ω) is the eigenfunction. This eigenvalue equation circumvents the full nonlinear equation (9) and thus, represents a significant simplification. The physical interpretation of (25) involves a cascade process in which the solid angle undergoes a creation- annihilation process ∅ with rate β0, Ω1 with rate β1, Ω2 with rate β2. Here, βi = | sin θ0 cosφ0 αi for i = 0, 1, 2 and α0 = 1. There is one annihilation process and two creation processes. These processes have relative weights that reflect the powerlaw decay of p(E). At the steady-state, the creation and the annihilation terms balance (see Appendix B), as reflected in the integrated form of (25) dΩ0 g(Ω0) ∣ sin θ0 cosφ0 ν−1−γ/2 1 + α ν−1−γ/2 2 − 1 . (27) To achieve a steady-state, βi < β0 for i = 1, 2 and therefore α ν−1−γ/2 i < 1. Since αi < 1, we have the lower bound ν > 1 + γ/2. We can immediately check that for elastic collisions, ν = 2 + γ/2 [56, 57] because α1 + α2 = 1, and therefore, we conclude the bounds 1 + γ/2 ≤ ν ≤ 2 + γ/2. The exponent ν varies continuously with the restitution coefficients rn and rt and the normalized moment of inertia q. This quantity must coincide with the value found for frictionless particles where tangential restitution is irrelevant (rt = −1) [54, 55, 58], but otherwise the exponent is distinct, as shown in Fig. 1. Also, the exponent ν increases monotonically with rn and |rt|. We conclude that the rotational degrees of motion do affect the power-law behavior (24). The azimuthal angle θ characterizes the fraction of energy stored in the rotational mode, cos2 θ = Ew/E with qw2. The angle distribution g(Ω) = (2π)−1f̃(cos θ) therefore captures the partition of energy into rotational and translational energies. We introduce the natural variable 0 ≤ x ≤ 1 defined by x = | cos θ| so that and present results for the angle distribution f(x) = 2f̃(cos θ). In equilibrium, energy is partitioned equally into all degrees of freedom and therefore geq(Ω) = (4π) −1 or equivalently, feq(x) = 1 (29) for 0 ≤ x ≤ 1. In particular, 〈x2〉 = 1/3. A. Simulation Methods We numerically studied the angle distribution f(x) by solving the linear eigenvalue equation (25) for the “angular” process (26) and by solving the full nonlinear Boltzmann equation (9) for the collision process (2). Both of these equations are solved using Monte Carlo simulations. The eigenvalue equation is solved by mimicking the angular process. Throughout the simulation, the value ν is fixed. There are N particles, each with a given polar angle. A particle with polar angle θ0 is picked at random and then, a random azimuthal angle φ0 is drawn. The polar angles θ1 and θ2 are then calculated according to (19). The original particle is annihilated with probability β0 and simultaneously, a new particle with angle θ1 is created with probability β1 and similarly, a second particle with angle θ2 is created with probability β2. Therefore, the number of particles may increase by one, remain unchanged, or decrease by one. The exponent ν is the value that keeps the total number of particles constant in the long time limit. The eigenvalue ν is calculated by repeating this simulation for various values of ν and then using the bisection method [59]. We present Monte Carlo simulations of 100 independent realizations with N = 107 particles. Driven steady-states are obtained by simulating the two competing processes of inelastic collisions and energy injection. In an inelastic collision, two particles are picked at random and also, the impact direction is chosen at random. The particle velocities are updated according to the collision law (5). Collisions are executed with probability proportional to the collision rate. Throughout this process, we keep track of the total energy loss. With a small rate, we augment the energy of a randomly selected particle by an amount equal to the loss total and subsequently, reset the total energy loss to zero. A fraction of the injected energy is rotational and the complementary fraction is translational. We draw this fraction according to the equilibrium distribution (29). We experimented with different angle distributions and found that the resulting stationary state did not change. Obtaining the distribution f(x) is generally challenging as it requires excellent statistics. The simulations are most efficient for Maxwell molecules because all possible collisions are equally likely. Therefore, for the full nonlinear Boltzmann equation (9), we present the angle distribution of the energetic particles only for the case γ = 0. For Maxwell molecules, the injection rate is 10−4 and the system size is N = 107. The corresponding values for hard spheres are 10−2 and N = 105. In all cases, the simulation results represent an average over 102 independent realizations. Unless noted otherwise, the simulation results are for maximally dissipative (rn = rt = 0) disks (q = 1/2). B. The Distribution of Total Energy The numerical simulations confirm several of our theoretical predictions. First, the energy distribution approaches a steady-state with a power-law high-energy tail. Second, the distribution of the total energy p(E) decays algebraically as in (24). Third, the exponent ν is in excellent agreement with the predictions of the eigenvalue equation. For Maxwell molecules, Monte Carlo simulation of the full nonlinear equation yields ν = 1.570± 0.005 whereas numerical solution of the eigenvalue equation (25) gives ν = 1.569 ± 0.005 (Fig. 2). For hard-spheres, where the simulation results are slightly less accurate, the corresponding values are ν = 2.065 ± 0.005 and ν = 2.060 ± 0.005 (Fig. 3). The behavior of the distribution of total energy is therefore qualitatively similar to the behavior in the no-rotation case [54, 55]. However, the quantitative behavior is different because the exponent ν does depend on the tangential restitution coefficient and the moment of inertia (Fig. 1). C. The Angle distribution The numerical simulations also confirm several of our theoretical predictions concerning the angle distribution. Extremely energetic particles have a universal distribution f(x). This distribution is independent of the energy, theory simulation FIG. 2: The tail of the energy distribution for driven Maxwell molecules. Shown are simulation results (solid line) and a line with the slope predicted by the theory (dashed line). The energy is normalized by the typical energy 10−4. theory simulation FIG. 3: The tail of the energy distribution for driven hard spheres. Shown are simulation results (solid line) and a line with the slope predicted by the theory (dashed line). 0 0.2 0.4 0.6 0.8 1 Angular Process Collision Process FIG. 4: The angle distribution f(x) obtained by Monte Carlo simulation of the angular process (26) (solid line) and the collision process (2) (dashed line) for Maxwell molecules. The special values x1, x2, and x3 discusses in the text are indicated by arrows. 0 0.2 0.4 0.6 0.8 1 =0, r =-0.9 =0.9, r =0.9, r FIG. 5: The angle distribution f(x) for various collision parameters (rn and rt) for Maxwell molecules. provided that the energy is sufficiently large. We had to probe only the most energetic particle out of roughly 103 particles to measure this distribution. For this reason, the linear analysis and the resulting eigenvalue equation are valuable because they allow for an accurate and efficient determination of the angle distribution of the energetic particles. We also verified that the distribution f(x) obeys the eigenvalue equation (25), as demonstrated in Fig. 4, where the simulations are compared to the solution of the angular process. The distribution f(x) has several noteworthy features. First, it is not uniform, implying the breakdown of en- ergy equipartition in a granular gas. Furthermore, this distribution is nonanalytic. It contains singularities and discontinuous derivatives. There are notable peaks in the distribution so that special values x and special ratios Ew/E are strongly preferred. The reason for these peaks is the fact that the polar angle is limited. For example, cos2 θ2 < 1/(1 + q) as seen by substituting cos θ0 = ±1 into (16) and (19). Consequently, there is a special ratio 1 + q with the corresponding special energy ratio Ew/E = x 1. This is the most pronounced peak in Fig. 4, 0 0.2 0.4 0.6 0.8 1 FIG. 6: The angle distribution f(x) for hard spheres. 0 0.2 0.4 0.6 0.8 1 FIG. 7: The angle distribution fall(x) of all particles for Maxwell molecules (solid line). Also shown for reference is the uniform equilibrium distribution (broken line). 2/3 = 0.81649. Numerically, we observe that the peak becomes more pronounced as the distribution is mea- sured at a finer scale, indicating that the distribution function diverges at this point. Similarly, there is another special ratio that corresponds to θ1 when cos θ0 = ±1, and unlike (30), this location depends on the tangential restitution, 1− ηt/q η2t /q + (1− ηt/q)2 . (31) Indeed, there is a barely noticeable cusp at x2 = 8/9 = 0.942809. Singularities may induce less pronounced “echo”-singularities. For example, using cos θ0 = x1 and φ0 = π/2 yields the special ratio 1 + ηt(1− 1/q) q[1− ηt(1− 1/q)]2 + [1 + ηt(1− 1/q)]2 . (32) There is a noticeable peak at the corresponding value x3 = 50/99 = 0.710669 in Fig. 4. We anticipate that as the transformation (19) is iterated, the strength of the singularities weakens and as a result there are discontinuous derivatives of increasing order, a subtle behavior that is difficult to measure. The location of the singularities varies with the collision parameters rn and rt and the moment of inertia q. In fact, the angle distribution is extremely sensitive to material properties as its shape changes dramatically with these parameters, see Fig. 5. The angle distribution also depends on the collision rate and it is much smoother for hard spheres, see Fig. 6. Since the collision rate vanishes for grazing collisions, φ = π/2, the associated singularities including in particular (32) are suppressed. Nevertheless, there is a pronounced jump at the special ratio given by (30) and there are also noticeable cusps. The angle distribution of all particles fall(x) ∝ dE P (E,Ω) is shown in Fig. 7. It is substantially different from f(x). Therefore, the energy distribution P (E,Ω) does not factorize in general and there are correlations between the solid angle and the total energy. Only for energetic particles does (23) hold. Moreover, fall(x) is much smoother in comparison with f(x) although there is a jump in the first derivative at the special ratio (30) showing that the angle distribution of all particles is also non-analytic, see Fig. 7. Generally, the angle distribution depends on energy and the deviation from a uniform distribution grows with energy. We also comment that lone measurement of the moment 〈x2〉 can be misleading. The angle distribution may very well have a value close to the equipartition value 〈x2〉eq = 1/3 but still, be very far from the equilibrium distribution. Indeed, in Fig. 4, 〈x2〉 ∼= 0.318, a value that barely differs from the equilibrium value, even though the corresponding distribution is far from uniform. The second moment may also differ substantially from the equipartition value and for example, 〈x2〉 = 0.202 when rn = 0.9 and rt = 0 (Fig. 5). We argue that the qualitative features of the angle distribution should be generic in granular materials. Colli- sions involving energetic particles must follow the linear cascade rules (20) with the angular transformations (19). The singularities are a direct consequence of these transformations and therefore should be generic. Measuring the parameter-sensitive distribution f(x) experimentally is challenging because a huge number of particles must be probed and the measurement has to be accurate. The distribution fall(x) provides a detailed probe of the partition of energy into rotational and translation motion. V. FREE COOLING We now consider freely cooling granular gases that evolve via purely collisional dynamics. Without energy input, all energy is eventually dissipated and the particles come to rest. This system has been studied extensively [1] for hard spheres with [33, 47] and without rotation [60]. We consider Maxwell molecules where in the absence of rotation an exact treatment is possible [24, 25, 27, 61, 62]. When γ = 0 the Boltzmann equation (9) simplifies ∂P (v,w) dvadwadvbdwb P (va,wa)P (vb,wb) (33) × [δ(v − v′a)δ(w −w′a) + δ(v − v′b)δ(w −w′b)− δ(v − va)δ(w −wa)− δ(v − vb)δ(w −wb)]. Consequently, the equations for the moments 〈vnwm〉 = dvdwP (v,w)vnwm close. A. The Temperatures Here, we consider only the translational temperature defined as the average translational energy, Tv = 〈Ev〉, and the rotational temperature, defined as the average rotational energy Tw = 〈Ew〉. These two temperatures are coupled through the linear equation λvv λvw λwv λww . (34) Appendix C details the derivation of the matrix of coefficients λvv = ηn(1− ηn) + ηt(1 − ηt) (35a) λvw = −2η2t /q (35b) λwv = −η2t /q (35c) λww = 2(ηt/q)(1− ηt/q). (35d) The two temperatures are coupled as long as ηt 6= 0 or alternatively, rt 6= −1. The solution of (34) is a linear combination of the two eigenvectors e−λ−t + C+ e−λ+t (36) with the constants C− and C+ set by the initial conditions, and c± = (λ± − λvv)/λvw. The eigenvalues are λvv + λww λvv − λww + λvwλwv . (37) The larger eigenvalue is irrelevant in the long time limit and therefore, e−λt (38) such that both temperatures decay with the same rate λ ≡ λ−. Of course, the total temperature also follows the same exponential decay, T = Tv + Tw ∼ e−λt. In this regime, the fraction of rotational energy is on average 1 + c− λ− λvv λ+ λvw − λvv . (39) The approach toward this value is exponentially fast and the relaxation time is inversely proportional to the difference in eigenvalues τ = 1/(λ+ − λ−). In equilibrium, Tw/T = 1/3 but for nonequilibrium granular gases the ratio varies. In Fig. 8 we plot the ratio of the average rotational energy to the total energy as a function of the coefficients of restitution. In accordance with our findings for driven steady-states, energy is not partitioned equally between all the degrees of freedom. 1 0 FIG. 8: The ratio of average rotational energy to total energy as a function of the coefficients of normal, rn, and tangential, rt, restitution. theory simulation FIG. 9: The scaling function underlying the energy distri- bution (solid line). The distribution was obtained using a Monte Carlo simulation with N = 107 particles. A dashed line with the slope predicted by the theory is also shown for reference. B. The Energy Distribution To study the full energy distribution, it is again convenient to make a transformation of variables from the velocity pair (v,w) to the total energy and the solid angle (E,Ω). The energy distribution is now time dependent and assuming that the temperature – T ∼ e−λt – is the characteristic energy scale we postulate the self-similar form P (E,Ω, t) → eλtΦ(Eeλt,Ω) (40) with the prefactor ensuring proper normalization, dz dΩΦ(z,Ω) = 1. We focus on the high-energy behavior where the linear equation (21) holds. By substituting the scaling form (40) into this linear equation and setting γ = 0, we find the integro-differential equation governing the scaling function λΦ(z,Ω) + λz Φ(z,Ω)= δ(Ω− Ω1) + δ(Ω− Ω2)− Φ(z,Ω0)δ(Ω− Ω0) .(41) We again write the multivariate energy distribution as a product Φ(z,Ω) → ψ(z)g(Ω) of the distribution of the total energy ψ(z) = dΩΦ(z,Ω) and the distribution of the solid angle g(Ω). This form is a solution of the equi-dimensional equation (41) when the distribution of the total energy decays as a power-law ψ(z) ∼ z−ν (42) at large energies, z → ∞. The angle distribution satisfies the eigenvalue equation dΩ0 g(Ω0) αν−11 δ(Ω− Ω1) + α 2 δ(Ω− Ω2)− [1− λ(ν − 1)]δ(Ω− Ω0) . (43) Of course, setting λ = 0, one recovers the steady-state equation (25) reflecting that the similarity solution is stationary. The factor 1 is replaced by the smaller factor 1 − λ(ν − 1) that accounts for the constant decrease in the number of particles at any given energy because of dissipation. Again, we have a nonlinear eigenvalue equation with the eigenvalue ν and the eigenfunction g(Ω). We solve this eigenvalue equation by performing a Monte Carlo simulation of the same angular process as described by (26) but with a different annihilation rate β0 = 1−λ(ν − 1). We compare the angle distribution predicted by (43) with the behavior of the energetic particles in the freely cooling gas. The numerical simulations of the inelastic collision process confirm the theoretical predictions. First, the energy distribution is self-similar as in (40) and the characteristic scale is proportional to the temperature. Second, the distribution of the total energy has a power-law tail, as displayed in Fig. 9 and the exponent ν is very close to the theoretical prediction (numerical simulations of the collision process gives ν = 2.98±0.05 while the eigenvalue equation yields ν = 2.92± 0.05). 0 0.2 0.4 0.6 0.8 1 Angular Process Collision Process FIG. 10: The angle distribution of the energetic particles. Shown are results for the collision process (solid line) and for the angular process (dashed line). 0 0.2 0.4 0.6 0.8 1 FIG. 11: The angle distribution of all particles for a freely cooling gas (solid line). Also shown for reference is the uni- form equilibrium distribution. The angle distribution deviates even more strongly from the uniform distribution with a very pronounced peak (see Fig. 10) because the dynamics are purely collisional. The singularities are weaker although the one at x1 given by (30) is clear. The agreement between the solution of the angular process and the Monte Carlo simulations is slightly worse than for driven systems because the statistics become prohibitive: now it is necessary to probe the most energetic out of roughly 106 particles to obtain the asymptotic angle distribution! The sharper power-law decay is responsible for this three order of magnitude increase: the cumulative distribution of total energy decays according dE′p(E′) ∼ E−µ with µ = ν − 1 about three times larger than before. Finally, the angle distribution of all particles deviates only slightly from a uniform distribution (see Fig. 11). We conclude that the behavior of the freely cooling gas is qualitatively similar to that found in driven steady-states. VI. CONCLUSIONS AND OUTLOOK The complete description of granular media with translational and rotational degrees of freedom requires the full bivariate distribution of energies. It is not sufficient to consider only the average kinetic energy of translations and rotations. Instead the full bivariate distribution is highly nontrivial. We have shown that in the limit of large particle energy, this distribution obeys a linear equation. Its solution can be written as a product of two distributions, one for the total energy, E = Ev + Ew, and one for the variable x = Ew/E, which captures the partition of the total energy between rotational and translational motion. The distribution of the total energy decays algebraically and the characteristic exponent depends on the collision parameters and the moment of inertia. The variable x is not uniformly distributed as in equilibrium. Instead the distribution f(x) is not analytic and displays a series of singularities of varying strengths. Remarkably, there are special preferred ratios of rotational-to-total energy. This violation of energy equipartition among different degrees of freedom is a direct consequence of the energy dissipation. The total energy and the variable x are correlated in general with the deviations from equilibrium increasing with energy. These two variable become uncorrelated only at extremely high-energies. We have studied both, the system which is driven at extremely high energies and displays a stationary energy cascade on energy scales below the driving one, and a freely cooling gas. In the latter gas the bivariate energy distribution is time dependent, reflecting the overall decrease of energy. Nevertheless, scaling the total energy with temperature, one finds a self-similar form for the distribution, which again factorizes in the high-energy limit. As in the driven system, the distribution of the total energy decays as a power law with, however, different exponents for the driven and the free cooling system. The angular distribution deviates even more from the uniform (equipartition) one in the cooling system. It should be straightforward to extend these results to three dimensions where the angular process takes place in three dimensions. In the limit of high energies one would again expect a limiting distribution for the partition angle Ew/E. Another possible extension refers to a more realistic law of friction, including Coulomb friction [39, 40]. Finally, it would be of interest to extend the analysis to other systems, where equipartition is violated. An example is a binary mixture, where the energy is shared unequally between the two components. Acknowledgments We thank the Kavli Institute for Theoretical Physics in University of California, Santa Barbara where this work was initiated. 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Flannery, Numerical Recipes, (Cambridge University Press, Cambridge, 1992). [60] S. E. Esipov and T. Pöschel, J. Stat. Phys. 86, 1385 (1997). [61] A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, J. Stat. Phys. 98, 743 (2000). [62] A. V. Bobylev and C. Cercignani, J. Stat. Phys. 106, 547 (2002). APPENDIX A: THE COLLISION RULES The total linear momentum v′a + v b = va + vb is conserved in the collision. The angular momenta of the two particles with respect to the point of contact, ω′i, are given by Iωa = Iwa +mR n̂× va (A1a) Iωb = Iwb −mR n̂× vb. (A1b) These are conserved, ω′i = ωi with i = a, b, because there is no torque at the point of contact. In inelastic collisions, the normal and tangential components of the relative velocity at the point of contact obey the collision law (4) where U = V +R n̂×W. It is convenient to introduce the momentum transfer δ, defined as follows: v′a = va−δ and v′b = vb+δ. Conservation of the angular velocity with respect to the point of contact and Eq. (A1) gives w′i = wi + n̂ × δ. In terms of δ, the difference in velocity at the point of contact is U′ = U− 2δ + 2 n̂× n̂× δ. Substituting this expression into the collision laws (4), the normal and the tangential components of δ are simply δ · n̂ = ηn U · n̂ (A2a) δ × n̂ = ηt U× n̂. (A2b) Consequently, the momentum transfer is δ = ηnU · n̂ n̂+ ηt(U−U · n̂ n̂) or explicitly, δ = ηn V · n̂ n̂+ ηt (V −V · n̂ n̂) + ηt R n̂×W (A3) We now have the explicit collision rules (5). APPENDIX B: PARTICLE NUMBER CONSERVATION In this appendix, we verify that the stationary solution is consistent with particle number conservation. Maxwell Molecules are considered for simplicity. It is straightforward to generalize this calculation to all γ and to free cooling. Our starting point is Eq. (21), specialized to Maxwell molecules, i.e. γ = 0, ∂P (E,Ω) δ(Ω− Ω1) + δ(Ω− Ω2)− P (E,Ω0)δ(Ω− Ω0) . (B1) As a first step we integrate this equation over the solid angle ∂p(E) − P (E,Ω0) . (B2) The power-law behavior (24) typically holds in a restricted energy range, El ≤ E ≤ Eu, where El and Eu are upper and lower cutoffs. In the driven case, the upper cutoff is set by the energy injection scale. Let N = dE p(E) be the total number of particles in this range. With the powerlaw decay (24), then N ∼ 1 ν − 1 E1−νl − E . (B3) To evaluate this time evolution of N , we substitute the product form (23) into (B2) and integrate over the energies in the aforementioned power-law range, dΩ0 g(Ω0) αν−11 + α 2 − 1 . (B4) Using Eq. (27), we confirm that the total number of particles is conserved, ∂N/∂t = 0. APPENDIX C: THE MATRIX COEFFICIENTS In an inelastic collision, the translational energy loss is ∆Ev = Ev − E′v with Ev = 12 (v a + v b ) and similarly, the rotational energy loss is ∆Ew = Ew −E′w with Ew = 12q(w b ). We can conveniently calculate these quantities by using v′a = va − δ, v′b = vb + δ, and w′i = wi + (1/qR)n̂× δ, and by expressing the momentum transfer δ using the natural coordinate system, δ = ηn Vnn̂+ ηt(Vt −W )t̂, ∆Ev = ηn(1− ηn)V 2n + ηt(1− ηt)V 2t − η2tW 2 + ηt(2ηt − 1)VtW (C1a) ∆Ew = −(η2t /q)V 2t + ηt(1− ηt/q)W 2 − ηt(1− 2ηt/q)VtW. (C1b) The rate of change of the respective temperatures equals 1/2 the average of this quantities, d 〈∆Ev〉 and 〈∆Ew〉. This is seen by multiplying (33) by 12v 2 and by integrating over the velocity. The averaging is with respect to the probability distribution functions of the two colliding particles. The cross-term vanishes, 〈VtW 〉 = 0, by symmetry. Using 〈V 2n 〉 = 2〈v2n〉 = 〈v2〉 = 2Tv and 〈W 2〉 = 2〈w2〉 = 4Tw/q we obtain the matrix elements (35). Introduction The Nonlinear Kinetic Theory The Linear Kinetic Theory Driven Steady-states Simulation Methods The Distribution of Total Energy The Angle distribution Free Cooling The Temperatures The Energy Distribution Conclusions and Outlook Acknowledgments References The Collision rules Particle Number Conservation The matrix coefficients

0704.1521

Cosmological Shock Waves in the Large Scale Structure of the Universe: Non-gravitational Effects

