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0704.1321

Reversed flow at low frequencies in a microfabricated AC electrokinetic pump

Reversed flow at low frequencies in a microfabricated AC electrokinetic pump Misha Marie Gregersen, Laurits Højgaard Olesen, Anders Brask, Mikkel Fougt Hansen, and Henrik Bruus MIC – Department of Micro and Nanotechnology, Technical University of Denmark DTU bldg. 345 east, DK-2800 Kongens Lyngby, Denmark (Dated: 29 Marts 2007) Microfluidic chips have been fabricated to study electrokinetic pumping generated by a low voltage AC signal applied to an asymmetric electrode array. A measurement procedure has been established and followed carefully resulting in a high degree of reproducibility of the measurements. Depending on the ionic concentration as well as the amplitude of the applied voltage, the observed direction of the DC flow component is either forward or reverse. The impedance spectrum has been thor- oughly measured and analyzed in terms of an equivalent circuit diagram. Our observations agree qualitatively, but not quantitatively, with theoretical models published in the literature. I. INTRODUCTION The recent interest in AC electrokinetic micropumps was initiated by experimental observations by Green, Gonzales et al. of fluid motion induced by AC electroos- mosis over pairs of microelectrodes [1, 2, 3] and by a the- oretical prediction by Ajdari that the same mechanism would generate flow above an electrode array [4]. Brown et al. [5] demonstrated experimentally pumping of elec- trolyte with a low voltage, AC biased electrode array, and soon after the same effect was reported by a num- ber of other groups observing flow velocities of the order of mm/s [6, 7, 8, 9, 10, 11, 12, 13]. Several theoretical models have been proposed parallel to the experimental observations [14, 15, 16]. However, so far not all aspects of the flow-generating mechanisms have been explained. Studer et al. [10] made a thorough investigation of flow dependence on electrolyte concentration, driving voltage and frequency for a characteristic system. In this work a reversal of the pumping direction for frequencies above 10 kHz and rms voltages above 2 V was reported. For a travelling wave device Ramos et al. [12] observed re- versal of the pumping direction at 1 kHz and voltages above 2 V. The reason for this reversal is not yet fully understood and the goal of this work is to contribute with further experimental observations of reversing flow for other parameters than those reported previously. An integrated electrokinetic AC driven micropump has been fabricated and studied. The design follows Studer et al. [10], where an asymmetric array of electrodes covers the channel bottom in one section of a closed pumping loop. Pumping velocities are measured in another sec- tion of the channel without electrodes. In this way elec- trophoretic interaction between the beads used as flow markers and the electrodes is avoided. In contrast to the soft lithography utilized by Studer et al., we use more well-defined MEMS fabrication techniques in Pyrex glass. This results in a very robust system, which exhibits sta- ble properties and remains functional over time periods extending up to a year. Furthermore, we have a larger electrode coverage of the total channel length allowing for the detection of smaller pumping velocities. Our im- proved design has led to the observation of a new phe- nomenon, namely the reversing of the flow at low voltages and low frequencies. The electrical properties of the fab- ricated microfluidic chip have been investigated to clarify whether these reflect the reversal of the flow direction. In accordance with the electrical measurements we propose and evaluate an equivalent circuit diagram. Supplemen- tary details related to the present work can be found in Ref. [17]. II. EXPERIMENTAL A. System design The microchip was fabricated for studies of the basic electrokinetic properties of the system. Hence, a simple microfluidic circuit was designed to eliminate potential side-effects due to complex device issues. The chip con- sists of two 500 µm thick Pyrex glass wafers anodically bonded together. Metal electrodes are defined on the bot- tom wafer and channels are contained in the top wafer, as illustrated schematically in Fig. 1(a). This construction ensures an electrical insulated chip with fully transparent channels. An electrode geometry akin to the one utilized by Brown et al. [5] and Studer et al. [10] was chosen. The translation period of the electrode array is 50 µm with electrode widths of W1 = 4.2 µm and W2 = 25.7 µm, and corresponding electrode spacings of G1 = 4.5 µm and G2 = 15.6 µm, see Fig. 1(d). Further theoretical investigations have shown that this geometry results in a nearly optimal flow velocity [16]. The total electrode array consists of eight sub-arrays each having their own connection to the shared contact pad, Fig. 1(b). This construction makes it possible to disconnect a malfunc- tioning sub-array. The entire electrode array has a width of 1.3 mm ensuring that the alignment of the electrodes and the 1.0 mm wide fluidic channels is not critical. A narrow side channel, Fig. 1(b), allows beads to be in- troduced into the part of the channel without electrodes, where a number of ruler lines with a spacing of 200 µm enable flow measurements by particle tracing, Fig. 1(c). An outer circuit of valves and tubes is utilized to con- http://arxiv.org/abs/0704.1321v1 FIG. 1: (a) Sketch of the fabricated chip consisting of two Pyrex glass wafers bonded together. The channels are etched into the top wafer, which also contains the fluid access ports. Flow-generating electrodes are defined on the bottom wafer. (b) Micrograph of the full chip containing a channel (white) with flow-generating electrodes (black) and a narrow side channel for bead injection (upper right corner). During flow measurements the channel ends marked with an asterisk are connected by an outer tube. The electrode array is divided into eight sub arrays, each having its own connection to the electrical contact pad. (c) Magnification of the framed area in panel (b) showing the flow-generating electrodes to the left and the measurement channel with ruler lines to the right. (d) Close up of an electrode array section with electrode transla- tion period of 50 µm. trol and direct electrolytes and bead solutions through the channels. During flow-velocity measurements, the inlet to the narrow side channel is blocked and to elimi- nate hydrostatic pressure differences the two ends of the main channel are connected by an outer teflon tube with an inner diameter of 0.5 mm. The hydraulic resistance of this outer part of the pump loop is three orders of mag- nitude smaller than the on-chip channel resistance and is thus negligible. The maximal velocity of the Poiseuille flow in the measurement channel section is denoted vpois, and the average slip velocity generated above the electrodes by electroosmosis is denoted vslip. To obtain a measurable vpois at as low applied voltages as possible, the electrode coverage of the total channel length is made as large Channel height H 33.6 µm Channel width w 967 µm Channel length Ltot 40.8 mm Channel length with electrodes Lel 16.0 mm Width of electrode array wel 1300 µm Narrow electrode gap G1 4.5 µm Wide electrode gap G2 15.6 µm Narrow electrode width W1 4.2 µm Wide electrode width W2 25.7 µm Electrode thickness h 0.40 µm Electrode surface area ([W1 + 2h]w) A1 4.84 × 10 −9 m2 Electrode surface area ([W2 + 2h]w) A2 25.63 × 10 −9 m2 Number of electrode pairs p 312 Electrode resistivity (Pt) ρ 10.6 × 10−8 Ωm Electrolyte conductivity (0.1 mM) σ 1.43 mS/m Electrolyte conductivity (1.0 mM) σ 13.5 mS/m Electrolyte permittivity ǫ 80 ǫ0 Pyrex permittivity ǫp 4.6 ǫ0 TABLE I: Dimensions and parameters of the fabricated mi- crofluidic system. as possible. In our system the total channel length is Ltot = 40.8 mm and the section containing electrodes is Lel = 16.0 mm, which ensures a high Poiseuille flow velocity, vpois = (3/4)(Lel/Ltot)vslip = 0.29 vslip [17]. The microfluidic chip has a size of approximately 16 mm × 28 mm and is shown in Fig. 1, and the device parameters are listed in Table I. B. Chip fabrication The flow-generating electrodes of e-beam evaporated Ti(10 nm)/Pt(400 nm) were defined by lift-off in 1.5 µm thick photoresist AZ 5214-E (Hoechst) using a negative process. The Ti layer ensures good adhesion to the Pyrex substrate. Platinum is electrochemically stable and has a low resistivity, which makes it suitable for the applica- tion. By choosing an electrode thickness of h = 400 nm, the metallic resistance between the contact pads and the channel electrolyte is at least one order of magnitude smaller than the resistance of the bulk electrolyte cover- ing the electrode array. In the top Pyrex wafer the channel of width w = 967 µm and height H = 33.6 µm was etched into the surface using a solution of 40% hydrofluoric acid. A 100 nm thick amorphous silicon layer was sputtered onto the wafer surface and used as etch mask in combination with a 2.2 µm thick photoresist layer. The channel pat- tern was defined by a photolithography process akin to the process used for electrode definition, and the wafer backside and edges were protected with a 70 µm thick etch resistant PVC foil. The silicon layer was then etched away in the channel pattern using a mixture of nitric acid and buffered hydrofluoric acid, HNO3:BHF:H2O = 20:1:20. The wafer was subsequently baked at 120◦C to harden the photoresist prior to the HF etching of the channels. Since the glass etching is isotropic, the chan- nel edges were left with a rounded shape. However, this has only a minor impact on the flow profile, given that the channel aspect ratio is w/H ≈ 30. The finished wafer was first cleaned in acetone, which removes both the pho- toresist and the PVC foil, and then in a piranha solution. After alignment of the channel and the electrode array, the two chip layers were anodically bonded together by heating the ensemble to 400◦C and applying a voltage difference of 700 V across the two wafers for 10 min. During this bonding process, the previously deposited amorphous Si layer served as diffusion barrier against the sodium ions in the Pyrex glass. Finally, immersing the chip in DI-water holes were drilled for the in- and outlet ports using a cylindrical diamond drill with a diameter of 0.8 mm. C. Measurement setup and procedures Liquid injection and electrical contact to the microchip was established through a specially constructed PMMA chip holder, shown in Fig. 2. Teflon tubing was fitted into the holder in which drilled channels provided a con- nection to the on-chip channel inlets. The interface from the chip holder to the chip inlets was sealed by O-rings. Electrical contact was obtained with spring loaded con- tact pins fastened in the chip holder and pressed against the electrode pads. The inner wires of thin coax cables were soldered onto the pins and likewise fastened to the holder. The pumping was induced by electrolytic solutions of KCl in concentrations ranging from c = 0.1 mM to 1.0 mM. The chip was prepared for an experiment by careful injection of this electrolyte into the channel and tubing system, after which the three valves to in- and outlets were closed. The electrical impedance spectrum of the microchip was measured before and after each series of flow measurements to verify that no electrode damaging had occurred during the experiments. If the Top PMMA plate Bottom PMMA plate Contact pins Aluminum holder Coax cables O-ringFitting FIG. 2: Chip holder constructed to connect external tubing and electrical wiring with the microfluidic chip. impedance spectrum had changed, the chip and the se- ries of performed measurements were discarded. Veloc- ity measurements were only carried out when the tracer beads were completely at rest before biasing the chip, and it was always verified that the beads stopped mov- ing immediately after switching off the bias. The steady flow was measured for 10 s to 30 s. After a series of mea- surements was completed, the system was flushed thor- oughly with milli-Q water. When stored in milli-Q water between experiments the chips remained functional for at least one year. D. AC biasing and impedance measurements Using an impedance analyzer (HP 4194 A), electrical impedance spectra of the microfluidic chip were obtained by four-point measurements, where each contact pad was probed with two contact pins. Data was acquired from 100 Hz to 15 MHz. To avoid electrode damaging by application of a too high voltage at low frequencies, all impedance spectra were measured at Vrms = 10 mV. The internal sinusoidal output signal of a lock-in am- plifier (Stanford Research SR830DSP) was used for AC biasing of the electrode array during flow-velocity mea- surements. The applied rms voltages were in the range from 0.5 V to 2 V and the frequencies between 0.5 kHz and 100 kHz. A current amplification was necessary to maintain the correct potential difference across the elec- trode array, since the overall chip resistance could be small (∼ 0.1 kΩ to 1 kΩ) when frequencies in the given in- terval were applied. The current through the microfluidic chip was measured by feeding the output signal across a small series resistor back into the lock-in amplifier. The lock-in amplifier was also used for measuring impedance spectra for frequencies below 100 Hz, which were beyond the span of the impedance analyzer. E. Flow velocity measurements After filling the channel with an electrolyte and ac- tuating the electrodes, the flow measurements were per- formed by tracing beads suspended in the electrolyte. Fluorescent beads (Molecular Probes, FluoSpheres F- 8765) with a diameter of 1 µm were introduced into the measurement section of the channel and used as flow markers for the velocity determination. A stereo microscope was focused at the beads, and with an at- tached camera pictures were acquired with time intervals of ∆t = 0.125 s to 1.00 s depending on the bead velocity. Subsequently, the velocity was determined by averaging over a distance of ∆x = 200 µm, i.e., v = ∆x/∆t. Only the fastest beads were used for flow detection, since these are assumed to be located in the vertical center of the channel. It should be noted that the use of fluorescent particles prevented an introduction of significant illumi- nation heating of the sample. 1 10 100 f [kHz] 1.0 V 1.5 V (Day 1) 1.5 V (Day 4) 1.5 V (Day 12) FIG. 3: Reproducible flow-velocities induced in a 0.1 mM KCl solution and observed at different days as a function of frequency at a fixed rms voltage of 1.5 V. A corresponding series was measured at Vrms = 1.0 V. Lines have been added to guide the eye. The limited number of acquired pictures led to an un- certainty of 5% in the determination of flow velocities, which corresponds to the movement of the tracer beads within 0.5 to 1 frame. Additionally, there is a statistical uncertainty on the vertical particle position in the chan- nel, which is estimated to introduce up to 10% error on the determined bead velocity. It is then assumed that the fastest beads are positioned within H/3 of the maximum of the Poiseuille flow profile. III. RESULTS In the parameter ranges corresponding to those pub- lished in the literature, our flow velocity measurements are in agreement with previously reported results. Us- ing a c = 0.1 mM KCl solution and driving voltages of Vrms = 1.0 V to 1.5 V over a frequency range of f = 1.1 kHz to 100 kHz, we observed among other mea- surement series the pumping velocities shown in Fig. 3. The general tendencies were an increase of velocity to- wards lower frequencies and higher voltages, and absence of flow above f ∼ 100 kHz. The measured velocities cor- responded to slightly more than twice those measured by Studer et al. [10] due to our larger electrode cover- age of the total channel. We observed damaging of the electrodes if more than 1 V was applied to the chip at a driving frequency below 1 kHz, for which reason there are no measurements at these frequencies. It is, however, plausible that the flow velocity for our chip peaked just below f ∼ 1 kHz. 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Vrms [V] s] 0.1 mM 0.4 mM 0.4 mM 0.5 mM 0.5 mM f = 1.0 kHz f [kHz] Vrms = 0.8 V 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Vrms [V] c = 0.4 mM CsAeff 2.30 µF 2.30 µF 0.55 µF 10 30 f [kHz] f = 1.0 kHz f = 12.5 kHz f = 21.0 kHz Vrms = 0.8 V FIG. 4: (a) Reversed flow observed for repeated measure- ments of two concentrations of KCl at 1.0 kHz. The inset shows that for a 0.4 mM KCl solution at a fixed rms volt- age of 0.8 V the flow direction remains negative, but slowly approaches zero for frequencies up to 50 kHz. (b) The theo- retical model presented in Ref. [19] predicts the trends of the experimentally observed velocity curves. The depicted graphs are calculated for a c = 0.4 mM solution and parameters cor- responding to the experiments (Table III) with ζeq = 160 mV. Additional curves have been plotted for slightly different pa- rameter values in order to obtain a closer resemblance to the experimental graphs, see Sec. IV. A. Reproducibility of measurements Our measured flow velocities were very reproducible due to the employed MEMS chip fabrication techniques and the careful measurement procedures described in Sec. II. This is illustrated in Fig. 3, which shows three velocity series recorded several days apart. The mea- surements were performed on the same chip and for the same parameter values. Between each series of measure- ments, the chip was dismounted and other experiments 0.1 1 10 102 103 104 105 106 107 108 f [Hz] |Z| (meas) |Z| (fit) θ (meas) θ (fit) |Z| ✲θ✛ FIG. 5: Bode plot showing the measured amplitude |Z| (right ordinate axis) and phase θ (left ordinate axis) of the impedance as a function of frequency over eight decades from 0.2 Hz to 15 MHz. The voltage was Vrms = 10 mV and the electrolyte concentration c = 1.0 mM KCl. The measure- ments are shown with symbols while the curves of the fitted equivalent diagram are represented by dashed lines. The mea- surement series obtained with the Impedance Analyzer consist of 400 very dense points while the series measured using the lock-in amplifier contains fewer points with a clear spacing. performed. However, it should be noted that a very slow electrode degradation was observed when a dozen of mea- surement series were performed on the same chip over a couple of weeks. B. Low frequency reversed flow Devoting special attention to the low-frequency (f < 20 kHz), low-voltage regime (Vrms < 2 V), not studied in detail previously, we observed an unanticipated flow reversal for certain parameter combinations. Fig. 4(a) shows flow velocities measured for a frequency of 1.0 kHz as a function of applied voltage for various electrolyte concentrations. It is clearly seen that the velocity series of c = 0.1 mM exhibits the known exclusively forward and increasing pumping velocity as function of voltage, whereas for slightly increased electrolyte concentrations an unambiguous reversal of the flow direction is observed for rms voltages below approximately 1 V. This reversed flow direction was observed for all fre- quencies in the investigated spectrum when the elec- trolyte concentration and the rms voltage were kept con- stant. This is shown in the inset of Fig. 4(a), where a velocity series was obtained over the frequency spectrum for an electrolyte concentration of 0.4 mM at a constant rms voltage of 0.8 V. It is noted that the velocity is nearly constant over the entire frequency range and tends to zero above f ∼ 20 kHz. ✻log |Z| log(ω) Rb Rel (a) (b) FIG. 6: (a) Equivalent circuit diagram. (b) Sketch of the impedance amplitude curve of the equivalent diagram. It con- sists of three plateaus, and four characteristic frequencies ωx, ω0, ωD and ωel (see Table II) that characterize the shape and may be utilized to estimate the component values. C. Electrical characterization To investigate whether the flow reversal was connected to unusual properties of the electrical circuit, we care- fully measured the impedance spectrum Z(f) of the mi- crofluidic system. Spectra were obtained for the chip containing KCl electrolytes with the different concentra- tions c = 0.1 mM, 0.4 mM and 1.0 mM. Fig. 5 shows the Bode plots of the impedance spec- trum obtained for c = 1.0 mM. For frequencies between f ∼ 1 Hz and f ∼ 103 Hz the curve shape of the impedance amplitude |Z| is linear with slope −1, after which a horizontal curve section follows, and finally the slope again becomes−1 for frequencies above f ∼ 106 Hz. Correspondingly, the phase θ changes between 0◦ and 90◦. From the decrease in phase towards low frequen- cies it is apparent that |Z| must have another horizontal curve section below f ∼ 1 Hz. When the curve is hori- zontal and the phase is 0◦ the system behaves resistively, while it is capacitively dominated when the phase is 90◦ and the curve has a slope of −1. Debye length λD Total electrode resistance Rel Total bulk electrolyte resistance Rb Total faradaic (charge transfer) resistance Rct Internal resistance in lock-in amplifier R Total measured resistance for ω → 0 Rx Total electrode capacitance Cel Total double layer capacitance Cdl Debye layer capacitance CD Surface capacitance Cs Debye frequency ωD Inverse ohmic relaxation time ω0 Inverse faradaic charge transfer time (primarily) ωx Characteristic frequency of electrode circuit ωel TABLE II: List of the symbols used in the equivalent circuit model. Rb Rb Rel Rel Rct Cdl Cdl Cel Cel ωD ωD ω0 ω0 mod meas mod meas meas mod meas mod meas mod meas mod meas [kΩ] [kΩ] [Ω] [Ω] [MΩ] [µF] [µF] [nF] [nF] [M rad s−1] [k rad s−1] 0.1 mM (A) 2.0 1.0 7.6 5 1.0 0.50 0.50 0.28 0.30 2.0 3.3 1.0 2.0 1.0 mM (A) 0.21 0.17 7.6 6 1.0 0.56 0.55 0.28 0.29 19.1 20.6 8.5 10.7 0.1 mM (B) 2.0 1.4 7.6 6 − 0.50 0.51 0.28 0.29 2.0 3.0 1.0 1.4 0.4 mM (B) 0.52 0.41 7.6 7 − 0.54 0.53 0.28 0.28 7.7 9.3 3.6 4.6 1.0 mM (B) 0.21 0.17 7.6 8 − 0.56 0.55 0.28 0.26 19.1 22.6 8.5 10.5 TABLE III: Comparison of measured (meas) and modeled (mod) values of the components in the equivalent diagram, Fig. 6. The measured values are given by curve fits of Bode plots, Fig. 5, obtained on two similar chips labeled A and B, respectively. The modeled values are estimated on basis of Table I and a particular choice of the parameters ζeq and Cs. This choice is not unique since different combinations can lead to the same value of Cdl. D. Equivalent circuit In electrochemistry the standard way of analyzing such impedance measurements is in terms of an equivalent circuit diagram [20]. The choice of diagram is not un- ambiguous [3]. We have chosen the diagram shown in Fig. 6(a) with the component labeling listed in Table II. Charge transport through the bulk electrolyte is repre- sented by an ohmic resistance Rb, accumulation of charge in the double layer at the electrodes by a capacitance Cdl, and faradaic current injection from electrochemical reac- tions at the electrodes by another resistance, the charge- transfer resistance Rct [16, 20]. Moreover, we include the ohmic resistance of the metal electrodes Rel, the mutual capacitance between the narrow and wide electrodes Cel, and a shunt resistance R ′ = 10 MΩ to represent the internal resistance of the lock-in amplifier. Finally, in electrochemical experiments at low fre- quency, the electrical current is often limited by diffusive transport of the reactants in the faradaic electrode reac- tion to and from the electrodes. This can be modeled by adding a frequency dependent Warburg impedance in series with the charge transfer resistance [20]. How- ever, because the separation between the electrodes is so small and the charge transfer resistance is so large, we are unable to distinguish the Warburg impedance in the impedance measurements and leave it out of the equiva- lent diagram. By fitting the circuit model to the impedance mea- surements we extract the component values listed in Ta- ble III. On the chip labeled B we were unable to measure the charge transfer resistance due to a minor error on the chip introduced during the bonding process. Fig. 6(b) il- lustrates the relation between component values and the impedance amplitude curve through four characteristic angular frequencies ω = 2πf . The inverse frequency Cdl primarily expresses the characteristic time for the faradaic charge transfer into the Debye layer. The characteristic time for charging the Debye layer through the electrolyte is given by ω−10 = Rb Cdl. The Debye fre- quency is ωD = 1/(RbCel), and finally ωel = 1/(RelCel) simply states the characteristic frequency for the on-chip electrode circuit in the absence of electrolyte. It is noted that the total DC-limit resistance R corresponds to the parallel coupling between R ′ and Rct. IV. DISCUSSION In the following we investigate to which extent the gen- eral theory of induced-charge (AC) electroosmosis can ex- plain our observations and experimental data. We first use the equivalent circuit component values extracted from the impedance measurements to estimate some im- portant electrokinetic parameters based on the Gouy– Chapman–Stern model [20], namely, the Stern layer ca- pacitance Cs, the intrinsic zeta potential ζeq on the elec- trodes and the charge transfer resistance Rct. Then we use this as input to the weakly nonlinear electro- hydrodynamic model presented in Ref. [19], which is an extension of the model in Ref. [16]. We compare theoret- ical values with experimental observations, and discuss the experimentally observed trends of the flow velocities. A. Impedance analysis The impedance measurements are performed at a low voltage of Vrms = 10 mV so it might be expected that Debye–Hückel linear theory applies (V . 25 mV). How- ever, since we only measure the potential difference be- tween the electrodes and we do not know the potential of the bulk electrolyte, we cannot say much about the exact potential drop across the double layer. Many electrode- electrolyte systems posses an intrinsic zeta potential at equilibrium ζeq of up to a few hundred mV. Indeed, the measured Cdl is roughly 10 times larger than predicted by Debye–Hückel theory, which indicates that the intrinsic zeta potential is at least ±125 mV. According to Gouy–Chapman–Stern theory the Cdl can be expressed as a series coupling of the compact Stern layer capacitance Cs and the differential Debye-layer ca- pacitance CD, , (1) where the two double-layer capacitances of an elec- trode pair are coupled in series through the electrolyte, and since the p electrode pairs are coupled in parallel, the effective area of the total double layer is Aeff = pA1A2/(A1+A2). A1 and A2 are the total surface areas exposed to the electrolyte of a narrow and wide electrode, respectively. For simplicity Cs is often assumed constant and independent of potential and concentration, while CD is given by the Gouy–Chapman theory as ζeqze . (2) Unfortunately, it is not possible to estimate the exact values of both Cs and ζeq from a measurement of Cdl, because a range of parameters lead to the same Cdl. We can, nevertheless, state lower limits as Cs ≥ 0.39 F/m and |ζeq| ≥ 175 mV for c = 0.1 mM or Cs ≥ 0.43 F/m and |ζeq| ≥ 125 mV at c = 1.0 mM. For the model values in Table III we used Eq. (1) with Cs = 1.8 F/m 2 and ζeq = 190 mV, 160 mV and 140 mV at 0.1 mM, 0.4 mM and 1.0 mM KCl, respectively, in ac- cordance with the trend often observed that ζeq decreases with increasing concentration, [21]. The bulk electrolyte resistance can be expressed as , (3) where σ is the conductivity, w is the width of the elec- trodes and p is the number of electrode pairs, see Table I, and 0.85 is a numerical factor computed for our particular electrode layout using the finite-element based program Comsol Multiphysics. Similarly, the mutual capaci- tance between the electrodes can be calculated as Cel = ǫw + ǫp(2wel − w) , (4) and the resistance Rel of the electrodes leading from the contact pads to the array is simply estimated from the resistivity of platinum and the electrode geometry. At frequencies above 100 kHz the impedance is dom- inated by Rb, Cel and Rel, and the Bode plot closely resembles a circuit with ideal components, see Fig. 5. Around 1 kHz we observe some frequency dispersion which could be due to the change in electric field line pattern around the inverse RC-time ω0 = 1/(RbCdl) [19]. Finally, below 1 kHz where the impedance is dominated by Cdl, the phase never reaches 90 ◦ indicating that the double layer capacitance does not behave as an ideal ca- pacitor but more like a constant phase element (CPE). This behavior is well known experimentally, but not fully understood theoretically [22]. B. Flow The forward flow velocities measured at c = 0.1 mM as a function of frequency, Fig. 3, qualitatively exhibit the trends predicted by standard theory, namely, the pump- ing increases with voltage and falls off at high frequency [4, 14]. More specifically, the theory predicts that the pumping velocity should peak at a frequency around the inverse RC-time ω0, corresponding to f ≈ 0.3 kHz, and decay as the inverse of the frequency for our applied driving volt- ages, see Fig. 11 in Ref. [16]. Furthermore, the velocity is predicted to grow like the square of the driving volt- age at low voltages, changing to V log V at large voltages [16, 19]. Experimentally, the velocity is indeed proportional to ω−1 and the peak is not observed within the range 1.1 kHz to 100 kHz, but it is likely to be just below 1 kHz. However, the increase in velocity between 1.0 V and 1.5 V displayed in Fig. 3 is much faster than V 2. That is also the result in Fig. 4(a) for c = 0.1 mM where no flow is observed below Vrms = 0.5 V, while above that voltage the velocity increases rapidly. For c = 0.4 mM and c = 0.5 mM the velocity even becomes negative at voltages Vrms ≤ 1 V. This cannot be explained by the standard theory and is also rather different from the re- verse flow that has been observed by other groups at larger voltages Vrms > 2 V and at frequencies above the inverse RC-time [10, 12, 13]. The velocity shown in the inset of Fig. 4(a) is re- markable because it is almost constant between 1 kHz and 10 kHz. This is unlike the usual behavior for AC electroosmosis that always peaks around the inverse RC- time, because it depends on partial screening at the elec- trodes to simultaneously get charge and tangential field in the Debye layer. At lower frequency the screening is almost complete so there is no electric field in the elec- trolyte to drive the electroosmotic fluid motion, while at higher frequency the screening is negligible so there is no charge in the Debye layer and again no electroosmosis. One possible explanation for the almost constant veloc- ity as a function of frequency could be that the amount of charge in the Debye layer is controlled by a faradaic elec- trode reaction rather than by the ohmic current running through the bulk electrolyte. Our impedance measure- ment clearly shows that the electrode reaction is negli- gible at f = 1 kHz and Vrms = 10 mV bias, but since the reaction rate grows exponentially with voltage in an Arrhenius type dependence, it may still play a role at Vrms = 0.8 V. However, previous theoretical investiga- tions have shown that faradaic electrode reactions do not lead to reversal of the AC electroosmotic flow or pumping direction [16]. Due to the strong nonlinearity of the electrode reaction and the asymmetry of the electrode array, there may also be a DC faradaic current running although we drive the system with a harmonic AC voltage. In the presence of an intrinsic zeta potential ζeq on the electrodes and/or the glass substrate this would give rise to an ordinary DC electroosmotic flow. This process does not necessar- ily generate bubbles because the net reaction products from one electrode can diffuse rapidly across the narrow electrode gap to the opposite electrode and be consumed by the reverse reaction. To investigate to which extent this proposition ap- plies, we used the weakly nonlinear theoretical model pre- sented in [19]. The model extends the standard model for AC electroosmosis by using the Gouy–Chapman–Stern model to describe the double layer, and Butler–Volmer reaction kinetics to model a generic faradaic electrode reaction [20]. The concentration of the oxidized and re- duced species in the diffusion layer near the electrodes is modeled by a generalization of the Warburg impedance, while the bulk concentration is assumed uniform, see Ref. [19] for details. The model parameters are chosen in accordance with the result of the impedance analysis, i.e., Cs = 1.8 F/m Rct = 1 MΩ, ζeq = 160 mV, as discussed in Sec. IVA. Further we assume an intrinsic zeta potential of ζeq = −100 mV on the borosilicate glass walls [21], and choose (arbitrarily) an equilibrium bulk concentration of 0.02 mM for both the oxidized and the reduced species in the electrode reaction, which is much less than the KCl electrolyte concentration of c = 0.4 mM. The result of the model calculation is shown in Fig. 4(b). At 1 kHz the fluid motion is dominated by AC electroosmosis which is solely in the forward direc- tion. However, at 12.5 kHz the AC electroosmosis is much weaker and the model predicts a (small) reverse flow due to the DC electroosmosis for Vrms < 1 V. Fig. 4(b) shows that the frequency interval with reverse flow is only from 30 kHz down to 10 kHz, while the mea- sured velocities remain negative down to at least 1 kHz. The figure also shows results obtained with a lower Stern layer capacitance Cs = 0.43 F/m 2 in the model, which turns out to enhance the reverse flow. In both cases, the reverse flow predicted by the theoret- ical model is weaker than that observed experimentally and does not show the almost constant reverse flow pro- file below 10 kHz. Moreover, the model is unable to ac- count for the strong concentration dependence displayed in Fig. 4(a). According to Ref. [18], steric effects give rise to a sig- nificantly lowered Debye layer capacitance and a poten- tially stronger concentration dependence when ζ exceeds 10 kBT/e ∼ 250 mV, which roughly corresponds to a driving voltage of Vrms ∼ 0.5 V. Thus, by disregarding these effects we overestimate the double layer capacitance slightly in the calculations of the theoretical flow velocity for Vrms = 0.8 V. This seems to fit with the observed ten- dencies, where theoretical velocity curves calculated on the basis of a lowered Cdl better resemble the measured curves. Finally, it should be noted that several electrode reac- tions are possible for the present system. As an example we mention 2H2O(l) +O2(aq) + 4e − ⇋ 4OH− . This re- action is limited by the amount of oxygen present in the solution, which in our experiment is not controlled. If this reaction were dominating the faradaic charge trans- fer, the value of Rct could change from one measurement series to another. V. CONCLUSION We have produced an integrated AC electrokinetic mi- cropump using MEMS fabrication techniques. The re- sulting systems are very robust and may preserve their functionality over years. Due to careful measurement procedures it has been possible over weeks to reproduce flow velocities within the inherent uncertainties of the velocity determination. An hitherto unobserved reversal of the pumping direc- tion has been measured in a regime, where the applied voltage is low (Vrms < 1.5 V) and the frequency is low (f < 20 kHz) compared to earlier investigated parameter ranges. This reversal depends on the exact electrolytic concentration and the applied voltage. The measured velocities are of the order −5 µm/s to −10 µm/s. Previ- ously reported studies of flow measured at the same pa- rameter combinations show zero velocity in this regime [10]. The reason why we are able to detect the flow reversal is probably our design with a large electrode coverage of the channel leading to a relative high ratio vpois/vslip = 0.29. Finally, we have performed an impedance character- ization of the pumping devices over eight frequency decades. By fitting Bode plots of the data, the measured impedance spectra compared favorably with our model using reasonable parameter values. The trends of our flow velocity measurements are ac- counted for by a previously published theoretical model, but the quantitative agreement is lacking. Most impor- tant, the predicted velocities do not depend on electrolyte concentration, yet the concentration seems to be one of the causes of our measured flow reversal, Fig. 4(a). This shows that there is a need for further theoretical work on the electro-hydrodynamics of these systems and in partic- ular on the effects of electrolyte concentration variation. Acknowledgments We would like to thank Torben Jacobsen, Department of Chemistry (DTU), for enlightening discussions about electrokinetics and the interpretation of impedance mea- surements on electrokinetic systems. [1] N. G. Green, A. Ramos, A. Gonzalez, H. Morgan and A. Castellanos, Phys. Rev. E 61(4), 4011 (2000). [2] A. Gonzalez, A. Ramos, N. G. Green, A. Castellanos and H. Morgan, Phys. Rev. E 61(4), 4019 (2000). [3] N. G. Green, A. Ramos, A. Gonzalez, H. Morgan and A. Castellanos, Phys. Rev. E 66, 026305 (2002). [4] A. Ajdari, Phys. Rev. E 61, R45 (2000). [5] A. B. D. Brown, C. G. Smith and A. R. Rennie, Phys. Rev. E 63, 016305 (2000). [6] V. Studer, A. Pépin, Y. Chen and A. Ajdari, Microelec- tron. Eng. 61-62, 915 (2002). [7] M. Mpholo, C. G. Smith and A. B. D. Brown, Sens. Ac- tuators B 92, 262 (2003). [8] D. Lastochkin, R. Zhou, P. Whang, Y. Ben and H.-C. Chang, J. Appl. Phys. 96, 1730 (2004). [9] S. Debesset, C. J. Hayden, C. Dalton, J. C. T. Eijkel and A. Manz, Lab Chip 4, 396 (2004). [10] V. Studer, A. Pépin, Y. Chen and A. Ajdari, The Analyst 129, 944 (2004). [11] B. P. Cahill, L. J. Heyderman, J. Gobrecht and A. Stem- mer, Phys. Rev. E 70, 036305 (2004). [12] A. Ramos, H. Morgan, N. G. Green, A. Gonzalez and A. Castellanos, J. Appl. Phys. 97, 084906 (2005). [13] P. Garćıa–Sánchez, A. Ramos, N. G. Green and H. Mor- gan, IEEE Trans. Dielect. El. In. 13, 670 (2006). [14] A. Ramos, A. Gonzalez, A. Castellanos, N. G. Green and H. Morgan, Phys. Rev. E 67, 056302 (2003). [15] N. A. Mortensen, L. H. Olesen, L. Belmon and H. Bruus, Phys. Rev. E 71, 056306 (2005). [16] L. H. Olesen, H. Bruus and A. Ajdari, Phys. Rev. E 73, 056313 (2006). [17] M. M. Gregersen, AC Asymmetric Electrode Microp- umps, MSc Thesis, MIC - Dept. of Micro and Nanotech- nology, DTU (2005), www.mic.dtu.dk/mifts [18] M. S. Kilic, M. Z. Bazant and A. Ajdari, Phys. Rev. E 75, 021502 (2007) and Phys. Rev. E 75, 021503 (2007). [19] L. H. Olesen, AC Electrokinetic micropumps, PhD The- sis, MIC - Dept. of Micro and Nanotechnology, DTU (2006), www.mic.dtu.dk/mifts [20] A. J. Bard and L. R. Faulkner, Electrochemical Methods, 2. ed. (Wiley, 2001). [21] B. J. Kirby and E. F. Hasselbrink, Electrophoresis 25, 187 (2004). [22] Z. Kerner and T. Pajkossy, Electrochim. Acta 46, 207 (2000).

0704.1322

Energy of 4-Dimensional Black Hole, etc

Energy of 4-Dimensional Black Hole Dmitriy Palatnik ∗ August 10, 2021 Abstract In this letter I suggest possible redefinition of mass density or stress-energy for a particle. Introducing timelike Killing vector and using Einstein identity E = mc , I define naturally density of mass for general case and calculate energy of black hole. 1 Introduction Standard definition of density [1], [3], [4], µ, for mass, γµ , (1) where γ is determinant of metric for spacial line element, lacks manifest dependence on speed of motion, v, or on gravitational interaction with other masses. Since this dependence is important in what follows below, I should redefine the mass density. Consider, first, flat spacetime with Cartesian coordinates. Matter distribution is specified by {µ;Ua}, field of density and field of 4-velocity, respectively. According to Einstein’s formula, m = m∗ , (2) for the mass field variation, δm, moving with speed, v, one may write, δm = δm∗ ; (3) Consider smooth distribution of 3-velocity, v, and assume that measuring of density takes place in frame in which matter locally at rest, i.e. v = 0. For differential of mass element measured experimentally by using Newton formula f = ma, one obtains, dm = dV 1− v2 ; (4) here µ∗ is density of mass, measured at rest. Formula (4) may be rewritten as dm = µ∗dV ; (5) where dΩ = cdtdV is 4-dimensional volume and ds2 = gabdx adxb. In third line of (5) I’ve transfered to general coordinate system with metric gab. There is an ambiguity in (5) because one might put there −g in stead of√ γ as well. As it’s shown below, −g seems to be more correct after accepting (with no proof) formula (15) for stress-energy; in order to get standard formula (10) for stress-energy, I take (5) as is. ∗The Waterford, 7445 N Sheridan Rd, Chicago, IL 60626-1818 [email protected] http://arxiv.org/abs/0704.1322v5 By definition (5) µ∗ is a (non-scalar) field independent of speed in difference with standard density of mass, µ, (1), which depends on speed and also not a scalar. Suppose, now, that mass, dm, is at rest in gravitational field g00 = 1 + 2 , where φ(x) is gravitational potential. From (5) it follows then, dm = µ∗ γdV , (7) where from (1), (7) follows µ∗ = µ g00. Using standard technique, i.e. formula δSm = (2c) −g Tbcδgbc , (8) where Sm = −mc ds ; (9) is action of matter, one obtains stress-energy for distribution of masses [1], . (10) Connection between standard density of mass and density measured at rest is µ = µ∗ . (11) Timelike Killing vector field, ξa, satisfies equations, ∇bξa +∇aξb = 0 ; (12) suppose, that spacetime is asymptotically flat and ξcξc → 1 on spacial infinity. If solution to (12) does exist, then, due to stress-energy symmetry T bc = T (bc) and conservation ∇cT bc = 0, current Jc = T cbξb conserves too: ∇c{T cbξb} = 0. Then, due to Gauss theorem conserving energy integral does exist, −gT 0bξb . (13) Accepting Einstein rule, E = mc2, as axiom, one might define mass element according to (13), as −gT 0cξc . (14) From (1), (10), (14) it follows formula for standard mass density: µ = µ∗ · U cξc , which in general case depends on additional factor U cξc. One might attempt to find another action rather than Sm = −mc ds, because as one observes, stress-energy should be taken as T ab = µ∗c , (15) in stead of (10), otherwise using (10) in (13) one would obtain infinite energy-mass for black hole. Besides, formulae (8), (9), (14) lead to inconsistent algebraic equation for stress-energy: T ab = T 0cξc UaU b It is most natural to change action (9), rather than formulae (8), (14). For case of Sz metric below, ξcU Note, that in formula (15) µ∗ is genuine scalar density of mass; for the element of mass (14) one would obtain, dm = µ∗ · U cξc . (16) In order to obtain (15) one should consider actions of type Sm = −mc Lm(α)ds, where Lm(α) is analog of lagrangian, depending only on α = U cξc. 2 Energy of Sz Black Hole Consider Schwarzschild solution, ds2 = 1− 2kM c2dt2 − 1− 2kM dr2 − r2(dθ2 + sin2 θdφ2) . (17) Solving (12) for metric (17), one obtains timelike Killing vector, 1− 2kM , 0, 0, 0 . (18) Substituting (14), (15), (17), (18) in (13), and using dV = drdθdφ, one gets, E = c2 r2dr sin θdθdφµ∗(r, θ, φ) . (19) Transfering to Cartesian coordinates, (r, θ, φ) → (x1, x2, x3), and introducing density of mass, µ∗ = Mδ(x 1)δ(x2)δ(x3) , (20) one obtains necessary expression for energy of black hole, E = Mc2. Note, that the same result could be derived by using formula (16). 3 Modification of Killing Equation Solutions to Killing eqs (12) do exist for restricted class of gravitational configurations; indeed, 4 components of Killing vector should satisfy 10 equations. Idea of following (i.e. formula (22) bellow) belongs to Boris Tsirelson. Consider, again, continuous matter distribution, specified by {µ∗ ; U c0}, scalar mass density field, µ∗(xc1), and 4-velocity field, U c0(xc1). Assume, that all other than matter fields are absent and only contribution of stress- energy is (15). Due to equations of motion of matter, stress-energy is divergence-free, ∇c0T c0c1 = 0, and symmetric. Then, current Jc0 = T c0c1ξc1 does have vanishing divergence, if T c0c1 [∇c0ξc1 +∇c1ξc0 ] = 0 . (21) From conservation of current, ∇c0Jc0 = 0, follows conservation of mass-energy (13). In stead of demanding implementation of Killing eqs (12) in order to have (21) satisfied, consider expression (15) and demand that Killing vector field satisfies following 4 equations: U c1 [∇c0ξc1 +∇c1ξc0 ] = 0 . (22) For Sz metric (17), solution to (22) with (g00) 2 , 0, 0, 0 , (23) is (18). Is it true that solving eqs (22) for empty space one may consider limit µ∗ → 0 with (23)? One could use (16) for computing mass-energy of black hole; result is same as above. One might find solution to (22) for Kerr black hole with metric [1], ds2 = 1− rgr dt2 + 2rgra sin2 θdtdφ − ρ dr2 − ρ2dθ2 r2 + a2 + sin2 θ sin2 θdφ2 ; (24) gab∂a∂b = r2 + a2 + sin2 θ 2rgra ∂t∂φ − 2 − 1 ∆ sin2 θ 1− rgr 2 , (25) where ρ2 = r2 + a2 cos2 θ ; (26) ∆ = r2 + a2 − rgr , (27) and using expression for 4-velocity, U c = g00 + 2φ̇g03 + (φ̇)2g33 , 0, 0, g00 + 2φ̇g03 + (φ̇)2g33  ; (28) here φ̇ ≡ dφ . That is, if to demand ξ1 = ξ2 = 0, then eqs (22) are equivalent to ∂rξ0 − 2Γ001ξ0 − 2Γ301ξ3 + φ̇{∂rξ3 − 2Γ013ξ0 − 2Γ313ξ3} = 0 ; (29) ∂θξ0 − 2Γ002ξ0 − 2Γ302ξ3 + φ̇{∂θξ3 − 2Γ023ξ0 − 2Γ323ξ3} = 0 . (30) Set of non-zero Christoffel symbols for metric (24) is Γ001, Γ 02, Γ 13, Γ 23, Γ 00, Γ 03, Γ 11, Γ 12, Γ 22, Γ Γ200, Γ 03, Γ 11, Γ 12, Γ 22, Γ 33, Γ 01, Γ 02, Γ 13, Γ −g = ρ2 sin θ . (31) The solutions of (29), (30) are,1 ξ(1)c0 = (g00, 0, 0, g03) ; ξ = (1, 0, 0, 0) ; (32) ξ(2)c0 = (g03, 0, 0, g33) ; ξ = (0, 0, 0, 1) ; (33) g00 = 1− ; g03 = sin2 θ ; g33 = − 2 + a2 + sin2 θ sin2 θ For energy of Kerr black hole use eqs (13), (15), (32); the result is E = c2 dr sin θdθdφµ∗(r; θ) g00 + φ̇g03 g00 + 2φ̇g03 + (φ̇)2g33 . (34) Transferring to Cartesian coordinates in case φ̇ = 0, according to transformation of spheroidal coordinates to Cartesian, specified in [5], (r, θ, φ) → (x, y, z), where r2 + a2 sin θ cosφ ; y = r2 + a2 sin θ sinφ ; z = r cos θ ; (35) and using (20), one recovers anew E = Mc2. Using second Killing vector, (33), one obtains conserving integral, Q = c2 dr sin θdθdφµ∗(r; θ) g03 + φ̇g33 g00 + 2φ̇g03 + (φ̇)2g33 . (36) 4 A Simple Theorem About Killing Vectors Here I should specify a theorem, claiming that if metric of spacetime doesn’t depend on coordinate xj , then spacetime has respective Killing vector ξc = gjc; number of Killing vectors for spacetime is equal to number of coordinates on which metric coefficients don’t depend. Proof. Rewrite Killing eqs (12) in form, ∂aξb + ∂bξa − 2Γcabξc = 0 . (37) Substitute ξc = gjc for specific j. Then eq (37) reads ∂jgab = 0 . (38) In case of eqs (22) one would have in stead of (38) equation Ua∂jgab = 0. One might even go step back and use eqs (21) then one would obtain equation UaU b∂jgab = 0 in stead of (38). 4-velocity, U a, should be taken along the trajectory of a particle. It’s understandable that if metric doesn’t depend on coordinate(s) xj , then does exist translational symmetry of the physical system in direction specified by that (or these) coordinate(s), which means that respective charges (energy, momenta) do conserve. 1 They are also solutions of original Killing eqs (12). 5 Acknowledgement I wish to thank Christian Network and Boris Tsirelson professor of Tel-Aviv University.2 I’m grateful for spirit of support from Catholic Church and Lubavich synagogue F.R.E.E. References [1] L D Landau E M Lifshits The Classical Theory of Fields Pergamon Press 1975 [2] L A Khalfin B S Tsirelson Foundations of Physics vol 22 No 7 1992 [3] Robert M Wald General Relativity U of Chicago Press 1984 [4] S Weinberg Gravitation and Cosmology John Wiley and Sons Inc NY 1972 [5] A Anabalon et al. arXiv: gr-qc/1009.3030v1 2 See, e.g. [2] 1 Introduction 2 Energy of Sz Black Hole 3 Modification of Killing Equation 4 A Simple Theorem About Killing Vectors 5 Acknowledgement

0704.1323

Multi-Higgs U(1) Lattice Gauge Theory in Three Dimensions

Multi-Higgs U(1) Lattice Gauge Theory in Three Dimensions Tomoyoshi Ono and Ikuo Ichinose Department of Applied Physics, Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, 466-8555 Japan Tetsuo Matsui Department of Physics, Kinki University, Higashi-Osaka, 577-8502 Japan (Dated: September 19, 2021) We study the three-dimensional compact U(1) lattice gauge theory with N Higgs fields numeri- cally. This model is relevant to multi-component superconductors, antiferromagnetic spin systems in easy plane, inflational cosmology, etc. For N = 2, the system has a second-order phase transition line c̃1(c2) in the c2(gauge coupling)−c1(Higgs coupling) plane, which separates the confinement phase and the Higgs phase. For N = 3, the critical line is separated into two parts; one for c2 . 2.25 with first-order transitions, and the other for c2 & 2.25 with second-order transitions. PACS numbers: 11.15.Ha, 05.70.Fh, 74.20.-z, 71.27.+a, 98.80.Cq There are many interesting physical systems involving multi-component (N -component) matter fields. Some- times they are associated with exact or approximate sym- metries like “flavor” symmetry. In some cases, the large- N analysis[1] is applicable and it gives us useful informa- tion. But the properties of the large-N systems may dif- fer from those at medium values of N that one actually wants to know. Study of the N -dependence of various systems is certainly interesting but not examined well. Among these “flavor” physics, the effect of matter fields upon gauge dynamics is of quite general inter- est in quantum chromodynamics, strongly correlated electron systems, quantum spins, etc.[2] In this let- ter, we shall study the three-dimensional (3D) U(1) gauge theory with multi-component Higgs fields φa(x) ≡ |φa(x)| exp(iϕa(x)) (a = 1, · · · , N). This model is of gen- eral interest, and knowledge of its phase structure, order of its phase transitions, etc. may be useful to get better understanding of various physical systems. These sys- tems include the following: N -component superconductor: Babaev[3] argued that under a high pressure and at low temperatures hydro- gen gas may become a liquid and exhibits a transition from a superfluid to a superconductor. There are two order parameters; φe for electron pairs and φp for pro- ton pairs. They may be treated as two complex Higgs fields (N = 2). In the superconducting phase, both φe and φp develop an off-diagonal long-range order, while in the superfluid phase, only the neutral order survives; lim|x|→∞〈φe(x)φp(0)〉 6= 0. p-wave superconductivity of cold Fermi gas: Each fermion pair in a p-wave superconductor has angular mo- mentum J = 1 and the order parameter has three com- ponents, Jz = −1, 0, 1. They are regarded as three Higgs fields (N = 3). As the strength of attractive force be- tween fermions is increased, a crossover from a super- conductor of the BCS type to the type of Bose-Einstein condensation is expected to take place[4]. Phase transition of 2D antiferromagnetic(AF) spin models: In the s = 1/2 AF spin models, a phase tran- sition occurs from the Neel state to the valence-bond solid state as parameters are varied. Senthil et al.[5] argued that the effective theory describing this transi- tion take a form of U(1) gauge theory of spinon (CP 1) field za(x) (|z1|2 + |z2|2 = 1). In the easy-plane limit (Sz = 0), |z1|2 = |z2|2 = 1/2 and so they are expressed by two Higgs fields as za = exp(iϕa)/ 2 (N = 2)[6]. Effects of doped fermionic holes (holons) to this AF spins are also studied extensively. The effective theory obtained by integrating out holon variables may be a U(1) gauge theory with N = 2 Higgs fields (with non- local gauge interactions). Kaul et al.[7] predicts that such a system exhibits a second-order transition, while numerical simulations of Kuklov et al.[8] exhibit a weak first-order transition. This point should be clarified in future study. Inflational cosmology: In the inflational cosmology[9], a set of Higgs fields is introduced to describe a phase transition and inflation in early universe. Plural Higgs fields are necessary in a realistic model[10]. The following simple consideration “predicts” the phase structure of the system. Among N phases ϕa(x) of the Higgs fields, the sum ϕ̃+ ≡ a ϕa couples to the gauge field and describes charged excitations, whereas the remaining N − 1 independent linear combinations ϕ̃i(i = 1, · · · , N − 1) describe neutral excitations. The latter N − 1 modes may be regarded as a set of N − 1 XY spin models. As the N = 1 compact U(1) Higgs model stays always in the confinement phase[11], we ex- pect N − 1 second-order transitions of the type of the XY model. Smiseth et al.[12] studied the noncompact U(1) Higgs models. A duality transformation maps the charged sec- tor into the inverted XY spin model. Thus they pre- dicted that the system exhibits a single inverted XY transition and N − 1 XY transitions. Their numerical study confirmed this prediction for N = 2. For N = 2, Kragset et al.[13] studied the effect of Berry’s phase term in the N = 2 compact Higgs model. They reported that Berry’s phase term sup- http://arxiv.org/abs/0704.1323v2 0.89 0.9 0.91 0.92 0.93 0.94 c 1 -1-0.5 0 0.5 1 1.5 2 �F�k���Q�S �F�k���P�U �F�k���P�Q (b)(a) �F�k���Q�S �F�k���P�U �F�k���P�Q FIG. 1: (a) System-size dependence of specific heat C for N = 2 at c2 = 0.4. (b) Scaling function η(x) for Fig.1(a). presses monopoles (instantons) and changes the second- order phase transitions to first-order ones. In this letter, we shall study the multi-Higgs models by Monte Carlo simulations. We consider the simplest form, i.e., the 3D compact lattice gauge theory without Berry’s phase; the Higgs fields φxa are treated in the London limit, |φxa| = 1. The action S consists of the Higgs coupling with its coefficient c1a (a = 1, . . . , N) and the plaquette term with its coefficient c2, x+µ,aUxµφxa +H.c. x,µ<ν (U †xνU x+ν,µUx+µ,νUxµ + H.c), (1) where Uxµ[= exp(iθxµ)] is the compact U(1) gauge field, µ, ν(= 1, 2, 3) are direction indices (we use them also as the unit vectors). We first study the N = 2 case with symmetric cou- plings c11 = c12 ≡ c1. We measured the internal energy E ≡ −〈S〉/L3 and the specific heat C ≡ 〈(S − 〈S〉)2〉/L3 in order to obtain the phase diagram and determine the order of phase transitions, where L3 is the size of the cubic lattice with the periodic boundary condition. In Fig.1(a), we show C at c2 = 0.4 as a function of c1. The peak of C develops as the system size is increased. The results indicate that a second-order phase transition occurs at c1 ≃ 0.91. By applying the finite-size-scaling (FSS) hypothesis to C in the form of C(c1, L) = L σ/νη(L1/νǫ), where ǫ = (c1 − c1∞)/c1∞ and c1∞ is the critical coupling at L → ∞, we obtained ν = 0.67, σ = 0.16, and c1∞ = 0.909. In Fig.1(b) we plot η(x), which supports the FSS. The above results for N = 2 are consistent with the “prediction” given above. The sum ϕ̃x+ ≡ ϕx1+ϕx2 cou- ples with the compact gauge field and generates no phase transition[11], while the difference ϕ̃x− ≡ ϕx1 − ϕx2 be- haves like the angle variable in the 3D XY model. The 3D XY model has a second-order phase transition with the critical exponent ν = 0.666...[14]. Our value of ν obtained above is very close to this value. However, it should be remarked that the simple separation of vari- ables in terms of ϕ̃± is not perfect due to the higher- order terms in the compact gauge theory. Nonetheless, 0 0.5 1 1.5 2 FIG. 2: Instanton density ρ for N = 2 at c2 = 0.4 as a function of c1. 0.5 1 1.5 2 2.5 3 3.5 4 c confinement Higgs �Fsecond order �Fcrossover FIG. 3: Phase diagram for N = 2. There are two phases, con- finement and Higgs, separated by second-order phase transi- tion line. There also exists a crossover line in the confinement phase separating dense and dilute instanton-density regions. our numerical studies strongly suggests that the phase transition for N = 2 belongs to the universality class of the 3D XY model. It is instructive to see the behavior of the instanton density ρ. We employ the definition of ρ in the 3D U(1) compact lattice gauge theory given by DeGrand and Toussaint[15]. ρ in Fig.2 decreases very rapidly near the phase transition point. This indicates that a “crossover” from dense to dilute instanton “phases” occurs simulta- neously with the phase transition. In other words, the observed phase transition can be interpreted as a con- finement(small c1)-Higgs(large c1) phase transition. In Fig.3, we present the phase diagram forN = 2 in the c2-c1 plane. There exists a second-order phase transition line separating the confinement and the Higgs phases. There also exists a crossover line similar to that in the 3D N = 1 U(1) Higgs model[11]. Let us turn to the N = 3 case. Among many pos- sibilities of three c1a’s, we first consider the symmetric case c11 = c12 = c13 ≡ c1. One may expect that there are two (N − 1 = 2) second-order transitions that may coincide at a certain critical point. Studying the N = 3 case is interesting from a general viewpoint of the critical phenomena, i.e., whether coincidence of multiple phase transitions changes the order of the transition. We stud- ied various points in the c2 − c1 plane and found that the order of transition changes as c2 varies. In Fig.4, we show E and ρ along c2 = 1.5 as a function of c1. Both quantities show hysteresis loops, which are signals of a first-order phase transition. In Fig.5, we present C at c2 = 3.0. The peak of C at around c1 ∼ 0.48 develops as L is increased, whereas E shows no discontinuity and hysteresis. Therefore, we conclude that the phase transi- tion at (c2, c1) ∼ (3.0, 0.48) is second order. In Fig.6(a), we present the phase diagram of the symmetric case for 0.548 0.55 0.552 0.554 0.556 c 1 0.548 0.55 0.552 0.554 0.556 0.015 0.025 (a) (b) FIG. 4: (a) Internal energy E and (b) instanton density ρ for N = 3 at c2 = 1.5 and L = 16. Both exhibit hysteresis loops indicating a first-order phase transition at c1 ≃ 0.551. 0.46 0.48 0.5 0.52 0.54 c 1 0.475 0.48 0.485 0.49 0.495 �F�k���Q�S �F�k���P�U �F�k���W �F�k���Q�S �F�k���P�U �F�k���W (a) (b) FIG. 5: (a) Specific heat for N = 3 at c2 = 3.0. (b) Close-up view near the peak. The peak develops as L increases. N = 3, where the order of transition between the confine- ment and Higgs phases changes from first (smaller c2) to second order (larger c2). In Fig.6(b) we present C along c1 = 0.2, which shows a smooth nondeveloping peak. ρ decreases smoothly around this peak. These results in- dicate crossovers at c2 ≃ 1.5. Then it becomes interesting to consider asymmetric cases, e.g., c11 6= c12 = c13. This case is closely related to a doped AF magnet. φ2 and φ3 correspond there to the CP 1 spinon field in the deep easy-plane limit, whereas φ1 corresponds to doped holes. This case is also relevant to cosmology because the order of Higgs phase transition in the early universe is important in the inflational cos- mology. Furthermore, one may naively expect that once a phase transition to the Higgs phase occurs at certain temperature T , no further phase transitions take place at lower T ’s even if the gauge field couples with other Higgs 0.5 1 1.5 2 2.5 3 3.5 �Fsecond order �Ffirst order �Fcrossover Higgs confinement 0 0.5 1 1.5 2 2.5 �F�k���Q�S �F�k���P�U �F�k���W c2 c10 (a) (b) FIG. 6: (a) Phase diagram for the N = 3 symmetric case. The phase transitions are first order in the region c2 . 2.25, whereas they are second order in the region c2 & 2.25. There exists a tricritical point at around (c2, c1) ∼ (2.25, 0.5). Crosses near c2 = 1.5 line show crossovers. (b) Specific heat for N = 3 at c1 = 0.2. It has a system-size independent smooth peak at which a crossover takes place. 0.3 0.35 0.4 0.45 0.5 c �F�k���Q�S �F�k���P�U �F�k���W 0.34 0.345 0.35 0.355 c11 0.460.48 0.5 0.520.540.56 �F�k���Q�S �F�k���P�U �F�k���W �F�k���Q�S �F�k���P�U �F�k���W (b) (c) FIG. 7: (a) Specific heat of the c1 = (1, 2, 2) model (N=3) at c2 = 1.0. (b,c) Close-up views of C near (b) c11 ∼ 0.35 and (c) c11 ∼ 0.52. bosons. However, our investigation below will show that this is not the case. Let us consider the case c12 = c13 = 2c11, which we call the c1 = (1, 2, 2) model, and focus on the case c2 = 1.0. As shown in Fig.7(a), C exhibits two peaks at c11 ∼ 0.35 and 0.52. Figs.7(b),(c) present the detailed behavior of C near these peaks, which show that the both peaks develop as L is increased. We conclude that both of these peaks show second-order transitions. This result is interpreted as the first-order phase transition in the symmetricN = 3 model is decomposed into two second-order transitions in the c1 = (1, 2, 2) model. Let us turn to the opposite case, c12 = c13 = 0.5c11, i.e., the c1 = (2, 1, 1) model at c2 = 1.0. One may expect that two second-order phase transitions appear as in the previous c1 = (1, 2, 2) model. However, the result shown in Fig.8 indicates that there exists only one second-order phase transition near c11 ∼ 1.08. The broad and smooth peak near c11 ∼ 0.85 shows no L dependence and we conclude that it is a crossover. This crossover is similar to that in the ordinary N = 1 gauge-Higgs system as we shall see by the measurement of ρ below. The orders of these transitions are understood as fol- lows: In the c1 = (1, 2, 2) model, as we increase c11, the two modes φxa(a = 2, 3) with larger c1a firstly be- come relevant and the model is effectively the symmetric N = 2 model. The peak in Fig.7(b) is interpreted as the second-order peak of this model. For higher c11’s, the gauge field is negligible due to small fluctuations, and the effective model is the N = 1 XY model of φx1. It gives the second-order peak in Fig.7(c). Similarly, in the c1 = (2, 1, 1) model, φx1 firstly becomes relevant. The ef- fective model is the N = 1 model, which gives the broad peak in Fig.8 as the crossover[11]. For higher c11’s, the effective model is the N = 2 symmetric model of φx2, φx3 and Uxµ, giving the sharp second-order peak in Fig.8. In Fig.9, we present ρ of the c1 = (1, 2, 2) and (2, 1, 1) models as a function of c11. ρ of the c1 = (1, 2, 2) model decreases very rapidly at around c11 ∼ 0.35, which is the 0.7 0.8 0.9 1 1.1 1.2 �F�k���Q�S �F�k���P�U �F�k���W FIG. 8: Specific heat of the c1 = (2, 1, 1) model at c2 = 1.0. 0.25 0.3 0.35 0.4 0.45 0.5 0.55 c110 0.5 1 1.5 (a) (b) c1=(1,2,2) model c1=(2,1,1) model FIG. 9: Instanton density ρ at c2 = 1.0 in the (a) c1 = (1, 2, 2) model and (b) c1 = (2, 1, 1) model. phase transition point in lower c11 region. On the other hand, at the higher phase transition point, c11 ∼ 0.52, ρ shows no significant changes. This observation indicates that the lower-c11 phase transition is the confinement- Higgs transition, whereas the higher-c11 transition is a charge-neutral XY -type phase transition. On the other hand, ρ of the c1 = (2, 1, 1) model de- creases rapidly at around c11 ∼ 0.85, where C exhibits a broad peak. This indicates that the crossover from the dense to dilute-instanton regions occurs there just like in the N = 1 case[11]. No “anomalous” behavior of ρ is ob- served at the critical point c11 ∼ 1.1, and therefore the phase transition is that of the neutral mode. We have also studied the symmetric case for N = 4, 5 at c2 = 0. Both cases show clear signals of first-order transitions at c1 ≃ 0.89(N = 4), 0.86(N = 5). On the other hand, at c2 = ∞, the gauge dynamics is “frozen” to Uxµ = 1 up to gauge transformations, so there remain N - fold independent XY spin models, which show a second- order transition at c1 ≃ 0.46. Thus we expect a tricritical point for general N > 2 at some finite c2 separating first- order and second-order transitions. Let us summarize the results. ForN = 2 there is a crit- ical line c̃1(c2) of second-order transitions in the c2 − c1 plane, which distinguishes the Higgs phase (c1 > c̃1) and the confinement phase (c1 < c̃1). This result is consis- tent with Kragset et al.[13]. For N = 3 there is a similar transition line, but the region 0 < c2 < c2c ≃ 2.25 is of second-order transitions while the region c2c < c2 is of first-order transitions. To study the mechanism of gener- ation of these first-order transitions, we studied the asym- metric cases and found two second-order transitions [in the c1 = (1, 2, 2) model] or one crossover and one second- order phase transition [in the c1 = (2, 1, 1) model]. The former case implies that two simultaneous second-order transitions strengthen the order to generate a first-order transition. Chernodub et al.[16] reported a similar gen- eration of an enhanced first-order transition in a related 3D Higgs model with singly and doubly charged scalar fields. We stress that the above change of the order is dynamical because (1) It depends on the value of c2, (2) Related 3D models, the CPN−1 and N -fold CP 1 gauge models, exhibit always second-order transitions (See the last reference of Ref.[2]). We thank Dr.K. Sakakibara for useful discussion. [1] See, e.g., S. Coleman, “Aspects of Symmetry” (Cam- bridge University Press 1985). [2] Y. Iwasaki, K. Kanaya, S. Sakai, and T. Yoshie, Phys. Rev. Lett.69, 21 (1992); G. Arakawa, I. Ichinose, T. Mat- sui, K. Sakakibara, Phys. Rev. Lett.94, 211601 (2005); S. Takashima, I. Ichinose, T. Matsui, Phys. Rev. B73, 075119 (2006). [3] E. Babaev, A. Sudbø, and N. W. Ashcroft, Nature 431, 666 (2004). [4] Y. Ohashi, Phys. Rev. Lett. 94, 050403 (2005). [5] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Science 303, 1490 (2004); T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Phys. Rev. B70, 144407 (2004). [6] Similar limit may be taken to relate the superconduc- tivity of ultracold fermionic atoms with spin J to the U(1) gauge model with N Higgs fields. J. Zhao, K. Ueda, and X. Wang, Phys. Rev. B74, 233102 (2006), consid- ered the SU(N) Hubbard model to describe the super- conductivity of fermionic atoms, which has a N = 2J+1- component order parameter. At large repulsion U and at the filling factor n = 1/N , the model becomes the U(1) gauge model with CPN−1 spins. A CPN−1 variable is parametrized as za = ρa exp(iϕa) with ρ2a = 1. In the symmetric limit, which is the easy-plane limit for N = 2, ρ2a = 1/N and za becomes a Higgs field. [7] R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, arXiv:cond-mat/0611536. [8] A. B. Kukulov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Ann. Phys. 321, 1602 (2006). [9] A. H. Guth, Phys. Rev. D23,347 (1981). [10] R. Allahverdi, K. Enqvist, J. Carcia-Bellido, and A. Mazumdar, arXiv:hep-ph/0605035. [11] S. Wenzel, E. Bittner, W. Janke, A.M.J. Schakel, and A. Schiller, Phys. Rev. Lett. 95, 051601(2005). [12] J. Smiseth, E. Smørgrav, and A. Sudbø, Phys. Rev. Lett. 93, 077002 (2004). [13] S. Kragset, E. Smørgrav, J. Hove, F. S. Nogueira, and A. Sudbø, Phys. Rev. Lett. 97, 247201 (2006). [14] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B63, 214503 (2001). [15] T. A. DeGrand and D. Toussaint, Phys. Rev. D22, 2478 (1980). [16] M. N. Chernodub, E.-M. Ilgenfritz, and A.Schller, Phys. Rev. B73, 100506 (2006). http://arxiv.org/abs/cond-mat/0611536 http://arxiv.org/abs/hep-ph/0605035

0704.1324

Identifying Dark Matter Burners in the Galactic center

Identifying Dark Matter Burners in the Galactic center Igor V. Moskalenko∗,† and Lawrence L. Wai∗∗,† ∗Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305 †Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94309 ∗∗Stanford Linear Accelerator Center, 2575 Sand Hill Rd, Menlo Park, CA 94025 Abstract. If the supermassive black hole (SMBH) at the center of our Galaxy grew adiabatically, then a dense "spike" of dark matter is expected to have formed around it. Assuming that dark matter is composed primarily of weakly interacting massive particles (WIMPs), a star orbiting close enough to the SMBH can capture WIMPs at an extremely high rate. The stellar luminosity due to annihilation of captured WIMPs in the stellar core may be comparable to or even exceed the luminosity of the star due to thermonuclear burning. The model thus predicts the existence of unusual stars, i.e. "WIMP burners", in the vicinity of an adiabatically grown SMBH. We find that the most efficient WIMP burners are stars with degenerate electron cores, e.g. white dwarfs (WD) or degenerate cores with envelopes. If found, such stars would provide evidence for the existence of particle dark matter and could possibly be used to establish its density profile. In our previous paper we computed the luminosity from WIMP burning for a range of dark matter spike density profiles, degenerate core masses, and distances from the SMBH. Here we compare our results with the observed stars closest to the Galactic center and find that they could be consistent with WIMP burners in the form of degenerate cores with envelopes. We also cross-check the WIMP burner hypothesis with the EGRET observed flux of gamma-rays from the Galactic center, which imposes a constraint on the dark matter spike density profile and annihilation cross-section. We find that the EGRET data is consistent with the WIMP burner hypothesis. New high precision measurements by GLAST will confirm or set stringent limits on a dark matter spike at the Galactic center, which will in turn support or set stringent limits on the existence of WIMP burners at the Galactic center. Keywords: black hole physics, dark matter, elementary particles, stellar evolution, white dwarfs, infrared, gamma rays PACS: 14.80.Ly, 95.30.Cq, 95.35.+d, 97.10.Cv, 97.10.Ri, 97.20.Rp, 98.35.Jk, 98.38.Jw, 98.70.Rz RESULTS The highest density “free space” dark matter regions occur for dark matter particles captured within the gravitational potential of adiabatically grown SMBHs. Any star close enough to such a SMBH can capture a large number of WIMPs during a short period of time. Annihilation of captured WIMPs may lead to considerable energy release in stellar cores thus affecting the evolution and appearance of such stars. Such an idea has been first proposed in [1] and further developed in [2] who applied it to main-sequence stars. An order-of-magnitude estimate of the WIMP capture rates for stars of various masses and evolution stages [3] lead us to the conclusion that WDs, fully burned stars without their own energy supply, are the most promising candidates to look for. WIMP capture by WDs or degenerate cores with envelopes located in a high density dark matter region has been discussed in detail in [4]. A high WIMP concentration in the stellar interior may affect the evolution and appearance of a star. The effects of WIMPs can be numerous, here we list only a few. The additional source of energy from WIMP pair-annihilation may cause convective energy transport from the stellar interior when radiative transport is not effective enough. In turn, this may inflate the stellar radius. On the other hand, WIMPs themselves may provide energy transport and suppress convection in the stellar core; this would reduce the replenishment of the thermonuclear burning region with fresh fuel. The appearance of massive stars and the bare WDs should not change, however. The former are too luminous, L∗ ∝ M4∗ , while the energy transport in the latter is dominated by the degenerate electrons. Here we discuss observational features of DM burners, and GLAST’s role in checking this hypothesis. There several possible ways to identify the DM burners: • The bare WDs burning DM should be hot, with luminosity maximum falling into the UV or X-ray band. The number of very hot WDs in the SDSS catalog [5] is small, just a handful out of 9316. This means that observation of a concentration of very hot WDs at the GC would be extremely unlikely unless they are “DM burners.” • Identification of DM burners may be possible by combining the data obtained by several experiments: – GLAST γ-ray measurements from the GC can be used to identify a putative DM spike at the SMBH, and also measure the annihilation flux from the spike. Identification of the DM spike requires a detection of a http://arxiv.org/abs/0704.1324v1 10-10 1.6 1.8 2 2.2 2.4 power-law index EGRET -3 s 103 104 105 106 Teff (K) L*/Lsun=1 R*/Rsun=10 FIGURE 1. Left: γ-ray flux vs. the DM central spike power-law index. The lines are shown for a series of annihilation cross sections 〈σv〉. Right: The visual K-band magnitude of DM burners at the GC without extinction vs. the effective surface temperature. point source at the GC (i.e. not extended) centered on the SMBH (i.e. with no offset), and a source spectrum matching a WIMP of a particular mass, which agrees with the “universal” WIMP mass as determined by any other putative WIMP signals (i.e. from colliders, direct detection, other indirect detection). – Direct measurement of the WIMP-nucleon scattering cross-section fixes the WIMP capture rate and thus the WIMP burner luminosity for a given degenerate core. – Determination of stellar orbits would allow a calculation of the WIMP burning rate by a particular star and, therefore, the proportion of its luminosity which is coming from the WIMP burning. – LHC measurements may provide information about the WIMP mass and interaction cross-sections. Figure 1 (left) shows the DM annihilation γ-ray flux from the central spike vs. DM density power-law index assuming 10γ’s above 1 GeV per annihilation and WIMP mass mχ = 100 GeV. The EGRET γ-ray flux from the GC Fγ(> 1 GeV) = 5× 10 −7 cm−2 s−1 [6]. Advances in near-IR instrumentation have made possible observations of stars in the inner parsec of the Galaxy [7, 8, 9]. The apparent K-band brightness of these stars is 14–17 mag while the extinction may be as large as 3.3 mag [10]. Assuming a central spike with index 7/3, the K-band brightness for bare Oxygen WDs with Teff ∼ 100,000 K and R∗/R⊙ ∼ 0.01 is about 22–23 mag not including extinction. A WIMP burning degenerate core with envelope may be cold enough to produce most of its emission in the IR band (Figure 1, right). For a given luminosity, the colder stars should necessarily have larger outer radii. A DM burner (w/envelope) with effective temperature Teff < 10,000 K and radius > 5R⊙ could have visual K-band magnitude mag > 10 (without extinction) and be visible with the current techniques. The horizontal dotted line (mag = 14) show the dimmest stars currently observed in the GC. I. V. M. acknowledges partial support from a NASA APRA grant. A part of this work was done at Stanford Linear Accelerator Center, Stanford University, and supported by Department of Energy contract DE-AC03-768SF00515. REFERENCES 1. P. Salati, and J. Silk, ApJ 338, 24–31 (1989). 2. A. Bouquet, and P. Salati, ApJ 346, 284–288 (1989). 3. I. V. Moskalenko, and L. L. Wai, arXiv: astro-ph/0608535 (2006). 4. I. V. Moskalenko, and L. L. Wai, ApJ 659, L29–L32 (2007). 5. D. J. Eisenstein et al., ApJS 167, 40–58 (2006). 6. H. A. Mayer-Hasselwander et al., Astron. Astrophys. 335, 161–172 (1998). 7. R. Genzel et al., Mon. Not. Royal Astron. Soc. 317, 348–374 (2000). 8. A. M. Ghez et al., ApJ 586, L127–L131 (2003). 9. A. M. Ghez et al., ApJ 620, 744–757 (2005). 10. G. H. Rieke, M. J. Rieke, and A. E. Paul, ApJ 336, 752–761 (1989). http://arxiv.org/abs/astro-ph/0608535 Results

0704.1325

Instabilities in the time-dependent neutrino disc in Gamma-Ray Bursts

Instabilities in the time-dependent neutrino disc in Gamma-Ray Bursts A. Janiuk1,2, Y. Yuan3, R. Perna4 & T. Di Matteo5 ABSTRACT We investigate the properties and evolution of accretion tori formed after the coalescence of two compact objects. At these extreme densities and temperatures, the accreting torus is cooled mainly by neutrino emission produced primarily by electron and positron capture on nucleons (β reactions). We solve for the disc structure and its time evolution by introducing a detailed treatment of the equation of state which includes photodisintegration of helium, the condition of β-equilibrium, and neutrino opacities. We self-consistently calculate the chemical equilibrium in the gas consisting of helium, free protons, neutrons and electron-positron pairs and compute the chemical potentials of the species, as well as the electron fraction throughout the disc. We find that, for sufficiently large accretion rates (Ṁ & 10M⊙/s), the inner regions of the disk become opaque and develop a viscous and thermal instability. The identification of this instability might be relevant for GRB observations. Subject headings: accretion, accretion discs – black hole physics – gamma rays: bursts – neutrinos 1. Introduction Gamma-Ray Bursts are commonly thought to be produced in relativistic ejecta that dissipate en- ergy by internal shocks however alternative ideas based on the Poynting flux dominated jets are also being proposed (for a review see e.g. Piran 2005; Meszaros 2006; Zhang 2007). The enormous power released during the gamma- ray burst explosion indicates that a relativistic phenomenon must be involved in creating GRBs (Narayan et al. 1992). The merger of two neutron stars or of a neutron star and a black hole (e.g. NS- NS or NS-BH) has been invoked e.g. by Paczyński (1986); Eichler et al. (1989); Paczyński (1991); 1Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland 2Present address: University of Nevada, Las Vegas, 4505 Maryland Pkwy, NV89154, USA 3 Center for Astrophysics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China 4JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, 440 UCB, Boulder, CO 80309, USA 5Physics Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15232, USA Narayan, Paczyński & Piran (1992), as well as the collapse of a massive star, the so called “collap- sar” scenario (e.g., Woosley 1993 and Paczyński 1998). In both cases a dense, hot accretion disk is likely to form around a newly born black hole (Witt et al. 1994). In the collapsar scenario, the collapsing envelope of the star accretes onto the newly formed black hole, while a transient debris disk is formed when the NS-NS or NS-BH binary merges (see e.g. Ruffert et al. 1997 for numerical simulations). The durations of GRBs, which range from mil- liseconds to over a thousand of seconds, are dis- tributed in two distinct peaks defining two main GRB classes: short (≤ 2 sec) and long (& 2 sec) bursts (Kouvelietou et al. 1993). For some long bursts, signatures of an accompanying supernova explosion have been detected in the afterglow spec- tra (Stanek et al. 2003; Hjorth et al. 2003), which strongly favors the “collapsar” interpretation for their origin. Furthermore, the GRB positions in- ferred from the afterglow observations are consis- tent with the GRBs being associated with the star forming regions in their host galaxies. For short bursts, several pieces of evidence from the analy- http://arxiv.org/abs/0704.1325v1 sis of the Swift satellite and follow-up observations (Gehrels et al. 2005; Fox et al. 2005; Villasenor et al. 2005) argue in favor of a binary merger model (Hjorth et al. 2005; Berger et al. 2005). In the merger scenario, the duration of the burst is comparable to the viscous timescale of the accretion disc whereas in the collapsar scenario, the external reservoir of stellar matter can feed the accretion torus for a much longer timescale. In general, accretion discs powering GRBs are ex- pected to have typical densities of the order of 1010−12 g cm−3 and temperatures of 1011 K within 10−20 Schwarzschild radii (RS = 2GM/c2). Thus, accretion proceeds with rates of a fraction to sev- eral solar masses per second. In this “hyper- accreting” regime, photons become trapped and are not efficient at cooling the disc. Neutrinos, however, are produced by weak interactions in the dense and hot plasma, releasing the gravitational energy of the accretion flow. These discs go under the name of “neutrino-dominated accretion flows”, or NDAFs. Over the last several years a number of stud- ies have investigated the structure of these discs (Popham, Woosley & Fryer 1999; Narayan, Piran & Kumar 2001; Kohri & Mineshige 2002; Di Mat- teo, Perna & Narayan 2002; Surman & McLaugh- lin 2004; Kohri, Narayan & Piran 2005; Chen & Beloborodov 2006; Gu, Liu & Lu 2006; Liu, Gu & Lu 2007). These models have employed the customary approximation of one-dimensional hy- drodynamics (Shakura & Sunyaev 1973), where the effects of MHD viscous stresses are described by the dimensionless parameter α, but have been limited to the steady-state approximation of con- stant Ṁ . Such an assumption is a good approx- imation when considering the collapsar scenario, where the burst duration is much longer than the viscous time, due to the continuous replenishing of the disc by the collapsing star envelope. However, even in this scenario, recent observations suggest that that engines are “long-lived” (past the torus feeding phase), requiring a time-dependent com- putation. In the merger scenario, on the other hand, a time-dependent calculation is necessary even for modeling the prompt phase of the burst, since the duration is set by the viscous timescale of the disc. Recently, a fully time-dependent calculation of the structure of such accretion discs has been pre- sented by Janiuk et al. (2004). The model was suitable for the torus being a results of either the gravitational collapse of a massive stellar core or the compact binary merger. However the struc- ture and evolution of the disk (like in many of the early calculations) was calculated under a num- ber of simplifying assumptions for the composition and the equation of state of the accreting matter. In this paper we improve upon our earlier results in several ways. Besides the requirement of time- dependent calculations, the high density and tem- perature regime in which the accreting gas lies, im- plies that both multi-dimensional numerical and semi-analytic calculations for such flows need to include the detailed microphysics. This includes photodisintegration of nuclei, the establishment of statistical equilibrium, neutronization, and the effects of neutrino opacities in the inner regions. Here, we introduce a detailed treatment of the equation of state, and calculate self-consistently the chemical equilibrium in the gas that consists of helium, free protons, neutrons and electron- positron pairs. We compute the chemical poten- tials of the species, as well as the electron fraction throughout the disk using the assumption of the equilibrium between the beta processes. Our EOS equations include, self-consistently, the contribu- tion of the neutrino trapping to the beta equilib- rium. Another important addition compared to our previous work (Janiuk et al. 2004) is the inclu- sion of photodisintegration of helium. The pres- ence of this term can affect the energy balance in the inner, opaque (to neutrinos as well as pho- tons) region of the flow and, as it will be shown, it eventually produces a thermal and viscous insta- bility in those regions. This is especially relevant since the GRB phenomenology requires a variable energy output. Other time-dependent disc studies of binary mergers or collapsars have been performed in 2D using hydrodynamical simulations (e.g. Mac- Fadyen & Woosley 1999; Ruffert & Janka 1999; Lee et al. 2002; Rosswog et al. 2004; Lee, Ramirez-Ruiz & Page 2005) and, most recently, in 3-D simulations (Setiawan et al. 2005). Also, MHD simulations of the GRB central engine have been performed, showing that the magnetic field possibly plays an important role in the genera- tion of a GRB jet (Proga et al. 2003; Fujimoto et al. 2006). The advantage of our calculations is that, whilst including all the relevant physics to calculate the equation of state, the structure and stability of the accretion disc, we are able to study a much larger range of parameter space and allow our calculations to evolve beyond what can be reached in higher dimensional calculation and comparable to at least the short-burst durations. The paper is organized as follows. In Sec- tion 2 we describe the basic assumptions of the model and the method used in the initial sta- tionary and subsequent time-dependent numerical simulations. In Section 3 we discuss the structure of the hyperaccreting disc for various values of the initial accretion rate, and study the time evolu- tion of its density and temperature, as well as the resulting neutrino lightcurve. We also discuss the physical origin of the instabilities in the disc, and we compare our model and results with the recent 2D and 3D simulations. We summarize our results in Section 4. 2. Neutrino-cooled accretion disks In this Section, we describe how we improve upon our previous time-dependent calculation (Ja- niuk et al. 2004) by computing self-consistently the equation of state of the extremely dense mat- ter by solving the balance of the β reaction rates. This allows us to determine the chemical poten- tials of electrons, protons and neutrons, as well as the electron fraction, in the initial disc configura- tion and throughout its evolution. 2.1. Initial disc configuration: 1-D Hydro- dynamics We start by considering a steady-state model of an accretion disc around a Schwarzschild black hole - formed as a remnant structure either after a compact binary merger, or in a collapsar after the birth of a black hole (for a recent calculation in Kerr spacetime see Chen & Beloborodov 2006). Throughout our calculations we use the vertically integrated equations and hence derive a vertically averaged disc structure. We write the surface den- sity of the disk as Σ = Hρ, where ρ is the density and where the disk half thickness (or disk height) is given by H = cs/ΩK . Here the sound speed is defined by cs = P/ρ and ΩK = GM/r3 is the Keplerian angular velocity with P the to- tal pressure. We note that, at very high accretion rates, the disc becomes moderately geometrically thick (H ∼ 0.5r) in regions where neutrino cooling becomes inefficient and advection dominates. Our ’slim disk’ approximation neglects terms ∼ (H/r)2 and assumes that the fluid is in Keplerian rota- tion. For the disc viscous stress we use the stan- dard α viscosity prescription of Shakura & Sun- yaev (1973) where the stress tensor is proportional to the pressure: τrϕ = −αP. (1) We adopt a value of α = 0.1 We set the inner radius of the disc at 3 RS, while the outer radius is at 50 RS. The initial mass of such a disc is about 0.35M⊙ for an accretion rate Ṁ = 1 M⊙/s. Throughout the calculations we adopt a black hole mass of M = 3M⊙. 2.2. The equation of state We assume that the torus consists of helium, electron-positron pairs, free neutrons and protons. The total pressure is contributed by all particle species in the disc, and the fraction of each species is determined by self-consistently solving the bal- ance of the beta reaction rates. In the equation of state we take into account the pressure due to the free nuclei and pairs, helium, radiation and the trapped neutrinos: P = Pnucl + PHe + Prad + Pν . (2) The component Pnucl includes free neutrons, pro- tons, and the electron-positron pair gas in beta equilibrium: Pnucl = Pe− + Pe+ + Pn + Pp (3) (~c)3 F3/2(ηi, βi) + βiF5/2(ηi, βi) where Fk are the Fermi-Dirac integrals of the or- der k, and ηe, ηp and ηn are the reduced chemi- cal potentials of electrons, protons and neutrons in units of kT , respectively (where ηi = µi/kT , also known as the degeneracy parameter, where µi the standard chemical potential) calculated from the chemical equilibrium condition (§ 2.3). The reduced chemical potential of positrons is ηe+ = −ηe − 2/βe and the relativity parameters of the species i are defined as βi = kT/mic Under the physical conditions in the torus, helium is generally non-relativistic and non- degenerate; therefore, its pressure is given by: PHe = nHekT, (5) where nHe is the number density of helium. This is defined as: nHe = nb(1 −Xnuc) , (6) and the fraction of free nucleons is given by Xnuc = 295.5ρ 11 exp(−0.8209/T11), (7) with T11 the temperature in unit of 10 11 K (e.g. Qian & Woosley 1996; Popham et al. 1999). The radiation pressure is given by: Prad = (kT )4 (~c)3 . (8) When neutrinos become trapped in the disc, the neutrino pressure is non-zero. Following the treat- ment of photon transport under the two-stream approximation (Popham & Narayan 1995; Di Mat- teo et al. 2002), we have (kT )4 3(~c)3 i=e,µ,τ (τa,νi + τs) + (τa,νi + τs) + 3τa,νi (kT )4 3(~c)3 b, (9) where τs is the scattering optical depth due to the neutrino scattering on free neutrons and pro- tons and τa,νe and τa,νµ are the absorptive opti- cal depths for electron and muon neutrinos, re- spectively (see§2.3). The contribution from tau neutrinos is the same as that from muon neutri- nos. These optical depths and neutrino absorption processes (which are the reverse of the emission processes) are discussed in more detail in the Ap- pendix. In the disc we have to consider both the neu- trino transparent and opaque regions, as well as the transition between the two. In the trans- parent case, the neutrinos are not thermalized and the chemical potential of neutrinos is negli- gible. On the other hand, when neutrinos are totally trapped, the chemical equilibrium condi- tion yields: µe + µp = µn + µν . The chemi- cal potential of neutrinos is a parameter depend- ing on how much neutrinos and anti-neutrinos are trapped, and assuming that the number densities of the trapped neutrinos and anti-neutrinos are the same, µν can be set to zero. In order to de- termine the distribution function of the partially trapped neutrinos, in principle one should solve the Boltzmann equation. To simplify this prob- lem, we use here a ”gray body” model, and we introduce a blocking factor b = i=e,µ,τ bi to de- scribe the extent to which neutrinos are trapped (see e.g. Sawyer 2003). In terms of this factor, we write the distribution function of neutrinos as f̃νi(p) = exp(pc/kT ) + 1 = bifνi , (0 ≤ bi ≤ 1). This simplified assumption is consistent with the two-stream approximation which we adopt here (Eq. 9). 2.3. Composition and chemical equilib- The equilibrium state of the gas in the accret- ing torus is completely determined by the chem- ical potentials of neutrons, protons and electrons (ηn, ηp, ηe), and the trapping factor of neutrinos (b) which is related to the optical depths of neu- trinos (cf. Eq. 9). For a given baryon number density, nb, temperature T , and a value for accre- tion rate Ṁ and viscous constant α, the chemical potentials, or equivalently the ratio of free protons x = np/nb, are determined from the condition of equilibrium between the transition reactions from neutrons to protons and from protons to neutrons. These reactions are: p + e− → n + νe (11) p + ν̄e → n + e+ (12) p + e− + ν̄e → n (13) n + e+ → p + ν̄e (14) n → p + e− + ν̄e (15) n + νe → p + e− (16) Therefore we have to calculate the ratio of protons that will satisfy the balance: np(Γp+e−→n+νe + Γp+ν̄e→n+e+ + Γp+e−+ν̄e→n) = nn(Γn+e+→p+ν̄e + Γn→p+e−+νe + Γn+νe→p+e−) .(17) The reaction rates are the sum of forward and backward rates and are given in the Appendix (see also Kohri, Narayan and Piran 2005). These are supplemented by two additional con- ditions: the conservation of the baryon number, nn +np = nb×Xnuc, and charge neutrality (Yuan 2005): ne = ne− − ne+ = np + n0e , (18) which says that the net number of electrons is equal to the number of free protons plus the num- ber of protons in helium: n0e = 2nHe = (1 −Xnuc) . (19) The number density of fermions under arbitrary degeneracy is determined by the following equa- tions: F1/2(ηi, βi) + βiF3/2(ηi, βi) Finally, the electron fraction is defined as: ne− − ne+ (Note that this is different from Ye = 1/(1 + nn/np), which is only valid for free n-p-e gas.) 2.4. Neutrino cooling The processes that are responsible for the neu- trino emission in the disc are electron-positron pair annihilation (e− + e+ → νi + ν̄i), bremsstrahlung (n + n → n + n + νi + ν̄i), plasmon decay (γ̃ → νe + ν̄e) and URCA process (reactions 11, 14 and 15). The first two processes produce neutrinos of all flavors, while the other produce only electron neutrinos and anti-neutrinos. The cooling rate due to pair annihilation is ex- pressed as: qe+e− = qνe + qνµ + qντ (22) where the cooling rates for all three neutrino fla- vors are calculated by means of Fermi-Dirac inte- grals and are given in the Appendix. The cooling rate due to nucleon-nucleon bremsstrahlung (in erg/cm3/s) is given by: qbrems = 3.35 × 1027ρ210T 5.511 , (23) where ρ10 is the baryon density in units of 10 g/cm3 and T11 is temperature in units of 10 11 K. The cooling rate due to the plasmon decay (in erg/cm3/s) is: qplasmon = 1.5×1032T 911γ6pe−γp(1+γp) 1 + γp where γp = 5.565 × 10−2 (π2 + 3(µe/kT )2)/3. The cooling rate due to the URCA reactions is given by the three emissivities: qurca = qp+e−→n+νe + qn+e+→p+ν̄e + qn→p+e−+ν̄e . The emissivities are given in the Appendix. Note that the blocking factor of the trapping neutrinos is used only for the emissivities of the URCA reactions. For simplicity, we neglect the blocking effects of neutrinos when calculating the emissivities for the electron-positron pair annihi- lation. Two reasons make this approximation rea- sonable: the emissivities for the electron-positron pair annihilation is much smaller than those of the URCA reactions, and the electron-positron pair annihilation does not change the electron fraction which sensitively affects the EOS. Each of the above neutrino emission process has a reverse process, which leads to neutrino absorp- tion. These are given by Equations 12, 13 and 16. Therefore we introduce the absorptive optical depths for neutrinos given by: τa,νi = qa,νi (26) where absorption of the electron neutrinos is de- termined by: qa,νe = q + qurca + qplasm + qbrems (27) and for the muon neutrinos: qa,νµ = q qbrems . (28) In addition, the free escape of neutrinos from the disc is limited by scattering. The scattering optical depth is given by: τs = τs,p + τs,n (29) = 24.28 × 10−5 H (Cs,pnp + Cs,nnn) where Cs,p = (4(CV − 1)2 + 5α2)/24, Cs,n = (1 + 5α2)/24, CV = 1/2 + 2 sin 2 θC, with α = 1.25 and sin2 θC = 0.23. The neutrino cooling rate is then given by Q−ν = i=e,µ τa,νi+τs 3τa,νi . (30) 2.5. Energy and momentum Conservation The hydrodynamic equations we solve to cal- culate the disc structure are the standard mass, energy and momentum conservation. Making use of the standard disk equations, the vertically integrated viscous heating rate (per unit area) over a half thickness H is given by: Ftot = 3GMṀ f(r) (31) where the Newtonian boundary condition is as- sumed: f(r) = 1 − rmin/r. Note that in the time-dependent calculations, instead of Eq. 31, we will solve the viscous diffusion equation (Eq. 44). Using mass and momentum conservation Ṁ = 4πρRHvr ≈ 6πνρH where vr ≈ (3ν)/(2r) and ν = (2Pα)/(3ρΩ) is the kinematic viscosity. The viscous heating rate can be written in terms of α, Q+visc = αΩHP. (32) Cooling in the disc is due to advection, radia- tion and neutrino emission. The advective cooling in a stationary disc is determined approximately = ΣvrT = qadv S , (33) where qadv ∝ d lnS/d ln r ∝ (d lnT/d ln r − (Γ3 − 1)d ln ρ/d ln r) ≈ const and we adopt the value of 1.0. The entropy density S is the sum of four components: S = Snucl + SHe + Srad + Sν . (34) The entropy density of the gas of free protons, neutrons and electron-positron pairs is given by: Snucl = Se− + Se+ + Sp + Sn (35) where (ǫi + Pi) − niηi (36) (~c)3 F3/2(ηi, βi) + βiF5/2(ηi, βi) is the energy density of electrons, positrons, pro- tons or neutrons, Pi is their pressure, given by Equation 4, ni are the number densities and ηi are the chemical potentials. The entropy density of helium is given by: SHe = nHe log(mHe − lognHe) for nHe > 0. The entropy density of radiation is Srad = 4 , (39) while for neutrinos we have Sν = 4 . (40) In the initial disc configuration we assume that qadv is approximately constant and of order of unity, but in the subsequent time-dependent evo- lution the advection term is calculated with the appropriate radial derivatives. For the case of photon and electron-positron pairs in the plasma the radiative cooling is equal 3Pradc 11σT 4 where we adopt the Rosseland-mean opacity κ = 0.4 + 0.64 × 1023ρT−3 [cm2g−1]. An important term in the cooling and heating balance in the disc is due to photodisintegration of α particles, with rate: Qphoto = qphotoH (42) where qphoto = 6.28 × 1028ρ10vr dXnuc and Xnuc is given by Equation 7. Finally, in order to calculate the initial stationary configuration, we solve the energy balance: Ftot = Q + Q−ν + Qphoto. 2.6. Time evolution After solving for the initial disc configuration, we allow the density and temperature to vary with time. We solve the time-dependent equations of mass and angular momentum conservation in the disc: 3r1/2 (r1/2νΣ) and the energy equation: 4 − 3χ 12 − 9χ 12 − 9χ (Q+ −Q−). where χ = (P − Prad)/P . The cooling term Q− consists of radiative and neutrino cooling, given by Equations (41) and (30). Advection is included in the energy equation via the radial derivatives. The cooling term due to photodisintegration of helium now must be proportional to the full time deriva- tive of Xnuc (cf. Eq. 43) : Qphoto ∝ vr ∂Xnuc ∂Xnuc . (46) 2.7. Numerical method The initial configuration of the disc is calcu- lated by means of the Newton-Raphson method, iterated with the hydrostatic equilibrium condi- tion. We interpolate over the matrix of pre- calculated results for the equation of state (pres- sure and entropy) and neutrino cooling rate ( the number of points is 1024x1024). The Fermi- Dirac integrals are calculated using the mixture of Gauss-Legendre and Gauss-Laguerre quadratures (Aparicio 1998). Having determined the initial radial profiles of density and temperature, as well as the other quantities at time t = 0, we start the time evolu- tion of the disc. We solve the set of Equations (44), (46) and (46) using the convenient change of vari- ables y = 2r1/2 and Ξ = yΣ, at fixed radial grid, equally spaced in y (see Janiuk et al. 2002 and ref- erences therein). The number of radial zones is set to 200, which we found to be an adequate resolu- tion. After determining the solutions for the first 100 time steps by the fourth-order Runge-Kutta method, we use the Adams-Moulton predictor- corrector method with an adaptive time step. The code used an explicit communication model that is implemented with the standardized MPI com- munication interface and can be run on multipro- cessor machines. We choose the no-torque inner boundary con- dition, Σin = Tin = 0 (see Abramowicz & Kato 1989). The outer boundary of the disc is done by adding an extra “dead-zone” to the computational domain, which accounts for the disk expansion and conservation of angular momentum. 3. Results We first analyze the pressure, entropy and neu- trino cooling rate distributions for a given tem- perature and baryon density in the gas. Then, we show the disc structure for a converged static disc model and finally we show examples of time evolution of the neutrino luminosity, density, tem- perature and electron fraction for given sets of pa- rameters. 3.1. EOS solutions for a given tempera- ture and density In Figure 1 and 2 we plot the results of the nu- merically calculated equation of state for the hot and dense matter. The plots show the dependence of the electron fraction, pressure, entropy and neu- trino cooling rate on temperature and density, re- spectively. In the upper panels, we show the neutrino cool- ing rate. At low temperatures, below T = mec 5 × 109K, there are almost no positrons and free nucleons. Therefore the neutrino emission pro- cesses switch off, and the cooling of the gas is either due to advection, or, when the matter be- comes transparent to photons, radiative cooling overtakes. For larger temperatures, the neutrino emission rate increases up to the temperature of about ∼ 5 × 1011 K. For very high tempera- tures, the optical depths for neutrinos increase very rapidly (τ ∝ T 5, see Eq. (7) in Di Matteo et al. 2002). Therefore the neutrino cooling rate decreases at high temperatures (Eq. 30). On the other hand, for a given temperature (e.g. T ∼ 1011 K), the neutrino cooling rate does not sensitively depend on density. It varies by one order of magni- tude in the range of 108 ≤ ρ ≤ 1014 g/cm3, where the optical depth is τ ∼ 100. The middle panels of Figures 1 and 2, show the entropy and pressure as a function of temperature and density. At low temperatures, the entropy of gas is not important. The highly degenerate electrons do not give contri- bution to the entropy, while they are a dominant term in the pressure, which is therefore indepen- dent of temperature up to T ∼ 5 × 1010 K. When the temperature increases, helium becomes disin- tegrated into free nucleons at energy comparable to the binding energy of helium, and after that the radiation (including photons and electron-positron pairs) contributes mainly to the total entropy and pressure. Therefore, both these quantities rise with temperature. At high densities, the entropy is dominated by neutrons. Finally, the electron fraction is shown in the bottom panel of Figures 1 and 2. At low temperatures, the electron fraction is equal to 0.5, it then decreases sharply as the helium nuclei become disintegrated. As the tem- perature further increases, positrons appear as the electrons become non-degenerate. The positron capture again increases the electron fraction (see Fig.1). The electron fraction changes significantly as a function of density for T > 1010K (in Fig.2, T = 1011K). At low densities, the torus consists of free neutrons and protons and Ye is close to 0.5 (see also Eq. 7). As density increases, Ye decreases to satisfy the beta-equilibrium among the free n-p- e gas. Above some density (when the temperature is high enough, e.g. for Fig. 2, ρHe ≈ 1013g cm−3) helium starts forming. Therefore Ye has a kink and stars rising steeply, asymptotically approach- ing 0.5 as the torus consists of plenty of ionized helium and some electrons to keep charge neutral- 3.2. The steady-state disc structure In Figures 3 and 4 we show the profiles of den- sity and temperature in the stationary accretion disk model for three accretion rates: 1 M⊙/s, 10 M⊙/s and 12 M⊙/s. In general, the temper- ature and density profiles both increase inward. However, for Ṁ = 12M⊙/s, a distinct branch of solutions is reached, which appears different than the so-called “NDAF” branch (see Kohri & Mi- neshige 2002). The density and temperature pro- files for this high accretion rate differ also from what was found in previous work (Di Matteo et al. 2002; Janiuk et al. 2004). Due to a more detailed equation of state, in which we allow for Fig. 1.— The dependence of the electron frac- tion (bottom), pressure (middle bottom), en- tropy (middle upper) neutrino cooling rate (up- per panel) on temperature, for the constant den- sity ρ = 1012 g/cm3. The accretion rate is Ṁ = 1 M⊙/s. The pressure and neutrino cooling are in cgs units and the entropy is in units of kB cm Fig. 2.— The dependence of the electron frac- tion (bottom), pressure (middle bottom), entropy (middle upper) and neutrino cooling rate (upper panel) on density, for the constant temperature T = 1011 K. The accretion rate is Ṁ = 1 M⊙/s. The pressure and neutrino cooling are in cgs units and the entropy is in units of kBcm a partial degeneracy of nucleons and electrons as well as neutrino trapping, our solutions reach den- sities as high as 1012 g/cm3 in the innermost radii of the disc. The temperature in this inner disc part is in the range 4×1010−1.25×1011 K, depending on the accretion rate. For the hottest disk model, a local peak in the density forms around 7 − 8RS, while below that radius the density decreases. Be- tween ∼ 3.5 and 7 RS, the plasma becomes much hotter and less dense than outside of this region. This means that the macroscopic state of the sys- tem is different here due to an abrupt change in the heat capacity. In order to check what is the reason for this transition, we investigate the pres- sure distribution in the disk. The profile of the pressure is shown in Fig. 5. The dominant term in the total pressure is due to the nucleons, while the radiation pressure (in- cluding electron-positron pairs) is always several orders of magnitude smaller. The neutrino pres- sure is large in the inner disc, once it gets opti- cally thick to neutrinos (i.e. for Ṁ ≥ 10M⊙/s). A significant contribution to the pressure is due to helium at densities high enough for helium to form, albeit at temperatures low enough such that its nuclei are not fully disintegrated. For the largest accretion rate shown, in the region of the temperature excess and inverse density gradient (3.5 − 7RS), the total pressure distribution flat- tens. The helium pressure is now vanishingly small due to the complete photodisintegration, and the nuclear pressure is slightly decreased due to the composition change: smaller number density of neutrons and larger number density of protons. The substantial contribution to the pressure is now given by the neutrinos (large optical depths; see below) and radiation pressure (increased number of electron-positron pairs). From the comparison of Figures 3, 4 and the bottom panel of Figure 5, it can be seen that the total pressure becomes locally correlated with temperature and anticor- related with density, thus consituting an unstable phase. In Figure 6 we show the neutrino optical depths due to scattering and absorption. The total opti- cal depth in the outer disc is typically dominated by scattering processes, while in the inner disc ab- sorption processes take over for very high accre- tion rates. For Ṁ = 1M⊙/s only the very in- ner disk radii have optical depth close to 1. For Fig. 3.— The baryon density as a function of the disc radius, calculated in the stationary so- lution. The accretion rate is Ṁ = 1 M⊙/s (solid line), Ṁ = 10 M⊙/s (long dashed line) and Ṁ = 12 M⊙/s (short dashed line) . Fig. 4.— The temperature as a function of the disc radius, calculated in the stationary solution. The accretion rate is Ṁ = 1 M⊙/s (solid line), Ṁ = 10 M⊙/s (long dashed line) and Ṁ = 12 M⊙/s (short dashed line) . Fig. 5.— The pressure components as a func- tion of the disc radius, calculated in the station- ary solution for the three accretion rate values: Ṁ = 1 M⊙/s (upper panel), Ṁ = 10 M⊙/s (mid- dle panel) and Ṁ = 12 M⊙/s (bottom panel) . The total pressure is marked by the solid line, and its components are: nuclear (gas) pres- sure (long dashed line), radiation pressure (short dashed line), helium pressure (dotted line) and neutrino pressure (dot-dashed line). Fig. 6.— The neutrino optical depths due to scat- tering (τs, solid line) and absorption (τa,e for elec- tron neutrinos, long dashed line, and τa,µ for muon neutrinos, short dashed line) as a function of ra- dius for Ṁ = 1M⊙/s (upper panel), Ṁ = 10M⊙/s (middle panel) and for Ṁ = 12M⊙/s (bottom panel). The sum of the three quantities is the total optical depth (τtot, dotted line). Ṁ = 12M⊙/s, in the radial strip of ∼ 3.5 − 7RS the disk is optically thick with absorptive optical depth for electron neutrinos exceeding the scatter- ing term and reaching values of the order of 100. 3.2.1. Composition and Chemical potentials In Figure 7 we show the distribution of the reduced chemical potentials of protons, electrons and neutrons throughout the disc. Reduced elec- tron chemical potentials much larger than unity (indicating strong electron degeneracy) are found in the inner disc parts for Ṁ = 10M⊙/s and Ṁ = 12M⊙/s, whereas for 1 M⊙/s electrons are only slightly degenerate. For the highest accretion rate, the maximum degeneracies correspond to the radius of the local peak in the density (cf. Fig. 3) and the excess of helium number density (cf. Fig. 8). Below this radius, the species become non- degenerate again, contributing to the increase of the electron fraction (cf. Fig. 9). In Figure 8 we plot the mass fraction of free nucleons as a function of radius for Ṁ = 12M⊙/s, 10 and 1M⊙/s. As the Figure shows, in the outer regions, Xnuc increases as the radius decreases, while the temperature and density increase (Fig. 3 and Fig. 4). Consistent with the behavior of Ye (Fig.2), Xnuc subsequently turns around (decreases) at radii where the den- sity is high enough for significant helium forma- tion. This trend is reversed sharply for highest accretion rates, when the temperature in the disk is high enough (Fig.4) for helium to be fully dis- sociated. In consequence, the number density of alpha particles increases at ∼ 7−12RS and sharply decreases at lower radii. A similar, but far less pro- nounced fluctuation in Xnuc is seen at smaller radii for the case of Ṁ = 10M⊙/s. For smallest accre- tion rate, 1M⊙/s, there is no helium throughout the disk. In Figure 9 we show the radial distribution of the electron fraction throughout the disc for 1, 10 and 12 M⊙/s. For the case of 1 M⊙/s (solid line), the electron fraction decreases inward in the disc as the electrons are captured by protons (in neutronization reactions). Once the electrons be- come non-degenerate, positrons appear, and the positron capture by neutrons again increases the electron fraction. For the hotter plasma (accre- tion rate of 10 M⊙/s, dashed line), consistently with the behavior discussed for Xnuc, helium nu- clei form as the density becomes high enough be- Fig. 7.— The chemical potentials of neutrons (ηn, solid line), electrons (ηe, dashed line) and protons (ηp, dotted line) as a function of the disc radius, calculated in the stationary solution. The accretion rate is Ṁ = 1 M⊙/s (triangles), Ṁ = 10 M⊙/s (squares) and Ṁ = 12 M⊙/s (cir- cles). Fig. 8.— The mass fraction of free nucleons as a function of radius for Ṁ = 1M⊙/s (solid line), Ṁ = 10M⊙/s (long dashed line) and Ṁ = 12 M⊙/s (short dashed line). low ∼ 20RS and Ye increases. For the accretion rate of 12 M⊙/s there is the sharp decrease in Ye, at ∼ 7−8RS, due to the sudden dissociation of he- lium. As helium is fully photo-dissociated, there is an almost equal number of neutrons and protons due to the balance of the electron and positron capture. This implies an electron fraction of 0.5. At the innermost radius, the temperature and den- sity drop due to the boundary condition, which affects the behaviour of both Ye and Xnuc. 3.2.2. Cooling and heating rates In Figure 10 we plot the rates of viscous heat- ing, advection and cooling due to neutrino emis- sion and photo-dissociation in the stationary disc. The accretion rates are Ṁ = 1 M⊙/s (upper panel), Ṁ = 10 M⊙/s (middle panel), and Ṁ = 12 M⊙/s (lower panel). For the highest accretion rates, in the innermost disc the neutrino cooling rate decreases substantially with respect to the cooling by photodissociacion. This is because the neutrinos are trapped in the disc due to a large opacity The smaller the accretion rate, the less important is the neutrino trapping effect. This implies that for an accretion rate of ≤ 10 M⊙/s neutrinos can escape from the innermost disc. The advective term is a couple orders of magni- tude smaller than the other terms. The photodis- sociacion term is negligible for an accretion rate of 1 M⊙/s, since there is no helium in the whole disc, and Qphoto is equal to zero by definition. For an accretion rate of 10 M⊙/s there is very little he- lium down to about 15-20 RS, and therefore Qphoto is much smaller than other terms. For the accre- tion rate of 12 M⊙/s , down to 6 − 10RS in the region of the disc of high density and maximum degeneracy, helium nuclei form. The nucleosyn- thesis of alpha particles leads to the plasma heat- ing instead of cooling, and therefore the relevant term in the energy balance has a negative value. Outward, above ∼ 10RS, there is some fraction of helium which can be photo-dissociated, so the cooling term due to this reaction is also important in the total energy balance. In the inner region helium is fully dissociated and Qphoto is equal to zero, increasing again only near the inner bound- ary due to the local density increase and decrease of temperature. Fig. 9.— The electron fraction as a function of the disc radius, calculated in the stationary solution. The accretion rate is Ṁ = 1 M⊙/s (solid line), Ṁ = 10 M⊙/s (long dashed line) and Ṁ = 12 M⊙/s (short dashed line). Fig. 10.— The heating and cooling rates due to photodissociation and neutrino emission (solid lines) as a function of radius for Ṁ = 1M⊙/s (up- per panel), Ṁ = 10M⊙/s (middle panel) and for Ṁ = 12M⊙/s (bottom panel). The other terms are: cooling rate due to advection (long dashed line) and viscous heating rate (short dashed line). 3.3. Stability analysis: instabilities at high-Ṁ The disc is thermally unstable if d logQ+/d logT > d logQ−/d logT . Then any small increase (de- crease) in temperature leads to a heating rate which is more (less) than the cooling rate, and as a consequence a further increase (decrease) of the temperature. The viscous instability, which appears when ∂Ṁ ∂Σ |Q+=Q− < 0, manifests itself in a faster (slower) evolution of an underdense (overdense) region. The instabilities can be con- veniently located in the surface density - temper- ature diagrams, in which the branch of thermal equilibrium solutions with a negative slope is not only unstable to the perturbations in the surface density, but it is also thermally unstable. In Figure 11 we show such stability curves for several radii in the disc. The criterion for a vis- cously stable disc is generally satisfied through- out the whole disc for Ṁ ≤ 10M⊙/s. How- ever, for larger accretion rates, there are unstable branches at the smallest radii. For Ṁ = 10M⊙/s, the disc becomes unstable below 5 RS, while for Ṁ = 12M⊙/s the instability strip is up to ∼ 7RS. Here helium is almost completely photodisinte- grated while the electrons and protons become non-degenerate again. For this high accretion rate, the electron fraction rises inward in the disc. Un- der these conditions, the energy balance is affected leading to the thermal and viscous instability, as demonstrated by the stability curves. This insta- bility will be discussed in more detail in Section 3.4. Time dependent solutions In this Section, we discuss how the tempera- ture, density, electron fraction and disk luminosity evolve with time. In Figures 12 and 13 we show the time evolution of density and temperature, when the initial accretion rate is 1 M⊙/s. These quan- tities exponentially decrease with time: ρ = ρ0(r) exp(−at) , (47) T = T0(r) exp(−bt) (48) where a ≈ 1.9 and b ≈ 0.085. The normalization of these relations depends on the radius, and for example for r = 6RS it is ρ0 = 2.2 × 1011 and T0 = 3.5× 1010. The exponential behaviour arises from the nature of energy equation (45). In Figure 14 we show the electron fraction as a function of time for several exemplary radial lo- cations in the disc, for the disc evolving from a starting accretion rate of 1M⊙/s. The fraction Ye is smaller in the inner disc radii, while outward, the electron fraction is over half an order of mag- nitude higher. Altogether, during the evolution of the system, the electron fraction constantly in- creases with time throughout the disc. The time-dependent neutrino luminosity of the disc is given by: Lν(t) = ∫ Rmax Q−ν (t)2πrdr (49) where Q−ν is given by Equation (30). In Figure 15 we show an example of such a lightcurve, for our standard model parameters (M = 3M⊙, α = 0.1), and Rmax = 50RS. The starting accretion rate was Ṁstart = 1 M⊙/s. At this accretion rate neutrinos can already escape from the accretion disc at the beginning of the evolution. For higher initial accretion rates, e.g 10 − 12M⊙/s, neutrinos are trapped in the inner- most disc, and, as a consequence, the neutrino luminosity is lower at the initial stages of disc evolution until the accretion rate drops to about ∼ 1M⊙/s. This result is qualitatively similar, al- beit it differs quantitatively, from what was ob- tained in Di Matteo et al. (2002) and Janiuk et al. (2004): in those calculations neutrino trap- ping was far more substantial even for a ’mod- erate’ accretion rate of Ṁ & 1M⊙/s. The differ- ence arises from the fact that here we calculate the neutrino opacities using the β reaction efficiencies, self-consistently with the equation of state. For an accretion rate of 1M⊙/s, the solution does not reach the viscously unstable branch. Ini- tially, the disc contains almost no alpha particles (cf. Fig. 8), which appear later on during the evo- lution and cooling of the plasma. The dynamical balance between the photodisintegration of helium and nucleosynthesis leads to an additional non- zero cooling/heating term in the energy equation and to only small amplitude flickering at the early stages of time-evolution. The situation is much more dramatic when the starting accretion rate is 12M⊙/s. In this case a Fig. 11.— The stability curves on the accretion rate vs. surface density plane, for several chosen radii in the disc: 3.39RS (solid line), 3.81RS (dot- ted line), 4.25RS (short dashed line), 5.19RS (long dashed line) and 8.60RS (dot-dashed line). Fig. 12.— The density as a function of time, for several chosen disc radii: 4.01, 6.04, 10.13, 20.7, 35.02, and 45.19 RS. The initial accretion rate is Ṁ = 1 M⊙/s. Fig. 13.— The temperature as a function of time, for several chosen disc radii: 4.01, 6.04, 10.13, 20.7, 35.02, and 45.19 RS. The initial accretion rate is Ṁ = 1 M⊙/s. Fig. 14.— The electron fraction as a function of time, for several chosen disc radii: 4.01, 6.04, 10.13, 20.7, 35.02, and 45.19 RS. The initial ac- cretion rate is Ṁ = 1 M⊙/s. large disc strip is viscously and thermally unstable and the most violent instability takes place around and below 7 − 12RS. In Figure 16 we show the behavior of the lo- cal accretion rate in the unstable disc, at sev- eral chosen locations within the instability strip. Near ∼ 12RS, the accretion rate varies due to the large and rapidly changing photodisintegra- tion term (locally, it can become larger than the neutrino cooling rate). This radius corresponds to the largest local value of the density of helium (cf. Figure 8 show- ing its starting model distribution), which is then being photodissociated. The photodissociacion process is the cause of the local rapid accretion rate changes. Then, inside from this highly vari- able strip, the accretion rate grows too fast to pre- serve the disc structure. This kind of behaviour occurs in the locally hotter and less dense region visible in the starting configuration e.g. in Figures 3 and 4, between 3.5 and 7 RS. In this region the helium is already totally photodissociated. Due to the growing accretion rate all the material is rapidly accreted onto the black hole and the in- nermost strip of the disc empties. After the inner strip is destroyed, the outer parts can still accrete onto the center. As they approach the black hole, their temperature and density grow and the above situation can repeat several times, until the whole disc is completely broken into rings and destroyed. These later in- jections of energy, with timescales dictated by the viscous timescale of each ring, can produce en- ergy flares following the main GRB activity. Our results therefore provide another physical mech- anism1 for the flare model recently proposed by Perna et al. (2006). In Figure 17 we show the neutrino lightcurve of the unstable disc. The instabilities due to photo- disintegration are reflected in oscillations of vari- able amplitude and millisecond timescale. This is of a particular interest if the neutrino annihila- tion provides the energy input for GRBs, however it should be pointed out that the oscillations ap- pearing in the presented lightcurve have a much 1In addition to the gravitational instability in the outer parts of the disk, which was hinted by the calculations of Di Matteo et al. (2002) and confirmed by those of Chen & Beloborodov (2006). Fig. 15.— The neutrino lightcurve, integrated over the disc surface. The initial accretion rate is Ṁ = 1 M⊙/s. Fig. 16.— The local accretion rate, as a function of time, for several chosen radial locations in the disc: 6.86, 7.73 and 12.85 RS. The starting accretion rate is Ṁ = 12 M⊙/s. smaller amplitude than the observed gamma ray variability. 4. Discussion 4.1. The unstable neutrino-opaque disc In our calculations we have shown that, for large accretion rates, the accreting torus becomes viscously and thermally unstable. We now discuss the physical origin of the instability. The unstable branch appears both in the steady-state solutions and in the subsequent time- dependent evolutions. In the steady-state case, for a chosen value of a constant accretion rate, this can be seen for instance by plotting the ra- dial profiles of density and temperature (cf. Figs. 3 and 4, where the distinct branch is found for the innermost radii), as well as by looking at the stability curves for a range of accretion rates at a chosen disc radius (cf. Fig. 11, where the un- stable inner disc radii exhibit a negative slope in the curve). In the time-dependent simulations, the unstable behavior is manifested by the highly variable accretion rate in certain strips of the disk and by the subsequent breaking of the disk inside from these variable strips (cf. Fig. 16). The in- stability arises from the fact that the accretion rate rises locally too fast to prevent the disc strip from emptying, as the material is supplied from outer strips at much slower rate than it is accreted inwards. The disc evolves unstably on a viscous timescale, τvisc = 1/(αΩ) × (r/H)2; for the radii shown in Figure 16, it is τvisc ≈ 0.05 s (note that the disc is rather thick, r/H ∼ 2.5, and therefore the viscous and thermal timescales are close to each other). Theoretically, in order to find again a stable solution, the disc would have to increase the local accretion rate up to about several tens of M⊙/s within one viscous timescale. However, this may not be possible if there is not enough material in the system to support much higher accretion rates during such violent oscillations. Therefore the system is unable to be stabilized and gets broken after a fraction of τvisc . In addi- tion, the dynamical instability is the source of the flickering of the local accretion rate at the edge of the unstable strip. Let us now discuss in more detail the physical reason driving this instability. In the inner part of the disc (below r ∼ 10RS for Ṁ = 12M⊙/s) there Fig. 17.— The neutrino luminosity, as a func- tion of time The starting accretion rate is Ṁ = 12 M⊙/s. Fig. 18.— The total pressure, as a function of time, for the chosen radial locations in the disc: 7.73 and 6.86 RS. The starting accretion rate is Ṁ = 12 M⊙/s. are two important processes, both of which are incorporated in our equation of state: photodisin- tegration of helium and neutrino trapping. As it was already mentioned in Sec. 3.2.1 and can be seen from Fig. 8, below ∼ 7 − 8RS helium in this disc is completely photodisintegrated. This part of the disc is also opaque to neutrinos, as we show in Fig. 6. These two mechanisms competitively influence the electron fraction in the disc (cf. Sec. 3.2, Fig. 6 and Fig. 9). Well outside the unstable strip, above ∼ 20RS, the electron fraction smoothly de- creases inwards as positrons appear because of the neutronization process. Then the scattering opti- cal depth for neutrinos becomes τs > 1, and the electron fraction increases again. After photodis- integration, the electron fraction decreases signif- icantly from almost 0.3 to much less than 0.1 due to electron capture. But again, when the disc becomes optically thick to absorption of electron neutrinos, the electron fraction gets higher and ap- proaches almost 0.5. The total pressure of sub-nuclear matter (cf. Fig 5) is mainly contributed by electrons, and therefore it is influenced by the changes in the electron fraction. In the narrow range of radii (6.8 < r/RS < 7.8), the pressure decreases due to photodisintegration. The sudden decrease of the pressure might drive the dynamical instabil- ity. (This picture is somewhat similar to that of the iron core collapse in the core collapse super- nova explosions: electron capture consumes most of the electrons and makes the EOS softer, conse- quently, it triggers the collapse of the iron core.) However, the transition from neutrino transpar- ent to opaque disc, and the increase of the elec- tron fraction due to the beta equilibrium (see also Yuan & Heyl 2005), are the reason for a steeper increase of the total pressure of the system. The same effect can also be observed in the time-dependent plot (Fig. 18), in which we show the pressure changes in the characteristic radii of the unstable part of the disc (cf. Fig. 16). The pressure decreases with time up to a radius R = 6.8RS, since the temperature and density gradually drop, as well as the neutrino opacities, so the electron fraction gets smaller. Then, in a strip between ∼ 6.8 and 12 RS, the pressure rises with time: in fact, when α particles appear, the electron fraction rises and matter locally piles up, thus increasing the pressure. At the border of these radii the disc breaks up, when the thermal-viscous instability induces an avalanche-like increase of the local accretion rate below ∼ 8RS. This happens because the increase in the pressure causes an excess in the local energy dissipation rate and the disc heats up, while at ad- jacent radii the pressure decreases and heating is insufficient. The system tries to compensate these temperature gradients by decreasing/ increasing the temperature in the outer/inner radius, respec- tively. But since in the unstable mode of the ther- mal balance this causes a further increase/decrease of density, the pressure drops further and the disc heats up in the outer radius, while cools down in the inner one. As long as it cannot find any sta- ble track of evolution, the emptying of the inner strip continues and finally the whole material is accreted towards the black hole or blown out. The radial extent of the unstable part depends on the initial accretion rate and in our model for Ṁ = 12M⊙/s it is up to ∼ 8RS, for Ṁ = 10M⊙/s it is up to ∼ 5RS, while for Ṁ = 1M⊙/s it is below ∼ 3.5RS. In the latter case, since the inner radius is located at ∼ 3RS, the instability hardly affects the disc. The extension of the instability strip de- pends also on the mass of the accreting compact object, and since for lower mass black holes the ac- cretion disc is generally denser, it reaches a density ∼ 1012 g/cm3 around ∼ 15RS. Above 25-30 RS, where the plasma is already optically thin and the evolution is stable, both the pressure and the accretion rate smoothly drop with time. The dominant source of cooling of the disc in this region is the neutrino emission (ad- vective cooling decreases as the disc transits from neutrino opaque to transparent). The photodis- integration term (if non-zero), is usually by 1-2 orders of magnitudes smaller than neutrino cool- ing, and in the inner disc, up to about 6.8 RS, there are no helium nuclei and the photodisinte- gration term is negligible, while at 7.5 RS it has a value of about Qphoto ∼ 1038 erg/s/cm2 with rapid fluctuations. These fluctuations induce the local accretion rate flickering (cf. Fig. 16), on a timescale and amplitude much smaller than for the viscous instability. 4.2. Comparison with previous work The neutrino dominated accretion flow has al- ready been studied in a number of papers, includ- ing both 1-D models and multi-D simulations. The steady-state 1-D models (e.g Popham et al. 1999; Kohri & Mineshige 2002) assumed the disk opti- cally thin to neutrinos, and neglected photodis- integration cooling. Di Matteo et al. (2002) took these two effects into account, and showed that the trapped neutrinos dominate the pressure in the inner region of the hyperaccreting disc, however their equation of state did not include the numeri- cal calculation of chemical equilibrium and did not incorporate the opacities directly in the EOS itera- tions (see also the time-dependent model of Janiuk et al. 2004). Kohri, Narayan & Piran (2005) con- sidered the neutrino opaque disk and the equilib- rium between neutrons and protons and calculated the number densities of species by numerically in- tegrating their distribution functions. However, these authors calculated the gas pressure from the ideal gas approximation, and neglected the contri- bution of helium to the pressure. In all of these papers the disc occurred to be stable against any kind of instability. On the other hand, in their recent work, Chen & Beloborodov (2006) find that the outskirts of the disk are gravitationally unstable. The approach used by these authors provides a detailed treat- ment of the microphysics which is very similar to ours; however, some differences between our work and theirs must be crucial to the development of viscous and thermal instabilities. One difference deals with the approximation made for the treat- ment of transition region between the neutrino- opaque and the transparent matter. In our work, we adopted a gray body model, i.e., we introduced the b factor to describe the distribution function (c.f. Eq. 10). This assumption is consistent with the two fluids approximation we have made, which has recently been studied numerically by Sawyer (2003), and shown to be appropriate for the con- ditions of these disks. On the other hand, Chen & Beloborodov (2006) smoothly connect the opti- cally thin and thick regimes by means of interpo- lation. A further difference lies in the description of the mass fraction of free nucleons. In this work we use an expression for Xnuc developed by Qian et al (1996), while in Chen & Beloborodov (2006) Xnuc is a function of Ye, which couples the nucle- osynthesis to the electron fraction. In our calculations we reach the range of densi- ties and temperatures where the nucleons start to become partially degenerate. This is accompanied by the neutrinos being more and more trapped in the gas and helium being destroyed by photodisso- ciation. As a result of these calculations, we found an additional, unstable branch of solutions for the disc thermal balance. This supports the recent results of 2-D simu- lations by Lee, Ramirez-Ruiz & Page (2005), who found the disc opaque to neutrinos to be thermally unstable. Their simulations showed that large cir- culations develop in the accretion flow. Setiawan, Ruffert and Janka (2005) found small fluctuations of the accretion rate and neutrino luminosity on the dynamical timescale, after the 10-20 msec of relaxation period (note, that in our calculations we start from the steady-state disc model at a given accretion rate, thus having no need for a relaxation to the quasi-steady configuration). The equation of state used in their work (see also Janka et al. 1999) is based on the work of Lattimer & Swesty (1991). Given the electron fraction, this EOS as- sumes the condition of nuclear statistical equilib- rium without neutrino trapping, but the evolution of the electron fraction is affected by the asym- metric neutrino emission from the hot and dense matter, which is called ‘neutrino leakage scheme’. The neutrino leakage scheme focuses on the ef- fects of the neutrino trapping on the net neutrino emissivities, not on the nuclear statistical equilib- rium. The equation of state used in the work of Rosswog et al. (2004) is temperature and compo- sition dependent, based on the relativistic mean field theory (Shen et al. 1998a,b), and the neu- trino cooling is accounted for by the multiflavor scheme (Rosswog & Liebendoerfer 2003). In our work, we use an equation of state based on the β equilibrium, including the contribution from the trapped neutrino, and neutrino trapping effects are accounted for by the appropriate opac- ities. It should be emphasized that most previous multi-D simulations neglected the effects of neu- trino trapping on the β equilibrium, as well as the contribution of the trapped neutrinos to the ther- modynamical properties of the dense matter. An- other difference between our treatment of the EOS and the previous numerical simulations is that we include the cooling of the photodisintegration of helium. Even though the original EOS of Lattimer & Swesty (1991) can provide detailed information about the composition of the dense matter, this information was not considered in order to keep the table of the EOS as small as possible (see e.g. Ruffert et al. 1996) just for numerical reasons. In this way, the disintegration cooling had not been investigated without the information on the com- position. Our results indicate that photodisinte- gration significantly affects the energy balance. 4.3. Limitations of our model We find the thermal-viscous instability to be an intrinsic property of the disc for extremely large torus densities (about 1012g cm−3) and high ac- cretion rates (Ṁ ≥ 10M⊙/s). This is seen both in the steady-state results (radial profiles of den- sity and temperature) and in the subsequent time evolution. Thermal and viscous instabilities have been studied in the case of standard accretion discs around compact objects (Lightman & Eardley 1974; Pringle 1977; Shakura & Sunyaev 1976). Two main physical processes that lead to disc instabilities were invoked to explain the time- dependent behavior of various objects: partial ionization of hydrogen in the discs of Dwarf No- vae (e.g. Meyer & Meyer-Hofmeister 1981; Smak 1984) and domination of radiation pressure in the X-ray transients (e.g Taam & Lin 1984). Such in- stabilities do not have to lead to a total disc break- down, but rather to a limit-cycle behavior, if only an additional (i.e. upper) stable branch of solu- tions can be found. This might be a hot state with a temperature above 104 K, or a slim disc, domi- nated by advection (Abramowicz et al. 1988). In our 1-D calculation the disc in the GRB central en- gine is not stabilized but rather breaks down into rings, as no stable solutions are reached (possibly, for even higher accretion rates again a stable part near the black hole could be formed - but these extremely high accretion rates would not be pro- duced by any compact merger scenario). There- fore, instead of a limit-cycle activity, what we find here are several dramatic accretion episodes on the viscous timescale. The remaining parts of the torus will subsequently accrete and, while ap- proaching the central black hole, will get hotter and denser, breaking at ∼ 7RS. Of course, it would be interesting to study whether such a violent instability would occur also in the 2D or 3D simulations. This is indeed likely to be the case, since as the multi-dimensional sim- ulations of accretion discs show, the instabilities derived first in 1D are still present in the hydro- dynamical simulations of flows with non-Keplerian velocity fields (e.g. Agol et al. 2001; Turner 2004; Ohsuga 2006). Possibly, the instability re- gion would be located at other (larger) radii if the calculations included the vertical structure of the disc: this is dependent on temperature and density, which above the meridional plane may be larger than the mean value considered in the ver- tically averaged model. We need to note that our 1D calculations do not take into account the possible effects of non- radial velocity components in the fluid. For exam- ple, the inverse composition gradient that leads to the disk instability, might be stabilized by ro- tation (e.g. Begelman & Meier 1982; Quataert & Gruzinov 2000). In the 2-D simulation of Lee et al (2005) the neutrino opaque disk exhibits circula- tions in the r-z direction. Such meridional circula- tions are known to be present in the Keplerian ac- cretion disks (e.g. Siemiginowska 1988), however it is unclear if they could always provide a stabiliz- ing mechanism for the thermal-viscous instability. Possibly, if the nonradial motions of the flow pro- vided a stronger stabilizing effect, the disk would exhibit oscillations in the viscous timescale, with- out breaking, similarly to the outbursts of Dwarf Nova disks. The assumption of the β equilibrium (justified, as the mixture of protons, electrons, neutrons and positrons is able to achieve the equilibrium con- ditions) might also have an effect on this result, as the equilibrium conditions reduce the heating and entropy in the gas. In fact, the β equilibrium condition which is satisfied in the innermost part of a hyperaccreting disc that is optically thick to neutrinos, is µn = µp +µe. Once the disc becomes transparent in its outer part, this condition is no longer valid. Analytically, it has been derived by Yuan (2005) that the condition for β equilibrium in this case is µn = µp + 2µe. 4.4. Observational consequences Our findings might be relevant for interpret- ing some recent observations. The flickering due to the photodisintegration of alpha particles may lead to a variable energy output on small (millisec- ond) timescales. The consequence of this may be variability in the gamma ray luminosity, although the changes in the local accretion rate may be spread by viscous effects (in the lightcurve Lν(t) integrated over the whole surface of the disc, the millisecond variability is somewhat smeared, and the amplitudes are not very large). Therefore the mass accreted by the black hole may not be vary- ing substantially, while some irregularity in the overall outflow could help produce internal shocks. The thermal-viscous instability, if accompa- nied by the disc breaking, may lead to the sev- eral episodic accretion events and several re- brightenings of the central engine on longer timescales, possibly detected in the later stages of the evolution. A similar kind of a long-term activity is possible also if the disk was not com- pletely broken, but exhibited some large accretion rate fluctuations on the viscous timescale. We thank Bożena Czerny, Pawe l Haensel and Daniel Proga for helpful discussions. We also thank the anonymous referee for detailed reports which helped us to improve our model and its presentation. This work was supported in part by grant 1P03D 00829 of the Polish State Com- mittee for Scientific Research and by NASA un- der grant NNG06GA80G. Y.-F. Y. is partially supported by Program for New Century Excel- lent Talents in University, and the National Natu- ral Science Foundation (10233030,10573016). RP acknowledges support from NASA under grant NNG05GH55G, and from the NSF under grant AST 0507571. A. Appendix The neutrino absorption and production rates in the beta processes for all participating particles at arbitrary degeneracy have been obtained in the previous works (Reddy, Prakash & Lattimer 1998; Yuan 2005). In the subnuclear dense matter with high temperatures, the nucleons are generally nondegenerate, therefore, the transition reaction rates from neutrons to protons and from protons to neutrons can be simplified as follows: Γp+e−→n+νe = |M |2 dEeEepe(Ee −Q)2fe(1 − befνe), (A1) Γp+e−←n+νe = |M |2 dEeEepe(Ee −Q)2(1 − fe)befνe , (A2) Γn+e+→p+ν̄e = |M |2 dEeEepe(Ee + Q) 2fe+(1 − befν̄e), (A3) Γn+e+←p+ν̄e = |M |2 dEeEepe(Ee + Q) 2(1 − fe+)befν̄e , (A4) Γn→p+e−+ν̄e = |M |2 dEeEepe(Q− Ee)2(1 − fe)(1 − befν̄e), (A5) Γn←p+e−+ν̄e = |M |2 dEeEepe(Q− Ee)2febefν̄e . (A6) Here Q = (mn −mp)c2, |M |2 is the averaged transition rate which depends on the initial and final states of all participating particles, for nonrelativistic noninteracting nucleons, |M |2 = G2F cos2 θC(1 + 3g2A), here GF ≃ 1.436×10−49 erg cm3 is the Fermi weak interaction constant, θC (sin θC = 0.231) is the Cabibbo angle, and gA = 1.26 is the axial-vector coupling constant. fe,νe are the distribution functions for electrons and neutrinos, respectively. The “chemical potential” of neutrinos is generally assumed to be zero. The factor be reflects the percentage of the partially trapped neutrinos. When neutrinos completely trapped, be = 1. The corresponding neutrino emissivities for the URCA reactions are given by: qp+e−→n+νe = |M |2 dEeEepe(Ee −Q)3fe(1 − befνe), (A7) qn+e+→p+ν̄e = |M |2 dEeEepe(Ee + Q) 3fe+(1 − befν̄e), (A8) qn→p+e−+ν̄e = |M |2 dEeEepe(Q− Ee)3(1 − fe)(1 − befν̄e). (A9) The emissivities due to the electron-positron pair annihilation, following the notation of Yakovlev et al (2001), is written as: qe−+e+→νi+ν̄i = C2+νi [8(Φ1U2 + Φ2U1) − 2(Φ−1U2 + Φ2U−1) + 7(Φ0U1 + Φ1U0) + 5(Φ0U−1 + Φ−1U0)] + 9C −νi [Φ0(U1 + U−1) + (Φ−1 + Φ1)U0] , (A10) where = 1.023 × 1023 erg cm−3 s−1, (A11) C+νi = C + C2Ai and C−νi = C − C2Ai , here CVi and CAi are the vector and axial-vector constants for neutrinos (CV e = 2 sin 2 θC + 0.5, CAe = 0.5, CV µ = CV τ = 2 sin 2 θC − 0.5 and CAµ = CAτ = −0.5). The dimensionless functions Uk and Φk (k= −1, 0, 1, 2) in the above equation can be expressed in terms of the Fermi-Dirac functions: U−1 = β3/2F1/2(ηe, βe) (A12) β3/2[F1/2(ηe, βe) + βeF3/2(ηe, βe)] (A13) β3/2[F1/2(ηe, βe) + 2βeF3/2(ηe, βe) + β eF5/2(ηe, βe)] (A14) β3/2[F1/2(ηe, βe) + 3βeF3/2(ηe, βe) + 3β eF5/2(ηe, βe) + β eF7/2(ηe, βe)]. (A15) Replacing ηe with ηe+ in Uk, we get the corresponding expressions for Φk. 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D, 72, 013007 Yuan Y., Heyl J.S., 2005, MNRAS, 360, 1493 Zhang B., 2007, astro-ph/0701520 This 2-column preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0509300 http://arxiv.org/abs/astro-ph/0701520 Introduction Neutrino-cooled accretion disks Initial disc configuration: 1-D Hydrodynamics The equation of state Composition and chemical equilibrium Neutrino cooling Energy and momentum Conservation Time evolution Numerical method Results EOS solutions for a given temperature and density The steady-state disc structure Composition and Chemical potentials Cooling and heating rates Stability analysis: instabilities at high-"705FM Time dependent solutions Discussion The unstable neutrino-opaque disc Comparison with previous work Limitations of our model Observational consequences Appendix

0704.1326

Complete integrable systems with unconfined singularities

Complete integrable systems with unconfined singularities V́ıctor Mañosa∗ Departament de Matemàtica Aplicada III, Control, Dynamics and Applications Group (CoDALab) Universitat Politècnica de Catalunya Colom 1, 08222 Terrassa, Spain [email protected] April 10th, 2007 Abstract We prove that any globally periodic rational discrete system in Kk (where K denotes either R or C,) has unconfined singularities, zero algebraic entropy and it is complete integrable (that is, it has as many functionally independent first integrals as the dimen- sion of the phase space). In fact, for some of these systems the unconfined singularities are the key to obtain first integrals using the Darboux-type method of integrability. PACS numbers 02.30.Ik, 02.30.Ks, 05.45.-a, 02.90.+p, 45.05.+x. Keywords: Singularity confinement, first integrals, globally periodic discrete systems, com- plete integrable discrete systems, discrete Darboux–type integrability method. Singularity confinement property in integrable discrete systems was first observed by Grammaticos, Ramani and Papageorgiou in [1], when studying the propagation of singu- larities in the lattice KdV equation xi+1j = x j+1+1/x j − 1/x j+1, an soon it was adopted as a detector of integrability, and a discrete analogous to the Painlevé property (see [2, 3] and references therein). It is well known that some celebrated discrete dynamical systems (DDS from now on) like the McMillan mapping and all the discrete Painlevé equations satisfy the singularity confinement property [1, 4]. In [5, p. 152] the authors write: “Thus singularity confinement appeared as a necessary condition for discrete integrability. However the suf- ficiency of the criterion was not unambiguously established”. Indeed, numerical chaos has been detected in maps satisfying the singularity confinement property [6]. So it is common knowledge that singularity confinement is not a sufficient condition for integrability, and some complementary conditions, like the algebraic entropy criterion have been proposed to ensure sufficiency [7, 8]. Corresponding Author: Phone: 00-34-93-727-8254; Fax: 00-34-93-739-8225. http://arxiv.org/abs/0704.1326v1 On the other hand a DDS can have a first integral and do not satisfy the singularity confinement property, as shown in the following example given in [9]: Indeed, consider the DDS generated by the map F (x, y) = (y, y2/x) which has the first integral given by I(x, y) = y/x. Recall that a first integral for a map F : dom(F ) ∈ Kk → Kk, is a K–valued function H defined in U , an open subset of dom(F ) ∈ Kk, satisfying H(F (x)) = H(x) for all x ∈ U . The above example shows that singularity confinement is not a necessary condition for integrability if “integrability” means the existence of a first integral. The first objective of this letter is to point out that more strong examples can be constructed if there are considerd globally periodic analytic maps. A map F : U ⊆ Kk → U is globally p–periodic if F p ≡ Id in U . Global periodicity is a current issue of research see a large list of references in [10, 11]. Indeed there exist globally periodic maps with unconfined singularities, since global periodicity forces the singularity to emerge after a complete period. However from [10, Th.7] it is know that an analytic and injective map F : U ⊆ Kk → U is globally periodic if and only if it is complete integrable, that is, there exist k functionally independent analytic first in U). Note that there is a difference between the definition of complete integrable DDS and the definition of complete integrable continuous DS: For the later case the number of functionally independent first integrals has to be just k− 1, which is the maximum possible number; see [13]. This is because the foliation induced by the k−1 functionally independent first integrals generically have dimension 1 (so this fully determines the orbits of the flow). Hence, to fully determine the orbits of a DDS, the foliation induced by the first integrals must have dimension 0, i.e. it has to be reduced to a set of points, so we need an extra first integral. In this letter we only want to remark that there exist complete integrable rational maps with unconfined singularities and zero algebraic entropy (Proposition 1), and that these unconfined singularities and its pre–images (the forbidden set) in fact play a role in the construction of first integrals (Proposition 2) for some globally periodic rational maps. Prior to state this result we recall some definitions. In the following F will denote a rational Given F : U ⊆ Kk → U , with F = (F1, . . . , Fk), a rational map, denote by S(F ) = {x ∈ Kk such that den(Fi) = 0 for some i ∈ {1, . . . , k}}, the singular set of F . A singularity for the discrete system xn+1 = F (xn) is a point x∗ ∈ S(F ). The set Λ(F ) = {x ∈ Kk such that there exists n = n(x) ≥ 1 for which Fn(x) ∈ S(x)}, is called the forbidden set of F , and it is conformed by the set of the preimages of the singular set. If F is globally periodic, then it is bijective on the good set of F , that is G = Kk \ Λ(F ) (see [11] for instance). Moreover G is an open full measured set ([12]). A singularity is said to be confined if there exists n0 = n0(x∗) ∈ N such that lim Fn0(x) exists and does not belong to Λ(F ). This last conditions is sometimes skipped in the literature, but if it is not included the “confined” singularity could re–emerge after some steps, thus really being unconfined. Rational maps on Kk extend to homogeneous polynomial maps on KP k, acting on homogeneous coordinates. For instance, the Lyness’ Map F (x, y) = (y, (a + y)/x), associ- ated to celebrated Lyness’ difference equation xn+2 = (a + xn+1)/xn, extends to KP Fp[x, y, z] = [xy, az 2 + yz, xz]. Let dn denote the degree of the n–th iterate of the extended map once all common factors have been removed. According to [7], the algebraic entropy of F is defined by E(F ) = lim log (dn)/n. The first result of the paper is Proposition 1. Let F : G ⊆ Kk → G be a globally p–periodic periodic rational map. Then the following statements hold. (a) F has k functionally independent rational first integrals (complete integrability). (b) All the singularities are unconfined. (c) The algebraic entropy of F is zero. Proof. Statement (a) is a direct consequence of [10, Th.7] whose proof indicates how to construct k rational first integrals using symmetric polynomials as generating functions. (b) Let x∗ ∈ S(F ), be a confined singularity of F (that is, there exists n0 ∈ N such that x{n0,∗} := limx→x∗ F n0(x∗) exists and x{n0,∗} /∈ Λ(F )). Consider ǫ ≃ 0 ∈ K k, such that x∗ + ǫ /∈ Λ(F ) (so that it’s periodic orbit is well defined). Set x{n0,∗,ǫ} := F n0(x∗ + ǫ). The global periodicity in Kk \Λ(F ) implies that there exists l ∈ N such that F lp−n0(x{n0,∗,ǫ}) = F lp(x∗ + ǫ) = x∗ + ǫ, hence F lp−n0(x{n0,∗,ǫ}) = lim x∗ + ǫ = x∗. But on the other hand F lp−n0(x{n0,∗,ǫ}) = F lp−n0(x{n0,∗}). Therefore x{n0,∗} ∈ Λ(F ), which is a contradiction. (c) Let F̄ denote the extension of F to KP k. F̄ is p–periodic except on the set of pre– images of [0, . . . , 0] (which is not a point of KP k), hence dn+p = dn for all n ∈ N (where dn denote the degree of the n–th iterate once all factors have been removed). Therefore E(F ) = limn→∞ log (dn)/n = 0. As an example, consider for instance the globally 5–periodic map F (x, y) = (y, (1+y)/x), associated to the Lyness’ difference equation xn+2 = (1 + xn+1)/xn, which is posses the unconfined singularity pattern {0, 1∞,∞, 1}. Indeed, consider an initial condition x0 = (ε, y) with |ε| ≪ 1, and y 6= −1, y 6= 0 and 1 + y + ε 6= 0 (that is, close enough to the singularity, but neither in the S(F ) nor in Λ(F )). Then x1 = F (x0) = (y, (1 + y)/ε), x2 = F (x1) = ((1 + y)/ε, (1 + y + ε)/(εy)), x3 = F (x2) = ((1 + y + ε)/(εy), (1 + ε)/y), and x4 = F (x3) = ((1 + ε)/y, ε), and finally x5 = F (x4) = x0. Therefore the singularity is unconfined since it propagates indefinitely. But the Lyness’ equation is complete integrable since it has the following two functionally independent first integrals [10]: H(x, y) = xy4 + p3(x)y 3 + p2(x)y 2 + p1(x)y + p0(x) I(x, y) = (1 + x)(1 + y)(1 + x+ y) Where p0(x) = x 3 + 2x2 + x, p1(x) = x 4 + 2x3 + 3x2 + 3x + 1, p2(x) = x 3 + 5x2 + 3x + 2, p3(x) = x 3+x2+2x+1. The extension of F to CP 2 is given by F̄ [x, y, z] = [xy, z(y+z), xz], which is again 5–periodic, hence dn = dn+5 for all n ∈ N, and the algebraic entropy is E(F ) = limn→∞ log (dn)/n = 0. More examples of systems with complete integrability, zero algebraic entropy and un- confined singularities, together with the complete set of first integrals can be found in [10]. The second objective of this letter is to notice that the unconfined singularities can even play an essential role in order to construct a Darbouxian–type first integral of some DDS, since they can help to obtain a closed set of functions for their associated maps. This is the case of some rational globally periodic difference equations, for instance the ones given by xn+2 = 1 + xn+1 , xn+3 = 1 + xn+1 + xn+2 , and xn+3 = −1 + xn+1 − xn+2 , (1) To show this role we apply the Darboux–type method of integrability for DDS (developed in [14] and [15, Appendix]) to find first integrals for maps. Set F : G ⊆ Kk → Kk. Recall that a set of functions R = {Ri}i∈{1,...,m} is closed under F if for all i ∈ {1, . . . ,m}, there exist functions Ki and constants αi,j, such that Ri(F ) = Ki j 6= 1. Each function Ki is called the cofactor of Ri. Very briefly, the method works as follows: If there exist a closed set of functions for F , say R = {Ri}i∈{1,...,m}, it can be tested if the function H(x) = i (x) gives a first integral for some values βi, just imposing H(F ) = H. In this letter, we will use the unconfined singularities of the maps associated to equations in (1) and its pre–images to generate closed set of functions. Proposition 2. Condider the maps F1(x, y) =(y, (1 + y)/x), F2(x, y, z) =(y, z, (1 + y + z)/x), and F3(x, y, z) =(y, z, (−1 + y − z)/x) associated to equations in (1) respectively. The fol- lowing statements hold: (i) The globally 5–periodic map F1 has the closed set of functions R1 = {x, y, 1 + y, 1+ x+ y, 1 + x}, which describe Λ(F1), and generates the first integral I1(x, y) = (1 + x)(1 + y)(1 + x+ y) (ii) The globally 8–periodic map F2 has the closed set of functions R2 = {x, y, z, 1 + y + z, 1 + x+ y + z + xz, 1 + x+ y}, which describe Λ(F2), and generates the first integral I2(x, y, z) = (1 + y + z)(1 + x+ y)(1 + x+ y + z + xz) (iii) The map F3 has the closed set of functions R3 = {x, y, z,−1 + y − z, 1 − x − y + z + xz,−1 + x− z − xy − xz + y2 − yz, 1− x+ y + z + xz,−1 + x− y}, which describe Λ(F3), and generates the first integral I3(x, y, z) = (−1 + y − z)(1 − x− y + z + xz)(1− x+ y + z + xz)(−1 + x− z − xy − xz + y2 − yz)(x− y − 1)/(x2y2z2). Proof. We only proof statement (ii) since statements (i) and (iii) can be obtained in the same way. Indeed, observe that {x = 0} is the singular set of F2. We start the process of characterizing the pre–images of the singular set by setting R1 = x as a “candidate” to be a factor of a possible first integral. R1(F2) = y, so {y = 0} is a pre–image of the singular set {R1 = 0}. Set R2 = y, then R2(F2) = z in this way we can keep track of the candidates to be factors of I4. In summary: R1 := x ⇒ R1(F2) = y, R2 := y ⇒ R2(F2) = z, R3 := z ⇒ R3(F2) = (1 + y + z)/x = (1 + y + z)/R1, R4 := 1 + y + z ⇒ R4(F2) = (1 + x+ y + z + xz)/x = (1 + x+ y + z + xz)/R1, R5 := 1 + x+ y + z + xz ⇒ R5(F2) = (1 + y + z)(1 + x+ y)/x = R4(1 + x+ y)/R1, R6 := 1 + x+ y ⇒ R6(F2) = 1 + y + z = R4. From this computations we can observe that R2 = {Ri}i=1,...,6 is a closed set under F2. Hence a natural candidate to be a first integral is I(x, y) = xαyβzδ(1 + y + z)γ(1 + x+ y)σ(1 + x+ y + z + xz)τ Imposing I(F2) = I, we get that I is a first integral if α = −τ , β = −τ , δ = −τ , γ = τ , and σ = τ . Taking τ = 1, we obtain I2. A complete set of first integrals for the above maps can be found in [10]. As a corollary of both the method and Proposition 2 we re–obtain the recently discovered second first integral of the third–order Lyness’ equations (also named Todd’s equation). This “second” invariant was already obtained independently in [10] and [16], with other methods. The knowledge of this second first integral has allowed some progress in the study of the dynamics of the third order Lyness’ equation [17]. Proposition 3. The set of functions R = {x, y, z, 1+ y+ z, 1+x+ y, a+x+ y+ z+xz} is closed under the map Fa(x, y, z) = (y, z, (a + y + z)/x) with a ∈ R, which is associated to the third order Lyness’ equation xn+3 = (a+ xn+1 + xn+2)/xn. And gives the first integral Ha(x, y, z) = (1 + y + z)(1 + x+ y)(a+ x+ y + z + xz) Proof. Taking into account that from Proposition 2 (ii) when a = 1, I2 is a first integral for F{a=1}(x, y, z), it seem that a natural candidate to be a first integrals could be Hα,β,γ(x, y, z) = (α+ y + z)(β + x+ y)(γ + x+ y + z + xz) for some constants α, β and γ. Observe that R1 := x ⇒ R1(Fa) = y, R2 := y ⇒ R2(Fa) = z, R3 := z ⇒ R3(Fa) = (a+ y + z)/x = K3/R1, where K3 = a+ y + z, at this point we stop the pursuit of the pre–images of the singularities because they grow indefinitely, and this way doesn’t seem to be a good way to obtain a family of functions closed under Fa. But we can keep track of the rest of factors in Hα,β,γ. R4 := α+ y + z ⇒ R4(Fa) = (a+ αx+ y + z + xz)/x, R5 := β + x+ y ⇒ R5(Fa) = β + y + z, R6 := γ + x+ y + z + xz ⇒ R6(Fa) = (γ + y + z)x+ (a+ y + z)(1 + y)/x. Observe that if we take α = 1, β = 1, and γ = a, we obtain R4(Fa) = R6/R1, R5(Fa) = R4 and R6(Fa) = K3(R5/R1). Therefore {Ri}i=1,...,6 is closed under Fa, furthermore Ha = (R4R5R6)/(R1R2R3) is such that Ha(Fa) = Ha In conclusion, singularity confinement is a feature which is present in many integrable discrete systems but the existence of complete integrable discrete systems with unconfined singularities evidences that is not a necessary condition for integrability (at least when “integrability” means existence of at least an invariant of motion, a first integral). However it is true that globally periodic systems are themselves “singular” in the sense that they are sparse, typically non–generic when significant classes of DDS (like the rational ones) are considered. Thus, the large number of integrable examples satisfying the the singularity confinement property together with the result in [9, p.1207] (where an extended, an not usual, notion of the singularity confinement property must be introduced in order to avoid the periodic singularity propagation phenomenon reported in this letter -see the definition of periodic singularities in p. 1204-) evidences that singularity confinement still can be considered as a good heuristic indicator of “integrability” and that perhaps there exists an interesting geometric interpretation linking both properties. However, although some alternative direc- tions have been started (see [18] for instance), still a lot of research must to be done in order to understand the role of singularities of discrete systems, their structure and properties in relation with the integrability issues. Acknowledgements. The author is partially supported by CICYT through grant DPI2005- 08-668-C03-01. CoDALab group is partially supported by the Government of Catalonia through the SGR program. The author express, as always, his deep gratitude to A. Cima and A. Gasull for their friendship, kind criticism, and always good advice. References [1] B. Grammaticos, A. Ramani, V.G. Papageorgiou. Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825–1828. [2] B. Grammaticos, A. Ramani. Integrability– and how to detect it, Lect. Notes Phys. 638 (2004), 31–94. [3] B. Grammaticos, A. Ramani. Integrability in a discrete world, Chaos, Solitons and Fractals 11 (2000), 7–18. [4] A. Ramani, B. Grammaticos, J. Hietarinta. Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829–1832. [5] Y. Otha, K.M. Tamizhmani, B. Grammaticos, A. Ramani. Singularity confinement and algebraic entropy: the case of discrete Painlevé equations. Physics Letters A 262 (1999), 152-157. [6] J. Hietarinta, C. Viallet. Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1998), 325–328. [7] M. Bellon, C. Viallet. Algebraic entropy, Comm. Math. Phys. 204 (1999), 425–437. [8] S. Lafortune, A. Ramani, B. Grammaticos, Y. Otha, K.M. Tamizhmani. Blending two discrete integrability criteria: singularity confinement and algebraic entropy. in “Bäcklund & Darboux Transformations: The Geometry of Soliton Theory”, A Coley et al. (Eds), CRM Proc. & Lect. Notes vol. 29, Amer. Math. Soc., Providence, RI, 2001, 299-311. arXiv:nlin.SI/0104020. [9] S. Lafortune, A. Goriely. Singularity confinement and algebraic integrability, J. Math. Physics 45 (2004), 1191–1197. [10] A. Cima, A. Gasull, V. Mañosa. Global periodicity and complete integrability of dis- crete dynamical systems, J. Difference Equations and Appl. 12 (7) (2006), 697-716. [11] A. Cima, A. Gasull, F. Mañosas. On periodic rational difference equations of order k, J. Difference Equations and Appl. 10 (6) (2004), 549–559. [12] J. Rubió–Massegú. On the existence of solutions for difference equations , To appear in J. Difference Equations and Appl. [13] A. Goriely. “Integrability and nonintegrability of dynamical systems”, Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. [14] A. Gasull, V. Mañosa. A Darboux–type theory of integrability for discrete dynamical systems, J. Difference Equations and Appl. 8 (12) (2002), 1171–1191. [15] A. Cima, A. Gasull, V. Mañosa. Dynamics of rational discrete dynamical systems via first integrals, Int. J. Bifurcation and Chaos. 16 (3) (2006) 631-645 [16] M. Gao, Y. Kato, M. Ito. Some invariants for kth–Order Lyness Equation, Applied Mathematics Letters 17 (2004), 1183–1189. [17] A. Cima, A. Gasull, V. Mañosa. Dynamics of the third order Lyness’ difference equa- tion, To appear in J. Difference Equations and Appl. arXiv:math.DS/0612407 (ex- tended version). [18] M.J. Ablowitz, R.G. Halburd, B. Herbst. On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905. http://arxiv.org/abs/nlin/0104020 http://arxiv.org/abs/math/0612407

0704.1327

On the largest prime factor of the Mersenne numbers

7 On the largest prime factor of the Mersenne numbers Kevin Ford Department of Mathematics The University of Illinois at Urbana-Champaign Urbana Champaign, IL 61801, USA [email protected] Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de México C.P. 58089, Morelia, Michoacán, México [email protected] Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109, Australia [email protected] Abstract Let P (k) be the largest prime factor of the positive integer k. In this paper, we prove that the series (log n)α P (2n − 1) is convergent for each constant α < 1/2, which gives a more precise form of a result of C. L. Stewart of 1977. http://arxiv.org/abs/0704.1327v1 1 Main Result Let P (k) be the largest prime factor of the positive integer k. The quantity P (2n− 1) has been investigated by many authors (see [1, 3, 4, 10, 11, 12, 14, 15, 16]). For example, the best known lower bound P (2n − 1) ≥ 2n+ 1, for n ≥ 13 is due to Schinzel [14]. No better bound is known even for all sufficiently large values of n. C. L. Stewart [15, 16] gave better bounds provided that n satisfies certain arithmetic or combinatorial properties. For example, he showed in [16], and this was also proved independently by Erdős and Shorey in [4], that P (2p − 1) > cp log p holds for all sufficiently large prime numbers p, where c > 0 is an absolute constant and log is the natural logarithm. This was an improvement upon a previous result of his from [15] with (log p)1/4 instead of log p. Several more results along these lines are presented in Section 3. Here, we continue to study P (2n − 1) from a point of view familiar to number theory which has not yet been applied to P (2n − 1). More precisely, we study the convergence of the series (logn)α P (2n − 1) for some real parameter α. Our result is: Theorem 1. The series σα is convergent for all α < 1/2. The rest of the paper is organized as follows. We introduce some notation in Section 2. In Section 3, we comment on why Theorem 1 is interesting and does not immediately follow from already known results. In Section 4, we present a result C. L. Stewart [16] which plays a crucial role in our argument. Finally, in Section 5, we give a proof of Theorem 1. 2 Notation In what follows, for a positive integer n we use ω(n) for the number of distinct prime factors of n, τ(n) for the number of divisors of n and ϕ(n) for the Euler function of n. We use the Vinogradov symbols ≫, ≪ and ≍ and the Landau symbols O and o with their usual meaning. The constants implied by them might depend on α. We use the letters p and q to denote prime numbers. Finally, for a subset A of positive integers and a positive real number x we write A(x) for the set A∩ [1, x]. 3 Motivation In [16], C. L. Stewart proved the following two statements: A. If f(n) is any positive real valued function which is increasing and f(n) → ∞ as n → ∞, then the inequality P (2n − 1) > n(log n)2 f(n) log log n holds for all positive integers n except for those in a set of asymptotic density zero. B. Let κ < 1/ log 2 be fixed. Then the inequality P (2n − 1) ≥ C(κ) ϕ(n) logn 2ω(n) holds for all positive integers n with ω(n) < κ log log n, where C(κ) > 0 depends on κ. Since for every fixed ε > 0 we have log logn n(log n)1+ε the assertion A above, taken with f(n) = (log n)ε for fixed some small posi- tive ε < 1− α, motivates our Theorem 1. However, since C. L. Stewart [16] gives no analysis of the exceptional set in the assertion A (that is, of the size of the set of numbers n ≤ x such that the corresponding estimate fails for a particular choice of f(n)), this alone does not lead to a proof of Theorem 1. In this respect, given that the distribution of positive integers n having a fixed number of prime factorsK < κ log log n is very well-understood starting with the work of Landau and continuing with the work of Hardy and Ramanu- jan [6], it may seem that the assertion B is more suitable for our purpose. However, this is not quite so either since most n have ω(n) > (1−ε) log logn and for such numbers the lower bound on P (2n − 1) given by B is only of the shape ϕ(n)(log n)1−(1−ε) log 2 and this is not enough to guarantee the convergence of series (1) even with α = 0. Conditionally, Murty and Wang [11] have shown the ABC-conjecture implies that P (2n − 1) > n2−ε for all ε > 0 once n is sufficiently large with respect to ε. This certainly implies the conditional convergence of series (1) for all fixed α > 0. Murata and Pomerance [10] have proved, under the Generalized Riemann Hypothesis for various Kummerian fields, that the in- equality P (2n − 1) > n4/3/ log logn holds for almost all n, but they did not give explicit upper bounds on the size of the exceptional set either. 4 Main Tools As we have mentioned in Section 3, neither assertion A nor B of Section 3 are directly suitable for our purpose. However, another criterion, implicit in the work of C. L. Stewart [16] and which we present as Lemma 2 below (see also Lemma 3 in [10]), plays an important role in our proof. Lemma 2. Let n ≥ 2, and let d1 < · · · < dℓ be all ℓ = 2 ω(n) divisors of n such that n/di is square-free. Then for all n > 6, #{p | 2n − 1 : p ≡ 1 (mod n)} ≫ log logP (2n − 1) where ∆(n) = max i=1,...,ℓ−1 di+1/di. The proof of C. L. Stewart [16] of Lemma 2 uses the original lower bounds for linear forms in logarithms of algebraic numbers due to Baker. It is interesting to notice that following [16] (see also [10, Lemma 3]) but us- ing instead the sharper lower bounds for linear forms in logarithms due to E. M. Matveev [9], does not seem to lead to any improvement of Lemma 2. Let 1 = d1 < d2 < · · · < dτ(n) = n be all the divisors of n arranged in increasing order and let ∆0(n) = max i≤τ(n)−1 di+1/di. Note that ∆0(n) ≤ ∆(n). We need the following result of E. Saias [13] on the distribution of positive integers n with “dense divisors”. Let G(x, z) = {n ≤ x : ∆0(n) ≤ z}. Lemma 3. The bound #G(x, z) ≍ x log z log x holds uniformly for x ≥ z ≥ 2. Next we address the structure of integer with ∆0(n) ≤ z. In what follows, as usual, an empty product is, by convention, equal to 1. Lemma 4. Let n = pe11 · · · p k be the prime number factorization of a positive integer n, such that p1 < · · · < pk. Then ∆0(n) ≤ z if and only if for each i ≤ k, the inequality pi ≤ z holds. Proof. The necessity is clear since otherwise the ratio of the two consecutive divisors j and pi is larger than z. The sufficiency can be proved by induction on k. Indeed for k = 1 it is trivial. By the induction assumption, we also have ∆(m) ≤ z, where m = n/pe11 . Remarking that p1 ≤ z, we also conclude that ∆(n) ≤ z. 5 Proof of Theorem 1 We put E = {n : τ(n) ≥ (log n)3}. To bound #E(x), let x be large and n ≤ x. We may assume that n > x/(log x)2 since there are only at most x/(log x)2 positive integers n ≤ x/(log x)2. Since n ∈ E(x), we have that τ(n) > (log(x/ log x))3 > 0.5(log x)3 for all x sufficiently large. Since τ(n) = O(x log x) (see [7, Theorem 320]), we get that #E(x) ≪ (log x)2 By the Primitive Divisor Theorem (see [1], for example), there exists a prime factor p ≡ 1 (mod n) of 2n − 1 for all n > 6. Then, by partial summation, n∈E(x) (log n)α P (2n − 1) n∈E(x) (log n)α ≤ 1 + (log t)α d#E(t) ≤ 1 + #E(x) #E(t)(log t)α ≪ 1 + t(log t)2−α Hence, (log n)α P (2n − 1) < ∞. (2) We now let F = {n : P (2n − 1) > n(log n)1+α(log log n)2}. Clearly, (logn)α P (2n − 1) n logn(log logn)2 < ∞. (3) From now on, we assume that n 6∈ E ∪ F . For a given n, we let D(n) = {d : dn+ 1 is a prime factor of 2n − 1}, D+(n) = max{d ∈ D(n)}. Since P (2n − 1) ≥ d(n)n+ 1, we have D+(n) ≤ (log n) (log logn)2. (4) Further, we let xL = e L. Assume that L is large enough. Clearly, for n ∈ [xL−1, xL] we have D +(n) ≤ L1+α(logL)2. We let Hd,L be the set of n ∈ [xL−1, xL] such thatD +(n) = d. We then note that by partial summation xL−1≤n≤xL n 6∈E∪F (log n)α P (2n − 1) d≤L1+α(logL)2 n∈Hd,L nd+ 1 d≤L1+α(logL)2 #Hd,L d≤L1+α(logL)2 #Hd,L We now estimate #Hd,L. We let ε > 0 to be a small positive number depending on α which is to be specified later. We split Hd,L in two subsets as follows: Let Id,L be the set of n ∈ Hd,L such that #D(n) > (logn) (log log n)2 > Lα+ε(logL)2, where M = M(ε) is some positive integer depending on ε to be determined later. Since D+(n) ≤ L1+α(logL)2, there exists an interval of length L1−ε which contains at least M elements of D(n). Let them be d0 < d1 < · · · < dM−1. Write ki = di − d0 for i = 1, . . . ,M − 1. For fixed d0, k1, . . . , kM−1, by the Brun sieve (see, for example, Theorem 2.3 in [5]), #{n ∈[xL−1, xL] : din + 1 is a prime for all i = 1, . . . ,M} (log(xL))M p|d1···dM i=1 di i=1 di xL(log logL) where we have used that ϕ(m)/m ≫ 1/ log log y in the interval [1, y] with y = yL = L 1+α(logL)2 (see [7, Theorem 328]). Summing up the inequality (6) for all d0 ≤ L 1+α(logL)2 and all k1, . . . , kM−1 ≤ L 1−ε, we get that the number of n ∈ Id,L is at most #Id,L ≪ xL(logL) M+2L1+αL(M−1)(1−ε) xL(logL) L(M−1)ε−α . (7) We now choose M to be the least integer such that (M − 1)ε > 2 + α, and with this choice of M we get that #Id,L ≪ . (8) We now deal with the set Jd,L consisting of the numbers n ∈ Hd,L with #D(n) ≤ M−1 (log n) (log log n)2. To these, we apply Lemma 2. Since τ(n) < (logn)3 and P (2n − 1) < n2 for n ∈ Hd,L, Lemma 2 yields log∆(n)/ log logn ≪ #D(n) ≪ (log n) (log log n)2. Thus, log∆(n) ≪ (logn) (log logn)3 ≪ (log xL) (log log xL) 3 ≪ Lα+ε(logL)3. Therefore ∆0(n) ≤ ∆(n) ≤ zL, where zL = exp(cL α+ε(logL)3) and c > 0 is some absolute constant. We now further split Jd,L into two subsets. Let Sd,L be the subset of n ∈ Jd,L such that P (n) < x 1/ logL L . From known results concerning the distribution of smooth numbers (see the corollary to Theorem 3.1 of [2], or [8], [17], for example), #Sd,L ≤ L(1+o(1)) log logL . (9) Let Td,L = Jd,L\Sd,L. For n ∈ Td,L, we have n = qm, where q > x 1/ logL L is a prime. Fix m. Then q < xL/m is a prime such that qdm+1 is also a prime. By the Brun sieve again, #{q ≤ xL/m : q, qdm+ 1 are primes} m(log(xL/m))2 ϕ(md) xL(logL) where in the above inequality we used the minimal order of the Euler function in the interval [1, xLL 1+α(logL)2] together with the fact that log(xL/m) ≥ log xL We now sum up estimate (10) over all the allowable values for m. An immediate consequence of Lemma 4 is that since ∆0(n) ≤ zL, we also have ∆0(m) ≤ zL for m = n/P (n). Thus, m ∈ G(xL, zL). Using Lemma 3 and partial summation, we immediately get m∈G(xL,zL) d(#G(t, zL)) #G(xL, zL) #G(t, zL) log zL + log zL t log t ≪ log zL log log xL ≪ L α+ε(logL)4, as L → ∞. Thus, #Td,L ≪ xL(logL) m∈Md,L xL(logL) 7Lα+ε L2−α−2ε , (11) when L is sufficiently large. Combining estimates (8), (9) and (11), we get #Hd,L ≤ #Jd,L +#Sd,L +#Td,L ≪ L2−α−2ε . (12) Thus, returning to series (5), we get that d≤L1+α(logL)2 L2−2α−2ε L2−2α−2ε Since α < 1/2, we can choose ε > 0 such that 2− 2α− 2ε > 1 and then the above arguments show that (log n)α P (2n − 1) ≪ 1 + L2−2α−ε which is the desired result. References [1] G. D. Birkhoff and H. S. Vandiver, ‘On the integral divisors of an−bn’, Ann. of Math. (2) 5 (1904), 173–180. [2] E. R. Canfield, P. Erdős and C. Pomerance, ‘On a problem of Oppen- heim concerning “factorisatio numerorum”’, J. Number Theory 17 (1983), 1–28. [3] P. Erdős, P. Kiss and C. Pomerance, ‘On prime divisors of Mersenne numbers’, Acta Arith. 57 (1991), 267–281. [4] P. Erdős and T. N. Shorey, ‘On the greatest prime factor of 2p− 1 for a prime p and other expressions’, Acta Arith. 30 (1976), 257–265. [5] H. Halberstam and H.-E. Richert, Sieve methods , Academic Press, London, 1974. [6] G. H. Hardy and S. Ramanujan, ‘The normal number of prime factors of an integer, Quart. Journ. Math. (Oxford) 48 (1917), 76-92. [7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford, 1979. [8] A. Hildebrand and G. Tenenbaum, ‘Integers without large prime fac- tors’, J. de Théorie des Nombres de Bordeaux , 5 (1993), 411–484. [9] E. M. Matveev, ‘An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II’, Izv. Ross. Akad. Nauk. Ser. Math. 64 (2000), 125–180; English translation Izv. Math. 64 (2000), 1217–1269. [10] L. Murata and C. Pomerance, ‘On the largest prime factor of a Mersenne number’, Number theory CRM Proc. Lecture Notes vol.36 , Amer. Math. Soc., Providence, RI, 2004, 209–218,. [11] R. Murty and S. Wong, ‘The ABC conjecture and prime divisors of the Lucas and Lehmer sequences’, Number theory for the millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 43–54. [12] C. Pomerance, ‘On primitive divisors of Mersenne numbers’, Acta Arith. 46 (1986), no. 4, 355–367. [13] E. Saias, ‘Entiers à diviseurs denses 1’, J. Number Theory 62 (1997), 163–191. [14] A. Schinzel, ‘On primitive prime factors of an − bn’, Proc. Cambridge Philos. Soc. 58 (1962), 555–562. [15] C. L. Stewart, ‘The greatest prime factor of an − bn’, Acta Arith. 26 (1974/75), no. 4, 427–433. [16] C. L. Stewart, ‘On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers’, Proc. London Math. Soc. (3) 35 (1977), 425–447. [17] G. Tenenbaum, Introduction to analytic and probabilistic number the- ory , Cambridge Univ. Press, 1995. Main Result Notation Motivation Main Tools Proof of Theorem ??

0704.1328

Developing the Galactic diffuse emission model for the GLAST Large Area Telescope

Developing the Galactic diffuse emission model for the GLAST Large Area Telescope Igor V. Moskalenko∗,†, Andrew W. Strong∗∗, Seth W. Digel‡,† and Troy A. Porter§ ∗Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305 †Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94309 ∗∗Max-Plank-Institut für extraterrestrische Physik, Postfach 1312, D-85741 Garching, Germany ‡Stanford Linear Accelerator Center, 2575 Sand Hill Rd, Menlo Park, CA 94025 §Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064 Abstract. Diffuse emission is produced in energetic cosmic ray (CR) interactions, mainly protons and electrons, with the interstellar gas and radiation field and contains the information about particle spectra in distant regions of the Galaxy. It may also contain information about exotic processes such as dark matter annihilation, black hole evaporation etc. A model of the diffuse emission is important for determination of the source positions and spectra. Calculation of the Galactic diffuse continuum γ-ray emission requires a model for CR propagation as the first step. Such a model is based on theory of particle transport in the interstellar medium as well as on many kinds of data provided by different experiments in Astrophysics and Particle and Nuclear Physics. Such data include: secondary particle and isotopic production cross sections, total interaction nuclear cross sections and lifetimes of radioactive species, gas mass calibrations and gas distribution in the Galaxy (H2, H I, H II), interstellar radiation field, CR source distribution and particle spectra at the sources, magnetic field, energy losses, γ-ray and synchrotron production mechanisms, and many other issues. We are continuously improving the GALPROP model and the code to keep up with a flow of new data. Improvement in any field may affect the Galactic diffuse continuum γ-ray emission model used as a background model by the GLAST LAT instrument. Here we report about the latest improvements of the GALPROP and the diffuse emission model. Keywords: gamma rays, cosmic rays, diffuse background, interstellar medium, gamma ray telescope PACS: 95.55.Ka, 95.85.Pw, 98.35.-a, 98.38.-j, 98.38.Cp, 98.58.Ay, 98.70.Sa, 98.70.Vc DISCUSSION AND RESULTS We give a very brief summary of GALPROP; for details we refer to the relevant papers [1]-[6] and a dedicated website. The propagation equation is solved numerically on a spatial grid, either in 2D with cylindrical symmetry in the Galaxy or in full 3D. The boundaries of the model in radius and height, and the grid spacing, are user-definable. Parameters for all processes in the propagation equation can be controlled on input. The distribution of CR sources can be freely chosen, typically to represent supernova remnants. Source spectral shape and isotopic composition (relative to protons) are input parameters. Cross-sections are based on extensive compilations and parameterizations [7]. The numerical solution is evolved forward in time until a steady-state is reached; a time-dependent solution is also an option. Starting with the heaviest primary nucleus considered (e.g., 64Ni) the propagation solution is used to compute the source term for its spallation products, which are then propagated in turn, and so on down to protons, secondary electrons and positrons, and antiprotons. In this way secondaries, tertiaries, etc., are included. Primary electrons are treated separately. The proton, helium, and electron spectra are normalized to data; all other isotopes are determined by the source composition and propagation. γ-rays and synchrotron emission are computed using interstellar gas data (for pion-decay and bremsstrahlung) and the interstellar radiation field (ISRF) model (for inverse Compton). The computing resources required by GALPROP are moderate by current standards. Recent extensions to GALPROP include • new detailed calculation of the ISRF [8, 9] • proper implementation of the anisotropic inverse Compton scattering using new ISRF (Figure 1, left) • interstellar gas distributions based on current HI and CO surveys [10, 11] • new parameterization of the π0 production in pp-collisions [12] which includes the diffraction dissociation • non-linear MHD wave – particle interactions (wave damping) [6] are included as an option • the kinetic energy range is now extended down to ∼1 keV http://arxiv.org/abs/0704.1328v1 0 10 20 30 40 50 60 70 80 90 Galactic latitude, degrees intermediate latitudes Eγ=2 GeV anti-GC l=180° FIGURE 1. Left: The ratio of anisotropic IC to isotropic IC for Galactic longitudes l = 0◦ and 180◦ vs. Galactic latitude. Right: γ-ray spectrum of inner Galaxy (330◦ < l < 30◦, |b| < 5◦) for an optimized model. Vertical bars: COMPTEL and EGRET data, heavy solid line: total calculated flux. This is an update of the spectrum shown in [5]. • the γ-ray calculations extend from keV to tens of TeV (e.g., Figure 1, right), and produce full sky maps as a function of energy; the output is in the FITS-format • gas mass calibration (XCO-factors) which can vary with position • a dark matter package to allow for propagation of the WIMP annihilation products and calculation of the corresponding synchrotron and γ-ray skymaps • GALPROP–DarkSUSY interface (together with T. Baltz) will become publicly available soon • a dedicated website has been developed (http://galprop.stanford.edu) The GALPROP code [1] was created with the following aims: (i) to enable simultaneous predictions of all relevant observations including CR nuclei, electrons and positrons, γ-rays and synchrotron radiation, (ii) to overcome the limitations of analytical and semi-analytical methods, taking advantage of advances in computing power, as CR, γ- ray and other data become more accurate, (iii) to incorporate current information on Galactic structure and source distributions, (iv) to provide a publicly-available code as a basis for further expansion. The first point is the most important: all data relating to the same system, the Galaxy, must have an internal consistency. For example, one cannot allow a model which fits secondary/primary ratios while not fitting γ-rays or not being compatible with the known interstellar gas distribution. There are many simultaneous constraints, and to find one model satisfying all of them is a challenge, which in fact has not been met up to now. Upcoming missions will benefit: GALPROP has been adopted as the standard for diffuse Galactic γ-ray emission for NASA’s GLAST γ-ray observatory, and is also made use of by the ACE, AMS, HEAT and Pamela collaborations. IVM is supported in part by NASA APRA grant, TAP is supported in part by the US Department of Energy. REFERENCES 1. A. W. Strong, and I. V. Moskalenko, ApJ 509, 212–228 (1998). 2. I. V. Moskalenko, and A. W. Strong, ApJ 493, 694–707 (1998). 3. A. W. Strong, I. V. Moskalenko, and O. Reimer, ApJ 537, 763–784 (2000). 4. I. V. Moskalenko, A. W. Strong, J. F. Ormes, and M. S. Potgieter, ApJ 565, 280–296 (2002). 5. A. W. Strong, I. V. Moskalenko, and O. Reimer, ApJ 613, 962–976 (2004). 6. V. S. Ptuskin et al, ApJ 642, 902–916 (2006). 7. S. G. Mashnik et al., Adv. Space Res. 34, 1288–1296 (2004). 8. I. V. Moskalenko, T. A. Porter, and A. W. Strong, ApJ 640, L155–L158 (2006). 9. T. A. Porter, A. W. Strong, and S. W. Digel, in preparation (2007). 10. P. M. W. Kalberla et al., Astron. Astrophys. 440, 775–782 (2005). 11. T. M. Dame, D. Hartmann, and P. Thaddeus, ApJ 547, 792–813 (2001). 12. T. Kamae et al., ApJ 647, 692–708 (2006). http://galprop.stanford.edu Discussion and Results

0704.1329

Prompt Emission of High Energy Photons from Gamma Ray Bursts

Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 30 October 2018 (MN LATEX style file v2.2) Prompt Emission of High Energy Photons from Gamma Ray Bursts Nayantara Gupta⋆ and Bing Zhang† Department of Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, USA Accepted 2007; Received 2007; in original form 2007 ABSTRACT Within the internal shock scenario we consider different mechanisms of high energy (> 1 MeV) photon production inside a Gamma Ray Burst (GRB) fireball and derive the expected high energy photon spectra from individual GRBs during the prompt phase. The photon spectra of leptonic and hadronic origins are compared within dif- ferent sets of parameter regimes. Our results suggest that the high energy emission is dominated by the leptonic component if fraction of shock energy carried by elec- trons is not very small (e.g. ǫ −3). For very small values of ǫ the hadronic emission component could be comparable to or even exceed the leptonic component in the GeV-TeV regime. However, in this case a much larger energy budget of the fireball is required to account for the same level of the observed sub-MeV spectrum. The fireballs are therefore extremely inefficient in radiation. For a canonical fireball bulk Lorentz factor (e.g. Γ = 400), emissions above ∼ 10 GeV are attenuated by two-photon pair production processes. For a fireball with an even higher Lorentz fac- tor, the cutoff energy is higher, and emissions of 10 TeV - PeV due to π0-decay can also escape from the internal shocks. The flux level is however too low to be detected by current TeV detectors, and these photons also suffer attenuation by external soft photons. GLAST LAT can detect prompt emission of bright long GRBs above 100 MeV. For short GRBs, the prompt emission can be only barely detected for nearby bright ones with relatively “long” durations (e.g. ∼ 1 s). With the observed high energy spectrum alone, it appears that there is no clean picture to test the leptonic vs. hadronic origin of the gamma-rays. Such an issue may be however addressed by collecting both prompt and afterglow data. A moderate-to-high radiative efficiency would suggest a leptonic origin of high energy photons, while a GRB with an ex- tremely low radiative efficiency but an extended high energy emission component would be consistent with (but not a proof for) the hadronic origin. Key words: Gamma Rays, Gamma Ray Bursts. 1 INTRODUCTION The study of Gamma Ray Bursts (GRBs) has been one of the most interesting areas in astrophysics in the past few years. Ongoing observational and theoretical investigations are disclosing the physical origin, characteristics of these objects as well as bringing new puzzles to us. EGRET detected high energy photons from five GRBs coincident with triggers from the BATSE instrument (Jones et al. 1996). GRB 940217 was detected by EGRET independent of BATSE trigger, which has extended emission and with the highest energy photon of 18GeV (Hurley et al. 1994). Gonzalez et al. (2003) discovered a distinct high energy component up to ⋆ [email protected] † [email protected] c© 2007 RAS http://arxiv.org/abs/0704.1329v3 2 Nayantara Gupta and Bing Zhang 200 MeV in GRB 941017 that has a different temporal evolution with respect to the low energy component. Although even higher energy gamma rays/neutrinos have not been firmly detected from GRBs yet, Atkins et al. (2000) have provided tentative evidence of TeV emission from GRB 970417A. For a long time, GRBs have been identified as potential sources of ultrahigh energy cosmic rays (Waxman 1995; Vietri 1997). Within the standard fireball picture (e.g. Mészáros 2006), there are about a dozen mechanisms that can produce GeV-TeV gamma-rays from GRBs (e.g. Zhang 2007). More theoretical and observational efforts are needed to fully understand high energy emission from GRBs. From the theoretical aspect, it is essential to investigate the relative importance of various emission components to identify the dominant mechanisms under certain conditions. The high energy photon spectra expected from GRBs during the prompt and the afterglow phases have been derived by various groups. In the scenario of external shock model the high energy photon spectra during the early afterglow phase due to synchrotron and synchrotron self Compton (SSC) emission by shock accelerated relativistic electrons and protons have been studied (Mészáros et al. 1994; Mészáros & Rees 1994; Panaitescu & Mészáros 1998; Wei & Lu 1998; Totani 1998; Chiang & Dermer 1999; Dermer et al. 2000a,b; Panaitescu & Kumar 2000; Sari & Esin 2001; Zhang & Mészáros 2001; Fan et al. 2007; Gou & Mészáros 2007). In the case of a strong reverse shock emission component, the SSC emission in the reverse shock region or the crossing inverse Compton processes between the forward and reverse shock regions are also important (Wang et al. 2001a,b; Pe’er & Waxman 2005). The discovery of X-ray flares in early afterglows in the Swift era (Burrows et al. 2005) also opens the possibility that scattering of the flaring photons from the external shocks can give strong GeV emission (Wang et al. 2006; Fan & Piran 2006). The effect of cosmic infrared background on high energy delayed γ-rays from GRBs has been also widely discussed in the literature (Dai & Lu 2002; Stecker 2003; Wang et al. 2004; Razzaque et al. 2004; Casanova et al 2007; Murase et al. 2007). The most important high energy emission component is believed to be emitted from the prompt phase. Swift early X-ray afterglow data suggest that the GRB prompt emission is of “internal” origin, unlike the external-origin afterglow emis- sion (Zhang et al. 2006, cf. Dermer 2007). The most widely discussed internal model of prompt emission is the internal shock model (Rees & Mészáros 1994). Within the internal shock model the spectrum of high energy photons expected during the prompt phase has been studied (Pilla & Loeb 1998; Fragile et al. 2004; Bhattacharjee & Gupta 2003; Razzaque et al. 2004; Pe’er & Waxman 2004; Pe’er et al. 2006). The various processes of high energy photon production in the internal shocks are electron synchrotron emission, SSC of electrons, synchrotron emission of protons, photon production through π0 decay produced in proton photon (pγ) interactions and radiations by secondary positrons produced from π+ decays. In this paper we consider all these processes self-consistently with a semi-analytical approach and study the relative importance of each component within the internal shock scenario. The derived photon spectra are corrected for internal optical depth for pair production, which is energy-dependent and also depends on various other parameters of GRBs e.g. their variability times, luminosities, the low energy photon spectra inside GRBs, and photon spectral break energies. If the electrons cool down by synchrotron and SSC emission to trans-relativistic energies, then they accumulate near a value of Lorentz factor of around unity. The accumulated electrons affect the high energy photon spectrum by direct-Compton scattering and other processes, which make the spectrum significantly different from the broken power laws considered in this work, see (Pe’er et al. 2005, 2006) for detailed discussions. In any case, for the values of parameters considered in the present paper this effect is not significant. GLAST’s (Gehrels & Michelson 1999) burst monitor (GBM) will detect photons in the energy range of 10keV to 25MeV and large area telescope (LAT) will detect photons in the energy range of 20MeV and 1000GeV. With a large field of view (> 2 sr for LAT), GLAST will detect high energy photons from many GRBs and open a new era of studying GRBs in the high energy regime. This is supplemented by AGILE (Longo et al. 2002), which is designed to observe photons in the energy range of 10-40 keV and 30MeV-50GeV and also has a large field of view. There are several other ground based detectors e.g. Whipple/VERITAS (Horan et al. 2007), Milagro (Atkins et al. 2004), which have been searching or will search for ∼ TeV photons from GRBs. De- tections or non-detections of high energy gamma rays from GRBs with space-based and ground-based detectors in the near future would make major steps in revealing the physical environment, bulk motion, mechanisms of particle acceleration and high energy photon production, photon densities, etc., of GRBs. 2 ELECTRON SYNCHROTRON RADIATION We define three reference frames: (i) the comoving frame or the wind rest frame is the rest frame of the outflowing ejecta expanding with a Lorentz factor Γ with respect to the observer and the central engine; (ii) the source rest frame is attached to the GRB central engine at a redshift z; and (iii) the observer’s frame is the reference frame of the observer on earth, which is related to the source rest frame by the redshift correction factor. We denote the quantities measured in the comoving frame with primes. The shock accelerated relativistic electrons lose energy by synchrotron radiation and SSC in the shock region. Assuming a power law distribution of fresh electrons accelerated from the internal shocks and considering a continuous injection of electrons during the propagation of the shocks, the relativistic primary electron number distribution in the comoving frame can be expressed as a broken power law in energy (Sari et al. 1998) c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 3 dNe(E E′e,m < E e < E E′e,c < E in the case of slow cooling, where E′e,m is the minimum injection energy of electrons and E e,c is the energy of an electron that loses its energy significantly during the dynamic time scale, known as the cooling energy of the electrons. If the electrons are cooling fast so that even the electrons with the minimum injection energy have cooled during the dynamical time scale, by considering continuous injection of electrons from the shock the comoving electron number distribution can be expressed as dNe(E E′e,c < E e < E E′e,m < E If the electrons cool down to sub-relativistic energies then they accumulate near electron Lorentz factor γ′e ∼ 1. This effect may distort the high energy photon spectrum by direct-Compton scattering (Pe’er et al. 2005, 2006), and we focus on the parameter regime where this effect is not significant. The energies in the source rest frame and the comoving frame are related as Ee ≃ ΓE where Γ is the average bulk Lorentz factor of the GRB fireball in the prompt phase. The expression for the minimum injection energy of electrons in the comoving frame is E′e,m = mec 2γ̄′pg(p) , where g(p) = p−2 for p >> 2 and g(p) ∼ 1/6 for p = 2 (Razzaque & Zhang 2007), mp, me are the masses of proton and electron, respectively, and γ̄′pmpc 2 is the average internal energy of protons in the comoving frame. We have assumed γ̄′p to be of the order of unity (in principle γ̄′p could be smaller than unity). The total internal energy is distributed among electrons, protons and the internal magnetic fields within the internal shocks. The fractions of the total energy carried by electrons, protons and internal magnetic fields are represented by ǫe, ǫp and ǫB , respectively, where ǫe + ǫp + ǫB = 1. We have assumed that all the electrons and protons are accelerated in internal shocks. In reality, the shock accelerated particles may be only a fraction of the total population and additional fractional parameters (ξe, ξp) may be introduced (e.g. Bykov & Mészáros 1996). In such a case, the following treatments are still generally valid by re-defining ǫ′e = ǫe/ξe and ǫ p = ǫp/ξp, while the relation ǫe + ǫp + ǫB = 1 still holds. The relativistic electrons lose their energy by synchrotron radiation and inverse Compton scattering (Panaitescu & Mészáros 1998; Sari & Esin 2001; Zhang & Mészáros 2001). The comoving cooling break energy in the relativistic electron spectrum can be derived by comparing the cooling and the dynamical time scales. The comoving cooling time scale t′cool of electrons is a convolution of the cooling time scales for synchrotron radiation t′syn and for inverse Compton (IC) scattering t t′cool t′syn . (3) We denote U as the internal energy density of the internal shock, and Ue, UB as the energy densities of electrons and magnetic fields, respectively. The energy density of the synchrotron radiation is Ue,syn = ηeǫeU (Sari & Esin 2001), where the radiation efficiency of electrons is ηe = [(E e,c/E 2−p, 1] for slow and fast cooling, respectively, and Le,IC Le,syn Ue,syn 1 + 4ηeǫe/ǫB denotes the relative importance between the IC and the synchrotron emission components1. Le,IC and Le,syn are the luminosities of radiations emitted in SSC and synchrotron emission of relativistic electrons respectively. The inverse of the cooling time scale of electrons can be expressed by the power divided by energy (E′e = meγ t′cool σe,Tβ (UB + Ue,syn) = σe,Tβ (1 + Ye) , (5) where σe,T is Thomson cross-section of electrons, βe ≃ 1 is the dimensionless speed of the relativistic electrons. The comoving dynamical time scale is t′dyn ≃ Γtv , where Γ is the average Lorentz factor of the GRB, and tv is the variability time in the source rest frame of the GRB, which denotes the variability time scale of the central engine. Throughout the paper, we assume that electron synchrotron radiation from the internal shocks is the mechanism that power the prompt gamma-ray emission in the sub-MeV band. However, for standard parameters within this scenario the cooling time scale of electrons is much shorter than the dynamical time scale of GRBs. As a result the flux density dNγ,s(Eγ,s) below the cooling break energy is proportional to E γ,s and cannot explain the harder spectral indices observed in many GRBs (Ghisellini et al. 2000). If the magnetic field created by internal shocks decays on a length scale much shorter then the comoving width of the plasma, then the resulting synchrotron radiation can explain 1 Strictly speaking, such a treatment is valid for the IC process in the Thomson regime. However, this is also a reasonable approximation if the peak of the spectral energy distribution of the IC component is in the Thomson regime, which is generally the case for the calculations performed in this paper. c© 2007 RAS, MNRAS 000, 1–?? 4 Nayantara Gupta and Bing Zhang some of the broadband GRB spectra observed by Swift (Pe’er & Zhang 2006). In this case the effective dynamical time scale is shorter by a factor of fc than its actual value. Hence, the ratio of the cooling and the dynamical time scale can be expressed as t′dyn t′cool = fc (6) at the cooling energy E′e = E e,c. The expression of the electron cooling energy in the comoving frame can be written as e,c = γ e,cmec = mec 2 3mec 4Γtvσe,T cUǫB(1 + Ye) = 530keV tv,−2Γ 2fc,2 Liso,51ǫB,−1(1 + Ye) . (7) Here and throughout the text the convention Qx = Q/10 x is adopted in cgs units. In the above expression Liso is the luminosity corresponding to the energy Eiso carried by all particles and the magnetic fields in the shocks. It is a fraction of the wind (outflow) luminosity Liso ∼ ηLw, where η is the efficiency of converting the kinetic energy of the wind to the shock internal energy. The luminosity Liso and internal energy U are related as U = Liso/(4πΓ 2c), where ris = Γ 2ctv is the internal shock radius. The synchrotron spectrum is a multi-segment broken power law (Sari et al. 1998) separated by several breaks, including the emission frequency from electrons with the minimum injection energy, the cooling break frequency, and the synchrotron self- absorption frequency (Rybicki & Lightman 1979). In the internal shocks, the magnetic field in the comoving frame can be expressed as (Zhang & Mészáros 2002) ≃ 4.4× 10 G(ξ1ǫB,−1) iso,51r is,13Γ 2 = 1.5× 10 (ξ1ǫB,−1Liso,51) Γ32tv,−2 where ξ is the compression ratio, which is about 7 for strong shocks. The synchrotron self absorption energy (Essa) in internal shocks can be expressed as (Li & Song 2004; Fan et al. 2005; cf. Pe’er & Waxman 2004) Essa ≃ 0.24 keVL γ,s,51Γ is,13 B = 0.69 keVL iso,51t v,−2 Γ2 (ξ1ǫB,−1) 1 + Ye where Lγ,s = Lisoǫeηe/(1 + Ye) is the isotropic gamma-ray luminosity due to synchrotron radiation. The cooling break energy E′e,c and the minimum injection energy E e,m of the electrons define two break energies in the synchrotron photon spectrum. The cooling break energy in the photon spectrum in the source rest frame is Eγ,c = Γ E′e,c )2 eB′c ≃ 1.9 × 10 tv,−2Γ 2fc,2 Liso,51ǫB(1 + Ye) 5 = 2.8eVtv,−2 (Liso,51ǫB,−1)3/2 Γ42fc,2 1 + Ye Notice that Eγ,c very sensitively depends on Γ and some other parameters so that it could become a large value when parameters change. For example, for B′ = 104G, Γ = 400, fc = 500, Liso = 10 51erg s−1, tv = 0.01s and ǫB = 0.1 we get Eγ,c ∼ 1.9 MeV. The break energy in the photon spectrum due to the minimum electron injection energy is Eγ,m = Γ E′e,m )2 eB′c ≃ 0.58 MeVΓ2 5 = 8.5MeV (ξ1ǫB,−1Liso,51) 2tv,−2) −1 (11) Assuming Essa < Eγ,m,s < Eγ,c,s the photon energy spectrum from synchrotron radiation of slow-cooling relativistic electrons is as follows dNγ,s(Eγ,s) dEγ,s γ,s Essa < Eγ,s ≤ Eγ,m,s 4/3+(p−3)/2 γ,m,s E −(p−3)/2 γ,s Eγ,m,s < Eγ,s ≤ Eγ,c,s 4/3+(p−3)/2 γ,m,s E γ,c,sE −(p−2)/2 γ,s Eγ,c,s ≤ Eγ,s In the case of slow-cooling electrons for very small values of ǫe (e.g. ∼ 10 −3, which is relevant when the hadronic emission component becomes important), the break in the photon spectrum due to the minimum injection energy of electrons goes below the synchrotron self absorption energy. The order in the spectral break energies becomes Eγ,m,s < Essa < Eγ,c,s, and the spectrum is also modified. The spectral indices of the electron synchrotron spectrum for different ordering of the spectral break energies are derived by Granot & Sari (2002). For Eγ,m,s < Eγ,s < Essa the spectral index of E dNγ,s(Eγ,s) dEγ,s is 7/2, and for Eγ,s < Eγ,m,s the spectral index is 3. The indices of the spectrum between Essa, Eγ,c,s and above Eγ,c,s remain as −(p− 3)/2 and −(p − 2)/2, respectively. When Essa is greater than both Eγ,m,s and Eγ,c,s their relative ordering becomes unimportant. In that case the spectral indices of E2γ,s dNγ,s(Eγ,s) dEγ,s are 7/2 between Eγ,m,s and Essa, and −(p− 2)/2 above Essa. Below Eγ,m,s the index is 3. For fast-cooling electrons the synchrotron photon energy spectrum for Essa < Eγ,c,s < Eγ,m,s is dNγ,s(Eγ,s) dEγ,s γ,s Essa < Eγ,s ≤ Eγ,c,s γ,c,sE γ,s Eγ,c,s < Eγ,s ≤ Eγ,m,s γ,c,sE (p−1)/2 γ,m,s E −(p−2)/2 γ,s Eγ,m,s ≤ Eγ,s c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 5 When the ordering of break energies in the photon spectrum becomes Eγ,c,s < Essa < Eγ,m,s the photon energy spectrum is dNγ,s(Eγ,s) dEγ,s γ,s Essa < Eγ,s ≤ Eγ,c,s γ,c,sE γ,s Eγ,c,s < Eγ,s ≤ Eγ,m,s γ,c,sE (p−1)/2 γ,m,s E −(p−2)/2 γ,s Eγ,m,s ≤ Eγ,s The total energy emitted in synchrotron radiation by relativistic electrons is Eisoηeǫe/(1+Ye). The normalisation constant for the synchrotron photon energy spectrum can be calculated from ∫ Eγ,max Eγ,min dNγ,s(Eγ,s) dEγ,s dEγ,s = Eiso (1 + Ye) The maximum electron energy Ee,max can be calculated by equating the acceleration time and the shorter of the dynamical and cooling time scales of the relativistic electrons. The expression of the acceleration time scale is t′acc = 2πζrL(E e)/c = 2πζE′e/eB ′c. Here rL(E e) is the Larmor radius of an electron of energy E e in a magnetic field B ′, ζ can be expressed as ζ ∼ βsh −2y, where βsh is the velocity of the shock in the comoving frame of the unshocked medium and y is the ratio of diffusion coefficient to the Bohm coefficient Rachen & Mészáros (1998). In ultra-relativistic shocks βsh ≈ 1 and numerical simulations for both parallel and oblique shocks gives ζ ∼ 1. With acc = min[t cool, t dyn] , (16) one can derive the maximum comoving electron energy e,max = min Liso,51ǫB(1 + Ye) , 14.3× 10 Γ2tv,−2B GeV (17) For electrons, the cooling term (first term in the bracket) always defines the maximum electron energy. The maximum synchrotron photon energy in the source rest frame can be then derived as Eγ,max = Γ E′e,max )2 eB′c = 0.48GeV t2v,−2 Liso,51ǫB,−1(1 + Ye) = 102GeV 1 + Ye This is used in eqn.(15) to define the normalization of the spectrum. The result has a very steep dependence on Γ. We also notice that B′ is not an independent parameter, but can be calculated from other parameters according to eqn.(8). For example, for Γ = 400, Liso = 10 51erg/s, tv = 0.01s and ǫB , ǫe ∼ 0.1, the magnetic field is of the order of 10 4G and the maximum photon energy becomes a few hundred GeV. 3 ELECTRON INVERSE COMPTON SCATTERING The relativistic electrons can be inverse Compton scattered by low energy synchrotron photons inside the GRB fireball and transfer their energy to high energy photons. Below, we derive the IC photon spectrum using the electron and synchrotron photon spectra. dNγ,i(Eγ,i) dEγ,i dNe(Ee) dEe × dNγ,s(Eγ,s) dEγ,s dEγ,s (19) The electron Lorentz factor (γe ′), IC and synchrotron photon energies (Eγ,i, Eγ,s) are related as Eγ,i ∼ γ Eγ,s, this can be used to simplify the above equation. The final expression for the IC photon spectrum considering slow cooling of electrons is dNγ,i(Eγ,i) dEγ,i γ,i Essa,i < Eγ,i ≤ Eγ,m,i 4/3+(p−3)/2 γ,m,i E −(p−3)/2 γ,i Eγ,m,i < Eγ,i ≤ Eγ,c,i 4/3+(p−3)/2 γ,m,i E γ,c,iE −(p−2)/2 γ,i Eγ,c,i < Eγ,i ≤ Eγ,K 4/3+(p−3)/2 γ,m,i E γ,c,iE (p−2)/2 γ,K E −(p−2) γ,i Eγ,K < Eγ,i Here Essa,i = γ Essa, Eγ,m,i = γ Eγ,m,s, and Eγ,c,i = γ Eγ,c,s, where γ e,m = E e,m/mec g(p)(mp/me)(ǫe/ǫp), γ e,c = E e,c/mec 2 are Lorentz factors corresponding to the minimum injection energy of electrons and the cooling break energy of electrons. In the case of fast cooling Eγ,m,i > Eγ,c,i and the IC photon spectrum has to be modified accordingly. dNγ,i(Eγ,i) dEγ,i γ,i Essa,i < Eγ,i ≤ Eγ,c,i γ,c,iE γ,i Eγ,c,i < Eγ,i ≤ Eγ,m,i γ,c,iE (p−1)/2 γ,m,i E −(p−2)/2 γ,i Eγ,m,i < Eγ,i ≤ Eγ,K γ,c,iE (p−1)/2 γ,m,i E (p−2)/2 γ,K E −(p−2) γ,i , Eγ,K < Eγ,i c© 2007 RAS, MNRAS 000, 1–?? 6 Nayantara Gupta and Bing Zhang In eqn.(21) the expressions for Essa,i, Eγ,c,i and Eγ,m,i are Essa,i = γ Essa, Eγ,c,i = γ Eγ,c,s and Eγ,m,i = γ′e,m Eγ,m,s. When EeEγ,s >> Γ 2m2ec 4 the cross section for IC scattering decreases as the scattering enters the Klein Nishina (KN) regime. A break in the photon spectrum at Eγ,i = Eγ,K appears when the Klein Nishina effect becomes important. We define a parameter κ = EeEγ,peak Γ2m2ec , where Eγ,peak = max[Eγ,c,s;Eγ,m,s]. The KN regime starts when κ = 1 (e.g. Fragile et al. 2004), Eγ,K = Γ2m2ec Eγ,peak = 2.5 GeV Eγ,peak,MeV In the KN regime the emissivity of electrons decreases by κ2, and the photon energy spectral index simply follows the electron energy spectral index, i.e. −(p− 2). The IC photon spectrum in eqn.(20) can be normalised as ∫ Eγ,max,i Eγ,m,i dNγ,i(Eγ,i) dEγ,i dEγ,i = Eiso ηeǫeYe 1 + Ye , (23) where Eγ,max,i = ΓE e,max due to the KN effect. 4 PROTON SYNCHROTRON RADIATION Relativistic protons lose energy by synchrotron radiation and photo-pion (π0, π+) production inside GRBs. They interact with the low energy photons in the GRB environment and pions are produced. There is a threshold energy for this interaction (pγ) to happen, EpEγ ≥ 0.3GeV 2Γ2, where Ep and Eγ are proton, photon energy in the source rest frame respectively. The π 0s decay to a pair of high energy photons, while the π+s decay to neutrinos and leptons. The threshold condition therefore suggests that the photon-pion related high energy spectrum is typically more energetic than the electron IC spectrum. We assume that the proton spectrum in the internal shocks can be expressed as a power law in proton energy. We consider a proton spectral index similar to electrons for our present discussion. Since protons are poor emitters, we only consider the scenario of slow-cooling in the comoving proton spectrum dNp(E E′p,m < E p < E E′p,c < E where E′p,m is the minimum injection energy of the protons and E p,c is break energy in the spectrum due to proton cooling. The minimum injection energy E′p,m = γ̄ 2g(p), where g(p) = p−2 for p ≫ 2 and g(p) ∼ 1/6 for p = 2. The cooling break energy can be derived by comparing the comoving and the cooling time scales. The inverse of the cooling time scale t′cool of a proton is t′cool t′syn The photo-pion cooling time scale t′π has been derived earlier in the context of estimation of neutrino fluxes from GRBs (Waxman & Bahcall 1997; Gupta & Zhang 2007). If fπ is the fraction of proton energy going to pion production in the ∆ res- onance of pγ interactions one has 1/t′π ∼ fπ/t dyn where the comoving time scale is 2 t′dyn = Γtv. The peak value of pγ interaction cross section at the ∆ resonance is σpγ = 5 × 10 −28cm2. This is much higher than the Thomson cross section for protons σp,T = σe,T , where σe,T = 6.625 × 10 −25cm2. We therefore neglect the IC process of protons. Substituting for t′syn and t π in eqn.(25), we get t′cool σp,Tβ where, β′p is dimensionless speed of relativistic protons. We use the general expression for fπ from Gupta & Zhang (2007) fπ(Ep) = f0 1.34α2−1 )α2−1 Ep < Epb 1.34α1−1 )α1−1 Ep > Epb where 0.9Liso,51 810Γ42tv,−2Eγ,peak,MeV 1 + Ye . (28) 2 In this definition, on average protons loose ∼ 20% energy in the time scale of t′π . Although it is not strictly the e-folding timescale usually used to define cooling, for order-of-magnitude estimates this is good enough. c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 7 In our present discussion α2 = (p + 2)/2 and α1 = (p + 1)/2. Eγ,peak,MeV is the peak energy in the electron synchrotron photon spectrum expressed in MeV, and Liso,51 is the GRB luminosity in unit of 10 51 erg s−1, which is the typical value for GRB luminosities. Epb = 0.3Γ 2/Eγ,peak,GeV GeV is the threshold proton energy for interaction with photons of energy Eγ,peak,GeV . For typically observed values of GRB parameters one has Epb ∼ 1 PeV. The break energy in the proton spectrum due to proton cooling can be calculated by comparing the comoving and cooling time scales of protons as discussed in the case of electrons in §2. We assume β′p ∼ 1 then for Ep < Epb the expression of cooling break energy in the comoving frame is p,c = σp,Tβ 2 cUǫB m2pc4 EpbΓtv 1.34α2−1 α2 + 1 108GeVfc,2 Γ2tv,−2 Liso,51ǫB 6t2v,−2 Epb(PeV)Γ2tv,−2 1.34α2−1 α2 + 1 where fc = . The synchrotron photon spectrum from relativistic protons is dNγ,ps(Eγ,ps) dEγ,ps −(p−3)/2 γ,ps Eγ,m,ps < Eγ,ps ≤ Eγ,c,ps γ,c,psE −(p−2)/2 γ,ps Eγ,c,ps < Eγ,ps The minimum injection energy in the photon spectrum from proton synchrotron radiation is related to that from electron synchrotron radiation as (Zhang & Mészáros 2001) Eγ,m,ps Eγ,m,s E′p,m E′e,m )2(me The cooling break energy in the photon spectrum from proton synchrotron radiation is the characteristic synchrotron photon energy for proton energy E′p,c. To normalize the proton synchrotron spectrum, it is important to find out the relative importance between proton synchrotron radiation and pγ interactions. Similar to the treatment of electrons, one can define Lp,pγ Lp,syn Ue,syn Ye . (32) where, Lp,pγ and Lp,syn are the luminosities of radiations emitted in pγ interactions and synchrotron emission of protons respec- tively. Notice that protons interact with the synchrotron emission of the electrons, so that Ye enters the problem. Eqn. (32) suggests that Yp is usually much greater than unity since σpγ ≫ σp,T . As a result, most of the proton energy is lost through pγ interaction rather than proton synchrotron radiation. The proton synchrotron photon spectrum can be normalised as ∫ Eγ,max,ps Eγ,m,ps Eγ,ps dNγ,ps(Eγ,ps) dEγ,ps dEγ,ps = Eiso 1 + Yp , (33) where ηp = E′p,c/E . The maximum proton synchrotron photon energy is derived by Eγ,max,ps = E′p,max , where E′p,max is again defined by comparing the comoving acceleration time with the shorter of the co- moving dynamical and cooling times scales p,max = min Liso,51ǫB(1 + Yp) , 1.4 × 10 Γ2tv,−2B TeV . (34) p,max = min ǫB,−1Liso,51 )1/2 Γ32tv,−2 1 + Yp (ξ1ǫB,−1Liso,51) TeV . (35) 5 π0 DECAY The relativistic protons interact with the low energy photons and photo-pions (π0,π+) are produced as a result. The probabilities of π0 and π+ production are 1/3 and 2/3, respectively. Pions subsequently decay, i.e. π0 → γγ and π+ → µ+νµ → νµν̄µνee As the cross section for the γγ interactions is much higher than the peak value of pγ interaction cross section, above the threshold energy of pair production γγ interactions are expected to dominate over pγ interactions. If the photon energy is 2mec ∼ 1 MeV in the comoving frame, then in the source rest frame it is of the order of a few hundred MeV as the Lorentz factors are typically of the order of few hundred for canonical GRBs. For example, for Γ = 400 the photons of energy 400 MeV can produce photo-pions by interaction with protons of minimum energy Ep ∼ 120 TeV. The π 0 typically carries 20% of the proton’s energy and the photons produced in π0 decay share its energy equally. Hence, the minimum energy of the photons produced from π0 decay is expected to c© 2007 RAS, MNRAS 000, 1–?? 8 Nayantara Gupta and Bing Zhang be ∼ 10%Ep ∼ 12 TeV. The photon spectrum produced from π 0 decay has been derived below using the proton spectrum defined in eqn.(24) and assuming the fraction fπ/3 of protons’ energy goes to π dNγ,π0(Eγ,π0) dEγ,π0 fπ(Eγ,π0) Eγ,π0 ≤ Eγ,π0,c Eγ,π0 > Eγ,π0,c where, Eγ,π0,c = 0.1Ep,c. For the expression for fπ , see eqn.(27), which contains a break energy. The break energy in the photon spectrum contained within fπ is Eγ,π0,b = 0.03Γ 2/ǫbr,GeV GeV assuming 10% of the proton’s energy goes to the photon produced via π0 decay. ǫbr is the break energy in the low energy photon spectrum (in the scenario of slowly cooling electrons it is the cooling break energy in the photon spectrum and for fast cooling electrons it is the photon energy corresponding to the minimum injection energy of electrons). The photon flux can be normalised in the following way γ,π0,max γ,π0,min Eγ,π0 dNγ,π0(Eγ,π0) dEγ,π0 dEγ,π0 = ǫpηpYp 1 + Yp where Eγ,π0,min = 30Γ GeV and Eγ,π0,max = 0.1Ep,max. Although high energy photons (∼ TeV) are absorbed by lower energy photons and e+e− pairs are produced, at extreme energies the pair production cross section decreases with increasing energy (Razzaque et al. 2004). Hence, ultrahigh energy photons can escape from the internal shocks for suitable parameters depending on the values of their various parameters and the low energy photon spectra. 6 SYNCHROTRON RADIATION OF POSITRONS PRODUCED IN π+ DECAY The shock accelerated protons may interact with the low energy photons to produce π+s along with π0s as discussed in the previous section. The π+s subsequently decay to muons and neutrinos. The energetic muons decay to positrons and neutrinos (pγ → π+ → µ+νµ → e +νµν̄µνe). The charged pions, muons and the positrons are expected to lose energy through synchrotron radiation and IC inside the shock region. As the Thomson cross section for positrons is much larger than pions or muons, they are expected to emit much more radiation compared to the heavier charged particles. On the other hand, since these positrons are very energetic, most IC processes happen in the Klein Nishina regime. We therefore neglect the contribution of the positron IC processes. The positron synchrotron spectrum produced in pγ interactions can be derived in the following way. The fraction of the protons’ energy tranferred to pions is denoted by fπ (eqn[27]). If we assume that the final state leptons share the pion’s energy equally then one fourth of the pion’s energy goes to the positron. The energy of the positron spectrum at the energy Ee+ can be expressed using the proton spectrum defined in eqn.(24) dN(Ee+) fπ(Ee+) Ee+ ≤ Ee+,c Ee+ > Ee+,c where, Ee+,c is the cooling break energy in the positron specrum and fπ(Ee+) = f0 1.34α2−1 )α2−1 Ee+ < Ee+b 1.34α1−1 )α1−1 Ee+ > Ee+b where f0 has been defined in eqn.(28), Ee+b = 0.05Epb , Epb = 0.3 GeVΓ 2/ǫbr,GeV , and ǫbr,GeV is the break energy in the photon spectrum as defined earlier. The positron spectrum in eqn.(38) can be normalised using the total energy carried by the positrons, e+,max e+,min dN(Ee+)(Ee+) dEe+ = ǫpηpYpEiso 1 + Yp The maximum and minimum positron energies are Ee+,max = 0.05Ep,max and Ee+,min = 15Γ GeV (which is ∼ 6TeV for Γ = 400). The synchrotron photon spectrum from the positrons can be subsequently derived using the same treatment for primary electrons as discussed in §2. The IC emission is in the KN regime and therefore not important. Also, photons having energies above a few hundred GeV are annihilated by lower energy photons as discussed in the following section. The relativistic muons produced in π+ decay lose energy by synchrotron radiation. We compare the decay and synchrotron energy loss time scales of the high energy muons. The maximum energies of positrons can be calculated in this way. If the muons decay before losing energy significantly high energy positrons are produced carrying approximately 5% of the initial proton’s energy. On the otherhand if the muons lose energy before they decay lower energy positrons are produced. These positrons radiate energy and produce lower energy photons. The muons initially carry approximately 10% of the relativistic protons’ energy hence, we expect the low energy photon flux produced by cooling of positrons is lower than that produced by relativistic electrons if ǫe and ǫp are comparable. c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 9 7 INTERNAL PAIR-PRODUCTION OPTICAL DEPTHS OF HIGH ENERGY PHOTONS Inside GRBs high energy photons interact with low energy photons to produce electron-positron pairs (e.g. Baring & Harding 1997; Lithwick & Sari 2001). The optical depth depends on the values of various parameters of the GRB fireball. We follow the approach discussed in Bhattacharjee & Gupta (2003) to derive internal optical depths of GRBs in detail. For two photons (a high energy photon γh and a low energy photon γl), the pair production cross section depends on the energies of the photons and the angle between their directions of propagation. The cross section is (Berestetskii et al. 1982) σγhγl (E , θ) = σT (1− β (3− β 1 + β′ 1− β′ (2− β where σT is the Thomson cross section, and β ′ = [1 − (E′γl,th/E )]1/2 is the center of mass dimensionless speed of the pair produced. The threshold energy of pair production with a high energy photon of energy E′γh is γl,th 2(mec E′γh(1− cosθ) For the photons with energy higher than the threshold energy, the pair production cross section decreases with increasing photon energy (Jauch & Rohrlich 1955; Razzaque et al. 2004). In the present work we calculate internal optical depths in different energy regimes using the cross sections with different energy dependences. The mean free path for γh γl interactions lγhγl can be calculated using the low energy photon spectrum. γhγlθ , θ) = γl,th dnγl(E dE′γl σγhγl (E , θ) (43) d(cosθ)(1− cosθ)l γhγlθ , θ) (44) where (E′γl dE′γl is the specific number density of low energy photons inside the GRB. The low energy photon spectrum is ob- servationally known, as revealed by gamma-ray detectors such as BATSE and Swift. Theoretically, it corresponds to the electron synchrotron component as discussed in §2, which is a broken power law spectrum separated by the synchrotron self absorption break, the minimum injection break and the cooling break. The low energy photon flux is related to the observed luminosity through ∫ E′γl,max E′γl,ssa dnγl (E dE′γl = Uγ = Lγ,iso 4πcris2Γ2 where Lγ,iso is the isotropic γ-ray luminosity. We have taken it to be equal to the luminosity of the synchrotron photons emitted by electrons: Lγ,iso = Le,syn = ǫeηeLiso . In eqn.(44) we have three variables: angle θ and photon energies E′γl , E . To simplify the integration in eqn.(44) we transform the integral with a new variable following Gould & Schreder (1967) E′γlE (1− cosθ) 2(mec2)2 E′γl,th = s0Θ (46) with s0 = (mec2)2 , and Θ = 1 (1− cosθ). As β′ = (1− 1/s)1/2, the pair production cross section can be expressed as a function of the new variable s. It is then possible to write eqn.(44) as −2 dnγl (E dE′γl Q[s0(E )] (47) where Q[s0(E ∫ s0(E sσ(s)ds , (48) and σ(s) = 16 σγhγl . For moderate values of s we use σ(s) ≃ 1 and for s >> 1 it can be approximated as σ(s) ≃ ln(s)/s. The expressions for Q[s0(E )] are (s20 − 1)/2 and s0(ln s0 − 1), respectively, in the two cases. Substituting for Q[s0(E )] in eqn.(47) we derive the final expression for l−1γhγl (E ). The internal optical depth τint(E ) is the ratio of comoving time scale and the mean time between two pair production interactions. c© 2007 RAS, MNRAS 000, 1–?? 10 Nayantara Gupta and Bing Zhang τint(E ) (49) The final photon energy spectrum to be observed on Earth from nearby GRBs (neglecting further attenuations with the infrared background and cosmic microwave background) can be obtained by correcting the original flux for the internal optical depth and the redshift z of the source dNγ,ob(Eγ,ob) dEγ,ob 4πd2z(1 + z) dNγ(Eγ) exp(−τint(Eγ)) , (50) where ΩΛ +Ωm(1 + z′)3 is the comoving distance of the source, H0 = 71km s −1 Mpc−1 is the Hubble constant, and ΩΛ = 0.73 and Ωm = 0.27 are adopted in our calculations. 8 PHOTON SPECTRUM FROM SECONDARY ELECTRONS AND POSITRONS The secondary pairs carry a significant fraction of energy in the primary spectrum, and this energy is re-radiated and converted to photons. A more realistic treatment should consider a photon-pair cascade process, which requires numerical calculations (Pe’er & Waxman 2004; Pe’er et al. 2006). Here instead we estimate the emission from the secondary pairs. We first calculate the photon energy spectra generated by different physical processes as discussed earlier. The photon spectra are then corrected for internal optical depths and subsequently the total energies carried by these photons are calculated by integrating the corrected photon energy spectra over photon energies. If we subtract the total energies carried by these high energy photons from their intial energies before including the effects of internal optical depths, we get the energies of the secondary e− and e+ produced in γγ in- teractions. These pairs are expected to have spectral indices similar to the high energy photons. With the knowledge of their spectral indices and the total energies carried by them the synchrotron photon spectra radiated by these secondary leptons are calculated. For the parameters adopted in this paper, it turns out that the emission contribution from the secondaries is below the emission level of the primaries, and hence, does not significantly modify the observed the spectrum. We therefore do not include this component in Figs.1-5, but caution that such a feedback process could be potentially important for the parameter regimes with high opacity. We refer to Pe’er & Waxman (2004) and Pe’er et al. (2006) for more detailed treatments of such cases. 9 SYNTHESIZED SPECTRA AND DETECTABILITY Using the procedure delineated above, we have calculated the broad-band emission spectrum from internal shocks for a wide range of parameter regimes. In particular we focus on the various high energy emission components discussed above and their relative significance. Our results are presented in Fig.1-5. In each set of calculations we have presented the internal optical depth after the final photon energy spectrum. For particles accelerated by ultra-relativistic shocks the spectral index is expected to be about 2.26 Lemoine & Pelletier (2003). Afterglow modeling suggests a larger scatter of p values for relativistic shocks, but p = 2.3 is close the mean value of the data (Panaitescu & Kumar 2002). In all our calculations, the spectral indices of relativistic electrons and protons are both assumed as p = 2.3. Figures 1-4 are the calculations for a typical long GRB with duration T90 = 20 s at redshift z = 1 (10 s in the source rest frame). Since we do not know the physical condition of the internal shocks from the first principles, we vary the parameter regime in a wide range. In each set of calculations, we design the parameters to make the electron synchrotron emission peaking at the sub-MeV range (∼ 0.36 MeV, 0.13 MeV, 0.6 MeV, 0.25 MeV for Figs.1-4, respectively), as suggested by the data. The global energetics of the GRB is also adjusted so that the gamma-ray luminosity in the sub-MeV range is about 1051 ergs s−1 as suggested by the observations. The variability time scale for these calculations is taken as tv = 0.01 s. The bulk Lorentz factor is adopted as Γ = 400 in Fig.1-3, as suggested by the recent early optical afterglow observations (Molinari et al. 2006). In order to check how Γ affects the spectra, we also calculate the case of Γ = 1000 for the parameter set of Fig.1, which is presented in Fig.4. In all the figures, the different components of the photon energy spectrum from a GRB for both electrons (e) and protons (p) are displayed with different line styles/colours. The observed energy fluxes E2γ,ob dNγ,ob(Eγ,ob) Eγ,ob in unit of ergs/cm2sec are plotted against the observed photon energy Eγ,ob(eV ). The green long dashed curves represent the synchrotron emission from the relativistic electrons. The short dashed curves (blue) represent the IC spectrum from energetic electrons; the dash-dotted curves (light blue) represents the synchrotron emission of the relativistic protons; the triple short dashed curves (orange) represent for the synchrotron emission of the relativistic positrons produced in π+ decays; the ultrahigh energy emission component from π0 decays c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 11 is shown by the double short dashed curves (black) in the extremely high energy regime. The thin black solid lines represent the synthesized spectra of various components without including the effect of pair production attenuation. Depending on parameters, the pair opacity becomes important in the GeV - TeV range. The thick black solid lines represent the final photon spectrum after including the internal optical depths. In order to check whether the predicted high energy components are detectable by GLAST, we also plot an indicative GLAST sensitivity threshold in the 100 MeV - 100 GeV energy range. The GLAST sensitivity estimate is based on the criterion of detecting at least a few photons in the band based on the average effective area and photon incoming zenith angle of LAT. Background is negligible for GRB detections. This gives a rough fluence threshold of ∼ 2× 10−7 erg cm−2 (B. Dingus, 2007, personal communication). The flux thresholds adopted in all the figures are therefore derived from the observed durations. For T90 = 20 s, this gives a flux threshold of ∼ 10 −8 erg cm−2 s−1. The sensitivity of VERITAS to photon above energy 200GeV has also been shown in our figures with pink dotted line. It is 2 × 10−8erg cm−2 s−1 (D. Horan, 2007, personal communication). Figure 1 is a standard “slow-cooling” leptonic-dominant case. The shock equipartition parameters are ǫe = 0.4 and ǫB = 0.2. The isotropic shock luminosity is Liso = 10 52 erg s−1. The slow cooling factor fc = 2500 is adopted, which suggests that the post-shock magnetic field decays on a length scale shorter than the comoving scale (Pe’er & Zhang 2006). The thick black line shown on the right side around 1015eV is the π0 component after including the effect of absorption due to pair production, indicating the reduction of pair opacity at high energies (Fig.1b, see also Razzaque et al. 2004). In this figure the break energies in the photon energy spectrum appear in the order of Essa < Eγ,m < Eγ,c in the electron synchrotron and IC spectral components. The spectral index of the photon energy spectrum is 4/3 between Essa and Eγ,m, −(p− 3)/2 between Eγ,m and Eγ,c, and −(p− 2)/2 above Eγ,c. Since ǫe is large, the leptonic components are many orders of magnitude stronger than the hadronic components. The value of Yp is much larger than 1, so that the proton synchrotron component is below the components due to π 0 decay and positron synchrotron radiation. We vary the values of the equipartition parameters (ǫe, ǫB , ǫp) and study the variations in the photon energy fluxes generated by various processes. The emission level of the electron IC spectral component decreases with decreasing ǫe (fixing ǫB) since Ye is decreasing. Moreover, as we decrease ǫe the minimum injection energy of electrons Eγ,m also decreases. In the slow cooling regimes, it is Eγ,c that defines the peak energy in the electron synchrotron spectrum, which could be adjusted to the sub-MeV range by adopting a suitable fc value. The change of Eγ,m therefore mainly affects the calculated internal optical depth. By lowering ǫe, we check the parameter regime where the hadronic component becomes comparable. Since eletrons are much more efficient emitters than protons, the parameter regime for the hadronic component to be comparable to the leptonic component in the high energy regime is ǫe/ǫp ∼ me/mp < 10 −3.3 A similar conclusion has been drawn for the external shocks (Zhang & Mészáros 2001). In Fig.2, with ǫe = 10 −3, ǫB = 0.05 and ǫp = 0.849. In order to adjust Eγ,c to the sub-MeV range, fc = 50000 is needed. In order to match the observed MeV emission flux by electron synchrotron, a large energy budget is needed due to a small ǫe: Eiso = 10 56 ergs and Liso = 10 55 erg s−1. Such a large energy budget has been suggested before (Totani 1998), but afterglow observations and modeling in the pre-Swift era have generally disfavored such a possibility (Panaitescu & Kumar 2002). In the Swift era, however, a large afterglow kinetic energy for some GRBs is not ruled out. For example, the bright afterglow of GRB 061007 demands a huge kinetic energy if the afterglow is produced by isotropic external shocks (Mundell et al. 2007; Schady et al. 2007). Modeling some X-ray afterglows below the cooling frequency requires a low ǫB and/or a large afterglow kinetic energy at least for some GRBs (Zhang et al. 2007). We therefore still consider such a possibility. In Fig.2, the break energy in the photon energy spectrum due to the minimum injection energy of electrons is below the synchrotron self absorption energy. The break energies appear in the order of Eγ,m < Essa < Eγ,c in the synchrotron and IC electron spectra. The spectral index of the photon energy flux is 7/2 between Eγ,m and Essa, −(p− 3)/2 between Essa and Eγ,c, and −(p− 2)/2 above Eγ,c. We can see that in the TeV energy regime beyond the maximum electron synchrotron energy, the positron synchrotron emission from π+ decay becomes dominant. Moreover, when ǫe is small, Ye is small, hence Yp becomes small. In this case the proton synchrotron component becomes comparable to the spectral components due to synchrotron radiation of the secondary positrons and π0 decays. The internal optical depth is plotted in Fig.2b, which peaks at a higher energy than that in Fig.1b. If the post shock magnetic field does not decay within a short distance (fc = 1), internal shocks are in the standard fast-cooling regime. We calculate such a case in Fig.3. The shock parameters are ǫe = 0.6, ǫB = 0.2, Liso = 10 52 erg s−1, Eiso = 10 53 erg. In this case the break energies appear as in the order of EC < Essa < Em. The photon energy spectral indices are 13/8, 1/2 and −(p− 2)/2, respectively, in the three energy regimes. The pair opacity depends on the bulk Lorentz factor. When Γ is large enough, the ultra-high energy photons would have lower 3 Proton energy loss and their contribution to high energy photon emission in the early afterglow phase has been studied earlier by Pe’er & Waxman (2005). Our results for the prompt emission phase are generally consistent with them. In order for the proton synchrotron component to be significant, even smaller ǫe (than 10−3) is demanded. Considering that photon-pion emission is more efficient than proton synchrotron emission, the condition ǫe/ǫp ∼ me/mp < 10 −3 can allow the hadronic components to be comparable to (but not dominant over) the leptonic components. c© 2007 RAS, MNRAS 000, 1–?? 12 Nayantara Gupta and Bing Zhang Figure 1. A leptonic-component-dominated slow cooling spectrum. (a) The different components of the photon energy spectrum from the internal shocks for the following parameters in the slow-cooling regime: Eiso = 10 53erg, Liso = 10 52erg/s, tv = 0.01s and fc = 2500. The thick solid black curve represents the final spectrum after including the effect of internal optical depths. The thin solid black curve represents the synthesized spectrum before including the effect of internal opical depths. The long dashed (green) curve is the electron synchrotron component; the short dashed (blue) curve is the electron IC component; the double short dashed (black) curve on the right side is for π0 decay component; the triple short dashed (orange) line represents the synchrotron radiation produced by positrons generated in π+ decays; the dash-dotted (light blue) line represents the proton synchrotron component. The tiny red horizontal line between 108 and 1011eV represents GLAST’s threshold. The pink dotted horizontal line above 2 × 1011eV represents the sensitivity of VERITAS experiment (b) Internal optical depths plotted against energy for the parameters adopted in (a). internal optical depth and may escape from the internal shocks (Razzaque et al. 2004). To test this, in Fig.4, we re-calculate with the parameter set for Fig.1, but increase Γ to 1000. The slow-cooling parameter fc is adjusted to 50 to maintain the sub-MeV energy peak. The results indeed suggest that the attenuation of the high energy photons is weaker. The observational breakthough in 2005 suggests that at least some short GRBs are low-fluence, nearby events that have a distinct progenitor than long GRBs (Gehrels et al. 2005; Bloom et al. 2006; Fox et al. 2005; Villasenor et al. 2005; Barthelmy et al. 2005; Berger et al. 2005). To check the prospect of detecting short GRB prompt emission with high energy detectors such as GLAST, we perform a calculation for the parameters of a short GRB in Fig.5. Due to their short durations, short GRB detections are favorable for high luminosity and relatively “long durations”. We therefore take an optimistic set of parameters with Liso = 10 51 erg s−1, T90 = 1 s, and z = 0.1. Other parameters include: Γ = 800, tv = 1 ms, ǫe = 0.4, ǫB = 0.2, ǫp = 0.4, fc = 50. The photon flux from synchrotron radiation of electrons peaks at 0.1MeV. Fig.5a suggests that the high energy component of such a burst is barely detectable by GLAST. The internal optical depth of this set of parameters does not grow to very large values (maximum 10), so that the attenuation signature is not significant in Fig.5a. The dip around several 1013 eV corresponds to the optical depth peak, above which the attenuated flux starts to rise. The abrupt drop at several 1014 eV corresponds to the disappearance of the electron IC component at high energies. c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 13 Figure 2. A slow-cooling spectrum with significant hadronic contribution (a) The spectra of various components. Parameters: ǫe = 10−3, ǫB = 0.05, ǫp = 0.849, tv = 0.01s, fc = 50000, Eiso = 10 56erg and Liso = 1055erg/s. Same line styles have been used as in Fig.1. (b) The corresponding internal optical depths. 10 CONCLUSIONS AND DISCUSSION We have calculated the broad-band spectrum of GRBs from internal shocks for a wide range of parameter regimes. We did not take into account the external attenuation of TeV photons by the infrared radiation background and that of the PeV photons by the cosmic microwave background. These external processes would further attenuate our calculated spectrum in high energy regimes, and reprocess the energy to delayed diffuse emission (Dai & Lu 2002; Stecker 2003; Wang et al. 2004; Razzaque et al. 2004; Casanova et al 2007; Murase et al. 2007). Such processes are not relevant for most of the calculations presented, however, since the internal attenuation already cuts the observed spectrum below TeV. They are however important for high Lorentz factor cases in which more high energy photons are leaked out of the internal shock region. The external attenuation is also prominant for high energy emission from the external reverse/forward shocks and the external IC processes related to X-ray flares. These processes have been extensively discussed in other papers (referenced in Introduction) and they are not discussed in this paper. For nearby GRBs (e.g. z < 0.3), TeV emission is transparent. It is possible that ground-based Cherenkov detectors such as VERITAS, Milagro would detect TeV gamma-rays from nearby energetic GRBs. In previous treatments of hadronic components from internal shocks (Fragile et al. 2004; Bhattacharjee & Gupta 2003), the shock accelerated protons are assumed to carry mp/me times more energy than electrons. This effectively fixed ǫe ∼ me/mp, which is not justified from the first principle. In this paper we have taken all the equipartition parameters ǫe, ǫp and ǫB as free parameters, and explore the relative importance of various components in different parameter regimes. The dominant hadronic c© 2007 RAS, MNRAS 000, 1–?? 14 Nayantara Gupta and Bing Zhang Figure 3. A leptonic-component-dominated fast-cooling spectrum. (a) The spectra of various components. Parameters: ǫe = 0.6, ǫB = 0.2, ǫp = 0.2, tv = 0.01s, fc = 1, Eiso = 1053erg and Liso = 1052erg/s. Same line styles have been used as in Fig.1. (b) The corresponding internal optical depths. component emission becomes interesting only when ǫe is extremely small. Given the same observed level of sub-MeV spectrum, the total energy budget of the GRB needs to be very large. Inspecting the calculated spectra for different parameter sets (Figs.1-4), one finds that there is no clean picture to test the leptonic vs. hadronic origin of the gamma-rays. Such an issue may be however addressed by collecting both prompt and afterglow data. A moderate-to-high radiative efficiency would suggest a leptonic origin of high energy photons, while a GRB with an extremely low radiative efficiency but an extended high energy emission component would be consistent with (but not a proof for) the hadronic origin. The prompt emission produced by leptons including the effect of pair production has been discussed by Pe’er & Waxman (2004); Pe’er et al. (2006). They calculated the emergent photon spectra for GRBs located at z = 1. The lower cut-off energy in the photon flux produced by leptons is determined by the synchrotron self absorption energy, the minimum injection energy or the cooling energy depending on the values of the various GRB parameters. Our leptonic-component-dominated cases are consistent with their results, although we do not explore cases with very high compactness. If the electrons cool down to trans-realtivistic energies then their high energy spectrum significantly deviates from broken power law (Pe’er et al. 2006, 2005). For our choice of values of the GRB parameters this effect is not important. Razzaque et al. (2004) estimated the internal optical depth for pair production and showed that at PeV energies the optical depth decreases with increasing photon energies. We have rederived the optical depths for values of GRB parameters. The results are generally consistent with (Razzaque et al. 2004) except that the growth of optical depth with increasing energy is more gradual before the optical depth peak. This is a result of including the whole low c© 2007 RAS, MNRAS 000, 1–?? Prompt Emission of High Energy Photons from Gamma Ray Bursts 15 Figure 4. The case of a higher Lorentz factor. (a) The spectra of various components. Parameters: Γ = 1000 and fc = 50. All the other parameters are the same as in Fig.1. (b) The corresponding internal optical depths. energy photon spectrum (rather than the threshold energy photons) for calculating the pair production optical depth. The optical depths depend on the cross section of γγ interactions, the low energy photon spectra, the various break energies in those spectra, luminosities, variability times and the GRB Lorentz factors. A change in values of any of these parameters may affect the values of the optical depths at various energies. For high bulk Lorentz factors, the π0 component may appear in the final spectra due to the reduced optical depths around PeV energy. However, these ultra-high energy photons will be immediately absorbed in the GRB neighborhood by cosmic microwave photons (Stecker 2003). The reradiated energy by the e+e− pairs would nonetheless contribute to the diffuse high energy γ-ray background (Casanova et al 2007). Upcoming γ-ray detectors have a good chance of detecting prompt emission from GRBs and reveal their physical nature during the prompt phase. Detection of the hadronic components is difficult but it would be possible to infer the dominance of these components by a coordinated broadband observational campaign if they are indeed important. More generally, detection or non detection of high energy photons in the prompt phase would constrain the values of various GRB parameters. In particular, the pair attenuation feature would help to constrain the bulk Lorentz factor of the fireball. Compared with EGRET, GLAST has a 10 times larger collecting area and a larger field of view. It is expected that GLAST LAT would detect high energy emission from a large number of bursts (mostly long GRBs and some bright, relatively “long” short GRBs), which will open a new era of studying GRBs in the GeV-TeV regime. On the other hand, it is difficult for VERITAS to detect prompt high energy gamma-rays even under the most optimistic conditions. High energy emissions from the external shock at the early afterglow phase for nearby GRBs may be c© 2007 RAS, MNRAS 000, 1–?? 16 Nayantara Gupta and Bing Zhang Figure 5. An energetic short GRB. (a) The spectra of various components. Parameters: ǫe = 0.4, ǫB = 0.2, ǫp = 0.4, tv = 0.001s, Γ = 800, fc = 50, Eiso = 1051erg and Liso = 1051erg/s. Same line styles have been used as in Fig.1. (b) The corresponding internal optical depths. the better targets for VERITAS and other TeV detectors. We thank Brenda Dingus, Deirdre Horan, Enwei Liang, Peter Mészáros, Jay Norris, Asaf Pe’er, Soeb Razzaque, and Dave Thompson for useful discussion/comments and/or technical support. We also thank the anonymous referee for a detailed report with important suggestions and comments. This work is supported by NASA under grants NNG05GC22G, NNG06GH62G and NNX07AJ66G. 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0704.1330

On the classification of Floer-type theories

To my son Philippe for his unbounded energy and optimism. On the classification of Floer-type theories. Nadya Shirokova. Abstract In this paper we outline a program for the classification of Floer-type theories, (or defining invariants of finite type for families). We consider Khovanov complexes as a local system on the space of knots introduced by V. Vassiliev and construct the wall-crossing morphism. We extend this system to the singular locus by the cone of this morphism and introduce the definition of the local system of finite type. This program can be further generalized to the manifolds of dimension 3 and 4 [S2], [S3]. http://arxiv.org/abs/0704.1330v1 Contents 1. Introduction. 2. Vassiliev’s and Hatcher’s theories. 2.1. The space of knots, coorientation. 2.2. Vassiliev derivative. 2.3. The topology of the chambers of the space of knots. 3. Khovanov homology. 3.1. Jones polynomial as Euler characterictics. Skein relation. 3.2. Reidemeister and Jacobsson moves. 3.3. Wall-crossing morphisms. 3.4. The local system of Khovanov complexes on the space of knots. 4. Main definition, invariants of finite type for families. 4.1. Some homological algebra. 4.2. Space of knots and the classifying space of the category. 4.3. Vassiliev derivative as a cone of the wall-crossing morphism. 4.4. The definition of a theory of finite type. 5. Theories of finite type. Further directions. 5.1. Examples of combinatorially defined theories. 5.2. Generalizations to dimension 3 and 4. 5.3. Further directions. 6. Bibliography. 1. Introduction. Lately there has been a lot of interest in various categorifications of classical scalar invariants, i.e. homological theories, Euler characteristics of which are scalar invariants. Such examples include the original instanton Floer homology, Euler characteristic of which, as it was proved by C.Taubes [T], is Casson’s invariant. Ozsvath-Szabo [OS] 3-manifold theory categorifies Turaev’s torsion, the Euler characteristic of their knot homologies [OS] is the Alexander polynomial. The theory of M. Khovanov categorifies the Jones polynomial [Kh] and Khovanov-Rozhansky theory categorifies the sl(n) invariants [KR] . The theory that we are constructing will bring together theories of V. Vassiliev, A. Hatcher and M. Khovanov, and while describing their results we will specify which parts of their con- structions will be important to us. The resulting theory can be considered as a ”categorification of Vassiliev theory” or a clas- sification of categorifications of knot invariants. We introduce the definition of a theory of finite type n and show that Khovanov homology theory in a categorical sense decomposes into a ”Taylor series” of theories of finite type. The Khovanov functor is just the first example of a theory satisfying our axioms and we believe, that all theories mentioned above will fit into our template. Our main strategy is to consider a knot homology theory as a local system, or a constructible sheaf on the space of all objects (knots, including singular ones), extend this local system to the singular locus and introduce the analogue of the ”Vassiliev derivative” for categorifications. By studying spaces of embedded manifolds we implicitly study their diffeomorphism groups and invariants of finite type. In his seminal paper [V] Vassiliev introduced finite type invariants by considering the space of all immersions of S1 into R3 and relating the topology of the singular locus to the topology of its complement via Alexander duality. He resolved and cooriented the discriminant of the space and introduced a spectral sequence with a filtration, which suggested the simple geometrical and combinatorial definition of an invariant of finite type, which was later interpreted by Birman and Lin as a ”Vassiliev derivative” and led to the following skein relation. If λ be an arbitrary invariant of oriented knots in oriented space with values in some Abelian group A. Extend λ to be an invariant of 1-singular knots (knots that may have a single singularity that locally looks like a double point ), using the formula λ( ) = λ(!) − λ(") Further extend λ to the set of n-singular knots (knots with n double points) by repeatedly using the skein relation. Definition We say that λ is of type n if its extension to (n + 1)-singular knots vanishes identically. We say that λ is of finite type if it is of type n for some n. Given the above formula, the definition of an invariant of finite type n becomes similar to that of a polynomial: its (n+1)st Vassiliev derivative is zero. It was shown that all known invariants are either of finite type, or are infinite linear combi- nations of those, e.g. in [BN1] it was shown that the nth coeffitient of the Conway polynomial is a Vassiliev invariant of order ≤ n. In this paper we are working with Khovanov homology, which will be our main example, however the latest progress in finding the combinatorial formula for the differential of the Ozsvath-Szabo knot complex [MOS], makes us hopeful that more and more examples will be coming. For the construction of the local system it is important to understand the topological type of the base. The topology of the connected components of the complement to the discriminant in the space of knots, called chambers, was studied by A. Hatcher and R.Budney [H], [B]. They introduced simple homotopical models for such spaces. Recall that the local system is well-defined on a homotopy model of the base, so Hatcher’s model is exactly what is needed to construct the local system of Khovanov complexes. Throughout the paper the following observation is the main guideline for our constructions: local systems of the classifying space of the category are functors from this category to the triangulated category of complexes. It would be very interesting to understand the relation between the Vassiliev space of knots is the classifying space of the category, whose objects are knots and whose morphisms are knot cobordisms. Our construction provides a Khovanov functor from the category of knots into the triangu- lated category of complexes. This allows us to translate all topological properties of the space of knots and Khovanov local system on it into the language of homological algebra and then use the methods of triangulated categories and homological algebra to assign algebraic objects to topological ones (singular knots and links). Recall that in his paper [Kh] M.Khovanov categorified the Jones polynomial, i.e. he found a homology theory, the Euler characteristics of which equals the Jones polynomial. He starts with a diagram of the knot and constructs a bigraded complex, associated to this diagram, using two resolutions of the knot crossing: 0-resolution 1-resolution The Khovanov complex then becomes the sum of the tensor products of the vector space V, where the homological degree is given by the number of 1’s in the complete resolution of the knot. The local system of Khovanov homologies on the Vassiliev’s space of knots can be considered as invariants of families of knots. The discriminant of Vassiliev’s space corresponds to knots with transversal self-intersection, i.e. moving from one chamber to another we change overcrossing to undercrossing by passing through a knot with a single double point. We study how the Khovanov complex changes under such modification and find the corresponding morphism. After defining a wall-crossing morphism we can extend the invariant to the singular locus by the cone of a morphism which is our ”categorification of the Vassiliev derivative”. Then we introduce the definition of a local system of finite type: the local system is of finite type n if for any selfintersection of the discriminant of codimension n, its n’th cone is an acyclic complex. The categorification of the Vassiliev derivative allows us to define the filtration on the Floer - type theories for manifolds of any dimension. In [S4] we prove the first finiteness result: Theorem [S4]. Restricted to the subcategory of knots with at most n crossing, Khovanov local system is of finite type n, n ≥ 3 and of type zero n = 0, 1, 2. This definition can be generalized to the categorifications of the invariants of manifolds of any dimension: we construct spaces of 3 and 4-manifolds by a version of a Pontryagin-Thom construction, consider homological invariants of 3 and 4-manifolds as local systems on these spaces and extend them to the discriminant. In subsequent papers our main example will be the Heegaard Floer homology [OS], the Euler characteristic of which is Turaev’s torsion. We show that local systems of such homological theories on the space of 3 - manifolds [S1] will carry information about invariants of finite type for families and information about the diffeomorphism group. We also have a construction [S2] for the refined Seiberg-Witten invariants on the space of parallelizable 4-manifolds. Acknowledgements. My deepest thanks go to Yasha Eliashberg for many valuable discus- sions, for inspiration and for his constant encouragement and support. I want to thank Maxim Kontsevich who suggested that I work on this project, for his attention to my work during my visit to the IHES and for many important suggestions. I want to thank graduate students Eric Schoenfeld and Isidora Milin for reading the paper and making useful comments. This paper was written during my visits to the IAS, IHES, MPIM and Stanford and I am grateful to these institutions for their exceptional hospitality. This work was partially supported by the NSF grant DMS9729992. 2. Vassiliev theory, invariants of finite type. 2.1. The space of knots, coorientation Vassiliev considered the space of all maps E = f : S1 → R3. This space is a space of functions, so it is an infinite-dimensional Euclidean space. It is linear, contractible, and consists of singular (D) and nonsingular(E - D) knots. The discriminant D forms a singular hypersurface in E and subdivides into chambers, corresponding to different isotopy types of knots. To move from one chamber to another one has to change one overcrossing to undercrossing, passing through a singular knot with one double point. The discriminant of the space of knots is a real hypersurface, stratified by the number of the double points, which subdivides the infinite-dimensional space into chambers, corresponding to different isotopy types of knots. Vassiliev resolved and cooriented the discriminant, so we can assume that all points of selfintersection are transversal, with 2n chambers adjacent to a point of selfintersection of the discriminant of codimension n. To study the topology of the complement to the discriminant, Vassiliev wrote a spectral sequence, calculating the homology of the discriminant and then related it to the homology of its complement via Alexander duality. His spectral sequence had a filtration, which suggested the simple geometrical and combinatorial definition of an invariant of finite type: an invariant is of type n if for any selfintersection of the discriminant of codimension (n+1) its alternated sum over the 2n+1 chambers adjacent to a point of selfintersection is zero. For our constructions it will be very important to have a coorientation of the discriminant, which was introduced by Vassiliev. Definition. A hypersurface in a real manifold is said to be coorientable if it has a non-zero section of its normal bundle, i.e. if there exists a continuous vector field which is not tangent to the hypersurface at any point and doesn’t vanish anywhere. So there are two sides of the hypersurface : one where this vector field is pointing to and the other is where it is pointing from. And there are two choices of such vector field. The coorientation of a coorientable hypersurface is the choice of one of two possibilities. For example, Mobius band in R3 is not coorientable. Vassiliev shows [V] that the discriminant of the space of knots has a coorientation, the conistent choice of normal directions. Recall that the nonsingular point ψ ∈ D of the discriminant is a map S1 → R3, gluing to- gether 2 distinct points t1, t2 of S 1, s.t. derivatives of the map ψ at those points are transversal. Coorientation of the discriminant. Fix the orientation of R3 and choose positively oriented local coordinates near the point ψ(t1) = ψ(t2). For any point ψ1 ∈ D close to ψ define the number r(ψ1) as the determinant: |t2, ψ1(t1)− ψ1(t2)) with respect to these coordinates. This determinant depends only of the pair of points t1, t2, not on their order. A vector in the space of functions at the point ψ ∈ D, which is transversal to the discriminant, is said to be positive, if the derivative of the function r along this vector is positive and negative, if this derivative is negative. This rule gives the coorientation of the hypersurface D at all its nonsingular points and also of any nonsingular locally irreducible component of D at the points of selfintersection of D. The consistent choice of the normal directions of the walls of the discriminant will give the ”directions” of the cobordisms (which are embedded into E× I) between knots of the space E. Note. It is interesting to compare this construction with the result of E.Ghys [Gh], who introduced a metric on the space of knots and 3-manifolds.) 2.3. The topology of the chambers of the space of knots. The study of the topology of the chambers of the space of knots was started by A. Hatcher [H], who found a simple homotopy models for these spaces. The main result is based on an earlier theorem regarding the topology of the classifying space of diffeomorphisms of an irreducible 3-manifold with nonempty boundary. In the following theorem A. Hatcher and D. McCullough answered the question posed by M. Kontsevich [K], regarding the finiteness of the homotopy type of the classifying space of the group of diffeomorphisms [HaM]: Theorem [HaM]. Let M be an irreducible compact connected orientable 3-manifold with nonempty boundary. Then BDiff(M, rel∂) has the homotopy type of a finite aspherical CW- complex. The proof of this theorem uses the JSJ-decomposition of a 3-manifold. When applied to knot complements, the JSJ-decomposition defines a fundamental class of links in S3, the ”knot generating links” (KGL). A KGL is any (n + 1)-component link L = (L0, L1, · · · , Ln) whose complement is either Seifert fibred or atoroidal, such that the n-component sub-link (L1, L2, · · · , Ln) is the unlink. If the complement of a knot f contains an incompressible torus, then f can be represented as a ‘spliced knot’ f = J�L in unique way, where L is an (n + 1)-component KGL, and J = (J1, · · · , Jn) is an n-tuple of non-trivial long knots. The spliced knot J�L is obtained from L0 by a generalized satellite construction. For any knot there is a representation of a knot as an iterated splice knot of atoroidal and hyperbolic KGLs. The order of splicing determines the ”companionship tree” of f , Gf , and is a complete isotopy invariant of long knots. Given a knot f ∈ K, denote the path-component of K containing f by Kf . The topology of the chambers Kf was further studied by R. Budney The main result of his paper [Bu] is the computation of the homotopy type of Kf if f is a hyperbolically-spliced knot ie: f = J�L where L is a hyperbolic KGL. The combined results can be summarized in the following theorem: Theorem [Bu, H]. If f = J�L where L is an (n+ 1)-component hyperbolic KGL, then Kf ⋍ S SO2 ×Af Af is the maximal subgroup of BL such that induced action of Af on K n preserves i=1KLi . The restriction map Af → Diff(S 3, L0) → Diff(L0) is faithful, giving an embedding Af → SO2, and this is the action of Af on SO2. This result completes the computation of the homotopy-type of K since we have the prior results: H1 If f is the unknot, then Kf is contractible. H2 If f is a torus knot, then Kf ≃ S H3 If f is a hyperbolic knot, then Kf ⋍ S 1 × S1 H4 If a knot f is a cabling of a knot g then Kf ⋍ S 1 ×Kg. B5 If the knot f is a connected sum of n ≥ 2 prime knots f1, f2, · · · , fn then Kf ⋍ ((C2(n)× i=1Kfi) /Σf . Here Σf ⊂ Sn is a Young subgroup of Sn, acting on C2(n) by permutation of the labellings of the cubes, and similarly by permuting the factors of the product i=1Kfi . The definition of Σf ⊂ Sn is that it is the subgroup of Sn that preserves a partition of {1, 2, · · · , n}, the partition being given by the equivalence relation i ∼ j ⇐⇒ Kfi = Kfj . B6 If a knot has a non-trivial companionship tree, then it is either a cable, in which case H4 applies, a connect-sum, in which case B5 applies or is hyperbolically spliced. If a knot has a trivial companionship tree, it is either the unknot, in which case H1 applies, or a torus knot in which case H2 applies, or a hyperbolic knot, in which case H3 applies. Moreover, every time one applies one of the above theorems, one reduces the problem of computing the homotopy-type of Kf to computing the homotopy-type of knot spaces for knots with shorter companionship trees, thus the process terminates after finitely-many iterations. For constructing a local system we need only the homotopy type of the chamber. The theorem of Hatcher and Budney provides us with a complete classification of homotopy types of chambers, corresponding to all possible knot types. 3. Khovanov’s categorification of Jones polynomial. 3.1. Jones polynomial as Euler characterictics. Skein relation. In his paper [Kh] M. Khovanov constructs a homology theory, with Euler characteristics equal to the Jones polynomial. He associated to any diagram D of an oriented link with n crossing points a chain complex CKh(D) of abelian groups of homological length (n+1), and proved that for any two diagrams of the same link the corresponding complexes are chain homotopy equivalent. Hence, the homology groups Kh(D) are link invariants up to isomorphism. His construction is as follows: given any double point of the link projection D, he allows two smoothings: 0-resolution 1-resolution If the the diagram has n double points, there are 2n possible resolutions. The result of each complete smoothing is the set of circles in the plane, labled by n-tuples of 1’s and 0’s: CKh(©, ...,© ︸ ︷︷ ︸ ntimes ) = V ⊗n The cobordisms between links, i.e., surfaces embedded in R3 × [0, 1], should provide maps between the associated groups. A surface embedded in the 4-space can be visualized as a sequence of plane projections of its 3-dimensional sections (see [CS]). Given such a presentation J of a compact oriented surface S properly embedded in R3 × [0, 1] with the boundary of S being the union of two links L0 ⊂ R 3 × {0} and L1 ⊂ R 3 × {1}, , Khovanov associates to J a map of cohomology groups θJ : Kh i,j(D0) → Kh i,j+χ(S)(D1), i, j ∈ Z The differential of the Khovanov complex is defined using two linear maps m : V ⊗ V → V and ∆ : V → V ⊗ V given by formulas : V ⊗ V v+ ⊗ v− 7→ v− v+ ⊗ v+ 7→ v+ v− ⊗ v+ 7→ v− v− ⊗ v− 7→ 0 → V ⊗ V v+ 7→ v+ ⊗ v− + v− ⊗ v+ v− 7→ v− ⊗ v− The differential in Khovanov complex can be informally described as ”all the ways of changing 0-crossing to 1-crossing”. Homological degree of the Khovanov complex in the number of 1’s in the plane diagram resolution. The sum of ”quantum” components of the same homological degree i gives the ith component of the Khovanov complex. One can see that the i-th differential di is the sum over ”quantum” components, it will map one of the quantum components in homological degree i to perhaps several quantum components of homological degree i+1. Khovanov theory can be considered as a (1+1) dimensional TQFT. The cubes, that are used in it’s definition come from the TQFT corresponding to the Frobenius algebra defined by V,m,∆. As we will see later, our constructions will give the interpretation of Khovanov local system as a topological D-brane and will suggest to study the structure of the category of topological D-branes as a triangulated category. We prove the following important property of the Khovanov’s complex: Theorem 1. Let k denote the kth crossing point of the knot projection D, then for any k the Khovanov’s complex C decomposes into a sum of two subcomplexes C = Ck0 ⊕ C 1 with matrix differential of the form d0 d0,1 Proof. Let Ck0 denote the subcomplex of C, consisting of vector spaces, which correspond to the complete resolutions of D, having 0 on the kth place. The differential d0 obtained by restricting d only to the arrows between components of Ck0 . We define C 1 the same way, by restricting to the complete resolutions of D, having 1 on the kth place. The only components of the differential, which are not yet used in our decomposition, are the ones which change 0-resolution on the kth place of Ck0 to 1 on the kth place in C 1 , we denote them d0,1. One can easily see from the definition of the Khovanov’s differential (which can be intuitivly described as ”all the ways to change 0-resolution in the ith component of the complex to the 1-resolution in the (i+1)st component”), that there is no differential mapping ith component of Ck1 to the (i+1)st component of C Mirror images and adjoints. Taking the mirror image of the knot will dualize Khovanov complex. So if we want to invert the cobordism between two knots, we should consider the ”dual” cobordism between mirror images of these knots. 3.2. Reidemeister and Jacobsson moves. A cobordism (a surface S embedded into R3 × [0, 1]) between knots K0 and K1 provide a morphism between the corresponding cohomology: FS : Kh i,j(D0) → Kh i,j+χ(S)(D1) where D0 and D1 are diagrams of the knots K0 and K1 and χ(S) is the Euler characteristic of the surface. We will distinguish between two types of cobordisms - first, corresponding to the wall crossing (and changing the type of the knot). And second, corresponding to nontrivial loops in chambers which will reflect the dependence of Khovanov homologies on the selfdiffeomorphisms of the knot, similar to the Reidemeister moves. In this paragraph we will discuss the second type of cobordisms. By a surface S in R4 we mean an oriented, compact surface S, possibly with boundary, properly embedded in R3 × [0, 1]. The boundary of S is then a disjoint union ∂S = ∂0S ⊔ −∂1S of the intersections of S with two boundary components of R3 × [0, 1]: ∂0S = (S ∩R 3 × {0}) −∂1S = (S ∩R 3 × {1}) Note that ∂0S and ∂1S are oriented links in R The surface S can be represented by a sequence J of plane diagrams of oriented links where every two consecutive diagrams in J are related either by one of the four Reidemeister moves or by one of the four moves birth, death, fusion described by Carter-Saito [CS]. To each Reidemeister move between diagrams D0 and D1 Khovanov [Kh] associates a quasi- isomorphism map of complexes C(D0) → C(D1). Given a representation J of a surface S by a sequence of diagrams, we can associate to J a map of complexes ϕJ : C(J0) → C(J1) Any link cobordism can be described as a one-parameter family Dt, t ∈ [0, 1] of planar dia- grams, called a movie. The Dt are link diagrams, except at finitely many singular points which correspond to either a Reidemeister move or a Morse modification. Away from these points the diagrams for various t are locally isotopic . Khovanov explained how local moves induce chain maps between complexes, hence homomorphisms between homology groups. The same is true for planar isotopies. Hence, the composition of these chain maps defines a homomorphism between the homology groups of the diagrams of links. In his paper [Ja] Jacobsson shows that there are knots, s.t. a movie as above will give a nontrivial morphism of Khovanov homology: Theorem [Ja] For oriented links L0 and L1, presented by diagrams D0 and D1, an oriented link cobordism Σ from L0 to L1, defines a homomorphism H(D0) → H(D1), invariant up to multiplication by -1 under ambient isotopy of Σ leaving ∂Σ setwise fixed. Moreover, this invariant is non-trivial. Jacobsson constructs a family of derived invariants of link cobordisms with the same source and target, which are analogous to the classical Lefschetz numbers of endomorphisms of man- ifolds. The Jones polynomial appears as the Lefschetz polynomial of the identity cobordism. From our perspective the Jacobsson’s theorem shows that the Khovanov local system will have nontrivial monodromies on the chambers of the space of knots. 3.3. Wall-crossing morphisms. In 3.2 we described what kind of modifications can occur in the cobordism, when we consider the ”movie” consisting only of manifolds of the same topological type. These modifications implied corresponding monodromies of the Khovanov complex. However, morphisms that are the most important for Vassiliev-type theories are the ”wall- crossing” morphisms. We will define them now (locally). Consider two complexes A• and B• adjacent to the generic wall of the discriminant. Recall, that the discriminant is cooriented ( 2.2). If B• is ”right” via coorientation (or ”further in the Ghys metric form the unknot) of A•, then we shift B•’s grading up by one and consider B•[1]: A•|B•[1] Note. In general, and this will be very important for us in subsequent chapters, if the complex K• is n steps (via the coorientation) away from the unknot, we shift its grading up by n. Thus adjacent complexes will have difference in grading by one (as above), defined by the coorientation. Now we want to understand what happens to the Khovanov complex when we change the kth over-crossing (in the knot diagram D) to an under-crossing. We will illustrate these changes on one of the Bar-Natan’s trademark diagrams (with his permission)[BN1]. By ”I” we mark the arrow , connecting components of the complex which will exchange places under wall-crossing morphisms when we change over-crossing to under-crossing for the self-intersection point 1. By ”II” when we do it for point 2 and ”III” when we do it for 3: 2 V {1} V ⊗2{2} ==zzzzzzzzzzzzzzzzzz V {1} <<yyyyyyyyyyyyyyyyyy V ⊗2{2} V ⊗3{3} V {1} wwwwwwww ;;wwwwwwww V ⊗2{2} ;;vvvvvvvvvvvvvvvvv d0 // J&K1 d1 // J&K2 d2 // J&K3 (0.1) Now recall the theorem proved in (3.1): for any k, where k is the number of crossings of the diagram D, the Khovanov complex can be split into the sum of two subcomplexes with the uppertriangular differential. Notice from the diagram above that when we change kth overcrossing to an undercrossing, 0 and 1-resolutions are exchanged , so A• = A•0 ⊕ A •[1] = B•0 [1] ⊕ B 1 [1], thus for every k we can define the wall-crossing morphism ω as follows: Theorem 2. The map defined as the identity on A•0 and as a trivial map on A ω : A•0 Id // B•0 [1] ω : A•1 ∅ // B•1 [1] is the morphism of complexes. Proof. From the Theorem 1 we know that for any crossing k the Khovanov complex can be decomposed as a direct sum with uppertriangular differential: d0 d0,1 It is an easy check that the wall-crossing morphism defined as above is indeed a morphism of complexes (i.e. it commutes with the differential): ω // B• ω // B• Since we defined the morphism as 0 on A•1, the diagram above becomes the following com- mutative diagram: Id // B•0 [1] Id // B•0 [1] 3.4. The local system of Khovanov complexes on the space of knots. In this paragraph we introduce the Khovanov local system on the space of knots. Definition. A local system on the locally connected topological space M is a fiber bundle over M, the sections of which are abelian groups. The fiber of the bundle depend continuously on the point of the base (such that the group structure on the set of fibers can be extended over small domains in the base). Any local system on M with fiber A defines a representation π1(M) → Aut(A). To any loop there corresponds a morphism of the fibers of the bundle over the starting point of the loop. The set of isomorphism classes of local systems with fiber A are in one-to-one corresondence with the set of such representations up to conjugation. For example any representation of an arbitrary group π in Aut(A) uniquely (up to isomorphism) defines a local system on the space K(π, 1) [GM]. Morphisms of local systems are morphisms of fiber bundles, preserving group structure in the fibers. Thus introducing the continuation functions (maps between fibers) over paths in the base will define a local system over the manifold M. Next we set up the Khovanov complexes as a local system on the space of knots. If we were doing it ”in coordinates”, we would introduce charts on the chambers of the space of knots and define our local system via transition maps, starting with some ”initial” point . This would be a very interesting and realistic approach, since the homotopy models for chambers are understood [H], [B], e.g. we would have just one chart for the chamber, containing the unknot (since that chamber is contractible), two for a torus knot, four for a hyperbolic one, etc. Then monodromies of the Khovanov local system along nontrivial loops in the chamber will be given via Jacobsson movies. It would be also very interesting to find a unique special point in every chamber of the space E and study monodromies of the local system with respect to this point. The candidate for such point is introduced in the works of J. O’Hara, who studied the minima of the electrostatical energy function of the knot [O’H]: E(K) = |(x− y)|−2dxdy It was shown that under some assumptions and for perturbation of the above functional, its critical points on the space of knots will provide a ’distinguished” point in the chamber. The first natural question for this setup is: which nontrivial loops in the chamber EK , corresponding to the knot K are distinguised by Khovanov homologies and which are not? However, assuming Khovanov’s theorem [Kh] (that his homology groups are invariants of the knot, independent on the choices made) and assuming also the results of Jacobsson [J] , it is enough for us to introduce the continuation maps, along any path γ in the chamber of the space of knots. These methods were developed by several authors (see [Hu]): Let K1 and K2 be two knots in the same chamber of the space E, let Ki be generic, and let γ = {Kt | t ∈ [0, 1]} be any path of equivalent objects in E from K1 to K2. Then a generic path γ induces a chain map F (γ) : CKh∗(K1)−→CKh∗(K2) called the “continuation” map, which has the following properties: • 1)Homotopy A generic homotopy rel endpoints between two paths γ1 and γ2 with associated chain maps F1 and F2 induces a chain homotopy H : HKh∗(K1)−→HKh∗+1(K2) ∂H +H∂ = F1 − F2 • 2)Concatenation If the final endpoint of γ1 is the initial endpoint of γ2, then F (γ2γ1) is chain homotopic to F (γ2)F (γ1). • 3)Constant If γ is a constant path then F (γ) is the identity on chains. These three properties imply that ifK1 andK2 are equivalent, thenHKh∗(K1) ≃ HKh∗(K2). (Khovanov’s theorem). This isomorphism is generally not canonical, because different homotopy classes of paths may induce different continuation isomorphisms on Khovanov homology (Jacobsson moves). However, since the loop is contractible, we do know that HKh∗(K) depends only on K, so we denote this from now on by HKh∗(K). We now define the restriction of the Khovanov local system to finite-dimensional subspaces of the space of knots. Note that in the original setting our complexes may have had different length. For example, the complex corresponding to the standard projection of the unknot will have length 1, however, we can consider very complicated ”twisted” projections of the unknot with an arbitrary large number of crossing points. The corresponding complexes will be quasiisomorphic to the original This construction resembles the definition of Khovanov homology introduced in [CK], [W]. They define Khovanov homology as a relative theory, where homology groups are calculated relative to the twisted unknots. When considering the restrictions of the Khovanov local system to the subcategories of knots with at most n crossings, we would like all complexes to be of length n+ 1. This can be achieved by ”undoing” the local system, starting with the knots of maximal crossing number n and then using the wall-crossing morphisms, define complexes of length (n + 1), quasiisomorphic to the original ones, in all adjacent chambers. We continue this process till it ends, when we reach the chamber containing unknot. Recall that Khovanov homology is defined for the knot projection (though is independent of it by Khovanov’s theorem). So we will consider a ramification of Vassiliev space, a pair, the embedding of the circle into R3 and its projection on (x,y)-plane. Then each chamber will be subdivided into ”subchambers” corresponding to nonsingular knot projections and the ”sub- discriminant” will consist of singular projections of the given knot. The local system, defined on such ramification will live on the universal cover of the base, the original Hatcher chamber corresponding to knot K and morphisms of the local system between ”subchambers” are given by Reidemaister moves. The composition of such moves may constitute the Jacobsson’s movie and will give nontrivial monodromies of the local system within the original chamber. Note. As we will see later, if one assines cones of Reidemeister morphisms to the walls of the ”subdiscriminant”, all such cones will be acyclic complexes. This statement in a different form was proved in the original Khovanov [Kh] paper. 4. The main definition, invariants of finite type for families. 4.1. Some homological algebra. We describe results and main definitions from the category theory and homological algebra which will be used in subsequent chapters. The standard references on this subject are [GM], [Th]. By constructing the local system of (3.4) we introduced the derived category of Khovanov complexes. The properties of the derived category are summarized in the axiomatics of the triangulated category, which we will discuss in this chapter. Definition. An additive category is a category A such that • Each set of morphisms Hom(A,B) forms an abelian group. • Composition of morphisms distributes over the addition of morphisms given by the abelian group structure, i.e. f ◦ (g + h) = f ◦ g + f ◦ h and (f + g) ◦ h = f ◦ h+ g ◦ h. • There exist products (direct sums) A×B of any two objects A,B satisfying the usual universal properties. • There exists a zero object 0 such that Hom(0, 0) is the zero group (i.e. just the identity morphism). ThusHom(0, A) = 0 = Hom(A, 0) for all A, and the unique zero morphism between any two objects is the one that factors through the zero object. So in an abelian category we can talk about exact sequences and chain complexes, and cohomology of complexes. Additive functors between abelian categories are exact (respectively left or right exact) if they preserve exact sequences (respectively short exact sequences 0 → A→ B → C or A→ B → C → 0). Definition. The bounded derived category Db(A) of an abelian category A has as objects bounded (i.e. finite length) A-chain complexes, and morphisms given by chain maps with quasi- isomorphisms inverted as follows. We introduce morphisms f for every chain map between complexes f : Xf → Yf , and g −1 : Yg → Xg for every quasi-isomorphism g : Xg → Yg. Then form all products of these morphisms such that the range of one is the domain of the next. Finally identify any combination f1f2 with the composition f1 ◦ f2, and gg −1 and g−1g with the relevant identity maps idYg and idXg . Recall that a triangulated category C is an additive category equipped with the additional data: Definition. Triangulated category is an additive category with a functor T : X → X[1] (where Xi[1] = Xi+1) and a set of distinguished triangles satisfying a list of axioms. The triangles include, for all objects X of the category: 1) Identity morphism X → X → 0 → X[1], 2) Any morphism f : X → Y can be completed to a distinguished triangle X → Y → C → X[1], 3) There is also a derived analogue of the 5-lemma, and a compatibility of triangles known as the octahedral lemma, which can be understood as follows: If we naively interprete property 1) as the difference X −X = 0, property 2) as C = X −Y , then the octahedron lemma says: (X − Y )− Z = C − Z = X − (Y − Z) When topological spaces considered up to homotopy there is no notion of kernel or cokernel. The cylinder construction shows that any map f : X → Y is homotopic to an inclusion X → cyl (f) = Y ⊔ (X × [0, 1])/f(x) ∼ (x, 1), while the path space construction shows it is also homotopic to a fibration. The cone Cf on a map f : X → Y is the space formed from Y ⊔ (X × [0, 1]) by identifying X × {1} with its image f(X) ⊂ Y , and collapsing X × {0} to a point. It can be considered as a cokernel, i.e. if f : X → Y is an inclusion, then Cf is homotopy equivalent to Y/X. Taking the ith cohomologyHi of each term, and using the suspension isomorphismHi(ΣX) ∼= Hi−1(X) gives a sequence Hi(X) → Hi(Y ) → Hi(Y,X) → Hi−1(X) → Hi−1(Y ) → . . . which is just the long exact sequence associated to the pair X ⊂ Y . Up to homotopy we can make this into a sequence of simplicial maps, so that taking the associated chain complexes we get a lifting of the long exact sequence of homology to the level of complexes. It exists for all maps f , not just inclusions, with Y/X replaced by Cf . If f is a fibration, Cf can act as the “kernel” or fibre of the map. If f : X →point, then Cf = ΣX, the suspension of the fibre X. Thus Cf acts as a combination of both cokernel and kernel, and if f : X → Y is a map inducing an isomorphism of homology groups of simply connected spaces then the sequence Hi(X) → Hi(Y ) → Hi(Cf ) → Hi−1(X) → Hi−1(Y ) → . . . implies H∗(Cf )=0. Then Cf homotopy equivalent to a point. Thus we can give the following definition. Definition. If X and Y are simplicial complexes, then a simplicial map f : X → Y , defines (up to isomorphism) an object in triangulated category, called the cone of morphism f, denoted Cf . C•X ⊕ C Y [1] with differential dCf = 0 dY [1] where [n ] means shift a complex n places up. Thus we can define the cone Cf on any map of chain complexes f : A • → B• in an abelian category A by the above formula, replacing C•X by A • and C•Y by B •. If A• = A and B• = B are chain complexes concentrated in degree zero then Cf is the complex {A → B}. This has zeroth cohomology h0(Cf ) = ker f , and h 1(Cf ) = coker f , so combines the two (in different degrees). In general it is just the total complex of A• → B•. So what we get in a derived category is not kernels or cokernels, but “exact triangles” A• → B• → C• → A• [ 1 ]. Thus we have long exact sequences instead of short exact ones; taking ith cohomology hi of the above gives the standard long exact sequence hi(A•) → hi(B•) → hi(C•) → hi+1(A•) → . . . The cone will fit into a triangle: u // B The “[1]” denotes that the map w increases the grade of any object by one. 4.2. Space of knots as a classifying space of the category. In this paragraph we will construct the Khovanov functor from the category of knots into the triangulated category of Khovanov complexes. Definition. The category of knots K is the category, the objects of which are knots, S1 → S3, morphisms are cobordisms, i.e. surfaces Σ properly embedded in R3 × [0, 1] with the boundary of Σ being the union of two knots K1 ⊂ R 3 × {0} and K2 ⊂ R 3 × {1}. We denote Kn the subcategory of knots with at most n crossings. (Recall that a knot’s crossing number is the lowest number of crossings of any diagram of the knot. ) Note that our cobordisms (morphisms in the category of knots) are directed via the coori- entation of the discriminant of the space of knots. Note that to reverse cobordism, we can consider the same cobordism between mirror images of the knots. Definition. The nerveN (C) of a category C is a simplicial set constructed from the objects and morphisms of C, i.e. points of N (C) are objects of C, 1-simplices are morphisms of C, 2-simplices are commutative triangles, 3-simplices are commutative tetrahedrons of C, etc. N (C) = (limN i(C)) The geometric realization of a simplicial set N (C) is a topological space, called the classi- fying space of the category C, denoted B(C). The following observation is the main guideline for our constructions: sheaves on the classi- fying space of the category are functors on that category [Wi]. Once we prove that the Vassiliev space of knots is a classifying space of the category K, our local system will provide a representation of the Khovanov functor. Let C be a category and let Set be the category of sets. For each object A of C let Hom(A,) be the hom functor which maps objects X to the set Hom(A,X). Recall that a functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A, ) for some object A of C. A representation of F is a pair (A,Ψ) where Ψ : Hom(A, ) → F is a natural isomorphism. If E - the space of knots, denote KE the category of knots, whose objects are points in E and morphisms Mor(x, y) = {γ : [0, 1] → X; s.t.γ(0) = x, γ(1) = y} and KK - subcategory corresponding to knots of the same isotopy type K. Proposition. The chamber EK of the space of long knots forK - unknot, torus of hyperbolic knot is the classifying space of the category KK . Proof. By Hatcher’s theorem [H] the chambers of the space of knots EK , corresponding to unknot, torus or hyperbolic knot are K(π, 1). By definition the space of long knots is E = {f : R1 → R3}, nonsingular maps which are standard outside the ball of large radius. If f1, f2 are vector equations giving knots K1,K2, then tf1 + (1 − t)f2 is a path in the mapping space, defining a knot for each value of t. The cobordism between two embeddings is given by equations in R3× I. All higher cobordisms can be contracted, since there are no higher homotopy groups in EK . So both the classifying space of the category and the chamber of the space of knots are K(π, 1) with the same π. They are the same as simplicial complexes. Note, that in the case of hyperbolic knots one can choose the distinguished point in the chamber - corresponding to the hyperbolic metric on the complement to the knot. 4.3. Vassiliev derivative as a cone of the wall-crossing morphism. To be able to construct a categorification of Vassiliev theory, we have to extend the local system, which we defined on chambers, to the discriminant of the space of knots. Recall that according to the axiomatics of the triangulated category, described in (4.1), we assign an new object to every morphism in the category: for a complex X = (Xi, dix) define a complex X[1] by (X[1])i = Xi+1, dX[1] = −dX For a morphism of complexes f : X → Y let f [1] : X[1] → Y [1] coincide with f component- wise. Let f : X → Y be a wall-crossing morphism. The cone of f is the following complex C(f): X → Y → Z = C(f) → X[1] C(f)i = X[1]i ⊕ Y i, dC(f)(x i+1, yi) = (−dXx i+1, f(xi+1)− dY y Recall, that we set up the local system on the space of knots (3.4) s.t. if the complex X• is n steps (via the coorientation) away from the unknot, we shift its grading up by n. So complexes in adjacent chambers will have difference in grading by one, defined by the coorientation. Thus, given a bigraded complex, associated to the generic wall of the discriminant, we get two natural specialization maps into the neighbourhoods, containing X• and Y •: So with any morphism f we associate the triangle: // Y • aaBBBBBBBB With any commutative cube u // • u // • (in the space of knots the above picture corresponds to the cobordism around the self- intersection of the discriminant of codimension two), we associate the map between cones, corresponding to the vertical and horisontal walls, and assign it to the point of their intersec- tion: Cuω // Cω Lemma. Given four chambers as above, the order of taking cones of morphisms is irrelevant, Cuω = Cωu. Proof. see [GM]. Consider a point of selfintersection of the discriminant of codimension n. There are 2n chambers adjacent to this point. Since the discriminant was resolved by Vassiliev [V], this point can be considered as a point of transversal selfintersection of n hyperplanes in Rn, or an origin of the coordinate system of Rn. Now our local system looks as follows. On chambers of our space we have the local system of Khovanov complexes, to any point t of the generic wall between chambers containing X• and Y • (corresponding to a singular knot), we assign the cone of the morphism X• → Y • (with the specialization maps from the cone to the small neighborhoods of t containing X• and Y •). To the point of codimention n we assign the nth cone, 2n-graded complex, etc. Definition. The Khovanov homology of the singular knot (with a single double point ) is a bigraded complex X• ⊕ Y •[1] with the matrix differential dCω = 0 dY [1] where X• is Khovanov complex of the knot with overcrossing, Y • is the Khovanov complex of the knot with undercrossing and ω is the wall-crossing morphism. In [S4] we give the geometric interpretation of the above definition. 4.4. The definition of a theory of finite type. Once we extended the local system to the singular locus, it is natural to ask if such an extension will lead to the categorification of Vassiliev theory. The first natural guess is that the theory, set up on some space of objects, which has quasi- isomorphic complexes on all chambers is a theory of order zero. Such theory will consist of trivial distinguished triangles as in (a) of the axiomatics of the triangulated category. When complexes, corresponding to adjacent chambers are quasiisomorphic, the cone of the morphism is an acyclic complex. Baby example of a theory of order 0. Let M be an n-dimensional compact oriented smooth manifold. Consider the space of func- tions on M. This is an infinite-dimensional Euclidean space. The chambers of the space will correspond to Morse functions on M, the walls of the discriminant - to simple degenerations when two critical points collide, etc. Let’s consider the Morse complex, generated by the crit- ical points of a Morse function on M. As it was shown by many authors, such complex is isomorphic to the CW complex, associated with M. Since we are calculating the homology of M via various Morse functions, complexes may vary, but will have the same homology and Euler characteristics. Then we can proceed according to our philosophy and assign cones of morphisms to the walls and selfintersections of the discriminant. Since complexes on the chambers of the space of functions are quasiisomorphic, all cones are acyclic. Now we can introduce the main definition of a Floer-type theory being of finite type n: Main Definition. The local system of (Floer-type) complexes, extended to the discriminant of the space of manifolds via the cone of morphism, is a local system of order n if for any selfintersection of the discriminant of codimension (n + 1), its (n+1)st cone is an acyclic complex. How one shows that an 2n-graded complex is acyclic? For example, if one introduces inverse maps to the wall-crossing morphisms and construct the homotopy H, s.t.: dH−Hd = I It is easy to check that the existence of such homotopy H implies, that the complex doesn’t have homology. Suppose dc = 0, i.e. c is a cycle, then: dHc−Hdc = dHc = I Example. Suppose some local system is conjectured to be of finite type 3. How one would check this? By our definition, we should consider 23 chambers adjacent to the every point of selfintersection of the discriminant of codimension 3, and 8 complexes, representing the local system in the small neighbourhood of this point. This will correspond to the following commutative cube: }}}}}}}} g |||||||| l // G• }}}}}}}} m // H• |||||||| Let’s write the homotopy equation in the matrix form. Consider dual maps f∗, g∗, ..., w∗. Then we get formulas for d and H as 8× 8 matrices: dA f g 0 a 0 0 0 0 dB 0 h 0 b 0 0 0 1 dD w 0 0 e 0 0 0 1 dC 0 0 0 c 0 0 0 0 dE k m 0 0 0 0 0 1 dF 0 l 0 0 0 0 0 1 dH n 0 0 0 0 0 0 1 dG dA 0 0 1 0 0 0 0 f∗ dB 0 0 0 0 0 0 g∗ 0 dD 0 0 0 0 0 0 h∗ w∗ dC 0 0 0 0 a∗ 0 0 0 dE 0 0 1 0 b∗ 0 0 k∗ dF 0 0 0 0 e∗ 0 m∗ 0 dH 0 0 0 0 c∗ 0l∗ h∗ dG After substituting these matrices into the equation dH − Hd = I we obtain the diagonal matrix which must be homotopic to the identity matrix: ff∗ + gg∗ + aa∗ 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 −”− 0 0 0 0 0 0 0 0 cc∗ + nn∗ + ll∗ Thus the condition for the local to be of finite type n can be interpreted as follows. For any selfintersection of the discriminant of codimension n + 1 consider 2n complexes, forming a commutative cube (representatives of the local system in the chambers adjacent to the self- intersection point). Then the naive geometrical interpretation of the local system being of finite type n is the following: each complex can be ”split” into n+1 subcomplexes, which map quasiisomorphically to n+ 1 neighbours, at least no homologies die or being generated. 5. Knots: theories of finite type. Further directions. 5.1. Examples of combinatorially defined theories. In the following table we give the examples of theories, which are the categorifications of classical invariants. All these theories fit into our framework and may satisfy the finitness condition. λ λ = χH∗(M) Jones polynomial Khovanov homology [Kh] Alexander polynomial Ozsvath-Szabo knot homology [OS2] sl(n) invariants Khovanov - Rozhansky homology [KhR] Casson invariant Instanton Floer homology [F] Turaev’s torsion Ozsvath-Szabo 3 manifold theory [OS1] Vafa invariant Gukov-Witten categorification [GW] Note, that the only theory which is not combinatorially defined is the original Instanton Floer homology [F]. The fact that it’s Euler characteristics is Casson’s invariant was proved by C.Taubes [T]. 5.2. Generalization to dimension 3 and 4. In our paper [S1] we generalized Vassiliev’s construction to the case of 3-manifolds. In [S2] we construct the space of parallelizable 4-manifolds and consider the paramentrized version of the Refined Seiberg-Witten invariant [BF]. a). The space of 3-manifolds and invariants of finite type. Note that all 3-manifolds are parallelizable and therefore carry spin-structures. Following Vassiliev’s approach to classification of knots, we constructed spaces E1 and E2 of 3-manifolds by a version of the Pontryagin-Thom construction. Our main results are as follows: Theorem [S1]. In E1 − D each connected component corresponds to a homeomorphism class of 3-dimensional framed manifold. For any connected framed manifold as above there is one connected component of E1 −D giving its homeomorphism type. Theorem [S1]. In E2 − D each connected component corresponds to a homeomorphism class of 3-dimensional spin manifold. For any connected spin manifold there is one connected component of E2 −D giving its homeomorphism type. By a spin manifold we understand a pair (M,θ) where M is an oriented 3-manifold, and θ is a spin structure on M . Two spin manifolds (M,θ) and (M ′, θ′) are called homeomorphic, if there exists a homeomorphism M →M ′ taking θ to θ′. The construction of the space naturally leads to the following definition: Definition. A map I : (M,θ) → C is called a finite type invariant of (at most) order k if it satisfies the condition: (−1)#L I(ML′) = 0 where L′ is a framed sublink of link L with even framings, L corresponds to the self-intersection of the discriminant of codimension k+1, #L′ - the number of components of L′, ML′ - spin 3-manifold obtained by surgery on L′. We introduced an example of Vassiliev invariant of finite order. Given a spin 3-manifold M3 we consider the Euler characteristic of spin 0-cobordism W . Denote by I(M,spin) = (sgn(W, spin) − 1)(mod2). Theorem [S1]. Invariant I(M,spin) is finite type of order 1. The construction of the space of 3-manifolds chambers of which correspond to spin 3- manifolds is important for understanding, which additional structures one needs in order to build the theory of finite-type invariants for homologically nontrivial manifolds. It suggests that one should consider spin ramifications of known invariants. In the following paper we will generalize our constructions and the main definition to the case of 3-manifolds. We will construct a local system of Ozsvath-Szabo homologies, extend it to the singular locus via the cone of morphism and find examples of theories of finite type. b). Stably parallelizable 4-manifolds. In this section we modify the previous construction [S1] to get the space of parallelizable 4-manifolds. By the definition the manifold is parallelizable if it admits the global field of frames, i.e. has a trivial tangent bundle. In the case of 4-manifolds this condition is equivalent to vanishing of Euler and the second Stieffel-Whitney class. In particular signature and the Euler characterictic of such manifolds will be 0. We will use the theorem of Quinn: Theorem Any punctured 4-manifold posesses a smooth structure. Recall also the result of Vidussi, which states that manifolds diffeomorphic outside a point have the same Seiberg-Witten invariants,so one cannot use them to detect eventual inequivalent smooth structures. Thus for the purposes of constructing the family version of the Seiberg- Witten invariants, it will be sufficient for us to consider “asymptotically flat” 4-manifolds, i.e. such that outside the ball BR of some large radius R they will be given as the set of common zeros of the system of linear equations (e.g. fi(x1, ...xn+4) = xi for i = 1, ...n.) By Gromov’s h-principle any smooth 4-manifold (with all of its smooth structures and met- rics) can be obtained as a common set of zeros of a system of equations in RN for sufficiently large N. Theorem [S2]. Any parallelizable smooth 4-manifold can be obtained as a set of zeros of n functions on the trivial (n+4)-bundle over Sn. Each manifold will be represented by |H1(M,Z2)⊕H 3(M,Z)| chambers. There is a theory which also fits into our template - Ozsvath-Szabo homologies for 3- manifolds, Euler characteristic of which is Turaev’s torsion. It would be interesting to show that this theory is also of finite type or decomposes as the Khovanov theory. c). Ozsvath-Szabo theory as triangulated category. In [S3] we put the theory developed by P.Ozsvath and Z. Szabo into the context of homo- logical algebra by considering a local system of their complexes on the space of 3-manifolds and extending it to the singular locus. We show that for the restricted category the Heegaard Floer complex CF∞ is of finite type one. For other versions of the theory we will be using the new combinatorial formulas, obtained in [SW]. Recall that the categorification is the process of replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors. On would hope that after establishing this correspondence, homological algebra will provide algebraic structures which one should assign to geometrical objects without going into the specifics of a given theory. One can see that this approach is very useful in topological category, in particular we will be getting knot and link invariants of Ozsvath and Szabo after setting up their local system on the space of 3-manifolds. Note Floer homology can be also considered as invariants for families, so it would be in- teresting to connect our work to the one of M.Hutchings [H]. His work can be interpreted as construction of local systems corresponding to various Floer-type theories on the chambers of our spaces. Then we extend them to the discriminant and classify according to our definition. 6.3. Further directions. 1. There is a number of immediate questions from the finite-type invariants story: a). What will substitute the notion of the chord diagram? What is the ”basis ” in the theories of finite type? b). What are the ”dimensions” of the spaces of theories of order n? 2. What is the representation-theoretical meaning of the theory of finite type? a). Is it possible to construct a ”universal” knot homology theory in a sense of T.Lee [L] ? b). Is it possible to rise such a ”universal” knot homology theory to the Floer-type theory of 3-manifolds? 3. There are ”categorifications” of other knot invariants: Alexander polynomial [OS], HOM- FLY polynomial [DGR ]. These theories also fit into our setting and it will interesting to show that they decompose into the series of theories of finite type or that their truncations are of finite type. 4. The next step in our program [S3] is the construction of the local system of Ozsvath-Szabo homologies on the space of 3-manifolds introduced in [S1]. We also plan to raise Khovanov theory to the homological Floer-type theory of 3-manifolds. 5. It should be also possible to generalize our program to the study of the diffeomorphism group of a 4-manifold by considering Gukov-Witten [GW] categorification of Vafa invariant on the moduli space constructed in [S2]. 5. Bibliography. [BF] Bauer S., Furuta M., A stable cohomotopy refinement of Seiberg-Witten invariants: I, II, math.DG/0204340. [BN1] Bar-Natan D.,On Khovanov’s categorification of the Jones polynomial, math.QA/0201043. [BN2] Bar-Natan D., Vassiliev and Quantum Invariants of Braids, q-alg/9607001. [Bu] R. Budney, Topology of spaces of knots in dimension 3, math.GT/0506524. [CJS] Cohen R., Jones J.,Segal G., Morse theory and classifying spaces, preprint 1995. [CKV] Champanerkar A., Kofman I., Viro O., Spanning trees and Khovanov homology, preprint. [D] Donaldson S., The Seiberg-Witten Equations and 4 manifold topology, Bull.AMS, v. 33, 1, 1996. [DGR] Dunfield N., Gukov S., Rasmussen J., The Superpolynomial for Knot Homologies , math.GT/0505662. 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[KhR] Khovanov M., Rozansky L.,Matrix factorizations and link homology, math.QA/0401268 [K] Kontsevich M., Feynmann diagrams and low-dimensional topology. First European Congress of Mathematics, Vol 2(Paris, 1992), Progr.Math.,120, Birkhauser,1994. [L] Lee T. An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants, q-alg/9601002. [MOS] Manolescu C., Ozsvath P., Sarkar S., A combinatorial description of knot Floer homology. , math.GT/0607691. [O’H] O’Hara J., Energy of Knots and Conformal Geometry , World Scientific Publishing (Jun 15 2000). http://arxiv.org/abs/math/0204340 http://arxiv.org/abs/math/0201043 http://arxiv.org/abs/q-alg/9607001 http://arxiv.org/abs/math/0506524 http://arxiv.org/abs/math/0505662 http://arxiv.org/abs/math/9909095 http://arxiv.org/abs/math/0401268 http://arxiv.org/abs/q-alg/9601002 http://arxiv.org/abs/math/0607691 [O] Ohtsuki T., Finite Type Invariants of Integral Homology 3-Spheres, J. Knot Theory and its Ramifications 5 (1996). [OS1] Ozsvath P., Szabo Z., Holomorphic disks and three-manifold invariants: properties and applications, GT/0006194. [OS2] Ozsvath P., Szabo Z., Holomorphic disks and knot invariants, math.GT/0209056. [R] D.Ruberman,A polynomial invariant of diffeomorphisms of 4-manifold, geometry and Topology, v.2, Proceeding of the Kirbyfest, p.473-488, 1999. [S1] Shirokova N., The Space of 3-manifolds, C.R. Acad.Sci.Paris, t.331, Serie1, p.131-136, 2000. [S2] Shirokova N., On paralelizable 4-manifolds and invariants for families, preprint 2005. [S3] Shirokova N., The constructible sheaf of Heegaard Floer Homology on the Space of 3-manifolds, in preparation. [S4] Shirokova N., The finiteness result for Khovanov homology, preprint 2006. [SW] Sarkar S., Wang J., A combinatorial description of some Heegaard Floer homologies, math.GT/0607777. [T] Taubes C.,Casson’s invariant and gauge theory, J.Diff.Geom., 31, (1990), 547-599. 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0704.1331

Siegel's theorem for Drinfeld modules

SIEGEL’S THEOREM FOR DRINFELD MODULES D. GHIOCA AND T. J. TUCKER Abstract. We prove a Siegel type statement for finitely generated φ- submodules of Ga under the action of a Drinfeld module φ. This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of P1 over a number field. 1. Introduction In 1929, Siegel ([Sie29]) proved that if C is an irreducible affine curve defined over a number field K and C has at least three points at infinity, then there are at most finitely many K-rational points on C that have integral coordinates. The proof of this famous theorem uses diophantine approximation along with the fact that certain groups of rational points are finitely generated; when C has genus greater than 0, the group in question is the Mordell-Weil group of the Jacobian of C, while when C has genus 0, the group in question is the group of S-units in a finite extension of K. Motivated by the analogy between rank 2 Drinfeld modules and elliptic curves, the authors conjectured in [GT06] a Siegel type statement for finitely generated φ-submodules Γ of Ga (where φ is a Drinfeld module of arbitrary rank). For a finite set of places S of a function field K, we defined a notion of S-integrality and asked whether or not it is possible that there are infinitely many γ ∈ Γ which are S-integral with respect to a fixed point α ∈ K. We also proved in [GT06] a first instance of our conjecture in the case where Γ is a cyclic submodule and α is a torsion point for φ. Our goal in this paper is to prove our Siegel conjecture for every finitely generated φ-submodule of Ga(K), where φ is a Drinfeld module defined over the field K (see our Theorem 2.4). We will also establish an analog (also in the context of Drinfeld modules) of a theorem of Silverman for nonconstant morphisms of P1 of degree greater than 1 over a number field (see our Theorem 2.5). We note that recently there has been significant progress on establish- ing additional links between classical diophantine results over number fields and similar statements for Drinfeld modules. Denis [Den92a] formulated analogs for Drinfeld modules of the Manin-Mumford and the Mordell-Lang 2000 Mathematics Subject Classification. Primary 11G50, Secondary 11J68, 37F10. Key words and phrases. Drinfeld module, Heights, Diophantine approximation. [email protected]; [email protected] http://arxiv.org/abs/0704.1331v1 2 D. GHIOCA AND T. J. TUCKER conjectures. The Denis-Manin-Mumford conjecture was proved by Scan- lon in [Sca02], while a first instance of the Denis-Mordell-Lang conjecture was established in [Ghi05] by the first author (see also [Ghi06b] for an ex- tension of the result from [Ghi05]). The authors proved in [GT07] several other cases of the Denis-Mordell-Lang conjecture. In addition, the first au- thor proved in [Ghi06a] an equidistribution statement for torsion points of a Drinfeld module that is similar to the equidistribution statement established by Szpiro-Ullmo-Zhang [SUZ97] (which was later extended by Zhang [Zha98] to a full proof of the famous Bogomolov conjecture). Breuer [Bre05] proved a special case of the André-Oort conjecture for Drinfeld modules, while special cases of this conjecture in the classical case of a number field were proved by Edixhoven-Yafaev [EY03] and Yafaev [Yaf06]. Bosser [Bos99] proved a lower bound for linear forms in logarithms at an infinite place associated to a Drinfeld module (similar to the classical result obtained by Baker [Bak75] for usual logarithms, or by David [Dav95] for elliptic logarithms). Bosser’s result was used by the authors in [GT06] to establish certain equidistribution and integrality statements for Drinfeld modules. Moreover, Bosser’s result is believed to be true also for linear forms in logarithms at finite places for a Drinfeld module (as was communicated to us by Bosser). Assuming this last statement, we prove in this paper the natural analog of Siegel’s theorem for finitely generated φ-submodules. We believe that our present paper provides additional evidence that the Drinfeld modules represent a good arithmetic analog in characteristic p for abelian varieties in characteristic 0. The basic outline of this paper can be summarized quite briefly. In Sec- tion 2 we give the basic definitions and notation, and then state our main results. In Section 3 we prove these main results: Theorems 2.4 and 2.5. 2. Notation Notation. N stands for the non-negative integers: {0, 1, . . . }, while N∗ := N \ {0} stands for the positive integers. 2.1. Drinfeld modules. We begin by defining a Drinfeld module. Let p be a prime and let q be a power of p. Let A := Fq[t], let K be a finite field extension of Fq(t), and let K be an algebraic closure of K. We let τ be the Frobenius on Fq, and we extend its action on K. Let K{τ} be the ring of polynomials in τ with coefficients from K (the addition is the usual addition, while the multiplication is the composition of functions). A Drinfeld module is a morphism φ : A→ K{τ} for which the coefficient of τ0 in φ(a) =: φa is a for every a ∈ A, and there exists a ∈ A such that φa 6= aτ 0. The definition given here represents what Goss [Gos96] calls a Drinfeld module of “generic characteristic”. We note that usually, in the definition of a Drinfeld module, A is the ring of functions defined on a projective nonsingular curve C, regular away from a closed point η ∈ C. For our definition of a Drinfeld module, C = P1 and η is the usual point at infinity on P1. On the other hand, every ring of regular SIEGEL’S THEOREM FOR DRINFELD MODULES 3 functions A as above contains Fq[t] as a subring, where t is a nonconstant function in A. For every field extension K ⊂ L, the Drinfeld module φ induces an action on Ga(L) by a∗x := φa(x), for each a ∈ A. We call φ-submodules subgroups of Ga(K) which are invariant under the action of φ. We define the rank of a φ-submodule Γ be dimFrac(A) Γ⊗A Frac(A). As shown in [Poo95], Ga(K) is a direct sum of a finite torsion φ-submodule with a free φ-submodule of rank ℵ0. A point α is torsion for the Drinfeld module action if and only if there exists Q ∈ A \ {0} such that φQ(α) = 0. The monic polynomial Q of minimal degree which satisfies φQ(α) = 0 is called the order of α. Since each polynomial φQ is separable, the torsion submodule φtor lies in the separable closure Ksep of K. 2.2. Valuations and Weil heights. Let MFq(t) be the set of places on Fq(t). We denote by v∞ the place in MFq(t) such that v∞( ) = deg(g) − deg(f) for every nonzero f, g ∈ A = Fq[t]. We letMK be the set of valuations on K. ThenMK is a set of valuations which satisfies a product formula (see [Ser97, Chapter 2]). Thus • for each nonzero x ∈ K, there are finitely many v ∈ MK such that |x|v 6= 1; and • for each nonzero x ∈ K, we have |x|v = 1. We may use these valuations to define a Weil height for each x ∈ K as (2.0.1) h(x) = max log(|x|v , 1). Convention. Without loss of generality we may assume that the nor- malization for all the valuations of K is made so that for each v ∈ MK , we have log |x|v ∈ Z. Definition 2.1. Each place in MK which lies over v∞ is called an infinite place. Each place in MK which does not lie over v∞ is called a finite place. 2.3. Canonical heights. Let φ : A → K{τ} be a Drinfeld module of rank d (i.e. the degree of φt as a polynomial in τ equals d). The canonical height of β ∈ K relative to φ (see [Den92b]) is defined as ĥ(β) = lim h(φtn(β)) Denis [Den92b] showed that a point is torsion if and only if its canonical height equals 0. For every v ∈MK , we let the local canonical height of β ∈ K at v be (2.1.1) ĥv(β) = lim log max(|φtn(β)|v , 1) 4 D. GHIOCA AND T. J. TUCKER Furthermore, for every a ∈ Fq[t], we have ĥv(φa(x)) = deg(φa) · ĥv(x) (see [Poo95]). It is clear that ĥv satisfies the triangle inequality, and also that∑ ĥv(β) = ĥ(β). 2.4. Completions and filled Julia sets. By abuse of notation, we let ∞ ∈ MK denote any place extending the place v∞. We let K∞ be the completion of K with respect to | · |∞. We let K∞ be an algebraic closure of K∞. We let C∞ be the completion of K∞. Then C∞ is a complete, algebraically closed field. Note that C∞ depends on our choice for ∞ ∈MK extending v∞. However, each time we will work with only one such place ∞, and so, there will be no possibility of confusion. Next, we define the v-adic filled Julia set Jφ,v corresponding to the Drin- feld module φ and to each place v of MK . Let Cv be the completion of an algebraic closure of Kv. Then | · |v extends to a unique absolute value on all of Cv. The set Jφ,v consists of all x ∈ Cv for which {|φQ(x)|v}Q∈A is bounded. It is immediate to see that x ∈ Jφ,v if and only if {|φtn(x)|v}n≥1 is bounded. One final note on absolute values: as noted above, the place v ∈ MK extends to a unique absolute value | · |v on all of Cv. We fix an embedding of i : K −→ Cv. For x ∈ K, we denote |i(x)|v simply as |x|v , by abuse of notation. 2.5. The coefficients of φt. Each Drinfeld module is isomorphic to a Drin- feld module for which all the coefficients of φt are integral at all the places inMK which do not lie over v∞. Indeed, we let B ∈ Fq[t] be a product of all (the finitely many) irreducible polynomials P ∈ Fq[t] with the property that there exists a place v ∈MK which lies over the place (P ) ∈MFq(t), and there exists a coefficient of φt which is not integral at v. Let γ be a sufficiently large power of B. Then ψ : A → K{τ} defined by ψQ := γ −1φQγ (for each Q ∈ A) is a Drinfeld module isomorphic to φ, and all the coefficients of ψt are integral away from the places lying above v∞. Hence, from now on, we assume that all the coefficients of φt are integral away from the places lying over v∞. It follows that for every Q ∈ A, all coefficients of φQ are integral away from the places lying over v∞. 2.6. Integrality and reduction. Definition 2.2. For a finite set of places S ⊂MK and α ∈ K, we say that β ∈ K is S-integral with respect to α if for every place v /∈ S, and for every morphisms σ, τ : K → K (which restrict to the identity on K) the following are true: • if |ατ |v ≤ 1, then |α τ − βσ|v ≥ 1. • if |ατ |v > 1, then |β σ|v ≤ 1. We note that if β is S-integral with respect to α, then it is also S′-integral with respect to α, where S′ is a finite set of places containing S. Moreover, the fact that β is S-integral with respect to α, is preserved if we replace SIEGEL’S THEOREM FOR DRINFELD MODULES 5 K by a finite extension. Therefore, in our results we will always assume α, β ∈ K. For more details about the definition of S-integrality, we refer the reader to [BIR05]. Definition 2.3. The Drinfeld module φ has good reduction at a place v if for each nonzero a ∈ A, all coefficients of φa are v-adic integers and the leading coefficient of φa is a v-adic unit. If φ does not have good reduction at v, then we say that φ has bad reduction at v. It is immediate to see that φ has good reduction at v if and only if all coefficients of φt are v-adic integers, while the leading coefficient of φt is a v-adic unit. We can now state our Siegel type result for Drinfeld modules. Theorem 2.4. With the above notation, assume in addition K has only one infinite place. Let Γ be a finitely generated φ-submodule of Ga(K), let α ∈ K, and let S be a finite set of places in MK . Then there are finitely many γ ∈ Γ such that γ is S-integral with respect to α. As mentioned in Section 1, we proved in [GT06] that Theorem 2.4 holds when Γ is a cyclic φ-module generated by a nontorsion point β ∈ K and α ∈ φtor(K) (see Theorem 1.1 and Proposition 5.6 of [GT06]). Moreover, in [GT06] we did not have in our results the extra hypothesis from Theo- rem 2.4 that there exists only one infinite place in MK . Even though we believe Theorem 2.4 is true without this hypothesis, our method for proving Theorem 2.4 requires this technical hypothesis. On the other hand, we are able to prove the following analog for Drinfeld modules of a theorem of Sil- verman (see [Sil93]) for nonconstant morphisms of P1 of degree greater than 1 over a number field, without the hypothesis of having only one infinite place in MK . Theorem 2.5. With the above notation, let β ∈ K be a nontorsion point, and let α ∈ K be an arbitrary point. Then there are finitely many Q ∈ A such that φQ(β) is S-integral for α. As explained before, in [GT06] we proved Theorem 2.5 in the case α is a torsion point in K. 3. Proofs of our main results We continue with the notation from Section 2. In our argument, we will be using the following key fact. Fact 3.1. Assume ∞ ∈ MK is an infinite place. Let γ1, . . . , γr, α ∈ K. Then there exist (negative) constants C0 and C1 (depending only on φ, γ1, . . . , γr, α) such that for any polynomials P1, . . . , Pr ∈ A (not all con- stants), either φP1(γ1) + · · ·+ φPr(γr) = α or log |φP1(γ1) + · · · + φPr(γr)− α|∞ ≥ C0 +C1 max 1≤i≤r (deg(Pi) log deg(Pi)). 6 D. GHIOCA AND T. J. TUCKER Fact 3.1 follows easily from the lower bounds for linear forms in logarithms established by Bosser (see Théorème 1.1 in [Bos99]). Essentially, it is the same proof as our proof of Proposition 3.7 of [GT06] (see in particular the derivation of the inequality (3.7.2) in [GT06]). For the sake of completeness, we will provide below a sketch of a proof of Fact 3.1. Proof of Fact 3.1. We denote by exp∞ the exponential map associated to the place ∞ (see [Gos96]). We also let L be the corresponding lattice for exp∞, i.e. L := ker(exp∞). Finally, let ω1, . . . , ωd be an A-basis for L of “successive minima” (see Lemma (4.2) of [Tag93]). This means that for every Q1, . . . , Qd ∈ A, we have (3.1.1) |Q1ω1 + · · ·+Qdωd|∞ = |Qiωi|∞. Let u0 ∈ C∞ such that exp∞(u0) = α. We also let u1, . . . , ur ∈ C∞ such that for each i, we have exp∞(ui) = γi. We will find constants C0 and C1 satisfying the inequality from Fact 3.1, which depend only on φ and u0, u1, . . . , ur. There exists a positive constant C2 such that exp∞ induces an isomor- phism from the ball B := {z ∈ C∞ : |z|∞ < C2} to itself (see Lemma 3.6 of [GT06]). If we assume there exist no constants C0 and C1 as in the conclu- sion of Fact 3.1, then there exist polynomials P1, . . . , Pr, not all constants, such that (3.1.2) φPi(γi) 6= α and | i=1 φPi(γi) − α|∞ < C2. Thus we can find y ∈ B such that |y|∞ = i=1 φPi(γi)− α|∞ and (3.1.3) exp∞(y) = φPi(γi)− α. Moreover, because exp∞ is an isomorphism on the metric space B, then for every y′ ∈ C∞ such that exp∞(y i=1 φPi(γi)−α, we have |y ′|∞ ≥ |y|∞. But we know that (3.1.4) exp∞ Piui − u0 φPi(γi)− α. Therefore | i=1 Piui − u0|∞ ≥ |y|∞. On the other hand, using (3.1.3) and (3.1.4), we conclude that there exist polynomials Q1, . . . , Qd such that Piui − u0 = y + Qiωi. SIEGEL’S THEOREM FOR DRINFELD MODULES 7 Hence | i=1Qiωi|∞ ≤ | i=1 Piui − u0|∞. Using (3.1.1), we obtain ∣∣∣∣∣ ∣∣∣∣∣ |Qiωi|∞ ≤ ∣∣∣∣∣ Piui − u0 ∣∣∣∣∣ ≤ max |u0|∞, |Piui|∞ ≤ C3 · |Pi|∞, (3.1.5) where C3 is a constant depending only on u0, u1, . . . , ur. We take logarithms of both sides in (3.1.5) and obtain degQi ≤ degPi + logC3 − log |ωi|∞ degPi + C4, (3.1.6) where C4 depends only on φ and u0, u1, . . . , ur (the dependence on the ωi is actually a dependence on φ, because the ωi are a fixed basis of “successive minima” for φ at ∞). Using (3.1.6) and Proposition 3.2 of [GT06] (which is a translation of the bounds for linear forms in logarithms for Drinfeld modules established in [Bos99]), we conclude that there exist (negative) constants C0, C1, C5 and C6 (depending only on φ, γ1, . . . , γr and α) such that ∣∣∣∣∣ φPi(γi)− α ∣∣∣∣∣ = log |y|∞ = log ∣∣∣∣∣ Piui − u0 − ∣∣∣∣∣ ≥ C5 + C6 degPi + C4 (degPi + C4) ≥ C0 + C1 degPi (degPi) , (3.1.7) as desired. � In our proofs for Theorems 2.5 and 2.4 we will also use the following state- ment, which is believed to be true, based on communication with V. Bosser. Therefore we assume its validity without proof. Statement 3.2. Assume v does not lie above v∞. Let γ1, . . . , γr, α ∈ K. Then there exist positive constants C1, C2, C3 (depending only on v, φ, γ1, . . . , γr and α) such that for any P1, . . . , Pr ∈ Fq[t], either φP1(γ1) + · · ·+ φPr(γr) = α or log |φP1(γ1) + · · ·+ φPr(γr)− α|v ≥ −C1 − C2 max 1≤i≤r (deg(Pi)) 8 D. GHIOCA AND T. J. TUCKER Statement 3.2 follows after one establishes a lower bound for linear forms in logarithms at finite places v. In a private communication, V. Bosser told us that it is clear to him that his proof ([Bos99]) can be adapted to work also at finite places with minor modifications. We sketch here how Statement 3.2 would follow from a lower bound for linear forms in logarithms at finite places. Let v be a finite place and let expv be the formal exponential map associated to v. The existence of expv and its convergence on a sufficiently small ball Bv := {x ∈ Cv : |x|v < Cv} is proved along the same lines as the existence and the convergence of the usual exponential map at infinite places for φ (see Section 4.6 of [Gos96]). In addition, (3.2.1) | expv(x)|v = |x|v for every x ∈ Bv. Moreover, at the expense of replacing Cv with a smaller positive constant, we may assume that for each F ∈ A, and for each x ∈ Bv, we have (see Lemma 4.2 in [GT06]) (3.2.2) |φF (x)|v = |Fx|v . Assume we know the existence of the following lower bound for (nonzero) linear forms in logarithms at a finite place v. Statement 3.3. Let u1, . . . , ur ∈ Bv such that for each i, expv(ui) ∈ K. Then there exist positive constants C4, C5, and C6 (depending on u1, . . . , ur) such that for every F1, . . . , Fr ∈ A, either i=1 Fiui = 0, or ∣∣∣∣∣ ∣∣∣∣∣ ≥ −C4 − C5 degFi As mentioned before, Bosser proved Statement 3.3 in the case v is an infinite place (in his result, C6 = 1 + ǫ and C4 = Cǫ for every ǫ > 0). We will now derive Statement 3.2 assuming Statement 3.3 holds. Proof. (That Statement 3.3 implies Statement 3.2.) Clearly, it suffices to prove Statement 3.2 in the case α = 0. So, let γ1, . . . , γr ∈ K, and assume by contradiction that there exists an infinite sequence {Fn,i} n∈N∗ 1≤i≤r such that for each n, we have (3.3.1) −∞ < log ∣∣∣∣∣ φFn,i(γi) ∣∣∣∣∣ < logCv. For each n ≥ 1, we let Fn := (Fn,1, . . . , Fn,r) ∈ A r. We view Ar as an r-dimensional A-lattice inside the r-dimensional Frac(A)-vector space Frac(A)r. In addition, we may assume that for n 6= m, we have Fn 6= Fm. Using basic linear algebra, because the sequence {Fn,i} n∈N∗ 1≤i≤r is infinite, we can find n0 ≥ 1 such that for every n > n0, there exist Hn, Gn,1, . . . , Gn,n0 ∈ SIEGEL’S THEOREM FOR DRINFELD MODULES 9 A (not all equal to 0) such that (3.3.2) Hn · Fn = Gn,j · Fj. Essentially, (3.3.2) says that F1, . . . ,Fn0 span the linear subspace of Frac(A) generated by all Fn. Moreover, we can choose theHn in (3.3.2) in such a way that degHn is bounded independently of n (e.g. by a suitable determinant of some linearly independent subset of the first n0 of the Fj). Furthemore, there exists a constant C7 such that for all n > n0, we have (3.3.3) degGn,j < C7 + degFn,i. Because i=1 φFn,i(γi) < Cv, equation (3.2.2) yields (3.3.4) ∣∣∣∣∣φHn φFn,i(γi) )∣∣∣∣∣ = |Hn|v · ∣∣∣∣∣ φFn,i(γi) ∣∣∣∣∣ Using (3.3.2), (3.3.4), and the fact that |Hn|v ≤ 1, we obtain ∣∣∣∣∣ φFn,i(γi) ∣∣∣∣∣ ∣∣∣∣∣φHn φFn,i(γi) )∣∣∣∣∣ ∣∣∣∣∣∣ φGn,j φFj,i(γi) )∣∣∣∣∣∣ (3.3.5) Since i=1 φFj,i(γi) < Cv for all 1 ≤ j ≤ n0, there exist u1, . . . , un0 ∈ Bv such that for every 1 ≤ j ≤ n0, we have expv(uj) = φFj,i(γi). Then Statement 3.3 implies that there exist constants C4, C5, C6, C8, C9 (de- pending on u1, . . . , un0), such that ∣∣∣∣∣∣ φGn,j φFj,i(γi) )∣∣∣∣∣∣ = log ∣∣∣∣∣∣ Gn,juj ∣∣∣∣∣∣ ≥ −C4 − C5 degGn,j ≥ −C8 − C9 degFn,i (3.3.6) where in the first equality we used (3.2.1), while in the last inequality we used (3.3.3). Equations (3.3.5) and (3.3.6) show that Statement 3.2 follows from Statement 3.3, as desired. � 10 D. GHIOCA AND T. J. TUCKER Next we prove Theorem 2.5 which will be a warm-up for our proof of Theorem 2.4. For its proof, we will only need the following weaker (but also still conjectural) form of Statement 3.2 (i.e., we only need Statement 3.3 be true for non-homogeneous 1-forms of logarithms). Statement 3.4. Assume v does not lie over v∞. Let γ, α ∈ K. Then there exist positive constants C1, C2 and C3 (depending only on v, φ, γ and α) such that for each polynomial P ∈ Fq[t], either φP (γ) = α or log |φP (γ)− α|v ≥ −C1 − C2 deg(P ) Proof of Theorem 2.5. The following Lemma is the key to our proof. Lemma 3.5. For each v ∈MK , we have ĥv(β) = limdegQ→∞ log |φQ(β)−α|v qd degQ Proof of Lemma 3.5. Let v ∈ MK . If β /∈ Jφ,v, then |φQ(β)|v → ∞, as degQ → ∞. Hence, if degQ is sufficiently large, then |φQ(β) − α|v = |φQ(β)|v = max{|φQ(β)|v , 1}, which yields the conclusion of Lemma 3.5. Thus, from now on, we assume β ∈ Jφ,v. Hence ĥv(β) = 0, and we need to show that (3.5.1) lim degQ→∞ log |φQ(β)− α|v qddegQ Also note that since β ∈ Jφ,v, then |φQ(β) − α|v is bounded, and so, lim supdegQ→∞ log |φQ(β)−α|v qd degQ ≤ 0. Thus, in order to prove (3.5.1), it suf- fices to show that (3.5.2) lim inf degQ→∞ log |φQ(β)− α|v qddegQ If v is an infinite place, then Fact 3.1 implies that for every polynomial Q such that φQ(β) 6= α, we have log |φQ(β)−α|∞ ≥ C0+C1 deg(Q) log deg(Q) (for some constants C0, C1 < 0). Then taking the limit as degQ → ∞, we obtain (3.5.2), as desired. Similarly, if v is a finite place, then (3.5.2) follows from Statement 3.4. � Theorem 2.5 follows easily using the result of Lemma 3.5. We assume there exist infinitely many polynomials Qn such that φQn(β) is S-integral with respect to α. We consider the sum log |φQn(β) − α|v qddegQn Using Lemma 3.5, we obtain that Σ = ĥ(β) > 0 (because β /∈ φtor). Let T be a finite set of places consisting of all the places in S along with all places v ∈MK which satisfy at least one of the following conditions: 1. |β|v > 1. 2. |α|v > 1. 3. v is a place of bad reduction for φ. SIEGEL’S THEOREM FOR DRINFELD MODULES 11 Therefore by our choice for T (see 1. and 3.), for every v /∈ T , we have |φQn(β)|v ≤ 1. Thus, using also 2., we have |φQn(β) − α|v ≤ 1. On the other hand, φQn(β) is also T -integral with respect to α. Hence, because of 2., then for all v /∈ T , we have |φQn(β) − α|v ≥ 1. We conclude that for every v /∈ T , and for every n, we have |φQn(β) − α|v = 1. This allows us to interchange the summation and the limit in the definition of Σ (because then Σ is a finite sum over all v ∈ T ). We obtain Σ = lim qddegQn log |φQn(β)− α|v = 0, by the product formula applied to each φQn(β)−α. On the other hand, we already showed that Σ = ĥ(β) > 0. This contradicts our assumption that there are infinitely many polynomials Q such that φQ(β) is S-integral with respect to α, and concludes the proof of Theorem 2.5. � Before proceeding to the proof of Theorem 2.4, we prove several facts about local heights. In Lemma 3.10 we will use the technical assumption of having only one infinite place in K. From now on, let φt = i=0 aiτ i. As explained in Section 2, we may assume each ai is integral away from v∞. Also, from now on, we work under the assumption that there exists a unique place ∞ ∈MK lying above v∞. Fact 3.6. For every place v of K, there exists Mv > 0 such that for each x ∈ K, if |x|v > Mv, then for every nonzero Q ∈ A, we have |φQ(x)|v > Mv. Moreover, if |x|v > Mv, then ĥv(x) = log |x|v + log |ad|v Fact 3.6 is proved in Lemma 4.4 of [GT06]. In particular, Fact 3.6 shows that for each v ∈ MK and for each x ∈ K, we have ĥv(x) ∈ Q. Indeed, for every x ∈ K of positive local canonical height at v, there exists a polynomial P such that |φP (x)|v > Mv. Then ĥv(x) = bhv(φP (x)) qd degP . By Fact 3.6, we already know that ĥv(φP (x)) ∈ Q. Thus also ĥv(x) ∈ Q. Fact 3.7. Let v ∈MK \{∞}. There exists a positive constant Nv, and there exists a nonzero polynomial Qv, such that for each x ∈ K, the following statements are true (i) if |x|v ≤ Nv, then for each Q ∈ A, we have |φQ(x)|v ≤ |x|v ≤ Nv. (ii) either |φQv(x)|v ≤ Nv, or |φQv(x)|v > Mv. Proof of Fact 3.7. This was proved in [Ghi07b]. It is easy to see that Nv := min 1≤i≤d satisfies condition (i), but the proof of (ii) is much more complicated. In [Ghi07b], the first author proved that there exists a positive integer dv such that for every x ∈ K, there exists a polynomial Q of degree at most dv such that either |φQ(x)|v > Mv, or |φQ(x)|v ≤ Nv (see Remark 5.12 which is 12 D. GHIOCA AND T. J. TUCKER valid for every place which does not lie over v∞). Using Fact 3.6 and (i), we conclude that the polynomial Qv := deg P≤dv P satisfies property (ii). � Using Facts 3.6 and 3.7 we prove the following important result valid for finite places. Lemma 3.8. Let v ∈ MK \ {∞}. Then there exists a positive integer Dv such that for every x ∈ K, we have Dv · ĥv(x) ∈ N. If in addition we assume v is a place of good reduction for φ, then we may take Dv = 1. Proof of Lemma 3.8. Let x ∈ K. If ĥv(x) = 0, then we have nothing to show. Therefore, assume from now on that ĥv(x) > 0. Using (ii) of Fact 3.7, there exists a polynomial Qv (depending only on v, and not on x) such that |φQv(x)|v > Mv (clearly, the other option from (ii) of Lemma 3.7 is not available because we assumed that ĥv(x) > 0). Moreover, using the definition of the local height, and also Fact 3.6, (3.8.1) ĥv(x) = ĥv(φQv(x)) qddegQv log |φQv(x)|v + log |ad|v qddegQv Because both log |φQv(x)|v and log |ad|v are integer numbers, (3.8.1) yields the conclusion of Lemma 3.8 (we may take Dv = q ddegQv(qd − 1)). The second part of Lemma 3.8 follows immediately from Lemma 4.13 of [Ghi07b]. Indeed, if v is a place of good reduction for φ, then |x|v > 1 (because we assumed ĥv(x) > 0). But then, ĥv(x) = log |x|v (here we use the fact that v is a place of good reduction, and thus ad is a v-adic unit). Hence ĥv(x) ∈ N, and we may take Dv = 1. � The following result is an immediate corollary of Fact 3.8. Corollary 3.9. There exists a positive integer D such that for every v ∈ MK \ {∞}, and for every x ∈ K, we have D · ĥv(x) ∈ N. Next we prove a similar result as in Lemma 3.8 which is valid for the only infinite place of K. Lemma 3.10. There exists a positive integer D∞ such that for every x ∈ K, either ĥv(x) > 0 for some v ∈MK \ {∞}, or D∞ · ĥ∞(x) ∈ N. Before proceeding to its proof, we observe that we cannot remove the assumption that ĥv(x) = 0 for every finite place v, in order to obtain the existence of D∞ in the statement of Lemma 3.10. Indeed, we know that in K there are points of arbitrarily small (but positive) local height at ∞ (see Example 6.1 from [Ghi07b]). Therefore, there exists no positive integer D∞ which would clear all the possible denominators for the local heights at ∞ of those points. However, it turns out (as we will show in the proof of Lemma 3.10) that for such points x of very small local height at ∞, there exists some other place v for which ĥv(x) > 0. SIEGEL’S THEOREM FOR DRINFELD MODULES 13 Proof of Lemma 3.10. Let x ∈ K. If x ∈ φtor, then we have nothing to prove (every positive integer D∞ would work because ĥ∞(x) = 0). Thus, we assume x is a nontorsion point. If ĥv(x) > 0 for some place v which does not lie over v∞, then again we are done. So, assume from now on that ĥv(x) = 0 for every finite place v. By proceeding as in the proof of Lemma 3.8, it suffices to show that there exists a polynomial Q∞ of degree bounded independently of x such that |φQ∞(x)|∞ > M∞ (with the notation as in Fact 3.6). This is proved in Theorem 4.4 of [Ghi07a]. The first author showed in [Ghi07a] that there exists a positive integer L (depending only on the number of places of bad reduction of φ) such that for every nontorsion point x, there exists a place v ∈ MK , and there exists a polynomial Q of degree less than L such that |φQ(x)|v > Mv . Because we assumed that ĥv(x) = 0 for every v 6= ∞, then the above statement yields the existence of D∞. � We will prove Theorem 2.4 by showing that a certain lim sup is positive. This will contradict the existence of infinitely many S-integral points in a finitely generated φ-submodule. Our first step will be a result about the lim inf of the sequences which will appear in the proof of Theorem 2.4. Lemma 3.11. Suppose that Γ is a torsion-free φ-submodule of Ga(K) gener- ated by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be a sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r and (Pm,i)1≤i≤r are distinct. Then for every v ∈MK , we have (3.11.1) lim inf log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i Proof. Suppose that for some ǫ > 0, there exists a sequence (nk)k≥1 ⊂ N such that i=1 φPnk,i (γi) 6= α and (3.11.2) log | i=1 φPnk,i (γi)− α|v i=1 q ddeg Pnk,i < −ǫ, for every k ≥ 1. But taking the lower bound from Fact 3.1 or Statement 3.2 (depending on whether v is the infinite place or not) and dividing through i=1 q ddeg Pnk,i , we see that this is impossible. � The following proposition is the key technical result required to prove Theorem 2.4. This proposition plays the same role that Lemma 3.5 plays in the proof of Theorem 2.5, or that Corollary 3.13 plays in the proof of Theorem 1.1 from [GT06]. Note that is does not provide an exact formula for the canonical height of a point, however; it merely shows that a certain limit is positive. This will suffice for our purposes since we only need that a certain sum of limits be positive in order to prove Theorem 2.4. Proposition 3.12. Let Γ be a torsion-free φ-submodule of Ga(K) generated by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be a 14 D. GHIOCA AND T. J. TUCKER sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r and (Pm,i)1≤i≤r are distinct. Then there exists a place v ∈MK such that (3.12.1) lim sup log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i Proof. Using the triangle inequality for the v-adic norm, and the fact that qddeg Pn,i = +∞, we conclude that proving that (3.12.1) holds is equivalent to proving that for some place v, we have (3.12.2) lim sup log | i=1 φPn,i(γi)|v∑r i=1 q ddeg Pn,i We also observe that it suffices to prove Proposition 3.12 for a subsequence (nk)k≥1 ⊂ N We prove (3.12.2) by induction on r. If r = 1, then by Corollary 3.13 of [GT06] (see also our Lemma 3.5), (3.12.3) lim sup deg P→∞ log |φP (γ1)|v qddegP = ĥv(γ1) and because γ1 /∈ φtor, there exists a place v such that ĥv(γ1) > 0, thus proving (3.12.2) for r = 1. Therefore, we assume (3.12.2) is established for all φ-submodules Γ of rank less than r and we will prove it for φ-submodules of rank r. In the course of our argument for proving (3.12.2), we will replace several times a given sequence with a subsequence of itself (note that passing to a subsequence can only make the lim sup smaller). For the sake of not cluster- ing the notation, we will drop the extra indices which would be introduced by dealing with the subsequence. Let S0 be the set of places v ∈MK for which there exists some γ ∈ Γ such that ĥv(γ) > 0. The following easy fact will be used later in our argument. Fact 3.13. The set S0 is finite. Proof of Fact 3.13. We claim that S0 equals the finite set S 0 of places v ∈ MK for which there exists i ∈ {1, . . . , r} such that ĥv(γi) > 0. Indeed, let v ∈ MK \ S 0. Then for each i ∈ {1, . . . , r} we have ĥv(γi) = 0. Moreover, for each i ∈ {1, . . . , r} and for each Qi ∈ Fq[t], we have (3.13.1) ĥv(φQi(γi)) = deg(φQi) · ĥv(γi) = 0. Using (3.13.1) and the triangle inequality for the local canonical height, we obtain that φQi(γi) SIEGEL’S THEOREM FOR DRINFELD MODULES 15 This shows that indeed S0 = S 0, and concludes the proof of Fact 3.13. � If the sequence (nk)k≥1 ⊂ N ∗ has the property that for some j ∈ {1, . . . , r}, we have (3.13.2) lim qddeg Pnk,j i=1 q ddeg Pnk,i then the inductive hypothesis will yield the desired conclusion. Indeed, by the induction hypothesis, and also using (3.13.2), there exists v ∈ S0 such (3.13.3) lim sup log | i 6=j φPnk,i (γi)|v i=1 q ddeg Pnk,i If ĥv(γj) = 0, then ∣∣∣φPnk,j(γj) is bounded as k → ∞. Thus, for large enough k, ∣∣∣∣∣ φPnk,i ∣∣∣∣∣ ∣∣∣∣∣∣ i 6=j φPnk,i ∣∣∣∣∣∣ and so, (3.13.3) shows that (3.12.2) holds. Now, if ĥv(γj) > 0, then we proved in Lemma 4.4 of [GT06] that (3.13.4) log |φP (γj)|v − q ddeg P ĥv(γj) is uniformly bounded as degP → ∞ (note that this follows easily from simple arguments involving geometric series and coefficients of polynomials). Therefore, using (3.13.2), we obtain (3.13.5) lim ∣∣∣φPnk,j(γj) i=1 q ddeg Pnk,i Using (3.13.3) and (3.13.5), we conclude that for large enough k, ∣∣∣∣∣ φPnk,i ∣∣∣∣∣ ∣∣∣∣∣∣ i 6=j φPnk,i ∣∣∣∣∣∣ and so, (3.13.6) lim sup i=1 φPnk,i i=1 q ddeg Pnk,i as desired. Therefore, we may assume from now on that there exists B ≥ 1 such that for every n, (3.13.7) max1≤i≤r q ddegPn,i min1≤i≤r q ddegPn,i ≤ B or equivalently, (3.13.8) max 1≤i≤r degPn,i − min 1≤i≤r degPn,i ≤ logq B 16 D. GHIOCA AND T. J. TUCKER We will prove that (3.12.2) holds for some place v by doing an analysis at each place v ∈ S0. We know that |S0| ≥ 1 because all γi are nontorsion. Our strategy is to show that in case (3.12.2) does not hold, then we can find δ1, . . . , δr ∈ Γ, and we can find a sequence (nk)k≥1 ⊂ N ∗, and a sequence of polynomials (Rk,i) k∈N∗ 1≤i≤r such that (3.13.9) φPnk,i (γi) = φRk,i(δi) and (3.13.10) ĥv(δi) < ĥv(γi) and (3.13.11) 0 < lim inf i=1 q ddegPnk,i i=1 q ddegRk,i ≤ lim sup i=1 q ddegPnk,i i=1 q ddegRk,i < +∞. Equation (3.13.9) will enable us to replace the γi by the δi and proceed with our analysis of the latter. Inequality (3.13.10) combined with Corollary 3.9 and Lemma 3.10 will show that for each such v, in a finite number of steps we either construct a sequence δi as above for which all ĥv(δi) = 0, or (3.12.2) holds for δ1, . . . , δr and the corresponding polynomials Rk,i, i.e. (3.13.12) lim sup log | i=1 φRk,i(δi)|v∑r i=1 q ddegRk,i Equation (3.13.11) shows that (3.12.2) is equivalent to (3.13.12) (see also (3.13.9)). We start with v ∈ S0 \ {∞}. As proved in Lemma 4.4 of [GT06], for each i ∈ {1, . . . , r} such that ĥv(γi) > 0, there exists a positive integer di such that for every polynomial Qi of degree at least di, we have (3.13.13) log |φQi(γi)|v = q ddegQiĥv(γi)− log |ad|v qd − 1 We know that for each i, we have limn→∞ degPn,i = +∞ because of (3.13.8). Hence, for each n sufficiently large, and for each i ∈ {1, . . . , r} such that ĥv(γi) > 0, we have (3.13.14) log |φPn,i(γi)|v = q ddeg Pn,iĥv(γi)− log |ad|v qd − 1 We now split the problem into two cases. Case 1. There exists an infinite subsequence (nk)k≥1 such that for every k, we have (3.13.15) ∣∣∣∣∣ φPnk,i ∣∣∣∣∣ = max 1≤i≤r ∣∣∣φPnk,i(γi) For the sake of not clustering the notation, we drop the index k from (3.13.15) (note that we need to prove (3.12.2) only for a subsequence). At SIEGEL’S THEOREM FOR DRINFELD MODULES 17 the expense of replacing again N∗ by a subsequence, we may also assume that for some fixed j ∈ {1, . . . , r}, we have (3.13.16) ∣∣∣∣∣ φPn,i(γi) ∣∣∣∣∣ ∣∣φPn,i(γi) ∣∣φPn,j (γj) for all n ∈ N∗. Because we know that there exists i ∈ {1, . . . , r} such that ĥv(γi) > 0, then for such i, we know |φPn,i(γi)|v is unbounded (as n → ∞). Therefore, using (3.13.16), we conclude that also |φPn,j (γj)|v is unbounded (as n→ ∞). In particular, this means that ĥv(γj) > 0. Then using (3.13.14) for γj, we obtain that lim sup log | i=1 φPn,i(γi)|v∑r i=1 q ddeg Pn,i = lim sup log |φPn,j (γj)|v∑r i=1 q ddeg Pn,i = lim sup qddegPn,j ĥv(γj)− log |ad|v qd−1∑r i=1 q ddeg Pn,i = lim qddeg Pn,j ĥv(γj)− log |ad|v qddeg Pn,j · lim sup qddegPn,j∑r i=1 q ddeg Pn,i (3.13.17) since qddeg Pn,j ĥv(γj)− log |ad|v qddegPn,j = ĥv(γj) > 0 and lim sup qddegPn,j∑r i=1 q ddeg Pn,i > 0 because of (3.13.8). Case 2. For all but finitely many n, we have (3.13.18) ∣∣∣∣∣ φPn,i(γi) ∣∣∣∣∣ < max 1≤i≤r ∣∣φPn,i(γi) Using the pigeonhole principle, there exists an infinite sequence (nk)k≥1 ⊂ N∗, and there exist j1, . . . , js ∈ {1, . . . , r} (where s ≥ 2) such that for each k, we have (3.13.19) |φPnk,j1 (γj1)|v = · · · = |φPnk,js (γjs)|v > max i∈{1,...,r}\{j1,...,js} |φPnk,i (γi)|v. Again, as we did before, we drop the index k from the above subsequence of N∗. Using (3.13.19) and the fact that there exists i ∈ {1, . . . , r} such that ĥv(γi) > 0, we conclude that for all 1 ≤ i ≤ s, we have ĥv(γji) > 0. Hence, using (3.13.14) in (3.13.19), we obtain that for sufficiently large n, we have (3.13.20) qddeg Pn,j1 ĥv(γj1) = · · · = q ddeg Pn,js ĥv(γjs). 18 D. GHIOCA AND T. J. TUCKER Without loss of generality, we may assume ĥv(γj1) ≥ ĥv(γji) for all i ∈ {2, . . . , s}. Then (3.13.20) yields that degPn,ji ≥ degPn,j1 for i > 1. We divide (with quotient and remainder) each Pn,ji (for i > 1) by Pn,j1 and for each such ji, we obtain (3.13.21) Pn,ji = Pn,j1 · Cn,ji +Rn,ji , where degRn,ji < degPn,j1 ≤ degPn,ji . Using (3.13.8), we conclude that degCn,ji is uniformly bounded as n→ ∞. This means that, at the expense of passing to another subsequence of N∗, we may assume that there exist polynomials Cji such that Cn,ji = Cji for all n. We let Rn,i := Pn,i for each n and for each i ∈ {1, . . . , r} \ {j2, . . . , js}. Let δi for i ∈ {1, . . . , r} be defined as follows: δi := γi if i 6= j1; and δj1 := γj1 + φCji (γji). Then for each n, using (3.13.21) and the definition of the δi and Rn,i, we obtain (3.13.22) φPn,i(γi) = φRn,i(δi). Using (3.13.8) and the definition of the Rn,i (in particular, the fact that Rn,j1 = Pn,j1 and degRn,ji < degPn,j1 for 2 ≤ i ≤ s), it is immediate to see (3.13.23) 0 < lim inf i=1 q ddegPn,i i=1 q ddegRn,i ≤ lim sup i=1 q ddeg Pn,i i=1 q ddegRn,i < +∞. Moreover, because of (3.13.22) and (3.13.23), we get that (3.13.24) lim sup log | i=1 φPn,i(γi)|v∑r i=1 q ddeg Pn,i if and only if (3.13.25) lim sup log | i=1 φRn,i(δi)|v∑r i=1 q ddegRn,i SIEGEL’S THEOREM FOR DRINFELD MODULES 19 We claim that if ĥv(δj1) ≥ ĥv(γj1), then (3.13.25) holds (and so, also (3.13.24) holds). Indeed, in that case, for large enough n, we have log |φRn,j1 (δj1)|v = q ddegRn,j1 ĥv(δj1)− log |ad|v qd − 1 ≥ qddeg Pn,j1 ĥv(γj1)− log |ad|v qd − 1 = log |φPn,j1 (γj1)|v log |φRn,ji (γji)|v, (3.13.26) where in the last inequality from (3.13.26) we used (3.13.20) and (3.13.14), and that for each i ∈ {2, . . . , s} we have degRn,ji < degPn,ji . Moreover, using (3.13.26) and (3.13.19), together with the definition of the Rn,i and the δi, we conclude that for large enough n, we have ∣∣∣∣∣ φRn,i(δi) ∣∣∣∣∣ = log ∣∣∣φRn,j1 (δj1) = qddeg Pn,j1 ĥv(γj1)− log |ad|v qd − 1 (3.13.27) Because Rn,j1 = Pn,j1 , equations (3.13.8) and (3.13.23) show that (3.13.28) lim sup qddegRn,j1∑r i=1 q ddegRn,i Equations (3.13.27) and (3.13.28) show that we are now in Case 1 for the sequence (Rn,i) n∈N∗ 1≤i≤r . Hence (3.13.29) lim sup log | i=1 φRn,i(δi)|v∑r i=1 q ddegRn,i as desired. Assume from now on that ĥv(δj1) < ĥv(γj1). Because v ∈ MK \ {∞}, using Corollary 3.9 and also using that if i 6= j1, then δi = γi, we conclude ĥv(γi)− ĥv(δi) ≥ Our goal is to prove (3.13.24) by proving (3.13.25). Because we replace some of the polynomials Pn,i with other polynomials Rn,i, it may very well be that (3.13.8) is no longer satisfied for the polynomials Rn,i. Note that in this case, using induction and arguing as in equations (3.13.2) through (3.13.6), we see that lim sup log | j=1 φRn,j (δj)|w∑r j=1 q ddegRn,j 20 D. GHIOCA AND T. J. TUCKER for some place w. This would yield that (see (3.13.22) and (3.13.23)) lim sup log | j=1 φPn,j (γj)|w∑r j=1 q ddeg Pn,j as desired. Hence, we may assume again that (3.13.8) holds. We continue the above analysis this time with the γi replaced by δi. Either we prove (3.13.25) (and so, implicitly, (3.13.24)), or we replace the δi by other elements in Γ, say βi and we decrease even further the sum of their local heights at v: ĥv(δi)− ĥv(βi) ≥ The above process cannot go on infinitely often because the sum of the local heights i=1 ĥv(γi) is decreased each time by at least . Our process ends when we cannot replace anymore the eventual ζi by new βi. Thus, at the final step, we have ζ1, . . . , ζr for which we cannot further decrease their sum of local canonical heights at v. This happens either because all ζi have local canonical height equal to 0, or because we already found a sequence of polynomials Tn,i for which (3.13.30) lim sup log | i=1 φTn,i(ζi)|v∑r i=1 q ddeg Tn,i Since (3.13.31) φPn,i(γi) = φTn,i(ζi), this would imply that (3.12.2) holds, which would complete the proof. Hence, we may assume that we have found a sequence (ζi)1≤i≤r with canonical local heights equal to 0. As before, we let the (Tn,i) n∈N∗ 1≤i≤r be the corresponding sequence of polynomials for the ζi, which replace the polynomials Pn,i. Next we apply the above process to another w ∈ S0 \{∞} for which there exists at least one ζi such that ĥw(ζi) > 0. Note that when we apply the above process to the ζ1, . . . , ζr at the place w, we might replace (at certain steps of our process) the ζi by (3.13.32) φCj (ζj) ∈ Γ. Because for the places v ∈ S0 for which we already completed the above process, ĥv(ζi) = 0 for all i, then by the triangle inequality for the local height, we also have φCj (ζj)  = 0. SIEGEL’S THEOREM FOR DRINFELD MODULES 21 If we went through all v ∈ S0 \ {∞}, and if the above process did not yield that (3.13.24) holds for some v ∈ S \ {∞}, then we are left with ζ1, . . . , ζr ∈ Γ such that for all i and all v 6= ∞, we have ĥv(ζi) = 0. Note that since ĥv(ζi) = 0 for each v 6= ∞ and each i ∈ {1, . . . , r}, then by the triangle inequality for local heights, for all polynomials Q1, . . . , Qr, we have (3.13.33) ĥv φQi(ζi) = 0 for v 6= ∞. Lemma 3.10 and (3.13.33) show that for all polynomials Qi, (3.13.34) D∞ · ĥ∞ φQi(ζi) We repeat the above process, this time for v = ∞. As before, we conclude that either (3.13.35) lim sup log | i=1 φTn,i(ζi)|∞∑r i=1 q ddeg Tn,i or we are able to replace the ζi by some other elements βi (which are of the form (3.13.32)) such that ĥ∞(βi) < ĥ∞(ζi). Using (3.13.34), we conclude that (3.13.36) ĥ∞(ζi)− ĥ∞(βi) ≥ Therefore, after a finite number of steps this process of replacing the ζi must end, and it cannot end with all the new βi having local canonical height 0, because this would mean that all βi are torsion (we already knew that for v 6= ∞, we have ĥv(ζi) = 0, and so, by (3.13.33), ĥv(βi) = 0). Because the βi are nontrivial “linear” combinations (in the φ-module Γ) of the γi which span a torsion-free φ-module, we conclude that indeed, the βi cannot be torsion points. Hence, our process ends with proving (3.13.35) which proves (3.13.24), and so, it concludes the proof of our Proposition 3.12. � Remark 3.14. If there is more than one infinite place in K, then we cannot derive Lemma 3.10, and in particular, we cannot derive (3.13.36). The idea is that in this case, for each nontorsion ζ which has its local canonical height equal to 0 at finite places, we only know that there exists some infinite place where its local canonical height has bounded denominator. However, we do not know if that place is the one which we analyze at that particular moment in our process from the proof of Proposition 3.12. Hence, we would not necessarily be able to derive (3.13.36). Now we are ready to prove Theorem 2.4. 22 D. GHIOCA AND T. J. TUCKER Proof of Theorem 2.4. Let (γi)i be a finite set of generators of Γ as a module over A = Fq[t]. At the expense of replacing S with a larger finite set of places of K, we may assume S contains all the places v ∈MK which satisfy at least one of the following properties: 1. ĥv(γi) > 0 for some 1 ≤ i ≤ r. 2. |γi|v > 1 for some 1 ≤ i ≤ r. 3. |α|v > 1. 4. φ has bad reduction at v. Expanding the set S leads only to (possible) extension of the set of S-integral points in Γ with respect to α. Clearly, for every γ ∈ Γ, and for every v /∈ S we have |γ|v ≤ 1. Therefore, using 3., we obtain γ ∈ Γ is S-integral with respect to α⇐⇒ |γ − α|v = 1 for all v ∈MK \ S. (3.14.1) Moreover, using 1. from above, we conclude that for every γ ∈ Γ, and for every v /∈ S, we have ĥv(γ) = 0 (see the proof of Fact 3.13). Next we observe that it suffices to prove Theorem 2.4 under the assump- tion that Γ is a free φ-submodule. Indeed, because A = Fq[t] is a principal ideal domain, Γ is a direct sum of its finite torsion submodule Γtor and a free φ-submodule Γ1 of rank r, say. Therefore, γ∈Γtor γ + Γ1. If we show that for every γ0 ∈ Γtor there are finitely many γ1 ∈ Γ1 such that γ1 is S-integral with respect to α − γ0, then we obtain the conclusion of Theorem 2.4 for Γ and α (see (3.14.1)). Thus from now on, we assume Γ is a free φ-submodule of rank r. Let γ1, . . . , γr be a basis for Γ as an Fq[t]-module. We reason by contradiction. φPn,i(γi) ∈ Γ be an infinite sequence of elements S-integral with respect to α. Because of the S-integrality assumption (along with the assumptions on S), we conclude that for every v /∈ S, and for every n we have log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i Thus, using the product formula, we see that lim sup log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i = lim sup log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i SIEGEL’S THEOREM FOR DRINFELD MODULES 23 On the other hand, by Proposition 3.12, there is some place w ∈ S and some number δ > 0 such that lim sup log | i=1 φPn,i(γi)− α|w∑r i=1 q ddeg Pn,i = δ > 0. So, using Lemma 3.11, we see that lim sup log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i v 6=w lim inf log | i=1 φPn,i(γi)− α|v∑r i=1 q ddeg Pn,i + lim sup log | i=1 φPn,i(γi)− α|w∑r i=1 q ddegPn,i ≥ 0 + δ Thus, we have a contradiction which shows that there cannot be infinitely many elements of Γ which are S-integral for α. � References [Bak75] A. Baker, Transcendental number theory, Cambridge University Press, Cam- bridge, 1975. [BIR05] M. Baker, S. I. Ih, and R. Rumely, A finiteness property of torsion points, 2005, preprint, Available at arxiv:math.NT/0509485, 30 pages. [Bos99] V. Bosser, Minorations de formes linéaires de logarithmes pour les modules de Drinfeld, J. Number Theory 75 (1999), no. 2, 279–323. [Bre05] F. Breuer, The André-Oort conjecture for products of Drinfeld modular curves, J. reine angew. Math. 579 (2005), 115–144. [Dav95] S. David, Minorations de formes linéaires de logarithmes elliptiques, Mem. Soc. Math. France 62 (1995), 143 pp. [Den92a] L. Denis, Géométrie diophantienne sur les modules de Drinfel′d, The arithmetic of function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 285–302. [Den92b] , Hauteurs canoniques et modules de Drinfel′d, Math. Ann. 294 (1992), no. 2, 213–223. [EY03] B. Edixhoven and A. 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Tucker, Equidistribution and integrality for Drinfeld mod- ules, submitted for publication, 29 pages, 2006. [GT07] , A dynamical version of the Mordell-Lang conjecture, submitted for pub- lication, 14 pages, 2007. [Poo95] B. Poonen, Local height functions and the Mordell-Weil theorem for Drinfel′d modules, Compositio Math. 97 (1995), no. 3, 349–368. [Sca02] T. Scanlon, Diophantine geometry of the torsion of a Drinfeld module, J. Num- ber Theory 97 (2002), no. 1, 10–25. [Ser97] J.-P. Serre, Lectures on the Mordell-Weil theorem, third ed., Aspects of Mathe- matics, Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a fore- word by Brown and Serre. [Sie29] C. L. Siegel, Über einige anwendungen diophantisher approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), 41–69. [Sil93] J. H. Silverman, Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793–829. [SUZ97] L. Szpiro, E. Ullmo, and S. Zhang, Equirépartition des petits points, In- vent. Math. 127 (1997), 337–347. [Tag93] Y. Taguchi, Semi-simplicity of the Galois representations attached to Drinfel′d modules over fields of “infinite characteristics”, J. Number Theory 44 (1993), no. 3, 292–314. [Yaf06] A. Yafaev, A conjecture of Yves André’s, Duke Math. J. 132 (2006), no. 3, 393–407. [Zha98] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165. Dragos Ghioca, Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1, E-mail address: [email protected] Thomas Tucker, Department of Mathematics, Hylan Building, University of Rochester, Rochester, NY 14627, E-mail address: [email protected] 1. Introduction 2. Notation 2.1. Drinfeld modules 2.2. Valuations and Weil heights 2.3. Canonical heights 2.4. Completions and filled Julia sets 2.5. The coefficients of t 2.6. Integrality and reduction 3. Proofs of our main results References

0704.1332

On the Exponential Decay of the n-point Correlation Functions and the Analyticity of the Pressure

On the Exponential Decay of the n-Point Correlation Functions and the Analyticity of the Pressure. Assane Lo April 1st, 2007 Abstract The goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n-point correlation functions for certain unbounded models of Kac type. The methods are based on estimating higher order derivatives of the solution of the Witten Laplacian equation on one forms associated with the hamiltonian of the system. We also provide a formula for the Taylor coefficients of the pressure that is suitable for a direct proof the analyticity. 1 Introduction In recent publications [66] we have given a generalization to the higher dimen- sional case of the exponential decay of the two-point correlation functions for models of Kac type. In this paper, we shall establish a weak exponential decay of the n-point correlation functions, and provide an exact formula suitable for a direct proof the analyticity of the pressure. Let Λ be a finite subset of Zd, and consider a Hamiltonian Φ of the phase space RΛ. We shall focus on the case where Φ = ΦΛ is given by ΦΛ(x) = + Ψ(x), (1) under suitable assumptions on Ψ. Recall that if 〈f〉 denote the mean value of f with respect to the Gibbs measure −Φ(x) the covariance of two functions g and h is defined by cov(g, h) = 〈(g − 〈g〉)(h− 〈h〉)〉 . (2) If one wants to have an expression of the covariance in the form cov(g, h) = 〈∇h ·w〉L2(Rn,Rn;e−Φdx) , (3) http://arxiv.org/abs/0704.1332v3 for a suitable vector field w we get, after observing that ∇h = ∇(h− 〈h〉), and integrating by parts, cov(g, h) = (h− 〈h〉)(∇Φ−∇) ·we−Φ(x)dx. (4) (Here we have assumed that g and h are functions of polynomial growth ). This leads to the question of solving the equation g − 〈g〉 = (∇Φ−∇) ·w. (5) Now, trying to solve this above equation with w = ∇f, we obtain the equation g − 〈g〉 = (−∆+∇Φ ·∇) f 〈f〉 = 0. The existence and smoothness of the solution of this equation were established in [8] (see also [66]) under certain assumptions on Φ. Now taking gradient on both sides of (6), we get ∇g = [(−∆+∇Φ ·∇)⊗ Id+HessΦ]∇f. (7) We then obtain the emergence of two differential operators: Φ := −∆+∇Φ ·∇ (8) Φ := A Φ ⊗ Id+HessΦ. (9) Here the tensor notation means that A Φ acts diagonally on the vector field solution to produce a system of equations. cov(g, h) = (1)−1 Φ ∇g ·∇h −Φ(x) dx. (10) The operators A Φ and A Φ are called the Helffer-Sjöstrand’s operators. These are unbounded operators acting on the weighted spaces L2(RΛ, e−Φdx) and L2(RΛ,RΛ, e−Φdx) respectively. The formula (10) was introduced by Helffer and Sjöstrand and in some sense is a generalization of Brascamp-Lieb inequality as already pointed out in [1]. The unitary transformation UΦ : L 2(RΛ) → L2(RΛ, e−Φdx) u 7−→ e will allow us to work with the unweighted spaces L2(RΛ) and L2(RΛ,RΛ) by converting the operators A Φ and A Φ into equivalent operators Φ = −∆+ ⊗ I+HessΦ. (12) respectively. The equivalence can be seen by observing that Φ = e −Φ/2 ◦A Φ ◦ e . (13) The operators W Φ and W Φ are unbounded operators acting on 2(RΛ) and L2(RΛ,RΛ) respectively. These are in fact, the euclidean versions of the Laplacians on zero and one forms respectively, already introduced by E. Witten [18] in the context Morse theory. The equivalence between the operators A Φ and Witten’s Laplacians was first observed by J. Sjöstrand [13] in 1996. 2 Higher Order Exponential Estimates We shall consider a Hamiltonian of the form Φ(x) = ΦΛ(x) = + Ψ(x), x ∈ RΛ. where |∂α∇Ψ| ≤ Cα, ∀α ∈ N . (14) g will denote a smooth function on RΓ with lattice support Sg = Γ (& Λ) . We shall identify g with g̃ defined on RΛ and shall assume that |∂α∇g| ≤ Cα ∀α ∈ N |Γ|. (15) As in [66] , we shall momentarily assume that Ψ is compactly supported in RΛ and g is compactly supported in RΓ but these assumptions will be relaxed later Let M be the diagonal matrix M = (δijρ(i))i,j∈Λ where ρ is a weight function on Λ satisfying e−λ ≤ ρ (i) ≤ eλ, if i ∼ j for some λ > 0. (16) Assume also that for every M as above, there exists δo ∈ (0, 1) such that −1HessΦ(x)Ma, a ≥ δoa , ∀x ∈ RΛ, ∀a ∈ RΛ. (17) For instance, the d−dimensional nearest neighbor Kac model ΦΛ(x) = ln cosh (xi + xj) satisfies this assumption for ν small enough. See [66] for details. The following theorem has been proved in [66]: Theorem 1 (A. Lo [66]) Let g be a smooth function with compact support on RΓ satisfying |∂α∇g| ≤ Cα ∀α ∈ N |Γ| (18) and Φ is as above. If f is the unique C∞−solution of the equation −∆f +∇Φ ·∇f = g − 〈g〉 〈f〉L2(µ) = 0, then ∑ f2xi(x)e 2κd(i,Sg) ≤ C ∀x ∈ RΛ. κ and C are positive constants. C could possibly depend on the size of the support of g but does not depend on Λ and f. We now propose to generalize this theorem to higher order derivatives. Proposition 2 If in addition to the assumptions of theorem 1, Φ satisfies the following growth condition: for κ > 0 as above, j,i1,...,ik∈Λ Φ2xjxi1 ...xik (x)e2κd({i1,...,ik},Sg) ≤ Ck ∀x ∈ R Λ, for k ≥ 2 (19) for some Ck > 0 not dependent on Λ and f , then for any k ≥ 1, f satisfies i1,...,ik∈Λ f2xi1 ...xik (x)e2κd({i1,...,ik},Sg) ≤ Ck,g ∀x ∈ R Λ (20) where Ck,g > 0 is a constant that depends on the size of the support of g but not on Λ and f. Proof. The case k = 1 being theorem 1, we assume for induction that the result is true when k is replaced by k̂ < k with k̂ ≥ 2. For k ≥ 2(see [8] for details), we have g, t1 ⊗ ...⊗ tk = (∇Φ ·∇−∆) f, t1 ⊗ ...⊗ tk f, t1 ⊗ ...⊗HessΦtj ⊗ ...⊗ tk A∪B={1,...,k},A∩B=∅ #B≤k−2 〈tA(∂x)∇Φ, tB(∂x)∇f〉 . In the right hand side of this last above equality, we have used the notation tJ(∂x)u := u, t1 ⊗ ...⊗ t#J Now fix i2, ..., ik ∈ Λ. Because ∇ f(x) → 0 as |x| → ∞ (see [66]), we consider xo ∈ R Λ that maximizes x 7−→ f2xi1 ...xik ρ2(i1, ..., ik) where ρ(i1, ..., ik) = e κd({i1,...,ik},Sg). Observe here that xo could possibly depend on i2, ..., ik ∈ Λ. Choose ρ(i1, ..., ik)fxi1 ...xik (xo) tj = eij if j = 2, ..., k Let M1 be the diagonal matrix M1 = (δsi1ρ(i1, ..., ik))si1 Mj = I if j 6= 1 (21) in particular, we have g,M1t1 ⊗ ...⊗Mktk = (∇Φ ·∇−∆) f,M1t1 ⊗ ...⊗Mktk f,M1t1 ⊗ ...⊗HessΦMjtj ⊗ ...⊗Mktk A∪B={1,...,k},A∩B=∅ #B≤k−2 〈tMA(∂x)∇Φ, tMB(∂x)∇f〉 tMA(∂x)u := f,M1tij1 ⊗ ...⊗M#Atij#A , ji ∈ A. As in [66], the function x 7−→ f(x),M1t1 ⊗ ...⊗Mktk achieves its maximum at xo. Using the notation ΦxiA = Φxiℓ1 ...xiℓr if A = {ℓ1, ...ℓr} ⊂ {1, ...k} , we therefore have gxi1 ...xik (xo)ρ(i1, ..., ik) 2fxi1 ...xik fxi1 ...xik (xo)fxsxi2 ...xik (xo)ρ(i1, ..., ik) 2Φxsxi1 (xo) fxi1 ... xs ...xik (xo)fxi1 ...xik (xo)ρ(i1, ..., ik) 2Φxsxij (xo) A∪B={1,...,k},A∩B=∅ #B≤k−2 ∇ΦxiA fxi1 ...xik (xo)ρ(i1, ..., ik) ,∇fxiB (xo) A∪B={1,...,k},A∩B=∅ #B≤k−2 ∇ΦxiA , ∇fxiB (xo)fxi1 ...xik (xo)ρ(i1, ..., ik) Equivalently gxi1 ...xik (xo)ρ(i1, ..., ik) 2fxi1 ...xik fxi1 ...xik (xo)fxs...xik (xo)ρ(i1, ..., ik) 2Φxsxi1 (xo) fxi1 ... xs ...xik (xo)fxi1 ...xik (xo)ρ(i1, ..., ik) 2Φxsxij (xo) A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik) fxiBxs(xo) A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik) Now taking summation over i2, ..., ik, we get i2,...,ik∈Λ gxi1 ...xik (xo)ρ(i1, ..., ik) 2fxi1 ...xik i2,...,ik∈Λ fxi1 ...xik (xo)fxs...xik (xo)ρ(i1, ..., ik) 2Φxsxi1 (xo) i2,...,ik∈Λ fxi1 ... xs ...xik (xo)fxi1 ...xik (xo)ρ(i1, ..., ik) 2Φxsxij (xo) i2,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik) 2fxiBxs(xo) i2,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik) Next, we propose to estimate each term of the right hand side of this above inequality. i2,...,ik∈Λ fxi1 ...xik (xo)fxs...xik (xo)ρ(i1, ..., ik) 2Φxsxi1 (xo) i2,...,ik∈Λ ∇fxi2 ...xik (xo),HessΦM1t1 i2,...,ik∈Λ M1∇fxi2 ...xik (xo),M 1 HessΦM1t1 i2,...,ik∈Λ 1 HessΦM1t1 i2,...,ik∈Λ i1,...ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) Similarly, it is easy to see that i2,...,ik∈Λ fxi1 ... xs ...xik (xo)fxi1 ...xik (xo)ρ(i1, ..., ik) 2Φxsxij (xo) ≥ (k − 1)δ0 i1,...ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) To estimate the last two terms, we have i2,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ∣ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik) 2fxiBxs(xo) i1,...,ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) i1,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ∣ΦxiAxsρ(i1, ..., ik)fxiBxs(xo) To estimate the second factor of the right hand side of this last above inequality, we have i1,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ∣ΦxiAxs(xo)ρ(i1, ..., ik)fxiBxs(xo) i1,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxs(xo)ρ(i1, ..., ik)fxiBxs(xo) i1,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 Φ2xiAxs (xo)ρ 2(i1, ..., ik) ρ2(i1, ..., ik)f xiBxs i1,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 Φ2xiAxs (xo)e 2κd({ij :j∈A},Sg) e2κd({ij :j∈B}∪{s},Sg)f2xiBxs ≤ Ck. This last inequality above follows from the induction assumption and that of Thus, i2,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxi1 ...xik (xo)ρ(i1, ..., ik) 2fxiBxs(xo) ≥ −Ck i1,...,ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) Similarly, we have i2,...,ik∈Λ A∪B={1,...,k},A∩B=∅ #B≤k−2 ΦxiAxsfxiBxs(xo)fxi1 ...xik (xo)ρ(i1, ..., ik) ≥ −Ck i1,...,ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) We then finally get i1,...,ik∈Λ gxi1 ...xik (xo)ρ(i1, ..., ik) 2fxi1 ...xik ≥ kδo i1,...ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) i1,...,ik∈Λ xi1 ...xik (xo)ρ(i1, ..., ik) i1,...,ik∈Λ xi1 ...xik (xo)ρ(i1, ..., ik) 2 = 0 then there is nothing to prove, otherwise we have, after using Cauchy-Schwartz and dividing by i1,...,ik∈Λ f2xi1 ...xik (xo)ρ(i1, ..., ik) i1,...ik∈Λ xi1 ...xik (xo)ρ(i1, ..., ik) i1,...,ik∈Λ g2xi1 ...xik + Ck,g ≤ Ck,g. � 3 Relaxing the Assumptions of Compact sup- As in [8], we consider the family cutoff functions χ = χε (22) (ε ∈ [0, 1]) in C∞o (R) with value in [0, 1] such that χ = 1 for |t| ≤ ε−1 ∣χ(k)(t) ∣ ≤ Ck for k ∈ N . We then introduce Ψε(x) = χε(|x|)Ψ(x) x ∈ R Λ (23) gε(x) = χε(|x|)g(x) x ∈ R . (24) A straightforward computation (see [66]) shows that Ψε(x) and gε(x).satisfy |∂α∇Ψε| ≤ Cα +Oα,Λ(ε), ∀α ∈ N |Λ|. (25) |∂α∇gε| ≤ Cα +Oα,Λ(ε), ∀α ∈ N , (26) and that M−1HessΦε(x)M ≥ δ , 0 < δ′ < 1. in the sense of (17) (27) It then only remains to check that j,i1,...,ik∈Λ Ψ2εxjxi1 ...xik (x)e2κd({i1,...,ik},Sg) ≤ Ck + Ok,Λ(ε) ∀x ∈ R Λ, ∀k ≥ 2 where Ck is a positive constant that does not depend on f and Λ. Ψε(x) = χε(r)Ψ(x) Let α be such that |α| ≥ 3. Using Leibniz’s formula, we have |∂αΨε| ≤ ∣∂βχε(r)∂ ∣ (29) ≤ |∂αχε(r)Ψ|+ |∂ β 6=0 ∣∂βχε(r)∂ ∣ . (30) Assuming that Ψ(0) = 0 and write Ψ(x) = x ·∇Ψ(sx)ds |∂αχε(r)Ψ(x)| ≤ ∣xj1∂ αχε(r)Ψxj1 (sx) ≤ C |r∂αχε(r)| . Now using the fact that χε(r) = Oα(ε), we have |∂αχε(r)Ψ(x)| = Oα,Λ(ε). Finally, using the fact that χε(r) = Oβ(ε) for every |β| ≥ 1, (31) it is then easy to see that β 6=0 χε(r)∂ = Oα,Λ(ε). (32) j,i1,...,ik∈Λ Ψ2εxjxi1 ...xik (x)e2κd({i1,...,ik},Sg) ≤ Ck,g+ Ok,Λ(ε) ∀x ∈ R Λ, ∀k ≥ 2. Now using the arguments developed in [8] (see also [66]) about the conver- gence of the corresponding solutions as ε → 0, we obtain: Proposition 3 If g(0) = Ψ(0) = 0, then Proposition 2 holds without the as- sumptions of compact support on Ψ and g. 4 The n-Point Correlation Functions The higher order correlation is defined as 〈g1, ..., gk〉 := 〈(g1 − 〈g1〉) ... (gk − 〈gk〉)〉 . (34) For simplicity we shall take k = 3 and Φ is as in proposition 2. Let g1, g2, and g3 be smooth functions satisfying (15) and fi i = 1, 2, 3 shall denote the unique solution of the system −∆fi +∇Φ ·∇fi = gi − 〈gi〉 L2(µ) 〈fi〉L2(µ) = 0. Recall that ∇fi = A (1)−1 Φ ∇gi. For an arbitrary smooth function c, it is easy to see that 〈c(x) (gi − 〈gi〉)〉 = 〈∇fi ·∇c〉 . A direct computation shows that 〈g1, g2,g3〉 = 〈∇f3 · (Hessf1)∇g2〉+ 〈∇f3 · (Hessg2)∇f1〉 + 〈∇f2 · (Hessf1)∇g3〉+ 〈∇f2 · (Hessg3)∇f1〉 . Let us now estimate each term of the right and side of this equality. Using Cauchy-Schwartz, and proposition 2, it is easy to see that |〈∇f3 · (Hessf1)∇g2〉| ≤ Ce −κ1d(Sg2 ,Sg1) |〈∇f3 · (Hessg2)∇f1〉| ≤ Ce −κ1d(Sg2 ,Sg1), |〈∇f2 · (Hessf1)∇g3〉| ≤ Ce −κ1d(Sg3 ,Sg1) |〈∇f2 · (Hessg3)∇f1〉| ≤ Ce −κ1d(Sg3 ,Sg1) Here the constants C only depends on the size of the support of the gi’s. and κ1 > 0. |〈g1, g2,g3〉| ≤ C −κ1d(Sg2 ,Sg1) + e−κ1d(Sg3 ,Sg1) If g1 = xi, g2 = xj , and g3 = xk, we obtain |〈(xi − 〈xi〉) (xj − 〈xj〉) (xk − 〈xk〉)〉| ≤ C e−κ1d(i,j) + e−κ1d(i,k) Thus if d > 1, we obtain this weak exponential decay of the truncated correla- tions in the sense that the exponential decay occurs as you simultaneously pull the spins away from a fixed one. Note that in the one dimensional case, we obtain a stronger exponential decay due to the fact that i ≤ j ≤ k =⇒ d(i, k) = d(i, j) + d(j, k). This was already pointed out in [8]. 5 The Analyticity of the Pressure In this section, we attempt to study a direct method for the analyticity of the pressure for certain classical convex unbounded spin systems. It is central in Statistical Mechanics to study the differentiability or even the analyticity of the pressure with respect to some distinguished thermodynamic parameters such as temperature, chemical potential or external field. In fact the analytic behavior of the pressure is the classical thermodynamic indicator for the absence or existence of phase transition. The most famous result on the analyticity of the pressure is the circle theorem of Lee and Yang [28]. This theorem asserts the following: consider a {−1, 1}−valued spin system with ferromagnetic pair interaction and external field h and regard the quantity z = eh as a complex parameter, then all zeroes of all partition functions (with free boundary condition), considered as functions of z lie in the complex unit circle. This theorem readily implies that the pressure is an analytic function of h in the region h > 0 and h < 0. Heilmann [29] showed that the assumption of pair interaction is necessary. A transparent approach to the circle theorem was found by Asano [30] and developed further by Ruelle [31],[32], Slawny [33], and Gruber et al [34]. Griffiths [35] and Griffiths- Simon [36] found a method of extending the Lee-Yang theorem to real-valued spin systems with a particular type of a priory measure. Newman [37] proved the Lee-Yang theorem for every a priory measure which satisfies this theorem in the particular case of no interaction. Dunlop [38],[39] studied the zeroes of the partition functions for the plane rotor model. A general Lee-Yang theorem for multicomponent systems was finally proved by Lieb and Sokal [40]. For further references see Glimm and Jaffe [41]. The Lee-Yang theorem and its variants depend on the ferromagnetic char- acter of the interaction. There are various other way of proving the infinite differentiability or the analyticity of the pressure for (ferromagnetic and non ferromagnetic) systems at high temperatures, or at low temperatures, or at large external fields. Most of these take advantage of a sufficiently rapid decay of correlations and /or cluster expansion methods. Here is a small sample of rele- vant references. Bricmont, Lebowitz and Pfister [42], Dobroshin [43], Dobroshin and Sholsman [44],[45], Duneau et al [46],[47],[48], Glimm and Jaffe [41],[49], Is- rael [50], Kotecky and Preiss [51], Kunz [52], Lebowitz [53],[54], Malyshev [55], Malychev and Milnos [56] and Prakash [57]. M. Kac and J.M. Luttinger [58] obtained a formula for the pressure in terms of irreducible distribution functions. We propose a new way of analyzing the analyticity of the pressure for certain unbounded models through a representation by means of the Witten Laplacians of the coefficients in the Taylor series expansion. The methods known up to now rely on complicated indirect arguments. 6 Towards the analyticity of the Pressure Let Λ be a finite domain in Zd (d ≥ 1) and consider the Hamiltonian of the phase space given by, Φ(x) = ΦΛ(x) = + Ψ(x), x ∈ RΛ. (36) where |∂α∇Ψ| ≤ Cα, ∀α ∈ N |Λ|, (37) HessΦ(x) ≥ δo, 0 < δo < 1. (38) Let g is a smooth function on RΓ with lattice support Sg = Γ. We identified with g̃ defined on RΛ by g̃(x) = g(xΓ) where x = (xi)i∈Λ and xΓ = (xi)i∈Γ (39) and satisfying |∂α∇g| ≤ Cα ∀α ∈ N |Γ| (40) Under the additional assumptions that Ψ is compactly supported in RΛ and g is compactly supported in RΓ, it was proved in [66] (see also [8]) that the equation −∆f +∇Φ ·∇f = g − 〈g〉 〈f〉L2(µ) = 0 has a unique smooth solution satisfying ∇kf(x) → 0 as |x| → ∞ for every k ≥ 1. Recall also that ∇f is a solution of the system (−∆+∇Φ ·∇)∇f +HessΦ∇f = ∇g in RΛ. (41) As in [66] and [8], these assumptions will be relaxed later on. ΦtΛ(x) = Φ(x) − tg(x), (42) where x = (xi)i∈Λ, and assume additionally that g satisfies Hessg ≤ C. (43) We consider the following perturbation θΛ(t) = log −ΦtΛ(x) . (44) Denote by dxe−Φ Λ(x) (45) < · >t,Λ= · dxe−Φ . (46) 7 Parameter Dependency of the Solution From the assumptions made on Φ and g, it is easy to see that there exists T > 0 such that or every t ∈ [0, T ), ΦtΛ(x) satisfies all the assumptions required for the solvability, regularity and asymptotic behavior of the solution f(t) associated with the potential ΦtΛ(x). Thus, each t ∈ [0, T ) is associated with a unique C∞−solution, f(t) of the equation f(t) = g − 〈g〉 L2(µ) 〈f(t)〉L2(µ) = 0. Hence, v(t) = ∇g (47) where v(t) = ∇f(t). Notice that the map t 7−→ v(t) is well defined and {v(t) : t ∈ [0, T )} is a family of smooth solutions on RΛ satisfying ∂αv(t) → 0 as |x| → ∞ ∀α ∈ N|Λ| and for each t ∈ [0, T ) and corresponding to the family of potential ΦtΛ : t ∈ [0, T ) . (48) Let us now verify that v is a smooth function of t ∈ (0, T ).We need to prove that for each t ∈ (0, T ), the limit v(t+ ε)− v(t) exists. Let vε(t) = v(t + ε)− v(t) We use a technique based on regularity estimates to get a uniform control of vε(t) with respect to ε. With ε small enough, we have 0 = −∆ v(t+ ε)− v(t) ∇Φt+ε ·∇v(t+ ε)−∇Φt ·∇v(t) HessΦt+εv(t+ ε)−HessΦtv(t) Equivalently, v(t + ε)− v(t) ∇Φt+ε ·∇ [v(t + ε)− v(t)] +HessΦt+ε v(t + ε)− v(t) HessΦt+ε −HessΦt v(t)− ∇Φt+ε −∇Φt ·∇v(t) = Hessgv(t) +∇g ·∇v(t) −∆vε(t) +∇Φt+ε ·∇vε(t) +HessΦt+εvε(t) = Hessgv(t) +∇g ·∇v(t) Let w(t) be the unique C∞−solution of the system −∆w(t) +∇Φt ·∇w(t) +HessΦtw(t) = Hessgv(t) +∇g ·∇v(t). (49) Recall that the unitary transformation UΦt+ε, allows us to reduce −∆vε(t) +∇Φt+ε ·∇vε(t) +HessΦt+εvε(t) = Hessgv(t) +∇g ·∇v(t) into ( |∇Φt+ε| ∆Φt+ε Vε +HessΦt+εVε = [Hessgv(t) +∇g ·∇v(t)] e−Φ t+ε/2 where Vε = vε(t)e−Φ t+ε/2. Remark 4 This unitary transformation already mentioned in the introduction was introduced in the proof of the existence of solution (see [66] ) to avoid work- ing with the weighted spaces L2(RΛ,RΛ, e−Φdx). The proof was based on Hilbert space method. The method consists of determining an appropriate function space and an operator which is a natural realization of the problem. In this particular problem, the function spaces to be considered are the Sobolev spaces BkΦ(R defined by BkΦ(R u ∈ L2(RΛ) : ZℓΦ∂ αu ∈ L2(RΛ) ∀ ℓ+ |α| ≤ k where These are subspaces of the well known Sobolev spaces W k,2(RΛ), k ∈ N. Taking scalar product with Vε on both sides of (51), we get ∇Φt+ε HessΦt+εVε ·Vεdx = [Hessgv(t) +∇g ·∇v(t)] e−Φ t+ε/2 ·Vεdx. Now using the uniform strict convexity on the left hand side and Cauchy- Schwartz on the right hand side, we obtain ‖Vε‖B0 ≤ Ct for small enough ε. (54) We then deduce that |∇Φt+ε| Vε = q̃ε (55) where q̃ε = [Hessgv(t) +∇g ·∇v(t)] e −Φt+ε/2+ ∆Φt+ε Vε −HessΦt+εVε (56) is bounded in B0 uniformly with respect to ε for ε small enough. Taking again scalar product with Vε on both sides of (55) and integrating by parts, we obtain ‖∇Vε‖ |∇Φt+ε| ≤ ‖q̃ε‖L2 ‖V ε‖L2 (57) It follows that Vε is uniformly bounded with respect to ε in B1 for ε small enough. Next, observe that |∇Φt| Vε = q̂ε (58) where q̂ε = q̃ε − |∇Φt+ε −∇Φt| (∇Φt+ε −∇Φt) ·∇Φt Vε (59) = q̃ε − ε2 |∇g| ε∇g ·∇Φt Vε (60) is uniformly bounded in B0 with respect to ε for small enough ε. Using regu- larity, it follows that for small enough ε, Vε is uniformly bounded in B2Φt with respect to ε.This implies that q̂ε is uniformly bounded in B Φt for ε small enough. Again, we can continue by a bootstrap argument to consequently get that for ε small enough, Vε is uniformly bounded in BkΦt with respect to ε for any k. V = w(t)e−Φ We have ( |∇Φt| V +HessΦtV = [Hessgv(t) +∇g ·∇v(t)] e−Φ . (61) Now combining this equation with (51), we obtain |∇Φt| (Vε −V) +HessΦt (Vε −V) = − [Hessgv(t) +∇g ·∇v(t)] e−Φ t/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ t+ε/2 |∇Φt| |∇Φt+ε| ∆Φt+ε +(HessΦt −HessΦt+ε)Vε. Now let us check that for small enough ε, the right hand side of (62) is O(ε) in B0. For the first term, we have − [Hessgv(t) +∇g ·∇v(t)] e−Φ t/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ t+ε/2 = [Hessgv(t) +∇g ·∇v(t)] e−Φ eεg/2 − 1 [Hessgv(t) +∇g ·∇v(t)] ge−Φ Thus for ε small enough ∥− [Hessgv(t) +∇g ·∇v(t)] e −Φt/2 + [Hessgv(t) +∇g ·∇v(t)] e−Φ t+ε/2 ≤ Cε. For the second term, we have |∇Φt| |∇Φt+ε| ∣∇Φt+ε ∣∇Φt+ε ∣∇Φt+ε ∣+ ε |∇g| Using now the fact that Vε is uniformly bounded in BkΦt with respect to ε for any k, we see that the second term of the right hand side of (62) is O(ε) in B0Φt . The last two terms of the right hand side of (62) are obviously O(ε) in B0Φt . From the same regularity argument as above, we get that Vε −V is O(ε) in B2Φt . Again iterating the regularity argument, we obtain that for small enough ε, Vε −V is O(ε) in BkΦt for every k. We have proved: Proposition 5 Under the above assumptions on Φ and g, there exists T > 0 so that for each t ∈ (0, T ), vε(t) converges to w(t) in C∞. Remark 6 The proposition establishes that v(t) is differentiable in t and is given by the unique C∞−solution w(t) of the system −∆w(t) +∇Φt ·∇w(t) +HessΦtw(t) = Hessgv(t)−∇g ·∇v(t). (63) Iterating this argument, we easily get that, v(t) is smooth in t ∈ (0, T ). Now we are ready for the following: 8 A Formula for the Taylor Coefficients First observe that for an arbitrary suitable function f(t) = f(t, w) < f(t) >t,Λ=< f ′(t) >t,Λ +cov(f, g). (64) Hence, < f(t) >t,Λ=< f ′(t) >t,Λ + < A (1)−1 (∇f) ·∇g >t,Λ . (65) Agf := A (1)−1 (∇f) ·∇g. (66) Thus, < f(t) >t,Λ=< f >t,Λ . (67) The linear operator +Ag will be denoted by Hg. To obtain a formula for the coefficients in the Taylor expansion of θΛ(t) = log −ΦtΛ(x) , (68) we first the derivatives of θΛ(t) in terms of Hg Λ(t) =< g >t,Λ=< g >t,Λ=< H gg >t,Λ; Λ(t) = < g >t,Λ=< A (1)−1 (∇g) ·∇g >t,Λ=< g >t,Λ; Λ (t) = (1)−1 (∇g) ·∇g >t,Λ=< (1)−1 (∇g) ·∇g (1)−1 (1)−1 (∇g) ·∇g ·∇g >t,Λ g >t,Λ . By induction it is easy to see that Λ (t) =< g >t,Λ=< H (n−1) g g >t,Λ (∀n ≥ 1) Next, we propose to find a simpler formula for θ Λ (t) that only involves Ag. Hgg = A (1)−1 Φt (∇g) ·∇g = Agg ∇f ·∇g + (1)−1 (1)−1 Φt (∇g) ·∇g ·∇g (69) where f satisfies the equation ∇f = A (1)−1 (∇g) . (70) With v(t) = ∇f, as before, we get ∇f ·∇g = A (1)−1 Φt (Hessgv(t) +∇g ·∇v(t)) ·∇g and H2g becomes H2gg = A (1)−1 (Hessgv(t) +∇g ·∇v(t)) +∇ (1)−1 (∇g) ·∇g (1)−1 2∇ (Agg) ·∇g = 2A2gg. Proposition 7 If θΛ(t) = log −Φt(x) where Φt(x) = ΦΛ(x)− tg(x) is as above then θ Λ (t), the nth− derivative of θΛ(t) is given by the formula Λ(t) =< g >t,Λ, and for n ≥ 1 Λ (t) = (n− 1)! < A g g >t,Λ . Proof. We have already established that Λ (t) =< H g g >t,Λ for n ≥ 1. It then only remains to prove that Hn−1g g = (n− 1)!A g g for n ≥ 1. The result is already established above for n = 1, 2, 3, . By induction, assume Hn−1g g = (n− 1)!A g g . if n is replaced by ñ ≤ n. g g = (n− 1)!An−1g g = (n− 1)! An−1g g +A An−1g g = (1)−1 An−2g g = ∇ϕn ·∇g where ∇ϕn = (1)−1 We obtain, ∇ϕn = A (1)−1 ∇An−2g g +Hessg∇ϕn +∇g ·∇ (∇ϕn) We then have An−1g g = ∇ϕn ·∇g (1)−1 ∇An−2g g +Hessg∇ϕn +∇g ·∇ (∇ϕn) (1)−1 ∇An−2g g +∇ (∇ϕn ·∇g) An−2g g +Ag An−2g g = AgHg = AgHg (n− 2)! H(n−2)g g (from the induction hypothesis) (n− 2)! (n−1) (n− 2)! (n− 1)!An−1g g (still by the induction hypothesis) = (n− 1)Ang g. Thus, Hng g = (n− 1)! (n− 1 + 1)A = n!Ang g Proposition 8 If g(0) = 0, then the formula Λ (t) = (n− 1)! < A g g >t,Λ, n ≥ 2 still holds if we no longer require Ψ and g to be compactly supported in RΛ. Proof. As in [8], consider the family cutoff functions χ = χε (71) (ε ∈ [0, 1]) in C∞o (R) with value in [0, 1] such that χ = 1 for |t| ≤ ε−1 ∣χ(k)(t) ∣ ≤ Ck for k ∈ N We could take for instance χε(t) = f(ε ln |t|) for a suitable f . We then introduce Ψε(x) = χε(|x|)Ψ, x ∈ R Λ (72) gε(x) = χε(|x|)g x ∈ R Γ (73) One can check that both Ψε(x) and gε(x) satisfies the assumptions made above on Ψ and g. Now consider the equation −∆fε +∇Φ ε ·∇fε = gε− < gε >t,Λ. (74) which implies −∆+∇Φtε ·∇ ⊗ vε +HessΦ εvε = ∇gε (75) where vε= ∇fε It was proved in [8] that vε = A (1)−1 ∇gε converges in C ∞ to A (1)−1 ∇g as ε → 0. Proposition 9 Let PΛ(t) = θΛ(t) be the finite volume Pressure. Denote by an (n ≥ 2) the nth Taylor coefficient. We have < An−1g g >Λ n |Λ| Remark 10 This formula for an gives a direction towards proving the analytic- ity of the pressure in the thermodynamic limit. In fact one only needs to provide a suitable Cn estimate for < An−1g g >Λ . 9 Some Consequences of the Formula for nth−Derivative of the Pressure. In the following, we shall additionally assume that ∇g(0) = 0, and ∇ΦtΛ(0) = 0 for all t ∈ [0, T ). When n = 1, we recall that A0gg = g, Λ(t) =< g >t,Λ and if v(t) = ∇f = A (1)−1 then we have −∆+∇ΦtΛ ·∇ ⊗ v(t) +HessΦtΛv(t) = ∇g Again the tensor notation means that (−∆+∇ΦtΛ ·∇) acts diagonally on the components of v(t). As in [8] v(t) is a solution of the equation g =< g >t,Λ +v(t) ·∇Φ Λ − divv(t). (76) Using the assumptions above, we have Λ(t) = < g >t,Λ = divv(t)(0). Similarly, the formula Λ (t) = (n− 1)! < A g g >t,Λ, implies that Λ (t) = (n− 1)!divvn(t)(0), where vn(t) = A (1)−1 An−1g g Acknowledgements. I would like to thank Professor. Haru Pinson and Pro- fessor. Tom Kennedy for accepting to discuss with me the ideas developed in this paper. 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Introduction Higher Order Exponential Estimates Relaxing the Assumptions of Compact support The n-Point Correlation Functions The Analyticity of the Pressure Towards the analyticity of the Pressure Parameter Dependency of the Solution A Formula for the Taylor Coefficients Some Consequences of the Formula for nth-Derivative of the Pressure.