Cosmological Shock Waves in the Large Scale Structure of the Universe: Non-gravitational Effects Hyesung Kang1, Dongsu Ryu2, Renyue Cen3, and J. P. Ostriker3 ABSTRACT Cosmological shock waves result from supersonic flow motions induced by hi- erarchical clustering of nonlinear structures in the universe. These shocks govern the nature of cosmic plasma through thermalization of gas and acceleration of nonthermal, cosmic-ray (CR) particles. We study the statistics and energetics of shocks formed in cosmological simulations of a concordance ΛCDM universe, with a special emphasis on the effects of non-gravitational processes such as radiative cooling, photoionization/heating, and galactic superwind feedbacks. Adopting an improved model for gas thermalization and CR acceleration efficiencies based on nonlinear diffusive shock acceleration calculations, we then estimate the gas thermal energy and the CR energy dissipated at shocks through the history of the universe. Since shocks can serve as sites for generation of vorticity, we also examine the vorticity that should have been generated mostly at curved shocks in cosmological simulations. We find that the dynamics and energetics of shocks are governed primarily by the gravity of matter, so other non-gravitational processes do not affect significantly the global energy dissipation and vorticity generation at cosmological shocks. Our results reinforce scenarios in which the intraclus- ter medium and warm-hot intergalactic medium contain energetically significant populations of nonthermal particles and turbulent flow motions. Subject headings: cosmic rays – large-scale structure of universe – methods: nu- merical – shock waves – turbulence 1Department of Earth Sciences, Pusan National University, Pusan 609-735, Korea: [email protected] 2Department of Astronomy & Space Science, Chungnam National University, Daejeon 305-764, Korea: [email protected] 3Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544-1001, USA: [email protected], [email protected] http://arxiv.org/abs/0704.1521v1 – 2 – 1. Introduction Astrophysical plasmas consist of both thermal particles and nonthermal, cosmic-ray (CR) particles that are closely coupled with permeating magnetic fields and underlying turbulent flows. In the interstellar medium (ISM) of our Galaxy, for example, an approx- imate energy equipartition among different components seems to have been established, i.e., εtherm ∼ εCR ∼ εB ∼ εturb ∼ 1 eV cm −3 (Longair 1994). Understanding the complex network of physical interactions among these components constitutes one of fundamental problems in astrophysics. There is substantial observational evidence for the presence of nonthermal particles and magnetic fields in the large scale structure of the universe. A fair fraction of X-ray clusters have been observed in diffuse radio synchrotron emission, indicating the presence of GeV CR electrons and µG fields in the intracluster medium (ICM) (Giovannini & Feretti 2000). Observations in EUV and hard X-ray have shown that some clusters possess excess radia- tion compared to what is expected from the hot, thermal X-ray emitting ICM, most likely produced by the inverse-Compton scattering of cosmic background radiation (CBR) pho- tons by CR electrons (Fusco-Femiano et al. 1999; Bowyer et al. 1999; Berghöfer et al. 2000). Assuming energy equipartition between CR electrons and magnetic fields, εCRe ∼ εB ∼ 0.01−0.1eV cm−3 ∼ 10−3−10−2εtherm can be inferred in typical radio halos (Govoni & Feretti 2004). If some of those CR electrons have been energized at shocks and/or by turbulence, the same process should have produced a greater CR proton population. Considering the ratio of proton to electron numbers, K ∼ 100, for Galactic CRs (Beck & Kraus 2005), one can expect εCRp ∼ 0.01 − 0.1εtherm in radio halos. However, CR protons in the ICM have yet to be confirmed by the observation of γ-ray photons produced by inelastic collisions between CR protons and thermal protons (Reimer et al. 2003). Magnetic fields have been also directly observed with Faraday rotation measure (RM). In clusters of galaxies strong fields of a few µG strength extending from core to 500 kpc or further were inferred from RM observations (Clarke et al. 2001; Clarke 2004). An upper limit of . µG was imposed on the magnetic field strength in filaments and sheets, based the observed limit of the RMs of quasars outside clusters (Kronberg 1994; Ryu et al. 1998). Studies on turbulence and turbulent magnetic fields in the large scale structure of the universe have been recently launched too. XMM-Newton X-ray observations of the Coma cluster, which seems to be in a post-merger stage, were analyzed in details to extract clues on turbulence in the ICM (Schuecker et al. 2004). By analyzing pressure fluctuations, it was shown that the turbulence is likely subsonic and consistent with Kolmogoroff turbulence. RM maps of clusters have been analyzed to find the power spectrum of turbulent magnetic fields in a few clusters (Murgia et al. 2004; Vogt & Enßlin 2005). While Murgia et al. (2004) – 3 – reported a spectrum shallower than the Kolmogoroff spectrum, Vogt & Enßlin (2005) argued that the spectrum could be consistent with the Kolmogoroff spectrum if it is bended at a few kpc scale. These studies suggest that as in the ISM, turbulence does exist in the ICM and may constitute an energetically non-negligible component. In galaxy cluster environments there are several possible sources of CRs, magnetic fields, and turbulence: jets from active galaxies (Kronberg et al. 2004; Li et al. 2006), ter- mination shocks of galactic winds driven by supernova explosions (Vök & Atoyan 1999), merger shocks (Sarazin 1999; Gabici & Blasi 2003; Fujita et al. 2003), structure formation shocks (Loeb & Waxmann 2000; Miniati et al. 2001a,b), and motions of subcluster clumps and galaxies (Subramanian et al. 2006). All of them have a potential to inject a similar amount of energies, i.e., E ∼ 1061 − 1062 ergs into the ICM. Here we focus on shock scenar- Astrophysical shocks are collisionless shocks that form in tenuous cosmic plasmas via col- lective electromagnetic interactions between gas particles and magnetic fields. They play key roles in governing the nature of cosmic plasmas: i.e., 1) shocks convert a part of the kinetic energy of bulk flow motions into thermal energy, 2) shocks accelerate CRs by diffusive shock acceleration (DSA) (Blandford & Ostriker 1978; Blandford & Eichler 1987; Malkov & Drury 2001), and amplify magnetic fields by streaming CRs (Bell 1978; Lucek & Bell 2000), 3) shocks generate magnetic fields via the Biermann battery mechanism (Biermann 1950; Kulsrud et al. 1997) and the Weibel instability (Weibel 1959; Medvedev et al. 2006), and 4) curved shocks generate vorticity and ensuing turbulent flows (Binney 1974; Davies & Widrow 2000). In Ryu et al. (2003) (Paper I), the properties of cosmological shock waves in the inter- galactic medium (IGM) and the energy dissipations into thermal and nonthermal components at those shocks were studied in a high-resolution, adiabatic (non-radiative), hydrodynamic simulation of a ΛCDM universe. They found that internal shocks with low Mach numbers of M . 4, which formed in the hot, previously shocked gas inside nonlinear structures, are responsible for most of the shock energy dissipation. Adopting a nonlinear DSA model for CR protons, it was shown that about 1/2 of the gas thermal energy dissipated at cos- mological shocks through the history of the universe could be stored as CRs. In a recent study, Pfrommer et al. (2006) identified shocks and analyzed the statistics in smoothed par- ticle hydrodynamic (SPH) simulations of a ΛCDM universe, and found that their results are in good agreement with those of Paper I. While internal shocks with lower Mach num- bers are energetically dominant, external accretions shocks with higher Mach numbers can serve as possible acceleration sites for high energy cosmic rays (Kang et al. 1996, 1997; Ostrowski & Siemieniec-Ozieblo 2002). It was shown that CR ions could be accelerated up – 4 – to ∼ Z × 1019eV at cosmological shocks, where Z is the charge of ions (Inoue et al. 2007). Ryu et al. (2007) (Paper II) analyzed the distribution of vorticity, which should have been generated mostly at cosmological shock waves, in the same simulation of a ΛCDM universe as in Paper I, and studied its implication on turbulence and turbulence dynamo. Inside nonlinear structures, vorticity was found to be large enough that the turn-over time, which is defined as the inverse of vorticity, is shorter than the age of the universe. Based on it Ryu et al. (2007) argued that turbulence should have been developed in those structures and estimated the strength of the magnetic field grown by the turbulence. In this paper, we study cosmological shock waves in a new set of hydrodynamic sim- ulations of large structure formation in a concordance ΛCDM universe: an adiabatic (non- radiative) simulation which is similar to that considered in Paper I, and two additional simulations which include various non-gravitational processes (see the next section for de- tails). As in Papers I and II, the properties of cosmological shock waves are analyzed, the energy dissipations to gas thermal energy and CR energy are evaluated, and the vorticity distribution is analyzed. We then compare the results for the three simulations to highlight the effects of non-gravitational processes on the properties of shocks and their roles on the cosmic plasmas in the large scale structure of the universe. Simulations are described in §2. The main results of shock identification and properties, energy dissipations, and vorticity distribution are described in §3, §4, and §5, respectively. Summary and discussion are followed in §6. 2. Simulations The results reported here are based on the simulations previously presented in Cen & Ostriker (2006). The simulations included radiative processes of heating/cooling, and the two sim- ulations with and without galactic superwind (GSW) feedbacks were compared in that pa- per. Here an additional adiabatic (non-radiative) simulation with otherwise the same setup was performed. Hereafter these three simulations are referred as “Adiabatic”, “NO GSW”, and “GSW” simulations, respectively. Specifically, the WMAP1-normalized ΛCDM cosmol- ogy was employed with the following parameters: Ωb = 0.048, Ωm = 0.31, ΩΛ = 0.69, h ≡ H0/(100 km/s/Mpc) = 0.69, σ8 = 0.89, and n = 0.97. A cubic box of comoving size 85 h−1Mpc was simulated using 10243 grid zones for gas and gravity and 5123 particles for dark matter. It allows a uniform spatial resolution of ∆l = 83 h−1kpc. In Papers I and II, an adiabatic simulation in a cubic box of comoving size 100 h−1Mpc with 10243 grid zones and 5123 particles, employing slightly different cosmological parameters, was used. The – 5 – simulations were performed using a PM/Eulerian hydrodynamic cosmology code (Ryu et al. 1993). Detailed descriptions for input physical ingredients such as non-equilibrium ioniza- tion/cooling, photoionization/heating, star formation, and feedback processes can be found in earlier papers (Cen et al. 2003; Cen & Ostriker 2006). Feedbacks from star formation were treated in three forms: ionizing UV photons, GSWs, and metal enrichment. GSWs were meant to represent cumulative supernova explosions, and modeled as outflows of sev- eral hundred km s−1. The input of GSW energy for a given amount of star formation was determined by matching the outflow velocities computed for star-burst galaxies in the sim- ulation with those observed in the real world (Pettini et al. 2002)(see also Cen & Ostriker 2006, for details). Figure 1 shows the gas mass distribution in the gas density-temperature plane, fm(ρgas, T ), and the gas mass fraction as a function of gas temperature, fm(T ), at z = 0 for the three simulations. The distributions are quite different, depending primarily on the inclusion of radiative cooling and photoionization/heating. GSW feedbacks increase the fraction of the WHIM with 105 < T < 107K, and at the same time affect the distribution of the warm/diffuse gas with T < 105. 3. Properties of Cosmological Shock Waves We start to describe cosmological shocks by briefing the procedure by which the shocks were identified in simulation data. The details can be found in Paper I. A zone was tagged as a shock zone currently experiencing shock dissipation, whenever the following three criteria are met: 1) the gradients of gas temperature and entropy have the same sign, 2) the local flow is converging with ~∇ · ~v < 0, and 3) |∆ log T | ≥ 0.11 corresponding to the temperature jump of a shock with M ≥ 1.3. Typically a shock is represented by a jump spread over 2 − 3 tagged zones. Hence, a shock center was identified within the tagged zones, where ~∇ ·~v is minimum, and this center was labeled as part of a shock surface. The Mach number of the shock center, M , was calculated from the temperature jump across the entire shock zones. Finally to avoid confusion from complex flow patterns and shock surface topologies associated with very weak shocks, only those portions of shock surfaces with M ≥ 1.5 were kept and used for the analysis of shocks properties. Figure 2 shows the locations of identified shocks in a two-dimensional slice at z = 0 in the GSW simulation. The locations are color-coded according to shock speed. As shown before in Paper I, external accretion shocks encompass nonlinear structures and reveal, in addition – 6 – to cluster complexes, rich topology of filamentary and sheet-like structures in the large scale structure. Inside the nonlinear structures, there exist complex networks of internal shocks that form by infall of previously shocked gas to filaments and knots and during subclump mergers, as well as by chaotic flow motions. The shock heated gas around clusters extends out to ∼ 5 h−1Mpc, much further out than the observed X-ray emitting volume. In the GSW simulation, with several hundred km s−1 for outflows, the GSW feedbacks affected most greatly the gas around groups of galaxies, while the impact on clusters with kT & 1 keV was minimal. In Figure 3 we compare shock locations in a region around two groups with kT ∼ 0.2 − 0.3 keV in the three simulations. It demonstrates that GSW feedbacks pushed the hot gas out of groups with typical velocities of ∼ 100 km s−1 (green points). In fact the prominent green balloons of shock surfaces around groups in Figure 2 are due to GSW feedbacks (see also Figure 4 of Cen & Ostriker 2006). In the left panels of Figure 4 we compare the surface area of identified shocks, normalized by the volume of the simulation box, per logarithmic Mach number interval, dS(M)/d logM (top), and per logarithmic shock speed interval, dS(Vs)/d log Vs (bottom), at z = 0 in the three simulations. Here S and Vs are given in units of (h −1Mpc)−1 and km s−1. The quantity S provides a measure of shock frequency or the inverse of the mean comoving distance between shock surfaces. The distributions of dS(M)/d logM for the NO GSW and GSW simulations are similar, while that for the Adiabatic simulation is different from the other two. This is mainly because the gas temperature outside nonlinear structures is lower without photoionization/heating in the Adiabatic simulation. As a result, external accretion shocks tend to have higher Mach number due to colder preshock gas. The distribution of dS(Vs)/d log Vs, on the other hand, is similar for all three simulations for Vs > 15 km s −1. For Vs < 15 km s −1, however, there are more shocks in the Adiabatic simulation (black points in Figure 3). Again this is because in the Adiabatic simulation the gas temperature is colder in void regions, and so even shocks with low speeds of Vs < 15 km s −1 were identified in these regions. The GSW simulation shows slightly more shocks than the NO GSW simulation around Vs ∼ 100 km s −1, because GSW feedbacks created balloon-shaped surfaces of shocks with typically those speeds (green points in Figure 3). For identified shocks, we calculated the incident shock kinetic energy flux, Fφ = (1/2)ρ1V where ρ1 is the preshock gas density. We then calculated the kinetic energy flux through shock surfaces, normalized by the volume of the simulation box, per logarithmic Mach num- ber interval, dFφ(M)/d logM , and per logarithmic shock speed interval, dFφ(Vs)/d logVs. In the right panels of Figure 4, we compare the flux at z = 0 in the three simulations. Once again, there are noticeable differences in dFφ(M)/d logM between the Adiabatic simulation and the other two simulations, which can be interpreted as the result of ignoring photoion- – 7 – ization/heating in the gas outside nonlinear structures in the Adiabatic simulation. GSW feedbacks enhance only slightly the shock kinetic energy flux for Vs ∼ 100−300 km s −1, as can be seen in the plot of dFφ(Vs)/d logVs. Yet, the total amount of the energy flux is expected to be quite similar for all three simulations. This implies that the overall energy dissipation at cosmological shocks is governed mainly by the gravity of matter, and that the inclusion of various non-gravitational processes such as radiative cooling, photoionization/heating, and GSW feedbacks have rather minor, local effects. We note that a temperature floor of Tfloor = TCBR was used for the three simulations in this work, while Tfloor = 10 4 K was set in paper I. It was because in Paper I only an adiabatic simulation was considered and the 104 K temperature floor was enforced to mimic the effect of photoionization/heating on the IGM. However we found that when the same temperature floor is enforced, the statistics of the current Adiabatic simulation agree excellently with those of Paper I. Specifically, the shock frequency and kinetic energy flux, dS(M)/d logM and dFφ(M)/d logM , for weak shocks with 1.5 ≤ M . 3 are a bit higher in the current Adiabatic simulation, because of higher spatial resolution. But the total kinetic energy flux through shock surfaces, Fφ(M > 1.5), agrees within a few percent. On the other hand, In Paper I we were able to reasonably distinguish external and internal shocks according to the preshock temperature, i.e., external shocks if T1 ≤ Tfloor and internal shocks if T1 > Tfloor. We no longer made such distinction in this work, since the preshock temperature alone cannot tell us whether the preshock gas is inside nonlinear structures or not in the simulations with radiative cooling. 4. Energy dissipation by Cosmological Shock Waves The CR injection and acceleration rates at shocks depend in general upon the shock Mach number, field obliquity angle, and the strength of the Alfvén turbulence responsible for scattering. At quasi-parallel shocks, in which the mean magnetic field is parallel to the shock normal direction, small anisotropy in the particle velocity distribution in the local fluid frame causes some particles in the high energy tail of the Maxwellian distribution to stream upstream (Giacalone et al. 1992). The streaming motions of the high energy particles against the background fluid generate strong MHD Alfvén waves upstream of the shock, which in turn scatter particles and amplify magnetic fields (Bell 1978; Lucek & Bell 2000). The scattered particles can then be accelerated further to higher energies via Fermi first order process (Malkov & Drury 2001). These processes, i.e., leakage of suprathermal particles into CRs, self-excitation of Alfvén waves, amplification of magnetic fields, and further acceleration of CRs, are all integral parts of collisionless shock formation in astrophysical plasmas. It – 8 – was shown that at strong quasi-parallel shocks, 10−4− 10−3 of the incoming particles can be injected into the CR population, up to 60% of the shock kinetic energy can be transferred into CR ions, and at the same time substantial nonlinear feedbacks are exerted to the underlying flow (Berezhko et al. 1995; Kang & Jones 2005). At perpendicular shocks with weakly perturbed magnetic fields, on the other hand, par- ticles gain energy mainly by drifting along the shock surface in the ~v× ~B electric field. Such drift acceleration can be much more efficient than the acceleration at parallel shocks (Jokipii 1987; Kang et al. 1997; Ostrowski & Siemieniec-Ozieblo 2002). But the particle injection into the acceleration process is expected to be inefficient at perpendicular shocks, since the transport of particles normal to the average field direction is suppressed (Ellison et al. 1995). However, Giacalone (2005) showed that the injection problem at perpendicular shocks can be alleviated substantially in the presence of fully turbulent fields owing to field line meandering. As in Paper I, the gas thermalization and CR acceleration efficiencies are defined as δ(M) ≡ Fth/Fφ and η(M) ≡ FCR/Fφ, respectively, where Fth is the thermal energy flux generated and FCR is the CR energy flux accelerated at shocks. We note that for gasdy- namical shocks without CRs, the gas thermalization efficiency can be calculated from the Rankine-Hugoniot jump condition, as follows: δ0(M) = eth,2 − eth,1 , (1) where the subscripts 1 and 2 stand for preshock and postshock regions, respectively. The second term inside the brackets subtracts the effect of adiabatic compression occurred at a shock too, not just the thermal energy flux entering the shock, namely, eth,1v1. At CR modified shocks, however, the gas thermalization efficiency can be much smaller than δ0(M) for strong shocks with large M , since a significant fraction of the shock kinetic energy can be transferred to CRs. The gas thermalization and CR acceleration efficiencies were estimated using the results of DSA simulations of quasi-parallel shocks with Bohm diffusion coefficient, self-consistent treatments of thermal leakage injection, and Alfvén wave propagation (Kang & Jones 2007). The simulations were started with purely gasdynamical shocks in one-dimensional, plane-parallel geometry, and CR acceleration was followed by solving the diffusion-convection equation explicitly with very high resolution. Shocks with Vs = 150 − 4500 km s −1 propagating into media of T1 = 10 4 − 106 K were considered. After a quick initial adjustment, the postshock states reach time asymptotic values and the CR modified shocks evolve in an approximately self-similar way with the shock structure broadening linearly with time (refer Kang & Jones 2007, for details). Given this self-similar nature of CR modified shocks, we calculated time asymptotic values of δ(M) and η(M) as the ratios of increases in the gas thermal and CR energies at shocks to the kinetic energy – 9 – passed through the shocks at the termination time of the DSA simulations. As in Eq. (1), the increase of energies due to adiabatic compression was subtracted. Figure 5 shows δ(M) and η(M) estimated from DSA simulations and their fittings for the cases with and without a preexisting CR component. The fitting formulae are given in Appendix A. Without a preexisting CR component, gas thermalization is more efficient than CR acceleration at shocks with M . 5. However, it is likely that weak internal shocks propagate through the IGM that contains CRs accelerated previously at earlier shocks. In that case, shocks with preexisting CRs need to be considered. Since the presence of preexisting CRs is equivalent to a higher injection rate, CR acceleration is more efficient in that case, especially at shocks with M . 5 (Kang & Jones 2003). In the bottom panel the efficiencies for shocks with PCR/Pg ∼ 0.3 in the preshock region are shown. For comparison, δ0(M) for shocks without CRs is also drawn. Both δ(M) and η(M) increase with Mach number, but η(M) asymptotes to ∼ 0.55 while δ(M) to ∼ 0.30 for strong shocks with M & 30. So about twice more energy goes into CRs, compared to for gas heating, at strong shocks. The efficiencies for the case without a preexisting CR component in the upper panel of Figure 5 can be directly compared with the same quantities presented in Figure 6 of Paper I. In Paper I, however, the gas thermalization efficiency was not calculated explicitly from DSA simulations, and hence δ0(M) for gasdynamic shocks was used. It represents gas thermalization reasonably well for weak shocks with M . 2.5, but overestimates gas thermalization for stronger CR modified shocks. Our new estimate for η(M) is close to that in Paper I, but a bit smaller, especially for shocks with M . 30. This is because inclusion of Alfvén wave drift and dissipation in the shock precursor reduces the effective velocity change experienced by CRs in the new DSA simulations of Kang & Jones (2007). A note of caution for η(M) should be in order. As outlined above, CR injection is less efficient and so the CR acceleration efficiency would be lower at perpendicular shocks, compared to at quasi-parallel shocks. CR injection and acceleration at oblique shocks are not well understood quantitatively. And the magnetic field directions at cosmological shocks are not known. Considering these and other uncertainties involved in the adopted DSA model, we did not attempt to make further improvements in estimating δ(M) and η(M) at general oblique shocks. But we expect that an estimate at realistic shocks with chaotic magnetic fields and random shock obliquity angles would give reduced values, rather than increased values, for η(M). So η(M) given in Figure 5 may be regarded as upper limits. By adopting the efficiencies in Figures 5, we calculated the thermal and CR energy fluxes dissipated at cosmological shocks, dFth(M)/d logM , dFth(Vs)/d logVs, dFCR(M)/d logM and dFCR(Vs)/d log Vs, using Fth = Fφδ(M) and FCR = Fφη(M), in the same way we – 10 – calculated dFφ(M)/d logM and dFφ(Vs)/d logVs in the previous section. We then integrated from z = 5 to z = 0 the shock kinetic energy passed and the thermal and CR energies dissipated through shock surfaces as follows: dYi(X) d logX Eth,0 ∫ z=0 dFi[X, z(t)] d logX dt, (2) where the subscript i ≡ φ, th, or CR stands for the kinetic, thermal, or CR energies fluxes, the variable X is either M or Vs, and Eth,0 is the total gas thermal energy at z = 0 inside the simulation box normalized by its volume. Figure 6 shows the resulting dYi(M)/d logM and dYi(Vs)/d logVs and their cumulative distributions, Yi(> M) and Yi(> Vs), for the GSW simulation. Weak shocks with M . 4 or fast shocks with Vs & 500 km s −1 are responsible most for shock dissipations, as already noted in Paper I. While the thermal energy generation peaks at shocks in the range 1.5 . M . 3, the CR energy peaks in the range 2, 5 . M . 4 if no preexisting CRs are included or in the range 1.5 . M . 3 if preexisting CRs of PCR/Pg ∼ 0.3 in the preshock region are included. With our adopted efficiencies, the total CR energy accelerated and the total gas thermal energy dissipated at cosmological shocks throughout the history of the universe are compared as YCR(M ≥ 1.5) ∼ 0.5Yth(M ≥ 1.5), when no preexisting CRs are present. With preexisting CRs in the preshock region, the CR acceleration becomes more efficient, so YCR(M ≥ 1.5) ∼ 1.7Yth(M ≥ 1.5), i.e., the total CR energy accelerated at cosmological shocks is estimated to be 1.7 times the total gas thermal energy dissipated. We note here again that these are not meant to be very accurate estimates of the CR energy in the IGM, considering the difficulty of modeling shocks as well as the uncertainties in the DSA model itself. However, they imply that the IGM and the WHIM, which are bounded by strong external shocks with high M and filled with weak internal shocks with low M , could contain a dynamically significant CR population. 5. Vorticity Generation at Cosmological Shock Waves Cosmological shocks formed in the large scale structure of the universe are by nature curved shocks, accompanying complex, often chaotic flow patterns. It is well known that vor- ticity, ~ω = ∇×~v, is generated at such curved oblique shocks (Binney 1974; Davies & Widrow 2000). In Paper II, the generation of vorticity behind cosmological shocks and turbulence dynamo of magnetic fields in the IGM were studied in an adiabatic ΛCDM simulation. In this study we analyzed the distribution of vorticity in the three simulations to assess quanti- tatively the effects of non-gravitational processes. Here we present the magnitude of vorticity – 11 – with the vorticity parameter τ(~r, z) ≡ tage(z)ω(~r, z) = tage(z) teddy(~r, z) , (3) where tage(z) is the age of the universe at redshift z. With teddy = 1/ω interpreted as local eddy turnover time, τ represents the number of local eddy turnovers in the age of the universe. So if τ ≫ 1, we expect that turbulence has been fully developed after many turnovers. Figure 7 shows fluid quantities and shock locations in a two-dimensional slice of (21.25 h−1Mpc)2, delineated by a solid box in Figure 2, at z = 0 in the GSW simulations. The region contains two clusters with kT ∼ 1− 2 keV in the process of merging. Bottom right panel shows that vorticity increases sharply at shocks. The postshock gas has a larger amount of vorticity than the preshock gas, indicating that most, if not all, of the vorticity in the simulation was produced at shocks. Figure 8 shows the gas mass distribution in the gas density-vorticity parameter plane, fm(ρgas, τ), (upper panel) and the gas mass fraction per logarithmic τ interval, dfm(τ)/d log τ , (bottom panel) for the three simulations. The most noticeable point in the upper panel is that vorticity is higher at the highest density regions with ρ̃ ≡ ρgas/〈ρgas〉 & 10 3 in the NO GSW and GSW simulations than in the Adiabatic simulation. This is due to the additional flow motions induced by cooling. Inclusion of GSW feedbacks, on the other hand, does not alter significantly the overall distribution in the gas density-vorticity parameter plane. The bottom panel indicates that cooling increased the mass fraction with large vorticity τ & 10, while reduced the mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the mass fraction with 1 . τ . 10, which corresponds to the gas in the regions outskirts of groups that expand further out due to GSWs (i.e., balloons around groups). But overall we conclude that the non-gravitational processes considered in this paper have limited effects on vorticity in the large scale structure of the universe. We note that the highest density regions in the NO GSW and GSW simulations have τ ∼ 30 on average. As described in details in Paper II, such values of τ imply that local eddies have turned over many times in the age of the universe, so that the ICM gas there has had enough time to develop magnetohydrodynamic (MHD) turbulence. So in those regions, magnetic fields should have grown to have the energy approaching to the turbulent energy. On the other hand, the gas with 1 . ρ̃ . 103, mostly in filamentary and sheet-like structures, has 0.1 . τ . 10. MHD turbulence should not have been fully developed there and turbulence growth of magnetic fields would be small. Finally in the low density void regions with ρ̃ . 1, vorticity is negligible with τ . 0.1 on average, as expected. – 12 – 6. Summary We identified cosmological shock waves and studied their roles on cosmic plasmas in three cosmological N-body/hydrodynamic simulations for a concordance ΛCDM universe in a cubic box of comoving size 85 h−1Mpc: 1) adiabatic simulation (Adiabatic), 2) simulation with radiative cooling and photoionization/heating (NO GSW), and 3) same as the second simulation but also with galactic superwind feedbacks (GSW). The statistics and energetics of shocks in the adiabatic simulation are in an excellent agreement with those of Paper I where an adiabatic simulation with slightly different cosmological parameters in a cubic box of comoving size 100 h−1Mpc was analyzed. Photoionization/heating raised the gas temperature outside nonlinear structures in the NO GSW and GSW simulations. As a result, the number of identified shocks and their Mach numbers in the NO GSW and GSW simulations were different from those in the Adi- abatic simulation. GSW feedbacks pushed out gas most noticeably around groups, creating balloon-shaped surfaces of shocks with speed Vs ∼ 100 km s −1 in the GSW simulation. How- ever, those have minor effects on shock energetics. The total kinetic energy passed through shock surfaces throughout the history of the universe is very similar for all three simula- tions. So we conclude that the energetics of cosmological shocks was governed mostly by the gravity of matter, and the effects non-gravitational processes, such as radiative cooling, photoionization/heating, and GSW feedbacks, were rather minor and local. We estimated both the improved gas thermalization efficiency, δ(M), and CR acceler- ation efficiency, η(M), as a function shock Mach number, from nonlinear diffusive shock simulations for quasi-parallel shocks that assumed Bohm diffusion for CR protons and in- corporated self-consistent treatments of thermal leakage injection and Alfvén wave propa- gation (Kang & Jones 2007). The cases without and with a preexisting CR component of PCR/Pg ∼ 0.3 in the preshock region were considered. At strong shocks, both the injection and acceleration of CRs are very efficient, and so the presence of a preexisting CR component is not important. At shocks with with M & 30, about 55 % of the shock kinetic energy goes into CRs, while about 30 % becomes the thermal energy. At weak shocks, on the other hand, without a preexisting CR component, the gas thermalization is more efficient than the CR acceleration. But the presence of a preexisting CR component is critical at weak shocks, since it is equivalent to a higher injection rate and the CR acceleration becomes more effi- cient with it. As a result, η(M) is higher than δ(M) even at shocks with M . 5. However, at perpendicular shocks, the CR injection is suppressed, and so the CR acceleration could be less efficient than at parallel shocks. Thus our CR shock acceleration efficiency should be regarded as an upper limit. With the adopted efficiencies, the total CR energy accelerated at cosmological shocks – 13 – throughout the history of the universe is estimated to be YCR(M ≥ 1.5) ∼ 0.5 Yth(M ≥ 1.5), i.e., 1/2 of the total gas thermal energy dissipated, when no preexisting CRs are present. With a preexisting CR component of PCR/Pg ∼ 0.3 in the preshock region, YCR(M ≥ 1.5) ∼ 1.7 Yth(M ≥ 1.5), i.e., the total CR energy accelerated is estimate to be 1.7 times the total gas thermal energy dissipated. Although these are not meant to be very accurate estimates of the CR energy in the ICM, they imply that the ICM could contain a dynamically significant CR population. We also examined the distribution of vorticity inside the simulation box, which should have been generated mostly at curved cosmological shocks. In the ICM, the eddy turn-over time, teddy = 1/ω, is about 1/30 of the age of the universe, i.e., τ ≡ tage/teddy ∼ 30. In filamentary and sheet-like structures, τ ∼ 0.1− 10, while τ . 0.1 in void regions. Radiative cooling increased the fraction of gas mass with large vorticity τ & 10, while reduced the mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the mass fraction with 1 . τ . 10. Although the effects of these non-gravitation effects are not negligible, the overall distribution of vorticity are similar for the three simulations. So we conclude that the non-gravitational processes considered in this paper do not affect significantly the vorticity in the large scale structure of the universe. HK was supported in part by KOSEF through Astrophysical Research Center for the Structure and Evolution of Cosmos (ARCSEC). DR was supported in part by a Korea Re- search Foundation grant (KRF-2004-015-C00213). RC was supported in part by NASA grant NNG05GK10G and NSF grant AST-0507521. The work of HK and DR was also supported in part by Korea Foundation for International Cooperation of Science & Technology (KICOS) through the Cavendish-KAIST Research Cooperation Center. A. Fitting Formulae for δ(M) and η(M) The gas thermalization efficiency, δ(M), and the CR acceleration efficiency, η(M), for the case without a preexisting CR component (in upper panel of Figure 5) are fitted as follows: for M ≤ 2 δ(M) = 0.92 δ0 (A1) η(M) = 1.96× 10−3(M2 − 1) (A2) for M > 2 δ(M) = (M − 1)n – 14 – a0 = −4.25, a1 = 6.42, a2 = −1.34, a3 = 1.26, a4 = 0.275 (A4) η(M) = (M − 1)n b0 = 5.46, b1 = −9.78, b2 = 4.17, b3 = −0.334, b4 = 0.570 (A6) The efficiencies for the case with a preexisting CR component (in bottom panel of Figure 5) are fitted as follows: for M ≤ 1.5 δ(M) = 0.90 δ0 (A7) η(M) = 1.025 δ0 (A8) for M > 1.5 δ(M) = (M − 1)n a0 = −0.287, a1 = 0.837, a2 = −0.0467, a3 = 0.713, a4 = 0.289 (A10) η(M) = (M − 1)n (A11) b0 = 0.240, b1 = −1.56, b2 = 2.80, b3 = 0.512, b4 = 0.557 (A12) Here δ0(M) is the gas thermalization efficiency at shocks without CRs, which was cal- culated from the Rankine-Hugoniot jump condition, (black solid line in Figure 5): δ0(M) = γ(γ − 1)M2R 2γM2 − (γ − 1) (γ + 1) (A13) γ + 1 γ − 1 + 2/M2 (A14) REFERENCES Beck, R., & Kraus, M. 2005, Astronomische Nachrichten, 326, 414 Bell A.R. 1978, MNRAS, 182, 147 Biermann, L. 1950, Z. 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A., & Jubelgas, M. 2006, MNRAS, 367, 113 Reimer, O., Pohl, M., Sreekumar, P., & Mattox, J. R. 2003, ApJ, 588, 155 Ryu, D., Kang, H., & Biermann, P. L. 1998, A&A, 335, 19 Ryu, D., Kang, H., Cho, J., & Das, S. 2007, in preparation (Paper II) Ryu, D., Kang, H., Hallman, E., & Jones, T. W. 2003, ApJ, 593, 599 (Paper I) Ryu, D., Ostriker, J. P., Kang, H., & Cen, R. 1993, ApJ, 414, 1 Sarazin, C. L. 1999, ApJ, 520, 529 – 17 – Schuecker, P., Finoguenov, A., Miniati, F., Böhringer, H., & Briel, U. G. 2004, A&A, 426, Subramanian, K., Shukurov, A., & Haugen, N. E. L. 2006, MNRAS, 366, 1437 Vogt, C. & Enßlin, T. 2005, A&A, 434, 67 Völk, H. J. & Atoyan, A. M. 1999, Astroparticle Phys., 11, 73 Weibel, E. S. 1959, Phys. Rev. Lett., 2. 83 This preprint was prepared with the AAS LATEX macros v5.2. – 18 – Fig. 1.— Top panels: Gas mass distribution in the gas density-temperature plane at z = 0 for the Adiabatic, NO GSW, and GSW simulations. Bottom panel: Gas mass fraction as a function of gas temperature at z = 0 for the three simulations. – 19 – Fig. 2.— Two-dimensional slice of (85 h−1Mpc)2 showing shock locations at z = 0 in the GSW simulation, which are color-coded according to shock speed as follows: black for Vs < 15 km s −1, blue for 15 ≤ Vs < 65 km s −1, green for 65 ≤ Vs < 250 km s −1, red for 250 ≤ Vs < 1000 km s −1, and magenta for Vs ≥ 1000 km s −1. A blown-up image of the box (dashed line) in the upper right corner is shown in Figure 3, while a blown-up image of the box (solid line) around two merging clusters is shown in Figure 7. – 20 – Fig. 3.— Two-dimensional slice of (21.25 h−1Mpc)2 showing shock locations at z = 0 in the Adiabatic, NO GSW and GSW simulations. The locations are color-coded according to shock speed. Two groups in the GSW simulation have kT ∼ 0.2− 0.3 keV. – 21 – Fig. 4.— Left panels: Inverse of the mean comoving distance between shock surfaces as a function of Mach number M (top) and shock speed Vs (bottom) at z = 0 for the Adiabatic (solid line), NO GSW (dashed line), and GSW (dotted line) simulations. Right panels: Kinetic energy flux per comoving volume passing through shock surfaces in units of 1040 ergs s−1 (h−1Mpc)−3 as a function of M (top) and Vs (bottom). Note that the bottom two panels have different ranges of abscissa. – 22 – Fig. 5.— Gas thermalization efficiency, δ(M), and CR acceleration efficiency, η(M), as a function of Mach number. Red and blue dots are the values estimated from numerical simulations based on a DSA model and red and blue lines are the fits. The top panel shows the case without preexisting CRs, while the bottom panel shows the case with preexisting CRs at a level of PCR/Pg ∼ 0.3 in the preshock region. Black solid line is for the gas thermalization efficiency for shocks without CRs. – 23 – 0 1 2 3 4 0 1 2 3 4 1.5 2 2.5 3 3.5 1.5 2 2.5 3 3.5 Fig. 6.— Left panels: Shock kinetic energy passed, dYφ (dotted line), thermal energy dis- sipated, dYth (dashed line), and CR energy dissipated, dYCR (solid line), through surfaces of cosmological shocks with Mach number between logM and logM + d(logM) (top) and through surfaces of cosmological shocks with shock speed between log Vs and log Vs+d(log Vs) (bottom), integrated from z = 5 to z = 0. Red and magenta lines are the CR energy for the cases without and with preexisting CRs, respectively. Blue and green lines are the thermal energy for the cases without and with preexisting CRs, respectively. The thermal energy expected to be dissipated at cosmological shocks without CRs (long dashed cyan line) is also plotted for comparison. Right panels: Cumulative energy distributions, Yi(> M) (top) and Yi(> Vs) (bottom), for Mach number greater than M and for shock speed greater than Vs. The energies are normalized by the gas thermal energy at z = 0 inside the simulation box – 24 – Fig. 7.— Two-dimensional slice of (21.25 h−1Mpc)2 around two merging clusters with kT ∼ 1−2 keV at z = 0 in the GSW simulation. Distributions of gas density (top left), temperature (top right), shock locations (bottom left), and vorticity (bottom right) are shown. In the gas density, temperature, and vorticity distributions, back, blue and red contours represent regions of low, middle, and high values, respectively. – 25 – Fig. 8.— Top panels: Gas mass distribution in the gas mass density-vorticity parameter plane at z = 0 for the Adiabatic, NO GSW, and GSW simulations. The vorticity parameter is defined as τ = ωtage(z), where ω = |~∇×~v| and tage(z) is the age of the universe at redshift z. Bottom panel: Gas mass fraction as a function of vorticity parameter at z = 0 for the three simulations. Introduction Simulations Properties of Cosmological Shock Waves Energy dissipation by Cosmological Shock Waves Vorticity Generation at Cosmological Shock Waves Summary Fitting Formulae for (M) and (M)

0704.1522

Outstanding Issues in Our Understanding of L, T, and Y Dwarfs

**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Outstanding Issues in Our Understanding of L, T, and Y Dwarfs J. Davy Kirkpatrick Infrared Processing and Analysis Center, California Institute of Technology Abstract. Since the discovery of the first L dwarf 19 years ago and the discovery of the first T dwarf 7 years after that, we have amassed a large list of these objects, now numbering almost six hundred. Despite making headway in understand- ing the physical chemistry of their atmospheres, some important issues remain unexplained. Three of these are the subject of this paper: (1) What is the role of “second parameters” such as gravity and metallicity in shaping the emergent spectra of L and T dwarfs? Can we establish a robust classification scheme so that objects with unusual values of log(g) or [M/H], unusual dust content, or unresolved binarity are easily recognized? (2) Which physical processes drive the unusual behavior at the L/T tran- sition? Which observations can be obtained to better confine the problem? (3) What will objects cooler than T8 look like? How will we know a Y dwarf when we first observe one? 1. Introduction In the last thirteen years, the sample of L and T dwarfs has grown from a paltry 1 (GD 165B, Becklin & Zuckerman 1988) to an astounding 586 at the time of this writing (see http://www.DwarfArchives.org). As the sample has grown, additional follow-up has further added to our knowledge of these objects. With this newfound knowledge major deficiencies have begun to show in our understanding of the physics shaping the spectra of L and T dwarfs. In this paper I discuss three important questions being asked by low mass star and brown dwarf researchers today. First, what role do “secondary” parameters such as gravity, metallicity, dust content, and unresolved binarity play in shaping emergent spectra? Second, why has the transition between L dwarfs and T dwarfs been so difficult to model? Third, what will objects cooler than type T8 look like? 2. Issue #1: Gravity, Metallicity, and Other “Secondary Parameter” Effects Astronomers have come to expect that spectral type can be used as a proxy for temperature. While this is true for main sequence stars classified from types O through M, is the same true for L and T dwarfs? Using measurements of trigonometric parallaxes (see Dahn et al. 2002; Tinney et al. 2003; Vrba et al. http://arxiv.org/abs/0704.1522v1 http://www.DwarfArchives.org 2004) and bolometric luminosities (see Reid et al. 1999; Golimowski et al. 2004), it can be shown that there is an amazingly linear correlation between optical spectral type and effective temperature for the entire L dwarf class. Effective temperature appears constant from early- through mid-T, then again shows a drop as a function of type for later T dwarfs. This is illustrated in the upper panel of Figure 1. Figure 1. Effective temperature plotted against spectral type. The top panel shows the relation between temperature and optical spectral type; the bottom panel shows temperature versus near-infrared type. See Kirkpatrick (2005) for details. The same correlation, however, is not seen for near-infrared L and T dwarf classifications. This is illustrated in the lower panel of Figure 1. Here the effective temperatures are essentially constant, albeit with a large scatter of ±200 K, from mid-L through mid-T types. This is somewhat surprising given that the near-infrared classification is loosely based on the optical classifica- tion (Reid et al. 2001; Testi et al. 2001; Geballe et al. 2002). However, these systems used only spectral indices for classification and not anchor points (pri- mary spectral standards) to serve as on-sky comparisons. This differs from the methodology used in the optical, where anchor points were selected and checked for self consistency across the entire wavelength region used for classification, not just over the areas used for measuring spectral indices (Kirkpatrick et al. 1999). In all fairness, the optical behavior of L dwarfs is much less complicated than the near-infrared behavior, as evidenced by the sometimes non-monotnic behavior of H- and K -band fluxes and shapes as a function of type (e.g., Fig. 3 of Geballe et al. 2002 and Fig. 10 of McLean et al. 2003), which is further reflected in the large spectral type error bars seen for some objects in the lower panel of Figure 1. Until very recently, setting up near-infrared L-dwarf spectral standards would have been difficult because of this scatter and the relatively sparse sampling of 1.0-2.5 µm spectra for each L dwarf subclass. What causes the large dispersion of morphologies seen in the near-infrared? As the optical spectra of L and T dwarfs do not show this degree of variation, we can use them to probe the influence of other parameters such as gravity, metallicity, and dust on the spectral shape at longer wavelengths. 2.1. Exploring the Effects Low Gravity The effects of lower gravity have been well studied in the opti- cal for late-M dwarfs. The earliest studies of gravity-dependent features were based on young objects in the Pleiades (Steele & Jameson 1995; Mart́ın et al. 1996) and ρ Ophiuchi (Luhman et al. 1997). At these ages (in the 1-100 Myr range) late-M dwarfs are substellar and are still contracting to their final radii; this means that they are both larger and less massive than old stars of the same effective temperature and hence will have lower surface gravity. Gravity- dependent features in the optical spectra of M dwarfs are well known as these are also the features used to distinguish luminosity classes (M dwarfs vs. M gi- ants) on the MK classification scheme. In the far red portion of the spectrum, lower gravity results in weaker alkali lines and weaker hydride bands. At later M types, TiO and VO bandstrengths are stronger at lower gravity as well, and this occurs because of the gravity (pressure) dependent nature of condensation (Lodders & Fegley 2002). The study of low-gravity effects in L dwarfs, on the other hand, has had a brief history because few young cluster brown dwarfs with L spectral types are known. Recent discoveries of field L dwarfs believed to be young, low-gravity brown dwarfs have, however, recently been reported (Kirkpatrick et al. 2006; Cruz et al. 2007; Cruz et al., this volume). With these optically typed, low-gravity late-M and L dwarfs, we can explore the effects of low gravity in the near-infrared. Figure 2 shows the 2MASS J−K colors for optically typed late-M through late-L dwarfs. Small dots with error bars are field dwarf fiducials, and objects with optical signatures of low gravity are denoted by triangles. Note that low-gravity objects tend to be redder in than normal dwarfs of the same type. An example of one low-gravity L0 in discussed in detail in Kirkpatrick et al. (2006). Features at J-band showing the hallmarks of low-gravity include retarded condensation of VO and weaker alkali lines. The redder J −K colors are believed to be due to a weakening of the collision-induced absorption by H2, which is a dominant opacity source at H− and particularly K −bands. Figure 2. J − K colors for a collection of M7-L8 dwarfs with optically determined spectral types. Each bin represents a full integral subtype: “M7” includes M7 and M7.5 dwarfs, “M8” includes M8 and M8.5 dwarfs, etc. For each group of objects, the median color in the group is plotted highest in the bin; colors falling farther from the median are plotted progressively farther down the y-axis. Grey circles show the color locations of objects optically classified as subdwarfs, and grey triangles show the color locations of objects whose optical spectra show the hallmarks of lower gravity. Low Metallicity The earliest studies illustrating effects of low metallicity on the spectra of ultra-cool dwarfs were by Gizis (1997), Schweitzer et al. (1999), and Lépine et al. (2003). Lower metallicity means that fewer metal+metal molecules will be formed relative to metal+hydrogen molecules. Thus in far optical spectra, absorption by metal hydrides will be increased relative to that of metallic oxides. How are the spectra of low-Z dwarfs affected at the transition from late-M to early-L? The hallmark of the M/L transition is the appearance of condensates, or rather the weakening of certain bands in the optical spectra as those molecules disappear into condensates. Metallicity is expected to play an important role in condensation (e.g., Lodders & Fegley 2002), with low-Z meaning that fewer heavier elements will be present to form condensates in the first place. Burgasser et al. (2007) have compiled a list of known subdwarfs and extreme subdwarfs of type M7 and later. (See Burgasser et al., this volume, for in-depth discussion of spectroscopic features.) Using optically classified objects from this list and new additions from Kirkpatrick et al. (in prep.) we find that they tend to have bluer J − K colors, in some cases extremely bluer colors, than normal late-M and L dwarfs of the same optical type (circles in Figure 2). This discrepancy is believed to be largely due to collision-induced absorption by H2, which is a far more dominant absorber at these wavelengths due to the decrease in abundance of metal species. Dust Content Despite the fact that low gravity tends to redden the near- infrared spectra and low metallicity tends to make them bluer, not all anoma- lously red near-infrared spectra appear to have low gravity nor do the anoma- lously blue ones appear to have low metallicity. An example of the first kind is 2MASS J22443167+2043433, whose color of J −K =2.45±0.16 makes it one of the reddest L dwarfs known. The optical spectrum of this object looks like that of a normal L6.5 dwarf, but the near- infrared spectrum shows enhanced flux at H- and K -bands and weak K I lines at J-band (McLean et al. (2003)). It is possible that this is a late-L analog to the low-gravity, early-L dwarfs discussed above. In this case the optical spectrum may be mimicking the spectrum of a normal slightly earlier L dwarf via weakening of the 7665/7699 Å K I resonance doublet, which is the primary shaper of the spectrum in the far red. However, it is also possible that thicker dust clouds and veiling could produce these same effects, although the physical explanation of the thicker dust relies on further undetermined physics (e.g., higher metallicity). An example of the second kind is 2MASS J17210390+3344160, whose J−K color of 1.14±0.03 is at least a half magnitude bluer than a normal, optically classified L3 dwarf. One possible explanation for this object is that it has lower metallicity than a standard field L dwarf, but not as low as the subdwarfs whose optical spectra clearly identify them as low-Z. In this case the near-infrared re- gion (because of collision induced absorption by H2) may be a more sensitive indicator of metallicity than any of the optical diagnostics. An alternate expla- nation is a reduction of condensates in the photosphere, leading to reduced flux at H- and K-bands relative to a normal L dwarf. This could be caused either by more efficient sedimentation (the physical process for which is unknown) or by reduced metallicity. Both scenarios are discussed at greater length in Cruz et al. (2007). For additional examples of blue L dwarfs and further discus- sion on possible physical scenarios, see Knapp et al. (2004), Chiu et al. (2006), and Cruz et al. (2007). Unresolved Binarity Another factor contributing to the larger scatter in near- infrared types may be unresolved binarity. Kirkpatrick et al. (in prep.) syn- thesize composite binaries using single dwarfs with types from late-M through late-T, and show that the optical spectral composites are virtually indistinguish- able from spectra of single objects. The largest deviations between the synthetic binaries and the spectra of single standards occur for late-L primaries with mid- to late-T secondaries. Looper et al. (in prep.) have done a similar analysis in the near-infrared. The largest deviations occur for the same set of hypothetical binaries, but the discrepancies are sometimes large enough to lead to peculiar looking spectra. See also Burgasser (2007) for other synthetic binary analysis in the near-infrared. 2.2. Disentangling these Effects through Classification One of our future challenges is finding spectral features that can reliably dis- tinguish between these four effects. Ideally we should establish standard near- infrared sequences of normal L dwarfs and optical/near-infrared sequences of L subdwarfs and low-gravity L dwarfs. L subdwarfs may be separable into sev- eral sequences to parallel overall metal content, but the construction of such sequences is hampered by a severe lack of current examples from which to draw standards. Also, nature should preclude L subdwarfs of lowest metallicity from currently existing in the Milky Way, as brown dwarfs of such low-Z (age≈10 Gyr) will have long ago cooled to type T or later. Figure 3 shows what a slice at L7 might look like for three of these near- infrared spectral sequences. The top spectrum shows an L7 that is anomalously red, due either to dust effects, low-gravity, or both. The bottom spectrum shows an L7 that is anomalously blue, probably due to low-metallicity effects. In the middle is a normal L7. The extreme nature of the variations is obvious. Inroads have already been made into formulating these new spectral se- quences and their classification schemes. Gorlova et al. (2003) and McGovern (2005) have obtained near-infrared spectra of young late-M dwarfs in clusters of various ages to quantify the effects of lower gravity at these temperatures. Allers et al. (2007) has devised a pair of indices at near-infrared wavelengths that can measure spectral type independent of gravity, and gravity independent of reddening, for late-M and early-L dwarfs. 3. Issue #2: The L/T Transition Trigonometric parallax measurements have revealed another effect not predicted a priori by the models. This is the ∼1.3 mag brightening of J-band seen between late-L and mid-T. This so called “J-band bump” is illustrated in Figure 4. High- resolution imaging studies of objects in the L/T transition region have revealed that the fraction of binaries with types between L7 and T3.5 is twice that seen at earlier or later types (Burgasser et al. 2006b). If all objects residing in the bump were binaries, then it might be possible to split the overluminosity of these joint systems between the components and substantially reduce the magnitude of the bump, as suggested by Liu et al. (2006) and Burgasser et al. (2006b). However, not all objects occupying the bump have been successfully split. The biggest offender, 2MASS J0559−1404 (see Figure 4), appears single to 0.′′05 with HST imaging (Burgasser et al. 2003c) and to 0.′′04 imaging with laser guide star adaptive optics at Keck (Gelino & Kulkarni 2005). Even if it eventually proves to be a tight, equal-magnitude binary, this reduces its absolute J-band magnitude by only 0.75 mag, meaning that the J-band bump still has an ampli- tude of 0.5-0.6 mag. On-going high-resolution radial velocity measurements as Figure 3. Overplot of three objects typed as ∼L7 in J-band. Shown are an unusually red L7 (top, light grey), a normal L7 (middle, dark grey), and a blue L7 (bottom, black). The spectra, from top to bottom, come from Looper (priv. comm.), Cruz (priv. comm.), and Burgasser et al. (2003a). The blue L7 has only partial wavelength coverage in this wavelength regime. All spectra have been normalized at 1.2 µm. well as on-going astrometric monitoring of 2MASS J0559−1404 will eventually reveal the truth about this object. Furthermore, some of the L/T transition binaries that have been split show the J-band brightness reversal themselves. Gizis et al. (2003) found the first sys- tem in which the secondary was brighter than the primary, although this observa- tion was done not at J-band but at the F1042W filter (∼ Y -band) of HST. Since then three other systems have been published (Liu et al. 2006; Burgasser et al. 2006b; Cruz et al. 2004). A fifth binary with brightness reversals in its compo- nents has been recently reported by Looper et al. (in prep.) and for this system the secondary is brighter than the primary by 0.5 mag at J . In summary, the “J-band bump” is real with an amplitude of at least half a mag. Regardless of the exact amplitude of the bump, we have come to learn that the relative lack of single objects at early-T is probably indicative of rapid evolution from late-L to mid-T (Burgasser et al. 2006b). The bump it- self my be due to a 1µm analog of the well-known 5µm “holes” in Jupiter’s atmosphere (Gillett et al. 1969; Westphal 1969). Using a scenario described in Ackerman & Marley (2001), it may be the formation of holes in the cloud cov- erage that is responsible for the extra flux near 1µm (Burgasser et al. 2002b). These holes would be relatively opacity- and cloud-free windows allowing pho- tons from deeper, warmer layers to escape. Orton et al. (1996) has shown that only ∼1% of Jupiter’s atmosphere contributes to the 5µm hole phenomenon, so the extent of cloud break-up may not have to be severe at the L/T boundary to account for the extra 1µm flux. Other possible explanations, including the gravity-dependent L/T transi- tion model of Tsuji & Nakajima (2003) and the sudden downpour model of Knapp et al. (2004), are discussed at greater length in Knapp et al. (2004) and Kirkpatrick (2005). See Barman (this volume) for further discussion of this and other outstanding issues being targetted by theorists. 4. Issue #3: Finding “Y” Dwarfs Current spectroscopic classifications for brown dwarfs run as late as T8, which corresponds to Teff ≈ 750K (Vrba et al. 2004; Golimowski et al. 2004). A total of three objects, all discovered using 2MASS (Burgasser et al. 2002a; Tinney et al. 2005), are typed either in the optical or near-infrared as T8 dwarfs (Burgasser et al. 2003b, 2006b). The two with measured trigonometric parallaxes lie at distances of only 5.7 and 9.1 pc (Vrba et al. 2004). This together with the fact that their limiting H-band brightnesses, 15.5< H <15.8, fall below the SNR=7 level of 2MASS (Skrutskie et al. 2006), suggest that we are missing later dwarfs simply because the detectability limit of 2MASS does not enable large enough volumes to be sampled for later types. What, then, might dwarfs cooler than T8 look like and what should they be called? If later objects are discovered with spectroscopic morphologies very similar to T dwarfs, the numbering should continue to T9, T10, or even be- yond. Only when a distinct change in spectroscopic morphology – such as the disappearance in the optical of the oxide bands at M/L boundary or the ap- pearance shortward of 2.5 µm of methane at the L/T boundary – should a new class be introduced. Finding these objects, which are commonly referred to as “Y dwarfs” after the suggestion of Kirkpatrick et al. (1999) and Kirkpatrick (2000), are one of the goals of current and planned surveys (see Pinfield et al. and Leggett contributions, this volume). Nature will eventually reveal what the spectra of Y dwarfs look like. In the meantime we are left to theoretical predictions to divine what the trigger at the T/Y boundary might be. Figure 5 shows a sequence of 0.6-2.5 µm model spectra below Teff=800K from Burrows et al. (2003). Inspection of these models suggests several possible triggers, among which are (1) the appearance of ammonia in the near-infrared near Teff ≈600K, (2) the disappearance of the alkali lines near Teff ≈500K, and (3) the end of the blueward trend of J − Ks color with type Figure 4. Absolute J-band magnitude versus spectral type, adapted from Burgasser et al. (2006b). Optical types are shown for L dwarfs and near- infrared types for T dwarfs. Objects known to be close doubles are encircled, two of which are shown both with their joint type and magnitude as well their individual component’s types and magnitudes (filled dots). The location of 2MASS J05591914−1404488 is singled out. near Teff ≈350K. Interestingly, the onset of water clouds near Teff ≈400-500K has no appreciable affect on the spectra. Acknowledgments. The author would like to acknowledge insightful dis- cussions with Katelyn Allers, Kelle Cruz, Dagny Looper, and Adam Burgasser during the preparation of this manuscript. 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0704.1523