0704.1333

A dynamical version of the Mordell-Lang conjecture for the additive group

A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP D. GHIOCA AND T. J. TUCKER Abstract. We prove a dynamical version of the Mordell-Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to the ones employed by Skolem, Chabauty and Coleman for studying diophantine equations. 1. Introduction Faltings proved the Mordell-Lang conjecture in the following form (see [Fal94]). Theorem 1.1 (Faltings). Let G be an abelian variety defined over the field of complex numbers C. Let X ⊂ G be a closed subvariety and Γ ⊂ G(C) a finitely generated subgroup of G(C). Then X(C)∩Γ is a finite union of cosets of subgroups of Γ. In particular, Theorem 1.1 says that an irreducible subvariety X of an abelian variety G has a Zariski dense intersection with a finitely generated subgroup of G(C) only if X is a translate of an algebraic subgroup of G. We also note that Faltings result was generalized to semiabelian varieties G by Vojta (see [Voj96]), and then to finite rank subgroups Γ of G by McQuillan (see [McQ95]). If we try to formulate the Mordell-Lang conjecture in the context of algebraic subvarieties contained in a power of the additive group scheme Ga, the conclusion is either false (in the characteristic 0 case, as shown by the curve y = x2 which has an infinite intersection with the finitely generated subgroup Z × Z, without being itself a translate of an algebraic subgroup of G2a) or it is trivially true (in the characteristic p > 0 case, because every finitely generated subgroup of a power of Ga is finite). Denis [Den92a] formulated a Mordell-Lang conjecture for powers of Ga in characteristic p in the context of Drinfeld modules. Denis replaced the finitely generated subgroup from the usual Mordell-Lang statement with a finitely generated φ-submodule, where φ is a Drinfeld module. He also strengthened the conclusion of the Mordell-Lang statement by asking that the subgroups whose cosets are contained in the intersection of the algebraic variety with the finitely generated φ-submodule be actually φ-submodules. The first author proved several cases of the Denis-Mordell-Lang conjecture in [Ghi05] and [Ghi06b]. In the present paper we investigate other cases of the Denis-Mordell-Lang con- jecture through methods different from the ones employed in [Ghi05]. In partic- ular, we prove the Denis-Mordell-Lang conjecture in the case where the finitely generated φ-module is cyclic and the Drinfeld modules are defined over a field of transcendence degree equal to one (this is our Theorem 2.5). Note that [Ghi05] and Key words and phrases. Drinfeld module, Polynomial Dynamics. The second author was partially supported by National Security Agency Grant 06G-067. http://arxiv.org/abs/0704.1333v1 2 D. GHIOCA AND T. J. TUCKER [Ghi06b] treat only the case where the transcendence degree of the field of definition is greater than one. One of the methods employed in [Ghi05] (and whose outcome was later used in [Ghi06b]) was specializations; hence the necessity of dealing with fields of transcendence degree greater than one. By contrast, the techniques used in this paper are more akin to those used in treating diophantine problems over number fields (see [Cha41], [Col85], or [BS66, Chapter 4.6], for example), where such specialization arguments are also not available. So, making a parallel between the classical Mordell-Lang conjecture and the Denis-Mordell-Lang conjecture, we might say that the papers [Ghi05] and [Ghi06b] deal with the “function field case”, while our present paper deals with the “number field case” of the Denis conjecture. Moreover, using specializations (as in [Hru98] and [Ghi05]), our Theorem 2.5 can be extended to Drinfeld modules defined over fields of arbitrary finite transcendence degree. We also note that recently there has been significant progress on establishing ad- ditional links between classical diophantine results over number fields and similar statements for Drinfeld modules. The first author proved in [Ghi06a] an equidis- tribution statement for torsion points of a Drinfeld module, which is similar to the equidistribution statement established by Szpiro-Ullmo-Zhang [SUZ97] (which was later extended by Zhang [Zha98] to a full proof of the famous Bogomolov conjec- ture). Bosser [Bos99] proved a lower bound for linear forms in logarithms at an infinite place associated to a Drinfeld module (similar to the classical result obtained by Baker [Bak75] for usual logarithms, or by David [Dav95] for elliptic logarithms). Bosser’s result was used by the authors in [GT06a] to establish certain equidistribu- tion and integrality statements for Drinfeld modules. Moreover, Bosser’s result is quite possibly true also for linear forms in logarithms at finite places for a Drinfeld module. Assuming this last statement, the authors proved in [GT06b] the analog of Siegel’s theorem for finitely generated φ-submodules. We believe that our present paper provides an additional proof of the fact that the Drinfeld modules represent the right arithmetic analog in characteristic p for abelian varieties in characteristic The idea behind the proof of our Theorem 2.5 can be explained quite simply. Assuming that an affine variety V ⊂ Gga has infinitely many points in common with a cyclic φ-submodule Γ, we can find then a suitable submodule Γ0 ⊂ Γ whose coset lies in V . Indeed, applying the logarithmic map (associated to a suitable place v) to Γ0 yields a line in the vector space C v. Each polynomial f that vanishes on V , then gives rise to an analytic function F on this line (by composing with the exponential function). Because we assumed there are infinitely many points in V ∩ Γ, the zeros of F must have an accumulation point on this line, which means that F vanishes identically on the line. This means that there is an entire translate of Γ0 contained in the zero locus of f . The inspiration for this idea comes from the method employed by Chabauty in [Cha41] (and later refined by Coleman in [Col85]) to study the intersection of a curve C of genus g, embedded in its Jacobian J , with a finitely generated subgroup of J of rank less than g. Our technique also bears a resemblance to Skolem’s method for treating diophantine equations (see [BS66, Chapter 4.6]). Alternatively, our results can be interpreted purely from the point of view of polynomial dynamics, as we describe the intersection of affine varieties with the iterates of a point in the affine space under polynomial actions on each coordinate. A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP3 In this paper we will treat the case of affine varieties embedded in Gga, while the polynomial action (on each coordinate of Gga) will always be given by Drinfeld modules. The more general problem of studying intersections of affine varieties with the iterates of a point in affine space under polynomial actions over number fields or function fields appears to be quite difficult. To our knowledge, very little about this question has been proven except in the case of multiplication maps on semiabelian varieties (see [Voj96] and [McQ95]). We refer the reader to Section 4 of Zhang’s notes [Zha06] for a number of algebraic dynamical conjectures that would generalize well-known arithmetic theorems for semiabelian varieties. Although these notes do not contain a dynamical analog of the Mordell-Lang conjecture, Zhang has indicated to us that it might be reasonable to conjecture that if ψ : Y −→ Y is a suitable morphism of a projective variety Y (one that is “polarized”, to use the terminology of [Zha06]), then the intersection of the ψ-orbit of a point β with a subvariety V must be finite if V does not contain a positive dimensional preperiodic subvariety. We briefly sketch the plan of our paper. In Section 2 we set the notation, describe the Denis-Mordell-Lang conjecture, and then state our main result. In Section 3 we prove this main result (Theorem 2.5), while in Section 4 we prove a couple of extensions of it (Theorems 4.1 and 4.2). 2. Notation and statement of our main result All subvarieties appearing in this paper are closed. 2.1. Drinfeld modules. We begin by defining a Drinfeld module. Let p be a prime and let q be a power of p. Let A := Fq[t], let K be a finite field extension of Fq(t), and let K be an algebraic closure of K. Let K sep be the separable closure of K inside K. We let τ be the Frobenius on Fq, and we extend its action on K. Let K{τ} be the ring of polynomials in τ with coefficients from K (the addition is the usual addition, while the multiplication is the composition of functions). A Drinfeld module is a morphism φ : A → K{τ} for which the coefficient of τ0 in φ(a) =: φa is a for every a ∈ A, and there exists a ∈ A such that φa 6= aτ The definition given here represents what Goss [Gos96] calls a Drinfeld module of “generic characteristic”. We note that usually, in the definition of a Drinfeld module, A is the ring of functions defined on a projective nonsingular curve C, regular away from a closed point η ∈ C. For our definition of a Drinfeld module, C = P1 and η is the usual point at infinity on P1. On the other hand, every ring of regular functions A as above contains Fq[t] as a subring, where t is a nonconstant function in A. For every field extension K ⊂ L, the Drinfeld module φ induces an action on Ga(L) by a∗x := φa(x), for each a ∈ A. We call φ-submodules subgroups of Ga(K) which are invariant under the action of φ. We define the rank of a φ-submodule Γ dimFq(t) Γ⊗A Fq(t). If φ1 : A→ K{τ}, . . . , φg : A→ K{τ} are Drinfeld modules, then (φ1, . . . , φg) acts on Gga coordinate-wise (i.e. φi acts on the i-th coordinate). We define as above the notion of a (φ1, . . . , φg)-submodule of G a; same for its rank. A point α is torsion for the Drinfeld module action if and only if there exists Q ∈ A \ {0} such that φQ(α) = 0. The set of all torsion points is denoted by φtor. 4 D. GHIOCA AND T. J. TUCKER 2.2. Valuations. Let MFq(t) be the set of places on Fq(t). We denote by v∞ the place in MFq(t) such that v∞( ) = deg(g) − deg(f) for every nonzero f, g ∈ A = Fq[t]. We let MK be the set of valuations on K. Then MK is a set of valuations which satisfies a product formula (see [Ser97, Chapter 2]). Thus • for each nonzero x ∈ K, there are finitely many v ∈MK such that |x|v 6= 1; • for each nonzero x ∈ K, we have |x|v = 1. Definition 2.1. Each place in MK which lies over v∞ is called an infinite place. Each place in MK which does not lie over v∞ is called a finite place. By abuse of notation, we let ∞ ∈MK denote any place extending the place v∞. For v ∈MK we let Kv be the completion of K with respect to v. Let Cv be the completion of an algebraic closure of Kv. Then | · |v extends to a unique absolute value on all of Cv. We fix an embedding of i : K −→ Cv. For x ∈ K, we denote |i(x)|v simply as |x|v, by abuse of notation. 2.3. Logarithms and exponentials associated to a Drinfeld module. Let v ∈ MK . According to Proposition 4.6.7 from [Gos96], there exists an unique formal power series expφ,v ∈ Cv{τ} such that for every a ∈ A, we have (2.1.1) φa = expφ,v a exp In addition, the coefficient of the linear term in expφ,v(X) equals 1. We let logφ,v be the formal power series exp−1 , which is the inverse of expφ,v. If v = ∞ is an infinite place, then expφ,∞(x) is convergent for all x ∈ C∞ (see Theorem 4.6.9 of [Gos96]). There exists a sufficiently small ball B∞ centered at the origin such that expφ,∞ is an isometry on B∞ (see Lemma 3.6 of [GT06a]). Hence, logφ,∞ is convergent on B∞. Moreover, the restriction of logφ,∞ on B∞ is an analytic isometry (see also Proposition 4.14.2 of [Gos96]). If v is a finite place, then expφ,v is convergent on a sufficiently small ball Bv ⊂ Cv (this follows identically as the proof of the analyticity of expφ,∞ from Theorem 4.6.9 of [Gos96]). Similarly as in the above paragraph, at the expense of replacing Bv by a smaller ball, we may assume expφ,v is an isometry on Bv. Hence, also logφ,v is an analytic isometry on Bv. For every place v ∈ MK , for every x ∈ Bv and for every polynomial a ∈ A, we have (see (2.1.1)) (2.1.2) a logφ,v(x) = logφ,v(φa(x)) and expφ,v(ax) = φa(expφ,v(x)). By abuse of language, expφ,∞ and expφ,v will be called exponentials, while logφ,∞ and logφ,v will be called logarithms. 2.4. Integrality and reduction. Definition 2.2. A Drinfeld module φ has good reduction at a place v if for each nonzero a ∈ A, all coefficients of φa are v-adic integers and the leading coefficient of φa is a v-adic unit. If φ does not have good reduction at v, then we say that φ has bad reduction at v. It is immediate to see that φ has good reduction at v if and only if all coefficients of φt are v-adic integers, while the leading coefficient of φt is a v-adic unit. All infinite places of K are places of bad reduction for φ. A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP5 2.5. The Denis-Mordell-Lang conjecture. Let g be a positive integer. Definition 2.3. Let φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. An algebraic (φ1, . . . , φg)-submodule of G a is an irreducible algebraic subgroup of G invariant under the action of (φ1, . . . , φg). Denis proposed in Conjecture 2 of [Den92a] the following problem, which we call the full Denis-Mordell-Lang conjecture because it asks for the description of the intersection of an affine variety with a finite rank φ-module (as opposed to only a finitely generated φ-module). Recall that a φ-module M is said to be a finite rank φ-module if it contains a finitely generated φ-submodule such that M/M ′ is a torsion φ-module. Conjecture 2.4 (The full Denis-Mordell-Lang conjecture). Let φ1 : A→ K{τ}, . . . , φg : A → K{τ} be Drinfeld modules. Let V ⊂ Gga be an affine variety defined over K. Let Γ be a finite rank (φ1, . . . , φg)-submodule of G a(K). Then there exist algebraic (φ1, . . . , φg)-submodules B1, . . . , Bl of G a and there exist γ1, . . . , γl ∈ Γ such that V (K) ∩ Γ = (γi +Bi(K)) ∩ Γ. In [Den92a], Denis showed that under certain natural Galois theoretical assump- tions, Conjecture 2.4 would follow from the weaker conjecture which would describe the intersection of an affine variety with a finitely generated φ-module. Since then, the case Γ is the product of the torsion submodules of each φi was proved by Scanlon in [Sca02], while various other instances of Conjecture 2.4 were worked out in [Ghi05] and [Ghi06b]. We note that Denis asked his conjecture also for t-modules, which includes the case of products of distinct Drinfeld modules acting on Gga. For the sake of simplifying the notation, we denote by φ the action of (φ1, . . . , φg) on Gga. We also note that if V is an irreducible affine subvariety of G a which has a Zariski dense intersection with a finite rank φ-submodule Γ of Gga, then the Denis- Mordell-Lang conjecture predicts that V is a translate of an algebraic φ-submodule of Gga by a point in Γ. In particular, if V is an irreducible affine curve, which is not a translate of an algebraic φ-submodule, then its intersection with any finite rank φ-submodule of Gga should be finite. In [Ghi05], the first author studied the Denis-Mordell-Lang conjecture for Drin- feld modules whose field of definition (for their coefficients) is of transcendence degree at least equal to 2. The methods employed in [Ghi05] involve specializa- tions, and so, it was crucial for the φ there not to be isomorphic with a Drinfeld module defined over Fq(t). In the present paper we will study precisely this case left out in [Ghi05] and [Ghi06b]. Our methods depend crucially on the hypothesis that the transcendence degree of the field generated by the coefficients of φi is one, since we use the fact that at each place v, the number of residue classes in the ring of integers at v is finite. The main result of our paper is describing the intersection of an affine subvariety V ⊂ Gga with a cyclic φ-submodule Γ of G Theorem 2.5. Let K be a function field of transcendence degree equal to one. Let φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. Let (x1, . . . , xg) ∈ G and let Γ ⊂ Gga(K) be the cyclic (φ1, . . . , φg)-submodule generated by (x1, . . . , xg). 6 D. GHIOCA AND T. J. TUCKER Let V ⊂ Gga be an affine subvariety defined over K. Then V (K) ∩ Γ is a finite union of cosets of (φ1, . . . , φg)-submodules of Γ. Using an idea from [Ghi06b], we are able to extend the above result to (φ1, . . . , φg)- submodules of rank 1 (see our Theorem 4.2) in the special case where V is a curve. 3. Proofs of our main results We continue with the notation from Section 2. Hence φ1, . . . , φg are Drinfeld modules. We denote by φ the action of (φ1, . . . , φg) on G a. Also, let (x1, . . . , xg) ∈ a(K) and let Γ be the cyclic φ-submodule of G a(K) generated by (x1, . . . , xg). Unless otherwise stated, V ⊂ Gga is an affine subvariety defined over K. We first prove an easy combinatorial result which we will use in the proof of Theorem 2.5. Lemma 3.1. Let Γ be a cyclic φ-submodule of Gga(K). Let Γ0 be a nontrivial φ- submodule of Γ, and let S ⊂ Γ be an infinite set. Suppose that for every infinite subset S0 ⊂ S, there exists a coset C0 of Γ0 such that C0 ∩ S0 6= ∅ and C0 ⊂ S. Then S is a finite union of cosets of φ-submodules of Γ. Proof. Since S is infinite, Γ is infinite, and thus Γ is torsion-free. Therefore, Γ is an infinite cyclic φ-module, which is isomorphic to A (as a module over itself). Hence, via this isomorphism, Γ0 is isomorphic to a nontrivial ideal I of A. Since A/I is finite (recall that A = Fq[t]), there are finitely many cosets of Γ0 in Γ. Thus, S contains at most finitely many cosets of Γ0. Now, let {yi+Γ0} i=1 be all of the cosets of Γ0 that are contained in S. Suppose (3.1.1) S0 := S \ (yi + Γ0) is infinite. Then using the hypotheses of this Lemma for S0, we see that there is a coset of Γ0 that is contained in S but is not one of the cosets (yi + Γ0) (because it has a nonempty intersection with S0). This contradicts the fact that {yi + Γ0} i=1 are all the cosets of Γ0 that are contained in S. Therefore S0 must be finite. Since any finite subset of Γ is a finite union of cosets of the trivial submodule of Γ, this completes our proof. � We will also use the following Lemma in the proof of Theorem 2.5. Lemma 3.2. Let θ : A → K{τ} and ψ : A → K{τ} be Drinfeld modules. Let v be a place of good reduction for θ and ψ. Let x, y ∈ Cv. Let 0 < rv < 1, and let Bv := {z ∈ Cv | |z|v < rv} be a sufficiently small ball centered at the origin with the property that both logθ,v and logψ,v are analytic isometries on Bv. Then for every polynomials P,Q ∈ A such that (θP (x), ψP (y)) ∈ Bv × Bv and (θQ(x), ψQ(y)) ∈ Bv ×Bv, we have logθ,v(θP (x)) · logψ,v(ψQ(y)) = logθ,v(θQ(x)) · logψ,v(ψP (y)). Proof. Since v is a place of good reduction for θ, all the coefficients of θQ are v-adic integers and thus, |θQ(θP (x))|v ≤ |θP (x)|v < rv (we use the fact that |θP (x)|v < rv < 1, and so, each term of θQ(θP (x)) has its absolute value at most equal to |θP (x)|v). Using (2.1.2), we conclude that Q · logθ,v(θP (x)) = logθ,v(θQP (x)) = logθ,v(θPQ(x)) = P · logθ,v(θQ(x)). A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP7 Similarly we obtain that Q · logψ,v(ψP (x)) = P · logψ,v(ψQ(x)). This concludes the proof of Lemma 3.2. � The following result is an immediate corollary of Lemma 3.2. Corollary 3.3. With the notation as in Theorem 2.5, assume in addition that x1 /∈ (φ1)tor. Let v be a place of good reduction for each φi. Suppose Bv is a small ball (of radius less than 1) centered at the origin such that each logφi,v is an analytic isometry on Bv. Then for each i ∈ {2, . . . , g}, the fractions λi := logφi,v ((φi)P (xi)) logφ1,v ((φ1)P (x1)) are independent of the choice of the nonzero polynomial P ∈ A for which φP (x1, . . . , xg) ∈ Bgv . The following simple result on zeros of analytic functions can be found in [Gos96, Proposition 2.1, p. 42]. We include a short proof for the sake of completeness. Lemma 3.4. Let F (z) = i=0 aiz i be a power series with coefficients in Cv that is convergent in an open disc B of positive radius around the point z = 0. Suppose that F is not the zero function. Then the zeros of F in B are isolated. Proof. Let w be a zero of F in B. We may rewrite F in terms of (z−w) as a power series F (z) = i=1 bi(z−w) i that converges in a disc Bw of positive radius around w. Let m be the smallest index n such that bn 6= 0. Because F is convergent in Bw, then there exists a positive real number r such that for all n > m, we have ∣∣∣ bnbm < rn−m. Then, for any u ∈ Bw such that 0 < |u − w|v < , we have |bm(u − w) m|v > |bn(u − w) n|v for all n > m. Hence |F (u)|v = |bm(u − w) m|v 6= 0. Thus F (u) 6= 0, and so, F has no zeros other than w in a nonempty open disc around w. � We are ready to prove Theorem 2.5. Proof of Theorem 2.5. We may assume V (K) ∩ Γ is infinite (otherwise the conclu- sion of Theorem 2.5 is obvisouly satisfied). Assuming V (K) ∩ Γ is infinite, we will show that there exists a nontrivial φ-submodule Γ0 ⊂ Γ such that each infinite sub- set of points S0 in V (K)∩Γ has a nonempty intersection with a coset C0 of Γ0, and moreover, C0 ⊂ V (K) ∩ Γ. Then Lemma 3.1 will finish the proof of Theorem 2.5. First we observe that Γ is not a torsion φ-submodule. Otherwise Γ is finite, contradicting our assumption that V (K) ∩ Γ is infinite. Hence, from now on, we assume (without loss of generality) that x1 is not a torsion point for φ1. We fix a finite set of polynomials {fj} j=1 ⊂ K[X1, . . . , Xg] which generate the vanishing ideal of V . Let v ∈MK be a place of K which is of good reduction for all φi (for 1 ≤ i ≤ g). In addition, we assume each xi is integral at v (for 1 ≤ i ≤ g). Then for each P ∈ A, we have φP (x1, . . . , xg) ∈ G a(ov), where ov is the ring of v-adic integers in Kv (the completion of K at v). Because ov is a compact space (we use the fact that K is a function field of transcendence degree 1 and thus has a finite residue field at v), we conclude that every infinite sequence of points φP (x1, . . . , xg) ∈ V (K)∩Γ contains a convergent subsequence in 8 D. GHIOCA AND T. J. TUCKER v. Using Lemma 3.1, it suffices to show that there exists a nontrivial φ-submodule Γ0 ⊂ Γ such that every convergent sequence of points in V (K)∩Γ has a nonempty intersection with a coset C0 of Γ0, and moreover, C0 ⊂ V (K) ∩ Γ. Now, let S0 be an infinite subsequence of distinct points in V (K) ∩ Γ which converges v-adically to (x0,1, . . . , x0,g) ∈ o v, let 0 < rv < 1, and let Bv := {z ∈ Cv | |z|v < rv} be a small ball centered at the origin on which each of the logarithmic functions logφi,v is an analytic isometry (for 1 ≤ i ≤ g). Since (x0,1, . . . , x0,g) is the limit point for S0, there exists a d ∈ A and an infinite subsequence {φd+Pn}n≥0 ⊂ S0 (with Pn = 0 if and only if n = 0), such that for each n ≥ 0, we have (3.4.1) ∣∣(φi)d+Pn (xi)− x0,i for each 1 ≤ i ≤ g. We will show that there exists an algebraic group Y0, independent of S0 and in- variant under φ, such that φd(x1, . . . , xg) + Y0 is a subvariety of V containing φd+Pn(x1, . . . , xg) for all Pn. Thus the submodule Γ0 := Y0(K) ∩ Γ will satisfy the hypothesis of Lemma 3.1 for the infinite subset V (K) ∩ Γ ⊂ Γ; this will yield the conclusion of Theorem 2.5. Using (3.4.1) for n = 0 (we recall that P0 = 0), and then for arbitrary n, we see (3.4.2) ∣∣(φi)Pn (xi) for each 1 ≤ i ≤ g. Hence logφi,v is well-defined at (φi)Pn (xi) for each i ∈ {1, . . . , g} and for each n ≥ 1. Moreover, the fact that (φi)Pn+d (xi) converges to a point in ov means that( (φi)Pn (xi) converges to a point which is contained in Bv (see (3.4.2)). Without loss of generality, we may assume (3.4.3) | logφ1,v (φ1)P1 (x1) | logφi,v (φi)P1 (xi) Using the result of Corollary 3.3, we conclude that for each i ∈ {2, . . . , g}, the following fraction is independent of n and of the sequence {Pn}n: (3.4.4) λi := logφi,v (φi)Pn (xi) logφ1,v (φ1)Pn (x1) Note that since x1 is not a torsion point for φ1, the denominator of λi (3.4.4) is nonzero. Because of equation (3.4.3), we may conclude that |λi|v ≤ 1 for each i. The fact that λi is independent of the sequence {Pn}n will be used later to show that the φ-submodule Γ0 that we construct is independent of the sequence {Pn}n. For each n ≥ 1 and each 2 ≤ i ≤ g, we have (3.4.5) logφi,v (φi)Pn (xi) = λi · logφ1,v (φ1)Pn (x1) For each i, applying the exponential function expφi,v to both sides of (3.4.5) yields (3.4.6) (φi)Pn (xi) = expφi,v λi · logφ1,v (φ1)Pn (x1) Since φd+Pn (x1, . . . , xg) ∈ V (K), for each j ∈ {1, . . . , ℓ} we have (3.4.7) fj (φd+Pn(x1, . . . , xg)) = 0 for each n. For each j ∈ {1, . . . , ℓ} we let fd,j ∈ K[X1, . . . , Xg] be defined by (3.4.8) fd,j (X1, . . . , Xg) := fj (φd(x1, . . . , xg) + (X1, . . . , Xg)) . A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP9 We let Vd ⊂ G a be the affine subvariety defined by the equations fd,j(X1, . . . , Xg) = 0 for each j ∈ {1, . . . , ℓ}. Using (3.4.7) and (3.4.8), we see that for each j ∈ {1, . . . , ℓ} we have (3.4.9) fd,j (φPn(x1, . . . , xg)) = 0 for each n, and so, (3.4.10) φPn(x1, . . . , xg) ∈ Vd(K). For each j ∈ {1, . . . , ℓ}, we let Fd,j(u) be the analytic function defined on Bv by Fd,j(u) := fd,j u, expφ2,v λ2 logφ1,v(u) , . . . , expφg ,v λg logφ1,v(u) We note, because of (3.4.3), and the fact that logφ1,v is an analytic isometry on Bv, that for each u ∈ Bv, we have (3.4.11) |λi · logφ1,v(u)|v = |λi|v · | logφ1,v(u)|v ≤ |u|v < rv. Equation (3.4.11) shows that λi · logφ1,v(u) ∈ Bv, and so, expφi,v λi · logφ1,v(u) well-defined. Using (3.4.6) and (3.4.9) we obtain that for every n ≥ 1, we have (3.4.12) Fd,j (φ1)Pn (x1) (φ1)Pn (x1) is a sequence of zeros for the analytic function Fd,j which has an accumulation point in Bv. Lemma 3.4 then implies that Fd,j = 0, and so, for each j ∈ {1, . . . , ℓ}, we have (3.4.13) fd,j u, expφ2,v λ2 logφ1,v(u) , . . . , expφg,v λg logφ1,v(u) For each u ∈ Bv, we let Zu := u, expφ2,v λ2 logφ1,v(u) , . . . , expφg,v λg logφ1,v(u) ∈ Gga(Cv). Then (3.4.13) implies that (3.4.14) Zu ∈ Vd for each u ∈ Bv. Let Y0 be the Zariski closure of {Zu}u∈Bv . Then Y0 ⊂ Vd. Note that Y0 is inde- pendent of the sequence {Pn}n (because the λi are independent of the sequence {Pn}n, according to Corollary 3.3). We claim that for each u ∈ Bv and for each P ∈ A, we have (3.4.15) φP (Zu) = Z(φ1)P (u). Note that for each u ∈ Bv, then also (φ1)P (u) ∈ Bv for each P ∈ A, because each coefficient of φ1 is a v-adic integer. To see that (3.4.15) holds, we use (2.1.2), which implies that for each i ∈ {2, . . . , g} we have expφi,v λi logφ1,v ((φ1)P (u)) = expφi,v λi · P · logφ1,v(u) = expφi,v P · λi logφ1,v(u) = (φi)P expφi,v λi logφ1,v(u) Hence, (3.4.15) holds, and so, Y0 is invariant under φ. Furthermore, since all of the expφi,v and logφi,v are additive functions, we have Zu1+u2 = Zu1 + Zu2 for every u1, u2 ∈ Bv. Hence Y0 is an algebraic group, which is also a φ-submodule of G Moreover, Y0 is defined independently of Γ. 10 D. GHIOCA AND T. J. TUCKER Let Γ0 := Y0(K)∩Γ. Because Y0 is invariant under φ, then Γ0 is a submodule of Γ. Because Y0 ⊂ Vd, it follows that the translate φd(x1, . . . , xg)+Y0 is a subvariety of V which contains all {φd+Pn(x1, . . . , xg)}n. In particular, the (infinite) translate C0 of Γ0 by φd(x1, . . . , xg) is contained in V (K)∩Γ. Hence, every infinite sequence of points in V (K)∩Γ has a nontrivial intersection with a coset C0 of (the nontrivial φ-submodule) Γ0, and moreover, C0 ⊂ V (K)∩Γ. Applying Lemma 3.1 thus finishes the proof of Theorem 2.5. � In the course of our proof of Theorem 2.5 we also proved the following statement. Theorem 3.5. Let Γ be an infinite cyclic φ-submodule of Gga. Then there exists an infinite φ-submodule Γ0 ⊂ Γ such that for every affine subvariety V ⊂ G a, if V (K) ∩ Γ is infinite, then V (K) ∩ Γ contains a coset of Γ0. Proof. Let v be a place of good reduction for φ; in addition, we assume the points in Γ are v-adic integers. Suppose that V (K) ∩ Γ is infinite. As shown in the proof of Theorem 2.5, there exists a positive dimensional algebraic group Y0, invariant under φ, and depending only on Γ and v (but not on V ), such that a translate of Y0 by a point in Γ lies in V . Moreover, Γ0 := Y0(K) ∩ Γ is infinite. Hence Γ0 satisfies the conclusion of Theorem 3.5. � 4. Further extensions We continue with the notation from Section 3: φ1, . . . , φg are Drinfeld modules. As usual, we denote by φ the action of (φ1, . . . , φg) on G a. First we prove the following consequence of Theorem 2.5. Theorem 4.1. Let V ⊂ Gga be an affine subvariety defined over K. Let Γ ⊂ G be a finitely generated φ-submodule of rank 1. Then V (K) ∩ Γ is a finite union of cosets of φ-submodules of Γ. In particular, if V is an irreducible curve which is not a translate of an algebraic φ-submodule, then V (K) ∩ Γ is finite. Proof. Since A = Fq[t] is a principal ideal domain, Γ is the direct sum of its finite torsion submodule Γtor and a free submodule Γ1, which is cyclic because Γ has rank 1. Therefore γ∈Γtor γ + Γ1, and so, V (K) ∩ Γ = γ∈Γtor V (K) ∩ (γ + Γ1) = γ∈Γtor (γ + (−γ + V (K)) ∩ Γ1) . Using the fact Γtor is finite and applying Theorem 2.5 to each intersection (−γ + V (K))∩ Γ1 thus completes our proof. � We use the ideas from [Ghi06b] to describe the intersection of a curve C with a φ-module of rank 1. So, let (x1, . . . , xg) ∈ G a(K), let Γ be the cyclic φ-submodule of a(K) generated by (x1, . . . , xg), and let Γ be the φ-submodule of rank 1, containing all (z1, . . . , zg) ∈ G a(K) for which there exists a nonzero polynomial P such that φP (z1, . . . , zg) ∈ Γ. Since all polynomials φP (for P ∈ A) are separable, we have Γ ⊂ G sep). A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP11 With the notation above, we prove the following result; this may be viewed as a Drinfeld module analog of McQuillan’s result on semiabelian varieties (see [McQ95]), which had been conjectured by Lang. Theorem 4.2. Let C ⊂ Gga be an affine curve defined over K. Then C(K) ∩ Γ is a finite union of cosets of φ-submodules of Γ. Before proceeding to the proof of Theorem 4.2 we first prove two facts which will be used later. The first fact is an immediate consequence of Theorem 1 of [Sca02] (the Denis-Manin-Mumford conjecture for Drinfeld modules), which we state below. Theorem 4.3 (Scanlon). Let V ⊂ Gga be an affine variety defined over K. Then there exist algebraic φ-submodules B1, . . . , Bℓ of G a and elements γ1, . . . , γℓ of φtor such that V (K) ∩ φtor = γi +Bi(K) ∩ φtor. Moreover, in Remark 19 from [Sca02], Scanlon notes that his proof of the Denis- Manin-Mumford conjecture yields a uniform bound on the degree of the Zariski closure of V (K)∩φtor, depending only on φ, g, and the degree of V . In particular, one obtains the following uniform statement for translates of curves. Fact 4.4. Let C ⊂ Gga be an irreducible curve which is not a translate of an algebraic φ-module of Gga. Then there exists a positive integer N such that for every y ∈ Gga(K), the set y + C(K) ∩ φtor has at most N elements. Proof. The curve C contains no translate of a positive dimensional algebraic φ- submodule of Gga, so for every y ∈ G a(K), the algebraic φ-modules Bi appear- ing in the intersection y + C(K) ∩ φtor are all trivial. In particular, the set( y + C(K) ∩φtor is finite. Thus, using the uniformity obtained by Scanlon for his Manin-Mumford theorem, we conclude that the cardinality of y + C(K) ∩ φtor is uniformly bounded above by some positive integer N . � We will also use the following fact in the proof of our Theorem 4.2. Fact 4.5. Let φ : A→ K{τ} be a Drinfeld module. Then for every positive integer D, there exist finitely many torsion points y of φ such that [K(y) : K] ≤ D. Proof. If y ∈ φtor, then the canonical height ĥ(y) of y (as defined in [Den92b]) equals 0. Also, as shown in [Den92b], the difference between the canonical height and the usual Weil height is uniformly bounded on K. Then Fact 4.5 follows by noting that there are finitely many points of bounded Weil height and bounded degree over the field K (using Northcott’s theorem applied to the global function field K). � We are now ready to prove Theorem 4.2. Proof of Theorem 4.2. Arguing as in the proof of Theorem 2.5, it suffices to show that if C is an irreducible affine curve (embedded in Gga), then C(K)∩ Γ is infinite only if C is a translate of an algebraic φ-submodule (because any translate of an algebraic φ-module intersects Γ in a coset of some φ-submodule of Γ). Therefore, from now on, we assume C is irreducible, that C(K) ∩ Γ is infinite, and that C is not a translate of an algebraic φ-submodule. We will derive a contradiction. 12 D. GHIOCA AND T. J. TUCKER Let z ∈ C(K) ∩ Γ. For each field automorphism σ : Ksep → Ksep that restricts to the identity on K, we have zσ ∈ C (Ksep) (because C is defined over K). By the definition of Γ, there exists a nonzero polynomial P ∈ A such that φP (z) ∈ Γ. Since φP has coefficients in K, we obtain φP (z σ) = (φP (z)) = φP (z). The last equality follows from the fact that φP (z) ∈ Γ ⊂ G a(K). We conclude that φP (z σ − z) = 0, and, thus, we have Tz,σ := z σ − z ∈ φtor. Moreover, Tz,σ ∈ (−z+C(K))∩φtor (because z σ ∈ C). Using Fact 4.4 we conclude that for each fixed z ∈ C(K)∩Γ, the set {Tz,σ}σ has cardinality bounded above by some number N (independent of z). In particular, this implies that z has finitely many Galois conjugates, so [K(z) : K] ≤ N . Similarly we have [K(zσ) : K] ≤ N ; thus, we may conclude that (4.5.1) [K (Tz,σ) : K] ≤ [K(z, z σ) : K] ≤ N2. As shown by Fact 4.5, there exists a finite set of torsion points w for which [K(w) : K] ≤ N2. Hence, recalling that N is independent of z, we see that the set (4.5.2) H := {Tz,σ} z∈C(K)∩Γ σ:Ksep→Ksep is finite. Now, since H is a finite set of torsion points, there must exist a nonzero poly- nomial Q ∈ A such that φQ(H) = {0}. Therefore, φQ(z σ − z) = 0 for each z ∈ C(K)∩Γ and each automorphism σ. Hence φQ(z) σ = φQ(z) for each σ. Thus, we have (4.5.3) φQ(z) ∈ G a(K) for every z ∈ C(K) ∩ Γ. Let Γ1 := Γ ∩ G a(K). Since Γ is a finite rank φ-module and G a(K) is a tame module (i.e. every finite rank submodule is finitely generated; see [Poo95] for a proof of this result), it follows that Γ1 is finitely generated. Let Γ2 be the finitely generated φ-submodule of Γ generated by all points z ∈ Γ such that φQ(z) ∈ Γ1. More precisely, if w1, . . . , wℓ generate the φ-submodule Γ1, then for each i ∈ {1, . . . , ℓ}, we find all the finitely many zi such that φQ(zi) = wi. Then this finite set of all zi generate the φ-submodule Γ2. Thus Γ2 is a finitely generated φ-submodule, and moreover, using equation (4.5.3), we obtain C(K) ∩ Γ = C(K) ∩ Γ2. Since Γ2 is a finitely generated φ-submodule of rank 1 (because Γ2 ⊂ Γ and Γ has rank 1), Theorem 4.1 finishes the proof of Theorem 4.2. � References [Bak75] A. Baker, Transcendental number theory, Cambridge University Press, Cambridge, 1975. [Bos99] V. Bosser, Minorations de formes linéaires de logarithmes pour les modules de Drinfeld, J. Number Theory 75 (1999), no. 2, 279–323. [BS66] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York, 1966. [Cha41] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur á l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885. [Col85] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. [Dav95] S. David, Minorations de formes linéaire de logarithmes elliptiques, Mem. Soc. Math. France 62 (1995), 143 pp. A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP13 [Den92a] L. Denis, Géométrie diophantienne sur les modules de Drinfel′d, The arithmetic of function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 285–302. [Den92b] , Hauteurs canoniques et modules de Drinfel′d, Math. Ann. 294 (1992), no. 2, 213–223. [Fal94] G. Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., no. 15, Academic Press, San Diego, CA, 1994, pp. 175–182. [Ghi05] D. Ghioca, The Mordell-Lang theorem for Drinfeld modules, Int. Math. Res. Not. (2005), no. 53, 3273–3307. [Ghi06a] D. Ghioca, Equidistribution for torsion points of a Drinfeld module, Math. Ann. 336 (2006), no. 4, 841–865. [Ghi06b] D. Ghioca, Towards the full Mordell-Lang conjecture for Drinfeld modules, submitted for publication, 6 pages, 2006. [Gos96] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer- Verlag, Berlin, 1996. [GT06a] D. Ghioca and T. J. Tucker, Equidistribution and integral points for Drinfeld modules, submitted for publication, 29 pages, 2006. [GT06b] , Siegel’s Theorem for Drinfeld modules, submitted for publication, 25 pages, 2006. [Hru98] E. Hrushovski, Proof of Manin’s theorem by reduction to positive characteristic, Model theory and algebraic geometry, Lecture Notes in Math., no. 1696, Springer, Berlin, 1998, pp. 197–205. [McQ95] M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143–159. [Poo95] B. Poonen, Local height functions and the Mordell-Weil theorem for Drinfel′d modules, Compositio Math. 97 (1995), no. 3, 349–368. [Sca02] T. Scanlon, Diophantine geometry of the torsion of a Drinfeld module, J. Number Theory 97 (2002), no. 1, 10–25. [Ser97] J.-P. Serre, Lectures on the Mordell-Weil theorem, third ed., Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, with a foreword by Brown and Serre. [SUZ97] L. Szpiro, E. Ullmo, and S. Zhang, Equirépartition des petits points, Invent. Math. 127 (1997), 337–347. [Voj96] P. Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), no. 1, 133–181. [Zha98] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165. [Zha06] S. Zhang, Distributions and heights in algebraic dynamics, prepint, Available at www.math.columbia.edu/~szhang/papers/dynamics.pdf, 66 pages, 2006. E-mail address: [email protected] Dragos Ghioca, Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1, E-mail address: [email protected] Thomas Tucker, Department of Mathematics, Hylan Building, University of Rochester, Rochester, NY 14627 1. Introduction 2. Notation and statement of our main result 2.1. Drinfeld modules 2.2. Valuations 2.3. Logarithms and exponentials associated to a Drinfeld module 2.4. Integrality and reduction 2.5. The Denis-Mordell-Lang conjecture 3. Proofs of our main results 4. Further extensions References

0704.1334

Fabrication of Analog Electronics for Serial Readout of Silicon Strip Sensors

Fabrication of Analog Electronics for Serial Readout of Silicon Strip Sensors E. Won,∗ J. H. Choi, and H. Ha Department of Physics, Korea University, Seoul 136-713, Korea H. J. Hyun, H. J. Kim, and H. Park Department of Physics, Kyungpook National University, Daegu 702-701, Korea (Received January 5 2006) Abstract A set of analog electronics boards for serial readout of silicon strip sensors was fabricated. A commercially available amplifier is mounted on a homemade hybrid board in order to receive analog signals from silicon strip sensors. Also, another homemade circuit board is fabricated in order to translate amplifier control signals into a suitable format and to provide bias voltage to the amplifier as well as to the silicon sensors. We discuss technical details of the fabrication process and performance of the circuit boards we developed. PACS numbers: 83.85.Gk, 84.30.Le, 84.30.Sk Keywords: amplifier,electronics, ASIC ∗Electronic address: [email protected]; Fax: +82-2-927-3292 http://arxiv.org/abs/0704.1334v2 mailto:[email protected] I. INTRODUCTION Over the last thirty years, there have been impressive developments in silicon strip sen- sors and their readout electronics in the field of elementary particle physics. They were first used more than twenty years ago for heavy flavour searches in fixed target experi- ments [1]. In particular, their potential use as high precision vertex detectors around high energy colliders, both for electron-positron and proton-antiproton machines, has initiated further development of high performance semiconductor detectors till now [2, 3, 4]. Sub- sequently, it became clear that much higher density electronics was required and it drove the construction of integrated circuit amplifiers in metal-oxide-silicon technology. Therefore, application specific integrated circuit (ASIC) technology has been heavily used in designing readout electronics for silicon sensors in the particle physics experiments [5, 6]. The inter- face electronics board between silicon sensors and readout ASIC chips is traditionally called hybrid boards and the experiment-specific hybrid boards have been produced for various experiments [6, 7, 8]. The design of such hybrid boards should consider cooling system for collider experiments and low electrical noise performance for detecting small signals. Recently, research activities on the high density readout electronics have been extended to the field of high resolution medical imaging [9] as well as charged particle trackers in the future particle physics program [10]. Therefore, it becomes clear that the knowledge and experience in fabricating hybrid board and reading out analog signals from it may be one of important items for the participation to such programs. In this respect, we discuss development of several different types of hybrid boards and related electronics board with the technology available domestically. Section II describes our first prototype hybrid board. The fabrication of detector bias and dc voltage delivery for the operation of ASIC chip, and the control logic translator system is described in section III. We also fabricated a specialized hybrid board in order to test the ASIC amplifier itself and it is described in section IV. Our latest design that mounts a 17 channel single-side silicon detector is described in section V. II. DESIGN OF HYBRID BOARD I In this section, we discuss the development of our first prototype of the hybrid board that mounts a commercially available high density ASIC amplifier, the VA chip [11, 12, 13]. It has in total 128 analog channels and each channel contains a charge-sensitive preamplifier, a shaper, a track-and-hold, and multiplexing capacity. In order to communicate to the VA chip, one has to wire-bond approximately 30 lines of various analog and digital signals from the VA chip to the hybrid board. Since the width of the VA chip is 5 mm, the layout size of 30 pads on the printed circuit board (PCB) also should be in the similar size in order to make good electric connections to the VA chip. It turns out that a pad width and the pitch of pads both should be on the order of 100 µm on the PCB in order to make a good ultrasonic wire-bonding to the VA chip. However, most of domestic, small-size vendors expressed difficulty in fabricating such fine structure PCB layout with their facility. Figure 1 shows our first attempt to fabricate high density pads on the PCB from a domestic vendor. A large rectangular hole is made on the left corner of the board where the sensor is to be mounted. This hole is placed in order to minimize the material for the future radioactive source or beam tests. The PCB is made with four layers where analog and digital grounds are routed in the same layer. One can also see a smaller, horizontally long rectangular pad (labeled as U2) near to the place for the VA chip (labeled as U1). This rectangular pad is for the R/C chip [14] as the detector to be mounted at the design stage is a dc-type sensor [15]. This complicates the detector biasing method quite significantly because the R/C chip we use is known to break down at 70 V and therefore a voltage division is made to provide full depletion bias voltage to sensors. A tin-lead alloy was used in order to cover all copper pads on the PCB. Later we realized that in some countries there are at least directives restricting the use of lead for such purpose and therefore we abandoned the use of tin-lead alloy completely. The use of tin-lead alloy resulted in significantly bad quality in the layout of the pad outlines. A microscope picture of the pads in Fig. 2 (a) illustrates the situation. The three horizontal lines in the figure represent bonding pads on the hybrid PCB. The average width of pads (thickness in vertical direction) is always less than 15 µm and is too narrow for any practical use. We labeled this first PCB board as the version 0.9. There is another problem in the version 0.9. The board was not flat and it prevented us from mounting silicon sensors as it does not provide mechanically stable configuration. This situation is also shown in Fig. 2 (c). After the fabrication of the version 0.9, there has been series of discussion with technicians from the vendor in order to identify source of these two problems. The problem with the poor quality of the pad outline is partially solved by modifying one of chemical etching processes in their PCB fabrication. We also use gold to cover all copper pads on the PCB and it partially helped in improving the quality of the bonding pad. The source of the non-flat structure of the board was due to improper handing of the PCB during the cooling process. After identifying sources of troubles mentioned above, we fabricated our second prototype hybrid board with minor modification as far as the design is concerned. A placeholder for a lemo connection to the analog signal output is made for debugging purpose in the second prototype. Figure 2 (b) shows the quality of the pitch for the second prototype board. Measurements showed that the width of pads is 110 µm which satisfies our specification. The trouble with the non-flatness of the board is also disappeared in the second prototype and it is clearly shown in Fig. 2 (d). We label the second prototype as the hybrid version 1.0. We note that in order this to be used in the real collider environment, it has to deal with the heat generated during the collision. One of solutions is to make the PCB with ceramic material but we did not investigate the possibility of making ceramic PCB at this time for a quick development of the board. There are in total twenty passive surface-mounted components soldered on the hybrid board. All components are a F -class which has ±1 % tolerance from their specification values. A conductive epoxy from Chemtronics CW2400 [16] is tested for a good ohmic contact with the gold pad on the hybrid board and is used in order to mount the VA chip on the hybrid board, as the bottom plate of the VA chip requires an electric contact. After through electrical tests, assembled hybrid boards are shipped to a local company [17] for an ultrasonic wire-bonding between the VA chip and the hybrid board. After the wire- bonding, the hybrid boards are delivered back to the laboratory and a readout setup is made to communicate with the hybrid board. A dc power supply is connected to a homemade electronics board in order to provide voltage and current sources to the hybrid board. We discuss the detailed design of this second homemade board in the following section. The control signals are generated from a commercially available field programmable gate array (FPGA) test board from Xilinx [18]. It has a SPARTAN XC3S200 on the board and a very high speed integrated circuit hardware description language (VHDL) firmware is written by us in order to generate LVCMOS control logic signals to be sent to the VA chip. The detailed time structure of these control signals may be found from the reference [12]. The indication of a successful communication with the VA chip may be the presence of a return signal from the VA chip. To be more specific, when all 128 channels are serially read out, there is a signal coming from the VA chip, indicating a serial data readout is completed. In the reference [12], it is referred as shift out and we confirmed that we were able to see this line became active-low, immediately after all 128 channels were read out. Figure 3 (a) and (b) show the behavior of the analog output and the shift out signals, captured in an oscilloscope from a commercially available VA evaluation board [13] that was tested by us, and from our homemade hybrid board version 1.0. The well-like signals in Fig. 3 (a) and (b) show the analog outputs from the evaluation board and our hybrid board version 1.0, respectively. Since the evaluation board we purchased has two VA chips on the board, the width of the well from the evaluation board in Fig. 3 (a) is twice larger than the width from the hybrid board version 1.0 in Fig. 3 (b). At this stage, both boards have no sensors mounted and therefore the analog outputs are pedestals only. The other signals in Fig. 3 (a) and (b) are shift out and should be active-low at the end of the serial readout of the VA chip. Such behavior can be clearly seen from the zoomed view in Fig. 3 (c) for the evaluation board and (d) for the hybrid board version 1.0, respectively. It appears that cross-talk from the clock signal at the edge to the analog output signal is somewhat worse in the hybrid board 1.0, as indicated in Fig. 3 (c) and (d). We attribute that it is originated from the ground routing issues in the PCB design or imperfect impedance matching but no conclusive statement can made at this moment without further study. III. DESIGN OF POWER, LOGIC TRANSLATORS AND CURRENT SOURCES In order to operate the VA chip, one has to provide several voltage and current sources, and non-standard control logic signals for serial readout and calibration purposes. Also, bias voltage for the silicon sensors has to be provided as well. In order to provide power, logic translators, and current sources (PLC), we developed another homemade electronics called PLC board. We started with a hand-soldered prototype which is shown in Fig. 4. There are four dc power lines connected to the PLC board: ±6.6 V for the main power that operates components mounted, +4 V for the sensor bias voltage, and +16 V for the extra bias for p-stop in the sensor we were planning to mount at the design stage. In order to bias the silicon sensors, a dc to dc converter is designed using the EMCO high voltage chip Q01-05 [19]. This model was chosen in order to provide positive and negative voltages simultaneously due to the fact that the R/C chip was used in the hybrid side. The VA chip control signals are originally generated from the outside of the PLC board and are fed into the PLC board as LVCMOS logic. The PLC receives the control signals and translate them into a new logic with logical one begin +1.5 V and zero −2.0 V. Once it is done, signals are transferred to the hybrid board. Another functionality in the PLC board is that the differential analog output from the VA chip is changed to a single-ended signal through an analog receiver. This may be the source of the noise that appears in following sections. After careful studies on this prototype, a PCB is fabricated and a picture of it is shown in Fig. 5. It has 6 layers and most of the components are chosen to be surface mountable type, in order to reduce the size of the board. The physical dimension is 84×84 mm. The analog and digital powers are now separated in the PCB version of the PLC board and it enables us to reduce the noise due to the digital clock. There are two square layouts on the PCB which are left blank in Fig. 5. Two EMCO high voltage chips are mounted on the back side of the board due to the mistakes in the design of the PCB. With this PLC board, the VA chip is tested in a calibration mode. An external coupling capacitor is connected to the calibration input and test pulses are generated in order to store electric charge to the capacitor. The channel to be tested is selected prior to the charge injection and the amplified signal comes out without serialization of the data. In this sense, the calibration is somewhat different from the serial signal readout from the sensor. Figure 6 shows the response of the VA chip for different test pulse values in mV. In this test, one MIP corresponds to 3 mV. A good linearity is achieved up to 7 MIPs, indicating a good performance of the VA chip with our assembled electronics. According to the VA chip specification, the dynamic range reaches to ± 10 MIPs but the goal of our study is to develop the electronic boards and therefore we did not test the full dynamic range of the VA chip in this study. IV. DESIGN OF HYBRID BOARD II Due to the fact that the delivery of sensors are behind the schedule, we decided to design another hybrid board that allows us to test the readability of the VA chip without real sensors. We label this one as VA-test hybrid board. In this board, the rectangular hole for the sensor mounting is removed, as indicated in Fig. 7. Instead, we place a set of wire-bond pads on the board near to the input sides of the VA chips to be mounted. One may see such configuration in Fig. 7, left side from the layouts for the VA chips. And then, direct wire- bonding from these pads to the input pads on the VA chip is made in order to inject electric charges to the VA chip. This is practically same method as the charge injection using the real sensor attached to the VA chip. This method is however somewhat different from the calibration mode mentioned in the previous section because in the calibration process, there is no holding of the charge inside of the VA chip and no serialization of the data is carried out. A lemo connector is prepared in order to inject electric charges using an external pulse generator and channel selection is made through hand-soldering to the pads to be tested, one at the time. Using this technique, one MIP “signal” is generated and the measured voltage output is shown in Fig. 8. The bump on the left corresponds to the pedestal of the entire electronics and the other bump on the right is one MIP signal. The measurement was done directly from the oscilloscope by measuring the voltage outputs from the hybrid board. From the fits to two bumps, the signal-to-noise ratio was measured to be 14. This is worse than the nominal values one may get with the silicon sensors. One of the reasons may be due to the fact that the electronics noise is larger. We discuss it in detail in the next section. The equivalent noise charge (ENC) is measured to be 1740 e− ENC with an 1 pF coupling capacitor and again this is a significantly worse value than the value in the specification, 180 + 7.5/pF e− ENC [13]. We discuss a possible reason for this in the next section. In this VA-test board, we also tested serial readout of multiple VA chips. For this test, two VA chips are daisy-chained and in total 256 channels are read out. We confirmed that the analog output behaves similar to described in Fig. 3 (a) where two VA chips were mounted on the evaluation board that we tested. We also injected a test pulse to the second VA chip and successfully read out signals from it. V. DESIGN OF HYBRID BOARD III AND BEAM TEST Based on the experience gained from studies described in previous sections, we designed our hybrid board version 2.0, shown in Fig. 9. This time, a 17 channel single-sided silicon detector (SSD) [15] is mounted. Also, in order to avoid the complexity in biasing the detector due to the R/C chip, we decide to make an array of surface-mount resistors and capacitors to compensate leakage current. The version 2.0 has in total 6 layers in the PCB including a power and a ground plane. One VA chip is hand-mounted with the conductive epoxy in the laboratory and all necessary wire-bonding processes are done from the company [17]. Note that there are only 17 channels that are wire-bonded from the sensor to the VA chip. All electrical tests are carried out and show no trouble. In order to measure real signal from the charged particle, a beam test is carried out at Korea Institute of Radiological and Medical Science (KIRAMS). A small proton cyclotron with an energy of 35 MeV is used for the test with the beam current ranging from 0.3 nA to 10 nA. Here, we discuss the performance of the electronics only and detailed performance of the sensor will be addressed in a separate paper. First, the width of the pedestal we measured at the laboratory is similar to the level at KIRAMS when the proton beam is present, indicating the electronics does not become noisy in environment such as the beam area. Then the detector is fully biased and the signal is read out using the data acquisition system we prepared. A trigger signal comes from a 30 ml liquid scintillator that is made of 10 % of BC501A and 90 % of mineral oil loading [21]. A 12 bit 40 MHz Versamodule Eurocard (VME) flash analog-to-digital converter (FADC) is used to digitize the analog output signal from the hybrid board. A VME CPU running a linux operating system collects data stored in a 4K word long buffer inside of the FADC. The ROOT [20] package is interfaced with a VME device driver that controls VME slave boards and the data are stored in a ROOT format for offline analysis. A clocked raw output from the hybrid board is shown in Fig. 10 (a). It has two sharp peaks and their positions correspond to the channels that were wire-bonded to the VA chip. These two peaks may correspond to the proton beams detected by the silicon strip sensor. However, we observe undershooting of the clocked analog outputs. The zoomed view of the peak is in Fig. 10 (b) with the clock signal shown in Fig. 10 (c). A clear undershooting exists over a clock period. It appears that the source of the undershooting may be from the driver chip on the PLC board that converts differential analog output from the VA chip to single-ended signal. It turns out that the speed of the driver chip was much slower than the clock speed of the VA chip operation which was 4 MHz for the beam test. This may explain a larger noise observed in the signal-to-ratio measurement shown in the previous section. VI. CONCLUSIONS A series of electronics boards are fabricated in order to serially read out small analog signals from high density sensors such as silicon strip detectors. A commercially available amplifier is mounted on a homemade hybrid board in order to receive analog signals from the detectors. Fabrication of the board, micro-patterning of pads for ultrasonic wire-bonding, and necessary wire-bonding on the hybrid board are carried out and tests show a good per- formance of the fabricated board. Also, a compact electronics board that provides necessary current, voltage source, and control logic is fabricated in order to communicate with the hybrid board. The linearity of the VA chip is studied in the calibration mode and a good response is achieved up to 7 MIPs. A VA-test hybrid board is fabricated and the signal-to- ratio is measured with somewhat worse ENC value then the specification value. We expose the electronics and silicon sensors in the proton beams and verified a good performance of the electronics in the beam environment. However, the noise value is higher than the nomi- nal value and there is undershooting in the analog output signal. It turns out that the speed of the driver chip on the PLC board may be too slow for our application. These problems will be examined in future and will be addressed in a separate paper. The electronics boards discussed in the paper may be used in applications including medical imaging and charged particle tracking system with no thermal dissipation is required. Fabrication of the hybrid board with a ceramic PCB will be studied in future. Acknowledgments This work is supported by grant No. R01-2005-000-10089-0 from the Basic Research Pro- gram of the Korea Science & Engineering Foundation and supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-C00258 and KRF-2005-070-C00032). EW is partially supported by a special startup grant from SK Corporation and by a Korea University Grant. [1] Proc. 3rd, 4th and 5th Europ. Symp. on Semiconductor Detectors, Munich, Nucl. Instr. and Meth. A 226 (1984); A 253 (1987); A 288 (1990). [2] F. Bedeschi, F. Bedeschi, S. Belforte, G. Bellettini, L. Bosisio, F. Cervelli, G. Chiarelli, R. DelFabbro, M. Dellorso, A. DiVirgillo, and E. Focardi, IEEE Trans. Nucl. Sci. NS-33 (1986). [3] M. Hazumi, Nucl. Instr. and Meth. A 379, 390 (1996). [4] A. Sill, Nucl. Instr. and Meth. A 447, 1 (2000). [5] G. Hall, Nucl. Instr. and Meth. A 541, 248 (2005). [6] M. Feuerstack-Raible, Nucl. Instr. and Meth. A 447, 35 (2000). [7] M. Tanaka, M. Hazumi, J. Ryuko, K. Sumisawa, and D. Marlow, Nucl. Instr. and Meth. A 432, 422 (1999). [8] A. Rahimi, K. E. Arms, K. K. Gan, M. Johnson, H. Kagan, C. Rush, S. Smith, R. Ter- Antonian, M. M. Zoeller, A. Ciliox, M. Holder, S. Nderitu, and M. Ziolkowski, International Journal of Modern Physics A, 20, 3805 (2005). [9] G. J. Royle, A. Papanestis, R. D. Speller, G. Hall, G. Iles, M. Raymond, E. Corrin, P. F. van der Stelt, N. Manthos, and F. A. Triantis, Nucl. Instr. and Meth. A 493, 176 (2002). [10] E. Won, H. Ha, S. K. Park, and Y. I. Kim, J. Korean Phys. Soc. 49, 52 (2006). [11] E. Nygard, Nucl. Instr. and Meth. A 301, 506 (1991). [12] O. Toker, Nucl. Instr. and Meth. A 340, 572 (1994). [13] IDEAS, Snaroya, Norway. [14] J. Fast for the CLEO collaborations, Nucl. Instr. and Meth. A 435, 9 (1999). [15] J. Lee, Dong Ha Kah, Hong joo Kim, Hwan Bae Park, and Jung Ho So, J. Korean Phys. Soc. 48, 850 (2006). [16] Chemtronics, Kennesaw, GA, USA. [17] LP electronics, Seoul, Korea. [18] Xilinx Corporation, San Jose, CA, USA. [19] EMCO High Voltage Corporation, Sutter Creek, CA, USA. [20] R. Brun and F. Rademakers, Nucl. Instr. and Meth. A 389, 81 (1997). [21] Private communication with H. J. Kim. FIG. 1: A picture of the hybrid board version 0.9. A large rectangular hole on left is prepared for future beam or radioactive source tests when a sensor is mounted on the hybrid board. U1 is the pad for the VA chip and U2 is for the R/C chip. FIG. 2: Zoomed views of hybrid board version 0.9 and version 1.0. Microscope picture of the bonding pad on the board is shown in (a) for the version 0.9 and in (b) for the version 1.0. They are in same scale and clear improvement in the quality of the layout can be seen in (b). Side views of the boards are shown in (c) for the version 0.9 and in (d) for the version 1.0. FIG. 3: Analog and digital output signals from the VA chip captured in the oscilloscope. A distorted well-shape signal in (a) and (b) are the analog outputs from the evaluation board provided by the company and from our hybrid board version 1.0, respectively. The evaluation board houses two VA chips and the width of the well in (a) reflects it. The other line in each figure is the digital control signal (shift out) indicating the end of the serial readout. (c) and (d) are zoomed views of (a) and (b) at the rising edge of the analog out. FIG. 4: A picture of the PLC board prototype with hand-wirings. A micro-connector at the bottom right is for the hybrid board, the one on the top-right is for the commercial evaluation board with a FPGA, and the one on the top left is for dc power connection. FIG. 5: A picture of the PLC board fabricated with a standard PCB process. The physical size is 84 × 84 mm. A micro-connector at the bottom right is for the hybrid board, the one on the top is for the commercial evaluation board with a FPGA, and the one on the left is for dc power connection. All the components are chosen to be surface mountable in order to reduce the size of the board. 0 2 4 6 8 10 12 14 16 18 20 Input Pulse (mV) FIG. 6: A response of the VA chip and hybrid board version 1.0 on test pulses when the VA chip is in the calibration mode. Here, an input test pulse of 3 mV corresponds to 1 MIP signal. FIG. 7: A picture of the VA-test board. This board is designed to mount in total four VA chips. Selected channels in the VA side can be wire-bonded to pads on the PCB for a direct delivery of current signals from an external pulse generator. -20 -10 0 10 20 30 400 Volt (mV) FIG. 8: Distribution of the measured analog output from the VA-test board when a test pulse of one MIP signal is injected. The signal distribution is on the right and the pedestal peak is shown on the left. The measurement was done directly from the oscilloscope by measuring the voltage outputs from the hybrid board. FIG. 9: A picture of the hybrid board version 2.0. A 17 channel single-sided silicon sensor is mounted on the left side. Surface-mounted resistor and capacitor arrays are placed in order to compensate dc current in order to replace the R/C chip. 0 200 400 600 800 1000 Time (25 ns) 0 10 20 30 40 50 60 Time (25 ns) FIG. 10: Analog output and clock signals from the hybrid board. The analog output from the VA chip over 128 channels is shown in (a) where the readout clock starts at 100 and ends at 950 counts in the horizontal axis. A zoomed view of the first peak in (a) is shown in (b) where an undershooting is clearly visible. The time synchronized clock signal fed into the VA chip is also shown in (c). INTRODUCTION DESIGN OF HYBRID BOARD I DESIGN OF POWER, LOGIC TRANSLATORS and CURRENT SOURCES DESIGN OF HYBRID BOARD II DESIGN OF HYBRID BOARD III and BEAM TEST CONCLUSIONS Acknowledgments References

0704.1335

On the reductive Borel-Serre compactification: $L^p$-cohomology of arithmetic groups (for large $p$)

arXiv:0704.1335v1 [math.AG] 11 Apr 2007 On the reductive Borel-Serre compactification: Lp-cohomology of arithmetic groups (for large p) Steven Zucker1 Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 USA2 Introduction The main purpose of this article is to give the proof of the following theorem, as well as some applications of the result. Theorem 1. Let M be the quotient of a non-compact symmetric space by an arithmetically-defined group of isometries, and MRBS its reductive Borel-Serre compactification. Then for p finite and sufficiently large there is a canonical iso- morphism H•(p)(M) ≃ H •(MRBS). Here, the left-hand side is the Lp-cohomology of M with respect to a (locally) invariant metric. Though it would be more natural to allow p =∞ in Theorem 1, this is not generally possible (see (3.2.2)). On the other hand, there is a natural mapping H• (M) → H• (M) when p < ∞, because M has finite volume. The definition of MRBS is recalled in (1.9). Theorem 1 can be viewed as an analogue of the so-called Zucker conjecture (in the case of constant coefficients), where p = 2: Theorem [L], [SS]. Let M be the quotient of a Hermitian symmetric space of non- compact type by an arithmetically-defined group of isometries, i.e., a locally sym- metric variety; let MBB its Baily-Borel Satake compactification. Then there is a canonical isomorphism H•(2)(M) ≃ IH where the right-hand side denotes the middle intersection cohomology of MBB. However, Theorem 1 is not nearly so difficult to prove, once one senses that it is true; it follows without much ado from the methods in [Z3] (the generalization to Lp, p 6= 2, of those of [Z1] for L2). As far as I know, the reductive Borel-Serre compactification was first used in [Z1,§4] (where it was called Y ). This space, a rather direct alteration of the manifold-with-corners constructed in [BS], was introduced there to facilitate the study of the L2-cohomology of M . It also plays a central role as the natural set- ting for the related weighted cohomology of [GHM]. It is a principal theme that MRBS is an important space when M is an algebraic variety over C, despite the 1Support in part by the National Science Foundation, through Grant DMS9820958 2e-mail address: [email protected] http://arxiv.org/abs/0704.1335v1 fact that MRBS is almost never an algebro-geometric, or even complex analytic, compactification of M . This work had its origin in my wanting to understand [GP]. It is convenient to formulate the latter before continuing with the content of this article. Let Y be a Hausdorff topological space. For any complex vector bundle E on Y , one has its Chern classes ck(E) ∈ H 2k(Y,Z). If we further assume that Y is connected, compact, stratified and oriented, then Hd(Y,Z) ≃ Z, where d is the dimension of Y ; the orientation picks out a generator ζY for this homology group, known as the fundamental class of Y . We shall henceforth assume that d is even, and we write d = 2n. Then, if one has positive integers ki for 1 ≤ i ≤ ℓ such that ki = n, one can pair ck1(E) ∪ · · · ∪ ckℓ(E) with ζY , and obtain what is called a characteristic number, or Chern number, of E. When Y is a C∞ manifold and E is a C∞ vector bundle, the Chern classes modulo torsion can be constructed from any connection ▽ in E, whereupon they get represented, via the de Rham theorem, by the Chern forms ck(E,▽) in H 2k(Y ). (For convenience, we will use and understand C-coefficients here and throughout the sequel unless it is specified otherwise.) If Y is compact, the Chern numbers can be computed by integrating ck1(E,▽) ∧ . . . ∧ ckℓ(E,▽) over Y . For stratified spaces Y , there is a lattice of intersection (co)homology theories, with variable perversity p as parameter, as defined by Goresky and MacPherson [GM1]. These range from standard cohomology as minimal object, to standard homology as maximal, and all coincide when Y is a manifold. They can all be defined as cohomology with values in some constructible sheaf whose restriction to the regular locus Y reg of Y is just CY reg . With mappings going in the direction of increasing perversity, we have the basic diagram (0.1) H•(Y ) → . . . → IH•p(Y ) → . . . → H•(Y ) ↓ ↑ ↑ H•(Y reg) ←− H•c (Y reg) ≃ H•(Y reg). When Y is compact, ζY lifts to a generator of IH (Y ) for all p. From now on, we writeM for Y reg, and start to view the situation in the opposite way, regarding Y as a topological compactification of the manifold M . For any vector bundle E on M , and bundle extension of E to E on Y , the functoriality of Chern classes imply that c•(E) 7→ c•(E) under the restriction mapping H •(Y ) H•(M). One might think of this as lifting the Chern class of E to the cohomology of Y , but one should be aware that ρ might have non-trivial kernel, so the lift may depend on the choice of E. The case where E = TM , the tangent bundle ofM , is quite fundamental. Finding a vector bundle on Y that extends TM is not so natural a question when Y has sin- gularities, and one is often inclined to forget about bundles and think instead about just lifting the Chern classes. When Y is a complex algebraic variety, one considers the complex tangent bundle T ′M of M . It is shown in [M] that for constructible Z- valued functions F on Y , there is a natural assignment of Chern homology classes c•(Y ;F ) ∈ H•(Y ), such that c•(Y, 1) recovers the usual Chern classes when Y is smooth. There has been substantial interest in lifting these classes to the lower intersection cohomology (as in the top row of (0.1)), best to cohomology (the most difficult lifting problem) for the reason mentioned earlier. Next, take for M a locally symmetric variety. For Y we might consider any of the interesting compactifications of M , which include: MBB , the Baily-Borel Satake compactification ofM as an algebraic variety [BB];MΣ, the smooth toroidal compactifications of Mumford (see [Mu]); MBS , the Borel-Serre manifold-with- corners [BS]; MRBS , the reductive Borel-Serre compactification. These fit into a diagram of compactifications: (0.2) MBS −→ MRBS MΣ −→ M If one tries to compare MBS and MΣ, one sees that there is a mapping (of compactifications of M) MBS → MΣ only in a few cases (e.g., G = SU(n, 1)). However, by a result of Goresky and Tai [GT, 7.3], (if Σ is sufficiently fine) there are continuous mappingsMΣ −→M RBS (seldom a morphism of compactifications) such that upon inserting them in (0.2), the obvious triangle commutes in the homotopy category. One thereby gets a diagram of cohomology mappings (0.3) H•(M) ←− H•(MRBS) ↑ ւ ↑ H•(MΣ) ←− H •(MBB); also, the fundamental classes inHd(MΣ,Z) andHd(M RBS ,Z) are mapped to ζMBB . Let E be a (locally) homogeneous vector bundle onM (an example of which is the holomorphic tangent bundle T ′M ). There always exists an equivariant connection on E, whose Chern forms are L∞ (indeed, of constant length) with respect to the natural metric on M . In [Mu], Mumford showed that the bundle E has a so-called canonical extension to a vector bundle EΣ on MΣ, such that these Chern forms, beyond representing the Chern classes of E in H•(M), actually represent the Chern classes of EΣ inH •(MΣ) (see our (3.2.4)). That served the useful purpose of placing these classes in a ring with Poincaré duality, and implied Hirzebruch proportionality for M . In [GP], Goresky and Pardon lift these classes to the cohomology of MBB , min- imal in the lattice of interesting compactifications of M , so these can be pulled back to the other compactifications in (0.2). (On the other hand, the bundles do not extend to MBB in any obvious way.) They achieve this by constructing another connection in E (see our (5.3.3)), one that has good properties near the singular strata of MBB , using features from the work of Harris and Harris-Zucker (see [Z5,App.B]). With this done, the Chern forms lie in the complex of controlled differential forms on MBB , whose cohomology groups give H•(MBB). In the case of the tangent bundle, the classes map to one of the MacPherson Chern homology classes in H•(M BB), viz., c•(M BB;χM ), where χM denotes the characteristic (i.e., indicator) function of M ⊂MBB [GP, 15.5]. In [GT, 9.2], an extension of E to a vector bundle ERBS toMRBS is constructed; this does not require M to be Hermitian. There, one finds the following: Conjecture A [GT, 9.5]. Let MΣ → M RBS be any of the continuous mappings constructed in [GT]. Then the canonical extension EΣ is isomorphic to the pullback of ERBS. In the absence of a proof of Conjecture A, we derive the “topological” analogue of Mumford’s result as a consequence of Theorem 1: Theorem 2. LetM be an arithmetic quotient of a symmetric space of non-compact type. Then the Chern forms of an equivariant connection onM represent c•(E in H•(MRBS). We point out that (0.3) and Conjecture A suggest that this is more basic in the Hermitian setting than Mumford’s result. Goresky and Pardon predict further: Conjecture B [GP]. The Chern classes of ERBS are the pullback of the classes in H•(MBB) constructed in [GP] via the quotient mapping MRBS →MBB. Our third main result is the proof of Conjecture B. The material of this article is organized as follows. In §1 we give a canon- ical construction of the bundle ERBS along the lines of [BS]. We next discuss Lp-cohomology, both in general in §2, then on arithmetic quotients of symmetric spaces in §3, achieving a proof of Theorem 1. We make a consequent observation in (3.3) that shows how Lp-cohomology can be used to provide definitions of map- pings between topological cohomology groups when it is unclear how to define the mappings topologically. In §4, we treat connections and the notion of Chern forms for a natural class of vector bundles on stratified spaces; this allows for the proof in §5 of both Theorem 2 and Conjecture B. This article was conceived while I was spending Academic Year 1998–99 on sabbatical at the Institute for Advanced Study in Princeton. I wish to thank Mark Goresky and John Mather for helpful discussions. 1. The Borel-Serre construction for homogeneous vector bundles In this section, we make a direct analogue of the Borel-Serre construction for the total space of a homogeneous vector bundle on a symmetric space, and then for any neat arithmetic quotient MΓ thereof. It defines a natural extension of the vector bundle to the Borel-Serre compactification of the space. That the bundle extends is clear, for attaching of a boundary-with-corners does not change homotopy type. Our construction retains at the boundary much of the group-theoretic structure. The construction is shown to descend to the reductive Borel-Serre compactification MRBSΓ , reproving [GT, 9.2]. (1.0) Convention. Whenever H is an algebraic group defined over Q, we also let H denote H(R), taken with its topology as a real Lie group, if there is no danger of confusion. (1.1) Standard notions. Let G be a semi-simple algebraic group over Q, and K a maximal compact subgroup of G, and X = G/K. (Note that this implies a choice of basepoint for X , namely the point x0 left fixed by K.) Let E = G ×K E be the homogeneous vector bundle on X determined by the representation of K on the vector space E. The natural projection π : E → X = G×K{0} is induced by the projection E → {0}, and isG-equivariant. For Γ ⊂ G(Q) a torsion-free arithmetic subgroup, letMΓ = Γ\X . Then Γ\E is the total space of a vector bundle EΓ overMΓ. (The subscript “Γ” was suppressed in the Introduction.) If P is any Q-parabolic subgroup of G, the action of P on X is transitive. Thus, one can also describe E → X as P ×KP E → P ×KP {0}, where KP = K ∩ P . (1.2) Geodesic action. Let UP denote the unipotent radical of P , and AP the lift to P associated to x0 of the connected component of the maximal Q-split torus Z of P/UP . Define the geodesic action of AP on E by the formula: (1.2.1) a ◦ (p, e) = (pa, e) whenever p ∈ P , e ∈ E and a ∈ AP ; this is well-defined because for k ∈ KP , (1.2.2) a ◦ (pk−1, ke) = (pk−1a, ke) = (pak−1, ke), as AP and KP commute. The geodesic action of AP commutes with the action of P on E , and it projects to the geodesic action of AP on X as defined in [BS, §3] (in [BS], the geodesic action is expressed in terms of Z, but the definitions coincide). (1.2.3) Remark. By taking E to be of dimension zero, the construction of Borel- Serre can be viewed as a case of ours above. As such, there is no real need to recall it separately. Conversely, a fair though incomplete picture of our construction can be seen by regarding E as simply a thickened version of X . (1.3) Corners. The simple roots occurring in UP set up an isomorphism AP ≃ (0,∞)r(P ), where r(P ) denotes the parabolic Q-rank of P . Let AP be the en- largement of AP obtained by transport of structure from (0,∞) r(P ) ⊂ (0,∞]r(P ). Define the corner associated to P : E(P ) = E×AP AP . There is a canonical mapping π(P ) : E(P )→ X(P ) = X ×AP AP . (1.3.1) Remark. ThoughX(P ) is contractible, and hence E(P ) is trivial, (1.2.1) does not yield a canonical trivialization of E(P ) over X(P ), because of the equivalence relation (1.2.2) determined by KP . Let ∞P denote the zero-dimensional AP -orbit in AP , which corresponds to (∞, ...,∞) ∈ (0,∞]r(P ). The face of E(P ) associated to P is (1.3.2) E(P ) = E ×AP {∞P} ≃ E/AP . It maps canonically to X/AP ≃ e(P ) ⊂ X(P ) (from [BS, 5.2]). There are geodesic projections implicit in (1.3.2), given by the rows of the commutative diagram (1.3.3) E(P ) −−−−→ E(P ) X(P ) −−−−→ e(P ) (1.4) Structure of E(P ). There is a natural P -action on E(P ), with AP acting trivially, projecting to the action of P on e(P ). We know that e(P ) is homogeneous under 0P (as in [BS, 1.1]), isomorphic to P/AP , which contains KP . We see that E(P ) is isomorphic to the homogeneous vector bundle on e(P ) determined by the representation of KP on E. (1.5) Compatibility. For Q ⊂ P , there is a canonical embedding of E(P ) in E(Q), given as follows. As in [BS, 4.3], write AQ = AP × AQ,P , with AQ,P ⊂ AQ de- noting the intersection of the kernels of the simple roots for AP . Then there is an embedding E(P ) = E ×AP AP ≃ (E ×AQ,P AQ,P )×AP AP ⊂ (E ×AQ,P AQ,P )×AP AP ≃ E ×AQ AQ = E(Q). Moreover, this projects to X(P ) ⊂ X(Q) via π(Q). (1.6) Hereditary property. If Q ⊂ P again, one can view E(Q) as part of the boundary of E(P ), in the same way that e(Q) is part of the boundary of e(P ). This is achieved by considering the geodesic action of AQ,P on E(P ) (AP acts trivially), and carrying out the analogue of (1.3). Thus, E(Q) ≃ E(P )/AQ,P = E(P )/AQ. (1.7) The bundle with corners. Using the identifications given in (1.5), we recall that one puts (1.7.1) X = X(P ) = e(P ), with P ranging over all parabolic subgroups of G/Q, including the improper one (G itself). With X endowed with the weak topology from the X(P )’s, this is the manifold-with-corners construction of Borel-Serre for X (see [BS, §7]). As such, it has a tautological stratification, with the e(P )’s as strata. We likewise put E = E(P ), with incidences given by (1.5), and endow it with the weak topology. There is an obvious projection onto X. Then E is a vector bundle over X that is stratified by the homogeneous bundles E(P ), given as in (1.4). (1.8) Quotient by arithmetic groups. We can see that G(Q) acts as vector bundle automorphisms on E over its action as homeomorphisms of X (given in [BS, 7.6]); also, as it is so for X , the action on E of any neat arithmetic subgroup Γ of G(Q) is proper and discontinuous (cf. [BS, 9.3]). Then EΓ = Γ\E is a vector bundle over MBSΓ = Γ\X . Let ΓP = Γ∩P . The action of ΓP (which is contained in 0P of (1.4)) commutes with the geodesic action of AP . The faces of EΓ are of the form E ′(P ) = ΓP \E(P ), and are vector bundles over the faces e′(P ) = ΓP \e(P ) of M Γ . By reduction theory [BS, §9] (but see also [Z5,(1.3)]), there is a neighborhood of e′(P ) in MBSΓ on which geodesic projection πP (from (1.3.3)) descends. The same is true for π̃P and E′(P ) (also from (1.3.3)). (1.9) The reductive Borel-Serre compactification. We recall the quotient space XRBS of X. With X given as in (1.7) above, one forms the quotient (1.9.1) XRBS = XP , (where XP = UP \e(P )) , where UP is, as in (1.2), the unipotent radical of P , and endows it with the quotient topology from X. Because UQ ⊃ UP whenever Q ⊂ P , X RBS is a Hausdorff space (see [Z1,(4.2)]). There is an induced action of G(Q) on XRBS , for which (1.9.1) is a G(Q)-equivariant stratification; G(Q) takes the stratum XP onto that of a conjugate parabolic subgroup, with P (Q) preserving XP . For any arithmetic group Γ ⊂ G(Q), one has a quotient mapping (1.9.2) q :MBSΓ →M Γ = Γ\X RBS = M̂P , with M̂P = ΓP \XP . (1.10) Descent of E to XRBS. Analogous to the description of X in (1.7), we have (1.10.1) E = E(P ) = E(P ), and the corresponding quotient (1.10.2) ERBS = (UP \E(P )) . We verify that ERBS is a vector bundle on XRBS . Since {X(P )} is an open cover of X (see (1.7.1)), it suffices to verify this for E(P )→ X(P ) for each P separately. Note that UP acts on E(P ) by the formula: u · (p, e, a) = (up, e, a), and this commutes with the action of KP ·AP . It follows that there is a canonical projection (1.10.3) UP \E(P )→ UP \X(P ). This gives a vector bundle on UP \X(P ) because UP ∩ (KP ·AP ) = {1}. Let XRBS(P ) be the image of X(P ) in XRBS , and ERBS(P ) be the image of E(P ) in ERBS . These differ from (1.10.3), for the UP quotient there is too coarse (for instance, there are no identifications on X or E in ERBS → XRBS). Rather, the pullback of (1.10.3) to XRBS(P ) is ERBS(P ). When Γ is a neat arithmetic group, ERBSΓ = Γ\E RBS is a vector bundle on MRBSΓ . This is verified in the same manner as (1.8). 2. Lp-cohomology By now, the notion of Lp-cohomology, with 1 ≤ p ≤ ∞, is rather well-established. The case of p = ∞, though, is visibly different from the case of finite p, and was neglected in [Z4]. Morally, Theorem 1 is about L∞-cohomology, but for technical reasons we will have to settle for Lp-cohomology for large finite p. It is our first goal to prove Theorem 1. (2.1) Preliminaries. Let M be a C∞ Riemannian manifold. For any C∞ dif- ferential form φ on M , its length |φ| is a non-negative continuous function on M . This determines a semi-norm: (2.1.1) ||φ||p = |φ(x)|p dVM (x) p if 1 ≤ p <∞; sup {|φ(x)| : x ∈M} if p =∞, where dVM (x) denotes the Riemannian volume density of M . One says that φ is Lp if ||φ||p is finite. (2.1.2) Definitions. Let w be a positive continuous real-valued function on the Riemannian manifold M . i) The [smooth] Lp de Rham complex with weight w is the largest subcomplex of the C∞ de Rham complex of M consisting of forms φ such that wφ is Lp, viz. (2.1.2.1) A•(p)(M ;w) = {φ ∈ A •(M) : wφ and wdφ are Lp}. ii) The [smooth] Lp-cohomology of M with weight w is the cohomology of (M ;w). It is denoted H• (M ;w). We note that in the above, there is a difference with the notation used elsewhere: for p 6=∞, w might be replaced with w p in (2.1.2.1). When w = 1, one drops the symbol for the weight. Note that the complex depends on w only through rates of the growth or decay of w at infinity. When M has finite volume, there are inclusions A• (M) →֒ A• (M) whenever 1 ≤ p < p′ ≤ ∞. The preceding extends to metrized local systems (cf. [Z1, §1]). Smooth functions are dense in the Banach space Lp for 1 ≤ p <∞, but not in L∞. We next recall the basic properties of Lp-cohomology. Let M be a compact Hausdorff topological space that is a compactification ofM . One defines a presheaf on M by the following rule (cf. [Z4, 1.9]): to any open subset V of M , one assigns (V ∩M ;w). Because M is compact (see (2.1.5, ii) below), the associated sheaf (M ;w) satisfies (2.1.3) A•(p)(M ;w) −→Γ(M,A•(p)(M ;w)). It follows from the definition that whenever q : M → M is a morphism of com- pactifications of M , one has for all p: (2.1.4) q∗A (p)(M ;w) ≃ A•(p)(M ;w). (2.1.5) Remarks. i) It is easy to see that the complex A• (M ;w) consists of fine sheaves if and only if for every covering of MBS there is a partition of unity subor- dinate to that covering consisting of functions f whose differential lies in A1 i.e., |df | is a bounded function on M . Thus, (2.1.4) is for q∗ (as written), not for Rq∗ in general. ii) Note that in general, the space of global sections of A• (M ;w), defined in the obvious way (or equivalently the restriction of A• (M ;w) toM) is A• (p),loc (M ;w) = A•(M). Without a compact boundary, there is no place to store the global bound- edness condition. The following fact makes for a convenient simplification: (2.1.6) Proposition. LetM be the interior of a Riemannian manifold-with-corners M (i.e., the metric is locally extendable across the boundary). Let A (p)(M ;w) be the sub-complex of A• (M ;w) consisting of forms that are also smooth at the boundary of M . Then the inclusion (p)(M ;w) →֒ A (p)(M ;w) is a quasi-isomorphism. � In other words, one can calculate H• (M ;w) using only forms with the nicest behavior along ∂M . Moreover, A (p)(M ;w) admits a simpler description; for that and the proof of (2.1.6), see (2.3.7) and (2.3.9) below. (2.2) The prototype. We compute a simple case of Lp-cohomology, one that will be useful in the sequel. (2.2.1) Proposition [Z4, 2.1]. Let R+ denote the positive real numbers, and t the linear coordinate from R. For a ∈ R, let wa(t) = e at. Then i) H0 (R+;wa) ≃ 0 if a > 0, C if a ≤ 0. ii) H1 (R+;wa) = 0 for all a 6= 0. Proof. Again, we carry this out here only for p =∞. First, (i) is obvious: it is just an issue of whether the constant functions satisfy the corresponding L∞ condition. To get started on (ii), proving that a complex is acyclic can be accomplished by finding a cochain homotopy operator B (lowering degrees by one), such that φ = dBφ + Bdφ. For the cases at hand (1-forms on R+), this equation reduces to φ = dBφ. When a < 0, one takes (2.2.2) B(φ)(t) = − g(x)dx when φ = g(t)dt (placing the basepoint at ∞ is legitimate, as g decays exponen- tially). We need to check that (2.2.2) lies in the L∞ complex. By hypothesis, |g(t)| ≤ Cw−a(t) for some constant C. This implies that |B(φ)(t)| ≤ |g(x)|dx ≤ C w−a(x)dx ∼ w−a(t) as t→∞. In other words, B(φ)(t)wa(t) ∼ 1, which is what we wanted to show. When a > 0, one takes instead (2.2.3) B(φ)(t) = g(x)dx, and shows that |B(φ)|(t) ∼ w−a(t), yielding the same conclusion about B(φ) as before. � (2.2.4) Remark. One can see that for a = 0, one is talking about H1 (R+), which is not even finite-dimensional (cf. [Z1, (2.40)]); H1 (R+) contains the linearly in- dependent cohomology classes of t−νdt, for all 0 ≤ ν ≤ 1. What was essential in the proof of (2.2.1) was that wa and one of its anti-derivatives had equal rates of growth or decay when a 6= 0. That is, of course, false for a = 0. (2.3) Further properties of Lp-cohomology. We begin with (2.3.1) Proposition (A Künneth formula for Lp-cohomology). Let I be the unit interval [0, 1], with the usual metric. Then for any Riemannian manifold N and weight w, the inclusion π∗ : A• (N ;w) →֒ A• (I×N ; π∗w) is a quasi-isomorphism; H•(p)(I ×N ; π ∗w) ≃ H•(p)(N ;w). Proof. The argument is fairly standard. The formula (2.2.3) defines an operator on forms on I. Because I has finite length, one has now (2.3.2) φ = Hφ+ dBφ+Bdφ, where H is—well—harmonic projection: zero on 1-forms, mean value on 0-forms. The differential forms on a product of two spaces decompose according to bidegree. On I ×N , denote the bidegree by (eI , eN ) (thus, for a non-zero form, eI ∈ {0, 1}). The exterior derivative on I×N can be written as d = dI+σIdN , where σI is given by (−1)eI . The operators in (2.3.2) make sense for Lp forms on I ×N , taking, for each q, forms of bidegree (1, q) to forms of bidegree (0, q), and we write them with a subscript “I”; thus, we have the identity (2.3.3) φ = HIφ+ dIBIφ+BIdIφ. It is clear that BIφ is L p whenever φ is. Note that σI anticommutes with BI . We can therefore write (2.3.3) as φ = HIφ+ dBIφ− σIdNBIφ+BIdφ−BIσIdNφ(2.3.4) = (HIφ+ dBIφ+BIdφ)− (σIdNBIφ+BIσIdNφ). Since σI and dN commute, the subtracted term equals (σIBI +BIσI)dNφ = 0, so (2.3.4) is just φ = (HIφ + dBIφ + BIdφ). This implies first that dBIφ is L p and then our assertion. � We next use a standard smoothing argument in a neighborhood of 0 ∈ R. To avoid unintended pathology, we consider only monotonic weight functions w. Given a smooth function ψ on R of compact support, let (2.3.5) (Ψf)(t) = (ψ ∗ f)(t) = ψ(x)f(t− x)dx = ψ(t− x)f(x)dx, defined for those t for which the integral makes sense. The discussion separates into two cases: i) w(t) is a bounded non-decreasing function of t. In this case, take ψ to be supported in R−. ii) Likewise, when w(t) blows up as t → 0+ take ψ to be supported in R+, and set f(x) = 0 for x ≤ 0. (2.3.6) Lemma. If f ∈ Lp(R+, w) (and ψ is chosen as above), then Ψf is also in Lp(R+, w). Proof. For p <∞, see [Z4, 1.5]. When p =∞, we consider each of the above cases. In case (i), we have: w(t)Ψf(t) = ψ(t− x)w(t)f(x)dx = ψ(t− x)w(x)f(x){w(t)w(x)−1}dx. By hypothesis, the integral involves only those x for which t < x, and there w(t)w(x)−1 ≤ 1. It follows that w(t)Ψf(t) is uniformly bounded. In case (ii), when w(t) blows up as t→ 0+ the argument is similar and is left to the reader. � We use (2.3.6) to prove: (2.3.7) Proposition. With w restricted as above, let A (p)(I;w) denote the sub- complex of A• (I;w) consisting of forms that are smooth at 0. Then the inclusion (p)(I;w) →֒ A (p)(I;w) is a quasi-isomorphism, with Ψ providing a homotopy inverse. Proof. There is a well-known homotopy smoothing formula, which is at bottom a variant of (2.3.2). We use the version given in [Z4,1.5], valid on the level of germs at 0: 1−Ψ = dE + Ed, E = (1−Ψ)B, with B as above. Our assertions follow immediately. � The behavior of w forces the value f(0) of a function f ∈ L∞(I;w)∩A(I) to be 0 precisely in case (ii) above. Thus we have: (2.3.8) Corollary. Write I for the closed interval [0, 1]. For the two cases pre- ceding (2.3.6), A•(∞)(I;w) ≈ A•(I) in case (i), A•(I, 0) in case (ii). There are several standard consequences and variants of (2.3.7) in higher dimen- sion. The simplest to state are (2.1.6) and its corollary; we now give the latter: (2.3.9) Proposition. LetM be the interior of a Riemannian manifold-with-corners M , and let A• (M) be the subcomplex of A• (M) consisting of forms that are smooth at the boundary. Then the inclusion A•(p)(M) →֒ A (p)(M) induces an isomorphism on cohomology. Thus the Lp-cohomology of M can be computed as the cohomology of A• (M), i.e., H• (M) ≃ H•(M). � Finally, we will soon need the following generalization of (2.3.1): (2.3.10) Proposition. Let wM and wN be positive functions on the Riemannian manifolds M and N respectively. Suppose that on the Riemannian product M ×N , one has in the sense of operators on Lp that d = dM ⊗ 1N + σM ⊗ dN , and that (N ;wN) is finite-dimensional. Then H•(p)(M ×N ;wM × wN ) ≃ H (p)(M ;wM)⊗H (p)(N ;wN ). Remarks. i) The condition on M × N is asserting that the forms on M ×N that have separate Lp exterior derivatives along M and along N are dense in the graph norm (cf. [Z1, pp.178–181] for some discussion of when this condition holds.) ii) When p = 2, the above proposition recovers only a special case of what is in [Z1, pp.180–181]; however, the full statement of the latter does generalize to all values of p, by a parallel argument. Proof of (2.3.10). The argument is similar to what one finds in [Z1,§2], which is for the case p = 2, though we cannot use orthogonal projection here. Let h• = h•p(N ;wN) be any space of cohomology representatives for H (N ;wN ); by hypothesis, h• is a finite-dimensional Banach space. It suffices to show that the inclusion (2.3.10.1) A•(p)(M ;wM )⊗ h p(N ;wN) →֒ A•(p)(M ×N ;wM × wN ) induces an isomorphism on cohomology. For each i, let Zi denote the closed forms in Ai = Ai (N ;wN). Then D dAi−1 is a complement to hi in Zi; it is automatically closed because of the finite- dimensionality of hi. By the Hausdorff maximal principle, there is a closed linear complement Ci to Zi in Ai (canonical complements exist when p = 2). Then the open mapping theorem of functional analysis (applied for the Lp graph norm on Ai), gives that the direct sum of Banach spaces, (2.3.10.2) hi ⊕Di ⊕ Ci, is boundedly isomorphic to Ai. With respect to this decomposition of Ai, dN breaks into the 0-mapping on Zi and an isomorphism di : Ci → Di+1. We can now obtain a cochain homotopy for A•. Let Bi denote the inverse of di−1, and B and d the respective direct sums of these. One calculates that dB+Bd is equal to 1− q, where q denotes projection onto h• with respect to (2.3.10.2). Adapting this formula to M × N runs a standard course. First, B defines an operator BN = 1M ⊗ B on M × N , and likewise does q. We have the identity 1− qN = dNBN +BNdN . Noting that dN commutes with σM and that σ M = 1M , we obtain (1− qN ) = σMdN (σMBN ) + (σMBN )σMdN , and likewise dM (σMBN ) + (σMBN )dM = 0. Adding, we get 1 − qN = dB̃ + B̃d, with B̃ = σMBN , and this gives what we wanted to know about (2.3.10.1), so we are done. � 3. Lp cohomology on the reductive Borel-Serre compactification In this section, we determine the cohomology sheaves of A• (MRBS) for large finite values of p, and compare the outcome to that of related calculations. (3.1) Calculations for MRBS , and the proof of Theorem 1. We first observe that (MRBS) is a complex of fine sheaves, for the criterion of (2.1.5, i) was verified in [Z1]. (The analogous statement on MBS is false unless M is already compact; indeed, this is why the space MRBS was introduced.) Let y ∈ UP \e(P ) ⊂ M RBS . The issue is local in nature, so it suffices to work with q̃ : MBSΓUP → XRBS , and therefore we lift y to ỹ ∈ XRBS . The fiber q̃−1(ỹ) is the compact nilmanifold NP = ΓUP \UP . Since NP is compact, neighborhoods of ỹ in XRBS give, via q̃−1, a fundamental system of neighborhoods of NP in M As in [Z1,(3.6)], the intersection with MΓUP of such a neighborhood is of the (3.1.1) A+ × V ×NP , where A+P ≃ (R +)r(P ) and V is a coordinate cell on M̂P (notation as in (1.9)). After taking the exponential of the A+ -variable, the metric is given, up to quasi-isometry, (3.1.2) dt2i + dv e−2αduα, where α runs over the roots in UP . By the Künneth formula (2.3.1), we may replace V by a point in (3.1.2); we are reduced to determining H• (A+P ×NP ), where the metric is dt2i + e−2αduα. The means of computing this runs parallel to the discussion in [Z1,(4.20)]. We consider the inclusions of complexes (3.1.3) A•(p)(A P ;wβ)⊗H β(uP ,C) A•(p)(A P ;wβ)⊗ ∧ β(uP ) ≃ A•(p)(A P ×NP ) UP →֒ A•(p)(A P ×NP ). Here, uP denotes the Lie algebra of UP , and (3.1.4) wβ(a) = a pβa−δ = apβ−δ = a p(β− δ (ai = e where δ denotes the sum of the positive Q-roots (cf. (3.1.9, ii) below). We can see that the contribution of δ (which enters because of the weighting of the volume form of NP ) is non-zero yet increasingly negligible as p→∞. The second inclusion in (3.1.3) is that of the “UP -invariant” forms. Note that this reduces considerations onNP to a finite-dimensional vector space, viz. ∧ •(uP ) Here, one is invoking the isomorphism (3.1.5) H•(NP ) ≃ H •(uP , C) for nilmanifolds, which is a theorem of Nomizu [N]. The exterior algebra decomposes into non-positive weight spaces for aP , which we write as (3.1.6) ∧•(uP ) ∧•β(uP ) The first inclusion in (3.1.3) is given by Kostant’s embedding [K,(5.7.4)] ofH•(uP ,C) in ∧•(uP ) ∗ as a set of cohomology representatives, and it respects aP weights. Our main Lp-cohomology computation is based on: (3.1.7) Proposition. For all p ≥ 1, the inclusions in (3.1.3) are quasi-isomorphisms. Proof. This is asserted in [Z1,(4.23),(4.25)] for the case p = 2. The proof given there was presented with p = 2 in mind, though there is no special role of L2 in it (cf. [Z3,(8.6)]). We point out that [Z1,(4.25)] is about the finite-dimensional linear algebra described above, and that the proof (4.23) of [Z1] goes through because the process of averaging a function over a circle (hence a nilmanifold, by iteration) is bounded in Lp-norm. As such, one sees rather easily that the proof carries over verbatim for general p, and (3.1.7) is thereby proved. � Remark. We wish to point out and rectify a small mistake in the argument in [Z1, §4], one that “corrects itself”. It is asserted that the second terms in (4.37) and (4.41) there vanish by Uj−1-invariance. This is false in general. However, the two expressions actually differ only by a sign, and they cancel, yielding the conclusion of (4.41). We next show how (3.1.7) yields the determination of H• (A+P × NP ). We may use the first complex in (3.1.3) for this purpose. The weights in (3.1.6) are non-positive, and (3.1.4) shows that once p is sufficiently large, wβ blows up expo- nentially in some direction whenever β 6= 0, and decays exponentially when β = 0. Applying (2.2.1) and the Künneth theorem, we obtain: (3.1.8) Corollary. For sufficiently large p < ∞, H• (A+P ×NP ) ≃ H 0(uP ,C) ≃ (3.1.9) Remark. i) We can specify what “sufficiently large” means, using (3.1.4). Write δ as a (non-negative) linear combination of the simple Q-roots: δ = Then we mean to take p > max{cβ}. ii) When p = ∞, one runs into trouble with the infinite-dimensionality of the unweighted H1 (R+) (see (2.2.4)). By using instead large finite p, we effect a perturbation away from the trivial weight, thereby circumventing the problem. There is a straightforward globalization of (3.1.8), which we now state: (3.1.10) Theorem. For sufficiently large p, the inclusion CMRBS → q∗A (p)(M BS) ≃ A•(p)(M is a quasi-isomorphism. � From this follows Theorem 1: (3.1.11) Corollary. For sufficiently large finite p, H• (M) ≃ H•(MRBS). (3.2) An example (with enhancement). Take first G = SL(2). Then M is a modular curve. There are only two distinct interesting compactifications (those in (0.2)): one is MBS , and the other is MRBS (which is homeomorphic to MBB and MΣ). A deleted neighborhood of a boundary point (cusp) of M BB is a Poincaré punctured disc ∆∗R = {z ∈ C : 0 < |z| < R}, with R < 1, with metric given in polar coordinates by ds2 = (r | log r|)−2(dr2 + (rdθ)2). Because R < 1, the metric is smooth along the boundary circle |z| = R. Setting u = log | log r| converts the metric to ds2 = du2+e−2udθ2 (recall (3.1.3)). One obtains from (2.3.1) and (3.1.7): (3.2.1) Proposition. Write ∆R for ∆ R ∪ {0}. Then for the Poincaré metric on H•(p)(∆ R) ≃ H •(∆R) ≃ C whenever 1 < p <∞. � (3.2.2) Remark. When p = 1, H2 (∆∗R) is infinite-dimensional, as is H (∆∗R); this follows from (2.2.4) and (3.1.7). By using a Mayer-Vietoris argument, in the same manner as [Z1,§5], we get that when M is a modular curve, we see that (M) is likewise infinite-dimensional. Thus the assertion in (3.1.11) fails to hold for p =∞, already when G = SL(2). Using the Künneth formula (2.3.10), it is easy to obtain the corresponding as- sertion for (∆∗R) (3.2.3) Corollary. For the Poincaré metric on (∆∗R) H•(p)((∆ n) ≃ H•(∆nR) ≃ C whenever 1 < p <∞. � Now, let M be an arbitrary locally symmetric variety. The smooth toroidal compactifications MΣ are constructed so that they are complex manifolds and the boundary is a divisor with normal crossings onMΣ. The local pictures ofM →֒MΣ are (∆∗)k × ∆n−k →֒ ∆n, for 0 ≤ k ≤ n. The invariant metric of M is usually not Poincaré in these coordinates, not even asymptotically. However, it is easy to construct other metrics which are. We will use a subscript “P” to indicate that one is using such a metric instead of the invariant one. We note that such a metric depends on the choice of toroidal compactification. The global version of (3.2.3) follows by standard sheaf theory: (3.2.4) Proposition. For a metric on M that is Poincaré with respect to MΣ, H•(p),P(M) ≃ H •(MΣ) whenever 1 < p <∞. � The above proposition actually gives a reinterpretation of the method in [Mu]. There, Mumford decided to work in the rather large complex of currents that also gives the cohomology of MΣ. However, he shows that the connection and Chern forms involved are “of Poincaré growth”, and that is equivalent to saying that they are L∞ with respect to any metric that is asymptotically Poincaré near the boundary of MΣ. One thereby sees that his argument for comparing Chern forms ([Mu, p.243], based on (4.3.4) below) in the complex of currents actually takes place in the subcomplex of Poincaré L∞ forms on M . Remark. For a convenient exposition of the growth estimates in the latter, see [HZ2,(2.6)]. Since there is in general no morphism of compactifications between MRBS and MΣ, the reader is warned that the comparison of their boundaries is a bit tricky (see [HZ1,(1.5),(2.7)] and [HZ2,(2.5)]). (3.3) On defining morphisms via Lp-cohomology. We give next an interesting consequence of Theorem 1. The space M has finite volume, so there is a canonical morphism (see (2.1)) (3.3.1) H•(p)(M)→ H (2)(M) whenever p > 2. For p sufficiently large, the left-hand side of (3.3.1) is naturally isomorphic to H•(MRBS). For p = 2, there is an analogous assertion: by the Zucker conjecture, proved in [L] and [SS] (see [Z2]), the right-hand side is naturally isomorphic to IH•m(M BB), intersection cohomology with middle perversity m of [GM1]. These facts transform (3.3.1) into the diagram (3.3.2) H•(MRBS) H•(MBB) → IH•m(M In other words, (3.3.3) Proposition. The mapping in (3.3.1) defines a factorization of the canon- ical mapping H•(MBB) −→ IH•m(M through H•(MRBS). A related assertion had been conjectured by Goresky-MacPherson and Rapoport, and was proved recently by Saper: (3.3.4) Proposition. Let h : MRBS → MBB be the canonical quotient mapping. Then there is a quasi-isomorphism Rh∗IC RBS ,Q) ≈ IC•m(M BB,Q), where IC denotes sheaves of intersection cochains. This globalizes to an isomorphism IH•m(M RBS ,Q) −→IH•m(M BB,Q), which un- derlies (3.3.3), enlarging the triangle into a commutative square defined over Q: H•(MRBS) −−−−→ IH•m(M H•(MBB) −−−−→ IH•m(M 4. Chern forms for vector bundles on stratified spaces In this section, we will treat the de Rham theory for stratified spaces that will be needed for the proof of Theorem 2. We also develop the associated treatment of Chern classes for vector bundles. (4.1) Differential forms on stratified spaces. Let Y be a paracompact space with an abstract prestratification (in the sense of Mather) by C∞ manifolds. Let S denote the set of strata of Y . If S and T are strata, one writes T ≺ S whenever T 6= S and T lies in the closure S of S. The notion of a prestratification specifies a system C of Thom-Mather control data (see [GM2, p. 42],[V1],[V2]), and that entails the following. For each stratum S of Y , there is a neighborhood NS of S in Y , a retraction πS : NS → S, and a continuous “distance function” ρS : NS → [0,∞) such that ρ (0) = S, subject to: (4.1.1) Conditions. Whenever T � S, put NT,S = NT ∩ S, πT,S = πT |NT,S , and ρT,S = ρT |NT,S . Then: i) πT (y) = πT,S(πS(y)) whenever both sides are defined, viz., for y ∈ NT ∩ −1NT,S; likewise ρT (y) = ρT,S(πS(y)). ii) The restricted mapping πT,S×ρT,S : N T,S → T×R +, where N◦T,S = NT,S−T , is a C∞ submersion. (The above conditions will be relaxed after (4.1.4) below.) A prestratified space is, thus, the triple (Y,S, C). Let Y ◦ denote the open stratum of Y . One understands that when S = Y ◦, one has NS = Y ◦, πS = 1Y ◦ and ρS ≡ 0. From (4.1.1), it follows that for all S ∈ S, πT,S|N◦ is a submersion; moreover, the closure S of S in Y is stratified by {T ∈ S : T � S}, and CS = {(πT,S, ρT,S) : T ≺ S} is a system of control data for We also recall the following (see [V2,Def. 1.4]): (4.1.2) Definition. A controlled mapping of prestratified spaces, f : (Y,S, C) → (Y ′,S′, C′), is a continuous mapping f : Y → Y ′ satisfying: i) If S ∈ S, there is S′ ∈ S′ such that f(S) ⊆ S′, and moreover, f |S is a smooth mapping of manifolds. ii) For S and S′ as above, f ◦ πS = πS′ ◦ f in a neighborhood of S. iii) For S and S′ as above, ρS′ ◦ f = ρS in a neighborhood of S. Let j : Y ◦ →֒ Y denote the inclusion. A subsheaf A•Y,C of j∗A Y ◦ , the complex of C-controlled C-valued differential forms on Y , is the sheafification of the following presheaf: for V open in Y , put (4.1.3) A•Y,C(V ) = {ϕ ∈ A •(V ∩ Y ◦) : ϕ|NS∩V∩Y ◦ ∈ imπ S for all S ∈ S}. (4.1.4) Remark. From (4.1.1, i), one concludes that the condition in (4.1.3) for T implies the same for S whenever S ≻ T , as (πT ) ∗ϕ = (πS) ∗(πT,S) We observe that the definition of A•Y,C is independent of the distance functions ρS . Indeed, all that we will need from the control data for most purposes is the collection of germs of πS along S. We term this weak control data (these are the equivalence classes implicit in [V2,Def. 1.3]). In this spirit, one has the notion of a weakly controlled mapping, obtained from (4.1.2) by discarding item (iii); cf. (5.2.2). The main role that ρS plays here is to specify a model for the link of S: (4.1.5) LS = π (s0) ∩ ρ for any s0 ∈ VS and sufficiently small ε > 0, but the link is also independent of C; besides, we will not need that notion in this paper. The following is well-known: (4.1.6) Lemma. Let Y be a space with prestratification. For any open covering V of Y , there is a partition of unity {fV : V ∈ V} subordinate to V that consists of C-controlled functions. � This is used in [V1, p. 887] to prove the stratified version of the de Rham theorem: (4.1.7) Proposition. Let C be a system of (weak) control data on Y . Then the complex A•Y,C is a fine resolution of the constant sheaf CY. � (4.1.8) Corollary. A closed C-controlled differential form on Y determines an element of H•(Y ). � (4.2) Controlled vector bundles. We start with a basic notion. (4.2.1) Definition. A C-controlled vector bundle on Y is a topological vector bundle E, given with local trivializations for all V in some open covering V of Y , such that the entries of the transition matrices are C-controlled. It follows from the definition that a C-controlled vector bundle determines a Cech 1-cocycle forV with coefficients inGL(r,A0Y,C). It thereby yields a cohomology class in H1(Y,GL(r,A0Y,C)). The latter has a natural interpretation: (4.2.2) Proposition. The set H1(Y,GL(r,A0Y,C)) is in canonical one-to-one cor- respondence with the set of isomorphism classes of vector bundles E of rank r on Y with ES = E|S smooth for all S ∈ S, together with a system {φS : S ∈ S}, {φT,S : S, T ∈ S} of germs of isomorphisms of vector bundles (total spaces) along each T ∈ S: i) φT : (πT ) ∗ET = ET ×T NT −→E|NT ,(4.2.2.1) ii) φT,S : (πT,S) ∗ET = ET ×T NT,S −→E|NT,S whenever T ≺ S, satisfying the compatibility conditions φT = φS ◦ φT,S. (4.2.2.2) Remark. Condition (ii) above is, of course, the restriction of (i) along S. If we use {ET : T ∈ S}, the stratification of E induced by S, then the natural projection ET ×T NT → ET gives weak control data for E. Thus we obtain from (4.2.2): (4.2.3) Corollary. A vector bundle E on Y is C-controlled (as in (4.2.1)) if and only if E admits weak control data such that the bundle projection E → Y is a weakly controlled mapping (as in (4.1)). � Proof of (4.2.2). Let ξ be a 1-cocycle for the open covering V of Y , with coefficients in GL(r,A0Y,C). Since the functions in A Y,C are continuous, ξ determines a vector bundle of rank r in the usual way; putting E0 for C r, one takes E = Eξ = {(E0 × Vα) : Vα ∈ V} modulo the identifications on Vαβ = Vα ∩ Vβ : (4.2.2.3) E0 × Vαβ →֒ E0 × Vα 1×ξαβ E0 × Vαβ →֒ E0 × Vβ It is a tautology that there exist isomorphisms (4.2.2.1) locally on the respective bases (Y ◦ or S), but we want it to be specified globally. Next, let (4.2.2.4) VS = {V ∈ V : V ∩ S 6= ∅}, V(S) = {V ∩ S : V ∈ VS}. Then V(S) is an open cover of S. By refining V, we may assume without loss of generality that ξαβ ∈ im (πS) ∗ on Vαβ ∩ NS whenever Vα, Vβ ∈ VS , and write ξαβ = (πS) ∗ ξSαβ. The bundle ES = E|S is constructed from the 1-cocycle ξ S. Let N ′S = NS ∩ {V : V ∈ VS}. The relation ξ = (πS) ∗ξS onN ′S determines a canonical isomorphism φS : E|N ′S −→(πS) ∗ES , for the local ones patch together; it is smooth on each stratum R ≻ S. One produces φS,T by doing the above for the restriction of E to S, along its stratum T . The consistency condition, φT = φS ◦ φT,S whenever T ≺ S, holds because of (4.1.4). Replacing V by any refinement of it, only serves to make N ′S smaller, so the germs of the pullback relations do not change. Also, we must check that the isomorphisms above remain unchanged when we replace ξ by an equivalent cocycle. Let ξ′αβ = ψβξαβψ α , where ψ is a 0-cochain for V with coefficients in GL(r,A Y,C). Without loss of generality again, we assume that ψ is of the form (πS) ∗ψS on N ′S. The isomorphism E(ξ′S) ≃ E(ξS) induced by ψ S then pulls back to the same for the restrictions of E(ξ′) and E(ξ) to N ′S, respecting the compatibilities. Thus, we have constructed a well-defined mapping from H1(Y,GL(r,A0Y,C)) to isomorphism classes of bundles on Y with pullback data along the strata. We wish to show that it is a bijection. Actually, we can invert the above construction explicitly. Given E, φT , etc., as in (4.2.2.1), let, for each T ∈ S, VT be a covering of T that gives a 1-cocycle ξ T for ET (as a smooth vector bundle on T ); N T a neighborhood of T , contained in NT , on which the isomorphisms φT and φT,S (for all S ≻ T ) are defined; V(T ) = π the corresponding covering of N ′S, on which (πT ) ∗ξT is a cocycle giving E|N ′ . Then {VT : T ∈ S} is a covering of Y , such that for all V ∈ V, E|V has been trivialized. We claim that the 1-cocycle for E, with respect to these trivializations, has coefficients in GL(r,A0Y,C). For Vα and Vβ in the same VT , we have seen already that ξαβ is in im(πT ) ∗. Suppose, then, that T ≺ S, and that Vα ∈ VT and Vβ ∈ VS have non-empty intersection. Then ξαβ is actually in im(πS) ∗, which one sees is a consequence of the compatibility conditions for (4.2.2.1), and our claim is verified. That we have described the inverse construction is easy to verify. � (4.3) Controlled connections on vector bundles. When we speak of a connection on a smooth vector bundle over a manifold, and write the symbol ▽ for it, we mean foremost the covariant derivative. Then, the difference of two connections is a 0-th order operator, given by the difference of their connection matrices with respect to any one frame. We can define the notion of a connection on a C-controlled vector bundle: (4.3.1) Definition. Let E be a C-controlled vector bundle on Y . A C-controlled connection on E is a connection ▽ on E|Y ◦ for which there is a covering V of Y such that for each V ∈ V, there is a frame of E|V such that the connection forms lie in A1V,C ⊗ End(E). Remark [added]. It is more graceful to define a controlled connection so as to be in accordance with (4.2.2.1): it is a system of connections {(ET ,▽T )}, with germs of isomorphisms (▽S)|NT,S = (πT,S) ▽T whenever T ≺ S. One sees that (4.1.1) and (4.3.1) imply that a C-controlled connection on E defines a usual connection on E|S for every S ∈ S. The next observation is evident from the definition: (4.3.2) Lemma. The curvature form Θ ∈ j∗(A Y◦ ⊗ End(E|Y◦)) of a C-controlled connection ▽ lies in A2Y,C ⊗ End(E). � It is also obvious that A•Y,C is closed under exterior multiplication. One can thus define for each k the Chern form ck(E,▽), a closed C-controlled 2k-form on Y , by the usual formula: ck(E,▽) = Pk(Θ, . . . ,Θ), where Pk is the appropriate invariant polynomial of degree k. By (4.1.8), ck(E,▽) defines a cohomology class in H2k(Y ). (4.3.3) Proposition. i) Every C-controlled vector bundle E on Y admits a C- controlled connection. ii) The cohomology class of ck(E,▽) in H 2k(Y ) is independent of the C-controlled connection ▽ on E. Proof. Let {φT , φT,S : T ≺ S} be the data defining a C-controlled vector bundle, as in (4.2.2.1), and N ′T ⊂ NT a domain for the isomorphisms involving ET . For each T , let ▽T be any smooth connection on ET , and (πT ) T the pullback connection on E|N ′ . Then V = {N ′T : T ∈ S} is an open covering of Y . Apply (4.1.6) to get a C-controlled partition of unity {fT } subordinate to V. Then ▽ = V is a C-controlled connection on E. This proves (i). The argument for proving (ii) is the standard one. For two connections on a smooth manifold, such as Y ◦, there is an identity: (4.3.4) ck(E,▽1)− ck(E,▽0) = dηk, where (4.3.4.1) ηk = k Pk(ω,Θt, . . . ,Θt)dt, ω = ▽1−▽0, ▽t = (1− t)▽0+ t▽1, and Θt denotes the curvature of ▽t. Now, if ▽0 and ▽1 are both C-controlled, one sees easily that ω and ▽t are likewise, and then so is ηk. It follows that (4.3.4) is an identity in A Y,C, giving (ii). We have been leading up to the following: (4.3.5) Theorem. Let E be a C-controlled vector bundle on the stratified space Y . Then the cohomology class in (4.3.3, ii) gives the topological Chern class of E in H2k(Y ); in particular, it is independent of the choice of C. Proof. This argument, too, follows standard lines. We start by proving the assertion when E is a line bundle L. On Y , there is the short exact exponential sequence (of sheaves): (4.3.5.1) 0→ ZY → A Y,C → (A ∗ → 1. The Chern class of L, c1(L), is then the image of any controlled Cech cocycle that determines L, under the connecting homomorphism (4.3.5.2) H1(Y, (A0Y ) ∗) −→ H2(Y,Z). To prove the theorem for line bundles, it is convenient to work in the double complex C•(A•Y,C), where C • denotes Cech cochains. It has differential D = δ + σd (i.e., Cech differential plus a sign σ = (−1)a times exterior derivative, where a is the Cech degree). On a sufficiently fine covering of Y we have a cochain giving L, ξ ∈ C1((A0Y,C) ∗) (if eα is the specified frame for L on the open subset Vα of Y , one has on Vα ∩ Vβ that eα = ξαβeβ), with δξ = 1, the connection forms ω ∈ C 0(A1Y,C), and λ = log ξ in C1(A0Y,C). We know by (4.3.5.2) above that δλ gives c1(L). The change-of-frame formula for connections gives δω + dλ = 0. Finally, the curvature (for a line bundle) is Θ = dω, so we wish to show that dω and δλ are cohomologous in the double complex. By definition, Dλ = δλ−dλ, and Dω = δω+dω = dω−dλ. This gives δλ− dω = D(λ− ω), and we are done. To get at higher-rank bundles, we invoke a version of the splitting principle. Let p : F(E)→ Y be the bundle of total flags for E. As E is locally the product of Y and a vector space, F(E) is locally on Y just Fr×Y , where Fr is a (smooth compact) flag manifold. As such, F(E) is a stratified space that is locally no more complicated than Y ; we take as the set of strata S̃ = p−1(S) = {p−1S : S ∈ S}. For weak control data, we deduce it from C in the same way it is done for E (see (4.2.2.2)): we take NFT = F(E|N ′T ), and use the natural projection F(E|N ) → F(ET ) induced by (4.2.2.1). It is standard that the vector bundle p∗E on F(E) decomposes (non-canonically) into a direct sum of line bundles: p∗E = 1≤j≤r Λj . (p ∗E is canonically filtered: Λ1 = F1 ⊂ F2 . . . Fr = p ∗E, with Λj ≃ Fj/Fj−1.) To obtain this, one starts by taking Λ1 to be the line bundle given at each point of F(E) by the one-dimensional subspace from the corresponding flag. Then, one splits the exact sequence (4.3.5.3) 0→ Λ1 → p ∗E → p∗E/Λ1 → 0, using a controlled metric on E. By that, we mean a metric that is a pullback via the isomorphisms (4.2.2.1); these can be constructed by the usual patching argument, using controlled partitions of unity (4.1.6). One obtains Λj , for j > 1, by recursion. We need a little more than that: (4.3.6) Lemma. i) The vector bundle p∗E is, in a tautological way, a controlled vector bundle on F(E). ii) The line bundles Λj are controlled subbundles of p Proof. We have that p∗E = E ×Y F, and its strata are (p ∗E)FT = ET ×T FT , for all T ∈ S. There is natural weak control data for p∗E that we now specify. By construction, we have a retraction (4.3.6.1) π(p∗ET ) : (p ∗E)|NFT = (p ∗E)|F|NT ≃ E|NT ×NT F|NT → ET ×T FT = (p ∗E)FT , induced by the weak control data for E (and thus also F), and likewise for the restriction to (p∗E)|NFT ,FS , when S ≻ T . These provide φFT and φFT ,FS (from (4.2.2.1)) respectively for p∗E, and (i) is proved. We show that Λ1 is preserved by φFT and φFT ,FS . (As before, we explain this only for the former, the other being its restriction to the strata.) Let pT : FT → T denote the restriction of p to FT . Since pT gives the flag manifold bundle associated to ET , (p TET ) contains a tautological line bundle, which we call Λ1,T . We have from the control data that (Λ1)|NFT ≃ (φFT ) ∗Λ1,T . We claim further that (4.3.6.1) takes Λ1|NFT to Λ1,T , as desired. The explicit formula for (4.3.6.1), obtained by unwinding the fiber products, is as follows. Let e be in the vector space ET,t, the fiber of ET over t ∈ T , and f a point the flag manifold of ET,t. Also, let n ∈ (πT ) −1(t). Then π(p∗ET )(e, f, n) = (e, f), which implies (ii) for j = 1. The assertion for j > 1 is obtained recursively. � We return to the proof of (4.3.5). Let ▽0 be the direct sum of C̃-controlled connections on each Λj ; and take ▽1 = p ▽, where ▽ is a C-controlled connection on E. Both ▽0 and ▽1 are C̃-controlled connections on F(E). By construction, ∗E,▽0) represents the k-th Chern class of p ∗E in H2k(F(E)). We then apply (4.3.3, ii) to obtain that ck(p ∗E,▽1) = p ∗ck(E,▽) represents p ∗ck(E) ∈ H 2k(F(E)). Since p∗ : H2k(Y ) −→ H2k(F(E)) is injective, it follows that ck(E,▽) represents ck(E) in H 2k(Y ), and (4.3.5) is proved. � 5. Proofs of Theorem 2 and Conjecture B In this section, we apply the methods of §4 in the case Y =MRBSΓ . (5.1) Control data for a manifold-with-corners. Let Y be a manifold-with- corners, with its open faces as strata. For each codimension one boundary stratum S, let φS : [0, 1]× S → Y define the collar NS of S in Y , so that {0} × S is mapped identically onto S. This determines partial control data (that is, without distance functions) for Y as follows. As NS , one takes φ([0, 1) × S), and as πS projection onto S. For a general boundary stratum T , write {S : S of codimension one, T ≺ S}. Let NT = {NS : S of codimension one, T ≺ S}; given the φS ’s above, this set is canonically diffeomorphic to [0, 1]r × T , where r is the codimension of T . Then NT is the subset of NT corresponding to [0, 1) r × T , in which terms πT is simply projection onto T . (5.2) Compatible control data. The existence of natural (partial) control data for MRBSΓ is, in essence, well-known, as is compatible control data for M Γ in the Hermitian case. We give a brief presentation of that here. This will enable us to determine that Conjecture B is true. The relevant notions are variants of (4.1.2). (5.2.1) Definition (see [GM2, 1.6]). Let Y and Y ′ be stratified spaces. A proper smooth mapping f : Y → Y ′ is said to be stratified when the following two condi- tions are satisfied: i) If S′ is a stratum of Y ′, then f−1(S′) is a union of connected components of strata of Y ; ii) Let T ⊂ Y be a stratum component as in (i) above. Then f |T : T → S ′ is a submersion. It follows that a stratified mapping f is, in particular, open. We assume henceforth, and without loss of generality, that all strata are connected. (5.2.2) Definition (cf. [V1: 1.4]). Let f : Y → Y ′ be a stratified mapping, with (weak) control data C for Y , and C′ for Y ′. We say that f is weakly controlled if for each stratum S of Y , the equation πS′ ◦ f = f ◦ πS holds in some neighborhood of S (here f maps S to S′). (5.2.3) Remark. Note that there is no mention of distance functions in (5.2.2). This is intentional, and is consistent with our stance in (4.1). (5.2.4) Lemma. Let f : Y → Y ′ be a stratified mapping. Given partial control data C for Y , there is at most one system of germs of partial control data C′ for Y ′ such that f becomes weakly controlled. Such C′ exists if and only if for all strata S of Y , there is a neighborhood of S (contained in NS) in which f(y) = f(z) implies f(πS(y)) = f(πS(z)). � When the condition in (5.2.4) is satisfied, one uses the formula πS′(f(y)) = f(πS(y)) to define C′, and we then write C′ = f∗C. In the usual manner, the mapping f determines an equivalence relation on Y , viz., y ∼ z if and only if f(y) = f(z). The condition on C thereby becomes: (5.2.5) y ∼ z ⇒ πS(y) ∼ πS(z) (near S). We will use the preceding for the stratified mappings MBSΓ →M Γ in general, and MRBSΓ →M Γ in the Hermitian case. The reason for bringing inM Γ is that it is a manifold-with-corners, and it also has natural partial control data. The boundary strata of X , the universal cover of MBSΓ , are the sets e(P ) of (1.3), as P ranges over all rational parabolic subgroups of G. Those of MBSΓ itself are the arithmetic quotients e′(P ) of e(P ), with P ranging over the finite set of Γ-conjugacy classes of such P . There are projections X → X/AP = e(P ), defined by collapsing the orbits of the geodesic action of AP to points. This extends to a P -equivariant smooth retraction (geodesic projection), given in (1.3.3): (5.2.6) X(P ) −−→ X(P )/AP = e(P ). Recall from (1.8) that there is a neighborhood of e′(P ) inMBSΓ on which geodesic projection onto e′(P ), induced by (5.2.6), is defined. We take the restriction of this geodesic projection over e′(P ) as the definition of πP in our partial control data C for MBSΓ . We have been leading up to: (5.2.7) Proposition. The quotient mapping MBSΓ →M Γ satisfies (5.2.5). Proof. The mappings e(P ) → e(P )/UP , as P varies, induce the mapping M MRBSΓ . It is a basic fact ([BS, 4.3]) that for Q ⊂ P , AQ ⊃ AP and πQ ◦ πP = πQ. This gives e(P )→ e(P )/UP . Since it is also the case that Q ⊂ P implies UQ ⊃ UP , we see that (5.2.5) is satisfied. � Only a little more complicated is: (5.2.8) Proposition. In the Hermitian case, the quotient mapping MRBSΓ → MBBΓ satisfies (5.2.5). Proof. When the symmetric space X is Hermitian, the P -stratum of XRBS , for each P , decomposes as a product: (5.2.8.1) e(P )/UP ≃ Xℓ,P ×Xh,P . This is induced by a decomposition of reductive algebraic groups over Q: P/UP = Gℓ,P ·Gh,P (cf. [Mu, p. 254]). Fixing Gh, one sees that the set of Q with Gh,Q = Gh (if non- empty) is a lattice, whose greatest element is a maximal parabolic subgroup P of G. The lattice is then canonically isomorphic to the lattice of parabolic subgroups R of Gℓ,P , whereby Q/UQ ≃ (R/UR)×Gh,P . (Thus Gh,Q = Gh,P . In the language of [HZ1,(2.2)] such Q are said to be subordinate to P .) The mapping MRBSΓ →M Γ is induced, in terms of (5.2.8.1), by (5.2.8.2) e(Q)/UQ → Xh,Q, for all Q; perhaps more to the point, the terms can be grouped by lattice, yielding (5.2.8.3) Xℓ,P ×Xh,P → Xh,P →֒ (Xh,P ) for P maximal (see [GT, 2.6.3]). One sees that (5.2.5) is satisfied. � We have thereby reached the conclusion: (5.2.9) Corollary. The natural partial control data for MBSΓ induces compatible partial control data for MRBSΓ and M Γ . � (5.3) Conjecture B. Let EΓ be a homogeneous vector bundle on MΓ, and E its extension to MRBSΓ from [GT] that was reconstructed in our §1. We select as partial control data C for MRBSΓ that given in (5.2.9). It is essential that the following hold: (5.3.1) Proposition. ERBSΓ is a controlled vector bundle on M Γ , with the π̃P ’s of (1.3.3) providing the weak control data. Proof. This is almost immediate from the construction in §1. Recall that the weak control data for MRBSΓ consists of the geodesic projections πP , defined in a neigh- borhood of M̂P . The vector bundle E Γ also gets local geodesic projections π̃P , induced from those of EBS, that are compatible with those of MRBSΓ because of (1.2). The same holds within the strata of these spaces, by (1.6). We see that the criterion of (4.2.3) is satisfied. � We proceed with a treatment of ▽GP, the connection on EΓ constructed in [GP]. For each maximal Q-parabolic subgroup of G, letMP be the corresponding stratum of MBB ; it is a locally symmetric variety for the group Gh,P . We also use “P” to label the strata: thus, we have for (4.1.2), πP : NP → MP , etc. Then ▽ GP can be defined recursively, starting from the strata of lowest dimension (Q-rank zero), and then increasing the Q-rank by one at each step. There is, first, the equivariant Nomizu connection for homogeneous vector bun- dles, whose definition we recall. Homogeneous vector bundles are associated bundles of the principal K-bundle: (5.3.2) κ : Γ\G −→MΓ. When we write the Cartan decomposition g = k ⊕ p, we note that (5.3.2) has a natural equivariant connection whose connection form lies in the vector space Hom(g, k); it is given by the projection of g onto k (with kernel p). This is known as the Nomizu connection. The homogeneous vector bundle EΓ onMΓ is associated to the principal bundle (5.3.2) via the representation K → GL(E). The connection induced on EΓ via k → gl(E) = End(E) is also called the Nomizu connection (of EΓ), and will be denoted ▽ No; its connection form is denoted θ ∈ g∗ ⊗ End(E). A K-frame for EΓ on an open subset O ⊂ MΓ is given by a smooth cross-section σ : O → κ−1(O) of κ; the resulting connection matrix is the pullback of θ via σ∗, an element of A1(O,End(E)). With that stated, we can start to describe ▽GP. For any maximal Q-parabolic P , one will be taking expressions of the form (5.3.3) ▽P = ψPP▽ P,No + Q,P (▽ [GP, 11.2]. Here, ▽P,No is the Nomizu connection for the homogeneous vector bundle onMP determined by the restrictionKh,P →֒ K → GL(E), and the functions {ψ Q � P} form a partition of unity onMP of a selected type, given in [GP, 3.5, 11.1.1]; the function ψ P is a cut-off function for a large relatively compact open subset VQ,P of MQ in M P , with ⋃ VQ,P =M and can be taken to be supported inside the neighborhood NQ,P of the partial control data when Q 6= P . Next, Φ∗Q,P indicates the process of parabolic induction from MQ to NQ,P , by means of πQ,P . It is defined as follows. Fix a maximal parabolic Q and a rep- resentation K → GL(E). The latter restricts, of course, to KQ = Kh,Q × Kℓ,Q, but through the Cayley transform, this actually extends to a representation λ of Kh,Q × Gℓ,Q. That allows one to define an action of all of Q on Eh,Q [GP, 10.1], which induces a Q-equivariant mapping (5.3.4) E = Q×KQ E −→ Gh,Q ×Kh,Q E = Eh,Q, given by (q, e) 7→ (gh, λ(gℓ)e) for q = ughgℓ ∈ UQGh,QGℓ,Q = Q. That in turn defines a UQ-invariant isomorphism of vector bundles homogeneous under Q: E ≃ π∗Q(Eh,Q). One then takes ▽GP to be ▽G in (5.3.3). Given any connection ▽h,Q on Eh,Q, the pullback connection ▽ = π∗Q(▽ h,Q) satisfies the same relation for its curvature form, viz., (5.3.5) Θ(▽) = π∗QΘ(▽ h,Q). It follows that the Chern forms of ▽GP are controlled on MBBΓ [GP, 11.6]. On the other hand, the connection itself is not. To proceed, weaker information about ▽GP suffices: (5.3.6) Proposition. With an appropriate choice of the functions ψ , the con- nection ▽GP is a controlled connection when viewed on MRBSΓ . Proof. TRBS1-2 his is not difficult. Recall from (4.3.1) that the issue is the exis- tence of local frames at each point of MRBSΓ , with respect to which the connection matrix is controlled. For each rational parabolic subgroup Q of G, we work in the corner X(Q). By (5.3.4), one gets local frames for E(Q), the restriction of E to X(Q) ⊂ X, from local frames for Eh,Q. We can write E(Q) as: (5.3.6.1) E(Q) ≃ UQ × AQ ×MQ ×KQ E. This also provides good variables for calculations. We note that Φ∗Q,P is independent of the UQ-variable. Likewise, ψ can be chosen to be a function of only (a,mKQ), constant on the compact nilmanifold fibers NP (i.e., the image of the UP -orbits). It follows by induction that ▽GP is controlled on MRBSΓ . � As we said, the Chern forms of ▽GP are controlled differential forms for MBBΓ , so are a fortiori controlled for MRBSΓ . It follows from (4.3.5) that (5.3.7) Proposition. c•(E GP) represents c•(E Γ ) ∈ H •(MRBSΓ ). � Thereby, Conjecture B is proved. (5.4) Theorem 2. Let ▽ctrl be any C-controlled connection on ERBSΓ , and ▽ No the equivariant Nomizu connection on EΓ. By (4.3.5) we know that ck(▽ ctrl) represents Γ ); we want to conclude the same for ck(▽ No). Toward that, we recall the standard identity on M satisfied by the Chern forms: (5.4.1) ck(▽ No)− ck(▽ ctrl) = dηk, which is a case of (4.3.4). The following is straightforward: (5.4.2) Lemma. i) A• is contained in A• (MRBSΓ ). ii) A G-invariant form on MΓ is L Proof. In terms of (3.1.1), a controlled differential form on MRBSΓ is one that is, for each given P , pulled back from V ⊂ M̂P . Such forms are trivially weighted by in the metric (3.1.2). It follows that a controlled form is locally L∞ on MRBSΓ . This proves (i). As for (ii), an invariant form has constant length, so is in particular L∞. � (5.4.3) Proposition. The closed forms ck(▽ No) and ck(▽ ctrl) represent the same class in H2k (MΓ). Proof. Since MRBSΓ is compact, a global controlled form on M Γ is globally L As such, (5.4.2) gives that the Chern forms for both ▽ctrl and ▽No are in the complex L• (MΓ). It remains to verify that ηk in (5.4.1) is likewise L ∞, for then the relation (5.4.1) holds in the L∞ de Rham complex A• (MΓ), so ck(▽ No) and ctrl) are cohomologous in the L∞ complex. By (4.3.4.1), it suffices to check that the difference ω = ▽No−▽ctrl is L∞. That can be accomplished by taking the difference of connection matrices with respect to the same local frame of ERBS , and for that purpose we use, for each Q, frames pulled back from M̂Q. For that, it is enough to verify the boundedness for ω in a neighborhood of every point of the boundary of MRBSΓ , and we may as well calculate on XRBS . Consider a point in the Q-stratum XQ of X RBS . As in (3.1.1), we can take as neighborhood base, intersected with X , sets that decompose with respect to Q as (5.4.3.1) NQ × A Q × V, with V open in XQ. In these terms, πQ is just projection onto V . As in (3.1.2) and (3.1.4), we use as coordinates (uα, a, v). We also decompose (see the end of (1.1)), (5.4.3.2) E ≃ Q×KQ E ≃ UQ ×A Q ×XQ ×KQ E. We obtain a canonical isomorphism E ≃ π∗QEQ, with EQ a homogeneous vector bundle on XQ. By (5.4.2, ii), the connection matrix of a connection that is pulled back from XQ, with respect to a local frame pulled back from XQ is L ∞, so we wish to do the same for the Nomizu connection. First, we have: (5.4.3.3) Lemma. Let Q̂ = Q/AQUQ and consider the diagram Q −−−−→ Q̂ −−−−→ XQ. Then Q ≃ Q̂×XQ X, the pullback of Q̂ with respect to πQ. Proof. We note that both Q̂ and Q are exhibited as principal KQ-bundles. To prove our assertion, it is simplest use the Langlands decomposition (of manifolds) Q ≃ Q̂ × AQ × UQ to yield the decomposition X ≃ XQ × AQ × UQ (cf. (5.4.3.1)). Q̂×XQ X ≃ Q̂×AQ × UQ ≃ Q. � It follows that if σ : O ⊆ XQ → Q̂ gives a local KQ-frame, then σ̃ : π Q (O) ⊆ X → Q ≃ Q̂ ×XQ X , defined by σ̃(x) = (σ(πQ(x)), x), gives the pullback frame π∗Qσ. In other words, π Qσ takes values in the principal KQ-bundle Q→ X that is the restriction of structure group of (5.3.2) from K to KQ. Let ▽No be the Nomizu connection on E . Recall that this is determined by (5.4.3.4) TX −→ g −→ k −→ End(E). As such, ▽No is not a KQ-connection. However, a frame for the restriction to X of the canonical extension E can be taken to be of the form σ̃ as above (cf. (1.10)). It follows that for x ∈ π−1 (O), the Nomizu connection is given by (5.4.3.5) TX,x →֒ q −→ k −→ End(E), where q denotes the Lie algebra of Q. This is a mapping that is of constant norm along the fibers of πQ. It follows that the connection matrix is L ∞. Therefore, we have: (5.4.3.6) Proposition. The connection difference ω is L∞. � This finishes the proof of (5.4.3). (5.4.4) Remark. The reader may find it instructive to compare, in the case of G = SL(2), the above argument to the one used in [Mu, pp. 259–260]. The two discussions, seemingly quite different, are effectively the same. We now finish the proof of Theorem 2 by demonstrating: (5.4.5) Proposition. ck(▽ No) and ck(▽ ctrl) represent the same class in H2k(MRBSΓ ). Proof. Because MΓ has finite volume, there is a canonical mapping H•(∞)(MΓ)→ H (p)(MΓ) for all p (see (3.3.1)). It follows from (5.4.3) that ck(▽ No) and ck(▽ ctrl) represent the same class in H2k (MΓ) for all p. Taking p sufficiently large, we apply Theo- rem 1 (i.e., (3.1.11)) to see that ck(▽ No) and ck(▽ ctrl) represent the same class in H2k(MRBSΓ ). � References [BB] Baily, W., Borel, A., Compactification of arithmetic quotients of bounded sym- metric domains. Ann. of Math. 84 (1966), 442–528. [BS] Borel, A., Serre, J.-P., Corners and arithmetic groups. Comm. Math. Helv. 4 (1973), 436–491. [GHM] Goresky, M., Harder, G., MacPherson, R., Weighted cohomology. In- vent. Math. 116 (1994), 139–213. [GM1] Goresky, M., MacPherson, R., Intersection homology, II. Invent. Math. 72 (1983), 77–129. [GM2] Goresky, M., MacPherson, R., Stratified Morse Theory. Springer-Verlag, 1988. [GP] Goresky, M., Pardon, W., Chern classes of modular varieties, 1998. [GT] Goresky, M., Tai, Y.-S., Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces. Amer. J. Math. 121 (1999), 1095–1151. [HZ1] Harris, M., Zucker, S., Boundary cohomology of Shimura varieties, II: Hodge theory at the boundary. Invent. Math. 116 (1994), 243–307. [HZ2] Harris, M., Zucker, S., Boundary cohomology of Shimura varieties, III: Co- herent cohomology on higher-rank boundary strata and applications to Hodge theory (to appear, Mem. Soc. Math. France). [K] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961), 329–387. [L] Looijenga, E., L2-cohomology of locally symmetric varieties. Compositio Math. 67 (1988), 3–20. [M] MacPherson, R., Chern classes for singular algebraic varieties. Annals of Math. 100 (1974), 423–432. [Mu] Mumford, D., Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42 (1977), 239–272. [N] Nomizu, K., On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. 59 (1954), 531–538. [SS] Saper, L., Stern M., L2-cohomology of arithmetic varieties. Ann. of Math. 132 (1990), 1–69. [V1] Verona, A., Le théorème de de Rham pour les préstratifications abstraites. C. R. Acad. Sc. Paris 273 (1971), 886–889. [V2] Verona, A., Homological properties of abstract prestratifications. Rev. Roum. Math. Pures et Appl. 17 (1972), 1109–1121. [Z1] Zucker, S., L2-cohomology of warped products and arithmetic groups. In- vent. Math. 70 (1982), 169–218. [Z2] Zucker, S., L2-cohomology and intersection homology of locally symmetric va- rieties, III. In: Hodge Theory: Proceedings Luminy, 1987, Astérisque 179–180 (1989), 245–278. [Z3] Zucker, S., Lp-cohomology and Satake compactifications. In: J. Noguchi, T. Oh- sawa (eds.), Prospects in Complex Geometry: Proceedings, Katata/Kyoto 1989. Springer LNM 1468 (1991), 317–339. [Z4] Zucker, S., Lp-cohomology: Banach spaces and homological methods on Rie- mannian manifolds. In: Differential Geometry: Geometry in Mathematical Physics and Related Topics, Proc. of Symposia in Pure Math. 54 (1993), 637–655. [Z5] Zucker, S., On the boundary cohomology of locally symmetric varieties. Viet- nam J. Math. 25 (1997), 279–318, Springer-Verlag.

0704.1336

The $^4$He total photo-absorption cross section with two- plus three-nucleon interactions from chiral effective field theory

7 The 4He total photo-absorption cross section with two- plus three-nucleon interactions from chiral effective field theory Sofia Quaglioni, Petr Navrátil Lawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USA Abstract The total photo-absorption cross section of 4He is evaluated microscopically using two- (NN) and three-nucleon (NNN) interactions based upon chiral effective field theory (χEFT). The calculation is performed using the Lorentz integral transform method along with the ab initio no-core shell model approach. An important feature of the present study is the consistency of the NN and NNN interactions and also, through the Siegert theorem, of the two- and three-body current operators. This is due to the application of the χEFT framework. The inclusion of the NNN interaction produces a suppression of the low-energy peak and enhancement of the high-energy tail of the cross section. We compare to calculations obtained using other interactions and to representative experiments. The rather confused experimental situation in the giant resonance region prevents discrimination among different interaction models. Key words: PACS: 25.20.Dc, 21.30.-x, 21.60.Cs, 27.10.+h Interactions among nucleons are governed by quantum chromodynamics (QCD). In the low-energy regime rele- vant to nuclear structure and reactions, this theory is non- perturbative, and, therefore, hard to solve. Thus, theory has been forced to resort to models for the interaction, which have limited physical basis. New theoretical develop- ments, however, allow us to connect QCD with low-energy nuclear physics. Chiral effective field theory (χEFT) [1,2] provides a promising bridge to the underlying theory, QCD. Beginning with the pionic or the nucleon-pion system [3] one works consistently with systems of increasing number of nucleons [4]. One makes use of spontaneous breaking of chiral symmetry to systematically expand the strong in- teraction in terms of a generic small momentum and takes the explicit breaking of chiral symmetry into account by expanding in the pion mass. Nuclear interactions are non- perturbative, because diagrams with purely nucleonic in- termediate states are enhanced [1,2]. Therefore, the chiral perturbation expansion is performed for the potential. The χEFT predicts, along with the nucleon-nucleon (NN) inter- action at the leading order, a three-nucleon (NNN) interac- tion at the next-to-next-to-leading order or N2LO [2,5,6], and even a four-nucleon (NNNN) interaction at the fourth Email addresses: [email protected] (Sofia Quaglioni), [email protected] (Petr Navrátil). order (N3LO) [7]. The details of QCD dynamics are con- tained in parameters, low-energy constants (LECs), not fixed by the symmetry, but can be constrained by exper- iment. At present, high-quality NN potentials have been determined at N3LO [8]. A crucial feature of χEFT is the consistency between the NN, NNN and NNNN parts. As a consequence, at N2LO and N3LO, except for two parame- ters assigned to two NNN diagrams, the potential is fully constrained by the parameters defining the NN interaction. The full interaction up to N2LO was first applied to the analysis of nd scattering [6] and later the N3LO NN poten- tial was combinedwith the availableNNN atN2LO to study the 7Li structure [9]. In a recent work [10] the NN potential at N3LO of Ref. [8] and the NNN interaction at N2LO [5,6] have been applied to the calculation of various properties of s- and mid-p-shell nuclei, using the ab initio no-core shell model (NCSM) [11,12], up to now the only approach able to handle the chiral NN+NNN potentials for systems beyond A = 4. In that study, a preferred choice of the two NNN LECs was found and the fundamental importance of the chiral NNN interaction was demonstrated for reproducing the structure of light nuclei. In the present work, we ap- ply for the first time the same χEFT interactions to the ab inito calculation of reaction observables involving the con- tinuum of the four-nucleon system. In particular, we study Preprint submitted to Elsevier 30 October 2018 http://arxiv.org/abs/0704.1336v1 the 4He total photo-absorption cross section. Experimental measurements of the α particle photo- disintegration suffer from a recurrent history of large discrepancies in the near-threshold region, where the 4He(γ, p)3H and the 4He(γ, n)3He break-up channels dom- inate the total photo-absorption cross section (we refer the reader to the reviews of available data in Refs. [13,14,15]). The latest examples date back to the past two years [15,16]. Of particular controversy is the height of the cross section at the peak, alternatively found to be either pronounced or suppressed with differences up to a factor of 2 between different experimental data. With the exception of [17], early evaluations of the 4He photo-disintegration [18,19,14] showed better agreement with the high-peaked experi- ments, and, ultimately, with those of Ref. [16]. The in- ability of these calculations to reproduce a suppressed cross section at low energy was often imputed to the semi- realistic nature of the Hamiltonian and, in particular, to the absence of the NNN force. The introduction of NNN in- teractions leads, indeed, to a reduction of the peak height, as it was recently shown in a calculation of the photo- absorption cross section with the Argonne V18 (AV18) NN potential augmented by the Urbana IX (UIX) NNN force [20]. A damping of the peak was also found using the correlated AV18 potential constructed within the unitary correlation operator method (UCOM) [21]. In both cases, however, the suppression is not sufficient to reach the low- lying data, and in particular those of Ref. [15]. The latter calculations represent a substantial step forward in the study of the 4He photo-disintegration. However, they still present a residual degree of arbitrariness in the choice of the NNN force to complement AV18 in the first case, or in the choice of the unitary transformation leading to the non-local phase-equivalent interaction in the second case. We note that the Illinois potential models have been found to be more realistic NNN partners of AV18 in the repro- duction of the structure of light p-shell nuclei [22]. From a fundamental point of view, it is therefore important to calculate the 4He photo-absorption cross section in the framework of χEFT theory, where NN and NNN potentials are derived in a consistent way and their relative strengths is well established by the order in the chiral expansion. When the wavelength of the incident radiation is much larger than the spatial extension of the system under con- sideration, the nuclear photo-absorption process can be de- scribed in good approximation by the cross section σγ(ω) = 4π ωR(ω) , (1) where ω is the incident photon energy and the inclusive response function R(ω) = 〈Ψf | D̂ |Ψ0〉 δ(Ef − E0 − ω) (2) is the sum of all the transitions from the ground state |Ψ0〉 to the various allowed final states |Ψf 〉 induced by the dipole operator: riY10(r̂i) . (3) In the above equations ground- and final-state energies are denoted by E0 and Ef , respectively, whereas τ i and r i = rir̂i represent the isospin third component and center of mass frame coordinate of the ith nucleon. This form of the transition operator includes the leading effects of the meson-exchange currents through the Siegert’s theorem. Additional contributions to the cross section (due to retar- dation, higher electric multiples, magnetic multiples) not considered by this approximation are found to be negligi- ble in the A = 2 [23] and A = 3 [24] nuclei, in particular for ω . 40 MeV. A similar behavior can be expected from a system of small dimensions like the 4He. Denoting by Ĥ the full Hamiltonian of the system, p i − p j V NNij + i<j<k V NNNijk , (4) wherem is the nucleon mass, V NNij is the sum of N 3LO NN and Coulomb interactions, and V NNNij is the N 2LO NNN force, we i) solve the many-body Schrödinger equation for the ground state |Ψ0〉, ii) obtain the response (2) by eval- uation [25,26] and subsequent inversion [27] of an integral transform with a Lorentzian kernel of finite width σI ∼ 10−20 MeV (z = E0 + σR + iσI), L(σR, σI) =− 〈ψ0|D̂ z − Ĥ D̂|ψ0〉 (ω − σR)2 + σ dω , (6) and iii) calculate the photo-absorption cross section in the long wave-length approximation using Eq. (1). Following these steps, a fully microscopic result for the 4He photo- absorption cross section can be reached through the use of efficient expansions over localized many-body states. In- deed, in the technique summarized by Eqs. (5-6) and known as Lorentz integral transform (LIT) method [28], the con- tinuum problem is mapped onto a bound-state-like prob- The present calculations are performed in the framework of the ab initio NCSM approach [11]. This method looks for the eigenvectors of Ĥ in the form of expansions over a complete set of harmonic oscillator (HO) basis states up to a maximum excitation of Nmax~Ω above the minimum energy configuration, where Ω is the HO parameter. The convergence to the exact results with increasing Nmax is accelerated by the use of an effective interaction derived, in this case, from the adopted NN and NNN χEFT po- tentials at the three-body cluster level [12]. The reliabil- ity of the NCSM approach combined with the LIT method was validated by comparing to the results obtained with the effective-interaction hyper-spherical harmonics (EIHH) technique [29] in a recent benchmark calculation [30]. A complete description of the NCSM approachwas presented, e.g., in Refs. [11,12]. Here, we emphasize some of the as- pects involved in a calculation of the effective interaction at the three-body cluster level in presence of a NNN poten- tial. We use the Jacobi coordinate HO basis antisymmen- trized according to the method described in Ref. [31]. The NCSM calculation proceeds as follows. First, we diagonalize the Hamiltonian with and without the NNN interaction in a three-nucleon basis for all relevant three-body channels. Second, we use the three-body solutions from the first step to derive three-body effective interactions with and without the NNN interaction. By subtracting the two effective in- teractions we isolate the NN and NNN contributions. This is needed due to a different scaling with particle number of the two- and the three-body interactions. The 4He effective interaction is then obtained by adding the two contribu- tions with the appropriate scaling factors [12]. Note that our effective interaction is model-space dependent. Conse- quently, we need both the effective interaction for the 4He ground state (JπT = 0+0), and the one for the 1−1 states, entering the LIT calculation. Indeed, due to the change of parity, the model-space size changes (Nmax → Nmax + 1). With the effective interactions replacing the interactions in the Hamiltonian (4), the four-nucleon calculations proceed as described in the text following Eq. (4). We start our discussion presenting the results obtained for the ground state of the α particle using two different values of the HO parameter, namely ~Ω = 22 and 28 MeV. This choice for the HO frequencies is driven by our final goal of evaluating the 4He photo-absorption cross section and providing an estimate for its theoretical uncertainty. Indeed, in the particular case of the 4He nucleus, frequen- cies in the range 12 ≤ ~Ω ≤ 28MeV allow to achieve a good description of both ground state and complex energy con- tinuum, as required in a calculation of response functions with the LIT method [30]. For all of the three observables examined in Fig. 1 the χEFT NN and NN+NNN interactions lead to very sim- ilar and smooth convergence patterns. In particular, an accurate convergence is reached starting from Nmax = 18, as we find independence from both model space and fre- quency. Although χEFT forces are known to present a relatively soft core, the use of effective interactions for both the NN and NNN forces is the essential key to this remark- able result. The summary of the extrapolated ground-sate properties is presented in Table 1. The present results for ground-state energy and point-proton radius with the N3LONN interaction are consistent with a previous NCSM evaluation (E0 = −25.36(4) MeV, 〈r 2 = 1.515(10) fm) obtained using a two-body effective interaction in a model space up to Nmax = 18 [35] and with that obtained by the hyper-spherical harmonic variational calculation of Ref. [36] (E0 = −25.38 MeV, 〈r 2 = 1.516 fm ) and by the Faddeev-Yakubovsky method [37] (E0 = −25.37 MeV). Finally, with the present choice for the LECs [10] the calculated binding-energy with inclusion of the NNN force is within few hundred KeV of experiment. This leaves room for additional effects expected from the inclusion -31.0 -30.0 -29.0 -28.0 -27.0 -26.0 -25.0 0 2 4 6 8 10 12 14 16 18 20 22 PSfrag replacements NN+NNN h̄Ω = 22 MeV h̄Ω = 28 MeV Fig. 1. (Color online) The 4He ground-state energy E0 [panel a)], point-proton root-mean-square radius 〈r2p〉 2 [panel b)] and total dipole strength 〈Ψ0|D̂ †D̂|Ψ0〉 [panel c)] obtained with the χEFT NN and NN+NNN interactions. Convergence pattern with respect to the model space truncation Nmax for ~Ω = 22 and ~Ω = 28 MeV. of the here missing N3LO NNN (not yet available) and NNNN interaction terms [38]. At the ground-state level, the inclusion of the NNN force affects mostly the energy, providing 3.21 MeV additional binding, while only a weak suppression of about 3.8% is found for the point-proton radius. That the total dipole strength follows the same pattern as the radius and is re- duced of 7.9% is not so surprising considering the approxi- mate relation between them [39]: 〈Ψ0|D̂ †D̂|Ψ0〉 ≃ 3(A− 1) 〈r2p〉 . (7) The latter expression, which is exact for the deuteron and the triton and for ground-state wave functions symmetric under exchange of the spatial coordinates of any pair of nu- cleons, represents a quite reasonable approximation for the α-particle and is found to be within 9% off our calculations with both the NN and NN+NNN χEFT potentials. As we Table 1 Calculated 4He ground-state energy E0, point-proton root-mean- square radius 〈r2p〉 2 , and total dipole strength 〈Ψ0|D̂ †D̂|Ψ0〉 ob- tained using the χEFT NN and NN+NNN interactions compared to experiment. The experimental value of the point-proton radius is deduced from the measured alpha-particle charge radius, 〈r2c〉 1.673(1) fm [32], proton charge radius, 〈R2p〉 2 = 0.895(18) fm [33], and neutron mean-square-charge radius, 〈R2n〉 = −0.120(5) fm 2 [34]. E0 [MeV] 〈r 2 [fm] 〈Ψ0|D̂ †D̂|Ψ0〉 [fm NN -25.39(1) 1.515(2) 0.943(1) NN+NNN -28.60(3) 1.458(2) 0.868(1) Expt. -28.296 1.455(7) - 10 20 30 40 50 60 100 150 200 18/19 16/17 14/15 12/13 0 40 80 120 160 200 PSfrag replacements σR [MeV] σI = 20 MeV Nmax = 18/19 h̄Ω = 22 MeV h̄Ω = 22 MeV h̄Ω = 28 MeV h̄Ω = 28 MeV h̄Ω = 28 MeV NN+NNN NN+NNN Fig. 2. (Color online) The LIT of the 4He dipole response as a func- tion of σR at σI = 20 MeV. Convergence pattern of the NN+NNN calculation with respect to the model-space truncation Nmax for ~Ω = 28 MeV (upper panel), and frequency dependence of the best (Nmax = 18/19) results with and without inclusion of the NNN force (lower panel). will see later, this also implies rather weak NNN effects on the 4He photo-absorption cross section at low energy. We turn now to the second part of our calculation, for which the ground state is an input. The actual evaluation of Eq. (5) is performed by applying the Lanczos algorithm to the Hamiltonian of the system, using as starting vector |ϕ0〉 = 〈Ψ0|D̂ †D̂|Ψ0〉 2 D̂|Ψ0〉 [26,30]. Indeed, the LIT can be written as a continued fraction of the elements of the resulting tridiagonal matrix, the so-called Lanczos coeffi- cients an and bn: L(σ) = 〈Ψ0|D̂ †D̂|Ψ0〉 (z − a0)− (z−a1)− (z−a2)− . (8) Due to the selection rules induced by the dipole opera- tor (3), for a given truncationNmax in the 0 +0 model space used to expand the ground state, a complete calculation of Eq. (8) requires an expansion of |ϕ0〉 over a 1 −1 space up to Nmax + 1. This is the oriigin of the even/odd no- tation for Nmax introduced to describe the convergence of the LIT in Fig. 2. The LITs obtained using the NN and NN+NNN χEFT interactions show, once again, conver- gence patterns very similar to each other. As an example, in the upper panel of Fig. (2) we show the model-space de- pendence of the LIT including the NNN force at ~Ω = 28 MeV. Thanks to the use of three-body effective interaction for both the NN and NNN terms of the potential, a sta- ble position and height of the peak in the low-σR region and satisfactory quenching of the oscillations in the tail are found for Nmax = 18/19. In this regard, our approach dif- 20 25 30 35 18/19 16/17 14/15 40 60 80 100 120 20 40 60 80 100 120 PSfrag replacements ω [MeV] Nmax = 18/19 h̄Ω = 22 MeV h̄Ω = 22 MeV h̄Ω = 28 MeV h̄Ω = 28 MeV h̄Ω = 28 MeV NN+NNN NN+NNN Fig. 3. (Color online) The 4He photo-absorption cross section as a function of the excitation energy ω. Convergence pattern of the NN+NNN calculation with respect to the model-space truncation Nmax for ~Ω = 28 MeV (upper panel), and frequency dependence of the best (Nmax = 18/19) results with and without inclusion of the NNN force (lower panel). fers from the one of Ref. [20], where the effective interaction (at the two-body cluster level) is constructed only for the NN potential, while the NNN force is taken into account as bare interaction. The bottom panel of Fig. 2 indicates that for Nmax = 18/19 we find also a fairly good agree- ment between the ~Ω = 22 and ~Ω = 28 MeV calculations, in particular below σR = 60 MeV, where for both NN and NN+NNN interactions the two curves are within 0.5% of each other. At higher σR the ~Ω = 22 MeV results present a weak oscillation (less than 5% in the range 60 ≤ σR ≤ 140 MeV) around the ~Ω = 28 MeV curves, and the dis- crepancy between the two frequencies becomes larger be- yond σR = 140 MeV, where the absolute value of the LIT is small. As we will see later, this small discrepancy will be propagated to the cross section by the inversion proce- dure [27], giving rise to the uncertainty of our calculations. As for the NNN effects at the level of the LIT, the shift of about 3 MeV in the position of the peak is due to the different ground-state energies for the NN and NN+NNN potentials. In addition one can notice a quenching of about 13% of the peak height. In analogy with Fig. 2, Fig. 3 shows the convergence be- havior of our results for the cross section. Starting from Nmax = 14/15 the calculated LIT’s are accurate enough to find stable inversions for the response function, and hence deriving the corresponding results for the cross section. The curves obtained for the NN+NNN interaction at the HO frequency value of ~Ω = 28 MeV are shown in the upper panel: the model space dependence is weak and the differ- ence between Nmax = 16/17 and 18/19 never exceeds 5% in the range from threshold to ω = 120 MeV. A somewhat larger discrepancy (less than 7%) is found by comparing the best results (Nmax = 18/19) for ~Ω = 22 MeV and 28 MeV. As with the LIT, the first oscillates slightly around the second, particularly in the tail of the cross section. We will use this discrepancy as an estimate for the theoretical uncertainty of our calculations. Note that both the NN and NN+NNN calculated cross sections are translated to the experimental threshold for the 4He photo-disintegration, Eth = 19.8 MeV (ω → ω+∆Eth, with ∆Eth being the dif- ference of the calculated and experimental thresholds). The same procedure will be applied later in the comparisonwith experimental data and different potential models. Under this arrangement, the position of the peak is not affected by the inclusion of the NNN force, while the relative differ- ence between the NN and NN+NNN cross sections varies almost linearly from −10% at threshold to about +25% at ω = 120 MeV. In particular, the peak height undergoes a 9% suppression and the two curves cross around ω = 40 MeV. In view of the inverse-energy-weighted integral of the cross-section (1), σγ(ω) dω = 4π2 〈Ψ0|D̂ †D̂|Ψ0〉 , (9) the mildness of the NNN force effects in the peak region is a consequence of the small reduction found for the total dipole strength. Considering in addition the approximate relation (7), we can infer a weak sensitivity of the cross section at low energy with respect to variations of the LECs in the NNN force, for which we have embraced the preferred choice suggested in Ref. [10]. We compare to experimental data in the region ω < 40 MeV, where corrections to the unretarded dipole approxi- mation are expected to be largely negligible and the relative uncertainty of our calculations is minimal. The data sets from Nilsson et al. [16] and Shima et al. [15] are chosen here as the latest examples of controversial experiments charac- terizing the 4He photo-effect since the 50’s (see reviews of available data in Refs. [13,14] and [15]). Note that in the upper panel of Fig. 4, we estimate the total cross section from the 4He(γ, n) measurements of Ref. [16] by assuming σγ(ω) ≃ 2σγ,n(ω). The latter assumption, which relies on the similarity of the (γ, p) and (γ, n) cross sections, pro- vides a sufficiently safe estimate of the total cross section below the three-body break-up threshold (ω = 26.1 MeV). At higher energies it represents a lower experimental bound for the total cross section, as in the energy range considered here the data of Nillsson et al. do not contain the contribu- tions of the 4He(γ, np)d and four-body break-up channels. Shima et al. [15] provide total photo-disintegration data obtained by simultaneous measurements of all the open channels. Finally, we show also an indirect determination of the photo-absorption cross section deduced from elas- tic photon-scattering on 4He by Wells et al. [40]. We find an overall good agreement with the photo-disintegration data from bremsstrahlung photons [16], which are consis- tent with the indirect measurements of Ref. [40], while we 20 25 30 35 20 25 30 35 PSfrag replacements 4He+γ →X ω [MeV] AV18+UIX NN+NNN χEFT NN χEFT NN+NNN Nilsson (‘07) Shima (‘05) Wells (‘92) Fig. 4. (Color online) The 4He photo-absorption cross section as a function of the excitation energy ω. Present NCSM results obtained using the χEFT NN and NN+NNN interactions compared to: (upper panel) the 4He(γ, n) data of Nilsson et al. [16] multiplied by a factor of 2, the total cross section measurements of Shima et al. [15], the total photo-absorption at the peak derived from Compton scattering via dispersion relations from Wells et al. [40]; (lower panel) the EIHH predictions for AV18, AV18+UIX [20] and UCOM [21]. The widths of the χEFT NN and χEFT NN+NNN curves reflect the uncertainties in the calculations (see text). reach only the last of the experimental points of Ref. [15]. The lower panel of Fig. 4 compares our present results with the prediction for the 4He photo-absorption cross sec- tion obtained in the framework of the EIHH approach [29] using the AV18, AV18+UIX [20] and UCOM [21] interac- tions. Interestingly, both the results with AV18 and χEFT NN interactions and those with AV18+UIX and χEFT NN+NNN forces show similar peak heights (∼ 3.2 mb and ∼ 3.0mb respectively), but different peak positions (partic- ularly for the first case) with an overall better agreement of the second set of curves. In this regard we notice that the α- particle ground-state properties obtained with AV18+UIX and the χEFT NN+NNN are very close to each other and to experiment. On the contrary, already at the ground-state level the two NN interactions are less alike as the 4He with the AV18 potential is more than 1 MeV less bound than with the N3LO NN potential, while they still yield to the same point-proton radius. A somewhat larger discrepancy is found close to threshold between the cross sections ob- tained with the χEFT NN+NNN and UCOM interactions. Beyond ω = 80 MeV, in the range not shown in the Fig- ure, the χEFT NN+NNN force leads to larger cross section values than AV18+UIX or UCOM, which yield to similar results in the region 45 ≤ ω ≤ 100 MeV. Keeping in mind that at such high energies the cross section is small and the uncertainty in our calculation larger, this effect can be re- lated in part to differences in the details and interplay of tensor and spin-orbit forces in the considered interaction models. At the same time corrections to the unretarded dipole operator play here a more important role. In conclusion we summarize our work. We have calcu- lated the total photo-absorption cross section of 4He us- ing the potentials of χEFT at the orders presently avail- able, the NN at N3LO and the NNN at N2LO. The micro- scopic treatment of the continuum problem was achieved by means of the LIT method, applied within the NCSM approach. Accurate convergence in the NCSM expansions is reached thanks to the use of three-body effective interac- tions. Our result shows a peak around ω = 27.8 MeV, with a cross section of 3 mb. The NNN force induces a reduction of the peak and an enhancement of the tail of the cross sec- tion. The fairly mild NNN effects are far from explaining the low-lying experimental data of Ref. [15] while moder- ately improve the agreement of the calculated cross section with the measurements of Nilsson et. al. [16]. In view of the overall good agreement between the χEFT NN+NNN and AV18+UIX calculations, the photo-absorption cross sec- tion at low energy appears to be more sensitive to change in the α-particle size, than to the details of the spin-orbit component of the NNN interaction. In this regard, a more substantial role of the NNN force can be expected in the photo-disintegration of p-shell nuclei, for which differences in the spin-orbit strength have crucial effects on the spec- trum [41,10]. Finally, the rather contained width of the the- oretical band embracing the χEFT NN+NNN, AV18+UIX and UCOM results within 15 MeV from threshold is re- markable compared to the large discrepancies still present among the different experimental data. Hence the urgency for further experimental activity to help clarify the situa- tion. Acknowledgments We would like to thank Winfried Lei- demann for supplying us with the computer code for the inversion of the LIT. We are also thankful to Giuseppina Orlandini and Sonia Bacca for useful discussions, and to Ian Thompson for critical reading of the manuscript. This work was performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Liv- ermore National Laboratory under contract No. W-7405- Eng-48. Support from the LDRD contract No. 04–ERD– 058 and from U.S. DOE/SC/NP (Work Proposal Number SCW0498) is acknowledged. References [1] S. Weinberg, Physica A 96 (1979) 327; J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) [2] S. Weinberg, Phys. Lett. B 251 (1990) 288; Nucl. Phys. B 363 (1991) 3. [3] V. Bernard, N. Kaiser, U.-G. Meißner, Int. J. Mod. Phys. E 4 (1995) 193. [4] C. Ordonez, L. Ray, U. van Kolck, Phys. Rev. Lett. 72 (1994) 1982; U. van Kolck, Prog. Part. Nucl. Phys. 43 (1999) 337; P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52 (2002) 339; E. Epelbaum, Prog. Part. Nucl. Phys. 57 (2006) 654. 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Skibinski, H. Witala, W. Gloeckle, E. Epelbaum, A. Nogga, H. Kamada, Acta Phys. Polon. B37 (2006) 2889. [39] L. L. Foldy, Phys. Rev. 107 (1957) 1303. http://arxiv.org/abs/nucl-th/0701038 http://arxiv.org/abs/nucl-th/0612016 [40] D. P. Wells, et al., Phys. Rev. C 46 (1992) 449. [41] S. Pieper, Nucl. Phys. A571 (2005) 516. References

0704.1337

Comment on "Mass and Width of the Lowest Resonance in QCD"

Comment on “Mass and Width of the Lowest Resonance in QCD” In a recent Letter [1], in which no use is made of QCD, I. Caprini, G. Colangelo, and H. Leutwyler (CCL) re- peated an unmentioned analysis of ππ scattering from 1973 [2], based on the Roy equations (REs), to make out a case for the existence of a scalar I = 0 reso- nance f0(441), listed in the PDG tables [3] as f0(600) and known as σ-meson. The primary aspect result- ing of the CCL analysis is the claimed model- and parametrization-independent determination of a σ-pole mass of (441+16 i 544+18 ) MeV implying unprece- dented small error bars. Moreover, the latter result is incompatible with very recent experimental findings, i.e., (500±30− i (264±30)) MeV [4] and (541±39− i (252± 42)) MeV [5, 6], as well as with a combined theoretical analysis yielding ((476–628)− i (226–346)) MeV [7]. The present comment will be devoted to complement a re- cent experimental discussion [4] of short-comings in the CCL analysis, by presenting theoretical arguments point- ing at a serious flaw in the theoretical formalism used by CCL, and also at the unlikeliness of their tiny er- ror bars in the σ mass and width. The simplest way to identify this flaw in Ref. [1] also present in the corre- sponding results [8] of S. Descotes-Genon and B. Mous- sallam (DM) on the scalar meson K∗0 (800) in the con- text of Roy-Steiner equations (RSEs), is to recall a warn- ing statement by G.F. Chew and S. Mandelstam (CM) from 1960 (see footnote 6 of Ref. [9]). CM state that if a strongly interacting particle with the same quantum numbers as a pair of pions should be found, then corre- sponding poles must be added to the double-dispersion representation, whether or not the new particle is inter- preted as a two-pion bound state. It should be empha- sized that this statement does not only apply to possi- ble bound-state (BS) poles of the S- or T-matrix in the physical sheet (PS) of the complex s-plane, but also to any kind of virtual BS poles and resonance poles in the unphysical sheet (US). This is justified from first princi- ples by reviewing briefly how dispersion relations (DRs) are to be derived on the basis of Cauchy’s integral for- mula t(s) = (2πi)−1 dz t(z)/(z − s) which holds for a function t(s) analytic in the domain encirculated by the closed integration contour. As the so-called “match- ing point” of CCL (and DM) is located in the US, the closed integration contour yielding the REs/RSEs must extend also to the US where the S- and T-matrix poles for scalar isoscalar ππ-scattering are found. Excluding these poles situated at sj (j = 1, . . . , n) from the inte- gration contour and assuming t(s → ∞) → 0 sufficiently fast one obtains the well known (here) unsubtracted DRs t(s) = rj/(s− sj)− dz Im[t(z)]/(s− z + iε), where L/R denotes the left-/right-hand cut, and rj is the residue of t(s) at the corresponding pole sj . According to CCL, REs/RSEs are twice-subtracted DRs yielding t(s) = t(s0) + (s− s0) t ′(s0) + (s0 − s) (s0 − sj)2(s− sj) (s0 − s) 2 Im[t(z)] (s0 − z + iε)2(s− z + iε) , (1) where the subtraction point s0 used by CCL appears to be the ππ threshold, as CCL perform the identification t(s0) = a 0 and t ′(s0) = (2a 0)/(12m π) with a 0 being S-wave scattering lengths for isospin I = 0, 2. It is now easy to see that the REs/RSEs considered by CCL and DM disregard the pole terms (PTs) in the DRs (yielding rj = 0), despite the presence of poles in the US that are claimed to exist by observing respective S-matrix zeros in the PS. As the s-dependence of the disregarded σ- and f0(980)-PTs in the vicinity of the ππ- and KK-theshold is clearly non-linear, it is to be expected on grounds of dispersion theory that the S-matrix poles predicted by CCL will not coincide with the actual ones to be deter- mined yet by CCL for self-consistency reasons. An anal- ogous statement applies to the results of DM. Moreover will the inclusion of PTs in REs/RSEs not only reinstate dispersion theoretic self-consistency, yet also yield a sig- nificant change in the resulting σ- andK∗ (800)-pole posi- tions, which unfortunately will enter now via the PTs as unknown parameters the REs/RSEs to be solved. Hence the inclusion of PTs in REs/RSEs will yield an uncer- tainty of pole positions which is likely to be of the order of the one estimated in Ref. [4] and therefore much larger than the error bars presently claimed by CCL and DM being even without taking into account PTs for at least two reasons clearly parametrization-dependent: (1) the extrapolation of the two particle phase space to the com- plex s-plane and below threshold invoked by CCL and DM is known to be speculative and even unphysical as it yields e.g. in the approach of DM scattering below the pseudo-threshold; (2) standard chiral perturbation theory (ChPT) disregarding (yet) non-perturbative PTs relates claimed values for scalar scattering lengths and their (too) tiny error bars entering REs/RSEs lacking (yet) PTs to scalar square radii the presently used (too) high values of which yield chiral symmetry breaking (ChSB) of the order of 6-8% being much larger than 3% as observed in Nature. A revision of the analysis of CCL and DM by taking into account PTs in REs/RSEs and ChPT would be highly desirable to reconcile their results with Refs. [4]-[7] and to improve the poor description of the resonance K∗(892) in the approach of DM. Frieder Kleefeld 1,2,3 1 Present address: Pfisterstr. 31, 90762 Fürth, Germany 2 Doppler & Nucl. Physics Institute (Dep. Theor. Phys.), Academy of Sciences of Czech Republic; collaborator of the CFIF, Instituto Superior Técnico, LISBOA, Portugal 3 Electronic address: [email protected] http://arxiv.org/abs/0704.1337v1 [1] I. Caprini et al., Phys. Rev. Lett. 96, 132001 (2006). [2] M. R. Pennington et al., Phys. Rev. D 7, 1429,2591 (1973). [3] W. M. Yao et al. [PDG], J. Phys. G 33, 1 (2006). [4] D. V. Bugg, J. Phys. G 34, 151 (2007). [5] M. Ablikim et al. [BES], Phys. Lett. B 598, 149 (2004). [6] D. V. Bugg, AIP Conf. Proc. 814, 78 (2006). [7] E. van Beveren et al., Phys. Lett. B 641, 265 (2006). [8] S. Descotes-Genon et al., Eur. Phys. J. C 48, 553 (2006). [9] G. F. Chew, S. Mandelstam, Phys. Rev. 119, 467 (1960). This work has been supported by the FCT of the Ministério da Ciência, Tecnologia e Ensino Superior of Portugal, under contracts POCI/FP/63437/2005, PDCT/FP/63907/2005 and the Czech project LC06002. Conversations with E. van Beveren, D. V. Bugg, J. Fischer, A. Moussallam, G. Rupp, M. D. Scadron and M. Znojil are gratefully acknowledged. References

0704.1338

True and Apparent Scaling: The Proximity of the Markov-Switching Multifractal Model to Long-Range Dependence

True and Apparent Scaling: The Proximity of the Markov-Switching Multifractal Model to Long-Range Dependence Ruipeng Liu a,b, T. Di Matteo b, Thomas Lux a aDepartment of Economics, University of Kiel, 24118 Kiel, Germany bDepartment of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, 0200 Canberra, Australia Abstract In this paper, we consider daily financial data of a collection of different stock market indices, exchange rates, and interest rates, and we analyze their multi-scaling properties by estimating a simple specification of the Markov-switching multifractal model (MSM). In order to see how well the estimated models capture the temporal dependence of the data, we estimate and compare the scaling exponents H(q) (for q = 1, 2) for both empirical data and simulated data of the estimated MSM models. In most cases the multifractal model appears to generate ‘apparent’ long memory in agreement with the empirical scaling laws. Key words: scaling, generalized Hurst exponent, multifractal model, GMM estimation 1 Introduction The scaling concept has its origin in physics but it is increasingly applied outside its traditional domain. In the literature ([1,2,3]) different methods have been proposed and developed in order to study the multi-scaling properties of financial time series. For more details on scaling analysis see [4]. Going beyond the phenomenological scaling analysis, the multifractal model of asset returns (MMAR) introduced by Mandelbrot et. al [5] provides a the- oretical framework that allows to replicate many of the scaling properties of financial data. While the practical applicability of MMAR suffered from its combinatorial nature and its non-stationarity, these drawbacks have been overcome by the introduction of iterative multifractal models (Poisson MF or Preprint submitted to Elsevier http://arxiv.org/abs/0704.1338v1 Markov-switching multifractal model (MSM) [6,7,8]) which preserves the hi- erarchical, multiplicative structure of the earlier MMAR, but is of much more ‘well-behaved’ nature concerning its asymptotic statistical properties. The at- tractiveness of MF models lies in their ability to mimic the stylized facts of financial markets such as outliers, volatility clustering, and asymptotic power- law behavior of autocovariance functions (long-term dependence). In contrast to other volatility models with long-term dependence [9], MSM models allow for multi-scaling rather than uni-scaling with varying decay exponents for all powers of absolute values of returns. One may note, however, that due to the Markovian nature, the scaling of the Markov-Switching MF model only holds over a limited range of time increments depending on the number of hierar- chical components and this ‘apparent’ power-law ends with a cross-over to an exponential cut-off. With this proximity to true multi-scaling, it seems worthwhile to explore how well the MSM model could reproduce the empirical scaling behaviour of finan- cial data. To this end, we estimate the parameters of a simple specification of the MSM model for various financial data and we assess its ability to replicate empirical scaling behaviour by also computing H(q) by means of the gener- alized Hurst exponent approach ([4,10,11]) and H by means of the modified R/S method [12] for the same data sets. We then proceed by comparing the scaling exponents for empirical data and simulated time series based on our estimated MSM models. As it turns out, the MSM model with a sufficient number of volatility components generates pseudo-empirical scaling laws in good overall agreement with empirical results. The structure of the paper is as follows: In Section 2 we introduce the multi- fractal model, the Generalized Hurst exponent (GHE) and the modified R/S approaches. Section 3 reports the empirical and simulation-based results. Con- cluding remarks and perspectives are given in Section 4. 2 Methodology 2.1 Markov-switching multifractal model In this section, we shortly review the building blocks of the Markov-switching multifractal process (MSM). Returns are modeled as [7,8]: rt = σt · ut (1) with innovations ut drawn from a standard Normal distribution N(0, 1) and instantaneous volatility being determined by the product of k volatility com- ponents or multipliers M t , M t ..., M t and a constant scale factor σ: σ2t = σ t , (2) In this paper we choose, for the distribution of volatility components, the binomial distribution: M t ∼ [m0, 2 −m0] with 1 ≤ m0 < 2. Each volatility component is renewed at time t with probability γi depending on its rank within the hierarchy of multipliers and it remains unchanged with probability 1− γi. The transition probabilities are specified by Calvet and Fisher [7] as: γi = 1− (1− γk) (bi−k) i = 1, . . . k, (3) with parameters γk ∈ [0, 1] and b ∈ (1,∞). Different specifications of Eq. (3) can be arbitrarily imposed (cf. [8] and its earlier versions). By fixing b = 2 and γk = 0.5, we arrive a relatively parsimonious specification: γi = 1− (1− γk) (2i−k) i = 1, . . . k. (4) This specification implies that replacement happens with probability of one half at the highest cascade level. Various approaches have been employed to estimate multifractal models. The parameters of the combinatorial MMAR have been estimated via an adaptation of the scaling estimator and Legendre transformation approach from statistical physics [13]. However, this approach has been shown to yield very unreliable results [14]. A broad range of more rigorous estimation methods have been developed for the MSM model. Calvet and Fisher (2001) ([6]) propose maximum likelihood estimation while Lux ([8]) proposes a Generalized Method of Moments (GMM) approach, which can be applied not only to discrete but also to continuous distributions of the volatility components. In this paper, GMM is used to estimate the two MSM model parameters in Eq. (2), namely: σ̂ and m̂0. 2.2 Estimation of scaling exponents Our analysis of the scaling behaviour of both empirical and simulated data uses two refined methods for estimating the time-honored Hurst coefficient: the estimation of generalized Hurst exponents from the structure function of various moments [4] and Lo’s modified R/S analysis that allows to correct for short-range dependence in the temporal evolution of the range [12]. 2.2.1 Generalized Hurst exponent approach The generalized Hurst exponent (GHE) method extends the traditional scal- ing exponent methodology, and this approach provides a natural, unbiased, statistically and computationally efficient estimator able to capture very well the scaling features of financial fluctuations ([10,11]). It is essentially a tool to study directly the scaling properties of the data via the qth order moments of the distribution of the increments. The qth order moments appear to be less sensitive to the outliers than maxima/minima and different exponents q are associated with different characterizations of the multi-scaling behaviour of the signal X(t). We consider the q-order moment of the distribution of the increments (with t = v, 2v, ..., T ) of a time series X(t): Kq(τ) = 〈| X(t+ τ)−X(t) |q〉 〈| X(t) |q〉 , (5) where the time interval τ varies between v = 1 day and τmax days. The gener- alized Hurst exponent H(q) is then defined from the scaling behavior of Kq(τ), which can be assumed to follow the relation: Kq(τ) ∼ )qH(q) . (6) Within this framework, for q = 1, H(1) describes the scaling behavior of the absolute values of the increments; for q = 2,H(2) is associated with the scaling of the autocorrelation function. 2.2.2 Lo’s modified R/S analysis Lo’s modified R/S analysis uses the range of a time series as its starting point: Formally, the range R of a time series {Xt}, t = 1, . . . , T is defined as: RT = max 1≤t≤T (Xt − X̄)− min 1≤t≤T (Xt − X̄). (7) Here, X̄ is the standard estimate of the mean. Usually the range is rescaled by the sample standard deviation (S), yielding the famous R/S statistic. Though this approach found wide applications in diverse fields, it turned out that no asymptotic distribution theory could be derived for H itself. Hence, no explicit hypothesis testing can be performed and the significance of point estimates H > 0.5 or H < 0.5 rests on subjective assessment. Luckily, the asymptotic distribution of the rescaled range itself under a composite null hypothesis excluding long-memory could be established by Lo (1991) [12]. Using this distribution function and the critical values reported in his paper, one can test for the significance of apparent traces of long memory as indicated by H 6= 0.5. However, Lo also showed that the distributional properties of the rescaled range are affected by the presence of short memory and he devised a modified rescaled range Qτ which adjusts for possible short memory effects by applying the Newey-West heteroscedasticity and autocorrelation consistent estimator in place of the sample standard deviation S: 1≤t≤T (Xt − X̄)− min 1≤t≤T (Xt − X̄) , (8) S2τ =S ωj(τ) i=j+1 (Xi − X̄)(Xi−j − X̄) ωj(τ) = 1− τ + 1 Under the null of no long term memory the distribution of the random variable VT = T −0.5Qτ converges to that of the range of a so-called Brownian bridge. Critical values of this distribution are tabulated in Lo (1991, Table II). 3 Results In this paper, we consider daily data for a collection of stock exchange indices: the Dow Jones Composite 65 Average Index (Dow) and NIKKEI 225 Av- erage Index (Nik) over the time period from January 1969 to October 2004, foreign exchange rates: British Pound to US Dollar (UK), and Australian Dollar to US Dollar (AU) over the period from March 1973 to February 2004, and U.S. 1 year and 2 years treasury constant maturity bond rates (TB1 and TB2, respectively) in the period from June 1976 to October 2004. The daily prices are denoted as pt, and returns are calculated as rt = ln(pt) − ln(pt−1) for stock indices and foreign exchange rates and as rt = pt−pt−1 for TB1 and We estimate the MSM model parameters introduced in Section 2 with a bi- nomial distribution of volatility components, that is M t ∼ [m0, 2−m0] and 1 ≤ m0 < 2 in Eq 2. This estimation is repeated for various hypothetical numbers of cascade levels (k = 5, 10, 15, 20). Table 1 presents these results for parameters m̂0 and σ̂. 1 Our estimation is based on the GMM approach 1 Note that the data have been standardized by dividing the sample standard de- viation which explains the proximity of the scale parameter estimates to 1. proposed by Lux [8] using the same analytical moments as in his paper. The numbers within the parentheses are the standard errors. We observe that the results for k > 10 are almost identical. In fact, analytical moment conditions in Lux [8] show that higher cascade levels make a smaller and smaller con- tribution to the moments so that their numerical values would stay almost constant. If one monitors the development of estimated parameters with in- creasing k, one finds strong variations initially with a pronounced decrease of the estimates which become slower and slower until, eventually a constant value is reached somewhere around k = 10 depending on individual time series. Based on the estimated parameters, we proceed with an analysis of simulated data from the pertinent MSM models. We first calculate the GHE for the empirical time series as well as for 100 simulated time series of each set of estimated parameters for q = 1 and q = 2. The values of the GHE are averages computed from a set of values corre- sponding to different τmax (between 5 and 19 days). The stochastic variable X(t) in Eq. (5) is the absolute value of returns, X(t) = |rt|. The second and seventh columns in Table 2 report the empirical GHEs, and values in the other columns are the mean values over the corresponding 100 simulations for dif- ferent k values: 5, 10, 15, 20, with errors given by their standard deviations. Boldface numbers are those cases which fail to reject the null hypothesis that the mean of the simulation-based Generalized Hurst exponent values equals the empirical Generalized Hurst exponent at the 5% level. We find that the exponents from the simulated time series vary across different cascade levels k. In particular, we observe considerable jumps from k = 5 to k = 10 for these values. In particular for the stock market indices, we find coincidence between the empirical series and simulation results for the scaling exponents H(2) for Dow and H(1) for Nik when k = 5. For the exchange rate data, we observe the simulations successfully replicate the empirical measurements of AU for H(1) when k = 10, 15, 20 and H(2) when k = 5; In the case of U.S. Bond rates, we find a good agreement for H(1) when k = 5 and for all k for TB1, and H(2) for TB2 when k = 5. Apparently, both the empirical data and the simulated MSM models are characterized by estimates of H(1) and H(2) much larger than 0.5 which are indicative of long-term dependence. While the empirical numbers are in nice agreement with previous literature, it is interesting to note that simulated data with k ≥ 10 have a tendency towards even higher estimated Hurst coefficients than found in the pertinent empir- ical records. 2 Since we know that the MSM model only has pre-asymptotic scaling, these results underscore that with a high enough number of volatility cascades, it would be hard to distinguish the MSM model from a ‘true’ long 2 We have checked if the generalized Hurst exponents approach is biased by com- puting H(1) and H(2) for random values generated by different random generators [11] with T = 9372 data points. We have found that H(1) = 0.4999 ± 0.009 and H(2) = 0.4995 ± 0.008. memory process. We have also performed calculations using the modified Rescaled range (R/S) analysis introduced by Lo [12,15,16,17,18,19,20], 3 whose results are reported in Tables 3 to 5. Table 3 presents Lo’s test statistics for both empirical and 1000 simulated time series for different values of k and for different trunca- tion lags τ = 0, 5, 10, 25, 50, 100. 4 We find that the values are varying with different truncation lags, and more specifically, that they are monotonically decreasing for both the empirical and simulation-based statistics. Table 4 re- ports the number of rejections of the null hypothesis of short-range dependence based on 95% and 99% confidence levels. The rejection numbers for each sin- gle k are decreasing as the truncation lag τ increases, but the proportion of rejections remains relatively high for higher cascade levels, k = 10, 15, 20. The corresponding Hurst exponents are given in Table 5. The empirical values of H are decreasing when τ increases. A similar behaviour is observed for the simulation-based H for given values of k. We also observe that the Hurst expo- nent values are increasing with increasing cascade level k for given τ . Boldface numbers are those cases which fail to reject the null hypothesis that the mean of the simulation-based Hurst exponent equals the empirical Hurst exponent at the 5% level. There are significant jumps between the values for k = 5 and k = 10 as reported in previous tables. Overall, the following results stand out: (1) There seems to be a good overall agreement between the empirical and simulated data for practically all series for levels k ≥ 10, while with a smaller number of volatility components (k = 5) the simulated MSM models have typically smaller estimated Hs than the corresponding empirical data, (2) the modified R/S approach would quite reliably reject the null of long memory for k = 5, but in most cases it would be unable to do so for higher numbers of volatility components, even if we allow for large truncation lags up to τ = 100. Results are also much more uniform than with the generalized Hurst technique which had left us with a rather mixed picture of coincidence of Hurst coefficients of empirical and simulated data. The fact, that according to Table 5, MSM model with 15 or more volatility components did always produce ‘apparent’ scaling in agreement with that of empirical data, is particular encouragingly. It contrasts with the findings reported in [19] on apparent scaling of estimated GARCH models whose estimated exponents did not agree with the empirical ones. 3 We also did a Monte Carlo study with 1000 simulated random time series in order to assess the bias of the pertinent estimates of H: for random numbers with sample size T = 9372 (comparable to our empirical records) we obtained a slight negative bias: H = 0.463 ± 0.024. 4 For τ = 0 we have the classical R/S approach. 4 Concluding Remarks We have calculated the scaling exponents of simulated data based on esti- mates of the Markov-switching multifractal (MSM) model. Comparing the generalized Hurst exponent values as well as Lo’s Hurst exponent statistics of both empirical and simulated data, our study shows that the MSM model captures quite satisfactorily the multi-scaling properties of absolute values of returns for specifications with a sufficiently large number of volatility compo- nents. Subsequent work will explore whether this encouraging coincidence of the scaling statistics for the empirical and synthetic data also holds for other candidate distributions of volatility components and alternative specifications of the transition probabilities. Acknowledgments T. Di Matteo acknowledges the partial support by ARC Discovery Projects: DP03440044 (2003) and DP0558183 (2005), COST P10 “Physics of Risk” project and M.I.U.R.-F.I.S.R. Project “Ultra-high frequency dynamics of fi- nancial markets”, T. Lux acknowledges financial support by the European Commission under STREP contract No. 516446. References [1] U. A. Müller, M. M. Dacorogna, R. B. Olsen, O. V. Pictet, M. Schwarz, C. Morgenegg, Journal of Banking and Finance 14, 1189-1208, (1990). [2] M. M. Dacorogna, R. Gençay, U. A. Müller, R. B. Olsen, O. V. Pictet, An Introduction to High Frequency Finance Academic Press, San Diego, (2001). [3] M. M. Dacorogna, U. A. Müller, R. B. Olsen, O. V. Pictet, Quantitative Finance 1, 198-201, (2001). [4] T. Di Matteo, Quantitative Finance 7, 21–36, No. 1, (2007). [5] B. Mandelbrot, A. Fisher, and L. Calvet, Cowles Foundation for Research and Economics Manuscript, (1997). [6] L. Calvet, and A. Fisher, Journal of Econometrics 105, 27–58, (2001). [7] L. Calvet, and A. Fisher, Journal of Financial Econometrics 84, 381–406, (2004). [8] T. Lux, Journal of Business and Economic Statistics in Press, (2007). [9] R. T. Baillie, T. Bollerslev, and H. Mikkelsen, Journal of Econometrics 74, 3–30, (1996). [10] T. Di Matteo, T. Aste, and M. Dacorogna, Physica A 324, 183–188, (2003). [11] T. Di Matteo, T. Aste, and M. Dacorogna, Journal of Banking and Finance 29, 827–851, (2005). [12] A. W. Lo, Econometrica 59, 1279–1313, (1991). [13] L. Calvet, and A. Fisher, Review of Economics and Statistics 84, 381–406, (2002). [14] T. Lux, International Journal of Modern Physics 15, 481–491, (2004). [15] T. C. Mills, Applied Financial Economics, 3, 303-306, (1993). [16] B. Huang and C. Yang, Applied Economic Letters, 2, 67-71, (1995). [17] C. Brooks, Applied Economic Letters, 2, 428-431, (1995). [18] T. Lux, Applied Economic Letters, 3, 701-706, (1996). [19] N. Crato and P. J. F. de Lima, Economics Letters, 45, 281-285, (1996). [20] Williger et al., Finance & Stochastics, 3, 1-13, (1999). Table 1 GMM estimates of MSM model for different values of k. k = 5 k = 10 k = 15 k = 20 m̂0 σ̂ m̂0 σ̂ m̂0 σ̂ m̂0 σ̂ Dow 1.498 0.983 1.484 0.983 1.485 0.983 1.487 0.983 (0.025) (0.052) (0.026) (0.044) (0.026) (0.042) (0.027) (0.044) Nik 1.641 0.991 1.634 0.991 1.635 0.991 1.636 0.991 (0.017) (0.036) (0.013) (0.028) (0.017) (0.036) (0.017) (0.037) UK 1.415 1.053 1.382 1.057 1.381 1.056 1.381 1.058 (0.033) (0.026) (0.029) (0.027) (0.036) (0.027) (0.038) (0.026) AU 1.487 1.011 1.458 1.013 1.457 1.014 1.458 1.014 (0.034) (0.066) (0.034) (0.061) (0.034) (0.066) ( 0.034) (0.065) TB1 1.627 1.041 1.607 1.064 1.607 1.064 1.606 1.067 (0.021) (0.032) (0.025) (0.024) (0.028) (0.024) (0.025) (0.024) TB2 1.703 1.040 1.679 1.068 1.678 1.079 1.678 1.079 (0.015) (0.036) (0.014) (0.029) (0.015) (0.032) (0.015) (0.034) Note: All data have been standardized before estimation. Table 2 H(1) and H(2) for the empirical and simulated data. H(1) H(2) Emp sim1 sim2 sim3 sim4 Emp sim1 sim2 sim3 sim4 Dow 0.684 0.747 0.849 0.868 0.868 0.709 0.705 0.797 0.813 0.812 (0.034) (0.008) (0.015) (0.021) (0.024) (0.027) (0.009) (0.015) (0.019) (0.022) Nik 0.788 0.801 0.894 0.908 0.908 0.753 0.736 0.815 0.824 0.824 (0.023) (0.008) (0.013) (0.019) (0.028) (0.021) (0.008) (0.013) (0.018) (0.024) UK 0.749 0.709 0.799 0.825 0.821 0.735 0.678 0.764 0.785 0.783 (0.023) (0.010) (0.018) (0.025) (0.026) (0.026) (0.010) (0.016) (0.021) (0.022) AU 0.827 0.746 0.837 0.860 0.857 0.722 0.705 0.790 0.808 0.808 (0.017) (0.009) (0.016) (0.022) (0.021) (0.024) (0.009) (0.015) (0.018) (0.018) TB1 0.853 0.856 0.909 0.915 0.911 0.814 0.783 0.826 0.832 0.829 (0.022) (0.035) (0.023) (0.026) (0.026) (0.027) (0.028) (0.020) (0.020) (0.020) TB2 0.791 0.866 0.920 0.924 0.919 0.778 0.781 0.823 0.827 0.822 (0.025) (0.029) (0.021) (0.022) (0.026) (0.029) (0.022) (0.017) (0.022) (0.023) Note: Emp refers to the empirical exponent values, sim1, sim2, sim3 and sim4 are the corresponding exponent values based on the simulated data for k = 5, k = 10, k = 15 and k = 20 respectively. The stochastic variable Xt is defined as |rt|. Bold numbers show those cases for which we cannot reject identity of the Hurst coefficients obtained for empirical and simulated data, i.e. the empirical exponents fall into the range between the 2.5 to 97.5 percent quantile of the simulated data. Table 3 Lo’s R/S statistic for the empirical and simulated data. τ = 0 τ = 5 τ = 10 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Dow 3.005 1.712 5.079 6.640 6.704 2.661 1.481 4.060 5.211 5.263 2.427 1.376 3.574 4.537 4.582 (0.381) (1.300) (1.769) (1.839) (0.329) (1.017) (1.333) (1.387) (0.305) (0.884) (1.133) (1.179) Nik 7.698 1.840 4.898 6.154 6.152 6.509 1.540 3.817 4.747 4.742 5.836 1.416 3.343 4.132 4.133 (0.425) (1.195) (1.520) (1.584) ( 0.355) (0.918) (1.147) (1.193) (0.325) (0.798) (0.984) (1.023) UK 6.821 1.544 4.599 6.047 6.175 5.912 1.370 3.815 4.918 5.008 5.333 1.286 3.405 4.337 4.408 (0.350) (1.200) (1.748) (1.848) (0.310) (0.972) (1.352) (1.417) (0.290) (0.854) (1.157) (1.207) AU 7.698 1.687 4.962 6.348 6.434 6.731 1.463 4.001 5.024 5.090 6.103 1.361 3.531 4.387 4.443 (0.386) (1.257) (1.742) (1.790) (0.333) (0.989) (1.315) (1.352) (0.309) (0.861) (1.117) (1.149) TB1 8.845 1.826 4.644 5.915 6.041 7.109 1.524 3.629 4.564 4.582 6.110 1.400 3.184 4.415 4.530 (0.398) (1.141) (1.425) (1.380) (0.330) (0.875) (1.074) (1.040) (0.302) (0.759) (0.921) (0.891) TB2 7.295 1.855 4.347 5.853 5.907 6.083 1.531 3.391 4.207 4.349 5.330 1.404 2.985 4.025 4.158 (0.413) (1.031) (1.215) (1.227) (0.339) (0.795) (0.928) (0.930) (0.310) (0.694) (0.804) (0.803) τ = 25 τ = 50 τ = 100 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Dow 2.042 1.237 2.877 3.580 3.616 1.736 1.153 2.385 2.909 2.941 1.464 1.098 1.965 2.338 2.366 (0.272) (0.694) (0.857) (0.893) (0.250) (0.560) (0.668) (0.696) (0.233) (0.443) (0.508) (0.530) Nik 4.760 1.260 2.692 3.285 3.279 3.941 1.169 2.246 2.701 2.698 3.220 1.113 1.868 2.204 2.203 (0.286) (0.631) (0.761) (0.788) (0.263) (0.514) (0.604) (0.623) (0.245) (0.412) (0.468) (0.482) UK 4.348 1.170 2.782 3.469 3.515 3.575 1.099 2.322 2.837 2.868 2.871 1.053 1.922 2.289 2.306 (0.262) (0.678) (0.876) (0.909) (0.244) (0.549) (0.680) (0.702) (0.228) (0.434) (0.513) (0.528) AU 5.035 1.224 2.848 3.474 3.516 4.130 1.142 2.362 2.830 2.861 3.281 1.089 1.947 2.280 2.302 (0.275) (0.676) (0.842) (0.866) (0.252) (0.544) (0.654) (0.672) (0.232) (0.429) (0.496) (0.508) TB1 4.580 1.245 2.571 2.961 2.971 3.514 1.156 2.148 2.442 2.449 2.649 1.101 1.790 2.004 2.006 (0.265) (0.598) (0.711) (0.685) (0.242) (0.484) (0.564) (0.542) (0.223) (0.384) (0.440) (0.417) TB2 4.129 1.249 2.432 2.762 2.786 3.250 1.162 2.052 2.305 2.320 2.502 1.109 1.731 1.915 1.921 (0.272) (0.554) (0.632) (0.630) (0.249) (0.456) (0.511) (0.507) (0.230) (0.369) (0.403) (0.398) Note: Emp stands for the empirical Lo’s statistic, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s statistics based on the corresponding 1000 simulated time series with pertinent k. Table 4 Number of rejections for Lo’s R/S statistic test. τ = 0 τ = 5 τ = 10 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ Dow 311 151 1000 1000 1000 1000 1000 1000 121 46 999 991 999 998 1000 1000 69 22 990 968 998 997 1000 995 Nik 433 253 1000 999 1000 1000 1000 1000 176 74 993 985 998 997 1000 999 98 36 983 963 997 991 999 993 UK 167 77 998 995 1000 999 999 998 74 22 991 976 998 997 998 997 41 7 982 943 996 990 997 992 AU 301 142 1000 999 999 999 1000 1000 116 39 997 990 998 994 1000 999 58 23 990 966 993 989 999 995 TB1 428 227 1000 1000 1000 999 999 999 146 55 993 976 997 991 998 996 75 24 976 934 990 970 996 989 TB2 453 256 999 995 998 997 1000 999 159 60 987 959 994 982 996 986 86 21 958 899 985 961 985 960 τ = 25 τ = 50 τ = 100 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ Dow 24 5 939 858 990 964 985 966 9 3 807 677 940 887 948 872 4 1 566 381 811 669 808 686 Nik 34 5 920 809 982 848 977 930 11 2 764 581 914 831 897 812 4 1 485 281 750 582 742 575 UK 11 1 929 843 982 942 979 953 4 1 789 630 919 840 926 843 1 1 541 327 783 632 774 640 AU 23 5 931 860 983 949 983 956 6 2 816 666 921 852 931 846 4 1 561 353 776 648 786 649 TB1 25 4 876 765 946 870 965 893 5 1 698 519 822 711 846 712 1 1 418 230 627 415 604 400 TB2 21 6 844 696 933 851 928 859 10 3 627 446 798 638 807 657 3 1 368 167 534 312 544 336 Note: k = 5, k = 10, k = 15 and k = 20 refer to the number of rejections at 95% (†) and 99% (‡) confidence levels (these intervals are given by [0.809, 1.862] and [0.721, 2.098], respectively) for the 1000 simulated time series. Table 5 Lo’s modified R/S Hurst exponent H values for the empirical and simulated data. τ = 0 τ = 5 τ = 10 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Dow 0.620 0.556 0.674 0.703 0.704 0.607 0.540 0.650 0.677 0.678 0.597 0.532 0.636 0.662 0.663 (0.024) (0.029) (0.030) (0.031) (0.024) (0.028) (0.029) (0.030) (0.024) (0.028) (0.028) (0.029) Nik 0.723 0.564 0.670 0.695 0.695 0.705 0.544 0.643 0.667 0.667 0.693 0.535 0.629 0.652 0.651 (0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.027) (0.028) UK 0.712 0.545 0.665 0.694 0.696 0.696 0.532 0.644 0.672 0.673 0.685 0.525 0.632 0.658 0.660 (0.025) (0.030) (0.033) (0.036) (0.025) (0.029) (0.032) (0.035) (0.025) (0.029) (0.031) (0.034) AU 0.726 0.555 0.673 0.700 0.701 0.711 0.539 0.650 0.674 0.676 0.700 0.531 0.636 0.660 0.661 (0.025) (0.029) (0.032) (0.032) (0.025) (0.028) (0.031) (0.031) (0.025) (0.028) (0.030) (0.030) TB1 0.746 0.565 0.670 0.689 0.691 0.721 0.547 0.642 0.660 0.661 0.704 0.535 0.627 0.644 0.645 (0.024) (0.028) (0.031) (0.029) (0.024) (0.028) (0.030) (0.028) (0.024) (0.028) (0.029) (0.028) TB2 0.724 0.567 0.662 0.679 0.680 0.704 0.545 0.634 0.650 0.652 0.689 0.536 0.620 0.636 0.637 (0.025) (0.028) (0.028) (0.028) (0.025) (0.027) (0.028) (0.028) (0.024) (0.027) (0.028) (0.027) τ = 25 τ = 50 τ = 100 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Dow 0.578 0.521 0.612 0.636 0.637 0.560 0.513 0.592 0.614 0.615 0.542 0.508 0.571 0.590 0.591 (0.024) (0.027) (0.027) (0.028) (0.023) (0.026) (0.026) (0.027) (0.023) (0.025) (0.025) (0.026) Nik 0.671 0.522 0.605 0.627 0.626 0.650 0.514 0.586 0.606 0.605 0.628 0.509 0.566 0.584 0.583 (0.025) (0.026) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.024) (0.025) (0.024) (0.025) UK 0.662 0.515 0.610 0.634 0.635 0.641 0.508 0.590 0.612 0.613 0.617 0.503 0.569 0.589 0.589 (0.025) (0.028) (0.029) (0.032) (0.024) (0.027) (0.028) (0.030) (0.024) (0.026) (0.026) (0.028) AU 0.679 0.520 0.612 0.634 0.635 0.657 0.512 0.592 0.612 0.613 0.631 0.507 0.571 0.588 0.589 (0.025) (0.027) (0.029) (0.029) (0.024) (0.026) (0.027) (0.027) (0.023) (0.025) (0.026) (0.026) TB1 0.672 0.522 0.603 0.619 0.621 0.642 0.514 0.583 0.597 0.598 0.610 0.509 0.563 0.575 0.576 (0.024) (0.027) (0.028) (0.027) (0.024) (0.026) (0.027) (0.026) (0.023) (0.025) (0.026) (0.024) TB2 0.661 0.520 0.597 0.611 0.612 0.633 0.514 0.578 0.591 0.592 0.604 0.509 0.559 0.571 0.571 (0.024) (0.027) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.023) (0.025) (0.025) (0.024) Note: Emp stands for the empirical value of Lo’s Hurst exponent, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s Hurst exponent based on the corresponding 1000 simulated time series with different k. Boldface numbers are those cases in which empirical Hs fall into the corresponding 2.5 to 97.5 percent quantiles of the 1000 simulation-based values of H. Introduction Methodology Markov-switching multifractal model Estimation of scaling exponents Results Concluding Remarks References

0704.1339

SkyMapper and the Southern Sky Survey - a resource for the southern sky

SkyMapper and the Southern Sky Survey a resource for the southern sky Stefan C. Keller, Brian P. Schmidt and Michael S. Bessell1 Research School of Astronomy and Astrophysics, Cotter Rd., Canberra, ACT 2611, Australia [email protected] Summary. SkyMapper is amongst the first of a new generation of dedicated, wide- field survey telescopes. The 1.3m SkyMapper telescope features a 5.7 square degree field-of-view Cassegrain imager and will see first light in late 2007. The primary goal of the facility is to conduct the Southern Sky Survey a six colour, six epoch survey of the southern sky. The survey will provide photometry for objects between 8th and 23rd magnitude with global photometric accuracy of 0.03 magnitudes and astrometry to 50 mas. This will represent a valuable scientific resource for the south- ern sky and in addition provide a basis for photometric and astrometric calibration of imaging data. 1 The SkyMapper Telescope The SkyMapper telescope is a 1.3m telescope currently under construction by the Australian National University’s Research School of Astronomy and Astrophysics in conjunction with Electro Optic Systems of Canberra, Aus- tralia. The telescope will reside at Siding Spring Observatory in central New South Wales, Australia. The telescope is a modified Cassegrain design with a 1.35m primary and a 0.7m secondary. Corrector optics are of fused silica construction for maximum UV throughput and a set of six interchangeable filters can be placed in the optical path. The facility will operate in an automated matter with minimal operator support. Further details on all aspects of our programme can be found in [2]. 2 Detectors and Filters The focal plane is comprised of 32 2k×4k CCDs from E2V, UK. Each CCD has 2048 × 4096, 15 micron square pixels. The devices are deep depleted, backside illuminated and 3-side buttable. They possess excellent quantum efficiency from 350nm-950nm (see Figure1), low read noise and near perfect cosmetics. http://arxiv.org/abs/0704.1339v1 2 Stefan C. Keller, Brian P. Schmidt and Michael S. Bessell Wavelength (Angstroms) Fig. 1. Spectral response of SkyMapper CCDs as measured in the laboratory. The shaded area encloses the range in response exhibited. 4000 6000 8000 Wavelength (Å) 10000 Fig. 2. The predicted throughput of the Southern Sky Survey filter set, excluding atmospheric absorption. The SkyMapper imager will utilise the recently developed STARGRASP controllers developed for the Pan-STARRS project by Onaka and Tonry et al. of the University of Hawaii [6]. Twin 16-channel controllers enable us to read out the array in 12 seconds with ∼4−5 electron read noise. Figure 2 shows the expected normalised throughput of our system. The filter set is based upon the Sloan Digital Sky Survey filter set with three important modifications:- the movement of the red edge of the u filter to the blue, the blue edge of the g filter to the red, and the introduction of an intermediate band v filter ( essentially a DDO38 filter ). At this time coloured glass fabrication of filters of these bandpasses offers the best solution for spatial uniformity compared to the competing interference film technology. Our filters are sourced from MacroOptica of Russia. SkyMapper 3 3 The Southern Sky Survey Performing the Southern Sky Survey is the primary preoccupation of the SkyMapper telescope. The survey will cover the 2π steradians of the southern hemisphere reaching g=23 at a signal-to-noise of 5 sigma. For stars brighter than g=18 we require global accuracy of 0.03 magnitudes and astrometry to better than 50 milli arc seconds. The survey’s six epochs are designed to capture variability on the time scales of days, weeks, months and years over the five year expected lifetime of the survey. The 5 sigma limits attained after one 110 second epoch and after the full six epochs are given in Table 1. In all bands we attain limits slightly deeper (∼ 0.5mag) than the Sloan Digital Sky Survey. Table 1. Southern Sky Survey limits (5 sigma) in AB magnitudes from multiple 110 second exposures u v g r i z 1 epoch 21.5 21.3 21.9 21.6 21.0 20.6 6 epochs 22.9 22.7 22.9 22.6 22.0 21.5 4 Global Photometric Calibration The greatest impediment to deriving accurate photometry from wide field imaging cameras is the accurate description of the illumination correction. The illumination correction corrects for geometry of the optics and inclusion of scattered light in the system (see Patat and Freudling these proceedings). During commissioning we will develop an illumination correction for each filter via dithered observations of a field. We will then rotate the instrument and repeat the dithered observations to ensure we rigourously understand the illumination correction for the system. We will establish six such reference fields at declinations of around -25◦ and spaced in right ascension. Each field will be 4.6 degrees square following the dither pattern. During the first year of operation we will perform the Five-Second Survey, a rapid survey in photometric conditions to provide all-sky standards between 8-16th magnitude. The Five-Second Survey will consist of a set of at least three images of a field in all filters. During Five-Second observing we will observe the two highest of our six reference fields every ninety minutes. This will ensure photometry is obtained on a highly accurate standard instrumental system. The Five-Second Survey will provide a network of photometric and astrometric standards to anchor the deeper main survey images. Furthermore, it enables the main survey to proceed in non-photometric conditions. 4 Stefan C. Keller, Brian P. Schmidt and Michael S. Bessell We will establish the six reference fields to include stars with photometry in the Walraven system [4]. As demonstrated by Pel & Lub (ibid), the Wal- raven system zeropoint is highly accurate: the closure solution over 2π in right ascension has rms of less than 1 millimag. In addition, the Walraven stars we have selected are spectrophotometric standards from the work of Gregg et al. [5]. The use of these standards will provide absolute flux calibration for our system. 5 A Filter Set for Stellar Astrophysics The majority of science goals identified for SkyMapper are based on the iden- tification of stellar populations. It was therefore fundamental to the science output of the telescope that we choose a filter set that offers optimal diagnos- tic power for the important stellar characteristics of effective temperature, surface gravity and metallicity. Below I will discuss some specific examples. Through an exploration of colour parameter space derived from model stellar atmospheres and filter bandpasses we arrived at the filter set shown in Figure 2. The filter set possesses two filters, u and v, distinctly either side of the Balmer Jump feature at 3646Å. Fig. 3. Precision of determined surface gravity from our filter set as a function of temperature and surface gravity (error bars show estimated uncertainties at each point for photometric uncertainties of 0.03mag in each filter). SkyMapper 5 5.1 Blue Horizontal Branch Stars In Figure 3 we show the uncertainty in the derived stellar surface gravity as a function of temperature for a range of surface gravities with photometric uncertainties of 0.03mag. per filter. In the case of A-type stars we expect to determine surface gravity to ∼ 10%. The sensitivity to surface gravity arises from the u−v colour which measure the Balmer Jump and the effect of H− opacity, both of which increase with surface gravity. It is at these temperatures that we find blue horizontal branch stars (BHBs). Due to their characteristic absolute magnitude BHBs are standard candles for the Galactic halo. A line of sight through the halo inevitably contains a mixture of local main-sequence A-type and blue straggler stars. However as is shown in Figure 3 the SkyMapper filter set enables us to clearly distinguish the BHBs of interest on the basis of their lower surface gravity. Simulations show that we will be able to derive a sample of BHBs to 130kpc with less than 5% contamination. 5.2 Extremely Metal-Poor Stars In the case of cooler stars (F0 and cooler) the u and v filters indicate the level of metal line blanketing blueward of ∼ 4000Å. Figure 4 shows the v−g, g−i colour-colour diagram for a range of metallicities and surface gravities. The v−g colour has a strong dependency on the metallicity and little dependency on the surface gravity hotter than K0 (g−i ∼ 1.7). log[Fe/H]=-1 Solar, logg=2 -5.4 Frebel et al. 2006 Fig. 4. v−g vs. g−i for stars of solar metallicity (dashed lines) and for a range of surface gravity (solid lines). Open circles are stars from the sample of [1] and the star symbol is HE1327-2326 from [3]. 6 Stefan C. Keller, Brian P. Schmidt and Michael S. Bessell This enables us to cleanly separate the extremely metal-poor stars in the halo from the vast bulk of the halo at [Fe/H]<-2. Our simulations show we should find of order 100 stars with [Fe/H]<-5. 6 Data Products and Their Possible Application to ESO Calibration The first SkyMapper data product will be the Five-Second Survey of 8-16th magnitude stars in the southern hemisphere. Two main survey data releases will follow. The first data release will occur when three images in each filter have been reduced for a field and the second (reaching 23rd magnitude in g) when the full set of six have been obtained and undergone quality control. The survey will provide sufficient density and spectral sampling of stan- dard stars to enable photometric calibration of any field imaged in any broad- band filter in the southern hemisphere. The largest source of dispersion in transformations between photometric systems is due to the lack of knowl- edge of the surface gravity and metallicity of the sample. The SkyMapper photometric system provides a prior on both these points of uncertainty. Consequently we will be able to provide improved transformations from our photometric system to any other system. Scheduled observations on, for in- stance VLT or VST, may then dispense with photometric standards and also proceed under non-photometric conditions. References 1. Cayrel, R. et al. A&A 416, 1117 (2004) 2. Keller, S. et al. PASA in press, astro-ph/0702511 (2007) 3. Frebel, A. et al. Nature 434, 871 (2005) 4. J.W. Pel & J. Lub: The Walraven System. In: The Future of Photometric, Spectrophotometric and Polarmetric Standardization, PASP in press 5. M. Gregg: Next Generation Spectral Library. In: http://lifshitz.ucdavis.edu/- mgregg/gregg/ngsl/ngsl.html 6. P. Onaka & J. Tonry: StarGrasp - Detector Controllers for Science and As- tronomy. In: http://www.stargrasp.org/ SkyMapper and the Southern Sky Survey a resource for the southern sky Stefan C. Keller, Brian P. Schmidt and Michael S. Bessell

0704.1340

Tautological classes on moduli spaces of curves with linear series and a push-forward formula when $\rho=0$

TAUTOLOGICAL CLASSES ON MODULI SPACES OF CURVES WITH LINEAR SERIES AND A PUSH-FORWARD FORMULA WHEN ρ = 0 DEEPAK KHOSLA Abstract. We define tautological Chow classes on the moduli space Gr triples consisting of a curve C, a line bundle L on C of degree d, and a linear system V on L of dimension r. In the case where the forgetful morphism to Mg has relative dimension zero, we describe the images of these classes in A1(Mg). As an application, we compute the (virtual) slopes of several different classes of divisors on Mg. Contents 1. Introduction 1 2. Statement of Theorem 3 2-A. A limit linear series moduli stack 3 2-B. Tautological Classes 4 2-C. A Push-Forward Formula 5 3. Applications 5 3-A. The Gieseker-Petri Divisor 5 3-B. Hypersurface Divisors 6 3-C. Syzygy Divisors 7 3-D. Secant Plane Divisors 9 4. Special Families of Curves 10 4-A. Definitions of Families 10 4-B. Computations on the Special Families 11 4-C. Pull-Back Maps on Divisors 11 5. Proofs of Lemmas 13 6. Appendix 21 References 25 1. Introduction The cone of effective divisors on a projective variety plays an important rôle in the understanding of its birational geometry. In the case of the moduli space Mg of genus-g curves, it has become apparent that the most interesting effective divisor classes are those that arise from the extrinsic geometry of curves in projective space. Indeed, in their pioneering work, Harris and Mumford [20] considered divisors of curves admitting a degree-d branched cover of P1, where d = (g + 1)/2. More Date: October 24, 2018. http://arxiv.org/abs/0704.1340v1 2 D. KHOSLA recently, work of Cukierman [2], Farkas-Popa [11], Khosla [21], and Farkas [10] [9] has found effective divisor classes of smaller slope that those considered by Mumford and Harris, and some of these classes have been used to improve on the best known bounds on n for which Mg,n is of general type for fixed g [9]. Although all of these divisors are described by conditions on the space of em- beddings of a curve in projective space, the techniques used to deal with them have been varied. In this paper, we put all of the above calculations into a unified frame- work and lay the ground for future work on the effective cone of Mg. Specifically we consider the moduli stack Grd(Mg) of genus-g curves together with a g d (linear series). The set of C-valued points consists of triples (C,L, V ), where C is a genus-g curve, L is a degree-d line bundle on C, and V ⊂ H0(L) is an (r + 1)-dimensional subspace. The forgetful morphism η : Grd(Mg) → Mg is representable, proper, and generically smooth of relative dimension ρ(g, r, d) = g − (r + 1)(g − d+ r) [23], [16], [5], [6] [14], [7], [25], [13]. (When ρ < 0, then η is not dominant.) If g, r, and d are chosen so that ρ = −1, then the image of η has a component of codimension 1, and it is this divisor that Eisenbud and Harris use to show that Mg is of general type when g ≥ 24 and g+1 is composite [8]. The closure of this divisor in the moduli space Mirrg of irreducible nodal curves may also be interpreted as the image of the virtual fundamental class of Grd(M g ) under the proper push-forward morphism η∗ : A∗(G g )) → A∗(M where Grd(M g ) is a partial compactification of G d(Mg) using torsion-free sheaves. If we now choose g, r, and d so that ρ = 0, we are led to the “second generation” of effective divisors on Mg. In this case, the morphism η : G d(Mg) → Mg is generically finite. Since the work of Cukierman [2], every interesting effective divisor class on Mg has been realized as the image under η of a divisor on G d(Mg). For example, the K3 locus in M10 [2], which was the first counterexample [11] to the Harris-Morrison slope conjecture [19], can be interpreted as the image under η of the divisor in G412(M10) of g 12’s which do not lie on a quadric. Again, the class of its closure in Mirr10 may be realized as the proper push-forward of the corresponding class in G412(M 10) under the morphism η∗ : A∗(G 10)) → A∗(M This latter class, in turn, is easily computed to be 2α− β − 6γ + η∗λ, where α, β, and γ are certain tautological classes defined on Grd(M g ). (See Sec- tions 2-B and 3-C.) In Section 2, we introduce a partial compactification Grd(M̃g) of G d(Mg), which is proper over an open substack M̃g of Mg that contains Mg and whose complement in Mg has codimension 2. We define tautological virtual codimension-1 Chow classes α, β, and γ on Grd(M̃g) and, in the case where ρ = 0, compute their images under the proper push-forward morphism η∗ : A3g−2(G d(M̃g)) → A3g−2(M̃g) = A 1(Mg). A PUSH-FORWARD FORMULA WHEN ρ = 0 3 This allows one to completely mechanically compute the slopes of all of the divisor classes on Mg that have thus far been studied. As examples, in Section 3 we study syzygy divisors, hypersurface divisors, Gieseker-Petri divisors, and secant plane divisors. In Section 4 we give the statements of a series of calculuations over special families of stable curves. These calculations assemble to give the main result. Finally Section 5 is devoted to the proofs of the lemmas stated in Section 4. This work was carried out for my doctoral thesis under the supervision of Joe Harris. I would like to thank Ethan Cotterill, Gavril Farkas, Johan de Jong, Martin Olsson, Brian Osserman, and Jason Starr for helpful conversations. 2. Statement of Theorem 2-A. A limit linear series moduli stack. Definition 2.1 ([24]). Let S be any scheme, and let g and n be non-negative integers. An n-pointed stable curve of genus g over S is a proper flat morphism π : X → S together with sections σ1 . . . , σn : S → X . Each geometric fiber Xs̄ must be a reduced, connected, 1-dimensional scheme such that (a) Xs̄ has only ordinary double points; (b) Xs̄ intersects the sections σ1 . . . , σn at distinct points p1, . . . pn that lie on the smooth locus of Xs̄; (c) the line bundle ωXs̄(p1 + · · ·+ pn) is ample; (d) dimH1(OXs̄) = g. Theorem 2.2 ([3],[24]). Let g and n be non-negative integers such that 2g − 2 + n > 0. The category Mg,n of families of n-pointed stable curves of genus g is an irreducible Deligne-Mumford stack that is proper, smooth, and of finite type over SpecZ. Definition 2.3 ([3]). Let k be an algebraically closed field, and let X be an n- pointed stable curve of over k. The dual graph ΓX of X is the following unoriented graph: (a) the set of vertices of ΓX is the set Γ X of irreducible components of X ; (b) the set of edges of ΓX is the set Γ X which is the union of the singular and marked points of X ; (c) an edge x ∈ Γ1X has for extremities the irreducible components on which x lies; Definition 2.4. An n-pointed stable curve X → S is tree-like if the dual graph of each geometric fiber is a tree. Proposition 2.5 ([22]). The category M̃g,n of families of tree-like curves is an open substack of Mg,n, whose complement has codimension 2. Definition 2.6. Let π : X → S be a smooth genus-g curve. A grd on X is the data of a line bundle L→ X of relative degree d together with a rank-r vector subbundle V of π∗L [26, Definition 4.2]. Proposition 2.7. Let π : X → S be a smooth genus-g curve. The étale sheafifica- tion of the functor (Sch/S) → (Sets) 4 D. KHOSLA given by T 7→ {grd’s on XT → T} is represented by a scheme Grd(X/S), proper over S. If Z is an irreducible compo- nent of Grd(X/S), then (1) dimZ ≥ dimS + ρ(g, r, d) In this way, one can construct a Deligne-Mumford stack Grd , representable and proper over Mg. In [22], Osserman and the author extend this construction to families of tree-like stable curves. That is, for X a tree-like stable curve over S of genus g, they construct an algebraic space Grd(X/S), proper over S and satisfying the same dimension lower bound (1). In addition, they prove that if S = Spec k, where k is an algebraically closed field, then there is an open substack of Grd(X/k) isomorphic to the space of refined limit linear series onX . In this way they construct a representable and proper Deligne-Mumford stack Grd over M̃g and hence over M̃g,n by pulling back. 2-B. Tautological Classes. Definition 2.8. Let g and n be as in Theorem 2.2 and let r and d be non-negative integers for which ρ(g, r, d) = 0. Let U ⊂ M̃g,n be the open substack over which η : Grd → M̃g,n is flat. Let [G d ] ∈ A3g−3+n+ρ(G d) be the class of the closure of η−1(U). Similarly, let [Crd] ∈ A3g−2+n+ρ(C d) be the class of the closure of (η ◦ π)−1(U) in the universal curve π : Crd → G In the following we work over M̃g,1 in order to be able to consistently define the universal line and vector bundles. There is a universal pointed quasi-stable curve Yrd → G d whose stabilization is the universal stable curve C d → G d . Let σ : G d → Y be the marked section. There is a universal line bundle L → Yrd of relative degree d together with a trivialization σ∗L ∼= OGr . On each geometric fiber, L has degree 1 on every exceptional curve, degree d on the pre-image of the component of stable curve containing the marked point, and degree 0 on the pre-image of all other components of the stable curve. There is a sub-bundle V →֒ π∗L which, over each point in Grd , is equal to the aspect of the limit linear g d on the component containing the marked point. Remark 2.9. By a theorem of Harer [17], for g ≥ 3, A3g−3(M̃g,1)Q = PicMg,1 ⊗Q = Qλ⊕Qδ0 ⊕Qδ1 ⊕ · · · ⊕Qδg−1 ⊕Qψ where λ and ψ are the first Chern classes of the Hodge and tautological bundles respectively, δ0 is the divisor of irreducible nodal curves, and δi is the divisor of unions of curves of genus i and g− i, where the marked point lies on the component of genus i. Definition 2.10. We define “codimension-1” cycle classes in A3g−3+ρ(G d) as fol- lows. α = π∗ c1(L) 2 ∩ [Yrd ] β = π∗ c1(L) · c1(ω) ∩ [Y γ = c1(V) ∩ [G A PUSH-FORWARD FORMULA WHEN ρ = 0 5 2-C. A Push-Forward Formula. We now state our main result. Theorem 2.11. Let g ≥ 1, and r, d ≥ 0, be integers for which ρ = g − (r + 1)(g − d+ r) = 0, and consider the map η : Grd → M̃g,1 η∗ : A3g−3(G d) → A3g−3(M̃g,1) is the proper push-forward morphism on corresponding Chow groups, then 6(g − 1)(g − 2) η∗α = 6(gd− 2g 2 + 8d− 8g + 4)λ + (2g2 − gd+ 3g − 4d− 2)δ0 (g − i)(gd+ 2ig − 2id− 2d)δi − 6d(g − 2)ψ, 2(g − 1) η∗β = 12λ− δ0 + 4 (g − i)(g − i− 1)δi − 2(g − 1)ψ, 2(g − 1)(g − 2) η∗γ = −(g + 3)ξ + 5r(r + 2) λ− d(r + 1)(g − 2)ψ (g + 1)ξ − 3r(r + 2) (g − i) iξ + (g − i− 2)r(r + 2) where 1! · 2! · 3! · · · r! · g! (g − d+ r)!(g − d+ r + 1)! · · · (g − d+ 2r)! ξ = 3(g − 1) + (r − 1)(g + r + 1)(3g − 2d+ r − 3) g − d+ 2r + 1 3. Applications In this section, we will apply Theorem 2.11 to various classes of divisors on Grd(M g ), where g, r, d are chosen so that ρ(g, r, d) = 0. We can parameterize such choices using integers r, s ≥ 1 and setting g = (r + 1)(s+ 1) and d = r(s+ 2). 3-A. The Gieseker-Petri Divisor. Petri’s theorem [14] states that if C is a gen- eral curve, then for all line bundles L on C, the natural map H0(L)⊗H0(KC ⊗ L ∨) → H0(KC) is injective. This implies that if g, r, and d are chosen so that ρ = 0, and (C,L) is a grd on a general curve C, then the natural map V ⊗H0(KC ⊗ L ∨) → H0(KC) 6 D. KHOSLA is an isomorphism. Away from a subset of codimension greater than 1, the sheaf π∗(ω ⊗ L ∨) on Grd(M g,1) is locally free, and the degeneracy locus of the map of vector bundles V ⊗ π∗(ω ⊗ L ∨) → π∗(ω) defines the Gieseker-Petri divisor in Grd . We compute its class as follows. By definition, c1(π∗(ω)) = λ. By Grothendieck-Riemann-Roch, c1(π∗L)− c1(R 1π∗L) = π∗ ch(L) · tdCr 1 + c1(L) + c1(L) c1(ω) c1(ω) 2 + κ Thus, c1π∗(ω ⊗ L ∨) = −c1(R 1π∗L) = − γ + λ. It follows that our degeneracy locus in Grd(M g,1) has class r + 1 (−α+ β) + (d+ 1− g)γ − rλ. It is easy to see that the slope of the image divisor in Mirrg,1 will be symmetric in r and s. Letting x = (r + 1) + (s + 1) and y = (r + 1)(s + 1) and applying Theorem 2.11, we find that the slope of the Gieseker-Petri divisor in Mirrg is 6(2x+ 7y2 + 7xy + xy2 + 12y + y3) y (4 + y) (y + 1 + x) 3-B. Hypersurface Divisors. Another natural substack in Grd is the locus of g which lie on a hypersurface of degree k; that is, grd’s (L, V ) for which the restriction Symk V → H0(L⊗k) has a non-trivial kernel. If ρ = 0 and the above two vector spaces have the same dimension, then this defines a virtual divisor in Grd . Namely, we look at the degen- eracy locus of the map of vector bundles Symk V → π∗(L Note that if d > g − 1, then Lk is always non-special when k ≥ 2. To compute the class of the degeneracy locus, observe first that c1(Sym k V) = r + k k − 1 A PUSH-FORWARD FORMULA WHEN ρ = 0 7 By Grothendieck-Riemann-Roch, c1(π∗L ⊗k) = π∗ ch(L⊗k) · tdCr 1 + kc1(L) + c1(L) c1(ω) c1(ω) 2 + κ β + λ. Applying Theorem 2.11, the image divisor in Mirrg has slope f(k, r, s) g(k, r, s) where f and g are (rather large) polynomials in k, r, and s, which are, in turn, related by the identity r + k = kr(s + 2)− (r + 1)(s+ 1) + 1. 3-C. Syzygy Divisors. Consider a basepoint-free grd (C,L, V ), so there is a map f : C → PV ∨. On PV ∨, we have the tautological sequence 0 → OPV ∨(−1) → V ∨ ⊗OPV ∨ → Q→ 0. For any i, consider the restriction map to C: H1(∧iQ∨ ⊗OPV ∨(2)) → H 0(∧if∗Q∨ ⊗ L⊗2). According to [10, Proposition 2.5], the map f fails Green’s property (Ni) if and only if this restriction map degenerates to a certain rank. In the case where the two vector spaces have the same dimension, f fails property (Ni) exactly when the restriction map is not an isomorphism. To globalize this, consider the tautological sequence on u : PV∨ → Grd : 0 → OPV∨(−1) → u ∗V∨ → Q→ 0. We will remove from Grd the closed substack, isomorphic to C d−1, of g ds with a basepoint, which has codimension greater than 1. Then there is a morphism f : Yrd → PV ∨ commuting with the projection to Grd . ✲ PV∨ Our restriction map now globalizes to i Q∨ ⊗OPV∨(2) → π∗ ∧ i f∗Q∨ ⊗ L⊗2. Note also that all the higher direct images of these two bundles vanish [10, Propo- sition 2.1]. Using the exact sequence 0 → ∧i+1Q∨ ⊗OPV∨(j) → ∧ i+1u∗V ⊗OPV∨(j) → ∧ iQ∨ ⊗OPV∨(j + 1) → 0 8 D. KHOSLA for j = 0, 1, we have ch(u∗(∧ iQ∨ ⊗OPV∨(2))) = ch(∧ i+1V ⊗ V)− ch∧i+2V r + 1 (r + 1)− r + 1 (r + 1) + r + 1 γ + · · · = (i+ 1) r + 2 + (r + 2) γ + · · · Applying Grothendieck-Riemann-Roch to π, we obtain that chπ∗(∧ if∗Q∨ ⊗ L⊗2) = π∗ f∗ ch∧iQ∨ exp(2c1(L)) · tdYr f∗ ch∧iQ∨ exp(2c1(L)) · tdCr One computes ch∧iQ∨ = r − 1 (u∗γ − ζ) r − 2 ζu∗γ + r − 2 r − 2 ζ2 + · · · , where ζ = c1(OPV∨(1)). It follows that chπ∗(∧ if∗Q∨ ⊗ L⊗2) = (2d+ 1− g) r − 1 r − 1 r − 2 r − 2 r − 1 (2d+ 1− g) r − 1 r − 2 In order for the two vector bundles to have the same dimension, therefore, we need (i+ 1) r + 2 = (2d+ 1− g) r − 1 which is achieved by setting r = (i + 2)s + 2(i + 1). The class of our degeneracy locus in Grd is r − 1 r − 2 r − 2 r − 1 −(r + 2) + (2d+ 1− g) r − 1 r − 2 A PUSH-FORWARD FORMULA WHEN ρ = 0 9 In Section 6 we prove that this locus is an actual effective divisor when i = 0 and 0 ≤ s ≤ 2. By Theorem 2.11 it follows that the slope of the image locus in Mirrg is 6f(i, t) t(i− 2)g(i, t) where f(i, t) = (24i2 + i4 + 16 + 32i+ 8i3)t7 + (4i3 + i4 − 16i− 16)t6 + (−13i2 − 7i3 + 12− i4)t5 + (−i2 − 14i− i4 − 24− 2i3)t4 + (2i2 + 2i3 − 6i− 4)t3 + (17i2 + i3 + 50i+ 41)t2 + (7i2 + 9 + 18i)t+ 2 + 2i, g(i, t) = (12i+ i3 + 8 + 6i2)t6 + (−4i+ i3 − 8 + 2i2)t5 + (−2− 11i− i3 − 7i2)t4 + (−i3 + 5i)t3 + (5i+ 1 + 4i2)t2 + (7i+ 11 + i2)t+ 2 + 4i, and t = s+ 1. 3-D. Secant Plane Divisors. Given a curve C in Pr, we can ask whether there is a k-plane meeting C in e points—that is, an e-secant k-plane. For example, an m-secant 0-plane is just an m-fold point of C. If (C,L, V ) is the associated grd, then this condition is described by saying that there is an effective divisor E on C of degree e such that the restriction map V → H0(LE) has a kernel of dimension at least r − k. To globalize this over Mirrg,1, we consider the relative Hilbert scheme of points on a family of curves. For a stable curve X over S, the functor (Sch/S) → (Sets) T 7→ {subschemes Σ ⊂ XT , finite of degree e over T} is represented by a scheme Hilbe(X/S), proper over S. Now let H d = Hilb e(Crd/G d), and consider the projection p : H d → G Σ ⊂ H d ×Grd C be the universal subscheme. The is a natural map of vector bundles p∗V → (π1)∗π 2L(Σ) d , and we are looking for the rank-(k+1) locus of this map. The class of the virtual degeneracy locus in Grd is, therefore, given by the Porteous formula as (3) p∗∆e−k−1,r−k c((π1)∗π 2L(Σ))/p ∗c(V) In order to get a locus of expected codimension one in Grd , we need that (e − k − 1)(r − k) = e+ 1 or ρ(e, r−k−1, r) = −1. Cotterill [1, Theorem 1] has proved that, in this case, one obtains an actual effective divisor on Grd . The computation (3) can, in principle, be carried out using the techniques in [28]. Cotterill [1] has made some progress 10 D. KHOSLA towards an explicit calculation; there remains, however, some work to be done. The final answer will have the form Pαα+ Pββ + Pγγ + Pλλ+ Pδ0δ0, where the coeffcients are rational functions in Q(r, s, e, k). 4. Special Families of Curves Our strategy for proving Theorem 2.11 will be to pull back to various families of stable curves over which the space of linear series is easier to analyze. In Section 4-A we define three families of pointed curves, and in Section 4-B we compute η∗ for these special families. In Section 4-C we compute the pull-backs of the standard divisor classes on M̃g,1 to the base spaces of each of our families. Assembling the results of these three sections, we compute η∗ over the whole moduli space. 4-A. Definitions of Families. Definition 4.1. Let i : M0,g →֒ M̃g,1 be the family of marked stable curves defined by sending a g-pointed stable curve (C, p1, . . . , pg) of genus 0 to the stable curve Ei, p0 of genus g, where Ei are fixed non-isomorphic elliptic curves, attached to C at the points pi, and p0 ∈ E1 is fixed as well. E1 E2 E3 Eg· · · · · · Figure 1. i(C, p1, . . . , pg) Definition 4.2. Let j : M̃2,1 →֒ M̃g,1 be the family of curves defined by sending a marked curve (C, p) to the marked stable curve (C ∪ C′, p0) where (C′, p′, p0) is a fixed Brill-Noether-general curve in M̃g−2,2, attached nodally to (C, p) at p′. C C′• Figure 2. j(C, p) A PUSH-FORWARD FORMULA WHEN ρ = 0 11 Definition 4.3. Fix Brill-Noether general curves (C1, p1) ∈ Mh,1 (C2, p2) ∈ Mg−h,1 and let C = C1 ∪C2 be their nodal union along the pi. Let kh : C1 →֒ M̃g,1 be the map sending p ∈ C1 to the marked curve (C, p). 4-B. Computations on the Special Families. Lemma 4.4. For the family i : M0,g →֒ M̃g,1 we have η∗α = η∗β = η∗γ = 0 Lemma 4.5. For the family j : M̃2,1 →֒ M̃g,1 we have η∗α = 2dN(d− 2g + 2) 3(g − 1) (3ψ − λ− δ1) + g − 1 (λ+ δ1 − 4ψ) η∗β = g − 1 (λ+ δ1 − 4ψ) η∗γ = 3(g − 1) (3ψ − λ− δ1), where N and ξ are defined in the statement of Theorem 2.11. Lemma 4.6. For the family kh : C1 →֒ M̃g,1 we have deg η∗α = −d deg η∗β = − 2(g − h)− 1 deg η∗γ = − r(r + 1) 4-C. Pull-Back Maps on Divisors. Lemma 4.7. Let ǫi be the class of the closure of the locus on M0,g of stable curves with two components, the component containing the first marked point having i marked points. (a) The classes ǫi are independent in H 2(M0,g;Q). (b) For the family i : M0,g →֒ M̃g,1 12 D. KHOSLA we have the following pull-back map on divisor classes. i∗λ = i∗ψ = i∗δ0 = 0 i∗δi = ǫi for i = 2, 3, . . . , g − 2 i∗δ1 = − (g − i)(g − i− 1) (g − 1)(g − 2) i∗δg−1 = − (g − i)(i− 1) g − 2 Lemma 4.8. For the family j : M̃2,1 →֒ M̃g,1 we have the following pull-back map on divisor classes. j∗λ = λ j∗ψ = 0 j∗δ0 = δ0 j ∗δi = 0 i = 1, 2, . . . , g − 3 j∗δg−2 = −ψ j ∗δg−1 = δ1 Lemma 4.9. For the family kh : C1 →֒ M̃g,1 we have the following pull-back map on divisor classes. deg k∗hλ = 0 deg k hψ = 2h− 1 deg k∗hδh = −1 deg k hδg−h = 1 deg k∗hδi = 0 i 6= h, g − h Proof of Theorem 2.11. Theorem 2.11 is now a consequence of the above lemmas. The main point is that the pull-backs of the classes η∗α, η∗β, and η∗γ to our special families coincide with the classes computed in Section 4-B. For example, to see this for j∗η∗γ, form the fiber the fiber square j∗Grd ✲ Grd M̃2,1 ✲ M̃g,1. Notice that although j is a regular embedding, j′ need not be. Nonetheless, ac- cording to Fulton [12, Chapter 6], there is a refined Gysin homomorphism j! : Ak(G d) → Ak−l(j ∗Grd), where l is the codimension of j, which commutes with push-forward: ! = j∗η∗. We need to check that j!c1(V) ∩ [G d ] = c1(j ′∗V) ∩ [j∗Grd ]. A PUSH-FORWARD FORMULA WHEN ρ = 0 13 Since j!c1(V) ∩ [G d ] = c1(j ′∗V) ∩ j![Grd ] [12, Proposition 6.3], it is enough to check that j![Grd ] = [j ∗Grd ]. Generalizing the dimension upper bound in [27, Corollary 5.9] to the multi-component case [22], we obtain dim j∗Grd = dimM̃2,1. This implies that the codimension of j′ is equal to that of j, so the normal cone of j′ is equal to the pull-back of the normal bundle of j, and the result follows. Now, for example, to compute η∗γ, write η∗γ = aλ− biδi + cψ Our goal is to solve for a, b0, b1, . . . , bg−1, c. Using Lemmas 4.6 and 4.9, we may solve for c and write bg−i in terms of bi. From Lemmas 4.4 and 4.7, we may further solve for b1, b2, . . . , bg−2 in terms of bg−1. It remains to determine a, b0, and bg−1. This is done by pulling back to M̃2,1, which has Picard number 3, and using Lemmas 4.5 and 4.8. The other push-forwards are computed similarly. � 5. Proofs of Lemmas In this section, we give proofs of the lemmas stated in Sections 4-B and 4-C. Proof of Lemma 4.4. If C0,g → M0,g is the universal stable curve, then i ∗C̃g,1 is formed by attaching M0,g ×Ei to C0,g along the marked sections σi : M0,g → C0,g. We have the following fiber square. i∗Crd ✲ i∗C̃g,1 i∗Grd ✲ M0,g By the Plücker formula for P1, given [C] ∈ M0,g, a limit linear series on i(C) must have the aspect (d− r − 1)pi + |(r + 1)pi| on each Ei. The line bundle L → i ∗Crd is, therefore, the pull-back from i ∗C̃g,1 of the bundle which is given by π∗2OE1dp on M0,g ×E1 and is trivial on all other components. Thus α = β = 0. The vector bundle V ⊂ π∗L is trivial with fiber isomorphic to H0(OE1(r + 1)p) ⊂ H 0(OE1dp) so γ = 0 as well. � Before proving Lemma 4.5 we state an elementary result in Schubert calculus. 14 D. KHOSLA Lemma 5.1. [16, p. 266] For integers r and d with 0 ≤ r ≤ d, let X = G(r,Pd) be the Grassmannian of r-planes in Pd. For integers 0 ≤ b0 ≤ b1 ≤ · · · ≤ br ≤ d− r, let σb = σbr ,...,b0 be the corresponding Schubert cycle of codimension bi. Let ζ = σ1,1,...,1,0 be the special Schubert cycle of codimension r. If k is an integer for which bi = dimX = (r + 1)(d− r), then ∫ ζk · σb = i=0(k − d+ r + ai)! 0≤i<j≤r (aj − ai), where ai = bi + i. Proof of Lemma 4.5. Since M2,1 is a smooth Deligne-Mumford stack, it is enough, by the moving lemma, to prove Lemma 4.5 for a family over a complete curve B →֒ M̃2,1 which intersects the boundary and Weierstrass divisors transversally. If π : C → B is the universal stable genus-2 curve and σ : B → C is the marked section, then j∗C̃g,1 is formed by attaching C to B×C ′ along the marked section Σ = σ(B) ⊂ C. We begin by assuming that B is disjoint from the closure of the Weierstrass locus W . In this case we claim that j∗Grd → B is a trivial N -sheeted cover of the form B × X , where is X a zero-dimensional scheme of length N . Indeed, for any curve (C, p) in M̃2,1 \W there are two (limit linear) grds on C with maximum ramification at p; the vanishing sequences are a1 = (d− r − 2, d− r − 1, . . . , d− 4, d− 3, d), a2 = (d− r − 2, d− r − 1, . . . , d− 4, d− 2, d− 1). If C is smooth, the two linear series are (d− r − 2)p+ |(r + 2)p| (d− r − 2)p+ |rp+KC |. There are analogous series on nodal curves outside the closure of the Weierstrass locus. In the case of irreducible nodal curves, the sheaves are locally free. For each of the two grds on C with maximum ramification at p, there are finitely many grds on C ′ with compatible ramification. Specifically, there are (2g − 2− d)N 2(g − 1) of type a1 and 2(g − 1) of type a2, for total of N limit linear series counted with multiplicity. Since C fixed, the cover j∗Grd → B is a trivial N -sheeted cover. A PUSH-FORWARD FORMULA WHEN ρ = 0 15 Consider a reduced sheet B1 ≃ B of type a1. (We assume for simplicity that the sheet is reduced—the computation is the same in the general case.) Then the universal line bundle L on j∗Crd is given as OC on C π∗2L1 on B1 × C for some line bundle L1 on C ′ of degree d. It follows that α = β = γ = 0 on B1. Next consider a sheet B2 ≃ B of type a2. Over B2×C ′ the universal line bundle L is isomorphic to π∗2L2 for some L2 of degree d on C ′. It remains to determine L over C. Now ωC(−2p) gives the correct line bundle for all [C] ∈ B2; however, it has the wrong degrees on the components of the singular fibers. As our first approximation to L on C we take ωC/B2(−2Σ) Let ∆ ⊂ C be the pull-back of the divisor on C of curves of the form C1 ∪C2, where the Ci have genus one, and the marked points lie on different components. Then ωC/B2(−2Σ +∆) has the correct degree on the irreducible components on each fiber. It remains only to normalize our line bundle by pull-backs from the base B2. In this case, L|C is required to be trivial along Σ since L is a pull-back from C ′ on the other component. If σ : B2 → C is the marked section, we let Ψ = σ∗ωC/B2 be the tautological line bundle on B2. Then σ∗OC∆ ∼= OB2 σ∗OCΣ ∼= Ψ It follows that on C, ωC/B2(−2Σ +∆)⊗ π ∗Ψ⊗−3 on C π∗2L2 on B2 × C Thus, if we let ω = c1(ωC/B2) σ = c1(OCΣ) δ = c1(OC∆) on C and let ψ = c1(Ψ) on B2, then c1(L) = ω − 2σ + δ − 3π∗ψ on C dπ∗2p on B2 × C For the relative dualizing sheaf ω j∗ eCg,1/B2 , we have c1(ωj∗C/B2) = ω + σ on C (2(g − 2)− 1)π∗2p on B2 × C 16 D. KHOSLA Figure 3. The morphism C → B2. To compute the products of these classes on j∗Crd , recall the following formulas on π : C2,1 → M2,1 π∗ω = 2 π∗δ = 0 π∗σ = 1 2 = 12λ− δ0 − δ1 π∗σ 2 = −ψ π∗δ 2 = −δ1 π∗(δ.σ) = 0 π∗(ω.δ) = δ1 π∗(σ.ω) = ψ. Then we compute α = π∗ c1(L) = 12λ− δ0 − 8ψ β = π∗ c1(L) · c1(ω) = 12λ− δ0 − 8ψ on B2. Since the marked point lies on C ′, V is trivial on B2, so γ = 0 on B2. Finally we consider the case where B (transversally) intersects the Weierstrass locus. In this case η : j∗Grd → B is the union of a trivial N -sheeted cover of B and a 1-dimensional scheme lying over each point of the divisor W . It will suffice to compute α and γ on Grd(j[C, p]) where C is a smooth genus-2 curve, and p ∈ C is a Weierstrass point. (Note that β is automatically zero.) There is a single grd on C with maximal ramification at p, namely (d− r − 2)p+ |(r + 2)p|, which has vanishing sequence (d− r − 2, d− r − 1, . . . , d− 4, d− 2, d). We claim that we only need to consider components of Grd(C ∪C ′) with this aspect on C. Indeed, any grd on C with ramification 1 less at p is still a subseries of |dp|. There will be finitely many corresponding aspects on C′ so that, as before, α = β = γ = 0 on these components. A PUSH-FORWARD FORMULA WHEN ρ = 0 17 It remains to consider the components of Grd(C ∪ C ′) where the aspect on C′ has ramification (0, 1, 2, 2, . . . , 2) or more at p′. We are reduced to studying the one-dimensional scheme S = Grd(C ′; p′, (0, 1, 2, 2, . . . , 2)). To simplify computations we specialize C′ to a curve which is the union of P1 with g− 2 elliptic curves E1, . . . , Eg−2 attached at general points p1, . . . , pg−2, and where the marked point p0 lies on E1, and the point of attachment p ′ lies on the P1. There will be two types of components of S: those on which the aspects on the E1 E2 Eg−2 · · · · · · Figure 4. The curve C′. Ei are maximally ramified at pi, and those on which the aspect on one Ei varies. Again, as in the proof of Lemma 4.4, we need only consider the latter case. Assume that for some i, the ramification at pi of the g d on Ei is one less than maximal. There are two possibilities: either the series is of the form (d− r − 1)pi + |rpi + q| for q ∈ Ei, which for q 6= pi imposes on the P 1 the ramification condition (1, 1, . . . , 1), or the grd is a subseries of (d− r − 2)pi + |(r + 2)pi| containing (d− r)pi + |rpi|, which generically imposes on the P1 the ramification condition (0, 1, 1, . . . , 1, 2). In the first case the components are parameterized by Ei, and we compute that α = −2 on each such irreducible component, irrespective of whether i = 1 or not. By Grothendieck-Riemann-Roch, γ = −1 when i = 1 and is zero otherwise. In the second case the grds are parameterized by a P 1. Because the line bundle is constant, α = 0. On each such P1, the vector bundle V may be viewed as the tautological bundle of rank r+1 on the Grassmannian of vector subspaces of a fixed vector space of dimension r+2 containing a subspace of dimension r. It follows that γ = −1 on each P1. Let X = G(r,Pd) be the Grassmannian of r-planes in Pd. Let ζ = σ1,1,...,1,0 18 D. KHOSLA be the special Schubert cycle of codimension r. Collecting our calculations, we have that on Grd(C ∪ C α = −2(g − 2) σ2,2,...,2,1,0 · σ1,1,...,1 · ζ = −2(g − 2) σ3,3,...,3,2,1 · ζ γ = − σ2,2,...,2,1,0 · (σ1,1,...,1 + σ2,1,1,...,1,0) · ζ σ2,2,...,2,1,0 · (σ1,0,0,...,0 · ζ) · ζ (σ3,2,2,...,2,1,0 + σ2,2,...,2,0 + σ2,2,...,2,1,1) · ζ (σ3,2,2,...,2,1,0 + ζ 2) · ζg−2 σ3,2,2,...,2,1,0 · ζ g−2 − From Lemma 5.1 we compute, −2d(2g − 2− d)N 3(g − 1) 3(g − 1) Since the class of the Weierstrass locus in M̃2,1 is 3ψ−λ−δ1, the lemma follows. � Proof of Lemma 4.6. Because the curves (Ci, pi) are Brill-Noether general, k a trivial N -sheeted cover of C1 of the form C1 ×X , where X is a zero-dimensional scheme of length N . Fix a sheet G ∼= C1 in k d ; this choice corresponds to aspects Vi ⊂ H 0(Ci, Li) where Li are degree-d line bundles on Ci. If (a0, a1, . . . , ar) is the vanishing sequence of V1 at p1, then we know that 0 = ρ(h, r, d)− (ai − i) = (r + 1)(d− r)− hr − r(r + 1), ai = (r + 1)d− r(r + 1)− hr. Let C1 be the blow-up of C1 × C1 at (p1, p1), E the exceptional divisor, and e its first Chern class. We may construct the universal curve k∗hC̃g,1 → C1 by attaching C1 × C2 to C1 along C1 × {p2} and the proper transform of C1 × {p1}. Over the sheet G, the universal line bundle L on k∗hC π∗2L1 ⊗OC1−dE ⊗ π 1 (dp1) A PUSH-FORWARD FORMULA WHEN ρ = 0 19 on C1 and π∗2L2(−dp2)⊗ π on C1 × C2. Thus c1(L) = dπ∗2p− de on C1 −dπ∗1p on C1 × C2. The relative dualizing sheaf ω eCg,1/C1 is isomorphic to π∗2ωC1 ⊗OCE ⊗ π 1OC1−p1 on C1 and π∗2ωC2(p2) on C1 × C2. We have c1(ω) = −π∗1p+ (2h− 2)π 2p+ e on C1 (2(g − h)− 1)π∗2p on C1 × C2. Thus, on G, degα = c1(L) 2 = −d2 deg β = c1(L) · c1(ω) = −d 2(g − j)− 1 The formulas for η∗α and η∗β now follow. To calculate γ on G, notice that it suffices to compute c1(V ′), where V ′ = V ⊗ L1(−dp1) is a sub-bundle of π1∗(π 2L1(−dE)). We claim there is a bundle isomorphism OC1(aj − d)p1 −→ V ′ To describe the map, pick a basis (σ0, σ1, . . . , σr) of V1 ⊂ H 0(L1) with σi vanish- ing to order ai at p1. Given local sections τi of OC1(ai − d)p1, let the image of (τ0, τ1, . . . , τr) be the section of V ′. This is clearly an isomorphism away from p1 and is checked to be an isomor- phism over p1 as well. Using (4), we have that on G, deg γ = deg V ′ = (ai − d) r(r + 1)− rh which finishes the proof of the lemma. � 20 D. KHOSLA Proof of Lemma 4.7. To prove the independence of the ǫi, consider the curves Bj →֒ M0,g for j = 2, 3, . . . , g−3 given by taking a fixed stable curve in ǫj and moving a marked point on the component with g − j marked points. Let B1 →֒ M0,g be the curve given by moving the first marked point along a fixed smooth curve.The intersection matrix (ǫi · Bj)  g − 1 0 0 0 · · · 0 0 0 −1 1 0 0 · · · 0 0 g − 3 0 −1 1 0 · · · 0 0 g − 4 . . . 0 0 0 0 · · · −1 1 3 0 0 0 0 · · · 0 −1 2  where the rows correspond to the Bj for j = 1, 2, . . . , g − 3, and the columns to the ǫi for i = 2, 3, . . . , g − 2. Since this matrix is non-singular, the first part of the lemma follows. To derive the formula for the pull-back, we follow Harris and Morrison [19, Section 6.F]. Let B be a smooth projective curve, π : C → B a 1-parameter family of curves in M0,g transverse to the boundary strata. Then π has smooth total space, and the fibers of π have at most two irreducible components. Let σi : B → C be the marked sections. Denote by Σi the image curve σi(B) in C. Then on B, δ1 = Σ δg−1 = Σ2j , where we are using D2 to denote π∗(D 2) for a divisor D on C. We now contract the component of each reducible fiber which meets the section Σ1. If Σj is the image of Σj under this contraction, then we have Σ21 = Σ Σ2j = (i − 1)ǫi The Σj are sections of a P 1-bundle, so 0 = (Σj − Σk) 2 = Σ k − 2Σj · Σk A PUSH-FORWARD FORMULA WHEN ρ = 0 21 (g − 2) 2≤j,k≤g k) = 2 2≤j,k≤g Σj · Σk It follows that δg−1 = (i − 1)(i− 2) g − 2 − (i− 1) (i− 1)(i− g) g − 2 Similarly, we can show that δ1 + δg−1 = i(i− g) g − 1 so the formula for δ1 follows as well. � The proofs of Lemmas 4.8 and 4.9 are straightforward, so we omit them. 6. Appendix In this section, we prove that the locus in Section 3-C defined for i = 0 and s = 2 is, in fact, a divisor in G624(M21). The case s = 1 is similar, and s = 0 is clear. We first establish the following result. Lemma 6.1. The space G624(M21) is irreducible. Proof. Note that a g624 is residual to a g 16; that is L ∈W 624(C) ⇐⇒ KC ⊗ L ∗ ∈W 216(C) for any smooth curve C of genus 21. Thus there is a dominant rational map V16,21 −→ G from the Severi variety of irreducible plane curves of degree 16 and genus 21. Since V16,21 is irreducible [18] and maps dominantly to G 24, so G 24 is irreducible. � Proposition 6.2. The substack Ẽ of G624(M21) defined in Section 3-C has codi- mension 1. Proof. Since G624(M21) is irreducible, it suffices to exhibit a smooth curve with a 24 not lying on a quadric. Let S = Bl21 P be the blow-up of P2 at 21 general points, and consider the linear system ∣∣∣13H − 2 Ej − 3 22 D. KHOSLA on S, where H is the hyperplane class and Ei are the exceptional divisors. A calculation using Macaulay 2 (see Proposition 6.3) shows that a general member C of ν is irreducible and smooth of genus 21. The series ∣∣∣6H − embeds S in P6 as the rank-2 locus of general 3×6 matrix of linear forms [4, Section 20.4]. The ideal of S is therefore generated by cubics, so S does not lie on quadric. It follows that C, which embeds in P6 in degree 24, does not lie on a quadric. Proposition 6.3. Let Σ be a set of 21 general points in P2 and let S = BlΣP be the blow-up of P2 at Σ. If H is the line class on S and E1, . . . , E21 are the exceptional divisors, then the linear system ∣∣∣13H − 2 Ej − 3 on S contains a smooth connected curve. Proof. We begin by showing that it is enough to exhibit a single set of 21 points over a finite field for which the above statement is true. Let Hilb 2 be the Hilbert scheme of k points in P2, and let Σk ⊂ Hilb 2 ×P2 be the universal subscheme. Let B ⊂ Hilb9Z P 2 ×Hilb12Z P be the irreducible open subset over which the composition Σ = π−11 Σ9 ∪ π 2 Σ12 ✲ Hilb9ZP 2 ×Hilb12Z P 2 ×P2 Hilb9ZP 2 ×Hilb12Z P is étale, where π1 and π2 are the obvious projections. Let π : S = BlΣ P B → B be the smooth surface over B whose fibers are blow-ups of P2 at 21 distinct points. If E9 and E12 are the exceptional divisors, let L = OS13H − 2E9 − 3E12. We may further restrict B to an open over which π∗L is locally free of rank at least C ⊂ Pπ∗L ×B S is the universal section, then the projection C → Pπ∗L is flat, so it suffices to find a single smooth fiber in order to conclude that the general fiber is smooth. To this end we use Macaulay 2 [15] and work over a finite field. A PUSH-FORWARD FORMULA WHEN ρ = 0 23 i1 : S = ZZ/137[x,y,z]; Following Shreyer and Tonoli [29], we realize our points in P2 as a determinental subscheme. i2 : randomPlanePoints = (delta,R) -> ( k:=ceiling((-3+sqrt(9.0+8*delta))/2); eps:=delta-binomial(k+1,2); if k-2*eps>=0 then minors(k-eps, random(R^(k+1-eps),R^{k-2*eps:-1,eps:-2})) else minors(eps, random(R^{k+1-eps:0,2*eps-k:-1},R^{eps:-2}))); i3 : distinctPoints = (J) -> ( singJ = minors(2, jacobian J) + J; codim singJ == 3); Let Σ9 and Σ12 be our subsets of 9 and 12 points, respectively. i4 : Sigma9 = randomPlanePoints(9,S); o4 : Ideal of S i5 : Sigma12 = randomPlanePoints(12,S); o5 : Ideal of S i6 : (distinctPoints Sigma9, distinctPoints Sigma12) o6 = (true, true) o6 : Sequence Their union is Σ. i7 : Sigma = intersect(Sigma9, Sigma12); o7 : Ideal of S i8 : degree Sigma o8 = 21 24 D. KHOSLA Next we construct the 0-dimensional subscheme Γ whose ideal consists of curves double through points of Σ9 and triple through points of Σ12. i9 : Gamma = saturate intersect(Sigma9^2, Sigma12^3); o9 : Ideal of S Let us check that Γ imposes the expected number of conditions (9 · 3+ 12 · 6 = 99) on curves of degree 13. i10 : hilbertFunction (13, Gamma) o10 = 99 Pick a random curve C of degree 13 in the ideal of Γ. i11 : C = ideal (gens Gamma * random(source gens Gamma, S^{-13})); o11 : Ideal of S We check that C is irreducible. i12 : # decompose C o12 = 1 To check smoothness, let Csing be the singular locus of C. i13 : Csing = (ideal jacobian C) + C; o13 : Ideal of S i14 : codim Csing o14 = 2 A double point will contribute 1 to the degree of Csing if it is transverse and more otherwise. Similarly, a triple point will contribute 4 to the degree of Csing if it is transverse and more otherwise. So for C to be smooth in the blow-up, we must have that degCsing = 9 + 4 · 12 = 57 A PUSH-FORWARD FORMULA WHEN ρ = 0 25 i15 : degree Csing o15 = 57 Definition 6.4. Let E be the effective codimension-1 Chow cycle which is the image of Ẽ under the map η : G624 → M Proposition 6.5. The class of E ⊂ Mirr21 is given as [E] = 2459λ− 377δ0. Proof. Applying Equation (2) from Section 3-C, [E] = η∗[Ẽ] = η∗(2α− β + λ− 8γ) 2459N where N is the degree of η. � Corollary 6.6. The slope conjecture is false in genus 21. Proof. Since < 6 + this is an immediate consequence of [11, Corollary 1.2]. � References [1] Ethan Cotterill, Enumerative geometry of curves with exceptional secant planes, Ph.D. thesis, Harvard University, 2007. [2] Fernando Cukierman and Douglas Ulmer, Curves of genus ten on K3 surfaces, Compositio Math. 89 (1993), no. 1, 81–90. [3] Pierre Deligne and David Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109. [4] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer- Verlag, New York, 1995, With a view toward algebraic geometry. [5] David Eisenbud and Joseph Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371–418. [6] David Eisenbud and Joseph Harris, On the Brill-Noether theorem, Algebraic geometry—open problems (Ravello, 1982), Lecture Notes in Math., vol. 997, Springer, Berlin, 1983, pp. 131– [7] David Eisenbud and Joseph Harris, A simpler proof of the Gieseker-Petri theorem on special divisors, Invent. Math. 74 (1983), no. 2, 269–280. [8] , The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), no. 2, 359–387. [9] Gavril Farkas, Koszul divisors on moduli spaces of curves, arXiv:math.AG/0607475. [10] , Syzygies of curves and the effective cone of Mg , to appear in Duke Mathematical Journal, arXiv:math.AG/0503498. [11] Gavril Farkas and Mihnea Popa, Effective divisors on Mg, curves on K3 surfaces and the Slope Conjecture, J. Algebraic Geom. 14 (2005), no. 2, 241–267, arXiv:math.AG/0305112. [12] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. 26 D. KHOSLA [13] William Fulton and Robert Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), no. 3-4, 271–283. [14] David Gieseker, Stable curves and special divisors: Petri’s conjecture, Invent. Math. 66 (1982), no. 2, 251–275. [15] Daniel R. Grayson and Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/. [16] Phillip Griffiths and Joseph Harris, On the variety of special linear systems on a general algebraic curve, Duke Math. J. 47 (1980), no. 1, 233–272. [17] John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221–239. [18] Joseph Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461. [19] Joseph Harris and Ian Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), no. 2, 321–355. [20] Joseph Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88, with an appendix by William Fulton. [21] Deepak Khosla, Moduli spaces of curves with linear series and the slope conjecture, Ph.D. thesis, Harvard University, 2005. [22] Deepak Khosla and Brian Osserman, A limit linear series moduli stack, in preparation. [23] Steven L. Kleiman and Dan Laksov, On the existence of special divisors, Amer. J. Math. 94 (1972), 431–436. [24] Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks Mg,n, Math. Scand. 52 (1983), no. 2, 161–199. [25] Robert Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307. [26] Brian Osserman, A limit linear series moduli scheme, to appear in Annales de l’Institut Fourier, arXiv:math.AG/0407496. [27] , Linked grassmannians and crude limit linear series, unpublished. [28] Ziv Ran, Cycle map on Hilbert schemes of nodal curves, Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, pp. 361–378. [29] Frank-Olaf Schreyer and Fabio Tonoli, Needles in a haystack: special varieties via small fields, Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., vol. 8, Springer, Berlin, 2002, pp. 251–279. Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712, USA E-mail address: [email protected] http://www.math.uiuc.edu/Macaulay2/ 1. Introduction 2. Statement of Theorem 2-A. A limit linear series moduli stack 2-B. Tautological Classes 2-C. A Push-Forward Formula 3. Applications 3-A. The Gieseker-Petri Divisor 3-B. Hypersurface Divisors 3-C. Syzygy Divisors 3-D. Secant Plane Divisors 4. Special Families of Curves 4-A. Definitions of Families 4-B. Computations on the Special Families 4-C. Pull-Back Maps on Divisors 5. Proofs of Lemmas 6. Appendix References

0704.1341

Equivariant symmetric bilinear torsions

Equivariant symmetric bilinear torsions ∗ Guangxiang Su † Abstract We extend the main result in the previous paper of Zhang and the au- thor relating the Milnor-Turaev torsion with the complex valued analytic torsion to the equivariant case. 1 Introduction Let F be a unitary flat vector bundle on a closed Riemannian manifold X. In [RS], Ray and Singer defined an analytic torsion associated to (X,F ) and proved that it does not depend on the Riemannian metric on X. Moreover, they conjectured that this analytic torsion coincides with the classical Reide- meister torsion defined using a triangulation on X (cf. [Mi]). This conjecture was later proved in the celebrated papers of Cheeger [C] and Müller [Mu1]. Müller generalized this result in [Mu2] to the case where F is a unimodular flat vector bundle on X. In [BZ1], inspired by the considerations of Quillen [Q], Bismut and Zhang reformulated the above Cheeger-Müller theorem as an equality between the Reidemeister and Ray-Singer metrics defined on the de- terminant of cohomology, and proved an extension of it to the case of general flat vector bundles over X. The method used in [BZ1] is different from those of Cheeger and Müller in that it makes use of a deformation by Morse functions introduced by Witten [W] on the de Rham complex. On the other hand, Turaev generalizes the concept of Reidemeister torsion to a complex valued invariant whose absolute value provides the original Reide- meister torsion, with the help of the so-called Euler structure (cf. [T], [FT]). It is natural to ask whether there exists an analytic interpretation of this Turaev torsion. Recently, Burghelea and Haller [BH1, BH2], following a suggestion of Müller, define a generalized analytic torsion associated to a nondegenerate symmetric bilinear form on a flat vector bundle over a closed manifold and make an explicit conjecture between this generalized analytic torsion and the Turaev torsion. Later this conjecture was proved by Su and Zhang [SZ]. Also Burghelea and Haller [BH3], up to a sign, proved this conjecture for odd dimensional manifolds, and comments were made how to derive the conjecture in full generality in their paper. ∗This work was partially supported by the Qiushi Foundation. †Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China. ([email protected]) http://arxiv.org/abs/0704.1341v4 In this paper, we will extend the main result in [SZ] to the equivariant case, which is closer in spirit to the approach developed by Bismut-Zhang in [BZ2]. The rest of this paper is organized as follows. In Section 2, we construct the equivariant symmetric bilinear torsions associated with equivariant nondegen- erate symmetric bilinear forms on a flat vector bundle. In Section 3, we state the main result of this paper. In Section 4, we provide a proof of the main result. Section 5 is devoted to the proofs of the intermediary results stated in Section 4. Since we will make substantial use of the results in [BZ1, BZ2, SZ], we will refer to [BZ1, BZ2, SZ] for related definitions and notations directly when there will be no confusion. 2 Equivariant symmetric bilinear torsions associated to the de Rham and Thom-Smale complexes In this section, for a G-invariant nondegenerate bilinear symmetric form on a complex flat vector bundle over an oriented closed manifold, we define two naturally associated equivariant symmetric bilinear forms on the equivariant determinant of the cohomology H∗(M,F ) with coefficient F . One constructed in a combinatorial way through the equivariant Thom-Smale complex associated to a equivariant Morse function, and the other one constructed in an analytic way through the equivariant de Rham complex. 2.1 Equivariant symmetric bilinear torsion of a finite dimensional complex Let (C, ∂) be a finite cochain complex (C, ∂) : 0 −→ C0 ∂0−→ C1 ∂1−→ · · · ∂n−1−→ Cn −→ 0,(2.1) where each Ci, 0 ≤ i ≤ n, is a finite dimensional complex vector space. Let each Ci, 0 ≤ i ≤ n, admit a nondegenerate symmetric bilinear form bi. We equip C with the nondegenerate symmetric bilinear form bC = i=0 bi. Let G be a compact group. Let ρ : G → End(C) be a representation of G, with values in the chain homomorphisms of C which preserve the bilinear form bC . In particular, if g ∈ G, ρ(g) preserves the Ci’s. Let Ĝ be the set of equivalence classes of complex irreducible representations of G. An element of Ĝ is specified by a complex finite dimensional vector space W together with an irreducible representation ρW : G→ End(W ). For W ∈ Ĝ, set CiW = HomG(W,C i)⊗W,(2.2) CW = HomG(W,C)⊗W.(2.3) Let ∂W be the map induced by ∂ on CW . Then (CW , ∂W ) : 0 −→ C0W ∂0,W−→ C1W ∂1,W−→ · · · ∂n−1,W−→ CnW −→ 0(2.4) is a chain complex. Thus we obtain the isotypical decomposition, (C, ∂) = W∈ bG (CW , ∂W ),(2.5) and the decomposition (2.5) is orthogonal. If E is a complex finite dimensional representation space for G, let χ(E) be the character of the representation. Put χ(C) = (−1)iχ(Ci), e(C) = (−1)idimCi, e(CW ) = (−1)idim(CiW ).(2.6) By (2.5), we get χ(C) = W∈ bG e(CW ) χ(W ) rk(W ) .(2.7) If λ is a complex line, let λ−1 be the dual line. If E is a finite dimensional complex vector space, set detE = Λmax(E).(2.8) detC = detCi )(−1)i detCW = detCiW )(−1)i .(2.9) By (2.5), we obtain detC = W∈ bG detCW .(2.10) For 0 ≤ i ≤ n, CiW is a vector subspace of Ci. Let bCiW be the induced symmetric bilinear form on CiW . let bdetCi be the symmetric bilinear form on detCiW induced by bCi , and let b(detCi )−1 be the dual symmetric bilinear form on (detCiW ) −1. Also we have symmetric bilinear forms bdetCW on detCW and bdetC on detC. det(C,G) = W∈ bG detCW .(2.11) Definition 2.1. We introduce the formal product bdet(C,G) = W∈ bG (bdetCW ) χ(W ) rk(W ) .(2.12) For W ∈ Ĝ, let xW , yW ∈ detCW , xW 6= 0, yW 6= 0. Set x = ⊕W∈ bGxW , y = ⊕ W∈ bGyW ∈ det(C,G). Then by definition, bdet(C,G)(x, y) = W∈ bG (bdetCW (xW , yW )) χ(W ) rk(W ) .(2.13) Tautologically, (2.13) is an identity of characters on G. In particular bdet(C,G)(x, y)(1) = W∈ bG bdetCW (xW , yW ).(2.14) In fact (2.14) just says that bdet(C,G)(1) = bdetC .(2.15) Of course, using the orthogonality of the χW ’s, knowing the formal product bdet(C,G) is equivalent to knowing the symmetric bilinear forms bdetCW . Clearly H(CW , ∂W ) = HomG(W,H(C, ∂)) ⊗W, H(C, ∂) = W∈ bG H(CW , ∂W ).(2.16) For W ∈ Ĝ, we define detH(CW , ∂W ) as in (2.9). Set det(H(C, ∂), G) = W∈ bG detH(CW , ∂W ).(2.17) For W ∈ Ĝ, there is a canonical isomorphism (cf. [KM] and [BGS, Section 1a)]) detCW ≃ detH(CW , ∂W ).(2.18) From (2.18), we get det(C,G) ≃ det(H(C, ∂), G).(2.19) Let bdetH(CW ,∂W ) be the symmetric bilinear form on detH(CW , ∂W ) corre- sponding to bdetCW via the canonical isomorphism (2.18). Definition 2.2. we introduce the formal product bdet(H(C,∂),G) = W∈ bG bdetH(CW ,∂W ) )χ(W ) rkW .(2.20) Tautologically, under the identification (2.19), bdet(C,G) = bdetH((C,∂),G).(2.21) By an abuse of notation, we will call the formal product bdet(C,G) a sym- metric bilinear form on det(C,G). 2.2 The Thom-Smale complex of a gradient field Let M be a closed smooth manifold, with dimM = n. For simplicity, we make the assumption that M is oriented. Let (F,∇F ) be a complex flat vector bundle overM carrying the flat connec- tion ∇F . We make the assumption that F carries a nondegenerate symmetric bilinear form bF . Let (F ∗,∇F ∗) be the dual complex flat vector bundle of (F,∇F ) carrying the dual flat connection ∇F ∗. Let f : M → R be a Morse function. Let gTM be a Riemannian metric on TM such that the corresponding gradient vector field −X = −∇f ∈ Γ(TM) satisfies the Smale transversality conditions (cf. [Sm]), that is, the unstable cells (of −X) intersect transversally with the stable cells. B = {x ∈M ;X(x) = 0}.(2.22) For any x ∈ B, let W u(x) (resp. W s(x)) denote the unstable (resp. stable) cell at x, with respect to −X. We also choose an orientation O−x (resp. O+x ) on W u(x) (resp. W s(x)). Let x, y ∈ B satisfy the Morse index relation ind(y) = ind(x) − 1, then Γ(x, y) =W u(x)∩W s(y) consists of a finite number of integral curves γ of −X. Moreover, for each γ ∈ Γ(x, y), by using the orientations chosen above, on can define a number nγ(x, y) = ±1 as in [BZ1, (1.28)]. If x ∈ B, let [W u(x)] be the complex line generated by W u(x). Set u, F ∗) = [W u(x)]⊗ F ∗x ,(2.23) u, F ∗) = x∈B, ind(x)=i [W u(x)]⊗ F ∗x .(2.24) If x ∈ B, the flat vector bundle F ∗ is canonically trivialized on W u(x). In particular, if x, y ∈ B satisfy ind(y) = ind(x) − 1, and if γ ∈ Γ(x, y), f∗ ∈ F ∗x , let τγ(f ∗) be the parallel transport of f∗ ∈ F ∗x into F ∗y along γ with respect to the flat connection ∇F ∗. Clearly, for any x ∈ B, there is only a finite number of y ∈ B, satisfying together that ind(y) = ind(x)− 1 and Γ(x, y) 6= ∅. If x ∈ B, f∗ ∈ F ∗x , set ∂(W u(x)⊗ f∗) = y∈B, ind(y)=ind(x)−1 γ∈Γ(x,y) nγ(x, y)W u(y)⊗ τγ(f∗).(2.25) Then ∂ maps Ci(W u, F ∗) into Ci−1(W u, F ∗). Moreover, one has ∂2 = 0.(2.26) That is, (C∗(W u, F ∗), ∂) forms a chain complex. We call it the Thom-Smale complex associated to (M,F,−X). If x ∈ B, let [W u(x)]∗ be the dual line to W u(x). Let (C∗(W u, F ), ∂) be the complex which is dual to (C∗(W u, F ∗), ∂). For 0 ≤ i ≤ n, one has Ci(W u, F ) = x∈B, ind(x)=i [W u(x)]∗ ⊗ Fx.(2.27) Let G be a compact group acting on M by smooth diffeomorphisms. we assume that the action of G lifts to F and preserves the flat connection of F . Then G acts naturally on H∗(M,F ). We assume that f and gTM are G- invariant. Then −X = −∇f is also G-invariant. We assume that it verifies the smale transversality conditions. Clearly B is G-invariant. Also if x ∈ B, g ∈ G, g (W u(x)) = ǫg(x)W u(gx), where ǫg(x) = +1 if g(W u(x)) has the same orientation as W u(gx), ǫg = −1 if not. Clearly g acts as a chain homomorphism on (C∗(W u, F ∗), ∂). The corresponding dual action of g on (C∗(W u, F ), ∂) is such that g (W u(x)∗) = ǫg(x)W u(gx)∗. Then g acts as a chain homomorphism on (C∗(W u, F ), ∂). Therefore g acts on H∗(C∗(W u, F ), ∂). 2.3 Equivariant Milnor symmetric bilinear torsion For x ∈ B, let bFx be a nondegenerate symmetric bilinear form on Fx. We assume that the bFx’s are G-invariant, i.e. for g ∈ G, x ∈ B = bFg(x).(2.28) The symmetric bilinear forms bFx ’s determine a G-invariant symmetric bilin- ear form on C∗(W u, F ) = x∈B [W u(x)]∗⊗Fx, such that the various [W u(x)]∗⊗ Fx are mutually orthogonal in C ∗(W u, F ), and that if x ∈ B, f, f ′ ∈ Fx, W u(x)∗ ⊗ f,W u(x)∗ ⊗ f ′ f, f ′ .(2.29) We construct the equivariant symmetric bilinear form bdet(C∗(Wu,F ),G) on det(C∗(W u, F ), G) as in Definition 2.1. Definition 2.3. The symmetric bilinear form on the determinant line of the cohomology of the Thom-Smale cochain complex (C∗(W u, F ), ∂), in the sence of Definition 2.2, is called the equivariant Milnor symmetric bilinear torsion and is denoted by b det(H∗(Wu,F ),G). Take g ∈ G. Set Mg = {x ∈M, gx = x}.(2.30) Since G is a compact group, Mg is a smooth compact submanifold of M . Let N be the normal bundle to Mg in M . By [BZ2, Proposition 1.13], we know that f |Mg is a Morse function on Mg, and X|Mg is a smooth section of TMg. For g ∈ G, set Bg = B ∩Mg.(2.31) Then Bg is the set of critical points of f |Mg . Definition 2.4. If x ∈ Bg, let indg(x) be the index of f |Mg at x. Let now bFx , b′Fx (x ∈ B) be two G-invariant nondegenerate symmetric bilinear forms on Fx. Let b det(H∗(Wu,F ),G), b ′M,−X det(H∗(Wu,F ),G) be the corresponding equivariant Milnor symmetric bilinear torsions. By [BZ2, Theorem 1.15] and [SZ, Proposition 2.5], we have the following theorem. Theorem 2.5. For g ∈ G, the following identity holds ′M,−X det(H∗(Wu,F ),G)(g) = b det(H∗(Wu,F ),G)(g) g log )])(−1)indg(x) (2.32) 2.4 Equivariant Ray-Singer symmetric bilinear torsion We continue the discussion of the previous subsection. However, we do not use the Morse function and make transversality assumptions. For any 0 ≤ i ≤ n, denote Ωi(M,F ) = Γ Λi(T ∗M)⊗ F , Ω∗(M,F ) = Ωi(M,F ).(2.33) Let dF denote the natural exterior differential on Ω∗(M,F ) induced from ∇F which maps each Ωi(M,F ), 0 ≤ i ≤ n, into Ωi+1(M,F ). The group G acts naturally on Ω∗(M,F ). Namely, if g ∈ G, s ∈ Ω∗(M,F ), gs(x) = g∗s(g −1x), x ∈M. Let gF be a G-invariant Hermitian metric on F . The G-invariant Rieman- nian metric gTM and gF determine a natural inner product 〈 , 〉g (that is, a pre-Hilbert space structure) on Ω∗(M,F ) (cf. [BZ1, (2.2)] and [BZ2, (2.3)]). Let dF∗g be the formal adjoint of d F with respect to 〈 , 〉g and Dg = dF +dF∗g . On the other hand gTM and the G-invariant symmetric bilinear form bF determine together a G-invariant symmetric bilinear form on Ω∗(M,F ) such that if u = αf , v = βg ∈ Ω∗(M,F ) such that α, β ∈ Ω∗(M), f, g ∈ Γ(F ), then 〈u, v〉b = (α ∧ ∗β)bF (f, g),(2.34) where ∗ is the Hodge star operator (cf. [Z]). Consider the de Rham complex (2.35) Ω∗(M,F ), dF : 0 → Ω0(M,F ) d → Ω1(M,F ) → · · · dF→ Ωn(M,F ) → 0. Let dF∗b : Ω ∗(M,F ) → Ω∗(M,F ) denote the formal adjoint of dF with respect to G-invariant the symmetric bilinear form in (2.34). That is, for any u, v ∈ Ω∗(M,F ), one has dFu, v u, dF∗b v .(2.36) Db = d F + dF∗b , D dF + dF∗b = dF∗b d F + dF dF∗b .(2.37) Then the Laplacian D2b preserves the Z-grading of Ω ∗(M,F ). As was pointed out in [BH1] and [BH2], D2b has the same principal symbol as the usual Hodge Laplacian (constructed using the inner product on Ω∗(M,F ) induced from (gTM , gF )) studied for example in [BZ1]. We collect some well-known facts concerning D2b as in [BH2, Proposition 4.1], where the reference [S] is indicated. Proposition 2.6. The following properties hold for the Laplacian D2b : (i) The spectrum of D2b is discrete. For every θ > 0 all but finitely many points of the spectrum are contained in the angle {z ∈ C| − θ < arg(z) < θ}; (ii) If λ is in the spectrum of D2b , then the image of the associated spectral projection is finite dimensional and contains smooth forms only. We refer to this image as the (generalized) λ-eigen space of D2b and denote it by Ω {λ}(M,F ). There exists Nλ ∈ N such that D2b − λ )Nλ∣∣∣ Ω∗{λ}(M,F ) = 0.(2.38) We have a D2b -invariant 〈 , 〉b-orthogonal decomposition Ω∗(M,F ) = Ω∗{λ}(M,F )⊕ Ω {λ}(M,F ) ⊥.(2.39) The restriction of D2b − λ to Ω∗{λ}(M,F ) ⊥ is invertible; (iii) The decomposition (2.39) is invariant under dF and dF∗b ; (iv) For λ 6= µ, the eigen spaces Ω∗{λ}(M,F ) and Ω {µ}(M,F ) are 〈 , 〉b- orthogonal to each other. For any a ≥ 0, set Ω∗[0,a](M,F ) = 0≤|λ|≤a Ω∗{λ}(M,F ).(2.40) Let Ω∗ [0,a] (M,F )⊥ denote the 〈 , 〉b-orthogonal complement to Ω∗[0,a](M,F ). Obviously, each Ω∗{λ}(M,F ) is a G-invariant subspace. By [BH2, (29)] and Proposition 2.6, one sees that (Ω∗ [0,a] (M,F ), dF ) forms a finite dimensional complex whose cohomology equals to that of (Ω∗(M,F ), dF ). Moreover, the G-invariant symmetric bilinear form 〈 , 〉b clearly induces a nondegenerate G-invariant symmetric bilinear form on each Ωi [0,a] (M,F ) with 0 ≤ i ≤ n. By Definition 2.2 one then gets a symmetric bilinear torsion det(H∗(Ω∗ [0,a] (M,F )),G) on detH∗(Ω∗ [0,a] (M,F ), dF ) = detH∗(Ω∗(M,F ), dF ). For any 0 ≤ i ≤ n, let D2b,i be the restriction of D2b on Ωi(M,F ). Then it is shown in [BH2] (cf. [S, Theorem 13.1]) that for any a ≥ 0, g ∈ G the following is well-defined, D2b,(a,+∞),i (g) = exp D2b,i [0,a] (M,F )⊥ )−s]) .(2.41) Definition 2.7. If g ∈ G, set bRSdet(H∗(M,F ),G)(g) = b det(H∗(Ω∗ [0,a] (M,F )),G)(g) D2b,(a,+∞),i )(−1)ii (2.42) by [BH2, Proposition 4.7], we know that bRS det(H∗(M,F ),G) does not depend on the choice of a ≥ 0, and is called the equivariant Ray-Singer symmetric bilinear torsion on detH∗(Ω∗(M,F ), dF ). 2.5 An anomaly formula for the equivariant Ray-Singer symmetric bilinear torsion We continue the discussion of the above subsection. Definition 2.8. Let θg(F, b F ) be the 1-form on Mg θg(F, b F ) = Tr g(bF )−1∇F bF .(2.43) ClearlyMg is a totally geodesic submanifold ofM . Let g TMg be the Rieman- nian metric induced by gTM on TMg. Let ∇TMg be the Levi-Civita connection on (TMg, g TMg ). Let e(TMg,∇TMg) be the Chern-Weil representative of the rational Euler class of TMg, associated to the metric preserving connection ∇TMg . Then e(TMg,∇TMg) = Pf if dimMg is even,(2.44) 0 if dimMg is odd. Let g′TM be another G-invariant metric and let ∇′TMg be the correspond- ing Levi-Civita connection on TMg. Let ẽ(TMg,∇TMg ,∇′TMg) be the Chern- Simons class of dimMg − 1 forms on Mg, such that TMg,∇TMg ,∇′TMg TMg,∇′TMg TMg,∇TMg .(2.45) Let b′F be another G-invariant nondegenerate symmetric bilinear form on Let b′RS det(H∗(M,F ),G) denote the equivariant Ray-Singer symmetric bilinear torsion associated to g′TM and b′F . By [SZ, Remark 6.4] and [BZ2, Theorem 2.7], we have the following exten- sion of the anomaly formula of [SZ, Theorem 2.9]. Theorem 2.9. If bF , b′F lie in the same homotopy class of nondegenerate symmetric bilinear forms on F , then for g ∈ G the following identity holds, (2.46) det(H∗(M,F ),G) det(H∗(M,F ),G) (g) = exp g log TMg,∇TMg · exp θg(F, b ′F )ẽ TMg,∇TMg ,∇′TMg Proof. Let bFl is a smooth one-parameter family of fiber wise non-degenerate symmetric bilinear forms on F and (gTMl , g l ) be a smooth family of metrics on TM,F . By [SZ, (6.4)], we have (2.47) b = e−tD (−1)ktk e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tk+1tD gdt1 · · · dtk + (−1)n+1tn+1 e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tn+2tD bdt1 · · · dtn+1, where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0, · · · , tk+1 ≥ 0 and Bb,g is defined in [SZ, (6.3)]. Proceeding as in [BZ1, Section 4], we first calculate the asymptotics as t→ 0 of Trs[g(b −1 ∂bFl exp(−tD2bl)]. Here the metric g TM will be fixed. By the same proof in [SZ, proposition 6.1], we have that as t→ 0+, g(bFl ) −1 ∂b e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tn+2tD dt1 · · · dtn+1 → 0. (2.48) Also, by [SZ, (6.22)], we have that for 1 ≤ k ≤ n, (t1, · · · , tk+1) ∈ ∆k, tkTrs g(bFl ) e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tk+1tD = 0.(2.49) So that by (2.47)-(2.49) we have that g(bFl ) −1 ∂b −tD2bl = lim g(bFl ) −1 ∂b −tD2gl (2.50) Now we assume that the nondegenerate symmetric bilinear form on F is fixed, and the metric gTMl on TM depends on l. Let ∗l be the Hodge star operator associated to gTMl . By [BZ1, (4.70), (4.74)], analogues of [SZ, (6.5), (6.24), (6.26), (6.27)] re- placing N by ∗−1l and [BGV, Chapter 6], we have that (2.51) g ∗−1 ∂∗TMl −tD2bl = lim g ∗−1 ∂∗TMl −tD2gl i, j=1 ei ∧ êj ∇ueiω F (ej) ωF , ω̂Fg − ω̂F 1≤i,j≤n )−1 ∂gTMl ei, ej ei ∧ êj  exp − Ṙl = lim g ∗−1l ∂∗TMl −tD2gl ∇TMl ϕTr 1≤i,j≤n )−1 ∂gTMl ei, ej ei ∧ êj  exp − Ṙl 1≤i,j≤n )−1 ∂gTMl ei, ej ei ∧ êj · exp − Ṙl ∧ ϕθg F, bF From (2.50), (2.51) and the calculations in [BZ1, Section 4], we get (2.46). The proof of Theorem 2.9 is completed. Q.E.D. 3 A formula relating equivariant Milnor and equiv- ariant Ray-Singer symmetric bilinear torsions In this section, we state the main result of this paper, which is an explicit comparison result between the equivariant Milnor symmetric bilinear torsion and equivariant Ray-Singer symmetric bilinear torsion. We assume that we are in the same situation as in Sections 2.2-2.4. By a simple argument of Helffer-Sjöstrand [HS, Proposition 5.1] (cf. [BZ1, Section 7b)]), we may and we well assume that gTM there satisfies the following property without altering the Thom-Smale cochain complex (C∗(W u, F ), ∂), (*): For any x ∈ B, there is a system of coordinates y = (y1, · · · , yn) centered at x such that near x, gTM = ∣∣dyi ∣∣2 , f(y) = f(x)− 1 ind(x)∑ ∣∣2 + 1 i=ind(x)+1 ∣∣2 .(3.1) By a result of Laudenbach [L], {W u(x) : x ∈ B} form a CW decomposition of M . For any x ∈ B, F is canonically trivialized over each cell W u(x). Let P∞ be the de Rham map defined by α ∈ Ω∗(M,F ) → P∞α = W u(x)∗ Wu(x) α ∈ C∗(W u, F ).(3.2) By the Stokes theorem, one has ∂P∞ = P∞d F .(3.3) Moreover, it is shown in [L] that P∞ is a Z-graded quasi-isomorphism, inducing a canonical isomorphism PH∞ : H ∗ (Ω∗(M,F ), dF → H∗ (C∗ (W u, F ) , ∂) ,(3.4) which in turn induces a natural isomorphism between the determinant lines, P detH∞ : detH ∗ (Ω∗ (M,F ) , dF → detH∗ (C∗ (W u, F ) , ∂) .(3.5) Also by [BZ2, Theorem 1.11], we know that P∞ commutes with G, and P is the canonical identification of the corresponding cohomology groups as G- spaces. Now let hTM be an arbitrary smooth metric on TM . By Definition 2.7, one has an associated equivariant Ray-Singer symmetric bilinear torsion bRS det(H∗(M,F ),G) on detH ∗(Ω∗(M,F ), dF ). From (3.5), one gets a well-defined equivariant symmetric bilinear form P detH∞ bRSdet(H∗(M,F ),G) (3.6) on detH∗(C∗(W u, F ), ∂). On the other hand, by Definition 2.3, one has a well-defined equivariant Mil- nor symmetric bilinear torsion b det(H∗(M,F ),G) on detH ∗(C∗(W u, F ), ∂), where X = ∇f is the gradient vector field of f associated to gTM . Let Mg = ∪mj=1Mg,j be the decomposition of Mg into its connected compo- nents. Clearly TrF [g] is constant on each Mg,j . Let N be the normal bundle to Mg in M . We identify N to the orthogonal bundle to TMg in TM |Mg . Take x ∈ Bg. Then g acts on TxM as a linear isometry. Also TMg = {Y ∈ TM |Mg , gY = Y }. Moreover g acts on N . Let e±iβ1 , · · · , e±iβq (0 < βj ≤ π) be the locally constant distinct eigenvalues of g|N . Then N splits orthogonally as Nβj . For 1 ≤ j ≤ q, g acts on Nβj as an isometry, with eigenvalues e±iβj . In particular, if e±iβj 6= −1, Nβj is even dimensional. Take x ∈ Bg. Since f is g-invariant, d2f(x) is also g-invariant. Therefore the decomposition TxM = TxMg ⊕ is orthogonal with respect to d2f(x). On TxMg, the index of d 2f(x)|TxMg×TxMg was already denoted indg(x). Let n+(βj)(x) (resp. n−(βj)(x)) be the num- ber of positive (resp. negative) eigenvalues of d2f(x)| βj . Then if e ±iβj 6= −1, n±(βj)(x) is even. Let ψ(TMg,∇TMg) be the Mathai-Quillen current ([MQ]) over TMg, asso- ciated to hTM , defined in [BZ1, Definition 3.6]. As indicated in [BZ1, Remark 3.8], the pull-back current X∗ψ(TMg,∇TMg ) is well-defined over Mg. The main result of this paper, which generalizes [SZ, Theorem 3.1] to the equivariant case. Theorem 3.1. For g ∈ G, the following identity in C holds, (3.7) P detH∞ det(H∗(M,F ),G) det(H∗(Wu,F ),G) (g) = exp θg(F, b F )X∗ψ TMg,∇TMg · exp (−1)indg(x) (n+(βj)(x)− n−(βj)(x)) 1− βj − 2Γ′(1) · Tr [g|Fx ] Remark 3.2. By proceeding similarly as in [BZ2, Section 5b)], in order to prove (3.7), we may well assume that hTM = gTM . Moreover, we may assume that bF , as well as the Hermitian metric hF on F , are flat on an open neighborhood of the zero set B of X. From now on, we will make these assumptions. 4 A proof of Theorem 3.1 We assume that the assumptions in Remark 3.2 hold. For any T ∈ R, let bFT be the deformed symmetric bilinear form on F defined bFT (u, v) = e −2TfbF (u, v).(4.1) Let dF∗bT be the associated formal adjoint in the sense of (2.36). Set DbT = d F + dF∗bT , D dF + dF∗bT = dF∗bT d F + dF dF∗bT .(4.2) Let Ω∗ [0,1],T (M,F ) be defined as in (2.40) with respect to D2bT , and let [0,1],T (M,F )⊥ be the corresponding 〈 , 〉bT -orthogonal complement. Let P [0,1] T be the orthogonal projection from Ω ∗(M,F ) to Ω∗ [0,1],T (M,F ) with respect to the inner product determined by gTM and gFT = e −2TfgF . Set (1,+∞) T = Id− P [0,1] Following [BZ2, (5.9)-(5.10)], we introduce the notations χg(F ) = TrF |Mg,j x∈Bg∩Mg,j (−1)indg(x),(4.3) χ̃′g(F ) = TrF |Mg,j x∈Bg∩Mg,j (−1)indg(x)ind(x), s [f ] = TrF |Mg,j x∈Bg∩Mg,j (−1)indg(x)f(x). Let N be the number operator on Ω∗(M,F ) acting on Ωi(M,F ) by multi- plication by i. By the technique developed in [SZ] and the corresponding results in [BZ2], we easily get the following intermediate results. The sketch of the proofs will be outlined in Section 5. Theorem 4.1. (Compare with [BZ2, Theorem 5.5] and [SZ, Theorem 3.3]) Let [0,1] T be the restriction of P∞ on Ω [0,1],T (M,F ), let P [0,1],detH T be the induced isomorphism on cohomology, then the following identity holds, [0,1],detH det(H∗(Ω∗ [0,1],T (M,F )),G) det(H∗(Wu,F ),G) χg(F )−eχ′g(F ) s [f ]T (4.4) Theorem 4.2. (Compare with [BZ2, Theorem 5.7] and [SZ, Theorem 3.4]) For any t > 0, gN exp −tD2bT (1,+∞) = 0.(4.5) Moreover, for any d > 0 there exist c > 0, C > 0 and T0 ≥ 1 such that for any t ≥ d and T ≥ T0, ∣∣∣Trs gN exp −tD2bT (1,+∞) ]∣∣∣ ≤ c exp(−Ct).(4.6) Theorem 4.3. (Compare with [BZ2, Theorem 5.8] and [SZ, Theorem 3.5]) For T ≥ 0 large enough, then [0,1] = χ̃′g(F ).(4.7) Also, D2bTP [0,1] = 0.(4.8) For the next results, we will make use the same notation for Clifford multi- plications and Berezin integrals as in [BZ1, Section 4]. Theorem 4.4. (Compare with [BZ2, Theorem 5.9] and [SZ, Theorem 3.6]) As t→ 0, the following identity holds, gN exp −tD2bT χg(F ) +O(t) if n is even,(4.9) TrF [g] L exp if n is odd. Theorem 4.5. (Compare with [BZ2, Theorem A.1] and [SZ, Theorem 3.7]) There exist 0 < α ≤ 1, C > 0 such that for any 0 < t ≤ α, 0 ≤ T ≤ 1 , then (4.10) ∣∣∣∣∣Trs gN exp − (tDb + T ĉ(∇f))2 TrF [g] L exp (−BT 2) F, bF ) ∫ B d̂f exp (−BT 2)− χg(F ) ∣∣∣∣∣ ≤ Ct. Theorem 4.6. (Compare with [BZ2, Theorem A.2] and [SZ, Theorem 3.8]) For any T > 0, the following identity holds, (4.11) lim gN exp tDb + ĉ(∇f) g|F |Mg,j 1− e−2T (1 + e−2T x∈B∩Mg,j (−1)indg(x)indg(x)− dimMg,je−2Tχ(Mg,j) g|F |Mg,j sinh(2T ) cosh(2T )− cos(βk) x∈B∩Mg,j (−1)indg(x)n−(βk)(x) g|F |Mg,j sinh(2T ) cosh(2T ) − cos(βk) dimNβkχ(Mg,j). Theorem 4.7. (Compare with [BZ2, Theorem A.3] and [SZ, Theorem 3.9]) There exist α ∈ (0, 1], c > 0, C > 0 such that for any t ∈ (0, α], T ≥ 1, then ∣∣∣∣∣Trs gN exp tDb + ĉ(∇f) − χ̃′g(F ) ∣∣∣∣∣ ≤ c exp(−CT ).(4.12) Clearly, we may and we will assume that the number α > 0 in Theorems 4.5 and 4.7 have been chosen to be the same. Next, we use above theorems to give a proof of Theorem 3.1. Since the process is similar to it in [SZ], so we refer to it for more details. First of all, by the anomaly formula (2.46), for any T ≥ 0, g ∈ G, one has (4.13) [0,1],detH detH∗(Ω∗ [0,1],T (M,F ),G) det(H∗(Wu,F ),G) [0,1],T (M,F )⊥∩Ωi(M,F ) )(−1)ii P detH∞ det(H∗(M,F ),G) det(H∗(Wu,F ),G) (g) exp TrF [g]fe TMg,∇TMg From now on, we will write a ≃ b for a, b ∈ C if ea = eb. Thus, we can rewrite (4.13) as (4.14) P detH∞ det(H∗(M,F ),G) det(H∗(Wu,F ),G)  ≃ log [0,1],detH detH∗(Ω∗ [0,1],T (M,F ),G) det(H∗(Wu,F ),G) (−1)ii log [0,1],T (M,F )⊥∩Ωi(M,F ) TrF [g]fe TMg,∇TMg Let T0 > 0 be as in Theorem 4.2. For any T ≥ T0 and s ∈ C with Re(s) ≥ n+ 1, set θg,T (s) = ts−1Trs gN exp −tD2bT (1,+∞) dt.(4.15) By (4.6), θg,T (s) is well defined and can be extended to a meromorphic function which is holomorphic at s = 0. Moreover, (−1)ii log [0,1],T (M,F )⊥∩Ωi(M,F ) ≃ − ∂θg,T (s) (4.16) Let d = α2 with α being as in Theorem 4.7. From (4.15) and Theorems 4.2-4.4, one finds that (4.17) ∂θg,T (s) = lim gN exp −tD2bT − a−1√ χg(F ) − 2a−1√ Γ′(1)− log d χg(F )− χ̃′g(F ) To study the first term in the right hand side of (4.17), we observe first that for any T ≥ 0, one has e−TfD2bT e Tf = (Db + T ĉ(∇f))2 .(4.18) Thus, one has N exp −tD2bT = Trs N exp −t (Db + T ĉ(∇f))2 .(4.19) By (4.19), one writes (4.20) gN exp −tD2bT − a−1√ χg(F ) ∫ √dT gN exp Db + t T ĉ(∇f) a−1 − χg(F ) gN exp − (tDb + tT ĉ(∇f))2 − a−1 χg(F ) In view of Theorem 4.5, we write (4.21) gN exp − (tDb + tT ĉ(∇f))2 − a−1 χg(F ) gN exp − (tDb + tT ĉ(∇f))2 TrF [g] L exp (tT )2 F, bF ) ∫ B d̂f exp −B(tT )2 χg(F ) TrF [g] L exp (tT )2 − a−1 F, bF ) ∫ B d̂f exp −B(tT )2 By [BZ1, Definitions 3.6, 3.12 and Theorem 3.18], one has, as T → +∞, (4.22) F, bF ) ∫ B d̂f exp −B(tT )2 F, bF (∇f)∗ψ TMg,∇TMg From [BZ1, (3.54)], [SZ, (3.35)] and integration by parts, we have (4.23) TrF [g] L exp (tT )2 − a−1 TrF [g] L exp (−BT ) + Ta−1 − T TrF [g]f exp (−BT ) TrF [g]f exp (−B0) . From Theorems 4.5, 4.6, [BZ1, Theorem 3.20], [BZ1, (7.72) and (7.73)] and the dominate convergence, one finds that as T → +∞, (4.24) gN exp − (tDb + tT ĉ(∇f))2 TrF [g] L exp (tT )2 F, bF ) ∫ B d̂f exp −B(tT )2 χg(F ) gN exp Db + t T ĉ(∇f) TrF [g] L exp F, bF ) ∫ B d̂f exp χg(F ) g|F |Mg,j 1− e−2t2 1 + e−2t x∈B∩Mg,j (−1)indg(x)indg(x)− dimMg,je−2t χ(Mg,j) g|F |Mg,j sinh(2t2) cosh(2t2)− cos(βk) x∈B∩Mg,j (−1)indg(x)n−(βk)(x) g|F |Mg,j sinh(2t2) cosh(2t2)− cos(βk) dimNβkχ(Mg,j) g|F |Mg,j x∈B∩Mg,j (−1)indg(x) (dimMg,j − 2indg(x))− χg(F ) g|F |Mg,j x∈B∩Mg,j (−1)indg(x)indg(x)− χ(Mg,j)dimMg,j 1 + e−2t 1− e−2t − g|F |Mg,j dimNβkχ(Mg,j)− x∈B∩Mg,j (−1)indg(x)n−(βk)(x) sinh(2t) cosh(2t)− cos(βk) On the other hand, by Theorems 4.6, 4.7 and the dominate convergence, we have that as T → +∞, (4.25) ∫ √Td gN exp Db + t T ĉ(∇f) a−1 − χg(F ) ∫ √Td gN exp Db + t T ĉ(∇f) − χ̃′g(F ) χ̃′g(F ) log (Td) + a−1 χg(F ) log (Td) g|F |Mg,j 1− e−2t2 1 + e−2t x∈B∩Mg,j (−1)indg(x)indg(x)− dimMg,je−2t χ(Mg,j) g|F |Mg,j sinh(2t2) cosh(2t2)− cos(βk) x∈B∩Mg,j (−1)indg(x)n−(βk)(x) g|F |Mg,j sinh(2t2) cosh(2t2)− cos(βk) dimNβkχ(Mg,j)− χ̃′g(F ) χ̃′g(F ) log (Td) + a−1 χg(F ) log (Td) + o(1) g|F |Mg,j x∈B∩Mg,j (−1)indg(x)indg(x)− χ(Mg,j)dimMg,j 1− e−2t g|F |Mg,j dimNβkχ(Mg,j)− x∈B∩Mg,j (−1)indg(x)n−(βk)(x) sinh(2t) cosh(2t)− cos(βk) χ̃′g(F )− χg(F ) log(Td) + Ta−1 + o(1). Combining (4.4), (4.14) and (4.20)-(4.25), one deduces, by setting T → +∞, (4.26) log P detH∞ det(H∗(M,F ),G) det(H∗(Wu,F ),G) − 2TrBgs [f ]T + χ̃′g(F )− χg(F ) log T − χ̃′g(F )− χg(F ) log π F, bF (∇f)∗ψ TMg,∇TMg TrF [g] L exp (−BT )−2 Ta−1+2T TrF [g]f exp (−BT ) TrF [g]f exp (−B0) g|F |Mg,j x∈B∩Mg,j (−1)indg(x)indg(x)− χ(Mg,j)dimMg,j 1 + e−2t 1− e−2t − 2e−2t 1− e−2t g|F |Mg,j dimNβkχ(Mg,j)− x∈B∩Mg,j (−1)indg(x)n−(βk)(x) sinh(2t) cosh(2t)− cos(βk) sinh(2t) cosh(2t)− cos(βk) χ̃′g(F )− χg(F ) log(Td)− 2a−1√ TrF [g]fe TMg,∇TMg 2a−1√ Γ′(1)− log d χ̃′g(F )− χg(F ) +o(1). By [BZ1, Theorem 3.20] and [BZ1, (7.72)], one has (4.27) lim TrF [g]f exp(−BT )− 2TTr s [f ] g|F |Mg,j x∈B∩Mg,j (−1)indg(x)indg(x)− χ(Mg,j)dimMg,j (4.28) lim TrF [g] L exp(−BT ) g|F |Mg,j x∈B∩Mg,j (−1)indg(x)indg(x)− χ(Mg,j)dimMg,j On the other hand, by [BZ1, (7.93)] and [BZ2, (5.55)], one has 1 + e−2t 1− e−2t − 2 e−2t 1− e−2t = 1− log π − Γ′(1),(4.29) (4.30) sinh(2t) cosh(2t)− cos(βk) sinh(2t) cosh(2t)− cos(βk) = − log(π)− 1 1− βk Also, by [BZ2, (5.64)], if x ∈ B ∩Mg, dimNβk − n−(βk)(x) [n+(βk)(x)− n−(βk)(x)] .(4.31) From (4.26)-(4.31), we get (3.7), which completes the proof of Theorem 3.1. 5 Proofs of the intermediary Theorems The purpose of this section is to give a sketch of the proofs of the intermediary Theorems. Since the methods of the proofs of these theorems are essentially the same as the corresponding theorem in [SZ], so we will refer to [SZ] for related definitions and notations directly when there will be no confusion, such as Bb,g, Ab,t,T , Ag,t,T , Ct,T , · · · . 5.1 Proof of Theorem 4.1 From Theorem 2.5 and [SZ, (4.44)] which in our situation we also have that P∞,T commutate with g ∈ G, one finds [0,1],detH det(H∗(Ω∗ [0,1],T (M,F )),G) det(H∗(Wu,F ),G) (g) = ∞,TP∞,T [0,1],T (M,F ) )(−1)i+1 (5.1) From [SZ, Propositions 4.4 and 4.5], one deduces that as T → +∞, (5.2) det ∞,TP∞,T [0,1],T (M,F ) = det ∞,TP∞,T [0,1],T (M,F ) (g) · det−1 [0,1],T (M,F ) = det (P∞,T eT ) P∞,T eT Ci(Wu,F ) (g) · det−1 Ci(Wu,F ) = det ))# ( π )N−n/2 ))∣∣∣∣ Ci(Wu,F ) · det−1 Ci(Wu,F ) From (5.1) and (5.2), one gets (4.4) immediately. The proof of Theorem 4.1 is completed. Q.E.D. 5.2 Proof of Theorem 4.2 The proof of Theorem 4.2 is the same as the proof of [SZ, Theorem 3.4] given in [SZ, Section 5]. 5.3 Proof of Theorem 4.3 Recall that the operator eT : C ∗(W u, F ) → Ω∗ [0,1],T (M,F ) has been defined in [SZ, (4.38)], and in the current case, we also have that eT commute with G. So by [SZ, Proposition 4.4], we have that for T ≥ 0 large enough, eT : C∗(W u, F ) → Ω∗ [0,1],T (M,F ) is an identification of G-spaces. So (4.7) follows. Also (4.8) was already proved in [SZ, Theorem 3.5]. 5.4 Proof of Theorem 4.4 In this section, we provide a proof of Theorem 4.4, which computes the asymp- totic of Trs[gN exp(−tD2bT )] for fixed T ≥ 0 as t → 0. The method is the essentially same as it in [SZ]. By [SZ, (6.4)], we have (5.3) b = e−tD (−1)ktk e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tk+1tD gdt1 · · · dtk + (−1)n+1tn+1 e−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tn+2tD bdt1 · · · dtn+1, where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0, · · · , tk+1 ≥ 0. Also, by the same proof of [SZ, Proposition 6.1], we have the following result. Proposition 5.1. As t→ 0+, one has gNe−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tn+2tD dt1 · · · dtn+1 → 0. (5.4) By [SZ, (6.22) and (6.23)], we have that for any 1 < k ≤ n, (t1, · · · , tk+1) ∈ tkTrs gNe−t1tD gBb,ge −t2tD2g · · ·Bb,ge−tk+1tD = 0,(5.5) while for k = 1, 0 ≤ t1 ≤ 1, (5.6) lim gNe−t1tD gBb,ge −(1−t1)tD2g = lim gNBb,ge −tD2g i, j=1 ei ∧ êj ∇ueiω F (ej) ωF , ω̂Fg − ω̂F · L exp So by [BZ2, (2.13)], and proceed as in [SZ, (6.26)-(6.28)], we have gNe−t1tD gBb,ge −(1−t1)tD2g = 0.(5.7) From (5.3), (5.4), (5.5), (5.7) and [BZ2, Theorem 5.9], one gets (4.9). The proof of Theorem 4.4 is completed. Q.E.D. 5.5 Proof of Theorem 4.5 In order to prove (4.10), one need only to prove that under the conditions of Theorem 4.5, there exists constant C ′′ > 0 such that (5.8)∣∣∣Trs gN exp − (tDb + T ĉ(∇f))2 − Trs gN exp − (tDg + T ĉ(∇f))2 F, bF F, gF )) ∫ B d̂f exp (−BT 2) ∣∣∣∣∣ ≤ C By [SZ, (7.8)], we have (5.9) e b,t,T = e g,t,T (−1)k −t1A2g,t,TCt,T e −t2A2g,t,T · · ·Ct,T e−tk+1A g,t,T dt1 · · · dtk + (−1)n+1 −t1A2g,t,TCt,T e −t2A2g,t,T · · ·Ct,T e−tn+2A b,t,T dt1 · · · dtn+1. By the same proof of [SZ, (7.21)], we have that there exists C1 > 0 such that for any t > 0 small enough and T ∈ [0, 1 ∣∣∣∣∣ −t1A2g,t,TCt,T e −t2A2g,t,T · · ·Ct,T e−tn+2A b,t,T dt1 · · · dtn+1 ∣∣∣∣∣ ≤ C1t. (5.10) Also by the same proof of [SZ, (7.23)], we have that there exists C2 > 0, 0 < d < 1 such that for any 1 < k ≤ n, 0 < t ≤ d, T ≥ 0 with tT ≤ 1, −t1A2g,t,TCt,T e −t2A2g,t,T · · ·Ct,T e−tk+1A g,t,T dt1 · · · dtk ∣∣∣∣ ≤ C2t, (5.11) while for k = 1 one has for any 0 < t ≤ d, T ≥ 0 with tT ≤ 1 and 0 ≤ t1 ≤ 1, by [BZ2, Proposition 9.3], we have (5.12)∣∣∣∣∣Trs −t1A2g,t,TCt,T e −(1−t1)A2g,t,T gωF (∇f) L exp (−BT 2) ∣∣∣∣∣ ≤ C2t. Now similar as [SZ, (7.25)], we have (5.13) gωF (∇f) L exp (−BT 2) F, gF F, bF )) ∫ B ∇̂f exp (−BT 2) . From (5.9)-(5.13), we get (5.8), which completes the proof of Theorem 4.5. Q.E.D. 5.6 Proof of Theorem 4.6 In order to prove Theorem 4.6, we need only to prove that for any T > 0, gN exp b,t,T gN exp g,t,T = 0.(5.14) By [SZ, (8.2) and (8.4)], there exists 0 < C0 ≤ 1, such that when 0 < t ≤ C0, one has the absolute convergent expansion formula (5.15) e b,t, T t − e g,t, T (−1)k −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T t dt1 · · · dtk, and that (−1)k −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T t dt1 · · · dtk(5.16) is uniformly absolute convergent for 0 < t ≤ C0. Proceed as in [SZ, Section 8], one has that for any (t1, · · · , tk+1) ∈ ∆k \ {t1 · · · tk+1 = 0}, (5.17) ∣∣∣∣Trs −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T ]∣∣∣∣ ≤ C3tk (t1 · · · tk)− g,t, T ∥∥∥∥∥ψe g,t, T ∥∥∥∥∥ for some positive constant C3 > 0. Also, by [SZ, (8.4)], (5.17) and the same assumption in [SZ] that tk+1 ≥ 1k+1 , one gets (5.18) −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk ≤ C4tk−n ∥∥∥∥ψe 2(k+1) g,t, T for some constant C4 > 0. From (5.15), (5.16), (5.18), [SZ, (8.9) and (8.10)] and the dominate conver- gence, we get (5.14), which completes the proof of Theorem 4.6. Q.E.D. 5.7 Proof of Theorem 4.7 In order to prove Theorem 4.7, we need only to prove that there exist c > 0, C > 0, 0 < C0 ≤ 1 such that for any 0 < t ≤ C0, T ≥ 1, ∣∣∣Trs gN exp b,t,T − Trs gN exp g,t,T )]∣∣∣ ≤ c exp(−CT ).(5.19) First of all, one can choose C0 > 0 small enough so that for any 0 < t ≤ C0, T > 0, by (5.15), we have the absolute convergent expansion formula (5.20) e b,t, T t − e g,t, T (−1)k −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T t dt1 · · · dtk, from which one has (5.21) Trs gN exp b,t,T − Trs gN exp g,t,T (−1)k −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk. Thus, in order to prove (5.19), we need only to prove (5.22) −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk −(t1+tk+1)A2 g,t, T −t2A2 g,t, T t · · ·C dt1 · · · dtk ≤ c exp(−CT ). By [SZ, (8.6)], we have for any t > 0, T ≥ 1, (t1, · · · , tk+1) ∈ ∆k \ {t1 · · · tk+1 = 0}, (5.23) Trs −(t1+tk+1)A2 g,t, T −t2A2 g,t, T t · · ·C = Trs −(t1+tk+1)A2 g,t, T t Ct,T −t2A2 g,t, T t Ct,T · · ·ψe −tkA2 g,t, T t Ct,T From (5.23), [SZ, (9.18) and (9.19)], one sees that there exists C5 > 0, C6 > 0 and C7 > 0 such that for any k ≥ 1, (5.24) −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk ≤ C5 (C6t)k from which one sees that there exists 0 < c1 ≤ 1, C8 > 0, C9 > 0 such that for any 0 < t ≤ c1 and T ≥ 1, one has (5.25)∣∣∣∣∣ −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk ∣∣∣∣∣ ≤ C8 exp (−C9T ) . On the other hand, for any 1 ≤ k < n, by proceeding as in (5.18), one has that for any 0 < t ≤ c1, T ≥ 1, (5.26) −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk ≤ C10tk−n ∥∥∥∥ψe 2(k+1) g,t, T for some constant C10 > 0. From (5.26) and [SZ, (9.23)], one sees immediately that there exists C11 > 0, C12 > 0 such that for any 1 ≤ k ≤ n− 1, 0 < t ≤ c1 and T ≥ 1, one has (5.27) −t1A2 g,t, T −t2A2 g,t, T t · · ·C −tk+1A2 g,t, T dt1 · · · dtk ≤ C11e−C12T . From (5.21), (5.25) and (5.27), one gets (5.19). The proof of Theorem 4.7 is completed. Q.E.D. References [BGS] J.-M. Bismut, H. Gillet and C. Soulé, Analytic torsions and holomorphic determinant line bundles I. Commun. Math. Phys. 115 (1988), 49-78. [BGV] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Opera- tors. Springer, Berline-Heidelberg-New York, 1992. [BZ1] J.-M. Bismut and W. Zhang, An Extension of a Theorem by Cheeger and Müller. Astérisque Tom. 205, Paris, (1992). [BZ2] J.-M. Bismut andW. Zhang, Milnor and Ray-Singer metrics on the equiv- ariant determinant of a flat vector bundle. Geom. Funct. Anal. 4 (1994), 136-212. [BH1] D. Burghelea and S. Haller, Torsion, as function on the space of repre- sentations. Preprint, math.DG/0507587. [BH2] D. Burghelea and S. Haller, Complex valued Ray-Singer torsion. Preprint, math.DG/0604484. [BH3] D. Burghelea and S. Haller, Complex valued Ray-Singer torsion II. Preprint, math.DG/0610875. [C] J. Cheeger, Analytic torsion and the heat equation. Ann. of Math. 109 (1979), 259-332. [FT] M. Farber and V. Turaev, Poincaré-Reidemeister metric, Euler structures and torsion. J. Reine Angew. Math. 520 (2000), 195-225. [HS] B. Helffer and J. Sjöstrand, Puis multiples en mécanique semi-classique IV: Etude du complexe de Witten. Comm. PDE 10 (1985), 245-340. [KM] F. F. Knudson and D. Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “div”. Math. Scand. 39 (1976), 19-55. [L] F. Laudenbach, On the Thom-Smale complex. Appendix in [BZ1]. [MQ] V. Mathai and D. Quillen, Superconnections, Thom classes, and equiv- ariant differential forms. Topology 25 (1986), 85-110. [Mi] J. Milnor, Whitehead torsion. Bull. Amer. Math. 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USSR-Izv. 34 (1990), 627-662. [W] E. Witten, Supersymmetry and Morse theory. J. Diff. Geom. 17 (1982), 661-692. [Z] W. Zhang, Lectures on Chern-Weil Theory and Witten Deformations, Nankai Tracts in Mathematics, Vol. 4. World Scientific, Singapore, 2001. http://arxiv.org/abs/math/0610577 Introduction Equivariant symmetric bilinear torsions associated to the de Rham and Thom-Smale complexes Equivariant symmetric bilinear torsion of a finite dimensional complex The Thom-Smale complex of a gradient field Equivariant Milnor symmetric bilinear torsion Equivariant Ray-Singer symmetric bilinear torsion An anomaly formula for the equivariant Ray-Singer symmetric bilinear torsion A formula relating equivariant Milnor and equivariant Ray-Singer symmetric bilinear torsions A proof of Theorem 3.1 Proofs of the intermediary Theorems Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ??