The orbit, mass, size, albedo, and density of (65489) Ceto/Phorcys: A tidally-evolved binary Centaur

The orbit, mass, size, albedo, and density of (65489) Ceto/Phorcys: A tidally-evolved binary Centaur W.M. Grundy1, J.A. Stansberry2, K.S. Noll3, D.C. Stephens4, D.E. Trilling2, S.D. Kern3, J.R. Spencer5, D.P. Cruikshank6, and H.F. Levison5. 1. Lowell Observatory, 1400 W. Mars Hill Rd., Flagstaff AZ 86001. 2. Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson AZ 85721. 3. Space Telescope Science Institute, 3700 San Martin Dr., Baltimore MD 21218. 4. Formerly at Dept. of Physics and Astronomy, Johns Hopkins University, Baltimore MD 21218; now at Dept. of Physics and Astronomy, Brigham Young University, N283 ESC Provo UT 84602. 5. Southwest Research Institute, 1050 Walnut St. #400, Boulder CO 80302. 6. NASA Ames Research Center, MS 245-6, Moffett Field CA 94035. In press in Icarus 2007 Primary contact: Will Grundy E-mail: [email protected] Voice: 928-233-3231 Fax: 928-774-6296 Running head: Binary Centaur (65489) Ceto/Phorcys Manuscript pages: 25 Figures: 5 Tables: 5 ABSTRACT Hubble Space Telescope observations of Uranus- and Neptune-crossing object (65489) Ceto/Phorcys (provisionally designated 2003 FX128) reveal it to be a close binary system. The mutual orbit has a period of 9.554 ± 0.011 days and a semimajor axis of 1840 ± 48 km. These values enable computation of a system mass of (5.41 ± 0.42) × 1018 kg. Spitzer Space Tele- scope observations of thermal emission at 24 and 70 µm are combined with visible photometry to constrain the system's effective radius (109 �11 km) and geometric albedo (0.084 �0.021 �0.014 ). We esti- mate the average bulk density to be 1.37 �0.66 �0.32 g cm , consistent with ice plus rocky and/or car- bonaceous materials. This density contrasts with lower densities recently measured with the same technique for three other comparably-sized outer Solar System binaries (617) Patroclus, (26308) 1998 SM165, and (47171) 1999 TC36, and is closer to the density of the saturnian irregular satellite Phoebe. The mutual orbit of Ceto and Phorcys is nearly circular, with an eccentricity � 0.015. This observation is consistent with calculations suggesting that the system should tidally evolve on a timescale shorter than the age of the solar system. Keywords: Centaurs, Kuiper Belt, Transneptunian Objects, Satellites. 1. Introduction Objects following unstable, giant-planet crossing orbits in the outer solar system have been termed Centaurs (e.g., Kowal et al. 1979; Elliot et al. 2005). They are thought to be bodies per- turbed from the transneptunian region, evolving on timescales of the order of ~10 years to either being ejected from the solar system, suffering catastrophic impacts, roasting in the heat of the Sun as comets, or getting parked in cold-storage in the Oort cloud (e.g., Duncan and Levison 1997; Holman 1997; Levison and Duncan 1997; Horner et al. 2004). Centaurs offer a sample from the transneptunian region that is closer and more accessible to Earth-bound observers. They are less accessible than comets, but could have surfaces more pristine than those of comets which have spent more time closer to the Sun. A typical member of the Centaur population has experienced numerous close encounters with one or more of the giant planets (e.g., Duncan et al. 1988; Levison and Duncan 1997). Such events might be expected to induce observable effects on the mutual orbits of binary Centaurs, when compared statistically with the mutual orbits of binary transneptunian objects (TNOs) in 1 In this paper we use the Deep Ecliptic Survey (DES) definition of Centaurs as non-resonant objects with perihelia between the orbits of Jupiter and Neptune (Elliot et al. 2005). The Minor Planet Center defines transneptunian dynamical classes somewhat differently, and would classify some DES Centaurs, including (65489) Ceto/Phorcys, as members of the scattered disk (e.g., Gladman et al. 2007). The Committee on Small Body Nomenclature of the International Astronomical Union has recently adopted a naming convention for objects on unstable, non-resonant, giant-planet-crossing orbits with semimajor axes greater than Neptune's. Befitting their Centaur-like transitional orbits between TNOs and comets, they are to be named for other hybrid and shape-shifting mythical creatures. Some 30 objects are currently known to fall into this category, although only Ceto/Phorcys and (42355) Typhon/ Echidna have been named according to the new policy, so far. The names (announced without explanation in IAUC 8778) come from classical Greek mythology. The primary body is named for Ceto (K����), an enormous sea monster, born of Gaia. English words like “cetacean” derive from her name. The secondary is named for Phorcys (�ó ��), her brother and husband, depicted as part man, part crab, and part fish. Together, they begot numerous terrible monsters, including the Gorgons. stable heliocentric orbits. For several years, Noll et al. (2004a, 2004b, 2006, 2007) have been carrying out a deep search for binaries among various dynamical classes of TNOs, most recently using the Advanced Camera for Surveys High Resolution Camera (ACS/HRC, Ford et al. 1996) aboard the Hubble Space Telescope (HST). An important result from this work is the finding that the dynamically cold “Classical” sub-population has relatively high binary rates compared with other transneptunian sub-populations (Stephens and Noll 2006). Recently, we (KSN, WMG, DCS, and HFL) have put greater emphasis on searching for binaries among the more dy- namically excited sub-populations, including the Centaurs. Prior to its untimely demise, we used ACS/HRC to search for satellites around 22 Centaurs (as per the DES definition), finding com- panions for two of them. The first, (42355) Ty- phon/Echidna (provisionally designated 2002 CR46), was discovered in 2006 January and a second, (65489) Ceto/Phorcys (provisionally designated 2003 FX128), was discovered in 2006 April (Noll et al. 2006). The latter system is the subject of this pa- The initial discovery of (65489) Ceto/Phorcys was 2003 March 22 at the Palomar 1.2 m Schmidt camera by C. Trujillo and the Near-Earth Asteroid Tracking (NEAT) team (MPEC 2003-H33). It was soon identified in a series of pre-discovery images dating back to 1987, enabling its elongated helio- centric orbit (shown in Fig. 1) to be determined, with osculating heliocentric elements: semimajor axis a� = 102 AU, inclination i� = 22 , and eccen- tricity e� = 0.82. The orbit crosses the orbits of both Uranus and Neptune and also extends far be- yond the solar wind radiation environment to an aphelion of about 186 AU, well beyond the 94 AU distance where Voyager 1 crossed (or was crossed by) the heliosphere termination shock (e.g., Decker et al. 2005) with important implications for radi- olytic processing of its surface (Cooper et al. 2003). 2. Hubble Space Telescope Observations and Analysis The binary nature of (65489) Ceto/Phorcys was discovered in HST ACS/HRC images ob- tained 2006 April 11 UT (see Fig. 2), as part of Cycle 14 program 10514 (led by KSN). Details of the clear filter combination and dithering techniques used in program 10514 are described by Noll et al. (2006). Follow up ACS/HRC images were obtained on May 6, 10, 30, and 31 UT as part of Cy- cle 14 program 10508 (led by WMG). The scheduling of these observations benefited from a statistical ranging Monte Carlo type of analysis inspired by the work of J. Virtanen and col- leagues (e.g., Virtanen et al. 2001, 2003). This approach enabled us to optimize the timing of Fig. 1: Plan view of the Solar System, showing the heliocentric orbit of (65489) Ceto/Phorcys (dotted curve) compared with orbits of the four gas giants (solid curves), with points indicating positions as of 2007 July. The orbit crosses the orbits of both Uranus and Neptune, but considerable time is also spent well beyond 94 AU from the Sun, the dis- tance at which Voyager 1 crossed the heliosphere termination shock (e.g., Decker et al. 2005).. -50 0 50 100 150 Distance from Sun (AU) subsequent observations for satellite orbit determination, within the limitations of HST schedul- ing and the restricted visibility windows imposed by the two-gyro mode in which HST is cur- rently operating. Our implementation of the technique is described in a forthcoming paper (Grundy et al. in preparation). The follow up observations used the F606W and F814W filters of the ACS/HRC camera. For each visit, eight 223 second exposures (four in each filter) were ob- tained, dithered over a four-point box pattern so that images with defects such as hot pixels or cosmic rays contaminating the flux of the binary system could be easily identified and excluded from further analysis. 2.1. Photometry and astrometry Data were processed through the standard CAL ACS pipeline at Space Telescope Science Institute (STScI), producing dark-subtracted, flat-fielded images (for details, refer to http:// www.stsci.edu/hst/acs/documents/handbooks/cycle16). Astrometry and pho- tometry were calculated from these images by fitting model point spread functions (PSFs) gener- ated by Tiny Tim (Krist and Hook 2004) and correcting the results for the geometric distortion arising from the separation between the ACS camera and the optical axis of the telescope. De- tails of the binary-PSF fitting procedure are described in Stephens and Noll (2006). The PyRAF program xytosky in the STSDAS package (http://www.stsci.edu/resources/soft- ware_hardware) was used to translate the pixel coordinates found for Ceto and Phorcys to their true right ascension and declination values. Accurate separations and position angles for the binary pair could then be calculated, and measurements from the multiple images combined to get the average relative sky plane astrometry for each visit in Table 1. Table 1. HST ACS/HRC observational circumstances and 1-� relative astrometry Average UT date and a ��b ��b (AU) (degrees) (arcsec) 2006/04/11 21.691 27.753 26.785 0.542 +0.0172 ± 0.002 +0.0833 ± 0.003 2006/05/06 7.447 27.813 26.881 0.812 �0.0475 ± 0.002 �0.0687 ± 0.001 2006/05/10 4.510 27.822 26.912 0.916 �0.0032 ± 0.001 +0.0860 ± 0.002 2006/05/30 0.217 27.870 27.129 1.440 +0.0362 ± 0.001 +0.0747 ± 0.001 2006/05/31 1.692 27.873 27.143 1.466 +0.0829 ± 0.002 +0.0295 ± 0.002 Fig. 2: Example Hubble Space Telescope ACS/HRC single frames of Ceto and Phorcys (centered on Ceto), tak- en 2006 April 11, May 5, May 10, May 30, and May 31 UT, from left to right. The HRC pixels are projected to sky-plane geometry with North up and East to the left, revealing the geometric distortion at the HRC focal plane. For scale, the mosaic of images is 2.0 × 0.4 arcsec. The left-most image was one of the binary discov- ery sequence of images, obtained through the CLEAR filter as part of program 10514. The other four images were obtained through the F814W filter as part of program 10508. The distance from the Sun to the target is r and from the observer to the target is �. The phase angle, the an- gular separation between the observer and Sun as seen from the target, is g. Relative right ascension �� and relative declination �� are computed as �� = (�2 − �1)cos(�) and �� = � 2 − �1, where subscripts 1 and 2 refer to Ceto and Phorcys, respectively. The PSF model binary solutions were also used to calculate individual magnitudes for Ceto and Phorcys. To get the magnitude of one object, we subtracted the model PSF of the other from each image and then used the PyRAF multidrizzle task in the STSDAS package to remove the ge- ometric distortion. For each resulting undistorted image, the total flux within a 3 pixel radius was measured and corrected to an aperture of 20 pixels (0.5 arcsec) computed for an undistorted model PSF. We then applied the Sirianni et al. (2005) AC05 (0.5 arcsec to ∞) aperture correc- tion coefficients, the photometric zero point for each filter, and the charge transfer efficiency cor- rection to get the final HST magnitude in each image. To compare the HST magnitudes with ground based results and to help constrain the size and albedo of (65489) Ceto/Phorcys, transformation coefficients were derived to convert the F606W and F814W magnitudes to Johnson-Cousins V and I magnitudes. The ACS and Johnson- Cousins filters are not exact counterparts, so assumptions had to be made about the spectral ener- gy distribution of Ceto/Phorcys. We applied a reddening coefficient to a solar spectrum until it had the same F606W�F814W color as (65489) Ceto/Phorcys. We then convolved this spectrum with the HST and Johnson-Cousins filters using the synphot task under STSDAS to get transfor- mation coefficients specific to Ceto/Phorcys. These coefficients were used to transform the HST magnitudes of Ceto and Phorcys in each image to the Johnson-Cousins system. For each visit the F606W, F814W, V, and I magnitudes were averaged to produce Table 2. Table 2. Photometry from Hubble Space Telescope ACS/HRC observations Observation Ceto Phorcys Date (UT) F606W F814W V I F606W F814W V I 2006/05/06 21.43 20.54 21.68 20.56 21.94 21.12 22.15 21.14 2006/05/10 21.35 20.53 21.57 20.55 21.94 21.10 22.18 21.13 2006/05/30 21.49 20.61 21.74 20.63 22.10 21.24 22.35 21.26 2006/05/31 21.49 20.63 21.72 20.65 22.07 21.24 22.29 21.26 Table note: Photometric 1-� uncertainties in F606W and F814W filters are about ±0.04 mag. Color transfor- mation uncertainties inflate the V and I uncertainties to about ±0.06 mag. Two filters were used in the follow-up observations to search for color differences between Ceto and Phorcys. The average F606W�F814W colors were 0.867 ± 0.028 and 0.836 ± 0.028 mag for Ceto and Phorcys, respectively. Ceto's measured color is slightly redder than that of Phorcys, but the difference is statistically insignificant. Our average V�I color for the system is 1.07 ± 0.04, corresponding to a spectral slope of 15.3 ± 1.5 (% rise per 100 nm relative to V), putting Ceto/Phorcys among the gray color clump of Centaurs, albeit at the red edge of that clump (e.g., Boehnhardt et al. 2003; Peixinho et al. 2003). Tegler et al. (2003) re- ported a B�V color for (65489) Ceto/Phorcys of 0.86 ± 0.03 and a V�R color of 0.56 ± 0.03, consistent with this result. No lightcurve variability has been reported for this system and our data show no evidence for it either. The average magnitudes from the 4 visits reported in Table 2 differ by more than the 1-� uncertainties, but that variability can be attributed to geometry. After removing photo- metric effects of the changing distances from the Sun to the target r and from the observer to the target � and also the effect of the phase angle g (by assuming generic dark-object phase behavior as represented by the H and G system of Bowell et al. 1989, with G = 0.15), we find that the largest deviation from the 4-visit mean for either object or filter is only 1.5-� (for Ceto during the May 10 UT visit in F606W). With only four random epochs and ±0.04 mag photometric un- certainties, the F606W data can only rule out a 0.15 mag sinusoidal lightcurve at about the 1-� confidence level. However, the observed F606W outlier is not mirrored in the F814W filter, as would be expected if a lightcurve from an elongated shape were responsible, so noise is the like- lier explanation for that point. At least one deviation as large as 1.5-� is to be expected among four sets of four measurements. 2.2. Orbit fitting and system mass Our methods for fitting mutual orbits to relative astrometry of binaries have been described previously (Noll et al. 2004a, 2004b, 2006, 2007). We use the downhill simplex “amoeba” algo- rithm (Nelder and Mead 1965; Press et al. 1992) to iteratively adjust a set of seven orbital ele- ments to minimize the �2 statistic of residuals between observed and predicted sky plane relative positions of the primary and secondary, accounting for light time delays and relative motion be- tween the observer and the binary pair. For previous orbit fits we had used the date and argu- ment of periapsis as two of the seven fitted orbital elements. This choice had originally been made because the orbit of the satellite of 1998 WW31 (Veillet et al. 2002) suggested to us that very high eccentricities might be the norm. Unfortunately, for circular orbits those two parame- ters become degenerate, and as eccentricity approaches zero, their use produces numerical insta- bilities which plagued our fits to the near-circular orbits of Pluto's satellites (Buie et al. 2006). Accordingly, we have since modified our orbit-fitting code to fit the following seven elements: period P, semimajor axis a, eccentricity e, inclination i, mean longitude at epoch �, longitude of ascending node �, and longitude of periapsis �. To estimate the uncertainty in each of these fit- ted elements, we used two different methods. First, we systematically varied each parameter around its best fit value, allowing the other six to adjust themselves to re-minimize �2, creating slices through the seven dimensional �2 space which enabling us to map out the 1-� confidence contour. Second, we fitted orbits to random sets of input astrometry data generated from the ob- served astrometry by adding Gaussian noise consistent with the astrometric uncertainties. The bundle of Monte Carlo orbits generated in this way provides an alternate measure of how well we have determined the orbital elements. Both methods gave very similar 1-� uncertainties on the fitted elements for the orbit of Phorcys around Ceto. The elements are tabulated in Table 3 and the projection of the orbit on the sky plane at the average time of the HST observations is shown in Fig. 3. Table 3. Orbital elements and derived parameters with 1-� uncertainties or limits Parameter Orbit 1 (�2 = 2.3) Orbit 2 (�2 = 2.4) Fitted orbital elements Period (days) P 9.557 ± 0.008 9.551 ± 0.007 Semimajor axis (km) a 1840 ± 41 1841 ± 47 Parameter Orbit 1 (�2 = 2.3) Orbit 2 (�2 = 2.4) Eccentricity e � 0.013 � 0.015 Inclinationa (deg) i 68.8 ± 2.9 116.6 ± 3.0 Mean longitudea at epochb (deg) � 23.0 ± 2.7 53.5 ± 5.2 Longitude of asc. nodea (deg) � 105.5 ± 3.7 134.6 ± 3.4 Longitude of periapsisa (deg) � 40 ± 360c 70 ± 360c Derived parameters System mass (1018 kg) Msys 5.40 ± 0.36 5.42 ± 0.42 Orbit pole right ascensiona (deg) � 15.5 ± 5.2 44.6 ± 5.2 Orbit pole declinationa (deg) � 21.2 ± 4.0 −26.7 ± 4.1 Referenced to J2000 equatorial frame. The epoch is Julian date 2453880 (2006 May 24 12:00 UT). The uncertainty of ±360 indicates that this parameter is unconstrained. It can be difficult to distin- guish between an orbit and its mirror image through the plane of the sky (e.g., Descamps 2005). Parallaxes from the differential motion of the Earth and the object will eventually enable a patient observer to break the ambiguity between an orbit and its mirror, as the observer's sky plane gradually rotates with respect to the orbit. More face-on orbits, with their smaller excursions of the satellite in the direction radial to the observer, are relatively slow to re- veal which solution is the real one. At the time of our observations, the Ceto/Phorcys orbit was tilted about 28 from face-on. We list both solu- tions, noting that �2 is slightly better for the first one, but does not ex- clude the second one. For both or- bits, the longitude of periapsis � is unconstrained, in that it can take any value from 0 to 360 without pushing �2 over the 1-� threshold. However, there is a shallow �2 minimum for � at about 40 for Orbit 1 and about 70 for Orbit 2. Neither orbit will be oriented edge-on to the Fig. 3: Projection of the orbit of Phorcys (gray oval) relative to Ce- to's location (black dot) onto the sky plane at the average time of the five HST observations. Points with 1-� error bars indicate relative astrometry between Ceto and Phorcys measured from the ensemble of dithered frames during each HST visit. Open circles are relative positions of Phorcys at the times of the observations as calculated from orbit 1 in Table 3. The spot at the origin and the open circles are scaled to the sizes of Ceto and Phorcys, respectively, derived in Section 4. inner solar system within the next century, so use of mutual events to obtain more detailed infor- mation about the system geometry (e.g., Noll et al. 2007) is a distant future prospect. The mutual orbit of Ceto/Phorcys can be compared with orbits of satellites of other small outer Solar System bodies (see Noll et al. 2007 Table 2). It is noteworthy that this is the least massive system with a near-circular orbit by some three orders of magnitude. All known semi- major axes for transneptunian binaries are larger except for that of Typhon/Echidna (Grundy et al. in preparation) and all periods are longer except for that of Charon's orbit about Pluto. Al- though these characteristics sound exceptional, we do not believe this system to be unusual. Closer systems such as this one are simply more difficult to discovery and to study. From the semimajor axis a and period P, we can compute the mass of the combined system Msys, according to 4�2 a3 , (1) where G is the gravitational constant which we take to be 6.6742 × 10-11 m3 s-2 kg-1. Since we have two possible orbits, we adopt values of P = 9.554 ± 0.011 days, a = 1840 ± 48 km, and Msys = (5.41 ± 0.42) × 1018 kg to encompass the 1-� uncertainties from both solutions. The semimajor axis is measured to much lower fractional precision than the period is and it is raised to a higher power in the mass equation, so it is the dominant source of uncertainty in the Ceto/Phorcys system mass. 3. Spitzer Space Telescope Observations and Analysis Spitzer Space Telescope ob- served (65489) Ceto/Phorcys in 2006 July with the Multiband Imag- ing Photometer for Spitzer (MIPS, Rieke et al. 2004) in its 24 and 70 µm bands, which have effective wavelengths of 23.68 and 71.42 µm. These observations were part of Cy- cle 3 program 30081 (led by JAS). Data were collected using the pho- tometry observing template, which is tailored for photometry of point sources. The observatory tracked the target during the observations, although the motion was negligible relative to the size of the point- spread function (PSF) at these wave- lengths. The target was imaged in both bands during two visits separat- ed by 37 hours, over which time (65489) Ceto/Phorcys moved 38 arcsec relative to the background stars, allowing us to identify Fig. 4: Spitzer Space Telescope MIPS images of Ceto+Phorcys at 24 µm (left) and 70 µm (right) from 2006 July 18 UT. Each image is 180 arcsec square, and is oriented North up, East left. The circles are centered at the ephemeris position of the object, and have diame- ters equal to four times the FWHM of the Spitzer PSF in each band. While the object is not observed at exactly the ephemeris position, the offsets are consistent in the data from this and the other epoch, the object motion vector relative to the background objects is as pre- dicted, and it is the only moving object visible at both epochs. Data from the second epoch were used to remove background sources from the first epoch, and vice versa. the target with certainty. Since the images from the two epochs have significant spatial overlap, we could use the image pairs to confirm that no strong background sources were present at the target position at either epoch and to subtract off the effect of weak background sources. The two visits would also help average over any possible thermal lightcurve variability. Images from the first epoch are shown in Fig. 4. We reduced the raw data and mosaicked them using the MIPS instrument team data analy- sis tools (Gordon et al. 2007). For the 24 µm data, basic processing included slope fitting, flat- fielding, and corrections for droop. All of these steps are currently implemented in the Spitzer Science Center (SSC) pipeline products. Additional corrections were made to remove readout offset (a jailbar pattern in the images), the effects of scattered light (which introduces a pointing- dependent background gradient and slightly degrades the sensitivity), and the application of a second-order flat field, derived from the data itself, to remove latents from previous observa- tions. The photometric repeatability of MIPS observations of moderately bright sources is better than 1% at 24 µm, and is 5% at 70 µm. The uncertainty in the absolute calibration of these bands is 4% and 8% respectively (Engelbracht et al. 2007; Gordon et al. 2007). For purposes of fitting models to our photometry, we use uncertainties that are the root-square-sum of the absolute cali- bration uncertainties and the measurement uncertainties determined from the images themselves. We adopt slightly larger 1-� calibration uncertainties, 5% and 10%, to account for uncertainties in color and aperture corrections. The widths of the filter bandpasses are about 25%, resulting in modest color corrections. We iteratively applied color corrections to our photometry, which con- verged to give a color temperature of 73.2 K, and color corrections of +2.6% and +8.6% at 24 and 70 µm, respectively (see Stansberry et al. 2007b). The uncertainties on the correction factors are perhaps a few percent of the factors themselves, and so are negligible for our study. We measured the flux density of (65489) Ceto/Phorcys using 9.96 and 29.6 arcsec diame- ter apertures (about 4 and 3 native pixels) at 24 and 70 µm; the PSF full-width at half-max is 6.5 and 20 arcsec in those bands. The apertures were positioned at the center-of-light centroid. Mo- saics were constructed using 1.245 arcsec and 4.925 arcsec pixels at 24 and 70 µm (about half the native pixel scale of those arrays). We applied aperture corrections of 1.91 and 1.85 to the 24 and 70 µm photometry to compute the total flux (see Gordon et al. 2007; Engelbracht et al. 2007 for details). Uncertainties in the aperture corrections are approximately 1%. Table 4 summa- rizes the circumstances of our Spitzer MIPS observations and the measured, color-corrected flux densities. Table 4. Spitzer Space Telescope MIPS observational circumstances and thermal fluxes Average UT date r � g 24 µm flux 70 µm flux and hour (AU) (degrees) (mJy) 2006/07/18 10.23 27.990 27.662 1.99 1.47 ± 0.02a 14.9 ± 1.0a 2006/07/19 19.46 27.993 27.686 2.00 1.48 ± 0.02a 14.5 ± 1.2a Adopted 27.991 27.674 2.00 1.473 ± 0.076b 14.69 ± 1.84b Errors reflect the signal to noise ratios in the observations, which were about 70 and 13, resulting from integra- tions of 900 and 2640 seconds, at 24 and 70 µm, respectively. The adopted 1-� flux errors include additional absolute calibration uncertainties. 4. Size, Albedo, and Density The thermal fluxes measured by Spitzer can be used to constrain the size of Ceto/Phorcys and also the albedo if they can be combined with suitable visible photometry. Tegler et al. (2003) reported a V magnitude of 20.70 ± 0.03 from Vatican Observatory 1.8 m telescope obser- vations when the system was at heliocentric distance r = 25.190 AU, geocentric distance � = 24.298 AU, and phase angle g = 1.09 . Assuming typical phase behavior for low albedo objects (G = 0.15 in the H and G system of Bowell et al. 1989; see also Sheppard and Jewitt 2002; Rousselot et al. 2005), we can convert the Tegler et al. V photometry to absolute magni- tude HV = 6.603 ± 0.030. Under the same assumptions, our HST photometry (in Tables 1 and 2) gives HV = 6.625 ± 0.030. Combining these, we adopt the weighted average HV = 6.614 ± 0.021. Thermal fluxes are used to to constrain an object's size via a variety of thermal models (e.g., Lebofsky and Spencer 1989). We used the approach described by Stansberry et al. (2006), which takes advantage of the existence of visual photometry plus two well-separated thermal in- frared wavelengths from Spitzer/MIPS to simultaneously fit the size, albedo, and beaming pa- rameter �, using the standard thermal model (STM), which assumes instantaneous thermal equi- librium with sunlight illuminating a spherical surface. We also assumed unit emissivity. A wide range of real-world complications such as rotation, pole orientation, surface roughness, and ther- mal inertia can strongly influence observable thermal fluxes relative to what would be predicted by the simple STM, but in general, the effects of these complications look spectrally similar to the effect of changing the beaming parameter �. If � is constrained by the data, the large model uncertainties in size and albedo which would otherwise flow from lack of knowledge of these other physical properties are greatly diminished. Using this model, we are able to match the ob- served thermal and visible photometry of (65489) Ceto/Phorcys with an effective radius Reff = 109 �11 km, along with beaming parameter � = 0.77 �0.08 �0.09 , and geometric albedo Ap = 0.084 �0.021 �0.014 . Uncertainties on the derived parameters were determined by means of Monte Carlo methods. Thousands of random sets of input fluxes were generated, consistent with the observational uncertainties. Each set resulted in slightly different values of the derived parame- ters, creating a population of solutions from which 1-� uncertainties could be estimated. Errors from possible lightcurve variations are small enough to be neglected. For example, with thermal observations at two effectively random times, the 1-� upper limit 0.15 mag lightcurve mentioned earlier would contribute about a 1.4% average radius error, well below our �11 % reported uncer- tainty. The derived value of � = 0.77 �0.08�0.09 is on the low side of what has been seen for other TNOs and Centaurs, suggesting that Ceto and Phorcys have some combination of relatively low thermal inertias, slow rotation rates, especially rough surfaces, or their poles oriented toward the Sun. For comparison, Stansberry et al. (2007a) found � values ranging from 0.6 to 2.3, with an aver- age of 1.3, for a sample of 23 TNOs and Centaurs detected with good signal precision at both 24 and 70 µm wavelengths by Spitzer/MIPS. Our 0.084 �0.021 �0.014 albedo of Ceto/Phorcys is consistent with albedos of other Centaurs. Stansberry et al. (2007a) report albedos for 11 other Centaurs (by the DES classification) detect- ed with good signal precision at both 24 and 70 µm by Spitzer/MIPS. That sample has an aver- age albedo of 0.06 and a standard deviation of 0.04. Considerable albedo diversity is also appar- ent among other small outer Solar System objects (e.g., Grundy et al. 2005; Cruikshank et al. 2007; Stansberry et al. 2007a), but correlations between albedo and dynamical class or color re- main poorly established, except for a general trend of progressively lower average albedos from TNOs to Centaurs to comet nuclei, and a trend of higher albedos for the redder Centaurs (Stans- berry et al. 2007a). By assuming the albedos of Ceto and Phorcys are similar to one another, we can use the 0.584 ± 0.030 average magnitude difference between them to work out their individual radii, ob- taining RCeto = 87 �9 km and RPhorcys = 66 �7 km. Computing the total system volume from these two radii, and combining that with the total system mass from the mutual orbit, we obtain an av- erage bulk density of 1.37 �0.66 �0.32 g cm -3. The assumption of equal albedos is consistent with the observation of indistinguishable colors for the two components, but we can also explore how sensitive we are to that assumption. Assume, for instance, that the albedo of Phorcys was actual- ly twice that of Ceto. We would then have overestimated the average bulk density by a factor of 9%. If Ceto's albedo were double that of Phorcys, we would have underestimated the average bulk density by 2%. Unless the albedo contrast between Ceto and Phorcys is very extreme in- deed, the density is relatively insensitive to our assumption of equal albedos. Our conclusions are comparably insensitive to possible lightcurve variations, as mentioned previously. Bulk densities of icy objects are often interpreted in terms of a rock fraction rock , (2) where ice is taken to be 0.937 g cm-3 for low temperature crystalline ice Ih (Lupo and Lewis 1979) and rock could range from 2.5 g cm-3 for hydrated silicates to 3.5 g cm-3 for anhydrous sili- cates (e.g., Schubert et al. 1986). For rock = 2.5 g cm-3, our average bulk density of Ceto/Phor- cys yields a rock fraction frock = 0.51 �0.29 �0.39 , a value intermediate between the rock-rich composi- tions of Triton and Pluto and the more ice-rich compositions of some of the smaller satellites of Saturn. However, this rock fraction calculation only makes sense in the absence of void space, or of other possible interior components, such as carbonaceous materials. We estimate the hydrostatic pressure at the cores of Ceto and Phorcys according to �G 2 R2 (3) (e.g., Stansberry et al. 2006) as 2.0 �0.6 and 1.1 �0.4 MPa, respectively. These pressures are insuf- ficient to squeeze much void space out of cold, particulate H2O ice. Laboratory experiments by Durham et al. (2005) show that at pressures in the 5 to 10 MPa range, appreciable compaction begins to occur in cold ice, but as much as 10% void space remains even at pressures as high as 100 MPa. If Ceto and Phorcys have remained cold since their accretion, we can anticipate rela- tively high fractions of void space within them, complicating interpretation of their densities in terms of ice and rock fractions. Bulk densities are known for several much larger outer Solar System bodies, including Pluto and Charon (2.03 ± 0.06 and 1.65 ± 0.06 g cm-3, respectively; Buie et al. 2006), Eris (2.26 ± 0.25 g cm-3; Brown 2006), and (136108) 2003 EL61 (3.0 ± 0.4 g cm-3; Rabinowitz et al. 2006). These higher densities are consistent with expectation for rock-rich and possibly differentiated objects which probably expe- rienced warm interior temperatures at some time in their histories. Two other small bi- nary TNOs have recently had their average bulk densities determined via the technique used in this paper. Stansberry et al. (2006) reported a bulk density of 0.50 �0.30 �0.20 g cm for (47171) 1999 TC36 and Spencer et al. (2006) reported a bulk density of 0.70 �0.32 �0.21 g cm-3 for (26308) 1998 SM165. These much lower densities require significant void space, even for pure ice compositions. Low densities compared with likely bulk compo- sitions have also been determined recently for the Trojan asteroid (617) Patroclus (Marchis et al. 2006) and for main belt as- teroids (e.g., (90) Antiope; Descamps et al. 2007) as well as for much smaller comet nuclei (e.g., A'Hearn et al. 2005; Davidsson and Gutiérrez 2006). However, the irregu- lar saturnian satellite Phoebe, with a size similar to that of Ceto+Phorcys, has a high- er bulk density of 1.630 ± 0.033 g cm-3 (Johnson and Lunine 2005). These diverse densities (shown in Figure 5) point to con- siderable variety in bulk compositions and interior structures among small outer Solar System bodies. Apart from bulk compositional differences, thermal history is another potential source of density diversity among TNOs. Objects in the outer parts of the protoplanetary disk could ini- tially accrete ice mostly in the amorphous phase, with a fluffy structure that includes consider- able void space (e.g., Jenniskens et al. 1998). Subsequent thermal pulses, from whatever source (impacts, tidal dissipation, Al26 decay, etc.), could crystallize amorphous ice. The crystallization would liberate additional heat as well as potentially reducing void space (e.g., Jewitt et al. 2007). The presence of varying amounts of NH3 could also lead to considerable diversity of thermal evolution and thus density (e.g., Leliwa-Kopystyński and Kossacki 2000). 5. Tidal evolution Since gravitational attraction falls off as the square of separation, Ceto and Phorcys both produce differential gravitational accelerations across the bodies of their partners, slightly dis- torting their equipotential figures. If their mutual orbit is eccentric, or either of their rotation states differs from the orbital rotation rate and orientation, this distortion will change over time, Fig. 5: Comparison of the average bulk density of (65489) Ceto/Phorcys (star) with densities for binary outer Solar System objects discussed in the text (black dots) and icy satellites of Saturn, Uranus, and Neptune (gray points, from Burns 1986 and McKinnon et al. 1995). For the binaries with unknown mass ratios, the radius shown is that of the primary, since it is likely to dominate the average density. The dashed curve is a theoretical bulk density for non- porous, pure H2O ice subject to self-compression at 77 K (Lupo and Lewis 1979). The dotted curve allows for poros- ity in pure, cold, granular H2O ice Ih, based on compaction data from Durham et al. (2005) as modeled by McKinnon et al. (2005). An expected trend of smaller objects having sys- tematically lower densities than their larger counterparts (Lacerda and Jewitt 2007) is not cleanly supported by the present data for (65489) Ceto/Phorcys. 100 1000 Radius (km) resulting in frictional dissipation of energy and leading to tidal evolution of their orbit and spin states. The rate of energy dissipation and tidal evolution depends on the specific tidal dissipation factor Q, which is a complicated function of interior structure and composition that describes the degree to which tidal friction is deposited in a given body. Q is approximately equal to 1/(2�), where � is the angle between the tidal bulge of the object and the line of centers between the two objects. Because Ceto and Phorcys have similar sizes and probably similar compositions, we as- sume that Q1 � Q2 � Q (where subscripts 1 and 2 to refer to Ceto and Phorcys, respectively). Typical Q values for solid planets and icy satellites range from 10 to 100 (e.g., Goldreich and Soter 1966; Yoder 1982; Hubbard 1984). We take Q = 100, to minimize dissipation and thus compute upper limits on tidal evolution timescales. Small solid objects require a correction to Q because their rigidity can be large compared to their self-gravity (Hubbard 1984). In this case, the small body may not be able to relax to hy- drostatic equilibrium over a tidal cycle, but instead acts elastically. Thus, the deformation of the body's surface is smaller than it would be in hydrostatic equilibrium as elastic forces tend to maintain the body's shape. The correction to Q takes the form Q' = Q �1�19�2 g R � , (4) where µ is the rigidity, is the density, R is the radius, and g is the surface gravity (Goldreich and Soter 1966). Values adopted for use in our calculations are tabulated in Table 5. The rigidi- ties of Ceto and Phorcys are, of course, unknown. If they are un-fractured monoliths, their rigidities could be as high as 1011 dyn cm-2, intermediate between values for solid ice and solid rock (Gladman et al. 1996). In this extreme case, the correction to Q would be large, with Q' = 3 × 106. At the other extreme, if Ceto and Phorcys are unconsolidated “sand piles” their rigidities could be as low as 107 dyn cm-2 (Yoder 1982). In this case, the correction to Q is small, giving Q' = 300. We use these bracketing values of Q' to explore the range of possible timescales for tidal spin-locking and orbital evolution in the Ceto/Phorcys system. Table 5. Physical parameters adopted. Parameter Ceto Phorcys Radius R 87 km 66 km Mass M 3.74 × 1018 kg 1.67 × 1018 kg Density 1.37 g cm-3 1.37 g cm-3 Surface gravity g 3.3 cm s-2 2.5 cm s-2 Table note: These values are derived from the effective radius Reff, system mass Msys, and magnitude difference, subject to the assumption of equal albedos and equal densities. Over time, tidal effects can synchonize the spin rate of Phorcys to its orbital period (as for the Earth's Moon, which keeps one face oriented toward Earth) and, on a longer timescale, syn- chronize Ceto's spin rate as well (analogous to the Pluto-Charon system, where both members keep the same face toward each other). The timescale for spin-locking Ceto (slowing its spin rate to match the mutual orbital period) is given by spin , 1 Q'��1 M1 a G M 2 , (5) where subscripts 1 and 2 refers to Ceto and Phorcys, respectively, M is mass, R is radius, and �� is the difference between an initial angular rotation rate and the final, synchronized rate (Goldreich and Soter 1966; Trilling 2000). The timescale for spin-locking Phorcys can be found simply by swapping all the subscripts in Equation 5. A typical rotation period for TNOs is around 10 hours (e.g., Trilling and Bernstein 2006). The current orbital period is much longer than this, so we set ��1 (and ��2) to correspond to this rate (one could equally start with the breakup rotation rate, in the 2 to 3 hour period range). We find that the spin-locking timescale for Ceto ranges from 106 to 1010 years and the timescale for spin-locking Phorcys is about a fac- tor of three shorter. We conclude that unless these bodies are rigid monoliths, Phorcys should be spin-locked to Ceto and Ceto to Phorcys as well, or nearly so. (Recall that if Q is smaller than our assumed maximum value of 100, the spin-lock timescale decreases, increasing the likelihood of spin-locking.) The eccentricity of the Ceto/Phorcys mutual orbit can also evolve due to tides raised on ei- ther object by the other. Following Goldreich and Soter (1966) we can write the contributions of the two terms as = � dedt � �� dedt � , (6) where (de/dt)1 accounts for the effect of tides raised on the primary by the secondary and (de/dt)2 accounts for tides raised on the secondary by the primary. The two contibutions can be ex- pressed as e � dedt �1 = 16 � GM 1 � R1 6.5 � sign �2�1�3n2 � (7) where n2 is the mean orbital motion of the secondary and e � dedt �2 = � �G M 13� R2 6.5 � . (8) Employing our previous result that Ceto and Phorcys should be spin-locked (or nearly so), �1 should be equal (or close) to n1 so (de/dt)1 is negative, implying that de/dt must also be negative (since (de/dt)2 is always negative) 2. A negative de/dt indicates circularization of the orbit. Using the values for Ceto and Phorcys in Table 5, we find 7×10�12�1.4×10�11 Equating e with �de and solving for dt to estimate the timescale to reduce an initial eccentricity to zero, we find that the circularization timescale ranges from 5 × 105 to 5 × 109 years for the 2 For bodies of equal density, (de/dt)2/(de/dt)1 � 1.5 (R1/R2), so (de/dt)2 always dominates, and the first term on the right hand side of Equation 6 is often ignored (e.g., Rasio et al. 1996; Trilling 2000). Thus, even when P1 < 2/3 P2, giving positive (de/dt)1, |(de/dt)1| can never be larger than |(de/dt)2|, so de/dt will always be negative. previously discussed range of Q'. Again, unless Ceto and Phorcys are rigid monoliths, this timescale is much less than the age of the Solar System (and, presumably, the age of the Ceto/ Phorcys binary) so we expect the system to have lost most (or all) of its primordial orbital eccen- tricity. The average density of (65489) Ceto/Phorcys is relatively high compared with two other small TNOs with eccentric mutual orbits (26308) 1998 SM165 (Spencer et al. 2006) and (47171) 1999 TC36 (Stansberry et al. 2006), leading us to ask whether the tidal evolution of the Ceto/Phorcys system could have driven internal temperatures high enough to enable ductile com- paction and loss of void space. What temperature would be required is not clear, but for crys- talline ice Ih, it must exceed 120 K, since ice Ih remains brittle and can maintain its pore space up to at least that temperature (e.g., Durham et al. 2005). This temperature is considerably higher than equilibrium temperatures in the Kuiper belt. Energy from both circularization and spin- locking would contribute to internal heating. The specific energy (i.e., per unit mass) available from a change in orbital semimajor axis is �Ecirc G M 1 �a initial �1�afinal �1� , (10) where ainitial and afinal are the initial and final semimajor axes. If the orbit changes by ~10% in the process of circularization, then the total energy deposited is GM1/(20af), and we may take af as the presently observed value of a. Rotational kinetic energy of a spinning body is I�2 where � is the angular rotation rate and I = 2 MR2 is the moment of inertia for a homogeneous spherical body. The specific energy deposited in each body via spin locking is thus �E spin ��initial 2��final 2� , (11) where �initial and �final are the initial and final rotation rates. Taking the initial spin periods to be 10 hours and final spin periods to be 9.5 days, we obtain an energy input smaller than that from circularization by a factor of a few. Combining the above parts, the total specific energy deposit- ed in the system is 2� �� initial final , (12) which is approximately 105 erg g-1. Lunine and Tittemore (1993) present an alternate derivation of the tidal heating of a synchronously rotating satellite. Rewriting their expression for energy deposition, we have �G M13� R2 ��G M 1 M2 e a � (13) which gives dE/dt ranging from 1010 to 1014 erg s-1 for our bracketing Q' assumptions. The total energy input is independent of rigidity and is around 1027 erg, or 105 erg g-1, in agreement with our derivation above. The heat capacity of cold water ice Ih depends on temperature, ranging from 3 × 106 erg g-1 K-1 at 30 K to over 107 erg g-1 K-1 at 60 K (Giauque and Stout 1936), so even if all 105 erg g-1 of heat were deposited instantaneously instead of over a 106 year or longer timescale, the tempera- ture rise would be much less than a degree, far too small for viscous creep to assist compaction and loss of pore space. Comparable arguments can be made to show that heat tidally deposited by any close encounter with a giant planet which does not completely disrupt the system will also produce a negligible temperature rise. However, if tidal heating were highly localized, as might happen if energy dissipation was confined to the intersections of large scale internal frac- tures, the local temperature rise could be higher. That the Ceto/Phorcys system is expected to be tidally spin-locked and circularized sug- gests that the spin axes of both objects could also have become aligned with the orbit pole. Such a configuration would help explain why no lightcurve has been reported for this object, and why there is no evidence for it in our ACS/HRC photometry in Table 2 nor in our Spitzer/MIPS pho- tometry in Table 4. If the spin axes are aligned with the orbit pole, which was tilted about 28 from the line of sight from Earth at the time of the observations (as seen in Fig. 3), a lightcurve observable when the system was near equator-on would be reduced in amplitude by the current geometry by a factor of sin(28 ), equal to 0.47. A more pole-on configuration is also consistent with the small beaming parameter �. Smaller values of � correspond to warmer surface tempera- tures, as would be expected when observing a hemisphere mostly in continuous sunlight. Unfor- tunately, we cannot use the assumption of aligned spin orientations to constrain the thermal iner- tia of the surfaces of Ceto and Phorcys, since observable thermal flux becomes independent of thermal inertia in pole-on geometry (e.g., Lebofsky and Spencer 1989). 6. Effect of encounters In its current heliocentric orbit, Ceto/Phorcys is most likely to encounter Uranus, since its perihelion is just inside the orbit of Uranus and its argument of perihelion is near 180 (see Fig. 1). Tidal disruption becomes probable for planetary encounters with closest approach dis- tances within = a � 3 M planetM , (14) where a is the binary's semimajor axis and Mplanet and Msys are the masses of the planet and the bi- nary system, respectively (Agnor and Hamilton 2006). For encounters between Ceto/Phorcys and Uranus, rtd is about 0.0045 AU, equivalent to 26 times the radius of Uranus or 360 times the Ceto/Phorcys separation a. For an encounter with Jupiter, rtd would be 0.013 AU, or 1000 times a. The probability that Ceto/Phorcys would have had an encounter closer than rtd with any of the giant planets on its way to being perturbed into its current orbit is not known, but by analogy with (42355) Typhon/Echidna (Noll et al. 2006), it is perhaps not surprising that the Ceto/Phor- cys binary survived the sequence of planetary encounters which put it on its present-day orbit. Although these encounters are unlikely to have been close enough to disrupt the binary, as evi- denced by the simple fact of its existence, they could possibly be expected to have had some ef- fect on the eccentricity of the binary orbit. If so, a future survey of comparable TNBs with orbits sampling the range from zero to non-zero eccentricity may enable statistical limits to be placed on their Q values. Trilling (2007) explores the possibility of binary disruption by encounters with other TNOs. For Ceto/Phorcys, the most likely time for such an encounter to have taken place would have been prior to perturbation into a short-lived Centaur orbit. The “hardness” H of the binary can be expressed as a function of the mass M3 of a third body being encountered as G M1 M 2 2 a M 2 (15) (Binney and Tremaine 1987; Trilling 2007) where M1 and M2 are the masses of Ceto and Phorcys (assuming both have the same density), a is the semimajor axis of their mutual orbit, and �v is the velocity dispersion of potential interlopers, which is unknown, because we do not know what heliocentric orbit the system was on prior to its becoming a Centaur. For H(M3) greater than uni- ty, the system is “hard” against disruption by interlopers of mass M3. Assuming a velocity dis- persion of 1 km s-1 (Jewitt et al. 1996), we find that H(M3) is unity for interlopers in the 3 km size range, assuming an average density of 1 g cm-3. For smaller interlopers, the Ceto/Phorcys system is “hard” but for larger interlopers, it is “soft” and could be disrupted. Durda and Stern (2000) estimated rates of impacts in the Kuiper belt as a function of impactor size, finding that on aver- age, an object the size of Ceto+Phorcys should be struck by one interloper of this size over the age of the solar system. Additional close, but non-impacting encounters would also be expected. A possible consequence could be that binaries at least as widely separated as Ceto and Phorcys (i.e., all currently known TNBs other than the near-contact binaries) represent a remnant popula- tion which has lost many of its members over the age of Solar System, but that even tighter bina- ries could be immune to disruption in the present collisional environment, and so could reveal the primordial binary abundance. This argument supports the ideas of Goldreich et al. (2002) and Petit and Mousis (2004) that closer binaries could be considerably more abundant than the comparatively widely separated ones currently being discovered by ground-based and even HST searches (e.g., Kern and Elliot 2006; Noll et al. 2007). 7. Conclusion Combined Hubble Space Telescope and Spitzer Space Telescope observations reveal the Centaur (65489) Ceto/Phorcys to be a binary system with a separation of 1840 ± 48 km. The average geometric albedo is 0.084 �0.021 �0.014 , consistent with radiometrically determined albedos of other Centaurs. If both components have equal albedos, as suggested by their shared colors, the radii of Ceto and Phorcys are 87 �9 and 66 �7 km, respectively. The average bulk density of the system is then 1.37 �0.66 �0.32 g cm -3, higher than recent densities for other comparably-sized binary outer Solar System objects, but smaller than the densities of much larger icy worlds such as Plu- to, Triton, and Eris. Despite probable recent encounters with gas giants, the mutual orbit of Ceto and Phorcys is nearly circular, consistent with calculations showing that the system should be- come tidally circularized and synchronized on timescales shorter than the age of the Solar Sys- Acknowledgments This work is based in part on NASA/ESA Hubble Space Telescope Cycle 14 program 10508 and 10514 observations. Support for these programs was provided by NASA through a grant from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We are especially grateful to Tony Roman and Tricia Royle at STScI for their rapid reactions in schedul- ing the HST follow-up observations. STSDAS and PyRAF are products of STScI. This work is also based in part on Spitzer Space Telescope Cycle 3 program 30081 observations. 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0704.1524