0704.1342

Two-pion-exchange contributions to the pp\to pp\pi^0 reaction

Two-pion-exchange contributions to the pp→ ppπ0 reaction Y. Kim(a,b), T. Sato(c), F. Myhrer(a) and K. Kubodera(a) (a) Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA (b) School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea (c) Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract Our previous study of the near-threshold pp → ppπ0 reaction based on a hybrid nuclear effective field theory is further elaborated by examining the momentum dependence of the relevant transition operators. We show that the two-pion exchange diagrams give much larger contributions than the one-pion exchange diagram, even though the former is of higher order in the Weinberg counting scheme. The relation between our results and an alternative counting scheme, the momentum counting scheme, is also discussed. http://arxiv.org/abs/0704.1342v2 In the standard nuclear physics approach (SNPA), a nuclear reaction amplitude is calculated with the use of the transition operator derived from a phenomenological Lagrangian and nuclear wave functions generated by a high-precision phenomenolog- ical NN potential. SNPA has been enormously successful in explaining a vast range of nuclear phenomena. Meanwhile, a nuclear chiral perturbation approach based on heavy-baryon chiral perturbation theory (HBχPT) is gaining ground as a powerful tool for addressing issues that cannot be readily settled in SNPA. HBχPT is a low- energy effective field theory of QCD, based on a systematic expansion in terms of the expansion parameter ǫ ≡ Q/Λχ ≪ 1, where Q is a typical energy-momentum involved in a process under study or the pion mass mπ, and the chiral scale Λχ ≃ 4πfπ ≃ 1 GeV. HBχPT has been applied with great success to low-energy processes includ- ing e.g., pion-nucleon scattering and electroweak reactions on a nucleon and in few- nucleon systems. Our present work is concerned with a HBχPT study of the near- threshold pp→ppπ0 reaction. A motivation of this study may be stated in reference to the generic NN→NNπ processes near threshold. Although HBχPT presupposes the small size of its expansion parameter Q/Λχ, the pion-production reactions involve somewhat large energy- and three-momentum transfers even at threshold. Therefore the application of HBχPT to the NN→NNπ reactions may involve some delicate aspects, but this also means that these processes may serve as a good test case for probing the limit of applicability of HBχPT. Apart from this general issue to be investigated, a specific aspect of the pp→ppπ0 reaction makes its study particularly interesting. For most isospin channels, the NN→NNπ amplitude near threshold is dominated by the pion rescattering diagram where the πN scattering vertex is given by the Weinberg-Tomozawa term, which represents the lowest chiral order contri- bution. However, a quantitatively reliable description of the NN→NNπ reactions obviously requires detailed examinations of the corrections to this dominant ampli- tude. Meanwhile, since the Weinberg-Tomozawa vertex does not contribute to the pion-nucleon rescattering diagram for pp→ppπ0, this reaction is particularly sensi- tive to higher chiral-order contributions and hence its study is expected to provide valuable information to guide us in formulating a quantitative description of all the NN→NNπ reactions (including the channels that involve a deuteron). The first HBχPT-based study of the near-threshold pp→ppπ0 reaction was made in Refs. [1, 2]. In HBχPT one naturally expects a small cross section for this reac- tion since, for s-wave pion production, the pion-nucleon vertex in the impulse ap- proximation (IA) diagram and the pion-rescattering vertex in the one-pion-exchange rescattering (1π-Resc) diagram arise from the next-to-leading-order (NLO) chiral la- grangian. A remarkable feature found in Refs. [1, 2] is that a drastic cancellation between the IA and 1π-Resc amplitudes leads to the suppression of the pp→ppπ0 amplitude far beyond the above-mentioned naturally expected level. This destruc- tive interference is in sharp contrast with the constructive interference reported in SNPA-based calculations [3, 4]. It is to be recalled that the pp→ppπ0 cross section obtained in Refs. [3, 4] was significantly smaller (by a factor of ∼5) than the experi- mental value [5]. The drastic cancellation between the IA and 1π-Resc terms found in the HBχPT calculations [1, 2] leads to even more pronounced disagreement be- tween theory and experiment. In this connection it is worth noting that, according to Lee and Riska [6], the heavy-meson (σ and ω) exchanges can strongly enhance the pp→ppπ0 amplitude. It is also to be noted that σ-meson-exchange introduced in many NN potentials is more properly described by correlated two-pion-exchange (see e.g., Refs. [7, 8]), and that there have been substantial developments in deriving a two-pion exchange NN potential using HBχPT, see e.g. [9]. These developments were conducive to a HBχPT study of two-pion-exchange (TPE) contributions to the pp→ppπ0 reaction [10, 11]. In the plane-wave approximation it was found [10] that TPE contributions are indeed very large (as compared to the 1π-Resc amplitude), a result that is in line with the finding in Ref.[6]. A subsequent DWBA calculation [11] indicates that this feature remains essentially unchanged when the initial- and final- state interactions are taken into account. More recent investigations [12, 13, 14, 15], however, have raised a number of important issues that call for further investigations, and the purpose of our present note is to address these issues. In Ref. [10], to be referred to as DKMS, were derived all the transition operators for pp→ppπ0 belonging to next-to-next-to-leading order (NNLO) in the Weinberg counting, and these operators were categorized into Types I ∼ VII, according to the patterns of the corresponding Feynman diagrams; see Figs. 2 - 5 in DKMS. Types I, II, III and IV belong to diagrams of the two-pion exchange (TPE) type, while Types V, VI and VII arise from diagrams of the vertex correction type. A notable feature pointed out in DKMS is that the contributions of Types II ∼ IV are by far the largest, and that they even exceed those of the 1π-Resc amplitude, which is formally of lower chiral order. On the other hand, the possibility of strong cancellation among the TPE diagrams was pointed out in Refs. [12, 13]. This motivates us to make here a further study of the behavior of the TPE diagrams.1 A remark is in order here on a counting scheme to be used. At the NN→NNπ threshold the nucleon three-momentum must change from the initial value p ∼√ mπmN to zero, entailing a rather large momentum transfer. To take this large 1For a brief report on this study, see Ref. [16]. momentum transfer into account, Cohen et al. [2] proposed a new counting scheme, to be called the momentum counting scheme (MCS); see Ref. [13] for a detailed review. In MCS the expansion parameter is ǫ̃ ≡ p/mN ≃ (mπ/mN)1/2, which is larger than the usual HBχPT expansion parameter ǫ ≃ mπ/mN . A study based on MCS [13] indicates that the 1π-Resc diagram for pp→ppπ0 is higher order in ǫ̃ (and hence less important) than a certain class of TPE diagrams, called “leading order loop diagrams”, and that MCS is consistent with the estimates of the TPE and other diagrams reported in DKMS. Furthermore, according to Hanhart and Kaiser (HK) [12], the “leading parts” (see below) of these MCS “leading order” diagrams exhibit exact cancellation among themselves;2 see also Lensky et al. [14]. Although these studies are illuminating, we consider it important to examine the behavior of the “sub-leading” parts (in MCS counting) of these TPE diagrams in order to see whether they can be still as large as indicated by the phenomenological success of the Lee-Riska heavy-meson exchange mechanism. In what follows we shall demonstrate that this is indeed the case. Analytic expressions for the pp→ppπ0 transition operators to NNLO in HBχPT were given in DKMS. Although these expressions are valid for arbitrary kinematics, we find it illuminating to concentrate here on their simplified forms obtained with the use of fixed kinematics approximation (FKA), wherein the energies associated with particle propagators are “frozen” at their threshold values. In FKA, the TPE operator corresponding to each of the above-mentioned Types I ∼ IV can be written ~Σ · ~k t(p, p′, x) (1) where ~p (~p ′) is the relative three-momentum in the initial (final) pp state (~p1−~p2 = 2~p, ~p ′1 − ~p ′2 = 2~p ′), ~k ≡ ~p − ~p ′, x = p̂ · p̂′, and ~Σ = 12(~σ1 − ~σ2). The function t(p, p ′, x) diverges as k → ∞, and it is useful to decompose t(p, p′, x) into terms that have definite k-dependence as k → ∞. It turns out [18] that t(p, p′, x) can be expressed as t(p, p′, x) k→∞∼ t1 gA/(8f |~k|+ t2 ln{|~k|2/Λ2} + t3 + δt(p, p ′, x), (2) where t3 is asymptotically k-independent, and δt(p, p ′, x) is O(k−1). For each of Types I ∼ IV, analytic expressions for ti’s (i = 1, 2, 3) can be extracted [18] from the amplitudes T given in DKMS [10]. The first term with t1 in eq.(2) is the leading 2HK [12] pointed out that the sign of the contribution of Type II in Ref. [10] should be reversed; we have confirmed the necessity of this correction. part in MCS discussed by HK [12], whereas the remaining terms, which we refer to as the “sub-leading” terms, were not considered by HK. The study of these sub-leading terms is an important theme in what follows. Table 1 shows the value of t1 for Type K (K= I ∼ IV) extracted from the results given in DKMS. The third row in Table 1 gives the ratio RK = TK/TResc, where TK is the plane-wave matrix element of T in eq.(1) for Type K (K=I ∼ IV) normalized by TResc, the plane-wave matrix element of the 1π-Resc diagram. The fourth row in Table 1 gives R ⋆K = T K/TResc, where T the plane-wave matrix element of T with the t1 term in eq.(2) subtracted. We can see from the table that the most divergent t1 terms of the TPE diagrams add up to zero, confirming the result of Ref. [12]. However, this does not necessarily mean that the TPE diagrams are unimportant, because we still need to examine the contributions of the “sub-leading” terms (the t2, t3 and δt terms) in eq.(2). Comparison of RK and R ⋆K indicates that the subtraction of the t1 term reduces the magnitude of TK drastically (except for Type I which has no t1 term), but the fact that |R ⋆K | is of the order of unity (Types I, II and IV) or larger than 1 (Type III) suggests that the TPE contributions can be quite important. The sum of the contributions of Types I ∼ IV R ⋆K (= RK) = −4.65 , (3) which indicates that, at least in plane-wave approximation, the TPE contributions are more important than the 1π-Resc contribution. Table 1: For the four types of TPE diagrams, K= I, II, III and IV, the second row gives the value of t1 defined in eq.(2), and the third row gives the ratio RK = TK/TResc, where TK is the plane-wave matrix element of T in eq.(1) for Type K, and TResc is the 1π-Resc amplitude. The last row gives R ⋆K = T K /TResc, where T K is the plane-wave matrix element of T in eq.(1) with the t1 term in eq.(2) subtracted. Type of diagrams : K = I II III IV (t1)K 0 1 1/2 −3/2 RK −.70 −6.54 −6.60 9.19 R ⋆K −.70 −0.82 −3.73 0.61 Next we investigate the behavior of the TPE diagrams as we go beyond the plane- wave approximation by using distorted waves (DW) for the initial- and final-state NN wave functions. For formal consistency we should use the NN potential derived from HBχPT, but we adopt here a “hybrid EFT” approach and use phenomenological potentials. A conceptual problem in adopting this hybrid approach is that, whereas the TPE transition operators derived in HBχPT are valid only for a momentum range sufficiently lower than Λχ∼1 GeV, a phenomenological NN potential can in principle contain any momentum components.3 To stay close to the spirit of HBχPT, we therefore introduce a Gaussian momentum regulator, exp(−p2/Λ2G), in the initial and final distorted wave integrals, suppressing thereby the high momentum components of the phenomenological NN potentials; this is similar to the MEEFT method used in Ref.[19]. ΛG should be larger than the characteristic momentum scale of the pp→ppπ0 reaction, p ≃ √mNmπ ≃ 360 MeV/c, but it should not exceed the chiral scale Λχ; in the present study we shall consider the range, 500 MeV< ΛG <1 GeV. As high- precision phenomenological NN potentials, we consider the Bonn-B potential [20], the CD-Bonn potential [21], and the Nijm93 potential of the Nijmegen group [22]. It is worth noting here that several groups [23, 24] have developed a systematic approach to construct from a phenomenological potential an effective NN potential, called Vlow−k, that resides within a model space which only contains momentum components below a specified cutoff scale Λlow−k. In this work we will use Vlow−k as derived by the Stony Brook group [24]. It is conceptually natural to use Vlow−k in conjunction with transition operators derived from HBChPT [25]. A problem however is Vlow−k [24], primarily meant for describing sub-pion-threshold phenomena, was obtained with the use of a rather low cutoff, Λlow−k ∼ 2 fm−1. This cutoff is perhaps too close to the characteristic momentum scale p ∼ 360 MeV/c for the pion production reaction. It therefore seems worthwhile to “rederive” Vlow−k employing a momentum cut-off higher than 2 fm−1 and use it in the present DWBA calculation. Below we will use Vlow−k generated from the CD-Bonn potential for Λlow−k= 4 and 5 fm −1. We remark that, as is well known, Vlow−k’s generated from any realistic phenomenological potentials lead to practically equivalent half-off-shell NN K-matrices and hence the same NN wave function. We evaluate the TPE contributions in DWBA for a typical case of Tlab = 281 MeV. Since the t1 terms in eq.(2) add up to zero, we drop the t1 terms in our cal- culation.4 Thus, in eq.(1), we use t⋆(p, p′, x) instead of t(p, p′, x), where t⋆(p, p′, x) is 3 A pragmatic problem associated with this conceptual issue is that, in a momentum-space calcu- lation of the matrix elements of the TPE operators sandwiched between distorted pp wave-functions generated by a phenomenological NN potential, the convergence of momentum integrations is found to be extremely slow [17, 18]. 4 Removing the t1 term lessens the severity of the convergence problem in our momentum inte- obtained from t(p, p′, x) by suppressing the t1 term. The partial-wave projected form of t⋆(p, p′, x) in a DWBA calculation is written as: J = − p2dp p′ 2dp′ dx ψ1S0(p ′) t⋆(p, p′, x) (p− p′x)ψ3P0(p) (4) Here ψα(p) is a distorted two-nucleon relative wave function in the α partial-wave (1S0 for the initial state and 3P0 for the final state) given by ψα(p) = cos(δα) δ(p− pon)/p2 + P Kα(p, pon) (E − Ep) , (5) where δα is the phase-shift for the α partial wave, and Kα(p, pon) is the partial- wave K-matrix pertaining to the asymptotic on-shell momentum pon. The plane-wave approximation corresponds to the use of the wave functions of the generic form: ψ(p) = δ(p− pon)/p2 . (6) We show in Table 2 the values of J , eq.(4), for the TPE operators of Types I ∼ IV, calculated at Tlab = 281 MeV, with the use of the Nijm93 potential of the Nijmegen group [22]5 and Vlow−k. For the Nijm93 potential case, we present the results for five different values of ΛG between 500 and 1000 MeV/c. For the Vlow−k case, the results for two choices of Λlow−k are shown: Λlow−k = 4 fm −1 and 5 fm−1. For comparison, the values of J corresponding to plane-wave approximation are also shown (bottom row). From Table 2 we learn the following: (1) The results for the Nijm93 potential with the gaussian cutoff ΛG are stable against the variation of ΛG within a reasonable range (500 - 1000 MeV/c); (2) There is semi-quantitative agreement between the results for the Nijm93 potential and those for Vlow−k; (3) A semi-quantitative agreement is also seen between the DWBA and PWBA calculations; (4) The feature found in the plane- wave approximation that the contributions of the TPE diagrams are more important than the 1π-Resc contribution remains unchanged in the DWBA calculation; the summed contribution of the TPE operators is larger (in magnitude) than that of 1π-Resc by a factor of 2∼3.5. gration mentioned in footnote 3. 5 We have checked the results obtained using the Bonn-B and CD-Bonn NN potentials are very similar to those for the Nijm93 potential case, which we show here as a representative case. Table 2: The values of J , eq.(4), corresponding to the TPE diagrams of Types I ∼ IV, evaluated in a DWBA calculation for Tlab = 281 MeV. The column labeled “Sum” gives the combined contributions of Types I ∼ IV, and the last column gives the value of J for 1π-Resc. For the Nijm93 potential case, the results for five different choices of ΛG are shown. For the case with Vlow−k, CD-4 (CD-5) represents Vlow−k generated from the CD- Bonn potential with a momentum cut-off Λlow−k = 4 fm −1 (5 fm−1). The last row gives the results obtained in plane-wave approximation. I II III IV Sum 1π−Resc VNijm : ΛG = 500MeV/c −0.11 −0.12 −0.55 0.08 −0.70 0.18 VNijm : ΛG = 600MeV/c −0.12 −0.12 −0.57 0.07 −0.74 0.20 VNijm : ΛG = 700MeV/c −0.12 −0.11 −0.57 0.06 −0.74 0.21 VNijm : ΛG = 800MeV/c −0.12 −0.11 −0.55 0.04 −0.74 0.22 VNijm : ΛG = 1000MeV/c −0.12 −0.10 −0.52 0.03 −0.71 0.23 Vlow−k (CD−4) −0.12 −0.09 −0.46 0.03 −0.65 0.23 Vlow−k (CD−5) −0.09 −0.06 −0.30 −0.01 −0.46 0.22 Plane−wave −0.06 −0.07 −0.30 0.05 −0.37 0.080 We now discuss the above results in the context of MCS [13]. A subtlety in MCS is that a loop diagram of a given order ν in ǫ̃ not only contains a contribution of order ν (“leading part”) but, in principle, can also involve contributions of higher orders in ǫ̃ (“sub-leading part”) due to the non-analytic functions generated by the loop integral. As mentioned, however, HK [12] considered only the leading part, which correspond to the t1 term in eq.(2). According to MCS, for the reaction pp → ppπ0, the loop diagrams corresponding to our Type II, III and IV diagrams belong to NLO in the ǫ̃ parameter, whereas those corresponding to Type I and the 1π-Resc tree diagram are next order in ǫ̃ (NNLO); see Table 11 in Ref. [13]. Meanwhile, as discussed earlier, the sum of the “leading parts” of the NLO diagrams vanishes, and therefore, in calculating J ’s in Table 2, we have dropped the t1 term contribution, retaining only the “sub-leading” parts of these NLO diagrams. This means that all the entries in Table 2 represent “sub-leading contributions” (NNLO) in MCS. If we look at Table 2 from this perspective, we note that the order-of-magnitude behavior of our numerical results is in rough agreement with MCS, although Type IV tends to be rather visibly smaller (in magnitude) than the others. However, it is striking that J for Type III is significantly (if not by an order of magnitude) larger than the other sub-leading contributions. (A similar feature was also seen in R⋆ in Table 1.) In view of the fact that Type III arises from crossed-box TPE diagrams [10], there is a possibility that the enhancement of the Type III diagrams may be related to the strong attractive scalar NN potential that is known to arise from TPE crossed-box-diagrams [7, 8]. We have studied the “sub-leading” parts, which are of NNLO in the momentum counting scheme (MCS) [13], of the TPE amplitudes for the pp→ppπ0 reaction in both PWBA and DWBA calculations. We have shown in fixed kinematics approximation (FKA) that, even though the leading parts of the TPE amplitudes cancel among themselves [12, 14], the contributions of the sub-leading parts are quite significant. They are in general comparable to the 1π-Resc amplitude, and the sub-leading part of the Type III diagrams is even significantly larger than the 1π-Resc diagram. The total contribution of the TPE diagrams is larger (in magnitude) than that of the 1π-Resc diagram by a factor of ∼5 (PWBA) or 2∼3 (DWBA). We have focused here on the TPE loop diagrams but, to obtain theoretical cross section for pp→ppπ0 that can be directly compared with the experimental value, we must consider the other diagrams discussed in DKMS as well as the relevant counter terms. These will be discussed in a forthcoming article [26] . The authors are indebted to Christoph Hanhart and Anders G̊ardestig for useful discussions. A helpful communication from Ulf Meissner is also grateful acknowl- edged. This work is supported in part by the US National Science Foundation, Grant No. PHY-0457014, and by the Japan Society for the Promotion of Science, Grant-in- Aid for Scientific Research (C) No.15540275 and Grant-in-Aid for Scientific Research on Priority Areas (MEXT), No. 18042003. References [1] B.-Y. Park, F. Myhrer, J.R. Morones, T. Meissner and K, Kubodera, Phys. Rev. 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Myhrer, unpublished notes (1999). http://arxiv.org/abs/nucl-th/0511054 http://arxiv.org/abs/nucl-th/0609007 http://arxiv.org/abs/nucl-th/0703012 http://arxiv.org/abs/nucl-th/0611051 [19] T.S. Park et al., nucl-th/0106025; nucl-th/0107012; Phys. Rev. C, 67 055206 (2003); K. Kubodera, nucl-th/0404027; K. Kubodera and T.-S. Park, Ann. Rev. Nucl. Part. Sci. 54, 19 (2004); M. Rho, nucl-th/0610003. [20] R. Machleidt, K. Holinde and C. Elster, Phys. Rep. 149, 1 (1987). [21] R. Machleidt, Phys. Rev. C, 63, 024001 (2001). [22] V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J. de Swart, Phys. Rev. C, 21, 861 (1980); Phys. Rev. C, 49, 2950 (1994). [23] E. Epelbaoum, W. Glöckle and U.-G. Meissner, Phys. Lett. B, 439, 1 (1998); E. Epelbaoum, W. Glöckle, A. Krüger and U.-G. Meissner, Nucl. Phys. A, 645, 413 (1999). [24] S. Bogner, T.T.S. Kuo and L. Coraggio, Nucl. Phys. A, 684, 4332c (2001); S. Bogner et al., Phys. Rev. C, 65, 051301 (R) (2002); S. Bogner, T.T.S. Kuo and A. Schwenk, Phys. 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0704.1343

Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields

HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS ISMAIL KOMBE Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi- Grushin vector fields : ∇γ = (∇x, |x| 2γ∇y) where γ > 0, ∇x and ∇y are usual gradient operators in the variables x ∈ Rm and y ∈ Rk, respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains. 1. Introduction This paper is concerned with Hardy and Rellich type inequalities with remainder terms for Baouendi-Grushin vector fields. Let x ∈ Rm, y ∈ Rk, γ > 0 and n = m + k, with m, k ≥ 1. Then the following Hardy type inequality for Baouendi-Grushin vector fields has been proved by Garofalo [G], (1.1) |∇xφ| 2 + |x|2γ |∇yφ| dxdy ≥ |x|2γ |x|2+2γ + (1 + γ2)2|y|2 )φ2dxdy where φ ∈ C∞0 (R m ×Rk \ {(0, 0)}) and Q = m+(1+ γ)k. Here, ∇xφ and ∇yφ denotes the gradients of φ in the variables x and y, respectively. A similar inequality with the same sharp constant (Q−2 )2 holds if Rn replaced by Ω and Ω contains the origin [D]. If γ = 0 then it is clear that the inequality (1.1) recovers the classical Hardy inequality in Rn (1.2) |∇φ(z)|2dz ≥ |φ(z)|2 where z = (x, y) ∈ Rm×Rk and the constant (n−2 )2 is sharp. There exists a large literature concerning with the Hardy inequalities and, in particular, sharp inequalities as well as their improved versions which have attracted a lot of attention because of their application to singular problems (See [BG], [PV], [BV], [GP], [CM], [VZ], [K1] and references therein). A sharp improvement of the Hardy inequality (1.2) was discovered by Brezis and Vázquez [BV]. They proved that for a bounded domain Ω ⊂ Rn (1.3) |∇φ(z)|2dz ≥ |φ(z)|2 dz + µ φ2dz, where φ ∈ C∞0 (Ω), ωn and |Ω| denote the n-dimensional Lebesgue measure of the unit ball B ⊂ Rn and the domain Ω respectively. Here µ is the first eigenvalue of the Laplace Date: April 09, 2007. Key words and phrases. Hardy inequality, Rellich inequality, Best constants, Baouendi-Grushin vector fields. AMS Subject Classifications: 26D10, 35H20. http://arxiv.org/abs/0704.1343v1 2 ISMAIL KOMBE operator in the two dimensional unit disk and it is optimal when Ω is a ball centered at the origin. In a recent paper Abdelloui, Colorado and Peral [ACP] obtained, among other things, the following improved Caffarelli-Kohn-Nirenberg inequality (1.4) |∇φ(z)|2|z|−2adz ≥ n− 2a− 2 |φ(z)|2 |z|2a+2 dz + C |∇φ|q|z|−aq where φ ∈ C∞0 (Ω), −∞ < a < , 1 < q < 2 and C = C(q, n,Ω) > 0. Motivated by these results, our first goal is to find improved weighted Hardy type inequalities for Baouendi-Grushin vector fields. It is well known that an important extension of Hardy’s inequality to higher-order deriva- tives is the following Rellich inequality (1.5) |∆φ(z)|2dz ≥ n2(n− 4)2 |φ(z)|2 where φ ∈ C∞0 (R n \ {0}), n 6= 2 and the constant n2(n−4)2 is sharp. Davies and Hinz [DH], among other results, obtained sharp weighted Rellich inequalities of the form (1.6) |∆φ(z)|2 dz ≥ C |φ(z)|2 for suitable values of α, β, p and φ ∈ C∞0 (R n \ {0}). In a recent paper, Tertikas and Zo- graphopoulos [TZ], among other results, obtained the following new Rellich type inequalities that connects first to second order derivatives: (1.7) |∆φ|2dz ≥ |∇φ|2 where φ ∈ C∞0 (R n \ {0}) and the constant n is sharp. Recently, Kombe [K2] obtained analogues of (1.6) and (1.7), and their improved versions on Carnot groups. Motivated by the above results, our second goal is to find sharp weighted Rellich type inequalities and their improved versions for Baouendi-Grushin vector fields in that they do not arise from any Carnot group. We should also mention that Kombe and Özaydin [KÖ] obtained (under some geometric assumptions) improved Hardy and Rellich inequalities on a Riemannian manifold that does not recover our current results. Analogue inequalities for the Greiner vector fields will be given in a forthcoming paper [K3]. 2. Notations and Back ground material In this section, we shall collect some notations, definitions and preliminary facts which will be used throughout the article. The generic point is z = (x1, ..., xm, y1, ..., yk) = (x, y) ∈ m × Rk with m, k ≥ 1, m + k = n. The sub-elliptic gradient is the n dimensional vector field given by (2.1) ∇γ = (X1, · · · , Xm, Y1, · · · , Yk) where (2.2) Xj = , j = 1, · · · , m, Yj = |x| , j = 1, · · · , k. The Baouendi-Grushin operator on Rm+k is the operator (2.3) ∆γ = ∇γ · ∇γ = ∆x + |x| 2γ∆y, HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS3 where ∆x and ∆y are Laplace operators in the variables x ∈ R m and y ∈ Rk, respectively (see [B], [G1], [G2]). If γ is an even positive integer then ∆γ is a sum of squares of C ∞ vector fields satisfying Hörmander finite rank condition: rank Lie [ X1, · · · , Xm, Y1, · · · , Yk] = n. The anisotropic dilation attached to ∆γ is given by δλ(z) = (λx, λ γ+1y), λ > 0, z = (x, y) ∈ Rm+k. The change of variable formula for the Lebesgue measure gives that d ◦ δλ(x, y) = λ dxdy, where Q = m+ (1 + γ)k is the homogeneous dimension with respect to dilation δλ. For z = (x, y) ∈ R m × Rk, let (2.4) ρ = ρ(z) := |x|2(1+γ) + (1 + γ)2|y|2 2(1+γ) By direct computation we get |∇γρ| = Let f ∈ C2(0,∞) and define u = f(ρ) then we have the following useful formula (2.5) ∆γu = |x|2γ f ′′ + We let Bρ = {z ∈ R n | ρ(z) < r}, Bρ̃ = {z ∈ R n | ρ̃(z, 0) < r} and call these sets, respectively, ρ-ball and Carnot-Carathéodory metric ball centered at the origin with radius r. The Carnot-Carathéodory distance ρ̃ between the points z andz0 is defined by ρ̃(z, z0) = inf{length(η) | η ∈ K} where the set K is the set of all curves η such that η(0) = z, η(1) = z0 and η̇(t) is in span{X1(η(t)), ..., Xm(η(t)), Y1(η(t)), ..., Yk(η(t))}. If γ is a positive even integer then Carnot-Carathéodory distance of z from the origin ρ̃(z, 0) is comparable to ρ(z). ( See [FGW] and [Be] for further details.) It is well known that Sobolev and Poincaré type inequalities are important in the study of partial differential equations, especially in the study of those arising from geometry and physics. In[FGW], Franchi, Gutierrez and Wheeden obtained the following Sobolev- Poincaré inequality for metric balls associated with Baouendi-Grushin type operators: (2.6) w1(B) |∇γφ| w1(z)dz w2(B) |φ(z)|qw2(z)dz where φ ∈ C∞0 (B) and the weight functions w1 and w2 satisfies some certain conditions. Here, c is independent of φ and B, 1 ≤ p ≤ q <∞ and w(B) = w(z)dz. If w1 = w2 = 1 then Monti [M] obtained the following sharp Sobolev inequality (2.7) |∇xφ| 2 + |x|2γ|∇yφ| Q−2dxdy where C = C(m, k, α) > 0. 4 ISMAIL KOMBE 3. Improved Hardy-type inequalities In this section we study improved Hardy type inequalities. These inequalities plays key role in establishing improved Rellich type inequalities. In the various integral inequalities below (Section 3 and Section 4), we allow the values of the integrals on the left-hand sides to be +∞. The following theorem is the first result of this section. Theorem 3.1. Let γ be an even positive integer, α ∈ R, −m < t < m , and Q+α− 2 > 0. Then the following inequality is valid (3.1) α|∇γρ| t|∇γφ| Q+ α− 2 α |∇γρ| ρα|∇γρ| tφ2dz for all compactly supported smooth function φ ∈ C∞0 (Bρ). Proof. Let φ = ρβψ ∈ C∞0 (Bρ) and β ∈ R \ {0}. A direct calculation shows that (3.2) ρα|∇γρ| t|∇γφ| 2dz = β2 ρα+2β−2|∇γρ| t+2ψ2dz ρα+2β−1|∇γρ| tψ∇γρ · ∇γψdz ρα+2β |∇γρ| t|∇γψ| Applying integration by parts to the middle term and using the following fact ρα+2β−1|∇γρ| = (Q+ α + 2β − 2)ρα+2β−2|∇γρ| yields (3.3) ρα|∇γρ| t|∇γφ| 2dz = f(β) ρα+2β−2|∇γρ| t+2ψ2dz + ρα+2β |∇γρ| t|∇γψ| where f(β) = −β2 − β(α + Q − 2). Note that f(β) attains the maximum for β = 2−α−Q and this maximum is equal to CH = ( Q+α−2 )2. Therefore we have the following (3.4) α|∇γρ| t|∇γφ| dz = CH α−2|∇γρ| 2−Q|∇γρ| t|∇γψ| It is easy to show that the weight functions w1 = w2 = ρ 2−Q|∇γρ| t satisfies the Mucken- houpt A2 condition for − < t < m . Therefore weighted Poincaré inequality holds (see [FGW], [Lu], [FGaW]) and we have ρ2−Q|∇γρ| t|∇γψ| 2dz ≥ ρ2−Q|∇γρ| tψ2dz ρα|∇γρ| tφ2dz where C is a positive constant and r2 is the radius of the ball Bρ. HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS5 We now obtain the desired inequality (3.5) ρα|∇γρ| t|∇γφ| 2dz ≥ CH ρα−2|∇γρ| t+2φ2dz + ρα|∇γρ| tφ2dz. Using the same method, we have the following weighted Hardy inequality which has a logarithmic remainder term. Similar results in the Euclidean setting can be found in [FT], [AR], [WW], [ACP]. Theorem 3.2. Let α ∈ R, t ∈ R, Q+ α− 2 > 0. Then the following inequality is valid (3.6) ρα|∇γρ| t|∇γφ| 2dz ≥ CH ρα−2|∇γρ| t+2φ2dz + ρα−2|∇γρ| t+2 φ (ln r for all compactly supported smooth function φ ∈ C∞0 (Bρ). Proof. We have the following result from (3.4): (3.7) ρα|∇γρ| t|∇γφ| 2dz = CH ρα−2|∇γρ| t+2φ2dz + ρ2−Q|∇γρ| t|∇γψ| Let ϕ ∈ C∞0 (Bρ) and set ψ(z) = (ln )1/2ϕ(z). A direct computation shows that (3.8) ρ2−Q|∇γρ| t|∇γψ| 2dz ≥ ρ−Q|∇γρ| t+2 ψ (ln r ρα−2|∇γρ| t+2 φ (ln r Substituting (3.8) into (3.7) which yields the desired inequality (3.6). � We now first prove the following weighted Lp-Hardy inequality which plays an important role in the proof of Theorem 3.3, Theorem 4.1 and Theorem 4.5. Theorem 3.3. Let Ω be either bounded or unbounded domain with smooth boundary which contains origin, or Rn. Let α ∈ R, t ∈ R, 1 ≤ p < ∞ and Q + α − p > 0. Then the following inequality holds (3.9) ρα|∇γρ| t|∇γφ| pdz ≥ Q+ α− p ρα|∇γρ| t |∇γρ| |φ|pdz for all compactly supported smooth functions φ ∈ C∞0 (Ω). Proof. Let φ = ρβψ ∈ C∞0 (Ω) and β ∈ R− {0}. We have |∇γ(ρ βψ)| = |βρβ−1ψ∇γρ+ ρ β∇γψ|. We now use the following inequality which is valid for any a, b ∈ Rn and p > 2, |a+ b|p − |a|p ≥ c(p)|b|p + p|a|p−2a · b where c(p) > 0. This yields ρα|∇γρ| t|∇φ|p ≥ |β|pρβp−p+α|∇γρ| p+t|ψ|p + p|β|p−2βρα+βp+1−p|∇γρ| p+t−2|ψ|p−2ψ∇ρ · ∇ψ. Integrating over the domain Ω gives 6 ISMAIL KOMBE (3.10) ρα|∇γρ| t|∇φ|pdx ≥ |β|p ρβp−p+α|∇γρ| t|ψ|pdz |β|p−2βρα+βp+1−p|∇γρ| p+t−2|ψ|p−2ψ∇ρ · ∇ψdz. Applying integration by parts to second integral on the right-hand side of (3.10) and using the fact that ∇γ(|∇γρ|) · ∇γρ = 0 then we get α|∇γρ| t|∇φ|pdx ≥ |β|p − |β|p−2β(βp− p+ α +Q) βp−p+α|∇γρ| p+t|ψ|pdz. We now choose β = p−Q−α to get the desired inequality (3.11) ρα|∇γρ| t|∇φ|pdz ≥ Q+ α− p ρα|∇γρ| t |∇γρ| |φ|pdz. Theorem (3.3) also holds for 1 < p < 2 and in this case we use the following inequality |a+ b|p − |a|p ≥ c(p) (|a|+ |b|)2−p + p|a|p−2a · b where c(p) > 0 (see [L]). � We now have the following improved Hardy inequality which is inspired by recent result of Abdellaoui, Colorado and Peral [ACP]. It is clear that if γ = t = 0 then our result recovers the inequality (1.4). Theorem 3.4. Let Ω ⊂ Rn be a bounded domain with smooth boundary which contains origin, 1 < q < 2, Q + α − 2 > 0, Q = m + (1 + γ)k and φ ∈ C∞0 (Ω) then there exists a positive constant C = C(Q, q,Ω) such that the following inequality is valid (3.12) ρα|∇γρ| t|∇γφ| 2dz ≥ CH |∇γρ| φ2dz + C |∇γφ| |∇γρ| where CH = Q+α−2 Proof. Let φ ∈ C∞0 (Ω) and ψ = ρ β where β ∈ R \ {0}. Then straightforward computation shows that |∇γφ| 2 −∇γ( ) · ∇γψ = Therefore |∇γφ| 2 −∇γ( ) · ∇γψ ρα|∇γρ| tdz = ρα|∇γρ| 2 |∇γρ| where we used the Jensen’s inequality in the last step. Applying integration by parts, we obtain HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS7 |∇γφ| 2 −∇γ( ) · ∇γψ ρα|∇γρ| tdz = |∇γφ| 2ρα|∇γρ| α + β (∆γ(ρ |∇γρ| tφ2dz ρα|∇γρ| t|∇γφ| + β(α+ β +Q− 2) |∇γρ| φ2dz. Therefore we have (3.13) ρα|∇γρ| t|∇γφ| 2dz ≥ −β(α + β +Q− 2) |∇γρ| 2 |∇γρ| We can use the following inequality which is valid for any w1, w2 ∈ R n and 1 < q < 2 (3.14) c(q)|w2| q ≥ |w1 + w2| q − |w1| q − q|w1| q−2〈w1, w2〉. Using the inequality (3.14), Young’s inequality and the weighted Lp-Hardy inequality (3.9), we get (3.15) 2 |∇γρ| 2 dz ≥ C |∇γφ| 2 |∇γρ| where C > 0. Substituting (3.15) into (3.13) then we obtain ρα|∇γρ| t|∇γφ| 2dz ≥ −β(α+β+Q−2) |∇γρ| φ2dz+C |∇γφ| 2 |∇γρ| Now choosing β = 2−α−Q then we have the following inequality ρα|∇γρ| t|∇γφ| 2dz ≥ Q + α− 2 |∇γρ| φ2dz+C |∇γφ| 2 |∇γρ| 4. Sharp Weighted Rellich-type inequalities The main goal of this section is to find sharp analogues of (1.6) and (1.7) for Baouendi- Grushin vector fields. We then obtain their improved versions for bounded domains. The proofs are mainly based on Hardy type inequalities. The following is the first result of this section. Theorem 4.1. (Rellich type inequality I) Let φ ∈ C∞0 (R m+k \ {(0, 0)}), Q = m+ (1 + γ)k and α > 2. Then the following inequality is valid (4.1) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 |∇γρ| φ2dz. Moreover, the constant (Q+α−4)2(Q−α)2 is sharp. 8 ISMAIL KOMBE Proof. A straightforward computation shows that (4.2) ∆γρ α−2 = (Q+ α− 4)(α− 2)ρα−4|∇γρ| Multiplying both sides of (4.2) by φ2 and integrating over Rn, we obtain φ2∆γρ α−2dz = ρα−2(2φ∆γφ+ 2|∇γφ| 2)dz. Since φ2∆γρ α−2dz = (Q+ α− 4)(α− 2) ρα−4|∇γρ| 2φ2dz. Therefore (4.3) (Q+ α− 4)(α− 2) ρα−4|∇γρ| 2φ2dz − 2 ρα−2φ∆γφdx = 2 ρα−2|∇γφ| Applying the weighted Hardy inequality (3.9) to the right hand side of (4.3), we get (4.4) − ρα−2φ∆γφdz ≥ ( Q+ α− 4 ρα−4|∇γρ| 2φ2dz. We now apply the Cauchy-Schwarz inequality to obtain (4.5) − ρα−2φ∆γφdz ≤ ρα−4|∇γρ| 2φ2dz )1/2( |∇γρ|2 |∆γφ| Substituting (4.5) into (4.4) yields the desired inequality (4.6) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 |∇γρ| φ2dz. It only remains to show that the constant C(Q,α) = (Q+α−4)2(Q−α)2 is the best constant for the Rellich inequality (4.1), that is (Q+ α− 4)2(Q− α)2 = inf |∆γf | |∇γρ|2 |∇γρ|2 f 2dz , f ∈ C∞0 (R n), f 6= 0 Given ǫ > 0, take the radial function (4.7) φǫ(ρ) = (Q+α−4 + 1 if ρ ∈ [0, 1], Q+α−4 +ǫ) if ρ > 1, where ǫ > 0. In the sequel we indicate B1 = {ρ(z) : ρ(z) ≤ 1} ρ-ball centered at the origin in Rn with radius 1. By direct computation we get (4.8) |∆γφǫ| |∇γρ|2 |∆γφǫ| |∇γρ|2 Bρ\B1 |∆γφǫ| |∇γρ|2 = A(Q,α, ǫ) +B(Q,α, ǫ) Bρ\B1 ρ−Q−2ǫ|∇γρ| HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS9 where B(Q,α, ǫ) = ( Q+ α− 4 + ǫ)2( − ǫ)2. (4.9) |∇γρ| φ2dz = |∇γρ| φ2dz + Bρ\B1 |∇γρ| = C(Q,α, ǫ) + Bρ\B1 ρ−Q−2ǫdz. Since Q+α−4 > 0 then A(Q,α, ǫ), and C(Q,α, ǫ) are bounded and we conclude by letting ǫ −→ 0. � Using the same argument as above and improved Hardy inequality (3.1), we obtain the following improved Rellich type inequality. Theorem 4.2. Let φ ∈ C∞0 (Bρ), Q = m+(1+γ)k and 4−Q < α < Q. Then the following inequality is valid (4.10) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 |∇γρ| (Q+ α− 4)(Q− α) 2C2r2 ρα−2φ2dz. Proof. We have the following fact from (4.3): (4.11) (Q+ α− 4)(α− 2) ρα−4|∇γρ| 2φ2dz − 2 ρα−2φ∆γφdx = 2 ρα−2|∇γφ| Applying the improved Hardy inequality (3.1) on the right hand side of (4.11), we get (Q + α− 4)(α− 2) ρα−4|∇γρ| 2φ2dz − 2 ρα−2φ∆γφdz Q+ α− 4 ρα−4|∇γρ| 2φ2dz + ρα−2φ2dz Now it is clear that, (4.12) ρα−2φ∆γφdz ≥ ( Q + α− 4 ρα−4|∇γρ| 2φ2dz Next, we apply the Young’s inequality to the expression − ρα−2φ∆φdz and we obtain (4.13) − ρα−2φ∆γφdz ≤ ǫ ρα−4|∇γρ| 2φ2dz + |∆γφ| |∇γρ|2 where ǫ > 0. Combining (4.13) and (4.12), we obtain |∆γφ| |∇γρ|2 −4ǫ2− (Q+α−4)(Q−α)ǫ ρα−4|∇γρ| 2φ2dz+ ρα−2φ2dz. 10 ISMAIL KOMBE Note that the quadratic function −4ǫ2 − (Q + α − 4)(Q − α)ǫ attains the maximum for (Q+α−4)(Q−α) and this maximum is equal to (Q+α−4)2(Q−α)2 . Therefore we obtain the desired inequality (4.14) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 |∇γρ| (Q+ α− 4)(Q− α) 2C2r2 ρα−2φ2dz. Arguing as above, and using the improved Hardy inequalities (3.2) and (3.4) we obtain the following Rellich type inequalities. Theorem 4.3. Let φ ∈ C∞0 (R m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then the following inequality is valid (4.15) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 ρα−4|∇γρ| 2φ2dz (Q+ α− 4)(Q− α) ρα−4|∇γρ| ln( r Theorem 4.4. Let φ ∈ C∞0 (R m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then the following inequality is valid (4.16) |∇γρ|2 |∆γφ| 2dz ≥ (Q+ α− 4)2(Q− α)2 |∇γρ| C(Q− α)(Q+ 3α− 8) |∇γφ| where Ω ⊂ Rn is a bounded domain with smooth boundary. We now have the following Rellich type inequality that connects first to second order derivatives. It is clear that if α = γ = 0 then our result covers the inequality (1.7). Theorem 4.5. (Rellich type inequality II) Let φ ∈ C∞0 (R m+k \ {(0, 0)}), Q = m+(1+ γ)k and 2 < α < Q. Then the following inequality is valid (4.17) |∆γφ| |∇γρ|2 (Q− α)2 |∇γφ| Furthermore, the constant C(Q,α) = is sharp. Proof. The proof of this theorem is similar to the proof Theorem (4.1). Using the same argument as above, we have the following from (4.3) (4.18) − ρα−2φ∆γφdx = ρα−2|∇γφ| (Q + α− 4)(α− 2) ρα−4|∇γρ| 2φ2dz. It is clear that (Q+ α− 4)(α− 2) > 0 and using the Hardy inequality (3.9) (p = 2, t = 0) we get HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS11 (4.19) − ρα−2φ∆γφdz ≥ Q+ α− 4 ρα−2|∇γφ| Let us apply Young’s inequality to expression − ρα−2φ∆γφ dz and we obtain (4.20) φ∆γφdz ≤ ǫ α−4|∇γρ| α |∆γφ| |∇γρ|2 Q + α− 4 α−2|∇γφ| α |∆γφ| |∇γρ|2 where ǫ > 0 and will be chosen later. Substituting (4.20) into (4.19) and rearranging terms, we get (4.21) |∆γφ| |∇γρ|2 −16ǫ2 (Q+ α− 4)2 ( Q− α Q+ α− 4 |∇γφ| Choosing ǫ = 1 (Q− α)(Q+ α− 4) which yields the desired inequality (4.22) α |∆γφ| |∇γρ|2 (Q− α)2 α |∇γφ| To show that constant is sharp, we use the same sequence of functions (4.7) and we get |∆γφǫ| |∇γρ|2 |∇γφǫ|2 (Q− α as ǫ −→ 0. Now, using the same argument as above and improved Hardy inequalities (3.1), (3.6) and (3.7) we obtain the following improved Rellich type inequalities. Theorem 4.6. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following inequality is valid (4.23) |∆γφ| |∇γρ|2 (Q− α)2 |∇γφ| (Q− α)(Q+ 3α− 8) 4C2r2 where C > 0 and r is the radius of the ball Bρ. Theorem 4.7. Let Ω be a bounded domain with smooth boundary ∂Ω. Let φ ∈ C∞0 (Ω), Q = m+ (1 + γ)k and 2 < α < Q. Then the following inequality is valid (4.24) |∆γφ| |∇γρ|2 dz ≥ ( |∇γφ| dz + C̃ |∇γφ| q(α−2) where C̃ = C(Q−α)(Q+3α−8) and C > 0. Theorem 4.8. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following inequality is valid (4.25) |∆γφ| |∇γρ|2 (Q− α)2 |∇γφ| dz + C(Q,α) ρα−4|∇γρ| (ln r where C(Q,α) = (Q−α)(Q+3α−8) 12 ISMAIL KOMBE References [ACP] B. Abdellaoui, D. Colorado, I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations 23 (2005), no. 3, 327-345. 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Ismail Kombe, Mathematics Department, Dawson-Loeffler Science &Mathematics Bldg, Oklahoma City University, 2501 N. Blackwelder, Oklahoma City, OK 73106-1493 E-mail address : [email protected] 1. Introduction 2. Notations and Back ground material 3. Improved Hardy-type inequalities 4. Sharp Weighted Rellich-type inequalities References

0704.1344

Resummation Effects in the Search of SM Higgs Boson at Hadron Colliders

UCRHEP-T428 MSUHEP-061208 Resummation E�e ts in the Sear h of SM Higgs Boson at Hadron Colliders Qing-Hong Cao Department of Physi s and Astronomy, University of California at Riverside, Riverside, CA 92521 Chuan-Ren Chen Department of Physi s and Astronomy, Mi higan State University, E. Lansing, MI 48824 Abstra t We examine the soft-gluon resummation e�e ts, in luding the exa t spin orrelations among the �nal state parti les, in the sear h of the Standard Model Higgs boson, via the pro ess gg → H → WW/ZZ → 4 leptons, at the Tevatron and the LHC. A omparison between the resummation and the Next-to-Leading order (NLO) al ulation is performed after imposing various kinemati s uts suggested in the literature for the Higgs boson sear h. For the H → ZZ mode, the resummation e�e ts in rease the a eptan e of the signal events by about 25%, as ompared to the NLO predi tion, and dramati ally alter various kinemati s distributions of the �nal state leptons. For the H → WW mode, the a eptan e rates of the signal events predi ted by the resummation and NLO al ulations are almost the same, but some of the predi ted kinemati al distributions are quite di�erent. Thus, to pre isely determine the properties of the Higgs boson at hadron olliders, the soft-gluon resummation e�e ts have to be taken into a ount. Ele troni address: q ao�u r.edu Ele troni address: r hen�pa.msu.edu http://arxiv.org/abs/0704.1344v2 mailto:[email protected] mailto:[email protected] I. INTRODUCTION Although Standard Model (SM) explains su essfully all urrent high energy physi s experimental data, the me hanism of ele toweak spontaneous symmetry breaking, arising from the Higgs me hanism, has not yet been tested dire tly. Therefore, sear hing for the Higgs boson (H) is one of the most important tasks at the urrent and future high energy physi s experiments. The negative result of dire t sear h at the LEP2, via the Higgsstrahlung pro ess e+e− → ZH , poses a lower bound of 114.1GeV on the SM Higgs boson mass (MH) [1℄. On the other hand, global �ts to ele troweak observables prefer MH . 200GeV at the 95% on�den e level [2℄, while the triviality arguments put an upper bound ∼ 1TeV [3℄. There is urrently an a tive experimental program at the Tevatron to dire tly sear h for the Higgs boson. The Large Hadron Collider (LHC) at CERN, s heduled to operate in late 2007, is expe ted to establish the existen e of Higgs boson if the SM is truly realized in Nature. At the LHC, the SM Higgs boson is mainly produ ed through gluon-gluon fusion pro ess indu ed by a heavy (top) quark loop. On e being produ ed, it will de ay into a fermion pair or ve tor boson pair. The strategy of sear hing for the Higgs boson depends on how it de ays and how large the de ay bran hing ratio is. If the Higgs boson is lighter than 130GeV, it mainly de ays into a bottom quark pair (bb̄). Unfortunately, it is very di� ult to sear h for the Higgs boson in this mode due to the extremely large Quantum Chromodynami s (QCD) ba kground at the LHC. However, the H → γγ mode an be used to dete t a Higgs boson with the mass below 150 GeV [4, 5℄ though the de ay bran hing ratio of this mode is quite small, ∼ O(10−3). If the Higgs boson mass (MH) is in the region of 130GeV to 2MZ (MZ being the mass of Z boson), the H → ZZ∗ mode is very useful be ause of its lean ollider signature of four isolated harged leptons. The H → WW (∗) mode is also important in this mass region be ause of its large de ay bran hing ratio. When MH > 2MZ , the de ay mode H → ZZ → ℓ+ℓ−ℓ′+ℓ′− is onsidered as the �gold-plated� mode whi h is the most reliable way to dete t the Higgs boson up to MH ∼ 600GeV be ause the ba kgrounds are known rather pre isely and the two on-shell Z bosons ould be re onstru ted experimentally. For MH > 600GeV, one an dete t the H → ZZ → ℓ+ℓ−νν̄ de ay hannel in whi h the signal appears as a Ja obian peak in the missing transverse energy spe trum. The dis overy of the Higgs boson relies on how well we understand the signals and its ba kgrounds, be ause one needs to impose optimal kinemati s uts to suppress the huge ba kgrounds and enhan e the signal to ba kground ratio (S/B). Many works have been done in the literature to al ulate the higher order QCD orre tions to the dominant pro- du tion pro ess of the Higgs boson gg → H [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18℄. In addition to determine the in lusive produ tion rate of the Higgs boson, an a urate predi - tion of kinemati s of the Higgs boson is very essential for the Higgs boson sear h. However, a �x order al ulation annot reliably predi t the transverse momentum (QT ) distribution of Higgs boson for the low QT region where the bulk events a umulate. This is be ause of large orre tions of the form ln(Q2/Q2T ) due to non- omplete an ellations of soft and ollinear singularities between virtual and real ontributions, where Q is the invariant mass of the Higgs boson. Therefore, one needs to take into a ount the e�e ts of the initial state multiple soft-gluon emissions in order to make a reliable predi tion on the kinemati distri- butions of the Higgs boson. One approa h to a hieve this is to in lude parton showering [19℄ whi h resums the universal leading logs in Monte Carlo event generators, e.g. HERWIG [20℄ and PYTHIA [21℄, whi h are ommonly used by experimentalists. The showering pro ess just depends on the initial state parton and the s ale of the hard pro ess being onsidered. The advantage is that it ould be in orporated into various physi s pro esses. Re ently, an approa h to mat h NLO matrix element al ulation and parton showing Monte Carlo generators, MC�NLO [22, 23℄, has been proposed. Another approa h is to in lude orre tly the soft-gluon e�e ts is to al ulate an analyti al result by using the Collins-Soper-Sterman (CSS) resummation formalism [24, 25, 26, 27℄ to resum these large logarithmi orre tions to all order in αs. However, in pra ti e the power of logarithms in luded in Sudakov exponent depends on whi h level the �xed order al ulation has been performed [28, 29, 30, 31, 32℄. It is very interesting to ompare the predi tions between parton showering and resumma- tion al ulation and detailed omparisons have been presented in Ref. [28, 33, 34, 35, 36℄ whi h on luded that all of the distributions are basi ally onsistent with ea h other, ex ept PYTHIA in the small QT region and HERWIG in the large QT region. In addition, the spin orrelation among the Higgs de ay produ ts has been proved to be ru ial to suppress the ba kgrounds [37, 38℄. Hen e, an a urate theoreti al predi tion, whi h in orporates the initial state soft-gluon resummation e�e ts and the spin orrelations among the Higgs de ay produ ts, is needed. In this paper, we present su h a al ulation and study the soft-gluon resummation (RES) e�e ts on various kinemati s distributions of �nal state parti les. Furthermore, we examine the impa t of the RES e�e ts on the Figure 1: Tree level Feynman diagram of pro ess gg → H → V1(→ ℓ1ℓ̄2)V2(→ ℓ3ℓ̄4). a eptan e rate of the signal events with various kinemati s uts (whi h were suggested in the literature [37, 39℄ for Higgs sear h) and ompare them with the leading order (LO) and NLO predi tions The paper is organized as follows. In Se . II, we present our analyti al formalism of the CSS resummation. In Se . III, we present the in lusive ross se tion of the signal pro ess for several ben hmark masses of the Higgs boson. In Se . IV, we study the pro ess gg → H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ for MH = 140GeV at the Fermilab Tevatron and for MH = 170GeV at the LHC. In Se . V, we examine the pro ess gg → H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− for MH = 140GeV and 200GeV, respe tively, and the pro ess gg → H → ZZ → ℓ+ℓ−νν̄ for MH = 600GeV, at the LHC. Our on lusions are given in Se . VI. II. TRANSVERSE MOMENTUM RESUMMATION FORMALISM At the hadron olliders, the SM Higgs boson is mainly produ ed via gluon-gluon fusion pro ess through a heavy quark triangle loop diagram, f. Fig 1, in whi h the e�e t of the triangle loop is repla ed by the e�e tive ggH oupling (denoted as the bold dot). Taking advantage of the narrow width of the Higgs boson, we an fa torize the Higgs boson pro- du tion from its sequential de ay. The resummation formula was already presented in Ref. The NLO Quantum Ele trodynami s (QED) and ele troweak (EW) orre tions to the Higgs de ay pro ess H → WW/ZZ → 4ℓ were al ulated in Ref. [40℄ and Ref. [41℄, respe tively. Re ently, the NLO QCD orre tion to the Higgs boson de ays H → WW/ZZ → 4q with hadroni four-fermion �nal states was al ulated in Ref. [42℄. Sin e the higher order orre tions for Higgs produ tion are dominated by the initial state soft-gluon resummation e�e ts, we fo us our attention on the RES e�e ts in this work. It is worth mentioning that the NLO QED orre tions to the Higgs boson de ay H → WW/ZZ → 4ℓ have been implemented in ResBos [43℄ program, and the phenomenologi al study of the ombined RES e�e ts and the QED orre tion will be presented elsewhere. [28℄. Here, we list some of the relevant formulas as follows, for ompleteness: dσ(h1h2 → H(→ V V → ℓ1ℓ2ℓ3ℓ4)X) dQ2dQ2TdydφHdΠ4 = σ0(gg → H) Q2ΓH/mH (Q2 −m2H)2 + (Q2ΓH/mH)2 M(H → V1V2 → ℓ1ℓ2ℓ3ℓ4) (2π)2 d2b eiQT ·bW̃gg(b∗, Q, x1, x2, C1,2,3)W̃ gg (b, Q, x1, x2) + Y (QT , Q, x1, x2, C4) where Q, QT , y, and φH are the invariant mass, transverse momentum, rapidity, and az- imuthal angle of the Higgs boson, respe tively, de�ned in the lab frame, and dΠ4 represents the four-body phase spa e of the Higgs boson de ay, de�ned in the Collin-Soper frame [44℄. In Eq. (1), |M(· · · )|2 denotes the matrix element square of the Higgs boson de ay and reads M(H → V1V2 → ℓ1ℓ2ℓ3ℓ4) 2G3Fm (q21 −m2V )2 +m2V Γ2V (q22 −m2V )2 +m2V Γ2V C+(p1 · p3)(p2 · p4) + C−(p1 · p4)(p2 · p3) where mV is the ve tor boson mass, qi(pi) denotes the momentum of the ve tor boson Vi (the lepton ℓi), and GF is the Fermi oupling onstant. Here, a212 + b a234 + b ± 4a12b12a34b34, where a12 and b12 respe tively denote the ve tor and axial ve tor omponents of the V ℓ1ℓ2 oupling, while a34 and b34 are the ones for V ℓ3ℓ4. For the W boson, mV = mW , and a = b = while for the Z boson, mV = mZ , and a = 4 sin θ2W − 1, b = −1 for Z → ℓ+ℓ−, a = 1, b = 1 for Z → νν̄, The dire tion of momentum pi is de�ned to be outgoing from the mother parti le. where θW is the weak mixing angle. In Eq. (1), the fun tion W̃gg sums over the soft gluon ontributions that grow as Q−2T × [1 or ln(Q2T/Q2)] to all order in αS, whi h ontains the singular part as QT → 0. The ontribution whi h is less singular than those in luded in W̃gg is al ulated order-by-order in αS and is in luded in the Y term. Therefore, we an obtain the NLO results by expending the above resummation formula, i.e. Eq. (1), to the α3S order. More details an be found in Ref. [43℄. In our al ulation, σ0 in ludes the omplete LO ontribution with �nite quark mass e�e ts [45, 46, 47, 48℄. It has been shown [9℄ that this pres ription approximates well the exa t NLO in lusive Higgs produ tion rate. For the numeri al evaluation, we hose the following set of SM input parameters [49℄: GF = 1.16637× 10−5GeV−2, α = 1/137.0359895, mZ = 91.1875GeV, αs(mZ) = 0.1186, me = 0.5109997MeV, mµ = 0.105658389GeV. Following Ref. [50℄, we derive the W boson mass as mW = 80.385GeV. Thus, the square of the weak gauge oupling is g2 = 4 2m2WGF . In luding the O(αs) QCD orre tions to W → qq̄′, we obtain the W boson width as ΓW = 2.093GeV and the de ay bran hing ratio of Br(W → ℓν) = 0.108 [51℄. In order to in lude the e�e ts of the higher order ele troweak orre tions, we also adapt the e�e tive Born approximation in the al ulation of the H → ZZ → 4 leptons mode by repla ing the sin2 θW in the Zℓℓ oupling by the e�e tive sin2 θ W = 0.2314, al ulated at the mZ s ale. III. INCLUSIVE CROSS SECTIONS For the mass of the Higgs boson being within the intermediate mass range, it will prin i- pally de ay into two ve tor bosons whi h sequentially de ay into either lepton or quark pairs. Leptons are the obje ts whi h an be easily identi�ed in the �nal state, so the di-lepton de ay mode is regarded as the �golden hannel� due to its lean signature and well-known ba k- ground. The drawba k is that the di-lepton mode su�ers from the small de ay bran hing ratio for the ve tor boson de ay (V → ℓℓ̄). For example, the bran hing ratio of Z → ℓ+ℓ− is only about 3.4%. Due to the huge QCD ba kgrounds, the purely hadroni de ay modes are not as useful for dete ting the Higgs boson. In this paper, we fo us on the purely leptoni de ays of the ve tor bosons in the H → Table I: In lusive ross se tions of gg → H → V V → 4ℓ at the Tevatron Run 2 and the LHC in the unit of fb, i.e. σ(gg → H)×Br(H → V V )×Br(V → ℓ1ℓ2)×Br(V → ℓ3ℓ4) for various Higgs boson masses. Here, ℓ and ℓ′ denote either e or µ. WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− ZZ → ℓ+ℓ−νν̄(= i=e,µ,τ νiν̄i) MH 140GeV 170GeV 140GeV 200GeV 600GeV Tevatron LHC LHC LHC LHC RES 13.1 891.1 11.0 17.7 6.3 NLO 11.5 848.9 10.5 16.4 5.6 LO 4.0 405.3 5.1 8.0 2.4 WW (∗) and H → ZZ(∗) modes. To over the intermediate mass range, we onsider the following ben hmark ases: (i) H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ (ℓ, ℓ′ = e or µ) for MH = 140GeV at the Femilab Tevatron Run 2 (a 1.96 TeV pp̄ ollider), and for MH = 170GeV at the LHC (a 14 TeV pp ollider); (ii) H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− (ℓ, ℓ′ = e or µ) for MH = 140 and 200GeV at the LHC; (iii) H → ZZ → ℓ+ℓ−νν̄ for MH = 600GeV at the LHC, where ℓ = e or µ, and ν = νe, νµ or ντ . All the numeri al results are al ulated by using ResBos [43℄. We adapt CTEQ6.1L parton distribution fun tion in the LO al ulation and CTEQ6.1M parton distribution fun tion [52℄ in the NLO and RES al ulations. The renormalization s ale (µR) and fa torization s ale (µF ) are hosen to be the Higgs boson mass in our al ulations, i.e. µR = µF = MH . The in lusive ross se tions for those ben hmark masses of the Higgs boson are sum- marized in Table I where di�erent sear hing hannels are onsidered. For omparison, we show the QT distributions al ulated by using the RES and NLO al ulations in Fig. 3(a). The RES al ulation is similar to that presented in Ref. [28, 29℄ with the known A and B [53, 54, 55, 56, 57℄, but with A g in luded, where [58℄ A(3)g = CACFNf (ζ(3)− + C3A( 11ζ(3) +C2ANf(− 7ζ(3) ) , (2) where CA = 3, CF = 4/3, Nf = 5 and the Riemann onstant ζ(3) = 1.202... . We also use the modi�ed parton momentum fra tions x1 and x2 to take into a ount the kinemati orre tions due to the emitted soft gluons [28℄, with x1 = mT e S and x2 = mT e 0 50 100 150 200 (GeV) = 140 GeV = 170 GeV = 200 GeV = 600 GeV Figure 2: Normalized distributions of transverse momentum of Higgs boson predi ted by RES al ulation at the LHC. where mT = Q2T +Q S is the enter-of-mass energy of the hadron ollider. We also adopt the mat hing pro edure des ribed in the Ref. [43℄ and the non-perturbation on- tribution W̃NP of BLNY form in the Ref. [59℄. In Fig. 2, we show the transverse momentum distributions of Higgs boson predi ted by RES al ulation at the LHC. As we see that the peak position is shifted to larger QT region and the shape be omes broader when the mass of Higgs be omes heavier. It is lear that the predi tion of NLO al ulation blows up in the QT → 0 region and the RES e�e ts have to be in luded to make a reliable predi tion on event shape distributions. In the NLO al ulation, it is ambiguous to treat the singularity of the QT distribution near QT = 0, see the dashed urve in Fig. 3(a). Before presenting our numeri al results, we shall explain how we deal with the singularity in the NLO al ulation when QT ∼ 0. In ResBos, we divide the QT phase spa e with a separation s ale Q T . We al ulate the QT singular part of real emission and virtual orre tion diagrams analyti ally and integrate the sum of these two parts up to Q T . By this pro edure, it yields a �nite NLO ross se tion, for integrating QT from 0 up to Q T , whi h is put into the QT = 0 bin of the NLO QT distribution (for bin width larger than Q T ). Sin e the separation s ale Q T is introdu ed in the theoreti al al ulation for te hni al reasons only and is not a physi al observable, the sum of both ontributions from QT > Q T and QT < Q T should not depend on Q T . As shown in Fig. 3(b), the NLO total ross se tion indeed does not depend on the hoi e of 0 50 100 150 200 (GeV) 0 2 4 6 8 (GeV) σ ( Q total σ ( Q )(a) (b) Figure 3: (a) Distribution of transverse momentum of Higgs boson, and (b) NLO total produ tion ross se tion of Higgs boson via gluon gluon fusion as MH = 170GeV at the LHC. T as long as it is not too large. We refer the readers to the Se . 3 and the Appendix of Ref. [43℄ for more details. In this study, we hoose Q T = 0.96GeV in our numeri al al ulations. As mentioned in the Introdu tion, MC�NLO, whi h mat hes NLO al ulations and par- ton showering Monte Carlo event generators, not only predi ts a reliable QT of the Higgs boson but also in ludes spin orrelations among the Higgs de ay produ ts. Therefore it is in- teresting to ompare the QT predi tions between MC�NLO and RES al ulations. In order to ompare the di�eren es in shape more pre isely, we show the QT distributions predi ted by MC�NLO and ResBos in Fig. 4 for MH = 140( 170, 200, 600)GeV. All distributions are normalized by the total ross se tions for the orresponding Higgs boson masses. The bottom part of ea h QT distribution plot presents the ratio between MC�NLO and ResBos. We note that for a light Higgs boson the distributions are onsistent in the peak region [34, 36℄, where the di�eren e is about 10%, but they are quite di�erent in the large QT region, say QT & 100GeV. For a heavy Higgs boson, e.g. MH = 600GeV, these two dis- tributions are very di�erent in the small QT region, and MC�NLO tends to populate more events in the small QT region, as ompared to ResBos. Sin e the Higgs boson is a s alar, the distributions of Higgs boson de ay produ ts just depend upon the Higgs boson's kinemati s. Therefore, the di�eren e in the QT distribution predi tions between MC�NLO and ResBos may prove to be ru ial for the pre ision measurements of the Higgs boson's properties. A further detailed study of the impa t of the QT di�eren e on the Higgs boson sear h is in MC@NLO 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) = 140 GeV = 200 GeV = 170 GeV = 600 GeV (a) (b) (c) (d) MC@NLO / RES Figure 4: Comparison of the QT distributions between ResBos and MC�NLO. order and will be presented elsewhere. IV. PHENOMENOLOGICAL STUDY OF THE H → WW MODE In the sear h for SM-like Higgs boson via H → WW (∗) mode, two s enarios of W boson de ay were onsidered in the literature [60, 61, 62℄: one is that both W bosons de ay leptoni ally, another is that one W boson de ays leptoni ally and another W boson de ays hadroni ally. Throughout this paper, we only on entrate on the di-lepton de ay mode, i.e. H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ , at the Tevatron and the LHC. The ollider signature, therefore, is two isolated opposite-sign harged leptons plus large missing transverse energy ( 6ET ) whi h originates from the two neutrinos. In this se tion, we �rst examine the RES e�e ts on various kinemati s distributions, and then show the RES e�e ts on the Higgs mass measurement. Finally, we study the RES e�e ts on the a eptan es of the kinemati s uts suggested in the literature for Higgs sear h. (a) (b) (c) Figure 5: Kinemati on�gurations of Higgs de ay (H → WW → ℓ+ℓ−νν̄) in the rest frame of H: (a) H → W++W + , (b) H → W − and ( ) H → W 0 . Here, +(−, 0) denotes the right-handed (left-handed, longitudinal) polarization state of the W boson. The long arrows denote the moving dire tions of the �nal-state leptons. The short bold arrows denote the parti les' spin dire tions. A. Basi kinemati s distributions For a heavy Higgs boson, the two ve tor bosons, whi h are generated from the spin-0 Higgs boson de ay, are predominantly longitudinally polarized, while the longitudinal and trans- verse polarization states are demo rati ally populated when the Higgs boson mass is near the threshold for de aying into the ve tor boson pair [63, 64℄. When 140GeV ≤ MH ≤ 170GeV, the transverse polarization modes ontribute largely. The two harged leptons in the �nal state have di�erent kinemati s be ause of the onservation of angular momentum, f. Fig. 5, therefore, one harged lepton is largely boosted and its momentum be omes harder while another be omes softer. Making use of these di�eren es, one an impose asymmetri trans- verse momentum (pT ) uts on the two harged leptons to suppress the ba kground. On the event-by-event basis, we arrange the two harged leptons in the order of transverse momen- tum: pLmaxT denotes the larger pT between the two harged leptons while p T is the smaller one. Fig. 6 shows the distributions of pLmaxT , p T and missing energy (6ET ) for MH = 140GeV at the Tevatron (�rst row) and for MH = 170GeV at the LHC (se ond row). Furthermore, in Fig. 7 we show the distributions of cos θLL, φLL and ∆YLL without imposing any kinemat- i s ut, where cos θLL is the osine of the opening angle between the two harged leptons, φLL is the azimuthal angle di�eren e between the two harged leptons on the transverse plane, and ∆YLL is the rapidity di�eren e of two harged leptons in the lab frame. Sin e we are mainly interested in the shapes of the kinemati s distributions, the urves shown in the �gures are all normalized by the orresponding total ross se tions. The solid urves present the distributions in luding the RES e�e ts, the dashed and dotted urves present 0 20 40 60 80 100 max (GeV) 0 20 40 60 80 (GeV) 0 20 40 60 80 100 /ET (GeV) 0 20 40 60 80 100 max (GeV) 0 50 100 (GeV) 0 50 100 150 200 /ET (GeV) (a) (b) (c) (d) (e) (f) Figure 6: Normalized distributions of the leading transverse momentum pLmax , softer transverse momentum pLT of the leptons, and the missing energy 6ET in gg → H → WW → ℓ+ℓ′−νℓν̄ℓ′ . The panels (a) to ( ) are for MH = 140GeV at the Tevatron, and (d) to (f) are for MH = 170GeV at the LHC. the distributions al ulated at the NLO and LO, respe tively. We note that the pT distributions of the harged leptons and the missing energy distri- butions are modi�ed largely by the RES e�e ts. This an be understood as follows. The two harged leptons prefer to move in the same dire tion due to the spin orrelation among the de ay produ ts of the Higgs boson, f. the distributions of cos θLL in Figs. 7(a) and (d). Hen e, one an approximately treat the Higgs boson de ay as �two-body� de ay, i.e. de ay- ing into two lusters as H → (ℓ+ℓ′−) (νℓν̄ℓ′). This is in analogy to the W boson produ tion and de ay in the Drell-Yan pro ess, ud̄ → W+ → ℓ+ν, whi h has been shown in Ref. [51℄ that the transverse momentum of lepton (pℓT ) is very sensitive to the transverse momentum of the W boson. The same sensitivity also applies to 6ET . As shown in Figs. 6( ) and (f), the lear Ja obian peak of the 6ET distribution around MH/2 in the LO al ulation is smeared in the NLO and RES al ulations. Furthermore, the 6ET distribution in the NLO and RES al ulations has a long tail due to the non-zero transverse momentum of the Higgs boson. Sin e the RES al ulation in ludes the e�e ts from multiple soft-gluon radiation, the 6ET -1 -0.5 0 0.5 1 0 1 2 3 -4 -2 0 2 4 -1 -0.5 0 0.5 1 0 1 2 3 -4 -2 0 2 4 (a) (b) (c) (d) (e) (f) Figure 7: Normalized distributions of cos θLL, φLL and ∆YLL in gg → H → WW → ℓ+ℓ′−νℓν̄ℓ′ : The panels (a) to ( ) are for MH = 140GeV at the Tevatron and (d) to (f) are for MH = 170GeV at the LHC. distribution near the Ja obian peak is further smeared in the RES al ulation as ompared to the NLO al ulation. When MH = 140GeV, only one W boson is on-shell and the two harged leptons do not move as lose as they do in the ase of MH = 170GeV (in whi h ase, both W bosons are on-shell). However the parallel on�guration is still preferred. The dominant ba kgrounds of the H → WW (∗) mode are from the W boson pair pro- du tion and top quark pair produ tion. The latter, as the redu ible ba kground, an be suppressed with suitable uts su h as jet-veto, but the former, as the irredu ible ba k- ground, still remains even after imposing the basi kinemati uts. In order to redu e this intrinsi ba kground, one needs to take advantage of the hara teristi spin orrelations of the harged leptons in the H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ de ay. For example, the distribution of the di�eren e in azimuthal angles of the harged leptons peaks at smaller value ( f. Figs. 7(b) and (e)) for the signal than that for the WW ontinuum produ tion ba kground [60, 62℄. We note that the RES e�e ts do not a�e t the cos θLL and φLL distributions very mu h, as shown in Figs. 7(a), (b), (d) and (e). To losely examine the di�eren e in their predi tions, we also present the ratio of the RES 10 20 30 40 50 60 max (GeV) RES/NLO RES/LO 0 10 20 30 40 (GeV) 0 10 20 30 40 50 60 70 /ET (GeV) 10 20 30 40 50 60 max (GeV) 0 10 20 30 40 50 (GeV) 0 20 40 60 80 /ET (GeV) (a) (b) (c) (d) (e) (f) Figure 8: Ratio of the Resummation ontribution to NLO and LO ontributions in gg → H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ . The panels (a) to ( ) are for MH = 140GeV at the Tevatron while (d) to (f) are for MH = 170GeV at the LHC. ontribution to the NLO and LO ontributions in Fig. 8. We note that the ratio is about one below the peak regions of pLmaxT , p T and 6ET , and be omes larger than one above the peak region, where both the LO and NLO ontributions drop faster than the RES ontribution does, whi h is onsistent with the results shown in Fig. 6. This uneven behavior indi ates that one annot simply use the leading order kinemati s with the onstant K-fa tor in luded to mimi the higher order quantum orre tions. We should stress that even though the NLO and RES al ulations in lude the same ontributions of the hard gluon radiation from initial states, the e�e ts of the multiple soft-gluon radiation ould ause more than 25% di�eren e between RES and NLO predi tions in the large pT and 6ET region. B. Higgs mass measurement In order to identify the signal events learly, it is ru ial to re onstru t the invariant mass of the Higgs boson. Unfortunately, one annot dire tly re onstru t the MH distribution in the H → WW mode due to the two neutrinos in the �nal state. Instead, both the transverse mass MT and the luster transverse mass MC [65℄, de�ned as 2pLLT 6ET (1− cos∆φ(pLLT , 6ET )), LL+ 6ET , (3) yield a broad peak near MH . In Eq. (3), p T (mLL) denotes the transverse momentum (invariant mass) of the two harged lepton system, and ∆φ(pLLT , 6ET ) is the di�eren e in azimuthal angles between pLLT and 6ET on the transverse plane. We note that the upper endpoint of MT distribution an learly re�e t the mass of Higgs boson, f. Figs. 9(a) and ( ). MT is insensitive to QT be ause it depends on QT in the se ond order, f. Eq. (3). Therefore, the position of the endpoint is only subje t to MH and ΓH . The latter e�e ts an be safely ignored be ause ΓH is very small (less than about 1.5GeV), for the Higgs boson mass less than 200GeV. The luster transverse mass MC also exhibits a lear Ja obian peak with a lear edge at MH , f. Figs. 9(b) and (d). But both the line shape and the Ja obian peak of MC distribution are modi�ed by the RES e�e ts be ause MC is dire tly related to 6ET whi h depends on QT in the �rst order. We suggest that one should use MT to extra t the mass of Higgs in H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ mode be ause the upper endpoint of the MT distribution is insensitive to high order orre tions. C. A eptan e study In order to separate the signal from its opious ba kgrounds, one needs to impose optimal uts to suppress ba kgrounds and enhan e the signal to ba kground ratio (S/B ) simulta- neously. The sele tion of the optimal uts highly depends on how well we understand the kinemati s of the signal and ba kground pro esses. As shown above, the RES e�e ts modify the distributions of transverse momentum of the harged leptons and the missing energy largely, therefore, it is important to study the RES e�e ts on the a eptan es of the kine- mati s uts. Here, we impose a set of kinemati s uts used by experimental olleagues in Refs. [37, 39℄. The orresponding a eptan es are summarized in Table II. • For the sear h for a 140 GeV Higgs boson at the Tevatron, we impose the following basi uts: pLmaxT > 15GeV , p T > 10GeV, |YL| < 2.0 , 6ET > 20GeV, (4) 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) 0 50 100 150 200 (GeV) (a) (b) (c) (d) Figure 9: Normalized distributions of the transverse mass MT and the luster mass MC in gg → H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ : (a) and (b) are for MH = 140GeV at the Fermilab Tevatron while ( ) and (d) are for MH = 170GeV at the LHC. and the optimal uts as follows: mLL < < MT < MH − 10GeV φLL < 2.0 rad , + 20GeV < HT < MH (5) where YL denotes the rapidity of harged lepton, and HT denotes the s alar sum of the transverse momenta of �nal state parti les, i.e. HT ≡ |peT | + |p T | + | 6ET |. The overall e� ien y of the uts is about 68% , 69% and 70% after imposing the basi uts (Eq. (4)) for RES, NLO and LO al ulations, respe tively, and about 44% for both RES and NLO al ulations and 46% for LO al ulation after imposing the optimal uts (Eq. (5)). • For the sear h of a 170 GeV Higgs boson at the LHC, we require the following basi uts: pLmaxT > 20GeV , p T > 10GeV, |YL| < 2.5 , 6ET > 40GeV, (6) Table II: A eptan e of gg → H → WW (∗) → ℓ+ℓ′−νℓν̄ℓ′ events after imposing the basi uts and the optimal uts for MH = 140GeV at the Tevatron and MH = 170GeV at the LHC. MH = 140GeV MH = 170GeV basi (Eq. (4)) optimal (Eq. (5)) basi (Eq. (6)) optimal (Eq. (7)) RES 0.68 0.44 0.61 0.19 NLO 0.69 0.44 0.61 0.19 LO 0.70 0.46 0.63 0.20 and the optimal uts: mLL < 80.0GeV , MH − 30.0GeV < MT < MH , φLL < 1.0 rad , θLL < 0.9 rad , |∆YLL| < 1.5 , (7) The ut e� ien y is about 61% for both RES and NLO al ulations, but about 63% for LO ontribution after imposing the basi ut (Eq. (6)). After imposing the optimal uts (Eq. (7)), the a eptan es of RES and NLO are about 19%, while LO is 20%. V. PHENOMENOLOGICAL STUDY OF THE H → ZZ MODE In the sear h for the SM Higgs boson, the H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− is an important dis overy hannel for a wide range of Higgs boson mass. The appearan e of four harged leptons with large transverse momenta is an attra tive experimental signature. This so- alled �gold-plated� mode provides not only a lean signature to verify the existen e of the Higgs boson but also an ex ellent pro ess to explore its spin and CP properties [66℄. In this se tion, we study three mass values of MH (140, 200 and 600GeV) at the LHC. For MH = 140GeV and 200GeV, we require the two Z bosons both de ay into harged leptons; for MH = 600GeV, we require one Z boson de ays into a harged lepton pair and another Z boson de ays into a neutrino pair, i.e. ℓ+ℓ−νν̄. In this se tion we �rst study the RES e�e ts on various kinemati s distributions and then examine the RES e�e ts on the a eptan es of the kinemati s uts. 0 20 40 60 80 100 max (GeV) 0 20 40 60 80 (GeV) 0 50 100 150 max (GeV) 0 20 40 60 80 100 (GeV) (a) (b) (c) (d) Figure 10: Normalized distributions of pLmax and pLT in gg → H → ZZ → ℓ+ℓ−ℓ′+ℓ′−: (a) and (b) are for MH = 140GeV; ( ) and (d) are for MH = 200GeV at the LHC. A. gg → H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− Similar to the H → WW (∗) mode, we also arrange the four harged leptons of the H → ZZ(∗) mode in the order of transverse momentum. We denote pLmaxT as the largest pT of the four harged leptons while p T the se ond leading pT . In Fig. 10, we show the distributions of pLmaxT and p T for MH = 140 and 200GeV, respe tively. Due to the similar kinemati s dis ussed in the H → WW (∗) mode, the shapes of the distributions of pLmaxT and pLT are hanged signi� antly by the RES e�e ts. The typi al feature is that the RES e�e ts shift the pT of the harged lepton to the larger pT region and, therefore, in rease the a eptan es of the kinemati s uts. The numeri al results will be shown later. Although one an measure the Higgs boson mass by re onstru ting the invariant mass of the four harged leptons, one still needs to re onstru t the Z bosons in order to suppress the ba kgrounds. The re onstru tion of the Z boson depends on the lepton �avors in the �nal state. In this study, we onsider two s enarios: di�erent �avor harged lepton pairs, i.e. H → 2e2µ, and four same �avor harged leptons, i.e. H → 4e( or 4µ). Hen e, we have two methods for re onstru ting the Z bosons: 1. Di�erent �avor harged lepton pairs (2e2µ): In this ase, it is easy to re onstru t the Z bosons be ause both ele tron and muon lepton �avors an be tagged. Using the �avor information, the Z bosons an be re onstru ted by summing over the same �avor opposite-sign leptons in the �nal state. 2. Four same �avor harged leptons (4e/4µ): If the �avors of four leptons are all the same, one needs to pursue some algorithms to re onstru t the Z boson mass. In our analysis, we �rst pair up the leptons with opposite harge. We require the pair whose invariant mass is losest to MZ to be the one generated from the on-shell Z boson, and the other pair is the one generated from another Z boson, whi h ould be on-sell or o�-shell. We name it as the minimal deviation algorithm (MDA) in this paper. In Fig. 11, we show the pT distributions of the re onstru ted Z boson for 140 and 200GeV, respe tively. When the �nal state lepton �avors are di�erent, one an re onstru ted the Z boson perfe tly by mat hing the lepton �avor. For the same �avor leptons, the re onstru ted Z boson distributions in the MDA are shown as the solid, dashed and dot-dashed urves for RES, NLO and LO, respe tively. Some points are worthy to point out as follow: • We note that the MDA an perfe tly re onstru t the distributions of true Z bosons, irregardless whether these two Z bosons are both on-shell or only one of them is on-shell. • When MH = 200GeV, both Z bosons are produ ed on-shell and boosted. The peak position of the transverse of momentum pZT is around (MH/2) 2 −m2Z ∼ 41GeV. For all the ases, the RES e�e ts hange the shape of pZT largely and shift the p T to the larger value region. It has been shown in Ref. [67℄ that angular orrelation between the two Z bosons from the Higgs de ay an be used to suppress the intrinsi ba kground from ZZ pair produ tion e� iently. One of the useful angular variables is the polar angle (θ∗Z) of the (ba k-to-ba k) Z boson momenta in the rest frame of the Higgs boson [67℄. As shown in Fig. 12, in the rest frame of Higgs boson, the ba k-to-ba k Z bosons like to lie in the dire tion perpendi ular to the z−axis, whi h is the moving dire tion of the Higgs boson in the lab frame. After being 0 20 40 60 80 100 (GeV) 0 20 40 60 80 100 (GeV) 0 20 40 60 80 100 (GeV) (a) (b) (c) Figure 11: Normalized transverse momentum of Z boson in gg → H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− at the LHC: (a) is the pT distributions of o�-shell Z boson for MH = 140GeV, (b) is the pT distributions of on-shell Z boson for MH = 140GeV and ( ) is the pT distributions of on-shell Z boson for MH = 200GeV. 0 0.5 1 1.5 0 0.5 1 1.5 (a) (b)M = 140 GeV M = 200 GeV Figure 12: Normalized polar angle of the (ba k-to-ba k) Z boson momenta distributions in the rest frame of the Higgs boson in gg → H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− at the LHC: (a) is for MH = 140GeV , (b) is for MH = 200GeV. boosted to the lab frame, two Z bosons will move lose to ea h other, .f. Fig. 13(b), where θZZ is the opening angle between the two Z bosons in the lab frame. Another interesting angular variable is the angle between the two on-shell Z boson de ay planes (φDP ) in the rest frame of the Higgs boson, whi h is shown in Fig. 13(a). The two Z bosons are re onstru ted as explained above. Sin e the angle θ∗Z and φDP are de�ned in the rest frame of the Higgs boson, the non-zero transverse momentum of the Higgs boson does not a�e t these two variables. Therefore, as learly shown in the �gures, all the distributions of the angular variables mentioned above are the same for the RES, NLO and LO al ulations. -1 -0.5 0 0.5 1 cos φ -1 -0.5 0 0.5 1 cos θ (a) (b) Figure 13: Normalized distributions of cosφDP and cos θZZ in the rest frame of the Higgs boson with mass 200GeV in gg → H → ZZ → ℓ+ℓ−ℓ′+ℓ′− at the LHC. B. gg → H → ZZ → ℓ+ℓ−νν̄ Although the �gold-plated� mode, H → ZZ → ℓ+ℓ−ℓ′+ℓ′−, is onsidered to be the most e�e tive hannel for the SM Higgs boson dis overy at the LHC, it su�ers from the small de ay bran hing of Z → ℓ+ℓ−. Moreover, the larger the Higgs mass be omes, the smaller the produ tion rate is. When the Higgs boson mass is larger than 600 GeV, the H → ZZ → ℓ+ℓ−νν̄ hannel may be ome important be ause the de ay bran hing ratio (Br) of H → ZZ → ℓ+ℓ−νν̄ is six times of the Br of H → ZZ → ℓ+ℓ−ℓ′+ℓ′−. The drawba k is that one annot re onstru t the Higgs mass from the �nal state parti les due to the presen e of two neutrinos. In this dis overy hannel, the missing transverse energy ( 6ET ) is ru ial to suppress the ba kground [37℄. The 6ET distribution is shown in Fig. 14(a) whi h exhibits a Ja obian peak around MH/2, and the soft-gluon resummation e�e ts smear the Ja obian peak and shift more events to the larger 6ET region. Similar to the H → WW mode, the kinemati s of this hannel is similar to the W boson produ tion and de ay in the Drell-Yan pro ess, therefore the shape of 6ET distribution hange signi� antly by the RES ontributions. The Higgs boson mass an be measured from the peaks of the distributions of the transverse mass MT and the luster mass MC , f. Eq. (3), as shown in Fig. 14(b) and ( ). Although the upper endpoint of MT is insensitive to high order orre tions as we mentioned in the study of H → WW (∗) mode, the Ja obian peak is smeared out by the width (ΓH) e�e ts of the Higgs boson. For MH = 600GeV, the total de ay width of the Higgs boson is about 0 100 200 300 400 500 /ET (GeV) 0 200 400 600 800 (GeV) 0 200 400 600 800 (GeV) (a) (b) (c) Figure 14: Normalized distributions of 6ET , MT and MC in gg → H → ZZ → ℓ+ℓ−νν hannel with MH = 600GeV at the LHC. Table III: A eptan e of the pro ess gg → H → ZZ → ℓ+ℓ−ℓ′+ℓ′− for MH = 140 (200)GeV and the pro ess gg → H → ZZ → ℓ+ℓ−νν̄ for MH = 600GeV after imposing uts. MH = 140GeV MH = 200GeV MH = 600GeV basi (Eq. 8) optimal (Eq. 9) basi (Eq. 8) optimal (Eq. 9) basi (Eq. 10) RES 0.53 0.15 0.67 0.14 0.55 NLO 0.54 0.12 0.67 0.11 0.56 LO 0.53 0 0.67 0 0.58 120 GeV, whi h is quite sizable and generates a noti eable smearing e�e t on the Ja obian peak. C. A eptan e study The dis overy potential of the H → ZZ → ℓ+ℓ−ℓ′+ℓ′− and H → ZZ → ℓ+ℓ−νν̄ modes has been studied in Ref. [37℄ after imposing the following uts: • For MH = 140GeV and 200GeV, the intermediate mass range, we impose the basi uts: pET > 7.0GeV, |YL| < 2.5, pLT > 20GeV, (8) and the optimal uts: , (9) where pET and YL are the transverse momentum and rapidity of ea h harged lepton, respe tively, and p T is the pT of the harder Z boson. • For MH = 600GeV, we require: pLT > 40GeV, |YL| < 2.5 , pLLT > 200GeV, 6ET > 150GeV, (10) where pLLT is the transverse momentum of the two harged lepton system. The numeri- al results of the a eptan es of the various uts are summarized in Table III. For the H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− mode, the RES and NLO ontributions have almost the same a eptan es after imposing the basi uts. However, after imposing the optimal uts the a eptan e of the RES ontribution is larger than the one of the NLO ontribution by 25%, and the LO ontribution is largely suppressed. For the H → ZZ → ℓ+ℓ−νν̄ mode, the a eptan es of the RES and NLO al ulations are similar to ea h other. VI. CONCLUSION The sear h for the SM Higgs boson is one of the major goals of the high energy physi s experiments at the LHC, and the ve tor boson de ay modes, H → WW (∗) or H → ZZ(∗), provide powerful and reliable dis overy hannels. The LHC has a great potential to dis- over the Higgs boson even with low luminosity (∼ 30 fb−1) during the early years of run- ning [37, 38, 68℄. In order to extra t the signal from huge ba kground events, we should have better theoreti al predi tions of the signal events as well as ba kground events. In this paper, we examine the soft gluon resummation e�e ts on the sear h of SM Higgs boson via the dominant produ tion pro ess gg → H at the LHC and dis uss the impa ts of the resummation e�e ts on various kinemati s variables whi h are relevant to the Higgs sear h. A omparison between the resummation e�e ts and the NLO al ulation is also presented. For H → WW (∗) → ℓ+ℓ−νν̄ mode, we study MH = 140GeV at the Tevatron and MH = 170GeV at the LHC. Due to the spin orrelations between the �nal state parti les, this pro ess is similar to the W boson produ tion and de ay in the Drell-Yan pro ess. The shapes of the kinemati s distributions are modi�ed signi� antly by RES e�e ts. For example, the e�e ts ould ause ∼ 50% di�eren e ompared to NLO al ulation in the transverse momentum distribution of the leading lepton (p T ), when MH = 170GeV. The Higgs boson mass annot be re onstru ted dire tly from the �nal state parti les be ause of two neutrinos. Therefore, the upper endpoint in the transverse mass distribution an be used to determine the mass of the Higgs boson, and we found that it is insensitive to the RES e�e ts. After imposing various kinemati s uts, the LO, NLO and RES al ulations yield similar a eptan e of the signal events. For the H → ZZ(∗) → ℓ+ℓ−ℓ′+ℓ′− mode, the so- alled �gold-plated� mode, we study MH = 140GeV and MH = 200GeV at the LHC in this paper. We pursue an algorithm, alled minimal deviation algorithm in this paper, to re onstru t the two Z bosons when the four harged leptons in the �nal state have the same �avors. The RES e�e ts hange the shapes of kinemati s signi� antly, e.g. pLmaxT and p T distributions. However, the variables φDP and θ Z , de�ned in the Higgs rest frame, are insensitive to RES e�e ts. After imposing the optimal kinemati s uts, the RES e�e ts ould in rease the a eptan e by 25% ompared to that of NLO al ulation while the LO ontribution is largely suppressed. When the Higgs boson is heavy (600GeV), we onsider the H → ZZ → ℓ+ℓ−νν̄ mode be ause of its larger de ay bran hing ratio, as ompared to the H → ZZ → ℓ+ℓ−ℓ′+ℓ′− mode. The shape of 6ET distribution, whi h is ru ial to suppress the ba kgrounds, is largely modi�ed be ause it is sensitive to the transverse momentum of the Higgs boson. In summary, we have presented a study of initial state soft-gluon resummation e�e ts on the sear h for the SM Higgs boson via gluon-gluon fusion at the LHC. The e�e ts not only signi� antly modify some of the kinemati distributions of the �nal state parti les, as ompared to the NLO and LO predi tions, but also enhan e the a eptan e of the signal events after imposing the kinemati uts to suppress the large ba kground events. Therefore, we on lude that the initial state soft-gluon resummation e�e ts should be taken into a ount as sear hing for the Higgs boson at the LHC. In addition, we note that the spin orrelations among the �nal state leptons ould be modi�ed by the ele troweak orre tions to the Higgs boson de ay. Therefore, we have implemented the NLO QED orre tion in the ResBos ode, and the phenomenologi al study will be presented in the forth oming paper. A knowledgments We thank Professor C.-P. Yuan for a riti al reading and useful suggestions. 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0704.1345

Triquark structure and isospin symmetry breaking in exotic Ds mesons

Triquark structure and isospin symmetry breaking in exotic Ds mesons S. Yasui and M. Oka Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan January 14, 2019 Abstract The color anti-triplet triquark qq̄q̄ is considered as a compact component in the tetraquark structure cqq̄q̄ of exotic Ds mesons. We discuss the mass spectrum and the flavor mixing of the triquarks by using the instanton induced interaction and the one-gluon exchange potentials. As a characteristic property of the triquark, we investigate the isospin violation. It is shown that the flavor 3 (isosinglet) and 6 (isotriplet) states may be strongly mixed and then are identified with Ds(2632). 1 Introduction Exotic Ds mesons attract much attention recently. The BaBar Collaboration [1] first announced the Ds(2317) with J π = 0+, whose mass lies approximately 160 MeV below the constituent quark model predictions [2, 3]. The decay width is less than 4.6 MeV. This state was confirmed by CLEO [4] and Belle [5]. It was followed by Ds(2460) with 1+, reported by CLEO [4], and Ds(2632) by SELEX [6]. At the same time, a charmonium candidate X(3872) was reported by Belle [7] and the other groups. These new mesons are novel because they do not fit to the quark model expectations and have caused further studies and speculations on their structures. In particular, it has been proposed that some of them are excited two-quark states [8, 9], chiral doublets in heavy quark limit [10], the molecular bound states [11, 12] or tetraquarks cqq̄q̄ (q = u, d and s) [13, 14, 15, 16, 17, 18, 19]. Here we briefly review the previous researches about the molecular and tetraquark pictures. Some suggest that the Ds(2317) may be a DK molecule state, and the Ds(2460) a D∗K molecule. Indeed the masses of the Ds(2317) and Ds(2460) are slightly below the thresholds of theD+K andD∗+K, respectively. The mass splitting 140 MeV between Ds and D∗s is almost the same as that between the Ds(2317) and Ds(2460). The ground state of Ds is the Ds(1969) with 0 − and the Ds(2112) with 1 −. Therefore, the Ds(2317) can be assigned to be the ground state in the 0+ sector, while the Ds(2460) can be identified http://arxiv.org/abs/0704.1345v3 also as a ground state in the 1+ sector. These properties are also explained in the chiral effective theory [10]. One of the prominent properties of the exotic Ds mesons is the isospin violating decay process Ds(2317) → Dsπ0 [1]. This process is considered to be realized by the virtual η emission and the η − π mixing, since the Ds(2317) is supposedly an isosinglet state.1 An anomalous branching ratio Γ(Ds(2632) → D0K+)/Γ(Ds(2632) → Dsη) ≃ 0.16 ± 0.06 [6] is also a very interesting problem. In the conventional cs̄ picture, the decay to the D0K+ is favored as compared with the decay to Dsη, since uū or dd̄ creation would be easier than ss̄. Maiani et al. [17] considered the tetraquark state cs̄dd̄, in which the isospin is maximally violated. This state has decay modes of [cs̄][dd̄] (Dsη) and [cd̄][ds̄] (D +K0), while the D0K+ decay is suppressed by the OZI forbidden process dd̄→ uū. This picture was also applied to study of the isospin violation in the decay process of X(3872) [18]. On the other hand, Chen and Li considered cs̄ss̄ [14]. They discussed that the decay to Dsη is dominant, while the decay to D 0K+ is suppressed by 1/Nc due to the OZI rule and the reduction of the amplitude due to the color matching to create a color singlet state. Liu and Zhu discussed that the Ds(2632) is assigned as an isosinglet member in flavor 15 [19]. Let us see the possibility of the isospin violated eigenstates [17, 18]. When the quarks have sufficiently large momenta, the asymptotic freedom suppresses the qq̄ creation pro- cess, and the flavor mixing interaction is less important. Then, the alignment in the diagonal components in the mass matrix realizes uū and dd̄ separately as eigenstates, which are mixed states of isosinglet and isotriplet states. In general, however, the flavor mixing term is in the order of ∼100 MeV [20, 21, 22, 23, 24], and much larger than the mass difference between u and d quarks, |mu −md| <∼ 5 MeV. Therefore, it is expected that the isospin breaking effect is too small to separate uū and dd̄. The purpose of this paper is to discuss the microscopic mechanism of the isospin viola- tion of the tetraquark for the open charm system, cqq̄q̄. It is noticed that the interaction between the light quarks q and the c quark is suppressed in the heavy quark limit. Thus, it is natural to consider that, in the first approximation, three light quarks are decoupled from the heavy quark. Therefore we consider states compound by the u, d and s quarks as triquarks or color non-singlet baryons. Here it must be noticed such a state cannot exist as an asymptotic state, but only in the bound state. We here consider a simple model with non-relativistic valence quarks under the influ- ence of the one-gluon exchange (OGE) and the instanton induced interaction (III). The mass spectroscopy of the triquark was first discussed in the diquark-triquark picture in the literature of the pentaquark [25, 26, 27], and further investigated in details in the OGE in- teraction [28, 29, 30]. Furthermore, the ’t Hooft interaction induced by the instanton was also used [31, 32, 33]. However, the effective interaction employed in [31, 32, 33] operates only in spin singlet and isosinglet channel in qq̄ pair, while the effective interaction used in [20, 21, 22, 23, 24] operates, not only in spin singlet and isosinglet channel, but also in spin triplet and isotriplet channel. It is known that the difference causes a discrepancy in 1Hayashigaki and Terasaki considers a possibility of the isotriplet state for Ds(2317) [16]. the meson mass spectrum [34]. Therefore it is an interesting problem to investigate the isospin violation of the triquark by using the effective interaction in [20, 21, 22, 23, 24]. The content of this paper is as follows. In Section 2, the flavor representation of the triquark, the III and OGE potential and the mass matrix are discussed. In Section 3, the isospin mixing is investigated by considering the ud quark mass difference. In Section 4, our discussion is summarized. 2 Quark model In the tetraquark picture of the exotic Ds mesons, the triquark is considered as a bound state composed by three light flavor quarks. The hamiltonian of the triquark is obtained only in light flavors space, since the interaction between the light and heavy quarks is suppressed in the OGE potential. This is also the case for the instanton induced interaction, since the heavy quark has no zero mode and free from the instanton vacuum [36, 37]. The flavor SU(3) multiplets of the triquark state qq̄q̄ are given as 3⊗ 3⊗ 3 = 3S ⊕ 3A ⊕ 6A ⊕ 15S. We write the subscripts of S and A according to the symmetry under the exchange of two anti-quarks. In Fig. 1, we show the weight diagrams of these multiplets. It is assumed that all the quarks and anti-quarks occupy the lowest energy single particle orbital, the s-wave orbital. In the following, we omit the subscripts in 6A and 15S for simplicity. The exotic states reported in experiments have the strangeness S = +1. Then, the isospin for each flavor multiplet is as follows; isosinglet for 3A, 3S and 15 , and isotriplet for 6 and 15 . Here the isospin components of 15 are distinguished by the superscript. It is straightforward to write down the flavor wavefunctions of these multiplets for S = +1. isosinglet   |3A〉 = 12 u(s̄ū− ūs̄)− d(d̄s̄− s̄d̄) |3S〉 = 12√2 2ss̄s̄+ u(s̄ū+ ūs̄) + d(d̄s̄+ s̄d̄) |150〉 = 1 2ss̄s̄− u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄) isotriplet |6〉 = 1 u(s̄ū− ūs̄) + d(d̄s̄− s̄d̄) |151〉 = 1 u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄) In the following discussion, we consider only the S = +1 sector. The triquark must belong to the color anti-triplet state, 3 S and 3 A, so that the tetraquark is a color singlet state. Then, the spin and color combination of the tri- quark is restricted by the Pauli principle. For example, the spin and color basis for the flavor 3A and 6 states with the spin J = 1/2 is {|λX〉, |ρY 〉}. Here, λ and ρ stands for the mixed states with λ- and ρ-symmetry in spin 1/2, and X and Y for color 3 S and 3 S S S 3 6 15 Figure 1: The weight diagram of the flavor SU(3) multiplets of the triquark qq̄q̄. respectively. On the other hand, the basis for the flavor 3S and 15 states with spin 1/2 is {|ρX〉, |λY 〉}. For spin J=3/2 state, we have |J=3/2 X〉 for 3A and 6, and |J =3/2 Y 〉 for 3S and 15. Now we discuss the hamiltonian of the triquark. The instanton induced interaction (III) has played very important role in the QCD vacuum in accompany with dynamical chiral symmetry breaking [20, 21, 22, 23, 24]. It induces the Kobayashi-Kondo-Maskawa- ’t Hooft (KKMT) interaction [35, 36, 37], which is given as 2Nf point-like vertex with the flavor anti-symmetric channel. In the quark model, the instanton effect has been discussed in the non-relativistic limit in the KKMT interaction. The OGE potential is also often used as an effective interaction [38]. Here we consider a hybrid model of the III and OGE potentials [20, 21, 22, 23, 24]. The hamiltonian is H = K + pIII III +H + (1− pIII)VOGE +Mmass + Vconf , (2) with the kinetic term K, the instanton induced interaction H III (i = 2 and 3 for the two- and three-body interactions), the OGE potential VOGE, the mass matrix Mmass and the confinement potential Vconf . The parameter pIII controls the ratio of the III and the OGE potentials. In the present discussion, we are interested in the isospin symmetry breaking, and not involved with the absolute masses of the tetraquarks. Therefore, we pick up only the III and OGE terms and the mass matrix; H̃ = pIII III +H + (1− pIII)VOGE +Mmass. (3) Since we do not solve the quark confinement dynamically, we just use a quark wave function from the harmonic oscillator potential with frequency ω. Concerning the III potential, the three body force in the three quark state, q1q2q3, is given in the flavor diagonal form as III = ~λ1 ·~λ2 + ~λ2 ·~λ3 + ~λ3 ·~λ1 dabcλ ~σ1 ·~σ2~λ1 ·~λ2 + ~σ2 ·~σ3~λ2 ·~λ3 + ~σ3 ·~σ1~λ3 ·~λ1 0 −〈ψ̄ψ〉 mq ω αs -0.2564 [GeV fm3] (0.25)3 [GeV3] 0.3837 [GeV] 0.5 [GeV] 1.319 Table 1: The parameter set from [21]. dabcλ 3 (~σ1 ·~σ2 + ~σ2 ·~σ3 + ~σ3 ·~σ1) ǫijkσ 3fabcλ δ(3)(~r1 − ~r2)δ(3)(~r2 − ~r3), with a coupling constant V 0 and the delta functions as a point-like three body interaction. We can deduce the two body instanton induced force, III = ~λi ·~λj + ~σi ·~σj~λi ·~λj δ(3)(~ri − ~rj), using the quark condensate 〈ψ̄ψ〉, where the coupling constant V (2)0 is given as 〈ψ̄ψ〉V (3)0 . The interactions in the q1q̄2q̄3 state are also obtained in a straightforward way. The OGE potential between the q1q2 pair is given as VOGE = 4παS ~λ1 ·~λ2 − ~σ1 ·~σ2 6m1m2 with a coupling constant αS. The first term is the electric interaction, and the second the magnetic interaction with spin dependence. However, we neglect the electric interaction, since in general it is sufficiently small as compared with the magnetic interaction. It should be noted that the magnetic interaction is switched off with a suppression of 1/mQ for the heavy-light quark pair (Qq) in the limit of the heavy mass. Therefore, it is understood that the triquark qq̄q̄ may exist as a compound unit in the tetraquark structure. As a summary, our interaction is sketched in Fig. 2. The parameter set in our interaction [21] is summarized in Table 1. From the III and OGE potentials, the energy spectrum of the triquark is obtained in the following way. By using the basis of the spin and color, {|λX〉, |ρY 〉, |ρX〉, |λY 〉}, we obtain the hamiltonian in matrix forms for flavor 3A, 3S, 6 and 15 representations, respectively. First we consider the III potential. For the flavor 3A and 3S states, the hamiltonian is given in the basis {|λX〉, |ρY 〉, |ρX〉, |λY 〉} by HIII(3) =  0 3 −3  0 I2 +  0 0 0 0 0 0 0 0  0 I3, (4) (c) qq (d) qq (e) qq (annihilation) (a) two-body (qq) (b) three-body (qqq) Figure 2: The diagram contributions for the III and OGE potentials. III: (a) the two-body interaction for qq̄ and (b) the three body interaction for qq̄q̄. OGE: (c) qq̄, (d) q̄q̄ and (e) qq̄ (annihilation). where I2 and I3 are the expectation values of the delta function for the point-like inter- action, I2 = 〈Ψ|δ(3)(r1 − r2)|Ψ〉 = for the two-body interaction, and I3 = 〈Ψ|δ(3)(r1 − r2)δ(3)(r2 − r3)|Ψ〉 = for the three-body interaction with the triquark spatial wavefunction Ψ. It should be mentioned that the 3A and 3S states are mixed due to the off-diagonal element in the two- body interaction in the III potential, since {|λX〉, |ρY 〉} belongs to 3A and {|ρX〉, |λY 〉} to 3S. Therefore, we may denote the mixed state as 3 in the following discussion. In the similar way, for the flavor 6 state, the basis {|λX〉, |ρY 〉} gives the matrix HIII(6) = 0 I2 + 0 I3, (7) and for the flavor 15 state, the basis {|ρX〉, |λY 〉} gives HIII(15) = 0 I2. (8) Second we consider the OGE potential. For the 3 state, we obtain the matrix in the basis of {|λX〉, |ρY 〉, |ρX〉, |λY 〉}, VOGE(3) =   I2. (9) Furthermore, for the 6 state, we obtain in the basis {|λX〉, |ρY 〉} VOGE(6) = I2, (10) and for the 15 state VOGE(15) = I2, (11) in the basis {|ρX〉, |λY 〉}. The mass differences among u, d and s quarks induce mixings between the flavor representations. In the basis of the flavor representation, {|3A〉, |3S〉, |15 0〉, |6〉, |151〉}, we easily obtain the mass part of the hamiltonian for S = +1 sector, as Mmass =  mu +md +ms 0 0 mu −md 0 0 mu+md + 2ms −mu+md2 +ms 0 mu−md√ 0 −mu+md +ms 0 0 −mu−md√2 mu −md 0 0 mu +md +ms 0 0 mu−md√ −mu−md√  .(12) The diagonal elements are isosinglet and isotriplet components, while the off-diagonal elements induce mixings between them. Note that the flavor representations with the same symmetry (A or S) are mixed. The 3A and 15 states are mixed with each other by the SU(3) symmetry breaking ( mu = md < ms). We also note that the isosinglet states (3A, 3S, 15 ) and the isotriplet states (6, 15 ) are also mixed due to the isospin symmetry breaking (mu < md) 2. We consider this interaction as a driving force for the isospin symmetry breaking in the next section. It should be noted that the Coulomb or electromagnetic interaction may also break isospin symmetry, which is not considered in this study. 3 Isospin mixing In general, the u − d quark mass difference is sufficiently small as compared with the energy splitting between the isosinglet and isotriplet states, and the isospin breaking can 2In the works in [31, 32, 33], the 3A and 6 states are mixed due to the mu = md 6= ms. As long as the isospin symmetry is not violated, however, we have no mixing between the 3A and 6 states. be neglected. However, in the triquark, we see that the isosinglet and isotriplet states sometimes happen to be degenerate and thus a large isospin mixing can occur. In this section, we investigate the mixing of the isosinglet and isotriplet states. For this purpose, we calculate the eigenenergies, E, of the hamiltonian (3). We choose the s quark mass ms = 0.48 GeV and the strength of the harmonic potential ω = 0.50 GeV in the following discussion. We take the parameter pIII as a free parameter. We present the binding energy spectrum of the triquark, ∆E = E − (mu +md +ms), for the OGE (pIII = 0) and III (pIII = 1) potentials in Fig. 3. The isosinglet and isotriplet states are shown by the solid and dashed lines, respectively. As the J = 3/2 states are heavier than J = 1/2, in the following discussion, we pay attention to the ground states with the spin J=1/2, the 3 and 6 multiplets. Let us see the result by the III potential. In SU(3) symmetric case, the ground state is the 3 state, which contains mainly the 3A component rather than the 3S component. On the other hand, the 3S component is mixed in the excited state in the 3 state. Now we break the SU(3) symmetry with keeping the isospin symmetry; mu = md < ms. The ground state is still the 3 state, and the first excited state is the 15 state, followed by the 15 and 6 states. The splitting between the 15 and 15 states makes the former lifted up as compared with the latter. This splitting comes from the fact that the mass matrix (12) mixes the 3S and 15 states. The same mixing pushes the 3 upward. On the other hand, in the OGE potential, the ground state is the 6 state, followed by the 3, 15 , and 15 states. It should be noticed that the flavor multiplets are different in the III and the OGE potentials. Especially the change of the ground state flavor is important for the isospin symmetry breaking as we see below. The reason that the 6 state is the ground state in the OGE can be understood by examining the annihilation diagram in Fig. 2(e). It vanishes for the usual color singlet meson qq̄, since the gluon (g) contained in the process qq̄ → g → qq̄ is a color octet state. In the triquark, however, the annihilation diagram does not vanish. This is because the gq̄ state contained in the process qq̄q̄ → gq̄ → qq̄q̄ remains color anti-triplet 3c due to the color decomposition, c ⊗ 3c = 3c ⊕ 6c ⊕ 15c. Note that the initial and final qq̄q̄ states are also color anti-triplet. The annihilation term increases the energy of the flavor 3 state, while it does not operate for the 6 state (see Eq. (1) ). Consequently, the 3 state is about 50 MeV above the 6 state in the OGE potential. Let us return to the discussion of the isospin symmetry breaking. We recall that the 3 state is isosinglet and the 6 state is isotriplet. Thus the mixing of the flavor multiplets are directly related to the isospin mixing. Explicitly, we plot the binding energies of the flavor multiplets as functions of the parameter pIII in Fig. 4. The solid lines indicate the isosinglet states, and the dashed lines the isotriplet states. We find that the isosinglet and isotriplet states become degenerate at A (pIII = 0.18) and B (pIII = 0.82). Now let us introduce the isospin symmetry breaking, namely the ud quark mass differ- ence, ∆m = md −mu ≃ 0.005 GeV [39]. The ∆m is comparable to the energy difference between the isosinglet and isotriplet states at A and B. There, the two degenerate states ΔE [GeV] (p =0) (p =1) SU(3) breaking SU(3) symmetry SU(3) breaking SU(3) symmetry III III isosinglet isotriplet Figure 3: The binding energies of the triquarks with J = 1/2 for the OGE (pIII = 0) and III (pIII = 1) potentials. 0 0.5 1 I=0 (3bar, 15bar) I=1 (6) I=1 (15bar) Figure 4: The binding energies of the various flavor multiplets with SU(3) breaking as functions of the parameter pIII . The bold-solid line indicates the isosinglet (3 and 15 ) states, the bold- dashed line isotriplet (6), and the thin-dashed line the isotriplet (15 ) states. Cf. Fig. 3. 0 0.1 0.2 0.3 0.4 uubar ddbar ssbar 0.6 0.7 0.8 0.9 1 uubar ddbar ssbar Figure 5: The ratios of the isosinglet (bold-solid line) and isotriplet (bold-dashed line) states as functions of the parameter pIII . (a) and (b) corresponds to the state A and B, respectively, in Fig. 4. The ratios of the uū (thin-solid line), dd̄ (thin-dashed line) and ss̄ (thin-dot-dashed line) are also shown. will split into two isospin mixed states which are orthogonal to each other. Here we choose one state at A. The ratios of isosinglet and isotriplet components are plotted as functions of the parameter pIII in Fig. 5(a). In the range of 0.16 < pIII < 0.20, we see a rapid change of the isosinglet (bold-solid line) and the isotriplet (bold-dashed line) components, hence the isospin is strongly mixed. In the same way at B, we also see an isospin mixing at pIII = 0.82 as shown in Fig. 5(b). However, in contrast to the case A, the isospin mixing at B occurs in a small range of the parameter pIII . This is understood from the mass matrix (12). At A, the isosinglet state is almost the 3A multiplet, while the isotriplet state is purely the 6 multiplet (see Fig.3). The mass matrix (12) induces the 3A and 6 multiplet mixing, namely the isospin violation, by mu−md. On the other hand, at B, the isosinglet state is changed to be the 15 state, while the isotriplet state is the same. In the mass matrix (12), however, there is no direct mixing between the 15 and 6 multiplets. They are mixed indirectly through the multi-step mixings of the 6 − 3A, 3A − 3S, and 3S − 15 . Therefore the isospin mixing at B is suppressed as compared to that at A. Here we recall the isospin violation in experimental observations. Maiani et al. con- sider Ds(2632) as the cs̄dd̄ state, which is an isospin mixed state [17]. In our analysis, the isospin mixing at A induces a mixing between the isosinglet (mostly 3A) and isotriplet (6) states. Hence, from Eq. (1), the mixed wavefunction, |3A〉 − |6〉 = −d(d̄s̄− s̄d̄), contains only the dd̄ component. On the other hand, at B, there is a mixing between the isosinglet (mostly 15 ) and the isotriplet (6) states. There, from Eq. (1), the mixed wavefunction |150〉 − |6〉 contains both of the uū and dd̄ components with the same fraction. Conse- quently, we see that the isospin mixed states, cs̄uū and cs̄dd̄, become separate eigenstates by the 3− 6 mixing rather than the 150 − 6 mixing. We also understand this result explicitly by looking at the fraction of uū, dd̄ and ss̄ components at A and B in Fig. 5(a) and (b), respectively. In Fig. 5(a), the dd̄ fraction (thin-dashed line) is overwhelming as compared with the uū fraction (thin-solid line) around pIII = 0.18. In contrast, in Fig. 5(b), the fraction of the uū and dd̄ components are almost the same at pIII = 0.82. Therefore, the isospin mixed state at A gives the cs̄dd̄, while the state at B does not. Thus, the discussion by Maiani et al. in [17] is proven to be possible as the the 3− 6 mixing. So far, we have discussed the isospin mixing by using the isospin basis of isosinglet and isotriplet. However, the isospin mixing is also investigated by basis {uū, dd̄}. Then the hamiltonian is generally given by (uū dd̄ uū m δ dd̄ δ m+ 2∆m . (13) The uū and dd̄ are eigenstates of this hamiltonian, if the flavor mixing term δ is much smaller than ∆m, and only the diagonal component is dominant. However, in general, δ is in the order of hundred MeV in the vacuum as we see the mass splitting of π − η. For the triquark with J = 1/2, due to the combination of spin and color, we have four uū-like states and also four dd̄-like states. In this basis, the hamiltonian is given by uū dd̄  m1 0 δ11 δ12 δ13 δ14 m2 δ21 δ22 δ23 δ24 m3 δ31 δ32 δ33 δ34 0 m4 δ41 δ42 δ43 δ44 δ11 δ12 δ13 δ14 m1+2∆m 0 δ21 δ22 δ23 δ24 m2+2∆m δ31 δ32 δ33 δ34 m3+2∆m δ41 δ42 δ43 δ44 0 m4+2∆m  where the diagonal uū−uū and dd̄−dd̄ parts are diagonalized in the spin and color spaces. The diagonalized energy, m1, m2, m3 and m4, are plotted as functions of the parameter pIII in Fig. 6(a). If the flavor mixing strength δij (i, j = 1, · · · , 4) are sufficiently small, the lowest uū- and dd̄-like states become eigenstates. As shown in Fig. 6(b), δ11 is so small as compared with ∆m around pIII = 0.18. There, the eigenstate become uū- and dd̄-like states, hence the isosinglet and isotriplets states are ideally mixed. This result is consistent with our discussion that the isospin mixing is caused by the 3−6mixing around pIII = 0.18. It should be noted that the contribution from the higher states is suppressed since the mixing is in the order of δij/(mk−m1) ≃ 0.1 (k ≥ 2) in the perturbation theory. Therefore the first order perturbation is sufficient for the present discussion. Lastly, we discuss the parameter dependence of the isospin mixing. We employ several free parameters, the s quark mass ms, the harmonic oscillator potential frequency ω and the parameter pIII for the OGE and III potentials. They may have some uncertainty due to the lack of the experimental information. However, one sees that the results are not 0 0.5 1 0 0.1 0.2 0.3 0.4 Figure 6: (a) The diagonal components of the uū − dd̄ matrix as functions of the parameter pIII . (b) δ11 as a function of pIII . Note the energy unit is given by MeV in (b). See the text. modified qualitatively by parameter change. As an example, we plot the size parameter b = 1/ mqω of the triquark wavefunction, which causes the 3 − 6 mixing at A, as a function of the parameter pIII for ms = 0.48 GeV and 0.58 GeV. We see that the b comes within a reasonable range 0.4 < b < 0.6 fm, and is not far from b = 0.5 fm [21]. This range is little affected by ms. Concerning the range of the parameter pIII , the obtained value pIII = 0.18 is smaller than the conventionally used value pIII ≃ 0.4 in hadron spectroscopy [21]. This observa- tion indicates that the OGE is more dominant than the instanton induced interaction in tetraquarks. It is noticed that the value pIII = 0.18 is not obtained dynamically, since the quark wave function is assumed to be Gaussian. The present study suggests that there would exist an essential mechanism to choose such pIII in charmed tetraquark. When the linear potential is used as a confinement potential, the quark wave function is modified from that of the harmonic oscillator potential, and the absolute values of the OGE and the instanton induced interaction are also modified. However, the ratio of both interactions is not changed, since both of the potentials are point-like interactions. In the present discussion, the isospin symmetry breaking is induced by the ratio of two interactions. Therefore our conclusion is not modified qualitatively. 4 Conclusion Possibility of isospin violation in the Ds tetraquark systems is examined in this paper. Tetraquarks are candidates of the exotic Ds mesons recently reported in experiment. We consider the energy spectrum of the triquark by using the non-relativistic quark model with the instanton induced interaction and the one-gluon exchange potentials. With taking the SU(3) symmetry breaking into account for S = +1 sector, we show that the flavor 3 (isosinglet) and the flavor 6 (isotriplet) representations form the ground states. Considering the isospin symmetry breaking by the quark mass difference, mu < md, it is 0 0.5 1 ms=0.48 [GeV] ms=0.58 [GeV] Figure 7: The b− pIII relation for the 3− 6 mixing. ms = 0.48 GeV (solid line) and 0.58 GeV (dashed line). See the text. shown that the 3 (isosinglet) and 6 (isotriplet) states may be mixed strongly with some range of the parameter pIII . There the isosinglet and the isotriplet states are ideally mixed, and one of the eigenstates is dominated by the dd̄ component. This result is also investigated by looking at the off-diagonal components in the uū − dd̄ matrix. Our conclusion supports the discussion given in [17, 18]. How do we experimentally confirm the picture given in this paper? The present mechanism of the isospin symmetry violation relies on the suppression of the flavor mixing interaction. Thus, at the ideal (maximal) mixing, the uū- and dd̄-like states are split by the diagonal part of the mass matrix, namely by 2∆m ∼ 10 MeV. Therefore the two states are expected to come close to each other. So far, due to the experimental restriction, only a few charged decay modes are observed, and they suggest a dd̄-like state, D+s (2630/cs̄dd̄), where its main decay mode is D+s (2630) → Dsη, while D+s (2630) → D0K+ is suppressed. The corresponding uū-like state will show different decay patterns. Therefore careful analyses of different charged modes of decays will reveal the nature of the isospin breaking. In particular, the decays into Dsπ 0 and D+K0 are two interesting modes. The present study suggests the possibility of the triquark à la “color non-singlet baryon”, which is a color non-singlet particle composed by three quarks. Although the triquark itself cannot exist asymptotically, it may appear as an effective degree of freedom in the exotic heavy mesons in heavy quark mass limit. It is considered in general that the color non-singlet light quark systems may exist by color neutralization with heavy quark spectator [30]. The triquark is a possible candidate among the color non-singlet quark systems, which can be examined by studying the tetraquark structure of exotic open charm mesons. The triquark would be also an interesting object in the lattice QCD simulation. Furthermore the triquark may be a relevant degree of freedom as a color non- singlet compound particle in the deconfinement phase such as the quark-gluon plasma and the quark matter. In order to understand such states in many aspects, it is important to study several properties, such as masses, decay widths and so forth. Acknowledgment We express our thanks to Dr. T. Shinozaki and Prof. S. Takeuchi for discussions. This work is supported by a Grant-in-Aid for Scientific Research for Priority Areas, MEXT (Ministry of Education, Culture, Sports, Science and Technology) with No. 17070002. References [1] B. Aubert, et al., [BABAR Collaboration], Phys. Rev. Lett. 90 242001 (2003). [2] S. Godfrey and N. Isgur, Phys. Rev. D32 189 (1985). [3] R. N. Cahn and J. D. Jackson, Phys. Rev. D68 037502 (2003). [4] D. Besson, et al., [CLEO Collaboration], Phys. Rev. D68 032002 (2003). [5] P. Krokovny, et al., [Belle Collaboration], Phys. Rev. Lett. 91 262002 (2003). [6] A. V. Evdokimov, et al., [SELEX Collaboration], Phys. Rev. Lett. 93 242001 (2004). [7] S.-K. Choi, et al., [Belle Collaboration], Phys. Rev. Lett. 91 262001(2003). [8] E. van Beveren and G. Rupp, Phys. Rev. Lett. 91, 012003 (2003). [9] T. Matsuki, T. Morii and K. Sudoh, Eur. Phys. J. A31, 701 (2007). [10] W. A. Bardeen, E. J. Eichten and C. T. Hill, Phys. Rev. D68 054024 (2003). [11] T. Barnes, F. E. Close and H. J. Lipkin, Phys. Rev. D68 054006 (2003). [12] A. P. Szczepaniak, Phys. Lett. B567 23 (2003). [13] H.-Y. Cheng and W.-S. Hou, Phys. Lett. B566 193 (2003). [14] Y.-Q. Chen and X.-Q. Li, Phys. Rev. Lett. 93 232001 (2004). [15] E. S. Swanson, Phys. Rep. 429 243 (2006). [16] A. Hayashigaki and K. Terasaki, Prog. Theor. Phys. 114 1191 (2006). [17] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D70 054009 (2004). [18] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D71 014028 (2005). [19] Y.-R. Liu, Shi-Lin Zhu, Y.-B. Dai and C. Liu Phys. Rev. D70 094009 (2004). [20] M. Oka and S. Takeuchi, Phys. Rev. Lett. 63 1780 (1989). [21] M. Oka and S. Takeuchi, Nucl. Phys. A524 649 (1991). [22] S. Takeuchi, Phys. Rev. Lett. 73 2173 (1994). [23] S. Takeuchi, Phys. Rev. D53 6619 (1996). [24] T. Shinozaki, M. Oka and S. Takeuchi, Phys. Rev. D71 074025 (2005). [25] M. Karliner and H. J. Lipkin, Phys. Lett. B575 249 (2003). [26] N. I. Kochelev, H.-J. Lee and V. Vento, Phys. Lett. B594 87 (2004). [27] H.-J. Lee, N. I. Kochelev and V. Vento, Phys. Lett. B610 50 (2005). [28] H. Hogaasen and P. Sorba, Mod. Phys. Lett. A19 2403 (2004). [29] P. Jiménez Delgado, Few Body Syst. 37 215 (2005). [30] R. L. Jaffe, Phys.Rev. D72, 074508 (2005). [31] V. Dmitrasinović, Phys. Rev. D70 096011 (2004). [32] V. Dmitrasinović, Phys. Rev. Lett. 94 162002 (2005). [33] V. Dmitrasinović, Mod. Phys. Lett. A21 533 (2006). [34] V. Dmitrasinovic, Phys. Rev. D71 094003 (2005). [35] M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. 45 1955 (1971). [36] G. ’t Hooft, Phys. Rev. D14 3432 (1976) [Erratum, ibid. 18 2199 (1978)]. [37] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B163 46 (1980). [38] A. De. Rújula, H. Georgi and S. L. Glashow, Phys. Rev. D12 147 (1975). [39] T. Nasu, M. Oka and S. Takeuchi, Phys. Rev. C68 024006 (2003); ibid. C69, 029903(E) (2004). Introduction Quark model Isospin mixing Conclusion

0704.1346

Prediction of future fifteen solar cycles

Prediction of future fifteen solar cycles K. M. Hiremath Indian Institute of astrophysics, Bangalore-560034, India [email protected] ABSTRACT In the previous study (Hiremath 2006a), the solar cycle is modeled as a forced and damped harmonic oscillator and from all the 22 cycles (1755-1996), long- term amplitudes, frequencies, phases and decay factor are obtained. Using these physical parameters of the previous 22 solar cycles and by an autoregressive model, we predict the amplitude and period of the future fifteen solar cycles. Predicted amplitude of the present solar cycle (23) matches very well with the observations. The period of the present cycle is found to be 11.73 years. With these encouraging results, we also predict the profiles of future 15 solar cycles. Important predictions are : (i) the period and amplitude of the cycle 24 are 9.34 years and 110 (±11), (ii) the period and amplitude of the cycle 25 are 12.49 years and 110 (± 11), (iii) during the cycles 26 (2030-2042 AD), 27 (2042-2054 AD), 34 (2118-2127 AD), 37 (2152-2163 AD) and 38 (2163-2176 AD), the sun might experience a very high sunspot activity, (iv) the sun might also experience a very low ( around 60) sunspot activity during cycle 31 (2089-2100 AD) and, (v) length of the solar cycles vary from 8.65 yrs for the cycle 33 to maximum of 13.07 yrs for the cycle 35. Subject headings: sunspots – solar cycle – prediction 1. Introduction Owing to proximity, the sun influences the earth’s climate and environment. Over- whelming evidence is building up that the solar cycle and related activity phenomena are correlated with the earth’s global climate and temperature, the sea surface temperatures of the three (Atlantic, Pacific and Indian) main ocean basins, the earth’s albedo, the galactic cosmic ray flux that in turn is correlated with the earth’s cloud cover and, Indian monsoon rainfall (Hiremath and Mandi 2004 and references there in; Georgieva et. al. 2005; Hire- math 2006b). The transient parts of the solar activity such as the flares and the coronal http://arxiv.org/abs/0704.1346v1 – 2 – mass ejections that are directed towards the earth create havoc in the earth’s atmosphere by disrupting the global communication, reducing life time of the earth bound satellites and, keep in dark places of the earth that are at higher latitudes by breaking the electric power grids. Owing to sun’s immense influence of space weather effects on the earth’s environment and climate, it is necessary to predict and know in advance different physical parameters such as amplitude and period of the future solar cycles. There are many predictions in the literature (Ohl 1966; Feynman 1982; Feynman and Gu 1986; Kane 1999; Hathaway, Wilson and Reichmann 1999; Badalyan, Obrido and Sykora 2001; Duhau 2003 Sello 2003; Maris, Poepscu and Besliu 2003; Euler and Smith 2004; Maris, Poepscu and Besliu 2004; Kaftan 2004; Echer et. al. 2004; Gholipour et. al., 2005; Schatten 2005; Li, Gao and Su 2005; Svaalgaard, Cliver and Kamide 2005; Chopra and Dabas 2006; Dikpati, Toma and Gilman 2006; Du 2006; Hathaway and Wilson 2006; Clilverd et. al., 2006; Tritakis and Vasilis 2006; Lantos 2006; Lundstedt 2006; Wang and Sheeley 2006; Choudhuri, Chatterjee and Jiang 2007; Javaraiah 2007) on the previous and future 24th solar cycles and beyond. Most of these studies mainly concentrate on prediction of the amplitude (maximum sunspot number during a cycle). However, prediction of period (length) of a solar cycle is also very important parameter and the present study fills that gap. Recently we modeled the solar activity cycle as a forced and damped harmonic oscillator that consists of both the sinusoidal and transient parts (eqn 1 of Hiremath 2006a). From the 22 cycles (1755-1996) sunspot data, the physical parameters (amplitudes, frequencies, phases and decay factors) of such a harmonic oscillator are determined. The constancy of the amplitudes and the frequencies of the sinusoidal part and a very small decay factor from the transient part suggests that the solar activity cycle mainly consists of persistent oscillatory part that might be compatible with long-period (∼ 22 yrs) Alfven oscillations. In the present study, with an autoregressive model and by using the physical parameters of 22 cycles, we predict the amplitudes and periods of future 16 solar cycles. Thus prediction from this study can be considered as a physical and precursor method. A Pth order autoregressive model relates a forecasted value xt of the time series X = [x0, x1, x2, ..., xt−1], as a linear combination of P past values xt = φ1xt−1 + φ2xt−2 + ...... + φpxt−p +Wt , where the coefficients φ1, φ2, ..., φp are calculated such that they minimize the uncorrelated random error terms, Wt. The routine is available in IDL software. Important condition for using an autoregressive model is that the series must be stationary such that it’s mean and standard deviation do not vary much with time. Hence, one can not apply autoregressive model directly to the observed sunspot series as it consists of near sinusoidal trends whose amplitudes and the standard deviations entirely different for different solar cycles. On the other hand, the derived physical parameters of the forced and damped – 3 – harmonic oscillator (Hiremath 2006a) for all the 22 solar cycles are stationary and, hence in the following, we use an autoregressive model to predict the future 15 solar cycles. The solution of the forced and damped harmonic oscillator (see the equation 1 of Hir- math 2006a) of the solar cycle consists of two parts : (i) the sinusoidal part that determines the amplitude and period of the solar cycle and, (ii) the transient part that dictates decay of the solar cycle from the maximum year and also determines bimodal structure of the sunspot cycle around the maximum years for some cycles. In the present study, we use physical parameters of the sinusoidal part only to predict amplitude and period of future cycles. 2. Results and conclusion Using past 22 cycles’ physical parameters, we construct the next (23rd) solar cycle and presented in Fig 1. Except decaying part of the solar cycle, one can notice that the predicted curve exactly matches with the observed curve. With this encouragement and from an autoregressive model, the physical parameters of future 16 solar cycles are computed and reconstructed solar cycles are presented in Fig 2. For the coming cycles 24-38, the results are summarized in Table 1. In Table 1, the first column represents the cycle number, the second column represents the year from minimum-minimum, the third column represents the period (length) of the solar cycle and, the last column represents the maximum sunspot number during a cycle. It is interesting to note that the amplitude of the cycle 24 is low compared to the amplitude of cycle 23 and is almost similar to average value computed from all of the predicted models (http://members.chello.be/j.janssens/SC24.html). Other interesting predictions are : (i) during the cycles 26, 27, 34. 37 and 38, the sun will experiences a very high solar activity, (ii) during cycle 31 (2087-2099 AD) the sun will experiences a very low sunspot activity and, (iii) length of the solar cycles vary from 8.65 yrs for the cycle 33 to maximum of 13.07 yrs for cycle 25. To conclude, the solar cycle is modeled as a forced and damped harmonic oscillator. From the previous 22 cycles sunspot data, the physical parameters such as the amplitudes, the frequencies and phases of such a harmonic oscillator are determined. The sinusoidal part of the forced and damped harmonic oscillator of previous solar cycles is considered for the prediction of future 16 cycles. With an autoregressive model and using previous 22 cycles parameters, coming 16 solar cycles are reconstructed from the predicted parameters. Important results of this prediction are : the amplitude of coming solar cycle 24 will be smaller than the present cycle 23 and around 2087-2099 AD, the sun will experiences a very low sunspot activity. http://members.chello.be/j.janssens/SC24.html – 4 – The author is thankful to Dr. Luc Dame and Dr. Javaraiah for the useful discussions. REFERENCES Badalyan, O. G., Obrido, V. N & Sykora, J. 2001, solar physics, 199, 421 Chopra, P & Dabas, R. S. 2006, in the Cospar Proceedings, Beijing Choudhuri, A. R., Chatterjee, P & Jiang, J. 2007, astro-ph/0701527 Clilverd, M. A., Clarke, E., Ulich, T., Rishberth, H & Jarvis, M. J., 2006. Space Weather, vol 4, S09005 Dikpati, M., Toma, G. D & Gilman, P. 2006, Geophys. Res. Lett., 33, L05102 Du, Z. L. 2006, A&A, 457, 309 Duhau, S. 2003, Sol. Phys., 213, 203 Echer, E., Rigozo, N. R., Nordemann, D. J. R & Vieira, I. E. A. 2004, Annales Geophysicae, 22, 2239 Feynman, J. 1982, JGR, 87, 6153 Feynman, J & Gu, X. Y. 1986, Rev. Geophys., 24, 650 Georgieva, K., Kiro, B., Javaraiah, J & Krasteva, R. 2005, Planet. Space. Sci., 53, 197 Gholipour, A., Lucas, C., Arabi, B. N & Shafiee, M. 2005, Solar. Terr. Phys, 67, 595 Javaraiah, J. 2007, MNRAS LETTERS, DOI: 10.1111/j.1745-3933.2007.00298.x Hathaway, D., Wilson, R. M & Reichmann, J. 1999, JGR, 104, 375 Hathaway, D & Wilson, R. M. 2006, Geophys. res. lett., 33, L18101 Hiremath, K, M & Mandi, P. I. 2004, New Astronomy, 9, 651 Hiremath, K. M. 2006a, A&A, 452, 591 Hiremath, K. M. 2006b, ILWS Workshop, Goa, India, Edts : Gopalswamy, N and Bhat- tacharya, A., p. 178 Kaftan, V. 2004, in the proceedings of IAU symp 223, p. 111 http://arxiv.org/abs/astro-ph/0701527 – 5 – Kane, R. P. 1999, Sol. Phys., 189, 217 antos, S. 2006, Sol. Phys., 236, 399 undstedt, H. 2006, Advance. Space. Res., 38, 862 Mari, S. G., Popescu, M. D., & Besliu, D. 2003, Romanian. Astron. Journ., 13, 139 Mari, S. G., Popescu, M. D., & Besliu, D. 2004, in the proceedings of IAU Symp 233 Li, K.J., Gao, P. X & Su, T.w. 2005, Chin. J. Astron. Astrophys., vol 5, No 5, 539 Ohl, A. I. 1966, Sol. Dannye, No. 12, 84 Sello, S. 2003, A & A, 410, 691 Schatten, K. 2005, Geophys. Res. Let, 32, L21106 Svalgaard, L., Cliver, E. W & Kamide, Y. 2005, Geophysical. Res. Let, 32, L001104 Vasilis, T., Helen, M & George, G. 2006, in the proceedings of AIP conference, 848, 154 ang, Y. M & Sheeley, Jr, N. R. 2006, Nature Phys, vol 2, Issue 6, 367 This preprint was prepared with the AAS LATEX macros v5.2. – 6 – Fig. 1.— Prediction of the future solar cycles. (a) The left plot illustrates the predicted (red continuous) curve over plotted on the observed (blue curve) sunspot cycle 23. The dashed red curves represent uncertainty in the Prediction. (b) The right plot illustrates predicted future 15 solar cycles. The red numbers over different solar cycle maximum are the cycle numbers. – 7 – Table 1. Predicted sunspot cycles Cycle Year Period Maximum Number Min-Min (Years) Number 23 1996.00-2007.73 11.73 136±14 24 2007.73-2017.07 9.34 110±11 25 2017.07-2029.56 12.49 110±11 26 2029.56-2041.50 11.94 157±16 27 2041.50-2053.51 12.00 180±18 28 2053.51-2064.30 10.80 140±14 29 2064.30-2075.01 10.71 149±15 30 2075.01-2086.79 11.78 118±12 31 2086.79-2097.95 11.16 63±6 32 2097.95-2108.84 10.89 108±11 33 2108.84-2117.49 8.65 128±13 34 2117.49-2126.92 9.43 170±17 35 2126.92-2139.99 13.07 139±14 36 2139.99-2151.74 11.75 159±16 37 2151.74-2163.19 11.45 187±19 38 2163.19-2175.48 12.29 187±19 Introduction Results and conclusion

0704.1347

Comparaison entre cohomologie cristalline et cohomologie \'etale $p$-adique sur certaines vari\'et\'es de Shimura

COMPARAISON ENTRE COHOMOLOGIE CRISTALLINE ET COHOMOLOGIE ÉTALE p-ADIQUE SUR CERTAINES VARIÉTÉS DE SHIMURA Sandra Rozensztajn Résumé. — Soit X un modèle entier en un premier p d’une variété de Shimura de type PEL, ayant bonne réduction associée à un groupe réductif G. On peut associer aux Zp-représentations du groupe G deux types de faisceaux : des cristaux sur la fibre spéciale de X, et des systèmes locaux pour la topologie étale sur la fibre générique. Nous établissons un théorème de comparaison entre la cohomologie de ces deux types de faisceaux. Abstract (Comparison between crystalline cohomology and p-adic étale cohomology on certain Shimura varieties) Let X be an integral model at a prime p of a Shimura variety of PEL type having good reduction, associated to a reductive group G. To Zp reprsententations of the group G can be associated two kinds of sheaves : crystals on the special fiber of X, and locally constant étale sheaves on the generic fiber. We establish a comparison between the cohomology of these two kinds of sheaves. 1. Introduction Considérons X un modèle entier d’une variété de Shimura de type PEL, défini sur une extension de Zp. Cette variété de Shimura correspond à un groupe réductif G, défini sur Z(p). On peut associer aux Z(p)-représentations du groupe G différents faisceaux : des systèmes locaux en Zp-modules sur la fibre générique de X , et des cristaux sur sa fibre spéciale. Nous établissons une comparaison entre la cohomologie étale du système local d’une part, et la cohomologie log-cristalline d’une extension du cristal à une compactification appropriée de X , pour une même représentation V du groupe G. Nous traitons ici le cas des variétés de Shimura unitaires et de celles de type Siegel. Dans le cas unitaire, nous obtenons un résultat qui tient compte de la torsion (théorème 6.3), dans le cas Siegel les résutats sont moins précis et ne sont valables qu’après tensorisation par Qp (théorème 6.4). L’intérêt de cette comparaison est que nous pouvons obtenir des renseignements sur le côté cristallin : des techniques de type complexe BGG, décrites par exemple dans [3], chapitre VI, permettent d’avoir http://arxiv.org/abs/0704.1347v1 2 SANDRA ROZENSZTAJN des renseignements sur la filtration de Hodge. On déduit alors des informations sur le côté étale, vu comme représentation galoisienne. La théorie de Hodge p-adique nous donne de tels théorèmes de comparaison entre cohomologie étale p-adique et log-cristalline dans le cas des coefficients constants, pour des schémas propres et possédant certaines propriétés de lissité. Ces résultats sont dûs entre autres à Tsuji ([15]) pour le cas où l’on considère les groupes de cohomologie après tensorisation par Qp, et à Tsuji et Breuil pour le cas où l’on tient compte de la torsion ([16] et [2], il y a alors des restrictions sur le degré des groupes de cohomologie que l’on peut étudier). Ces théorèmes sont rappelés dans le paragraphe 6.1. Le principe de notre méthode est de considérer la cohomologie à coefficients constants de la variété abélienne universelle sur X et de ses puissances, et d’en déduire la com- paraison qui nous intéresse en découpant les groupes de cohomologie des faisceaux considérés dans les groupes de cohomologie à coefficients constants à l’aide de cer- taines correspondances algébriques. Le premier problème est que de tels théorèmes de comparaison n’étant valables que sur des schémas propres, nous devons supposer l’existence (prouvée dans certains cas seulement) de compactifications non seulement de X , mais aussi des variétés de Kuga-Sato, et plus précisément un système projectif de telles compactifications. La partie 2 explique précisément dans quelle situation nous nous pla cons, ainsi que les propriétés des compactifications que nous utilisons. Le deuxième problème est que toutes les représentations de G ne donnent pas des faisceaux dont la cohomologie puisse être découpée par des correspondances algébriques dans la cohomologie de la variété abélienne universelle. On détermine dans la partie 3 quelles sont les représentations de G qui donnent des faisceaux que l’on peut atteindre de cette fa con, qui sont les seuls pour lesquels nous obtenons un résultat. Cette partie utilise fortement la structure des représentations du groupe réductif G, ce qui explique que l’on soit obligé de faire une description cas particulier par cas particulier. Nous expliquons la construction des faisceaux ainsi que l’action de ces correspon- dances algébriques dans la partie 4. Enfin l’énoncé et la preuve du théorème principal occupent la partie 6. C’est ici qu’apparâıt une différence entre les cas unitaire et Siegel : en effet le point-clé de la preuve est la compatibilité de l’action des correspondances algébriques que nous considérons avec les théorèmes de comparaison à coefficients constants. Le cas Siegel utilise la compatibilité de cet isomorphisme de comparaison avec les structures produit sur les groupes de cohomologie, ce qui n’est démontré que dans le cas rationnel et non dans le cas de torsion. 2. Les objets considérés 2.1. Variétés de Shimura de type PEL. — 2.1.1. Les données. — On se donne B une Q-algèbre simple finie, munie d’une invo- lution positive notée ∗ (c’est-à-dire que trB/Q(xx ∗) > 0 pour tout x non nul de B), V un module de type fini sur B, muni d’une forme bilinéaire (, ) telle que pour tous v et w dans V, et tout b dans B on ait (bv, w) = (v, b∗w). On notera 2g la dimension de V sur Q. On fixe dans toute la suite un nombre premier p, et on fera l’hypothèse que p > 2g. Le rôle de cette hypothèse est expliqué au paragraphe 4.3.1. On suppose que B est non ramifié en p, c’est-à-dire que BQp est un produit d’algèbres de matrices sur des extensions non ramifiées de Qp. On se donne un Z(p)-ordre OB dans B qui devient un ordre maximal de BQp après tensorisation par Zp, et stable par l’involution de B. On se donne aussi V un OB-réseau de V autodual. Le fait que V soit autodual implique en particulier que la forme bilinéaire induite sur V est non dégénérée. Soit C l’anneau des endomorphismes B-linéaires de V. On définit le groupe G par G(R) = {g ∈ (C ⊗ R)∗, ∃µ ∈ R∗, ∀v, w ∈ V ⊗ R, (gv, gw) = µ(v, w)}, pour toute Z(p)-algèbre R. On se donne un R-homomorphisme d’algèbres h : C → C∞ = C ⊗Q R tel que h(z)∗ = h(z̄), et la forme (v, h(i)w) soit définie positive sur V∞ = V⊗QR. On associe à h le morphisme µh : C ∗ → GC, qui définit la filtration de Hodge sur VC, c’est-à-dire la décomposition V = Vz ⊕V1, où µh(z) agit par z sur Vz et par 1 sur V1. Le corps dual associé à ces données est le corps E(G, h) qui est le corps de définition de la classe d’isomorphisme de Vz comme B-représentation. C’est le sous-corps de C engendré par les tr(b), b ∈ B agissant sur Vz. 2.1.2. Deux cas particuliers. — Dans la suite nous nous intéresserons uniquement à deux cas particuliers : le cas Siegel et le cas unitaire. Le cas Siegel correspond à la situation où B est réduit à Q. La variété de Shimura associée est alors la variété modulaire de Siegel. Le cas unitaire correspond au cas où B est une extension qua- dratique imaginaire de Q. La forme alternée (, ) est alors la partie imaginaire d’une forme hermitienne sur V, qu’on peut voir comme un B-espace vectoriel de dimension moitié. 2.1.3. Le problème de modules. — On peut associer aux données de Shimura prcédentes un problème de modules, tel que décrit dans [10], dont on rappelle ici l’essentiel. Fixons Kp un sous-groupe compact ouvert de G(A f ). On considère le foncteur des OE(G,h) ⊗ Z(p)-schémas dans les ensembles, qui à S associe l’ensemble à équivalence près des quadruplets (A, λ, i, η), où A est un schéma abélien A sur S, muni d’une polarisation λ première à p, et d’une flèche i : OB → End(A) ⊗ Z(p) qui est un morphisme d’algèbres à involution, l’involutin sur End(A) ⊗ Z(p) étant l’involution de Rosati donnée par λ, et η est une structure de niveau. Enfin on suppose que OB agit sur Lie(A) comme sur Vz, c’est-à-dire que det(b,Lie(A)) = det(b,Vz) pour tout b ∈ OB . La structure de niveau consiste en ce qui suit : on considère le A f -module de Tate de A, c’est un A f -faisceau lisse sur S. Soit s un point géométrique de S, une structure de niveau consiste en une Kp-orbite η d’isomorphismes VAp → H1(As,A de B-modules munis d’une forme alternée, et qui soit fixée par π1(S, s). 4 SANDRA ROZENSZTAJN Deux quadruplets (A, λ, i, η) et (A′, λ′, i′, η′) sont dit équivalents s’il existe une isogénie première à p de A vers A′, commutant à l’action de OB, transformant η en η′, et λ en un multiple scalaire (dans Z∗(p)) de λ Ce foncteur est représentable, par un schéma quasi-projectif et lisse M sur OE ⊗ Z(p) pourvu que l’on choisisse K p suffisamment petit. Notons A le schéma abélien universel sur M. Pour les cas Siegel et unitaire, on a le résultat suivant (lemme 7.2 de [10]) : Lemme 2.1. — OM ⊗ V et H 1(A/M)∨ sont localement isomorphes comme OB- modules munis d’une forme alternée. 2.1.4. La situation géométrique considérée. — On obtient alors la situation suivante : NotonsK le complété en une place v|p de E(G, h), etO son anneau des entiers. Notons On = O/̟ n+1, où ̟ est une uniformisante de O. C’est l’anneau des vecteurs de Witt de longueur n sur le corps résiduel de O puisque p est non ramifié dans E(G, h). Posons S = SpecO, et Sn = SpecOn. D’une fa con générale, on notera avec un indice n la réduction d’un O-schéma modulo ̟n+1 On notera X = MO, c’est donc un schéma lisse sur S, muni d’un schéma abélien A, provenant du schéma abélien universel sur la variété de Shimura. De plus, on a un morphisme OB → End(A) ⊗Z Z(p). On note f : A → X , et fs : A s → X les morphismes structuraux. 2.2. Existence de compactifications. — Nous aurons besoin d’utiliser aussi des compatifications de X , ainsi que du schéma abélien universel A et de ses puissances. Ces compactifications ont été décrites en détail dans le cas Siegel ([3]), pour le cas unitaire la construction détaillée n’est écrite que pour GU(2, 1), c’est-à-dire les sur- faces modulaires de Picard (voir [11] pour la compactification de la base et [14] pour celle du schéma abélien universel), même si leur existence dans le cas unitaire général ne pose pas de problèmes. Nous résumons dans ce paragraphe les seules propriétés de ces compactifications que nous utilisons. 2.2.1. Compactifications de la base. — Nous avons besoin tout d’abord de compactifi- cation du modèle entier de la variété de Shimura. En considérant des compactifications toröıdales (décrites dans [3], chapitre IV pour le cas Siegel, et dans [11] pour le cas de GU(2, 1)), on obtient l’existence d’un schéma X vérifiant la propriété suivante : Propriété 1. — il existe un schéma X propre et lisse sur S, contenant X comme ouvert dense, tel que le complémentaire de X dans X est un diviseur à croisements normaux relatifs. On fixera dans la suite une fois pour toute une telle compactificationX . On peut re- marquer que les résultats ne dépendent en fait pas du choix de X parmi l’ensemble des compactifications toröıdales : deux compactifications toröıdales sont toujours compa- rables, au sens où il existe une troisième qui les domine toutes les deux, ce qui permet de voir que les groupes de cohomologie décrits en 5.2.5 ne dépendent pas de ce choix de compactification. 2.2.2. Compactifications du schéma abélien universel. — On se donne X propre lisse sur S, muni d’un diviseur à croisements normaux relatifs D, et on note X l’ouvert complémentaire. Pour chaque s, on appelle bonne compactification de As une com- pactification As de As, telle que f−1s (X \X) est un diviseur à croisements normaux relatifs. Considérons la des compactifications toröıdales « lisses » des As (voir [3] dans le cas Siegel, [14] dans le cas de GU(2, 1)), on obtient pour tout s ≥ 1, une famille de bonnes compactifications As de As, vérifiant les deux propriétés suivantes : Propriété 2. — 1. Pour toute isogénie u de As, il existe deux bonnes compacti- fications As1 et As2, et un morphisme As1 → As2 prolongeant u. 2. Étant donné deux bonnes compactificationsAs1 etAs2, il en existe une troisième As3 et des X-morphismes As3 → As1 et As3 → As2 induisant l’identité sur A Dans le cas Siegel, nous utiliserons encore une propriété supplémentaire de la famille des compactifications toröıdales : Propriété 3. — Si L est un faisceau symétrique sur As, il existe des entiers a et b, et une compactification de la famille As tels que le faisceau (O(2)⊗a ⊗L)⊗b se prolonge en un faisceau sur As. De plus, on peut choisir a et b premiers à p. Cette construction fait l’objet du chapitre VI du livre [3]. Elle n’est détaillée que pour le cas où le faisceau symétrique ample considéré est O(2), mais cela s’adapte au cas d’un faisceau symétrique ample quelconque, pour lequel on prendra donc O(2)⊗a ⊗ L, avec a assez grand pour que le faisceau soit ample. 3. Représentations de G 3.1. Z(p)-représentations. — On note Rep(G) la catégorie des représentations de G sur un Z(p)-module libre de type fini, et RepQ(G) celles des représentations de G sur un Q-espace vectoriel de dimension finie. Notons V0 ∈ Rep(G) la duale de la représentation standard de G, c’est-à-dire la duale du réseau V défini au paragraphe 2.1.1. 3.2. L’algèbre des endomorphismes. — Les représentations de G de la forme ∧•Vs0 , pour s ≥ 1, jouent un rôle particulier : les faisceaux que nous allons leur associer dans la section 4 ont une interprétation géométrique. Nous définissons une sous-algèbre de l’algèbre End(∧•Vs0 ) des endomorphismes G-linéaires de ∧ •Vs0 , formé de morphismes ayant aussi une interprétation géométrique qui sera décrite dans le paragraphe 4.2.1. L’objectif est de pouvoir découper dans les ∧•Vs0 des représentations irréductibles de G à l’aide de cette algèbre d’endomorphismes, en s’inspirant des constructions de Weyl. Dans le cas unitaire, on définit pour tout s ≥ 1 une sous-Z(p)-algèbre E(A)s de End(∧•Vs0). C’est l’algèbre engendrée par l’action de Ms(Z) sur V 0 , muni de la mul- tiplication opposée, et par l’action de OB sur V0. 6 SANDRA ROZENSZTAJN Dans le cas Siegel, on note E(C)s la sous-Z(p)-algèbre de End(∧ •Vs0) engendrée par l’action de Ms(Z) sur V 0 , et par les opérations suivantes. On note Z(p)(1) la représentation du groupe symplectique correspondant à l’action du groupe sur Z(p) par le multiplicateur. Observons que V0 = V(−1). On note u ∈ ∧2V20 (1) l’élément provenant de la forme bilinéaire sur V, et pour tous 1 ≤ i < j ≤ s, on note ui,j l’image de u par l’application ∧ 2V20 (1) → ∧ 2Vs0(1) induite par l’application V20 → V 0 consistant à placer les deux facteurs V0 aux places i et j. Le cup-produit par ui,j définit une application ϕi,j : ∧ •Vs0(−1) → ∧ •Vs0 qui envoie chaque ∧kVs0(−1) dans ∧ k+2Vs0 . L’application duale de ϕi,j permet de définir une application ψi,j : ∧ •Vs0 → ∧ •Vs0 (−1). Enfin on note θi,j = ϕi,j ◦ ψi,j , qui est donc un endomorphisme de ∧ •Vs0 . On définit alors E(C)s comme l’algèbre engendrée par l’action deMs(Z) et les θi,j , 1 ≤ i < j ≤ s. On notera Es pour désigner indifféremment E(A)s et E(C)s. On dira que u ∈ Es est un projecteur homogène (de degré t) si son image est contenue dans ∧tVs0 ⊂ ∧ •Vs0 . On donne des définitions similaires pour les éléments de Es ⊗ Q agissant sur ∧•(V0 ⊗Q) 3.3. Représentations atteignables. — On note Repa(G) la sous-catégorie de Rep(G) formée des représentations isomorphes à une somme directe de représentations de G de la forme im q, où q est un projecteur homogène de Es agissant sur un ∧ •Vs0 . Si V ∈ Repa(G), on note t(V ) le plus grand degré des projecteurs homogènes qui apparaissent dans la définition de V . On définit de fa con similaire RepaQ(G). Comme nous n’obtenons des résultats que pour les représentations de G qui sont dans Repa(G), il va s’agir de voir que cette sous-catégorie n’est pas trop petite, et qu’il n’est donc pas trop restrictif de s’y limiter. C’est l’objet des paragraphes suivants. 3.4. Poids p-petits. — Supposons notre groupe réductif G déployé sur un certain corps E. Les représentations irréductibles de G sur E sont paramétrées par l’ensemble des poids dominants, une fois fixé un tore maximal et un système de racines positives. Si a est un poids dominant, on note VE(a) la représentation irréductible de G sur E de plus haut poids a. On définit comme dans [9], II.3.15 ce qu’est un poids dominant p-petit. La propriété qui nous intéresse ici est la propriété suivante (voir [13], 1.9) : si le poids a est p-petit, alors il existe à homothétie près un unique réseau dans VE(a) qui est stable sous l’action de Dist(G), l’algèbre des distributions de G sur OE,(v), pour v une place de E divisant p. Cela a donc un sens de définir V (a) comme la représentation irréductible de G sur OE,(v) de plus haut poids a. 3.5. Description de Repa(G) dans le cas unitaire. — On se place ici dans le cas unitaire. Le groupe G est alors un groupe unitaire relatif à un corps E quadratique imaginaire, qui correspond à l’algèbre B du paragraphe 2.1.1, donc G est de la forme GU(g), et il est déployé sur E. On a donc GE −→ GL(g)E ⊗Gm,E . L’ensemble des représentations sur E irréductibles de G est paramétré par les g+1- uplets (a1, . . . , ag; c) d’entiers, avec a1 ≥ · · · ≥ ag et ai = c (mod 2), une fois choisi un isomorphisme entre GE et GL(g)E ×Gm,E. Notons i et j les deux morphismes de E dans E, le choix de l’isomorphisme revient à en privilégier un des deux. La représentation V0⊗E correspond à la somme de deux représentations irréductibles V1 et V2, V1 de plus haut poids (1, 0, . . . , 0; 1) et V2 de plus haut poids (0, . . . , 0,−1; 1). V1 est l’espace propre associé à la valeur propre i(x) de l’endomorphisme u(x), pour tout x dans E, et V2 est l’espace propre associé à la valeur propre j(x). 3.5.1. Description de RepaQ(G). — Notons Rep E(G) l’ensemble des V ⊗QE, où V ∈ RepaQ(G), et V0 = V0 ⊗ E. Proposition 3.1. — RepaE(G) contient toutes les représentations qui sont de la forme V (a)⊕V (a∗), où V (a) est la représentation irréductible de plus haut poids (a1, . . . , ag; c) avec ag ≥ 0 et c = ai = s, et a ∗ est le poids (−ag, . . . ,−a1; c). Lemme 3.2. — Soit a = (a1, . . . , ag; c) un poids dominant de G, tel que ag ≥ 0 et ai, et V (a) la représentation irréductible associée. Alors il existe un élément Ca de QSs (où s = ai) tel que V (a) = CaV 1 et V (a ∗) = CaV Démonstration. — Regardons d’abord V (a) comme une représentation de GLg, en oubliant l’action du multiplicateur. Comme expliqué dans [6], 15.5, il existe un Ca ∈ QSs idempotent tel que V (a) = CaV 1 , V (a) et V1 étant vues toutes deux comme des représentations deGLg. Il faut voir ensuite que l’égalité tient aussi comme représentations de GL(g)E × Gm,E, donc que le multiplicateur agit de la même fa con sur les deux. Or il agit par x 7→ xc sur V (a), et par x 7→ xs sur V ⊗s1 , et on a s = c. Lemme 3.3. — Soit s ≥ 0. Il existe un projecteur q dans E(A)s ⊗ Q commutant à l’action du groupe des permutations Ss tel que l’image de q agissant sur ∧ •V s0 est V ⊗s0 . Démonstration. — ∧•V s0 = ⊕0≤i1≤2g,...,0≤is≤2g ∧ i1 V0 ⊗ · · · ⊗ ∧ isV0. Considérons un entier m non nul, vj la matrice diagonale [(1, . . . , 1,m, 1, . . . )] avec un m en j-ème position. L’espace propre correspondant à la valeur propre m est la somme des termes pour lesquels ij = 1. Soit pj le projecteur sur cet espace propre. Les pj commutent, leur produit est donc un projecteur q sur l’intersection des images, c’est-à-dire les termes pour lesquels chaque ij est égal à 1, c’est-à-dire V 0 . De plus, q commute bien à l’action de Ss. Enfin on utilise que V0 = V1 ⊕ V2, V 0 est donc égal à une somme de termes de la forme V ⊗x1 ⊗ V 2 avec x+ y = s. Lemme 3.4. — Il existe un élément q′ dans le centre de E(A)s⊗Q dont la restriction de l’action à V ⊗s0 est un projecteur sur V 1 ⊕ V Démonstration. — Fixons z ∈ OE . z agit par i(z) sur V1 et par j(z) sur V2. Notons i(z) = a et j(z) = b. Choisissons z de sorte que les axby soient tous distincts. Alors V ⊗x1 ⊗ V 2 est (dans V 0 ) l’espace propre associé à la valeur propre a 8 SANDRA ROZENSZTAJN Soit P le polynôme Πr+t=s(X − a rbt). Il est à coefficients entiers, et c’est le po- lynôme minimal de l’action de u(z) sur V ⊗s0 . Soit Q = Πr+t=s,r 6=0,t6=0(X−a rbt). Alors P = QT où T = (X − as)(X − bs), et Q et T sont premiers entre eux. Il existe donc des polynômes U et V (à coefficients rationnels), tels que UQ+V T = 1. Alors l’action de u(1− UQ)(z) sur V ⊗s0 est un projecteur sur V 1 ⊕ V 2 , qu’on note q On peut maintenant prouver la proposition : soit a comme dans l’énoncé, et s = ai. Posons P = Caq ′q, alors P ∧• V s0 est la représentation V (a) ⊕ V (a ∗). En effet, notons que Ca, q ′ et q commutent par construction, donc P est un projecteur. On a q ∧• V s0 = V 0 , q ′q ∧• V s0 = V 1 ⊕ V 2 , Caq ′q ∧• V s0 = CaV 1 ⊕ CaV 2 . Or 1 = V (a), et CaV 2 = V (a Une fois décrites les représentations qui sont dans RepaE(G), il faut maintenant retrouver quelle est la Q-forme de ces représentations qui est dans RepaQ(G). V1 et V2 sont naturellement isomorphes, et Gal(E/Q) agit sur V0 = V1⊕V2 par (x, y) 7→ (ȳ, x̄). Son action sur V (a)⊕V (a∗) peut être décrite par la même formule, ce qui nous permet d’obtenir les représentations qui sont dans RepaQ(G). 3.5.2. Description de Repa(G). — Proposition 3.5. — Repa(G) contient toutes les représentations qui sont de la forme (V (a)⊕ V (a∗))Gal(E/Q), où V (a) est la représentation irréductible de plus haut poids (a1, . . . , ag; c) avec ag ≥ 0 et c = ai = s, et a ∗ est le poids (−ag, . . . ,−a1; c), a et a∗ sont p-petits, et 2g < p, et Gal(E/Q) agit sur (V (a) ⊕ V (a∗)) comme décrit au paragraphe précédent. Il suffit de voir que si les conditions données sont vérifiées, on peut prendre des dénominateurs premiers à p dans les lemmes du paragraphe 3.5.1. Lemme 3.6. — On se place comme dans le lemme 3.2. Alors si ai < p, Ca est dans Z(p)Ss. Démonstration. — Ca est de la forme (1/n)C a, où C a est dans ZSs, et n est l’entier tel que C′a = nC′a. Or n divise s! (voir [6], 4.2), donc 1/n ∈ Z(p) si s < p. Lemme 3.7. — Dans le cadre du lemme 3.3, on peut choisir q dans E(A)s dès que p > 2g. Démonstration. — Lorsque on écrit pj comme un polynôme en vj , les dénominateurs qui apparaissent sont les différences entre les valeurs propres de vj , qui sont les m pour 0 ≤ i ≤ 2g. Si 2g < p, on peut prendre un m dont l’image dans Z/pZ est un générateur de Z/pZ∗, de sorte que les mi −mi sont tous premiers à p. Lemme 3.8. — Dans le lemme 3.4, on peut prendre q′ dans E(A)s dès que p > s. Démonstration. — Écrivons donc UQ+V T = c, avec U et V à coefficients entiers et c entiers, et étudions les facteurs premiers de c. On obtient c = Q(as)Q(bs). Il s’agit donc de trouver z ∈ OE tel que c soit premier à p (et que aucun des a rbt, r > 0, t > 0, r + t = s ne soit égal à as ou à bs). On a Q(as) = as(s−1)/2Π1≤t≤s−1(a t − bt), et Q(bs) = bs(s−1)/2Π1≤t≤s−1(b t − at). Supposons d’abord que p est inerte dans E. Alors OE/p est isomorphe à Fp2 . La conjugaison dans OE se traduit par x 7→ x p dans OE/p. Choisissons donc x un générateur de F∗ , alors un z relevant x convient. Supposons maintenant p décomposé dans E. Alors OE/p est égal à Fp × Fp, et la conjugaison dans OE échange les deux facteurs dans OE/p. Choisissons un x dans F tel que xi 6= 1 pour tout i entre 1 et s, et prenons u et v dans F∗p tels que u/v = x. Alors si z est un relèvement de (u, v), il convient. 3.6. Description de Repa(G) dans le cas Siegel. — Dans le cas Siegel, la descrip- tion de Repa(G) est faite dans l’article [12], 5.1. On obtient toutes les représentations de plus haut poids p-petit, à l’action du centre près. 4. Les faisceaux 4.1. Constructions fonctorielles. — 4.1.1. Cas étale. — Soit x un point géométrique de XK . À chaque représentation du groupe fondamental de la variété π1(XK , x) correspond un système local sur XK . Considérons le faisceau constant Zp sur A, et F = R 1fK∗Zp(1), où fK : AK → XK est le morphisme structural. F correspond à la représentation standard de G sur Zp, autrement dit à un morphisme π1(XK , x) → G(Zp). On peut donc associer par composition un système local à toute représentation sur Zp de G, et ceci de fa con fonctorielle. On note F(V ) le système local associé à la représentation V . On définit de même le foncteur Fn(V ), qui à V associe un fibré en Z/p nZ-modules, vérifiant F(V )⊗Zp Z/p nZ = Fn(V ). On observe que F(V0) = R 1f∗Zp, et plus généralement F(∧ tVs0) = R tfs,K∗Zp, où fs,K est le morphisme structural A K → XK . 4.1.2. Cas des fibrés à connexion. — Proposition 4.1. — Il existe un foncteur F de Rep(G) dans l’ensemble des OX- modules à connexion intégrable sur X, et pour tout n un foncteur Fn de Rep(G) dans l’ensemble des OXn -modules à connexion intégrable sur Xn, ces deux foncteurs étant compatibles. Ici compatible, signifie que Fn(V/̟ n+1) = F(V )/̟n+1. On ne va faire la construc- tion que sur OX , la construction sur OXn s’obtenant par des méthodes similaires. Notons H1(A) = (R )∨. Introduisons T = Isom(OX ⊗ V ,H1(A)), les iso- morphismes devant respecter la structure de B-module et la forme alternée à une constante près. C’est un torseur sur X sous l’action (à droite) de G, en effet les deux faisceaux en question sont localement isomorphes, comme expliqué dans le lemme 2.1. Soit V ∈ Rep(G), on note F(V ) le faisceau des sections du fibré T ×G V . C’est un faisceau deOX -modules quasi-cohérent. De même un morphisme entre représentations se transforme en morphisme entre faisceaux. Ce fibré est muni d’une connexion qui provient de la connexion de Gauss-Manin sur H1(A). 10 SANDRA ROZENSZTAJN Notons Hi(As) = Rifs∗Ω , on a alors F(∧tVs0) = H t(As). De même on notera Hi(Asn) = R ifs∗Ω 4.2. Action de Es. — 4.2.1. Traduction géométrique. — Comme Es est une sous-algèbre de End(∧ •Vs0), par fonctorialité de F et F , on a donc aussi des morphismes de Z(p)-algèbres Es End(R•fs,∗Zp) et Es acris → End(H•(As)). On va donner une interprétation géométrique de ces deux morphismes. Soit G ⊂ Es la partie formée des éléments suivants : les matrices de déterminant non nul, dans le cas unitaire les éléments non nuls de OB, dans le cas Siegel les opérations θi,j , 1 ≤ i < j ≤ s définies au paragraphe 3.2. L’ensemble G, qu’on appellera ensemble des éléments géométriques de Es, engendre Es comme Z(p)-algèbre. Soit u ∈ G qui provient d’une matrice ou, dans le cas unitaire, d’un élément de OB . Alors u provient d’un élément de End(Vs0), qui agit naturellement sur A s/X , donc sur R•fs,∗Zp et H •(As) par aét(u) et acris(u) respectivement. Soit P le faisceau de Poincaré sur A × A, pour 1 ≤ i < j ≤ s on note Pi,j le faisceau sur As obtenu en tirant P par le morphisme de projection sur les i-ièmes et j-ièmes facteurs As → A×A. Alors l’opération consistant à faire le cup-produit par la première classe de Chern de Pi,j correspond à aét(ψi,j) et acris(ψi,j), l’opération duale correspond à aét(ϕi,j) et acris(ϕi,j), comme expliqué dans [12]. Par fonctorialité des constructions précédentes, on a, pour une représentation V de la forme V = q(∧•Vs0), q étant un projecteur de Es : F(V ) = aét(q)R •fs,K∗Zp, et F(V ) = acris(q)H •(As). On notera encore acris et aét les morphismes naturels de Es vers End(H •(Asn)) et End(R•fs∗Z/p nZ) respectivement. 4.2.2. Conséquence sur les faisceaux à connexion. — Lemme 4.2. — Pour tout V ∈ Repa(G) la connexion sur F(V ) et sur Fn(V ) est quasi-nilpotente. Démonstration. — Notons que les opérations élémentaires commutent à la connexion sur H•(As) induite par la connexion de Gauss-Manin, de sorte que, en reprenant les notations précédentes, F(V ) est stable par la connexion de H•(As). Comme la connexion de Gauss-Manin sur H•(As) est quasi-nilpotente, c’est aussi le cas pour la connexion sur F(V ). Comme X est lisse sur S, chaque Xn est un relèvement de X0 qui est lisse sur Sn. Les faisceaux Fn(V ), qui sont des OXn -modules cohérents munis d’une connexion intégrable et quasi-nilpotente, définissent donc des cristaux sur (X0/Sn)cris, ainsi que dans (Xm/Sn)cris pour tout m ≤ n. On peut donc voir Fn comme un foncteur de Repa(G) vers la catégorie des cristaux sur (X0/Sn)cris. Avec cette interprétation les Hi(Asn) s’identifient aux R ifs,cris∗OAs /Sn . Lemme 4.3. — Pour tout V ∈ Repa(G), les faisceaux F(V ) et Fn(V ) sont locale- ment libres sur X et Xn respectivement. En effet c’est le cas pour les Hn(As). 4.3. Prolongement des cristaux. — L’objectif est de construire un foncteur F de Repa(G) vers l’ensemble des fibrés localement libres munis d’une connexion à pôles logarithmiques le long de X \X intégrable et quasi-nilpotente, qui prolonge F . 4.3.1. Unicité du prolongement. — Lemme 4.4. — Soit E un fibré localement libre sur X muni d’une connexion intégrable et quasi-nilpotente. S’il existe un prolongement de E en un fibré localement libre sur X muni d’une connexion à pôles logarithmiques le long de X \X intégrable et quasi-nilpotente, alors il est unique, et de plus tout prolongement de E en un fibré cohérent sur X muni d’une connexion ayant les mêmes propriétés est aussi localement libre (et donc égal au prolongement précédent). De plus, si E1 et E2 sont deux tels faisceaux admettant des prolongements, et u : E1 → E2 est un morphisme horizontal, u admet un unique prolongement horizontal entre les prolongements des faisceaux. Démonstration. — Soit E un tel prolongement cohérent. Regardons tout d’abord EK . D’après [4], il existe au plus un prolongement de EK en un fibré muni d’une connexion à pôles logarithmiques, qui est l’extension canonique de Deligne, ce prolongement est donc nécessairement EK . Le faisceau E est alors uniquement déterminé. En effet, notons X ′ la réunion de X et XK dans X, j l’inclusion de X ′ dans X, et E ′ le faisceau qui cöıncide avec E sur X et avec EK sur XK . Alors pour des raisons de codimension, et le faisceau E étant cohérent, E = j∗(E ′). En particulier, tous les prolongements cohérents munis de connexion sont égaux en tant que faisceaux, donc si l’un est localement libre, tous le sont. Enfin, E étant localement libre, sa connexion est entièrement déterminée par sa restriction à EK . Pour l’existence du prolongement des morphismes, cela provient de la fonctorialité de l’extension canonique de Deligne. Corollaire 4.5. — Le fibré Hi(As), muni de la connexion de Gauss-Manin, ne dépend pas du choix de la compactification As. De plus, étant données deux compac- tifications A de A et As de As, pour tout i, Hi(As) et ∧iH1(A)s sont égaux comme sous-faisceaux de (X → X)∗H i(As) munis d’une connexion à pôles logarithmiques. Démonstration. — En effet, il suffit pour pouvoir appliquer le lemme précédent de vérifier que les ∧iH1(A)s sont localement libres, il suffit donc de voir que H1(A) est localement libre. Cela se déduit des résultats de [8], qu’on peut appliquer car on a supposé que dimX A < p. Le cas général se déduit de l’identité précédente. On notera H (As) ce faisceau à connexion. Lemme 4.6. — Hi(Asn) est localement libre sur Xn, et ne dépend pas de la com- pactification As. 12 SANDRA ROZENSZTAJN Démonstration. — En effet, le faisceau H (Asn) est obtenu à partir de H (As) par changement de base. On notera dans la suite H (Asn) pour ce faisceau. 4.3.2. Prolongement de l’action de Es. — Il s’agit maintenant de prolonger le mor- phismeEs acris → End(H•(As)) en un morphisme de Z(p)-algèbresEs alog-cris → End(H (As)). La restriction res : End(H (As)) → End(H•(As)) est injective. Pour construire alog-cris, il suffit donc de vérifier que l’image de acris est contenue dans l’image de res. Comme acris est un morphisme de Z(p)-algèbres, il suffit de vérifier que l’image par acris d’une partie génératrice de Es est contenue dans l’image de res. Il s’agit donc de vérifier que l’action des éléments de G sur les H•(As) se prolonge en une action sur les H (As). Soit u ∈ G. Supposons d’abord que u soit une matrice, ou (dans le cas unitaire) un élément de OB . Alors u agit sur A s par une isogénie. D’après la propriété 2, il existe donc deux compactifications As1 et As2, et un morphisme u ′ : As1 → As2 prolongeant l’action de u. Alors u′ fournit l’élément de End(H (As)) voulu. Supposons maintenant qu’on est dans le cas Siegel et que u est de la forme θi,j . Il suffit de montrer que les morphismes acris(ϕi,j) et acris(ψi,j) se prolongent en éléments de End(H (As)). Il existe, d’après la propriété 3, une compactification As telle que le faisceau (O(2)⊗a⊗Pi,j) ⊗b se prolonge en un faisceau L surAs, avec a et b premiers à p. Il existe aussi une compactification As telle que le faisceau O(2)⊗c se prolonge en un faisceau L′ sur As , avec c premier à p. Alors l’action de 1 (cL)) est dans End(H (As)), l’action de − a (c1(L ′)) aussi, et l’action de 1 (c1(L)) − (c1(L ′)) prolonge celle de acris(ϕi,j). Pour acris(ψi,j), on fait le même raisonnement, en utilisant la dualité de Poincaré. On note encore alog-cris pour le morphismeEs → End(H (Asn)) obtenu par réduction. 4.3.3. Définition de F . — Si V ∈ Repa(G), on veut définir F(V ) comme le faisceau localement libre muni d’une connexion à pôles logarithmiques intégrable et quasi- nilpotente sur X prolongeant F(V ). Au vu du paragraphe 4.3.1, il suffit de montrer l’existence de ce prolongement, son unicité et le fait que la construction est fonctorielle étant alors automatiques. Soit V = q(∧•Vs0), où q est un projecteur de Es. Il suffit de poser F(V ) = alog-cris(q)(H (As)). On note Fn(V ) la réduction modulo ̟ n+1 de F(V ). Munissons Sn de la log- structure triviale, et Xn de la log-structure provenant du diviseur à croisements nor- maux (X\X)n. Alors Fn définit un foncteur de Rep a(G) vers la catégorie des cristaux sur (X0/Sn) cris, en effet cette catégorie est équivalente à celle des OXn -modules munis d’une connexion à pôles logarithmiques intégrable et quasi-nilpotente, Xn étant un relèvement de X0 log-lisse sur Sn. 5. Structures sur les groupes de cohomologie 5.1. Cas étale. — Notons K la clôture algébrique de K, et Γ = Gal(K/K). Le groupe Hmét (XK ,Fn(V )) est naturellement muni d’une action de Γ car le faisceau Fn(V ) est défini sur XK . On aura besoin du lemme suivant pour comparer l’action de Galois surHmét (XK ,Fn(V )) et sur la cohomologie de As Lemme 5.1. — Soit f : Z → T un morphisme de schémas, F un faisceau constant sur Z (Z/pnZ ou Zp). Soit q agissant sur H •(Z) = R•f∗F et sur H •(Z, F ) de fa con compatible avec la suite spectrale de Leray. On suppose que q agit comme un projec- teur, dont l’image est entièrement contenue dans Hs(Z). Notons V = qH•(Z), alors pour tout m on a Hm(T, V ) = qHm+s(Z, F ). Démonstration. — En effet considérons la suite spectrale de Leray pour calculer la cohomologie de F sur Z. On lui applique q, on obtient toujours une suite spectrale convergente car q est un projecteur. D’autre part qHm(T,Hi(Z)) = Hm(T, qHi(Z)), toujours parce que q est un projecteur. La suite spectrale obtenue a une seule colonne non nulle, dont les termes sont les Hm(T, qHs(Z)), et aboutit à qHm+s(Z, F ), d’où le résultat. Notons encore aét les morphismesEs → End(H ét(A ,Zp)) et Es → End(H ét(A ,Z/pnZ)). Supposons que V = q(∧•Vs0), q étant un projecteur homogène de degré t, alors on a Hmét (XK ,Fn(V )) = aét(q)H ét (A ,Z/pnZ). Comme Es agit par des correspon- dances algébriques définies sur K sur la cohomologie de As, l’action de Γ commute à l’action de Es, et la structure galoisienne obtenue sur H ét (XK ,Fn(V )) est compatible à celle sur la cohomologie de As 5.2. Cas cristallin. — 5.2.1. Les modules de Fontaine-Laffaille. — On note MF tor la catégorie suivante. Les objets sont les O-modules M de longueur finie, muni d’une filtration FiliM décroissante, telle que Fil0M =M et Filp−1M = 0, et pour tout i, un φi : Fil iM →M O-semi-linéaire, vérifiant φi|Fili+1M = pφi+1, et i imφi = M . Les morphismes res- pectent la filtration et commutent aux φi. 5.2.2. La catégorie MF (φ). — On introduit la catégorieMF (φ) des K-espaces vec- toriels munis d’une filtration décroissante et d’un Frobenius. Les objets sont les K- espaces vectoriels de dimension finie M , muni d’une filtration décroissante Fil et de l’action d’un Frobenius φ semi-linéaire par rapport au Frobenius σ de K. Les morphismes doivent commuter au Frobenius, et respecter la filtration. 5.2.3. Calculs dans un cas particulier. — On se place dans le cas suivant : on a un log-schéma Z qui est propre, et lisse sur S muni de la log-structure triviale. Soit n un entier positif. On note Sn = SpecOn, muni de la log-structure triviale. Si (Z,M) est un schéma sur SpecO, on note (Zn,Mn) le changement de base à Sn. Si E est un cristal sur le site ((Zm,Mm)/Sn)) cris, on noteraH cris((Zm,Mm)/Sn, E) pourHi(((Zm,Mm)/Sn)) cris, E), etH cris((Zm,Mm)/Sn) pourH cris((Zm,Mm)/Sn,OZm/Sn). 14 SANDRA ROZENSZTAJN On omettra la mention de la log-structure si cela ne cause pas de confusion. Re- marquons que pour tout m ≤ n, les Hicris((Zm,Mm)/Sn) ne dépendent pas de m, on notera Hicris((Z,M)/Sn) leur valeur commune. Enfin on note H cris((Z,M)/S) = Hicris((Z,M)/Sn). On a les deux résultats suivants : Proposition 5.2. — Pour tout 0 ≤ i ≤ p − 2, Hicris((Z,M)/Sn) est un module de Fontaine-Laffaille. Pour tout i ≥ 0, Hicris((Z,M)/S)⊗K est un élément de MF (φ). Démonstration. — La preuve de la première partie de la proposition est identique à celle de l’article [5], qui traite le cas où Z est muni de la log-structure triviale. Notons S′n le log-schéma dont le schéma sous-jacent est le même que Sn, et dont la log-structure provient de N → On, 1 7→ 0. Il s’agit de la même log-structure que celle définie dans [7], paragraphe 3.4. Notons (Z ′,M ′) le log-schéma déduit de (Z,M) par le changement de base S′n → Sn. Alors H cris((Z,M)/Sn) et H cris((Z ′,M ′)/S′n) sont canoniquement isomorphes pour tout i. Les Hicris((Z ′,M ′)/S′n) sont munis d’un Fro- benius et d’un opérateur de monodromie, définis dans [7], paragraphe 3. L’opérateur de monodromie est ici nul, (Z ′,M ′) provenant par changement de base de (Z,M) qui est log-lisse sur Sn. H cris((Z ′,M ′)/S′) ⊗ K, et donc aussi Hicris((Z,M)/S) ⊗ K est ainsi naturellement muni d’une structure d’élément de MF (φ). 5.2.4. Action des endomorphismes sur le prolongement des cristaux. — NotonsH (As/Sn) la limite des Hicris(A n/Sn), pour les As de notre famille de compactifications. Alors : Proposition 5.3. — Le morphisme Hicris(A n/Sn) → H (As/Sn) est un isomor- phisme pour toute compactification As et pour tout n. Démonstration. — Il suffit pour cela de voir que tout morphisme entre compactifica- tions qui est l’identité sur As induit un isomorphisme entre les groupes de cohomolo- Considérons la suite spectrale de Leray : E 2 = H i(Xn/Sn, R jfscris∗OAsn/Sn) ⇒ Hi+j(Asn/Sn) Un morphisme entre deux compactifications induit un morphisme de suites spec- trales, qui est un isomorphisme sur les E 2 , donc aussi sur l’aboutissement. On cherche à définir un morphisme de Z(p)-algèbres Es → End(H (As/Sn)). On a la suite spectrale suivante, qu’on appellera encore suite spectrale de Leray, qui provient de n’importe quelle compactification As de As : 2 = H i(Xn/Sn,H (Asn)) ⇒ H (As/Sn) Proposition 5.4. — Il existe un unique morphisme alog-cris de Z(p)-algèbres Es → End(H (As/Sn)) tel que l’action de Es sur les H (Asn) et sur les H (As/Sn) donnée par alog-cris soit compatible à la suite spectrale de Leray. Démonstration. — On commence par définir l’image de l’ensemble G des éléments géométriques de Es, et on montre ensuite que l’on peut prolonger en un morphisme de Z(p)-algèbres. Pour définir l’image d’un élément de G, on fait exactement comme dans le pa- ragraphe 4.3.2. Il faut voir que le choix fait est unique. Cela provient du fait que l’action de u ∈ G sur les termes E 2 de la suite spectrale ne dépend pas des choix faits, comme expliqué en 4.3.2, et de la compatibilité de l’action du prolongement à la suite spectrale de Leray. Il reste à voir que l’action des éléments de G se prolonge en un morphisme d’algèbres Es → End(H (As/Sn)). Cela provient encore une fois de la compatibilité avec la suite spectrale de Leray, et du fait que Es → End(H (As/Sn)) est un morphisme d’algèbres. 5.2.5. La cohomologie des cristaux. — PosonsHicris(X/S,F(V )) = lim←− Hicris(Xn/Sn,Fn(V )). On a le résultat suivant : Proposition 5.5. — Pour tout V ∈ Repa(G), pour tout i, Hicris(X/S,F(V )) ⊗ K est un élément de MF (φ). Pour tout V ∈ Repa(G) homogène de degré t, pour tout i tel que i+ t ≤ p−2, pour tout n, Hicris(Xn/Sn,Fn(V )) est un élément de MF Démonstration. — En effet, soit V = q(∧•Vs0), avec q homogène de degré t, on a alors de fa con similaire au lemme 5.1 les égalitésHicris(X/S,F(V ))⊗K = acris(q)H log-cris(A s/S)⊗ K et Hicris(Xn/Sn,Fn(V )) = acris(q)H cris(A n/Sn). Il reste à voir que l’action de Es par alog-cris respecte les structures d’élément de MF (φ), et que l’action de E(A)s respecte les structures de module de Fontaine-Laffaille, ce qui se voir sur les éléments géométriques. 6. Théorème de comparaison 6.1. Le cas des faisceaux constants. — Notons RepZp(Γ) la catégorie des Zp- représentations de type fini de Γ = Gal(K/K). Nous avons un foncteur contrava- riant et pleinement fidèle : Vcris : MF tor → RepZp(Γ) qui est défini par Vcris(M) = Hom(M,Acris,∞). L’anneau Acris est défini dans [1], 6.3, et Acris,∞ = Acris ⊗ Qp/Zp. L’anneau Acris est muni d’une filtration décroissante et d’un Frobenius, et les homo- morphismes que l’on considère doivent être compatibles à la filtration et à l’action du Frobenius. Soit Z un schéma propre et lisse sur SpecOK , et D un diviseur à croisements normaux relatifs de Z, U l’ouvert complémentaire de D. On munit Z de la log- structure M définie par le diviseur D, et SpecOK de la log-structure triviale. Proposition 6.1. — Pour 0 ≤ m ≤ p− 2, on a un isomorphisme canonique compa- tible à l’action de Galois : Vcris(H cris((Z,M)/Sn)) = H ét (UK ,Z/p 16 SANDRA ROZENSZTAJN Proposition 6.2. — Pour tout m, il existe un isomorphisme canonique qui respecte l’action de Γ, la filtration et le Frobenius : γm : Bcris ⊗O H cris((Z,M)/S) −→ Bcris ⊗Qp H ét (UK ,Qp) Démonstration. — Le résultat de la proposition 6.1 provient de travaux de Breuil ([2]) et Tsuji ([16]). Ces résultats s’appliquent dans un cadre beaucoup plus général que celui considéré ici, et décrivent une comparaison entre la cohomologie étale de UK et la cohomologie de (Z ′,M ′)/En. Ici (Z ′,M ′) est obtenu comme dans le paragraphe 5.2.3 par changement de base de (Z,M) de Sn à S n. En est le log-schéma dont le schéma sous-jacent est SpecOn〈u〉, l’enveloppe à puissances divisées de l’algèbreOn[u] des polynômes en l’indéterminée u, muni de la log-structure associée à N → On〈u〉, 1 7→ u. Dans notre cas particulier, on a une relation simple entre la cohomologie de (Z ′,M ′)/En et celle de (Z ′,M ′)/S′n, donnée par H cris((Z ′,M ′)/En) = On〈u〉 ⊗ Hmcris((Z ′,M ′)/S′n), qui nous permet d’obtenir le résultat de la proposition 6.1. Pour la version rationnelle 6.2, le résultat provient de résultats de Tsuji ([15], voir aussi [17]). Comme dans le cas de torsion, la situation se simplifie par rapport au cas général, du fait qu’ici la monodromie agissant sur Hmcris((Z,M)/S)⊗K est nulle. 6.2. Les théorèmes. — Théorème 6.3. — Dans le cas unitaire, soit V ∈ Repa(G), et m tel que m+ t(V ) ≤ p−2.Hm (XK ,Fn(V )) est muni d’une action du groupe de Galois Γ,H cris(Xn/Sn,Fn(V )) est muni d’une structure de module de Fontaine-Laffaille, et on a un isomorphisme : Vcris(H cris(Xn/Sn,Fn(V ))) = H ét (XK ,Fn(V )) Notons qu’on peut déduire de ce théorème comme dans l’article [2], paragraphe 4.2, une comparaison entre les parties de torsion de lim Hmét (XK ,Fn(V )) et de lim←− Hmcris(Xn/Sn,Fn(V )), ainsi qu’une comparaison entre leurs parties libres. Théorème 6.4. — Dans le cas unitaire et Siegel, soit V ∈ Repa(G), il existe un isomorphisme γ : Bcris ⊗O H log-cris(X/S,F(V )) −→ Bcris ⊗Zp H ét (XK ,F(V )) Le point essentiel de la preuve dans les deux cas est le résultat suivant : Lemme 6.5. — Soit u ∈ E(A)s. u agit sur H (As/Sn) et sur H ,Z/pnZ) (m ≤ p − 2) de fa con compatible avec l’isomorphisme Vcris. Soit u ∈ Es, u agit sur (As/S)⊗Q et sur Hmét (A ,Qp) de fa con compatible avec l’isomorphisme γm du théorème 6.2. Démonstration. — Il suffit de montrer la compatibilité des actions pour l’ensemble des éléments géométriques G de Es, puisqu’ils engendrent Es comme Z(p)-algèbre. Soit u un élément de Es provenant d’une matrice de déterminant non nul, ou d’un élément non nul de OB. Son action sur la cohomologie provient d’une isogénie de As, qu’on notera encore u. D’après la propriété 2 u se prolonge en un mor- phisme entre deux compactifications u : As1 → As2. D’où par fonctorialité de Vcris, Vcris(u : H cris((A 2)n/Sn) → H cris((A 1)n/Sn) = (u ∨ : Hmét (A ,Z/pnZ)∨ → Hmét (A ,Z/pnZ)∨), ce qui est bien la compatibilité voulue. De même, on a aussi la compatibilité pour l’action sur Hmét (UK ,Qp) et H cris((X,M)/S)⊗K. Dans le cas Siegel, il faut aussi considérer les éléments de la forme θi,j . Il s’agit donc de voir que l’action des ϕi,j et des ψi,j sur H ét (UK ,Qp) et H cris((X,M)/S)⊗K est compatible. Cela provient du fait que l’isomorphisme de comparaison 6.2 fait correspondre les classes de Chern ([15]) et est compatible aux structures produit sur les groupes de cohomologie et à la dualité de Poincaré ([17]). Démonstration des théorèmes 6.3 et 6.4. — Montrons le théorème 6.3. Soit V ∈ Repa(G), on peut supposer qu’il existe un entier s, et un projecteur q dans E(A)s de degré t, tels que V = q(∧•Vs0). Le théorème de comparaison s’applique car m+ t ≤ p− 2, et nous donne un isomorphisme Vcris(H (As/Sn)) = H ét (A ,Z/pnZ)∨. Appliquons q : comme l’action de E(A)s commute à Vcris on a donc un isomor- phisme : Vcris(alog-cris(q)H (As/Sn)) = (aét(q)H ét (A ,Z/pnZ))∨. D’après le lemme 5.1, cela donne : Vcris(H log-cris(X,Fn(V ))) = H ét (XK ,Fn(V )) La preuve du théorème 6.4 est identique. Remarque 6.6. — On voit apparâıtre dans le lemme 6.5 le point qui explique pour- quoi on n’a pas de résultats de comparaison prenant en compte la torsion pour le cas Siegel : la compatibilité de l’isomorphisme de comparaison à coefficients constants avec la dualité de Poincaré et les structures produits n’est actuellement montrée que dans le cas rationnel (même s’il est vraisemblable qu’elle soit vraie aussi dans le cas de torsion, en introduisant des limitations sur le degré des groupes de cohomolo- gie considérés). Enfin on peut remarquer que si on se limite aux représentations qui peuvent être obtenues à l’aide uniquement des éléments de Es provenant de Ms(Z), on peut prendre en compte la torsion pour le cas Siegel. Références [1] C. Breuil – « Topologie log-syntomique, cohomologie log-cristalline et cohomologie de Cech », Bull. Soc. Math. France 124 (1996), p. 587–647. [2] , « Cohomologie étale de p-torsion et cohomologie cristalline en réduction semi- stable », Duke Math. 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Sandra Rozensztajn, IRMA, Université Louis Pasteur, 7 rue René-Descartes, 67084 Strasbourg Cedex, France • E-mail : [email protected] 1. Introduction 2. Les objets considérés 3. Représentations de G 4. Les faisceaux 5. Structures sur les groupes de cohomologie 6. Théorème de comparaison Références

0704.1348

Large portfolio losses: A dynamic contagion model

Large portfolio losses: A dynamic contagion model The Annals of Applied Probability 2009, Vol. 19, No. 1, 347–394 DOI: 10.1214/08-AAP544 c© Institute of Mathematical Statistics, 2009 LARGE PORTFOLIO LOSSES: A DYNAMIC CONTAGION MODEL By Paolo Dai Pra, Wolfgang J. Runggaldier, Elena Sartori and Marco Tolotti University of Padova, University of Padova, University of Padova, and Bocconi University and Scuola Normale Superiore Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investi- gate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large devi- ation principle we compute the limiting distributions of the system and determine the time evolution of the credit quality indicators of the firms, deriving moreover the dynamics of a global financial health indicator. We finally describe a suitable version of the “Central Limit Theorem” useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis. 1. Introduction. 1.1. General aspects. The main purpose of this paper is to describe prop- agation of financial distress in a network of firms linked by business rela- tionships. Once the model for financial contagion has been described, we quantify the impact of contagion on the losses suffered by a financial insti- tution holding a large portfolio with positions issued by the firms. A firm experiencing financial distress may affect the credit quality of business partners (via direct contagion) as well as of firms in the same sector (due to an information effect). We refer to direct contagion when the actors on the market are linked by some direct partner relationship (e.g., firms in a borrowing-lending network). Reduced-form models for direct contagion can be found—among others—in Jarrow and Yu [27] for counterparty risk, Davis and Lo [13] for infectious Received March 2007; revised April 2008. AMS 2000 subject classifications. 60K35, 91B70. Key words and phrases. Credit contagion, credit crisis, interacting particle systems, large deviations, large portfolio losses, mean field interaction, nonreversible Markov pro- cesses, phase transition. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2009, Vol. 19, No. 1, 347–394. This reprint differs from the original in pagination and typographic detail. http://arxiv.org/abs/0704.1348v3 http://www.imstat.org/aap/ http://dx.doi.org/10.1214/08-AAP544 http://www.imstat.org http://www.ams.org/msc/ http://www.imstat.org http://www.imstat.org/aap/ http://dx.doi.org/10.1214/08-AAP544 2 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI default, Kiyotaki and Moore [28], where a model of credit chain obligations leading to default cascade is considered and Giesecke and Weber [23] for a particle system approach. Concerning the banking sector, a microeconomic liquidity equilibrium is analyzed by Allen and Gale [1]. Information effects are considered in information-driven default models; here the idea is that the probability of default of each obligor is influenced by a “not perfectly” observable macroeconomic variable, sometimes also referred to as frailty. This dependence increases the correlation between the default events. For further discussions on this point see Schönbucher [33] as well as Duffie et al. [16] and Collin-Dufresne et al. [7]. 1.2. Purpose and modeling aspects. We propose in this paper a direct contagion model which is constructed in a general modeling framework where information effects could also be included. In addition to modeling contagion, with the approach that we shall develop we intend also to find a way to explain what is usually referred to as the clustering of defaults (or credit crises), meaning that there is evidence—looking at real data—of periods in which many firms end up in financial distress in a short time. A standard methodology to reproduce this real-world effect is to rely on macroeconomic factors as indicators of business cycles. These factor models seem to explain a large part of the variability of the default rates. What these models do not explain is above all clustering: as Jarrow and Yu in [27] argue, “A default intensity that depends linearly on a set of smoothly varying macroeconomic variables is unlikely to account for the clustering of defaults around an economic recession.” A second issue that we would like to capture is—in some sense—more “fundamental” and refers to the nature of a credit crisis. We shall propose a model where the general “health” of the system is described by endogenous financial indicators, endogenous in the sense that its dynamics depends on the evolution of the variables of the system. Our aim is to show how a credit crisis can be described as a “microeconomic” phenomenon, driven by the propagation of the financial distress through the obligors. Our model is to be considered within the class of reduced-form models and is based on interacting intensities. The probability of having a default somewhere in the network depends also on the state of the other obligors. The first papers on interacting intensities appear to be those by Jarrow and Yu [27], and Davis and Lo [13] on infectious default. In our perspective the idea of a network where agents interact leads natu- rally to the literature of particle systems used in statistical mechanics. This point of view is quite new in the world of financial mathematics especially when dealing with credit risk management. Among some very recent papers we would like to mention the works by Giesecke and Weber [23], and [24] for an interacting particle approach, the papers by Frey and Backhaus [19] LARGE PORTFOLIO LOSSES 3 on credit derivatives pricing and Horst [26] on cascade processes. More de- veloped is the use of particle and dynamical systems in the literature on financial market modeling. It has been shown that some of these models have “thermodynamic limits” that exhibit similar features compared to the limiting distributions (in particular when looking at the tails) of market returns time series. For a discussion on financial market modeling see the survey by Cont [9] and the paper by Föllmer [18] that contains an inspiring discussion on interacting agents. Another reason to focus on particle systems is that they allow to study a credit crisis as a microeconomic phenomenon and so provide the means to explain phenomena such as default clustering that are difficult to explain by other means. In fact, interacting particle systems may exhibit what is called phase transition in the sense that in the limit, when the number N of particles goes to infinity, the dynamics may have multiple stable equilibria. The effects of phase transition for the system with finite N can be seen on different time-scales. On a long time-scale we expect to observe what is usually meant by metastability in statistical mechanics: the system may spend a very long time in a small region of the state space around a stable equilibrium of the limiting dynamics and then switch relatively quickly to another region around a different stable equilibrium. This switch, of which the rigorous analysis will be postponed to future work, occurs on a time-scale proportional to ekN for a suitable k > 0, that could be unrealistic for financial applications. The model we propose exhibits, however, a different feature that can be interpreted as a credit crisis. For certain values of the initial condition the system is driven toward a symmetric equilibrium, in which half of the firms are in good financial health. After a certain time that depends on the initial state, the system is “captured” by an unstable direction of this symmetric equilibrium, and moves toward a stable asymmetric equilibrium; during the transition to the asymmetric equilibrium, the volatility of the system increases sharply, before decaying to a stationary value. All this occurs at a time-scale of order O(1) (i.e., the time-scale does not depend on N ). 1.3. Financial application. As already mentioned in Section 1.1, the ap- plied financial aim of this paper is to quantify the impact of contagion on the losses suffered by a financial institution holding a large portfolio with positions issued by the firms. In particular, we aim at obtaining a dynamic description of a risky portfolio in the context of our contagion model. The standard literature on risk management usually focuses on static models allowing to compute the distribution of a risky portfolio over a given fixed time-horizon T . For a recent paper that introduces a discussion relating to static and dynamic models see Dembo, Deuschel and Duffie [14]. 4 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI We shall consider large homogeneous portfolios. Attention to large ho- mogeneous portfolios becomes crucial when looking at portfolios with many small entries. Suppose a bank is holding a credit portfolio with N = 10,000 open positions with small firms; it is quite costly to simulate the dynamics of each single firm, taking into account all business ties. If the firms are supposed to be exchangeable, in the sense that the losses that they may cause to the bank in case of financial distress depend on the single firm only via its financial state indicator, it is worth evaluating a homogeneous model where N goes to infinity and then to look for “large-N” approximations. This apparently restrictive assumption may be easily relaxed by considering many homogeneous groups within the network (in this context see also [19]). We shall provide formulas to compute quantiles of the probability of excess losses in the context of our contagion model; we shall in fact determine the entire portfolio loss distribution. Other credit risk related quantities can also be computed, as we shall briefly mention at the end of Section 4. We conclude this section by noticing that in recent years the challenging issue of describing the time evolution of the loss process connected with port- folios of many obligors has received more and more attention. Applications can be found, for example, in the literature dealing with pricing and hedging of risky derivatives such as CDOs, namely Collateralized Debt Obligations (see, e.g., the papers by Frey and Backhaus [20], Giesecke and Goldberg [22] and Schönbucher [34]). We believe that our paper may be considered as an original contribution to the modeling of portfolio loss dynamics: to our knowledge, this is the first attempt to apply large deviations on path spaces (i.e., in a dynamic fashion) for finance or credit management purposes. For a survey on existing large deviations methods applied to finance and credit risk see Pham [31]. 1.4. Methodology. Our interacting particle system, which describes the firms in the network, will be Markovian, but nonreversible. Usually, when the dynamics admit a reversible distribution, this distribution can be found explicitly by the detailed balance condition [see (6) below]. In the model we propose in this paper, and that will be introduced in Section 2, no reversible distribution exists. This makes it difficult to find an explicit formula for the stationary distribution. For this reason we have not pursued the “static” approach consisting in studying the N → +∞ asymptotics of the stationary distribution. We shall rather proceed in a way that in addition allows to obtain nonequilibrium properties of the system dynamics. First we study the N →∞ limiting distributions on the path space. To this effect we shall derive an appropriate law of large numbers based on a large deviations principle. We then study the possible equilibria of the limiting dynamics. This study leads to considering different domains of attraction corresponding to each of the stable equilibria. Finally, we study the finite volume approximations LARGE PORTFOLIO LOSSES 5 (for finite but large N ) of the limiting distribution via a suitable version of the Central Limit Theorem that allows to analyze the fluctuations around this limit. As a consequence of the different domains of attraction of the limiting dynamics one obtains for finite N and on ordinary time-scales an interesting behavior of the system that has an equally interesting financial interpretation, which was already alluded to at the end of Section 1.2. This behavior will also be documented by simulation results. Our interaction model is characterized by two parameters indicating the strength of the interactions. Phase transition occurs in an open subset of the parameter space, whose boundary is a smooth curve (critical curve) that we determine explicitly. We shall derive the Central Limit Theorem in a fixed time-interval [0, T ] for every value of the parameters. We do not consider in this paper the Central Limit Theorem in the case when the time-horizon T depends on N itself; it will be dealt with elsewhere. When T grows with N we expect the behavior to depend more strongly on the parameters. In the case when the parameters belong to the uniqueness region (the complement of the closure of the region where phase transition occurs) we believe that the Central Limit Theorem should be uniform in time, while in the phase transition region the Central Limit Theorem should extend to any time-scale strictly smaller than the metastability scale (which grows exponentially in N ). On the critical curve one expects a critical time-scale (of order N ) at which large and non-Gaussian fluctuations are observed. For real applications the interaction parameters have to be calibrated to market data. In this paper we do not consider the issue of calibration but rather present some simulation results of the loss behavior for different values of the parameters. The outline of the paper is as follows. The more detailed description of the model will be given in Section 2. Section 3 is devoted to stating the main limit theorems on the stochastic dynamics, in particular a law of large numbers and a central limit theorem. The financial application, in particular to large portfolio losses with specific examples, will be described in Section 4. Section 5 contains the proofs of the results stated in Sections 3 and 4. A Conclusions section completes the paper. 2. The model. 2.1. A mean-field model. In this section we describe a mean-field interac- tion model. What characterizes a mean-field model—within the large class of particle systems—is the absence of a “geometry” in the configuration space, meaning that each particle interacts with all the others in the same way. This “homogeneity” assumption is clearly rather restrictive; neverthe- less this kind of framework has been proposed by authors in different fields. Among the others we quote Frey and Backhaus [19] for a credit risk model 6 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI and Brock and Durlauf [4] for their contribution to the Social Interaction models. These models are used to capture the interaction of agents when facing any kind of decision problems. As pointed out in [19], if we are con- sidering a large group of firms belonging to the same sector (e.g., the energy sector), then the ability of generating cash flows and the capacity of rais- ing capital from financial institutions may be considered as “homogeneous” characteristics within the group (and this assumption is quite common in practice); we moreover recall that the final aim of this work is to study ag- gregate quantities for a large economy such as the expected global health of the system and large portfolio losses as well as related quantities. These considerations allow us to avoid the (costly) operation of modeling a fully heterogeneous set of firms. Other approaches, different from the mean-field one, have also been pro- posed in the literature: Giesecke and Weber have chosen a local-interaction model (the Voter model1) assuming that each particle interacts with a fixed number d of neighbors; it may be argued that the hypothesis that each firm has the same (constant) number of partners is rather unrealistic. Cont and Bouchaud (see [10]) suggest a random graph approach, meaning that the connections are randomly generated with some distribution functions. The philosophy behind our model can be summarized as follows: • We introduce only a small number of variables that, however, have a simple economic interpretation. • We define dynamic rules that describe interaction between the variables. • We keep the model as simple as possible; in particular, as we shall see, we define it in such a way that it has some symmetry properties. On one hand this may make the model less adherent to reality; on the other it leads to exact computations and still allows to show what basic features of the model produce phenomena such as clustering of defaults, phase transition, etc. More generally, it allows to show how, contrary to most models relying on macroeconomic factors, the “health” of the system can here be described by endogenous financial indicators so that a credit crisis can be viewed as a microeconomic phenomenon. Consider a network of N firms. The state of each firm is identified by two variables, that will be denoted by σ and ω [(σi, ωi) is the state of the ith firm]. The variable σ may be interpreted as the rating class indicator : a low value reflects a bad rating class, that is, a higher probability of not being able to pay back obligations. The variable ω represents a more fundamental indicator of the financial health of the firm and is typically not directly 1The Voter model assumes—roughly speaking—that the variable σi ∈ {−1,1} is more likely to take a positive value if the majority of the nearest neighbors of i are in a positive state and vice versa. LARGE PORTFOLIO LOSSES 7 observable. It could, for example, be a liquidity indicator as in Giesecke and Weber [23] or the sign of the cash balances as in Çetin et al. [5]. The important fact is that, while there is usually a strong interaction between σi and ωi, the nonobservability of ω makes it reasonable to assume that ωi cannot directly influence the rating indicators σj for j 6= i. In this paper we assume that the two indicators σi, ωi can only take two values, that we label by 1 (“good” financial state) and −1 (financial distress). In the case of portfolios consisting of defaultable bonds, we may then refer to the rating class corresponding to σ = −1 also as “speculative grade” and that corresponding to σ = +1 as “investment grade.” Although the restriction to only two possible values may appear to be unrealistic, we believe that many aspects of the qualitative behavior of the system do not really depend on this choice. On the other hand, modulo having more complex formulae, the results below can be easily extended to the case when these variables take an arbitrary finite number of values. In our binary variable model we are naturally led to an interacting in- tensity model, where we have to specify the intensities or rates (inverse of the average waiting times) at which the transitions σi 7→ −σi and ωi 7→ −ωi take place. If we neglect direct interactions between the ωi’s, and we make the mean-field assumption that the interaction between different firms only depends on the value of the global financial health indicator we are led to consider intensities of the form σi 7→ −σi with intensity a(σi, ωi,mσN ), ωi 7→ −ωi with intensity b(σi, ωi,mσN ), where a(·, ·, ·) and b(·, ·, ·) are given functions. Since both financial health and distress tend to propagate, we assume that a(−1, ωi,mσN ) is increasing in both ωi and m N , and a(1, ωi,m N ) is decreasing. Similarly, b(σi,−1,m and b(σi,1,m N ) should be respectively increasing and decreasing in their variables. The next simplifying assumption is that the intensity a(σi, ωi,m N) is actually independent of m N , that is, of the form a(σi, ωi). Although this assumption amounts to a rather mild computational simplification, it allows to show that aggregate behavior (phase transition, etc.) may occur even in absence of a direct interaction between rating indicators. Although a model of this generality could be fully analyzed, we make the following choice of the intensities, inspired by spin-glass systems, to make 8 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI the model depend on only a few parameters: σi 7→ −σi with intensity e−βσiωi , ωi 7→ −ωi with intensity e−γωim Here β and γ are positive parameters which indicate the strength of the corresponding interaction. Put differently, we are considering a continuous- time Markov chain on {−1,1}2N with the following infinitesimal generator: Lf(σ,ω) = e−βσiωi∇σi f(σ,ω) + e−γωjm N∇ωj f(σ,ω),(3) where ∇σi f(σ,ω) = f(σi, ω) − f(σ,ω) (analogously for ∇ωi ), and where the jth component of σi is σij = σj , for j 6= i, −σi, for j = i. The rest of the paper is devoted to a detailed analysis of the above model. We conclude this subsection with some general remarks on the model we have just defined. Remark 2.1. • We have viewed the variable σ as a rating class indicator. Contrary to the standard models for rating class transitions, our rating indicator σ is not Markov by itself, but it is Markov only if paired with ω. This property is in line with empirical data and with recent research in the field of credit migration models. It is in fact well documented that real data of credit migration between rating classes exhibit a “non-Markovian” behavior. For a discussion on this topic see, for example, Christensen et al. [6]. In that paper the authors propose a hidden Markov process to model credit migration. The basic criticism to Markovianity is the fact that the probability of being downgraded is higher for firms that have been just downgraded. In order to capture this issue, the authors consider an “excited” rating state (e.g., B∗ from which there is a higher probability to be downgraded compared to the standard state B). This point of view is not far from ours, even though the mechanism of the transition is different. The downgrade to σ = −1 is higher when (σ = 1, ω = −1) compared to (σ = 1, ω = 1). • In our model, unlike other rating class models, we do not introduce a de- fault state for firms; it could be identified as a value for the pair (σ,ω) for which the corresponding intensities are identically zero, that is, a(σ,ω,m N ) = b(σ,ω,m N ) = 0 for all values of m N . This would have the effect of intro- ducing a “trap state” for the system, changing drastically the long-time LARGE PORTFOLIO LOSSES 9 behavior. Even in case of defaultable firms, however, our model could be meaningful up to a time-scale in which the fraction of defaulted firms is small. • With a choice of the intensities as in (2) we introduce a form of symmetry in our model, whereby the values σ = −1 and σ = +1 for the rating indica- tor turn out to be equally likely. One could, however, modify the model in order to make the value σ = −1 less (more) likely than the value σ = +1 and this could, for example, be achieved by letting the intensity for ωi be of the form eωiφ(m ), where φ is an increasing, nonlinear and noneven function. A possible “prototype” choice would be φ(x) = γ(x−K)+ + δ with γ, δ > 0 and K ∈ (0,1). Note that with this latter choice we have φ≥ 0 so that the value ωi = +1 (and hence also σi = +1) becomes more likely. Such an asymmetric setup might be more realistic in financial appli- cations but, besides leading to more complicated derivations, it depends also on the specific application at hand. Since, as already mentioned, we want to study a model that is as simple as possible and yet capable of producing the basic features of interest, in this paper we concentrate on the “symmetric choice” in (2). The large deviation approach to the Law of Large Numbers developed in Sections 3.1 and 3.2 can be adapted to the asymmetric setup (see Remark 3.5) with no essential difference. On the other hand, our proof of the Central Limit Theorem in Section 3.3 may require more regularity on the function φ above. We leave this point for further investigation. 2.2. Invariant measures and nonreversibility. Mean-field models as the one we propose in this paper have already appeared, mostly in the statistical mechanics literature (see in particular [12] and [8], from which we borrow many of the mathematical tools). However, unlike what happens for the models in the cited references, we now show that our model is nonreversible. This implies that an explicit formula for the stationary distribution and its N →∞ asymptotics is not available. It is thus appropriate to follow a more specifically dynamic approach to understand the long-time behavior of the system. As already mentioned, we shall thus first study the N →∞ limit of the dynamics of the system, obtaining limit evolution equations. Then we study the equilibria of these equations. This is not necessarily equivalent to studying the N →∞ properties of the stationary distribution µN . However, as we shall show later in this paper, this provides rather sharp information on how the system behaves for t and N large. The operator L given in (3) defines an irreducible, finite-state Markov chain. It follows that the process admits a unique stationary distribution µN , that is, a distribution such that, for each function f on the configuration space of (σ,ω), µN (σ,ω)Lf(σ,ω) = 0.(4) 10 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI This distribution reflects the long-time behavior of the system, in the sense that, for each f and any initial distribution, E[f(σ(t), ω(t))] = µN (σ,ω)f(σ,ω). The stationarity condition (4) is equivalent to [µN (σ i, ω)eβσiωi − µN (σ,ω)e−βσiωi ] [µN (σ,ω i)eγωim N − µN (σ,ω)e−γωim N ] = 0 for every σ,ω ∈ {−1,1}N . Simpler sufficient conditions for stationarity are the so-called detailed bal- ance conditions. We say that a probability ν on {−1,1}2N satisfies the de- tailed balance condition for the generator L if ν(σi, ω)eβσiωi = ν(σ,ω)e−βσiωi and ν(σ,ωi)eγωim N = ν(σ,ω)e−γωim for every σ,ω. When the detailed balance conditions (6) hold, we say the sys- tem is reversible: the stationary Markov chain with generator L and marginal law ν has a distribution which is left invariant by time-reversal. In the case (6) admits a solution, they usually allow to derive the stationary distribution explicitly. This is not the case in our model. We have in fact: Proposition 2.2. The detailed balance equations (6) admit no solution, except at most for one specific value of N . Proof. By way of contradiction, assume a solution ν of (6) exists. Then one easily obtains ∇σi log ν(σ,ω) = −2βσiωi, ∇ωi log ν(σ,ω) = −2γωim which implies ∇ωi ∇σi log ν(σ,ω) = 4βσiωi, ∇σi ∇ωi log ν(σ,ω) = 4N−1γωiσi. This is not possible since ∇ωi ∇σi log ν(σ,ω)≡∇σi∇ωi log ν(σ,ω). � LARGE PORTFOLIO LOSSES 11 3. Main results: law of large numbers and Central Limit Theorem. In this section we state the results concerning the dynamics of the system (σi[0, T ], ωi[0, T ]) i=1 in the limit as N → ∞. Note that for each value of N we are considering a Markov process with generator (3). Thus, it would be more accurate to denote by (σ i [0, T ], ω i [0, T ]) the trajectories of the variables related to the ith firm in the system with N firms. For convenience, we consider a fixed probability space (Ω,F , P ) where all D([0, T ])-valued processes σ i [0, T ], ω i [0, T ] are defined, and the following conditions are satisfied: • for each N ≥ 1 the processes (σ(N)i [0, T ], ω i [0, T ]) i=1 are Markov pro- cesses with infinitesimal generator (3); • for each N ≥ 1 the {−1,1}2-valued random variables (σ(N)i (0), ω i (0)) are independent and identically distributed with an assigned law λ. This last assumption on the initial distribution is stronger than what we actually need to prove the results below; however, it allows to avoid some technical aspects in the proof, that we consider not essential for the purposes of the paper. The other point, concerning the fact of realizing all processes in the same probability space, is not a restriction; we are not making any assumption on the dependence of processes with different values of N , so this joint realization is always possible. Its main purpose is to allow to state a strong law of large numbers. Our approach proceeds according to the following three steps, to which correspond the three subsections below, namely: (i) look for the limit dynamics of the system (N →∞); (ii) study the equilibria of the limiting dynamics; (iii) describe the “finite volume approximations” (for large but finite N ) via a central limit-type result. 3.1. Deterministic limit: large deviations and law of large numbers. In what follows D([0, T ]) denotes the space of right-continuous, piecewise con- stant functions [0, T ] → {−1,1}, endowed with the Skorohod topology (see [17]). Let (σi[0, T ], ωi[0, T ]) i=1 ∈D([0, T ])2N denote a path of the process in the time-interval [0, T ] for a generic T > 0. If f(σi[0, T ], ωi[0, T ]) is a function of the trajectory of the variables related to a single firm, one is interested in the asymptotic behavior of empirical averages of the form f(σi[0, T ], ωi[0, T ]) =: f dρN , where ρN is the sequence of empirical measures δ(σi[0,T ],ωi[0,T ]). 12 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI We may think of ρN as a (random) element of M1(D([0, T ]) × D([0, T ])), the space of probability measures on D([0, T ])×D([0, T ]) endowed with the weak convergence topology. Our first aim is to determine the limit of f dρN as N →∞, for f con- tinuous and bounded; in other words we look for the weak limit limN ρN in M1(D([0, T ]) × D([0, T ])). This corresponds to a law of large numbers with the limit being a deterministic measure. This limit, being an element of M1(D([0, T ])×D([0, T ])), can be viewed as a stochastic process, and rep- resents the dynamics of the system in the limit N →∞. The fluctuations of ρN around this deterministic limit will be studied in Section 3.3 below, and this turns out to be particularly relevant in the risk analysis of a portfolio (Section 4). The result we actually prove is a large deviation principle, which is much stronger than a law of large numbers. We start with some preliminary no- tions letting, in what follows, W ∈M1(D([0, T ])×D([0, T ])) denote the law of the {−1,1}2-valued process (σ(t), ω(t)) such that (σ(0), ω(0)) has distri- bution λ, and both σ(·) and ω(·) change sign with constant intensity 1. For Q ∈M1(D([0, T ])×D([0, T ])) let H(Q|W ) := dQ log , if Q≪W and log dQ ∈L1(Q), +∞, otherwise, denote the relative entropy between Q and W . Moreover, ΠtQ denotes the marginal law of Q at time t, and t := γ σΠtQ(dσ, dτ). For a given path (σ[0, T ], ω[0, T ]) ∈D([0, T ]) ×D([0, T ]), let Nσt (resp. Nωt ) be the process counting the jumps of σ(·) [resp. ω(·)]. Define F (Q) = (1− e−βσ(t)ω(t))dt+ (1− e−ω(t)γ t )dt σ(t)ω(t−)dNσt + ω(t)γ whenever (NσT +N T )dQ<+∞, and F (Q) = 0 otherwise. Finally let I(Q) :=H(Q|W )− F (Q). We remark that, if (NσT +N T )dQ = +∞, then H(Q|W ) = +∞ (this will be shown in Section 5, Lemma 5.4) and thus also I(Q) = +∞. LARGE PORTFOLIO LOSSES 13 Proposition 3.1. For each Q ∈ M1(D([0, T ]) × D([0, T ])), I(Q) ≥ 0, and I(·) is a lower-semicontinuous function with compact level-sets [i.e., for each k > 0 one has that {Q : I(Q) ≤ k} is compact in the weak topology]. Moreover, for A,C ⊆M1(D([0, T ])×D([0, T ])) respectively open and closed for the weak topology, we have lim inf logP (ρN ∈A) ≥− inf I(Q),(8) lim sup logP (ρN ∈C) ≤− inf I(Q).(9) This means that the distributions of ρN obey a large deviation principle (LDP) with rate function I(·) (see, e.g., [15] for the definition and funda- mental facts on LDP). The proof of Proposition 3.1 is given in Section 5 and follows from ar- guments similar to those in [12]. Various technical difficulties are due to unboundedness and noncontinuity of F , which are related to the nonre- versibility of the model. The key step to derive a law of large numbers from Proposition 3.1 is given in the following result, whose proof is also given in Section 5. In what follows, for q ∈M1({−1,1}2) a probability on {−1,1}2, we define mσq := σ,ω=±1 σq(σ,ω), that can be interpreted as the expected rating under q. Proposition 3.2. The equation I(Q) = 0 has a unique solution Q∗. Moreover, if qt ∈M1({−1,1}2) denotes the marginal distribution of Q∗ at time t, then qt is the unique solution of the nonlinear (McKean–Vlasov) equation = Lqt, t∈ [0, T ], q0 = λ, where Lq(σ,ω) = ∇σ[e−βσωq(σ,ω)] +∇ω[e−γωmσq q(σ,ω)](11) with (σ,ω) ∈ {−1,1}2. From Propositions 3.1 and 3.2, it is easy to derive the following strong law of large numbers. 14 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Theorem 3.3. Let Q∗ ∈ M1(D([0, T ]) × D([0, T ])) be the probability given in Proposition 3.2. Then ρN → Q∗ almost surely in the weak topology. Proof. Let Q∗ be the unique zero of the rate function I(·) as given by Proposition 3.2. Let BQ∗ be an arbitrary open neighborhood of Q ∗ in the weak topology. By the upper bound in Proposition 3.1, we have limsup logP (ρN /∈BQ∗)≤− inf Q/∈BQ∗ I(Q)< 0, where the last inequality comes from lower semicontinuity of I(·), com- pactness of its level sets and the fact that I(Q) > 0 for every Q 6=Q∗. In- deed, if infQ/∈BQ∗ I(Q) = 0, then there exists a sequence Qn /∈BQ∗ such that I(Qn) → 0. By the compactness of the level sets there exists then a sub- sequence Qnk → Q̄ /∈BQ∗ . By lower semicontinuity it then follows I(Q̄) ≤ lim inf I(Qnk) = 0 which contradicts I(Q)> 0 for q 6=Q∗. By the above in- equality we thus have that P (ρN /∈BQ∗) decays to 0 exponentially fast. By a standard application of the Borel–Cantelli lemma, we obtain that ρn →Q∗ almost surely. � 3.2. Equilibria of the limiting dynamics: phase transition. Equation (10) describes the dynamics of the system with generator (3) in the limit as N → +∞. In this section we determine the equilibrium points, or stationary (in t) solutions of (10), that is, solutions of Lqt = 0 and, more generally, the large time behavior of its solutions. First of all, it is convenient to reparametrize the unknown qt in (10). Let q be a probability on {−1,1}2. Note that each f :{−1,1}2 → R can be written in the form f(σ,ω) = aσ + bω + cσω + d. It follows that q is completely identified by the expectations mσµ := σ,ω=±1 σq(σ,ω), mωµ := σ,ω=±1 ωq(σ,ω),(12) mσωµ := σ,ω=±1 σωq(σ,ω). In particular, if q = qt, the marginal of Q ∗ appearing in Proposition 3.2, then we write mσt for m , and similarly for mωt ,m t . In order to rewrite (10) in terms of the new variables mσt ,m t , observe that ṁσ = σ,ω=±1 σq̇t(σ,ω) = σ,ω=±1 σLqt. LARGE PORTFOLIO LOSSES 15 On the other hand, a straightforward computation shows that, for every probability q, σ,ω=±1 σLq = 2sinh(β)mωq − 2cosh(β)mσq , giving ṁσt = 2sinh(β)m t − 2cosh(β)mσt . By making similar computations for mωt ,m t , it is shown that (10) can be rewritten in the following form: ṁσt = 2sinh(β)m t − 2cosh(β)mσt , ṁωt = 2sinh(γm t )− 2cosh(γmσt )mωt ,(13) ṁσωt = 2sinh(β) + 2sinh(γm t − 2(cosh(β) + cosh(γmσt ))mσωt , with initial condition mσ0 =m λ , m λ . Note that m t does not appear in the first and in the second equation in (13); this means that the differential system (13) is essentially two-dimensional: first one solves the two-dimensional system (on [−1,1]2) (ṁσt , ṁ t ) = V (m t ),(14) with V (x, y) = (2 sinh(β)y − 2cosh(β)x,2 sinh(γx)− 2y cosh(γx)), and then one solves the third equation in (13), which is linear in mσωt . Note also that to any (mσ∗ ,m ∗ ) satisfying V (m ∗ ) = 0, there corresponds a unique mσω∗ := sinh(β)+mσ∗ sinh(γm cosh(β)+cosh(γmσ∗ ) such that (mσ∗ ,m ∗ ) is an equilibrium (stable solution) of (13). Moreover, ifmσt →mσ∗ as t→ +∞, then mσωt →mσω∗ . Thus, to discuss the equilibria of (13) and their stability, it is enough to analyze (14) and for this we have the following proposition, where by “linearly stable equilibrium” we mean a pair (x̄, ȳ) such that V (x̄, ȳ) = 0, and the linearized system (ẋ, ẏ) =DV (x̄, ȳ)(x− x̄, y − ȳ) is stable, that is, the eigenvalues of the Jacobian matrix DV (x̄, ȳ) have all negative real parts. Theorem 3.4. (i) Suppose γ ≤ 1 tanh(β) . Then (14) has (0,0) as a unique equilibrium solution, which is globally asymptotically stable, that is, for every initial condition (mσ0 ,m 0 ), we have (mσt ,m t ) = (0,0). (ii) For γ < 1 tanh(β) the equilibrium (0,0) is linearly stable. For γ = 1 tanh(β) the linearized system has a neutral direction, that is, DV (0,0) has one zero eigenvalue. 16 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI (iii) For γ > 1 tanh(β) the point (0,0) is still an equilibrium for (14), but it is a saddle point for the linearized system, that is, the matrix DV (0,0) has two nonzero real eigenvalues of opposite sign. Moreover (14) has two linearly stable solutions (mσ∗ ,m ∗ ), (−mσ∗ ,−mω∗ ), where mσ∗ is the unique strictly positive solution of the equation x= tanh(β) tanh(γx),(15) mω∗ = tanh(β) mσ∗ .(16) (iv) For γ > 1 tanh(β) , the phase space [−1,1]2 is bipartitioned by a smooth curve Γ containing (0,0) such that [−1,1]2 \ Γ is the union of two disjoint sets Γ+,Γ− that are open in the induced topology of [−1,1]2. Moreover (mσt ,m t ) = (mσ∗ ,m ∗ ), if (m 0 ) ∈ Γ+, (−mσ∗ ,−mω∗ ), if (mσ0 ,mω0 ) ∈ Γ−, (0,0), if (mσ0 ,m 0 ) ∈ Γ. Proof. See Section 5. � Remark 3.5. The results in this section are specific to our model with the symmetry properties as induced by the specification of the intensities in (2). With an asymmetric setup such as described in Remark 2.1, (15) becomes x= tanh(β) tanh(φ(x)) thus allowing more flexibility in the position of the equilibria. In particular, by letting φ(x) = γ(x−K)+ + δ, while still having three equilibria, we may choose their relative position by suitably choosing the values for γ,K, δ. Notice that in this way we also increase the number of parameters in our model. 3.3. Analysis of fluctuations: Central Limit Theorem. Having established a law of large numbers ρN →Q∗, it is natural to analyze fluctuations around the limit, that is, the rate at which ρN converges to Q ∗ and the asymptotic distribution of ρN −Q∗. To study the asymptotic distribution of ρN −Q∗ there are at least the following two possible approaches: (i) An approach based on a functional central limit theorem using a result in [2] that relates large deviations with the Central Limit Theorem (see [35], Chapter 3, for some results in this direction). LARGE PORTFOLIO LOSSES 17 (ii) A weak convergence-type approach based on uniform convergence of the generators (see [17]). In this paper we shall follow an approach of the second type; more pre- cisely we shall provide a dynamical interpretation of the law of large numbers discussed in Theorem 3.3. Let ψ :{−1,1}2 → R, and define ρN (t) by ψdρN (t) := ψ(σi(t), ωi(t)). In other words, ρN (t) is the marginal of ρN at time t and we also have N (t) =m ρN (t) . Note that, for each fixed t, ρN (t) is a probability on {−1,1}2, and so, by the considerations leading to (12), it can be viewed as a three- dimensional object. Thus (ρN (t))t∈[0,T ] is a three-dimensional flow. A simple consequence of Theorem 3.3 is the following convergence of flows: (ρN (t))t∈[0,T ] → (qt)t∈[0,T ] a.s.,(17) where the convergence of flows is meant in the uniform topology. Since the flow of marginals contains less information than the full measure of paths, the law of large numbers in (17) is weaker than the one in Theorem 3.3. However, the corresponding fluctuation flow N(ρN (t)− qt))t∈[0,T ] is also a finite-dimensional flow, and it allows for a very explicit characteriza- tion of the limiting distribution. The following theorem gives the asymptotic behavior of this fluctuation flow; its proof is given in Section 5. Theorem 3.6. Consider the following three-dimensional fluctuation pro- cess: xN (t) := N(mσρN (t) −m yN (t) := N(mωρN (t) −m zN (t) := N(mσωρN (t) −m Then (xN (t), yN (t), zN (t)) converges as N →∞, in the sense of weak con- vergence of stochastic processes, to a limiting three-dimensional Gaussian process (x(t), y(t), z(t)) which is the unique solution of the following linear stochastic differential equation: dx(t) dy(t) dz(t)  =A(t)  dt+D(t) dB1(t) dB2(t) dB3(t) (18) 18 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI where B1,B2,B3 are independent, standard Brownian motions, A(t) = 2 − cosh(β) −γmωt sinh(γmσt ) + γ cosh(γmσt ) sinh(γmσt ) + γm t cosh(γm t )− γmσωt sinh(γmσt ) sinh(β) 0 − cosh(γmσt ) 0 0 −(cosh(β) + cosh(γmσt )) D(t)D∗(t) −mσωt sinh(β) + cosh(β) 0 0 −mωt sinh(γmσt ) + cosh(γmσt ) −mσt sinh(β) +mωt cosh(β) mσt cosh(γmσt )−mσωt sinh(γmσt ) −mσt sinh(β) +mωt cosh(β) mσt cosh(γm t )−mσωt sinh(γmσt ) −mσωt sinh(β) + cosh(β)−mω sinh(γmσt ) + cosh(γmσt ) and (x(0), y(0), z(0)) have a centered Gaussian distribution with covariance matrix 1− (mσλ)2 mσωλ −mσλmωλ mωλ −mσλmσωλ mσωλ −mσλmωλ 1− (mωλ)2 mσλ −mσωλ mωλ mωλ −mσλmσωλ mσλ −mσωλ mωλ 1− (mσωλ )2 .(19) Theorem 3.6 guarantees that, for each t > 0, the distribution of (xN (t), yN (t), zN (t)) is asymptotically Gaussian, and provides a method to compute the limiting covariance matrix. Indeed, denote by Σt the covariance matrix of (x(t), y(t), z(t)). A simple application of Itô’s rule to (18) shows that Σt solves the Lyapunov equation =A(t)Σt + ΣtA(t) ∗ +D(t)D∗(t).(20) In order to solve (20), it is convenient to interpret Σ as a vector in R3×3 = 3⊗R3. To avoid ambiguities, for a 3×3 matrix C we write vec(C) whenever we interpret it as a vector. It is easy to check that (20) can be rewritten as follows d(vec(Σt)) = (A(t)⊗ I + I ⊗A(t)) vec(Σt) + vec(D(t)D∗(t)),(21) where “⊗” denotes the tensor product of matrices. Equation (21) is linear, so its solution can be given an explicit expression and can be computed after LARGE PORTFOLIO LOSSES 19 having solved (13). More importantly, the behavior of Σt for large t can be obtained explicitly as follows. A. Case γ < 1 tanh(β) . In this case we have shown in Theorem 3.4 that the solution (mσt ,m t ) of (13) converges to (0,0, tanh(β)) as t→ +∞. In particular, one immediately obtains the limits A := lim A(t), DD∗ := lim D(t)D∗(t).(22) A direct inspection (see the Appendix) shows that A has three real strictly negative eigenvalues. Moreover, the eigenvalues of the matrix A × I + I × A are all of the form λi + λj where λi and λj are eigenvalues of A, and therefore they are all strictly negative. It follows from (21) that limt→+∞ Σt = Σ where vec(Σ) = −(A⊗ I + I ⊗A)−1 vec(DD∗).(23) B. Case γ > 1 tanh(β) . Also in this case, by Theorem 3.4, the limit (mσt ,m exists. Disregarding the exceptional case in which the initial condition of (13) belongs to the stable manifold Γ introduced in Theorem 3.4(iv), the limit above equals either (mσ∗ ,m ∗ ), or (−mσ∗ ,−mω∗ ,mσω∗ ), depending on the initial condition, where (mσ∗ ,m ∗ ) are obtained by Theorem 3.4(iii). In both cases one obtains as in (22) the limits A and DD∗, and we show in the Appendix that also in this case the eigenvalues of A are real and strictly negative, so that limt→+∞ Σt = Σ is obtained as in (23). C. Case γ = 1 tanh(β) . In this case, as shown in the Appendix, the limiting matrix A is singular; it follows that the limit limt→+∞ Σt does not exist, as one eigenvalue of Σt grows polynomially in t. This means that, for critical values of the parameters, the size of normal fluctuations around the deterministic limit grows in time. Similarly to what is done in [8] for reversible models, it is possible to determine the critical long-time behavior of the fluctuation by a suitable space–time scaling in the model, giving rise to nonnormal fluctuations. More precisely, one can show the following convergence in distribution: N1/4(m·ρN ( Nt)−m·( N→∞−→ Z where Z is non-Gaussian. This result is contained in [32]. We now state an immediate corollary of Theorem 3.6 concerning the fluc- tuations of the global health indicator; this will be used in the next section on large portfolio losses. 20 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Corollary 3.7. As N →∞ we have that N [mσρN (t) −m converges in law to a centered Gaussian random variable Z with variance V (t) = Σ11(t),(24) where Σ(t) solves (20) and mσt solves (13). We conclude this section with the following: Remark 3.8. The evolution equation (20) for the covariance matrix Σt is coupled with the McKean–Vlasov equation (13), and their joint behavior exhibits interesting aspects even before the system gets close to the stable fixed point. In particular, in the case γ > 1 tanh(β) , if the initial condition is sufficiently close to the stable manifold Γ, the system (13) spends some time close to the symmetric equilibrium (0,0) before drifting to one of the stable equilibria. A closer look at (20) shows that when the system is close to the neutral equilibrium, the covariance matrix Σ grows exponentially fast in time, causing sharp peaks in the variances. This is related to the credit crisis mentioned in the Introduction. A more detailed discussion on this point is given in the next section, in relation with applications to portfolio losses. 4. Portfolio losses. We address now the problem of computing losses in a portfolio of positions issued by the N firms. A rather general modeling framework is to consider the total loss that a bank may suffer due to a risky portfolio at time t as a random variable defined by LN (t) = iLi(t). Different specifications for the single (marginal) losses Li(t) can be chosen accounting for heterogeneity, time dependence, interaction, macroeconomic factors and so on. A punctual treatment of this general modeling framework can be found in the book by McNeil, Frey and Embrechts [29]. For a com- parison with the most widely used industry examples of credit risk models see Frey and McNeil [21], Crouhy, Galai and Mark [11] or Gordy [25]. The same modeling insights are also developed in the most recent literature on risk management and large portfolio losses analysis; see [14, 19, 23, 26] for different specifications. In this paper we adopt the point of view of Giesecke and Weber [23]. The idea is to compute the aggregate losses as a sum of marginal losses Li(t), of which the distribution is supposed to depend on the realization of the variable σi, that is, on the rating class. In particular, conditioned on the realization of σ, the marginal losses will be assumed to be independent and identically distributed (the independence condition can be weakened; see LARGE PORTFOLIO LOSSES 21 Example 4.4 below). More precisely, we assume given a suitable conditional distribution function Gx, x ∈ {−1,1}, namely Gx(u) := P (Li(t) ≤ u|σi(t) = x)(25) where the first and second moments are well defined, namely l1 :=E(Li(t)|σi(t) = 1)<E(Li(t)|σi(t) = −1) =: l−1(26) v1 := Var(Li(t)|σi(t) = 1), v−1 := Var (Li(t)|σi(t) = −1).(27) The inequality in (26) specifies that we expect to lose more when in financial distress. The aggregate loss of a portfolio of volume N at time t is then defined as LN (t) = Li(t). We recall the definition of the global health indicatorsm N (t) := i=1 σi(t), and mσt := σ dqt where qt solves the McKean–Vlasov equation [see (10)]. We also introduce a deterministic time function, which will be seen to represent an “asymptotic” loss when the number of firms goes to infinity, namely L(t) = (l1 − l−1) mσt + (l1 + l−1) .(28) We state now the main result of this section. Theorem 4.1. Assume Li(t) has a distribution of the form (25). Then for t ∈ [0, T ] with generic T > 0 and for any value of the parameters β > 0 and γ > 0, we have LN (t) −L(t) → Y ∼N(0, V̂ (t)) in distribution, where L(t) has been defined in (28) and V̂ (t) = (l1 − l−1)2V (t) (1 +mσt )v1 (1−mσt )v−1 ,(29) with V (t) as defined in (24). Proof. See Section 5. � 22 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Remark 4.2. The Gaussian approximation in Theorem 4.1 leads in particular to P (LN (t) ≥ α) ≈N NL(t)−α V̂ (t) .(30) By the symmetry of the model, the above Gaussian approximation for the losses is appropriate for a wide (depending on N ) range of values of α. If we modify the model to become asymmetric as discussed in Remark 2.1 and, more precisely, we modify it so that σ = −1 becomes much less likely than σ = +1, then for a “realistic” value of N , the number of firms with σi = −1 could be too small for the Gaussian approximation to be sufficiently precise. One could then rather consider a Poisson-type approximation instead. We shall now provide examples illustrating possible specifications for the marginal loss distributions where, without loss of generality, we assume a unitary loss (e.g., loss due to a corporate bond) when a firm is in the bad state. We start with a very basic example where we assume that the marginal losses (when conditioned on the value of σ) are deterministic. This means that the riskiness of the loss portfolio is related only to the number of firms in financial distress and so we can use directly the results of Section 3, in particular of Corollary 3.7. Example 4.3. Suppose that marginal losses are described as follows: Li(t) = 1, if σi(t) = −1, 0, if σi(t) = 1. On the other hand LN (t) = 1− σi(t) Recalling that m N (t) = i σi(t), by Corollary 3.7 [see also (30)], we can compute various risk measures related to the portfolio losses such as the following Var-type measure: P (LN (t) ≥ α) = P N −NmσN (t) N (t) ≤ N − 2α (−2α+ (1−mσt )N√ V (t) (−2α+ 2L∞(t)N√ V (t) where L∞(t) := limN→∞ LN (t) = limN→∞ 1−σi(t) 1−mσt LARGE PORTFOLIO LOSSES 23 Fig. 1. Excess loss in a large portfolio (N = 10,000) for different values of the parameters γ and β compared with the independence case. Looking at a portfolio of N = 10,000 small firms, we compute the excess loss probability for different values of the parameters β,γ comparing them with the benchmark case where there is no interaction at all, that is, where β = γ = 0 (“independence case”). In Figure 1 we show the cumulative prob- ability of having excess losses for the same portfolios. In this figure we see that, when the dependence increases, variance and risk measures increase as well. More general specifications are already suggested in the existing literature. For example, one could consider the losses to depend also on a random exogenous factor Ψ; more precisely, the marginal losses Li(t) are independent and identically distributed conditionally to the realizations of the σi(t)’s and of Ψ. The conditional distributions Gx(u) := P (Li(t) ≤ u|σi(t) = x,Ψ) are random variables, as well as the corresponding moments l1, l−1, v1, v−1. In particular in the following example we apply our approach to a very tractable class of models, the “Bernoulli mixture models.” This kind of mod- eling has been used in the context of cyclical correlations, that is in models where exogenous factors are supposed to characterize the evolution of the in- dicator of defaults (the classical factor models). In the context of contagion- based models this class was first introduced by Giesecke and Weber in [23]. 24 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Example 4.4 (Bernoulli mixture models). Assume that the marginal losses Li(t) are Bernoulli mixtures, that is, Li(t) = 1, with probability P (σi(t),Ψ), 0, with probability 1−P (σi(t),Ψ), where the mixing derives not only from the rating class indicator σi(t) of firm i, but also from an exogenous factor Ψ ∈ Rp that represents macroeconomic variables that reflect the business cycle and thus allow for both contagion and cyclical effects on the rating probabilities. Notice that, with the above specification, the quantities defined in (26) and (27) now depend on the random factor Ψ, that is, l1 = P (1,Ψ), v1 = P (1,Ψ)(1−P (1,Ψ)) and analogously for l−1, v−1. Consequently, the asymptotic loss function L(t) as well as the variance of the Gaussian approximation V̂ (t) defined in (28) and (29) are also functions of Ψ. With a slight abuse of notation we shall write Lψ(t) [respectively V̂ψ(t)] for the asymptotic loss (variance) at time t given that Ψ = ψ. Next we give a possible expression for the mixing distribution for P (σ,Ψ) that is in line with existing models on contagion. Let a and bi, i= 1,2, be nonnegative weight factors. Let us assume for simplicity that Ψ ∈ R is a Gamma distributed random variable. Define then P (σ,Ψ) = 1− exp −aΨ− b1 This specification follows the CreditRisk+ modeling structure, even though in the standard industry examples direct contagion is not taken into account. Notice that the factor 1−σ increases the probability of default for the firms in the bad rating class (σ = −1). Using (30) we have that P (LN (t) ≥ α) ≈ NLψ(t)−α NV̂ψ(t) dfΨ(ψ), where fΨ is the density function of the Gamma random variable Ψ. In Figure 2 we plot the excess loss probability in the case where a= 0.1, b1 = 1, b2 = 0.5 and β = 1.5 is supposed to be fixed. We compare different specifications for Ψ and γ. In particular we consider the following cases: Ψ = 4.5, γ = 0.6; Ψ = 4.5, γ = 1.1; Ψ ∼ Γ(2.25,2), γ = 1.1. The shape of the excess losses suggests that the loss may be sensibly higher in the case of high uncertainty about the value of the macroeconomic factor [Ψ ∼ Γ(2.25; 2)] and in the case of high level of contagion (γ = 1.1). Notice that in all three situations we are in the subcritical case, since the critical value for γ is γc = 1/ tanh(β) ≃ 1.105. This also implies that the equilibrium value is the same in the three situations and depends only on Ψ. LARGE PORTFOLIO LOSSES 25 Fig. 2. Loss amount in a large portfolio (N = 10,000) in the case of marginal losses which (depending on the rating class) are distributed as Bernoulli random variables for which the parameter depends on Ψ. Remark 4.5. Notice that the asymptotic loss distribution in the above Bernoulli mixture model does not only depend on a mixing parameter as in standard Bernoulli mixture models but, via L(t), it depends also on the value mσt of the asymptotic average global health indicator. Moreover, com- pared to Giesecke and Weber [23], we are able to quantify the time-varying fluctuations of the global indicator mσ ρN (t) . We shall see that this may sen- sibly influence the distribution of losses in particular when looking at two different time horizons T1 and T2 before and after a credit crisis. Remark 4.6. Further examples may be considered, in particular when the distribution of the marginal losses Li(t) depends on the entire past trajectory of the rating indicator σi, taking, for example, into account how long the firm has been in the bad state. Instead of depending simply on σi(t), the distribution of Li(t) could then be made dependent on Si(t) := ((1−σi(s))/2)ds≥δt} with δ ∈ (0,1), which is equal to 1 if firm i has spent a fraction δ of time in the bad state. Corresponding to (32) we would then Li(t) = 1, with probability P (Si(t),Ψ), 0, with probability 1− P (Si(t),Ψ). This model is not a straightforward extension of Example 4.4. In fact the theory developed above, in particular the Central Limit Theorem result in 26 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Section 3.3, does not appear to be strong enough to handle it. For this purpose an approach based on a functional central limit theorem that was alluded to at the beginning of Section 3.3 would be more appropriate. This, however, goes beyond the scope of the present paper. Let us point out that in the examples above we have considered only the problem of computing large portfolio losses which led to examples where we computed (approximately) the quantiles P (LN (t) ≥ α) where α is a (large) integer. From here, one could then compute the probability that the loss ratio LN (t) belongs to a given interval and this would then allow to compute (approximately) for our contagion model also other quantities in a risk- sensitive environment. In any case notice that Theorem 4.1 provides the entire asymptotic distribution for the portfolio losses. In the previous examples we have described large portfolio losses at a predetermined time horizon T for different specifications of the conditional loss distribution. In what follows, we shall describe in more detail how the phenomenon of a credit crisis may be explained in our setting and how this issue may influence the quantification of losses. This dynamic point of view on risk management that accounts for the possibility of a credit crisis in the market, is one of the main contributions of this work. As one could expect, the possibility of having a credit crisis is related to the existence of particular conditions on the market, more precisely to certain levels of interaction between the obligors (i.e., the parameters β and γ) and certain values of the state variables describing the rating classes and the fundamentals (i.e., σ and ω). 4.1. Simulation results. To illustrate the situation we shall now present some simulation results. We shall proceed along two steps: the first one relates more specifically to the particle system, the second to the portfolio losses. Step 1 (Domains of attraction). In Section 3.2 we have characterized all the equilibria of the system depending on the values of the parameters. In particular we have shown that for supercritical values, by which we mean γ > 1 tanh(β) , there are two asymmetric equilibrium configurations in the space (mσ,mω) that, for our symmetric model, are symmetric to one another and are defined as (mσ∗ ,m ∗ ) and (−mσ∗ ,−mω∗ ). In particular, Theorem 3.4 allows to characterize their domains of at- traction, that is, the sets of initial conditions that lead the trajectory to one of the equilibria, and we shall denote them by Γ+ and Γ−. Numerical simulations provide diagrams as in Figure 3. LARGE PORTFOLIO LOSSES 27 Fig. 3. Domains of attraction Γ+ for (mσ∗ ,m ∗ ) and Γ − for (−mσ∗ ,−m ∗ ) and their boundary Γ for β = 1 and varying γ. Here the critical value for γ is γc := 1/ tanh(β) ≃ 1.313. Step 2 (Credit crises). We show results from numerical simulations that detect the crises when the values of the parameters are supercritical and the initial conditions are “near” the boundary of the domains of attraction, that is, near Γ. Given the symmetry of our model, the behavior of the system will be perfectly symmetric when starting in either Γ+ or Γ−, but the typical credit crisis corresponds to what happens in Γ−, so that below we shall illustrate this latter case. The analysis in an asymmetric model would be analogous. In Figure 4 we have plotted a trajectory starting in (mσ0 ,m 0 ) ∈ Γ− but near the boundary. It can be seen that the path moves toward (mσ,mω) = (0,0) and then leaves it decaying to the stable equilibrium. Concerning the time evolution, we see in Figure 5 that, for an initial condition in Γ− and near the boundary, the variable mσt (the same would happen also with mωt that for clarity is not plotted) is first attracted to the unstable value zero, around which it spends a long time before moving to the stable equilibrium value mσ∗ . This can be explained, in financial terms, as follows: Suppose that at the initial time the market conditions are such that (mσ,mω) are in Γ− but close to the curve Γ. Then for a while the system moves close to the stable manifold Γ toward (0,0), until it gets “captured” 28 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Fig. 4. Domains of attraction Γ+ for (mσ∗ ,m ∗ ) and Γ − for (−mσ∗ ,−m ∗ ) and phase dia- gram of (mσt ,m t ) with initial conditions (m 0 ) = (0.6,−0.85) when β = 1 and γ = 2.3 [here γc = 1/ tanh(β) ≃ 1.313]. by the unstable direction of the equilibrium point (0,0). Since the system configuration belongs to Γ−, the new stable equilibrium that the system is attracted to is given by (−mσ∗ ,−mω∗ ). This situation represents (in a stylized manner) what we intend as a credit crisis: the state (0,0) may be considered as a “credit bubble,” the decay toward the stable equilibrium mimics a credit crisis (i.e., a crash in the credit market). As soon as the system moves away from (0,0), the uncertainty (volatility) increases quickly and the credit quality indicators move to the stable con- figuration changing completely the picture of the market (the speed of the convergence depends on the level of interaction). This situation is also well illustrated by the loss probability computed before and after the crisis (i.e., in certain time instants T1 and T2). In Figure 6 we see the excess probability of suffering a loss larger than x for the case of Example 4.4 with an exogenous parameter Ψ ∼ Γ(2.25; 2). One can see that before the crisis both the expected loss and the variance may be underestimated as well as the corresponding risk measures. Put differently, a model that does not distinguish between stable and unstable equilibria LARGE PORTFOLIO LOSSES 29 Fig. 5. Trajectory of mσt and V (t) with initial conditions m 0 =−0.5, m 0 = 0.395 when β = 1.5 and γ = 2.1 [here γc = 1/ tanh(β) ≃ 1.105]. We have marked by (∗) the time hori- zons T1 = 2 and T2 = 10 before and after the crisis where in Figure 6 we shall compute the excess loss probabilities. (does not take credit crises into account) may underestimate the excess loss probability, since it does not recognize in the given situation the possibility of a sudden crash. Finally we mention the fact that for different levels of interaction we can distinguish between a smoothly varying business cycle and a crisis. When β and γ, the parameters describing the level of interaction, are sufficiently small, the business cycle (described in our simple model by the proportion of firms in the rating classes) evolves smoothly and the induced variance (level of uncertainty about the number of bad rated firms) is lower compared to the crisis case. In Figure 7 we show this fact for two levels of β and γ, both supercritical. 5. Proofs. 5.1. Proofs of Propositions 3.1 and 3.2. One of the main tools in this proof is the Girsanov formula for Markov chains. Since a Markov chain is a functional of the multivariate point process that counts the jumps between all pairs of states, this formula can be derived from the corresponding Gir- sanov formula for point processes (see, e.g., [3], Section 4.2). We state it here for completeness. 30 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Fig. 6. Excess probability of losses in a portfolio of N = 10,000 obligors, β = 1.5 and γ = 2.1 computed in T1 = 2 and T2 = 10, namely before and after the crisis in the case of Example 4.4 with Ψ ∼ Γ(2.25; 2) [here γc = 1/ tanh(β) ≃ 1.105]. Fig. 7. Trajectories of mσt and V (t) for different levels of interaction, that is, letting β and γ vary. In the case of higher values we really see a crisis and a corresponding peak in the uncertainty in the market. In the case of smaller values the number of bad rated firms decreases smoothly to a new equilibrium, that is, toward a bad business cycle. The critical values for γ are, respectively, 1/ tanh(1.5) ≃ 1.105 and 1/ tanh(0.9) ≃ 1.396. LARGE PORTFOLIO LOSSES 31 Proposition 5.1. Let S be a finite set, and (X(t))t∈[0,T ], (Y (t))t∈[0,T ] two S-valued Markov chains with infinitesimal generators, respectively, Lf(x) = y 6=x Lx,y[f(y)− f(x)], Mf(x) = y 6=x Mx,y[f(y)− f(x)]. Assume X(0) and Y (0) have the same distribution, and denote by PX and PY the law of the two processes on the appropriate set of trajectories in the time-interval [0, T ]. Assume that whenever Mx,y = 0 also Lx,y = 0. Then PX ≪ PY , and (x([0, T ])) = exp y 6=x(t) (Mx(t),y −Lx(t),y)dt+ Lx(t−),x(t) Mx(t−),x(t) where x(t−) := lims↑t x(s), log = 1 and Nt is the counting process that counts the jumps of the trajectory x([0, T ]). In what follows we denote by PN the law on the path space of (σ[0, T ], ω[0, T ]) ∈ (D([0, T ]))2N under the interacting dynamics, with initial condi- tions such that (σ i (0), ω i (0)) i=1 are independent and identically dis- tributed with an assigned law λ (see beginning of Section 3). As in Section 3.1 we let W ∈ M1(D([0, T ]) × D([0, T ])) denote the law of the {−1,1}2- valued process (σ(t), ω(t)) such that (σ(0), ω(0)) has distribution λ, and both σ(·) and ω(·) change sign with constant rate 1. By W⊗N we mean the product of N copies of W . We begin with some preliminary lemmas. Lemma 5.2. dW ⊗N (σ[0, T ], ω[0, T ]) = exp[NF (ρN (σ[0, T ], ω[0, T ]))],(33) where F is the function defined in (7). Proof. Let (N t (i)) i=1 be the multivariate counting process which counts the jumps of σi for i= 1, . . . ,N , and (N t (i)) i=1 be the multivariate counting process which counts the jumps of ωi for i= 1, . . . ,N . Since each jump of the trajectory (σ[0, T ], ω[0, T ]) is counted by exactly one of the above counting processes, Proposition 5.1 applied to this case yields dW ⊗N (σ[0, T ], ω[0, T ]) 32 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI = exp (1− e−βσi(t)ωi(t))dt+ log e−βσi(t −)ωi(t −) dN t (i) (1− e−γωi(t)m ρN (t))dt log e −γωi(t−)mσ ρN (t −) dN t (i) Since, with probability 1 with respect to W ⊗N , there are no simultaneous jumps, we have log e−βσi(t −)ωi(t −) dN t (i)= −β (−σi(t))ωi(t)dNσt (i) log e −γωi(t−)m ρN (t −) dN t (i)= −γ (−ωi(t))mσρN (t) dN t (i), from which (33) follows easily after having observed that, W⊗N almost surely, (NσT +N T )dρN <+∞, and that simultaneous jumps of σ and ω do not occur under dW ⊗N . � The main problem in the proof of Proposition 3.1 is related to the fact that the function F in (7) is neither continuous nor bounded. The following technical lemmas have the purpose of circumventing this problem. In what follows, we let Q ∈M1(D[0, T ]2) : (NσT +N T )dQ<+∞ .(34) We first define, for r > 0 and Q∈ I , Fr(Q) = (r− e−βσ(t)ω(t))dt+ (r− e−ω(t)γ t )dt (βσ(t)ω(t−)− log r)dNσt(35) (ω(t)γ − log r)dNωt LARGE PORTFOLIO LOSSES 33 Note that F = F1. Moreover, Lemma 5.2 can be easily extended to show dW ⊗Nr (σ[0, T ], ω[0, T ]) = exp[NFr(ρN (σ[0, T ], ω[0, T ]))],(36) where Wr is the law of the {−1,1}2-valued process σ(t), ω(t) such that (σ(0), ω(0)) has distribution λ, and both σ(·) and ω(·) change sign with constant rate r. Lemma 5.3. For 0< r ≤ min(e−β , e−γ), Fr is lower semicontinuous on I . For r ≥max(eβ, eγ), Fr is upper semicontinuous. Proof. By definition of weak topology the fact that the map (r− e−βσ(t)ω(t))dt+ (r− e−ω(t)γ t )dt is continuous is rather straightforward (since Q-expectations of bounded continuous functions in D([0, T ]) are continuous in Q). Thus we only have to deal with the term (βσ(t)ω(t−)− log r)dNσt (ω(t)γ − log r)dNωt We show that for 0< r≤ min(e−β , e−γ) the expression in (37) is lower semi- continuous in Q ∈ I . This shows that Fr is lower semicontinuous. The case r ≥max(eβ, eγ) is treated similarly. For ε > 0 consider the function ϕε :D[0, T ]→ R defined by ϕε(η) := , if η(t) jumps for some t ∈ (0, ε], 0, otherwise. Given η ∈ D([0, T ]) we define η(s) for s > T by letting η(s) ≡ η(T ). Then, letting θt denote the shift operator, we have that, for t ∈ [0, T ], θtη is the element of D([0, T ]) given by θtη(s) := η(t+ s). Consider now two functions f, g :{−1,1}2 → R, and define fε, gε :D[0, T ]2 → R by fε(σ[0,T ], ω[0,T ]) := inf{f(σ(t), ω(t)) : t ∈ (0, ε)}, and similarly for gε. Then define Φε(σ[0,T ], ω[0,T ]) := fε(θtσ, θtω)ϕε(θtσ)dt+ gε(θtσ, θtω)ϕε(θtω)dt. The key to the continuation of the proof below are the following two proper- ties of Φε. These properties are essentially straightforward, and their proofs are omitted: 34 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI • Φε is continuous and bounded on {(σ[0,T ], ω[0,T ]) :NσT +NωT <+∞}. • Suppose f, g ≥ 0. Then, assuming σ[0,T ], ω[0,T ] have a finite number of jumps, Φε(σ[0,T ], ω[0,T ]) increases when ε ↓ 0 to f(σt− , ωt−)dN g(σt− , ωt−)dN Therefore by monotone convergence f(σt− , ωt−)dN g(σt− , ωt−)dN = sup Φε(σ[0,T ], ω[0,T ])dQ. In particular, the map f(σt− , ωt−)dN g(σt− , ωt−)dN is lower semicontinuous on I . Now, for r≤ min(e−β , e−γ), the function f(σ,ω) =−βσω− log r is nonnega- tive. As for the function g, that should be −ω(t)γQt − log r, we notice that it is not a function of (σ,ω), but rather a function of (σ,ΠtQ), thus depending explicitly on t and Q. However, due to its boundedness and the fact that γ is continuous in Q uniformly in t, σ, the argument above applies with minor modifications thus leading to the conclusion of the proof. � Lemma 5.4. Let Q∈M1(D([0, T ])2) be such that H(Q|W )<+∞. Then Q ∈ I . The same result applies if Wr replaces W . Proof. By the entropy inequality (see (6.2.14) in [15]) NσT dQ≤ log T dW +H(Q|W ). But NσT has Poisson distribution under W , so T dW <+∞. By applying the same argument to NωT , the proof is completed. This proof extends with no modifications to the case r 6= 1. � Lemma 5.5. The function I(Q) :=H(Q|W )−F (Q) is lower semicontinuous on M1(D[0, T ]2). LARGE PORTFOLIO LOSSES 35 Proof. It is well known (see [15], Lemma 6.2.13) that the entropy H(Q|W ) is lower semicontinuous in Q in all of M1(D([0, T ])2). Moreover, by definition, F (Q) < +∞ for every Q, and so we have H(Q|W ) = I(Q) whenever H(Q|W ) = +∞. Since, by Lemma 5.4, H(Q|W ) = +∞ for Q /∈ I , we are left to prove the following two statements: (i) I(Q) is lower semicontinuous in I . (ii) If H(Q|W ) = +∞ and Qn →Q weakly, then I(Qn)→ +∞. The following key identity, which holds for r > 0, is a simple consequence of the definition of relative entropy and of the Girsanov formula for Markov chains. H(Q|Wr) =H(Q|W ) + =H(Q|W ) + 2T (r− 1) + log r (NσT +N T )dQ. In particular, by Lemma 5.4, we have that H(Q|W )<+∞ ⇐⇒ H(Q|Wr)< +∞. A simple consequence of (38) is then the following: I(Q) =H(Q|Wr)− Fr(Q),(39) where the difference in (39) is meant to be +∞ whenever H(Q|Wr) = +∞ [which is equivalent to H(Q|W ) = +∞]. We are now ready to prove (i) and (ii). To prove (i) it is enough to choose r ≥max(eβ , eγ) and use Lemma 5.3. Moreover, for the same choice of r, the stochastic integrals in (35) are nonpositive, so Fr(Q) ≤ 2Tr. Therefore, if H(Q|W ) = +∞ and Qn→Q, lim inf I(Qn) ≥ lim infH(Qn|Wr)− 2Tr = +∞, where the last equality follows from lower semicontinuity of H(·|Wr) and H(Q|Wr) = +∞. Thus (ii) is proved. � Lemma 5.6. The function I(Q) has compact level sets, that is, for every k > 0 the set {Q : I(Q) ≤ k} is compact. Proof. Choosing, as above, r ≥max(eβ , eγ), we have that Fr(Q) ≤ 2Tr for every Q. Thus, by (39), {Q : I(Q) ≤ k} ⊆ {Q :H(Q|Wr)≤ k+ 2Tr}. Since (see [15], Lemma 6.2.13) the relative entropy has compact level sets, {Q : I(Q) ≤ k} is contained in a compact set. Moreover, by lower semiconti- nuity of I , {Q : I(Q)≤ k} is closed, and this completes the proof. � 36 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Lemma 5.7. For every r > 0 there exists δ > 1 such that lim sup exp[δNFr(ρN )]dW r <+∞. Proof. We give the proof for r = 1; the modifications for the general case are obvious. The proof consists of rather simple manipulations. The idea can be summarized as follows. If δ = 1, then by Lemma 5.2, exp[δNF (ρN )] is the Radon–Nikodym derivative of PN with respect to W ⊗N , and therefore has expectation 1. For δ > 1, we write δF (ρN ) = F1(ρN ) +F2(ρN ) in such a way that F2 is bounded and exp[NF1(ρN )] is a Radon–Nikodym derivative of a probability with respect to W⊗N . More specifically, observe that, using δNF (ρN ) = (δ − δe−βσi(t)ωi(t))dt+ δβσi(t)ωi(t t (i) (δ − δe−γωi(t)m ρN (t))dt δγωi(t)m ρN (t −) dN t (i) (1− e−δβσi(t)ωi(t))dt+ δβσi(t)ωi(t)dN t (i) (1− e−δγωi(t)m ρN (t))dt+ δγωi(t)m ρN (t) t (i) (δ − δe−βσi(t)ωi(t) − (1− e−δβσi(t)ωi(t)))dt (δ − δe−γωi(t)m ρN (t) − (1− e−δγωi(t)m ρN (t)))dt =NF1(ρN ) +NF2(ρN ), where NF1(ρN ) := (1− e−δβσi(t)ωi(t))dt+ δβσi(t)ωi(t)dN t (i) (1− e−δγωi(t)m ρN (t))dt+ δγωi(t)m ρN (t) t (i) LARGE PORTFOLIO LOSSES 37 NF2(ρN ) := (δ − δe−βσi(t)ωi(t) − (1− e−δβσi(t)ωi(t)))dt (δ − δe−γωi(t)m ρN (t) − (1− e−δγωi(t)m ρN (t)))dt. Note that exp[NF1(ρN )] has the same form of exp[NF (ρN )] after having replaced β by δβ. In particular, exp[NF1(ρN )]dW ⊗N = 1. Moreover, it is easy to see that F2(ρN )≤ T (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ). Putting all together, we obtain exp[δNF (ρN )]dW ≤ exp[NT (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ)] exp[NF1(ρN )]dW = exp[NT (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ)], from which the conclusion follows easily. � Completing the proof of Proposition 3.1. It remains to show the upper and the lower bounds (9) and (8). We prove them separately; our main tool is the Varadhan Lemma in the version in [15], Lemmas 4.3.4 and 4.3.6. We deal first with the upper bound (9). Take r≥ max(eβ , eγ), so that the function Fr in (35) is upper semicontinuous. Denote by PN the distribution of ρN under PN , and by WN its distribution under W⊗Nr . By (36) (Q) = exp[NFr(Q)].(40) By Sanov’s theorem (Theorem 6.2.10 in [15]), the sequence of probabilities WN satisfies a large deviation principle with rate function H(Q|Wr). Since Fr is upper semicontinuous and satisfies the superexponential estimate in Lemma 5.7, we can apply Lemma 4.3.6 in [15], together with identity (39), to obtain the upper bound (9). The lower bound (8) is proved similarly, by taking 0< r≤ min(e−β , e−γ), so that Fr becomes lower semicontinuous, using (40) again and Lemma 4.3.4 in [15]. � The remaining part of this section is devoted to the proof of Proposition 3.2. It mainly consists in giving an alternative representation of the rate function I(Q). 38 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI Let now Q ∈M1(D([0, T ]) ×D([0, T ])). We associate with Q the law of a time-inhomogeneous Markov process on {−1,1}2 which evolves according to the following rules: with intensity e−βσω, with intensity exp σ,τ∈{−1,1} σΠtQ(σ, τ) −γωmσ ΠtQ = e−γ and with initial distribution λ. We denote by PQ the law of this process. In other words, PQ is the law of the Markov process on {−1,1}2 with initial distribution λ and time-dependent generator LQt f(σ,ω) = e−βσω∇σf(σ,ω) + e −γωmσ ΠtQ∇ωf(σ,ω). Lemma 5.8. For every Q ∈M1(D([0, T ])×D([0, T ])) such that I(Q)< +∞, we have I(Q) =H(Q|PQ). Proof. We begin by observing that, since by assumption I(Q) <∞, we have H(Q|W )<+∞ and so by Lemma 5.4 it follows that Q ∈ I , which implies that the integrals below are well defined. Using again Girsanov’s formula for Markov chains in Proposition 5.1, we obtain (σ[0, T ], ω[0, T ])dQ (1− e−βσ(t)ω(t))dt+ (1− e−γω(t) σΠtQ(dσ, dτ))dt (−βσ(t−)ω(t−))dNσt −γω(t−) σΠt−Q(dσ, dτ) (1− e−βσ(t)ω(t))dt+ (1− e−γω(t) σΠtQ(dσ,dτ))dt σ(t)ω(t)dNσt + γ σΠtQ(dσ, dτ) (1− e−βσ(t)ω(t))dt+ (1− e−ω(t)γ t )dt LARGE PORTFOLIO LOSSES 39 σ(t)ω(t)dNσt + ω(t)γ = F (Q). Finally, just observe that I(Q) = dQ log dQ log dQ log =H(Q|PQ). � Completing the proof of Proposition 3.2. By properness of the relative entropy [H(µ|ν) = 0 ⇒ µ = ν], from Lemma 5.8 we have that the equation I(Q) = 0 is equivalent to Q= PQ. Suppose Q∗ is a solution of this last equation. Then, in particular, qt := ΠtQ ∗ = ΠtP Q∗ . The marginals of a Markov process are solutions of the corresponding forward equation that, in this case, leads to the fact that qt is a solution of (10). This differential equation, being an equation in finite dimension with locally Lipschitz coeffi- cients, has at most one solution in [0, T ]. Since PQ is totally determined by the flow qt, it follows that equation Q= P Q has at most one solution. The existence of a solution follows from the fact that I(Q) is the rate function of a LDP, and therefore must have at least one zero, indeed, by (8) with A= M1(D[0, T ] ×D[0, T ]), we get infQ I(Q) = 0. Since I is lower semicon- tinuous, this inf is actually a minimum. � 5.2. Proof of Theorem 3.4. We first observe that the square [−1,1]2 is stable for the flow of (14), since the vector field V (x, y) points inward at the boundary of [−1,1]2. It is also immediately seen that the equation V (x, y) = 0 holds if and only if x= tanh(β) tanh(γx) and y = 1 tanh(β) x. More- over a simple convexity argument shows that x= tanh(β) tanh(γx) has x= 0 as unique solution for γ ≤ 1 tanh(β) , while for γ > 1 tanh(β) a strictly positive so- lution, and its opposite, bifurcate from the null solution. We have therefore found all equilibria of (14). We now remark that (14) has no cycles (periodic solutions). Indeed, sup- pose (xt, yt) is a cycle of period T . Then by the Divergence Theorem [V1(xt, yt)ẋt + V2(xt, yt)ẏt]dt= divV (x, y)dxdy,(41) where V1, V2 are the components of V and C is the open set enclosed by the cycle. But a simple direct computation shows that divV (x, y)< 0 in all of [−1,1]2, so that (41) cannot hold. 40 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI It follows by the Poincaré–Bendixon theorem that every solution must converge to an equilibrium as t→ +∞. This completes the proof of (i). The matrix of the linearized system is DV (0,0) = −2cosh(β) 2 sinh(β) 2γ −2 from which also (ii) and (iii) are readily shown. It remains to show (iv). For γ > 1 tanh(β) , we let vs be an eigenvector of the negative eigenvalue of DV (0,0). By the Stable Manifold Theorem (see Section 2.7 in [30]), the set of initial conditions that are asymptotically driven to (0,0) form a one- dimensional manifold Γ that is tangent to vs at (0,0). Since any solution converges to an equilibrium point, and solutions starting in Γc cannot cross Γ (otherwise uniqueness would be violated), the remaining part of statement (iv) follows. 5.3. Proof of Theorem 3.6. Proof. One key remark is the fact that the stochastic process (mσ ρN (t) ρN (t) ρN (t) ) is a sufficient statistic for our model; in this context this means that its evolution is Markovian. This can be proved by checking that if we apply the generator L in (3) to a function of the form ϕ(mσ ρN (t) ρN (t) ρN (t) ), then we obtain again a function of (mσ ρN (t) ρN (t) ρN (t) ). A long but straightforward computation actually gives Lϕ(mσρN (t),m ρN (t) ,mσωρN (t)) = [KNϕ](m ρN (t) ,mωρN (t),m ρN (t) where KNϕ(ξ, η, θ) (j,k)∈{−1,1}2 [jξ + kη + jkθ + 1] e−βjk ξ − 2 j,η, θ − 2 − ϕ(ξ, η, θ) + e−γξk ξ, η− 2 k,θ− 2 − ϕ(ξ, η, θ) This implies that KN is the infinitesimal generator of the three-dimensional Markov process (mσ ρN (t) ρN (t) ρN (t) ). Note now that (xN (t), yN (t), zN (t)) is obtained from (mσ ρN (t) ρN (t) ρN (t) ) through a time dependent, linear invertible transformation. We call Tt this transformation, that is, Tt(ξ, η, θ) = ( N(ξ −mσt ), N(η −mωt ), N(θ −mσωt )) LARGE PORTFOLIO LOSSES 41 (the dependence onN of Tt is omitted in the notation). Therefore (xN (t), yN (t), zN (t)) is itself a (time-inhomogeneous) Markov process, whose infinitesimal generator HN,t can be obtained from (42) as follows: HN,tf(x, y, z) =KN [f ◦ Tt](T−1t (x, y, z)) + [f ◦ Tt](T−1t (x, y, z)). A simple computation gives then HN,tf(x, y, z) (j,k)∈{−1,1}2 + jmσt + km t + jkm t + 1 e−βj k x− 2√ j, y, z − 2√ − f(x, y, z) + e−γ(x/ x, y− 2√ k,z − 2√ −f(x, y, z) Nṁσt fx(x, y, z)− Nṁωt fy(x, y, z)− Nṁσωt fz(x, y, z), where fx stands for , and similarly for the other derivatives. At this point we compute the asymptotics of HN,tf(x, y, z) as N → +∞, assum- ing f :R3 → R a C3 function with compact support. First of all we make a Taylor expansion of terms like x− 2√ j , y, z − 2√ − f(x, y, z) = − 2√ fx(x, y, z)− fz(x, y, z)(44) fxx(x, y, z) + fzz(x, y, z) + fxz(x, y, z) + o e−γ(x/ N) = 1− γ .(45) Note that, since all derivatives of f are bounded, the remainder in (44) is ) uniformly in (x, y, z) ∈ R3. Moreover, the remainder in (45) is o( 1√ uniformly for x in a compact set. Therefore, since f has compact support, when we use (44) and (45) to replace the corresponding terms in (43), we obtain remainders whose bounds are uniform in R3. When (44) and (45) are plugged into (43), all terms of order N coming from the sum over (j, k) ∈ 42 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI {−1,1}2 are canceled by the terms Nṁσt fx(x, y, z) − Nṁωt fy(x, y, z) −√ Nṁσωt fz(x, y, z). It follows then by a straightforward computation that t∈[0,T ] x,y,z∈R3 |HN,tf(x, y, z)−Htf(x, y, z)|= 0, where Htf(x, y, z) = 2{fx[−x cosh(β) + y sinh(β)] + fy[−γxmωt sinh(γmσt ) + γx cosh(γmσt )− y cosh(γmσt )] + fz[x sinh(γm t ) + γxm t cosh(γm − γxmσωt sinh(γmσt )− z cosh(β)− z cosh(γmσt )] + fxx[−mσωt sinh(β) + cosh(β)] + fyy[−mωt sinh(γmσt ) + cosh(γmσt )] + fzz[−mσωt sinh(β) + cosh(β) −mωt sinh(γmσt ) + cosh(γmσt )] + 2fxz[−mσt sinh(β) +mωt cosh(β)] + 2fyz[m t cosh(γm t )−mσωt sinh(γmσt )]} is the infinitesimal generator of the linear diffusion process (18). Using Theo- rem 1.6.1 in [17], the proof is completed if we show that (xN (0), yN (0), zN (0)) converges as N → +∞, in distribution to (x(0), y(0), z(0)). This last state- ment follows by the standard Central Limit Theorem for i.i.d. random vari- ables; indeed, by assumption, (σi(0), ωi(0)) are independent with law λ, and (19) is just the covariance matrix under λ of (σ(0), ω(0), σ(0)ω(0)). It should be pointed out that Theorem 1.6.1 in [17] does not deal explicitly with time-dependent generators, as is the case here. To fix this point it is enough to introduce an additional variable, τ(t) := t, and consider the process α(t) := (x(t), y(t), z(t), τ(t)), whose generator is time-homogeneous. This argument, together with the fact that the convergence of HN,tf(x, y, z) to Htf(x, y, z) is uniform in both (x, y, z) and t, completes the proof. � 5.4. Proof of Theorem 4.1. We start with a technical lemma. Lemma 5.9. For t∈ [0, T ] we have the convergence in distribution j lσj(t) −L(t) →X ∼N (l1 − l−1)2V (t) where L(t) is defined in (28) and V (t) in (24). LARGE PORTFOLIO LOSSES 43 Proof. Define, for x ∈ {−1,1}, the quantity ANx (t) as the number of σi that, at a given time t, are equal to x. We may then write AN1 (t) −1(t) . Recall moreover that for N →∞, mσN (t) →mσt . We then have j lσj(t) −L(t) 1 (t) + l−1A −1(t) −L(t) N (t) + l−1 1−mσN (t) −L(t) (l1 + l−1) (l1 − l−1) N (t)− (l1 − l−1) mσt − (l1 + l−1) (l1 − l−1) N (t)−m →X ∼N (l1 − l−1)2V (t) where the last convergence follows from Corollary 3.7 noticing that m N (t) = ρN (t) Proof of Theorem 4.1. We have to check that LN (t) −L(t) → Y ∼N(0, V̂ (t)), where V̂ (t) is defined in (29). Separating the firms according to whether their σj(t) is +1 or −1, j Lj(t) −L(t) j:σj(t)=1 Lj(t) + j:σj(t)=−1Lj(t) −L(t) We then add and subtract j lσj(t) to obtain j:σj(t)=1 (Lj(t)− l1) j:σj(t)=−1(Lj(t)− l−1) j lσj(t) −L(t) Since we have only independence conditionally on σ(t), we need to check whether the CLT still applies. Let us show the convergence of the corre- sponding characteristic functions: LN (t)−NL(t)√ 44 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI j:σj(t)=1 (Lj(t)− l1)√ j:σj(t)=−1(Lj(t)− l−1)√ j lσj(t) −NL(t)√ ∣σ(t) The last of the three terms is measurable with respect to the sigma algebra generated by σ(t) so that we can take it out from the inner expectation. Be- cause of the conditional independence we can separate the remaining terms in the product of conditional expectations: j:σj(t)=1 (Lj(t)− l1)√ ∣σ(t) j:σj(t)=−1(Lj(t)− l−1)√ ∣σ(t) By conditional independence, j:σj(t)=1 (Lj(t)− l1)√ ∣σ(t) AN1 (t) Lj(t)− l1√ ∣σ(t) 1− v1 )]AN1 (t) where the last equality follows because l1 and v1 are the first two conditional moments of Lj(t). Recalling that AN1 (t) converges almost surely to we have 1− v1 )]AN1 (t) = lim 1− v1 AN1 (t) AN1 (t) )]AN1 (t) = exp 1 +mσt The same argument holds for the terms where σj(t) = −1. Since −1(t) 1−mσt , we have 1− v−1 AN−1(t) AN−1(t) −1(t) = exp 1−mσt LARGE PORTFOLIO LOSSES 45 Finally, recall from Lemma 5.9 that lσj (t) −NL(t) converges to X ∼N(0, (l1−l−1)2V (t) ), so that j lσj(t) −NL(t)√ = exp (l1 − l−1)2V (t) Thus, denoting by E[· · · |σ(t)] the inner conditional expectation in (48), we have shown that E[· · · |σ(t)] = exp (l1 − l−1)2V (t) 1 +mσt × exp 1−mσt = exp V̂ (t) By the Dominated Convergence Theorem, taking the limit as N → +∞ in (48), we can interchange the limit with the outer expectation, and the proof is completed. � 6. Conclusions and possible extensions. In this paper we have described propagation of financial distress in a network of firms linked by business relationships. We have proposed a model for credit contagion, based on interacting par- ticle systems, and we have quantified the impact of contagion on the losses suffered by a financial institution holding a large portfolio with positions issued by the firms. Compared to the existing literature on credit contagion, we have proposed a dynamic model where it is possible to describe the evolution of the indica- tors of financial distress. In this way we are able to compute the distribution of the losses in a large portfolio for any time horizon T , via a suitable version of the central limit theorem. The peculiarity of our model is the fact that the changes in rating class (the σ variables) are related to the degree of health of the system (the global indicator mσ). There is a further characteristic of the firms that is summarized by a second variable ω (a liquidity indicator) and that describes the ability of the firm to act as a buffer against adverse news coming from the market. The evolution of the pair (σ,ω) depends on two parameters β and γ, which indicate the strength of the interaction. The fact that our model leads to endogenous financial indicators that de- scribe the general health of the systems has allowed us to view a credit crisis as a microeconomic phenomenon. This has also been exemplified through simulation results. 46 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI The model we have proposed in this paper exhibits some phenomena having interesting financial interpretation. There are many extensions that could make the model more flexible and realistic, allowing also calibration to real data. One of them, concerning the symmetry of the model, has already been mentioned in Remark 2.1. Other more substantial extensions are the following: • In real applications, the variable σ denoting the rating class is not binary; one could extend the model by taking σ to be valued in a finite, totally ordered set. • One could assume the fundamental values ωi to be R+-valued, and evolv- ing according to the stochastic differential equation dωi(t) = ωi(t)[f(m N (t))dt+ g(m N (t))dBi(t)] + dJi(t), where f and g are given functions, the Bi(·) are independent Brownian motions, and Ji(·) is a pure jump process whose intensity is a function of ωi(t) and m N (t). • An interesting extension of the above model consists in letting the func- tions a(·, ·, ·) and b(·, ·, ·) in (1) be random rather than deterministic; in particular they may depend on (possibly time-dependent) exogenous macroeconomic variables. • The mean-field assumption may be weakened by assuming that the rate at which ωi changes depends on an i-dependent weighted global health of the form N,i := where J : [0,1]2 → R is a function describing the interaction between pairs of firms. In other words, the ith firm “feels” the information given by the rating of the other firms in a nonuniform way. Other generalizations could be useful, in particular to introduce inhomogene- ity in the model. In principle, the extensions listed above could be treated by the same techniques used in this paper. APPENDIX: THE EIGENVALUES OF THE MATRIX A IN THEOREM 3.6 We begin by writing down explicitly the limit matrix A: − cosh(β) −γ sinh(γm cosh(γmσ∗ ) sinh(γmσ∗ ) + γ cosh(γm sinh(γmσ∗ ) + γm ∗ cosh(γm ∗ ) + γ sinh(β) +mσ∗ sinh(γm cosh(β) + cosh(γmσ∗ ) sinh(γmσ∗ ) LARGE PORTFOLIO LOSSES 47 sinh(β) 0 − cosh(γmσ∗ ) 0 0 −(cosh(β) + cosh(γmσ∗ )) where for the first term in the second row we have used (16). By direct computation, one shows that the eigenvalues of A are given by the following expressions: λ1 = −2(cosh(β) + cosh(γmσ∗ )), λ2 = − cosh(β) + cosh(γmσ∗ ) (cosh(β)− cosh(γmσ∗ )) sinh(β) cosh(γmσ∗ ) ,(49) λ3 = − cosh(β) + cosh(γmσ∗ ) (cosh(β)− cosh(γmσ∗ )) sinh(β) cosh(γmσ∗ ) Note that these eigenvalues are all real, and that clearly λ1, λ2 < 0. Moreover, λ3 < 0 if and only if < cosh2(γmσ∗ )(50) where γc = tanh(β) (a) If γ < γc, then by part (i) in Theorem 3.4 we have m ∗ = 0. In this case (50) holds, because < 1 = cosh2(γ · 0). In this case the matrix A has three different real eigenvalues, all strictly negative. (b) If γ = γc, we still have m ∗ = 0, but it is immediately seen that λ3 = 0. (c) Finally, if γ > γc, set y = γm ∗ ; by (15) we have mσ∗ = tanh(γmσ∗ ) ⇔ y = tanh(y).(51) Then (50) is equivalent to showing that < cosh2(y)(52) 48 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI and from (51) we obtain tanh(y) sinh(y) cosh(y)< cosh(y)< cosh2(y) because y/ sinh(y)< 1 and cosh(y)< cosh2(y), since y = γmσ∗ > 0 if γ > γc. Then, in this case too, the matrix A has three different real eigenvalues, all strictly negative. Acknowledgment. The authors would like to acknowledge the extremely careful reading of the paper and the useful suggestions made by an anony- mous referee. REFERENCES [1] Allen, F. and Gale, D. (2000). Financial contagion. Journal of Political Economy 108 1–33. [2] Bolthausen, E. (1986). Laplace approximations for sums of independent random vectors. Probab. Theory Related Fields 72 305–318. MRMR836280 [3] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York. MRMR636252 [4] Brock, W. A. and Durlauf, S. N. (2001). 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[34] Schönbucher, P. (2006). Portfolio losses and the term structure of loss transition rates: a new methodology for the pricing of portfolio credit derivatives. Working paper, ETH Zürich. [35] Tolotti, M. (2007). The impact of contagion on large portfolios. Modeling aspects. Ph.D. thesis, Scuola Normale Superiore. http://www.ams.org/mathscinet-getitem?mr=MR838085 http://www.ams.org/mathscinet-getitem?mr=MR1407254 http://ssrn.com/abstract=678966 http://www.ams.org/mathscinet-getitem?mr=MR2224986 http://www.ams.org/mathscinet-getitem?mr=MR2175089 http://www.ams.org/mathscinet-getitem?mr=MR1083151 http://www.ams.org/mathscinet-getitem?mr=MR2384674 50 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI P. Dai Pra W. J. Runggaldier E. Sartori Dipartimento di Matematica Pura ed Applicata University of Padova 63, Via Trieste I-35121-Padova Italy E-mail: [email protected] [email protected] [email protected] M. Tolotti Istituto di Metodi Quantitativi Bocconi University 25, Via Sarfatti I-20136 Milano Italy Scuola Normale Superiore Italy E-mail: [email protected] Department of Applied Mathematics University of Venice Venice Italy E-mail: [email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] Introduction General aspects Purpose and modeling aspects Financial application Methodology The model A mean-field model Invariant measures and nonreversibility Main results: law of large numbers and Central Limit Theorem Deterministic limit: large deviations and law of large numbers Equilibria of the limiting dynamics: phase transition Analysis of fluctuations: Central Limit Theorem Portfolio losses Simulation results Proofs Proofs of Propositions 3.1 and 3.2 Proof of Theorem 3.4 Proof of Theorem 3.6 Proof of Theorem 4.1 Conclusions and possible extensions Appendix: The eigenvalues of the matrix A in Theorem 3.6 Acknowledgment References Author's addresses