GLRT-Optimal Noncoherent Lattice Decoding

IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 1 GLRT-Optimal Noncoherent Lattice Decoding Daniel J. Ryan†‡∗, Iain B. Collings‡ and I. Vaughan L. Clarkson⋆ †School of Electrical and Information Engineering, University of Sydney, AUSTRALIA, Phone: +612 9372 4465, Fax: +612 9372 4490, Email: [email protected] ‡Wireless Technologies Laboratory, CSIRO ICT Centre, AUSTRALIA, Phone: +612 9372 4120, Fax: +612 9372 4490, Email: [email protected] ⋆School of Information Technology and Electrical Engineering, University of Queensland, AUSTRALIA, Phone: +617 3365 8834, Fax: +617 3365 4999, Email: [email protected] Abstract This paper presents new low-complexity lattice-decoding algorithms for noncoherent block detection of QAM and PAM signals over complex-valued fading channels. The algorithms are optimal in terms of the generalized likelihood ratio test (GLRT). The computational complexity is polynomial in the block length; making GLRT-optimal noncoherent detection feasible for implementation. We also provide even lower complexity suboptimal algorithms. Simulations show that the suboptimal algorithms have performance indistinguishable from the optimal algorithms. Finally, we consider block based transmission, and propose to use noncoherent detection as an alternative to pilot assisted transmission (PAT). The new technique is shown to outperform PAT. Index Terms Noncoherent detection, lattice decoding, wireless communications. I. INTRODUCTION Noncoherent transmission of digital signals over unknown fading channels has recently received significant attention especially for the case of the block-fading channel model. Applications include recovery from deep fades in pilot-symbol assisted modulation based schemes, eavesdropping, and non-data-aided channel estimation. Noncoherent transmission is particularly applicable to systems exhibiting small coherence intervals where the use of training signals would result in a significant loss in throughput. Recently, some elegant information-theoretic results have been derived for noncoherent single and multiple-antenna systems under the assumption of Rayleigh fading, for I. V. L. Clarkson was on study leave throughout 2005 at the Dept. of Electrical & Computer Engineering, University of British Columbia, Canada. This paper has appeared in part at VTC-Spring 2006 and ICASSP 2006. http://arxiv.org/abs/0704.1524v1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 2 example [1, 2]. Information theoretic aspects of noncoherent transmission were considered in [3] which concluded that at low SNR and for small coherence intervals there is a significant capacity penalty by using training. Under this noncoherent detection regime, it has been shown by numerical simulation that standard modulation techniques such as quadrature amplitude modulation (QAM) can achieve near-capacity in the single-antenna noncoherent block Rayleigh-fading channel [4]. This paper focuses on noncoherent receiver design for the block fading channel. A great deal of work has been performed on partially coherent receivers such as pilot-symbol assisted modulation (PSAM) [5, 6], per-survivor techniques [7], and coupled estimators [8]. However the challenge remains to develop high-performance, low- complexity, fully noncoherent receivers. Various suboptimal algorithms have been proposed for block-based noncoherent detection. For slowly fading channels, a blind phase recovery approach was proposed in [9] for noncoherent detection of differentially encoded QAM [10] where the attenuation was assumed to be known exactly at the receiver. In [11], a suboptimal technique for PSK was proposed which involved forming a number of equally spaced channel phase estimates. An extension to multi-amplitude constellations was also presented, where every sequence of symbol amplitudes is considered, and then the PSK technique is applied to determine the phase of the symbols. Unfortunately, the complexity of this suboptimal approach is still exponential in the sequence length, albeit with a smaller base. Recently, lattice decoding algorithms have been applied to noncoherent and differential detection. For PSK over temporally-correlated Rayleigh fading channels, a form of lattice decoding (namely sphere-decoding) can be applied since it turns out that the detection metric is Euclidean [12]. Lattice decoding techniques have also been used for differential detection of diagonal space-time block codes over Rayleigh fading channels, by approximating the decision metric with a Euclidean metric [13, 14]. In [15], we presented simulation results for another lattice projection approach for suboptimal PAM and QAM detection. Unfortunately, each of these algorithms require complexity exponential with the block length to guarantee that the optimal estimate is found [16, 17]. Practical implementation considerations demand that low complexity algorithms be developed. For the case of the constant envelope PSK constellation, a detection algorithm with complexity O(T logT ) was developed in [18, 19] (where T is the block length), which can provide the optimal data estimate over an unknown noncoherent fading channel, in terms of the Generalized Likelihood Ratio Test (GLRT). The GLRT is equivalent to joint ML estimation of a continuous valued channel parameter and discrete-valued data parameters. This approach was generalized in [20], where they outlined a general graph-based approach which involved forming a spanning tree. Specific details were presented for the cases of QAM over a phase noncoherent channel (i.e. known channel amplitude) [20, 21], and for PSK over a fading channel with coding [22]. The challenge remains to develop efficient IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 3 algorithms for optimal noncoherent sequence detection of multi-amplitude constellations over fading channels. In this paper we propose a new GLRT-optimal noncoherent lattice decoding approach for QAM and PAM symbols which has complexity polynomial in the block length. We start by considering detection of M -ary PAM over real- valued fading channels (which we shall term real-PAM). We show that the GLRT-optimal codeword estimate is the closest codeword (or lattice point) in angle to the line described by the received vector. We propose an algorithm that searches along the line, and chooses the best codeword estimate from this search. We provide a theorem that bounds the search to a segment of the line, limiting the number of codewords that need to be considered. We show how the search can be done in an iterative manner, and that the resulting complexity of the algorithm is O(T logT ). We then consider the more practical case of M -ary QAM detection over complex-valued fading channels, and show that in this case the GLRT-optimal codeword estimate is the closest codeword in angle to a plane described by the received vector. We propose an algorithm that searches across the plane, and chooses the best codeword estimate from this search. We provide a theorem that bounds the search to a segment of the plane. We show how the extent of the search can be further reduced by exploiting the rotational symmetry of the constellation. The resulting plane search algorithm can be performed with complexity of order O(T 3). We also present new suboptimal noncoherent QAM detection algorithms with even lower complexity; by com- bining a channel phase estimator with our fast real-PAM algorithm. We propose using O(T ) instances of the real-PAM algorithm. This approach therefore has complexity of order O(T 2 logT ). Simulations indicate that there is a negligible performance loss compared to GLRT, when using this suboptimal technique. Finally, we also propose a pilot-assisted version of our new reduced-search noncoherent lattice-decoding algo- rithms. The pilot symbol is used to remove the ambiguities inherent with noncoherent detection. Our approach obtains improved performance compared with standard pilot assisted transmission [6], while maintaining the same data rate. II. SYSTEM MODEL A. Signal Model We define a codebook C(X , T ) as the set of all possible sequences of T transmitted symbols, x = [x1, . . . , xT ] such that each xt is in some constellation X . For an M -ary PAM constellation, X = {±1,±3, . . . ,±(M − 1)}. For QAM, X is a subset of the Gaussian (complex) integers with odd real and imaginary components. For example, for an M2-ary square QAM constellation, X = { x |Re {x} ∈ X ′, Im {x} ∈ X ′} where X ′ , {±1,±3, . . . ,±(M − 1)}. Thus each codebook C(X , T ) is a set of lattice points drawn from a subset of the unit lattice of RT or CT . We consider block fading channels and assume that the channel h is constant for at least T symbols as in [1–4, IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 4 11]. We will consider narrowband fading channels where h is either real-valued and complex-valued channels. Thus we can write the received codeword y = [ y1, . . . , yT ] ′ as follows, y = hx+ n (1) where n = [ n1, . . . , nT ] ′ is a vector of additive white Gaussian noise. B. Detection The noncoherent detection problem is to estimate x based on y without knowledge of the channel and in the absence of training data. The log-likelihood function of the maximum likelihood (ML) detector (of both channel and data) is given by L(y;x, h) = −‖y − hx‖ where constant factors have been discarded and ‖·‖ represents the Euclidean norm. For a given codeword hypothesis x̂, the likelihood function is maximized by choosing where (·)† denotes Hermitian transpose. Hence, the ML estimate of x conditioned on the corresponding channel estimate, is given by opt = arg max x̂∈C(X ,T ) y; x̂, = arg max x̂∈C(X ,T ) ∣x̂†y This is the Generalized Likelihood Ratio Test (GLRT) [23] considered in [11, 20]. Note that (4) is equivalently given by opt = arg max x̂∈C(X ,T ) ∣x̂†y = arg max x̂∈C(X ,T ) cos2 θ(x̂,y) (6) where θ(x,y) is the principal angle between x and y [24]. Thus x̂opt can be found by searching the points of C(X , T ) to find the one closest in angle to y. For QAM, we can also obtain a geometric interpretation of (4) by expressing the complex vectors in R2T . We will use the underscore notation x to denote the mapped version of x as follows, x = [ Re {x1} Im {x1} . . . Re {xT } Im {xT }] ′ (7) IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 5 and denote the real-valued codebook as CR(X , T ) = {x | x ∈ C}. For M 2-ary square QAM, we therefore have CR(X , T ) = C(X ′, 2T ) where X ′ is an M -ary PAM constellation. We also define Y ∈ R2T×2 as a basis for the subspace yC mapped into the real space R2T ; that is Re {y1} Im {y1} . . . Re {yT } Im {yT } − Im {y1} Re {y1} . . . − Im {yT } Re {yT } . (8) Note that the columns of Y are orthogonal. The projection matrix P(y) ∈ R2T×2T is defined as P(y) , such that P(y)x = arg min v∈YR2 ‖x− v‖ . That is, the vector P(y)x is the projection of x onto the subspace YR2. Now, it can be easily shown that = arg max x̂:x̂∈C(X ,T ) cos2 θ(x̂,P(y)x̂) Thus the GLRT-optimal data estimate x̂opt, corresponds to the x̂ ∈ CR(X , T ) closest in angle to the plane YR It is important to note that two forms of ambiguity exist for this noncoherent detection problem. The first is the well-known phase ambiguity which occurs for any constellation that is invariant to a particular phase rotation. For example, for square QAM constellations the following four optimal channel estimate and codeword pairs have the same likelihood: (ĥopt, x̂opt), (−ĥopt,−x̂opt), (−iĥopt, ix̂opt) and (iĥopt,−ix̂opt); corresponding to the four π/2 rotations of the constellation. We will assume that this type of ambiguity can be resolved, for example, by using the phase of the last symbol from the previous codeword [4], or by using differential encoding [10]. The second type of ambiguity we call a divisor ambiguity and arises when there are multiple points in C(X , T ) that lie on the same 1-dimensional (real or complex) subspace e.g. [1, 1, 1] and [3, 3, 3] for 4-ary real-PAM with T = 3. This produces a lower bound on the noncoherent block detection error rate as discussed and analyzed in [25]. III. REDUCED SEARCH SPACE In this section we show that the GLRT-optimal data estimate x̂opt, can be found without testing all the elements of C(X , T ). In the previous section we established that x̂opt is the codeword closest in angle to a particular subspace, so it naturally makes sense to define a ‘nearest neighbor set’ of the subspace and search within that set. The subspace of interest has basis vector y and passes through the origin. We show that the nearest neighbor set for this subspace contains x̂opt. This implies that low complexity decoding algorithms can be developed, based on finding this particular nearest neighbor set, and searching it. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 6 Definition 1: We define NN(v) to be the point, or set of points, in C(X , T ) closest to the arbitrary point v (i.e., the nearest neighbor to v). That is, d is an element of NN(v) if ‖v − d‖ 6 ‖v − z‖ for all z ∈ C(X , T ). Of course, usually NN(v) will have a single element, and in this case we can write NN(v) = d. Definition 2: We define N (C(X , T ),y) to be the nearest neighbor subset of the codebook C(X , T ), corresponding to the subspace with basis vector y, passing through the origin. That is, u ∈ N (C(X , T ),y) if and only if there exists some λ such that NN(λy) = u. Note that from a geometrical perspective, it is useful to think of λ as being equivalent to the inverse of a channel estimate; implying that a point u is in the nearest neighbor set if there is a channel estimate ĥ such that the distance |y− ĥu| is smaller than for any other point. Consequently we define λ̂opt , (ĥopt)−1 as the reciprocal of the optimal channel estimate. The following property of the GLRT-optimal codeword estimate x̂opt, allows us to reduce the set of codewords which need to be tested, to a small subset of the |X | possible codewords (where |·| denotes set cardinality). Note that an equivalent result was presented in [20], however the geometrical interpretation of our formulation is more apparent; and is important when developing our new search algorithms later. Property 1: opt ∈ N (C(X , T ),y). Proof: Consider the case where x̂opt /∈ N (C(X , T ),y). From Definition 2 this implies that there exists some x̂ ∈ N (C(X , T ),y) such that ||λ̂opty−x̂|| < ||λ̂opty−x̂opt|| however this would imply L(y; x̂, ĥopt) > L(y; x̂opt, ĥopt) from (2) and hence we have a proof by contradiction. IV. PAM DETECTION FOR REAL-VALUED FADING CHANNELS This section presents a low complexity algorithm for GLRT-optimal noncoherent PAM detection over real-valued channels. Practically, such channels arise in baseband transmission (eg. multi-level PCM), or in certain bandpass systems where phase and frequency are separately estimated by a phase-locked loop. We first present a theorem that we will use to reduce the number of codewords that need to be examined, even beyond the limitations imposed by Property 1. Note that in this real-valued channel case, the subspace of interest (defined by y) actually reduces to a line, yR. The theorem implies that only a limited extent of the line needs to be searched; and that the extent depends on the largest value of y. We then propose a fast low-complexity iterative algorithm to perform the search. In the sequel, we will extend the algorithm to complex-valued channels. Later, we will directly incorporate the algorithm from this section into an extremely low complexity suboptimal algorithm for noncoherent detection over the more commonly encountered complex-valued channels. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 7 A. Limiting the Search Space Theorem 1: For noncoherent detection of M -ary PAM codewords of length T over a real-valued fading channel ∣λ̂optyt ∣ 6 M + T − 2 for all t = 1, 2, . . . , T . Proof: Define n , argmaxt {|λ̂ optyt|}. Note that if |λ̂ optyn| 6 M then the theorem is satisfied. Now, consider the alternative case when |λ̂optyn| > M . Rearranging the GLRT-optimal channel estimate in (3) gives (x̂opt)′(y − ĥoptx̂opt) = 0 and hence (x̂opt)′(λ̂opty − x̂opt) = 0. (10) We will use this property to bound λ̂opt. Using (2) and the fact that C(X , T ) contains all possible sequences { x | xt ∈ X ∀t }, the elements of x̂ opt can be determined on an element-wise basis as t = argmin ∣λ̂optyt − x ∣. (11) For the case we are considering where |λ̂optyn| > M , it follows that since the largest PAM constellation values are ±(M − 1), that x̂optn = sgn {yn}(M − 1) where sgn is the signum function. We now substitute (11) into (10) to bound λ̂opt, which gives x̂optn λ̂optyn − x̂ λ̂optyt − x̂ . (12) Now (11) and the symmetry of the PAM constellation implies that sgn = sgn {λ̂optyt}. Moreover, since t ∈ {±1,±3, . . . ,±(M − 1)} it follows form the definition of λ̂ opt that −1 < λ̂optyt− x̂ t < 1 for all x̂ t except ±(M − 1). More generally, for all x̂optt , sgn (λ̂optyt − x̂ t ) > −1 and hence x̂ λ̂optyt − x̂ > −|x̂ Substituting this into (12) gives x̂optn λ̂optyn − x̂ and hence ∣λ̂optyn x̂optn + 6 M − 1 + (T − 1)(M − 1) M − 1 = M + T − 2. Therefore, since |λ̂optyt| 6 |λ̂ optyn| for all t the theorem is proved. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 8 B. PAM Algorithm In this Section we use Property 1 and Theorem 1 to develop a low-complexity algorithm for real-PAM detection. This algorithm reduces the number of codewords for which the decision metric is evaluated to order MT , which is much smaller than the set of all possible MT codewords that would be considered by an exhaustive search. Furthermore, we demonstrate how the algorithm can be implemented in an iterative manner so that the complexity is O(T logT ). Property 1 implies that x̂opt can be found by calculating the metric in (4) for only those x̂ ∈ N (C(X , T ),y), i.e. for only those x̂ ∈ C for which the line yR passes through its Euclidean nearest neighbor region. Furthermore, Theorem 1 implies that only a finite segment of the line need be considered. We have demonstrated such a search in Figure 1, which shows the positive axes for 8-ary PAM with T = 2, where the shaded regions indicate the nearest neighbor regions of the points which need to be searched. The specifics of the algorithm are as follows. First, for ease of notation we modify the received codeword y by changing the signs of all negative elements in y. This will mean that the corresponding (modified) x̂opt will now have all positive elements. The true (original) GLRT estimate of x can be obtained by applying the reverse sign changes to x̂opt. Observe that we can do this without loss of generality since the PAM constellation is symmetric around zero. Definition 3: We define P (x̂) to be the range of λ such that x̂ is the nearest neighbor to λy, within the limits 0 < λ < λmax , (M + T − 2)/maxt {yt} (where the limits are due to Theorem 1 and the fact that all xt are greater than zero for the modified received codeword). Formally, P (x̂) , {λ | x̂ ∈ NN(λy), λ ∈ (0, λmax)}. Note that each non-empty P (x̂) corresponds to a distinct interval of the line yR. The proposed algorithm proceeds by enumerating these non-empty P (x̂)’s, by first enumerating their boundary points along the line yR. We then sort the boundary points so that the decision metrics for the corresponding x̂ can be calculated in an iterative manner. For real-PAM, the boundary values of λ can be shown to be given by νt,b = for all t = 1, . . . , T and b = 2, 4, . . . ,M − 2 such that 0 < νt,b < λ max (where the values of b come from the regular boundaries in the positive half of the PAM constellation X ). We use V0 to denote the set of all (νt,b, t) pairs. We then sort the elements of V0 in ascending order of their νt,b value, and append the value (λ max, 0) to the end of the ordered set (since this is the outer boundary of the segment of the line yR which needs to be searched, according to Theorem 1; where the second element of the pair is arbitrarily set to 0 since it is not needed in the algorithm). We denote the newly ordered set by V , and index it by k, (i.e. we denote its kth element by (νk , tk)). These ordered values are shown on the example case of Figure 1, where the values of νk denote the distance along the line yR where the line crosses from one nearest neighbor region into the next. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 9 We now show how V can be used to enumerate the codewords which need to be searched, and show how to calculate the corresponding decision metrics in an iterative manner. The algorithm can be visualized geometrically as searching along a segment of the line yR, by iterating in k. Whenever the line crosses from the nearest neighbor region of one lattice point (codeword) to the nearest neighbor region of another, we calculate the metric for the new lattice point. Note that in this context, the value of tk indicates the dimension of the boundary that is going to be crossed (in the T -dimensional space) when leaving the kth segment of the line. The iterative search starts with the codeword x̂(1) = 1 , [ 1, 1, . . . , 1]′; which has a corresponding decision metric L(1) = (1′y)2/ ‖1‖2, where L(x̂) , (x̂′y)2/ ‖x̂‖2 is the likelihood function in (4). We will use the symbol λ̂ as a marker for the most likely codeword, and we initially set it to λ̂ = ν1/2 (i.e. during the iteration process, λ̂ will be updated whenever a codeword is found to have a higher likelihood than any previously searched codeword, and the value of λ̂ will be chosen such that NN(λ̂) gives the new codeword). The iteration proceeds by noting that each time a nearest neighbor boundary is crossed, only one element of the T -dimensional nearest neighbor codeword vector changes (since for real-PAM, the boundaries are straight lines, orthogonal to one of the dimensions, and parallel to all the others). Therefore the kth codeword which needs to be considered, is calculated from the (k − 1)th codeword, on an element-wise basis as follows: x̂(k)p = (k−1) p , for p 6= tk−1 p + 1, for p = tk−1. We define αk , (x̂ (k))′y and βk , ∥x̂(k) , and hence L(x̂(k)) = α2k/βk is the decision metric for the kth codeword considered. The values αk and βk are calculated iteratively as follows, αk = αk−1 + 2ytk−1 (14) βk = βk−1 + 4x̂tk−1 + 4. (15) If L(x̂(k)) improves on the previous best codeword estimate then we update λ̂ in the interior of P (x̂(k)), by setting λ̂ = (νk + νk−1)/2. Once all segments of the line have been searched, we have x̂opt = NN(λ̂y). Pseudo-code for the algorithm is given in Table III. The complexity of the algorithm is a function of the number of intersection points νt,b, i.e., NI , |V0|, where |·| denotes set cardinality. NI is upper bounded by (M/2− 1)T , however in general it will be much less than this due to the restricted line search implied by Theorem 1, as shown by simulation in Section VII. The sorting of V0 can be performed using standard sorting techniques in O(NI logNI) [26]. The updates (13), (14) and (15) have complexity O(1), and the final calculation of x̂opt is of order T . Thus the overall complexity is dominated by the IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 10 sorting operation, and hence the complexity of the algorithm is of order O(T logT ); a significant improvement compared with an exhaustive search over all MT possible codewords in the codebook C(X , T ). V. GLRT-OPTIMAL QAM DETECTION FOR COMPLEX-VALUED FADING CHANNELS This section presents a low complexity algorithm for GLRT-optimal noncoherent QAM detection over complex- valued fading channels. Similarly to the the real-PAM case, we first present a theorem that we will use to reduce the number of codewords that need to be examined, beyond the limitations imposed by Property 1. In the complex- valued channel case, the subspace of interest is the plane YR2, where Y was defined in (8). The theorem implies that only a limited extent of the plane needs to be searched; and that the extent depends on the largest element in y. We then propose a fast low-complexity algorithm to perform the search for QAM. We also show how PAM detection over complex-valued channels can be viewed as a special case of the QAM algorithm. A. Limiting the Search Space Theorem 2: For noncoherent detection of M2-ary QAM codewords of length T over a complex-valued fading channel λ̂optyt ∣ 6 M + 2T − 2, and λ̂optyt ∣ 6 M + 2T − 2, for all t = 1, 2, . . . , T . Proof: Define the point v , λ̂opty, along with its corresponding real-valued representation v, as in (7). Also define n , argmaxt {|vt|}. Note that if |vn| 6 M then |Re {λ̂ optyt}| 6 M and |Im {λ̂ optyt}| 6 M for all t and the theorem is satisfied. Now, consider the alternative case when |vn| > M . Similarly to the real-PAM case, rearranging the GLRT-optimal channel estimate in (3) gives (x̂opt)†(y − ĥoptx̂opt) = 0 and hence (x̂opt)†(λ̂opty − x̂opt) = 0. (16) It follows that λ̂opt = ‖x̂opt‖ |y†x̂opt| Combining this with the the fact that for any vector u ∈ CT , the real-valued representation of the complex scalar y†u is Y′u, we obtain the real-valued representation of λ̂opt as opt = IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 11 and therefore v = Yλ̂opt = It follows that v′x = ‖x‖ , i.e. )′(v − x̂opt) = 0. (17) Using (2) and the fact that C(X , T ) contains all possible sequences { x | xt ∈ X ∀t }, the elements of x̂ opt can be determined on an element-wise basis as t = arg min x∈X ′ |vt − x| (18) for all t = 1, . . . , 2T where we recall from Section II-A that X ′ = {±1,±3, . . . ,±(M − 1)}. We now substitute (18) into (17) which gives x̂optn vn − x̂ vt − x̂ . (19) This is similar to (12) in the proof of Theorem 1. By following through the subsequent steps in the proof of Theorem 1, and keeping in mind that the dimensions of the vectors are now of dimension 2T , we obtain |vn| 6 M +2T − 2 which implies that |Re {λ̂optyt}| < M + 2T − 2 and |Im {λ̂ optyt}| < M + 2T − 2 for all t = 1, . . . , T . B. QAM Algorithm In this Section we use Property 1 and Theorem 2 to develop a low-complexity algorithm for QAM detection. Property 1 implies that x̂opt can be found by calculating the metric in (4) for only those x̂ ∈ N (C(X , T ),Y), i.e. for only those x̂ ∈ C for which the plane YR2 passes through its Euclidean nearest neighbor region. Furthermore, Theorem 2 implies that only a finite region of the plane need be considered. Conceptually, this is a direct extension of the real-PAM case shown in Figure 1 (considered previously). The difference being that Figure 1 shows the line yR, but we now have a plane YR2. Also the number of orthogonal dimensions doubles when considering complex-valued channels. We demonstrated this complex-valued channel QAM case in Figure 2 which is a two dimensional plot in the plane YR2. The parallel lines (at various angles) are the boundaries arising from the QAM constellation, and the shaded region indicates the nearest neighbor regions of codewords which need to be searched. The QAM search algorithm we present here, follows the same principles as the real-PAM algorithm of Section IV-B, where instead of working with boundary points of line segments, we need to work with boundary edges of planar regions. The specifics of the algorithm are as follows. First, for ease of notation we modify the received codeword y by multiplying it by the complex scalar y∗m/|ym|, where m = argmaxt |yt|. This will mean that the mth element of y will be real-valued and positive. The true IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 12 (original) GLRT-optimal estimate of the channel can be be obtained by applying the reverse phase rotation to ĥopt, while the optimality of the new GLRT-optimal codeword estimate is unaffected. Hence Theorem 2 implies that the search over the plane Yλ is reduced to the segment of the plane for which |λ1|, |λ2| < λ max where λmax , (M +2T − 2)/|ym|. Furthermore, as discussed in Section II-B because of the π/2 phase ambiguity in square QAM constellations, there are four GLRT-optimal inverse channel estimates ±λ̂opt,±iλ̂opt (with corresponding phase ambiguous GLRT-optimal codeword estimates). Hence, we only need to consider the square region of the plane S = { λ | λ1 ∈ (0, λ max), λ2 ∈ [0, λ max) } (20) since exactly one of ±λ̂opt,±iλ̂opt will exist in this region of the plane. Note that S is the shaded region in Figure 2 (mentioned previously). Similarly to the real-PAM case we make the following definition. Definition 4: We define P (x̂) to be the range of λ ∈ S such that x̂ is the nearest neighbor to Yλ. Formally, P (x̂) , {λ | x̂ ∈ NN(Yλ), λ ∈ S }. Note that each non-empty P (x̂) corresponds to a distinct region of the plane YR2. The proposed algorithm proceeds by enumerating these non-empty P (x̂)’s, by first enumerating their boundary vertices in the plane. These vertices are found by calculating the intersection of all the constellation-point boundary lines in the plane (e.g. as shown in Figure 2). The vertices are then used to calculate an interior-point inside each of the nearest neighbor regions in the shaded square S. The respective nearest neighbor codeword is calculated for each interior-point, and then it is only these points for which the likelihood metrics are calculated. Clearly, this is a significantly reduced search space compared with the space of all possible codewords. For QAM the vertices of the nearest-neighbor regions in the plane YR2 can be found by first noting that, since t can be given in on an element-wise basis as in (18), P (x̂) can be written as P (x̂) , { λ | xt = arg min x∈X ′ |(Yλ)t − x|, λ ∈ S} where (Yλ)t is the tth element of Yλ and we recall that X ′ = {±1,±3, . . . ,±(M − 1)}. This can be written as the feasible region for the set of linear inequalities corresponding to the nearest neighbor region boundaries in X ′ for each element of x̂t, as P (x̂) = { λ | l(x̂t) 6 (Yλ)t 6 u(x̂t), λ ∈ S} where l(x̂t) and u(x̂t) are the upper and lower nearest neighbor boundaries in the constellation X ′. For t /∈ {2m, 2m− 1} they take on values in the set {0,±2, . . . ,±(M − 2),±∞}. For t ∈ {2m, 2m− 1} we must IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 13 consider intersections with the boundary of S, and therefore in this case l(x̂t) and u(x̂t) take on values in the set {0,±2, . . . ,±(M − 2),±(M + 2T − 2)}. By including the square boundary of the region S, all non-empty P (x̂) are closed simply connected sets on the plane R2. Therefore, since P (x̂) is formed from linear inequalities it is a convex polygon in R2. For each P (x̂), denote B(x̂) as its polygonal boundary and V (x̂) as the vertices of the polygon. We now propose a method that enumerates all the vertices V (x̂) for all non-empty P (x̂), and then uses these vertices to generate a point in the interior of all P (x̂), which is then used to obtain a unique codeword via finding the nearest neighbor codeword to that point. Consider the set of points { ν ± ǫµ |ν ∈ V (x̂)}. If µ is some vector that is not parallel to any side of the polygon P (x̂), and if ǫ is chosen sufficiently small, then at least one point in this set will be in the interior of P (x). Since the received symbol is subject to AWGN, and is therefore irrational with probability one, it follows that the arbitrary choice of µ , [ 1 1 ]′ will almost surely guarantee this, given that ǫ > 0 is chosen sufficiently small. In practice, simply setting ǫ to some small positive constant will be sufficient to ensure that a point in the interior of P (x̂) is enumerated. However, in Appendix A we present a technique to perform this in a strictly optimal fashion with complexity per vertex of O(T ). Since the vertices are shared by adjacent P (x̂), each vertex is only required to be enumerated once. We define the set of all vertices within or on the boundary of S as V = { ν | ν ∈ V (x̂), P (x̂) 6= ∅ }. The set V can be enumerated as the the intersections of the lines Y t,1ν1 + Y t,2ν2 = b and Y t′,1ν1 + Y t′,2ν2 = b ′, for all pairs of t, t′ and for all nearest neighbor boundaries b, b′ in X ′. That is Y t,1 Y t,1 Y t,2 Y t,2 for all t = 1, 2, . . . , 2T − 1, t′ = t + 1, t + 2, . . . , 2T , and b, b′ ∈ B(t), where B(t) , {0,±2, . . . ,±(M − 2)} if t /∈ {2m, 2m− 1} and for symbol indices t ∈ {2m, 2m− 1} where we consider the square boundary B(t) , {0,±2, . . . ,±(M − 2),±(M + 2T − 2)}. To enumerate a point in each P (x̂), for each vertex ν enumerated we calculate the points on the plane λ+ , ν+ǫµ, and λ− , ν+ǫµ. Then for each of these two points, if it is in the square S, we calculate the corresponding codewords NN(λ+Y) and/or NN(λ−Y) and the decision metrics in (4). Pseudo-code is provided in Table IV. The complexity of the algorithm is a function of the number of codewords examined, NC , which is in turn a function of the number of vertices calculated. The number of vertices calculated in (21) corresponding to the intersections between lines in where b, b′ is a boundary of X ′ and both b and b′ are non-zero is T (2T − 1)[(M − 1)2 − 1]; for which at most two codewords are generated for a quarter of these intersections. For the intersections IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 14 of the boundaries of X ′ and the square S there are 2(2T − 2)(M − 1)2 intersections, which for one quarter of these intersections one codeword is generated. For the vertices at (0, 0) and (λmax, λmax) one codeword is generated. Hence the total number of codewords examined is at most T (2T − 1) [(M − 1)2 − 1] + (2T − 2)(M − 1)2 + 2. (22) Since the complexity of each codeword and decision metric calculation is of order T then the overall complexity is of order M2T 3 (which is linear in the constellation size M2) a significant improvement over an exhaustive search over all M2T possible codewords in the codebook C(X , T ). A further reduction in computational expense, without any loss in optimality, can be achieved by enumerating only one out of each set of four phase ambiguous vertices. The technique is not presented here due to space constraints, however the number of non-zero vertices examined is reduced by a factor of 4 and 1/3 of the matrix inverse calculations in (21) are avoided. C. PAM Over Complex Channels PAM detection over complex fading channels can be viewed as a special case of complex-channel QAM, where there is zero imaginary component in the constellation. In this case, the search over the plane YR2 can be restricted by extending the proof of Theorem 2. To do this, we note that the condition in (16) holds, which implies that (x̂opt)′(Re {λ̂opty} − x̂opt) = 0 since x̂opt is always real-valued. The rest of the proof follows to give the result that |Re {λ̂optyt}| 6 M + T − 2. This fact combined with Property 1 and the π phase ambiguity of PAM constellations, implies that we only consider codewords x̂ = NN(Yλ) for λ in the region S = { λ | 0 < λ1 < λ max = (M + T − 2)/|ym| }. The specifics of the M -ary PAM algorithm are the same as for the M2-ary QAM case, with the exception that the calculation of (21) to obtain the vertices in the interior of the (21) is only performed for all t = 1, 3, . . . , 2T −1, t′ = t + 2, t+ 4, . . . , 2T , and b, b′ ∈ B(t), where B(t) , {0,±2, . . . ,±(M − 2)} if t 6= 2m− 1 and for B(t) , {0,±2, . . . ,±(M − 2),M + T − 2} if t = 2m− 1. The total number of codewords searched can be shown to be at most T (T − 1) [(M − 1)2 − 1] + (T − 1)(M − 1) + 1. (23) Since the complexity of each codeword and decision metric calculation is of order T then the overall complexity is O(T 3). In the following section, we will see that a simple suboptimal approach can achieve even lower complexity with near-optimal performance. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 15 VI. SUBOPTIMAL ALGORITHMS FOR COMPLEX-VALUED FADING CHANNELS In this section, we propose even lower complexity suboptimal algorithms for detection of QAM and PAM over complex-valued fading channels. We directly use the GLRT-optimal algorithm for real-PAM from Section IV as the basis for the algorithms. A. Suboptimal PAM algorithm Since for PAM constellations, all constellation points lie along the real line in the complex plane, a suboptimal phase estimation technique combined with our GLRT-optimal algorithm for real-valued fading channels should be sufficient to provide near-optimal performance. This effectively reduces the search over the whole plane YR2 for the GLRT-optimal case, to a search over a single line at the given estimated phase angle. We use the power-law estimator [27] which, for constellations exhibiting a rotational symmetry of π radians, is simply φ̂PL , y2t (24) where ∠ refers to the complex argument. Detection is performed by first rotating the received codeword y according to this estimate, and then detecting Re{e−jφ̂PLy} using the GLRT-optimal algorithm of PAM over a real-valued fading channel. B. Suboptimal QAM algorithm Here we propose a suboptimal algorithm, which reduces the overall algorithmic complexity to O(T 2 logT ) by using O(T ) instances of the PAM detection algorithm presented in Section IV. Instead of enumerating the intersections of lines on the (λ1, λ2)-plane, as we did in Section V-B, here we propose to use a modified version of the nearest-neighbor real-PAM line-search algorithm for L lines of the type presented in Section IV. We generate these lines emanating from the origin into S (the shaded region in Figure 2), evenly spaced in angle. Of course, this does not guarantee that we fully enumerate N (C(X ), T ) since a finite number of radiating lines can not completely cover a plane, however, we will see by simulation in Section V-B that the performance is close to the optimal. As in the optimal case, we multiply y by y∗m/|ym| so that ym will be real-valued and positive. In this suboptimal QAM case, this implies that we only examine points on the plane Yλ for λ = [ λ1 λ2 ] ′ satisfying 0 < |λ1|, |λ2| < λmax0 , where λ 0 , (M + 2T − 2)/|ym|. The L directions of the lines with respect to the direction of positive λ1 have angles Φ , {φ1, . . . , φL} where φℓ = (ℓ − 1)π/(2L). For each angle φℓ, we perform a nearest neighbor line search for the line with basis vector [ cosφℓ sinφℓ ], as proposed in the suboptimal PAM algorithm in Section VI-A. The search is performed for IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 16 the segment of the line λy where λ ∈ R and 0 < λ < λmaxℓ where λ l , λ 0 /max{cosφℓ, sinφℓ}. In this case the lines searches are performed for blocks of length 2T . There is of course a modification required to update the codeword metrics in terms of complex numbers. The first line search performed is for φ1 = 0, and hence the line search is over λy1 = λy. In this case the intervals of the line P (x̂) are defined as, P (x̂) , λ | x̂ ∈ NN(λy ), λ ∈ (0, λmaxl ) Hence the algorithm works by enumerating and calculating the metric for all x̂ ∈ C(X ′, 2T ) for which P (x̂) is non-empty. In this case the set V0 of boundary points of the regions P (x̂) is enumerated by calculating νt,b = b/|yt| for all t = 1, . . . , 2T and b = 2, 4, . . . ,M − 2, (which are the nearest neighbor boundaries in the positive half of the constellation X ′), and storing only those values of (νt,b, t) such that νt,b < λ ℓ . The set of ordered boundary points V is again obtained by sorting, and (λmaxℓ , 0) is appended to V as the extent of the search. Recall that (νk, tk) are the kth elements of V . The search through the codewords is initialized to the first codeword for the which the line segment passes through, which is given by x̂(1) = s where s , sgn . The likelihood update variables are initialized to α = (x̂(1))†y and β = ‖x̂(1)‖2. To regenerate the optimal codeword, the values of λ and φ are initialized to λ = ν1/2 and φ = φ1 = 0. The (k + 1)th codeword considered, x̂(k+1), is calculated from the kth codeword as (k+1) + 2stk . (25) To update the decision metric we define αk , (x̂ (k))†y and βk , ‖x̂ ‖2, and hence L(x̂(k)) = |αk| 2/βk is the decision metric for the kth codeword considered. The values αk are updated as follows, If tk is odd, then αk is updated as αk−1 + 2stk−1y(tk−1+1)/2, tk−1 odd αk−1 − 2istk−1ytk−1/2, tk−1even. The values of βk are updated according to βk = βk−1 + 4stk−1 x̂tk−1 + 4. (27) If L(x̂(k)) = |αk| 2/βk improves on the best codeword estimate then we store λ = (νk + νk−1)/2 and φ = φℓ. To start the next line search, y is multiplied by e 2L and the line search is then performed again for the new value of y. When all line searches have been performed, we calculate x̂opt = NN(λejφy) for the original y. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 17 Pseudo-code is provided in Table V. The significantly reduced algorithmic complexity compared to the GLRT-optimal algorithm is governed by the number of line searches and the complexity of each line search. Since there are L phases, each performing a version of the real-PAM line-search algorithm of Section IV for the case M -ary PAM detection of 2T symbols. Thus NC 6 L(2T (M/2−1)+1). From Section V-B we have noted that the number of codewords in N (C(X , T ),y) is of order M2T 2 and thus L must be O(T ) for it to be possible that the majority of N (C(X , T ),y) is enumerated. Hence, if L is increased proportionally to T , the overall complexity of the algorithm is O(T 2 log T ). Note that however, the improved computational performance of the algorithm is largely due to being able to choose L small, which corresponds to avoiding examining a significant number of the x̂ with associated P (x̂) being so small as to imply that x̂ is not relatively close in angle to the plane YR2. We will see via simulation in Section VII that small L (e.g. L = 4 for T = 7 16-QAM detection) can achieve near-optimal performance. VII. SIMULATION RESULTS We now present simulation results to demonstrate the performance of the new PAM and QAM noncoherent reduced search lattice-decoding algorithms. Simulations are performed to obtain the codeword error rate (CER) as a function of SNR for noncoherent detection of 8-ary PAM and 16-ary square QAM. For both case, the simulations are performed for codeword lengths of T = 3 and 7 over a block Rayleigh fading channel where h is i.i.d. circularly symmetric complex Gaussian with unit variance. We have assumed that the phase ambiguities have been removed within each codeword, (for example, by the use of differential encoding [10]). Figure 3 presents results for 8-ary PAM for the GLRT-optimal plane search algorithm from Section V-C and the suboptimal phase-estimator plus line-search algorithm from Section VI-A. We also compare with the suboptimal grid-search algorithm proposed in [20] and the quantization based receiver proposed in [11]. For the grid-search algorithm we use uniformly spaced channel phase estimates and the channel attenuation estimates are chosen uniformly from the CDF of the Rayleigh fading channel distribution. For fairness the number of channel attenuation estimates is adjusted so that the total number of channel estimates was kept equal to the maximum number of codeword estimates that potentially could be produced by our GLRT-optimal algorithm. Best results are obtained for choosing the channel phase estimates as 0 and π/2, and hence the kth channel amplitude estimate is given by |ĥ(k)|2 = − log(1 − k/(1 + ⌈NC/L⌉)). For the quantization-based receiver (QBR) considered in [11], all possible sequences of (positive) amplitude levels are produced, and the sign of each symbol is then determined by symbol-wise coherent detection using uniformly spaced channel phase estimates (a channel amplitude estimate is not required since the signal amplitude is assumed known). For QBR, we again use the channel phase estimates 0 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 18 and π/2. Figure 4 presents the CER as a function of SNR, for 16-QAM transmission. Results are shown for the GLRT- optimal QAM algorithm given in Section V-B and the suboptimal algorithm given in Section VI-B. We also compare with the grid-based algorithm, where best performance for a fixed number of codeword estimates was obtained using L = 4 channel phase estimates, which we also use for QBR. For both the PAM and QAM cases we see that the suboptimal line-search algorithms provide negligible per- formance loss compared to the GLRT-optimal algorithm. For the case of T = 3, where QBR is computationally possible, there is a noticeable performance loss. As discussed in Section II-B, divisor ambiguities result in a lower bound on the CER. Expressions for these lower bounds were provided in [25] and are also shown in the figure. Clearly, for high SNR, both of our GLRT-optimal algorithms and both suboptimal algorithms detection achieve these bounds for both PAM and QAM. As noted in [11], there is an inherent suboptimality introduced by quantizing the unbounded channel attenuation by employing a grid-search approach, and hence the performance is clearly inferior. Also, although QBR achieves near-optimal performance for T = 3, since the complexity of QBR increases exponentially with T is not possible to produce curves for T = 7. In Table I we present the relative computational complexities of the algorithms for the simulations in terms of the average number of codewords examined. The numbers in brackets indicate the number of codewords examined by the search if the restrictions on the search region provided by Theorems 1 and 2 are not applied (and are therefore slightly greater than the worst case values given in (22) and (23)). We see that the suboptimal phase-estimator plus line-search approaches examine far fewer codewords yet obtains near-optimal performance, and that the complexity of QBR quickly becomes infeasible with increasing T . GLRT-Optimal Phase Estimator QBR Grid Reduced Search + Line Search Search 8-PAM T = 3 132.3 (173) 7.3 (10) 128 174 8-PAM T = 7 772.6 (1093) 16.4 (22) 32768 1094 16-QAM T = 3 52.6 (87) 22.9 (28) 108 88 16-QAM T = 7 311.8 (439) 52.9 (60) 8748 440 TABLE I NUMBER OF CODEWORDS EXAMINED FOR NONCOHERENT PAM AND QAM DETECTION VIII. REDUCED AMBIGUITY TRANSMISSION In this section we extend our new noncoherent detection algorithm to pilot assisted transmission (PAT) systems [6]. Unlike, standard PAT we propose to use the pilot symbol for noncoherent ambiguity resolution, rather than IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 19 simply for channel estimation. We propose to replace the pilot symbol of PAT with a symbol generated in the following way. Two bits are allocated for resolving the π/2 phase ambiguity of square QAM, and the remaining bits in the symbol are allocated to parity, remove divisor ambiguities and improve error performance. Therefore, this scheme has the same data rate as PAT and can be compared directly. With parity check bits in the codeword, we can now even further reduce the search space of our reduced search GLRT lattice-decoding algorithm by only considering codewords which satisfy a parity check. This significantly reduces the ambiguity problem. We will denote this parity-aided transmission scheme as reduced ambiguity (RA) transmission. An arbitrarily chosen parity check scheme might reduce the number of divisor ambiguities, however since the metric (4) has a geometric interpretation it may be possible to design other parity-check schemes which both resolve ambiguities and optimize performance by providing a minimum angular separation between codewords. The resolution of ambiguities can be achieved, at least for 16-QAM, by using the following parity-check scheme. Two parity bits p1, p2 are calculated from the data bits {d1, d2, . . . , d2(T−1)} as follows, p1 ≡ 1 + 4(T−1) dt (28) p2 ≡ 1 + 2(T−1) d2t (29) where ≡ denotes equality in GF (2). They are then mapped to the upper right-hand quadrant of the QAM constellation of the first (pilot) symbol in the codeword as follows: (00) ֌ 1 + j, (01) ֌ 1 + 3j, (11) ֌ 3 + 3j and (10) ֌ 3 + j. Effectively this means the first two bits of the first symbol of each codeword is chosen such that x1, x2 > 0, which removes the π/2 phase ambiguity, and the other two bits are parity bits, which in this case can be shown to completely remove the divisor ambiguities (see Appendix B). Figure 5 presents the bit error rate (BER) as a function of SNR for detection of 16-QAM transmitted over a block independent phase-noncoherent AWGN channel. Again we have assumed that the phase ambiguities have been removed within each codeword. Results are shown for three codeword lengths T = 3, 5, 7. The figure shows curves for our new RA reduced-search GLRT-optimal algorithm, and compares them to standard PAT. Both schemes use a single pilot symbol per codeword; which for the RA scheme is generated as described above, and for PAT it is a symbol which has energy equal to the average energy per symbol. For PAT, the GLRT estimate of the channel (based on the pilot symbol) is used to perform GLRT-optimal data detection, while for RA lattice decoding we use our reduced search GLRT-optimal algorithm. Note that for PAT, the BER is independent of the codeword length T since it is a symbol-by-symbol detection scheme, whereas for RA lattice decoding the BER decreases as T increases since it is a sequence detection scheme. Clearly our scheme outperforms PAT increasingly with T . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 20 Figure 6 shows the CER for the scenario of Figure 5. This serves to highlight even further the benefit from our lattice (sequence) decoding approach compared with PAT. For PAT, since bit errors occur independently on a symbol-by-symbol basis, the CER increases with T . However, for RA lattice decoding the CER decreases. Also the figure highlights the advantage of using pilot symbols, compared with fully noncoherent transmission, by observing that the SNR range is significantly lower than for Figures 3 and 4. IX. CONCLUSION In this paper we developed polynomial-time lattice-decoding algorithms for noncoherent block detection of PAM and QAM. Faster suboptimal algorithms for QAM were also presented which have excellent agreement with the optimal algorithms. A reduced ambiguity transmission scheme was introduced which was shown to outperform pilot assisted transmission over the phase noncoherent channel. APPENDIX A. Strictly Optimal Calculation of Interior Points For each non-empty region P (x̂), there exists a vertex ν ∈ V (x̂) and small scalars ν+, ν− > 0, such that either ν+ , ν + [ ν+ 0 ]′ or ν− , ν + [ ν− 0 ]′ is in the interior of P (x̂). Suppose the first case is true. Now, the line ν+ γ[ 1 0 ]′intersects an edge of the boundary of P (x), and we will call this intersection point µ. We propose to choose ν+ = γ > 0 so that ν+ is the midpoint of ν and µ. Defining ut as the tth element of u = Y ν, where Y is defined by the original received vector y, we can calculate ν follows, Note that almost surely y 6= 0. Similarly, using the line ν − γ[ 1 0 ]′ we calculate ν− = γ > 0 as This process will in general always calculate a point in each non-empty P (x̂). However, to avoid calculation problems we first rotate ν by ym/|ym|, so that the vectors [ ν + 0 ]′ and [ ν− 0 ]′ are not parallel to any of the edges of P (x̂) (e.g. those that are part of S). This rotation is later reversed, so that the points calculated are in the original coordinates. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 21 B. Removal of Ambiguities in 16-ary QAM In this section we show that the proposed RA pilot symbol approach (using parity checks, as discussed in Section VIII) totally removes both the phase and divisor ambiguities otherwise inherent in a noncoherent detection system (as discussed in Section II-A). We start by recalling that the proposed parity scheme involves calculating the parity bits from the data bits dt for t = 1, . . . , 4T as follows. p1 ≡ 1 + 4(T−1) dt p2 ≡ 1 + 2(T−1) d2t (30) where ≡ denotes equality in GF (2). The data and parity bits are then mapped to the symbols as shown in Table II, where we recall from the definition in (7) that x2t−1 = Re {xt} and x2t = Im {xt}. d2t−1d2t xt 00 −3 01 −1 p1p2 x1 (pilot symbol) 00 1 + i 01 1 + 3i 11 3 + 3i 10 3 + i TABLE II MAPPING OF DATA AND PARITY BITS. Since x1 is constrained to have positive real and imaginary components, the phase ambiguity has been removed. It remains to show that all divisor ambiguities have also been removed. To do this, we first define the associates of a Gaussian integer g to be the elements of the set A(g) = {g, gi,−g,−gi}. We also denote A(g)T to be a codeword of length T composed of only elements of A(g). For 16-QAM, it can be easily shown that a necessary condition for a divisor ambiguity to exist is that there exists codewords x(1) ∈ A(g1) T , x(2) ∈ A(g2) T for some g1, g2 ∈ X , {1 + i, 3 + 3i, 3 + i, 1 + 3i} such that g1 6= g2. For a codeword x and some g ∈ X , we define N1, N2, N3 and N4 as the number of occurrences in a codeword of each of the four possible rotations of g in the codeword, that is g, gi,−g and −gi respectively. Noting that the phase ambiguity has been removed (since x1 is constrained to have positive real and imaginary components), a sufficient condition for two codewords to be unambiguous is that there exists some t, such that the tth symbols from the two codewords are in different quadrants of the complex plane. It follows then, that a sufficient condition for two codewords to be unambiguous is that they do not have the same values of N1 to N4. We now use this property on N1 to N4 to show that for arbitrary T , it is not possible for two ambiguous codewords x(1) ∈ A(g1) T , x(2) ∈ A(g2) T , to satisfy the parity check (30) for any g1, g2 ∈ X such that g1 6= g2. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 22 We consider each g ∈ X (where we have previously defined X = {1+ i, 3+ 3i, 3+ i, 1+ 3i}) in turn, showing that all codewords x ∈ A(g)T that satisfy the parity check, are distinguishable in phase from all parity-satisfying codewords considered up to that point. For 16-QAM this process involves considering the four Gaussian integers 1 + i, 3 + 3i, 3 + i and 1 + 3i in turn, as detailed in the following four cases. Define xD to be the data codeword component of x, i.e. x = [x2 . . . xT ] • Case xD ∈ A(1 + i)T−1: In this case, we show that there does not exist any x ∈ A(1 + i)T that satisfies the parity check. Using Table II, the bits (d4ℓ−3 . . . d4ℓ) are mapped to the symbol xℓ = x2ℓ−1 + ix2ℓ ∈ x D in the following way: (1111) ֌ 1 + i, (0111) ֌ −1 + i, (0101) ֌ −1 − i and (1101) ֌ 1 − i. Clearly from (29), p2 ≡ 1, and therefore the pilot symbol x1 will be either 1 + 3i or 3 + 3i. It follows that x /∈ A(1 + i) • Case xD ∈ A(3 + 3i)T−1: In this case, we show the conditions under which a codeword x ∈ A(3 + 3i)T satisfies the parity check. The associated bit mappings are (1010) ֌ 3 + 3i, (0010) ֌ −3 + 3i, (0000) ֌ −3− 3i and (1000) ֌ 3 − 3i. Clearly, p1 ≡ 1 +N2 +N4 and p2 ≡ 1. Therefore, 1 + 3i, if (p1p2) = (01) i.e. if N2 6≡ N4, 3 + 3i, if (p1p2) = (11) i.e. if N2 ≡ N4. Furthermore it follows that x ∈ A(3 + 3i)T only if N2 ≡ N4. • Case xD ∈ A(3 + i)T−1: In this case, we show the conditions under which a codeword x ∈ A(3 + i)T satisfies the parity check, and show that under these conditions there does not exist any ambiguous codeword from A(3+3i)T , i.e. from the previous case. The bit mappings are (1011) ֌ 3+i, (0110) ֌ −1+3i, (0001)֌ −3−i and (1100) ֌ 1−3i. In this case, p1 ≡ 1 + N1 + N3 and p2 ≡ 1 + N1 + N2 + N3 + N4 ≡ 1 + T − 1 ≡ T . If T is odd, then p2 ≡ 1 and therefore x1 ∈ {1 + 3i, 3 + 3i} and therefore x /∈ A(3 + i) T . If T is even then p2 ≡ 0 and p1 ≡ 1 +N1 +N3 ≡ N2 +N4. Therefore 3 + i if (p1p2) = (10) i.e. if N2 6≡ N4, 1 + i, if (p1p2) = (00) i.e. if N2 ≡ N4. It follows that x ∈ A(3 + i)T only if N2 6≡ N4 and T is even. Recall that in the previous case, valid parity satisifying codewords only occurred if N2 ≡ N4. Therefore an ambiguity will not occur between two codewords x ∈ A(3 + i)T and x(1) ∈ A(3 + 3i)T since they will be distinguishable in phase. • Case xD ∈ A(1 + 3i)T−1: In this case, we show the conditions under which a codeword x ∈ A(1+3i)T satisfies the parity check, and show IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 23 that under these conditions there does not exist any ambiguous codeword from either A(3+3i)T or A(3+ i)T , i.e. from the previous two cases. The bit mappings are (1110) ֌ 1+3i, (0011)֌ −3+ i, (0100)֌ −1− 3i and (1001) ֌ 3 − i. Here, p1 ≡ 1 + N1 + N3 and p2 ≡ T . If T is even, then p2 ≡ 0 and therefore x1 ∈ {1 + i, 3 + i} and no ambiguity occurs. If T is odd then p1 ≡ 1 +N2 +N4 and p2 ≡ 1. Therefore, 1 + 3i if (p1p2) = (01) i.e. if N2 6≡ N4, 3 + 3i, if (p1p2) = (11) i.e. if N2 ≡ N4. It follows that x ∈ A(1 + 3i)T only if N2 6≡ N4 and T is odd. Clearly, these conditions are different to those to the previous two cases and therefore no ambiguities exist. REFERENCES [1] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999. [2] L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inform. Theory, vol. 48, pp. 359–383, Feb. 2002. [3] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inform. Theory, vol. 49, no. 4, pp. 951–963, Apr. 2003. [4] R.-R. Chen, R. Koetter, D. 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IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 25 1 begin 2 s := sgn y; // Store sign of each yt 3 y := s ◦ y; // Make each yt positive 4 Bmax := M + T − 2; 5 m := argmaxt {yt}; 6 λmax := (M + 2T − 2)/|ym|; // Search region: 0 < λ < λ 7 V0 := ∅; // Calculate and store P (x) boundary points 8 for t := 1 to T do 9 for all b ∈ {2, 4, . . . ,M − 2} do 10 ν := b/yt; 11 if ν < λmax ; 12 V0 := {V, (ν, t)}; 13 else break; 14 end for all; 15 end for; 16 V := sort(V0); // Sort V0 in ascending order of ν 17 V := {V, (λmax, 0)}; 18 x̂ := [ 1 1 . . . 1 ]′; // Initialize data estimate 19 α := x̂′y; // Initialize likelihood terms 20 β := ‖x̂‖2; 21 L := α2/β; 22 λ := V(1, 1)/2; 23 for k := 1 to |V| − 1 do // Iteratively examine likelihoods 24 t := V(k, 2); 25 α := α + 2yt; // Update likelihood terms 26 β := β + 4x̂t + 4; 27 x̂t := x̂t + 2; // Update x 28 if α2/β > L // If better x found 29 L := α2/β; // Update likelihood 30 λ := (V(k, 1) + V(k + 1, 1))/2; // Store point in P (x̂) 31 end if; 32 end for; 33 return x̂opt := s ◦ NN(λy); TABLE III M -ARY REAL-PAM NONCOHERENT LATTICE DECODING ALGORITHM IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 26 1 begin 2 m := arg maxt {|yt|}; 3 y := (y∗m/|ym|)y; // Rotate y so that ym is purely real 4 Bmax := M + 2T − 2; 5 λmax := Bmax/|ym|; // Search boundary (Thm. 2). 6 x̂best := NN((ǫ+ iǫ)y); // Codeword near origin 7 Lmax := L(x̂best); // L’hood L(x) , / ‖x‖2 8 B := {2, 4, . . . ,M − 2}; // Positive NN boundaries // Calculate only intersection points (i.e. vertices) in // first quadrant using (21) by reducing number of // NN boundaries B1,B2 and then rotating. 9 for t := 1 to 2T − 1 do 10 B1 := B; 11 if t ∈ {2m − 1, 2m} then B1 := {B1, B max}; 12 for t′ := t + 1 to 2T do 13 B2 := B; 14 if t′ ∈ {2m − 1, 2m} then B2 := {B2, B max}; 15 S := Y t′,1 Y t′,2 ; // Matrix in (21) 16 for all b1 ∈ B1 17 for all b2 ∈ B2 // Calculate intersection point; 18 ν := Real-To-Complex(S[ b1 b2 ] 19 ν := Rotate-To-First-Quadrant(ν); 20 for all s ∈ {−1, 1} 21 λ := ν + s(ǫ + iǫ); // Point in some partition // Check that λ is in reduced search region 22 if 0 < Re {λ} < λmax and 0 6 Im {λ} < λmax then 23 x̂ := NN(λy); // Calculate NN 24 if L(x̂) > Lmax // If better x found 25 x̂best := x̂; // Update codeword estimate 26 Lmax := L(x̂); // Update likelihood 27 end if; 28 end if; 29 end for all; 30 end for all; 31 end for all; 32 end for; 33 end for; 34 return x̂opt := x̂best; TABLE IV M2-ARY SQUARE QAM NONCOHERENT LATTICE DECODING ALGORITHM IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 27 1 begin 2 L := 0; // Initialize likelihood 3 λmax0 := (M + 2T − 2)/maxt |yt|; 4 for ℓ = 1 to L then 5 // Search region: 0 < λ < λmax (Theorem 2) 6 λmax := λmax0 /min ˛cos ℓπ ˛sin ℓπ 7 V0 = ∅; // Calculate and store P (x) boundary points 8 for t = 1 to 2T then 9 for all b ∈ {2, 4, . . . ,M − 2} then 10 ν := b/|y 11 if ν < λmax ; 12 V0 := {V0, (ν, t)}; 13 else break; 14 end for all; 15 end for; 16 V := sort(V0); // Sort V0 in ascending order of ν 17 V := {V, (λmax, 0)}; 18 s := sgn{y}; 19 x̂ := s; // Initialize data estimate 20 α := x̂†y; // Initialize likelihood terms 21 β := ‖x̂‖2; 22 if α2/β > L // If better x found 23 L := α2/β; // Update likelihood 24 λ := V(1)/2; 25 φ := ℓπ/(2L); 26 end if; 27 for k := 1 to |V| − 1 do // Iteratively examine likelihoods 28 t := V(k, 2); 29 if t′ is odd then 30 α := α + 2sty(t+1)/2; 31 else 32 α := α − 2istyt/2; 33 end if 34 β := β + 4s 35 x̂t := x̂t + 2st; 36 if |α|2/β > L // If better x found 37 L := |α|2/β; // Update likelihood 38 λ := (V(k, 1) + V(k + 1, 1))/2; // Store point in P (x̂) 39 φ := ℓπ/(2L); 40 end if; 41 end for; 42 y := ye 2L ; // Rotate y for next line search 43 end for; 44 return x̂opt := NN(λejφy); TABLE V SUBOPTIMAL M2-ARY SQUARE QAM MULTIPLE LINE-SEARCH NONCOHERENT DETECTION ALGORITHM IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 28 0 2 4 6 8 10 12 Fig. 1. Illustration of noncoherent detection of 8-ary PAM for T = 2. The dots are all the (two dimensional) PAM codewords in the positive quarter-plane, and the angled line is yR, for an example received codeword y. The shaded regions indicate the nearest neighbor regions of points which need to be searched. That is, they are in N (C(X , T ),y) (from Property 1), and they correspond to values of λ less than λmax = (M + T − 2)/maxt |yt| = M/maxt |yt| (from Theorem 1). −6 −4 −2 0 2 4 6 = Real{λ} Fig. 2. Plot of partitions P (x̂) on the R2 plane for 16-ary QAM detection of a sequence of length T = 3 for the received vector y = [ −0.1076 − 0.4728i, −0.7002 − 0.0968i, −1.1228 + 0.4955i ]. The bold square corresponds to the search boundary S . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 29 20 30 40 50 60 70 SNR (dB) New GLRT−Optimal Reduced−Search Phase−Estimator + Line Search Grid Search [20] QBR (exponential complexity) Ambiguity lower bound Fig. 3. Plot of Codeword Error Rate (CER) as a function of SNR for an 8-ary PAM system. 20 30 40 50 60 70 SNR (dB) New GLRT−Optimal Reduced−Search Phase−Estimator + Line Search (Section VI.B) Grid Search [20] QBR (exponential complexity) Ambiguity Lower Bound Fig. 4. Plot of Codeword Error Rate (CER) as a function of SNR for a 16-ary square QAM system. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED NOV. 2006) 30 10 12 14 16 18 20 22 SNR (dB) RA T=3 RA T=5 RA T=7 Fig. 5. Comparison of Bit Error Rate (BER) as a function of SNR for 16-QAM for PAT versus RA transmission. 10 12 14 16 18 20 22 SNR (dB) PAT T=7 PAT T=5 PAT T=3 RA T=3 RA T=5 RA T=7 Fig. 6. Comparison of Codeword Error Rate (CER) as a function of SNR for 16-QAM for PAT versus RA transmission. Introduction System Model Signal Model Detection Reduced Search Space PAM Detection For Real-Valued Fading Channels Limiting the Search Space PAM Algorithm GLRT-Optimal QAM Detection For Complex-Valued Fading Channels Limiting the Search Space QAM Algorithm PAM Over Complex Channels Suboptimal Algorithms for Complex-Valued Fading Channels Suboptimal PAM algorithm Suboptimal QAM algorithm Simulation Results Reduced Ambiguity Transmission Conclusion Appendix Strictly Optimal Calculation of Interior Points Removal of Ambiguities in 16-ary QAM References

0704.1525

Distributions of H2O and CO2 ices on Ariel, Umbriel, Titania, and Oberon from IRTF/SpeX observations