0704.1349

Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients

CARLEMAN ESTIMATES AND UNIQUE CONTINUATION FOR SECOND ORDER PARABOLIC EQUATIONS WITH NONSMOOTH COEFFICIENTS HERBERT KOCH AND DANIEL TATARU Abstract. In this work we obtain strong unique continuation results for variable coefficient second order parabolic equations. The coefficients in the principal part are assumed to satisfy a Lipschitz condition in x and a Hölder C 3 condition in time. The coefficients in the lower order terms, i.e. the potential and the gradient potential, are allowed to be unbounded and required only to satisfy mixed norm bounds in scale invariant L x spaces. 1. Introduction The evolution of the understanding of the strong unique continuation prob- lem for second order parabolic equations mirrors and is closely related to the corresponding strong unique continuation problem for second order elliptic equations. Consequently, we begin with a brief overview of the latter problem. To a second order elliptic operator ∆g = ∂ig ij∂j and potentials V , W in R we associate the elliptic equation (1.1) −∆gu = W∇u+ V u Given a function u ∈ L2loc(Rn) and x0 ∈ Rn we say that u vanishes of infinite order at x0 if there exists R so that for each integer N we have (1.2) B(x0,r) |u|2 dx ≤ c2Nr2N , r < R The elliptic strong unique continuation property ESUCP has the form Let u be a solution to (1.1) which vanishes of infinite order at x0. Then u(x) = 0 for x in a neighborhood of x0. ESUCP ESUCP type results go back to the pioneering work of Carleman [5] in dimension n = 2, later extended to higher dimension in by Aronszajn and collaborators [3], [4]. Their results apply to Lipschitz metrics g but only mildly unbounded potentials V and W . A key ingredient in their approach was to obtain a class of weighted L2 estimates which were later called Carleman estimates. The simplest Carleman estimate has the form ‖|x|−τu‖L2 . ‖|x|2−τ∆u‖L2 The first author was supported in part by the DFG grant KO1307/5-3. The second author was supported in part by the NSF grant DMS-0301122. Part of the work was done while the first author was supported by the Miller Institute for Basic Research in Science. http://arxiv.org/abs/0704.1349v2 and holds uniformly for τ away from ±(n−2 + N). This restriction is related to the spectrum of the spherical Laplacian. Adding some extra convexity to the |x|−τ weight makes the above estimate more robust and allows one to also use it in the variable coefficient case. The role played by the convexity was further clarified and explained by Hörmander [12], [13], who introduced the pseudoconvexity condition for weights as an almost necessary and sufficient condition in order for the Carleman estimates to hold. The problem becomes more difficult if one seeks to work with unbounded potentials V in or near the scale invariant L 2 space. There the L2 Carle- man estimates are insufficient. Instead the key breakthrough was achieved in Jerison-Kenig [15], where the L2 Carleman estimates are replaced by Lp estimates of the form ‖|x|−τu‖ . ‖|x|−τ∆u‖ Relevant to the present paper is also the alternative proof of this result which was given by Jerison [14], taking advantage of Sogge’s [25] spectral projection bounds for the spherical Laplacian. In the case of operators with smooth variable coefficients Lp Carleman estimates were first obtained by Sogge [26], [27]. Working with gradient potentials in the scale invariant space W ∈ Ln intro- duces an added layer of difficulty. There not even the Lp Carleman estimates can hold. Wolff’s solution to this in [29] is a weight osculation argument, which allows one to taylor the weight in the Carleman estimate to the solution u, producing estimates of the form ‖e−τφ(x)u‖ + ‖e−τφ(x)W∇u‖ . ‖eτφ(x)∆u‖ where the choice of φ depends on both u, W and τ . Finally, the authors’s article [17] combines the ideas above into a nearly optimal scale invariant ESUCP result for the elliptic problem, with (i) a Lipschitz metric g, (ii) an L 2 potential V , and (iii) an almost Ln gradient potential W . The present paper is the counterpart of [17] for the parabolic strong unique continuation problem. We consider the second order backwards parabolic operator (1.3) P = ∂t + ∂kg kl(t, x)∂l in R × Rn and potentials V,W1,W2. To these we associate the parabolic equation (1.4) Pu = V u+W1∇xu+∇x(W2u) Given a function u ∈ L2loc and (t0, x0) ∈ R × Rn we say that u vanishes of infinite order at (t0, x0) if there exists R so that for each integer N we have (1.5) B(x0,r) |u|2 dxdt ≤ c2Nr2N , r < R Alternatively we may only require that x→ u(t0, x) vanishes of infinite order at x0, i.e. (1.6) B(x0,r) |u(t0, x)|2 dx ≤ c2Nr2N , r < R The two conditions (1.5) and (1.6) are largely equivalent provided that the coefficients gkl have some uniform regularity as t → t0. However, our as- sumptions in this article are not strong enough to guarantee this, therefore we consider the two separate cases. Now we can define the strong unique continuation property SUCP(I) : Let u be a solution to (1.4) which vanishes of infinite order at (t0, x0). Then u(t0, x) = 0 for x in a neighborhood of x0. SUCP(I) and the slightly stronger variant Let u be a solution to (1.4) so that x→u(t0, x) vanishes of infinite order at x0. Then u(t0, x) = 0 for x in a neighborhood of x0. SUCP(II) The study of unique continuation for parabolic equations began with early work of Mizohata [21] and Yamabe [30], followed by Saut-Scheurer [23]; Lp Carleman estimates were first obtained by Sogge [24]. The study of the parabolic strong unique continuation problem began with work of Lin [20] who considered SUCP(II) for the heat equation with W = 0 and V bounded and time independent. This continued with work of Chen [6] and Poon [22]. Fernández [11], and Escauriaza, Fernández and Vessella [7] con- sidered SUCP(II) under various assumptions on the coefficients and pointwise bounds for W = 0 and V . It is a consequence of Alessandrini and Vessella [1] that SUCP(I) andSUCP(II) are equivalent under weak assumptions on the coefficients, and they derived SUCP(II) in [2] for bounded W and V . The article of Poon [22] contributed to clarifying the correct form of the L2 Carleman estimates for the parabolic strong unique continuation problem in Escauriaza and Fernández’s work [9]. In the simplest form, these have the ‖t−τ− 8t u‖L2 . ‖t−τ+ 8t (∂t +∆)u‖L2 and hold uniformly with respect to τ away from (2n + N)/4. This restriction is connected with the spectral properties of the Hermite operator. The Lp spectral projection bounds for the Hermite operator were indepen- dently obtained by Thangavelu [28] and Kharazdhov [16]; see also the sim- plified proof in the authors’s paper [19]. These bounds were essential in the proof of Lp Carleman inequalities for the heat operator of Escauriaza [8] and Escauriaza and Vega [10] which yield SUCP(I) when g = In, W = 0 and V ∈ L1L∞ + L∞Ln/2. Our aim is to prove that SUCP(I) respectively, SUCP(II) hold under sharp scale invariant assumptions on the metric g and Lp conditions on the potentials V and W1,W2. The contribution of this work is comparable to [17] for the elliptic problem: We study almost optimal conditions on (1) the coefficients g (2) the potential V (3) the gradient potentials Wj The combination of rough variable coefficients and Lp conditions on the po- tential seems to be new. Also, to the best of our knowledge this is the first result on unique continuation for parabolic problems under Lp conditions on the coefficients of the gradient term. For simplicity we always assume that t0 = 0, x0 = 0. For SUCP(I) it is natural1 to consider a larger class of operators P which have the form (1.7) P = + ∂kg kl∂l + dkl∂l + ∂ld where (gkl), (dkl) and (ekl) are real valued and (gkl) and (ekl) are symmetric. Then simple scale invariant assumptions for the coefficients would be (1.8) ‖d‖L∞ + ‖(t+ x2) 2∂xd‖L∞ + ‖t∂td‖L∞ ≪ 1 Here and in the sequel d stands for a generic coefficient of the form gkl − δkl, dkl and ekl. For V and W1,2 we could consider conditions of the form (1.9) ‖V ‖L1L∞+L∞Ln/2 ≪ 1, (1.10) ‖W1,2‖L2L∞+L∞Ln ≪ 1. Here and in the sequel we use the notation LpLq = L The situation is however more complex and we may take (1.8) to (1.10) only as guidelines. We will have to strengthen (1.8) to include some dyadic summability. on the other hand we are able to weaken the time differentiability to a C 3 Holder condition on small time scales. We are also able to slightly weaken (1.9) almost to uniform bounds on dyadic sets. However, we are unable to use mixed norms for W1 and W2, and we restrict ourselves to a summable Ln+2 norm in dyadic sets. To state our assumptions on g, V , W1 and W2 we consider a double infinite dyadic partition of the space, (1.11) R+ × Rn = where (1.12) Aij = {(t, x) ∈ R+ ×Rn | e−4i−4 ≤ t ≤ e−4i, ej ≤ 1+ 2|x|t− 2 ≤ ej+1}. Consider the subset of indices (1.13) A = {(i, j) : j ≤ 2i+ 2} defining a partition of the cylinder Q = [0, 1]× B(0, 1) 1This becomes clearer later on after a change of coordinates and conjugation with respect to a Gaussian weight Define (1.14) A(τ) = {(i, j) ∈ A : 4i ≥ ln τ + 1, j ≤ 1 ln τ + 2} which corresponds to a partition of the cut parabola Qτ = {(t, x) : |x|2 ≤ τt ≤ 1}. t = τ−1 |x| = 1 Figure 1. The cut parabola We also consider a decomposition of Q into dyadic time slices Ai = [e −4i−4, e−4i]× B(0, 1) and a similar partition of the cut parabola Qτ into the sets Aτi = Ai ∩Qτ Given a function space X and 1 ≤ q < ∞ we introduce the Banach spaces lq(A, X) with norms ‖V ‖q lq(A,X) i,j∈A ‖V ‖q X(Aij) In a similar manner we define the spaces l∞(A, X). Within the sets Aij we define the modulus of continuity (mij) in time mij(ρ) = e 4iρ+ e (2i−j)ρ and denote by C t the space of continuous functions with finite seminorm = sup t1,t2,x |u(t1, x)− u(t2, x)| mij(|t1 − t2|) For the reader’s convenience we note that within Aij we have e 4i ≈ t−1 and (2i−j) ≈ (t+ |x|2)− 13 . For the coefficients of the operator P in (1.7) we change the condition (1.8) (1.15) sup ‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij) ≪ 1. where we note that ej−2i ≈ (t + x2) 12 in Aij . The pointwise bound for g − In in (1.15), namely (1.16) sup ‖g − In‖l1(A(τ),L∞) ≪ 1 is not really needed for our results. It can be always obtained from the other bounds after a change of coordinates. This is discussed in the appendix. The assertion (1.15) is satisfied for g ∈ Lipx∩C t provided that g(0, 0) = In. Indeed by scaling we may assume that the Lipx ∩ C t norm is small therefore it suffices to compute ‖(t + |x|2) 3‖l1(A(τ),L∞) ≤ i≥ln τ j≤ln(τ)/2+2 (j−2i) . 1. For the potential V we consider: ‖V ‖l∞(A,L1L∞+L∞Ln/2) ≪ 1 for n > 2 ‖V ‖l∞(A,L1L∞+LpLp′ ) ≪ 1 for n > 2, 1 ≤ p <∞, ‖V ‖l∞(A,L1L∞+L2L1) ≪ 1 for n = 1 (1.17) where p′ in the second line is the dual exponent. In addition we require that (1.18) sup ‖χiV ‖L1L∞+L∞Ln/2 ≪ 1 n > 2 with the obvious modifications for n = 1, 2, where χi is the characteristic function of the set {(t, x) : e−4i−4 ≤ t ≤ e−4i, t−1/2|x| ≤ i}. Both (1.17) and (1.18) are fulfilled if V ∈ L1L∞ + L∞Ln/2 with small norm. Finally for the gradient potentials W1,2 we introduce the summability con- dition with respect to time slices (1.19) sup ‖W1,2‖Ln+2(Aτ ) ≪ 1. As a consequence of this we note the uniform bound (1.20) sup ‖W1,2‖Ln+2(Ai) ≪ 1. Now we can state our main results. Theorem 1. Let P be as in (1.7) with coefficients satisfying (1.15). As- sume that the potentials V and W1,2 satisfy (1.17), (1.18) and (1.19). Then SUCP(I) holds at (0, 0) for H1 functions u satisfying (1.4). It is part of the conclusion that the trace of u at t = 0 exists near x = 0. The assumptions on the operator seem to be too weak to imply existence of a trace in general. More precisely shall prove ‖u(t, .)‖L2(B(0,1/8)) . e− for some δ > 0. The C 3 Hölder regularity in time for the metric g seems so be new, improv- ing the C 2 Hölder regularity in [9]. It is not clear to the authors whether this condition is optimal or not. We may replace the assumptions by stronger translation invariant assump- tions, (1.21) ‖g‖Lipx + ‖g‖ (1.22) ‖V ‖L1L∞+L∞Ln/2 ≪ 1 (1.23) ‖W1,2‖Ln+2(Ai) . 1 Then we also obtain a stronger conclusion. Theorem 2. Let P be as in (1.3) with coefficients as in (1.21). Assume that the potentials V and W satisfy (1.22) respectively (1.23). Then SUCP(II) holds at (0, 0) for H1 functions u satisfying (1.4). We remark that the condition (1.21) is really too strong, and that with some additional work (see Remark 2.3) one can bring it almost to the level of (1.15). Precisely, it suffices to replace (1.15) by (1.24) sup ‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖ t (Aij) where the slightly stronger time continuity modulus m2ij is given by m2ij(ρ) = e 4i−2jρ+ e (2i−j)ρ However, we cannot keep the additional terms in (1.7), because we need to be able to meaningfully talk about the trace of the solution at time t = 0. Both theorems are consequences of quantitative estimates, which also imply weak unique continuation under the assumptions of Theorem 2: Let u be a solution to (1.4) for which u(t0, .) vanishes in the closure of an open set. Then (u(t0, .) vanishes in a neighborhood of the closure. If u satisfies the assumptions and vanishes in an open set U , then it vanishes in the time slices t = t0 in an open neighborhood of the closure of Ut0 = {(x : (t, x) ∈ U}. Theorems 1 and 2 are nontrivial consequences of a Carleman inequality. To state a first version of the Carleman inequality we introduce an additional fam- ily B(τ) of sets which is a partition of the cylinder [0, τ−1)×B(0, 1), consisting (1.25) Aij , ln τ ≤ 4i ≤ τ 1/2, 0 ≤ j ≤ ln τ/2 + 2, (1.26) [e−4i−4, e−4i]× B(0, e−2iτ 1/2), 4i > τ (1.27) Aij , ln τ ≤ 4i, ln τ/2 ≤ j ≤ 2i. This partition is coarser than the partition of the same cylinder into the sets Aij . This is the reason why we need the assumption (1.18). More precisely Assumptions (1.17) and (1.18) imply (1.28) ‖V ‖l∞(B(τ),L1L∞+L∞Ln/2) ≪ 1. Theorem 3. Let τ0 ≫ 1, ε > 0 and P as in (1.7) with coefficients satisfying (1.15). Suppose that W1,2 satisfY (1.19) with constants depending on ε and τ0. Then there exists C > 0 such that for all τ ≥ τ0 the following is true: Suppose that v ∈ L2(H1) is compactly supported in [0, 8τ−1) × B(0, 2) and that it vanishes of infinite order near (0, 0). Then we can find a function φ ∈ C∞([0, 8τ−1]×B(0, 1)\{0, 0}) and h ∈ C∞(R+) which satisfy (1.29) τ ≤ h′ ≤ (1 + ε)τ (1.30) ∣∣∣∣φ(x, t)− h(− ln t)− x )∣∣∣∣ ≤ ε such that ‖eφv‖ l2(B(τ),L∞L2∩L2L n−2 )) ≤ C‖eφ(P +W1∇ +∇W2)v‖ l2(B(τ),L1L2+L2L n+2 ) for n ≥ 3, respectively ‖eφv‖l2(B(τ),L∞L2∩Lp′Lq′ ) ≤ C‖e φ(P +W1∇+∇W2)v‖l2(B(τ),L1L2+LpLq), for n = 2, 1 , 2 < p, and ‖eφv‖l2(B(τ),L∞L2∩L4L∞) ≤ C‖eφ(P +W1∇ +∇W2)v‖l2(B(τ),L1L∞+L4/3L1) for n = 1. The statement of the Carleman inequality is involved for several reasons. The weight t−τe−|x| 2/8t, which works for the constant coefficient case, has to be modified so that it has more convexity in order to handle variable coefficients, spatial localization and the gradient potential. However the polynomial growth in time (imposed by the assumption of vanishing of infinite order) limits the available amount of convexity; this is the origin of the l1 summability in (1.16), (1.20), and to a lesser extend of (1.18). If W = 0 then the Carleman inequality holds for a large explicit class of weights eφ. This cannot be possibly true for general gradient potentials. Instead, we are only able to prove that there exists some weight function φ, which now depends on τ , u andW , for which the uniform Carleman inequality holds. This strategy goes back to the seminal work of T. Wolff [29] and has been used by the authors for the elliptic problem [17]. The partition Aij is much finer than the dyadic decomposition in t only, which would correspond to the dyadic decomposition in the elliptic case. We are only able to localize the estimates to the sets Aij if we make the weight function sufficiently convex. We can do this for many Aij, but not for all of them. The sets (1.26) correspond directly to the assumption (1.18). We need to have control of the L1L∞+L∞Ln/2 norm of V in sets which are not smaller than those of the partition in (1.25), (1.26) and (1.27). We have stated Theorem 3 in a simpler form which suffices to derive Theo- rem 1 and Theorem 2. However the full estimate we prove is stronger in that it also contains precise L2 bounds. These are essential for the localization and perturbations techniques we use. The strategy of the proof is the same as in [17]: (1) We construct families of pseudoconvex weights and derive L2 Carle- man inequalities. The convexity of weights determines the space-time localization scales and the admissible size of perturbations. (2) We enhance the above L2 Carleman inequalities to include Lp esti- mates. Due to the L2 localization it suffices to do this in small sets. This allows us to use perturbation arguments starting from the case of the heat equation with the weight t−τe−|x| 2/8t. (3) Lp estimates for the spectral projections to spherical harmonics im- ply the Lp Carleman inequalities in the elliptic case. Here spectral projection for the Hermite operator play a similar role. (4) Finally we include Wolff’s osculating argument into the scheme in order to handle the gradient potentials. The efficiency of this part depends on the flexibility in the choice of the weight functions. The complexity of the weights and the L2 Carleman estimates comes mainly from the geometry of the classical harmonic oscillator. Orbits are contained in a sphere in R2n. The projection down in the x space is a ball, where frequency variables have a different behavior in radial and angular directions and near the boundary of the the ball. It turns out that our analytic estimates reflect these features. 2. Proof of Theorem 1 and 2 In this section we prove Theorem 1 and 2 assuming Theorem 3. The relation between Carleman estimates and unique continuation is fairly straightforward in the elliptic case. In the parabolic situation the argument is less direct due to the more complex geometry of the level sets of the weight functions. It is a standard consequence of a localized energy inequality that for the parabolic equation (1.4) u(t) and its gradient can be controlled by L2 norms of u. Proposition 2.1. Let n ≥ 3 and suppose that v solve the parabolic equation vt + ∂ka kl∂lv =W1∇v +∇(W2v) + V v on the space-time cylinder Q = [0, 2]×B(0, 2) with a ∈ Lip uniformly elliptic ‖W1,2‖Ln+2(Q) + ‖V ‖L1L∞+L∞Ln/2 ≪ 1 F extτ F intτ 1/16τ 211/42δ Figure 2. The sets Eδ, F τ and F 0≤t≤1 ‖v(t)‖L2(B(0,1)) + ‖∇xv‖L2([0,1]×B(0,1)) . ‖v‖L2(Q). If n = 2 then the same statement is true with L∞Ln/2 replaced by LpLq with 1 ≤ p < ∞, 1 < q ≤ ∞ and 1 = 1. Similarly, if n = 1 we have to replace it by L4L∞. Given our assumptions (1.15), (1.17), (1.18), (1.19) and (1.20) we can apply Proposition 2.1 rescaled in sets of the form [t0, 2t0] × B(x0, t1/20 ), which are subsets of the Aij. Summing up with respect to such sets contained in a parabolic cube Qr = [0, r 2]×B(0, r) we obtain the following consequence. Corollary 2.2. The following estimate holds under the assumptions of Theo- rem 1 2u‖L∞L2(Qr) + ‖t 2∇u‖L2(Qr) . ‖u‖L2(Q2r) Proof of Theorem 1. We choose τ ≥ τ0 and 0 < δ ≪ τ−1/2 and introduce the [0, 2δ2]×B(0, 2δ) [0, δ2]× B(0, δ) F extτ = [0, 2/τ ]×B(0, 2) [0, 1/τ ]×B(0, 1) F intτ =[1/32τ, 1/16τ ]× B(0, 1/4) Our strategy will be to truncate u in Eδ and F τ and to apply Theorem 3 to the truncated function in order to obtain a good bound on u in F intτ . Let η be a cutoff function supported in [0, 2)× B(0, 2) and identically 1 in [0, 1]×B(0, 1). For δ ≪ τ−1/2 we define vδ(t, x) = (1− η(t/δ2, x/δ))η(τt/8, x)u(x, t) which satisfies (P +W∇)vδ = V vδ − [P +W∇, η(t/δ2, x/δ)]u+ [P +W∇, η(τt/8, x)]u. The second term on the right hand side is supported in Eδ and the third one in F extτ . We apply Theorem 3 to vδ. One should keep in mind that the corresponding weight φ depends on δ but that the bounds we prove are uniform with respect to δ. We normalize the function h by h(0) = 0. We have to control the size of φ in the sets Eδ, F τ and F τ . Due to (1.29) we have τs ≤ h(s) ≤ 2τs, s ≥ 0 By (1.30) we obtain a rough polynomial bound in δ (2.1) eφ ≤ t−2τe− +ε(τ+x ) ≤ t1/2c(τ)δ−4τ−1 in Eδ. M = sup{eh(− ln(t))e− +ε(τ+x ) : (t, x) ∈ F extτ } By (1.29) the supremum is attained at a point (t0, x0) with ≤ 8τt0 ≤ 1 and |x0| = 1. A simple computation also shows that sup{t−1/2eh(− ln(t))e− +ε(τ+x ) : (t, x) ∈ F extτ } . τ 1/2M. Then M dominates eφ in F extτ : (2.2) eφ ≤M, t−1/2eφ . τ 1/2M in F extτ . Next we need to bound eφ from below in F intτ in terms of M , (2.3) inf F intτ eφ ≥ e To see this we compute for (t, x) ∈ F intτ and sufficiently small ε: φ(t, x)− φ(t0, x0) ≥ h(− ln t)− h(− ln t0) + − 2ε(τ + 1 − 20ε)τ ≥ 1 and use (2.1) and (2.2). Theorem 3 applied to vδ yields ‖eφvδ‖ l2(B(τ),L2L n−2 ∩L∞L2) .‖eφV vδ‖ l2(B(τ),L2L n+2+L1L2) + ‖eφ[P +W∇, η(t/δ2, x/δ)]u‖ l2(B(τ),L2L n+2+L1L2) + ‖eφ[P +W∇, η(τt, x)]u‖ l2(B(τ),L2L n+2 +L1L2) (2.4) By Hölder’s inequality we have ‖eφV vδ‖ l2(B(τ),L2L n+2+L1L2) .‖V ‖l∞(B(τ),L1L∞+L∞Ln/2)‖eφvδ‖ l2(B(τ),L2L n−2 ∩L∞L2) Due to the smallness in (1.28) we can absorb this term on the left hand side of the inequality. We calculate the first commutator fδ = [P +W∇, η(t/δ2, x/δ)]u + ∂kg kl∂l + 2 dkl∂l +W∇)η(t/δ2, x/δ) + 2δ−1gkl(∂kη)(t/δ 2, x/δ)∂lu By (2.1) we have ‖eφfδ‖ l2(B(τ),L2L n+2 +L1L2) . c(τ)δ−4τ−1‖t 2 fδ‖ l2(B(τ),L2L n+2 +L1L2) For the W term we use (1.19) and Holder’s inequality. For term involving dkl we bound the L1L2 norm in terms of an L∞L2 norm, using (1.15) which implies that the pointwise bound for dkl is summable with respect to dyadic time regions. For the remaining terms we simply bound the L1L2 norm in terms of the L2 norm. This yields ‖eφfδ‖ l2(B(τ),L2L n+2+L1L2) .c(τ)δ−4τ−2(‖u‖L2(Eδ) + ‖(∂xη)(t/δ 2, x/δ)t 2∇u‖L2 + ‖(∂xη)(t/δ2, x/δ)t 2u‖L∞L2) Then we can apply a straightforward modification of Corollary 2.2 on the Eδ scale to finally obtain ‖eφ[P +W∇, η(t/δ2, x/δ)]u‖ l2(B(τ),L2L n+2 +L1L2) . c(τ)δ−4τ−2‖u‖L2(Eδ). Similarly we can estimate the second commutator ‖eφ[P +W∇, η(τt, x)]u‖ l2(B(τ),L2L n+2 +L1L2) .Mτ 1/2‖u‖L2(F extτ ). Hence by inequality (2.4) we get (2.5) ‖eφvδ‖ l2(B(τ),L2L n−2∩L∞L2) . Mτ 1/2‖u‖L2(F extτ ) + c(τ)δ −4τ−2‖u‖L2(Eδ). Within F intτ we have vδ = u. Then by (2.3) we obtain (2.6) ‖u‖L∞L2(F intτ ) . τ 1/2e− τ‖u‖L2(F extτ ) + c(τ)δ −4τ−2‖u‖L2(Eδ). Also by the vanishing of infinite order the second term tends to zero as δ → 0. Hence as δ → 0 we obtain (2.7) ‖u‖L∞L2(F intτ ) . τ 1/2e− τ‖u‖L2(F extτ ) For 0 < t≪ 1 we choose τ = 1 to obtain ‖u(t, .)‖L2(B(0,1/4)) . t−1/2e− 32t . This completes the proof of Theorem 1. � Proof of Theorem 2. We extend the potentials V and W by zero to negative time, and gkl(t, x) = gkl(0, x) for t < 0. By definition, possibly after rescaling, we have u(0, .) ∈ L2(B(0, 2)). We now solve the mixed problem (2.8) ut + ∂kg kl∂lu = 0 for t < 0 and |x| ≤ 2 with the boundary condition u(t, x) = 0 if |x| = 2 and t < 0 and the obvious initial condition to obtain an extension of u to negative. The heat kernel for (2.8) satisfies Gaussian estimates. In particular we obtain from (1.6) for all positive integers N with a constant cN possibly differing from (1.6) (2.9) Br(0) |u|2dxdt . c2Nr2N . We seek to prove that the bound (2.7) still holds in this context. The difficulty is that we only know that u vanishes of infinite order at (0, 0) for negative time. To account for this we shift the time up, t→ t+2δ. Arguing as in the previous proof we obtain (2.6) with u replaced by u(t+ 2δ, x). Letting δ → 0 by (2.9) we obtain (2.7) and conclude as above. Remark 2.3. If one wants to prove Theorem 2 under the weaker assumptions on g in (1.24) then the origin needs to be avoided in the above argument. Hence the time translation needs to be accompanied by a spatial translation, namely uδ(t, x) = u(t+ 2δ 2, x− 8τδe1) This translation places the image of the origin, or better of the cube [0, τδ2]× B(0, 4τδ), within the region {τt < x2}. But in this region the conjugated operator Pψ, introduced later, is elliptic so only pointwise bounds for g are needed for the Carleman estimates. 3. L2 bounds in the flat case and the Hermite operator In this section we prove the simplest possible L2 Carleman estimate for the constant coefficient backward parabolic equation ∂tu+∆xu = f This serves as a good pretext to introduce the class of weight functions which is later modified for the variable coefficient case. We also describe the change of coordinates which turns the backward par- abolic operator into a forward parabolic equation for the Hermite operator H . In this way we are able to relate the L2 Carleman estimates for the heat operator to spectral information for H . Proposition 3.1. Let u ∈ L2 with compact support away from 0. Then (3.1) ‖t−τ− 8t u‖L2 ≤ ‖t−τ+ 8t (∂t +∆)u‖L2 uniformly with respect to τ away from (2n+ N)/4. Proof. In R+ × Rn we introduce new coordinates (s, y) ∈ R× Rn defined by (3.2) t = e−4s x = 1 e−2sy = −4e−4s − e−2sy Hence in the new coordinates our operator becomes 4t(∂t +∆x) = − − 2y ∂ If we conjugate it by tn/4e− 8t = e−nse− 2 we obtain 4t1+n/4e− 8t (∂t +∆x)t −n/4e 8t = − ∂ +∆y − y2 =: −∂s −H =: −P0 where H is the Hermite operator H = −∆y + y2 Then it is natural to define the new functions v(s, y) = e−nse− 2 u(e4s, e2sy), g(s, y) = e(−n−4)se− 2 f(e4s, e2sy) which are related by P0v = g ⇐⇒ (∂t +∆)u = f In the new coordinates, the bound (3.1) becomes (3.3) ‖e4τsv‖L2 . ‖e4τsP0v‖L2. Denoting w = e4τsv, we conjugate e4τsP0v = e 4τsP0e −4τsw = (−∂s −H + 4τ)w and the above bound becomes (3.4) ‖w‖L2 . ‖(−∂s −H + 4τ)w‖L2. Since ∂s and H − 4τ commute we expand ‖(−∂s −H + 4τ)w‖2L2 = ‖∂sw‖2L2 + ‖(H − 4τ)w‖2L2 ≥ d(4τ, n+ N)2‖w‖2L2. Note the spectral gap, which is essential in order to obtain strong unique continuation results. � For later use we also record the following slight generalization of the above result. For expediency this is stated in the (y, s) coordinates, i.e. in the form of an analogue of (3.3). Proposition 3.2. Let h be an increasing, convex, twice differentiable function so that d(h′,N) + h′′ ≥ 1 (3.5) ‖(1 + h′′)1/2eh(s)v‖L2 + ∥∥∥∥min (1 + h′′)1/2 1 + h′ eh(s)Hv . ‖eh(s)(∂s −H)v‖L2 for all compactly supported v ∈ L2. Proof. After the substitution w = eh(s)v the bound (3.5) becomes (3.6) ‖(1 + h′′)1/2w‖2L2 + ∥∥∥∥min (1 + h′′)1/2 1 + h′ . ‖(∂s −H + h′(s))w‖2L2. and we obtain the L2 estimate through expanding the term on the right hand side with respect to its selfadjoint and skewadjoint part: ‖(∂s −H + h′(s))w‖2L2 =‖∂sw‖2L2 + ‖(−H + h′)w‖2L2 + ‖(h′′)1/2w‖2L2 (d(h′,N)2 + h′′)‖w(s)‖2L2ds To complete the proof we observe that for each s we have ‖Hw(s)‖L2 . ‖(−H + h′)w(s)‖L2 + h′(s)‖w(s)‖L2 4. Resolvent bounds for the Hermite operator As seen in the previous section, the spectral properties of the Hermite op- erator play an essential role even in the simplest L2 Carleman estimates for the heat equation. In this section we take a look at L2 and Lp bounds for its spectral projectors and its resolvent. The spectrum of H is n+ 2N, and its eigenfunctions are the Hermite func- tions defined by uα = cα(∂y − y)αe− 2 , Huα = (n+ 2|α|)uα As |α| increases, so does the multiplicity of the eigenvalues. We denote the spectral projectors by Πλ for λ ∈ n + 2N. We consider both the spectral projectors and the resolvent of H and obtain both Lp and localized L2 bounds. 4.1. Weighted L2 bounds. We consider two parameters 1 ≤ d, R . λ We denote BR = {y : |y| < R}, , Bjd = {y : |yj| < d}, j = 1, ..., n By χR, respectively χ d we denote bump functions in BR, respectively B d which are smooth on the corresponding scales. Proposition 4.1. The spectral projectors Πλ satisfy the localized L 2 bounds (4.1) R− 4‖χRΠλf‖L2 +R− 4‖χR∇Πλf‖L2 . ‖f‖2L2, respectively (4.2) d− 2‖|Dj| dΠλf)‖L2 . ‖f‖L2 Proof. The inequality (4.1) is trivial unless R ≪ λ 12 . To prove it in dimension n = 1 we only need to consider the case when f is a Hermite function, f = Πλf = hλ in which case it follows from the pointwise bound 4 |h′λ(x)|+ λ 4 |hλ(x)| . ‖hλ‖L2, |x| ≤ In dimension n = 1 (4.2) follows by interpolation from (4.1) with R = d. This extends trivially to higher dimension by separation of variables. It remains to prove (4.1) in higher dimensions. Summing up (4.2) with d = R over j we obtain the bound 2‖|D| 2χRΠλf‖L2 . ‖f‖2L2 For |x| . R ≪ λ 12 we have |ξ|2 ≈ λ in the characteristic set ofH−ℜz, therefore the above norm should essentially control the left hand side of (4.1). For later use we prove a slightly more general result, which in particular concludes the proof of (4.1). (4.3) λ 4‖v‖L2+λ− 4‖∇v‖L2. ‖|D| 2 v‖L2+‖|y| 2 v‖L2+‖(H−λ)v‖ yL2+∇L2+λ Indeed the norm on the right is equivalent to 4v‖L2 + ‖(H + λ)− 2 (H − λ)v‖L2 & λ 4‖v‖L2 + λ− 2v‖L2 In our case we apply (4.3) to v = R− 2χRΠλf . Then 2 v‖L2 . ‖Πλf‖L2 while (H − λ)v = 2R− 2∇(∇χRΠλf) +R− 2∆χRΠλf which yields ‖(H − λ)v‖ yL2+∇L2+λ 2‖f‖L2 To state the corresponding resolvent bounds we define the spaces X̃2(z) by ‖u‖X̃2(z) = (1+ |ℑz|) 2‖u‖L2+‖(H−z)u‖ yL2+∇L2+|z| 2‖|Dj| du‖L2 and the corresponding dual spaces X̃∗2 (z). These spaces are larger than the corresponding “elliptic” spaces, (4.4) ‖v‖X2(z) . |z| 2‖v‖L2 + ‖yv‖L2 + ‖∇v‖L2 On the other hand by extending the bound (4.3) to complex λ we obtain a counterpart of (4.1), namely (4.5) R− 4‖χRu‖L2 +R− 4‖χR∇u‖L2 . ‖u‖X2(z) Finally, the result of (4.2) can be written in the following dual forms (4.6) ‖Πλf‖X2(λ) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ) The localized L2 resolvent bounds have the form Proposition 4.2. Let n ≥ 2, z ∈ C with dist(z, n + 2N) & 1, and 1 ≤ d ≤ R ≪ ℜz. Then (4.7) ‖u‖X̃2(z) . ‖(H − z)f‖X̃∗2 (z) where the d component of norms is omitted in dimension n = 1. Proof. We first note that the bounds (4.6) almost imply (4.7) up to a loga- rithmic divergence. They do imply easily a bound for higher powers of the resolvent for z away from the spectrum of H , (4.8) ‖(H − z)−1−kf‖X̃2(z) . (1 + |ℑz|) −k‖f‖X̃∗2 (z), k ≥ 1. as well as (4.9) ‖u‖L2 . (1 + |ℑz|)− 2‖(H − z)u‖X̃∗2 (z). Hence it remains to show that (4.10) ‖u‖X̃2(z) . (1 + |ℑz|) 2‖u‖L2 + ‖(H − z)u‖X̃∗2 (z). Using a positive commutator technique we first prove a one dimensional estimate. For this we define the one dimensional skewadjoint pseudodifferential operator2 Qr = iOp w(χ(yr−1)χ(ξ|y|−1)) where χ is a mollified signum function which satisfies χ′(x) = , |x| ≤ 2 Its properties are summarized in the following 2As defined the symbol of Q is not smooth at 0. However, any smooth modification in the ball {x2 + ξ2 < r2} will do. Lemma 4.3. a) Qr is bounded in L p for 1 ≤ p ≤ ∞ uniformly for r ≥ 1. b) Qr is also bounded in X̃2(z) uniformly with respect to z ∈ C and r ≥ 1. c) Qr satisfies the commutator estimate (4.11) r−1‖|D| 2χru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 〈(H − z)u,Qu〉 Proof. a) The Lp boundedness is straightforward and is left for the reader. b) For the X̃2(z) boundedness we consider first the d terms, which without any loss in generality we can write in the form 2‖(d2 +D2) 4χdu‖L2 Since Q is bounded in L2 it suffices to prove the commutator bound ‖[Qr, (d2 +D2) 4χd]u‖L2 . ‖u‖L2 But this is easily verified using the pdo calculus. Next we consider the term ‖(H − z)u‖ yL2+∇L2+|z| for which it suffices to prove the commutator bound ‖[Qr, H ]u‖ yL2+∇L2+|z| . ‖u‖L2 or equivalently, by duality, ‖[Qr, H ]u‖L2 . ‖yu‖L2 + ‖∇u‖L2 + ‖|z| 2u‖L2 This follows again from the pseudodifferential calculus. c) Since Qr is skewadjoint we have the identity 〈(H − z)u,Qru〉 = 〈[H,Qr]u, u〉+ ℑz〈iQru, u〉 therefore it suffices to insure that (4.12) 〈[H,Qr]u, u〉 & ‖u‖2X̄2(r) +O(‖u‖ For this we compute the commutator [H,Q], [H,Q] = Opw({ξ21 + y21, χ(y1r−1)χ(ξ1|y1|−1)}) +O(1)L2→L2 = Opw(2r−1χ′(y1r −1)ξ1χ(ξ1|y1|−1)) +O(1)L2→L2 r−1(χ1r) 2(ξ21 + r 2 . r−1χ′(y1r −1)ξ1χ(ξ1|y1|−1) + 1 and the conclusion follows by Garding’s inequality. � We return to the proof of the proposition. By separation of variables the bound (4.11) extends to higher dimensions and gives r−1‖|Dj| 2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + 2ℜ〈(H − z)u,Qjru〉 where Qjr the higher dimensional analogue of Qr with respect to the j variable. TheX2(z) boundedness ofQr also extends easily to higher dimension. Hence by Cauchy-Schwartz we obtain r−1‖|Dj| 2χjru‖2L2 . (1 + |ℑz|)‖u‖2L2 + ‖(H − z)u‖X∗2 (z)‖u‖X2(z) To conclude the proof of the estimate (4.10) it remains to show that ‖(H − z)u‖ 2L2+yL2+∇L2 . ‖(H − z)u‖X∗2 (z) which follows by duality from (4.4). The final step in the L2 resolvent bounds is to replace the y′ derivatives by angular derivatives. Let ∇⊥ = y|y| ∧∇ be the angular derivative and |D⊥| the corresponding fractional derivative. We split the coordinates into y = (y1, y ′) and use the notation ′ for coordi- nates and derivatives in the obvious sense. For 1 ≤ d ≤ R ≤ λ we define the sector BR,d = {R < |y1| < 2R, |y′| ≤ d} and χR,d a bump function inBR,d. Then we define the function spaceX2(λ,R, d) ‖u‖2X2(λ,R,d) = ‖u‖ L2 +R −1/2λ−1/4‖∇(χR,du)‖2L2 + R−1/2λ1/4‖χR,du‖2L2 + d−1/2‖|D⊥| 2χR,du‖2L2 and X∗2 (λ,R, d) as its dual. Lemma 4.4. Suppose that n ≥ 2 and 1 ≤ d ≤ R. Then ‖u‖X2(λ,R,d) ≈ R− 4‖χR,du‖L2 +R− 4‖∇χR,du‖L2 + d− 2‖|D′| 2χR,du‖L2 Proof. Within BR,d the angular derivatives are close to the y ′ derivatives, namely |D⊥u| . |D′u|+ |∇u|, |D′u| . |D⊥u|+ |∇u|. This implies the corresponding bounds for L2 norms, and the conclusion follows by interpolation. � ¿From the above lemma we obtain ‖u‖X2(λ,R,d) . ‖u‖X̃2(λ,R,d) Hence, we may replace X̃2 by X2 in (4.7) and (4.6): Corollary 4.5. a) For λ in the spectrum of H we have (4.13) ‖Πλf‖X2(λ,R,d) . ‖f‖L2, ‖Πλf‖L2 . ‖f‖X∗2 (λ,R,d) b) For z away from the spectrum of H and 1 ≤ d ≤ R . ℜz we have (4.14) ‖(H − z)−1−kf‖X2(λ,R,d) . (1 + |ℑz|)−k‖f‖X∗2 (λ,R,d), k ≥ 0 4.2. The Lp bounds of the resolvent. The Lp bounds for the spectral projectors and the resolvent were proved in [16], [28] (see also [10]). For the sake of completeness we also present them here in a simpler manner following the approach in [19]. We refer the reader to the same paper for further results. We consider pairs of exponents satisfying (4.15) where the range for p is (4.16) p ≥ 4 for n = 1, p > 2 for n = 2, p ≥ 2 for n ≥ 3. This leads to the following range3 for q: (4.17) q ∈ [2,∞] for n = 1, q ∈ [2,∞) for n = 2, q ∈ [2, 2n ] for n ≥ 3. The dual exponents are denoted by p′ and q′ as usual. Proposition 4.6. Let q be as in (4.17). Then a) The spectral projectors Πλ satisfy ‖Πλ‖Lq′→L2 . 1, ‖Πλ‖L2→Lq . 1, n ≥ 2 ‖Πλ‖Lq′→L2 . λ p , ‖Πλ‖L2→Lq . λ− p , n = 1 (4.18) b) For z away from n+ N the resolvent (H − z)−1 satisfies ‖(H − z)−1‖Lq′→Lq . (1 + |ℑz|) p′ , n ≥ 1(4.19) Outline. To revisit the Lp bounds associated to the spectral projectors we recall the approach in [19]. The first step there is to establish pointwise bounds for the Schrödinger evolution4 (4.20) ‖eitH‖L1→L∞ . (sin t)− This immediately (see also [18]) leads to Strichartz estimates for the solution to the inhomogeneous equation ivt −Hv = g, v(0) = v0 namely (4.21) ‖v‖Lp([0,2π];Lq) . ‖v0‖L2 + ‖g‖Lp′([0,2π];Lq′) where (p, q) are as described in (4.15), (4.16). To obtain (4.18) we apply (4.21) to v = e−iλtΠλu, which yields L 2 → Lp bounds, and hence by duality and selfadjointness all estimates of (4.18) for 3The exponent q = ∞ is actually allowed in the spectral projection bounds in dimension n = 2. However, it is not allowed in any of the resolvent bounds. 4These bounds are very robust and are in effect established in [19] for a much larger class of operators n ≥ 2. The case n = 1 can be dealt with directly using the pointwise bounds for the Hermite functions. We note a consequence of the bounds (4.18), namely (4.22) ‖(H − z)−1−k‖Lq′→Lq . (1 + |ℑz|) , n ≥ 2, k ≥ 1 which is obtained by interpolating between q = 2 and q = 2n Similarly we get (4.23) ‖(H − z)−1‖Lq′→L2 . (1 + |ℑz|) p′ , n ≥ 2. Then we apply (4.21) to v(x, t) = χ(t)e−iztu(x), g = χ′(t)e−iztu(x) + χ(t)e−izt(H − z)u where χ is a unit bump function on an interval of size (1+ |ℑz|)−1. This yields ‖u‖Lq . (1 + |ℑz|) p‖u‖L2 + (1 + |ℑz|) p′ ‖(H − z)u‖Lq′ . ‖(H − z)u‖Lq′ . concluding the proof of (4.19) for n ≥ 2. The case n = 1 is a variation on the same theme. � 4.3. Combining the estimates. Here we combine the L2 and the Lp com- ponents in the resolvent bounds: Proposition 4.7. For z away from n+ 2N the resolvent (H − z)−1 satisfies (4.24) ‖(H − z)−1‖Lq′→X2(ℜz,R,d) . (1 + |ℑz|) 2 , n ≥ 2, (n, q) 6= (2,∞) with the obvious modification for n = 1. Proof of Proposition 4.7. Taking into account the bounds (4.19) and (4.23), it remains to prove the estimate ‖u‖X̃2(ℜz,R,d) .(1 + |ℑz|) p′ ‖(H − z)u‖Lq′ + (1 + |ℑz|) p‖u‖Lq + (1 + |ℑz|) 2‖u‖L2 But this follows from (4.12) in the same way as for Proposition 4.2 since the operator Q is bounded in Lq. � 5. Lp estimates in the flat case and parametrix bounds In this section we begin with the mixed norm LpLq Carleman estimates in the simplest case, i.e. with constant coefficients and a polynomial weight. These were proved in [8] except for the endpoint which was obtained later in [10] . After a conformal change of coordinates and conjugation with respect to the exponential weight the Carleman estimates reduce to proving LpLq estimates for a parametrix K for ∂t −H + τ . In this article we need a stronger version of these bounds, where we add in localized L2 norms. In a simplified form, Escauriaza-Vega’s result in [10] has the form: Theorem 4. [10] Let p and q be as above. Then ‖t−τe− 8t u‖L∞(L2)∩Lp(Lq) ≤ ‖t−τe− 8t (∂t +∆)u‖L1(L2)+Lp′(Lq′ ), for all u with compact support in Rn × [0,∞) vanishing of infinite order at (0, 0) uniformly with respect to 4τ with a positive distance from integers. One can write the estimate in the (s, y) coordinates using the same trans- formation as in Section 3: (5.1) ‖eτsv‖L∞(L2)∩Lp(Lq) . ‖eτs(∂s +H)v‖L1(L2)+Lp′ (Lq′ ) Setting w = eτsv this becomes (5.2) ‖w‖L∞(L2)∩Lp(Lq) . ‖(∂s +H − τ)w‖L1(L2)+Lp′ (Lq′ ) Denoting by Πλ the spectral projection onto the λ eigenspace ofH we obtain a parametrix K for (∂t −H + τ), K(∂t +H − τ) = I where the s-translation invariant kernel of K is K(s) = s(τ−λ)1s(τ−λ)<0 Since w decays at ±∞ we have w = K(∂s +H − τ)w therefore (5.2) can be rewritten in the form (5.3) ‖Kf‖L∞(L2)∩Lp(Lq) . ‖f‖L1(L2)+Lp′ (Lq′ ) The main result of this section is an improvement of (5.2), namely Proposition 5.1. Assume that τ is away from n + N and that 1 ≤ d ≤ R . τ (5.4) ‖Kf‖L∞(L2)∩Lp(Lq)∩L2X2(τ,R,d) . ‖f‖L1(L2)+Lp′ (Lq′ )+L2X∗2 (τ,R,d) Proof. We work in dimension n ≥ 2; some obvious adjustments are needed in dimension n = 1, which is slightly easier. We consider four endpoints: A: The L1L2 → L∞L2 bound follows easily since the projectors Πλ are L2 bounded. B: The L1L2 → LpLq bound. Here it suffices to prove ‖K(.)f‖LptLqx . ‖f‖L2 Splitting f into spectral projections and using (4.18) we obtain ‖K(t)f‖Lq . e−|(λ−τ)t|‖Πλf‖L2 For |t| ≥ 1 we can use Cauchy-Schwartz to obtain ‖K(t)f‖Lq∩X2(R,d) . e−c|t|‖f‖L2 which suffices for all q. For |t| ≤ 1 we consider the most difficult case p = 2 and compute ‖K(t)f‖2L2([−1,1],Lq) . e−|(λ−τ)t|‖Πλf‖L2 |λ− τ | + |µ− τ | ‖Πλf‖L2‖Πµf‖L2 0≤i,j 2−i−j |λ−τ |≈2i ‖Πλf‖L2 |µ−τ |≈2i+j ‖Πµf‖L2 |λ−τ |≈2i ‖Πλf‖2L2 |µ−τ |≈2i+j ‖Πµf‖2L2 .‖f‖2L2 C: The L1L2 → X2(τ, R, d) bound for K follows in the same way from ‖Πu‖X2(τ,R,d) . ‖u‖L2. D: The Lp + L2X∗2 (τ, R, d) → L∞L2 bound for K is equivalent to the L1L2 → LpLq ∩X2(τ, R, d) bound for K∗. By reversing time this is seen to be the same as the L1L2 → LpLq bound for K. E: The Lp + L2(X∗2 (R, d)) → LpLq ∩ L2X2(R, d) bound. Using (4.18) and (4.13) directly yields ‖K(s)‖ n+2+X∗2 (R,d)→L n−2 ∩X2(R,d) e|s(τ−λ)| . s−1e−cs Similarly we obtain ‖K(s)‖ n−2 ∩X2(R,d) 2 e−cs, ‖K(s)‖ (R,d) 2 e−cs Interpolation with the L2 estimate gives ‖K(s)‖Lq′→Lq . s ‖K(s)‖Lq′→X2(R,d) . s 2 , ‖K(s)‖X∗2 (R,d)→Lq . s If p > 2 then the Hardy-Littlewood Sobolev inequality implies ‖K ∗ f‖LpLq . ‖f‖Lp′Lq′ . ‖K ∗ f‖L2X2(R,d) . ‖f‖Lp′Lq′ , ‖K ∗ f‖LpLq . ‖f‖L2X∗2 (R,d). With obvious changes the analysis is similar if n = 1, 2. It remains to prove the L2 → L2 type bounds, namely ‖Kf‖L2X2(R,d) . ‖f‖L2X∗2 (R,d) (n = 1, 2) respectively L2(X2(R,d)∩L n−2 ) . ‖f‖ L2(X∗2 (R,d)+L n+2 ) (n > 2) For this, following an idea in [10], we consider a dyadic frequency decompo- sition in time. By the Littlewood-Paley theory it suffices to prove the bound for a single dyadic piece at frequency 2j, namely (5.5) ‖Sj(Ds)Kf‖ L2(X2(R,d)∩L n−2 ) . ‖f‖ L2(X∗2 (R,d)+L n+2 ) (n > 2) and its one and two dimensional counterpart. Taking a time Fourier transform we can write (for f ∈ S(Rn)) ŜjK(σ)f = s(2 λ− τ − iσ Πλf = s(2 −jσ)(H − τ − iσ)−1f therefore by the inversion formula (SjK)(t)f = eitσs(2−jσ)(H − τ − iσ)−1fdσ =− t−2 (s(2−jσ)(H − τ − iσ)−1)fdσ Hence using the resolvent bounds (4.19) and (4.22) and the first line for |t| ≤ 2−j and the second line for |t| ≥ 2−j we obtain ‖SjK(t)‖ (R,d) n−2 ∩X2(R,d) 1 + 22jt2 and the similar estimate in one and two dimensions. The bound on the right is integrable in t, therefore (5.5) follows. � 6. Modified weights and pseudoconvexity The main result of this section, Theorem 5 is a considerable improvement of Section 3. The weights t−τ in Section 3, while easy to use, satisfy merely a degenerate pseudoconvexity condition, in the sense that the selfadjoint and the skewadjoint parts of the operator in (3.4) commute. This is in contrast to strong pseudoconvexity where one obtains better L2 bounds from the positivity of the commutator. A perturbation argument easily implies an L2 Carleman estimate for variable coefficients as soon as g = In+O(t). However, even arbi- trarily small perturbations of g from In at t = 0 destroy the pseudoconvexity. To obtain results for general variable coefficients we need a more robust weight with additional convexity. A good way of doing this is by adding convexity in t and by using a weight of the form eh(− ln t) with a convex function h. Then we obtain for the heat operator the strength- ened L2 estimates of Proposition 3.2. The assumption of vanishing of infinite order forces us to work with functions h with at most linear growth at infinity. This in turn limits the convexity of h, and hence the gain from the convexity in the L2 bounds. These Carleman inequalities with the weight e−h(ln t) are more stable with respect to perturbations. They can be obtained for coefficients satisfying (6.1) |g − In|+ (t + |x|2)|∂tg|+ (t+ |x|2)1/2|∇g| . (t+ |x|2)ε. with suitable functions h. It is not difficult to weaken (6.1) almost to our condi- tions (1.15) and (1.16). This venue was pursued by Escauriaza and Fernández In this paper we seek to obtain Lp Carleman inequalities and also to handle Lp gradient potentials. Both require good spatial and temporal localization, which depends on the strength of the L2 estimates. The weights eh(− ln t) seem to be insufficient for this purpose. Consequently we consider a larger class of weights of the form eh(− ln t)+φ(xt −1/2,− ln(t)) having some additional convexity in y = xt−1/2. Here we think of φ essen- tially as a function of y with a milder dependence on s = − ln t. Obtaining pseudoconvexity is not entirely straightforward because the Hamilton flow for the Hermite operator H is periodic so no nonconstant function of y can be convex along its orbits. We note that the projection of the orbits to the y space are ellipses of size O( τ ) where τ is the energy, centered at 0. Hence we can choose φ to be convex in y for |y| ≪ τ . We compensate the lack of convexity of φ when |y| ≈ τ by the s convexity of h. To elaborate this idea we explain the precise setup. Let δ1 be small positive constant. We begin with constants {αij}A (see (1.13) and (1.14) for the notation) which control the regularity of the coefficients5 gkl − δkl, dkl and ekl of P given by (1.7) as in (1.15). (6.2) δ1αij = ‖d‖L∞(Aij) + ej−2i‖d‖Lipx(Aij) + ‖d‖Cmijt (Aij). The condition (1.15) guarantee that for all τ ≥ 1 (6.3) ‖αij‖l1(A(τ)) ≤ 1. ‖αij‖l∞(A) ≤ 1. We first adjust the αij ’s upward so that they vary slowly and do not con- centrate in irrelevant regions. This readjustment depends on the choice of the parameter τ . Lemma 6.1. Let αij be a sequence satisfying (6.3). Then for each τ ≫ 1 there exists a double sequence (εij)A(τ) with the following properties: 5denoted generically by d here and later (1) For each (i, j) ∈ A(τ) αij ≤ εij (2) We have εij ∈ l1(A(τ)), ‖εij‖l1(A(τ)) . 1. (3) The sequence εij is slowly varying, | ln εi1j1 − ln εi2j2| ≤ (|i1 − i2|+ |j1 − j2|), (i1, j1), (i2, j2) ∈ A(τ). (4) The sequence (εi) defined by j:(i,j)∈A(τ) εij, i ≥ ln τ. has the following properties (6.4) | ln εi1 − ln εi2 | ≤ |i2 − i1|, εij . εi, εi[ln τ/2]+2 ≈ εi (5) For each i ≥ ln τ there exists an unique 0 ≤ j(i) ≤ [ln τ/2]+2 with the following properties: (6.5) εij(i) ≈ εi. εij ≤ e−jτ−1/2 if 0 ≤ j ≤ j(i), j(i) 6= 0 εij > e −jτ−1/2 if j(i) < j ≤ [ln τ/2] + 2. (6.6) We shall see that j(i) is an important threshold. If j ≥ j(i) then we can localize our estimates to the corresponding Aij and even to smaller sets. On the other hand, we cannot localize to sets smaller than (6.7) Bi0 = j≤j(i) Proof. To fulfill the conditions (1)-(4) we simply mollify the αij, ε̃ij = max (k,l)∈A(τ) (|i−k|+|j−l|), ε̃i = [ln τ/2]+2∑ ε̃ij. For the last part of (4) we redefine ε̃ij := ε̃ij + e |j−([ln τ/2]+2)|ε̃i This also increases ε̃i by a fixed factor. For (5) we begin with a preliminary guess for j(i) which we call j0(i) ∈ R+. We consider three cases. j0(i) = ln τ/2 if ε̃i < τ − ln(ε̃iτ 1/2) if τ−1 ≤ ε̃i < τ−1/2, 0 if τ−1/2 ≤ ε̃i. We define (6.8) εij := max{ε̃ij, 2e− |j−j0(i)|ε̃i}, which is still slowly varying because ε̃i is slowly varying. We define j(i) according to (6.6). It is uniquely determined since the se- quence εij is slowly varying compared to e 2 . Since ε̃ij is slowly varying we must have ε̃ij ≤ ε̃i/2. This allows us to conclude that for j close to j0(i), the second term in (6.8) is larger than the first one, εij = 2e |j−j0(i)|ε̃i for |j − j0(i)| ≤ 2. If j0(i) = 0 then εi0 = 2ε̃i ≥ 2τ−1/2 and hence j(i) = 0. If 0 < j0(i) < ln τ/2 then for |j − j0(i)| ≤ 2 we have εij = 2e |j−j0(i)|e−j0(i)τ−1/2 therefore j0(i)− 2 < j(i) ≤ j0(i). If j0(i) = ln τ/2 then for |j − j0(i)| ≤ 2 we εij ≤ 2e− |j−j0(i)|e−j0(i)τ−1/2 and we arrive again at j0(i)−2 ≤ j(i). In all three cases we have |j0(i)−j(i)| ≤ 2 therefore (6.5) holds. We observe that εi . ε̃i. � The sequence (εij)A(τ) is used to describe the amount of spatial convexity needed in the region Aij , which will be reflected in the construction of φ below. The partial sums εi measure the amount of s-convexity needed in [i, i+1]. The purpose of part (5) above is to correlate the two amounts in a region where they have the same strength (where j is close to j(i)). Our weights have the form (6.9) ψ(s, y) = h(s) + φ(s, y) Their choice is described in the next two lemmas: Lemma 6.2. Let τ and (εi) be as in Lemma 6.1. Then there is a convex function h with the following properties: (1) h′ ∈ [τ, 2τ ]. (2) h′′(s) + dist(h′(s),N) > 1 (3) εiτ . h ′′(s) . εiτ + 1 for s ∈ [i, i+ 1]. (4) |h′′′| . h′′. The proof of the lemma is fairly straightforward and uses only the fact that (εi) is slowly varying and summable. The second part is needed in order to avoid the eigenvalues of the Hermite operator. Lemma 6.3. Let τ , (εij) and (εi) be as in Lemma 6.1. Then there exists a smooth spherically symmetric function φ : R× Rn → R with the following properties: (1) (Bounds) The function φ is supported in |y| ≤ 2τ 1/2 and satisfies (6.10) 0 ≤ φ(s, y) . εiτ, |∂sφ(s, y))|+ |∂2sφ(s, y)| . εiτ (6.11) l,k=0 (1 + |y|)k|D1+ky ∂lsφ| . ǫiτ 1/2 for i ≤ s ≤ i+ 1, (2) (Monotonicity) (6.12) ∂rφ(s, y) ≈ εiτ 2 for (s, y) ∈ Aij , (i, j) ∈ A(τ), j ≥ j(i) + 1 (3) (Convexity) (6.13) (1 + |y|)∂2rφ(s, y) ≈ εijτ 2 in Aij, (i, j) ∈ A(τ). Proof. Let φj(y) = e2j + |y|2, j ≥ 0. We fix a smooth partition of unity 1 = η(s− i) and define ln aj(s) = η(s− i) ln εij . These functions satisfy the bounds (6.14) aj(s) ≈ εij, i ≤ s ≤ i+ 1, |a′j |, |a′′j |, |a′′′j | . aj. Their sum satisfies a(s) := [ln τ/2]+2∑ aj(s) ≈ εi, i ≤ s ≤ i+ 1. We define φ(s, y) = τ 2χ(|y|τ−1/2) [ln τ/2]+2∑ aj(s)φj(|y|) where χ is a smooth function supported in [−2, 2] and identically 1 in [−3 We verify the properties: 0 ≤ φ(s, y) . a(s)τ . εiτ, i ≤ s ≤ i+ 1. The remaining part of (6.10) follows from (6.14). Estimate (6.11) is a conse- quence of (6.14) and |(1 + |y|)kDk+1φj(y)| . 1, 0 ≤ k ≤ 3. The upper bound from (6.12) is covered by (6.11) and the lower one follows ∂rφ(s, y) & τ j≤j(i) aj ≈ τ 1/2 j(i)∑ εij ∼ εiτ 1/2 in Aij with j ≥ j(i) where we use εij(i) ≈ εi. The assertion (6.13) follows from immediate bounds on second derivatives of the φj. � Our aim in this section is to prove L2 Carleman estimates for the variable coefficient operator P with the exponential weight ψ(− ln t ψ(s, y) = h(s) + δ2φ(s, y). where δ2 is a small constant and h and φ are as in in Lemma 6.2 and 6.3. The calculations are involved. For a first orientation we outline the key part of the argument for the constant coefficient heat equation. Using the change of coordinates of Section 3 we transform the problem to weighted estimates for the operator P0 = ∂s +H and the exponential weight e ψ(s,y). This translates to obtaining bounds from below for the conjugated operator P0,ψ = e ψ(s,y)P0e −ψ(s,y). Lemma 6.4. Let τ be large enough. Let ψ be as in (6.9) with h, φ as in the above two Lemmas 6.2,6.3 with δ2 ≪ 1 . Then the operator P0,ψ satisfies the bound from below ‖(h′′) 2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2) . ‖P0,ψv‖2L2 for all functions v supported in {|y|2 ≤ 9τ} where the weights aint, a⊥ are defined by a4int = εij(1 + |y|)−1τ 2 , a4⊥ = 1j≥j(i)εi(1 + |y|)−1τ 2 in Aij . Proof. We decompose P0,ψ into its selfadjoint and its skewadjoint part P0,ψ = L 0,ψ + L where (6.15) Lr0,ψ := −∆y + y2 − ψs − ψ2y , Li0,ψ := ∂s + ψy∂ + ∂ψy. Expansion of the norm gives (6.16) ‖(Lr0.ψ + Li0,ψ)v‖2L2 = ‖Lr0,ψv‖2L2 + ‖Li0,ψv‖2L2 + 〈[Lr0,ψ, Li0,ψ]v, v〉 The conclusion of the lemma follows from the commutator bound (6.17) 〈[Lr0,ψ, Li0,ψ]v, v〉 & ‖(h′′) 2 v‖2 + δ2τ−1(‖a2int∇v‖2 + ‖a2⊥∇⊥v‖2) The commutator is explicitly computed [Lr0,ψ, L 0,ψ] = ψss + 4ψyψyyψy − 4∂ψyy∂ − 4yψy + 4ψyψsy −∆2ψ Since δ2 ≪ 1 the first term has size h′′(s). The second one is nonnegative since ψ is convex for |y|2 < 9τ . The Hessian of the radial function ψ can be written in the form (6.18) ψyy = ψrr One can see that the radial and angular derivatives carry different weights. Our construction of φ guarantees that ψrr . , ψyy & ψrrIn hence the weight ψrr can be used for all derivatives. For the size of the two weights we have ψrr ≈ a4int, & a4⊥ This gives the last two terms in (6.17). It remains to see that the remaining terms in the commutator are negligible compared to the first term on the right hand side of (6.17). For this we use the bound (6.11) to conclude that in Aij we have | − 4yψy + 4ψyψsy −∆2ψ| . δ2εiτ . δ2h′′ To switch to operators with variable coefficients it is convenient to extend the weights to the full space and to regularize them. Precisely we shall assume a4int(s, y) ≈ εiτ in Aij if |y|2 ≥ τ a4int(s, y) ≈ εij(1 + |y|)−1τ 2 in Aij if |y|2 ≤ τ. (6.19) Observe that the two cases above match since εi ≈ εij in the region where y2 ≈ τ . We also introduce a modification a of aint which is used to include the effect of the spectral gap in regions where we have very little convexity: a4(s, y) ≈ 1 + εiτ in Aij if |y|2 ≥ τ a4(s, y) ≈ 1 + εij(1 + |y|)−1τ 2 in Aij if |y|2 ≤ τ. (6.20) Finally we choose a⊥ with the properties supp a⊥ ⊂ {Aij : j(i)− 1 ≤ j ≤ ln τ} a4⊥(s, y) .εi(1 + |y|)−1τ 2 in Aij a4⊥(s, y) ≈εi(1 + |y|)−1τ 2 in Aij if j(i) ≤ j ≤ ln τ − 1 (6.21) The bounds for the weights from above are assumed to remain true after applying powers of the differential operators ∂s, ∂y and y∂y to them. Consider now a the more general class of operators P with real variable coefficients given by (1.7). We repeat the change of coordinates and write in the (s, y) coordinates: 4e−4sP = − ∂ − 2y ∂ + ∂ig ij∂j + yid ij∂j + ∂id ijyj + yie This further leads to 4e−(n+4)s− 2 Pens+ 2 = −P̃ where P̃ is given by − ∂igij∂j − yi(gij − 2δij + 2dij + eij)yj − yi(gij − δij + dij)∂j − ∂i(gij − δij)yj We rewrite it in the generic form P̃ = P0 − ∂d∂ − ydy − yd∂ − ∂dy with P0 = ∂s +H . To simplify as much as possible the proof of the main L2 Carleman estimate we introduce a stronger condition on the regularity of the coefficients: |d|+ 〈y〉(|dy|+ τ− 2 |dyy|+ τ−1|dyyy|+ τ− 2 |ds|) . δ1εij in Aij |d|+ 〈y〉(|dy|+ τ− 2 |dyy|+ τ−1|dyyy|+ τ− 2 |ds|) . δ1 (6.22) This improved regularity will be gained later on by regularizing the coefficients. We are now in the position to formulate the Carleman estimate. Proposition 6.5. Let τ be large enough and δ1 ≪ δ2 ≪ 1. Let ψ be as in (6.9) with h, φ as in Lemmas 6.2,6.3. Assume that the coefficients g − In, d and e satisfy (6.22). Then the following L2 Carleman estimate holds for all functions u supported in {y ≤ 9τ}: j=0,1,2 2‖a2eψDju‖+ τ− 12‖a2⊥eψD⊥u‖ . ‖eψP̃ u‖.(6.23) Proof. After conjugation Pψ := e ψ(s,y)P̃ e−ψ(s,y) we decompose Pψ into its selfadjoint and its skewadjoint part Pψ = L ψ + L which for y2 < 9τ can be expressed in the generic form (see also (6.15)): Lrψ = L 0,ψ + ∂d∂ + τd Liψ = L 0,ψ + τ 2 (d∂j + ∂jd) with d satisfying (6.22). Then (6.23) follows from (6.24) j=0,1,2 τ−j(δ2‖a2intDjv‖2 + 〈h′′〉 2Djv‖2) + δ2τ−1‖a2⊥∇⊥v‖2 . ‖P̃ψv‖2. The proof will consist of three steps. Step 1: First we show that for v supported in {|y|2 ≤ 9τ} we have (‖a2int∇v‖2+‖a2⊥∇⊥v‖2)+‖(h′′) 2v‖2+‖Lrψv‖2 . ‖P̃ψv‖2+ δ1‖a2intv‖2 (6.25) We compute ‖Pψv‖2L2 = ‖Lrψv‖2L2 + ‖Liψv‖2L2 + 〈[Lrψ, Liψ]v, v〉 We expand the commutator [Lrψ, L ψ] = [L 0,ψ, L 0,ψ] + [∂d∂ + τd, L 0,ψ] + τ 2 [Lrψ, d∂j + ∂jd] The main contribution in (6.25) comes from the first commutator, for which we use (6.17) to obtain the terms on the left side of (6.25). The second commutator is estimated by |〈[M r, Li0,ψ]v, v〉| . δ1(‖a2intv‖2 + τ−1‖a2int∇v‖2) and the second term on the right is negligible since δ1 ≪ δ2. Indeed, we write [∂d∂ + τd, Li0,ψ] = −∂kqkl∂l + r where the coefficients q, r have the generic form q = ds + ψydy + ψyyd+ dψyy, r = ∂d∂∆ψ + τ(ds + ψydy) Using the bounds (6.22) for d and (6.11) for φ we estimate |q| . δ1τ−1a4int, |r| . δ1a4int. Finally, the third commutator is estimated in a similar fashion. We write it in the form 2 [Lrψ, d∂j + ∂jd] = −∂kqkl∂l + r where the coefficients q, r have the generic form q = τ 2 (dy + ddy), r = τ 2 (∆dy + ∂d∂dy + dψys + dψyyψy + τddy) Using (6.22) and (6.11) we obtain the same bounds for q and r as in the previous case. This concludes the proof of (6.25). Step 2: We use an elliptic estimate to show that for functions v supported in {|y|2 ≤ 9τ} we have (6.26) δ2 τ−j‖a2intDjv‖2+τ−1‖a2⊥D⊥v‖2 +‖(h′′) 2v‖2+‖Lrψv‖2L2 . ‖P̃ψv‖2 The elliptic bound ‖D2v‖+ τ‖v‖ . τ 2‖Dv‖+ ‖(−∆− h′(s))v‖ can easily proven by a Fourier transform. It implies ‖D2v‖+ τ‖v‖ . τ 2‖Dv‖+ ‖(H − h′(s))v‖+ ‖y2v‖, We can replace H − h′(s) by Lrψ due to the pointwise estimate |(Lrψ − (H − h′(s)))v| . δ1(|D2v|+ τ 2 |Dv|+ τ |v|) Then (6.26) follows from (6.25). Step 3: Here we use the spectral gap condition to improve our bound when h′′ ≪ 1 and show that (6.26) implies (6.23). It suffices to show that if h′′(s) < 1 ‖v‖L2 + τ−1‖D2v‖ . ‖Lrψv‖L2 Indeed, let s ∈ [i, i + 1] so that h′′(s) < 1 . Then h′ has a positive distance from the integers. Also εi . 1 which implies that at time s we must have |g − In| . δ1τ−1, |Dg| . δ1τ−1, |ψr| . δ2τ− Hence we may think of Lrψ as a small perturbation of H − h′(s) and compute ‖v‖+ τ−1‖D2v‖ . ‖(H − h′(s))v‖ . ‖Lrv‖+ (δ1 + δ22)‖v‖+ δ1τ−1‖D2v‖ where the last two terms on the right are negligible compared to the left hand side. The proof of the proposition is concluded. We want to reformulate the previous result in a more symmetric fashion. To do this we weaken the estimates slightly by using a coarser partition of the space. We distinguish three cases for i corresponding to the value of j(i) in Lemma 6.1 (v). Definition 6.6. We define the partition Bij as follows. (1) If j(i) = 0 (which corresponds to εi & τ 2 ) we set Bij = Aij, b ≈ a, b⊥ ≈ a⊥ (2) If 0 < j(i) < [ln τ/2+2] (which corresponds to τ−1 . εi . τ 2 ) we set Bij = Aij, b ≈ a, b⊥ = a⊥ j ≥ j(i) respectively Bi0 = j<j(i) Aij , b ≈ a|Aij(i) b⊥ = 0 on Bi0 (3) If j(i) = [ln τ/2 + 2] (which corresponds to τ−1 . εi) we set Bi0 = [ln τ/2]+2⋃ Aij, b = 1, b⊥ = 0 on Bi0. Heuristically the definition of the Bij partition is motivated by the fact that in regions Aij with j < j(i) the weight φ is ineffective, i.e. it changes by at most O(1). Thus the convexity there is useless, and instead we rely directly on localized bounds for the Hermite operator. Since the εij are slowly varying b . a and b⊥ . a⊥ and we may replace the a’s by b’s in the above proposition. To provide some bounds on the size of b and b⊥ we introduce a function 1 ≤ r(s) ≤ τ 12 which is smooth and slowly varying on the unit scale in s so r(s) ≈ ej(i) s ∈ [i, i+ 1] This describes the region where b is tapered off and b⊥ = 0. Precisely, consider two cases corresponding to the three cases above. (1) If r(s) ≈ 1 then we have the bounds Mτ(1 + r)− 2 . b4(r, s) .Mτ(1 + r)−1 b4⊥(r, s) .Mτ(1 + r) (6.27) where the parameter M ≥ 1 is defined by M ≈ ε(s)τ 12 . (23) If r(s) ≫ 1 then τr(s)− 2 (r(s) + r)− 2 . b4(r, s) . τr(s)−1(r(s) + r)−1 b4⊥(r, s) . τr(s) −1(r(s) + r)−1 (6.28) with approximate equality when r . r(s) and approximate equality on the right when r = τ By slightly changing b and b⊥ we may and do assume that the functions b and b⊥ are smooth with controlled derivatives. Thus b and b⊥ are smooth on the unit scale in s and on the dyadic scale in y, and their derivatives satisfy the bounds (6.29) |bs|+ (rs + r)|br|+ (rs + r)2|brr| . b, r2 < 9τ (6.30) |b⊥s|+ (rs + r)|b⊥r|+ (rs + r)2|b⊥rr| . b⊥ + b r2 < 9τ In addition we have (6.31) supp b⊥r ⊂ {r > rs} Using the functions b and b⊥ we define the Banach space X 2 with norm = ‖bv‖2L2 + τ−1/2‖b⊥D Then the symmetrized version of Proposition 6.5 has the form Proposition 6.7. Assume that the coefficients of P satisfy (6.22). Let ψ be as in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman estimate holds for all functions u supported in {y ≤ 9τ}: (6.32) ‖eψ(s,y)u‖X02 . ‖e ψ(s,y)Pu‖(X02 )∗ Proof. Conjugating with respect to the exponential weight, the bound (6.32) is rewritten in the form (6.33) ‖v‖X02 . ‖Pψv‖(X02 )∗ Observing that D2⊥ = −y−2∆Sn−1 we introduce the operator Q = Q(|y|, (−∆Sn−1) 2 ), q(r, λ) = (b4(r) + r−2τ−1b4⊥(r)λ Then the inequality (6.33) can be written as ‖Qv‖L2 . ‖Q−1Pψv‖L2 whereas inequality (6.23) implies ‖Q2w‖L2 . ‖Pψw‖L2. Hence it is natural to apply (6.23) to the function w = Q−1v, which solves Pψw = Q −1Pψv +Q −1[Q,Pψ]w Thus (6.33) would follow provided that the commutator term is small, ‖Q−1[Q,Pψ]w‖L2 ≪ ‖b2w‖L2 + τ−1/2‖b2∇w‖L2 + τ−1‖b2D2yw‖L2 Unfortunately a direct computation shows that the smallness fails when j is close to j(i) even in the flat case, i.e. with Pψ replaced by P0,ψ = ∂s −∆+ y2 − ψs − ψ2y To remedy this we introduce an additional small parameter δ and use it to define a modification Qδ of Q. We modify r(s) to rδ(s) defined by rδ(s) −2 = δ8r(s)−2 + δ2τ−1 and use it to define the function bδ(r, s) 4 = δ−12τ(r2 + rδ(s) We can still compare it with b, bδ(r, s) 4 . δ−4b(r, s)4 Its usefulness lies in the fact that it is larger than b exactly in the region where the commutator term above is not small. The modification Qδ of Q has symbol qδ(r, s, λ) = q(r, s, λ) + bδ(r, s) = (b 4(r, s) + r−2τ−1b4⊥(r, s)λ 4 + bδ(r, s) which satisfies (6.34) q ≤ qδ . δ−1q We claim that it satisfies the bound (6.35) ‖Q−1δ [Qδ, Pψ]w‖L2 . (δ + c(δ)δ1) j=0,1,2 2‖b2Djw‖L2 Suppose this is true. Then we fix δ sufficiently small, and for δ1 small enough we apply (6.23) to w = Q−1δ v. By (6.35) he commutator term in the equation for w can be neglected, and we obtain ‖Q2w‖L2 . ‖Q−1δ Pψv‖L2 which by (6.34) implies that ‖Qv‖L2 . δ−1‖Q−1Pψv‖L2 It remains to prove (6.35). I. We first calculate the commutator in the flat case, i.e. with Pψ replaced by P0,ψ. Due to the spherical symmetry the only contribution comes from the radial part of the Laplacian and the s derivative. Hence using polar coordinates we compute Q−1δ [Qδ, P0,ψ] = Q Qδrr + Qδr + 2Qδr∂r −Qδs Then is suffices to verify that on the symbol level we have (6.36) |qδrr|+ r−1|qδr|+ |qδs|+ τ 1/2|qδr| . δqδb2(1 + τ−1r−2λ2) I.(1). We begin with the q component of qδ. Using (6.29) and (6.30) one obtains |qrr|+ r−1|qr|+ |qs|+ τ 1/2|qr| . (r(s) + r)−1τ Thus it remains to show that (r(s) + r)−1τ 2 q . δqδb 2(1 + τ−1r−2λ2) Optimizing with respect to λ it suffices to consider the cases λ = 0 respectively λ = rτ 2 , where the above inequality becomes (r(s) + r)−1τ 2 (b+ b⊥) . δ(bδ + b)b or equivalently (b+ b⊥)b 1 . δ(bδ + b)b which is true since by (6.27) and (6.28) we have b⊥b1 . b 1 while b1 . δbδ. I.(2). Next we consider the bδ component of qδ, for which it suffices to prove (6.37) |bδrr|+ r−1|bδr|+ |bδs|+ τ 1/2|bδr| . δbδb2 I.(2).(a). For the s derivative we compute (r2δ(s))s r2 + r2δ(s) therefore we want to show that (r2δ(s))s . δ(r 2 + r2δ(s))b We optimize the right hand side with respect to r. The minimum is attained when r2 = min{r2δ(s), τ}. We need to consider two cases: I.(2).(a).(i). If rδ(s) . τ 2 then r2δ(s) ≈ δ−8r2(s) and τ > δ−8r2(s). Hence using the estimate from below in (6.27) and (6.28) we obtain b4(rδ) & δ Then the above bound for r = rδ follows since |(rδ(s)−2)s| . rδ(s)−2. I.(2).(a).(ii). If rδ(s) & τ 2 then by6 (6.28) we evaluate b2(τ 2 ) ≈ τ 14 r(s)− 12 . Then the above bound becomes δ8(r(s)−2)s . δr 4 r(s)− Since |(r(s)−2)s| . r(s)−2 it suffices to show that δ8r(s)−2 . δr−2δ τ 4 r(s)− The worst case is r(s)2 = δ6τ , rδ(s) = δ −2τ when it is verified directly. I.(2).(b). For the r derivatives the last term is the worst. Since r2 + r2δ(s) we want to show that 2 r . δ(r2 + r2δ(s))b Optimizing with respect to r the worst case is when r2 = min{r2δ(s), τ}. I.(2).(b).(i). If r2δ(s) . τ then rδ(s) ≈ δ−4r(s) therefore for r = rδ(s) the above relation becomes 2 . δ−3r(s)b2(rδ(s)) which follows from the bound from below in (6.27) and (6.28). I.(2).(b).(ii). If r2δ(s) & τ then as before we evaluate b 2 ) ≈ τ 14 r(s)− 12 and rewrite the above bound as τ . δr2δ(s)τ 4 r(s)− The right hand side is smallest either when rδ(s) = τ 2 and r(s) = δ4τ when rδ(s) = δ 2 and r(s) = τ 2 . In both cases the inequality is easily verified. II. Now we deal with the general case, which we treat as a perturbation. Since we do not care about the dependence of the constants on δ to keep the notations simple we include bδ in b and work with Q instead of Qδ. Thus in the computations below we allow the implicit constants to depend on δ. Suppose that A is a pseudodifferential operator of order 1 and let η be any Lipschitz function. Then (6.38) ‖[A, η]f‖L2 . ‖f‖L2. We write q(λ) = b+ (b4 + r−2b4⊥λ 4 − b =: b+ q1(λ). Even though q1 has order , we treat it as an operator of order 1 and estimate 〈λ〉k−1|q1(k)(λ)| . 6the equality holds on the right when r2 = τ Hence, for each r we obtain the bound on the sphere Sn−1 (6.39) ‖[Q, η]f‖L2 . ‖η‖Lip(Sn−1)‖f‖L2. As a consequence, it also follows that (6.40) ‖[Q, η∇jθ]f‖L2 . ‖η‖Lip(Sn−1)‖∇jθf‖L2. where ∇θ stands for the vector fields xi∂j − xj∂i generating the tangent space of Sn−1. To use these bounds we write the difference Pψ − P 0ψ in polar coordinates, Pψ − P 0ψ = P 0θ ∂2r + P 1θ ∂r + P 2θ where P θ are spherical differential operators of order j. Modulo zero homoge- neous coefficients which are polynomials in xr−1 we can write P 0θ = d, P θ = dr −1∇θ + τ 2d+ dy P 2θ = dr −2∇2θ + (τ 2d+ dy)r −1∇θ + τd + τ where d stands for coefficients satisfying (6.22). In the support of b⊥ we have a ≈ b therefore our regularity assumptions on d show that for fixed r we have ‖d‖L∞ + ‖d‖Lip(Sn−1) . δb4τ−1 The coefficients involving dy satisfy better Lipschitz bounds and are neglected in the sequel. We expand the commutator [Q,Pψ] = j=0,1,2 r + P θ (Qrr + 2Qr∂r) + P Using the trivial b−1 bound for Q−1 and (6.17), (6.40) we estimate the first term, j=0,1,2 ‖Q−1[Q,P jθ ]∂ r w‖L2 . b−1δb4τ−1 j=0,1,2 2‖Djw‖L2 This is bounded by the right hand side in (6.35) since b2⊥ . rτ The second term in the commutator is estimated by ‖Q−1P 0θ (Qrr + 2Qr∂r)w‖L2 . δ(‖Qrrw‖L2 + ‖Qr∂rw‖L2) This is bounded by the right hand side in (6.35) provided that |qrr|+ τ− 2 |qr| . b2(1 + τ− 2 r−1λ) which follows from (6.36). The third term in the commutator is treated simi- larly. This concludes the proof of the proposition. � To conclude our study of the L2 Carleman estimates we need to also pay some attention to elliptic estimates. The conjugated operator Pψ is elliptic in the region {y2+ ξ2 ≥ 4τ}. Precisely, in this region we have the symbol bound |Lrψ(s, y, ξ)| & y2 + ξ2 Consequently, we can improve our estimates in this region. We consider a smooth symbol ae(y, ξ) with the following properties supp ae ⊂ {y2 + ξ2 ≥ 8τ} ae(y, ξ) = (y 2 + ξ2) 2 in {y2 + ξ2 ≥ 9τ} We define the space X2 with norm (6.41) ‖v‖2X2 = ‖v‖ + ‖awe (y,D)v‖2 The dual space X∗2 has norm (6.42) ‖f‖2X∗2 = inf{‖f1‖ (X02 ) ∗ + ‖f2‖2; f = f1 + awe (y,D)f2} We note that due to the elliptic bound for high frequencies, we also have the dual bounds (6.43) τ− 2‖bDv‖ . ‖v‖X2 , ‖∇f‖X∗2 . ‖b Then our final L2 Carleman estimate is Theorem 5. Assume that the coefficients of P satisfy (6.22). Let ψ be as in (6.9) with h, φ as in Lemmas 6.2,6.3. Then the following L2 Carleman estimate holds for all functions u for which the right hand side is finite: (6.44) ‖eψ(s,y)u‖X2 . ‖eψ(s,y)Pu‖X∗2 Proof. We first prove the result using the stronger assumption (6.22) on the coefficients. After conjugation we have to show that (6.45) ‖v‖X02 . ‖Pψv‖X∗2 We consider two overlapping smooth cutoff symbols χi = χi(y 2 + ξ2) and χe = χε(y 2+ξ2). The interior one χi is supported in {y2+ξ2 ≤ 7τ} and equals 1 in {y2 + ξ2 ≥ 6τ}. The exterior one χe is supported in {y2 + ξ2 ≥ 4τ} and equals 1 in {y2 + ξ2 ≤ 5τ}. We need the following bounds for χi and χe: Lemma 6.8. a) The operator χi(x,D) satisfies the bound (6.46) ‖χi(x,D)f‖(X02 )∗ . ‖f‖X∗2 b) The operators χi(x,D) and χ e (x,D) satisfy the following commutator estimates: (6.47) ‖b−1[χi(x,D), Pψ]v‖ . τ− 4‖bv‖+ ‖χev‖ (6.48) ‖[χwe , Pψ]v‖ . τ 8‖bv‖ Proof. a) By duality the bound (6.46) is equivalent to ‖χi(x,D)v‖X2 . ‖v‖X02 We have ‖ae(x,D)χi(x,D)v‖ . τ−N‖f‖ since the supports of the symbols (1 − χe(x, ξ)) and ae(x, ξ) are O(τ 2 ) sepa- rated. Then it remains to show that ‖χi(x,D)v‖X02 . ‖v‖X02 which is fairly straightforward and is left for the reader. b) We now consider the bound (6.47). Commute first χi with ∂s+H−h′(s). We have [χi(x,D), ∂s +H − h′(s)] = [χi(x,D), H ] Since χe = 1 in the support of ∇x,ξχi and the Poisson bracket of χi and x2+ξ2 vanishes, by standard pdo calculus we obtain ‖[χi(x,D), ∂s +H − h′(s)]v‖ . ‖χev‖+ τ−N‖v‖ The difference Pψ − (∂s +H − h′(s)) can be expressed in the form Pψ − (∂s +H − h′(s)) = ∂g∂ + τ 2 (g∂ + ∂g) + τg where the function g satisfies the bounds |g|+ 〈y〉|gy|+ 〈y〉|gyy| . εi These lead to an estimate for fixed s ∈ [i, i+ 1], ‖b−1[χi(x,D), Pψ − (∂s +H − h′(s))]v‖ . εiτ 2‖〈y〉−1b−1v‖ Then (6.47) follows since 2 〈y〉−1 . τ− Finally, the proof of the estimate (6.48) is similar but simpler. We continue with the proof of the proposition. For the nonelliptic part we apply (6.32) to the function χi(x,D)v which is supported in {y2 < 9τ}. This gives ‖χi(x,D)v‖X02 . ‖χi(x,D)Pψv‖(X02 )∗ + ‖[χi(x,D), Pψ]v‖L2 For the first term on the right we use the bound (6.46) while for the second we use (6.47). This yields (6.49) ‖χi(x,D))v‖X02 . ‖Pψv‖X∗2 + τ 4‖bv‖+ ‖χev‖ On the other hand for the estimate in the elliptic region we compute (6.50) 〈(χwe )2v, Pψv〉 = 〈χwe v, Lrψχwe v〉+ 〈χwe v, [χwe , P rψ]v〉 For the first term we split Lrψ into H−h′ plus a perturbation. Using pointwise bounds for the coefficients of Pφ we obtain |Lrψv − (H − h′)v| . δ1((τ + y2)|v|+ τ 2 |Dv|+ |D2v|) which shows that 〈χwe v, Lrψχwe v〉 = 〈χwe v, (H − h′)χwe v〉+O(δ1〈χwe v, (H + τ)χwe v〉) The symbol of H − h′ is elliptic in the support of χe, therefore a standard elliptic argument yields 〈χwe v, (H + τ)χwe v〉 . 〈χwe v, (H − h′)χwe v〉+ Cτ−N‖v‖2 for a large constant C. This further gives 〈χwe v, (H + τ)χwe v〉 . 〈χwe v, Lrψχwe v〉+ Cτ−N‖v‖2 Returning to (6.50), we obtain c〈χwe v, (H + τ)χwe v〉 ≤ −〈(χwe )2v, Pψv〉+ 〈χwe v, [χwe , P rψ]v〉+ Cτ−N‖v‖2 We use (6.48) and then the Cauchy-Schwartz inequality to obtain 〈χwe v, (H + τ)χwe v〉 . ‖(H + τ)− 2Pψv‖2 + τ− 4‖bv‖2 The first term on the right is properly controlled due to the straightforward estimate ‖(H + τ)− 2f‖ . ‖f‖X∗2 Hence combining the above inequality with (6.49) we obtain ‖χ(ix,D))v‖X02 + ‖(H + τ) 2χwe (x,D)v‖ . ‖Pψv‖X∗2 + τ 4‖bv‖+ ‖χwe (x,D)v‖ The last two terms on the right are negligible compared to the left hand side, therefore we obtain (6.51) ‖v‖X02 + ‖(H + τ) 2χwe (x,D)v‖ . ‖Pψv‖X∗2 Then (6.45) follows since χe = 1 in the support of ae. It remains to show that the assumption (6.22) on the coefficients for (6.7) can be replaced by the weaker condition (1.15). This is a direct consequence of (6.43) combined with the following regularization result: Lemma 6.9. Let d be a function which satisfies (1.15). Then there is an approximation g1 of it satisfying (6.22) so that |g − g1| . b2τ−1 Proof. First we transfer (1.15) to the (s, y) coordinates. A short computation yields the equivalent form (6.52) ‖d‖L∞(Aij) + ej‖d‖Lipy(Aij) + ‖g‖Cmijt (Aij) . εij where the new continuity modulus m̃ij is given by m̃ij(ρ) = ρ+ e Within Aij we regularize d in y on the δy = τ 2 scale and in s on the δs = e 4 scale, d1 = S (Dy)S (Ds)d These localized regularizations are assembled together using a partition of unit corresponding to Aij. In Aij we compute |d− d1| . εij(e−jδy +mij(δs)) ≈ εij(e−jτ− 2 + e− 4 ) . ε 4 . b2τ−1 while |∂sg1| . εij mij(δs) ≈ εije−jτ The bounds for higher order derivatives of g1 follow trivially due to the fre- quency localization. � 7. Lp Carleman estimates for variable coefficient operators The variable coefficient counterpart of Proposition 4 uses the more convex weights constructed in Section 6. For convenience we write it in the (s, y) coordinates. Let τ >> 1 and B(τ) be as in (1.25),(1.26) and (1.27). We define the function space X through its norm (7.1) ‖v‖X := ‖v‖X2 + ‖v‖l2(B(τ);L∞t L2x) + ‖v‖l2(B(τ);LptLqx) where (p, q) is an arbitrary Strichartz pair, with X2 as defined in (6.41). Its (pre)dual space has the norm ‖f‖X∗ = inf f=f1+f2+f3 ‖f1‖X∗2 + ‖f2‖L1L2 + ‖f3‖Lp′Lq′(7.2) Then we have the following improvement of Theorem 5. Theorem 6. There exists ψ as in (6.9) with h and φ as in Lemma 6.2 and 6.3. Then the following estimate holds for all compactly supported sufficiently regular functions u. (7.3) ‖eψu‖X . ‖eψP̃ u‖X∗ The relation between ψ and the partition B(τ) remains a bit mysterious at this level. If we replace it by the empty partition then the statement remains true for all ψ with h and φ as in Lemma 6.2. The same is true for a partition into time slices of size 1. The convexity properties of φ allow a localization to the finer partition Bij as in 6.6 (and, as we shall soon see, to an even finer partition). q It is possible to choose φ and h so that the partition (Bij) is finer than the one defined by B(τ). We assume in the sequel that ψ has been chosen with these properties. Proof. As usual this is equivalent to proving a bound from below for the con- jugated operator, (7.4) ‖v‖X . ‖Pψv‖X∗ The main step in the proof is to produce a parametrix for Pψ. The key point is that the parametrix is allowed to have a fairly large L2 error. This is because L2 errors can be handled by Theorem 5. The advantage in having a large L2 error is that it permits to localize the parametrix construction to relatively small sets, on which we can freeze the coefficients and eventually reduce the problem to the case of the Hermite operator. The properties of the parametrix are summarized in the following Proposition 7.1. a) Under the assumptions of the theorem there exists a parametrix T for Pψ with the following properties: (7.5) ‖Tf‖X . ‖f‖X∗ (7.6) ‖PψTf − f‖X∗2 . ‖f‖X∗ b) The same result holds with Pψ replaced by P We first use the proposition to conclude the proof of the Theorem. Let Pψv = f + g, ‖f‖X∗2 + ‖g‖L1L2+Lp′Lq′ ≈ ‖Pψv‖X∗ With T as in part (a) of the proposition we set w = v − Tg, Pψw = f + g − PψTg By (7.5) we can bound Tg in X , therefore it suffices to bound w in X . On the other hand by (7.6) we obtain ‖Pψw‖X∗2 . ‖f‖X∗2 + ‖g − PψTg‖X∗2 . ‖Pψv‖X∗ It remains to show that ‖w‖X . ‖Pψw‖X∗2 By Theorem 5 we can estimate the X2 norm of w and replace this with the weaker bound (7.7) ‖w‖L∞L2∩LpLq . ‖w‖X2 + ‖Pψw‖X∗2 This is proved using a duality argument and the parametrix T for P ∗ψ given by part (b) of the proposition. For f ∈ X∗ we write 〈w, f〉 =〈w, P ∗ψTf〉+ 〈w, f − P ∗ψTf〉 =〈Pψw, Tf〉+ 〈w, f − P ∗ψTf〉 Using both (7.5) and (7.6) with Pψ replaced by P ψ we obtain |〈w, f〉| . (‖Pψw‖X∗2 + ‖w‖X2)‖f‖X∗ and (7.7) follows. This concludes the proof of Theorem 6. � It remains to prove the Proposition 7.1. Proof of Proposition 7.1. The strategy for the proof is simple: On sufficiently small sets we can approximate the problem by one with constant coefficients and the properties of the parametrix follow from Section 4. We use a partition of unity to construct a global parametrix from local ones. We obtain L2 errors (1) Commuting cutoff functions with the operator. Hence the partition has to be sufficiently coarse. (2) Approximating the variable coefficient operator by constant coefficient operators. Hence the partition has to be sufficiently fine. To elaborate on this we define the notion of a local parametrix: Definition 7.2 (Local parametrix). Given a convex set B we call T a (B-) local parametrix for Pψ if for all f supported in B (7.8) ‖Tf‖X . ‖f‖X∗ , (7.9) ‖PψTf − f‖X∗2 . ‖f‖X∗ and Tf is supported in 2B. If T is a parametrix and η is supported on 2B, η = 1 on B then ψT is a local parametrix, but with constants depending on the commutator of Pψ and η. Vice verse, if (Bj) is a covering, (ηj) a subordinate partition of 1 and Tj are local parametrices then is a global parametrix, because (7.8) is obtained by summation, and Tj(ηjf)− f = (PψTjηjf − ηjf provided ∑ ‖ηjf‖2X∗ . ‖f‖2X∗ and its adjoint ‖u‖2X . ‖ηju‖2X . This is obvious for the L2 part and has to be checked for the other part. This strategy of constructing local parametrices leads, if it is possible, to estimates which are stronger than in Proposition 7.1 and Theorem 6, because we may replace the function space X by l2X(Bj) respectively. l 2X∗(Bj). In the first part of the proof we study the localization, and in the second part we provide the local parametrices. 7.1. Localization scales. Here we introduce a localization scale which is finer than the Bij partition of the space, and show that it suffices to construct the parametrix in each of these smaller sets. Precisely, the sets Bkij introduced below are the smallest sets to which one can localize the L2 estimates for the operator Pψ. The choice of their size is not yet apparent at this point, but will become clear in the very last step of the proof, where we estimate the commutator of Pψ with cutoff functions on such sets. We consider three cases depending on the size of εi. (1) If εi ≤ τ−1 then we use Bi0 as it is. (2) If τ−1 ≤ εi ≤ τ− 2 then we partition the set Bi0 into time slices B i0 of thickness δs = b−2i0 . (3) If τ−1 ≤ εi and j 6= 0 then we partition Bij into subsets Bkij which have the time scale, radial scale and angular scale given by δs = b−2ij , δy = τ 2 b−2ij , δy ⊥ = τ 2 b−2ij,⊥ This gives a decomposition of the space R× Rn = We also consider a subordinated partition of unity Suppose that in each set Bkij we have a parametrix T ij satisfying (7.5) and (7.6). Then we define the global parametrix T by T kijχ We have by an iterated application of Minkowski’s inequality ‖χkijf‖l2(L1L2+Lp′Lq′) . ‖f‖L1L2+Lp′Lq′ and the dual bound T kijχ ijf‖L∞L2∩LpLq . ‖T kijχkijf‖l2(L∞L2∩LpLq). Hence (7.5) for T would follow if we proved that vkij‖X2 . ‖vkij‖l2X2 where vkij = T ijf are supported in B ij . This is trivial by orthogonality for the L2 component of the X2 norm. It is also straightforward for the elliptic part since the kernel of the operator in the following sense: Let χ0 ∈ C∞(R) be supported in [−3/2, 3/2], identically 1 in [−1, 1]. We define χe(x, ξ) = χ0(5− (x2 + ξ2)/τ 2). Then χwe v ij‖X2 . ‖χwe vkij‖l2X2 and the adjoint estimate holds since the kernel of χwe is rapidly decreasing beyond the τ− 2 scale, which is much smaller than the smallest possible spatial size for Bkij, namely δy & τ It remains to consider the angular part of the X2 norm, which is best de- scribed using the spherical multiplier Q appearing in Proposition 6.7. The symbol of Q is smooth with respect to λ on the τ 2 rb2b−2⊥ scale therefore its kernel is rapidly decaying on the angular scale δθ = τ− 2 r−1b−2b2⊥ which cor- responds to δy⊥ = τ 2 b2⊥b −2. But by (3) and (6.5) this can be no larger than 8 which is again much smaller than the smallest possible spatial size for Bkij . Thus orthogonality arguments still apply. Then we have [P0, χ ij]u = 2χijy uy + χ yyu+ χ We claim that the right hand side is negligible in the estimate. For this it suffices to verify that |χijy | ≪ τ− 2 b2ij , |χijyy| ≪ b2ij , |χijs | ≪ b2ij The last relation is trivial. For the first two we consider three cases. (1) If j = 0 and εj < Cτ −1 then we need no spatial truncation. We are allowed to truncate at |y| > Cτ 12 to separate the elliptic region, though. (2) If j = 0 and εj > Cτ −1 then |χijy | . e−j(i), |χijyy| . e−2j(i) while b4i0 = a ij(i) = εij(i)τ 2 e−j(i) ≈ Ce−2j(i)τ (3) Otherwise, |χijy | . e−j , |χijyy| . e−2j while b4ij = a ij = εijτ 2 e−j & Ce−2j(i)τ The results can be summarized by saying that it suffices to construct local parametrices in the sets Bkij. 7.2. Freezing coefficients. Our first observation is that restricting the result in the proposition to a single region Bkij allows us to freeze the weights b, b⊥ in the X2 norms. Next we are interested in freezing the coefficients of P . We consider the same three cases as above: The case εi ≤ τ−1. In this case we are localized to Bi0 = [i, i+ 1]× Rn and we have bi0 ≈ 1, εij . τ−1 By (6.2) the second relation leads to |d| . τ−1 Then using also (6.43) we can estimate the terms involving d in the expression (6.22) for Pψ, (7.10) ‖∂d∂v‖X∗2 + τ‖dv‖X∗2 + τ 2‖(d∂ + ∂d)v‖X∗2 . ‖v‖X2 Hence without any restriction of generality we can assume that d = 0 in Pψ, which corresponds to taking g = In. We also observe that in this case we have |φ| . 1, |φy| . τ− Then we can also drop the φ component of ψ. Finally, since |hss| . 1 we can replace h by its linearization at some point in the corresponding s region. Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs. We note that the separation of τ from integers is no longer needed due to the localization to unit s intervals. The case τ−1 ≤ εi. In this case we are localized to a region of the form Bki0 = [s0, s0 + e j(i)τ− 2 ]×B(0, ej(i)) and we have b4i0 ≈ τe−2j(i), εij . e−j(i)τ− The second relation leads to |d| . e−j(i)τ− Then (7.10) is still valid, so we can assume again that d = 0 in Pψ. We also observe that in this case we have |φ| . 1, |φy| . e−j(i) Then we can also drop the φ component of ψ. Finally, since |hss| . e−j(i)τ− we can replace h by its linearization at some point in the corresponding s region. Conclusion: It suffices to prove the result when d = 0, ψ(y, s) = τs. The case τ−1 ≤ εi. In this case we are localized to a region of the form Bkij = [s0, s0 + τ ij ]× B(y0, τ− ij ), |y0| ≈ ej and we have b2ij ≈ ε Using (6.2) it follows that |g(s, y)− g(s0, y0)| . ε Arguing as before, this allows us to freeze d within Bkij. However, we note that we are no longer allowed to replace d by 0. Next we turn our attention to the weight function ψ. First we have |hss| . εjτ . εijτ 2 e−j which allows us to replace h by its linearization in s at s0. Secondly, we claim that we can replace φ by its linearization at y0. In the radial direction we have weaker localization but a stronger bound |φrr| . εijτ 2 e−j In the transversal direction we have better localization but a weaker bound, |φyy| . εiτ 2 e−j . The first bound allows us to obtain the relation |φ2y(y, s)− φ2y(y0, s0)| . ε 2 ≈ b2ij Using also the second bound we can write (φy(y, s)− φy(y0, s0))∂y = νr∂r + ν⊥∂⊥ where the coefficients νr and ν⊥ are smooth on the B ij scale and satisfy the bounds |νr| . τ− 2 b2ij , |νr| . τ− 2 b2ij,⊥ Conclusion: It suffices to prove the result when d = g(s0, y0), ψ(y, s) = τs + cy, |c| . εiτ Additional simplification in the highly localized case. Given the above simpli- fications we need to work with a constant coefficient operator Pψ which has the form Pψ = −∂t +H − τ + ∂d∂ + c∂, |d| . εij, |c| ≤ εi We diagonalize the second order part with a linear change of variables to obtain Pψ = −∂t +H − τ + c∂ +O(εij)y2 We can freeze the last term at y0 and add it into τ . To deal with c we make the change of variable y → y − (s− s0)c Then our operator becomes P̃ψ = −∂t +∆− (y − c(s− s0))2 + τ and the s− s0 terms are negligible due to the s localization. Conclusion: We can assume without any restriction in generality that g = In and ψ = τs. 7.3. The localized parametrix. We begin with the global parametrix K constructed in Section 5. Then we define the parametrix TB in B by TB = χ2BK, B = B and show that it satisfies (7.8) and (7.9). The Lp part of (7.8) follows directly from (5.4). It remains to prove the X2 part, ‖TBf‖X2 . ‖f‖L1L2+Lp′Lq′ The elliptic part of the X2 bound, namely ‖awe (x,D)χ2BKf‖L2 . ‖f‖L1L2+Lp′Lq′ , is obtained by an argument which is similar to the one beginning with (6.50). For the rest we consider two cases. i) If j = 0 then B is a ball, and we can use (5.4) directly with R = d. ii) If j > 0 then B is contained in a sector B ⊂ BR,d but may be shorter than R. This is why we can use (5.4) for the angular part of the X2 norm, but not for the L2 part. However, the L2 part can be always obtained by taking advantage of the time localization, ‖bijTBf‖L2 . ‖TBf‖L∞L2 . ‖f‖L1L2+Lp′Lq′ It remains to consider the error estimate (7.9). We have f − (∂s −H + τ)TBf = [χ2B, ∂s −H + τ ]Kf = [χ2B, ∂s −H + τ ]χ4BKf But arguing as above χ4BK0f satisfies the same X2 bound as χ2BK0f . Hence it suffices to show that [χ2B , ∂s −H + τ ] : X2 → X∗2 This is where the dimensions of the set B are essential; they are chosen to be minimal so that the above property holds. We have [χ2B, ∂s −H + τ ] = −∂sχ2B + (∂yχ2B)∂y + ∂y(∂yχ2B) = −∂sχ2B + (∂rχ2Br)∂r + ∂r(∂rχ2Br) + (∂⊥χ2By)∂⊥ + ∂⊥(∂⊥χ2B) For the first factor we use the bound |∂sχ2B| . b2ij For the radial derivatives of χ2B we combine (6.43) with |∂rχ2B| . τ− 2 b2ij Finally, for the angular derivatives we use the angular H 2 norm in X2 and the bound |∂rχ2B| . τ− 2 b2ij⊥ 8. The gradient term In this section we consider the full problem, i.e. involving also the gradient potential W . Ideally one might want to have a stronger version of Theorem 6 which includes additional bounds for the gradient, more precisely for ‖eψ(s,y)∇u‖L2 But such bounds cannot hold, for this would imply that one can improve the Lp indices in a restriction type theorem. To overcome this difficulty we proceed as in [17], using Wolff’s osculation Lemma. Wolff’s idea is that by varying the weight one can ensure concentration in a sufficiently small set, in which the gradient potential term is only as strong as the potential term. Thus we still obtain a one parameter family of Carleman estimates, but with the weight depending not only on the parameter but also on the function we apply the estimate to. Given a gradient potential W satisfying (1.19), we first readjust the param- eters εij, εi constructed in Lemma 6.1 in order to insure that we have the additional condition ‖W‖Ln+2(Aτi ) ≪ εi Then we begin with the spherically symmetric weights ψ constructed in Sec- tion 6 and modify them as follows: (8.1) Ψ(s, y) = ψ(s, y) + δk(s, y) where the perturbation k is supported in {|y| ≤ 9τ} and is subject to the following conditions: (8.2) |∂αs ∂βy ∂ ⊥k(s, y)| . εiτ 2 s ∈ [i, i+ 1] Here δ is a sufficiently small parameter. In order to prove the strong unique continuation result in the presence of the gradient potential W we need the following modification of Theorem (6): Theorem 7. Assume that (1.15) holds. Then for each τ > 0 and W subject ‖W‖Ln+2(Aτi ) ≤ εi and each function u vanishing of infinite order at (0, 0) and ∞ there exists a perturbation k as in (8.2) so that (8.3) ‖eΨu‖X+‖eΨW∇u‖X∗+‖eΨ∇(Wu)‖X∗+τ 2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗ Here and in the sequel we will omit indices for W . After returning to the (x, t) coordinates and taking (1.19) into account this implies Theorem 3. The reader should note that the choice of φ depends on both u and W . This is essential since for fixed φ (8.3) cannot hold uniformly for all u and W . Proof. Up to a point the proof follows the steps which were discussed in detail before. We outline the main steps: STEP 1: Show that the L2 Carleman estimate (6.23) holds with ψ replaced by Ψ for all perturbations k as in (8.2). The new conjugated operator PΨ is obtained from Pψ after conjugating with respect to the weight e k(y,s). This adds a few extra components to the selfadjoint and skewadjoint parts, LrΨ = L ψ + k y + ks + 2ky(ψy + d) LiΨ = L ψ − ky(1 + d)∂ − ∂ky(1 + d) Observing that we can write k2y + ks + 2ky(ψy + d) = τd, ky(1 + d) = τ with d as in (6.22) we conclude that the conjugated operator PΨ retains the same form as Pψ, therefore the proof of (6.23) rests unchanged. STEP 2: Show that the symmetric L2 Carleman estimate (6.23) holds with ψ replaced by Ψ for all perturbations k as in (8.2). Since PΨ has the same form as Pψ, this argument is identical. STEP 3: Show that the symmetric mixed L2 ∩ Lp Carleman estimate in Theorem 6 holds with ψ replaced by Ψ for all perturbations k as in (8.2). Since PΨ has the same form as Pψ, this argument is also identical. STEP 4: Decompose W into a low and a high Hermite-frequency part, W =Wlow +Whigh, Wlow = χ i (x,D)W where the smooth symbol χ1i (x, ξ) is supported in {x2 + ξ2 ≤ 81τ} and equals 1 in the region {x2 + ξ2 ≤ 64τ}. Then we show that the high frequency part of W satisfies the desired estimates for all perturbations k as in (8.2), namely (8.4) ‖eΨWhigh∇v‖X∗ +‖∇Whighv‖X∗ + τ 2‖eΨWhighv‖X∗ . ‖eΨu‖X‖W‖Ln+2 After conjugation this becomes ‖Whigh∇v‖X∗ + ‖∇Whighv‖X∗ + τ 2‖Whighv‖X∗ . ‖v‖X‖W‖Ln+2 We only consider the first term on the left. The second one is equivalent by duality, and the third one is similar but simpler. We divide v into two components, v = (1− χ1e)v + χ1ev where the smooth symbol χ1e(x, ξ) is supported in {x2 + ξ2 ≥ 9τ} and equals 1 in the region {x2 + ξ2 ≥ 10τ}. For the high frequency component of v we use the H1 part of the X2 norm to estimate ‖Whigh∇χ1ev‖ 2(n+2) . ‖Whigh‖Ln+2‖∇χ1ev‖L2 . ‖W‖Ln+2‖v‖X For the low frequency component of v it is still possible to estimate directly the high frequency of the output, ‖χ1e(Whigh∇(1− χ1e)v)‖H−1 .τ− 2‖Whigh∇(1− χ1e)v‖L2 .‖Whigh‖Ln+2τ− 2‖∇(1− χ1e)v‖ 2(n+2) .‖W‖Ln+2‖v‖ 2(n+2) Finally, the last remaining part has a much better L2 estimate, ‖(1− χ1e)(Whigh∇(1− χ1e)v)‖ . τ−N‖W‖Ln+2‖v‖ which is due to the unbalanced frequency localizations of the two factors. Due to the estimate (8.4), it suffices to prove (8.3) withW replaced byWlow. This allows us to replace the term ∇(Wlowv) by ∇(Wlowv) =Wlow∇v + (∇Wlow)v where we can estimate ‖∇Wlow‖Ln+2 . τ 2‖W‖Ln+2 Hence without any restriction in generality we can drop the third term in (8.3) and show that we can choose the perturbation k so that (8.5) ‖eΨW∇u‖X∗ + τ 2‖eΨWu‖X∗ . ‖eΨ(x)P̃ u‖X∗ STEP 5: Show that, given u and W , we can choose the perturbation k so that (8.5) holds. At this stage we no longer need the full X∗ norm for the W terms, it suffices instead to consider the L 2(n+2) n+4 norm. Begin with the unperturbed integral Fψdxdt, Fψ = |eψW∇u| 2(n+2) n+4 + |τ 2 eψWu| 2(n+2) We can select a subset I of R consisting of time intervals of length 1 with unit separation at least 8 so that Fψdxdt . Fψdxdt By a small abuse of notation we label Ii, Ii ⊂ [i− 1, i+ 1] We define a family of perturbations k depending on parameters bi, σi by k(y, s) = 100χ3τIi + χ2Iiχ|y|2≤τ (biy + σi(s− i)) |bi| ≤ τ 2 , |σi| ≤ τ Due to the choice of the intervals Ii it is easy to see that after changing the weight ψ to Ψ we retain the concentration to a dilate of I, FΨdxdt . FΨdxdt The choice of the parameters bi, σi can be made independently for each i. We consider two cases. i) Suppose εi . τ 2 . Then the choice of the parameters is irrelevant since in 3Ii we can estimate ‖eψW∇u‖ 2(n+2) 2‖eψWu‖ 2(n+2) . ‖W‖Ln+2(‖eΨ∇u‖L2 + τ 2‖eΨu‖L2) . (τ−1/2‖eΨ∇u‖L2 + ‖eΨu‖L2) . ‖u‖X2 ii) Suppose εi ≫ τ− 2 . Then we need to choose the parameters bi, σi in a favorable manner. This choice is made using Wolff’s Lemma: Lemma 8.1 (Wolff’s Lemma [29]). Let µ be a measure in Rn and B a convex set. Then one can find bk ∈ B and disjoint convex sets Ek ⊂ Rn so that the measures exbkµ are concentrated in Ek,∫ exbkdµ & exbkdµ and ∑ |Ek|−1 & |B| We apply the lemma for the measures dµi = 13IiFψ In our case we have Bi = δεi [−τ, τ ]×B(0, τ , |Bi| ≈ εn+1i τ Hence we can find parameters bki and σ i and convex sets E i ⊂ 3Ii×B(0, 3τ so that the corresponding measures FΨk are concentrated in E i with |Eki |−1 & εn+1i τ At the same time we have |W |n+2dxdt . εn+2i Hence we can choose k so that∫ |W |n+2dxdt . εiτ− 2 |Eki |−1 which by Holder’s inequality leads to (8.6) ‖W‖ 2 (Eki ) Denoting this index k by k(i) we can write ‖eΨW (∇, τ 2 )u‖ 2(n+2) . ‖eΨW (∇, τ 2 )u‖ 2(n+2) n+4 (Ii) . ‖eΨW (∇, τ 2 )u‖ 2(n+2) n+4 (E . ‖W (∇, τ 2 )(eΨu)‖ 2(n+2) n+4 (E Decomposing the function v = eΨu into low and high frequencies we further estimate ‖eΨW (∇, τ 2 )u‖ 2(n+2) . ‖W (∇, τ 2 )(1− χ1i (x,D))(eΨu)‖ 2(n+2) n+4 (Ii) + ‖W (∇, τ 2 )χ1i (x,D)(e 2(n+2) n+4 (E The first term on the right is estimated as in Step 4, ‖W (∇, τ 2 )(1− χ1i (x,D))(eΨu)‖ 2(n+2) n+4 (Ii) . ‖W‖l∞Ln+2‖(1− χ1i (x,D))(eΨu)‖H1 . ‖W‖l∞Ln+2‖eΨu‖X It is only for the second term on the right that we need to use (8.6): ‖W (∇, τ 2 )χ1i (x,D)(e 2(n+2) n+4 (E . ‖W‖ l∞i L ‖(∇, τ 2 )χ1i (x,D)(e 2(n+2) n+4 (E . ‖eΨu‖ 2(n+2) . ‖eΨu‖X The proof of the Theorem is concluded. � Appendix A. The change of coordinates Suppose that the coefficients g satisfy (1.15). In this section we verify that we can change coordinates so that (1.15) and (1.16) are both satisfied. Due to the anisotropic character of the equation we must leave the time variable unchanged and consider changes of coordinates which have the form (t, x) → (s, y), s = t, y = χ(t, x). The expression for the operator P in the new coordinates is P = ∂t + ∂kg̃ kl(t, y)∂l + d̃kl∂l where the new coefficients g̃, d̃ are computed using the chain rule, g̃kl = d̃kl = Dχ−1lm(∂ There is a price to pay for this, namely in the new coordinates we obtain lower order terms which cannot be treated perturbatively. Instead we obtain coefficients d̃k which have the same regularity and size as g1 − In. The Lipschitz condition (1.15) ensures that g has a limit at (0, 0) so we assume that g is continuous. After a linear change of coordinates we may and do choose g with g(0, 0) = In. Again by (1.15) this implies (A.1) |g(t, x)− In| ≪ 1. Proposition A.1. Let g be a metric which satisfies (1.15) with g(0, 0) = In. Then there is change of coordinates (t, y) = (t, χ(t, x)) which is close to the identity (A.2) ‖∂xχ− In‖L∞ ≪ 1 and has regularity (A.3) ‖(t+ x2)−1/2(t∂t)α((t+ x2)1/2∂x)βχ‖l1(A(τ);L∞) ≪ 1, 2 ≤ 2α + |β| ≤ 4 so that in the new coordinates both functions g̃ and d̃ satisfy (1.15), while g̃−In and d̃ satisfy (1.16). Proof. Consider the covering of the [0, 2]× B(0, 2) = ∪Aij with an associated smooth partition of unity ηij . We can assume that the functions ηij satisfy (A.4) |∂αt ∂βxηij | . cαβt−α(t + x2)− We choose the points (ti, xij) = (e −4i, e−2i+j) ∈ Aij. and insure that ηij = 1 near (ti, xij). By (1.15) we have (A.5) sup (i,j)∈A(τ) |g(ti, xij)−g(ti, xi,(j−1))|+ |g(ti, xij)−g(ti+1, x(i+1),j)| ≪ 1. Within a fixed set Aij we consider the linear map defined by the matrix χij = g −1/2(ti, xij). It transforms the coefficients at (ti, xij) to the identity and has the desired properties within Aij . We assemble the maps defined by χij using the partition of unity, χ(t, x) = ηij(t, x)χijx. ∇χ(t, x)− In = (∇ηij)χijx+ ηij(χij − In), Let (t, x) ∈ Ai0,j0. Since ∇ηij = 0 we have ∇xχ(t, x)− In = ηij(t,x)>0 ∇xηij(t, x)(χij − χi0,j0) + ηij(χij − In). The first term on the right hand is small by (A.5) (for χij) and the second one by (A.1) therefore the smallness of ∇χ− In follows. For the second order spatial derivatives we write D2xχ(t, x) = D2ηij(t, x)χijx+ 2Dxηij(t, x)χij D2xηij(t, x)(χij − χi0,j0)x+ 2Dxηij(χij − χi0,j0). Hence by (A.4) and (A.5) (again for χij) we obtain ‖(|x|2 + t)1/2D2xχ‖l1(A(τ);L∞) ≪ 1. ∂tχ(t, x) = ∂tηij(χij − χi0j0)x gives the desired bound for the time derivative. A similar computation yields the bound for the higher order derivatives in (A.3). Consider now the new metric g̃. Since both Dχ and (Dχ)−1 are Lipschitz on the dyadic scale with l1(A(τ)) summability, from (1.15) for g we easily obtain (1.15) for g̃. In addition, our construction insures that g̃(ti, xij) = In. This in turn leads to the bound ‖g̃ − In‖L∞(Aij) . ‖g̃‖Lipx(Aij) + ‖g̃‖Cmijt (Aij) which shows that (1.15) for g̃ implies (1.16) for g̃ − In. It remains to consider the lower order terms. From ∂tχ we obtain coefficients d̃ of the form d̃ = t ∂tηijχij Within Ai0,j0 this gives d̃ = t ∂tηij(t, x)(χij − χi0j0) The functions t∂tηij(t, x) are bounded and smooth on the dyadic scale, while the l1(A(τ)) summability comes from the χij−χi0j0 factor due to (A.5). Hence both (1.15) and (1.16) are satisfied. The contribution of ∂2xχ to d̃ has the form (∂xχ) −1(∂2xχ)g There is no singularity at x = 0 since χ is linear in x for x2 ≪ t. Then from (A.2) and (A.3) we obtain |d̃| . with added l1(A(τ)) summability inherited from ∂2xχ. This is better than (1.16), and in effect this term can be included inW and treated perturbatively. The bound (1.16) is also easy to obtain from the similar bounds for g and derivatives of χ. References [1] Giovanni Alessandrini and Sergio Vessella. Local behaviour of solutions to parabolic equations. Comm. Partial Differential Equations, 13(9):1041–1058, 1988. [2] Giovanni Alessandrini and Sergio Vessella. Remark on the strong unique continuation property for parabolic operators. Proc. Amer. Math. Soc., 132(2):499–501 (electronic), 2004. [3] N. Aronszajn. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9), 36:235–249, 1957. [4] N. Aronszajn, A. Krzywicki, and J. Szarski. A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat., 4:417–453 (1962), 1962. [5] T. Carleman. 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[18] Herbert Koch and Daniel Tataru. Dispersive estimates for principally normal pseudo- differential operators. Comm. Pure Appl. Math., 58(2):217–284, 2005. [19] Herbert Koch and Daniel Tataru. Lp eigenfunction bounds for the Hermite operator. Duke Math. J., 128(2):369–392, 2005. [20] Fang-Hua Lin. A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math., 43(1):127–136, 1990. [21] Sigeru Mizohata. Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., 31:219–239, 1958. [22] Chi-Cheung Poon. Unique continuation for parabolic equations. Comm. Partial Differ- ential Equations, 21(3-4):521–539, 1996. [23] Jean-Claude Saut and Bruno Scheurer. Unique continuation for some evolution equa- tions. J. Differential Equations, 66(1):118–139, 1987. [24] C. D. Sogge. A unique continuation theorem for second order parabolic differential operators. Ark. Mat., 28(1):159–182, 1990. [25] Christopher D. Sogge. 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Mathematisches Institut der Universität Bonn, Beringstr.1, 53115 Bonn, Germany E-mail address : [email protected] Department of Mathematics, University of California, Berkeley, CA 94720 E-mail address : [email protected] 1. Introduction 2. Proof of Theorem 1 and 2 3. L2 bounds in the flat case and the Hermite operator 4. Resolvent bounds for the Hermite operator 4.1. Weighted L2 bounds 4.2. The Lp bounds of the resolvent 4.3. Combining the estimates 5. Lp estimates in the flat case and parametrix bounds 6. Modified weights and pseudoconvexity 7. Lp Carleman estimates for variable coefficient operators 7.1. Localization scales. 7.2. Freezing coefficients 7.3. The localized parametrix. 8. The gradient term Appendix A. The change of coordinates References

0704.1350

Specialized computer algebra system for application in general relativity

Specialized computer algebra system for application in general relativity S.I. Tertychniy Abstract: A brief characteristic of the specialized computer algebra system GRGEC intended for symbolic computations in the field of general relativity is given. The code GRGEC constitute a full-fledged programming system intended for application in the field of the general relativity and adjacent areas of the differential geometry and the classical field theory. Written mostly in the lisp dialect known as standard lisp, it is realized, structurally, as the top layer upon the universal computer algebra system Reduce. The latter is utilized as the primary tool for execution of the general kind symbolic mathematical calculations. The code infrastructure includes, in particular, the user inter- face based on the interpreter of the so called language of problem specification which models the natural language in its simplified version adapted to the description of the notions and relationships taking place in the application field. The collection of algorithms implementing the set of data objects and the rules of operations with them models the most important notions and relationships (equations) established in the relevant areas of the physics and the geometry. One could note in this respect implementation of the calcu- lus of exterior forms, the spinor algebra tools, the major elements of the tensor calculus. (All these techniques operate with separate object compo- nents, no abstract index methods have been implemented). The application specific algorithms enable one, in particular, to handle various bases in folia- tions of exterior forms connected with the metric structure, the connection, the curvature with its irreducible constituents and invariants, the equations connecting the above objects such as Cartan equations, Bianchi equations, various algebraic identities, the field equations of the gravity theory (Ein- stein equations). The handling of a number of the classical field has been implemented including electromagnetic field, massless spinor field, massive spinor fields, massless scalar field, conformally invariant scalar field, massive scalar field and others. It is worth noting also the feasibility to manipu- late with Newman-Penrose spin coefficients, Lanczos representation of the conformal curvature, Rainich theory of the coupling of electromagnetic and gravitational fields, Killing vectors and more. GRGEC is currently available free of charge at http://grg-ec.110mb.com http://arxiv.org/abs/0704.1350v1 http://grg-ec.110mb.com

0704.1351

Rigidly rotating dust solutions depending upon harmonic functions

Rigidly rotating dust solutions depending upon harmonic functions Stefano Viaggiu Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, I-00133 Roma, Italy E-mail: [email protected] (or: [email protected]) August 4, 2021 Abstract We write down the relevant field equations for a stationary axially symmetric rigidly rotating dust source in such a way that the general solution depends upon the solution of an elliptic equation and upon harmonic functions. Starting with the dipole Bonnor solution, we built an asymptotically flat solution with two curvature singularities on the rotational axis with diverging mass. Apart from the two point singu- larities on the axis, the metric is regular everywhere. Finally, we study a non-asymptotically flat solution with NUT charge and a massless ring singularity, but with a well-defined mass-energy expression. PACS numbers: 04.20.-q, 04.20.Jb, 04.40.Nr Introduction The problem of building a physically admissible metric for an isolated ro- tating body is still long an unresolved problem [1, 9]. In fact, in order to obtain a physically reasonable source, many restrictions must be imposed (energy conditions, regularity, reasonable equation of state). In particular, Einstein’s equations for a rotating body with a perfect fluid source seem not to be integrable. A remarkable exception is given by dust pressureless stationary axially symmetric spacetimes. In a remarkable paper [10], Wini- cour showed that Einstein’s equations for a stationary axially symmetric dust source with differential rotation can be reduced to quadratures. These http://arxiv.org/abs/0704.1351v2 equations contain as a subclass the van Stockum one [11] of rigidly rotating matter. The first asymptotically flat solution of the van Stockum class can be found in [12]. The author in [12] shows that the van Stockum class of solutions that are not cylindrically symmetric cannot exsist in the Newto- nian theory. The solution in [12] has a curvature singularity with diverging mass. Further, the technique named ”displace, cut, reflect” [13] to obtain rotating discs immersed in rotating dust must be noted. Unfortunately, this method generates distributional exotic matter on the z = 0 plane. In this paper, starting from the Lewis [14, 15] form of the metric, we write down the equations for stationary axially symmetric rigidly rotating space- times in a co-moving reference frame in such a way that the general solution depends upon the solution of an elliptic equation and upon harmonic func- tions. The class of solutions contains the van Stockum line element for a suitable choice of the harmonic function. In this context, starting from the dipole Bonnor solution [12], we obtain an asymptotically flat solution with two curvature singularities on the rotation axis and showing simalar properties to the Bonnor solution. In section 1 we derive the basic equations. In section 2 we present our solution. Section 3 collects some final remarks and conclusions. In the ap- pendix we derive a non asymtotically flat solution with a NUT charge and a well-defined mass-energy expression. 1 Basic Equations Our starting point is the Lewis [14] line element for a stationary axisym- metric space-time: ds2 = ev(ρ,z) dρ2 + dz2 + L(ρ, z)dφ2 + 2m(ρ, z)dtdφ − f(ρ, z)dt2, (1) where x4 = t is the time coordinate, x1 = ρ is the radial coordinate in a cylindrical system, x2 = z is the zenithal coordinate and x3 = φ is the azimuthal angular coordinate on the plane z = 0. Also, t ∈ (−∞,∞) , ρ ∈ (0,∞) , z ∈ (−∞,∞) , φ ∈ [0, 2π). (2) Further, the root square of the determinant of the 2-metric spanned by the Killing vectors ∂t, ∂φ is |det g(2)| = fL+m2 = W (ρ, z). (3) Expression (3) characterizes the measure of the area of the orbits of the isometry group. In the vacuum, the field equations for (1) imply thatW (ρ, z) is harmonic, i.e. W,α,α = 0, where subindices denote partial derivative and a summation with respect to α = ρ, z is implicit. Therefore, the function W (ρ, z) can be chosen as a coordinate. Looking for regular solutions on the axis, the simplest assumption can be made by setting W = ρ. In this way, the van Stockum line element emerges by taking a dust source. However, this is not the most general choice. Thanks to the gauge freedom, we can fL+m2 = ρ2H(ρ, z), (4) where H(ρ, z) is a sufficiently regular function to be specified by the field equations. We consider a perfect fluid Tµν = (E + P )uµuν + Pgµν , with E being the mass-energy density , P the hydrostatic pressure and uµ the 4-velocity of the fluid. We consider a co-moving reference frame: , uφ = uρ = uz = 0. (5) Denoting with Rµν the Ricci tensor, the relevant field equations are Rzz −Rρρ = 0, (6) Rρρ +Rzz = (P − E)ev , (7) Rρz = 0, (8) Rφφ = T − Tφφ , (9) Rtφ = T − Ttφ , (10) Rtt = − T + Ttt , (11) where T = 3P − E. Equation (9) involves a second-order partial equation for L(ρ, z), while (10) and (11) give second order equations for m(ρ, z) and f(ρ, z) respectively. Thanks to (4), equations (9)-(11) are not independent. Therefore, from (4), we can express L(ρ, z) in terms of (f,m,H). Putting this expression in (9) and using equations (10) and (11), we obtain the following compatibility equation: 4HPev = H,α,α − H,ρ. (12) In what follows we study dust solutions for which P = 0. Setting ρ2H = F 2, equation (12) becomes F,α,α = 0. Thus the compatibility condition for (9)- (11) requires that F (ρ, z) be a harmonic function. Conversely, with F (ρ, z) no more harmonic, the line element (1) is appropriate to describe spacetimes with non-vanishing pressure P . For F = ρ the van Stockum line element is regained together with the Papapetrou form of the metric [15]. Equations (6) and (8) are linear first-order equations involving v,ρ and v,z and they permit us to calculate v,ρ, v,z in terms of (f,m,F ). By applying the integrability condition (v,ρ,z = v,z,ρ) for the equations so obtained, we read f,zF,ρ = F,zf,ρ. (13) Finally, when expressions for v,ρ, v,z are put in (7), we obtain F,zf,z = −f,ρF,ρ. (14) Excluding the case F = const (it can be see that this leads to the trivial solution E = 0), we have f = const. We naturally choose f = 1. Therefore our system of equations is F,α,α = 0, (15) m,α,α − m,αF,α = 0, (16) e−vm2,α , (17) v,ρ = m2zF,ρ −m2,ρF,ρ + 4FF,zF,z,ρ − 4FF,ρF,z,z − 2m,ρm,zF,z 2FF 2,α ,(18) v,z = 4FF,zF,z,z + 4FF,ρF,z,ρ − F,zm2,z + F,zm2,ρ − 2m,ρm,zFρ 2FF 2,α ,(19) L+m2 = F 2. (20) First of all, for non-expanding spacetimes, the shear qik = [ui;k + uk;i] vanishes identically for (15)-(20), and therefore our system of equations de- scribes rigidly rotating sources in a co-moving reference frame. When F = ρ, equation (16) is invariant under the transformation z → z+a (a a constant) and a solution can be expanded as i m(ρ, z + ai). Setting F 6= ρ, if F (ρ, z),m(ρ, z) are solutions, then also F (ρ, z+a),m(ρ, z+a) are, and thus the solutions cannot be expanded. Note that we have identified (ρ, z) with the radial and the zenithal coordinate respectively in a cylindrical coordinate system. According to this assump- tion, some conditions must be imposed. Firstly, by setting E = 0, the metric must reduce to the standard flat expression ds2 = dρ2 + dz2 + ρ2dφ2 − dt2. Therefore, limE→0 F = ρ. Further, looking for regular spacetimes on the rotation axis , the norm of the space-like Killing vector ∂φ must be van- ishing (except at isolated points) at ρ = 0, i.e. limρ→0 L = 0. Finally, for asymptotically flat spacetimes, at spatial infinity F (ρ, z) looks as follows: F = ρ+ o(1). 2 Generating an asymptotically flat solution Our starting point is the Bonnor dipole solution [12] F = ρ,m = (ρ2+z2) We can obtain a solution of (16) by taking the map ρ → F (ρ, z), z → G(ρ, z), where F = ρ 1 + bc (ρ2+z2) , G = z 1− bc (ρ2+z2) with c ≥ 0, being b a constant. Therefore we get the solution F = ρ (ρ2 + z2) , (21) cρ2[ρ2 + z2 + bc] ρ2 + z2[(ρ2 + z2) + 2bcρ2 + b2c2 − 2bcz2] v = ln(α) + γ(ρ2 + z2 + bc) [(ρ2 + z2) + 2ρ2bc+ b2c2 − 2z2bc]4 c2e−vβ∆ [(ρ2 + z2) + 2ρ2bc+ b2c2 − 2z2bc]4 β = α(ρ2 + z2) = (ρ2 + z2) + c2b2 − 2ρ2cb+ 2z2cb, γ = (ρ2 − 8z2)(ρ2 + z2)2 + 2ρ4cb+ 18ρ2z2bc+ ρ2c2b2 + 16z4bc− 8z2c2b2, ∆ = (ρ2 + 4z2)(ρ2 + z2) 2 − 8z4cb+ 4z2c2b2 − 6ρ2z2cb+ ρ2c2b2 + 2ρ4cb. Solution (21) is asymptotically flat. Note that the map ρ → F (ρ, z) , z → G(ρ, z) is not bijective, i.e. is not a diffeomorphism. Concerning the features of (21), they depend on the sign of the constant b. For b > 0, apart from ρ = 0, z = ± bc, our solution is regular everywhere. At these two points, we have curvature singularities with properties close to the ρ = 0, z = 0 singularity of the dipole Bonnor solution (see [12]). In particular, the mass-energy diverges at these points. Otherwise, the energy density E(ρ, z) is integrable. For b = 0, we regain the Bonnor solution. Finally, for b < 0, the two point singularities disappear, but emerges a curvature ring singularity for z = 0, ρ = |b|c with diverging mass. Inde- pendently on the parameter b, at spatial infinity the metric reduces to the standard expression in asymptotical cylindrical coordinates, and so also by setting c = 0 (E = 0). Note that, because of the non-invertibility of the map between the Bonnor solution and solution (21), the curvature singularity at the origin of [12] is shifted in the two curvature singularities of (21) (for b > 0) on the rotation axis. As a final consideration, it must be noted that there exists for (21) a finite non singular region about the origin. Thus, our solution could be matched, in principle, with some asymptotically flat vacuum solution. We do not enter in this discussion, but only mention this possibility. 3 Conclusions and final remarks We have studied stationary axially symmetric rigidly rotating dust space- times in terms of harmonic functions. In [16] Bonnor found the general solution for charged dust with zero Lorentz force in terms of harmonic func- tions. However, the use of such kind of functions in [16] is different from the one in our paper. In fact, in our paper harmonic functions appear in equation (4) thanks to the gauge freedom, while in [16] the function F (ρ, z) is chosen to be the cylindrical polar coordinate ρ. In the Bonnor paper, harmonic functions arise in order to obtain the most general solution for equation (16) with F = ρ. In this case, all the solutions of the equation m,α,α − 1ρm,ρ = 0 are given by taking a generic harmonic function η(ρ, z), with m = ρηρ. In a similar way, another harmonic function is introduced when charged dust comes in action. Therefore, a direct relation between does not exist between the harmonic function F (ρ, z) and η(ρ, z) of [16]. Also, the paper [17] must be noticed in which charged dust solutions are given in terms of Bessel functions of first and second kind and hyperbolic functions. Also in the paper [17] the condition F = ρ is retained. In this paper, section two, starting with the dipole Bonnor solution, we build a class of asymptotically flat solutions containing the Bonnor one as a subclass by a suitable choice of the functions F (ρ, z), G(ρ, z). Obviously, it must be noted that not all the harmonic functions generate physically sen- sible solutions. For a physically sensible solution we mean a regular (apart from isolated singularities) asymptotically flat solution. Generally, it is a simple matter to verify that, if F (ρ, z) = ρ,m(ρ, z) is a regular differen- tiable solution for (16), then also m(F (ρ, z), G(ρ, z)) is a solution for (16) with F (ρ, z) harmonic being G(ρ, z) the harmonic conjugate to F (ρ, z) i.e. Fρ = Gz, Fz = −Gρ. Further, in order to generate a new asymptotically flat and regular solution on the axis starting with a seed solution with these two properties, we must build a non-bijective (not a diffeomorphism) map ρ → F (ρ, z), z → G(ρ, z) such that limE→0 F = ρ , limE→0G = z, and limρ→0 L = limρ→0 F = 0 (apart from isolated points) and such that at spa- tial infinity the functions F (ρ, z), G(ρ, z) look as follows: F = ρ+o(1) , G = z + o(1). Concerning isolated singularities, no general conclusions can be made. The functions F (ρ, z), G(ρ, z) of section two satisfy all the conditions mentioned above. Finally, in the appendix we present a non-asymptotically flat solution not obtained from a seed solution with the technique discussed above and there- fore it represents an ad hoc solution. In particular, it is possible to build ad hoc solutions for (16) by setting F (ρ, z) = ρ(1 + c ρ2+z2 ) (with c a constant) and m(ρ, z) a homogeneous function such that m,α,α − m,ρρ = 0. Appendix We consider the following solution: F = ρ ρ2 + z2 , m = c ρ2 + z2 , (22) ln[ρ4 − 2cρ2 + 2ρ2z2 + c2 + z4 + 2cz2] ln[c+ ρ2 + z2]− 2 ln[ρ2 + z2] , E = c2e−v [c+ ρ2 + z2] with c a constant. When c > 0, the solution (22) has a curvature ring sin- gularity when (ρ, z) = ( c, 0) The axis is regular for z > 0, while it shows a conical (no curvature) singularity when z ≤ 0. Further, for z < 0 there is a region where closed time-like curves (CTC) appear resulting in a vioaltion of causality. However, it is possible to take a simple coordinate transformation found in [18], i.e. τ = t+ 2cφ giving the whole rotational axis free of coni- cal singularities. Unfortunately, we are forced to introduce a periodic time coordinate τ and therefore once again CTC appear. Therefore, the problem of violation of causality cannot be avoided with an opportune coordinate transformation. The spacetime asymptotically reads the expression appropriate for asymp- totic NUT metrics [19, 20] with NUT charge q given by c = 2q. We can estimate the mass inside an infinite cylinder of radius R by means of the integral M(R) = EFevdφ. (23) The integral (23) is well defined everywhere. Thus, for the mass we get the formula M(R) = 2π2c2 1 +R− R2 + 1 . (24) Because of the non-asymptotical flatness, the solution (22) is not interesting in an astrophysical context. However, the natural arena for this solution is in the extra relativistic context given by non-Abelian gauge theories or in the low energy string theory [21, 22] where, in order to obtain supersymmetries, NUT charge comes in action. References [1] Neugebauer G and Meinel R 1993 Astrophys. J. 414 L97 [2] Senovilla J M M 1987 Class. Quantum Grav. 4 L 115 [3] Senovilla J M M 1992 it Class. Quantum Grav. 9 L 167 [4] Wahlquist M D 1968 Phys. Rev. 172 1291 [5] Kramer D 1985 Class. Quantum Grav. 2 L 135 [6] Stephani M 1988 J. Math. Phys. 29 1650 [7] Stewart J M and Ellis GFR 1968 J. Math. Phys. 9 1072 [8] Herlt E 1988 it Gen. Rel. Grav. 20 635 [9] Lukacs B et alt. 1983 Gen. Rel. Grav. 15 567 [10] Winicour J 1975 J. Math. Phys. 16 1805 [11] Stockum V 1937 Proc. Roy. Soc. Eddim. 57 135 [12] Bonnor W B 1977 J. Phys. A: Math. Gen. 10 1673 [13] Vogt D and Letelier P S 2006 Preprint astro-ph/0611428 (To appear in IJMPD) [14] Lewis T 1932 Proc. Roy. Soc. Lond. 136 176 [15] Papapetrou V A 1953 Ann. Phys., Lpz 6 12 [16] Bonnor W B 1980 J. Phys. A: Math. Gen. 13 3465 [17] Georgiou A 2001 Proc. Royal Soc.: Math. Phys. Sci. 457 1153 http://arxiv.org/abs/astro-ph/0611428 [18] Misner CW 1963 J. Math. Phys. 4 924 [19] Newman E, Tamburino L and Unti T 1963 J. Math. Phys. 4 915 [20] Dadhich N and Turakulov Z Y 2002 Class. Quantum Grav. 19 2765 [21] Radu E 2003 Phys. Rev. D 67 084030 [22] Johnson C V and Myers R C 1994 Phys. Rev. D 50 6512 Basic Equations Generating an asymptotically flat solution Conclusions and final remarks

0704.1352

The Green function estimates for strongly elliptic systems of second order

THE GREEN FUNCTION ESTIMATES FOR STRONGLY ELLIPTIC SYSTEMS OF SECOND ORDER STEVE HOFMANN AND SEICK KIM Abstract. We establish existence and pointwise estimates of fundamental so- lutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain Ω ⊆ Rn, n ≥ 3, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation. 1. Introduction In this article, we study Green’s functions (or Green’s matrices) of second order, strongly elliptic systems of divergence type in a domain Ω ⊂ Rn with n ≥ 3. In particular, we treat the Green matrix in the entire space, usually called the fundamental solution. We shall prove that if a given elliptic system has the property that all weak solutions of the system are locally Hölder continuous, then it has the Green’s matrix in Ω. (For example, if coefficients of the system belong to the space of VMO introduced by Sarason [19], then it will enjoy such a property). For such elliptic systems, we study standard properties of the Green’s matrix including pointwise bounds, Lp and weak Lp estimates for Green’s matrix and its derivatives, For the scalar case, i.e., a single elliptic equation, the existence and properties of Green’s function was studied by Littman, Stampacchia, and Weinberger [14] and Grüter and Widman [11]. In this article, we follow the approach of Grüter and Widman in constructing Green’s matrix. The main technical difficulties arise from lack of Harnack type inequalities and the maximum principle for the systems. The key observation on which this article is based is that even in the scalar case, one can get around Moser’s Harnack inequality [18] or maximum principle but instead rely solely on De Giorgi-Nash type oscillation estimates [5] in constructing and studying properties of Green’s functions. From this point of view, this article provides a unified approach in studying Green’s function for both scalar and systems of equations. We should point out that there has been some study of Green’s matrix for systems with continuous coefficients, notably by Fuchs [7] and Dolzmann-Müller [6]. Our existence results and interior estimates of Green’s function will include theirs, since as is well known, weak solutions of systems with uniformly continuous (or VMO) coefficients enjoy local Hölder estimates. On the other hand, we have not attempted to replicate their boundary estimates, which depend in particular on having a C1 boundary. Our method does not require boundedness of the domain nor regularity of the boundary in constructing Green’s matrices, while the methods 2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35J45. Key words and phrases. Green’s function, fundamental solution, second order elliptic system. http://arxiv.org/abs/0704.1352v2 2 S. HOFMANN AND S. KIM of Fuchs [7] and Dolzmann-Müller [6] require both boundedness and regularity of the domain at the very beginning. We note that a scalar elliptic equation with complex coefficients can be identified as an elliptic system with real coefficients satisfying a special structure, and thus our results apply in particular to complex perturbations of a scalar real equation. In the complex coefficients setting, the main results of Section 3 in our paper can be also obtained by following the method of Auscher [2]. The estimates of the present paper will be applied to the development of the layer potential method for equations with complex coefficients in [1]. The organization of this paper is as follows. In Section 2, we define the prop- erty (H), which is essentially equivalent to De Giorgi’s oscillation estimates in the scalar case, and introduce a function space Y 0 (Ω) which substitutes W 0 (Ω) in constructing Green’s functions; they are identical if Ω is bounded but in general, 0 (Ω) is a larger space and is more suitable for our purpose. In Section 3, we study Green’s functions defined in the entire space, which are usually referred to as the fundamental solutions. The main result is that for a system whose coeffi- cients are close to those of a diagonal system, the fundamental solution behaves very much like that of a single equation. In Section 4, we study Green’s matrices in general domains, including unbounded ones. We also study the boundary behav- ior of Green’s matrices when the boundary of domain satisfies a measure theoretic exterior cone condition, called the condition (S). We prove in particular that if the coefficients of the system are close to those of a diagonal system, then again the boundary behavior of its Green’s function is much like that of a single equation. In section 5, we discuss the Green’s matrices of the strongly elliptic systems with VMO coefficients. By following the same techniques already developed in the pre- vious two sections, we construct the Green’s matrix in general domains including the entire space. One subtle difference is that in this VMO coefficients case, one should play with a localized version of property (H) since basically, the regularity of weak solutions of the systems with VMO coefficients is inherited from the systems with constant coefficients when the scale is made small enough. Therefore, all the estimates for the Green’s matrix stated in this section are only meaningful near a pole. Finally, we would like to mention that when n = 2, the method used in this article breaks down in several places and for that reason we plan to treat the two dimensional case in a separate paper. 2. Preliminaries 2.1. Strongly elliptic systems. Throughout this article, the summation con- vention over repeated indices shall be assumed. Let L be a second order elliptic operator of divergence type acting on vector valued functions u = (u1, . . . , uN )T defined on Rn (n ≥ 3) in the following way: (2.1) Lu = −Dα(A αβ Dβu), where Aαβ = Aαβ(x) (α, β = 1, . . . , n) are N by N matrices satisfying the strong ellipticity condition, i.e., there is a number λ > 0 such that (2.2) A ij (x)ξ α ≥ λ |ξ| |ξiα| 2, ∀x ∈ Rn GREEN FUNCTION ESTIMATES 3 We also assume that A ij are bounded, i.e., there is a number Λ > 0 such that (2.3) i,j=1 α,β=1 ij (x)| 2 ≤ Λ2, ∀x ∈ Rn. If we write (2.1) component-wise, then we have (2.4) (Lu)i = −Dα(A ij Dβu j), ∀i = 1, . . . , N. The transpose operator of tL of L is defined by (2.5) tLu = −Dα( αβDβu), where tAαβ = (Aβα)T (i.e., tA ij = A ji ). Note that the coefficients ij satisfy (2.2), (2.3) with the same constants λ,Λ. In the sequel, we shall use the notation − f := 1 f (assuming 0 < |S| < ∞), where S is a measurable subset of Rn and |S| denotes the Lebesgue measure of measurable S. Definition 2.1. We say that the operator L satisfies the property (H) if there exist µ0, H0 > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) satisfy (2.6) )n−2+2µ0 , 0 < r < s ≤ R. Similarly, we say that the transpose operator tL satisfies the property (H) if corre- sponding estimates hold for all weak solutions u of tLu = 0 in BR. Lemma 2.2. Let (aαβ(x))nα,β=1 be coefficients satisfying the following conditions: There are constants λ0,Λ0 > 0 such that for all x ∈ R (2.7) aαβ(x)ξβξα ≥ λ0 |ξ| , ∀ξ ∈ Rn; ∣aαβ(x) ≤ Λ20. Then, there exists ǫ0 = ǫ0(n, λ0,Λ0) such that if (2.8) ǫ2(x) := ij (x) − a αβ(x)δij < ǫ20, ∀x ∈ R then the operator L associated with the coefficients A ij satisfies the condition (H) with µ0 = µ0(n, λ0,Λ0), H0 = H0(n,N, λ0,Λ0) > 0. Proof. See e.g., [12, Proposition 2.1]. � Lemma 2.3. Suppose that the operator L satisfies the following Hölder property for weak solutions: There are constants µ0, C0 > 0 such that all weak solutions u of Lu = 0 in B2R = B2R(x0) satisfy the estimate (2.9) [u]Cµ0 (BR) ≤ C0R where [f ]Cµ(Ω) denotes the usual C µ(Ω) semi-norm of f ; see [10] for the definition. Then, the operator L satisfies the property (H) with µ0 and H0 = H0(n,N, λ,Λ, C0). 4 S. HOFMANN AND S. KIM Proof. We may assume that r < s/4; otherwise, (2.6) is trivial. Denote ur = − We may assume, by replacing u by u − us, if necessary, that us = 0. From the Caccioppoli inequality, (2.9), and then the Poincaré inequality, it follows ≤ Cr−2 |u− u2r| ≤ Cr−2 |u(x) − u(y)| dy dx ≤ Cr−2[u]2Cµ0 (B2r)(2r) 2µ0 |B2r| ≤ Cr n−2+2µ0 [u]2Cµ0(Bs/2) ≤ C(r/s)n−2+2µ0s−2 ≤ C(r/s)n−2+2µ0 The proof is complete. � Lemma 2.4. Assume that the operator L satisfies the property (H). Then, the operator L satisfies the Hölder property (2.9). Moreover, for any p > 0, there exists Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) satisfy (2.10) ‖u‖L∞(Br) ≤ (R− r)n/p ‖u‖Lp(BR) , ∀r ∈ (0, R). Proof. From a theorem of Morrey [17, Thoerem 3.5.2], the property (H), and the Caccioppoli inequality, it follows that (2.11) [u]2Cµ0(BR) ≤ CR 2−n−2µ0 ‖Du‖ L2(B3R/2) ≤ CR−n−2µ0 ‖u‖ L2(B2R) Then, by a well known averaging argument (see e.g., [12]) we derive (2.12) ‖u‖L∞(BR/2) ≤ C where C = C(n,N, λ,Λ, µ0, H0) > 0. For the proof that (2.12) implies (2.10), we refer to [9, pp. 80–82]. � 2.2. Function spaces Y 1,2(Ω) and Y 0 (Ω). Definition 2.5. For an open set Ω ⊂ Rn (n ≥ 3), the space Y 1,2(Ω) is defined as the family of all weakly differentiable functions u ∈ L2 (Ω), where 2∗ = 2n , whose weak derivatives are functions in L2(Ω). The space Y 1,2(Ω) is endowed with the ‖u‖Y 1,2(Ω) := ‖u‖L2∗ (Ω) + ‖Du‖L2(Ω) . We define Y 0 (Ω) as the closure of C c (Ω) in Y 1,2(Ω), where C∞c (Ω) is the set of all infinitely differentiable functions with compact supports in Ω. We note that in the case Ω = Rn, it is well known that Y 1,2(Rn) = Y see e.g., [15, p. 46]. By the Sobolev inequality, it follows that (2.13) ‖u‖L2∗ (Ω) ≤ C(n) ‖Du‖L2(Ω) , ∀u ∈ Y 0 (Ω). Therefore, we have W 0 (Ω) ⊂ Y 0 (Ω) and W 0 (Ω) = Y 0 (Ω) when Ω has finite Lebesgue measure. From (2.13), it follows that the bilinear form (2.14) 〈u,v〉 GREEN FUNCTION ESTIMATES 5 defines an inner product on H := Y 0 (Ω) N . Also, it is routine to check that H equipped with the inner product (2.14) is a Hilbert space. Definition 2.6. We shall denote by H the Hilbert space Y 0 (Ω) N with the inner product (2.14). We denote := 〈u,u〉 = ‖Du‖L2(Ω) . We also define the bilinear form associated to the operator L as B(u,v) := ij Dβu By the strong ellipticity (2.2), it follows that the bilinear form B is coercive; i.e, (2.15) B(u,u) ≥ λ 〈u,u〉 3. Fundamental matrix in Rn Throughout this section, we assume that the operators L and tL satisfy the property (H). The main goal of this section is to construct the fundamental matrix of the the operator L in the entire Rn, where n ≥ 3. Since Y 1,2(Rn) = Y we have, as in Definition 2.5, ‖u‖L2∗ (Rn) ≤ C(n) ‖Du‖L2(Rn) , ∀u ∈ Y 1,2(Rn). We note that W 1,2(Rn) ⊂ Y 1,2(Rn) ⊂ W loc (R n). Unless otherwise stated, we employ the letter C to denote a constant depending on n, N , λ, Λ, µ0, H0, and sometimes on an exponent p characterizing Lebesgue classes. It should be under- stood that C may vary from line to line. 3.1. Averaged fundamental matrix. Our approach here is based on that in [11]. Let y ∈ Rn and 1 ≤ k ≤ N be fixed. For ρ > 0, consider the linear functional u 7→ − Bρ(y) uk. Since (3.1) Bρ(y) ≤ Cρ(2−n)/2 ‖u‖L2∗(Rn) ≤ Cρ (2−n)/2 ‖u‖ Lax-Milgram lemma implies that there exists a unique vρ = vρ;y,k ∈ H such that (3.2) ij Dβv ρ Dαu i = − Bρ(y) uk, ∀u ∈ H. Note that (2.15), (3.2), and (3.1) imply that λ ‖vρ‖ ≤ B(vρ,vρ) ≤ Cρ (2−n)/2 ‖vρ‖H , and thus we have (3.3) ‖Dvρ‖L2(Rn) = ‖vρ‖H ≤ Cρ (2−n)/2. We define the “averaged fundamental matrix” Γρ( · , y) = (Γ jk( · , y)) j,k=1 by (3.4) Γ jk( · , y) = v ρ = v ρ;y,k. Note that we have (3.5) ij DβΓ jk( · , y)Dαu i = − Bρ(y) uk, ∀u ∈ H, 6 S. HOFMANN AND S. KIM and equivalently (α ↔ β , i ↔ j). (3.6) ij Dβu ik( · , y) = − Bρ(y) uk, ∀u ∈ H. In the sequel, we shall denote by L∞c (Ω) the family of all L ∞ functions with compact supports in Ω. For a given f ∈ L∞c (R n)N consider a linear functional (3.7) w 7→ f ·w, which is bounded on H since (3.8) ≤ ‖f‖ n+2 (Rn) n−2 (Rn) ≤ C ‖f‖ n+2 (Rn) Therefore, by Lax-Milgram lemma, there exists u ∈ H such that (3.9) ij Dβu f iwi, ∀w ∈ H. In particular, if we set w = vρ in (3.9), then by (3.6), we have (3.10) ik( · , y)f i = − Bρ(y) Moreover, by setting w = u in (3.9), it follows from (3.8) that (3.11) ‖Du‖L2(Rn) ≤ C ‖f‖L2n/(n+2)(Rn) . 3.2. L∞ estimates for averaged fundamental matrix. Let u ∈ H be given as in (3.9). We will obtain local L∞ estimates for u in BR(x0), where x0 ∈ R n and R > 0 are fixed but arbitrary. Fix x ∈ BR(x0) and 0 < s ≤ R. We decompose u as u = u1 + u2, where u1 ∈ W 1,2(Bs(x)) N is the weak solution of tLu1 = 0 in Bs(x) satisfying u1 = u on ∂Bs(x); i.e., u1 − u ∈ W 0 (Bs(x)). Then, for 0 < r < s, we have Br(x) Br(x) |Du1| Br(x) |Du2| )n−2+2µ0 Bs(x) |Du1| Bs(x) |Du2| )n−2+2µ0 Bs(x) Bs(x) |Du2| Since u2 ∈ W 0 (Bs(x)) N is a weak solution of tLu2 = f in Bs(x), we have Bs(x) |Du2| ≤ C ‖f‖ L2n/(n+2)(Bs(x)) For given p > n/2, choose p0 ∈ (n/2, p) such that µ1 := 2− n/p0 < µ0. Then (3.12) ‖f‖ n+2 (Bs(x)) ≤ ‖f‖ Lp0(Bs(x)) 1+2/n−2/p0 ≤ C ‖f‖ Lp0(Rn) s n−2+2µ1 . Therefore, after combining the above inequalities, we have for all r < s ≤ R Br(x) )n−2+2µ0 Bs(x) + Csn−2+2µ1 ‖f‖ Lp0(Rn) . GREEN FUNCTION ESTIMATES 7 By a well known iteration argument (see e.g., [8, Lemma 2.1, p. 86]), we have Br(x) )n−2+2µ1 BR(x) + Crn−2+2µ1 ‖f‖ Lp0(Rn) )n−2+2µ1 + Crn−2+2µ1 ‖f‖ Lp0(Rn) , (3.13) for all 0 < r < R and x ∈ BR(x0). From (3.13) it follows (see, e.g. [12]) (3.14) [u]2Cµ1 (BR(x0)) ≤ C R−(n−2+2µ1) ‖Du‖ L2(Rn) + ‖f‖ Lp0(Rn) Note that since u ∈ H, we have L2(BR(x0)) ≤ ‖u‖ (BR(x0)) ≤ CR2 ‖Du‖ L2(Rn) . Consequently, we have L∞(BR/2(x0)) ≤ CR2µ1 [u]2Cµ1 (BR(x0)) + CR −n ‖u‖ L2(BR(x0)) R2−n ‖Du‖ L2(Rn) +R 2µ1 ‖f‖ Lp0(Rn) + CR2−n ‖Du‖ L2(Rn) ≤ CR2−n ‖f‖ L2n/(n+2)(Rn) + CR 2µ1 ‖f‖ Lp0(Rn) , where we used the inequality (3.11) in the last step. Therefore, if f is supported in BR(x0), then (3.12) yields (recall µ1 = 2− n/p0) (3.15) ‖u‖L∞(BR/2(x0)) ≤ CR 2−n/p0 ‖f‖Lp0(BR(x0)) ≤ CR 2−n/p ‖f‖Lp(BR(x0)) . Now, (3.10) implies that for ρ < R/2, we have, by setting x0 = y in (3.15), (3.16) BR(y) ik( · , y)f Bρ(y) |u| ≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀p > n/2 provided that f is supported in BR(y). Therefore, by duality, we see that (3.17) ‖vρ‖Lq(BR(y)) ≤ CR 2−n+n/q, ∀q ∈ [1, n ), ∀ρ ∈ (0, R/2), where vρ = vρ;y,k is as in (3.4). Fix x 6= y and let r := 2 |x− y|. If ρ < r/2, then since vρ ∈ W 1,2(Br(x)) N and satisfies Lvρ = 0 weakly in Br(x), it follows from Lemma 2.4 that (3.18) |vρ(x)| ≤ Cr −n ‖vρ‖L1(Br(x)) ≤ Cr −n ‖vρ‖L1(B3r(y)) ≤ Cr Since ρ, y, k are arbitrary, we have obtained the following estimates. (3.19) |Γρ(x, y)| ≤ C |x− y| , ∀ρ < |x− y| /3. 3.3. Uniform weak-L n−2 estimates for Γ ρ( · , y). We claim that the following estimate holds: (3.20) Rn\BR(y) |Γρ( · , y)| n−2 ≤ CR−n, ∀R > 0, ∀ρ > 0. If R > 3ρ, then by (3.19) we have Rn\BR(y) |Γρ(x, y)| n−2 dx ≤ C Rn\BR(y) |x− y| dx ≤ CR−n. 8 S. HOFMANN AND S. KIM Next, we consider the case R ≤ 3ρ. Let vTρ be the k-th column of the averaged fundamental matrix Γρ( · , y) as in (3.4). From (3.3), we see that ‖vρ‖L2∗ (Rn\BR(y)) ≤ ‖vρ‖L2∗ (Rn) ≤ ‖Dvρ‖L2(Rn) ≤ Cρ (2−n)/2. and thus (3.20) also follows in the case when R ≤ 3ρ. Now, let At = {x ∈ R n : |Γρ(x, y)| > t} and choose R = t−1/(n−2). Then, |At \BR(y)| ≤ t At\BR(y) |Γρ( · , y)| n−2 ≤ Ct− n−2 t n−2 = Ct− n−2 . Obviously, |At ∩BR(y)| ≤ CR n = Ct− n−2 . Therefore, we obtained that for all t > 0, we have (3.21) |{x ∈ Rn : |Γρ(x, y)| > t}| ≤ Ct− n−2 , ∀ρ > 0. 3.4. Uniform weak-L n−1 estimates for DΓρ( · y). Let vρ be as before. Fix a cut-off function η ∈ C∞(Rn) such that η ≡ 0 on BR/2(y), η ≡ 1 outside BR(y), and |Dη| ≤ C/R. If we set u := η2vρ, then by (3.2) ij Dβv ij Dβv ρDαη, which together with (3.19) implies that if R > 6ρ, then Rn\BR(y) |Dvρ| ≤ CR−2 BR(y)\BR/2(y) ≤ CR2−n. On the other hand, if R ≤ 6ρ, then (3.3) again implies Rn\BR(y) |Dvρ| |Dvρ| ≤ Cρ2−n ≤ CR2−n. Therefore, we have (3.22) Rn\BR(y) |DΓρ( · , y)| ≤ CR2−n, ∀R > 0, ∀ρ > 0. Next, let At = {x ∈ R n : |DxΓ ρ(x, y)| > t} and choose R = t−1/(n−1). Then |At \BR(y)| ≤ t At\BR(y) |DΓρ( · , y)| ≤ Ct− and |At ∩BR(y)| ≤ CR n = Ct− n−1 . We have thus find that for all t > 0, we have (3.23) |{x ∈ Rn : |DxΓ ρ(x, y)| > t}| ≤ Ct− n−1 , ∀ρ > 0. 3.5. Construction of the fundamental matrix. First, we claim (3.24) ‖DΓρ( · , y)‖Lp(BR(y)) ≤ CpR 1−n+n/p, ∀ρ > 0, ∀p ∈ (0, n Let vρ be as before. Note that BR(y) |Dvρ| BR(y)∩{|Dvρ|≤τ} |Dvρ| BR(y)∩{|Dvρ|>τ} |Dvρ| ≤ τp |BR|+ {|Dvρ|>τ} |Dvρ| GREEN FUNCTION ESTIMATES 9 By using (3.23), we estimate {|Dvρ|>τ} |Dvρ| ptp−1 |{|Dvρ| > max(t, τ)}| dt ≤ Cτ− ptp−1 dt+ C ptp−1−n/(n−1) dt 1− p/(p− n τp−n/(n−1). By optimizing over τ , we get (3.25) BR(y) |Dvρ| ≤ CR(1−n)p+n, from which (3.24) follows. If we utilize (3.21) instead of (3.23), we obtain a similar estimates for Γρ( · , y) (3.26) ‖Γρ( · , y)‖Lp(BR(y)) ≤ CpR 2−n+n/p, ∀ρ > 0, ∀p ∈ (0, n Let us fix q ∈ (1, n ). We have seen that for all R > 0, there exists some C(R) < ∞ such that ‖Γρ( · , y)‖W 1,q(BR(y)) ≤ C(R), ∀ρ > 0. Therefore, by a diagonalization process, we obtain a sequence {ρµ} and Γ( · , y) loc (R n)N×N such that limµ→∞ ρµ = 0 and that (3.27) Γρµ( · , y) ⇀ Γ( · , y) in W 1,q(BR(y)) N×N , ∀R > 0, where we recall that ⇀ denotes weak convergence. Then, for any φ ∈ C∞c (R n)N , it follows from (3.5) ij DβΓjk( · , y)Dαφ i = lim ij DβΓ jk ( · , y)Dαφ = lim Bρµ (y) φk = φk(y). (3.28) Let vTρ be the k-th column of Γ ρ( · , y) as before, and let vT be the corresponding k-th column of Γ( · , y). Then, for any g ∈ L∞c (BR(y)) N , (3.26) yields (3.29) v · g = lim vρµ · g ≤ CpR 2−n+n/p ‖g‖Lp′(BR(y)) , where p′ denotes the conjugate exponent of p ∈ [1, n ). Therefore, we obtain (3.30) ‖Γ( · , y)‖Lp(BR(y)) ≤ Cp R 2−n+n/p, ∀p ∈ [1, n By a similar reasoning, we also have by (3.24) (3.31) ‖DΓ( · , y)‖Lp(BR(y)) ≤ Cp R 1−n+n/p, ∀p ∈ [1, n Also, with the aid of (3.20) and (3.22), we obtain Rn\BR(y) |Γ( · , y)| ≤ CR−n,(3.32) Rn\BR(y) |DΓ( · , y)| ≤ CR2−n.(3.33) 10 S. HOFMANN AND S. KIM In particular, (3.32), (3.33) imply that (3.34) ‖Γ( · , y)‖Y 1,2(Rn\Br(y)) ≤ Cr 1−n/2, ∀r > 0. Moreover, arguing as before, we see that the estimates (3.32) and (3.33) imply |{x ∈ Rn : |Γ(x, y)| > t}| ≤ Ct− n−2 , ∀t > 0(3.35) |{x ∈ Rn : |DxΓ(x, y)| > t}| ≤ Ct n−1 ∀t > 0.(3.36) Next, we turn to pointwise bounds for Γ( · , y). Let vT be the k-th column of Γ( · , y). For each x 6= y, denote r = 2 |x− y|. Then, it follows from (3.34) and (3.28) that v is a weak solution of Lv = 0 in Br(x). Therefore, by Lemma 2.4 and (3.30) we find (3.37) |v(x)| ≤ Cr−n ‖v‖L1(Br(x)) ≤ Cr −n ‖v‖L1(B3r(y)) ≤ Cr from which it follows (3.38) |Γ(x, y)| ≤ C |x− y| , ∀x 6= y. 3.6. Continuity of the fundamental matrix. From the property (H), it follows that Γ( · , y) is Hölder continuous in Rn \ {y}. In fact, (2.11) together with (3.28) and (3.33) implies (3.39) |Γ(x, y)− Γ(z, y)| ≤ C |x− z| µ0 |x− y| 2−n−µ0 if |x− z| < |x− y| /2. Moreover, by the same reasoning, it follows from (2.11) and (3.22) that for any given compact set K ⋐ Rn \ {y}, the sequence {Γρµ( · , y)} µ=1 is equicontinuous on K. Also, by Lemma 2.4 and (3.20), we find that there are CK < ∞ and ρK > 0 such that (3.40) ‖Γρ( · , y)‖L∞(K) ≤ CK ∀ρ < ρK for any compact K ⋐ R n \ {y} . Therefore, we may assume, by passing if necessary to a subsequence, that (3.41) Γρµ( · , y) → Γ( · , y) uniformly on K, for any compact K ⋐ Rn \ {y} . We will now show that Γ(x, · ) is also Hölder continuous in Rn \ {x}. Denote by tΓσ( · , x) the averaged fundamental matrix associated to tL, the transpose of L. Since each column of Γρ( · , y) and tΓσ( · , x) belongs to H, we have by (3.5), Bρ(y) tΓσkl( · , x) = ij DβΓ jk( · , y)Dα tΓσil( · , x) ji Dα tΓσil( · , x)DβΓ jk( · , y) = − Bσ(x) lk( · , y). (3.42) By the same argument as appears in Sec. 3.5, we obtain a sequence {σν} ν=1 tending to 0 such that tΓσν ( · , x) converges to tΓ( · , x) uniformly on any compact subset of n \ {x}, where tΓ( · , x) is a fundamental matrix for tL satisfying all properties stated in Sec. 3.5. By (3.42), we find that gklµν := − Bρµ (y) tΓσνkl ( · , x) = − Bσν (x) lk ( · , y). From the continuity of Γ lk ( · , y), it follows that for x, y ∈ R n with x 6= y, we have gklµν = lim Bσν (x) lk ( · , y) = Γ lk (x, y) GREEN FUNCTION ESTIMATES 11 and thus by (3.41) we obtain gklµν = lim lk (x, y) = Γlk(x, y). On the other hand, (3.27) yields gklµν = lim Bρµ (y) tΓσνkl ( · , x) = − Bρµ (y) tΓkl( · , x) and thus it follows from the continuity of tΓkl( · , x) that gklµν = lim Bρµ (y) tΓkl( · , x) = tΓkl(y, x). We have thus shown that Γlk(x, y) = tΓkl(y, x), ∀k, l = 1, . . . , N, ∀x 6= y, which is equivalent to say (3.43) Γ(x, y) = tΓ(y, x)T , ∀x 6= y. Therefore, we have proved the claim that Γ(x, · ) is Hölder continuous in Rn \ {x}. So far, we have seen that there is a sequence {ρµ} tending to 0 such that ρµ( · , y) → Γ( · , y) in Rn \ {y}. However, by (3.42), we obtain lk(x, y) = limν→∞ Bσν (x) lk( · , y) = limν→∞ Bρ(y) tΓσνkl ( · , x) Bρ(y) tΓkl( · , x) = − Bρ(y) Γlk(x, · ), (3.44) i.e., we have the following representation for the averaged fundamental matrix: (3.45) Γρ(x, y) = − Bρ(y) Γ(x, z) dz. Therefore, by the continuity, we obtain (3.46) lim ρ(x, y) = Γ(x, y), x 6= y. 3.7. Properties of fundamental matrix. We record what we obtained so far in the following theorem: Theorem 3.1. Assume that operators L and tL satisfy the property (H). Then, there exists a unique fundamental matrix Γ(x, y) = (Γij(x, y)) i,j=1 (x 6= y) which is continuous in {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ(x, · ) is locally integrable in Rn for all x ∈ Rn and that for all f = (f1, . . . , fN)T ∈ C∞c (R n)N , the function u = (u1, . . . , uN )T given by (3.47) u(x) := Γ(x, y)f (y) dy belongs to Y 1,2(Rn)N and satisfies Lu = f in the sense (3.48) ij Dβu f iφi, ∀φ ∈ C∞c (R n)N . Moreover, Γ(x, y) has the property (3.49) ij DβΓjk( · , y)Dαφ i = φk(y), ∀φ ∈ C∞c (R n)N . 12 S. HOFMANN AND S. KIM Furthermore, Γ(x, y) satisfies the following estimates: ‖Γ( · , y)‖Y 1,2(Rn\Br(y)) + ‖Γ(x, · )‖Y 1,2(Rn\Br(x)) ≤ Cr 2 , ∀r > 0,(3.50) ‖Γ( · , y)‖Lp(Br(y)) + ‖Γ(x, · )‖Lp(Br(x)) ≤ Cpr 2−n+ n p , ∀p ∈ [1, n ),(3.51) ‖DΓ( · , y)‖Lp(Br(y)) + ‖DΓ(x, · )‖Lp(Br(x)) ≤ Cpr 1−n+ n p , ∀p ∈ [1, n ),(3.52) |{x ∈ Rn : |Γ(x, y)| > t}|+ |{y ∈ Rn : |Γ(x, y)| > t}| ≤ Ct− n−2 ,(3.53) |{x ∈ Rn : |DxΓ(x, y)| > t}|+ |{y ∈ R n : |DyΓ(x, y)| > t}| ≤ Ct n−1 ,(3.54) |Γ(x, y)| ≤ C |x− y| , ∀x 6= y,(3.55) |Γ(x, y)− Γ(z, y)| ≤ C |x− z| µ0 |x− y| 2−n−µ0 if |x− z| < |x− y| /2,(3.56) |Γ(x, y)− Γ(x, z)| ≤ C |y − z| µ0 |x− y| 2−n−µ0 if |y − z| < |x− y| /2,(3.57) where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0. Proof. Let Γρ(x, y) and Γ(x, y) be constructed as above. We have already seen that Γ is continuous in {(x, y) ∈ Rn × Rn : x 6= y} and satisfies all the properties (3.49) – (3.57). By using the Lax-Milgram lemma as in Sec. 3.1, we find that for all f ∈ C∞c (R n)N , there is a unique u ∈ Y 1,2(Rn)N satisfying ij Dβu f ivi, ∀v ∈ Y 1,2(Rn)N . If we set vi = Γ ki(x, · ) above, then (3.5) together with (3.43) implies that (3.58) ki(x, · )f ji Dα ik( · , x)Dβu j = − Bρ(x) Assume that f is supported in BR(x) for some R > 0. Then, by (3.27) and (3.43) we have ki(x, · )f i = lim BR(x) ki(x, · )f Γki(x, · )f By the same argument which lead to (3.14) in Section 3.2, we find that u is Hölder continuous. Therefore, (3.47) follows by taking the limits in (3.58). Now, it only remains to prove the uniqueness. Assume that Γ̃(x, y) is another ma- trix such that Γ̃ is continuous on {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ̃(x, · ) is locally integrable in Rn for all x ∈ Rn and that for all f ∈ C∞c (R n)N , ũ(x) := Γ̃(x, y)f (y) dy belongs to Y 1,2(Rn) and satisfies Lu = f in the sense of (3.48). Then by the uniqueness in H = Y 1,2(Rn)N , we must have u = ũ. Therefore, for all x ∈ Rn we (Γ− Γ̃)(x, ·)f = 0, ∀f ∈ C∞c (R n)N , and thus we have Γ ≡ Γ̃ in {(x, y) ∈ Rn × Rn : x 6= y}. � Theorem 3.2. Assume that the operators L and tL satisfy the property (H). If f ∈ (L n+2 (Rn) ∩ L loc(R n))N for some p > n/2, then there exists a unique u in GREEN FUNCTION ESTIMATES 13 Y 1,2(Rn)N such that (3.59) ij Dβu f ivi, ∀v ∈ Y 1,2(Rn)N . Moreover, u is continuous and has the following representation: (3.60) uk(x) = Γki(x, y)f i(y) dy, k = 1, . . . , N, where (Γki(x, y)) k,i=1 is the fundamental matrix of L. Proof. Since f ∈ L n+2 (Rn)N , the same argument as appears in Sec. 3.1 implies that there is u ∈ Y 1,2(Rn)N satisfying (3.59). If we set vi = Γ ki(x, · ) in (3.59), then (3.2) implies that (3.61) ki(x, · )f ji Dα ik( · , x)Dβu j = − Bρ(x) Next, note that (3.20), (3.26), and the assumption f i ∈ L n+2 (Rn) ∩ L loc(R n) for some p > n/2, imply that ki(x, · )f i = lim B1(x) ki(x, · )f Rn\B1(x) ki(x, · )f B1(x) Γki(x, · )f Rn\B1(x) Γki(x, · )f Γki(x, · )f (3.62) Finally, by the same argument which lead to (3.14) in Sec. 3.2, we find that u is Hölder continuous, and thus (3.60) follows from (3.61) and (3.62). � Corollary 3.3. Suppose that f = (f1, . . . , fN )T has a bound (3.63) |f(x)| ≤ C(1 + |x|)−(1+n/2+ǫ) ∀x ∈ Rn for some ǫ > 0. Then, u = (u1, . . . , uN)T given by (3.60) is a unique Y 1,2(Rn)N solution of Lu = f in Rn in the sense of (3.59). Proof. Note that (3.63) implies f ∈ (L n+2 (Rn) ∩ Lp(Rn))N . � Theorem 3.4. Assume that L and tL satisfy the property (H). If f ∈ Y 1,2(Rn)N satisfies Df ∈ L loc(R n)N×n for some p > n, then (3.64) fk(x) = DαΓki(x, · )A ij Dβf j , k = 1, . . . , N, where (Γki(x, y)) k,i=1 is the fundamental matrix of L. Proof. We denote by tΓρ the averaged fundamental matrix of tL. Recall that columns of tΓρ belong to H. Then, by (3.5) we have ji Dα ik( · , x)Dβf j = − Bρ(x) 14 S. HOFMANN AND S. KIM As in (3.62), the assumption Df ∈ L loc(R n)N for p > n, together with (3.22) and (3.24) yields Bρ(x) fk = lim B1(x) Rn\B1(x) ji Dα ik( · , x)Dβf B1(x) Rn\B1(x) ji Dα tΓik( · , x)Dβf ij DαΓki(x, · )Dβf (3.65) where we used (3.43) in the last step. By the Morrey’s inequality [17], f is contin- uous and thus (3.64) follows from (3.65). � Corollary 3.5. Assume that L, tL, L̃, and tL̃ satisfy the property (H). Denote by Γ and Γ̃ the fundamental matrices of L and L̃, respectively. If the coefficients A of L and à ij of L̃ are Hölder continuous, then (3.66) Γ̃lm(x, y) = Γlm(x, y) + DαΓli(x, · )(A ij − à ij )DβΓ̃jm( · , y), x 6= y. Proof. We denote by Γρ and Γ̃ρ (ρ < |x− y| /4) the averaged fundamental matrices of L and L̃ respectively. Recall that columns of Γρ and Γ̃ρ belong to H. Moreover, since we assume that the coefficients are Hölder continuous, the standard elliptic theory, (3.38), and (3.45) implies thatDΓρ(x, · ) andDΓ̃ρ( · , y) are locally bounded. Therefore, by setting f j = Γ̃ jm( · , y) in (3.64) we have (3.67) Γ̃ lm(x, y) = DαΓli(x, · )A ij DβΓ̃ jm( · , y), Next, set f j = Γ lj(x, · ) and apply (3.64) with L replaced by tL̃ to get lm(x, y) = t̃Γmi(y, · ) ij DβΓ lj(x, · ). By using (3.43) and interchanging indices (α ↔ β, i ↔ j), we obtain (3.68) Γ lm(x, y) = li(x, · )à ij DβΓ̃jm( · , y). Now, set r = |x− y| /4 and split the integral (3.67) into three pieces (recall ρ < r) Br(x) Br(y) Rn\(Br(x)∪Br(x)) DαΓli(x, · )A ij DβΓ̃ jm( · , y). Since we assume that the coefficients are Hölder continuous, it follows from the stan- dard elliptic theory that DΓ(x, · ) and DΓ̃( · , y) are continuous (and thus bounded) on Br(y) and Br(x) respectively. Moreover, (3.45) implies DΓ̃ρ( · , y) → DΓ̃( · , y) uniformly on Br(x) as ρ → 0. Therefore, as in (3.65), we may take the limit ρ → 0 in (3.67) to get Γ̃lm(x, y) = DαΓli(x, · )A ij DβΓ̃jm( · , y), GREEN FUNCTION ESTIMATES 15 Similarly, by taking the limit ρ → 0 in (3.68), we obtain Γlm(x, y) = DαΓli(x, · )à ij DβΓ̃jm( · , y). The proof is complete. � Remark 3.6. We note that in terms of matrix multiplication (3.60) is written as u(x) = Γ(x, y)f (y) dy, where both u,f are understood as column vectors. Also, (3.66) reads Γ̃(x, y) = Γ(x, y) + DαΓ(x, · )(A αβ − Ãαβ)DβΓ( · , y). 4. Green’s matrix in general domains 4.1. Construction of Green’s matrix. In this section, we shall construct the Green’s matrix in any open, connected set Ω ⊂ Rn, where n ≥ 3. To construct the Green’s matrix in Ω, we need to adjust arguments in Section 3. Henceforth, we shall denote Ωr(y) := Ω ∩Br(y) and dy := dist(y, ∂Ω). Also, as in Section 3, we use the letter C to denote a constant depending on n, N , λ, Λ, µ0, H0, and sometimes on an exponent p characterizing Lebesgue classes. It is routine to check that for any given y ∈ Ω and 1 ≤ k ≤ N , the linear functional u 7→ − Ωρ(y) uk is bounded on H = Y 0 (Ω) N . Therefore, by Lax-Milgram lemma, there exists a unique vρ = vρ;y,k ∈ H such that (4.1) ij Dβv ρ Dαu i = − Ωρ(y) uk, ∀u ∈ H. Note that as in (3.3), we have (4.2) ‖Dvρ‖L2(Ω) = ‖vρ‖H ≤ C |Ωρ(y)| We define the “averaged Green’s matrix” Gρ( · , y) = (G jk( · , y)) j,k=1 by jk( · , y) = v ρ = v ρ;y,k. Note that as in (3.5), we have (4.3) ij DβG jk( · , y)Dαu i = − Ωρ(y) uk, ∀u ∈ H. Next, observe that as in (3.7)–(3.10), for any given f ∈ L∞c (Ω) N , there exists a unique u ∈ H such that ik( · , y)f i = − Ωρ(y) Moreover, as in (3.11), we have ‖Du‖L2(Ω) ≤ C ‖f‖L2n/(n+2)(Ω) . Also, by following the argument as appears in Section 3.2, we find that if f is supported in BR(y), then we have ‖u‖L∞(BR/4(y)) ≤ CR 2−n/p ‖f‖Lp(BR(y)) , ∀R < dy, ∀p > n/2. 16 S. HOFMANN AND S. KIM Therefore, as in (3.16), for any f ∈ L∞c (BR(y)), R < dy, we have BR(y) ik( · , y)f ≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀ρ < R/4, ∀p > n/2. Therefore, as in (3.17), we see that if R < dy, then ‖Gρ( · , y)‖Lq(BR(y)) ≤ CR 2−n+n/q, ∀ρ < R/4, ∀q ∈ [1, n Then, by following the lines in (3.18)–(3.19), we obtain |Gρ(x, y)| ≤ C |x− y| if |x− y| < dy/2, ∀ρ < |x− y| /3. Next, we shall derive an estimate corresponding to (3.22). Let η ∈ C∞(Rn) be a cut-off function such that 0 ≤ η ≤ 1, η ≡ 1 outside BR/2(y), η ≡ 0 on BR/4(y), and |Dη| ≤ C/R, where R ≤ dy . By setting u = η 2vρ ∈ H in (4.1), we obtain η2 |Dvρ| ≤ CR−2 BR/2(y)\BR/4(y) ≤ CR−2 BR/2(y)\BR/4(y) |x− y| 2(2−n) = CR−2R4−n = CR2−n, ∀ρ < R/12. (4.4) Therefore, we have (r = R/2) (4.5) Ω\Br(y) |DGρ( · , y)| ≤ Cr2−n, ∀ρ < r/6, ∀r < dy/2. On the other hand, (4.2) implies that if ρ ≥ r/6, then (4.6) Ω\Br(y) |DGρ( · , y)| |DGρ( · , y)| ≤ C |Ωρ(y)| n ≤ Cr2−n. Therefore, by combining (4.5) and (4.6), we obtain (4.7) Ω\Br(y) |DGρ( · , y)| ≤ Cr2−n, ∀r < dy/2, ∀ρ > 0. From the estimate (4.7), which corresponds to (3.22), we can derive an estimate corresponding to (3.24) as follows. By following the lines between (3.22) and (3.23), we obtain (4.8) |{x ∈ Ω : |DxG ρ(x, y)| > t}| ≤ Ct− n−1 , ∀ρ > 0 if t > (dy/2) Then, by following lines (3.24)–(3.25), we find (set τ = (R/2)1−n) (4.9) BR(y) |DGρ( · , y)| ≤ CRp(1−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0, Now, we will derive estimates corresponding (3.20) and (3.26). Let η be the same as in (4.4). Note that (4.4) and (4.7) implies that for R < dy, (4.10) |D(ηvρ)| η2 |Dvρ| ≤ CR2−n, ∀ρ < R/12. Since ηvρ ∈ H = Y 0 (Ω), it follows from (4.10) and (2.13) that (4.11) Ω\Br(y) 2∗ ≤ Cr−n, ∀r < dy/2, ∀ρ < r/6. GREEN FUNCTION ESTIMATES 17 On the other hand, if ρ ≥ r/6, then (4.2) implies Ω\Br(y) 2∗ ≤ C |Dvρ| )2∗/2 ≤ C |Ωρ| ≤ Cr−n. (4.12) Therefore, by combining (4.11) and (4.12), we obtain (4.13) Ω\Br(y) |Gρ( · , y)| 2 ≤ Cr−n, ∀r < dy/2, ∀ρ > 0. As in Section 3.3, the above estimate (4.13) yields (4.14) |{x ∈ Ω : |Gρ(x, y)| > t}| ≤ Ct− n−2 , ∀ρ > 0 if t > (dy/2) Then, as we argued in (4.9), we find (set τ = (R/2)2−n) (4.15) BR(y) |Gρ( · , y)| ≤ CRp(2−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0, Now, observe that (4.9) and (4.15) in particular imply that (4.16) ‖Gρ( · , y)‖W 1,p(Bdy (y)) ≤ C(dy) for some p ∈ (1, ), uniformly in ρ. Therefore, from (4.16) together with (4.7) and (4.13), it follows that there exist a sequence {ρµ} tending to 0 and functions G( · , y) and G̃( · , y) such that ρµ( · , y) ⇀ G( · , y) in W 1,p(Bdy (y)) N×N and(4.17) ρµ( · , y) ⇀ G̃( · , y) in Y 1,2(Ω \Bdy/2(y)) N×N as µ → ∞.(4.18) Since G( · , y) ≡ G̃( · , y) on Bdy(y) \Bdy/2(y), we shall extend G( · , y) to entire Ω by setting G( · , y) = G̃( · , y) on Ω \ Bdy(y) but still call it G( · , y) in the sequel. Moreover, by applying a diagonalization process and passing to a subsequence, if necessary, we may assume that (4.19) Gρµ( · , y) ⇀ G( · , y) in Y 1,2(Ω \Br(y)) N×N as µ → ∞, ∀r < dy. We claim that the following holds: (4.20) ij DβGjk( · , y)Dαφ i = φk(y), ∀φ ∈ C∞c (Ω) To see (4.20), write φ = ηφ+(1− η)φ, where η ∈ C∞c (Bdy (y)) is a cut-off function satisfying η ≡ 1 on Bdy/2(y). Then, (4.3), (4.17), and (4.19) yield φk(y) = lim Ωρµ (y) ηφk + lim Ωρµ (y) (1 − η)φk = lim ij DβG jk (·, y)Dα(ηφ i) + lim ij DβG jk (·, y)Dα((1− η)φ ij DβGjk( · , y)Dα(ηφ ij DβGjk( · , y)Dα((1 − η)φ ij DβGjk( · , y)Dαφ i as desired. Next, we claim thatG( · , y) = 0 on ∂Ω in the sense that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy, we have (1− η)G( · , y) ∈ Y 0 (Ω) N×N . 18 S. HOFMANN AND S. KIM To see this, it is enough to show that (4.21) (1 − η)Gρµ( · , y) ⇀ (1− η)G( · , y) in Y 1,2(Ω)N×N as µ → ∞, for (1 − η)Gρµ( · , y) ∈ Y 0 (Ω) N×N for all µ ≥ 1 and Y 0 (Ω) is weakly closed in Y 1,2(Ω) by Mazur’s theorem. To show (4.21), we note that (4.19) yields (1− η)Gkl( · , y)φ = Gkl( · , y)(1− η)φ = lim kl ( · , y)(1− η)φ = lim (1− η)G kl ( · , y)φ, ∀φ ∈ L n+2 (Ω), D((1 − η)Gkl( · , y)) · ψ = − Gkl( · , y)Dη ·ψ + DGkl( · , y) · (1 − η)ψ = − lim kl ( · , y)Dη · ψ + limµ→∞ kl ( · , y) · (1− η)ψ = lim D((1 − η)G kl ( · , y)) · ψ, ∀ψ ∈ L 2(Ω)N . By using the same duality argument as in (3.29), we derive the following esti- mates that correspond to (3.30)–(3.36): ‖G( · , y)‖Lp(Br(y)) ≤ Cp r 2−n+n/p, ∀r < dy, ∀p ∈ [1, ),(4.22) ‖DG( · , y)‖Lp(Br(y)) ≤ Cp r 1−n+n/p, ∀r < dy, ∀p ∈ [1, ),(4.23) ‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr 1−n/2, ∀r < dy/2,(4.24) |{x ∈ Ω : |G(x, y)| > t}| ≤ Ct− n−2 , ∀t > (dy/2) 2−n,(4.25) |{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct n−1 , ∀t > (dy/2) 1−n.(4.26) Also, we obtain pointwise bound and Hölder continuity estimate for G( · , y) corresponding to (3.38) and (3.39), respectively, as follows. Denote by vT the k-th column of G( · , y) and set R := d̄x,y/2, where (4.27) d̄x,y := min(dx, dy, |x− y|). Since v is a weak solution of Lu = 0 in B3R/2(x) ⊂ Ω \ BR/2(y), it follows from (2.10) and (4.24) that |v(x)| ≤ CR(2−n)/2 ‖v‖L2∗(Ω\BR/2(y)) ≤ CR which in turn implies that (4.28) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|). In particular, we have (4.29) |G(x, y)| ≤ C |x− y| if |x− y| < dx/2 or |x− y| < dy/2. Similarly, it follows from (2.11) and (4.24) that (4.30) [v]2Cµ0(BR(x)) ≤ CR 2−n−2µ0 B3R/2(x) ≤ CR2(2−n−µ0). Therefore, we find that (4.31) |G(x, y)−G(z, y)| ≤ C |x− z| µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2, where d̄x,y is given by (4.27). GREEN FUNCTION ESTIMATES 19 Denote by tGσ( · , x) the averaged Green’s matrix of tL in Ω with a pole at x ∈ Ω. Observe that we have an identity corresponding to (3.42). (4.32) − Ωρ(y) tGσkl( · , x) = − Ωσ(x) lk( · , y). Let tG( · , x) be a Green’s matrix of tL in Ω with a pole at x ∈ Ω that is obtained by a sequence {σν} ν=1 tending to 0. Then, by a similar argument as appears in Section 3.6, we obtain (4.33) Glk(x, y) = tGkl(y, x), ∀k, l = 1, . . . , N, ∀x, y ∈ Ω, x 6= y, which is equivalent to say (4.34) G(x, y) = tG(y, x)T , ∀x, y ∈ Ω, x 6= y. Using (4.34), we find that G(x, · ) satisfies the estimates corresponding to (4.22)– (4.26) and (4.31). Moreover, by following the lines (3.44)–(3.45) and using (4.32) we obtain (4.35) Gρ(x, y) = − Ωρ(y) G(x, z) dz. Therefore, by the continuity, we find (4.36) lim ρ(x, y) = G(x, y), ∀x, y ∈ Ω, x 6= y. Finally, we summarize what we obtained so far in the following theorem. Theorem 4.1. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω) for x ∈ Ω; we set dx = ∞ if Ω = R n. Assume that operators L and tL satisfy the property (H). Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y)) i,j=1 (x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈ C∞c (Ω) N , the function u = (u1, . . . , uN )T given by (4.37) u(x) := G(x, y)f (y) dy belongs to Y 0 (Ω) N and satisfies Lu = f in the sense (4.38) ij Dβu f iφi, ∀φ ∈ C∞c (Ω) Moreover, G(x, y) has the properties that (4.39) ij DβGjk( · , y)Dαφ i = φk(y), ∀φ ∈ C∞c (Ω) and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy, (4.40) (1− η)G( · , y) ∈ Y 0 (Ω) N×N . Furthermore, G(x, y) satisfies the following estimates: ‖G( · , y)‖Lp(Br(y)) ≤ Cp r 2−n+n/p, ∀r < dy, ∀p ∈ [1, ),(4.41) ‖G(x, · )‖Lp(Br(x)) ≤ Cp r 2−n+n/p, ∀r < dx, ∀p ∈ [1, ),(4.42) ‖DG( · , y)‖Lp(Br(y)) ≤ Cp r 1−n+n/p, ∀r < dy, ∀p ∈ [1, ),(4.43) ‖DG(x, · )‖Lp(Br(x)) ≤ Cp r 1−n+n/p, ∀r < dx, ∀p ∈ [1, ),(4.44) 20 S. HOFMANN AND S. KIM ‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr 1−n/2, ∀r < dy/2,(4.45) ‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr 1−n/2, ∀r < dx/2,(4.46) |{x ∈ Ω : |G(x, y)| > t}| ≤ Ct− n−2 , ∀t > (dy/2) 2−n,(4.47) |{y ∈ Ω : |G(x, y)| > t}| ≤ Ct− n−2 , ∀t > (dx/2) 2−n,(4.48) |{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct n−1 , ∀t > (dy/2) 1−n.(4.49) |{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct n−1 , ∀t > (dx/2) 1−n,(4.50) (4.51) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|), |G(x, y)−G(z, y)| ≤ C |x− z| µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2,(4.52) |G(x, y)−G(x, z)| ≤ C |y − z| µ0 d̄2−n−µ0x,y if |y − z| < d̄x,y/2,(4.53) where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0. Proof. LetGρ(x, y) andG(x, y) be constructed as above. We have already seen that G is continuous on {(x, y) ∈ Ω× Ω : x 6= y} and satisfies all the properties (4.39) – (4.53). Also, as in the proof of Theorem 3.1, we find that for all f ∈ C∞c (Ω) there is a unique u ∈ (Y 0 (Ω) ∩ C(Ω)) N satisfying ij Dβu f ivi, ∀v ∈ Y 0 (Ω) If we set vi = G ki(x, · ) above, then by (4.3) and (4.34), we find (4.54) ki(x, · )f ji Dα ik( · , x)Dβu j = − Ωρ(x) Fix r < dx/2. By (4.17), (4.18), and (4.34), we have ki (x, · )f i = lim Br(x) ki (x, · )f Ω\Br(x) ki (x, · )f B1(x) Gki(x, · )f Ω\B1(x) Gki(x, · )f Gki(x, · )f Therefore, (4.37) follows by taking the limits in (4.54). By proceeding as in the proof of Theorem 3.1, we also derive the uniqueness of Green’s matrix in Ω. � 4.2. Boundary regularity. Let Σ be any subset of Ω and u be aW 1,2(Ω) function. Then we shall say u = 0 on Σ (in the sense of W 1,2(Ω)) if u is a limit in W 1,2(Ω) of a sequence of functions in C∞c (Ω \ Σ). We shall denote ΣR(x) := ∂Ω ∩ BR(x) for any R > 0. We shall abbreviate ΩR = ΩR(x) and ΣR = ΣR(x) if the point x is well understood in the context. Lemma 4.2 (Boundary Poincaré inequality). Assume that |BR \ Ω| ≥ θ |BR| for some θ > 0. Then, for any u ∈ W 1,2(ΩR) satisfying u = 0 on ΣR, we have the following estimate: (4.55) ‖u‖L2(ΩR) ≤ R ‖Du‖L2(ΩR) . GREEN FUNCTION ESTIMATES 21 Proof. Since u = 0 in ΣR, we may extend u to a W 1,2(BR) function by setting u = 0 in S := BR \Ω. Note that Du = 0 in S. Then the lemma follows from (7.45) in [10, p. 164]. � Lemma 4.3 (Boundary Caccioppoli inequality). Let the operator L satisfy condi- tions (2.2), (2.3). Suppose u is a W 1,2(ΩR) N solutions of Lu = 0 in ΩR satisfying u = 0 on ΣR. Then, we have (4.56) ‖Du‖L2(Ωr) ≤ ‖u‖L2(ΩR) , ∀0 < r < R, where C = C(n,N, λ,Λ) > 0. Proof. It is well known. � Definition 4.4. We say that Ω satisfies the condition (S) at a point x̄ ∈ ∂Ω if there exist θ > 0 and Ra ∈ (0,∞] such that (4.57) |BR(x̄) \ Ω| ≥ θ |BR(x̄)| , ∀R < Ra. We say that Ω satisfies the condition (S) uniformly on Σ ⊂ ∂Ω if there exist θ > 0 and Ra such that (4.57) holds for all x̄ ∈ Σ. Definition 4.5. Let Ω satisfy the condition (S) at x̄ ∈ ∂Ω. We shall say that an operator L satisfies the property (BH) if there exist µ1, H1 > 0 such that if u ∈ W 1,2(ΩR(x̄)) N is a weak solution of the problem, Lu = 0 in ΩR(x̄) and u = 0 on ΣR(x̄), where R < Ra, then u satisfies the following estimates: (4.58) Ωr(x̄) )n−2+2µ1 Ωs(x̄) , ∀0 < r < s ≤ R. Lemma 4.6. There exists ǫ0 = ǫ0(n, λ0,Λ0) > 0 such that if the coefficients of the operator L in (2.1) satisfies (2.8) in Lemma 2.2, then L satisfies the property (BH) with µ1 = µ1(n, λ0,Λ0, θ) > 0 and H1 = H1(n,N, λ0,Λ0, θ) > 0. Proof. Throughout the proof, we shall abbreviate Ωr = Ωr(x̄) for any r > 0, Σr = Σr(x̄), the point x̄ ∈ ∂Ω to be understood. For any s ≤ R < Ra, let vi (i = 1, . . . , N) be a unique W 1,2(Ωs) solution of L0v i = 0 in Ωs satisfying vi − ui ∈ W 0 (Ωs), where L0v i = −Dα(a αβDβv We claim that there exist µ2(n, λ0,Λ0, θ) > 0 and C(n, λ0,Λ0, θ) > 0 such that the following estimate holds: (4.59) )n−2+2µ2 , ∀0 < r < s. We first note that we may assume that r ≤ s/8; otherwise (4.59) becomes trivial. Since each vi satisfies vi = 0 on Σs, it follows from Theorem 8.27 [10, pp. 203–204] and Theorem 8.25 [10, pp. 202–203] that there is µ2 = µ2(n, λ0,Λ0, θ) > 0 and C = C(n, λ0,Λ0, θ) > 0 such that (4.60) osc vi ≤ Crµ2s−µ2 sup |vi| ≤ Crµ2s−µ2−n/2‖vi‖L2(Ωs/2). 22 S. HOFMANN AND S. KIM In particular, the estimate (4.60) implies vi(x̄) = lim vi(x) = 0. Then, Lemma 4.3 and Lemma 4.2 imply that for all i = 1, . . . , N (recall r < s/8) ≤ Cr−2 |vi|2 = Cr−2 |vi − vi(x̄)|2 ≤ Crn−2 )n−2+2µ2 |vi|2 )n−2+2µ2 |Dvi|2, and thus we have proved the claim. Next, note that w := u− v belongs to W 0 (Ωs) N and thus it satisfies aαβDβw (aαβδij −A ij )Dβu Therefore, we have (4.61) ≤ (λ−1 ‖ǫ‖L∞) where ǫ(x) is as defined in (2.8). By combining (4.59) and (4.61), we obtain )n−2+2µ2 + C0 ‖ǫ‖ , ∀0 < r < s. Now, choose a µ1 ∈ (0, µ2). Then, from a well known iteration argument (see, e.g., [8, Lemma 2.1, p. 86]), it follows that there is ǫ0 such that if ‖ǫ‖L∞ < ǫ0, then (4.58) holds. � Theorem 4.7. Let the operator L satisfy the properties (H) and (BH). Assume that Ω satisfies the condition (S) at x̄ ∈ ∂Ω with parameters θ,Ra. Let x ∈ Ω such that |x− x̄| = dx ≤ R/2, where R < Ra is given. Then, any weak solution u of Lu = 0 in ΩR(x̄) satisfying u = 0 on ΣR(x̄), we have (4.62) |u(x)| ≤ CdµxR 1−n/2−µ ‖Du‖L2(ΩR(x̄)) , dx := dist(x, ∂Ω), where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1). Proof. The proof is an adaptation of a technique due to Campanato [4]. In this proof, we shall use the notation ux,r := − Ωr(x) u. Also, we shall abbreviate d = dx. Observe that (4.63) Ωd(x) = Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄). We may assume that R > 3d so that Ω2d(x) ⊂ ΩR(x̄); otherwise 2d ≤ R ≤ 3d and (4.62) follows from Lemma 2.4. We estimate u(x) by |u(x)| ≤ |u(x)− ux,2d|+ |ux,2d − ux̄,2d|+ |ux̄,2d| := I + II + III. We shall estimate I first. For any r1 < r2 ≤ 2d, we estimate (4.64) |ux,r1 − ux,r2 | ≤ 2 |u(z)− ux,r1| + 2 |u(z)− ux,r2 | Note that since Bd(x) ⊂ Ω, we have |Ωr(x)| ≥ Cr n, ∀r ≤ 2d. GREEN FUNCTION ESTIMATES 23 Therefore, by integrating (4.64) over Ωr1(x) with respect to z, we estimates (4.65) |ux,r1 − ux,r2 | ≤ Cr−n1 |u− ux,r1 | |u− ux,r2 | Since u = 0 on ΣR(x̄), we may extend u to BR(x̄) as a W 1,2 function by setting u = 0 on BR(x̄) \Ω. Therefore, by a version of Poincaré inequality (see, e.g. (7.45) in [10, p. 164]), we have for all r ≤ 2d, (4.66) |u− ux,r| |u− ux,r| ≤ Cr2 = Cr2 Therefore, by (4.65) and (4.66), we obtain (4.67) |ux,r1 − ux,r2 | ≤ Cr−n1 Ωr1(x) + r22 Ωr2 (x) Next, we claim that the following estimate holds: (4.68) Ωr(x) )n−2+2µ ΩR(x̄) , ∀r ≤ 2d. We first consider the case when r ≤ d. Note that in this case, we have Ωr(x) = Br(x) and Ωd(x) = Bd(x). Since L satisfies (H), it follows from (4.63) that (4.69) Ωr(x) )n−2+2µ Ωd(x) )n−2+2µ Ω2d(x̄) On the other hand, since L satisfies (BH), it follows from (4.58) that (4.70) Ω2d(x̄) )n−2+2µ ∫ ΩR(x̄) By combining (4.69) and (4.70), we obtain (4.68). Next, consider the case when d < r. In this case, we have Ωr(x) ⊂ Ω2r(x̄), and thus it follows from (4.58) Ωr(x) Ω2r(x̄) )n−2+2µ ΩR(x̄) We proved the claim (4.68). Now, by using (4.68), we estimates (4.67) as follows (recall r1 < r2 ≤ 2d): (4.71) |ux,r1 − ux,r2 | ≤ Cr−n1 (r 1 + r 2−n−2µ ΩR(x̄) For any r ≤ 2d, set r1 = r2 −(i+1) and r2 = r2 −i in (4.71) to get ∣ux,r2−(i+1) − ux,r2−i ≤ Cr2µ2−2µ(i+1)R2−n−2µ ΩR(x̄) Therefore, for 0 ≤ j < k, we obtain ∣ux,r2−k − ux,r2−j ∣ux,r2−(i+1) − ux,r2−i ≤ Crµ 2−µ(i+1) R1−n/2−µ ‖Du‖L2(ΩR(x̄)) = C2−jµrµR1−n/2−µ ‖Du‖L2(ΩR(x̄)) . (4.72) 24 S. HOFMANN AND S. KIM By setting r = 2d, j = 0, and letting k → ∞ in (4.72), we obtain (4.73) I = |u(x) − ux,2d| ≤ Cd µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) . Next, we estimate III. Since |Br(x̄) ∩Bd(x)| ≥ Cr n for r ≤ 2d, we have (4.74) |Ωr(x̄)| ≥ Cr n, ∀r ≤ 2d. Also, as in (4.66), we have for all r ≤ 2d (recall u ≡ 0 on BR(x̄) \ Ω) (4.75) |u− ux̄,r| |u− ux̄,r| ≤ Cr2 = Cr2 Therefore, as in (4.67) we have for r1 < r2 ≤ 2d, |ux̄,r1 − ux̄,r2 | ≤ Cr−n1 Ωr1(x̄) + r22 Ωr2 (x̄) Then, by using the property (BH), we obtain (c.f. (4.72), (4.73)) (4.76) |û(x̄)− ux̄,2d| ≤ Cd µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) , where û(x̄) := limk→∞ ux̄,2−kr. (note that (4.72) implies û(x̄) exists). It follows from (4.74), (4.55), and (4.58) that for any r ≤ 2d, we have |ux̄,r| Ωr(x̄) ≤ Cr−n Ωr(x̄) ≤ Cr2−n Ωr(x̄) ≤ Cr2−n )n−2+2µ ΩR(x̄) = Cr2µR2−n−2µ ΩR(x̄) and thus that û(x̄) = 0. Therefore, by (4.76) we obtain (4.77) III = |ux̄,2d| = |û(x̄)− ux̄,2d| ≤ Cd µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) . Finally, we estimate II. (4.78) |ux,2d − ux̄,2d| ≤ 2 |u(z)− ux,2d| + 2 |u(z)− ux̄,2d| By integrating (4.78) over Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄) with respect to z, we estimate |ux,2d − ux̄,2d| ≤ Cd−n Ω2d(x) |u− ux,2d| Ω2d(x̄) |u− ux̄,2d| ≤ Cd2−n Ω2d(x) Ω2d(x̄) ≤ Cd2µR2−n−2µ ΩR(x̄) (4.79) where we have used (4.66), (4.75), (4.68), and (4.58). Therefore, by combining (4.73), (4.77), and (4.79), we obtain (4.62). � Theorem 4.8. Let the operators L, tL satisfy the properties (H) and (BH). Assume that Ω satisfies the condition (S) uniformly on ∂Ω with parameters θ,Ra. Denote Rx,y := min(|x− y| , 4Ra). GREEN FUNCTION ESTIMATES 25 Then the Green matrix G(x, y) satisfies |G(x, y)| ≤ CdµxR 1−n/2−µ x,y d 1−n/2 y if dx ≤ Rx,y/8,(4.80) |G(x, y)| ≤ CdµyR 1−n/2−µ x,y d 1−n/2 x if dy ≤ Rx,y/8,(4.81) where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1). As a conse- quence, we have G( · , y) = 0, G(x, · ) = 0 on ∂Ω in the usual sense. Proof. We only need to prove (4.80), for (4.81) will then follow from (4.34). Set R = Rx,y/4, r = dy/2, and choose x̄ ∈ ∂Ω such that |x− x̄| = dx. Then, since dy ≤ |x− y|+ dx ≤ |x− y| , we have |y − x̄| ≥ |x− y| − dx ≥ |x− y| ≥ R+ r, and thus, ΩR(x̄) ⊂ Ω\Br(y). Now, we apply Theorem 4.7 with u = G( · , y). Then, by (4.62) and (4.24), we obtain |G(x, y)| ≤ CdµxR 1−n/2−µ ‖DG( · , y)‖L2(Ω\Br(y)) ≤ Cd 1−n/2−µ x,y d 1−n/2 The proof is complete. � Remark 4.9. We note that in the scalar case, the maximum principle yields (see [11, Theorem 1.1]) (4.82) G(x, y) ≤ C |x− y| , ∀x 6= y ∈ Ω. Then, by the boundary Caccioppoli inequality, we have (c.f. (4.4)–(4.7)) Ω\Br(y) |DG( · , y)| ≤ Cr2−n, ∀r > 0. Therefore, in the scalar case we don’t need to require that r < dy/2 (or r < dx/2) in the proof of Theorem 4.8 and we may as well set r = |x− y| /2 to get G(x, y) ≤ CdµxR 1−n/2−µ x,y |x− y| 1−n/2 if dx ≤ Rx,y/8, G(x, y) ≤ CdµyR 1−n/2−µ x,y |x− y| 1−n/2 if dy ≤ Rx,y/8. In particular, if G(x, y) is the Green’s function on Rn+, then we obtain G(x, y) ≤ Cdµx |x− y| 2−n−µ if dx ≤ |x− y| /8, G(x, y) ≤ Cdµy |x− y| 2−n−µ if dy ≤ |x− y| /8, for ∂Rn+ satisfies the condition (S) with θ = 1/2 and Ra = ∞. 5. Remarks on VMO coefficients case Definition 5.1 (Sarason [19]). For a measurable function f defined on Rn, we shall denote fx,r = − Br(x) f and for 0 < δ < ∞ we define (5.1) Mδ(f) := sup Br(x) ∣f − fx,r ∣ ; M0(f) := lim Mδ(f). We shall say that f belongs to VMO if M0(f) = 0. 26 S. HOFMANN AND S. KIM Definition 5.2. We say that the operator L satisfies the property (H)loc if there exist µ0, H0, Rc > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) with R ≤ Rc satisfy (5.2) )n−2+2µ0 , 0 < r < s ≤ R. Similarly, we say that the transpose operator tL satisfies the property (H)loc if corresponding estimates hold for all weak solutions u of tLu = 0 in BR with R ≤ Rc. Lemma 5.3. Let the coefficients of the operator L in (2.1) satisfy the conditions (2.2) and (2.3). If the coefficients belong to VMO in addition, then L satisfies the property (H)loc. Proof. It is well known that if the coefficients are uniformly continuous, then L satisfies the property (H)loc; see e.g. [8, pp. 87–89]. Essentially, the same proof carries over to the VMO coefficients case. One only needs to make a note of the following two facts. First, a theorem of Meyers [16] implies that there is some p = p(n,N, λ,Λ) > 2 such that if u is a weak solution of Lu = 0 in BR(x), then Br(x) B2r(x) , ∀r < R/2. Secondly, note that the John-Nirenberg theorem [13] implies that Br(x) ∣f − fr,x ≤ C(n, q)Mδ(f), ∀r < c(n)δ, ∀q ∈ (0,∞), where Mδ(f) is defined as in (5.1). For the details, we refer to [3, pp. 47–48]. � In the rest of this section, we shall assume that the operators L and tL satisfy the property (H)loc with parameters µ0, H0, Rc. We shall denote (5.3) rx := min(dx, Rc), r̄x,y := min(d̄x,y, Rc), where dx = dist(x, ∂Ω) and d̄x,y is as in (4.28). It is routine to check that all estimates appearing in Section 4.1 remain valid if dx, d̄x,y are replaced by rx, r̄x,y, respectively. Therefore, we have the following theorem: Theorem 5.4. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω) for x ∈ Ω; we set dx = ∞ if Ω = R n. Assume that operators L and tL satisfy the prop- erty (H)loc. Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y)) i,j=1 (x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈ C∞c (Ω) N , the function u = (u1, . . . , uN )T given by (5.4) u(x) := G(x, y)f (y) dy belongs to Y 0 (Ω) N and satisfies Lu = f in the sense (5.5) ij Dβu f iφi, ∀φ ∈ C∞c (Ω) Moreover, G(x, y) has the properties that (5.6) ij DβGjk( · , y)Dαφ i = φk(y), ∀φ ∈ C∞c (Ω) GREEN FUNCTION ESTIMATES 27 and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy, (5.7) (1− η)G( · , y) ∈ Y 0 (Ω) N×N . Furthermore, G(x, y) satisfies the following estimates: For rx, ry, r̄x,y as in (5.3), ‖G( · , y)‖Lp(Br(y)) ≤ Cp r 2−n+n/p, ∀r < ry, ∀p ∈ [1, ),(5.8) ‖G(x, · )‖Lp(Br(x)) ≤ Cp r 2−n+n/p, ∀r < rx, ∀p ∈ [1, ),(5.9) ‖DG( · , y)‖Lp(Br(y)) ≤ Cp r 1−n+n/p, ∀r < ry , ∀p ∈ [1, ),(5.10) ‖DG(x, · )‖Lp(Br(x)) ≤ Cp r 1−n+n/p, ∀r < rx, ∀p ∈ [1, ),(5.11) ‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr 1−n/2, ∀r < ry/2,(5.12) ‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr 1−n/2, ∀r < rx/2,(5.13) |{x ∈ Ω : |G(x, y)| > t}| ≤ Ct− n−2 , ∀t > (ry/2) 2−n,(5.14) |{y ∈ Ω : |G(x, y)| > t}| ≤ Ct− n−2 , ∀t > (rx/2) 2−n,(5.15) |{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct n−1 , ∀t > (ry/2) 1−n,(5.16) |{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct n−1 , ∀t > (rx/2) 1−n,(5.17) (5.18) |G(x, y)| ≤ Cr̄2−nx,y , ∀x, y ∈ Ω, |G(x, y)−G(z, y)| ≤ C |x− z| µ0 r̄2−n−µ0x,y if |x− z| < r̄x,y/2,(5.19) |G(x, y)−G(x, z)| ≤ C |y − z| µ0 r̄2−n−µ0x,y if |y − z| < r̄x,y/2,(5.20) where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0. Remark 5.5. Dolzmann-Müller [6] derived a global estimate (5.21) |G(x, y)| ≤ C |x− y| ∀x, y ∈ Ω, x 6= y, assuming that Ω is a bounded C1 domain. We have not attempted to derive the corresponding estimate here. However, we would like to point out that the constant C in their estimate depends on the domain (e.g., the diameter of the domain and also some characteristics of ∂Ω) while our interior estimate (5.18) does not. References [1] Alfonseca, A.; Auscher, P.; Axelsson, A.; Hofmann, S.; Kim, S. Analyticity of layer potentials and L2 Solvability of boundary value problems for divergence form elliptic equations with complex L∞ coefficients. preprint. [2] Auscher, P. Regularity theorems and heat kernel for elliptic operators. J. London Math. Soc. (2) 54 (1996), no. 2, 284–296. [3] Auscher, P.; Tchamitchian, Ph. Square root problem for divergence operators and related topics. Astérisque No. 249 (1998) [4] Campanato, S. Equazioni ellittiche del II◦ ordine espazi L(2,λ). (Italian) Ann. Mat. Pura Appl. (4) 69 (1965) 321–381. [5] De Giorgi, E. Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli re- golari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. [6] Dolzmann, G.; Müller, S. Estimates for Green’s matrices of elliptic systems by Lp theory. Manuscripta Math. 88 (1995), no. 2, 261–273. 28 S. HOFMANN AND S. KIM [7] Fuchs, M. The Green matrix for strongly elliptic systems of second order with continuous coefficients. Z. Anal. Anwendungen 5 (1986), no. 6, 507–531. [8] Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton, NJ, 1983. [9] Giaquinta, M. Introduction to regularity theory for nonlinear elliptic systems. Birkhäuser Verlag, Basel, 1993. [10] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order, Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001. [11] Grüter, M.; Widman, K.-O. The Green function for uniformly elliptic equations. Manuscripta Math. 37 (1982), no. 3, 303–342. [12] Hofmann, S.; Kim, S. Gaussian estimates for fundamental solutions to certain parabolic systems. Publ. Mat. Vol. 48 (2004), pp. 481-496. [13] John, F.; Nirenberg, L. On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961) 415–426. [14] Littman, W.; Stampacchia, G.; Weinberger, H. F. Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 43–77. [15] Malý, J.; Ziemer, W. P. Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence, RI, 1997. [16] Meyers, N. G. An Lp-estimate for the gradient of solutions of second order elliptic diver- gence equations, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189–206. [17] Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Springer-Verlag New York, Inc., New York 1966 [18] Moser, J. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961) 577–591. [19] Sarason, D. Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975), 391–405. (S. Hofmann) Mathematics Department, University of Missouri, Columbia, Missouri 65211, United States of America E-mail address: [email protected] (S. Kim) Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia E-mail address: [email protected] 1. Introduction 2. Preliminaries 2.1. Strongly elliptic systems 2.2. Function spaces Y1,2() and Y1,20() 3. Fundamental matrix in Rn 3.1. Averaged fundamental matrix 3.2. L estimates for averaged fundamental matrix 3.3. Uniform weak-Lnn-2 estimates for bold0mu mumu (,y) 3.4. Uniform weak-Lnn-1 estimates for Dbold0mu mumu (y) 3.5. Construction of the fundamental matrix 3.6. Continuity of the fundamental matrix 3.7. Properties of fundamental matrix 4. Green's matrix in general domains 4.1. Construction of Green's matrix 4.2. Boundary regularity 5. Remarks on VMO coefficients case References

0704.1353

Supporting Knowledge and Expertise Finding within Australia's Defence Science and Technology Organisation

Supporting Knowledge and Expertise Finding within Australia's Defence Science and Technology Organisation Supporting Knowledge and Expertise Finding within Australia's Defence Science and Technology Organisation Paul Prekop DSTO Fern Hill, Department of Defence, Canberra ACT 2600 [email protected] Abstract This paper reports on work aimed at supporting knowledge and expertise finding within a large Research and Development (R&D) organisation. The paper first discusses the nature of knowledge important to R&D organisations and presents a prototype information system developed to support knowledge and expertise finding. The paper then discusses a trial of the system within an R&D organisation, the implications and limitations of the trial, and discusses future research questions. 1. Introduction This paper describes work undertaken to support knowledge and expertise finding within Australia's Defence Science and Technology Organisation (DSTO). DSTO is a government funded research and development (R&D) organisation, with a very broad, applied R&D program focused primarily within the defence and national security domains. DSTO employs approximately 1900 engineers and scientists across a wide range of academic disciplines (about 30% of staff hold PhDs), within seven sites throughout Australia. Like most other large R&D organisations [1, 2] and professional services firms, DSTO is a project-centric organisation; projects are formed to address specific questions or problems, or to develop specific products. The nature of the outcomes of the projects undertaken by DSTO varies considerably, and can range from academic papers and technical reports, through to prototype and working system development, and to professional services and consulting engagements. The work described in this paper is part of an ongoing knowledge management improvement program aimed at exploring: Methods to allow staff to build and maintain wide and detailed awareness of DSTO's past, current and planned projects; Methods to enable staff to locate other staff with relevant skills, interests, abilities or experience; Low cost (in terms of time and effort) methods to support the development of communities of interest, and less-formal collaboration and sharing within the organisation; Organisational cultural and behavioural issues that may act as barriers to effective knowledge and expertise sharing. A prototype information system, the Automated Research Management System (ARMS), was developed to explore approaches to addressing these issues. Section 2 discusses the nature of knowledge and knowledge management within the R&D environment, and describes the types of support for knowledge and expertise-finding needed within organisations such as DSTO. Section 3 describes ARMS and how it supports knowledge and expertise finding within DSTO. Section 4 outlines the ARMS trial and trial methodology, and Section 5 discusses the results of two studies undertaken as part of the ARMS trial. Finally, Section 6 discusses the implications and limitations of the work undertaken so far and describes potential areas for future work. 2. The Nature of Knowledge and Expertise within R&D Organisations 2.1. Theoretical Background The main theoretical idea underpinning this work is that the knowledge important to an organisation, or that makes it unique or gives it a competitive advantage, is embedded in key elements that make up the organisation [3–5]. According to [3], this knowledge is embedded in three key organisational elements – the members of the organisation, the tools used within the organisation, and the tasks performed by the organisation. For many ©1530-1605/07 $20.00 Commonwealth of Australia 2007 Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 organisations, organisationally important knowledge is embedded within skills, experiences, expertise and competencies of the individuals that make up the organisation [1, 3, 6]. This is particularly true for R&D organisations [1] and other professional services organisations [5]. As well as people, significant organisational knowledge is embedded within the tools the organisation uses, including specialised physical hardware used as part of a manufacturing process, for example, through to conceptual or intellectual tools such as consulting or analysis frameworks [3, 7]. The third key element identified by [3] is the tasks performed by the organisation. Tasks reflect an organisation's goals, intention and purpose [3, 5, 7]. For R&D, engineering and other professional services organisations, key knowledge is also embedded within the products or other kinds of outcome the organisation produces. The development of products or other kinds of outcome uniquely combines together the organisation's staff, tools and tasks to address a particular question or problem, or to develop some kind of product, and can be seen as uniquely embedding the application of the organisation's collective knowledge, skills, experiences and expertise within a particular domain, to address a particular question or problem or to develop some kind of product [1, 7, 9, 10]. As discussed in [11], knowledge management is centred on two, potentially limiting, philosophical foundations. The first is the idea that tacit and explicit knowledge are two distinctly different forms or types of knowledge, rather than simply being a dimension along which all knowledge exists. The underlying assumption that tacit and explicit knowledge are different leads to the conclusion that a key goal of knowledge management is the codification of tacit knowledge into explicit knowledge [12]. However, as [11] points out, not all knowledge can (or should) be codified, and any knowledge management approaches that rely on the codification of knowledge are likely to fail. The second philosophical foundation that knowledge management rests on is the data – information – knowledge continuum: the idea that information is in some way better data and that knowledge is in some way better information. As discussed in [11], this leads to knowledge management approaches that focus only on capturing and managing some form of codified knowledge [13], while ignoring data and information that could provide equal or even greater value to users. However, the view that knowledge important to an organisation is embedded in the key elements that make up the organisation potentially provides an approach to knowledge management that doesn't rest on these two potentially limiting foundations. The focus of a knowledge management approach that accepts the embeddedness of knowledge as important becomes one of finding and devising methods, systems and approaches that in some way index and expose the core entities within the organisation that hold the embedded knowledge, rather than focusing on codification and managing the codified knowledge. The importance of the knowledge embedded in the different organisational entities will vary with the nature of the organisation and the nature of the work performed by the organisation. The following section discusses the kinds of knowledge important within an industrial R&D organisation, and the kinds of entities the knowledge is likely to be embedded within. 2.2. Knowledge and Expertise within R&D Organisations For industrial R&D organisations (and many other knowledge intensive firms [14]), the key knowledge that makes the organisation unique is embedded in the experience, expertise and competencies of the engineers and scientists that make up the organisation and the organisation's collective project history – the past products the organisation has developed, or the past problems or questions it has addressed [1, 10]. Within most industrial R&D organisations, the products developed or projects undertaken require the application of collective individual knowledge, skills, experience and expertise in unique ways. As a result, products and projects can be seen as an embedding the application of the organisation's collective knowledge, skills, experience and expertise, within a particular domain, to address a particular question or problem [1, The information and knowledge created as part of past projects can provide important insights into finding solutions to current problems, or gaining an understanding of how similar problems have been solved in the past. Project histories can also support problem reformulation, and can offer some help in validating proposed solutions [15]. Past projects are also important because they can provide a link back to the staff who contributed. This in turn can provide valuable insight into the knowledge, experience and expertise that individual staff may have [1, 8, 15]. Colleagues act not only as important sources of information, and pointers to other sources of information [16], but most importantly they provide an interactive think along function [17, 18]. As discussed previously, the goal of this work was to develop ways of improving knowledge and Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 expertise finding. The approach taken by the work described in the following section was to develop an information system (ARMS) to act as a rich, interactive model of the embedded knowledge within the organisation – in particular, the current and past projects within the organisation, and the expertise, skills and experience of the staff that make up the organisation. 3. The ARMS Prototype The Automated Research Management System (ARMS) is a prototype, web based, information system developed to explore approaches to supporting knowledge and expertise finding within DSTO. ARMS holds information about the key R&D entities relevant to DSTO – staff and projects, as well as formal and informal project outputs (academic papers, technical reports, design documents, data collections, and so on). Within ARMS, these entities are organised around the organisation's hierarchical structure, and around Themes – collections of taxonomic descriptors used to describe the client and scientific domains that DSTO works within. The key R&D entities and their relationships are shown in Figure 1. StaffStaff ThemeTheme ProjectProject OutputOutput Author Of Created ForDescribed By Manager Of Contributes To UnitUnitSiteSite Described By Interested In Located At Exists In Exists In Role In Head Of Role InMember Of Figure 1. Conceptual domain model 3.1. The Users' Perspective As discussed in Section 2, the key knowledge, experience, expertise and competency of an industrial R&D organisation exists in its project history and the collective expertise, skills, experience and knowledge of its scientists and engineers. ARMS supports knowledge and expertise finding by exposing both projects (and the outputs associated with projects) and staff as richly interlinked first class objects (unlike many similar systems [19], where staff are either not included, or simply included as author labels associated with the entities held by the system). From a user’s perspective, ARMS exists as a collection of dynamically generated web pages, with each R&D entity having its own web page that pulls together all the information related to that entity. Staff pages contain basic staff information, including: contact information, site location and organisational affiliation, and descriptions and links to the projects and project outputs the staff member has contributed to. This information is drawn from existing organisational information systems. Staff pages can also contain optional information, including staff descriptions of current and past project work, staff photo and basic biographical information, and current interests and work. Project pages contain project descriptions (abstract, overview, background, themes, etc), project milestones, planned deliverables, information about the project's relationship to other past current and planned projects, and the project's status. Project pages also contain information and links to the staff that have contributed to the project, as well as information about and links to the project's outputs. All this information is obtained from existing information systems. End users are also able to add richer project description information, as well as any kinds of additional outputs to the project's home page. Output pages contain basic metadata describing the output (title, abstract, publication details, document type and so on) and the documents that make up the output (for example MS-Word files, data files, image files, etc). Output pages contain information and links to the staff who contributed to the output, as well as information and links to the project the output was developed for. Output information contained within ARMS is extracted from an existing publication management system. End users are also able to add additional project outputs to any project page. ARMS provides multiple entry points into the data. Users can access staff, project and output entities directly via the staff, project and output browsing pages. Each of these pages provides filterable lists of the staff, project and output entities contained within ARMS. In addition to the browsing pages, users can also enter ARMS via a representation of the organisation's structure (the Unit entities shown in Figure 1), or via descriptions of the organisation's R&D program (the Themes entities shown in Figure 1). Each DSTO unit has a corresponding unit page that holds basic contact information for the unit, the unit’s head, administrative contacts for the unit, and the Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 unit’s structure (for example the groups within a branch), the staff who are members of that unit, and the tasks associated with that unit. This information is extracted from several different existing information systems. In addition, end users can optionally insert additional information describing units. Units can be directly entered via the unit browse page, an interactive form of the organisation’s hierarchical structure. DSTO's R&D program is represented as a collection of Themes, a combination of taxonomy descriptors (based around DEFTEST [20]) that cover the core science and technology (S&T) areas of DSTO, and taxonomy descriptors that cover the key client areas DSTO works within. S&T and client Themes are used to describe the staff, project and output information held within ARMS. Each Theme has an automatically generated home page that aggregates all the projects, staff and outputs described by the Theme. In many ways the Theme home pages can be seen as aggregating everything the organisation 'knows' about a particular Theme area – that is all the staff, projects and outputs related to that Theme. Themes can be directly entered via the Theme browse page, an interactive and structured collection of the Themes held by ARMS. An important part of ARMS is rich hyperlinking between the various entities that contextualises the information held by ARMS. Staff, for example, are contextualised by the projects they contribute (or have contributed) to and the outputs they have contributed to. Projects are contextualised by the staff that contribute to them, and the part of the organisation they were performed by. The intent of the contextualisation is to allow users to infer richer meanings based on explicit relationships present in the data. The utility of this approach, especially in terms of expertise finding, is discussed in Section 5.2. In addition to the browsing functions, ARMS also includes a search function that support free text searching over all the information held by ARMS (fields associated with each entity as well as full document text), field searching (for example, searching explicitly by unit name, or project number), as well as Theme searching. The different search types can be combined with Boolean operators (AND and OR) to form complex queries. 3.2. Technical Perspective From a technical perspective, ARMS consists of a centralised repository that holds information extracted from existing corporate information systems, as well as a small amount of information specifically created to support ARMS. Access to the information and functions provided by ARMS is via a web based interface that supports the functions described in Section 3.1, as well as a Web Services1 interface that provides dynamic programmatic access to the core functions and information (see Figure 2). The Web Services interface was provided to support dynamic access to by ARMS by specialised applications, such as collaborative Microsoft SharePoint portals [22] and other specialised web sites. Taxonomy Repository Domain Logic Unstructured Repository Sem i-structured Repository Search Engine W eb Interface W eb Services Interface Staff System Project Management System Publication Repository W rapperW rapper W rapper Collaborative SharePoint Collaborative SharePoint Collaborative SharePoint Portal Specialised W eb Sites S&T Staff M anager Figure 2. ARMS logical architecture Almost all of the information used by ARMS was drawn from existing corporate information systems as well as less-formal corporate information collections such as spreadsheets and intranet sites. The re-used corporate information generally needed to be cleaned and reformatted before it could be used. Most of the data cleaning issues encountered were common to other data warehousing projects [23], and included a lack of common record identifiers across the different systems, conflicting data values across the different systems, missing and erroneous data, and name and type conflicts. The data cleaning and re-formatting functions were encapsulated in series of customer wrappers that were developed for each of the core corporate applications and other corporate information sources. As discussed in Section 3.1, ARMS provided a search function over all the information held. The search engine combined free text searching over the unstructured information held by ARMS (generally reports, papers, presentation and other project outcomes) with searching over the structured and 1 An overview of Web Services can be found in [21]. Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 semi-structured information held. Free text search was provided by a commercially available text search engine; searching over structured and semi-structured information was provided by a custom developed search engine. 4. The ARMS Trial To validate the concepts underpinning ARMS and its utility in supporting expertise and knowledge finding within DSTO, a prototype version of the system was evaluated within two DSTO divisions from June until November 2004. The ARMS prototype (as described in [24]) was fully developed and fully populated with relevant staff, project and output information. The data held by ARMS was kept up-to-date throughout the trial period. During the trial period ARMS was made available to staff within the two divisions selected (Division A and Division B). These divisions were selected because together they reflect a good cross section of the staff, organisational structures, and research programs within DSTO. Division A was split across five sites, and generally had a multi-disciplinary, professional services focus. At the time of the trial it contained approximately 80 staff. Division B was split across three sites, and generally had a computer science/software engineering base, with a strong R&D focus. At the time of the trial, it contained approximately 110 staff. Table 1. ARMS Trial Studies Study Date Collection Methods Study One: Organisational Wide Focus Groups May Focus Groups Study Two: Stake- holder Interviews June/July Unstructured and semi-structured interviews Study Three: Concept and Implementation Survey July Survey questions Study Four: Project Seeking Experiment September Data seeking experiment Study Five: Utility of Usage Study November Semi-structured interviews and survey questions Over the trial period, five different studies were undertaken (see Table 1). Each of the studies aimed to explore the utility of ARMS from various perspectives. This paper discusses two of the key studies, Study Three and Study Five, both of which measured the utility of ARMS from the perspective of R&D staff. The Concept and Implementation Survey (Study Three) was undertaken after ARMS had been available in the two trial divisions for approximately 1½ months. The survey was sent to all staff in the two trial divisions who had used ARMS at least once during the trial period. Of the 75 divisional staff who had used ARMS at least once, 23 staff responded. Respondents were spread across three sites; 40% from Site A, 52% from Site B, and 8% from Site C. The respondents represented a good cross section of the organisation’s management structure, with 17% of respondents holding organisational unit management positions (group or branch), and 40% of respondents having project management responsibility. Overall the sample reflected a good cross-section of the organisation, with slight over-sampling of respondents from Site B. The results of this study are discussed in Section 5. The Utility of Usage Study (Study Five) was undertaken towards the end of the ARMS trial. The most frequent users of ARMS were identified, and invited to participate in the study. Of the users selected, 10 agreed to participate. Study respondents were spread across three sites; 70% from Site A, 20% from Site B and 10% from Site C. Around half of the respondents held project management responsibility. Overall, the sample in this study reflected a good cross section of the organisation, except for the lack of respondents holding organisational unit management positions, and an over-sampling of respondents from Site A. Study Five was run as a set of survey questions and semi-structured interviews. Participants were asked to list the functions they used ARMS to perform over the previous month. The interviews discussed how participants used ARMS, its utility to them, and how ARMS compared to alternative approaches they may have tried. The interviews were recorded, and the recordings transcribed. The interviews ran for an average of 50 minutes each. 5. Results and Discussion 5.1. ARMS Usage Studies Three and Five recorded how ARMS was used by participants over the previous month. Table 2, below, lists the functions ARMS was used to perform and the percentage of users who used ARMS to perform the listed function. One of the most striking features of Table 2 is the dramatic changes in some of the ways ARMS was used over the trial period. (Study Three was undertaken Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 after ARMS had been available for 1½ months; Study Five was undertaken toward the end of the 6 month trial). Table 2. Overall ARMS Usage Use of ARMS Study Three Study Five 1. Exploring the system; for example seeing what information it holds and exploring how I could use it. 96% 20% 2. Building an awareness of DSTO's projects and staff; for example seeing what work a group is doing, or what work is going on within a Work Area, or seeing what a particular person has been working on recently. 61% 10% 3. Finding out about a particular project; for example who is working on it, who the project manager is, or finding the various papers, reports and other outputs associated with the project. 56% 40% 4. Finding out about a person; for example their contact information, what they are working on, what papers and reports they have written or what outputs they have been involved in, or finding out about their interests or experience. 52% 50% 5. Finding out about a particular research area; for example who is interested in the area, which tasks contribute to the research area, or what work has been produced in the research area. 35% 20% 6. Finding out about a particular organisational unit; for example who is in the unit, what tasks the unit is responsible for, or finding contact information for the unit. 30% 0% 7. Finding out about a particular Client Area; for example who is working in, which tasks contribute to the Client Area, or what work has been produced for the Client Area. 22% 0% 8. Finding a particular formal or informal output. 13% 30% 9. Searching for a person with particular skills, interests, experience or abilities. 13% 10% 10. Other 4% 0% The largest change was the drop in exploratory use of ARMS between the two studies. This change is likely to be a result of users building an understanding of the functions and features of ARMS as the study progressed. Once an understanding of ARMS (based on exploring the system) was developed they either moved to non-use, if they felt ARMS provided no value, or they moved to using specific features and functions of ARMS. Using ARMS to build and maintain an awareness of the work being performed within DSTO (Question 2), and using ARMS to find out about particular organisational units (Question 6) and client areas (Question 7) also dropped over the study period. The drop in using ARMS to perform these functions is likely to be due to two key factors. The semi-structured interviews undertaken as part of Study Five revealed that awareness (Question 2) is something that is built for a specific purpose, for example moving to a new organisational unit or moving into a new research area, and then is maintained by actively being involved in the organisational unit or research area. Once an overall awareness has been built, it is maintained by more direct information seeking activities, for example finding out about a staff member, or a project, or hunting up specific project outputs. The changes in the results for Questions 6 and 7 are due to similar reasons, with participants initially using ARMS to build a general awareness of organisational units and client areas, and then maintaining that awareness by more direct information seeking methods. The second factor likely to affect the use of ARMS to build and maintain awareness is a fundamental limit inherent within the ARMS trial. As discussed previously, ARMS was fully populated and maintained with data drawn from only two trial divisions, not the whole organisation. As a result the value of ARMS in providing a rich awareness to participants was limited to only the two trial divisions. This limitation and its likely impact on the overall trial result is discussed in more detail in Section 6.1. 5.2. Finding People, Projects and Outputs One set of functions that ARMS was regularly used to perform over the study period was finding people, projects and outputs (Questions 3, 4 and 8 respectively). As shown by Table 2, finding people was constantly the most frequently used function of ARMS. The semi-structured interviews undertaken as part of Study Five revealed that participants who used ARMS to find people were generally looking for colleagues who have similar research interests or worked in similar areas. The semi-structured interviews revealed that users who employed ARMS in this way, were, in general, seeking out connections with other colleagues in order to share ideas, discuss problems/issues, have work reviewed, gain insights and Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 different perspectives on a common/shared problems, and perhaps even re-use and build on the work already completed by colleagues. These findings confirm much of the previous research (described in Section 2) describing the information seeking behaviours and needs of scientists and engineers. Overall, participants reported that the information available via ARMS in most cases allowed them to identify colleagues they felt would be worth contacting. However, there were some limitations in the data ARMS provided. One key limitation was the scope of the information available via ARMS. As discussed previously, ARMS only contained information for the two trial divisions. This limited the number of colleagues users could learn about. The second key limitation with the staff data available via ARMS was that individual staff expertise, experience and research interests, in general, could only be derived by understanding the context surrounding the staff member (this is discussed in Section 5.3). While in most cases the surrounding context provided users with sufficient information to infer a staff member's expertise, experience and research interests, it did mean that users were generally unable to directly search for staff with particular expertise, experience and research interests but instead had to search for staff via outputs or projects, and infer staff relevance based on the relationships between staff and the projects and outputs they contributed to. An interesting question explored as part of the semi- structured interviews was the relationship between the kinds of information available via ARMS, and the kinds of information available via the user's social networks. Most respondents acknowledged the importance their social network plays in being able to find colleagues with specific expertise, experience and research interests. However, they generally found that there were limits in the coverage of their social network; in particular many respondents felt their social network often didn't extend into other organisational sites, or into other organisational units. In comparison, they felt that the information held by ARMS was more complete and offered greater coverage of all parts of the organisation, and they viewed ARMS as a way of filling gaps in their social network. The second key group of functions regularly used throughout the trial was finding projects and outputs. The semi-structured interviews revealed that when searching for projects and outputs, participants were generally looking either to find relevant and useful colleagues, or were explicitly looking for information related to past projects. As discussed in Section 2, for engineers in particular, descriptions of past projects and the formal and informal products of projects (reports, papers, designs, data sets, meeting minutes, and so on) are important because they offer insights and approaches to solving past problems that may be useful for solving current problems. The semi- structured interviews revealed that participants were using ARMS in this way. As well as ARMS, participants could obtain project and formal publication information from two existing organisational information systems, a Project Management System (PMS) and a Publications Repository (PR). The basic project and publication information held by ARMS was obtained from these two systems, and in most cases ARMS held little additional information. The key value added by ARMS was the rich contextualisation of the information, together with existing staff information to provide multiple entry point into the information sought, and to allow rich relationships and deeper meaning of the information to be inferred. This is discussed in more detail in the following section. 5.3. Contextualisation and Navigation As shown by Figure 1, the information contained within ARMS is richly interlinked. The rich interlinking helps to contextualise this information, making it easier for ARMS users to develop a deeper understanding of the information held within ARMS, as well as providing users with multiple entry points into the information, and a navigation model drawn from the users' domains. The semi-structured interviews undertaken as part of Study 5 showed that by knowing, for example, the background of key staff involved in a project, users are able to infer more about the likely directions, perspectives or methodologies a project may use. Or by knowing about the project that created a particular output, users are able to build a richer understanding of the output's meaning. The rich interlinking within ARMS also provided users with multiple entry points into the information held by ARMS, and provided them with natural navigation paths drawn from their domain. The semi- structured interviews revealed that almost all users reported that the richly contextualised nature of ARMS helped them navigate ARMS, allowing them to enter ARMS from many different directions, often using only incomplete information as a starting point. For example, several users reported browsing ARMS by starting with a vague awareness that a staff member may be doing work that might be interesting or relevant to them. By using ARMS, they were able to Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 move from the individual's staff page to their project pages – to find out about the work being performed. From project pages, they would move to output pages, to gain a deeper insight of the work being performed, or they would move to Theme pages, to find related work, staff or outputs. Other users reported a similar approach using projects as a starting point to find people or to find other related projects. This form of navigation also allows for far richer forms of accidental information discovery [25], and provides users with the ability to develop rich mental models of the organisation's past, planned and current work program. 5.4. Cultural Impact As discussed in Section 1, one of the goals of the work described in this paper was to address potential barriers to information sharing within DSTO, including behavioural and cultural issues, as well as a lack of low cost methods, systems or processes to support information sharing within DSTO. Both Study Three and Study Five attempted to measure the likely influence ARMS would have on these barriers. The results are shown in Table 3; questions were measured on a 7 point, end anchored scale, with 1 meaning strongly disagree, and 7 meaning strongly agree. Table 3. Cultural Impact Question Study Three Study Five If ARMS held information about everyone in the DSTO and all DSTO's past, current and planned work: 1. It would be easy to find out what is going on in DSTO? x = 5.55 s = .80 x = 6.11 s = .78 2. It would encourage people to work together rather than compete with one another? x = 4.61 s = 1.16 x = 4.58 s = .88 3. It would encourage people to share the information they have, or their knowledge and expertise? x = 4.83 s = 1.07 x = 4.67 s = .87 The most interesting insight from Table 3 is the different perceptions respondents had of the role ARMS could play in lowering the cost of sharing within the organisation (Question 1), versus the ability of ARMS (or any information system) to influence the behavioural and cultural issues that affect information and knowledge sharing within the organisation (Questions 2 and 3). As discussed previously, almost all of the information included within ARMS was drawn from existing organisational information systems. As a result, users were able to build a basic understanding of the past, current and planned work within the organisation, and the skills, interests and experience of staff without requiring the staff described by ARMS to actively make the information available. The strong results for Question 1 across both studies, and the positive insights gathered by the semi-structured interviews, show that, within the trial organisation, the approach of populating ARMS with existing corporate information did lower one of the barriers to sharing by providing a low cost method for 'finding out what is going on in the organisation'. However, the semi-structured interviews undertaken as part of Study Five revealed that, while being able to find out what was going on in the organisation is a necessary first step towards supporting sharing and collaboration, by itself, it isn't sufficient [14] to significantly impact existing behavioural and cultural issues affecting sharing within DSTO. Many factors affect sharing, including: organisational incentives (for example, rewards, time, encouragement) and disincentives; individual behaviours; the need to derive value from sharing and collaborating; organisational structures; and administrative barriers. 6. Conclusions This paper has described ARMS, an information system aimed at supporting knowledge and expertise finding within DSTO. ARMS was trialled within two divisions of DSTO over a six month period. During this time, five different studies were undertaken; this paper has reported on two of the key studies aimed at measuring the utility of ARMS from the perspective of R&D staff. The work described in this paper has three key implications. The first is that, within the trial organisation, ARMS provided considerable value in supporting the information and knowledge management needs of R&D staff. In particular, ARMS provided a mechanism by which R&D staff could find colleagues that had similar research interests, or had expertise or insights that could help with a particular problem. ARMS also provided a rich corporate memory function, allowing R&D staff to browse the organisation’s previous work. As discussed in the literature (see Section 2), finding staff, and finding previous work are two key information and knowledge needs for staff within industrial R&D organisations. The rich interlinking of the information held by ARMS helped to improve the usability of the Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 information held by providing multiple entry points into the information, and intuitive navigations paths that matched the user's model of the domain. The rich interlinking also allowed users to see the information held by ARMS in a wider context, and this often enabled them to build a much richer understanding of the information. By reusing existing corporate information, and not requiring R&D staff to add information to ARMS to expose their skills, expertise or experience, or to expose the past and current work within the organisation, ARMS helped overcome one of the key cultural and behavioural barriers toward sharing within DSTO. While ARMS provided the necessary first step towards improving such sharing, ARMS (or any IT system) is still likely to have a limited impact on entrenched organisational cultural and behavioural barriers [26]. 6.1. Limitations The work described in this paper has been applied only within one R&D organisation. While much of the data collected suggests that the information and knowledge seeking needs of DSTO's scientists and engineers is similar to other R&D organisations reported in the literature, the information and knowledge management problems faced by DSTO may be relatively unique, and as a result, the value of an information system like ARMS to other R&D organisations may be different. A second key limitation of this work was the scope of the information made available within ARMS over the trial period. As discussed previously, only the information related to the two trial divisions was made available within ARMS. The data collected as part of the studies showed that, in many ways, this limited the overall utility of ARMS, especially in supporting information seeking outside of the two divisions. 6.2. Future Work A significant feature the current implementation of ARMS lacks is automatically generated, easy to navigate descriptions that describe the skills, expertise and experience of individual staff. While ARMS users were generally able to infer and derive the likely skills, expertise and experience of individual staff (as described Section 5.3), the lack of automatically derived descriptions of skills, expertise and experience of individual staff limited the ability of users to directly search and browse for staff by skills, expertise and experience. Previous work [27, 28] has shown positive results in deriving descriptions of individual's expertise from descriptions of the work they perform, or the roles they hold. Given that ARMS already strongly relates well described projects and outputs to individual staff, it would be possible to automatically derive or infer descriptions of individual staff expertise from the information already contained within ARMS. By drawing together staff, projects and outputs, Theme pages act as a potential community of practice hub because they provide wide awareness of an individual's particular expertise or interests, and provide resources relevant to that particular area (projects and outputs and their descriptions). Potentially, the Theme home pages could be expanded to include additional functions to encourage the creation of community – for example, via richer resources, and richer interaction methods (cf. [14]). However, due to time constraints imposed on the development of the ARMS prototype, this approach has not yet been explored. 7. Acknowledgements The author acknowledges the valuable assistance of Dr Mark Burnett, Mr Chris Chapman, Ms Phuong La, and Ms Jemma Nguon, in developing the ARMS prototype, and further acknowledges the assistance of Mr Justin Fidock with some of the evaluation studies. 8. References [1] T. J. Allen, Managing the Flow of Technology: Transfer and the Dissemination of Technology Information within the R&D Organisation. Cambridge, Mass: MIT Press, 1977. [2] S. Hirsh and J. Dinkelacker, "Seeking Information in Order to Produce Information: An Empirical Study at Hewlett Packard Labs," Journal of the American Society for Information Science and Technology, vol. 55, pp. 807–817, 2004. [3] L. Argote and P. Ingram, "Knowledge Transfer: A basis for Competitive Advantage in Firms," Organisational Behaviour and Human Decision Processes, vol. 82, pp. 150– 169, 2000. [4] R. Cowan, P. A. David, and D. Foray, "The Explicit Economics of Knowledge Codification and Tacitness," Industrial and Corporate Change, vol. 9, pp. 211–253, 2000. [5] W. H. Starbuck, "Learning by Knowledge-Intensive Firms," Journal of Management Studies, vol. 29, pp. 713– 740, 1992. Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 [6] T. Davenport and L. Prusak, Working Knowledge: How Organisations Manage What They Know. Boston: Harvard Business School Press, 1998. [7] J. L. Cummings and B. S. Teng, "Transferring R&D knowledge: the key factors affecting knowledge transfer success," Journal of Engineering and Technology Management vol. 20, pp. 39–68, 2003. [8] M. Hertzum and A. M. Pejtersen, "The information- seeking practices of engineers: Searching for documents as well as people," Information Processing and Management, vol. 36, pp. 761–778, 2000. [9] H. 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Berends, "Knowledge Sharing and distributed cognition in industrial research," presented at Third European Conference on Organisational Knowledge, Learning and Capabilities (OKLC02), Athens, Greece, 2002. [18] D. W. McDonald and M. S. Ackerman, "Just Talk to Me: A Field Study of Expertise Location," presented at ACM Conference on Computer Supported Collaborative Work (CSCW'98), Seattle, WA, 1998. [19] J. Grudin, "Enterprise Knowledge Management and Emerging Technologies," presented at Thirty-Ninth Annual Hawaii International Conference on Systems Science (HICSS-39), Waikoloa, Hawaii, 2006. [20] "DEFTEST: Defence Technological and Scientific Thesaurus," Defence Reports Section, Department of Defence, Canberra, Australia. 1988. [21] F. Curbera, M. Duftler, R. Khalaf, W. Nagy, N. Mukhi, and S. Weerawarana, "Unravelling the Web Services Web An Introduction to SOAP, WSDL, and UDDI," IEEE Internet Computing, vol. March–April, pp. 86–93, 2002. [22] P. Prekop, P. La, M. Burnett, and C. Chapman, "Using Web Services to Enable Reuse of DSTO Corporate Data within Microsoft Sharepoint: A Case Study," Defence Science and Technology Organisation (DSTO), Edinburgh, South Australia, tech note DSTO-TN-0672, July 2005. [23] E. Rahm and H. H. Do, "Data Cleaning: Problems and Current Approaches," IEEE Data Engineering Bulletin, vol. 23, pp. 3–13, 2000. [24] P. Prekop, M. Burnett, and C. Chapman, "The Prototype Automated Research Management System (ARMS)," Defence Science and Technology Organisation (DSTO), Edinburgh, South Australia, technical note DSTO-TN-0540, February 2004. [25] A. Foster and N. Ford, "Serendipity and Information Seeking: an Empirical Study," Journal of Documentation, vol. 59, pp. 321–340, 2003. [26] J. Swan, S. Newell, and M. Robertson, "Knowledge Management – When will people management enter the debate?" presented at Thirty-Third Annual Hawaii International Conference on System Science (HICSS-33), Waikoloa, Hawaii, 2000. [27] M. Maybury, R. D. Amore, and D. House, "Awareness of Organisation Expertise," International Journal of Human- Computer Interaction, vol. 14, pp. 199–217, 2002. [28] A. Vivacquan and H. Lieberman, "Agents to Assist in Finding Help," presented at ACM Conference on Human Factors in Computing Systems (CHI2000), 2000. Proceedings of the 40th Hawaii International Conference on System Sciences - 2007 Select a link below Return to Main Menu Return to Previous View

0704.1354

Reply to Comment on ``An Improved Experimental Limit on the Electric Dipole Moment of the Neutron''

Reply to Comment on “An Improved Experimental Limit on the Electric Dipole Moment of the Neutron” C.A. Baker,1 D.D. Doyle,2 P. Geltenbort,3 K. Green,1, 2 M.G.D. van der Grinten,1, 2 P.G. Harris,2 P. Iaydjiev∗,1 S.N. Ivanov†,1 D.J.R. May,2 J.M. Pendlebury,2 J.D. Richardson,2 D. Shiers,2 and K.F. Smith2 Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, UK Institut Laue-Langevin, BP 156, F-38042 Grenoble Cedex 9, France (Dated: December 11, 2018) PACS numbers: 13.40.Em, 07.55.Ge, 11.30.Er, 14.20.Dh Our Letter [1] places a new experimental limit on the electric dipole moment (EDM) of the neutron. The Com- ment [2] points out that we did not explicitly include in our analysis the effect of the Earth’s rotation, which shifts all of the frequency ratio measurements Ra to lower (higher) values by 1.3 ppm when the B0 field is upwards (downwards). However, this effect is essentially indis- tinguishable from other effects that can shift Ra, and all such shifts were compensated for in [1] by using ex- perimentally determined values of Ra0 (which we call Ra0↓ and Ra0↑, respectively, for the two polarities of B0), where 〈∂Bz/∂z〉V = 0. We turn now to the details. Naively, one would ex- pect that the crossing point of the lines in Fig. 2 of [1] (which lies at Ra − 1 = 5.9 ± 0.8 ppm) would have 〈∂Bz/∂z〉V = 0, with its ordinate yielding the true EDM. However, what were referred to in [1] as horizon- tal quadrupole fields (involving ∂Bx/∂y etc.) shift these lines towards the right. A difference in the strengths of these quadrupolar fields upon B0 reversal leads to a differential shift in Ra, and thus to a vertical displace- ment of the crossing point. The Earth’s rotation mim- ics this behavior precisely, by moving the B0-down (- up) line leftwards (rightwards). Thus, where quadrupole fields are mentioned in [1], one might better read this as “quadrupole fields and Earth-rotation effects combined”. The “quadrupole shift” listed in Table 1 of [1] simply rep- resents the move from the crossing point to the average of the EDM values determined (independently) by the measured Ra0↓ and Ra0↑ values. The shift measurements are described (rather than just “mentioned”) in [1]. First, the strongest constraint arises from a study of the depolarization of the neutrons as a function of Ra, and thus, effectively, as a function of 〈∂Bz/∂z〉V . Neutrons of different energies have different heights of their centers of mass, and thus the T2 spin re- laxation is maximized when〈(∂Bz/∂z) 2〉V is minimized. The values of Ra−1 at which the polarization product α was found to peak were (5.7± 0.2, 5.9± 0.2) ppm for B0 ∗On leave from Institute of Nuclear Research and Nuclear Energy, Sofia, Bulgaria †On leave from Petersburg Nuclear Physics Institute, Russia up, down respectively. In the presence of the dipole in the region of the door of the storage chamber [1], the point for B0 down (up) at which 〈(∂Bz/∂z) 2〉V is minimized is 0.2 ppm higher (lower) than the point Ra0↓ (Ra0↑). These data provide direct, independent measurements for each B0 polarity of the actual values Ra0 at which 〈∂Bz/∂z〉V = 0, taking into account any and all shift mechanisms, known or unknown, acting on Ra. Since these depolarization results are drawn from the EDM data themselves, they cannot be described as “ex post facto”. We conclude from our data that the differen- tial quadrupole shift and Earth rotation effect cancel to within 15% in our apparatus. The fact that the resulting dn values ((−0.6 ± 2.3,−0.9 ± 2.3) × 10 −26 e cm for B0 up, down respectively) agree so well with each other gives added confidence in the experimental results overall. Second, after about 60% of the data had been taken, a bottle of variable height was used to measure the profile of the magnetic field within the storage volume. Ex- trapolation of these data to the EDM bottle (which does include a small correction due to Earth’s rotation) yields Ra0↑ −Ra0↓ = (1.5± 1.0) ppm. Our data show no evidence for changes in the relevant long-term B-field properties from the periodic disassem- bly of the magnetic shields. Since the publication of [1], we have improved our fit- ting procedure to take full account of correlations be- tween the quadrupole and dipole corrections, and to in- clude explicitly the effect of the Earth’s rotation. The results yield new net shifts (to be compared with those listed in Table 1 of [1]) for the dipole and combined quadrupole/Earth rotation effects of (−0.46,+0.30) × 10−26 e cm respectively, with a net uncertainty of 0.37× 10−26 e cm for both. In combination with the other ef- fects discussed in [1] this yields an overall systematic cor- rection to the crossing point of (0.20±0.76)×10−26 e cm for the second analysis of [1]. The final value for the EDM from this analysis is then (−0.4± 1.5(stat)± 0.8(syst))× 10−26 e cm, implying |dn| < 2.8× 10 −26 e cm (90% CL), identical to the previous limit from this analysis. The Comment asserts incorrectly that the Ra−1 values averaged to zero in the first analysis of [1]. By choice of the applied ∂Bz/∂z, they averaged to 8.9 ppm for both B0 polarities. Since any net differential shifts in Ra have been shown to be small, this analysis need not be altered. http://arxiv.org/abs/0704.1354v1 In conclusion, the overall limit of |dn| < 2.9 × 10 −26 e cm (90% CL) remains unchanged. [1] C. Baker et al., Phys. Rev. Lett. 97, 131801 (2006). [2] S.K. Lamoreaux and R. Golub, Phys. Rev. Lett., 98, 149101, (2007).

0704.1355

Lowest Landau Level of Relativistic Field Theories in a Strong Background Field

Lowest Landau Level of Relativistic Field Theories in a Strong Background Field Xavier Calmet1 a and Martin Kober2 1 Université Libre de Bruxelles, Service de Physique Théorique, CP225, Boulevard du Triomphe (Campus plaine), B-1050 Brussels, Belgium. 2 Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany. Abstract. We consider gauge theories in a strong external magnetic like field. This situation can appear either in conventional four-dimensional theories, but also naturally in extra-dimensional the- ories and especially in brane world models. We show that in the lowest Landau level approximation, some of the coordinates become non-commutative. We find physical reasons to formal problems with non-commutative gauge theories such as the issue with SU(N) gauge symmetries. Our con- struction is applied to a minimal extension of the standard model. It is shown that the Higgs sector might be non-commutative whereas the remaining sectors of the standard model remain commuta- tive. Signatures of this model at the LHC are discussed. We then discuss an application to a dark matter sector coupled to the Higgs sector of the standard model and show that here again, dark matter could be non-commutative, the standard model fields remaining commutative. PACS. 11.10.Nx – 12.60.Fr Gauge theories formulated on non-commutative spaces have received a lot of attention over the last decade. The main reason is that they were discov- ered to appear in a certain limit of string theory[1,2, 3]. Non-commuting coordinates do appear generically whenever one studies a physical system in an exter- nal background field in the first Landau level approx- imation. This phenomenon was discovered by Landau in 1930 [4]. A textbook example is an electron in a strong magnetic field. In the framework of string the- ory, the effective low energy four dimensional theory describing strings ending on a brane in the presence of a strong external background field is shown to be non-commutative[1,2,3]. It is notoriously difficult to construct a non-commutative version of the standard model. There are different approaches in the literature [5,6]. The main difficulty is to obtain the right gauge symme- tries i.e. SU(N) groups necessary to describe the stan- dard model. The issue here is that the commutator of two non-commutative Lie algebra valued gauge trans- formations is not a gauge transformation unless one chooses U(N) Yang-Mills symmetries and the funda- mental, anti-fundamental or adjoint representations. This no-go theorem can be avoided if one considers the enveloping algebra. However this may not seem very natural. This is the motivation for the present work. We shall study relativistic field theories in a strong external potential to identify the physical rea- a Email: [email protected] son for this technical problem. We shall start from clas- sical gauge theories formulated on a regular space-time which are perfectly well behaved and renormalizable theories and consider them in a strong external field, then we shall consider their first Landau Level. Differ- ent field theories have been considered in the lowest Landau level approximation leading to more or less exotic non-commutative gauge theories, see e.g. [7] or [8]. However, we wish to consider physical situations which lead to the kind of non-commutative gauge the- ories found in [1,2,3] to understand the physical reason for some of the pathologies of these theories. Further- more our construction allows us to consider models where only certain sectors of the theory are noncom- mutative. Let us first consider a charged scalar field in a strong magnetic field. The action is given by (D̄µφ) ∗(D̄µφ) − V (φ∗φ) (1) F̄µν F̄ where D̄µ = ∂µ + iqAµ and F̄µν = −i[D̄µ, D̄ν]. This theory is gauge invariant under U(1) gauge transfor- mations and is renormalizable. Let us now study this theory in the limit of a strong external magnetic field. We consider quantum fluctuations Aµ aroundCµ which is the background field which corresponds to the con- stant magnetic field. We then have D̄µ = ∂µ + iqAµ + http://arxiv.org/abs/0704.1355v2 Alternatives Parallel iqCµ = Dµ + iqCµ and the action becomes (Dµφ) ∗(Dµφ)− V (φ∗φ) (2) −iqφ∗CµDµφ+ iq(Dµφ)∗Cµφ+ q2φ∗CµCµφ µν − 1 µν − 1 where Cµν = ∂µCν − ∂νCµ. To be more specific we shall pick Cµ = (0, , 0) which leads to a con- stant magnetic field of magnitude B in the z-direction. Note that a strong external magnetic like field does not imply that the quantum fluctuation is strongly cou- pled to the scalar field. The action (2) has a remaining gauge invariance: δφ = iαφ, δAµ = ∂µα and the back- ground field is kept invariant. The classical canonical momenta of the center of mass of particle φ is given by πµ = pµ + qAµ + qCµ. The first quantization of the classical Hamiltonian implies that the coordinates and the spatial compo- nents of the canonical momentum do not commute: [xi, πj ] = ih̄δij . (3) We now express the canonical momentum in terms of the kinematical one and find [xi, pj+qAj+qBǫjkxk] = ih̄δij for i, j ∈ {1, 2}. Let us now consider the limit√ B ≫ m and |Cµ| ≫ |Aµ|. In this limit the terms involving the kinematical momentum pj and the po- tential Aj can be neglected: [xi, xj ] = ih̄ ǫij ≡ iθij . (4) This means that the scalar field is non-commutative in the x − y plane. It should be noted that our re- sult is not a gauge artifact, the very same result would be obtained if we had chosen e.g. the Landau gauge Cµ = (0, By, 0, 0). Furthermore, it is easy to see that since Lorentz covariance is explicitly broken by the background field new Lorentz violating vertices involv- ing the gauge boson Aµ will be generated through its interaction with the background field. In particu- lar three gauge bosons and four gauge bosons vertices which are typical of non-commutative gauge theories are generated. We have just shown that in the limit B, the coordinates x and y of the scalar field do not commute, let us rename them x̂ and ŷ. In the limit m ≪ B, local gauge transformations of the scalar field involve non-commuting coordinates: δαφ = iα(t, x̂, ŷ, z)φ(t, x̂, ŷ, z), in order to build a gauge in- variant action, the gauge boson has to transform ac- cording to δαAµ(x̂) = ∂µα(x̂)+i[α(x̂), Aµ(x̂)]. The low energy action is then given by (Dµφ(x̂)) ∗(Dµφ(x̂))− V (φ(x̂)∗φ(x̂))(5) Fµν (x̂)F µν(x̂) with Fµν = −i[Dµ(x̂), Dν(x̂)]. Using the Weyl quanti- zation procedure, it is easy to replace the non-commuting coordinates in the argument of the field φ by commuting ones (Dµφ) ∗ ⋆ (Dµφ)− V (φ∗ ⋆ φ) (6) Fµν ⋆ F where the star product is given by f ⋆ g = fei∂iθ with θij = h̄ ǫij for i, j ∈ {1, 2} and θµν = 0 in the time and z-directions. It should be noted to our derivation that it is not specific to a scalar field theory since the important point comes from the equations of motion which are the Klein-Gordon equations. Since every component of a spinor field satisfies the Klein-Gordon equations, our result applies to spinor field as well. Our first result is that the action (2) is very identical to a U(1) non- commutative gauge theory with a non-commutativity in the x − y plane. We find that a non-commutative gauge theory is very closely related to a commutative gauge theory in a strong external field in the limit that the mass of the particle is small compared to the external background field. It is well known that the ac- tion we started from is well behaved at the quantum level and in particular that it is renormalizable. On the other hand the non-commutative action (6) is not renormalizable and suffers from UV/IR mixing. This is a strong hint that the issues with the quantum field calculations involving the action (6) should disappear in the limit where more and more Landau levels are included in the calculations. However, we should point out that the naive limit B → 0 which would corre- spond to a vanishing external field implies an infinite non-commutative parameter. The limits B → 0 and√ B ≫ m do not commute. This is clearly another kind of UV/IR mixing and probably the origin of UV/IR mixing in the quantized version of the theory. We can now push our analysis further and consider Yang-Mills theories instead of a simple U(1) theory. To be very concrete let us consider a SU(2) Yang- Mills theory. In that case there are three gauge po- tentials B1µ, B µ and B µ. We see that the same pro- cedure as the one outlined in this work leads to two canonical momenta, one for each of the components of the doublet φ = (φ1, φ2). To be very precise let us consider the canonical momentum πi of the particle described by the field φ1 and π which corresponds to the particle φ2. It is clear that it only depends on B3µ since the generator T 3 is the only diagonal one. However T 1 and T 2 are not diagonal and thus B1µ and B2µ do not contribute to the canonical momenta. One finds πi + gB3i + gD3i, where B3i is the fluctuation around the strong external field D3i and −gB3i−gD3i. Let us now assume that the non- vanishing components of the strong external field D3i are given by Eǫijx j , we find [xi, pj+gB j +gEǫjkx ih̄δij and [xi, pj − gB3j − gEǫjkxk] = ih̄δij for i, j ∈ {1, 2} let us now consider the limit E ≫ m and X. Calmet and M. Kober Lowest Landau Level of Relativistic Field Theories in a Strong Background Field |D3j | ≫ |B3j | one finds [xi, xj ] = ih̄ 1gE ǫ ij and simul- taneously [xi, xj ] = −ih̄ 1 ǫij which is clearly incon- sistent. In other words, there is no non-commutative SU(2) theory equivalent to a SU(2) gauge theory re- stricted to its first Landau Level. However if we had started from a U(N) gauge group, one of the generators would be proportional to the identity matrix and we could have chosen the strong external field in the direc- tion of the identity matrix and obtained a consistent non-commutative algebra. In that case the first Lan- dau level of a gauge theory can be described in terms of a dual non-commutative gauge theory as long as the external strong field is chosen in the direction of the identity matrix. This is the physical origin of the for- mal problem with SU(N) gauge invariance mentioned at the beginning of this work. Furthermore, it should be stressed that the commu- tative action (2) is not gauge invariant under regular gauge transformations for U(N) (N>1) gauge groups unless the background field transforms as well: δAµ = ∂µα + i[α,Aµ] and δCµ = i[α,Cµ]. Note that we are using a different convention than in the background quantization technique where the background field trans- forms as a gauge field whereas the quantum fluctuation transforms homogeneously [9]. The subtlety only ap- pears for N>1. This suggests a generalization of non- commutative gauge transformations to δAµ = ∂µα+ iα ⋆ Aµ − iAµ ⋆ α (7) δCµ = iα ⋆ Cµ − iCµ ⋆ α, (8) where we set g = 1. It is also suggestive that the background field which is closely related to the non- commutative parameter through an equation such as eq. (4) should be introduced in the action: (Dµφ) † ⋆ (Dµφ)− V (φ† ⋆ φ) (9) −iφ† ⋆ Cµ ⋆ Dµφ+ i(Dµφ)† ⋆ Cµ ⋆ φ +φ† ⋆ Cµ ⋆ C µφ− 1 Fµν ⋆ F µν − 1 Cµν ⋆ C Fµν ⋆ C µν − 1 Cµν ⋆ F In the sequel we shall however restrict our consider- ations to U(1) gauge theories where this subtlety is irrelevant. Let us now apply this idea to physics beyond the standard model. If the U(1) external field we are con- sidering couples only to one specie of particle we would have a reason to explain why only a certain sector of the model is non-commutative. It is tempting to iden- tify the scalar field we have introduced with the Higgs field of the standard model. However, the Higgs field of the standard model is charged under SU(2) × U(1) and this would lead to a SU(2) non-commutative the- ory which is as explained previously not consistent for fields which are Lie algebra valued. Furthermore, it is not possible to gauge the standard model Higgs dou- blet under a new U(1) without affecting its charge as- signment under the standard model gauge group. How- ever, there has been a growing interest [10,11,12,13, 14,15,16] for particles which are not charged under the gauge group of the standard model or almost decou- pling from the action of the standard model. Further- more, scalar singlets are interesting dark matter candi- dates [17] and could explain why the Higgs boson of the standard model has not yet been discovered [18]. Let us consider the coupling of the action (1) to the standard model and we assume that φ is a SU(3)×SU(2)×U(1)Y singlet, but that it is charged under a new U(1)E gauge group under which standard model particles are sin- glets. We shall call this new particle the e-photon. Let us assume that the e-photon has a vacuum expecta- tion which fills the universe which will single out a preferred direction in space-time. There are different model building options which will affect the precise form of the non-commutative tensor θµν . For exam- ple, the e-photon and φ could for example be living in extra-dimensions and the standard model confined to a brane in which case the non-commutativity could be in three dimensions. If the new degrees of freedom are confined to live in four dimensions then we would have non-commutativity in only two-dimension in the plane perpendicular to the direction of the external strong field. Let us consider the scalar sector of the theory. We (D̄µφ) ∗(D̄µφ)−m2φφ∗φ (10) −λφ(φ∗φ)2 + (DµH)†(DµH) −m2HH†H − λH(H†H)2 + λφ∗φH†H where H is the Higgs doublet of the standard model and φ is the new scalar singlet charged under the new U(1)E interaction. The SU(2)× U(1) symmetry of the standard model has to be spontaneously broken, i.e. the doublet acquires a vacuum expectation value and using the unitary gauge, one has H = (0, h+ v) where v2 = −m2H/(2λH). However, we have two options for the U(1)E gauge symmetry. Let us first consider the case where the extra U(1) is not spontaneously bro- ken, in other words φ does not acquire a vacuum ex- pectation value. In that case the scalar potential is given by V [h, φφ∗] = −2m2Hh2 + λH(h + v)4 + λ(h + v)2φφ∗+m2φφφ ∗+λφφφ ∗φφ∗. It is easy to show that the e-photon and the usual photon do not mix. Further- more there is no coupling between the e-photon and the fermions of the standard model. This new long range force is thus not in conflict with experiments. The new charged scalars are protected by the exact U(1)E symmetry and thus dark matter candidates. Furthermore, although the carrier of the new force in the dark matter sector are massless, the bounds on a fifth force in the dark matter sector [19] do not ap- ply to our model because the e-photon does not cou- ple to regular matter. Let us now assume that the e- photon has some vacuum expectation value such that it correspond to a strong magnetic-type field in the z-direction and consider this model in the first Lan- dau Level approximation. We find that the scalars φ have non-commuting coordinates in that limit. The Alternatives Parallel non-commuting coordinates can be removed at the expense of introducing a star product V [h, φ ⋆ φ∗] = −2m2Hh2 +λH(h+ v)4 +λ(h+ v)2φ ⋆ φ∗ +m2φφ ⋆φ∗ + λφφ⋆φ ∗⋆φ⋆φ∗, i.e. the only non-commutative interac- tion are those which involve the field φ and obviously the e-photon. If its mass is low enough, this dark mat- ter candidate could be produced at the LHC through the decomposition of a Higgs boson. The decay rate Γ (Higgs → φφ∗) = 1 is basically commutative. However, the self-interaction of the φ- mesons and of the e-photon are non-commutative and the non-commutative nature of this sector could be checked by searching for the usual characteristic non- commutative self-interactions of the e-photon. The cross section for the dark matter candidate at the Tevatron and LHC corresponds to the one of a singlet added to the standard model in the commutative case and there should thus be a clear signal. However, detecting the non-commutative nature of the dark matter sector at a hadron collider will be a difficult task since one would have to search for the typical self-interactions of the e-photon. However, one would expect that the background field will impact the distribution of dark matter in our universe which would allow to identify a preferred direction in space-time. Another option is to assume that the remaining U(1)E is spontaneously broken by a vacuum expecta- tion value of the φ-mesons in which case the e-photon acquires a mass. In that scenario, the φ-mesons are not dark matter candidates. However, the residual degree of freedom after U(1)E symmetry breaking, which we call σ, will mix with the standard model Higgs bo- son. One finds: hphys = cosα h + sinα σ, σphys = cosα σ − sinα h where α, the mixing angle, is de- termined by the scalar potential. When we consider the model in a strong external potential correspond- ing to a strong magnetic-like field in the z-direction and in the lowest Landau level limit, we find that both scalar fields are non-commuting in the x − y plane whereas the remaining fields of the standard model are commutative. In that case the only sector of the theory which would exhibit a non-commutative nature is the scalar potential sector. The new non- commutative operators are vhh ⋆ φ ⋆ φ, vφh ⋆ h ⋆ φ and h ⋆ h ⋆ φ ⋆ φ. Because of the trace property of the star product ( d4xf ⋆g = d4xg⋆f = d4xfg), the inter- actions of the two scalar degrees of freedom with the fermions and gauge bosons of the standard model are commutative. Conclusions: We have shown that the phenomenon discovered by Landau in 1930 appears in relativistic field theories. We find physical reasons to the formal problems with non-commutative gauge theories such as the issue with SU(N) gauge symmetries. We apply our construction to a minimal extension of the stan- dard model and show that the Higgs sector might be non-commutative whereas the remaining sectors of the standard model remain commutative. We discuss the signatures of this model at the LHC. We then dis- cuss an application to a dark matter sector coupled to the Higgs sector of the standard model and show that here again, dark matter could be non-commutative, the standard model fields remaining commutative. Acknowledgments: This work was supported in part by the IISN and the Belgian science policy office (IAP V/27). References 1. A. Connes, M. R. Douglas and A. S. Schwarz, JHEP 9802, 003 (1998). 2. N. Seiberg and E. Witten, JHEP 9909, 032 (1999). 3. V. Schomerus, JHEP 9906, 030 (1999). 4. L. Landau, Z. Phys. 64, 629 (1930). 5. X. Calmet, B. Jurco, P. Schupp, J. Wess and M. Wohlgenannt, Eur. Phys. J. C 23, 363 (2002). 6. M. Chaichian, A. Kobakhidze and A. Tureanu, Eur. Phys. J. C 47, 241 (2006). 7. R. Jackiw, Nucl. Phys. Proc. Suppl. 108, 30 (2002) [Phys. Part. Nucl. 33, S6 (2002 LNPHA,616,294- 304.2003)]. 8. P. A. Horvathy, Annals Phys. 299, 128 (2002). 9. S. Weinberg, “The quantum theory of fields. Vol. 2: Modern applications,” Cambridge, UK: Univ. Pr. (1996) 489 p. 10. A. Hill and J. J. van der Bij, Phys. Rev. D 36, 3463 (1987). 11. J. J. van der Bij, Phys. Lett. B 636, 56 (2006). 12. X. Calmet, Eur. Phys. J. C 28, 451 (2003). 13. X. Calmet, Eur. Phys. J. C 32, 121 (2003). 14. X. Calmet and J. F. Oliver, Europhys. Lett. 77, 51002 (2007) [arXiv:hep-ph/0606209]. 15. B. Patt and F. Wilczek, arXiv:hep-ph/0605188. 16. D. O’Connell, M. J. Ramsey-Musolf and M. B. Wise, Phys. Rev. D 75, 037701 (2007). 17. J. McDonald, Phys. Rev. D 50, 3637 (1994). 18. J. J. van der Bij and S. Dilcher, Phys. Lett. B 638, 234 (2006). 19. M. Kesden and M. Kamionkowski, Phys. Rev. Lett. 97, 131303 (2006); Phys. Rev. D 74, 083007 (2006). http://arxiv.org/abs/hep-ph/0606209 http://arxiv.org/abs/hep-ph/0605188

0704.1356

Mechanical and dielectric relaxation spectra in seven highly viscous glass formers

Mechanical and dielectric relaxation spectra in seven highly viscous glass formers U. Buchenau∗ Institut für Festkörperforschung, Forschungszentrum Jülich Postfach 1913, D–52425 Jülich, Federal Republic of Germany (Dated: April 2, 2007) Published dielectric and shear data of six molecular glass formers and one polymer are evaluated in terms of a spectrum of thermally activated processes, with the same barrier density for the retardation spectrum of shear and dielectrics. The viscosity, an independent parameter of the fit, seems to be related to the high-barrier cutoff time of the dielectric signal, in accordance with the idea of a renewal of the relaxing entities after this critical time. In the five cases where one can fit accurately, the temperature dependence of the high-barrier cutoff follows the shoving model. The Johari-Goldstein peaks, seen in four of our seven cases, are describable in terms of gaussians in the barrier density, superimposed on the high-frequency tail of the α-process. Dielectric and shear measurements of the same substance find the same peak positions and widths of these gaussians, but in general a different weight. PACS numbers: 64.70.Pf, 77.22.Gm I. INTRODUCTION The publications of Kia Ngai deal with more sub- stances and more measurement techniques than the work of any other scientist in the field of undercooled liquids. Within the past three decades, whenever a new develop- ment appeared, he was the quickest to appreciate it, ana- lyze it and bring it to the general attention, thus speeding up the progress substantially. Many scientists in the field share his conviction that the flow process in highly viscous liquids can only be understood by combining all possible techniques for its study1,2,3,4,5,6. The present paper evaluates recently published7,8 broadband shear and dielectric relaxation data on seven glass formers. The mechanical shear data were obtained with a new technique9 which allows to cover a large dynamical range. Samples for dielectric and shear measurements were taken from the same charge, and the temperature sensors of both measurements were calibrated to each other. The data show a striking similarity of G′′(ω) and ǫ′′(ω) on the right hand side of the α-peak, a similarity which is sometimes perturbed by the secondary Johari-Goldstein peak10,11. The similarity suggests a common origin of the α-peak in dielectrics and shear. The fact that the shear peak appears at a higher frequency than the dielectric peak is explainable in terms of the viscosity, which in a compliance treatment12 is a free parameter. In order to identify the elementary processes with ther- mally activated jumps over an energy barrier V , one can use a recent translation13 of the textbook12 retardation spectrum L(ln τ) into a barrier density function l(V ). We will argue that the compliance barrier density of the shear equals the electric dipole moment barrier density lǫ(V ) of the dielectric data. The next section (section II) explains and motivates this approach in more detail. The results of the data treatment are described in section III. They are discussed and compared to other approaches in section IV. Section V summarizes and concludes the paper. II. THE BARRIER DENSITY FUNCTIONS FOR SHEAR AND DIELECTRICS The choice of a retardation spectrum for the shear is motivated by a surprising coincidence, which is more or less visible in the data of all seven substances. We show the two examples where it is most clearly seen in Fig. 1 and Fig. 2. Fig. 1 compares G′′(ω) and −ǫ′′(ω) at the same tem- perature for the type-A glass former PPE. PPE, 1,3- Bis(3-phenoxyphenoxy)benzene, is a diffusion pump oil with the commercial name Santovac-5P, a molecule con- sisting of five phenyl rings connected by four oxygens to form a short chain. In terms of the classification proposed by the Bayreuth group14, it is a type-A glass former, a -1 0 1 2 3 4 5 6 7 G'' - " scaled Santovac-5P (PPE) 254 K log( /s-1) FIG. 1: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in a double-log scale for PPE at 254 K. http://arxiv.org/abs/0704.1356v1 glass former which shows no or at least no pronounced secondary Johari-Goldstein peak. The (negative) ǫ′′(ω)- data have been scaled to coincide with the G′′(ω)-data on the right hand side of the peak. One finds good agree- ment betweenG′′(ω) and−ǫ′′(ω) as soon as the frequency is two decades higher than the one of the peak in G′′(ω). In addition, Fig. 1 shows a change of slope of the α- tail, from an ω−1/2- to an ω−1/3-behavior. The tendency is seen most clearly in the dielectric data, but it is also in the shear data; their fit improves markedly if one allows for a ω−1/3-component. This might be the influence of a hidden Johari-Goldstein peak, an explanation which has been favored for glycerol on the basis of pressure, aging and chemical series measurements (for a review, see Ngai and Paluch15). But there is also impressive experimen- tal evidence for a limiting ω−1/3-behavior in shear com- pliance data of type-A molecular glass formers16, which show the so-called Andrade17 creep, J(t) ∼ t1/3, in the short-time limit. Therefore we will fit our data in terms of a sum of an ω−β-term (with β as fit parameter) and an ω−1/3-term, dominating at high frequency. Toward lower frequency, the common slope terminates for the shear data already at a higher frequency (a shorter time) than for the dielectric data. The natural explana- tion for this is that the parallel between dielectrics and shear is in fact between shear compliance and dielectric susceptibility. In a comparison of these two quantities, the shear compliance starts to deviate from the dielectric susceptibility as soon as the viscous flow sets in. This suggests a treatment of the shear data in terms of a re- tardation spectrum, with the viscosity as an independent parameter12. The good agreement at the right-hand side of the α- peak is found to be a general feature of all seven mea- sured substances, as long as there is no disturbing in- fluence from the secondary Johari-Goldstein peak10,11. This is seen in Fig. 2, which shows that the good agreement between G′′ and −ǫ′′ disappears as the peaks merge. The substance, tripropylene glycol (TPG), -2 -1 0 1 2 3 4 5 6 7 1 G'' 188 K G'' 192 K - " 188 K scaled - " 192 K scaled tripropylene glycol (TPG) log( /s-1) FIG. 2: Comparison of G′′(ω) to a properly scaled −ǫ′′(ω) in a double-log scale for TPG at 188 and 192 K. C6H20O4, is still a molecule and not yet a polymer (the whole series from the small molecule propylene glycol to long-chain polypropylene glycol is well-investigated by dielectrics18,19,20). TPG itself has been studied under aging21 and under pressure22. It consists of three con- nected propylene groups, with a large dielectric moment and a pronounced Johari-Goldstein peak (which is absent or at least much less pronounced in propylene glycol18). Fig. 2 shows another tendency which will be evaluated quantitatively in this paper, namely a much stronger in- crease of the imaginary quantities in the α-peak region with temperature than in the Johari-Goldstein peak re- gion. If one wants to decompose a measured relaxation into a spectrum of exponential decays in time, one can choose between two equivalent possibilities12, the relaxation spectrum in which the elementary exponential relaxators add to decrease the modulus or the retardation spectrum in which they add to increase a susceptibility. In princi- ple, the choice is not crucial, because the two spectra can be calculated from each other. Here, we choose the retar- dation spectrum, with the viscosity η as an independent variable. For this choice, one has the textbook expressions12 for the real and imaginary parts of the complex frequency- dependent shear compliance J ′(ω) = Jg + L(ln τ) 1 + ω2τ2 d(ln τ) (1) J ′′(ω) = L(ln τ) 1 + ω2τ2 d(ln τ) + , (2) where τ is the relaxation time and L(ln τ) is the weight of this relaxation time in the retardation spectrum. Jg, the glass compliance, is the inverse of the infinite frequency modulus G∞. In an energy landscape picture23, one reckons with thermally activated jumps over the energy barrier be- tween two neighboring minima. In fact, one very of- ten finds a broad secondary relaxation peak (the Johari- Goldstein peak10,11) below the α-peak of the flow pro- cess. This peak follows the Arrhenius relation in the glass phase, indicating that it stems from local thermally activated jumps. For a jump over an energy barrier of height V , the Arrhenius relation for the relaxation time τV reads τV = τ0e V/kBT , (3) where τ0 = 10 −13 s and T is the temperature. For a spectrum of thermally activated jumps, one defines13 the barrier density function ls(V ) ls(V ) = L(V/kBT + ln τ0). (4) The index s stands for the shear. With this definition, the complex shear compliance equations (1) and (2) trans- form into J ′(ω) = Jg + Jg ls(V ) 1 + ω2τ2V dV (5) J ′′(ω) = Jg ls(V ) 1 + ω2τ2V . (6) The dielectric susceptibility can also be described24,25 in terms of a dielectric barrier density function lǫ(V ) ǫ′(ω)− ǫ∞ ǫ(0)− ǫ∞ lǫ(V ) 1 + ω2τ2V dV (7) ǫ′′(ω) ǫ(0)− ǫ∞ lǫ(V ) 1 + ω2τ2V dV. (8) Here ǫ(0) is the static dielectric susceptibility, ǫ∞ is the real part of ǫ(ω) in the GHz range (larger than n2, the square of the refractive index, because of vibrational contributions6). The above definitions of eqs. (5-8) imply a normaliza- tion of both ls(V ) and lǫ(V ) with ls(V )dV = J0e − Jg , (9) where J0e is the recoverable compliance of the steady- state flow12, and lǫ(V )dV = 1. (10) The dielectric α-peak occurs always at a lower fre- quency than the shear one and seems to coincide with the heat capacity and the structural relaxation peaks1,2,3,4,5,6. Below, we will adopt the view that the left side of the dielectric peak marks the disappearance and renewal of the relaxing entities. In order to describe this decay, one needs to multiply the barrier density of the energy landscape with an ap- propriate cutoff function at a cutoff barrier Vc. Here, we will assume that the relaxing entities decay exponentially in time with the critical relaxation time τc. With the Ar- rhenius relation τc = τ0 exp(Vc/kBT ), this translates into a double-exponential cutoff c(V ) = exp(− exp((V − Vc)/kBT )). (11) Equations (6) and (8) show that a Johari-Goldstein peak in G′′(ω) or ǫ′′(ω) at the peak frequency ω1 cor- responds to a peak in l(V ) at a peak barrier V1 = kBT ln(1/ω1τ0). We will see that the Johari-Goldstein peaks are reasonably well described by gaussians in l(V ). To describe both the α-peak and the Johari-Goldstein- peak in terms of a barrier density, ls(V ) and lǫ(V ) will be fitted by the form l(V ) = (aβe βV/kBT + a1/3e V/3kBT + a1e −γ1(V−V1) )c(V ). The first two terms describe the high-frequency tail of the α-process, the third term the Johari-Goldstein peak (if there is one; in three of our seven examples, it is not needed). The dimensionless parameter β determines the slope ω−β at the beginning of the α-tail in the double-log plot of Fig. 1. Instead of using the three prefactors aβ , a1/3 and a1 as fit parameters, it is better to use the correspond- ing weights wβ , w1/3 and w1 in the integral over the barriers, equs. (9) and (10). A type-A glass former without Johari-Goldstein peak with w1 = 0 is char- acterized by the two dimensionless parameters β and b2 = w1/3/(wβ + w1/3), at least as far as the form of its spectrum is concerned. β and b2 have the advantage to be reasonably temperature-independent. With this prescription, one can fit the ǫ′′(ω) of a type-A glass former with two temperature-independent parame- ters, β and b2, and two temperature-dependent parame- ters, ∆ǫ = ǫ(0)− ǫ∞ and Vc. Their temperature depen- dence is a decrease with increasing temperature, which -2 -1 0 1 2 3 4 5 log( /s-1) 254 K 262 K 274 K FIG. 3: (a) Data and fit of ǫ′′(ω) in a double-log scale for TPE between 254 and 274 K (b) the same for G(ω). is well fitted by an appropriate power law ∆ǫ(T ) = ∆ǫ(Tg) , (13) where Tg is the glass temperature. Similarly, one de- scribes the decrease of Vc with the exponent γV and the one of G∞ with γG. The strategy of our evaluation is to fit l(V ) to the dielectric data, and then use the same spectral form to describe the shear. The fit of the shear data re- quires three additional temperature-dependent parame- ters, Jg, J e and η. Again, it is worthwhile to look for combinations which might turn out to be temperature- independent. One of them is the ratio J0e − Jg , (14) which appears in the normalization of the shear spec- trum, eq. (9). A second interesting possibility is not to fit the viscosity η, but the ratio fjc = f0Jgη , (15) where τc is the Arrhenius relaxation time of the terminal barrier Vc. As we will show in the discussion, one can argue that the ratio fjc should be 2 for a renewal of the relaxing entities within the critical time τc. In the case of a type-B glass former, Fig. 2 shows that one needs another dimensionless parameter, because the weight of the Johari-Goldstein peak is different in the two quantities. In Fig. 2, the Johari-Goldstein peak is more prominent in the shear signal, but this varies from substance to substance. 205 210 215 220 225 230 235 240 fit data from V DC704 temperature (K) FIG. 4: Fit values of G∞ in DC704 as a function of tem- perature. The continuous line is the temperature dependence Vc/∆v (with ∆v = 0.057 nm 3) expected from the shoving model26. III. DATA EVALUATION A. The three type-A glass formers Three of our seven substances, TPE, DC704 and PPE, happen to have no or at least only a rather weak Johari- Goldstein peak. Let us begin with TPE. TPE stands for triphenylethylene, C20H16, a rather flexible molecule with three phenyl rings attached to a central C = C double bond. Fig. 3 (a) shows data and fit for ǫ′′(ω) in a double-log plot, Fig. 3 (b) the ones for G(ω). The dielectric data in Fig. 3 (a) are well fitted with only the first two terms of eq. (12), without any Johari-Goldstein peak. β and b2 turn out to be temperature-independent within experimental accuracy. One gets a good fit for the shear data in Fig. 3 (b), tak- ing over β, b2 and the cutoff barrier Vc from the fit of the dielectric data at the given temperature and fitting G∞, f0 and fjc. G∞ is temperature-dependent, but f0 and fjc are again temperature-independent within the exper- imental accuracy, thus justifying our choice of variables. The parameters and their temperature dependence are listed in Table I. The temperature exponents γV and γG of the critical barrier Vc and the infinite frequency shear modulus G∞ are the same within their error bars (about 5 % for γV and about 10 % for γG). This shows the validity of the shoving model26, according to which the energy barrier of the α-process should be proportional to the infinite frequency shear modulus G∞. The shoving model postu- lates that the α-process happens when the local energy concentration exceeds the product G∞∆v, where ∆v is a volume expansion. The same results, maybe even a bit clearer because of the stronger dielectric signals, are obtained for the two other type-A glass formers PPE and DC704. Again, the fit parameters are listed in Table I. In particular, the ω−1/3-contribution is much better seen, as illustrated in Fig. 1 for PPE. In DC704, again a diffusion pump oil (1,3,3,5-tetramethyl-1,1,5,5-tetraphenyl-trisiloxane, a rather large molecule) we have the additional advan- tage of a large temperature range of the measurement, from 209 to 239 K. As in TPE, we find temperature- independent parameters β, b2, f0 and fjc. Again, we find the shoving model26 confirmed in both glass form- ers. In DC704, one even sees the curvature of both curves (see Fig. 4), which justifies our temperature exponent Ansatz, eq. (13). Table I comprises the fit parameters for these three type-A glass formers. Note that our formalism allows to describe both shear and dielectric data over the whole temperature range with eleven temperature-independent parameters. glass former TPE DC704 PPE Tg (K) 249 211 244 ∆ǫ 0.0491 0.257 2.011 γǫ 1.85 2.26 1.90 β 0.77 0.85 1.04 b2 0.18 0.27 0.215 Vc(Tg) (eV) 0.767 0.639 0.755 γV 4.3 4.6 4.6 G∞(Tg) (GPa) 1.38 1.80 1.27 γG 4.7 4.2 4.7 f0 1.65 2.38 2.22 fjc 2.5 2.45 2.05 TABLE I: Parameters of the three type-A glass formers. Up- per part ǫ(ω), lower part G(ω). B. The four type-B glass formers In the type-B glass formers DHIQ, PB20, Squalane and TPG, one needs to fit a Johari-Goldstein peak on top of the high-frequency tail of the α-process. This is illus- trated in Fig. 5 for our first type-B example, squalane. Squalane, C30H62, is a short chain molecule with 24 carbon atoms in the backbone and 6 attached CH3- groups, rather polymerlike. It has a strong and well- separated Johari-Goldstein peak (see Fig. 5), much bet- ter visible in the shear data than in the dielectric data. The dielectric dipole moment is very weak. Nevertheless, it is possible to fit both sets of data with the same retar- dation spectrum, attaching a substantially higher weight -3 -2 -1 0 1 2 3 4 5 6 168 K 174 K 180 K fits squalane log( /s-1) FIG. 5: Data and fits of (a) ǫ′′(ω) (b) G(ω) in squalane. to the Johari-Goldstein peak in the shear (see Table II). In this substance, it is not possible to fit the shear data with a temperature-independent parameter f0; one has to postulate a rather strong increase of f0 with increasing temperature f0(T ) = f0(Tg) + f 0(T − Tg), (16) but one can keep the parameter fjc constant (see Table PB20 is a true polymer, relatively short (5000 g/mol), composed of 80 % 1,4-polybutadiene monomers and 20 % 1,2-polybutadiene monomers. The results look very similar to those of squalane, and the resulting fit param- eters in Table II are in fact close to those of squalane. Even more than squalane, it has the polymer feature of a relatively slow decrease of ǫ′′(ω) at low frequency, ex- plainable in terms of chain modes with long relaxation times12. This is illustrated in Fig. 6, which shows the deviation between fit and data at low frequency. As a consequence, the resulting parameters have a larger er- ror bar in squalane and polybutadiene than in the two molecular substances TPG and DHIQ. In particular, the deviations between γV and γG do not demonstrate a fail- ure of the shoving model. TPG is a much more favorable case, with a very large dipole moment and no problems at the cutoff barrier. As Fig. 7 (a) shows, our spectrum of eq. (12) provides beau- tiful fits over a large temperature range. One needs to take the temperature dependence of the Johari-Goldstein peak position V1 into account. Our fit found V1 = 0.295 , (17) a bit smaller shift than the one found in aging experiments21. Fig. 7 (b) shows that the shear data are well described in terms of the dielectric retardation spectrum. There is a small temperature dependence of f0, but fjc is again a temperature-independent constant. The shoving model is found to be well fulfilled (see Fig. 8). -2 -1 0 1 2 3 4 5 6 7 fit polybutadiene 180 K log( /s-1) FIG. 6: Data and fit of ǫ′′(ω) in polybutadiene at 180 K. Finally, DHIQ, decahydroisoquinoline, C9H17N , is best described as two cyclohexanol rings sharing one C−C-bond, one of the two rings having anNH replacing a CH2-group. In this case, the Johari-Goldstein peak is very prominent in the dielectric data27,28, comparable to the one in G(ω). The dipole moment is large; both Vc and G∞ can be determined with high accuracy. Again, their temperature exponents γV and γG agree within the error bars (see Table II), in agreement with the shov- ing model26. Since both are exceptionally large (DHIQ is very fragile, m=158 in Angell’s scheme29), their good agreement provides a strong argument for the validity of the model. In Table II, the Johari-Goldstein peak is characterized by the weight of the peak w(T ) = a1 (π/γ1) = a1FWHM (π/4 ln 2) (18) which shows a Boltzmann factor behavior w(T ) = w(Tg) exp(−Ea(1/kBT − 1/kBTg)), (19) with a formation energy Ea which is on the average 2/3 of the peak position V1. IV. DISCUSSION The preceding section presented a quantitative descrip- tion of the α- and the β-process in dielectrics and shear -3 -2 -1 0 1 2 3 4 5 6 7 182 K 188 K 194 K 200 K 206 K 212 K 218 K 224 K fits TPG log( /s-1) FIG. 7: Data and fits of (a) ǫ′′(ω) (b) G(ω) in TPG. glass former Squalane PB20 TPG DHIQ Tg (K) 167 176 184 175 ∆ǫ(Tg) 0.0155 0.132 23.3 1.707 γǫ 2.1 0.0 1.51 0.0 β 0.6 0.44 0.85 0.4 b2 0.2 0.2 0.22 0.2 Vc(Tg) (eV) 0.517 0.54 0.63 0.635 γV 3.2 3.8 3.0 6.4 G∞(Tg) (GPa) 1.33 1.63 2.69 3.1 γG 2.5 2.7 2.8 6.3 f0(Tg) 2.4 3.25 6.7 1.56 f ′0 (1/K) 0.4 0.41 -0.04 0.25 fjc 2.7 2.4 2.5 2.0 V1 (eV) 0.27 0.28 0.32* 0.32 FWHM(eV) 0.135 0.170 0.154 0.16 ws(Tg) 0.56 0.55 0.15 0.50 wǫ(Tg) 0.03 0.23 0.02 0.48 Ea (eV) 0.25 0.12 0.19 0.25 *average value, see eq. (17) TABLE II: Parameters of the four type-B glass formers. Up- per part G(ω), middle part ǫ(ω), lower part Johari-Goldstein peak parameters for both. for seven different glass formers, a description which is based on the concept of isolated and independent ther- mally activated jumps in the energy landscape. The de- scription allows for a reasonable fit of the temperature dependence in terms of temperature-independent param- eters. The number of parameters is not small; one needs eleven or twelve parameters for a type-A glass former (depending on whether f0 is temperature-independent or not, see Table I and II) and five additional parameters for the description of the β- or Johari-Goldstein peak (see Table II). Nevertheless, the exercise is not completely meaning- less. One does indeed get meaningful quantitative in- formation, which is impossible to obtain otherwise. The 180 185 190 195 200 205 210 215 220 225 230 fit data from V temperature (K) FIG. 8: Fit values ofG∞ in TPG as a function of temperature. The continuous line is the temperature dependence Vc/∆v (with ∆v = 0.037 nm3) expected from the shoving model26. first and rather important one is the probable equality of the retardation spectra of shear and dielectrics (but with a different weight of the Johari-Goldstein peak), an infor- mation which one can guess from the raw data (see Figs. 1 and 2), but which requires a full fit for its quantitative check. The second and equally important quantitative infor- mation concerns the dimensionless ratio fjc between the terminal dielectric relaxation time τc and the product of the viscosity with the total retardation compliance, eq. (15). The seven fitted values lie between 2 and 2.7 (average value 2.37). This indicates a general relation between the dielectric terminal time and the viscosity. Since the dielectric terminal time seems to coincide with the structural lifetime5, it is probably also the lifetime of the double-well potentials which are responsible for the retardation spectrum. Question: What do we expect for the ratio fjc if this is indeed the case? To answer this question, consider a constant applied shear stress. After the time τc, all the double-well potentials of the spectrum would have reached thermal equilibrium, giving their full contribu- tion to the compliance. From this consideration, if we renew them at the time τc, we would naively expect them to be able to give their contribution again after this time, yielding fjc = 1. But this answer is not correct. To get the correct an- swer, one must consider the difference between energy and free energy in these double-well potentials. To keep the argument simple, let us restrict ourselves to the spe- cial case of a symmetric double-well; it applies as well to the asymmetric case. If the double-well is initially symmetric and if it couples to the shear stress σ with a coupling constant v (the coupling constant has the dimension of a volume), then the asymmetry ∆ under the stress is σv. One well has the energy −σv/2, the other has the energy +σv/2. In thermal equilibrium, the population of the two wells is given by their Boltzmann factors. It is easy to calculate the energy U of the equilibrated system in the limit of a small stress U = − . (20) This is the energy transported to the heat bath in the equilibration of the relaxing entity after switching on the stress. The free energy F is F = − , (21) only half of the energy itself. If one thinks about it, the reason is clear: spending the energy, one has spanned an entropic spring by the population difference in the two wells. If one removes the stress slowly, one gets half the energy back. But if the double-well potential decays, one gets nothing back. The contribution of the relaxing entity to the compli- ance is given by the second derivative of the free energy with respect to the stress. But if we now deal with the effect of a renewal of the double-well potential on the vis- cosity, we have to count the energy. This means we spend twice as much energy under a constant stress as the one calculated above in our first oversimplified picture. And this means the viscosity must be a factor of 2 smaller, which implies fjc = 2. This is reasonably close to the fitted values in Table I and II. A third quantitative conclusion of the present study is a surprising agreement with the conclusions of Plazek et al16 from their recoverable shear compliance experi- ments. If one takes the parameters of Table I to calculate the recoverable compliance, one gets curves which closely resemble those reported by them. Obviously, it is exper- imentally much easier to detect the Andrade creep17 in creep experiments than in dynamical ones. If one calcu- lates f0 from their data, one finds values between 1.5 and 2.3, similar to those in Table I. Here, however, a word of caution is in place. Our data, taken as they are, do not imply a limited recoverable shear compliance. In fact, they are well fitted by the BEL model30, which has a divergent recoverable compliance. The values in the two tables stem from the assumption that the two retardation spectra of dielectrics and shear (at least as far as the α-peak is concerned) are the same. The same is true for the fourth conclusion, the valid- ity of the shoving model26. The fitted G∞-values were obtained under the same assumption. Finally, the Johari-Goldstein peak increases its height with increasing temperature. The increase follows a Boltzmann factor, with a formation energy of about two thirds of the barrier height at the center of the peak. V. SUMMARY AND CONCLUSIONS Dielectric and shear relaxation data in seven highly viscous liquids, most of them molecular liquids, were evaluated in terms of a barrier density of independent thermally activated relaxation centers. Three of the sub- stances are type-A glass formers without or with only a very small Johari-Goldstein peak, four of them show a pronounced Johari-Goldstein peak. The most important conclusion is the probable equal- ity of the dielectric and shear retardation spectra, guessed from the raw data and confirmed by a quanti- tative fit. The difference in the peak positions is due to the influence of the viscosity. The Johari-Goldstein peak has different weight in dielectrics and shear. The second important conclusion concerns the viscos- ity. It seems probable that the viscosity results from the constant renewal of the double-well potentials in the sample within the terminal dielectric relaxation time. Our data support earlier recovery compliance results by Plazek et al16, according to which one has an Andrade17 creep J ∼ t1/3 at short times in type-A glass formers (glass formers without Johari-Goldstein peak). They further support the shoving model26, which pos- tulates a proportionality between the infinite frequency shear modulus and the Arrhenius barrier of the terminal relaxation time. Acknowledgement: The author is deeply thankful to Kristine Niss and Bo Jakobsen for communicating their beautiful data to him, to Niels Boye Olsen and Tage Christensen for enlightening discussions and to Jeppe Dyre for constant encouragement and a lot of helpful advice. ∗ Electronic address: [email protected] 1 N. O. Birge and S. R. Nagel, Phys. Rev. Lett. 54, 2674 (1985); N. O. Birge, Phys. Rev. B 34, 1631 (1986) 2 K. L. Ngai and R. W. Rendell, Phys. Rev. B 41, 754 (1990) 3 I. Chang and H. Sillescu, J. Chem. Phys. 101, 8794 (1997) and further references therein 4 K. Schröter and E. Donth, J. Non-Cryst. Solids 307-310, 270 (2002) 5 U. Buchenau, M. Ohl and A. Wischnewski, J. Chem. Phys. 124, 094505 (2006) 6 U. Buchenau, R. Zorn, M. Ohl and A. Wischnewski, cond-mat/0607056 and Phil. Mag. 2006 (in press) 7 K. Niss, B. Jakobsen and N. B. Olsen, J. Chem. Phys. 123, 234510 (2005) 8 B. Jakobsen, K. Niss and N. B. Olsen, J. Chem. Phys. 123, 234511 (2005) 9 T. Christensen and N. B. Olsen, Rev. Sci. Instrum. 66, 5019 (1995) 10 G. P. Johari and M. Goldstein, J. Chem. Phys. 53, 2372 (1970) 11 G. P. Johari and M. Goldstein, J. Chem. Phys. 55, 4245 (1971) 12 D. J. Ferry, ”Viscoelastic properties of polymers”, 3rd ed., John Wiley, New York 1980 13 U. Buchenau, Phys. Rev. B 63, 104203 (2001) 14 A. Kudlik, Ch. Tschirwitz, S. Benkhof, T. Blochowicz and E. Rössler, Europhys. Lett. 40, 649 (1997) 15 K. L. Ngai and M. Paluch, J. Chem. Phys. 120, 857 (2004) 16 D. J. Plazek, C. A. Bero and I.-C. Chay, J. Non-Cryst. Solids 172-174, 181 (1994) 17 E. N. da C. Andrade, Proc. Roy. Soc. A 84, 1 (1910) 18 C. Leon, K. L. Ngai and C. M. Roland, J. Chem. Phys. 110, 11585 (1999) 19 J. Mattsson, R. Bergman, P. Jacobsson and L. Börjesson, Phys. Rev. Lett. 90, 075702 (2003) 20 J. Mattsson, R. Bergman, P. Jacobsson and L. Börjesson, Phys. Rev. Lett. 94, 165701 (2005) 21 J. C. Dyre and N. B. Olsen, Phys. Rev. Lett. 91, 155703 (2003) 22 S. Pawlus, S. Hensel-Bielowska, K. Grzybowska, J. Ziolo and M. Paluch, Phys. Rev. B 71, 174107 (2005) 23 M. Goldstein, J. Chem. Phys. 51, 3728 (1968) 24 M. Pollak and G. E. Pike, Phys. Rev. Lett. 28, 1449 (1972) 25 K. S. Gilroy and W. A. Phillips, Phil. Mag. B 43, 735 (1981) 26 J. C. Dyre, N. B. Olsen and T. Christensen, Phys. Rev. B 53, 2171 (1996) 27 R. Richert, K. Duvvuri and L.-T. Duong, J. Chem. Phys. 118, 1828 (2003) 28 M. Paluch, S. Pawlus, S. Hensel-Bielowska, E. Kaminska, D. Prevosto, S. Capaccioli, P. A. Rolla and K. L. Ngai, J. Chem. Phys. 122, 234506 (2005) 29 R. Böhmer, K. L. Ngai, C. A. Angell and D. J. Plazek, J. Chem. Phys. 99, 4201 (1993) 30 A. J. Barlow, A. Erginsav and J. Lamb, Proc. Roy. Soc. A 309, 473 (1969) mailto:[email protected] http://arxiv.org/abs/cond-mat/0607056

0704.1357

Computational and experimental imaging of Mn defects on GaAs (110) cross-sectional surface

Computational and experimental imaging of Mn defects on GaAs (110) cross-sectional surface A. Stroppa,1, 2, ∗ X. Duan,1, 2, † M. Peressi,1, 2, ‡ D. Furlanetto,3, 4 and S. Modesti3, 4 1Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014 Trieste, Italy 2CNR-INFM DEMOCRITOS National Simulation Center, via Beirut 2-4, 34014 Trieste, Italy 3CNR-INFM TASC National Laboratory, Area Science Park, 34012 Trieste, Italy 4Dipartimento di Fisica and Center of Excellence for Nanostructured Materials, CENMAT, Università di Trieste, via A. Valerio 2, 34127 Trieste, Italy (Dated: August 10, 2021) Abstract We present a combined experimental and computational study of the (110) cross-sectional surface of Mn δ-doped GaAs samples. We focus our study on three different selected Mn defect configu- rations not previously studied in details, namely surface interstitial Mn, isolated and in pairs, and substitutional Mn atoms on cationic sites (MnGa) in the first subsurface layer. The sensitivity of the STM images to the specific local environment allows to distinguish between Mn interstitials with nearest neighbor As atoms (IntAs) rather than Ga atoms (IntGa), and to identify the finger- print of peculiar satellite features around subsurface substitutional Mn. The simulated STM maps for IntAs, both isolated and in pairs, and MnGa in the first subsurface layer are consistent with some experimental images hitherto not fully characterized. PACS numbers: 73.20.-r,73.43.Cd,68.37.Ef http://arxiv.org/abs/0704.1357v1 I. INTRODUCTION Mn-doped GaAs1,2,3,4 has attracted considerable attention among the diluted magnetic semiconductors for its possible application in the emerging field of spintronic.5,6,7 Although other materials such as ferromagnetic metals and alloys, Heusler alloys, or magnetic oxides seem to be promising candidates for spintronic devices, the diluted magnetic semiconduc- tors and Mn-doped GaAs in particular are of tremendous interest in that they combine magnetic and semiconducting properties and allow an easy integration with the well es- tablished semiconductor technology. Besides possible spintronic applications, characterizing and understanding the properties of Mn defects in GaAs is a basic research problem which is still debated. The growth conditions and techniques affect the solubility of Mn in GaAs, which is in general rather limited, and its particular defect configurations, thus determining the mag- netic properties of the samples.8,9,10,11,12 The highest Curie temperature Tc reachable for Mn-doped GaAs up to few years ago was 110 K,13 rather low for practical technological purposes. Intense efforts have been pursued in the last years in order to understand the physics of this material and to improve its quality and efficiency. Out-equilibrium growth techniques1,5 have enabled to increase the solubility of Mn and the Curie temperature; post-growth annealing of epitaxial samples at temperatures only slightly above the growth temperature has been particularly successfull.9,10,14 Nowadays, δ-doping is used as an al- ternative to the growth of bulk MnxGa1−xAs, 15,16 allowing to obtain locally high dopant concentrations and, remarkably, an important enhancement of Tc, up to about 250 K. 17,18 For further improvements it is essential to investigate the different configurations of Mn impurities and their effect on the magnetic properties of the system. The most common and widely studied Mn configuration is substitutional in the cation sites (MnGa), with Mn acting as a hole-producing acceptor.17 To a less extent, Mn can also occupy interstitial sites, in particular tetrahedral ones. In such a case, it is expected to strongly modify the magnetic properties, acting as an electron-producing donor and hence destroying the free holes and hindering ferromagnetism.19 Interstitials have not been fully characterized up to now, although their existence has been suggested in different situations.8,9,10,11,14,20,21,22,23,24,25,26 For instance, the enhancement of the Curie temperature after post-growth annealing has been attributed to the reduction of interstitial defects with their out diffusion towards the surface.11 It has been suggested that interstitial sites are highly mobile and could be immobilized when adjacent to substitutional MnGa, thus forming compensated pairs with antiferromagnetic coupling. 27 A first identifi- cation of interstitial Mn dates back to almost fifteen years ago by electron paramagnetic resonance (EPR).20 Very recently EPR spectra from variously doped and grown samples of Mn-doped epitaxial GaAs have allowed to identify the presence of ionized Mn interstitials at concentrations as low as 0.5%, although not providing details about the specific local en- vironment of the interstitial site.28 Recent X-ray absorption near edge structure (XANES) and extended x-ray absorption fine structure (EXAFS) spectra in Mn δ-doped GaAs samples suggest that Mn occupy not only substitutional Ga sites but also interstitial sites, mainly in case of Be co-doping.29 Cross-sectional Scanning Tunneling Microscopy (XSTM) allows a direct imaging of the electronic states and can be used to characterize the impurities near the cleavage surface.30 In recent years several XSTM studies of Mn-doped GaAs samples have been performed but without a complete consensus on the defects characterization.22,31,32,33,34,35,36,37,38 We stress that most of XSTM studies mainly concern MnxGa1−xAs alloys and have identified mainly substitutional Mn defects. δ-doped samples have been investigated by Yakunin et al.,35 who pointed out the advantage that in such samples it is easy to discriminate Mn related defects from other defects. From the theoretical point of view, numerical works have been also focused mainly on the simulation of XSTM images of substitutional impurities on uppermost surface layers.31,33,34,35 A complete and detailed investigation of interstitial impurities as they can appear on the exposed cleaved surface is still lacking thus preventing the possibility of a comprehensive interpretation of all the available experimental XSTM images. Mn δ-doped (001) GaAs samples recently grown at TASC Laboratory in Trieste and analyzed with XSTM on the (110) cleavage surface have shown several Mn related features (see Fig. 1). Some of them have already be studied by other groups, like the asymmetric cross-like (or butterfly-like) structures marked by A in Fig. 1(a), attributed to Mn acceptors a few atomic layers below the surface.34 Some other features, such those marked by B, or those of Fig 1(b), have not been yet assigned to specific Mn configurations. In order to identify the kind of Mn defects that cause them we have performed new density functional simulation of cross-sectional XSTM images focusing on three selected defect configurations not yet fully studied, but whose presence cannot be excluded in real samples. In particular, we focus our attention on interstitial surface configurations, both individual as well as in pairs. We have also considered MnGa on the first layer below the surface and compared all the simulations with the experimental maps. II. EXPERIMENTAL DETAILS Mn δ-doped samples were grown by molecular beam epitaxy on GaAs(001) in a facility which includes a growth chamber for III-V materials and a metallization chamber. After the growth of a Be doped buffer at 590oC and of an undoped GaAs layer 50 nm thick at 450oC with an As/Ga beam pressure ratio of 15, the samples were transferred in the metallization chamber where a submonolayer-thick Mn layer was deposited at room temperature at the rate of 0.003 monolayer/s. An undoped GaAs cap layer was subsequently grown at 450oC. This procedure was repeated in order to have three δ-doped Mn layer of 0.01, 0.05 and 0.2 monolayers in the same sample. During the transfers and the Mn deposition the vacuum was always better than 2×10−8 Pa. The 0.1 mm thick wafers containing the Mn layers were cleaved in situ in a ultra high vacuum STM system immediately prior to image acquisition to yield atomically flat, electronically unpinned {110} surfaces containing the [001] growth direction and the cross section of the δ-doped layer. The XSTM image presented in Fig. 1 and the others shown in this paper have been acquired from a δ-doped Mn layer of 0.2 monolayers with W tips. The densities of the features observed by XSTM near each Mn layer were approximately proportional to the Mn coverage of the δ-doped layer in the range 0.01-0.2 monolayer. No trace of contaminants was observed by in situ x-ray photoemission spectroscopy after the transfer in the metallization, after the Mn deposition, and after the transfer in the growth chamber. For these two reasons we attribute the features observed by XSTM to the Mn atoms, and not to defects or contaminants caused by the growth interruption and transfers between the chambers. The density of the defects caused by these steps should not depend on the Mn coverage, contrary to what we observe. Moreover, a sample was grown with the same procedure described above, including the transfers between the chambers, but without the Mn deposition. The photoluminescence spectra of this sample are undistinguishable from that of a good undoped GaAs epitaxial layer grown without transfers between the chambers. This confirms that the transfers do not introduce an appreciable amount of defects. III. THEORETICAL APPROACH Our numerical approach is based on spin-resolved Density Functional Theory (DFT) using the ab-initio pseudopotential plane-wave method PWscf code of the Quantum ESPRESSO distribution.39 Cross-sectional surfaces are studied using supercells with slab geometries, according to a scheme previously used,40 with 5 atomic layers and a vacuum region equivalent to 8 atomic layers. Mn dopants are on one surface, whereas the other is passivated with hydrogen. For a single Mn impurity we use a 4×4 in-plane periodicity corresponding to distances between the Mn atom and its periodic images of 15.7 Å along the [11̄0] and 22.2 Å along [001]. No substantial changes in the XSTM images have been found using a 6×4 periodicity, which has been instead routinely used when considering interstitial complexes. Other details on technicalities can be found in Ref. 41. In our study, we have mainly focused on the Local Spin Density Approximation (LSDA) for the exchange-correlation functional. An ultrasoft pseudopotential is used for Mn atom, considering semicore 3p and 3s states kept in the valence shell while norm-conserving pseu- dopotentials have been considered for Ga and As atoms. The 3d-Ga electrons are considered as part of core states.42,43 Tests beyond LSDA (with Generalized Gradient Correction and LSDA+U methods) have not shown any substantial difference in the features of the XSTM maps. As a further check, we have also simulated ionized substitutional MnGa (with charge state equal to 1−) on surface and in the first subsurface layer as well as ionized interstitial Mn (with charge state equal to 2+) on surface layer. Neither the former nor the latter simulated XSTM maps show significant differences with respect to the neutral cases. We address the reader to a future pubblication for details.44 The XSTM images are simulated using the model of Tersoff-Hamann,45,46 where the tunneling current is proportional to the Local Density of States (LDOS) at the position of the tip, integrated in the energy range between the Fermi energy Ef and Ef + eVb, where Vb is the bias applied to the sample with respect to the tip. The position of the Fermi level is relevant for the XSTM images. In general, Ef strongly depends on the concentration of dopants: this is contrivedly large in our simulations even in the case of a single Mn dopant per supercell. Therefore to overcome this problem we fix Ef according to the experimental indications: in order to account for the p-doping in the real samples, we set Ef close to the Valence Band Maximum (VBM). The VBM in the DOS of the Mn-doped GaAs can be exactly identified by aligning the DOS projected onto surface atoms far from the impurity with the one of the clean surface. In any case, the comparison between experiments and simulations must be taken with some caution, due to the possible differences in the details entering in the determination of the XSTM image, such as tip-surface separation, precise value of the bias voltage and position of Ef , surface band gap. IV. SURFACE MN INTERSTITIALS We first focus on interstitial dopant configurations, IntAs and IntGa. Throughout this work we have considered only tetrahedral interstitial position, since it is known from bulk calculations that the total energy corresponding to the hexagonal interstitial site is higher by more than 0.5 eV.11,48,49 The tetrahedral interstitial site in the ideal geometry has four nearest-neighbor (NN) atoms at a distance equal to the ideal host bond length d1 and six next-nearest-neighbor (NNN) atoms at the distance d2 = d1, which are As(Ga) atoms for IntGa(As), respectively. At the ideal truncated (110) surface, the numbers of NNs and NNNs reduce to three (2 surface atoms and 1 subsurface atom) and four (2 surface atoms and 2 subsurface atoms) instead of four and six respectively. In the uppermost panels of Fig. 2 we show a ball and stick side and top view of the relaxed IntAs and IntGa configurations. In the relaxed structure, due to symmetry breaking because of the surface and the consequent buckling of the outermost surface layers, the NN and NNN bond lengths are no longer equal. Furthermore, some relaxed NNs bond lengths turn out to be longer than NNNs ones. In the following, we do not distinguish among NN and NNN atoms: they are simply referred as neighbor surface or subsurface atoms, as shown in the Figure. The two relaxed configurations slightly differ in energy, by ∼ 130 meV/Mn atom, in favour of IntGa. This is at variance with the bulk case studied in the literature, where it has been found that IntAs is favoured: for neutral state, the energy difference is actually so small (5 meV/Mn atom)48 that it is not meaningful, but it goes up to 350 meV in case of interstitial Mn with 2+ charge state.11 After optimization of the atomic positions, sizeable displacements from the ideal zinc blende positions occur for the Mn impurities and their surface and subsurface neighbors; small relaxations effects are still present in the third layer, in both configurations. In IntAs, with respect to the ideal (110) surface plane, Mn relaxes outward by ∼ 0.06 Å and Assurf (Assubsurf) move upwards (downwards). On the other hand, the Ga atoms (both on surface and subsurface) are shifted towards the bulk. In IntGa, Mn relaxes inward by ∼ 0.32 Å; the Gasurf and Gasubsurf atoms are displaced downwards and the Assurf (Assubsurf) atom moves upwards (downwards). The interatomic distances between Mn and the nearest atoms are in general longer by more than 2-3 % than ideal values (details in Ref. 41). The simulated XSTM images of IntAs (left) and IntGa (right) configurations at negative and positive bias voltages (from − 2.0 V to +2.0 V) are shown in the lower panels of Fig. 2. In IntAs, Mn appears as an additional bright spot at negative bias voltage (Vb=−1 V), slightly elongated in the [001] direction and located near the center of the surface unit cell identified by surface As atoms. The Assurf atoms close to Mn appear less bright than the others. These features are similar changing Vb from −1 to −2 V. In the empty states image at Vb=1 V Mn appears again as an elongated bright spot. The underlying cation lattice is only barely visible at this bias voltage. The very bright XSTM feature originates from the Mn d minority states and a strong peak of Gasurf majority states.50 At Vb=2 V, this feature is still well visible, as well as another region brighter than the underlying cationic sublattice in correspondence of Assurf atoms neighbor to Mn, suggesting a contribution coming from the hybridization between Mn-d and Assurf -p states. In IntGa configuration, at negative voltage, Mn appears as an almost circular bright spot located in between two surface As atoms adjacent along the [001] direction. At positive bias voltages, the two Gasurf atoms neighbor to Mn appear very bright with features extending towards Mn in a “v”-shaped form and the atoms in the neighborhood also look brighter than normal. These features remain visible by increasing the positive bias voltage up to 2 eV. Remarkably the empty states images of Mn are quite different for the two interstitial configurations, making them clearly distinguishable by XSTM analysis. Some features in the experimental XSTM images appear as bright spots both at positive and negative bias voltages. These spots lie along the [001] Ga rows and between the [1-10] Ga columns at positive bias voltage (see Fig. 1(b)). Their location with respect to the surface Ga lattice and the comparison with the simulated images allow to identify them as IntAs Mn atoms. The numerical simulation gives easily informations on the magnetic properties of the system. The total and absolute magnetization, calculated from the spatial integration of the difference and the absolute difference respectively between the majority and minority electronic charge distribution, are different in the two configurations: 4.23 and 4.84 µB for IntAs and 3.41 and 4.71 µB for IntGa respectively. These differences indicate in both cases the presence of region of negative spin-density and a clear dependence of the induced magnetization on the local Mn environment. The individual atomic magnetic moments can be calculated as the difference between the majority and minority atomic-projected charges. In IntAs, Mn magnetic moment is 3.96 µB, almost integer, corresponding to the presence of a gap in the Mn-projected minority density of states. Mn magnetization is slightly lower in IntGa (3.67 µB). In both cases they are significantly larger compared to the bulk case, indicating a surface induced enhancement. The analysis of spin-polarization induced by interstitial Mn on its nearest neighbors shows in both cases an antiferromagnetic Mn–Ga coupling and a smaller ferromagnetic Mn–As coupling: more precisely, the magnetic moments induced on surface Ga atoms neighbors to Mn are equal to −0.14 and −0.17 µB in IntAs and IntGa respectively, whereas those induced on surface or subsurface As atoms neighbors to Mn are positive and at most equal to 0.05 µB. We address the reader to Ref. for further details. In the experimental images of Mn δ-doped GaAs samples we often observe two spots close one each other at a distance of about 8 Å, as reported in Fig. 3 (larger panel). The simulated image of two IntAs atoms separated by a clean surface unit cell along (110), partially superimposed, reproduces the main features of this experimental image, and it is basically a superposition of images of individual IntAs (elongated bright spot each one, with major axis along the [001] direction, and a surrounding darker region). V. SUBSTITUTIONAL MN DEFECTS IN THE FIRST SUBSURFACE LAYER Another typical feature present in the experimental XSTM maps is a bright spot visible at positive bias voltages with two satellite features forming a triangular structure, as shown in Fig. 1(a) (feature B) and in Fig. 4 in the lower panels. This feature seems similar to that caused by the arsenic antisite defect (As on Ga) in GaAs.51,52 However in the arsenic antisite defects the satellites are visible only at negative sample bias, while the defect that we observe in the Mn layers shows satellite only in the positive bias images. On the other hand there is a clear resemblance of the defect B (Fig. 1(a) and Fig. 4) with the simulated image of a substitutional MnGa atom in the first subsurface layer shown in the panels partially superimposed to the experimental images. It can be seen at Vb < 0 a deformation of the surface As rows in correspondence of the Mn impurity below, and, even more remarkably, the peculiar satellite bright features on two neighboring surface As stoms at Vb > 0 giving rise to a triangular-shaped image. Therefore we attribute the defect B to substitutional Mn Ga atoms in the first subsurface layer. Finally, we discuss our findings in comparison with some relevant results present in the literature. The comparison of our simulations with those of Sullivan et al.33 is possible only for the isolated MnGa in the first subsurface layer at negative bias voltage: in such a case the simulated images show similar features. The corresponding image at positive bias is not reported and other configurations are not comparable. The XSTM imaging of substitutional Mn is reported with more details by Mikkelsen et al.,31,32 where both the simulated maps for surface and subsurface MnGa and the experimental ones attributed to this impurity configuration are shown at negative and positive bias, thus allowing for a more complete comparison. The images for MnGa in the first subsurface layer have a good resemblance with ours, a part from the satellite features that we have identified at positive bias on neighbor As atoms which are not present in their images, neither in the simulated nor in the experimental one. More precisely, we notice that their simulated surface area is too small to make such satellite features visible. The simulated images for surface MnGa are also similar to ours and, like ours, not corresponding to any experimental feature.44 This leads to the conclusion that the presence of substitutional Mn in the first layer of the exposed surface is very unlikely. Mikkelsen et al. reported also the simulation of surface interstitial Mn in their Fig. 3(d),32 that according to our understanding on the basis of the symmetry planes should correspond to IntGa, although not explicitely indicated. Their images are similar to ours for the same configuration. They rule out the presence of interstitials since these images are not compatible with experiments, at variance with our findings concerning IntAs. It should be noted however that we observe the IntAs features in the experimental samples only in the first few hours after the sample cleavage. They disappear for longer times, probably because of surface contamination or diffusion. Kitchen et al.36,37 report experimental images for Mn adatoms at the GaAs (110) surface with highly anisotropic extended star-like feature, attributed to a single surface Mn acceptor. Interestingly, these images are compatible with our simulated surface MnGa, not show here. A resemblance with our empty state image for IntAs (see Fig. 2 at Vb=+2 V) is instead only apparent because the mirror symmetry plane is different. An anisotropic, crosslike feature in XSTM image is reported also Yakunin et al.34 and, from comparison with an envelope-function, effective mass model and a tight-binding model, it is attributed to a hole bound to an individual Mn acceptor lying well below the surface. We observe similar feature of different sizes (see Fig. 1), the smallest of them are those reported in Fig. 4, that we identify as MnGa in the first subsurface layer. A part from different details, our simulated images for surface and subsurface MnGa are compatible with such crosslike features, although experimental and simulated images reported therein concern substitutional impurities located more deeply subsurface than those we have considered. Crosslike features are observed even at very short Mn-Mn spatial separations.35 VI. CONCLUSIONS We have reported a combined experimental and first-principles numerical study of XSTM images of the (110) cross-sectional surfaces of Mn δ-doped GaAs samples. We suggest an identification of three typical configurations observed in the experimental sample on the basis of a comparison of numerical prediction and observed images both at negative and positive applied bias. (i) Some structures observed can be identified as surface Mn interstitial with As nearest neighbors, on the basis of their position with respect to the surface lattice and the comparison with the simulated images. At variance, there is no evidence in the experimental samples of Mn interstitial with Ga nearest neighbors, whose XSTM imaging according to our numerical simulations would correspond to very different features. (ii) Besides isolated configurations, also pairs of Mn interstitials with As nearest neighbors are clearly observed and identified. (iii) Subsurface substitutional MnGa atoms in the first subsurface layer can also be unambigously identified in the experimental images by a main bright spot corresponding to the dopant and from peculiar satellite features on two neighboring As atoms which are clearly observed in the experimental images and predicted by simulations. VII. ACKNOWLEDGMENTS Computational resources have been partly obtained within the “Iniziativa Trasversale di Calcolo Parallelo” of the Italian CNR-Istituto Nazionale per la Fisica della Materia (CNR- INFM) and partly within the agreement between the University of Trieste and the Consorzio Interuniversitario CINECA (Italy). We thank A. Franciosi, S. Rubini and coworkers for the preparation of the sample and fruitful comments and discussions; A. Debernardi for his help in the pseudopotential generation and for useful discussions. Ball and stick models and simulated images are obtained with the package XCrySDen.53 ∗ Electronic address: [email protected]; Presently at: Institute of Material Physics, University of Vienna, Sensengasse 8/12, A-1090 Wien, Austria and Center for Computational Materials Science (CMS), Wien, Austria † Presently at School of Physics, The University of Sydney, NSW 2006 Australia ‡ Electronic address: [email protected] 1 K. Takamura et al., J. Appl. 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Tersoff and D.R. Hamann, Phys. Rev. B 31, 805 (1985). 47 However, we have checked that a variation of ∼ ± 0.3 eV in the bias voltage considered in the simulated images does not affect their basic features. 48 J. Mašek and F. Máca, Phys. Rev. B 69, 165212 (2004). 49 J.X. Cao, X.G. Gong, and R.Q. Wu, Phys. Rev. B 72, 153410 (2005). 50 A. Stroppa, Nuovo Cim. 29, 315 (2006). 51 R.M. Feenstra, J.M. Woodall and G.D. Pettit, Phys. Rev. Lett. 71, 1176 (1993). 52 G. Mathieu et al., Appl. Phys. Letters 82, 712 (2003). 53 A. Kokalj, Comp. Mater. Sci., 2003, Vol. 28, p. 155. Code available from http://www.xcrysden.org/. http://www.pwscf.org http://www.xcrysden.org/ FIG. 1: (a) Experimental (110) XSTM image of a 0.2 monolayer Mn δ-doped layer in GaAs at a sample bias voltage of 1.7 eV. This image has not been corrected for the drift of the sample. (b) XSTM image of a Mn related structure at the bias voltage of −1.4 eV (left) and +1.9 eV (right). The white lines show the [001] Ga atomic rows. FIG. 2: Isolated Mn interstitial dopants on GaAs(110) surface, with As nearest neighbors (IntAs, left) and Ga nearest neighbors (IntGa, right). Upper panels: ball-and-stick model of the relaxed surface, top and side view. Only the three topmost layers are shown in the side view. Black spheres are Mn, white spheres are As, grey spheres are Ga. Lower panels: simulated XSTM images at occupied states and empty states respectively, for different bias voltages. FIG. 3: Smaller superimposed panel: simulated XSTM image of a pair of IntAs on GaAs(110) surface with a relative distance of ∼ 8 Å along the [11̄0] direction at a bias voltage Vb=−2 V. The larger panel shows an experimental image compatible with the simulation. FIG. 4: Upper smaller superimposed panels simulated XSTM image of a subsurface MnGa on GaAs(110) at negative (left) and positive (right) bias voltages. The lower panels show correspond- ing experimental images of the structure B (see Fig. 1(a)) taken at sample bias voltages of −1.4 V (left) and +1.8 V (right) that are compatible with the simulations, performed with voltages of −1 V and +1 V. Fig. 1 Fig. 2 Fig. 3 Fig. 4 Introduction Experimental details Theoretical approach Surface Mn interstitials Substitutional Mn defects in the first subsurface layer Conclusions Acknowledgments References

0704.1358

Distance preserving mappings from ternary vectors to permutations

Distance preserving mappings from ternary vectors to permutations Jyh-Shyan Lin, Jen-Chun Chang, Rong-Jaye Chen,∗Torleiv Kløve † November 2, 2018 Abstract Distance-preserving mappings (DPMs) are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of the same or longer length such that every two distinct vectors are mapped to permutations with the same or even larger Hamming dis- tance than that of the vectors. In this paper, we propose a construc- tion of DPMs from ternary vectors. The constructed DPMs improve the lower bounds on the maximal size of permutation arrays. Key words: distance-preserving mappings, distance-increasing mappings, permutation arrays, Hamming distance 1 Introduction A mapping from the set of all q-ary vectors of length m to the set of all permutations of {1, 2, . . . , n} is called a distance-preserving mapping (DPM) if every two distinct vectors are mapped to permutations with the same or even larger Hamming distance mutual than that of the vectors. A distance- increasing mapping (DIM) is a special DPM such that the distances are strictly increased except when that is obviously not possible. DPMs and ∗Jyh-Shyan Lin, Jen-Chun Chang, and Rong-Jaye Chen are with the Dept. of Com- puter Science and Inform. Engineering, National Taipei University, Taipei, Taiwan. †Torleiv Kløve is with the Department of Informatics, University of Bergen, Bergen, Norway. http://arxiv.org/abs/0704.1358v1 DIMs are useful for the construction of permutation arrays (PAs) which are applied to various applications, such as trellis code modulations and power line communications [7], [8], [9], [10], [11], [13], [21], [22], [23], [24]. All DPMs and DIMs proposed so far are from binary vectors: [2], [3], [4], [5], [6], [12], [14], [15], [16], [17], [19], [20]. In this paper we propose a general construction method to construct DPMs or DIMs from ternary vectors. By using this method, we construct DIMs for n = m + 2 for m ≥ 3, DPMs for n = m+ 1 for m ≥ 9, and DPMs for n = m for m ≥ 13. The paper is organized as follows. In the next section we introduce some notations and state our main results. In Section 3 we introduce a general recursive construction of DPMs and DIMs. In Sections 4 and 5 we introduce mappings that can be used to start the recursion in the three cases we con- sider. Finally, in an appendix, we give explicit listings of the values of some mappings that are used as building blocks to construct the mappings given in Sections 4 and 5. 2 Notations and main results Let Sn denote the set of all n! permutations of Fn = {1, 2, . . . , n}. A per- mutation π : Fn → Fn is represented by an n-tuple π = (π1, π2, . . . , πn) where πi = π(i). Let Z denote the set of all ternary vectors of length n. The Hamming distance between two n-tuples a = (a1, a2, . . . , an) and b = (b1, b2, . . . , bn) is denoted by dH(a,b) and is defined as dH(a,b) = |{j ∈ Fn : aj 6= bj}|. Let Fn,k be the set of injective functions from Z to Sn+k. Note that Fn,k is empty if (n + k)! < 3n. For k ≥ 0, let Pn,k be the set of functions in Fn,k such that dH(f(x), f(y)) ≥ dH(x,y) for all x,y ∈ Zn . These mappings are called distance preserving mappings (DPM). For k ≥ 1, let In,k be the set of functions in Fn,k such that dH(f(x), f(y)) > dH(x,y) (1) for all distinct x,y ∈ Zn . These mappings are called distance increasing mappings (DIM). Our main result is the following theorem. Theorem 1 a) In,2 is non-empty for n ≥ 3. b) Pn,1 is non-empty for n ≥ 9. c) Pn,0 is non-empty for n ≥ 13. The proof of the theorem is constructive. A relatively simple recursive method is given (in the next section) to construct a mapping of length n+1 from a mapping of length n. Explicit mappings that start the recursion in the three cases are given in last part of the paper, including the appendix. An (n, d) permutation array (PA) is a subset of Sn such that the Hamming distance between any two distinct permutations in the array is at least d. An (n, d; q) code is a subset of vectors (codewords) of length n over an alphabet of size q and with distance at least d between distinct codewords. One construction method of PAs is to construct an (n, d′)-PA from an (m, d; q) code using DPMs or DIMs. More precisely, if C is an (m, d; q) code and there exists an DPM f from Zmq to Sn, then f(C) is an (n, d) PA. If f is DIM, then f(C) is an (n, d+1) PA. This has been a main motivation for studying DPMs. Let P (n, d) denote the largest possible size of an (n, d)-PA. The exact value of P (n, d) is still an open problem in most cases, but we can lower bound this value by the maximal size of a suitable code provided a DPM (or DIM) is known. Let Aq(n, d) denote the largest possible size of an (n, d) code over a code alphabet of size q. In [5], Chang et al. used this approach to show that for n ≥ 4 and 2 ≤ d ≤ n, we have P (n, d) ≥ A2(n, d − 1). In [17], Chang further improved the bound to P (n, d) ≤ A2(n, d− δ) for n ≥ nδ and δ + 1 ≤ d ≤ n where δ ≥ 2 and nδ is a positive integer determined by δ, e.g. n2 = 16. From Theorem 1 we get the following bounds. Theorem 2 a) For n ≥ 5 and 2 ≤ d ≤ n, we have P (n, d) ≥ A3(n− 2, d− 1). b) For n ≥ 10 and 2 ≤ d ≤ n, we have P (n, d) ≥ A3(n− 1, d). c) For n ≥ 13 and 2 ≤ d ≤ n, we have P (n, d) ≥ A3(n, d). Bounds on A2(n, d) and A3(n, d) have been studied by many researchers, see e.g. [18, Ch.5] and [1]. In general, the lower bounds on P (n, d) obtained from use of ternary codes are better than those obtained from binary codes. For example, using Chang’s bound [17], we get P (16, 5) ≥ A2(16, 3) ≥ 2720, whereas Theorem 2 gives P (16, 5) ≥ A3(16, 5) ≥ 19683. Similarly, we get P (16, 9) ≥ A2(16, 7) ≥ 36 and P (16, 9) ≥ A3(16, 9) ≥ 243. 3 The general recursive construction. For any array u = (u1, u2, . . . , un), we use the notation ui = ui. We start with a recursive definition of functions from Zn to Sn+k. For f ∈ Fn,k, define g = H(f) ∈ Fn+1,k as follows. Let x = (x1, x2, . . . , xn) ∈ Z and f(x) = (ϕ1, ϕ2, . . . , ϕn+k). Suppose that the element n+ k− 4 occurs in position r, that is ϕr = n + k − 4. Then g(x|0)n+k+1 = n+ k + 1, g(x|0)i = ϕi otherwise; g(x|1)r = n+ k + 1, g(x|1)n+k+1 = n+ k − 4, g(x|1)i = ϕi otherwise; if n is odd or xn < 2, then g(x|2)n+k = n + k + 1, g(x|2)n+k+1 = ϕn+k, g(x|2)i = ϕi otherwise; if n is even and xn = 2, then g(x|2)n+k−1 = n+ k + 1, g(x|2)n+k+1 = ϕn+k−1, g(x|2)i = ϕi otherwise. We note that g(x|a)i 6= f(x)i for at most one value of i ≤ n+ k. For f ∈ Fm,k, we define a sequence of functions f ∈ Fn,k, for all n ≥ m, recursively by fm = f and fn+1 = H(fn) for n ≥ m. Theorem 3 If fm ∈ Pm,k where k ≥ 0, m is odd, and fm(x)m+k 6∈ {m+ k − 4, m+ k − 3} for all x ∈ Z then fn ∈ Pn,k for all n ≥ m. Theorem 4 If fm ∈ Im,k, where k > 0 and m is odd, and fm(x)m+k 6∈ {m+ k − 4, m+ k − 3} for all x ∈ Z then fn ∈ In,k for all n ≥ m. Proof: We prove Theorem 4; the proof of Theorem 3 is similar (and a little simpler). The proof is by induction. First we prove that g = fm+1 ∈ Im+1,k. Let x,y ∈ Zm f(x) = (ϕ1, ϕ2, . . . , ϕm+k), ϕr = m+ k − 4, f(y) = (γ1, γ2, . . . , γm+k), γs = m+ k − 4. We want to show that dH(g(x|a), g(y|b)) > dH((x|a), (y|b)) if (x|a) 6= (y|b). First, consider x = y and a 6= b. Since ϕm+k 6= m + k − 4, it follows immediately from the definition of g that dH(g(x|a), g(x|b)) ≥ 2 > 1 = dH((x|a), (x|b)). For x 6= y, we want to show that dH(g(x|a), g(y|b))− dH(f(x), f(y)) ≥ dH(a, b) (2) for all a, b ∈ Z3 since this implies dH(g(x|a), g(y|b)) ≥ dH(f(x), f(y)) + dH(a, b) > dH(x,y) + dH(a, b) = dH((x|a), (y|b)). The condition (2) is equivalent to the following. m+k+1∑ (∆g,i −∆f,i) ≥ dH(a, b), (3) where ∆g,i = dH(g(x|a)i, g(y|b)i) ∆f,i = dH(f(x)i, f(y)i), and where, for technical reasons, we define ∆f,n+k+1 = 0. The point is at most three of the terms ∆g,i − ∆f,i are non-zero. We look at one combination of a and b in detail as an illustration, namely a = 1 and b = 2. Then g(x|a)i = f(x)i and g(y|b)i = f(y)i and so ∆g,i = ∆f,i for all i ≤ m+ k + 1, except in the following three cases i f(x)i f(y)i g(x|a)i g(y|b)i) r m+ k − 4 γr m+ k + 1 γr m+ k ϕm+k γm+k ϕm+k m+ k + 1 m+ k + 1 − − m+ k − 4 γm+k i ∆f,i ∆g,i ∆g,i −∆f,i r 0 or 1 1 0 or 1 m+ k 0 or 1 1 0 or 1 m+ k + 1 0 1 1 Note that we have used the fact that γm+k 6= m + k − 4. We see that∑ (∆g,i −∆f,i) ≥ 1 = dH(a, b). The other combinations of a and b are similar. This proves that fm+1 = g ∈ Im+1,k. Now, let h = H(g) = fm+2. A similar analysis will show that h ∈ Im+2,k. We first give a table of the last three symbols in h(x|a1a2) as these three symbols are the most important in the proof. Let ϕs = m + k − 3. By assumption, s < m+ k. a1a2 h(x|a1a2)m+k h(x|a1a2)m+k+1 h(x|a1a2)m+k+2 00 ϕm+k m+ k + 1 m+ k + 2 10 ϕm+k m+ k − 4 m+ k + 2 20 m+ k + 1 ϕm+k m+ k + 2 01 ϕm+k m+ k + 1 m+ k − 3 11 ϕm+k m+ k − 4 m+ k − 3 21 m+ k + 1 ϕm+k m+ k − 3 02 ϕm+k m+ k + 2 m+ k + 1 12 ϕm+k m+ k + 2 m+ k − 4 22 m+ k + 2 ϕm+k m+ k + 1 In addition, h(x|1a2)r = m+ k + 1 and h(x|a11)s = m+ k + 2. Note that we have used the fact that ϕm+k 6= m + k − 3 here, since if we had ϕm+k = m+ k − 3, then we would for example have had h(x|01)m+k = m+ k + 2. From the table we first see that dH(h(x|a1a2), h(x|b1b2)) > dH(a1a2, b1b2) if a1a2 6= b1b2. For example h(x|10) and h(x|21) differ in positions r, s, m+k, m+ k+ 1 and m+ k+ 2. As another example, h(x|02) and h(x|22) differ in positions m+ k and m+ k + 1. Next, consider dH(h(x|a1a2), h(y|b1b2)) for x 6= y. We see that dH(h(x|a1a2)i, h(y|b1b2)i) ≥ dH(f(x)i, f(y)i) for i < m+ k: from the table above, we can see that dH(h(x|a1a2)m+kh(x|a1a2)m+k+1h(x|a1a2)m+k+2, h(y|b1b2)m+kh(y|b1b2)m+k+1h(y|b1b2)m+k+2) ≥ dH(ϕm+k, γm+k) + dH(a1a2, b1b2). As an example, let a1a2 = 10 and b1b2 = 02. Then h(x|10)m+k, h(x|10)m+k+1, h(x|10)m+k+2 = ϕm+k, m+ k − 4, m+ k + 2 h(y|02)m+k, h(y|02)m+k+1, h(y|02)m+k+2 = γm+k, m+ k + 2, m+ k + 1. The distance between the two is 2 (if ϕm+k = γm+k) or 3 (otherwise). The other combinations of a1a2 and b1b2 are similar. From this we can conclude that h ∈ Im+2,k in a similar way we showed that g ∈ Im+1,k above. Further, we note that h(x|a1a2)m+k+2 6∈ {(m+ 2) + k − 4, (m+ 2) + k − 3}. Therefore, we can repeat the argument and, by induction, obtain fn ∈ In,k for all n ≥ m. A function F is given by an explicit listing in the appendix. It belongs to I3,2 and satisfy F (x)5 6∈ {1, 2}. This, combined with Theorem 4, proves Theorem 1 a). 4 Proof of Theorem 1, second part To prove Theorem 1 b), using Theorem 3, we need some f ∈ P9,1 such that f(x)10 6∈ {6, 7} for all x ∈ Z . (4) An extensive computer search has been unsuccessful in coming up with such a mapping. However, an indirect approach has been successful. The approach is to construct f from two simpler mappings found by computer search. For a vector ρ = (ρ1, ρ2, . . . , ρn) and a set X ⊂ {1, 2, . . . , n}, let ρ\X denote the vector obtained from ρ by removing the elements with subscript in X . For example, (ρ1, ρ2, ρ3, ρ4, ρ5, ρ6)\{1,5} = (ρ2, ρ3, ρ4, ρ6). By computer search we have found mappings G ∈ F5,2 and H ∈ F4,2 that satisfy the following conditions a) for every x ∈ Z5 , 6 ∈ {G(x)1, G(x)2, G(x)3}, b) for every x ∈ Z5 , 7 ∈ {G(x)4, G(x)5, G(x)6}, c) for every distinct x,y ∈ Z5 dH(G(x)\{7}, G(y)\{7}) ≥ dH(x,y), d) for every u ∈ Z4 , 1 ∈ {H(u)1, H(u)2, H(u)3}, e) for every distinct u,v ∈ Z4 dH(H(u)\{5,6}, H(v)\{5,6}) ≥ dH(u,v). The mappings G and H are listed explicitly in the appendix. We will now show how these mappings can be combined to produce a mapping f ∈ P9,1 satisfying (4). Let x ∈ Z9 . Then x = (xL,xR), where xL ∈ Z and xR ∈ Z . Let (ϕ1, ϕ2, ϕ3, ϕ4, ϕ5, ϕ6, ϕ7) = G(xL), (γ1, γ2, γ3, γ4, γ5, γ6) = H(xR) + (4, 4, 4, 4, 4, 4). We note that Condition d) implies that γ5 ≥ 6 and γ6 ≥ 6. Similarly, Conditions a) and b) imply that ϕ7 ≤ 5. Define ρ = (ρ1, ρ2, . . . , ρ10) as follows. ρi = γ5 if 1 ≤ i ≤ 3 and ϕi = 6, ρi = γ6 if 4 ≤ i ≤ 6 and ϕi = 7, ρi = ϕi if 1 ≤ i ≤ 6 and ϕi ≤ 5, ρi = ϕ7 if 7 ≤ i ≤ 9 and γi−6 = 5, ρi = γi−6 if 7 ≤ i ≤ 10 and γi−6 ≥ 6. In ρ, swap 1 and 6 and also swap 2 and 7, and let the resulting array be denoted by π. More formally, πi = 1 if ρi = 6, πi = 2 if ρi = 7, πi = 6 if ρi = 1, πi = 7 if ρi = 2, πi = ρi otherwise. Then define f(x) = π. We will show that f has the stated properties. We first show that π ∈ S10. We have ϕ ∈ S7 and γ is a permutation of (5, 6, 7, 8, 9, 10). In particular, 5,6, and 7 appear both in ϕ and γ. The effect of the first line in the definition of ρ is to move another elements (γ5) into the position where ϕ has a 6. Similarly, the second line overwrites the 7 in ρ, and the fourth line overwrites the 5 in γ. The definition of ρ is then the concatenation of the six first (overwritten) elements of ϕ and the five first (overwritten) elements of γ. Therefore, ρ contains no duplicate elements, that is, ρ ∈ S10. The element 1 in ρ must be either in one of the first six positions, coming from ϕ, or in one of the positions 7− 9 (if ϕ7 = 1). Similarly, the element 2 must be in one of the first nine positions of ρ. Therefore, both 6 and 7 must be among the first nine elements of π, that is π10 6∈ {6, 7}. Finally, we must show that f is distance preserving. Let x 6= x′, and let the arrays corresponding to x′ be denoted by ϕ′, γ′, ρ′ and π′. By assumption, dH(x,x ′) = dH(xL,x L) + dH(xR,x ≤ dH(ϕ\{7}, ϕ \{7}) + dH(γ\{5,6}, γ \{5,6}). (5) For 1 ≤ i ≤ 6 we have dH(ϕi, ϕ i) ≤ dH(ρi, ρ i). (6) If ϕi = ϕ i this is obvious. Otherwise, we may assume without loss of gener- ality that ϕ′i < ϕi and we must show that ρi 6= ρ i. If ϕi ≤ 5, then ρ′i = ϕ i < ϕi = ρi. If ϕi = 6, then ρ′i = ϕ i ≤ 5 and ρi = γ5 ≥ 6. If ϕi = 7, then 4 ≤ i ≤ 6 and so ϕ i 6= 6. Hence ρ′i = ϕ i ≤ 5 and ρi = γ6 ≥ 6. This completes that proof of (6). A similar arguments show that for 7 ≤ i ≤ 10 we have dH(γi−6, γ i−6) ≤ dH(ρi, ρ i), (7) and that for 1 ≤ i ≤ 10 we have dH(ρi, ρ i) ≤ dH(πi, π i). (8) Combining (5)–(8), we get dH(x,x ′) ≤ dH(ϕ\{7}, ϕ \{7}) + dH(γ\{5,6}, γ \{5,6}) ≤ dH(ρ, ρ ′) ≤ dH(π, π Hence, f is distance preserving. 5 Proof of Theorem 1, last part The construction of a mapping f ∈ P13,0 which proves Theorem 1 c) is similar to the construction in the previous section. However, the construction is more involved and contains several steps. We will describe the constructions and properties of the intermediate mappings. The details of proofs are similar to the proof in the previous section and we omit these details. We start with three mappings R, S ∈ F3,2 and T ∈ F4,2. These were found by computer search and are listed explicitly in the appendix. They have the following properties: • for every x ∈ Z3 , 1 ∈ {R(x)1, R(x)2, R(x)3}, • for every x ∈ Z3 , R(x)5 6= 5, • for every distinct x,y ∈ Z3 dH(R(x)\{4,5}, R(y)\{4,5}) ≥ dH(x,y), • for every x ∈ Z3 , 2 ∈ {S(x)1, S(x)2, S(x)3}, • for every x ∈ Z3 , S(x)5 6= 1, • for every distinct x,y ∈ Z3 dH(S(x)\{4,5}, S(y)\{4,5}) ≥ dH(x,y), • for every x ∈ Z4 , 2 ∈ {T (x)1, T (x)2, T (x)3}, • for every x ∈ Z4 , T (x)6 6= 1, • for every distinct x,y ∈ Z4 dH(T (x)\{5,6}, T (y)\{5,6}) ≥ dH(x,y). These mappings are used as building blocks similarly to what was done in the previous section. Construction of U ∈ F6,2 Let x ∈ Z6 and let (ϕ1, ϕ2, ϕ3, ϕ4, ϕ5) = R(x1, x2, x3), (γ1, γ2, γ3, γ4, γ5) = S(x4, x5, x6) + (3, 3, 3, 3, 3). Define ρ = (ρ1, ρ2, . . . , ρ8) as follows. ρi = γ5 if 1 ≤ i ≤ 4 and ϕi = 5, ρi = ϕi if 1 ≤ i ≤ 4 and ϕi 6= 5, ρi = ϕ5 if 5 ≤ i ≤ 8 and γi−4 = 4, ρi = γi−4 if 5 ≤ i ≤ 8 and γi−4 6= 4. In ρ, swap 1 and 7 and also swap 5 and 8, and let the resulting array be U(x). It has the following properties: • for every x ∈ Z6 , 7 ∈ {U(x)1, U(x)2, U(x)3}, • for every x ∈ Z6 , 8 ∈ {U(x)5, U(x)6, U(x)7}, • for every distinct x,y ∈ Z6 dH(U(x)\{4,8}, U(y)\{4,8}) ≥ dH(x,y). Construction of V ∈ F7,2 Let x ∈ Z7 and let (ϕ1, ϕ2, ϕ3, ϕ4, ϕ5) = R(x1, x2, x3), (γ1, γ2, γ3, γ4, γ5, γ6) = T (x4, x5, x6, x7) + (3, 3, . . . , 3). Define ρ = (ρ1, ρ2, . . . , ρ8, ρ9) as follows. ρi = γ6 if 1 ≤ i ≤ 4 and ϕi = 5, ρi = ϕi if 1 ≤ i ≤ 4 and ϕi 6= 5, ρi = ϕ5 if 5 ≤ i ≤ 9 and γi−4 = 4, ρi = γi−4 if 5 ≤ i ≤ 9 and γi−4 6= 4. In ρ, swap 2 and 5, and let the resulting array be V (x). It has the following properties: • for every x ∈ Z7 , 1 ∈ {V (x)1, V (x)2, V (x)3}, • for every x ∈ Z7 , 2 ∈ {V (x)5, V (x)6, V (x)7}, • for every distinct x,y ∈ Z7 dH(V (x)\{4,9}, V (y)\{4,9}) ≥ dH(x,y). Construction of f ∈ P13,0 Let x ∈ Z13 and let (ϕ1, ϕ2, . . . , ϕ8) = U(x1, x2, . . . , x6), (γ1, γ2, . . . , γ9) = V (x7, x8, . . . , x13) + (4, 4, . . . , 4). Define ρ = (ρ1, ρ2, . . . , ρ13) as follows. ρi = γ4 if 1 ≤ i ≤ 3 and ϕi = 7, ρi = ϕi if 1 ≤ i ≤ 3 and ϕi 6= 7, ρi = γ9 if 4 ≤ i ≤ 6 and ϕi+1 = 8, ρi = ϕi+1 if 4 ≤ i ≤ 6 and ϕi+1 6= 8, ρi = ϕ4 if 7 ≤ i ≤ 9 and γi−6 = 5, ρi = γi−6 if 7 ≤ i ≤ 9 and γi−6 6= 5, ρi = ϕ8 if 10 ≤ i ≤ 13 and γi−5 = 6, ρi = γi−5 if 10 ≤ i ≤ 13 and γi−5 6= 6. In ρ, swap 1 and 9 and also swap 2 and 10, and let the resulting array be f(x). Then f ∈ P13,0 and f(x)13 6∈ {9, 10}. References [1] A.E. Brouwer, Heikki O. Hämäläinen, Patric R.J. Österg̊ard, N.J.A. Sloane, “Bounds on mixed binary/ternary codes”, IEEE Trans. on In- form. Theory, vol. 44, no. 1, pp. 140–161, Jan. 1998. [2] J.-C. Chang, “Distance-increasing mappings from binary vectors to per- mutations”, IEEE Trans. on Inform. Theory, vol. 51, no. 1, pp. 359–363, Jan. 2005. [3] J.-C. Chang, “New algorithms of distance-increasing mappings from bi- nary vectors to permutations by swaps”, Designs, Codes and Cryptog- raphy, vol. 39, pp. 335–345, Jan. 2006. 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Vinck, “Coding and modulation for powerline communications”, A.E.Ü . Int. J. Electron. Commun., vol. 54, no. 1, pp. 45–49, Oct. 2000. [22] A. J. H. Vinck and J. Häring, “Coding and modulation for power-line communications”, Proc. Int. Symp. Power Line Communication, Lim- erick, Ireland, Apr. 57, 2000. [23] A. J. H. Vinck, J. Häring, and T. Wadayama, “Coded M-FSK for power- line communications”, Proc. IEEE Int. Symp. Information Theory, Sor- rento, Italy, June 2000, p. 137. [24] T. Wadayama and A. J. H. Vinck, “A multilevel construction of permu- tation codes”, IEICE Trans. Fundamentals Electron., Commun. Comp. Sci., vol. 84, pp. 2518–2522, 2001. Appendix Listing of the elements x ∈ Z3 and the corresponding values of F (x) ∈ S5. (0,0,0)(1,2,3,4,5), (0,0,1)(1,2,5,4,3), (0,0,2)(1,2,3,5,4), (0,1,0)(4,2,3,1,5), (0,1,1)(4,2,5,1,3), (0,1,2)(5,2,3,1,4), (0,2,0)(1,4,3,2,5), (0,2,1)(1,4,5,2,3), (0,2,2)(1,5,3,2,4), (1,0,0)(2,3,1,4,5), (1,0,1)(2,5,1,4,3), (1,0,2)(2,3,1,5,4), (1,1,0)(2,3,4,1,5), (1,1,1)(2,5,4,1,3), (1,1,2)(2,3,5,1,4), (1,2,0)(4,3,1,2,5), (1,2,1)(4,5,1,2,3), (1,2,2)(5,3,1,2,4), (2,0,0)(3,1,2,4,5), (2,0,1)(5,1,2,4,3), (2,0,2)(3,1,2,5,4), (2,1,0)(3,4,2,1,5), (2,1,1)(5,4,2,1,3), (2,1,2)(3,5,2,1,4), (2,2,0)(3,1,4,2,5), (2,2,1)(5,1,4,2,3), (2,2,2)(3,1,5,2,4) Listing of the elements x ∈ Z5 and the corresponding values of G(x) ∈ S7. (0,0,0,0,0)(6,1,2,7,3,4,5), (0,0,0,0,1)(6,3,2,7,1,5,4), (0,0,0,0,2)(6,3,2,7,4,5,1), (0,0,0,1,0)(6,2,1,7,5,3,4), (0,0,0,1,1)(6,1,2,7,5,3,4), (0,0,0,1,2)(6,3,2,7,5,4,1), (0,0,0,2,0)(6,1,2,7,3,5,4), (0,0,0,2,1)(6,3,1,7,2,5,4), (0,0,0,2,2)(6,3,1,7,4,5,2), (0,0,1,0,0)(6,2,5,7,1,4,3), (0,0,1,0,1)(6,2,5,7,3,4,1), (0,0,1,0,2)(6,3,5,7,4,1,2), (0,0,1,1,0)(6,2,5,7,1,3,4), (0,0,1,1,1)(6,5,1,7,2,3,4), (0,0,1,1,2)(6,2,5,7,4,3,1), (0,0,1,2,0)(6,4,5,7,1,3,2), (0,0,1,2,1)(6,2,4,7,1,5,3), (0,0,1,2,2)(6,1,5,7,4,2,3), (0,0,2,0,0)(6,4,2,7,3,1,5), (0,0,2,0,1)(6,3,4,7,2,1,5), (0,0,2,0,2)(6,3,2,7,4,1,5), (0,0,2,1,0)(6,4,1,7,5,3,2), (0,0,2,1,1)(6,5,4,7,2,3,1), (0,0,2,1,2)(6,5,2,7,4,3,1), (0,0,2,2,0)(6,4,1,7,5,2,3), (0,0,2,2,1)(6,5,4,7,3,2,1), (0,0,2,2,2)(6,5,3,7,4,2,1), (0,1,0,0,0)(6,1,3,2,7,5,4), (0,1,0,0,1)(6,3,2,4,7,5,1), (0,1,0,0,2)(6,3,2,5,7,4,1), (0,1,0,1,0)(6,4,2,5,7,3,1), (0,1,0,1,1)(6,2,1,5,7,3,4), (0,1,0,1,2)(6,5,2,1,7,4,3), (0,1,0,2,0)(6,2,1,3,7,5,4), (0,1,0,2,1)(6,3,1,4,7,5,2), (0,1,0,2,2)(6,5,2,3,7,4,1), (0,1,1,0,0)(6,3,5,2,7,1,4), (0,1,1,0,1)(6,2,3,5,7,4,1), (0,1,1,0,2)(6,3,5,2,7,4,1), (0,1,1,1,0)(6,2,5,4,7,3,1), (0,1,1,1,1)(6,2,5,1,7,3,4), (0,1,1,1,2)(6,3,5,1,7,4,2), (0,1,1,2,0)(6,2,5,3,7,1,4), (0,1,1,2,1)(6,5,1,3,7,2,4), (0,1,1,2,2)(6,4,5,3,7,2,1), (0,1,2,0,0)(6,5,4,2,7,1,3), (0,1,2,0,1)(6,4,3,2,7,1,5), (0,1,2,0,2)(6,5,3,2,7,4,1), (0,1,2,1,0)(6,4,2,1,7,3,5), (0,1,2,1,1)(6,5,4,1,7,3,2), (0,1,2,1,2)(6,3,4,5,7,2,1), (0,1,2,2,0)(6,5,4,3,7,1,2), (0,1,2,2,1)(6,5,4,3,7,2,1), (0,1,2,2,2)(6,4,3,1,7,2,5), (0,2,0,0,0)(6,4,1,5,3,7,2), (0,2,0,0,1)(6,3,2,4,1,7,5), (0,2,0,0,2)(6,3,2,5,4,7,1), (0,2,0,1,0)(6,1,4,2,5,7,3), (0,2,0,1,1)(6,2,4,1,5,7,3), (0,2,0,1,2)(6,3,2,1,5,7,4), (0,2,0,2,0)(6,4,2,3,5,7,1), (0,2,0,2,1)(6,1,2,3,5,7,4), (0,2,0,2,2)(6,3,1,5,4,7,2), (0,2,1,0,0)(6,3,5,2,1,7,4), (0,2,1,0,1)(6,5,1,4,2,7,3), (0,2,1,0,2)(6,3,5,2,4,7,1), (0,2,1,1,0)(6,1,5,4,2,7,3), (0,2,1,1,1)(6,2,5,1,3,7,4), (0,2,1,1,2)(6,4,5,1,2,7,3), (0,2,1,2,0)(6,2,5,3,1,7,4), (0,2,1,2,1)(6,5,1,3,2,7,4), (0,2,1,2,2)(6,3,5,1,4,7,2), (0,2,2,0,0)(6,5,3,4,1,7,2), (0,2,2,0,1)(6,5,1,4,3,7,2), (0,2,2,0,2)(6,5,3,2,4,7,1), (0,2,2,1,0)(6,4,2,1,3,7,5), (0,2,2,1,1)(6,5,4,1,3,7,2), (0,2,2,1,2)(6,5,3,1,4,7,2), (0,2,2,2,0)(6,5,4,3,1,7,2), (0,2,2,2,1)(6,5,4,3,2,7,1), (0,2,2,2,2)(6,5,2,3,4,7,1), (1,0,0,0,0)(2,6,1,7,3,5,4), (1,0,0,0,1)(1,6,3,7,2,5,4), (1,0,0,0,2)(3,6,2,7,1,4,5), (1,0,0,1,0)(4,6,1,7,5,3,2), (1,0,0,1,1)(4,6,2,7,5,3,1), (1,0,0,1,2)(1,6,2,7,5,3,4), (1,0,0,2,0)(4,6,1,7,5,2,3), (1,0,0,2,1)(4,6,1,7,3,2,5), (1,0,0,2,2)(3,6,4,7,1,2,5), (1,0,1,0,0)(2,6,5,7,4,1,3), (1,0,1,0,1)(2,6,3,7,1,4,5), (1,0,1,0,2)(1,6,3,7,5,4,2), (1,0,1,1,0)(2,6,5,7,1,3,4), (1,0,1,1,1)(4,6,5,7,2,3,1), (1,0,1,1,2)(1,6,2,7,4,3,5), (1,0,1,2,0)(4,6,5,7,1,2,3), (1,0,1,2,1)(4,6,5,7,3,2,1), (1,0,1,2,2)(1,6,5,7,4,3,2), (1,0,2,0,0)(2,6,4,7,3,1,5), (1,0,2,0,1)(5,6,3,7,2,1,4), (1,0,2,0,2)(5,6,3,7,4,1,2), (1,0,2,1,0)(2,6,4,7,5,1,3), (1,0,2,1,1)(2,6,4,7,5,3,1), (1,0,2,1,2)(5,6,3,7,4,2,1), (1,0,2,2,0)(3,6,1,7,5,2,4), (1,0,2,2,1)(1,6,5,7,3,2,4), (1,0,2,2,2)(5,6,1,7,4,2,3), (1,1,0,0,0)(3,6,5,4,7,1,2), (1,1,0,0,1)(3,6,2,5,7,4,1), (1,1,0,0,2)(3,6,5,2,7,4,1), (1,1,0,1,0)(4,6,1,2,7,3,5), (1,1,0,1,1)(3,6,2,4,7,5,1), (1,1,0,1,2)(4,6,3,1,7,5,2), (1,1,0,2,0)(4,6,1,2,7,5,3), (1,1,0,2,1)(4,6,1,3,7,5,2), (1,1,0,2,2)(4,6,2,3,7,5,1), (1,1,1,0,0)(4,6,5,2,7,1,3), (1,1,1,0,1)(5,6,2,4,7,1,3), (1,1,1,0,2)(4,6,3,5,7,1,2), (1,1,1,1,0)(5,6,2,1,7,3,4), (1,1,1,1,1)(4,6,5,1,7,3,2), (1,1,1,1,2)(3,6,4,1,7,5,2), (1,1,1,2,0)(4,6,5,3,7,1,2), (1,1,1,2,1)(4,6,5,3,7,2,1), (1,1,1,2,2)(5,6,1,3,7,4,2), (1,1,2,0,0)(5,6,3,2,7,1,4), (1,1,2,0,1)(5,6,3,4,7,1,2), (1,1,2,0,2)(5,6,3,2,7,4,1), (1,1,2,1,0)(5,6,4,1,7,3,2), (1,1,2,1,1)(5,6,3,4,7,2,1), (1,1,2,1,2)(5,6,4,1,7,2,3), (1,1,2,2,0)(5,6,4,2,7,3,1), (1,1,2,2,1)(5,6,4,3,7,1,2), (1,1,2,2,2)(5,6,4,3,7,2,1), (1,2,0,0,0)(3,6,5,4,1,7,2), (1,2,0,0,1)(4,6,1,5,3,7,2), (1,2,0,0,2)(3,6,5,2,4,7,1), (1,2,0,1,0)(4,6,1,2,5,7,3), (1,2,0,1,1)(4,6,2,1,5,7,3), (1,2,0,1,2)(4,6,3,1,5,7,2), (1,2,0,2,0)(4,6,1,3,5,7,2), (1,2,0,2,1)(4,6,2,3,5,7,1), (1,2,0,2,2)(5,6,1,3,4,7,2), (1,2,1,0,0)(4,6,5,2,1,7,3), (1,2,1,0,1)(5,6,2,4,1,7,3), (1,2,1,0,2)(4,6,3,5,1,7,2), (1,2,1,1,0)(5,6,2,1,3,7,4), (1,2,1,1,1)(4,6,5,1,3,7,2), (1,2,1,1,2)(5,6,2,1,4,7,3), (1,2,1,2,0)(4,6,5,3,1,7,2), (1,2,1,2,1)(4,6,5,3,2,7,1), (1,2,1,2,2)(5,6,2,3,4,7,1), (1,2,2,0,0)(5,6,3,2,1,7,4), (1,2,2,0,1)(5,6,3,4,1,7,2), (1,2,2,0,2)(5,6,3,2,4,7,1), (1,2,2,1,0)(5,6,4,1,3,7,2), (1,2,2,1,1)(5,6,3,4,2,7,1), (1,2,2,1,2)(5,6,4,1,2,7,3), (1,2,2,2,0)(5,6,4,2,3,7,1), (1,2,2,2,1)(5,6,4,3,1,7,2), (1,2,2,2,2)(5,6,4,3,2,7,1), (2,0,0,0,0)(2,1,6,7,3,5,4), (2,0,0,0,1)(1,5,6,7,3,4,2), (2,0,0,0,2)(2,3,6,7,4,5,1), (2,0,0,1,0)(2,4,6,7,3,5,1), (2,0,0,1,1)(4,2,6,7,5,3,1), (2,0,0,1,2)(3,1,6,7,2,4,5), (2,0,0,2,0)(4,1,6,7,5,2,3), (2,0,0,2,1)(4,1,6,7,3,2,5), (2,0,0,2,2)(3,1,6,7,5,4,2), (2,0,1,0,0)(3,2,6,7,4,1,5), (2,0,1,0,1)(4,2,6,7,3,1,5), (2,0,1,0,2)(3,2,6,7,1,4,5), (2,0,1,1,0)(2,5,6,7,1,3,4), (2,0,1,1,1)(4,5,6,7,2,3,1), (2,0,1,1,2)(2,3,6,7,5,4,1), (2,0,1,2,0)(4,5,6,7,1,2,3), (2,0,1,2,1)(4,5,6,7,3,2,1), (2,0,1,2,2)(3,1,6,7,4,2,5), (2,0,2,0,0)(2,5,6,7,4,1,3), (2,0,2,0,1)(5,3,6,7,2,1,4), (2,0,2,0,2)(5,3,6,7,4,1,2), (2,0,2,1,0)(3,4,6,7,2,1,5), (2,0,2,1,1)(3,4,6,7,2,5,1), (2,0,2,1,2)(5,3,6,7,4,2,1), (2,0,2,2,0)(3,4,6,7,5,1,2), (2,0,2,2,1)(3,4,6,7,5,2,1), (2,0,2,2,2)(5,1,6,7,4,2,3), (2,1,0,0,0)(3,5,6,4,7,1,2), (2,1,0,0,1)(4,1,6,5,7,3,2), (2,1,0,0,2)(3,5,6,2,7,4,1), (2,1,0,1,0)(4,1,6,2,7,5,3), (2,1,0,1,1)(4,2,6,1,7,5,3), (2,1,0,1,2)(4,3,6,1,7,5,2), (2,1,0,2,0)(4,1,6,3,7,5,2), (2,1,0,2,1)(4,2,6,3,7,5,1), (2,1,0,2,2)(5,1,6,3,7,4,2), (2,1,1,0,0)(4,5,6,2,7,1,3), (2,1,1,0,1)(5,2,6,4,7,1,3), (2,1,1,0,2)(4,3,6,5,7,1,2), (2,1,1,1,0)(5,2,6,1,7,3,4), (2,1,1,1,1)(4,5,6,1,7,3,2), (2,1,1,1,2)(5,2,6,1,7,4,3), (2,1,1,2,0)(4,5,6,3,7,1,2), (2,1,1,2,1)(4,5,6,3,7,2,1), (2,1,1,2,2)(5,2,6,3,7,4,1), (2,1,2,0,0)(5,3,6,2,7,1,4), (2,1,2,0,1)(5,3,6,4,7,1,2), (2,1,2,0,2)(5,3,6,2,7,4,1), (2,1,2,1,0)(5,4,6,1,7,3,2), (2,1,2,1,1)(5,3,6,4,7,2,1), (2,1,2,1,2)(5,4,6,1,7,2,3), (2,1,2,2,0)(5,4,6,2,7,3,1), (2,1,2,2,1)(5,4,6,3,7,1,2), (2,1,2,2,2)(5,4,6,3,7,2,1), (2,2,0,0,0)(3,5,6,4,1,7,2), (2,2,0,0,1)(4,1,6,5,3,7,2), (2,2,0,0,2)(3,5,6,2,4,7,1), (2,2,0,1,0)(4,1,6,2,5,7,3), (2,2,0,1,1)(4,2,6,1,5,7,3), (2,2,0,1,2)(4,3,6,1,5,7,2), (2,2,0,2,0)(4,1,6,3,5,7,2), (2,2,0,2,1)(4,2,6,3,5,7,1), (2,2,0,2,2)(5,1,6,3,4,7,2), (2,2,1,0,0)(4,5,6,2,1,7,3), (2,2,1,0,1)(5,2,6,4,1,7,3), (2,2,1,0,2)(4,3,6,5,1,7,2), (2,2,1,1,0)(5,2,6,1,3,7,4), (2,2,1,1,1)(4,5,6,1,3,7,2), (2,2,1,1,2)(5,2,6,1,4,7,3), (2,2,1,2,0)(4,5,6,3,1,7,2), (2,2,1,2,1)(4,5,6,3,2,7,1), (2,2,1,2,2)(5,2,6,3,4,7,1), (2,2,2,0,0)(5,3,6,2,1,7,4), (2,2,2,0,1)(5,3,6,4,1,7,2), (2,2,2,0,2)(5,3,6,2,4,7,1), (2,2,2,1,0)(5,4,6,1,3,7,2), (2,2,2,1,1)(5,3,6,4,2,7,1), (2,2,2,1,2)(5,4,6,1,2,7,3), (2,2,2,2,0)(5,4,6,2,3,7,1), (2,2,2,2,1)(5,4,6,3,1,7,2), (2,2,2,2,2)(5,4,6,3,2,7,1) Listing of the elements x ∈ Z4 and the corresponding values ofH(x) ∈ S6. (0,0,0,0)(1,2,3,4,5,6), (0,0,0,1)(1,2,3,6,4,5), (0,0,0,2)(1,2,3,5,4,6), (0,0,1,0)(1,4,2,6,5,3), (0,0,1,1)(1,4,2,3,6,5), (0,0,1,2)(1,4,2,5,6,3), (0,0,2,0)(1,3,4,6,5,2), (0,0,2,1)(1,3,4,5,6,2), (0,0,2,2)(1,3,4,2,6,5), (0,1,0,0)(1,5,3,4,6,2), (0,1,0,1)(1,2,5,3,4,6), (0,1,0,2)(1,5,3,2,4,6), (0,1,1,0)(1,5,2,4,6,3), (0,1,1,1)(1,5,2,3,4,6), (0,1,1,2)(1,4,5,2,6,3), (0,1,2,0)(1,3,5,4,6,2), (0,1,2,1)(1,5,4,3,6,2), (0,1,2,2)(1,5,4,2,6,3), (0,2,0,0)(1,6,3,4,5,2), (0,2,0,1)(1,2,6,3,4,5), (0,2,0,2)(1,6,3,2,4,5), (0,2,1,0)(1,6,2,4,5,3), (0,2,1,1)(1,6,2,3,4,5), (0,2,1,2)(1,4,6,2,5,3), (0,2,2,0)(1,3,6,4,5,2), (0,2,2,1)(1,6,4,3,5,2), (0,2,2,2)(1,6,4,2,5,3), (1,0,0,0)(4,1,3,5,6,2), (1,0,0,1)(4,1,3,6,5,2), (1,0,0,2)(4,1,3,2,6,5), (1,0,1,0)(3,1,2,4,6,5), (1,0,1,1)(3,1,2,6,4,5), (1,0,1,2)(3,1,2,5,4,6), (1,0,2,0)(2,1,4,5,6,3), (1,0,2,1)(2,1,4,3,6,5), (1,0,2,2)(2,1,4,6,5,3), (1,1,0,0)(6,1,3,4,5,2), (1,1,0,1)(4,1,5,3,6,2), (1,1,0,2)(6,1,3,2,4,5), (1,1,1,0)(3,1,5,4,6,2), (1,1,1,1)(6,1,5,3,4,2), (1,1,1,2)(6,1,5,2,4,3), (1,1,2,0)(2,1,5,4,6,3), (1,1,2,1)(6,1,4,3,5,2), (1,1,2,2)(6,1,4,2,5,3), (1,2,0,0)(5,1,3,4,6,2), (1,2,0,1)(4,1,6,3,5,2), (1,2,0,2)(5,1,3,2,4,6), (1,2,1,0)(5,1,2,4,6,3), (1,2,1,1)(5,1,6,3,4,2), (1,2,1,2)(5,1,6,2,4,3), (1,2,2,0)(2,1,6,4,5,3), (1,2,2,1)(5,1,4,3,6,2), (1,2,2,2)(5,1,4,2,6,3), (2,0,0,0)(4,2,1,5,6,3), (2,0,0,1)(4,2,1,3,6,5), (2,0,0,2)(4,2,1,6,5,3), (2,0,1,0)(3,4,1,5,6,2), (2,0,1,1)(3,4,1,6,5,2), (2,0,1,2)(3,4,1,2,6,5), (2,0,2,0)(2,3,1,4,6,5), (2,0,2,1)(2,3,1,6,4,5), (2,0,2,2)(2,3,1,5,4,6), (2,1,0,0)(6,2,1,4,5,3), (2,1,0,1)(6,2,1,3,4,5), (2,1,0,2)(4,5,1,2,6,3), (2,1,1,0)(3,5,1,4,6,2), (2,1,1,1)(6,4,1,3,5,2), (2,1,1,2)(6,5,1,2,4,3), (2,1,2,0)(6,3,1,4,5,2), (2,1,2,1)(2,5,1,3,4,6), (2,1,2,2)(6,3,1,2,4,5), (2,2,0,0)(5,2,1,4,6,3), (2,2,0,1)(5,2,1,3,4,6), (2,2,0,2)(4,6,1,2,5,3), (2,2,1,0)(3,6,1,4,5,2), (2,2,1,1)(5,4,1,3,6,2), (2,2,1,2)(5,6,1,2,4,3), (2,2,2,0)(5,3,1,4,6,2), (2,2,2,1)(2,6,1,3,4,5), (2,2,2,2)(5,3,1,2,4,6) Listing of the elements x ∈ Z3 and the corresponding values of R(x) ∈ S5. (0,0,0)(1,2,3,5,4), (0,0,1)(1,4,3,5,2), (0,0,2)(1,5,3,4,2), (0,1,0)(1,2,4,5,3), (0,1,1)(1,4,2,5,3), (0,1,2)(1,5,4,3,2), (0,2,0)(1,2,5,4,3), (0,2,1)(1,4,5,3,2), (0,2,2)(1,3,5,4,2), (1,0,0)(4,1,3,5,2), (1,0,1)(5,1,3,4,2), (1,0,2)(2,1,3,5,4), (1,1,0)(3,1,4,5,2), (1,1,1)(5,1,4,3,2), (1,1,2)(2,1,4,5,3), (1,2,0)(4,1,5,3,2), (1,2,1)(5,1,2,4,3), (1,2,2)(2,1,5,4,3), (2,0,0)(4,2,1,5,3), (2,0,1)(5,4,1,3,2), (2,0,2)(2,5,1,4,3), (2,1,0)(3,2,1,5,4), (2,1,1)(3,4,1,5,2), (2,1,2)(3,5,1,4,2), (2,2,0)(4,3,1,5,2), (2,2,1)(5,3,1,4,2), (2,2,2)(2,3,1,5,4) Listing of the elements x ∈ Z3 and the corresponding values of S(x) ∈ S5. (0,0,0)(2,1,3,4,5), (0,0,1)(2,4,3,1,5), (0,0,2)(2,5,3,1,4), (0,1,0)(2,1,4,5,3), (0,1,1)(2,4,1,5,3), (0,1,2)(2,5,4,1,3), (0,2,0)(2,1,5,4,3), (0,2,1)(2,4,5,1,3), (0,2,2)(2,3,5,1,4), (1,0,0)(4,2,3,1,5), (1,0,1)(5,2,3,1,4), (1,0,2)(1,2,3,5,4), (1,1,0)(3,2,4,1,5), (1,1,1)(5,2,4,1,3), (1,1,2)(1,2,4,5,3), (1,2,0)(4,2,5,1,3), (1,2,1)(5,2,1,4,3), (1,2,2)(1,2,5,4,3), (2,0,0)(4,1,2,5,3), (2,0,1)(5,4,2,1,3), (2,0,2)(1,5,2,4,3), (2,1,0)(3,1,2,5,4), (2,1,1)(3,4,2,1,5), (2,1,2)(3,5,2,1,4), (2,2,0)(4,3,2,1,5), (2,2,1)(5,3,2,1,4), (2,2,2)(1,3,2,5,4) Listing of the elements x ∈ Z4 and the corresponding values of T (x) ∈ S6. (0,0,0,0)(2,4,3,1,5,6), (0,0,0,1)(2,4,3,6,1,5), (0,0,0,2)(2,4,3,5,1,6), (0,0,1,0)(2,1,4,6,5,3), (0,0,1,1)(2,1,4,3,6,5), (0,0,1,2)(2,1,4,5,6,3), (0,0,2,0)(2,3,1,6,5,4), (0,0,2,1)(2,3,1,5,6,4), (0,0,2,2)(2,3,1,4,6,5), (0,1,0,0)(2,5,3,1,6,4), (0,1,0,1)(2,4,5,3,1,6), (0,1,0,2)(2,5,3,4,1,6), (0,1,1,0)(2,5,4,1,6,3), (0,1,1,1)(2,5,4,3,1,6), (0,1,1,2)(2,1,5,4,6,3), (0,1,2,0)(2,3,5,1,6,4), (0,1,2,1)(2,5,1,3,6,4), (0,1,2,2)(2,5,1,4,6,3), (0,2,0,0)(2,6,3,1,5,4), (0,2,0,1)(2,4,6,3,1,5), (0,2,0,2)(2,6,3,4,1,5), (0,2,1,0)(2,6,4,1,5,3), (0,2,1,1)(2,6,4,3,1,5), (0,2,1,2)(2,1,6,4,5,3), (0,2,2,0)(2,3,6,1,5,4), (0,2,2,1)(2,6,1,3,5,4), (0,2,2,2)(2,6,1,4,5,3), (1,0,0,0)(1,2,3,5,6,4), (1,0,0,1)(1,2,3,6,5,4), (1,0,0,2)(1,2,3,4,6,5), (1,0,1,0)(3,2,4,1,6,5), (1,0,1,1)(3,2,4,6,1,5), (1,0,1,2)(3,2,4,5,1,6), (1,0,2,0)(4,2,1,5,6,3), (1,0,2,1)(4,2,1,3,6,5), (1,0,2,2)(4,2,1,6,5,3), (1,1,0,0)(6,2,3,1,5,4), (1,1,0,1)(1,2,5,3,6,4), (1,1,0,2)(6,2,3,4,1,5), (1,1,1,0)(3,2,5,1,6,4), (1,1,1,1)(6,2,5,3,1,4), (1,1,1,2)(6,2,5,4,1,3), (1,1,2,0)(4,2,5,1,6,3), (1,1,2,1)(6,2,1,3,5,4), (1,1,2,2)(6,2,1,4,5,3), (1,2,0,0)(5,2,3,1,6,4), (1,2,0,1)(1,2,6,3,5,4), (1,2,0,2)(5,2,3,4,1,6), (1,2,1,0)(5,2,4,1,6,3), (1,2,1,1)(5,2,6,3,1,4), (1,2,1,2)(5,2,6,4,1,3), (1,2,2,0)(4,2,6,1,5,3), (1,2,2,1)(5,2,1,3,6,4), (1,2,2,2)(5,2,1,4,6,3), (2,0,0,0)(1,4,2,5,6,3), (2,0,0,1)(1,4,2,3,6,5), (2,0,0,2)(1,4,2,6,5,3), (2,0,1,0)(3,1,2,5,6,4), (2,0,1,1)(3,1,2,6,5,4), (2,0,1,2)(3,1,2,4,6,5), (2,0,2,0)(4,3,2,1,6,5), (2,0,2,1)(4,3,2,6,1,5), (2,0,2,2)(4,3,2,5,1,6), (2,1,0,0)(6,4,2,1,5,3), (2,1,0,1)(6,4,2,3,1,5), (2,1,0,2)(1,5,2,4,6,3), (2,1,1,0)(3,5,2,1,6,4), (2,1,1,1)(6,1,2,3,5,4), (2,1,1,2)(6,5,2,4,1,3), (2,1,2,0)(6,3,2,1,5,4), (2,1,2,1)(4,5,2,3,1,6), (2,1,2,2)(6,3,2,4,1,5), (2,2,0,0)(5,4,2,1,6,3), (2,2,0,1)(5,4,2,3,1,6), (2,2,0,2)(1,6,2,4,5,3), (2,2,1,0)(3,6,2,1,5,4), (2,2,1,1)(5,1,2,3,6,4), (2,2,1,2)(5,6,2,4,1,3), (2,2,2,0)(5,3,2,1,6,4), (2,2,2,1)(4,6,2,3,1,5), (2,2,2,2)(5,3,2,4,1,6) Introduction Notations and main results The general recursive construction. Proof of Theorem ??, second part Proof of Theorem ??, last part