Distributions of H2O and CO2 ices on Ariel, Umbriel, Titania, and Oberon from IRTF/SpeX observations W.M. Grundy1,2, L.A. Young1,3, J.R. Spencer3, R.E. Johnson4, E.F. Young1,3, and M.W. Buie2 1. Visiting/remote observer at the Infrared Telescope Facility, which is operated by the University of Hawaii under contract from NASA. 2. Lowell Observatory, 1400 W. Mars Hill Rd., Flagstaff AZ 86001. 3. Southwest Research Institute, 1050 Walnut St., Boulder CO 80302. 4. Univ. of Virginia Dept. Engineering Physics, 116 Engineer’s Way, Charlottesville VA 22904. Published in 2006: Icarus 184, 543-555. Abstract We present 0.8 to 2.4 µm spectral observations of uranian satellites, obtained at IRTF/SpeX on 17 nights during 2001-2005. The spectra reveal for the first time the presence of CO2 ice on the surfaces of Umbriel and Titania, by means of 3 narrow absorption bands near 2 µm. Several additional, weaker CO2 ice absorptions have also been detected. No CO2 absorption is seen in Oberon spectra, and the strengths of the CO2 ice bands decline with planetocentric distance from Ariel through Titania. We use the CO2 absorptions to map the longitudinal distribution of CO2 ice on Ariel, Umbriel, and Titania, showing that it is most abundant on their trailing hemispheres. We also examine H2O ice absorptions in the spectra, finding deeper H2O bands on the leading hemispheres of Ariel, Umbriel, and Titania, but the opposite pattern on Oberon. Potential mechanisms to produce the observed longitudinal and planetocentric distributions of the two ices are considered. KEYWORDS: Ices; satellites of Uranus; surfaces of satellites; infrared observations; spectroscopy. 1. Introduction The discovery of CO2 ice on the surface of the uranian satellite Ariel (Grundy et al. 2003) raised a number of questions regarding its origin and ultimate fate. Is it a relic of Ariel’s primordial volatile inventory? Was it delivered by more recent impactors? Or is it being produced in situ in chemical reactions enabled by energetic radiation? How long does exposed CO2 ice survive on the satellite’s surface, and what processes act on it? How stable is it against sublimation and sputtering? Does it participate in radiolytic chemistry, such as envi- sioned by Delitsky and Lane (1997, 1998), Hudson and Moore (2001), and Johnson (2001)? Questions such as these could potentially be addressed by means of a survey of CO2 ice as a function of longitude and planetocentric distance among the major uranian satellites, the basic physical properties of which are listed in Table 1. The results of such a survey of the four largest satellites are presented in this paper. Table 1. Major satellites of Uranus. Satellite Distance from Uranus (105 km) Orbital Period (days) Radiusa Massb (1023 g) Surface gravity (cm s-2) Visual full disk albedoc Miranda 1.30 1.41 236 0.66 7.9 0.307 Ariel 1.91 2.52 579 13.5 26.9 0.350 Umbriel 2.66 4.14 585 11.7 22.9 0.189 Titania 4.36 8.71 789 35.3 37.8 0.232 Oberon 5.81 13.46 761 30.1 34.7 0.205 aRadii are from Thomas (1988). bMasses are from Jacobson et al. (1992). cFull disk albedos at 0.55 µm and 1˚ phase are from Karkoschka (1997) Table II. 2. Observations and Reduction We obtained infrared spectroscopy of uranian satellites on seventeen nights during 2001-2005 at NASA’s Infrared Telescope Facility (IRTF) on Mauna Kea, as tabulated in Table 2. The data were collected using the short cross-dispersed mode of the SpeX spectrograph (Rayner et al. 1998, 2003), which covers the 0.8 to 2.4 µm wavelength range in five spectral orders, recorded simultaneously on a 1024 × 1024 InSb array. Urani- an satellite observations were done at an average airmass of 1.26. Maximum airmasses were 1.61, 1.77, 1.57, and 1.46 for Ariel, Umbriel, Titania, and Oberon, respectively. When observing at higher airmasses, we used an image rotator to keep SpeX’s slit oriented near the parallactic angle on the sky-plane, to minimize wavelength-dependent slit losses from differential atmospheric refraction. During 2001-2002 we used the 0.5 arcsec slit, and during 2003-2005 we used the 0.3 arcsec slit, resulting in measured spectral resolutions (λ/∆λ for unresolved OH airglow emission lines) between 1300 and 1400 and between 1600 and 1700, respect- ively. All spectra were collected as pairs, with the telescope being offset along the slit between two locations, re- ferred to as A and B beams. Pairs of A and B spectral frames were subtracted prior to extraction using the Horne (1986) optimal extraction algorithm as implemented by M.W. Buie et al. at Lowell Observatory (e.g., Buie and Grundy 2000). Table 2. Circumstances of observations UT date of observation mid-time Sky conditions and H band image size Sub-observer longitude ( )˚ Sub-observer latitude ( )˚ Phase angle ( )˚ Total integration (min) Ariel 2001/07/05.59 Cirrus, 1.1” 233.8 -23.1 1.91 50 2001/07/08.61 Clear, 1.1” 304.8 -23.2 1.79 48 2002/07/16.55 Partly cloudy, 0.7” 294.8 -19.3 1.65 140 2002/07/17.56 Partly cloudy, 0.8” 79.8 -19.4 1.60 108 2003/08/05.50 Clear, 0.6” 200.0 -15.9 0.95 84 2003/08/09.51 Clear, 0.6” 53.6 -16.0 0.75 156 2003/09/07.40 Clear, 0.6” 219.8 -17.2 0.70 90 2003/10/04.24 Clear, 0.6” 93.5 -18.1 1.88 108 2003/10/08.33 Clear, 0.9” 316.6 -18.2 2.03 132 2004/07/15.50 Clear, 0.6” 159.9 -11.1 1.99 112 Umbriel 2001/07/07.59 Cirrus, 1.0” 219.8 -23.0 1.83 52 2004/07/05.59 Thin cirrus, 0.8” 216.2 -10.9 2.32 80 2004/07/16.60 Thin cirrus, 0.8” 92.1 -11.1 1.95 74 2004/07/27.48 Clear, 0.6” 317.6 -11.4 1.51 184 2005/09/18.39 Clear, 0.6” 159.9 - 9.4 0.85 196 Titania 2001/07/06.56 Cirrus, 0.9” 237.0 -23.0 1.87 56 2001/07/07.55 Cirrus, 1.0” 277.8 -23.0 1.83 36 2003/10/08.24 Clear, 0.9” 98.0 -18.1 2.03 64 2004/07/15.60 Clear, 0.6” 213.9 -11.1 1.98 64 2005/10/13.38 Clear, 0.6” 299.6 -10.2 1.93 106 Oberon 2001/07/06.60 Cirrus, 0.9” 164.0 -23.0 1.87 32 2001/07/08.55 Clear, 1.1” 216.2 -23.1 1.79 48 2005/10/13.27 Clear, 0.6” 110.6 -10.2 1.93 124 Uranian satellite observations were interspersed between observations of nearby solar analog reference stars HD 210377 and HD 214572. We also observed more distant, but better-known solar analogs 16 Cyg B, BS 5968, BS 6060, HD 219018, and SA 112-1333. Those observations showed that HD 214572 is a reasonable solar analog in this spectral range, and that HD 210377 was also usable as a solar analog after correcting for its effective temperature, which is a few hundred K cooler than the sun. From the solar analog observations ob- tained each night we computed telluric extinction and corrected the star and satellite observations to a common airmass. We then divided the airmass-corrected satellite spectra by the airmass-corrected solar analog spectra. This operation eliminated most instrumental, stellar, and telluric spectral features, and produced spectra pro- portional to the satellite disk-integrated albedos. Cancellation of telluric features was sometimes imperfect near 1.4 and 1.9 µm, where strong and narrow telluric H2O vapor absorptions make sky transparency especially variable in time. Additional spurious features originate in scattered light from Uranus, such as at 0.94 µm. These could be either positive or negative, depending whether Uranus was closer to the object or sky beam position, and only affected Umbriel and Ariel spectra for longitudes near 0 or 180˚. Examples of final, nor- malized albedo spectra are shown in Fig. 1. Wavelength calibration was derived from telluric airglow emission lines, primarily those of OH, ex- tracted from the satellite frames. The dispersion ranges from 0.00023 to 0.00054 µm per pixel from the shortest to longest wavelengths. The wavelength uncertainty is no more than a pixel in regions having abundant sky emission lines, which include the wavelengths where the satellites exhibit absorption features analyzed in this paper. Our spectra could not be photometrically calib- rated, since variable slit losses from different tracking time scales combine with variable focus and seeing to undermine photometric fidelity of comparisons between the faint satellites and the much brighter sol- ar analog stars. However, other sources of photo- metry exist for the uranian satellites in our wavelength range (e.g., Karkoschka 1997) and these could be used to scale the spectra to an albedo scale, enabling the use of quantitative radiative transfer models. 3. Analysis The spectra of all four satellites in Fig. 1 look broadly similar to one another. A gentle reddish spectral slope extends from 0.8 to about 1.4 µm, followed by a water ice absorption complex between 1.45 and 1.7 µm, and two more H2O ice absorptions between 1.9 and 2.2 µm and from 2.25 µm to the limit of our spectral cov- erage at 2.4 µm. The H2O ice bands are weakest in the spectrum of Umbriel, the satellite with the lowest al- bedo of the four (Karkoschka 1997). Lower albedos and weaker H2O bands typically go together, since the same dark materials (probably carbonaceous, in the case of uranian satellites) which suppress albedos are also effective at masking H2O ice absorptions. Although this dark material has a profound effect on the satellites’ albedos, H2O ice could still be the most abundant surface material on all four satellites, considering that thermal segregation of H2O ice, as seen on the jovian satellites, is improbable at the low surface temperatures of the uranian satellites (Spencer 1987). Impact gardening probably leads to intimate mixing between dark and Figure 1: Example spectra of uranian satellites, normalized at J band and offset upward with zero levels indicated by tick marks on the ordinate axis. Gaps around 1.4 and 1.85 µm coincide with high atmospheric opacity. The Ariel spectrum is an average of all spectra with subsolar longitudes between 210˚ and 330˚. The Umbriel spectrum is from 2005/09/18 UT and the Titania and Oberon spectra are both from 2005/10/13 UT. icy components, enabling even small quantities of dark material to significantly suppress surface albedos (Clark 1981). 3.1. Water ice The shapes of the H2O ice absorption bands and their depths relative to one another, especially between 1.50 and 1.57 µm and around 1.65 µm, confirm that most of the H2O ice on all four satellites is crystalline at the ∼mm depths sampled by these wavelengths (e.g., Grundy et al. 1999; Hansen and McCord 2004). Amorph- ous ice lacks 1.57 and 1.65 µm absorption bands and exhibits somewhat different shapes for the 1.5 and 2 µm bands (Grundy and Schmitt 1998; Schmitt et al. 1998). The H2O features in the satellite spectra are consistent with simple Hapke models (e.g., Hapke 1993) having no amorphous ice at all, although we can include up to 10 to 20% amorphous ice before the discrepancies become conspicuous. This predominance of crystalline ice is not surprising, since it is the more thermodynamically stable, lower energy phase. Crystalline H2O has also been reported on other outer solar system surfaces at comparable and greater heliocentric distances, both icy satellites and trans-neptunian objects. In fact, everywhere that the phase of H2O ice on these objects’ surfaces has been identified, it is crystalline (e.g., Buie and Grundy 2000; Bauer et al. 2002; Grundy and Young 2004; Jewitt and Luu 2004). Under certain circumstances, energetic charged particles or UV photons can disrupt the crystal structure of ice (e.g., Kouchi and Kuroda 1990; Johnson 2000; Moore and Hudson 1992; Johnson and Quickenden 1997), but evidently this effect is so inefficient that ice on the uranian satellites is not amorphous at mm depths. Nature does tend to find its way to thermodynamically favored states such as crystallinity, in the case of H2O ice. Thermal recrystallization goes as e , where EA is an activation energy, k is the Boltzmann constant, and T is the temperature. Laboratory studies of crystallization rates of pure H2O ice at much warmer temperat- ures (Delitsky and Lane 1998; Jenniskens et al. 1998), if extrapolated over many orders of magnitude, suggest that thermally activated crystallization could be extremely slow at uranian satellite surface temperatures. However, even if the low thermal crystallization rates implied by that extrapolation are correct, the presence of impurities in the ice, or of non-thermal energy inputs from less energetic charged particles, photons, or even micrometeorites may help overcome thermal limits (e.g., Brown et al. 1978). Sputtering might also remove H2O more efficiently than it is amorphized. Consequently, we do not subscribe to the idea proposed by Bauer et al. 2002 and by Jewitt and Luu (2004) that the existence of crystalline ice on the surfaces of these bodies im- plies recent heating episodes. The H2O ice absorption bands of jovian and saturnian satellites tend to be deeper on their leading hemi- spheres (e.g., Clark et al. 1984; Grundy et al. 1999; Cruikshank et al. 2005). Similar behavior is exhibited by the uranian satellites, except for Oberon, as shown in Fig. 2. Integrated areas of the 1.5 µm water ice absorp- tion complex were computed by normalizing each spectrum to a line fitted to continuum wavelengths on either side of the absorption band (1.29 to 1.32 and 1.72 to 1.75 µm, in this example), then integrating one minus the normalized spectrum over the band interval (1.42 to 1.72 µm). Ariel, with the most complete longitudinal sampling, shows a sinusoidal pattern of variation of its integrated H2O ice band, with maximum H2O absorp- tion coinciding with the leading hemisphere (90˚ longitude in the left-handed IAU system). Fewer longitudes were observed for the other satellites, but Umbriel and Titania both show more H2O absorption on their leading hemispheres. The trend for Oberon appears to be reversed, but confirming observations are needed. Magnetospheric charged particle bombardment can drive sputtering as well as radiolytic destruction of H2O ice, and could be responsible for the observed leading-trailing asymmetries. The magnetic field of Uranus ro- tates rigidly, at the same frequency as Uranus spins on its axis, with a period of 17.9 hours. The satellites orbit more slowly (see Table 1), so the magnetosphere overtakes them from behind and its charged particles prefer- entially strike their trailing hemispheres (Cheng et al. 1986; Lanzerotti et al. 1987). Magnetospheric charged particles spiral along the magnetic field lines at gyroradii inversely proportional to the strength of the magnetic field. The dipole component of the field diminishes as the inverse cube of the distance, so gyroradii grow as the cube of the distance from Uranus. For protons, the dominant ions in the uranian magnetosphere (Bridge et al. 1986), the energies at which gyroradii match the radii of the satellites range from ∼100 keV for Miranda down to <1 keV for Titania, spanning the ‘dominant’ ∼10 keV energy range for magnetospheric protons (Johnson 2000). That these protons have gyroradii larger than the radii of the satellites in the outer parts of the system suggests that outer satellites should receive more isotropicially distributed proton bombardment. This tendency is moderated by the fact that the net plasma flow past the satellites, arising from the greater speed of the urani- an magnetic field relative to the Keplerian motion of the satellites, is faster further from Uranus. This net plasma flow increases from 6 up to 39 km sec-1 from Miranda to Titania. However, these speeds correspond to proton energies only 0.2 to 8 eV, far below 10 keV energies. Although 10 keV protons circle field lines in or- bits larger than the radii of satellites from Umbriel on out, the orbits are executed relatively quickly compared with the rate at which the magnetic field sweeps past the satellites. Orbital periods for protons are inversely Figure 2: Variation of integrated area of the 1.5 µm H2O ice band complex as a function of sub-solar/sub-viewer longitude, showing large cyclical variations with more H2O ice absorption on the leading hemispheres, at least for satellites with better longitude coverage, where sine fits to the data are plotted as dashed curves. Data are duplicated outside the 0˚ to 360˚ interval to better show the periodic trends. proportional to magnetic field strength, being 0.3, 1, 3, and 13 seconds, from Miranda through Titania. Com- bining these gyroperiods with net plasma flow rates, each successive proton orbit advances by 2, 14, 60, and 500 km, relative to satellites Miranda through Titania. Only at Titania and beyond does a 10 keV proton ad- vance far enough between successive loops to begin to have much chance of looping around and hitting the satellite’s leading hemisphere. Computing the actual three dimensional spatial distribution of impacts on a satellite’s surface as a function of proton energy is beyond the scope of this paper. It depends on the substantial tilt and offset of the dipole field of Uranus as well as the distribution of particle pitch angles, which could vary seasonally. However, the simple arguments presented here suggest that particles in the 10 keV range will mostly hit trailing hemispheres of satellites, and can only influence the leading hemispheres of the more distant ones. Additionally, magnetospheric plasma densities decline rapidly with planetocentric distance, more rapidly than is compensated by the increase in net plasma flow relative to satellite orbital motion (or at least they did during the Voyager 2 encounter in 1986, near the time of southern summer solstice, e.g., Bridge et al. 1986; Cheng et al. 1991). Overall magnetospheric plasma sputtering rates should thus be lower on the more distant satellites. For both of these reasons, we would expect magnetospheric sputtering to produce diminishing lead- ing-trailing contrasts with planetocentric distance. This expectation is consistent with results from sinusoidal fits to the observed longitudinal variation in H2O bands, which give fractional variations 93 ± 4%, 69 ± 47%, and 34 ± 13% for Ariel, Umbriel, and Titania, respectively. Another mechanism capable of producing leading-trailing asymmetries in the satellites’ H2O ice bands is magnetospheric delivery of carbonaceous dust particles. Dark particles from the uranian rings which become electrically charged will experience a Lorentz force from the magnetic field lines sweeping past them. This force will tend to accelerate them such that they spiral slowly outward, eventually culminating in low speed collisions onto the trailing hemispheres of the satellites, primarily the inner ones, so this mechanism could also contribute to the observed decline of leading-trailing spectral contrasts with planetocentric distance. The addi- tion of these dark particles to the satellites’ trailing hemispheres would suppress their albedos and their ice ab- sorption bands, by reducing the fraction of photons able to escape from the surface at all wavelengths, as op- posed to just the wavelengths at which ice absorbs. A possible way to establish limits on the supply of carbon- aceous grains via this mechanism could involve comparison of visible wavelength albedos between leading and trailing hemispheres. Voyager images do not reveal dramatic global leading-trailing albedo patterns, but a detailed comparison, controlling for geological age by selecting specific regions, would be needed to properly address this question. Such a study is beyond the scope of this paper. Yet another mechanism which could produce leading-trailing asymmetries diminishing with planetocentric distance is impact cratering. In general, leading hemispheres suffer more collisions with debris from outside of the Uranus system (Zahnle et al. 2001, 2003), and gravitational focusing by Uranus causes the inner satellites to be cratered more frequently and with higher average impact velocities. Large impacts could dredge up cleaner ice from below the surfaces of the satellites, resulting in deeper H2O ice bands on leading hemispheres. Dust-sized impactors could sputter away ice and deliver exogenous, non-icy materials, creating the opposite spectral effect. Relatively little is known about the size distribution, composition, and net effect of smaller im- pactors on icy surfaces in the Uranus system, but the dust counter on NASA’s New Horizons1 spacecraft should provide useful information about the dust population in the ecliptic plane at large heliocentric distances. Oberon is the most distant major satellite. It spends part of its time outside of the uranian magnetosphere in the solar wind environment (Cheng et al. 1991). This difference between Oberon and the other uranian satel- 1. NASA’s first New Frontiers mission, bound for Pluto and the Kuiper belt (Stern and Cheng 2002, Stern and Spencer 2003). lites offers a possible explanation for its H2O ice bands following a different longitudinal pattern. Closer in, magnetospheric plasma may be the dominant remover of H2O ice, chiefly from trailing hemispheres, while fur- ther out, micrometeoroid impacts may increase in relative importance, preferentially sputtering ice from lead- ing hemispheres. 3.2. Carbon dioxide ice Three narrow CO2 ice absorption bands are seen in some of the spectra around 2 µm. They are most appar- ent in Ariel spectra, where they were previously noted Grundy et al. (2003). Here we report that these bands also appear, although less prominently, in spectra of Umbriel and Titania. Figure 3 shows the CO2 ice ab- sorptions, and a general trend of more CO2 ice absorp- tion on the satellites closer to Uranus (toward the bot- tom of the figure): no CO2 absorption is evident in Oberon’s spectrum, the strongest CO2 absorptions are barely visible in Titania spectra, and the absorptions become progressively stronger in spectra of Umbriel and Ariel. Two additional CO2 ice absorptions at 1.578 and 1.610 µm also appear to be present in spectra of Ariel and possibly Umbriel and Titania. These absorp- tions are attributable to 21 + 22 + 3 and 1 + 42 + 3 transitions in CO2 ice, respectively (Quirico and Schmitt 1997; Gerakines et al. 2005). The 21 + 22 + 3 band looks like it is superimposed on the longer wavelength shoulder of a broader dip. That feature appears to be a stellar absorption that is slightly deeper in the sun than in our solar analogues, resulting in incomplete cancellation. Additional, very weak, unidentified CO2 ice absorptions from the laboratory spectra of Hansen (2005) are also visible in the regions enlarged in Fig. 3. In general, SpeX does not quite spectrally resolve the cores of these CO2 ice absorp- tions. Our spectra show no apparent absorption features at 2.134 µm, where the ‘forbidden’ 23 overtone of CO2 is reported by Bernstein et al. (2005) for CO2 molecules mixed into H2O ice (this behavior was also noted by B. Schmitt, personal communication 2002). The absence of this band couples with the wavelengths and the nar- row profiles of the other CO2 bands to strongly favor pure CO2 ice, in the sense of CO2 molecules being associated with their own kind, as opposed to being isolated in H2O ice. That the CO2 molecules are not isolated in H2O ice is perhaps not surprising. There is some uncer- Figure 3: Enlarged views of the 1.6 and 2 µm regions of the same satellite spectra as in Fig. 1, showing narrow CO2 ice absorption features superimposed on the broad H2O ice bands. Wavelengths of additional weak, unidentified CO2 ice absorptions in the laboratory data of Hansen (1997, 2005) are indicated by vertical line segments. An arbitrarily scaled Hapke model spectrum (e.g., Hapke 1993) based on those laboratory data is included at the bottom of each panel to give an idea of the sorts of band shapes expected for pure CO2 ice, albeit at a warmer temperature (150 K). An arrow marks the location of the ‘forbidden’ 23 transition (Bernstein et al. 2005). tainty about how long such a thermodynamically unstable situation would last. Laboratory evidence suggests that, at least at higher temperatures, H2O:CO2 ices mixed at the molecular level can reorganize themselves over time into segregated zones of pure phases, and thus lose the 23 absorption feature (Bernstein et al. 2005). It is unclear how rapidly such a reorganization could progress at the uranian satellites’ surface temperatures, but the apparent ability of H2O to maintain its crystal structure against amorphization hints that such a phenomenon might not be out of the question. The spatial distribution of CO2 ice can place valuable constraints on its origin and evolution. Fig. 4 explores the longitudinal and planetocentric distribution of CO2 ice on the four satellites by plotting the sums of the in- tegrated areas of the three strongest CO2 ice absorptions near 2 µm from each spectrum versus sub-observer longitude. The integrated areas for the three absorptions were computed using 1.957 to 1.962 µm and 1.969 to 1.974 µm for the continuum and 1.962 to 1.969 µm for the band interval of the 21 + 3 CO2 band. For the 1 + 22 + 3 band we used 2.002 to 2.008 µm and 2.015 to 2.020 µm for the continuum and 2.008 to 2.015 µm for the band interval. For the 42 + 3 band we used 2.062 to 2.068 µm and 2.072 to 2.078 µm for the continuum and 2.068 to 2.072 µm for the band interval. These measurements show more CO2 ice absorption on the trailing hemispheres of Ariel, Umbriel, and Ti- tania, as well as declining CO2 absorption with distance from Uranus. This distribution contrasts somewhat Figure 4: Variation of total integrated area of the three strongest CO2 ice bands near 2 µm. Sine fits are plotted as dashed curves. Both the integrated areas and their longitudinal variation decline with planetocentric distance. No CO2 absorption was detected in the Oberon spectra. with reports from Galileo/NIMS observations of Europa, Ganymede, and Callisto (McCord et al. 1997, 1998; Hibbitts et al. 2000, 2003). Those observations of the 4.25 µm 3 fundamental band showed CO2 to be present on both leading and trailing hemispheres, although more abundant at the center of the trailing hemisphere of Callisto. Also, the 3 band is shifted in wavelength from where it occurs in pure CO2 ice, suggesting that in the jovian system CO2 is trapped or bound up in some other material. The absorption was also observed to be geo- graphically associated with darker materials (Hibbitts et al. 2000, 2003). Pure CO2 ice in the jovian system would not be thermally stable, adding weight to the interpretation that it is complexed with some other, less volatile material. The planetocentric trend was the opposite of what we see in the uranian system: the 4.25 µm band depth increases with distance from Jupiter (McCord et al. 1998). The 3 band of carbon dioxide has also recently been detected in saturnian satellite spectra by Cassini/VIMS (e.g., Buratti et al. 2005; Clark et al. 2005). As in the jovian system, the band appears to be shifted from the wavelength of the pure CO2 ice trans- ition, and may be associated with dark material there, as well. It is important to realize that the 4.25 µm 3 fundamental absorption detected by NIMS and VIMS is in- trinsically much stronger than the overtone and combination bands we are observing near 2 µm, by a factor of a thousand in absorption coefficients (e.g., Hansen 1997, 2005). Accordingly, our detection of much the weak- er bands requires considerably more condensed CO2 than is indicated by the jovian and saturnian satellite ob- servations. Simple radiative transfer models (e.g., Hapke 1993; Grundy et al. 2003) can be used to get an idea of the quantities of CO2 ice required by our observations. A CO2 ice glaze uniformly covering Ariel’s entire trailing hemisphere would have to be at least 5 µm thick to match the observed absorption bands. It would need to be thicker if the CO2 ice were more geographically restricted. If the CO2 ice were as segregated as possible from other materials, in a checkerboard-like pattern, at least 5% of the trailing hemisphere of Ariel would have to be composed of CO2 ice. In the following sub-sections, we consider possible sources and sinks of CO2 on the satellites’ surfaces, es- timating rates where possible, and considering what longitudinal or planetocentric trends various processes might lead to in light of the observed CO2 ice distribution. We begin with processes which can mobilize or eliminate CO2 ice. 3.2.1. Processes which move or destroy CO2 If the CO2 ice on the satellite surfaces is not trapped in a less volatile material such as H2O ice or a carbon- aceous residue (as is suggested by the non-detection of the 23 band and by the absence of wavelength shifts relative to pure CO2 ice measured in the laboratory), we can use its known vapor pressure curve (Brown and Ziegler 1980) and sticking coefficient (Weida et al. 1996) to consider its stability against sublimation over the course of a seasonal cycle in the uranian system. We used a thermophysical model (e.g., Spencer and Moore 1992) to compute diurnal temperature histories as functions of latitude and season on the satellites, assuming plausible values of thermal inertia , bolometric bond albedo AB, and emissivity . From the computed temper- ature histories and the CO2 ice vapor pressure curve, we computed sublimation histories for each latitude band. Integrating these histories over time, we arrived at seasonally averaged sublimation rates as a function of latit- ude. Inverting these rates gives the time to sublimate one gram of CO2 ice per square centimeter. Examples are shown in Fig. 5. The CO2 sublimation rate is proportional to its vapor pressure, which is an extremely steep function of tem- perature at uranian satellite surface temperatures, so local seasonal sublimation rates tend to be dominated by the seasonal maximum temperatures achieved. Polar regions, which are bathed in decades of continuous sun- light each summer because of the extreme obliquity (98˚), experience the highest peak temperatures and most rapid seasonal sublimation rates. This latit- ude dependence of seasonal sublimation rates will tend to drive accumulation of CO2 at equatorial latitudes. But even at the satellites’ equators, appreciable quantities of CO2 ice could sublimate on time scales much shorter than the age of the Solar Sys- tem. Of the four satellites, Ariel spins fast- est on its axis, with a 2.5 day period. Its faster rotation reduces the day-night tem- perature contrast, so for any set of thermo- physical parameters, Ariel’s equatorial di- urnal peak temperatures remain lower than on Umbriel, Titania, and Oberon, with their longer periods of 4.1, 8.7, and 13.5 days, respectively. As a result, CO2 ice is most stable against sublimation at low latitudes on the faster rotating, inner satellites. The extremely strong temperature de- pendence makes sublimation rates highly sensitive to assumed model parameters. For instance, increasing AB from 0.5 to 0.6 reduces absorbed sunlight by 20%, leading to a few K drop in maximum temperatures. This small temperature drop drives an order of magnitude increase in the time scale to sublimate 1 g cm-2 of CO2 ice, as shown in Fig. 5. Although we as- sumed plausible values for albedos, thermal inertias, emissivities, etc., in our computations of temperature his- tories, the extreme temperature sensitivity makes the sublimation numbers highly uncertain. Our models pro- duce peak temperatures in the 70 to 80 K range. Peak temperatures from Voyager 2 thermal emission measure- ments were as high as 85 K (Hanel et al. 1986). If CO2 ice temperatures get that high, sublimation will proceed much more rapidly than we have estimated in Fig. 5. The thermal emission observed by Voyager is dominated by the darkest regions which absorb the most sunlight as well as topographic lows that absorb thermal radi- ation from surrounding areas. Any CO2 ice in these warmest regions would be mobilized very rapidly. The net result of thermal mobilization will be that CO2 ice migrates away from polar latitudes toward equat- orial latitudes on relatively short time scales, of the order of hundreds to millions of years. In the equatorial re- gions, the ice will tend to accumulate in local cold traps: regions of higher albedo, higher thermal inertia, high- er emissivity, and/or higher elevation. There its texture will slowly evolve toward larger particles with rounded shapes, or even a compacted crust overlaying more porous ice below (e.g., Clark et al. 1983; Spencer 1987; Eluszkiewicz et al. 1998; Grundy and Stansberry 2000; Titus et al. 2001). The above sublimation estimates do not account for the fate of CO2 molecules after they are mobilized. Be- ing unlikely to collide with other gas molecules, they will generally follow ballistic trajectories. Some will es- cape, while others will fall back to the surface and stick elsewhere. We estimate escaping fractions following Chamberlain and Hunten (1987, Eq. 7.1.5). As with sublimation rates, the fraction escaping is extremely sens- itive to temperature, and escape rates from the warmer poles greatly exceed escape rates from the cooler equat- orial belts. For the thermal parameters described above, we find maximum polar escape fractions ranging from Figure 5: Estimates of time in Earth years to sublimate one gram of CO2 per cm2 as a function of latitude for satellites as labeled by their initials. North and South latitudes mirror each other, so half of each curve is omitted to reduce clutter. On the left are models with  = 104 erg cm-2 s-1/2 K-1. Models with  = 2 × 104 erg cm-2 s-1/2 K-1 are on the right. Solid curves are for AB = 0.5, dotted curves for AB = 0.6, and dashed curves for AB = 0.7. Emissivity  = 0.9 was assumed for all models. 1 × 10-9 (for Titania with AB = 0.7) up to 7 × 10-4 (for Umbriel with AB = 0.5). For equatorial latitudes, peak temperatures are lower and so maximum escape fractions are lower as well, from 1 × 10-10 to 3 × 10-4 for the same two cases, with  = 10,000 erg cm-2 s-1/2 K-1. For any particular set of thermal parameters, Umbriel has the highest escape fraction, followed by Ariel, because of their lower masses and gravitational accelerations. Since these two satellites show the most CO2 ice, the escape of sublimated CO2 to space does not seem to govern the planetocentric distribution of CO2. Even for the highest equatorial escape rates of any of our models, thermal escape of CO2 would be extremely slow. For Umbriel with  = 10,000 erg cm-2 s-1/2 K-1 and AB = 0.5, the time to sublimate 1 g cm-2 at the equator is about 105 years, of which 3 × 10-4 g escape to space, so the time to lose 1 g cm-2 to space is about 3 × 108 years. For other thermal model assumptions and for other satellites, the time scales are even longer, much longer than the age of the solar system in most cases. Charged particle sputtering can mobilize CO2, and will act at a much higher rate for CO2 ice than it does for less volatile H2O ice, assuming the two ices are segregated from one another (e.g., Johnson et al. 1983). Using equation 3.27 from Johnson (2000), with values from Table 3.4 in that book, we estimate that for typical urani- an satellite surface temperatures and magnetospheric proton energies, the sputtering rate for CO2 ice should be some 50 to 100 times higher than for H2O ice. Cheng et al. (1991) estimate seasonal average sputtering rates for H2O molecules from uranian satellite surfaces in their Table IV. We multiply their H2O sputtering rates by 50 to 100 to get an idea of possible magnetospheric proton CO2 sputtering rates for the uranian satellites, find- ing values in the 0.6 to 1.4 × 10-8 g cm-2 yr-1 range for Ariel and Umbriel, dropping to 2 to 4 × 10-10 g cm-2 yr-1 for Titania and as low as 0.5 to 1 × 10-10 g cm-2 yr-1 for Oberon. As before, these values can be inverted to get an idea of how long it would take for one gram of CO2 ice to be sputtered from a square centimeter. For Ariel and Umbriel, these time scales are in the 100 Myr range, increasing to more than a billion years for Titania and Oberon. Sputtered molecules are ejected with more kinetic energy than thermally mobilized molecules generally have, so they can more readily escape a satellite’s gravitational well. The energy spectrum of sputtered CO2 molecules is uncertain, but crude estimates can be made by assuming it is similar to the energy distribution of sputtered H2O molecules, which goes approximately as U'(E + U')-2, where U' is a measure of binding energy and E is the kinetic energy of ejection (e.g., Johnson 1998). The appropriate value of U' for CO2 ice is un- known. Assuming an H2O-like U' = 0.05 eV (Haring et al. 1984) the fraction of molecules sputtered from the surfaces of the satellites at greater than escape velocity would be 0.85, 0.40, 0.44, 0.26, and 0.29 from Miranda through Oberon. If U' is as low as the 0.009 eV value reported by Haring et al. (1984) for CO ice, the escape fractions would be 0.51, 0.11, 0.13, 0.06, and 0.07. These values are generally smaller than the Cheng et al. (1991) estimated escape fractions for H2O molecules, since CO2 molecules are more massive (44 AMU as op- posed to 18 AMU for H2O). Combining the escape fractions with sputtering rates as estimated in the previous paragraph, we find time scales for one gram of CO2 ice to be sputtered and lost to space to be in the few hun- dred million year range for Ariel and Umbriel, but longer than the age of the solar system for Titania and Oberon. Sputtering will gradually remove CO2 ice from the low latitude cold traps where thermal mobilization drives it, at least on the trailing hemispheres of the inner satellites, where magnetospheric sputtering rates are highest and the molecules are less tightly bound, gravitationally, because the satellites are less massive (see Table 1). The fact that CO2 ice is observed to be most abundant in exactly these regions where it should be re- moved most rapidly suggests sputtering by magnetospheric protons is not controlling the distribution of CO2 ice on the satellites’ surfaces, and that whatever is controlling that distribution acts on shorter time scales. Sputtering by solar UV photons could also mobilize CO2 on satellite surfaces (e.g., Johnson 1990). UV sputtering should affect all longitudes and all uranian satellites equivalently, so although it may be important for increasing mobility of CO2 molecules, it cannot explain the observed spatial distribution of CO2 ice. Anoth- er source of energy which can lead to sputtering is micrometeorite impacts, which, if they come from outside of the Uranus system, preferentially strike the satellites’ leading hemispheres (Zahnle et al. 2003). The abund- ance of dust-sized impactors at 20 AU is not well known, but could be sufficient to play a role in mobilizing CO2 ice from the leading hemispheres (e.g., Eviatar and Richardson 1986). Finally, CO2 molecules can be destroyed by energetic UV or charged particle radiation. Gerakines and Moore (2001) estimated G value of 1.1 and 8.1 CO2 molecules destroyed per 100 eV incident as protons or UV photons, respectively, via the reaction path energy + CO2  CO + O. A UV photon with wavelength shorter than 2275 Å can dissociate a CO2 molecule (Gerakines et al. 1996; Delitsky and Lane 1998). From SORCE spectra (e.g., Rottman et al. 2004), we estimate about 0.07% of the solar flux to be at or below that wavelength. Accounting for the geometry of the Uranus system, we obtain seasonal average UV fluxes ranging from 0.5 through 0.8 erg cm-2 s-1 for equatorial through polar latitudes on the uranian satellites. These fluxes correspond to destruction of the order of 1018 CO2 molecules, or ∼10 g cm-2 yr-1, if all of the UV photons are absorbed within the CO2 ice. This is a very high destruction rate, compared with other processes we have considered. Magnetospheric charged particles can also dissociate CO2 molecules. Using the (Gerakines and Moore 2001) G value of 1.1 molecule per 100 eV, along with dose rates from Cheng et al. (1991, Figure 23), we es- timate radiolytic destruction rates in the 10-10 to 10-7 g cm-2 yr-1 range for Oberon through Miranda. These rates are much lower than the photolytic destruction rates estimated in the previous paragraph, but are comparable to the magnetospheric sputtering rates estimated previously. Of course, not all CO and O resulting from photolysis and radiolysis of CO2 will be lost. Some will recom- bine to make new CO2 molecules while others will react with water or other locally available materials to form new molecules, some of which could be quite complex (e.g., Benit et al. 1988; Delitsky and Lane 1997, 1998). Indeed, it seems likely that a carbon cycle much like the one envisioned by Johnson et al. (2005) for the jovian system also operates on the surfaces of the uranian satellites, with carbon atoms repeatedly recycled between CO2 and other molecules such as carbonic acid (H2CO3) and formaldehyde (H2CO). The relatively high destruction and loss rates estimated in this section, especially from UV photolysis as well as from sputtering and radiolysis, suggest that there must be a recent, or more likely an ongoing source for the observed CO2 ice. Accordingly, we next turn our attention to possible sources of CO2. 3.2.2. Possible sources of CO2 Out-gassing from satellite interiors is a possible source for CO2. Ariel, with its relatively deep CO2 absorp- tion bands, also shows clear evidence of past geologic activity (Plescia 1987), but Umbriel, with the next strongest CO2 bands, shows no evidence for recent activity, at least not on the hemisphere observed by Voy- ager (Plescia 1989). Titania, with still less observed CO2, may have some relatively youthful provinces, but its surface is mostly ancient (Zahnle et al. 2003). Although Voyager was unable to view more than about half of each satellite’s surface, the presence of CO2 ice does not seem to correlate particularly well with the existence of less-cratered, younger regions or with evidence of recent geological activity. Either could be an indicator of ongoing or recent release of volatiles from the satellites’ interiors. An out-gassing source for the observed CO2 ice thus does not look especially promising. Impactors could deliver CO2 to the uranian satellites, or could release it from their sub-surfaces. If exogenic delivery or impact exhumation were dominated by frequent, smaller impactors, they would tend to distribute CO2 through the Uranus system according to a consistent pattern, and the balance between delivery and loss rates would govern the observed planetocentric and longitudinal distributions. Infrequent, larger impactors could produce a more random distribution pattern, in which a local high abundance of CO2 might simply point to a single recent impact event. Zahnle et al. (2003) estimated cometary cratering rates on the uranian satellites. Owing to gravitational focusing, cratering rates per unit area increase with proximity to Uranus, being something like a factor of 4 or 5 higher on Ariel than on Oberon, and impactors strike with average speeds a factor of almost 2 higher on Ariel than on Oberon (Zahnle et al. 2003). Impactors from outside the Uranus sys- tem would preferentially strike the satellites’ leading hemispheres, so the observation that CO2 ice is predomin- antly on the trailing hemispheres would require mobilization and redistribution, followed by preferential re- moval from the leading hemispheres, possibly by dust impact driven sputtering. CO2 molecules could form through radiolytic action on the satellites’ surfaces, with oxygen coming from the ubiquitous H2O ice, and carbon arriving via implantation of magnetospheric ions (Strazzulla et al. 2003). A magnetospheric source of carbon ions would neatly explain the observed distribution of CO2 ice. However, the Voyager 2 spacecraft did not detect carbon ions in the magnetosphere during its Uranus flyby in 1986 (Bridge et al. 1986; Krimigis et al. 1986), making a magnetospheric source of carbon ions seems improbable. However, magnetospheric delivery of larger carbonaceous particles from the uranian rings remains a possibility, as men- tioned previously. There is also no shortage of dark and presumably carbonaceous material already on the satellites’ surfaces, as evidenced by their low visible albedos. Recent investigations of radiolysis at interfaces between carbon- aceous materials and H2O ice (Mennella et al. 2004; Gomis and Strazzulla 2005) show that CO2 can be pro- duced from purely local materials, even when the carbon and oxygen are not initially present in a single phase. The low albedos of the satellites’ surfaces, along with temperatures too cold for thermal segregation of H2O ice as well as abundant craters indicative of impact gardening, suggest relatively intimate mixing between H2O ice and dark particles. In one experiment by Gomis and Strazzulla (2005), H2O ice deposited on a carbonaceous substrate was irradiated at 80 K with 200 keV Ar+ ions. CO2 was efficiently produced at the interface. It would be interesting to see if proton, electron, or even UV irradiation produces similar results. Any energy source capable of breaking bonds in H2O ice or in the dark carbonaceous materials, and mobilizing radicals containing O or C should be able to contribute to this process. Whether or not radiolysis can produce CO2 rapidly enough to replace CO2 lost to UV photolysis is an im- portant question to consider. Prior to saturation in the Gomis and Strazzulla (2005) experiment, each 200 keV Ar ion produced about 30 CO2 molecules, or 7 keV per CO2 molecule produced. This is orders of magnitude more energy than the ∼12 eV per CO2 molecule destroyed according to the photolytic G value from Gerakines and Moore (2001). From these numbers, radiolytic production rates would be far below photolytic destruction rates. For production not to be overwhelmed by photolytic destruction, much more efficient production path- ways are necessary. It is possible that lower energy protons or electrons are more effective at producing CO2 from the available precursors, and it is also possible that the expected granular texture of the ice and carbon- aceous material enhances production of CO2, because of its large surface area. Additionally, photolysis may not be a purely destructive process. We are not aware of laboratory studies of photolytic production of CO2 from H2O ice and carbonaceous precursors at uranian satellite surface temperatures, but it probably can occur (e.g., Bernstein et al. 1995; Wu et al. 2002). Photolytic production would not, however, be able to explain the observed longitudinal distribution of CO2 ice. Another possibility is that more of the satellites’ surfaces are in- volved in production than in destruction. Such a configuration is plausible, since CO2 will tend to accumulate in localized cold traps, owing to its high thermal mobility, unlike the global distribution of the non-volatile pre- cursors: H2O ice and carbonaceous solids. As noted previously, the spectra are consistent with CO2 being con- fined to a relatively small fraction of the surface area, as little as 5%. 3.2.3. Discussion The planetocentric and longitudinal distributions of CO2 and H2O ices in the uranian system offer valuable clues regarding their origin and fate. Having considered candidate sources and sinks on the satellite surfaces, we now review the trends expected from these mechanisms and compare them with the observational data, which shows CO2 ice to be most abundant on the inner satellites, and especially on their trailing hemispheres, and H2O ice to be more abundant on leading hemispheres from Ariel through Titania, but on the trailing hemi- sphere of Oberon. Sublimation mobilizes CO2 relatively rapidly at seasonal maximum uranian satellite surface temperatures, which are higher at polar latitudes, so CO2 accumulates at low latitudes. Relatively little is lost to space. At low latitudes, sublimation rates are much lower, but still rapid enough to drive CO2 ice to accumulate in local cold traps, and to evolve in texture. Sublimation does not depend on longitude, and has no real effect on the much less volatile H2O ice. Voyager images of Umbriel show a bright equatorial feature (‘Wunda’) near the center of the trailing hemisphere. It is tempting to speculate that CO2 ice is concentrated there, however, similar low-lat- itude, bright features are not evident on the trailing hemispheres of Ariel or Titania. Impactors preferentially strike leading hemispheres and inner satellites. Impacts could dredge up fresh H2O ice from the subsurfaces, primarily of the leading hemispheres of the inner satellites, consistent with the obser- vations of deeper H2O ice bands there. An impact source of CO2 could be consistent with the observed planeto- centric trend, but not with the observed longitudinal trends. An impact loss mechanism could explain the ob- served CO2 longitudinal trends, but not the planetocentric trend. Impacts apparently cannot explain the ob- served CO2 ice distribution, at least not without the help of additional mechanisms. Magnetospheric charged particle sputtering preferentially mobilizes CO2 and H2O ices from the trailing hemispheres of the inner satellites, exactly those regions where CO2 ice is observed. Sputtering rates are estim- ated to be well below sublimation rates for CO2, but are probably the dominant sink for H2O ice. A much great- er fraction of sputtered CO2 molecules should escape to space, compared with sublimated CO2 molecules. Cor- relations with indicators of recent geological activity in Voyager images are not evident, arguing against an out-gassing source of CO2 from the satellite interiors. Radiolytic production of CO2 driven by magnetospheric irradiation should act most rapidly on the trailing hemispheres of the inner satellites, because magnetospheric densities are higher closer to Uranus. The longit- udinal and planetocentric pattern of radiolytic production is consistent with the observed distribution of CO2, and appears to be the best candidate source for CO2 on the satellites’ surfaces. The fact that our estimated rates of radiolytic production are considerably below the estimated photolytic destruction rate is troubling. This dis- crepancy may point to problems in our estimated production rates or could imply that much more surface area is involved in production than is actually covered with CO2 ice, in order to maintain an equilibrium. The idea that CO2 ice may only cover a small fraction of the surface area of the satellites’ trailing himespheres is at least consistent with the spectral evidence. Finally, it seems probable that CO2 ice on the uranian satellites’ surfaces participates in a radiolytic and photolytic carbon cycle (e.g., Johnson et al. 2005). If so, carbonic acid is also likely to be present, and might be detectable by means of its characteristic infrared absorption bands. With hydrogen being more readily lost to space than heavier carbon and oxygen, chemistry at the satellites’ surfaces probably tends to be oxidizing and acidic. Such a surface environment may help explain why NH3, an alkali, has not been convincingly detected, despite its presence being suggested by geomorphologic evidence (Croft and Soderblom 1991; Kargel et al. 1991). A tentative report of NH3 in a spectrum of Miranda (Bauer et al. 2002) has yet to be confirmed. If NH3 were confirmed to be present on the surface of a uranian satellite, the chemical implications would be quite re- markable. 4. Conclusion Seventeen nights of IRTF/SpeX observations of the 4 largest uranian satellites reveal the existence of CO2 ice on Umbriel and Titania for the first time, as well as confirming its presence on Ariel. The observations also provide information about the longitudinal and planetocentric distributions of CO2 and H2O ices. These distri- butions likely result from equilibria between source and sink mechanisms, and can tell us something about the relative importance of the processes involved. Magnetospheric sputtering removal of H2O ice from trailing hemispheres of Ariel, Umbriel, and Titania is consistent with the observation of deeper H2O ice absorption bands on the leading hemispheres of those satellites, with diminishing leading-trailing asymmetries with plan- etocentric distance. For Oberon, magnetospheric sputtering is greatly reduced, and the leading-trailing asym- metry appears to be reversed. The deeper H2O ice bands on Oberon’s trailing hemisphere may point to an ef- fect of micrometeoroid sputtering that only becomes important where magnetospheric effects become negli- gible. Radiolytic production of CO2, driven by magnetospheric charged particle bombardment, appears most consistent with the observation of deeper CO2 ice absorption bands on the trailing hemispheres of Ariel, Um- briel, and Titania, as well as declining CO2 absorption with planetocentric distance and its absence from the surface of Oberon. Thermally-driven processes such as grain growth, volatile transport, sintering, and solar gardening can be expected to influence the texture and distribution of CO2 ice on the surfaces of the uranian satellites. The sea- sonally-averaged distribution of sunshine should drive CO2 ice away from polar latitudes. Low latitude regions which radiatively cool fastest will cold-trap CO2 ice, and if enough ice accumulates to raise the albedos of these regions, the process will be self-reinforcing. Uppermost surfaces of CO2 ice particles can radiatively cool faster than subsurface depths where solar energy is absorbed, so sublimation will tend to preferentially remove CO2 from below the uppermost surface. Loss to space driven primarily by sputtering by UV photons and mag- netospheric charged particles limits the residency time of CO2 ice in these cold traps. Additional limits on CO2 accumulation come from UV photolysis, which is expected to break CO2 bonds at a relatively high rate, lead- ing to a chemical cycle in which carbon, oxygen, and hydrogen are repeatedly combined into new species and broken apart again, forming a complex mixture which should include CO2 as well as species like carbonic acid and formaldehyde. More detailed comparisons between leading hemispheres (which see relatively little mag- netospheric radiolysis) and trailing hemispheres should be able to shed more light on the relative importance of radiolysis and photolysis. If radiolysis by magnetospheric particles is indeed the source of CO2 ice on the uranian satellites, produc- tion rates on Miranda should be even higher than on Ariel. However, Miranda’s lower gravity will allow a much larger fraction of sublimated and sputtered CO2 molecules to escape. Spectral observations of Miranda could provide useful constraints on the balance between sublimation and radiolytic production. ACKNOWLEDGMENTS: We are grateful to W. Golisch, D. Griep, P. Sears, S.J. Bus, J.T. Rayner, and K. Crane for assistance with the IRTF and with SpeX, R.S. Bussmann for contributing to the reduction pipeline, M.R. Showalter for the Rings Node’s on-line ephemeris services, and NASA for its support of the IRTF. We especially thank K.A. Tryka for sharing with us an unpublished manuscript on NH3 and CO2 stability in the ur- anian system, which contained many useful ideas for modeling the fate of CO2 ice on the satellites’ surfaces. Scientific discussions with B. Schmitt and M.P. Bernstein also provided useful ideas. We thank the free and open source software communities for empowering us with many of the tools used to complete this project, notably Linux, the GNU tools, LATEX, FVWM, Tcl/Tk, TkRat, and MySQL. We acknowledge the significant cultural role and reverence for the summit of Mauna Kea within the indigenous Hawaiian community and are honored to have had the opportunity to observe there. Finally, we are grateful for funding from NSF grants AST-0407214 and AST-0085614 and from NASA grants NAG5-4210, NAG5-10497, NAG5-12516, and NNG04G172G, without which this work could not have been done. References Bauer, J.M., T.L. Roush, T.R. Geballe, K.J. Meech, T.C. Owen, W.D. Vacca, J.T. Rayner, and K.T.C. Jim 2002. The near infrared spectrum of Miranda: Evidence of crystalline water ice. Icarus 158, 178-190. Benit, J., J.P. Bibring, and F. Rocard 1988. Chemical irradiation effects in ices. Nucl. Instr. and Methods in Phys. Res. B32, 349-353. Bernstein, M.P., S.A. 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0704.1526