0704.1359

Hamiltonian Quantum Dynamics With Separability Constraints

Hamiltonian Quantum Dynamics With Separability Constraints Nikola Burić ∗ Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia. November 9, 2018 Abstract Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quan- tum system form a special submanifold of PH. We analyze the Hamil- tonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce non- linearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamil- ton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dy- namical system with mixed phase space. Possible physical realizations of the separability constraints are discussed. PACS: 03.65.-w ∗e-mail: [email protected] http://arxiv.org/abs/0704.1359v1 1 Introduction Classical and quantum descriptions of a physical system that is considered as composed of interacting subsystems have radically different features. The typical feature of quantum dynamics is the creation of specifically quantum correlations, the entanglement, among the subsystems. On the other hand, the typical property of classical description is the occurrence of chaotic or- bits and fractality of the phase space portrait, which can be considered as typically classical type of correlations between the subsystems. The type of correlations introduced by the dynamical entanglement does not occur in the classical description, and likewise, the type of correlations introduced by the chaotic orbits with fractal structures does not occur in the quantum description. This intriguing complementarity of the two descriptions repre- sents a problem that is expected to be solved by a detailed formulation of the correspondence principle. Comparison of typical features of classical and quantum mechanics is fa- cilitated if the same mathematical framework is used in both theories. It is well known, since the work of Kibble [1],[2],[3], that the quantum evolution, determined by the linear Schroedinger equation, can be represented using the typical language of classical mechanics, that is as a Hamiltonian dynamical system on an appropriate phase space, given by the Hilbert space geometry of the quantum system. This line of research was later developed into the full geometric Hamiltonian representation of quantum mechanics. [4]-[12]. Such geometric formulation of quantum mechanics has recently inspired nat- ural definitions of measures of the entanglement [13], and has been used to model the spontaneous collapse of the state vector [14],[15], and dynamics of decoherence [16]. It is our goal to use the geometric Hamiltonian formulation of quantum mechanics to study the relation between the dynamical entanglement and typical qualitative properties of Hamiltonian dynamics. Motivated by the fact that the Schroedinger equation can always be considered as a Hamil- tonian dynamical system, and that for Hamiltonian systems the definitions and properties of the dynamical chaos are well understood, we shall seek for a formal condition that when imposed on the Hamiltonian system repre- senting the Schroedinger equation of the compound quantum system renders the Hamiltonian dynamics nonintegrable and chaotic. It is well known that the linear Schroedinger equation of quantum mechanics represents always an integrable Hamiltonian dynamical system, irrespective of the dynamical symmetries of the system. This is in sharp contrast with the Hamiltonian formulation of classical systems, where enough symmetry implies integrabil- ity and the lack of it implies the chaotic dynamics. Linearity of the quantum Hamiltonian dynamics, and the consequent integrability, is introduced in the Hamiltonian formulation by a very large dimensionality of the phase space of the quantum system. This high dimensionality can be considered as a conse- quence of two reasons. For a single quantum system, say a one dimensional particle in a potential, linear evolution and with it the principle of state superposition require infinite dimensional phase space of the Hamiltonian formulation. If the classical mechanical model is linear, say the harmonic os- cillator, the quantum Hamiltonian dynamics can be exactly describe on the reduced finite-dimensional phase space, the real plane in the case of the har- monic oscillator. The other related reason that increases the dimensionality of the quantum phase space compared to the classical model is in the way the state space of the compound systems are formed out of the components state spaces in the two theories. In order to represents the entangled states as points of the quantum phase space the dimensionality of the quantum phase space is much larger than just the sum of the dimensions of the components phase spaces. The points in the Cartesian product of the components phase spaces represent the separable quantum states and form a subset of the full quantum phase space. Needles to say, although the separable states are the most classical-like states of the compound system, they still are quantum states with nonclassical properties like nonzero dispersion of some subsys- tem’s variables. Our main result will be that when the quantum dynamics, represented as a Hamiltonian system, is constrained on the manifold of sepa- rable quantum states the relation between the symmetry and the qualitative properties of the dynamics such as integrability or chaotic motion is reestab- lished. Thus, suppression of dynamical entanglement is enough to enable manifestations of the qualitative differences in dynamics of quantum systems and the relation between integrability and symmetry, traditionally related with classical mechanical models. In order to study the relation between the dynamical entanglement, sep- arability and the properties of Hamiltonian formulation of the quantum dy- namics we shall use, in this paper, the simplest quantum system that displays the dynamical entanglement, that is a system of two interacting qubits: H = ωσ1 + ωσ2 + µxσx 2 + µyσy 2 + µzσz 2, (1) where σx,y,z i are the three Pauli matrices of the i-th qubit, and satisfy the usual SU(2) commutation relations. In particular we shall compare the dy- namics of the system (1) in the case µz 6= 0, µx = µy = 0 with the case when µx 6= 0, µz = µy = 0. The former case is symmetric with respect to SO(2) rotations around z-axis and the later lacks this symmetry. Besides its simplicity, the systems of the form (1) are of considerable current interest because the hamiltonian of the universal quantum processor is of this form [17],[18]. Various lines of research, during the last decade, improved the under- standing of the relation between dynamical entanglement and properties of the dynamics. Strong impetus to the study of all aspects of quantum entan- glement came from the theory of quantum computation [18]. Quantization of classical non-integrable systems, and various characteristic properties of resulting quantum systems, have been studied for a long time [19]. The de- pendence of the dynamical entanglement, between a quantum system and its environment, on the qualitative properties of the dynamics of the system was studied indirectly, within the theory of environmental decoherence [20]. The relation between the rates of dynamical entanglement and the qualitative properties of the dynamics in the semi-classical regime was initiated in the reference [21] and various aspects of this relations have been studied since [22]-[31]. The relation between the symmetry of the genuinely quantum sys- tem (1) and the degree of dynamical entanglement was studied in reference [32]. As we shall see, our present analyzes is related to the quoted works, but the relation between the dynamical entanglement and symmetry is here approached from a very different angle The structure of the paper is as follows. We shall first recapitulate the necessary background such as: the complex symplectic and Riemannian ge- ometry of CP n; Hamiltonian formulation on CP n of the quantum dynamics; geometric formulation of the set of separable pure states and Hamiltonian for- mulation of the constrained dynamics. In parallel with the general reminder, the explicit formulas for the system of two interacting qubits will be given. These are then applied, in section 3, to the study the qualitative properties of the separability constrained dynamics for the qubits systems. The main results are summarized and discussed in section 4. There we also discuss a model of an open quantum system with dynamics that clearly differentiates between the symmetric and the nonsymmetric systems. 2 Geometry of the state space CP n Hamiltonian formulation of quantum mechanics is based on the fact that the scalar product of vectors |ψ > in the Hilbert space of a quantum system can be used to represent the linear Schroedinger equation of quantum mechanics in the form of Hamilton’s equations. The canonical phase space structure of this equations is determined by the imaginary part of the scalar product, and the Hamilton’ s function is given by the quantum expectation < ψ|H|ψ > of the quantum hamiltonian. However, due to phase invariance and arbitrary normalization the proper space of pure quantum states is not the Hilbert space used to formulate the Schroedinger equation, but the projective Hilbert space which is the manifold to be used in the Hamiltonian formulation of quantum mechanics. In general, the resulting Hamiltonian dynamical system is infinite-dimensional, but we shall need the general definitions only for the case of quantum system with finite-dimensional Hilbert space, like the finite collection of qubits, in which case the quantum phase space is also finite-dimensional. We shall first review the definition of the complex projective space CP n, and then briefly state the basic definitions and recapitulate the formulas which are needed for the Hamiltonian formulation of the quantum dynamics on the state space and its restriction on the separable state subset. The general reference for the mathematical aspects of complex differential geometry is [33]. All concepts and formulas will be illustrated using the system of two interacting qubits. Differential geometry of the state space CP n is discussed by viewing it as a real 2n dimensional manifold endowed with complex, Riemannian and symplectic structure. In the case ofCP n this three structures are compatible. 2.1 Definition and intrinsic coordinates of CP n States of a collection of N = n + 1 qubits are represented using normalized vectors of the complex Hilbert space CN . Since all quantum mechanical predictions are given in terms of the Hermitian scalar product on CN , and this is invariant under multiplication by a constant (vector independent) phase factor, the states of the quantum system are actually represented by equivalence classes of vectors in CN . Two vectors ψ1 and ψ2 are equivalent: ψ2 ∼ ψ1 if there is a complex scalar a 6= 0 such that ψ2 = aψ1. This set of equivalence classes defines the complex projective space: CP n :≡ (Cn+1 − 0)/ ∼. It is the state space of the system of N qubits. Global coordinates (c1, . . . cN) of a vector in CN that represent an equivalence class [ψ], that is an element of CP n, are called homogeneous coordinates on CP n. The complex projective space is topologically equivalent to S2n+1/S1, where the 2n+ 1-dimensional sphere comes from normalization and the circle S1 takes care of the unimportant overall phase factor. The projective space CP n is locally homeomorphic with Cn. Intrinsic coordinates on CP n are introduced as follows. A chart Uµ consists of equiv- alence classes of all vectors in (Cn+1 − 0) such that cµ 6= 0. In the chart Uµ the local ( so called inhomogeneous) coordinates ζν, ν = 1, 2 . . . n are given ζν = ξν (ν ≤ µ− 1), ζν = ξν+1 (ν > µ), (2) where ξν = cν/cµ ν = 1, 2, . . . µ− 1, µ+ 1, . . . n+ 1. (3) The coordinates ζνµ(c) and ζ µ′(c) of a point c which belongs to the domain where two charts Uµ and Uµ′ overlap are related by the following holomorphic transformation ζνµ′(c) = (c )ζνµ(c) (4) As an illustration consider the system of two qubits. The Hilbert space is H = H1 ⊗H2 = C2 ⊗ C2 = C4. As a basis we can choose the set of separable vectors | ↑↑>, | ↑↓>, | ↓↑>, | ↓↓> or any other four orthogonal vectors. The coordinates of a vector in C4 with respect to a basis are denoted (c1, c2, c3, c4). The corresponding projective space is CP 3 ≡ S7/S1. At least two charts are needed to define the intrinsic coordinates over all CP 3. Consider first all vectors with a nonzero component along |1 >= | ↑↑> that is c1 6= 0, i.e. all vectors except the vector | ↓↓>. Then the numbers ξν1 are defined as ξ11 = c 1/c1 = 1, ξ21 = c 2/c1, ξ31 = c 3/c1, ξ41 = c 4/c1 and finally the three intrinsic coordinates (ζ11 , ζ 1 , ζ 1 ) are given by relabelling of ξν1 : ζ 1 = ξ 1 , ζ 1 = ξ 1 , ζ 1 = ξ 1. To coordinatize the vector |4 >= | ↓↓> we need another chart. Quantum mean values of linear operators on C4 are indeed reduced to functions on CP 3. For example, consider the following Hamiltonian operator H = ωσz ⊗ 1+ ω1⊗ σz + µσx ⊗ σx (5) In the separable bases the normalized quantum expectation < ψ|H|ψ > / < ψ|ψ > is given by the following function of (c1, c2 . . . , c̄4) 2ω(c1c̄1 − c4c̄4) + µ(c̄2c3 + c̄3c2 + c̄1c4 + c̄4c1) c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4 . (6) In the intrinsic coordinates ζ1, ζ2, ζ3 and their conjugates this expression is given by 2ω(1− ζ3ζ̄3) + µ(ζ̄1ζ2 + ζ̄2ζ1 + ζ3 + ζ̄3) 1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3 . (7) We shall also analyze the following Hamiltonian H = ωσz ⊗ 1+ ω1⊗ σz + µσz ⊗ σz , (8) whose normalized mean value is given by 2ω(c1c̄1 − c4c̄4) + µ(c1c̄1 + c4c̄4 − c2c̄2 − c3c̄3) c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4 . (9) The corresponding function on CP 3 is, in the intrinsic coordinates, given by ω(1− ζ3ζ̄3) + µ(1 + ζ3ζ̄3 − ζ1ζ̄1 − ζ2ζ̄2) 1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3 . (10) 2.1.1 Submanifold of separable states Consider two quantum systems A and B with the corresponding Hilbert spaces HA and HB. Taken together, the systems A and B form another quantum system. The statistics of measurements that could be performed on this compound system requires that the Hilbert space of the compound system is given by the direct product HAB = HA ⊗ HB. The space of pure states of the compound system is the projective Hilbert space PHAB. In the case of finite dimensional state spaces PHn+1A = CP n and PHm+1A = CPm the state space of the compound system is CP (m+1)(n+1)−1. Vectors inHAB of the form ψA ⊗ ψB where ψA/B ∈ HA/B are called separable. The corresponding separable states form the (m + n)-dimensional submanifold CPm × CP n embedded in CP (m+1)(n+1)−1. In the case of two qubits the submanifold of the separable states CP 1 × CP 1 forms a quadric in the full state space CP 3, given in terms of the homogeneous coordinates (c1, c2, c3, c4) of CP 3 by the following formula c1c4 = c2c3. (11) In terms of the intrinsic coordinates ζ1, ζ2, ζ3, in the chart with c1 6= 0, i.e. ξ1 = 1, the equation (11) is ζ1ζ2 = ζ3. (12) 2.2 Complex structure on CP n Consider a complex manifold M with complex dimension dimC M = n (in particular CP n ). We can look at M as a real manifold with dimR M = 2n. The real coordinates (x1, . . . x2n) are related to the holomorphic (ζ1, . . . ζn) and anti-holomorphic (ζ̄1, . . . ζ̄n) coordinates via the following formulas: (xν + ıxν+n)/ 2 = ζν, ν = 1, 2, . . . n, (xν − ıxν+n)/ 2 = ζ̄ν, ν = 1, 2, . . . n, (13) qν ≡ xν = (ζν + ζ̄ν)/ 2, ν = 1, 2, . . . n, pν ≡ xν+n = (ζ̄ν − ζ̄ν)/ 2, ν = 1, 2, . . . n. (14) The tangent space TxM is spanned by 2n vectors: , . . . , . . . } (15) or by the basis , . . . , . . . }. (16) An almost complex structure on a real 2n-dimensional manifold is given by a (1, 1) tensor J satisfying J2 = 1, i.e. Jac J b = −δab . Locally, the almost complex structure J is given in the real coordinates by the following matrix , (17) where 1 is n-dimensional unit matrix. If the real 2n manifold is actually a complex manifold, like in our case, the almost complex structure is defined globally and is called the complex structure. 2.3 Riemannian structure on CP n Hermitian scalar product induces a complex Euclidean metric on CN . The metric induced on CP n is the Fubini-Study metric, and is given, in (ζ, ζ̄) coordinates, using an n× n matrix with following entries gµ,ν̄(ζ, ζ̄) = δµ,ν(1 + ζζ̄)− ζµζ̄ν (1 + ζζ̄)2 , µ, ν = 1, 2 . . . n, (18) where ζζ̄ ≡ ∑nµ ζµζ̄µ. The Fubini-Study metric in (ζ, ζ̄) coordinates is then given by 2n × 2n matrix G(ζ, ζ̄) = 1 0 gµ,ν̄ gµ̄,ν 0 . (19) In the real coordinates the Fubini-Study metric is given by the standard transformation formulas Gi,j(q, p̄) = Gk,l(ζ(q, p), ζ̄(q, p)) , (20) where we used Z = (ζ1, . . . ζ̄n and X = (q 1 . . . pn). In the example of two qubits the Fubini-Study metric on CP 3 is 0 0 0 (1+ζζ̄)−ζ1ζ̄1 (1+ζζ̄)2 −ζ1ζ̄2 (1+ζζ̄)2 −ζ1ζ̄3 (1+ζζ̄)2 0 0 0 −ζ (1+ζζ̄)2 (1+ζζ̄)−ζ2ζ̄2 (1+ζζ̄)2 −ζ2ζ̄3 (1+ζζ̄)2 0 0 0 −ζ (1+ζζ̄)2 −ζ3ζ̄2 (1+ζζ̄)2 (1+ζζ̄)−ζ3ζ̄3 (1+ζζ̄)2 (1+ζζ̄)−ζ1ζ̄1 (1+ζζ̄)2 −ζ2ζ̄1 (1+ζζ̄)2 −ζ3ζ̄1 (1+ζζ̄)2 0 0 0 −ζ1ζ̄2 (1+ζζ̄)2 (1+ζζ̄)−ζ2ζ̄2 (1+ζζ̄)2 −ζ3ζ̄2 (1+ζζ̄)2 0 0 0 −ζ1ζ̄3 (1+ζζ̄)2 −ζ2ζ̄3 (1+ζζ̄)2 (1+ζζ̄)−ζ3ζ̄3 (1+ζζ̄)2 0 0 0 Transformation to the real coordinates, by application of the formula (20), gives −p1p2+q1q2 −p1p3+q1q3 1q2−p2q1 p1q3−p3q1 −p1p2+q1q2 p2p3+q2q3 p2q1−p1q2 2q3−p3q2 −p1p2+q1q2 p2p3+q2q3 p3q1−p1q3 p3q2−p2q3 2q1−p1q2 p3q1−p1q3 −p1p2+q1q2 −p1p3q1q3 p1q2−p2q1 3q2−p2q3 −p1p2+q1q2 −p2p3+q2q3 p1q3−p3q1 p2q3−q2p3 0 −p1p3−q1q3 −p2p3+q2q3 where a = (p1)2+(p2)2+(p3)2+(q1)2+(q2)2+(q3)2+2, b = (p1)2+(p3)2+(q1)2+(q3)2+2. Obviously, G is positive definite and symmetric. 2.4 Symplectic structure on CP n The Hermitian scalar product on CN is also used to define the symplectic structure on CN and this induces the symplectic structure on CP n. The symplectic structure is the closed nondegenerate two form Ω on CP n, which is, in (ζ, ζ̄) coordinates given by ω = ıg(ζ, ζ̄)µ,ν̄dζ µ ∧ ζ̄ν (23) where gµ,ν̄ is the Fubini-Study metric (18). In real coordinates, the symplectic structure is given by Ω(q, p) = JG(q, p) where G(q, p) is given by (20) and J by (17). The symplectic form on the two qubits state space is in the real bases given by the product of matrices (17) and (22). The results is 2q1+p1q2 −p3q1+p1q3 p1p2+q1q2 p1p3+q1q3 p2q1−p1q2 2q3−p3q2 p2p1+q1q2 p2p3+q2q3 p3q1−p1q3 p3q2−p2q3 1p3+q1q3 p2p3+q2q3 −p2p1+q1q2 −p1p3+q1q3 1q2−p2q1 p1q3−p3q1 −p1p2+q1q2 −p2p3+q2q3 p2q1−p1q2 2q3−p3q2 −p1p3+q1q3 −p2p3+q2q3 p3q1−p1q3 p3q2−p2q3 3 Quantum Hamiltonian dynamical system on CP n The Schroedinger equation on CN is in some basis {|ψi >, i = 1, 2 . . .N} given by: =< ψj |H|ψi > cj. (25) In the real coordinates this equation assumes the form of a Hamiltonian dynamical system on R2N with a global gauge symmetry corresponding to the invariance |ψ >→ exp(ix)|ψ >. Reduction with respect to this symmetry results in the Hamiltonian system onCP n, considered as a real manifold with the symplectic structure given by (23). The Hamilton equation on CP n, that are equivalent to the Schroedinger equation (25), are = 2Ωl,k∇kH(x), (26) where Ωl,k is the inverse of the symplectic form, and H(x) is given by the normalized quantum expectation of the Hamilton’s operator < ψ|H|ψ > / < ψ|ψ > expressed in terms of the real coordinates (14). For example, the hamiltonian (7) is given in terms of the real coordinates qi ≡ xi, pi ≡ xi+n, i = 1, . . . n by [2− (p3)2 − (q3)2] + µ (p1p2 + q1q2 + 2q3). (27) and the symmetric hamiltonian (9) is given by [2−(p3)2−(q3)2]−µ [(p1)2+(p2)2+(q1)2+(q2)2−(p3)2−(q3)2−2] (28) The Hamilton’s equations (26) with the hamiltonian (27) and the sym- plectic form (24) assume the following form q̇1 = −2ωp1 + µp2 − µ(p3q1 + p1q3)/ q̇2 = −2ωp2 + µp2 − µ(p3q2 + p2q3)/ q̇3 = −4ωp3 − 2µp3q3 ṗ1 = 2ωq1 − µq2 + µ(q3q1 − p1p3)/ ṗ2 = 2ωq2 − µq1 + µ(q3q2 − p2p3)/ ṗ3 = 4ωq3 + µ((q3)2 − (p3)2 − 2)/ 2. (29) The equations of motion with the symmetric hamiltonian (28) on CP 3 are quite simple q̇1 = −2(ω + µ)p1 q̇2 = −2(ω + µ)p2 q̇2 = −4ωp3 ṗ1 = 2(ω + µ)q1 ṗ2 = 2(ω + µ)q1 ṗ3 = 4ωq3. (30) 3.1 Quantum Hamiltonian system with imposed sep- arability constraints Dynamics of a constrained Hamiltonian system is usually described by the method of Lagrange multipliers [34],[35]. Consider a Hamiltonian system given by a symplectic manifold M with the symplectic form Ω and the Hamilton’s function H on M. Suppose that besides the forces described by H the dynamics of the system is affected also by forces whose sole effect is to constrain the motion on a submanifold N ∈ M determined by a set functional relations f1(q, p) = . . . fk(q, p) = 0 (31) The method of Lagrange multiplies assumes that the dynamics on N is de- termined by the following set of differential equations Ẋ = Ω(∇X,∇H ′), H ′ = H + , λjfj (32) which should be solved together with the equations of the constraints (31). The Lagrange multipliers λj are functions of (p, q) that are to be determined from the following, so called compatibility, conditions. ḟl = Ω(∇fl,∇H ′) (33) onN . The equations (33) uniquely determine the functions λ1(p, q), . . . λk(p, q) if and only if the matrix of Poison brackets {fi, fj} = Ω(∇fi,∇fj) is nonsin- gular. If this is the case then all constraints (31) are called primary, and N is symplectic manifold with the symplectic structure determined by the so called Dirac-Poison brackets {F1, F2}′ = {F1, F2}+ {fi, F1}{fi, fj}−1{fj, F2} (34) As we shall see, this is the case in the examples of pairs of interacting qubits constrained on the manifold of separable states that we shall analyze. On the other hand, if some of the compatibility equations do not contain multipliers, than for that constrain ḟj = {fj, H} = 0, which represents an additional constraint. These are called secondary constraints, and they must be added to the system of original constraints (31). If this enlarged set of constraints is functionally independent one can repeat the procedure. At the end one either obtains a contradiction, in which case the original problem has no solution, or one obtains appropriate multipliers λk such that the system (33) is compatible. In the later case the solution for λk might not be unique in which case the orbits of (32) and (31) are not uniquely determined by the initial conditions. Let us apply the formalism of Lagrange multipliers on the system of two interacting qubits additionally constrained to remain on the manifold of separable pure state. The real and imaginary parts of (12) give the two constraints in terms of real coordinates (q1, q2, q3, p1, p2, p3) f1 = p 1p2 − q1q2 + 2q3, f2 = 2q3 − p2q1 − p1q2 (35) The compatibility conditions (33) assume the following form ḟ1 = Ω(∇f1,∇H) + λ2Ω(∇f1,∇f2) = 0, ḟ2 = Ω(∇f2,∇H) + λ1Ω(∇f2,∇f1) = 0. (36) where Ω is the symplectic form (24) and Ω(∇f1,∇H) = Ωa,b∇af1∇bH . The matrix of Poisson brackets {fi, fj} on N is 0 [2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8 −[2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8 0 and is nonsingular. Thus the compatibility conditions can be solved for the Lagrange multipliers λ1(q, p), λ2(q, p), λ1 = 4µ 4p1p2q1q2 + [(q1)2 − 2][2 + (p2)2 − (q2)2] + (p1)2[(q2)2 − (p2)2 − 2] [2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2 λ2 = 8µ (p1)2p2q2 − p2q2[(q1)2 − 2] + p1q1[2 + (p2)2 − (q2)2] [2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2 Finally, the dynamics of the constrained system is described by the equa- tions (32) and (31) with λ1(q, p), λ2(q, p) and f1(q, p), f2(q, p) given by (38) and (35). For the Hamiltonian (27) the resulting equations of motion for q1, q2, p1, p2 are q̇1 = −4µp 1q1q2 + 2ωp1[2 + (p2)2 + (q2)2] 2 + (p2)2 + (q2)2 q̇2 = −4µp 2q1q2 − 2ωp2[2 + (p1)2 + (q1)2] 2 + (p1)2 + (q1)2 ṗ1 = 2µq2[(q1)2 − (p1)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2] 2 + (p2)2 + (q2)2 ṗ2 = 2µq1[(q2)2 − (p2)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2] 2 + (p1)2 + (q1)2 . (39) The same procedure for the symmetric hamiltonian (28) results with the following equations of motion q̇1 = 2µp1[(p2)2 + (q2)2 − 2)]− 2ωp1[2 + (p2)2 + (q2)2] 2 + (p2)2 + (q2)2 q̇2 = 2µp2[(p1)2 + (q1)2 − 2]− 2ωp2[(2 + (p1)2 + (q1)2] 2 + (p1)2 + (q1)2 ṗ1 = −2µq1[(q2)2 + (p2)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2] 2 + (p2)2 + (q2)2 ṗ2 = −2µq2[(q1)2 + (p1)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2] 2 + (p1)2 + (q1)2 . (40) There are also the equations expressing q̇3 and ṗ3 in terms of q1, q2, p1, p2, but the solutions of these are already given by the constraints. 3.2 Qualitative properties of the constrained dynam- ics of two interacting qubits In this section we present the results of numerical analyzes of the qualita- tive properties of the dynamics generated by the constrained equations (40) and (39), corresponding to the quantum Hamiltonians (28) with the SO(2) symmetry and (27) without such symmetry. It is well known that any quantum system is integrable when considered as the Hamiltonian dynamical system on the symplectic space H, and that the reduction on the symplectic manifold PH preserves this property. This is simply a consequence of the form of the quantum Hamiloton’s function, which is always defined as the mean value of the Hamiltonian operator. Contrary to the case of classical Hamiltonian systems, the symmetry of the physical system has no relevance for the property of integrability in the Hamiltonian formulation of the Schroedinger equation. We illustrate this fact, in figures -1.50 -0.75 0.00 0.75 1.50 -0.4 -0.2 0.0 0.2 0.4 -0.30 -0.15 0.00 0.15 0.30 -0.30 -0.15 -0.30 -0.15 0.00 0.15 0.30 -0.30 -0.15 Figure 1: Projections on (q1, p1) plane of a typical orbit for the hamiltonian systems (28) (a) and (27) (b) on CP 3 and on the submanifold of separable states (c) for (40) and d for (39). The values of the parameters are ω = 1 and µ = 1.7 1a,b, by projections on (q1, p1) plane of a typical orbit for the symmetric and nonsymmetric hamiltonians of the pair of qubits. The motion on CP 3 in the symmetric case has further degeneracy compared with the nonsymmetric case, but both cases generate integrable, regular Hamiltonian dynamics. On the other hand, the qualitative properties of the dynamics constrained by the separability conditions, are quite different. Typical orbits in the sym- metric and nonsymmetric cases are illustrated in figure 1c,d. Symmetric dynamics constrained by separability is still regular, while the nonsymmetric Hamiltonian generates the constrained dynamics with typical chaotic orbits. This is further illustrated in figures 2, where we show Poincaré surfaces of section, defined by q2 = 0, p2 > 0 and H(p1, q1, p2, q2) = h for different values of the coupling µ. Obviously, the constrained system displays the transition from predominantly regular to predominantly chaotic dynamics, with all the intricate structure of the phase portrait, characteristic for typical Hamilto- nian dynamical systems. Thus, we can conclude that the quantum system constrained on the manifold of separable state behaves as typical classical Hamiltonian systems. If there is enough symmetry, i.e. enough integrals of motion, the constrained dynamics is integrable, otherwise the constrained quantum dynamics is that of typical chaotic Hamiltonian system. -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 Figure 2: Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3, (b) µ = 1.7 4 Summary and discussion We have studied Hamiltonian formulation of quantum dynamics of two in- teracting qubits. Hamiltonian dynamical system on the state space CP 3 as the phase space, is integrable irrespective of the different symmetries of the quantum system. We have then studied the dynamics of the quantum Hamiltonian system constrained on the manifold of separable states. The main result of this analyzes, and of the paper, is that the quantum Hamilto- nian system without symmetry generates nonintegrable chaotic dynamics on the set of separable states, while the constrained symmetric dynamics gives an integrable system. It is important to bare on mind that neither the system nor the separable states that lie on an orbit of the constrained system have an underlining classical mechanical model. Thus, forcing a non-degenerate quantum system to remain on the manifold of separable states is enough to generate a dynamical system with typical properties of Hamiltonian chaos. Our analyzes of the separability constrained quantum dynamics has been rather formal. In order to inquire into possible interpretation of our results we need a model of a physical realization of the separability constraints. To this end we consider an open quantum system of two interacting qubits, whose dynamics satisfies the Markov assumption [36], and we choose a Hermitian Lindblad operator of the following form L = l11σ ⊗ σ2+σ2− + l12σ1+σ1− ⊗ σ2−σ2+ + l21σ1−σ1+ ⊗ σ2+σ2− + l22σ1−σ1+ ⊗ σ2−σ2+ i,j=1 li,j|i >< j|1 ⊗ |i >< j|2 (41) where |1 >≡ | ↑> and |2 >≡ | ↓>. The dynamics of a pure state of the open system under the action of a Hamiltonian H and the Linblad γL is described by the following stochastic nonlinear Schroedinger equation [36],[37] |dψ > = −iH|ψ > dt+ γ (L− < ψ|L|ψ >)2|ψ > dt + γ(L− < ψ|L|ψ >)|ψ > dW, (42) where dW is the increment of complex Wiener c-number process W (t). The equation (42) represent a diffusion process on a complex Hilbert space, and is central in the ”Quantum State Diffusion” (QSD) theory of open quantum systems [37]. It has been used to study the systems of in- teracting qubits in various environments for example in [32],[38], and the effect of the Linblad operator (41) on the entanglement between two qubits was considered in [16]. The influence of the non-Hamiltonian terms of drift (proportional to γ2) and the diffusion (proportional to γ)), with the Linblad operator of the form (41), is to drive an entangled state towards one of the separable states with the corresponding probability. This process occurs on the time scale proportional to γ−1. So, for large γ there occurs an almost instantaneous collapse of an entangled state into a separable one. We believe that with a proper choice of the parameters li,j the long term dynamics of a pure state described by (42) can have the same qualitative properties as the separability constrained quantum dynamics. In particular, the difference between the qualitative properties of symmetric and nonsymmetric systems, reflected in the constrained Hamiltonian system, should also manifest in the dynamics of (42) for a proper choice of li,j. This expectations are supported by figures 3, which illustrate the dynamics of (< σ1x >,< σ y >) for the Hamil- tonian operators (5) and (8) as calculated using the constrained Hamiltonian equations (39) and (40) (figures 3b and 3a ), or the QSD equation (42) (fig- ures 3d and 3c) for a particular choice of li,j and large γ = 5. Of course, the choice of optimal values for li,j should be according to some criterion, which is the problem we are currently investigating. -1.0 -0.5 0.0 0.5 1.0 1.0 d) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Figure 3: Figures illustrate the dynamics of (< σx >,< σy >) for the constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c) and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 = 0.21, l2,1 = 0.215, l2,2 = 0.205. The pair of coupled qubits, analyzed in this paper, is the simplest quan- tum system exhibiting dynamical entanglement. We intend to investigate the effects of suppression of the dynamical entanglement in systems with spacial degrees of freedom, obtained by quantization of classically chaotic systems, for example a pair of coupled nonlinear oscillators. In this case, the Hamiltonian formulation of the quantum dynamics requires an infinite- dimensional phase space, and the analyzes of the separability constrained dynamics is more complicated. However, it wold be interesting to compare the dynamics obtained by separability constraints with that of some more standard semi-classical approximation. Acknowledgements This work is partly supported by the Serbian Min- istry of Science contract No. 141003. I should also like to acknowledge the support and hospitality of the Abdus Salam ICTP. References [1] T.W.B. Kibble, Commun.Math.Phys. 64 (1978) 73. [2] T.W.B. Kibble, Commun.Math.Phys. 65 (1979) 189. [3] T.W.B. Kibble and S. Randjbar-Daemi, J.Phys.A. 13 (1980) 141. [4] A. Heslot, Phys.Rev.D 31 (1985) 1341. [5] S. Weinberg, Phys.Rev.Lett, 62 (1989) 485. [6] S. Weinberg, Phys.Rev.Lett, 63 (1989) 1115. [7] S. Weinberg, Ann.Phys. 194 (1989) 336. [8] D.C. Brody and L.P. Hughston, Phys.Rev.Lett. 77 (1996) 2851. [9] D.C. Brody and L.P. Hughston, Phys.Lett. A 236 (1997) 257. [10] D.C. Brody and L.P. Hughston, Proc.R.Soc.London, 455 (1999) 1683. [11] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. Springer, Berlin, 1999. [12] I. Bengtsson and K. Žyczkowski, Geometry of Quantum States, Cam- bridge Uni. Press, Cambridge UK, 2006. [13] D.C. Brody and L.P. Hughston, J. Geom. Phys. 38 (2001) 19. [14] L.P. Hughston, Proc.R.Soc.London A, 452 (1996) 953. [15] S.L. Adler and T.A. Brun, J.Phys.A: Math. Gen. 34 (2001) 4797. [16] A. Belenkiy, S. Shnider and L. Horwitz, arXiv:quant-ph/0609142v1. [17] D. Deutch, A. Barenco and A. Ekert, Proc. Royal Soc. London A, 449 (1995) 669. [18] M. A. Nielsen and I.L. Chuang , Quantum Computation and Quantum Information, Cambridge Uni. Press, Cambridge UK ,2001. [19] F. Haake, Quantum Signatures of Chaos, Springer-Verlag, Berlin, 2000. http://arxiv.org/abs/quant-ph/0609142 [20] W.H. Zurek,Rev.Mod.Phys. 73 (2003) 715. [21] K. Funruya, M. C. Nemes and G.O. Pellegrino, Phys.Rev.Lett., 80 (1998) 5524. [22] P.A. Miller and S. Sarkar, Phys. Rev.E, 60 (1999) 1542. [23] B. Georgeot and D.L.Shepelyansky, Phys.Rev.E, 62 (2000) 6366. [24] B. Georgeot and D.L. Shepelyansky, Phys. Rev. E 62 (2000) 3504. [25] P. Zanardi, C. Zalka and L. Faoro, Phys.Rev. A 62, (2000) 030301. [26] A. Lakshminarayan, Phys.Rev.E, 64 (2001) 036207. [27] H. Fujisaki, T. Miyadera and A. Tanaka, Phys.Rev.E 67 (2003) 066201. [28] J.N. Bandyopathyay and A. Lakshminarayan, Phys.Rev.E, 69 (2004) 016201. [29] X. Wang, S. Ghose, B.C. Sanders and B. Hu, Phys.Rev.E, 70 (2004) 016217. [30] F. Mintert, A. R.R. Carvalho, M. Kuś and A. Buchleitner, Phys.Rep. 415 (2005) 207. [31] M. Novaes, Ann. Phys. (NY), 318 (2005) 308. [32] N. Burić, Phys.Rev.A, 73 (2006) 052111. [33] S. Kobajayashi and K. Nomizu K, Foundations of Differential Geometry, Wiley, New York, 1969. [34] P.A.M. Dirac, Can.J.Math. 2 (1950) 129. [35] V.I. Arnold, V.V. Kozlov and A.I. Neisthadt, Dynamical Systems III, Springer, Berlin, 1988. [36] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Sys- tems. Oxford Uni. Press, Oxford, 2001. [37] I.C. Percival, 1999. Quantum State Difussion, Cambridge Uni. Press, Cambridge UK, 1999. [38] N. Buric, Phys.Rev. A 72 (2005) 042322. FIGURE CAPTIONS Figure 1 Projections on (q1, p1) plane of a typical orbit for the hamil- tonian systems (28) (a) and (27) (b) on CP 3 and on the submanifold of separable states (c) for (40) and d for (39). The values of the parameters are ω = 1 and µ = 1.7 Figure 2 Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a)µ = 1.1, (b) µ = 1.3, (c) µ = 1.5 and (d) µ = 1.7 Figure 3 Figures illustrate the dynamics of (< σx >,< σy >) for the constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c) and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 = 0.21, l2,1 = 0.215, l2,2 = 0.205. Introduction Geometry of the state space CPn Definition and intrinsic coordinates of CPn Submanifold of separable states Complex structure on CPn Riemannian structure on CPn Symplectic structure on CPn Quantum Hamiltonian dynamical system on CPn Quantum Hamiltonian system with imposed separability constraints Qualitative properties of the constrained dynamics of two interacting qubits Summary and discussion

0704.1360

Planck Length and Cosmology

November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- Modern Physics Letters A c© World Scientific Publishing Company Planck Length and Cosmology Xavier Calmet Service de Physique Théorique, CP225 Boulevard du Triomphe B-1050 Brussels Belgium [email protected] Received (Day Month Year) Revised (Day Month Year) We show that an unification of quantum mechanics and general relativity implies that there is a fundamental length in Nature in the sense that no operational procedure would be able to measure distances shorter than the Planck length. Furthermore we give an explicit realization of an old proposal by Anderson and Finkelstein who argued that a fundamental length in nature implies unimodular gravity. Finally, using hand waving arguments we show that a minimal length might be related to the cosmological constant which, if this scenario is realized, is time dependent. Keywords: General Relativity; Quantum Mechanics; Cosmology. PACS Nos.: 98.80.-k, 04.20.-q. 1. Introduction The idea that a unification of quantum mechanics and general relativity implies the notion of a fundamental length is not new1. However, it has only recently been established that no operational procedure could exclude the discreteness of space-time on distances shorter than the Planck length2. This makes the case for a fundamental length of the order of the Planck length much stronger. It seems reasonable to think that any quantum description of general relativity will have to include the fact that measurement of distance shorter than the Planck length are forbidden. It is notoriously difficult to build a quantum theory of gravity. Besides technical difficulties the lack of experimental guidance, the Planck length being so miniscule lP ∼ 10−33cm, is flagrant. In this work we shall however argue that a fundamental length in nature, even if it is as small as the Planck scale may have dramatic impacts on our universe. In particular, we will argue that it may be related to the vacuum energy, i.e. dark energy and thus account for roughly 70% of the energy of the universe. We shall first present our motivation for a minimal length which follows from quantum mechanics, general relativity and causality. We will then argue that a fun- http://arxiv.org/abs/0704.1360v1 November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- 2 Xavier Calmet damental length in nature may lead to unimodular gravity. If this is the case, the cosmological constant is an integration parameter and is thus arbitrary. Finally, we shall consider argument based on spacetime quantization to argue that the cosmo- logical constant might not be actually constant but might be time dependent. 2. Minimal Length from Quantum Mechanics and General Relativity We first review the results obtained in ref.2. We show that quantum mechanics and classical general relativity considered simultaneously imply the existence of a minimal length, i.e. no operational procedure exists which can measure a distance less than this fundamental length. The key ingredients used to reach this conclusion are the uncertainty principle from quantum mechanics, and gravitational collapse from classical general relativity. A dynamical condition for gravitational collapse is given by the hoop conjecture3: if an amount of energy E is confined at any instant to a ball of size R, where R < E, then that region will eventually evolve into a black hole. We use natural units where ~, c and Newton’s constant (or lP ) are unity. We also neglect numerical factors of order one. From the hoop conjecture and the uncertainty principle, we immediately deduce the existence of a minimum ball of size lP . Consider a particle of energy E which is not already a black hole. Its size r must satisfy r ∼> max [ 1/E , E ] , (1) where λC ∼ 1/E is its Compton wavelength and E arises from the hoop conjecture. Minimization with respect to E results in r of order unity in Planck units or r ∼ lP . If the particle is a black hole, then its radius grows with mass: r ∼ E ∼ 1/λC . This relationship suggests that an experiment designed (in the absence of gravity) to measure a short distance l << lP will (in the presence of gravity) only be sensitive to distances 1/l. Let us give a concrete model of minimum length. Let the position operator x̂ have discrete eigenvalues {xi}, with the separation between eigenvalues either of order lP or smaller. For regularly distributed eigenvalues with a constant separation, this would be equivalent to a spatial lattice. We do not mean to imply that nature implements minimum length in this particular fashion - most likely, the physical mechanism is more complicated, and may involve, for example, spacetime foam or strings. However, our concrete formulation lends itself to detailed analysis. We show below that this formulation cannot be excluded by any gedanken experiment, which is strong evidence for the existence of a minimum length. Quantization of position does not by itself imply quantization of momentum. Conversely, a continuous spectrum of momentum does not imply a continuous spec- trum of position. In a formulation of quantum mechanics on a regular spatial lattice, with spacing a and size L, the momentum operator has eigenvalues which are spaced November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- Xavier Calmet 3 by 1/L. In the infinite volume limit the momentum operator can have continuous eigenvalues even if the spatial lattice spacing is kept fixed. This means that the displacement operator x̂(t)− x̂(0) = p̂(0) t does not necessarily have discrete eigenvalues (the right hand side of (2) assumes free evolution; we use the Heisenberg picture throughout). Since the time evolu- tion operator is unitary the eigenvalues of x̂(t) are the same as x̂(0). Importantly though, the spectrum of x̂(0) (or x̂(t)) is completely unrelated to the spectrum of the p̂(0), even though they are related by (2). A measurement of arbitrarily small displacement (2) does not exclude our model of minimum length. To exclude it, one would have to measure a position eigenvalue x and a nearby eigenvalue x′, with |x− x′| << lP . Many minimum length arguments are obviated by the simple observation of the minimum ball. However, the existence of a minimum ball does not by itself preclude the localization of a macroscopic object to very high precision. Hence, one might attempt to measure the spectrum of x̂(0) through a time of flight experiment in which wavepackets of primitive probes are bounced off of well-localised macroscopic objects. Disregarding gravitational effects, the discrete spectrum of x̂(0) is in princi- ple obtainable this way. But, detecting the discreteness of x̂(0) requires wavelengths comparable to the eigenvalue spacing. For eigenvalue spacing comparable or smaller than lP , gravitational effects cannot be ignored, because the process produces min- imal balls (black holes) of size lP or larger. This suggests a direct measurement of the position spectrum to accuracy better than lP is not possible. The failure here is due to the use of probes with very short wavelength. A different class of instrument, the interferometer, is capable of measuring dis- tances much smaller than the size of any of its sub-components. Nevertheless, the uncertainty principle and gravitational collapse prevent an arbitrarily accurate mea- surement of eigenvalue spacing. First, the limit from quantum mechanics. Consider the Heisenberg operators for position x̂(t) and momentum p̂(t) and recall the stan- dard inequality (∆A)2(∆B)2 ≥ − 1 (〈[Â, B̂]〉)2 . (3) Suppose that the position of a free mass is measured at time t = 0 and again at a later time. The position operator at a later time t is x̂(t) = x̂(0) + p̂(0) . (4) We assume a free particle Hamiltonian here for simplicity, but the argument can be generalized2. The commutator between the position operators at t = 0 and t is [x̂(0), x̂(t)] = i , (5) November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- 4 Xavier Calmet so using (3) we have |∆x(0)||∆x(t)| ≥ t . (6) We see that at least one of the uncertainties ∆x(0) or ∆x(t) must be larger than of order t/M . As a measurement of the discreteness of x̂(0) requires two position measurements, it is limited by the greater of ∆x(0) or ∆x(t), that is, by t/M , ∆x ≡ max [∆x(0),∆x(t)] ≥ , (7) where t is the time over which the measurement occurs andM the mass of the object whose position is measured. In order to push ∆x below lP , we take M to be large. In order to avoid gravitational collapse, the size R of our measuring device must also grow such that R > M . However, by causality R cannot exceed t. Any component of the device a distance greater than t away cannot affect the measurement, hence we should not consider it part of the device. These considerations can be summarized in the inequalities t > R > M . (8) Combined with (7), they require ∆x > 1 in Planck units, or ∆x > lP . (9) Notice that the considerations leading to (7), (8) and (9) were in no way specific to an interferometer, and hence are device independent. In summary, no device subject to quantum mechanics, gravity and causality can exclude the quantization of position on distances less than the Planck length. 3. Minimal Length and Unimodular Gravity General relativity is a scaleless theory: SGR = −gR(g) (10) varying this action with respect to the metric gµν leads to the well-known Einstein equations. The action (10) is invariant under general coordinate transformations and this may seem at odd with the notation of a minimal or fundamental length in nature. This may suggest that a quantum mechanical description of general relativity will fix the measure of Einstein-Hilbert action −g to some constant linked to the fundamental length. In that case one is led to unimodular gravity: SGR = d4xR(g) (11) with the constraint −g = constant which implies that only variation of the metric which respect this contraint may be considered. This is basically the argument made by Anderson and Finkelstein 4 in favor of a unimodular theory of gravity. November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- Xavier Calmet 5 There may be different ways to implement a minimal length in a theory, but we shall concentrate on one approach based on a noncommutative spacetime which indeed leads to a unimodular theory of gravity. Positing a noncommutative relation between e.g. x and y implies ∆x∆y ≥ |θxy| ∼ l2, with [x̂, ŷ] = iθxy and where l is the minimal length introduced in the theory. This also implies that a spacetime volume is quantized ∆V ≥ l4. One of the motivations to consider a noncommutative spacetime is that the non- commutative relations for the coordinates imply the existence of a minimal which can be thought of being proportional to the square root of the vacuum expecta- tion value of θµν i.e. lmin ∼ θ. If this length is fundamental it should not de- pend on the observer. Assuming the invariance of this fundamental length, one can show that there is a class of spacetime symmetries called noncommutative Lorentz transformations5 which preserve this length. It has recently been shown6, that there are also general coordinate transformations ξµ(x̂) that leave the canonical noncom- mutative algebra invariant and thus conserve the minimal length: [x̂µ, x̂ν ] = iθµν , (12) where θµν is constant and antisymmetric. They are of the form: ξµ(x̂) = θµν∂νf(x̂), where f(x̂) is an arbitrary field. The Jacobian of these restricted coordinate trans- formations is equal to one. This implies that the four-volume element is invariant: d4x′ = d4x. These noncommutative transformations correspond to volume preserv- ing diffeomorphisms which preserve the noncommutative algebra. A canonical non- commutative spacetime thus restricts general coordinate transformations to volume preserving coordinate transformations. These transformations are the only coordi- nate transformations that leave the canonical noncommutative algebra invariant. They form a subgroup of the unimodular transformations of a classical spacetime. The version of General Relativity based on volume-preserving diffeomorphism is known as the unimodular theory of gravitation7. Unimodular gravity here ap- pears as a direct consequence of spacetime noncommutativity defined by a constant antisymmetric θµν . One way to formulate gravity on a noncommutative spacetime has been presented in refs.6. Our approach might not be unique, but if the non- commutative model is reasonable, it must have a limit in which one recovers the commutative unimodular gravity theory in the limit in which θµν goes to zero. For small θµν we thus expect SNC = d4xR(gµν) +O(θ), (13) where R(gµν) is the usual Ricci scalar Once matter is included, one finds the fol- lowing equations of motion: Rµν − gµνR = −8πG(T µν − gµνT λλ) +O(θ). (14) These equations do not involve a cosmological constant and the contribution of vacuum fluctuations automatically cancel on the right-hand side of eq.(14). As done November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- 6 Xavier Calmet in e.g. ref.13 we can use the Bianchi for R and the equations of motion for T = −8πGT λλ and find: Dµ(R+ T ) = 0 (15) which can be integrated easily and give R + T = −Λ, where Λ is an integration constant. It can then be shown that the differential equations (14) imply Rµν − 1 gµνR− Λgµν = −8πGT µν −O(θ), (16) i.e. Einstein’s equations20 of General Relativity with a cosmological constant Λ that appears as an integration constant and is thus uncorrelated to any of the parameters of the action (13). As we have shown, one needs to impose energy conservation and the Bianchi identities to derive eq.(16) from eq.(14). Because any solution of Einstein’s equations with a cosmological constant can, at least over any topologically R4 open subset of spacetime, be written in a coordinate system with g = −1, the physical content of unimodular gravity is identical at the classical level to that of Einstein’s gravity with some cosmological constant13. 4. Cosmological implications of spacetime quantization We now come to the link between a fundamental length and cosmology and rephrase the arguments developed in refs.14,15,16,17,18 within the framework of a fundamen- tal length. It has been shown that the quantization of an unimodular gravity action proposed by Henneaux and Teitelboim12, which is an extension of the action de- fined in eq. (13), leads to an uncertainty relation between the fluctuations of the volume V and those of the cosmological constant Λ: δV δΛ ∼ 1 using natural units, i.e. ~ = lp = c = mp = 1. Now if spacetime is quantized, as it is the case for noncommuting coordinates, we expect the number of cells of spacetime to fluctu- ate according to a Poisson distribution, δN ∼ N , where N is the number of cells. This is however obviously an assumption which could only be justified by a complete understanding of noncommutative quantum gravity. It is then natural to assume that the volume fluctuates with the number of spacetime cells δV = δN . One finds δV ∼ V and thus Λ ∼ V − 12 , i.e., we obtain an effective cosmological constant which varies with the four-volume as obtained in a different context in refs.19,14,15,16. In deriving this result, we have assumed as in refs.14,15 that the fluctuation are around zero as explained below. A minimal length thus leads leads to a vacuum energy density ρ ρ ∼ 1√ . (17) Here we assume that the scale for the quantization of spacetime is the Planck scale. A crucial assumption made in refs.14,15,16 as well is that the value of cosmo- logical constant fluctuates around zero. This was made plausible by Baum21 and Hawking22 using an Euclidean formulation of quantum gravity. November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- Xavier Calmet 7 Now the question is really to decide what we mean by the four-volume V . If this is the four-volume related to the Hubble radius RH as in refs. 14,15,16 then this model predicts ρ ∼ (10−3eV )4 which is the right order for today’s energy density, it is however not obvious what is the equation of state for this effective cosmological constant. The choice V = R4H might be ruled out because of the equation of state of such a dark energy model as shown in ref.23 in the context of holographic dark energy which leads to similar phenomenology. However if we assume that the four- volume is related to the future event horizon as suggested by M. Li24, again in the context of holographic dark energy, then we get an equation of state which is compatible with the data w = −0.903 + 1.04z which is precisely the equation of state for the holographic dark energy obtained in ref.24. Details will appear in a forthcoming publication. 5. Conclusions We have argued that an unification of quantum mechanics and general relativity implies that there is a fundamental length in Nature in the sense that no opera- tional procedure would be able to measure distances shorter than the Planck length. Further we give an explicit realization of an old proposal by Anderson and Finkel- stein who had argued that a fundamental length in nature would imply unimodular gravity. Finally, using hand waving arguments we show that a minimal length might be related to the cosmological constant, which if this scenario is realized, is time dependent and thus only effectively a constant. Much more work remains to be done to establish this connection. It would be interesting to related the time de- pendence of the cosmological constant to that of other parameters of the standard model such as the fine-structure constant. Indeed as argued in refs.25 if one of the parameters of the standard model, such as a gauge coupling, a mass term or any other cosmological parameter, is time dependent, it is quite natural to expect that the remaining parameters of the theory will be time dependent as well. Acknowledgments I would like to thank Professor Xiao-Gang He and the Physics Department of the National Taiwan University for their hospitality during my stay at NTU. I am grateful to Professors Xiao-Gang He and Pauchy W-Y. Hwang for their invitation to present this work at the CosPA 2006 meeting. This work was supported in part by the IISN and the Belgian science policy office (IAP V/27). References 1. C. A. Mead, Phys. Rev. 135, B849 (1964). 2. X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004); Int. J. Mod. Phys. D 14, 2195 (2005); X. Calmet, arXiv:hep-th/0701073. 3. K. S. Thorne, Nonspherical gravitational collapse: A short review, in J. R . Klauder, Magic Without Magic, San Francisco 1972, 231–258. http://arxiv.org/abs/hep-th/0701073 November 27, 2018 16:18 WSPC/INSTRUCTION FILE procCOSPAcal- 8 Xavier Calmet 4. J. L. Anderson and D. Finkelstein, Am. J. Phys. 39, 901 (1971), see also D. R. Finkel- stein, A. A. Galiautdinov and J. E. Baugh, J. Math. Phys. 42, 340 (2001). 5. X. Calmet, Phys. Rev. D 71, 085012 (2005). 6. X. Calmet and A. Kobakhidze, Phys. Rev. D 72, 045010 (2005); Phys. Rev. D 74, 047702 (2006); X. Calmet, arXiv:hep-th/0510165, to appear in Europhys.Lett. 7. A. Einstein, Siz. Preuss. Acad. Scis., (1919); “Do Gravitational Fields Play an essential Role in the Structure of Elementary Particle of Matter,” in The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity by A. Einstein et al (Dover, New York, 1923), pp. 191–198, English translation. 8. J. J. van der Bij, H. van Dam and Y. J. Ng, Physica 116A, 307 (1982). 9. M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984). 10. F. Wilczek, Phys. Rept. 104, 143 (1984). 11. W. Buchmuller and N. Dragon, Phys. Lett. B 207, 292 (1988). 12. M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195 (1989). 13. W. G. Unruh, Phys. Rev. D 40, 1048 (1989). 14. Y. J. Ng and H. van Dam, Int. J. Mod. Phys. D 10, 49 (2001). 15. Y. J. Ng, Mod. Phys. Lett. A 18, 1073 (2003). 16. M. Ahmed, S. Dodelson, P. B. Greene and R. Sorkin, Phys. Rev. D 69, 103523 (2004). 17. R .D. Sorkin, in Relativity and Gravitation: Classical and Quantum (Proceedings of the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), edited by J.C. D’Olivo, E. Nahmad-Achar, M. Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche, (World Scientific, Singapore, 1991, pp. 150-173. 18. R. D. Sorkin, Int. J. Th. Phys. 36: 2759–2781 (1997). 19. W. Chen and Y. S. Wu, Phys. Rev. D 41, 695 (1990) [Erratum-ibid. D 45, 4728 (1992)]. 20. A. Einstein, “The Foundation Of The General Theory Of Relativity,” Annalen Phys. 49, 769 (1916). 21. E. Baum, Phys. Lett. B 133, 185 (1983). 22. S. W. Hawking, Phys. Lett. B 134, 403 (1984). 23. S. D. H. Hsu, Phys. Lett. B 594, 13 (2004). 24. M. Li, Phys. Lett. B 603, 1 (2004). 25. X. Calmet and H. Fritzsch, Eur. Phys. J. C 24, 639 (2002); Phys. Lett. B 540, 173 (2002); arXiv:hep-ph/0211421; Europhys. Lett. 76, 1064 (2006). http://arxiv.org/abs/hep-th/0510165 http://arxiv.org/abs/hep-ph/0211421

0704.1361

A Dynamic Algorithm for Blind Separation of Convolutive Sound Mixtures

A Dynamic Algorithm for Blind Separation of Convolutive Sound Mixtures Jie Liu , Jack Xin∗, and Yingyong Qi † Abstract We study an efficient dynamic blind source separation algorithm of convolutive sound mixtures based on updating statistical information in the frequency domain, and minimizing the support of time domain demixing filters by a weighted least square method. The permutation and scaling indeterminacies of separation, and concatenations of signals in adjacent time frames are resolved with optimization of l1× l∞ norm on cross-correlation coefficients at multiple time lags. The algorithm is a direct method without iterations, and is adaptive to the environment. Computations on recorded and synthetic mixtures of speech and music signals show excellent performance. Keywords: Convolutive Mixtures, Indeterminacies, Dynamic Statistics Update, Optimization, Blind Separation. ∗Department of Mathematics, UC Irvine, Irvine, CA 92697, USA. Qualcomm Inc, 5775 Morehouse Drive, San Diego, CA 92121, USA. http://arxiv.org/abs/0704.1361v1 1 Introduction Blind source separation (BSS) methods aim to extract the original source sig- nals from their mixtures based on the statistical independence of the source signals without knowledge of the mixing environment. The approach has been very successful for instantaneous mixtures. However, realistic sound signals are often mixed through a media channel, so the received sound mixtures are linear convolutions of the unknown sources and the channel transmission functions. In simple terms, the observed signals are unknown weighted sums of the signals and its delays. Separating convolutive mixtures is a challenging problem especially in realistic settings. In this paper, we study a dynamic BSS method using both frequency and time domain information of sound signals in addition to the independence assumption on source signals. First, the convolutive mixture in the time domain is decomposed into instantaneous mixtures in the frequency domain by the fast Fourier transform (FFT). At each frequency, the joint approx- imate diagonalization of eigen-matrices (JADE) method is applied. The JADE method collects second and fourth order statistics from segments of sound signals to form a set of matrices for joint orthogonal diagonalization, which leads to an estimate of de-mixing matrix and independent sources. However, there remain extra degrees of freedom: permutation and scaling of estimated sources at each frequency. A proper choice of these parameters is critical for the separation quality. Moreover, the large number of samples of the statistical approach can cause delays in processing. These issues are to be addressed by utilizing dynamical information of signals in an optimiza- tion framework. We propose to dynamically update statistics with newly received signal frames, then use such statistics to determine permutation in the frequency domain by optimizing an l1 × l∞ norm of channel to channel cross-correlation coefficients with multiple time lags. Though cross channel correlation functions and related similarity measure were proposed previ- ously to fix permutation [13], they allow cancellations and may not measure similarity as accurately and reliably as the norm (metric) we introduced here. The freedom in scaling is fixed by minimizing the support of the esti- mated de-mixing matrix elements in the time domain. An efficient weighted least square method is formulated to achieve this purpose directly in con- trast to iterative method in [17]. The resulting dynamic BSS algorithm is both direct and adapted to the acoustic environment. Encouraging results on satisfactory separation of recorded sound mixtures are reported. The paper is organized as follows. In section 2, a review is presented on frequency domain approach, cumulants and joint diagonalization prob- lems, and indeterminacies. Then the proposed dynamic method is presented, where objective functions of optimization, statistics update and efficient computations are addressed. Numerical results are shown and analyzed to demonstrate the capability of the algorithm to separate speech and music mixtures in both real room and synthetic environments. Conclusions are in section 3. 2 Convolutive Mixture and BSS Let a real discrete time signal be s(k) = [s1(k), s2(k), · · · , sn(k)], k a discrete time index, such that the components si(k) (i = 1, 2, · · · , n), are zero-mean and mutually independent random processes. For simplicity, the processing will divide s into partially overlapping frames of length T each. The in- dependent components are transmitted and mixed to give the observations xi(k): xi(k) = aij(p) sj(k − p), i = 1, 2, · · · , n; (2.1) where aij(p) denote mixing filter coefficients, the p-th element of the P - point impulse response from source i to receiver j. The mixture in (2.1) is convolutive, and an additive Gaussian noise may be added. The sound signals we are interested in are speech and music, both are non-Gaussian [1]. We shall consider the case of equal number of receivers and sources, especially n = 2. An efficient way to decompose the nonlocal equation (2.1) into local ones is by a T-point discrete Fourier transform (DFT) [2], Xj(ω, t) = τ=0 xj(t+ τ) e−2π Jωτ , where J = −1, ω is a frequency index, ω = 0, 1/T, · · · , (T − 1)/T , t the frame index. Suppose T > P , and extend aij(p) to all p ∈ [0, T − 1] by zero padding. Let Hij(ω) denote the matrix function obtained by T-point DFT of aij(p) in p, Sj(ω, t) the T-point DFT of sj(k) in the t-th frame. If P ≪ T , then to a good approximation [17]: X(ω, t) ≈ H(ω)S(ω, t), (2.2) where X = [X1, · · · ,Xn]Tr, S = [s1, · · · , sn]Tr, Tr is short for transpose. The components of S remains independent of each other, the problem is converted to a blind separation of instantaneous mixture in (2.2). Note that P is on the order of 40 to 50 typically, while T is 256 or 512, so the assumption P ≪ T is reasonable. 2.1 Instantaneous Mixture and JADE Let us briefly review an efficient and accurate method, so called joint approx- imate diagonalization eigen-matrices (JADE) [6] for BSS of instanteneous mixture. There are many other approaches in the literature [4], e.g. info- max method [1] which is iterative and based on maximizing some informa- tion theoretical function. JADE is essentially a direct method for reducing covariance. We shall think of S as a random function of t, and suppress ω de- pendence. First assume that by proper scaling E[|Sj(t)|2] = 1, j = 1, · · · , n. It follows from independence of sources that (′ conjugate transpose): E[S(t)S(t) ] = In, RX ≡ E[X(t)X(t) ] = HH , (2.3) the latter identity is a factorization of the Hermitian covariance matrix of the mixture. However, there is non-uniqueness in the ordering and phases of columns of H. Suppose that (1) the mixing matrix H is full rank; (2) the Sj(t)’s are independent at any t; (3) the process S(t) is stationary. Let W be a matrix such that In = WRXW = WHH ′W ′, W is called a whitening matrix. Then WH is an orthogonal matrix, denoted by U . Multiplying W from the left onto (2.2), one finds that: Z(t) ≡ WX(t) = US(t). (2.4) The 4th order statistics are needed to determine U . The 4th-order cumulant of four mean zero random variables is: Cum[a, b, c, d] = E(abcd)−E(ab)E(cd)−E(ac)E(bd)−E(ad)E(bc), (2.5) which is zero if a, b, c, d split into two mutually independent groups. For source vector S, Cum[Si, Sj, Sk, Sl] = kurti δijkl, kurti = Cum[Si, Si, Si, Si] is the kurtosis. If kurti 6= 0, the i-th source is called kurtic. Kurtosis is zero for a mean zero Gaussian random variable. The last assumption of JADE is that (4) there is at most one non-kurtic source. Define cumulant matrix set QZ(M) from Z in (2.4) as the linear span of the Hermitian matrices Q = (qij) satisfying (∗ complex conjugate): qij = k,l=1 Cum(Zi, Z j , Zk, Z l )mlk, 1 ≤ i, j ≤ n, (2.6) where matrix M = (mij) = ele k, el being the unit vector with zero com- ponents except the l-th component equal to one. Equations (2.4) and (2.6) imply that (up is the p-th column of U): (kurtp u pMup)upu p, ∀ M, (2.7) or Q = UDU , D = diag(kurt1u 1Mu1, · · · , kurtnu nMun). Hence, U is the joint diagonalizer of the matrix set QZ(M). Once U is so determined, the mixing matrix H = W−1U . It can be shown [6] using identity (2.7) that the joint diagonalizer of QZ(M) is equal to U up to permutation and phase, or up to a matrix multiplier P where P has exactly one unit modulus entry in each row and column. Such a joint diagonalizer is called essentially equal to The algorithm of finding the joint diagonalizer is a generalization of Jacobi method or Givens rotation method [9]. As the cumulant matri- ces are estimated in practice, exact joint diagonalizer may not exist, in- stead, an approximate joint diagonalizer, an orthogonal matrix V , is sought to maximize the quantity: C(V,B) = r=1 |diag(V ′Br V )|2, where B = {B1, B2, · · · , Bn2} is a set of basis (or eigen) matrices of QZ(M), |diag(A)|2 is the sum of squares of diagonals of a matrix A. Maximizing C(V,B) is same as minimizing off diagonal entries, which can be achieved in a finite number of steps of Givens rotations. The costs of joint diagonalization is roughly n2 times that of diagonalizing a single Hermitian matrix. Though stationarity is assumed for the theoretical analysis above, JADE turns out to be quite robust even when stationarity is not exactly satisfied for signals such as speech or music. 2.2 Dynamic Method of Separating Convolutive Mixture For each frequency ω, equation (2.2) is a BSS problem of instantaneous mixtures. The speech or music signals in reality are stationary over short time scales and nonstationary over longer time scales, which depend on the production details. For speech signals, human voice is stationary for a few 10 ms, and becomes non-stationary for a time scale above 100 ms due to envelope modulations [8, 13]. The short time stationarity permits FFT to generate meaningful spectra in equation (2.2) within each frame. For a sampling frequency of 16,000 Hertz, each frame of 512 points lasts 32 ms. The mixing matrix H may depend on t over longer time scales, denoted by H = H(ω, t), unless the acoustic environment does not change as in most synthetic mixing. A demixing method with potential real time application should be able to capture the dynamic variation of mixing. Our approach consists of four steps. Step I is to find an initialization for H(ω, t). After receiving the initial nT frames of mixtures, compute their FFT and obtain X(ω, t), t = 1, 2, · · · , nT , to collect nT samples at each discrete frequency. For each ω, perform JADE, and estimate the mixing matrix denoted by H0(ω). To ensure a good statistical estimate, nT is on the order of 80 to 100, and may be properly reduced later. Step I gives separated components of signals over all frequencies. How- ever, such JADE output has inderterminacies in amplitude, order and phase. This benign problem for instantaneous mixtures becomes a major issue when one needs to assemble the separated individual components. For example, the permutation mismatches across frequencies can degrade the quality of separation seriously. Step II is to use nonstationarity of signals to sort out a consistent order of separated signals in the frequency domain. Such a method for batch processing was proposed in [13]. A separation method requiring the entire length of the signal is called batch processing. The sorting al- gorithm of [13] proceeds as follows. (1) Estimate the envelope variation by a moving average over a number of frames (beyond stationarity time scale) for each separated frequency component. The envelope is denoted by Env(ω, t, i), where i is the index of separated components. (2) Com- pute a similarity measure equal to the sum of correlations of the envelopes of the separated components at each frequency. The similarity measure is sim(ω) = i 6=j ρ(Env(ω, t, i),Env(ω, t, j)), where ρ(·, ·) is the normal- ized correlation coefficients (see (2.9)) involving time average over the en- tire signal length to approximate the ensemble average so the t dependence drops out. (3) Let ω1 be the one with lowest similarity value where sep- aration is the best. The ω1 serves as a reference point for sorting. (4) At other frequencies ωk (k = 2, 3, · · ·), find a permutation σ to maxi- i=1 ρ(Env(ωk, t, σ(i)), j=1 Envs(ωj, t, i)), among all permutations of 1, 2, · · · , n. Here Envs denotes the sorted envelopes in previous frequen- cies. (5) Permute the order of separated components at the k-th frequency bin according to σ in step (4), and define Envs(ωk, t, i). Repeat (4) and (5) until k = T . We shall modify the above sorting method in three aspects. The first is to use segments of signal instead of the entire signal to compute statistics (correlations) to minimize delay in processing. The second is to use correla- tion coefficients of separated signals at un-equal times or multiple time lags in step (2) to better characterize the degree of separation. Moreover, we no- tice that the similarity measure of [13] as seen above is a sum of correlation coefficients of potentially both signs, and so can be nearly zero due to can- cellations even though each term in the sum is not small in absolute value. We introduce an l1 × l∞ norm below to characterize more accurately chan- nel similarity by taking sum of absolute values of correlation coefficients and maximum of time lags. The third is to simplify the maximization problem on σ to avoid comparing correlations with summed envelopes at all previ- ous frequencies. We also do not use envelopes of signals inside correlation functions. The reason is that the smoothing nature of envelope operation reduces the amount of oscillations in the signals and may yield correlation values less accurate for capturing the degree of independence. Specifically, let ŝi(ω, t) = ai(ω, t) e jφi(ω,t) be the i-th separated signal at frequency ω, where ai(ω, t) = |ŝi(ω, t)|, φi the phase functions, t the frame index. The correlation function of two time dependent signals over M frames is: cov(a(ω, t), b(ω′, t)) = M−1 a(ω, t)b∗(ω′, t)−M−2 a(ω, t) b∗(ω′, t), (2.8) and the (normalized) correlation coefficient is: ρ(a(ω, t), b(ω′, t)) = cov(a(ω, t), b(ω′, t)) cov(a(ω, t), a(ω, t)) cov(b(ω′, t), b(ω′, t)) . (2.9) From speech production viewpoint, frequency components of a speech signal do not change drastically in time, instead are similarly affected by the motion of the speaker’s vocal chords. The correlation coefficient is a natural tool for estimating coherence of frequency components of a speech signal. A similar argument may be applied to music signals as they are produced from cavities of instruments. Now with M = nT in (2.8), define C(ω) = i 6=j k∈{−K0,...,K0} |ρ(|ŝi(ω, t)|, |ŝj(ω, t− k)|)|, for ω ∈ [ωL, ωU ] (2.10) with some positive integer K0. Find ω1 between ωL and ωU to minimize C(ω). With ω1 as reference, at any other ω, find the permutation σ to maximize: σ = argmax k∈{−K0,...,K0} |ρ(|ŝi(ω1, t)|, |ŝσ(i)(ω, t− k)|)|. (2.11) Notice that the objective functions in (2.10)-(2.11) are exactly the l1 × l∞ norms over the indices i(j) and k. Multiple time lag index k is to accomodate the translational invariance of sound quality to the ear. Maximizing over k helps to capture the correlation of the channels, and sum of i (j) reflects the total coherence of a vector signal. Step III fixes the scaling and phase indeterminacies in ŝ(ω, t). Each row of the de-mixing matrix H−10 (ω) may be multiplied by a complex number λi(ω) (i = 1, 2 · · · , n) before inverse FFT (ifft) to reconstruct demixing ma- trix h(0)(τ) in the time domain. The idea is to minimize the support of each row of the inverse FFT by a weighted least square method. In other words, we shall select λi’s so that the entries of ifft(H 0 )(τ) ≡ h(0)(τ) are real and nearly zero if τ ≥ Q for some Q < T , Q as small as possible, T being the length of FFT. Smaller Q improves the local approximation, or accuracy of equation (2.2). To be more specific, using H−10,i (ω) to denote the i-th row vector of H−10 (ω), we can explicitly write the equation to shorten the support of inverse FFT: ifft(λi(ω)H 0,i (ω))(τ) = 0 (2.12) in terms of the real and imaginary parts of λi(ω) for ω = 0, 1/T, ..., (T−1)/T . Those real and imaginary parts are the variables and the equations are linear. Now, we let τ run from q to T − 1. If we want small support, q should be small, then there are more equations than unkowns. So we multiply a weight to each equation and minimize in the least square sense. Equation (2.12) for larger τ is multiplied by a larger weight in the hope that the value of the left hand side of (2.12) will be closer to zero during the least square process. If we choose the weighting function to be the exponential function βτ for some β > 1, then the above process can be mathematically written as [λi(0), ..., λi((T − 1)/T )] = argmin |βτ ifft(λi(ω)H−10,i (ω))(τ)| 2 (2.13) where H−10,i (ω) is the i-th row vector of H 0 (ω). A few comments are in order. First, since the mixing matrix H0(ω) is the FFT of a real matrix, we impose that H0(ω) = H0(1−ω)∗. So, supposing T is even, we only need to apply JADE to obtain H0(ω) for ω = 0, 1/T, ..., 1/2; H0(0) and H0(1/2) will automatically be real. When fixing the freedom of scaling in each ω, we choose λ(0) and λ(1/2) real, and λ(ω) = λ(1−ω)∗ for other ω. Second, to fix the overall scaling and render the solution nontrivial, we set λ(0) = 1. Third, the weighted least square problem (2.13) can be solved by a direct method or matrix inversion (chapter 6 in [9]). Note that when n = 2, among the 2(T − q) equations from (2.12) with τ = q, ..., T − 1, there are T − 1 variables including λi(1/2), the real and imaginary parts of λi(ω) for ω = 1/T, ..., 1/2−1/T . So, we can make roughly half of h i (τ) ≈ 0, the best one can achieve in general. Separated signals, denoted by s̃(0)(t), are then produced, for t ∈ [0, nT ], t the frame index. The last step IV is to update h(0)(τ) when δnT ≪ nT many new frames of mixtures arrive. The steps I to III are repeated using frames from δnT +1 to δnT + nT , to generate a new time domain demixing matrix h (1)(τ), τ ∈ [0, T − 1], and separated signal s̃(1)(τ), τ ∈ [T (nT −∆nT ) + 1, T (nT + δnT )] with T the size of one frame. We use τ here instead of t because in the most part of the paper, t is the frame index. Now, s̃(1)(τ) and s(0)(τ) share a common interval of size T∆nT . On this common interval, s̃ (1)(τ) and s(0)(τ) will be the same if we are doing a perfect job and if the ordering of s̃(1) is consistent with that of s(0). In order to determine the ordering of s̃(1)(τ), we compute ρ i (τ), s j (τ − k) on this common interval with different k and i, j = 1, ..., n. Then we determine the permutation σ of the components of s̃(1)(t) by minimization: σ = argmax k∈{−K1,...,K1} i (τ), s̃ (τ − k) ∣ (2.14) with some constant K1. After doing the necessary permutation of s̃ (1), the separated signals are then extended to the extra frames δnT +nT by concate- nating the newly separated δnT many frames of s̃ (1) with those of s̃(0). The continuity of concatenation is maintained by requiring that maxτ |h(k)ii (τ)|’s (i = 1, 2, · · · , n) are invariant in k, where k = 1, 2, · · ·, labels the updated filter matrix in time. The procedure repeats with the next arrival of mixture data, and is a direct method incorporating dynamic information. Because sorting order depends only on the relative values of channel correla- tions, we observed in practice that the maxk∈{−K.,...,K.} in equations (2.10), (2.11), (2.14) may be replaced by k=−K. , with a different choice of K. value. The maxk∈{−K.,...,K.} is a more accurate characterization however. 2.3 Adaptive Estimation and Cost Reductions Cumulants and moments are symmetric functions in their arguments [15]. For example when n = 2, there are 16 joint fourth order cumulants from (2.5), however, only six of them need to be computed, the others follow from symmetry. Specifically, among the 16 cumulants: Q(1) = Cum(y1, y 1 , y 1 , y1), Q(2) = Cum(y1, y 1 , y 1, y2) Q(3) = Cum(y1, y 1 , y 2 , y1), Q(4) = Cum(y1, y 1 , y 2, y2) Q(5) = Cum(y1, y 2 , y 1 , y1), Q(6) = Cum(y1, y 2 , y 1, y2) Q(7) = Cum(y1, y 2 , y 2 , y1), Q(8) = Cum(y1, y 2 , y 2, y2) Q(9) = Cum(y2, y 1 , y 1 , y1), Q(10) = Cum(y2, y 1 , y 1, y2) Q(11) = Cum(y2, y 1 , y 2 , y1), Q(12) = Cum(y2, y 1 , y 2, y2) Q(13) = Cum(y2, y 2 , y 1 , y1), Q(14) = Cum(y2, y 2 , y 1, y2) Q(15) = Cum(y2, y 2 , y 2 , y1), Q(16) = Cum(y2, y 2 , y 2, y2) we have the relations: Q(2) = Q(3)∗ = Q(5)∗ = Q(9), Q(4) = Q(6) = Q(11) = Q(13), Q(7) = Q(10)∗, Q(8) = Q(15) = Q(12)∗ = Q(14)∗, where ∗ is complex conjugate. For N samples, we only need to compute the following six 1×N vectors Y1 = (y 1, ..., y 1 ), Y2 = (y 2, ..., y Y3 = (y 2, ..., y 2 ), Y4 = (y 1 , ..., y Y5 = (y 2 , ..., y 2 ), Y6 = (y 2 , ..., y then all the 4th order and 2nd order statistical quantities can be recon- structed. For example, Q(1) = Y4·Y Tr4 − (2 sum(Y4) sum(Y4) + sum(Y1) (sum(Y1) ∗)) (2.15) where sum(Yi) is the summation of the N components of Yi. As formula (2.5) suggests, cumulants are updated through moments when δnT early samples are replaced by the same number of new samples. As δnT is much less than the total number of terms nT in the empirical estimator of expectation, the adjustment costs 2δnT flops for each second moments and 6δnT flops for each joint fourth order moment. The con- tributions of the early samples are subtracted from the second and fourth moments, then the contributions of the new samples are added. The cu- mulant update approach is similar to cumulant tracking method of moving targets ([12] and references therein). Due to dynamical cumulants update, the prewhitening step at each fre- quency is performed after cumulants are computed from X(ω). This is dif- ferent from JADE [6] where the prewhitening occurs before computing the commulants. This way, it is more convenient to make use of the previous cumulant information and updated X(ω). Afterward, we use the multilin- earity of the cummulants to transform them back to the commulants of the prewhitened X(ω), before joint diagonalization. It is desirable to decrease nT to lower the number of samples for cu- mulants estimation. However, this tends to increase the variance in the estimated cumulants, and render estimation less stable in time. Numeri- cal experiments indicated that with nT as low as 40, the separation using overlapping frames is still reliable with reasonable quality. It is known [8] that the identity of a speaker is carried by pitch (per- ception of the fundamental frequency in speech production) which varies in the low frequency range of a few hundred Hertz. We found that instead of searching among all frequencies for the reference frequency ω1 in step II, it is often sufficient to search in the low frequency range. The smaller searching range alleviates the workload in sorting and permutation correcting. This is similar to a feature oriented method, see [16, 18, 3] among others. 2.4 Experimental Results The proposed algorithm with adaptivity and cost reduction considerations was implemented in Matlab. The original code of JADE by J.-F. Cardoso is obtained from a open source (http://web.media.mit.edu/∼paris/ ) main- tained by P. Smaragdis. Separation results with both dynamic and batch processing of three different types of mixtures are reported here: (1) real room recorded data; (2) synthetic mixture of speech and music; (3) synthetic mixture of speech and speech noise. They will be called case (1), (2) and (3) in the following discussion. The values of the parameters used in the three cases are listed in Table 1. In the table, ”nT (dyn.)” is the initial value of nT in dynamic processing and ”nT (bat.)” is the nT in batch processing. Other than nT , dynamic and batch process share the same parameters. The frame size is T , ”overlap” is the overlapping percentage between two successive frames, δnT and ∆nT are as in step IV, K0 and K1 are from (2.11) and (2.14), β is in (2.13), and q is the lower limit of τ in (2.12). Note that the values of ωL and ωU from (2.10) are not listed in the table. In our computation, we use the following two choices (A) ωL = 0, ωU = 1/2. (B) ωL = ωU = 4/T , namely fixing reference frequency ω1 = 4/T . http://web.media.mit.edu/~paris/ For the three cases reported in this paper, both choices work and generate very similar results. As a consequence, we will only plot the results of the first choice. The first choice is more general while the second is motivated by the pitch range of speech signal and is computationally more favorable. However, we do not know precisely the robustness of the latter. case T overlap nT δnT ∆nT K0 K1 β q nT (dyn.) (bat.) (1) 512 0% 100 20 30 4 10 1.04 2 200 (2) 256 50% 100 20 40 15 20 1.04 2 160 (3) 256 50% 100 20 40 10 20 1.04 2 160 Table 1: Parameters used in both dynamic and batch processing. For a quantitative measure of separation in all three cases, we compute the maximal correlation coefficient over multiple time lags: ρ̄(a, b) = max k∈{−K2,...,K2} |ρ(a(τ), b(τ + k))| (2.16) with ρ defined in (2.9). The ρ̄ is computed for the mixtures, the sources and the separated signals for both batch and dynamic processing. An exception is the lack of sources in case (1). We choose K2 = 20 in all the computations. The results are listed in Table 2 which shows that the ρ̄ values of the mixtures are much larger than those of the dynamically separated signals, which are on the same order as the ρ̄ values of the batch separated signals. In the synthetic cases (2) and (3), the ρ̄ values of the batch separated signals are on the same order of the ρ̄ values of the source signals or 10−2. In cases (2) and (3), we use the ratio ρ̄(x, s1)/ρ̄(x, s2) to measure the relative closeness of a signal x to source signals s1 and s2. Table 3 lists these ratios for x being the separated signals by dynamic and batch methods with A and B denoting the two ways of setting the reference frequency ω1. The outcomes are similar no matter x = s̃1 or x = s̃2 (first or second separated signal) in either dynamic or batch cases and either way of selecting the reference frequency ω1. In case (1), the recorded data [13] consists of 2 mixtures of a piece of mu- sic (source 1) and a digit (one to ten) counting sentence (source 2) recorded in a normal office size room. The sampling frequency is 16 kHz, and 100 k data points are shown in Fig. 1. The signals last a little over 6 seconds. The result of dynamic BSS algorithm is shown in Fig 2. As a comparison, we show in Fig. 3 result of batch processing of steps I to III of the algorithm ρ̄(·, ·) of 3 cases mixture dyn. separation bat. separation sources (1)-A 0.8230 0.0269 0.0160 N/A (1)-B 0.8230 0.0225 0.0159 N/A (2)-A 0.6240 0.0503 0.0673 0.0201 (2)-B 0.6240 0.0182 0.0600 0.0201 (3)-A 0.4613 0.0351 0.0378 0.0243 (3)-B 0.4613 0.0267 0.0677 0.0243 Table 2: Values of the correlation coefficient ρ̄(x, y), (x, y) being either the two mixtures or the two sources or the two separated signals by dynamic and batch methods. The A and B in the first column denote the two different ways of selecting the reference frequency ω1. ρ̄(x, s1)/ρ̄(x, s2) case(2) case(3) x= dyn. s̃1(A) 4.5899 4.5096 x= dyn. s̃2(A) 0.1086 0.2852 x= dyn. s̃1(B) 5.3083 5.8411 x= dyn. s̃2(B) 0.0494 0.2799 x= bat. s̃1(A) 15.0912 1.4632 x= bat. s̃2(A) 0.0760 0.1665 x= bat. s̃1(B) 6.2227 25.8122 x= bat. s̃2(B) 0.0636 0.1719 Table 3: Ratios of ρ̄(x, s1) and ρ̄(x, s2), x being a separated signal on the first column by dynamic or batch method, s1 and s2 are source signals. The ratio measures the relative closeness of x to s1 and s2. If the ratio is larger (smaller) than one, x is closer to s1 (s2). The A and B in the first column denote the two different ways of selecting ω1. with nT = 200. The batch processing gives a clear separation upon listen- ing to the separated signals. The dynamic processing is comparable. The filter coefficients in the time domain hij(τ) at the last update of dynamic processing are shown in Fig. 4. Due to weighted least square optimization in step III, they are localized and oscillatory with support length Q close to half of the FFT size T . For cases (2) and (3), we show the envelopes of the absolute values of the mixtures or the separated signals. The signal envelope was computed using the standard procedure of amplitude demodulation, i.e., lowpass filtering the rectified signal. The filter was an FIR filter with 400 taps and the cutoff frequency was 100 Hz. Signal envelopes help to visualize and compare source and processed signals. We have normalized all the envelopes so that the maximum height is 1. The values of aij in (2.1), which are used to synthetically generate the mixtures, are shown in Fig. 5 (see [19, (8)]). Fig. 6 and Fig. 7 show the mixtures and separated signals of case (2). Fig. 8 and Fig. 9 show the mixtures and separated signals of case (3). In view of these plots, Table 2 and Table 3, separation is quite satisfactory, which is also confirmed by hearing the separated signals. The processing time in MATLAB on a laptop can be a factor of 5 to 8 above the real time signal duration, however, the time is expected to be closer to real time with the computation is executed by Fortran or C directly or with additional cost reduction techniques. A breakdown of time consumption in the algorithm shows that 40% of the processing is spent on computing cumulants, 30 % on sorting in frequency and time domains, 15% on fixing scaling functions, 3% on joint diagonalization, the rest on other operations such as computing lower order statistics, FFT, IFFT etc. 3 Conclusions A dynamic blind source separation algorithm is proposed to track the time dependence of signal statistics and to be adaptive to the potentially time varying environment. Besides an efficient updating of cumulants, the method made precise the procedure of sorting permutation indeterminacy in the fre- quency domain by optimizing a metric (the l1 × l∞ norm) on multiple time lagged channel correlation coefficients. A direct and efficient weighted least square approach is introduced to compactify the support of demixing fil- ter to improve the accuracy of frequency domain localization of convolutive mixtures. Experimental results show robust and satisfactory separation of real recorded data and synthetic mixtures. An interesting line of future work will be concerned with various strategies to reduce computational costs. 4 Acknowledgements The work was partially supported by NSF grants ITR-0219004, DMS-0549215, NIH grant 2R44DC006734; the CORCLR (Academic Senate Council on Re- search, Computing and Library Resources) faculty research grant MI-2006- 07-6, and a Pilot award of the Center for Hearing Research at UC Irvine. References [1] A. Bell, T. Sejnowski, An Information-Maximization Approach to Blind Separation and Blind Deconvolution, Neural Computation, 7(1995), pp 1129–1159. [2] P. Brémaud, “Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis”, Springer-Verlag, 2002. [3] T. Chan, C. Wong, Total variation blind deconvolution, IEEE Transac- tions Image Processing, 7(1998), pp 370-375. [4] S. Choi, A. Cichocki, H. Park, S. Lee, Blind Source Separation and Independent Component Analysis: A Review, Neural Information Pro- cessing -Letters and Reviews, Vol. 6, No. 1, 2005, pp 1-57. [5] J-F. Cardoso, Blind signal separation: statistical principles, Proceed- ings of IEEE, V. 9, No. 10, pp 2009-2025, 1998. [6] J-F. Cardoso, A. Souloumiac, Blind Beamforming for Non-Gaussian Signals, IEEE Proceedings-F, vol. 140, no. 6, pp 362-370, 1993. [7] J-F. Cardoso, A. Souloumiac, Jacobi angles for simultaneous diagonal- ization, SIAM J. Matrix Analysis, vol. 17, pp 161-164, 1996. [8] L. Deng, D. O’Shaughnessy, “Speech Processing — A Dynamic and Optimization-Oriented Approach”, Marcel Dekker Inc., New York, 626 pages, 2003. [9] G. Golub, C. Van Loan, “Matrix Computations”, John Hopkins Uni- versity Press, 1983. [10] S. Greenberg, W. Ainsworth, A. Popper, R. Fay, Speech Processing in the Auditory Systems, Springer Handbook of Auditory Research, Chapters 7 and 8, Springer, 2004. [11] M. Kawamoto, K. Matsuoka, N. Ohnishi, A method of blind separa- tion for convolved non-stationary signals, Neurocomputing 22(1998), pp 157-171. [12] T. Liu, J. Mendel, Cumulant-based subspace tracking, Signal Processing, 76(1999), pp 237-252. [13] N. Murata, S. Ikeda, A. Ziehe, An approach to blind separation based on temporal structure of speech signals, Neurocomputing 41(2001), pp 1-24. [14] A. Nandi, eds, “Blind Estimation Using Higher-Order Statistics”, Kluwer Academic Publishers, 1999. [15] C. Nikias, A. Petropulu, “Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework”, Prentice-Hall Signal Processing Series, ed. A. Oppenheim, 1993. [16] S. Osher and L. Rudin, Feature-Oriented Image Enhancement Using Shock Filters, SIAM J. Numer. Analysis, Vol. 27, No. 4, pp 919-940, 1990. [17] L. Parra, C. Spence, Convolutive Blind Separation of Non-Stationary Sources, IEEE Transactions on Speech and Audio Processing, Vol. 8 (2000), No. 5, pp 320 –327. [18] Y. Qi, J. Xin, A Perception and PDE Based Nonlinear Transformation for Processing Spoken Words, Physica D 149 (2001),143-160. [19] K. Torkkola, Blind separation of convolved sources based on information maximization, Neural Networks Signal Processing, VI(1996), pp 423- Figure Captions Fig 1: Case (1), two recorded signals in a real room where a speaker was counting ten digits with music playing in the background. Fig 2: Case (1) with choice A, separated digit counting sentence (bottom) and background music (top) by the proposed dynamic method. Choice B gives similar results. Fig 3: Case (1) with choice A, separated digit counting sentence (bottom) and background music (top) by batch processing using the proposed steps I to III. Choice B gives similar results. Fig 4: Case (1) with choice A, the localized and oscillatory filter coefficients in the time domain at the last frame of dynamic processing. Choice B gives similar results. Fig 5: The weights aij used in generating synthetic mixtures of cases (2) and (3), as proposed in [19]. Fig 6: Case (2), the synthetic mixtures are generated by a female voice and a piece of instrumental music. Fig 7: Case (2) with choice A, the envelopes of the separated signals from mixtures whose envelopes are in Fig. (6). The small amplitude portion of the music is well recovered. Choice B gives similar results. Fig 8: Case (3), the synthetic mixtures of a female voice and a speech noise with signal to noise ratio equal to −3.8206 dB. The x1 plot shows a speech in a strong noise, the valley structures in the speech signal are filled by noise. Fig 9: Case (3) with choice A, the envelopes of the separated signals, noise (top) and speech (bottom). The envelopes of the two mixtures are in Fig. 8. The strongly noisy x1 in Fig. 8 has been cleaned, the valleys in the envelope re-appeared. Choice B gives an even better result. 0 1 2 3 4 5 6 7 8 9 10 Recorded mixture signals in a real room 0 1 2 3 4 5 6 7 8 9 10 Figure 1: 0 1 2 3 4 5 6 7 8 9 10 Separated source of background music by proposed dynamic method 0 1 2 3 4 5 6 7 8 9 10 Separated source of counting digits (1 to 10) by proposed dynamic method Figure 2: 0 1 2 3 4 5 6 7 8 9 10 Separated source of background music by the batch method 0 1 2 3 4 5 6 7 8 9 10 Separated source of counting digits (1 to 10) by the batch method Figure 3: 0 100 200 300 400 500 τ = 1:512 0 100 200 300 400 500 τ = 1:512 0 100 200 300 400 500 τ = 1:512 0 100 200 300 400 500 τ = 1:512 Figure 4: 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Figure 5: 0.5 1 1.5 2 2.5 3 3.5 4 envelopes of the two components of the sound mixture Figure 6: 0.5 1 1.5 2 2.5 3 3.5 4 envelope dynamic batch exact 0.5 1 1.5 2 2.5 3 3.5 4 dynamic batch exact Figure 7: 0.5 1 1.5 2 2.5 3 3.5 4 envelopes of the two components of the sound mixture Figure 8: 0 0.5 1 1.5 2 2.5 3 3.5 4 envelope dynamic batch exact 0 0.5 1 1.5 2 2.5 3 3.5 4 dynamic batch exact Figure 9: Introduction Convolutive Mixture and BSS Instantaneous Mixture and JADE Dynamic Method of Separating Convolutive Mixture Adaptive Estimation and Cost Reductions Experimental Results Conclusions Acknowledgements

0704.1362

Fast recursive filters for simulating nonlinear dynamic systems

Fast recursive filters for simulating nonlinear dynamic systems J. H. van Hateren [email protected] Netherlands Institute for Neuroscience, Royal Netherlands Academy of Arts and Sciences, Amsterdam, and Institute for Mathematics and Computing Science, University of Groningen, The Netherlands Abstract A fast and accurate computational scheme for simulating nonlinear dynamic systems is presented. The scheme assumes that the system can be represented by a combination of components of only two different types: first-order low-pass filters and static nonlinearities. The parameters of these filters and nonlinearities may depend on system variables, and the topology of the system may be complex, including feedback. Several examples taken from neuroscience are given: phototransduction, photopigment bleaching, and spike generation according to the Hodgkin-Huxley equations. The scheme uses two slightly different forms of autoregressive filters, with an implicit delay of zero for feedforward control and an implicit delay of half a sample distance for feedback control. On a fairly complex model of the macaque retinal horizontal cell it computes, for a given level of accuracy, 1-2 orders of magnitude faster than 4th-order Runge-Kutta. The computational scheme has minimal memory requirements, and is also suited for computation on a stream processor, such as a GPU (Graphical Processing Unit). 1 Introduction Nonlinear systems are ubiquitous in neuroscience, and simulations of concrete neural systems often involve large numbers of neurons or neural components. In particular if model performance needs to be compared with and fitted to measured neural responses, computing times can become quite restrictive. For such applications, efficient computational schemes are necessary. In this article, I will present such a highly efficient scheme, that has recently been used for simulating image processing by the primate outer retina (van Hateren 2006, 2007). The scheme is particularly suited for data-driven applications, where the time step of integration is dictated by the sampling interval of the analog-to-digital or digital-to-analog conversion. It assumes that the system can be decomposed into components of only two types: static nonlinearities and first-order low-pass filters. Interestingly, these components are also the most common ones used in neuromorphic VLSI circuits (Mead 1989). In the scheme presented here, the components need not have fixed parameters, but are allowed to depend on the system state. They are arranged in a possibly complex topography, typically involving several feedback loops. The efficiency of the scheme is produced by using very fast recursive filters for the first-order low-pass filters. I will show that it is best to use slightly different forms of the filter algorithm for feedforward and feedback processing loops. No attempt is made to rigorously analyze convergence or optimality of the scheme, which would anyway be difficult to do for arbitrary nonlinear systems. The scheme should therefore be viewed as a practical solution, that works well for the examples I give in this article, but may need specific testing and benchmarking on new problems. The scheme I present here can be efficiently implemented on stream processors. Recently there has been growing interest in using such processors for high performance computing (e.g., Göddeke et al. 2007, Ahrenberg et al. 2006, Guerrero-Rivera et al. 2006). In particular the arrival of affordable graphical processing units (GPUs) with raw computating power more than an order of magnitude higher than that of CPUs is driving this interest (see http://www.gpgpu.org). Current GPUs typically have about 100 processors that can work in parallel on data in the card’s memory. Once the data and the (C-like) programs are loaded into the card, the card computes essentially independently of the CPU. Results can subsequently be uploaded to the CPU for further processing. GPUs are especially suited for simulating problems, such as in retinal image processing, that can be written as parallel, local operations on a two-dimensional grid. Stream processors are, unlike CPUs, data driven and not instruction driven. They process the incoming data as it becomes available, and therefore usually need algorithms with fixed, or at least predictable computing times. The processing scheme I present in this article has indeed a fixed computing time. Moreover, it has low computational cost and low memory requirements, because it only deals with current and previous values of input, state variables, and output. The output is produced without delays that are not part of the model, that is, at the same time step as the current input, and the scheme is thus also suited for real-time applications. The article is organized as follows. First, I will present a fairly complete overview of methods to simulate a first-order low-pass filter with a minimal recursive filter. Subsequently, I will give several examples of how specific neural systems - in particular several subsystems of retinal processing and spike generation following the Hodgkin-Huxley equations - can be decomposed into suitable components. Computed results of the various forms of recursive filters are compared with benchmark calculations using a standard Matlab solver. It is shown that for a practical, fairly complex model the most efficient algorithm (modified Tustin) outperforms a conventional 4th-order Runge-Kutta integration by 1-2 orders of magnitude. Finally, I will discuss the merits and limits of the approach taken here. 2 Discrete simulation of a first-order low-pass filter Much of the material presented in this section is not new. However, I found that most of it is scattered throughout the literature, and I will therefore give a fairly complete overview. Table 1 summarizes the filters and their properties. In the continuous time domain, the equation =+ , (1) describes a first-order low-pass filter transforming an input function x(t) into an output function y(t), where τ is the time constant, and the coefficient in front of x is chosen such that the filter has unit DC gain: y=x if the input is a constant. In the examples below, I will usually write this equation in the standard form yxy −=ɺτ . (2) Fourier transforming this equation gives as the transfer function of this filter )( , (3) where the tilde denotes Fourier transforms. The impulse response of the filter is .0for 0 0for teth t τ τ (4) We will assume now that )(tx is only available at discrete times ∆= ntn , as )( ∆= nxxn , and that we only require )(ty at the same times, as )( ∆= nyyn . Here ∆ is the time between samples. Conforming with the most common integration schemes, we will further assume that for calculating the current value of the output only the current value of the input, the previous value of the output, and possibly the previous value of the input are available. We therefore seek real coefficients a1, b0, and b1 such that 11011 −− ++−= nnnn xbxbyay (5) produces an output close to that expected from Eq. (2). The indices and signs of the coefficients are chosen here in such a way that they are consistent with common use in the digital processing community for describing IIR (infinite impulse response) or ARMA (auto- regressive, moving average) filters that relate the z-transforms of input and output (Oppenheim and Schafer 1975). I will not use the z-transform formalism here, but only note that Fourier transforming Eq. (5) and using the shift theorem gives ∆−∆− ++−= ωω inn nn exbxbeyay 101 , (6) and therefore a transfer function nω , (7) where the operator )exp(1 ∆−=− ωiz represents a delay of one sample. The coefficients a1, b0, and b1 are not independent because of the additional constraint that the filter of Eq. (2) has unit DC gain. A constant input c must then produce a constant output c, thus Eq. (5) yields cbcbcac 101 ++−= and therefore 1101 =++− bba . (8) Because representing a general continuous system as in Eq. (2) by a discrete system as in Eq. (5) can only be approximate (note that Eqs. 3 and 7 cannot be made identical), there is no unique choice for the coefficients a1, b0, and b1. Below I will give an overview of several possibilities, mostly available in the literature, and discuss their appropriateness for the computational scheme to be presented below. The first three methods discussed below, forward Euler, backward Euler, and the Trapezoidal rule, are derived from general methods for approximating derivatives. The further methods discussed are more specialized, dealing specifically with Eq. (2) and differing with respect to how the input signal is assumed to behave between the sampled values. 2.1 Forward Euler Forward Euler (Press et al. 1992) is quite often used in neural simulations. Applied to Eq. (2) it amounts to the approximation τ/)( 11111 ∆−+=∆+≈ −−−−− nnnnnn yxyyyy ɺ , (9) hence we get the recurrence equation ./' with )'/1()'/11( 11 +−= −− ττ nnn xyy (10) Here as well as below I will use 'τ , which is τ normalized by the sample distance, to keep the equations concise. Eq. (10) suffers from two major problems: first, it is not very accurate, and even unstable for small 'τ (Press et al. 1992), and second, it produces an implicit delay of 2/∆ for centered samples. The second problem is illustrated in Fig. 1. Figure 1A shows a starting sinusoid, where the filled circles give the function values at the sampling times. The continuous function of Fig. 1A can subsequently be filtered by Eq. (2) using a standard integration routine (Matlab ode45) at a time resolution much better than ∆ (obviously, in this simple case the result could have been obtained analytically, but we will encounter other examples below where this is not possible). Fig. 1B shows the result (continuous line). When the samples of the sinusoid are processed by Eq. (10), the result lags by half a sampled distance (red open circles in Fig. 1B). 2.2 Backward Euler Backward Euler (Press et al. 1992) applied to Eq. (1) yields τ/)(11 ∆−+=∆+≈ −− nnnnnn yxyyyy ɺ , (11) hence nnn xyy )]1'/(1[)]1'/('[ 1 +++= − τττ . (12) Backward Euler is stable (Press et al. 1992) and slightly more accurate than forward Euler, but suffers from the problem that it produces an implicit delay of 2/∆− for centered samples, that is, a phase advance. Fig. 1C illustrates this, where the continous curve is the correct result (identical curve as the black curve in Fig. 1B), and the red open circles give the result of applying Eq. (12). 2.3 Trapezoidal rule The trapezoidal rule (also known as Crank-Nicholson, Rotter and Diesmann 1999) is equivalent to the bilinear transformation and Tustin’s method in digital signal processing (Oppenheim and Schafer 1975). It combines forward and backward Euler: τ/)()( 112 1 ∆−+−+=∆++≈ −−−−− nnnnnnnnn yxyxyyyyy ɺɺ , (13) and leads to 11 )]5.0'/(5.0[)]5.0'/(5.0[)]5.0'/()5.0'[( −− +++++−= nnnn xxyy ττττ . (14) The method is stable, accurate, and produces a negligible implicit delay (Fig. 1D). Figure 1. (A) Starting sinusoid (continuous line) and function values at the sample times (filled circles, 16 samples per period). The function equals 1 at times earlier than shown. (B) Continuous line: sinusoid of (A) filtered by Eq. (2) with τ'=16, computed with Matlab ode45; red open circles: result of filtering the samples of (A) with Eq. (10), the recurrence equation that follows from forward Euler. Output samples lag by approximately half a sample distance. (C) As (B), for backward Euler (Eq. (12)). Output samples lead by approximately half a sample distance. (D) As (B), for Trapezoidal (Eq. (14)). 2.4 Exponential Euler A method that has gained some popularity in the field of computational neuroscience (for example in the simulation package Genesis, Bower and Beeman 1998) is sometimes called Exponential Integration (MacGregor 1987, Rotter and Diesmann 1999) or Exponential Euler (Moore and Ramon 1974, Rush and Larsen 1978, Butera and McCarthy 2004). It assumes that the input is approximately constant, namely equal to 1−nx , on the interval from ∆− )1(n to ∆n . Equation (1) then has the exact solution (see e.g. appendix C.6 of Rotter and Diesmann 1999) '/1 )1( − − −+= nnn xeyey ττ . (15) This method is closely related to forward Euler, as a comparison of Eqs. (10) and (15) shows: for large 'τ (time constant large compared with the sample distance), the factors '/11)'/1exp( ττ −≈− and '/1)'/1exp(1 ττ ≈−− approximate those of forward Euler. The exponential Euler method is stable, and more accurate than forward Euler for small 'τ . However, it has the same implicit delay of 2/∆ as forward Euler (not shown). 2.5 Zero-Order Hold (ZOH) When using analog-to-digital and digital-to-analog converters, a choice has to be made for the assumed signal values between the sample times. A simple practical choice is to keep the value of the last sample until a new sample arrives. This is called a zero-order hold (ZOH), and for a sampled sinusoid it assumes the continuous line shown in Fig. 2A. It involves an implicit delay of 2/∆ . Digitally filtering the samples of a ZOH system can compensate for this delay by assuming that a unit sample at 0nn = (black line and filled circle in Fig. 2B) represents a block as shown by the dashed red line in Fig. 2B. The coefficients a1, b0, and b1 for approximating Eq. (2) by Eq. (5) can be readily obtained from the response to this pulse; these coefficients then also apply to an arbitrary input signal, because the filter is linear and time-invariant. For samples 20 +≥ nn , the present and previous input are zero, thus the terms with b0 and b1 do not contribute. Because Eq. (4) shows that the output must decline exponentially, we find '/1/1 ττ −∆− ==− eea . For sample 0nn = , the previous input and output are zero, thus the terms with a1 and b1 do not contribute. We then find b0 from the Figure 2. (A) Zero-Order Hold sampling model, where the sample values (dots) taken from a function (dashed line) are hold until a new sample arrives (continuous line). (B) A unit sample (black line and filled circle) is assumed here to represent a block in the previous inter-sample interval (red dashed line) (C) Continuous line: sinusoid of Fig. 1A filtered by Eq. (2) with τ'=16, computed with Matlab ode45; red open circles: result of filtering the samples of Fig. 1A with Eq. (17), the recurrence equation that follows from the ZOH processing scheme (i.e., assumed pulse shape of (B)). convolution of the block s(t) (dashed line in Fig. 2B) with the pulse response h(t) of the filter, evaluated at sample 0nn = ′−∆−′− −=−=′⋅=′′−′= ∫∫ 0 111 eetdetdttpthb t . (16) With Eq. (8) we then find 01 011 =−+= bab . The recurrence equation therefore is nnn xeyey )1( '/1 ττ − − −+= . (17) Note that the difference with Eq. (15) is that here the current input sample, xn, is used, where in Eq. (15) it is the previous input sample, xn-1. Whereas Eq. (15) implies a delay of 2/∆ , the present scheme has a delay of 2/∆− , i.e., a phase advance (see Fig. 2C). The filter in Eq. (17) is a special case of a general scheme of representing linear filters by using the matrix exponential (e.g., Rotter and Diesmann 1999, where it is called Exact Integration). Such filters are consistent with assuming a ZOH, and therefore imply a delay of 2/∆− . Although Rotter and Diesmann (1999) do not use a ZOH but a function representation using Dirac δ-functions, a delay is implied by the choice of integration interval in their Eq. (3), which excludes the previous input sample and fully includes the present input sample. Had the integration interval been chosen symmetrical, the δ-functions at the previous and present input samples would each have contributed by one half, leading to a scheme with )(5.0 1 nn xx +− as input, and therefore an implicit delay of 0. 2.6 First-Order Hold (FOH) Another choice for the assumed function values between samples is the first-order hold (FOH), where sample values are connected by straight lines. It assumes that a unit sample at 0nn = (black line and filled circle in Fig. 3A) represents a triangular pulse as shown by the dashed red line in Fig. 3A. The method is also called the triangular or ramp-invariant approximation, and is in fact equivalent to assuming that a function can be represented by B- splines of order one (Unser 1999, 2005). A general derivation of the recurrence relation, also valid for the more general lead-lag system xxyy xy +=+ ɺɺ ττ of which Eq. (2) is a special case, is given by Brown (2000). A simple, alternative derivation goes similarly as given above for the ZOH. For samples 20 +≥ nn , the present and previous input are zero, and we again Figure 3. (A) A unit sample (black line and filled circle) is assumed here to represent linear interpolation in the previous and next inter-sample intervals (red dashed line) (B) Continuous line: sinusoid of Fig. 1A filtered by Eq. (2) with τ'=16, computed with Matlab ode45; red open circles: result of filtering the samples of Fig. 1A with Eq. (19), the recurrence equation that follows from the FOH processing scheme (i.e., assumed pulse shape of (A)). find '/11 τ−=− ea . For sample 0nn = , the previous input and output are zero, and now b0 equals ττ ττ ′+′−=′ −=′′−′= ∫∫ 0 1)1( etdttpthb t . (18) With Eq. (8) we then find )/1exp()1(1 011 τττ ′−′+−′=−+= bab . The recurrence equation therefore is '/1'/1 '/1 ))1(()1( − − ′+−′+′+′−+= nnnn xexeyey τττ ττττ . (19) Fig. 3B illustrates that the FOH has a negligible implicit delay. 2.7 Centered Step-Invariant The centered step-invariant approximation (e.g., Thong and McNames 2002) is not often used, and is given here only for completeness; its performance is similar to that of FOH and Trapezoidal. It assumes that a unit sample at 0nn = represents a block that is, contrary to the regular zero-order hold, centered on the sample time. This is equivalent to assuming that a function can be represented by B-splines of order zero (Unser 1999). As before, we must have τ−=− ea , and for b0 we get )2/(1/ −=′⋅=′′−′= ∫∫ etdetdttpthb . (20) With Eq. (8) we then find )/1exp()2/(1exp(1 011 ττ ′−−′−=−+= bab . The recurrence equation therefore is '/1)2/(1)2/(1 '/1 )()1( − −′−′− − −+−+= nnnn xeexeyey ττττ . (21) This method also has a negligible implicit delay (not shown). 2.8 Modified Tustin’s method Below I will show that for implementing nonlinear feedback systems, a delay of 2/∆− is in fact favourable. One possibility is to use the ZOH for obtaining such a delay, but a modification of Tustin’s method (the Trapezoidal rule discussed above) is at least as good, and has coeffients that are simpler to compute. Whereas the Trapezoidal rule has no appreciable implicit delay, because it weighs the present and previous inputs equally (b0=b1), it can be given a 2/∆− delay by combining these weights to apply to the present input only: nnn xyy )]5.0'/(1[)]5.0'/()5.0'[( 1 +++−= − τττ . (22) The method is evaluated along with the other methods in the remainder of this article, and will be shown to work very well for feedback systems. To my knowledge, this modification of Tustin’s method has not been described in the literature before. 3 Relationship between recursive schemes for first-order low-pass filters A Taylor expansion of the various forms of 1a− gives FOHand ZOH, Euler,lexponentiafor ... 1 +′′ −==− ′− τea , (23) , Euler forwardfor 11 τ ′ −=− a (24) Eulerbackwardfor ... 1)/11/(1)1'/(' −=′+=+=− τττa , (25) Tustin. modified and lTrapezoidafor ... ) ... 1()5.0'/()5.0'( −=+−=− ττττττ (26) Compared to the theoretical exponential decline, Eq. (4), the exponential Euler, ZOH, and FOH are fully correct, the forward and backward Euler schemes are correct only up to the factor with )/1( τ ′ , whereas Trapezoidal and modified Tustin are correct up to the factor with 2)/1( τ ′ . The accuracy of the latter is related to the fact that )5.0'/()5.0'( +− ττ is a first-order Padé approximation of )'/1exp( τ− (Bechhoefer 2005). Note that in the limit of ∆>>τ , all algorithms use approximately the same weight for the previous output sample, namely τ ′− /11 . With respect to the weights acting on the input, the algorithms presented above can be divided into three groups, depending on the implicit delay they carry (see Table 1). If only the previous input sample is used (forward and exponential Euler), there is a delay of 2/∆ , if only the present input sample is used (backward Euler, ZOH, and modified Tustin’s method) there is a delay of 2/∆− , and if both the previous and present input samples are used (Trapezoidal and FOH), there is no delay. Below we will only analyze the groups with delays 2/∆− and 0. The coeffients b0 of the group with the phase advance (delay 2/∆− ) can be expanded as ZOHfor ... =−= ′− τeb , (27) Eulerbackwardfor ...- )/11( )1'/(1 320 τττττ =+=b , (28) Tustin, modifiedfor ... - ) ... - ()5.0'/(1 τττττ (29) where we find that ZOH and modified Tustin are more similar to each other than to backward Euler. Finally, the coeffients of the FOH can be compared with those of Trapezoidal: for FOH ... ... −′+′−=′+′−= − ττττ τeb (30) lTrapezoidafor ... ) ... ()5.0'/(5.0 (31) for FOH ... ... 1)(1()1( −′+−′=′+−′= − ττττ τeb (32) lTrapezoidafor ... )5.0'/(5.0 τb (33) The coefficients start to differ in the factor with 2)/1( τ ′ . We will see in the examples below that FOH and Trapezoidal perform very similarly on concrete problems. 4 Examples of nonlinear dynamic systems In this section I will provide several examples of nonlinear dynamic systems that are well suited to be simulated using autoregressive filters of the type discussed above. I will show for these examples how the systems can be rearranged to contain only static nonlinearities and first-order low-pass filters. Furthermore, I will compare the results of several of the algorithms presented above with an accurate numerical benchmark, and discuss the speed and accuracy of the various possibilities. 4.1 Phototransduction: coupled nonlinear ODEs An example of a system where coupled nonlinear differential equations can be represented by a feedback system is the phototransduction system in the cones of the vertebrate retina. I will concentrate here on the main mechanism, which provides gain control and control of temporal bandwidth (van Hateren 2005). For the present purpose, a suitable form is given by XCX β−+= )1/(1 4ɺ (34) CCXC τ/)( −=ɺ . (35) The variable β is linearly related to the light intensity, and can be considered as the input to the system. The variable X represents the concentration of an internal transmitter of the cone, and can be considered as the output of the system because it regulates the current across the cone’s membrane. The variable C is an internal feedback variable, proportional to the intracellular Ca2+ concentration. We will now rewrite the equations such that they get the form of Eq. (2): /1 and /1 with )1/( 4 XCqXɺ (36) . CXCC −=ɺτ (37) By defining a time constant βτ (actually not a constant, because it varies with β ) and an auxiliary variable q, we see that both equations formally take a form similar to Eq. (2), where q now has the role of input to Eq. (36), with the factor )1/(1 4C+ as a gain. We can thus represent these equations by the system diagram shown in Fig. 4A. The boxes containing a τ there represent unit-gain first-order low-pass filters. From the system diagram it is clear that the divisive feedback uses its own result after that has progressed through two low-pass filters and a static nonlinearity. The following describes the algorithm associated with Fig. 4A: • assume an initial steady state with 0ββ = , and obtain initial values of all variables by solving (analytically or numerically) Eqs. (36) and (37) for 0=Xɺ and 0=Cɺ • repeat for each time step o read β as input o compute a1, b0, and b1 for βτ β /1= , and update X by low-pass filtering it, taking )1/()/1( 4C+β as input to the filter o use a precomputed a1, b0, and b1 for Cτ to update C by low-pass filtering it, taking X as input to the filter o write X as output /β τ =1/β C1+ τC /β τ =1/β C1+ τC 0 100 200 time (ms) 10 100 1000 10-10 forward Euler exponential Euler Trapezoidal First-Order Hold backward Euler Zero-Order Hold modified Tustin 1/ [(ms) ]∆ -1 Figure 4. (A) System diagram of Eqs. (36) and (37). Boxes containing a ‘τ’ are unit-gain first-order low-pass filters, possibly depending on input or state variables (e.g., βτ depends on β). The other boxes represent static nonlinearities given by the function definition inside the box. (B) Scheme equivalent to (A), where the required phase advance of one sample distance ( ∆ ) for the feedback is obtained by using two low-pass filters of type −τ that each provide a 2/∆− delay (i.e., a 2/∆ phase advance). The box to the right represents a 2/∆ delay to compensate for the phase advance of βτ . (C) Thin black line: response X of Eqs. (36) and (37), using τC=3 ms, to ))2sin(9.01(0 tfπββ += for t≥0 and β= β0 for t<0, with β0=0.025 (ms) -1 and f=10 Hz, computed with Matlab ode45; dashed red line: result of filtering with the scheme of (B), with ∆=1 ms and using the modified Tustin’s method for −τ . (D) Root-mean-square (rms) error between the output when using the various recursive filters for the scheme of (B) and the result of ode45 at its maximum accuracy setting. Input as in (C). The thin straight lines are an aid for judging the scaling behaviour of the various methods, and have slopes of -1 and -2 in double-logarithmic coordinates. Note that βτ is obtained from the current value of β . In principle, it might have been based partly on the previous value of β as well, because β changes in the interval between previous and current sample. However, for βτ significantly larger than ∆ , this is expected to be a second-order effect, and the changing time constant is therefore treated in the simplest possible way, as described in the algorithm above. Because at each time step only the result of C that was obtained at the previous time step can be used in the division by )1( 4C+ , the feedback path would effectively get an (implicit) extra delay of ∆ if calculated following this scheme. Such an extra delay will affect the results (and in extreme cases may lead to spurious oscillations), which can only be minimized by choosing ∆ rather small. However, there is a way to alleviate this problem. As we have seen above, several of the autoregressive schemes have an implicit delay of 2/∆− . Because there are two low-pass filters concatenated in the feedback loop, using such a scheme will produce a total delay of ∆− , exactly compensating for the implicit delay ∆ of the feedback. In other words, the divisor used at the point of divisive feedback will have the correct, current time. Because the forward low-pass filter, βτ , has a delay of 2/∆− , we need to compensate that if we require that the output of the system has the right phase. (This may not always be necessary, especially not when the system is part of a larger system, where it would be more convenient to correct the sum of all delays at the final output.) The required delay of 2/∆ can be approximated by linear interpolation, i.e., a recurrence equation nnn xxy 5.05.0 1 += − . The linear interpolation implies a slight low-pass filtering of the signal, and is therefore only accurate if the sampling rate is sufficiently high compared with the bandwidth of the signal. We can then replace the scheme of Fig. 4A by the one of Fig. 4B, where the symbol −τ indicates that we are using filters with a 2/∆− delay (see Table 1). How well do the recursive schemes of Section 2 perform on this problem? To evaluate that, the thin black line in Fig. 4C shows the response X of Eqs. (36) and (37) to a sinusoidal modulation of β , computed using the Matlab routine ode45 at high time resolution and high precision settings. The dashed red line shows the result when using the scheme of Fig. 4B with the modified Tustin’s method used for −τ with ∆=1 ms. How the accuracy depends on ∆ is evaluated in Fig. 4D, which shows the rms (root-mean-square) deviation from the ode45 benchmark as a function of ∆ , not only for the modified Tustin’s method, but also for most of the other schemes. To get a fair comparison, the diagram of Fig. 4A was used for schemes with implicit delays 0 and 2/∆ , where for the latter an explicit delay of 2/∆− was added as a final stage. As is clear, the ZOH and especially the modified Tustin’s method are superior. They scale more favourably as a function of 1/∆, and for a given level of accuracy it is sufficient to use a ∆ at least an order of magnitude larger than for the other schemes. They compute therefore at least an order of magnitude faster. Because of the simplicity and speed of computing the coefficients of the modified Tustin’s method, this appears to be the scheme to be recommended for this type of feedback system. Note, however, that this scheme is only accurate when τ is at least a few times larger than ∆ (Eqs. 26 and 29), and breaks down completely for 1<′τ (with -a1 even becoming negative for 5.0<′τ ). 4.2 Photopigment bleaching: dynamics on different time scales For an example of a stiff set of differential equations, we will look at the dynamics of photopigment bleaching in human cones (Mahroo and Lamb 2004, Lamb and Pugh 2004, van Hateren and Snippe 2007). For the present purpose, a suitable form of the equations is RRRBIR τ/])1([ −−−=ɺ (38) RB ττ / −=ɺ . (39) Here I is a (scaled) light intensity, R is the (normalized) amount of photopigment excited by light, and B the (normalized) amount of bleached photopigment. The rate by which excited pigment is bleached is governed by first-order kinetics ( Rτ/1 ), whereas the reconversion of bleached pigment to excitable pigment is governed by rate-limited dynamics (Mahroo and Lamb 2004): the second term in the right-hand-side of Eq. (39) is consistent with first-order kinetics for small B, but saturates for large B. Eqs. (38) and (39) form a stiff set of equations, because the time constants s104.3 3−⋅=Rτ and s25=Bτ differ substantially. Through the factor )1( RB −− , bleaching provides a slow gain control, controlling the sensitivity of the eye in bright light conditions. Rewriting the equations into the form of Eq. (2) gives RRBIRR −−−= )1(ɺτ (40) . / and with RbBBb (41) This processing scheme is depicted in Fig. 5A, where bτ and Bg at time nt are derived from B at time 1−nt . Note that the phase advance of −τ is sufficient for the loop involving bτ , but only provides half of the required phase advance for the direct loop. Fig. 5B shows a 1−B−R RgΒτb 0 500 1000 time (ms) forward Euler exponential Euler Trapezoidal First-Order Hold backward Euler Zero-Order Hold modified Tustin 10-12 10-10 10 100 10001 1/ [(ms) ]∆ -1 Figure 5. (A) System diagram of Eqs. (40) and (41). (B) Thin black line: response R of Eqs. (40) and (41), using τR=3.4 ms and τB=25 s, to ))2sin(9.01(10 3 tfI π+= − for t≥1 ms, I=10-5 for t<0, and I=10-5+(10-3-10-5)t for 0≤t<1 ms, with f=10 Hz, computed with Matlab ode45; dashed red line: result of filtering with the scheme of (A), with ∆=1 ms and using the modified Tustin’s method for −τ . (C) Root-mean-square (rms) error between the various recursive filters used for the scheme of (A) and the result of ode45 at its maximum accuracy setting. Input as in (B). The thin straight line has a slope of -1 in double-logarithmic coordinates. benchmark calculation using ode45, and the result of using the scheme of Fig 5A with the modified Tustin’s method. The stimulus I steps at t=0 from 10-5 to a sinusoidal modulation around 10-3. Because an instantaneous step contains considerable power in its high-frequency components, using a recursive filter with a rather course ∆ causes significant aliasing, which in this particular example would noticeably affect the response right after the step. To reduce the effect of aliasing, the step was assumed here to take 1 ms, that is, there is a linear taper between t=0 and 1 ms. Fig. 5C compares the rms error of the various schemes as a function of ∆ . Again, the ZOH and the modified Tustin’s method perform best, despite the fact that there is no complete compensation of the feedback delay. 4.3 Spiking neurons: Hodgkin-Huxley equations As a final example of a highly nonlinear system with fast dynamics, we will look at the Hodgkin-Huxley equations for spike generation (Hodgkin and Huxley 1952). Following the formulation by Gerstner and Kistler (2002, Chapter 2.2) these equations are given by Eqs. (42)-(45): IEugEungEuhmguC +−−−−−−= )()()( LLK Naɺ , (42) where u is the membrane potential (in mV, defined relative to the resting potential), C the membrane capacitance (taken as 1 µF/cm2), the input variable I is externally applied current, and the other terms represent membrane currents (consisting of a sodium, potassium, and leakage current). The membrane currents are given by the reversal potentials for the ions (in mV, defined relative to the resting potential: 115Na =E , 12K −=E , and 6.10L =E ), by conductances (in mS/cm2, 120Na =g , 36K =g , and 3.0L =g ), and by variables n, m, and h, describing the gating of the ion channels by the membrane potential nnn nn βα −−= )1(ɺ (43) mmm mm βα −−= )1(ɺ (44) . )1( hhh hh βα −−=ɺ (45) The rate constants α and β are functions of u, the form of which was determined empirically by Hodgkin and Huxley (1952): ]1)1.01/[exp()01.01.0( −−−= uunα , ]80/exp(125.0 un −=β , ]1)1.05.2/[exp()1.05.2( −−−= uumα , ]18/exp(4 um −=β , )20/exp(07.0 uh −=α , and ]1)1.03/[exp(1 +−= uhβ . Rewriting the equations into the form of Eq. (2) gives CRgnghmgR EgEnghEmgI uIIRu and )/(1 with (46) )/( and )/(1 with nnnnnn βααβατ (47) )/( and )/(1 with mmmmmm βααβατ (48) )/( and )/(1 with hhhhhh βααβατ (49) This processing scheme is depicted in Fig. 6A. The feedback is partly additive (through the gated current eI , which acts as a strong positive feedback during the rising phase of the spike, and as a negative feedback during the potassium-driven after-hyperpolarization), partly multiplicative (through the input resistance eR , which drops considerably during the spike, and is the main cause of the absolute refractory period of the neuron), and partly through the time constant eτ , causing fast dynamics during the spike. Note that the system contains, for each of the three feedback variables, two low-pass filters in series ( eτ and the one belonging to either n, m, or h), thus we can fully utilize the phase advance of −τ as in the example on phototransduction. Figures 6C and D show a benchmark calculation using ode45 of the response (black line) to a current input as shown in Fig. 6B. This stimulus is again tapered at the beginning to reduce aliasing. Some tapering is realistic, because normally the axon of a spiking neuron (where spiking starts) will not be driven by instantaneous current steps, but only by band-limited currents because of low-pass filtering by the cell body and dendrites. Figure 6C shows the result of using the scheme of Fig. 6A with Trapezoidal (obviously without the 2/∆ processing block), and Fig. 6D with the modified Tustin’s method. Fig. 6E compares the rms error of the various schemes as a function of ∆ . Again, the ZOH and the R (n,m,h)[I+I (n,m,h)]e e (n,m,h) (u )- (u )-h∞ (u )- (u )-m∞ (u )- (u )-n∞ n m h - forward Euler exponential Euler Trapezoidal First-Order Hold backward Euler Zero-Order Hold modified Tustin 10 100 10001 1/ [(ms) ]∆ -1 0 100 200 time (ms) 0 100 200 time (ms) 0 100 200 time (ms) Figure 6. (A) System diagram of Eqs. (46) - (49). (B) Driving current density I, with I=0 for t<0, )/5.0(sin 0 0 ttII π= for 0≤t<t0 ms, and )))(5.0(sin5.01( 0 0 ttfII −−= π for t≥t0, with t0=10 ms a taper, f=10 Hz, and I0=12 µA/cm 2. (C) Thin black line: response u of Eq. (46) to the stimulus defined at (B), computed with Matlab ode45; dashed red line: result of filtering with the scheme of (A), with ∆=1/32 ms and using Trapezoidal for τ . (D) Thin black line: as in (C); dashed red line: result of filtering with the scheme of (A), with ∆=1/32 ms and using the modified Tustin’s method for −τ . (E) Root-mean-square (rms) error between the various recursive filters used for the scheme of (A) and the result of ode45 at its maximum accuracy setting. Input as in (B). The thin straight lines have slopes of -1 and -2 in double-logarithmic coordinates. modified Tustin’s method perform best. In particular the modified Tustin’s method provides accurate results: even at a course ∆=1/2 ms it misses no spikes in the example of Fig. 6, and the timing precision of the spikes is in the order of 0.1∆. This contrasts with, for instance, a scheme like Trapezoidal, which needs ∆ at least as small as 1/32 ms in order not to miss spikes, and has a timing precision of the spikes in the order of 10∆. 4.4 When to use −τ or 0τ Two of the examples given above involve feedback with exactly two low-pass filters in the forward and backward branches of the feedback loop. For these schemes low-pass filters with phase advance are clearly useful. However, for other topologies this is not necessarily the case. Fig. 7 shows a few examples. When concatenating low-pass filters and static nonlinearities (Fig. 7A), zero-delay filters 0τ may be used, as an alternative to using −τ and performing delay correction at a later stage. In a feedforward structure as shown in Fig. 7B, a zero-delay filter must be used. Similarly, if a feedback scheme contains more than two low- pass filters, some of the filters need to be zero-delay (Fig. 7C). If a system contains a feedback loop with only one low-pass filter in either the feedforward or feedback branch, a filter −τ can only provide half of the required phase advance. In those situations, as in the example on photopigment bleaching given above, it is still helpful to use −τ , in addition to making ∆ sufficiently small. In principle, a phase advance (a delay of 2/∆− ) might be added by implementing it as a linear extrapolation 15.05.1 −−= nnn xxy . However, I have not tested such a scheme, which might have stability problems. Finally, if a feedback loop contains no low-pass filters at all, it is in fact identical to a static nonlinearity, and can usually be treated analytically, or via a precomputed look-up table. 4.5 Comparison with a 4th-order Runge-Kutta integration scheme Although the present article focusses on simple autoregressive filters working on data with a given step size, it is interesting to compare the performance of the scheme with a standard integration method, such as 4th-order Runge-Kutta (RK4; Press et al. 1992). Figure 8 shows the results for RK4 and the modified Tustin’s method, applied to a fairly complex model of the macaque retinal horizontal cell (van Hateren 2005). This model consists of cones connected to horizontal cells in a feedback circuit, and constitutes a cascade of a static nonlinearity, two nonlinear (divisive) feedback loops, and a subtractive feedback loop. All 1 NL1 τ2 NL2 NL τ0 NL τ1 Figure 7. (A) Concatenation of low-pass filters and nonlinearities (NL), where zero-delay low-pass filters can be used. (B) In a feedforward loop as shown, a zero-delay low-pass filter should be used. (C) In a feedback loop, the total delay compensation needs to match the implicit delay ∆ of the computational feedback scheme. loops contain, in various configurations, low-pass filters and static nonlinearities. For details, such as parameter values and the differential equations involved, see van Hateren (2005). The inset in Fig. 8A shows the response of the model horizontal cell to a 40 ms light flash (horizontal bar) of contrast 2 given on a background of 100 td (see van Hateren 2005 for details on the stimulus). The vertical scale bar denotes 2 mV. This model was computed either using modified Tustin for the components (as in the examples in this article), or using RK4 for the entire set of differential equations. It should be stressed that this use of RK4 is different from the use of integrators, such as forward Euler, earlier in this article, where each low-pass filter was integrated separately. Here the RK4 algorithm is used, in the conventional way, on the entire model at once. All root-mean-square (rms) errors are calculated relative to the result of modified Tustin at a step size of 0.1 µs. Identical results were obtained when calculating all errors relative to RK4 at 0.1 µs, be it that errors then saturate at (i.e., do not go below) 4.7·10-6 because of the limited accuracy of RK4 at 0.1 µs. Figure 8A shows the rms error of RK4 and modified Tustin. For all step sizes shown, modified Tustin outperforms RK4. The different scaling behaviour is indicated by the two lines with slopes of -1 and -2 on the double-logarithmic coordinates. As argued by Morrison et al. (2007), in many situations the most interesting measure of performance of an integration method is the computing time required to achieve a given accuracy. This is shown in Fig. 8B for the two methods considered here. For this calculation the step size of modified Tustin was adjusted such that the accuracy of the result matched one of the RK4 calculations, and the corresponding computing times of the methods are plotted. Depending on accuracy, modified Tustin is typically 1-2 orders of magnitude faster than RK4. It should be noted that the calculation at the largest rms error already required a step size for modified Tustin (2.5 ms) that brought it well out of the range where the condition that the step modified Tustin modified Tustin 1/ [(ms) ]∆ -1 102 103 104101 10-10 rms error (mV) 10-110-5 10-3 Figure 8. (A) Root-mean-square (rms) error of computing the response (inset, vertical bar = 2 mV) to a 40 ms light flash (horizontal bar inset) of the macaque retinal horizontal cell model of van Hateren (2005). Both a 4th-order Runge-Kutta scheme (RK4, fixed time step, routines rkdumb/rk4 of Numerical Recipes, Press et al. 1992; the input is an analytical block function according to the horizontal bar) and modified Tustin were implemented in a double-precision Fortran90 program (Intel compiler, Linux, 3.0 GHz Xeon). Errors are calculated relative to the result of modified Tustin at a time step ∆=0.1 µs. The straight lines have slopes of -1 and -2 on double-logarithmic coordinates. (B) Computing times for RK4 and modified Tustin at matched rms error. For the four sets of data points the time steps ∆ for (RK4, modified Tustin) are (1 µs, 70 µs), (10 µs, 230 µs), (0.1 ms, 0.7 ms), and (1 ms, 2.5 ms). Ratios of computing times are 250, 70, 20, and 6. The straight lines have slopes of -1 and -0.5 on double-logarithmic coordinates. size should be a few times smaller than τ (Eqs. 26 and 29) is valid, because the fastest low- pass filters in the model have time constants of 3-4 ms (van Hateren 2005). Nevertheless, even under these conditions modified Tustin is approximately 6 times faster than RK4 at the same accuracy. 5 Discussion The fast recursive scheme presented in this article is particularly suited for situations where computing time is restrictive, for example when large arrays of neurons need to be computed. The scheme is fast, because each component is updated at each time step with only a few floating point operations. The examples given show that it is already quite accurate with fairly large time steps. It accomplishes this by computing feedback in a way that makes use of the fact that several autoregressive implementations of first-order low-pass filters produce an implicit phase advance of half a sample distance. The computational scheme is associated with a simple diagrammatic representation, that makes it relatively easy to get an intuitive understanding of the dynamics and of the processing flow, and allows for convenient symbolic manipulation (e.g., rearranging modules into equivalent schemes). Because the τ of the low-pass filters may depend on input and system variables, the filter coefficients may require updating at each time step. This may constitute a significant part of the computational load. Fortunately, the coefficients for the Trapezoidal rule (for 0τ ) and the modified Tustin’s method (for −τ ) can be obtained with only a few floating-point operations. These schemes also give results at least as accurate as any of the other schemes, and therefore should be considered as first choice. The present scheme is primarily intended for nonlinear filtering. It could be used for arbitrary linear filtering as well, because any linear filter can be approximated by a parallel arrangement of a number of low-pass filters with different weights and time constants. However, I have not tested how well the present scheme performs on such arrangements, and it seems likely that there are better ways to deal with arbitrary linear filters. One possibility is to use the matrix exponential (Rotter and Diesmann 1999), which is particularly suited when the signal consists of (or can be approximated by) point processes, as is common in calculating networks of spiking neurons. The matrix exponential can also be viewed as equivalent to a ZOH model and then needs a ∆/2 compensation depending on whether it is used in a feedforward branch or is used as part of a nonlinear feedback branch. Another possibility is to use canned routines, like c2d in Matlab, that provide coefficients for a recursive discrete system corresponding to any rational continuous transfer function. For a linear filter that is part of a nonlinear feedforward loop, the c2d routines using FOH or Tustin’s method are required, whereas ZOH is required when the linear filter is part of a feedback loop and a phase advance is wanted. All calculations presented in this article were done with double precision arithmetic. For strongly stiff problems, such a precision is indeed necessary because of the large difference in time constants; the time step needs to be small enough to accommodate the shortest time constant, but such a short time step results in considerable error build-up in the processing of the largest time constant if single-precision arithmetic is used. However, I found that for the examples discussed in this article, single precision arithmetic already gives quite accurate results. This is of interest, because using single precision may accelerate computation, depending on processor architecture. Moreover, stream processors such as present-day GPUs may not yet support double-precision arithmetic (although double precision can be readily emulated, Göddeke et al. 2007, and GPUs with double precision are announced for the end of 2007). I found that simulating the response of a large array of cones using the cone model of van Hateren and Snippe (2007), of which the examples of Sections 4.1 and 4.2 are part, provides performance one to two orders of magnitude higher on current GPUs than on current CPUs. Such performance is of interest for developing and testing models of the human retina (van Hateren, 2007) and also for using light adaptation in human cones as an algorithm for rendering and compression high-dynamic range video (van Hateren, 2006). Acknowledgments I thank Sietse van Netten and Herman Snippe for comments on the manuscript. References Ahrenberg L., Benzie P., Magnor M., Watson J. (2006) Computer generated holography using parallel commodity graphics hardware. Optics Express 14:7636-7641 Bechhoefer J. (2005) Feedback for physicists: a tutorial essay on control. Rev. Mod. Phys. 77:783-838 Bower J.M., Beeman D. (1998) The book of GENESIS: Exploring realistic neural models with the GEneral NEural SImulation System, 2nd edn. New York: Springer-Verlag (TELOS) Brown, K. S. (2000). Lead-lag algorithms. http://www.mathpages.com/home/kmath198/kmath198.htm Butera R.J., McCarthy M.L. (2004) Analysis of real-time numerical integration methods applied to dynamic clamp experiments. J. Neural Eng. 1:187-194 Gerstner W., Kistler W.M. (2002) Spiking neuron models. Cambridge: Cambridge University Press Göddeke D., Robert Strzodka R., Turek S. (2007) Performance and accuracy of hardware- oriented native-, emulated- and mixed-precision solvers in FEM simulations. International Journal of Parallel, Emergent and Distributed Systems, in press Guerrero-Rivera R., Morrison A., Diesmann M., Pearce, T.C. (2006) Programmable logic construction kits for hyper-real-time neuronal modeling. Neural Comp. 18:2651-2679 Hodgkin A.L., Huxley A.F. (1952) A quantitative description of membrane current and it sapplication to conduction and excitation in nerve. J. Physiol. 117:500-544 Lamb T.D., Pugh E.N. (2004) Dark adaptation and the retinoid cycle of vision. Progr. Ret. Eye Res. 23:307-380 MacGregor R.J. (1987) Neural and brain modeling. San Diego: Academic Press. Mahroo O.A.R., Lamb T.D. (2004) Recovery of the human photopic electroretinogram after bleaching exposures: estimation of pigment regeneration kinetics. J. Physiol. 554:417-437 Mead C. (1989) Analog VLSI and neural systems. Reading, Mass.: Addison-Wesley Moore J.W., Ramon F. (1974) On numerical integration of the Hodgkin and Huxley equations for a membrane action potential. J. Theor. Biol. 45:249-273 Morrison A., Straube S., Plesser H.E., Diesmann M. (2007) Exact subthreshold integration with continuous spike times in discrete-time neural network simulations. Neural Comp. 19:47-79 Oppenheim A.V., Schafer R.W. (1975) Digital signal processing. Englewood Cliffs, NJ: Prentice Hall Press W.H., Teukolsky S.A., Vetterling, W. T., Flannery, B. P. (1992) Numerical recipes in Fortran. New York: Cambridge University Press Rotter S., Diesmann M. (1999) Exact digital simulation of time-invariant linear systems with applications to neuronal modeling. Biol. Cybern. 81:381-402 Rush S., Larsen H. (1978) A practical algorithm for solving dynamic membrane equations. IEEE Trans. Biomed. Eng. 36:389:392 Thong T., McNames J. (2002) Transforms for continuous time system modeling. Proceedings 45th IEEE Midwest Symposium on Circuits and Systems, August 4-7, 2002, Tulsa, Oklahoma, pp. II-408 – II-411. Unser M. (1999) Splines - A perfect fit for signal and image processing. IEEE Signal Process. Mag. 16:22-38 Unser M. (2005) Cardinal exponential splines: Part II - Think analog, act digital. IEEE Trans. Signal Process. 53:1439-1449 van Hateren, J.H. (2005) A cellular and molecular model of response kinetics and adaptation in primate cones and horizontal cells. J. Vision 5:331-347 van Hateren, J.H. (2006) Encoding of high dynamic range video with a model of human cones. ACM Transactions on Graphics 25:1380-1399 van Hateren, J.H. (2007) A model of spatiotemporal signal processing by primate cones and horizontal cells. J. Vision 7(3):3, 1-19 van Hateren, J.H., Snippe, H.P. (2007) Simulating human cones from mid-mesopic up to high-photopic luminances. J. Vision 7(4):1, 1-11 Table 1 Autoregressive filters approximating yxy −=ɺτ by 11011 −− ++−= nnnn xbxbyay , with sample distance ∆ , and ∆≡′ /ττ Scheme forward Euler backward Euler Trapezoidal rule exponential Euler Zero-Order Hold First-Order Hold modified Tustin’s method also known as • Tustin’s method • Bilinear transformation • Crank-Nicholson • exponential integration • step-invariant approximation • Exact Integration • ramp-invariant approximation • triangular rule -a1 (weight of yn-1, previous output) )'/11( τ− )1'/(' +ττ )5.0'/()5.0'( +− ττ '/1 τ−e '/1 τ−e '/1 τ−e )5.0'/()5.0'( +− ττ (weight of xn, present input) - )1'/(1 +τ )5.0'/(5.0 +τ - '/11 τ−− e '/11 τττ −′+′− e )5.0'/(1 +τ (weight of xn-1, previous input) '/1 τ - )5.0'/(5.0 +τ '/11 τ−− e - '/1)1( τττ −′+−′ e - implicit delay 2/∆ 2/∆− 0 2/∆ 2/∆− 0 2/∆− symbol 0τ −τ remarks can be unstable preferred choice for feedforward preferred choice for feedback

0704.1363

Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on $S^n$

arXiv:0704.1363v1 [math.DG] 11 Apr 2007 7 Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on Sn M. Boucetta Faculté des Sciences et Techniques BP 549 Marrakech Morocco Email: [email protected] ∗ Abstract. We compute the eigenvalues with multiplicities of the Lichnerow- icz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere Sn. The spaces of symmetric eigentensors are explicitly given. Mathematical Subject Classification (2000):53B21, 53B50, 58C40 Key words: Lichnerowicz Laplacian 1 Introduction Let (M, g) be a Riemannian n-manifold. For any p ∈ IN, we shall denote by Γ(⊗pT ∗M), Ωp(M) and SpM the space of covariant p-tensor fields on M , the space of differential p-forms on M and the space of symmetric covariant p-tensor fields on M , respectively. Note that Γ(⊗0T ∗M) = Ω0(M) = S0M = C∞(M, IR), Ω(M) = Ωp(M) and S(M) = Sp(M). Let D be the Levi-Civita connection associated to g; its curvature tensor field R is given by R(X, Y )Z = D[X,Y ]Z − (DXDY Z −DYDXZ) , ∗Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche Scientifique PROTARS III. http://arxiv.org/abs/0704.1363v1 and the Ricci endomorphism field r : TM −→ TM is given by g(r(X), Y ) = g(R(X,Ei)Y,Ei), where (E1, . . . , En) is any local orthonormal frame. For any p ∈ IN, the connectionD induces a differential operatorD : Γ(⊗pT ∗M) −→ Γ(⊗p+1T ∗M) given by DT (X, Y1, . . . , Yp) = DXT (Y1, . . . , Yp) = X.T (Y1, . . . , Yp)− T (Y1, . . . , DXYj , . . . , Yp). Its formal adjoint D∗ : Γ(⊗p+1T ∗M) −→ Γ(⊗pT ∗M) is given by D∗T (Y1, . . . , Yp) = − DEiT (Ei, Y1, . . . , Yp), where (E1, . . . , En) is any local orthonormal frame. Recall that, for any differential p-form α, we have dα(X1, . . . , Xp+1) = (−1)j+1DXjα(X1, . . . , X̂j, . . . , Xp+1). (1) We denote by δ the restriction of D∗ to Ω(M) ⊕ S(M) and we define δ∗ : Sp(M) −→ Sp+1(M) by δ∗T (X1, . . . , Xp+1) = DXjT (X1, . . . , X̂j, . . . , Xp+1). Recall that the operator trace Tr : Sp(M) −→ Sp−2(M) is given by TrT (X1, . . . , Xp−2) = T (Ej, Ej , X1, . . . , Xp−2), where (E1, . . . , En) is any local orthonormal frame. The Lichnerowicz Laplacian is the second order differential operator ∆M : Γ(⊗ pT ∗M) −→ Γ(⊗pT ∗M) given by ∆M(T ) = D ∗D(T ) +R(T ), where R(T ) is the curvature operator given by R(T )(Y1, . . . , Yp) = T (Y1, . . . , r(Yj), . . . , Yp) {T (Y1, . . . , El, . . . , R(Yi, El)Yj, . . . , Yp) + T (Y1, . . . , R(Yj, El)Yi, . . . , El, . . . , Yp)} , where (E1, . . . , En) is any local orthonormal frame and, in T (Y1, . . . , El, . . . , R(Yi, El)Yj, . . . , Yp), El takes the place of Yi and R(Yi, El)Yj takes the place of Yj. This differential operator, introduced by Lichnerowicz in [15] pp. 26, is self- adjoint, elliptic and respects the symmetries of tensor fields. In particular, ∆M leaves invariant S(M) and the restriction of ∆M to Ω(M) coincides with the Hodge-de Rham Laplacian, i.e., for any differential p-form α, ∆Mα = (dδ + δd)(α). (2) We have shown in [6] that, for any symmetric covariant tensor field T , ∆M (T ) = (δ ◦ δ ∗ − δ∗δ)(T ) + 2R(T ). (3) Note that if T ∈ S(M) and gl denotes the symmetric product of l copies of the Riemannian metric g, we have (Tr ◦∆M)T = (∆M ◦ Tr)T, (4) ∆M(T ⊙ g l) = (∆MT )⊙ g l, (5) where ⊙ is the symmetric product. The Lichnerowicz Laplacian acting on symmetric covariant tensor fields is of fundamental importance in mathematical physics (see for instance [9], [20] and [22]). Note also that the Lichenrowicz Laplacian acting on symmetric covariant 2-tensor fields appears in many problems in Riemannian geometry (see [3], [5], [19]...). On a Riemannian compact manifold, the Lichnerowicz Laplacian ∆M has discrete eigenvalues with finite multiplicities. For a given Riemannian com- pact manifold, it may be an interesting problem to determine explicitly the eigenvalues and the eigentensors of ∆M on M . Let us enumerate the cases where the spectra of ∆M was computed: 1. ∆M acting on C ∞(M, |C): M is either flat toris or Klein bottles [4], M is a Hopf manifolds [1]; 2. ∆M acting on Ω(M): M = S n or P n( |C) [10] and [11], M = |CaP 2 or G2/SO(4) [16] and [18], M = SO(n + 1)/SO(2) × SO(n) or M = Sp(n+ 1)/Sp(1)× Sp(n) [21]; 3. ∆M acting on S 2(M) and M is the complex projective space P 2( |C) [22]; 4. ∆M acting on S 2(M) and M is either Sn or P n( |C) [6] and [7]; 5. Brian and Richard Millman give in [2] a theoretical method for com- puting the spectra of Lichnerowicz Laplacian acting on Ω(G) where G is a compact semisimple Lie group endowed with the Killing form; 6. Some partial results where given in [12]-[14]. In this paper, we compute the eigenvalues and we determine the spaces of eigentensors of ∆M acting on S(M) in the case where M is the Euclidian sphere Sn. Let us describe our method briefly. We consider the (n+ 1)-Euclidian space IRn+1 with its canonical coordinates (x1, . . . , xn+1). For any k, p ∈ IN, we denote by SpHδk the space of symmetric covariant p-tensor fields T on IR satisfying: 1. T = 1≤i1≤...≤ip≤n+1 Ti1,...,ipdxi1⊙. . .⊙dxip where Ti1,...,ip are homogeneous polynomials of degree k; 2. δ(T ) = ∆IRn+1(T ) = 0. The n-dimensional sphere Sn is the space of unitary vectors in IRn+1 and the Euclidian metric on IRn+1 induces a Riemannian metric on Sn. We denote by i : Sn →֒ IRn+1 the canonical inclusion. For any tensor field T ∈ Γ(⊗pT ∗IRn+1), we compute i∗(∆IRn+1T )−∆Sn(i and get a formula (see Theorem 2.1). Inspired by this formula and having in mind the fact that i∗ : SpHδk −→ S pSn is injective and its image is dense in SpSn (see [10]), we give, for any k, a direct sum decomposition of SpHδk composed by eigenspaces of ∆Sn. Thus we obtain the eigenvalues and the spaces of eigentensors with its multiplicities of ∆Sn acting on S(S n) (see Section 4). Note that the eigenvalues and the eigenspaces of ∆Sn acting on Ω(S n) was computed in [10] by using the representation theory. In [11], I. Iwasaki and K. Katase recover the result by a method using the restriction of harmonic tensor fields and a result in [8]. The formula obtained in Theorem 2.1 combined with the methods developed in [10] and [11] permit to present those results in a more precise form (see Section 3). 2 A relation between ∆IRn+1 and ∆Sn We consider the Euclidian space IRn+1 endowed with its canonical coordinates (x1, . . . , xn+1) and its canonical Euclidian flat Riemannian metric < , >. We denote by D be the Levi-Civita covariant derivative associated to < , >. We consider the radial vector field given by −→r = For any p-tensor field T ∈ Γ(⊗pT ∗IRn+1) and for any 1 ≤ i < j ≤ p, we denote by i−→r ,jT the (p− 1)-tensor field given by i−→r ,jT (X1, . . . , Xp−1) = T (X1, . . . , Xj−1, −→r ,Xj, . . . , Xp−1), and by Tri,jT the (p− 2)-tensor field given by Tri,jT (X1, . . . , Xp−2) = T (X1, . . . , Xi−1, El, Xi, . . . , Xj−2, El, Xj−1, . . . , Xp−2), where (E1, . . . , En+1) is any orthonormal basis of IR n+1. Note that Tri,jT = 0 if T is a differential form and Tri,jT = TrT if T is symmetric. For any permutation σ of {1, . . . , p}, we denote by T σ the p-tensor field T σ(X1, . . . , Xp) = T (Xσ(1), . . . , Xσ(p)). For 1 ≤ i < j ≤ p, the transposition of (i, j) is the permutation σi,j of {1, . . . , p} such that σi,j(i) = j, σi,j(j) = i and σi,j(k) = k for k 6= i, j. Let T denote the set of the transpositions of {1, . . . , p}. The sphere i : Sn →֒ IRn+1 is endowed with the Euclidian metric. Theorem 2.1 Let T be a covariant p-tensor field on IRn+1. Then, i∗(∆IRn+1T ) = ∆Sni ∗T + i∗ p(1− p)T + (2p− n + 1)L−→r T − L−→r ◦ L−→r T T σ +O(T ) where O(T ) is given by O(T )(X1, . . . , Xp) = 2 < Xi, Xj > Tri,j(X1, . . . , X̂i, . . . , X̂j, . . . , Xp) DXj(i−→r ,jT )(X1, . . . , X̂j, . . . , Xp), where X̂ designs that X is deleted. Proof. The proof is a massive computation in a local orthonormal frame using the properties of the Riemannian embedding of the sphere in the Eu- clidian space. We choose a local orthonormal frame of IRn+1 of the form (E1, . . . , En, N) such that Ei is tangent to S n for 1 ≤ i ≤ n and N = 1 −→r where r = x21 + . . .+ x For any vector field X on IRn+1, we have DXN = (X− < X,N > N) , (6) DNX = [N,X ] + (X− < X,N > N). (7) Let ∇ be the Levi-Civita connexion of the Riemannian metric on Sn. We have, for any vector fields X, Y tangent to Sn, DXY = ∇XY− < X, Y > N. (8) Let T be a covariant p-tensor field on IRn+1 and (X1, . . . , Xp) a family of vector fields on IRn+1 which are tangent to Sn. A direct calculation using the definition of the Lichnerowicz Laplacian gives ∆IRn+1(T )(X1, . . . , Xp) = D ∗D(T )(X1, . . . , Xp) −EiEi.T (X1, . . . , Xp) + 2 Ei.T (X1, . . . , DEiXj, . . . , Xp) +DEiEi.T (X1, . . . , Xp)− T (X1, . . . , DDEiEiXj , . . . , Xp) T (X1, . . . , DEiDEiXj, . . . , Xp) −2 T (X1, . . . , DEiXl, . . . , DEiXj, . . . , Xp) −N.N.T (X1, . . . , Xp) + 2 N.T (X1, . . . , DNXj , . . . , Xp) +DNN.T (X1, . . . , Xp)− T (X1, . . . , DDNNXj , . . . , Xp) T (X1, . . . , DNDNXj, . . . , Xp)− 2 T (X1, . . . , DNXl, . . . , DNXj , . . . , Xp). (6)-(8) make it obvious that DDEiEiXj = ∇∇EiEiXj− < ∇EiEi, Xj > N − [N,Xj ] (9) (Xj− < Xj, N > N), DEiDEiXj = ∇Ei∇EiXj − (< Ei,∇EiXj > +Ei. < Ei, Xj >)N < Ei, Xj > Ei, (10) DNDNX = [N, [N,X ]] + [N,X ] + ( )(X− < X,N > N) N. < X,N > N. (11) By (8)-(10), we get easily, in restriction to Sn, Ei.T (X1, . . . , DEiXj , . . . , Xp) +DEiEi.T (X1, . . . , Xp) T (X1, . . . , DDEiEiXj, . . . , Xp)− T (X1, . . . , DEiDEiXj , . . . , Xp) Ei.T (X1, . . . ,∇EiXj , . . . , Xp) +∇EiEi.T (X1, . . . , Xp) T (X1, . . . ,∇∇EiEiXj, . . . , Xp)− T (X1, . . . ,∇Ei∇EiXj , . . . , Xp) Xj .T (X1, . . . , N , . . . , Xp) + p(n + 1)T (X1, . . . , Xp)− nLNT (X1, . . . , Xp). On other hand, also by using (8), we have T (X1, . . . , DEiXl, . . . , DEiXj, . . . , Xp) = T (X1, . . . , DEiXl, . . . ,∇EiXj, . . . , Xp)− T (X1, . . . , DXjXl, . . . , N , . . . , Xp) = T (X1, . . . ,∇EiXl, . . . ,∇EiXj, . . . , Xp)− T (X1, . . . , N , . . . ,∇XlXj, . . . , Xp) T (X1, . . . , DXjXl, . . . , N , . . . , Xp) = T (X1, . . . ,∇EiXl, . . . ,∇EiXj , . . . , Xp) T (X1, . . . , DXjXl, . . . , N , . . . , Xp)− T (X1, . . . , N , . . . , DXlXj , . . . , Xp) < Xl, Xj > T (X1, . . . , N , . . . , N , . . . , Xp). So we get, in restriction to Sn, since DNN = 0 ∆IRn+1(X1, . . . , Xp)−∇ ∗∇T (X1, . . . , Xp) = p(n + 1)T (X1, . . . , Xp)− nLNT (X1, . . . , Xp)− 2 DXj(iN,jT )(X1, . . . , X̂j, . . . , Xp) < Xl, Xj > T (X1, . . . , N , . . . , N , . . . , Xp)−N.N.T (X1, . . . , Xp) N.T (X1, . . . , DNXj, . . . , Xp)− T (X1, . . . , DNDNXj , . . . , Xp) T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp). Remark that, in restriction to Sn, the following equality holds DXj (iN,jT )(X1, . . . , X̂j, . . . , Xp) = DXj (i−→r ,jT )(X1, . . . , X̂j, . . . , Xp). Now by using (7) and (11) and by taking the restriction to Sn, we have N.T (X1, . . . , DNXj, . . . , Xp) = N.T (X1, . . . , [N,Xj], . . . , Xp) + 2 )T (X1, . . . , Xj, . . . , Xp) N.T (X1, . . . , Xj, . . . , Xp)− 2 N(< Xj , N >)T (X1, . . . , N , . . . , Xp) = N.T (X1, . . . , [N,Xj], . . . , Xp)− 2pT (X1, . . . , Xp) + 2pN.T (X1, . . . , Xj, . . . , Xp) N(< Xj , N >)T (X1, . . . , N , . . . , Xp). T (X1, . . . , DNDNXj, . . . , Xp) = T (X1, . . . , [N, [N,Xj], . . . , Xp)− 2 N(< Xj, N >)T (X1, . . . , N , . . . , Xp). T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp) = T (X1, . . . , [N,Xi], . . . , [N,Xj ], . . . , Xp) + p(p− 1) T (X1, . . . , Xp) T (X1, . . . , Xi, . . . , [N,Xj], . . . , Xp) + T (X1, . . . , [N,Xi], . . . , Xj , . . . , Xp). So we get, in restriction to Sn −N.N.T (X1, . . . , Xp) + 2 N.T (X1, . . . , DNXj , . . . , Xp) T (X1, . . . , DNDNXj, . . . , Xp)− 2 T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp) = −LN ◦ LNT (X1, . . . , Xp) + 2pLNT (X1, . . . , Xp)− p(1 + p)T (X1, . . . , Xp). The curvature of Sn is given by R(X, Y )Z =< X, Y > Z− < Y,Z > X and r(X) = (n− 1)X. Hence, a direct computation gives that the curvature operator is given by R(T )(X1, . . . , Xp) = p(n− 1)T (X1, . . . , Xp) + 2 T σ(X1, . . . , Xp) < Xi, Xj > T (X1, . . . , El, . . . , El, . . . , Xp). Finally, we get i∗(∆IRn+1T ) = ∆Sni ∗T + i∗ (p(1− p)T + (2p− n)LNT − LN ◦ LNT T σ +O(T ) One can conclude the proof by remarking that i∗(LNT ) = i ∗(L−→r T ) and i ∗(LN ◦LNT ) = −i ∗(L−→r T )+ i ∗(L−→r ◦L−→r T ). Q.E.D. Corollary 2.1 Let α be a differential p-form on IRn+1. Then i∗(∆IRn+1α) = ∆Sni ∗α + i∗ (2p− n+ 1)L−→r α− L−→r ◦ L−→r α− 2di−→r α Corollary 2.2 Let T be a symmetric p-tensor field on IRn+1. Then i∗(∆IRn+1T ) = ∆Sni ∗T + i∗ 2p(1− p)T + (2p− n + 1)L−→r T − L−→r ◦ L−→r T − 2δ∗(i−→r T ) + 2Tr(T )⊙ <,> where ⊙ is the symmetric product. 3 Eigenvalues and eigenforms of ∆Sn acting on Ω(Sn) In this section, we will use corollary 2.1 and the results developed in [10] to deduce the eigenvalues and the spaces of eigenforms of ∆Sn acting on Ω ∗(Sn). We recover the results of [10] and [11] in a more precise form. Let ∧pHk be the space of all coclosed harmonic homogeneous p-forms of degree k on IRn+1. A differential form α belongs to ∧pHk if δ(α) = 0 and α can be written 1≤i1<...<ip≤n+1 αi1...ipdxi1 ∧ . . . ∧ dxip , where αi1...ip are harmonic polynomial functions on IR n+1 of degree k. For any α ∈ ∧pHk, we have L−→r α = di−→r α + i−→r dα = (k + p)α. (12) We have (see [10]), ∧pHk −→ Ω p(Sn) is injective and its image is dense. For any α ∈ ∧pHk, we put ω(α) = α− di−→r α. (13) Lemma 3.1 We get a linear map ω : ∧pHk −→ ∧ pHk which is a projector, i.e., ω ◦ ω = ω. Moreover, Kerω = d(∧p−1Hk+1), Imω = ∧ pHk ∩Keri−→r , and hence ∧pHk = ∧ pHk ∩Keri−→r ⊕ d(∧ p−1Hk+1). The following lemma is an immediate consequence of Corollary 2.1 and (12). Lemma 3.2 1. For any α ∈ ∧pHk ∩Keri−→r , we have ∗α = (k + p)(k + n− p− 1)i∗α. 2. For any α ∈ d(∧p−1Hk+1), we have ∗α = (k + p)(k + n− p+ 1)i∗α. Remark 3.1 We have (k+ p)(k+n− p− 1) = (k′ + p)(k′ +n− p+1) ⇔ k = k′ +1 and n = 2p. The following Table gives explicitly the spectra of ∆Sn and the spaces of eigenforms with its multiplicities . The multiplicity was computed in [11]. Table I p The eigenvalues The space of eigenforms Multiplicity p = 0 k(k + n− 1), k ∈ IN ∧0Hk (n+k−2)!(n+2k−1) k!(n−1)! 1 ≤ p ≤ n, (k + p)(k + n− p− 1), ω(∧pHk) (n+k−1)!(n+2k−1) p!(k−1)!(n−p−1)!(n+k−p−1)(k+p) n 6= 2p k ∈ IN∗ (k + p)(k + n− p+ 1), d(∧p−1Hk+1) (n+k)!(n+2k+1) (p−1)!k!(n−p)!(n+k−p+1)(k+p) k ∈ IN 1 ≤ p ≤ n, (k + p)(k + p+ 1) n = 2p k ∈ IN ω(∧pHk+1)⊕ d(∧ p−1Hk+1) 2(2p+k)!(2p+2k+1) p!(p−1)!k!(k+p+1)(k+p) 4 Eigenvalues and eigentensors of ∆Sn acting on S(Sn) This section is devoted to the determination of the eigenvalues and the spaces of eigentensors of ∆Sn acting on S(S Let SpPk be the space of T ∈ S p(IRn+1) of the form 1≤i1≤...≤ip≤n+1 Ti1...ipdxi1 ⊙ . . .⊙ dxip, where Ti1...ip are homogeneous polynomials of degree k. We put SpHδk = S pPk ∩Ker∆IRn+1 ∩Kerδ and S pHδ0k = S pHδk ∩KerTr. In a similar manner as in [10] Lemma 6.4 and Corollary 6.6, we have SpPk = S pHδk ⊕ (r 2SpPk−2 + dr 2 ⊙ Sp−1Pk−1), (14) SpHδk −→ S is injective and its image is dense in SpSn. Now, for any k ≥ 0, we proceed to give a direct sum decomposition of SpHδk consisting of eigenspaces of ∆Sn and, hence, we determine completely the eigenvalues of ∆Sn acting on S p(Sn). This will be done in several steps. At first, we have the following direct sum decomposition: SpHδk = S pHδ0k ⊕ Sp−2lHδ0k ⊙ <,> l, (15) where <,>l is the symmetric product of l copies of <,>. The task is now to decompose SpHδ0k as a sum of eigenspaces of ∆Sn and get, according to (5), all the eigenvalues. This decomposition needs some preparation. Lemma 4.1 Let T ∈ SpPk and h ∈ IN ∗. Then we have the following formu- 1. δ∗(i−→r T )− i−→r δ ∗(T ) = (p− k)T ; 2. δ∗(h)(i−→r T )− i−→r δ ∗(h)(T ) = h(p− k + h− 1)δ∗(h−1)(T ); 3. δ∗(i−→r hT )− i−→r hδ∗(T ) = h(p− k − h+ 1)i−→r h−1T, where i−→r ︷ ︸︸ ︷ i−→r ◦ . . . ◦ i−→r and δ ∗(h) = ︷ ︸︸ ︷ δ∗ ◦ . . . ◦ δ∗ . Proof. The first formula is easily verified and the others follow by induction on h. Q.E.D. Now, we will construct two linear maps pδ∗ : S pPk −→ S pPk for k ≤ p, and p−→r : S pPk −→ S pPk for k ≥ p satisfying: 1. pδ∗ ◦ pδ∗ = pδ∗ , Kerpδ∗ = i−→r (S p+1Pk−1), Impδ∗ = Kerδ ∗ ∩ SpPk; 2. p−→r ◦ p−→r = p−→r , Kerp−→r = δ ∗(Sp−1Pk+1), Imp−→r = Keri−→r ∩ S The procedure is to put, for T ∈ SpPk, pδ∗(T ) = αsi−→r sδ∗(s)(T ), and p−→r (T ) = ∗(s)(i−→r sT ), and find (α0, . . . , αk) and (β0, . . . , βp) such that the required properties are satisfied. A straightforward computation using Lemma 4.1 gives δ∗(pδ∗(T )) = (αs − (s+ 1)(k − p− s− 2)αs+1)i−→r sδ∗(s+1)(T ), i−→r (p−→r (T )) = (βs − (s+ 1)(p− k − s− 2)βs+1)δ ∗(s)(i−→r s+1T ). Hence, we define pδ∗ and p−→r as follows: pδ∗(T ) = αsi−→r sδ∗(s)(T ) α0 = 1 and αs − (s+ 1)(k − p− s− 2)αs+1 = 0 for 1 ≤ s ≤ k − 1; p−→r (T ) = ∗(s)(i−→r β0 = 1 and βs − (s+ 1)(p− k − s− 2)βs+1 = 0 for 1 ≤ s ≤ p− 1. From this definition and by using Lemma 4.1, one can check easily that pδ∗ and p−→r satisfy the required properties. On other hand, it is easy to check that we have, for any symmetric tensor field T on IRn+1, ∆IRn+1(i−→r T ) = i−→r ∆IRn+1(T ) + 2δT, (16) δ(i−→r T ) = i−→r δ(T )− Tr(T ), (17) Tr(δ∗(T )) = −2δ(T ) + δ∗(Tr(T )), (18) Tr(i−→r T ) = i−→r Tr(T ). (19) From these formulas and from (3), one deduce easily that pδ∗(S pHδ0k ) ⊂ SpHδ0k and p−→r (S pHδ0k ) ⊂ S pHδ0k and thus one get the following direct sum decompositions: SpHδ0k = S pHδ0k ∩Kerδ ∗ ⊕ i−→r Sp+1Hδ0k−1 , if k ≤ p, (20) SpHδ0k = S pHδ0k ∩Keri−→r ⊕ δ Sp−1Hδ0k+1 , if k ≥ p. (21) These decompositions are far for being sufficient and, in order to obtain a more sharp direct sum decompositions of SpHδ0k , we need the following lemma. Lemma 4.2 1. For k < p, i−→r : S pPk −→ S p−1Pk+1 is injective. 2. For k > p, δ∗ : SpPk −→ S p+1Pk−1 is injective. 3. For k = p, Kerδ∗ = Keri−→r . Proof. 1. Let T ∈ SpPk such that i−→r T = 0. The second formula in Lemma 4.1 gives, for any h ≥ 1, i−→r δ ∗(h)(T ) = −h(p− k + h− 1)δ∗(h−1)(T ). Since δ∗(h)(T ) = 0 for h ≥ k + 1 and h(p − k + h − 1) 6= 0 for any h ≥ 1, we get form this relation that δ∗(h−1)(T ) = 0 for any h ≥ 1, in particular for h = 1, we get T = 0. 2. The same argument as 1. using the third formula in Lemma 4.1. 3. Let T ∈ SpPp such that i−→r T = 0. From Lemma 4.1, we get i−→r δ ∗(T ) = 0. Since δ∗(T ) ∈ Sp+1Pp−1 and from 1. we deduce that δ ∗(T ) = 0 and hence Keri−→r ⊂ Kerδ ∗. The same argument using Lemma 4.1 and 2. will give the other inclusion. Q.E.D. By combining (20) and (21) with Lemma 4.2, we obtain the following lemma. Lemma 4.3 We have: 1. if k < p SpHδ0k = Sp+lHδ0k−l ∩Kerδ 2. if k > p SpHδ0k = Sp−lHδ0k+l ∩Keri−→r 3. If k = p, for any 0 ≤ l ≤ p, Sp+lHδ0p−l ∩Kerδ = δ∗l Sp−lHδ0p+l ∩Keri−→r SpHδ0p = Sp+lHδ0p−l ∩Kerδ Sp−lHδ0p+l ∩Keri−→r Now, we use Corollary 2.2 to show that the decompositions of SpHδ0k given in Lemma 4.3 are composed by eigenspaces of ∆Sn. Theorem 4.1 We have: 1. If k ≤ p, for any 0 ≤ q ≤ k and any T ∈ i−→r (k−q) Sp+k−qHδ0q ∩Kerδ ∗T = ((k + p)(n + p+ k − 2q − 1) + 2q(q − 1)) i∗T ; 2. If k ≥ p, for any 0 ≤ q ≤ p and for any T ∈ δ∗(p−q) SqHδ0k+p−q ∩Keri−→r ∗T = ((k + p)(n+ p+ k − 2q − 1) + 2q(q − 1)) i∗T. Proof. 1. Let T = i−→r (k−q)(T0) with T0 ∈ S p+k−qHδ0q ∩ Kerδ ∗. We have from Corollary 2.2 ∗T = i∗ 2p(p− 1)T + (n− 2p− 1)L−→r T + L−→r ◦ L−→r T +2δ∗(i−→r T )− 2Tr(T )⊙ <,> We have TrT = 0, L−→r = (k + p)T and L−→r ◦ L−→r T = (k + p) Moreover, by using Lemma 4.1, we have 2δ∗(i−→r T ) = 2δ ∗(i−→r (k−q+1)T0) δ∗(T0)=0 = 2(k − q + 1)(p+ k − q − q − k + q − 1 + 1)i−→r (k−q)T0 = 2(k − q + 1)(p− q)T. Hence ∗T = (2p(p−1)+(n−2p−1)(k+p)+(k+p)2+2(p−q)(k−q+1))i∗T. One can deduce the desired relation by remarking that 2p(p− 1) + 2(p− q)(k − q + 1) = 2(k + p)(p− q) + 2q(q − 1). 2. This follows by the same calculation as 1. Q.E.D. From the fact that SpHδk −→ S is injective and its image is dense in SpSn, from (15), and from Lemma 4.3 and Theorem 4.1, note that we have actually proved that the eigenvalues of ∆Sn acting on S pSn belongs to {(k + p− 2l)(n+ p+ k − 2l − 2q − 1) + 2q(q − 1), k ∈ IN, 0 ≤ l ≤ [ ], 0 ≤ q ≤ min(k, p− 2l) Our next goal is to sharpen this result by computing dimSpHδ0k ∩Kerδ k ≤ p and dimSpHδ0k ∩Keri−→r if k ≥ p. Lemma 4.4 We have the following formulas: 1. dimSpHδk = dimS pPk − dimS pPk−2 − dimS p−1Pk−1 + dimS p−1Pk−3, 2. dimSpHδ0k = dimS pHδk − dimS p−2Hδk , 3. dim(SpHδ0k ∩Kerδ ∗) = dimSpHδ0k − dimS p+1Hδ0k−1 (k ≤ p), 4. dim(SpHδ0k ∩Keri−→r ) = dimS pHδ0k − dimS p−1Hδ0k+1 (k ≥ p). Note that we use the convention that SpPk = S pHδk = S pHδ0k = 0 if k < 0 or p < 0. Proof. 1. The formula is a consequence of (14), the relation (r2SpPk−2) ∩ (dr 2 ⊙ Sp−1Pk−1) = r 2(dr2 ⊙ Sp−1Pk−3) and the fact that dr2 ⊙ . : SpPk −→ S p+1Pk+1 is injective. 2. The formula is a consequence of (15). 3. The formula is a consequence of (20) and Lemma 4.2. 4. The formula is a consequence of (21) and Lemma 4.2. Q.E.D. A straightforward calculation using Lemma 4.4 and the formula dimSpPk = (n+ p)! (n+ k)! gives dimSpHδ0k ∩ Kerδ ∗ if k ≤ p and dimSpHδ0k ∩ Keri−→r if k ≥ p. We summarize the results on the following Table. Table II Space Dimension Conditions on k and p S0Hδ0k ∩Keri−→r (n+ k − 2)!(n+ 2k − 1) k!(n− 1)! k ≥ 0 SpHδ00 ∩Kerδ (n+ p− 2)!(n+ 2p− 1) p!(n− 1)! p ≥ 0 S1Hδ0k ∩Keri−→r (n + k − 3)!k(n + 2k − 1)(n+ k − 1) (n− 2)!(k + 1)! k ≥ 1 SpHδ01 ∩Kerδ (n + p− 3)!p(n+ 2p− 1)(n+ p− 1) (n− 2)!(p+ 1)! p ≥ 1 SpHδ0k ∩Kerδ (n + k − 4)!(n+ p− 3)!(n+ p+ k − 2) k!(p+ 1)!(n− 1)!(n− 2)! (n− 2)(n+ 2k − 3)(n+ 2p− 1)(p− k + 1) 2 ≤ k ≤ p SpHδ0k ∩Keri−→r (n + k − 3)!(n+ p− 4)!(n+ p+ k − 2) (k + 1)!p!(n− 1)!(n− 2)! (n− 2)(n+ 2k − 1)(n+ 2p− 3)(k − p+ 1) k ≥ p ≥ 2 Remark 4.1 Note that, for n = 2, we have dim(SpHδ0k ∩Kerδ ∗) = 0 for 2 ≤ k ≤ p, dim(SpHδ0k ∩Keri−→r ) = 0 for k ≥ p ≥ 2. For simplicity we introduce the following notations. (k, l, q) ∈ IN3, 0 ≤ l ≤ [ ], 0 ≤ k ≤ p− 2l, 0 ≤ q ≤ k (k, l, q) ∈ IN3, 0 ≤ l ≤ [ ], k > p− 2l, 0 ≤ q ≤ p− 2l V kq,l = i−→r Sp−2l+k−qHδ0q ∩Kerδ ⊙ <,>l for (k, l, q) ∈ S0, W kq,l = δ ∗(p−2l−q) SqHδ0p−2l+k−q ∩Keri−→r ⊙ <,>l for (k, l, q) ∈ S1. Let us summarize all the results above. Theorem 4.2 1. For n = 2, we have: (a) The set of the eigenvalues of ∆S2 acting on S pS2 is (k + p− 2l)(p+ k − 2l + 1), k ∈ IN, 0 ≤ l ≤ [ (b) The eigenspace associated to the eigenvalue λ(k, l) = (k + p − 2l)(k + p− 2l + 1) is given by Vλ(k,l) = min(l,[ k V k−2a0,l−a ⊕ V k+1−2a 1,l−a if 0 ≤ k ≤ p− 2l min(l,[ k W k−2a0,l−a ⊕W k+1−2a 1,l−a if k > p− 2l; (c) The multiplicity of λ(k, l) is given by m(λ(k, l)) = 2(min(l, [ ]) + 1)(1 + 2p+ 2k − 4l). 2. For n ≥ 3, we have: (a) The set of the eigenvalues of ∆Sn acting on S pSn is {(k + p− 2l)(n + p+ k − 2l − 2q − 1) + 2q(q − 1), k ∈ IN, 0 ≤ l ≤ [ ], 0 ≤ q ≤ min(k, p− 2l) (b) The space SpHδk = ( (k,l,q)∈S0 V kq,l)⊕ ( (k,l,q)∈S1 W kq,l) is dense in SpSn and, for any (k, q, l) ∈ S0 (resp. (k, q, l) ∈ S1), V kq,l (resp. W q,l) is a subspace of the eigenspace associated to the eigenvalue (k + p− 2l)(n+ p+ k − 2l − 2q − 1) + 2q(q − 1); (c) The dimensions of V kq,l and W q,l are given in Table II since dimV kq,l = dim Sp−2l+k−qHδ0q ∩Kerδ for (k, l, q) ∈ S0, dimW kq,l = dim SqHδ0p−2l+k−q ∩Keri−→r for (k, l, q) ∈ S1. References [1] E. Bedford and T. Suwa, Eigenvalues of Hopf manifolds, American Mathemaical Society, Vol. 60 (1976), 259-264. [2] B. L. Beers and R. S. Millman, The spectra of the Laplace-Beltrami operator on compact, semisimple Lie groups, Amer. J. Math., 99 (4) (1975), 801-807. [3] M. Berger and D. Ebin, Some decompositions of the space of symmet- ric tensors on Riemannian manifolds, J. Diff. Geom., 3 (1969), 379-392. [4] M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Math., Vol 194, Springer Verlag (1971). [5] A. Besse, Einstein manifolds, Springer-Verlag, Berlin-Hiedelberg-New York (1987). [6] M. Boucetta , Spectre des Laplaciens de Lichnerowicz sur les sphères et les projectifs réels, Publicacions Matemàtiques, Vol. 43 (1999), 451-483. [7] M. Boucetta , Spectre du Laplacien de Lichnerowicz sur les projectifs complexes, C. R. Acad. Sci. Paris, t. 333, Série I, (2001), 571-576. [8] S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl., 54 (1975), 259-289. [9] G. W. Gibbons and M. J. Perry, Quantizing gravitational instantons, Nuclear Physics B, Vol. 146, Issue I (1978), 90-108. [10] A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on Sn and P n( |C), Osaka J. Math., 15 (3) (1978), 515-546. [11] I. Iwasaki and K. Katase, On the spectra of Laplace operator on ∧∗(Sn), Proc. Japan Acad., 55, Ser. A (1979), 141-145. [12] E. Kaneda, The spectra of 1-forms on simply connected compact ir- reducible Riemannian symmetric spaces, J. Math. Kyoto Univ., 23 (1983), 369-395 and 24 (1984), 141-162. [13] A. Lévy-Bruhl-Laperrière, Spectre de de Rham-Hodge sur les formes de degré 1 des sphères de IRn (n ≥ 6), Bull. Sc. Math., 2e série, 99 (1975), 213-240. [14] A. Lévy-Bruhl-Laperrière, Spectre de de Rham-Hodge sur l’espace projectif complexe, C. R. Acad. Sc. Paris 284 Série A (1977), 1265-1267. [15]A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Etude Sci. Publ. Math., 10 (1961). [16] K. Mashimo, Spectra of Laplacian on G2/SO(4), Bull. Fac. Gen. Ed. Tokyo Univ. of Agr. and Tech. 26 (1989), 85-92. [17] K. Mashimo, On branching theorem of the pair (G2, SU(3)), Nihonkai Math. J., Vol. 8 No. 2 (1997), 101-107. [18] K. Mashimo, Spectra of the Laplacian on the Cayley projective plane, Tsukuba J. Math., Vol. 21 No. 2 (1997), 367-396. [19] R. Michel, Problème d’analyse géométrique liés à la conjecture de Blaschke, Bull. Soc. Math. France, 101 (1973), 17-69. [20] K. Pilch and N. Schellekens, Formulas of the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces, J. Math. Phys., 25 (12) (1984), 3455-3459. [21] C. Tsukamoto, The sepctra of the Laplace-Beltrami operator on SO(n + 2)/SO(2) × SO(n) and Sp(n + 1)/Sp(1) × Sp(n), Osaka J. Math. 18 (1981), 407-226. [22] N. P. Warner, The spectra of operators on |CP n, Proc. R. Soc. Lond. A 383 (1982), 217-230.

0704.1364

CCD BV survey of 42 open clusters

Astronomy & Astrophysics manuscript no. 6588 c© ESO 2018 October 23, 2018 CCD BV survey of 42 open clusters G. Maciejewski and A. Niedzielski Centrum Astronomii Uniwersytetu Mikołaja Kopernika, ul. Gagarina 11, Pl-87100 Toruń, Poland e-mail: [email protected] Received 10 January 2006 / Accepted 10 January 2006 ABSTRACT Aims. We present results of a photometric survey whose aim was to derive structural and astrophysical parameters for 42 open clusters. While our sample is definitively not representative of the total open cluster sample in the Galaxy, it does cover a wide range of cluster parameters and is uniform enough to allow for simple statistical considerations. Methods. BV wide-field CCD photometry was obtained for open clusters for which photometric, structural, and dynamical evolution parameters were determined. The limiting and core radii were determined by analyzing radial density profiles. The ages, reddenings, and distances were obtained from the solar metallicity isochrone fitting. The mass function was used to study the dynamical state of the systems, mass segregation effect and to estimate the total mass and number of cluster members. Results. This study reports on the first determination of basic parameters for 11 out of 42 observed open clusters. The angular sizes for the majority of the observed clusters appear to be several times larger than the catalogue data indicate. The core and limiting cluster radii are correlated and the latter parameter is 3.2 times larger on average. The limiting radius increases with the cluster’s mass, and both the limiting and core radii decrease in the course of dynamical evolution. For dynamically not advanced clusters, the mass function slope is similar to the universal IMF slope. For more evolved systems, the effect of evaporation of low-mass members is clearly visible. The initial mass segregation is present in all the observed young clusters, whereas the dynamical mass segregation appears in clusters older than about log(age) = 8. Low-mass stars are deficient in the cores of clusters older than log(age) = 8.5 and not younger than one relaxation time. Key words. open clusters and associations: general; stars: evolution 1. Introduction Open clusters are not trivial stellar systems, and their dynami- cal evolution is not yet fully understood. Most of them are not very populous assemblages of a few hundred stars. The least massive clusters do not last longer than a few hundred Myr (Bergond et al. 2001). The dynamics of more massive and pop- ulous clusters is driven by internal forces to considerable de- gree, which leads to evaporation of low-mass members and to a mass segregation effect. Moreover, cluster member stars in- cessantly evolve along stellar evolution paths, which makes an open cluster a vivid system evolving in time; hence, star clus- ters are considered excellent laboratories of stellar evolution and stellar-system dynamics (Bonatto & Bica 2005). To obtain a complete picture of a cluster, it is necessary to study not only its most dense region (center) but also the ex- panded and sparse coronal region (halo). As wide-field CCD imaging of open clusters is usually difficult, the majority of studies published so far are based on observations of the cen- tral, most populous, and relatively dense core region. Nilakshi et al. (2002) have presented the first, to our knowledge, results of an extensive study of spatial structure of 38 rich open clus- Send offprint requests to: G. Maciejewski ters based on star counts performed on images taken from the Digital Sky Survey (DSS). Bonatto & Bica (2005) and Bica & Bonatto (2005) analyzed over a dozen open clusters in detail using 2MASS photometry. In the former paper, the possible existence of a fundamental plane of several open clusters pa- rameters was suggested. More recently, Sharma et al. (2006) published results of studies concerning cores and coronae evo- lution of nine open clusters based on projected radial profiles analysis. In this paper a sample of 42 northern open clusters of linear diameters, distances, ages, and number of potential members from a wide range is investigated in detail based on wide-field BV CCD photometry. The basic parameters and CCD photom- etry of 11 clusters were obtained for the first time. This paper is organized as follows. In Sect. 2 the sample selection, observations, and data reduction are described. In Sect. 3 the radial structure of clusters under investigation is presented based on star counts. Results of color-magnitude- diagram fitting are given in Sect. 4. The mass functions of tar- get clusters are analyzed in Sect. 5. The obtained photometric parameters for individual clusters under investigation and the reliability of the results are discussed in Sect. 6, while in Sect. 7 http://arxiv.org/abs/0704.1364v1 2 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters Table 1. List of observed open clusters with redetermined equatorial and Galactic coordinates. Name Coordinates J2000.0 l b (hhmmss±ddmmss) (◦) (◦) King 13 001004+611215 117.9695 −1.2683 King 1 002204+642250 119.7626 1.6897 King 14 003205+630903 120.7486 0.3612 NGC 146 003258+632003 120.8612 0.5367 Dias 1 004235+640405 121.9639 1.2130 King 16 004345+641108 122.0949 1.3263 Berkeley 4 004501+642305 122.2377 1.5216 Skiff J0058+68.4 005829+682808 123.5814 5.6060 NGC 559 012935+631814 127.2008 0.7487 NGC 884 022223+570733 135.0659 −3.5878 Tombaugh 4 022910+614742 134.2071 1.0815 Czernik 9 023332+595312 135.4172 −0.4869 NGC 1027 024243+613801 135.7473 1.5623 King 5 031445+524112 143.7757 −4.2866 King 6 032750+562359 143.3584 −0.1389 Berkeley 9 033237+523904 146.0621 −2.8275 Berkeley 10 033932+662909 138.6158 8.8785 Tombaugh 5 034752+590407 143.9374 3.5924 NGC 1513 040946+492828 152.5955 −1.6243 Berkeley 67 043749+504647 154.8255 2.4896 Berkeley 13 045552+524800 155.0851 5.9244 Czernik 19 045709+284647 174.0986 −8.8321 Berkeley 15 050206+443043 162.2580 1.6187 NGC 1798 051138+474124 160.7028 4.8463 Berkeley 71 054055+321640 176.6249 0.8942 NGC 2126 060229+495304 163.2169 13.1294 NGC 2168 060904+241743 186.6426 2.2061 NGC 2192 061517+395019 173.4298 10.6469 NGC 2266 064319+265906 187.7759 10.3003 King 25 192432+134132 48.8615 −0.9454 Czernik 40 194236+210914 57.4762 −1.1003 Czernik 41 195101+251607 62.0054 −0.7010 NGC 6885 201140+263213 65.5359 −3.9766 IC 4996 201631+373919 75.3734 1.3158 Berkeley 85 201855+374533 75.7257 0.9812 Collinder 421 202310+414135 79.4299 2.5418 NGC 6939 203130+603922 95.8982 12.3012 NGC 6996 205631+443549 85.4401 −0.5039 Berkeley 55 211658+514532 93.0267 1.7978 Berkeley 98 224238+522316 103.8561 −5.6477 NGC 7654 232440+613451 112.7998 0.4279 NGC 7762 234956+680203 117.2100 5.8483 the relations between structural and dynamical parameters are presented and discussed. Sect. 8 contains the final conclusions. 2. Observations and reduction In this survey the cluster diameter and location on the sky were the main criteria of target selection. In the first step we need the New catalog of optically visible open clusters and candi- dates by Dias et al. (2002) to select Galactic clusters with ap- parent diameters ranging from 5 to 20 arcmin and a declina- tion larger than +10◦. The former limitation guaranteed that the entire cluster with its possible extended halo would fit in the instrument’s field of view. The latter one comes from the observatory location and eliminates potential targets that can- not be observed at elevations higher than 45◦. We found 295 open clusters fulfilling these criteria. In the second step, all small and relatively populous clusters were rejected from the sample. In these clusters stellar images are blended, making some portion of stars undetectable due to a considerable seeing of about 5′′(FWHM) at the observing loca- tion. To avoid poorly populated objects, hardly distinguishable from the stellar background, the minimal number of potential cluster members was set for 20. Moreover, to obtain at least 2 mag of the main sequence coverage, only clusters for which the brightest stars were brighter than 16 mag in V were selected, since the limiting magnitude was estimated as 18.5–19.5 mag. All of the selected clusters were also visually inspected on DSS images. Finally, the sample of 62 open clusters was adopted. We preferred previously unstudied open clusters, for which no ba- sic parameters were available in the literature, and these clus- ters were observed with higher priority. In this paper we present results for 42 open clusters, which are listed in Table 1. The collected photometric data for 20 unstudied clusters deny their cluster nature, suggesting that they constitute only an accidental aggregation of stars on the sky. These objects will be discussed in a forthcoming paper (Maciejewski & Niedzielski 2007, in preparation) where extensive, detailed analysis of ev- ery object will be presented. 2.1. Observations Observations were performed with the 90/180 cm Schmidt- Cassegrain Telescope located at the Astronomical Observatory of the Nicolaus Copernicus University in Piwnice near Toruń, Poland. A recently upgraded telescope was used in Schmidt imaging mode with a correction plate with a 60 cm diameter and a field-flattening lens mounted near the focal plane to com- pensate for the curvature typical of Schmidt cameras. The telescope was equipped with an SBIG STL-11000 CCD camera with a KAI-11000M CCD detector (4008 × 2672 pixels × 9 µm). The field of view of the instrument was 72 ar- cmin in declination and 48 arcmin in right ascension with the scale of 1.08 arcsec per pixel. The camera was equipped with a filter wheel with standard UBVR Johnson-Cousins filters. The 2 × 2 binning was used to increase the signal-to-noise ratio. Observations were carried out between September 2005 and February 2006 (see Table 2 for details). A set of 4 expo- sures in B and V filters was acquired for each program field: 2 long (600 s) and 2 short (60 s) exposures in every filter. For open clusters containing very bright stars, 2 extra very short (10 s) exposures in each filter were obtained. One of Landolt’s (1992) calibration fields was observed several times during each night, at wide range of airmasses. The field was observed between succeeding program exposures, in practice every hour. G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 3 Table 2. Calibration (aV , aB, bV , bB, a(V−B), and b(V−B)) and atmospheric extinction (kV and kB) coefficients for individual nights. Date kV aV aB a(B−V) kB bV bB b(B−V) 05.09.2005 0.3072 -0.0922 0.1876 1.2799 0.4648 20.4407 20.5982 0.1573 06.09.2005 0.4614 -0.0886 0.1825 1.2694 0.6387 20.4518 20.5750 0.1230 07.09.2005 0.5689 -0.0782 0.1775 1.2579 0.6790 20.5841 20.5501 -0.0349 08.09.2005 0.5346 -0.0997 0.1752 1.2749 0.7134 20.5139 20.6531 0.1394 04.10.2005 0.1804 -0.1030 0.1754 1.2784 0.3136 20.1073 20.2526 0.1452 05.10.2005 0.2448 -0.0990 0.1946 1.2935 0.3767 20.2757 20.3897 0.1140 06.10.2005 0.3447 -0.0911 0.1682 1.2593 0.4640 20.4129 20.4985 0.0855 07.10.2005 0.3621 -0.0975 0.2251 1.3227 0.4879 20.4133 20.3996 -0.0137 08.10.2005 0.1810 -0.0830 0.1906 1.2736 0.3014 20.1319 20.2151 0.0832 23.02.2006 0.3968 -0.1023 0.1458 1.2642 0.7820 20.9667 20.7216 0.5354 26.02.2006 0.1911 -0.1281 0.1142 1.2512 0.3658 21.0104 21.2547 0.2419 2.2. Data reduction and calibration The collected observations were reduced with the software pipeline developed for the Semi-Automatic Variability Search1 sky survey (Niedzielski et al. 2003, Maciejewski & Niedzielski 2005). CCD frames were processed with a standard proce- dure including debiasing, subtraction of dark frames, and flat- fielding. The instrumental coordinates of stars were transformed into equatorial ones based on positions of stars brighter than 16 mag extracted from the Guide Star Catalog (Lasker et al. 1990). The instrumental magnitudes in B and V bands were corrected for atmospheric extinction and then transformed into the standard system. The preliminary analysis, including determining the width of stellar profiles, calculating of the atmospheric extinction co- efficients in both filters, and determining the transformation equations between instrumental magnitudes and the standard ones, was performed based on observations of Landolt fields. The mean FWHM of the stellar profiles was calculated for each Landolt field frame acquired during one night. The aperture ra- dius used for photometric measurements was calculated as 3σ of the maximum mean FWHM obtained from the Landolt field observed during a night, and in practice it was between 6 and 8 arcsec. The atmospheric extinction coefficients kV and kB were de- termined for each night from 6–8 observations of the adopted Landolt field at airmasses X between 1.6 and 3.2. Typically more than 1000 stars in V and 750 in B were detected in ev- 1 http://www.astri.uni.torun.pl/˜gm/SAVS ery Landolt field frame and the extinction coefficient in a given filter was determined for each star from changes in its raw in- strumental magnitudes with X. The median value was taken as the one best representing a night. The values of the atmospheric extinction coefficients for individual nights are listed in Table 2. The raw instrumental magnitudes braw, vraw of stars in the Landolt field were corrected for the atmospheric extinction, and instrumental magnitudes outside the atmosphere b, v were cal- culated as b = braw − kbX , (1) v = vraw − kvX . (2) Next, the mean values of instrumental magnitudes outside the atmosphere were calculated for every star. In every Landolt field there were about 30 standard stars that were used to deter- mine coefficients in the calibration equations of the form: V − v = aV(b − v) + bV , (3) B − b = aB(b − v) + bB , (4) B − V = a(V−B)(b − v) + b(V−B) , (5) where B, V are standard magnitudes and b, v are the mean in- strumental ones corrected for the atmospheric extinction. The detailed list of transformation coefficients for each night is pre- sented in Table 2. The final list of stars observed in all fields contains equa- torial coordinates (J2000.0), V magnitude, and (B–V) color in- dex. The files with data for individual open clusters are avail- able on the survey’s web site2. 3. Radial structure Analysis of the radial density profiles (RDP) is a commonly used method for investigating cluster structure. It loses infor- mation on 2-dimensional cluster morphology but it provides a uniform description of its structure with a few basic parameters instead. Defining the cluster’s center is essential for the RDP analysis. Since the coordinates of clusters as given in Dias et al. (2002) were found in several cases to be different from the actual ones, we started with redetermination of the centers for all the open clusters in our sample. 3.1. Redetermination of central coordinates Our algorithm for redetermining the central coordinates started with the approximated coordinates taken from the compilation by Dias et al. (2002) or from a tentative approximate position when the catalogue data were found to be inconsistent with the cluster position as seen on DSS charts. To determine the center position more accurately, two per- pendicular stripes (20 arcmin long and 3–6 arcmin wide, de- pending on cluster size) were cut along declination and right as- cension starting from the approximate cluster center, and stars were counted within every stripe. The histogram of star counts 2 http://www.astri.uni.torun.pl/˜gm/OCS 4 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters Table 3. Structural parameters obtained from the King profile Name rlim rcore f0 fbg (′) (′) stars arcmin2 stars arcmin2 (1) (2) (3) (4) (5) King 13 11.8 3.3±0.3 5.02±0.27 3.09±0.11 King 1 12.3 2.1±0.1 5.16±1.92 1.76±0.05 King 14 9.0 2.3±0.4 2.84±0.31 4.97±0.09 NGC 146 2.7 1.2±0.3 5.61±0.80 4.87±0.16 Dias 1 2.3 0.3±0.1 13.29±6.33 2.96±0.07 King 16 8.8 1.9±0.2 4.70±0.27 3.25±0.06 Berkeley 4 3.1 1.3±0.3 2.84±0.40 3.57±0.09 Skiff J0058+68.4 10.9 3.8±0.4 3.66±0.20 2.04±0.09 NGC 559 14.5 2.3±0.2 6.72±0.32 2.38±0.09 NGC 884 10.1 5.8±1.3 1.08±0.11 0.94±0.09 Tombaugh 4 5.6 1.1±0.1 12.91±0.53 1.62±0.07 Czernik 9 3.3 0.8±0.1 6.36±0.59 1.43±0.08 NGC 1027 10.3 3.3±0.5 1.63±0.13 0.81±0.05 King 5 10.9 2.4±0.2 5.81±0.29 1.96±0.09 King 6 10.9 3.6±0.4 1.55±0.09 0.62±0.04 Berkeley 9 7.3 1.2±0.1 3.88±0.18 1.16±0.03 Berkeley 10 8.3 1.3±0.1 6.36±0.39 1.28±0.07 Tombaugh 5 11.8 2.2±0.4 3.75±0.35 2.41±0.10 NGC 1513 9.2 3.7±0.6 2.47±0.20 1.04±0.09 Berkeley 67 5.2 1.9±0.1 3.53±0.16 0.92±0.04 Berkeley 13 6.1 1.4±0.1 5.20±0.32 2.13±0.06 Czernik 19 5.5 1.4±0.2 4.12±0.30 1.41±0.07 Berkeley 15 7.6 1.4±0.1 5.74±0.28 1.53±0.05 NGC 1798 9.0 1.3±0.1 9.55±0.28 3.14±0.05 Berkeley 71 3.3 1.2±0.2 6.04±0.58 1.99±0.09 NGC 2126 10.0 1.9±0.3 1.78±0.15 0.93±0.04 NGC 2168 9.8 4.8±0.5 2.27±0.16 1.03±0.08 NGC 2192 4.6 1.4±0.2 2.19±0.18 0.57±0.04 NGC 2266 5.9 1.2±0.1 7.69±0.50 2.32±0.08 King 25 6.3 2.3±0.3 4.93±0.34 1.36±0.13 Czernik 40 8.5 2.3±0.3 8.44±0.53 3.40±0.18 Czernik 41 5.6 1.7±0.2 3.96±0.28 1.65±0.09 NGC 6885 8.6 2.4±0.3 2.74±0.21 2.66±0.08 IC 4996 2.2 1.2±0.4 3.27±0.58 4.61±0.14 Berkeley 85 5.0 1.5±0.2 4.84±0.43 2.94±0.09 Collinder 421 6.1 1.1±0.3 2.67±0.46 0.93±0.07 NGC 6939 15.2 2.2±0.1 6.92±0.22 2.62±0.06 NGC 6996 2.1 0.9±0.3 3.58±0.78 3.66±0.10 Berkeley 55 6.0 0.7±0.1 7.63±0.67 1.45±0.04 Berkeley 98 4.6 2.1±0.3 4.00±0.30 5.40±0.08 NGC 7654 11.2 5.0±0.5 4.39±0.21 3.36±0.20 NGC 7762 9.5 2.4±0.2 5.06±0.29 1.45±0.08 was built along each stripe with a bin size of 1.0 arcmin for the cluster with a diameter larger than 10 arcmin and 0.5 arcmin for the smaller ones. The bin with the maximum value in both coordinates was taken as the new cluster center. This proce- dure was repeated until the new center position became stable, usually a few times. The accuracy of the new coordinates was determined by the histogram’s bin size and was assumed to be 1 arc min typically. The new equatorial and Galactic coordinates are listed in Table 1. 3.2. Analysis of radial density profiles The RDPs were built by calculating the mean stellar surface density in concentric rings, 1 arcmin wide, centered on the re- determined cluster center. If Ni denotes the number of stars counted in the ith ring of the inner radius Ri and outer Ri+1 the stellar surface density ρi can be expressed as π(R2i+1 − R . (6) The density uncertainty in each ring was estimated assuming the Poisson statistics. The basic structural parameters were de- rived by least-square fitting the two-parameter King (1966) sur- face density profile ρ(r) = fbg + rcore , (7) where f0 is the central density, fbg the density of the stellar background in a field, and rcore the core radius defined as the distance between the center and the point where ρ(r) becomes half of the central density. The RDPs and the fitted King profiles are shown in Fig. 1 where the densities were normalized (after background density subtraction) to the central value. As one can note, all clusters can be described by the King profile reasonably well, and even for relatively small objects, no significant systematic deviation is noticeable. The RDP of Berkeley 67 indicates the presence of a strong background gradient, so the background level was artificially straightened for the profile fitting procedure. In sev- eral cases (NGC 146, Dias 1, Berkeley 4, and NGC 7654), the RDPs were cut off at r smaller than expected. That is because these clusters were located in a field centered on another open cluster and were observed serendipitously. The RDPs were also used to determine the limiting radius rlim, the radius where cluster’s outskirts merge with the stellar background. This is not a trivial task and properly determin- ing rlim is important for further investigations. Therefore a uni- form algorithm was developed and applied to all clusters. In its first step the boundary density level ρb was calculated for every RDP as ρb = fbg + 3σbg , (8) where σbg denotes the background density error derived from the King profile fit. Next, moving from the cluster center (r = 0 arcmin) outwards, the first point below ρb was sought. When this ith point was encountered, the algorithm was checked to see if farther-out points were also located below ρb. If this con- dition was fulfilled, the limiting radius was interpolated as the crossing point between the boundary density level ρb and the line passing through the (i–1)th and ith points. When farther- out (at least two) points following the ith point were located above the boundary density level ρb, the algorithm skipped the ith point and continued seeking the next point located below ρb, and the procedure was repeated. As the formal error of rlim determination, one half of RDP bin size was taken, i.e. 0.5 arc min. Due to the limited field of view for several clusters (for instance King 13, King 16, NGC 884, NGC 1027, King 6, G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 5 1 King 13 r (arcmin) King 1 r (arcmin) King 14 r (arcmin) NGC 146 r (arcmin) Dias 1 r (arcmin) King 16 r (arcmin) 1 Berkeley 4 r (arcmin) Skiff J0058+68.4 r (arcmin) NGC 559 r (arcmin) NGC 884 r (arcmin) Tombaugh 4 r (arcmin) Czernik 9 r (arcmin) 1 NGC 1027 r (arcmin) King 5 r (arcmin) King 6 r (arcmin) Berkeley 9 r (arcmin) Berkeley 10 r (arcmin) Tombaugh 5 r (arcmin) 1 NGC 1513 r (arcmin) Berkeley 67 r (arcmin) Berkeley 13 r (arcmin) Czernik 19 r (arcmin) Berkeley 15 r (arcmin) NGC 1798 r (arcmin) 1 Berkeley 71 r (arcmin) NGC 2126 r (arcmin) NGC 2168 r (arcmin) NGC 2192 r (arcmin) NGC 2266 r (arcmin) King 25 r (arcmin) 1 Czernik 40 r (arcmin) Czernik 41 r (arcmin) NGC 6885 r (arcmin) IC 4996 r (arcmin) Berkeley 85 r (arcmin) Collinder 421 r (arcmin) 0 10 20 NGC 6939 r (arcmin) 0 10 20 NGC 6996 r (arcmin) 0 10 20 Berkeley 55 r (arcmin) 0 10 20 Berkeley 98 r (arcmin) 0 10 20 NGC 7654 r (arcmin) 0 10 20 NGC 7762 r (arcmin) Fig. 1. Radial density profiles normalized to the central value after background level subtraction. NGC 1513, NGC 2168, NGC 6885, NGC 6939, NGC 7654, and NGC 7762) our determination of rlim may in fact represent a lower limit. The results of the RDP analysis (limiting radius rlim, core radius rcore, central density f0, and background level fbg) are listed in Table 3 in Cols. 2, 3, 4, and 5, respectively. 4. The color–magnitude diagrams The collected V and (B−V) data allowed us to construct color– magnitude diagrams (CMDs) for all observed clusters. Since the field of view was wide and the majority of clusters oc- curred relatively small, the CMD for the cluster region could be decontaminated for the field stars’ contribution. While in general it is impossible to point out individual cluster members based only on photometry, the contribution from field stars can be removed from the cluster CMD in a statistical manner. The algorithm applied to our data was based on ideas presented in Mighell et al. (1996) and discussed in Bica & Bonato (2005). Two separate CMDs were built: one for the cluster and one for an offset field. The offset field was defined as a ring of the inner radius rlim+1 arcmin and the outer radius was set as large as possible to fit within the observed CCD frame but avoiding contribution from other clusters, typically 15–19 arcmin. Both CMDs were divided into 2-dimensional bins of ∆V = 0.4 mag and ∆(B − V) = 0.1 mag size (both values being fixed after a series of tests, as a compromise between resolution and the star numbers in individual boxes). The number of stars within each box was counted. Then the cleaned cluster CMD was built by subtracting the number of stars from the corresponding off- set box from the number of stars in a cluster box. The latter number was weighted with the cluster to offset field surface ra- tio. Knowing the number of cluster stars occupying any given box on clean CMD, the algorithm randomly chose the required number of stars with adequate V magnitude and B − V color index from the cluster field. Finally, the list of stars in aech 6 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters Table 4. Astrophysical parameters obtained from isochrone fitting. Name log(age) E(B − V) (M − m) d Rlim Rcore (mag) (mag) (kpc) (pc) (pc) (1) (2) (3) (4) (5) (6) (7) King 13 8.4 0.86+0.14 −0.12 15.49 +0.55 −0.58 3.67 +1.37 −1.30 12.6 −4.5 3.54 +1.32 −1.26 King 1 9.6 0.76+0.09 −0.09 12.53 +0.32 −0.51 1.08 +0.23 −0.33 3.9 −1.2 0.67 +0.14 −0.20 King 14 7.0 0.58+0.10 −0.10 13.75 +0.85 −0.73 2.46 +1.35 −0.94 6.5 −2.5 1.62 +0.89 −0.62 NGC 146 7.6 0.56+0.07 −0.07 13.97 +0.50 −0.61 2.80 +0.84 −0.89 2.2 −0.7 0.94 +0.28 −0.30 Dias 1 7.1 1.08+0.13 −0.11 14.49 +1.04 −0.57 1.69 +1.21 −0.58 1.2 −0.4 0.17 +0.12 −0.06 King 16 7.0 0.89+0.10 −0.13 14.18 +0.72 −0.87 1.92 +0.88 −0.85 4.9 −2.2 1.09 +0.50 −0.48 Berkeley 4 7.1 0.83+0.08 −0.08 14.53 +1.14 −0.69 2.46 +1.86 −0.86 2.2 −0.8 0.94 +0.71 −0.33 Skiff J0058+68.4 9.1 0.85+0.12 −0.13 13.48 +0.57 −0.49 1.48 +0.55 −0.50 4.7 −1.6 1.63 +0.61 −0.55 NGC 559 8.8 0.68+0.11 −0.12 13.79 +0.39 −0.66 2.17 +0.56 −0.82 9.2 −3.5 1.48 +0.38 −0.56 NGC 884 7.1 0.56+0.06 −0.06 14.08 +0.43 −0.57 2.94 +0.75 −0.87 8.6 −2.5 5.00 +1.27 −1.47 Tombaugh 4 9.0 1.01+0.08 −0.10 14.81 +0.59 −0.37 2.17 +0.78 −0.58 3.5 −1.0 0.67 +0.24 −0.18 Czernik 9 8.8 1.05+0.12 −0.14 14.35 +0.36 −0.76 1.66 +0.41 −0.70 1.6 −0.7 0.39 +0.10 −0.17 NGC 1027 8.4 0.41+0.12 −0.11 11.34 +0.35 −0.53 1.03 +0.25 −0.34 3.1 −1.0 1.00 +0.24 −0.33 King 5 9.1 0.67+0.09 −0.10 13.82 +0.32 −0.61 2.23 +0.46 −0.77 7.1 −2.4 1.56 +0.32 −0.54 King 6 8.4 0.53+0.12 −0.11 11.17 +0.55 −0.47 0.80 +0.29 −0.25 2.6 −0.8 0.85 +0.31 −0.27 Berkeley 9 9.6 0.79+0.08 −0.08 12.03 +0.34 −0.52 0.82 +0.18 −0.25 1.7 −0.5 0.29 +0.06 −0.09 Berkeley 10 9.0 0.71+0.10 −0.08 13.46 +0.70 −0.40 1.79 +0.80 −0.46 4.3 −1.1 0.70 +0.31 −0.18 Tombaugh 5 8.4 0.80+0.08 −0.10 13.10 +0.38 −0.40 1.33 +0.31 −0.37 4.6 −1.3 0.85 +0.20 −0.24 NGC 1513 7.4 0.76+0.13 −0.18 12.96 +0.76 −1.16 1.32 +0.67 −0.72 3.5 −1.9 1.41 +0.71 −0.77 Berkeley 67 9.0 0.90+0.09 −0.08 13.86 +0.60 −0.37 1.64 +0.61 −0.41 2.5 −0.6 0.90 +0.34 −0.22 Berkeley 13 9.0 0.66+0.15 −0.14 14.01 +1.05 −0.80 2.47 +1.82 −1.07 4.4 −1.9 1.02 +0.75 −0.44 Czernik 19 7.4 0.67+0.08 −0.08 14.07 +0.73 −0.56 2.50 +1.13 −0.78 4.0 −1.2 1.05 +0.47 −0.32 Berkeley 15 8.7 1.01+0.15 −0.16 15.28 +0.36 −0.46 2.69 +0.71 −0.96 6.0 −2.1 1.12 +0.30 −0.40 NGC 1798 9.2 0.37+0.10 −0.09 13.90 +0.26 −0.63 3.55 +0.64 −1.22 9.3 −3.2 1.36 +0.25 −0.47 Berkeley 71 9.0 0.81+0.08 −0.08 15.08 +0.65 −0.30 3.26 +1.30 −0.73 3.1 −0.7 1.11 +0.44 −0.25 NGC 2126 9.1 0.27+0.11 −0.12 11.02 +0.64 −1.03 1.09 +0.45 −0.52 3.2 −1.5 0.61 +0.25 −0.29 NGC 2168 7.9 0.28+0.15 −0.16 10.49 +1.14 −1.00 0.84 +0.68 −0.42 2.4 −1.2 1.12 +0.91 −0.56 NGC 2192 9.3 0.04+0.11 −0.14 12.11 +0.53 −0.42 2.50 +0.86 −0.81 3.3 −1.1 1.02 +0.35 −0.33 NGC 2266 9.0 0.00+0.09 −0.09 12.24 +0.50 −0.30 2.81 +0.88 −0.66 4.8 −1.1 0.95 +0.30 −0.22 King 25 8.8 1.36+0.11 −0.13 15.03 +0.46 −0.93 1.45 +0.44 −0.67 2.7 −1.2 0.99 +0.30 −0.45 Czernik 40 8.9 0.99+0.13 −0.14 15.52 +0.42 −0.38 3.09 +0.89 −0.97 7.7 −2.4 2.03 +0.59 −0.64 Czernik 41 8.7 1.28+0.14 −0.17 14.64 +0.42 −0.87 1.36 +0.40 −0.65 2.2 −1.1 0.69 +0.20 −0.33 NGC 6885 7.1 0.66+0.14 −0.25 12.06 +1.03 −1.48 1.01 +0.72 −0.65 2.5 −1.6 0.69 +0.50 −0.45 IC 4996 7.0 0.58+0.05 −0.07 12.86 +0.53 −0.64 1.63 +0.50 −0.53 1.1 −0.3 0.57 +0.18 −0.19 Berkeley 85 9.0 0.77+0.14 −0.15 13.61 +0.47 −0.85 1.76 +0.57 −0.80 2.5 −1.2 0.76 +0.25 −0.35 Collinder 421 8.4 0.64+0.11 −0.12 12.08 +0.48 −0.48 1.05 +0.33 −0.34 1.8 −0.6 0.33 +0.10 −0.11 NGC 6939 9.1 0.38+0.18 −0.10 12.15 +0.56 −0.72 1.56 +0.64 −0.59 6.9 −2.6 0.98 +0.40 −0.37 NGC 6996 8.3 0.84+0.10 −0.12 13.49 +0.41 −0.66 1.50 +0.40 −0.57 0.9 −0.4 0.40 +0.10 −0.15 Berkeley 55 8.5 1.74+0.10 −0.11 15.81 +0.40 −0.51 1.21 +0.31 −0.39 2.1 −0.7 0.26 +0.07 −0.08 Berkeley 98 9.4 0.13+0.11 −0.11 13.26 +0.25 −0.38 3.73 +0.67 −1.05 5.0 −1.4 2.29 +0.41 −0.65 NGC 7654 7.0 0.73+0.14 −0.16 13.11 +1.18 −1.12 1.48 +1.24 −0.78 4.8 −2.5 2.13 +1.78 −1.12 NGC 7762 9.3 0.66+0.08 −0.09 11.52 +0.42 −0.75 0.78 +0.20 −0.30 2.2 −0.8 0.54 +0.14 −0.20 cleaned cluster box was saved and used for constructing the decontaminated CMD. The photometric parameters, such as distance modulus, reddening, and age of the target clusters, were derived by fit- ting a set of theoretical isochrones of sollar metallicity (Bertelli at al. 1994) to the decontaminated CMDs. For every isochrone of a given age, a grid of χ2 was calculated for a number of ob- served distance moduli and reddenings in steps of 0.01 mag. The isochrone with the lowest χ2 value was chosen as the final result. The resolution of the isochrone set was assumed to be the cluster age uncertainty, i.e. 0.1 in log(age). A map of scaled chi-square statistics ∆χ2 for the best-fit isochrone was prepared to estimate the uncertainties of E(B − V) and (M − m). Here, ∆χ2 was defined as χ2 − χ2min χ2min/ν , (9) where χ2min is the minimum χ 2 and ν the number of degrees of freedom (equal 2 in this case, Burke at al. 2004). The projection of the ∆χ2 = 1.0 contour on the parameter axes was taken as the 1-σ error. The decontaminated CMDs for individual clusters are pre- sented in Fig. 2 where the best-fit isochrones are also shown. The parameters such as log(age), reddening, and distance mod- ulus obtained for investigated clusters are listed in Table 4 in G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 7 King 13 B-V (mag) King 1 B-V (mag) King 14 B-V (mag) NGC 146 B-V (mag) Dias 1 B-V (mag) King 16 B-V (mag) Berkeley 4 B-V (mag) Skiff J0058+68.4 B-V (mag) NGC 559 B-V (mag) NGC 884 B-V (mag) Tombaugh 4 B-V (mag) Czernik 9 B-V (mag) NGC 1027 B-V (mag) King 5 B-V (mag) King 6 B-V (mag) Berkeley 9 B-V (mag) Berkeley 10 B-V (mag) Tombaugh 5 B-V (mag) NGC 1513 B-V (mag) Berkeley 67 B-V (mag) Berkeley 13 B-V (mag) Czernik 19 B-V (mag) Berkeley 15 B-V (mag) NGC 1798 B-V (mag) Berkeley 71 B-V (mag) NGC 2126 B-V (mag) NGC 2168 B-V (mag) NGC 2192 B-V (mag) NGC 2266 B-V (mag) King 25 B-V (mag) Czernik 40 B-V (mag) Czernik 41 B-V (mag) NGC 6885 B-V (mag) IC 4996 B-V (mag) Berkeley 85 B-V (mag) Collinder 421 B-V (mag) 0 1 2 NGC 6939 B-V (mag) 0 1 2 NGC 6996 B-V (mag) 0 1 2 Berkeley 55 B-V (mag) 0 1 2 Berkeley 98 B-V (mag) 0 1 2 NGC 7654 B-V (mag) 0 1 2 NGC 7762 B-V (mag) Fig. 2. Decontaminated CMDs for individual clusters. The best-fitted isochrones are drawn with solid lines. Cols. 2, 3, and 4, respectively. The distances were calculated under the assumption of the total-to-selective absorption ratio of R = 3.1 and are listed in Col. 5. The linear sizes of limiting radii Rlim and core radii Rcore are also listed in Cols. 6 and 7, respectively. 5. Mass functions The first step towards deriving the cluster mass function (MF) was to build the cluster’s luminosity functions (LF) for the core, halo, and overall regions separately. We used 0.5 mag bins. Another LF was built for an offset field starting at r = rlim + 1 arcmin and extending to the edge of the clean field on a frame. The LF of the offset field was subtracted, bin by bin, from every region LF, taking the area proportion into account, and this way the decontaminated LF was derived. The resulting LFs were converted into MFs using the respective isochrone. The derived mass functions φ(m) for the overall cluster region, defined as the number of stars N per mass unit, are plotted as functions 8 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters Table 5. Astrophysical parameters obtained from the mass function analysis. Name χ χcore χhalo Nevolved Mturno f f Ntot Mtot Ncore Mcore (stars) (M⊙) (stars) (M⊙) (stars) (M⊙) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) King 13 2.23 ± 1.23 0.63 ± 1.62 2.57 ± 1.02 14 3.36 44538 15598 1178 757 King 1 2.43 ± 1.56 −3.46 ± 0.44 3.58 ± 1.81 45 1.16 10520 3305 188 177 King 14 1.16 ± 0.38 0.63 ± 0.17 1.35 ± 0.63 0 14.77 1436 906 316 244 NGC 146 1.16 ± 0.66 − − 0 7.71 1066 624 − − Dias 1 1.07 ± 0.38 − − 0 5.27 426 248 − − King 16 1.12 ± 0.37 0.66 ± 0.27 1.26 ± 0.39 0 11.90 1084 709 244 261 Berkeley 4 0.75 ± 0.54 − − 0 12.81 479 467 − − Skiff J0058+68.4 −0.38 ± 0.66 −0.85 ± 1.54 0.36 ± 0.86 76 1.79 1053 851 232 214 NGC 559 1.31 ± 0.43 0.02 ± 0.33 2.09 ± 0.66 28 1.97 7286 3170 585 395 NGC 884 −0.05 ± 0.28 −0.81 ± 0.23 0.49 ± 0.22 1 15.24 341 1103 104 732 Tombaugh 4 0.53 ± 2.51 −6.03 ± 2.95 2.42 ± 0.47 6 1.75 3407 1744 102 164 Czernik 9 1.71 ± 1.58 − − 7 2.38 1424 559 − − NGC 1027 1.51 ± 0.36 0.47 ± 0.80 1.83 ± 0.39 0 3.34 1946 833 127 89 King 5 1.52 ± 0.30 0.06 ± 0.09 1.77 ± 0.42 22 1.88 5933 2313 417 227 King 6 1.74 ± 0.39 1.44 ± 0.32 1.58 ± 0.47 0 3.35 1172 465 321 141 Berkeley 9 1.94 ± 1.27 −5.58 ± 1.85 3.40 ± 2.05 4 1.24 2097 697 10 14 Berkeley 10 1.27 ± 0.55 −0.66 ± 0.75 2.45 ± 0.69 12 1.97 2641 1121 67 61 Tombaugh 5 1.31 ± 0.38 0.65 ± 0.27 1.82 ± 0.22 7 3.33 2750 1287 257 163 NGC 1513 1.55 ± 0.20 0.54 ± 0.36 1.27 ± 0.20 0 9.16 2813 1317 365 295 Berkeley 67 −1.66 ± 1.43 −2.61 ± 0.86 −1.02 ± 3.31 6 1.86 112 140 31 43 Berkeley 13 1.87 ± 0.96 1.67 ± 2.72 2.84 ± 1.29 7 1.92 1960 717 983 370 Czernik 19 1.14 ± 0.18 0.48 ± 0.18 0.87 ± 0.22 0 9.76 811 502 186 171 Berkeley 15 1.10 ± 1.09 0.00 ± 1.67 1.43 ± 1.30 12 2.21 2574 1230 170 124 NGC 1798 3.13 ± 0.57 −1.13 ± 1.43 4.69 ± 1.29 28 1.74 23209 6932 202 229 Berkeley 71 −1.06 ± 1.83 − − 9 1.92 232 256 − − NGC 2126 1.14 ± 0.41 0.73 ± 0.86 1.20 ± 0.57 10 1.82 901 395 223 106 NGC 2168 0.93 ± 0.20 0.82 ± 0.51 0.89 ± 0.16 0 4.25 1421 849 339 216 NGC 2192 −4.59 ± 2.12 −9.50 ± 3.35 −3.12 ± 1.14 23 1.54 78 107 5 8 NGC 2266 1.58 ± 0.91 0.15 ± 0.52 2.28 ± 0.96 12 1.93 3570 1392 312 167 King 25 1.73 ± 1.00 0.43 ± 1.38 2.62 ± 1.84 9 2.31 6075 2336 467 278 Czernik 40 2.41 ± 0.62 0.49 ± 1.28 3.37 ± 1.13 71 2.13 47503 15790 1066 597 Czernik 41 0.77 ± 0.70 −2.22 ± 2.83 2.72 ± 0.54 5 2.56 1161 635 24 44 NGC 6885 1.68 ± 0.27 0.70 ± 0.38 1.66 ± 0.42 0 6.78 711 1641 250 141 IC 4996 0.87 ± 0.45 − − 0 14.47 347 304 − − Berkeley 85 1.50 ± 0.73 −1.86 ± 0.99 1.99 ± 0.79 10 1.94 4082 1618 40 54 Collinder 421 1.04 ± 0.37 0.97 ± 0.42 1.32 ± 0.36 6 3.14 424 233 153 81 NGC 6939 0.96 ± 0.46 0.19 ± 0.48 1.56 ± 0.44 48 1.80 5154 2363 786 391 NGC 6996 1.73 ± 0.57 − − 3 3.72 664 277 − − Berkeley 55 0.91 ± 1.54 −1.53 ± 0.81 2.34 ± 2.00 4 3.00 1466 795 45 85 Berkeley 98 2.52 ± 1.18 0.94 ± 2.60 3.22 ± 1.35 29 1.41 6158 1967 1024 421 NGC 7654 1.42 ± 0.15 1.11 ± 0.13 1.89 ± 0.37 0 9.23 6163 3133 2351 1491 NGC 7762 −0.12 ± 0.33 −0.35 ± 0.78 0.24 ± 0.36 28 1.47 1106 616 186 113 of stellar mass m in Fig. 3, where the standard relations of the logφ(m) = log = −(1 + χ) log m + b0 , (10) fitted to the data for each cluster, are also shown. The error bars were calculated assuming the Poisson statistics. The values of the MF slope parameters χ for overall clusters regions are listed in Col. 2 of Table 5. This procedure was applied to objects with rlim > 4 ′. For smaller ones, core and halo regions were not separated to avoid small number statistics. The resulting fit parameters χcore and χhalo are collected in Table 5 in Cols. 3 and 4, respectively. The completeness of our photometry was estimated by adding a set of artificial stars to the data. It was defined as a ratio of the number of artificial stars recovered by our code and the number of artificial stars added. To preserve the original region crowding, the number of artificial stars was limited to 10% of the number of actually detected stars found in the orig- inal images within a given magnitude bin. The completeness factor was calculated for every magnitude bin. The obtained completeness factor was close to 100% for stars brighter than 17 mag in all clusters and decreased for fainter stars more or less rapidly depending on the stellar density in a given field. The faint limit of the LM was set individually for each field af- G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 9 King 13 log (m) King 1 log (m) King 14 log (m) NGC 146 log (m) Dias 1 log (m) King 16 log (m) Berkeley 4 log (m) Skiff J0058+68.4 log (m) NGC 559 log (m) NGC 884 log (m) Tombaugh 4 log (m) Czernik 9 log (m) NGC 1027 log (m) King 5 log (m) King 6 log (m) Berkeley 9 log (m) Berkeley 10 log (m) Tombaugh 5 log (m) NGC 1513 log (m) ) Berkeley 67 log (m) ) Berkeley 13 log (m) ) Czernik 19 log (m) ) Berkeley 15 log (m) ) NGC 1798 log (m) Berkeley 71 log (m) NGC 2126 log (m) NGC 2168 log (m) NGC 2192 log (m) NGC 2266 log (m) King 25 log (m) Czernik 40 log (m) Czernik 41 log (m) NGC 6885 log (m) IC 4996 log (m) Berkeley 85 log (m) Collinder 421 log (m) 0 0.5 1 NGC 6939 log (m) 0 0.5 1 NGC 6996 log (m) 0 0.5 1 Berkeley 55 log (m) 0 0.5 1 Berkeley 98 log (m) 0 0.5 1 NGC 7654 log (m) 0 0.5 1 NGC 7762 log (m) Fig. 3. The mass functions for individual clusters with the standard relation (Eq. 10) fitted (solid lines). ter careful inspection of the observed faint-end range (typically 18–19 mag) with the completeness factor lower than 50%. The derived cluster parameters allowed us to estimate the total mass Mtot, total number of stars Ntot, core mass Mcore, and the number of stars within the core Ncore for each cluster. These quantities were calculated by extrapolating the MF from the turnoff down to the H-burning mass limit of 0.08 M⊙ using the method described in Bica & Bonatto (2005). If the value of χ was similar or greater than that of the universal initial mass function (IMF), χIMF = 1.3±0.3 (Kroupa 2001), the mass func- tion was extrapolated with given χ to the mass of 0.5 M⊙ and then with χ = 0.3 down to 0.08 M⊙. For the lower actual values of χ, the MFs were extrapolated with the actual value within the entire range from the turnoff mass down to 0.08 M⊙. The contribution of the evolved stars was included in the cluster’s total mass by multiplying their actual number Nevolved (Col. 5 in Table 5) by the turnoffmass Mturno f f (Col. 6 in Table 5). The total number of cluster’s stars Ntot, total cluster’s mass Mtot, number of stars in the core Ncore, and the core mass Mcore are given in Cols. 7, 8, 9, and 10 of Table 5, respectively. To describe the dynamical state of a cluster under investi- gation, the relaxation time was calculated in the form trelax = 8 ln N tcross , (11) where tcross = D/σV denotes the crossing time, N is the total number of stars in the investigated region of diameter D, and σV is the velocity dispersion (Binney & Tremaine 1987) with a typical value of 3 km s−1 (Binney & Merrifield 1998). The calculations were performed separately for the overall cluster and core region. The cluster dynamic evolution was described by the dynamical-evolution parameter τ, defined as trelax , (12) which was calculated for the core and the overall cluster sep- arately. The difference between MF slopes of the core and corona ∆χ = χhalo−χcore can be treated as the mass segregation measure. These quantities were used for a statistical description of the cluster’s sample properties. 10 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 7 8 9 log (age) 0 0.5 1 E(B-V) (mag) 10 12 14 16 (M-m) (mag) 0 5 10 15 r (arcmin) Fig. 4. Reliability of the obtained results. The data obtained in this paper (vertical axes) are plotted versus the literature ones (horizontal axes). 6. Reliability of results and comparison with previous studies To test the reliability of the results of age, reddening, distance modulus, and apparent diameter determination presented in this paper, we compared them with the available catalogue data taken from the WEBDA3 open cluster data base (Mermilliod 1996). Our determinations of basic cluster parameters are plot- ted against the results of the previous studies of 30 clusters in Fig. 4. The ages of clusters show excellent agreement with the literature data. The least-square-fitted linear relation for the 30 clusters is log(ageour) = (0.993 ± 0.009) log(age) (13) with a correlation coefficient of 0.90 and fits the perfect match line within the error. As shown in Fig. 4b, satisfactory reliability of our E(B−V) determination was also achieved. Only two clusters come sig- nificantly off the line of perfect match. The least-square-fitted linear relation for the 28 clusters is E(B − V)our = (0.99 ± 0.04)E(B− V) (14) with a correlation coefficient of 0.88. The distance moduli de- termined in this study, Fig. 4c, are very similar to the literature data as well. The best-fit linear relation for the 27 clusters is (M − m)our = (0.99 ± 0.01)(M − m) (15) with the correlation coefficient of 0.90. All three relations prove that our results are reliable. As displayed in Fig. 4d, the literature values of apparent radii of open clusters are considerably (4–5 times) lower for the majority of the clusters under investigation. 3 http:///www.univie.ac.at/webda 0 1 2 3 4 5 Rcore (pc) 0 1 2 3 4 5 Rcore (pc) Fig. 5. Relation between limiting and core radii. See text for description. 7. Statistical considerations The sample of 42 open clusters studied in detail within this sur- vey is by no means complete. Since it was defined by celestial coordinates, estimated sizes, and richness of potential objects, as well as non-availability of previous CCD studies, it is defini- tively not representative of the total open cluster sample in the Galaxy. However the sample covers quite a wide range of clus- ters parameters and is uniform enough to perform simple sta- tistical analysis. Even though 20 clusters out of 62 covered by this sur- vey were found not to be real does not necessarily mean that ∼30% of clusters in Dias et al. (2002) are doubtful. Such a high frequency of accidental star density fluctuations is defi- nitely caused by our selection criteria. 7.1. Limiting and core radii From their analysis based on DSS images of 38 open clus- ters Nilakshi et al. (2002) concluded that the angular size of the coronal region is about 5 times the core radius, hence Rlim ≈ 6Rcore. Bonatto & Bica (2005) reported a similar re- lation between the core and limiting radii based on their study of 11 open clusters. Bica & Bonatto (2005) used data for 16 clusters to find that Rlim = (1.05 ± 0.45) + (7.73 ± 0.66)Rcore. More recently, Sharma et al. (2006) determined core and limit- ing radii of 9 open clusters using optical data and presented the relation Rlim = (3.1 ± 0.5)Rcore with the correlation coefficient of 0.72. The best fit obtained using the data reported in this study gives Rlim = (3.1 ± 0.2)Rcore with the correlation coeffi- cient of 0.74 (Fig. 5 a). Although the correlation is quite strong, Rlim may vary for individual clusters between about 2Rcore and 7Rcore (Fig. 5a). The obtained relation is quite different from the one ob- tained in the papers mentioned above, except for Sharma et al. (2006) who used observations gathered with a wide-field Schmidt telescope similar to ours. The field of view in sur- veys by Nilakshi et al. (2002), Bonatto & Bica (2005), and Bica & Bonatto (2005) was wider with a radius of 1–2◦. That sug- gests that our determinations of the limiting radius for some extensive clusters are underestimated due to a limited field of view (see Sect. 3.2). However, it has to be pointed out that the methods of determining the cluster limiting radius differ con- siderably, and sometimes the adopted definition is not clear. G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 11 We also note that our open clusters’ size determinations differ from many in the literature at the level of angular diameters (Fig. 4d). One of the reasons for such an inconsistency may be the difference in the content of the cluster samples used by different authors who frequently use non uniform photometric data. Finally, as Sharma et al. (2006) notes, open clusters ap- pear to be larger in the near-infrared than in the optical data. To illustrate the difference between results, we again plot our data in Fig. 5b as in Fig. 5a, together with the literature determinations taken from the following papers: open squares denote results from Bica & Bonatto (2005) and Bonatto & Bica (2005), open circles those of Nilakshi et al. (2002), and open triangles Sharma et al. (2006). It is clear that data coming from optical investigations fit each other. We also note that the litera- ture data contain no determination of sizes for clusters smaller than 2 pc. 7.2. Structural parameters The data gathered within this survey show that the limiting ra- dius correlates with the cluster’s total mass (Fig. 6a). We ob- tained the relation log Rlim = (0.39±0.07) log Mtot − (0.6±0.2) with a moderate correlation coefficient of 0.70. This result indi- cates that clusters with large diameters and small total masses do not form bound systems. On the other hand, small massive clusters are dissolved by the internal dynamics (Bonatto & Bica 2005). As one could expect, the core radius is also related to the cluster’s total mass (Fig. 6b). The obtained least-square linear relation is log Rcore = (0.32±0.08) log Mtot − (0.99±0.25) with a weak correlation coefficient of 0.53. As shown in Figs. 6c and d, both radii tend to decrease in the course of the dynamical evolution. For the limiting ra- dius, the obtained relation is log Rlim = (−0.13 ± 0.04) logτ + (0.66 ± 0.05) with a weak correlation coefficient of 0.47. This suggests that dynamical evolution makes a cluster smaller due to dissolving coronae. The dynamical evolution of the core ra- dius is more visible. The least-square fitted linear relation is log Rcore = (−0.16 ± 0.04) logτ + (0.10 ± 0.05) with a corre- lation coefficient of 0.51. Moreover, the core radius is mod- erately correlated with the dynamical-evolution parameter of the core τcore. The obtained relation, plotted in Fig. 6e, is log Rcore = (−0.15 ± 0.03) logτcore + (0.36 ± 0.08) with a cor- relation coefficient of 0.66. The last two relations indicate that the dynamical evolution of both, overall cluster and core, tends to reduce the core radius. No relation of Rlim and Rcore with cluster age or mass segregation was noted in the investigated sample. To investigate the relative size of halos, the concentration parameter c, defined as c = (Rlim/Rcore), was plotted against other parameters. The concentration parameter seems to be re- lated to cluster age, as shown in Fig. 6f. For clusters younger than about log(age) = 9, it tends to increase with cluster age (log c = (0.11 ± 0.03) log(age) − (0.38 ± 0.23) with a correla- tion coefficient of 0.56). Nilakshi et al. (2002) notes a decrease in the size of halos for older systems. In Fig. 6g the concentration parameter is plotted against mass segregation parameter ∆χ. As no relation is seen, one can 2 3 4 log Mtot 2 3 4 log Mtot -1 0 1 2 3 log τ -1 0 1 2 3 log τ 0 1 2 3 4 5 log τcore 7 8 9 log (age) 0 2 4 6 8 Fig. 6. Relations between structural parameters with each other. See text for details. conclude that there is no low concentrated clusters with log c < 0.5 with a high value of ∆χ > 3. 7.3. Mass function slopes The mass function slopes of the overall cluster, core, and halo were sought for relations with other parameters. As displayed in Fig. 7a, the mass function slope of the overall cluster re- gion and the cluster age are not strictly related. However, a deficit in low-mass members occurs in clusters older than log(age) = 9.0. We also investigated relations between the mass function slopes and the dynamical-evolution parameter τ – Fig. 7b. Bonatto & Bica (2005) report a relation between the overall χ and τ in the form of χ(τ) = χ0 − χ1 exp(−τ0/τ), sug- gesting that the MF slopes decrease exponentially with τ. Our results confirm this relation, and the least-square fit was ob- 12 G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 7 8 9 log (age) -1 0 1 2 3 log τ 7 8 9 log (age) -1 0 1 2 3 log τ 0 1 2 3 4 5 log τcore 0 2 4 6 8 7 8 9 log (age) -1 0 1 2 3 log τ Fig. 7. Relations between mass function slopes and other clus- ters parameters. See text for details. tained with χ0 = 1.34±0.13, χ1 = 9.9±2.3, and τ0 = 450±130 (a correlation coefficient was 0.81). It is worth noting that the obtained value of χ0 is almost identical to χIMF . In Fig. 7c the relation between χcore and the log(age) is presented. It is clear that χcore decreases rapidly with cluster age for clusters older than log(age) = 8.5. This suggests that evaporation of the low-mass members from cluster cores does not occur in clusters younger than log(age) = 8.5. The cores of clusters older than log(age) = 8.5 are dynamically evolved and deprived of low-mass stars. As one can see in Fig. 7d, χcore does not correlate with τ. However, χcore tends to decrease with τ, which indicates that low-mass-star depleted cores appear in dynamically evolved clusters with log τ > 1. In Fig. 7e we show that χcore and τcore are related, and the relation is similar to the one for χ and τ – χcore = (0.44 ± 0.27) − (8.1 ± 1.1) exp(− 2790±630 τcore ). Such an evolution of χcore was also reported by Bica & Bonatto (2005). Our fit indicates, however, that the initial χcore(0) = 0.44 ± 0.27 is much lower than the value of 1.17 ± 0.23 obtained by these authors. This suggests that χcore is significantly lower than χIMF for dynami- cally young systems. As displayed in Fig. 7f, χcore is correlated with the mass segregation parameter ∆χ. The least-square-fitted relation χcore = (−0.83 ± 0.10)∆χ + (1.33 ± 0.33) with the correlation coefficient of −0.83 indicates that – as one could expect – χcore decreases with the increase in mass segregation. Finally, in Figs. 7g and h relations between χhalo and log(age) or τ were plotted, respectively. The MF slopes of the coronal regions of clusters younger than log(age) = 8.9 tend to increase with age. That relation was marked with the least- square-fitted dashed line for which the correlation coefficient is 0.61. Bifurcation occurs for older clusters and χhalo becomes either very high or low as compared to the mean value. This suggests that the clusters were observed in different stages of dynamical evolution. In clusters with higher values of χhalo, the mechanism of dynamical mass segregation is more effi- cient than the evaporation of low-mass members from halos. Clusters with low values of χhalo are dynamical evolved sys- tems devoid of low-mass stars in the overall volume. Then, χhalo seems to decrease with τ for log τ > 2. That suggests that in general the dynamical evolution of a cluster halo is driven by the dynamical evolution of the overall system (we obtained χhalo = (2.0±0.2)− (12.3±6.5) exp(− 700±350 ) with the correla- tion coefficient of 0.69). It is also worth noting that the average χhalo for dynamically not evolved clusters is larger than χIMF . 7.4. Mass segregation The mass segregation ∆χ is most prominent for clusters older than about log(age) = 8 (Fig. 8a). The mean values of χhalo and χcore differ significantly for clusters with log(age) < 8, and ∆χ , 0 even for very young clusters. This suggests the existence of the initial mass segregation within the protostellar gas cloud. We plotted ∆χ in Fig. 8b as a function of the dynamical- evolution parameter τ. No relation can be seen. However, one can note that strong mass segregation occurs in cluster older then their relaxation time, i.e. log τ > 0. As displayed in Fig. 8c, mass segregation seems to be related to the core dynamical-evolution parameter τcore. Although a strict rela- tion is not present, one can see that clusters with dynamically evolved cores (log τcore > 3) reveal a strong mass segregation effect. 8. Conclusions Wide-field CCD photometry in B and V filters was collected for 42 open clusters and the basic structural and astrophysical parameters were obtained. Eleven cluster under investigation were studied for the first time. G. Maciejewski and A. Niedzielski: CCD BV survey of 42 open clusters 13 7 8 9 log (age) -1 0 1 2 3 log τ 0 1 2 3 4 5 log τcore Fig. 8. Evolution of the mass segregation measure in time. See text for details. A simple statistical analysis of our sample of open clusters leads to the following conclusions: – The angular sizes of most of the observed open clusters ap- peared to be several times larger than the catalogue data indicate. – A correlation exists between core and limiting radii of open clusters. The latter seem to be 2-7 times larger, with average ratio of 3.2. The limiting radius tends to increase with the cluster’s mass. Both limiting and core radii decrease in the course of the dynamical evolution. Moreover, core radius decreases with the core dynamical-evolution parameter. – The relative size of a cluster halo (in units of the core ra- dius) tends to increase with cluster age for systems younger than log(age) = 9. Among clusters with a strong mass- segregation effect, there are no systems with small halos. – The MF slope of the overall cluster region is related to the dynamical-evolution parameter with the relation found in Bica & Bonatto (2005). For clusters with log τ < 2, the MF slope is similar to the slope of the universal IMF. For clusters with log τ > 2 (older than about log(age) = 9), the results of evaporation of the low-mass members are seen, and χ reaches an extremely low value for clusters with log τ = 3. – The MF slope of the core region is smaller than the uni- versal value even for very young clusters, while the mass function slope of the corona is larger. This indicates the ex- istence of the initial mass segregation. The dynamical mass segregation appears in clusters older than about log(age) = – A strong deficiency of low-mass stars appears in cores of clusters older than log(age) = 8.5 and not younger than one relaxation time. Acknowledgements. We thank the anonymous referee for remarks that significantly improved the paper. This research is partially sup- ported by UMK grant 369-A and has made use of the WEBDA data base operated at the Institute for Astronomy of the University of Vienna, SIMBAD data base, as well as The Guide Star Catalogue- II, which is a joint project of the Space Telescope Science Institute and the Osservatorio Astronomico di Torino. Space Telescope Science Institute is operated by the Association of Universities for Research in Astronomy, for the National Aeronautics and Space Administration under contract NAS5-26555. The participation of the Osservatorio Astronomico di Torino is supported by the Italian Council for Research in Astronomy. Additional support is provided by the European Southern Observatory, Space Telescope European Coordinating Facility, the International GEMINI project, and the European Space Agency Astrophysics Division. 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Milone & J.-C. Mermilliod, ASP Conf. Ser., 90, 475 Mighell, K. J., Rich, R. M., Saha, M., & Falls, S. M. 1996, AJ, 111, Niedzielski, A., Maciejewski, G., & Czart, K. 2003, Acta Astron., 53, Nilakshi, Sagar, R., Pandey, A. K., & Mohan, V. 2002, A&A, 383, 153 Sharma, S., Pandey, A. K., Ogura, K., et al. 2006, AJ, 132, 1669 Introduction Observations and reduction Observations Data reduction and calibration Radial structure Redetermination of central coordinates Analysis of radial density profiles The color–magnitude diagrams Mass functions Reliability of results and comparison with previous studies Statistical considerations Limiting and core radii Structural parameters Mass function slopes Mass segregation Conclusions

0704.1365

Geometry and Dynamics of Quantum State Diffusion

Geometry and Dynamics of Quantum State Diffusion Nikola Burić ∗ Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia. November 26, 2018 Abstract Riemannian metric on real 2n-dimensional space associated with the equation governing complex diffusion of pure states of an open quantum system is introduced and studied. Examples of a qubit under the influence of dephasing and thermal environments are used to show that the curvature of the diffusion metric is a good indicator of the properties of the environment dominated evolution and its stability. PACS: 03.65.Yz ∗e-mail: [email protected] http://arxiv.org/abs/0704.1365v1 1 Introduction The states of an open quantum system are commonly described by a density matrix ρ̂. In many cases, the evolution of ρ̂(t) is governed by a master equation of the Linblad form [1],[2] for the density matrix ρ̂(t) dρ̂(t) = −i[Ĥ, ρ̂] + 1 [L̂lρ̂, L̂ l ] + [L̂l, ρ̂L̂ l ], (1) where the Linblad operators L̂l describe the influence of the environment. The equation (1) represents the general form of an evolution equation for a quantum system which satisfies Markov property. However, this theoretical approach to the dynamics of open quantum systems is not unique. In real experiments it is often useful to understand and model the dynamics of pure quantum states [3],[4],[5]. Indeed, the evolution of an open system can be described directly in terms of the dynamics of the system’s pure state. The corresponding evolution equation is a stochastic modification of the unitary Schroedinger equation. In fact, the density matrix ρ̂ can be written, in different but equivalent ways, as a convex combination of pure states. Each of these results in a stochastic differential equation for |ψ(t) > in the Hilbert spaceH. Such stochastic Schroedinger equations (SSE) are called stochastic unravelling [6],[7],[2] of the Linblad master equation for the reduced density matrix ρ̂(t). There are many different forms of nonlinear and linear SSE that have been used in the context of open systems [8],[3], [2],[6], [7] or suggested as fundamental modifications of the Schroedinger equation [9] [10],[11],[6] [12],[13]. They are all consistent with the requirement that the solutions of (1) and of SSE satisfy ρ̂(t) = E[|ψ(t) >< ψ(t)|]. (2) where E[|ψ(t) >< ψ(t)|] is the expectation with respect to the distribution of the stochastic process |ψ(t) >. The advantages of the description in terms of the pure states and SSE over the description by ρ̂ are twofold. On the practical side, the computations are much more practical, as soon as the size of the Hilbert space is moderate or large [14]. On the theoretical side, the stochastic evolution of pure states provides valuable insides which can not be inferred from the density matrix approach [15],[16],[6],[2],[5],[17]. There are two main approaches to the unravelling of the Linblad master equation: the method of quantum state diffusion [6] and the relative state method [3], [2], with specific advantages associated with each of the methods. The relative state method is usually used do describe the situations when the measurement is the dominant interaction with the environment. The method offers particular flexibility in that the master equation can be unravelled into different stochastic equations conditioned on the results of measurement. On the other hand the correspondence between the QSD equations and the Linblad master equations is unique, and is not related to a particular mea- surement scheme, or the form of the Markov environment. The resulting SSE is always of the form of a diffusion process on the Hilbert space of pure states, which is its main property to be explored in this paper. We shall concentrate on the unique unravelling of the master equation given by the quantum state diffusion equation, and explore the fact that it represents a diffusion process. QSD equation is the unique unravelling of (1) which preserves the norm of the state vector and has the same invariance as (1) under the unitary transformations of the environment operators {L̂l}, [6]. The equation is given by the following formula: |dψ > = −iĤ|ψ > dt 2 < L̂ l > L̂l − L̂ l L̂l− < L̂ l >< L̂l > |ψ(t) > dt (L̂l− < Ll >)|ψ(t) > dWl (3) where <> denotes the quantum expectation in the state |ψ(t) > and dWl are independent increments (indexed by l) of complex Wiener c-number processes Wl(t). The equation (3) represent a diffusion process on a complex vector space. We shall utilize the diffusion matrix of this process to define a Riemannian metric on the corresponding real space. We shall then study the properties of this diffusion metric as a field fixed by the environment and in relation to the stochastic evolution of the state vector, for different types of the environment. It will be shown, using examples of the dephasing and thermal environments and the measurement of an observable, that the curvature of the diffusion metric is a good indicator of the properties of the environment dominated evolution and its stability. We shall see that the curvature maxima of the diffusion metric coincide with the states that are preferred by the particular type of the environment. Furthermore, if the maxima are sharp and positive the stochastic dynamics governed by the environment and a Hamiltonian perturbation that does not commute with L̂l, is likely to be attracted to the state with the maximal (positive) curvature. On the other hand the states that correspond to the negative values of the curvature are unstable. Our analyzes of the QSD equation, and the results, are strictly related to the fact that the equation represent a norm-preserving diffusion process, and in this sense are applicable to the stochastic modifications of the Schroedinger equation that describe a norm-preserving diffusion on the Hilbert space of pure states, like the QSD equation and, for example, the equations of the spontaneous collapse models [13]. The structure of the paper is as follows. We shall first discuss, in the next section, a way to relate a Riemannian metric on a real space R2n to a complex diffusion process on Cn. Then, in section 3, we shall apply this procedure to define the Riemannian metric associated with QSD, and than the properties of this metric for various types of environments will be studied. Finally, in section 4, we shall summarize and discuss our results. 2 Riemannian metric of a complex diffusion Using the following notation f(|ψ >) = −iĤ |ψ > (4) 2 < ψ|L̂†l |ψ > L̂l − L̂ l L̂l− < ψ|L̂ l |ψ >< ψ|L̂l|ψ > |ψ >, B(|ψ >)dW = (L̂l− < ψ|Ll|ψ >)|ψ > dWl. (5) the QSD equation (3) assumes the standard form of a stochastic differen- tial equation (SDE) for an n-dimensional autonomous (stationary) complex diffusion process: d|ψ >= f(|ψ >)dt+B(|ψ >)dW. (6) |ψ(t) > and f(|ψ(t) >) are complex vectors of complex dimension n, and dW are differential increments of an m-dimensional complex Wiener process: E[dWl] = E[dWldWl′ ] = 0, E[dWldW̄ l′ ] = δl,l′dt, l = 1, 2 . . .m, (7) where E[·] denotes the expectation with respect to the probability distribu- tion given by the (m-dimensional) process W , and W̄l is the complex conju- gate of Wl. B(|ψ >) is n × m matrix, where m is at most n2 − 1, and the diffusion matrix is G = BB†. (8) Thus, G(|ψ >) is Hermitian and nonnegative-definite. Notice that, unlike the case of a general SDE, the dissipative part of the drift (4) and the diffusion term (5) are determined by the same operators L̂l, and related in such a way that the diffusion equation preserves the norm of the state vector. The complex n-dimensional equation (3) generates 2n-dimensional real diffusion. Let us introduce the following real n dimensional vectors (ψ̄ − ψ), q = (ψ̄ + ψ) (q + ip), ψ̄ = (q − ip), (9) and a 2n dimensional vector X = (q, p). Similarly, we introduce real and imaginary parts of the vector f and order them as components of a 2n real vector F = (fR, f I), and introduce real and imaginary parts of the incre- ments of the complex m-dim Wiener process dW by dWi = (dW i + idW 2, i = 1, 2, . . .m (10) It is easily checked that the real and the imaginary parts are increments of a real 2m-dimensional process, i.e. E(dWRi dW j ) = E(dW j ) = δi,jdt, E(dW j ) = 0. (11) With this notation we have dψ̄ + dψ dψ̄ − dψ . (12) Substitution of the complex equation (3) and its complex conjugate, leads to the following 2n dimensional real SDE: fR(p, g) f I(p, q) BR −BI BI BR , (13) The matrix B of dimension 2n× 2m B = 1√ BR −BI BI BR , (14) where (B)ij = (B R)ij + i(B I)ij (15) gives the diffusion matrix G for the real 2n dimensional diffusion described by the process (13) G = BBT = 1 (BR)(BR)T + (BI)(BI)T (BR)(BI)T − (BI)(BR)T (BI)(BR)T − (BR)(BI)T (BI)(BI)T + (BR)(BR)T We can write the matrix G in terms of real and imaginary components of the n× n complex matrix G = BB† as follows G = 1 GR GI −GI GR where −GI = (GI)T , since the matrix G is Hermitian. Furthermore, one can see that, besides the equalities between the entries corresponding to the symmetry of the matrix, there are other equalities (G)i,j = (G)i+n,j+n, i, j = 1, 2 . . . n (18) The matrix G is symmetric and nonnegative, but it could be singular. However, the matrix Diag{1/2, 1/2, . . . , 1/2}+ G gives a Riemannian metric on the real 2n dimensional vector space. The factor 1/2 of the Euclidian part is chosen in order that the Euclidian norm of a vector corresponding to a complex n-vector of unit norm is also unity. Once the diffusion metric Diag{1/2, 1/2, . . . , 1/2} + G is calculated the standard formulas [18] give the connection coefficients Γkµν of the Levi-Civita connection for this metric in terms of the coefficients gµν = δµν/2 + (G)µν Γkµν = gkλ(∂µgλν + ∂νgλµ − ∂λgµν) (19) Curvature tensor, Ricci tensor and the scalar curvature of the diffusion metric are also given by the standard formulas [18]: Rkλµν = ∂µΓ νλ − ∂νΓkµλ + Γ µη − Γ νη, (20) Ricµν = R µλν , R = gµνRicµν . (21) Before we present the results of calculations of the diffusion metric and its curvature for different types of environments, we would like to consider briefly real representation of the QSD equation in the case when the Linblad operators are Hermitian. This includes, for example, the dephasing environ- ment or measurement, or the primary QSD [19], [6] and other fundamental stochastic modifications of the Schroedinger equation [9],[10],[12]. The goal of this digression is to point out to the connection between the general QSD equation (3) and some other stochastic modifications of the Schroedinger equation that have the form of a norm-preserving diffusion equation, and that consequently the construction of the diffusion metrics and its proper- ties are applicable to these equations also. In the case of Hermitian Linblad operators the real representation of (3) assumes a specially simple and illumi- nating form. Applying the same derivation as from equation (9) to equation (13) one obtains the following: dpi = −Hijqjdt+ (2 < L > Lij − (L2)ij− < L2 > δij)pjdt (Lij− < L > δij)pjdWR + (Lij− < L > δij)qjdW I , (22) where we have, for reasons of simplicity, included only one Linblad operator and the summation over repeated indexes is assumed. Noticing that for an arbitrary linear operator B Bijqj = δij ∂ < B > , Bijpi = δij ∂ < B > equation (23) becomes dpi = − ∂ < H > ∂ < L > dWR + ∂ < L > , (24) where ∆2L =< L2 > − < L >2. There is an analogous equation for dqi. The two sets of equations represent a diffusion process on R2n, consisting of the drift given by a Hamiltonian dynamical system on R2n with the Hamilton’s function < H > and the dissipative part determined by ∆2L =< L2 > − < L >2 and the diffusion term determined by< L >. The drift and the diffusion are such that the norm of the vectors in R2n is preserved. Furthermore, the equations are invariant under a global gauge transformation corresponding to the multiplication of vectors |ψ > by a phase factor. Takeing into the account the norm invariance and the global phase symmetry the equations can be written as a diffusion equation on the phase space S2n−1/S1 of the following form dX = Ω∇ < H > dt+∇(∆2L)dt+ 1√ ∇ < L > dW (25) where ∇ and Ω∇ are the gradient and the skew gradient on S2n−1/S1, and X denotes the set of 2n− 2 coordinates on the reduced phase space S2n−1/S1. Equations like (25) have been analyzed as candidates for a description of the spontaneous state reduction in [13], or in the case L̂ = Ĥ in [12]. 3 QSD metric and qualitative properties of dynamics Application of formula (16) gives for the case (5) of the QSD equation an explicit procedure for calculation of the diffusion metric coefficients gij, in terms of the coefficients of the Linblad operators and the coefficients of the state ψ > in some bases |ψ >= ∑i ci|i >. The components of the diffusion matrix G = BB† are given by Bkk′(c, c̄) = (Ll− < Ll > 1)kj(Ll†− < Ll† > 1)k′j′cj c̄′j (26) where: < Ll >= ss′ Lss′c sc̄s. Expressing ci, c̄i in terms of x1 . . . x2n xi = (c̄i + ci)/ 2 i = 1, . . . n −1(c̄i − ci)/ 2 i = n+ 1, . . . 2n, (27) separating of GR and GI and substituting in (16) finally gives the 4n2 entries of the real matrix G. We shall study the diffusion metric for the following three types of envi- ronments: (a) dephasing environment; (b) the environment corresponding to measurement of an observable and (c) thermal environment. The first two are represented by Hermitian and the third one by a non-Hermitian Linblad operators. The main geometrical object which we shall study are the diffusion metric norm of a state vector and its scalar curvature. In order to illustrate how these objects depend on the environment we shall use the simplest but important quantum system, namely a single qubit. The system operators can be expressed as combinations of the Pauli sigma matrices σ̂x, σ̂y, σ̂z, a state |ψ > of unit norm is determined by < σ̂x >,< σ̂y >,< σ̂z > or by the spherical angles (θ, φ) given by < σ̂z > = cos(θ) < σ̂x > = sin(θ) cos(φ) < σ̂y > = sin(θ) sin(φ), (28) The environment operators are [20],[3] L̂ = µσ̂+σ̂− (29) for the dephasing and L̂ = µ1σ̂+ + µ2σ̂− (30) for the thermal environment, with µ1 and µ2 proportional to the temperature, and finally for the measurement of, say, σ̂z the Linblad operator is just L̂ = µσ̂z. (31) The formulas for the entries gij of the diffusion metrics in terms of the coordinates x1, x2, x3, x4 in the three considered cases can be conveniently written using the following notation: s = x1x2 + x3x4, a = x1x4 − x2x3. (32) Because many of the metric entries are repeated, it is more convenient to present them in a list rather than to write down the corresponding matrices. Using the notation (32), the entries of the metrics in the three considered cases are: For dephasing: g11 = 1/2 + (µ 2/16)d2 (2 + d2 )2, g12 = (µ 2/16)sd2 (2 + d2 ), g13 = 0 g14 = (µ 2/16)ad2 (2 + d2 ), g22 = 1/2 + (µ 2/16)d4 , g23 = −g14 g24 = 0, g33 = g11, g34 = g21, g44 = g22; (33) For the thermal environment: g11 = 1/2 + d + (2 + d2 ]/16, g12 = s[d (2 + d2 )µ1 + (2 + d µ2]/16, g1,3 = 0 g14 = a[d (2 + d2 )µ1 + (2 + d µ2]/16, g22 = 1/2 + d + (2 + d2 g13 = 0, g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22; (34) 3,0 1 3,0 1 3,0 1 3,0 1 3,0 1 3,0 1 Figure 1: Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3, (b) µ = 1.7 and for the measurement of σ̂z g11 = 1/2 + (g 2/16)d12(2 + d2 g12 = (g 2/16)s(d2 − 2)(d2 + 2), g13 = 0 g14 = a(d ), g22 = 1/2 + (g 2/16)d2 (2 + d2 g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22. (35) These formulas are used to compute the diffusion metric norm and the scalar curvature as functions of the state parameters θ and φ. We shall first consider the dependence of the stated properties of the diffusion metric on the type of the environment and the coupling strengths µ, µ1, µ2 and then analyze the relation between these properties and the stochastic dynamics of the state vectors. 0,0 0,5 1,0 1,5 2,0 2,5 3,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 D (c) 0,0 0,5 1,0 1,5 2,0 2,5 3,0 0,0 0,2 0,4 0,6 0,8 1,0 0 0.5 1 1.5 2 2.5 3 0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 2: Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3, (b) µ = 1.7 In Figures 1 and 2 we illustrate the diffusion metric norm and curva- ture considered as functions on the sphere of states fixed by the type of environment and the value of the corresponding coupling µ, µ1, µ2. Consider first Figure 1. The first row (fig. (a),(c),(e)) represent the diffusion metric norm and the second row (fig. (b),(d),(f)) the curvature for the three types of the environments and for some typical fixed values of the corresponding coupling strengths. The curvature is not constant, and can be positive or negative depending on the state vector and on the coupling strength. The maxima of the curvature can be sharp like in the cases of the dephasing and measurement of σz. On the other hand, in the thermal case the maxima is surrounded by a large neighborhood of states with almost maximal value of the curvature. Thus, the curvature has a sharp maxima only at the states which are clearly favored by the environment. If there are no such states the curvature maximum differs very little from the neighboring values. The curvature minima are at the states that are like repellers for the environment dominated dynamics. Dependence of the curvature maxima and the norm on the coupling strength is illustrated in Figure 2 for the dephasing and the thermal en- vironments. The most important information from these Figures is that in the dephasing and measurement ( not shown) cases there are clearly sharp values of corresponding coupling strength where the curvature maxima goes from negative to positive values. Also, we see that the curvature minima are negative for all values of the coupling strength. We shall now study the relation between the sign of the curvature maxima and a stability of the stochastic dynamics of the state vector. The relation will not be analyzed in a mathematically rigorous way using an appropriate notion of the stochastic stability and considering the evolution of the met- ric as a stochastic process governed by the process |ψ(t) >. Instead, our strategy is to compute the curvature along different sample paths and see if the path remains near the state corresponding to the curvature maxima. We do such computations for the evolution governed by the environment and an additional fixed small hamiltonian, and we pay special attention to the case when the Linblad operators and the hamiltonian do not commute. The computations are repeated for the values of the coupling to the environment slightly above and below the critical value when the curvature maxima is zero. If the Hamiltonian perturbation is zero the sample paths that started near a maximum of the curvature remain near this maximum. For very small added Hamiltonian part and for a fixed value of the coupling to the environment, the sample paths of the system could wonder away from the maximum or could remain near it. In the former case we shall say that the stochastic dynamics is unstable and in the later case it is stable. The relevant computations are illustrated in Figures 3 and 4. In the case of the dephasing environment (or the measurement of σ̂z), when the maxima of the curvature are sharply picked, Figures 3 clearly illus- trate that positive curvature maxima correspond to the stability and negative to instability in the above mentioned sense. On the other hand, in the ther- mal case, the dynamics is always unstable even if there is no Hamiltonian perturbation. This is illustrated in Figure 4. We can conclude that the dif- fusion metric curvature provides us with a clear picture of the qualitative properties of the system’s dynamics under strong influence of the environ- ment. It is well known that if the Linblad operators are Hermitian and commute with the hamiltonian, than the attractors of the stochastic QSD dynamics are the common eigenstates of the Hamiltonian and the Linblad operators [6],[13]. The curvature maxima coincide with the eigenstates of the Linblad 1,0 -1,0 1,0-0,9 0 400 800 1200 0 400 800 1200 1,0 -1,0 1,0-0,9 σ yσx Figure 3: Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3, (b) µ = 1.7 1.0 -1.0 1.0-0.9 0 200 400 600 800 100012001400 Figure 4: Poincaré sections for the separability constrained non-symmetric quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3, (b) µ = 1.7 operators, and consequently with the eigenstates of the hamiltonian. The probability of convergence to one of the attractors is, in this case, determined solely by the distance of the initial state from the attractor eigenstate, that is by the quantum mechanical transition probability, and does not depend on the parameters of the Hamiltonian and stochastic terms. The sign of the curvature maxima has no effect on this probability. This is the reason why we expected that the relevance of the sign of the curvature maxima on the stochastic stability is manifested if the Linblad operator and the Hamiltonian perturbation do not commute. This expectation is qualitatively confirmed, as we described and illustrated in Figures 3 and 4, by numerical computa- tions. Observations of numerical sample paths, when the Linblad and the Hamiltonian operators do not commute, are enough to establish the quali- tative connection between the maxima of the curvature and the stability of small domains near the maxima. Finally, our treatment of the relation between the geometry of the dif- fusion and the stability of the stochastic dynamics is rather heuristic. We treated the diffusion metric as a given field on R2n (determined by the Lin- blad operators), and we numerically studied the paths of the stochastic pro- cess |ψ(t) > in relation to the sign of the curvature maxima. However, the problem of stability versus the properties of the diffusion metric should be formulated and studied using the appropriate notions of stochastic stability [21],[22]. Nevertheless, we think that the numerical evidence strongly indi- cates that there is a clear relation between the sign and the shape of the curvature maxima and the systems dynamical stability. 4 Summary and discussion According to the view of QSD theory, evolution of a state of an open quan- tum system is a diffusion process governed by a complex stochastic differen- tial equation on the Hilbert space of the system. The diffusion term of the QSD evolution equation explicitly depends on the operators modelling the environment and on the current state vector of the system. We have stud- ied the Riemannian metric associated with the diffusion term in the QSD equation. The metric is defined on the real 2n-dimensional space (here n is the complex dimension of the Hilbert space) and is directly related to the properties of the Linblad operators of the environment. We have shown that the scalar curvature of the metric has local maxima at states that are favored by the corresponding environment. The curvature at different points, and in particular its local maxima, can be be negative or positive depending on the strength of the coupling to the environment. Also, the sharpness of the curvature maxima reflects the type of the environment. We have shown that there is a sense in which the sign of the curvature maxima is related to the stability of the corresponding state under the addition of a small perturba- tion that does not commute with the considered Linblad operator. If the environment type and the coupling strength are such that the curvature has sharp positive maxima, than the corresponding state is likely to attract the states of the system whose evolution is governed by the environment and a Hamiltonian that do not necessarily commute. On the other hand, if the curvature maxima are negative, the corresponding states are dynamically unstable under a small Hamiltonian perturbation that does not commute with the Linblad operators. In conclusion, the curvature of the diffusion metric is a relatively easy to calculate, and a very good indicator of what the environment dominated dynamics of the system would look like. The QSD equation describes the evolution of a pure quantum state using the Hilbert space of the quantum system, but, because it is norm-preserving, it gives also an equation on the state space, namely on the space of rays of the Hilbert space. The Riemannian metric associated with the diffusion on R2n gives a Hermitian modification of the Fubini-Study metric on CP n−1. It is common to consider the complex projective manifold with the associated Fubini-Study metric as the proper framework for the geometry of quantum states [23],[24], so the modification of the metric due to the diffusion should also be formulated within this framework. The examples that we have analyzed in this paper are restricted on a single qubit under the influence of various types of environments. It would be interesting to analyze the properties of the diffusion metric in the case of coupled gubits, and in particular to see what is the curvature at the entangled states. Probably the proper framework for such analyzes is the formulation on CP n−1, mentioned in the previous paragraph, because the entangled states then have characteristic geometric interpretation [23]. Acknowledgements This work is partly supported by the Serbian Min- istry of Science contract No. 141003. I should also like to acknowledge the support and hospitality of the Abdus Salam ICTP. References [1] Lindblad G 1976 Commun. Math. Phys. 48 119. [2] Breuer H-P and Petruccione F 2001 The Theory of Open Quantum Systems ( Oxford: Oxford University Press. ) [3] Carmichael H J 1983. An Open Systems Approach to Quantum Optics (Berlin: Springer-Verlag, Berlin) [4] Pashkin Yu A et al...2003 Nature 421 823 [5] Buric N 2005 Phys.Rev. A 72 042322 [6] Percival I C 1999 Quantum State Difussion (Cambridge: Cambridge Uni. Press.) [7] Belavkin V P 1999 Rep.Math.Phys. 43 405 [8] Gardiner C W and Zoller P 2000 Quantum Noise ( Berlin: Springer- Verlag) [9] Pearle P 1993 Phys.Rev.A 48 913 [10] Bassi A and Chirardi G 2003 Phys.Rep. 379 257 [11] Gisin N 1989 Helv.Phys.Acta 62 363 [12] Hughston L P 1996 Proc.R.Soc.Lond. A 452 953 [13] Adler S L and Brun T A 2001 J.Phys.A: Math. Gen. 34 4797 [14] Schack R, Brun T A and Percival I C 1996. Phys.Rev.A. 53 2696 [15] Wiseman H M and Milburn G Phys. Rev.A 47642 [16] Molmer K, Castin Y and Dalibar J 1993 J. Opt. Soc. Am. B 10 524 [17] Burić N 2006 Phys.Rev.A 73 052111 [18] Kobajayashi S and Nomizu K 1969 Foundations of Differential Geometry ( New York: Wiley) [19] Percival I C 1995 Proc.R.Soc. A 451 503 [20] Mintert F, Carvalho A R, Kus M and Buchleitner A 2005 Phys. Rep. 415 207 [21] Khas’minski R Z 1980 Stochastic Stability of Differential Equations (Alphen aan der Rijn: Sijthoff and Noordhoff) [22] Arnold L 1998 Random Dynamical Systems ( Berlin: Springer Verlag) [23] Brody D C and Hughston L P 2001 J.Geom. Phys. 38 19 [24] Bengtsson I and Žyczkowski K 2006 Geometry of Quantum States (Cam- bridge: Cambridge University Press) FIGURE CAPTIONS Figure 1 Diffusion metric norm (a,c,e) and curvature (b,d,f) as functions of state parametrized by (θ, φ), for dephasing environment with µ = 0.6 (a,b); measurement of σ̂z with µ = 1. (c,d) and thermal environment with µ1 = 2, µ2 = 1 (e,f). Figure 2 Diffusion metric norm (a,c) and curvature (b,d) as functions of θ for different values of the parameters µ or µ1, µ2, and the maximum over (θ, φ) of the curvature as a function of µ (e) or µ1 = 2µ2 = 2µ(f) . Figures a,b,e corespond to the dephasing and c, d, f to the thermal environment. Figure 3Diffusion metric curvature (a,c) along the corresponding stochas- tic path illustrated in b,d for the dephasing environment and µ = 0.3 when maxR < 0 (a,b), and µ = 0.5 when maxR > 0(c,d). The small Hamiltonian perturbation is 0.01σ̂x. Figure 4 Diffusion metric curvature (a) along a stochastic path (b) for the thermal environment and µ1 = 2µ2 = 1.6. The Hamiltonian part is zero. Introduction Riemannian metric of a complex diffusion QSD metric and qualitative properties of dynamics Summary and discussion

0704.1366

Investigation of transit-selected exoplanet candidates from the MACHO survey

Astronomy & Astrophysics manuscript no. macho c© ESO 2018 November 8, 2018 Investigation of transit-selected exoplanet candidates from the MACHO survey⋆ S. D. Hügelmeyer1, S. Dreizler1, D. Homeier1, and A. Reiners1,2 1 Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 2 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany Received <date> / Accepted <date> ABSTRACT Context. Planets outside our solar system transiting their host star, i. e. those with an orbital inclination near 90◦, are of special interest to derive physical properties of extrasolar planets. With the knowledge of the host star’s physical parameters, the planetary radius can be determined. Combined with spectroscopic observations the mass and therefore the density can be derived from Doppler- measurements. Depending on the brightness of the host star, additional information, e. g. about the spin-orbit alignment between the host star and planetary orbit, can be obtained. Aims. The last few years have witnessed a growing success of transit surveys. Among other surveys, the MACHO project provided nine potential transiting planets, several of them with relatively bright parent stars. The photometric signature of a transit event is, however, insufficient to confirm the planetary nature of the faint companion. The aim of this paper therefore is a determination of the spectroscopic parameters of the host stars as well as a dynamical mass determination through Doppler-measurements. Methods. We have obtained follow-up high-resolution spectra for five stars selected from the MACHO sample, which are consistent with transits of low-luminosity objects. Radial velocities have been determined by means of cross-correlation with model spectra. The MACHO light curves have been compared to simulations based on the physical parameters of the system derived from the radial velocities and spectral analyses. Results. We show that all transit light curves of the exoplanet candidates analysed in this work can be explained by eclipses of stellar objects, hence none of the five transiting objects is a planet. Key words. Stars: planetary systems - Eclipses - Techniques: radial velocities 1. Introduction After the first detections of planets outside our solar system (Wolszczan & Frail 1992; Mayor & Queloz 1995), an inten- sive search with various methods began (see Schneider 2002, for an overview) resulting in currently more than 200 planets (http://exoplanet.eu/). Most of these exoplanet detections have been performed via the radial velocity (RV) method where the “wobble” of the parent star caused by the planet is measured by spectral line shifts. Since these effects are very small for low- mass planets in orbits of tens to hundreds of days, the determi- nation of orbital period, phase, inclination, eccentricity, and RV amplitude demands RV accuracies of a few meters per second (Marcy et al. 2000). Meanwhile, alternative methods for planet detections have also been successfully applied. The first four microlensing plan- ets have been detected (Bond et al. 2004; Udalski et al. 2005; Beaulieu et al. 2006; Gould et al. 2006), possible first direct im- ages of extra-solar planets were published (Chauvin et al. 2004, 2005a,b; Neuhäuser et al. 2005; Biller et al. 2006), and the num- ber of detections due to transit searches is steadily increasing (McCullough et al. 2006; Bakos et al. 2007; O’Donovan et al. 2006; Collier Cameron et al. 2007). The transit method is of special interest, since it permits the derivation of additional physical parameters of the planet, e. g. the radius can be measured either indirectly via the ra- ⋆ Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme ID 075.C-0526(A) dius of the host star or directly via detection of the sec- ondary eclipse as observed with the Spitzer Space Telescope (Charbonneau et al. 2005; Deming et al. 2005). If combined with a radial velocity variation measurement, the mass and mean density can be determined, revealing constraints for the plane- tary structure. Furthermore, transiting systems allow us to inves- tigate the atmospheres of the planets (Charbonneau et al. 2002; Vidal-Madjar et al. 2004) as well as the spin-orbit-alignment be- tween the rotational axis of the host star and the planetary orbit (Wolf et al. 2007; Gaudi & Winn 2007; Winn et al. 2006). Drake & Cook (2004, hereafter DC) published a list of nine restrictively selected, transiting planet candidates from the MACHO project (Alcock et al. 1992). Only transit light curves with no indication of gravitational distortion and only those with clear U-shaped transit events were considered. De-reddened colours as well as light curve fitting provide a good estimate of the companion radius. Only companions below 3 Jupiter radii were selected. Based on high-resolution spectra, the orbital velocities of five potential host stars of exoplanet candidates have been measured. We analysed the RV measurements together with MACHO transit light curves in order to determine the system parameters complemented by a spectral analysis. The paper is organized as follows: In the next section, we shortly describe the spectroscopic observations and the spectral analysis as well as the Doppler-measurements. In section 3, we describe the results of the individual systems and summarise in section 4. http://arxiv.org/abs/0704.1366v1 http://exoplanet.eu/ 2 S. D. Hügelmeyer et al.: Investigation of transit-selected exoplanet candidates Table 1. Orbital elements, rotation velocities, and stellar parameters for all five analysed systems. Components c and d of MACHO 118.18272.189 and component b of MACHO 120.22041.3265 are not visible in the spectra. PMACHO is taken from Drake & Cook (2004) and P denotes the period derived using the light curves and RV measurements. K is the semi-amplitude of the RV variations, V0 the system velocity, and i the orbital inclination. In case of systems with elliptical orbits, e is the eccentricity and ω the longitude of the periastron. Furthermore, the mass MRV is given in cases where the RV amplitude of two components is known. Then T RVeff and R are calculated for zero- and terminal-age main sequence models. T SAeff is the effective temperature derived from the spectral analyses. In cases where just T SAeff from the spectral analyses is known, M SA and R are derived masses and radii from evolution models. All values in this table relate to the assumption of zero-age main sequence stars. MACHO ID P PMACHO K V0 i e ω M RV T RVeff T eff M [days] [days] [km s−1] [km s−1] [◦] [◦] [M⊙] [K] [K] [M⊙] [R⊙] 118.18272.189 a – 1.9673 0.00 −25.51± 0.03 – – – – – 5800 – – b – – 0.00 +05.46± 0.03 – – – – – 5800 – – c 3.9346 – – – (90.0) – – 0.41 3730 – – 0.38 d 3.9346 – – – – – 0.41 3730 – – 0.38 118.18407.57 a 4.7972 2.3986 78.84± 0.10 −20.48± 0.07 84.0 – – 1.27 6430 6200 – 1.23 b – – 0.00 −08.39± 0.03 – – – – 6600 – – c 4.7972 2.3986 89.68± 0.09 −20.48± 0.06 – – 1.11 5980 6200 – 1.04 118.18793.469 a 4.0744 2.0372 75.81± 0.18 −56.30± 0.11 85.6 0.041 89.94 0.90 5140 5400 – 0.81 b 4.0744 2.0372 83.67± 0.25 −56.30± 0.14 0.82 5070 5400 – 0.76 120.22041.3265 a 5.4083 5.4083 22.18± 0.06 −24.00± 0.04 89.8 0.108 19.98 – – 6200 1.19 1.15 b 5.4083 5.4083 114.90 −24.00± 0.04 – – (3340) (0.23) 0.28 402.47800.723 a 8.5496 4.2748 75.91± 0.04 +00.40± 0.04 85.8 – – 1.26 6400 6400 – 1.22 b – – 0.00 −26.40± 0.04 – – – – 5800 – – c 8.5496 4.2748 68.09± 0.07 +00.40± 0.07 – – 1.40 6820 6400 – 1.37 2. Observations and analyses In period 75 we secured three spectra for each of the five brightest candidates. We used ESO’s Fibre-fed Extended Range Échelle Spectrograph (FEROS) mounted on the 2.2 m telescope at La Silla, Chile. The spectrograph provides a spectral resolu- tion of R ∼ 48 000 and covers a wavelength range from 3500 Å to 9200 Å. The instrumental specifications list a RMS velocity error of ∼ 25 m s−1. This is sufficient to detect faint low-mass star companions and distinguish them from sub-stellar compan- ions, which was the primary aim of the observations. The sec- ondary aim is to use the spectra for a spectral analysis in order to derive the stellar parameters of the host stars. The observations of the five targets have been performed between August 19 and September 16, 2005. For each object we have obtained three spectra with exposure times between 2400 s and 3500 s, depending on the brightness of the object. The signal-to-noise ratio is ∼ 10. The data were reduced using the FEROS Data Reduction System (DRS). The échelle spectra were bias and flat field cor- rected and wavelength calibrated. The latter calibration was ad- ditionally quality checked by cross-correlating the observation with a sky line spectrum. The spectra were then corrected by ap- plying relative wavelength shifts. Barycentric and Earth rotation velocity influences to the wavelengths are accounted for auto- matically by the DRS. For the determination of the radial velocities we used the extracted FEROS spectra and synthetic spectra of main se- quence model stars calculated from LTE model atmospheres us- ing PHOENIX (Hauschildt et al. 1999) version 14.2. Both spectra were normalised and relative fluxes were interpolated on a log- arithmic wavelength scale. A cross-correlation (CC) between a model with Teff = 5600 K and observation was performed be- tween 5000 Å and 7000 Å. The CC was implemented using a grid with 200 steps of ∆ log λ/[Å] = 2.2 · 10−6 in each di- rection. This method turned out to be robust against the use of different model spectra. We could identify up to three spectro- scopic components in our data. Each of the peaks in the CC was then fitted by a Gaussian and the position of the maximum of the fit gives the radial velocity. The errors of the RV measurements were calculated from the standard deviation of the Gaussian plus the accuracy limit of FEROS of 25 m s−1. These RV errors are in the range between 50 and 350 m s−1. The CC function was also used to determine the projected rotation velocities of the stars. We therefore applied a solar spec- trum as template convolved with rotational profiles following Gray (2005). This method allows to derive stellar radii in bi- naries, assuming a synchronised orbit. In this analysis, the deter- mined rotational velocity v sin i is of the order of the uncertainty in most cases, which, due to the low signal-to-noise ratio, is about 5 km s−1. These derived radii are consistent with the ones obtained from main sequence models (see Table 1). Additional constraints for the radii of the binary components visible in the spectra can therefore not be derived. In order to spectroscopically identify the components of the analysed systems, we again used the PHOENIXmodel grid which ranges from 4000 K to 6800 K in Teff and from −1.5 to 0.0 in relative solar metallicity [Fe/H]. It should be noted that this is not sufficient for a detailed abundance determination, which was not the aim of this work. The surface gravity is kept constant at log g = 4.5. Knowing the RV of the individual components of a system, the models were gauss-folded to the resolution of the ob- servation and shifted to their position in the observed spectrum. Depending on the number of spectral components, all possible combinations of model spectra were tested for each observed spectrum. A χ2-test was used to identify the best fitting models. Given the low signal-to-noise ratio of the spectra, we estimate an uncertainty of about 400 K in effective temperature. In cases where we know the RV amplitudes for two compo- nents (MACHO 118.1407.57, 118.18793.469, 402.47800.723), S. D. Hügelmeyer et al.: Investigation of transit-selected exoplanet candidates 3 M sin i is known for both components. Assuming i = 90◦ for the first iteration, we determined radii and effective tempera- tures (T RVeff in Table 1) from interpolation of the Geneva model tracks (Schaerer et al. 1993) assuming zero-age main sequence (ZAMS) or terminal-age main sequence (TAMS) stars. We then applied the eclipsing binary simulation software Nightfall1 with the derived stellar and orbital parameters from the previous step as input and calculated a best model fit to the R-band light curve and radial velocity measurements simultaneously. We used the third light contribution and the inclination as free parameters and calculated χ2-values for the light curve fits assuming ZAMS and TAMS stars. In a second iteration, we repeated the fit with the now known inclination (see Fig. 3). For these three systems the so derived effective temperature can be compared to the one of the spectral analyses (T SAeff in Table 1). Deviations are within our estimated uncertainties and show the overall consistency of our main-sequence solution. In the other two cases (MACHO 120.22041.3265 and MACHO 118.18272.189), the effective temperature from the spectral analysis was used to derive masses and radii of each components, again assuming ZAMS and TAMS stars. In the light curve simulations for the MACHO R-band photometry we varied the inclination and the radius R2 of the potential transiting object, assuming ZAMS and TAMS primary stars. 3. Results We will present results for the five targeted MACHO objects for which we found an orbital solution that explains the detected transits and the measured radial velocities. In Fig. 1 we show fit- ted light curves to the photometric MACHO data (bottom pan- els in the plots) and RV curve fits to the Doppler-measurements (asterisks in the top panel of the plots). The dashed lines are for circular orbits and the solid lines show a best fit ellipti- cal orbit. Fig. 2 again shows Nightfall light curve solutions to the photometric data as well as the RV fits to the Doppler- measurements. Here circular orbits reproduce the observations best. All stars were assumed to be on the ZAMS for the fits in Figs. 1 and 2. χ2-contour plots for both ZAMS and TAMS stars are depicted in Fig. 3. The inclination of the orbital plane and the third light contribution (bottom three plots) and the radius R2 (top two plots) of the potential transiting objects were treated to vary. A list of the orbital parameters PMACHO (period given by Drake & Cook 2004), the derived period P in our analyses, the RV amplitude K, the system velocity V0, the orbital inclination i, and in cases of systems with elliptical orbits, e the eccentric- ity and ω the longitude of the periastron as well as the mass, effective temperature, and radius is shown in Table 1. MACHO 120.22041.3265 MACHO 120.22041.3265 is the only system in our sample with just one component visible in the spectra. Spectral analysis yields Teff = 6200 K and indicates a low metallicity ([Fe/H] = −1.0). The fit of a sinusoidal to the RVs folded to a period of 5.4083 d (DC, dashed curve in Fig. 1) differs from the RV mea- surement at a phase of 0.87 by ∼ 10 km s−1. A better fit is pro- vided by an elliptical orbit with an eccentricity of e = 0.108, a longitude of periastron of ω = 19.98, and an orbital semi- amplitude of K = 22.18 km s−1. For such a system the radius and mass of the secondary is R2 = 0.3 R⊙ and M2 = 0.23 M⊙ for a ZAMS and R2 = 0.5 R⊙ and M2 = 0.25 M⊙ for a TAMS 1 http://www.hs.uni-hamburg.de/DE/Ins/Per/Wichmann/Nightfall.html Fig. 1. Radial velocity and light curve fits for systems with ellip- tical orbits. The dashed lines show best-fit sinusoidals while the solid lines show best-fit eccentric orbits. Component a is plotted in black, component b in grey. The system velocity for the circu- lar orbit is shown by the thin line, and for the elliptical orbit by the thick dotted line. The solutions shown are calculated assum- ing ZAMS stars. The error bars for the RV measurements are of the size of the symbols. primary (see Fig. 3), clearly indicating an M dwarf companion. With these parameters, the system is very similar to OGLE-TR- 78 (Pont et al. 2005). We used equation (6.2) of Zahn (1977) to calculate an esti- mate for the circularisation time of the system. Due to the low mass ratio q = M2/M1, we find a circularisation time of the order of the Hubble time even for this close binary system. MACHO 118.18793.469 Two spectral components could be identified, each with Teff = 5400 K and a highly subsolar ([Fe/H] = −1.5) metallicity. For the light curve and RV fits with Nightfall we used the RV amplitudes of the two stars to derive masses by the procedure described in the previous section. A reasonable fit to the RV measurements folded to twice the period of DC can be achieved with sinusoidals (dashed curves in Fig. 1), i. e. assuming a cir- cular orbit for the two components. However, an improved fit can be achieved by fitting the light curve and radial veloci- ties in Nightfall at the same time to an elliptical orbit (solid 4 S. D. Hügelmeyer et al.: Investigation of transit-selected exoplanet candidates Fig. 2. Radial velocity and light curve fits for systems with cir- cular orbits. Component a is plotted in black, component b in grey, and component c in a lighter grey. The solutions shown are calculated assuming ZAMS stars. The error bars for the RV measurements are of the size of the symbols. curves in Fig. 1). The best fit is achieved with a small eccen- tricity of e = 0.041 and a periastron longitude of ω = 89.94◦. By varying the third light and the inclination, we construct the χ2-map shown in Fig. 3. As suggested by the spectral analysis of MACHO 118.18793.469, the lowest χ2-value is found for a third light of zero. The inclination is 85.6◦ for the ZAMS and 82.2◦ for the TAMS. The low depths of the transit is therefore due to a grazing eclipse. This is also supported by the V-shape of the best-fit model. The best-fit light curve model shows different transit depths. This is an indicator for two transits in one orbital period caused by two stars of slightly different size. MACHO 118.18407.57 Three components are visible in the CCs of the three spectra, one of which shows RV variations below 1 km s−1. Therefore, this component is a third component, either in a wider orbit or physically unrelated to the other two. Component a and c show RV changes of over 100 km s−1. They can be well fitted with sinusoidals of twice the period given by DC, i. e. 4.7972 d. If the photometric data are phased accordingly, we then see both transits where the transit depths are reduced due to third light of component b. For the light curve simulation we once more used the RV amplitudes of a and c to get the masses and varied the incli- nation and third light coming from component b. The effective temperatures and radii of the components are interpolated from the Geneva evolution tracks assuming young stars on the ZAMS and older stars on the TAMS. The contribution of component b meets the expectations from the spectral analyses (Teff = 6200 K for components a and c and Teff = 6600 K for component b, also see Fig. 3). The inclination of the system is 84◦ assuming that the stars are on the ZAMS and 79.5◦ for the TAMS. The system shows different transit depths, as MACHO 118.18793.469 does. MACHO 402.47800.723 The second brightest object in the sample shows three compo- nents in the spectra. Components a and c are best fitted by a model with Teff = 6400 K, b has Teff = 5800 K. As in the case of MACHO 118.18407.57, the masses are derived from the ra- dial velocities. The RV measurements of a and c are well fitted assuming a circular orbit with twice the period of DC. The third component only shows small RV variations and therefore seems to have a larger period than the other two. The fractional third light contribution for this component is ∼ 1/3 and an inclination of the eclipsing system is 85.8◦ assuming stars on the ZAMS and 82.3◦ for the TAMS (see Fig. 3). We again see transit depth dif- ferences. Due to the high signal-to-noise ratio of the light curve, these are quite obvious and amount to ∆R = 0.015. This observa- tion is also expected from the orbital semi-amplitude differences. MACHO 118.18272.189 Each of the three FEROS spectra displays two components. The spectral analysis reveals that both components have a similar ef- fective temperature (Teff = 5800 K) and a subsolar metallicity of [Fe/H] = −0.5. The cross-correlation shows that component b has a constant RV of ∼ 5.46 km s−1 within the above men- tioned statistical errors. Component a shows RV variations of ∼ 3.5 km s−1. Folding the RV measurements to the orbital pe- riod given by DC, one sees that the two components visible in the spectrum cannot be responsible for the transit in the light curve since one RV point is very close to the transit. However, here the two components should almost have the same RV. This is clearly not present in the data. The same is the case if we dou- ble the period (see Fig. 2). Thus, we exclude the scenario that the two visible components are responsible for the transit. S. D. Hügelmeyer et al.: Investigation of transit-selected exoplanet candidates 5 In another plausible scenario, we treat component b as third light and assume that component a is eclipsed by a low mass object not visible in the spectra. However, if we fit a sinusoidal to the RV points, in our solution the star would move away from the observer after the transit while it should do the opposite. We can therefore discard this scenario. One could argue that the variations in RV measured for com- ponent a is just caused by systematic errors and that the eclipse visible in the light curve is caused by a planet orbiting a with- out causing any noticeable RV changes. We have fitted this sce- nario taking the light from component b into account and found a radius of R2 = 0.25 R⊙ assuming that a is on the ZAMS and R2 = 0.35 R⊙ for a being on the TAMS (see Fig. 3). These val- ues, however, seem unrealisticly high for planets and we can re- ject the 3-body scenario. Finally, one scenario that can explain both the transit light curve and the measured RVs is a four body system consisting of the two G stars which are visible in the spectra and two M dwarfs invisible in the spectra. Here the two faint components or- bit each other in twice the period from DC and eclipse each other twice. We assume an inclination of 90◦ and two low-mass stars of equal size. The effective temperature of the eclipsing bodies was derived from the transit depth of the MACHO R-band light curve using blackbody fluxes for all four components. The tran- sit depth is reduced by the light of components a and b. The RV variations of component a can in this scenario be explained by the reflex motion of a to the orbit of the binary M star system with a much larger period. We therefore do not observe a correla- tion between the transits and the RV. This scenario is underlined by the fact that the two RV measurements in Fig. 2 at periods of ∼ 1.0 and ∼ 1.3, which have approximately the same RVs, are from two spectra only taken one day apart, while the third RV value comes from a spectrum 26 days later. Component b would be in a very wide orbit or physically unrelated to the other three stars. 4. Summary For none of the five analysed MACHO-candidates a planetary or brown dwarf companion could be identified. We therefore con- firm the speculation of DC that due to the depths of the transits in the photometric data the objects would be low-mass stars rather than sub-stellar objects. From the five candidates, we found one grazing eclipse of two nearly identical G stars (MACHO 118.18793.469), two blends of deep eclipses of G stars with a significant third light contribution (MACHO 118.18407.57 and MACHO 402.47800.723), one binary star with a G type primary and an M dwarf secondary (MACHO 120.22041.3265) and one rather complicated, blended system with four stars, of which each two are nearly identical (G and M type). With this work we could show that also for deep transit surveys for extrasolar planets, follow-up observations to weed out false positives are efficiently possible with moderate effort. After all, our results once again underline the need for spec- troscopic follow-up of transit planet candidates as already shown by Bouchy et al. (2005) and Pont et al. (2005) for the OGLE sur- vey and Torres et al. (2004) in the case of a blend scenario. Acknowledgements. We would like to thank the referee for very useful com- ments. S.D.H. gratefully acknowledges the support of the German-Israeli Foundation for Scientific Research and Development grant I-788-108.7/2003. A.R. has received research funding from the European Commission’s Sixth Framework Programme as an Outgoing International Fellow (MOIF-CT-2004- 002544). This paper utilizes public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University. References Alcock, C., Axelrod, T. S., Bennett, D. 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0704.1367

On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi)

ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE (WITH AN APPENDIX BY EDOARDO SERNESI) FLAMINIO FLAMINI*, ANDREAS LEOPOLD KNUTSEN** AND GIANLUCA PACIENZA*** Abstra t. Under natural hypotheses we give an upper bound on the dimension of families of singular urves with hyperellipti normalizations on a surfa e S with pg > 0 via the study of the asso iated families of rational urves in S . We use this result to prove the existen e of nodal urves of geometri genus 3 with hyperellipti normalizations, on a general K3 surfa e, thus obtaining spe i� 2-dimensional families of rational urves in S . We give two in�nite series of examples of general, primitively polarized K3s su h that their Hilbert squares ontain a P or a threefold birational to a P -bundle over a K3. We dis uss the onsequen es on the Mori one of the Hilbert square. 1. Introdu tion For any smooth surfa e S, the Hilbert s heme S[n] parametrizing 0-dimensional length n sub- s hemes of S is a smooth 2n-dimensional variety whose inner geometry is naturally related to that of S. For instan e, if ∆ ⊂ S[n] is the ex eptional divisor, that is, the ex eptional lo us of the Hilbert-Chow morphism µ : S[n] → Symn(S), then irredu ible (possibly singular) rational urves not ontained in ∆ roughly orrespond to irredu ible (possibly singular) urves on S with a g1n′ on their normalizations, for some n′ ≤ n (see � 2.1 for the pre ise orresponden e when n = 2). One of the features of this paper is to show how ideas and te hniques from one of the two sides of the orresponden e makes it possible to shed light on problems naturally arising on the other side. If S is a K3 surfa e, S[n] is a hyperkähler manifold ( f. [31, 2.2℄) and rational urves play a fundamental r�le in the study of the (birational) geometry of S[n]. Indeed a result due to Huybre hts and Bou ksom [32, 11℄ implies in parti ular that these urves govern the ample one of S[n] (we will re all the pre ise statement below and in � 6.1). The presen e of a Pn ⊂ S[n] gives rise to a birational map (the so- alledMukai �op [41℄) to another hyperkähler manifold and, for n = 2, all birational maps between hyperkähler fourfolds fa tor through a sequen e of Mukai �ops [12, 30, 60, 62℄. Moreover, as shown by Huybre hts [32℄, uniruled divisors allow to des ribe the birational Kähler one of S[n] (see � 7 for the pre ise statement). For hyperkähler fourfolds pre ise numeri al and geometri properties of the rational urves that are extremal in the Mori one have been onje tured by Hassett and Ts hinkel [25℄. The s ope of this paper, and the stru ture of it as well, is twofold: we �rst devise general methods and tools to study families of urves with hyperellipti normalizations on a surfa e S, mostly under the additional hypothesis that pg(S) > 0, in � 2-� 4. Then we apply these to obtain on rete results in the ase of K3 surfa es, in � 5-� 7. In parti ular, we have tried to develop a systemati way to 2000 Mathemati s Subje t Classi� ation : Primary 14H10, 14H51, 14J28. Se ondary 14C05, 14C25, 14D15, 14E30. (*) and (***) Member of MIUR-GNSAGA at INdAM "F. Severi". (**) Resear h supported by a Marie Curie Intra-European Fellowship within the 6th European Community Frame- work Programme. (***) During the last part of the work the author bene�tted from an "a ueil en délégation au CNRS". http://arxiv.org/abs/0704.1367v1 2 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA produ e rational urves on S[2] by showing the existen e of nodal urves on S with hyperellipti normalizations. To give an overview of the paper, we hoose to start with the se ond part. Let (S,H) be a general, smooth, primitively polarizedK3 surfa e of genus p = pa(H) ≥ 2. We have [2])R ≃ R[Y ]⊕ R[P1∆], where P1∆ is the lass of a rational urve in the ruling of the ex eptional divisor ∆ ⊂ S[2], and Y := {ξ ∈ S[2]|Supp(ξ) = {p0, y}, with p0 ∈ S and y ∈ C ∈ |H|}, where p0 and C are hosen. One has that P ∆ lies on the boundary of the Mori one and by the result of Huybre hts and Bou ksom [32, 11℄ mentioned above, if the Mori one is losed, then also the other boundary is generated by the lass of a rational urve. If X ∼alg aY − bP1∆ is an irredu ible urve in S[2], di�erent from a �ber of ∆, then we de�ne a/b to be the slope of the urve. Thus, the lower the slope is, the loser is X to the boundary of the Mori one. Des ribing the Mori one NE(S[2]) amounts to omputing slope(NE(S[2])) := inf slope(X) | X is an irredu ible urve in S[2] and, if the Mori one is losed, then slope(NE(S[2])) = sloperat(NE(S [2])), where sloperat(NE(S [2])) := inf slope(X) | X is an irredu ible rational urve in S[2] (See � 6.1, 6.2 and 6.3 for further details.) If now C ∈ |mH| is an irredu ible urve of geometri genus pg(C) ≥ 2 and with hyperellipti normalization, let g0(C) ≥ pg(C) be the arithmeti genus of the minimal partial desingularization of C that arries the g12 (see � 2.1 and � 6.2). By the uni ity of the g 2, C de�nes a unique irredu ible rational urve RC ⊂ S[2] with lass RC ∼alg mY − (g0(C)+12 )P ∆, f. (6.11). (This formula is also valid if RC is asso iated to a given g 2 on the normalization of an irredu ible rational or ellipti urve C.) Thus, the higher g0(C) (or pg(C)) is, and the lower m is, the lower is the slope of RC . This motivates the sear h for urves on S with hyperellipti normalizations of high geometri genus, thus �unexpe ted� from Brill-Noether theory. It is well-known that there exist �nitely many (nodal) rational urves, a one-parameter family of (nodal) ellipti urves, and a two-dimensional family of (nodal) urves of geometri genus 2 in |H| (see � 5). Every su h family yields in a natural way a two-dimensional family of irredu ible rational urves in S[2], f. � 2. Also note that, by a result of Ran [46℄, the expe ted dimension of a family of rational urves in a symple ti fourfold, when e a posteriori also of a family of urves with hyperellipti normalizations lying on a K3, equals two ( f. Lemma 5.1). In [22, Examples 2.8 and 2.10℄ we found two-dimensional families of nodal urves of geometri genus 3 in |H| having hyperellipti normalizations when pa(H) = 4 or 5. In this paper we generalize this: Theorem 5.2. Let (S,H) be a general, smooth, primitively polarized K3 surfa e of genus p = pa(H) ≥ 4. Then the family of nodal urves in |H| of geometri genus 3 with hyperellipti normal- izations is nonempty, and ea h of its irredu ible omponents is two-dimensional. The proof takes the whole � 5 and relies on a general prin iple of onstru ting urves with hy- perellipti normalizations on general K3s outlined in Proposition 5.11: �rst onstru t a marked K3 surfa e (S0,H0) of genus p su h that |H0| ontains a family of dimension ≤ 2 of nodal (possibly redu ible) urves with the property that a desingularization of some δ > 0 of the nodes is a limit of a hyperellipti urve in the moduli spa e Mp−δ of stable urves of genus p−δ and su h that this family is not ontained in a higher-dimensional su h family. Then onsider the parameter spa e Wp,δ of pairs ((S,H), C), where (S,H) is a smooth, primitively marked K3 surfa e of genus p and C ∈ |H| is a nodal urve with at least δ nodes. Now map (the lo al bran hes of) Wp,δ into Mp−δ by partially ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 3 normalizing the urves at δ of the nodes and mapping them to their respe tive lasses. The existen e of the parti ular family in |H0| ensures that the image of this map interse ts the hyperellipti lo us Hp−δ ⊂ Mp−δ. A dimension ount then shows that the dimension of the parameter spa e I ⊂ Wp,δ onsisting of ((S,H), C) su h that a desingularization of some δ > 0 of the nodes of C is a limit of a hyperellipti urve is at least 21. Now the dominan e on the 19-dimensional moduli spa e of primitively marked K3 surfa es of genus p follows as the dimension of the spe ial family on S0 was The te hni al di� ulties in the proof of Proposition 5.11 mostly arise be ause the urves in the spe ial family on S0 may be redu ible (in fa t, as in all arguments by degeneration, in pra ti al appli ations they will very often be). Therefore we need to partially desingularize families of nodal urves, and this tool is provided in Appendix A by E. Sernesi. Moreover, we need a areful study of the Severi varieties of redu ible nodal urves on K3s, and here we use results of Tannenbaum [55℄. Given Proposition 5.11, the proof of Theorem 5.2 is then a omplished by onstru ting a suitable (S0,H0) in Proposition 5.19 with |H0| ontaining a desired two-dimensional family of spe ial urves, with δ = p− 3, and then showing that the urves in the spe ial family on S0 in fa t deform to urves with pre isely δ nodes on the general S in Lemma 5.20. As will be dis ussed below, showing that the spe ial family on S0 is not ontained in a family of higher dimension of urves with the same property, is quite deli ate. We also show that the asso iated rational urves in S[2] over a threefold, f. Corollary 5.3, and that g0 = pg = 3, f. Remark 5.23. Turning ba k to the des ription of NE(S [2]), this shows that the lass of the asso iated rational urves in S[2] is Y − 3 P1∆, so that we obtain ( f. Corollary 6.27): (6.28) sloperat(NE(S [2])) ≤ 1 In Propositions 7.2 and 7.7 we present two in�nite series of examples of general primitively polarized K3 surfa es (S,H) of in�nitely many degrees su h that S[2] ontains either a P2 (these examples were shown to us by B. Hassett) or a threefold birational to a P1-bundle over a K3 and �nd the two-dimensional families of urves with hyperellipti normalizations in |H| orresponding to the lines and the �bres respe tively. In parti ular, these examples show that the bound (6.28) an be improved for in�nitely many degrees of the polarization. Namely, for any n ≥ 6 and d ≥ 2, we get: (7.4) sloperat(NE(S [2])) ≤ 2 2n−9 if p = pa(H) = n 2 − 9n+ 20; (7.9) sloperat(NE(S [2])) ≤ 1 if p = pa(H) = d Nevertheless, to our knowledge, (6.28) is the �rst non-trivial bound valid for any genus p of the polarization. The proofs of Propositions 7.2 and 7.7 are again by deformation, but unlike the proof of Proposition 5.11, we now deform S 0 of a spe ial K3 surfa e S0. The idea is to start with a spe ial quarti surfa e S0 ⊂ P3 su h that S[2]0 ontains a P2 or a threefold birational to a P1-bundle over itself, perform the standard involution on S 0 to produ e a new su h and then deform S 0 keeping the new one by keeping a suitable polarization on the surfa e that is di�erent from OS0(1). Here we use results on deformations of symple ti fourfolds by Hassett and Ts hinkel [25℄ and Voisin [57℄. By a result proved in [22℄, any irredu ible urve C ∈ |H| with hyperellipti normalization must satisfy g0(C) ≤ p+22 , where p = pa(H) ( f. Theorem 6.16 and (6.17)). It is then natural to ask whether this inequality a tually ensures the existen e of su h urves. We all this �The hyperellipti 4 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA existen e problem� and we see that a positive solution to this problem would yield a bound on the slope of rational urves that is mu h stronger than the ones obtained above, f. (6.25). In this sense, Theorem 5.2 is hopefully only the �rst step towards stronger existen e results. The study of urves on S with hyperellipti normalizations is not the only way to obtain bounds on the slope of the Mori one of S[2]. In fa t, an irredu ible urve C ∈ |mH| with a singular point x of multipli ity multx(C) yields an irredu ible urve in S with lass mY − (1/2)multx(C) (see the proof of Theorem 6.18). In parti ular, if p = pa(H), one has the bound ( f. Theorem 6.21) (6.22) slope(NE(S[2])) ≤ p−1 , obtained by using well-known results on Seshadri onstants on S. This bound is stronger than (6.28) but weaker than the bounds on the slope of the Mori one obtained from (7.4) and (7.9). Moreover, one relatively easily sees that the best bound one an obtain by Seshadri onstants is in any ase weaker than (7.4) and (7.9) and also weaker than the ones one ould obtain by solving �The hyperellipti existen e problem�, f. (6.25). In any ase, note that (6.22), (7.4) and (7.9) show that the bounds tend to zero as the degree of the polarization tends to in�nity, that is, (6.23) inf slope(NE(S[2])) | S is a proje tive K3 surfa e and likewise for sloperat(NE(S [2])). All the families of urves in |H| with hyperellipti normalizations we have seen above have in fa t dimension equal to two, the expe ted one. Moreover, a ru ial point in the proof of Theorem 5.2 is to bound the dimensions of families of irredu ible urves with hyperellipti normalizations on the spe ial K3 surfa e S0. This brings us over to the des ription of the �rst part of this paper. The problem of bounding the dimension of spe ial families of urves on surfa es, like in our ase of urves with hyperellipti normalizations, is interesting in its own, may be studied for larger lasses of surfa es, and may lead to further appli ations in other ontexts. Whereas methods from adjun tion theory have proved very useful for the study of smooth hyperellipti urves on surfa es [51, 53, 10℄, these methods do not extend to the ase of singular urves, where in fa t very little seems to be known. Even in the relevant ase of nodal urves on smooth surfa es, whose parameter spa es (the so- alled Severi varieties) have re eived mu h attention over the years and have been studied also in relation with moduli problems (see e.g. [49℄ for P2 and [21℄ for surfa es of general type), the dimension of their sublo i onsisting of urves with hyperellipti normalizations is not determined. The pre ise question we address is whether there exists an upper bound on the dimension of families of irredu ible urves on a proje tive surfa e with hyperellipti normalizations. One easily sees that, if the anoni al system of the surfa e is birational, then no urve with hyperellipti normalization an move, f. e.g. [33℄. On the other hand, taking any surfa e S admitting a (generi ally) 2 : 1 map onto a rational surfa e R and pulling ba k the families of rational urves on R, we obtain families of arbitrarily high dimensions of urves on S having hyperellipti normalizations. Moreover, the in�nite series of examples in Proposition 7.2 of general, primitively polarized K3 surfa es (S,H) su h that S[2] ontains a P2 shows that one annot even hope, in general, to �nd a bound in the simplest ase of Pi ard number one: in fa t, the (3m− 1)-dimensional family of rational urves in |OP2(m)| yields a (3m− 1)-dimensional family of irredu ible urves in |mH| having hyperellipti normalizations, f. � 7.1. Nevertheless, for a large lass of surfa es, it is possible to derive a geometri onsequen e on the family V , when its dimension is greater than two: ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 5 Theorem 4.6'. Let S be a smooth, proje tive surfa e with pg(S) > 0. Let V be a redu ed and irredu ible s heme parametrizing a �at family of irredu ible urves on S with hyperellipti normal- izations (of genus ≥ 2) su h that dim(V ) ≥ 3. Then the algebrai equivalen e lass [C] of the urves parametrized by V has a de omposition [C] = [D1] + [D2] into algebrai ally moving lasses su h that [D1 + D2] ∈ V . Moreover the rational urves in S[2] orresponding to the irredu ible urves parametrized by V over only a (rational) surfa e R ⊂ S[2]. In fa t we prove a stronger result, f. Theorem 4.6, that in parti ular relates the de omposition [C] = [D1] + [D2] to the g 2s on the normalizations of the urves parametrized by V . This additional point will in fa t be the ru ial one in our appli ation in the proof of Theorem 5.2. An immediate orollary is that the �naïve� dimension bound one may hope for, thinking about the fa t that rational urves in S[2] arising from urves on S of geometri genera ≤ 2 move in dimension at most two, is in fa t true under additional hypotheses on V , f. Corollary 4.7. These are satis�ed if e.g. the Néron-Severi group of S is of rank 1 and generated by the lass of a urve in V , and seem quite natural, taking into a ount the examples of large families mentioned above. The idea of the proof of Theorem 4.6 is rather simple and geometri and illustrates well the ri h interplay between the properties of urves on S and those of subvarieties of S[2]. The proof relies on the following two fundamental results: The �rst is Mori's bend-and-break te hnique (see Lemma 2.10 for the pre ise version we need), whi h gives a breaking into redu ible members of a family of rational urves of dimension ≥ 3 overing a surfa e. The se ond is a suitable version of Mumford's well-known theorem on 0- y les on surfa es with pg > 0 ( f. Corollaries 3.2 and 3.4). The onsequen e of parti ular interest to us is that any threefold in S[2] an only arry a two-dimensional overing family of rational urves when pg(S) > 0, f. Proposition 3.6. Combining those two ingredients, we see that any family satisfying the hypotheses of Theorem 4.6 yields a family of rational urves in S[2] of the same dimension ≥ 3, that an therefore only over a surfa e in S[2], on whi h we an apply bend-and-break to produ e a redu ible member. Then we have to show that we an also produ e a de omposition of the urves on S into algebrai ally moving lasses, and this is arried out in Proposition 4.3. Beside the appli ation in the proof of Theorem 5.2, we hope that Theorem 4.6 and the ideas behind its proof will �nd more appli ations. One is a Reider-like result for families of singular urves with hyperellipti normalizations obtained in [33℄, where also more examples are given. The paper is organized as follows. We go from the more general results to those pe uliar to the ase of K3 surfa es. We start in � 2 with the orresponden e between urves with hyperellipti normalizations on any smooth surfa e S and rational urves on S[2] and prove other preliminary results, before turning to the bend-and-break lemma for families of rational urves overing a surfa e in S[2]. The version of Mumford's theorem we need for our purposes is proved in � 3, and then rephrased in terms of rational quotients. Then we prove (a stronger version of) Theorem 4.6' in � 4. We then turn to K3 surfa es and prove Theorem 5.2 along the lines of the degeneration argument sket hed above. Se tion 6, apart from some known fa ts on the Hilbert s heme of points on a K3 surfa e, ontains the omputation of the lasses of rational urves in S[2] asso iated to urves in S with rational, ellipti or hyperellipti normalizations, as explained in � 2.1. The relation between the existen e of su h a urve, its singular Brill-Noether number (an invariant introdu ed in [22℄) and the slope of the Mori one of S[2] is also dis ussed, as well as the relation between the slope of the Mori one and Seshadri onstants. We end the paper presenting the two series of examples of general K3 surfa es whose Hilbert square ontains a P2 (respe tively a threefold birational to a P1-bundle over a K3) and dis ussing the numeri al properties of a line (respe tively a �bre) in it, as well as those of 6 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA the asso iated singular urves in S with hyperellipti normalizations. In Appendix A by E. Sernesi the reader will �nd a general result about partial desingularizations of families of nodal urves. A knowledgements. The authors thank L. Caporaso, O. Debarre, A. Iliev and A. Verra for useful dis ussions related to these problems. We are extremely grateful to: C. Ciliberto, for many valuable onversations and helpful omments on the subje t and for having pointed out some mistakes in a preliminary version of this paper; B. Hassett, for having pointed out the examples behind Proposition 7.2; E. Sernesi, for many helpful onversations and for his Appendix A. We �nally express our gratitude to the Department of Mathemati s, Università "Roma Tre" and to the Institut de Re her he Mathématique Avan ée, Université L. Pasteur et CNRS, where parts of this work have been done, for the ni e and warm atmosphere as well as for the kind hospitality. 2. Rational urves in S[2] Let S be a smooth, proje tive surfa e. In this se tion we gather some basi results that will be needed in the rest of the paper. We �rst des ribe the natural orresponden e between rational urves in S[2] and urves on S with rational, ellipti or hyperellipti normalizations. Then, in � 2.2, we apply Mori's bend-and-break te hnique to rational urves in Sym2(S) overing a surfa e. Re all that we have the natural Hilbert-Chow morphism µ : S[2] → Sym2(S) that resolves Sing(Sym2(S)) ≃ S. The µ-ex eptional divisor ∆ ⊂ S[2] is a P1-bundle over S. The Hilbert- Chow morphism gives an obvious one-to-one orresponden e between irredu ible urves in S[2] not ontained in ∆ and irredu ible urves in Sym2(S) not ontained in Sing(Sym2(S)). We will therefore often swit h ba k and forth between working on S[2] and Sym2(S). 2.1. Irredu ible rational urves in S[2] and urves on S. Let T ⊂ S × S[2] be the in iden e variety, with proje tions p2 : T → S[2] and pS : T → S. Then p2 is �nite of degree two, bran hed along ∆ ⊂ S[2]. (In parti ular, T is smooth as ∆ is.) Let X ⊂ S[2] be an irredu ible rational urve not ontained in ∆. We will now see how X is equivalent to one of three sets of data on S. Let νX : X̃ ≃ P1 → X be the normalization and set X ′ := p−12 (X) ⊂ T . By the universal property of blowing up, we obtain a ommutative square: (2.1) C̃X p2|X′ // X, de�ning the urve C̃X , ν̃X and f . In parti ular, ν̃X is birational and C̃X admits a g 2 (i.e., a 2 : 1 morphism onto P1, given by f ), but may be singular, or even redu ible. Set ν̃ := pS |X′ ◦ν̃X : C̃X → S. Assume �rst that C̃X is irredu ible. We set CX := ν̃(C̃X) ⊂ S. Sin e X 6⊂ ∆, CX is a urve. As C̃X arries a g12, it is easily seen that also the normalization of CX does, that is, CX has rational, ellipti or hyperellipti normalization. Moreover, it is easily seen that ν̃ : C̃X → CX is generi ally of degree one. Indeed, for general x ∈ CX , as x 6∈ pS(p−12 (∆)), we an write (pS |X′)−1(x) = {(x, x+ y1), . . . , (x, x+ yn)}, where n := deg ν̃. By de�nition of p2, and sin e X ′ = p−12 (X), we must have that ea h (yi, x + yi) ∈ X ′, for i = 1, . . . , n, and ea h ouple ((x, x+yi), (yi, x+yi)) is the pushdown by ν̃X of an element of the g 2 on C̃X . Hen e, ea h ouple (x, yi) is the pushdown by the normalization morphism of an element of the indu ed g on the normalization of CX . Sin e x has been hosen general, x 6∈ Sing(CX), so that we must have n = 1, as laimed. ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 7 In parti ular, by onstru tion, ν̃ : C̃X → CX is a partial desingularization of CX , in fa t, it is the minimal partial desingularization of CX arrying the g 2 in question (whi h is unique, if pg(CX) ≥ 2). We have therefore obtained: (I) the data of an irredu ible urve CX ⊂ S together with a partial normalization ν̃ : C̃X → CX with a g 2 on C̃X (unique, if pg(CX) ≥ 2), su h that ν̃ is minimal with respe t to the existen e of the g Next we treat the ase where C̃X is redu ible. In this ase, it must onsist of two irredu ible smooth rational omponents, C̃X = C̃X,1 ∪ C̃X,2, that are identi�ed by f . If ν̃ does not ontra t any of the omponents, set CX,i := ν̃(C̃X,i) ⊂ S and nX,i := deg ν̃| eCX,i , for i = 1, 2. We therefore obtain: (II) the data of a urve CX = nX,1CX,1 + nX,2CX,2 ⊂ S, with nX,i ∈ N, CX,i an irredu ible, rational urve, a morphism ν̃ : C̃X = C̃X,1 ∪ C̃X,2 → CX,1 ∪ CX,2 (resp. ν̃ : C̃X → CX,1 if CX,1 = CX,2) that is nX,i : 1 on ea h omponent and where C̃X,i is the normalization of CX,i, and an identi� ation morphism f : C̃X,1 ∪ C̃X,2 ≃ P1 ∪ P1 → P1. If ν̃ ontra ts one of the two omponents of C̃X , say C̃X,2, to a point xX ∈ S (it is easily seen that it annot ontra t both), then µ(X) ⊂ Sym2(S) is of the form {xX + CX}, for an irredu ible urve CX ⊂ S, whi h is ne essarily rational. It is easily seen that CX = ν̃(C̃X,1) and deg ν̃| eCX,1 = 1, so that we obtain: (III) the data of an irredu ible rational urve CX ⊂ S together with a point xX ∈ S. Note that in all ases (I)-(III), the support of the urve CX on S is simply (2.2) Supp(CX) = one-dimensional part of {x ∈ S | x ∈ Supp(ξ) for some ξ ∈ X} and the set is already purely one-dimensional pre isely unless we are in ase (III) with xX 6∈ C. Conversely from the data (I), (II) or (III) one re overs an irredu ible rational urve in S[2] not ontained in ∆. Indeed, in ase (I) (resp. (II)), the g12 on C̃X (respe tively, the identi� ation f ) indu es a P1 ⊂ Sym2(C̃X) and this is mapped to an irredu ible rational urve in Sym2(S) by the natural omposed morphism Sym2(C̃X) ν̃(2) // Sym2(CX) // Sym2(S). The irredu ible rational urve X ⊂ S[2] is the stri t transform by µ of this urve. In ase (III), X ⊂ S[2] is the stri t transform by µ of {xX + CX} ⊂ Sym2(S). We see that the data (III) orrespond pre isely to rational urves of type {x0 + C} ⊂ Sym2(S), where x0 ∈ S is a point and C ⊂ S is an irredu ible rational urve. Moreover, it is easily seen that the data (II) orrespond pre isely to the images by α : C̃1 × C̃2 ≃ P1 × P1 −→ C1 + C2 ⊂ Sym2(S), resp. α : Sym2(C̃) ≃ P2 −→ Sym2(C) ⊂ Sym2(S), of irredu ible rational urves in |n1F1 + n2F2| for n1, n2 ∈ N, resp. |nF | for an integer n ≥ 2, where Pic(C̃1 × C̃2) ≃ Z[F1] ⊕ Z[F2], resp. Pic(Sym2(C̃)) ≃ Z[F ], and C1, C2, resp. C, are irredu ible rational urves on S and �˜� denotes normalizations. The data of type (II) will however not be studied more in this paper, where we will fo us on the other two, mostly on (I). Note that an irredu ible rational urve X ⊂ Sym2(S) arising from rational (resp. ellipti ) urves C as in ase (I) moves in Sym2(C), whi h is a surfa e birational to P2 (resp. an ellipti ruled surfa e), 8 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA and a urve X ⊂ Sym2(S) of the form {xX +C} moves in the threefold {S +C}, whi h is birational to a P1-bundle over S, and ontains Sym2(C). At the same time, it is well-known that if kod(S) ≥ 0, then rational urves on S do not move and ellipti urves move in at most one-dimensional families. This follows for instan e from the following general result (that we will later need in the ase pg = 2). Lemma 2.3. Let S be a smooth, proje tive surfa e with kod(S) ≥ 0 ontaining an n-dimensional irredu ible family of irredu ible urves of geometri genus pg. Then n ≤ pg and if equality o urs, then either the family onsists of a single smooth rational urve; or kod(S) ≤ 1 and n ≤ 1; or kod(S) = 0. Proof. This is �folklore�. For a proof see [33℄. � As a onsequen e, if kod(S) ≥ 0, then rational urves in Sym2(S) arising from rational or ellipti urves on S move in families of dimension at most two in Sym2(S). On the other hand, irredu ible rational urves X ⊂ Sym2(S) arising from urves on S with hyperellipti normalizations of geometri genus pg ≥ 2 (ne essarily of type (I)), move in a family whose dimension equals that of the family of urves with hyperellipti normalizations in whi h C ⊂ S moves (by uni ity of the g 2). Apart from some spe ial ases, it is easy to see that the onverse is also true: Lemma 2.4. Let {Xb}b∈B be a one-dimensional irredu ible family of irredu ible rational urves in Sym2(S) overing a (dense subset of a) proper, redu ed and irredu ible surfa e Y ⊂ Sym2(S) that does not oin ide with Sing(Sym2(S)) ∼= S. Then C = CXb in S for every b ∈ B (notation as above) if and only if either Y = Sym2(C0), with either C0 ⊂ S an irredu ible rational urve and C ≡ nC0 for n ≥ 1, or C0 = C ⊂ S an irredu ible ellipti urve; or Y = C + C ′ := {p + p′ | p ∈ C, p′ ∈ C ′}, with C an irredu ible rational urve and C ′ ⊂ S any irredu ible urve; or Y = C1 + C2, with C1, C2 ⊂ S irredu ible rational urves and C = n1C1 + n2C2 for n1, n2 ∈ N. Proof. The "if" part is immediate. For the onverse, we treat the three ases (I)-(III) separately. If C is as in (I), then learly Y ⊂ Sym2(C), so that Y = Sym2(C) and C must be either rational or ellipti , as Y is uniruled. If C = n1C1 + n2C2 as in (II), then either C1 = C2 =: C0 and again Y = Sym 2(C0), or C1 6= C2 and Y = C1 + C2. Finally, if C is as in (III), then, for every b ∈ B, we have {Xb}b∈B = {xb+C}b∈B for some xb ∈ S, and the {xb}b∈B de�ne the desired urve C ′. We note that by Lemma 2.3 also the rational urves in Sym2(S) arising from singular urves of geometri genus 2 on S move in at most two-dimensional families. We will see below that this is a general phenomenon, under some additional hypotheses. We will fo us our attention on urves with hyperellipti normalizations (of genus pg ≥ 2) in Se tions 4-7. 2.2. Bend-and-break in Sym2(S). Let V ⊆ Hom(P1,Sym2(S)) be a redu ed and irredu ible sub- s heme (not ne essarily omplete). We onsider the universal map (2.5) PV := P 1 × V // Sym2(S) and assume that the following two onditions hold: (2.6) For any v ∈ V, ΦV (P1 × v) 6⊆ Sing(Sym2(S)) ≃ S; and (2.7) ΦV is generi ally �nite ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 9 (the latter just means that V indu es a �at family of rational urves in Sym2(S) of dimension dim(V )). Set (2.8) RV := im(ΦV ), the Zariski losure of im(ΦV ) in Sym 2(S). It is the (irredu ible) uniruled subvariety of Sym2(S) overed by the urves parametrized by V . In the language of [35, Def. 2.3℄, RV is the losure of the lo us of the family ΦV . Note that dim(RV ) ≥ 2 if dim(V ) ≥ 1 by (2.7). Moreover ( f. e.g. [24, Prop. 2.1℄), (2.9) dim(RV ) ≤ 3 if kod(S) ≥ 0. When RV is a surfa e, using Mori's bend-and-break te hnique we obtain the following result. In the statement we underline the fa t that the breaking an be made in su h a way that, for general ξ, η ∈ RV , two omponents of the redu ible (or non-redu ed) member at the border of the family pass through ξ and η, respe tively. This will be entral in our appli ations (Proposition 4.3 and � 5, where we prove Theorem 5.2). We give the proof be ause we ould not �nd in literature pre isely the statement we will need. Lemma 2.10. Assume that dim(V ) ≥ 3 and dim(RV ) = 2. Let ξ and η be any two distin t general points of RV . Then there is a urve Yξ,η in RV su h that Yξ,η is algebrai ally equivalent to (ΦV )∗(P v) and either (a) there is an irredu ible nonredu ed omponent of Yξ,η ontaing ξ and η; or (b) there are two distin t, irredu ible omponents of Yξ,η ontaing ξ and η, respe tively. Proof. Sin e dim(V ) ≥ 3 by assumption, by (2.7) we an pi k a one-dimensional smooth subs heme B = Bξ,η ⊂ V parametrizing urves in V su h that (ΦV )∗(P1 × v) ontains both ξ and η, for every v ∈ B. We therefore have a family of rational urves: (2.11) ΦB := (ΦV )|B : P1 ×B −→ RV . and two marked (distin t) points x, y ∈ P1 su h that ΦB(x×B) = ξ and ΦB(y ×B) = η, su h that ea h ΦB(P 1 × v) is non onstant, for any v ∈ B; in parti ular ΦB(P1 ×B) is a surfa e. As in the proofs of [36, Lemma 1.9℄ and [35, Cor. II.5.5℄, let B be any smooth ompa ti� ation of B. Consider the surfa e P1 × B. Let 0 ∈ B denote a point at the boundary, P10 the �bre over 0 of the proje tion onto the se ond fa tor and x0, y0 ∈ P10 ⊂ P1 ×B the orresponding marked points. By the Rigidity Lemma [36, Lemma 1.6℄, ΦB annot be de�ned at the point x0, as in the proof of [36, Cor. 1.7℄, and the same argument works for y0. Therefore, to resolve the indetermina ies of the rational map ΦB : P 1 × B − − → RV , we must at least blow up P1 × B at the points x0 and y0. Now let W be the blow-up of P1 × B su h that ΦB : W −→ RV is an extension of ΦB , that is, we have a ommutative diagram P1 ×B //___ RV . Let Ex0 := π −1(x0) and Ey0 := π −1(y0). Note that neither of these an be ontra ted by ΦB , for otherwise ΦB itself would be de�ned at x0 or y0. Therefore the urve ΦB(Ex0) has an irredu ible omponent Γξ ontaining ξ and the urve ΦB(Ey0) has an irredu ible omponent Γη ontaining η and by onstru tion, Γξ+Γη ⊆ ΦB∗(π−1(P1× 0)) and the latter is the desired urve Yξ,η. The two ases (a) and (b) o ur as Γξ = Γη or Γξ 6= Γη, respe tively. � 10 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA 3. Rationally equivalent zero- y les on surfa es with pg > 0 In this se tion we extend to the singular ase a onsequen e of Mumford's result on zero- y les on surfa es with pg > 0 ( f. [42, Corollary p. 203℄) and reformulate the results in terms of rational quotients. 3.1. Mumford's Theorem. The main result of this subse tion, whi h we prove in detail for the reader's onvenien e, relies on the following generalization of Mumford's result ( f. [58, Chapitre 22℄ and referen es therein, for a detailed a ount). Theorem 3.1. (see [58, Prop. 22.24℄) Let T and Y be smooth proje tive varieties. Let Z ⊂ Y × T be a y le of odimension equal to dim(T ). Suppose there exists a subvariety T ′ ⊂ T of dimension k0 su h that, for all y ∈ Y , the zero- y le Zy is rationally equivalent in T to a y le supported on T ′. Then, for all k > k0 and for all η ∈ H0(T,ΩkT ), we have [Z]∗η = 0 in H0(Y,ΩkY ) where, as ostumary, [Z]∗η denotes the di�erential form indu ed on Y by the orresponden e Z. Mumford's original �symple ti � argument and the theorem above yield the following result (see [42, Corollary p. 203℄). Corollary 3.2. Let S be a smooth, irredu ible proje tive surfa e with pg(S) > 0 and Σ ⊂ S[n] a redu ed, irredu ible (possibly singular) omplete subs heme su h that µ(Σ) 6⊂ Sing(Symn(S)), where µ : S[n] → Symn(S) is the Hilbert-Chow morphism. If there exists a subvariety Γ ⊂ Symn(S) su h that dim(Γ) ≤ 1, Γ 6⊂ Sing(Symn(S)) and all the zero- y les parametrized by µ(Σ) are rationally equivalent to zero- y les supported on Γ, then dim(Σ) ≤ n. Proof. Let π : Σ̃ → Σ ⊂ S[n] be the desingularization morphism of Σ. Let Z = Λπ ⊂ Σ̃× S[n] be the graph of π. Then Z ∼= Σ̃, so that codim(Z) = dim(S[n]), as in Theorem 3.1. By assumption, µ(Σ) parametrizes zero- y les of length n on S that are all rationally equivalent to zero- y les supported on Γ, with dim(Γ) ≤ 1. Sin e µ(Σ) is not ontained in Sing(Symn(S)) by assumption, µ|Σ : Σ → µ(Σ) is birational. If Γ′ denotes the stri t transform of Γ under µ, we get that dim(Γ′) ≤ 1. We an apply Theorem 3.1 with Z = Y = Σ̃, T = S[n] and T ′ = Γ′. Thus, for ea h k > 1 and for ea h η ∈ H0(Ωk ), [Z]∗η = 0 in H0(Σ̃,Ωk Let ω ∈ H0(S,KS) be a non-zero 2-form on S. As in [42, Corollary℄, we de�ne: ω(n) := p∗i (ω) ∈ H0(Sn,Ω2Sn) where Sn is the nth- artesian produ t and pi is the natural proje tion onto the i fa tor, 1 ≤ i ≤ n. The form ω(n) is Sym(n)-invariant and, sin e we have that µ is surje tive, this indu es a anoni al 2-form ω µ ∈ H0(S[n],Ω2S[n]) (see [42, �1℄, where ω µ = ηµ in the notation therein). From what we observed above, [Z]∗(ω µ ) = 0 as a form in H 0(Σ̃,Ω2 ). Consider (Symn(S))0 := xi | xi 6= xj , 1 ≤ i 6= j ≤ n and such that ω(xi) ∈ Ω2S,xi is not 0 Then (Symn(S))0 ⊂ Symn(S) is an open dense subs heme that is isomorphi to its preimage via µ in S[n]. For ea h ξ ∈ (Symn(S))0, ξ is a smooth point and πn : S n → Symn(S) ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 11 is étale over ξ. Thus, the 2-form ω(n) ∈ H0(Sn,Ω2Sn) is non-degenerate on the open subset (Sn)0 of points in the preimage of (Symn(S))0, i.e. it de�nes a non-degenerate skew-symmetri form on the tangent spa e of (Sn)0. Let π0n := πn|(Sn)0 ; sin e π0n : (Sn)0 → (Sym n(S))0 is étale, there exists a 2-form 0 ∈ H0((Symn(S))0,Ω2(Symn(S))0) su h that ω(n) = π∗n(ω 0 ) and ω 0 is also non-degenerate. Therefore, the maximal isotropi subspa es 0 (ξ) are n-dimensional. Now Σ ⊂ S[n] and Σ ∩ µ−1((Symn(S))0) 6= ∅, sin e µ(Σ) 6⊂ Sing(Symn(S)) by assumption. Sin e Σ is redu ed, let ξ ∈ Σ ∩ µ−1((Symn(S))0) be a smooth point. Then, sin e Σsmooth = π−1(Σsmooth), by abuse of notation we still denote by ξ ∈ Σ̃ the orresponding point. We know that [Z]∗ω[n]µ (ξ) = 0 in the tangent spa e Tξ(Σ̃). Sin e ξ ∈ Σsmooth ∩ µ−1((Symn(S))0) ⊂ (Symn(S))0, then [Z]∗(ω µ ) = ω 0 |Σsmooth∩µ−1((Symn(S))0). This implies dim(Σ) ≤ n. � 3.2. The property RCC and rational quotients. Re all that a variety T (not ne essarily proper or smooth) is said to be rationally hain onne ted (RCC, for brevity), if for ea h pair of very general points t1, t2 ∈ T there exists a onne ted urve Λ ⊂ T su h that t1, t2 ∈ Λ and ea h irredu ible omponent of Λ is rational (see [35℄). Furthermore, by [16, Remark 4.21(2)℄, if T is proper and RCC, then ea h pair of points an be joined by a onne ted hain of rational urves. Also re all that, for any smooth variety T , there exists a variety Q, alled the rational quotient of T , together with a rational map (3.3) f : T −− → Q, whose very general �bres are equivalen e lasses under the RCC-equivalen e relation (see, for in- stan e, [16, Theorem 5.13℄ or [35, IV, Thm. 5.4℄). In this language, an equivalent statement of Corollary 3.2 is: Corollary 3.4. Let S be a smooth, proje tive surfa e with pg(S) > 0. If Y ⊂ S[n] is a omplete subvariety of dimension > n not ontained in Exc(µ), then any desingularization of Y has a rational quotient of dimension at least two. Proof. Let Ỹ be any desingularization of Y and Q its rational quotient. Up to resolving the indeter- mina ies of f : Ỹ −− → Q, we may assume that f is a proper morphism whose very general �bre is a RCC-equivalen e lass, so that in parti ular ea h �bre is RCC (see [35, Thm. 3.5.3℄). If dim(Q) = 0, it follows that Ỹ (so also Y ) is RCC, ontradi ting Corollary 3.2. If dim(Q) = 1, then by utting Ỹ with dim(Y )− 1 general very ample divisors, we get a urve Γ′ that interse ts every �bre of f . Every point of Ỹ is onne ted by a hain of rational urves to some point on Γ′. We thus obtain a ontradi tion by Corollary 3.2 (with Γ the image of Γ′ in Sym2(S)). � Let now RV be the variety overed by a family of rational urves in Sym 2(S) parametrized by V , as de�ned in (2.8), R̃V be any desingularization of RV and QV be the rational quotient of R̃V . Of ourse dim(QV ) ≤ dim(RV )− 1, as RV is uniruled by onstru tion. Lemma 3.5. If dim(V ) ≥ dim(RV ), then dim(QV ) ≤ dim(RV )−2 (for any desingularization R̃V of RV ). In parti ular, if dim(V ) ≥ 2 and dim(RV ) = 2, then any desingularization of RV is a rational surfa e. 12 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA Proof. With notation as in � 2.2, we have dim(PV ) ≥ dim(RV ) + 1, so that the general �bre of ΦV is at least one-dimensional, f. (2.5). This means that, if ξ is a general point of RV , there exists a family of rational urves in RV passing through ξ, of dimension ≥ 1. Of ourse the same is true for a general point of R̃V . Thus, the very general �bre of f in (3.3) has dimension at least two, when e dim(QV ) ≤ dim(RV ) − 2. The last statement follows from the fa t that any smooth surfa e that is RCC is rational ( f. [35, IV.3.3.5℄). � Combining Corollary 3.4 and Lemma 3.5, we then get: Proposition 3.6. If pg(S) > 0 and dim(V ) ≥ 2, then either (i) RV is a surfa e with rational desingularization; or (ii) dim(V ) = 2, RV is a threefold and any desingularization of RV has a two-dimensional rational quotient. Proof. By (2.9), dim(RV ) = 2 or 3. If dim(RV ) = 2, then (i) holds by Lemma 3.5. If dim(RV ) = 3, then dim(QV ) = 2 by Corollary 3.4. Hen e dim(V ) = 2 by Lemma 3.5 and (ii) holds. � Remark 3.7. Let S be a smooth, proje tive surfa e with pg(S) > 0 and let Y ⊂ S[2] be a uniruled threefold di�erent from Exc(µ), where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism. Take a overing family {Cv}v∈V of rational urves on Y . By Corollary 3.4 the family must be two-dimensional (see Lemma 3.5). Then the urves in the overing family yield, via the orrespon- den e des ribed in � 2.1, urves on S with rational, ellipti or hyperellipti normalizations, and the orresponden e is one-to-one in the hyperellipti ase. We therefore see that we must be in one of the following ases: (a) S ontains an irredu ible rational urve Γ and Y = {ξ ∈ S[2] | Supp(ξ) ∩ Γ 6= ∅}; (b) S ontains a one-dimensional irredu ible family {E}v∈V of irredu ible ellipti urves and Y = {ξ ∈ E[2]v }v∈V ; ( ) S ontains a two-dimensional, irredu ible family of irredu ible urves with hyperellipti nor- malizations, not ontained in a higher dimensional irredu ible family, and Y is the lo us overed by the orresponding rational urves in S[2]. (Note that in fa t ase (b) an only o ur for kod(S) ≤ 1 by Lemma 2.3 and ase ( ) only when |KS | is not birational. The latter fa t is easy to see, f. e.g. [33℄.) In the ase of K3 surfa es, uniruled divisors play a parti ularly important r�le [32, �5℄, f. � 7. Now all ases (a)-( ) above o ur on a general, proje tive K3 surfa e with a polarization of genus ≥ 6. In fa t, ases (a) and (b) o ur on any proje tive K3 surfa e sin e it ne essarily ontains a one-dimensional family of irredu ible, ellipti urves and a zero-dimensional family of rational urves, by a well-known theorem of Mumford (see the proof in [38, pp. 351-352℄ or [2, pp. 365-367℄). Case ( ) o urs on a general primitively polarized K3 surfa e of genus p ≥ 6 by Corollary 5.3 below with a family of urves of geometri genus 3. In addition to this, in Proposition 7.7 we will see that there is another threefold as in ( ) arising from urves of geometri genus > 3 in the hyperplane linear system on general proje tive K3 surfa es of in�nitely many degrees. Moreover, there is not a one-to-one orresponden e between families as in (a), (b) or ( ) above and uniruled threefolds in S[2]. In fa t, in Proposition 7.2 we will see that there is a two-dimensional family of urves with hyperellipti normalizations, as in ( ), in the hyperplane linear systems on general K3 surfa es of in�nitely many degrees whose asso iated rational urves over only a P2 in S[2]. ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 13 4. Families of urves with hyperellipti normalizations The purpose of this se tion is to study the dimension of families of urves on a smooth proje tive surfa e S with hyperellipti normalizations. We �rst remark that it is not di� ult to see that if |KS | is birational, then the dimension of su h a family is for ed to be zero (see e.g. [33℄). At the same time it is easy to �nd obvious examples of surfa es, even with pg(S) > 0, with large families of urves with hyperellipti normalizations, namely surfa es admitting a �nite 2 : 1 map onto a rational surfa e. (For examples of su h ases, see e.g. [26, 27, 28, 29, 48, 51, 53, 10℄ to mention a few.) In these ases one an pull ba k the families of rational urves on the rational surfa e to obtain families of urves on S with hyperellipti normalizations of arbitrarily high dimensions. Moreover, in Proposition 7.2 below we will see that even a general, primitively polarized K3 surfa e (S,H), for in�nitely many degrees, ontains a P2 in its Hilbert square, whi h is not ontained in ∆ (but the surfa e is not a double over of a P2, by generality). Therefore, by the orresponden e in � 2.1, S ontains large families of urves with hyperellipti normalizations. One an see that in all these examples of large families the algebrai equivalen e lass of the members breaks into nontrivial e�e tive de ompositions. For example, in the mentioned K3 ase of Proposition 7.2, we will see that the urves in |OP2(n)| in P2 ⊂ S[2] orrespond to urves in |nH|. In this se tion we will see that this is a general phenomenon, with the help of Lemma 2.10. To this end, let V be a redu ed and irredu ible s heme parametrizing a �at family of urves on S all having onstant geometri genus pg ≥ 2 and hyperellipti normalizations. Let ϕ : C → V be the universal family. Normalizing C we obtain, possibly restri ting to an open dense subs heme of V , a �at family ϕ̃ : C̃ → V of smooth hyperellipti urves of genus pg ≥ 2 ( f. [56, Thm. 1.3.2℄). Let ωeC/V be the relative dualizing sheaf. As in [37, Thm. 5.5 (iv)℄, onsider the morphism γ : C̃ → P(ϕ̃∗(ωeC/V )) over V . This morphism is �nite and of relative degree two onto its image, whi h we denote by PV . We thus obtain a universal family ψ : PV → V of rational urves mapping to Sym2(S), as in (2.5), satisfying (2.6) and (2.7). (Stri tly speaking, (2.5) denoted a universal family of maps, whereas it now denotes a universal family of urves.) To summarize, re alling (2.8), we have (4.1) C̃ // PV // RV Also note that (4.1) is ompatible with the orresponden e of ase (I) in � 2.1, in the sense that, for general v ∈ V , we have (using the same notation as in � 2.1) (4.2) π(ϕ̃−1(v)) = pS(p 2 Xv) = (pS)∗(p 2 Xv) = CXv , with Xv = µ ΦV (ψ −1(v)) ⊂ S[2], where µ is the Hilbert-Chow morphism (in parti ular, pS and p2 are the �rst and se ond proje tions, respe tively, from the in iden e variety T ⊂ S × S[2]). Note that the se ond equality in (4.2) follows as pS is generi ally one-to-one on the urves in question, as we saw in � 2.1. This will be entral in the proof of the next result. We now apply Lemma 2.10 to �break� the urves on S. Proposition 4.3. Let S be a smooth, proje tive surfa e and V and RV as above. Assume that dim(V ) ≥ 3 and dim(RV ) = 2 and let [C] be the algebrai equivalen e lass of the members parametrized by V . Then there is a de omposition into two e�e tive, algebrai ally moving lasses [C] = [D1] + [D2] 14 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA su h that, for general ξ, η ∈ RV , there are e�e tive divisors D′1 ∼alg D1 and D′2 ∼alg D2 su h that ξ ⊂ D′1 and η ⊂ D′2 and [D′1 +D′2] ∈ V , where V is the losure of V in the omponent of the Hilbert s heme of S ontaining V . Proof. For general ξ, η ∈ RV , both being supported at two distin t points on S, let B = Bξ,η ⊂ V be as in the proof of Lemma 2.10 and B be any smooth ompa ti� ation of B. By abuse of notation, we will onsider ξ and η as being points in S[2]. By (the proof of) Lemma 2.10, using the Hilbert- Chow morphism, there is a �at family {Xb}b∈B of urves in the surfa e µ−1∗ (RV ) ⊂ S[2] (where µ is the Hilbert-Chow morphism as usual) parametrized by B, su h that, for general b ∈ B, Xb is an irredu ible rational urve and (4.4) CXb = (pS)∗(p 2 (Xb)) = π(ϕ̃ −1(b)), with notation as in � 2.1 ( f. (4.2)). In parti ular, {CXb}b∈B is a one-dimensional nontrivial subfamily of the family {CXv}v∈V given by V . Moreover, for some b0 ∈ B \B, we have Xb0 ⊇ Yξ + Yη, where Yξ and Yη are irredu ible rational urves (possibly oin iding) su h that ξ ∈ Yξ and η ∈ Yη. Also note that Yξ, Yη 6⊂ ∆ ⊂ S[2]. Pulling ba k to the in iden e variety T ⊂ S × S[2], we obtain a �at family {X ′b := p 2 (Xb)}b∈B of urves in T , su h that (4.5) X ′b0 := p 2 (Xb) ⊇ p 2 (Yξ) + p 2 (Yη) =: Y ξ + Y Note that the family {X ′b}b∈B is in fa t a family of urves in the in iden e variety T0 ⊂ S×µ−1∗ (RV ), whi h is a surfa e ontained in T . Sin e pS maps this family to a family of urves overing (an open dense subset of) S, by (4.4), we see that (pS)|T0 is surje tive, in parti ular generi ally �nite. Thus, hoosing ξ and η general enough, we an make sure they lie outside of the images by p2 of the �nitely many urves ontra ted by (pS)|T0 . Hen e q −1(Yξ) and q −1(Yη) are not ontra ted by pS . Therefore, re alling (4.4) and (4.5) and letting b′ ∈ B be a general point, we get C ∼alg (pS)∗X ′b′ ∼alg (pS)∗X ′b0 ⊇ (pS)∗Y ξ + (pS)∗Y η ⊇ Dξ +Dη, where Dξ := p(q −1Yξ) and Dη := p(q −1Yη). By onstru tion we have Dξ ⊃ ξ and Dη ⊃ η, viewing ξ and η as length-two subs hemes of S. (Note that Dξ and Dη are not ne essarily distin t.) Possibly after adding additional omponents to Dξ and Dη , we an in fa t assume that C ∼alg (pS)∗X ′b′ = Dξ +Dη, with Dξ and Dη not ne essarily redu ed and irredu ible. Sin e this onstru tion an be repeated for general ξ, η ∈ RV and the set {x ∈ S | x ∈ Supp(ξ) for some ξ ∈ RV } is dense in S, as the urves parametrized by V over the whole surfa e S, the obtained urves Dξ and Dη must move in an algebrai system of dimension at least one. By onstru tion, Dξ + Dη lies in the border of the family ϕ : C → V of urves on S, and as su h, [Dξ +Dη ] lies in the losure of V in the omponent of the Hilbert s heme of S ontaining V . Moreover, as the number of su h de ompositions is �nite (as S is proje tive and the divisors are e�e tive), we an �nd one de omposition [C] = [D1] + [D2] holding for general ξ, η ∈ RV . � The next two results are immediate onsequen es: Theorem 4.6. Let S be a smooth, proje tive surfa e with pg(S) > 0. Then the following onditions are equivalent: (i) S[2] ontains an irredu ible surfa e R with rational desingularization, su h that R 6= µ−1∗ (C1+ C2), µ ∗ (Sym 2(C)) for rational urves C,C1, C2 ⊂ S and R 6⊂ Exc(µ), where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism; ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 15 (ii) S ontains a �at family of irredu ible urves with hyperellipti normalizations of geometri genus pg ≥ 3, parametrized by a redu ed and irredu ible s heme V su h that dim(V ) ≥ 3. Furthermore, if any of the above onditions holds, then (a) the rational urves in S[2] that orrespond to the irredu ible urves parametrized by V , over only the surfa e R in S[2]; and (b) the algebrai equivalen e lass [C] of the urves parametrized by V has an e�e tive de om- position [C] = [D1] + [D2] into algebrai ally moving lasses su h that, for general ξ, η ∈ R, there are e�e tive divisors D′1 ∼alg D1 and D′2 ∼alg D2 su h that ξ ⊂ D′1, η ⊂ D′2 and [D′1 + D 2] ∈ V , where V is the losure of V in the omponent of the Hilbert s heme of S ontaining V . Proof. Assume (ii) holds. By Proposition 3.6 we have that RV ⊂ Sym2(S) is a surfa e with rational desingularization, so that (i) holds. Assume now that (i) holds. Then R arries a family of rational urves of dimension n ≥ 3. By Lemma 2.4 and the assumptions in (i), this yields an n-dimensional family of urves on S that have rational, ellipti or hyperellipti normalizations. From Lemma 2.3, we get (ii). Finally, assume that these onditions hold. Then (a) follows from Proposition 3.6 again, where R is the proper transform via µ of the surfa e RV therein; �nally, (b) follows from Proposition 4.3. � Corollary 4.7. Let S be a smooth, proje tive surfa e with pg(S) > 0 and V be a redu ed, irredu ible s heme parametrizing a �at family of irredu ible urves with hyperellipti normalizations (of geometri genus ≥ 2). Denote by [C] the algebrai equivalen e lass of the members of V . If [C] has no de omposition into e�e tive, algebrai ally moving lasses, then dim(V ) ≤ 2. In parti ular, Corollary 4.7 holds when e.g. NS(S) = Z[C]. The examples with the double overs of smooth rational surfa es and the result in Proposition 7.2 mentioned above, show that the results above are natural. The statement in Theorem 4.6(b) shows that in fa t the length-two zero-dimensional s hemes on the urves in the family orresponding to the elements of the g 2s on their normalization, are in fa t �generi ally ut out� by moving divisors in a �xed algebrai de omposition of the lass of the members in the family. This reminds of the nowadays well-known results of Reider and their generalizations [47, 8, 9℄. In fa t, Theorem 4.6(b) an be used to prove a Reider-like result involving the arithmeti and geometri genera of the urves in the family, f. [33℄. Moreover, the pre ise statement in Theorem 4.6(b) will be ru ial in the next se tion, where we will prove existen e of urves with hyperellipti normalizations by degeneration methods. 5. Nodal urves of geometri genus 3 with hyperellipti normalizations on K3 surfa es In the rest of the paper we will fo us on the existen e of urves with �Brill-Noether spe ial� hyperellipti normalizations (i.e. of geometri genera > 2) and in this se tion we will see that Theorem 4.6(b) is parti ularly suitable to prove existen e results by degeneration arguments. To do this and to dis uss some onsequen es on S[2], we will in the rest of the paper fo us on K3 surfa es, whi h in fa t were one of our original motivations for this work. We start with the following observation ombining a result of Ran, already mentioned in the Introdu tion, with the results from the previous se tion. Lemma 5.1. Let S be a smooth, proje tive K3 surfa e and L be a globally generated line bundle of se tional genus p ≥ 2 on S. Let |L|hyper ⊆ |L| be the subs heme parametrizing irredu ible urves in |L| with hyperellipti normalizations. 16 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA Then, any irredu ible omponent of |L|hyper has dimension ≥ 2, with equality holding if L has no de omposition into moving lasses. Proof. Any n-dimensional omponent of |L|hyper yields an n-dimensional family of irredu ible rational urves in S[2]. By [46, Cor. 5.1℄, we have n ≥ 2. The last statement follows from Corollary 4.7. � The main aim of this se tion is to apply Theorem 4.6(b) to prove: Theorem 5.2. Let (S,H) be a general, smooth, primitively polarized K3 surfa e of genus p = pa(H) ≥ 4. Then the family of nodal urves in |H| of geometri genus 3 with hyperellipti normal- izations is nonempty, and ea h of its irredu ible omponents is two-dimensional. In [22℄ we studied whi h linear series may appear on normalizations of irredu ible urves on K3 surfa es. To do so, we introdu ed a singular Brill-Noether number ρsing(pa, r, d, pg) whose negativity, when Pic(S) ≃ Z[H], ensures non-existen e of urves in |H|, with pa = pa(H) and of geometri genus pg, having normalizations admitting a g d (we will return to this in � 6.3 below). Moreover, in [22, Examples 2.8 and 2.10℄, we already gave examples of nodal urves with hyperellipti normalizations with geometri genus 3 and arithmeti genus 4 or 5. Theorem 5.2 shows that this is a general phenomenon. The proof will be given in the remainders of this se tion. Moreover, we will also determine the dimension of the lo us overed in S[2] by the rational urves asso iated to urves in a omponent of the family: Corollary 5.3. Let (S,H) be a general, smooth, primitively polarized K3 surfa e of genus p = pa(H) ≥ 6. Then the subs heme of |H| parametrizing nodal urves of geometri genus 3 with hyper- ellipti normalizations ontains a two-dimensional omponent V su h that dim(RV ) = 3. This orollary in parti ular shows that all three ases in Remark 3.7 o ur on a general K3 surfa e. In � 6.2-6.3 we will both ompute the lasses of the orresponding rational urves in S[2] (see (6.26)) and dis uss some of the onsequen es of Theorem 5.2 on the Mori one of S[2]. Before starting on the proof of Theorem 5.2, we re all that, for any smooth surfa e S and any line bundle L on S, su h that |L| ontains smooth, irredu ible urves of genus p := pa(L), and any positive integer δ ≤ p, one denotes by V|L|,δ the lo ally losed and fun torially de�ned subs heme of |L| parametrizing the universal family of irredu ible urves in |L| having δ nodes as the only singularities and, onsequently, geometri genus pg := p − δ. These are lassi ally alled Severi varieties of irredu ible, δ-nodal urves on S in |L|. It is nowadays well-known, as a dire t onsequen e of Mumford's theorem on the existen e of nodal rational urves on K3 surfa es (see the proof in [38, pp. 351-352℄ or [2, pp. 365-367℄) and standard results on Severi varieties, that if (S,H) is a general, primitively polarized K3 surfa e of genus p ≥ 3, then the Severi variety V|H|,δ is nonempty and regular, i.e. it is smooth and of the expe ted dimension p− δ, for ea h δ ≤ p ( f. [55, Lemma 2.4 and Theorem 2.6℄; see also e.g. [15, 20℄). The regularity property follows from the fa t that, sin e by de�nition V|L|,δ parametrizes irredu ible urves, the nodes of these urves impose independent onditions on |L| ( f. [15, 20℄ and [55, Remark 2.7℄). From equisingular deformation theory, this implies that suitable obstru tions to some lo ally trivial deformations are zero. In other words, it implies �rst that, for any δ′ > δ, V|L|,δ′ ⊂ V |L|,δ (see [52, Anhang F℄, [59℄ and [50, Thm. 4.7.18℄ for P2 and [55, � 3℄ forK3s). Furthermore, if [C] ∈ V|L|,δ+k, k > 0, is a general point of an irredu ible omponent, the fa t that the nodes impose independent onditions allows to learly des ribe what V |L|,δ looks like lo ally around the point [C]: it is the union of smooth bran hes through [C], ea h bran h orresponding to a hoi e of δ "marked" (or "assigned") nodes among the δ+ k nodes of C, and these bran hes interse t transversally at [C]; moreover, the other k "unassigned" nodes of C disappear when one deforms [C] in the orresponding bran h of V |L|,δ (see [52, Anhang F℄, [59℄ and [49, � 1℄ for P and [55, � 3℄ for K3s). ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 17 The situation is slightly di�erent for redu ible, nodal urves in |L|. Sin e they appear in the proof of Theorem 5.2, we also have to take are of this ase. To this end, we de�ne the �degenerated� version of V|L|,δ by W|L|,δ := C ∈ |L| | C, not ne essarily irredu ible, has only nodes(5.4) as singularities and at least δ nodes For the same reasons as above, W|L|,δ is a lo ally losed subs heme of |L|. Note that (5.5) W|L|,δ = ∪δ′≥δV|L|,δ′ if all the urves in |L| are irredu ible, whi h is a partial ompa ti� ation of V|L|,δ. Let [C] ∈ W|L|,δ. Choosing any subset {p1, . . . , pδ} of δ of its nodes, one obtains a pointed urve (C; p1, . . . , pδ), where p1, . . . , pδ are also alled themarked (or assigned) nodes of C ( f. [55, De�nitions 3.1-(ii) and 3.6-(i)℄). Re all that there exists an algebrai s heme, whi h we denote by (5.6) B(C; p1, p2, . . . , pδ), lo ally losed in |L|, representing the fun tor of in�nitesimal deformations of C in |L| that preserve the marked nodes, i.e. the fun tor of lo ally trivial in�nitesimal deformations of the pointed urve (C; p1, . . . , pδ) ( f. [55, Proposition 3.3℄, where we have identi�ed the s hemes therein with their proje tions into the linear system |L|). In other words, B(C; p1, p2, . . . , pδ) is the lo al bran h of W|L|,δ around [C] ∈W|L|,δ, orresponding to the hoi e of the δ marked nodes. We have: Theorem 5.7. ( f. [55, Theorem 3.8℄) Let (C; p1, . . . , pδ) be as above. Assume that the general element of |L| is a smooth, irredu ible urve and that the partial normalization of C at the δ marked nodes p1, . . . , pδ is a onne ted urve. Then B(C; p1, p2, . . . , pδ) is smooth at the point [(C; p1, p2, . . . , pδ)] of dimension dim(|L|)− δ. Proof. This follows from [55, Theorem 3.8℄ sin e, by our assumptions, the pointed urve (C; p1, . . . , pδ) is virtually onne ted in the language of [55, De�nition 3.6℄. � For the proof of Theorem 5.2 we need to re all other fundamental fa ts. We �rst de�ne, for any globally generated line bundle L of se tional genus p := pa(L) ≥ 2, on a K3 surfa e S, and any integer δ su h that 0 < δ ≤ p− 2, the lo us in the Severi variety V|L|,δ, (5.8) V hyper |L|,δ := C ∈ V|L|,δ | its normalization is hyperellipti Observe that in parti ular, for any p ≥ 3, one always has V hyper|L|,p−2 = V|L|,p−2 6= ∅ and, by regularity of V|L|,p−2, this is smooth and of dimension two. Let Mg be the moduli spa e of smooth urves of genus g, whi h is quasi-proje tive of dimension 3g−3 for g ≥ 2. Denote by Mg its Deligne-Mumford ompa ti� ation. Then Mg is the moduli spa e of stable, genus g urves. Let Hg ⊂ Mg denote the lo us of hyperellipti urves, whi h is known to be an irredu ible variety of dimension 2g − 1 (see e.g. [1℄) and Hg ⊂ Mg be its ompa ti� ation. Moreover, re all from [23, Def.(3.158)℄ that a nodal urve C (not ne essarily irredu ible) is stably equivalent to a stable urve C ′ if C ′ is obtained from C by ontra ting to a point all smooth rational omponents of C meeting the other omponents in only one or two points. 18 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA As above, we de�ne the degenerated version of V hyper |L|,δ by hyper |L|,δ := C ∈W|L|,δ | there exists a desingularization C̃ of δ of the(5.9) nodes of C, su h that C̃ is stably equivalent to a (stable) urve C ′ with [C ′] ∈ Hpa(L)−δ Note that, by de�nition, any su h C̃ is onne ted. Similarly as in (5.5), we have: (5.10) W hyper |L|,δ = ∪δ′≥δV hyper |L|,δ if all the urves in |L| are irredu ible. Theorem 5.2 will be a dire t onsequen e of the next three results, Propositions 5.11 and 5.19 and Lemma 5.20. The entral degeneration argument is given by the following: Proposition 5.11. Let p ≥ 3 and δ ≤ p− 2 be positive integers. Assume there exists a smooth K3 surfa e S0 with a globally generated, primitive line bundle H0 on S0 with pa(H0) = p and su h that hyper |H0|,δ (S0) 6= ∅ and dim(W hyper |H0|,δ (S0)) ≤ 2. Then, on the general, primitively marked K3 surfa e (S,H) of genus p, W hyper |H|,δ (S) is nonempty and equidimensional of dimension two. Proof. Let Bp be the moduli spa e of primitively marked K3 surfa es of genus p. It is well-known that Bp is smooth and irredu ible of dimension 19, f. e.g. [2, Thm.VIII 7.3 and p. 366℄. We let b0 = [(S0,H0)] ∈ Bp. Similarly as in [5℄, onsider the s heme of pairs (5.12) Wp,δ := (S,C) | [(S,H)] ∈ Bp and [C] ∈W|H|,δ(S) and the natural proje tion (5.13) π : Wp,δ −→ Bp. (The fa t that Wp,δ is a s heme, in fa t a lo ally losed s heme, follows from the already mentioned proof of Mumford's theorem on the existen e of nodal rational urves as in [38, pp. 351-352℄ or [2, pp. 365-367℄.) Note that for general [(Sb,Hb)] = b ∈ Bp we have π−1(b) = ∪δ′≥δV|Hb|,δ′(Sb) by (5.5) (as Pic(Sb) ≃ Z[Hb]), so that π−1(b) is nonempty, equidimensional and of dimension g := p− δ, by the regularity property re alled above. In parti ular, π is dominant. Observe that Wp,δ is singular in odimension one, so in parti ular it is not normal. For brevity, let W := Wp,δ and let C f→ W be the universal urve. As in Theorem A.1, (i) and (ii), in Appendix A, there exists a ommutative diagram // W, where α is a �nite, unrami�ed morphism de�ning a marking of all the δ-tuples of nodes of the �bres of f ( f. Theorem A.1, with V = W, E(δ) = W(δ)). Pre isely, by using notation as in Theorem A.1, if for w ∈ W the urve C(w) has δ + τ nodes, τ ∈ Z+, α−1(w) onsists of elements, sin e any ηw ∈ α−1(w) parametrizes an unordered, marked δ-tuple of the δ + τ nodes of C(w). Let ηw ∈ W(δ). Then ηw is represented by a pointed urve (C; p1, p2, . . . , pδ), where (S,C) ∈ W and where p1, p2, . . . , pδ are δ marked nodes on C. ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 19 Let W(S,H) (resp. W(δ)(S,H)) be the �bre of π (resp. of α ◦ π) over [(S,H)] ∈ Bp, and let α(S,H) : W(δ)(S,H) −→ W(S,H) be the indu ed morphism. For ηw ∈ W(δ)(S,H) as above, we have (5.14) T[ηw](W(δ)(S,H)) ∼= T[(C;p1,p2,...,pδ)](B(C; p1, p2, . . . , pδ)), where B(C; p1, p2, . . . , pδ) is as in (5.6). Indeed, sin e α is �nite and unrami�ed, then also α(S,H) is. Therefore, it su� es to onsider the image of the di�erential dα(S,H)[ηw ]. The latter is given by �rst-order deformations of C in S (equivalently in |H|) that are lo ally trivial at the δ marked nodes; these are pre isely given by T[(C;p1,p2,...,pδ)](B(C; p1, p2, . . . , pδ)) ( f. [55, Remark 3.5℄). Let W̃(δ) be the smooth lo us of W(δ). By Theorem 5.7 and by (5.14), together with the fa t that Bp is smooth, W̃(δ) ontains all the pairs (S,C) with δ marked nodes on C, su h that |C| is globally generated (i.e. its general element is a smooth, irredu ible urve) and the partial normalization of C at these marked nodes is a onne ted urve. More pre isely, by the proof of Mumford's theorem on the existen e of nodal rational urves on K3 surfa es, as in [38, pp. 351-352℄ or [2, pp. 365-367℄), any irredu ible omponent of W(δ) has dimension ≥ 19 + p − δ = 19 + g; furthermore, by (5.14), dim(T[ηw ](W(δ)(S,H))) = g, where ηw represents (S,C) with C with the δ marked nodes. It also follows that W(δ) is smooth, of dimension 19 + g at these points. If we restri t C to W̃(δ), from Theorem A.1, (iv) and (v), we have a ommutative diagram W̃(δ) // W, where α̃ = α|fW(δ) and where f̃ is the �at family of partial normalizations at δ nodes of the urves parametrized by α(W̃(δ)) (in the notation of Theorem A.1 in Appendix A, f̃ = f in (v) and C̃ = C in (iii) and (iv)). There is an obvious rational map W̃(δ) //___ Mg, de�ned on the open dense subs heme W̃ ⊂ W̃(δ) su h that, for ηw ∈ W̃0(δ), C̃(ηw) is stably equivalent to a stable urve of genus g. Set ψ := c|fW0 . By de�nition, for any ηw ∈ W̃0(δ), the map ψ ontra ts all possible smooth rational omponents of C̃(ηw) meeting the other omponents in only one or two points and maps the resulting stable urve into its equivalen e lass in Mg. Pi k any C0 ∈ W hyper|H0|,δ (S0) and let w0 = [(S0, C0)] ∈ W be the orresponding point. Now |H0| is globally generated and the normalization of C0 at some δ nodes satisfying the onditions in (5.9) is a onne ted urve. Therefore, letting ηw0 ∈ α−1(w0) be the point orresponding to marking these δ nodes, we have that ηw0 ∈ W̃0(δ) and the map c is de�ned at ηw0 . Let Ṽ ⊆ W̃0 be the irredu ible omponent ontaining ηw0 ; then, as proved above, dim(Ṽ) = 19+g. By assumption, ψ(Ṽ) ∩Hg 6= ∅. Hen e, for any irredu ible omponent K ⊆ ψ(Ṽ) ∩Hg, we have (5.15) dim(K) ≥ dim(ψ(Ṽ)) + dim(Hg)− dim(Mg) = dim(ψ(Ṽ)) + 2− g. 20 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA Pi k any K ontaining ψ(ηw0) and let I ⊆ ψ−1|eV (K) be any irredu ible omponent ontaining ηw0 . Sin e the general �bre of ψ|eV has dimension dim(Ṽ)− dim(ψ(Ṽ)) = 19 + g − dim(ψ(Ṽ)), from (5.15) we have dim(I) = dim(K) + 19 + g − dim(ψ(Ṽ))(5.16) ≥ dim(ψ(Ṽ)) + 2− g + 19 + g − dim(ψ(Ṽ)) = 21. Consider now (5.17) π ◦ (α̃|I) : I −→ Bp. Sin e, by assumption, the �bre over b0 = [(S0,H0)] is at most two-dimensional, we on lude from (5.16) that π◦(α̃|I) is dominant, that all the �bres are pre isely two-dimensional and that dim(I) = 21. This shows that W hyper |H|,δ 6= ∅ for general [(S,H)] ∈ Bp and Lemma 5.1 implies that in fa t any irredu ible omponent of W hyper |H|,δ (S) has dimension two. � Remark 5.18. In parti ular, Lemma 5.1, Proposition 5.11 and [22, Examples 2.8 and 2.10℄ prove Theorem 5.2 for p = 4 and 5. We next onstru t the desired spe ial primitively marked K3 surfa e: Proposition 5.19. Let d ≥ 2 and k ≥ 1 be integers. There exists a K3 surfa e S0 with Pic(S0) = Z[E]⊕ Z[F ]⊕ Z[R] and interse tion matrix  E2 E.F E.R F.E F 2 F.R R.E R.F R2 0 d k d 0 k k k −2 and su h that the following onditions are satis�ed: (a) |E| and |F | are ellipti pen ils; (b) R is a smooth, irredu ible rational urve. ( ) H0 := E +F +R is globally generated, in parti ular the general member of |H0| is a smooth, irredu ible urve of arithmeti genus p := 2k + d; (d) the only e�e tive de ompositions of H0 are H0 ∼ E + F +R ∼ (E + F ) +R ∼ (E +R) + F ∼ (F +R) + E. Proof. Sin e the latti e has signature (1, 2), then, by a result of Nikulin [43℄ (see also [39, Cor. 2.9(i)℄), there is a K3 surfa e S0 with that as Pi ard latti e. Performing Pi ard-Lefs hetz re�e tions on the latti e, we an assume that H0 is nef, by [2, VIII, Prop. 3.9℄. Straightforward al ulations on the Pi ard latti e rules out the existen e of e�e tive divisors Γ satisfying Γ2 = −2 and Γ.E < 0 or Γ.F < 0, or Γ2 = 0 and Γ.H0 = 1. Hen e (a) and ( ) follow from [48, Prop. 2.6 and (2.7)℄. Similarly one omputes that if Γ > 0, Γ2 = −2 and Γ.R < 0, then Γ = R, proving (b). Similarly, (d) is proved by dire t al ulations using the nefness of E, F and H0 and re alling that by Riemann-Ro h and Serre duality a divisor D on a K3 surfa e is e�e tive and irredu ible only if D2 ≥ −2 and D.N > 0 for some nef divisor N . � The following result, together with (5.10) and Proposition 5.11, now on ludes the proof of Theo- rem 5.2 and Corollary 5.3. From Remark 5.18, we need only onsider p ≥ 6. Lemma 5.20. Let p ≥ 6 be an integer. There exists a smooth K3 surfa e S0 with a globally generated, primitive line bundle H0 on S0 with p = pa(H0) su h that (a) W hyper |H0|,p−3(S0) 6= ∅; ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 21 (b) dim(W hyper |H0|,p−3(S0)) = 2; ( ) there exists a omponent of W hyper |H0|,p−3(S0) whose general member deforms to a urve [Ct] ∈ hyper |Ht|,p−3(St), for general [(St,Ht)] ∈ Bp; (d) for general [(St,Ht)] ∈ Bp, the two-dimensional irredu ible omponent Vt ⊆ V hyper|Ht|,p−3(St) given by ( ), satis�es dim(RVt) = 3 (with notation as in � 2.2). Proof. Set k = 1 if p is even and k = 2 if p is odd and let d := p− 2k ≥ 2. Consider the marked K3 surfa e (S0,H0) in Proposition 5.19. We will onsider two general smooth ellipti urves E0 ∈ |E| and F0 ∈ |F | and urves of the form C0 := E0 ∪ F0 ∪R, with transversal interse tions and a desingularization (5.21) C̃0 = Ẽ0 ∪ F̃0 ∪ R̃→ C0 of the δ := p − 3 = d + 2k − 3 nodes marked in Figure 1 below, that is, all but one of ea h of the interse tion points E0 ∩ F0, E0 ∩R and F0 ∩R. E 0 F 0 −−−−−−−−−−−−−−−−> E 0 F 0 partial normalization k points k points k=1,2 k=1,2 d points Figure 1. The urves C0 and C̃0 Then [C0] ∈W hyper|H0|,p−3, as C̃0 is stably equivalent to a union of two smooth ellipti urves interse ting in two points ( f. [23, Exer ise (3.162)℄), proving (a). Clearly the losure of the family we have 22 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA onstru ted is isomorphi to |E| × |F | ≃ P1 × P1, and is therefore two-dimensional. Denote by W0 ⊂W hyper|H0|,p−3 this two-dimensional subs heme. We will now show that any irredu ible omponent W of W hyper |H0|,p−3 has dimension ≤ 2. A entral observation, whi h will be used together with Theorem 4.6(b), will be that, with the above hoi es of k, we have (5.22) E.H0 = F.H0 = d+ k = p− k is odd. We start by onsidering families of redu ible urves. These are all lassi�ed in Proposition 5.19(d). If the general element in W is of the form D ∪R, for D ∈ |E +F |, then in order to have a partial desingularization D̃ ∪ R̃ to be (degenerated) hyperellipti , we must have deg(D̃ ∩ R̃) = 2, so that we must desingularize 2(k − 1) of the interse tion points of D ∩R. Finally, as pa(D̃ ∪ R̃) = 3, we must have pa(D̃) = 2. Therefore W ⊆ WD × {R} ≃ WD, where WD ⊂ |D| is a subfamily of irredu ible urves of geometri genus ≤ 2. It follows that dim(W ) ≤ dim(WD) ≤ 2, by Lemma 2.3. If the general element in W is of the form D ∪ E, for D ∈ |F + R|, then in order to have a partial desingularization D̃ ∪ R̃ that is (degenerated) hyperellipti , we must have deg(D̃ ∩ Ẽ) = 2. If the proje tion W → |E| is dominant, this means that g12(D̃) ⊆ |f∗E|| eD, where f : S̃ → S denotes the omposition of blow-ups of S that indu es the partial desingularization D̃∪ R̃→ D∪R. But this would mean that |f∗E|| eD, whi h is base point free on D̃, is omposed with the g 2(D̃), a ontradi tion, as deg(O eD(f ∗E)) = E.D = E.H0 is odd by (5.22). Therefore, the proje tion W → |E| is not dominant, when e dim(W ) ≤ dim(|D|) = 1 D2+1 = k ≤ 2, as desired. By symmetry, the ase where the general element in W is of the form D ∪ F , for D ∈ |E +R| is treated in the same way. Finally, we have to onsider the ase of a family W ⊆ |H0| of irredu ible urves. In this ase assume dim(W ) ≥ 3, and let C be a general urve parametrized by W . Then by Theorem 4.6 (b), there exists an e�e tive de omposition into moving lasses H0 ∼M +N su h that 2(C̃) ⊆ |f∗M || eC , |f ∗N || eC , where f : S̃ → S denotes the su ession of blow ups of S that indu es the normalization C̃ → C. From Proposition 5.19(d) we see that we must have 2(C̃) ⊆ |f∗E|| eC , or |f ∗F || eC , whi h means that either |f∗E|| eC or |f ∗F || eC is omposed with the g 2(C̃), again a ontradi tion, as both have odd degree by (5.22). We have therefore proved (b). To prove ( ) we will show that any [C0] ∈W hyper|H0|,p−3 in the two-dimensional, irredu ible omponent W0 onsidered above in fa t deforms to a urve [Ct] ∈W hyper|Ht|,p−3(St), for general [(St,Ht)] ∈ Bp, that has pre isely δ = p− 3 nodes ( f. (5.10)). To this end, denote by S → Bp the universal family of K3 surfa es, f̃ : C̃ → W̃(δ) and I ⊂ W̃(δ) as in the proof of Proposition 5.11, and let ϕ : C̃I → I be the restri tion of f̃ . Sin e the �ber over [(S0,H0)] of I → Bp as in (5.17) ontains an open, dense subset of P1 × P1, we an �nd a smooth, irredu ible urve B ⊂ I satisfying: for x ∈ B general, ϕ−1(x) is a (partial) desingularization of δ = p−3 of the nodes of a urve in W|Ht|,δ(St) ( f. (5.4)), for general [(St,Ht)] ∈ Bp, and ϕ −1(x) ∈ H3 ⊂ M3; moreover B ontains a point x0 ∈ I su h that ϕ−1(x0) is C̃0 as in (5.21), for C0 general in W0. Let ϕB : C̃B → B be the indu ed universal urve. Sin e the dualizing sheaf of ϕ−1B (x0) = C̃0 is globally generated (as ea h omponent interse ts the others in two points), we in fa t have, possibly after substituting B with an open neighbourhood of x0, a morphism γB : C̃B → P(ϕ̃∗(ωeC/B)) over ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 23 B that is 2 : 1 on the general �bre ϕ−1B (x) and ontra ts the rational omponent R̃ of ϕ B (x0) and maps the two ellipti urves Ẽ0 and F̃0 ea h 2 : 1 onto (di�erent) P s ( f. (5.21) and Figure 1). Let ν : C̃′B → C̃B be the normalization and // P(ϕ̃∗(ωeCB/B the Stein fa torization of γB ◦ ν. In parti ular, γ2 is �nite of degree two onto its image. Moreover, ν◦ϕB : C̃′B → B is a �at family whose general �ber (ν◦ϕB)−1(x) is a desingularization of ϕ B (x) ∈ C̃B. Let pg be the geometri genus of this general �bre. Let D ⊂ C̃′B be the stri t transform via γ1 of the losure of the bran h divisor of γ2 on the smooth lo us of C̃ B . By Riemann-Hurwitz, for general x ∈ B, we have D.ϕ B (x) = 2pg + 2, whereas D.ϕ−1B (x0) ≥ 8, as the urve γ1(ϕ B (x0)) ontains two smooth ellipti urves, ea h being mapped 2 : 1 by γ2 onto (di�erent) P s. This implies pg = 3. Sin e, for general x ∈ B, we have pg ≤ pa(ϕ−1B (x)) = p − δ = 3, we �nd that ϕ B (x) is smooth. This means that the general urve in W|Ht|,δ(St), for (St,Ht) ∈ Bp general, has pre isely δ = p− 3 nodes, proving ( ). To prove (d), again we onsider the morphism (up to possibly restri ting I as above) γI : CI → P(ϕ∗(ωCI/I)) over I whi h, apart some possible ontra tions of rational omponents in spe ial �bres over I, is relatively 2 : 1 onto its image. We have a natural morphism h : CI → S, indu ing a natural map Φ : im(γI)−− → Sym2(S), whose domain has nonempty interse tion with every �bre over Bp. Let R := im(Φ). Then R ∩ Sym2(St) = RVt , for general [(St,Ht)] ∈ Bp. One easily sees that {Sym2(E′)}E′∈|E| ∪ {Sym2(F ′)}F ′∈|F | ⊆ R ∩ Sym2(S0). Sin e the two varieties on the left are threefolds, we have dim(Φ−1(ξ0)) = 0 for general ξ0 ∈ R ∩ Sym2(S0) ⊂ R. Therefore, for general ξ ∈ R, we have dim(Φ−1(ξ)) = 0, so that dim(R) = dim(CI) = dim(I) + 1 = 22, when e dim(RVt) = 22− dim(Bp) = 3. � Remark 5.23. For general [(St,Ht)] ∈ Bp the obtained urves in the last proof have in fa t δ = p−3 non-neutral nodes ( f. [22, �3℄). In fa t a desingularization of less than p− 3 nodes of Ct admits no 2s, as learly a desingularization of less than p − 3 nodes of C0 is not stably equivalent to a urve in the hyperellipti lo us H3 ⊂ M3. 6. On the Mori one of the Hilbert square of a K3 surfa e In this se tion we �rst summarize entral results on the Hilbert square of a K3 surfa e and show how to ompute the lass of a rational urve in S[2]. Then we dis uss the relations between the existen e of urves on S and the slope of the Mori one of S[2], that is, the one of e�e tive lasses in N1(S [2])R. In parti ular, we show how to dedu e the bound (6.28) from Theorem 5.2 and (6.22) from known results about Seshadri onstants. Finally, we dis uss the relation between the existen e of a urve on S with given singular Brill-Noether number and the slope of the Mori one of S[2]. 6.1. Preliminaries on S[2] for a K3 surfa e. Re all that for any smooth surfa e S we have (6.1) H2(S[2],Z) ≃ H2(S,Z)⊕ Ze, where ∆ := 2e is the lass of the divisor parametrizing 0-dimensional subs hemes supported on a single point (see [7℄). So we may identify a lass in H2(S,Z) with its image in H2(S[2],Z). When S is a K3 surfa e the ohomology group H2(S[2],Z) is endowed with a quadrati form q, alled 24 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA the Beauville-Bogomolov form, su h that its restri tion to H2(S,Z) is simply the up produ t on S, the two fa tors H2(S,Z) and Ze are orthogonal with respe t to this form and q(e) = −2. The de omposition (6.1) indu es an isomorphism (6.2) Pic(S[2]) ≃ Pic(S)⊕ Z[e], and ea h divisor D on S orresponds to the divisor on S[2], by abuse of notation also denoted by D, onsisting of length-two subs hemes with some support on D. Given a primitive lass α ∈ H2(S[2],Z), there exists a unique lass wα ∈ H2(S[2],Q) su h that α.v = q(wα, v), for all v ∈ H2(S[2],Z), and one sets (6.3) q(α) := q(wα). We denote also by ρα ∈ H2(S[2],Z) the orresponding primitive (1, 1)- lass su h that ρα = cwα, for some c > 0 (for further details, we refer the reader to [25℄). If now Pic(S) = Z[H], then the Néron-Severi group of S[2] has rank two. We may take as generators of N1(S [2])R the lass P ∆ of a rational urve in the ruling of the ex eptional divisor ∆ ⊂ S[2], and the lass of the urve in S[2] de�ned as follows {ξ ∈ S[2]|Supp(ξ) = {p0, y} | y ∈ Y }, where Y is a urve in |H| and p0 is a �xed point on S. By abuse of notation, we still denote the lass of the urve in S[2] by Y . Note that we always have that (6.4) P ∆ lies on the boundary of the Mori one. Indeed, the urve P1∆ is ontra ted by the Hilbert-Chow morphism S [2] → Sym2(S), so that the pull-ba k of an ample divisor on Sym2(S) is nef, but zero along P1∆. Therefore, des ribing the Mori one NE(S[2]) amounts, by (6.4), to omputing (6.5) slope(NE(S[2])) := inf | aY − bP1∆ ∈ N1(S[2]) is e�e tive, a, b ∈ Q+ We will also all the (possibly in�nite) number a/b asso iated to an irredu ible urve X ∼alg aY −bP1∆ with a > 0 and b ≥ 0, the slope of the urve X and denote it by slope(X). Thus, the smaller slope(X) is, the nearer is X to the boundary of NE(S[2]). By a general result due to Huybre hts [32, Prop. 3.2℄ and Bou ksom [11℄, a divisor D on S[2] is ample if and only if q(D) > 0 and D.R > 0 for any (possibly singular) rational urve R ⊂ S[2]. As a onsequen e, if the Mori one is losed then the boundary (whi h remains to be determined) is generated by the lass of a rational urve (the other boundary is generated by P1∆, by (6.4)). This means that one would have slope(NE(S[2])) = sloperat(NE(S [2])), where (6.6) sloperat(NE(S [2])) := inf | aY − bP1∆ ∈ N1(S[2]) is the lass of a rational urve, a, b ∈ Q+ (A priori, one only has slope(NE(S[2])) ≤ sloperat(NE(S[2])).) Hassett and Ts hinkel [25℄ make a pre ise predi tion on the geometri and numeri al properties of su h extremal rational urves in S[2]. Indeed, a ording to their onje tures [25, p. 1206 and Conj. 3.6℄, the extremal ray R has to be generated either by the lass of a line inside a P2, su h that q(R) = −5 as in (6.3), or by the lass of a rational urve that is a �bre of a P1-bundle over a K3 surfa e and su h that q(R) = −2 or −1 ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 25 6.2. The lasses of rational urves in S[2]. Assume that Pic(S) = Z[H] with pa(H) = pa ≥ 2. Let X ⊂ S[2] be an irredu ible rational urve. Let CX ⊂ S be the orresponding urve as in � 2.1 and assume that CX ∈ |mH| with m ≥ 1. (In parti ular, m ≥ 2 if we are in ase (II)). We an write X ∼alg a1Y + a2P1∆. Sin e X.H = m(2pa − 2), Y.H = 2pa − 2 and P1∆.H = 0 by the very de�nition of H as a divisor in S[2], and Y.e = 0 and P1∆.e = −2, we obtain, de�ning g0(X) := X.e − 1, (6.7) X ∼alg mY − (g0(X) + 1 To ompute g0(X), onsider the diagram (2.1). Sin e ν XOX(∆) ≃ (ν∗XOX(e))⊗2, the double over f is de�ned by ν∗XOX(∆). By Riemann-Hurwitz we therefore get (6.8) g0(X) = pa(C̃X). Note that in the ases (II) and (III) in the orresponden e in � 2.1, X.e = g0(X) + 1 is pre isely the length of the interse tion s heme C̃X,1 ∩ C̃X,2, where C̃X = C̃X,1 ∪ C̃X,2. In ase (III), sin e ν̃ : C̃X → S ontra ts one of the two omponents of C̃X to a point xX ∈ S, we obtain that (6.9) g0(X) = multxX (CX)− 1 (if CX is of type (III)). One an he k that for all divisors D in S[2] one has X.D = q(wX ,D) with (6.10) wX := mH − (g0(X) + 1 e ∈ H2(S[2],Q). In parti ular, 2wX ∈ H2(S[2],Z). From (6.5) and (6.7) we see that sear hing for irredu ible rational urves in (or at least �near�) the boundary of the Mori one of S[2], or with negative square q(X), amounts to sear hing for irredu ible urves in |mH| with (partial) hyperellipti normalizations of high genus ( ase (I)), or to irredu ible rational urves in |mH| with high multipli ity at a point ( ase (III)), or to irredu ible rational urves on S with some orresponden e between some overings of their normalizations ( ase (II)). Moreover, we should sear h for urves with as low m as possible. Now m ≥ 2 in ase (II), as remarked above. Moreover, any rational urve in |H| on a general S is nodal, by a result of Chen [13, Thm. 1.1℄ (the same is also onje tured for rational urves in |mH| for m > 1, see [14, Conj. 1.2℄), so that g0(X) ≤ 1 if CX is of type (III) in these ases, by (6.9). Hen e, we see that the most natural andidates are irredu ible urves in |H| with hyperellipti normalizations. By the above, an irredu ible urve C ∈ |mH| with hyperellipti normalization de�nes, by the uni ity of the g 2, a unique irredu ible rational urve X = RC ⊂ S[2] with lass (6.11) RC ∼alg mY − (g0(C) + 1 where g0(C) := g0(RC) is well-de�ned as (6.12) g0(C) := the arithmeti genus of a minimal partial desingularization of C admitting a g (For example, if C is nodal, then we simply take the desingularization of the non-neutral nodes of C, f. [22, �3℄). From (6.5) we then get (6.13) slope(NE(S[2])) ≤ 2m g0(C) + 1 pg(C) + 1 , if there exists a C ∈ |mH| with hyp. norm. and, by (6.3) and (6.10), (6.14) q(RC) = 2m 2(pa − 1)− (g0(C) + 1) ≤ 2m2(pa − 1)− (pg(C) + 1) 26 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA In parti ular, the higher g0(C) (or pg(C)) is - thus the more �unexpe ted� the urve on S is from a Brill-Noether theory point of view - the lower is the bound on the slope of NE(S[2]) and the more negative is the square q(RC) in S 6.3. The invariant ρsing, Seshadri onstants, the �hyperellipti existen e problem� and the slope of the Mori one. In [22℄ we introdu ed a singular Brill-Noether invariant (6.15) ρsing(pa, r, d, g) := ρ(g, r, d) + pa − g, in order to study linear series on the normalization of singular urves. Pre isely, we proved Theorem 6.16. Let S be a K3 surfa e su h that Pic(S) ≃ Z[H] with pa := pa(H) ≥ 2. Let C ∈ |H| and C̃ → C be a partial normalization of C, su h that g := pa(C̃). If ρsing(pa, r, d, g) < 0, then C̃ arries no g Proof. One easily sees that the proof of [22, Thm. 1℄ also holds for a partial normalization of C. � For r = 1 and d = 2, we have (6.17) ρsing(pa, 1, 2, g) < 0 ⇔ g > pa + 2 In parti ular, a onsequen e of Theorem 6.16 is the following: Theorem 6.18. Let S be a smooth, proje tive K3 surfa e with Pic(S) ≃ Z[H] and pa := pa(H) ≥ 2. Let Y and P1∆ be the generators of N1(S [2])R with notation as in � 6.1. If X ∈ N1(S[2])Z with X ∼alg Y − kP1∆, then k ≤ Proof. We an assume thatX is an irredu ible urve. Then, pre isely as in the ase of a rational urve, X orresponds either to the data of an irredu ible urve C ∈ |H| on S, with a partial normalization C̃ admitting a 2 : 1 morphism onto the normalization X̃ of X, or to the data of an irredu ible urve C ∈ |H| on S together with a point x0 := xX ∈ S. (The ase orresponding to ase (II) in � 2.1 does not o ur, sin e the oe� ient of Y is one, pre isely as in the ase of a rational X explained above.) In the latter ase µ(X) = {x0 + C} ⊂ Sym2(S), where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism as usual, and one easily omputes k = (1/2)multx0(C) as in the rational ase above. Sin e learly multx0(C) ≤ 2 if pa = 2 and multx0(C) ≤ 3 if pa = 3, we have k ≤ in these two ases. If pa ≥ 4, then from dim |H| − 3 − (pa − 4) = 1 and the fa t that being singular at a given point imposes at most three independent onditions on |H|, we an �nd an irredu ible urve C ′ ∈ |H|, di�erent from C, singular at x0, and passing through at least pa − 4 points of C. Therefore 2pa − 2 = H2 = C ′.C ≥ multx0(C ′) ·multx0(C) + pa − 4 ≥ 2multx0(C) + pa − 4, when e multx0(C) ≤ (pa + 2)/2, so that k ≤ (pa + 2)/4. In the �rst ase, then, pre isely as in the rational ase above, (6.19) k = pa(C̃) + 1 − pg(X) from Riemann-Hurwitz. By Brill-Noether theory on X̃ , it follows that C̃ arries a g1d, with d ≤ 2⌊pg(X) + 3 By Theorem 6.16 we have ρsing(pa(C), 1, d, pa(C̃)) ≥ 0, when e pa(C̃) ≤ d−1+pa(C)/2. The desired result now follows. � ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 27 By the proof of Theorem 6.18 we see that if C ∈ |mH| is an irredu ible urve and x0 ∈ C, then the lass of the orresponding urve µ−1∗ {x0+C} ⊂ S[2] is given by mY − (1/2)multx0(C)P1∆. Hen e slope(NE(S[2])) ≤ inf C∈|mH| multx(C) = inf C∈|mH| multx(C) It follows that (6.20) slope(NE(S[2])) ≤ ε(H) pa − 1 where ε(H) := inf multx(C) (and the in�mum is taken over all irredu ible urves C ⊂ S passing through x) is the (global) Seshadri onstant of H ( f. [17, � 6℄, [18℄ or [4℄). These onstants are very di� ult to ompute. The only ase where they have been omputed on general K3 surfa es is the ase of quarti surfa es, where one has ε(H) = 2 by [3℄, yielding the bound slope(NE(S[2])) ≤ 1. As a omparison, the bound one gets from (6.13) using the singular urves of genus two in |H| is slope(NE(S[2])) ≤ 2/3. However, it is well-known that ε(H) ≤ H2 on any surfa e, see e.g. [54, Rem. 1℄. Hen e, by (6.20) we obtain Theorem 6.21. Let (S,H) be a primitively polarized K3 surfa e of genus pa := pa(H) ≥ 2 su h that Pic(S) ≃ Z[H]. Then ( f. (6.5)) (6.22) slope(NE(S[2])) ≤ ε(H) pa − 1 pa − 1 In parti ular, (6.22) shows that there is no lower bound on the slope of the Mori one of S[2] of K3 surfa es, as the degree of the polarization tends to in�nity, that is, (6.23) inf slope(NE(S[2])) | S is a proje tive K3 surfa e The same fa t about sloperat(NE(S [2])) will follow from (7.4) and (7.9) below. Note that one always has ε(H) > ⌊ H2⌋ − 1 under the hypotheses of Theorem 6.21. Indeed, if ε(H) < H2, then there is an x ∈ S and an irredu ible urve C su h that ε(H) = C.H multx(C) , see e.g. [44, Cor. 2℄. Sin e one easily omputes dim |H ⊗ I(⌊ H2⌋−1) x | ≥ 2, we an �nd a D ∈ |L| su h that D 6⊇ C, multx(D) ≥ ⌊ H2⌋ − 1 and D passes through at least one additional point of C. Thus ε(H) = multx(C) multx(C) ≥ multx(C) ·multx(D) + 1 multx(C) > multx(D) ≥ ⌊ H2⌋ − 1, as desired. It follows that (6.24) pa − 1 2pa − 2⌋ − 1 pa − 1 , for (S,H) as in Theorem 6.21, showing that there is a natural limit to how good a bound one an get on slope(NE(S[2])) by using Seshadri onstants. The bound in (6.22) is not (ne essarily) obtained by rational urves in S[2]. However, the presen e of pg(X) in (6.19) above tends to indi ate that the better bounds will be obtained by rational urves in S[2]. (Of ourse, if the Mori one is losed, then the bound will indeed be obtained by rational urves, as explained at the end of � 6.1.) In fa t, the bound (6.22) above will be improved, for in�nitely many values of H2, in Propositions 7.2 and 7.7 below by rational urves. We now return to the study of irredu ible rational urves in S[2] and to sloperat(NE(S [2])). Given Theorem 6.16 and (6.17), a natural question to ask is the following: 28 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA Hyperellipti existen e problem (HEP). For 3 ≤ pg ≤ pa+22 , does there exist a singular urve in |H| with hyperellipti normalization of geometri genus pg? By (6.13) we have that a positive solution to (HEP) for �maximal� pg = ⌊ pa + 2 ⌋ =⇒(6.25) sloperat(NE(S [2])) ≤ if pa is even; if pa is odd and, by (6.14), the q-square of the asso iated rational urves would be mu h less than what predi ted by Hassett and Ts hinkel [25, Conj. 3.1℄. Moreover, the bounds in (6.25) would be mu h stronger than the bound given by the right hand inequality in (6.22), and even stronger than the best bounds one ould obtain from Seshadri onstants ( ompare the left hand side inequality in (6.22) with (6.24)). It is natural to try to solve (HEP) using nodal urves, as one has better ontrol of their deformations and their parameter spa es (the Severi varieties onsidered in � 5). After the positive answer to the hyperellipti existen e problem for the spe i� values pg = 3 and pa = 4, 5 in [22, Examples 2.8 and 2.10℄, Theorem 5.2 gives the �rst examples, at least as far as we know, of positive answers to the hyperellipti existen e problem for primitively polarized K3 surfa es of any degree. In Remark 5.23 we showed that pg(C) = g0(C) = 3 for these onstru ted urves C ∈ |H| ( f. (6.12)), so that the lasses of the asso iated rational urves RC ⊂ S[2] are, using (6.10), (6.26) wRC = H − 2e, q(wRC ) = q(RC) = 2p− 10 ≥ −2. Moreover, using (6.13), Theorem 5.2 yields ( f. (6.6)): Corollary 6.27. Let (S,H) be a general, primitively polarized K3 surfa e of genus pa(H) ≥ 4. Then (6.28) sloperat(NE(S [2])) ≤ 1 Note that the existen e of nodal urves of geometri genus 2 in |H|, whi h was already known and followed from the nonemptiness of the Severi varieties on general K3 surfa es, as explained in the beginning of � 5, leads to the less good bound of . Therefore, again as far as we know, (6.28) is the �rst �nontrivial� bound on the slope of rational urves holding for all degrees of the polarization. As already mentioned, for in�nitely many degrees of the polarization we will in fa t improve this bound in Propositions 7.2 and 7.7 below. Remark 6.29. One may also look for irredu ible singular urves with hyperellipti normalizations in |mH|, m ≥ 2. In [22, Corollary 4℄, we also proved that, apart from some spe ial numer- i al ases (where we were not able to on lude), the negativity of ρsing(pa(mH), 1, 2, g) implies the non-existen e of irredu ible nodal urves in |mH| with hyperellipti normalizations. A posi- tive solution to the hyperellipti existen e problem for singular urves in |mH| would then pro- vide an even better bound on the slope of the Mori one. Namely, one would for instan e get slope(NE(S[2])) ≤ 4/[m(pa(H) + 4)] for even pa. Whereas we tend to believe that the nonnegativity of ρsing should imply existen e of urves with hyperellipti normalizations for the spe i� values of pa and g in a primitive linear system |H| on a general K3, we are not sure what to expe t for urves in |mH| when m > 1. For instan e, the degeneration methods to prove existen e as in the proof of Theorem 5.2 will ertainly get more di� ult, be ause the irredu ibility of the obtained urves after deformation is not automati ally ensured. ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 29 Remark 6.30. We do not know whether there will always be omponents in |H|hyper (whenever nonempty) of singular urves with hyperellipti normalizations su h that the singularities of the general member are as ni e as possible, that is, all nodes and all non-neutral [22, �3℄. 7. P2s and threefolds birational to P1-bundles in the Hilbert square of a general K3 surfa e We now give an in�nite series of examples of general, primitively polarized K3 surfa es (S,H), of in�nitely many degrees su h that S[2] ontains either a P2 or a threefold birational to a P1-bundle, thus showing both possibilities o urring in Proposition 3.6. Both series of examples are similar to Voisin's onstru tions in [57, � 3℄. The idea is to start with a smooth quarti surfa e S0 su h that S 0 ontains an �obvious� P or threefold birational to a P1- bundle over S0, use the involution on the quarti to produ e another su h P or uniruled threefold, and then deform S0 keeping the latter one and loosing the �rst one in the Hilbert square. We remark that the question of existen e of P2s in S[2] when S is K3 is a very interesting problem be ause of the following fa t: a P2 in S[2] gives rise to a birational map from S[2] onto another hyperkähler fourfold, and onversely any birational transformation X −− → X ′ between proje tive, symple ti fourfolds an be fa torized into a �nite sequen e of Mukai �ops ( f. [41, Thm. 0.7℄), by [60, Thm. 2℄, see also [12, 30, 62℄. Therefore, in the ase of a K3 surfa e, if S[2] ontains no P2s, then S[2] admits no other birational model than itself. Also uniruled divisors have an in�uen e on the birational geometry of a hyperkähler manifold X. Indeed, Huybre hts proved in [32, Prop. 4.2℄ that a lass α in the losure of the positive one CX lies in the losure of the birational Kähler one BKX if and only if q(α,D) ≥ 0, for all uniruled divisors D ⊂ X. (Re all that the positive one CX is the onne ted omponent of {α ∈ H1,1(X,R) : q(α) ≥ 0} ontaining the one KX of all Kähler lasses of X, and the birational Kähler one BKX equals by de�nition ∪f :X−−→X′f∗KX′ , where f is a bimeromorphi map onto another hyperkähler manifold X ′). 7.1. P2s in S[2]. The �rst nontrivial ase, the ase of degree 10, is parti ularly easy, so we begin with that one. Example 7.1. (Hassett) Let S ⊂ P6 be a general K3 surfa e of degree 10. By [40℄ the surfa e S is a omplete interse tion S = G ∩ T ∩ Q, where G := Grass(2, 5) is the Grassmannian of lines in P4 embedded in P9 by its Plü ker embedding, T is a general 6-dimensional linear subspa e of P9, and Q is a hyperquadri in P9. Set Y := G∩T . Then Y is a Fano 3-fold of index 2. Let F (Y ) be its variety of lines. It is lassi ally known (see e.g. [19℄ for a modern proof) that F (Y ) ∼= P2. Then we may embed this plane in S[2] by mapping the point orresponding to a line [ℓ] to ℓ ∩Q. By generality, S does not ontain any line, so that this map is a morphism. The onstru tion behind the following result, generalizing the previous example, was shown to us by B. Hassett. Proposition 7.2. Let (S,H) be a general primitively polarized K3 surfa e of degree H2 = 2(n2 − 9n+ 19), for n ≥ 6. Then S[2] ontains a P2. The lass wℓ ∈ H2(S[2],Q) orresponding to a line ℓ ⊂ P2 is (7.3) wℓ = H − 2n − 9 In parti ular (7.4) sloperat(NE(S [2])) ≤ 2 2n− 9 . 30 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA Moreover the urves C ⊂ S with hyperellipti normalizations asso iated to the lines ℓ ⊂ P2 ⊂ S[2] lie in |H|, have geometri genus pg = 2n− 10, and ρsing(pa(C), 1, 2, pg) = n(n− 13) + 42 ≥ 0. Proof. Consider the latti e ZF ⊕ ZG with interse tion matrix F 2 F.G G.F G2 , n ≥ 6. Sin e it has signature (1, 1), then, by a result of Nikulin [43℄ (see also [39, Cor. 2.9(i)℄), there is an algebrai K3 surfa e S0 with the given Pi ard latti e. Performing Pi ard-Lefs hetz re�e tions on the latti e, we an assume that G is nef, by [2, VIII, Prop. 3.9℄. By Riemann-Ro h and Serre duality, we have G > 0 and F > 0. Straightforward omputations on the Pi ard latti e rules out the existen e of divisors Γ satisfying Γ2 = −2 and Γ.F ≤ 0 or Γ.G ≤ 1; or Γ2 = 0 and Γ.F = 1 or Γ.G = 1, 2. By [48℄ it follows that both |F | and |G| are base point free, ϕ|F | : S0 → P2 is a double over and ϕ|G| : S0 → P3 is an embedding onto a smooth quarti not ontaining lines. As explained in � 4, S ontains a P2 arising from the double over. If ℓ0 is a line on the P , the orresponding lass in H2(S 0 ,Q) is wℓ0 = 2F − 3e, whi h oin ides with the orresponding integral lass ρℓ0 ( f. [25, Example 5.1℄). As S0 is a quarti surfa e not ontaing lines, S 0 admits an involution ι : S 0 → S 0 ; ξ 7→ (ℓξ ∩ S0) \ ξ, by [6, Prop. 11℄, where ℓξ is the line determined by ξ, and the sign \ means that we take the residual subs heme. The orresponding involution on ohomology is given by ( f. e.g. [45, (4.1.6)-(4.1.7)℄) v 7→ q(G− e, v) · (G− e)− v. The involution sends the P2 into another P2, and the orresponding lass asso iated to a line on it is (7.5) q(G− e, 2F − 3e) · (G− e)− (2F − 3e) = 2((n − 3)G− F )− (2n− 9)e. In order to obtain a generalK3 with the desired property we now deform S 0 . Pre isely, we onsider a general deformation of S 0 su h that (i) e remains algebrai and (ii) ι(P 2) is preserved. Deformations satisfying (i) form a ountable union of hyperplanes in the deformation spa e of S 0 , whi h is smooth and of dimension 21, and may be hara terized as those of the form S[2], where S is a K3 surfa e (see [7, Thm. 6 and Rem. 2℄). Deformations preserving ι(P2) an be hara terized as those preserving the image in H2(S[2],Z) of the lass of the line in ι(P2) as an algebrai lass (see [25, Thm. 4.1 and Cor. 4.2℄ or [57℄), that is, using (7.5), those deformations keeping H := (n− 3)G−F ∈ Pic(S[2]0 ), or, equivalently, H ∈ Pic(S), by (6.2). As H2 = [(n− 3)G−F ]2 = 2(n2 − 9n+19) ≥ 2 for n ≥ 6 and H is primitive, those deformations form a divisor in the 20-dimensional spa e of deformations keeping e algebrai , by [34, Thm. 14℄. We therefore obtain a 19-dimensional spa e of deformations of S 0 , whose general member is S where (S,H) is a general primitively polarized (algebrai ) K3 surfa e of degree H2 = 2(n2−9n+19), n ≥ 6, and S[2] ontains a plane. The lass wℓ ∈ H2(S[2],Q) orresponding to the line ℓ is as in (7.3), yielding (7.4). As S is general, it does not ontain smooth rational urves, so that the P2 is not of the form C [2], for a smooth rational urve C on S. By Lemma 2.4, the lines in the P2 in S[2] give rise to a two-dimensional family V of urves on S with hyperellipti normalizations, so that RV = µ(P where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism. By (7.3) we have ℓ.H = H2, so that, by the very de�nition of the divisor H in H2(S[2],Z), the lines in the P2 orrespond to urves C ∈ |H|. Comparing (6.10) and (7.3), we see that g0(C) = 2n − 10, f. (6.12). Now we note that the general line in the P2 is not tangent to ∆ = 2e. (Indeed, this follows by deformation sin e in S 0 we have ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 31 that ι(P2) ∩ ∆ is a smooth plane sexti , sin e we have a omposite map S0 → P2 → ι(P2) that is �nite of degree two, when e rami�ed along a smooth sexti , as S0 is a smooth K3.) Therefore we have pg(C) = 2n− 10. We ompute ρsing = n(n− 13) + 42 ≥ 0 (re all that n ≥ 6). � The examples ontained in the above proposition is interesting in several regards. Noti e �rst that q(ℓ) = −5/2, f. (6.3), in a ordan e with the predi tion in [25, Conj. 3.6℄. The proposition shows in parti ular that the orresponden e in Remark 3.7 is not one-to-one and also shows that the ase dim(V ) = dim(RV ) = 2 of Proposition 3.6 a tually o urs. The result also gives nontrivial examples of urves in |H| with hyperellipti normalizations and positively answers the hyperellipti existen e problem for pa = n 2−9n+20 and pg = 2n−10, n ≥ 6. Moreover (7.4) shows that there is no lower bound on sloperat(NE(S [2])) as the degree of the polarization tends to in�nity. The same follows from (7.9) in Proposition 7.7 below. Both the bounds (7.4) and (7.9) below in fa t yield better bounds on slope(NE(S[2])) than (6.22). Finally, the oni s on the P2 give a �ve-dimensional family V (2) of irredu ible urves with hyperel- lipti normalizations on S. Of ourse this family has obvious non-integral members, orresponding to non-integral oni s. More generally, for any m ≥ 3, the (3m−1)-dimensional family of nodal rational urves in |OP2(m)| ( f. [15, Thm. 1.1℄) yields orresponding families V (m) of urves in |mH| with hyperellipti normalizations with dimV (m) = 3m − 1 ≥ 5 and dim(RV ) = 2, showing in parti ular that the ase dim(V ) > dim(RV ) = 2 of Proposition 3.6 a tually o urs. In the ase of the oni s, we ompute pg = 4n− 19 as above and as pa(2H) = 4n2 − 36n+ 77, we get ρsing = 4n(n− 11) + 117 ≥ −3 in these ases. This does not ontradi t [22, Thm. 1℄. 7.2. Threefolds birational to P1-bundles in S[2]. We start with an expli it example in the spe ial ase of a quarti surfa e. Example 7.6. In the ase of a general quarti S in P3 we an �nd a P1-bundle over S in S[2], arising from the two-dimensional family of hyperplane se tions of geometri genus two. In fa t, taking the tangent plane through the general point of S we get a nodal urve of geometri genus 2. We obtain in this way a family V of nodal urves with hyperellipti normalizations in the hyperplane linear system. This family is parametrized by an open subset of S, and the lo us in S[2] overed by the asso iated rational urves is birational to a P1-bundle over this open subset. To see this, set Cp := (S ∩ TpS), and let C̃p be the normalization of Cp. Note that the g 2 on C̃p, viewed on Cp, is given by the pen il of lines in TpS through the node p. If, for two distin t points p, q ∈ S, the g12s on C̃p and C̃q had two ommon points, say x and y (so that the map ΦV in (2.5) sends (p, x+ y) and (q, x+ y) to the same point x+ y in Sym2(S)), then the line TpS ∩ TqS, whi h is bitangent to S, would also pass through x and y. This is absurd, as deg(S) = 4. By (6.10), the lass w ∈ H2(S[2],Q) orresponding to the urves of geometri genus 2 is w = H− 3 when e q(w) = −1/2, as predi ted by [25, Conj. 3.6℄. Moreover, performing the usual involution on the quarti , we send the onstru ted uniruled threefold to another one, with orresponding �bre lass given by e, so that it simply is the P1-bundle ∆ over S. This shows that also our original threefold was smooth, so in fa t a P1-bundle over S. We now give an in�nite series of examples of general K3s whose Hilbert squares ontain threefolds birational to P1-bundles. Proposition 7.7. Let (S,H) be a general primitively polarized K3 surfa e of degree H2 = 2(d2−1), for d ≥ 2. Then S[2] ontains a threefold birational to a P1-bundle over a K3 surfa e. The lass wf ∈ H2(S[2],Q) orresponding to a �bre is (7.8) wf = H − de ∈ H2(S[2],Z). 32 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA In parti ular (7.9) sloperat(NE(S [2])) ≤ 1 Moreover the urves C ⊂ S with hyperellipti normalizations asso iated to the �bres of the threefold lie in |H|, have geometri genus pg = 2d− 1, and ρsing(pa(C), 1, 2, pg) = d(d − 4) + 4 ≥ 0. Proof. This time we start with the latti e ZF ⊕ ZG with interse tion matrix F 2 F.G G.F G2 , d ≥ 2. As in Proposition 7.2 one easily shows that there is an algebrai K3 surfa e S0 with Pic(S0) = ZF ⊕ ZG and that ϕ|G| : S0 → P3 is an embedding onto a smooth quarti not ontaining lines and F is a smooth, irredu ible rational urve. (Note that F [2] = P2 and performing the same pro edure on this plane as in the proof of Proposition 7.2, one gets pre isely the same series of examples as above.) We now onsider the divisor F ⊂ S[2]0 , de�ned as the length-two s hemes with some support along F . One easily sees that this is a threefold birational to a P1-bundle over S0 and that the lass in 0 ,Z) orresponding to the �bres f is ρf = F , f. [25, Example 4.6℄. The involution on the quarti sends this threefold to another threefold birational to a P1-bundle over S0 and the orresponding lass of the �bres is (7.10) q(G− e, F ) · (G− e)− F = dG− F − de. Note that this threefold satis�es the onditions in [25, Thm. 4.1℄ by [25, Example 4.6℄, so that, as in the previous example, we an deform S 0 , keeping e algebrai and H := dG − F . We thus obtain a 19-dimensional spa e of deformations of S 0 , whose general member is S , where (S,H) is a general, primitively polarized (algebrai ) K3 surfa e of degree H2 = 2(d2−1) ≥ 6 and S[2] ontains a threefold birational to a P1-bundle, again over a K3 surfa e (see also [25, Thm. 4.3℄). The unique lass wf ∈ H2(S[2],Q) orresponding to a �bre f is as in (7.8) and yields (7.9). By (7.8) we have f.H = H2, so that, by the very de�nition of the divisorH inH2(S[2],Z), the �bres f of Y orrespond to urves C ∈ |H|. Comparing (6.10) and (7.8), we see that g0(C) = 2d − 1 ≥ 3, f. (6.12). As in the proof of Proposition 7.2, one an see that the general �bre of Y is not tangent to ∆ = 2e, so that in fa t we have pg(C) = 2d−1. In parti ular, Y is not one of the obvious uniruled threefolds arising from the rational urves on S, or the one-dimensional families of ellipti urves on S. A omputation shows that ρsing = d(d − 4) + 4 ≥ 0. � Again, a few omments are in order. The square of the lass of the �bres of the uniruled threefolds onstru ted above is q(f) = −2, as predi ted in [25, Conj. 3.6℄. The obtained family V of urves on S with hyperellipti normalizations has dim(V ) = 2 and dim(RV ) = 3, showing that also this ase of Proposition 3.6 a tually o urs. This family gives nontrivial examples of urves in |H| with hyperellipti normalizations and positively answers the hyperellipti existen e problem for pa = 2(d 2 − 1) and pg = 2d − 1 for every d ≥ 2. Note that the ase d = 2 is the ase des ribed in [22, Example 2.8℄. Referen es [1℄ E. Arbarello, M. Cornalba, Su una ongettura di Petri, Comment. Math. Helveti i 56 (1981), 1�38. [2℄ W. Barth, K. Hulek, C. Peters, A. 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Flaminio Flamini, Dipartimento di Matemati a, Università degli Studi di Roma "Tor Vergata", Viale della Ri er a S ienti� a, 00133 Roma, Italy. e-mail flamini�mat.uniroma2.it. Andreas Leopold Knutsen, Dipartimento di Matemati a, Università Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. e-mail knutsen�mat.uniroma3.it. Gianlu a Pa ienza, Institut de Re her he Mathématique Avan ée, Université L. Pasteur et CNRS, rue R. Des artes - 67084 Strasbourg Cedex, Fran e. e-mail pa ienza�math.u-strasbg.fr. EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES 35 Appendix A. PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES EDOARDO SERNESI In this Appendix we show how to onstru t simultaneous partial desingularizations of families of nodal urves, generalizing a well known pro edure of simultaneous total desingularization, as des ribed in [4℄. We work over an algebrai ally losed �eld k of hara teristi 0. For every morphism X → Y , and for every y ∈ Y , we denote by X(y) the s heme-theoreti �bre of y. Theorem A.1. Let f : C // V be a �at proje tive family of urves, with C and V algebrai s hemes, su h that all �bres have at most ordinary double points (nodes) as singularities. Let δ ≥ 1 be an integer. Then there is a ommutative diagram: with the following properties: (i) α is �nite and unrami�ed, the square is artesian, and q is an étale over of degree δ. (ii) The left triangle de�nes a marking of all δ-tuples of nodes of �bres of f . In parti ular f ′ parametrizes all urves of the family f having ≥ δ nodes and, for ea h η ∈ E(δ), Dδ(η) ⊂ C′(η) is a set of δ nodes of the urve C′(η). (iii) The diagram is universal with respe t to properties (i) and (ii). Pre isely, if Ẽ ×V C is a diagram having the properties analogous to (i) and (ii), then there is a unique fa torization // E(δ) su h that q̃ and f̃ are obtained by pulling ba k q and f ′ by ϕ. If moreover E(δ) is normal, then the above diagram an be enlarged as follows: Work done during a visit to the Institut Mittag-Le�er (Djursholm, Sweden), whose support is gratefully a knowl- edged. I am grateful to F. Flamini, A. L. Knutsen and G. Pa ienza for a epting this note as an Appendix to their paper, and to F. Flamini for some useful remarks. 36 EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES where: (iv) β is a birational morphism su h that, for ea h η ∈ E(δ), the restri tion: β(η) : C(η) // C′(η) is the partial normalization at the nodes Dδ(η). (v) The omposition f̄ := f ′ ◦ β is �at. Proof. Consider the �rst relative otangent sheaf T . Sin e all �bres of f are nodal, T1 ommutes with base hange ([3, Lemma 4.7.5℄ or [5℄), thus on every �bre C(v), v ∈ V , it restri ts to T1 , whi h is the stru ture sheaf of the s heme of nodes of C(v). It follows that we have C/V = OE for a losed subs heme E ⊂ C supported on the nodes of the �bres of f . Consider the omposition fE : E ⊂ C By onstru tion it follows that fE is �nite and unrami�ed. Now �x δ ≥ 1 and onsider the �bre produ t: E ×V · · · ×V E︸ ︷︷ ︸ Sin e fE is �nite and unrami�ed, it follows from [1, Exp.1, Prop. 3.1℄, and by indu tion on δ (see [3, Lemma 4.7.11(i)℄), that we have a disjoint union de omposition: E ×V · · · ×V E = ∆ where ∆ is the union of all the diagonals, and Eδ onsists of all the ordered δ-tuples of distin t points of E mapping to the same point of V ; moreover the natural proje tion morphism Eδ // V is �nite and unrami�ed. There is a natural a tion of the symmetri group Σδ on Eδ that ommutes with the proje tion to V . We denote the quotient Eδ/Σδ by E(δ). Sin e the omposition Eδ // E(δ) // V is �nite and unrami�ed and the �rst morphism is an étale over, the morphism α : E(δ) → V is �nite and unrami�ed. Note that if, for a losed point v ∈ V , C(v) has δ+ t nodes as the only singularities, with t > 0, then α−1(v) has degree . Now let Dδ = {(η, e) : e ∈ Supp(η)} ⊂ E(δ) ×V E EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES 37 Then the �rst proje tion de�nes the tautologi al family: (A.2) Dδ ⊂ E(δ) ×V E ⊂ E(δ) ×V C whi h is an étale over of degree δ. The �bre Dδ(η) is the δ-tuple parametrized by η, for ea h η ∈ E(δ)2. We therefore have the following diagram: where we have denoted by C ′ = E(δ) ×V C. The �bres of f ′ are all the urves of the family f having ≥ δ nodes. For ea h η ∈ E(δ) the divisor Dδ(η) ⊂ C′(η) marks the set of δ nodes parametrized by η. This proves (i) and (ii). (iii) follows from the fa t that α : E(δ) → V is the relative Hilbert s heme of degree δ of fE : E → V , and (A.2) is the universal family. Assume that E(δ) is normal. Then we an normalize C lo ally around Dδ as in [4, Theorem 1.3.2℄, to obtain a birational morphism β having the required properties (iv) and (v). � A typi al example of the situation onsidered in the theorem is when V parametrizes a omplete linear system of urves on an algebrai surfa e. If the morphism fE is self-transverse of odimension 1 (see [3, De�nition 4.7.13℄) then the Severi variety of irredu ible δ-nodal urves is nonsingular and of odimension δ, and E(δ) is nonsingular (see [3, Lemma 4.7.14℄), so that the theorem applies and the simultaneous partial desingularization exists. This happens for example for the linear systems of plane urves [3, Proposition 4.7.17℄. Referen es [1℄ Revêtements étales et groupe fondamentale, Séminaire de Géométrie Algébrique du Bois Marie 1960-61 (SGA1), Le ture Notes in Math. 224. Springer, Berlin, 1971. [2℄ R. Hartshorne, Algebrai Geometry, Graduate Texts in Mathemati s 52. Springer-Verlag, New York- Heidelberg, 1977. [3℄ E. Sernesi, Deformations of Algebrai S hemes, Grundlehren der Mathematis hen Wissens haften 334. Springer-Verlag, Berlin, 2006. [4℄ B. Tessier, Résolution simultanée I, II. Le ture Notes in Math. 777, 71�146. Springer, Berlin, 1980. [5℄ J. Wahl, Deformations of plane urves with nodes and usps, Amer. J. of Math. 96 (1974), 529�577. Edoardo Sernesi, Dipartimento di Matemati a, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. e-mail sernesi�mat.uniroma3.it. If δ = 1 then E(1) = E and D1 ⊂ E ×V E is the diagonal. 1. Introduction 2. Rational curves in S[2] 2.1. Irreducible rational curves in S[2] and curves on S 2.2. Bend-and-break in Sym2(S) 3. Rationally equivalent zero-cycles on surfaces with pg>0 3.1. Mumford's Theorem 3.2. The property RCC and rational quotients 4. Families of curves with hyperelliptic normalizations 5. Nodal curves of geometric genus 3 with hyperelliptic normalizations on K3 surfaces 6. On the Mori cone of the Hilbert square of a K3 surface 6.1. Preliminaries on S[2] for a K3 surface 6.2. The classes of rational curves in S[2] 6.3. The invariant sing, Seshadri constants, the ``hyperelliptic existence problem'' and the slope of the Mori cone 7. P2s and threefolds birational to P1-bundles in the Hilbert square of a general K3 surface 7.1. P2s in S[2] 7.2. Threefolds birational to P1-bundles in S[2] References Appendix A. References

0704.1368

Anisotropy and Magnetic Field Effects on the Genuine Multipartite Entanglement of Multi-Qubit Heisenberg {\it XY} Chains

Anisotropy and Magnetic Field Effects on the Genuine Multipartite Entanglement of Multi-Qubit Heisenberg XY Chains Chang Chi Kwong and Ye Yeo Department of Physics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Abstract It has been shown that, for the two-qubit Heisenberg XY model, anisotropy and magnetic field may together be used to produce entanglement for any finite temperature by adjusting the external magnetic field beyond some finite critical strength. This interesting result arises from an analysis employing the Wootters concurrence, a computable measure of entanglement for two-qubit states. Recently, Mintert et al. proposed generalizations of Wootters concurrence for multipartite states. These MKB concurrences possess a mathematical property that enables one to understand the origin of this characteristic behavior. Here, we first study the effect of anisotropy and magnetic field on the multipartite thermal entanglement of a four-qubit HeisenbergXY chain using the MKB concurrences. We show that this model exhibits characteristic behavior similar to that of the two- qubit model. In addition, we show that this can again be understood using the same mathematical property. Next, we show that the six-qubit Heisenberg XY chain possesses properties necessary for it to have the characteristic behavior too. Most importantly, it is possible to directly measure the multipartite MKB concurrences of pure states. This may provide an experimental verification of our conjecture that for a Heisenberg XY chain of any even number of qubits, it is always possible to obtain non-zero genuine multipartite entanglement at any finite temperature by applying a sufficiently large magnetic field. PACS numbers: 03.67.Mn, 03.65.Ud http://arxiv.org/abs/0704.1368v2 I. INTRODUCTION The one-dimensional Heisenberg models have been extensively studied in solid state physics (see references in [1]). Interest in these models has been revived lately by sev- eral proposals for realizing quantum computation [2] and information processing [3] using quantum dots (localized electron spins) as qubits. An intriguing phenomenon in such quan- tum systems with more than one component is entanglement. It refers to the non-classical correlations that exist among these components. Due to this nature, quantum entanglement has recently been recognized as an indispensable physical resource for performing classically impossible information processing tasks; such as quantum computation, teleportation [4], key distribution [5] and superdense coding [6]. In any physically realistic consideration of quantum-information-processing device, it is important to take into account the fact that it would be in thermal contact with some heat bath. Consequently, entanglement in interacting Heisenberg spin systems at finite temperatures has been investigated by a number of authors (see, e.g., Ref.[7] and references therein). The state of a typical solid state system at thermal equilibrium (temperature T ) is χ = e−βH/Z, where H is the Hamiltonian, Z = tr e−βH is the partition function, and β = 1/kT , where k is the Boltzmann’s constant. The entanglement associated with the thermal state χ is referred to as the thermal entanglement [1]. To achieve a quantitative understanding of the role of entanglement in the field of quan- tum information science, it is necessary to quantify the amount of entanglement that is associated with a given state. Several entanglement measures have been proposed. One famous example is the Wootters concurrence [8]. Consider a two-qubit state ρ, the Wootters concurrence CW [ρ] ≡ max{λ1 − λ2 − λ3 − λ4, 0}, (1) where λk (k = 1, 2, 3, 4) are the square roots of the eigenvalues in decreasing order of magnitude of the spin-flipped density-matrix operator R = ρ(σy⊗σy)ρ∗(σy⊗σy), the asterisk indicates complex conjugation. Wootters concurrence is closely related to the entanglement of formation [9]. It appears as an auxillary function that has to be evaluated when computing the entanglement of formation. Recently, Mintert, Kús and Buchleitner (MKB) [10, 11] proposed generalizations of Wootters concurrence for multipartite quantum systems, which can be evaluated efficiently for arbitrary mixed states. This is an extremely important development in our investigation of multipartite entanglement, a subject we have yet to achieve a complete understanding. With the availability of such a good and computable measure of entanglement for systems of two qubits, many thoroughly analyzed the thermal entanglement in two-qubit Heisenberg models in terms of thermal concurrence [1, 12, 13, 14, 15, 16, 17]. The recent breakthroughs in the experimental physics of double quantum dot (see, e.g., Ref.[18]) shows that these studies are worthwhile pursuits. Interestingly, Kamta et al. [15] showed that the anisotropy and the magnetic field may together be used to control the extent of thermal concurrence in a two-qubit Heisenberg XY model and, especially, to produce entanglement for any finitely large T , by adjusting the external magnetic field beyond some finitely large critical strength. Such robustness is absent in the case of two-qubit Heisenberg XX model. A natural question is whether thermal entanglement of a Heisenberg XY chain with more qubits has similar behavior. This question is not only of practical importance but also of fundamental importance (see, for instance, Ref.[19]). In this paper, we provide answers for Heisenberg XY chains with any even number of qubits by analyzing the MKB concurrences associated with their thermal states. Our paper is organized as follows. Since the results of MKB [10, 11] is crucial to our analysis, we will present them in the following section. However, we do so explicitly in the context of direct relevance to our paper, i.e., n-qubits. Specifically, when n = 2, the MKB concurrence coincides with the Wootters concurrence. It thus allows us to understand the special property of the thermal entanglement of the two-qubit Heisenberg XY model, in the light of a mathematical property of the MKB concurrences (see Eq.(11)). This together with a discussion of the two-qubit Heisenberg XY model [15] will be given in Section III. We show, in Section IV, that the thermal entanglement of the four-qubit Heisenberg XY chain has behavior similar to the two-qubit case. Namely, anisotropy and magnetic field may together be used to produce genuine multipartite entanglement [20, 21] for any finitely large T by adjusting the external magnetic field beyond some finitely large critical strength. This can again be understood in the light of the same mathematical property of MKB con- currences. Our results agree with those in Ref.[22], which employed bipartite entanglement measure. In general, it is not sufficient to only consider bipartite entanglement. For instance, the Greenberger-Horne-Zeilinger (GHZ) state [23] is a state with genuine multipartite en- tanglement but yields zero entanglement between one particle and any other particle. On the other hand, the W state [24] is one where every particle is entangled with every other particle, but it has no genuine multipartite entanglement [10, 20]. Multipartite entangle- ment surely is more interesting. Therefore, our study complements theirs. However, we must emphasize that our analysis more importantly demonstrates that the thermal entan- glement of the Heisenberg XY chains have the characteristic behavior due to two reasons. First, in the presence of a finitely large external magnetic field, the ground state of the model has non-zero genuine multipartite entanglement. Second, for non-zero temperatures, this ground state is the highest weight state (with weight ≈ 1, i.e., the thermal state is almost a pure state) when a finitely large enough magnetic field is applied. In Section V, we show that this is indeed the case for the six-qubit Heisenberg XY chain. We thus have firm mathematical basis (Eq.(11)) to establish that the thermal entanglement in this case will exhibit similar characteristic behavior. Most important of all, the MKB concurrences may be directly measured for multipartite pure states [25, 26]. It is thus possible to experi- mentally verify our conjecture that for any even n, it is always possible to obtain non-zero genuine multipartite entanglement at any finite temperature by applying a sufficiently large magnetic field. Further discussions of this possibility and a summary of our results will be presented in the concluding Section VI. II. THE MKB CONCURRENCES Consider an n-qubit pure state |ψ〉 ∈ H1⊗H2⊗· · ·⊗Hn, the MKB concurrence is defined as the expectation value of a Hermitean operator A that acts on two copies of the state [10]: C[|ψ〉〈ψ|] ≡ 〈ψ| ⊗ 〈ψ|A|ψ〉 ⊗ |ψ〉. (2) In general, A could have the form {sji=±} ps1is2i ···sniP ⊗ P (2)s2i ⊗ · · · ⊗ P , (3) where ps1is2i ···sni ≥ 0, (Π+0 +Π Π−1 (4) are projectors onto the symmetric and antisymmetric subspaces of Hj ⊗ Hj. Here, Π±0 ≡ (|00〉 ± |11〉)(〈00| ± 〈11|), Π±1 ≡ (|01〉 ± |10〉)(〈01| ± 〈10|) and {|0〉, |1〉} is an orthonormal basis of Hj. The summation in Eq.(3) is performed over the set {sji = ±}+, which contains all n-long strings of +’s and −’s with even number of −’s. The superscript + indicates that the string with n +’s is not included in the sum. This is because for separable states, its expectation value in the symmetric twofold copy is non-zero. Terms with odd number of − ’s are naturally excluded in the sum since their expectation values in the twofold copy states is always zero. By choosing the value of all the ps1is2i ···sni ’s in Eq.(3) to be 4, an entanglement monotone Cn can be obtained. The resulting operator An can equivalently be written as 4(I − P (1)+ ⊗ + ⊗ · · ·⊗P + ) [26]. The concurrence Cn of an n-qubit pure state |ψ〉 can then be written Cn[|ψ〉〈ψ|] = 21−n/2 (2n − 2)〈ψ|ψ〉 − Trρ2i . (5) The above summation runs over all (2n − 2) reduced density operators ρi of the state |ψ〉. Cn accounts for all possible types of entanglement in a state and takes the value zero if and only if the state is fully separable. For even number n of qubits, it is possible to define an MKB concurrence C(n) that detects multipartite entanglement, by choosing the operator A = A(n) ≡ 2nP (1)− ⊗ P − ⊗ · · · ⊗ P We note that when n = 2, C(2)[|ψ〉〈ψ|] = |〈ψ∗|σy ⊗ σy|ψ〉| = CW [|ψ〉〈ψ|]. (6) That is, the MKB concurrence C(2) coincides with the Wootters concurrence. And, for n = 4, we have C(4)[|ψ〉〈ψ|] = |〈ψ∗|σy ⊗ σy ⊗ σy ⊗ σy|ψ〉|, (7) which is also an entanglement monotone [11]. There is obviously no equivalent definition for the case of odd number of qubits since the expectation value of A(n) are always zero for odd n. The MKB concurrence for a mixed state ρ of n qubits can be obtained via the convex roof construction: C[ρ] ≡ inf piC[|ψi〉〈ψi|], ρ = pi|ψi〉〈ψi| , (8) where the infimum is taken over all possible pure state decompositions of ρ. To evaluate C[ρ], consider the spectral decompositions ρ = i |φ̃i〉〈φ̃i| and A = α |χ̃α〉〈χ̃α|, where the eigenstates |φ̃i〉 and |χ̃α〉 are subnormalized such that their norms squared are the eigenvalues corresponding to the states. If r is the rank of the operator A, it is possible to define r complex symmetric matrices T α with elements T αjk ≡ 〈φ̃j| ⊗ 〈φ̃k|χ̃α〉. And, Eq.(8) becomes C[ρ] = inf |[V T αV T ]ii|2 (9) where the infimum is now taken over the set of left unitary matrices V . It can be shown that the following inequality holds [10]: C[ρ] ≥ inf |[V τV T ]ii|. (10) Here, the matrix τ is defined to be α zαT α in terms of arbitrary complex numbers zα satisfying only the condition that α |zα|2 = 1. An algebraic solution of the inequality Eq.(10) is given in Ref.[10] to be max{0, λ1 − j>1 λj} where λi’s are singular values of τ written in decreasing order. For C(n), when A = A(n) is of rank 1, T 1 = τ and the lower bound in Eq.(10) turns out to be the exact value of C(n). In general, an optimization over zα is also necessary to obtain the optimal lower bound for C[ρ]. However, it is possible to obtain a good approximation to C[ρ] by approximating τ with a matrix whose elements [10] τij ≈ 〈φ̃1| ⊗ 〈φ̃1|A|φ̃i〉 ⊗ |φ̃j〉 〈φ̃1| ⊗ 〈φ̃1|A|φ̃1〉 ⊗ |φ̃1〉 , (11) where |φ̃1〉 is the eigenstate of ρ with the largest eigenvalue. This is the mathematical property of MKB concurrences that will play a critical role in our understanding of the characteristic behavior of the thermal entanglement of Heisenberg XY models. We will first illustrate this explicitly with the two-qubit Heisenberg XY chain in the next section. III. TWO-QUBIT HEISENBERG XY MODEL The Hamiltonian H2 for the anisotropic two-qubit Heisenberg XY model in an external magnetic field Bm ≡ ηJ (η is a real number) along the z axis is (1 + γ)Jσx1 ⊗ σx2 + (1− γ)Jσy1 ⊗ σ 1 ⊗ I2 + I1 ⊗ σz2), (12) where Ij is the identity matrix and σ j , σ j , σ j are the Pauli matrices at site j = 1, 2. The parameter −1 ≤ γ ≤ 1 measures the anisotropy of the system and equals 0 for the isotropic XX model [12] and ±1 for the Ising model [14]. (1 + γ)J and (1 − γ)J are real coupling constants for the spin interaction. The model is said to be antiferromagnetic for J > 0 and ferromagnetic for J < 0. The thermal concurrence associated with the thermal state χ2, Eq.(16), can be derived from Eq.(17). It is invariant under the substitutions η −→ −η, γ −→ −γ, and J −→ −J . Therefore, we restrict our considerations to η ≥ 0, 0 ≤ γ ≤ 1, and J > 0. The eigenvalues and eigenkets of H2 are given by [15] H2|Φ0〉 = B|Φ0〉, H2|Φ1〉 = J |Φ1〉, H2|Φ2〉 = −J |Φ2〉, H2|Φ3〉 = −B|Φ3〉, (13) where B ≡ B2m + γ 2J2 = η2 + γ2J , |Φ0〉 = (B +Bm)2 + γ2J2 [(B +Bm)|00〉+ γJ |11〉], |Φ1〉 = 1√ [|01〉+ |10〉], |Φ2〉 = [|01〉 − |10〉], |Φ3〉 = 1√ (B − Bm)2 + γ2J2 [(B − Bm)|00〉 − γJ |11〉]. (14) The Wootters concurrence associated with the eigenkets, |Φ0〉 and |Φ3〉, are given by γ√ η2+γ2 Hence, they represent entangled states when γ 6= 0. We note that when η = 0, |Φ0〉 and |Φ3〉 reduce to (|00〉 + |11〉)/ 2 and (|00〉 − |11〉)/ 2 respectively, so that the eigenstates are the four maximally entangled Bell states: |Ψ0Bell〉, |Ψ1Bell〉, |Ψ2Bell〉, and |Ψ3Bell〉. And, in the limit of large η, CW [|Φ0〉〈Φ0|] = CW [|Φ3〉〈Φ3|] ≈ γη−1, (15) only going to zero asympotically when η is infinitely large. In contrast, when γ = 0, |Φ0〉 = |00〉 and |Φ3〉 = |11〉 are product states with eigenvalues ηJ and −ηJ respectively, though |Φ1〉 and |Φ2〉 remain the same [12]. For the above system in thermal equilibrium at temperature T , its state is described by the density operator wi|Φi〉〈Φi| [e−βB|Φ0〉〈Φ0|+ e−βJ |Φ1〉〈Φ1|+ eβJ |Φ2〉〈Φ2|+ eβB|Φ3〉〈Φ3|], (16) where the partition function Z2 = 2 cosh βB + 2 cosh βJ , the Boltzmann’s constant k ≡ 1 from hereon, and β = 1/T . After some straightforward algebra, we obtain eβJ , e−βJ , 2γ2J2 sinh2 βB + 2γJ sinh2 βB sinh βB, 2γ2J2 sinh2 βB − 2γJ sinh2 βB sinh βB. (17) In the zero-temperature limit, i.e., β −→ ∞, at which the system is in its ground state, Eq.(16) reduces to the following three possibilities. (a) 0 ≤ η < 1− γ2: [eβJ |Φ2〉〈Φ2|+ eβB|Φ3〉〈Φ3|] −→ |Φ2〉〈Φ2|, (18) with Z2 = e βJ+eβB. Equation (17) gives CW [χ2] = 1, its maximum value, in agreement with the fact that |Φ2〉 is a maximally entangled Bell state. (b) η = 1− γ2: χ2 −→ [|Φ2〉〈Φ2|+ |Φ3〉〈Φ3|]. (19) From Eq.(17), the above equally weighted mixture has CW [χ2] = (1− γ). (20) (c) η > 1− γ2: χ2 −→ |Φ3〉〈Φ3|, (21) and Eq.(17) yields accordingly CW [χ2] = η2 + γ2 . (22) Therefore, for a given γ, ηcritical = 1− γ2 marks the point of quantum phase transition (phase transition taking place at zero temperature due to variation of interaction terms in the Hamiltonian of a system [1]). For values of γ other than γ = 1 , there is a sudden increase or decrease in C[χ2] at ηcritical, depending on whether γ > 13 or γ < , before decreasing to zero asymptotically, as η is increased beyond the critical value ηcritical [15]. Here, we focus on the behavior of the model at non-zero temperatures and subject to magnetic field of appropriate strengths. At non-zero temperatures, due to mixing, CW [χ2] decreases to zero as the temperature T is increased beyond some critical value. In fact, as T → ∞, the statistical weights wi → 1/4 for all i and CW [χ2] → 0. However, for a large but finite T , 1 + e−2βB + e−β(B−J) + e−β(B+J) can always be made as close to unity as possible by increasing the strength η of the external magnetic field. That is, when η is large enough, only |Φ3〉 contributes significantly to the thermal state χ2. The η required for this to occur depends on T , larger η for higher T . This is always possible for finite T . Next, we note that in the limit of large η, λ1 ≈ λ2 ≈ 0 ≈ λ4, λ3 ≈ γη−1. (24) It follows that CW [χ2] ≈ γη−1 ≈ CW [|Φ3〉〈Φ3|]. (25) Hence, for a finitely large T , the thermal concurrence of the system in a large enough magnetic field is very well approximated by the concurrence of |Φ3〉 in the same magnetic field. In other words, the entanglement associated with the thermal state χ2 in this case is mainly due to that associated with the eigenstate |Φ3〉. In summary, for any non-zero T and an appropriate η, |Φ3〉 is the highest weight eigen- state. It follows from Eq.(11) that this is the state which will mainly contribute to the MKB concurrence C(2)[χ2] or equivalently the Wootters concurrence (see Eq.(6)). An important point to note here is that |Φ3〉 has non-zero concurrence as long as η is finite (see Eq.(15)). Another is that γ 6= 0, otherwise |Φ3〉 will be a product state with no entanglement. These clearly explain the characteristic robustness of the thermal entanglement of the two-qubit Heisenberg XY model described in Ref[15] - the system at finite T can always be entangled provided large enough magnetic field is applied. They also identify the necessary character- istic features for a model to exhibit such behavior. We shall illustrate that this is indeed the case for the four-qubit Heisenberg XY chain in the next section. IV. FOUR-QUBIT HEISENBERG XY MODEL The Hamiltonian Hn for an anisotropic n-qubit Heisenberg XY chain in an external magnetic field Bm = ηJ along the z-axis is [(1 + γ)σxj σ j+1 + (1− γ)σ j+1 + ησ j ], (26) where the periodic boundary condition σαn+1 = σ 1 (α = x, y, z) applies. Like in the two-qubit model, we consider the case when η ≥ 0, 0 ≤ γ ≤ 1, and J > 0. In this section, we consider n = 4. After some straightforward algebra, we obtain the eigenvalues and eigenvectors of H4. Firstly, we present H4|Φ15〉 = −ω+J |Φ15〉, (27) where [η2 + 2(1 + γ2)]± [η2 + 2(1 + γ2)]2 − 8η2, (28) |Φ15〉 = N−Ω (Ω 1 |0000〉+ Ω−2 |0011〉+ Ω−3 |0101〉+ Ω−2 |0110〉 +Ω−2 |1001〉+ Ω−3 |1010〉+ Ω−2 |1100〉+ |1111〉), (29) with N±Ω ≡ 1/ 1 + (Ω±1 ) 2 + 4(Ω±2 ) 2 + 2(Ω±3 ) Ω±1 = (2η ± ω+)(ω+2 − 8)− 8γ2(η ± ω+) Ω±2 = 2η ± ω+ Ω±3 = ± 2η ± ω+ . (30) |Φ0〉, which can be obtained from |Φ15〉 by substituting Ω−i (i = 1, 2, 3) with Ω+i or Ω−i → Ω+i , satisfies H4|Φ0〉 = ω+J |Φ0〉. The MKB concurrences, as defined in Section II, for |Φ15〉 are given by C(4)[|Φ15〉〈Φ15|] = 2 Ω 1 + 2(Ω 2 + (Ω−3 ) 1 + (Ω−1 ) 2 + 4(Ω−2 ) 2 + 2(Ω−3 ) C4[|Φ15〉〈Φ15|] = 7[2(Ω−2 ) 2 + (Ω−3 ) 2]2 + 2[4(Ω−1 ) 2 − Ω−1 + 4][2(Ω−2 )2 + (Ω−3 )2] + 7(Ω−1 )2 1 + (Ω−1 ) 2 + 4(Ω−2 ) 2 + 2(Ω−3 ) In the limit of large η, ω+ ≈ 2η + 2γ 4γ2 − γ4 Ω−1 ≈ 2γ2 − γ4 Ω−2 ≈ − γ3 − 4γ Ω−3 ≈ 4γ − 3γ3 . (32) It follows that C(4)[|Φ15〉〈Φ15|] ≈ 8γ2 − 4γ4 , (33) C4[|Φ15〉〈Φ15|] ≈ 24γ − 9γ3 . (34) We note that |Φ15〉 is a state with genuine four-partite entanglement [27], which remains non-zero even for large η and going to zero only in the asymptotic limit of infinite magnetic field. Secondly, we have H4|Φ14〉 = −[(α+ + α−)γ + 2]J |Φ14〉, H4|Φ13〉 = −[(α+ + α−)γ − 2]J |Φ13〉, (35) where η2 + 4γ2 ± η , (36) |Φ14〉 = 1 + (α−)2 (−α−|0001〉+ α−|0010〉 − α−|0100〉+ |0111〉 +α−|1000〉 − |1011〉+ |1101〉 − |1110〉), |Φ13〉 = 1 1 + (α−)2 (−α−|0001〉 − α−|0010〉 − α−|0100〉+ |0111〉 −α−|1000〉+ |1011〉+ |1101〉+ |1110〉). (37) Corresponding to these states are |Φ1〉 and |Φ2〉, which can be derived from |Φ14〉 and |Φ13〉 respectively by −α− → α+. They satisfy H4|Φ1,2〉 = [(α+ + α−)γ ± 2]J |Φ1,2〉. The MKB concurrences for |Φ14〉 are given by C(4)[|Φ14〉〈Φ14|] = 2α 1 + (α−)2 C4[|Φ14〉〈Φ14|] = 1 + (α−)2 3 + 8(α−)2 + 3(α−)4 . (38) Hence, |Φ14〉 is also a state with genuine four-partite entanglement [20, 21]. It reduces to a W state when γ = 0. Thirdly, we have H4|Φ3,4〉 = ηJ |Φ3,4〉 and H4|Φ11,12〉 = −ηJ |Φ11,12〉, where |Φ3,4〉 = 1 (|0001〉 ± |0010〉 − |0100〉 ∓ |1000〉), |Φ11,12〉 = 1 (|0111〉 ± |1011〉 − |1101〉 ∓ |1110〉). (39) These states belong to the family of W states which are entangled but do not contain genuine four-partite entanglement [20]. Fourthly, by substituting ω+ in Eq.(30) with ω− we obtain the corresponding ∆±1 , ∆ 2 and ∆ 3 in terms of which we express |Φ10,11〉 = N±∆(∆ 1 |0000〉+∆±2 |0011〉+∆±3 |0101〉+∆±2 |0110〉 +∆±2 |1001〉+∆±3 |1010〉+∆±2 |1100〉+ |1111〉). (40) Here, N±∆ ≡ 1/ 1 + (∆±1 ) 2 + 4(∆±2 ) 2 + 2(∆±3 ) 2. They satisfy H4|Φ10,11〉 = ±ω−J |Φ10,11〉. Lastly, we have the following four degenerate eigenstates with eigenvalue zero: |Φ6〉 = 1√ (|0011〉 − |1100〉) , |Φ7〉 = 1√ (|0101〉 − |1010〉) , |Φ8〉 = (|0110〉 − |1001〉) , |Φ9〉 = 1 (|0011〉 − |0110〉 − |1001〉+ |1100〉) . (41) The eigenstates |Φ6,7,8〉 belong to the family of GHZ states that contain only genuine four- partite entanglement, while |Φ9〉 is a product state of two Bell states. As in the case of the two-qubit model, we construct the thermal state of the four-qubit Heisenberg XY chain as follows: wi|Φi〉〈Φi| = e−βH4 . (42) Here, the partition fuction Z4 = 4 + 4 cosh βηJ + 2 cosh β[(α + + α−)γ + 2]J +2 cosh β[(α+ + α−)γ − 2]J + 2 cosh βω+J + 2 cosh βω−J. (43) At non-zero temperatures the state of the system becomes a mixture of the energy eigenstates with statistical weights w15 = , w14 = eβ[(α ++α−)γ+2]J , · · · (44) A. Zero Temperature In the β → ∞ limit, [eβ[(α ++α−)γ+2]J |Φ14〉〈Φ14|+ eβω+J |Φ15〉〈Φ15|], (45) with Z4 = e β[(α++α−)γ+2]J + eβω +J . Like in the 2-qubit Heisenberg XY model, there are thus two possible lowest energy states, namely |Φ14〉 or |Φ15〉, depending on the strength of the applied magnetic field. As η is increased from zero, there are in general two instances when (α+ + α−)γ + 2 = ω+. We let η1 and η2 denote the solutions. Their dependence on the anisotropy of the system are plotted in Fig. 1. As γ is increased from zero, both η1 and η2 become smaller and converge to zero when γ = 1. In fact, for the Ising model (γ = 1), both |Φ14〉 and |Φ15〉 are the lowest energy states of the system. The system state is an equal mixture of both states: [|Φ14〉〈Φ14|+ |Φ15〉〈Φ15|]. (46) But, once the external magnetic field is turned on, |Φ15〉 becomes the only ground state of the model regardless of the strength of the field. It follows that for 0 < γ < 1, depending on η we have the density operator of the system |Φ15〉〈Φ15| 0 ≤ η < η1 [|Φ15〉〈Φ15|+ |Φ14〉〈Φ14|] η = η1 |Φ14〉〈Φ14| η1 < η < η2 [|Φ14〉〈Φ14|+ |Φ15〉〈Φ15|] η = η2 |Φ15〉〈Φ15| η > η2 FIG. 1: The two transition η’s for the system at zero temperature plotted against the anisotropy We can therefore calculate the MKB concurrrences of the system at zero temperature (β → ∞) for the different regions of magnetic field strength. Plots of C(4) and C4 against magnetic field for different values of anisotropy γ are shown in Fig. 2 and 3. For both concurrences, there are sharp changes at the transition magnetic fields due to quantum phase transition. FIG. 2: The MKB concurrence C(4), which is an entanglement monotone, plotted against magnetic field η for different values of anisotropy γ. In general, there are sharp changes in C(4) at the two transition fields η1 and η2. B. Non-zero Temperatures In general, as the temperature of a system is increased, its density operator becomes closer to the maximally mixed state, 1 I, where n is the dimension of the Hilbert space FIG. 3: The MKB concurrence C4 plotted against η for different values of γ. Similar to Fig.2, there are sharp changes of the concurrence at the two transition magnetic fields. and I is the identity operator. There thus exist critical temperatures Tc beyond which the MKB concurrences of the system become zero, like in the two-qubit case (see Fig. 4). The existence of Tc’s is guaranteed by the fact that a state becomes separable when it is sufficiently close to 1 I [28]. FIG. 4: The points where the MKB concurrence C(4) equals zero are plotted for two different values of γ. The region above each curve is the region where C(4) = 0. On the other hand, it can also be seen from Fig. 4 that it is always possible to have non-zero entanglement in the system by applying a sufficiently large magnetic field to it. In particular, even when T > Tc, one could reintroduce entanglement into the system by increasing η. This is always possible for finite temperatures as long as γ 6= 0. Hence, the thermal entanglement associated with the four-qubit Heisenberg XY model exhibits the same characteristic robustness as in the two-qubit case. We note that, in contrast to the two-qubit case (see Fig. 3 in Ref.[15]), each of our graphs in Fig. 4 has two “singular turning points”. This is due to the fact that there are two transition η’s instead of one in the two-qubit model. In order to understand this robustness, we draw inspiration from the two-qubit case. We observe that for any finite temperature T , the statistical weight w15 when η is large enough, is given by w15 = exp βω+J = where ξ ≡ 1 + e−2βω+J + 4e−βω+J + 4e−βω+J cosh βηJ + 2e−βω+J cosh β[(α+ + α−)γ + 2]J + 2e−βω +J cosh β[(α+ + α−)γ − 2]J + 2e−βω+J cosh βω−J . It can be shown that the large η behavior of w15 is independent of γ. Furthermore, by increasing η appropriately, ξ can be made to be as close as possible to unity. This results in |Φ15〉 being the only state that significantly contributes to the thermal state χ4. In the light of Eq.(11), we may conclude C(4)[χ4] ≈ C(4)[|Φ15〉〈Φ15|]. (49) That this is indeed the case has been established numerically (see Table I). Since the MKB concurrence C(4) can be determined exactly, it is calculated for states χ4 and |Φ15〉 under different combinations of η and T (some results are shown in Table I). The two values for each combination of η and T are then compared. For a given temperature, the two values, C(4)[χ4] and C(4)[|Φ15〉〈Φ15|], agree when large enough magnetic field is applied. Hence, for γ 6= 0, the revival and robustness of entanglement in the Heiseberg four-qubit model can be understood, as in the two-qubit case, in terms of the large η behaviors of both the MKB concurrence C(4)[|Φ15〉〈Φ15|] (Eq.(15)) and the statistical weight w15 (Eq.(48)). We may conclude, when w15 ≈ 1, that the entanglement associated with χ4 is of the genuine four- partite kind [20, 21]. The total entanglement as measured by C4 thus also undergoes revival since genuine four-partite entanglement is only one kind of the entanglement measured by C4. In contrast, for γ = 0 and large η, the ground state is the product state |1111〉. This is an important distinction between the four-qubit XX model and the XY model. Consequently, T η =0 η =100 η =1000 χ4 |Φ15〉〈Φ15| χ4 |Φ15〉〈Φ15| χ4 |Φ15〉〈Φ15| 1 0 1 0.0000180069 0.0000180069 1.79177×10−7 1.79177×10−7 5 0 1 0.0000180068 0.0000180069 1.79177×10−7 1.79177×10−7 10 0 1 0.0000174316 0.0000180069 1.79177×10−7 1.79177×10−7 50 0 1 0 0.0000180069 1.79175×10−7 1.79177×10−7 100 0 1 0 0.0000180069 1.07513×10−7 1.79177×10−7 TABLE I: A comparison between C(4)[χ4] and C(4)[|Φ15〉〈Φ15|] for some combinations of η and T . no revival of entanglement is observed for the case of four-qubit isotropic Heisenberg XX chain. We may apply the above analysis employing C(n) to study Heisenberg XY chains with even number n of qubits. Firstly, we determine if the ground state |Φg〉 remains genuinely multipartite entangled at large η. Secondly, we determine if the ground state statistical weight wg can be made very close to 1 at large η. Through Eq.(11), these two properties together imply that for any finite temperature, nonzero genuine multipartite entanglement can always be obtained by applying a large enough magnetic field. We show that the six- qubit Heisenberg XY chain has both properties in the next section. V. SIX-QUBIT HEISENBERG XY MODEL AND BEYOND In this section, we show that the six-qubit Heisenberg XY chain (with γ 6= 0) does indeed possess the characteristic properties, which enable the model to have non-zero thermal entanglement at any given finite temperature when subject to an external magnetic field of appropriate strength. To this end, we determine the eigenstate |Φg〉 of H6 whose eigenvalue Eg = −λJ is the minimum when η is large. Here, 3(2 + 2γ2 + η2) + 2κ+ 2 (4γ2 + η2)((3 + γ2 + η2) + κ) (50) with κ ≡ γ4 + (−3 + η2)2 + 2γ2(3 + η2). And, |Φg〉 = N(|000000〉Θ1 + |000011〉Θ2 + |000101〉Θ3 + |000110〉Θ4 + |001001〉Θ4 + |001010〉Θ3 +|001100〉Θ2 + |001111〉Θ5 + |010001〉Θ3 + |010010〉Θ4 + |010100〉Θ3 + |010111〉Θ6 +|011000〉Θ2 + |011011〉Θ7 + |011101〉Θ6 + |011110〉Θ5 + |100001〉Θ2 + |100010〉Θ3 +|100100〉Θ4 + |100111〉Θ5 + |101000〉Θ3 + |101011〉Θ6 + |101101〉Θ7 + |101110〉Θ6 +|110000〉Θ2 + |110011〉Θ5 + |110101〉Θ6 + |110110〉Θ7 + |111001〉Θ5 + |111010〉Θ6 +|111100〉Θ5 + |111111〉Θ8) (51) {24J2 − (λ− 3Jη)(λ− Jη)}Θ8 + 2J2(λ− Jη)ζ + 8J2τ 2Jγ(λ+ 3Jη) Θ2 = − (λ + 3Jη)Θ1 Θ3 = − (λ + 3Jη)Θ8 + (λ− Jη)τ + 2J2ζ 3(2− γ2)Θ8 + (λ− Jη)ζ + 2τ Θ5 = − (λ− 3Jη)Θ8 3Θ8 + τ Θ7 = − (λ− 3Jη)Θ8 − 2J2ζ Θ8 = λ 4 − 2J2λ2(2 + 2γ2 + η2) + J4{η2(η2 − 12) + 4γ2(η2 + 4)}, Θ21 +Θ 8 + 3(2Θ 2 + 2Θ 4 + 2Θ 5 + 2Θ , (52) τ = −8Jλη{λ2 − J2(6− 2γ2 + η2)}, (53) ζ = 4λ{λ2(γ2 − 1)− J2(4γ4 − 9η2 − 4γ2 + γ2η2)}. (54) The MKB concurrence C(6) is calculated for this state, giving C(6)[|Φg〉〈Φg|] = 2N2|Θ1Θ8 + 6Θ2Θ5 + 6Θ3Θ6 + 3Θ4Θ7|. (55) In the limit of large η, C(6)[|Φg〉〈Φg|] ≈ 2γ3η−3. (56) It can again be shown that |Φg〉 is a genuine six-partite entangled state [20, 21]. In addition, the statistical weight wg of the ground state |Φg〉 can be shown numerically to be close to unity when an appropriately large magnetic field is applied. The ground state |Φg〉 therefore possesses the two desired properties for the system to have non-zero genuine six- partite entanglement at any finite temperature, provided an appropriate magnetic field is applied. We conjecture that the robustness of genuine multipartite entanglement is a general property of Heisenberg XY chain with even number of particles. This conjecture can be experimentally tested, as will be discussed in the next section. VI. CONCLUSIONS In this study, we have investigated in detail the origin of the robustness of genuine mul- tipartite entanglement in two-, four-, and six-qubit Heisenberg XY models. Two important properties possessed by the ground states |Φg〉 of these models, which enable them to ex- hibit the characteristic robustness, were identified. The first property, namely the statistical weight wg associated with |Φg〉 can be made very close to unity by applying a large enough magnetic field, allows us to use Eq.(11) and conclude that the MKB concurrence of the thermal state χ equals that of the ground state. It follows that we could obtain fairly ac- curately the MKB concurrence of χ (a mixed state) by calculating the MKB concurrence of |Φg〉 (a pure state) in this case. The second property being that |Φg〉 remains genuinely multipartite entangled under such a magentic field, then guarantees that there is non-zero genuine multipartite entanglement for any finite temperature as long as sufficiently large magnetic field is applied. These properties allow us to extend our study to XY chains with any even number of qubits. Heisenberg XY chains with odd number of qubits were not considered in this study because C(n) = 0 for odd n. However, one could similarly study the robustness of other kinds of entanglement employing the other MKB concurrences. In fact, the applicability of our analysis is not only restricted to the Heisenberg XY chains. As long as the ground state of a system can be made to dominate the thermal state at any finite temperature by adjusting some parameters of the system Hamiltonian, Eq.(11) tells us that the MKB concurrence of the thermal state is given by that of the ground state. So, if in addition, the ground state remains multipartite entangled under these conditions, similar robustness of multipartite entanglement is expected to be observed. Therefore, our analysis can be used to identify possible candidates for realization of quantum computation at finite temperatures. The MKB concurrences C(n) and Cn can be directly measured for pure states, if two copies of the states are available [25, 26]. This is obviously due to the fact that the MKB concurrences are defined in terms of expectation values of Hermitean operators. Indeed, this has lead to the direct measurement of Wootters concurrence, previously thought to be not directly measurable, for pure states in laboratory [25]. Since we are interested in the region where the ground state |Φg〉 is the only state that contributes significantly to the thermal state χ, the state χ is “almost pure” and therefore it is possible to measure the MKB concurrences of χ with a high degree of success. The results obtained here can thus be experimentally verified. In conclusion, we have established a rather general method of identifying systems that exhibit robustness of multipartite entanglement at finite temperatures. Our analysis rest on the mathematical property of the MKB concurrences, namely Eq.(11). 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Greenberger, in Proceedings of Squeezed States and Quantum Uncertainty, edited by D. Han, Y. S. Kim, and W. W. Zachary (NASA Conf. Publ. 3135, 73, 1992). [25] S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert and A. Buchleitner, Nature (London) 440, 1022 (2006). [26] L. Aolita and F. Mintert, Phys. Rev. Lett. 97, 050501 (2006). [27] It turns out that C(4)[|Ψ〉〈Ψ|] 6= 0 is only a sufficient condition for |Ψ〉 to have genuine mul- tipartite entanglement. However, in this case, it can be shown that |Φ15〉 possesses genuine four-partite entanglement [20, 21]. [28] K. Zyckowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A. 58, 883 (1998). http://arxiv.org/abs/quant-ph/0506073 Introduction The MKB Concurrences Two-qubit Heisenberg XY Model Four-qubit Heisenberg XY model Zero Temperature Non-zero Temperatures Six-qubit Heisenberg XY model and beyond Conclusions References

0704.1369

Double Helicity Asymmetry of Inclusive pi0 Production in Polarized pp Collisions at sqrt(s)=62.4GeV

Double Helicity Asymmetry of Inclusive π0 Production in Polarized pp Collisions at√ s = 62.4 GeV K. Aoki for the PHENIX Collaboration Department of Physics, Kyoto University, Kyoto, Kyoto, Japan, 606-8502 Abstract. The proton spin structure is not understood yet and there has remained large uncertainty on ∆g, the gluon spin contribution to the proton. Double helicity asymmetry (ALL) of π0 production in polarized pp collisions is used to constrain ∆g. In this report, preliminary results of ALL of π0 in pp collisions at s = 62.4 GeV measured by PHENIX experiment in 2006 is presented. It can probe higer x region than the previously reported π0ALL at s = 200 GeV thanks to the lower center of mass energy. Keywords: Spin, Proton spin structure PACS: 14.20.Dh, 13.85.Ni INTRODUCTION The so-called “proton spin crisis,” initiated by the results from the polarized deep inelastic scattering experiments, has triggered wide effort towards the understanding of proton spin. Despite the wide efforts, there has remained large uncertainty on ∆g, the gluon spin contribution to the proton. RHIC, the world’s first polarized proton- proton collider, provides us an opportunity to directly probe gluons in the proton. Double helicity asymmetry (ALL) of inclusive π0 production in polarized pp collisions is sensitive to ∆g because π0 production is dominated by gluon-gluon and quark-gluon interactions in the measured pT range. PHENIX has previously reported π0ALL in pp collisions at s = 200 GeV [1] which is based on the data taken in 2005 (Run5) and it indicates that ∆g is not large.[2] But a large uncertainty remains for large Bjorken x (> 0.1) and more statistics are needed. During the run in 2006 (Run6), one-week data taking was performed at s = 62.4 GeV. Spin rotator commissioning was successful and we had longitudinally polarized collisions. [3] Even in this short data taking with a small integrated luminosity of 60 nb−1 and the average polarization of 48%, the data has a big advantage to cover the larger x region thanks to the lower center of mass energy. According to a pertubative QCD (pQCD) calculation, collisions at s = 62.4 GeV has ∼ 300 times larger cross-section than that at s = 200 GeV at fixed xT = 2pT/ s. It corresponds to 10 times larger statistics than the previously reported π0ALL which is based on the integrated luminosity of 1.8 pb−1 with average polarization of 47%. ALL is defined as ALL = σ++−σ+− σ+++σ+− http://arxiv.org/abs/0704.1369v1 where σ++(+−) is the production cross-section in like (unlike) helicity collisions. Ex- perimentally, ALL is calculated as ALL = |PB||PY | N++−RN+− N+++RN+− , R = where PB(Y ) denotes the beam polarization, N ++(+−) is the π0 yield and L++(+−) is the luminosity in like (unlike) helicity collisions. R is the relative luminosity. EXPERIMENT The stable polarization direction of RHIC beam is transverse. Then it is rotated to get longitudinally polarized collisions just before the PHENIX interaction point. PHENIX local polarimeter[3] confirms that the beam is longitudinal by measuring AN of forward neutrons. PHENIX has Beam-Beam Counter (BBC) which covers 3.0 < |η| < 3.9 and Zero Degree Calorimeter (ZDC) which covers very forward angle (±2mrad).[4] These two detectors serve as independent luminosity measure. We used BBC counts to measure relative luminosity R in equation (2) and its uncertainty is estimated by comparing to ZDC counts. It is found to be δR = 1.3×10−3. This corresponds to δALL = 2.8×10−3 which is less than the statistical uncertainty. PHENIX has the ability to clearly identify π0 through its gamma decay by using an Electro-Magnetic Calorimeter (EMCal) which covers the central rapidity region (|η| < 0.35) and half in azimuth angle. [4] PHENIX also has an excellent gamma triggering capability (the threshold is 0.8 GeV or 1.4 GeV) which makes high-statistics π0 measurement feasible.[5] EMCal based trigger without coincidence with BBC is used because the collision trigger efficiency based on BBC is low at s = 62.4 GeV. The systematic uncertainty is evaluated by the bunch shuffling technique,[6] and it is found to be negligible. ALL CALCULATION π0ALL (Aπ LL) is calculated by subtracting A LL from A π0+BG LL . A π0+BG LL is the asymmetry for the diphoton invariant-mass range of 112 MeV/c2-162 MeV/c2 (under the π0 peak). ABGLL is the asymmetry for the range of 177 MeV/c 2-217 MeV/c2 (higher side band). Figure 1 shows the diphoton invariant mass spectra. The lower mass peak corresponds to background from hadrons and cosmic particles, which induce EMCal clusters with more complicated structure, each of them are then splitted on several ones. This peak roughly corresponds to two EMCal cell separation between two clusters, which moves to higher mass with increasing cluster pair pT . Since we used EMCal based trigger without coincidence with collision trigger at s = 62.4 GeV, the cosmic background is prominent unlike in data at s = 200 GeV. The contribution of such background under π0 peak is negligible in the measured pT range. Since it does affect the lower side band, the Aπ LL estimation was done based only on the higher side band. The subtraction is FIGURE 1. Diphoton invariant mass spectra. done by using the following formula. LL − rABGLL where r is the background fraction. RESULTS Figure 2 shows the Run 6 results of π0ALL as a function of pT . ALL is consistent with zero over the measured pT region. Detailed offline analysis on beam polarization is not provided yet by the RHIC polarimeter group. Thus online values are used and systematic uncertainty of 20% is assigned for a single beam polarization measurement. It introduces scaling uncertainty of 40% on ALL. Theory curves based on pQCD using four proton spin models are also shown.[7] The theory is based on pQCD; thus it is important to test pQCD applicability at s = 62.4 GeV. To test pQCD applicability, analysis on π0 cross- section is on-going. With our cross section result, we will be able to discuss our ALL result further by comparing with pQCD calculations. Figure 3 shows the Run 6 results of π0ALL as a function of xT together with Run 5 results. A clear statistical improvement can be seen in the large xT region. SUMMARY During the RHIC run in 2006, π0ALL at s= 62.4 GeV was measured with the PHENIX detector. Preliminary results of π0ALL at s = 62.4 GeV with integrated luminosity of 60 nb−1 and the average polarization of 48% are presented. There is a clear statistical improvement in the large xT regin compared to the Run5 preliminary results at 200 GeV with integrated luminosity of 1.8pb−1 and the average polarization of 47%. To extract the gluon spin contribution to the proton, it is important to test pQCD applicability at s = 62.4 GeV. Analysis on cross-section is on-going to test pQCD at this energy. With our cross section result, we will be able to discuss our ALL result further by comparing with pQCD calculations. FIGURE 2. π0ALL as a function of pT . The error bar denotes statistical uncertainty. Gray band denotes systematic error from relative luminosity. FIGURE 3. π0ALL as a function of xT . REFERENCES 1. K. Boyle, AIP Conf. Proc. 842, 351–353 (2006), nucl-ex/0606008. 2. M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D74, 014015 (2006), hep-ph/0603213. 3. M. Togawa, et al., RIKEN Accel. Prog. Rep. to be published 40 (2007). 4. K. Adcox, et al., Nucl. Instrum. Meth. A499, 469–479 (2003). 5. K. Okada, et al., RIKEN Accel. Prog. Rep. 36, 248 (2003). 6. S. S. Adler, et al., Phys. Rev. Lett. 93, 202002 (2004), hep-ex/0404027. 7. B. Jager, A. Schafer, M. Stratmann, and W. Vogelsang, Phys. Rev. D67, 054005 (2003), hep-ph/0211007. nucl-ex/0606008 hep-ph/0603213 hep-ex/0404027 hep-ph/0211007 Introduction Experiment ALL calculation Results SUMMARY

0704.1370

Time dependence of joint entropy of oscillating quantum systems

Time dependence of joint entropy of oscillating quantum systems Özgür ÖZCAN,1, ∗ Ethem AKTÜRK,2, † and Ramazan SEVER3, ‡ 1Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey 2Department of Physics, Hacettepe University, 06800, Ankara,Turkey 3Department of Physics, Middle East Technical University, 06531, Ankara,Turkey Abstract The time dependent entropy (or Leipnik’s entropy) of harmonic and damped harmonic oscillators is extensively investigated by using time dependent wave function obtained by the Feynman path integral method. Our results for simple harmonic oscillator are in agrement with the literature. However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time for certain values of damping factor. According to the results, the envelop of the joint entropy curve increases with time monotonically. This results is the general properties of the envelop of the joint entropy curve for quantum systems. Keywords: Path integral, joint entropy, simple harmonic oscillator, damped harmonic oscillator, negative joint entropy PACS numbers: 03.67.-a, 05.30.-d, 31.15.Kb, 03.65.Ta ∗E-mail: [email protected] †E-mail: [email protected] ‡E-mail: [email protected] http://arxiv.org/abs/0704.1370v3 mailto:[email protected] mailto:[email protected] mailto:[email protected] I. INTRODUCTION The investigation of time dependent entropy of the quantum mechanical systems attracts much attention in recent years. For both open and closed quantum systems, the different information-theoretic entropy measures have been discussed [2, 3, 4]. In contrast, the joint entropy [5, 6] can also be used to measure the loss of information, related to evolving pure quantum states [7]. The joint entropy of the physical systems which are named MACS (maximal classical states) were conjectured by Dunkel and Trigger [8]. According to Ref. [8], the joint entropy of the quantum mechanical systems increase monotonically with time but this results are not sufficient for simple harmonic oscillator [9]. The aim of this study is to calculate the complete joint entropy information analytically for simple harmonic and damped harmonic oscillator systems. This paper is organized as follows. In section II, we explain fundamental definitions needed for the calculations. In section III, we deal with calculation and results for harmonic oscillator systems. Moreover, we obtain the analytical solution of Kernel, wave function in both coordinate and momentum space and its joint entropy. We also obtain same quantities for damped harmonic oscillator case. Finally, we present the conclusion in section IV. II. FUNDAMENTAL DEFINITIONS We deal with a classical system with d = sN degrees of freedom, where N is the particle number and s is number of spatial dimensions [8]. We assume that the density function g(x, p, t) = g(x1, ..., xd, p1, ..., pd, t) which is the non-negative time dependent phase space density function of the system has been normalized to unity, dxdpg(x, p, t) = 1. (1) The Gibbs-Shannon entropy is described by S(t) = − 1 dxdpg(x, p, t)ln(hdg(x, p, t)), (2) where h = 2πh̄ is the Planck constant. Schrödinger wave equation with the Born interpre- tation [10] is given by = Ĥψ. (3) The quantum probability densities are defined in position and momentum spaces as |ψ(x, t)|2 and |ψ̃(p, t)|2, where |ψ̃(p, t)|2 is given as ψ̃(p, t) = dxe−ipx/h̄ (2πh̄)d/2 ψ(x, t). (4) Leipnik proposed the product function as [8] gj(x, p, t) = |ψ(x, t)|2|ψ̃(p, t)|2 ≥ 0. (5) Substituting Eq. (5) into Eq. (2), we get the joint entropy Sj(t) for the pure state ψ(x, t) or equivalently it can be written in the following form [8] Sj(t) = − dx|ψ(x, t)|2 ln |ψ(x, t)|2 − dp|ψ̃(p, t)|2 ln |ψ̃(p, t)|2 − − ln hd. (6) We find time dependent wave function by means of the Feynman path integral which has form [11] K(x′′, t′′; x′, t′) = ∫ x′′=x(t′′) x′=x(t′) Dx(t)e S[x(t)] ∫ x′′ Dx(t)e L[x,ẋ,t]dt. (7) The Feynman kernel can be related to the time dependent Schrödinger’s wave function K(x′′, t′′; x′, t′) = ψ∗n(x ′, t′)ψn(x ′′, t′′). (8) The propagator in semiclassical approximation reads K(x′′, t′′; x′, t′) = ∂x′∂x′′ Scl(x ′′, t′′; x′, t′) Scl(x ′′,t′′;x′,t′). (9) The prefactor is often referred to as the Van Vleck-Pauli-Morette determinant [12, 13]. The F (x′′, t′′; x′, t′) is given by F (x′′, t′′; x′, t′) = ∂x′∂x′′ Scl(x ′′, t′′; x′, t′) . (10) III. CALCULATION AND RESULTS A. Simple Harmonic Oscillator (SHO) To get the path integral solution for the SHO, we must calculate its action function. The Lagrangian of the system is given by L(x, ẋ, t) = (ẋ2 − 1 ω2x2) (11) Following a straightforward calculation, it is given by: S(xcl(t ′′), xcl(t ′)) = 2 sinωt [(x′′2cl + x cl) cosωt− 2x′clx′′cl] (12) with t = t′′ − t′ and x′cl = x0, x′′cl = x. Substituting Eq. (9) into Eq. (7), we obtain the Feynman kernel [11]: K(x, x0; t) = ( 2πh̄i sinωt 2 exp{−mω [(x2 + x0 2) cotωt− 2x0x sinωt ]}. (13) By the use of the Mehler-formula 2+y2)/2 )2Hn(x)Hn(y) = 1− z2 4xyz − (x2 + y2)(1 + z2) 2(1− z2) ] (14) where Hn is Hermite polynomials, we can write the Feynman kernel defining x ≡ mω/h̄x0, mω/h̄x and z = e−iωT K(x, x0; t) = e−itEn/h̄Ψ∗(x0)Ψ(x) (15) with energy-spectrum and wave-functions: En = h̄ω(n+ ), (16) Ψn(x) = ( 22nπh̄n!2 x) exp(− x2). (17) Time dependent wave function of the SHO is defined as Ψ(x, t) = K(x, x0; t)Ψ(x0, 0)dx0. (18) It can be written as Ψ(x, t) = − iωt e−2iωt + αᾱe−iωt where x̄ or ᾱ is mean of the Gaussian curve. The probability density has |Ψ(x, t)|2 = − (α− ᾱ cosωt)2 where α = x. Thus it can be written as |Ψ(x, t)|2 = (x− x̄ cosωt)2 This has been shown in Fig.1. In momentum space, the probability density has the form |ψ̃(p, t)|2 = mωπh̄ [ −p2 mωx̄2 cos 2ω(t)− 1 − 2px̄ sinω(t) . (22) The joint entropy of harmonic oscillator becomes Sj(t) = ln x̄2 sin2 ω(t). (23) In Fig.2, the joint entropy of this system was plotted by using Mathematica in three dimen- sion. As known from fundamental quantum mechanics and classical dynamics, displacement of simple harmonic oscillator from equilibrium depends on harmonic functions (e.g sine or cosine function). Therefore, other properties of the SHO systems indicate the same har- monic behavior. If the frequency of the SHO is sufficiently small, the system shows the same behavior as the free particle[8]. As seen from Fig.3 and Fig.4, envelop of the sinusoidal curve is also monotonically increase with omega and constant with time at constant omega, respectively. When the frequency increases, the joint entropy of this system indicates a fluc- tuation with increasing amplitude with time. If t goes to zero, it is important that Eq.(20) is in agreement with following general inequality for the joint entropy: Sj(t) ≥ ln( ) (24) originally derived by Leipnik for arbitrary one-dimensional one-particle wave functions. B. Damped Harmonic Oscillator (DHO) The DHO is very important physical system in all physical systems defining an interaction with its environment. The Lagrangian of the DHO is given by L(x, ẋ, t) = eγt ẋ2 − m ω2x2 + j(t)x) . (25) Damped free particle kernel is K(x, t; x0, 0) = ( γmeγt/2 4πih̄ sinh 1 ( iγmeγt/2 4h̄ sinh 1 (x− x0)2 . (26) The DHO kernel has the form [14] K(x, t; x0, 0) = ( mωeγt/2 2πih̄ sinhωt Scl(x, x0, t) , (27) or explicitly K(x, t; x0, 0) = ( mωeγt/2 2πih̄ sinωt (ax2 + 2bx20 + 2xx0c+ 2xd+ 2x0e− f) . (28) Where the coefficients a, b, c, d, f are [14] a = (−γ + ω cotωt)eγt, (29) b = ( + ω cotωt), (30) c = (− sinωt eγt), (31) m sinωt j(t′)eγt ′/2 sinωt′dt′, (32) m sinωt j(t′)eγt ′/2 sinω(t− t′)dt′, (33) j(t′)j(s)eγ(s+t ′/2) sinω(t− t′) sinωsdsdt′. (34) The wave function ψn(x, 0) and energy eigenvalues become ψn(x, 0) = N0Hn(α0x) exp h̄ω0 (36) where Hn(x) is the Hermite polynomial of order n and the coefficients are α0 = ( )1/2, N0 = (2nn! π)1/2 . (37) The time dependent wave function is obtained as [14] ψn(x, t) = dx0K(x, t; x0, 0)ψ(x, 0) (2nn!)1/2 cot−1× + cotωt+ f exp[−(Ax2 + + 2Bx)]Hn[D(x− E)]. (38) To simplify the evaluation, we set j(t) = 0. Such that kernel and wave function of the DHO [15] become K(x, t; x0, 0) = ( mωeγt/2 2πih̄ sinωt γ(x20 − eγtx2) + sinωt × [(x20 + x2eγt) cosωt− 2eγt/2xx0] where ω = (ω20 − γ2/4)1/2 and ψn(x, t) = (2nn!)1/2 cot−1 + cotωt Hn[Dx] exp[−Ax2]. (40) Where D, A and N are D(t) = αeγt/2 η(t) sinωt , (41) η2(t) = cosωt+ csc2 ωt, (42) A(t) = η2(t) sin2 ωt − cotωt+ γ/2ω + cotωt η2 sin2 ωt , (43) N(t) = )1/4 exp(γt η(t)(sinωt)1/2 . (44) The ground state wave function is given by ψ0(x, t) = N(t) exp cot−1 + cotωt exp[−A(t)x2]. (45) So the probability distribution in coordinate space becomes |ψ0(x, t)|2 = N(t)2 exp[−2A′(t)x2] (46) where A′ is defined by A′(t) = η2(t) sin2 ωt . (47) The probability density in coordinate space is shown in Fig.5 and Fig.6 for the different values of γ. The probability density in momentum space can be written easily |ψ0(p, t)|2 = N(t)2 2A(t)A(t)†h̄ A′(t) A(t)A(t)† . (48) The time dependent joint entropy can be obtained from Eq. (2) as Sj(t) = N(t) 2A′(t) (lnN(t)2 − 2A(t)A(t)† N(t)2 2A(t)A(t)† − ln 2π.(49) The joint entropy depends on damping factor γ. When γ → 0, all the above results are converged to simple harmonic oscillator. However, when the γ 6= 0, the joint entropy has remarkably different features of the SHO. As can be seen in Fig.7 and Fig.8, the joint entropy of the DHO has very interesting properties. One of the most important properties of the joint entropy is the probability of taking values for small γ values. As we know from literature the joint entropy must be positive and monotonically increase.However, this system has different properties from literature because of periodically discontinuity of the joint entropy. On the other hand, envelop of this curve is also monotonically increase with time for large γ. As can be shown these results, the envelop of the joint entropy curves has general properties as monotonically increase for quantum systems. Thus, we have found that the joint entropy is depend on properties of investigated system. IV. CONCLUSION We have investigated the joint entropy for explicit time dependent solution of one- dimensional harmonic oscillators. We have obtained the time dependent wave function by means of Feynmann Path integral technique. Our results show that in the simple harmonic oscillator case, the joint entropy fluctuated with time and frequency. This result indicates that the information periodically transfer between harmonic oscillators. On the other hand, in the DHO case, the joint entropy shows a remarkable smooth discontinuities with time. It also depends on choice of initial values of parameter i.e. ω. These results can be explained as the information exchange between harmonic oscillator and system which is supplied damping. But the information exchange appears in certain values of time for damping. If the damping factor increases, the information entropy has not periodicity anymore. Moreover, for certain values of the damping factor, the transfer of information between systems is exhausted. V. ACKNOWLEDGEMENTS This research was partially supported by the Scientific and Technological Research Coun- cil of Turkey. [1] E. Aydiner, C. Orta and R.Sever, E-print:quant-ph/0602203 [2] W.H. Zurek, Phys. Today 44(10), 36 (1991). [3] R. Omnes, Rev. Mod. Phys. 64, 339 (1992). [4] C. Anastopoulos, Ann. Phys. 303, 275 (2003). [5] R. Leipnik, Inf. Control. 2, 64 (1959). [6] V.V.Dodonov, J.Opt. B: Quantum Semiclassical Opt. 4, S98 (2002). [7] S. A. Trigger, Bull. Lebedev Phys. Inst. 9, 44 (2004). [8] J. Dunkel and S. A. Trigger, Phys. Rev.A71, 052102 (2005). [9] P. Garbaczewski, Phys. Rev. A 72, 056101 (2005). [10] M. Born, Z. Phys. 40, 167 (1926). [11] R.P. Feynmann, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, USA (1965). [12] D.C. Khandekar, S.V. Lawande, K.V. Bhagwat, Path-Integral Methods and Their Applica- tions, World Scientific, Singapore (1993). [13] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 3rd Edition (2004). [14] C.I. Um, K.H. Yeon and T.F.George,Physics Reports, 362,63-192 (2002). [15] K.H. Yeon and C.I. Um, Phys. Rev. A 36,11 (1987). http://arxiv.org/abs/quant-ph/0602203 ÈΨ@x,tDÈ2 FIG. 1: |Ψ(x, t)|2 versus time and coordinate 0 0.05 0.1 0.15 S jHtL FIG. 2: The 3D graph of joint entropy of simple harmonic oscillator. 0 0.2 0.4 0.6 0.8 1 FIG. 3: The joint entropy of simple harmonic oscillator versus ω . 0 5 10 15 20 FIG. 4: The joint entropy of simple harmonic oscillator versus time. ÈΨ0@x,tDÈ FIG. 5: The probability function as a function of time and coordinate at γ = 0.1. ÈΨ0@x,tDÈ FIG. 6: The probability function as a function of time and coordinate at γ = 0.5. FIG. 7: The 3D graph of the joint entropy of damped harmonic oscillator for damping factor(γ) at ω0 = 2. FIG. 8: The 3D graph of the joint entropy of damped harmonic oscillator for damping factor(γ) at ω0 = 1. Introduction Fundamental Definitions CALCULATION AND RESULTS Simple Harmonic Oscillator (SHO) Damped Harmonic Oscillator (DHO) Conclusion Acknowledgements References

0704.1371

Tight bound on coherent states quantum key distribution with heterodyne detection

info.eps Tight bound on coherent states quantum key distribution with heterodyne detection Jérôme Lodewyck1, 2 and Philippe Grangier2 Thales Research and Technologies, RD 128, 91767 Palaiseau Cedex, France Laboratoire Charles Fabry de l’Institut d’Optique, CNRS UMR 8501, Campus Universitaire, bâtiment 503, 91403 Orsay Cedex, France We propose a new upper bound for the eavesdropper’s information in the direct and reverse reconciliated coherent states quantum key distribution protocols with heterodyne detection. This bound is derived by maximizing the leaked information over the symplectic group of transformations that spans every physical Gaussian attack on individual pulses. We exhibit four different attacks that reach this bound, which shows that this bound is tight. Finally, we compare the secret key rate obtained with this new bound to the homodyne rate. I. INTRODUCTION Continuous variable quantum key distribution (CVQKD) is an alternative to single photon “discrete” QKD that encodes key information in variables with a continuum of degrees of freedom. Such variables include the quadratures X and P of a mode of the electromagnetic field. A CVQKD protocol using these quadratures has been introduced in [1]. It consists in sending a train of pulsed coherent states modulated in X and P with a Gaussian distribution (Alice’s module), and in quadrature measurements with an homodyne detection upon reception (Bob’s module). Then, Bob’s continuous data are used as the basis for constructing a secret binary encryption key, in a classical informa- tion process called “reverse reconciliation” (RR). The security of the RRCVQKD homodyne protocol has first been proven against individual Gaussian attacks [1], and later extended to individual or finite-size non-Gaussian attacks [2]. More recently, new security proofs covering collective, Gaussian and non-Gaussian attacks [3, 4] have appeared. In the homodyne protocol [1], Bob randomly chooses to measure either X or P , and then announces his choice. Another possible approach, proposed in [5] by Weed- brook and co-workers, is that Bob measures both X and P quadratures of each incoming coherent state, by sepa- rating them on a 50-50 beam-splitter TB. This detection, called “heterodyne” is represented in Fig. 1. Notably, in this protocol, Bob does not need to randomly switch his measurement basis. Alice FIG. 1: Heterodyne protocol. Bob measure both quadratures X and P of the incoming mode B′. A generic eavesdropping strategy involves a transformation S on Alice’s mode and two vacuum ancillary modes. In [5], the authors proposed a bound on the informa- tion acquired by the eavesdropper (Eve) in the hetero- dyne protocol, under the hypothesis of individual Gaus- sian attacks. They also considered a possible attack based on feed-forward (see below for details), that they however found to be suboptimal with respect to their proposed bound. Therefore, a gap remained between the apparently most stringent bound and the best known at- tack, which is surprising in the simple scenario of individ- ual Gaussian attacks. Later on, in [6], the same authors conjectured that their proposed attack is indeed optimal, and so that tighter bounds should apply to Eve’s infor- mation. However, no such tighter bound was proposed so far. In this article, we propose a new bound for individual Gaussian attacks on the CVQKD heterodyne protocols, tighter than the bound of [5]. In addition, we explic- itly present a series of attacks which are optimal with respect to this bound, closing the gap between the best known attacks and the tightest known bound. The ar- ticle is organized as follows: after introducing notations in section II, the new bounds are derived in section III. In section IV, we extend the technique used to estab- lish these bounds to obtain new results about the homo- dyne protocols. Specifically, we will show that any ho- modyne attack using quantum memory is optimal, and that in some cases this optimality can be reached without quantum memory. Section V is devoted to another tech- nique, based on symplectic invariants, which allows us to derive again the new heterodyne bounds from another point of view. Then section VI describes four optimal attacks with respect to the new heterodyne bound, and section VII concludes our study by discussing practical advantages of the heterodyne protocol. II. NOTATIONS In the case of Gaussian attacks, the channel linking Alice to Bob is fully characterized by its transmission in intensity T (possibly greater than 1 for amplifying chan- nels), and its excess noise ǫ above the shot noise level [7], such that the total noise measured by Bob is (1+T ǫ)N0, where N0 is the shot noise variance appearing in the http://arxiv.org/abs/0704.1371v1 Heisenberg relation 〈X2〉〈P 2〉 ≥ N20 . Alternatively we will make use of the total added noise referred to the in- put χ defined by χ = 1/T+ǫ−1. These parameters might depend on the quadrature considered, in which case we will add a subscript indicating this quadrature (e.g. χP ). The quantum channel is considered to be probed by Eve with the help of ancillary modes. To index these modes, we will note XM and PM the quadra- tures of mode M , and write down the 2n quadra- tures of n modes M1, . . . ,Mn by the vector Q = (XM1 , . . . , XMn , PM1 , . . . , PMn). The total Gaussian state of the system is then represented by its covariance matrix γ of components γi,j = 〈QiQj〉. In the heterodyne protocol, we note B the modes mea- sured by Bob, and B′ the incoming beam, on which we will base our demonstrations. This mode is coupled with two ancillas on which Eve respectively measures X and P (Fig. 1). In [5], the authors bounded the conditional variance VXB′ |XE1 and VPB′ |PE2 of the mode B ′ knowing Eve’s measurement results by V minB′|E = T (1 + χV ) N0, (1) where (V − 1)N0 is Alice’s modulation variance. This is basically the homodyne RR bound derived in [1, 8] applied to each quadrature X and P . Assuming that Eve does not control the beam-splitter TB, this bound then leads to the minimal conditional variance of mode B knowing Eve’s measurement by adding the shot noise unit and the intensity decrease in- troduced by the beam-splitter TB: VB|E = VB′|E +N0 . (2) Then the information acquired by Eve results from VB|E : IBE = 2× with VB = T (V + χ) + 1 where the factor 2 reflects the double quadrature mea- surement. A similar reasoning on IAB finally gives the secret rate ∆I = IAB − IBE . This rate is shown to be greater than the homodyne rate for any channel parame- ter, giving advantage to the heterodyne detection scheme. Several explicit attacks against the heterodyne proto- col have been considered. In [5], the authors propose an eavesdropping strategy based on heterodyne measure- ment and feed-forward (Fig. 2-1), which they numeri- cally show to be suboptimal with respect to bound (1). In [9], Namiki et al. introduce an eavesdropping strat- egy against the homodyne RRCVQKD protocol based on a cloning machine (Fig. 2-2). In this attack, Eve can measure both X and P quadratures of each coher- ent state, then requiring no quantum memory. The price to pay is that this attack is no more optimal with re- spect to the homodyne bound (1). In fact, the search for quantum-memory-less homodyne attacks is very sim- ilar to finding attacks on the heterodyne protocol because Alice Alice Alice Alice Bob EPR pair EPR pair 1/2(4) FIG. 2: Our general results are illustrated by considering four optimal attacks against the heterodyne protocol. In the feed- forward attack (1), Eve taps a fraction 1 − TE of the signal on which she makes an heterodyne measurement. Then she translates Bob’s quadratures according to her measurement results modified by a gain gE. In the cloning attack (2), Eve amplifies the signal sent by Alice with a phase insensitive am- plifier, and taps the amplified beam. Quantum teleportation (3) consist in a making the incoming beam interfere with a part of an EPR pair. X and P are measured in the output arms of the interferometer. Bob’s quadratures are then trans- lated according to Eve’s results. Finally, in the entangling cloner attack (4), Eve tap a fraction 1 − TE of the incoming signal, while introducing some excess noise with the help of an EPR pair. The joint measurement of the tapped signal and the other part of the EPR pair optimally exploits both tapped signal and EPR noise correlations. both schemes require that Eve measuresX and P on each channel symbol. Therefore, the cloning attack is yet an- other sub-optimal attack against the heterodyne protocol when considering bound (1). We shall prove in this ar- ticle that these two attacks are optimal with respect to the new bounds we derive for the heterodyne protocol, as well as two other attacks based on EPR entanglement. III. NEW BOUNDS BASED ON THE IWASAWA SYMPLECTIC DECOMPOSITION To derive the new bounds on heterodyne detection pro- tocols, we will use the symplectic formalism which de- scribes all physically possible Gaussian individual trans- formations on a set of n modes. The real symplectic group is defined by the set of linear transformations of the quadrature vectorQ, which 2n×2nmatrix S satisfies SβST = β, with β = −In 0 , (3) where In is the n × n identity matrix. The main idea of our demonstration is to use a proper parameterization that spans all symplectic transformations applied to the modes going through the quantum channel, hence all pos- sible attacks, and to compute the best information Eve can obtain when these parameters vary. The real symplectic group is a n(2n + 1) parameters space for which various parameterizations – or decom- positions – exist. We choose the Iwasawa decomposi- tion [10], which uniquely factorizes any 2n× 2n symplec- tic matrix S as the product of 3 special matrices: 0 D−1 where B + iF is a n × n unitary matrix, D is diagonal with strictly positive components, A is lower triangular with all diagonal terms set to 1, and ATC is symmetric. In our study, we consider 3 modes, depicted in Fig. 1. The first one, noted B′ is send from Alice to Bob, who performs an heterodyne measurement upon reception. Eve makes this mode interact with two ancillary modes E1 and E2 initially in the vacuum state, and then mea- sures X on E1, and P on E2. To respect the symmetry of this problem, we only consider symplectic transforma- tions S that do not mix X and P quadratures: . (5) As B, D and A are invertible, expanding S yields F = C = 0, and B orthogonal. We recall that the columns of the orthogonal matrix B, as well as its rows, form an orthonormal basis. With this form, the Iwasawa decom- position has a physical meaning in terms of linear optical components. Namely, any symplectic transformation is composed of an orthogonal transformation which is it- self a composition of rotations (i.e. beam-splitters) and reflections (i.e. π-phase shifts), 1-mode squeezers, and feed-forward. Finally, we can write the Iwasawa decomposition cor- responding to our attack model: 1 0 0 a 1 0 b c 1 s1 0 0 0 s2 0 0 0 s3 b1 b2 b3 b4 b5 b6 b7 b8 b9  (6) 1 −a δ 0 1 −c 0 0 1 s−11 0 0 0 s−12 0 0 0 s−13 b1 b2 b3 b4 b5 b6 b7 b8 b9 with δ = ac − b. The orthogonal matrix B can be pa- rameterized by 3 real parameters (Euler angles) plus a binary parameter (the sign of the determinant). This leaves 9 real and one discrete symplectic parameters to characterize the matrix S. By expanding S and using orthogonality properties of B, we can express channel parameters as functions of these symplectic parameters: TX = t X with tX = S1,1 = s1b1 TP = t P with tP = S4,4 = − ab4 2 + S1,3 2 + S4,6 We note that s1 and b1 are equivalent to channel pa- rameters TX and χX. As we are looking for the best attack for given channel parameters, we will consider s1 and b1 fixed. Our attacks are then characterized by 7 real and one discrete parameters. The input covariance matrix γi is diagonal with di- agonal terms (V, 1, 1, V, 1, 1)N0. The output covariance matrix is expressed as γ = SγiST. From S and γ, we obtain Eve’s noises and conditional variances 2 + S2,3 r2 + 1 (rb1 + b4)2 2 + S6,6 1− b21 − b24 VXB′ |XE1 = γ1,1 − γ1,2γ2,1 r2 + 1 (V χPE2 + 1)(χXE1 (V + χXE1 )(χPE2 VPB′ |PE2 = γ4,4 − γ4,6γ6,4 r2 + 1 (V χXE1 + 1)(χPE2 (V + χPE2 )(χXE1 where r = as1/s2. All these quantities only depend on parameters r and b4: our parameter space drops to 2 parameters. Then, we will require that the attack leaves channel parameters symmetric in X and P , i.e. TX = TP ≡ T and χX = χP ≡ χ. The former relation univocally fixes δ = s3(b1(s 1 − 1)/s1 + ab4/s2)/b7, and the later fixes r: b1b4(1 − s21) + σ (1− b21 − b24)ρ 1− b21 . (9) where σ = ±1 and ρ = (s21 − 1)(1 − s21(2b21 − 1)). In terms of channel parameters, ρ = (Tχ)2 − (1− T )2. For a symmetric channel, the Heisenberg inequality requires that ρ ≥ 0 [11], therefore r is well defined for any attack that can be made symmetric. Finally, we are left with only one parameter, b4, such that b 4 < 1−b21 = χ/(1+χ), and the sign σ. For RR, the information Eve acquires is given by IBE = VXB |XE1 VPB |PE2 IBE is maximum when (VXB′ |XE1 + 1)(VPB′ |PE2 + 1) is minimum. For direct reconciliation (DR), for which the key is distilled from Alice’s data, the information Eve acquires is given by the Shannon formula: IAE = V + χXE1 1 + χXE1 V + χPE2 1 + χPE2 We find that both mutual informations IAE and IBE have an extremum at b4 = σ 1− s21(2b21 − 1)− b21(s 1 − 1) s21 + 1 We check numerically that in the quantum regime de- fined by ǫ ≤ 2, this extremum is indeed the absolute maximum. For this value of b4, we compute Eve’s noise and conditional variance as functions of channel param- eters: = χPE2 ≡ χmin T (2− ǫ)2 2− 2T + T ǫ+ +1 (11) VXB′ |XE1 = VPB′ |PE2 ≡ V B′|E = V χE + 1 V + χE N0 (12) These expressions form the new bounds for direct and reverse reconciliated heterodyne protocols. As they are obtained for the same value of b4, any attack that reaches bound (11) (i.e. optimal for DR) also reaches bound (12) (i.e. optimal for RR). IV. APPLICATION TO THE HOMODYNE DETECTION PROTOCOL In this section, we will show that bound (1) on the homodyne protocol can also be derived from the Iwa- sawa symplectic decomposition. In the homodyne pro- tocol, Eve stores the quantum states of mode E1 and mode E2 in quantum memories, waiting for Bob’s mea- surement basis disclosure. After this, Eve can measure the same quadrature Q = X or P chosen by Bob on both modes. The information acquired by Eve in the RR homodyne protocol is deduced from the conditional vari- ance on Bob’s measurement knowing the quadrature Q of modes E1 and E2, which can be computed from the output covariance matrix γ: VQB′ |QE1 ,QE2 = det(γQ) det(γE) , (13) where γQ is the restriction of γ to the quadrature Q, and γE is the restriction of γ to the quadrature Q of Eve’s modes E1 and E2. By expanding the Iwasawa decompo- sition of the symplectic transformation S decribing the attack, and by using orthogonality properties of matrix B, we can express this conditional variance as: VQB′ |QE1 ,QE2 = TQ′(V χQ′ + 1) N0, (14) where Q′ = P or X is the quadrature not measured by Bob. This conditional variance coincides with the homo- dyne bound (1). It is important to note that contrary to the heterodyne conditional variance which depends on symplectic parameters r and b4 as shown by equa- tions (8), the homodyne conditional variance (14) only depends on channel parameters, but no other symplectic parameter characterizing the attack The DR case is treated similarly, by considering the covariance matrix γAE that gathers the modulation value chosen by Alice (XA, PA) and modes E1 and E2 owned by Eve. By expanding the Iwasawa decomposition of S, we find VQA|QE1 ,QE2 = det(γAEQ ) det(γE) (V − 1)(1 + χQ′) V χQ′ + 1 N0, (15) which yields IAE = (V − 1)N0 VQA|QE1 ,QE2 V + χhom 1 + χhom with χhom = 1/χQ′. This expression matches the highest bound for the information acquired by Eve in the DR homodyne protocol established in [12]. It depends only on channel parameters, and not on the other symplectic parameters. Therefore, we have shown any attack against the DR or RR homodyne protocols 1. that do not mix quadratures X and P 2. that can be performed with two ancillary modes 3. in which all ancillas are initially vacuum states 4. in which Eve measures the same quadrature as Bob on all of her ancillary modes with the help of quan- tum memories is optimal for the channel parameters it can reproduce. In particular, the entangling cloner attack introduced in [1] and the assymetric cloning attack studied in [9] are optimal homodyne attacks ; for channels with no ex- cess noise (ǫ = 0), the beam-splitting attack is optimal. Equations (14) and (15) show that the optimality of any attack that verify conditions 1–4 holds even for attacks yielding dissymmetric channel parameters (i.e. TX 6= TP and χX 6= χP ). In fact, conditions 1–3 do not hamper the generality of the attacks we consider. Indeed, condition 1. is not restrictive as one can reduce any symplectic matrix to the block diagonal form (5) by means of local Gaussian operations [13]. We will see in section VII that numerical simulations show that condition 2. is in fact not neces- sary. Finally, condition 3. is also not restrictive because one can include the preparation of a non-vacuum initial state from vacuum states inside the symplectic transfor- mation describing the attack, eventually by making use of extra ancillary modes. The fact that the heterodyne attack scheme breaks condition 4. is the reason why, in general, the hetero- dyne bound (12) is higher than the homodyne bound (1), thus imposing more stringent constraints on Eve’s infor- mation. However, for some particular values of the chan- nel parameters, these two bounds coincide. Namely, this happens when χ = 1− T + T/V 2/T − 1/V for RR. For DR, bound (11) is equal to its homodyne counterpart = 1/χ when χ = 1− 1/T with T ≥ 1. For these channel parameters, an optimal heterodyne attack is also an optimal homodyne attack, but without the need for quantum memories, therefore lowering the technological requirements for the eavesdropper. We recall that like all the results presented in this pa- per, the optimality of any homodyne attack is to be un- derstood in the context of individual Gaussian attacks. However, since the homodyne bound (1) is proven secure against the larger class of individual and finite-size Gaus- sian and non-Gaussian attacks [2], we can say that any Gaussian individual attack that fulfills conditions 1–4 is optimal among that extended class of attacks. Security proofs of the homodyne protocol against collective at- tacks require the use of the Holevo entropy [3, 4], then the results presented here do not apply to this general class of attacks. V. PROOF BASED ON SYMPLECTIC INVARIANTS It is possible to derive the heterodyne bound (12) from another technique that does not require the Iwa- sawa decomposition. This technique is based on the fact that the output covariance matrix γ issues from some symplectic transformation S applied to the initial co- variance matrix γi. Since γi is diagonal with diagonal terms (V, 1, 1, V, 1, 1)N0, this property simply states that (V, 1, 1)N0 are the symplectic eigenvalues of the output covariance matrix γ. In other word, finding the best at- tack for RR amounts to minimizing the conditional vari- ance of Bob’s measurement knowing Eve’s measurement over the set of covariance matrices with symplectic eigen- values (V, 1, 1)N0. In terms of symplectic transforma- tions, Heisenberg relations on the three modes we con- sider require that all symplectic eigenvalues are greater than N0. Therefore, covariance matrices with eigenval- ues (V, 1, 1)N0 are covariance matrices that are compat- ible with Heisenberg relations and an input modulation of variance V N0. Since symplectic eigenvalues are usually hard to ex- press analytically, we will rather use symplectic invari- ants, which are totally equivalent to symplectic eigenval- ues. For a three mode state, there exist three symplectic invariants ∆j,3 with j = 1, 2, 3 defined as the sum of the determinant of all 2j × 2j sub-matrices of γ which diag- onal is on the diagonal of γ [14]. Applied to the input covariance matrix γi, these invariants read ∆1,3 = V 2 + 2 (16) ∆2,3 = 2V 2 + 1 (17) ∆3,3 = V 2 (18) We will now express the symplectic invariants as func- tions of the components of the output covariance matrix γ. For this purpose, we write this matrix as VB′ cm cn 0 0 0 cm VEm c 0 0 0 cn c VEn 0 0 0 0 0 0 VB′ cn cm 0 0 0 cn VEn c 0 0 0 cm c VEm N0 (19) where m stands for “measured” and n for “not mea- sured”, and VB′ = T (V + χ). This notation assumes that the attack does not mix quadratures X and P , and that swapping measurements of modes E1 and E2 would not change Eve’s information. The later assumption is backed by results of section III where we found that the optimal heterodyne attack yields to equal variances for X and P measurements. From equation (19) we can compute symplectic invariants as ∆1,3 = 2c 2 + V 2B′ + 4x+ 2y ∆2,3 = c 4 + 2c2V 2B′ + 4c 2x− 4cVB′x+ 4x2 − 2c2y +2V 2B′y + 4xy + y 2 − 4cz − 2VB′z ∆3,3 = (−2cx+ VB′(c2 − y) + z)2, where we introduced variables x = cmcn, y = VEmVEn, z = VEmc n + VEnc With these variables, Eve’s conditional variance yields VB′|E = VB′ − VB′ − z + σ′ z2 − 4yx2 N0, σ ′ = ±1 Since y > 0, we will only consider σ′ = 1 because it gives more information to Eve. Using the invariance of symplectic invariants, we univocally fix x and z 2(1− c2 − y) + V 2 − V 2B′ z = VB′y − V − c2(VB′ + c) + 2(1− y) + V 2 − V 2B′ as well as c, as a function of channel parameters V − VB′ . (20) Consequently, the heterodyne conditional variance VB′|E only depends on channel parameters and y. Then, we notice that y appears in the homodyne conditional vari- VB′|E1,E2 = y − c2 N0. (21) Therefore, the homodyne bound (1) contraints y by c2 ≤ y ≤ c2 + T (V χ+ 1) (22) We numerically find that VB′|E is a decreasing function of y, therefore the highest value for y must be considered to bound Eve’s information. In fact, using the results of section IV stating that any attack on the homodyne protocole is optimal (the covariance matrix (19) fulfills conditions 1–4), we can say that the only possible value for y is indeed c2+T (V χ+1). Now, VB′|E only depends on channel parameters, and we can check that it coincides with bound (12). In conclusion, we have shown another technique for de- riving bound (12). This technique is slightly less general than the Iwasawa decomposition because it assumes that the optimal attack respects the symmetry of the prob- lem. Furthermore, it does not cover the DR protocol. Yet, it enables to find bound (12) from more fundamen- tal Heisenberg-like properties. VI. OPTIMAL ATTACKS We shall now exhibit four optimal attacks against the heterodyne protocol with repect to bounds (11) and (12), depicted in Fig. 2. The existence of such optimal attacks show that the bounds we derived are tight: it is not pos- sible to further reduce the estimation Alice and Bob can make about Eve’s information. The first optimal attack we consider is the feed forward attack introduced in [5]. The symplectic matrix associated with this attack is 1 gE 0 0 1 0 0 0 1 1 0 0 0 −1√ 1−TE 0√ 0 0 1 and S is obtained from S by replacing the first line in the leftmost matrix by [1, 0,−gE]. Using equations (7) which link coefficients of the symplectic transformation matrix to channel parameters T and ǫ, we can see that to faithfully reproduce these channel parameters, Eve must choose: g2E = ǫT, TE = 4 ǫ(2− 2T + T ǫ)) (2 + T ǫ)2/T (2− ǫ) (2 + T ǫ) With these parameters, we can check from the compo- nents of S injected in equations (8) that this attack reaches bounds (11) and (12). Quantum teleportation is represented by the symplec- tic matrix 1 gE 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 SEPRX with SEPRX = 1 0 0 1 0 0 0 s−1 0 0 0 s Here, S is obtained by using [1, 0, gE] as the first line of the leftmost matrix, and changing s → 1/s. The second from left matrix simply swaps the 1st and 3rd modes to respect our mode order convention. Channel parameters fix s and gE : g2E = 2T, s 1− T + T ǫ− T ǫ(2− 2T + T ǫ) and noise computation shows that this attack is optimal. Then, the entangling cloner attack is represented by the symplectic matrix SecX = 1 0 0 1−TE 0√ 0 0 1 SEPRX To fake channel parameters, Eve must choose TE = T and s4 + 1 Once again, noise and conditional variance computations from the components of this matrix yield bounds (11) and (12). Finally, the cloning attack studied in [9] is also op- timal. This can be checked by verifying that the con- ditional variance of equation (44) in [9] coincides with bound (12). For this attacks, the authors show that in order to reproduce channel parameters, Eve must choose TE = T (1− ǫ/2) and G = 1− ǫ/2 For no excess noise (ǫ = 0), all these attacks are equiv- alent to beam-splitting attacks. VII. DISCUSSION We first discuss the generality of the model shown on Fig. 1, on which we built our proofs. This model assumes that Eve’s attack only involves two modes, but one can imagine that Eve could use and measure more modes, also carrying some information about Alice and Bob’s 0 0.2 0.4 0.6 0.8 1 β = 0.87 Previous heterodyne New heterodyne Homodyne 0 0.2 0.4 0.6 0.8 1 β = 1.0 Previous heterodyne New heterodyne Homodyne 0 0.05 0.1 0.15 0.2 FIG. 3: Effective information rate for typical experimental parameters: V = 11, ǫ = 0.02 and a perfect error correction β = 1 (top) or a constant reconciliation efficiency β = 0.87 (bottom). The new bound on heterodyne protocol provides more secret information than the homodyne protocol or the previous heterodyne bound. We can see from the bottom graph inlet that with these parameters, the new heterodyne provides secret information for every channel transmission. However, in practice, the reconciliation efficiency β drops as the distance rises, then limiting the range of the protocol. transmission. To tackle this problem, we can imagine that Eve concentrates all the information her modes bear into a single mode for each quadratures, by iterative con- structive interferences between her modes using beam- splitters. Since local operations using beam-splitters on Eve’s modes do not alter the conditional variance VB′|E , any attack on n modes for each quadrature X and P can be mapped to an equivalent attack, where Eve only needs to measure one mode for each quadrature. Therefore, it seems reasonable to assess that it is useless for Eve to introduce extra modes that in the end do not provide any information about Alice and Bob’s data. This tech- nique is illustrated in [9], where the authors consider the interference of the two modes owned by Eve in the assy- metric attack against the homodyne protocol, and show that this interference enables Eve to measure only one mode without loosing information. To back this argument, we performed numerical sim- ulations that give 2n modes to Eve, with n = 1, 2, 5. In these simulations, 107 attacks are tested by generat- ing random symplectic transformations parameterized by the Iwasawa decomposition. It shows that the two main results of this paper hold with more that two modes for Eve, namely that any attack using quantum memories on the homodyne protocol is optimal, and that the informa- tion Eve can get on the heterodyne protocol is bounded by (12). We complete our study by discussing practical ad- vantages of the heterodyne scheme over the homodyne scheme, when considering that a classical error correction with limited efficiency β has to be applied to experimen- tal data to obtain a secret key [15]. In this picture, the practical key rate becomes ∆Ieff = βIAB − IBE , result- ing in a bit loss of ∆I −∆Ieff = (1− β)IAB . Because for a given efficiency β the mutual information IAB of the heterodyne scheme is higher, this protocols suffers from greater key loss than the homodyne scheme. When con- sidering bound (1), this loss was rapidly fatal. However, with the new bound (12), we can see from Fig. 3 that the heterodyne scheme recovers its advantage. Still, there are two other practical drawbacks to the heterodyne protocol. First, for a given distance, the sig- nal to noise ratio (SNR) of the transmission is lower be- cause of Bob’s heterodyning beam-splitter. Since the rec- onciliation efficiency is an increasing function of the SNR, this effect lowers the final key rate. Because of this, both heterodyne and homodyne protocols feature an equiva- lent key rate. For example, for T = 0.25 (corresponding to 25 km), ǫ ≃ 0, 02 and V ≃ 11N0 [16], the homodyne scheme achieves β = 0.87 and the heterodyne scheme β = 0.80, both yielding to a few 0.01 bits per symbol. Second, Alice and Bob need to reconcile twice as much data as for the homodyne case. This effect also low- ers the final key rate when, as experimentally observed, computing speed limits the experimental repetition rate. However, on-going work on reconciliation at low SNR may take advantage of the high effective key rate of the heterodyne protocol. In conclusion, we have derived new bounds for individ- ual attacks on the direct and reverse reconciliated QKD protocols with heterodyne detection. These new bounds offer a higher secret key rate than previous bounds. We have shown that the feed-forward attack, the cloning attack, the quantum teleportation and the entangling cloner all achieve these bounds, then closing the gap be- tween best known bounds and best known attacks. On the other hand, the behaviour of these new bounds with respect to non-Gaussian [2] and collective attacks [3, 4] remains an open question. We thank Frédéric Grosshans, Raul Garćıa-Patrón and Nicolas Cerf for fruitfull discussions. We acknowledge support from the SECOQC European Integrated Project. J.L. acknowledges support from IFRAF. [1] F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. Cerf, and P. Grangier, Nature 421, 238 (2003). [2] F. Grosshans and N. J. Cerf, Phys. Rev. Lett. 92, 047905 (2004). [3] M. Navascues, F. Grosshans, and A. Aćın, Phys. Rev. Lett. 97, 190502 (2006). [4] R. Garcia-Patron and N. J. Cerf, Phys. Rev. Lett. 97, 190503 (2006). [5] C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, Phys. Rev. Lett. 93, 170504 (2004). [6] C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, Phys. Rev. A 73, 022316 (2006). [7] J. Lodewyck, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, Phys. Rev. A 72, 050303(R) (2005). [8] F. Grosshans, N.Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Quantum Inf. Comput. 3, 535 (2003). [9] R. Namiki, M. Koashi, and N. Imoto, Phys. Rev. A 73, 032302 (2006). [10] Arvind, B. Dutta, N. Mukunda, and R.Simon, e-print quant-ph/9509002 (1995). [11] C. M. Caves, Phys. Rev. D 26, 1817 (1982). [12] F. Grosshans and P. Grangier, Phys. Rev. Let. 88, 057902 (2002). [13] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). [14] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 73, 032345 (2006). [15] M. Bloch, A. Thangaraj, and S. W. McLaughlin (2005), e-print cs.IT/0509041. [16] A higher modulation variance V could be used to in- crease the SNR, thus compensating for the SNR decrease due to Bob’s heterodyne measurement. However, a higher modulation variance also increases the information IBE . Then, the effective secret rate ∆Ieff features a maximum for a certain value of V , which turns out to be compara- ble for both heterodyne and homodyne protocols for the given channel parameters.

0704.1372

Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

GAUSSIAN ESTIMATES FOR FUNDAMENTAL SOLUTIONS OF SECOND ORDER PARABOLIC SYSTEMS WITH TIME-INDEPENDENT COEFFICIENTS SEICK KIM Abstract. Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on Rn. In particular, in the case when n = 2 they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations. 1. Introduction In 1967, Aronson [1] proved Gaussian upper and lower bounds for the fundamen- tal solutions of parabolic equations in divergence form with bounded measurable coefficients. To establish the Gaussian lower bound Aronson made use of the Har- nack inequality for nonnegative solutions which was proved by Moser in 1964 (see [17]). Related to Moser’s parabolic Harnack inequality, we should mention Nash’s earlier paper [18] where the Hölder continuity of weak solutions to parabolic equa- tions in divergence form was established. In 1985, Fabes and Stroock [10] showed that the idea of Nash could be used to establish a Gaussian upper and lower bound on the fundamental solution. They showed that actually such Gaussian estimates could be used to prove Moser’s Harnack inequality. We note that Aronson also ob- tained Gaussian upper bound estimates of the fundamental solution without using Moser’s Harnack inequality. In [2], Auscher proposed a new proof of Aronson’s Gaussian upper bound esti- mates for the fundamental solution of second order parabolic equations with time- independent coefficients. His method relies crucially on the assumption that the coefficients are time-independent and thus it does not exactly reproduce Aronson’s result, which is valid even for the time-dependent coefficients case. However, his method is interesting in the sense that it carries over to equations with complex coefficients provided that the complex coefficients are a small perturbation of real coefficients. Along with this direction, Auscher, McIntosh and Tchamitchian also showed that the heat kernel of second order elliptic operators in divergence form with complex bounded measurable coefficients in the two dimensional space has a Gaussian upper bound (see [3] and also [5]). We would like to point out that a parabolic equation with complex coefficients is, in fact, a special case of a system of parabolic equations. From this point of 2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35K40. Key words and phrases. Gaussian estimates, a priori estimates, parabolic system. http://arxiv.org/abs/0704.1372v1 2 SEICK KIM view, Hofmann and the author showed that the fundamental solution of a parabolic system has an upper Gaussian bound if the system is a small perturbation of a di- agonal system, which, in particular, generalized the result of Auscher mentioned above to the time-dependent coefficients case (see [12]). However, the above men- tioned result of Auscher, McIntosh and Tchamitchian regarding the heat kernel of two dimensional elliptic operators with complex coefficients does not follow directly from our result. One of the main goals of this article is to provide a proof that weak solutions of the parabolic system of divergence type with time-independent coefficients associ- ated to an elliptic system in two dimensions enjoy the parabolic local boundedness property and to show that its fundamental solution has a Gaussian upper bound. More generally, we show that if weak solutions of an elliptic system satisfy Hölder estimates at every scale, then weak solutions of the corresponding parabolic system with time-independent coefficients also satisfies similar parabolic Hölder estimates from which, in particular, the parabolic local boundedness property follows easily. Also, such an argument allows one to derive Hölder continuity estimates for weak solutions of parabolic equations with time-independent coefficients directly from De Giorgi’s theorem [7] on elliptic equations, bypassing Moser’s parabolic Harnack inequality. In fact, this is what Auscher really proved in the setting of complex coefficients equations by using a functional calculus method (see [2] and also [4], [5]). Even in those complex coefficients settings, we believe that our approach is much more straightforward and thus appeals to wider readership. Finally, we would like to point out that in this article, we are mainly interested in global estimates and that we do not attempt to treat, for example, the systems with lower order terms, etc. However, let us also mention that, with some extra technical details, our methods carry over to those cases as well as to the systems of higher order; see e.g. [4], [5] for the details, and also Remark 3.5. The remaining sections are organized in the following way. In Section 2 we give notations, definitions, and some known facts. We state the main results in Section 3 and give the proofs in Section 4. 2. Notation and definitions 2.1. Geometric notation. (1) Rn = n-dimensional real Euclidean space. (2) x = (x1, · · · , xn) is an arbitrary point of R (3) X = (x, t) denotes an arbitrary point in Rn+1, where x ∈ Rn and t ∈ R. (4) Br(x) = {y ∈ R n : |y − x| < r} is an open ball in Rn with center x and radius r > 0. We sometimes drop the reference point x and write Br for Br(x) if there is no danger of confusion. (5) Qr(X) = (y, s) ∈ Rn+1 : |y − x| < r and t− r2 < s < t . We sometimes drop the reference point X and write Qr for Qr(X). (6) Q∗r(X) = (y, s) ∈ Rn+1 : |y − x| < r and t < s < t+ r2 (7) Qr,s(X) = {(y, s) ∈ Qr(X)}; i.e., Qr,s(X) = Br(x) × {s} if s ∈ (t − r 2, t) and Qr,s(X) = ∅ otherwise. We sometimes drop the reference point X and write Qr,s for Qr,s(X). (8) For a cylinder Q = Ω× (a, b) ⊂ Rn+1, ∂PQ denotes its parabolic boundary, namely, ∂PQ = ∂Ω × (a, b) ∪ Ω × {a}, where ∂Ω is the usual topological boundary of Ω ⊂ Rn and Ω is its closure. GAUSSIAN ESTIMATES 3 2.2. Notation for functions and their derivatives. (1) For a mapping from Ω ⊂ Rn to RN , we write f(x) = (f1(x), . . . , fN (x))T as a column vector. (2) fQ = f , where |Q| denotes the volume of Q. (3) ut = ∂u/∂t. (4) Dxiu = Diu = uxi = ∂u/∂xi. (5) Du = (ux1 , . . . , uxn) T is the spatial gradient of u = u(x, t). (6) For f = (f1, . . . , fN )T , Df = (Df1, . . . , DfN); that is Df is the n × N matrix whose i-th column is Df i. 2.3. Function spaces. (1) For Ω ⊂ Rn and p ≥ 1, Lp(Ω) denotes the space of functions with the following norms: ‖u‖Lp(Ω) = |u(x)| and ‖u‖L∞(Ω) = ess supΩ |u| . (2) Cµ(Ω) denotes the space of functions that are Hölder continuous with the exponent µ ∈ (0, 1], and [u]Cµ(Ω) = sup x 6=x′∈Ω |u(x)− u(x′)| |x− x′| µ < ∞. (3) The Morrey space M2,µ(Ω) is the set of all functions u ∈ L2(Ω) such that ‖u‖M2,µ(Ω) = sup Bρ(x)⊂Ω Bρ(x) (4) C P (Q) denotes the space of functions defined on Q ⊂ R n+1 such that [u]Cµ (Q) = sup X 6=X′∈Q |u(X)− u(X ′)| dP (X,X ′)µ where dP (X,X ′) = max |x− x′| , |t− t′| 2.4. Elliptic and parabolic systems and their adjoints. Definition 2.1. We say that the coefficients A ij (x) satisfy the uniform ellipticity condition if there exist numbers ν0,M0 > 0 such that for all x ∈ R n we have (2.1) αβ(x)ξβ , ξα ≥ ν0 |ξ| αβ(x)ξβ ,ηα ≤ M0 |ξ| |η| , where we used the following notation. (1) For α, β = 1, . . . , n, Aαβ(x) are N ×N matrices with (i, j)-entries A ij (x). (2) ξα = (ξ α, · · · , ξ T and |ξ| αβ(x)ξβ ,ηα α,β=1 i,j=1 ij (x)ξ We emphasize that we do not assume that the coefficients are symmetric. 4 SEICK KIM Definition 2.2. We say that a system of N equations on Rn α,β=1 Dxα(A ij (x)Dxβu j) = 0 (i = 1, . . . , N) is elliptic if the coefficients satisfy the uniform ellipticity condition. We often write the above system in a vector form (2.2) Lu := α,β=1 αβ(x)Dβu) = 0, u = (u 1 . . . , uN )T . The adjoint system of (2.2) is given by (2.3) L∗u := α,β=1 (Aαβ)∗(x)Dβu where (Aαβ)∗ = (Aβα)T , the transpose of Aβα. Definition 2.3. We say that a system of N equations on Rn+1 uit − α,β=1 Dxα(A ij (x)Dxβu j) = 0 (i = 1, . . . , N) is parabolic if the (time-independent) coefficients satisfy the uniform ellipticity condition. We often write the above system in a vector form (2.4) ut − Lu := ut − α,β=1 αβ(x)Dβu) = 0. The adjoint system of (2.4) is given by (2.5) ut + L ∗u := ut + α,β=1 (Aαβ)∗(x)Dβu where (Aαβ)∗ = (Aβα)T , the transpose of Aβα. 2.5. Weak solutions. In this article, the term “weak solution” is used in a rather abusive way. To avoid unnecessary technicalities, we may assume that all the coef- ficients involved are smooth so that all weak solutions are indeed classical solutions. However, this extra smoothness assumption will not be used quantitatively in our estimates. This is why we shall make clear the dependence of constants. (1) We say that u is a weak solution of (2.2) in Ω ⊂ Rn if u is a (classical) solution of (2.2) in Ω and u, Du ∈ L2(Ω). (2) We say that u is a weak solution of (2.4) in a cylinder Q = Ω×(a, b) ⊂ Rn+1 if u is a (classical) solution of (2.2) in Q and u, Du ∈ L2(Q), u(·, t) ∈ L2(Ω) for all a ≤ t ≤ b, and supa≤t≤b ‖u(·, t)‖L2(Ω) < ∞. 2.6. Fundamental solution. By a fundamental solution (or fundamental matrix) Γ(x, t; y) of the parabolic system (2.4) we mean anN×N matrix of functions defined for t > 0 which, as a function of (x, t), is a solution of (2.4) (i.e., each column is a solution of (2.4)), and is such that Γ(x, t; y)f(y) dy = f(x)(2.6) GAUSSIAN ESTIMATES 5 for any bounded continuous function f = (f1, . . . , fN)T , where Γ(x, t; y)f (y) de- notes the usual matrix multiplication. 2.7. Notation for estimates. We employ the letter C to denote a universal con- stant usually depending on the dimension and ellipticity constants. It should be understood that C may vary from line to line. We sometimes write C = C(α, β, . . .) to emphasize the dependence on the prescribed quantities α, β, . . .. 2.8. Some preliminary results and known facts. Lemma 2.4 (Energy estimates). Let u be a weak solution of (2.4) in QR = QR(X). Then for 0 < r < R, we have t−r2≤s≤t |u(·, s)| (R− r)2 Proof. See e.g., [14, Lemma 2.1, p. 139]. � Lemma 2.5 (Parabolic Poincaré inequality). Let u be a weak solution of (2.4) in QR = QR(X). Then there is some constant C = C(n,M0) such that |u− uQR | ≤ CR2 Proof. See e.g., [19, Lemma 3]. � Lemma 2.6. Let Q2R = Q2R(X0) be a cylinder in R n+1. Suppose u ∈ L2(Q2R) and there are positive constants µ ≤ 1 and M such that for any X ∈ QR and any r ∈ (0, R) we have Qr(X) ∣u− uQr(X) ≤ M2rn+2+2µ. Then u is Hölder continuous in QR with the exponent µ and [u]Cµ (QR) ≤ C(n, µ)M . Proof. See e.g., [15, Lemma 4.3, p. 50]. � Definition 2.7 (Local boundedness property). We say that the system (2.4) sat- isfies the local boundedness property for weak solutions if there is a constant M such that all weak solutions u of (2.4) in Q2r(X) satisfy the estimates Qr(X) |u| ≤ M |Q2r| Q2r(X) Similarly, we say that the adjoint system (2.5) satisfies the local boundedness prop- erty if the corresponding estimates hold for weak solutions u of (2.5) in Q∗2r(X). Theorem 2.8 (Theorem 1.1, [12]). Assume that the system (2.4) and its adjoint system (2.5) satisfy the local boundedness property for weak solutions. Then the fundamental solution of the system (2.4) has an upper bound (2.7) |Γ(x, t; y)|op ≤ C0t −n/2 exp k0 |x− y| where |Γ(x, t; y)|op denotes the operator norm of the fundamental matrix Γ(x, t; y). Here, C0 = C0(n, ν0,M0,M) and k0 = k0(ν0,M0). 6 SEICK KIM 3. Main results Definition 3.1. We say that an elliptic system (2.2) satisfies the Hölder estimates for weak solutions at every scale if there exist constants µ0 > 0 and H0 such that all weak solutions u of the system in B2r = B2r(x0) satisfy the following estimates (3.1) [u]Cµ0 (Br) ≤ H0r −(n/2+µ0) ‖u‖L2(B2r) . Similarly, we say that a parabolic system (2.4) satisfies Hölder estimates for weak solutions at every scale if there exist constants µ1 > 0 and H1 such that all weak solutions u of the system in Q2r = Q2r(X0) satisfy the following estimates (3.2) [u]Cµ1 ≤ H1r −(n/2+1+µ1) ‖u‖L2(Q2r) . Remark 3.2. Elliptic systems with constant coefficients satisfy the above property, and in that case, the ellipticity condition (2.1) can be weakened and replaced by the Legendre-Hadamard condition. De Giorgi’s theorem [7] states that the property is satisfied if N = 1. The property is also satisfied if n = 2 and it is due to Morrey (see Corollary 3.6). Some other examples include, for instance, a certain three dimensional elliptic system which was studied by Kang and the author in [13]. We shall prove the following main results in this paper: Theorem 3.3. If an elliptic system (2.2) satisfies the Hölder estimates for weak solutions at every scale, then the corresponding parabolic system (2.4) with time- independent coefficients also satisfies the Hölder estimates for weak solutions at every scale. Theorem 3.4. Suppose that the elliptic system (2.2) and its adjoint system (2.3) defined on Rn both satisfy the Hölder estimates for weak solutions at every scale with constants µ0, H0. Let Γ(x, t; y) be the fundamental solution of the parabolic system (2.4) with the time-independent coefficients associated to the elliptic system (2.2). Then Γ(x, t; y) has an upper bound (3.3) |Γ(x, t; y)|op ≤ C0t −n/2 exp k0 |x− y| where C0 = C0(n, ν0,M0, µ0, H0) and k0 = k0(ν0,M0). Here, |Γ(x, t; y)|op denotes the operator norm of fundamental matrix Γ(x, t; y). Remark 3.5. We would like to point out that (3.3) is a global estimate. Especially, the bound (3.3) holds for all time t > 0. Suppose that the elliptic system (2.2) and its adjoint system (2.3) enjoy the Hölder estimates for weak solutions up to a fixed scale R0; that is, there is a number R0 > 0 such that if u is a weak solution of either (2.2) or (2.3) in Br = Br(x) with 0 < r ≤ R0, then u is Hölder continuous and satisfies [u]Cµ0 (Br) ≤ H0r −(n/2+µ0) ‖u‖L2(B2r) . Then, the statement regarding the bound (3.3) for the fundamental solution should be localized as follows: For any given T > 0, there are constants k0 = k0(ν0,M0) and C0 = C0(n, ν0,M0, µ0, H0, R0, T ) such that (3.3) holds for 0 < t ≤ T . Corollary 3.6. Let Γ(x, t; y) be the fundamental solution of the parabolic system (2.4) with time-independent coefficients associated to an elliptic system (2.2) defined on R2. Then Γ(x, t; y) has an upper bound (3.3) with the constants C0, k0 depending only on the ellipticity constants ν0,M . GAUSSIAN ESTIMATES 7 Proof. First, let us recall the well known theorem of Morrey which states that any two dimensional elliptic system (2.2) with bounded measurable coefficients satisfies the Hölder estimates for weak solutions at every scale, with the constants µ0, H0 depending only on the ellipticity constants (see, [16, pp. 143–148]). Next, note that the ellipticity constants ν0,M0 in (2.1) remain unchanged for à ij (x) = A ji (x). Therefore, the corollary is an immediate consequence of Theorem 3.4. � Remark 3.7. In fact, the converse of Theorem 2.8 is also true (see [12, Theorem 1.2]). Therefore, in order to extend the above corollary to the parabolic system with time-dependent coefficients, one needs to show that the system satisfies the local boundedness property for weak solutions. Unfortunately, we do not know whether it is true or not if the coefficients are allowed to depend on the time variable. If n ≥ 3, it is not true in general, even for the time-independent coefficients case since there is a famous counter-example due to De Giorgi (see [8]). 4. Proof of Main Results 4.1. Some technical lemmas and proofs. Lemma 4.1. If u is a weak solution of the parabolic system with time-independent coefficients (2.4) in QR = QR(X0), then ut ∈ L 2(Qr) for r < R and satisfies the estimates (4.1) ‖ut‖L2(Qr) ≤ C(R − r) −1 ‖Du‖L2(QR) . In particular, if u is a weak solution of (2.4) in Q2r, then the above estimates together with the energy estimates yield (4.2) ‖ut‖L2(Qr) ≤ Cr −2 ‖u‖L2(Q2r) . Proof. We first note that if the coefficients are symmetric, (i.e., A ij = A ji ) this is a well known result; a proof for such a case is found, for example, in [14, pp. 172–181] or in [9, pp. 360–364]. However, the standard proof does not carry over to the non-symmetric coefficients case and for that reason, we provide a self-contained proof here. Fix positive numbers σ, τ such that σ < τ ≤ R. Let ζ be a smooth cut-off function such that ζ ≡ 1 in Qσ, vanishes near ∂PQτ , and satisfies 0 ≤ ζ ≤ 1 and |ζt|+ |Dζ| ≤ C(τ − σ)−2. Note that on each slice Qτ,s, we have ut −Dα(A αβDβu) · ζ2ut ζ2 |ut| αβDβu, Dαut αβDβu, Dαζut 8 SEICK KIM Therefore, we find by using the Cauchy-Schwarz inequality that ζ2 |ut| ζ2 |Du| |Dut|+ C ζ |Du| |Dζ| |ut| ζ2 |Dut| ζ2 |Du| ζ2 |ut| Thus we have (4.3) ζ2 |ut| ζ2 |Dut| ζ2 |Du| Since ut also satisfies (2.4), the energy estimates yield (4.4) ζ2 |Dut| (τ − σ)2 This is the part where we exploit the assumption that the coefficients are time- independent. Combining (4.3) and (4.4), we have (τ − σ)2 (τ − σ)2 If we set ǫ = (τ − σ)2/2C0, we finally obtain (τ − σ)2 Here, we emphasize that C is a constant independent of σ, τ . Then by a standard iteration argument (see e.g. [11, Lemma 3.1, pp. 161]), we have (4.5) (R− r)2 for 0 < r < R. The proof is complete. � Lemma 4.2. If u is a weak solution of the parabolic system with time-independent coefficients (2.4) in Q2r = Q2r(X0), then Du(·, s),ut(·, s) ∈ L 2(Qr,s) for all s ∈ [t0 − r 2, t0], and satisfy the following estimates uniformly in s ∈ [t0 − r 2, t0]. ‖Du(·, s)‖L2(Qr,s) ≤ Cr −2 ‖u‖L2(Q2r) ,(4.6) ‖ut(·, s)‖L2(Qr,s) ≤ Cr −3 ‖u‖L2(Q2r) .(4.7) Proof. By the energy estimates applied to ut we obtain (4.8) sup t0−r2≤s≤t0 |ut(·, s)| Q3r/2 On the other hand, the estimates (4.5) and the energy estimates (this time, applied to u itself) yield Q3r/2 Q7r/4 .(4.9) Combining (4.8) and (4.9) together, we have the estimates (4.7). GAUSSIAN ESTIMATES 9 Next, assume that u is a weak solution of (2.4) in Q4r = Q4r(X0). Let ζ be a smooth cut-off function such that ζ ≡ 1 in Qr, vanishes near ∂PQ2r, and satisfies (4.10) 0 ≤ ζ ≤ 1 and |ζt|+ |Dζ| ≤ Cr−2. Note that on each slice Q2r,s, we have Q2r,s ut −Dα(A αβDβu) · ζ2u Q2r,s ζ2ut · u+ Q2r,s αβDβu, Dαu αβDβu, Dαζu Using the ellipticity condition and the Cauchy-Schwarz inequality, we find Q2r,s ζ2 |Du| Q2r,s ζ2 |ut| |u|+ C Q2r,s ζ |Du| |Dζ| |u| Q2r,s ζ2 |ut| Q2r,s ζ2 |u| Q2r,s Q2r,s ζ2 |Du| Then by (4.10), (4.7), and the energy estimates, for all s ∈ [t0 − r 2, t0], we have Q2r,s Q2r,s Q2r,s (4.11) If we set ǫ = r2, then the above estimates (4.11) now become from which the estimates (4.6) follows by a well known covering argument. � Lemma 4.3. Assume that the elliptic system (2.2) satisfies the Hölder estimates for weak solutions at every scale with constants µ0, H0. Let u be a weak solution of the inhomogeneous elliptic system (4.12) Dα(A αβ(x)Dβu) = f in B2 = B2(x0), where f belongs to the Morrey space M2,λ(B2) with λ ≥ 0. Then, for any γ ≥ 0 with γ < γ0 = min(λ + 4, n + 2µ0) (we may take γ = γ0 if γ0 < n) there exists C = C(n, ν0,M0, µ0, H0, λ, γ) such that u satisfies the following local estimates (4.13) Br(x) + rγ−2 ‖f‖ M2,λ(B2) uniformly for all x ∈ B1 = B1(x0) and 0 < r ≤ 1. Moreover, if γ < n, then u belongs to the Morrey space M2,γ(B1) and (4.14) ‖u‖M2,γ (B1) ≤ C ‖u‖L2(B2) + ‖Du‖L2(B2) + ‖f‖M2,λ(B2) 10 SEICK KIM Proof. First, we note that the property (3.1) implies that for all 0 < ρ < r and x ∈ Rn, we have Bρ(x) ≤ C ·H0 )n−2+2µ0 Br(x) In the light of the above observation, the estimates (4.13) is quite standard and is found, for example, in [11, Chapter 3]. Then, by Poincaré inequality we have (4.15) Br(x) ∣u− uBr(x) ≤ Crγ L2(B2) + ‖f‖ M2,λ(B2) uniformly for all x ∈ B1 = B1(0) and 0 < r ≤ 1. It is well known that if γ < n, then the estimates (4.15) yield (4.14) (see e.g. [11, Chapter 3]). � 4.2. Proof of Theorem 3.3. Let u be a weak solution of (2.4) in a cylinder Q4 = Q4(0). We rewrite (2.4) as Lu = ut. By Lemma 4.2, we find that ut(·, s) is in L2(Q2,s) and satisfies ‖ut(·, s)‖L2(Q2,s) ≤ C ‖u‖L2(Q4) for all − 4 ≤ s ≤ 0. Therefore, we may apply Lemma 4.3 with f = ut and λ = 0, and then apply Lemma 4.2 to find that for all x ∈ B1(0) and 0 < r ≤ 1, we have Br(x) |Du(·, s)| ≤ Crγ−2 ‖Du(·, s)‖ L2(Q2,s) + ‖ut(·, s)‖ L2(Q2,s) ≤ Crγ−2 ‖u‖ L2(Q4) uniformly in s ∈ [−4, 0] (4.16) for all γ < min(4, n+ 2µ0). By Lemma 2.5 and then by (4.16) we find that for all X = (x, t) ∈ Q1 and r ≤ 1 Qr(X) ∣u− uQr(X) ≤ Cr2 Br(x) |Du(y, s)| dy ds ≤ Cr2+γ ‖u‖ L2(Q4) (4.17) Note that if n ≤ 3, then we may write γ = n + 2µ for some µ > 0. In that case, (4.17) now reads (4.18) Qr(X) ∣u− uQr(X) ≤ Crn+2+2µ ‖u‖ L2(Q4) for all X ∈ Q1 and r ≤ 1. Therefore, if n ≤ 3, then Lemma 2.6 yields the estimates (4.19) [u]Cµ (Q1/2) ≤ C ‖u‖L2(Q4) . We have thus shown that in the case when n ≤ 3, any weak solution u of (2.4) in a cylinder Q4 = Q4(0) satisfies the above a priori estimates (4.19) provided that the associated elliptic system satisfies the Hölder estimates for weak solutions at every scale. The general case is recovered as follows. For given X0 = (x0, t0) and r > 0, let us consider the new system (4.20) ut − L̃u := ut − α,β=1 Dα(à αβ(x)Dβu) = 0, where Ãαβ(x) = Aαβ(x0 + rx). Note that the associated elliptic system L̃u = 0 also satisfies the Hölder estimates for weak solutions at every scale. Moreover, the ellipticity constants ν0,M0 remain the same for the new coefficients à αβ . Let u be GAUSSIAN ESTIMATES 11 a weak solution of (2.4) in Q4r(X0). Then ũ(X) = ũ(x, t) := u(x0 + rx, t0 + r is a weak solution of (4.20) in Q4(0) and thus ũ satisfies the estimates (4.19). By rescaling back to Q4r(X0), the estimates (4.19) become (4.21) [u]Cµ (Qr/2) ≤ Cr−(n/2+1+µ) ‖u‖L2(Q4r) . Thus, when n ≤ 3, the theorem now follows from a well known covering argument. In the case when n ≥ 4, we invoke a bootstrap argument. For the sake of simplicity, let us momentarily assume that 4 ≤ n ≤ 7. Let u be a weak solution of (2.4) in Q8 = Q8(0). Let us fix X0 = (x0, t0) ∈ Q2(0) and observe that ut also satisfies the system (2.4) in Q4(X0). Thus, by a similar argument that led to (4.16), we find that for all x ∈ B1(x0) and 0 < r ≤ 1 we have (4.22) Br(x) |Dut(·, s)| ≤ Crγ−2 ‖ut‖ L2(Q4(X0)) uniformly in s ∈ [t0 − 4, t0], for all γ < 4 (we may take γ = 4 if n > 4). Then, by (4.14) in Lemma 4.3, Lemma 4.1, and Lemma 4.2 we conclude that (4.23) ‖ut(·, s)‖M2,γ (B1(x0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [t0 − 4, t0]. Since the above estimates (4.23) hold for all X0 = (x0, t0) ∈ Q2(0), we find that, in particular, ut(·, s) belongs to M 2,γ(B2(0)) for all −4 ≤ s ≤ 0, and satisfies (4.24) ‖ut(·, s)‖M2,γ (B2(0)) ≤ C ‖u‖L2(Q8(0)) for all s ∈ [−4, 0], where we also used (4.7) of Lemma 4.2. The above estimates (4.24) for ut now allows us to invoke Lemma 4.3 with f = ut and λ = γ. Then, by Lemma 4.3 and Lemma 4.2, we find that for all x ∈ B1(0) and 0 < r ≤ 1, we have Br(x) |Du(·, s)| ≤ Crγ−2 ‖u‖ L2(Q8(0)) uniformly in s ∈ [−4, 0] for all γ < min(γ+4, n+2µ0). Since we assume that n ≤ 7, we may write γ = n+2µ for some µ > 0. By the exactly same argument we used in the case when n ≤ 3, we derive the estimates [u]Cµ (Q1/2) ≤ C ‖u‖L2(Q8) , and the theorem follows as before. Finally, if n ≥ 8, we repeat the above process; if u is a weak solution of (2.4) in Q16(0), then ut(·, s) is in M 2,γ(B1(0)) for all γ < 8 and so on. The process cannot go on indefinitely and it stops in k = [n/4] + 1 steps. The proof is complete. � 4.3. Proof of Theorem 3.4. The proof is based on Theorem 2.8, the proof of which, in turn, is found in [12]. By Theorem 2.8, we only need to establish the local boundedness property for weak solutions of the parabolic system (2.4) and for those of its adjoint system (2.5). From the hypothesis that the elliptic system (2.2) satisfies the Hölder estimates for weak solutions at every scale, we find, by Theorem 3.3, that the parabolic system (2.4) with the associated time-independent coefficients also satisfies the Hölder estimates for weak solutions at every scale; that is, there exist some constants µ > 0 and C, depending on the prescribed quantities, such that if u is a weak solution of (2.4) in Q4r(X), then it satisfies the estimates [u]Cµ (Q2r) ≤ Cr −(n/2+1+µ) ‖u‖L2(Q4r) . 12 SEICK KIM Let us fix Y ∈ Qr = Qr(X). Then, for all Z ∈ Qr(Y ) ⊂ Q2r(X), we have (4.25) |u(Y )| ≤ |u(Z)|+dP (Y, Z) µ · [u]Cµ (Q2r) ≤ |u(Z)|+Cr −(n/2+1) ‖u‖L2(Q4r) . By averaging (4.25) over Qr(Y ) with respect to Z, we derive (note |Qr| = Cr |u(Y )| ≤ Cr−(n+2) ‖u‖L1(Qr(Y )) + Cr −(n/2+1) ‖u‖L2(Q4r) . Since Y ∈ Qr is arbitrary, we find, by Hölder’s inequality, that u satisfies ‖u‖L∞(Qr) ≤ Cr −(n/2+1) ‖u‖L2(Q4r) for some constant C = C(n, ν0,M0, µ0, H0). To finish the proof, we also need to show that if u is a weak solution of the adjoint system (2.5) in Q∗4r = Q 4r(X), then it satisfies the local boundedness property (4.26) ‖u‖L∞(Q∗r) ≤ Cr−(n/2+1) ‖u‖L2(Q∗ The verification of (4.26) requires only a slight modification of the previous argu- ments (mostly, one needs to replace Qr by Q r and so on), but it is rather routine and we skip the details. � References 1. Aronson, D. G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967) 890–896. 2. Auscher, P. Regularity theorems and heat kernel for elliptic operators. J. London Math. Soc. (2) 54 (1996), no. 2, 284–296. 3. Auscher, P.; McIntosh, A.; Tchamitchian, Ph. Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152 (1998), no. 1, 22–73. 4. Auscher, P.; Qafsaoui, M. Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177 (2000), no. 2, 310–364. 5. Auscher, P.; Tchamitchian, Ph. Square root problem for divergence operators and related topics. Astérisque No. 249 (1998) 6. Davies, E. B. Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109 (1987), no. 2, 319–333. 7. De Giorgi, E. Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43. 8. De Giorgi, E. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. (Italian) Boll. Un. Mat. Ital. (4) 1 (1968), 135–137. 9. Evans, L. C. Partial differential equations. American Mathematical Society, Providence, RI, 1998. 10. Fabes, E. B.; Stroock, D. W. A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327–338. 11. Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press:Princeton, NJ, 1983. 12. Hofmann, S.; Kim, S. Gaussian estimates for fundamental solutions to certain parabolic systems. Publ. Mat. 48 (2004), 481–496. 13. Kang, K.; Kim, S. On the Hölder continuity of solutions of a certain system related to Maxwell’s equations. SIAM J. Math. Anal. 34 (2002), no. 1, 87–100 (electronic). 14. Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N. Linear and quasilinear equations of parabolic type. American Mathematical Society: Providence, RI, 1967. 15. Lieberman, G. M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996 16. Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Springer-Verlag New York, Inc., New York, 1966 17. Moser, J. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964) 101–134; Correction: Comm. Pure Appl. Math. 20 (1967) 231–236. GAUSSIAN ESTIMATES 13 18. Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931–954. 19. Struwe, M. On the Hölder continuity of bounded weak solutions of quasilinear parabolic sys- tems. Manuscripta Math. 35 (1981), no. 1-2, 125–145. Mathematics Department, University of Missouri, Columbia, Missouri 65211 E-mail address: [email protected] 1. Introduction 2. Notation and definitions 2.1. Geometric notation 2.2. Notation for functions and their derivatives 2.3. Function spaces 2.4. Elliptic and parabolic systems and their adjoints 2.5. Weak solutions 2.6. Fundamental solution 2.7. Notation for estimates 2.8. Some preliminary results and known facts 3. Main results 4. Proof of Main Results 4.1. Some technical lemmas and proofs 4.2. Proof of Theorem ?? 4.3. Proof of Theorem ?? References

0704.1373

A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations

appor t de r ech er ch e Thème COM INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations Laurent Burgy — Laurent Réveillère — Julia Lawall — Gilles Muller N° ???? Février 2007 http://arxiv.org/abs/0704.1373v1 Unité de recherche INRIA Futurs Parc Club Orsay Université, ZAC des Vignes, 4, rue Jacques Monod, 91893 ORSAY Cedex (France) Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 60 19 66 08 A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations Laurent Burgy , Laurent Réveillère , Julia Lawall , Gilles Muller Thème COM — Systèmes communicants Projets Phoenix et Obasco Rapport de recherche n° ???? — Février 2007 — 16 pages Abstract: The secure and robust functioning of a network relies on the defect-free implementation of network applications. As network protocols have become increasingly complex, however, hand-writing network message processing code has become increasingly error-prone. In this paper, we present a domain-specific language, Zebu, for describing protocol message formats and related processing constraints. From a Zebu specification, a compiler automatically generates stubs to be used by an application to parse network messages. Zebu is easy to use, as it builds on notations used in RFCs to describe protocol grammars. Zebu is also efficient, as the memory usage is tailored to application needs and message fragments can be specified to be processed on demand. Finally, Zebu-based applications are robust, as the Zebu compiler automatically checks specification consistency and generates parsing stubs that include validation of the message structure. Using a mutation analysis in the context of SIP and RTSP, we show that Zebu significantly improves application robustness. Key-words: Langage métier, protocoles réseau, analyze de mutation Une approche langage pour améliorer la robustesse de l’implémentation de protocoles réseaux applicatifs Résumé : Pour être sûr et robuste, le fonctionnement d’un réseau doit reposer sur des implémentations d’applications sans faille. Les protocoles réseau étant de plus en plus complexes, écrire manuellement le code qui prend en charge leurs messages devient de plus en plus difficile et sujet à erreurs. Dans ce papier, nous présent un langage métier, Zebu, pour décrire le format des messages d’un protocole réseau et les contraintes de traitement associées. D’une spécification Zebu, un compilateur génère automati- quement des talons à utiliser par une application pour l’analyze grammaticale de messages réseau. Zebu is simple d’usage, utilisant les mêmes notations que celles utilisées dans les RFCs pour décrire les grammaires de protocoles. Zebu est efficace, l’implantation mémoire étant calquée sur les besoins de l’application et les fragments du message pouvant être traités à la demande. Enfin, les applications basées sur Zebu sont robustes, le compilateur vérifiant la consistence de la spécification et les talons générés étant incluant la validation de la structure du message. En utilisant une analyze de mutation dans le contexte de SIP et RTSP, nous montrons que Zebu améliore de manière significative la robustesse des applications. Mots-clés : DSL, network protocols, mutation analysis A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations 3 Contents 1 Introduction 3 2 Issues in developing network protocol parsers 4 2.1 ABNF formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Hand-writing parsers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Using parser generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Integrating a parser with an application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Robust Parser Development with Zebu 7 3.1 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Annotating an ABNF specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 The Zebu compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Developing an application with Zebu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Experiments 10 4.1 Robustness evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Related Work 14 6 Conclusion 14 1 Introduction In the Internet era, many applications, ranging from instant messaging clients and multimedia players to HTTP servers and proxies, involve processing network protocol messages. A key part of this processing is to parse messages as they are received from the network. As message parsing represents the front line of interaction between the application and the outside world, the correctness of the parser is critical; any bugs can leave the application open to attack [24]. In the context of in-network application such as proxies, where achieving high throughput is essential, parsing must also be efficient. Implementing a correct and efficient network protocol message parser, however, is a difficult task. The syntax of network protocol messages is typically specified in a RFC (Request for Comments) using a variant of BNF known as ABNF (Augmented BNF ) [6]. Such a specification amounts to a state machine, which for efficiency is often implemented in an unstructured way using gotos. The resulting code is thus error-prone and difficult to maintain. Furthermore, some kinds of message processing may not use all fragments of the message. For example, a router normally only uses the header fields that describe the message destination, and ignores the header fields that describe properties of the message body [23]. It is thus desirable, for efficiency, to defer the parsing of certain message fragments to when their values are actually used. In this case, complex parsing code may end up scattered throughout the application. In the programming languages community, parsers have long been constructed using automated parser generators such as yacc [15]. Nevertheless, such tools are not suitable for generating parsers for network protocol messages, as the grammars provided in RFCs are often not context free, and such tools provide no support for deferring the parsing of some message fragments. Thus, parsers for network protocol messages have traditionally been implemented by hand. This situation, however, is becoming increasingly impractical, given the variety and complexity of protocols that are continually being developed. For example, the Gaim instant messaging client parses more than 10 different instant messaging protocols [10]. The message grammar in the IMAP RFC is about 500 lines of ABNF, and includes external references to others RFCs. SIP (Session Initiation Protocol) [26], which is mainly used in telephony over IP, has a multitude of variants and extensions, implying that SIP parsers must be tolerant of minor variations in the message structure and be extensible. Incorrect or inefficient parsing makes the application vulnerable to denial of service attacks, as illustrated by the “leading slash” vulnerability found in the Flash HTTP Web server [24]. In our experiments (Section 4), we have crashed the widely used SER parser [23] for SIP via a stream of 2416 incorrect messages, sent within 17 seconds. RR n° 0123456789 4 Burgy, Réveillère, Lawall & Muller To address the growing complexity of network protocol messages and the inadequacy of standard tools, some parser generators have recently been developed that specifically target the kinds of complex data layouts found in network protocol messages. These tools include DATASCRIPT [3] and PacketTypes [18] for binary protocols, and PADS [9], GAPA [5] and binpac [22] for both binary and text-based protocols. However, none of these approaches accepts ABNF as the input language, and thus, the RFC specification must be translated to another formalism, which is tedious and error prone. Furthermore, such approaches have mainly targeted application protocol analyzers, which parse a fixed portion of the message and then proceed to some analysis phase. Thus, they do not provide fine-grained control over the time when parsing occurs. While these approaches relieve some of the burden of implementing a network protocol message parser, there still remains a gap between these tools and the needs of applications. We propose to directly address the issues of correctness and efficiency at the parser generator level. To this end, we present a domain-specific language, Zebu, for describing HTTP-like text-based protocol message formats and related processing constraints. Zebu is an extension of ABNF, implying that the programmer can simply copy a network protocol message grammar from an RFC to begin developing a parser. Zebu extends ABNF with annotations indicating which message fields should be stored in data structures, and other semantic information, such as the type of the value expressed by a field, constraints on the range of its value, and whether certain fields are mandatory or optional. Fields can additionally be declared as lazy, which gives control over the time when the parsing of a field occurs. A Zebu specification is then processed by a compiler that generates stubs to be used by an application to process network messages. Based on the annotations, the Zebu compiler implements domain-specific optimizations to reduce the memory usage of a Zebu based application. Besides efficiency, Zebu also addresses robustness, as the compiler performs many consistency checks, and generates parsing stubs that validate the message structure. This paper In this paper, we present the Zebu language and an assessment of its performance and robustness in the context of the SIP and RTSP (Real Time Streaming Protocol [27]) network protocols. Our contributions are as follows: � We introduce a declarative language, named Zebu, for describing protocol message formats and related processing constraints. Zebu builds on the ABNF notation typically used in RFCs to describe protocol grammars. � We have defined a test methodology based on a mutation analysis for evaluating the robustness improve- ment induced by Zebu. � We have applied our test methodology existing and Zebu-based SIP and RTSP parsers. While the Zebu- based parsers reject 100% of the invalid mutated messages, none of the existing parsers that we have tested detects more than about 25% of the injected mutants. � Finally, we show that the added safety and robustness provided by Zebu does not significantly impact performance. Indeed, our performance evaluation shows that a Zebu-based parser can be as efficient on average as a hand-crafted one. The rest of this paper is organized as follows. Section 2 discusses specific characteristics of network protocol message parsing code, illustrating its inherent complexity. Section 3 introduces the Zebu language, and describes the verification of specifications and the generation of parsing stub functions. Section 4 assesses the robustness and performance of Zebu-based parsers. Section 5 described related work and Section 6 concludes. 2 Issues in developing network protocol parsers To illustrate the growing complexity of network protocol messages and the inadequacy of existing approaches to creating the associated parsers, we consider the SIP protocol [26]. The SIP message syntax is similar to that of other recent text-based protocols such as HTTP and RTSP. A SIP message begins with a line indicating whether the message is a request (including a protocol method name) or a response (including a return code). A sequence of required and optional headers then follows. Finally, a SIP message can include a body containing the payload. Widely used SIP parsers include that of the SIP Express Router (SER) [23] and the oSIP library [20] used e.g., in the open PBX Asterisk [29]. Both parsers are hand-written. INRIA A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations 5 Request-Line = Method SP Request-URI SP SIP-Version CRLF 1 Method = INVITEm / ACKm / OPTIONSm / BYEm 2 / CANCELm / REGISTERm 3 / extension-method 4 INVITEm = %x49.4E.56.49.54.45 ; INVITE in caps 5 Request-URI = SIP-URI / SIPS-URI / absoluteURI 6 SIP-Version = "SIP" "/" 1*DIGIT "." 1*DIGIT 7 extension-method = token 8 [...] CSeq = "CSeq" HCOLON 1*DIGIT LWS Method 10 LWS = [*WSP CRLF] 1*WSP ; linear whitespace 11 SWS = [LWS] ; sep whitespace 12 HCOLON = *( SP / HTAB ) ":" SWS 13 Figure 1: Extract of the ABNF of the message syntax from the SIP RFC 3261 We first present an extract of the ABNF specification of the SIP message grammar, and then describe the difficulty of hand-writing the corresponding parser. We next consider to what extent these difficulties are addressed by existing parser generation tools, and describe the issues involved in integrating a parser with a network application. 2.1 ABNF formalism An extract of the ABNF specification of the SIP message grammar is shown in Figure 1. Lines 1 to 8 define the structure of a request line, which appears at the beginning of a message, and lines 10 to 13 define the structure of the CSeq header field, which is used to identify the collection of messages making up a single transaction. An ABNF specification consists of a set of derivation rules, each defining a set of alternatives, separated by /. An alternative is a sequence of terminals and nonterminals. Among the terminals, a quoted string is case insensitive. Case sensitive strings must be specified as an explicit sequence of character codes, as in the INVITEm rule (line 5). ABNF includes a general form of repetition, n*m X, that indicates that at least n and at most m occurrences of the terminal or nonterminal X must be present. ABNF also defines shorthands such as n* for n*∞, *n for 0*n, * for 0*∞ and n for n*n. Therefore, 1*DIGIT in the CSeq rule (line 10) represents a sequence of digits of length at least 1. Brackets are used as a shorthand for 0*1. 2.2 Hand-writing parsers The specification of the CSeq header in Figure 1 amounts to only four lines of ABNF. However, implementing parsing based on such an ABNF specification efficiently in a general-purpose language such as C or C++ often requires many lines of code. For example, SER and oSIP contain about 200 and 340 lines of C and C++ code, respectively, specifically for parsing the CSeq header. This CSeq-specific code includes operations for reading individual characters from the message, operations for transitioning in a state machine according to the characters that are read, calls to various generic header parsing operations, and error checking code. Among the complexities encountered is the fact that, as shown in Figure 1, a CSeq header value can stretch over multiple lines if the continuation line begins with a space or horizontal tab (WSP). In addition to the constraints described by the ABNF specification, the parser developer has to take into account constraints on the message structure that are informally specified in the text of the RFC. For example, the CSeq header includes a CSeq number expressed as any sequence of at least one digit (1*DIGIT) and a CSeq method (Method). The SIP RFC states that the CSeq number must be an unsigned integer that is less than 231 and that the CSeq method must be the same as the method specified in the request line. However, existing hand-written implementations do not always check all these requirements. For example, oSIP converts the CSeq number to an integer without performing any verification. If the CSeq number contains any non-numeric characters, the result is a meaningless value. 2.3 Using parser generators PADS and binpac use a type-declaration like format for specifying message grammars, while GAPA uses a BNF-like format. Both of these formats require reorganizing the information in the ABNF specification. We take PADS as a concrete example. Figure 2 shows a PADS specification corresponding to the four lines of ABNF describing the CSeq header. This specification is in the spirit of the HTTP specification provided by the PADS developers [21]. RR n° 0123456789 6 Burgy, Réveillère, Lawall & Muller bool chkCseqMethod (request_line_t r, Cseq_t c) { 1 return ( r.method == c.method ); 2 Ptypedef Puint16_FW(:3:) Cseq_number_t : 5 CSeq_t x => { 100 <= x && x < 699 }; 6 Pstruct wsp_crlf_t { 8 PString_ME(:"(\\s|\\t)* \\r\\n":) wsp; 9 }; 10 POpt wsp_crlf_t o_wsp_crlf_t; 12 Pstruct lws_t { 14 o_wsp_crlf_t wsp_crlf; 15 PString_ME(:"(\\s|\\t)+":) wsp; 16 }; 17 POpt lws_t sws_t; 19 Pstruct hcolon_t { 21 PString_ME(:"(\\s|\\t)*":) sp_or_htab; 22 ’:’; sws_t sws; 23 }; 24 Pstruct CSeq_t { 26 PString_ME(:"[Cc][Ss][Ee][Qq]":) name; 27 hcolon_t hcolon; 28 CSeq_number_t number; 29 lws_t lws; 30 method_t method; 31 }; 32 Precord Pstruct SIP_msg { 34 request_line_t request_line; 35 [...] CSeq_t cseq: checkCSeqMethod(request_line,cseq); 37 [...] }; 39 Figure 2: PADS specification of the SIP RFC 3261 A PADS specification describes both the grammar and the data structures that will contain the result of parsing the message. Thus, the rules of the ABNF specification are translated into what amount to structure declarations in PADS. As a PADS structure must be declared before it is used, the rule ordering is often forced to be different than that of the ABNF specification. For example, in the ABNF specification, the CSeq nonterminal is defined before the LWS, SWS, and HCOLON nonterminals, while in the PADS specification, the structure corresponding to the CSeq nonterminal is defined afterwards (line 26). PADS also does not implement the same default parsing strategies as ABNF, and thus e.g., case insensitive strings must be specified explicitly using regular expressions (line 27). Similarly, translating SIP whitespace into PADS requires writing many lines of specifications (lines 8-19), including regular expressions. Finally, the PADS specification must express the various constraints contained in the RFC text. Although PADS allows the developer to define constrained types (lines 5 to 6), which are used here in the case of the CSeq number (line 29), non-type constraints such as the relationship between the method mentioned in the request line and the method mentioned in the CSeq header must be implemented by arbitrary C code (lines 1 to 2 and line 37). Of these issues, probably the most difficult for the programmer is to convert ABNF specifications to regular expressions. Regular expressions for even simple ABNF specifications are often complex and voluminous. For example, a regular expression for a URI has been published that is 45 lines of code [1]. While a tool has been developed to convert an ABNF specification to a regular expression [1], in the PADS, GAPA, and binpac specifications that we have seen, the regular expressions appear to have been written by hand, and sometimes do not capture all of the constraints specified by the RFC. 2.4 Integrating a parser with an application The ease of integrating a parser with an application depends on whether the parser parses the fields needed by the application, and whether the result of this parsing is stored in appropriate data structures. We consider the issues that arise when using the handwritten oSIP and SER parsers, and when using a parser generated by a tool such as PADS. INRIA A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations 7 oSIP parses the fixed set of required SIP header fields, and separates the rest of the message into pairs of a header field name and the corresponding raw unparsed data. Applications that do not use all of the information in the required header fields incur the time cost of parsing this information and the space cost of storing the result (see Section 4). Applications that use the many SIP extensions must parse these header fields themselves. The former increases the application time and space requirements, which can be critical in the case of in-network applications such as proxies, while the latter leaves the application developer on his own to develop complex parsing code. SER provides more fine-grained parsing than oSIP, as it parses only those header fields that are requested by the application. By default, however, SER only gives direct access to the top-level subfields of a header, such as the complete URI. To extract, e.g., only the host portion of a URI, the programmer must intervene. One approach is for an application built using SER to reparse the subfield, to obtain the desired information. SER applications are written using a domain-specific language targeted towards routing, which does not provide string-matching facilities. Nevertheless, SER provides an escape from this language, allowing a SER application to invoke an arbitrary shell script. A SER application can thus invoke a script written in a language such as Perl to extract the desired information. This approach, however, incurs a high performance penalty for forking a new process, as we show in Section 4, and compromises the safety benefits of using the SER language. Another approach is to use the SER extension framework, which, à la Apache [2], allows integrating new modules into the parsing process. Although efficient, this approach requires the programmer to write low level C code that conforms to rather contorted requirements. Again, incorrect behavior inside a module may compromise the robustness of the whole application. Finally, parser generators such as PADS allow the developer to construct the parser such that it parses only as much of the message as is needed. However, the generated data structures directly follow the specified parsing rules, implying that accessing message fields often requires long chains of structure field references. Furthermore, all of the parsed data is stored, which increases the memory footprint. 3 Robust Parser Development with Zebu We now present the Zebu language for describing HTTP-like text-based protocol message formats and related processing constraints. Zebu is based on ABNF, as found in RFCs, and extends it with annotations indicating which message fields should be stored in data structures and other semantic attributes. These annotations ex- press both constraints derived from the protocol RFC and constraints that are specific to the target application. From a Zebu specification, a compiler automatically generates stubs to be used by the application to process network messages. The features of Zebu are driven by the kinds of information that an application may want to extract from a network protocol message. We first consider the features that are needed to do this processing robustly and efficiently, and then present the corresponding annotations that the programmer must add to the ABNF specification so that the Zebu compiler can generate the appropriate stub functions. Finally, we describe the Zebu compiler, which performs both verification and code generation, and the process of constructing an application with Zebu. 3.1 Issues A HTTP-like text-based network message consists of a command line, a collection of header fields, and a message body. The command line indicates whether the message is a request or a response, and identifies basic information such as the version of the protocol and the method of a request message. A header field specifies a protocol-specific key and an associated value, which may be composed of a number of subfields. Finally, the message body consists of free text whose structure is typically not specified by the protocol. Thus, decomposing it further falls out of the scope of Zebu. From the contents of a message, an application may need to determine whether the message is a request or a response, to detect the presence of a particular header field, or to extract command line or header field subfields. Each of these operations involves retrieving a command line or header field, and potentially accessing its contents. In a HTTP-like text-based protocol, each command line or header field normally occupies one or more complete lines, where each line after the first begins with a special continuation character. Thus, as exemplified by the very efficient SIP parser SER, a parser can be constructed in two levels: a top-level parser that simply scans each line of the message until it reaches the desired command line or header field, and a collection of dedicated parsers that process each type of command line or header field. The dedicated parsers must respect RR n° 0123456789 8 Burgy, Réveillère, Lawall & Muller message sip3261 { 1 request { 2 ; Request only 3 requestLine = Method:method SP Request-URI:uri SP SIP-Version 4 ; Constraints that apply only for the CSeq of a request 6 header CSeq { CSeq.method == requestLine.method } 7 ; Constraints that apply only for the Max-Forwards of a request 9 header Max-Forwards { mandatory } 10 response { 14 ; Response only 15 statusLine = SIP-Version SP Status-Code:code SP Reason-Phrase:rphrase 16 [...] 17 enum Method = INVITEm / ACKm / OPTIONSm / BYEm / CANCELm / REGISTERm / extension-method 20 extension-method = token 21 INVITEm = %x49.4E.56.49.54.45 ; INVITE in caps 22 [...] 23 struct Request-URI = SIP-URI / SIPS-URI / absoluteURI { lazy } 25 [...] 26 uint16 Status-Code = Informational / Redirection / Success / Client-Error / Server-Error 28 / Global-Failure / extension-code 29 uint16 Global-Failure = "600" ; Busy Everywhere 30 / "603" ; Decline 31 / "604" ; Does not exist anywhere 32 / "606" ; Not Acceptable 33 uint16 extension-code = 3DIGIT { extension-code >= 100 && extension-code <= 699 } 34 [...] 35 ; Header CSeq 37 header CSeq = 1*DIGIT:number as uint32 LWS Method:method 38 ; Header Max-Forwards 40 header Max-Forwards = 1*DIGIT:value as uint32 { mandatory } 41 ; Header To 43 header To { "to" / "t" } = ( name-addr / addr-spec:uri ) *( SEMI to-param ) { mandatory 44 name-addr = [ display-name ] LAQUOT addr-spec:uri RAQUOT 45 struct addr-spec = SIP-URI / SIPS-URI / absoluteURI { lazy } 46 [...] 48 Figure 3: Excerpt of the Zebu Specification for the SIP protocol both the ABNF specification and any constraints specified informally in the RFC. To avoid reparsing already parsed message elements for each requested parsing operation, the parser should save all parsed data in data structures for later use, ideally in the format desired by the application. This analysis suggests that to enable the Zebu compiler to generate a useful and efficient parser, the pro- grammer must annotate the ABNF specification obtained from an RFC with the following information: (1) An indication of the nonterminal representing the entry point for parsing each possible command line and header field. (2) A specification of any constraints on the message structure that are informally described by the RFC. (3) An indication of the message subfields that will be used by the application. The first two kinds of annotations are generic to the protocol, and can thus be reused in generating parsers for multiple applications. The third kind of annotation is application-specific. This kind of annotation can be viewed as a simplified form of the action that can be specified when using yacc and other similar parser generators, in that it allows the programmer to customize the memory layout used by the parser to the specific needs of the application. 3.2 Annotating an ABNF specification We present the three kinds of annotations required by Zebu, using as an example an extract of the Zebu specification of a SIP parser, as shown in Figure 3. Parser entry points The Zebu programmer annotates the rule for parsing the command line of a request message with requestLine, the rule for parsing the command line of a response message with statusLine, and the rules for parsing each kind of header with header. Because a command line or header field cannot contain INRIA A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations 9 another command line or header field, the nonterminals for these lines are no longer useful. In the case of a command line, the nonterminal is simply dropped. Thus, for example, the ABNF rule for the Request-Line nonterminal (Figure 1, line 1) is transformed into the following Zebu rule (c.f., Figure 3, line 4): requestLine = Method SP Request-URI SP SIP-Version In the case of a header field, the description of the key is moved from the right-hand side of the rule to the left, where it replaces the nonterminal, resulting in a rule whose structure is suggestive of a key-value pair. For example, the ABNF CSeq rule on line 10 of Figure 1 is reorganized into the following Zebu rule (c.f., Figure 3, line 38) header CSeq = 1*DIGIT LWS Method (The delimiter HCOLON is also dropped, as it is a constant of the protocol). Some header fields, such as the SIP To header field, can be represented by any of a set of keys. In this case, the header is given a name, which is followed by the ABNF specification of the possible variants, in braces, as shown in line 44 of Figure 3. As in ABNF, the matching of the header key, and any other string specified by a Zebu grammar, is case insensitive. RFC constraints The text of the RFC for a protocol typically indicates how often certain header fields may appear, whether header fields can be modified, and various constraints on the values of the header subfields. The Zebu programmer must annotate the corresponding ABNF rules with these constraints. Constraints are specified in braces at the end of a grammar rule. Possible atomic constraints are that a header field is mandatory (mandatory) and that a header field can appear more than once in a message (multiple). For example, in the SIP specification, the header To is specified to be mandatory and read-only (line 44). More complex constraints can be expressed using C-like boolean expressions. For example, in Section 2.2, we noted that in a request message, the method mentioned in the command line must be the same as the method mentioned in the CSeq header. This constraint is described in line 7. Some constraints on header fields are specific to either request or response messages. Accordingly, the Zebu programmer must group the request line and its associated constraints in a request block, and the status line and its associated constraints in a response block. In the case of SIP, the request block (lines 2-12) indicates that for the CSeq header the method must be the same as the method in the request line (line 7), and that the Max-Forwards header is mandatory (line 10). The constraints in the response block (lines 14-18) have been elided. Subfields used by the application The parsing functions generated by the Zebu compiler create a data structure for each command line or header field that is parsed. By default, this data structure contains only the type of the command line or header field and a pointer to its starting point in the message text. When the application will use a certain subfield of the command line or message header, the Zebu programmer can annotate the nonterminal deriving this subfield with an identifier name. This annotation causes the Zebu compiler to create a corresponding entry in the enclosing command line or header field data structure. For example, in line 4, the Zebu programmer has indicated that the application needs to use the method in the command line, which is given the name method, and the URI, which is given the name uri. By default, a subfield is just represented as a pointer to the start of its value in the message text. This is the case of method and uri in our example. Often, however, the application will need to use the value in some other form, such as an integer. The Zebu programmer can additionally specify a type for a named value, either at the nonterminal reference or at its definition. For example, in line 38 the CSeq number is specified as being a uint32. Nonterminals can also be specified as structures (struct), unions (union), and enumerations (enum). A structure collects all derived named subfields. As illustrated in the case of Request-URI (line 25), a structure may even be used in the case of an alternation, when the application does not need to know from what element of the alternation a named entry is derived. A union, in contrast, records which alternation was matched and in each case only includes subfields derived from the given alternation. Finally, an enumeration is a special case of union in which the only information that is recorded is the identity of the matching alternation; the matched data is not stored. In line 20, for example, Method is specified as being an enumeration, because the application only needs to know whether the method of the message is one of the standard ones or an extension method, but does not need to know the identity of the extension method in the latter case. An application may use the information in certain subfields only in some exceptional cases. The Zebu constraint lazy allows the programmer to specify that a specific subfield should not be parsed until requested by the application. For example, in the SIP specification, Request-URI has this annotation (line 25). RR n° 0123456789 10 Burgy, Réveillère, Lawall & Muller 3.3 The Zebu compiler The Zebu compiler verifies the consistency of the ABNF specification and the annotations added by the pro- grammer, and then generates stub functions allowing an application to parse the command line and header fields and access information about the parsed data. The Zebu compiler is around 3700 lines of OCaml code. A run-time environment defining various utility functions is also provided, and amounts to around 700 lines of C code. Verifications Although RFCs are widely published and form the de facto standard for many protocols, we have found some errors in RFC ABNF specifications. These are simple errors, such as typographical errors, but still they complicate the process of translating an ABNF specification into code, whether done by hand or using a parser generator. The Zebu compiler thus checks basic consistency properties of the ABNF specification: that there is no omission (i.e., each referenced rule is defined), that there is no double definition, and that there are no cycles. Additionally, the annotations provided by the Zebu programmer must be consistent with the ABNF specifica- tion. For example, in line 30, the nonterminal Global-Failure is annotated with uint16. This non-terminal is specified to be an alternation of strings, and thus the Zebu compiler checks that each element of this alternation represents an unsigned integer that is less than 216. Code generation An application does not use the data structures declared in a Zebu specification directly, but instead uses stub functions generated by the Zebu compiler. The use of stub functions allows parsing to be carried out lazily, so that only as much data is parsed as is needed to fulfill the request of a given stub function call. As illustrated in Figure 4a, stub functions are generated for determining the type of a message (request or response), for parsing the command line and the various headers, for accessing individual header subfields, and for managing the parsing of subfields designated as lazy. The names of these stub functions depend on the specific structure of the grammar, but follow a well-defined schema that facilitates their use by the application developer. The parsing functions generated by the Zebu compiler use the two-level parsing strategy described in Sec- tion 3.1. Header-specific parsers use the PCRE [12] library for matching the regular expression of a header value that has been derived from the ABNF specification. The parsing functions contain run-time assertions that check the constraints specified in the RFC. Once a header is parsed and checked, its named subfields, if any, are converted to the specified types and stored in the data structure associated with the header. The values of the named subfields can then be accessed using the “get” stub functions. 3.4 Developing an application with Zebu The developer defines the application logic as an ordinary C program, using the stub functions to access information about the message content. Figure 4b illustrates the implementation of an application that extracts the host information from the URI stored in the From header field of an INVITE message. This kind of operation is useful in, e.g. an intrusion detection system, which searches for certain patterns of information in network messages. The application uses the stubs generated from the SIP message grammar specification to access the re- quired information. The application initially uses the functions sip3261_Method_getType and sip3261_- RequestLine_getMethod to determine whether the current message is an INVITE request (line 6). If so, it uses the function sip3261_parse_headers to parse the From header field (line 8), and then the functions sip3261_- header_From_getUri and sip3261_get_header_From to extract the URI (line 9). Line 46 of the Zebu SIP specification indicates that the parsing of the URI should be lazy, so the function sip3261_Lazy_Addr_spec_- getParsed is used to force the parsing of this subfield (line 10). After a check that the host name is present (line 13), its value is extracted using the function sip3261_Option_Str_getVal in line 14. Overall, due to the annotations in the Zebu specification, stub functions are available to access exactly the message fragments needed by the application. Similarly, memory usage is limited to the application’s declared needs. 4 Experiments A robust network application must accept valid messages, to provide continuous service, and reject invalid network messages, to avoid corrupting its internal state. As the parser is the front-line in the treatment of INRIA A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations11 // Init extern sip3261 sip3261_init(); // Top level parser extern void sip3261_parse(sip3261, char *, int); // Generic parser for headers extern void sip3261_parse_headers(sip3261, E_Headers); // Dedicated parser for addr-spec extern void sip3261_parse_addr_spec(T_Lazy_addr_spec); // Accessors extern T_bool sip3261_isRequest(sip3261); extern T_bool sip3261_isResponse(sip3261); extern T_RequestLine sip3261_get_RequestLine(sip3261); extern T_header_From sip3261_get_header_From(sip3261); extern T_Method sip3261_RequestLine_getMethod(T_RequestLine); extern T_MethodEnum sip3261_Method_getType(T_Method); extern T_Str sip3261_Method_getValue(T_Method); extern T_Lazy_addr_spec sip3261_header_From_getUri(T_header_From); extern T_addr_spec sip3261_Lazy_Addr_spec_getParsed(T_Lazy_addr_spec); extern T_Option_Str sip3261_Addr_spec_gethost(T_addr_spec); extern T_Str sip3261_Option_Str_getVal(T_Option_Str); [...] 4a. Generated stubs sip3261 msg = sip3261_init(); sip3261 msg = sip3261_parse(msg, buf, len); 1 // Process only request messages 2 if (sip3261_isRequest(msg)) { 3 // Filter INVITE methods 4 T_RequestLine requestLine = sip3261_get_RequestLine(msg); 5 if (sip3261_Method_getType(sip3261_RequestLine_getMethod(requestLine)) == E_INVITEm) { 6 // We parse only the header From 7 sip3261_parse_headers(msg, E_HEADER_FROM); 8 T_Lazy_addr_spec l_addr_spec = sip3261_header_From_getUri(sip3261_get_header_From(msg)); 9 sip3261_parse_addr_spec(l_addr_spec); 10 T_Option_Str host = sip3261_Lazy_Addr_spec_getParsed(l_addr_spec); 11 // host may be undefined in some cases, check it and log its value 12 if (sip3261_Option_Str_isDefined(host)) { 13 mylog(sip3261_Option_Str_getVal(host)); 14 }}} 15 4b. Application logic Figure 4: Fragment of a Zebu-based SIP message statistics reporting application network messages, it has a key role to play in providing this robustness. In this section, we evaluate the robustness improvement offered by Zebu, by comparing the reaction of Zebu-based parsers and a variety of existing parsers to valid and invalid network messages. Our experiments are based on a mutation analysis technique. For SIP, we compare with the oSIP and SER parsers previously described in Section 2. For RTSP, we use the parser in the widely used VLC media player and streaming server [32], and the parser provided by the LiveMedia library 1. Figure 5 shows the sizes of the ABNF and Zebu specifications of the message grammars for SIP and RTSP. The Zebu specification is longer, because it includes rules that are mentioned only by reference to another RFC in the original SIP and RTSP specifications. Figure 5 also shows the number of lines of code in the oSIP, SER, VLC, and LiveMedia parser implementations. Protocol ABNF size Zebu spec size Parser Parser size SIP 700 (approx) 1081 oSIP 11982 SER 13277 RTSP 200 (approx) 330 VLC 1200 (approx) LiveMedia 1000 (approx) Figure 5: The sizes of the SIP and RTSP message grammars, and the sizes of existing parsers. Sizes in lines of code. In the rest of this section, we first introduce mutation analysis, and then compare the robustness of existing SIP and RTSP parsers with that of the corresponding Zebu-based parsers. Finally, we evaluate the performance of Zebu, showing that Zebu-based parsers are often as efficient as hand-written ones. 1LiveMedia: Streaming Media, http://www.livemediacast.net/ RR n° 0123456789 12 Burgy, Réveillère, Lawall & Muller 4.1 Robustness evaluation Mutation analysis is a fault-based testing technique for unit-level testing [7]. Traditional mutation testing involves introducing small changes, i.e., mutations, in program source code, to determine whether a given test suite is sufficient to distinguish between correct and incorrect programs. In our case, however, we are interested in assessing the robustness of the program, i.e., the parser, and thus we introduce mutations into the test data, i.e., the network messages, rather than into the program source code. We use mutation rules both to generate invalid messages and to generate valid messages that have properties that are known to be challenging for network protocol message parsers. A robust parser should reject the invalid messages and accept the valid ones. To generate invalid messages, we have defined a set of mutation rules for messages based on ABNF structure: � Mutations on the characters set. Message literals are derived from a fixed set of possible characters. The first, middle, or last character of a message literal is replaced with any character outside the valid set. � Mutations on repetitions. As described in Section 2.1, ABNF offers a generic mechanism of repetition. Mutants are chosen to describe an invalid number of repetitions. � Mutations based on constraints. Protocol specifications include additional constraints not specified in the message grammar about the values of header subfields. For example, the response code of a SIP response is not only an unsigned integer of three digits, but its value must also be less than 699 (see Figure 3). Mutants are chosen that violate these constraints. To generate valid but problematic messages, we have extended our character set mutation rule to create messages of the form suggested by the SIP Torture Test Message RFC [28]. This RFC describes a set of valid SIP messages that test corner cases in SIP implementations. To compare the robustness of Zebu-based applications to applications based on hand-crafted parsers, we consider the parsing of the principal fields of a network protocol message. For SIP, these fields are the command line and the six mandatory header fields, while for RTSP they are the command line and the header fields Transport, CSeq and UserAgent. We drive each of the parsers listed in Figure 5 using minimal applications that request access to these fields. The Zebu-based applications log-Zebu-SIP and log-Zebu-RTSP, for SIP and RTSP respectively, consist of a few lines of C code that log statistical information about incoming messages. These applications use the stubs generated by the Zebu compiler to access network messages, analogous to the code illustrated in Figure 4b. The SER application, log-SER is written using the SER configuration language to access the information in the various fields. The other applications, log-oSIP using oSIP, log-VLC using VLC, and log-LiveMedia using LiveMedia, are written in C using the appropriate API functions provided by the given parser. Invalid messages In our first set of tests, we apply our mutation rules to SIP and RTSP messages, generating a stream of invalid messages, which we then send to each of the SIP and RTSP applications, respectively. As shown in Figure 6, while the Zebu-based applications detect every mutant as representing an invalid message, none of the hand-crafted parsers detects more than about 25% of the injected mutants. This situation may have a critical impact. In the case of SIP for example, we have crashed SER via a stream of 2416 incorrect messages, sent within of 17 seconds. Because SER is widely used for telephony, which is a critical service, the ability to crash the server is unacceptable. Mutation sites Injected Mutants Detected Mutants % detected Mutants log-Zebu-SIP 5976 100.0% SIP log-oSIP 81 5976 1020 17.1% log-SER 1512 25.3% log-Zebu-RTSP 2730 100.0% RTSP log-VLC 19 2730 4 0.1% log-LiveMedia 748 27.4% Figure 6: Mutation coverage for invalidSIP and RTSP messages Valid messages While message parsers should detect erroneous messages as early as possible to preserve the robustness of the applications that use them, they also must correctly parse valid messages. The SIP Torture Test Message RFC [28] describes a set of valid SIP messages that test corner cases in SIP implementations. Guided by this RFC, we have extended our character set mutation rule to generate mutants that are valid SIP messages but are designed to torture a SIP implementation. Figure 7 shows that up to about 4% of the valid INRIA A Language-Based Approach for Improving the Robustness of Network Application Protocol Implementations13 Mutation sites Injected Mutants Rejected Mutants % rejected Mutants log-Zebu-SIP 0 0.0% SIP log-oSIP 18 549 21 3.9% log-SER 2 0.4% Figure 7: Mutation coverage for valid SIPmessages messages are rejected by hand-crafted SIP parsers. By comparison, the Zebu-based SIP parser strictly follows the message grammar. We have tried an analogous experiment with the RTSP applications, but the VLC and LiveMedia parsers are quite lax in their parsing of the message elements, such as the URI, that are covered by the SIP Torture Test RFC, and thus all three applications accept all of the mutated messages. 4.2 Performance Evaluation We now compare the performance of Zebu-based parsers to that of hand-crafted ones. Our results are only for SIP, which is the most demanding in terms of performance. For our experiments, we have implemented four versions of the SIP message statistics reporting application described in Section 3.4. In each case, the application records the host information of the URI stored in the From header field of an INVITE message. The first version (inv-SER-module) is implemented as a dedicated SER module to obtain full access to the internal data structures of SER. The second version (inv-SER-exec) is written using the configuration language of SER and relies on the escape mechanism provided by SER to invoke sed to extract the host information, as described in Section 2.4. The third version (inv-oSIP) is implemented using a few lines of C code on top of the oSIP SIP stack. The last version (inv-Zebu) is the Zebu-based application depicted in Figure 4b. Our application illustrates the case where an application such as a intrusion detection system needs to access a fragment of a header subfield. To explore the effect that various kinds of messages have on the parsing performance for such an application, we consider a collection of INVITE messages, which are relevant to our application, and an example of a non-INVITE message, which is not. Among the INVITE messages, in INVITE1 the From header field contains only the URI subfield and an required tag subfield; all of the other subfields, which are optional, are omitted. This entails the minimal processing for a message that is relevant to the application. The remaining INVITE messages, INVITE2 and INVITE3, show the effect of varying the position of the From header field. In INVITE2, the From header field is the first of 34 header fields, while in INVITE2 it is the last of 34 header fields. The non-INVITE message is a BYE and has 7 headers. inv-SER-module inv-SER-exec inv-oSIP inv-Zebu Message size Cycles Ratio Cycles Ratio Cycles Ratio Cycles Ratio INVITE1 697 13 788 1 7 593 550 551 182 703 13 51 054 4 INVITE2 1 734 13 595 1 8 803 456 648 276 275 20 80 270 6 INVITE3 1 734 32 045 1 10 015 827 313 - - 133 164 4 BYE 334 10 252 1 10 765 1 105 773 10 6 037 0.6 Figure 8: Performance of SIP applications (time in cycles, ratio as compared to inv-SER-module) Our experiments were performed using a Pentium III (1GHz) as the server, which is stressed by a bi-processor Xeon 3.2Ghz client. Figure 8 compares the parsing time for each of the applications to that of SER-module, which has the fastest parser among the existing parsers that we tested. SER uses the efficient two-level parsing strategy described in Section 3.1, to parse only the header fields that are relevant to the application. The parsing done by inv-SER-module is particularly efficient in the case of INVITE messages, as the information required by the application is already available in the SER internal data structures. The parsing done by inv-SER-exec is roughly as efficient as that done by inv-SER-module for the non-INVITE message. The parsing done by inv-SER-exec for the INVITE messages, on the other hand, is up to 648 times slower, because it forks a sed process. Despite the bad performance in this case, the use of the configuration language of SER remains relevant, because it provides ease of programming and safety, which are not provided by the use of a SER module. The parsing done by inv-oSIP is over 13 times slower than the parsing done by inv-SER-module for INVITE messages and over 10 times slower for the non-INVITE message. In both cases, oSIP parses the six required SIP headers (plus two more required headers in the case of a REGISTER message) and stores pointers to the starting point of each sub-field. As the application requests information about the INVITE header field, oSIP RR n° 0123456789 14 Burgy, Réveillère, Lawall & Muller additionally copies the subfields into a data structure that is provided to the application, roughly doubling the execution time. No results are presented for inv-oSIP for INVITE3, because oSIP crashes on this message. Finally, while Zebu follows the same two-level parsing strategy as SER, the parsing done by inv-Zebu is significantly slower than the parsing done by inv-SER-module for the INVITE messages, because Zebu checks the URI more rigorously than SER. On the other hand, Zebu is significantly more efficient than SER for the non-INVITE messages. SER is directed towards routing applications, and thus it always parses the Via header, which is essential in the routing process, although irrelevant to our application. Thus, Zebu provides better performance in such cases by being more closely tailored to the needs of the application, and retains safety, which is lost in SER when using the module approach. 5 Related Work Parser generators such as DATASCRIPT [3], PacketTypes [18], PADS [9], GAPA [5] and binpac [22] have been recently developed to address the growing complexity of network protocol messages. However, as described in Section 2, these tools do not fulfill all the requirements of network application developers. APG [17] is a parser generator that accepts ABNF directly. Semantic actions are specified via callback functions rather than annotations on the grammar. We have found the use of such callback functions to be somewhat heavyweight, in our experience in using APG. Furthermore, APG is not specific to HTTP-like text-based protocols, and thus cannot implement the two-level parsing strategy outlined in Section 3.1, which we have found (Section 4) essential to obtaining good performance. Domain-specific languages have been used successfully in various application domains including operating systems [16, 19] and networks [11, 13]. Several of these languages have explicitly targeted improving system robustness. The Devil language, in the domain of device-driver development, provides high-level abstractions for specifying the code for interacting with the device, and performs a number of compile-time and (optional) run- time verifications to check that the specifications are consistent [25]. The language Promela++ for specifying network protocols, can be translated automatically both into the model checking language Promela [14] and into efficient C code, thus easing the development of a protocol implementation that is both verified and efficient Mutation analysis has been used to test the robustness of other software components, such as operating systems [8], network intrusion detection systems [31], and databases [30]. Our work is most similar to the work on network intrusion detection systems, which also mutates network protocol messages. 6 Conclusion In this paper, we have presented the Zebu declarative language for describing protocol message formats and related processing constraints. Zebu builds on the ABNF notations typically used in RFCs to describe protocol grammars. In evaluating Zebu, we have particularly focused on analyzing the improvement in robustness that it provides. For this, we have defined a test methodology based on a mutation analysis that injects errors into network messages. We have applied our test methodology to SIP and RTSP servers by comparing existing parsers with Zebu-generated ones. The results of our experiments show that nearly 4 times more erroneous messages are detected by the Zebu- based parser than by widely-used hand-written parsers. In the case of SIP, we were able to crash the widely used SER parser [23] via a stream of 2416 incorrect messages, sent within a space of 17 seconds. Because SER is used for telephony, which is a critical service, the ability to crash the server is unacceptable. We have also found valid messages that are not accepted by the SER and oSIP parsers, which can similarly have a critical impact. Finally, we have shown that the added safety and robustness provided by Zebu does not significantly impact performance. 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Peterson. binpac: a yacc for writing application protocol parsers. In Proceedings of the 6th ACM SIGCOMM on Internet measurement, pages 289–300, Rio de Janeiro, Brazil, Oct. 2006. RR n° 0123456789 16 Burgy, Réveillère, Lawall & Muller [23] A. Pelinescu-Onciul, J. Janak, and J. Kuthan. SIP express router (SER). IEEE Network Magazine, 17(4):9, July/August 2003. [24] X. Qie, R. Pang, and L. L. Peterson. Defensive programming: Using an annotation toolkit to build DoS- resistant software. In 5th Symposium on Operating System Design and Implementation (OSDI 2002), pages 45–60, Boston, MA, USA, Dec. 2002. [25] L. Réveillère and G. Muller. Improving driver robustness: an evaluation of the Devil approach. In The International Conference on Dependable Systems and Networks, pages 131–140, Göteborg, Sweden, July 2001. IEEE Computer Society. [26] Rosenberg, J. et al. SIP: Session Initiation Protocol. RFC 3261, IETF, June 2002. 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INRIA Unité de recherche INRIA Futurs Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN 0249-6399 Introduction Issues in developing network protocol parsers ABNF formalism Hand-writing parsers Using parser generators Integrating a parser with an application Robust Parser Development with Zebu Issues Annotating an ABNF specification The Zebu compiler Developing an application with Zebu Experiments Robustness evaluation Performance Evaluation Related Work Conclusion

0704.1374

A Close Look at Star Formation around Active Galactic Nuclei

A Close Look at Star Formation around Active Galactic Nuclei R. I. Davies, F. Mueller Sánchez, R. Genzel, L.J. Tacconi, E.K.S. Hicks, S. Friedrich, Max Planck Institut für extraterrestrische Physik, Postfach 1312, 85741, Garching, Germany A. Sternberg School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel ABSTRACT We analyse star formation in the nuclei of 9 Seyfert galaxies at spatial resolutions down to 0.085′′, corresponding to length scales of order 10 pc in most objects. Our data were taken mostly with the near infrared adaptive optics integral field spectrograph SINFONI. The stellar light profiles typically have size scales of a few tens of parsecs. In two cases there is unambiguous kinematic evidence for stellar disks on these scales. In the nuclear regions there appear to have been recent – but no longer active – starbursts in the last 10-300Myr. The stellar luminosity is less than a few percent of the AGN in the central 10 pc, whereas on kiloparsec scales the luminosities are comparable. The surface stellar luminosity density follows a similar trend in all the objects, increasing steadily at smaller radii up to ∼ 1013 L⊙ kpc −2 in the central few parsecs, where the mass surface density exceeds 104 M⊙ pc −2. The intense starbursts were probably Eddington limited and hence inevitably short-lived, implying that the starbursts occur in multiple short bursts. The data hint at a delay of 50–100Myr between the onset of star formation and subsequent fuelling of the black hole. We discuss whether this may be a consequence of the role that stellar ejecta could play in fuelling the black hole. While a significant mass is ejected by OB winds and supernovae, their high velocity means that very little of it can be accreted. On the other hand winds from AGB stars ultimately dominate the total mass loss, and they can also be accreted very efficiently because of their slow speeds. Subject headings: galaxies: active — galaxies: nuclei — galaxies: Seyfert — galaxies: starburst — infrared: galaxies 1. Introduction During recent years there has been increasing evidence for a connection between active galactic nuclei (AGN) and star formation in the vicinity of the central black holes. This subject forms the central topic of this paper, and is discussed in Sections 4 and 5. A large number of studies have addressed the issue of star formation around AGN. Those which have probed closest to the nucleus, typically on scales of a few hundred parsecs, have tended to focus on Seyferts – notably Seyfert 2 galaxies – since these are the closest examples (Sarzi et al. 2007; Asari et el. 2007; 1Based on observations at the European Southern Observatory VLT (60.A-9235, 070.B-0649, 070.B-0664, 074.B-9012, 076.B- 0098). http://arxiv.org/abs/0704.1374v2 – 2 – González Delgado, et al. 2005; Cid Fernandes et al. 2004; González Delgado, et al. 2001; Gu et al. 2001; Joguet et al. 2001; Storchi-Bergmann et al. 2001; Ivanov et al. 2000). The overall conclusion of these studies is that in 30–50% cases the AGN is associated with young (i.e. age less than a few 100 Myr) star formation. While this certainly implies a link, it does not necessarily imply any causal link between the two phenomena. Instead, it could more simply be a natural consequence of the fact that both AGN and starburst require gas to fuel them. And that in some galaxies this gas has fallen towards the nucleus, either due to an interaction or secular evolution such as bar driven inflow. One aspect which must be borne in mind when interpreting such results, and which has been pointed out by Knapen (2004), is the discrepancy in the scales involved. AGN and starburst phemonena occur on different temporal and spatial scales; and observations are sensitive to scales that are different again. For example, star formation has typically been studied on scales of several kiloparsecs down to a few hundred parsecs. In contrast, accretion of gas onto an AGN will occur on scales much less than 1 pc. Similarly, the shortest star formation timescales that most observations are sensitive to are of order 100 Myr to 1 Gyr. On the other hand, in this paper we show that the active phase of star formation close around a black hole is typically rather less than 100 Myr. Correspondingly short accretion timescales for black holes are reflected in the ages of jets which, for a sample of radio galaxies measured by Machalski et al. (2007), span a range from a few to 100 Myr. In Seyfert galaxies the timescales are even shorter, as typified by NGC 1068 for which Capetti et al. (1999) estimate the age of the jets to be only ∼ 0.1 Myr. That the putative causal connection between AGN and starbursts might occur on relatively small spatial scales and short timescales can help us to understand why no correlation has been found between AGN and (circum-)nuclear starbursts in general. It is simply that the circumnuclear activity on scales greater than a few hundred parsecs is, in most cases, too far from the AGN to influence it, or be strongly influenced by it (cf Heckman et al. 1997). In this paper we redress this imbalance. While the optical spectroscopy pursued by many authors allows a detailed fitting of templates and models to the stellar features, we also make use of established star formation diagnostics and interpret them using starburst population synthesis models. Observing at near infrared wavelengths has brought two important advantages. The optical depth is 10 times less than at optical wavelengths, and thus our data are less prone to the effects of extinction which can be significant in AGN. And we have employed adaptive optics to reach spatial resolutions of 0.1–0.2′′, bringing us closer to the nucleus. Applying these techniques, we have already analysed the properties of the nuclear star formation in a few objects (Davies et al. 2004a,b, 2006; Mueller Sánchez et al. 2006). Here we bring those data together with new data on 5 additional objects. Our sample enables us to probe star formation in AGN from radii of 1 kpc down to less than 10 pc. Our aim is to ascertain whether there is evidence for star formation on the smallest scales we can reach; and if so, to constrain its star formation history. Ultimately, we look at whether there are indications that the nuclear starburst and AGN are mutually influencing each other. In §2 we describe the sample selection, observations, data reduction, PSF estimation, and extraction of the emission and absorption line morphologies and kinematics. In §3 we discuss the observational diagnostics and modelling tools. Brief analyses of the relevant facets of our new data for the individual objects are provided in Appendix A, where we also summarise results of our previously published data, re-assessing them where necessary to ensure that all objects are analysed in a consistent manner. The primary aims of our paper are addressed in §4 and §5. In §4 we discuss global results concerning the existence and recent history of nuclear star formation for our whole sample. In §5 we discuss the implications of nuclear starbursts on the starburst-AGN connection. Finally, we present our conclusions in §6. – 3 – 2. Sample, Observations, Data Processing 2.1. Sample Selection The AGN discussed in this paper form a rather heterogeneous group. They include type 1 and type 2 Seyferts, ULIRGs, and even a QSO, and do not constitute a complete sample. In order to maximise the size of the sample, we have combined objects on which we have already published adaptive optics near infrared spectra with new observations of additional targets. Source selection was driven largely by technical considerations for the adaptive optics (AO) system, namely having a nucleus bright and compact enough to allow a good AO correction. This is actually a strength since it means that 7 of the 9 AGN are in fact broad line objects – as given either by the standard type 1 classification or because there is clear broad (FWHM > 1000km s−1) Brγ emission in our spectra. Fig. 1 shows broad Brγ in K-band spectra of 3 AGN that are not usually classified as broad line galaxies. This is in contrast to most other samples of AGN for which star formation has been studied in detail, and avoids any bias that might arise from selecting only type 2 Syeferts. That there may be a bias arises from the increasing evidence that the obscuration in perhaps half of type 2 AGN lies at kpc scales rather than in the nucleus, which may be caused by spatially extended star formation in the galaxy disk (Brand et al. 2007; Martinez-Sansigre et al. 2006; Rigby et al. 2006). Such AGN do not fit easily into the standard unification scheme (and perhaps should not really be considered type 2 objects). Because broad lines can be seen in the infrared, we know that we are seeing down to the nuclear region and hence our results are not subject to any effects that this might otherwise introduce. It is exactly broad line AGN for which little is known about the nuclear star formation, because the glare of the AGN swamps any surrounding stellar light in the central arcsec. As a result, most studies addressing star formation close to AGN have focussed on type 2 Seyferts. Adaptive optics makes it possible to confine much of the AGN’s light into a very compact region, and to resolve the stellar continuum around it. The use of adaptive optics does give rise to one difficulty when attempting to quantify the results in a uniform way, due to the different resolutions achieved – which is a combination of both the distance to each object (i.e. target selection) and the AO performance. As a result, the standard deviation around the logarithmic mean resolution of our sample (excluding NGC 2992, see Section A) of 22 pc is a factor of 3. However, this has enabled us to study the centers of AGN across nearly 3 orders of magnitude in spatial scale, from 1 kpc in the more distant objects to only a few parsecs in the nearby objects with the best AO correction. 2.2. Observations & Reduction A summary of our observations is given in Table 1. A description of observations and processing of the new data is given below. Data for IRAS 05189-2524 and NGC 1068 were taken in December 2002 at the VLT with NACO, an adaptive optics near infrared camera and long slit spectrograph (Lenzen et al. 2003; Rousset et al. 2003). Since IRAS 05189-2524 is nearly face on, there is no strongly preferred axis and the slit was oriented north- south; for NGC 1068 two orientations were used, north-south and east-west. In all cases the slit width was 0.086′′, yielding a nominal resolution of R ∼ 1500 with the wide-field camera (pixel scale 0.054′′) and medium resolution grism. The galaxy was nodded back and forth along the slit by 10′′ to allow sky subtraction. For IRAS 05189-2524, 12 integrations of 300 sec were made; for NGC 1068 12 integrations of 300 sec were made at one position angle, and 14 frames of 200 sec at the other. All data were reduced and combined, using – 4 – standard longslit techniques in IRAF, to make the final H-band spectrum. Data for NGC 7469, NGC 2992, NGC 1097, NGC 1068, and NGC 3783 were taken during 2004–2005 at the VLT with SINFONI, an adaptive optics near infrared integral field spectrograph (Eisenhauer et al. 2003; Bonnet et al. 2004). Data were taken with various gratings covering the H and K bands either separately (R ∼ 4000) or together (R ∼ 1500). The pixel scales were 0.125′′×0.25′′ or 0.05′′×0.1′′, depending on the trade-offs between field of view, spatial resolution, and signal-to-noise ratio. Individual exposure times are in the range 50–300 sec depending on the object brightness. Object frames were interspersed with sky frames, usually using the sequence O-S-O-O-S-O, to facilitate background subtraction. The data were processed using the dedicated spred software package (Abuter et al. 2006), which provides similar processing to that for longslit data but with the added ability to reconstruct the datacube. The data processing steps are as follows. The object frames are pre-processed by subtracting sky frames, flatfielding, and correcting bad pixels (which are identified from dark frames and the flatfield). The wavemap is generated, and edges and curvature of the slitlets are traced, all from the arclamp frame. The arclamp frame is then reconstructed into a cube, which is checked to ensure that the calibration is good. The pre-processed object frames are then also reconstructed into cubes, spatially shifted to align them using the bright nucleus as a reference, and combined. In some cases the final cube was spatially smoothed using a 3 × 3 median filter. Estimation of the spatial resolution (see below) was always performed after this stage. In some cases, the strong near-infrared OH lines did not subtract well. With longer exposure times this is to be expected since the timescale for variation of the OH is only 1–2 mins. If visual inspection of the reconstructed cubes showed signs of over- or under-subtraction of the OH lines, these cubes were reprocessed using the method described in Davies (2007a). Standard star frames are similarly reconstructed into cubes. Telluric correction and flux calibration were performed using B stars (K-band) or G2V stars (H-band). In addition, flux calibration was cross-checked in 3′′ apertures using 2MASS data, and in smaller 1–3′′ apertures using broad-band imaging from NACO or HST NICMOS. Agreement between cubes with different pixel scales, and also with the external data, was consistent to typically 20%. 2.3. PSF Estimation There are a multitude of ways to derive the point spread function (PSF) from adaptive optics data, five of which are described in Davies (2007b). With AGN, it is usually possible to estimate the PSF from the science data itself, removing any uncertainty about spatial and temporal variations of the PSF due to atmospheric effects. Typically one or both of the following methods are employed on the new data presented here. If a broad emission line is detected, this will always yield a measure of the PSF since the BLR of Seyfert galaxies has a diameter that can be measured in light days. Alternatively, the non-stellar continuum will provide a sufficiently good approximation in all but the nearest AGN since at near infrared wavelengths it is expected to originate from a region no more than 1–2 pc across. In every case we have fit an analytical function to the PSF. Since the Strehl ratio achieved is relatively low, even a Gaussian is a good representation. We have used a Moffat function, which achieves a better fit because it also matches the rather broad wings that are a characteristic of partial adaptive optics correction. The PSF measured for NGC 3227, which is shown in Fig. 1 of Davies et al. (2006), can be considered typical. If one applies the concept of ‘core plus halo’ to this PSF, then the Gaussian fit would represent just the core while the Moffat fit the entire ‘core plus halo’. Integrating both of these functions indicates that about 75% – 5 – of the flux is within the ‘core’, and it is thus this component which dominates the PSF. In this paper, a more exact representation of the PSF is not needed since we have not performed a detailed kinematic analysis, and we have simply used the Moffat to derive a FWHM for the spatial resolution. The resolutions achieved are listed in Table 1. 2.4. Emission/Absorption Line Characterisation The 2D distribution of emission and absorption features has been found by fitting a function to the continuum-subtracted spectral profile at each spatial position in the datacube. The function was a convolu- tion of a Gaussian with a spectrally unresolved template profile – in the case of emission lines it was an OH sky emission line, and for stellar absorption features we made use of template stars observed in the same con- figuration (pixel scale and grism). A minimisation was performed in which the parameters of the Gaussian were adjusted until the convolved profile best matched the data. During the minimisation, pixels in the data that consistently deviated strongly from the data were rejected. The uncertainties were boot-strapped using Monte Carlo techniques, assuming that the noise is uncorrelated and the intrinsic profile is well represented by a Gaussian. The method involves adding a Gaussian with the derived properties to a spectral segment that exhibits the same noise statistics as the data, and refitting the result to yield a new set of Gaussian parameters. After repeating this 100 times, the standard deviation of the center and dispersion were used as the uncertainites for the velocity and line width. The kinematics were further processed using kinemetry (Krajnović et al. 2006). This is a parameteri- sation (i.e. a mathematical rather than a physical model) of the 2D field. As such, beam smearing is not a relevant issue to kinemetry, which yields an analytical expression for the observed data. Of course, when the coefficients of this expression are interpreted or used to constrain a physical model, then beam smearing should be considered. Mathematically, the kinemetry procedure fits the data with a linear sum of sines and cosines with various angular scalings around ellipses at each radius. We have used it for 3 purposes: to de- termine the best position angle and axis ratio for the velocity field, to remove high order noise from the raw kinematic extraction, and to recover the velocity and dispersion radial profiles. In all of the cases considered here, the kinematic centre of the velocity field was assumed to be coincident with the peak of the non-stellar continuum. In addition, the uniformity of the velocity field permitted us to make the simplifying assumption of a single position angle and axis ratio – i.e. there is no evidence for warps or twisted velocity contours. We then derived the position angle and inclination of the disk by minimising the A1 and B3 parameters respectively (see Krajnović et al. 2006 for a description of these). The rotation curves were recovered by correcting the measured velocity profile for inclination. We have assumed throughout the paper that the dispersion is isotropic, and hence no inclination correction was applied to the dispersion that was measured. The innermost parts of the kinematics derived as above are of course still affected by beam smearing. In general, the central dispersion cannot necessarily be taken at face value since it may either be artificially increased by any component of rotation included within the beam size, or decreased if neighbouring regions within the beam have a lower dispersion. In the galaxies we have studied, there are two aspects which mitigate this uncertainty: the rotation speed in the central region is much less than the dispersion and so will not significantly alter it; and when estimating the central value we consider the trend of the dispersion from large radii, where the effect of the beam is small, to the center. For the basic analyses performed here, we have therefore adopted the central dispersion at face value. More detailed physical models for the nuclear disks, which properly account for the effects of beam smearing, will be presented in future publications. Lastly, we emphasize that the impact of the finite beam size on the derived rotation curve does not affect – 6 – our measurement of the dynamical mass. The reason is that, for all the dynamical mass estimates we make, the mass is estimated at a radius much large than the FWHM of the PSF – as can be seen in the relevant figures. 3. Quantifying the Star Formation In this section we describe the tools of the trade used to analyse the data, and which lead us to the global results presented in Section 4. Specific details and analyses for individual objects can be found in Appendix A. We use the same methods and tools for all the objects to ensure that all the data are analysed in a consistent manner. Perhaps the most important issue is how to isolate the stellar continuum, which is itself a powerful diagnostic. In addition, we use three standard and independent diagnostics to quantify the star formation history and intensity in the nuclei of these AGN. These are the Brγ equivalent width, supernova rate, and mass-to-light ratio. Much of the discussion concerns how we take into account the contribution of the AGN when quantifying these parameters. We also consider what impact an incorrect compensation could have on interpretation of the diagnostics. We model these observational diagnostics using the stellar population and spectral synthesis code STARS (e.g. Sternberg 1998; Sternberg et al. 2003; Förster Schreiber et al. 2003; Davies et al. 2003, 2005). This code calculates the distribution of stars in the Hertzsprung-Russell diagram as a function of age for an assumed star formation history. We usually assume an exponentially decaying star formation rate, which has an associated timescale τSF. Spectral properties of the cluster are then computed given the stellar population present at any time. We note that the model output of STARS is quantitatively similar to that from version 5.1 of Starburst99, which unlike earlier versions does include AGB tracks (Leitherer et al. 1999; Vazquez & Leitherer 2005). As discussed in detail below, particular predictions of STARS include, for ages greater than 10 Myr: equivalent widths of WCO2−0 ∼ 12Å and WCO6−3 ∼ 4.5Å, and H-K color of 0.15 mag. The equivalent quantities predicted by Starburst99 v5.1 are WCO2−0 ∼ 11Å and WCO6−3 ∼ 5Å, and H-K color of 0.2 mag. 3.1. Isolating the stellar continuum For small observational apertures a significant fraction of the K-band (and even H-band) continuum can be associated with non-stellar AGN continuum. The AGN contribution can be estimated from a simple measurement of the equivalent width of a stellar absorption feature. We use CO 2-0 2.29µm in the K-band or CO 6-3 1.62µm in the H-band. Although the equivalent widths WCO2−0 and WCO6−3 vary considerably for individual stars, the integrated values for stellar clusters span only a rather limited range. This was shown by Oliva et al. (1995) who measured these values for elliptical, spiral, and star-forming (Hii) galaxies. We have plotted their measurements of these two absorption features in the left-hand panel of Fig. 2, together with the equivalent widths of giant and supergiant stars from Origlia et al. (1993). In STARS we use empirically determined equivalent widths from library spectra (Förster Schreiber 2000) to compute the time-dependent equivalent width for an entire cluster of stars. Results for various star formation histories are shown in the centre and right panels of Fig. 2, for WCO6−3 and WCO2−0 respectively. Typical values are WCO6−3 = 4.5Å and WCO2−0 = 12Å. The dashed box in the left panel shows that the – 7 – locus of 20% deviation from each of these computed values is consistent with observations. That the Hii galaxies have slightly higher WCO2−0 can be understood because these are selected to have bright emission lines and hence are strongly biassed towards young stellar ages – often corresponding to the maximum depth of the stellar features that occurs at 10 Myr due to the late-type supergiant population. It may be this bias for galaxies selected as ‘starbursts’, and the similarity of the CO depth for starbursts of all other ages, that led Ivanov et al. (2000) to conclude that there is no evidence for strong starbursts in Seyfert 2 galaxies. Similarly, an estimate of the dilution can be found from the Nai 2.206µm line. Fig. 7 of Davies et al. (2005) shows that for nearly all star formation histories the value WNa I remains in the range 2–3Å. Our conclusion here is that within a reasonable uncertainty of ±20% (see Fig. 2), one can assume that the intrinsic equivalent width of the absorption – most notably CO – features of any stellar population that contains late-type stars is independent of the star formation history and age. For a stellar continuum diluted by additional non-stellar emission, the fraction of stellar light is fstellar = Wobs/Wint where Wobs and Wint are the observed and intrinsic equivalent widths of the CO features discussed above. Thus, we are able to correct the observed continuum magnitude for the contribution associated with the 3.2. Stellar colour and luminosity Our data cover both the H and K-bands – hence the reason for using both WCO6−3 and WCO2−0. In order to homogenize the dataset, we need to convert H-band stellar magnitudes to K-band. The STARS computation in Fig. 3 shows that this conversion is also independent of the star formation history, being close to H −K = 0.15 mag (no extinction) for all timescales and ages. This result is supported empirically by photometry of elliptical and spiral galaxies performed by Glass (1984). For ellipticals H −K ∼ 0.2–0.25, and for spirals H − K ∼ 0.2–0.3. Some of the difference between the data and models could be due to extinction since H −K = (H −K)0 + (AH −AK); and for AV = 1, AH −AK = 0.08. However, at the level of precision required here, the 5–10% difference between model and data can be considered negligible. To convert from absolute magnitude to luminosity we use the relation MK = −0.33 − 2.5 logLK where LK is the total luminosity in the 1.9–2.5µm band in units of bolometric solar luminosity (1L⊙ = 3.8 × 1026 W), and as such different from the other frequently used monochromatic definition with units of the solar K-band luminosity density (2.15 × 1025 Wµm−1). We then use STARS to estimate the bolometric stellar luminosity Lbol. The relation between Lbol and LK is shown in the right panel of Fig. 3. The dimensionless ratio Lbol/LK depends on the age and the exponential decay timescale of the star formation. However, the range spanned is only 20–200 for ages greater than 10 Myr. Thus even if the star formation history cannot be constrained, a conversion ratio of Lbol/LK ∼ 60 will have an associated uncertainty of only 0.3 dex. In general we are able to apply constraints on the star formation age, and so our errors will be accordingly smaller. – 8 – 3.3. Specific Star Formation Diagnostics Graphs showing how the diagnostics vary with age and star formation timescale are shown in Fig. 4. 3.3.1. Brγ equivalent width Once the stellar continuum luminosity is known, an upper limit to the equivalent width of Brγ asssociated with star formation can be found from the narrow Brγ line flux. In some cases it is possible to estimate what fraction of the narrow Brγ might be associated with the AGN. This can be done both morphologically, for example if the line emission is extended along the galaxy’s minor axis; and/or kinematically, for example if the line shows regions that are broader, perhaps with FWHM a few hundred km s−1, suggestive of outflow. Even if acounting for the AGN contribution is not possible, one may be able to set interesting upper limits or even rule out continuous star formation scenarios, and put a constraint on the time since the star formation was active. This can be seen in the lefthand panel of Fig. 4, which shows for example that for ages less than 109 yrs, continuous star formation scenarios will always have WBrγ > 12 Å. 3.3.2. Supernova rate We estimate the type ii (core collapse) supernova rate νSN from the radio continuum using the relation (Condon 1992): LN (W Hz −1) = 1.3 × 1023 ν−α(GHz) νSN (yr where LN is the non-thermal radio continuum luminosity, ν is frequency of the observation and α ∼ 0.8 the spectral index of the non-thermal continuum. This relation was derived for Galactic supernova remnants; but a similar one, differing only in having a coefficient of 1.1 × 1023, was derived by Huang et al. (1994) for M 82. For the 5 GHz non-thermal radio continuum luminosity of Arp 220 (176 mJy, Anantharamaiah et al. 2000) it would lead to a supernova rate of 2.9 yr−1, comfortably within the 1.75–3.5yr−1 range estimated by Smith et al. (1998) based on the detection of individual luminous radio supernovae. This, therefore, seems a reasonable relation to apply to starbursts. We have to be careful, however, to take into account any contribution from the AGN to the radio continuum. Our premise for the nuclei of Seyfert galaxies is that if the nuclear radio continuum is spatially resolved (i.e. it has a low brightness temperature) and does not have the morphology of a jet, it is likely to originate in extended star formation. At the spatial scales of a few parsec or more that we can resolve, emission from the AGN will be very compact. As a result, we can use the peak surface brightness to estimate the maximum (unresolved) contribution from an AGN. Wherever possible, we use radio continuum observations at a comparable resolution to our data to derive the extended emission; and observations at higher resolution to estimate the AGN contribution. Details of the data used in each case are given in the relevant sub-sections for each object in Appendix A. In addition, we exclude any emission obviously associated with jets, for example as in NGC 1068. To use νSN as a diagnostic, we normalise it with respect to the stellar K-band luminosity. This gives the ratio 1010νSN/LK , for which STARS output is drawn in Fig. 4. – 9 – 3.3.3. Mass-to-light ratio Models indicate that the ratio M/LK of the stellar mass to K-band luminosity should be an excellent diagnostic since, for ages greater than 10 Myr, it increases monotonically with age as shown in Fig. 4. However, in practice estimating the stellar mass is not entirely straightforward. In many cases it is only practicable to derive the dynamical mass. It may be possible to estimate and hence correct for the molecular gas mass based on millimetre CO maps, but these are scarce at sufficiently high spatial resolution and are associated with their own CO-to-H2 conversion uncertainties. We also note that it is often not possible to separate the ‘old’ and ‘young’ stellar populations. The best one can do is estimate the overall mass-to-light ratio, and argue that this is an upper limit to the true ratio for the young population. While there inevitably remains uncertainty on the true ratio, the limit is often sufficient to apply useful constraints on the age of the ‘young’ population. Our estimates of the dynamical mass are based wherever possible on the stellar kinematics, since the gas kinematics can be perturbed by warps, shocks, and outflows. We begin by estimating the simple Keplerian mass assuming that the stars are supported by ordered rotation at velocity Vrot = Vobs/ sin i in a thin plane. However, the stellar kinematics in all the galaxies exhibit a significant velocity dispersion indicating that a considerable mass is supported by random rather than ordered motions. Thus the simple Keplerian mass is very much an underestimate, and any estimate of the actual mass is associated with large uncertainties – see for example Bender et al. (1992), who derive masses of spheroidal systems. As stated in Section 2, we assume that the random motions are isotropic. Our relation for estimating the mass enclosed within a radius R is then M = (V 2rot + 3σ 2)R/G. where σ is the observed 1-dimensional velocity dispersion. We note that when taking rotation into account in estimating the masses of spheroids with various density profiles, Bender et al. (1992) also use a factor 3 between the V and σ terms in their Appendix B. Despite the complexities involved, within the unavoidable uncertainties (a factor 2–3), their relation gives the same mass as that above. Although this uncertainty appears to be quite large, it does not impact the results and conclusions in this paper since we are concerned primarily with order-of-magnitude estimates when considering mass surface densities. 4. Properties of Nuclear Star Formation In the following section we bring to together the individual results (detailed in Appendix A) to form a global picture. It is possible to do this because all the data have been analysed in a consistent manner, using the tools described in Section 3 to compare in each object the same diagnostics to the same set of stellar evolutionary synthesis models. We note that the discussion that follows is based on results for 8 of the AGN we have observed. As explained in Appendix A we exclude NGC 2992 because we are not able to put reliable constraints on the properties of the nuclear star formation. Despite this, there are indications that at higher spatial resolution one should expect to find a distinct nuclear stellar population as has been seen in other AGN. – 10 – Size scale Tracing the stellar features rather than the broad-band continuum, we have in all cases resolved a stellar population in the nucleus close around the AGN. While this should not be unexpected if the stellar distribution follows a smooth r1/4 or exponential profile, we have in several cases been able to show that on scales of < 50 pc there is in fact an excess above what one would expect from these profiles. This suggests that in general we are probing an inner star forming component. Fig. 5 shows normalised azimuthally averaged stellar luminosity profiles for the AGN. These have not been corrected for a possible old underlying population, nor has any deconvolution with the PSF been performed. Nevertheless, it is still clear that the stellar intensity increases very steeply towards the nucleus. In 6 of the 8 galaxies shown, the half-width at half-maximum is less than 50 pc. The remaining 2 galaxies are the most distant in the sample, and the spatial resolution achieved does not permit a size measurement on these scales. We may conclude that the physical radial size scale of the nuclear star forming regions in Seyfert galaxies does not typically exceed 50 pc. Stellar Age For 8 of the AGN studied here, we have been able to use classical star formation diagnostics based on line and continuum fluxes as well as kinematics to constrain the ages of the inner star forming regions. The resulting ages should be considered ‘characteristic’, since in many cases there may simultane- ously be two or more stellar populations that are not co-eval. For example, if a bulge population exists on these small spatial scales, it was not usually possible to account for the contamination it would introduce. While this would have little effect on WBrγ , it could impact on M/LK more strongly, increasing the inferred age. The ages we find lie in the range 10–300Myr, compelling evidence that it is common for there to be relatively young star clusters close around AGN. Intriguingly, we also find rather low values of WBrγ : typically WBrγ . 10 Å (see Table 3). This indicates directly that there is currently little or no on-going star formation. Coupled with the relatively young ages, we conclude that the star formation episodes are short-lived. One may speculate then that the star formation is episodic, recurring in short bursts. The scale of the bursts and time interval between them would certainly have an impact on the fraction of Seyfert nuclei in which observational programmes are able to find evidence for recent star formation. Nuclear Stellar Disks The first evidence for nuclear stellar disks came from seeing limited optical spectroscopy, for which a slight reduction in σ∗ was seen for some spiral galaxies (Emsellem et al. 2001; Márquez et al. 2003; Shapiro et al. 2003). And there is now a growing number of spiral galaxies – more than 30 – in which the phenomenon has been observed, suggesting that they might occur in 30% or more of disk galaxies (Emsellem 2006c). The σ∗-drop has been interpreted by Emsellem et al. (2001) as arising from a young stellar population that is born from a dynamically cold gas component, and which makes a significant contribution to the total luminosity. This appears to be borne out by N-body and SPH simulations of iso- lated galaxies (Wozniak et al. 2003), which suggest that although the entire central system will slowly heat up with time, the σ∗-drop can last for at least several hundred Myr. Indeed, preliminary analysis of optical integral field data for NGC 3623 suggest that the stellar population responsible for the σ∗-drop cannot be younger than 1 Gyr (Emsellem 2006b). Our results provide strong support for the nuclear disk interpretation. In previous work (Davies et al. 2006; Mueller Sánchez et al. 2006), we had argued that in both Circinus and NGC 3227 the inner distributions were disk-like, albeit thickened. We have now found much more direct evidence for this phenomenon in NGC 1097 and NGC 1068. In both of these galaxies, we have spatially resolved a σ∗-drop and an excess – 11 – stellar continuum over the same size scales. In NGC 1097 this size was ∼0.5′′, corresponding to about 40 pc. For NGC 1068 these effects were measured out to ∼1′′, equivalent to 70 pc. These are not the scale lengths of the disks, but simply the maximum radius to which we can detect them. In both cases the mean mass surface densities are of order Σ =(1-3)×104 M⊙ pc −2. For an infinitely large thin self gravitating stellar disk, one can use the expression σ2z = 2πGΣz0 to estimate the scale height. Although this may not be entirely appropriate, we use it here to obtain a rough approximation to the scale heights, which are 5–20 pc. Thus while the disks appear to be flattened, they should still be considered thick since the radial extent is only a few times the scale height. The impact of nuclear starbursts on the central light profile of galaxies was considered theoretically more than a decade ago by Mihos & Hernquist (1994). They performed numerical simulations of galaxy mergers to study the mass and luminosity profiles of the remnants, taking gas into account, and estimating the star formation rate using a modified Schmidt law. They found that there should be a starburst in the nucleus which would give rise to an excess stellar continuum above the r1/4 profile of the older stars in the merged system. Several years ago, compact nuclei were found to be present in a significant fraction of spiral galaxies (Balcells et al. 2003) as well as Coma cluster dwarf ellipticals (Graham & Guzmán 2003). More recently, nuclei with a median half-light radius of 4.2 pc have been found in the majority of early-type members of the Virgo Cluster (Côté et al. 2006); and traced out to ∼1′′, equivalent to ∼ 100 pc, in some of the ‘wet’ merger remnants in that cluster (Kormendy et al. 2007). While the nuclear starbursts in these latter cases are caused by a merger event, whereas those we are studying arise from secular evolution as gas from the galaxy disk accretes in the nucleus, there appear to be many parallels in the phenomenology of the resulting starbursts. Star Formation Rate It is possible to estimate the bolometric luminosity Lbol∗ of the stars from their K- band luminosity LK even if one knows nothing about the star formation history. As discussed in Section 3, this would result in an uncertainty of about a factor 3. The diagnostics in Table 3 and discussions in Appendix A enable us to apply some constraints to the characteristic age of the star formation. Because continuous star formation is ruled out by the low WBrγ , we have assumed exponential decay timescales of τSF = 10–100Myr. We have then used STARS to estimate the average star formation rates. In order to allow a meaningful comparison between the objects, the rates have been normalised to the same area of 1 kpc2. These are the rates given in Table 3. They are calculated simply as the mass of stars produced divided by the entire time since the star forming episode began. Because τSF is shorter than the age, the average includes both active and non-active phases of the starburst. Indeed, for τSF = 10 Myr one would expect the star formation rate during the active phases to be at least a factor of a few, and perhaps an order of magnitude, greater. The table shows that on scales of a few hundred parsecs one might expect a few ×10 M⊙ yr −1 kpc−2, while on scales of a few tens of parsecs mean rates reaching ∼ 100 M⊙ yr −1 kpc−2 should not be unexpected; and correspondingly higher – up to an order of magnitude, see Fig. 6 – during active phases. An obvious question is why there should be such vigorous star formation in these regions. Star formation rates of 10–100M⊙ yr −1 kpc−2 are orders of magnitude above those in normal galaxies and comparable to starburst galaxies. The answer may lie in the Schmidt law and the mass surface densities we have estimated in Table 3. Fig. 7 shows these surface densities at the radii over which they were estimated, revealing a trend towards higher densities on smaller scales and values of a few times 104 M⊙ pc −2 in the central few tens of parsecs. The global Schmidt law, as formulated by Kennicutt (1998), states that the star formation rate depends on the gas surface density as ΣSFR ∝ Σ gas. If one assumes that 10–30% of the mass in our AGN is – 12 – gas, then this relation would predict time-averaged star formation rates in the range 10–100M⊙ yr −1 kpc−2, as have been observed. That the high star formation rates may simply be a consequence of the high mass surface densities is explored futher by Hicks et al. (in prep.). Stellar Luminosity As a consequence of the high star formation rates, the stellar luminosity per unit area close around the AGN is very high in these objects. Despite this, because the star formation is occurring only in very small regions, the absolute luminosities are rather modest. This can be seen in Fig. 8 which shows the bolometric luminosity of the stars as a fraction of the entire bolometric luminosity of the galaxy. We have calculated a range for the ratio Lbol∗/LK appropriate for each galaxy based on the ages in Table 3 for different τSF. Because we assume that all the K-band stellar continuum is associated with the young stars, we have adopted the lower end of each range in an attempt to minimise possible overestimation of Lbol∗. The resulting values for the ratio used span 30–130, within a factor of 2 of the ‘baseline’ value of 60 given in Section 3. In the central few tens of parsecs, young stars contribute a few percent of the total. But integrated over size scales of a few hundred parsecs, this fraction can increase to more than 20%. On these scales, the star formation is energetically significant when compared to the AGN. Such high fractions imply that on the larger scales the extinction to the young stars must be relatively low. On the other hand, on the smallest scales where in absolute terms the stellar luminosity is small, there could in general be considerable extinction even at near infrared wavelengths. In this paper we have not tried to account for extinction since it is very uncertain. The primary effect of doing so would simply be to increase the stellar luminosity above the values discussed here. Fig. 9 shows the stellar bolometric luminosity Lbol∗ integrated as a function of radius. All the curves follow approximately the same trend, with the luminosity per unit area increasing towards smaller scales and approaching 1013 L⊙ kpc −2 in the central few parsecs. This appears to be a robust trend and will not change significantly even with large uncertainties of a factor of a few. It is remarkable that the luminosity density of 1013 L⊙ kpc −2 is that estimated by Thompson et al. (2005) for ULIRGs, which they modelled as optically thick starburst disks. The main difference between the ULIRG model and the starbursts close around AGN is the spatial scales on which the starburst occurs. Based on this model, they argued that ULIRGs are radiating at the Eddington limit for a starburst, defined as when the radiation pressure on the gas and dust begins to dominate over self-gravity. The limiting luminosity-to-mass ratio was estimated to be ∼ 500 L⊙/M⊙ by Scoville (2003). He argued that in a star cluster, once the upper end of the main sequence was populated, the radiation pressure would halt further accretion on to the star cluster and hence terminate the star formation. Following Thompson et al. (2005), we apply this definition to the entire disk rather than a single star cluster. For the 1013 L⊙ kpc −2, this implies a mass surface density of 2 × 104 M⊙ pc −2. Comparing these quantities to the AGN we have observed, we find that on scales of a few tens of parsecs they are an order of magnitude below the Eddington limit. On the other hand, we have already seen that the low WBrγ indicates that there is little on-going star formation and hence that the starbursts are short-lived. This is important because short-lived starbursts fade very quickly. As shown in Fig. 6, for a decay timescale of τSF = 10 Myr, Lbol∗ will have decreased from its peak value by more than an order of magnitude at an age of 100 Myr. Thus it is plausible – and probably likely – that while the star formation was active, the stellar luminosity was an order of magnitude higher. In this case the starbursts would have been at, or close to, their Eddington limit at that time. The luminosity-to-mass ratio of 500 L⊙/M⊙ associated with the Eddington limit is in fact one that all young starbursts would exceed if, beginning with nothing, gas was accreted at the same rate that it was converted into stars. That, however, is not a realistic situation. A more likely scenario, shown in Fig. 10, is – 13 – that the gas is already there in the disk. In this case, a starburst with a star-forming timescale of 100 Myr could never exceed 100 L⊙/M⊙. To reach 500 L⊙/M⊙, the gas would need to be converted into stars on a timescale . 10 Myr. This timescale is independent of how much gas there is. Thus, for a starburst to reach its Eddington limit, it must be very efficient, converting a significant fraction of its gas into stars on very short ∼ 10 Myr timescales. This result is consistent with the prediction of the Schmidt law, which states that disks with a higher gas surface density will form stars more efficiently. The reason is that the star formation efficiency is simply SFE = ΣSFR/Σgas ∝ Σ gas. Thus, from arguments based solely on the Schmidt law and mass surface density, one reaches the same conclusion that the gas supply would be used rather quickly and the lifetime of the starburst would be relatively short. Summarising the results above, a plausible scenario could be as follows. The high gas density leads to a high star formation rate, producing a starburst that reaches its Eddington limit for a short time. Because the efficiency is high, the starburst can only be active for a short time and then begins to fade. Inevitably, one would expect that the starburst is then dormant until the gas supply is replenished by inflow. This picture appears to be borne out by the observations presented here. 5. Starburst-AGN Connection In the previous sections we have presented and discussed evidence that in general there appears to have been moderately recent star formation on small spatial scales around all the AGN we have observed. Fig. 11 shows the first empirical indication of a deeper relationship between the star formation and the AGN. In this figure we show the luminosity of the AGN, both in absolute units of solar luminosity and also in relative units of its Eddington luminosity LEdd, against the age of the most recent known nuclear star forming episode. Since the AGN luminosity is not well known, we have made the conservative assumption that it is equal to half the bolometric luminosity of the galaxy – as may be the case for NGC 1068 (Pier et al. 1994, but see also Bland-Hawthorne et al. 1997). To indicate the expected degree of uncertainty in this assertion we have imposed errorbars of a factor 2 in either direction, equivalent to stating that the AGN luminosity in these specific objects is likely to be in the range 25–100% of the total luminosity of the galaxy. The Eddington luminosity is calculated directly from the black hole mass, for which estimates exist for these galaxies from reverberation mapping, the MBH − σ∗ relation, maser kinematics, etc. These are listed in Table 2. For the age of the star formation, we have plotted the time since the most recent known episode of star formation began, as given in Table 3. For galaxies where a range of ages is given, we have adopted these to indicate the uncertainty; the mean of these, ∼ ±30%, has been used to estimate the uncertainty in the age for the rest of the galaxies. We note that these errorbars reflect uncertainties in characterising the age of the star formation from the available diagnostics and also in the star formation timescale τSF. However, there are still many implicit assumptions in this process, and we therefore caution that the actual errors in our estimation of the starburst ages may be larger than that shown. Conceding this, we do not wish to over-interpret the figure. Keeping the uncertainties in mind, Fig. 11 shows the remarkable result that AGN which are radiating at lower efficiency . 0.1 L/LEdd are associated with younger . 50–100Myr starbursts; while those which are more efficient & 0.1 L/LEdd have older & 50–100Myr starbursts. If one were to add to this figure the Galactic Centre – which is known to have an extremely low luminosity (L/LEdd < 10 −5; Ozernoy & Genzel 1996; Baganoff et al. 2003) and to have experienced a starburst 6 ± 2 Myr ago (Paumard et al. 2006) – it would be consistent with the categories above. The inference is that either there is a delay between the onset of starburst activity and the onset of AGN activity, or star formation is quenched once the black hole has become active. – 14 – In Section 4 we argued that the starbursts are to some extent self-quenching: that very high star forma- tion efficiencies are not sustainable over long periods. In addition, an intense starburst will provide significant heat input to the gas, which is perhaps partially responsible for the typically high gas velocity dispersions in these regions (Hicks et al., in prep.). This itself could help suppress further star formation. Heating by the AGN could also contribute to this process, and has been proposed as the reason why the molecular torus is geometrically thick (Pier & Krolik 1992; Krolik 2007). It is also used to modulate star formation (at least on global scales) in semi-analytic models of galaxy evolution (Granato et al. 2004; Springel et al. 2005). While this is certainly plausible, it does not explain either why the star formation in some galaxies with a lower luminosity AGN has already ceased, nor why none of the AGN associated with younger starbursts are accreting efficiently. Instead we argue for the former case above, that efficient fuelling of a black hole is associated with a starburst that is at least 50–100Myr old. It may be because of such a delay between AGN and starburst activity that recent star formation is often hard to detect close to AGN: the starburst has passed its most luminous (very young) age, and is in decline while the AGN is in its most active phase (see Fig. 6). This does not necessarily imply that the a priori presence of a starburst is required before an AGN can accrete gas – although it seems inevitable that one will occur as gas accumulates in the nucleus. Nor does it imply that all starbursts will result in fuelling a black hole; indeed it is clear that there are many starbursts not associated with AGN. As we argue below, the crucial aspect may be the stellar ejecta associated with the starburst; and in particular, not just the mass loss rate, but the speed with which the mass is ejected. Winds from OB stars In the Galactic Center, Ozernoy & Genzel (1996) proposed that it is the recent starburst there that is limiting the luminosity of the black hole. In this scenario, mechanical winds from young stars – both the outflow and the angular momentum of the gas (which is a consequence of the angular momentum of the stars themselves) – hinder further inflow. The authors argued that almost none of the gas flowing into the central parsec reached the black hole because of outflowing winds from IRS 16 and He i stars in that region. Detailed modelling of the Galactic Center region as a 2-phase medium was recently performed by Cuadra et al. (2006). They included both the fast young stellar winds with velocities of 700 km s−1 (Ozernoy et al. 1997) and the slower winds of ∼ 200 km s−1 (Paumard et al. 2001); and also took into account the orbital angular momentum of the stars (Paumard et al. 2001; Genzel et al. 2003), which had a strong influence on reducing the accretion rate. They found that the average accretion rate onto the black hole was only ∼ 3 × 10−6 M⊙ yr −1, although an intermittent cold flow superimposed considerable variability onto this. In contrast, the hypothetical luminosity Ozernoy & Genzel (1996) estimate that Sgr A∗ would have if it could accrete all the inflowing gas, would be 5 × 1043 erg s−1, typical of Seyfert galaxies. In principle this process could be operating in other galaxy nuclei where there has been a starburst which extends to less than 1 pc from the central black hole. However, it cannot explain the timescale of the delay we have observed, which is an order of magnitude greater than the main sequence lifetime of OB and Wolf-Rayet stars. Winds from AGB stars Stars of a few (1–8 M⊙) solar masses will evolve on to the asymptotic giant branch (AGB) at the end of their main sequence lifetimes. The timescale for stars at the upper end of this range to reach this phase is ∼ 50 Myr, comparable to the delay apparent in Fig. 9. Since AGB stars are known to have high mass-loss rates, of order 10−7–10−4 M⊙ yr −1 at velocities of 10–30km s−1 (Winters et al. 2003), they may be prime candidates for explaining the delay between starburst and AGN activity. To quantify this, we consider how much of the mass in the wind could be accreted by the central supermassive black – 15 – hole. The Bondi parameterisation of the accretion rate onto a point particle for a uniform spherically sym- metric geometry is given by (Bondi 1952) 2πG2 M2 ρ (V 2 + c2s) where M is the mass of the point particle moving through a gas cloud, V is the velocity of the particle with respect to the cloud, ρ is the density of the cloud far from the point particle, and cs is the sound speed. This approximation is still used to quantify accretion on to supermassive black holes in models of galaxy evolution (Springel et al. 2005), even though it may be significantly inaccurate for realistic (e.g. turbulent) media (Krumholz et al. 2006). Here, it is sufficient to provide an indication of the role that stellar winds may play in accretion onto a central black hole. The density of the stellar wind at a distance R from the parent star is given by ρwind = Ṁwind 4R2 Vwind In our case, R is the distance from the star to the black hole. One would therefore expect that the accretion rate on to the black hole could be written as (see also Melia 1992) ṀBH ∼ G2 M2BH Ṁwind + c2s) 3/2 Vwind R2 This equation shows that ṀBH ∝ V . We have implicitly assumed that Vwind is greater than the orbital velocity Vorb of the star from which it originates. This is not the case for AGB winds, and so one reaches the limiting case of ṀBH ∝ V , where for the galaxies we have observed Vorb ∼ 50–100km s −1. This is still at least an order of magnitude less than the winds from OB and Wolf-Rayet stars. Thus, even though the mass loss rates from individual OB and Wolf-Rayet stars are similar to those of AGB stars, the AGB winds will fuel a black hole much more efficiently. However, for slow stellar winds that originate close to a 107 M⊙ black hole, the equation breaks down because the conditions of uniformity and spherically symmetry are strongly violated. Indeed, the apparent accretion rate exceeds the outflow rate – implying that essentially the entire wind can be accreted. For AGB wind velocities of 10–30 km s−1, the maximum radius at which the entire wind from a star in Keplerian orbit around a 107 M⊙ black hole will not exceed the escape velocity from that orbit (i.e. Vwind + Vorb < Vesc) is around 10–70pc. We adopt the middle of this range, 40 pc, as the characteristic radius within which it is likely that a significant fraction, and perhaps most, of the AGB winds are accreted onto the black hole. Fig. 9 indicates that the stellar luminosity within this radius is ∼ 2 × 109 L⊙. It is this luminosity that has been used to scale the STARS model (for τSF = 10 Myr and an age of 100 Myr) in Fig. 6, and so one can also simply read off the mass loss from the figure. The mass loss rate for such winds peaks at about 0.1 M⊙ yr −1 and then tails off proportionally to the K-band luminosity, leading to a cumulative mass lost of 2× 107 M⊙ after 1 Gyr (although most of the loss occurs actually occurs within half of this timespan). This mass loss rate is sufficient to power a Seyfert nucleus for a short time. A typical Seyfert with MBH ∼ 10 7 M⊙ requires 0.02 M⊙ yr −1 to radiate at the Eddington limit. Even for the short bursts we have modelled, Fig. 6 shows that this can be supplied by AGB winds for starburst ages in the range 50–200Myrs. We note that taking an AGB star luminosity of 104 L⊙ (which is at the high end of the likely average, Nikolaev & Weinberg 1997) we then find that there are ∼ 2 × 105 AGB stars close enough to the black hole to contribute to accretion. In order to provide at least 0.02 M⊙ yr −1, the typical mass loss rate per star must – 16 – exceed 10−7 M⊙ yr −1, which is the lower limit of the range measured for Galactic AGB stars given above. Thus the mass losses and rates estimated here appear to be plausible. The low speed of these winds means they will not create much turbulence. We quantify this by consid- ering their total mechanical energy 1 mv2 integrated over the same timespan, which is ∼ 1045 J. These two quantities – gas mass ejected and mechanical energy – are compared to those for supernovae below. Supernovae Type ii supernovae are the stellar outflows most able to create turbulence in the interstellar medium, since they typically eject masses of ∼ 5 M⊙ at velocities of ∼ 5000 km s −1 (Chevalier 1977). Each supernova therefore represents a considerable injection of mechanical momentum and energy into the local environment. A large number of compact supernova remnants are known, for example in M 82 and Arp 220, and are believed to have expanded into dense regions with nH ∼ 10 3–104 cm−3 (Chevalier & Fransson 2001). These authors argue that such remnants become radiative when they reach sizes of ∼ 1 pc, at which point the predicted expansion velocity will have slowed to ∼ 500 km s−1. By this time, the shock front will have driven across ∼ 1000 M⊙ of gas. When integrated over the age of the starburst, even for low supernova rates – e.g. the current rate within 30 pc of the nucleus of NGC 3227 is ∼ 0.01 yr−1 (Davies et al. 2006) – this represents a substantial mass of gas that has been affected by supernova remnants. The STARS model we have constructed in Fig. 6 indicates that typically one could expect ∼ 106 supernovae to occur as a result of one of the short-lived starbursts; and that most of these will occur around 10–50Myr after the beginning of the starburst. For a decay timescale of the star formation rate that is longer than τSF = 10 Myr, this timespan will increase. Hence, supernovae may also play a role in causing the observed delay between starburst and AGN activity. STARS calculates the mass loss and mass loss rates using a very simple scheme, assuming that a star ejects all of its lost mass at the end of its life on a stellar track. Thus, it does not calculate the mass lost from supernovae explicitly, rather the combined mass lost from OB winds and supernovae which is much higher. We therefore adopt the ∼ 5 M⊙ per supernova given above, which yields a total ejected mass of ∼ 8 × 106 M⊙. This is about 40% of that released by AGB winds. However, since this gas is ejected at high speed and ṀBH ∝ V , the efficiency with which it can be accreted onto the black hole is extremely low. This can also be seen in the total mechanical energy of ∼ 1050 J, which is several orders of magnitude greater than for AGB winds. In fact the total mechanical energy exceeds the binding energy of the nuclear region, which is of order 1048 J (assuming 108 M⊙ within 40 pc). As a result, it is highly likely that supernova cause some fraction of the gas to be permanently expelled. Indeed, superwinds driven by starbursts are well known in many galaxies. This is not important as long as sufficient gas either remains to fuel the AGN, or more is produced by stellar winds – which, as we have argued above, appears to be the case for AGB stars. 6. Conclusions We have obtained near infrared spectra of 9 nearby active galactic nuclei using adaptive optics to achive high spatial resolution (in several cases better than 10 pc). For 7 of these, integral field spectroscopy with SINFONI allows us to reconstruct the full 2-dimensional distributions and kinematics of the stars and gas. Although the individual AGN are very varied, we have analysed them in a consistent fashion to derive: the stellar K-band luminosity, the dynamical mass, and the equivalent width of the Brγ line. We have combined these with radio continuum data from the literature, which has been used to estimate the supernova rate. We have used these diagnostics to constrain STARS evolutionary synthesis models and hence characterize – 17 – the star formation timescales and ages of the starbursts close around AGN. Our main conclusions can be summarised as follows: • The stellar light profiles show a bright nuclear component with a half-width at half-maximum of less than 50 pc. In a number of cases these nuclear components clearly stand out above an inward extrapolation of the profile measured on larger scales. In addition, there are 2 cases which show kinematical evidence for a distinct stellar component, indicating that the nuclear stellar populations most probably exist in thick nuclear disks. The mean mass surface densities of these disks exceeds 104 M⊙ pc • There is abundant evidence for recent star formation in the last 10–300Myr. But the starbursts are no longer active, implying that the star formation timescale is short, of order a few tens of Myr. While the starbursts were active, the star formation rates would have been much higher than the current rates, reaching as high as 1000 M⊙ kpc −2 in the central few tens of parsecs (comparable to ULIRGs, but on smaller spatial scales). These starbursts would have been Eddington limited. Due to the very high star forming efficiency, the starbursts would have also exhausted their fuel supply on a short timescale and hence have been short-lived. It therefore seems likely that nuclear starbursts are episodic in nature. • There appears to be a delay of 50–100Myr (and in some cases perhaps more) between the onset of star formation and the onset of AGN activity. We have interpreted this as indicating that the starburst has a significant impact on fuelling the central black hole, and have considered whether outflows from stars might be responsible. While supernovae and winds from OB stars eject a large mass of gas, the high velocity of this gas means that its accretion efficiency is extremely low. On the other hand, winds from AGB stars ultimately dominate the total mass ejected in a starburst; and the very slow velocities of these winds mean they can be accreted onto the black hole very efficiently. The authors thank all those who assisted in the observations, and also the referee for a thorough review of the paper. This work was started at the Kavli Institute for Theoretical Physics at Santa Barabara and as a result was supported in part by the National Science Foundation under Grant No. PHY05-51164. RD aknowledges the interesting and useful discussions he had there with Eliot Quartaert, Norm Murray, Julian Krolik and Todd Thompson. Facilities: Keck:II (NIRSPAO, NIRC2), VLT:Yepun (NACO, SINFONI). A. Individual Objects This appendix contains specific details on the individual objects. We summarize our published results from near infrared adaptive optics spectroscopy of individual objects, and present a brief analysis of the new data for several other objects. The aim of re-assessing the data for Mkn 231 that has already been published is to ensure that it is analysed using STARS in a manner that is consistent with the new data. For NGC 7469, we make a significant update of the analysis using new data from integral field spectroscopy. In general, for objects with new data, we provide only the part of the analysis relevant to understanding star formation around the AGN. Our intention is that a complete analysis for each object will be presented in future publications. – 18 – Our analyses are restricted to the nuclear region. Since there is no strict universal definition of what comprises the ‘nuclear region’, we explicitly state in Table 3 the size of the region we study in each galaxy. The table also presents a summary of the primary diagnostics. The way in which these have been derived, and their likely uncertainties, has been discussed in some detail already in Section 3. As such, the description of these methods is not repeated, and in this Section we discuss only issues that require special attention. A.1. Summary of Star Forming Properties of Galaxies already Studied A.1.1. Mkn 231 A detailed analysis of the star formation in the nucleus of Mkn 231 at a resolution of about 0.18′′ (150 pc) was given in Davies et al. (2004b). Here we summarize only the main points; no new data is presented, but the analysis is updated using STARS to make it consistent with the other objects studied in this paper. The presence of stellar absorption features across the nucleus demonstrates the existence of a significant population of stars. The radial distribution and kinematics indicate they lie, like the gas (Downes & Solomon 1998), in a nearly face-on disk. Davies et al. (2004b) found that the dynamical mass imposed a strong constraint on the range of acceptable starburst models, yielding an upper limit to the age of the stars of around 120 Myr. Re-assessing the mass-to-light ratio using STARS models suggests that for the increased mass required by a more face-on orientation (i = 10◦) an upper age of 250 Myr is also possible, depending on the star formation timescale. However, either a small change of only a few degrees to the inclination (to i = 15◦), or a relatively short star formation timescale of 10 Myr would reduce the limit to the ∼100 Myr previously estimated. This is more consistent with the extremely high supernova rate. The stellar luminosity, found from the dilution of the CO absorption (Davies et al. 2004b), indicates that stars within 1′′ (800 pc) of the nucleus contribute 25–40% of the bolometric luminosity of the galaxy. Similarly, within 200 pc, stars comprise 10–15% of Lbol. The age, star formation rate, and size scale (disk scale length of 0.18–0.2′′) are all consistent with high resolution radio continuum imaging (Carilli et al. 1998). A.1.2. Circinus Star formation in the central 16 pc of Circinus was addressed by Mueller Sánchez et al. (2006). The diagnostics given in Table 3 are taken from this reference. We used the depth of the CO 2-0 bandhead to estimate the stellar luminosity, combined with the narrow Brγ flux (which we argued originated in star forming regions rather than the AGN narrow line region) and the radio continuum, to constrain starburst models. The conclusion was that the starburst was less than 80 Myr old and was already decaying. On these scales it contributes 1.4% of Lbol, or more if extinction is considered. A similar nuclear star formation intensity was estimated by Maiolino et al. (1998), who were also able to study Circinus on larger scales. They found that the luminosity of young stars within 200 pc of the AGN was of order 1010L⊙, and hence comparable to the AGN. – 19 – A.1.3. NGC 3227 An analysis similar to that for Circinus was performed on NGC 3227 by Davies et al. (2006), and the diagnostics given in Table 3 are taken from this reference. In this case we were able to make estimates of and correct for contributions of: (1) the narrow line region to Brγ, because there were clear regions along the minor axis that had higher dispersion; (2) the AGN to the radio continuum, by estimating the maximum contribution from an unresolved source; and (3) the bulge stars to the stellar luminosity, by extrapolating the radial profile of the bulge to the inner regions. The STARS models yielded the result that in the nucleus, star formation began approximately 40 Myr ago and must have already ceased. At the resolution of 0.085′′, the most compact component of stellar continuum had a measured FWHM of 0.17′′, suggesting an intrinsic size scale of ∼ 12 pc. Young stars within 30 pc of the AGN (i.e. more than just the most compact region) have a luminosity of ∼ 3 × 109L⊙, which is ∼ 20% of the entire galaxy. A.2. Star Forming Properties of Galaxies with New Data A.2.1. NGC 7469 Star formation on large scales in NGC 7469 has been studied by Genzel et al. (1995). They found that within 800 pc of the nucleus, a region that includes the circumnuclear ring, the luminosity from young stars was ∼ 3 × 1011L⊙, about 70% of the galaxy’s bolometric luminosity. This situation is similar to that in Mkn 231. On smaller scales, the nuclear star formation in NGC 7469 was directly resolved by Davies et al. (2004a) on a size scale of 0.15–0.20′′ (50–65 pc) FWHM. An analysis of the longslit data, similar to that for Mkn 231, was made – making use of stellar absorption features, kinematics, and starburst models. We estimated that the age of this region was no more than 60 Myr under the assumption that the fraction of stellar light in the K-band in the central 0.2′′ was 20–30%. Our new integral field SINFONI observations of NGC 7469 at a spatial resolution of 0.15′′ (measured from both the broad Brγ and the non-stellar continuum profiles, see Section 2) are used here to make a more accurate estimate of the nuclear K-band luminosity. They enable us to provide a short update to the detailed analysis in Davies et al. (2004a). The SINFONI data show that the equivalent width of the 2.3µm CO 2-0 is WCO2−0 = 1.8Å in a 0.8 aperture and 0.9Å in a 0.2′′ aperture. The corresponding K-band magnitudes are K = 10.4 and K = 11.8 respectively. If one takes the intrinsic equivalent width of the 2.3µm CO 2-0 bandhead to be 12Å (see Section 3), one arrives at a more modest value of 8% for the stellar fraction of K-band continuum in the 0.2′′ aperture. The stellar K-band luminosity in this region is then 6 × 107 L⊙. Comparing this to the dynamical mass in Davies et al. (2004a) yields a mass-to-light ratio of M/LK ∼ 0.6 M⊙/L⊙. Previously, extrapolation from a 37 mas slit to a filled aperture had led to an underestimation of the total magnitude but an overestimation of the stellar contribution. Fortuitously, these uncertainties had compensated each other. The same analysis for the 0.8′′ aperture yields a K-band stellar luminosity of 3 × 108 L⊙ and hence M/LK ∼ 1.6 M⊙/L⊙. The K-band datacube yields estimates of the upper limit to WBrγ of 17Å and 11Å in 0.2 ′′ and 0.8′′ apertures respectively. This has been corrected for dilution of the stellar continuum (as described in Section 3) but not for a possible contribution to the narrow Brγ from the AGN. Hence the actual WBrγ corresponding to only the stellar line and continuum emission will be less than these values – indicating that the star formation is unlikely still to be on-going. We estimate the age of the star formation using the STARS models in Fig. 4. Within the 0.2′′ aperture – 20 – this gives 100 Myr, comparable to our original estimate. Such a young age is supported by radio continuum measurements. With a 0.2′′ beam, Colina et al. (2001) reported that the unresolved core flux in NGC 7469 was 12 mJy at 8.4 GHz. With much higher spatial resolution of 0.03′′, Sadler et al. (1995) reported an upper limit to the unresolved 8.4 GHz continuum of 7 mJy. We assume that the difference of 5 mJy is due to emission extended on scales of 10–60 pc which is resolved out of one beam but not the other. As discussed in Section 3, star formation is a likely candidate for such emission. In this case, we would estimate the supernova rate to be ∼ 0.1 yr−1 and the ratio 1010 νSN/LK ∼ 3. This is likely to be a lower limit since there was only an upper limit on the core radio flux density. For a ratio of this order, even allowing for some uncertainty, Fig 4 implies an age consistent with no more than 100 Myr. Within the 0.8′′ aperture, which we adopt in Table 3, continuous star formation is inconsistent with WBrγ . For a star formation timescale of τSF = 100 Myr, the mass-to-light ratio implies an age of 190 Myr, just consistent with the measured value of WBrγ = 11Å. If some of the narrow Brγ is associated with the AGN rather than star formation, then a shorter star formation timescale is required. For τSF = 10 Myr, the ratio M/LK yields an age of 110 Myr. A.2.2. IRAS 05189-2524 Fig. 12 shows the H-band spectrum integrated across two segments of the NACO slit, located on either side of the nucleus. It shows that even away from the nucleus, the depth of the stellar absorption features is only a few percent. We have therefore decomposed the data into the stellar and non-stellar parts using both the stellar absorption features and the spectral slope of the continuum. The latter method has been shown to work for well sampled data by Davies et al. (2004a). The rationale is that the hot dust associated with the AGN will be much redder than the stellar continuum. An AGN component is also expected to be unresolved for a galaxy at the distance (170 Mpc) of IRAS 05189-2524. The spectral slope was determined by fitting a linear function to the spectrum at each spatial position along the slit. It is plotted as a function of position in Fig. 13, showing a single narrow peak. A Gaussian fit to this yields a spatial resolution of 0.12′′ (100 pc) FWHM. The stellar continuum, also shown in Fig. 13, has been determined by summing the four most prominent absorption features: CO 4-1, Si I, CO 5-2, CO 6-3. While a Gaussian is not an optimal fit to this profile, it does yield an aproximate size scale, which we find to be 0.27′′ FWHM. Quadrature correction with the spatial resolution yields an intrinsic size of 0.25′′ (200 pc). As a cross-check, in the figure we have compared the sum of these two components to the full continuum profile. The good match indicates that the decomposition appears to be reasonable. Remarkably, the 200 pc size of the nuclear stellar light is very similar to that of the 8.44 GHz radio con- tinuum map of Condon et al. (1991). With a beam size of 0.50′′×0.25′′, they resolved the nuclear component to have an intrinsic size of 0.20′′×0.17′′. In constrast to radio sources which are powered by AGN and have brightness temperatures Tb ≫ 10 5 K, the emission here is resolved and has a low brightness temperature of ∼ 4000 K. This implies a star forming origin. Using their scaling relations further suggests that the flux density corresponds to a supernova rate of ∼ 1 yr−1. As described in Section 3, we have estimated the stellar luminosity by comparing the H-band spectrum to a template star to correct for dilution. We used HR 8465 a K 1.5 I star for which the equivalent width of CO 6-3 is 4.2Å, within the 4–5Å range predicted by STARS in Fig. 2. By extrapolating from the spatial profiles along the slit we have estimated the integrated equivalent width within a 1.1′′ aperture, for which Scoville et al. (2000) gave an H-band magnitude of 11.83. Using all four features above we find for the – 21 – template W = 14.4Å and for IRAS 05189-2524 WCO6−3 = 6.7Å. This implies that in the central 1.1 approximately 45% of the H-band continuum originates in stars. Using the colour conversion H −K = 0.15 from Fig. 3 (see Section 3) we find a K-band magnitude for the stars of 12.55 mag and hence a K-band stellar luminosity of 2×109 L⊙. Putting these results together we derive a ratio of supernova rate to K-band stellar luminosity of νSN[yr −1]/LK [10 10L⊙] ∼ 5. Applying corrections for extinction and an AGN contribution would tend to decrease this ratio. As a second diagnostic we use WBrγ . We estimate the dilution of the K-band continuum via two methods. Firstly, we measure WNaI = 0.3Å, indicating a stellar fraction of 0.10–0.15. A consistency check is provided by the H-band dilution, which we extrapolate to the K-band using blackbody functions for the stars and dust assuming characteristic temperatures of 5000 K and 1000 K respectively. This method suggests the K-band stellar fraction is around ∼ 0.14. Hence correcting the directly measured equivalent width of the narrow Brγ for the non-stellar continuum yields WBrγ = 4–5Å. Since IRAS 05189-2524 is close to face-on (Scoville et al. 2000), it is not straightforward to make a reliable estimate of the dynamical mass. Nevertheless, requiring νSN/LK to be high while WBrγ is low already puts significant constraints on the star formation history. Thus, although the star formation has probably ended, the age is unlikely to be greater than 100 Myr, and could be as low as 50 Myr where νSN/LK peaks. For such ages the ratio Lbol/LK is in the range 100–150. Hence for the young stars within 0.55 (450 pc) of the nucleus we find Lbol ∼ (2–3) × 10 11 L⊙, about 20% of Lbol for the galaxy. A.2.3. NGC 2992 The spatial resolution of the K-band data for NGC 2992 has been estimated from both the broad Brγ and the non-stellar continuum (see Section 2 and 3). The two methods yield symmetric PSFs, with FWHMs of 0.32′′ and 0.29′′ respectively, corresponding to 50 pc. The CO 2-0 equivalent width of ∼ 3Å implies a stellar fraction of ∼ 0.25 within a radius of 0.4′′, and hence a stellar luminosity of LK = 3.5 × 10 7 L⊙. Unlike IRAS 05189-2524, the radio continuum in NGC 2992 is quite complex. Much of the extended emission on scales of a few arcsec appears to originate from a superbubble, driven either by the AGN or by a nuclear starburst. On the other hand, most of the nuclear emission seems to be unresolved. With a beam size of 0.34′′×0.49′′, Wehrle & Morris (1988) measured the unresolved flux to be 7 mJy at 5 GHz. At a resolution better than 0.1′′, Sadler et al. (1995) reported a 2.3 GHz flux of 6 mJy. Based on this as well as non-detections at 1.7 GHz and 8.4 GHz, they estimated the core flux at 5 GHz to be <6 mJy. Taking a flat spectral index, as indicated by archival data (Chapman et al. 2000), one might expect the 5 GHz core flux to be not much less than 6 mJy, leaving room for only ∼ 1 mJy in extended emission in the central 0.5′′. If we assume this difference can be attributed to star formation, it implies a supernova rate of ∼ 0.003 yr−1 and hence 1010 νSN/LK ∼ 1. Fig. 4 shows that a ratio of this order is what one might expect for ages up to 200 Myr. However, given the uncertainty it does not impose a significant constraint. It is also difficult to quantify what fraction of the narrow Brγ is associated with star formation. This is made clear in Fig. 14 which shows that the morphology of the line (centre left panel) does not follow that of the stars (far left). In addition, particularly the south-west side is associated with velocities that are bluer than the surrounding emission, indicative of motion towards us. The western edge also exhibits high velocity dispersion. Taken together, these suggest that we may be seeing outflow from the apex of an ionisation cone – 22 – with a relatively large opening angle. This interpretation would tend to support the hypothesis that the radio bubble has been driven by the AGN. The stellar continuum appears to trace an inclined disk, the north west side of which is more obscured (Fig. 14). However, the velocity dispersion is high, exceeding 150 km s−1 across the whole field (Fig. 15). This is similar to the 160 km s−1 reported by Nelson & Whittle (1995) from optical spectroscopy, and suggests that we are seeing bulge stars. To analyse the radial luminosity profile we have fitted it with both an r1/4 law and exponential profile. The fits in Fig. 16 were optimised at radii r > 0.5′′ and then extrapolated inwards, convolved with the PSF. Whether one could claim that there is excess continuum in the nucleus depends on the profile fitted. The r1/4 law provides a stronger constraint since it is more cuspy, and suggests there is no excess. Although this evidence is inconclusive, Fig. 15 suggests that there is some kinematic evidence favouring the existence of a distinct nuclear stellar population. This comes in the form of a small unresolved drop in dispersion at the centre, similar to those in NGC 1097 and NGC 1068. While the evidence in NGC 2992 is not compelling, the dispersion is consistent with there being an equivalent – but fainter – nuclear disk on a scale of less than our resolution of 50 pc. In general it seems that the K-band light we are seeing is dominated by the bulge, and we are therefore unable to probe in detail the inner region where it seems that more recent star formation has probably occurred. Thus, although the available data suggest there has likely been recent star formation in the nucleus of NGC 2992, the only strong constraint we can apply is that continuous star formation in the central arcsec over the last Gyr can be ruled out since it would require WBrγ > 10–15Å. We therefore omit NGC 2992 from the discussion and analysis in Sections 4 and 5. A.2.4. NGC 1097 In NGC 1097, the first evidence for recent star formation near the nucleus was in the form of a reduction in the stellar velocity dispersion. Emsellem et al. (2001) proposed this could be explained by the presence of a dynamically cold nuclear disk that had recently formed stars. Direct observations of a spiral structure in the central few arcsec, from K-band imaging (Prieto et al. 2005) and [N ii] streaming motions (Fathi et al. 2006), have since confirmed this idea. However, some issues remain open, such as why there are three spiral arms rather than the usual two, and why gas along one of them appears to be outflowing. Our data, at a resolution of 0.25′′ measured from the H-band non-stellar continuum, also reveal the same spiral structure. Indeed, we find that it is traced by the morphology of the CO bandhead absorption as well as by the 2.12µm H2 1-0 S(1) line. Interestingly, 1-0 S(1) emission is stronger where the stellar features are weaker. This suggests that obscuration by gas and dust plays an important role. Fig. 18 shows that an r1/4 law, typical of stellar bulges, with effective radius Reff = 0.5 ′′ is a good fit to the stellar radial profile at 0.5′′ < r < 1.8′′. It therefore seems reasonable to argue that at these radii it is only the gas that lies in a disk. In this picture the spiral structure in the stellar continuum arises solely due to extinction of the stars behind the disk. Extrapolating this fit, convolved with the PSF, to the nucleus indicates that at r < 0.5′′ there is at least 25% excess stellar continuum. There could be much more, given that it coincides with a change in the dominant kinematics. For NGC 1097 we parameterized the kinematics of the gas and stars quantitatively using kinemetry. Based on the uniformity of the velocity field, we made the simplifying assumption that across the central 4′′ the gas lies in a single plane whose centre is coincident with the peak of the non-stellar emission. We were then able to derive the position angle and inclination of the disk (see Section 2). The 2D kinematics of the stars – 23 – is traced via the CO2-0 absorption bandhead, and that of the gas through the 1-0S(1) emission line. These independently yielded similar parameters: both gave a position angle of −49◦ and their inclinations were 43◦ and 32◦ respectively. These are fully consistent with values found by other authors (Storchi-Bergmann et al. 2003; Fathi et al. 2006). The resulting rotation curves and velocity dispersions are shown in Fig. 19. The residuals, which can be seen in the velocity field of the gas but not the stars, and their relation to the spiral structure described above will be discussed elsewhere (Davies et al. in prep). The important result here is that at our spatial resolution, we find that the central stellar dispersion is σ∗ = 100 km s −1, less than the surrounding 150 km s−1 and also less than that in the seeing limited spectra of Emsellem et al. (2001). In the same region we find that the rotation velocity of the gas starts to decrease rapidly, and its dispersion increases from σgas ∼ 40 km s −1 to ∼ 80 km s−1. Fig. 19 also shows that while the kinematics of the stars and gas are rather different at large (> 0.5′′) radii, they are remarkably similar at radii < 0.5′′. This certainly provides a strong indication that in the nuclear region the stars and gas are coupled, most likely in a (perhaps thick) disk; and that the stars in this disk, which are bright and hence presumably young, give rise to the excess stellar continuum observed. Evidence for a recent starburst has been found by Storchi-Bergmann et al. (2005) through optical and UV spectra. They argued that a number of features they observed could only arise from an 106 M⊙ instanta- neous starburst, which occurred a few Myr ago and is reddened by AV = 3 mag of extinction. Using STARS we have modeled this starburst as a 106 M⊙ burst beginning 8 Myr ago with an exponential decay timescale of 1 Myr. The age we have used is a little older to keep the Brγ equivalent width low; and at this age, the model predicts WBrγ = 4Å. As Fig. 20 shows, the observed Brγ is weak, although perhaps slightly resolved. Corrected for the non-stellar continuum, we measure only WBrγ ∼ 1Å. However, the bulge population may account for a significant fraction of the K-band stellar continuum. Correcting also for this could increase WBrγ to 2–5Å, consistent with that of the model – assuming that the Brγ is associated with the starburst rather than the AGN. To within a factor of a few, the scale of the model starburst is also consistent with that measured: In the central 0.5′′ we measure a Brγ flux of 2 × 10−19 W m−2, compared to that predicted by the model of 5× 10−19 W m−2. Given the uncertainties – factors of a few – both in the parameters of the starburst model and also in the corrections we have applied to the data, we consider this a good agreement. We cannot constrain the starburst further due to its compactness. Storchi-Bergmann et al. (2005) found that it was occurring in the central 0.2′′, whereas our resolution is only 0.25′′. The Brγ emission is confined to the central 0.4–0.5′′, although its size is hard to measure due to its weakness with respect to the stellar absorption features. In this region the K-band stellar luminosity is 4.5× 106 L⊙. To estimate the dynamical mass we use the mean kinematics of the stars and gas, i.e. Vrot = 40 km s −1 (corrected for inclination) and σ = 90 km s−1 (this is the central value, which is least biassed by bulge stars), yielding 1.4× 108 M⊙. This is actually dominated by the black hole, which has a mass of (1.2±2)×108 M⊙ (Lewis & Eracleous 2006). The difference between these implies a mass of gas and stars of ∼ 2× 107 M⊙, although with a large uncertainty. The associated mass-to-light ratio is M/LK ∼ 4. On its own, this implies that over the relatively large area that it encompasses, the maximum characteristic age for the star formation is a few hundred Myr. If one speculates that star formation has been occurring sporadically for this timescale, then the starburst seen by Storchi-Bergmann et al. (2005) is the most recent active episode. In order to make a rough estimate of the supernova rate in the central region we make use of measure- ments reported by Hummel et al. (1987). They find an unresolved component (size < 0.1′′) with 5 GHz flux density 3.5 ± 0.3 mJy, but at lower resolution there is a 4.1 ± 0.3 mJy component of size 1′′. As discussed in Section 3 we assume that the difference – albeit with only marginal significance – of 0.6 ± 0.4 mJy is due to star formation in the central region, which implies a supernova rate of 6 × 10−4 yr−1 and hence – 24 – 1010 νSN/LK ∼ 1.3, a value consistent with rather more recent star formation. Indeed, when compared to Fig. 4, this and the low WBrγ imply a young age and short star formation timescale. For τSF = 10 Myr the age is 60–70 Myr; for an instantaneous burst of star formation, the age would be ∼ 10 Myr, broadly consistent with that of Storchi-Bergmann et al. (2005). Thus, although our data do not uniquely constrain the age of the starburst in the nucleus of NGC 1097, they do indicate that recent star formation has occurred; and they are consistent with a very young compact starburst similar to that derived from optical and UV data. A.2.5. NGC 1068 Evidence for a stellar core in NGC 1068 with an intrinsic size scale of ∼ 45 pc was first presented by Thatte et al. (1997). Based on kinematics measured in large (2–4′′) apertures, they assumed the core was virialized and estimated a mass-to-light ratio based on this assumption leading to an upper limit on the stellar age of 1600Myr. Making a reasonable correction for an assumed old component lead to a younger age of 500 Myr. Stellar kinematics from optical integral field spectra (Emsellem et al. 2006a; Gerssen et al. 2006) show evidence for a drop in the stellar velocity dispersion in the central few arcsec to σ∗ ∼ 100 km s −1, inside a region of higher 150–200km s−1 dispersion (presumably the bulge). Our near infrared adaptive optics data are able to fully resolve the inner region where σ∗ drops, as shown in Fig. 21. As for NGC 1097, the velocity distribution of the stars was derived through kinemetry, again making use of the uniformity of the stellar velocity field to justify the simplifying assumption that the position angle and inclination do not change significantly in the central 4′′. The derived inclination of 40◦ and position angle of 85◦ are quantitatively similar to those found by other authors in the central few to tens of arcseconds (Emsellem et al. 2006a; Gerssen et al. 2006; Garćıa-Lorenzo et al. 1999). The uniformity of the stellar kinematics is in contrast to molecular gas kinematics, as traced via the 1-0 S(1) line, which are strongly perturbed and show several distinct structures superimposed. These are too complex to permit a comparably simple analysis and will be discussed, together with the residuals in the stellar kinematics in a future work (Mueller Sánchez et al. in prep). The crucial result relevant here is that at our H-band resolution of 0.10′′ we find that σ∗ reduces from 130 km s−1 at 1–2′′ to only 70 km s−1 in the very centre. That there is in the same region an excess in the stellar continuum is demonstrated in Fig. 22. Here we show the radial profile of the stellar continuum from both SINFONI integral field spectra out to a radius of 2′′ and NACO longslit spectra out to 5′′ (350 pc). At radii 1–5′′, corresponding roughly to the region of high stellar dispersion measured by Emsellem et al. (2006a), the profile is well matched by an r1/4 law, as one might expect for a bulge. At radii r < 1′′ – the same radius at which we begin to see a discernable reduction in the stellar dispersion – the stellar continuum increases by as much as a factor 2 above the inward extrapolation of the profile, indicating that there is extra emission. As for NGC 1097, the combined signature of dynamically cool kinematics and excess emission is strong evidence for a nuclear disk which has experienced recent star formation. We can make an estimate of the characteristic age of the star formation in the central arcsec based on the mass-to-light ratio in a similar way to Thatte et al. (1997). Because the stars appear to lie in a disk, we estimate the dynamical mass as described in Section 3 from the stellar kinematics, using the rotation velocity and applying a correction for the dispersion. The stellar rotation curve is essentially flat at 0.1–0.5′′, with V∗ = 45 km s −1 (corrected for inclination). We also take σ∗ = 70 km s −1, which is the central value and – 25 – hence least biassed by the high dispersion bulge stars. These lead to a mass of 1.3× 108 M⊙ within r = 0.5 (35 pc), and a mean surface density of 3×104 M⊙ pc −2. Correcting for the non-stellar continuum, the H-band magnitude (which the behaviour of σ∗ indicates is dominated by the disk emission) in the same region is 11.53 mag. For H − K = 0.15 mag (Fig. 4), we find LK = 4.3 × 10 7 L⊙ and hence M/LK = 3 M⊙/L⊙. If no star formation is on-going, this implies a characteristic age of 200–300Myr fairly independent of the timescale (for τSF . 100 Myr, see Fig. 4) on which stars were formed. We note that this is significantly younger than the age estimated by Thatte et al. (1997) primarily because their mass was derived using a higher σ∗ corresponding to the bulge stars. The assumption of no current star formation is clearly demonstrated by the Brγ map in Fig. 23. Away from the knots of Brγ, which are associated with the coronal lines and the jet rather than possible star formation, the equivalent width is WBrγ ∼ 4Å. This is significantly less than that for continuous star formation of any age. Thus, while it seems likely that star formation has occurred in the last few hundred Myr, it also seems an unavoidable conclusion that there is no current star formation. To complete our set of diagnostics for NGC 1068, we consider also the radio continuum. This is clearly dominated by phenomena associated with the AGN and jets, and our best estimate of the flux density away from these features is given by the lowest contour in maps such as Figure 1 of Gallimore et al. (2004). From this we estimate an upper limit to the 5 GHz continuum associated with star formation of 128 mJy within r < 0.5′′. However, converting to a supernova rate and comparing to the K-band stellar luminosity yields a limit that is not useful, being an order of magnitude above the largest expected values. A.2.6. NGC 3783 At near infrared wavelengths, the AGN in NGC 3783 is remarkably bright. Integrated over the central 0.5′′ less than 4% of the K-band continuum is stellar. In addition, the broad Brackett lines are very strong and dominate the H-band. Both of these phenomena are immediately clear from the H- and K-band spectra in Fig. 24. However, it does mean that the spatial resolution can be measured easily from both the non-stellar continuum and the broad emission lines (see Section 2). We find the K-band PSF to be symmetrical with a FWHM of 0.17′′. Due to the ubiquitous Brackett emission in the H-band we were unable to reliably trace the stellar absorption features and map out the stellar continuum. Instead we have used the CO 2-0 bandhead at 2.3µm even though the dilution at the nucleus itself is extreme. The azimuthally averaged radial profile is shown in Fig. 25 together with the PSF for reference. At radii from 0.2′′–1.6′′ (the maximum we can measure) the profile is well fit by an r1/4 de Vaucouleurs law with Reff = 0.6 ′′ (120 pc). As has been the case previously, at smaller radii we find an excess that here is perhaps marginally resolved. Thus a substantial fraction of the near infrared stellar continuum in the central region is likely to originate in a population of stars distinct from the bulge. We were unable to measure the stellar kinematics due to the limited signal-to-noise. Instead, we used the molecular gas kinematics to estimate the dynamical mass. As before, we used kinemetry to derive the position angle of −14◦ and the inclination in the range 35–39◦. This orientation is consistent with the larger (20′′) scale isophotes in the J-band 2MASS image and implies that in NGC 3783 there is no significant warp on scales of 50 pc to 4 kpc. A small inclination is also consistent with its classification as a Seyfert 1. Adopting these values, the resulting rotation curve is shown in Fig. 26. At very small radii the rising rotation curve may be the result of beam smearing across the nucleus. At r > 0.2′′, the falling curve suggests that – 26 – the rotation is dominated by the central (r < 0.2′′) mass, perhaps the supermassive black hole. We estimate the dynamical mass within a radius of 0.3′′ (60 pc), corresponding to the point where the excess continuum begins and also where the rotation curve appears to be unaffected by beam smearing. Taking Vrot = 60 km s −1 and σ = 35 km s−1 we derive a dynamical mass of Mdyn = 1.0 × 10 8 M⊙. The black hole mass of 3 × 107 M⊙ (from reverberation mapping, Peterson et al. 2004) is only 30% of this, and so cannot be dominating the dynamics on this scale unless its mass is underestimated. With respect to this, we note that Peterson et al. (2004) claim the statistical uncertainty in masses derived from reverberation mapping is about a factor 3. Alternatively, there may be a compact mass of gas and stars at r < 0.3′′. However, including σ in the mass estimate implicitly assumes that the dispersion arises from macroscopic motions. On the other hand, because we are observing only the hot H2, it is possible that the dispersion is dominated by turbulence arising from shocks or UV heating of clouds that generate the 1-0 S(1) emission – issues that are discussed in more detail by Hicks et al. (in prep.). In this case we will have overestimated the dynamical mass. Excluding σ from the mass estimation yields Mdyn = 5 × 10 7 M⊙. We consider these two estimates as denoting the maximum range of possible masses. Subtracting MBH then gives a mass of stars and gas in the range (2–7)×107 M⊙, implying a mass surface density of 1700–6000M⊙ pc −2 and M/LK = 0.6–2.1M⊙/L⊙. Based on these ratios alone, Fig 4 indicates that the characteristic age of the star formation may be as low as ∼ 70 Myr, although it could also be an order of magnitude greater. Without additional diagnostics we cannot discriminate further. We are unable to use Brγ as an additional constraint on the star formation history. Its morphology and velocity field are similar to that of [Sivi], and rather different from the 1-0 S(1). It shows an extension to the north which appears to be outflowing at > 50 km s−1 (Fig 27) – perhaps tracing an ionisation cone. Since the Brγ resembles the [Sivi], it is reasonable to conclude that it too is associated with the AGN rather than star formation. Thus the equivalent width of Brγ (with respect to the stellar continuum) of WBrγ = 30Å represents an upper limit to that associated with star formation. The radio continuum in the nucleus of NGC 3783 has been measured with several beam sizes at 8.5 GHz. For a beam of 1.59′′ × 0.74′′, Morganti et al. (1999) found it was unresolved with a flux density of 8.15 ± 0.24 mJy. With a smaller ∼ 0.25′′ beam, Schmitt et al. (2001) measured a total flux density of 8.0 mJy dominated by an unresolved component of 7.7± 0.05mJy. At smaller scales still of ∼ 0.03′′ corresponding to 6 pc, Sadler et al. (1995) placed an upper limit on the 8.5 GHz flux density of 7 mJy. Taken together, these results imply that there is some modest 8.5 GHz radio continuum of 0.7–1 mJy extended on scales of 0.3–1′′. Based on this we estimate a supernova rate as described in Section 3 of ∼ 0.007 yr−1, and hence a ratio 1010 νSN/LK ∼ 2. Given that the unresolved radio continuum on the smallest scales is an upper limit, the extended component may be stronger and hence the true νSN/LK ratio may be greater than that estimated here. Fig 4 then puts a relatively strong limit of ∼ 50 Myr on the maximum age of the star formation. This age is fully consistent with that above associated with our lower mass estimate. The value of WBrγ < 30Å above does not impose additional constraints, although we note that if the Brγ flux associated with star formation is only a small fraction of the total then it would imply that the timescale over which the star formation was active is no longer than a few times ∼ 10 Myr. 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Table of Observations Object Banda Res.b (′′) Date Instrument Mkn 231c H 0.176 May ’02 Keck, NIRC2 NGC 7469c K 0.085 Nov ’02 Keck, NIRSPAO K 0.15 Jul ’04 VLT, SINFONI Circinusc K 0.22 Jul ’04 VLT, SINFONI NGC 3227c K 0.085 Dec ’04 VLT, SINFONI IRAS 05189-2524 H 0.12 Dec ’02 VLT, NACO NGC 2992 K 0.30 Mar ’05 VLT, SINFONI NGC 1097 H 0.245 Oct ’05 VLT, SINFONI NGC 1068 H 0.10 Oct ’05 VLT, SINFONI H 0.13 Dec ’02 VLT, NACO NGC 3783 K 0.17 Mar ’05 VLT, SINFONI aBand used for determining the quantitative star formation proper- ties. NGC 1097, NGC 1068, and NGC 3783 were actually observed in H and K bands. bSpatial resolution (FWHM) estimated from the data itself, using the methods described in Section 2. cReferences to detailed studies of individual objects: Mkn 231 (Davies et al. 2004b), NGC 7469 (Davies et al. 2004a), Circinus, (Mueller Sánchez et al. 2006), NGC 3227 (Davies et al. 2006). – 33 – Table 2. Summary of basic data for AGN Object Classificationa Distance logLbol b logMBH Ref.c Mpc for MBH Mkn 231 ULIRG, Sy 1, QSO 170 12.5 7.2 1 NGC 7469 Sy 1 66 11.5 7.0 2 Circinus Sy 2 4 10.2 6.2 3 NGC 3227 Sy 1 17 10.2 7.3 4 IRAS 05189-2524 ULIRG, Sy 1 170 12.1 7.5 1 NGC 2992 Sy 1 33 10.7 7.7 5 NGC 1097 LINER, Sy 1 18 10.9 8.1 6 NGC 1068 Sy 2 14 11.5 6.9 7 NGC 3783 Sy 1 42 10.8 7.5 2 a Classifications are taken primarily from the NASA/IPAC Extragalactic Database. In addition, we have labelled as Seyfert 1 those for which we have observed broad (i.e. FWHM > 1000 km s−1) Brγ; see also Fig 1. b Calculated in the range 8–1000µm from the IRAS 12–100µm flux densities; with an additional correction for optical and near-infrared luminosity in cases where appropriate. c References for black hole masses: (1) Dasyra et al. (2006); (2) Peterson et al. (2004); (3) Greenhill et al. (2003); (4) Davies et al. (2006); (5) Woo & Urry (2002); (6) Lewis & Eracleous (2006); (7) Lodato & Bertin (2003) Table 3. Measured & Derived Properties of the Nucleia Object radius logLK∗ Σdyn WBrγ Mdyn/LK 10 10νSN/LK age 〈SFR〉 ′′ pc 104M⊙ pc −2 Å M⊙/L⊙ yr −1L−1 Myr M⊙ yr −1 kpc−2 Mkn 231b 0.6 480 9.3 9.8 0.9 — 3.1 20 120–250 25–50 NGC 7469 0.4 128 8.5 8.7 1.0 11 1.6 3 110–190 50–100 Circinus 0.4 8 6.2 7.5 17 30 23 1.5 80 ∼ 70 NGC 3227c 0.4 32 7.8 8.0 3.7 4 1.9 2.2 40 ∼ 380 IRAS 05189-2524 0.55 450 9.3 — — 4 — 5 50–100 30–70 NGC 2992d 0.4 64 7.5 — — <12 — 1 — — NGC 1097e 0.25 22 6.7 8.2 1.3 1 4.5 1.4 8 ∼ 80 NGC 1068 0.5 35 7.6 8.1 3.4 4 3.0 <20 200–300 90–170 NGC 3783f 0.3 60 7.5 7.3 0.2 <30 0.6 2 50–70 30–60 a The methods used to measure these quantities (within the radii given) are described in Section 3. Specific issues associated with individual objects are discussed in Appendix A. b Mdyn depends strongly on even small changes to the inclination; here it is given for i = 10 ◦. Correcting Mdyn for an estimate of the gas mass given in Downes & Solomon (1998) yields M/LK = 2.3M⊙/L⊙. c The best star formation models indicate that M/LK is much less than the limit given here using the dynamical mass. d It is likely that much of the narrow Brγ in the nuclear region here is associated with an ionisation cone. In addition, the high stellar velocity dispersion, even on the smallest scales we have been able to measure, suggests that the K-band light is dominated by the bulge. e Correcting WBrγ for the old stellar population would probably yield a value in the range 2–5Å. Even on this scale the dynamical mass is dominated by the supermassive black hole. Both Σdyn and M/LK are estimated after subtracting MBH. f Much of the Brγ here is outflowing and hence associated with the AGN. Mdyn is derived from gas kinematics as described in the text. Both Σdyn and M/LK are estimated after subtracting MBH. – 35 – Fig. 1.— K-band spectra showing broad Brγ emission in 3 AGN which are not usually classified as Seyfert 1. Top: NGC 3227 (0.25′′ aperture); Middle: IRAS 05189−2524 (1′′ aperture); Bottom: NGC 2992 (0.5′′ aper- ture). The most prominent emission and absorption features are marked. Fig. 2.— Left: equivalent width of the CO 2-0 and CO 6-3 features for various stars and galaxies. The late- type supergiant stars (skeletal star shapes) and giant stars (open star shapes) are taken from Origlia et al. (1993). The galaxies (denoted ‘E’ for elliptical, ‘S’ for spiral, and ‘H’ for star forming Hii galaxy) are from Oliva et al. (1995). The dashed box encloses the region for which there is no more than 20% deviation from each of the values WCO2−0 = 12Å and WCO6−3 = 4.5Å. Centre and right: calculated WCO6−3 and WCO2−0 respectively from STARS for several different star formation histories. Each line is truncated when the cluster luminosity falls below 1/15 of its maximum. In each case, the dashed lines show typical values adopted, and the dotted lines denote a range of ±20%. – 36 – Fig. 3.— Left: H-K colour of star clusters with different star formation timescales and ages, as calculated by STARS. Right: Ratio of bolometric to K-band luminosity. Although the range of 20–200 initially appears large, the uncertainty on an intermediate value of 60 is only 0.3 dex. This is small compared to the range of interest in the paper, which is several orders of magnitude. Fig. 4.— Various diagnostics calcuated with STARS for several star formation timescales, as functions of age: Brγ equivalent width, supernova rate, and mass-to-light ratio. Note that all are normalised to the K-band stellar continuum; and that LK is the total luminosity in the 1.9–2.5µm band in units of bolometric solar luminosity (1Lbol = 3.8 × 10 26 W ), rather than the other frequently used monochromatic definition which has units of the solar K-band luminosity density. – 37 – Fig. 5.— Size scales of nuclear star forming regions. The profile has been determined from the CO absorption features in the H or K band, which are approximately independent of star formation history (see text for details). For longslit data, spatial profiles have been averaged; for integral field data, azimuthally averaged proifles are shown. For NGC1068, data at two different pixel scales are shown (corresponding to the solid and dashed brown lines). The horizontal dotted line is drawn at half-maximum height, to assist in estimating size scales by eye. – 38 – Fig. 6.— A STARS star formation model based on the main characteristics of the observed starbursts, which is illustrative of a ‘typical’ nuclear starburst that we have observed. The scaling is fixed as 2× 109 M⊙ (typical of that within a 30 pc radius; Fig. 9) at an age of 100 Myr (the typical age in Table 3). The star formation timescale is τSF = 10 Myr to reproduce a low Brγ equivalent width. The panels are, from top left: (a) star formation rate (SFR); (b) number of ionising photons (QLyc, proportional to the Brγ luminosity); (c) supernova rate (SNR); (d) bolometric luminosity (Lbol); (e) K-band luminosity (LK); (f) cumulative number of supernovae; (g) cumulative mass that has been recycled back into the ISM by supernovae and winds; (h) mass loss rate from stars; (i) rate at which the lost mass can in principle be accreted onto a central supermassive black hole (due to its outflow speed; see text for details). In the last two panels, the mass loss is split into that due to OB and Wolf-Rayet stars and supernovae (dotted lines), and that due to late-type and AGB stars (dashed lines). Stellar mass loss in STARS is accounted for at the end of each star’s life as the difference in mass between the original (ZAMS) stellar mass and the remnant mass as the end-product of its stellar evolution, as described in Sternberg (1998). For this reason, the mass loss rates from OB and Wolf-Rayet stars do not appear explicitly in panel (h) at very young cluster ages. – 39 – Fig. 7.— Mean enclosed mass surface density as a function of radius. The points are from data given in Table 3; the dashed lines represent the mass models derived for NGC 7469 and Mkn 231 (Davies et al. 2004a,b). The galaxies all follow the same trend towards increasing densities in the central regions. Fig. 8.— Integrated bolometric luminosity of the young stars Lbol∗ as a fraction of that of the galaxy Lbol−galaxy, plotted as a function of radius. Lbol∗ is calculated from the stellar LK or LH assuming that, on the generally small scales here, all the near infrared stellar continuum originates in the young stars. On 10 pc scales the contribution of young stars is at most a few percent of the galaxy’s total luminosity, while on kpc scales it may be significant and hence comparable to the AGN luminosity. For NGC 1068, data at two different pixel scales are shown (corresponding to the solid and dashed brown lines). – 40 – Fig. 9.— Integrated bolometric luminosity of young stars Lbol∗ (see Fig. 8) as a function of radius. For comparison, the dotted line has constant surface brightness. For NGC 1068, data at two different pixel scales are shown (corresponding to the solid and dashed brown lines). Fig. 10.— Luminosity to mass ratio calculated by STARS as a function of age for different star formation timescales. The dotted lines show how the ratio would vary if gas were fed in to a cluster at the same rate as it was converted into stars. The solid lines assume that the gas is present at the start, but at the end has all been processed in stars. A cluster can only exceed 500 L⊙/M⊙ for a timescale of ∼ 10 Myr. – 41 – Fig. 11.— Graph showing how the luminosity of an AGN might be related to the age of the most recent episode of nuclear star formation. On the left is shown the luminosity in solar units; on the right, it is with respect to the Eddington luminosity for the black hole. Generally the luminosity of the AGN is not well known and so we have approximated it by 0.5Lbol, and adopted an uncertainty of a factor 2. The starburst age refers to our best estimate of the most recent episode of star formation within the central 10–100pc, as given in Table 3. See the text for details of the adopted uncertainties. – 42 – Fig. 12.— Spectrum (in grey) of IRAS 05189-2524, integrated over two 0.22′′ wide sections centered ±0.27′′ either side of the nucleus, and which have been shifted to match their velocities. Overplotted in black is a fit to the continuum, comprising spectra of various supergiant stars. The main absorption and emission features have been identified. Fig. 13.— Spatial profiles of non-stellar (left), stellar (centre), and total (right) continuum for IRAS 05189- 2524 (1′′ = 800 pc). The first two have been derived at each point along the spatial extent of the slit from the spectral slope and the stellar absorption features respectively. A comparison of their sum (dashed line in right panel) to the total continuum indicates that the decomposition appears to be reasonable. – 43 – Fig. 14.— Maps of NGC 2992 (1′′ = 160 pc). From left to right: stellar continuum, narrow Brγ, Brγ velocity (−150 to +150 km s−1), and Brγ dispersion (0 to 200 km s−1). For reference, on each panel are superimposed contours of the Brγ flux (8, 16, 32, and 64% of the peak). The symbol plotted on the map of the line flux indicates the centre of the broad Brγ and non-stellar emission. The narrow Brγ emission extends far more to the north west than the stellar continuum. And, particularly on the south western edge, it exhibits a blue shifted velocity and high dispersion. All these are consistent with an interpretation as the apex of an ionisation cone. Fig. 15.— Radial profiles of velocity and dispersion for the stars in NGC 2992 (1′′ = 160 pc). The 2D maps were analysed using the kinemetric technique described by Krajnović et al. (2006) which yielded a position angle of 24◦ and an inclination of ∼ 40◦, not dissimilar to the isophotal values of 30◦ and 50◦ respectively. The rotation curve has been corrected for the inclination. – 44 – Fig. 16.— Radial profile of the stellar continuum in NGC 2992 (1′′ = 160 pc), derived from isophotal analysis. Solid circles denote the stellar continuum (i.e. already corrected for the non-stellar component). Overplotted with triangles are an r1/4 law (top panel) and an exponential profile (bottom panel). The profiles were fitted at radii r > 0.5′′ and extrapolated inwards, convolved with the PSF which is shown as open squares. Both fits are equally good at r > 0.5′′, but only the exponential suggests there might be excess continuum at the centre, arising from a distinct stellar population. This is therefore inconclusive. – 45 – Fig. 17.— Spectrum of NGC 1097 (thick grey line), extracted within an aperture of radius 0.2′′ and scaled arbitrarily. Overdrawn (thin black line) is a match to the stellar continuum constructed from template spectra of several late-type supergiant stars and a blackbody function representing the non-stellar component. Notably, Brγ in the nucleus is extremely weak even in the nucleus. Fig. 18.— Radial profile of the stellar continuum in NGC 1097 (1′′ = 80 pc). The solid circles denote the stellar continuum (i.e. already corrected for the non-stellar component). The triangles denote an r1/4 profile fitted to radii r > 0.5′′ and extrapolated inwards. This model has been convolved with the PSF, shown as open squares for comparison. Note that even though an exponential profile might match the data equally well, an r1/4 profile provides a stronger constraint on whether there is excess continuum at the centre. – 46 – Fig. 19.— Radial profiles of velocity and dispersion for the gas and stars in NGC 1097 (1′′ = 80 pc). The 2D maps were created by convolving template spectra (i.e unresolved line profile for the gas, stellar template for the stars) with a Gaussian and minimising the difference with respect to the galaxy spectrum at each spatial pixel. These were then analysed using the kinemetric technique described by Krajnović et al. (2006) which yielded the same position angle of −49◦ for the gas and stars, and similar inclinations of 32◦ and 43◦ respectively. The rotation curve has been corrected for the inclination. Fig. 20.— Maps of K-band non-stellar continuum (left), stellar continuum (centre), and Brγ line flux (right) for the central few arcsec of NGC 1097 (1′′ = 80 pc). In each case, the centre (as defined by the non-stellar continuum) is marked by a crossed circle. The colour scale is shown on the right, as percentage of the peak in each map. – 47 – Fig. 21.— Radial profiles of velocity and dispersion for stars in NGC 1068 (1′′ = 70 pc). The 2D maps were then analysed using kinemetry Krajnović et al. (2006), yielding an inclination of 40◦ and a position angle of 85◦. The rotation curve has been corrected for the inclination. Also plotted for comparison are azimuthally averaged radial profiles of the H-band stellar luminosity and the PSF. Fig. 22.— Radial profiles of the H-band stellar luminosity in NGC 1068 (1′′= 70 pc) from NACO longslit data at position angle 0◦ and 90◦ (green and red symbols; squares and triangles deonte opposite sides of the nucleus). To trace the profile to larger radius, open symbols are each the mean of 9 points. Superimposed are SINFONI data from Fig. 21 (blue crosses, flux scaled to match). The dashed line denotes an r1/4 law with Reff = 1.5 ′′ to match the outer profile. At r < 1′′, the stellar continuum reveals an excess above the inward extrapolation of this profile. – 48 – Fig. 23.— Maps of the central few arcsec of NGC 1068 (1′′ = 70 pc): H-band non-stellar continuum (far left) and stellar continuum (centre left); also Brγ line flux (center right) and Brγ equivalent width (far right). In each case, the centre (as defined by the non-stellar continuum) is marked by a crossed circle. The colour scale is shown on the right, as percentage of the peak in each map (and also as WBrγ in Å). – 49 – Fig. 24.— H- and K-band spectra of the central 1′′ of NGC 3783, with the prominent emission and absorption features labelled. In the H-band it is challenging to measure the stellar absorption due to the very strong brackett emission from the AGN’s broad line region. Instead we have used the K-band CO 2-0 bandhead even though the dilution at this wavelength is extreme. – 50 – Fig. 25.— Radial profile of the stellar continuum in NGC 3783 (1′′ = 200 pc). The solid circles denote the stellar continuum (i.e. already corrected for the non-stellar component). The triangles denote an r1/4 profile fitted to radii 0.2 < r < 1.6′′ and extrapolated inwards. This model has been convolved with the PSF, shown as open squares for comparison. Note that even though an exponential profile might match the data equally well, an r1/4 profile provides a stronger constraint on whether there is excess continuum at the centre. Fig. 26.— Rotation curve derived from the the H2 1-0 S(1) velocity field in NGC 3783 (1 ′′ = 200 pc). Also shown is the dispersion as a function of radius. The velocity field was analysed using kinemetry (Krajnović et al. 2006) which yielded a major axis of about −14◦ and an inclination in the range 35–39◦. The drop in velocity at r < 0.15′′ maybe due to beam smearing across the nucleus. – 51 – Fig. 27.— Images of the central 2′′ of NGC 3783 (1′′ = 200 pc). Top row from left: non-stellar continuum, H2 1-0 S(1) line flux, H2 1-0 S(1) velocity. Bottom row from left: narrow Brγ line flux, [Sivi] line flux, [Sivi] velocity. The Brγ velocity field is similar to that of [Sivi] and shows an outflow of >50 km s−1 to the north. This is in contrast to the 1-0 S(1) velocity field which traces rotation. Introduction Sample, Observations, Data Processing Sample Selection Observations & Reduction PSF Estimation Emission/Absorption Line Characterisation Quantifying the Star Formation Isolating the stellar continuum Stellar colour and luminosity Specific Star Formation Diagnostics Br equivalent width Supernova rate Mass-to-light ratio Properties of Nuclear Star Formation Starburst-AGN Connection Conclusions Individual Objects Summary of Star Forming Properties of Galaxies already Studied Mkn 231 Circinus NGC 3227 Star Forming Properties of Galaxies with New Data NGC 7469 IRAS 05189-2524 NGC 2992 NGC 1097 NGC 1068 NGC 3783