Proof of the Labastida-Marino-Ooguri-Vafa Conjecture

9 Proof of the Labastida-Mariño-Ooguri-Vafa Conjecture Kefeng Liu and Pan Peng Abstract Based on large N Chern-Simons/topological string duality, in a series of papers [38, 22, 20], J.M.F. Labastida, M. Mariño, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invari- ants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera. Contents 0 Introduction 3 0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Labastida-Mariño-Ooguri-Vafa conjecture . . . . . . . . . . . . 4 0.3 Main ideas of the proof . . . . . . . . . . . . . . . . . . . . . . 6 0.4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Preliminary 9 1.1 Partition and symmetric function . . . . . . . . . . . . . . . . 9 1.2 Partitionable set and infinite series . . . . . . . . . . . . . . . 11 2 Labastida-Mariño-Ooguri-Vafa conjecture 12 2.1 Quantum trace . . . . . . . . . . . . . . . . . . . . . . . . . . 12 http://arxiv.org/abs/0704.1526v3 2.2 Quantum group invariants of links . . . . . . . . . . . . . . . . 14 2.3 Chern-Simons partition function . . . . . . . . . . . . . . . . . 17 2.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Hecke algebra and cabling 22 3.1 Centralizer algebra and Hecke algebra representation . . . . . 22 3.2 Quantum dimension . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Cabling Technique . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Proof of Theorem 1 28 4.1 Pole structure of quantum group invariants . . . . . . . . . . . 28 4.2 Symmetry of quantum group invariants . . . . . . . . . . . . . 30 4.3 Cut-and-join analysis . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Framing and pole structures . . . . . . . . . . . . . . . . . . . 38 5 Proof of Theorem 2 41 5.1 A ring characterizes the partition function . . . . . . . . . . . 41 5.2 Multi-cover contribution and p-adic argument . . . . . . . . . 43 5.3 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Concluding Remarks and Future Research 47 6.1 Duality from a mathematical point of view . . . . . . . . . . . 47 6.2 Other related problems . . . . . . . . . . . . . . . . . . . . . . 48 A Appendix 50 A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 0 Introduction 0.1 Overview For decades, we have witnessed the great development of string theory and its powerful impact on the development of mathematics. There have been a lot of marvelous results revealed by string theory, which deeply relate different aspects of mathematics. All these mysterious relations are connected by a core idea in string theory called “duality”. It was found that string theory on Calabi-Yau manifolds provided new insight in geometry of these spaces. The existence of a topological sector of string theory leads to a simplified model in string theory, the topological string theory. A major problem in topological string theory is how to compute Gromov- Witten invariants. There are two major methods widely used: mirror symme- try in physics and localization in mathematics. Both methods are effective when genus is low while having trouble in dealing with higher genera due to the rapidly growing complexity during computation. However, when the target manifold is Calabi-Yau threefold, large N Chern-Simons/topological string duality opens a new gate to a complete solution of computing Gromov- Witten invariants at all genera. The study of large N Chern-Simons/topological string duality was orig- inated in physics by an idea that gauge theory should have a string theory explanation. In 1992, Witten [46] related topological string theory of T ∗M of a three dimensional manifold M to Chern-Simons gauge theory on M . In 1998, Gopakumar and Vafa [11] conjectured that, at large N , open topo- logical A-model of N D-branes on T ∗S3 is dual to closed topological string theory on resolved conifold O(−1) ⊕ O(−1) → P1. Later, Ooguri and Vafa [38] showed a picture on how to describe Chern-Simons invariants of a knot by open topological string theory on resolved conifold paired with lagrangian associated with the knot. Though large N Chern-Simons/topological string duality still remains open, there have been a lot of progress in this direction demonstrating the power of this idea. Even for the simplest knot, the unknot, Mariño-Vafa formula [33, 26] gives a beautiful closed formula for Hodge integral up to three Chern classes of Hodge bundle. Furthermore, using topological vertex theory [1, 27, 28], one is able to compute Gromov-Witten invariants of any toric Calabi-Yau threefold by reducing the computation to a gluing algorithm of topological vertex. This thus leads to a closed formula of topological string partition function, a generating function of Gromov-Witten invariants, in all genera for any toric Calabi-Yau threefolds. On the other hand, after Jones’ famous work on polynomial knot invari- ants, there had been a series of polynomial invariants discovered (for example, [15, 9, 19]), the generalization of which was provided by quantum group the- ory [43] in mathematics and by Chern-Simons path integral with the gauge group SU(N) [45] in physics. Based on the large N Chern-Simons/topological string duality, Ooguri and Vafa [38] reformulated knot invariants in terms of new integral invari- ants capturing the spectrum of M2 branes ending on M5 branes embedded in the resolved conifold. Later, Labastida, Mariño and Vafa [22, 20] refined the analysis of [38] and conjectured the precise integrality structure for open Gromov-Witten invariants. This conjecture predicts a remarkable new al- gebraic structure for the generating series of general link invariants and the integrality of infinite family of new topological invariants. In string theory, this is a striking example that two important physical theories, topological string theory and Chern-Simons theory, exactly agree up to all orders. In mathematics this conjecture has interesting applications in understanding the basic structure of link invariants and three manifold invariants, as well as the integrality structure of open Gromov-Witten invariants. Recently, X.S. Lin and H. Zheng [30] verified LMOV conjecture in several lower degree cases for some torus links. In this paper, we give a complete proof of Labastida-Mariño-Ooguri-Vafa conjecture for any link (We will briefly call it LMOV conjecture). First, let us describe the conjecture and the main ideas of the proof. The details can be found in Sections 4 and 5. 0.2 Labastida-Mariño-Ooguri-Vafa conjecture Let L be a link with L components and P be the set of all partitions. The Chern-Simons partition function of L is given by ZCS(L; q, t) = ~A∈PL W ~A(L; q, t) sAα(x α) (0.1) for any arbitrarily chosen sequence of variables xα = (xα1 , x 2 , . . . , ) . In (0.1), W ~A(L) is the quantum group invariants of L labeled by a sequence of partitions ~A = (A1, . . . , AL) ∈ PL which correspond to the irreducible rep- resentations of quantized universal enveloping algebra Uq(sl(N,C)), sAα(x is the Schur function. Free energy is defined to be F = logZCS . Use plethystic exponential, one can obtain ~A 6=0 f ~A(q d, td) (xα)d , (0.2) where (xα)d = (xα1 ) d, (xα2 ) d, . . . Based on the duality between Chern-Simons gauge theory and topological string theory, Labastida, Mariño, Ooguri, Vafa conjectured that f ~A have the following highly nontrivial structures. For any A, B ∈ P, define the following function MAB(q) = χA(Cµ)χB(Cµ) ℓ(µ)∏ (q−µj/2 − qµj/2) . (0.3) Conjecture (LMOV). For any ~A ∈ PL, (i). there exists P ~B(q, t) for ∀ ~B ∈ PL, such that f ~A(q, t) = |Bα|=|Aα| P ~B(q, t) MAαBα(q). (0.4) Furthermore, P ~B(q, t) has the following expansion P ~B(q, t) = Q∈Z/2 N ~B; g,Q(q −1/2 − q1/2)2g−2tQ . (0.5) (ii). N ~B; g,Q are integers. For the meaning of notations, the definition of quantum group invariants of links and more details, please refer to Section 2. This conjecture contains two parts: • The existence of the special algebraic structure (0.5). • The integrality of the new invariants N ~B; g,Q. If one looks at the right hand side of (0.5), one will find it very interesting that the pole of f ~A in (q −1/2− q1/2) is actually at most of order 1 for any link and any labeling partitions. However, by the calculation of quantum group invariants of links, the pole order of f ~A might be going to ∞ when the degrees of labeling partitions go higher and higher. This miracle cancelation implies a very special algebraic structure of quantum group invariants of links and thus the Chern-Simons partition function. in Section 2.4.2, we include an example in the simplest setting showing that this cancelation has shed new light on the quantum group invariants of links. 0.3 Main ideas of the proof In our proof of LMOV conjecture, there are three new techniques. • When dealing with the existence of the conjectured algebraic structure, one will encounter the problem of how to control the pole order of (q−1/2 − q1/2). We consider the framed partition function Z(L; q, t, τ) of Chern-Simons invariants of links which satisfies the following cut- and-join equation ∂Z(L; q, t, τ) i,j≥1 ijpαi+j ∂pαi ∂p + (i+ j)pαi p ∂pαi+j Z(L; q, t, τ) . Here pαn = pn(x α) are regarded as independent variables. However, a deeper understanding of this conjecture relies on the fol- lowing log cut-and-join equation ∂F (L; q, t, τ) i,j≥1 ijpαi+j ∂pαi ∂p + (i+ j)pαi p ∂pαi+j + ijpαi+j This observation is based on the duality of Chern-Simons theory and open Gromov-Witten theory. The log cut-and-join equation is a non- linear ODE systems and the non-linear part reflects the essential recur- sion structure of Chern-Simons partition function. The miracle cance- lation of lower order terms of q−1/2 − q1/2 occurring in free energy can be indicated in the formulation of generating series of open Gromov- Witten invariants on the geometric side. A powerful tool to control the pole order of (q−1/2 − q1/2) through log cut-and-join equation is developed in this paper as we called cut-and- join analysis. An important feature of cut-and-join equation shows that the differential equation at partition1 (d) can only have terms obtained from joining2 while at (1d), non-linear terms vanishes and there is no joining terms. This special feature combined with the degree analysis will squeeze out the desired degree of (q−1/2 − q1/2). • We found a rational function ring which characterizes the algebraic structure of Chern-Simons partition function and hence (open) topo- logical string partition function by duality. A similar ring characterizing closed topological string partition func- tion firstly appears in the second author’s work on Gopakumar-Vafa conjecture [39, 40]. The original observation in the closed case comes from the structure of R-matrix in quantum group theory and gluing algorithm in the topological vertex theory. However, the integrality in the case of open topological string the- ory is more subtle than the integrality of Gopakumar-Vafa invariants 1Here, we only take a knot as an example. For the case of links, it is then a natural extension. 2Joining means that a partition is obtained from combining two rows of a Young dia- gram and reforming it into a new partition, while cutting means that a partition is obtained by cutting a row of a Young diagram into two rows and reforming it into a new partition. in the closed case. This is due to the fact that the reformulation of Gromov-Witten invariants as Gopakumar-Vafa invariants in the closed case weighted by power of curve classes, while in the open case, the gen- erating function is weighted by the labeling irreducible representation of Uq(sl(N,C)). This subtlety had already been explained in [22]. To overcome this subtlety, one observation is that quantum group in- variants look Schur-function-like. This had already been demonstrated in the topological vertex theory (also see [31]). We refine the ring in the closed case and get the new ring R(y; q, t) (cf. section 5.1). Corre- spondingly, we consider a new generating series ofN ~B; g,Q, T~d, as defined in (5.1). • To prove T~d ∈ R(y; q, t), we combine with the multi-cover contribution and p-adic argument therein. Once we can prove that T~d lies in R, due to the pole structure of the ring R, the vanishing of N ~B; g,Q has to occur at large genera, we actually proved Q∈Z/2 N ~B; g,Q(q −1/2 − q1/2)2gtQ ∈ Z[(q− 2 − q 2 )2, t± 2 ] . The paper is organized as follows. In Section 1, we introduce some basic notations about partition and generalize this concept to simplify our calcu- lation in the following sections. Quantum group invariants of links and main results are introduced in Section 2. In Section 3, we review some knowledge of Hecke algebra used in this paper. In Section 4 and 5, we give the proof of Theorem 1 and 2 which answer LMOV conjecture. In the last section, we discuss some problems related to LMOV conjecture for our future research. 0.4 Acknowledgments The authors would like to thank Professors S.-T. Yau, F. Li and Z.-P. Xin for valuable discussions. We would also want to thank Professor M. Mariño for pointing out some misleading parts. K. L. is supported by NSF grant. P. P. is supported by NSF grant DMS-0354737 and Harvard University. Before he passed away, Professor Xiao-Song Lin had been very interested in the LMOV conjecture, and had been working on it. We would like to dedicate this paper to his memory. 1 Preliminary 1.1 Partition and symmetric function A partition λ is a finite sequence of positive integers (λ1, λ2, · · · ) such that λ1 ≥ λ2 ≥ · · · . The total number of parts in λ is called the length of λ and denoted by ℓ(λ). We use mi(λ) to denote the number of times that i occurs in λ. The degree of λ is defined to be |λ| = If |λ| = d, we say λ is a partition of d. We also use notation λ ⊢ d. The automorphism group of λ, Aut λ, contains all the permutations that permute parts of λ while still keeping it as a partition. Obviously, the order of Aut λ is given by |Autλ| = mi(λ)! . There is another way to rewrite a partition λ in the following format (1m1(λ)2m2(λ) · · · ) . A traditional way to visualize a partition is to identify a partition as a Young diagram. The Young diagram of λ is a 2-dimensional graph with λj boxes on the j-th row, j = 1, 2, ..., ℓ(λ). All the boxes are put to fit the left-top corner of a rectangle. For example (5, 4, 2, 2, 1) = (12245) = . For a given partition λ, denote by λt the conjugate partition of λ. The Young diagram of λt is transpose to the Young diagram of λ: the number of boxes on j-th column of λt equals to the number of boxes on j-th row of λ, where 1 ≤ j ≤ ℓ(λ). By convention, we regard a Young diagram with no box as the partition of 0 and use notation (0). Denote by P the set of all partitions. We can define an operation “ ∪ ” on P. Given two partitions λ and µ, λ ∪ µ is the partition by putting all the parts of λ and µ together to form a new partition. For example (1223) ∪ (15) = (122235). Using Young diagram, it looks like ∪ = . The following number associated with a partition λ is used throughout this paper, jmj(λ)mj(λ)! , κλ = λj(λj − 2j + 1) . It’s easy to see that κλ = −κλt . (1.1) A power symmetric function of a sequence of variables x = (x1, x2, ...) is defined as follows pn(x) = xni . For a partition λ, pλ(x) = ℓ(λ)∏ pλj (x). It is well-known that every irreducible representation of symmetric group can be labeled by a partition. Let χλ be the character of the irreducible rep- resentation corresponding to λ. Each conjugate class of symmetric group can also be represented by a partition µ such that the permutation in the conju- gate class has cycles of length µ1, . . . , µℓ(µ). Schur function sλ is determined sλ(x) = |µ|=|λ| χλ(Cµ) pµ(x) (1.2) where Cµ is the conjugate class of symmetric group corresponding to partition 1.2 Partitionable set and infinite series The concept of partition can be generalized to the following partitionable set. Definition 1.1. A countable set (S,+) is called a partitionable set if 1). S is totally ordered. 2). S is an Abelian semi-group with summation “ + ”. 3). The minimum element 0 in S is the zero-element of the semi-group, i.e., for any a ∈ S, 0+ a = a = a+ 0. For simplicity, we may briefly write S instead of (S,+). Example 1.2. The following sets are examples of partitionable set: (1). The set of all nonnegative integers Z≥0; (2). The set of all partitions P. The order of P can be defined as follows: ∀λ , µ ∈ P, λ ≥ µ iff |λ| > |µ|, or |λ| = |µ| and there exists a j such that λi = µi for i ≤ j − 1 and λj > µj. The summation is taken to be “ ∪ ” and the zero-element is (0). (3). Pn. The order of Pn is defined similarly as (2): ∀ ~A, ~B ∈ Pn, ~A ≥ ~B i=1 |Ai| > i=1 |Bi|, or i=1 |Ai| = i=1 |Bi| and there is a j such that Ai = Bi for i ≤ j − 1 and Aj > Bj. Define ~A ∪ ~B = (A1 ∪ B1, ..., An ∪ Bn). ((0), (0), ..., (0)) is the zero-element. It’s easy to check that Pn is a partitionable set. Let S be a partitionable set. One can define partition with respect to S in the similar manner as that of Z≥0: a finite sequence of non-increasing non-minimum elements in S. We will call it an S-partition, (0) the zero S-partition. Denote by P(S) the set of all S-partitions. For an S-partition Λ, we can define the automorphism group of Λ in a similar way as that in the definition of traditional partition. Given β ∈ S, denote by mβ(Λ) the number of times that β occurs in the parts of Λ, we then have AutΛ = mβ(Λ)! . Introduce the following quantities associated with Λ, ℓ(Λ)! |AutΛ| , θΛ = (−1)ℓ(Λ)−1 uΛ . (1.3) The following Lemma is quite handy when handling the expansion of generating functions. Lemma 1.3. Let S be a partitionable set. If f(t) = n≥0 ant n, then β 6=0, β∈S Aβ pβ(x) Λ∈P(S) aℓ(Λ)AΛ pΛ(x) uΛ , where ℓ(Λ)∏ pΛj , AΛ = ℓ(Λ)∏ AΛj . Proof. Note that β∈S, β 6=0 Λ∈P(S), ℓ(Λ)=n ηΛuΛ . Direct calculation completes the proof. 2 Labastida-Mariño-Ooguri-Vafa conjecture 2.1 Quantum trace Let g be a finite dimensional complex semi-simple Lie algebra of rank N with Cartan matrix (Cij). Quantized universal enveloping algebra Uq(g) is generated by {Hi, X+i, X−i} together with the following defining relations: [Hi, Hj] = 0 , [Hi, X±j] = ±CijX±j , [X+i, X−j] = δij −Hi/2 i − q i − q 1−Cij∑ (−1)k 1− Cij 1−Cij−k ±i X±jX ±i = 0 , for all i 6= j , where {k}q = 2 − q k2 2 − q 12 , {k}q! = {i}q , and { {a}q · {a− 1}q · · · {a− b+ 1}q {b}q! The ribbon category structure associated with Uq(g) is given by the fol- lowing datum: 1. For any given two Uq(g)-modules V and W , there is an isomorphism ŘV,W : V ⊗W →W ⊗ V satisfying ŘU⊗V,W = (ŘU,W ⊗ idV )(idU ⊗ŘV,W ) ŘU, V⊗W = (idV ⊗ŘU,W )(ŘU, V ⊗ idW ) for Uq(g)-modules U , V , W . Given f ∈ HomUq(g)(U, Ũ), g ∈ HomUq(g)(V, Ṽ ), one has the following naturality condition: (g ⊗ f) ◦ ŘU, V = ŘeU, eV ◦ (f ⊗ g) . 2. There exists an element K2ρ ∈ Uq(g), called the enhancement of Ř, such that K2ρ(v ⊗ w) = K2ρ(v)⊗K2ρ(w) for any v ∈ V , w ∈ W . 3. For any Uq(g)-module V , the ribbon structure ΘV : V → V associated to V satisfies Θ±1V = trV Ř V,V . The ribbon structure also satisfies the following naturality condition x ·ΘV = ΘeV · x. for any x ∈ HomUq(g)(V, Ṽ ). Definition 2.1. Given z = i xi⊗yi ∈ EndUq(g)(U ⊗V ), the quantum trace of z is defined as follows trV (z) = tr(yiK2ρ)xi ∈ EndUq(g)(U). 2.2 Quantum group invariants of links Quantum group invariants of links can be defined over any complex simple Lie algebra g. However, in this paper, we only consider the quantum group invariants of links defined over sl(N,C)3 due to the current consideration for large N Chern-Simons/topological string duality. Roughly speaking, a link is several disconnected S1 embedded in S3. A theorem of J. Alexander asserts that any oriented link is isotopic to the closure of some braid. A braid group Bn is defined by generators σ1, · · · , σn−1 and defining relation: σiσj = σjσi , if |i− j| ≥ 2 ; σiσjσi = σjσiσj , if |i− j| = 1 . Let L be a link with L components Kα, α = 1, . . . , L, represented by the closure of an element of braid group Bm. We associate each Kα an irreducible representation Rα of quantized universal enveloping algebra Uq(slN), labeled by its highest weight Λα. Denote the corresponding module by VΛα. The j-th strand in the braid will be associated with the irreducible module Vj = VΛα , if this strand belongs to the component Kα. The braiding is defined through the following universal R-matrix of Uq(slN ) R = q Hi⊗Hj positive root β expq[(1− q−1)Eβ ⊗ Fβ] . 3In the following context, we will briefly write slN . Here (Cij) is the Cartan matrix and expq(x) = k(k+1) x {k}q! Define braiding by Ř = P12R, where P12(v ⊗ w) = w ⊗ v. Now for a given link L of L components, one chooses a closed braid representative in braid group Bm whose closure is L. In the case of no confusion, we also use L to denote the chosen braid representative in Bm. We will associate each crossing by the braiding defined above. Let U , V be two Uq(slN)-modules labeling two outgoing strands of the crossing, the braiding ŘU,V (resp. Ř V,U) is assigned as in Figure 1. PSfrag replacements U VŘU,V PSfrag replacements U VŘ−1V,U Figure 1: Assign crossing by Ř. The above assignment will give a representation of Bm on Uq(g)-module V1 ⊗ · · · ⊗ Vm. Namely, for any generator, σi ∈ Bm4, define h(σi) = idV1 ⊗ · · · ⊗ ŘVi+1,Vi ⊗ · · · ⊗ idVN . Therefore, any link L will provide an isomorphism h(L) ∈ EndUq(slN )(V1 ⊗ · · · ⊗ Vm) . For example, the link L in Figure 2 gives the following homomorphism h(L) = (ŘV, U ⊗ idU)(idV ⊗Ř−1U, U)(ŘU, V ⊗ idU) . PSfrag replacements Figure 2: A braid representative for Hopf link Let K2ρ be the enhancement of Ř in the sense of [41], where ρ is the half-sum of all positive roots of slN . The irreducible representation Rα is labeled by the corresponding partition Aα. Definition 2.2. Given L labeling partitions A1, . . . , AL, the quantum group invariant of L is defined as follows: W(A1,...,AL)(L) = qd(L) trV1⊗···⊗Vm(h(L)) , where d(L) = − ω(Kα)(Λα,Λα + 2ρ) + lk(Kα,Kβ)|Aα| · |Aβ| , and lk(Kα,Kβ) is the linking number of components Kα and Kβ. A sub- stitution of t = qN is used to give a two-variable framing independent link invariant. Remark 2.3. In the above formula of d(L), the second term on the right hand side is meant to cancel not important terms involved with q1/N in the definition. It will be helpful to extend the definition to allow some labeling partition to be the empty partition (0). In this case, the corresponding invariants will be regarded as the quantum group invariants of the link obtained by simply removing the components labeled by (0). 4In the case of σ−1 , use Ř−1 Vi,Vi+1 instead. A direct computation5 shows that after removing the terms of q N , qd(L) can be simplified as α=1 κAαw(Kα)/2 · t α=1 |A α|w(Kα)/2 . (2.1) Example 2.4. The following examples are some special cases of quantum group invariants of links. (1). If the link involved in the definition is the unknot ©, WA(©; q, t) = trVA(idVA) is equal to the quantum dimension of VA which will be denoted by dimq VA. (2). When all the components of L are associated with the fundamental representation, i.e., the labeling partition is the unique partition of 1, the quantum group invariant of L is related to the HOMFLY polynomial of the link, PL(q, t), in the following way: W( ,··· , )(L; q, t) = tlk(L) 2 − t 12 2 − q 12 PL(q, t) . (3). If L is a disjoint union of L knots, i.e., L = K1 ⊗K2 ⊗ · · · ⊗ KL , the quantum group invariants of L is simply the multiplication of quan- tum group invariants of Kα W(A1,...,AL)(L; q, t) = WAα(Kα; q, t) . 2.3 Chern-Simons partition function For a given link L of L components, we will fix the following notations in this paper. Given λ ∈ P, ~A = (A1, A2, . . . , AL), ~µ = (µ1, µ2, . . . , µL) ∈ PL. Let x = (x1, ..., xL) where xα is a series of variables xα = (xα1 , x 2 , · · · ) . 5It can also be obtained from the ribbon structure. The following notations will be used throughout the paper: [n]q = q 2 − q 2 , [λ]q = ℓ(λ)∏ [λj ]q , z~µ = zµα , | ~A| = (|A1|, ..., |AL|) , ‖ ~A ‖= |Aα| , ℓ(~µ) = ℓ(µα) , ~At = (A1)t, . . . , (AL)t , χ ~A (~µ) = χAα(Cµα) , s ~A(x) = sAα(x Denote by 1~µ = (1|µ 1|, · · · , 1|µL|) . (2.2) Chern-Simons partition function can be defined to be the following gen- erating function of quantum group invariants of L, ZCS(L) = 1 + ~A 6=0 W ~A(L; q, t)s ~A(x). Define free energy F = logZ = ~µ 6=0 F~µp~µ (x) . (2.3) Here in the similar usage of notation, p~µ(x) = pµα(x We rewrite Chern-Simons partition function as ZCS(L) = 1 + ~µ6=0 Z~µp~µ(x) where Z~µ = χ ~A(~µ) W ~A. (2.4) By Lemma 1.3, we have F~µ = Λ∈P(PL), |Λ|=~µ∈PL θΛZΛ . (2.5) 2.4 Main results 2.4.1 Two theorems that answer LMOV conjecture Let P ~B(q, t) be the function defined by (0.4), which can be determined by the following formula, P ~B(q, t) = | ~A|=|~B| f ~A(q, t) χAα(Cµ)χBα(Cµ) ℓ(µα)∏ q−µj/2 − qµj/2 Theorem 1. There exist topological invariants N ~B; g,Q ∈ Q such that expan- sion (0.5) holds. Theorem 2. Given any ~B ∈ PL, the generating function of N ~B; g,Q, P ~B(q, t), satisfies (q−1/2 − q1/2)2P ~B(q, t) ∈ Z q−1/2 − q1/2 , t±1/2 It is clear that Theorem 1 and 2 answered LMOV conjecture. Moreover, Theorem 2 implies, for fixed ~B, N ~B; g,Q vanishes at large genera. The method in this paper may apply to the general complex simple Lie algebra g instead of only considering sl(N,C). We will put this in our future research. If this is the case, it might require a more generalized duality picture in physics which will definitely be very interesting to consider and reveal much deeper relation between Chern-Simons gauge theory and geometry of moduli space. The extension to some other gauge group has already been done in [42] where non-orientable Riemann surfaces is involved. For the gauge group SO(N) or Sp(N), a more complete picture had appeared in the recent work of [6, 7] and therein a BPS structure of the colored Kauffman polynomials was also presented. We would like to see that the techniques developed in our paper extend to these cases. The existence of (0.5) has its deep root in the duality between large N Chern-Simons/topological string duality. As already mentioned in the introduction, by the definition of quantum group invariants, P ~B(q, t) might have very high order of pole at q = 1, especially when the degree of ~B goes higher and higher. However, LMOV-conjecture claims that the pole at q = 1 is at most of order 2 for any ~B ∈ PL. Any term that has power of q−1/2−q1/2 lower than −2 will be canceled! Without the motivation of Chern- Simons/topological string duality, this mysterious cancelation is hardly able to be realized from knot-theory point of view. 2.4.2 An application to knot theory We now discuss applications to knot theory, following [22, 20]. Consider associating the fundamental representation to each component of the given link L. As discussed above, the quantum group invariant of L will reduce to the classical HOMFLY polynomial PL(q, t) of L except for a universal factor. HOMFLY has the following expansion PL(q, t) = p2g+1−L(t)(q 2 − q 2 )2g+1−L . (2.6) The lowest power of q−1/2 − q1/2 is 1 − L, which was proved in [29] (or one may directly derive it from Lemma 4.1). After a simple algebra calculation, one will have F( ,..., ) = t−1/2 − t1/2 q−1/2 − q1/2 p̃2g+1−L(t)(q 2 − q 2 )2g+1−L . (2.7) Lemma 4.3 states that p̃1−L(t) = p̃3−L(t) = · · · = p̃L−3(t) = 0 , which implies that the pk(t) are completely determined by the HOMFLY polynomial of its sub-links for k = 1− L, 3− L, . . . , L− 3. Now, we only look at p̃1−L(t) = 0. A direct comparison of the coeffi- cients of F = logZCS immediately leads to the following theorem proved by Lickorish and Millett [29]. Theorem 2.5 (Lickorish-Millett). Let L be a link with L components. Its HOMFLY polynomial PL(q, t) = pL2g+1−L(t) 2 − q )2g+1−L satisfies pL1−L(t) = t 2 − t )L−1 L∏ pKα0 (t) where pKα0 (t) is HOMFLY polynomial of the α-th component of the link L with q = 1. In [29], Lickorish and Millett obtained the above theorem by skein anal- ysis. Here as the consequence of higher order cancelation phenomenon, one sees how easily it can be achieved. Note that we only utilize the vanishing of p̃1−L. If one is ready to carry out the calculation of more vanishing terms, one can definitely get much more information about algebraic structure of HOMFLY polynomial. Similarly, a lot of deep relation of quantum group invariants can be obtained by the cancelation of higher order poles. 2.4.3 Geometric interpretation of the new integer invariants The following interpretation is taken in physics literature from string theo- retic point of view [22, 38]. Quantum group invariants of links can be expressed as vacuum expec- tation value of Wilson loops which admit a large N expansion in physics. It can also be interpreted as a string theory expansion. This leads to a ge- ometric description of the new integer invariants N ~B; g,Q in terms of open Gromov-Witten invariants (also see [20] for more details). The geometric picture of f ~A is proposed in [22]. One can rewrite the free energy as −1ℓ(~µ) λ2g−2+ℓ(~µ)Fg,~µ(t)p~µ . (2.8) The quantities Fg,~µ(t) can be interpreted in terms of the Gromov-Witten in- variants of Riemann surface with boundaries. It was conjectured in [38] that for every link L in S3, one can canonically associate a lagrangian submanifold CL in the resolved conifold O(−1)⊕O(−1) → P1 . The first Betti number b1(CL) = L, the number of components of L. Let γα, α = 1, . . . , L, be one-cycles representing a basis for H1(CL, Z). Denote by Mg,h,Q the moduli space of Riemann surfaces of genus g and h holes embedded in the resolved conifold. There are hα holes ending on the non- trivial cycles γα for α = 1, . . . , L. The product of symmetric groups Σh1 × Σh2 × · · · × ΣhL acts on the Riemann surfaces by exchanging the hα holes that end on γα. The integer N ~B; q,t is then interpreted as N ~B; q,t = χ(S ~B(H ∗(Mg,h,Q))) (2.9) where S ~B = SB1 ⊗ · · · ⊗ SBL , and SBα is the Schur functor. The recent progress in the mathematical definitions of open Gromov- Witten invariants [18, 23, 24] may be used to put the above definition on a rigorous setting. 3 Hecke algebra and cabling 3.1 Centralizer algebra and Hecke algebra representa- We review some facts about centralizer algebra and Hecke algebra represen- tation and their relation to the representation of braid group. Denote by V the fundamental representation of Uq(slN) 6. Let {K±1i , Ei, Fi : 1 ≤ i ≤ N − 1} be the standard generators of the quantized universal enveloping algebra Uq(slN). Under a suitable basis {X1, ..., XN} of V , the fundamental repre- sentation is given by the following matrices Ei 7−→ Ei,i+1 Fi 7−→ Ei+1,i Ki 7−→ q−1/2Ei,i + q1/2Ei+1,i+1 + i 6=j 6We will reserve V to denote the fundamental representation of Uq(slN ) from now on. where Ei,j denotes the N × N matrix with 1 at the (i, j)-position and 0 elsewhere. Direct calculation shows K2ρ(Xi) = q −N+1−2i 2 Xi (3.1) 2N Ř(Xi ⊗Xj) = q−1/2Xi ⊗Xj, i = j , Xj ⊗Xi, i < j , Xj ⊗Xi + (q−1/2 − q1/2)Xi ⊗Xj, i > j . The centralizer algebra of V ⊗n is defines as follows Cn = EndUq(slN )(V ⊗n) = x ∈ End(V ⊗n) : xy = yx, ∀y ∈ Uq(slN) Hecke algebra Hn(q) of type An−1 is the complex algebra with n − 1 generators g1, ..., gn−1, together with the following defining relations gigj = gjgi , |i− j| ≥ 2 gigi+1gi = gi+1gigi+1 , i = 1, 2, ..., n− 2, (gi − q−1/2)(gi + q1/2) = 0 , i = 1, 2, ..., n− 1. Remark 3.1. Here we use q−1/2 instead of q to adapt to our notation in the definition of quantum group invariants of links. Note that when q = 1, the Hecke algebra Hn(q) is just the group algebra CΣn of symmetric group Σn. When N is large enough, Cn is isomorphic to the Hecke algebra Hn(q). A very important feature of the homomorphism h : CBn −→ Cn is that h factors through Hn(q) via 2N σi 7→ gi 7→ q− 2N h(σi) . (3.2) It is well-known that the irreducible modules Sλ (Specht module) ofHn(q) are in one-to-one correspondence to the partitions of n. Any permutation π in symmetric group Σn can express as a product of transpositions π = si1si2 · · · sil . If l is minimal in possible, we say π has length ℓ(π) = l and gπ = gi1gi2 · · · gil . It is not difficult to see that gπ is well-defined. All of such {gπ} form a basis of Hn(q). Minimal projection S is an element in Cn such that SV ⊗n is some irre- ducible representation Sλ. We denote it by pλ. The minimal projections of Hecke algebras are well studied (for example [14]), which is a Z(q±1)-linear combination {q 12gi}. 