0704.1375

Decrease of entanglement by local operations in the D\"ur-Cirac method

Decrease of entanglement by local operations in the Dür–Cirac method Yukihiro Ota,1, 2, ∗ Motoyuki Yoshida,1, † and Ichiro Ohba1, 2, 3, ‡ Department of Physics, Waseda University, Tokyo 169–8555, Japan Advanced Research Institute for Science and Technology, Waseda University, Tokyo 169–8555, Japan Kagami Memorial Laboratory for Material Science and Technology, Waseda University, Tokyo 169–0051, Japan (Dated: October 27, 2018) One cannot always obtain information about entanglement by the Dür–Cirac (DC) method. The impracticality is attributed to the decrease of entanglement by local operations in the DC method. We show that, even in 2–qubit systems, there exist states whose entangled property the DC method never evaluates. The class of such states in 2–qubit systems is completely characterized by the value of the fully entangled fraction. Actually, a state whose fully entangled fraction is less than or equal is always transformed into a separable state by local operations in the DC method, even if it has negative partial transposition. PACS numbers: 03.67.–a, 03.67.Mn, 03.65.Ca I. INTRODUCTION Quantum mechanics has a quite different mathemati- cal and conceptual structure from that of classical me- chanics. Quantum entanglement vividly illustrates this point [1]. Investigation into the character of entangle- ment is necessary for not only the deep understanding of quantum theory but also its application. Indeed, entan- glement is regarded as a key concept of quantum infor- mation processing [2]. The classification and quantification of bipartite entan- glement (i.e., entanglement between two subsystems in a total quantum system) are well established [3, 4, 5, 6, 7, 8, 9, 10]. In particular, the positive partial transposition criterion (PPT) [3, 4, 5, 6] is very useful, because one can readily obtain a sufficient condition for an entangled state, a necessary condition for a separable state (i.e., a state with no quantum correlation) [11], or a necessary condition for a distillable state [10], by linear algebra. The situation becomes more complicated as the num- ber of subsystems in a total system increases. In 3–qubit systems, for example, there are two inequivalent classes of entanglement. By stochastic local operations and clas- sical communication [12], a Greenberger–Horn–Zeilinger (GHZ) state cannot be transformed into a W state, and vice versa [13]. However, multiparticle entanglement can play an important role in quantum protocol (e.g., quan- tum telecloning [14]) and quantum computing. More- over, its classification will be useful for deeply under- standing quantum phase transitions in condensed mat- ter physics [15]. Thus, research into multiparticle entan- glement is a crucial and popular issue in both quantum physics and quantum information theory. Various attempts to classify and quantify multiparti- cle entanglement have been made [16, 17, 18, 19, 20, 21, ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] 22, 23]. Among them, Dür and Cirac [17] proposed a systematic way of classifying multiparticle entanglement in N–qubit systems. Hereafter, we call it the Dür–Cirac (DC) method. The main idea is that, using a sequence of local operations, one can transform an arbitrary density matrix of an N–qubit system into a state whose entan- gled property is easily examined. It should be noted that entanglement cannot increase through local operations. Accordingly, if the density matrix transformed by local operations is entangled, then the original density matrix represents an entangled state. However, one cannot always obtain an entangled prop- erty by the DC method. In our previous paper [24], we suggested that there exists an impracticality in the DC method through an example. In this letter, we reveal the possibility that one can- not obtain the desired information on entanglement by the DC method, though it is a very simple and effec- tive method for examining multiqubit entanglement. We show that there is such a possibility even in 2–qubit systems. The most important quantity in our discus- sion is a fully entangled fraction [7, 10]. Our main re- sult is that, in 2–qubit systems, one can never determine whether a quantum state is entangled or not through the DC method if the fully entangled fraction is less than or equal to 1 . Then, we completely characterize the class of the state in 2–qubit systems whose entangled property is never obtained by the DC method. The impracticality of the DC method is due to the decrease of entanglement by local operations in the DC method. Additionally, we investigate what parts of the local operations reduce the entanglement in 2–qubit system. The letter is organized as follows. We briefly review the DC method in section II. Then, we illustrate the impracticality of the DC method through an example in a 2–qubit system, and show the relation to the local operations in section III. After that, we show our main results in section IV. Our results are shown only in 2– qubit systems, but they clearly reveal the limitation of the DC method. Section V is devoted to a summary. http://arxiv.org/abs/0704.1375v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] II. REVIEW OF THE DC METHOD We briefly review the DC method [16, 17, 18, 19]. Its main idea is that, using a sequence of local operations, one can transform an arbitrary density matrix of an N– qubit system into a state whose property of entanglement is easily examined. First, we explain how to specify a bipartition of the system concerned. We divide an N–qubit system into two subsystems, system A and system B, as follows. Let us consider a set of binary numbers, {ki}Ni=1 (ki = 0, 1). When ki is equal to 0 (1), the ith qubit is in system A (B). We always set k1 = 0; the first qubit is al- ways in system A. Representing the number by a binary, i=2 ki2 i−2, a partition is specified in the N–qubit system if an integer k(∈ [1, 2N−1 − 1]) is chosen; we call such a partition the bipartition k. The authors in Ref. [16, 17, 18, 19] introduced a special family of density matrices as follows: ρN = λ 0 〉〈Ψ 0 |+ λ 0 〉〈Ψ 2N−1−1 |Ψ+j 〉〈Ψ j |+ |Ψ j 〉〈Ψ , (1) where the coefficients λ±0 and λj are real and positive, and λ+0 +λ ∑2N−1−1 j=1 λj = 1 because trρN = 1. These coefficients are related to the information on an arbitrary density matrix of an N–qubit system, as shown below. The generalized GHZ state [16, 17, 18, 19] in an N–qubit system |Ψ±j 〉 is defined as follows: |Ψ±j 〉 = (|0j〉 ± |1̄〉) (0 ≤ j ≤ 2N−1 − 1), (2) where j ≡ i=2 ji2 i−2 for the binary number ji (= 0, 1), |0j〉 ≡ |0〉1 ⊗ i=2 |ji〉i and |1̄〉 ≡ |1〉1 ⊗ i=2 |1− ji〉i. The symbol ̄ means a bit–flip of j: ̄ = 2N−1−1− j. We write the computational basis for the ith qubit as |0〉i and |1〉i ( i〈0|0〉i = 1, i〈1|1〉i = 1, and i〈0|1〉i = 0). The sub- scription i(= 1, 2, . . .N) is the label of the qubit. We can easily find the generalized GHZ states are the elements of an orthonormal basis of the Hilbert space correspond- ing to the N–qubit system. Note that the convention of generalized GHZ states (2) is slightly different from the corresponding one in Ref. [16, 17, 18, 19], but such a dif- ference doesn’t matter in our discussion. We summarize the several useful properties of ρN . The compact consequences for partial transposition with re- spect to any bipartition are known [16, 17, 18, 19]. First, ρN has positive partial transposition (PPT) with re- spect to a bipartition k if and only if ∆ ≤ 2λk, where ∆ = |λ+0 − λ 0 |. On the other hand, ρN has negative partial transposition (NPT) with respect to a biparti- tion k if and only if ∆ > 2λk. Furthermore, the authors in Ref. [17, 18] proved the theorems about multiparticle entanglement. Among them, we explain an important one [17]. We concentrate on two qubits, for example the ith and jth qubits, in an N–qubit system. Let us con- sider all possible bipartitions, Pij under which the ith and jth qubits belong to different parties. The theorem is that ρN has NPT with respect to ∀k ∈ Pij if and only if the maximal entangled states between the ith and jth qubits can be distilled. The most important result in Ref. [16, 17] is that an arbitrary density matrix, ρ of an N–qubit system, can be transformed into ρN by local operations, and local operations cannot increase entanglement. Accordingly, if ρN is an entangled state with respect to a bipartition, ρ is also such a state. Moreover, according to the theo- rem explained at the end of the above paragraph, if ρN has NPT with respect to ∀k ∈ Pij , the maximal entan- gled state between the ith and jth qubits can be distilled from ρN . Then, one should be able to distill the max- imal entangled state between such qubits from ρ. This result implies that one can know the sufficient condition for the distillability of ρ for an arbitrary N . Note that, through the PPT criterion, one can only obtain the neces- sary condition for the distillability in an N -qubit system when N > 2 [6]. Under the local operations, the coefficients λ±0 and λj of ρN are given by the following relations: 0 = 〈Ψ 0 |ρ|Ψ 0 〉, 2λj = 〈Ψ |ρ|Ψ+ 〉+〈Ψ− |ρ|Ψ− 〉. (3) Consequently, one can systematically treat the evalua- tion of multiparticle entanglement as a task for bipar- tite entanglement, because it is only necessary to cal- culate some specific matrix elements of ρ. In addition, this point will be useful for investigating entanglement in experiments [19]. III. LOCAL OPERATIONS IN THE DC METHOD As shown in the previous section, one can readily evalu- ate the information of multiparticle entanglement by the DC method. However, the desired information about en- tanglement isn’t always obtained. Let us illustrate such an impractical case by an example in a 2–qubit system. One can easily find that, by the PPT criterion, the fol- lowing density matrix has NPT (i.e., entangled): |Ψ+0 〉〈Ψ |Ψ+1 〉〈Ψ |Ψ−1 〉〈Ψ |Ψ+1 〉〈Ψ 1 |+ |Ψ 1 〉〈Ψ . (4) One needs only to calculate ∆ and 2λ1 to apply the DC method to a 2–qubit system. According to Eq. (3), one can readily obtain the following results for ρf : ∆ = 2λ1 = . Then, it is not possible to determine whether ρf is entangled or not, because ∆ = 2λ1. We will show that the above problem should be at- tributed to the decrease of entanglement by local opera- tions in the DC method. Let us explain Dür and Cirac’s explicit expressions to clarify this point. The local oper- ations in the DC method are sequence of the following three steps. First, we perform the following probabilistic unitary operator on an arbitrary density operator of an N–qubit system: L1ρ = 1 , (5) where W1 = i=1 σ x and σ x = |0〉i〈1|+ |1〉i〈0|. Note j, j′=0 |Ψ+j 〉〈Ψ |+ µ+− |Ψ+j 〉〈Ψ |Ψ−j 〉〈Ψ |+ µ−− |Ψ−j 〉〈Ψ where µσσ jj′ s are the matrix elements of ρ for the gener- alized GHZ states (σ, σ′ = ±). As a result of this opera- tion, the terms corresponding to |Ψ+j 〉〈Ψ | and |Ψ−j 〉〈Ψ are vanishing because W1|Ψ±j 〉 = ±|Ψ The following probabilistic unitary operators are nec- essary for the second step: Llρ = (l = 2, 3, . . . , N), (7) where Wl = σ z ⊗σ(l)z and σ(i)z = |0〉i〈0|− |1〉i〈1|. Equa- tion (7) is a local operation with respect to the first and lth qubit. Note that we abbreviate the identity opera- tors for the other qubits in Wl. In the second step, we perform l=2 Ll on the result of the first step. By this operation, the terms corresponding to |Ψ±j 〉〈Ψ | (j 6= j′) are vanishing because Wl|Ψ±j 〉 = (−1)jl |Ψ j 〉. In this stage, the resultant state is a diagonal form with respect to the generalized GHZ states. Finally, we perform the local random phase–shift, Lr on the result of the second step: Lrρ = 2π δ(Φ− 2π)Rφ ρR†φ, (8) where Rφ = i=1 R (i)(φi), R (i)(φi)|0〉i = eiφi |0〉i, R(i)(φi)|1〉i = |1〉i, and Φ = i=1 φi. Note that Lr |Ψ±0 〉〈Ψ 0 | = |Ψ 0 〉〈Ψ 0 | and Lr |Ψ j 〉〈Ψ j | = (|0j〉〈0j| + |1̄〉〈1̄|) (j 6= 0). After the final step, we can find that the resultant state is equivalent to Eq. (1). Now, let us go back to Eq. (4). We only need to per- form L1 on ρf to transform it into the form of Eq. (1): L1ρf = 12 |Ψ 0 〉〈Ψ 0 | + 14 |Ψ 1 〉〈Ψ 1 | + 14 |Ψ 1 〉〈Ψ 1 |. Obvi- ously, the resultant state is separable. It implies that the entanglement decreases by the local operation L1. In the subsequent section, we will characterize the class of the quantum states in a 2–qubit system whose entangled property is not obtained by the DC method due to its decrease by the local operations. IV. LIMITATION OF THE DC METHOD IN 2–QUBIT SYSTEMS We attempt to reveal the class of the quantum states whose entangled property is not obtained by the DC method. In this section, we focus on the case N = 2 because its entanglement structure is well known. Let us first introduce an important quantity for our consideration: F(ρ) = max 〈Ψ+0 |(U ⊗ V ) ρ (U ⊗ V ) †|Ψ+0 〉, (9) where U and V are unitary operators on the Hilbert spaces for the first and second qubits, respectively. Equa- tion (9) is called a fully entangled fraction [7, 10]. We show that the value of a fully entangled fraction plays an important role in determining whether the DC method works or not. According to the DC method, the sufficient condition for an entangled state in a 2–qubit system is ∆ > 2λ1. Using trρ = 1 and Eq. (3), we readily obtain the following relation: ∆ > 2λ1 ⇐⇒ 〈Ψ+0 |ρ|Ψ 0 〉 > or 〈Ψ−0 |ρ|Ψ 0 〉 > .(10) The right–hand side of Eq. (10) implies F(ρ) > 1 . Note that |Ψ−0 〉 = (I(1) ⊗ σ z )|Ψ+0 〉, where I(1) = |0〉1〈0| + |1〉1〈1|. Summarizing the above argument, we obtain the following statements: ∆ > 2λ1 =⇒ F(ρ) > , (11) F(ρ) ≤ 1 =⇒ ∆ ≤ 2λ1. (12) Accordingly, we obtain the following conclusion. Let us consider the density matrix in a 2–qubit system which has NPT; it is an entangled state. However, if its fully entangled fraction is less than or equal to 1 , then it is not possible to determine whether such a state is entangled or not by the DC method. Actually, Eq. (4) is just such an example. Next, we investigate the density matrix ρ whose fully entangled fraction is greater than 1 . In general, the con- dition F(ρ) > 1 does not imply 〈Ψ±0 |ρ|Ψ 0 〉 > 12 . How- ever, the following statement is always true: ∃Ũ ⊗ Ṽ s.t. |Ψ+0 〉 = Ũ ⊗ Ṽ |ψ̃〉, (13) where Ũ and Ṽ are unitary operators on the Hilbert spaces for the first and second qubits, respectively, and |ψ̃〉 is the maximally entangled state that satisfies 〈ψ̃|ρ|ψ̃〉 = F(ρ). Consequently, using the above local unitary operator, we obtain 〈Ψ+0 |ρ̃|Ψ 0 〉 > , (14) ρ̃ = (Ũ ⊗ Ṽ ) ρ (Ũ ⊗ Ṽ )†. (15) According to Eqs. (11) and (14), we obtain the following statement: F(ρ) > 1 =⇒ ∃Ũ ⊗ Ṽ s.t. ∆̃ ≡ |λ̃+0 − λ̃ 0 | > 2λ̃1, where λ̃±0 = 〈Ψ 0 |ρ̃|Ψ 0 〉 and 2λ̃1 = 〈Ψ 1 |ρ̃|Ψ 1 〉 + 〈Ψ−1 |ρ̃|Ψ 1 〉. The local unitary transformed state ρ̃ is en- tangled if ∆̃ > 2λ̃1; one can obtain the entangled prop- erty of ρ̃ by the DC method. On the other hand, the original density matrix ρ is related to ρ̃ through the local unitary operator Ũ⊗ Ṽ from Eq. (15); ρ̃ is equivalent to ρ with respect to entanglement. Therefore, one can obtain the entangled property of a density matrix whose fully entangled fraction is greater than 1 by the DC method with a suitable local unitary operator. Let us show an example for such a case. We consider a Bell–diagonal state. Such a state is defined by as follows: ρBD = |+ µ− , (17) where µ±j ≥ 0 and j=0(µ j + µ j ) = 1. Note that our example in Ref. [24] was a special case of Eq. (17). We can show that ρBD has NPT if and only if |µ+0 − µ 0 | > µ 1 + µ 1 or |µ 1 − µ 1 | > µ 0 + µ 0 . (18) According to tr ρBD = 1 and Eq. (18), if one of µ j s (σ = ±) is at least greater than 1 , then ρBD has NPT, and vice versa. In addition, we easily obtain the following relation: F(ρBD) = max σ=±, j=0, 1 µσj . (19) Then, if ρBD is entangled, F(ρBD) is greater than 12 . In this case, we can obtain the information of the entangle- ment for the Bell–diagonal state in a 2–qubit system by the DC method with a suitable local operator. Note that one only needs to use Lr to transform ρBD into ρN . Finally, we consider whether, through the DC method with appropriate local unitary operators, we can obtain the entangled property of the quantum state whose fully entangled fraction is less than or equal to 1 . It should be noted that the converse statement of Eq. (16) can be easily shown. Therefore, we conclude that one never ob- tains the entangled property for a density matrix whose fully entangled fraction is less than or equal to 1 by the DC method, even if one uses local unitary operators. In summary, we have completely classified the states in 2–qubit systems whose entangled property is not ob- tained by the DC method, or by the DC method with lo- cal unitary operators. The limitation of the method is de- termined by the value of the fully entangled fraction. If it is greater than 1 , we can always obtain the desired infor- mation on entanglement by the DC method with suitable local unitary operators. Otherwise, we never obtain it. The impracticality of the DC method is attributed to the decrease of entanglement by the local operations. Note that the Bell–diagonal state is entangled if F(ρBD) > 12 . Moreover, we can easily find L1 ρBD = L2 ρBD = ρBD and LrρBD = ρN . Accordingly, in 2–qubit systems, the crucial decrease of entanglement occurs in L1 and L2. V. SUMMARY We have shown that one cannot always obtain an en- tangled property by the DC method, even in 2–qubit systems. The most important quantity in our discussion is a fully entangled fraction. One can never determine whether a quantum state is entangled or not through the DC method, if the fully entangled fraction is less than or equal to 1 . On the other hand, one can make such a determination by the DC method with suitable local uni- tary operators, if the fully entangled fraction is greater than 1 The impracticality of the DC method is attributed to the decrease of entanglement by the local operations. Ac- tually, from Eqs. (4) and (5), we have easily shown that L1ρf is separable, even if ρf is entangled. The Bell– diagonal state (17) is invariant under L1 and L2; we only need to use Lr for transforming it into the form of Eq. (1). In addition, the Bell–diagonal state which is entangled has a fully entangled fraction greater than 1 . Therefore, the crucial decrease of entanglement for examining it by the DC method occurs in L1 and L2 in 2–qubit systems. Finally, we would like to comment on the case of mul- tiqubit systems. The DC method has been proposed as a systematic estimation of multiparticle entanglement. Therefore, it is necessary to study the limitation of the method in N–qubit systems when N > 2. However, the situation will be more complicated in this case. Nev- ertheless, the results in this letter can hint at a solu- tion. Namely, we will consider the following question: (i) Is it possible to obtain the entangled property of Bell– diagonal states in N–qubit systems, ρBD = 2N−1−1 j 〉〈Ψ j |+ µ j 〉〈Ψ , (20) by the DC method with suitable local unitary operators? (ii) How are fragile quantum states with respect to en- tanglement, for example ρf , under the local operations characterized in N–qubit systems? We think the above questions are related to the decrease of quantum entan- glement under local operations and decoherence. In ad- dition, our examination of the above questions will lead to the understanding of the structure of quantum states in N–qubit systems. Acknowledgments The authors acknowledge H. Nakazato for valu- able discussions. This research is partially supported by a Grant–in–Aid for Priority Area B (No. 763), MEXT, by the 21st Century COE Program (Physics of Self-Organization Systems) at Waseda University from MEXT, and by a Waseda University Grant for Special Research Projects (Nos. 2004B–872 and 2007A–044). [1] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, Dordrecht, 1995). [2] The Physics of Quantum Information: Quantum Cryp- tography, Quantum Teleportation, Quantum Computa- tion, edited by D. Bouwmeester, A. Ekert, and A. Zeilinger (Springer, Berlin, 2000). [3] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). [4] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996). [5] P. Horodecki, Phys. Lett. A 232, 333 (1997). [6] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998). [7] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [8] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997). [9] V. Vedral and M.B. Plenio, Phys. Rev. A 57, 1619 (1998). [10] G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, Berlin, 2001). [11] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [12] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and A. V. Thapliyal, Phys. Rev. A 63, 012307 (2001). [13] W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000). [14] M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, Phys. Rev. A 59, 156 (1999). [15] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002). [16] W. Dür, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999). [17] W. Dür and J. I. Cirac, Phys. Rev. A 61, 042314 (2000). [18] W. Dür and J. I. Cirac, Phys. Rev. A 62, 022302 (2000). [19] W. Dür and J. I. Cirac, J. Phys. A 34, 6837 (2001). [20] A. 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0704.1376

Compton-thick AGN and the Synthesis of the Cosmic X-ray Background: the Suzaku Perspective

Compton-thick AGN and the Synthesis of the Cosmic X-ray Background: the Suzaku Perspective Roberto Gilli1,∗), Andrea Comastri1, Cristian Vignali2 and Günther Hasinger3 1INAF - Osservatorio Astronomico di Bologna, Bologna, Italy 2Dipartimento di Astronomia, Università degli Studi di Bologna, Bologna, Italy 3Max-Planck-Institut für extraterrestrische Physik, Garching, Germany (Received ) We discuss the abundance of Compton-thick AGN as estimated by the most recent population synthesis models of the cosmic X-ray background. Only a small fraction of these elusive objects have been detected so far, in line with the model expectations. The advances expected by the broad band detectors on board Suzaku are briefly reviewed. §1. Introduction Despite extensive observational efforts, the population of heavily obscured, Compton- thick AGN remains elusive, especially at high redshifts, preventing a complete census of accreting supermassive black holes (SMBHs). While Compton-thick (CT) nuclei were shown to hide in about half of local Seyfert 2 galaxies (Risaliti et al. 1999, Guainazzi et al. 2005), observations of heavily obscured objects beyond z ∼ 0.1 are very sparse (see Comastri 2004 for a review) and their abundance can be con- strained only by indirect arguments. One argument is the comparison between the mass function of local SMBHs with the one expected if they accreted most of their mass during past AGN phases (Marconi et al. 2004). Another, and probably more stringent, argument is the residual emission at 30 keV in the spectrum of the cos- mic X-ray background (XRB), which is left after removing the contribution from the better known population of less obscured, Compton-thin AGN. The residual 30 keV XRB emission can indeed be modeled by assuming a population of CT AGN as large as that of moderately absorbed ones over a broad range of redshifts and luminosities (see Gilli, Comastri & Hasinger 2007, hereafter GCH07, for a recent work). In particular, this residual emission is mostly filled by “mildly” CT objects (defined as those with 1024 < NH < 10 25 cm−2) in which the direct, primary emis- sion is visible above ∼ 10 keV, rather than by “heavily” CT objects (NH > 10 cm−2), in which only reflected radiation is visible at high energy, and are therefore significantly less luminous than the formers at 30 keV. As a consequence, the number of heavily CT AGN is poorly constrained even by population synthesis models and is generally assumed to be similar (equal in GCH07) to that of mildly CT objects, as suggested by the results of Risaliti et al. (1999). Because of the strong selection effects due to absorption, only a small percentage of CT sources have been observed ∗) E-mail: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1376v1 2 R. Gilli, A. Comastri, C. Vignali and G. Hasinger Fig. 1. Left : The fractions of CT AGN in the GCH07 baseline model as a function of the 2-10 keV and 15-200 keV limiting fluxes compared to those observed in the CDFS (Tozzi et al. 2006) and in the Swift/BAT catalog (Markwardt et al. 2005). The upper Swift/BAT point is corrected for incompleteness. Note the steep increase expected at fluxes below the current sensitivities and the identical CT fraction in the two bands at extremely faint fluxes, where all AGN should be detected. Right : The predicted 20-40 keV AGN counts (see Section 2) normalized to an Euclidean Universe and compared with those measured by INTEGRAL (Beckmann et al. 2006). in current X-ray surveys. In the Chandra Deep Field South (CDFS) only about 5% of the detected AGN have been identified as CT candidates (Tozzi et al. 2006). In the recent INTEGRAL/IBIS and Swift/BAT surveys performed above 10 keV, where the absorption bias is less effective, a higher fraction is observed (∼ 10− 15%, Markwardt et al. 2006, Beckmann et al. 2006). This is already remarkable, if one bears in mind that X-ray surveys above 10 keV are still limited to very bright fluxes (∼ 10−11 erg cm−2 s−1). As shown in Fig. 1 left, these small observed fractions are in good agreement with those expected if CT AGN are intrinsically as abundant as moderately obscured ones (see GCH07), and are predicted to increase dramatically at fluxes below the current sensitivity limits. §2. Uncertainties on the number of Compton-thick AGN Since the overall abundance of CT AGN is estimated by subtracting from the XRB spectrum the contribution of Compton-thin sources, it is imperative to estimate the latter at the best of present knowledge. The modeling presented in GCH07 took into account a detailed characterization of the average AGN X-ray spectra, including dispersion, and cosmological evolution, but is nonetheless worth exploring the parameter space to some extent and check how different assumptions may affect the estimated CT number. In the baseline model presented by GCH07, a Gaussian distribution in the AGN primary continuum was considered, with average spectral Compton-thick AGN and the Synthesis of the Cosmic X-ray Background 3 INTEGRAL JEM-X INTEGRAL IBIS/SPI Chandra INTEGRAL JEM-X INTEGRAL IBIS/SPI Chandra INTEGRAL JEM-X INTEGRAL IBIS/SPI Chandra Fig. 2. The spectrum of the X-ray background. Most of the datapoints are described in GCH07. Here we add the recent 5-200 keV measurement by INTEGRAL (Churazov et al. 2007) and the 1-7 keV measurement by Chandra (Hickox & Markevitch 2006). Model curves based on GCH07 are also plotted: the upper magenta curves show the total contribution from AGN (plus galaxy clusters), according to different assumptions on the average AGN spectral slope and number of CT objects as labeled (see text); the lower black curves represent the corresponding CT contribution. slope and dispersion of 〈Γ 〉 = 1.9 and σΓ = 0.2, respectively, in agreement with the observed distributions (Mateos et al. 2005). In Fig. 2 we show the effects of assuming an average spectral index 〈Γ 〉 = 1.8 with the same dispersion. A sligthly harder average spectral powerlaw is in principle sufficient to saturate the XRB emission with Compton-thin AGN, leaving little room for CT sources. Indeed, when adding as many CT AGN as in the baseline model, the 30 keV XRB emission measured by HEAO-1, and recently confirmed within 10% by INTEGRAL (Churazov et al. 2007), is exceeded if 〈Γ 〉 = 1.8. Furthermore, the baseline model appears to be in much better agreement with other observational constraints, such as the spectral distributions observed in different AGN samples and the observed numbers of CT AGN (see GCH07). The model predictions have been further compared with the AGN counts in the 20-40 keV band recently estimated by Beckmann et al. (2006). The situation (Fig. 1 right) is similar to that shown in Fig.2 for the XRB, although the constraints are less stringent. While a model with 〈Γ 〉 = 1.8 would imply a small number of CT AGN, the baseline model provides a good match to the data with a relative ratio of one between Compton-thick and Compton-thin AGN at all redshifts. 4 R. Gilli, A. Comastri, C. Vignali and G. Hasinger This assumed ratio appears more in line with current observations both in the local (Risaliti et al. 1999) and in the distant Universe (Martinez-Sansigre et al. 2006). §3. The Suzaku perspective To date, about 40 local AGN have been shown to be CT through X-ray obser- vations (Comastri 2004) and their number is expected to increase significantly in the next future. Indeed, CT AGN candidates in current INTEGRAL and Swift surveys can be easily flagged as such if their X-ray flux above 10 keV is much larger than their soft X-ray flux, which is often available from archival X-ray data. Like BeppoSAX in the past years, Suzaku is now carrying on board detectors which are sensitive in the broad 0.5-50 keV band and are therefore the ideal instruments to determine the X-ray spectral energy distribution of bright, nearby CT objects. In EAO-1 we have obtained Suzaku observations of 2 CT candidates selected from the INTEGRAL and Swift AGN catalogs (two further candidates will be observed in EAO-2). One of the two objects indeed proved to be CT, while the other turned out to be heavily obscured but still Compton thin (see Comastri et al. this volume, for a more detailed discussion). Suzaku observations of additional CT candidates selected above 10 keV are being performed by other groups (e.g. Ueda et al., this volume). Eventually, once these programs are put together to get sufficient object statistics, the fraction of CT AGN in the local Universe will be determined to better accuracy. In particular, new mildly CT AGN should be revealed in significant numbers, and a few spectra of heavily CT AGN, which went undetected by BeppoSAX, may be also obtained. Acknowledgements We are grateful to Mike Revnivtsev and Volker Beckmann for providing the INTEGRAL XRB spectrum and 20-40 keV AGN counts, respectively. RG, AC and CV acknowledge financial support from the Italian Space Agency (ASI) under the contract ASI–INAF I/023/05/0. References 1) V. Beckmann, et al., ApJ 652 (2006), 126. 2) E. Churazov, et al., A&A (2007), in press (astro-ph/0608250). 3) A. Comastri, Supermassive Black Holes in the Distant Universe (A.J. Barger, Dordrecht, 2004), p. 245. 4) R. Gilli, A. Comastri and G. Hasinger, A&A 463 (2007), 79. (GCH07) 5) M. Guainazzi, et al., A&A 444 (2005), 119 6) R. Hickox & M. Markevitch, ApJ 645 (2006), 95. 7) A. Marconi, et al., MNRAS 351 (2004), 169. 8) C.B. Markwardt, et al., ApJ 633 (2005), L77. 9) A. Martinez-Sansigre, et al., MNRAS 370 (2006), 1479 10) S. Mateos, et al., A&A 444 (2005), 79. 11) G. Risaliti, R. Maiolino and M. Salvati, ApJ 522 (1999), 157. 12) P. Tozzi, et al., A&A 451 (2006), 457. http://arxiv.org/abs/astro-ph/0608250 Introduction Uncertainties on the number of Compton-thick AGN The Suzaku perspective

0704.1377

Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate

Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate Hiroki Saito1, Yuki Kawaguchi2, and Masahito Ueda2,3 Department of Applied Physics and Chemistry, The University of Electro-Communications, Tokyo 182-8585, Japan Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan Macroscopic Quantum Control Project, ERATO, JST, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: October 31, 2018) The spin vortices are shown to be created through the Kibble-Zurek (KZ) mechanism in a quantum phase transition of a spin-1 ferromagnetic Bose-Einstein condensate, when the applied magnetic field is quenched below a critical value. It is shown that the magnetic correlation functions have finite correlation lengths, and magnetizations at widely separated positions grow in random directions, resulting in spin vortices. We numerically confirm the scaling law that the winding number of spin vortices is proportional to the square root of the length of the closed path, and for slow quench, proportional to τ with τQ being the quench time. The relation between the spin conservation and the KZ mechanism is discussed. PACS numbers: 03.75.Mn, 03.75.Lm, 73.43.Nq, 64.60.Ht I. INTRODUCTION Spontaneous symmetry breaking in a phase transition produces local domains of an order parameter. If do- mains are separated by such a long distance that they cannot exchange information, local domains grow ini- tially with random phases and eventually give rise to topological defects when they overlap. This scenario of topological-defect formation in continuous-symmetry breaking is known as the Kibble-Zurek (KZ) mecha- nism [1, 2], which originally predicted the cosmic-string and monopole formation in the early Universe [1], and has since been applied to a wide variety of systems. Ex- perimentally, the KZ mechanism has been examined in liquid crystals [3, 4], superfluid 4He [5] and 3He [6, 7], an optical Kerr medium [8], Josephson junctions [9, 10], and superconducting films [11]. Recently, spontaneous magnetization in a spinor Bose- Einstein condensate (BEC) has attracted much interest as a new system for studying the KZ mechanism [12, 13, 14, 15]. In the experiment performed by the Berkeley group [12], a BEC of F = 1 87Rb atoms are prepared in the m = 0 state, where F is the hyperfine spin and m is its projection on the direction of the magnetic field. By quench of the magnetic field, say in the z direction, magnetization appears in the x-y plane. Since the spinor Hamiltonian is axisymmetric with respect to the z axis, the magnetization in the x-y direction breaks the U(1) symmetry in the spin space. Thus, local domain forma- tion is expected to lead to topological defects — spin vortices — through the KZ mechanism. However, the origin of the spin vortices observed after the quench in the Berkeley experiment [12] cannot be at- tributed to the KZ mechanism. In fact, in Ref. [12], the spin correlation extends over the entire system (at least in the x direction) and the domains are not independent with each other. We have shown in Ref. [13] that the ori- gin of the observed spin vortices is initial spin correlation due to the residual m = ±1 atoms, which forms domain structure followed by spin vortex creation [16]. In order to realize the KZ mechanism in this system, i.e., in order to ensure that the magnetic domains grow independently, the size of the system must be much larger than the spin correlation length and the long-range correlation in the initial spin state must be absent. The aim of the present paper is to show that under these conditions spin vortices are generated through the KZ mechanism. In the present paper we will consider 1D-ring and 2D- disk geometries. We will show that in the 1D ring the average spin winding number after the quench is propor- tional to the square root of the system size, which is in agreement with the KZ prediction [2]. In 2D the winding number along a path with radius R is also proportional to R1/2 as long as R is much larger than the vortex spac- ing, while it is proportional to R for small R. When the magnetic field is quenched slowly, the winding num- ber is shown to be proportional to τ Q with τQ being the quench time. This power law can be understood by Zurek’s simple discussion [2]. The spinor BEC is different from the other systems in which the KZ mechanism has been observed, in that the total spin is conserved when the quadratic Zeeman en- ergy q is negligible. This fact is seemingly incompatible with the KZ postulate, since the magnetic domains must be correlated with each other so that the total magnetiza- tion vanishes. We will show that for q = 0 small magnetic domains are aligned to cancel out the local spin averaged over the correlation length, and that they are indepen- dent with each other over a greater length scale; the spin conservation is thus compatible with the KZ mechanism. The present paper is organized as follows. Section II analyzes spontaneous magnetization of a spin-1 BEC and the resultant magnetic correlation functions using the Bogoliubov approximation. Section III performs numeri- cal simulations of the dynamics of quenched BECs in 1D and 2D, and shows that the KZ mechanism does emerge in the present system. Section IV provides conclusions. http://arxiv.org/abs/0704.1377v1 II. BOGOLIUBOV ANALYSIS OF A QUENCHED FERROMAGNETIC BOSE-EINSTEIN CONDENSATE A. Hamiltonian for the spin-1 atoms We consider spin-1 bosonic atoms with mass M con- fined in a potential Vtrap(r). The noninteracting part of the Hamiltonian is given by Ĥ0 = ψ̂†m(r) 2 + Vtrap(r) ψ̂m(r), where ψ̂m(r) annihilates an atom in magnetic sublevel m of spin at a position r. The interaction between atoms with s-wave scattering is described by Ĥint = 2(r) + c1F̂ :, (2) where the symbol :: denotes the normal order and ρ̂(r) = ψ̂†m(r)ψ̂m(r), (3) F̂ (r) = ψ̂†m(r)fmm′ ψ̂m′(r), (4) with f = (fx, fy, fz) being the spin-1 matrices. The in- teraction coefficients in Eq. (2) are given by 4πh̄2 a0 + 2a2 , (5a) 4πh̄2 a2 − a0 , (5b) where aS is the s-wave scattering lengths for two colliding atoms with total spin S. When magnetic field B is applied, the linear Zeeman effect rotates the spin around the direction of B at the Larmor frequency. Since Ĥ0 and Ĥint are spin-rotation invariant and we assume the uniform magnetic field, the linear Zeeman term has only a trivial effect on spin dy- namics — uniform rotation of spins about B — which is therefore ignored. The quadratic Zeeman effects for an F = 1 87Rb atom is described by Ĥq = ψ̂†m(r) (B · f)2 ψ̂m′(r), (6) where µB is the Bohr magneton and Ehf > 0 is the hy- perfine splitting energy between F = 1 and F = 2. The total Hamiltonian is given by the sum of Eqs. (1), (2), and (6), Ĥ = Ĥ0 + Ĥq + Ĥint. (7) B. Time evolution in the Bogoliubov approximation In the initial state, all atoms are prepared in them = 0 state. We study the spin dynamics of the system using the Bogoliubov approximation with respect to this initial state. For simplicity, we assume Vtrap = 0 in this section. In the Bogoliubov approximation, the BEC part in the field operator is replaced by a c-number. In the present case, we write the m = 0 component of the field operator ψ̂0(r) = e −ic0nt/h̄ n+ δψ̂0(r) , (8) where n is the atomic density. We expand ψ̂±1(r) as ψ̂±1(r) = e −ic0nt/h̄ eik·r â±1,k, (9) where V is the volume of the system and â±1,k is the annihilation operator of an atom in the m = ±1 state with wave vector k. Keeping only up to the second order of δψ̂0(r) and ψ̂±1(r) in the Hamiltonian, we obtain the Heisenberg equation of motion for â±1,k as dâ±1,k(t) = (εk + q + c1n)â±1,k(t) + c1nâ ∓1,−k(t), where εk = h̄ 2k2/(2M) and q = µ2BB 2/(4Ehf). The mag- netic field is assumed to be applied in the z direction. The solution of Eq. (10) is obtained as â±1,k(t) = − i εk + q + c1n â±1,k(0) ∓1,−k(0), (11) where (εk + q)(εk + q + 2c1n). (12) When Ek is imaginary, the corresponding modes are dynamically unstable and grow exponentially. Since c1 < 0 and q > 0 for F = 1 87Rb atoms, the exponential growth occurs for q < 2|c1|n ≡ qc. (13) This critical value of q agrees with the phase boundary between the polar phase and the broken-axisymmetry phase [17, 18]. When q ≤ qc/2, the wave number of the most unstable mode is kmu = ± , (14) and when qc/2 < q < qc, kmu = 0. C. Fast quench We consider the situation in which q is much larger than the other characteristic energies for t < 0, and q is suddenly quenched below qc at t = 0. During t < 0, the time evolution in Eq. (11) is â±1,k(t) ≃ e−iqt/h̄â±1,k(0), and the m = ±1 state remains in the vacuum state. For t > 0, we obtain the time evolution of the density of the m = ±1 component as ±1(r, t)ψ̂±1(r, t) 16|Ek|2 e2|Ek|t/h̄,(15) where the expectation value is taken with respect to the vacuum state of the m = ±1 component. In the sec- ond line of Eq. (15), we have kept the unstable modes alone with k < kc ≡ 2M(qc − q)/h̄ by assuming that |Ek|t/h̄ ≫ 1. This result indicates that the m = ±1 components grow exponentially after the quench. Since the operator ψ̂0 in Eq. (4) is replaced by the Bogoliubov approximation, the magnetization oper- ator F̂+ = F̂ − = F̂x + iF̂y has the form, F̂+(r) = 1(r) + ψ̂−1(r) . (16) Using Eq. (11), the time evolution of the correlation func- tion is calculated to be F̂+(r, t)F̂−(r ′, t) εk + q eik·(r−r (17a) qc − q − εk e2|Ek|t/h̄+ik·(r−r ′), (17b) where in the second line we have kept the unstable modes alone. From the exponential factor in Eq. (17b), we see that the sum is contributed mostly from k around the mode with maximum |Ek|. The denominator in the summand of Eq. (17b) is much smoother than the exponential fac- tor if q is not close to qc, and then we approximate εk with εmu = h̄ 2k2mu/(2M) in the denominator. We expand 2|Ek|t/h̄ around kmu in the exponent as 2|Ek|t ξ2∆k2 − 1 Ξ4∆k4 +O(∆k6), where ∆k = k − kmu. It is clear that τ sets the time scale for the exponential growth. The magnetization is observed when it sufficiently grows, i.e., t ∼ τ . Replacing the summation with the Gaussian integral in Eq. (17b), we find that ξ represents the correlation length. For q < qc/2, kmu is given by Eq. (14), and , (19) qc − 2q . (20) For qc/2 < q < qc, kmu = 0 and q(qc − q) , (21) 2q − qc q(qc − q) . (22) At q = qc/2, Eqs. (20) and (22) vanish, and the ∆k 4 term in Eq. (18) becomes important, with Ξ = 4 2M2q2c . (23) We first consider a 1D system with the periodic bound- ary condition, i.e., the 1D ring geometry. We assume that the radius of the ring R is much larger than the domain size, and the curvature of the ring does not affect the dynamics. For q < qc/2, the magnetic correlation function is cal- culated to be F̂+(θ, t)F̂−(θ ′, t) cos[kmuR(θ − θ′)] ×et/τ−τR 2(θ−θ′)2/(tξ2), (24) where τ and ξ are given by Eqs. (19) and (20), and θ and θ′ are azimuthal angles. For qc/2 < q < qc, we obtain F̂+(θ, t)F̂−(θ ′, t) qc − q et/τ−τR 2(θ−θ′)2/(tξ2) with Eqs. (21) and (22). At q = qc/2, the correlation function reads F̂+(θ, t)F̂−(θ ′, t) qc − q τR4(θ − θ′)4 R2(θ − θ′)2 τR4(θ − θ′)4 ,(26) where Γ is the Gamma function and 0F2(a, b, z) = Γ(a)Γ(b) Γ(a+ j)Γ(b + j) is the generalized hypergeometric function. Equa- tion (26) is shown in Fig. 2(a), where Ξ gives a char- acteristic width of the correlation function. Next, we consider the 2D geometry. For qc/2 < q < qc, and then kmu = 0, the integral can be performed analytically, giving F̂+(r, t)F̂−(r ′, t) 2πξ2t qc − q et/τ−τ |r−r ′|2/(tξ2), where τ and ξ are given in Eqs. (21) and (22). For other q, we can perform only the angular integral as F̂+(r, t)F̂−(r ′, t) qc − q − εmu kJ0(k|r − r′|)e2|Ek|t/h̄dk, where J0 is the Bessel function. If the exponential factor is much sharper than the Bessel function around kmu, the correlation function (29) is approximated to be ∝ J0(kmu|r − r′|) [14, 15]. As shown above, the correlation function (17b) has a finite correlation length, and the magnetization at posi- tions widely separated from each other grow with inde- pendent directions in the x-y plane. Thus, the growth of the magnetic domains is expected to leave topological defects through the KZ mechanism. D. Slow quench In the previous sections, we have assumed that the magnetic field is suddenly quenched to the desired value at t = 0 and q is held constant for t > 0. We assume here that for t > 0 the magnetic field is gradually quenched q(t) = qc . (30) The magnetic correlation can be estimated to be F̂+(r, t)F̂−(r ′, t) dk exp 2|Ek(t)|t dt+ ik · (r − r′) . (31) Since we are interested in the vicinity of the critical point where correlation starts to grow, we expand |Ek(t)| around kmu = 0 and keep the terms up to the order of k2. For the 1D ring, we obtain F̂+(θ, t)F̂−(θ ′, t) ∝ ef(t)−R 2(θ−θ′)2/ξ2Q , (32) and for the 2D geometry, F̂+(r, t)F̂−(r ′, t) ∝ ef(t)−|r−r ′|2/ξ2 Q , (33) where f(t) = tan−1 τQ − t 1− 2t , (34) t(τQ − t) . (35) For t≪ τQ, f(t) can be expanded as f(t) = , (36) and from f(t) ∼ 1, the time scale for magnetization is given by Q . (37) Substitution of tQ into Eq. (35) yields Q . (38) The same power law is obtained in Ref. [14]. It is interesting to note that the results (37) and (38) are easily obtained also by the simple discussion by Zurek [2]. Since q(t) depends on time, τ and ξ given in Eqs. (21) and (22) are time dependent, and hence they are regarded as the growth time and correlation length at each instant of time. The local magnetization is de- veloped after a time tQ has elapsed such that τ(tQ) ∼ tQ. (39) Using τ(t) = t(τQ − t) , (40) we obtain tQ in Eq. (37). Substituting this tQ into ξ2(t) = τQ − 2t t(τQ − t) yields Eq. (38). III. NUMERICAL SIMULATIONS AND THE KIBBLE-ZUREK MECHANISM A. Gross-Pitaevskii equation with quantum fluctuations The multicomponent Gross-Pitaevskii (GP) equation is obtained by replacing the field operators ψ̂m with the macroscopic wave function ψm in the Heisenberg equa- tion of motion: ∇2 + Vtrap + q + c0ρ F∓ψ0 ± Fzψ±1 , (42a) ∇2 + Vtrap + c0ρ (F+ψ1 + F−ψ−1) , (42b) where ρ and F are defined using ψm instead of ψ̂m in Eqs. (3) and (4). The wave function is normalized as |ψm|2 = N, (43) with N being the number of atoms in the condensate. Suppose that all atoms are initially in the m = 0 state. It follows then from Eq. (42a) that ψ±1 will remain zero in the subsequent time evolution. This is because quan- tum fluctuations in the transverse magnetization that trigger the growth of magnetization are neglected in the mean-field approximation. We therefore introduce an ap- propriate initial noise in ψ±1 so that the mean-field ap- proximation reproduces the quantum evolution. Let us write the initial state as ψ±1(r) = eik·ra±1,k(0), (44) where a±1,k are c-numbers. We assume that the c- number amplitudes a±1,k(0) are stochastic variables whose average values vanish, 〈a±1,k(0)〉avg = 0, (45) where by 〈· · · 〉avg we denote the statistical average over an appropriate probability distribution. The linear ap- proximation of the GP equation with respect to a±1,k gives the same time evolution as Eq. (11), in which the operators are replaced by the c-numbers. We thus obtain F+(r, t)F−(r ′, t) = εk + q e−ik·(r−r ′)|a1,k(0)|2 +eik·(r−r ′)|a−1,−k(0)|2 . (46) Comparing Eq. (46) with Eq. (17a), we find that they have the same form if the variance of a±1,k(0) satisfies 〈|a±1,k(0)|2〉avg = for all k. 0 100 200 300 t [ms] q=qc/2 0 π 2π |F+| /ρ arg F+ q=qc/ 20 0 π 2π |F+| /ρ arg F+ (b) (c) FIG. 1: (Color online) (a) Time evolution of the auto cor- relation function given in Eq. (49) for the 1D ring geometry. (b) Magnitude of the normalized magnetization |F+|/ρ (solid curve, left axis) and direction of the magnetization argF+ (dashed curve, right axis) at t = 70 ms for q = 0 and (c) for q = qc/2. The radius of the ring is R = 50 µm, the atomic density is n = 2.8 × 1014 cm−3, and the number of atoms is N = 106. In the following, we will perform numerical simulation of spontaneous magnetization using the GP equation and show that the ensuing dynamics exhibits defect formation similar to the KZ mechanism. As the initial state of the m = ±1 wave functions, we use Eq. (44) with a±1,k(0) = αrnd + iβrnd, (48) where αrnd and βrnd are random variables following the normal distribution p(x) = 2/π exp(−2x2). Equation (48) then satisfies Eqs. (45) and (47). B. 1D ring geometry Let us first investigate the 1D ring system. Experi- mentally this geometry can be realized, e.g., by an opti- cal trap using a Laguerre-Gaussian beam [19]. We reduce the GP equation (42) to 1D by assuming that the wave function ψm depends only on the azimuthal angle θ. The average density of atoms is assumed to be n = 2.8× 1014 cm−3. When the radius of the ring is R = 50 µm and the radius of the small circle is 2 µm, the total number of atoms is N ≃ 106. Figure 1 illustrates a single run of time evolution for an initial state given by Eqs. (44) and (48). Figure 1 (a) shows time evolution of the auto correlation function defined by F̄ (t) = |F+(θ, t)|2 ρ2(θ, t) . (49) For both q = 0 and q = qc/2, the transverse magneti- zation grows exponentially with a time constant ∼ τ = h̄/qc ≃ 8 ms. Snapshots of the transverse magnetization at t = 70 ms are shown in Figs. 1 (b) and 1 (c) for q = 0 and q = qc/2, respectively. We define the spin winding number as 2i|F+|2 , (50) which represents the number of rotation of the spin vector in the x-y plane along the circumference, and of course w is an integer. The spin winding numbers are w = 7 in Fig. 1 (b) and w = −1 in Fig. 1 (c). Figure 2 (a) shows the ensemble average of the nor- malized correlation function, 〈Fcorr(δθ)〉avg = dθF+(θ)F−(θ + δθ) dθρ(θ)ρ(θ + δθ) , (51) at t = 70 ms. For q = qc/2, the correlation function has the characteristic width of ∼ Ξ in Eq. (23), indicating that the ring is filled with magnetic domains with an av- erage size of ∼ Ξ. According to the KZ theory, the mag- netic domains with random directions give rise to the spin winding, which is estimated to be w ∼ (R/Ξ)1/2. This R dependence of w is clearly seen in Fig. 2 (b). The ensem- ble average of the winding number, 〈w〉avg , vanishes due to the random nature of the initial noise, and the square root of its variance, 〈w2〉1/2avg, should be regarded as a typ- ical winding number. The variance is expected to obey the χ2 distribution with 1000 degrees of freedom, and hence we show the 95% confidence interval to estimate the statistical errors in Fig. 2. As shown in the inset of Fig. 2 (b), the typical winding number changes in time, since the ferromagnetic energy is converted to the kinetic energy and the system exhibits complicated dynamics. The situation is different for q = 0, in which the cor- relation function oscillates with a Gaussian envelope as shown in Fig. 2 (a). This form of the correlation function gives us the answer to the question as to how the KZ mechanism manifests itself in spin conserving systems. The finite correlation length for q = 0 indicates that the spin is conserved not only globally but also locally, that is, the locally integrated spin over the correlation length |δr|<∼ξ F (r + δr)dδr, (52) is held to be zero for any r. This local spin conservation is due to formation of staggered domain or helical spin structures whose periodic length is much smaller than −π/2 0 π/2 q=qc/2 10 100 1000 R [µm] q=qc/2 0 100 t [ms] q=qc/2 FIG. 2: (Color online) (a) Numerically obtained correlation function given in Eq. (51) at t = 70 ms (solid curves), and theoretical fits (dashed curves) from Eqs. (24) and (26). Other parameters are the same as those in Fig. 1. (b) R dependence of the variance of the spin winding number, where the number of atoms is related to R as N = 106×R [µm] /50. The dashed lines are semi-log fits to the numerical data. The inset shows the time dependence of 〈w2〉avg for R = 50 µm. The data in (a) and (b) are averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars in (b) represent the 95% confidence interval of the χ2 distribution. ξ. Thus, the neighboring domains tend to have opposite magnetizations to cancel out the spin locally, and the domains far from each other grow independently; the spin conservation and the KZ mechanism are thus compatible. The oscillation in the correlation function originates from the fact that the most unstable modes have nonzero wave numbers ±kmu. Each correlated region of size ∼ ξ = [8h̄2/(Mqc)] 1/2 contains spin waves of eikmuRθ and e−ikmuRθ. If there is an imbalance between these modes, the winding number monotonically increases or decreases in each region of ∼ ξ. This is the reason why 〈w2〉avg is larger for q = 0 than for q = qc/2 in Fig. 2 (b). It follows from this consideration that for kmuξ ≫ 1 the winding 0.1 1 1−2q/qc slope = 3/2 FIG. 3: (Color online) Dependence of the variance of the spin winding number on q. Except for q, the parameters are the same as those in Fig. 1. The dashed line is proportional to (1−2q/qc) 3/2. The plots show the averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars represent the 95% confidence inter- val of the χ2 distribution. number is proportional to w ∼ kmuξ = kmu 1− 2q , (53) where Eqs. (14) and (20) are used. Figure 3 shows the averaged variance of the winding number versus 1−2q/qc. For small q, 〈w2〉avg is proportional to (1 − 2q/qc)3/2, in agreement with Eq. (53). When q is close to qc/2, the spin winding within the correlated region, kmuξ, becomes small, and then the winding number reduces to the value shown in Fig. 2 (b), i.e., 〈w2〉avg ≃ 4. We next discuss the results of simulations of slow quench as in Eq. (30). Since the winding number for the slow quench is small compared with the fast quench, we take a large ring of R = 400 µm. Figure 4 shows the variance of the winding number as a function of the quench time. We can clearly see that 〈w2〉avg has a power law of τ Q within the statistical error, which is in agree- ment with ξ−1Q ∼ τ Q , with ξQ being given in Eq. (38). Thus, the present system follow the quench-time scaling of Zurek [2]. We note that, in the slow quench, the wind- ing number converges to an almost constant value for varying quench time τQ, as shown in the inset of Fig. 4. This is because little excess energy other than for exciting spin vortices is available for the slow quench. C. 2D disk geometry When the confinement in the z direction is tight, the system is effectively 2D. For simplicity, we ignore the den- sity dependence in the z direction, and assume that the τQ[s] slope=−1/3 0 100 200 300 400 t [ms] 0.8 s 1.6 s 3.2 s 6.4 s FIG. 4: (Color online) Dependence of the variance of the spin winding number at t = 400 ms on the quench time τQ, where q is varied as in Eq. (30). The radius of the ring is R = 400 µm, the atomic density is n = 2.8 × 1014 cm−3, and the number of atoms is N = 8 × 106. The dashed line is proportional . The inset shows time evolution of 〈w2〉avg. The data are averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars represent the 95% confidence interval of the χ2 distribution. 2D GP equation has the same form as Eq. (42). We as- sume that the wave function vanishes at the wall located at (x2 + y2)1/2 = Rw = 100 µm, and that the potential is flat inside of the wall. Then the density n = 2.8× 1014 cm−3 is almost constant except within the healing length {3/[8πn(a0 + 2a2)]}1/2 ≃ 0.16 µm near the wall. When the thickness in the z direction is ≃ 1 µm, the number of atoms is N ≃ 107. Such a system will be realized using an optical sheet and a hollow laser beam. The initial state of ψ0 is a stationary solution of the GP equation, and the initial state of ψ±1 is given by Eq. (44) with random variables (48). Figure 5 (a) shows time evolution of the autocorrelation function of the transverse magnetization, F̄ (t) = |F+(r, t)|2 ρ2(r, t) , (54) which grows exponentially with the same time constant as that in Fig. 1, and saturates for t >∼ 100 ms. Snapshots of |F+(r)| and argF+(r) at t = 100 ms are shown in Figs. 5 (b) and 5 (c). We see that |F+(r)| at t >∼ 100 ms contains many holes, around which the spin direction rotates by 2π. Since this topological spin structure consists of singly-quantized vortices in the m = ±1 states filled by atoms in the m = 0 state, it is called the “polar-core vortex.” We can estimate the spin healing length ξs by equating the kinetic energy h̄ 2/(2Mξ2s ) with 0 100 200 300 t [ms] q=0 q=qc/2 (c) q = q / 2 (b) q = 0 t = 50 ms t = 100 mst = 75 ms t = 200 ms t = 50 ms t = 100 mst = 75 ms t = 200 ms FIG. 5: (Color) (a) Time evolution of the autocorrelation function given in Eq. (54) for the 2D disk geometry. The radius of the disk is Rw = 100 µm, the atomic density is n = 2.8× 10 14 cm−3, and the number of atoms is N = 107. (b) Profiles of the magnetization |F+| (upper) and its direction argF+ (lower) for q = 0 and (c) for q = qc/2. The size of each panel is 200 µm ×200 µm. the energy of magnetization |q − qc|, giving 2M |q − qc| . (55) This length scale is ξs ≃ 1.7 µm for q = 0 and ξs ≃ 2.4 µm for q = qc/2, which are in good agreement with the sizes of the vortex cores in Figs. 5 (b) and 5 (c). In 2D, the correlation function is defined by 〈Fcorr(δr)〉avg = drF+(r)F−(r + δr) drρ(r)ρ(r + δr) , (56) which are shown in Figs. 6 (a) and 6 (b). We find that as in the 1D case the most unstable wave length is reflected in the shape of the spin correlation function (56), and the characteristics of these correlation functions in the ra- 1 10 100 R [µm] q=qc/2 (a) q = 0 (b) q = q / 2c -50 500 x [µm] FIG. 6: (Color) (a) Spin correlation function defined in Eq. (56) at t = 100 ms for q = 0 and (b) for q = qc/2. (c) The variance of the winding number along the circumfer- ence of the circle of radius R. The dashed lines and dotted lines are proportional to R and R2, respectively. In (a)-(c) the parameters are the same as those in Fig. 5, and the data are averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars in (c) represent the 95% confidence interval of the χ2 distribution. dial direction are similar to those in 1D shown in Fig. 2. For q = 0, the mean distance between spin vortices in Fig. 5 (b) is not determined by the correlation length (the whole width of the concentric pattern in Fig. 6 (a)) but by ∼ k−1mu, i.e., the width of the concentric rings in Fig. 6 (a). On the other hand, for q = qc/2, the density of spin vortices is determined by the correlation length, i.e., the size of the blue circle ≃ 30 µm in Fig. 6 (b). The staggered concentric correlation for q = 0 suggests that the spin is conserved locally within the region of the cor- relation length, and domains at a distance larger than the correlation length grow independently, while conserving the total spin. The spin winding number for 2D is defined as w(R) = 2i|F+|2 (F−∇F+ − F+∇F−) · dr, where C(R) is a circle with radius R < Rw located at the center of the system. Figure 6 (c) shows the R depen- dence of the ensemble average of w2(R), where the radius of the system is fixed to Rw = 100 µm and the data are taken at t = 100 ms. It should be noted that 〈w2(R)〉avg is proportional to R for large R, as expected from the KZ theory [2], while it is proportional to R2 for small R. This R2 dependence is due to the fact that the probabil- ity P for a spin vortex to be in the circle is proportional to πR2. The variance of the winding number is therefore 0(1−P )+12P/2+(−1)2P/2 ∝ R2, if the probability that two or more vortices enter the circle is negligible. This condition is met when the density of spin vortices times πR2 is much smaller than unity, and hence the radius R at which the crossover from 〈w2(R)〉avg ∝ R to ∝ R2 occurs is larger for q = qc/2 than for q = 0. As in 1D, nonzero kmu enhances the winding of magnetization, and the winding number is larger for q = 0 than for q = qc/2. Figures 5 (b) and 5 (c) obviously show that the density of spin vortices is uniform when the size of the system is large enough. The number of spin vortices in a radius R is therefore proportional to R2. If the topological charge of each spin vortex, +1 or −1, was chosen at random, the net winding number along the circle of radius R, i.e., the difference between the numbers of +1 and −1 vor- tices would be proportional to R. However, from Fig. 6 (c), the winding number is proportional to R1/2 for large R, consistent with the KZ mechanism. The topological charge of each spin vortex is thus not at random but anticorrelated to each other to reduce the net winding number. Figure 7 shows the result of the slow quench for 2D, where q(t) is given by Eq. (30). The winding number fol- lows the scaling law, 〈w2〉avg ∝ τ−1/3Q , as predicted from Eq. (38), indicating that Zurek’s discussion is applicable also to 2D. In order to obtain this scaling law, we must specify the time at which the winding number is taken, since the spin winding number decays in time, as shown in the inset of Fig. 7. From the scaling law in Eq. (37), we specify the time to take the winding number as = const., (58) which is indicated by the arrows in the inset of Fig. 7. IV. CONCLUSIONS In this paper, we have studied the dynamics of a spin-1 BEC with a ferromagnetic interaction after quench of the applied magnetic field in an attempt to investigate spon- taneous defect formation in the spinor BEC. We have analyzed the magnetization triggered by quantum fluc- tuations using the Bogoliubov approximation, and per- formed numerical simulations of the GP equation with initial conditions that simulate quantum fluctuations. We have shown that the correlation functions of the magnetization have finite correlation lengths (Figs. 2, 6 (a), and 6 (b)), and therefore magnetic domains far from τQ [s] slope = -1/3 0 100 200 300 400 t [ms] τQ=1 2 4 8 [s] FIG. 7: (Color online) (a) Variance of the spin winding num- ber versus the quench time τQ for the 2D disk geometry, where q is varied as in Eq. (30). The inset shows time evolution of 〈w2〉avg. The plots are taken at the times when t/τ constant is satisfied, which are shown by the arrows in the inset. The dashed line is proportional to τ . The radius of the disk is Rw = 400 µm and the closed path for taking the winding number is R = 320 µm. The atomic density is n = 2.8×1014 cm−3 and the number of atoms is N = 1.6×108. The data are averages over 1000 runs of simulations for dif- ferent initial states produced by random numbers. The error bars represent the 95% confidence interval of the χ2 distribu- tion. each other grow in random directions. We find that topo- logical defects — spin vortices — emerge through the KZ mechanism. We have confirmed that the winding num- ber along the closed path is proportional to the square root of the length of the path (Figs. 2 (b) and 6 (c)), indicating that the topological defects are formed from domains with random directions of magnetizations. Even when the total magnetization is conserved for q = 0, the winding number has the same dependence on the length of the path (Fig. 2 (b)). This is due to the fact that domains within the correlation length tend to be aligned in such a manner as to cancel out local mag- netization, and consequently the total magnetization is conserved. Thus, the neighboring domains have local cor- relation, while domains far from each other are indepen- dent, which makes the KZ mechanism compatible with the total spin conservation. The formation of the local correlation also creates topological defects as well as the KZ mechanism, and the winding number exhibits the q dependence as shown in Fig. 3. When the magnetic field is quenched in finite time τQ as in Eq. (30), the winding number has been shown to be proportional to τ Q (Figs. 4 and 7). This τQ depen- dence of the winding number can be understood from Zurek’s simple discussion [2]: the domains are frozen at which the spin relaxation time becomes the same order of elapsed time. In the Berkeley experiment [12], the system is an elon- gated quasi-2D geometry, and not suitable for testing the KZ mechanism. The KZ mechanism should apply to the system in which the size of the system in the x direction is made much larger. In this case, the har- monic potential may affect the scaling law, which mer- its further study. Moreover in the experiment, from the analysis in Ref. [13], there are some initial atoms in the m = ±1 components with long-range correlation, which play a role of seeds for large domains and hinder the ob- servation of the KZ mechanism. If the residual atoms in the m = ±1 components is eliminated completely, mag- netization is triggered by quantum fluctuations as shown in the present paper. Another way to remove the effect of the residual atoms may be applying random phases to the m = ±1 states to erase the initial correlation. Note added. After our work was completed, the preprint by Damski and Zurek [20] appeared, which per- forms 1D simulations of the quench dynamics of a spin-1 Acknowledgments This work was supported by Grants-in-Aid for Scien- tific Research (Grant Nos. 17740263 and 17071005) and by the 21st Century COE programs on “Coherent Op- tical Science” and “Nanometer-Scale Quantum Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan. MU acknowledges support by a CREST program of the JST. [1] T. W. B. Kibble, J. Phys. A 9, 1387 (1976). [2] W. H. Zurek, Nature (London) 317, 505 (1985); Phys. Rep. 276, 177 (1996). [3] I. Chuang, R. Durrer, N. Turok, and B. Yurke, Science 251, 1336 (1991). [4] M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Sri- vastava, Science 263, 943 (1994). [5] P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. Mc- Clintock, and C. D. H. Williams, Nature (London) 368, 315 (1994); M. E. Dodd, P. C. Hendry, N. S. Lawson, P. V. E. McClintock, and C. D. H. Williams, Phys. Rev. Lett. 81, 3703 (1998). [6] V. M. H. Ruutu, V. B. Eltsov, A. J. Gill, T. W. B. Kibble, M. Krusius, Yu. G. Makhlin, B. Plaçais, G. E. Volovik, and W. Xu, Nature (London) 382, 334 (1996); V. M. Ruutu, V. B. Eltsov, M. Krusius, Yu. G. Makhlin, B. Plaçais, and G. E. Volovik, Phys. Rev. Lett. 80, 1465 (1998). [7] C. Bäuerle, Yu. M. Bunkov, S. N. Fisher, H. Godfrin, and G. R. Pickett, Nature (London) 382, 332 (1996). [8] S. Ducci, P. L. Ramazza, W. González-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999). [9] R. Carmi, E. Polturak, and G. Koren, Phys. Rev. Lett. 84, 4966 (2000). [10] R. Monaco, J. Mygind, and R. J. Rivers, Phys. Rev. Lett. 89, 080603 (2002). [11] A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett. 91, 197001 (2003). [12] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature (London) 443, 312 (2006). [13] H. Saito, Y. Kawaguchi and M. Ueda, Phys. Rev. A 75, 013621 (2007). [14] A. Lamacraft, cond-mat/0611017. [15] M. Uhlmann, R. Schützhold, and U. R. Fischer, cond-mat/0612664. [16] H. Saito, Y. Kawaguchi and M. Ueda, Phys. Rev. Lett. 96, 065302 (2006). [17] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. -J. Mies- ner, A. P. Chikkatur, and W. Ketterle, Nature (London) 396, 345 (1998). [18] K. Murata, H. Saito, and M. Ueda, Phys. Rev. A 75, 013607 (2007). [19] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997). [20] B. Damski and W. H. Zurek, arXiv:0704.0440. http://arxiv.org/abs/cond-mat/0611017 http://arxiv.org/abs/cond-mat/0612664 http://arxiv.org/abs/0704.0440

0704.1378

Triangulated categories without models

TRIANGULATED CATEGORIES WITHOUT MODELS FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND Abstract. We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are fundamental tools in both algebra and topology. In algebra they often arise as the stable category of a Frobenius cat- egory ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which belong, up to exact equivalence, to one of these two families will be termed algebraic and topo- logical, respectively. We borrow this terminology from [Kel06, 3.6] and [Sch06]. Algebraic triangulated categories are generally also topological, but there are many well-known examples of topological triangulated categories which are not algebraic. In the present paper we exhibit examples of triangulated categories which are neither algebraic nor topological. As far as we know, these are the first examples of this kind. Even worse (or better, depending on the perspective), our examples do not even admit non-trivial exact functors to or from algebraic or topological triangulated categories. In that sense, the new examples are completely orthogonal to previously known triangulated categories. Let (R,m) be a commutative local ring with m = (2) 6= 0 and m2 = 0. Examples of this kind of rings are R = Z/4, or more generally R = W2(k) the 2-typical Witt vectors of length 2 over a perfect field k of characteristic 2. There are also examples which do not arise as Witt vectors, for instance the localization of the polynomial ring Z/4[t] at the prime ideal (2). We denote by F(R) the category of finitely generated free R-modules. Theorem 1. The category F(R) has a unique structure of a triangulated category with identity translation functor and such that the diagram is an exact triangle. Given an object X in an algebraic triangulated category T and an exact triangle −→ A −→ C −→ ΣA, 1991 Mathematics Subject Classification. 18E30, 55P42. Key words and phrases. Triangulated category, stable model category. The first author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract. http://arxiv.org/abs/0704.1378v2 2 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND the equation 2 · 1C = 0 holds, compare [Kel06, 3.6] and [Sch06]. Since the ring R satisfies 2 · 1R 6= 0, the triangulation of the category F(R) is not algebraic. We cannot rule out the possibility of a topological model for F(R) as easily: the classical example of A = S the sphere spectrum in the stable homotopy category shows that the morphism 2 · 1C can be nonzero in this more general context. Nevertheless, F(R) is not topological either, which follows from Theorem 2. Here we call an exact functor between triangulated categories trivial if it takes every object to a zero object. Theorem 2. Every exact functor from F(R) to a topological triangulated category is trivial. Every exact functor from a topological triangulated category to F(R) is trivial. Acknowledgements. We are grateful to Bernhard Keller for helpful conversations on the results of this paper, and to Amnon Neeman, who suggested the possibility of constructing a triangulated structure on F(Z/4) by using Heller’s theory [Hel68]. In the original version of this note the first author alone constructed the tri- angulation of the category F(Z/4) and proved that it does not admit any model. The second author joined the project later by providing a simpler and more general proof that the triangulation is not topological. The third author’s contribution was an old preprint on the example considered in Remark 8, which provided some guidance for the other results. The triangulated categories. Let T be an additive category and let Σ: T be a self-equivalence that we call translation functor. A candidate triangle (f, i, q) in (T,Σ) is a diagram (3) A −→ ΣA, where if , qi, and (Σf)q are zero morphisms. A morphism of candidate triangles (α, β, γ) : (f, i, q) → (f ′, i′, q′) is a commutative diagram // B′ // C′ // ΣA′ The category of candidate triangles is additive. The mapping cone of the morphism (α, β, γ) is the candidate triangle B ⊕A′ β f ′ // C ⊕B′ // ΣA⊕ C′ −Σf 0 Σα q′ // ΣB ⊕ ΣA′. A homotopy (Θ,Φ,Ψ) from (α, β, γ) to (α′, β′, γ′) is given by morphisms // B′ // C′ // ΣA′ such that β′ − β = Φi+ f ′Θ, γ′ − γ = Ψq + i′Φ, Σ(α′ − α) = Σ(Θf) + q′Ψ. TRIANGULATED CATEGORIES WITHOUT MODELS 3 We say in this case that the morphisms are homotopic. The mapping cones of two homotopic morphisms are isomorphic. A contractible triangle is a candidate triangle such that the identity is homotopic to the zero morphism. A homotopy (Θ,Φ,Ψ) from 0 to 1 is called a contracting homotopy. Any morphism from or to a contractible triangle is always homotopic to zero. A triangulated category is a pair (T,Σ) as above together with a collection of candidate triangles, called distinguished or exact triangles, satisfying the follow- ing properties. The family of exact triangles is closed under isomorphisms. The candidate triangle (4) A −→ A −→ 0 −→ ΣA, is exact. Any morphism f : A → B in T can be extended to an exact triangle like (3). A candidate triangle (3) is exact if and only if its translate −→ ΣA −→ ΣB, is exact. Any commutative diagram // ΣA // B′ // C′ // ΣA′ whose rows are exact triangles can be extended to a morphism whose mapping cone is also exact. This non-standard set of axioms for triangulated categories is equivalent to the classical one, see [Nee01], and works better for the purposes of this paper. Now we are ready to prove Theorem 1. Proof of Theorem 1. Given an object X in F(R) we consider the candidate triangle X2 defined as (5) X −→ X. We are going to prove that the category F(R) has a triangulated category struc- ture with identity translation functor where the exact triangles are the candidate triangles isomorphic to the direct sum of a contractible triangle and a candidate triangle of the form (5). The family of exact triangles is closed under isomorphisms by definition. The candidate triangle (4) is contractible, and hence exact. The ring R is a quotient of a discrete valuation ring with maximal ideal generated by 2, see [Coh46, Corollary 3]; therefore any morphism f : A→ B in F(R) can be decomposed up to isomorphism 1 0 0 0 2 0 0 0 0  : A =W ⊕X ⊕ Y −→W ⊕X ⊕ Z = B. Then f is extended by the direct sum of (5) and the contractible triangle W ⊕ Y // W ⊕ Z // Y ⊕ Z // W ⊕ Y. 4 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND The translate of a contractible triangle is also contractible, and the triangle (5) is invariant under translation. This proves that the translate of an exact triangle is exact. Translating a candidate triangle six times yields the original one, therefore if a candidate triangle has an exact translate then the original candidate triangle is also exact. We say that a candidate triangle A → A is a quasi-exact triangle if −→ B, is an exact sequence of R-modules. The exact triangles are all quasi-exact. Now we are going to show that any diagram of candidate triangles (6) A // B′ // C′ // A′ with exact rows can be completed to a morphism with exact mapping cone. Suppose that the upper row in (6) is contractible and the lower row is quasi- exact. Since f ′α = βf then f ′αq = 0; since C is projective, there exists γ′ : C → C′ such that q′γ′ = αq. Let (Θ,Φ,Ψ) be a contracting homotopy for the upper row. Then γ = γ′ + (i′β − γ′i)Φ completes (6) to a morphism of candidate triangles. If the upper row in (6) is quasi-exact and the lower row is contractible then (6) can also be completed to a morphism. This can be shown directly, but it also follows from the previous case since we have a duality functor HomR(−, R) : F(R) −→ F(R)op, which preserves contractible triangles and quasi-exact triangles. Here we use that R is injective as an R-module, see [Lam99, Example 3.12]. If the upper and the lower rows in (6) areX2 and Y2, respectively, then γ = β+2δ extends (6) to a morphism of candidate triangles for any δ : X → Y . This proves that any diagram like (6) with exact rows can be completed to a morphism ϕ = (α, β, γ). Now we have to check that the completion can be done in such a way that the mapping cone is exact. Suppose that the upper and the lower rows are X2 ⊕ T and Y2 ⊕ T ′, respectively, with T and T ′ contractible. The morphism ϕ is given by a matrix of candidate triangle morphisms ϕ11 ϕ12 ϕ21 ϕ22 : X2 ⊕ T −→ Y2 ⊕ T where ϕij = (αij , βij , γij). Here ϕ12, ϕ21 and ϕ22 are homotopic to 0 since either the source or the target is contractible, therefore the mapping cone of ϕ is isomorphic to the mapping cone of ϕ11 0 : X2 ⊕ T −→ Y2 ⊕ T which is the direct sum of the mapping cone of ϕ11 and two contractible triangles, T ′ and the translate of T . TRIANGULATED CATEGORIES WITHOUT MODELS 5 We can suppose that α11 = 1 0 0 0 2 0 0 0 0  : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y. Moreover, as we have seen above we can take γ11 = β11 + 2δ for 0 0 0 0 1 0 0 0 0  : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y. We have 2β11 = 2α11, therefore β11 = α11 + 2Φ for some Φ: X → Y . Now we observe that (δ,Φ, 0) is a homotopy from ϕ11 to ζ = (α11 + 2δ, α11 + 2δ, α11 + 2δ), so the mapping cone of ϕ11 is isomorphic to the mapping cone of ζ. The mapping cone of ζ is clearly the direct sum of five candidate triangles, namely M2, N2, M2 (once again), P2, and the mapping cone of the identity 1 : L2 → L2, which is contractible. Therefore the mapping cone of ζ is exact, and also the mapping cone of ϕ11, ψ and ϕ. It remains to show the uniqueness claim in Theorem 1. In any triangulation, all contractible candidate triangles are exact [Nee01, 1.3.8]. The triangle X2 is a finite direct sum of copies of R2. Hence every triangulation of (F(R), Id) which contains R2 contains all the exact triangles which we considered above. Two triangulations with the same translation functor necessarily agree if one class of triangles is con- tained in the other, so there is only one triangulation in which R2 is exact. This completes the proof. � Remark 7. The exact triangles in F(R) can be characterized more intrinsically as follows. Let T be a quasi-exact triangle, which we can regard as a Z/3-graded chain complex of free R-modules with H∗(T ) = 0. As T is free we have a short exact sequence 2T →֒ T ։ 2T, and the resulting long exact sequence in homology reduces to an isomorphism σ : H∗(2T ) → H∗−1(2T ). As the grading is 3-periodic we can regard σ 3 as an automorphism of H∗(2T ). We claim that T is exact if and only if σ 3 = 1. One direction is straightforward: if T is contractible then H∗(2T ) = 0, and if T = X2 then Hi(2T ) = 2X for all i and σ is the identity. The converse is more fiddly and we will not go through the details. It would be nice to give a proof of Theorem 1 based directly on this definition of exactness, but we do not know how to do so. Remark 8. Let k be a field of characteristic 2. The same arguments as in the proof of Theorem 1 show that the category F(k[ε]/ε2) of finitely generated free modules over the algebra k[ε]/ε2 of dual numbers admits a triangulation with the identity translation functor and such that the diagram k[ε]/ε2 −→ k[ε]/ε2 −→ k[ε]/ε2 −→ k[ε]/ε2 is an exact triangle. However, this triangulated category is both algebraic and topological, and hence, from our current perspective, less interesting. Indeed F(k[ε]/ε2) is an algebraic and topological triangulated category for any field k. The translation functor Σ = τ∗ is the restriction of scalars along the k-algebra automorphism τ : k[ε]/ε2 → k[ε]/ε2 with τ(ε) = −ε, and (9) k[ε]/ε2 −→ k[ε]/ε2 −→ k[ε]/ε2 −→ τ∗k[ε]/ε2 6 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND is an exact triangle. An algebraic model for this triangulated category was obtained by Keller in [Kel05]. Keller’s model is a differerential graded (dg) k-category. Here we exhibit an alternative model, which is a dg k-algebra A such that F(k[ε]/ε2) is exact equivalent to the category of compact objects in the derived categoryD(A) of dg (right) A-modules. This shows that F(k[ε]/ε2) is both algebraic and topological. Let A = k〈a, u, v, v−1〉/I be the free graded k-algebra generated by a, u, v and v−1 in degrees |a| = |u| = 0 and |v| = −1 modulo the two-sided homogeneous ideal I generated by a2, au+ ua+ 1, av + va and uv + vu. The differential d : A→ A is determined by d(a) = u2v, d(u) = 0, d(v) = 0, and the Leibniz rule. The ungraded algebra H0(A) is isomorphic to the dual num- bers k[ε]/ε2, where ε = [u] is the homology class of the cycle u. The graded algebra H∗(A) is determined by this isomorphism since [v] is a unit in degree −1 such that ε · [v] + [v] · ε = 0. We claim that the 0-dimensional homology functor H0 : D c(A) −→ F(k[ε]/ε2) is an equivalence of categories, where the left hand side is the full subcategory of those dg A-modules whose H0 is finitely generated over k[ε]/ε Let M be any dg A-module and let [x] ∈ H0(M) be a homology class with [x] · ε = 0. We choose a representing cycle x and an element y with d(y) = xu; then the element z = yuv−xa is a cycle with x = zu−d(ya), so [x] = [z] ·ε in homology. So every homology class which is annihilated by ε is also divisible by ε, which proves that H0(M) is a free k[ε]/ε 2-module. Moreover, the translation functor in D(A) is the usual shift of complexes M 7→M [1] and the natural isomorphism τ∗H0(M) ∼= H0(M [1]) = H−1(M) is given by [x] 7→ [xv]. The universal case of this is M = A{x, y}, the free graded right module over the underlying graded algebra of A with |x| = 0 and |y| = 1. We can endow M with a dg A-module structure with d(x) = 0 and d(y) = xu, so that M is just the mapping cone of the chain map A → A. The cycle z = yuv − xa ∈ M gives a quasiisomorphism A→M . Using this, we obtain an exact triangle −→ A[1] in Dc(A) which maps to the exact triangle (9). The rest of the proof that H0 is an exact equivalence from Dc(A) to F(k[ε]/ε2) is relatively straightforward, and we omit it. We still owe the proof that the triangulated category F(R) does not admit non- trivial exact functors to or from a topological triangulated category. For this pur- pose we introduce two intrinsic properties that an object A of a triangulated cate- gory may have. A Hopf map for an object A is a morphism η : ΣA → A which satisfies 2η = 0 and such that for some (hence any) exact triangle (10) A −→ ΣA TRIANGULATED CATEGORIES WITHOUT MODELS 7 we have iηq = 2 · 1C . An object which admits a Hopf map will be termed hopfian. We note that the class of hopfian objects is closed under isomorphism, suspension and desuspension. If F is an exact functor with natural isomorphism τ : ΣF ∼= FΣ and η : ΣA → A a Hopf map for A, then the composite F (η)τ : ΣF (A) −→ F (A) is a Hopf map for F (A). We call an object E exotic if there exists an exact triangle (11) E −→ ΣE for some morphism h : E → ΣE. We note that the class of exotic objects is closed under isomorphism, suspension and desuspension. Every exact functor takes exotic objects to exotic objects. Every object of the triangulated category F(R) of Theorem 1 is exotic. We remark without proof that the morphism h which makes (11) exact is unique and natural for morphisms between exotic objects. We show below that h is of the form h = 2ψ for an isomorphism ψ : E → ΣE. Remark 12. The integer 2 plays a special role in the definition of exotic objects, which ultimately comes from the sign which arises in the rotation of a triangle. In more detail, suppose that there is an exact triangle (13) E −→ ΣE for some integer n. We claim that if E is nonzero, then n ≡ 2 mod 4 and 4 ·1E = 0, so that the triangle (13) equals the ‘exotic’ triangle (11) with n = 2. Indeed, we can find a morphism ψ : E → ΣE which makes the diagram // ΣE // ΣE commute, and ψ is an isomorphism. We have nψ = h = −nψ which gives 2nψ = 0. Since ψ is an isomorphism, this forces 2n · 1E = 0. Exactness of (13) lets us choose a morphism f : E → E with 2 · 1E = n · f . But then 4 · 1E = n 2f2 = 0. So if n is divisible by 4, then E = 0. If n is odd, then E is anhihilated by 4 and the odd number n2, so also E = 0. Hopf maps are incompatible with the property of being exotic in the sense that these two classes of objects are orthogonal. Proposition 14. Let T be a triangulated category, A a hopfian object and E an exotic object. Then the morphism groups T(A,E) and T(E,A) are trivial. In par- ticular, every exotic and hopfian object is a zero object. Proof. Let η : ΣA → A be a Hopf map. Given any morphism f : E → A there exists g : E → C such that (f, f, g) is a morphism from (11) to (10), and hence if = 2g = iηqg = iη(Σf)h = iη(Σf)2ψ = 0. Here we use the notation of Remark 12 for n = 2 and the fact that 2η = 0. Moreover, (10) is exact, so f = 2f ′ for some f ′ : E → A. This equation follows for any morphism f : E → A, hence f is divisible by any power of 2, but 4 · 1E = 0, so f = 0. The proof of T(A,E) = 0 is similar. Alternatively, we can reduce this statement to the previous one by observing that the properties of being exotic and hopfian 8 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND are self-dual. In other words, an object E is exotic in a triangulated category T if and only if E is exotic as an object of the opposite category Top with the opposite triangulation, and similarly for Hopf maps. � Proposition 15. Every object of a topological triangulated category is hopfian. Proof. We can assume that the topological triangulated category is HoM for a stable model category M. We use that for every object A of HoM there exists an exact functor F : HoSp → HoM from the stable homotopy category which takes the sphere spectrum to an object isomorphic to A. Here Sp is the category of ‘sequential spectra’ of simplicial sets with the stable model structure of Bousfield and Friedlander [BF78, Sec. 2]. To construct F we let X be a cofibrant-fibrant object of the model category M which is isomorphic to A in the homotopy category HoM. The universal property of the model category of spectra [SS02, Thm. 5.1 (1)] provides a Quillen adjoint functor pair Hom(X,−) whose left adjoint X∧ takes the sphere spectrum S to X , up to isomorphism. The left derived functor of the left Quillen functor X ∧ − : Sp → M is exact and can serve as the required functor F . Since exact functors preserve Hopf maps it thus suffices to treat the ‘universal example’, i.e., to exhibit a Hopf map for the sphere spectrum as an object of the stable homotopy category. The stable homotopy class η : ΣS → S of the Hopf map from the 3-sphere to the 2-sphere precisely has this property, hence the name. In more detail, we have an exact triangle −→ S/2 −→ ΣS in the stable homotopy category, where S/2 is the mod-2 Moore spectrum; then the morphism 2 · 1S/2 factors as iηq, and moreover 2η = 0. � In topological triangulated categories, something a little stronger than Proposi- tion 15 is true in that Hopf maps can be chosen naturally for all objects. However, we don’t need this and so we omit the details. Now we can give the Proof of Theorem 2. Every object of the triangulated category F(R) is exotic and every object of a topological triangulated category is hopfian. So an exact functor from one type of triangulated category to the other hits objects which are both exotic and hopfian. But such objects are trivial by Proposition 14. � Remark 16. The only special thing we use in the proof of Theorem 2 about topolog- ical triangulated categories is that therein every object has a Hopf map. Hopf maps can also be obtained from other kinds of structure that were proposed by different authors in order to ‘enrich’ or ‘enhance’ the notion of a triangulated category. So our argument also proves that the triangulated category F(R) of Theorem 1 does not admit such kinds of enrichments, and every exact functors to or from such enriched triangulated categories is trivial. For example, if T is an algebraic trian- gulated category, then for some (hence any) exact triangle (10) we have 2 · 1C = 0; so the zero map is a Hopf map. Another example of such extra structure is the notion of a triangulated derivator, due to Grothendieck [Gro90], and the closely related notions of a stable homotopy TRIANGULATED CATEGORIES WITHOUT MODELS 9 theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram categories in the sense of Franke [Fra96]. In each of these settings, the stable homotopy category is the underlying category of the free example on one gener- ator (the sphere spectrum). We do not know a precise reference of this fact for triangulated derivators, but we refer to [Cis02, Cor. 4.19] for the ‘unstable’ (i.e., non-triangulated) analog. In Franke’s setting the universal property is formulated as Theorem 4 of [Fra96]. These respective universal properties in the enhanced context provide, for every object A, an exact functor (Ho Sp)cp → T which takes the sphere spectrum S to A, up to isomorphism. This functors sends the classical Hopf map for the sphere spectrum to a Hopf map for A. Another kind of structure which underlies many triangulated categories is that of a stable infinity category as investigated by Lurie in [Lur06]. The appropriate uni- versal property of the infinity category of spectra is established in [Lur06, Cor. 17.6], so again every object of the homotopy category of any stable, presentable infinity category has a Hopf map. References [BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisim- plicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II. Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. [Cis02] D.-C. Cisinski, Propriétés universelles et extensions de Kan dérivées, Preprint (2002). [Coh46] I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. [Fra96] J. Franke, Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, K-theory Preprint Archives #139 (1996). http://www.math.uiuc.edu/K-theory/ [GM03] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. [Gro90] A. Grothendieck, Dérivateurs, manuscript, around 1990, partially available from http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html [Hel68] A. Heller, Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968), 28–63. [Hel88] A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, vi+78 pp. [Hel97] A. Heller, Stable homotopy theories and stabilization, J. Pure Appl. Algebra 115 (1997), 113-130. [Hov99] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. [Kel05] B. Keller, On triangulated orbit categories, Doc. Math. 74 (2005), 551–581. [Kel06] B. Keller, On differential graded categories, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, vol. II, European Mathematical Society, 2006, pp. 151–190. [Lam99] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New-York, 1999. [Lur06] J. Lurie, Derived algebraic geometry I: Stable infinity categories. \protect\vrule width0pt\protect\href{http://arxiv.org/abs/math/0608228}{math.CT/0608228} [Nee01] A. Neeman, Triangulated Categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. [Sch06] S. Schwede, Algebraic versus topological triangulated categories, Extended notes of a talk given at the ICM 2006 Satellite Workshop on Triangulated Categories, Leeds, UK, http://www.math.uni-bonn.de/people/schwede/leeds.pdf, 2006. [SS02] S. Schwede and B. Shipley, A uniqueness theorem for stable homotopy theory, Math. Z. 239 (2002), 803-828. 10 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND Universitat de Barcelona, Departament d’Àlgebra i Geometria, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain E-mail address: [email protected] Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany E-mail address: [email protected] Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK E-mail address: [email protected] Introduction Acknowledgements The triangulated categories References

0704.1379

U-max-Statistics

U -max-Statistics W. Lao∗ and M. Mayer† Abstract In 1948, W. Hoeffding introduced a large class of unbiased estima- tors called U -statistics, defined as the average value of a real-valued k-variate function h calculated at all possible sets of k points from a random sample. In the present paper we investigate the corresponding extreme value analogue, which we shall call U -max-statistics. We are concerned with the behavior of the largest value of such function h in- stead of its average. Examples of U -max-statistics are the diameter or the largest scalar product within a random sample. U -max-statistics of higher degrees are given by triameters and other metric invariants. Keywords: random diameter, triameter, spherical distance, extreme value, U -statistics, Poisson approximation 1 Introduction U -statistics form a very important class of unbiased estimators for distri- butional properties such as moments or Spearman’s rank correlation. A U -statistic of degree k with symmetric kernel h is a function of the form U(ξ1, . . . , ξn) = h(ξi1 , · · · , ξik), where the sum stretches over J = {(i1, . . . , ik) : 1 ≤ i1 < · · · < ik ≤ n}, ξ1, . . . , ξn are random elements in a measurable space S and h is a real-valued Borel function on Sk, symmetric in its k arguments. In his seminal paper, Hoeffding [8] defined U -statistics for not necessarily symmetric kernels and for ∗Institute of Stochastics, University of Karlsruhe, Englerstrasse 2, Karlsruhe, 76128 Germany †Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 54, CH-3012 Bern, Switzerland. Supported by Swiss National Foundation Grant No. 200021-103579 http://arxiv.org/abs/0704.1379v1 random points in d-dimensional Euclidean space Rd. Later the concept was extended to arbitrary measurable spaces. Since 1948, most of the classical asymptotic results for sums of i.i.d. random variables have been formulated in the setting of U -statistics, such as central limit laws, strong laws of large numbers, Berry-Esséen type bounds and laws of the iterated logarithm. The purpose of this article is to investigate the extreme value analogue of U -statistics, i.e. Hn = max h(ξi1, . . . , ξik). A typical example of such U -max-statistic is the diameter of a sample of points in a metric space, obtained by using the metric as kernel. Grove and Markvorsen [6] introduced an infinite sequence of metric invariants general- izing the notion of diameter to “triameter”, “quadrameter”, etc. on com- pact metric spaces. Their k-extent is the maximal average distance between k points, which is an example for a U -max-statistic of arbitrary degree k. Other examples are the largest surface area or perimeter of a triangle formed by point triplets, or the largest scalar product within a sample of points in The key to our results is the observation that for all z ∈ R, the U -max- statistic Hn does not exceed z if and only if Uz vanishes, where 1{h(ξi1, . . . , ξik) > z}. The random variable Uz counts the number of exceedances of the threshold z and is a normalized U -statistic in the usual sense. We approximate its distribution with the help of a Poisson approximation result for the sum of dissociated random indicator kernel functions by Barbour et al. [3], which enables us to determine the distribution ofHn up to some known error. In or- der to deduce the corresponding limit law for Hn, the behavior of the upper tail of the distribution of h must be known, which often requires compli- cated geometric computations. Denote by ‖ · ‖ the Euclidean norm. The general results are used to derive limit theorems for the following settings: largest interpoint distance and scalar product of a sample of points in the d-dimensional closed unit ball Bd = {x ∈ Rd : ‖x‖ ≤ 1}, where the directions of the points have a density on the surface Sd−1 of Bd and are independent of the norms; smallest spherical distance of a sample of points with density on d−1; largest perimeter of all triangles formed by point triplets in a sample of uniformly distributed points on the unit circle S. 2 Poisson approximation for U-max-statistics The following result is easily derived from Theorem 2.N for dissociated indi- cator random variables from Barbour et al. [3]. We use the convention that improper sums for k = 1 equal zero. Theorem 2.1. Let ξ1, . . . , ξn be i.i.d. S-valued random elements and h : Sk → R a symmetric Borel function. Putting pn,z = P {h(ξ1, . . . , ξk) > z} , λn,z = pn,z, τn,z(r) = p n,zP {h(ξ1, . . . , ξk) > z, h(ξ1+k−r, ξ2+k−r, . . . , ξ2k−r) > z} , we have, for any n ≥ k and any z ∈ R, |P {Hn ≤ z} − exp{−λn,z}| (2.1) ≤ (1− exp{−λn,z}) k − r τn,z(r) Clearly the result can be reformulated as well for the minimum value of the kernel by replacing h with −h. One of the main applications of this theorem consists in determining a suitable sequence of transformations zn : T → R with T ⊂ R, such that both the right hand side of (2.1) converges to zero as n→ ∞ for all z = zn(t), t ∈ T , and the limits of exp{−λn,zn(t)} are non-trivial for all t ∈ T . The usual choice is T = [0,∞). One way to achieve this goal is based on the following two remarks and will eventually lead to the well known Poisson limit theorem of Silverman and Brown [12], originally proved by a suitable coupling. Remark 1. As already Silverman and Brown [12] stated, pn,z ≤ τn,z(1) ≤ · · · ≤ τn,z(k) = 1. Remark 2. If the sample size n tends to infinity, then the error (2.1) is asymptotically pn,zn k−1 + τn,z(r)n and for k > 1 the sum is dominating, see [3, p. 35]. Remark 3. The symmetry condition on h can be avoided if h is symmetrized h∗(x1, . . . , xk) = max j1,...,jk h(xj1 , . . . , xjk), where the maximum is taken over all permutations of 1, . . . , k. The conditions stated in [12] suffice to ensure that Theorem 2.1 provides a non-trivial Weibull limit law. Corollary 2.2 (Silverman-Brown limit law [12]). In the setting of Theo- rem 2.1, if for some sequence of transformations zn : T → R with T ⊂ R, the conditions λn,zn(t) = λt > 0 (2.2) n2k−1pn,zn(t)τn,zn(t)(k − 1) = 0 (2.3) hold for all t ∈ T , then P {Hn ≤ zn(t)} = exp{−λt} (2.4) for all t ∈ T . Remark 4. Condition (2.2) implies pn,zn(t) = O(n−k) and by Remarks 1 and 2 we obtain for (2.4) the rate of convergence n−1 + n2k−rpn,zn(t)τn,zn(t)(r) with upper bound O(n2k−1pn,zn(t)τn,zn(t)(k − 1)). (2.5) If k > 2, it is sometimes useful to replace (2.3) by the weaker requirement n2k−rpn,zn(t)τn,zn(t)(r) = 0 (2.6) for each r ∈ {1, . . . , k−1}, a fact that follows immediately from Theorem 2.1 and Remark 2. Appel and Russo [2] obtained a Weibull limit law similar to Corollary 2.2 for bivariate h. They assume that the upper tail of the distribution of h(ξ1, x) does not depend on x for almost all x ∈ S, which implies that (2.2) and (2.3) hold. However, this condition is fulfilled only in very rare settings, e.g. for uniformly distributed points on Sd−1. 3 Largest interpoint distance The asymptotic behavior of the range of a univariate sample can be deter- mined by classical extreme value theory, see e.g. [5, Sec. 2.9]. The largest interpoint distance Hn = max 1≤i<j≤n ‖ξi − ξj‖ within a sample of points in Rd is a natural and consistent generalization of the range to spatial data. Matthews and Rukhin [10] derived its limiting behavior for a normal sample, a work which has been generalized by Henze and Klein [7] to a sample of points with symmetric Kotz distribution. Appel et al. [1] found corresponding limit laws in the setting of uniformly distributed points in 2-dimensional compact sets, which are not too smooth near the endpoints of their largest axes. They also provided bounds for the limit law of the diameter of uniformly distributed points in ellipses and the unit disk. The exact limit distribution for the disk and in more general settings was found independently by Lao [9] and Mayer and Molchanov [11]. Lao [9] used Theorem A of [12] to obtain the exact limit law for the diameter of a uniform sample in Bd. The results in [11] rely on a combination of geometric considerations and blocking techniques and yield e.g. the special case of Theorem 3.1 for spherically symmetric distributions. In what follows, we denote by 〈·, ·〉 the scalar product, by µd−1 the (d−1)- dimensional Hausdorff measure and by Γ and B the complete Gamma and Beta functions. Theorem 3.1. Let ξ1, ξ2, . . . be i.i.d. points in B d, d ≥ 2, such that ξi = ‖ξi‖Ui, i ≥ 1, where Ui and ‖ξi‖ are independent and Ui ∈ Sd−1. Assume that the distribution function F of 1− ‖ξ1‖ satisfies s−αF (s) = a ∈ (0,∞) for some α ≥ 0. Further assume that U1 has a density f with respect to µd−1 and that f(x)f(−x)µd−1(dx) ∈ (0,∞). Then n2/γ(2−Hn) ≤ t = 1− exp for t > 0, where γ = (d− 1)/2 + 2α 2 a2Γ2(α + 1) Γ(d+1 + 2α) f(x)f(−x)µd−1(dx). The rate of convergence for t < ∞ is O(n− d−1+4α ). Remark 5. Spherically symmetric distributed points have independent and uniformly distributed directions and hence [11, Th. 4.2] follows immediately from Theorem 3.1 with f(x)f(−x)µd−1(dx) = 2πd/2 The special case α = 1 and a = d yields the limit law for the diameter of a sample of uniformly distributed points in Bd, see [9] or [11]. Remark 6. If ‖ξi‖ = 1 almost surely, then α = 0 and a = 1. For instance, if Ui are uniformly distributed on S d−1, then for t > 0 n4/(d−1)(2−Hn) ≤ t = 1− exp 2d−3Γ(d 2Γ(d+1 see [2] or [11]. Another example appears if Ui has the von Mises-Fisher distribution of dimension d ≥ 2 with density fF (x) = Cd(κ) exp {κ〈µ, x〉} for x ∈ Sd−1, where µ ∈ Sd−1 represents the mean direction and κ > 0 is the concentration parameter. The normalizing constant Cd(κ) is given by Cd(κ) = κd/2−1 (2π)d/2Id/2−1(κ) where Iν denotes the modified Bessel function of the first kind of order ν. fF (x)fF (−x)µd−1(dx) = C2d(κ) 2πd/2 the corresponding limit law follows immediately. A key part of the proof of Theorem 3.1 is the asymptotic tail behavior of the distribution of the distance between two i.i.d. points. Lemma 3.2. If the conditions of Theorem 3.1 hold, then s−γP {‖ξ1 − ξ2‖ ≥ 2− s} = σ1. Proof. Let η1 and η2 be independent random variables with distribution F and denote by βx the smaller central angle between U2 and x ∈ Sd−1. The cosine theorem yields P {‖ξ1 − ξ2‖ ≥ 2− s} = P ‖ξ1‖2 + ‖ξ2‖2 + 2‖ξ1‖‖ξ2‖ cos β−U1 ≥ (2− s)2 cos β−U1 ≥ (2− s)2 − (1− η1)2 − (1− η2)2 2(1− η1)(1− η2) and by expansion of cos β−U1 about 0 we obtain for sufficiently small s P {‖ξ1 − ξ2‖ ≥ 2− s} = P |β−U1| ≤ 2(s̃− η1 − η2) 2 , η1 + η2 ≤ s̃ , (3.1) where |s̃ − s| ≤ C1s2 for some finite C1, thus s̃/s → 1 as s ↓ 0. Lebesgue’s differentiation theorem (see e.g. [4, Th. 2.9.5]) implies that |β−x| ≤ 2(s̃− y) (4(s̃− y)) d−12 = µd−1(B d−1)f(−x) = π Γ(d+1 f(−x) (3.2) for µd−1-almost every x ∈ Sd−1 and any y ∈ [0, s̃]. Integration over all x ∈ Sd−1 with respect to f yields (s̃− y)− |β−U1| ≤ 2(s̃− y) where Γ(d+1 f(x)f(−x)µd−1(dx), and hence with (3.1) P {‖ξ1 − ξ2‖ ≥ 2− s} (s̃− η1 − η2) 2 1{η1 + η2 ≤ s̃} ) = c. If α = 0, then P {ηi = 0} = a, i = 1, 2, and thus s̃−γP {‖ξ1 − ξ2‖ ≥ 2− s} = ca2 = σ1. If α > 0, P {‖ξ1 − ξ2‖ ≥ 2− s} ∫ s̃−y1 (s̃− y1 − y2) 2 dF (y2)dF (y1) and substituting vi = yi/s̃, i = 1, 2, yields P {‖ξ1 − ξ2‖ ≥ 2− s} = ca2α2 ∫ 1−v1 (1− v1− v2) 2 (v1v2) α−1dv2dv1. By Dirichlet’s Formula, the double integral equals Γ2(α)Γ(d+1 Γ(d+1 + 2α) and the proof is complete. Proof of Theorem 3.1. Plugging the transformation zn(t) = 2− tn−2/γ , t > 0 into Corollary 2.2 and using the tail probabilities given in Lemma 3.2, we P {‖ξ1 − ξ2‖ > zn(t)} = tγ , t > 0. Hence condition (2.2) holds for all t > 0. The more extensive part of the proof aims to show that (2.3) holds. Let βx and β x be the smaller central angles between U2 and x ∈ Sd−1 and between U3 and x ∈ Sd−1. Further let η1, η2 and η3 be independent random variables with distribution F . Put sn = tn −2/γ . Following the proof of Lemma 3.2 P {‖ξ1 − ξ2‖ > zn(t), ‖ξ1 − ξ3‖ > zn(t)} |β−U1| ≤ 2s n , |β ′−U1| ≤ 2s n , ηi ≤ sn, i = 1, 2, 3 |β−x| ≤ 2s f(x)µd−1(dx)1{ηi ≤ sn, i = 1, 2, 3} ≤ CE(sd−1n 1{ηi ≤ sn, i = 1, 2, 3}), (3.3) where the last step follows from (3.2) and C is a suitable finite positive constant. If α = 0, then P {ηi = 0} = a, i = 1, 2, 3, and we obtain n3P {‖ξ1 − ξ2‖ > zn(t), ‖ξ1 − ξ3‖ > zn(t)} ≤ Ca3 lim n3sd−1n = Ca 3td−1 lim n−1 = 0. If α > 0, we derive from (3.3) that P {‖ξ1 − ξ2‖ > zn(t), ‖ξ1 − ξ3‖ > zn(t)} sd−1n dF (y3)dF (y2)dF (y1) and substituting vi = yi/sn, i = 1, 2, 3, yields n3P {‖ξ1 − ξ2‖ > zn(t), ‖ξ1 − ξ3‖ > zn(t)} ≤ Ca3 lim sd−1+3αn = Ca 3td−1+3α lim d−1+4α = 0. The rate of convergence is determined via (2.5). 4 Largest scalar product Besides the Euclidean metric, the scalar product is another symmetric kernel on Rd × Rd. The behavior of its largest value Hn = max 1≤i<j≤n 〈ξi, ξj〉 within a sample of points in Bd is determined in the next result. Theorem 4.1. Let ξ1, ξ2, . . . be i.i.d. points in B d, d ≥ 2, such that ξi = ‖ξi‖Ui, i ≥ 1, where Ui and ‖ξi‖ are independent and Ui ∈ Sd−1. Assume that the distribution function F of 1− ‖ξ1‖ satisfies s−αF (s) = a ∈ (0,∞) for some α ≥ 0. Further assume that U1 has a square-integrable density f on Sd−1 with respect to µd−1. Then n2/γ(1−Hn) ≤ t = 1− exp for t > 0, where γ = (d− 1)/2 + 2α 2 a2Γ2(α + 1) Γ(d+1 + 2α) f 2(x)µd−1(dx). The rate of convergence for t < ∞ is O(n− d−1+4α ). Lemma 4.2. If the conditions of Theorem 4.1 hold, then s−γP {〈ξ1, ξ2〉 ≥ 1− s} = σ2. Proof. If βx is the smaller central angle between U2 and x ∈ Sd−1 and η is distributed as 1− ‖ξ1‖‖ξ2‖, then P {〈ξ1, ξ2〉 ≥ 1− s} = P {cos βU1 ≥ (1− s)/(1− η), η ≤ s} . Expanding cos βU1 about 0 yields for all sufficiently small s P {〈ξ1, ξ2〉 ≥ 1− s} = P |βU1| ≤ (2(s̃− η)) 2 , η ≤ s̃ , (4.1) where |s̃−s| ≤ C1s2 for some finite C1, and thus s̃/s → 1 as s ↓ 0. Lebesgue’s differentiation theorem (see e.g. [4, Th. 2.9.5]) implies that |βx| ≤ (2(s̃− y)) (2(s̃− y)) d−12 = µd−1(B d−1)f(x) = Γ(d+1 f(x). (4.2) for µd−1-almost every x ∈ Sd−1 and any y ∈ [0, s̃]. Integration over all x ∈ Sd−1 with respect to f yields (s̃− y)− |βU1| ≤ (2(s̃− y)) Γ(d+1 f 2(x)µd−1(dx), and by (4.1) we obtain P {〈ξ1, ξ2〉 ≥ 1− s} (s̃− η) 2 1{η ≤ s̃} If α = 0, then P {η = 0} = a2 and hence s−γP {〈ξ1, ξ2〉 ≥ 1− s} = ca2 = σ2. If α > 0, then P {〈ξ1, ξ2〉 ≥ 1− s} equals asymptotically, as s ↓ 0, to (1−s̃)/y1 (s̃− 1 + y1y2) 2 dF (1− y2)dF (1− y1). By substituting v1 = (1− y1)/s̃ and v2 = (1− y2)/(1− (1− s̃)/y1)) the last expression equals asymptotically, as s ↓ 0, to ca2α2 1− v1 1− s̃v1 s̃− 1 + (1− s̃v1)(1− s̃v2 1− v1 1− s̃v1 vα−11 v 2 dv2dv1. Hence s−γP {〈ξ1, ξ2〉 ≥ 1− s} = ca2α2 (1− v1) +αvα−11 dv1 (1− v2) 2 vα−12 dv2 = ca2α2B((d+ 1)/2 + α, α)B((d+ 1)/2, α) = σ2. Proof of Theorem 4.1. An application of Corollary 2.2 yields, together with the transformation zn = 1− tn−2/γ , t > 0, and Lemma 4.2 the limit P {〈ξ1, ξ2〉 ≥ zn(t)} = hence (2.2) holds for any t > 0 and it remains to check (2.3). Put sn = tn and let βx and β x be the smaller central angles between U2 and x ∈ Sd−1 and between U3 and x. Following the proof of Lemma 4.2 P {〈ξ1, ξ2〉 ≥ zn(t), 〈ξ1, ξ3〉 ≥ zn(t)} |βU1 | ≤ (2sn) 2 , |β ′U1| ≤ (2sn) 2 , ‖ξi‖ ≥ zn(t), i = 1, 2, 3 |βx| ≤ (2sn) f(x)µd−1(dx)1{‖ξi‖ ≥ zn(t), i = 1, 2, 3} ≤ CE(sd−1n 1{‖ξi‖ ≥ zn(t), i = 1, 2, 3}), (4.3) where the last step follows from (4.2) and C is a suitable finite positive constant. If α = 0, then P {‖ξi‖ = 1} = α, i = 1, 2, 3, and hence n3P {〈ξ1, ξ2〉 ≥ zn(t), 〈ξ1, ξ3〉 ≥ zn(t)} ≤ Ca3 lim n3sd−1n = Ca 3td−1 lim n−1 = 0. If α > 0, then (4.3) is bounded from above by sd−1n dF (y3)dF (y2)dF (y1) and substituting vi = yi/sn, i = 1, 2, 3, yields finally n3P {〈ξ1, ξ2〉 ≥ zn(t), 〈ξ1, ξ3〉 ≥ zn(t)} ≤ Ca3 lim sd−1+3αn = Ca 3td−1+3α lim d−1+4α = 0, and the rate of convergence is determined via (2.5). 5 Smallest spherical distance A nice application of Theorem 4.1 comes from the field of directional statis- tics. The following theorem determines the limiting behavior of the smallest spherical distance Sn = min 1≤i<j≤n within i.i.d. points U1, U2, . . . on S d−1, where βi,j denotes the smaller of the two central angles between Ui and Uj . In other words, Sn equals the smallest central angle formed by point pairs within the sample. Theorem 5.1. Let U1, U2 . . . be i.i.d. points on S d−1, d ≥ 2, having square- integrable density f on Sd−1 with respect to µd−1. Then n2/(d−1)Sn ≤ t = 1− exp for any t > 0, where Γ(d+1 f 2(x)µd−1(dx) The rate of convergence is O(n− 12 ) for finite t. If the points are uniformly distributed on Sd−1, then Theorem 5.1 applies f 2(x)µd−1(dx) = 2πd/2 If the points on Sd−1 follow the von Mises-Fisher distribution as introduced in Section 3, then f 2F (x)µd−1(dx) = C d(κ)/Cd(2κ). In dimension 2, Sn equals the minimal spacing, i.e. the smallest arc length between the “order” statistics. Proof of Theorem 5.1. Clearly, the relation cos βi,j = 〈Ui, Uj〉 holds for all pairs of i and j between 1 and n. Since the cosine function is continuous and monotone strictly decreasing on (0, π) and by the fact that 2 arccos(1− s) = it follows that n2/(d−1)Sn ≤ t = lim 1≤i<j≤n βi,j ≤ tn−2/(d−1) = lim 1≤i<j≤n βi,j ≤ arccos 1− t2n−4/(d−1)/2 = lim 1≤i<j≤n 〈Ui, Uj〉 ≥ 1− t2n−4/(d−1)/2 Theorem 4.1 yields the proof with α = 0 and a = 1. 6 Largest perimeter Finally we present a result for a U -max-statistic of degree 3, namely the limit law for the largest value 1≤i<j<ℓ≤n peri(Ui, Uj , Uℓ) of the perimeter peri(Ui, Uj , Uℓ) of all triangles formed by triplets of indepen- dent and uniformly distributed points U1, U2, . . . on the unit circle S. The random triameter (see [6]) of the sample is the largest perimeter up to a factor 3, hence the limit law for the triameter of U1, U2, . . . can be derived immediately. Theorem 6.1. If U1, U2, . . . are independent and uniformly distributed points on S, then 3−Hn) ≤ t = 1− exp for all t > 0 and for finite t the rate of convergence is O(n− 12 ). Lemma 6.2. If U1, U2, U3 are independent and uniformly distributed points on S, then peri(U1, U2, U3) ≥ 3 Proof. Clearly, peri(x1, x2, x3) is maximal for x1, x2, x3 being the vertices of an equilateral triangle on S, which has perimeter 3 3. If β1 and β2 are the angles (measured counter-clockwise) between U1 and U2 and between U2 and U3 respectively. By rotational symmetry, β1 and β2 are independent and uniformly distributed on [0, 2π]. The cosine theorem yields for sufficiently small s peri(U1, U2, U3) ≥ 3 (2− 2 cosβ1) 2 + (2− 2 cosβ2) + (2− 2 cos(2π − β1 − β2)) 2 ≥ 3 3− s, β1, β2 ∈ [2π/3± cs] , (6.1) where cs = C1 s and C1 is a suitable finite positive constant. If η1 and η2 are independent and uniformly distributed on [−cs, cs], then the last expression equals (2− 2 cos(2π/3 + η1)) 2 + (2− 2 cos(2π/3 + η2)) + (2− 2 cos(2π/3− η1 − η2)) 2 ≥ 3 P {β1 ∈ [2π/3± cs]}2 . By series expansion, (6.1) equals 2(cs/π) η21 + η 2 + (η1 + η2) 2 ≤ 8s̃/ (6.2) = 2(cs/π) −η1/2± (4s̃/ 3− 3η21/4) = π−2 s̃/33/4 s̃/33/4 (4s̃/ 3− 3y2/4) 2dy = where |s̃ − s| ≤ C2s3/2 for some finite C2, and the proof follows by the fact that s̃/s → 1 as s ↓ 0. Proof of Theorem 6.1. Plugging into Corollary 2.2 the transformation zn(t) = 3 3− tn−3 together with the result of Lemma 6.2 yields P {peri(U1, U2, U3) > zn(t)} = hence (2.2) is satisfied for all t > 0. Condition (2.3) does not hold, so we use the weaker requirement (2.6) to replace (2.3), i.e. we need to show that n5P {peri(U1, U2, U3) > zn(t), peri(U1, U4, U5) > zn(t)} = 0 (6.3) n4P {peri(U1, U2, U3) > zn(t), peri(U1, U2, U4) > zn(t)} = 0. (6.4) For (6.3), we follow the proof of Lemma 6.2. In addition, denote by β ′1 and β the random angles between U1 and U4 and between U4 and U5 respectively. It follows immediately by rotational symmetry, that β1, β2, β 1 and β 2 are in- dependent and uniformly distributed on [0, 2π]. With the help of Lemma 6.2 we check (6.3) by n5P {peri(U1, U2, U3) > zn(t), peri(U1, U2, U4) > zn(t)} ≤ C1 lim n5P {peri(U1, U2, U3) > zn(t)}2 = C2t2 lim n−1 = 0, where C1 and C2 are suitable finite positive constants. To show (6.4) we follow the proof of Lemma 6.2 and introduce the random variable η3, which is independent of η1 and η2 and uniformly distributed on [−cs, cs]. For suitable finite positive constants C3, C4 and C5 P {peri(U1, U2, U3) > zn(t), peri(U1, U2, U4) > zn(t)} ≤ C3c3sP η2, η3 ∈ −η1/2± (4s̃/ 3− 3η21/4) = C4c s̃/33/4 s̃/33/4 −y/2± (4s̃/ 3− 3y2/4) dy = C5s̃ and with s = tn−3 and s/s̃ → 1 as s → 0 n4P {peri(U1, U2, U3) > zn(t), peri(U1, U2, U4) > zn(t)} ≤ C5t3/2 lim 2 = 0. Hence (6.4) holds and the rate of convergence is determined by Remark 2. Acknowledgements The authors would like to thank Prof. Dr. N. Henze and Prof. Dr. I. Molchanov for their invaluable help concerning this and many other problems. References [1] M. J. B. Appel, C. A. Najim, and R. P. Russo. Limit laws for the diameter of a random point set. Adv. Appl. Probab., 34:1–10, 2002. [2] M. J. B. Appel and R. P. Russo. Limiting distributions for the maximum of a symmetric function on a random point set. J. Theor. Probab., 19:365–375, 2006. [3] A. D. Barbour, L. Holst, and S. Janson. Poisson Approximation. Claren- don Press, Oxford, 1992. [4] H. Federer. Geometric Measure Theory. Springer, Berlin, 1969. [5] J. Galambos. The Asymptotic Theory of Extreme Order Statistics. Wi- ley, New York, 1978. [6] K. Grove and S. Markvorsen. Curvature, triameter, and beyond. Bull. Amer. Math. Soc., 27:261–265, 1992. [7] N. Henze and T. Klein. The limit distribution of the largest interpoint distance from a symmetric Kotz sample. J. Multiv. Anal., 57:228–239, 1996. [8] W. Hoeffding. A class of statistics with asymptotically normal distribu- tion. Ann. Math. Statist., 19:293–325, 1948. [9] W. Lao. The limit law of the maximum distance of points in a sphere in Rd. Technical report, University of Karlsruhe, Karlsruhe, Germany, 2006. [10] P. C. Matthews and A. L. Rukhin. Asymptotic distribution of the normal sample range. Ann. Appl. Probab., 13:454–466, 1993. [11] M. Mayer and I. Molchanov. Limit theorems for the diameter of a random sample in the unit ball. Extremes. To appear. [12] B. W. Silverman and T. C. Brown. Short distances, flat triangles and poisson limits. J. Appl. Probab., 15:815–825, 1978. Introduction Poisson approximation for U-max-statistics Largest interpoint distance Largest scalar product Smallest spherical distance Largest perimeter

0704.1380

Dependence of exciton transition energy of single-walled carbon nanotubes on surrounding dielectric materials

Dependence of exciton transition energy of single-walled carbon nanotubes on surrounding dielectric materials Y. Miyauchia, R. Saitob, K. Satob, Y. Ohnoc, S. Iwasakic, T. Mizutanic, J. Jiangd, S. Maruyamaa∗ Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan Department of Physics, Tohoku University and CREST, Sendai 980-8578, Japan Department of Quantum Engineering, Nagoya University, Nagoya 464-8603, Japan and Center for High Performance Simulation and Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-7518, USA (Dated: November 19, 2018) We theoretically investigate the dependence of exciton transition energies on dielectric constant of surrounding materials. We make a simple model for the relation between dielectric constant of environment and a static dielectric constant describing the effects of electrons in core states, σ bonds and surrounding materials. Although the model is very simple, calculated results well reproduce experimental transition energy dependence on dielectric constant of various surrounding materials. PACS numbers: 78.67.Ch; 78.67.-n; 71.35.-y I. INTRODUCTION Photoluminescence (PL) of single-walled carbon nan- otubes (SWNTs) has been intensively studied for elu- cidating their unusual optical and electronic properties due to one dimensionality1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. Since both of electron-electron repulsion and electron- hole binding energies for SWNTs are considerably large compared with those for conventional three-dimensional materials, the Coulomb interactions between electron- electron and electron-hole play an important role in optical transition of SWNTs17,18,19,20,21,22. Optical transition energies of SWNTs are strongly affected by the change of environment around SWNTs such as bundling23, surfactant suspension7,14,24 and DNA wrapping25. Lefebvre et al.7 reported that the tran- sition energies for suspended SWNTs between two pil- lars fabricated by the MEMS technique are blue-shifted relative to the transition energies for micelle-suspended SWNTs. Ohno et al.14 have compared the PL of sus- pended SWNTs directly grown on a grated quartz sub- strate using alcohol CVD technique6 with SDS-wrapped SWNTs2. The energy differences between air-suspended and SDS-wrapped SWNTs depend on (n,m) and type of SWNTs [type I ((2n+m) mod 3 = 1) or type II ((2n+m) mod 3 = 2)26,27,28]. Recently, Ohno et al. studied E11 transition energies of SWNTs in various surrounding materials with different dielectric constant, κenv 29. Observed dependence of E11 on κenv for a (n,m) nanotube showed a tendency that can be roughly expressed as E11 = E 11 +A env (1) where E∞11 denotes a transition energy when κenv is in- finity, Aexpnm is the maximum value of an energy change ∗Corresoponding author. FAX: +81-3-5841-6421. E-mail: [email protected] (S. Maruyama) of E11 by κenv, and α is a fitting coefficient in the or- der of 1, respectively. At this stage, the reason why the experimental curve follows Eq.(1) is not clear. In the previous theoretical studies of excitonic transi- tion energies for SWNTs17,19,21,22, a screening effect of a surrounding material is mainly described using a static dielectric constant κ. However, since κ consists of both κenv and screening effect by nanotube itself, κtube, ex- perimental dependence of transition energies on dielectric constants of environment can not directly compared with calculations17,19,21,22 using the static dielectric constant κ. In this study, we make a simple model for the relation between κenv and κ. The calculated results of excitons for different κenv reproduced well the experimental tran- sition energy dependence on dielectric constant of various surrounding materials. II. THEORETICAL METHOD A. Exciton transition energy Within the extended tight-binding model21,22,28, we calculated transition energies from the ground state to the first bright exciton state by solving the Bethe- Salpeter equation, [E(kc)− E(kv)]δ(k c,kc)δ(k v,kv) v,kckv) Ψn(kckv) = ΩnΨ where kc and kv denote wave vectors of the conduc- tion and valence energy bands and E(kc) and E(kv) are the quasi-electron and quasi-hole energies, respec- tively. Ωn is the energy of the n-th excitation of the exciton (n = 0, 1, 2, · · · ), and Ψn(kckv) are the excitonic wavefunctions. The kernel K(k v,kckv) describes the Coulomb interaction between an electron and a hole. De- tails of the exciton calculation procedure is the same as presented in Refs21,22,28. http://arxiv.org/abs/0704.1380v1 The exciton wavefunction |Ψnq > with a center-of-mass momentum q(= kc − kv) can be expressed as |Ψnq >= Znkc,(k−q)vc c(k−q)v|0 >, (3) where Zn kc,(k−q)v is the eigenvector of the n-th (n = 0, 1, 2, · · · ) state of the Bethe-Salpeter equation, and |0〉 is the ground state. Due to momentum conservation, the photon-excited exciton is an exciton with q ≈ 0 for par- allel excitations to the nanotube axis. In this Letter, we calculate the n = 0 state of q = 0 exciton for each (n,m) SWNT. B. Dielectric screening effect In our calculation, the unscreened Coulomb potential V between carbon π orbitals is modeled by the Ohno potential19. We consider the dielectric screening effect within the random phase approximation (RPA). In the RPA, the static screened Coulomb interaction W is ex- pressed as17 W = V/κǫ(q), (4) where ǫ(q) is the dielectric function describing effects of the polarization of the π bands. κ is a static dielec- tric constant describing the effects of electrons in core states, σ bonds, and surrounding materials. In the cal- culation, we directly calculate only the polarization for the π band, and the effects of electrons in core states, σ bands, and surrounding materials are represented by a single constant κ. In the most accurate expression, the inhomogeneous and nonlocal dielectric response of the nanotube itself and the surrounding materials should be considered. However, it is not easy within extended tight binding method. In this study, instead of treating the complicated dielectric response including surround- ing materials, we make a simple model for a relation be- tween the static dielectric constant κ and κenv to obtain the E11 dependence on κenv. C. Relationship between κ and κenv Figure 1 shows a schematic view for the model relation- ship between κ and κenv. Here we consider the screening effect related to κ as a linear combination of the screening of nanotube itself and the surrounding material Ctube κtube , (5) where κtube is the dielectric constant within a nanotube except for the π bands, and Ctube and Cenv are coef- ficients for the inside and outside of a nanotube, re- spectively. As shown in Eq.(1), the transition energies observed in the experiment29 indicate that there is a FIG. 1: Schematic of the connection of the net dielectric con- stant κ and the dielectric constant of the surrounding material κenv and the nanotube itself κtube limit value19 when κenv → ∞. Hence, when κenv → ∞, Cenv/κenv can be removed from Eq.(5), and 1/κ is expressed by the limit value as Ctube κtube κ∞tube , (κenv → ∞) (6) where κ∞tube is the limit value of the net dielectric con- stant κ when κenv is infinity. Since electric flux lines through inside of the nanotube remain even when κenv → ∞, we assume there is a certain value of κ (κ∞tube) that corresponds to the situation when dielectric screen- ing by surrounding material is perfect and only dielectric response of the nanotube itself contributes to the net screening effect. Replacing Ctube/κtube by κ tube, Eq.(5) is modified as κ∞tube . (7) Next, we imagine that the SWNT is placed in the vac- uum, which corresponds to κ = κvac and κenv = 1, and then Cenv can be expressed as Cenv = κ∞tube , (8) where κvac is the static dielectric constant not for the vac- uum, but for the situation that the nanotube is placed in the vacuum. We now express κ as a function of κenv through two parameters κ∞tube and κ vac, whose values can be estimated from the following discussions. In the pre- vious papers17,19,21,22, κ value is put around 2 to obtain a 0 20 40 0 50 100 1 2 3 FIG. 2: (a) The E11 energy for a (9, 8) SWNT as a function of κ. (b) δE11 dependence on κenv. Inset in (a) shows the E11 dependence up to κ = 100. In (b), circles denote the experi- mental data and solid curves denote the calculated results of Eq.(10) for κvac = 1.0 (black), 1.5 (red) and 2.0 (blue). good fit with experiments for SWNTs with surrounding materials. Jiang et al.21 have compared the calculated results with the results for the two photon absorption experiments10, and obtained the best fit using κ = 2.22 for SWNTs in a polymer matrix. Here, since κvac is for nanotubes without surrounding materials, κvac should be less than about 2 and close to 1 due to vacancy of inside of the tubes. With regard to κ∞tube, according to the ex- perimental results7,14,29, transition energy change due to change of surrounding materials is at most 30-100meV. Fig.2(a) shows the calculated E11 energy dependence on κ for a (9,8) SWNT in a small κ region, while the inset shows the E11 dependence up to κenv = 100. As shown in Fig.2(a), variation of κ that yields the transition en- ergy change of 30 to 100 meV is about 1 to 3 when κ is around 2. Therefore, the value of κ∞tube should be around 2 to 3 and that of κvac should be around 1 to 2. D. Dependence of excitation energy on κenv As shown in Fig.2(a), the calculated E11 energies de- crease with increasing κ. This is mainly due to the fact that the self energy (e-e repulsion) always exceeds to the e-h binding energy and that the both interactions (e-e and e-h) decrease with increasing κ. The E11 al- most linearly depends on κ around the small κ region. We checked that the linear dependence is universal for all (n,m)’s for diameters more than 0.7 nm. Assuming the linear dependence, variation of the excitation energy δE11 ≡ E11 − E11(κenv = 1) for the small κ region is approximated by δE11 = −Anm(κ− κ vac), (9) where Anm is the gradient of δE11 near the small κ region for each (n,m) type. After we transform κ using the relationship of Eq.(5), Eq.(9) is modified as δE11 = −Anm(κ tube − κ vac)( κenv − 1 κenv + (κ tube − κ vac)/κvac Anm(κ tube − κ vac) corresponds to the maximum value of δE11 when κenv → ∞, which corresponds to the value of coefficient Aexpnm in the fitting curve of Eq.(1). For (9, 8) SWNT, the fitted value to the calculated results for Anm is 33 meV and A nm obtained by the fit to the experiment29 using Eq.(1) is 36 meV, and κ∞tube − κ should be around 1. The values for κ∞tube and κ are consistent with the values conventionally used for SWNTs in dielectric materials17,19,21,22. III. RESULTS AND DISCUSSION Figure 2(b) compares δE11 for a (9,8) SWNT depend- ing on κenv by the experiment (solid circles) and the cal- culated results (lines) for κvac = 1, 1.5, 2.0 using Eq.(10). As shown in Fig.2(b), the qualitative shape of theoretical curves are in good agreement with the experiment and not affected so much by the change of κvac. Since the ex- act value of κvac is unknown, we hereafter set κ vac = 1.5 for each (n,m) SWNT. For the (9, 8) SWNT in Fig.2(b), κ∞tube = 2.7 and κ vac = 1.5 are fitting values. These values are consistent with the discussion in the previous section. After setting κvac = 1.5, Eq.(10) turns to be δE11 = −Anm(κenv − 1) κenv/(κ tube − κ vac) + 1/1.5 . (11) Thus, we express δE11 as a function of κenv with one parameter (κ∞tube − κ vac). Figure 3(a) shows the calculated values of Anm for each (n,m)’s. Family pattern of (2n + m = const.) family is drawn with the 2n + m values by dotted lines. We found a slight diameter dependence and relatively large chiral angle dependence of Anm for type II SWNTs (blue) compared with type I SWNTs (red). The type II SWNTs with larger chiral angles tend to have larger value of Anm. For a convenient use of Eq.(10), we give a fitting function of Anm meV as Anm = A+Bdt + (C +D/dt) cos 3θ, (12) which gives the average (maximum) error of ±2meV(8meV) for type I, and ±2meV(5meV) for type II SWNTs. The fit curve is shown in Fig.3(a) by solid lines. Here dt (nm) is the diameter of nanotube and θ is the chiral angle26. The values of (A, B, C, D) are (36, -4, 0, 0) and (33, -3, 6, 7) for type I and for type II SWNTs, respectively. In order to expand our result to many (n,m) SWNTs, we need a function to describe (κ∞tube−κ vac). It is impor- tant to note that κ∞tube should depend on the diameter. 0.8 1.2 1.6 d t (nm) 19 22 34 37 2n+m=20 35 38 0 0.5 1 FIG. 3: (a) Calculated values of Anm for each (n,m) SWNT. Open (red) and solid (blue) circles correspond to type I and type II SWNTs, respectively. Solid lines denote the fit curve by Eq.(12). (b) κ∞tube−κ vac vs 1/d2t . The values of κ tube−κ are obtained by the fit of Eq.(10) to the experimental data for each (n,m). An exact function should be calculated by taking into ac- count the Coulomb interaction considering induced sur- face charge at the boundary of the nanotube and sur- rounding materal for an e-e or e-h pair for each (n,m) SWNT. Instead of calculating this complicated function, here we roughly estimate the (κ∞tube − κ vac) as a simple function of diameter dt, since (κ tube−κ vac) should depend on the cross section of a SWNT. As shown in Fig.3(b), (κ∞tube − κ vac) is roughly proportional to 1/d2t , (κ∞tube − κ vac) = , (13) with the coefficient E = 1.5±0.3 nm2. Here (κ∞tube−κ is obtained by the fit using Eq.(10) and Anm calculated for each chirality. Fig.3(b) clearly shows that our cal- culated Anm well describes the chiral angle dependence of δE11 and that the remaining diameter dependence is understood by (κ∞tube − κ vac) through 1/d2t . This 1/d dependence implies that κ∞tube depends on the volume of inner space of the nanotube. Although the number of experimental data available for the fit is small and se- lection of this function is arbitrary to some extent, it is reasonable that 1/κ∞tube increase with the increase of the diameter, since 1/κ∞tube corresponds to the Coulomb in- teraction through the inner space of the nanotube. In order to find an accurate form of the function, future ex- periments and theoretical studies are definitely needed. Figure 4 shows δE11 as a function of κenv for (a) the ex- periment and (b) the calculation using Eq.(11) and (13). Fig.4(c) compares δE11 for the experiment and that for the calculation with the same κenv values. The same sym- bols for an (n,m) are used in three figures of Fig.4. De- tails of experimental data will be published elsewhere29. Although our treatment is very simple, the calculated (a) (b) (8,6) (9,4) (8,7) (9,7) (10,5) (11,3) (12,1) (9,8) (10,6) 60 40 20 0 δE11(theory) (meV) - - -0 20 40 0 20 40 FIG. 4: The transition energy dependence plotted as a func- tion of κenv. (a) experiment and (b) calculated results are indicated by (a) symbols and (b) solid curves. In (b), (n,m) for each curve is indicated by a symbol on the curve. (c) Comparison of δE11 for the experiment (δE11(experiment)) and calculation (δE11(theory)). A dotted line indicates the line of δE11(experiment) = δE11(theory). Open (red) and solid (blue) symbols correspond to type I and type II SWNTs, respectively. The data in the dotted circle are the data for κenv = 1.9 29 (see text). curves for various (n,m) SWNTs well reproduce the ex- perimentally observed tendency for each (n,m) SWNT, and the degree of difference between each (n,m) type is also in good agreement with the experiment. As shown in Fig.4(c), δE11(theory) is in a good agreement with δE11(experiment) except for several points indicated by a dotted circle in the figure, which correspond to a case for the smallest κenv = 1.9 (hexane) except for κenv = 1 (air) in the experimental data29. The value of κenv = 1.9 for hexane is adopted as the dielectric constant for the material, in which the dipole moments of liquid hexane are not aligned perfectly even in the presence of the elec- tric field. Since κenv = 1.9 is a macroscopic value, a local dielectric response might be different from the averaged macroscopic response. If the local dielectric constant near SWNTs becomes large (for example, κenv ≈ 3), the fitting of Fig.4(c) becomes better. We expect that the dipole moments of a dielectric material might be aligned locally for a strong electric field near an exciton, which makes the local dielectric constant relatively large. This will be an interesting subject for exciton PL physics. Since the difference of Anm between each (n,m) type de- creases with increasing the diameter, it is predicted that the amount of variation due to the change of κenv mostly depend on diameter in the larger diameter range. Thus a PL experiment for nanotubes with large diameters would be desirable for a further comparison. IV. SUMMARY In summary, the dependence of exciton transition en- ergies on dielectric constant of surrounding materials are investigated. We proposed a model for the relation be- tween dielectric constant of the environment and a static dielectric constant κ in the calculation. Although the model is quite simple, calculated results well reproduce the feature of experimentally observed transition energy dependence on dielectric constant of various surrounding materials, and various dt and θ. Acknowledgments Y.M. is supported by JSPS Research Fellowships for Young Scientists (No. 16-11409). 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0704.1381

Mpemba effect and phase transitions in the adiabatic cooling of water before freezing

Mpemba effect and phase transitions in the adiabatic cooling of water before freezing S. Esposito∗, R. De Risi, and L. Somma Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Complesso Universitario di Monte S. Angelo, Via Cinthia, I-80126 Napoli, Italy An accurate experimental investigation on the Mpemba effect (that is, the freezing of initially hot water before cold one) is carried out, showing that in the adiabatic cooling of water a relevant role is played by supercooling as well as by phase transitions taking place at 6 ± 1oC, 3.5 ± 0.5oC and 1.3 ± 0.6oC, respectively. The last transition, occurring with a non negligible probability of 0.21, has not been detected earlier. Supported by the experimental results achieved, a thorough theoretical analysis of supercooling and such phase transitions, which are interpreted in terms of different ordering of clusters of molecules in water, is given. A well-known phenomenon such as that of the freezing of water has attracted much interest in recent times due to some counter-intuitive experimental results [1] and the apparent lacking of a generally accepted physical inter- pretation of them [2], [4], [3], [5]. These results consist in the fact that, many times, initially hot water freezes more quickly than initially cold one, a phenomenon which is now referred to as the Mpemba effect (for a short his- torical and scientific survey see the references in [3]). The observations sound counter-intuitive when adopting the naive, simple view according to which initially hot water has first to cool down to the temperature of the initially cold one, and then closely follow the cooling curve of the last one. The effect takes place even for not pure water, with solutions or different liquids (the original Mpemba observation occurred when he tried to make an ice cream). Several possible physical phenomena, aimed to explain such observations, have been proposed in the literature, mainly pointing out that some change in water should occur when heated [2] [5]. However, such explanations cannot be applied if some precautions are taken during the experiments (whilst the Mpemba effect has been observed even in these cases) and, in any case, calculations do not seem to support quantitatively the appearance of the effect (see references in [3]). Some novel light has been introduced in the discussion, in our opinion, in Ref. [3], where the Mpemba effect has been related to the occurrence of supercooling both in preheated and in non-preheated water. Initially hot water seems to supercool to a higher local temperature than cold water, thus spontaneously freezing earlier. As a consequence, such a scenario, apparently supported by experimental investigations, points toward a statistical explanation of the effect, neither the time elapsed nor the effective freezing temperature being predictable. Here, we prefer to face the problem by starting from ∗Corresponding author, [email protected] what is known about the freezing process, rather than the cooling one. In general it is known that, for given values of the ther- modynamic quantities (for example the volume and the energy), a physical system may exist in a state in which it is not homogeneous, but it breaks into two or more homogeneous parts in mutual equilibrium between them. This happens when stability conditions are not fulfilled, so that a phase transition occurs; it is, for example, just the case of water that, at the pressure p of 1 atm and at temperature T of 0oC, becomes unstable. When liquid water is cooled, the average velocities of its molecules decreases but, even if the temperature goes down to 0oC (the fixed temperature where liquid and solid phases coexist) or lower, this is not a sufficient con- dition for freezing to start. In fact, in order that ice begins to form, first of all some molecules of the liquid water should arrange in a well-defined order to form a minimum crystal and this, in the liquid state, may hap- pen only randomly. Second, such starting nucleus has to attract further molecules in the characteristic loca- tions of the crystalline structure of ice, by means of the interaction forces of the nucleus with the non-ordered molecules in the liquid. Nucleation and crystal growth processes are both favored at temperatures lower than 0oC, so that supercooling of liquid water is generally re- quired before its effective freezing. In fact, in pure water, only molecules in the liquid with statistically lower veloci- ties can arrange the initial nucleus and, furthermore, only slow moving molecules are able to join that cluster and put their kinetic energies into potential energy of bond formation. When ice begins to form, these molecules are removed from those attaining to the given Maxwell dis- tribution for the liquid water, so that the average speed becomes larger, and the temperature of the system rises to 0oC (obviously, the temperature is set at the value where the continuing exchange of molecules is equal in terms of those joining and those leaving the formed crys- tal surface). Thus supercooling is, de facto, a key ingredient in the freezing process, although supercooled water exists in a state of precarious equilibrium (water is in a metastable http://arxiv.org/abs/0704.1381v1 state). Minor perturbations such as impurities or other can trigger the sudden appearance of the stable crys- talline phase for the whole liquid mass, again with the release of the entire crystallization heat (melting heat) which increases the temperature of the freezing liquid to the normal 0oC one. In general, when a system is in a metastable state, sooner or later it will pass to another stable state. In water, density and entropy fluctuations favor the forma- tion of crystallization nuclei but, if the liquid constitutes a stable state, such nuclei are always unstable and will disappear with time being. However, since the fluctua- tions become more pronounced the lower the tempera- ture, if water is supercooled, for sufficiently large nuclei they will result to be stable and grew with time, becom- ing freezing centers. The starting of the phase transition is thus determined by the probability of appearance of those nuclei, and the reported Mpemba effect could be simply a manifestation of this process. We have calculated just this probability P as function of the absolute temperature T of the metastable phase (the one at which the nucleus is in equilibrium with the liquid), obtaining the following result [11]: (T − T∗)2 . (1) Here T∗ is the equilibrium temperature of the liquid-solid phase, α is a dimensionless normalization factor and β is a constant whose expression is given by 16πτ3v2 3Q2kT∗ , (2) where τ is the surface tension, v the molecular volume of the crystallization nucleus, Q the molecular heat of the transition from the metastable phase to the nucleus phase, and k is the Boltzmann constant. Just to give an idea of the macroscopic value of the constant β, let us note that τ3v2 = W 3 is the cube of the work done by the surface forces, and by assuming that Q ∼ kT∗ we may write: Wsurf , (3) that is the constant β is ruled by the ratio Wsurf/Q. The probability P has a minimum at the liquid-solid equilibrium temperature T∗ and increases for decreasing temperature, as expected. From the formulae above it is clear that the probability for nucleation, and thus the onset of the freezing process as well, is enhanced if the work done by the surface forces (or the surface tension itself) is lowered in some way. In normal daily conditions when a commercial refrigerator is employed, this is eas- ily induced in two simple ways: either by the presence of impurities, when solutions (such as an ice cream so- lution, as in the Mpemba case) are used as the freezing V= 20 cm3 V= 50 cm3 V= 65 cm3 V= 80 cm3 PSC 1 0.28 0 0.46 Tc = −8 oC Tc = −14 oC Tc = −22 oC Tc = −26 PSC 0.75 0.50 0 0.11 TABLE I: Probabilities for the occurrence of supercooling for different volumes V of the sample and different temperatures of the cryostat Tc. liquid instead of pure water, or by fluctuations of the ex- ternal pressure or temperature, caused in the commercial refrigerator itself. This explains why no appreciable su- percooling is observed in normal situations. Obviously, the most direct way to induce freezing in supercooled water is to introduce an external body in it, in order to directly lower the surface tension. We have thus performed an accurate experimental in- vestigation, accounting for a total of about one hundred runs, aimed to clarify the phenomenology of the Mpemba effect and its interpretation. In the first part of our ex- periments we have tested all the above qualitative pre- dictions about supercooling, by studying the cooling and freezing of tens of cm3 of normal water in a commer- cial refrigerator, in daily operation conditions. The key point, in fact, is not to obtain the most favorable physical conditions, employing sophisticated setups, but rather to reproduce the Mpemba conditions, that is adiabatic cooling (with commercial refrigerators) of not extremely small quantities of water. We have used an Onofri re- frigerator for the cooling of double distilled water and a NiCr-Ni thermocouple as a temperature sensor (Leybold 666193), interfaced with a Cassy Lab software for data acquisition. For fixed temperatures of the cryostat we have indeed observed supercooling in our samples, with the freezing occurring just along the lines predicted above. In par- ticular, during the supercooling phase we have induced a number of small perturbations in our samples, namely, variations of external pressure or temperature, mechan- ical perturbations or introduction of an external macro- scopic body (a glass thermometer held at the same tem- perature of the sample). In all these cases we have regis- tered the sudden interruption of the supercooling phase and a practically instantaneous increase of the temper- ature to the value of 0oC, denoting the starting of the freezing process. Conversely, if no perturbation is in- duced (or takes places) the water reached an equilibrium with the cryostat at temperatures up to about −30oC (lasting also for several thousands of seconds). We have then verified that when the freezing process started from the supercooling phase, the Mpemba effect took place with a probability in agreement with that re- ported in Ref. [3]. In about half (with a total probability of 0.47) of the runs performed we have detected a supercooling phase. FIG. 1: Left: Cooling curves for V = 20 cm3 and Tc = −8 Right: The fitted time duration of the phase transition at 3.5oC as function of the volume V of the samples, for different temperatures Tc of the cryostat. In Table I we report the observed probability PSC for the occurrence of supercooling for different volumes V of the water sample and for different temperatures Tc of the cryostat. We find the data to be fitted by a straight line, denoting (in the range considered) a linear decreasing of PSC for decreasing temperatures of the cryostat and for increasing volumes of the samples, this probability reaching the maximum PSC = 1 for Tc = 0 oC (and V = An interesting feature of what we have observed is the sensible appearance of iced water in our samples. In fact, when supercooling did not occur, the ice started to form around the walls of the beaker, while the inner parts were still in a liquid form, as usually expected. Instead the im- mediate freezing of supercooled water involved the whole sample, this showing a very peculiar symmetric form. We have used cylindrical beakers with the temperature sensor in their periphery, near the walls; the observed structure was a pure radial (planar) one, with no liquid water and radial filaments of ice from the center of the beakers to the walls (in one case we have been also able to take a low resolution picture of this, before its destruc- tion outside the refrigerator). However, although supercooling plays a relevant role in the manifestation of the Mpemba effect, the things are made more complicated by the occurrence of other statis- tical effects before the temperature of the water reaches the value of 0oC. This comes out when an accurate mea- surement of the cooling curves is performed (some exam- ples of what we have obtained during the second part of our experiments are reported in Fig. 1). According to a simple naive model, the heat exchange from the water sample (at initial temperature T0) to the cryostat (at fixed temperature Tc) is described by the equation C dT = δ (Tc − T0) dt, (4) where C and δ are the thermal capacity and the heat conductivity of the water, respectively. Thus by solving the differential equation in (4), the following expression Tc = −8± 2 V= 20 cm3 V= 50 cm3 V= 65 cm3 V= 80 cm3 ∆t1 (s) 7± 1 ∆t2 (s) 11± 6 220± 100 500± 170 630 ± 160 ∆t3 (s) 12± 6 70± 30 Tc = −14± 2 V= 20 cm3 V= 50 cm3 V= 65 cm3 V= 80 cm3 ∆t1 (s) 37± 1 ∆t2 (s) 8± 3 130 ± 80 480± 160 500± 60 ∆t3 (s) 7± 4 Tc = −22± 1 V= 20 cm3 V= 50 cm3 V= 65 cm3 V= 80 cm3 ∆t1 (s) 63 ± 1 7± 1 ∆t2 (s) 170± 100 130± 70 ∆t3 (s) Tc = −26± 1 V= 20 cm3 V= 50 cm3 V= 65 cm3 V= 80 cm3 ∆t1 (s) 3.5± 0.7 ∆t2 (s) 3± 1 320 ± 70 210 ± 170 ∆t3 (s) 200 ± 70 1± 1 TABLE II: Time duration of the phase transitions at 6oC (∆t1), 3.5 oC (∆t2) and 1.3 oC (∆t3) for different volumes V of the sample and different temperatures of the cryostat Tc. for the temperature as function of time t is obtained: T = Tc − (Tc − T0) e −t/τ , (5) where τ = C/δ is a time constant measuring the cool- ing rate of the sample. However, although the overall dependence of T on time is that expressed by Eq. (5), our experimental data clearly reveal the presence of three transition points before freezing (or supercooling), where τ changes its value. This transitions occur at tempera- tures T1 = 6±1 oC, T2 = 3.5±0.5 oC and T3 = 1.3±0.6 with a probability of P1 = 0.11, P2 = 0.84 and P3 = 0.21, respectively. The time duration ∆t of each phase tran- sition, during which the temperature keeps practically constant [12], depends on the volume of the sample and on the temperature of the cryostat. The data we have collected are summarized in Table II. For the phase transition at T2 these data show a linear increase of ∆t2 with Tc and a quadratic one with V ; in Fig. 1 we give the fitting curves corresponding to best fit function ∆t2 = (a+bTc)V 2. Instead, for the other two phase tran- sitions no sufficient data are available in order to draw any definite conclusion on the dependence on V and Tc, though ∆t1 and ∆t3 appear to be shorter than ∆t2. The occurrence of these phase transitions is likely re- lated to the formation of more or less ordered structures in water, resulting from the competition between long- range density ordering and local bond ordering maxi- mizing the number of local bonds [8]. The anomalous density maximum at about 4oC (which we observe here at T2 = 3.5 ± 0.5 oC) is, for example, explained just in Tc = −8 oC Tc = −14 oC Tc = −22 oC Tc = −26 τ1 (s) 600± 110 680± 100 1000 ± 110 950 ± 190 τ2 (s) 1080 ± 260 1060 ± 170 530± 90 570± 3 τ3 (s) 1590 ± 930 1520 ± 730 270± 50 220 ± 80 τ4 (s) 620± 480 500± 180 150± 30 640 ± 490 TABLE III: Time constants τ1 (T < T1), τ2 (T1 < T < T2), τ3 (T2 < T < T3), τ4 (T > T3) of the cooling curves before and after the three phase transitions detected, for different temperatures of the cryostat Tc. term of this: as water is cooled, the local specific volume increases due to the progressive increase in tetrahedral order, so that the entropy, that always decreases upon cooling, at 4oC becomes anticorrelated with the volume, resulting in an inversion (from positive to negative) of the thermal expansion coefficient and a corresponding density maximum [9]. Similar explanations in terms of different ordering could apply also to the other two tran- sitions we have observed, but an exhaustive discussion of them, which would require more experimental data, is beyond the scope of this Letter. We only note that, while the first transition at T1 = 6 ± 1 oC seems related to the effect observed in Ref. [10] at 8oC, to the best of our knowledge no other author has reported the one at T3 = 1.3 ± 0.6 oC (which, as mentioned, occurs with an appreciable probability of 0.21). The observed mean values of the four time constants of the cooling curves, before and after the three phase transitions, are reported in Table III for different val- ues of Tc. All the time constants are approximately in- dependent on the volume V, in disagreement with the naive model discussed above which predicts an increase of τ with the thermal capacity. Instead they depend lin- early on Tc, showing a negative slope for τ1 and positive ones for τ2, τ3, τ4 and a finite value for Tc = 0 oC. Note that (in the naive model) the ratios of the different time constants, at fixed volumes, give the (inverse) ratios of the heat conductivities in the different ordered phases (all these ratios decrease with the cryostat temperature), which are directly related to microscopic quantities like the size and average velocity of the ordered clusters of molecules in water. Coming back to the Mpemba effect, it is easy to see that Eq. (5) predicts that, for constant τ , initially hot water reaches the freezing point later than initially cold water. However, from what just discussed, in general this could be no longer true if the time constant changes its value during the cooling process (the slope of the cooling curves changes), or phase transitions before freezing oc- cur (with time durations sufficiently long/short). In ad- dition to these effects, the reaching of the freezing point does not automatically guarantees the effective starting of the freezing process, since relevant supercooling may take place, thus statistically causing the freezing of ini- tially hot water before cold one. From the data we have collected we have verified that, for given V and Tc, in many cases no inversion between the cooling curves happens before the freezing point, ir- respective of the change in the value of τ or the time duration of the phase transitions. Nevertheless we have as well realized that this is mainly due to the not very large difference between the initial temperatures of the samples, and in few cases (among those studied by our- selves) it cannot be applied, the largest effect causing the inversion being the phase transition at T2. In conclusion our experimental results, and their in- terpretation reported here, clearly point out the statisti- cal nature of the Mpemba effect (as already realized in [3]), whose explanation is given in terms of transitions between differently ordered phases in water and super- cooling. The very detection of such phenomena seems to require the cooling to be adiabatic (as fulfilled in our experiment, as well as in those performed by other au- thors [3]), since for non adiabatic processes (for example, in fused salt) the coexistence of local solid nuclei in the liquid phase has been observed [13]. An unexpected novel transition at T3 = 1.3 ± 0.6 has been as well detected with a non negligible proba- bility, calling for further accurate investigation in order to achieve a more complete understanding of the unique properties of water. Acknowledgements: Interesting discussions with G. Salesi and M. Villa are kindly acknowledged. [1] E.B. Mpemba, Cool. Phys. Educ. 4, 172 (1969). [2] G.S. Kell, Am. J. Phys. 37, 564 (1969) [3] D. Auerbach, Am. J. Phys. 63, 882 (1995). [4] B. Wojciechowski, I. Owczarek and G. Bednarz, Crystatl. Res. Tech. 23, 843 (1988). [5] J.I. Katz, preprint arXiv:physics/0604224. [6] See, for example, H.B. Callen, Thermodynamics (Wiley, New York, 1960). [7] L.D. Landau and E.M. Lifshitz, Statistical Physics (Perg- amon, Oxford, 1980). [8] H. Tanaka, Phys. Rev. Lett. 80, 5750 (1998). [9] P.G. Debenedetti and H.E. Stanley, Physics Today, June 2003, 40. [10] K. Kotera, T. Saito and T. Yamanaka, Phys. Lett. A 345, 184 (2005). [11] We do not give the details of such calculations; the inter- ested reader may follow those reported in section 162 of Ref. [7] for a similar case. [12] In some cases we have been able to observe also a van der Waals-like profile of T (t) at the transition point (metastable state), instead of only the mean constant value of T . [13] We are indebted with M. Villa for having pointed out this to us. http://arxiv.org/abs/physics/0604224

0704.1382

Effects of atomic interactions on Quantum Accelerator Modes

Effects of atomic interactions on Quantum Accelerator Modes. Laura Rebuzzini,1, 2, ∗ Roberto Artuso,1, 3, 4 Shmuel Fishman,5 and Italo Guarneri1, 2, 3 Center for Nonlinear and Complex Systems and Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy. Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Ugo Bassi 6, 27100 Pavia, Italy. Istituto Nazionale di Fisica della Materia, Unità di Como, Via Valleggio 11, 22100 Como, Italy. Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy. Physics Department, Technion, Haifa 32000, Israel. We consider the influence of the inclusion of interatomic interactions on the δ-kicked accelerator model. Our analysis concerns in particular quantum accelerator modes, namely quantum ballistic transport near quantal resonances. The atomic interaction is modelled by a Gross-Pitaevskii cubic nonlinearity, and we address both attractive (focusing) and repulsive (defocusing) cases. The most remarkable effect is enhancement or damping of the accelerator modes, depending on the sign of the nonlinear parameter. We provide arguments showing that the effect persists beyond mean-field description, and lies within the experimentally accessible parameter range. PACS numbers: 05.45.Mt, 03.75.-b, 42.50.Vk Keywords: Quantum Accelerator Modes (QAMs) are a manifesta- tion of a novel type of quantum ballistic transport (in mo- mentum), that has been recently observed in cold atom optics [1]. In these experiments, ensembles of about 107 cold alkali atoms are cooled in a magnetic-optical trap to a temperature of a few microkelvin. After releasing the cloud, the atoms are subjected to the joint action of the gravity acceleration and a pulsed potential periodic in space, generated by a standing electromagnetic wave, far-detuned from any atomic transitions. The external optical potential is switched on periodically in time and the period is much longer than the duration of each pulse. For values of the pulse period near to a resonant integer multiple of half of a characteristic time TB (the Talbot time [2]), typical of the kind of atoms used, a consider- able fraction of the atoms undergo a constant accelera- tion with respect to the main cloud, which falls freely under gravity and spreads diffusively. The non-interacting model is a variant of the well- known quantum kicked rotor (KR) [3], in which the ef- fects of a static force, produced by the earth gravitational field, are taken into account. The linear potential term breaks invariance of the KR hamiltonian under space translations. Such an invariance may be recovered by moving to a temporal gauge, where momentum is mea- sured w.r.t. the free fall: this transformation gets rid of the linear term and the new hamiltonian, expressed in dimensionless units, reads Ĥ(t′) = (p̂+ gt′)2 + k cos(x̂) δ(t′ − tτ). (1) where p̂ and x̂ are the momentum and position opera- tor, k and τ are the strength and the temporal period of the external kicking potential, g is the gravity accelera- tion. The relationship between the rescaled parameters and the physical ones, denoted by primes, is k = k′/~, 0.005 0.015 0.025 0.035 0.045 -500 0 500 1000 0.005 0.015 0.025 0.035 0.045 600 800 1000 0.005 0.015 0.025 -1000 0 1000 2000 3000 0.005 0.015 0.025 2200 2400 2600 2800 3000 FIG. 1: (Color online) The probability distribution at times t = 25 (1st row) and 45 (2nd row). ((Red) line: lin- ear case (u = 0), (purple) triangles/(green) circles: focus- ing/defocusing nonlinearity (u = ∓1.25)). In the right col- umn enlargements of mode are shown; the position of the mode, predicted by (2) is marked by the (blue) vertical dot- ted line. τ = ~τ ′G2/M = 4πτ ′/TB, η = Mg ′τ ′/~G and g = η/τ , where η is the momentum gain over one period, G is twice the angular wavenumber of the standing wave of the driving potential and M is the mass of the atom. Symmetry recovery allows to decompose the wavepacket into a bundle of independent rotors (whose space coordinate is topologically an angle): this Bloch-Wannier fibration plays an important role in the http://arxiv.org/abs/0704.1382v1 2000 2500 3000 2000 2500 3000 2000 2500 3000 FIG. 2: (Color online) The Husimi function of the QAM at time t = 45, in the repulsive (a), linear (b) and attractive case theory of QAMs [4]. QAMs appear when the time gap between kicks ap- proaches a principal quantum resonance, i.e. τ = 2πl+ǫ, with l integer and |ǫ| small. The key theoretical step is that in this case the quantum propagator may be viewed as the quantization of a classical map, with |ǫ| playing the role of an effective Planck’s constant [4]: QAMs are in correspondence with stable periodic orbits of such pseudo-classical area-preserving map. We refer the reader to the original papers for a full account of the theory, we just mention a few remarkable points: stable periodic orbits are labelled by their action winding num- ber w = j/q, which determines the acceleration of the QAM w.r.t. the center of mass distribution . (2) The modes are sensitive to the quasimomentum (Bloch index induced by spatial periodicity), being enhanced at specific, predictable values [4]; also the size of the elliptic island around the pseudoclassical stable orbit plays an important role (if the size is small compared to |ǫ| the mode is not significant [4]). We consider in this letter the role of atomic interactions in such a system; namely evolution is determined by a nonlinear Schrödinger equation with a cubic nonlinearity: iψ̇(x, t′) = Ĥ(t′) + u|ψ(x, t′)|2 ψ(x, t′), (3) where u is the rescaled nonlinear parameter, whose sign describes an attractive (negative)/repulsive (positive) atomic interaction. We will come back to its connec- tion with physical units in the end of the paper. The 0.005 0.015 0.025 -2.5 0 2.5 -2.5 0 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 FIG. 3: (Color online) Maximum height reached by the mode at time t = 45 as a function of u in position (a) and momen- tum (b) representation. In the inset the exponential decrease of hmax for positive u in shown in a semi-logarithmic plot. condensate wave function is normalized to unity. The dynamics does not only acquire in this way a qualitative novel form, but, due to the nonlinear term, Bloch de- composition into independent rotors breaks down. The main scope of this letter will be to numerically scrutinize how QAMs are still present in the modified system, and explore how nonlinearity modifies their features. In the end we will briefly comment upon some stability issues, by showing that a more refined description, including loss of thermalized particles, does not destroy the scenario we get from a mean field description. Our analysis will be restricted to QAMs correspond- ing to fixed points of period q = 1 of the pseudoclassical map; the numerical analysis of nonlinear evolution has been performed by using standard time-splitting spec- tral methods [5]. There are several physical parameters characterizing the system: g, τ , k and u. Here we mainly address the role of nonlinearity u: we fix k = 1.4, l = 1, ǫ = −1, τη ≃ 0.4173, and choose as the initial state a symmetric coherent state centered in the stable fixed point of the pseudoclassical map (x0 ≃ 0.3027, p0 = 0), whose corresponding winding number is zero. A quite remarkable feature appears when we com- pare results for opposite nonlinearity signs (keeping the strength |u| fixed), see fig.(1). As in the linear system, the wave packet splits into two well-separated components: the accelerator mode (whose acceleration is still compati- ble with (2)) and the remaining part, which moves under two competitive contributions, the free fall in the gravi- tation field, and the recoil against the accelerating part. Note that for the present choice of the parameters, the former contribution is negligible compared to the second. We remark some features, that are common to what we observed for a choice of other parameter values: the distribution around the accelerator mode is more peaked and narrower in the presence of attractive nonlinearity; the opposite happens in the case of a repulsive interac- 0.005 0.015 0.025 0.035 0.045 2 (b) -3000 -2000 -1000 0 1000 2000 3000 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 FIG. 4: (Color online) The probability distribution at t = 45 for strong nonlinearity (u = ±3); the initial states (see text) are shown in the insets (line and symbols as in fig.1). tion. This can also be appreciated from a Husimi repre- sentation of the modes (see fig.(2)). While for repulsive interactions the spreading of the distribution, together with peak damping, seems to de- pend monotonically on the nonlinearity strength, the attractive case exhibits more complicated features (see fig.(3)). Enhancement of the accelerator mode is only observed for small nonlinearities, while a striking feature appears at larger values of |u|, namely the accelerator mode is suppressed (see fig.(4a)). The intuitive expla- nation of this result is that strong focusing nonlinearity opposes to the separation of the wave packet into two parts; indeed, in the case of exact resonance (namely τ = 2π), the mode is absent, so the whole wave freely falls without splitting and then the maximum height of the wave, plotted vs u as in fig.(3a), is found to mono- tonically increase to the left towards a saturation value. While the behavior shown in fig.(3) has been observed for a variety of other parameter choices, we mention that more complex, strongly fluctuating behaviour was some- times observed at large focusing nonlinearities. In all such cases a bad correspondence between the quantum and the pseudoclassical dynamics was also observed, al- ready in the linear case. We remark that the mode damping is sensitive to the choice of the initial state, as shown in fig.(4). While a gaussian initial wave packet leads to the mentioned QAM suppression, we may tailor a QAM enhancing ini- tial condition as follows: we take the quasimomentum β0 that in the linear case dominates the mode (here β0 = π/τ−η/2 ≃ 0.5551 [4]) and we drop from the initial 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 FIG. 5: (Color online) The quasi-momentum distribution function (thick line) for |u| = 0.5 (a) and |u| = 3 (b) at different times (1)t = 5, (2)t = 25 and (3)t = 45. Pur- ple(dark)/green(light) lines refer to attractive/repulsive in- teractions. gaussian all components with |β − β0| > 0.15. As quasi- momentum is the fractional part of momentum, this leads to the comb like state of fig.(4b). Even through quasimo- mentum is not conserved due to nonlinearity, the QAM is strongly enhanced with respect to the linear case and the recoiling part is almost cancelled. Another way of looking at the nonlinear evolution with techniques that are proper in the linear setting is to con- sider the distribution function over quasimomenta, de- fined by f(β, t) = |〈n+ β|ψ(t)〉| . (4) This distribution is stationary under linear evolution, its shape being determined by the choice of the initial state. We consider the evolution of a gaussian wave packet (for which the linear f is essentially a constant - the hor- izontal red line of fig.(5)), and probe the effect of nonlin- earities of both signs. Typical results are as in fig.(5): the effect of attractive (repulsive) nonlinearity is to enhance (lower) the distribution around a value β̄ ≃ 0.4. No devi- ation occurs for quasimomentum β0 (marked by vertical lines), whose wave function, according to fig.(4b), closely follows the linear pseudo-classical island. Again the β̄ peak of the focusing case is suppressed for large focusing nonlinearities. To make sure that our findings may be experimentally significant we discuss some stability issues: the first con- cerns decay properties of the QAMs. It is known that linear modes decay due to quantum tunnelling out of pseudoclassical islands [6]: we checked that, on the avail- 0 20 40 0 10 20 30 FIG. 6: (Color online) (a) Probability inside the island for |u|=3 (symbols and line as in fig.(1)). (b) The mean number of non-condensed particles vs the number of kicks, for u equal to 0.1, 0.5, 0.75, 1, 2, 5, 7 and 10 (starting from below); 12 terms in the sum (5) are considered. able time scale, the nonlinear decay behaves in a similar way. In fig.(6a) the probability inside the classical is- land is shown as a function of time for the initial state of fig.(4b); it has been calculated integrating the Husimi distribution of each β-rotor fiber over the island area and summing the contributions of different rotors. However in the condensate regime there is another pos- sible mechanism that might completely modify the for- mer picture, namely depletion of the condensate due to proliferation of noncondensed, thermal particles. A stan- dard technique to estimate the growth of the number of thermal particles is provided by the formalism of Castin and Dum [7], which has been employed in similar con- texts in [8]. To the lowest order in the perturbation ex- pansion and in the limit of zero temperature T → 0, the number of non-condensed particles is given by: 〈δN̂(t)〉 = 〈vk(t)|vk(t)〉 (5) where vk(t) is one of the mode functions of the system. The modal functions (uk(t), vk(t)) are pairs of functions that represent the time-dependent coefficients of the de- composition, in terms of annihilation and creation op- erators, of the equation of motion for the field operator describing the thermal excitations above the condensate. They describe the spatial dependence of these excitations and propagate by modified Bogoliubov equations. Our findings (see fig.(6b)) are consistent with a poly- nomial growth of noncondensed particles, namely in our parameter region (and within the time scale we typically consider) no exponential instability takes place. This is consistent with recent experimental work [9], where 87Rb atom condensate has been used to explore QAMs. In [9], a condensate of 50000 Rb atoms with repulsive interac- tions is realized. In the case of a ”cigar shaped” trap, the relationship between the number of atoms in the conden- sate N and the effective 1-d nonlinear coupling constant u is, in our units, N = ua2 /2a0 [10], where a0 is the 3-dimensional scattering length and a⊥ ≫ a0 is the ra- dial extension of the wave function. Using the parameter values of the experiment [9], one finds N ≃ 105 ·u and so N ∼ 50000 corresponds to u ∼ 0.5. Therefore our range of parameters includes the experimental accessible one. We have investigated effects of atomic interactions, in the form of a cubic nonlinearity, on the problem of quan- tum accelerator modes: in particular we have charac- terized the consequences of both attractive and repul- sive interaction; we have also provided evidences that the modes are not strongly unstable when reasonable pa- rameters are chosen. We thank G. S. Summy for providing us with details of his work. This work has been partially supported by the MIUR-PRIN 2005 project ”Transport properties of classical and quantum systems”. S.F. acknowledges sup- port by the Israel Science Foundation (ISF), by the US- Israel Binational Science Foundation (BSF), by the Min- erva Center of Nonlinear Physics of Complex Systems, by the Shlomo Kaplansky academic chair. I.G. acknowl- edges hospitality by the Institute of Theoretical Physics at the Technion where part of this work was done. ∗ Electronic address: [email protected] [1] M.K.Oberthaler, R.M.Godun, M.B. D’Arcy, G.S. Summy and K. Burnett, Phys. Rev. Lett. 83, 4447 (1999); R.M. Godun, M.B. D’Arcy, M.K. Oberthaler, G.S. Summy and K. Burnett, Phys. Rev. A 62, 013411 (2000). [2] M.V.Berry and E.Bodenschatz, J.Mod.Opt. 46, 349 (1999). [3] G. Casati, B.V. Chirikov, F.M. Izrailev and J. Ford, Vol. 93 of Lectures Notes in Physics, edited by G. Casati and J. Ford (Spriger, Berlin, 1979). [4] S.Fishman, I.Guarneri and L.Rebuzzini, Phys.Rev.Lett. 89, 084101 (2002); J.Stat.Phys. 110, 911; I.Guarneri, L.Rebuzzini and S.Fishman, Nonlinearity 19, 1141 (2006). [5] A.D.Bandrauk and H. Shen, J. Phys. A: Math. Gen. 27, 7147 (1994). [6] M. Sheinman, S. Fishman, I. Guarneri and L. Rebuzzini, Phys. Rev. A 73, 052110 (2006). [7] Y.Castin and R.Dum, Phys.Rev. A 57, 3008 (1998); Phys.Rev.Lett. 79, 3553 (1998). [8] S.A. Gardiner, D. Jaksch, R. Dum, J.I. Cirac and P. Zoller, Phys.Rev.A,62, 023612 (2000); C. Zhang, J. Liu, M.G. Raizen and Q. Niu, Phys. Rev. Lett. 92, 054101 (2004). [9] G. Behinaein, V. Ramareddy, P. Ahmadi and G. S. Summy, Phys. Rev. Lett. 97, 244101 (2006) [10] D. S. Petrov, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. Lett. 85, 3745 (2000). mailto:[email protected]

0704.1383

How far is it to a sudden future singularity of pressure?

How far is it to a sudden future singularity of pressure? Mariusz P. Da̧browski∗ Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Tomasz Denkiewicz† Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and Fachbereich Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany Martin A. Hendry‡ Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK (Dated: November 12, 2018) We discuss the constraints coming from current observations of type Ia supernovae on cosmo- logical models which allow sudden future singularities of pressure (with the scale factor and the energy density regular). We show that such a sudden singularity may happen in the very near future (e.g. within ten million years) and its prediction at the present moment of cosmic evolution cannot be distinguished, with current observational data, from the prediction given by the standard quintessence scenario of future evolution. Fortunately, sudden future singularities are characterized by a momentary peak of infinite tidal forces only; there is no geodesic incompletness which means that the evolution of the universe may eventually be continued throughout until another “more serious” singularity such as Big-Crunch or Big-Rip. PACS numbers: 98.80.Cq; 98.80.Hw Over the past decade observations of high-redshift Type Ia supernovae (SNIa) have provided strong evi- dence that the expansion of the universe is accelerating, driven in the standard paradigm by some form of dark energy[1, 2]. Current data[2] continue to leave open the possibility that dark energy exists in the form of phan- tom energy, which may violate all energy conditions [3]: the null (̺c2 + p ≥ 0), weak (̺c2 ≥ 0 and ̺c2 + p ≥ 0), strong (̺c2 + p ≥ 0 and ̺c2 + 3p ≥ 0), and dominant energy (̺c2 ≥ 0, −̺c2 ≤ p ≤ ̺c2) conditions (where c is the speed of light, ̺ is the mass density in kgm−3 and p is the pressure). Phantom matter may dominate the universe in the future and drive it towards a Big-Rip (BR) singularity in which all matter will be dissociated by gravity [4]. This is dramatically different from the standard picture of future cosmic evolution which sug- gests an asymptotically empty de-Sitter state driven by the cosmological constant or quintessence [5] and leading to the violation of the strong energy condition only. Phantom-driven scenarios have encouraged the study of other exotic possibilities for the future evolution of the universe. One of these possibilities appears in those models which do not assume any explicit form for the equation of state p = p(̺), leaving the evolution of the energy density and pressure unconstrained. This free- dom may result in a so-called sudden future singularity (SFS) of pressure [6] which violates only the dominant energy condition. The nature of a sudden future singu- larity is different from that of a standard Big-Bang (BB) singularity, and also from a Big-Rip singularity, in that it does not exhibit geodesic incompletness and the cosmic evolution may eventually be extended beyond it [7, 8]. The only physical characteristic of these singularities is a momentarily infinite peak of the tidal forces in the uni- verse. In more general models this peak may also appear in the derivatives of the tidal forces. It is interesting to note that these types of singularity are in a way simi- lar to yet another type, which were termed finite density singularities [9]. However, the crucial difference is that finite density singularities occur as singularities in space rather than in time, which means that even at the present moment of cosmic evolution they could exist somewhere in the Universe [10]. We will not discuss in detail finite density singularities in this paper since they basically ap- pear in cosmological models without homogeneity. On the other hand, it is worth mentioning that the sudden future singularities are quite generic since they may arise in both homogeneous [11] and inhomogeneous [12] mod- els of the universe. In order to obtain a sudden future singularity consider the simple framework of an Einstein-Friedmann cosmol- ogy governed by the standard field equations , (1) p = − c , (2) where the energy-momentum conservation law ˙̺ = −3 ȧ , (3) is trivially fulfilled due to the Bianchi identity. Here a(t) is the scale factor, G is the gravitational constant, and the curvature index k = 0,±1. What is crucial to ob- tain a sudden future singularity is that no link between http://arxiv.org/abs/0704.1383v2 the energy density and pressure (the equation of state) is specified. This allows us to integrate (3) only by quadra- tures as ̺a3 = exp 3p(t′) c2̺(t′) ln a(t′) p(t′) ̺(t′) ln a(t′)dt′ Of course (4) reduces to the standard expression for energy conservation, ̺a3(w+1) = const., provided a barotropic equation of state, p = w̺c2 for constant w, is assumed. (The condition for phantom models, for ex- ample, is w < −1). From equations (1)-(2) one can easily see that a pres- sure singularity p → ∓∞ occurs when the acceleration ä → ±∞, no matter that the value of the energy density ̺ and the scale factor a(t) are regular. Since in that case | p |> ̺, it is clear that the dominant energy condition is violated. This condition can be achieved if the scale factor takes the form [6] a(t) = as [1 + (1− δ) ym − δ (1− y)n] , y ≡ with the appropriate choice of the constants δ, ts, as,m, n. Moreover, we can see that the r-th derivative of the scale factor (5) is given by a(r) = as m(m− 1)...(m− r + 1) (1− δ) ym−r + (−1)r−1δ n(n− 1)...(n− r + 1) (1− y)n−r and is related to the appropriate pressure derivative p(r−2). Thus, in general, it is possible that one has a pressure derivative p(r−2) singularity which accompanies the blow-up of the r-th derivative of the scale factor a(r). Observationally this could be manifested in, for example, the blow-up of the characteristics known as statefinders , such as jerk, snap etc. [14]. The pressure derivative sin- gularity p(r−2) appears when r − 1 < n < r r = integer , (7) and for any r ≥ 3 it fulfills all energy conditions. These singularities are called generalized sudden future singu- larities (GSFS) and are possible, for example, in theories with higher-order curvature quantum corrections [13]. Let us now return to the case of r = 2, for which 1 < n < 2 and we obtain sudden future singularity models of pressure (and obviously all of its higher derivatives) which lead to violation of the dominant energy condition. In such models, expressed in terms of the scale factor (5), the evolution begins with the standard BB singularity at t = 0 for a = 0, and finishes at SFS for t = ts where a = as ≡ a(ts) is a constant. (Note that we have changed the original parametrization of Ref. [6] for the scale factor (5) using A = δas). The standard Friedmann limit (i.e. models without an SFS) of (5) is achieved when δ → 0; hence δ becomes the “non-standardicity” parameter of SFS models. Ad- ditionally, notwithstanding Ref. [6] and in agreement with the field equations (1)-(2), we assume that δ can be both positive and negative leading to a deceleration or an acceleration (cf. (6)) of the universe, respectively. It is important to our discussion that the asymptotic behaviour of the scale factor (5) close to the BB singu- larity at t = 0 is given by a simple power-law aBB = y simulating the behaviour of flat k = 0 barotropic fluid models with m = 2/[3(w+1)] . This allows us to preserve all the standard observed characteristics of early universe cosmology – such as the cosmic microwave background, density perturbations, nucleosynthesis etc. – provided we choose an appropriate value of m. On the other hand, close to an SFS the asymptotic behaviour of the scale factor is non-standard, aSFS = as [1− δ (1− y)n ], show- ing that aSFS = as for t = ts (i.e. y = 1) at the SFS. Notice that one does not violate the energy conditions if the parameter m lies in the range 0 < m ≤ 1 (w ≥ −1/3), (8) This range of values is, in fact, equivalent to a standard (neither quintessence-like nor phantom-like) evolution of the universe. However, with no adverse impact on the field equations (1)-(2), one could also extend the val- ues of m to lie in the complementary ranges [7] m > 1 (i.e. −1 < w < −1/3) for quintessence, and m < 0 (i.e. w < −1) for phantom, although these ranges may lead to violation of the strong and weak energy conditions respectively. We will next calculate the luminosity distance as a function of redshift, and hence the redshift-magnitude relation, for SFS models. This will allow us to estab- lish whether these models are a realistic possibility for the future evolution of the universe, and more specifi- cally whether current cosmological observations of high redshift supernovae are consistent with values of the con- stant n in the range 1 < n < 2, as required in order that the scale factor will display an SFS (or, more generally, a GSFS for r − 1 < n < r). We will then explore the range of values for the other SFS model parameters which are consistent with current observational constraints on standard cosmology, and thus determine limits on how far into the future an SFS might occur. In fact, as we will see below, we need to consider only two further pa- rameters: δ and y0 = t0/tS , where t0 is the current age of the Universe in the SFS model. Notice that, in view of (8), it is reasonable to take m = 2/3 as for the standard dust-dominated evolution. This implies that, at early times, our SFS model reduces to the Einstein-de-Sitter universe. We proceed within the framework of Friedmann cos- mology, and consider an observer located at r = 0 at coordinate time t = t0. The observer receives a light ray emitted at r = r1 at coordinate time t = t1. We then have a standard null geodesic equation 1− kr2 , (9) with the scale factor a(t) given by (5). Using (5) again, the redshift is given by 1 + z = a(t0) a(t1) δ + (1− δ) ym0 − δ (1− y0) δ + (1− δ) ym1 − δ (1− y1) n , (10) where y0 = y(t0) and y1 = y(t1). The luminosity distance is defined as DL = r1a(t0) (1 + z) . (11) Neglecting extinction and k−corrections, the observed and absolute magnitudes of a source at redshift z and luminosity distance DL are related by m(z) = M − 5 log10 H0 + 25 + 5 log10 DL(z), (12) which, with the help of the equation (9), (10) and (11), allows a redshift-magnitude relation for SFS cosmologi- cal models to be constructed. It is obvious that equation (9) has to be integrated numerically in order to estab- lish the relation between t0 and t1, which can then be inserted into (10) and (11) to constrain the SFS model parameters. As a first step we determine the dependence on the SFS model parameters of the Hubble law, which replaces equation (12) for z ≈ 0, i.e. cz ≈ H0DL, where H0(kms Mpc−1) = 3.09× 1019 t0(sec)y0 m (1− δ) ym−10 + nδ (1− y0) δ + (1− δ) ym0 − δ (1− y0) is the present value of the Hubble parameter, which we can take as 72kms−1Mpc−1 [1]. Similarly we could derive an expression, in terms of the SFS model parameters, for the deceleration param- eter q0 = −(äa/ȧ2)0. However, in order to search the parameter space for models which are admissible by cur- rent observations, we write the product of H0 and q0 as q0H0 = − = (14) m(m− 1)(1− δ)ym−20 − δn(n− 1) (1− y0) m(1− δ)ym−10 + nδ (1− y0) In order to obtain an accelerated universe at the present moment of the evolution, this product should be nega- tive. Fig. 1 shows an example plot of the product H0q0 as a function of δ and y0, with the other parameters fixed atm = 2/3, n = 1.9993, t0 = 13.2457 Gyr. From the plot we see that there are large regions of the parameter space which admit cosmic acceleration. We have explored the parameter space further with various configurations of m,n, δ, y0, t0, q0, and H0, and obtained the general con- clusion that there is a large class of SFS models which are compatible with current acceleration. H0*q0 FIG. 1: Parameter space (H0q0, δ, y0) for fixed values of m = 2/3, n = 1.9993, t0 = 13.3547 Gyr of the sudden future singularity models. There are large regions of the parameter space which admit cosmic acceleration. Out of these admissible models we then searched for those which are compatible with the redshift-magnitude relation (12) observed for recent SNIa data [2], and hence with the derived parameters of the standard ‘Concor- dance cosmology’ (CC). We were able to identify SFS models that are in remarkably tight agreement with cur- rent SNIa data. As an illustrative example Fig. 2 shows luminosity distance as a function of redshift for the CC model with H0 = 72kms −1Mpc−1, Ωm0 = 0.26 and ΩΛ0 = 0.74, and an SFS model with parametersm = 2/3, y0 = 0.99936, δ = −0.471, n = 1.9999. We see that the SFS model mimics the CC model very closely over a wide range of redshifts. In particular, it is clear that recent SNIa data from the Tonry at al. ‘Gold’ sample [1] and SNLS sample [2] cannot yet discriminate between the CC and SFS models. Taking the current age of the universe in the SFS model to be equal to the age of the CC model, i.e. t0 = 13.6Gyr, we find that the time to the sudden singularity is ts−t0 ≈ 8.7Myr, which is amazingly close to the present epoch. In that context there is no wonder that these singularities are called “sudden”. We have also checked that the larger the value of r in (7) the later in future a GSFS appears. It means that the strongest of these singularities which violates the dominant energy condition (i.e. an SFS) is more likely to become reality. Our remark about the effect of the sudden pressure singularity seems in agreement with the result of Ref. FIG. 2: The distance modulus µL = m − M for the con- cordance cosmology (CC) model with H0 = 72kms −1Mpc−1, Ωm0 = 0.26, ΩΛ0 = 0.74 (dashed curve) and sudden fu- ture singularity (SFS) model for m = 2/3, n = 1.9999, δ = −0.471, y0 = 0.99936 (solid curve). Also shown are the ‘Gold’ (open circles) and SNLS (filled circles) SNIa data. Taking the age of the SFS model to be equal to that of the CC model, i.e. t0 = 13.6 Gyr, one finds that an SFS is possible in only 8.7 million years. [16] which showed that the dominant energy condition is now violated and that it became violated quite recently (at redshift z ∼ 0.2). Of course this violation may also be due to phantom energy [3]. In conclusion, we have shown that a sudden future singularity may happen in the comparatively near fu- ture (e.g. within ten million years) and its prediction at the present moment of cosmic evolution cannot be distinguished, with current observational data, from the prediction given by the standard quintessence scenario of future evolution in the Concordance Model. Fortu- nately, sudden future singularities are characterized by a momentary peak of infinite tidal forces only; there is no geodesic incompletness which means that the evolution of the universe may eventually be continued beyond the SFS until another “more serious” singularity such as a Big-Crunch or a Big-Rip. One could then consider, more generally, a scale factor of the form [7, 15] a(t) = A+ [(as −A)−D(tr − ts)p − Etos] ym (15) − (A+Dtpr) (1− y)n +D(tr − tsy)p + Etosyo , where the constants m, o, p, A,D,E are chosen so that the universe begins with a Big-Bang at t = 0 where a = 0, next faces a sudden future singularity at t = ts where a(ts) = as, and then eventually continues to a Big-Rip at t = tr where a(tr) → ∞. All of the matter sources may be involved since the constants in (15) can be taken as: 0 < m ≤ 1 (quintessence), p < 0 (phantom), and o > 1 (standard positive matter pressure). Whether the universe will end in a Big-Rip or a Big- Crunch is an open question. Moreover, unlike a sudden future singularity, both a Big-Rip and Big-Crunch sin- gularity would represent the real end of the universe. Fortunately, as was shown in Refs. [4, 17], a Big-Rip singularity is not possible in the very near future: in or- der to reach it one must wait about the same time as the current age of the universe. Apart from that, it is still possible to avoid it due to a negative tension brane contribution in a turnaround cyclic cosmology [18]. ACKNOWLEDGEMENTS M.P.D. and T.D. acknowledge the support of the Polish Ministry of Education and Science grant No 1 P03B 043 29 (years 2005-2007). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] [1] J.L. Tonry et al., Astroph. J. 594, 1 (2003); M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). [2] P. Astier et al., Astron. Astrophys. 447, 31 (2006); A.G. Riess et al., astro-ph/0611572; J.D. Neill et al. astro-ph/0701161; J. Guy et al. astro-ph/0701828; M.W. Wood-Vasey et al. astro-ph/0701041. [3] R.R. Caldwell, Phys. Lett. B 545, 23 (2002); M.P. Da̧browski, T. Stachowiak and M. Szyd lowski, Phys. Rev. D 68, 103519 (2003); P.H. Frampton, Phys. Lett. B 562 (2003), 139; H. Štefančić, Phys. Lett. B586, 5 (2004); S. Nojiri and S.D. Odintsov, Phys. Lett. B595, 1 (2004). [4] R.R. Caldwell, M. Kamionkowski, and N.N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003). [5] R.R. Caldwell, R. Dave, and P.J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998). [6] J.D. Barrow, Class. Quantum Grav. 21, L79 (2004). [7] L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D70, 121503(R) (2004); L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D74, 064030 (2006). [8] M.P. Da̧browski and A. Balcerzak, Phys. Rev. D73, 101301(R) (2006). [9] M.P. Da̧browski, Journ. Math. Phys. 34, 1447 (1993). [10] M.P. Da̧browski and M.A. Hendry, Astroph. Journ. 498, 67 (1998); R.K. Barrett, C.A. Clarkson, Class. Quantum Grav. 17, 5047 (2000). [11] J.D. Barrow and Ch. Tsagas, Class. Quantum Grav. 22, 1563 (2005). [12] M.P. Da̧browski, Phys. Rev. D71, 103505 (2005). [13] S. Nojiri, S.D. Odintsov and S. Tsujikawa, Phys. Rev. D 71,063004 (2005). [14] U. Alam et al., Mon. Not. R. Astron. Soc. 344, 1057 (2003), M.P. Da̧browski, Phys. Lett. B 625, 184 (2005). [15] C. Cattoen and M. Visser, Class. Quantum Grav. 22, 4913 (2005). [16] J. Santos et al., astro-ph/0702728. [17] A.A. Starobinsky, Grav. Cosmol. 6, 157 (2000). [18] L. Baum and P.H. Frampton, Phys. Rev. Lett. 98, 071301 (2007). mailto:[email protected] mailto:[email protected] mailto:[email protected] http://arxiv.org/abs/astro-ph/0611572 http://arxiv.org/abs/astro-ph/0701161 http://arxiv.org/abs/astro-ph/0701828 http://arxiv.org/abs/astro-ph/0701041 http://arxiv.org/abs/astro-ph/0702728

0704.1384

Generalizing circles over algebraic extensions

7 Generalizing circles over algebraic extensions T. Recio∗ J.R. Sendra∗† L.F. Tabera∗‡ C. Villarino∗† August 29, 2021 Abstract This paper deals with a family of spatial rational curves that were introduced in [4], under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametriza- tion (simplifying the coefficients of the rational functions, when pos- sible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line (“the K-axis”) in a n-degree finite algebraic extension K(α) ≈ Kn under the transformation at+b : K(α) → K(α). The aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via K-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many K-rational points, passing through these points at in- finity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section. ∗The authors are partially supported by the project MTM2005-08690-CO2-01/02 “Min- isterio de Educación y Ciencia”. †Partially supported by CAM-UAH2005/053 “Dirección General de Universidades de la Consejeŕıa de Educación de la CAM y la Universidad de Alcalá”. ‡L.F.Tabera also supported by a FPU research grant. http://arxiv.org/abs/0704.1384v1 1 Introduction The problem of obtaining a real parametrization of a rational planar curve given by a complex parametrization has been studied –from an algorithmic point of view– in [9]. There, the problem is reduced to determining that a certain curve obtained after manipulating the given parametrization is a real line or a real circle. From a real parametrization of this circle (or line), a real parametrization of the original curve is then achieved. This auxiliary circle is found by an analogous to Weil descente’s method [16] applied to the complex parametrization of the originally given curve. In [4], the same approach has been extended to the general case of planar or spatial rational curves C given by a parametrization over K(α), where α is an algebraic element over K. In order to obtain, whenever possible, a parametrization over K of C, another rational curve, with remarkable properties, is associated to C. In [4] it is shown that this associated curve is, in the relevant cases, a generalization of a circle, in the sense we will discuss below, deserving to be named hypercircle. The simplest hypercircles should be the circles themselves. We can think of the real plane as the field of complex numbers C, an algebraic extension of the reals R of degree 2. Analogously, we can consider a characteristic zero base field K and an algebraic extension of degree n, K(α). Let us identify K(α) as the vector space Kn, via the choice of a suitable base, such as the one given by the powers of α. This is the framework in which hypercircles are defined. Now let us look to the different, equivalent, ways of defining a common circle on the real plane, with the purpose of taking the most convenient one for generalization. The first definition of a circle is the set of points in the real plane that are equidistant from a fixed point. This approach does not extend well to more general algebraic extensions, because we do not have an immediate notion of metric over Kn. On the other hand, algebraically, a real planar circle is a conic such that its homogeneous degree two form is x2 + y2 and such that it contains an infinite number of real points. Even if we will prove in Section 6 that we can show an analogous definition for a hypercircle, this is not an operative way to start defining them. Finally, from another point of view, we see that circles are real rational curves. This means that there are two real rational functions (φ1(t), φ2(t)) whose image cover almost all the points of the circle. For instance, the circle x2+y2 = 1 is parametrized by φ(t) = ( t ). Every proper (almost one- to-one [14]) rational parametrization of a circle verifies that φ1(t) + iφ2(t) = ∈ C(t) \ C, which defines a conformal mapping u : C → C. Moreover, if we identify C with R2, the image of the real axis (t, 0) under u is exactly the circle parametrized by φ(t). Conversely, let u(t) = at+b ∈ C(t) be a unit of the near-ring C(t) under the composition operator (see [17]). If c 6= 0 and d/c /∈ R then, the closure of the image by u of the real axis is a circle. Otherwise, it is a line. This method to construct circles generalizes easily to algebraic extensions. Namely, let u(t) = at+b be a unit of K(α)(t) (i.e. verifying that ad − bc 6= 0). Let us identify K(α) with Kn and let u be the u : K(α) ≈ Kn → K(α) ≈ Kn t 7→ u(t) Then, the Zariski-closure of the image of the axis (t, 0, . . . , 0) under the map u is a rational curve in Kn. These curves are, by definition, our hypercircles. Roughly speaking, it happens (see [4]) that a parametrization over K of the hypercircle associated to a given rational curve C (whose parametrization we want to simplify) can be used to get –in a straightforward manner– a parametrization of C over K. As pointed in [4], it seems that, due to the geometric properties of hypercircles, it is algorithmically simpler to obtain such parametrization for this type of curves than it is for C. In fact, it is shown in [10] how to get this in some cases. Therefore, the reparametrization problem is behind our increasing interest in the study of hypercircles on its The structure of this paper is as follows. In Section 2 we formally intro- duce the notion of hypercircle. We study the influence on a hypercircle when adding and multiplying the defining unit u(t) by elements of K(α), reduc- ing the affine classification of hypercircles to those defined by some simpler units. Next we characterize the units associated to lines. In Section 3 we show how to transform, projectively, a hypercircle into the rational normal curve (see [6]). From this, we derive the main geometric properties of hy- percircles (smoothness, degree, affine equivalence, etc.) and we reduce the study of hypercircles to the subclass of primitive hypercircles (See Definition 3.5). In Section 4 the behavior of hypercircles at infinity is analyzed, showing its precise and rich structure. In Section 5, exploiting the stated geometric features, we present ad hoc parametrization and implicitization methods for hypercircles. In Section 6 we characterize hypercircles among curves of de- gree equal to the dimension of the ambient affine space, passing through the prescribed points of infinity described in Section 4 and having infinitely many rational points. Finally, Section 7 is devoted to show how the insight gained throughout this paper can be applied to derive heuristics for solving the problem of simplifying the parametrization of curves with coefficients involving algebraic elements. Throughout this paper the following notation and terminology will be used. • K will be a field of characteristic zero, K ⊆ L a finite algebraic extension of degree n and F the algebraic closure of K. • α will be a primitive element of L over K. • u(t) will be a unit under composition of L(t). That is, u(t) = at+b ad− bc 6= 0. Its inverse −dt+b is denoted by v(t). • For u(t) = at+b and c 6= 0, M(t) = tr + kr−1t r−1 + · · · + k0 ∈ K[t] denotes the minimal polynomial of −d/c over K. • We will denote as m(t) the polynomial obtained by dividing M(t) by ct+ d. That is, m(t) = ct+ d = lr−1t r−1 + lr−2t r−2 + · · ·+ l0 ∈ L[t]. • Sometimes we will represent u(t) as u(t) = (at+ b)m(t) p0(t) + p1(t)α + · · ·+ pn−1(t)α where pi(t) ∈ K[t]. • By {σ1 = Id, σ2, . . . , σs}, s ≥ n we will denote the group of K- automorphisms of the normal closure of K ⊆ L. • We will represent by {α1 = α, . . . , αn} the conjugates of α. We assume without loss of generality that σi(α) = αi for i = 1, . . . , n. 2 Definition and First Properties In this section we begin with the formal definition of a hypercircle. Definition 2.1. Let u(t) be a unit in L(t), where L = K(α). Let u(t) = φi(t)α where φi(t) ∈ K(t), for i = 0, . . . , n − 1. The α-hypercircle U generated by u(t) is the rational curve in Fn parametrized by φ(t) = (φ0(t), . . . , φn−1(t)). Observe that the expansion of u(t) in powers of α is unique, because {1, α, . . . , αn−1} is a basis ofK(α)(t) as aK(t)−vector space. The parametriza- tion can be obtained by rationalizing the denominator as follows: suppose given the unit u(t) = at+b , c 6= 0 (remark that, if c = 0, it is straightforward to obtain φ(t)), and the extension K ⊆ K(α). Let M(t) be the minimal polynomial of −d/c over K. Compute the quotient m(t) = ∈ K(α)[t] and develop the unit as at+ b ct+ d (at+ b)m(t) p0(t) + p1(t)α + · · ·+ pn−1(t)α where pi(t) ∈ K[t]. From this, φ(t) = p0(t) , . . . , pn−1(t) is the parametriza- tion associated to u(t). Remark that gcd(p0(t), . . . , pn−1(t),M(t)) = 1. More- over, it is clear that F(φ0(t), . . . , φn−1(t)) = F(t). So this parametrization is proper in F, and it follows from the results in [1] that alsoK(φ0(t), . . . , φn−1(t)) = K(t). Example 2.2. Let us consider the algebraic extension Q ⊆ Q(α), where α3 + 2α+ 2 = 0. The unit t−α has an associated hypercircle parametriced by φ(t) = t3 + 2t+ 2 t3 + 2t− 2 t3 + 2t− 2 t3 + 2t− 2 A picture of the spatial real curve is shown in Figure 1 As it stands, the definition of a hypercircle U depends on a given unit u(t) ∈ L(t) and on a primitive generator α of an algebraic extension L. In what follows we will analyze the effect on U when varying some of these items, searching for a simple representation of a hypercircle to ease studying its geometry. First notice that, given a unit u(t) ∈ L(t) and two different primitive elements α and β of the extension K ⊆ L, we can expand the unit in Figure 1: A hypercircle in R3 two different ways u(t) = i=0 α iφi(t) = i=0 β iψi(t). The hypercircles Uα ≃ (φ0(t), . . . , φn−1(t)) and Uβ ≃ (ψ0(t), . . . , ψn−1(t)) generated by u(t) are different curves in Fn, see Example 2.3. Nevertheless, let A ∈ Mn×n(K) be the matrix of change of basis from {1, α, . . . , αn−1} to {1, β, . . . , βn−1}. Then, A(φ0(t), . . . , φn−1(t)) t = (ψ0(t), . . . , ψn−1(t)) t. That is, it carries one of the curve onto the other. Thus, Uα and Uβ are related by the affine trans- formation induced by the change of basis and, so, they share many important geometric properties. In the sequel, if there is no confusion about the algebraic extension and the primitive element, we will simply call U a hypercircle. Example 2.3. Let us consider the algebraic extension Q ⊆ Q(α), where α4 + 1 = 0. Let us take the unit u(t) = t−α . By normalizing u(t), we obtain the parametrization φ(t) associated to u(t): φ(t) = t4 − 1 t4 + 1 t4 + 1 t4 + 1 t4 + 1 This hypercircle Uα is the zero set of {X1X2 − X3X0 − X3, X 1 + X 2X2, X1X0+X2X3−X1, X 0 +X3X1− 1}. Now, we take β = α 3+1, instead of α, as the primitive element of Q(α) = Q(β). The same unit u(t) generates the β-hypercircle Uβ parametrized by ψ(t) = t4 + 2t3 − 2t2 + 2t− 1 t4 + 1 −6t3 + 4t2 − 2t t4 + 1 6t3 − 2t2 t4 + 1 t4 + 1 which is different to Uα; note that ψ(1) = (1,−2, 2,−1) that does not satisfy the equation X20 +X3X1 − 1 = 0 of Uα. On the other hand it is well known that a given parametric curve can be parametrized over a given field S by different proper parametrizations, pre- cisely, those obtained by composing to the right a given proper parametriza- tion by a unit in S(t). In this way, we have a bijection between α-hypercircles and the equivalence classes of units of K(α)(t) under the equivalence relation “u ∼ v iff u(t) = v(τ(t)) for a unit τ(t) ∈ K(t)” (fixing the correspondence, between a unit in K(α)(t) and a hypercircle, by means of the expansion of the unit in terms of powers of α). More interesting is to analyze, on a hypercircle defined by a unit u(t), the effect of composing it to the left with another unit τ(t) ∈ K(α)(t), that is, of getting τ(u(t)). For instance, τ(t) could be τ(t) = t + λ or τ(t) = λt, or τ(t) = 1/t, with λ ∈ K(α)∗. Every unit is a sequence of compositions of these three simpler cases, for instance, when c 6= 0, we have t 7−→ ct 7−→ ct + d 7−→ ct + d bc− ad ct+ d bc− ad ct+ d at + b ct+ d = u(t). Therefore, studying their independent effect is all we need to understand completely the behavior of a hypercircle under left composition by units. For circles, adding a complex number to the unit that defines the circle correspond to a translation of the circle. Multiplying it by a complex number acts as the composition of a rotation and a dilation. And the application τ(t) = 1/t gives an inversion. The following lemma analyzes what happens in the general case. Lemma 2.4. Let U be the α-hypercircle generated by u(t), and λ = i ∈ K(α)∗, where λi ∈ K. Then, 1. λ + u(t) is a unit generating the hypercircle obtained from U by the translation of vector (λ0, . . . , λn−1). 2. λu(t) is a unit generating the hypercircle obtained from U by the affine transformation over K given by the matrix of change of basis from B⋆ = {λ, λα, . . . , λαn−1} to B = {1, α, . . . , αn−1}. Proof. To prove (1), let φ(t) = (φ0(t), . . . , φn−1(t)) ∈ K(t) n be the parametriza- tion of U obtained from u(t). Then, λ+u(t) = i=0 (λi +φi(t))α i generates the hypercircle parametrized by (λ0 + φ0(t), . . . , λn−1 + φn−1(t)) ∈ K(t) which is the translation of U of vector (λ0, . . . , λn−1). For the second asser- tion, let φ⋆(t) ∈ K(t)n be the parametrization of the hypercircle associated to the unit λu(t). The rational coordinates φ⋆i (t) of φ ⋆(t) are obtained from the matrix A = (ai,j) ∈ Mn×n(K) of change of basis from B ⋆ to B, for i, j = 0, . . . , n− 1. Indeed, λu(t) = φi(t)λα φi(t) ajiφi(t) Then φ⋆(t)t = A φ(t)t. Finally, the following lemma uses the previous results to transform affinely one hypercircle into another one whose unit is simpler. Lemma 2.5. Let u(t) = at+b be a unit and U its associated hypercircle. 1. If c = 0 then U is affinely equivalent over K to the line generated by u⋆(t) = t. 2. If c 6= 0 then U is affinely equivalent over K to the hypercircle U⋆ generated by u⋆(t) = 1 t+d/c Proof. This lemma follows from Lemma 2.4, taking into account that u(t) is obtained from u⋆(t) by the following composition: u⋆(t) 7→ λ1u ⋆(t) 7→ λ1u ⋆(t) + λ2 = u(t) with suitable λ1, λ2, u ⋆. If c = 0, then λ1 = 6= 0 and λ2 = for u⋆(t) = t. Analogously, if c 6= 0, then u(t) is obtained from u⋆(t) = 1 t+d/c taking λ1 = bc−ad 6= 0 and λ2 = Therefore the (affine) geometry of hypercircles can be reduced to those generated by a unit of type 1 (then we say the unit is in reduced form). The simplest hypercircle of this kind is given by 1 , when d ∈ K. It is the line parametrized by ( 1 , 0, . . . , 0). In the complex case, Moebius transfor- mations defining lines are precisely those given either by a polynomial unit in t (i.e. a unit without t at the denominator) or by a unit such that the root of the denominator is in R. The same property holds for hypercircles. Theorem 2.6. Let U be the α-hypercircle associated to u(t). Then, the following statements are equivalent: 1. U is a line. 2. U is associated to a polynomial unit. 3. The root of the denominator of every non polynomial unit generating U belongs to K. 4. U is polynomially parametrizable (over F). 5. U has one and only one branch (over F ) at infinity. 6. U is polynomially parametrizable over K. 7. U has one and only one branch (over K ) at infinity. Proof. (1)⇔ (2). By definition, we know that hypercircles have a parametriza- tion overK. Thus, if U is a line, it can be parametrized as (a0t+b0, . . . , an−1t+ bn−1), where ai , bi ∈ K. Therefore, u(t) = (∑n−1 i=0 aiα i=0 biα i is a polynomial unit associated to U . Conversely, let u(t) = at+ b ∈ L(t), a 6= 0, be a polynomial unit associated to U . Then U is the line parametrized by P(t) = (a0t + b0, . . . , an−1t + bn−1) ∈ K[t] n, where a = i=0 ai α i and i=0 bi α (2) ⇔ (3). Let u(t) = at+b be a polynomial unit associated to U , and let u⋆(t) be another non polynomial unit associated to U . Then, u⋆(t) = u(τ(t)), where τ(t) is a unit of K (t). Therefore, the root of u⋆(t) belongs to K. Conversely, by Lemma 2.5, (3) implies (1), and we know that (1) implies (2). (3) ⇔ (4). Indeed, (3) implies (2) and therefore (4). Conversely, let u(t) be a non-polynomial unit generating U , and let φ(t) = (φi)i=1,...,n ∈ K(t) the associated parametrization of U . Then, φ(t) is proper, φi(t) = pi(t) deg(pi) ≤ deg(M) and gcd(p0(t) . . . pn−1(t),M(t)) = 1. Thus, the fact that U admits a polynomial parametrization, implies, by Abhyankar-Manocha- Canny’s criterion of polynomiality (see [8]), that the denominator M(t) is either constant or has only one root. Now, M(t) can not be constant, since it is a minimal polynomial. Thus, M has only one root, and since it is irreducible, it must be linear. Moreover, since M ∈ K[t], its root is an element in K. (4) ⇔ (5) This is, again, the geometric version of Abhyankar-Manocha- Canny’s criterion. Same for (6) ⇔ (7). (4) ⇔ (6) Obviously (6) implies (4). Conversely, if we have a polynomial parametrization over F, it happens [2] that any proper parametrization must be either polynomial or in all its components the degree of the numerator must be smaller or equal than the degree of the denominator and, then, this denominator has only one single root over F. So, since the parametrization φ(t) induced by the unit is proper, and by hypothesis U is polynomial, then φ(t) must be either polynomial (in which case we are done because φ(t) is over K) or its denominator M(t) has a single root a ∈ F. Now, reasoning as above one gets that a ∈ K. So, a change of parameter, such as t 7→ 1+as turns φ(t) into a K-polynomial parametrization. As a corollary of this theorem, we observe that a parabola can never be a hypercircle, since it is polynomially parametrizable, but it is not a line. Nevertheless, it is easy to check that the other irreducible conics are indeed hypercircles for certain algebraic extensions of degree 2. 3 Main Geometric Properties. This section is devoted to the analysis on the main geometric properties of hypercircles. The key idea, when not dealing with lines, will be to use the reduction to units of the form u(t) = 1 , where d /∈ K (see Lemma 2.5). Theorem 3.1. Let U be the α-hypercircle associated to the unit u(t) = at+b K(α)(t) and let r = [K(−d) : K]. Then, 1. there exists an affine transformation χ : Fn −→ Fn defined over K such that the curve χ(U) is parametrized by χ̃(t) = , . . . , , 0, . . . , 0 2. there exists a projective transformation ρ : P(F) −→ P(F) , defined over K, such that the curve ρ(U) is the rational normal curve of degree r in P(F) , parametrized by ρ̃(t : s) = [sr : sr−1t : · · · : str−1 : tr : 0 : · · · : 0]. Proof. For the case of lines the result is trivial. By Lemma 2.5, we can consider that U is the hypercircle associated to u(t) = 1 and r ≥ 2. Let M(t) = tr + kr−1t k−1 + · · ·+ k0 ∈ K[t], m(t) = i=0 lit i ∈ L[t], as indicated in Section 1 and, since the numerator of u(t) is 1, it holds that m(t) =∑n−1 i=0 pi(t)α i, pi(t) ∈ K[t]. Also, note that both M(t) and the denominator of u(t) are monic, and hence lr−1 = 1. First of all, we prove that there are exactly r polynomials in {pi(t), i = 0, . . . , n − 1} ⊂ K[t] being linearly independent. For this purpose, we observe that the coefficients of m(t), {1, lr−2, . . . , l0} ⊂ L, are linearly independent over K. Indeed, from the equalityM(t) = (t+d)m(t), one has that lr−i = (−d) i−1+(−d)i−2kr−1+· · ·+ kr−i+1, for i = 2, . . . , r. So, {1, lr−2, . . . , l0} ⊂ L are K–linearly independent, since otherwise one would find a non-zero polynomial of degree smaller than r vanishing at −d. Now, let ~li = (li,0, . . . , li,n−1) t be the vector of coordinates of li in the base {1, α, . . . , α n−1}. Then, {~1,~lr−2, . . . ,~l0} ⊂ K n are K–linearly independent. Moreover, since (p0(t), . . . , pn−1(t)) t = ~1tr−1+~lr−2t r−2+· · ·+~l0, there are r polynomials pij , 0 ≤ i1 < · · · < ir ≤ n− 1, linearly independent. By simplicity, we assume w.l.o.g. that the first r polynomials are linearly independent. Observe that this is always possible through a permutation matrix. The new curve, that we will continue denoting by U , is not, in general, a hypercircle. In this situation, we proceed to prove (1) and (2). In order to prove (1), let A ∈ Mn−r×r(K) be the matrix providing the linear combinations of the n − r last polynomials in terms of the first r polynomials; i.e. (pr(t), . . . , pn−1(t)) t = A(p0(t), . . . , pr−1(t)) t. Now, given the bases B = {1, . . . , tr−1} and B⋆ = {p0(t), . . . , pr−1(t)}, let M ∈ Mr×r(K) be the transpose matrix of change of bases from B to B⋆. Finally, the n× n matrix M Or,n−r −A In−r defines, under the previous assumptions, the affine transformation χ. Note that if r = n then Q = M. The proof of (2) is analogous to (1). Now, let consider the basis B = {1, . . . , tr−1, tr} and B⋆ = {p0(t), . . . , pr−1(t),M(t)}. Let A ∈ Mn−r×r+1(K) be the matrix providing the linear combinations of the n−r last polynomials in terms of basis B⋆; i.e. (pr(t), . . . , pn−1(t)) t = A(p0(t), . . . , pr−1(t),M(t)) Let M ∈ Mr+1×r+1(K) be the transpose matrix of change of bases from B to B⋆. Finally, the n + 1× n + 1 matrix M Or+1,n−r −A In−r defines, under the previous assumptions, the projective transformation ρ. Note that if r = n then Q = M. As a direct consequence, we derive the following geometric properties of hypercircles. Corollary 3.2. In the hypothesis of Theorem 3.1 1. U defines a curve of degree r. 2. U is contained in a linear variety of dimension r and it is not contained in a variety of dimension r − 1. 3. U is a regular curve in P(F) 4. The Hilbert function of U is equal to its Hilbert polynomial and hU(m) = mn+ 1. Proof. All these properties are well known to hold for the rational normal curve of degree r e.g. [6], [7], [15]). In the following theorem, we classify the hypercircles that are affinely equivalent over K. We will assume that the denominator of the generat- ing units are not constant. The case where the units are polynomials are described in Theorem 2.6. Theorem 3.3. Let Ui, i = 1, 2, be α-hypercircles associated to ui(t) = ait+bi and let Mi(t) be the minimal polynomial of −di over K. Then, the following statements are equivalent: 1. U1 and U2 are affinely equivalent over K. 2. There exists a unit τ(t) ∈ K(t) such that it maps a root (and hence all roots) of M1(t) onto a root (resp. all roots) of M2(t). Proof. First of all note that, because of Theorem 2.6, the result for lines is trivial. For dealing with the general case, we observe that, by Lemma 2.5, we can assume that ui(t) = 1/(t + di). Next, suposse that U1 and U2 are affinely equivalent over K. By Theorem 3.1, statement (1), [K(d1) : K] = [K(d2) : K] = r and the curves U 1 := χ(U1) and U 2 := χ(U2) parametrized by χ̃1(t) = ( M1(t) , . . . , t M1(t) ) and χ̃2(t) = ( M2(t) , . . . , t M2(t) respectively, are affinely equivalent over K; note that, for simplicity we have omitted the last zero components in these parametrizations. There- fore, there exists A = (ai,j) ∈ GL(r,K) and ~v ∈ Mr×1(K), such that ϕ(t) := A χ̃1(t) t+~v parametrizes U⋆2 . In consequence, since ϕ(t) and χ̃2(t) are proper parametrizations of the same curve, there exists a unit τ(t) ∈ K(t) such that ϕ(t) = χ̃2(τ(t)). Then, considering the first component in the above equality, one gets that (a1,1 + · · ·+ a1,rt r−1 + v1M1(t))M2(τ(t)) =M1(t). Now, substituting t by −d1, we obtain (a1,1 + · · ·+ a1,r(−d1) r−1 + v1M1(−d1))M2(τ(−d1)) =M1(−d1) = 0. Note that a1,1 + · · ·+ a1,r(−d1) r−1 6= 0, because [K(d1) : K] = r. Also, note that τ(−d1) is well defined, because −d1 does not belong to K. This implies that M2(τ(−d1)) = 0. So, τ(−d1) is a root of M2(t). Conversely, let τ(t) = k1t+k2 k3t+k4 ∈ K(t) be a unit that maps the root γ of M1(t) onto the root β of M2(t), i.e. τ(γ) = β. This relation implies that K(γ) = K(β) and that deg (M1(t)) = deg (M2(t)) = r. Therefore, because of Theorem 3.1, it is enough to prove that the curves U⋆1 := χ(U1) and U⋆2 := χ(U2) are affinely equivalent over K. Recall that U i is parametrized by ϕi(t) := χ̃(t) = Mi(t) , . . . , t Mi(t) ; here again, we omit the last zero components of the parametrization. In order to prove the result, we find an invertible matrix A ∈ GL(r,K) and a vector ~v ∈ Mr×1(K), such that Aϕt1(t)+~v = ϕ 2(τ(t)). For this purpose, we consider the polynomialM(t) = M2(τ(t))(k3t + k4) r ∈ K[t]. Now, since τ(t) is a unit of K(t), and the roots of M2(t) are not in K, one gets that deg(M) = deg(M2) = r. Moreover, since γ is a root of M(t), and taking into account that M1(t) is the minimal polynomial of γ over K and that deg(M) = r = deg(M1), one has that there exists c ∈ K∗ such that M(t) = cM1(t). Now, in order to determine A and ~v, let us substitute τ(t) in the i-th component of ϕ2(t): τ(t)i M2(τ(t)) τ(t)i(k3t+ k4) M2(τ(t))(k3t+ k4)r (k1t + k2) i(k3t+ k4) cM1(t) Since numerator and denominator in the above rational function have the same degree, taking quotients and remainders, ϕ2(t) can be expressed as (ϕ2(τ(t)))i=1,...,r = (vi + ai,1 + · · ·+ ai,rt M1(t) )i=1,...,r, for some vi, ai,j ∈ K. Take A = (ai,j) and ~v = (vi). Then, A(ϕ1(t)) t + ~v = (ϕ2(τ(t)) t. Finally, let us see that A is regular. Indeed, suppose that A is singular and that there exists a non trivial linear relation λ1F1 + · · · + λrFr = ~0, where Fi denotes the i-th row of A. This implies that( M2(t) + · · ·+ λr M2(t) ◦ τ(t) = λ1v1 + · · · + λrvr is constant, which is impossible because λ1+···+λrt M2(t) is not constant and τ(t) is a unit of K(t). For two true circles, there is always a real affine transformation relating them. We have seen that this is not the case of hypercircles. However, for algebraic extensions of degree 2 (where the circle case fits), we recover this property for hypercircles that are not lines. Corollary 3.4. Let K(α) be an extension of degree 2. Then all α-hypercircles, that are not lines, are affinely equivalent over K. Proof. By Lemma 2.5, we may assume that the hypercircles are associated to units of the form 1 . Now, we consider two α-hypercircles not being lines, namely, let Ui be the α-hypercircle associated to for i = 1, 2, and di 6∈ K. Let di = λi + µiα, with λi, µi ∈ K and µi 6= 0. Then, the unit τ(t) = τ0 + τ1t ∈ K[t] where τ0 = µ2λ1−µ1λ2 and τ1 = , verifies that τ(−d1) = −d2. By Theorem 3.3, U1 and U2 are affinely equivalent over K. In Corollary 3.2 we have seen that the degree of a hypercircle is given by the degree of the field extension provided by the pole of any non polynomial generating unit. Lines are curves of degree one, a particular case of this phenomenon. Now, we consider other kind of hypercircles of degree smaller than n. This motivates the following concept. Definition 3.5. Let U be an α-hypercircle. If the degree of U is [K(α) : K], we say that it is a primitive hypercircle. Otherwise, we say that U is a non- primitive hypercircle. Regarding the complex numbers as an extension of the reals, lines may be considered as circles when we define them through a Moebius transforma- tion. Lines are the only one curves among these such that its degree is not [C : R]. The situation is more complicated in the general case. Apart from lines, which have been thoroughly studied in Theorem 2.6, there are other non-primitive hypercircles. This is not a big challenge because, as we will see, non-primitive hypercircles are primitive on another extension. Moreover, these cases reflect some algebraic aspects of the extension K ⊆ K(α) = L in the geometry of the hypercircles. Actually, we will see that there is a cor- respondence between non-primitive hypercircles and the intermediate fields of K ⊆ L. More precisely, let U be a non-primitive hypercircle associated to u(t) = 1 , where r = [K(d) : K] < [L : K] = n. In this case, we have the algebraic extensions K ⊆ K(d) ( L. We may consider u(t) as a unit either in the extension K ⊆ K(d) with primitive element d or in K(d) ( L with primitive element α. In the first case, u(t) defines a primitive hypercircle in Fr. In the second case, as u(t) is a K(d) unit, it defines a line. The analysis of U can be reduced to the case of the primitive hypercircle associated to u(t) in the extension K ⊆ K(d). Theorem 3.6. Let U be the non-primitive hypercircle associated to u(t) = ∈ K(α)(t). Let V be the hypercircle generated by the unit 1 in the extension K ⊆ K(d). Then, there is an affine inclusion from Fr to Fn, defined over K, that maps the hypercircle V onto U . Proof. Taking into account Lemma 2.5, we may assume that u(t) = 1 . Let φ(t) = (φ0(t), . . . , φn−1(t)) ∈ K(t) n be the parametrization of U , obtained from u(t), with respect to the basis B = {1, α, . . . , αn−1}. Similarly, let ψ(t) = (ψ0(t), . . . , ψr−1(t)) ∈ K r(t) be the parametrization of the hypercircle V, associated to u(t), with respect to the basis B⋆ = {1, d, . . . , dr−1}, where r = [K(d) : K]. The matrix D = (dji) ∈ Mn×r(K) whose columns are the co- ordinates of di with respect to B induces a K-linear transformation χ : Fr 7→ Fn that maps V onto U . Indeed, as u(t) = i=0 ψi(t)d j=0 φj(t)α j, one has that ψi(t)d ψi(t) dj,iα dj,iψi(t) φj(t)α Then φ(t)t = D ψ(t)t. Moreover, χ is one to one, because rank(D) = r. As a consequence of this theorem, every hypercircle is affinely equivalent, over K, to a primitive hypercircle. Therefore, the study of hypercircles can be reduced to the study of primitives hypercircles. 4 Properties at Infinity of a Hypercircle Circles have a very particular structure at infinity, namely, they pass through the cyclic points, i.e. [±i : 1 : 0], which are related to the minimal polynomial defining the circle as a hypercircle as remarked in the introduction. In this section, we will see that a similar situation occurs for more general primitive hypercircles. More precisely, let U be the primitive hypercircle defined by the unit u(t) = at+b . By Corollary 3.2, U is a parametric affine curve of degree n. So, there are at most n different points in the hyperplane at infinity. Let φ(t) = (φ0(t), . . . , φn−1(t)) be the parametrization of U generated by u(t); recall that φi(t) = pi(t) . Thus, projective coordinates of the points attained by φ(t) are given by [p0(t) : · · · : pn−1(t) : M(t)]. Now, substituting t by every conjugate σ(−d) of −d, we obtain [p0(σ(−d)) : · · · : pn−1(σ(−d)) : 0] = [σ(p0(−d)) : · · · : σ(pn−1(−d)) : 0] We prove next that these points are the points of the hypercircle at infinity. Lemma 4.1. Let U be a primitive hypercircle associated to the unit u(t) = . The n points at infinity are Pj = [σj(p0(−d)) : · · · : σj(pn−1(−d)) : 0], 1 ≤ j ≤ n where σj are the K-automorphisms of the normal closure of L = K(α) over Proof. First of all, observe that gcd(p0, . . . , pn−1,M) = 1, and hence Pj are well defined. Moreover, pi(−d) 6= 0, for every i ∈ {0, . . . , n − 1}, since pi(t) ∈ K[t] is of degree at most n and, thus, if pi(−d) = 0, then pi(t) = c ∈ K and the hypercircle would be contained in a hyperplane. But this is impossible since U is primitive (see Corollary 3.2). It remains to prove that they are different points. Suppose that two different tuples define the same projective point. We may suppose that P1 = Pj. P1 verifies that i=0 pi(−d)α i = (−ad + b)m(−d) 6= 0 and Pj verifies that∑n−1 i=0 pi(σj(−d))α i = (aσj(−d) + b)m(σj(−d)) = 0. Thus, Pj is contained in the projective hyperplane i=0 α iXi = 0, but not P1. Hence, P1 6= Pj. Let us check that, as in the case of circles, the points at infinity of prim- itive α-hypercircles do not depend on the particular hypercircle. Theorem 4.2. For a fixed extension K ⊆ K(α) of degree n, the set of points at the infinity P = {P1, . . . , Pn} of any primitive hypercircle does not depend on the particular α-hypercircle U , but only on the algebraic extension and on the primitive element α. Moreover, the set P is characterized by the following property: {X0 + αjX1 + · · ·+ α j Xn−1 = 0} ∩ U = P \ {Pj}, where αj = σj(α) are the conjugates of α in F, 1 ≤ j ≤ n, and U is the projective closure of U . Proof. Let U be the primitive α-hypercircle generated be a unit u(t) = at+b U has the projective parametrization [p0(t) : · · · : pn−1(t) : M(t)]. Let Pj = [σj(p0(−d)) : · · · : σj(pn−1(−d)) : 0]. Its evaluation in the equation of hyperplane X0 + αkX1 + . . .+ α k Xn−1, yields: σj(pi(−d))α k = σk σ−1k ◦ σj(pi(−d))α (a(σ−1k ◦ σj(−d)) + b)m(σ k ◦ σj(−d)) If j = k, the previous expression equals σk ((−ad + b)m(−d)) 6= 0. If j 6= k, then σ−1k ◦ σj(−d) is a conjugate of −d, different from −d, because −d is a primitive element. So m(σ−1k ◦ σj(−d)) = 0. In order to show that this point does not depend on a particular hyper- circle, take the n hyperplanes X0 + αkX1 + · · ·+ α k Xn−1 = 0, k = 1 . . . n. Every point at infinity of a hypercircle is contained in exactly n− 1 of those hyperplanes. Also, any of these hyperplanes contains exactly n− 1 points at infinity of the hypercircle. One point at infinity may be computed by solving the linear system given by any combination of n−1 hyperplanes. The matrix of the linear system is a Vandermonde matrix, each row depending on the corresponding αk, so there is only one solution. Remark 4.3. Notice that this theorem provides a n-simplex combinatorial structure of the points at infinity of any primitive hypercircle. The following result shows that the points at infinity can be read directly from the minimal polynomial of α. Proposition 4.4. Let Mα(t) be the minimal polynomial of α over K. Let mα(t) = Mα(t) i ∈ K(α)[t], where ln−1 = 1. Then, the points at infinity of every primitive α-hypercircle are [l0 : l1 : · · · : ln−2 : ln−1 : 0] and its conjugates. Proof. We consider the symmetric polynomial r(x, y) = Mα(x)−Mα(y) . Substi- tuting (x, y) by (t, α) we obtain that r(t, α) = Mα(t)−Mα(α) Mα(t) = mα(t). That is, mα(t) is symmetric in t and α. Take now the hypercircle induced by the unit 1 mα(t) Mα(t) . By Lemma 4.1, we already know that one point at infinity is [p0(α) : · · · : pn−1(α) : 0], where mα(t) = pi(t)α i. By symmetry, i=0 pi(t)α i=0 pi(α)t i. That is, pi(α) = li. Thus, the points at infinity are [l0 : l1 : · · · : ln−2 : 1 : 0] and its conjugates. Next result deals with the tangents of a hypercircle at infinity, and it explains again why parabolas can not be hypercircles. Proposition 4.5. The tangents to a primitive hypercircle at the points at infinity are not contained in the hyperplane at infinity. Proof. Let U be the primitive α-hypercircle generated by at+b , and [p0(t) : · · · : pn−1(t) :M(t)] the projective parametrization generated by the unit. In the proof of Lemma 4.1, we have seen that pn−1(t) is not identically 0, because pn−1(−d) 6= 0. So, we can dehomogenize w.r.t. the variable Xn−1, obtaining the affine parametrization ( p0(t) pn−1(t) , . . . , pn−2(t) pn−1(t) pn−1(t) ) of U on another affine chart. We have to check that the tangents to the curve at the intersection points with the hyperplane Xn−1 = 0 are not contained in this hyperplane. The points of C in the hyperplane Xn−1 = 0 are obtained by substituting t by σ(−d). The last coordinate of the tangent vector is M ′(t)pn−1(t)−M(t)p n−1(t) pn−1(t)2 We evaluate this expression at σ(−d). M(σ(−d)) = 0 and, as all its roots are different in F, M ′(σ(−d)) 6= 0. We also know that σ(pn−1(−d)) 6= 0. Hence, the last coordinate of the tangent vector is non-zero. Thus, the tangent line is not contained in the hyperplane at infinity. Finally, we present a property of hypercircles that can be derived from the knowledge of its behavior at infinity. We remark a property of circles stating that given three different points in the plane, there is exactly one circle passing through them (which is a line if they are collinear). The result is straightforward if we recall that there is only one conic passing throught five points. In the case of circles, we have the two points at infinity already fixed, so, given three points in the affine plane there will only be a conic (indeed a circle if it passes through the cyclic points at infinity) through them. Even if hypercircles are curves in n-space, surprisingly, the same occurs for hypercircles. We are going to prove that, given 3 different points in Kn, there is ex- actly one hypercircle passing through them. If the points are not in general position, the resulting hypercircle needs not to be a primitive one. First, we need a lemma that states what are the points over K of the hypercircle that are reachable by the parametrization. Lemma 4.6. Let U be the α–hypercircle, non necessarily primitive, associ- ated to u(t) = at+b with induced parametrization Φ(t). Φ(K) = U ∩Kn \ {ā} with a = i=0 aiα i, ā = (a0, . . . , an−1). Proof. We already know that Φ(t) is proper and, obviously, Φ(K) ⊆ U ∩Kn, also, ā is not reachable by Φ(t), since otherwise one would have that a = u(λ) for some λ, and this implies that ad − b = 0, which is impossible since u(t) is a unit. In order to prove the other inclusion, write as before φi(t) = pi(t) where M(t) is the minimal polynomial of −d over K. Then, we consider the ideal I over F[t, X̄ ] generated by (p0(t)−X0M(t), . . . , pn−1(t)−Xn−1M(t)), where X̄ = (X0, . . . , Xn−1), and the ideal J = I + (ZM(t)− 1) ⊆ F[Z, t, X̄]. Let I1 be the first elimination ideal of I; i.e. I1 = I ∩ F[X̄ ] and let J2 be the second elimination ideal of J ; i.e. J2 = J ∩ F[X̄]. Observe that I ⊆ J and therefore I1 ⊆ J2. Note that U = V (J2); i.e. U is the variety defined by J2 over F. Thus U ⊆ V (I1). Now, let us take x̄ ∈ (U ∩ K n) \ {ā}. Then x̄ ∈ V (I1). Observe that, by construction, the leading coefficient of pi(t) − XiM(t) w.r.t. t is ai − Xi. Therefore, since x̄ 6= ā one has that at least one of the leading coefficients of the polynomials in I w.r.t. t does not vanish at x̄. Thus, applying the Extension Theorem (see Theorem 3, pp. 117 in [5]), there exists t0 ∈ F such that (t0, x̄) ∈ V (I). This implies that pi(t0) − xiM(t0) = 0 for i = 1 . . . n − 1. Let us see that M(t0) 6= 0. Indeed, if M(t0) = 0 then pi(t0) is also zero for every index and therefore gcd(p0(t), . . . , pn−1(t),M(t)) 6= 1, which is impossible. Hence Φ is defined at t0 and Φ(t0) = x̄. To end up, we only need to show that t0 ∈ K. For this purpose, we note that the inverse of Φ(t) is given by P (X̄) = i + b∑ Xiαi − a Now, since x̄ 6= ā one deduces that P (x̄) is well defined, and the only pa- rameter value generating x̄ is t0 = P (x̄). Hence, the gcd of the polynomi- als pi(t) − xiM(t) is a power of (t − t0). Thus, taking into account that pi,M ∈ K[t], one deduces that t0 ∈ K. Finally, it only remains to state that ā is generated when t takes the value of the infinity of K. But this follows taking Φ(1/t) and substituting by t = 0. Proposition 4.7. Let Xi = (Xi0, . . . , Xi,n−1) ∈ K n ⊆ Fn, 1 ≤ i ≤ 3 be three different points. Then, there exists only one α–hypercircle passing through them. Proof. Let Yi = j=0 Xijα j ∈ K(α), 1 ≤ i ≤ 3. Consider the following linear homogeneous system in a, b, c, d: b = Y1d, a+ b = Y2(c+ d), a = Y3c Observe that, if the three points are different, there is only one projective solution, namely [a : b : c : d] where a = Y1Y3 − Y3Y2, b = Y1Y2 − Y1Y3, c = Y1 − Y2, d = Y2 − Y3. Take the unit u(t) = at+b . It verifies that u(0) = Y1, u(1) = Y2, u(∞) = Y3. Then, the hypercircle associated to u passes through X1, X2, X3. In order to prove that this hypercircle is unique, let v be the unit associated to a hypercircle passing through the three points and ψ(t) the parametrization induced by v(t). By Lemma 4.6, as Xi ∈ K n, the point Xi is reached for a parameter value ti in K∪{∞}. So, there are three values t1, t2, t3 ∈ K∪{∞} such that v(ti) = Yi. Let τ(t) ∈ K(t) be the unique unit associated to the transformation of the projective line P(F) into itself given by τ(0) = t1, τ(1) = t2, τ(∞) = t3. Then v(τ(t)) = u(t) and both units represents the same hypercircle. 5 Parametrization and Implicitation of a Hy- percircle In this section, we will provide specific methods to parametrize and implic- itate hypercircles. These methods show the power of the rich structure of hypercircles, simplifying problems that are usually much harder in general. Given a unit u(t) defining U , it is immediate to obtain a parametrization of U (see Section 2). If U is given by implicit equations (as it is usually the case in Weil’s descente method), the next proposition shows how to parametrize it. Proposition 5.1. The pencil of hyperplanes X0+X1α+ · · ·+Xn−1α n−1 = t parametrizes the primitive α–hypercircle U . Proof. Let I be the implicit ideal of U . Note that, since U is K−rational it is K-definable, and hence a set of generators of I can be taken inK[X0, . . . , Xn−1]. Let u(t) be any unit associated with U and (φ0(t), . . . , φn−1(t)) the induced parametrization. Let v(t) be the inverse unit of u(t), u(v(t)) = v(u(t)) = t. Then (φ0(v(t)), . . . , φn−1(v(t))) = (ψ0(t), . . . , ψn−1(t)) = Ψ(t) is another parametrization of U which is no more defined over K but over K(α). The later parametrization is in standard form [10], that is ψi(t)α φi(t)α ◦ v(t) = u ◦ v(t) = t. This implies that the pencil of hyperplanes Ht ≡ X0+X1α+· · ·+Xn−1α n−1−t parametrizes U . Indeed, if Ψ(t) is defined, Ht ∩U consists in n− 1 points at infinity of U (Theorem 4.2) and Ψ(t) itself. We deduce that ψi(t)−Xi belongs to the ideal I +Ht, which has a set of generators in K(α)(t)[X0, . . . , Xn−1]. So, the parametrization Ψ(t) can be computed from I. Notice that the obtained parametrization Ψ(t) has coefficients over K(α). Thus, it is not the parametrization induced by any associated unit u(t). The interest of obtaining a unit associated to a hypercircle is that it helps us to solve the problem of reparametrizing a curve over an optimal field extension of K, see [4]. There, it is shown that given a parametrization Ψ(t) ∈ K(α)r of a curve there is a hypercircle associated to it. Any unit associated to the hypercircle reparametrizes the original curve over K. To get a parametrization φ(t) over K or, equivalently, a unit u(t) associated to U , we refer to [10]. In addition, note that the proof of Proposition 4.7 shows how to construct a unit associated to a hypercircle, when points over K are known, and therefore a parametrization of it. The inverse problem, computing implicit equations of a hypercircle from the parametrization induced by an associated unit, can be performed using classic implicitation methods. However, the special structure of hypercircles provides specific methods that might be more convenient. Proposition 5.2. Let U be a hypercircle associated to the unit u(t), and let v(t) be the inverse of u(t). Let ri(X0, . . . , Xn−1) s(X0, . . . , Xn−1) where ri, s ∈ K[X0, . . . , Xn−1]. Then, the ideal of U is the elimination ideal with respect to Z: I(U) = (r1(X̄), . . . , rn(X̄), s(X̄)Z − 1) ∩ F[X0, . . . , Xn−1]. Proof. Let u(t) = at+b , then v(t) = −dt+b . Now, consider ξi(X0, . . . , Xn−1)α ηi(X0, . . . , Xn−1)α where ξi, ηj ∈ K(X0, . . . , Xn−1) and ηi = ri(X0,...,Xn−1) s(X0,...,Xn−1) . The map ξ : Fn −→ Fn, ξ = (ξ0, . . . , ξn−1) is birational and its inverse is η = (η0, . . . , ηn−1). Indeed: ηi(ξ0(X̄), . . . , ξn−1(X̄))α i = v αjξj(X̄) is an equality in K(α)(X0, . . . , Xn−1). We deduce that ξ0(X0, . . . , Xn−1), . . . , ξn−1(X0, . . . , Xn−1) It is clear that U is the image of the line L ≡ {X1 = 0, . . . , Xn−1 = 0} under the map ξ, U = ξ(L). The set of points where ξ is not defined is the union of the hyperplanes i=0 σj(α) iXi + σj(d) = 0, 1 ≤ j ≤ n. The intersection of these hyperplanes with L is the set of points (−σ(d)j, 0, . . . , 0), 1 ≤ j ≤ n. Thus, for a generic p ∈ L, ξ(p) is defined and belongs to U . The result is similar for the inverse map η. The set of points where η is not defined is the union of the hyperplanes i=0 σj(α) iXi − σj(a) = 0, 1 ≤ j ≤ n. These n hyperplanes intersect U in at most one affine point, see Proposition 5.1. So, for a generic p ∈ U , η(p) is again defined and belongs to L. Let us compute now the points X̄ such that η(X̄) is defined, but it does not belong to the domain of ξ. If X̄ is such a point, then σj(α) iηi(X̄) + σj(d) = 0. As ηi is defined over K, applying σj to the definition of η, we obtain that σj(v) σj(α) = −σj(d) But σj(v) = −σj(d)t+σj (b) t−σj(a) . It follows from Lemma 4.6 that the value −σj(d) cannot be reached, even in F. Thus, the image of η is contained in the domain of ξ. We are ready to prove the theorem, by verifying that the set U \ {s = 0}, which is just eliminating a finite number of points in U , is the set of points X̄ such that ri(X̄) = 0, i ≥ 1 and s(X̄) 6= 0. If X̄ ∈ U\{s = 0}, then η is defined and η(X̄) = (η0(X̄), 0, . . . , 0). Hence ηi(X̄) = ri(X̄) = 0. Conversely, if X̄ is a point such that ri(X̄) = 0 and s(X̄) 6= 0, then η(X̄) is defined and belongs to L. It is proven that ξ is defined in η(X̄), so X̄ = ξ(η(X̄)) ∈ ξ(L) = U . The thesis of the theorem follows taking the Zariski closure of U \ {s = 0}. This method to compute the implicit equations of U is not free from elimination techniques, as it has to eliminate the variable Z. However, it has the advantage that it yields already an ideal in F[X0, . . . , Xn−1] defined over K and such that it describes a non trivial variety containing the hypercircle. Namely, (r1(X̄), . . . , rn−1(X̄)) are polynomials over K whose zero set contains the hypercircle. The following example shows that the elimination step is necessary in some cases. Example 5.3. Let Q ⊆ Q(α) be the algebraic extension defined by α3+α2− 3 = 0. Let us consider the unit u(t) = (2+α)t+α t+1−α . Its inverse is v(t) = (α−1)t+α t−2−α A parametrization of U is φ(t) = 2t3 + 6t2 + 7t+ 3 t3 + 4t2 + 5t− 1 t3 + 6t2 + 9t+ 2 t3 + 4t2 + 5t− 1 t2 + 4t+ 1 t3 + 4t2 + 5t− 1 A Gröbner basis of the ideal of the curve is I := {x21 − x2x0 − x2x1 − x1 + x2, x0x1 − x2x0 − 3x 2 − 2x1 + 4x2, x20 − 3x2x1 − 2x0 + 2x1 + 3x2 − 2}. Then, proposition 5.2 states that this ideal is I = (r1(x0, x1, x2), r2(x0, x1, x2), s(x0, x1, x2)Z − 1) ∩ F[x0, x1, x2] where r1 = 2 − 8x2 + 4x2x0 + 6x 2x0 + 17x2x1 + x2x 0 + 3x1 − 3x 1x2 + x 0 − x 0x1 + 4x0x1 − 12x 2 − 8x 1 + 9x 2 + 3x 1 − 3x 0 − 9x0x1x2, r2 = −2 − 7x2 + 4x2x0 − x2x1 + 8x1 − 2x0 − 2x0x1 + 6x 2 − 2x 1 + x s = 9x32 + 6x 2x0 − 12x 2 + 5x2x0 − 17x2 − 3x 1x2 − 9x0x1x2 + x2x 0 + 24x2x1 + 3x31 + 8x0 + 4x0x1 − 5x 0 − x 0x1 + 5x1 − 9x 1 − 7 + x But, if we take J = (r1, r2), then J ( I. The saturation of J with respect to I is J : I∞ = (x21 − x0x2 − x1x2 − 2x1 + 3x2 + 1, x0x1 − x0x2 − 3x 2 − x0 − 2x1 + 2x2 + 2, x 0 − 3x1x2 − 4x0 + 3x2 + 4) This ideal corresponds to the union of the line −αx0 +3x2 = −2α (α + α2)x0 −3x1 = −3 + 2α + 2α and its conjugates. Next theorem shows an alternative method to implicitate a hypercircle without using any elimination techniques. It is based on properties of the normal rational curve of degree n. Theorem 5.4. Let ϕ(t) = ( q0(t) , . . . , qn−1(t) ) be a proper parametrization of a primitive hypercircle U with coefficients in F. Let I be the homogeneous ideal of the rational normal curve of degree n in P(F) given by a set of homogeneous generators h1(Ȳ ), . . . , hr(Ȳ ). Let Q ∈ Mn+1×n+1(F) be the matrix that carries {q0(t), . . . , qn−1(t), N(t)} onto {1, t, . . . , t n}. Let fi(X̄) = hi Q0jXj , . . . , QnjXj , 1 ≤ i ≤ r. Then {f1, . . . , fr} is a set of generators of the homogeneous ideal of U . Proof. If the parametrization is proper, {q0(t), . . . , qn−1(t), N(t)} is a basis of the polynomials of degree at most n. This follows from the fact shown in Corollary 3.2 that a primitive hypercircle is not contained in any hyperplane. Note that a projective point X̄ belongs to U if and only if Q(X̄) belongs to the rational normal curve, if and only if hi(Q(X̄)) = 0, 1 ≤ i ≤ r. Remark 5.5. • It is well known that the set of polynomials {YiYj−1−Yi−1Yj | 1 ≤ i, j ≤ n} is a generator set of I (see [6]). • Notice that it is straightforward to compute Q from the parametrization. Therefore, we have an effective method to compute the implicit ideal of the projective closure of U . The affine ideal of U can be obtained by dehomogenization Xn = 1. • If the parametrization is given by polynomials over an algebraic exten- sion K(β) of K, then the coefficients of fi belongs to K(β). Moreover, if we write fi(X̄) = j=0 fij(X̄)β j, with fij ∈ K[X̄ ], then, {fij} is a set of generators over K of the hypercircle U . • In practice, this method is much more suited to compute an implicita- tion of a hypercircle than the method presented in Proposition 5.2. Example 5.6. The implicit equations of a hypercircle can be computed by classical implizitation methods, for example Gröbner basis or with the two methods presented in Proposition 5.2 and Theorem 5.4. Here, we present two cases that show the practical behavior of these methods. The first example considers the algebraic extension Q ⊆ Q(α), where α4 + α2 − 3 and the unit (1−α3)t+α2 t+1+2α−3α2 . The parametrization of the hypercircle is given by t4 + 15t3 + 22t2 + 101t− 195 t4 + 10t3 − 17t2 − 366t+ 233 , φ1 = −11t3 − 73t2 + 65t− 114 t4 + 10t3 − 17t2 − 366t+ 233 2t3 + 57t2 − 25t− 59 t4 + 10t3 − 17t2 − 366t+ 233 , φ3 = −t4 − 6t3 + 4t2 + 17t− 56 t4 + 10t3 − 17t2 − 366t+ 233 The second example starts from the extension Q ⊆ Q(β), where β is such that β4+3β+1 = 0. Here, the unit defining U is u = (1+β−β2)t+1+β3 t+1+β2−β3 and the parametrization induced by u(t) is t4 + 11t3 + 47t2 + 95t+ 72 t4 + 13t3 + 62t2 + 126t+ 81 , ψ1 = t4 + 7t3 + 15t2 + 17t+ 9 t4 + 13t3 + 62t2 + 126t+ 81 −t4 − 10t3 − 31t2 − 23t t4 + 13t3 + 62t2 + 126t+ 81 , ψ3 = t3 + 13t2 + 42t+ 36 t4 + 13t3 + 62t2 + 126t+ 81 The running times for computing the implicit ideal (using a Mac Xserver with 2 processors G5 2.3 GHz, 2 Gb RAM Maple 10) are Example 1 Example 2 Gröbner basis method 0.411 0.332 Proposition 5.2 2.094 2.142 Theorem 5.4 0.059 0.021 We refer the interested reader to [11] for a brief discussion and compari- son of the running times of these algorithms. 6 Characterization of Hypercircles In the introduction, we defined algebraically a circle as the conic such that its homogeneous part is x2 + y2 and contains an infinite number of real points. The condition on the homogeneous part is equivalent to impose that the curve passes through the points at infinity [±i : 1 : 0]. Analogously, hypercircles are regular curves of degree n with infinite points over the base field passing through the points at infinity described in Theorem 4.2. The following result shows that this is a characterization of these curves. Theorem 6.1. Let U ⊆ Fn be an algebraic set of degree n such that all whose components are of dimension 1. Then, it is a primitive α-hypercircle if and only if it has an infinite number of points with coordinates in K and passes through the set of points at infinity characterized in Theorem 4.2. Proof. The only if implication is trivial. For the other one, let U ⊆ Fn be an algebraic set of pure dimension 1 and degree n passing through P = {P1, . . . , Pn}, the n points at infinity of a primitive α-hypercircle. Suposse that U has infinite points with coordinates in K. Then, we are going to prove that U is irreducible. Let W be an irreducible component of U with infinite points in K. Note that, since W is irreducible and contains infinitely many points over K, the ideal I(W) over F is generated by polynomials over K (see Lemma 2 in [3]). Let q be any point at infinity of W; then q ∈ P . As W is K-definable it follows that W also contains all conjugates of q. Thus, P is contained in the set of points at infinity of W. It follows that W is of degree at least n; since W ⊆ U , U = W. Therefore, U is irreducible and I(U) is generated by polynomials with coefficients over K. Now, consider the pencil of hyperplanes Ht ≡ X0+X1α+· · ·+Xn−1α n−1−t, where t takes values in F. Notice that Ht ∩ P = {P2, . . . , Pn}. Thus, P1 ∈ U \Ht so, for all t, U 6⊆ Ht. Moreover, for every point p = (p0, . . . , pn−1) ∈ U , t(p) = i=0 piα i ∈ F is such thatH t(p)∩U = {p, P2, . . . , Pn}. The cardinal of {t(p) | t ∈ U} is infinite, since otherwise, by the irreducibility of U , it would imply that there is a t0 such that U ⊆ Ht0 , which is impossible. So, for generic t, the intersection is H t ∩ U = {p(t), P2, . . . , Pn}. Let us check that the coordinates of p(t) are rational functions in K(α)(t). Take the ideal I(U) of U . The ideal of p(t) (as a point in F(t)n) is I +Ht, defined over K(α)(t). The reduced Gröbner basis of the radical I +Ht is of this kind (X0 − ψ0, . . . , Xn−1 − ψn−1) and it is also defined over K(α)(t)[X0, . . . , Xn−1]. Hence, (ψ0, . . . , ψn−1) is a K(α)- parametrization of U . Thus, since U is irreducible, it is rational. Moreover∑n−1 i=0 (ψi(t))α i = t and the parametrization is proper. As the curve is rational and has an infinite number of points over K, it is parametrizable over K (it follows, for example from the results in [14]). Let u(t) be a unit such that Ψ◦u(t) = (φ0(t), . . . , φn−1(t)) is a parametrization over K, where φi(t) ∈ K(t) i=0 φi(t)α i = u(t). We conclude that U is the hypercircle associated to the unit u(t). Remark that a parametric curve, definable over K and with a regular point over K, is parametrizable over the same field; for this, it is enough to K-birationally project the curve over a plane, such that the K-regular point stays regular on the projection, and then apply the results in [14]. Then, a small modification of the proof above, yields the following: Theorem 6.2. Let U ⊆ Fn be a 1-dimensional irreducible algebraic set of degree n, definable over K . Then, it is a primitive α-hypercircle if and only if it has a regular point with coordinates in K and passes through the set of points at infinity characterized in Theorem 4.2. 7 An Application As mentioned in the introduction, hypercircles play an important role in the problem of the optimal-algebraic reparametrization of a rational curve (see [3], [4], [10] [12], [13] for further details). Roughly speaking, the problem is as follows. Given a rational K–definable curve C by means of a proper rational parametrization over K(α), decide whether C can be parametrized over K and, in the affirmative case, find a change of parameter transforming the original parametrization into a parametrization over K. In [4], a K– definable algebraic variety in Fn, where n = [K(α) : K], is associated to C. This variety is called the associated Weil (descente) parametric variety. In [4], it is proved that this Weil variety has exactly one one-dimensional component iff C is K–definable (which is our case) and, in this case, C can be parametrized over K iff this one-dimensional component is a hypercircle. Moreover, if it is a hypercircle a proper rational parametrization over K of the hypercircle generates the change of parameter one is looking for; namely its generating unit. In the following example, we illustrate how to use the knowledge of the geometry of hypercircles to help solving the problem. Suppose given the parametric curve C ≃ (η1(t), η2(t)) = (−2t4 − 2t3)α− 2t4 6α2t2 + (4t3 − 2)α+ t4 − 8t −2t4α 6α2t2 + (4t3 − 2)α + t4 − 8t where α is algebraic over Q with minimal polynomial x3 + 2. We follow Weil’s descente method presented in [4] to associate a hypercircle to C. The method consists in writing ηi( j=0 tjα qij(t0,t1,t2) N(t0,t1,t2) . In this situation C is Q−definable if and only if V = V (q11, q12, q21, q22) \ V (N) is of dimension 1. Moreover, C is Q-parametrizable if and only if the one- dimensional component of V is an α-hypercircle. For this example, the equa- tions of V are: V = V (2t30t2 − 4t 2 + 3t 1 + 2t 1t2 + 2t0t 2 + 2t 1t2 − t 0t1 + 6t0t1t 2,−6t 0t1t2 + t40+2t0t 1−8t0t 2−2t0t 0t2−4t1t 2−12t 2, 12t 1−9t0t1t 2−4t0t 2t20t1t2 + 4t 2 − 4t0t 2, 9t0t 2 − 9t 2 − 2t 0t2 − 2t 1t2 + 6t0t1t 2 − 2t 2 + t 0t1 − 2t21t2−2t0t 2, 6t 2+12t 0t1−2t0t 1t2−2t 2+8t1t 2, 6t 2+9t0t1t 0t1t2+4t 2+8t0t 2, 18t2t 1+36t 2t1+14t 0t2+32t 1t2+12t0t1t 7t20t1+14t 1t2+14t0t 2, 6t0t 1t2+2t0t 1t2+ t 0t1+2t 2− 8t1t 2+12t 2t0, 9t 0t2t1− 36t42t1 − 4t 0t2 − 4t 1t2 + 12t0t1t 2 − 4t 2 + 2t 0t1 − 4t 1t2 − 4t0t 2, 6t 1 + 48t 36t42t0−11t 0t1+6t 1+14t0t 1t2−22t 2+64t1t 2, 3t 1t0+6t0t1t 2+2t0t 0t1t2− 2t21t 2 + 2t0t 2, 27t 1 − 27t0t 2 − 9t 2 + 9t 2t1 − 2t 0t2 − 2t 1t2 + 6t0t1t 2 − 2t t20t1 − 2t 1t2 − 2t0t 2, 6t 0 + 12t 2t1 − 5t0t1t 2 + 2t 2, t0t 2t1 + 2t Thus the main point is to verify that this curve is a hypercircle. If V is a hypercircle, then its points at infinity must be as in Theorem 6.1. So, let us first of all check whether this is the case. The set of generators of the defining ideal form a Gröbner basis with respect to a graded order, thus to compute the points at infinity we take the set of leading forms of these polynomials. Leading forms= {t40 − 2t0t 1 − 6t 0t1t2 − 12t 2 − 8t0t 2, 2t 0t2 − 4t 2 +3t 2t31t2 + 6t0t1t 2, 9t0t 2 − 9t 2, 12t 1 − 9t0t1t 2 + 6t 2, 6t 2 + 12t 2, 6t 9t0t1t 2, 18t2t 1+36t 2t1, t0t 2t1+2t 2, 6t0t 1t2+12t 2t0, 9t 0t2t1−36t 2t1, 6t 48t21t 2 − 36t 2t0, 3t 1t0 + 6t0t1t 2, 27t 1 − 27t0t 2, 6t 0 + 12t The solutions of this system, after dehomogenizing {t2 = 1}, are t0 = t21, t 1 + 2 = 0. That is, the points at infinity are of the form [α i : αi : 1 : 0], = x2 + αx + α2. Thus, by Proposition 4.4, the points at infinity of V remind those of an α-hypercircle. Now, following Proposition 5.1, we may try to parametrize V by the pencil of hyperplanes t0 + αt1 + α 2t2 − t. Doing so, we obtain the parametrization (α2 + 2αt+ t2)t 3αt+ α2 + 3t2 −1/2α2t3 3αt+ α2 + 3t2 −1/2αt2(t+ α) 3αt+ α2 + 3t2 Remark that this parametrization can also be computed by means of inverse computation techniques as described in [13]. Then, by direct computation, we observe that the parametric irreducible curve defined by this parametrization is of degree 3, passes through the point (0, 0, 0) and this point is regular. Moreover, it is Q-definable, since it is the only 1-dimensional component of V (see [4]), which is, by construction, a Q-definable variety. It follows from 6.2 that it is a hypercircle. Then, from this parametrization, the algorithm presented in [10] com- putes a unit u(t) = 2 2t+α2 associated to V. So, V is the hypercircle associated to u(t) and C is parametrizable over Q. In particular, the parametrization of V associated to u(t) is 2t3+1 2t3+1 2t3+1 . Moreover, the unit u(t) gives the change of parameter we need to compute a parametrization of C over the base field (see [4]), namely: η (u(t)) = References [1] Alonso C., Gutierrez J. and Recio T. (1994). A Rational Function De- composition Algorithm by Near-Separated Polynomials. Journal of Sym- bolic Computation vol. 19 pp. 527-544. [2] Andradas C. and Recio T. (2006). Plotting Missing Points and Branches of Real Parametric Curves. Applicable Algebra in Engineering, Commu- nication and Computing. Special issue on Algebraic Curves (To appear) [3] Andradas C., Recio T. and Sendra J.R. (1997). Relatively Optimal Reparametrization Algorithm Through Canonical Divisors. Proc. ISSAC 1997, ACM Press pp. 349-356. [4] Andradas C., Recio T. and Sendra J.R. (1999). Base Field Restriction Techniques for Parametric Curves. Proc. ISSAC 1999, ACM Press pp. 17-22. [5] Cox D., Little J. and O’Shea D. (1997). Ideals, Varieties, and Algo- rithms. Springer-Verlag. [6] Harris J. (1992). Algebraic Geometry, a First Course. Springer-Verlag. [7] Hartshorne R. (1977). Algebraic Geometry. Graduate Texts in Mathe- matics, no. 52. Springer-Verlag. [8] Manocha D. and Canny J.F. (1991). Rational Curves with Polynomial Parametrizations. Computer Aided Design vol 23, no. 9 pp. 645-652. [9] Recio T. and Sendra J.R. (1997). Real Reparametrizations of Real Curves. Journal of Symbolic Computation vol. 23 pp. 241-254. [10] Recio T., Sendra J.R. and Villarino C. (2004). From Hypercircles to Units. Proc. ISSAC 2004, ACM Pres pp. 258-265 [11] Recio T., Sendra J.R., Tabera L.F. and Villarino C. (2006). Fast Com- putation of the Implicit Ideal of a Hypercircle Proc. Algebraic Geometry and Geometric Modeling pp. 113-115. [12] Sendra J.R. and Villarino C. (2001). Optimal Reparametriztion of Poly- nomial Algebraic Curves. International Journal of Computational Ge- ometry and Applications vol. 11/4 pp. 439–453. [13] Sendra J.R. and Villarino C. (2002). Algebraically Optimal Reparametrizations of Quasi-Polynomial Algebraic Curves. Journal of Algebra and its Applications vol. 1/1 pp. 51–74. [14] Sendra J.R. and Winkler F. (1991). Symbolic Parametrization of Curves. J. Symbolic Computation 12/6, 607-631 [15] Walker R. J. (1950). Algebraic Curves. Princeton University Press, Princeton. [16] Weil A. (1959). Adeles et Groupes Algebriques. Seminaire Bourbaki vol. [17] http://www.algebra.uni-linz.ac.at/Nearrings/ http://www.algebra.uni-linz.ac.at/Nearrings/ Introduction Definition and First Properties Main Geometric Properties. Properties at Infinity of a Hypercircle Parametrization and Implicitation of a Hypercircle Characterization of Hypercircles An Application

0704.1385

Decreasing families of dynamically determined intervals in the power-law family

7 Decreasing families of dynamically determined intervals in the power-law family Waldemar Pa luba∗ Institute of Mathematics Warsaw University Banacha 2 02-097 Warsaw, Poland e-mail: [email protected] Abstract We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family x 7→ −|x|α+a. The critical order α is an arbitrary real number α > 1. The range of the parameter a is confined to an interval (1, aα) of length depending on the critical order. We prove that in every power-law family there is a unique parameter pα corresponding to the kneading sequence RLRRRLRC. Subsequently, we obtain monotonicity results concerning ratios of all intervals labeled by infinite post-critical orbit in the case of the kneading sequence RLRL... This extends the results from [9], via refinement of the tools based on special properties of power-law mappings in non-euclidean metric. Mathematics Subject Classification (2000): Primary 37D05. 1 Introduction In this paper we continue our work done in [9] on families of unimodal quasi- quadratic maps of the form fa(x) = −|x| α + a, with a real parameter a and an arbitrary – in general non-integer – fixed exponent α > 1 . ∗Partially supported by a KBN grant no. 2 PO3A 010 22. http://arxiv.org/abs/0704.1385v1 The problem of monotone behaviour of the dynamics in such a family has been first successfully solved for the strictly quadratic case α = 2 . The tools initially developed for the quadratic case (see e.g [2], [6], [10], also [11], and an independent attempt, partly relying on numerical evidence in [1]) were broadly generalized in the work of Kozlovski-Shen-van Strien, see [3]. There are also very interesting recent results by G.Levin, concerning uniqueness of appearance of periodic orbits of given multiplier in the quadratic family z2 + c. Not only was he able to give a simple proof of Douady-Hubbard- Sullivan theorem (cf.[4]), but he could continue somewhat beyond the hyper- bolic domains in the Mandelbrot set also, see [5]. In this work, we focus on questions closely related to these of Levin’s, though only orbits of periods 2 or 4 appear here. In return, working with real variable tools, we can do all critical degrees, integer or non-integer, indiscriminately. Despite of a great deal of progress achieved in the aforementioned papers, and in other works as well, virtually all those developments are inherently limited to the case of integer critical degrees. Non-integers clearly require a fresh and different approach. For any real number α > 1 the power-law map x 7→ |x|α has negative Schwarzian derivative, and hence it expands the non-euclidean lengths. This observation has long become one of the key tools in one-dimensional dynamics. However, the power-law is not just a negative Schwarzian map. It is a homogeneous map, and in the Poincaré metric with the element dt on the positive half-line (0,∞), it is nothing but a linear map acting as multiplication by the coefficient α, once we set the origin of the Poincaré coordinates at 1. This simple fact is rather hard to make use of in a direct way, but carries some strong consequences that can be applied in a dynamical setting. In our previous paper on this subject (see [9]), we introduced the tech- nique of indirect use of linearity of the power-law map in the non-euclidean metric and exemplified its usefulness in dynamics. There, we studied maps in the one-parameter quasi-quadratic family fa with the kneading sequence RRR . . . , that is for the value of the parameter a smaller than 1 . For the infinite decreasing family of intervals with endpoints labeled by the succes- sive points of the post-critical orbit we proved that the ratio of any two such intervals is a function monotone in parameter a . This means, we studied the situation which arises before the orbit of the critical point becomes a super- stable orbit of period 2 . It is clear that this period 2 super-sink situation arises only once in our family. In the current work, we further develop our tools in order to examine the case of some parameters greater than 1, where the length of the interval (1, aα) to which those parameters are confined depends on the critical order α > 1 . In particular we shall be able to deal with kneading sequences of the form RLRL . . . , proving monotonicity of the ratios, in the respective decreas- ing families, of intervals delimited by the post-critical orbit. As a step in the build-up of the above techniques, we shall also establish uniqueness of the period 8 super-sink, corresponding to the kneading sequence RLRRRLRC, in every power-law family (even when it does not admit a holomorphic ex- tension!); uniqueness of the period 4 super-sink RLRC is elementary and follows along the way. 2 Notation and preliminaries To begin with, we set some notation in conformance with that of [9]. The names non-euclidean and Poincaré we be used interchangeably. The Poincaré coordinate of a point x in an oriented, open interval (p, q) will be denoted by pp,q(x) = ln q − x and respectively pp,∞(x) = ln(x− p); also pp,−∞(x) = ln |x− p|. To single out the non-euclidean metric on the half-line, which turns the mapping h into a linear map, we will coin the term nonlinearity of an interval for the length of this interval measured in the Poincaré metric on (0,∞). Under this convention, the integral of nonlinearity of h over an interval (p, q) equals, up to a multiplicative constant, to the nonlinearity of the domain of integration. Given an orientation preserving homeomorphism ϕ : (p, q) → (r, s) we shall observe the ‘bar’ notation for its counterpart in the non-euclidean co- ordinates, i.e. the mapping ϕp,q : R → R defined by the formula ϕp,q(t) = pr,s p−1p,q(t) The non-euclidean push of ϕ at a point x ∈ (p, q) is, by definition, the quantity pr,s (ϕ(x)) − pp,q(x) . By the strength of a push we mean its absolute value. By ϕ+p,q and ϕ p,q we denote the finite or infinite limits ϕ+p,q = lim (ϕ(t) − t) ϕ−p,q = lim (ϕ(t) − t) , provided they exist. When ϕ is the restriction of the homogeneous map h(x) = xα to an inter- val (p, q) ⊆ (0,∞) we shall always put h or h in place of ϕ or ϕ respectively. For a fixed exponent α > 1, let fa = −|x| α + a , and the successive points of the orbit of the critical point will simply be denoted by na = f a (0) . Moreover, homogeneity of the power-law map allows for the linear change of coordinates, na 7→ na/1a , so that we can set 0a = 0, 1a = 1 and the dependence on the parameter a turns into the dependence on the value of 2a in these new coordinates. For a > 1 this rescaled value of 2a is in the interval (−1, 0) – so long as the post-critical orbit does not escape to infinity – and the quantity p0,1(|2a|), which for obvious reason we will denote by ā , is increasing simultaneously with a. Throughout this work, this very quantity will be chosen as our new parameter, and it is always tacitly assumed that the rescaling na 7→ na/1a has been done. We now record several observations concerning one-dimensional non-euclidean coordinates. Below, they are stated as propositions, verifiable by elementary calculations derived directly from the definition of the Poincaré metric. Proposition 2.1 For any x ∈ (−∞, 0) the following two Poincaré coordi- nates coincide p0,−∞(x) = px,1(0). Proof. We have p0,−∞(x) = ln (−x) = ln = px,1(0). � Proposition 2.2 For any x ∈ (−1, 0) the following two Poincaré coordi- nates coincide p1,−1(x) = px,−x p−1x,1 (p0,−1(x)) Proof. The identity in question is tantamount to ln x−1 = ln c−x 1 + x x + c , (2.1) where c = p−1x,1(p0,−1(x)), i.e. px,1(c) = p0,−1(x). But this last equality means , and further c = x2, so that (2.1) follows. � Given a point x ∈ (−∞, 0), we then pick a point y ∈ (x, 0). We shall let the point x vary, by which we mean a choice of another point x̃ ∈ (−∞, 0). The discrepancy in the non-euclidean coordinate will be denoted by ∆ϑ = p0,−∞(x̃) − p0,−∞(x). A broader version of Proposition 2.1 is the following Proposition 2.3 In the above notation we have px̃,0 p−1x̃,1 (px,1(y) + ∆ϑ) − px,0(y) = p1,−∞(x̃) − p1,−∞(x). Proof. We have ∆ϑ = ln x̃ and p1,−∞(x̃) − p1,−∞(x) = ln . Denote c = p−1x̃,1(px,1(y) + ∆ϑ), a point characterized by c− x̃ 1 − c y − x 1 − y . (2.2) We will be done once we show c−x̃ = 1−x̃ c− x̃ y − x 1 − x 1 − x̃ . (2.3) From (2.2) we get x̃ = 1−y and (2.3) can now be checked im- mediately. � Proposition 2.1 is what we get of Proposition 2.3, in place of subtracting two infinite terms, when we set y = 0. We generalize Proposition 2.2 in a similar way. Suppose we are given a point x ∈ (−1, 0), and a point y ∈ (x,−x). Again, we let the point x vary by choosing a new point x̃ ∈ (−1, 0). The discrepancies in the appropriate Poincaré coordinates of the two points will be denoted by ∆t = p1,−1(x̃) − p1,−1(x) , and by ∆θ = p0,−1(x̃) − p0,−1(x) respectively. A statement parallel to Proposition 2.3 is the following Proposition 2.4 In the above notation we have px̃,−x̃ p−1x̃,1 (px,1(y) + ∆θ) − px,−x(y) = ∆t. Proof. The point c = p−1x̃,1(px,1(y) + ∆θ) satisfies = y−x · 1+x , which can be transformed into c− x̃ 1 − y y − x 1 + x̃ 1 − x̃ 1 − x 1 − x̃ . (2.4) We will be done if we show that c−x̃ · x+y = x̃−1 · 1+x , which is the same as x̃ + c c− x̃ y + x y − x 1 + x̃ x̃− 1 1 + x . (2.5) Since x̃+c = 1 + 2x̃ , equation (2.5) follows immediately from (2.4). � Proposition 2.5 Suppose x, x′ ∈ (0, 1) and y ∈ (x, 1). Let y′ be such a point in (x′, 1) that px′,1(y ′) = px,1(y). Then p1,0(x ′) − p1,0(x) = py′,0(x ′) − py,0(x). (2.6) Proof. The point y′ is chosen in such a way that y = y−x , or 1−x = 1−x Identity (2.6) is now immediate. � 3 The period 4 super-sink In this short section we describe the behavior of the point 4a when we let the parameter ā vary in such a range, that 3a ∈ (0, 1) and the point 4a stays within the interval (2a,−2a). Let a positive number t be the Poincaré coordinate of 2a in the oriented interval (1,−1), and we set g(t) = p2a,−2a(4a). The following theorem holds true. Theorem 3.1 The inverse function g−1 : R → R+ is strictly increasing, and g′(t) > 1. In particular, the value g(t) = 0, corresponding to the super-stable orbit with the kneading sequence RLRC is assumed only once. Proof. Consider a pair of admissible parameter values ā and ā′, i.e. such that the orbits 2a, 3a, 4a (and respectively 2a′ , 3a′, 4a′) satisfy the restrains on the dynamics we set above. Then ∆t = p1,−1(2a′) − p1,−1(2a), while ∆g = p2 (4a′) − p2a,−2a(4a). (3.1) Applying Proposition 2.4 to this case we get (p−12 ,1(p2a,1(4a) + (p0,−1(2a′) − p0,−1(2a)))) − p2a,−2a(4a) = ∆t, (3.2) so, because of monotonicity of the coordinate functions, we only need to establish that ,1(4a′) − p2a,1(4a) > p0,−1(2a′) − p0,−1(2a). (3.3) This inequality becomes clear once we split the procedure leading from point 2a (respectively 2a′) to 4a (respectively to 4a′) into three steps. In the first step, we act on the interval (0,−1) by the restriction of the power-law map. Thus, due to negative Schwarzian derivative, the initial discrepancy (p0,−1(2a′) − p0,−1(2a)) in the Poicaré coordinates gets increased. So we see (3a′) − p1,2a(3a) ≥ p0,−1(2a′) − p0,−1(2a). In the second step, we turn the interval (0,−1) over, onto the interval (1, 2a), or onto (1, 2a′) respectively, and then we truncate the image at the critical point 0. This cut-off only increases the Poincaré coordinate of ev- ery point, which after the turnover landed in (1, 0), because we now read the Poincaré coordinate in the interval (1, 0) rather than in a larger domain (1, 2a), or (1, 2a′) respectively. Moreover, the increase in the Poincaré coordi- nate inflicted by cutting the domain interval short, is in the case of point 3a smaller then in the case of 3a′ . This is so, because the endpoint 2a is closer to the critical point, while the endpoint 2a′ is further away to the left, so of two corresponding points with identical Poincaré coordinate within the respec- tive domain intervals (with the other endpoint at 1), the gain in the latter situation is larger than in the former. But instead of equal coordinates, we have even better inequality p1,2 (3a′) > p1,2a(3a), which further enlarges the gain. Thus, in this second step, made of the turnover followed by truncation, the initial discrepancy grows even further and so p1,0(3a′) − p1,0(3a) > p1,2 (3a′) − p1,2a(3a). In the last step, we again act by a negative Schwarzian map stretching the discrepancy between the Poincaré coordinates yet further, and finally we make the turnover onto (1, 2a), and respectively onto (1, 2a′), to arrive at (3.3). Therefore ∆g > ∆t and the proof is complete. � 4 The period 8 super-sink In the previous section we have established that, when we vary the param- eter ā , the position of the point 4a within the interval (2a,−2a) changes monotonically, with the derivative greater than 1. It clearly follows from the proof, that this derivative actually stays bounded away from 1, in a way that depends on the critical order α . In section 5 we will study in detail the case of p2a,−2a(4a) < 0, and describe the behavior of the intervals delimited by the post-critical orbit with the kneading sequence RLRL . . . . In here, we will focus on these admissible parameters ā , for which p2a,−2a(4a) > 0 and p4a,−4a(8a) ≤ 0 , i.e. we are past the (unique) parameter corresponding to RLRC, but we do not cover the critical point yet another time. From now on, we are making our choice of the parameter subject to this restriction. We shall see that, as long as the above condition on the dynamics is satisfied, the movement of the point 8a is also monotone in pa- rameter, and in the non-euclidean metric in (4a,−4a) this point moves with the derivative strictly positive. It will follow that the RLRRRLRC super- stable orbit appears uniquely in every power-law family. It is a subject of an ongoing work, that goes beyond the scope of this paper, to examine whether a claim analogous to that of Theorem 3.1 can be fully extended to larger set of parameters. In our current case, the scheme of the argument we used to prove Theorem 3.1, alone will not suffice, and a more delicate technique must be employed. Yet, some understanding of the way Poincaré coordinates vary remains an important component. Since, due to the more intricate dynamics, the re- quired property of the non-euclidean coordinates becomes less self-evident, we state it as a separate lemma. The points x, y, z below will correspond to the points 2a, 4a, 8a of the post-critical orbit. The origin of the sum- mands, which do not have equivalent in the statement of Proposition 2.4 will be explained later, in the course of the proof of Theorem 4.1 below. Here, we only indicate that the last term has to do with the limit strength of a non-euclidean push. Lemma 4.1 Suppose we are given two triples of points, (x, y, z) and (x̃, ỹ, z̃), satisfying the following conditions: (i) x, x̃ ∈ (0,−1) and p0,−1(x̃) > p0,−1(x) , (ii) y ∈ (0,−x), ỹ ∈ (0,−x̃) and px̃,1(ỹ) ≥ px,1(y) + (p0,−1(x̃) − p0,−1(x)) , (iii) z ∈ (y, 0], z̃ ∈ (ỹ,−ỹ) and pỹ,x̃(z̃) ≥ py,x(z) + (p0,−x̃(ỹ) − p0,−x(y)) + ln ỹ−x̃ − ((px̃,1(ỹ) − px,1(y)) − (p0,−1(x̃) − p0,−1(x))) . Then pỹ,−ỹ(z̃) > py,−y(z) . Proof. It is immediate to check that for arbitrary y, ỹ ∈ (0, 1) one has pỹ,−1(0) = py,−1(0) + (p0,1(ỹ) − p0,1(y)) + ln 1 + ỹ 1 + y − (p−1,1(ỹ) − p−1,1(y)) . (4.1) We now assume ỹ > y, and allowing z 6= 0 we verify, that for any z ∈ [0, y) the following generalization of (4.1) holds pỹ,−ỹ(p ỹ,−1(py,−1(z) + (p0,1(ỹ) − p0,1(y)) + ln 1 + ỹ 1 + y −(p−1,1(ỹ) − p−1,1(y)))) ≥ py,−y(z). (4.2) In order to see this, notice that (p0,1(ỹ) − p0,1(y)) + ln 1 + ỹ 1 + y − (p−1,1(ỹ) − p−1,1(y)) = ln and denote c = p−1ỹ,−1(py,−1(z) + ln ), which means c−ỹ −ỹ−c = z−y c− ỹ 1 + ỹ 1 + z z − y 1 + ỹ . (4.3) We will be done if we show that c−ỹ −ỹ−c ≥ z−y , being equivalent to 2ỹ +1 ≥ y+z or ỹ . The last inequality follows from (4.3), once we recall ỹ ≥ y. In the next step we extend formula (4.2), allowing x̃ 6= −1. Assuming 1 > −x̃ > ỹ > y > z ≥ 0, we will now show that pỹ,−ỹ(p ỹ,x̃(py,x̃(z) + (p0,−x̃(ỹ) − p0,−x̃(y)) ỹ − x̃ y − x̃ − (px̃,1(ỹ) − px̃,1(y)))) > py,−y(z). (4.4) We emphasize that the inequality in formula (4.4) is always sharp, even for z = 0. This time, we set c = py,x̃(z) + (p0,−x̃(ỹ) − p0,−x̃(y)) + ln ỹ − x̃ y − x̃ − (px̃,1(ỹ) − px̃,1(y)), (4.5) which means c− ỹ x̃− c z − y x̃− z −x̃− ỹ −x̃− y 1 − ỹ 1 − y We transform this identity into x̃− ỹ c− ỹ − 1 = x̃− z z − y x̃ + ỹ x̃ + y 1 − y 1 − ỹ and further into c− ỹ x̃− ỹ x̃− z z − y x̃ + ỹ x̃ + y 1 − y 1 − ỹ We will be done if we show pỹ,−ỹ(c) > py,−y(z), i.e. −ỹ−c > z−y , which is equivalent to ỹ , and so it is enough to verify that x̃− ỹ x̃− z z − y x̃ + ỹ x̃ + y 1 − y 1 − ỹ z − y This inequality can be rewritten as x̃− z z − y x̃ + ỹ x̃− ỹ x̃ + y 1 − y 1 − ỹ z − y x̃− ỹ or (recall that y < z, x̃ < 0, ỹ > 0) (x̃− z)(x̃ + ỹ)y(1 − y) < (x̃ + y)(1 − ỹ)(yx̃− zỹ), and further x̃y(1 + x̃)(ỹ − y) < z(ỹ − y)(x̃ỹ + x̃y − x̃ + yỹ). To conclude, we cancel out (ỹ − y), and observe that x̃y + x̃ỹ − x̃ + yỹ > x̃(1 + x̃). (4.6) This is so because (4.6) boils down to the inequality x̃2+x̃(2−y−ỹ)−yỹ < 0, which is elementarily true for all x̃ ∈ (−1, 0) and y, ỹ ∈ (0, 1). For completion of the proof we now consider an arbitrary point x ∈ (0, x̃), such that y < −x. We consider the movement of x-variable from position x to x̃ and apply Proposition 2.4 twice, first to the induced movement of y-variable, then to the consequent movement of z-variable. By virtue of that Proposition, we see that points ŷ and ẑ, determined by the identities px̃,1(ŷ) = px,1(y) + (p1,−1(x̃) − p1,−1(x)) pŷ,x̃(ẑ) = py,x(z) + (p1,−1(x̃) − p1,−1(x)) satisfy ŷ < ỹ and pŷ,−ŷ(ẑ) = py,−y(z) + (p1,−1(x̃)− p1,−1(x)) > py,−y(z). Thus obviously ln ỹ−x̃ > ln ỹ−x̃ ŷ−x̃ If ẑ ≥ 0, i.e. pŷ,−ŷ(ẑ) ≤ 0, then keeping x̃ fixed, we then apply formula (4.4) with ŷ, ẑ in place of y, z, to the effect of yet further increase of the Poincaré coordinate of z̃ compared to that of ẑ (and so of z itself), measured within respective symmetric y-domains. In case of pŷ,−ŷ(ẑ) > 0 the image of point z has already past the midpoint of the (varying) symmetric y-domain interval while y-variable has been changing from y to ŷ. Again, we then keep x̃ fixed, to move the y-variable further, from ŷ to ỹ. This time, application of formula (4.4) can induce some decrease in the Poincaré coordinate of the outcome – the resulting point p−1ỹx̃ (c), with c as in (4.5), can divide the y- domain interval (ỹ,−ỹ) in smaller proportion than ẑ did in (ŷ,−ŷ). Anyway, due to sharp inequality in (4.4) for all z such that py,−y(z) < 0, the midpoint could only be attained from the other side. In other words, inequality (4.4) guarantees that the derivative of the induced z-movement, measured in the respective Poincaré coordinates, is positive (and actually bounded away from 0) as long as the values assumed by the z-variable are non-positive. Thus, in particular the value 0 can be attained only once, and so if we put a point ẑ with py,−y(ẑ) > 0 into the formula at the left-hand side of inequality (4.4), we necessarily end up with a point on the same side of 0. Because the starting point z was on the other side, the lemma holds in this case too. � With lemma 4.1 in place, we are in the position to state and prove the main result of this section. Theorem 4.1 In the power-law family fa : x 7→ −|x| α +a, with α > 1, there exists unique parameter a = a(α) corresponding to the kneading sequence RLRRRLRC. Proof. In the course of the proof we make use of the tools developed in section 2 of [9], where we pointed to some consequences of homogeneity of the power-law mappings. In particular, we had Lemma 2.1 there, asserting that for any two points q, q̃ ∈ (0, 1) one has q,1 − h q̃,1 = (p0,1(h(q)) − p0,1(h(q̃))) − (p0,1(q) − p0,1(q̃)). (4.7) Speaking colloquially, identity (4.7) tells, that when we move the endpoint of an interval (1, q) in (1, 0) towards the critical point, then an extra gain in the Poincaré coordinate, coming from the successive action of the power-law map, is just enough to make up for the loss (measured in non-euclidean met- ric in (1, q) and (1, q̃) respectively) suffered because of the simultaneously increased strength of the limit non-euclidean push towards that moving end- point. Other propositions and lemmas of section 2 of [9] served to establish, that this limit situation, corresponding to Poincaré coordinate close to −∞, is essentially the worst possible, and when we consider an interior point of a definite Poincaré coordinate rather than the limit case, then the balance of gains vs. losses is in our favor (”we are never in the red”). We will be sending upon those properties when necessary, without reproducing them in this paper. Proceeding similarly to what we did in the proof of Theorem 3.1, we split the procedure leading from 4a to 8a , and respectively from 4a′ to 8a′ , into several steps. First, we increase ā to ā′. Theorem 3.1 yields, in particular, that p0,−2 (4a′) − p0,−2a(4a) > 0. Next, we act upon 4a′ , and 4a, by the map h, and under the action of h the above discrepancy gets enlarged. This is so, because due to homogeneity, we may for the purpose of performing this step, tentatively set each of the endpoints, −2a′ and respectively −2a, at 1. Then each of the Poincaré coordinates p0,−2 (4a′), p0,−2a(4a), is transformed by same, fixed negative Schwarzian map h0,1. In the following step, we turn each of the intervals (0, h(2a′)), (0, h(2a)) over, and stretch them onto (1, 3a′) and respectively (1, 3a). The image of 4a′ is 5a′ , and by the so far described steps, it is clear that p1,3 (5a′)−p1,3−a(5a) > p0,−2 (4a′)−p0,−2a(4a). By the truncation argument from the proof of Theorem 3.1, we know that p1,0(3a′) − p1,0(3a) > ā − a . In particular, the nonlinearity of the interval (1, 3a′) is larger than that of (1, 3a). Now, we act by the homogeneous map h again. Notice, that unlike in the case of h0,2 , this time the mapping h1,3 does not coincide with h1,3a . Anyway, we can still claim that in this step the discrepancy of the respective Poincaré coordinates grows again, i.e p1,h(3 )(h(5a′)) − p1,h(3a)(h(5a)) > p1,3a′ (5a′) − p1,3a(5a) . (4.8) To this end, we invoke Propositions 2.5 and 2.4 of [9]. From the former, it follows that the strength of the non-euclidean push generated by h restricted to some domain, is a monotone function of the nonlinearity of that domain, when measured for a fixed Poincaré coordinate within the varying domain. From the latter, we derive that when the domain stays fixed, the strength of the non-euclidean push of h is monotone in the Poincaré coordinate of the argument. We have noticed already that the nonlinearity of (1, 3a) is in- creasing in parameter ā, and also that p1,3 (5a′) > p1,3a(5a) , so the principle of monotone behaviour of the strength of non-euclidean push can be applied to the triples of points we consider. This immediately implies the desired increase in the discrepancy of appropriate Poincaré coordinates, as stated in (4.8). Making the next step, we turn the obtained triples (1, h(5a′), h(3a′)) and (1, h(5a), h(3a)) over, onto (2a′, 6a′ , 4a′), and respectively onto (2a, 6a, 4a), and then truncate them at the critical point 0. In the proof of Theorem 3.1, as well as in a step above, we were satisfied to ascertain that this truncation increases the Poincaré coordinates discrepancy, which in current step would yield p2 ,0(6a′) − p2a,0(6a) > p2a′ ,4a′ (6a′) − p2a,4a(6a), because by Theorem 3.1 we know that p2 (4a′) > p2a,−2a(4a). To proceed further, one more observation is needed. It is fairly clear that we have following lower bound on the increase of the Poincaré coordinates discrepancy, generated by the cut-off at 0: ,0(6a′) − p2a,0(6a) > p2a′ ,4a′ (6a′) − p2a,4a(6a) + ln 4a′ − 2a′ 4a − 2a . (4.9) The equality in (4.9) is the limit case, attained for infinitesimally short in- tervals placed at the left-hand endpoints, i.e. when p2a,4a(6a) → −∞ and simultaneously p2 (6a′) → −∞. For non-infinitesimal intervals satisfying (6a′) > p2a,4a(6a), the same argument as in the proof of Theorem 3.1 obviously yields sharp inequality in (4.9), and so the growth of the discrep- ancy gained in the cut-off step is strictly larger than the logarithmic term. In the following step we once more act by homogeneous map h, and because h0,2 coincides with h0,2a , the same argument as before gives ,0)(h(6a′)) − ph(2a),0(h(6a)) > p2a′ ,0(6a′) − p2a,0(6a) . (4.10) This adds yet an extra amount to the discrepancy we consider. We again turn the intervals (h(2a′), 0) and (h(2a), 0) over and stretch them onto (1, 3a′) and (1, 3a), with 6a′ going onto 7a′ and 6a going onto 7a respectively. It remains to examine what happens in the last step, when we act by the respective (non- coinciding!) restrictions of h to the obtained intervals, before we eventually return onto (2a′, 4a′) and onto (2a, 4a) by linear rescaling. This is what we need Lemma 4.1 for. In what follows we verify its assumptions are fulfilled in our setting. In this last step we perform, the strength of the non-euclidean push induced by h| (1,3 ) , measured at 7a′ , can be greater than the respective strength of h| (1,3a) at 7a. This means that the discrepancy accumulated in all the so far steps can now diminish. However, the identity (4.7) provides a bound from the above on the amount of possible loss. To see this, we recall ā′ − ā = p0,−1(2a′) − p0,−1(2a) < p1,0(3a′) − p1,0(3a) , (4.11) and according to 4.7 we have ,1(4a′) − p2a,1(4a)) − (p1,0(3a′) − p1,0(3a)) = (h ) (4.12) We know that p3 ,1(7a′) > p3a,1(7a) and the interval (1, 3a′) has larger non- linearity than (1, 3a), so we are in a position to invoke Propositions 2.5 and 2.4 of [9] once more. By them we have ,1(7a′) − p3a,1(7a)) − (p4a′ ,2a′ (8a′) − p4a,2a(8a)) < (h ). (4.13) The inequalities (4.9), (4.11) and (4.13) put together, provide for fulfillment of condition (iii) of Lemma 4.1, with the points x, y and z assuming values 2a, 4a and 8a, as indicated before the statement of the lemma. Now the claim of Theorem 4.1 follows directly from Lemma 4.1, and so we are done. � We complete this section explicitly recording one extra property, which we actually proved along the way. Denote the variable τ = p4a,−4a(8a) and let γ = p2a,−2a(4a). From the proofs of Theorem 4.1 and Lemma 4.1 there immediately follows Corrolary 4.1 The function γ = γ(τ) : R− → R+ is strictly increasing in τ , with the derivative γ′(τ) bounded away from 0 and +∞. 5 RLRLRLRL . . . In this section we let the parameter ā vary in a range such that the kneading sequence is RLRL . . . . From Theorem 3.1 it follows immediately that the range of admissible ā’s is always a half-line (−∞, ā1), with the specific value of ā1 depending on the critical order α. Upholding the normalization 0a = 0 , 1a = 1 we have set before, this means the post-critical orbit begins with 2a ∈ (0,−1) , 3a ∈ (1, 0) and 4a ∈ (2a, 0). Then, we get two sequences of nested intervals, the odds: (1, 3a), (3a, 5a), (5a, 7a) . . . , and the evens: (0, 2a), (2a, 4a), (4a, 6a) . . . . In terms of multipliers, we either have a period 2 orbit with negative multiplier, or this periodic orbit had turned into a repeller and, by bifurcation, there was born a period 4 attracting periodic orbit with positive multiplier. In what follows, we shall see that the ratios of consecutive intervals within each of the two decreasing families are functions strictly monotone in parameter ā. Moreover, the initial increase of the parameter, i.e. ā′ − ā, does not eventually vanish, but a definite part of it is preserved through all the steps. This will further provide, with some extra work, for monotonicity of the multipliers, also in the case of repelling period 2 orbit. This is a work in preparation. The remaining part of this paper is devoted to the proof of the following claim. Theorem 5.1 For ā ∈ (−∞, ā1) and for all non-negative integers n, the ratio functions rne = |(2n + 4)a − (2n + 2)a| |(2n + 2)a − (2n)a| and rno = |(2n + 5)a − (2n + 3)a| |(2n + 3)a − (2n + 1)a| (5.1) are strictly increasing in ā. Moreover, when the parameter increases from ā to ā′, then for every n ∈ Z+ the induced discrepancy of the Poincaré coordinates satisfies p(n+2) ((n + 4)a′) − p(n+2)a,na((n + 4)a) > (p1,−1(2 a) − p1,−1(2a)) . (5.2) Proof. As before, we divide the procedure into steps. Once we cover the most delicate step, which turns out to be the passage from (5a′ , 7a′) to (6a′ , 8a′), we will be in a position to continue inductively. We begin by moving the initial point 2a to a new position 2a′ , with ā ′ > ā. Then, by the truncation argument from the proof of Theorem 3.1, we have (p1,0(3a′) − p1,0(3a)) > ā ′ − ā = ∆ā > ∆t, (5.3) where we denoted ∆t = (p1,−1(2 a) − p1,−1(2a)). Since we then act by the homogeneous map h, by (4.7) we get ,1(4a′) − p2a,1(4a) = (p1,0(3a′) − p1,0(3a)) + (h ,1) . (5.4) Passing from 3a′ to 4a′ , we cannot directly apply the truncation argument again, because in this step the Poincaré coordinate p1,2 (0) of the cut-off point decreases (cf. Proposition 2.1). That can be fixed by decomposing the step in two, and simultaneous use of Proposition 2.4, identity (4.7) and truncation. According to (5.3) and Proposition 2.4, ,1(p1,0(3a′)) − p2a,−2a p−12a,1(p1,0(3a)) > ∆t. (5.5) Truncation at 0 obviously gives ,1(p1,0(3a′)) − p2a,0 p−12a,1(p1,0(3a)) > ∆t. (5.6) Then, to the Poincaré coordinate of the point (p−12 ,1(p1,0(3a′)), read in the domain (2a′ , 1), we add the extra gain of (h ,1). The non-euclidean length of this same extra interval, read in the domain (2a′ , 0) rather than in (2a′ , 1), is of course larger, because of truncation. Thus ,0(4a′) − p2a,0(4a) > ∆t + (h ,1) . (5.7) Doing the homogeneous mapping again, by (4.7) and (5.7) we get ,1(5a′) − p3a,1(5a) = h2a′ ,0(p2a′ ,0(4a′)) − h2a,0(p2a,0(4a)) > > ∆t + (h ,1) + (h 4a,2a ) . (5.8) Now, similarly to the final step in the proof of Theorem 4.1, we can argue that the so far acquired gain in the Poincaré coordinate is enough to make up for possible losses in the next two steps. This is fairly clear. The interval (3a′ , 1) has larger nonlinearity than (3a, 1), and p3 ,1(5a′) > p3a,1(5a), so Propositions 2.5 and 2.4 of [9] do apply when we act by h | (1,3 ) and h | (1,3a). Therefore, in this step the discrepancy (p3 ,1(5a′) − p3a,1(5a)) can only be diminished by an amount smaller than (h ,1), yielding (6a′) − p4a,2a(6a) > ∆t + (h 4a,2a ), (5.9) By (5.7), the nonlinearity of (4a′, 2a′) is larger than that of (4a, 2a), and also (6a′) > p4a,2a(6a). Thus, when we act by h | (4 ), and respectively by h | (4a,2a), we certainly do not lose more than (h 4a,2a ) in the outgoing discrepancy. Hence, by ( 5.9) (7a′) − p5a,3a(7a) > ∆t. (5.10) We can now make a shortcut towards completion of the current cycle. The nonlinearity of (3a′ , 1) is larger than that of (3a, 1) and p3 ,1(5a′) > p3a,1(5a), which in turn gives that the nonlinearity of (5a′ , 3a′) is larger than that of (5a, 3a). Also p5 (7a′) > p5a,3a(7a), so we can apply the argument about monotonicity of the strength of the non-euclidean push, which we recalled in the proof of Theorem 4.1, immediately arriving at (8a′) − p6a,4a(8a) > p5a′ ,3a′ (7a′) − p5a,3a(7a) > ∆t. (5.11) However, the above argument alone turns out to be insufficient, when we want to do further iterates. To obtain an inequality which we could use inductively at all steps, we need more subtle understanding at this particular stage of our procedure. Here we go. From (5.3) and Proposition 2.5 it follows that (p3a,1(5a)),0 (3a′) − p5a,0(3a) > ∆ā , (5.12) so by p3 ,1(5a′) > p3a,1(5a) we have ,0(3a′) − p5a,0(3a) > ∆ā. (5.13) By the same argument applied to (5a′ , 3a′) rather than (1, 3a′), we get (p5a,3a (7a)),0 (3a′) − p7a,0(3a) > ∆ā. (5.14) Now we do the homogeneous mapping h, and rescale the image onto (1, 2a′). The image of 3a′ is 4a′ , and by a version of the truncation argument alike that used before in the step leading from (1, 3a′) to (2a′ , 4a′), we use ,1(p1,0(3a′))) − p2a,0(p (p1,0(3a))) > ∆t (5.15) and (5.14) to get (h1,0(p1,0(p (p5a,3a (7a))))),0 (4a′) − p8a,0(4a) > ∆t. (5.16) This is so, because (5.14) implies ph(p−1 (p5a,3a (7a))),0 (h(3a′)) − ph(7a),0(h(3a)) > ∆ā, (5.17) and when we consider the interval (c, d), where d = p−12 1(p2a,1(4a)+∆ā), and the point c is defined so that pc,1(d) − p8a,1(4a) = ∆ā (5.18) then, according to Proposition 2.5 applied to the domain (1, 2a′) in place of (0, 1), and with the points d and c singled out, we see that point c divides the interval (d, 2a′) at the same proportion as 8a divided (4a, 2a). Re-applying Proposition 2.5 to the domain (0, 2a′) with the same singled out pair of points, we further see that pc,0(d) > p8a,0(4a) + ∆t, (5.19) because by Proposition 2.4 p2 ,0(d) − p2a,0(4a) > ∆t. Recalling (5.17) and taking into account that p2 ,1(4a′) > p2 ,1(d), which in turn gives p1,4 (0) > p1,d(0), we can now do the standard truncation argument, cutting-off at 0 to arrive at (5.16). This formula could do for the iterative procedure if we cared only for some, indefinite growth. To obtain definite growth, claimed in the statement of Theorem 5.1, we need to work harder. In the next step of the proof, we will see that the extra amount of ∆t in formula (5.16) allows us to move 7a towards the endpoint by at least that much. To this end, we again consider the interval (5a′ , 3a′), but this time the point within we single out, is point e determined by (e) = p5a,3a(7a) + ∆t. (5.20) From (5.14), using Proposition 2.1 with points 0, 3a′ and e in place of 1, 0 and x respectively, or by a direct check, one gets pe,0(3a′) − p7a,0(3a) > ∆ā− ∆t. (5.21) Doing the homogeneous mapping, we have ph(e),0(h(3a′)) − ph(7a). 0(h(3a)) > ∆ā− ∆t. (5.22) Again, we consider an interval (f, d), where d has same meaning as above, and point f is defined by pf,1(d) − p8a,1(4a) = ∆ā− ∆t. (5.23) From (5.18) and (5.23), it follows by Proposition 2.1 that pd,−∞(c)−pd,−∞(f) = ∆t, and again by this same proposition pc,0(d) − pf,0(d) = ∆t. Hence, by (5.19), we have pf,0(d) > p8a,0(4a). This, and (5.22) lead to (p5a,3a (7a)+∆t)),0 (4a′) > p8a,0(4a). (5.24) We can describe what we have found so far in the following way. We move the parameter up, from ā to ā′. In the odd family, we see 3a moving to 3′a by more than ∆ā. Consequently, the non-euclidean coordinate of 5a vary, within its dynamically determined base interval, by at least ∆t, plus an additional increment which is sufficient to make up for the increased – due to larger nonlinearity of the new new domain intervals – strength of the non-euclidean push backwards. In the next odd return we do not let 7a move all the way to its new position 7a′ at once. Instead, we first only add ∆t to its Poincaré coordinate. This corresponds to starting from the point e in the already fully enlarged domain (5a′ , 3a′), rather than from 7a′. We have just seen that not only is the nonlinearity of (e, 3a′) larger than that of (7a, 3a), but the nonlinearity of (ê, 4a′) is larger than that of (8a, 4a) also. Here ê is the dynamical successor of e on the even side. This latter estimate from the below on the the nonlinearity, turns out to be fundamental for the prospective iterates. Recall we defined e by (5.20) so as to have p5 (e) = p5a,3a(7a) + ∆t. The same way we derived (5.11) from (5.10) we also get (ê) − p6a,4a(8a) > ∆t. (5.25) This will be needed, when it comes to definite growth in both odd and even family. But now, for points e and ê we have stronger input: in both cases, we know that the nonlinearity of the remaining part of the base interval in- creased. Therefore, we will now be able to proceed pretty much like in the initial step, that led from (1, 3a′) to (2a′ , 4a′), rather than use the earlier de- scribed shortcut. Similarly to that initial step, we again want to know that the surplus exceeding ∆t in (5.25) will make up for possible loss, inflicted by increased nonlinearity of (5a′ , 7a′), upon next return to (5a′ , 7a′). However, we have to overcome a serious obstacle. Formula (4.7) we previously used to that goal, holds true only so long as the critical point is the endpoint corre- sponding to non-euclidean +∞. This is of course not the case for (5a, 3a), nor for all other intervals in our odd and even families, except for the initial ones. For intervals not bounded by the critical point, we only know mono- tonicity of the strength of non-eucliean push and this, in general, does not give control over an amount of the gain in Poincaré coordinates discrepancy. Composing h mappings over two arbitrary, successive domains, yet worsens the the situation. Fortunately, all this can be fixed with (5.21) and (5.24) in place. Increased nonlinearity of that part of a domain interval which bounds us away from the endpoint, provides an effective replacement for the critical endpoint. In particular, we will see that the gain in the non-euclidean coor- dinates discrepancy is even better than that in formula (4.7) . This is why we have striven for those nonlinearity inequalities. As soon as we are over with the part which takes 7a to e, the remaining part, in which we move e further to 7a′ , will require only an easy estimate. All the above holds true for even successors, ê and eventually 8a′ , as well. With one extra observation to make, we will be able to do arbitrarily long iterates, preserving the ∆t discrepancy all along the way. To carry out the above described strategy, we recall that in Proposition 2.2 of [9] we gave an explicite formula for the strength of non-euclidean push, which turns out to be |pr,s(ϕ(x)) − pp,q(x)| = |ϕ x,q + ϕ p,x| (5.26) We also noticed there, that for the homogeneous mapping h restricted to some interval, the quantities h and h depend solely on the nonlinearity of that domain interval. By monotonicity of the strength of the non-euclidean push as a function of the nonlineatity of the domain, also the limit values, h and h , behave monotonically. By all the above, taking (5.21) into account, we have (ê) − p6a,4a(8a) = h5a′ ,3a′ (p5a,3a(7a) + ∆t) − p6a,4a(8a) = ),h(3 )(h(e)) − ph(5a),h(3a)(h(7a)) > ∆t + (h ,e − h 5a,7a ) (5.27) The sign at the superscript of h in (5.27) depends only on an orientation of the domain, so (5.27) provides a better estimate than we could derive from (4.7), if the endpoint 3a′ coincided with the critical point. Doing the successive h-map step on the even side, because of (5.24), we get in the same ),h(4 )(h(ê))−ph(6a),h(4a)(h(8a)) > ∆t+ (h 5a,7a ) + (h ,ê−h 6a,8a (5.28) These are formulas analogous to (5.7) and (5.8), and what we want now, is a similar estimate where the input is 7a′ and 8a′ , rather than e and ê. To move from e to 7a′ we could simply invoke Lemma 2.4 of [9]. However, there is no generalization of that lemma which could be used over two unrelated domains. We need to be a bit more careful, and use the dynamical relation between an interval and its image. Doing the mapping h, by homogeneity and (4.7) we have ),0(h(7a′)) − ph(5 ),0(h(e)) = (p5 ,0(7a′) − p5 ,0(e)) + (h (5.29) Before we do another mapping h, we take the image over onto (1,−∞), so that h(3a′) goes onto (4a′), and cut off at 0. Because of this truncation ,0(8a′) − p6 ,0(ê) > (p5 ,0(7a′) − p5 ,0(e)) + (h ,e). (5.30) Now, acting by homogeneous map, we get ),0(h(8a′)) − ph(6 ),0(h(ê)) > (p5 ,0(7a′) − p5 ,0(e)) + ,e) + (h ,ê) (5.31) We neglect a positive summand (p5 ,0(7a′)− p5 ,0(e)) and truncate at h(4a′) to arrive at ),h(4 )(h(8a′)) − ph(6 ),h(4 )(h(ê)) > (h ,e) + (h (5.32) Similarly, neglecting a positive summand at (5.29), followed by cutting off at h(3a′) leads to ),h(3 )(h(7a′)) − ph(5 ),h(3 )(h(e)) > (h ,e). (5.33) The inequalities (5.27) and (5.33), in conjunction with (5.28) and (5.32), give ),h(3 )(h(7a′)) − ph(5a),h(3a)(h(7a)) > ∆t + (h 5a,7a ) (5.34) and also ),h(4 )(h(8a′))−ph(6a),h(4a)(h(8a)) > ∆t+(h 5a,7a 6a,8a (5.35) which are the desired estimates. Now, by the same argument which led from (5.7) and (5.8), through (5.9) to (5.10), we can see that (5.34) and (5.35) imply (11a′) − p9a,7a(11a) > ∆t. (5.36) In the same way we obtained (5.11) from (5.10), we can also derive (12a′) − p10a,8a(12a) > ∆t. (5.37) We have completed the second cycle. Those were necessary to initialize the inductive procedure. We are now in a position to do the final argument, which can be used repeatedly. We believe that, because all the elaborate notation of the first two cycles is already in place, it will be more instructive to present this argument in detail as the next cycle, rather than in general terms. It will be obvious that what we do, is tantamount to the inductive step. We pick a point ǫ ∈ (9a′ , 7a′), such that (ǫ) = p9a,7a(11a) + ∆t. (5.38) We will prove that the nonlinearity of (ǫ, 7a′) is larger than that of (11a, 7a), and simultaneously the nonlinearity of (ǫ̂, 8a′) is larger than that of (12a, 8a). As before, ǫ̂ stands for the dynamical successor of ǫ on the even side. This will permit to bypass the non-critical endpoint obstacle in the next cycle, the way we did earlier, with e and ê. To show this nonlinearity increase, we proceed in several steps. First, in (5a′ , e) we find a point β = p ,e(p5a,7a(9a)). Then, in (β, e) we find ε, such that pβ,e(ε) = p9a,7a(11a) + ∆t. We use the the fact that Poincaré coordinate of the point e, compared to that of 7a, is already moved by ∆t towards the endpoint, to ascertain that the nonlinearity of (ε, e) is larger than nonlinearity of (11a, 7a). This is so, because according to Proposition 2.5, pβ,3 (e) = p9a,3a(7a) + ∆t, and the nonlinearity of (e, 3a′) is larger than nonlinearity of (7a, 3a); we have |(β, e)| |(e, 3a′)| = (exp ∆t) |(9a, 7a)| |(7a, 3a)| > (exp ∆t) |(β, δ)| |(δ, e)| , (5.39) where δ ∈ (β, e) is a point such that |(δ,e)| |(e,3 |(11a,7a)| |(7a,3a)| . Thus, pe,β(ε) > pe,β(δ), and consequently pε,3 (e) > pδ,3 (e) or, in other words, |(ε,e)| |(e,3 |(δ,e)| |(e,3 . Since (e, 3a′) has larger nonlinearity than (7a, 3a), the nonlinearity of (ε, e) must be larger than nonlinearity of (11a, 7a). Next, we do the map- ping h and consider the situation on the even side. The interval (ε, 3a′) has larger nonlinearity than (11a, 3a) and pε,3 (e) > p11a,3a(7a), so by principles of monotonicity of the strength of non-euclidean push in nonlinearty of the domain, as well as in the coordinate of the point, the action of hε,3 makes pε̂,4 (ê) > p12a,4a(8a), where ε̂ is the dynamic successor of ε. Because (ê, 4a′) has larger nonlinearity than (8a, 4a), it follows that the nonlinearity of (ε̂, ê) is also larger than that of (12a, 8a). By the same two principles applied to pβ,e(ε), we get pβ̂,ê(ε̂) > p10a,8a(12a) + ∆t, but because of the proved nonlinearity increases, we can also claim that ph(β),h(e)(h(ε)) − ph(9a),h(7a)(h(11a)) > ∆t + (h β,ε − h 9a,11a ) and h(β̂),h(ê) (h(ε̂))− ph(10a),h(8a)(h(12a)) > ∆t+ (h β,ε − h 9a,11a ) + (h β̂,ε̂− h 10a,12a We are through with the first part of the inductive step. Now, our immediate plan is to move e to 7a′ , then β up to 9a′ , and to replace ε by ǫ, keeping all the above gains untouched, both on the odd and on the even side. Having done all that, we will easily be able to move ǫ to 11a′, to complete the procedure. Denote by λ the point determined by p5 (λ) = p5a,7a(9a), and let τ ∈ (λ, 7a′) be such, that pλ,7 (τ) = p9a,7a(11a) + ∆t. The interval (5a′, 7a′) has larger nonlinearity than (5a′ , e), so consequently (λ, 7a′) has larger non- linearity than (β, e), and (τ, 7a′) has larger nonlinearity than (ε, e). Thus (h(τ), h(7a′)) has larger nonlinearity than (h(ε), h(e)). The distance of 8a′ to the critical point 0 is smaller than similar distance for the point ê, so the truncation argument after cutting of at 0, implies that (τ̂ , 8a′) has larger nonlinearity than (ε̂, ê) and, in turn, larger than (12a, 8a). Again, we in- crease the intervals in question, choosing 9a′ in place of λ, and replacing τ by ǫ. Then, of course, (ǫ, 7a′) has yet larger nonlinearity, so (h(ǫ), h(7a′)) has larger nonlinearity than (h(τ), h(7a′)) and , after truncation, (ǫ̂, 8a′) has larger nonlinearity than (τ̂ , 8a′). It immediately implies ),h(7 )(h(ǫ)) − ph(9a),h(7a)(h(11a)) > ∆t + (h ,ǫ − h 9a,11a ), (5.40) ph(10 ),h(8 )(h(ǫ̂))−ph(10a),h(8a)(h(12a)) > ∆t+(h 9a,11a β̂,ε̂−h 10a,12a (5.41) This is what we aimed at. By the same argument that earlier let us replace e by 7a′ and ê by 8a′ to derive formulas (5.34) and (5.35), we can now replace ǫ by 11a′ and ǫ̂ by 12a′ , arriving at ),h(7 )(h(11a′)) − ph(9a),h(7a)(h(11a)) > ∆t + (h 9a,11a ), (5.42) ph(10 ),h(8 )(h(12a′)) − ph(10a),h(8a)(h(12a)) > ∆t + (h 9a,11a ) + (h 10a,12a ). (5.43) Similarly to (5.36) and (5.37), we also get (15a′)−p13a ,11a(15a) > ∆t, and p14a′ ,12a′ (16a′)−p14a ,12a(16a) > ∆t. This completes the inductive step. The claim of the theorem follows imme- diately. � References [1] Dragan, V., Jones, A., Stacey, P., Repeated radicals and the real Fatou theorem, Austral. Math. Soc. Gaz. 29 (2002), 259–268. [2] Graczyk, J., Świa̧tek, G., Induced expansion for quadratic polynomials, Ann. Sci. Ecole Norm. Sup. 29(1996), 399–482. [3] Kozlovski, O., Shen, W., van Strien, S., Rigidity for real polynomials, to appear in Ann. Math. (2007). [4] Levin, G., On explicit connections between dynamical and parameter spaces, J. Anal. Math. 91 (2003), 297–327. [5] Levin, G.,Multipliers of periodic orbits of quadratic polynomials and the parameter plane, preprint (2007). [6] Lyubich, M., Dynamics of quadratic polynomials. I, II. Acta Math. 178 (1997), no. 2, 185–247, 247–297. . [7] de Melo, W., van Strien, S., One-Dimensional Dynamics, Springer, Berlin 1993. [8] Milnor, J., Thurston, W., Iterated Maps of the Interval, In: Dynamical Systems, Lect. Notes Math. 1342, Springer 1988, 465–563. [9] Pa luba, W., A Case of Monotone Ratio Growth for Quadratic-Like Map- pings, Bull. Pol. Acad. Sci. Math. 52 (2004), pp. 381–393. [10] Shishikura, M., Yoccoz puzzles, τ−functions and their applications, un- published. [11] Tsujii, M., A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems 20 (2000), pp. 925–933. Introduction Notation and preliminaries The period 4 super-sink The period 8 super-sink RLRLRLRL…

0704.1386

Disorder effect on the Friedel oscillations in a one-dimensional Mott insulator

Disorder effect on the Friedel oscillations in a one-dimensional Mott insulator Y. Weiss, M. Goldstein and R. Berkovits The Minerva Center, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel The Friedel oscillations resulting from coupling a quantum dot to one edge of a disordered one- dimensional wire in the Mott insulator regime, are calculated numerically using the DMRG method. By investigating the influence of the disorder on the Friedel oscillations decay we find that the effect of disorder is reduced by increasing the interaction strength. This behavior is opposite to the recently reported influence of disorder in the Anderson insulator regime, where disorder led to a stronger decay of the Friedel oscillations. PACS numbers: 73.21.Hb, 73.21.La, 71.45.Lr Properties of one-dimensional systems incorporating disorder and electron-electron interactions are the sub- ject of many recent theoretical and experimental studies. It is well known that when the interactions are not too strong, the addition of disorder turns the metallic system into an Anderson insulator (AI). However, for strong in- teractions (i.e., when the clean system is a Mott insula- tor) the exact effect of disorder depends on its strength, and in general is not completely understood. While for clean systems the Mott insulator (MI) phase is a well studied problem1, the addition of disorder opens a few questions, which have attracted several studies in the last decade2,3,4,5,6. Specifically, since most studies on disordered one- dimensional wires concentrate on either the AI or the MI phases, a full comparison between the two regimes is still lacking. Nevertheless, a qualitatively different behavior between these two regimes was demonstrated in a few cases. For example, the effect of interactions on the per- sistent current in one-dimensional disordered rings was calculated in previous works7,8, and an important differ- ence between the AI and MI phases was found. While for strong interactions and weak disorder (MI phase) the persistent current was reduced, for strong disorder (AI phase) an increase of the current was found. However, the exact diagonalization techniques which were used in these studies, are applicable only for very small system sizes. The definition of a disordered MI phase should be re- fined, since when the disorder is strong, i.e., when the random potential felt by the electron is much larger than any other energy scale in the problem, the MI state is destroyed. For a weak disorder, however, it was shown in several studies that the Mott energy gap vanishes only when a finite disorder is introduced, so that below this critical disorder the MI phase is stable9,10. Usually this is not the case for a MI consisting of spinless particles, since an Imry-Ma type of argument11 shows that the long range order is destroyed even for an infinitesimal disorder12. Yet, for a finite sized mesoscopic sample, the Imry-Ma length scale might be a few orders of magnitude larger than the sample’s size, so that the effective ground state for a weak enough disorder remains a MI one. Such finite one dimensional wires coupled to dots have been recently manufactured, and signatures of a charge den- sity wave in strong magnetic fields have been observed13. Increasing the disorder above a critical strength changes the MI state either to a Mott Glass or to an AI5,6. In this paper we investigate the influence of interac- tions on the Friedel oscillations (FO) in a disordered one- dimensional wire, and compare this behavior between the AI and the MI regimes. We study interacting spinless electrons confined to a 1D wire which can be in either its AI or MI phases. Without disorder, it is known that in order to get a MI phase the repulsive e-e interactions should be strong enough, while for weaker interactions the wire is described by the Tomonaga-Luttinger liquid (TLL) theory. The MI phase, for strong interactions, appears for spinless 1D electrons as a 2kF charge den- sity wave (CDW). When disorder is included, the TLL phase switches into an AI state. However, the finite size CDW state is expected, as noted above, to remain stable against the application of a weak enough disorder, i.e. to remain a MI state. For example, previous numerical simulations have presented the long range order of such a weakly disordered CDW2. In order to verify the exis- tence of the CDW order in the presence of disorder for the length scales considered, one should check the elec- tron density of the entire system. The behavior of the FO decay length in the presence of disorder in the AI phase, were discussed in a recent paper14. It was shown that the effect of disorder on the FO decay length can be described by an additional exponential term e−x/ξ, where ξ is a characteristic de- cay length. For a constant strength of (weak) disorder ξ decreases as the interactions increase, i.e. the disor- der effect is enhanced with increasing interactions. Ar- guing that ξ is a good approximation of the mobility localization length, it was found that it is in good accor- dance with theoretical predictions made using the TLL framework15,16,17, which are suitable for the weak inter- actions regime. However, for the CDW phase umklapp processes are important, and the above considerations are not applicable. In order to calculate the decay length of the weakly dis- ordered CDW wire, we use a method similar to the one used for the TLL regime. We couple the wire to a quan- tum dot with a single level from one end, and study the http://arxiv.org/abs/0704.1386v1 electrons density change in the sites nearby. The density change, which have shown Friedel oscillations with a 2kF wave vector and a power law decay in the metallic case (TLL), should now present 2kF oscillations with an expo- nential decay, since the underlying lattice state (CDW) is an insulator. For the clean case we will show that the exponential decay length scales as the CDW correlation length, ζ, as predicted18. However, in the disordered case we find an additional decay factor due to the disorder, as in the TLL case14. By calculating this decay length we are able to present a clear picture of the dependence of the decay length due to disorder on interactions, in both the AI and MI regimes. While the decay length of the FO due to disorder is monotonically decreasing as inter- action increases for the AI phase, for the MI phase it is monotonically increasing. The origin of the difference between these two regimes will be explained. The system under investigation is the strong electron electron interactions regime of a one-dimensional wire with disorder. The wire is modeled by a one-dimensional lattice of spinless fermions, moving in a random on-site potential, and experiencing nearest neighbor repulsive in- teractions. The Hamiltonian is Ĥwire = ǫj ĉ j ĉj − t j ĉj+1 + h.c.) (1) j ĉj − j+1 ĉj+1 − where ǫj are the random on-site energies, taken from a uniform distribution in the range [−W/2,W/2], I is the nearest neighbor interaction strength, and t, which is the hopping matrix element between nearest neighbors, sets the energy unit scale. ĉ j (ĉj) is the creation (annihila- tion) operator of a spinless electron at site j in the wire, and a positive background is included in the interaction term. Such a (clean) wire undergoes a phase transition at I = 2t between TLL and CDW. In order to study the CDW and the weakly disordered CDW phase the inter- action strength is taken to be strong, i.e. I > 2t. We now introduce a quantum dot with a single orbital at one end of the wire, by adding the following term to the Hamiltonian: Ĥdot = ǫ0ĉ 0ĉ0 − V (ĉ 0ĉ1 + h.c.) (2) +I(ĉ 0ĉ0 − 1ĉ1 − where ǫ0 describes the dot energy level. As in Ref. 14, we take ǫ0 ≫ W and V = t. The Hamiltonian Ĥ was diagonalized using a finite-size DMRG method14,19, and the occupation of the lattice sites were calculated, for different values of W and I. The dot energy level was taken to be ǫ0 = 10t. The size of the wire was up to L = 300 sites, which is both long enough due to the exponential decay of the calculated quantities, and still short enough to maintain the CDW order for the disorder strengths taken (W/t = 0.1 and 0.2). During the renormalization process the number of particles in the system is not fixed, so that the results describe the experimentally realizable situation of a finite section of a 1D wire which is coupled to a dot and to an external electron reservoir20. Yet, the calculated density remains close to half filling in all the calculated scenarios (even in the presence of disorder) since the interaction term contains a positive background, and the calculation is done for µ = 0. We start with the case in which no disorder is consid- ered (W = 0), so that the ground state of the CDW is twofold degenerate. This degeneracy is broken, however, once a pinning impurity, denoted by ǫ 0 → 0 +, is cou- pled to one end of the wire, and the wire shows a 2kF modulation20. The particle density of such a state, in the j-th site of the wire, will be denoted by N0j . When the pinning impurity is replaced by a dot level with ǫ0 ≫ ǫ the particle density in the wire (to be denoted as Nj) is changed by an oscillating 2kF term. 2kF oscillations in the density difference were also obtained in the TLL phase (Ref. 14), where the density without the quantum dot is flat, and the deviation of the population from this flat density once the lead is coupled to the dot shows Friedel oscillations. Here one should notice that the ref- erence state (without the dot) does not have a flat parti- cle distribution, but rather has a CDW 2kF oscillations. Coupling the dot results in a new CDW state, which has also 2kF oscillations, but with a different amplitude. The difference between these two states has a 2kF oscillation, which has an exponential decay from its value at the edge of the wire. 5 10 15 20 25 30 -0.05 FIG. 1: Typical oscillations for a clean sample with L = 280 for a CDW with I/t = 2.5. The upper panel shows Nj (cir- cles) and N0j (squares), and the lower panel presents their difference ∆Nj . In order to calculate the density difference between the cases when the quantum dot is coupled or uncoupled to the wire, one defines the density change in site j as ∆Nj = Nj −N j , (3) and studies the dependence of ∆Nj on j for different parameters. A typical result of Nj vs. N j , and the re- sulting ∆Nj , showing the 2kF oscillations caused by the dot orbital at j = 0, is presented in Fig. 1. When W 6= 0, the CDW ground state is no longer de- generate, and the infinitesimal pinning impurity is not required. The disorder itself pins the CDW to different places on the lattice, with the ability to break the long range order of the clean CDW by localized solitons, with a density which depends on the disorder strength2. Yet, when a dot level with ǫ0 ≫ W is connected to one side of the wire, the local effect in its vicinity is stronger than the pinning caused by the disorder. This results in a change of the particle density near the dot, and this change de- creases with distance. It turns out that the definition of ∆Nj in Eq. (3) is suitable for the disordered case as well, since it cancels out the disorder pinning effects which are the same for the two cases, isolating the density fluctua- tions created by the dot. A typical picture of ∆Nj for a disordered CDW sample is presented in Fig. 2. Whereas the upper panel shows the density of the two similar systems, one which is coupled to the quantum dot and the other is not, the lower panel presents the difference between these two densities, and the decay of the oscillations can be clearly seen. 5 10 15 20 25 30 FIG. 2: Typical oscillations for a single disordered sample with L = 280, W = 0.1 and ǫ0 = 10, for a CDW with I = 3. The upper panel shows Nj (circles) and N j (squares), and the lower panel presents their difference ∆Nj . Since the CDW is an insulating phase, the decay of the 2kF oscillations without disorder is supposed to be exponential and the characteristic length is the CDW correlation length2, i.e., ∝ exp(−x/ζ). In Fig. 3 such an exponential decay of ∆Nj is shown on a semi-log scale for various interaction strengths. An exact Bethe Ansatz solution18 of our model gives the relation between the correlation length and the interaction as ζ ∼ exp(π/ I/(2t)− 1). (4) The correlation lengths extracted from the DMRG re- sults are presented with a fit to the exact formula in the inset of Fig. 3. As can be seen, for I not very close to the TLL-CDW transition point (which occurs at I = 2t), the results fit the theory very well. 10 20 30 40j 0.0001 2 3 4 5I I=2.5 FIG. 3: The oscillations decay in the CDW regime for various interaction strengths and without disorder (note the semi-log scale). As the interaction increases, the correlation length decreases and the decay is faster. Inset: the inverse correla- tion length of the CDW state for various interaction strengths (symbols) fitted to the theory prediction Eq. (4). For W 6= 0, ∆Nj is averaged over 100 realizations, for which we expect a sampling error of the order of one percent. Assuming that the disorder adds another expo- nential term to the oscillations decay, which is thus pro- portional to exp(−x/ζ − x/ξ), there are two competing length scales - the decay length due to disorder (ξ) vs. the correlation length (ζ). For strong interactions and weak disorder ζ ≪ ξ so that the disorder effect is hardly seen, but increasing the disorder or decreasing the in- teraction strength should result in a combination of the two exponential decays. The DMRG results, presented in Fig. 4, show the disorder effect on the oscillations de- cay. For I = 2.5 and I = 3 one can see faster decay for the disordered samples with W = 0.1. For stronger interaction larger disorder is required in order to affect the decay. Similarly to the AI phase, the extra decay length can be extracted by fitting, for each value of I, the W 6= 0 curve multiplied by ex/ξ to the W = 0 one. Such a rescaling is presented in the inset of Fig. 4. As can be seen in Fig. 5, the decay length extracted for the disordered MI regime increases as a function of the interaction strength (for 2t < I <∼ 3.5t), an opposite behavior to the AI case (I < 2t). Results for stronger values of I are not shown, since for too strong interactions 10 20 30 40j 0.001 10 20 30 40j 0.001 FIG. 4: The decay of the oscillations of a disordered CDW with I = 2.5, 3 and 3.5 (top to bottom, note the semi-log scale). The lines correspond to the clean sample result, and the symbols to the averaged disordered data. For W = 0.1 (circles) the disorder effect is clearly seen for I = 2.5 and I = 3 but not for I = 3.5 in which ξ is much larger than the correlation length ζ. For W = 0.2 (squares) ξ is small enough to affect the decay even for I = 3.5. Inset: multiplying ∆Nj by ex/ξ collapses the disordered data on the clean curves. 0 1 2 3 4 300 300 600 600 900 900 1200 1200 1500 1500 2 2.5 3 3.5 4 FIG. 5: The decay length due to disorder (ξ) in the TLL (I < 2t) and in the CDW (I > 2t) phases as a function of the interaction strength. The data for the TLL phase was taken from Ref. 14. Inset: zoom into the CDW regime. the correlation length is very small, and thus the estimate of ξ is less accurate. These results point out that as the interaction strength increases in the MI phase, the disorder effect decreases. In the AI phase, on the other hand, the disorder effect is enhanced with increasing interactions. The difference between these two behaviors results from the difference in the ground states of the two phases in the clean case. In our model there is a competition between the kinetic energy (the hopping term) and the potential (the inter- action). The hopping term prefers the existence of a flat particle distribution whereas the interaction term prefers a CDW-like form. For different values of I the results of that competition are different: for I < 2t (the TLL phase) the hopping term wins, and the distribution is flat, while for I > 2t (the CDW phase) a CDW starts to form. Inside the clean TLL phase, as I increases, the CDW fluctuations are stronger. Yet, the average density profile in the ground state remains flat because of the hopping term. But when disorder is introduced, the flat density state becomes less favorable than a state with a fluctu- ating density, the latter being preferred by both the dis- order and the interactions. For a constant disorder, as the interactions become stronger, these fluctuations are enhanced, so the disorder effect increases. In the CDW phase, on the other hand, without dis- order, the interaction wins over the hopping, and the ground state has a CDW form. Turning on the disorder might change the particle distribution, e.g. by allowing an electron to move into a site with lower on-site energy, but this results in raising the interaction energy. As the interaction strength gets stronger, the probability of such a process decreases, so that the actual effect of the dis- order is getting weaker. In conclusion, while the decay length of the 2kF os- cillations envelope due to disorder is monotonically de- creasing in the AI phase, we have shown that it is mono- tonically increasing in the disordered MI phase. The dif- ference between these two regimes is explained by the difference between the ground states of the clean sam- ples in each case. In the AI phase the pure ground state is flat, and both the disorder and the interactions try to introduce fluctuations in it. In the MI phase, on the other hand, the pure ground state oscillates with a 2kF wave vector, and these oscillations are enhanced by the interactions and reduced by the disorder. As a result, the disorder effect (for a constant disorder strength) is getting weaker as the interactions are enhanced. Acknowledgments Support from the Israel Academy of Science (Grant 877/04) is gratefully acknowledged. 1 N. F. Mott, Metal Insulator Transitions (Taylor and Fran- cis, London, 1990) 2 H. Pang, S. Liang and J. F. Annett, Phys. Rev. Lett. 71, 4377 (1993). 3 S. Fujimoto and N. Kawakami, Phys. Rev. B 54, R11018 (1996). 4 M. Mori and H. Fukuyama, J. Phys. Soc. Jpn. 65, 3604 (1996). 5 E. Orignac, T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 83, 2378 (1999). 6 T. Giamarchi, P. Le Doussal and E. Orignac, Phys. Rev. B 64, 245119 (2001). 7 M. Abraham and R. Berkovits, Phys. Rev. Lett. 70, 1509 (1993). 8 G. Bouzerar, D. Poilblanc, and G. Montambaux, Phys. Rev. B 49, 8258 (1994). 9 M. Ma, Phys. Rev. B 26, 5097 (1982). 10 A. W. Sandvik, D. J. Scalapino and P. Henelius, Phys. Rev. B 50, 10474 (1994). 11 Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975). 12 R. Shankar, Int. J. Mod. Phys. B 4, 2371 (1990). 13 K. J. Thomas, D. L. Sawkey, M. Pepper, W. R. Tribe, I. Farrer, M. Y. Simmons and D. A. Ritchie, J. Phys.: Condens. Matter 16, L279 (2004). 14 Y. Weiss, M. Goldstein and R. Berkovits, Phys. Rev. B 75, 064209 (2007). 15 W. Apel, J. Phys. C 15, 1973 (1982). 16 Y. Suzumura and H. Fukuyama, J. Phys. Soc. Jpn. 52, 2870 (1983). 17 T. Giamarchi, Quantum Physics in One Dimension (Ox- ford University Press, New York, 2003). 18 H. J. Mikeska and A. K. Kolezhuk, in Quantum Mag- netism, Lecture Notes in Physics 645, edited by U. Schollwöck, J. Richter, D. J. J. Farnell and R. F. Bishop (Springer-Verlag, 2004). 19 S. R. White, Phys. Rev. B 48, 10345 (1993). 20 Y. Weiss, M. Goldstein and R. Berkovits, J. Phys.: Con- dens. Matter 19, 086215 (2007).