3.2 Quantum dimension 3.2.1 Explicit formula An explicit formula for quantum dimension of any irreducible representation of Uq(slN) can be computed via decomposing V ⊗n into permutation modules. A composition of n is a sequence of non-negative integer b = (b1, b2, . . .) such that ∑ bi = n . We will write it as b � n. The largest j such that bj 6= 0 is called the end of b and denoted by ℓ(b). Let b be a composition such that ℓ(b) ≤ N . DefineMb to be the subspace of V ⊗n spanned by the vectors Xj1 ⊗ · · · ⊗Xjn (3.3) such that Xi occurs precisely bi times. It is clear thatM b is anHn(q)-module and is called permutation module. Moreover, by explicit matrix formula of {Ei, Fi, Ki} acting on V under the basis {Xi}, we have Mb = {X ∈ V ⊗n : Ki(X) = q− bi−bi+1 2 X} . The following decomposition is very useful V ⊗n = b�n, ℓ(b)≤N Let A be a partition and VA the irreducible representation labeled by A. The Kostka number KAb is defined to be the weight of VA in M b, i.e., KAb = dim(VA ∩Mb) . Schur function has the following formulation through Kostka numbers sA(x1, . . . , xN) = b�|A|, ℓ(b)≤N By (3.1), K2ρ is acting on M b as a scalar j=1 q (N+1−2j)bj . Thus dimq VA = trVA idVA b�|A| dim(VA ∩Mb) (N+1−2j)bj 2 , . . . , q− (3.4) By (1.2), dimq VA = |µ|=|A| χA(Cµ) 2 , . . . , q− |µ|=|A| χA(Cµ) ℓ(µ)∏ t−µj/2 − tµj/2 q−µj/2 − qµj/2 . (3.5) Here in the last step, we use the substitution t = qN . 3.2.2 An expansion of the Mariño-Vafa formula Here we give a quick review about Mariño-Vafa formula [33, 26] for the convenience of Knot theorist. For details, please refer [26]. Let Mg,n denote the Deligne-Mumford moduli stack of stable curves of genus g with n marked points. Let π :M g,n+1 →M g,n be the universal curve, and let ωπ be the relative dualizing sheaf. The Hodge bundle E = π∗ωπ is a rank g vector bundle over M g,n. Let si : M g,n → M g,n+1 denote the section of π which corresponds to the i-th marked point, and let Li = s iωπ. A Hodge integral is an integral of the form 1 · · ·ψjnn λ 1 · · ·λkgg where ψi = c1(Li) is the first Chern class of Li, and λj = cj(E) is the j-th Chern class of the Hodge bundle. Let Λ∨g (u) = u g − λ1u+ · · ·+ (−1)gλg be the Chern polynomial of E∨, the dual of the Hodge bundle. Define Cg,µ(τ) = − −1ℓ(µ) |Aut(µ)| [τ(τ + 1)]ℓ(µ)−1 ℓ(µ)∏ ∏µi−1 a=1 (µiτ + a) (µi − 1)! Mg,l(µ) Λ∨g (1)Λ g (−τ − 1)Λ∨g (τ)∏ℓ(µ) i=1 (1− µiψi) Note that C0,µ(τ) = − −1ℓ(µ) |Aut(µ)| [τ(τ + 1)]ℓ(µ)−1 ℓ(µ)∏ ∏µi−1 a=1 (µiτ + a) (µi − 1)! M0,l(µ) ∏ℓ(µ) i=1 (1− µiψi) . (3.6) The coefficient of the leading term in τ is: −1ℓ(µ) |Autµ| · |µ|ℓ(µ)−3 . (3.7) The Mariño-Vafa formula gives the following identity: pµ(x) u2g−2+ℓ(µ)Cg,µ(τ) = log ρ)sA(x) (3.8) where qρ = (q−1/2, q−3/2, · · · , q−n+1/2, · · · ). ρ)sA(y)q Gµpµ. |Λ|=µ ℓ(Λ)∏ χAα(Λ sAα(q 2 (3.9) = Gµ(0) + |Λ|=µ Ω 6=(1Λ),A χA(Λ) χA(Ω) 2 − q− 12 )|µ| |Λ|=µ χA(Λ) A (3.10) The third summand of the above formula gives the non-vanishing leading term in τ which is equal to (3.7). Therefore, we have: |Λ|=µ χA(Λ) A 6= 0 (3.11) for ∀p ≥ |µ|+ ℓ(µ)− 2. 3.3 Cabling Technique Given irreducible representations VA1 , ..., VAL to each component of link L. Let |Aα| = dα, ~d = (d1, ..., dL). The cabling braid of L, L~d, is obtained by substituting dα parallel strands for each strand of Kα, α = 1, . . . , L. Using cabling of the L gives a way to take trace in the vector space of tensor product of fundamental representation. To get the original trace, one has to take certain projection, which is the following lemma in [30]. Lemma 3.2 ([30], Lemma 3.3). Let Vi = SiV ⊗di for some minimal projec- tions, Si = Sj if the i-th and j-th strands belong to the same knot. Then trV1⊗···Vm (h(L)) = trV ⊗n(h(L(d1,...,dL)) ◦S1 ⊗ · · · ⊗Sn) . where m is the number of strands belonging to L, n = α=1 dαrα, rα is the number of strands belong to Kα, the α-th component of L. 4 Proof of Theorem 1 4.1 Pole structure of quantum group invariants By an observation from the action of Ř on V ⊗ V , we define X̃(i1,...,in) = q #{(j,k)|j<k,ij>ik} 2 Xi1 ⊗ · · · ⊗Xin . {X̃(i1,...,in)} form a basis of V ⊗n. By (3.2), ∀gj ∈ Hn(q), we have 2gjX̃(...,ij,ij+1,...) = X̃(...,ij+1,ij ,...), ij ≤ ij+1 , qX̃(...,ij+1,ij ,...) + (1− q)X̃(...,ij,ij+‘,...), ij ≥ ij+1. (4.1) Lemma 4.1. Let © be the unknot. Given any ~A = (A1, . . . , AL) ∈ PL, W ~A(L; q, t) W ~A(©⊗L; q, t) ξKα(t) dα , (4.2) where |Aα| = dα, Kα is the α-th component of L, and ξKα(t), α = 1, . . . , L, are independent of ~A. Proof. Choose β ∈ Bm such that L is the closure of β, the total number of crossings of L~d is even and the last L strands belongs to distinct L com- ponents of L. Let rα be the number of the strands which belong to Kα. dαrα is equal to the number of components in the cabling link L~d. Y = tr dα(rα−1) . (4.3) Y is both a central element of EndUq(slN ) V ⊗(d1+...+dL) and a Z[q±1]-matrix under the basis {X̃(i1,...,in)}. On the other hand, p ~A = pA1 ⊗ ...⊗ pAL is a Z (q±1)-matrix under the basis {X̃(i1,...,in)}. By Schur lemma, we have p ~A ◦ Y = ǫ · p ~A , where ǫ ∈ Z(q±1) is an eigenvalue of Y . Since Z[q±1] is a UFD for transcen- dental q, ǫ must stay in Z[q±1]. By Lemma 3.2, trV ⊗n(p ⊗ · · · ⊗ p⊗rL ◦ L~d) = trV ⊗(d1+···dL)(p ~A ◦ Y) = ǫ · tr ⊗(d1+···dL) (p ~A) = ǫ ·W ~A(© ⊗L) . (4.4) The definition of quantum group invariants of L gives W ~A(L; q, t) = q κAαw(Kα)/2t dαw(Kα)/2 · ǫ ·W ~A(© ⊗L; q, t) (4.5) When q → 1, L~d reduces to an element in symmetric group Σn of ‖ ~d ‖ cycles. Moreover, when q → 1, the calculation is actually taken in individual knot component while the linking of different components have no effect. By Example 2.4 (1) and (3), we have W ~A(© ⊗L; q, t) = WAα(©; q, t) . This implies W ~A(L; q, t) W ~A(©⊗L; q, t) WAα(Kα; q, t) WAα(©; q, t) . (4.6) Let’s consider the case when K is a knot. A is the partition of d associated with K and Kd is the cabling of K. Each component of Kd is a copy of K. q → 1, Kd reduces to an element in Σdr. To calculate the Y , it is then equivalent to discussing d disjoint union of K. Say K has r strands. Consider Y0 = trV ⊗(r−1) K . The eigenvalue of Y0 is then PK(1,t)tw(K) , where PK(q, t) is the HOMFLY polyno- mial for K. Denote by ξK(t) = PK(1, t), we have WA(K; q, t) WA(©; q, t) = ξK(t) |A| . (4.7) Combined with (4.6), the proof is completed. 4.2 Symmetry of quantum group invariants Define φ~µ (q) = ℓ(µα)∏ /2 − qµαj /2) . Comparing (2.3) and (0.2), we have F~µ = χ ~A (~µ/d) z~µ/d qd, td d · z~µ/d χ ~A(~µ/d) P ~B(q d, td) MAαBα(q φ~µ/d(q z~µ/d χ ~B(~µ/d)P ~B(q d, td) = φ~µ(q) d · z~µ/d χ ~B(~µ/d)P ~B(q d, td) , φ~µ(q) d · z~µ/d χ ~B(~µ/d)P ~B(q d, td) . (4.8) Apply Möbius inversion formula, P ~B(q, t) = χ ~B(~µ) φ~µ(q) F~µ/d(q d, td) (4.9) where µ(d) is the Möbius function defined as follows µ(d) = (−1)r, if d is a product of r distinct prime numbers; 0, otherwise. To prove the existence of formula (0.5), we need to prove: • Symmetry of P ~B(q, t) = P ~B(q−1, t). • The lowest degree q−1/2 − q1/2 in P ~B is no less than −2. Combine (4.8), (2.5) and (2.4), it’s not difficult to find that the first property on the symmetry of P ~B follows from the following lemma. Lemma 4.2. W ~At(q, t) = (−1)‖ ~A‖W ~A(q −1, t). Proof. The following irreducible decomposition of Uq(slN) modules is well- known: V n = dBVB , where dB = χB(C(1n)). Let d ~A = α=1 dAα. Combined with Lemma 3.2 and eigenvalue of Y in Lemma 4.1 and (2.1), we have W( ,..., )(L~d) = | ~A|=~d W ~A(L)d ~A | ~A|=~d d ~A q κAαw(Kα)/2 · t |Aα|w(Kα)/2ǫ ~A · dimq VAα , (4.10) Where ǫ ~A is the eigenvalue of Y on α=1 VAα as defined in the proof of Lemma 4.1. Here if we change Aα to (Aα)t, we have κ(Aα)t = −κAα , which is equivalent to keep Aα while changing q to q−1. Note that W( ,..., )(L~d) is essentially a HOMFLY polynomial of L~d by Example 2.4. From the expansion of HOMFLY polynomial (2.6) and Exam- ple 2.4 (2), we have W( ,..., )(L~d ; q −1, t) = (−1) |Aα|W( ,..., )(L~d ; q, t) . (4.11) However, one can generalize the definition of quantum group invariants of links in the following way. Note that in the definition of quantum group invariants, the enhancement of Ř, K2ρ, acts on Xi (see (3.1)) as a scalar (N+1−2i). We can actually generalize this scalar to any zαi where α cor- responds the strands belonging to the α-th component (cf. [43]). It’s not difficult to see that (4.10) still holds. The quantum dimension of VAα thus becomes sAα(z 1 , . . . , z N) obtained in the same way as (3.4). We rewrite the above generalized version of quantum group invariants of links as W ~A(L; q, t; z), where z = {zα}. (4.11) becomes W( ,..., )(L~d ; q −1, t;−z) = (−1) |Aα|W( ,..., )(L~d ; q, t; z) (4.12) Now, combine (4.12), (4.10) and (4.11), we obtain d ~At q κ(Aα)tw(Kα)/2 · t |(Aα)t|w(Kα)/2ǫ ~At(q −1; −z) s(Aα)t(−zα) = (−1) α |Aα| · d ~A q κAαw(Kα)/2 · t |Aα|w(Kα)/2ǫ ~A(q; z) sAα(z (4.13) Note the following facts: sAt(−z) = (−1)ℓ(A)sA(z) , (4.14) d ~At = d ~A . (4.15) where the second formula follows from χAt(Cµ) = (−1)|µ|−ℓ(µ)χA(Cµ) . (4.16) Apply (4.14) and (4.15) to (4.13). Let zαi = q (N+1−2i), then using −z in- stead of z is equivalent to substitute q by q−1 while keeping t in the definition of quantum group invariants of links. This can be seen by comparing zαi = q −N−2i+1 ℓ(µ)∏ t−µj/2 − tµj/2 q−µj/2 − qµj/2 pµ(−zα) = (−1)ℓ(µ)pµ(zα) . Therefore, we have ǫ ~At(q −1, t) = ǫ ~A(q, t) . (4.17) By the formula of quantum dimension, it is easy to verify that W ~At(© L; q, t) = (−1)‖ ~A‖W ~A(© L; q−1, t) (4.18) Combining (4.4), (4.17) and (4.18), the proof of the Lemma is then com- pleted. By Lemma 4.2, we have the following expansion about P ~B P ~B(q, t) = Q∈Z/2 N ~B,g,Q(q −1/2 − q1/2)2g−2N0tQ for some N0. We will show N0 ≤ 1. Let q = eu. The pole order of (q−1/2 − q1/2) in P ~B is the same as pole order of u. Let f(u) be a Laurent series in u. Denote degu f to be the lowest degree of u in the expansion of u in f . Combined with (4.8), N0 ≤ 1 follows from the following lemma. Lemma 4.3. degu F~µ ≥ ℓ(~µ)− 2. Lemma 4.3 can be proved through the following cut-and-join analysis. 4.3 Cut-and-join analysis 4.3.1 Cut-and-join operators Let τ = (τ1, · · · , τL), substitute W ~A(L; q, t; τ) =W ~A(L; q, t) · q α=1 κAατα/2 in the Chern-Simons partition function, we have the following framed parti- tion function Z(L; q, t, τ) = 1 + ~A 6=0 W ~A(L; q, t, τ) · s ~A(x) . Similarly, framed free energy F (L; q, t, τ) = logZ(L; q, t, τ) . We also defined framed version of Z~µ and F~µ as follows Z(L; q, t, τ) = 1 + ~µ6=0 Z~µ(q, t, τ) · p~µ(x) , F (L; q, t, τ) = ~µ6=0 F~µ(q, t, τ)p~µ(x) . One important fact of these framing series is that they satisfy the cut-and- join equation which will give a good control of F~µ. Define exponential cut-and-join operator E i,j≥1 ijpi+j ∂pi∂pj + (i+ j)pipj ∂pi+j , (4.19) and log cut-and-join operator L i,j≥1 ijpi+j ∂pi∂pj + (i+ j)pipj ∂pi+j + ijpi+j . (4.20) Here {pi} are regarded as independent variables. Schur function sA(x) is then a function of {pi}. Schur function sA is an eigenfunction of exponential cut-and-join with eigenvalue κA [8, 12, 49]. Therefore, Z(L; q, t, τ) satisfies the following ex- ponential cut-and-join equation ∂Z(L; q, t, τ) EαZ(L; q, t, τ) , (4.21) or equivalently, we also have the log cut-and-join equation ∂F (L; q, t, τ) LαF (L; q, t, τ) . (4.22) In the above notation, Eα and Lα correspond to variables {pi} which take value of {pi(xα)}. (4.22) restricts to ~µ will be of the following form: |~ν|=|~µ|, ℓ(~ν)=ℓ(~µ)±1 α~µ~νF~ν + nonlinear terms , (4.23) where α~µ~ν is some constant, ~ν is obtained by cutting or jointing of ~µ. Recall that given two partitions A and B, we say A is a cutting of B if one cuts a row of the Young diagram of B into two rows and reform it into a new Young diagram which happens to be the Young diagram of A, and we also say B is a joining of A. For example, (7, 3, 1) is a joining of (5, 3, 2, 1) where we join 5 and 2 to get 7 boxes. Using Young diagram, it looks like =⇒ cut=⇒ . In (4.23), cutting and joining happens only for the α-th partition. 4.3.2 Degree of u By (2.5), it is easy to see through induction that for two links L1 and L2, F( ,··· , )(L1 ⊗ L2) = 0 . (4.24) For simplicity of writing, we denote by F( ,··· , )(L) = F ◦(L) Note that when we put the labeling irreducible representation by the fundamental ones, quantum group invariants of links reduce to HOMFLY polynomials except for a universal factor. Therefore, if we apply skein re- lation, the following version of skein relation can be obtained. Let positive crossing used later appear between two different components, K1 and K2, of the link. Denote by lk = lk(K1,K2), then: − F ◦ = t−lk+ 2 (q− 2 − q 2 )F ◦ (4.25) We want to claim that degu F ◦(L) ≥ L− 2 , (4.26) where L is the number of components of L. Firstly, for a knot K, a simple computation shows that: degu F (K) = −1 . Using induction, we may assume that claim (4.26) holds for L ≤ k. When L = k + 1, by (4.25), − F ◦ = 1 + degu ≥ k − 1 . (4.27) However, if < k − 1 , this will imply that in the procedure of unlinking L, the lowest degree term of u in F ◦ are always the same. However, this unlinking will lead to F ◦ equal to 0 due to (4.24), which is a contradiction! This implies that if the number of components of L1 is greater than the number of components of L2, we always have degu F ◦(L1) > degu F ◦(L2) Therefore, we proved the claim (4.26). 4.3.3 Induction procedure of cut-and-join analysis Let µi = (µi1, · · · , µiℓi). We use symbol Ẑ~µ = Z~µ · z~µ; F̂ = F~ν · z~µ . Notice the following fact from the definition of quantum group invariants: Ẑ(µ1,··· ,µL)(L) = Ẑ(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL )(L~µ) , (4.28) F̂(µ1,··· ,µL)(L) = F̂(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL)(L~µ) . (4.29) Let τ = (τ1, 0, · · · , 0). Similar as (3.9), we give the following formula: F~µ = |Λ|=~µ ℓ(Λ)∏ χAα(Λ W ~A(q, t)q = F~µ(0) + |Λ|=~µ Ω 6=(1Λ),A χA(Λ) χA(Ω) ZΩ(0)κ + Z1Λ(0) |Λ|=~µ χA(Λ) A (4.30) Let g ~µ (τα) be the degree of τα in the coefficient of the lowest degree of u in F~µ. By (3.11) and (4.30), if |µα| > 1, we have degτα g ~µ = |µ α|+ ℓ(µα)− 2 > 0 . By induction, assume that if |µα| ≤ d, degu F~µ ≥ ℓ(~µ) − 2 and hence by (3.11), degτα g ~µ (τα) = |µα|+ℓ(µα)−2. Combined with (4.29), if |λα| = d+1 and ℓ(λα) > 1, we have: degu F~λ = ℓ( ~λ)− 2 . Without loss of generality, assume µ1 = (d, 1), we will just consider the cut-and-join equation for τ1: ∂F(µ1,··· ,µL) (d+1),µ2··· ,µL ℓ(~ν)=ℓ(~µ)+1 β~νF~ν where β~ν are some constants and ∗ represents some non-linear terms in the cut-and-join equation. The crucial observation of this non-linear terms is that its degree in u is equal to ℓ(~µ)− 2. Comparing the degree in u on both sides of the equation, we have: degu F (d+1),µ2,··· ,µL ) = 1 + ℓ(µα)− 2 . This completes the induction. The proof of Lemma 4.3 follows immediately. Define F̃~µ = φ~µ(q) , Z̃~µ = φ~µ(q) Lemma 4.3 directly implies follows: Corollary 4.4. F̃ are of the following form: F̃~µ(q, t) = finitely many nα aα(t) [nα]2 + polynomial. Remark 4.5. Combine the above Corollary, (2.7) and (2.5), we have: F̃( ,..., )(q, t) = + polynomial. (4.31) 4.4 Framing and pole structures Consider δn = σ1 · · ·σn−1. LetSA be the minimal projection fromHn → HA, χA(Cµ)SA We will apply a lemma of Aiston-Morton [2] in the following computation: δnnSA = q κASA . (d1), . . . , (dL) , . . . , Due to the cabling formula to the length of partition (4.28) and (4.29), we can simply deal with all the color of one row without loss of generality. Take framing τα = nα+ and choose a braid group representative of L such that the writhe number of Lα is nα. Denote by ~τ = (τ1, . . . , τL), L; q, t;~τ χ ~A(C~d)W ~A(L; q, t)q α=1 κAα α dαnαTr χ ~A(C~d)q dα ⊗Lα=1 SAα dαnαTr χ ~A(C~d)(δd1 ⊗ · · · ⊗ δdL)⊗ α=1 SAα dαnαTr L~d · ⊗ α=1δα ·P ⊗ · · · ⊗P(dL) Here, P means that in the projection, we use qdα, tdα instead of using q, t. If we denote by L ∗Q~d = L~d · δ~d ·P ⊗ · · · ⊗P(dL) we have L; q, t; = H(L ∗Q~d) . (4.32) Here H is the HOMFLY polynomial which is normalized as H(unknot) = 2 − t− 12 2 − q− 12 With the above normalization, for any given link L, we have [1]L · H(L) ∈ Q [1]2, t± Substituting q by qdα in the corresponding component, it leads to the follows: [dα] · Ẑ~d(L; q, t;~τ) ∈ Q [1]2, t± . (4.33) On the other hand, given any two frames ~ω = (ω1, . . . , ωL), Ẑ~µ(L; q, t; ~ω) = χ ~A(C~µ)W ~A(L; q, t; ~ω) χ ~A(C~µ) χ ~A(C~ν) Ẑ~ν(L; q, t)q κAαωα . Exchange the order of summation, we have the following convolution formula: Ẑ~µ(L; q, t; ~ω) = Ẑ~ν(L; q, t) χ ~A(C~µ)χ ~A(C~ν)q κAαωα . (4.34) Return to (4.33). This property holds for arbitrary choice of nα, α = 1, . . . , L. By convolution formula, the coefficients of possible other poles vanish for arbitrary integer nα. q nα is involved through certain polynomial relation, which implies the coefficients for other possible poles are simply zero. Therefore, (4.33) holds for any frame. Consider Ẑ~d(L), where each component of L is labeled by a young diagram of one row. As discussed above, this assumption does not lose any generality. Let K be a component of L labeled by the color (c). If we multiply Z̃~d(L) by [c]2, we call it normalizing Z̃~d(L) w.r.t K. Similarly, we call [c]2F̃~d(L) normalizing F̃~d(L) w.r.t K. By (4.31), after normalizing F̃( ,..., )(L) w.r.t any component of L, one will obtain a polynomial in terms of [1]2 and t± 2 . Note that Z̃~d = Aut |Λ| Therefore, if we normalizing on both side w.r.t K, the equality still holds. Applying (4.32), one has [c]2F̃~d(L; q, t) ∈ Q [1]2, t± . (4.35) We thus proved the following proposition: Proposition 4.6. Notation as above, we have: [dα] · Ẑ~d(L; q, t) ∈ Q[[1] 2, t± 2 ]; (4.36) 2F̃~d(L; q, t) ∈ Q [1]2, t± , ∀α . (4.37) For a given ~d, denote by D~d the gcd{d1, . . . , dL}. Proposition 4.6 implies that the principle part of F̃ are possible summations of , where k divides all dα’s, or equivalently, k|D~d. Similar as above, choose a braid group element representative for L such that its α-th component has writhe number nα and choose frame as ~τ = (n1 + , . . . , nL + Ẑ~d(L; q, t;~τ) = χ ~A(C~d)W ~A(L; q, t)q κAα(nα+ dαnαTr χ ~A(C~d)q D~d ⊗Lα=1 SAα α dαnαTr(L~d · ⊗ ⊗Lα=1 P (D~d) (dα/D~d) α dαnαẐ~d/D~d (L~d · ⊗ ; qDµ, tDµ) . (4.38) This implies that Ẑ~d is a rational function of q D~d and t± D~d . Passing this to F̃~d, we have F̃~d is a rational function of [D~d] 2 and t± D~d . As discussed above, the principle part of F̃~d are summations of where k|D~d. Combining them, we have F̃~d(L; q, t) = H~d/D~d (tD~d) [D~d] + polynomial in [D~d] 2 and t± D~d . (4.39) Once again, due to arbitrary choice of nα, we know the above pole structure of F̃~d holds for any frame. Now we will show H~d/D~d (t) only depends on ~d/D~d and L. If one checks (4.38), one finds the computation also follows if 1/D~d is replaced by 1/k for any k|D~d. By induction, we obtain that H~d/D~d(t) only depends on ~d/D~d and Combining (4.29), we proved the following Proposition: Proposition 4.7. Notations are as above. Assume L is labeled by the color ~µ = (µ1, . . . , µL). Denote by D~µ is the greatest common divisor of {µ11, . . . , µ1ℓ(µ1), . . . , µij, . . . , µLℓ(µL)}. F̃~µ has the following structure: F̃~µ(q, t) = H~µ/D~µ(t [D~µ]2 + f(q, t) , where H~µ/D~µ(t) only depends on ~µ/D~µ and L, f(q, t) ∈ Q [1]2, t± Remark 4.8. In Proposition 4.7, it is very interesting to interpret in topo- logical string side that H~µ/D~µ(t) only depends on ~µ/D~µ and L. The principle term is generated due to summation of counting rational curves and indepen- dent choice of k in the labeling color k · ~µ/D~µ. This phenomenon simply tells us that contributions of counting rational curves in the labeling color k ·~µ/D~µ are through multiple cover contributions of ~µ/D~µ. 5 Proof of Theorem 2 5.1 A ring characterizes the partition function Let ~d = (d1, ..., dL), y = (y 1, . . . , yL). Define T~d = ~B=~d s ~B(y)P ~B(q, t) . (5.1) By the calculation in Appendix A.1, one will get T~d = q A∈P(Pn),‖A‖=~d/k θAWA(q k, tk)sA(z k) (5.2) However, T~d = | ~B|=~d s ~B(y)P ~B(q, t) Q∈Z/2 | ~B|=~d N ~B; g,Qs ~B(y) (q1/2 − q−1/2)2g−2tQ . Denote by Ω(y) the space of all integer coefficient symmetric functions in y. Since Schur functions forms a basis of Ω(y) over Z, N ~B; g,Q ∈ Z will follow from ∑ | ~B|=~d N ~B; g,Qs ~B(y) ∈ Ω(y) . Let v = [1]2q. It’s easy to see that [n] q is a monic polynomial of v with integer coefficients. The following ring is very crucial in characterizing the algebraic structure of Chern-Simons partition function, which will lead to the integrality of N ~B; g,Q. R(y; v, t) = a(y; v, t) : a(y; v, t) ∈ Ω(y)[v, t±1/2], b(v) = q ∈ Z[v] If we slightly relax the condition in the ring R(y; v, t), we have the following ring which is convenient in the p-adic argument in the following subsection. M(y; q, t) = f(y; q, t) : f ∈ Ω(y)[q±1/2, t±1/2], b(v) = q ∈ Z[v] Given f(y; q,t) ∈ M(y; q, t), if f(y; q, t) is a primitive polynomial in terms of q±1/2, t±1/2 and Schur functions of y, we call f(y; q,t) is primitive. 5.2 Multi-cover contribution and p-adic argument Proposition 5.1. T~d(y; q, t) ∈ R(y; v, t). Proof. Recall the definition of quantum group invariants of links. By the formula of universal R-matrix, it’s easy to see that W ~A(L) ∈ M(y; q, t) . Since we have already proven the existence of the pole structure in LMOV conjecture, Proposition 5.1 will be naturally satisfied if we can prove T~d(y; q, t) ∈ M(y; q, t) . (5.3) Before diving into the proof of (5.3), let’s do some preparation. For ∀A = ( ~A1, ~A2, ...) ∈ P(PL) , define A(d) = ~A1, ..., ~A1︸ ︷︷ ︸ , ~A2, ..., ~A2︸ ︷︷ ︸ , . . . Lemma 5.2. If θA = , d > 1 and gcd (c, d) = 1, then for any r|d, we can find B ∈ P(PL) such that A = B(r). Proof. Let ℓ = ℓ(A), we have (−1)ℓ(A)−1 (−1)ℓ(A)−1(ℓ(A)− 1)! |AutA| ~A1, ..., ~A1︸ ︷︷ ︸ , . . . , ~An, ..., ~An︸ ︷︷ ︸ so ℓ(A) = m1 + ...+mn. Note that m1, m2, · · · , mn Let η = gcd(m1, m2, · · · , mn). We have A = Ã(η), where ~A1, . . . , ~A1︸ ︷︷ ︸ , . . . , ~An, . . . , ~An︸ ︷︷ ︸ By Corollary A.4 , ℓ |uA and gcd(c, d) = 1, one has d|η. We can take B = ). This completes the proof. Remark 5.3. By the choice of η in the above proof, we know | ~Aα| is divisible by η for any α. By Lemma 5.2 and (5.2), we know T~d is of form f(y; q, t)/k, where f ∈ M(y; q, t), k|~d. We will show k is in fact 1. Given r f(y; q,t) where f(y; q, t) ∈ M(y; q, t) is primitive, define f(y; q, t) = Ordp . (5.4) Lemma 5.4. Given A∈ P , p any prime number and fA(y; q, t) ∈ M(y; q, t), we have θA(p)fA(p)(y; q, t)− θAfA(y p; qp, tp) ≥ 0 . Proof. Assume θA = pr ·a , where gcd(p r · a, b) = 1, p ∤ a. In (5.5), minus is taken except for one case: p = 2 and r = 0, in which the calculation is very simple and the same result holds. Therefore, we only show the general case which minus sign is taken. By Lemma 5.2, one can choose B ∈ P(PL) such that A = B(p r). Note that fA = f Let s = ℓ(B) and ~B1, . . . , ~B1︸ ︷︷ ︸ , . . . , ~Bk, . . . , ~Bn︸ ︷︷ ︸ Since ℓ(A(p)) = p · ℓ(A), by Theorem A.2, θA(p) − = Ordp p · ℓ(A) pr+1s pr+1s1, · · · , pr+1sk prs1, · · · , prsk (5.5) ≥ 2(r + 1)− (1 + r) > 0 . (5.6) By Theorem A.6, fA(p)(y; q, t)− fA(yp; qp, tp = Ordp pr+1a fB(y; q, t) pr+1 − fB(yp; qp, tp)p ≥ 0 . Apply the above two inequalities to θA(p)fA(p)(y; q, t)− θAfA(y p; qp, tp) = Ordp θA(p) − fA(p)(y; q, t) + fA(p)(y; q, t)− fA(yp; qp, tp The proof is completed. Apply the above Lemma, we have θA(p)WA(p)(q, t)sA(p)(z)− θAWA(q p, tp)sA(z ≥ 0 . (5.7) Φ~d (y; q, t) = A∈P(Pn), ‖A‖=~d θAWA(q, t)sA(z) . By Lemma 5.2, one has B : ‖ B ‖= p ~d and Ordp(θB) < 0 A(p) : ‖ A ‖= ~d Therefore, By (5.7), (y; q, t)− Φ~d (y p; qp, tp) ‖A‖=~d θA(p)WA(p)(q, t)sA(p)(z)− θAWA(q p, tp)sA(z ≥ 0 . We have thus proven the following Lemma. Lemma 5.5. For any prime number p and ~d, (y; q, t)− 1 Φ~d (y p; qp, tp) ≥ 0 . For any p|~d, by (5.2), T~d = q A∈P(PL), ‖A‖=~d/k θAWA(q k, tk)sA(z k|~d, p∤k Φ~d/k(y n; qn, tn) + k|~d, p∤k µ(pk) Φ~d/(pk)(y pk; qpk, tpk) k|~d, p∤k Φ~d/k(y k; qk, tk)− 1 Φ~d/(pk)(y pk; qpk, tpk) By Lemma 5.5, Ordp T~d ≥ 0 . (5.8) By the arbitrary choice of p, we prove that T~d ∈ M(y; q, t), hence T~d ∈ R(y; v, t) . The proof of Proposition 5.1 is completed. 5.3 Integrality By (4.9) and Propositions 4.7, (note that φ~µ/d(q d, td) = φ~µ(q, t).) P ~B(q, t) = χ ~B(~µ) φ~µ(q) F~µ/d(q d, td) χ ~B(~µ) F̃~µ/d(q d, td) χ ~B(~µ) d|D~µ H~µ/D~µ(t [D~µ]2 + polynomial χ ~B(~µ)δ1,D~µ H~µ/D~µ(t [D~µ]2 + polynomial , where δ1,n equals 1 if n = 1 and 0 otherwise. It implies that P ~B is a rational function which only has pole at q = 1. In the above computation, we used a fact of Möbius inversion, = δ1,n . Therefore, for each ~B, Q∈Z/2 N ~B; g,Q(q −1/2 − q1/2)2gtQ ∈ Q[(q−1/2 − q1/2)2, t± 2 ] . On the other hand, by Proposition 5.1, T~d ∈ R(y; v, t) and Ordp T~d ≥ 0 for any prime number p. We have | ~B|=~d N ~B; g,Qs ~B(y) ∈ Ω(y) . This implies N ~B; g,Q ∈ Z. Combine the above discussions, we have Q∈Z/2 N ~B; g,Q(q −1/2 − q1/2)2gtQ ∈ Z[(q−1/2 − q1/2)2, t±1/2] . The proof of Theorem 2 is completed. 6 Concluding Remarks and Future Research In this section, we briefly discuss some interesting problems related to string duality which may be approached through the techniques developed in this paper. 6.1 Duality from a mathematical point of view Let p = (p1, . . . ,pL), where pα = (pα1 , p 2 , . . . , ) . Defined the following gen- erating series of open Gromov-Witten invariants Fg,~µ(t, τ) = where ω is the Kähler class of the resolved conifold, τ is the framing parameter t = e ω, and e = tQ . Consider the following generating function F (p; u, t; τ) = u2g−2+ℓ(~µ)Fg,~µ(t; τ) pαµα . It satisfies the log cut-and-join equation ∂F (p; u, t; τ) LαF (p; u, t; τ) . Therefore, duality between Chern-Simons theory and open Gromov-Witten theory reduces to verifying the uniqueness of the solution of cut-and-join equation. Cut-and-join equation for Gromov-Witten side comes from the degen- eracy and gluing procedure while uniqueness of cut-and-join system should in principle be obtained from the verification at some initial value. How- ever, it seems very difficult to find a suitable initial value. For example, in the case of topological vertex theory, cut-and-join system has singularities when the framing parameter takes value at 0, −1, ∞ while these points are the possible ones to evaluate at. One solution might be through studying Riemann-Hilbert problem on controlling the monodromy at three singularity points. When the case goes beyond, the situation will be even more compli- cated. A universal method of handling uniqueness is required for the final proof of Chern-Simons/topological string duality conjecture. A new hope might be found in our development of cut-and-join analysis. In the log cut-and-join equation (4.22), the non-linear terms reveals the im- portant recursion structure. For the uniqueness of cut-and-join equation, it will appear as the vanishing of all non-linear terms. We will put this in our future research. 6.2 Other related problems There are many other problems related to our work on LMOV conjecture. We briefly list some problems that we are working on. Volume conjecture was proposed by Kashaev in [16] and reformulated by [34]. It relates the volume of hyperbolic 3-manifolds to the limits of quantum invariants. This conjecture was later generalized to complex case [35] and to incomplete hyperbolic structures [13]. The study of this conjecture is still staying at a rather primitive stage [36, 17, 48, 10]. LMOV conjecture has shed new light on volume conjecture. The cut-and- join analysis we developed in this paper combined with rank-level duality in Chern-Simons theory seems to provide a new way to prove the existence of the limits of quantum invariants. There are also other open problems related to LMOV conjecture. For example, quantum group invariants satisfy skein relation which must have some implications on topological string side as mentioned in [20]. One could also rephrase a lot of unanswered questions in knot theory in terms of open Gromov-Witten theory. We hope that the relation between knot theory and open Gromov-Witten theory will be explored much more in detail in the future. This will definitely open many new avenues for future research. A Appendix Here we carry out the calculation in Section 5.1. T~d = | ~B|=~d s ~B(y)P ~B(q, t) | ~B|=~d s ~B(y)χ ~B(~µ) φ~µ(q) F~µ/k qk, tk k|~µ ,|~µ|=~d p~µ(y) φ~µ(q) Λ∈P(P ), |Λ|=~µ/k θΛZΛ(q k, tk) k|~µ ,|~µ|=~d ‖Λ‖=~µ/k k)pΛ(y k)ZΛ(q k, tk) k|~µ ,|~µ|=~d ‖Λ‖=~µ/k k)pΛ(y ℓ(Λ)∏ χ~Aβ( W~Aβ(q |~Aβ |=|~Λβ |=dβ/k (−1)ℓ(Λ)−1 ℓ(Λ)∏ φ−1~Λβ (qk)p~Λβ(y χ~Aβ ( W~Aβ (q k, tk) k|~µ ,|~µ|=~d A=(~A1,...), ‖A‖=~d/k (−1)ℓ(A)−1 ℓ(A)∏ W~Aβ (q k, tk) χ~Aβ( φ−1~Λβ (qk)p~Λβ(y A∈P(PL), ‖A‖=~d/k θAWA(q k, tk)sA(z where pn(z) = pn(y) · pn(xi = qi−1) . Lemma A.1. p is prime, r ≥ 1. Then pr−1a pr−1b ≡ 0 mod (p2r) Proof. Consider the ratio pr−1a pr−1b (a−b)pr+k k∏pr−1b (a−b)pr−1+k gcd(k,p)=1, 1≤k≤prb (a− b)pr + k ≡ 1 + pr(a− b) gcd(k,p)=1, 1≤k≤prb mod (p2r) . Ap(n) = 1≤k≤n, gcd(k,p)=1 Therefore, the proof of the lemma can be completed by showing rb) = for some c, d such that gcd(d, p) = 1. If gcd(k, p) = 1, there exist αk, βk such that αkk + βkp r = 1 . Bp(n) = 1≤k≤n,gcd(k,p)=1 By the above formula, rb) ≡ bAp(pr) mod (pr) ≡ bBp(pr) mod (pr) , k − p pr−1∑ pr(pr + 1) pr−1(pr−1 + 1) p2r−1(p− 1) Here, we have 2r − 1 ≥ r since r ≥ 1. The proof is then completed. The following theorem is a simple generalization of the above Lemma. Theorem A.2. i=1 ai = a, p is prime, r ≥ 1, then pra1, · · · , pran pr−1a1, · · · , pr−1an ≡ 0 mod (p2r) Proof. We have pra1, · · · , pran pr−1a1, · · · , pr−1an pr(a− i=1 ai) pr−1(a− i=1 ai) pr−1ak ≡ 0 mod (p2r) . In the last step, we used the Lemma A.1. The proof is completed. Lemma A.3. We have gcd(a, b) Proof. Notice that ( gcd(a, b) gcd(a, b) However, gcd(a, b) gcd(a, b) = 1 , gcd(a, b) This direct leads to the following corollary. Corollary A.4. a = a1 + a2 + · · ·+ an. Then gcd(a1, a2, · · · , an) a1, a2, · · · , an Lemma A.5. a, r ∈ N, p is a prime number, then r − apr−1 ≡ 0 mod (pr) . Proof. If a is p, since pr−1 ≥ r, the claim is true. If gcd(a, p) = 1, by Fermat theorem, ap−1 ≡ 1 mod (p). We have r − apr−1 = apr−1 (kp+ 1)p r−1(p−1) − 1 pr−1∑ (kp)i ≡ 0 mod (pr) . Here in the last step, we used Lemma A.3. A direct consequence of the above lemma is the following theorem. Theorem A.6. Given f(y; q, t) ∈ M(y; q, t), we have f(y; q, t)p r+1 − f(yp; qp, tp)pr ≥ r + 1 . References [1] M. Aganagic, A. Klemm, M. Mariño and C. Vafa, “The topological vertex”, arxiv.org: hep-th/0305132. [2] A.K. Aiston and H.R. Morton, “Idempotents of Hecke algebras of type A”, J. Knot Theory Ramif. 7 (1998), 463-487. [3] D. Bar-Natan, “On the Vassiliev knot invariants”, Topology 34 (1995) 423. [4] J.S. 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Introduction Overview Labastida-Mariño-Ooguri-Vafa conjecture Main ideas of the proof Acknowledgments Preliminary Partition and symmetric function Partitionable set and infinite series Labastida-Mariño-Ooguri-Vafa conjecture Quantum trace Quantum group invariants of links Chern-Simons partition function Main results Hecke algebra and cabling Centralizer algebra and Hecke algebra representation Quantum dimension Cabling Technique Proof of Theorem 1 Pole structure of quantum group invariants Symmetry of quantum group invariants Cut-and-join analysis Framing and pole structures Proof of Theorem 2 A ring characterizes the partition function Multi-cover contribution and p-adic argument Integrality Concluding Remarks and Future Research Duality from a mathematical point of view Other related problems Appendix