0704.1387

Description of the Scenario Machine

DESCRIPTION OF THE SCENARIO MACHINE V.M. Lipunov1, K.A. Postnov2, M.E. Prokhorov3, A.I. Bogomazov4 Sternberg astronomical institute, Universitetskij prospect, 13, 119992, Moscow, Russia ABSTRACT We present here an updated description of the “Scenario Machine” code. This tool is used to carry out a population synthesis of binary stars. Previous version of the descrip- tion can be found at http://xray.sai.msu.ru/∼mystery//articles/review/contents.html; see also (Lipunov et al. 1996b,c). Subject headings: binaries: close — binaries: general 1. Basic equations and initial distributions We use the current scenario of evolution of bi- nary stellar systems based upon the original ideas that appeared in the papers by Paczyn’ski (1971); Tutukov & Yungelson (1973), van den Heuvel & Heise (1972) (see also re- view by van den Heuvel (1994)). The sce- nario for normal star evolution was joined with the ideas of neutron star evolution (see pi- oneer works by Shvartsman (1970, 1971a,b); Illarionov & Sunyaev (1975); Shakura (1975); Bisnovatyi-Kogan & Komberg (1976), Lipunov & Shakura (1976), Savonije & van den Heuvel (1977)). This joint scenario has allowed to construct a two-dimensional classification of possible states of binary systems containing NS (Kornilov & Lipunov (1983a,b); Lipunov (1992)). According to this classification, we will distinguish four basic evolutionary stages for a normal star in a binary system: I — A main sequence (MS) star inside its Roche lobe (RL); II — A post-MS star inside its RL; III — AMS or post-MS star filling its RL; the mass is transferred onto the companion. 1E-mail: [email protected] 2E-mail: [email protected] 3E-mail: [email protected] 4E-mail: [email protected] IV — A helium star left behind the mass-transfer in case II and III of binary evolution; may be in the form of a hot white dwarf (for M ≤ 2.5M⊙), or a non-degenerate helium star (a Wolf-Rayet-star in case of initial MS mass > 10M⊙). The evolution of single stars can be represented as a chain of consecutive stages: I→ II→ compact remnant; the evolution of the most massive single stars probably looks like I → II → IV → compact remnant. The component of a binary system can evolve like I→ II→ III→ IV→ compact remnant. In our calculations we choose the distributions of initial binary parameters: mass of the primary zero age main sequence component (ZAMS), M1, the binary mass ratio, q = M2/M1 < 1, the orbital separation a. Zero initial eccentricity is assumed. The distribution of binaries by orbital separa- tions can be taken from observations (Krajcheva et al. 1981; Abt 1983), f(log a) = const, max(10R⊙,RL[M1]) < a ≤ 106R⊙; Especially important from the evolutionary point of view is how different are initial masses of the components (see e.g. Trimble (1983)). We have parametrized it by a power-law shape, assuming the primary mass to obey Salpeter’s power law: f(M) = M−2.351 , 0.1M⊙ < M1 < 120M⊙, (2) http://arxiv.org/abs/0704.1387v2 http://xray.sai.msu.ru/~mystery//articles/review/contents.html f(q) ∼ qαq , q = M2/M1 < 1; (3) We should note that some apparently reason- able distributions – such as both the primary and secondary mass obeying Salpeter’s law, or “hier- archical” distributions involving the assumption that the total binary mass and primary’s mass are distributed according to the Salpeter mass func- tion – all yield essentially flat-like distributions by the mass ratio (i.e. with our parameter αq ≃ 0). We assume that the neutron star is formed in the core collapse of the pre-supernova star. Masses of the young neutron stars are randomly distributed in the range MminNS – M NS . Initial NS masses are taken to be in the range MNS = 1.25 − 1.4M⊙. The range of initial masses of the young NSs was based on the masses of the neu- tron star in the B1913+16 binary system and of the radio pulsar in the J0737-3039 binary. In the B1913+16 system (the Hulse-Taylor pulsar, radio pulsar + neutron star) the mass of the neutron star, which is definitely not accreting matter from the optical donor, is MNS = 1.3873 ± 0.0006M⊙ (Thorsett & Chakrabarty 1999; Wex et al. 2000; Weisberg & Taylor 2003). The mass of the pul- sar in the J0737-3039 system (radio pulsar + radio pulsar), which likewise does not accrete from an optical companion, is MPSR = 1.250 ± 0.010M⊙ (Lyne et al. 2004). The mass range of stars pro- ducing neutron stars in the end of their evolution is assumed to be Mn – Mb; stars with initial masses M > Mb are assumed to leave behind black holes; Mn is taken to be equal to 10M⊙ in most cases (but in general this parameter is free). Note that according to some stellar models, a very massive star (≈ 50 − 100M⊙) can leave behind a neutron star as a remnant due to very strong mass loss via powerful stellar wind, so we account for this possibility in the corresponding models. We take into account that the collapse of mas- sive star into a neutron star can be asymmetrical, so that the newborn neutron star can acquire an additional, presumably randomly oriented in space kick velocity w (see Section 4 below for more de- tails). The magnetic field of rotating compact ob- jects (neutron stars and white dwarfs) largely define the evolutionary stage of the compact object in a binary system (Shvartsman (1970); Davidson & Ostriker (1973); Illarionov & Sunyaev (1975)), so we use the general classification of mag- netic rotating compact objects (see e.g. Lipunov (1992)) in our calculations. The initial magnetic dipole moment of the newborn neutron star is taken according to the distribution f(logµ) ∝ const, 1028 ≤ µ ≤ 1032G cm3, (4) The initial rotational period of the newborn neu- tron star is assumed to be ∼ 10 ms. It is not definitely clear as yet whether the mag- netic field of neutron stars decays or not (see for a comprehensive review Chanmugam (1992)). We assume that the magnetic fields of neutron stars decays exponentially on a timescale of td (usually we take this parameter to be equal to 108, 5 · 107 and 107 years). A radio pulsar is assumed to be “switched on” until its period P (in seconds) has reached the “death-line” defined by the relation µ30/P d = 0.4, where µ30 is the dipole magnetic moment in units of 1030 G cm3. We assumed that magnetic fields of neu- tron stars decay exponentially to minimal value Bmin = 8 · 107 G and do not decay further: B0 exp(−t/td), t < td ln(B0/Bmin), Bmin, t ≥ td ln(B0/Bmin). Parameters B0 and td in equation (5) are the ini- tial field strength and the field decay time. We also assume that the mass limit for neutron stars (the Oppenheimer-Volkoff limit) is MOV = 2.0M⊙ (in general, it is a free parameter in the code; it depends on equation of state of the mate- rial of the neutron star). The most massive stars are assumed to leave be- hind black holes after the collapse, provided that the progenitor mass before the collapse has a mass Mcr. The masses of the black holes are calcu- lated as Mbh = kbhMPreSN , where the parameter kbh = 0.0 − 1.0, MPreSN is the mass of the pre- supernova star. We consider binaries with M1 ≥ 0.8M⊙ with a constant chemical (solar) composition. The pro- cess of mass transfer between the components is treated as conservative when appropriate, that is the total angular momentum of the binary system is assumed to be constant. If the accretion rate from one component to another is sufficiently high (say, the mass transfer occurs on a timescale few times shorter than the thermal Kelvin-Helmholtz time for the normal companion) or a compact ob- ject is engulfed by a giant companion, the com- mon envelope stage of binary evolution can begin (Paczynski 1976; van den Heuvel 1983). Other cases of non-conservative evolution (for example, stages with strong stellar wind or those where the loss of binary angular momentum oc- curs due to gravitational radiation or magnetic stellar wind) are treated using the well known prescriptions (see e.g. Verbunt & Zwaan (1981); Rappaport et al. (1982); Lipunov & Postnov (1988)). 2. Evolutionary scenario for binary stars Significant discoveries in the X-ray astronomy made during the last decades stimulated the as- tronomers to search for particular evolutionary ways of obtaining each type of observational ap- pearance of white dwarfs, neutron stars and black holes, the vast majority of which harbours in bi- naries. Taken as a whole, these ways costitute a general evolutionary scheme, or the “evolutionary scenario”. We follow the basic ideas about stel- lar evolution to describe evolution of binaries both with normal and compact companions. To avoid extensive numerical calculations in the statistical simulations, we treat the continu- ous evolution of each binary component as a se- quence of a finite number of basic evolutionary states (for example, main sequence, red super- giant, Wolf-Rayet star, hot white dwarf, etc.), at which stellar parameters significantly differ from each other. The evolutionary state of the binary can thus be determined as a combination of the states of each component, and is changed once the more rapidly evolving component goes to the next evolutionary stage. At each such stage, we assume that the star does not change its physical parameters (mass, ra- dius, luminosity, the rate and velocity of stellar wind, etc.) that have effect on the evolution of the companion (especially in the case of compact magnetized stars). Every time the faster evolving component passes into the next stage, we recalcu- late its parameters. Depending on the evolution- ary stage, the state of the slower evolving star is changed or can remain unchanged. With some ex- ceptions (such as the common envelope stage and supernova explosion), states of both components cannot change simultaneously. Whenever possible we use analytical approximations for stellar pa- rameters. Prior to describing the basic evolutionary states of the normal component, we note that unlike single stars, the evolution of a binary compo- nent is not fully determined by the initial mass and chemical composition only. The primary star can fill its Roche lobe either when it is on the main sequence, or when it has a (degenerate) he- lium or carbon-oxygen core. This determines the rate of mass transfer to the secondary compan- ion and the type of the remnant left behind. We will follow Webbink (1979) in treating the first mass exchange modes for normal binary compo- nents, whose scheme accounts for the physical state of the star in more detail than the simple types of mass exchange (A, B, C) introduced by Kippenhahn & Weigert (1967). We will use both notations A, B, C and D (for very wide systems with independently evolving companions) for evo- lutionary types of binary as a whole, and Web- bink’s notations for mass exchange modes for each component separately [Ia], [Ib], [IIa], [IIb], [IIIa], 3. Basic evolutionary states of normal stars The evolution of a binary system consisting ini- tially of two zero-age main sequence stars can be considered separately for each components until a more massive (primary) component fills its Roche lobe. Then the matter exchange between the stars begins. The evolutionary states of normal stars will be denoted by Roman figures (I-IV), whereas those of compact stars will be marked by capital letters (E, P, A, SA ...). We divide the evolution of a nor- mal star into four basic stages, which are signifi- cant for binary system evolution and bear a clear physical meaning. We will implicitly express the mass and radius of the star and the orbital semi- major axis in solar units (m ≡ M/M⊙, r ≡ R/R⊙, a ≡ A/R⊙), the time in million years, the lumi- nosities in units of 1038 erg s−1, the wind velocities in units of 108 cm s−1 and the accretion rates Ṁ onto compact objects in units of 10−8M⊙ yr unless other units are explicitely used. 3.1. Main sequence stars At this stage, the star is on the zero-age main sequence (ZAMS) and its size is much smaller than the Roche lobe radius. The time the star spends on the main sequence is the core hydrogen burning time, tH , which depends on the stellar mass only (Iben & Tutukov 1987): 1.0 + 0.95 79 , m ≥ 79.0, 103.9−3.8 logm+log 2 m, 79 > m ≥ 10, 2400m−2.16, 10 > m ≥ 2.3, 104m−3.5, m < 2.3, The radius of the ZAMS star is assumed to be 100.66 logm+0.05, m > 1.12, m, m ≤ 1.2, (7) and its luminosity is logL = −5.032 + 2.65 logm, (α) −4.253 + 4.8 logm, (β) −4.462 + 3.8 logm, (γ) −3.362 + 3.0 logm, (δ) −3.636 + 2.7 logm, (ǫ) here we assume the next indication: (α), m < 0.6; (β), 0.6 ≤ m < 1.0; (γ), 1.0 ≤ m < 10.0; (δ), 10 ≤ m < 48.0; (ǫ), m ≥ 48.0. The initial mass of the primary and the mode of the first mass exchange (which is determined by the initial orbital period and masses of the com- ponents; see Webbink (1979)) determine the mass and the type of the core that will be formed during stage I. For example, for single stars and primaries in “type C” binaries that fill its Roche lobe having a degenerate core, we use the expressions 0.1mmax, (α) 0.446 + 0.106mmax, (β) 0.24m0.85max, (γ) min(0.36m0.55max, 0.44m max), (δ) 0.44m0.42max, (ǫ) ≈ MCh, (ζ) 0.1m1.4max, (η) where mmax is the maximum mass the star had during the preceding evolution. We assume the next indication in this formula: (α), mmax < 0.8; (β), 0.8 ≤ mmax < 2.3; (γ), 2.3 ≤ mmax < 4.0; (δ), 4.0 ≤ mmax < 7.5; (ǫ), 7.5 ≤ mmax < 8.8; (ζ), 8.8 ≤ mmax < 10.0; (η), mmax ≥ 10.0. A main-sequience star accreting matter during the first mass transfer will be treated as a rapidly rotating “Be-star” with the stellar wind rate dif- ferent from what is expected from a single star of the same mass (see below). 3.2. Post main-sequence stars The star leaves the main sequence and goes to- ward the red (super)giant region. The star still does not fill its Roche lobe. The duration of this stage for a binary component is not any more a function of the stellar mass only (as in the case of single stars), but also depends on the initial binary type (A, B, or C) (see Iben & Tutukov (1985, 1987)): tII = 0, (α), 2tKH , (β), 6300m−3.2, (γ), tHe, (δ), In type A systems, the primaries fill their Roche lobes when they belong to the main sequence. We assume the next indication in this formula: (α), type A; (β), type B excluding mode [IIIA]; (γ), types C, D and mode [IIIA], mmax < 5; (δ), types C, D and mode [IIIA], mmax > 5. The radius of the post-MS star rapidly increases (on the thermal time scale) and reaches the char- acteristic giant values. The star spends the most time of helium burning with such large radius. In the framework of our approximate description, we take the radius of the giant star to be equal to the maximum value, which depends strongly on the mass of its core and is calculated according to Webbink’s mass transfer modes as follows (see also Iben & Tutukov (1985, 1987)): rII = 3000m4c, (α) 1050(mc − 0.5)0.68, (β) 10m0.44c , (γ) We assume the next indication in this formula: (α), mode [IIIA] or [IIIB] with He core; (β), mode [IIIB] with CO or ONeMg core; (γ), modes [I] or [II]. Formula (11) depicts stars with mass ≤ 10M⊙. This maximum radius can formally exceed the Roche lobe size; in such cases we put it equal to 0.9RL during the stage II. The most sensi- tive to this crude approximation are binaries with compact companions, which can lead, for exam- ple, to the overestimation of the number of ac- creting neutron stars observed as X-ray pulsars. However, these stages are less important for our analysis than the stages at which the optical star fills its Roche lobe. A more detailed treatment of normal star evolution (given, for example, by Pols & Marinus (1994)) can reduce such uncer- tainties. Luminosities of giants are taken from de Jager (1980): log lII = 15.92m6c (1.0+m4c)(2.512+3.162mc) , (α) 10−4.462+3.8 logm, (β) 10−3.362+3.0 logm, (γ) 10−3.636+2.7 logm, (δ) We assume the next indication in this formula: (α), m < 23.7, mc < 0.7; (β), m < 23.7, mc > 0.7; (γ), 48 > m > 23.7; (δ), m > 48. Radii of (super)giants are determined by us- ing the effective temperature Teff and luminosi- ties. Typical effective temperatures are taken from Allen (1973): logTeff = 4.50, (α) 3.60, (β) 3.70, (γ) We assume the next indication in this formula: (α), m > 10.0; (β), m < 10.0, type C or D; (γ), m < 10.0, type A or B. So, we calculate RII with mass higher than 10M⊙ calculate using formula rII = exp {2.3(0.5 log lii − 2 logTeff + 9.7)}. 3.3. Roche lobe overflow At this stage the star fills its Roche lobe (RL) and mass transfer onto the companion occurs. The mass transfer first proceeds on the thermal time scale (see extensive discussion of this approx- imaiton in van den Heuvel (1994)) tKH ∼ 30m2r−1∗ (L/L⊙)−1, (15) The common envelope stage (CE) may be formed if the Roche lobe overflow occurs in the type C system (where the primary has a well de- veloped core) even for q < 1; otherwise (for type B systems) we use the condition q ≤ qcr = 0.3 for the CE stage to occur. Radius of the star at the Roche lobe filling stage is taken to be that of the equivalent Roche lobe radius (Eggleton 1983): a(1− e) 0.49q2/3 0.6q2/3 + ln(1 + q1/3) , (16) Here a is the binary orbital separation and e is the orbital eccentricity, q is the arbitrary mass ratio. For q < 0.6 a more precise approximation can be used: = 0.4622 1 + q . (17) A star filling its Roche lobe has quite differ- ent boundary conditions in comparison with sin- gle stars. The stellar radius at this stage is limited by the Roche lobe. If the stellar size exceeds the Roche lobe, the star can lose matter on a time scale close to the dynamical one until its radius becomes smaller than the new Roche lobe size. Now we consider how the RL-filling star loses matter. Let the star be in equilibrium and Req(M) = RL(M) at the initial moment of time. When a fraction of mass δm is transported to the companion, the mass ratio q and semi-major axis a of the binary changes depending on the mass transfer mode assumed (see below for details). The RL size then becomes equal to RL(M − δm). On the other hand, mass loss disturbs equi- librium of the star (hydrodynamical and ther- mal). The hydrodynamical equilibrium is restored on the dynamical timescale td ∝ GM/R3 )−1/2 The stellar radius changes to a value Rad(M − δm) (where “ad” means adiabatic), which can be bigger or smaller than the equilibrium radius Req(M − δm) of the star. The thermal equilib- rium establishes on the thermal time scale TKH ≈ GM2/RL, so after that the stellar radius relaxes to the equilibrium value Req(M − δm). Relations RL(M − δm), Rad(M − δm) and Req(M−δm) determine the mode of mass transfer during the RL overflow stage. Following Webbink (1985), one usually introduces the logarithmic derivative ζ = d lnR/d lnM (R ∝ M ζ). It lo- cally fits the real dependence R(M). Three values of ζ are relevant: d lnRL d lnM ζad = d lnRad d lnM , (18) ζeq = d lnReq d lnM where “L”, “ad” and “eq” correspond to the values of radii discussed above. Three possible cases are considered depending on ζi: 1. If ζad < ζL, the star cannot be inside its RL regardless of the mass loss rate (dM/dt < 0). Such stars lose their matter in hydrodynami- cal time scale. Mass loss rate is limited only by the speed of sound near the inner La- grangian point L1. ζeq is unimportant be- cause the size of the star becomes bigger and bigger than RL. The equilibrium is impossi- 2. ζeq < ζL < ζad. The star losing mass cannot be in thermal equilibrium, because otherwise its size would exceed RL. Nevertheless, in this case Rad < RL. So the hydrodynamical equilibrium is established. As a result, the star loses mass on thermal time scale. 3. ζL < ζad, ζeq . In this case the size of the star losing mass becomes smaller than its RL. The evolutionary expansion of the star or the binary semi-major axis decrease due to orbital angular momentum loss via magnetic stellar wind (MSW) or gravitational radia- tion (GW) support the permanent contact of the star with RL. The star then loses mass on a time scale dictated by ots own evolu- tionary expansion or on a time scale corre- sponding to the orbital angular momentum loss. For non-degenerate stars, Req increases mono- tonically with M . On the other hand, the expo- nent ζad is determined by the entropy distribution over the stellar radius which is different for stars with radiative and convective envelopes. It can be shown that stars with radiative envelopes should shrink in response to mass loss, while those with convective envelopes should expand 5. Therefore, stars with convective envelopes in bi- naries should generally have a higher mass loss rate than those with radiative envelopes under other equal conditions. The next important fac- tor is the dependence RL(M). It can found by substituting one of the relations a(M) (see below) into equation (16) or into equation (17) and dif- ferentiating it with respect to M . For example, assuming the conservative mass exchange when the total mass and the orbital angular momen- tum of the system do not change, one readily gets that the binary semi-major axis decreases when the mass transfer occurs from the more massive to the less massive component; RL decreases of the primary correspondingly. When the binary mass ratio reaches unity, the semi-major axis takes on a minimal value. In contrast, if the less massive star loses its mass conservatively the system expands. In that case the mass transfer can be stable. If more massive component with radiative en- velope fills its RL, the mass transfer proceeds on thermal time scale until the masses of the compo- nents become equal 6. The next stage of the first mass exchange poceeds in more slower (nuclear) time scale. If the primary has convective enve- lope, the mass transfer can proceed much faster on a time scale intermediate between the thermal and hydrodynamical one, and probably on the hy- drodynamical time scale. In that case the fast stage of the mass exchange ends when the mass of the donor decreases to ∼ 0.6 mass of the sec- ondary companion (Tutukov & Yungelson 1973). Further mass transfer should proceed on the evo- lutionary time scale. 5The adiabatic convection in stellar envelope can be de- scribed by the polytropic equation of state P ∝ ρ5/3, simi- lar to non-relativistically degenerate white dwarfs. For such equation of state the mass-radius relation becomes inverse: R ∝ M−1/3. For non-degenerate stars with convective en- velopes this relation holds approximately. 6The mass exchange can stop earlier if the entire envelope is lost and the stellar core is stripped (the core has other values ζad and ζeq). The process of mass exchange strongly depends on stellar structure at the moment of the RL over- flow. The structure of the star in turn depends on its age and the initial mass. The moment of the RL overflow is determined by the mass of the components and by the initial semi-major axis of the system. to calculate a diagram in the M − a (or M − Porb) plane which allows us to conclude when the primary in a binary with given initial pa- rameters fills its RL, what is its structure at that moment and what type of the first mass exchange is expected. We use the diagram calculated by Webbink (1979) (see also the description of mod- ern stellar wind scenarios below). We distinguish different sub-stages of the RL overflow according to the characteristic timescales of the mass transfer: stage III: This is the most frequent case for the first mass transfer phase. The primary fills its RL and the mass transfer proceeds faster than evolutionary time scale (if outer layers of the star are radia- tive, then it is thermal time scale, if outer layers are convective, then time scale is shorter, up to hydrodynamical time scale). This stage comes to the end when mass ratio in the system changes (“role-to-role transition”), i.e. when the mass of the donor (mass losing) star is equal to mass of second companion (for radiative envelopes) or 0.6 of the mass of the second companion (for convec- tive envelopes). This stage also stops if the donor star totally lost its envelope. stage IIIe: This is the slow (evolutionary driven) phase of mass transfer. We assume it to occur in short- period binaries of type A. However, it is not ex- cluded that it may occur after the mass reversal during the first stage of mass exchange for binaries of type B (see van den Heuvel (1994)), e.g. as in wide low-mass X-ray binaries. stage IIIs This is the specific to super-accreting com- pact companions substage of fast mass transfer at which matter escapes from the secondary compan- ion carrying away its orbital angular momentum. Its duration is equal to tIIIs = tKH q(1 + q) 2− q − 2q2 , (19) q = Ma/Md < 0.5, here and below subscripts “a” and “d” refer to the accreting and donating mass star, respectively. For systems with small mass ratios, q < 0.5, this timescale corresponds to an effective q-time short- ening of the thermal time for the RL-overflowing star. stages IIIm,g At these stages, the mass transfer is controlled by additional losses of orbital angular momentum Jorb caused by magnetic stellar wind or gravita- tional wave emission. The characteristic time of the evolution is defined as τJ = −(Jorb/J̇orb), and in the case of MSW is (see Verbunt & Zwaan (1981); Iben & Tutukov (1987)) τMSW = 4.42 a5mxλ (m1 +m2)2m4op , (20) Here mop denotes mass of the low-mass optical star (0.3 < m < 1.5) that is capable of producing an effective magnetic stellar wind (because only such stars have outer convective envelopes which are prerequisit for effecftive MSW), λ is a numer- ical parameter of order of unity. We have used the mass-radius relation r ≈ m for main sequence stars in deriving this formula. The upper limit of the mass interval and empirical braking law for main-sequence G-stars are taken from Skumanich (1972), the lower limit is determined by absence of cataclysmic variables with orbital periods Porb ≈ 3h (Verbunt 1984; Mestel 1952; Kawaler 1988; Tout & Pringle 1992; Zangrilli et al. 1997). We use λ = 1 (see for details Kalogera & Webbink (1998)). The time scale of the gravitational wave emis- sion is τGW = 124.2 m1m2(m1 +m2) × (21) (1− e2)−7/2, Wether the evolution is governed by MSW or GW is decided by which time scale (τMSW or τGW ) turns out to be the shortest among all ap- propriate evolutionary time scales. stage IIIwd This is a special case where the white dwarf overflows its RL. This stage is encountered for very short period binaries (like Am CVn stars and low- mass X-ray binaries like 4U 1820-30) whose evolu- tion is controlled by GW or MSW. The mass trans- fer is calculated using the appropriate time scale (GW or MSW). The radius of the white dwarf in- creases with mass RWD ∝ M−1/3. This fact, how- ever, does not automatically imply that the mass transfer is unstable, since the less massive WD fills its RL first. It can be shown that the mass exchange is always stable in such systems if the mass ratio q < 0.8. This condition always holds in WD+NS and WD+BH systems. WD loses its matter until its mass decreases to that of a huge Jupiter-like planet (∼ a few 10−3M⊙), where the COulomb interaction reverses the mass-radius re- lation R(M). Such a planet can approach the sec- ondary companion of the system due to GW emis- sion until the tidal forces destroy it completely. The matter of the planet can fall onto the sur- face of the second companion ir form a long-living disk around it. If the second star is a neutron star and its rotation had been spun up by accretion such that a millisecond radio pulsar appeared, the planet can be evaporated by relativistic particles emmited by the pulsar (Paczynski & Sienkiewicz 1983; Joss & Rappaport 1983; Kolb et al. 1998; Kalogera & Webbink 1998). The mass loss rate at each of the III-stages is calculated according to the relation Ṁ = ∆M/τi, (22) where ∆M is the a priori known mass to be lost during the mass exchange phase (e.g. ∆M = M1− (M1 +M2)/2 in the case of the conservative stage III, or ∆M = M1−Mcore(M1) in case of III(e,m,g) or CE) and τi is the appropriate time scale. The radius of the star at stage III is assumed to follow the RL radius: Rdl (Md(t)) = Rd(Md(t)). (23) 3.4. Wolf-Rayet and helium stars In the process of mass exchange the hydrogen envelope of the star can be lost almost completely, so a hot white dwarf (for m ≤ 2.5), or a non- degenerate helium star (for higher masses) is left as a remnant. The life-time of the helium star is determined by the helium burning in the stellar core (Iben & Tutukov 1985) tHe = 1658m−2, (α) 1233m−3.8, (β) 0.1tH , (γ) 6913m−3.47, (δ) ≃ 10, (ǫ) 0.1tH , (ζ) We assume the next indication in this formula: (α), m < 1.1, modes [IIA-IIF]; (β), m > 1.1, mmax < 10, modes [IIA-IIF]; (γ), mmax > 10, modes [IIA-IIF]; (δ), mode [IIIA]; (ǫ), mode [IIIC]; (ζ), modes [IIIB,D,E] and type D. If the helium (WR) star fills its Roche lobe (a relatively rare so-called “BB” case of evolu- tion; Delgado & Thomas (1981); see discussion in van den Heuvel (1994)), the envelope is lost and a CO stellar core is left with mass 1.3 + 0.65(m− 2.4),m ≥ 2.5, 0.83m0.36,m < 2.5, The mass-radius dependence in this case is (Tutukov & Yungelson 1973) rWR = 0.2m 0.6. (26) 3.5. Stellar winds from normal stars The effect of the normal star on the compact magnetized component is largely determined by the rate Ṁ and the velocity of stellar wind at in- finity v∞, which is assumed to be v∞ = 3vp ≈ 1.85 m/r, (27) where vp is the escape velocity at the stellar sur- face. For “Be-stars” (i.e. those stars at the stage “I” that increased its mass during the first mass ex- change), the wind velocity at the infinity is taken to be equal to the Keplerian velocity at the stellar surface: GM/R ≈ 0.44 m/r . (28) The lower stellar wind velocity leads to an effec- tive increase of the captured mass rate by the sec- ondary companion to such “Be-stars”. The stellar wind mass loss rate at the stage “I” is calculated as (de Jager 1980) ṁ = 52.3αwl/v∞, (29) Here αw = 0.1 is a numerical coefficient (in gen- eral, we can treat it as free parameter). For giant post-MS stars (stage “II”) we assume v∞ = 3vp and for massive star we take the max- imum between the stellar wind rate given by de Jager’s formula and that given by Lamers (1981) ṁ = max(52.3αw , 102.33 l1.42r0.61 m0.99 ), (30) M ≥ 10M⊙; For red super-giants we use Reimers’s formula (Kudritzki & Reimers 1978): ṁ = max(52.3αw , 1.0 ), (31) M ≥ 10M⊙; For a Wolf-Rayet star the stellar wind loss rate can significantly increase (up to 10−5M⊙ year We parametrize it as ṀWR = kWRMWR/tHe, (32) where the numerical coefficient is taken to be kWR = 0.3 (in general, it can be changed if nec- essary). The mass loss in other stages (MS, (su- per)giant) is assumed to be limited by 10% of the mass of the star at the beginning of the stage. 3.6. Change of binary parameters: mass, semi-major axis and eccentricity The duration of any evolutionary stage is de- termined by the more rapidly evolving compo- nent ∆t = min(∆t1,∆t2). On the other hand, based on the evolutionary considerations we are able to calculate how the mass of the faster evolv- ing star changes (e.g. due to the stellar wind or RL overflow), that is we can estimate the quantity ∆M = Mi −Mf . Then we set the characteristic mass loss rate at this stage as Ṁo = ∆M/∆t, (33) Next, we should calculate the change of mass for the slower evolving companion, the orbital semi- major axis and the eccentricity. 3.7. Mass change The mass of the star loosing matter is calcu- lated as Mf = Mi − Ṁo ×∆t, (34) Accordingly, the mass of the accreting star is Mf = Mi + Ṁc ×∆t, (35) where Ṁc is the accretion rate of the captured matter. For stages without RL overflow the accretion rate of the captured stellar wind matter is ṁc = 3.8×10−2 a(v2w + 0.19(m1 +m2)/a) ṁo . At stages where RL overflow ocurs and both components are normal (non-degenerate), we will assume that the accretor can accomodate mass at the rate determined by its thermal time, i.e. ṁc = ṁo tKH(donor) tKH(accretor) . (37) This means that the evolution can not be fully conservative, especially during the first mass transfer where the primary component usually has a shorter thermal time scale. The mass increase rate by compact accretors is assumed to be limited by the critical Edding- ton luminosity (see, however, the possible hyper accretion stage discussed below): LEdd = 4πGMmp ≈ 1.3 · 1038 ×m erg/s (38) (σT is the Thomson cross-section) at the stopping radius Rstop for the accreted matter (see, e.g., de- tailed discussion in Lipunov (1992)). This corre- sponds to the critical accretion rate Ṁcr = Rstop . (39) Thus, the mass of the accreting compact star at the end of the stage is determined by the relation Mf = Mi +min(Ṁc, Ṁcr)×∆t . (40) 3.8. Semi-major axis change The binary separation a changes differently for various mass exchange modes. First, we introduce a measure of non-conservativeness of the mass ex- change as the ratio between the mass change of the accretor and the donor: β ≡ −(M iaccr −Mfaccr)/(M idonor −M donor). (41) If the mass exchange is conservative (β = 1, i.e. Ma + Md = const) and one can neglect the angular momenta of the components, the orbital momentum conservation implies M iaM . (42) In a more general case of quasi-conservative mass transfer 0 ≤ β < 1, the orbital separation changes differently depending on the specific an- gular momentum carried away from the system by the escaping matter (see van den Heuvel (1994) for more detail). We treat the quasi-conservative mass transfer by assuming the isotropic mass loss mode in which the matter carries away the spe- cific orbital angular momentum of the accreting component (ja) J̇out = (1− β)Ṁcja . (43) From here we straightforwardly find 1 + qi 1 + qf 1 + β 1 + β )3+2/β . (44) In this formula q = Maccr/Mdonor and the non-conservative parameter β is set to be the minimal value between β = 1 and the ra- tio TKH(donor)/TKH(accr) (TKH(donor) and TKH(accr) is the thermal time of donor and ac- cretor, respectively). When no matter is captured by the secondary companion without additional losses of angu- lar momentum (the so-called “absolutely non- conservative case”), which relates to the spherical- symmetric stellar wind from one component, we use another well-known formula M i1 +M . (45) In this case the orbital separation always increases. When the orbital angular momentum is carried away by GW or MSW with no RL overflow, the following approximate formulas are used: (1−∆t/τMSW )1/4, for MSW, (1−∆t/τGW )1/5, for GW, In a special case of a white dwarf filling its RL (stage “IIIwd” above), assuming a stable (i.e. where d lnRwd/d lnM = d lnRRL/d lnM) conser- vative mass transfer with account for the mass- radius relation Rwd ∝ M−1/3wd , the orbital separa- tion must increase according to the equation )−2/3 , (47) where mi is the initial mass of the WD donor and mf is its mass at the end of the mass transfer. 3.9. The change of eccentricity Tidal interaction between components, as well as the orbital angular momentum loss due to MSW or GR decrease the eccentricity of the binary sys- tem. The tidal interaction is essential in very close binaries or even during the common envelope stage. MSW is effective only in systems with low- mass late-type main sequence stars (see above), GW losses become significant only in short-period binaries. The tidal interaction conserves the orbital an- gular momentum which implies the relation a(1− e2) = const. (48) It seems that the orbit becomes a circle faster than major semi-axis of the orbit decreases during common envelope stage. We suppose that tcyr = 1/3tCE. We accept that tcyr for RL-filling stars is equal to its Kelvin-Helmholtz time. Detached systems change their eccentricity during the next character time (Zahn 1975; Press & Teukolsky 1977; Zahn 1989b; Zahn & Bouchet 1989b) tcyr = tKH 1 + e )3/2( . (49) here tKH ≈ GM2/RL is thermal time of the star, R is radius of the star, RL is its RL size. For sys- tems which consist of two normal stars we choose minimal value of tcyr. Resonances at very high ec- centricities are not taken into account (Mardling 1995a,b). Orbits of the systems with MSW become circu- lar during tcyr = τMSW (see 46). In case of GW analytical exact solutions were obtained for a(t) and e(t) (Peters & Mathews 1963; Peters 1964). 4. Special cases: supernova explosion and common envelope Supernova explosion in a binary is treated as an instantaneous mass loss of the exploding star. The additional kick velocity can be imparted to the newborn neutron star due the collapse asym- metry (see below for discussion). In this case the eccentricity and semi-major axis of the binary af- ter the explosion can be straightforwardly calcu- lated (Boersma 1961) (see necessary formulas also in Grishchuk et al. 2001). Briefly, we use the fol- lowing scheme. 1. First, velocities and locations of the compo- nents on the orbit prior to the explosion are calculated; 2. then the mass of the exploding star Mpr - Mremnant is changed and the arbitrarily di- rected kick velocity w is added to its orbital velocity; 3. after that the transition to the new system’s barycenter is performed (at this point the spatial velocity of the new center of mass of the binary is calculated); 4. in this new reference frame the new total en- ergyE′tot and the orbital angular momentum J ′orb are computed; if the new total energy is negative, the new semi-major axis a′ and ec- centricity e′ are calculated by using the new J ′orb end E tot; if the total energy is positive (that is, the binary is unbound) spatial ve- locities of each component are calculated. The kick velocity w distribution is taken in the Maxwellian form: f(w) ∼ w 0 . (50) We suppose that the absolute value of the ve- locity that can be added during the formation of a black hole depends on the mass loss by the col- lapsing star, the value of the parameter w0 during the BH formation is defined as wbh0 = (1− kbh)w0. (51) An effective spiral-in of the binary compo- nents occurs during the common envelope (CE) stage. This complicated process (introduced by Paczynski (1976)) is not fully understood as yet, so we use the conventional energy consideration to find the binary system parameters after the CE by introducing a parameter αCE = ∆Eb/∆Eorb, where ∆Eb = Egrav − Ethermal is the binding en- ergy of the ejected envelope matter and ∆Eorb is the drop in the orbital energy of the system during the spiral-in phase (van den Heuvel 1994). This parameter measures the fraction of the system’s orbital energy that comes during the spiral-in pro- cess to the binding energy (gravitational minus thermal) of the ejected common envelope. Thus GMaMc − GMaMd GMd(Md −Mc) where Mc is the mass of the core of the mass- losing star with the initial mass Md and radius Rd (which is simply a function of the initial separation ai and the initial mass ratio Ma/Md, where Ma is the mass of the accreting star). On the CE stage the luminosity of the accret- ing star can reach the Eddington limit so that the further increase of the accretion rate can be pre- vented by radiation pressure. This usually hap- pens at accretion rates Ṁ ≃ 10−4−10−5M⊙ yr−1. However, Chevalier (1993) suggested that when the accretion rate is higher (Ṁ ≃ 10−2− 10−3M⊙ yr−1), the energy is radiated away not by high- energy photons only, but also by neutrinos (see also Zeldovich et al. (1972) and the next section). On the typical time scale for the hyper accretion stage of 102 yr, up to ∼ 1M⊙ of matter may be incident onto the surface of the neutron star. 5. Three regimes of mass accretion by neutron stars A considerable fraction of observed neutron stars have increased their masses in the course of their evolution, or are still increasing their masses (e.g., in X-ray sources). But how large can this mass increase be? It is clear that the only origin of a mass increase is accretion. It is evident that the overall change in the mass of a neutron star is determined not only by the accretion rate, but also by the duration of the accretion stage: Ṁdt = ṀTa, (53) where Ṁ is the mean accretion rate and Ta is the lifetime of the accretion stage. We emphasize that, in the case under consideration, the accretion rate is the amount of matter falling onto the surface of the neutron star per unit time, and can differ signifcantly from the values indicated by the clas- sical Bondi-Hoyle formulas. Three regimes of ac- cretion are possible in a close binary containing a neutron star: ordinary accretion, super-accretion, and hyper-accretion. 5.1. Ordinary accretion The ordinary accretion regime is realized when all matter captured by the gravitational field of the neutron star falls onto its surface. This is possible only if the radiation pressure and electromagnetic forces associated with the magnetic field of the star and its rotation are small compared to the gravita- tional force. In this case, the increase in the mass will be precisely determined by the gas dynamics of the accretion at the gravitational-capture ra- dius or, if the donor fills its Roche lobe, by the binary mass ratio and the evolutionary status of the optical component. In this case, the accretor is observed as an X-ray source with luminosity Lx = Ṁ , (54) where Mx and R∗ are the mass and radius of the neutron star. The accretion rate Ṁ is determined by the Bondi-Hoyle formula Ṁ = πR2Gρv , (55) where RG is the gravitational-capture radius of the neutron star, v is the velocity of the gas flow relative to the neutron star, and ρ is the density of the gas. The X-ray luminosity of the accretor Lx and its other main parameters can be used to estimate the mass ∆M accumulated during the accretion phase: LxR∗Ta , (56) 5.2. Super-accretion Regime of super accretion was considered, for instance, in paper Lipunov (1982d). Despite the absence of detailed models for supercritical disk accretion (supercritical accretion is realized pre- cisely via an accretion disk), it is possible to esti- mate the main characteristics of the process – the accretion rate, magnetosphere radius, and evolu- tion equations. Accretion is considered to be su- percritical when the energy released at the radius where the accretion exceeds the Eddington limit: Rstop > LEdd = 1.38×1038(Mx/M⊙) erg s−1, where Rstop is either the radius of the neutron star or the magnetosphere radius RA. For strongly magnetized neutron stars with magnetic fields B ≫ 108 G, all matter arriving at the magnetosphere is accreted onto the mag- netic poles, where the corresponding gravitational energy is released. If the black body temperature T , roughly estimated as SσT 4 = Ṁ , (58) is higher than 5 × 109 K (S is the area of the base of the accretion column), most of the energy will escape from the neutron star in the form of neutrinos, and, hence, will not hinder accretion (Zeldovich et al. 1972; Basko & Sunyaev 1975). In this case, the rate at which the neutron star accumulates mass will be Ṁ ≃ Ṁcrit ≫ Ṁcrit, (59) For lower temperatures there should be an up- per limit on the accretion rate equal to the stan- dard Eddington limit. 5.3. Hyper-accretion A considerable fraction of neutron stars in bi- nary systems pass through the common-envelope stage in the course of their evolution. In this case, the neutron star is effectively immersed in its opti- cal companion, and for a short time (102-104 yrs) spirals-in inside a dense envelope of the compan- ion. The formal accretion rate estimated using the Bondi-Hoyle formulas is four to six orders of magnitude higher than the critical rate and, as was suggested by Chevalier (1993), this may re- sult in hyper-accretion, when all the energy is car- ried away by neutrinos for the reasons described above. There are currently no detailed theories for hyper-accretion or the common-envelope stage. The amount of matter accreted by the neutron star can be estimated as ∫ Thyper Ṁdt ≃ (60) (Mopt −Mcore) where Thyper is the duration of the hyper-accretion stage, RG is the gravitational-capture radius of the neutron star, a is the initial semi-major axis of the close binary orbit, Mcore is the mass of the core of the optical star, and Mopt and Mx are the total masses of the optical star and of the neu- tron star at the onset of the hyper-accretion stage. The mass of the neutron star can increase during the common envelope stage as much as ∼ 1M⊙ (Bogomazov et al. 2005), up to MOV . Such NSs collapse into black holes. 6. Mass accretion by black holes If the black hole has formed in the binary sys- tem, its X-ray luminosity is Lx = µṀc 2, (61) where µ = 0.06 for Schwarzschild black hole and µ = 0.42 (maximum) for extremal Kerr BH. We use the Bondi-Hoyle formulas to estimate the accretion rate onto BH. Powerful X-ray radiation is able to originate only if an accretion disc has formed around the black hole (Karpov & Lipunov 2001). For spher- ically symmetric accretion onto a black hole the X-ray luminosity is insignificant. A very low stel- lar wind velocity is necessary to form an accretion disc (Lipunov 1992) V < Vcr ≈ (62) ≈ 320(4η)1/4m3/8T−1/410 R 8 (1 + tan 2 β)−1/2, where η is averaged over the z-coordinate dynamic viscous coefficient, m = Mx/M⊙, Mx is the rela- tivistic star mass, T10 = T/10, T is the orbital period in days, R8 = Rmin/10 8cm, Rmin is the minimal distance from the compact object up to which free Keplerian motion is still possible and β is the accretion disk axis inclination angle with respect to the radial direction. For black holes Rmin = 3Rg, where Rg = 2GMbh/c 7. Accretion induced collapse and com- pact objects merging WD and NS are degenerate configurations, which have upper limit of their mass (the Chan- drasekhar and Oppenheimer-Volkov limits corre- spondingly). The Chandrasekhar limit depends on chemical composition of the white dwarf MCh = 1.44M⊙, He WD, 1.40M⊙, CO WD, 1.38M⊙, ONeMg WD, If the mass of WD becomes equal to MCh, the WD loses stability and collapses. The col- lapse is accompanied by the powerful thermonu- clear burst observed as a type Ia supernova. Col- lapses of He and CO WDs leave no remnants (Nomoto & Kondo 1991). The outcome of the collapse of a ONeMg WD is not clear. It can lead to the formation of a neutron star (the accretion induced collapse, AIC). Some papers come to the different conclusion about the result of AICs (see e.g. Garcia-Berro & Iben (1994); Ritossa et al. (1996)). The question about the NS formation during the WD collapse remains undecided. Nev- ertheless, some NSs could have been formed from AIC WD (van Paradijs 1997). In the “Scenario Machine” code the possibility of NS formation during ONeMg WD is optional. The merging of two compact objects in a binary WD system (e.g., due to the GW losses) should likely to be similar to AIC. During the merging of two helium WDs, one object with a mass of less than MCh (for example, 0.5M⊙ + 0.6M⊙ → 1.1M⊙) can form. At the same time, if the total mass of the components exceeds 1.0 − 1.2M⊙, a thermonuclear burning can happen. It is likely that the merging of a ONeMg WD with another WD can form a NS. For the typical NS mass ≈ 1.4M⊙, the binary NS+NS merging event can produce a black hole. Massive (≈ 2.8M⊙) neutron star can be formed only if the NS equation of state is very hard and MOV ≃ 2.8− 3.0M⊙. BH+BH merging should produce a rapidly ro- tating black hole with the mass equal to the total mass of the coalescing binary. 8. Additional scenarios of stellar wind from massive stars 8.1. Evolutionary scenario B In the end of 1980s and in the beginning of 1990s the series of new evolutionary tracks were calculated. The authors used new tables of opaci- ties (Rogers & Iglesias 1991; Kurucz 1991), new cross-sections in nuclear reactions (Landre et al. 1990) and new parameters of convection in stars (Stothers & Chin 1991). For stars with M < 10M⊙ those tracks proved to be almost coincident with previous calculations. More massive stars had much stronger stellar winds. A massive star loses up to 90% of its ini- tial mass in the main-sequence, supergiant, and Wolf-Rayet stages via stellar wind. There- fore, the presupernova mass in this case can be ≈ 8 − 10M⊙, essentially independent of the initial mass of the star (de Jager et al. 1988; Nieuwenhuijzen & de Jager 1990; Schaller et al. 1992). 8.2. Evolutionary scenario C The papers mentioned above were criticised and in 1998 a new version of the evolutionary scenario was developed (Vanbevern et al. 1998). The stel- lar wind loss rates were corrected taking into ac- count empirical data about OB and WR stars. Here we list the main equations of this scenario (all results concern only with the stars with initial mass M0 > 15M⊙). log Ṁ = 1.67 logL− 1.55 logTeff − 8.29, (α) logL+ logR− logM − 7.5, (β) 0.8 logL− 8.7, (γ) logL− 10, (δ) We assume the next indication in this formula: (α), H burning in the core; (β), giant, M0 ≥ 40M⊙; (γ), giant, M0 < 40M⊙; (δ), Wolf-Rayet star. Note that in this subsection we used the mass M is in M⊙, the luminosity L is in L⊙ and the ra- dius R is in R⊙. With these new calculations, the Webbink diagram described above changed signif- icantly Vanbevern et al. (1998). In this scenario, the total mass loss by a star is calculated using the formula ∆M = (M −Mcore), (65) where Mcore is the stellar core mass (66). If the maximum mass of the star (usually it is the initial mass of a star, but the mass transfer in binary systems is able to increase its mass above its initial value) Mmax > 15M⊙, the mass of the core in the main sequence stage is determined using (66α), and in giant and supergiant stages using (66β). In the Wolf-Rayet star stage (helium star), ifMWR < 2.5M⊙ and Mmax ≤ 20M⊙ it is described using (66γ), if MWR ≥ 2.5M⊙ and Mmax ≤ 20M⊙ as (66δ), if Mmax > 20M⊙ as (66ǫ) mcore = 1.62m0.83opt (α) 10−3.051+4.21 lgmopt−0.93(lgmopt) 0.83m0.36WR (γ) 1.3 + 0.65(mWR − 2.4) (δ) mcore = 3.03m 0.342 opt (ǫ) These evolutionary scenarios have some pecu- liar properties with respect to the classical sce- nario. One of them is that the strong stellar wind from massive stars leads to a rapid and significant increase of the system’s orbit and such stars can- not fill its RL at all. There are three observational facts that conflict with the strong stellar wind scenarios: 1. A very high Ṁ is a problem by itself. The observers calculate this quantity for most of OB and WR stars using the emission measure EM ∝ n2edl. The estimate of Ṁ using EM is maximal for homogenous wind. However, there are evidences (see e.g. Cherepashchuk et al. (1984)) that stellar winds of massive stars are strongly “clumpy”. In this case the real Ṁ must be 3-5 times less. This note is especially important for the scenario with high stellar wind. 2. Very massive Wolf-Rayet stars do exist. There are at least three double WR+OB sys- tems including very massive WR-stars: CQ Cep 40M⊙, HD 311884 48M⊙ and HD92740 77M⊙ (Cherepashchuk et al. 1996). Such heavy WR stars are at odds with the as- sumed high mass loss rate. 3. In the semi-detached binary system RY Sct (W Ser type) the mass of the primary com- ponent is ≈ 35M⊙ (Cherepashchuk et al. 1996). This mass is near the limit (Vanbevern et al. 1998) beyond which the star, according to the high mass loss sce- nario, cannot fill its RL. 8.3. Evolutionary Scenario W The evolutionary scenario W is based on the stellar evolution calculations by Woosley et al. (Woosley et al. (2002), Fig. 16), which repre- sents the relationship between the mass of the relativistic remnant and the initial mass of the star. We included into population-synthesis code two models with W-type stellar winds, which we label Wb and Wc. In models Wb and Wc, the mass-loss rates were computed as in scenario B and scenario C, respectively. The use of these models to calculate the wind rate in a scenario based on Woosley’s diagram (Woosley et al. (2002), Fig. 16) is justified by the fact that scenarios B and C are based on the same nu- merical expressions for the mass-loss rates from Schaller et al. (1992); Vanbevern et al. (1998); Nieuwenhuijzen & de Jager (1990) that were used by Woosley et al. (2002). 9. The “Ecology” of Magnetic Rotators One of the most important achievements in as- trophysics in the end of the 1960s was the real- ization that in addition to “ordinary” stars, which draw energy from nuclear reactions, there are ob- jects in the Universe whose radiation is caused by a strong gravitational and magnetic field. The well- known examples include neutron stars and white dwarfs. The property that these objects have in common is that their astrophysical manifestations are primarily determined by interaction with the surrounding matter. In the early 1980s, this approach led to the cre- ation of a complete classification scheme involv- ing various regimes of interaction between neu- tron stars and their environment, as well as to the first Monte Carlo simulation of the NS evolution (Lipunov 1984). In addition to NSs, this scheme has been shown to be applicable to other types of magnetized rotating stars. By virtue of the relationship between the grav- itational and electromagnetic forces, the NS in various states can manifest itself quite differently from the astronomical point of view. Accordingly, this leads to the corresponding classification of NS types and to the idea of NS evolution as a gradual changing of regimes of interaction with the envi- ronment. The nature of the NS itself turns out to be important also when constructing the classifica- tion scheme. This indicates that there should be a whole class of quite different objects which have an identical physical nature. To develop the theory describing properties of such objects (in a sense, it should establish “ecological” links between differ- ent objects), it proved to be convenient to use sym- bolic notations elaborated for the particular case of NS. We start this subsection with recollecting the magnetic rotator formalism (mainly according to the paper Lipunov (1987)). 9.1. A Gravimagnetic Rotator We call any gravitationally-bounded object having an angular momentum and intrinsic mag- netic field by the term “gravimagnetic rotator” or simply, rotator. In order to specify the intrin- sic properties of the rotator, three parameters are sufficient – the mass M , the total angular mo- mentum J = Iω (I is the moment of inertia and ω is the angular velocity), and the magnetic dipole moment µ. Given the rotator radius R0, one can express the magnetic field strength at the poles B0 by using the dipole moment B0 = 2µ/R 0. The angle β between the angular moment J and the magnetic dipole moment µ can also be of impor- tance: β = arccos(Jµ). 9.2. The Environment of the Rotator We assume that the rotator is surrounded by an ideally conductive plasma with a density ρ∞ and a sound velocity a∞ at a sufficiently far dis- tance from the rotator. The rotator moves rela- tive to the environment with a velocity v∞. Un- der the action of gravitational attraction, the sur- rounding matter should fall onto the rotator. A rotator without a magnetic field would capture a stationary flow of matter, Ṁc, which can be es- timated using the Bondy-Hole-Lyttleton formu- lae (Bondi & Hole 1944; Bondi 1952; McCrea 1953): Ṁc = δ (2GM)2 (a2∞ + v ρ∞, (67) where δ is a dimensionless factor of the order of unity. When one of the velocities, a∞ or v∞, far exceeds the other, the accretion rate is determined by the dominating velocity, and can be written in a convenient form as (55). In the real astrophysical situation, the param- eters of the surrounding matter at distances R ≫ RG can be taken as conditions at infinity As already noted, the matter surrounding a NS or a WD is almost always in the form of a high-temperature plasma with a high conduc- tivity. Such accreting plasma must interact effi- ciently with the magnetic field of the compact star (Amnuel’ & Guseinov 1968). Hence, the interac- tion between the compact star and its surround- ings cannot be treated as purely gravitational and therefore the accretion is not a purely gas dynamic process. In general, such interaction should be described by the magneto hydrodynamical equa- tions. This makes the already complicated pic- ture of interaction of the compact star with the surrounding medium even more complex. The following classification of magnetic ro- tators is based on the essential characteristics 7RG = of the interaction of the plasma surrounding them with their electromagnetic field. This approach was proposed by Shvartsman (1970) who distinguished three stages of interaction of magnetic rotators: the ejection stage, the propeller stage, which was later rediscovered by Illarionov & Sunyaev (1975) and named as such, and the accretion stage. Using this ap- proach, Shvartsman (1971a) was able to pre- dict the phenomenon of accreting X-ray pulsars in binary systems. New interaction regimes dis- covered later have led to a general classifica- tion of magnetic rotators (Lipunov 1982a, 1984; Kornilov & Lipunov 1983a). It should be noted that the interaction of the magnetic rotator with the surrounding plasma is not yet understood in detail. However, even the first approximation reveals a multitude of interac- tion models. To simplify the analysis, we assume the electromagnetic part of the interaction to be independent of the accreting flux parameters, and vice versa. Henceforth, we shall assume in almost all cases that the intrinsic magnetic field of a rotator is a dipole field (Landau & Lifshiz 1971): (1 + 3 sin2 θ)1/2, (68) This is not just a convenient mathematical sim- plification. We will show that the magnetoplasma interaction takes place at large distances from the surface of the magnetic rotator, where the dipole moment makes the main contribution. Moreover, the collapse of a star into a NS is known to “cleanse” the magnetic field. Indeed, the conser- vation of magnetic flux leads to a decrease of the ratio of the quadrupole magnetic moment q to the dipole moment µ in direct proportion to the radius of the collapsing star, q/µ ∝ R. It should be emphasized, however, that the con- tribution of the quadrupole component to the field strength at the surface remains unchanged. The light cylinder radius is the first important characteristric of the rotating magnetic field: , (69) where c is the speed of light. A specific property of the field of the rotating magnetic dipole in vacuum is the stationarity of the field inside the light cylinder and formation of magneto dipole radiation beyond the light cylin- der. The luminosity of the magnetic dipole radia- tion is equal to (Landau & Lifshiz 1971): sin2 β = kt ω, (70) where kt = sin2 β. This emission exerts a corresponding braking torque Km = − µ2 sin2 β ω3cω, (71) leading to a spin down of the rotator. Although magnetic dipole radiation from pulsars do not ex- ists, almost all models predict energy loss quantity near this value. At the same time we do not take into account possibly complicated angular depen- dence of such loss. 9.3. The Stopping Radius Now we consider qualitatively the effect of the electromagnetic field of a magnetic rotator on the accreting plasma. Consider a magnetic rotator with a dipole magnetic moment µ, , rotational fre- quency ω, and mass M . At distances R ≫ RG the surrounding plasma is characterized by the following parameters: density ρ∞, sound veloc- ity a∞ and/or velocity v∞ relative to the star. The plasma will tend to accrete on to the star un- der the action of gravitation. The electromagnetic field, however, will obstruct this process, and the accreting matter will come to a stop at a certain distance. Basically, two different cases can be considered: • When the interaction takes place beyond the light cylinder, Rstop > Rl. This case first considered by Shvartsman (1970, 1971a). In this case the magnetic rotator generates a relativistic wind consisting of a flux of differ- ent kinds of electromagnetic waves and rel- ativistic particles. The form in which the major part of the rotational energy of the star is ejected is not important at this stage. What is important is that both relativistic particles and magnetic dipole radiation will transfer their momentum and hence exert pressure on the accreting plasma. Indeed, random magnetic fields are always present in the accreting plasma. The Larmor radius of a particle with energy ≪ 1010 eV mov- ing in the lowest interstellar magnetic field ∼ 10−6 G is much smaller than the charac- teristic values of radius of interaction, so the relativistic wind will be trapped by the mag- netic field of the accreting plasma and thus will transfer its momentum to it. Thus, a relativistic wind can effectively im- pede the accretion of matter. A cavern is formed around the magnetic rotator, and the pressure of the ejected wind Pm at its bound- ary balances the ram pressure of the accret- ing plasma Pa: Pm(Rstop) = Pa(Rstop), (72) This equality defines a characteristic size of the stopping radius, which we call the Shvartzman radius RSh. • The accreting plasma penetrates the light cylinder Rstop < Rl. The pressure of the ac- creting plasma is high enough to permit the plasma to enter the light cylinder. Since the magnetic field inside the light cylinder de- creases as a dipole field, the magnetic pres- sure is given by , (73) Matching this pressure to the ram pressure of the accreting plasma yields the Alfven ra- dius RA. The magnetic pressure and the pressure of the relativistic wind can be written in the following convenient form: , R ≤ Rl, 4πR2c , R > Rl, We introduce a dimensionless factor kt such that the power of the ejected wind is Lm = kt ω, (75) Assuming kt = 1/2 we get for R = Rl a continuous function Pm = Pm(R). The accreting pressure of plasma outside the capture radius is nearly constant, and hence gravitation does not affect the medium pa- rameters significantly. In contrast, at dis- tances inside the gravitational capture ra- dius RG the matter falls almost freely and exerts pressure on the “wall” equal to the dy- namical pressure. For spherically symmetric accretion we obtain , R > RG, , R ≤ RG, Here we used the continuity equation Ṁc = 4πR2Gρ∞v∞. When presented in this form, the pressure Pa is a continuous function of distance. Summarizing, for the stopping radius we get Rstop = Ra, Rstop ≤ Rl, RSh, Rstop > Rl, The expressions for the Alfven radius are: 2µ2G2M2 Ṁcv5 , RA > RG, , RA ≤ RG, and for the Shvartzman radius: RSh = 2G2m2ω4 Ṁcv5∞c , RSh > RG, 9.4. The Stopping Radius in the Super- critical Case The estimates presented above for the stopping radius were obtained under the assumption that the energy released during accretion does not ex- ceed the Eddington limit, so we neglected the re- verse action of radiation on the accretion flux pa- rameters. Now, we turn to the situation where one can- not neglect the radiation pressure. Consider this effect after Lipunov (1982b). Suppose that the accretion rate of matter captured by the magnetic rotator is such that the luminosity at the stopping radius exceeds the Eddington limit (see equation We shall assume, following Shakura & Sunyaev (1973), that the radiation sweeps away exactly that amount of matter which is needed for the accretion luminosity of the remaining flux to be of the order of the Eddington luminosity at any radius: Ṁ(R) = Ledd, (80) This yields Ṁ(R) = Ṁc , Rs = Ṁc, (81) where Rs is a spherization radius (where the accre- tion luminosity first reaches the Eddington limit), and κ designates the specific opacity of matter. Using the continuity equation, the ram pressure of the accreting plasma is now obtained as another function of the radial distance Pa ≈ ρv2 ≈ Ṁ(R) R−3/2, R ≤ Rs, in contrast to the subcritical regime when Pa ∝ R−5/2. Matching Pa and Pm (see the previous section) for the supercritical case gives , (83) RSh = , (84) The critical accretion rate Ṁcr is defined by the boundary of the inequality Ṁc ≥ Ṁcr, (85) and, correspondingly, is Ṁcr = Rst. (86) The dependence of the Alfven radius on the ac- cretion rate is such that the Alfven radius (be- yond the capture radius) slightly decreases with increasing accretion rate as Ṁ−1/6, while it de- cerases below the capture radius as Ṁ−2/7 and attains its lowest value for the critical accretion rate Ṁc > Ṁcr, beyond which it is independent of the external conditions. We also note that in the supercritical regime, the pressure of the accreting plasma increases more slowly (as R−3/2) when approaching the magnetic rotator than the pressure of the relativis- tic wind (as R−2) ejected by it. This means that in the supercritical case a cavern may exist even below the capture radius. The estimates presented here, of course, are most suitable for the case of disk accretion. In fact, the supercritical regime seems to emerge most frequently under these conditions. This can be simply understood. Indeed, the accretion rate is proportional to the square of the capture radius Ṁc ∝ R2G. At the same time, the angular momen- tum of the captured matter is also proportional to R2G. Hence, at high accretion rates the formation of the disk looks natural. 9.5. The Effect of the Magnetic Field Apparently, the magnetic field of a star be- comes significant only when the stopping radius exceeds the radius of the star, Rst > Rx. We take the Alfven radius RA for Rst, since it is the small- est of the two quantities RA and RSh. Hence, we can estimate the lowest value of magnetic field of a star which will influence the flow of matter µmin = 4G2M2x , RA > RG, , RA ≤ RG Ṁc ≤ Ṁcr, 2GMxR , Ṁc > Ṁcr, The case RA ≥ RG, Ṁc ≥ Ṁcr is considered most frequently, and for this case we get the fol- lowing numerical estimations µmin = 10 x G cm 3, (88) or, equivalently, Bmin = 10 x G, (89) Most presently observed NS have magnetic fields ∼ 1012 G and dipole moments ∼ 1030 G cm3, so the magnetic field must necessarily be taken into account when considering interaction of matter with these stars. 9.6. The Corotation Radius The corotation radius is another important characteristics of a magnetic rotator. Suppose that an accreting plasma penetrates the light cylinder and is stopped by the magnetic field at a certain distance Rst given by the balance between the static magnetic field pressure and the plasma pressure. Suppose that the plasma is “frozen” in the rotator’s magnetic field. This field will drag the plasma and force it to rotate rigidly with the angular velocity of the star. The matter will fall on to the stellar surface only if its rotational ve- locity is smaller than the Keplerian velocity at the given distance Rst: ωRst < GMx/Rst, (90) Otherwise, a centrifugal barrier emerges and the rapidly rotating magnetic field impedes the ac- cretion of matter (Shvartsman 1970; Pringle & Rees 1972; Davidson & Ostriker 1973; Lamb 1973; Illarionov & Sunyaev 1975). The latter authors assumed that if ωRst ≫ GMx/Rst, the mag- netic field throws the plasma back beyond the capture radius. They called this effect the “pro- peller” regime. In fact, matter may not be shed (Lipunov 1982a), but it is important to note that a stationary accretion is also not possible. The corotation radius is thus defined as Rc = (GMx/ω 2)1/3 ∼ 2.8×108m1/3x (P/1s)2/3 cm, where P is the rotational period of the star. If Rst < Rc , rotation influences the accretion insignificantly. Otherwise, a stationary accretion is not possible for Rst > Rc. 9.7. Nomenclature The interaction of a magnetic rotator with the surrounding plasma to a large extent depends on the relation between the four characteristic radii: the stopping radius, Rst, the gravitational capture radius, RG, the light cylinder radius, Rl, and the corotation radius, Rc. The difference between the interaction regimes is so significant that the mag- netic rotators behave entirely differently in differ- ent regimes. Hence, the classification of the inter- action regimes may well mean the classification of magnetic rotators. The classification notation and terminology is described below and summarized in Table 1, based on paper by Lipunov (1987). Naturally, not all possible combinations of the characteristic radii can be realized. For example, the inequality Rl > Rc is not possible in princi- ple. Furthermore, some combinations require un- realistically large or small parameters of magnetic rotators. Under the same intrinsic and external conditions, the same rotator may gradually pass through several interaction regimes. Such a pro- cess will be referred to as the evolution of a mag- netic rotator. We describe the classification by considering an idealized scenario of evolution of magnetic rota- tors. Suppose the parameters ρ∞, v∞ and Ṁc of the surrounding medium remain unchanged. We shall also assume for a while a constancy of the ro- tator’s magnetic moment µ. Let the potential ac- cretion rate Ṁc at the beginning be not too high, so that the reverse effect of radiation pressure can be neglected, Ṁc ≤ Ṁcr. We also assume that the star initially rotates at a high enough speed to provide a powerful relativistic wind. Ejectors (E). We shall call a magnetic rotator an ejecting star (or simply an ejector E) if the pres- sure of the electromagnetic radiation and ejected relativistic particles is so high that the surround- ing matter is swept away beyond the capture ra- dius or radius of the light cylinder (if Rl > RG). Ejector: RSh > max(Rl, RG), (92) It follows from here that Pm ∝ R−2 while the accretion pressure within the capture radius is Pa ∝ R5/2 i.e. increases more rapidly as we ap- proach an accreting star. Consequently, the radius of a stable cavern must exceed the capture radius (Shvartsman 1970). It is worth noting that the reverse transition from the propeller (P) stage to the ejector (E) stage is non-symmetrical and occurs at a lower period (see below). This means that to switch a pulsar on is more difficult than to turn it off. This is due to the fact that in the case of turning-on of the pulsar the pressures of plasma and relativis- tic wind must be matched at the surface of the light cylinder, not at the gravitational capture ra- dius. In fact, the reverse transition occurs under the condition of equality of the Alfven radius to the radius of the light cylinder (RA = Rl). It should be emphasized that, as mentioned by (Shvartsman 1970), relativistic particles can be formed also at the propeller stage by a rapidly rotating magnetic field (see also Kundt (1990)). Propellers (P). After the ejector stage, the propeller stage sets in under quite general condi- tions, when accreting matter at the Alfven sur- face is hampered by a rapidly rotating magnetic field of the magnetic rotator. In this regime the Alfven radius is greater than the corotation ra- dius, RA > Rc. A finite magnetic viscosity causes the angular momentum to be transferred to the accreting matter so that the rotator spins down. Until now, the propeller stage is one of the poorly investigated phenomena. However, it is clear that sooner or later the magnetic rotator is spin down enough for the rotational effects to be of no im- portance any longer, and the accretion stage sets Accretors (A). In the accretion stage, the stopping radius (Alfven radius) must be smaller than the corotation radius RA < Rc. This is the most thoroughly investigated regime of interac- tion of magnetic rotators with accreting plasma. Examples of such systems span a wide range of bright observational phenomena from X-ray pul- sars, X-ray bursters, low-mass X-ray binaries to most of the cataclysmic variables and X-ray tran- sient sources. Georotators (G). Imagine that the star be- gins rotating so slowly that it cannot impede the accretion of plasma, i.e. all the conditions mentioned in the previous paragraph are sat- isfied. However, matter still can not fall on to the rotator’s surface if the Alfven radius is larger than the gravitational capture radius (Illarionov & Sunyaev 1975; Lipunov 1982c). This means that the attractive gravitational force of the star at the Alfven surface is not signifi- cant. A similar situation occurs in the interaction of solar wind with Earth’s magnetosphere. The plasma mainly flows around the Earth’s magne- tosphere and recedes to infinity. This analogy ex- plains the term “georotator” used for this stage. Clearly, a georotator must either have a strong Table 1: Classification of neutron stars and white dwarfs. Abbrevi- Type Characteristic Accretion Well known ation radii relation rate observational appearances E Ejector Rst > RG Ṁ ≤ Ṁcr Radio pulsars Rst > Rl P Propeller Rc > Rst Ṁc ≤ Ṁcr ? Rst ≤ RG Rst ≤ Rl A Accretor Rst ≤ RG Ṁc ≤ Ṁcr X-ray pulsars Rst ≤ Rl G Georotator RG ≤ Rst Ṁc ≤ Ṁcr ? Rst ≤ Rc M Magnetor Rst > a Ṁc ≤ Ṁcr AM Her, polars, Rc > a ? soft gamma repeaters, anomalous X-ray pulsars SE Super- Rst > Rl Ṁc > Ṁcr ? ejector SP Super- Rc < Rst Ṁc > Ṁcr ? propeller Rst ≤ Rl SA Super- Rst ≤ Rc Ṁc > Ṁcr ? accretor Rst ≤ RG magnetic field or be embedded in a strongly rar- efied medium. Magnetors (M). When a rotator enters a bi- nary system, it may happen that its magneto- sphere engulfs the secondary star. Such a regime was considered by Mitrofanov et al. (1977) for WD in close binary systems called polars due to their strongly polarized emission. In the case of NS, magnetors M may be realized only under the extreme condition of very close binaries with no matter within the binary separation. Supercritical interaction regimes. So far, we have assumed that the luminosity at the stop- ping surface is lower than the Eddington limit. This is fully justified for G and M regimes since gravitation is not important for them. For types E, P, and especially A, however, this is not always true. The critical accretion rate for which the Ed- dington limit is achieved is Ṁcr = 1.5× 10−6R8M⊙ yr−1, (93) where R8 ≡ Rst/108 cm is the stopping radius (Schwartzman radius or Alfven radius, see above). We stress here that the widely used condition of supercritical accretion rate Ṁ & 10−8(M/M⊙)M⊙ yr is valid only for the case of non-magnetic NS, where Rst ≈ 10 km coincides with the stellar ra- dius. In reality, for a NS with a typical magnetic field of 1011 − 1012 G, the Alfven radius reaches 107 − 108 cm, so much higher accretion rates are required for the supercritical accretion to set in. The electromagnetic luminosity released at the NS surface, however, will be restricted by Ledd, and most of the liberated energy may be carried away by neutrinos (Basko & Sunyaev 1975) (see also section about hyper accretion). Most of the matter in the dynamic model of supercritical accretion forms an outflowing flux covering the magnetic rotator by an opaque shell (Shakura & Sunyaev 1973). The following three additional types are distinguished, depending on the relationship between the characteristic radii: superejector (SE), superpropeller (SP) and super- accretor (SA). 9.8. A Universal Diagram for Gravimag- netic Rotators The classification given above was based on re- lations between the characteristic radii, i.e. quan- tities which cannot be observed directly. This drawback can be removed if we note that the light cylinder radius Rl, Shvartzman radius RSh and corotation radius Rc are functions of the well- observed quantity, rotational period of the mag- netor p. Hence, the above classification can be reformulated in the form of inequalities for the ro- tational period of a magnetic rotator. One can introduce two critical periods pE and pA such that their relationship with period p of a magnetic rotator specifies the rotator’s type: p < pE , → E or SE, pE ≤ p < pA, → P or SP, p > pA, → A, SA, G or M, The values of pE and pA can be determined from Table 2 which defines the basic nomencla- ture, and are functions of the parameters v∞, Ṁc, µ and Mx. The parameters p and µ character- ize the electromagnetic interaction, while Ṁc de- scribes the gravitational interaction. Instead of Ṁc we introduce the potential accretion luminos- ity L L ≡ Ṁc , (95) The physical sense of the potential luminosity is quite clear: the accreting star would be ob- served to have this luminosity if the matter for- mally falling on the gravitational capture cross- section were to reach its surface. Approximate expressions for critical periods (Lipunov 1992) are: 0.42v 38 s, (α) 38 s, (β) 1.4 · 10−2m−1/9µ4/930 s, (γ) We assume the next indication in this formula: (α), Ṁc ≤ Ṁcr, p ≤ pGL; (β), Ṁc ≤ Ṁcr, p > pGL; (γ), Ṁc > Ṁcr. 38 , s, (α) 1.2m−5/7µ 38 , s, (β) 0.17m−2/3µ 30 , s, (γ) We assume the next indication in this formula: (α), Ṁc ≤ Ṁcr and RA > RG; (β), Ṁc ≤ Ṁcr and RA ≤ RG; (γ), Ṁc > Ṁcr. Here a new critical period pGL was introduced from the condition RG = Rl: pGL = 4πGMx ≈ 500mxv−27 s, (98) Treating the rotator’s magnetic dipole moment µ and Mx as parameters, we find that an over- whelming majority of the magnetor’s stages can be shown on a “p-L” diagram (Lipunov 1982a). The quantity L also proves to be convenient be- cause it can be observed directly at the accretion stage. 9.9. The Gravimagnetic Parameter By expecting the expression for the stopping radius in the subcritical regime (Ṁc ≤ Ṁcr) one can note that the magnetic dipole moment µ and the accretion rate Ṁc always appear in the same combination, , (99) as was noticed by Davies & Pringle (1981). The parameter y characterizes the ratio between the gravitational and magnetic “properties” of a star and will, therefore, be called the gravimagnetic pa- rameter. Two magnetic rotators having quite dif- ferent magnetic fields, subjected to different ex- ternal conditions but with identical gravimagnetic parameters, have similar magnetospheres, as long as the accretion rate is quite low (Ṁc ≤ Ṁcr). Otherwise, the flux of matter near the stopping radius no longer depends on the accretion rate at a large distance. In fact, the number of independent parameters can be further reduced (see e.g. Lipunov (1992)) by introducing the parameter Ṁcv∞ , (100) Table 2: Parameter of the evolution equation of a magnetic rotators. Parameter Regime E, SE P, SP A SA G M Ṁ 0 0 Ṁc Ṁc(RA/Rs) 0 Ṁc κt ∼ 2/3 . 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3 ∼ 1/3 Rt Rl Rm Rc Rc RA a Plotting the rotator’s period p versus Y - parameter we can draw a somewhat less obvious but more general classification diagram than the “p-L” diagram discussed above. This permits us to show on a single plot the rotators with key parameters Ṁc, µ and v∞ spanning a very wide range. In the case of supercritical accretion, another characteristic combination is found in all the ex- pressions: , (101) In analog to the subcritical “p-Y” diagram, a supercritical “p-Ys” diagram can be drawn. 10. Evolution of Magnetic Rotators The evolution of a magnetic rotator, which de- termines its observational manifestations, involves the slow changing of the regimes of its interaction with the surrounding medium. Such an approach to the evolution was developed in the 1970s by Shvartsman (1970); Bisnovatyi-Kogan & Komberg (1976); Illarionov & Sunyaev (1975); Shakura (1975); Wickramasinghe & Whelan (1975), Savonije & van den Heuvel (1977) and others. Three stages were mostly considered in these pa- pers: ejector, propeller and accretor. All these stages can be described by a unified evolutionary equation. 10.1. The evolution equation Analysis of the nature of interaction of a mag- netized star with the surrounding plasma allows us to write an approximate evolution equation for the angular momentum of a magnetic rotator in the general form (Lipunov 1982a): = Ṁksu − κt , (102) where ksu is a specific angular momentum applied by the accretion matter to the rotator. This quan- tity is given by ksu = (GMxRd) 1/2, Keplerian disk accretion, G, wind accretion in a binary, ∼ 0, a single magnetic rotator. (103) where Rd is the radius of the inner disk edge, Ω is the rotational frequency of the binary sys- tem, and ηt ≈ 1/4 (Illarionov & Sunyaev 1975). The values of dimensionless factor κt, characteris- tic radius Rt and the accretion rate Ṁ in different regimes are presented in Table 2. The evolution equation (102) is approximate. In practice, the situation with propellers and su- perpropellers is not yet clear. In Table 2 Rm is the size of a magnetosphere whose value at the propeller stage is not known accurately and which may differ significantly from the standard expres- sions for the Alfven radius. 10.2. The equilibrium period The evolution equation presented above indi- cates that an accreting compact star must en- deavor to attain an equilibrium state in which the resultant torque vanishes (Davidson & Ostriker 1973; Lipunov & Shakura 1976). This hypothe- sis is confirmed by observations of X-ray pulsars. By equating the right-hand side of equation (103) to zero, we obtain the equilibrium period: peq ≈ 7.8π κt/ǫ2(GMx) −5/7y−3/7 s, (α) peq = A/BwL 10 s, (β) (104) where A ≈ 5× 10−4(3κt)µ230I−145 m−1x s yr−1, and Bw ≈ 5.2×10−6R26m 0 /(10 2/3m2x)I 45 Ṁ−6η s yr L37 = L/10 37, T10 = T/10 days; (α), disk accre- tion; (β), quasi-spherical accretion. Alternatively: peq ≈ 1.0L−3/737 µ 30 s, disk, peq = 10η ×(m2/30 /(102/3m2x))−1/2× ×L−137 T 10 µ30 s, stellar wind, (105) Let us turn to the case of disk accretion. The above model of the spin-up and spin-down torques possesses an unexpected property. The equilib- rium period obtained by setting the torque to zero is connected with the critical period pA through a dimensionless factor: peq(A) = 2 pA, (106) The parameters κt and ε must be such that peq > pA. Since κt ≈ ε ≈ 1, the equilibrium period in the case of disk accretion is close to the critical period, pA, separating accretion stage A and the propeller stage P . In the case of the supercritical accretion the equilibrium period is determined by formula (Lipunov 1982b): peq(SA) ≃ 0.17µ2/330 m−1/9x s, (107) 10.3. Evolutionary Tracks The evolution of NS in binaries must be stud- ied in conjunction with the evolution of nor- mal stars. This problem was discussed qualita- tively by Bisnovatyi-Kogan & Komberg (1976); Savonije & van den Heuvel (1977), Lipunov (1982a) and other. We begin with the qualitative analysis presented in the latter of these paper. The most convenient method of analysis of NS evolution is using the ”p-L” diagram. It should be recalled that L is just the potential accretion luminosity of the NS. This quantity is equal to the real luminosity only at the accretion stage. In Figure 1 we show the evolutionary tracks of a NS. As a rule, a NS in a binary is born when the companion star belongs to the main sequence (loop-like track). During the first 105 − 107 years, the NS is at the ejector stage, and usually it is not seen as a radiopulsar since its pulse radia- tion is absorbed in the stellar wind of the normal star. The period of the NS increases in accordance with the magnetic dipole losses. After this, the matter penetrates into the light cylinder and the NS passes first into the propeller stage and then into the accretor stage. By this time, the normal star leaves the main sequence and the stellar wind strongly increases. This results in the emergence of a bright X-ray pulsar. The period of the NS stabilizes around its equilibrium value. Finally, the normal star fills the Roche lobe and the accre- tion rate suddenly increases; the NS moves first to the right and then vertically downward in the ”p-L” diagram. In other words, the NS enters the supercritical stage SA (superaccretor) and its spin period tends to a new equilibrium value (see equa- tion (107)). After the mass exchange, only the helium core of the normal star is left (a WR star in the case of massive stars), the system becomes detached and the NS returns back to the propeller or ejec- tor state. Accretion is still hampered by rapid NS rotation. This is probably the reason underlying the absence of X-ray pulsars in pairs with Wolf- Rayet stars (Lipunov 1982c). Since the helium star evolves on a rather short time-scale (≈ 105 yr), the NS does not have time to spindown con- siderably: after explosion of the normal star, the system can be disrupted leaving the old NS as an ejector, i.e. as a high-velocity radio pulsar. The “loop-shaped” track discussed above can be written in the form: • I+E → I+P → II+P → II+A → III+SA → IV+P → E+E (recycled pulsar) → . . . • I+E → I+P → II+A → III+SA → IV+E → (recycled ejector) → IV+P → E+E (re- cycled pulsar) → . . . • Another version of the evolutionary track of a NS formed in the process of mass exchange within a binary system is: III+SE → III+SP → IV+P → E+E → . . . The overall lifetime of a NS in a binary system depends on the lifetime of the normal star and on the parameters of the binary system. However, Fig. 1.— Tracks of NS on the period (p) - gravimagnetic parameter (Y) diagram: track of a single NS (vertical line) and of a NS in a binary system (looped line). For the second track, possible observational appearances of the NS are indicated. the number of transitions from one stage to an- other during the time the NS is in the binary is proportional to the magnetic field strength of the Figure 2 demonstrates the effect of NS mag- netic field decay (track (a) with and (b) without magnetic field decay). The first track illustrates the common path which results in the production of a typical millisecond pulsar. 10.4. Evolution of Magnetic Rotators in Non-circular Orbits So far we have considered evolution of a mag- netic rotator related to single rotators or those entering binary system with circular orbits. This approximation was appropriate for the gross anal- ysis of binary evolutionary scenario performed by Kornilov & Lipunov (1983a,b). This approx- imation is further justified by the fact that the tidal interaction in close binaries leads to or- bital circularization in a short time (Hut 1981). However, the more general case of a binary with eccentric orbit must be considered for fur- ther analysis. It is especially important because many of the currently observed X-ray pulsars, as well as radio pulsars with massive companions PSR B1259-63 (Johnston et al. 1992) and PSR B0042-73 (Kaspi et al. 1994), are in highly ec- centric orbits around massive companions. Pre- viously, such studies have been performed by Gnusareva & Lipunov (1985); Prokhorov (1987). Orbital eccentricity necessarily emerges after the first supernova explosion and mass expulsion from the binary system. In massive binaries with long orbital periods & 10 days, the eccentricity may be well conserved until the second episode of mass exchange (Hut 1981). Here, we concentrate on the evolutionary consequences of eccentricity. 10.5. Mixing types of E-P-A binary sys- tems with non-zero orbital eccentric- The impact of eccentricity on the observed properties of X-ray pulsars has been consid- ered in many papers (Amnuel’ & Guseinov 1971; Shakura & Sunyaev 1973; Pacini & Shapiro 1975; Lipunov & Shakura 1976). The most important consequence of orbital eccentricity for the evolu- tion of rotators can be understood without de- tailed calculations, and suggests the existence of two different types of binary systems separated by a critical eccentricity, ecr (Gnusareva & Lipunov 1985). Consider an ideal situation when a rotator en- ters a binary system with some eccentricity. The normal star (no matter how) supplies matter to the compact magnetized rotator. We assume that all the parameters of the binary system (binary separation, eccentricity, masses, accretion rate, etc.) are stationary and unchanged. Then a crit- ical eccentricity ecr appears such that at e > ecr the rotator is not able to reach the accretion state in principle. Let the rotator be rapid enough ini- tially to be at the ejector (E) state. With other parameters constant, the evolution of such a star is determined only by its spindown. The star will gradually spindown to such a state that when pass- ing close to the periastron where the density of the surrounding matter is higher, the pulsar will “choke” with plasma and pass into the propeller regime. Therefore, for a small part of its life the ro- tator will be in a mixed EP-state, being in the pro- peller state at periastron and at the ejector state close to apastron. The subsequent spindown of the rotator leads most probably to the propeller state along the entire orbit. This is due to the fact that the pressure of matter penetrating the light cylinder Rl increases faster than that caused by relativistic wind and radiation, as first noted by Shvartsman (1971a). So it proves to be much harder for the rotator to pass from the P state to the E state than from E to P state (see the follow- ing section). The rotator will spindown ultimately to some period, pA, at which accretion will be possible dur- ing the periastron passage. Accretion, in contrast, will lead to a spin-up of the rotator, so that it reaches some average equilibrium state character- ized by an equilibrium period peq defined by the balance of accelerating and decelerating torques averaged over the orbital period. If the eccentric- ity was zero, the rotator would be in the accretion state all the time. By increasing the eccentricity and keeping the periastron separation between the stars unchanged, we increase the contribution of the decelerating torque over the orbital period and thus decrease peq. At some ultimate large enough eccentricity ecr the equilibrium period will be less than the critical period pA permitting the transi- Fig. 2.— The period-gravimagnetic parameter diagram for NS in binary systems. (a) with NS magnetic field decay (the oblique part of the track corresponds to “movement” of the accreting NS along the so-called “spin-up” line), (b) a typical track of a NS without field decay in a massive binary system. tion from the propeller state to the accretion state at apastron to occur. The rotational torque ap- plied to the rotator, averaged over orbital period, vanishes, and in this sense the equilibrium state is achieved, but the rotator periodically passes from the propeller state to the accretion state. Thus, X-ray pulsars with unreachable full-orbit accretion state must exist. This means that from the observational point of view such binaries will be observed as transient X-ray sources with sta- tionary parameters for the normal component. Typically, the evolutionary track of a rotator in an eccentric binary is • E → PE → P → AP, e > ecr • E → PE → P → AP → A, e < ecr this may be the principal formation channel of transient X-ray sources. 10.6. Ejector-propeller hysteresis As mentioned earlier, the transition of the ro- tator from the ejector state to the propeller state is not symmetrical. Here we consider this effect in more detail. In terms of our approach, we must study the dependence of Rst on Ṁc. To find Rst, we must match the ram pressure of the accreting plasma with that caused by the relativistic wind or by the magnetosphere of the rotator. This depen- dence Rst(Ṁc) will be substantially different for rapidly (Rl < RG) and slowly (Rl > RG) rotat- ing stars (see Figure 3). One can see that in the case of a fast rotator, an interval of Ṁc appears where three different values of Rst are possible, the upper value R1 corresponding to the ejector state and the bottom value R3 to the propeller state; the intermediate value R2 is unstable. This means that the rotator’s state is not determined solely by the value of Ṁc , but also depends on previous behavior of this value. Now consider a periodic changing of Ṁc caused, for example, by the rotator’s motion along an ec- centric orbit, and large enough for the rotator to transit from the ejector state to the propeller state and vice versa. Initially, the rotator is in the ejec- tor state. By approaching the normal star, the accretion rate Ṁc increases and reaches a criti- cal value ṀEP , where the equilibrium points R1 (stable point corresponding to the ejector state) and R2 (unstable) approach RG (upper kink), where they merge (see Figure 2). After that only one equilibrium point remains in the system, the stopping radius Rst jumps from ≈ RG down to Fig. 3.— Dependence of the stopping radius Rst on the modified gravimagnetic radius Ry for two possible relations between the light cylinder radius Rl and the gravitation capture radius RG: Rl > RG (left-hand panel) and Rl < RG (right-hand panel; here the ejector-propeller hysteresis becomes possible). R3 < Rl, and the rotator changes to the propeller state. As Ṁc decreases further along the orbit and reaches the critical value ṀEP once again, the reverse transition from propeller to ejector does not occur. The transition only occurs when Ṁc reaches another critical value, ṀPE < ṀEP , where the unstable point R2 meets the stable pro- peller point R3, and the stopping radius Rst jumps from ∼ Rl up to R1 > RG. It should be noted that for fast enough rotators, a situation is pos- sible when the step down from the ejector state occurs in such a manner that the stopping ra- dius Rst < Rc and the rotator passes directly to the accretion state. The reverse transition always passes through the propeller stage: A → P → E. In principle, transitions from the ejector state to supercritical states SP or SA are also possible (Prokhorov 1987). In the case of slow rotators (Rl > RB) ), the “E-P” hysteresis is not possible, and transitions between these states are symmet- rical. 10.7. E-P transitions for different orbits Orbital motion of the rotator around the nor- mal companion in an eccentric binary draws a hor- izontal line on the ”p-Y” diagram, with the begin- ning at a point corresponding to Ṁc(ap), and the end at a point corresponding to Ṁc(aa) (here ap and aa are the periasrton and apastron distances, respectively). The length of this segment is de- termined by the eccentricity. Since Y ∝ Ṁc , the rotator moves along this segment from left to right and back as it revolves from the apastron to the periastron. At each successive orbital period, this line slowly drifts up to larger periods. The evolu- tion of this system is thus determined by the or- der the critical lines on the diagram are crossed by this “line”. It is seen from the ”p-Y” diagram that the regions with and without hysteresis are sepa- rated by a certain value of the parameter Y = Yk. Since Y ∝ Ṁc ∝ 1/r2 , four different situations are possible depending on the relationship of the bi- nary orbital separation a with critical value acr, corresponding to Yk (see e.g. Lipunov (1992); Lipunov et al. (1996b)). 1. ap > acr. In this case no hysteresis oc- curs and transitions E–P and the reverse take place in symmetrical points of the orbit. The rotator passes the following sequence of stages: E → EP → P → . . . Here EP means a mixed state of the rotator at which it is in the ejector state during one part of the orbit, and in the propeller state during the rest of the orbital cycle. 2. aa > acr > ap. In this case, the hysteresis occurs at the beginning of the mixed EP- state (state EPh), but as the rotator slows down the hysteresis gradually decays and disappears. E-P transition for this system is: E → EPh → EP → P → . . . 3. aa < acr. The hysteresis is possible in prin- ciple, but the shape of the transition depends on the eccentricity. Suppose a pulsar has spun down so much that the first transition from the ejector to the propeller occurred at the periastron. If the eccentricity were smal e < ecr, (do not confuse this ecr with the critical eccentricity introduced in the pre- vious section) the reverse transition to the ejector state would not occur even at the apastron, and the evolutionary path would be E → P → . . . 4. If e > ecr , the track is E → EPh → P → . . . It should be noted that just after the first EP transition (as well as before the last), the system spends a finite time in the E and P states at every revolution. The value of ecr can be expressed through the orbital parameters as ecr ≃ cr − a1/7a cr + a , (108) To conclude, we note that the hysteresis during the ejector-propeller transition may be possible for single radio pulsars also. For example, when the pulsar moves through a dense cloud of interstellar plasma, the pulses can be absorbed. The radio pulsar turns on again when it comes out from the cloud. The hysteresis amplitude for single pulsars can be high enough because of small relative ve- locities of the interstellar gas and the pulsar, so that RG ≫ Rl. 11. Summary The “Scenario Machine” is the numerical code for theoretical investigations of statistical proper- ties of binary stars, i.e. this is population synthesis code (Lipunov et al. 1996b). 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Basic equations and initial distributions Evolutionary scenario for binary stars Basic evolutionary states of normal stars Main sequence stars Post main-sequence stars Roche lobe overflow Wolf-Rayet and helium stars Stellar winds from normal stars Change of binary parameters: mass, semi-major axis and eccentricity Mass change Semi-major axis change The change of eccentricity Special cases: supernova explosion and common envelope Three regimes of mass accretion by neutron stars Ordinary accretion Super-accretion Hyper-accretion Mass accretion by black holes Accretion induced collapse and compact objects merging Additional scenarios of stellar wind from massive stars Evolutionary scenario B Evolutionary scenario C Evolutionary Scenario W The ``Ecology'' of Magnetic Rotators A Gravimagnetic Rotator The Environment of the Rotator The Stopping Radius The Stopping Radius in the Supercritical Case The Effect of the Magnetic Field The Corotation Radius Nomenclature A Universal Diagram for Gravimagnetic Rotators The Gravimagnetic Parameter Evolution of Magnetic Rotators The evolution equation The equilibrium period Evolutionary Tracks Evolution of Magnetic Rotators in Non-circular Orbits Mixing types of E-P-A binary systems with non-zero orbital eccentricity Ejector-propeller hysteresis E-P transitions for different orbits Summary

0704.1388

Why do some intermediate polars show soft X-ray emission? A survey of XMM-Newton spectra

Draft version October 29, 2018 Preprint typeset using LATEX style emulateapj v. 10/09/06 WHY DO SOME INTERMEDIATE POLARS SHOW SOFT X-RAY EMISSION? A SURVEY OF XMM-Newton SPECTRA. P.A. Evans and Coel Hellier Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK Draft version October 29, 2018 ABSTRACT We make a systematic analysis of the XMM-Newton X-ray spectra of intermediate polars (IPs) and find that, contrary to the traditional picture, most show a soft blackbody component. We compare the results with those from AM Her stars and deduce that the blackbody emission arises from reprocessing of hard X-rays, rather than from the blobby accretion sometimes seen in AM Hers. Whether an IP shows a blackbody component appears to depend primarily on geometric factors: a blackbody is not seen in those that have accretion footprints that are always obscured by accretion curtains or are only visible when foreshortened on the white-dwarf limb. Thus we argue against previous suggestions that the blackbody emission characterises a separate sub-group of IPs which are more akin to AM Hers, and develop a unified picture of the blackbody emission in these stars. Subject headings: accretion, accretion discs – novae, cataclysmic variables – X-rays: binaries. 1. INTRODUCTION Intermediate polars (IPs) – interacting binaries with a magnetic white-dwarf primary – have traditionally been noted for their hard X-ray emission. This arises as the magnetic field of the white dwarf disrupts the accre- tion disc and channels material towards the magnetic polecaps. This material forms stand-off shocks, below which it cools via free-free interactions, producing hard X-rays. However, a growing number of systems have been shown to emit a distinct blackbody component in softer X-rays (e.g. Mason et al. (1992); Haberl et al. (1994); de Martino et al. (2004)), reminiscent of the soft component prominent in the X-ray spectra of many AM Her stars (also known as polars). These systems are similar to IPs but the white-dwarf has a magnetic field strong enough to prevent an accretion disc from forming at all. In these systems, the soft blackbody component is though to arise from a heated polecap surrounding the accretion column (See Warner (1985); Hellier (2001) for a review of these objects). Currently it is unclear why the blackbody component is seen in some IPs and not others. Haberl & Motch (1995) suggested that there are two distinct classes of IP, with the ‘soft’ systems being evolutionary progenitors of polars. They argued that the ‘hard IPs’ may have larger and cooler polecaps, pushing the soft emission into the EUV and explaining the difference in the spectra. We present here a study of XMM-Newton X-ray data of 12 IPs, aimed at discovering why some IPs show a blackbody component while others don’t. Our method is similar to that of Ramsay & Cropper (2004) (hereafter RC04) who analysed the XMM-Newton data of twenty- one polars, which enables us to compare the IPs with the polars. 2. OBSERVATIONS AND DATA ANALYSIS The XMM-Newton observatory (Jansen et al. 2001) was launched in 1999, and we have obtained observa- 1 Current address: Department of Physics and Astronomy, Uni- versity of Leicester, Leicester, LE1 7RH, UK TABLE 1 The XMM observations of IPs analysed in this paper. Object ObsID Date References AO Psc 0009650101 2001-06-09 1,2 EX Hya 0111020101 2000-07-01 1,2 0111020201 2000-07-01 1,2 FO Aqr 0009650201 2001-05-12 1,2,3 GK Per 0154550101 2002-03-09 2,4 HT Cam 0144840101 2003-03-24 2,5,6 PQ Gem 0109510301 2002-10-08 2,7 NY Lup 0105460301 2000-09-07 2,8 UU Col 0201290201 2004-08-21 9 V1223 Sgr 0145050101 2003-04-13 1,2 V405 Aur 0111180401 2001-10-05 2,10 V2400 Oph 0105460601 2001-08-30 2 WX Pyx 0149160201 2003-05-20 11 References. — (1) Cropper et al. (2002), (2) Evans & Hellier (2005b), (3) Evans et al. (2004), (4) Vrielmann et al. (2005) (5) de Martino et al. (2005), (6) Evans & Hellier (2005a), (7) Evans, Hellier & Ram- say (2006), (8) Haberl et al. (2002), (9) de Martino et al. (2006) (10) Evans & Hellier (2004), (11) Schlegel (2005). tions of twelve IPs from the public archive. We ana- lyzed the data from the EPIC-MOS and pn instruments (Turner et al. 2001; Strüder et al. 2001) which provide high-throughput, medium-resolution spectroscopy across the 0.2–12 keV energy range. The higher resolution RGS instruments (den Herder et al. 2001) have only 20 per cent of the effective area of the MOS cameras and the data are not used here. A summary of the observations used is given in Table 1. We re-ran the pipeline processing for these observations using xmm-sas v7.0.0. The observations of GK Per, NY Lup and V2400 Oph suffered from pile-up, and thus only the wings of the PSF were included in the source extrac- tion. The MOS-1 observation of EX Hya was so badly piled up that we excluded it from our analysis. RC04 used only the EPIC-pn data as it was better cal- ibrated than the EPIC-MOS data at soft energies. Us- ing the better calibrations of xmm-sas v.7 we extracted http://arxiv.org/abs/0704.1388v1 2 Evans & Hellier spectra from all three EPIC instruments. Response ma- trices were created for each spectrum, using the xmm-sas rmfgen and arfgen tasks. We then modelled the spectra using xspec v11. For each star, all model parameters were tied between the EPIC instruments, except for the normalisation which we allowed to vary in order to com- bat the effects of cross-calibration uncertainties. Although IP spectra can vary considerably over the spin cycle, for the majority of the systems in this paper, we do not have enough geometric information to identify phase regions when the hard/soft components are best presented to us (as RC04 did), so we extracted spectra covering the entire observation. Note that the results of our spectroscopy are thus weighted averages from across the spin cycle; this was taken into account when inter- preting our results . To reproduce the hard component we used the strat- ified accretion column model of Cropper et al. (1999). This models the spectrum in terms of the white dwarf mass (MWD) and specific accretion rate (i.e. accretion rate per unit area, ṁ), from which it calculates the temperature and density profile of the column. This is then divided into 100 bins, evenly distributed in veloc- ity space, each bin emitting as an optically thin plasma (a mekal). To the stratified column model, we added narrow Gaussians for the 6.4-keV iron fluorescence line and the 0.547 keV Oxygen vii photoionisation line where necessary. We then applied to this emission a simple pho- toelectric absorber. For most systems this did not give an acceptable fit, so we added either one or two partial- covering absorbers as necessary. Next, we added a blackbody component to the mod- els. Since absorption at the densities of the partial- covering components (typically ∼ 1023 cm−2) will com- pletely smother any soft X-ray emission and thus be re- dundant with model normalisation, the blackbody com- ponent was absorbed only by the simple absorption, which was of order 1019–1021 cm−2. For some systems the addition of a blackbody did not improve the fit. For these systems we manually raised the blackbody normalisation until it significantly reduced the fit quality, thus finding an upper limit. Since this will be temperature dependent, we did this for blackbody temperatures of 40, 60 and 80 eV. We quote, in Table 2, the f-test statistic to judge the the significance of adding the blackbody component. However, this test will produce false positives in the pres- ence of calibration systematics. We have thus estimated the systematics by fitting a model optimised for the MOS data to the pn data (allowing only the normalisation to change) and recording the change in χ2(=∆χ2system). We claim the presence of a blackbody only if it improves the χ2 by more than ∆χ2system. This method is more con- servative than using the f-test alone. We include this estimate of the systematics in all the errors quoted in this paper. Details of the fits are given in Table 3. The ṁ was un- constrained for every system, so is not given. We do not quote errors on the partial-covering absorbers as they do not affect the softness ratio. The ratio is sensitive, how- ever, to the metal abundance in the column, as there is a forest of iron L lines in the 0.5–1.2 keV range, affecting the model fit at the soft end. For all twelve systems we then calculated the flux from the hard and soft components. Following RC04 we de- fined the softness ratio as Fs/4Fh, where Fs and Fh are the fluxes of the soft and hard components respectively. The factor of four arises because the hard component is optically thin and thus radiates isotropically, whereas the hard component is optically thick. Where the blackbody- emitting region is seen foreshortened, the observed ratio will be an underestimate. The softness ratios are shown in Figs. 1 and 2. We show the observed ratio, the ratio of unabsorbed fluxes over the 0.2–12 keV range, and the ratio of unabsorbed fluxes calculated over all energies. These bolometric fluxes and softness ratios are given in Table 4. For the systems with no detectable soft component we show the upper limit calculated for a 60-eV blackbody, and present the fluxes and ratios for a range of blackbody temperatures in Table ??. 3. RESULTS We show the spectra for the systems with a blackbody component in Fig. 3, and for those without in Fig. 4. For the latter we have also shown the upper limit determined for a 60-eV blackbody component. For FO Aqr, AO Psc, V1223 Sgr and HT Cam we found no evidence for a soft component, in agree- ment with previous observations, (see Norton et al. (1992); Hellier et al. (1996); Beardmore et al (2000); Evans & Hellier (2005a) respectively). 3.1. V405 Aur The XMM observation of V405 Aur contains system- atic discrepancies between the two EPIC-MOS instru- ments below 0.4 keV. However, when processed under sas 7.0 these are at a much lower level than when Evans & Hellier (2004) analysed the data, and we have made no allowance for these discrepancies in the fit. Note also that as there is no pn data for V405 Aur, we have no estimate of the effects of systematics discussed in Sec- tion 2, so our errors are likely to be underestimates. The best-fitting blackbody temperature was kT = 64.78+0.81 −1.11 eV. This is significantly higher than the 40±4 eV reported by Evans & Hellier (2004) analysing the same observation, however they used two mekals to fit the hard component whereas we used the stratified column model. Since the calibration has also changed since Evans & Hellier (2004), we analysed our better- calibrated data using their model, and found a fit in agreement with theirs. This demonstrates that the re- sults are somewhat model dependent; the stratified col- umn model is likely to be the more physically realis- tic. Fitting the hard component with a single, high temperature plasma, Haberl et al. (1994) found a black- body temperature of 49–64 eV (from ROSAT data) and de Martino et al. (2004) found 73 ± 14 eV (using Bep- poSAX ). Our fitted hydrogen column of 3.46+0.41 −0.31 × 10 20 cm−2 agrees with that of de Martino et al. (2004) [(4±2)×1020 cm−2] but not with those of Haberl et al. (1994) or Evans & Hellier (2004) who reported (5.7 ± 0.3) × 1020 and (10.6+0.9 −1.2) × 10 20 cm−2 respectively. However, the fitted column will depend on the emission model used, so some discrepancy is expected. Soft X-rays from IPs 3 TABLE 2 Fit statistics for each star with and without a blackbody. Star χ2 (dof) χ2 (dof) (with bb) f-test ∆χ2system χ (No bb) (with bb) AO Psc 3372.74 (2888) 3772.71 (2884) 0.93 700 0.03 FO Aqr 1462 (1864) 1444 (1860) 1.6×10−4 168 18 HT Cam 1325 (1273) 1310 (1269) 4.1×10−3 79 16 V1223 Sgr 3840.8 (3128) 3840.6 (3124) 0.99 143 0.2 EX Hya 14045 (4876) 10158 (4873) < 10−99 1284 3887 GK Per 17040 (4079) 5024 (4075) < 10−99 84 12016 NY Lup 902 (699) 669 (695) 8×10−14 40 233 PQ Gem 20435 (2439) 2940 (2435) < 10−99 144 17495 UU Col 1676 (817) 910 (813) < 10−99 76 766 V2400 Oph 1019 (1003) 953 (999) 9×10−14 50 66.15 V405 Aur 17626 (997) 1146 (994) < 10−99 * 16480 WX Pyx 594 (477) 495 (473) 8×10−18 52 99 Note. — The f-test gives the probability that no blackbody is present, making no allowance for systematics. The ∆χ2system is the change in χ 2 in fitting the same model to the MOS and pn cameras, thus giving an estimate of the systematic errors. The last column is the improvement in χ2 when a blackbody is added. We consider this significant if it exceeds ∆χ2system. ∗ There was no pn data for V405 Aur, so ∆χ2system was not estimated. TABLE 3 Spectral components used in the fitted models. Star wabs nH blackbody kT Part Abs (1) Part Abs (2) MWD Abundance (1020cm−2) (eV) (nH , Cv Frc) (nH , Cv Frc) (M⊙) (solar) V405 Aur 3.46 (+0.41, −0.31) 64.78 (+0.81, −1.11 ) 17, 0.49 3.0, 0.63 0.40 (+0.05, −0.06) 0.069 (+0.024, −0.021) GK Per 23.3 (+2.0, −1.9) 62 (±2) 23, 0.74 4.7, 0.45 0.92 (+0.39, −0.13) 0.21 (+0.14, −0.07) NY Lup 7.8 (±3.9) 104 (+21, −23) 14, 0.49 0.38, 0.71 0.96 (+0.40, −0.55) 0.68 (+0.51, −0.59) V2400 Oph 7.0 (+2.9, −4.9) 117 (+33, −44) 11, 0.52 0.61, 0.53 0.69 (+0.06, −0.24) 0.33 (+0.12, −0.10) PQ Gem 0 (+0.30) 47.6 (+2.9, −1.4) 42, 0.60 3.4, 0.56 0.70 (+0.16, −0.14) < 0.08 EX Hya 9.76 (+2.2, −0.86) 31.0 (+1.3, -2.4) 75, 0.35 4.0, 0.29 0.449 (+0.005, −0.013) 0.514 (+0.01, −0.0029) UU Col 0 (+0.59) 73 (+20, −9) 10, 0.34 1.23 (+0.17, −0.29) 0.66 (+1.0, -0.62) WX Pyx 8.4 (+3.8, −2.9) 82 (+11, −15) 1.4 (+0, −0.09) < 2.87 FO Aqr 0 (+2.1) 21, 0.80 6.4, 0.98 1.19 (+0.11, −0.31) 0.31 (+0.20, −0.23) AO Psc 3.89 (+0.69, −1.44) 14, 0.62 1.8, 0.75 0.594 (+0.13, −0.040) 0.362 (+0.20, −0.064) HT Cam 3.86 (+0.81, -0.88) 0.687 (+0.094, −0.061) 0.52 (+0.24, −0.11) V1223 Sgr 1.03 (+0.36, −0.52) 13, 0.46 1.3, 0.63 1.046 (+0.049, −0.012) 0.398 (+0.090, -0.049) Note. — The column density of the partial absorption is given in units of 1022 cm−2. Errors are quoted to the same power of ten as the corresponding parameter. TABLE 4 The unabsorbed, bolometric fluxes of the soft and hard components for those systems which show blackbody emission. Object Fh,bol Fs,bol Ratio (erg s−1 cm−2) (erg s−1 cm−2) V405 Aur 5.1× 10−11 (+3.6, −1.1) 4.3× 10−11 (+2.4, −1.2) 0.211 (+0.018, -0.038) GK Per 1.20× 10−9 (+0.25, −0.06) 2.29× 10−10 (+0.95, −0.62) 4.8× 10−2 (+1.8, −1.3) NY Lup 4.15× 10−11 (+12.7, −0.18) 4.3× 10−12 (+9.9, −1.4) 2.6× 10−2 (+1.6, −1.1) V2400 Oph 9.2× 10−11 (+4.1, −1.9) 3.3× 10−12 (+2.1, −1.5) 8.9× 10−3 (+4.6, −2.8) PQ Gem 1.07× 10−10 (+0.40, −0.23) 1.33× 10−11 (+0.13, −0.17) 3.11× 10−2 (+0.74, −0.83) EX Hya 3.95× 10−10 (+0.26, −0.20) 1.59× 10−10 (+3.47, −0.62) 1.00 (+0.48, −0.20) WX Pyx 7.51× 10−12 (+0.35, −0.93) 6.0× 10−13 (+5.9, −2.9) 2.00× 10−2 (+2.38, −0.93) UU Col 6.81× 10−12 (+2.29, −0.80) 3.04× 10−13 (±0.98) 1.12× 10−2 (±0.45) Note. — The ratio is defined as in Fig. 1. Errors are given to the same power of ten as the values. 4 Evans & Hellier Fig. 1.— Softness ratios of the IPs observed with XMM, defined as Fs/4Fh, where Fs and Fh are the fluxes of the soft blackbody and hard plasma components respectively, calculated over the 0.2–12 keV energy range covered by XMM. Upper panel : Ratios calculated from the spectral fits. Lower panel : The ratios calculated from the spectral fits, after the absorption components were removed. Fig. 2.— As for Fig. 1, but with the effects of absorption removed and Fs and Fh extended over all energies. The ratios for the polars given in RC04 are also shown (hollow squares); RC04 did not quote errors. The uppermost six systems are the polars in which RC04 found no blackbody component. We have found an upper limit for these systems as we did for the IPs. The dashed line corresponds to a softness ratio of 0.5; systems with a higher softness ratio exhibit a ‘soft excess’. Soft X-rays from IPs 5 TABLE 5 The unabsorbed, bolometric fluxes from the systems with no detectable soft X-ray component, and the upper limit of the softness ratio, for a range of temperatures. Object Fh,bol Ratio40eV Ratio60eV Ratio80eV (erg s−1 cm−2) FO Aqr 2.71× 10−10(+0.65, −0.18) < 4.6× 10−4 < 1.4× 10−4 < 8.0× 10−5 AO Psc 1.51× 10−10(+0.09, −0.11) < 4.3× 10−3 < 1.1× 10−3 < 6.2× 10−4 HT Cam 8.48× 10−12(+0.53, −0.38) < 2.5× 10−2 < 5.3× 10−3 < 2.6× 10−3 V1223 Sgr 2.96× 10−10( ±0.13) < 8.5× 10−4 < 3.5× 10−4 < 2.3× 10−4 3.2. GK Per A soft blackbody component was necessary to model the XMM spectrum of GK Per as previously found by Vrielmann et al. (2005). They reported a blackbody temperature of 59.6 ± 0.2 eV absorbed by a column of (3.2±0.2)×1021 cm−2. Our temperature of 62±2 eV and column of (2.3± 0.2) cm−2 are very similar, though not formally in agreement. Note that Vrielmann et al. (2005) parameterised the hard emission using a bremsstrahlung component and a mekal, supporting our assertion above that these results are model dependent. 3.3. NY Lup Haberl et al. (2002) analysed this XMM observation of NY Lup (=RXJ154814) and found a soft component with a blackbody temperature of 84–97 eV and a column density of (11.7–15.5)×1020 cm−2. Our values of kTbb = 104+21 −23 eV and nH = (7.8± 3.9)× 10 21 cm−2 agree. 3.4. V2400 Oph V2400 Oph was identified as a soft IP by de Martino et al. (2004), who analysed a BeppoSAX ob- servation and reported a blackbody temperature of 103± 10 eV and absorption column (46+12 −13)×10 20 cm−2. We find a blackbody temperature of 117+33 −44 eV, in agreement with this result, but a slightly lower absorption column of (7.0+2.9 −4.9)× 10 20 cm−2. This is probably because de Mar- tino used a single mekal and a single partial-covering absorber to model the hard emission, whereas we used the stratified column model and two partial covering ab- sorbers. 3.5. PQ Gem PQ Gem was the first IP found to have a soft-X-ray component (Mason et al. 1992). This component was also present in the XMM data, with a best-fitting black- body temperature of 47.6+2.9 −1.4 eV, in agreement with the 46+12 −23 eV of Duck et al. (1994) from ROSAT data, and 56+12 −14 eV of de Martino et al. (2004) from BeppoSAX data. The fitted column density goes to zero, which is likely to be an artefact of fitting a complex absorption with too simple a model. We quote an upper limit of 3× 1019 cm−2 based on the phase-resolved modelling of Evans et al. (2006). 3.6. EX Hya The best-fitting model for EX Hya used a blackbody component, which has not been previously reported in this system. However even with this component, the procedure outlined in Section 2 resulted in a poor fit =2.1). A possible reason for this is our choice of ab- sorption model. We have used a cold absorber in our models since the data do not warrant the extra param- eters in ionised absorption models, even though one ex- pects any absorbing material (e.g. the accretion curtains) to be ionised. We therefore tried various ionised absorp- tion models, but gained only a minor improvement to the fit. We thus reverted to the cold absorber model for consistency with the rest of this paper. We also tried us- ing phase-resolved spectroscopy, in case the poor fit was the result of averaging phase-variant parameters, how- ever this still did not yield an acceptable fit. We have nonethless included our results for EX Hya, for complete- ness, but due to the poor fit, we do not much place much weight on the EX Hya data when considering our results. As the distance to EX Hya is known (64.5±1.2 pc: Beuermann et al. (2003)), we can determine the size of the accretion footprint from the soft X-ray flux. Table 4 gives this as (1.59+3.47 −0.62) × 10 −10 ergs cm−2 s−1 with a temperature of 31.0+1.3 −2.4 eV, from which we compute an emitting area (8.4+29.8 −4.2 ) × 10 13 cm2. Suleimanov et al (2005) gave the mass of the white dwarf in EX Hya as 0.5±0.05M⊙, thus the observed blackbody emitting area in EX Hya covers (7.3+29.3 −4.0 ) × 10 −4 of the white-dwarf surface. 3.7. UU Col UU Col was identified as a soft IP by Burwitz et al. (1996). de Martino et al. (2006) have recently confirmed this with a detailed analysis of the XMM observation. They reported a blackbody temperature of 49.7+5.6 −2.9 eV, which is lower than our value of 73+20 −9 eV, however in their model the blackbody is absorbed by the partial cov- ering absorber, and no simple absorber is present. 3.8. WX Pyx The XMM observation of WX Pyx, the only X-ray observation of this star to date, has a relatively low sta- tistical quality. It was previously analysed by Schlegel (2005) who did not report looking for a blackbody com- ponent. However, we find that adding a blackbody does significantly improve the fit. 3.9. Comparison with the polars In Fig. 2 we have plotted the softness ratios of both the IPs and the polars (from RC04). For the polars which RC04 reported not to have a blackbody, we obtained the spectra as extracted an calibrated by RC04 (Ramsay, private communication), and fitted them in the same way as the IPs (Section 2) to obtain an upper limit. 6 Evans & Hellier Fig. 3.— The EPIC-pn spectra of the eight IPs for which the best-fitting models contain a blackbody component. The solid line shows the hard component; the broken line the blackbody. For V405 Aur we have shown the MOS-1 spectrum, since the pn camera did not collect any data. The chief difference in the two distributions is that while several polars show a softness ratio > 0.5, no IP can be confirmed to do this, and it can be excluded for all but EX Hya – for which our results are uncertain (Section 3.6). The ‘soft excess’ in polars is believed to arise due to ‘blobby accretion’ (e.g. Kuijpers & Pringle (1982)). In this model, dense blobs of matter penetrate into the white dwarf photosphere and the energy is ther- malised to a blackbody. Whether such accretion occurs in IPs has not been widely discussed in the literature. Hellier & Beardmore (2002) suggested that viscous interactions in an accre- tion disc would destroy blobs, although Vrielmann et al. (2005) interpreted flares in the lightcurve of GK Per as resulting from the accretion of blobs. Our findings sug- gest that blobby accretion is not significant in IPs. 4. DISCUSSION The ‘polar’ class of magnetic cataclysmic variable has long been known to be characterised by a soft black- body component (e.g. King & Watson (1987)). This Soft X-rays from IPs 7 Fig. 4.— The EPIC-pn spectra of the four IPs for which the best-fitting model does not contain a blackbody component. The solid line shows the best-fitting model. The broken line shows the upper limit to a blackbody component, given a temperature of 60 eV. For FO Aqr we show the MOS-1 data, as the signal-to-noise ratio of the pn data is worse. is considered to arise from the white-dwarf surface, heated either by reprocessing of hard X-rays from the accretion column, or by thermalisation of blobs of accretion (e.g. Kuijpers & Pringle (1982)). In con- trast, IPs were thought to lack this component (e.g. King & Lasota (1990)). However, observations with ROSAT found a blackbody component in some IPs, lead- ing Haberl & Motch (1995) to suggest that there were two spectrally distinct classes of IP. This raised the ques- tion of why. To address this we have conducted a systematic survey of the spectral characteristics of the IPs observed with XMM-Newton, which has much greater spectral coverage and throughput than ROSAT. We find that, of twelve IPs analysed, eight show a soft blackbody component while four do not. This suggests that a blackbody is a normal component of IPs, and hence of accretion onto magnetic white dwarfs, and that the spectra differ only in degree. We thus ask what causes the differing visibility of the soft component. There does not appear to be any correlation with the white-dwarf mass [see Cropper et al. (1999); Ezuka & Ishida (1999); Ramsay (2000); Suleimanov et al (2005) for mass estimates], nor any obvious correlation with the orbital period. In polars, systems with higher magnetic field strengths appear to have higher softness ratios (e.g. Ramsay et al. (1994)). Of the IPs in our sample showing polarisation, and thus known to have a relatively strong field (5–20 MG), all (PQ Gem, V405 Aur and V2400 Oph) show a blackbody component, while the four stars showing no blackbody emission (FO Aqr, AO Psc, HT Cam & V1223 Sgr) do not show polarisation. We give a possible explanation for this after discussing the role of absorp- tion. We first consider the simple absorber, which is proba- bly of interstellar origin. The detection of a blackbody component in most systems shows that interstellar ab- sorption is not sufficient to extinguish the soft emission. Further, the systems with no detected blackbody compo- nent do not have higher interstellar columns than those with a blackbody (Table 3), so this absorption cannot explain the differing visibility of the soft component. We thus turn to the partial-covering absorption, which in IPs is predominantly caused by the accretion cur- tains crossing the line of sight. Here we find that the systems where the lightcurves are dominated by deep absorption dips owing to the accretion curtains (FO Aqr, V1223 Sgr and AO Psc; see Beardmore et al. (1998); Beardmore et al (2000); Hellier et al. (1991) re- spectively) do tend to be those which lack a blackbody component. In contrast, systems showing a blackbody component, such as V405 Aur, NY Lup, EX Hya and V2400 Oph, tend to be systems where the lightcurves suggest that the accretion curtains do not hide the accre- tion footprints (see Evans & Hellier (2004); Haberl et al. (2002); Allan et al. (1998); Hellier & Beardmore (2002) respectively). We thus suggest that the major reason why some IPs don’t show a blackbody component is simply that the heated region near the accretion footprint is hidden by the accretion curtains, while in other IPs it is not, the difference being the result of the system inclination and the magnetic colatitude (see Fig. 5). Coupled with this is 8 Evans & Hellier to foreshortening No BB seen BB seen BB seen Little BB owing Fig. 5.— The factors that affect blackbody emission in an IP. a) When the upper magnetic pole is on the visible face, blackbody emission will only be seen if the inclination is such that the heated accretion region is visible above the accretion curtains. b) When the lower pole is on the visible face, it will likely be too foreshortened for us to detect blackbody emission. c) In UU Col the magnetic axis is highly inclined, so the foreshort- ening seen in b) is reduced and blackbody emission is seen. the effect of foreshortening, such that an optically thick heated region will not produce much blackbody emission if it is only seen while on the white-dwarf limb, rather than in the middle of the face. A proper investigation of this idea would need knowl- edge of the size and location of the accretion footprints and of the surrounding heated polecaps, so that we could estimate the difference absorbing columns of different spectral components, and how these vary with spin-cycle phase. However, this information is not known for the majority of IPs. The softness ratio might conceivably also vary with parameters such as accretion rate and white-dwarf mass, which are again only poorly known. However, as a test of our ideas, we can outline how they might apply to the remaining systems in our sam- ple which we did not consider when forming the model, namely HT Cam, GK Per, PQ Gem and UU Col. In PQ Gem the accretion curtains do cause an ab- sorption dip when they obscure the accretion footprints. However, the geometry of this star is relatively well de- termined (Potter et al. 1997; Mason 1997; Evans et al. 2006) and it appears that the heated polecap is graz- ingly visible above the accretion curtain for part of the cycle; thus it shows both an absorption dip and a soft blackbody, and is on the boundary between the two cases illustrated in the upper panel of Fig. 5. UU Col also shows an absorption dip when the ac- cretion curtains obscure the upper pole, and also shows blackbody emission. de Martino et al. (2006) proposed that the blackbody emission comes from the lower pole, viewed when that pole is closest to us (lowest panel of Fig. 5). We thus suggest that UU Col has an abnor- mally high inclination of the magnetic dipole, such that the lower pole is not foreshortened as much as in other IPs where no blackbody component is seen. V405 Aur is another system that appears to have a highly inclined dipole, such that blackbody emission from the lower pole is significant, leading in that system to a double-peaked soft-X-ray lightcurve (Evans & Hellier 2004). In contrast to all the other IPs, the XMM data of GK Per reported here were collected during an outburst. Fig. 6.— Schematic diagram of GK Per in outburst. Accretion occurs from all azimuths, resulting in a circular blackbody-emitting region (dark ring). As can be seen, even when the accretion cur- tains lie across our line-of-sight, part of this region is unobscured. Hellier et al. (2004) have argued that during outburst the accretion occurs from all azimuths, forming a complete accretion ring at the poles. As illustrated in Fig. 6, this means that some portion of the heated polecap is likely to be visible ‘behind’ the magnetic pole, where accretion does not normally occur. Thus in GK Per in outburst we see a system with both strongly absorbed X-ray emission (from in front of the magnetic pole) and a blackbody component. Lastly, we consider HT Cam. This shows very lit- tle sign of absorption, and its lightcurve can explained without any absorption effects (Evans et al. 2006). Yet it shows no blackbody emission, in apparent contradic- tion to our model. However, as previously suggested by de Martino et al. (2006) and Evans et al. (2006), it ap- pears that HT Cam has an exceptionally low accretion rate (partly accounting for the lack of absorption). If so, it could be that the blackbody component is simply too cool to be detected in the XMM bandpass. We note that the blackbody temperature in EX Hya, the other star in our sample below the period gap, is lower than in the others (Table 3), and that in HT Cam might be lower still. 5. SUMMARY We have analysed data from XMM observations of 12 intermediate polars and find that a soft blackbody com- ponent is a common feature of their X-ray spectra. We suggest that in the systems showing no blackbody emis- sion the heated accretion polecaps are largely hidden by the accretion curtains, or are only visible when on the white dwarf limb and highly foreshortened. Thus IPs with lightcurves dominated by absorption dips caused by the passage of accretion curtains across the line of sight tend to show no blackbody emission. Further, these are also the systems least likely to show polarisation, since the cyclotron-emitting column will also be obscured by the accretion curtains, or would be beamed away from us if the accretion region were on the white-dwarf limb. After comparing the blackbody emission seen in IPs with that seen in polars, we conclude that the blobby emis- sion responsible for soft X-ray excesses in polars does not occur in IPs. ACKNOWLEDGEMENTS We thank Gavin Ramsay for providing us with the spectra of the polars with no detectable soft component. 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0704.1389

Gutzwiller description of non-magnetic Mott insulators: a dimer lattice model

Gutzwiller description of non-magnetic Mott insulators: a dimer lattice model. Michele Fabrizio,1,2 1 International School for Advanced Studies (SISSA) and CNR-INFM-Democritos National Simulation Centre, Via Beirut 2-4, I-34014 Trieste, Italy 2 The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O. Box 586, I-34014 Trieste, Italy (Dated: September 12, 2021) We introduce a novel extension of the Gutzwiller variational wavefunction able to deal with insulators that escape any mean-field like description, as for instance non-magnetic insulators. As an application, we study the Mott transition from a paramagnetic metal into a non-magnetic Peierls, or valence-bond, Mott insulator. We analyze this model by means of our Gutzwiller wavefunction analytically in the limit of large coordination lattices, where we find that: (1) the Mott transition is first order; (2) the Peierls gap is large in the Mott insulator, although it is mainly contributed by the electron repulsion; (3) singlet-superconductivity arises around the transition. PACS numbers: 71.10.-w, 71.10.Fd, 71.30.+h I. INTRODUCTION Among the theoretical tools devised to deal with strongly correlated metals close to a Mott metal-to- insulator transition (MIT), the simplest one likely is the variational approach introduced in the 60ths’ by Gutzwiller1,2 to describe itinerant ferromagnetism and narrow band conductors. In its original version, the Gutzwiller variational wavefunction has the form |ΨG〉 = P |φ〉 = PR |φ〉, (1) where |φ〉 is an uncorrelated wavefunction for which Wick’s theorem holds, PR an operator at site R, and both |φ〉 and PR have to be determined by minimizing the variational energy. The role of the operator PR is to modify, according to the on-site interaction, the weights of the local electronic configurations with respect to their values in the uncorrelated wavefunction. In spite of its simplicity, the Gutzwiller wavefunction is quite effective in capturing physical properties that supposedly identify strongly correlated metals, as for in- stance the large increase of the effective mass.3 However, since the dependence upon the distance |R−R′| of inter- site correlations are still determined by the uncorrelated wavefunction, while the local operators PR just affect the amplitudes, the Gutzwiller wavefunction can describe a Mott insulator either if PR suppresses completely charge fluctuations, that provides a very poor description of an insulator, or if |φ〉 itself is insulating. The latter case can be stabilized within the original Gutzwiller approach only when |φ〉 is an admissible Hartree-Fock solution of the Hamiltonian. As an example let us consider a sin- gle band model at half-filling, for instance the Hubbard model H = − RR′,σ tRR′ c nR,↑ nR,↓, where c and c creates and annihilates, respectively, an electron with spin σ =↑, ↓ at site R and nR,σ = . This Hamiltonian admits at the mean-field level two possible phases, one paramagnetic, 〈nR,↑〉 = 〈nR,↓〉, and the other magnetic, 〈nR,↑〉 6= 〈nR,↓〉. The latter is the only one that can eventually describe an insulator. In the Hubbard model the action of the oper- ator P is to increase the weight of singly occupied sites at expenses of doubly occupied and empty sites, in or- der to minimize the Coulomb repulsion U . Evidently, even when the repulsion is very strong, hence the model is a Mott insulator, a realistic wavefunction should still allow for charge fluctuations responsible for the super- exchange, that survives even deep inside the Mott phase. However, any paramagnetic uncorrelated wavefunction, for instance the Fermi sea, is unable to generate any super-exchange and necessarily leads to a non-realistic Mott insulator where configurations with empty or dou- bly occupied sites are fully suppressed.3 The only way to generate super-exchange is to assume a magnetically ordered |φ〉, which is also the only insulating wavefunc- tion accessible within Hartree-Fock. However a magnetic state might not always be the right choice, especially if magnetism is sufficiently frustrated. Recently, an improved version of the Gutzwiller wave- function has been proposed,4 in which additional inter- site correlations are provided by density-density Jastrow factors, namely |ΨG〉 → exp vR,R′ nR nR′ |ΨG〉, (2) where nR is the site R occupation number and vR,R′ variational parameters. This novel class of wavefunctions has the capability to disentangle charge from other de- grees of freedom, hence is more suitable to capture Mott localization, as it has indeed been shown.4,5,6 However, unlike the conventional Gutzwiller wavefunction (1), the Gutzwiller-Jastrow wavefunction (2) can only be dealt with numerically by variational Monte Carlo, which is in- herently limited to finite-size systems, albeit quite large.7 An alternative approach, that is closely related to recently proposed extensions of Dynamical Mean Field Theory (DMFT) from the original single-site formula- http://arxiv.org/abs/0704.1389v1 tion8 to a cluster one9,10,11,12,13, is to consider a vari- ational wavefunction of the same form as (1) but de- fined on a lattice with non-primitive unit cells. In this case, the operator PR acts on all the available electronic configurations of the lattice sites belonging to the non- primitive cell. The advantage is that in this way one may include additional short-range correlations without losing the property of the wavefunction to be analytically man- ageable, at least in infinite-coordination lattices. The ob- vious disadvantage is that this wavefunction could bias the variational solution towards translational-symmetry breaking. Within this scheme, the variational problem becomes generically equivalent to optimize a Gutzwiller wave- function for a multi-band Hamiltonian. There have been recently an amount of attempts to extend the Gutzwiller wavefunction to multi-orbital models that in- clude further complications like for instance Coulomb ex- change14,15,16,17. In this paper we introduce a further extension that is capable to generate inter-site correla- tions as the super-exchange for paramagnetic wavefunc- tions, otherwise missed by the conventional Gutzwiller approach. This novel class of wavefunctions also allows to explore new kinds of variational solutions. Specifi- cally, there are interesting examples of correlated models where the Mott insulating phase escapes any Hartree- Fock mean-field treatment, in other words can not be represented by a single Slater determinant. A very sim- ple case, that we will explicitly consider throughout this work, is a Peierls insulator, namely a short-range valence- bond crystal, in which pairs of nearest neighbor sites are strongly bound into a singlet configuration, leading to a state that is simply a collection of spin-singlets. Such a Mott insulating state is not accessible by Hartree-Fock theory, just because each singlet is itself not express- ible as a Slater determinant, nor by the conventional Gutzwiller approach, which, as mentioned, gives a poor description of paramagnetic insulators. The paper is organized as follows. In Section II we present the variational wavefunction and discuss under which conditions it can be deal with analytically. In Sec- tion III we discuss how to build up the wavefunction in the case in which the basic unit of the lattice model is a dimer. Next, in Section IV, we solve the variational problem for a specific lattice model of dimers. Conclu- sions are given in Section V. II. THE VARIATIONAL WAVEFUNCTION In this Section, we introduce an extension of the Gutzwiller wavefunction (1) which is particularly con- venient to perform analytical calculation in the limit of infinite-coordination lattices.14 Let us consider a generic multi-band Hamiltonian. Each lattice site R contains several orbitals that give rise to a bunch of electronic con- figurations which we denote individually as |Γ;R〉. The most general operator PR can be chosen of the form: λ(R)ΓΓ′ |Γ;R〉〈Γ′;R|, (3) where λ(R)ΓΓ′ are variational parameters. In general PR needs not to be hermitean, namely for Γ 6= Γ′ it is not required that λ(R)∗ΓΓ′ = λ(R)Γ′Γ. Indeed, as we shall see, the non-hermitean character plays a very important role. We further assume that the Wick’s theorem holds for the uncorrelated wavefunction, hence that |φ〉 is either a Slater determinant or a BCS wavefunction. It was realized by Bünemann, Weber and Gebhard14 that average values of operators on the Gutzwiller wave- function (1) can be analytically computed in infinite co- ordination lattices provided the following two constraints are imposed on PR: 〈φ| P† |φ〉 = 〈φ|φ〉 = 1, (4) 〈φ| P† CR |φ〉 = 〈φ| CR |φ〉, (5) where CR is the local single-particle density-matrix op- erator, with elements c R,α cR,β and c R,α c , α labeling single-particle states, while c and c create and an- nihilate, respectively, an electron at site R in state α. The first constraint, Eq. (4), does not actually limit the variational freedom, since PR is defined up to a nor- malization factor. On the contrary, the latter constraint, Eq. (5), may reduce the variational freedom, although it seems not in a relevant manner, at least in all cases that we have so far investigated. We notice that Eq. (5) is not the same as imposing 〈φ| P† CRPR |φ〉 = 〈φ| CR |φ〉, (6) unless P commutes with CR, which is a further con- straint to be imposed on PR. This actually is the only case that has been hitherto considered, see e.g. Refs. 14 and 15. However, as we shall see, there are interesting models which force to abandon the supplementary con- dition (6), which is anyway unnecessary.17 By means of Wick’s theorem, the left-hand side of (5) includes a disconnected term 〈φ| P† |φ〉 〈φ| CR |φ〉 = 〈φ| CR |φ〉, where the right-hand side follows from (4), plus con- nected terms that are obtained by selecting in all possible ways a pair of single-fermion operators from P† , av- eraging on |φ〉 what remains, and finally averaging the two single-fermion operators with those of CR. There- fore, imposing (5) means that the sum of all connected terms vanishes, whatever is the element of the single- particle density-matrix. In other words, the operator that is left after taking out from P† any pair of single- fermion operators has null average on |φ〉. In turns, this also implies that, when averaging on |φ〉 P† multi-particle operators at different sites, the only con- nected terms that survive are those that involve four or R’+ + . . . =R’R R FIG. 1: Graphical representation of the average on |φ〉 of PR, drawn as a box, times a generic multi-particle oper- ator at site R′, drawn as a circle. Lines that join the two operators represent the average of two single-fermion opera- tors, one at R and the other at R′. The dots include all terms where the two sites are joined by more than four lines. The important thing to notice is the absence of terms in which the two sites are connected by two lines. more single-fermion operators of P† , that are repre- sented graphically in Fig. 1 as lines coming out of P† This property of PR turns out to be extremely useful in infinite-coordination lattices. In this limit, the con- tribution to the average value on |φ〉 of terms in which more than two fermionic lines come out of P† can be shown to vanish14, which simplifies considerably all cal- culations. For instance, the average value on (1) of any local operator OR becomes 〈φ| P†OR P |φ〉 = 〈φ| P†R OR PR|φ〉, (7) which also implies, taking OR = 1, that the variational wavefunction (1) is normalized. In addition, the average value of the inter-site single-particle density matrix turns out to be 〈φ| P† c† R,α cR′,β P |φ〉 = 〈φ| P† Z(R)αγ Z(R′) 〈φ| c† R,γ cR′,δ |φ〉 Z(R)αγ ∆(R′) 〈φ| c† ∆(R)αγ Z(R′) 〈φ| c R,γ cR′,δ |φ〉 ∆(R)αγ ∆(R′) 〈φ| c |φ〉, (8) 〈φ| P† c† P |φ〉 = 〈φ| P† R,α PRP Z(R)αγ Z(R′)βδ 〈φ| c†R,γ c Z(R)αγ ∆(R′)βδ 〈φ| c†R,γ cR′,δ |φ〉 ∆(R)αγ Z(R′)βδ 〈φ| cR,γ c ∆(R)αγ ∆(R′)βδ 〈φ| cR,γ cR′,δ |φ〉, (9) where the matrices Z and ∆ are determined by inverting the following set of equations 〈φ| P† R,α PR cR,β |φ〉 (10) Z(R)αγ 〈φ| c†R,γ cR,β |φ〉 ∆(R)αγ 〈φ| cR,γ cR,β |φ〉, (11) 〈φ| P† |φ〉 (12) Z(R)αγ 〈φ| c†R,γ c ∆(R)αγ 〈φ| cR,γ c |φ〉, . (13) Näıvely speaking, it is as if, when calculating the inter- site density matrix, a fermionic operator transforms ef- fectively into Z(R)αβ c ∆(R)αβ cR,β, (14) namely that a particle turns into a particle or a hole with probabilities Z and ∆, respectively. Although all the above expressions are strictly valid only in infinite- coordination lattices, it is quite common to use the same formulas also to evaluate average values on the Gutzwiller wavefunction in finite-coordination lattices. This ap- proximation is refereed to as the Gutzwiller approxima- tion1,2,18,19,20, and is known to be equivalent to the sad- dle point solution within the slave-boson technique.21 We conclude by noting that the constraint (5) turns out to be useful also when the variational wavefunction (1) is applied to Anderson impurity models. In this case the operator P acts only on the electronic configurations |Γ〉 of the impurity, namely λΓΓ′ |Γ〉〈Γ′|. If we impose 〈φ| P† P |φ〉 = 1, (15) 〈φ| P† P Cimp |φ〉 = 〈φ| Cimp |φ〉, (16) where Cimp is the single-particle density matrix of the impurity, then, for any operator of the conduction bath, Obath, and because of (4) and (5), the following result holds 〈φ| P† Obath P |φ〉 = 〈φ| Obath |φ〉. (17) A. Some formal definitions In order to perform actual calculations, it is convenient to introduce some notations. We define a matrix Fα with elements (Fα)Γ1Γ2 = 〈Γ1;R| cR,α |Γ2;R〉, as well as its hermitean conjugate, F †α, where we assumed that the definition of the local configurations is the same for all sites. It follows that Fα = δαβ I, Fα Fβ + Fβ Fα = 0, where I is the identity. Next, we introduce the uncor- related occupation-probability matrix, P0(R), with ele- ments (P0(R))Γ1Γ2 = 〈φ|Γ1;R〉 〈Γ2;R|φ〉, (18) that satisfies 1 = Tr (P0(R)) , 〈φ| c† R,α cR,β |φ〉 = Tr P0(R)F 〈φ| c† |φ〉 = Tr P0(R)F Analogously, the variational parameters that define P λ(R)Γ1Γ2 , are interpreted as elements of a matrix λ(R). With these definitions, Eqs. (4) and (5) become 〈φ| P† |φ〉 = Tr P0(R)λ(R) † λ(R) 〈φ| P† |φ〉 = Tr P0(R)λ(R) † λ(R)F †α Fβ = 〈φ| c† R,α cR,β |φ〉, 〈φ| P† R,α c |φ〉 = Tr P0(R)λ(R) † λ(R)F †α F = 〈φ| c† that suggests to introduce a variational occupation- probability matrix P (R) = P0(R)λ(R) † λ(R) with ma- trix elements (P (R))Γ1Γ2 = (P0(R))Γ1Γ3 λ(R) λ(R)Γ4Γ2 , that must satisfy Tr (P (R)) = 1, (20) P (R)F †α Fβ = 〈φ| c† R,α cR,β |φ〉, (21) P (R)F †α F = 〈φ| c† |φ〉, , (22) Eqs. (20), (21) and (22) replace the constraints (4) and (5). With these definitions, the matrices Z and ∆, see Eqs. (11) and (13), are obtained by solving P0(R)λ(R) † F †α λ(R)Fβ Z(R)αγ Tr P0(R)F ∆(R)αγ Tr P0(R)Fγ Fβ , (23) P0(R)λ(R) † F †α λ(R)F Z(R)αγ Tr P0(R)F ∆(R)αγ Tr P0(R)Fγ F . (24) The above equations simplify if one uses the natural basis, namely the single-particle basis that diagonalizes the density-matrix, 〈φ| c† R,α cR,β|φ〉 = n(R)α δαβ , 〈φ| c† R,α c |φ〉 = 0. In this case Z(R)αβ = P0(R)λ(R) † F †α λ(R)Fβ n(R)β , (25) ∆(R)αβ = P0(R)λ(R) † F †α λ(R)F 1− n(R)β . (26) Moreover, if one constructs the states |Γ;R〉 so that P0(R) is diagonal (P0(R))ΓΓ′ = δΓΓ′ P0(R; Γ), (P (R))Γ1Γ2 = P0(R; Γ1) λ(R)Γ3Γ2 , (27) B. The variational problem We are now in position to settle up the variational problem. We consider a generic tight-binding Hamilto- H = − R,αcR′,β E(R)ΓΓ′ |Γ;R〉〈Γ′;R|, (28) where α and β stem for spin, orbital and lattice site in the chosen unit cell, and the hermitean matrix E(R) with elements E(R)ΓΓ′ may be also unit-cell dependent. The average value of this Hamiltonian on the Gutzwiller wave- function (1) in the limit of infinite coordination lattices or, in finite coordination ones, within the Gutzwiller ap- proximation, is Evar = − Z(R)αγ Z(R′) 〈φ| c† Z(R)αγ ∆(R′) 〈φ| c† ∆(R)αγ Z(R′) 〈φ| c R,γ cR′,δ |φ〉 ∆(R)αγ ∆(R′) 〈φ| c P0(R)λ(R) † E(R)λ(R) ≡ Ehop + Eint. (29) The last term depends only on the local properties of the uncorrelated wavefunction |φ〉, specifically on the occupa- tion probabilities P0(R). Therefore, for any given choice of P0(R), the optimal |φ〉 that minimizes the variational energy is the ground state of the Hamiltonian Hvar = − Z(R)αγ Z(R′) Z(R)αγ ∆(R′) ∆(R)αγ Z(R′) R,γ cR′,δ ∆(R)αγ ∆(R′) µ(R)αβ c R,αcR,β ν(R)αβ c +H.c. , (30) where the parameters µ(R)αβ and ν(R)αβ are Lagrange multipliers to be determined by imposing that the ground state has indeed the chosen P0(R). The last task is to find the values of the variational parameters λ(R)ΓΓ′ as well as of P0(R) for which the variational energy (29) is minimum. We note that the variational Hamiltonian (30) that has to be solved may include also inter-site pairing terms, which are absent in the original Hamiltonian (28). Analogously to other more conventional variational ap- proaches, like Hartree-Fock theory, it is common to in- terpret the single-particle spectrum of the variational Hamiltonian (30) as an approximation of the true co- herent spectrum of quasi-particles.22 III. THE MODEL Let us now apply the variational wavefunction to spe- cific models that are inspired by the valence-bond crys- tal example we mentioned in the introduction, and where the off-diagonal elements of the operator PR as well as its non-hermitean character do play an important role. Since the operator PR is built out of purely local prop- erties, namely the available on-site electronic configura- tions plus a variational guess for the uncorrelated on-site single-particle density-matrix, a lot of preliminary results can be extracted without even specifying how lattice-sites are coupled together. Therefore we start our analysis from defining some local properties and later we will con- sider a specific lattice model. A. The isolated dimer The basic unit of the model we are going to investigate consists of a dimer with Hamiltonian Hdimer = −t⊥ 1σc2σ +H.c. (ni − 1)2 ≡ H⊥ +HU , (31) where 1 and 2 refer to the two sites of the dimer and ni, i = 1, 2, is the on-site occupation number. It is more convenient to work in the basis of the even (bonding) and odd (anti-bonding) combinations, defined through ceσ = (c1σ + c2σ) , coσ = (c1σ − c2σ) . and use this basis to built the available electronic con- figurations, which we will denote as |n,Γ〉, with n that refers to the number of electrons. The empty and the fourfold occupied dimer states are denoted as |0〉 and |4〉, respectively, while the singly-occupied states as |1, e(o), σ〉 = c† e(o)σ and the states with 3 electrons as |3, e(o), σ〉 = c† e(o)σ o(e)↑ c o(e)↓ |0〉. There are six doubly-occupied configurations. Two are spin-singlets with two electrons in the even or in the odd orbital, |2, e〉 and |2, o〉, respectively. When each orbital is singly occupied, the two electrons form either a spin triplet, |2, 1, Sz〉 with Sz = −1, 0, 1, or a spin singlet, |2, 0〉. Since we are not going to consider variational so- lutions that break spin-SU(2) symmetry, it is convenient to define the projector operators |1, e(o)〉〈1, e(o)| = |1, e(o), σ〉〈1, e(o), σ|, |3, e(o)〉〈3, e(o)| = |3, e(o), σ〉〈3, e(o), σ|, |2, 1〉〈2, 1| = Sz=−1 |2, 1, Sz〉〈2, 1, Sz|. The isolated-dimer ground state in the subspace with two electrons is |Ψ〉 = cos θ√ 1↑c2↓ + c 2↑c1↓ sin θ√ 1↑c1↓ + c 2↑c2↓ (cos θ + sin θ) |2, e〉 (cos θ − sin θ) |2, o〉, where tan 2θ = 4t⊥/U and has energy + 4t2⊥. (32) |Ψ〉 can be always rewritten in the form of a Gutzwiller wavefunction. First of all, we needs to choose an uncor- related wavefunction |φ〉. A natural choice might be the ground state at U = 0, namely |2, e〉. Indeed |Ψ〉 can be written as |Ψ〉 = P |2, e〉, where P = |Ψ〉〈2, e| = 1√ (cos θ + sin θ) |2, e〉〈2, e| (cos θ − sin θ) |2, o〉〈2, e|. (33) and obviously satisfies both (4) and (5). Another possibility, that we are also going to con- sider in what follows, is to use an uncorrelated wavefunc- tion that corresponds to a dimer in which the two sites are only coupled by an intersite singlet-Cooper pairing, namely with 〈c†1↑c 2↓〉 = 〈c 1↓〉 6= 0. In this case |φ〉 = 1 |0〉+ |2, e〉 − |2, o〉 − |4〉 and, once again, the true ground state can be written as |Ψ〉 = |Ψ〉〈φ| |φ〉 ≡ P |φ〉. Already at this stage one can appreciate how important is the role of the off-diagonal elements in P , especially for large U/t⊥. B. The non-isolated dimer: variational density matrix When the dimer is coupled to the rest of the system, in order to built the operator P we need to specify an un- correlated local single-particle density matrix based on a variational guess of the uncorrelated wavefunction |φ〉. A simple guess would be a magnetic wavefunction in which the two sites of each dimer have opposite magnetization. This choice is also the only one admitted by an Hartree- Fock decomposition of the interaction term HU . How- ever, a magnetic wavefunction is not the most suitable choice to reproduce the limit of isolated dimers, which is a collection of singlets. Alternatively, one can consider a paramagnetic |φ〉 that has built in the tendency of each dimer to lock into a spin-singlet. This can be accomplished in two ways that do not exclude each other. The first is to assume an uncorrelated wavefunction with a huge splitting between even and odd orbitals, namely with 〈φ| c†eσceσ |φ〉 ≫ no = 〈φ| c†oσcoσ |φ〉. This implies that, among the doubly-occupied configura- tions of each dimer, mainly the spin-singlet |2, e〉 survives in the uncorrelated wavefunction. The latter can then be turned into the isolated dimer configuration by an ap- propriate Gutzwiller operator P , as shown before. The other possibility is to include Cooper pairing correlations in the singlet channel ∆SC = 〈φ| c†1↑c 2↓ + c 1↓ |φ〉. In this case, the isolated dimer can be recovered by as- suming a very strong pairing ∆SC ≃ 1 and suppressing, through P , configurations with none or two singlet-pairs. Note that both ne − no and ∆SC do not appear by a mean-field decoupling of HU , so that a variational wave- function with such correlations built in can not be sta- bilized within Hartree-Fock theory. Here the role of P becomes crucial. Therefore, let us assume for |φ〉 a BCS-wavefunction defined such that 〈φ| c†1σc1σ |φ〉 = 〈φ| c 2σc2σ |φ〉 = , (34) 〈φ| c†1σc2σ |φ〉 = 〈φ| c 2σc1σ |φ〉 = , (35) 〈φ| c†1↑c 2↓ |φ〉 = 〈φ| c 1↓ |φ〉 = , (36) with real ∆SC . In the even/odd basis this translates into 〈φ| c†eσceσ |φ〉 = , (37) 〈φ| c†oσcoσ |φ〉 = , (38) 〈φ| c† e↓ |φ〉 = −〈φ| c o↓ |φ〉 = , (39) where ne + no = n. As previously mentioned, the cal- culations simplify considerably in the natural basis, that is derived in the Appendix for this particular choice of density matrices. As a particular application, we assume hereafter that the model is half-filled, namely ne +n0 = 2. The density matrix of the operators in the natural basis, d e(o)σ e(o)σ is, by Eqs. (A.3) and (A.4), 〈φ| d† e(o)σ e(o)σ |φ〉 = 1 where δ2 +∆2 . (40) The two angles θe and θo, that are defined by Eq. (A.2) and identify the unitary transformation from the original to the natural basis, are given by θe = θ and θo = θ−π/2, where tan 2θ = . (41) We note that, for q → 1/2, the uncorrelated wavefunc- tion describes an insulator where charge fluctuations are completely suppressed since each natural orbital is fully occupied. It is obvious that, if our choice of the varia- tional wavefunction is correct, then the optimal uncor- related wavefunction must asymptotically acquire q = 1/2 for U → ∞. The expression in the natural basis of the hopping, Eq. (A.6), and interaction, Eq. (A.7), operators can be derived through (A.5) and have a relatively simple ex- pression at half-filling: H⊥ = −t⊥ 2 cos 2θ |4̃〉〈4̃| − |0̃〉〈0̃| + cos 2θ |3̃〉〈3̃| − |1̃〉〈1̃| − sin 2θ |1̃〉〈3̃|+H.c. 2 sin 2θ |0̃〉〈2̃,+|+ |4̃〉〈2̃,+|+H.c. |0̃〉〈0̃|+ |4̃〉〈4̃| − |0̃〉〈4̃| − |4̃〉〈0̃| |2̃,+〉〈2̃,+|+ |2̃,−〉〈2̃,−|+ |2̃, 0〉〈2̃, 0| |1̃〉〈1̃|+ |3̃〉〈3̃| , (43) where we have defined |1̃(3̃)〉〈1̃(3̃)| = |1̃(3̃), e〉〈1̃(3̃), e|+ |1̃(3̃), o〉〈1̃(3̃), o|, |1̃〉〈3̃| = |1̃, e〉〈3̃, e|+ |1̃, o〉〈3̃, o|, |2̃,±〉 = 1√ |2̃, e〉 ± |2̃, o〉 and denoted the local configurations in the natural basis as |ñ,Γ〉 to distinguish them from the analogous ones in the original representation. C. The Gutzwiller operator P The most general Gutzwiller operator P should include at least all the projectors |ñ,Γ〉〈ñ,Γ| as well as all the off-diagonal operators |ñ,Γ〉〈ñ′,Γ′| that appear in the lo- cal Hamiltonian, Eqs. (42) and (43). As we mentioned before, our expectation is that the uncorrelated wave- function which better connects to the large-U Mott in- sulator should be identified by q → 1/2, in which locally only the configurations |3̃〉 and |4̃〉 are occupied with non- negligible probability. This suggests that P must include at least those off-diagonal operators that would turn |4̃〉 into the isolated dimer ground state, namely |0̃〉〈4̃| and |2̃,+〉〈4̃|. The latter forces to include also |1̃〉〈3̃|, as we shall see. Therefore we assume for P the following variational ansatz: λnΓ |ñ,Γ〉〈ñ,Γ|+ λ13 |1̃〉〈3̃| +λ04 |0̃〉〈4̃|+ λ2+ 4 |2̃,+〉〈4̃|, (44) with real λ’s. We define P (n,Γ) = λ2nΓ P0(ñ,Γ), (45) for all n 6= 3, 4, while, for n = 3, 4, P (3) = λ23 + λ P0(3̃), (46) P (4) = λ24 + λ 04 + λ P0(4̃). (47) Then the conditions Eqs. (4) and (5) read P (n,Γ) = 1, (48) nP (n,Γ) = 2 + 4q, (49) λ13 λ1 P0(3̃)P0(1̃) = 2λ2+ 4 λ2+ P0(4̃)P0(2̃,+). (50) Here P0(ñ,Γ) are the occupation probabilities in the nat- ural basis of the uncorrelated wavefunction. Specifically P0(ñ,Γ) = gen,Γ )en ( )4−en where gen,Γ is the degeneracy of the configuration. Eq. (50) guarantees that the anomalous averages 〈φ| P† P d† e(o)↑d e(o)↓ |φ〉 vanish in the natural basis, and explains why we have included |1̃〉〈3̃| in (44). It is convenient to rewrite P (4) P0(4̃) ui for i = 4, 04, 2 + 4, P (3) P0(3̃) ui for i = 3, 13, where u23 + u 13 = 1, which can be satisfied by choosing u3 = cosψ and u13 = sinψ, and u 2+4 = 1. The latter parameters can be expressed by means of another unit vector v = (v1, v2, v3), through (u4 + u04) , cos 2θ√ (u4 − u04)− sin 2θ u2+4, sin 2θ√ (u4 − u04) + cos 2θ u2+4, In terms of all the variational parameters, the P (n,Γ)’s, θ, q, ψ and v, the average values per dimer of the in- teraction, HU , and intra-dimer hopping, H⊥, are readily found to be EU = 〈φ| P†HU P |φ〉 = P (3) + P (1) P (0) + U P (2,+) + P (2,−) + P (2, 0) v22 + v P (4), (52) E⊥ = 〈φ| P†H⊥ P |φ〉 = −2 t⊥ δ∗, (53) where the actual correlated values of the hybridization and of the anomalous average are 2 δ∗ = 〈φ| P (ne − no) P |φ〉 = 4 v1 v2 P (4) + cos (2θ + 2ψ) P (3) −2 cos 2θ P (0)− cos 2θ P (1), (54) 2∆∗ = 〈φ| P e↓ − c o↓ +H.c. P |φ〉 = 4 v1 v3 P (4) + sin (2θ + 2ψ) P (3) −2 sin2θ P (0)− sin 2θ P (1), (55) We note that δ∗ and ∆∗ are mutually exclusive, namely the choice of parameters that maximizes one of the two, makes the other vanishing. Upon the action of P , the single fermion operators in the Nambu spinor representation transform effectively e(o)↑ e(o)↓ Z +∆ e−i β τ2 e(o)↑ e(o)↓ , (57) where τi, i = 1, 2, 3, are the Pauli matrices that act on the Nambu spinor components, tanβ = . (58) and, finally, 〈φ| P† d† e(o)σ e(o)σ 1− 4q2 P (0)P (1) + P (1)P (2,+) P (1)P (2,−) + 1 P (1)P (2, 0) P (1)P (2, 1) + cosψ P (3)P (2, 1) + cosψ P (3)P (2, 0) + cosψ P (3)P (2,−) + cosψ P (3)P (2,+) v1 cosψ + v2 cos (2θ + ψ) + v3 sin (2θ + ψ) P (4)P (3) , (59) 〈φ| P† d† e(o)↑ P d e(o)↓ |φ〉 1− 4q2 P (3)P (2,−) P (3)P (2,+) P (3)P (2, 1) P (3)P (2, 0) v1 sinψ − v2 sin (2θ + ψ) +v3 cos (2θ + ψ) P (4)P (3) , (60) with real Z and ∆. Therefore, if the dimers are cou- pled one to another by the single particle hopping term R 6=R′ i,j=e,o τ3 ΨR′,j , (61) where R,i↑, cR,i↓), and Ψ its hermitean conjugate, the uncorrelated wave function |φ〉 minimizes the effective hopping Tvar = (Z +∆) R 6=R′ i,j=e,o −2 i β τ2 Ψ under the condition that the local density matrix satisfies Eqs.(37)-(39). One can readily show that this amounts to find the ground state |φ〉 of the variational Hamiltonian Hvar = (Z +∆) R 6=R′ i,j=e,o τ3 ΨR′,j R,e τ3 ΨR,e −Ψ R,o τ3 ΨR,o R,e τ1 ΨR,e −Ψ R,o τ1 ΨR,o , (63) with µ3 and µ1 such that 〈φ| Hvar |φ〉 + 4q µ3 cos(2θ + 2β) + 4q µ1 sin(2θ + 2β), is maximum. Before we consider specific lattice models, it is worth re-deriving within this variational scheme the isolated- dimer ground-state energy (32) at half-filling. For that purpose, we take all P (n,Γ) zero but P (4) = 1. The variational energy is simply Evar = EU + E⊥ = U v22 + v − 4 t⊥ v1 v2. The minimum under the constraint v · v = 1 is obtained for v3 = 0 and exactly reproduces (32). We note that the minimum energy is independent on θ, namely there exists a continuous family of variational solutions with equal energy parametrized by θ. However, in spite of the fact that the uncorrelated wavefunction may describe a superconductor, the actual value of the anomalous aver- age ∆∗ = 0. IV. A LATTICE MODEL OF DIMERS As a particular application, let us consider the follow- ing lattice model H = − tRR′ c R,iσcR′,iσ +H.c. (nR,i − 1)2 <RR′> R,1σcR,2σ +H.c. (ǫk − t⊥) c†k,eσck,eσ + (ǫk + t⊥) c (nR,i − 1)2 , (64) where nR,i = and ǫk is the band dis- persion induced by tRR′ , with half-bandwidth D. The Hamiltonian (64) represents two Hubbard models cou- pled by a single-particle hopping t⊥, each model being defined on a lattice with coordination number z. As we mentioned, the variational results that we have so far derived are rigorous strictly speaking only if z → ∞, although, in the spirit of the Gutzwiller approximation, they can be used for generic z as well. If U ≫ D, t⊥, (64) describes at half-filling a Mott insu- lator which may be magnetic at t⊥ ≪ D, but is certainly non-magnetic at t⊥ ≫ D, where the ground state reduces essentially to a collection of singlets. For instance, in the case of a Bethe lattice with nearest neighbor hopping, the transition is at t⊥ = D/ 8, value that is going to decrease if frustration is included. If U is small and the Fermi surface is not nested, then the model is metallic for t⊥ ≤ D and is a band insulator otherwise. In fact, in the absence of nesting there is generically a finite win- dow of t⊥ values in which, upon increasing U , the model undergoes a transition from a paramagnetic metal into a non-magnetic Mott insulator, and this is just the case we are going to consider in what follows. The same model have been recently studied by Fuhrmann, Heilmann and Monien using DMFT23 and by Kancharla and Okamoto24 using DMFT and cluster DMFT, respectively, that gives us the opportunity to directly check the accuracy of our wavefunction. The variational Hamiltonian (63) of the model (64) has a very simple expression, Hvar = ǫk∗ − µ3 τ3 − µ1 τ1 ǫk∗ + µ3 τ3 + µ1 τ1 , (65) where ǫk∗ = (Z +∆) ǫk. The variational single-particle spectrum has the conven- tional BCS form with eigenvalues Eek = (ǫk∗ − µ3)2 + µ21, Eok = (ǫk∗ + µ3) + µ21, hence, for any µ1 6= 0, has a gap equal to 2µ1. On the contrary, when µ1 = 0, the spectrum is gapless for |µ3| ≤ D, otherwise is gaped. The Lagrange multipliers µ1 and µ2 are obtaining by maximizing Ehop = − Eek + Eok + 4q µ3 cos(2θ + 2β) + 4q µ1 sin(2θ + 2β). (66) In terms of (66), (52) and (53) the variational energy is Evar = Ehop + EU + E⊥, (67) and depends on eight independent variational parame- ters. We have solved numerically the variational problem at fixed t⊥/D = 0.5 as function of U/D. To simplify cal- culations, we have assumed for the band dispersion ǫk either a flat or a semi-circular density of states, although both would give rise to nesting that could stabilize mag- netic phases, which we do not take into account. How- ever, from the point of view of the paramagnetic-metal to paramagnetic-insulator transition, this choice is not influential. We find that the variational solution displays a first order phase transition at Uc ≃ 2.05 D for a flat density of states, as shown by the behavior of the variational en- ergy in Fig. 2. This result agrees almost quantitatively with the DMFT calculation23 obtained with a semicir- cular density of states, that also predicts a first order transition with a coexistence region between U ≃ 1.5 D and 1.8 D at the same value of t⊥ = 0.5 D. We note that the energy is everywhere finite and vanishes like 1/U for large U , see the inset of Fig. 2. The asymptotic behavior UEvar/D 2 ∼ −7/6 is compatible with second order per- turbation theory in t and t⊥ using as zeroth-order state a collection of dimers, as explained below. In Fig. 3 we show the behavior across the transition of the three con- tribution to the energy, namely EU , E⊥ and Ehop. We FIG. 2: Variational energy Evar in units of D and for a flat non-interacting density of states, as function of U/D for t⊥ = D/2. At Uc ≃ 2.05 D a first order transition occurs. The inset shows the asymptotic value of UEvar/D FIG. 3: The different contributions to the variational energy, EU , E⊥ and Ehop. find that the transition is accompanied by an energy loss in Ehop, but a gain in both E⊥ and EU . In order to characterize physically the two phases, in Fig. 4 we plot the values of µ1 and µ3 across the tran- sition. Since µ1 = 0, within our numerical accuracy, and |µ3| < D, the phase at U < Uc is gapless hence metallic, see the behavior of the density of states (DOS) drawn in Fig. 5. On the contrary, on the U > Uc side FIG. 4: The behavior of the parameters µ1 and µ3 in units of D as function of U/D, see Eq. (65). of the transition, µ1 6= 0, that implies a finite gap in the single-particle variational spectrum, see Fig. 5. In the gaped phase at U > Uc the spectrum looks like the one of a Peierls insulator with a very large hybridization gap, not consistent with the bare value of t⊥. In reality this gap is, more properly, the Mott-Hubbard gap. Indeed the DOS has weight both below and above the chemical potential, suggestive of asymmetric Mott-Hubbard side- bands. Moreover, as we are going to discuss below, the actual difference between the occupations of the bond- ing and anti-bonding bands, which we denoted as 2δ∗ in Eq. (54), decreases with U , unlike the single-particle gap, see Fig. 7. This behavior is reminiscent of what has been found by Biermann et al.25 as an attempt to understand the physics of VO2. The other quantities that identify the variational spec- trum are Z and ∆, shown in Fig. 6. We see that Z is decreasing with U but reaches a finite value Z = 1/4 for U → ∞. On the contrary, ∆ = 0 for U < Uc, while ∆ 6= 0 for U > Uc and increases monotonically to reach asymptotically the same value 1/4 for large U . Further insights can be gained by the average values of the intra-dimer hopping and pairing, Eqs. (54) and (55), drawn in Fig. 7. As we mentioned, the intra-dimer hy- bridization 2δ∗ is monotonically decreasing with U , apart from the jump at the first order transition. More inter- estingly, around the transition the variational solution is characterized by a sizeable BCS order parameter ∆∗, which does not follow the behavior of the BCS coupling µ1 present in the variational Hamiltonian. Indeed, while µ1 is zero within our numerical accuracy for U < Uc, yet a non negligible ∆∗ develops just before the transi- tion, see Fig. 7. Moreover, although µ1 starts already U/D = 2.0 U/D = 0 U/D = 2.05 U/D = 3.0 −2 0.5 FIG. 5: The variational single-particle spectrum for the even, i.e. bonding, band, solid lines, and odd, i.e. anti-bonding, one, dashed lines, across the transition for a non-interacting semi-circular density of states. large for U > Uc and increases monotonically with U , see Fig. 4, the actual order parameter ∆∗ is apprecia- ble only near the transition and fastly decreases with U to very tiny values. Also interesting is that, besides the intra-dimer superconducting order parameter, also an inter-dimer one arises, that can be for instance defined through ∆̃∗ = tRR′ 〈φ| P† Ψ†R,i τ1 ΨR′,i P |φ〉 FIG. 6: The behavior of the parameters Z and ∆. FIG. 7: The average values of the intra-dimer hopping, δ∗, and pairing, ∆∗. 4ǫk=0 where V is the number of sites. We find that ∆̃∗ has actually the opposite sign of ∆∗, and both closely follow each other, rapidly decreasing with U , see Fig. 8. Even though ∆∗ and ∆̃∗ are everywhere finite for U > Uc, sug- gestive of a superconducting phase that survives up to very large U , we believe that superconductivity, it it oc- curs at all, may appear only very close to the transition, where the value of the order parameter is larger. Indeed, FIG. 8: The intra-dimer, circles, and, with reversed sign, the inter-dimer, triangles, superconducting order parameters. the optimal solution with finite ∆∗ and ∆̃∗ and another solution in which both are forced to be zero are practi- cally degenerate within our numerical accuracy for large U . Moreover, since the variational values of the order parameter are extremely small but close to Uc, the in- clusion of quantum fluctuations, for instance in the form of a Jastrow factor as in Eq. (2), would likely suppress superconductivity leading to a bona fide insulating wave- function. Unfortunately, we can not compare this result with the DMFT analyses of Refs. 23,24, where superconductivity has not been looked for.26 A. Large U limit In order to appreciate qualities and also single out defects of the variational wavefunction, it is worth dis- cussing the large-U solution. To leading order in 1/U , one can assume all P (n,Γ) = 0 but P (3) and P (4). The two constraints Eqs. (48) and (49) can be solved by defin- P (3) = 2− 4q ≡ 4d, P (4) = 1− 4d, with d ≪ 1, namely q → 1/2, while Eq. (50) is already satisfied since P (1) = P (2,+) = 0. Moreover, one readily recognizes that the variational solution asymptotically tends to acquire ψ ≃ β → π/4, θ → 0 and v1 ≫ v2, v3, which is indeed what we find by numerical minimization. It then follows that ∆ → 1 1− 4q2 4d(1− 4d) → 1 This implies that µ3 → 0, hence that µ1 is determined by maximizing Ehop = − k∗ + µ 1 + 4q µ1, which leads to µ1 = ǫ2/4d and Ehop = −4 ǫ2 d, (68) where At leading order, the variational energy per dimer is therefore Evar = −4 ǫ2 d− 4 t⊥ v1 v2 + U v22 + 2U d, that is minimized by d = ǫ2/U 2, v1 ≃ 1 and v2 ≃ 2t⊥/U , and takes the value Evar = − = − 1 tRR′ tR′R . (69) In the case of a flat density of states δ (ǫ− ǫk) = θ (D − |ǫ|) , and with t⊥ = 0.5 D we recover the numerical result Evar ≃ −7/6 D2/U , see Fig. 2. We note that, in spite of the hybridization δ∗ ∼ t⊥/U being small, the single-particle gap of the variational spectrum 2µ1 ≃ U is large, as one should expect in a Mott insulator. Coming back to the large-U value of the variational energy (69), one can readily realize that it coincides with the second order correction in tRR′ to the energy of the state |Ψ〉 = R,1↑c R,2↓ + c R,2↑c which is just a collection of singlets. In other words, in spite of being non-magnetic, our variational wavefunc- tion is able to reproduce the correct super-exchange be- tween dimers. This is a remarkable property that ac- tually derives from the square-root dependence upon d of Ehop, see Eq. (68). If we considered a more conven- tional Gutzwiller operator P commuting with the single- particle density matrix, that amounts to further impose δ2∗ +∆ ∗, we would find Ehop ∝ d, implying a transition into an unrealistic insulator with d = 0 above a critical U . The obvious defect of the wavefunction is that, since it emphasizes strongly the role of individual dimers, the hopping among dimers, although finite for any U , is under-estimated with respect to the intra-dimer one. Therefore we do not expect the wavefunction to be particularly accurate for small t⊥/D. anti−bonding bonding FIG. 9: The non-interacting density of states of the lattice of dimers. The bonding and anti-bonding state of each dimer give rise to two bands that overlap, leading to a metallic phase in the absence of interaction. V. CONCLUSIONS In this work we have proposed an extension of the Gutzwiller variational approach to account for correlated models which display metal-insulator transitions into Mott insulators that escape any simple single-particle descriptions, like the Hartree-Fock approximation. The wavefunction has still the same form as the conventional Gutzwiller wavefunction, |ΨG〉 = P |φ〉 = PR |φ〉, with R identifying unit cells that may also be non- primitive ones, with the novel feature that the opera- tor PR is non-hermitean and does not commute with the local single-particle density matrix. In essence, this prop- erty realizes a variational implementation of a Schrieffer- Wolff transformation27, although only restricted to the lattice sites within each unit cell. We have shown that, by slightly reducing the variational freedom, this wavefunc- tion, like the conventional Gutzwiller wavefunction,14 al- lows for an extension of the Gutzwiller approximation to evaluate average values, approximation that becomes exact in the limit of infinite coordination lattices. As an application, we have considered the Mott tran- sition into a Peierls, or valence-bond, insulator, namely an insulator that is adiabatically connected to a collec- tion of independent dimers. Such an insulator can not be described by Hartree-Fock simply because the singlet configuration of each dimer is not a Slater determinant. Specifically, we have considered the hypothetical situa- tion shown in Fig. 9, where the splitting between the bonding and antibonding orbitals of each dimer is as- sumed not to be sufficient to lead, in the absence of in- teraction, to a band insulator. When interaction is taken into account, in the form of an on-site repulsion U , one expects, above a critical U , a transition from the metal into a Mott insulator. If magnetism is prevented, for instance by a sufficiently large splitting between bond- ing and anti-bonding orbitals and/or by frustration, the Mott insulator is non-magnetic. We have shown that our wavefunction overcomes the difficulties of Hartree- Fock theory and allows to study, albeit variationally, this transition. In particular we find that: (i) at the variational level the Mott transition is first order; (ii) the variational spectrum inside the Mott insulator looks similar to that of a Peierls insulator with a large hybridization gap, namely a large splitting be- tween bonding and anti-bonding bands. In reality the gap is the Mott-Hubbard gap and the actual difference between the occupations of the bonding and anti-bonding bands, is small; (iii) inter-site singlet-superconductivity appears around the transition. While (i) and (ii) are presumably true, as they have been also found by more rigorous calculations23,25,28, the emergence of superconductivity might be an artifact of the variational wavefunction.26 Nevertheless, the possi- ble occurrence of superconductivity is quite suggestive. It is known for instance that two-leg Hubbard ladders with nearest neighbor hopping display dominant super- conducting fluctuations with the same symmetry that we find variationally29, although at half-filling they always describe non-magnetic spin-gaped insulators30,31 because of nesting. Moreover, the uncorrelated wavefunction |φ〉 is quite similar to the wavefunctions used in Refs. 32,33 to simulate t-J ladders. It would be surprising and inter- esting if this tendency towards superconductivity turned into a true symmetry breaking instability in higher di- mensionality, as suggested by our analysis, which we think it is worth deserving further investigations. Note added: During the completion of this work, we be- came aware of a recent extension of slave-boson technique whose saddle-point solution closely resembles our varia- tional approach.34 Indeed the two conditions we impose on the Gutzwiller operator, Eqs. (4) and (5), are in one- to-one correspondence with the constraints identified in Ref. 34 within the slave-boson formalism. Acknowledgments We are grateful to C. Castellani and E. Tosatti for their helpful comments and suggestions. We also thanks A. Georges for useful discussions in connection with Ref. 34. APPENDIX: THE NATURAL BASIS Let us assume that, in the Nambu-spinor representa- the uncorrelated wavefunction has the following density matrices Ĉe = ne/2 ∆SC/2 ∆SC/2 1− ne/2 , Ĉo = no/2 −∆SC/2 −∆SC/2 1− no/2 (A.1) The natural orbitals are obtained by the unitary trans- formation e(o)↑ = cos θe(o) ce(o)↑ + sin θe(o) c e(o)↓ e(o)↓ = cos θe(o) ce(o)↓ − sin θe(o) c e(o)↑, where tan 2θe = ne − 1 , tan 2θo = no − 1 , (A.2) and posses a diagonal density matrix with the non- vanishing elements given by 〈φ| d† e(o)σ e(o)σ |φ〉 = 1 + qe(o), (A.3) where qe(o) = (ne(o) − 1)2 +∆2SC . (A.4) In the natural basis we introduce states that have the same formal expression as in the original basis but are built with d-operators, and denote them as |ñ,Γ〉. The transformation rules from these states to the original ones |0〉 = cos θe cos θo|0̃〉+ sin θe sin θo |4̃〉 +cos θe sin θo |2̃, o〉+ sin θe cos θo |2̃, e〉, |1, e(o), σ〉 = cos θo(e) |1̃, e(o), σ〉 + sin θo(e) |3̃, e(o), σ〉, |2, e(o)〉 = cos θe cos θo |2̃, e(o)〉+ cos θe(o) sin θo(e) |4̃〉 − sin θe(o) cos θo(e) |0̃〉 − sin θe(o) sin θo(e) |2̃, o(e)〉, |2, 1, Sz〉 = |2̃, 1, Sz〉, (A.5) |2, 0〉 = |2̃, 0〉, |3, e(o), σ〉 = cos θo(e) |3̃, e(o)〉 − sin θo(e) |1̃, e(o)σ〉, |4〉 = cos θe cos θo|4̃〉+ sin θe sin θo |0̃〉 − cos θe sin θo |2̃, e〉 − sin θe cos θo |2̃, o〉. The inverse transformation is obtained by letting θe(o) → −θe(o). The hopping operator in the original representation can be written as H⊥ = −2t⊥ 1σc2σ +H.c. = −2t⊥ c†eσceσ − c†eσceσ = −2t⊥ |3, o, σ〉〈3, o, σ| − |3, e, σ〉〈3, e, σ| +|1, e, σ〉〈1, e, σ| − |1, o, σ〉〈1, o, σ| +2 |2, e〉〈2, e| − 2 |2, o〉〈2, o| , (A.6) while the interaction operator as (ni − 1)2 = |0〉〈0|+ |4〉〈4| n=1,3 |n, e, σ〉〈n, e, σ|+ |n, o, σ〉〈n, o, σ| |2, e〉+ |2, o〉 〈2, e|+ 〈2, o| +2 |2, 0〉〈2, 0| . (A.7) Their expression in the natural basis can be obtained by the transformation rules (A.5). 1 M. C. Gutzwiller, Phys. Rev. 134, A923 (1964). 2 M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965). 3 W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970). 4 M. Capello, F. Becca, M. Fabrizio, S. Sorella, and E. Tosatti, Phys. Rev. Lett. 94, 026406 (2005). 5 M. Capello, F. Becca, S. Yunoki, M. Fabrizio, and S. Sorella, Phys. Rev. B 72, 085121 (2005). 6 M. Capello, F. Becca, M. Fabrizio, and S. Sorella, Super- fluid to Mott-insulator transition in Bose-Hubbard models (2006), 7 S. Sorella, Phys. Rev. B 71, 241103 (2005). 8 A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 9 A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 62, R9283 (2000). 10 D. Senechal, D. Perez, and M. Pioro-Ladriere, Phys. Rev. Lett. 84, 522 (2000). 11 M. Potthoff, M. Aichhorn, and C. Dahnken, Phys. Rev. Lett. 91, 206402 (2003). 12 T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys. 77, 1027 (2005). 13 G. Kotliar, S. Y. Savrasov, G. Pálsson, and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001). 14 J. Bünemann, W. Weber, and F. Gebhard, Phys. Rev. B 57, 6896 (1998). 15 C. Attaccalite and M. Fabrizio, Phys. Rev. B 68, 155117 (2003). 16 Q.-H. Wang, Z. D. Wang, Y. Chen, and F. C. Zhang, Phys- ical Review B (Condensed Matter and Materials Physics) 73, 092507 (pages 4) (2006). 17 M. Ferrero, Ph.D. thesis, SISSA-Trieste (2006). 18 W. Metzner and D. Vollhardt, Phys. Rev. Lett. 59, 121 (1987). 19 W. Metzner and D. Vollhardt, Phys. Rev. B 37, 7382 (1988). 20 F. Gebhard, Phys. Rev. B 41, 9452 (1990). 21 G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986). 22 J. Bünemann, F. Gebhard, and R. Thul, Phys. Rev. B 67, 075103 (2003). 23 A. Fuhrmann, D. Heilmann, and H. Monien, Physical Re- view B 73, 245118 (2006). 24 S. S. Kancharla and S. Okamoto, Band insulator to Mott insulator transition in a bilayer Hubbard model (2007), URL http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0703728. 25 S. Biermann, A. Poteryaev, A. I. Lichtenstein, and A. Georges, Physical Review Letters 94, 026404 (pages 4) (2005). 26 There are however recent DMFT calculations that seem to support the existence of a superconducting region around the transition. [A. Privitera, M. Schiro’, M. Capone and C. Castellani, unpublished (2007).]. 27 J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966). 28 G. Moeller, V. Dobrosavljević, and A. E. Ruckenstein, Phys. Rev. B 59, 6846 (1999). 29 M. Fabrizio, Phys. Rev. B 48, 15838 (1993). 30 S. P. Strong and A. J. Millis, Phys. Rev. B 50, 9911 (1994). 31 D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, Phys. Rev. B 53, 8521 (1996). 32 G. Sierra, M. A. Mart́ın-Delgado, J. Dukelsky, S. R. White, and D. J. Scalapino, Phys. Rev. B 57, 11666 (1998). 33 S. Sorella, G. B. Martins, F. Becca, C. Gazza, L. Capriotti, A. Parola, and E. Dagotto, Phys. Rev. Lett. 88, 117002 (2002). 34 F. Lechermann, A. Georges, G. Kotliar, and O. Parcollet, unpublished (2007). http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0703728

0704.1390

Velocity oscillations in actin-based motility

Velocity oscillations in actin-based motility Azam Gholami1, Martin Falcke1, Erwin Frey2 1Hahn-Meitner-Institute, Dept. Theoretical Physics, Glienicker Str. 100, 14109 Berlin, Germany 2Arnold Sommerfeld Center for Theoretical Physics and Center of NanoScience, Ludwig-Maximilians-Universität, Theresienstr. 37, 80333 München, Germany (Dated: June 28, 2021) We present a simple and generic theoretical description of actin-based motility, where polymer- ization of filaments maintains propulsion. The dynamics is driven by polymerization kinetics at the filaments’ free ends, crosslinking of the actin network, attachment and detachment of filaments to the obstacle interfaces and entropic forces. We show that spontaneous oscillations in the velocity emerge in a broad range of parameter values, and compare our findings with experiments. PACS numbers: 05.20.-y, 36.20.-r, 87.15.-v Force generation by semiflexible polymers is versatilely used for cell motility. The leading edge of lamellipodia of crawling cells [1] is pushed forward by a polymerizing actin network and bacteria move inside cells by riding on a comet tail of growing actin filaments [2, 3]. In vivo sys- tems are complemented by in vitro assays using plastic beads and lipid vesicles [4, 5, 6]. The defining feature of semiflexible polymers is the order of magnitude of their bending energy which is in the range of kBT . They un- dergo thermal shape fluctuations and the force exerted by the filaments against an obstacle arises from elastic and entropic contributions [7, 8]. Mathematical models have quantified the force gener- ated by actin filaments growing against obstacles [7, 8, 9]. The resisting force depends on the obstacle which is pushed. In case of pathogens, it has a small com- ponent from viscous drag of the moving obstacle but consists mainly of the force exerted by actin filaments bound to the surface of the bacteria and pulling it back- wards [10, 11]. The tethered ratchet model [12] is a math- ematical formulation of these experimental findings in terms of the dynamics of the number of attached and de- tached polymers. The starting point of our approach will be the dynamics of the distributions of the free length of both polymer populations. Actin polymerization in living cells and extracts is con- trolled by a complex molecular network [3]. Nucleation of new filaments, capping of existing ones, exchange of ADP for ATP on actin monomers, buffering of monomers etc. all contribute to that control and have been modeled [12, 13, 14]. Our goal is not to model the full complex- ity of that biochemical network. Rather we focus on the core process of force generation and force balance ensuing from the interplay between bound pulling filaments and polymerizing pushing filaments, the transition between these two groups and the motion of the whole force gen- erating configuration. This is motivated by recent ob- servations of complex dynamics in simple reconstituted systems: the velocity of beads or pathogens propelled by actin polymerization may oscillate [16, 17, 18]. Our goal is to describe the dynamics of such biochemically compressed under tension viscous force cross-linked actin network cross-linkers obstacle brush FIG. 1: (color online) Schematic representation of an ensem- ble of actin filaments oriented at ϑ = 0 with respect to the normal n̂ of an obstacle interface, which may either be a cell membrane or a bacterium. While attached filaments are un- der tension and pull the interface back, detached filaments are compressed, elongate by polymerization with rate kon and push the interface forward. All filaments in the brush are firmly anchored in a cross-linked network, whose front advances with velocity vg reducing the free length l of the filaments. Attached filaments detach with stress dependent rate kd and detached filaments attach with constant rate ka. vo is the interface velocity in the extracellular medium, and x is the distance between the front of the network and the interface. simpler systems and find a robust microscopic descrip- tion for oscillation mechanisms, which may then be con- trolled by higher order processes. Such a study is meant to complement investigations based on a continuum ap- proach [17, 18]. We consider a fixed number N of actin filaments [19] firmly anchored into a rigid cross-linked network, which advances with velocity vg; for an illustration see Fig. 1. Filaments of variable length l are either attached to the obstacle interface via a protein complex or detached from it, with time-dependent number distributions denoted by Na(l, t) and Nd(l, t), respectively. In the detached state, filaments polymerize at a velocity vp(l, x), which depends on both the polymer length l and the distance x between rigid support and obstacle. Transitions between the two filament populations occur with a constant attachment rate ka and a stress-dependent detachment rate kd [20]. This results in the evolution equations vg(l)− vp Nd = −ka Nd + kd Na , (1a) vg(l) Na = ka Nd − kd Na . (1b) The right hand side of Eq. 1 describes attachment and detachment process. The second term on the left hand side accounts for the gain and loss of attached and de- tached polymers due to the dynamics of the polymer mesh, growing with velocity vg, and the polymerization kinetics of the filaments in the brush. The correction factor l/x in front of vg is due to the fact that for bent polymers the rigid network swallows by this amount more in contour length than for straight filaments. This factor is equal to 1 for l ≤ x. Processes contributing to the growth of the rigid poly- mer mesh are entanglement and crosslinking of filaments in the brush. Both imply a vanishing vg for l → 0, since short polymers do not entangle and crosslinking proteins are unlikley to bind to them. At the same time vg can not grow without bound but must saturate at some value vmaxg due to rate limitations for crosslinker binding. This suggests to take the following sigmoidal form vg(l) = v g tanh(l/l̄) , (2) with a characteristic length scale l̄. The polymerization rate is proportional to the prob- ability of a gap of sufficient size d (≈ 2.7 nm) between the polymer tip and the obstacle for insertion of an actin monomer [7]. This implies an exponential dependence of vp on the force Fd by which the polymer pushes against the obstacle, vp(l, x) = v p exp [−d · Fd(l, x)/kBT ] . (3) Here, vmaxp ≈ 500 nm s−1 [7] is the free polymerization velocity. For the entropic force Fd we use the results ob- tained in Ref. [8] for D = 2, 3 spatial dimensions, where we take the accepted value of `p ≈ 15 µm [21] for the persistence length of F-actin. The dynamics of the distance x between grafted end of the filament and the obstacle interface (see Fig. 1) is given by the difference of the average vg and the velocity of the obstacle ∂tx = − dl vg(l) [Na(l, t) +Nd(l, t)] (4) dl [Na(l, t) Fa(l, x) +Nd(l, t) Fd(l, x)] , where ζ is an effective friction coefficient of the obsta- cle. The force Fa(l, x) acting on the obstacle interface results from the compliance of the filaments attached to it by some linker protein complex, which we model as springs with spring constant kl and zero equilibrium length. This complex has a nonlinear force-extension re- lation which we approximate by a piece-wise linear func- tion; for details see the supplementary material. Let R‖ ≈ l[1− l(D− 1)/4`p] be the equilibrium length of the filament. Then, the elastic response of filaments expe- riencing small compressional forces (x ≤ R‖) is approxi- mated by a spring constant k‖ = 12kBT `2p/(D−1)l4 [22]. For small pulling forces (x ≥ R‖), the linker-filament complex acts like a spring with an effective constant keff = klk‖/(kl + k‖). In the strong force regime, the force-extension relation of the filament is highly nonlinear and diverges close to full stretching [23]. Therefore, only the linker will stretch out. The complete force-extension relation is captured by −k‖(x−R‖) , x ≤ R‖ , −keff(x−R‖) , R‖ < x < l , −kl(x− l)− keff(l −R‖) , x ≥ l . Finally, we specify the force-dependence of the detach- ment rate by kd = k d exp [−d · Fa(l, x)/kBT ] . (6) Here, k0d ≈ 0.5 s −1 [12] is the detachment rate in the absence of forces and we have followed Ref. [20]. Eq. 1a has a singularity at vp(ls) = vg(ls)ls/x since the coefficient of the derivative of Nd with respect to l is zero at ls. To illustrate the key physical features at that singularity, we start with the simple equation ∂tNd−∂l[vg(l)l/x−vp(l, x)]Nd = 0 with x kept constant. Then those parts of the distribution of Nd with l < ls will grow and catch up with ls since vg(l)l/x−vp(l, x) is posi- tive there, while the parts with l > ls will shorten towards ls. As a consequence the whole distribution will become concentrated at ls. To quantify this heuristic argument we expand vg(l)l/x − vp(l, x) up to linear order around ls like v1(l− ls) and use the method of characteristics to solve the equation. Starting initially with a Gaussian dis- tribution we obtain Nd(l, t) = c(t) exp[−(l− l̄(t))2/w(t)2] with c(t) = c0 exp(v1t), l̄(t) = ls + (l̄0 − ls) exp(−v1t) and w(t) = w0 exp(−v1t). This shows that Nd evolves to a monodisperse distribution which is localized around ls. Its width decreases exponentially with time while its height grows exponentially. The time scale for this con- traction is given by [∂l(vgl/x− vp)]−1. Since the same kind of singularity also occurs in the full set of dynamic equations, Eqs. 1, we may readily infer that Na and Nd evolve into delta-functions with that dynamics. This is well supported by simulations, and allows us to continue with the ansatz Nd(l, t) = nd(t) δ(l − ld(t)) , (7a) Na(l, t) = na(t) δ(l − la(t)) . (7b) It defines the dynamic variables nd(t), ld(t), na(t) and la(t). Upon inserting Eqs. 7 into Eqs. 1 and Eq. 4, we obtain the following set of ordinary differential equations ∂tld(t) = vp(ld, x)− vg(ld) + kd (la − ld) , (8a) ∂tla(t) = − vg(la) + ka (ld − la) , (8b) ∂tna(t) = −kd(la, x) na(t) + ka nd(t) , (8c) ∂tx(t) = [na(t) Fa(la, x) + nd(t) Fd(ld, x)] [vg(la) na(t) + vg(ld) nd(t)] , (8d) where nd(t) = N − na(t) since we keep the total number of filaments fixed. The values of many parameters in the dynamics can be estimated using known properties of actin filaments. We choose the linker spring constant kl ≈ 1 pN nm−1 [12] and assume N = 200 [12] filaments to be crowded behind the obstacle. A realistic value of the drag coefficient ζ is 10−3 pN s nm−1 but results did not change qualitatively for a range from 10−5 pN s nm−1 to 1 pN s nm−1. We have numerically solved Eqs. 8 in both D = 2 and D = 3 dimensions, and found the dynamic regimes shown in Fig. 2: stationary states and oscillations. The existence of an oscillatory regime is very robust against changes of parameters within reasonable limits including the spatial dimension. We checked robustness against changes in the parameter values for the number of poly- mers N , l̄ (see Eq. 2), kl, vmaxp and k d, in addition to the examples shown in Fig. 2. In general, we find that oscil- lations occur for vmaxg . 500 nm s −1 and within a range of values for ka. Note that the oscillatory region in pa- rameter space depends on the orientation ϑ of filaments with respect to the obstacle surface, i.e. oscillating and non-oscillating sub-populations of filaments may coexist in the same network. Oscillations appear with finite amplitude and period at the lower boundary of the oscillatory region; compare the example shown in Fig. 3a. The stationary state changes stability slightly inside the oscillatory regime and oscil- lations set in with a finite period. That is compatible with oscillations appearing by a saddle node bifurcation of limit cycles. The upper boundary of the oscillatory region is determined by a Hopf bifurcation. An exam- ple of an oscillation close to that bifurcation is shown in Fig. 3b. More details on the phase diagram will be published elsewhere [15]. We start with the description of oscillations in the phase with vg > vp, i.e., decreasing lengths x, la and ld; see Fig. 3. Then the magnitude of pulling and pushing forces increases due to their length-dependence. When the pushing force becomes too strong, an avalanche-like detachment of attached filaments is triggered and the ob- stacle jerks forward; compare the steep rise in ld, la and x FIG. 2: (color online) Phase diagram of Eqs. 8a - 8d outlining stationary and oscillatory regimes with ζ = 10−3 pN s nm−1 for (a-c) and (a) D = 2, ϑ = 0 , (b) D = 2, ϑ = π/4, (c) D = 3, ϑ = 0 and (d) D = 3, ϑ = 0, ζ = 10−5 pN s nm−1. l̄=100 nm, all other parameter values are specified in the text. 1000 1500 2000 2500 0 200 400 600 800 1000 time (s) na(t) la(t) x(t)ld(t) (a) 0 10 20 30 40 50 60 70 time (s) na(t) la(t) x(t)ld(t) (b) FIG. 3: (color online) x , la , ld (in nm) and na as a function of time, as obtained from a numerical solutions of Eqs. 8a - 8d with vmaxg = 300 nm s −1 and (a) ka = 0.143 s −1 (b) ka = 3.49 s −1. D = 3, l̄=100 nm in both panels. shown in Fig. 3. That causes a just as sudden drop of the pushing force. With low pushing force now, polymeriza- tion accelerates and increases the length of detached fila- ments. The restoring force of attached filaments is weak in this phase due to their small number. Hence, despite of not so strong pushing forces, the obstacle moves for- ward. In the meantime, some detached filaments attach to the surface such that the average length and number of attached filaments increases as well. When the detached filaments are long enough to notice the presence of the obstacle interface, they start to buckle. This, in turn, in- creases the pushing force and slows down the polymeriza- tion velocity. Therefore, the graft velocity now exceeds the polymerization velocity and the average lengths of attached and detached filaments start to decrease again and the cycle starts anew. The period of oscillations is dependent on the parameter values. It reduces from 240 s in Fig. 3a to 13 s in Fig. 3b as ka increases from 0.143 s−1 to 3.49 s−1 at vmaxg = 300 nm s The oscillations in x correspond to the saltatory mo- tion of the obstacle in the lab frame and the oscillations of its velocity since vg stays essentially constant. An il- lustration is shown in Fig. 4 for a given set of parameters which leads to oscillations with periods of the order of 100 s and velocity of the order of 0.7 µm s−1. This is in good agreement with the results of experiments on os- cillatory Listeria propulsion [16]. The period of velocity oscillations with beads propelled by actin polymerization differs from those of Listeria by one order of magnitude (8− 15 min [18]). Periods of that length can be obtained within our model upon using values for ka close to the lower boundary of the oscillatory regime. 0 50 100 150 200 250 300 350 400 time (s) (a)speed displacement 0 50 100 150 200 250 300 350 400 time (s) (b)speed displacement FIG. 4: (color online) Velocity and displacement of the obsta- cle as a function of time with (a) ka = 0.9 s −1, (b) ka = 1 s vmaxp = 750 nm s −1, vmaxg = 75 nm s −1, k0d = 0.1 s l̄=100 nm ζ = 10−3 pN s nm−1 and D = 3 in both panels. We have also studied the system when the network is oriented at an angle ϑ = π/4. In this case, the spring constant of the attached filaments parallel to n̂ for D = 2 reads k−1‖ (ϑ) = 4` + e−�/2 − 1 + cos 2ϑ( 1 e−2� − e−�/2) − cos2 ϑ(e−�/2 − 1)2]/kBT , where � = l/lp and R‖(ϑ) = l(1 − l/4`p) cosϑ [22]. For the pushing force of a filament grafted at ϑ = π/4, we use the results of the factorization approximation given in Ref. [8], which is well valid for a stiff filament like actin. A numerical solu- tion of Eqs. 8a-8d results in the phase diagram shown in Fig. 2(b) with the adapted forms of Fd and Fa. The main effect is that one needs higher values for the attachment rates and lower values for vg to obtain oscillations. In summary, we have presented a simple and generic theoretical description of oscillations arising from the interplay of polymerization driven pushing forces and pulling forces due to binding of actin filaments to the obstacle. The physical mechanism for such oscillations relies on the load-dependence of the detachment rate and the polymerization velocity, mechanical restoring forces and eventually also on the cross-linkage and/or entangle- ment of the filament network. The oscillations are very robust with respect to changes in various parameters, i.e. are generic in this model. Therefore, complex biochem- ical regulatory systems supplementing the core process described here may rather stabilize motion and suppress oscillations than generate them. Oscillations of the velocity were observed during propulsion of pathogens by actin polymerization. There, the core mechanism described here is embedded into a more complex control of polymerization, which e.g. also comprises nucleation of new filaments and capping of ex- isting ones. Hence, the study presented here can not be expected to fully capture all features of such processes. Our results still agree well with respect to velocity spike amplitudes and periods in Listeria experiments reported in Refs. [16, 17]. The velocity in between spikes appears to be smaller in experiments than in our simulations. This may be accounted for in our model by including capping of filaments upon dissociation from the obsta- cle. Periods may also become longer when capping and nucleation were included since it would take longer to restore the pushing force after the avalanche like rup- ture of attached filaments. Altogether, qualitative and quantitative comparison with experiments suggests that our model may be a promising candidate for a robust mechanism of velocity oscillations in actin-based bacte- ria propulsion. We thank R. Straube and V. Casagrande for inspiring discussions. E.F. acknowledges financial support of the German Excellence Initiative via the program ”Nanosys- tems Initiative Munich (NIM)”. A.G. acknowledges fi- nancial support of the IRTG ”Genomics and Systems Biology of Molecular Networks” of the German Research Foundation. [1] D. Bray, Cell Movements, 2nd ed, Garland, New York. [2] J. Plastino and C. Sykes, Curr. Opin. Cell Biol, 17, 62 (2005). [3] E. Gouin, M.D. Welch, P. Cossart, Curr. Opin. Microbiol, 8, 35 (2005). [4] T.P. Loisel, R. Boujemaa, D. Pantaloni, M.F. 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It has been shown, however, that a variable number of filaments is not required for propulsion; see e.g. W.M. Brieher, M. Coughlin and T.J. Mitchison, J. Cell Biol. 165 233 (2004) [20] E. Evans and K. Ritchie, Biophys. J. 76, 2439 (1999). [21] A. Ott et al., Phys. Rev. E 48, R1642 (1993) ; L. LeGoff et al., Phys. Rev. Lett. 89, 258101 (2002). [22] K. Kroy and E. Frey, Phys. Rev. Lett. 77, 306 (1996). [23] J.F. Marko and E.D. Siggia, Macromol., 28, 8759 (1995). References

0704.1391

Path integrals for stiff polymers applied to membrane physics

Path integrals for stiff polymers applied to membrane physics D.S. Dean(1) and R.R. Horgan(2) (1) IRSAMC, Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France (2) DAMTP, CMS, University of Cambridge, Cambridge, CB3 0WA, UK e-mail:[email protected], [email protected] (Dated: 11 April 2007) Path integrals similar to those describing stiff polymers arise in the Helfrich model for membranes. We show how these types of path integrals can be evaluated and apply our results to study the ther- modynamics of a minority stripe phase in a bulk membrane. The fluctuation induced contribution to the line tension between the stripe and the bulk phase is computed, as well as the effective inter- action between the two phases in the tensionless case where the two phases have differing bending rigidities. I. INTRODUCTION Recently their has been much interest in the effective interactions between components of membranes of different composition. These effective interactions can be direct basic interactions such as electrostatic and van der Waals forces. However the fact that the membrane fluctuates also leads to effective interactions which are due to how the different components, or inclusions, alter the membrane fluctuations. Coarse grained models, based on the Helfrich model[1, 2], of multicomponent membrane describes the membrane in terms local mechanical properties, for instance the bending rigidity κb, the Gaussian rigidity κg or the spontaneous curvature [10, 11, 12, 13, 14, 15, 16]. The effective, fluctuation mediated, interaction between regions of differing rigidity can be computed using a cumulant expansion giving the effective pair-wise component of the interaction between two regions. This term is of order and is the analogue of the pair-wise component of van der Waals forces. However when |δκb/g| is large then this two-body, or dilute, approximation will break down and a full N -body calculation is needed. We should expect the dilute approximation to break down reasonably frequently as experimentally measured values of κb for commonly occurring lipid types vary from 3 to 30kBT . In this paper we show how the full N -body calculation for a system with spatially varying rigidity and elasticity, can be carried out for stripe geometries of the type shown in Figure 1. Within this geometry we can compute the contribution to the line tension between the two phases due to membrane fluctuations. This can be seen as a renormalization of the line tension already present due to basic interactions such as van der Waals, electrostatic and steric forces. In addition, in some cases, we can evaluate the effective interaction between the two interfaces as a function of their separation l. They key point in our calculation is that we convert the usual functional integral into a path integral where the direction in which the physical parameters change is treated like a fictitious time variable within the path integral formalism. The authors have already applied this approach to electrostatic problems where it has proved to be efficient for carrying out computations for films [7], interfaces [8] and in cylindrical geometries such as lipid-tubules [9]. The paper is organized as follows. In section II we describe the model and show how it can be analyzed using path integrals which are mathematically identical to those arising for stiff polymers. In section III we present the results of our computations. We start with an analysis of bulk homogeneous membranes and show how some standard results can be recovered using our path integral formalism and then how the method can be applied to a striped membrane system. The formalism is then used to compute the membrane fluctuation–induced contribution to the line tension between two phases. In section IV the fluctuation–induced Casimir force in a striped system is calculated for typical physical situations. In section V we conclude with a discussion of our results. The generalized Pauli-van Vleck formula used to evaluate the path integrals is described in detail in Appendix A. Some aspects of this approach have been previously described in the literature [4, 5, 6], however our approach is slightly different and self-contained. Hence for the sake of completeness (and because the results seem to be relatively unknown) we include this detailed description of the approach. http://arxiv.org/abs/0704.1391v1 LBULK (0) BULK (0)STRIPE κ µ κ µ κ µ0 0 0 0 FIG. 1: Schematic diagram of striped membrane configuration with mechanical parameters (rigidity and elasticity indicated. II. THE MODEL In the Monge gauge the Helfrich model for a membrane whose height fluctuations are denoted by h is given by + µ (∇h)2 . (1) In this model we have neglected the Gaussian rigidity κg and so for ease of notation we denote the bending rigidity simply by κ. In this form of the Helfrich model there is no spontaneous curvature and it is implicitly assumed that the fluctuations of h are small. The term µ can be interpreted as a local elastic or surface energy. When µ is constant it can be interpreted as a surface tension. The integral in x = (z, y) in Equation (1) is over the projected are of the membrane Ap. The physical area of the membrane is larger than Ap due to fluctuations and we denote it by A = Ap + ∆A, where ∆A is the excess area due to fluctuations. In the limit of small height fluctuations (i.e. to quadratic order in h) the excess area is given by d2x (∇h)2 . (2) The canonical partition function can be written as a functional integral over the height field h d[h] exp(−βH), (3) where β = 1/kBT , T is the canonical temperature and kB Boltzmann’s constant. If the mechanical parameters κ and µ only vary with the coordinate z then we can express h in terms of its Fourier decomposition in the direction y writing h(z, y) = h̃(z, k) exp(iky). (4) We have imposed periodic boundary conditions in the y direction and thus have k = 2πn/L, where L is the width of the system and n is an integer. Note that we are assuming that the interfaces between differing phases are straight and and thus lie at constant values of z. This is a realistic assumption if the bare (in the absence of height fluctuations) line tension γ0 is positive and large. We will in fact see that the renormalization of the line tension γ between phases due to the height fluctuations is positive and thus this assumption remains valid (and is in fact reinforced) by height fluctuations. In the limit of large L the sum over modes can be written as . (5) We now find that the Hamiltonian decomposes as Hk, (6) where dzκ(z) ∂2h̃(z,−k) ∂2h̃(z, k) +(2κ(z)k2+µ(z)) ∂h̃(z,−k) ∂h̃(z, k) +(µ(z)k2+κ(z))h̃(z, k)h̃(z,−k). The field h is real and so we have the relation h̃(z, k) = h̃(z,−k). The full functional integral for the partition function Z can then be written as Θk, (8) where d[X ] exp a2(k, t) + a1(k, t) + a0(k, t) . (9) In the above, the coefficients are given by a2(k, t) = βκ(t) a1(k, t) = β(2κ(t)k 2 + µ(t)) a0(k, t) = β(µ(t)k 2 + κ(t)), (10) and l is the length of the system (in the z direction). The above path integral is that arising in an elastic model of a semi-flexible polymer [3] (in one dimension) where κ = a2/β is the rigidity of the polymer, the term proportional to a1 represents the elastic energy and the term proportional to a0 represents an external harmonic potential. In this model the length of the polymer is not fixed, in contrast to the worm-like chain model where the magnitude of the tangent vector is fixed. The method of evaluation of the above type of path integral is given in Appendix A and we find that it takes the form K(X,Y; t) = (2π)− 2 [det (B(t)))] TAI(t)X− TAF (t)Y +X TB(t)Y , (11) where the initial condition vector is X = (X,U) with X = X(0) and U = dX/ds|s=0, and the final condition vector is Y = (Y, V ) with Y = X(t) and V = dX/ds|s=t. When the coefficients ak are independent of t the classical action can be written as a combination of surface terms (using the equation of motion) as Scl(X,Y) = . (12) The above expression is in general rather complicated but can be used to determine the matrices AF , AI and B. Also, when the coefficients are independent of time, the time reversed trajectories have the same weight as the original trajectories. Thus the path integral going from (X,U) to (Y, V ) in time t has the same value of the path integral going from (Y,−V ) to (X,−U) in time t. Mathematically this means that AF = SAIS, (13) BT = SBS, (14) where 0 − 1 . (15) In the limit of large t the propagator K should factorize, as it is dominated by the lowest eigenfunction, and so we should find that B(t) → 0 as t → ∞. This can be verified by explicit calculation in the cases considered here. III. CALCULATIONS FOR BULK AND STRIPED SYSTEMS To start with we will show how the path integral formalism introduced here reproduces some standard results concerning bulk systems. We consider a bulk system of projected length l and projected width L; we thus have a projected area Ap = Ll. Periodic boundary conditions are imposed in both directions z and y. The free energy is given by F = −kBT ln(Θk), (16) where dXKk(X,X, l). (17) Using Equation (A17) and the notation developed in the appendix we find Θk = det (B(k, l))) 2 det (AI(k, l) +AF (k, l)− 2B(k, l))− 2 . (18) The classical equation of motion in this case has solutions X(t) = a cosh(pt) + b sinh(pt) + c cosh(qt) + d sinh(qt), (19) where p = k, (20) q = (k2 +m2) 2 , (21) where we have defined , (22) and so m is an inverse length scale. The expressions for AF AI and B can be computed using computer algebra but they simplify in the (thermodynamic) limit l → ∞. We define A∗I/F = lim AI/F , (23) and find that A∗I = βκ pq(p+ q) pq pq (p+ q) . (24) We also find that for large l B(l) → 0, and det (B(l)) ≈ pq(p+ q)2 exp (−(p+ q)l) . (25) The extensive part of the bulk free energy is thus k + (k2 +m2) , (26) where we have introduced the ultra-violet cut-off length scale a which corresponds to the lipid size. The infra-red cut-off scale (where needed) is given by L (the lateral size of the system). The excess area of the system is given by , (27) and in the tensionless limit where µ = 0 we find the well known result . (28) In the case where µ 6= 0 we find that sinh−1 . (29) This gives ≈ kBT , (30) when a ≪ 1/m. For a striped geometry where the length of the bulk phase is l0, and large and that of the minority phase is l we find that for this composite striped (hence the superscript s in what follow) system we have, as l0 → ∞, k (l, l0) = dXdYKk(X,Y, l)K k (X,Y, l0) det(B(0)(l0)) [det(B(l))] F +AI(l) )]− 1 I +AF (l)−B T (l)(A F +AI(l)) −1B(l) , (31) where the superscript (0) refers to the bulk phase and the absence of this superscript refers to the minority phase. In the limit l → ∞ the above expression simplifies giving k (l, l0) ≈ det(B(0)(l0)) [det(B(l))] det(A , (32) where we have used Equation (13). In order to compute the free energy cost of the interface between the two phases we subtract the separate bulk free energies for large systems of size l0, corresponding to the bulk, and of size l, corresponding to the minority phase, from that of a large striped system composed of length l0 of the bulk phase and l of the minority phase. This free–energy difference is ∆F = −kBT k (l, l0) k (l0)Θk(l) , (33) which gives ∆F = −kBT det(A F ) det(A det(A I + A . (34) The above expression is in general quite complicated but when κ and κ0 are non-zero, then at large k the eigenvalues q (given by Equation (21)) in the stripe phase becomes asymptotically equal to q0, the corresponding eigenvalue in the bulk phase. We thus find that the sum in Equation (34) is ultra-violet divergent and is dominated by the term 1−∆2/4 , (35) where κ− κ0 κ+ κ0 . (36) We may interpret this result as the existence of a height fluctuation induced line tension γhf between the two phases (note the factor of a half as there are two interfaces) given by γhf = 1−∆2/4 . (37) Thus the dominant contribution to the fluctuation induced line tension between the two phases comes from the miss– match in their bending rigidities. We also remark that it does not depend on m and is only dependent on κ and κ0 0 0.2 0.4 0.6 0.8 1 FIG. 2: The function I(∆) defined in Equation (38), where ∆ = (κ− κ0)/(κ+ κ0). Note that I(∆) = I(−∆). through ∆2, which is a symmetric function of the two rigidities. As an example the fluctuation induced line tension between two phases whose rigidity differs by a factor of 10 has an energy of about 0.5 kBT per lipid at the interface. The correction terms to γhf are UV convergent and, after some manipulation, γ can be expressed as a power series in ma/π as γhf = 1−∆2/4 I(∆)− 1 (1−∆2/4) . (38) I(∆) is shown in Figure 2 for ∆ > 0 and is a non-negative function of ∆ with I(0) = 0 and I(∆) = I(−∆). From Figure 2 we see that I(∆) has a maximum value I(∞) ∼ 0.04. For µ = 10−2N/m, κ = 25kBT ∼ 10−19J and a = 10−9m, we find ma = (µ/κ)a ∼ 0.32 and so ma/π ∼ 0.1. The correction to the leading term due to non-zero µ is thus expected to be certainly less than O(1%). IV. APPLICATIONS In this section we discuss the application of the theory developed above to two cases in the system with a stripe as shown in Figure 1. These cases are distinguished by the values of the masses in the two regions, m0 = (µ0/κ0), (µ/κ), which control the relationship of the surface to bending energies. In general, the boundary conditions satisfied by the system will vary and will determine the precise way in which our formalism is applied and the form taken by the relevant Helfrich action. We consider two cases that might be thought of as extreme situations and are chosen to show how the results are markedly different depending on the exact situation. We concentrate on computing the Casimir force across the stripe which can be interpreted as a force between the opposing interfaces. The Casimir free energy, FC(l), is therefore normalized to F = 0 in the limit l → ∞. We have FC(l) = F (l, l0) − lim F (l, l0) |l+l0=constant , (39) where F (l, l0) = − kBT k (l, l0) , (40) and Θ k (l, l0) is defined in Equation (31). It is understood that the total volume of the system is held fixed by imposing l + l0 = constant, and that l0 is large compared with any system-specific length scale. −1 −0.5 0 0.5 1 FIG. 3: The function C(∆) defined in Equation (44), where ∆ = (κ− κ0)/(κ+ κ0). Note that C(∆) 6= C(−∆). A. m0 = m = 0 This case corresponds to an untethered membrane which is tensionless, as is the case for a membrane in the presence of the various lipid species in solution. Upon a change in the physical area of the membrane, A, lipid molecules can leave or enter meaning that any area change costs no free energy. Then we have µ0 = µ = 0. In this case, the choice for the general solution to the classical equations of motion is not given by Equation (19) but by X(t) = a cosh(pt) + b sinh(pt) + c t cosh(pt) + d t sinh(pt) . (41) The method follows the manipulations of section III, and appendix A. We find that FC(l,∆) = 1 + a2(l,∆)e −2kl + a4(l,∆)e , (42) where a2(l,∆) = (1−∆2/4) 1−∆/2 1 + ∆/2 a4(l,∆) = 16(1−∆2/4)2 . (43) We note that all terms are invariant under ∆ → −∆ except the first term in a2(l), and hence the Casimir free energy is not invariant under this transformation in this case. On dimensional grounds we have FC(l,∆) = C(∆)kBT , (44) and C(∆) is shown in Figure 3. Since C(∆) < 0 for ∆ 6= 0, the Casimir force is attractive and is given by fC(l,∆) = − C(∆)kBT C(∆)kBT , (45) with |C(∆)| / 0.4. 0 0.2 0.4 0.6 0.8 1 FIG. 4: The function G(Γ) defined in Equation (48), where Γ = (µ− µ0)/(µ+ µ0). Note that G(Γ) = G(−Γ). B. m0,m > 0 This case is for non-zero µ0 and µ. The projected area is constant but any change in the area results in a free energy change. In practice, the typical observed values of µ give ma ∼ 0.3 (see section III) and consequently the width of the stripe, l, satisfies l ≪ 1/m. As shown in section III the free energy has a contribution that corresponds to the energy of the interfaces between the stripe and the bulk medium and includes the UV divergent part of the sum over modes. The remaining, l-dependent, terms are UV convergent and are cut-off but exponential factors on a scale k / 1/l. Thus, the Casimir free energy excluding the interface energy gets contributions only from low mode number k ≪ 1/m and so we can approximate the eigenvalues in Equations (20) and (21) by p = k , q = m . (46) Also, only leading terms in eql will survive; the others will be suppressed by factors of e−ml. Following the derivation of the previous subsection, we find a result that is independent of κ0 and κ, as one might expect on dimensional grounds for µ ≫ κ/l2. We get FG(l,Γ) = 1− Γ2e−2kl , (47) where Γ = (µ − µ0)/(µ + µ0). The leading corrections are suppressed by the factor (1/ml)2. For the values of m considered this is of order (a/l)2 ∼ 1/N2, where N is the number of lipid molecules across the strip. Similarly to before we may write FG(l,Γ) = G(Γ)kBT (ml)2 . (48) G(Γ) is symmetric under Γ → −Γ, and is shown in Figure 4. We see that |G(Γ)| / 0.2 and, being negative for Γ 6= 0, gives rise to an attractive Casimir force of maximum magnitude 0.2kBT/l The more general case where m ∼ 1/l is very much more complicated and the expressions cannot be presented here but need to be investigated computationally in the three parameter µ, κ, l space. We have verified that this analysis is feasible but complicated and so we have chosen not to present it in this paper. However, any more general result for the Casimir force will interpolate between the two extremes presented here and so we would expect a leading contribution to behave like −ckBT/l2 with c / 1. V. CONCLUSION In this paper we have shown how the formalism developed in earlier work [9] can be applied to the more general case of higher derivative Gaussian energy functions such as apply to the path integral analysis of stiff polymers and the Helfrich model for membranes. Some aspects of this approach have been previously described in the literature [4, 5, 6]. However, our approach is slightly different and self-contained. Hence, for the sake of completeness (and because the results seem to be relatively unknown) we have included a detailed description of this approach. In particular, we have shown how to generalize the Pauli-van Vleck formula for the evolution kernel of all theories of this type. As a model system we have considered a toroidal lipid membrane with one very large circumference and the other finite of length L, with a stripe of width l wrapped around the finite circumference and of different, minority, lipid type to the bulk, majority, type. This is shown schematically in Figure 1. This geometry imposes periodic boundary conditions on the system. We have shown how to compute, in general, two important energies in this system, namely the energy, or line tension, associated with the lipid-lipid interface and the Casimir force between the interfaces, as a function of width l. An important controlling parameter, m, has the dimensions of a mass and is given by m2 = µ/κ. We have presented these calculations explicitly for the cases where m = 0 and ma ∼ 1, l ≫ 1/m, where a is the inter-lipid spacing. These correspond, respectively, to the cases where the actual area, A, or the projected area, Ap, is conserved. In the latter case, we calculate the mean excess area of the system ∆A/Ap in section III. In both cases, the interface energy is positive and the Casimir force attractive as can be seen from Equations (35),(44), (48) and the associated figures. Our general result is that in appropriate dimensionful units the energy coefficient is ckBT where, at maximum, c ∼ 1. This is to be compared with the natural bending rigidity with lies in the range 5kBT ≤ κ ≤ 100kBT . The general case where l ∼ 1/m is complicated and long, and although we have the results we have not presented an explicit analysis in the µ, κ, l parameter space because of the complexity. However, there is no computational or algebraic impediment to carrying this out. In terms of relevance to the physical system we might consider two scenarios in the two-lipid model discussed here. Either the minority lipid can be dissolved in the majority lipid to form a homogeneous phase for the mixture, or the minority lipid can precipitate out of solution and form a pure minority phase, the stripe in our idealized case, within the pure majority phase. Which situation is stable is, of course, decided by a competition between the free energies of the configurations which is in turn dependent on the boundary conditions imposed. However, it is clear that the attractive Casimir force will tend to reduce the stripe width l, presumably by evaporation of minority lipid from the interface into solution. The interface energy is constant throughout such a process but will always tend to minimize the interface length. A stability analysis, however, requires a computation of the free energy of the mixed phase which our calculation does not address. However, as has been discussed in [16], the suppression of lipid mode fluctuations by confining the membrane in a stack will change the free-energy of both configurations and so can affect their stability; an effect which can be analyzed by our methods. APPENDIX A: THE GENERALIZED PAULI-VAN VLECK FORMULA In this appendix we show how generalized quadratic path integrals can be evaluated giving a generalization of the Pauli-van Vleck formula. The treatment is very close to that of [6] and is based on the Chapman-Kolmogorov decomposition of the path integral. We consider the following path integral K(X,Y; t) = X(t)=Y X(0)=X d[X ] exp (−S[X ]) , (A1) where S is a quadratic action which will have the general form S[X ] = . (A2) In general the coefficients ak can be time dependent but for the problems related to membranes studied here we will only require the results for ak constant. The usual Wiener measure occurring in path integrals has N = 1 and the corresponding path integral is that for standard Brownian motion, or a free particle, with a0 = 0. If a0 6= 0 in this case, then the path integral corresponds to that of a simple harmonic oscillator with a0 = mω 2 and a1 = m thus relating the coefficients ai, i = 0, 1 to the mass m and frequency of the oscillator. The path integrals arising in section II are, of course, for the case N = 2 which, as mentioned above, also arises for the path integrals of stiff or semi-flexible polymers. We also refer the reader to the approach of [4] which is based on an eigenfunction expansion method for the case N = 2. Now one must state how the initial and final points of the path integral should be specified. The presence of the term dNX/dsN means that the paths that contribute to the path integral are ones where the derivatives dN−1X/dsN−1 and lower must be continuous. The path integral should therefore be specified in terms of the vector X = (X,X(1), X(2) · · ·X(N−1)) where we have used the notation X(k) = dkX/dsk. We can now decompose the path integral using the Chapman-Kolmogorov formula K(X,Z; t+ t′) = dYK(X,Y; t)K(Y,Z; t′). (A3) This decomposition ensures the continuity of the path X(t) up to its N − 1th derivative. The classical path is given by the one that minimizes the action: δX(s) = 0, (A4) with the boundary conditions on the end points X(0) = X and X(t) = Y. This gives a total of 2N boundary conditions (N from each end). The equation for the classical path can be written as δX(s)δX(s′) Xcl(s ′) = 0, (A5) which is a linear differential equation of order 2N . For instance, when the ak are constant it reads (−1)kak Xcl(s) = 0. (A6) There are thus 2N linearly independent solutions to this equation and their coefficients are linearly related to the 2N conditions for the end points. The classical action is a quadratic form in the initial and final condition vectors X and Y and we can write Scl(X,Y) = TAI(t)X+Y TAF (t)Y − 2XTB(t)Y , (A7) where we have used the subscripts I and F to denote the initial and final coordinates. We now write the path X(s) as X(s) = Xcl(s) + x(s), where the boundary conditions imply that x(0) = 0 and x(t) = 0. The path integral can now be written as K(X,Y; t) = exp (−Scl(X,Y)) d[x] exp dsds′x(s′) δX(s)δX(s′) = Q(t) exp (−Scl(X,Y)) , (A8) where we formally write can write Q(t) = det δX(s)δX(s′) , 0 ≤ s, s′ ≤ t. (A9) The above functional determinant can be evaluated using an eigenfunction expansion, however for higher order operators this quickly becomes impractical. Instead, we return to the Chapman Kolmogorov formula Equation (A3) and pursue its consequences using the formal result Equation (A8). Explicitly carrying out the intermediate integration over Z, we find that K(X,Z; t+ t′) = (2π) 2 Q(t)Q(t′) det (AI(t) +AF (t AI(t)−B(t)(AI(t′) +AF (t))−1BT (t) AF (t ′)− BT (t′)(AI(t′) +AF (t))−1B(t′) TB(t)(AI(t ′) +AF (t)) −1B(t′)Z , (A10) Now comparing the quadratic forms and prefactors we find the following relations: AI(t+ t ′) = AI(t)−B(t)(AI(t′) +AF (t))−1BT (t) (A11) AF (t+ t ′) = AF (t ′)−BT (t′)(AI(t′) +AF (t))−1B(t′) (A12) B(t+ t′) = B(t)(AI(t ′) +AF (t)) −1B(t′) (A13) Q(t+ t′) = (2π) 2 Q(t)Q(t′) det (AI(t) +AF (t 2 . (A14) In [6] it is pointed out that relation Equation (A14) above can be used to derive a differential equation for Q. However, a more rapid way of finding Q is to note that taking the determinant of both sides of Equation (A13) gives det (B(t+ t′)) = det (B(t)) det (B(t′)) det (AI(t) +AF (t , (A15) and using this relation we find that by direct substitution into Equation (A14) that the solution for Q is Q(t) = (2π)− 2 [det (B(t))] 2 . (A16) The generalized form of the Pauli-van Vleck formula may thus be written in the familiar form (for N = 1 and its generalization to higher dimensions) K(X,Y; t) = (2π)− 2 det ∂Xi∂Yj exp (−Scl(X,Y)) . (A17) [1] W. Helfrich,Z. Naturforsch. 28c, 693 (1973). [2] D. Boal, Mechanics of the cell ( Cambridge University Press, Cambridge, 2002). [3] R. Harris and J. Hearst, J. Chem. Phys. 44, 2595 (1966); M. Doi and S.F. Edwards, The theory of polymer dynamics (Clarendon Press, Oxford, 1986). [4] H. Kleinert, J. Math. Phys. 27, 3003 (1986). [5] J.Z. Simon, Phys. Rev. D 41, 3720 (1990). [6] D.A. Smith, J. Phys. A 34 4507 (2001). [7] D.S. Dean and R.R. Horgan, Phys. Rev. E 65, 061603, (2002). [8] D.S. Dean and R.R. Horgan, Phys. Rev. E 69, 061603 (2004). [9] D.S. Dean and R.R. Horgan, Phys. Rev. E 71, 041907, (2005); ibid, J. Phys. C. 17, 3473, (2005). [10] S. Leibler,J. Phys. (Paris) 47, 506 (1986). [11] T. Taniguchi, Phys. Rev. Lett. 76, 4444 (1996). [12] Y. Jiang, T. Lookman, and A. Saxena, Phys. Rev. E, 61, R57 (2000). [13] F. Divet, G. Danker, and C. Misbah, Phys. Rev. E, 72 041901 (2005). [14] R.R. Netz, J. Phys. I France 7, 833 (1997). [15] R.R. Netz and P. Pincus,Phys. Rev. E 52, 4114 (1995). [16] D.S. Dean and M. Manghi, Phys. Rev. E, 74, 021916 (2006). Introduction The model Calculations for bulk and striped systems Applications m0=m=0 m0,m > 0 Conclusion The generalized Pauli-van Vleck formula References

0704.1392

Measuring CP violation in Bs->phi phi with LHCb

Measuring CP violation in B0 → φφ with LHCb J. F. Libby on behalf of the LHCb Collaboration∗ University of Oxford Sensitivity studies to the CP -violating parameters of the decay B0s → φφ with the LHCb exper- iment are presented. The decay proceeds via a b → sss̄ gluonic-penguin quark transition, which is sensitive to contributions from beyond the Standard Model particles. A time-dependent angular analysis of simulated data leads to an expected statistical uncertainty of 6◦ on any new physics induced CP -violating phase for a sample corresponding to 2 fb−1 of integrated luminosity. The expected precision on sin 2β from the related decay B0 → φK0S is also discussed. I. INTRODUCTION The amplitude for the decay B0s → φφ is dominated by gluonic-penguin quark transitions b → sss̄ (Fig. 1). Gluonic-penguin processes are sensitive to beyond the Standard Model particles that contribute within the loop. The e+e− B-factories have measured sin 2β in nine B0 gluonic penguin decay modes, such as B0 → φK0S and B0 → η′K0S [1]. All the measurements of sin 2β from these modes have values below that measured in b → cc̄s transitions, but no individual measurement shows a significant deviation. The decay B0s → φφ is predicted to have a CP - violating phase less than 1◦ within the SM [2]. The dependence on Vts in both the mixing and decay am- plitudes leads to a cancellation of the B0s -mixing phase. Therefore, if any significant CP -violation is measured in B0s → φφ decays it is an unambiguous signature of new physics. The decay is of a pseudoscalar meson to two vector mesons, which leads to the final state being a CP -even and CP -odd admixture. Therefore, a time- dependent angular analysis is required to extract the CP - violating parameters of the decay. The paper is organized as follows. Section II contains a brief description of the LHCb experiment. The predicted event yields and background estimations are described FIG. 1: The main diagram contributing to the decay B0s → ∗Electronic address: [email protected] in Section III. The CP sensitivity study is presented in Section IV. The LHCb prospects with the related mode B0 → φK0S are discussed in Section V. The conclusions are given in Section VI. II. THE LHCb EXPERIMENT The Large Hadron Collider (LHC) collides protons at a centre-of-mass energy of 14 TeV. The LHC pro- duces 1012 bb̄ quark pairs per nominal year of data-taking (107 s) when operating at an instantaneous luminosity of 2×1032 cm−2s−1.1 The LHCb spectrometer [3, 4] instru- ments one forward region about the pp collision point. The forward geometry captures approximately one-third of all B hadrons produced and increases the probability of both B hadrons from the bb̄ pairs being within the acceptance, which improves the efficiency of flavour tag- ging. A silicon vertex detector, with sensors perpendicular to the beam axis, is situated close to the interaction region in a secondary vacuum. The detector provides accurate determination of primary and secondary vertices leading to a proper-time resolutions of approximately 40 fs in hadronic B-decays such as B0s → φφ. Tracking stations either side of a 1.2 T dipole magnet produce momentum measurements with an accuracy of a few parts per mille. There are two Ring-Imaging Čerenkov detectors, with 3 different radiators, that allow identification of K± from π± over the momentum range 1 to 100 GeV/c. In addition, the detector includes an electromagnetic calorimeter, a hadron calorimeter and a muon detector. These components are critical for identifying large trans- verse momentum, pT , electrons, photons, hadrons and muons from B-hadron decay in the initial hardware stage (Level-0) of the LHCb trigger. The Level-0 trigger re- duces the 40 MHz collision rate to 1 MHz. All data is then transferred from the detector to a dedicated CPU farm where the Higher Level Trigger (HLT) algorithms are performed. Initially an association between the high 1 This luminosity optimises the number of single interactions per bunch crossing. http://arxiv.org/abs/0704.1392v1 mailto:[email protected] hmBsT Entries 4808 Mean 5.369 RMS 0.02572 Bs mass [GeV/c2] 4.8 5 5.2 5.4 5.6 5.8 hmBsT Entries 4808 Mean 5.369 RMS 0.02572 m Bs true FIG. 2: The B0s mass of candidates reconstructed in the signal (blue) and inclusive bb̄ simulation samples (red) before trigger selections. The normalisations are arbitrary. pT objects that satisfy the Level-0 trigger and tracks with large impact parameter to the primary vertex is sought. If this is successful, exclusive and inclusive HLT algo- rithms are executed resulting in a 2 kHz output rate to disk. III. EVENT SELECTION AND BACKGROUND ESTIMATION The estimation of B0s → φφ yields and background has been performed on simulated data. The simulation of the pp collisions and subsequent hadronisation is per- formed by the PYTHIA generator [5]. The decay of any B hadrons produced is simulated by EVTGEN [6] and the detector response is performed by GEANT4 [7]. The re- sulting data are processed by the complete LHCb recon- struction software. A dedicated sample containing events with a B0s → φφ decay is used to evaluate the selection efficiency. Minimum bias events are selected rarely by the Level-0 trigger and the HLT. Therefore, an inclusive sample of 34 million events containing bb̄ quark pairs is used to estimate the background. Due to the very large number of bb̄ pairs that will be produced at LHCb, the inclusive bb̄ sample only corresponds to approximately 15 minutes of data taking at the nominal instantaneous luminosity; this leads to large uncertainties on the back- ground estimates. The selection process begins with the identification of φ candidates reconstructed from two oppositely charged kaons. Particle identification and pT requirements are placed on the K± and they must be consistent with pro- duction at a common vertex. The mass of the K+K− must be within 20 MeV/c2 of the φ mass; the mass in- terval corresponds to approximately ±3σmφ , where σmφ is the mass resolution. B-meson candidates are reconstructed in events with two or more φ candidates with a pT > 1.2 GeV/c 2. The B-meson candidates must be consistent with production at a common vertex and this vertex must be well sepa- rated from the primary vertex. The B0s candidate mass distribution is shown in Fig. 2 for signal and background samples. Events within 40 MeV/c2 of the B0s mass are considered as signal; the mass interval corresponds to approximately ±3σmB , where σmB is the mass resolu- tion. Once Level-0 and HLT trigger selections have been applied there are 4000 signal events expected in every 2 fb−1 of integrated luminosity, which corresponds to one nominal year of LHC operation. The event yield is calculated assuming the measured branching fraction B(B0s → φφ) = (14 −5(stat.) ± 6(syst.)) × 10 −6 [8]; the measured value lies within theoretically predicted range [9, 10]. The background remaining in the bb̄ inclusive simulation sample is found to consist of combinatoric B0s candidates. The background-to-signal ratio is bounded to lie between 0.4 to 2.1 at the 90% confidence level, with a central value of 0.9. IV. B0s → φφ CP SENSITIVITY The magnitude of any new physics induced CP -phase, φNP , is extracted from a time-dependent analysis of the differential cross section with respect to the three transversity angles defined in Fig 3. The amplitude for the decay can be written in terms of three helicity am- plitudes Hλ(t) where λ = 0,±1. The helicity amplitudes are related to the transversity basis by A0(t) = H0(t), A||(t) = (H+1 +H−1) and A⊥(t) = (H+1 −H−1). The amplitudes A0 and A|| are CP even and A⊥ is CP odd. The differential cross section is then given by: dΓ(t) dχd cos θ1d cos θ2 = |A0(t)| 2f1(χ, θ1, θ2)|A||(t)| 2f2(χ, θ1, θ2) + |A⊥(t)| 2f3(χ, θ1, θ2) + ℑ(A∗0(t)A⊥(t))f4(χ, θ1, θ2) + ℜ(A 0(t)A||(t))f5(χ, θ1, θ2) + ℑ(A ||(t)A⊥(t))f6(χ, θ1, θ2) (1), where fi (i = 1 − 6) are even angular functions as re- quired by Bose symmetry [11]. The time-dependent fac- tors of these angular functions are sensitive to any CP - violating phase. In principle the CP -violating phase can χ=ϕ1+ ϕ2 FIG. 3: A schematic of the definition of the transversity angles θ1, θ2 and χ. accep Entries 4663 Mean 2.675 RMS 1.343 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 accep Entries 4663 Mean 2.675 RMS 1.343 Acceptabnce Proper time (ps) FIG. 4: The variation of the acceptance as a function of the proper-time, τ . The acceptance function fit to the simulated data is ǫ(τ ) = 0.084τ 0.027+τ3 be different between the three transversity amplitudes. However, to simplify the analysis it has been assumed to be equal. Significant fine-tuning of the phases would be required for no effects to be observed if new physics induced phases are present. The time-dependent terms also depend on the strong phase differences δ||,0 between A(t)⊥ and A(t)||,0, the relative magnitudes of the three amplitudes and the mass (lifetime) differences between the B0s mass eigenstates, ∆ms (∆Γs). Simulated data are generated to follow the differential distribution given in Eqn. 1 with the value of φNP cho- sen to be 0.2 rad. The strong phases are assumed to be δ|| = 0 and δ0 = π; these values are motivated by näıve factorization [12]. The magnitudes of the transversity amplitudes are set to the values measured in the anal- ogous channel for B0 decays B0 → K∗φ [13, 14]. The value of ∆ms is taken to be 17 ps −1 [15] and ∆Γs/Γ is set to be 0.15, compatible with current experimental con- straints [16] and theoretical expectations [17]. The signal sample size has a mean of 4000 events corresponding to an integrated luminosity of 2 fb−1. The following experimental effects are also simulated. • The background is assumed to be at level 90% of the signal with flat transversity angle and mass distributions, and an exponential lifetime distribu- tions. • The tagging power, ǫ(1− 2ω), where ǫ is the tag- ging efficiency and ω is the mistag rate, is assumed to be 9% which has been found in simulation stud- ies of other B0s hadronic decays [18]. This is signif- icantly better than that for B0 modes because the kaon associated with the B0s hadronisation is also used. • The proper-time acceptance measured from the signal simulation sample is shown in Fig. 4. The reduced acceptance for short lifetimes is the result of trigger and selection requirements on the impact parameters of the B daughters. • The proper time and B0 mass resolutions are estimated to be 40 fs and 12 MeV/c2, respectively. These resolutions are estimated from the signal simulation sample used for the selection studies. • The angular acceptance and resolution are as- sumed to be flat and to have negligible effect, re- spectively; this assumption is motivated by the studies of related channel B0s → Jψφ [19]. phi_s -0.4 -0.2 0 0.2 0.4 0.6 0.8 Prob 1 Constant 1.373± 20.59 Mean 0.006382± 0.1941 Sigma 0.004403± 0.1002 Prob 1 Constant 1.373± 20.59 Mean 0.006382± 0.1941 Sigma 0.004403± 0.1002 phi_s -0.4 -0.2 0 0.2 0.4 0.6 0.8 A RooPlot of "phi_s" φNP (rad) FIG. 5: The distribution of fitted value of φNP for 500 simu- lated B0s → φφ experiments. Five-hundred samples of signal and background events are generated and a maximum likelihood fit is performed on each one. The parameters φNP , δ||, δ0 and the mag- nitude of the transversity amplitudes are extracted from the fit; all other parameters are fixed. The distribution of the fitted value of φNP for these 500 experiments is shown in Fig 5. The average error on φNP is 0.1 rad (5.7◦). The pull distribution of the true value subtracted from the fitted value divided by the uncertainty is nor- Sets of 500 experiments are produced varying the input parameters assumed. The variation of the uncertainty on φNP as a function of the B(B s → φφ), signal-to- background ratio and ∆Γs/Γs is given in Table I. The variation of the assumed B leads to the expected statisti- cal scaling of the uncertainty. The uncertainty on φNP is not degraded significantly until the background-to-signal TABLE I: The variation of the uncertainty on φNP as a function of the branching fraction, background-to-signal ra- tio (B/S) and ∆Γs/Γs. B (×10−5) σ(φNP ) 0.35 13◦ 0.7 8.1◦ 1.4 5.7◦ 2.1 4.6◦ B/S σ(φNP ) 0 5.5◦ 0.9 5.7◦ 2 6.1◦ 5 7.2◦ ∆Γs/Γs σ(φNP ) 0.05 7.2◦ 0.15 5.7◦ 0.05 4.9◦ ratio is greater than three. Increasing the value of ∆Γs/Γ leads to a reduction in the uncertainty because enhanced interference from the lifetime difference among the am- plitudes increases the sensitivity to φNP . The values of φNP , the proper-time resolution and the relative magni- tudes of the transversity amplitudes are also varied; all these had negligible effect on the sensitivity to φNP . V. B0 → φK0S CP SENSITIVITY Simulation studies of the decay B0 → φK0S have also been performed. The expected yield per 2 fb−1 of in- tegrated luminosity is 800 events. These yields do not include K0S daughters without measurements in the sili- con vertex detector, approximately two-thirds ofK0S from these decays, which are not reconstructed in the current HLT algorithms. Algorithms to perform this reconstruc- tion are currently being developed. The background-to- signal ratio is estimated to be 2.4 from the bb̄ inclusive simulation. A time-dependent analysis of the B0 → φK0S is re- quired to extract the sensitivity to sin 2β. A toy sim- ulation study is performed to extract the sensitivity to sin 2β. The tagging power is assumed to be 5% and the proper-time resolution is taken to be 60 fs from the simu- lated signal sample. A 10% K+K− S-wave contribution is also included in the fit. The uncertainty on sin 2β is expected to be 0.32 for a data sample corresponding to 2 fb−1 of integrated luminosity. VI. CONCLUSIONS Sensitivity studies to CP -violation in the decay B0s → φφ have been presented. A sample of data corresponding to an integrated luminosity of 2 fb−1 gives an uncertainty of 6◦ on any new physics induced CP phase. Varying the assumptions used within reasonable ranges changes the predicted statistical uncertainty between 4◦ and 13◦. The largest statistical uncertainty results from decreasing the branching fraction by a factor of four. The measurement of sin 2β from the B0 → φK0S has also been investigated. The sensitivity is expected to be 0.32 with a data set corresponding to 2 fb of integrated luminosity; this is of the same order as the current sensitivity of the e+e− B-factories [20, 21]. Therefore, in conclusion, B0s → φφ is the most sensitive mode with which to study gluonic- penguin B decays with LHCb. [1] E. Barberio et al. (Heavy Flavour Averaging Group), http://www.slac.stanford.edu/xorg/hfag . [2] M. Raidal, Phys. Rev. Lett 89, 231803 (2002). [3] LHCb Technical Proposal, LHCb Collaboration, CERN/LHCC 98-4 (1998). [4] LHCb Reoptimized Detector Design and Performance, LHCb Collaboration, CERN/LHCC 2003-040 (2003). [5] T. Sjöstrand, L. Lönnblad and S. Mrenna, hep-ph/0108264. [6] D. Lange, Nucl. Instr. and Meth. A462 (2001) 152. [7] S. Agostinelli et al. (GEANT4 Collaboration), Nucl. In- str. and Meth. A506 (2003) 250. [8] D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 95, 031801 (2005). [9] Y.-H. Chen et al., Phys. Rev. D 59, 074003 (1999). [10] X. Q. Li, G. R. Lu and Y. D. Yang, Phys. Rev. D 68, 114015 (2005) [Erratum-ibid D 71, 019902 (2005)]. [11] B. de Paula et al., LHCb-2007-047. [12] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34, 103 (1987). [13] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 93 231804 (2004). [14] K.-F. Chen et al. (Belle Collaboration), Phys. Rev. Lett. 94 221804 (2005). [15] A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 97 242003 (2006). [16] G. Weber, these proceedings. [17] C. Tarantino, hep-ph/0702235. [18] M. Calvi, O. Leroy and M. Musy, LHCb-2007-058. [19] L. Fernandez, CERN-THESIS-2006-042. [20] K.-F. Chen et al. (Belle Collaboration), hep-ex/0608039. [21] B. Aubert et al. (BABAR Collaboration), hep-ex/0607112. http://www.slac.stanford.edu/xorg/hfag http://arxiv.org/abs/hep-ph/0108264 http://arxiv.org/abs/hep-ph/0702235 http://arxiv.org/abs/hep-ex/0608039 http://arxiv.org/abs/hep-ex/0607112

0704.1393

A Panchromatic Study of the Globular Cluster NGC 1904. I: The Blue Straggler Population

A Panchromatic Study of the Globular Cluster NGC 1904. I: The Blue Straggler Population 1 B. Lanzoni1,2, N. Sanna3, F.R. Ferraro1, E. Valenti4, G. Beccari2,5,6, R.P. Schiavon7, R.T. Rood7, M. Mapelli8, S. Sigurdsson9 1 Dipartimento di Astronomia, Università degli Studi di Bologna, via Ranzani 1, I–40127 Bologna, Italy 2 INAF–Osservatorio Astronomico di Bologna, via Ranzani 1, I–40127 Bologna, Italy 3 Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, via della Ricerca Scientifica, 1, I–00133 Roma, Italy 4 European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile 5 Dipartimento di Scienze della Comunicazione, Università degli Studi di Teramo, Italy 6 INAF–Osservatorio Astronomico di Collurania, Via Mentore Maggini, I–64100 Teramo, Italy 7 Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA, 22904 8 University of Zürich, Institute for Theoretical Physics, Winterthurerstrasse 190, CH-8057 Zurich 9 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802 30 March, 07 ABSTRACT By combining high-resolution (HST-WFPC2) and wide-field ground based (2.2m ESO-WFI) and space (GALEX) observations, we have collected a multi- wavelength photometric data base (ranging from the far UV to the near infrared) of the galactic globular cluster NGC1904 (M79). The sample covers the entire cluster extension, from the very central regions up to the tidal radius. In the present paper such a data set is used to study the BSS population and its radial distribution. A total number of 39 bright (m218 ≤ 19.5) BSS has been detected, http://arxiv.org/abs/0704.1393v1 – 2 – and they have been found to be highly segregated in the cluster core. No signifi- cant upturn in the BSS frequency has been observed in the outskirts of NGC 1904, in contrast to other clusters (M 3, 47 Tuc, NGC 6752, M 5) studied with the same technique. Such evidences, coupled with the large radius of avoidance es- timated for NGC 1904 (ravoid ∼ 30 core radii), indicate that the vast majority of the cluster heavy stars (binaries) has already sunk to the core. Accordingly, ex- tensive dynamical simulations suggest that BSS formed by mass transfer activity in primordial binaries evolving in isolation in the cluster outskirts represent only a negligible (0–10%) fraction of the overall population. Subject headings: Globular clusters: individual (NGC1904); stars: evolution – binaries: close - blue stragglers 1. INTRODUCTION Blue straggler stars (BSS) appear brighter and bluer than the Turn-Off (TO) point along an extension of the Main Sequence in color-magnitude diagrams (CMDs) of stellar popula- tions. Hence, they mimic a young stellar population, with masses larger than the normal cluster stars (this is also confirmed by direct mass measurements; e.g. Shara, Saffer & Livio 1997). BSS are thought to be objects that have increased their initial mass during their evolution, and two main scenarios have been proposed for their formation (e.g., Bailyn 1995): the collisional scenario suggests that BSS are the end-products of stellar mergers induced by collisions (COL-BSS), while in the mass-transfer scenario BSS form by the mass-transfer activity between two companions in a binary system (MT-BSS), possibly up to the complete coalescence of the two stars (Mateo et al. 1990; Pritchet & Glaspey 1991; Bailyn & Pinsonneault 1995; Carney Latham; Tian et al. 2006; Leigh, Sills & Knigge 2007). Hence, understanding the origin of BSS in stellar clusters provides valuable insight both on the binary evolution processes and on the effects of dynamical interactions on the (otherwise normal) stellar evolution. The MT formation scenario has by recently received further sup- port by high-resolution spectroscopic observations, which detected anomalous Carbon and Oxygen abundances on the surface of a number of BSS in 47 Tuc (Ferraro et al. 2006a). However the role and relative importance of the two mechanisms are still largely unknown. 1Based on observations with the NASA/ESA HST, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555. Also based on GALEX observations (program GI-056) and WFI observations collected at the European Southern Observatory, La Silla, Chile, within the observing programs 62.L-0354 and 64.L-0439. – 3 – To clarify the BSS formation and evolution processes we studying the BSS radial distri- bution over the entire cluster extension in a number of galactic globular clusters (GCs). We completed such studies in 5 GCs: M 3 (Ferraro et al. 1997), 47 Tuc (Ferraro et al. 2004), NGC 6752 (Sabbi et al. 2004), ω Cen (Ferraro et al. 2006b), and M 5 (Lanzoni et al. 2007, see also Warren, Sandquist & Bolte 2006). Apart from ωCen where mass segregation pro- cesses have not yet played a major role in altering the initial BSS distribution, the BSS are always highly concentrated in the cluster central regions. Moreover, in M 3, 47 Tuc, NGC 6752, and M 5 the BSS fraction decreases at intermediate radii and rises again in the outskirsts of the clusters, yielding a bimodal distribution. Preliminary evidences of such a bimodality have been found also in M 55 by Zaggia, Piotto & Capaccioli (1997). Recent dynamical simulations (Mapelli et al. 2004, 2006; Lanzoni et al. 2007) have been used to interpret the observed trends and have shown that a significant fraction ( >∼ 50%) of COL- BSS is required to account for the observed BSS central peaks. In addition, a fraction of 20-40% MT-BSS is needed to reproduce the outer increase observed in these clusters. The case of ω Cen is reproduced by assuming that the BSS population in this cluster is com- posed entirely of MT-BSS. These results demonstrate that detailed studies of the BSS radial distribution within GCs are very powerful tools for better understanding the BSS formation channels and for probing the complex interplay between dynamics and stellar evolution in dense stellar systems. In this paper we present multi-wavelength observations of NGC 1904. These observa- tions are part of a coordinated project aimed at properly characterize the UV excess of old stellar aggregates as globular clusters, in terms of their hot stellar populations, like Hori- zontal Branch (HB) and Extreme HB stars, post-Asymptotic Giant Branch stars, BSS, etc. From integrated light measurements obtained with UIT (see Dorman, O’Connell & Rood 1995), NGC 1904 was known to be relatively bright in the UV, and it was selected as a prime target in both our high-resolution (using HST) and wide-field (using GALEX) UV surveys. We have obtained a large set of data: (i) high-resolution ultraviolet (UV) and optical images of the cluster center have been secured with the WFPC2 on board HST; (ii) complementary wide-field observations covering the entire cluster extension have been obtained in the UV and optical bands by using the far- and near-UV detectors on board the Galaxy Evolution Explorer (GALEX) satellite and with ESO-WFI mounted at the 2.2m ESO telescope, re- spectively. The combination of these datasets allowed a study of the structural properties of NGC 1904 (thus leading to an accurate redetermination of the center of gravity and the surface density profile), and of the radial distribution of the evolved stellar populations (in particular the BSS and horizontal branch star distributions have been derived over the en- tire cluster extension). While a companion paper (Schiavon et al. 2007, in preparation) will focus on the morphology and the structure of the HB, the present paper is devoted to the – 4 – BSS population. 2. OBSERVATIONS AND DATA ANALYSIS 2.1. The data sets The present study is based on a combination of different photometric data sets: 1. The high-resolution set – It consists of a series of UV, near UV and optical images of the cluster center obtained with HST-WFPC2 with two different pointings. In both cases the Planetary Camera (PC, the highest resolution instrument with 0.′′046 pixel−1) has been pointed approximately on the cluster center to efficiently resolve the stars in the highly crowded central regions; the three Wide Field Cameras (WFC with resolution 0.′′1 pixel−1) have been used to sample the surrounding regions. Observations in Pointing A (Prop. 6607, P.I. Ferraro) have been performed through filters F160BW (far-UV), F336W (approximately an U filter) and F555W (V ), for a total exposure time texp = 3300, 4400, and 300 sec, respec- tively. Pointing B is a set of public HST-WFPC2 observations (Prop. 6095, P.I. Djorgovski) obtained through filters F218W (mid-UV), F439W (B) and F555W (V ). Because of the dif- ferent orientations of the four cameras, this data set is complementary to the former (with the PC field of view in common), thus offering full coverage of the innermost regions of the cluster (see Figure 1). The combined photometric sample is ideal for efficiently studying both the hot stellar populations (as the BSS and the HB stars) and the cool red giant branch (RGB) population, and to guarantee a proper combination with the wide-field data set (see below). The photometric reduction of both the high-resolution sets was carried out using RO- MAFOT (Buonanno et al. 1983), a package developed to perform accurate photometry in crowded fields and specifically optimized to handle under-sampled Point Spread Functions (PSFs; Buonanno & Iannicola 1989), as in the case of the HST-WFC chips. The standard procedure described in Ferraro et al. (1997, 2001) was adopted to derive the instrumen- tal magnitudes and to calibrate them to the STMAG system by using the zero-points of Holtzman et al. (1995). The magnitude lists were finally cross-correlated in order to obtain a combined catalog. 2. The wide-field set - A complementary set of wide-field U, B, and I images was secured by using the Wide Field Imager (WFI) at the 2.2m ESO-MPI telescope, during an observing run in January 1999 (Progr. ID: 062.L-0354, PI: Ferraro). A set of WFI V images (Progr. ID: 064.L-0255) was also retrieved from the ESO-STECF Science Archive. Additional deep wide-field images were obtained in the UV band with the satellite GALEX – 5 – (GI-056, P.I. Schiavon) through the FUV (1350–1750 Å) and NUV (1750–2800 Å) detectors. With a global field of view (FoV) of 34′ × 34′, the WFI observations cover the entire cluster extension. There is also full coverage of the cluster in the UV thanks to the large GALEX FoV, which is approximately 1 deg in diameter and includes theWFI FoV (see Figure 2, where the cluster is roughly centered on WFI CCD #2). However, because of the low resolution of the instrument (4′′ and 6′′ in the FUV and NUV channels, respectively), GALEX data have been used to sample only the external cluster regions not covered by HST. The raw WFI images were corrected for bias and flat field, and the overscan regions were trimmed using IRAF2 tools (mscred package). Standard crowded field photometry, including PSF modeling, was carried out independently on each image using DAOPHOTII/ALLSTAR (Stetson 1987). For each WFI chip a catalog listing the instrumental U, B, V, and I magnitudes was obtained by cross-correlating the single-band catalogs. Several hundred stars in common with Kravtsov et al. (1997), Stetson (2000), and Ferraro et al. (1992) have been used to transform the instrumental U , B, V , and I magnitudes to the Johnson/Cousins photometric system. As for the WFI data, also for GALEX observations standard photometry and PSF fitting were performed independently on each image using DAOPHOTII/ALLSTAR. A combined FUV-NUV catalog was then obtained by cross-correlating the single-band catalogs. 2.2. Astrometry and homogenization of the catalogs The HST, WFI, and GALEX catalogs have been placed on the absolute astrometric system by adopting the procedure already described in Ferraro et al. (2001, 2003). The new astrometric Guide Star Catalog (GSC-II3) was used to search for astrometric standard stars in the WFI FoV, and a cross-correlation tool specifically developed at the Bologna Observatory (Montegriffo et al. 2003, private communication) has been employed to obtain an astrometric solution for each WFI chip. Several hundred GSC-II reference stars were found in each chip, thus allowing an accurate absolute positioning of the stars. Then, we used more than 3000 and 1500 bright WFI stars in common with the HST and GALEX samples, respectively, as secondary astrometric standards, so as to place all the catalogs on the same absolute astrometric system. We estimate that the global uncertainties in the 2IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Associa- tion of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 3Available at http://www-gsss.stsci.edu/Catalogs/GSC/GSC2/GSC2.htm. http://www-gsss.stsci.edu/Catalogs/GSC/GSC2/GSC2.htm – 6 – astrometric solution is of the order of ∼ 0.′′2, both in right ascension (α) and declination (δ). Once placed on the same coordinate system, the catalogs have been cross-correlated and the stars in common have been used to transform all the magnitudes in the same photometric system. In particular, the HST STMAG magnitudes have been converted to the WFI ones by using the stars in common between the two samples in the optical bands. Then, the GALEX FUV and NUV instrumental magnitudes have been calibrated onto the HST m160 and m218 magnitudes, respectively using the stars in common between the GALEX and HST samples. At the end of the procedure a homogeneous master catalog of magnitudes and absolute coordinates of all the stars included in the HST, WFI, and GALEX samples was finally produced. 2.3. Center of gravity and definition of the samples Once the absolute positions of individual stars have been obtained, the center of gravity Cgrav of NGC 1904 has been determined by averaging the coordinates α and δ of all stars lying in the PC FoV, following the iterative procedure described in Montegriffo et al. (1995, see also Ferraro et al. 2003, 2004). In order to correct for spurious effects due to incompleteness in the very inner regions of the cluster, we considered two samples with different limiting magnitudes (V < 19 and V < 20), and we computed the barycenter of stars for each sample. The two estimates agree within ∼ 1′′, setting Cgrav at α(J2000) = 05 h 24m 11.s09, δ(J2000) = −24o 31′ 29.′′00. The newly determined center of gravity is located at ∼ 7′′ south- est (∆α = 7.′′3, ∆δ = −2′′) from that previously derived by Harris (1996) on the basis of the surface brightness distribution. In order to reduce spurious effects in the most crowded regions of the cluster due to the low resolution of the WFI and GALEX observations, we considered only the HST data for the inner 85′′ from the center, this value being imposed by the geometry of the combined WFPC2 FoVs (see Figure 1). Thus, in the following we define as HST sample the ensemble of all the stars observed with HST at r ≤ 85′′ from Cgrav, and as External sample all the stars detected with WFI and/or GALEX at r > 85′′, out to ∼ 1100′′ (see Figure 2). The CMDs of the HST and External samples in the (V, B − V ) planes are shown in Figure 3. Note that only the data suitable for the study of the BSS population will be considered in the following, while those obtained through filters F160BW and FUV on board HST and GALEX, respectively, will be used in a forthcoming paper specifically devoted to the analysis the HB properties (Schiavon et al. 2007). – 7 – 2.4. Density profile Considering all the stars brighter than V = 20 in the combined HST+External catalog (see Figure 3), we have determined the projected density profile of NGC 1904 by direct star counts over the entire cluster extension. Following the procedure already described in Ferraro et al. (1999a, 2004), we have divided the entire sample in 31 concentric annuli, each centered on Cgrav and split in an adequate number of sub-sectors (quadrants for the annuli totally sampled by the observations, octants elsewhere). The number of stars lying within each sub-sector was counted, and the star density was obtained by dividing these values by the corresponding sub-sector areas. The stellar density in each annulus was then obtained as the average of the sub-sector densities, and the standard deviation was estimated from the variance among the sub-sectors. The radial density profile thus derived is plotted in Figure 4, and the average of the three outermost (r > 8.′3) surface density measures has been adopted as the background contribution (corresponding to 0.95 arcmin−2). Figure 4 also shows the mono-mass King model that best fits the derived density profile, with the corresponding values of the core radius and concentration being rc ≃ 9. ′′7 (with a typical error of ∼ ±2′′) and c = 1.71, respectively (hence, the tidal radius is rt ≃ 500 ′′ ≃ 50 rc). These values are in good agreement with those quoted by Harris (1996, rc = 9. ′′6 and c = 1.72), Trager, Djorgovski & King (1993, rc = 9. ′′55 and c = 1.72), and McLaughlin & van der Marel (2005, rc = 10. ′′3 and c = 1.68), derived from the surface brightness profile, and they confirm that NGC 1904 has not yet experienced core collapse. By assuming a distance modulus (m−M)0 = 15.63 (distance d ∼ 13.37 kpc, Ferraro et al. 1999b), the derived value of rc corresponds to ∼ 0.65 pc. By summing the luminosities of stars with V ≤ 20 observed within ∼ 4′′, we estimate that the extinction-corrected central surface brightness of the cluster is µV,0(0) ≃ 16.20 mag/arcsec in good agreement with Harris (1996, µV,0 = 16.23), Djorgovski (1993, µV,0 = 16.15), and McLaughlin & van der Marel (2005, µV,0 = 16.18). Following the procedure described in Djorgovski (1993, see also Beccari et al. 2006), we derive log ν0 ≃ 3.97, where ν0 is the central luminosity density in units of L⊙/pc 3 (for comparison, log ν0 = 4.0 in Harris 1996; Djorgovski 1993; McLaughlin & van der Marel 2005). 3. THE BSS POPULATION OF NGC 1904 3.1. BSS selection At UV wavelengths BSS are among the brightest objects in a GC, and RGB stars are particularly faint. By combining these advantages with the high-resolution capability of HST, – 8 – the usual problems associated with photometric blends and crowding in the high density central regions of GCs are minimized, and BSS can be most reliably recognized and separated from the other populations in the UV CMDs. For these reasons our primary criterion for the definition of the BSS sample is based on the position of stars in the (m218, m218 − B) plane (see also Ferraro et al. 2004, for a detailed discussion of this issue). In order to avoid incompleteness bias and the possible contamination from TO and sub-giant branch stars, we have adopted a limiting magnitude m218 = 19.5, roughly corresponding to 1 magnitude brighter than the cluster TO. The resulting BSS selection box in the UV CMD is shown in Figure 5. Once selected in the UV CMD, all the BSS lying in the field in common with the optical-HST sample have been used to define the selection box in the (V, B − V ) and (V, U − V ) planes. The limiting magnitude in the V band is V ≃ 18.9, and the adopted BSS selection boxes in these planes are shown in Figures 3 and 6 (only stars not observed in HST-Pointing B are shown in the latter). With these criteria we have identified 39 BSS in NGC 1904: 37 in the HST sample (32 from HST-Pointing B, and 5 from HST-Pointing A) and 2 in the External sample (r > 85′′), the most distant lying at r ≃ 270′′ (∼ 4.′5) from the cluster center (see Figure 2). All candidate BSS have been confirmed by visual inspection, evaluating the quality and the precision of the PSF fitting. This procedure significantly reduces the possibility of introducing spurious objects, such as blends, background galaxies, etc., in the sample. The coordinates and magnitudes of all the identified BSS are listed in Table 1. In order to study the radial distribution of BSSs, one needs to compare their number counts as a function of radius with those of a population assumed to trace the radial density distribution of normal cluster stars. We chose to use HB stars for that purpose, given their high luminosities and relatively large number. Thanks to the (essentially blue) HB morphology, such a population can be easily selected in all CMDs, and the adopted selection boxes, designed to include the bulk of HB and the few post-HB stars, are shown in Figures 5–6. In order to be conservative, a few stars lying within the adopted HB selection boxes in the optical bands, but not detected in the UV filters (GALEX-NUV channel and HST- F218W filter), have been excluded from the following analysis. However slightly different boxes or the inclusion of these stars in the sample have no effects on the results. With these criteria we have identified 249 HB stars (197 at r ≤ 85′′ from the HST sample, and 52 at 85′′ < r ≤ rt from the External sample). – 9 – 3.2. BSS mass distribution The position of BSS in the CMD can be used to derive a ”photometric ” estimate of their masses through the comparison with theoretical isochrones. We did this in the (V, B − V ) plane, where 34 BSS (32 from the HST-Pointing B and 2 from the External sample) out of the 39 identified in the cluster have been measured. A set of isochrones of appropriate metallicity (Z = 6 × 10−4) has been extracted from the data-base of Cariulo, Degl’Innocenti & Castellani (2003) and transformed into the ob- servational plane by adopting a reddening E(B − V ) = 0.01 (Ferraro et al. 1999b). The 12 Gyr isochrone nicely reproduces the main cluster population, while the region of the CMD populated by the BSS is well spanned by a set of isochrones with ages ranging from 1 to 6 Gyr (see Figure 7). Thus, the entire dataset of isochrones available in this age range (stepped at 0.5 Gyr) has been used to derive a grid linking the BSS colors and magnitudes to their masses. Each BSS has been projected on the closest isochrone and a value of its mass has been derived. As shown in the lower panel of Figure 7, BSS masses range from ∼ 0.95 to ∼ 1.6M⊙, and both the mean and the median of distribution correspond to 1.2 M⊙. The TO mass turns out to be MTO = 0.8M⊙. 3.3. The BSS radial distribution The radial distribution of BSS identified in NGC 1904 has been studied following the same procedure previously adopted for other clusters (see references in Ferraro 2006; Beccari et al. 2006). In Figure 8 we compare the BSS cumulative radial distribution to that of HB stars. The two distributions are obviously different, with the BSS being more centrally concentrated than HB stars. A Kolmogorov-Smirnov test gives a ∼ 7×10−4 probability that they are extracted from the same population, i.e. the two populations are different at more than 3σ level. For a more quantitative analysis, the surveyed area has been divided into 6 concentric annuli, the first roughly corresponding to the core radius (r = 10′′), and the others chosen in order to sample approximately the same fraction of the cluster luminosity out to the tidal radius (rt ≃ 500 ′′). The luminosity in each annulus has been calculated by integrating the surface density profile shown in Figure 4. The number of BSS and HB stars (NBSS and NHB, respectively), as well as the fraction of sampled luminosity (Lsamp) measured in each annulus are listed in Table 2 and have been used to compute the population ratio NBSS/NHB and the – 10 – specific frequencies (see Ferraro et al. 2003): Rpop = (Npop/N (Lsamp/L tot ) , (1) with pop = BSS, HB. The resulting radial trend of RHB over the surveyed area is essentially constant, with a value close to unity (see Figure 9). This is just what expected on the basis of the stellar evolu- tion theory, which predicts that the fraction of stars in any post-main sequence evolutionary stage is strictly proportional to the fraction of the sampled luminosity (Renzini & Fusi Pecci 1988). In contrast the BSS show a completely different radial distribution: as shown in Figure 9, the specific frequency RBSS is highly peaked at the cluster center decreases to a minimum at r ≃ 12 rc and remains approximately constant outwards. The same behavior is clearly visible also in Figure 10, where the population ratio NBSS/NHB is plotted as a function of r/rc. 3.4. Dynamical simulations Following the same approach as Mapelli et al. (2004, 2006) and Lanzoni et al. (2007), we have used a Monte-Carlo simulation code (originally developed by Sigurdsson & Phinney 1995) in order to reproduce the observed radial distribution and to derive some clues about the BSS formation mechanisms. Such a code follows the dynamical evolution of N BSS within a background cluster, taking into account the effects of both dynamical friction and distant encounters. Since stellar collisions are most probable in the central high-density regions of the clusters, in the simulations we define COL-BSS those objects with initial positions ri ∼ rc. Since primordial binaries most likely evolve in isolation if they orbit in the cluster outskirts, we identify as MT-BSS those BSS having ri ≫ rc. Within these defintions, in any given run we assume that a certain fraction of the N simulated BSS is made of COL-BSS and the remaining fraction of MT-BSS. The initial positions ri of the two types of BSS are randomly generated within the appropriate radial range (ri ∼ rc for COL-BSS, and ri ≫ rc for the others) following a flat distribution, according to the fact that the number of stars in a King model scales as dN = n(r) dV ∝ r−2πr2dr ∝ dr. Their initial velocities are randomly extracted from the cluster velocity distribution illustrated in Sigurdsson & Phinney (1995), and an additional natal kick is assigned to COL-BSS to account for the recoil induced by the three-body encounters that trigger the merger and produce the BSS (see, e.g., Sigurdsson, Davies & Bolte 1994; Davies, Benz & Hills 1994). Each BSS has characteristic mass M and maximum lifetime tlast. We follow their dynamical evolution in the (fixed) gravitational potential for a time ti (i = 1, N), where each ti is a – 11 – randomly chosen fraction of tlast. At the end of the simulation we register the final positions of BSS, and we compare their radial distribution with the observed one. The percentage of COL- and MT-BSS is changed and the procedure repeated until a reasonable agreement between the simulated and the observed distributions is reached. For a more detailed discussion of the procedure and the ranges of values appropriate for the input parameters we refer to Mapelli et al. (2006). Here we only list the assumptions made in the present study: – the background cluster has been approximated with a multi-mass King model, deter- mined as the best fit to the observed profile4. The cluster central velocity dispersion is set to σ = 3.9 km s−1 (Dubath, Meylan & Mayor 1997), and, assuming 0.5M⊙ as the average mass of the cluster stars, the central stellar density is nc = 3 × 10 4 pc−3 (Pryor & Meylan 1993); – BSS masses have been fixed toM = 1.2M⊙ (see Section 3.2) and characteristic lifetimes tlast ranging between 1.5 and 4 Gyr have been considered; – COL-BSS have been distributed with initial positions ri ≤ rc and have been given a natal kick velocity of 1× σ; – initial positions ranging between 5 rc and rt have been considered for MT-BSS in dif- ferent runs; – in each simulation we have followed the evolution of N = 10, 000 BSS. The simulated radial distribution that best reproduces the observed one (with a reduced χ2 ≃ 0.1) is shown in Figure 10 and is obtained by assuming that the totality of BSS is made of COL-BSS. In the best-fit case the BSS characteristic lifetime is tlast ≃ 1.5 Gyr, but a variation between 1 and 4 Gyr of this parameter still leads to a very good agreement (χ2 ≃ 0.2–0.3) with the observations. For the sake of comparison, in Figure 10 we also show the results of the simulations obtained by assuming a percentage of MT-BSS ranging from 10% to 40% (see lower and upper boundaries of the gray region, respectively)5. As can be 4By adopting the same mass groups as those of Mapelli et al. (2006), the resulting value of the King dimensionless central potential is W0 = 10 5Note that a population of 40% MT-BSS was needed in order to reproduce the bimodal distribution observed in M 3, 47 Tuc and NGC 6752 (Mapelli et al. 2006), and 10% was found to be the appropriate percentage of MT-BSS in the case of M 5 (Lanzoni et al. 2007). – 12 – seen, while a population of 10% MT-BSS is still marginally consistent with the observations, larger percentages systematically overestimate the BSS population at r >∼ 5 rc. Increasing the BSS mass up to 1.5M⊙ does not change this conclusion. By assuming 12 Gyr for the age of NGC 1904, we have used the simulations and the dynamical friction timescale (from, e.g., Mapelli et al. 2006) for 1.2M⊙ stars to estimate the radius of avoidance ravoid of the cluster, i.e., the radius within which all these stars are expected to have already sunk to the cluster core because of mass segregation processes. We find that ravoid ∼ 30 rc (i.e., ∼ 300 ′′), which corresponds to a significant fraction of the entire cluster extension. This evidence is consistent with the fact that the simulated MT-BSS appear to be a negligible fraction of the overall BSS population. 4. DISCUSSION We have studied the brightest portion (m218 ≤ 19.5) of the BSS population in NGC 1904. We have found a total of 39 objects, with a high degree of segregation in the cluster center. Approximately 38% of the entire BSS population is found within the cluster core, while only ∼ 13% of HB stars are counted in the same region. This indicates a significant overabundance of BSS in the center, as also confirmed by the fact that the BSS specific frequency RBSS within rc is roughly 3 times larger than expected for a normal (non-segregated) population on the basis of the sampled light (see Figure 9). The peak value is in good agreement with what is found in the case of M 3, 47 Tuc, NGC 6752 and M 5 (see Ferraro et al. 2004; Sabbi et al. 2004; Lanzoni et al. 2007). Unlike these clusters, no significant upturn of the distribution at large radii has been detected in NGC 1904. We emphasize that the absence of an external upturn in the BSS radial distribution is not an effect of low statistics. In the case of NGC 6752, where a similar amount of BSS (34) has been detected, the BSS radial distribution is clearly bimodal (Sabbi et al. 2004). This can be seen also in Figure 11, where the two distributions are directly compared. They nicely agree within r ∼ 12rc, but the fraction of BSS in NGC 6752 rises again at larger distances from the center, despite the smaller number of BSS observed in this cluster compared to NGC 1904. Extensive dynamical simulations have been used to derive some hints about the BSS formation mechanisms. Even if admittedly crude, this approach has been successfully used to demonstrate that the external rising branch of the BSS radial distribution observed in M 3, 47 Tuc, NGC 6752 and M 5 cannot be due to COL-BSS originated in the core and then kicked out in the outer regions: hence, a significant fraction (20-40%) of the overall population is – 13 – required to be made of MT-BSS in these clusters (Mapelli et al. 2006; Lanzoni et al. 2007). By using the same simulations to interpret the (flat) BSS radial distribution of NGC 1904, we found that only a negligible percentage (0–10%) of MT-BSS is needed. However, we emphasize that if a rising peripheral BSS frequency is absent (as in the case of NGC 1904) our simple approach cannot distinguish between BSS created by MT (and then segregated into the cluster core by the dynamical friction) and COL-BSS created by collisions inside the core. On the other hand, the negligible fraction of peripheral MT-BSS found in NGC 1904 is in agreement with the quite large value of the radius of avoidance estimated for this cluster (ravoid ≃ 30 rc), which indicates that all the heavy stars (binaries) within this radial distance have had enough time to sink to the core and are therefore not expected in the cluster outskirts. Such a radial distance corresponds to 0.6 rt, i.e., it represents a significant fraction of the cluster extension (only 1% of the cluster light is contained between ravoid and rt), and hence only a small fraction of the massive objects are expected to be unaffected by the dynamical friction). In all the other studied cases, ravoid is significantly smaller: ravoid ∼ 0.2 rt (Mapelli et al. 2006; Lanzoni et al. 2007). In turn, this suggests that at least a fraction of the BSS population that we now observe in the cluster center are primordial binaries which have sunk to the core because of the dynamical friction process, and mixed with those that formed through stellar collisions. Only systematic surveys of physical and chemical properties for a large number of BSS in different environments (see examples in De Marco et al. 2005; Ferraro et al. 2006a) can definitively identify the formation processes of these stars. This research was supported by Agenzia Spaziale Italiana under contract ASI-INAF I/023/05/0, by the Istituto Nazionale di Astrofisica under contract PRIN/INAF 2006, and by the Ministero dell’Istruzione, dell’Università e della Ricerca. RTR is partially funded by NASA through grant number HST-GO-10524 from the Space Telescope Science Institute. REFERENCES Bailyn, C. D., & Pinsonneault, M. H. 1995, ApJ, 439, 705 Bailyn, C. D. 1995, ARA&A, 33, 133 Beccari, G., Ferraro, F. 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R., Piotto, G., & Capaccioli, M., 1997, A&A, 327, 1004 This preprint was prepared with the AAS LATEX macros v5.2. – 17 – Fig. 1.—Map of the combined HST sample. The light solid and dotted lines delimit the FoVs of Pointing B and A, respectively. Star positions are plotted with respect to the center of gravity Cgrav derived in Section 2.3: α(J2000) = 05 h 24m 11..s09, δ(J2000) = −24o 31′ 29.′′00. The positions of all BSS identified in this sample are marked with heavy dots and the concentric annuli used to study their radial distribution (cfr. Table 1) are also shown. The inner and outer annuli correspond to r = rc = 10 ′′ and r = 85′′, respectively. – 18 – Fig. 2.— Map of the External sample. The light solid and dotted lines delimit the WFI and the GALEX FoVs, respectively. The two BSS detected in the External sample are marked as heavy dots, and the concentric annuli used to study their radial distribution are shown as heavy circles. The inner annulus is at 85′′ and corresponds to the most external one in Figure 1. The heavy dashed circle marks the tidal radius of the cluster (rt ≃ 500 – 19 – Fig. 3.— (V, B − V ) CMDs of the HST (Pointing B) and External samples. The hatched regions (V ≥ 20) indicate the stars not used to derive the cluster surface density profile. The adopted BSS and HB selection boxes are shown, and all the identified BSS are marked with the empty circles. – 20 – Fig. 4.— Observed surface density profile (dots and error bars) and best-fit King model (solid line). The radial profile is in units of number of stars per square arcsec. The dotted line indicates the adopted level of the background, and the model characteristic parameters (core radius rc, concentration c, dimensionless central potential W0), as well as the χ 2 value of the fit are marked in the figure. The location of the cluster tidal radius is marked by the arrow. The lower panel shows the residuals between the observations and the fitted profile at each radial coordinate. – 21 – Fig. 5.— CMD of the ultraviolet (Pointing B) HST sample. The adopted magnitude limit and selection box used for the definition of the BSS population (empty circles) are shown. The two solid triangles correspond to BSS-38 and 39 found in the External Sample, with UV magnitudes obtained through the GALEX NUV detector. The selection boxes adopted for HB and post-HB stars are also shown. – 22 – Fig. 6.— (V, U − V ) CMD of the HST (Pointing A) sample (only stars not observed in Pointing B are plotted). The adopted BSS and HB selection boxes are shown, and all the identified BSS and HB stars are marked with the empty circles and squares, respectively. – 23 – Fig. 7.— Upper panel: zoomed (V, B − V ) CMD of the BSS region; the 34 BSS measured in this plane are shown. The set of isochrones ranging from 1 to 6 Gyr (stepped by 0.5 Gyr) from Cariulo, Degl’Innocenti & Castellani (2003) data base used to derive BSS masses is also shown. Lower panel: derived mass distribution for the BSS shown in the upper panel. – 24 – Fig. 8.— Cumulative radial distribution of BSS (solid line) and HB (dashed line) stars as a function of the projected distance from the cluster center for the combined HST+External sample. The location of the cluster tidal radius is marked by the arrow. – 25 – Fig. 9.— Radial distribution of the BSS (dots) and HB (gray regions) specific frequencies, as defined in equation (1), and as a function of the radial distance in units of the core radius. The vertical size of the gray regions correspond to the error bars. – 26 – Fig. 10.— Radial distribution of the population ratio NBSS/NHB as a function of r/rc (dots with error bars), compared with the simulated distribution (solid line and triangles) obtained by assuming 100% of COL-BSS. The results of the simulations obtained by assuming a percentage of MT-BSS ranging from 10% to 40% (lower and upper boundaries of the gray region, respectively) are also shown. – 27 – Fig. 11.— Radial distribution of the population ratio NBSS/NHB for NGC 1904 (filled circles) and NGC 6752 (open circles) plotted as a function of the radial distance in core radius units. – 28 – Table 1. The BSS population in NGC1904 Name RA DEC m218 U B V I [degree] [degree] BSS-1 81.048797100 -24.526391100 19.11 18.64 18.66 18.45 - BSS-2 81.047782800 -24.526527400 18.65 18.12 18.21 17.94 - BSS-3 81.047540400 -24.526005300 18.85 18.34 18.38 18.22 - BSS-4 81.048954000 -24.525188200 17.94 17.75 17.87 17.78 - BSS-5 81.047199200 -24.525797600 17.58 17.47 17.40 17.35 - BSS-6 81.045528300 -24.525876800 18.64 18.24 18.28 18.12 - BSS-7 81.044296100 -24.526362200 19.13 18.77 18.76 18.55 - BSS-8 81.048506700 -24.524266800 19.18 18.55 18.64 18.39 - BSS-9 81.041556100 -24.526972400 19.35 18.65 18.86 18.49 - BSS-10 81.045827300 -24.525062700 18.06 17.35 17.17 16.96 - BSS-11 81.045467700 -24.525147900 17.84 17.44 17.56 17.41 - BSS-12 81.047088300 -24.524375900 18.80 18.20 18.01 17.80 - BSS-13 81.046548400 -24.524483700 19.43 18.77 18.67 18.38 - BSS-14 81.045121300 -24.524748300 19.28 18.33 18.65 18.21 - BSS-15 81.045883300 -24.524278600 18.97 18.42 18.30 18.13 - BSS-16 81.046164500 -24.522567300 18.41 17.72 17.63 17.42 - BSS-17 81.044313700 -24.523304200 18.66 18.46 18.44 18.27 - BSS-18 81.047157500 -24.521982700 18.49 18.45 18.41 18.32 - BSS-19 81.043418100 -24.523247200 17.88 18.16 17.68 17.55 - BSS-20 81.046665900 -24.520192000 19.49 18.71 18.92 18.60 - BSS-21 81.044991400 -24.520127500 18.93 18.45 18.44 18.21 - BSS-22 81.046157800 -24.519245100 19.49 18.92 19.06 18.74 - BSS-23 81.049326100 -24.521621900 18.08 - 18.01 17.99 - BSS-24 81.049155900 -24.520629500 18.41 - 17.98 17.87 - BSS-25 81.047244500 -24.517052600 19.41 18.97 19.11 18.82 - BSS-26 81.051592200 -24.523146600 18.67 - 18.33 18.21 - BSS-27 81.050476100 -24.517107000 19.01 18.65 18.54 18.39 - BSS-28 81.060767000 -24.520983900 18.92 - 18.59 18.45 - BSS-29 81.068117100 -24.517558600 19.39 - 18.91 18.60 - BSS-30 81.043233100 -24.533852000 17.34 - 16.75 16.65 - BSS-31 81.044917900 -24.540315000 18.34 - 17.48 17.22 - BSS-32 81.038520800 -24.540891600 17.77 17.46 17.30 17.20 - BSS-33 81.045196533 -24.515874863 - 17.91 - 17.84 - BSS-34 81.032196045 -24.512256622 - 18.75 - 18.41 - BSS-35 81.037574768 -24.525493622 - 18.87 - 18.58 - BSS-36 81.041069031 -24.518457413 - 18.80 - 18.64 - BSS-37 81.045227051 -24.517648697 - 18.86 - 18.68 - BSS-38 81.056510925 -24.449676514 19.04† 18.78 18.56 18.35 18.08 – 29 – Table 1—Continued BSS-39 81.058883667 -24.555763245 19.39† 19.14 19.05 18.73 18.34 Note. — † Note that, while the header of the column referes to HST-F218W magnitudes, those of BSS-38 and -39 have been obtained with the NUV channel of GALEX and transformed to the m218 scale as described in Section 2.2. ri re NBSS NHB L samp/L 0 10 15 34 0.14 10 20 10 45 0.18 20 40 7 62 0.22 40 85 5 56 0.23 85 150 1 34 0.13 150 500 1 18 0.10 Table 2: Number of BSS and HB stars, and fraction of luminosity sampled in the 6 concentric annuli used to study the BSS radial distribution of NGC 1904 (ri and re correspond to the internal and external radius of each considered annulus, in arcsec). INTRODUCTION OBSERVATIONS AND DATA ANALYSIS The data sets Astrometry and homogenization of the catalogs Center of gravity and definition of the samples Density profile THE BSS POPULATION OF NGC 1904 BSS selection BSS mass distribution The BSS radial distribution Dynamical simulations DISCUSSION

0704.1394

Calculating Valid Domains for BDD-Based Interactive Configuration

Calculating Valid Domains for BDD-Based Interactive Configuration Tarik Hadzic, Rune Moller Jensen, Henrik Reif Andersen Computational Logic and Algorithms Group, IT University of Copenhagen, Denmark [email protected],[email protected],[email protected] Abstract. In these notes we formally describe the functionality of Calculating Valid Domains from the BDD representing the solution space of valid configu- rations. The formalization is largely based on the CLab [1] configuration frame- work. 1 Introduction Interactive configuration problems are special applications of Constraint Satisfaction Problems (CSP) where a user is assisted in interactively assigning values to variables by a software tool. This software, called a configurator, assists the user by calculating and displaying the available, valid choices for each unassigned variable in what are called valid domains computations. Application areas include customising physical products (such as PC’s and cars) and services (such as airplane tickets and insurances). Three important features are required of a tool that implements interactive configu- ration: it should be complete (all valid configurations should be reachable through user interaction), backtrack-free (a user is never forced to change an earlier choice due to incompleteness in the logical deductions), and it should provide real-time performance (feedback should be fast enough to allow real-time interactions). The requirement of obtaining backtrack-freeness while maintaining completeness makes the problem of calculating valid domains NP-hard. The real-time performance requirement enforces further that runtime calculations are bounded in polynomial time. According to user- interface design criteria, for a user to perceive interaction as being real-time, system response needs to be within about 250 milliseconds in practice [2]. Therefore, the cur- rent approaches that meet all three conditions use off-line precomputation to generate an efficient runtime data structure representing the solution space [3,4,5,6]. The chal- lenge with this data structure is that the solution space is almost always exponentially large and it is NP-hard to find. Despite the bad worst-case bounds, it has nevertheless turned out in real industrial applications that the data structures can often be kept small [7,5,4]. 2 Interactive Configuration The input model to an interactive configuration problem is a special kind of Constraint Satisfaction Problem (CSP) [8,9] where constraints are represented as propositional formulas: http://arxiv.org/abs/0704.1394v1 Definition 1. A configuration model C is a triple (X,D,F ) where X is a set of vari- ables {x0, . . . , xn−1}, D = D0 × . . . × Dn−1 is the Cartesian product of their finite domains D0, . . . , Dn−1 and F = {f0, ..., fm−1} is a set of propositional formulae over atomic propositions xi = v, where v ∈ Di, specifying conditions on the values of the variables. Concretely, every domain can be defined as Di = {0, . . . , |Di| − 1}. An assign- ment of values v0, . . . , vn−1 to variables x0, . . . , xn−1 is denoted as an assignment ρ = {(x0, v0), . . . , (xn−1, vn−1)}. Domain of assignment dom(ρ) is the set of vari- ables which are assigned: dom(ρ) = {xi | ∃v ∈ Di.(xi, v) ∈ ρ} and if dom(ρ) = X we refer to ρ as a total assignment. We say that a total assignment ρ is valid, if it satisfies all the rules which is denoted as ρ |= F . A partial assignment ρ′, dom(ρ′) ⊆ X is valid if there is at least one total assign- ment ρ ⊇ ρ′ that is valid ρ |= F , i.e. if there is at least one way to successfully finish the existing configuration process. Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or blue), the size (small, medium, or large) and the print (”Men In Black” - MIB or ”Save The Whales” - STW). There are two rules that we have to observe: if we choose the MIB print then the color black has to be chosen as well, and if we choose the small size then the STW print (including a big picture of a whale) cannot be selected as the large whale does not fit on the small shirt. The configuration problem (X,D,F ) of the T- shirt example consists of variables X = {x1, x2, x3} representing color, size and print. Variable domains are D1 = {black ,white, red , blue}, D2 = {small ,medium , large}, and D3 = {MIB , STW }. The two rules translate to F = {f1, f2}, where f1 = (x3 = MIB) ⇒ (x1 = black ) and f2 = (x3 = STW ) ⇒ (x2 6= small). There are |D1||D2||D3| = 24 possible assignments. Eleven of these assignments are valid configurations and they form the solution space shown in Fig. 1. ♦ (black , small ,MIB) (black , large, STW ) (red , large,STW ) (black ,medium,MIB) (white,medium,STW ) (blue,medium,STW ) (black ,medium,STW ) (white, large ,STW ) (blue, large, STW ) (black , large,MIB) (red ,medium,STW ) Fig. 1. Solution space for the T-shirt example 2.1 User Interaction Configurator assists a user interactively to reach a valid product specification, i.e. to reach total valid assignment. The key operation in this interaction is that of computing, for each unassigned variable xi ∈ X \dom(ρ), the valid domainD i ⊆ Di. The domain is valid if it contains those and only those values with which ρ can be extended to be- come a total valid assignment, i.e. Dρi = {v ∈ Di | ∃ρ ′ : ρ′ |= F ∧ρ∪{(xi, v)} ⊆ ρ The significance of this demand is that it guarantees the user backtrack-free assignment to variables as long as he selects values from valid domains. This reduces cognitive effort during the interaction and increases usability. At each step of the interaction, the configurator reports the valid domains to the user, based on the current partial assignment ρ resulting from his earlier choices. The user then picks an unassigned variable xj ∈ X \ dom(ρ) and selects a value from the calculated valid domain vj ∈ D j . The partial assignment is then extended to ρ ∪ {(xj , vj)} and a new interaction step is initiated. 3 BDD Based Configuration In [5,10] the interactive configuration was delivered by dividing the computational ef- fort into an offline and online phase. First, in the offline phase, the authors compiled a BDD representing the solution space of all valid configurations Sol = {ρ | ρ |= F}. Then, the functionality of calculating valid domains (CV D) was delivered online, by efficient algorithms executing during the interaction with a user. The benefit of this ap- proach is that the BDD needs to be compiled only once, and can be reused for multiple user sessions. The user interaction process is illustrated in Fig. 2. InCo(Sol, ρ) 1: while |Solρ| > 1 2: compute D ρ = CVD(Sol, ρ) 3: report D to the user 4: the user chooses (xi, v) for some xi 6∈ dom(ρ), v ∈ D 5: ρ← ρ ∪ {(xi, v)} 6: return ρ Fig. 2. Interactive configuration algorithm working on a BDD representation of the so- lutions Sol reaches a valid total configuration as an extension of the argument ρ. Important requirement for online user-interaction is the guaranteed real-time expe- rience of user-configurator interaction. Therefore, the algorithms that are executing in the online phase must be provably efficient in the size of the BDD representation. This is what we call the real-time guarantee. As the CV D functionality is NP-hard, and the online algorithms are polynomial in the size of generated BDD, there is no hope of pro- viding polynomial size guarantees for the worst-case BDD representation. However, it suffices that the BDD size is small enough for all the configuration instances occurring in practice [10]. 3.1 Binary Decision Diagrams A reduced ordered Binary Decision Diagram (BDD) is a rooted directed acyclic graph representing a Boolean function on a set of linearly ordered Boolean variables. It has one or two terminal nodes labeled 1 or 0 and a set of variable nodes. Each variable node is associated with a Boolean variable and has two outgoing edges low and high. Given an assignment of the variables, the value of the Boolean function is determined by a path starting at the root node and recursively following the high edge, if the associated variable is true, and the low edge, if the associated variable is false. The function value is true, if the label of the reached terminal node is 1; otherwise it is false. The graph is ordered such that all paths respect the ordering of the variables. A BDD is reduced such that no pair of distinct nodes u and v are associated with the same variable and low and high successors (Fig. 3a), and no variable node u has iden- tical low and high successors (Fig. 3b). Due to these reductions, the number of nodes u v u x x x (a) (b) Fig. 3. (a) nodes associated to the same variable with equal low and high successors will be converted to a single node. (b) nodes causing redundant tests on a variable are eliminated. High and low edges are drawn with solid and dashed lines, respectively in a BDD for many functions encountered in practice is often much smaller than the number of truth assignments of the function. Another advantage is that the reductions make BDDs canonical [11]. Large space savings can be obtained by representing a col- lection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are shared. Due to the canonicity, two BDDs are identical if and only if they have the same root. Consequently, when using this representation, equivalence checking between two BDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any Boolean operation on two BDDs can be carried out in time proportional to the product of their size. The size of a BDD can depend critically on the variable ordering. To find an optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for choosing an ordering is to locate dependent variables close to each other in the order- ing. For a comprehensive introduction to BDDs and branching programs in general, we refer the reader to Bryant’s original paper [11] and the books [12,13]. 3.2 Compiling the Configuration Model Each of the finite domain variables xi with domain Di = {0, . . . , |Di| − 1} is encoded by ki = ⌈log|Di|⌉ Boolean variables x 0, . . . , x . Each j ∈ Di, corresponds to a binary encoding v0 . . . vki−1 denoted as v0 . . . vki−1 = enc(j). Also, every combina- tion of Boolean values v0 . . . vki−1 represents some integer j ≤ 2 ki − 1, denoted as j = dec(v0 . . . vki−1). Hence, atomic proposition xi = v is encoded as a Boolean ex- pression xi0 = v0 ∧ . . . ∧ x = vki−1. In addition, domain constraints are added to forbid those assignments to v0 . . . vki−1 which do not translate to a value in Di, i.e. where dec(v0 . . . vki−1) ≥ |Di|. Let the solution space Sol over ordered set of variables x0 < . . . < xk−1 be repre- sented by a Binary Decision Diagram B(V,E,Xb, R, var), where V is the set of nodes u, E is the set of edges e and Xb = {0, 1, . . . , |Xb| − 1} is an ordered set of variable indexes, labelling every non-terminal node u with var(u) ≤ |Xb| − 1 and labelling the terminal nodes T0, T1 with index |Xb|. Set of variable indexes Xb is constructed by taking the union of Boolean encoding variables i=0 {x 0, . . . , x } and ordering them in a natural layered way, i.e. xi1j1 < x iff i1 < i2 or i1 = i2 and j1 < j2. Every directed edge e = (u1, u2) has a starting vertex u1 = π1(e) and ending vertex u2 = π2(e). R denotes the root node of the BDD. Example 2. The BDD representing the solution space of the T-shirt example introduced in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables: x1, x2 and x3, whose domain sizes are four, three and two, respectively. Each variable is repre- sented by a vector of Boolean variables. In the figure the Boolean vector for the vari- able xi with domain Di is (x i , x i , · · ·x i ), where li = ⌈lg |Di|⌉. For example, in the figure, variable x2 which corresponds to the size of the T-shirt is represented by the Boolean vector (x02, x 2). In the BDD any path from the root node to the terminal node 1, corresponds to one or more valid configurations. For example, the path from the root node to the terminal node 1, with all the variables taking low values represents the valid configuration (black , small ,MIB). Another path with x01, x 1, and x 2 taking low values, and x12 taking high value represents two valid configurations: (black ,medium,MIB) and (black ,medium, STW ), namely. In this path the variable x03 is a don’t care variable and hence can take both low and high value, which leads to two valid configurations. Any path from the root node to the terminal node 0 corresponds to invalid configura- tions. ♦ 4 Calculating Valid Domains Before showing the algorithms, let us first introduce the appropriate notation. If an index k ∈ Xb corresponds to the j + 1-st Boolean variable x j encoding the finite domain variable xi, we define var1(k) = i and var2(k) = j to be the appropriate mappings. Now, given the BDD B(V,E,Xb, R, var), Vi denotes the set of all nodes u ∈ V that are labelled with a BDD variable encoding the finite domain variable xi, i.e. Vi = {u ∈ V | var1(u) = i}. We think of Vi as defining a layer in the BDD. We define Ini to be the set of nodes u ∈ Vi reachable by an edge originating from outside the Vi layer, i.e. Ini = {u ∈ Vi| ∃(u ′, u) ∈ E. var1(u ′) < i}. For the root node R, labelled with i0 = var1(R) we define Ini0 = Vi0 = {R}. We assume that in the previous user assignment, a user fixed a value for a finite domain variable x = v, x ∈ X , extending the old partial assignment ρold to the current Fig. 4. BDD of the solution space of the T-shirt example. Variable xji denotes bit vj of the Boolean encoding of finite domain variable xi. assignment ρ = ρold ∪ {(x, v)}. For every variable xi ∈ X , old valid domains are denoted as Dρoldi , i = 0, . . . , n− 1. and the old BDD B ρold is reduced to the restricted BDD, Bρ(V,E,Xb, var). The CV D functionality is to calculate valid domains D for remaining unassigned variables xi 6∈ dom(ρ) by extracting values from the newly restricted BDD Bρ(V,E,Xb, var). To simplify the following discussion, we will analyze the isolated execution of the CV D algorithms over a given BDD B(V,E,Xb, var). The task is to calculate valid domains V Di from the starting domains Di. The user-configurator interaction can be modelled as a sequence of these executions over restricted BDDs Bρ, where the valid domains are Dρi and the starting domains are D The CV D functionality is delivered by executing two algorithms presented in Fig. 5 and Fig. 6. The first algorithm is based on the key idea that if there is an edge e = (u1, u2) crossing over Vj , i.e. var1(u1) < j < var1(u2) then we can include all the values from Dj into a valid domain V Dj ← Dj . We refer to e as a long edge of length var1(u2) − var1(u1). Note that it skips var(u2)− var(u1) Boolean variables, and therefore compactly represents the part of a solution space of size 2var(u2)−var(u1). For the remaining variables xi, whose valid domain was not copied by CV D − Skipped, we execute CV D(B, xi) from Fig. 6. There, for each value j in a domain D we check whether it can be part of the domain Di. The key idea is that if j ∈ Di then there must be u ∈ Vi such that traversing the BDD from u with binary encoding of j CV D − Skipped(B) 1: for each i = 0 to n− 1 2: L[i]← i+ 1 3: T ← TopologicalSort(B) 4: for each k = 0 to |T | − 1 5: u1 ← T [k], i1 ← var1(u1) 6: for each u2 ∈ Adjacent[u1] 7: L[i1]← max{L[i1], var1(u2)} 8: S ← {}, s← 0 9: for i = 0 to n− 2 10: if i+ 1 < L[s] 11: L[s]← max{L[s], L[i+ 1]} 12: else 13: if s+ 1 < L[s] S ← S ∪ {s} 14: s← i+ 1 15: for each j ∈ S 16: for i = j to L[j] 17: V Di ← Di Fig. 5. In lines 1-7 the L[i] array is created to record longest edge e = (u1, u2) orig- inating from the Vi layer, i.e. L[i] = max{var1(u ′) | ∃(u, u′) ∈ E.var1(u) = i}. The execution time is dominated by TopologicalSort(B) which can be implemented as depth first search in O(|E|+ |V |) = O(|E|) time. In lines 8-14, the overlapping long segments have been merged in O(n) steps. Finally, in lines 15-17 the valid domains have been copied in O(n) steps. Hence, the total running time is O(|E|+ n). CV D(B, xi) 1: V Di ← {} 2: for each j = 0 to |Di| − 1 3: for each k = 0 to |Ini| − 1 4: u← Ini[k] ′ ← Traverse(u, j) 6: if u ′ 6= T0 7: V Di ← V Di ∪ {j} 8: Return Fig. 6. Classical CVD algorithm. enc(j) denotes the binary encoding of number j to ki values v0, . . . , vki−1. If Traverse(u, j) from Fig. 7 ends in a node different then T0, then j ∈ V Di. will lead to a node other than T0, because then there is at least one satisfying path to T1 allowing xi = j. Traverse(u, j) 1: i← var1(u) 2: v0, . . . , vki−1 ← enc(j) 3: s← var2(u) 4: if Marked[u] = j return T0 5: Marked[u]← j 6: while s ≤ ki − 1 7: if var1(u) > i return u 8: if vs = 0 u← low(u) 10: else u← high(u) 12: if Marked[u] = j return T0 13: Marked[u]← j 14: s← var2(u) Fig. 7. For fixed u ∈ V, i = var1(u), Traverse(u, j) iterates through Vi and returns the node in which the traversal ends up. When traversing with Traverse(u, j) we mark the already traversed nodes ut with j, Marked[ut] ← j and prevent processing them again in the future j-traversals Traverse(u′, j). Namely, if Traverse(u, j) reached T0 node through ut, then any other traversal Traverse(u′, j) reaching ut must as well end up in T0. Therefore, for every value j ∈ Di, every node u ∈ Vi is traversed at most once, leading to worst case running time complexity of O(|Vi| · |Di|). Hence, the total running time for all variables is O( i=0 |Vi| · |Di|). The total worst-case running time for the two CV D algorithms is thereforeO( i=0 |Vi|· |Di|+ |E|+ n) = O( i=0 |Vi| · |Di|+ n). References 1. Jensen, R.M.: CLab: A C++ library for fast backtrack-free interactive product configuration. http://www.itu.dk/people/rmj/clab/ (2007) 2. Raskin, J.: The Humane Interface. Addison Wesley (2000) 3. Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in dynamic CSPs-application to configuration. Artificial Intelligence 1-2 (2002) 199–234 ftp://fpt.irit.fr/pub/IRIT/RPDMP/Configuration/. 4. Madsen, J.N.: Methods for interactive constraint satisfaction. Master’s thesis, Department of Computer Science, University of Copenhagen (2003) 5. Hadzic, T., Subbarayan, S., Jensen, R.M., Andersen, H.R., Møller, J., Hulgaard, H.: Fast backtrack-free product configuration using a precompiled solution space representation. In: PETO Conference, DTU-tryk (2004) 131–138 6. Møller, J., Andersen, H.R., Hulgaard, H.: Product configuration over the internet. In: Pro- ceedings of the 6th INFORMS Conference on Information Systems and Technology. (2002) 7. Configit Software A/S. http://www.configit-software.com (online) 8. Tsang, E.: Foundations of Constraint Satisfaction. Academic Press (1993) 9. Dechter, R.: Constraint Processing. Morgan Kaufmann (2003) 10. Subbarayan, S., Jensen, R.M., Hadzic, T., Andersen, H.R., Hulgaard, H., Møller, J.: Com- paring two implementations of a complete and backtrack-free interactive configurator. In: CP’04 CSPIA Workshop. (2004) 97–111 11. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers 8 (1986) 677–691 12. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design. Springer (1998) 13. Wegener, I.: Branching Programs and Binary Decision Diagrams. Society for Industrial and Applied Mathematics (SIAM) (2000) Calculating Valid Domains for BDD-Based Interactive Configuration Tarik Hadzic, Rune Moller Jensen, Henrik Reif Andersen

0704.1395

Higgs and Z' Phenomenology in B-L extension of the Standard Model at LHC

arXiv:0704.1395v1 [hep-ph] 11 Apr 2007 Preprint typeset in JHEP style - HYPER VERSION Higgs and Z ′ Phenomenology in B − L extension of the Standard Model at LHC W. Emam and S. Khalil Center for Theoretical Physics at the British University in Egypt, Sherouk City, Cairo 11837, Egypt. Faculty of Science, Ain Shams University, Cairo 11566, Egypt. Abstract: The phenomenology of the low scale U(1)B−L extension of the standard model and its implications at LHC is presented. In this model, an extra gauge boson corresponding to B−L gauge symmetry and an extra SM singlet scalar (heavy Higgs) are predicted. We show a detailed analysis of both heavy and light Higgses decay and production in addition to the possible decay channels of the new gauge boson. We find that the cross sections of the SM-like Higgs production are reduced by ∼ 20% − 30%, while its decay branching ratios remain intact. The extra Higgs has relatively small cross sections and the branching ratios of Z ′ → l+l− are of order ∼ 20% compared to ∼ 3% of the SM resuls. Hence, the search for Z ′ is accessible via a clean dilepton signal at LHC. Keywords: Low scale B − L, Higgs production, Higgs decays, Z ′ gauge boson. http://arxiv.org/abs/0704.1395v1 http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. B − L extension of the SM 2 2.1 Symmetry breaking 2 2.2 Higgs sector 3 3. Higgs Production and Decay at Hadron Colliders 5 3.1 Higgs Production 5 3.2 Higgs Decay 9 4. Z ′ decay in B − L extension of the SM 12 5. Light H ′ Scenario 14 6. Conclusions 14 1. Introduction The Standard Model (SM) of elementary particles has been regarded only as a low energy effective theory of the yet-more-fundamental theory. Several attempts have been proposed to extend the gauge symmetry of the SM via one or more U(1) gauge symmetries beyond the hypercharge gauge symmetry, U(1)Y [1–3]. The evidence for non-vanishing neutrino masses, based on the apparent observation of neutrino oscillation, strongly encourages this type of extensions. In this class of models [1, 2], three SM singlet fermions arise quite naturally due to the anomaly cancellation conditions. These three particles are accounted for right handed neutrinos, and hence a natural explanation for the seesaw mechanism is obtained. A low scale B − L symmetry breaking, based on the gauge group GB−L ≡ SU(3)C × SU(2)L × U(1)Y × U(1)B−L, has been considered recently [2]. It was shown that this model can account for the current experimental results of the light neutrino masses and their large mixing. Therefore, it can be considered as one of the strong candidates for minimal extensions of the SM. In addition, one extra neutral gauge boson corresponding to B − L gauge symmetry and an extra SM singlet scalar (extra Higgs) are predicted. In fact, the SM Higgs sector can be generally extended by adding extra singlet scalars without enlarging its gauge symmetry group [4, 5]. In Ref.[2], it has been emphasized that these new particles may have significant impact on the SM phenomenology, hence lead to interesting signatures at Large Hadron Collider (LHC). – 1 – The aim of this paper is to provide a comprehensive analysis for the phenomenology of such TeV scale extension of the SM, and its potential discovery at the LHC. The production cross sections and the decay branching ratios of the SM like Higgs, H, and the extra Higgs boson H ′ are analyzed. We also consider the decay branching ratios of the extra gauge boson, Z ′. We show that the cross sections of the Higgs production are reduced by ∼ 20%− 30% in the interesting mass range of ∼ 120− 250 GeV relative to the SM predictions. However, its decay branching ratios remain intact. In addition, we find that the extra Higgs (∼ TeV) is accessible at LHC, although it has relatively small cross sections. We also examine the availability of the decay channel H ′ → HH, which happens to have very small partial decay width. Concerning the Z ′ gauge boson, the branching ratios of Z ′ → l+l− are found to be of order ∼ 20% compared to ∼ 3% of the SM BR(Z → l+l−). This paper is organized as follows. In section 2 we review the Higgs mechanism and symmetry breaking within the minimal B − L extension of the SM. We also discuss the mixing between the SM-like Higgs and the extra Higgs boson. Section 3 is devoted for the phenomenology of the two Higgs particles. The production cross sections and decay branching ratios of these Higgs particles at LHC are presented. In section 4 we study the decay of the extra gauge boson Z ′. In section 5 we briefly discuss the scenario of very light Higgs. Finally we give our concluding remarks in section 6. 2. B − L extension of the SM 2.1 Symmetry breaking The fermionic and kinetic sectors of the Lagrangian in the case of B − L extension are given by LB−L = i l̄Dµγµl + i ēRDµγµeR + i ν̄RDµγµνR µν − 1 µν − 1 µν . (2.1) The covariant derivative Dµ is different from the SM one by the term ig ′′YB−LCµ, where g is the U(1)B−L gauge coupling constant, YB−L is the B−L charge, and Cµν = ∂µCν−∂νCµ is the field strength of the U(1)B−L. The YB−L for fermions and Higgs are given in Table 1. particle l eR νR q φ χ YB−L −1 −1 −1 1/3 0 2 Table 1: B − L quantum numbers for fermions and Higgs particles The Higgs and Yukawa sectors of the Lagrangian are given by LB−L = (Dµφ)(Dµφ) + (Dµχ)(Dµχ)− V (φ, χ) λe l̄φeR + λν l̄φ̃νR + λνR ν̄ RχνR + h.c. . (2.2) – 2 – Here, λe, λν and λνR refer to 3 × 3 Yakawa matrices. The interaction terms λν lφ̃νR and λνR ν̄ RχνR give rise to a Dirac neutrino mass term: mD ≃ λνv and a Majorana mass term: MR = λνRv ′, respectively. The U(1)B−L and SU(2)L × U(1)Y gauge symmetries can be spontaneously broken by a SM singlet complex scaler field χ and a complex SU(2) doublet of scalar fields φ, respectively. We consider the most general Higgs potential invariant under these symmetries, which is given by V (φ, χ) = m21φ †φ+m22χ †χ+ λ1(φ †φ)2 + λ2(χ +λ3(χ †χ)(φ†φ), (2.3) where λ3 > −2 λ1λ2 and λ1, λ2 ≥ 0, so that the potential is bounded from below. For non-vanishing vacuum expectation values (vev’s), we require λ23 < 4λ1λ2 , m 1 < 0 and m22 < 0. The vev’s, |〈φ〉| = v/ 2 and |〈χ〉| = v′/ 2, are then given by 1 − 2λ3m22 λ23 − 4λ1λ2 , v′2 = −2(m21 + λ1v2) Depending on the value of the λ3 coupling, one can have v ′ ≫ v or v′ ≈ v. Therefore, the symmetry breaking scales, v and v′, can be responsible for two different symmetry breaking scenarios. In our analysis we take v = 246 GeV and constrain the other scale, v′, by the lower bounds imposed on the mass of the extra neutral gauge boson. After the B−L gauge symmetry breaking, the gauge field Cµ (will be called Z ′ in the rest of the paper) acquires the following mass: m2Z′ = 4g ′′v′2. (2.4) The experimental search for Z ′ at CDF experiment leads to mZ′ >∼ O(600) GeV. However, the strongest limit comes from LEP II [6]: mZ′/g ′′ > 6TeV . (2.5) This implies that v′ >∼ O(TeV). Moreover, if the coupling g ′′ is < O(1), one can still obtain mZ′ >∼ O(600) GeV. 2.2 Higgs sector In addition to the SM complex SU(2)L doublet, another complex scalar singlet arise in this class of models. Out of these six scalar degrees of freedom, only two physical degrees of freedom, (φ, χ), remain after the B − L and electroweak symmetries are broken. The other four degrees of freedom are eaten by Z ′, Z and W± bosons. The mixing between the two Higgs scalar fields is controlled by the coupling λ3. In fact, one finds that for positive λ3 , the B−L symmetry breaking scale, v′, becomes much higher than the electroweak symmetry breaking scale, v. In this case, the SM singlet Higgs, φ, and the SM like Higgs, χ, are decoupled and their masses are given by 2λ1v, Mχ = ′. (2.6) – 3 – For negative λ3, however, theB−L breaking scale is at the same order of the the electroweak breaking scale. In this scenario, a significant mixing between the two Higgs scalars exists and can affect the SM phenomenology. This mixing can be represented by the following mass matrix for φ and χ: M2(φ, χ) = vv′ λ2v . (2.7) Therefore, the mass eigenstates fields H and H ′ are given by cos θ − sin θ sin θ cos θ , (2.8) where the mixing angle θ is defined by tan 2θ = |λ3|vv′ λ1v2 − λ2v′2 . (2.9) The masses of H and H ′ are given by m2H,H′ = λ1v 2 + λ2v (λ1v2 − λ2v′2)2 + λ23v2v′2. (2.10) We call H andH ′ as light and heavy Higgs bosons, respectively. In our analysis we consider a maximum mixing between the two Higgs bosons by taking |λ3| ≃ λmax1 λmax2 , where λmax1 and λmax2 are given by λmax1 = m2H +m 4m2Hm + 1 + 1 λmax2 = m2H +m 4m2Hm + 1− 1 , (2.11) and the maximum mixing angle is then given by tan 2θ = λmax1 λ λmax1 v 2 − λmax2 v′2 . (2.12) By considering the maximum mixing and fixing v = 246 GeV and v′ = 1 TeV, we have reduced the number of free parameters of this model into just two, namely mH and mH′ . In Figure 1, we present the maximum mixing as a function of the light Higgs mass, mH for mH′ = 500 GeV and 1 TeV. Due to the mixing between the two Higgs bosons, the usual couplings among the SM- like Higgs, H, and the SM fermions and gauge bosons are modified. In addition, there are new couplings among the extra Higgs, H ′, and the SM particles: gHff = i cos θ, gH′ff = i sin θ, gHV V = −2i cos θ, gH′V V = −2i sin θ, gHZ′Z′ = 2i sin θ, gH′Z′Z′ = −2i cos θ, gHνRνR = −i sin θ, gH′νRνR = i cos θ. (2.13) – 4 – 100 200 300 400 500 600 700 800 900 1000 θ2cos (GeV)Hm =500(GeV)H’m =1000(GeV)H’m Figure 1: H −H ′ mixing angle as function of mH for m′H = 500 GeV and 1 TeV. The Higgs self couplings are give by gH3 = 6i(λ1v cos 3 θ − v′ cos2 θ sin θ), gH′3 = 6i(λ2v ′ cos3 θ + v cos2 θ sin θ), gH4 = 6iλ1 cos gH′4 = 6iλ2 cos gHH′2 = 2i( v cos3 θ + λ3v ′ cos2 θ sin θ − 3λ2v′ cos2 θ sin θ), gH2H′ = 2i( v′ cos3 θ − λ3v cos2 θ sin θ + 3λ1v cos2 θ sin θ), gH2H′2 = iλ3 cos 4 θ. (2.14) These new couplings lead to a different Higgs phenomenology from the well known one, predicted by the SM. The detailed analysis of Higgs bosons in this class of models and their phenomenological implications, like their productions and decays at the LHC, will be discussed in the next section. 3. Higgs Production and Decay at Hadron Colliders 3.1 Higgs Production At the LHC, two 7-TeV proton beams with a center-of-mass energy of 14 TeV and a luminosity of 1034cm−2s−1 will collide with each other. The machine is expected to start running early 2008. The detection of the SM Higgs boson is the primary goal of the LHC project. – 5 – H, H ′ H, H ′ H, H ′ H, H ′ Figure 2: The dominant Higgs boson production mechanisms in hadronic collisions. At hadron colliders, the two Higgs bosons couple mainly to the heavy particles: the massive gauge bosons Z ′, Z and W± and the heavy quarks t, b. The main production mechanisms for Higgs particles can be classified into four groups [7]: the gluon–gluon fusion mechanism[8], the associated Higgs production with heavy top or bottom quarks[9], the associated production with W/Z/Z ′ bosons[10], and the weak vector boson fusion processes[11]: gg → H (3.1) gg, qq̄ → QQ̄+H, (3.2) qq̄ → V +H (3.3) qq → V ∗V ∗ → qq +H. (3.4) The Feynman diagrams of these processes are displayed in Figure 2. The cross sections of the Higgs production in these four mechanisms are directly proportional to the the Higgs couplings with the associated particles. In case of the gluon–gluon fusion mechanism the Higgs production is mediated by triangular loops of heavy quarks. Thus, the cross section of this process is proportional to the Higgs coupling with the heavy quark mass. In case of B − L extension of the SM, the production cross sections for the light Higgs, H, and the heavy Higgs, H ′, can be approximated as σH ∝ α2s cos2 θ , (3.5) σH′ ∝ α2s sin2 θ , (3.6) where the first bracket is due to the coupling QQH(H ′), while the second bracket corre- sponds to an approximated loop factor. As can be seen from Equations 3.5 and 3.6, the – 6 – H, H ′ Figure 3: Feynman diagrams for Higgs production in association with heavy quarks in hadronic collisions, pp → qq̄, gg → QQ̄H , at LO. cross section of the light Higgs production is reduced respect to the SM one by the factor of cos2 θ. On the other hand, the heavy Higgs production is suppressed by two factors: the small sin θ, and the large mH′ . Therefore, the the heavy Higgs production is typically less than that of the light Higgs by two orders of magnitudes, i.e., ≃ sin θ cos θ2 ≃ O(10−2). (3.7) Now, we consider the mechanism of Higgs production in association with heavy quark pairs, Equation 3.2. In addition to the Feynman diagram shown in Figure 2, a set of other diagrams that also contribute to this process is given in Figure 3. Note that although this process shares the same coupling with the gluon-gluon fusion process, the leading order expression of its cross section indicates that it is less by one order of magnitude, for mH(H′) < 1 TeV. Furthermore, the typical ratio of σ(gg → H ′QQ̄) to σ(gg → HQQ̄) is of order (sin θ/ cos θ)2 ≃ O(0.1). Finally, we study the Higgs production in association with W/Z/Z ′ bosons and in the weak vector boson fusion processes, Equations 3.3 and 3.4 respectively. In B−L extension of the SM, the cross sections of these channels are proportional to the mass of the gauge boson and the mixing angle θ of the two Higgs bosons: V ≡ W/Z : σH ∝ cos2 θ × × Loop function, (3.8) σH′ ∝ sin2 θ × × Loop function. (3.9) In case of V ≡ Z ′, The production is enhanced by the HZ ′Z ′ coupling arising with mZ′ . However, it is suppressed by a large value of v ′ and the mass of the virtual gauge boson(s), mZ′ : V ≡ Z ′ : σH ∝ sin2 θ × (g′′Y × Loop function, (3.10) σH′ ∝ cos2 θ × (g′′Y × Loop function. (3.11) From these equations, one can observe that the relative ratio between the light Higgs production associated with W/Z and Z ′ gauge bosons is given by σH(W/Z)/σH(Z – 7 – [pb]σ (GeV)Hm H→gg Hqq→qq WH→qq ZH→qq Z’H→qq Ht t→,ggqq H+X)[pb]→(ppσ =14 TeVs MRST/NLO =1 TeVH’m Figure 4: The cross sections of the light Higgs production as function of mH : 100 GeV ≤ mH ≤ 1 TeV, for mH′ = 1 TeV. cos2 θ/ sin2 θ×g′′2/g2(g′2). Therefore, σH(W/Z) can be larger than σH(Z ′) by one order of magnitude at most. In contrary, the situation is reversed for the heavy Higgs production and one finds that σ′H(Z ′) > σ′H(W/Z), which confirms our earlier discussion. The cross sections for the Higgs bosons production in these channels (Equations 3.1- 3.4) have been calculated using the FORTRAN codes: HIGLU, HQQ, V2HV, and VV2HV, respectively [12]. Extra subroutines have been added to these programs for the new cou- plings associated with the two higgs scalars and the extra gauge boson [12]. As inputs, we use v = 246 GeV, v′ = 1 TeV, and center of mass energy s = 14 TeV. We also fix the mass of the extra gauge boson at mZ′ = 600 GeV. The cross sections for the light Higgs boson production are summarized in Figure 4. as functions of the light Higgs mass with mH′ = 1 TeV. Figure 5, on the other hand, represents the heavy Higgs productions as functions of mH′ with mH = 200 GeV. As shown in Figure 4, the salient feature of this low scale B − L extension is that all cross sections of the light Higgs production are reduced by about 25 − 35% in the interesting mass range: mH < 250 GeV. As in the SM, the main contribution to the production cross section comes from the gluon-gluon fusion mechanism with a few tens of pb. The next relevant contribution is given by the Higgs production in the weak vector boson mechanism, Equation 3.4. This contribution is at the level of a few pb, as estimated above. Furthermore, the production associated with Z/W is dominant over the production associated with Z ′ for mH < 300 GeV. Now, we analyze the production of the heavy Higgs. It turns out that its cross sections are smaller than the light Higgs ones. As shown in Figure 5, all these cross sections are scaled down by factor O(10−2), which is consistent with the result obtained in Equation 3.7. Unlike the light Higgs scenario, the production associated with Z ′ is dominant over the – 8 – [pb]σ (GeV)H’m H’→gg H’qq→qq WH’→qq ZH’→qq Z’H’→qq H’t t→,ggqq H’+X)[pb]→(ppσ =14 TeVs MRST/NLO =200 GeVHm Figure 5: The cross sections of the heavy Higgs production as function of mH′ : 300 GeV ≤ mH′ ≤ 1 TeV, for mH = 200 GeV. production associated with Z/W in agrement with our previous prediction. 3.2 Higgs Decay The Higgs particle tends to decay into the heaviest gauge bosons and fermions allowed by the phase space. The Higgs decay modes can be classified into three categories: Higgs decays into fermions (Figure 6), Higgs decays into massive gauge bosons (Figure 7), and Higgs decays into massless gauge bosons (Figure 8). H, H ′ Figure 6: The Feynman diagram for the Higgs boson decays into fermions. The decay widths into fermions are directly proportional to the Hff couplings Γ(H −→ ff) ≈ mH cos2 θ, (3.12) Γ(H ′ −→ ff) ≈ mH′ sin2 θ. (3.13) – 9 – H, H ′ V H, H ′ H, H ′ Figure 7: Diagrams for the Higgs boson decays into massive gauge bosons. H, H ′ H, H ′ H, H ′ Figure 8: Loop induced Higgs boson decays into a) two photons (Zγ) and b) two gluons. H, H ′ H, H ′ W Figure 9: Diagrams for the three–body decays of the Higgs boson into tbW final states. In case of the top quark, three-body decays into on-shell and off-shell states (Figure 9) were taken into consideration. On the the hand, the decay widths into massive gauge bosons V = Z ′, Z,W are directly proportional to the HV V couplings. This includes two-body, three-body, and four-body decays V ≡ W/Z : ΓH ≈ cos2 θ, ΓH′ ≈ sin2 θ, (3.14) V ≡ Z ′ : ΓH ∝ sin2 θ, ΓH′ ∝ cos2 θ. (3.15) As shown in Figure 8, the massless gauge bosons are not directly coupled to the Higgs bosons, but they are coupled via W, charged fermions, and quark loops. This implies that – 10 – 100 200 300 400 500 600 700 800 900 1000 (GeV)Hm Figure 10: The branching ratios of the light Higgs decay as function of mH for mH′ = 1 TeV. the decay widths are in turn proportional to the HV V and Hff couplings, hence they are relatively suppressed. From the above Equations, one finds that all decay widths of the light Higgs are proportional to cos2 θ, except the new decay mode of Z ′Z ′. Furthermore, this channel has a very small contribution to the total decay width. Therefore, the light Higgs branching ratios (the ratios between the partial decay widths and the total decay width) have small dependence on the mixing parameter θ. Thus, it is expected to see no significant difference between the results of the light Higgs branching ratios in this model of B − L extension and the SM ones. On the other hand, the heavy Higgs branching ratios have relevant dependence on θ. The decay widths and branching ratios of the Higgs bosons in these channels have been calculated using the FORTRAN code: HDECAY with extra subroutines for the new couplings associated with the two higgs scalars and the extra gauge boson [12, 13]. As in the Higgs production analysis, we use the following inputs: v = 246 GeV, v′ = 1 TeV, mZ′ = 600 GeV, and c.m. energy s = 14 TeV. The decay branching ratios of the light and heavy Higgs bosons are shown in Figures 10 and 11, respectively, as functions of the Higgs masses. As expected, the branching ratios of the light Higgs are very close to the SM ones. In the “low mass”range: 100 GeV < MH < 130 GeV, the main decay mode is H → bb̄ with a branching ratio of ∼ 75 − 50% . The decays into τ+τ−and cc̄ pairs come next with branching ratios of order ∼ 7 − 5% and ∼ 3 − 2%, respectively. The γγ and Zγ decays are rare, with very small branching ratios. In the “High mass ”range: mH > 130 GeV, the WW , ZZ, and to some extent the tt̄ decays give the dominant contributions. The Z ′Z ′ decay arises for quite large Higgs mass with a small branching ratio <∼ 1%. – 11 – 200 400 600 800 1000 1200 1400 1600 1800 2000 (GeV)H’m Figure 11: The branching ratios of the heavy Higgs decay as function of mH′ for mH = 200 GeV Regarding the heavy Higgs decay branching ratio, one finds that H ′ → WW and ZZ are the dominant decay modes, with a branching ratio of ∼ 70% and ∼ 20%, respectively. To a lower extent, the tt̄ and Z ′Z ′ account for the remaining branching ratios. Note that these two decay modes are in particular sensitive to the running mixing angles. Thus, they have the behaviors shown in Figure 11. The other modes give very tiny contributions and hence they are not shown in this figure. It is useful to mention that the heavy Higgs may decay to a pair of the lighter Higgs. The partial decay width of this channel, which can be expressed by Γ(H ′ −→ HH) ≈ 4m2H′ , (3.16) is suppressed by the tiny gH2H′ coupling (Equation 2.14) and the relatively large mH′ . In fact, the resulting branching ratio of this decay mode is at the level of 10−8, and hence does not appear in Figure 11. 4. Z ′ decay in B − L extension of the SM In this section we study the decay of the extra gauge boson predicted by the B−L extension of the SM at LHC. In fact, there are many models which contain extra gauge bosons [6, 14]. These models can be classified into two categories depending on whether or not they arise in a GUT scenario. In some of these models, the Z ′ and the SM Z are not true mass eigenstates due to mixing. This mixing induces the couplings between the extra Z ′ boson and the SM fermions. However, there is a stringent experimental limit on the mixing parameter. In our model of B−L extension of the SM, there is no tree-level Z−Z ′mixing. – 12 – 500 600 700 800 900 1000 1100 1200 1300 1400 (GeV)Z’m l→Z’ s,sc,cb b→Z’ t t→Z’ Figure 12: The decay branching ratios of the extra gauge boson Z ′ as function of mZ′ . Nevertheless, the extra B − L Z ′ boson and the SM fermions are coupled through the non-vanishing B − L quantum numbers. The interactions of the Z ′ boson with the SM fermions are described by LZ′int = B−L g ′′ Z ′µ fγ µf. (4.1) The decay widths of Z ′ → f f̄ are then given by [6] Γ(Z ′ → l+l−) ≈ (g′′Y lB−L) Γ(Z ′ → qq̄) ≈ (g′′Y , q ≡ b, c, s Γ(Z ′ → tt̄) ≈ (g′′Y (4.2) Figures 12 shows the decay branching ratios of Z ′ as a function of mZ′ . Contrarily to the SM Z decay, the branching ratios of Z ′ → l+l− are relatively high compared to Z ′ → qq̄. This is due to the fact that |Y lB−L| = 3|Y B−L|. Thus, one finds BR(Z ′ → l+l−) ≃ 20% compared with BR(Z → l+l−) ≃ 3%. Therefore, searching for Z ′ can be easily accessible via a clean dilepton signal, which can be one of the first new physics signatures to be observed at the LHC. – 13 – 5. Light H ′ Scenario In this section we discuss the possibility of having mH′ <∼ mH and the phenomenological implications of this scenario. As shown in section two, the mass of the non-SM Higgs mH′ receives a dominant contribution from the vev of the B − L symmetry breaking v′ and the self coupling λ2. The Z ′ searches and the neutrino masses impose a lower limit on v′: v′ >∼ 1 TeV. The self coupling λ2 is essentially unconstrained parameter. If λ2 ∼ O(1), then mH′ is of order TeV as assumed in the previous sections. There are two other interesting possibilities which have recently received some attention in the literature. The first one corresponds to the case of λ1v 2 ∼ λ2v′2, i.e., λ2 ∼ O(10−2). Therefore, one finds mH ∼ mH′ and the mixing angle is given by θ ∼ π/4. Hence, the two Higgs H and H ′ couple similarly to the fermion and gauge fields, giving the same production cross section and decay branching ratio. Therefore, the distinguish between H and H ′ at LHC in this type of models is rather difficult. This scenario is usually known as intense Higgs coupling [15]. The second possibility concerns the case of λ2 <∼ 10 −3, in which one obtains mH′ ≪ mH . In fact, LEP and Tevatron direct searches do not exclude a light Higgs boson with a mass below 60 GeV. Such light Higgs may have escaped experimental detection due to the suppression of its cross sections. Therefore, a window with a very light Higgs mass still exist. Having λ2 <∼ 10 −3 implies that λ3 is also less than 10 −3. In this respect, the Higgs masses are approximately given by λ1v, (5.1) mH′ ≃ O ≃ O(10−2)GeV, (5.2) and the coupling gHH′H′ in Equation 2.14 becomes very small. Thus, the decay H → H ′H ′ is not comparable to the decay into other SM particles. The phenomenology of this scenario, derived from different SM extensions, has been studied in details [4],[16]. In addition, this light scalar particle has been considered as an interesting candidate for dark matter [17]. 6. Conclusions In this paper we have considered the TeV scale B − L extension of the SM. We provided a comprehensive analysis for the phenomenology of the SM like Higgs, the extra Higgs scalar, and the extra gauge boson predicted in this model, with special emphasize on their potential discovery at the LHC. We have shown that the cross sections of the SM-like Higgs production are reduced by ∼ 20% − 30% in the mass range of ∼ 120 − 250 GeV compared to the SM results. On the other hand, the implications of the B − L extension to the SM do not change the decay branching ratios. Moreover, we found that the extra Higgs has relatively small cross sections, but it is accessible at LHC. Finally, we showed that the branching ratios of Z ′ → l+l− are of order ∼ 20% compared to ∼ 3% of the SM BR(Z → l+l−). Hence, searching for Z ′ is accessible via a clean dilepton signal at LHC. – 14 – Acknowledgment This work is partially supported by the ICTP under the OEA-project-30. References [1] R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980); R. E. Marshak and R. N. Mohapatra, Phys. Lett. B 91, 222 (1980); C. Wetterich, Nucl. Phys. B 187, 343 (1981); A. Masiero, J. F. Nieves and T. Yanagida, Phys. Lett. B 116, 11 (1982); R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 27, 254 (1983); R. E. Marshak and R. N. Mohapatra, S. S. Rao, W. Buchmuller, C. Greub and P. Minkowski, Phys. Lett. B 267, 395 (1991). [2] S. Khalil, arXiv:hep-ph/0611205. [3] D. G. Cerdeno, A. Dedes and T. E. J. Underwood, JHEP 0609 (2006) 067 [arXiv:hep-ph/0607157]; W. F. Chang, J. N. Ng and J. M. S. Wu, arXiv:hep-ph/0701254. [4] D. O’Connell, M. J. Ramsey-Musolf and M. B. Wise, Phys. Rev. D 75, 037701 (2007) [arXiv:hep-ph/0611014]. [5] A. Datta and A. Raychaudhuri, Phys. Rev. D 57, 2940 (1998) [arXiv:hep-ph/9708444]. [6] M. Carena, A. Daleo, B. A. Dobrescu and T. M. P. Tait, Phys. Rev. D 70, 093009 (2004). [7] A. Djouadi, arXiv:hep-ph/0503172. [8] H. M. Georgi, S. L. Glashow, M. E. Machacek and D. V. Nanopoulos, Phys. Rev. Lett. 40, 692 (1978). [9] R. Raitio and W. W. Wada, Phys. Rev. D 19, 941 (1979); J. N. Ng and P. Zakarauskas, Phys. Rev. D 29, 876 (1984); D. A. Dicus and S. Willenbrock, Phys. Rev. D 39, 751 (1989). [10] S. L. Glashow, D. V. Nanopoulos and A. Yildiz, Phys. Rev. D 18 (1978) 1724; J. Finjord, G. Girardi and P. Sorba, Phys. Lett. B 89, 99 (1979); E. Eichten, I. Hinchliffe, K. D. Lane and C. Quigg, Rev. Mod. Phys. 56, 579 (1984) [Addendum-ibid. 58, 1065 (1986)]. [11] D. A. Dicus and S. S. D. Willenbrock, Phys. Rev. D 32, 1642 (1985); R. N. Cahn and S. Dawson, Phys. Lett. B 136, 196 (1984) [Erratum-ibid. B 138, 464 (1984)]; W. Kilian, M. Kramer and P. M. Zerwas, Phys. Lett. B 373, 135 (1996) [arXiv:hep-ph/9512355]. [12] The FORTRAN codes: HIGLU, HQQ, V2HV, and VV2HV have been written by M. Spira and can be found at http://people.web.psi.ch/spira/proglist.html. The modefied version was made by W. Emam and can be found at http://www.bue.edu.eg/centres/ctp/people/codes/proglist.html. [13] A. Djouadi, J. Kalinowski and M. Spira, Comput. Phys. Commun. 108, 56 (1998) [arXiv:hep-ph/9704448]. [14] J. L. Hewett and T. G. Rizzo, Phys. Rep. 183, 139 (1989); M. Cvetic and S. Godfrey, arXiv:hep-ph/9504216; A. Leike, Phys. Rept. 317, 143 (1999). [15] E. Boos, A. Djouadi, M. Muhlleitner and A. Vologdin, Phys. Rev. D 66, 055004 (2002) [arXiv:hep-ph/0205160]. [16] M. Krawczyk, P. Mattig and J. Zochowski, Eur. Phys. J. C 19, 463 (2001). [arXiv:hep-ph/0009201]. [17] C. Boehm, J. Orloff and P. Salati, Phys. Lett. B 641, 247 (2006) [arXiv:astro-ph/0607437]. – 15 –

0704.1396

What Can be Learned Studying the Distribution of the Biggest Fragment ?

WHAT CAN BE LEARNED STUDYING THE DISTRIBUTION OF THE BIGGEST FRAGMENT ? E. BONNET , F. GULMINELLI , B. BORDERIE , N. LE NEINDRE M.F. RIVET The INDRA and ALADIN Collaborations: 1Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud 11, F-91406 Orsay-Cedex, France. 2GANIL, DSM-CEA/IN2P3-CNRS, B.P.5027, F14076 Caen-Cedex, France, France. 3LPC, IN2P3-CNRS, ENSICAEN et Université de Caen, F-14050 Caen-Cedex, France. In the canonical formalism of statistical physics, a signature of a first order phase transition for finite systems is the bimodal distribution of an order parameter. Previous thermodynamical studies of nuclear sources produced in heavy-ion col- lisions provide information which support the existence of a phase transition in those finite nuclear systems. Some results suggest that the observable Z1 (charge of the biggest fragment) can be considered as a reliable order parameter of the transition. This talk will show how from peripheral collisions studied with the INDRA detector at GSI we can obtain this bimodal behaviour of Z1. Getting rid of the entrance channel effects and under the constraint of an equiprobable distribution of excitation energy (E∗), we use the canonical description of a phase transition to link this bimodal behaviour with the residual convexity of the en- tropy. Theoretical (with and without phase transition) and experimental Z1 −E correlations are compared. This comparison allows us to rule out the case without transition. Moreover that quantitative conparison provides us with information about the coexistence region in the Z1−E ∗ plane which is in good agreement with that obtained with the signal of abnormal fluctuations of configurational energy (microcanonical negative heat capacity). 1 Introduction It is well known that a Liquid-Gas phase transition (PT) occurs in van der Waals fluids. The similarity between inter-molecular and nuclear interactions leads to a qualitatively similar equation of state which defines the spinodal and coexistence zones of the phase diagram. That is why we expect a “Liquid-Gas like” PT for nuclei. The order parameter is a scalar (one dimension) and is, in this case, the density of the system (more precisely the density difference between the ordered and disordered phase). The energy is also an order parameter because the PT has a latent heat. When an homogeneous system enters the spinodal region of the phase diagram, its en- tropy exhibits a convex intruder along the order parameter(s) direction(s). The system becomes unstable and decomposes itself in two phases. For finite systems, due to surface energy effects, we expect a residual convexity for the system entropy after the transition leading directly to a bimodal distribution (accumulation of statistics for large and low values) of the order parameter. The challenge is to select an observable connected to the theoretical order parameter of the transition, and to explore sufficiently the phase dia- gram to populate the coexistence region and its neighbourhood. Quasi-projectile sources produced in (semi-)peripheral collisions cover a large range of dissipation and conse- quently permit this sufficient exploration. Several theoretical 1,2 and experimental works 3,4,5 show that the biggest fragment has a specific behaviour in the fragmentation process. In particular its size is correlated to the excitation energy (E∗) of the sources. We can reasonably explore whether the Z1-E experimental plane shows a bimodal pattern. Other experimental signals obtained with Bormio 2007: XLV International winter meeting on Nuclear Physics 1 http://arxiv.org/abs/0704.1396v1 multifragmentation data can be correlated with the presence of a phase transition in hot nuclei. Indeed, abnormal fluctuations of configurational energy (AFCE) 6,7 can be related to the negative heat capacity signal 8, and the fossil signal of spinodal decom- position 9 can illustrate the density fluctuations occurring when the nuclei pass through the spinodal zone 10. These two signals are not direct ones and need some hypotheses and/or high statistics. In this work we will present the study of the bimodality signal which is expected to be more robust and direct. We will also show that its observation reinforces the conclusions extracted from the two previous signals. The idea is to show experimentally that the biggest fragment charge, Z1, can be a reliable observable to the order parameter of the PT. After an introduction of the canonical ensemble, we explain the procedure of renormalization which allows to get rid of entrance channel and data sorting effects. Then, comparing experimental and canonical (E,Z1) distributions, we will show that the observed signal of bimodality is related to the abnormal convexity of the entropy of the system. At the end, we propose a localisation of the coexistence zone deduced from a comparison between experimental data and the canonical description of a PT. 2 Canonical description of first-order phase transition. Let us consider an observable E, known on average, free to fluctuate. The least biased distribution will be a Boltzmann-Gibbs distribution (def. 1) 11. If this observable is an order parameter of the system we have to distinguish two cases: with and without phase transition. Pcanβ (E) = e S(E)−βE with Zcanβ = dE e S(E)−βE (1) S (E) ∼ S (Eβ) + (E− Eβ) (E− Eβ) For a one phase system (PT is not present), the microcanonical entropy, S(E)=log W(E) where W(E) is the number of microstates associated to the value of E, is concave every- where. We can perform on it a saddle point approximation (eq. 2) around the average value of E, Eβ , meaning that the canonical distribution has a simple gaussian shape (eq. 3). (s.g.) (E) = 2πσ2E (E− Eβ) with σ2E = − P(s.g.)(E∗,Z1) = 2πdetΣ ~xΣ−1~x, ~x = E∗ − Eβ Z1 − Zβ , Σ = σ2E ρ σEσZ ρ σEσZ σ The parameters of this gaussian are directly linked to the characteristics of the entropy. In the same way we can define the minimum biased two dimensional distribution for the (E∗,Z1) observables leading to a 2D simple gaussian distribution 12 (def. 4). Parameters of this function gathered in the variance-covariance matrix are also deduced from the curvature matrix of the 2D microcanonical entropy 12,13. P(d.g.)(E∗,Z1) = Nliq × P (s.g.) liq (E ∗,Z1) + Ngaz × P (s.g.) gaz (E ∗,Z1) (5) When a system passes through a phase transition and enters in the spinodal region, the homogeneous system has a convex intruder in its microcanonical entropy along the order Bormio 2007: XLV International winter meeting on Nuclear Physics 2 parameter(s) direction(s) 14. Instabilities occur and, due to the finite size of the system, the surface energy effects cause the non-additivity of the entropy leading at the end of the PT to a residual convex entropy for the two-phase system even at equilibrium. We cannot describe anymore the microcanonical entropy with a single saddle point approx- imation but we can introduce a double saddle point approximation. In this case the canonical distribution of the (E∗,Z1) observables can be described as the sum of two 2D simple gaussian distributions, one for each phase (def. 5) 12,13. In the canonical ensemble, the energy distribution P canβ (E ∗) as well as the two- dimensional distribution P canβ (E ∗, Z1) are conditioned by the number of available states expS with a Boltzmann factor ponderation. The convex intruder in S leads to a bi- modality in the distribution 12. Experimentally, this relation is not so clear: the weight of the different states has no reason to be exponential and the measured distribution (E∗) is modified by a factor gexp(E ∗) which is determined in a large part by en- trance channel effects and data sorting : P exp(E∗, Z1) = e S(E∗,Z1)gexp(E ∗). The relative population of the different values of the E∗ distribution looses its thermostatistic meaning (P exp(E∗) ∝ gexp(E ∗)P canβ (E ∗)eβE ). We cannot therefore directly compare experimen- tal and canonical distributions and deduce entropy properties of the system. Pexpω (E ∗,Z1) = ω(E)× P exp(E∗,Z1) (6) with ω(E∗) = P(exp)(E∗,Z1) dZ1 2.1 Renormalization method. In 12, a method was proposed to get rid of the experimental effects. Assuming that the experimental bias gexp(E ∗) affects the Z1 distribution only through its correlation with the deposited energy E∗ (phase space dominance), a renormalization of the (E∗,Z1) distribution under the constraint of an equiprobable distribution of E∗ (eq. 6) allows to be E∗-shape independent. If the system passes through a PT and the correlation between E∗ and Z1 is not a one-to-one correspondence, it could reflect a residual convex intruder of the entropy. 2.2 Spurious bimodality In principle one can ask whether the renormalization procedure given by eq. 6 can create spurious bimodality. This does not seem to be the case for different schematic models 12 but cannot be excluded a priori. Another ambiguity arises from the fact that a physical bimodality can be hidden by the renormalization procedure if the correlation between Z1 and E∗ is too strong. Bimodality can be also difficult to spot if the energy interval is not wide enough. For these reasons in the following we will compare the two canonical cases (with and without transition) with the experimental distribution, to check the validity of the obtained signal. 3 Data selection and first observation of Z1 distributions. Data used in this present work are 80 MeV/A Au+Au reactions performed at the GSI facility and detected with the INDRA 4π multidetector. We focus on peripheral and semi- peripheral collisions to study quasi-projectile sources (forward part of each event). To perform thermostatistical analyses, we select a set of events with a dynamically compact Bormio 2007: XLV International winter meeting on Nuclear Physics 3 20 40 60 80 2000 Source (MeV/A) *E 1 2 3 4 5 6 7 8 9 10 (E (exp)P (MeV/A) *E 1 2 3 4 5 6 7 8 9 10 (E (exp)ω P (MeV/A) *E 1 2 3 4 5 6 7 8 9 10 Figure 1. Upper part : left : experimental distribution of the argest size fragment (Z1) of source events; right : experimental correlation between Z1 and the excitation energy (E ∗). Lower part : left : ex- perimental reweighted correlation between Z1 and the excitation energy; right : excitation energy (E experimental distribution of source events in black squares; the open red circles show this distribution after the renormalization process. For this, we keep only E∗ bins with a statistics greater than 100. configuration for fragments, to reject dynamical events which are always present in heavy- ion reactions at intermediate energies. We require in addition a constant size of the sources to avoid size evolution effects in the bimodality signal 13,15. We evaluate the excitation energy using a standard calorimetry procedure 16,17. We compute the energy balance event-by-event in the centre of mass of the QP sources calculated with fragments only to minimize the effect of pre-equilibrium particles. Afterwards we keep only particles emitted in the forward part of the QP sources and double their contribution, assuming an isotropic emission. In figure 1 information on the experimental Z1 and E ∗ observables is shown, the latter covering a range between roughly 1 and 8 MeV/A (lower-right part). Spinodal zone limits obtained with the AFCE signal are around 2.5 and 5.8 MeV/A for this set of data 13. The shape of the distribution P exp(E∗, Z1) (upper right part) shows the dominance of low dissipation-large Z1 events and reflects the cross-section distribution and data selection. If we look at the corresponding Z1 distribution (upper left part) we do not see any clear signal of bimodality: a large part of statistics is around 65-70, and only a shoulder is visible around 30-40. This particular shape could reflect Bormio 2007: XLV International winter meeting on Nuclear Physics 4 the lack of statistics for the ”gas-like” events. If we apply the renormalization procedure (eq. 6) we obtain (lower left part) a P expω (E,Z1) distribution which has a double humped shape, tending to prove that this procedure can reveal bimodality. 10 20 30 40 50 60 70 80 90 (exp)ωP (s.g.)ωP (d.g.)ωP > 1 <Z 20 30 40 50 60 70 10 20 30 40 50 60 70 80 90 [1.25 , 3.00[ [3.00 , 6.25[ [6.25 , 9.75[ 10 20 30 40 50 60 70 [3.00,6.25[∈ *E Figure 2. Upper part: left: Largest size fragment (Z1) experimental reweighted distribution (black squares with error bars); the blue dashed curve corresponds to the best solution obtained by comparing data and a single gaussian function (concave entropy, no PT), the red continuous curve corresponds to the best solution obtained by comparing data and a double gaussian function (convex entropy, PT); right: microcanonical sampling (fixed E∗) of the mean, the RMS and the skewness of the Z1 distri- butions. For each bin of E∗ (upper X axis), RMS (colored squares-left Y axis) and skewness (colored triangles-right Y axis) are plotted as a function of the mean value (Z1-lower X axis); the two vertical dashed lines delimit the evaluated experimental spinodal zone where a quantitative comparison between data and PT case is performed. Lower part: left: same reweighted distribution of Z1 as above (black curve); the three other distributions correspond to the three regions delimited by the vertical dashed lines (from left to right E∈[1.25,3.00[,[3.00,6.25[ and [6.25,9.75[); right: best solutions obtained after the 2D comparison between data and canonical PT case; results are plotted for the Z1 axis projection; the two solutions correspond to two different ranges of Z1 where fits have been performed Z1∈[25,55] (dashed curve) and Z1∈[10,79] (continuous curve). The corresponding parameter values are listed in table 1. 4 Canonical-Experimental comparisons. To confirm that the two-hump distribution of Z1 signals a convex intruder in the under- lying entropy, in this section we compare the experimental reweighted distributions with Bormio 2007: XLV International winter meeting on Nuclear Physics 5 Parameters [10,79] [25,55] Errors (%) Ē∗liq 2,10 1,67 23 2,09 1,66 23 60,2 62,2 3 12,9 9,85 4 Ē∗gaz 7,11 6,81 4 3,07 2,97 3 21,1 23,8 12 σZgaz 15,2 18,8 2 ρ -0,906 -0,860 4 Nliq/Ngaz 1,12 0,66 52 Ndof 605 387 - χ2 1488 646 - χ2/Ndof 2,459 1,669 - Table 1. Parameters values for the two best reproductions of data by the double gaussian function P (d.g.) for two ranges in Z1 [10,79] (first column) et [25,55] (second column); the third column gives relative errors computed with the previous values; Ē∗,Z̄1,σE∗ ,σZ1 stand respectively for centroids and RMS in the two directions (E,Z1) of each phase (liquid and gas). The ρ parameter is the correlation factor ]-1,1[ between Z1 and E ∗ and the ratio Nliq/Ngaz indicates the repartition of statistics between the two phases. The three last lines give the number of degrees of freedom and the absolute and normalized χ2 estimator values. the analytic expectation for a system exhibiting or not a first order PT. We apply the same renormalization to P (s.g.) (E) and P (d.g.) (E) and try to reproduce the data. We focus on the projection on the Z1 axis to perform the fit. The results are shown in the upper left part of fig. 2. The scatter points with errors bars correspond to the data; the continuous (respectively dashed) curve corresponds to the best solution obtained for the double (respectively simple) reweighted gaussian. We can clearly distinguish the two be- haviours, the no-transition case can not curve itself in the Z1=40-50 region and can only reproduce one phase. The fact that data are reproduced with the functional describing a first order transition allows us to associate the experimental bimodality signal to a genuine convexity of the system entropy. This confirms also that the Z1 observable is linked to the order parameter of the transition. To obtain more quantitative information we have to better localize the spinodal region. To do this, we look at the second and third moments of the Z1 distribution for each bin of E ∗. Their evolution is plotted on the upper right part of fig. 2 as a function of the mean value of Z1 (lower X axis) and E (upper X axis). The squares (left Y axis) stand for the sigma (σ) of the distribution and the triangles (right Y axis) for the skewness (skw). σ shows a maximum in the range 30-40 for < Z1 >. This maximum of fluctuations signs the core of the spinodal zone which corresponds to the hole in P expω (E,Z1) distribution. All values of Z1, for a given E∗, are more or less equivalent. In the same region the skewness changes sign, illustrat- ing the change in the distribution of asymmetry, with a value close to zero when the distribution approaches a normal one. The two vertical dashed lines on the plot delimit three regions (E∈[1.25,3.00[,[3.00,6.25[ and [6.25,9.75[) and the three corresponding Z1 reweighted distributions are plotted in the lower left part of the same figure. The middle one, flat and broad, is very close to the behavior expected for a critical distribution 2 and illustrates the effect of an energy constraint on the order parameter distribution. If we had made a thinner range, we would have approached the microcanonical case. We Bormio 2007: XLV International winter meeting on Nuclear Physics 6 select the region E∈[3.00,6.25[ to compare the two reweighted distributions P expω (E,Z1) and P (d.g.) ω (E,Z1) (eq. 5). The best solutions obtained after this 2D fit procedure are shown in the lower right part of figure 2 and table 1. They correspond to two ranges of Z1 where fits are performed ([10,79] and [25,55]). These two best solutions are shown for the projection on the Z1 axis. (MeV/A) *E 2 4 6 8 10 2 / Tk 2σ> s<A 2 10×) s / Z max (Z Figure 3. Microcanonical sample (fixed E∗) of the fluctuations of normalized FO kinetic energy (open circles) and largest fragment charge (full squares). T, As and Zs stand respectively for the temperature, the mass and charge of the source. Using two different ranges for the Z1 range allows us to estimate the sensitivity of the different parameters. The description of the two phases, given by a set of four parameters for each phase, can be summarized as follows: the average characteristics of the phases, given by Ē∗, Z̄1, are well defined. The ratio between the populations of the liquid and gas phase strongly depends on the interval used to perform the fit. In the two cases the normalized estimator, χ2, is good. Concerning the E∗ axis, the values obtained for the liquid and gas phase centroids reflect the location of the coexistence zone, and are consistent with the location of the spinodal zone obtained with the AFCE signal with the same set of events 13,18. We can further explore the coherence between the two signals by looking at the fluctuations associated to Z1 and to the Freeze-Out configurational kinetic energy 8: we observe in fig. 3 that their evolution with excitation energy has a similar behaviour and exhibits a maximum for E∼5MeV/A. This observation shows that we can consistently characterize the core of the spinodal zone with the maximum fluctuations of different observables connected to the order parameter of the phase transition. Bormio 2007: XLV International winter meeting on Nuclear Physics 7 5 Conclusion and outlook. In this contribution we have shown that, taking into account the dynamics of the entrance channel and sorting effects with a renormalization procedure, the distribution of the largest size fragment (Z1) of each event shows a bimodal pattern. The comparison with an analytical estimation assuming the presence (the absence) of a phase transition, shows that the experimental signal can be unambiguously associated to the case where the system has a residual convex intruder in its entropy. This link makes the Z1 observable a reliable order parameter for the PT in hot nuclei. A bijective relation between the order of the transition and the bimodality signal has been proposed in 12 and analyses on data are in progress. References 1. R. Botet et al., Phys. Rev. E 62 (2000) 1825. 2. F. Gulminelli et al., Phys. Rev. C 71 (2005) 054607. 3. J. D. Frankland et al. (INDRA and ALADIN collaborations), Phys. Rev. C 71 (2005) 034607. 4. J. D. Frankland et al. (INDRA and ALADIN collaborations), Nucl. Phys. A 749 (2005) 102. 5. M. Pichon et al. (INDRA and ALADIN collaborations), Nucl. Phys. A 279 (2006) 6. N. Le Neindre, thèse de doctorat, Université de Caen (1999), http://tel.ccsd.cnrs.fr/tel-00003741. 7. M. D’Agostino et al., Phys. Lett. B 473 (2000) 219. 8. P. Chomaz et al., Nucl. Phys. A 647 (1999) 153. 9. G. Tăbăcaru et al., Eur. Phys. J. A 18 (2003) 103. 10. P. Chomaz et al., Phys. Rep. 389 (2004) 263 and references therein. 11. R. Balian, Cours de physique statistique de l’École Polytechnique, Ellipses (1983). 12. F. Gulminelli (2007), Nucl. Phys. A, in press. 13. E. Bonnet, thèse de doctorat, Université Paris-XI Orsay (2006), http://tel.ccsd.cnrs.fr/tel-00121736. 14. D. H. E. Gross, Microcanonical Thermodynamics-Phase Transitions in “Small” Sys- tems, Singapore-World Scientific, 2001. 15. E. Bonnet et al. (INDRA and ALADIN collaborations) (2007) in preparation. 16. D. Cussol et al., Nucl. Phys. A 561 (1993) 298. 17. E. Vient et al. (INDRA Collaboration), Nucl. Phys. A 700 (2002) 555. 18. N. Le Neindre et al. (INDRA and ALADIN collaborations), Nucl. Phys. A (2007) submitted. Bormio 2007: XLV International winter meeting on Nuclear Physics 8 10 20 30 40 50 60 70 80 90 Source

0704.1397

The p-adic generalized twisted (h,q)-euler-l-function and its applications

The p-adic Generalized Twisted (h, q)-Euler-l-Function and Its Applications Mehmet Cenkci Department of Mathematics, Akdeniz University, 07058-Antalya, Turkey [email protected] Abstract : The main purpose of this paper is to construct the p-adic twisted (h, q)-Euler-l- function, which interpolates generalized twisted (h, q)-Euler numbers associated with a primitive Dirichlet character χ. This is a partial answer for the open question which was remained in [13]. An application of this function leads general congruences systems for generalized twisted (h, q)- Euler numbers associated with χ, in particular, Kummer-type congruences for these numbers are obtained. Keywords : p-adic q-Volkenborn integration, Euler numbers and polynomials, Kummer con- gruences. MSC 2000 : 11B68, 43A05, 11S80, 11A07. 1. Introduction Let N, Z, Q, R and C denote, respectively, sets of positive integer, integer, rational, real and complex numbers as usual. Let p be a fixed odd prime number and x ∈ Q. Then there exists integers m, n and νp (x) such that x = p νp(x)m/n and p does not divide either m or n. Let |·|p be defined such that |x|p = p −νp(x) and |0|p = 0. Then |·|p is a valuation on Q which satisfies the non-Archimedean property |x+ y|p 6 max |x|p , |y|p Completion ofQ with respect to |·|p is denoted by Qp and called the field of p-adic rational numbers. But Qp itself is not complete with respect to |·|p. Cp is the completion of the algebraic closure of Qp and Zp = x ∈ Qp : |x|p 6 1 is called the ring of p-adic rational integers (see [14], [17]). Let d be a fixed positive odd integer and let X = Xd = lim←− Z/dpNZ , X1 = Zp, 0<a<dp (a,p)=1 a+ dpNZp a+ dpNZp = x ∈ X : x ≡ a moddpN where N ∈ N and a ∈ Z with 0 6 a < dpN ([1], [5], [12], [18]). When talking about q-extensions, q can variously be considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, we normally assume that |q| < 1. If q ∈ Cp, we assume that |1− q|p < p −1/(p−1) so that for |x|p 6 1, we have q x =exp(xlogq) ([1], [2], [5], [6], http://arxiv.org/abs/0704.1397v1 [7]). We use the notations [x]q = 1− qx and [x] 1− (−q) 1 + q We say that f is uniformly differentiable function at a point a ∈ X, and denote this property by f ∈ UD (X), if the quotient of the differences f (x)− f (y) has a limit l = f ′ (a) as (x, y)→ (a, a). For f ∈ UD (X), the p-adic invariant q-integral on X was defined by Iq (f) = f (t) dµq (t) = lim [dpN ]q dpN−1 f (a) qa (cf. [5], [7]), where for any positive integer N a+ dpNZp [dpN ]q (cf. [5], [6], [7]). The concept of twisted has been applied by many authors to certain functions which interpolate certain number sequences. In [15], Koblitz defined twisted Dirichlet L-function which interpolates twisted Bernoulli numbers in the field of complex numbers. In [20], Simsek constructed a q- analogue of the twisted L-function interpolating q-twisted Bernoulli numbers. Kim et.al. [12] derived a p-adic analogue of the twisted L-function by using p-adic invariant integrals. By using the definition of h-extension of p-adic q-L-function which is constructed by Kim [11], Simsek [22, 23] and Jang [4] defined twisted p-adic generalized (h, q)-L-function. In [18], Satoh derived p-adic interpolation function for q-Frobenius-Euler numbers. Simsek [21] gave twisted extensions of q-Frobenius-Euler numbers and their interpolating function q-twisted l-series. In [1], Cenkci et.al. constructed generalized p-adic twisted l-function in p-adic number field. Recently, Kim and Rim [13] defined twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. In that paper, they raised the following question: Find a p-adic analogue of the q-twisted l- function which interpolates E (h,1) n,ξ,q,χ, the generalized twisted q-Euler numbers attached to χ [8], [10]. In a forthcoming paper, Rim et.al. [16] answered this question by constructing partial (h, q)-zeta function motivating from a method of Washington [24, 25]. In this paper, we construct p-adic generalized twisted (h, q)-Euler-l-function by employing p-adic invariant measure on p-adic number field. This is the answer of the part of the question posed in [13]. This way of derivation of p-adic generalized twisted (h, q)-Euler-l-function is different from that of [16], and leads an explicit integral representation for this function. As an application of the derived integral representation, we obtain general congruences systems for generalized twisted q-Euler numbers associated with χ, containing Kummer-type congruences. 2. Generalized Twisted q-Euler Numbers In this section, we give a brief summary of the concepts p-adic q-integrals and Euler numbers and polynomials. Let UD (X) be the set of all uniformly differentiable functions on X. For any f ∈ UD (X), Kim defined a q-analogue of an integral with respect to a p-adic invariant measure in [5, 7] which was called p-adic q-integral. The p-adic q-integral was defined as follows: Iq (f) = f (t) dµq (t) = lim [dpN ]q dpN−1 f (a) qa. Note that I1 (f) = lim Iq (f) = f (t) dµ1 (t) = lim dpN−1 f (a) is the Volkenborn integral (see [17]). The Euler zeta function ζE (s) is defined by means of ζE (s) = 2 for s ∈ C with Re(s) > 1 (cf. [8]). For a Dirichlet character χ with conductor d, d ∈ N, d is odd, the l-function associated with χ is defined as ([8]) l (s, χ) = 2 χ (k) (−1) for s ∈ C with Re(s) > 1. This function can be analytically continued to whole complex plane, except s = 1 when χ = 1; and when χ = 1, it reduces to Euler zeta function ζE (s). In [9], (h, q)-extension of Euler zeta function is defined by E,q (s, x) = [2]q [k + x] with s, h ∈ C, Re(s) > 1 and x 6=negative integer or zero. (h, q)-Euler polynomials are defined by the p-adic q-integral as E(h)n,q (x) = q(h−1)t [t+ x] q dµ−q (t) , for h ∈ Z. E n,q (0) = E n,q are called (h, q)-Euler numbers. In [9], it has been shown that for n ∈ Z, n > 0 E,q (−n, x) = E n,q (x) , thus we have E(h)n,q (x) = [2]q qhk [k + x] from which the following entails: E(h)n,q (x) = (1− q) 1 + qh+j In [8, 9], (h, q)-extension of the l-function associated with χ is defined by l(h)q (s, χ) = [2]q χ (k) (−1) for h, s ∈ C with Re(s) > 1. The negative integer values of s are determined explicitly by l(h)q (−n, χ) = E n,q,χ, for n ∈ Z, n > 0 where E n,q,χ are the generalized (h, q)-Euler numbers associated with χ defined E(h)n,q,χ = χ (t) q(h−1)t [t] q dµ−q (t) = [2]q χ (k) (−1) qhk [k] Now assume that q ∈ Cp with |1− q|p < 1. From the definition of p-adic invariant q integral on X, Kim [8] defined the integral I−1 (f) = lim Iq (f) = f (t) dµ−1 (t) (2.1) for f ∈ UD (X). Note that I−1 (f1) + I−1 (f) = 2f (0) , (2.2) where f1 (t) = f (t+ 1). Repeated application of last formula yields I−1 (fn) = (−1) I−1 (f) + 2 n−1−j f (j) , (2.3) with fn (t) = f (t+ n). Let Tp = Cpn = lim Z/pnZ, where Cpn = w ∈ X : wp is the cyclic group of order pn. For w ∈ Tp, let φw : Zp → Cp denote the locally constant function defined by t→ w For f (t) = φw (t) e zt, we obtain φw (t) e ztdµ−1 (z) = wez + 1 using (2.1) and (2.2), and χ (t)φw (t) e ztdµ−1 (t) = 2 χ (i)φw (i) e wdedz + 1 using (2.1) and (2.3) (cf. [8]). As a consequence, the twisted Euler numbers and generalized twisted Euler numbers associated with χ can respectively be defined by wez + 1 , and 2 χ (i)φw (i) e wdedz + 1 En,w,χ from which tnφw (t) dµ−1 (t) = En,w, and χ (t) tnφw (t) dµ−1 (t) = En,w,χ follow. Twisted extension of (h, q)-Euler zeta function is defined by E,q,w (s, x) = [2]q wkqhk [k + x] with h, s ∈ C, Re(s) > 1 and x 6=negative integer or zero. For n ∈ Z, n > 0 and h ∈ Z, this function gives E,q,w (−n, x) = E n,q,w (x) , where E n,q,w (x) are the twisted q-Euler polynomials defined as E(h)n,q,w (x) = q(h−1)tφw (t) [x+ t] q dµ−q (t) = [2]q wkqhk [k + x] by using p-adic invariant q-integral on X in the fermionic sense (cf. [13], [16]). The following expressions for twisted (h, q)-Euler polynomials can be verified from the defining equalities: E(h)n,q,w (x) = (1− q) 1 + wqh+j , (2.4) E(h)n,q,w (x) = [2]qd qhawa (−1) n,qd,wd , (2.5) where n, d ∈ N with d is odd. From (2.4), the twisted (h, q)-Euler polynomials can be determined explicitly. A few of them are 0,q,w (x) = 1 + q 1 + wqh 1,q,w (x) = 1 + q 1 + wqh 1 + wqh+1 2,q,w (x) = 1 + q (1− q) 1 + wqh 1 + wqh+1 1 + wqh+2 For x = 0, E n,q,w (0) = E n,q,w are called twisted (h, q)-Euler numbers. Thus we can write E(h)n,q,w (x) = qxj [x] j,q,w. Let χ be a Dirichlet character of conductor d with d ∈ N and d is odd. Then the generalized twisted (h, q)-Euler numbers associated with χ are defined as E(h)n,q,w,χ = χ (t) q(h−1)tφw (t) [t] q dµ−q (t) . These numbers arise at the negative integer values of the twisted (h, q)-Euler-l-function which is defined by l(h)q,w (s, χ) = [2]q χ (k) (−1) wkqhk with h, s ∈ C, Re(s) > 1. Indeed, for n ∈ Z, n > 0 and h ∈ Z, we have l(h)q,w (−n, χ) = E n,q,w,χ (cf. [13], [16]). We conclude this section by stating the distribution property for generalized twisted (h, q)-Euler numbers associated with χ, which will take a major role in constructing a measure in the next section. For n, d ∈ N with d is odd, we have E(h)n,q,w,χ = [2]qd qhawaχ (a) (−1) n,qd,wd 3. p-adic Twisted (h, q)-l-Functions In this section we first focus on defining a p-adic invariant measure, which is apparently an important tool to construct p-adic twisted (h, q)-Euler-l-function in the sense of p-adic invariant q-integral. We afterwards give the definition of p-adic twisted (h, q)-Euler-l-function, together with Witt’s type formulas for twisted and generalized twisted (h, q)-Euler numbers. Throughout, we assume that ξ is the rth root of unity with (r, pd) = 1, where p is an odd prime and d is an odd natural number. If (r, pd) = 1, it has been known that |1− ξ|p > 1 (see [15], [19]) and ξ lies in the cyclic group Cpn = w : wp . The following theorem plays a crucial role in constructing p-adic generalized twisted (h, q)-Euler-l-function on X. Theorem 3.1 Let q ∈ Cp with |1− q|p < p −1/(p−1) and ξ is the rth root of unity with |1− ξ|p > 1. For N ∈ Z, n ∈ Z, n > 0, let µ n,ξ,q be defined as n,ξ,q a+ dpNZp ξaqhaE n,qdp Then µ n,ξ,q extends uniquely to a measure on X. Proof. In order to show that µ n,ξ,q is a measure on X, we need to show that it is a distribution and is bounded on X. To show it is a distribution on X, we check the equality n,ξ,q a+ idpN + dpN+1Zp n,ξ,q a+ dpNZp Beginning the calculation from right hand side yields n,ξ,q a+ idpN + dpN+1Zp dpN+1 a+idpN ξa+idp qh(a+idp n,qdp a+ idpN dpN+1 ξaqha [p] n,(qdpN ) ,(ξdpN ) ξaqhaE n,qdp n,ξ,q a+ dpNZp where we have used (2.5). To present boundedness, we use equation (2.4) to expand the polynomial E n,qdp so that n,ξ,q a+ dpNZp (1− q) n (−1) ξaqha 1 + ξdp N+jdpN Now, since d is an odd natural number and p is an odd prime, we have N+jdpN 1, so by induction on j, we obtain n,ξ,q a+ dpNZp for a constant M . This is what we require, so the proof is completed. Let χ be a Dirichlet character with conductor d. Then we can express the generalized twisted (h, q)-Euler numbers associated with χ as an integral over X, by using the measure µ n,ξ,q. Lemma 3.2 For n ∈ Z, n > 0, we have χ (t) dµ n,ξ,q (t) = E n,q,ξ,χ. Proof. From the definition of p-adic invariant integral, we have χ (t) dµ n,ξ,q (t) = lim dpN−1 χ (c) ξcqhcE n,qdp Writing c = a+ dm with a = 0, 1, . . . , d− 1 and m = 0, 1, 2, . . ., we get χ (t) dµ n,ξ,q (t) = [d] [2]qd χ (a) (−1) ξaqha × lim [2]qd (qd)p n,(qd)p ,(ξd)p = [d] [2]qd χ (a) (−1) ξaqhaE n,qd,ξd Assuming χ (0) = 0 and by the fact that χ (d) = 0, last expression equals E n,ξ,q,χ, and the proof is completed. Since it is impossible to have a non-zero translation-invariantmeasure on X, µ n,ξ,q is not invariant under translation, but satisfies the following: Lemma 3.3 For a compact-open subset U of X, we have n,ξ,q (pU) = [p] [2]qp n,ξp,qp (U) . Proof. Let U = a+ dpNZp be the compact-open subset of X. Then n,ξ,q (pU) = µ n,ξ,q pa+ dpN+1Zp dpN+1 ξpaqhpaE n,qdp dpN+1 [2]qp [2]qp (qp)dp n,(qp)dp ,(ξp)dp [2]qp n,ξp,qp a+ dpNZp = [p] [2]qp n,ξp,qp (U) , which is the desired result. Next, we give a relation between µ n,ξ,q and µ−q. Lemma 3.4 For any n ∈ Z, n > 0, we have n,ξ,q (t) = q (h−1)tξt [t] q dµ−q (t) . Proof. From the definition of µ n,ξ,q and expansion of twisted (h, q)-Euler polynomials, we have n,ξ,q a+ dpNZp (1− q) n (−1) ξaqha 1 + ξdp N+jdpN By the same method presented in [7], we obtain n,ξ,q a+ dpNZp (1− q) n (−1) ξaqha 1 + q ξaq(h−1)a [a] q (−1) qa = q(h−1)aξa [a] q lim 1−(−qdpN ) 1−(−q) = q(h−1)aξa [a] q lim a+ dpNZp We thus have n,ξ,q (t) = q (h−1)tξt [t] q dµ−q (t) , the desired result. Let ω denote the Teichmüller character mod p. For an arbitrary character χ and n ∈ Z, let χn = χω −n in the sense of product of characters. For t ∈ X∗ = X − pX, we set 〈t〉q = [t]q /ω (t). Since 〈t〉q − 1 < p−1/(p−1), 〈t〉 q is defined by exp slogp 〈t〉q for |s|p 6 1, where logp is the Iwasawa p-adic logarithm function ([3]). For |1− q|p < p −1/(p−1), we have 〈t〉 q ≡ 1 modpN We now define p-adic generalized twisted (h, q)-Euler-l-function. Definition 3.5 For s ∈ Zp, p,q,ξ (s, χ) = (h−1)tξtdµ−q (t) . The values of this function at non-positive integers are given by the following theorem: Theorem 3.6 For any n ∈ Z, n > 0, p,q,ξ (−n, χ) = E n,q,ξ,χn − χn (p) [p] [2]qp n,qp,ξp,χn Proof. p,q,ξ (−n, χ) = (h−1)tξtdµ−q (t) = χn (t) [t] (h−1)tξtdµ−q (t) χn (t) dµ n,ξ,q (t) = χn (t) dµ n,ξ,q (t)− χn (t) dµ n,ξ,q (t) n,q,ξ,χn − χn (p) [p] [2]qp n,qp,ξp,χn where Lemma 3.2, Lemma 3.3 and Lemma 3.4 are used. This theorem will be mainly used in the next section, where certain applications of p-adic generalized twisted (h, q)-Euler-l-function are given. 4. Kummer Congruences for Generalized Twisted (h, q)-Euler Numbers This section is devoted to an application of the p-adic generalized twisted (h, q)-Euler-l-function to an important number theoretic concept, congruences systems. In particular, we derive Kummer- type congruences for generalized twisted (h, q)-Euler numbers by using p-adic integral representa- tion of p-adic generalized twisted (h, q)-Euler-l-function and Theorem 3.6. In the sequel, we assume that q ∈ Cp with |1− q|p < 1. Then q ≡ 1 (modZp). For t ∈ X we have [t]q ≡ t (modZp), thus 〈t〉q ≡ 1 (modpZp). For a positive integer c, the forward difference operator ∆c acts on a sequence {am} by ∆cam = am+c − am. The powers ∆ c of ∆c are defined by ∆0c =identity and for any positive integer k, ∆ c = ∆c ◦∆ c . Thus ∆kcam = am+jc. For simplicity in the notation, we write n,q,ξ,χn n,q,ξ,χn − χn (p) [p] [2]qp n,qp,ξp,χn Theorem 4.1 For n ∈ Z, n > 0 and c ≡ 0 (mod (p− 1)), we have n,q,ξ,χn modpkZp Proof. Since ∆kc is a linear operator, by Theorem 3.6 we have n,q,ξ,χn = ∆kc l p,q,ξ (−n, χ) = ∆ (h−1)tξtdµ−q (t) (h−1)tξtdµ−q (t) (h−1)tξt q − 1 dµ−q (t) . Now, 〈t〉q ≡ 1 (modpZp), which implies that 〈t〉 q ≡ 1 (modpZp) since c ≡ 0 (mod (p− 1)), and thus q − 1 modpkZp Therefore ∆kc l p,q,ξ (−n, χ) ≡ 0 modpkZp from which the result follows. Theorem 4.2 Let n and n′ be positive integers such that n ≡ n′ (mod (p− 1)). Then, we have n,q,ξ,χn n′,q,ξ,χn′ (modpZp) . Proof. Without loss of generality, let n > n′. Then p,q,ξ (−n, χ)− l p,q,ξ (−n ′, χ) = (h−1)tξt q − 1 dµ−q (t) . Since n− n′ ≡ 0 (mod (p− 1)), we have 〈t〉 q − 1 ≡ 0 (modpZp), which entails the result. Acknowledgment: This work was supported by Akdeniz University Scientific Research Projects Unit. References [1] M. Cenkci, M. Can, V. Kurt, p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers, Advan. Stud. Contemp. Math. 9 No. 2 (2004) 203–216. [2] M. Cenkci, M. Can, Some results on q-analogue of the Lerch zeta function, Advan. Stud. Contemp. Math. 12 No. 2 (2006) 213–223. [3] K. Iwasawa, Lectures on p-adic L-Functions, Ann. of Math. Studies, Vol. 74, Princeton Uni- versity Press, Princeton, N. J., 1972. [4] L.-C. Jang, On a q-analogue of the p-adic generalized twisted L-functions and p-adic q- integrals, J. Korean Math. Soc. 44 No. 1 (2007) 1–10. [5] T. Kim, On a q-analogue of p-adic log gamma functions and related integrals, J. Number Theory 76 (1999) 320–329. [6] T. Kim, On p-adic q-L-functions and sums of powers, Discrete Math. 252 (2002) 179–187. [7] T. Kim, q-Volkenborn integration, Russian J. Math. Phys. 9 (2002) 288–299. [8] T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = −1, J. Math. Anal. Appl. (2006) doi:10.1016/j.jmaa.2006.09.27. [9] T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465. [10] T. Kim, On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. 329 (2007) 1472– 1481. [11] T. Kim, A new approach to q-zeta functions, arXiv:math.NT/0502005. [12] T. Kim, L.-C. Jang, S.-H. Rim, H.-K. Pak, On the twisted q-zeta functions and q-Bernoulli polynomials, Far East J. Appl. Math. 13 No. 1 (2003) 13–21. [13] T. Kim, S.-H. Rim, On the twisted q-Euler numbers and polynomials associated with basic q-l-functions, arXiv:math.NT/0611807. [14] N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Graduate Texts in Mathe- matics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. [15] N. Koblitz, A new proof of certain formulas for p-adic L-functions, Duke Math. J. 46 No. 2 (1979) 455–468. [16] S.-H. Rim, Y. Simsek, V. Kurt, T. Kim, On p-adic twisted Euler (h, q)-l-functions, arXiv:math.NT/0702310. [17] A. M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics, Vol. 198, Springer-Verlag, New York, 2000. [18] J. Satoh, q-analogue of Riemann’s ζ-function and q-Euler numbers, J. Number Theory 31 (1989) 346–362. [19] K. Shiratani, On a p-adic interpolating function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Math. 39 (1985) 113–125. [20] Y. Simsek, On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers, J. Korean Math. Soc. 40 No. 6 (2003) 963–975. http://arxiv.org/abs/math/0502005 http://arxiv.org/abs/math/0611807 http://arxiv.org/abs/math/0702310 [21] Y. Simsek, q-analogue of the twisted l-series and q-twisted Euler numbers, J. Number Theory 110 No. 2 (2005) 267–278. [22] Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl. 324 (2006) 790–804. [23] Y. Simsek, On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers, Russian J. Math. Phys. 13 No. 3 (2006) 340–348. [24] L. C. Washington, A note on p-adic L-functions, J. Number Theory 8 (1976) 245–250. [25] L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, New York, 1997. Introduction Generalized Twisted q-Euler Numbers p-adic Twisted ( h,q) -l-Functions Kummer Congruences for Generalized Twisted ( h,q) -Euler Numbers

0704.1398

Global coronal seismology

arXiv:0704.1398v1 [astro-ph] 11 Apr 2007 GLOBAL CORONAL SEISMOLOGY I. BALLAI Solar Physics and Space Plasma Research Centre (SP2RC), Department of Applied Mathematics, University of Sheffield, Hounsfield Road, Hicks Building, Sheffield, S3 7RH, U.K. ([email protected]) Received ; accepted Abstract. Following the observation and analysis of large-scale coronal wave-like distur- bances, we discuss the theoretical progress made in the field of global coronal seismol- ogy. Using simple mathematical techniques we determine average values for magnetic field together with a magnetic map of the quiet Sun. The interaction between global coronal waves and coronal loops allows us to study loop oscillations in a much wider context, i.e. we connect global and local coronal oscillations. Keywords: Sun: magnetic field, Sun: waves, Sun: coronal seismology 1. Introduction The possibility of using waves propagating in solar atmospheric plas- mas to infer quantities impossible to measure (magnetic field, trans- port coefficients, fine structuring, etc.) became a reality after high cadence observation of oscillatory motion was made possible by space and ground-based telescopes. These observations combined with the- oretical models allow to develop a new branch of solar physics called coronal seismology. Pioneering studies by Uchida (1970), Roberts, Ed- win and Benz (1984), Aschwanden et al. (1999) and Nakariakov et al. (1999), have formed the basis of a very promising and exciting field of solar physics. Traditionally, the terminology of coronal seismology was used mainly to describe the techniques involving waves propagating in coronal loops. Since then, this word has acquired a much broader significance and the technique is generalised to acquire information about the mag- netic solar atmosphere (De Pontieu, Erdélyi and James, 2004; Erdélyi, 2006). Coronal seismology uses waves which are localized to a par- ticular magnetic structures, therefore it would be necessary to label these seismic studies as local coronal seismology. After the discovery of large-scale wave-like disturbances, such as EIT waves, X-ray waves, etc., it became necessary to introduce a new terminology, i.e. global coronal seismology where the information is provided by global waves propagating over very large distances, sometimes comparable to the solar radius. Although this may seem a separate subject, in reality these c© 2018 Kluwer Academic Publishers. Printed in the Netherlands. ballai_SP.tex; 27/10/2018; 6:13; p.1 http://arxiv.org/abs/0704.1398v1 2 BALLAI two aspects of coronal seismology are very much linked. A global wave generated by sudden energy releases (flares, CMEs) can interact with active region loops or prominences and localized loop or prominence waves and oscillations are emerging so, there must be a link between the generating source and flare-induced waves in coronal loops. Global waves have been known since the early 1960s. Although it is still not known how the release of energy and energized particles will transform into waves, today it is widely accepted that these distur- bances are similar to the circularly expanding bubble-like shocks after atomic bomb explosion or shock waves which follows the explosion of a supernova. Thanks to the available observational facilities, global waves were observed in a range of wavelengths in different layers of the solar atmosphere. A pressure pulse can generate seismic waves in the solar photosphere propagating with speeds of 200 – 300 kms−1 (Kosovichev and Zharkova, 1998; Donea et al., 2006). Higher up, a flare generates very fast super-alfvénic shock waves known as Moreton waves (Moreton and Ramsey, 1960), best seen in the wings of Hα images, propagat- ing with speeds of 1000 – 2000 kms−1. In the corona, a flare or CME can generate an EIT wave (Thompson et al., 1999) first seen by the SOHO/EIT instrument or an X-ray wave seen in SXT (Narukage et al., 2002). There is still a vigourous debate how this variety of global waves are connected (if they are, at all). Co-spatial and co-temporal investigations of various global waves have been carried out but without a final widely accepted result being reached. The present study deals with the properties of EIT waves, therefore some characteristics of these waves will be given below. Unambiguous evidence for large-scale coronal impulses initiated dur- ing the early stage of a flare and/or CME has been provided by the Extreme-ultraviolet Imaging Telescope (EIT) observations onboard SOHO and by TRACE/EUV. EIT waves propagate in the quiet Sun with speeds of 250 – 400 km s−1 at an almost constant altitude. At a later stage in their propagation EIT waves can be considered a freely propa- gating wavefront which is observed to interact with coronal loops (see, e.g. Wills-Davey and Thompson, 1999). Using TRACE/EUV 195 Å observations, Ballai, Erdélyi and Pintér (2005) have shown that EIT waves (seen in this wavelength) are waves with average periods of the order of 400 seconds. Since at this height, the magnetic field can be considered vertical, EIT waves were interpreted as fast MHD waves. ballai_SP.tex; 27/10/2018; 6:13; p.2 Global coronal seismology 3 2. Coronal Global EIT Waves and their Applications The observations of EIT waves propagating in the solar corona allowed us to shed light on some elementary properties of coronal global EIT waves, however, the available observational precision does not permit us yet to determine more characteristics of these waves. One of the un-answered problems related to EIT waves is connected to their propagation. The core of the problem resides in the lack of detection of an EIT wave with every flare or CME. This could be explained partly by the poor temporal resolution of the SOHO satellite (the only satellite giving full-disk EUV images at the moment) where frames are available with a low cadence, therefore EIT waves generated near the limb simply cannot be recorded. In general, EIT waves seen by SOHO are generated by sources which are located near the centre of the solar disk. EUV images provided by TRACE are much better to use, although the field of view of this instrument is limited. Since EIT waves propagate over a large area of the solar surface (at a certain altitude) they are dispersive. Other ingredients to be considered are the stratification of the medium and the inhomogeneous character of the plasma. All of these factors influence the propagation of coronal EIT waves. Another plausible explanation for the absence of EIT waves asso- ciated with every flare or CME might be that EIT waves diffuse very rapidly, i.e. they become evanescent in a short time after their launch. This means that only those EIT waves could be observed which propa- gate as guided (trapped) waves. The MHD equations in a gravitationally- stratified plasma allows as a solution the magnetoacoustic/magnetogravitational waves of growing amplitude with time (upward propagating waves) and decreasing amplitude with time (downward propagating waves). Trapped EIT waves might arise as a combination of magnetoacoustic and magnetogravitational waves propagating in opposite directions. Further investigations of the possibility of trapping spherical waves in a dissipative medium are needed. 2.1. Interaction of EIT waves with other coronal magnetic entities In this subsection we enumerate a few possible phenomena arising from the interaction of global EIT waves with coronal magnetic entities. According to the classical picture, EIT waves collide with coronal loops resulting in a multitude of modes generated in loops either in the form of standing oscillations or propagating waves. Both types of waves have the general property that they decay very rapidly in a few wavelengths ballai_SP.tex; 27/10/2018; 6:13; p.3 4 BALLAI or periods (see e.g. Nakariakov et al., 1999; Aschwanden et al., 2002). This damping was later used to diagnose the magnetic field inside coronal loops (Nakariakov et al., 1999), transport coefficients for slow waves or global fast waves, sub-structuring, heating function, etc. In coronal loops we consider only the transversal generation of waves, i.e. waves and oscillations are triggered by the interaction of EIT waves and coronal loops. From the EIT wave point of view, a coronal loop (similar to an active region or coronal hole) is an entity with a stronger magnetic field (at least one order of magnitude) than the medium in which they propagate (quite Sun). Therefore, beside transferring energy to coronal loops, EIT waves can be scattered, reflected, and refracted (Terradas and Ofman, 2004). Without claiming completeness, we can draw a few conditions that could influence the appearance of coronal loop oscillations: - height of the loop: since EIT waves propagate at certain heights in the solar corona, it is likely that not all loops will interact with the global waves. Schrijver, Aschwanden and Title (2002) pointed out that only those loops will be affected by EIT waves whose heights exceed 60 – 150 Mm. This means that cool, low-lying loops will not interact with EIT waves. - the height of the interaction between EIT waves and coronal loops: this factor simply means that it is easier to generate oscillations in a loop if the interaction point between the EIT wave and coronal loop is closer to the apex of the loop rather than the footpoint. - Orientation of the loop: if the front of the EIT wave is perpendicular to the plane of the coronal loop the interaction between the EIT waves and the coronal loop occurs in two points at the same time. If the loop is stiff enough, a standing oscillation can be easily excited. If the front is not perpendicular, the collision between the EIT wave and loops occurs in two points delayed in time by τ = s cosα/vEIT , where s is the distance between the footpoints, α is the attack angle, and vEIT is the propagation speed of the EIT wave. In this case, standing modes can be excited only in very special cases. Another important element is the orientation of the coronal loop with respect to the vertical axis (inclination). - distance between the flaring site and coronal loop (or energy of EIT waves): During their propagation, EIT waves are losing energy due to the geometrical damping (dilatation of the front) and due to some physical damping effects. Therefore it might happen that the energy of an EIT wave originating from a distant flare is not enough to dislocate the loop. - radius of the loop and the density contrast (or Alfvén speed contrast): a massive loop is much harder to dislocate than a thin loop. The ratio ballai_SP.tex; 27/10/2018; 6:13; p.4 Global coronal seismology 5 between the densities in the loop and its environment is known to influence the amplitude of oscillations. In order to describe quantitatively the interaction between EIT waves and coronal loops, we suppose a medium in which the coronal loop is situated, for simplicity, in a magnetic-free medium (in fact this constraint can be relaxed and the result is obvious) retains its identity and does not disperse or fragment. The tube is considered thin, i.e. its radius is small relative to other geometrical scales of the problem. During the wave propagation we suppose a quasi-static pressure balance to be maintained at all times. An EIT wave colliding with a coronal loop exerts a force which will need to work against two forces, one being the elastic force of the tube represented by the magnetic tension of the tube and inertia of the fluid element which needs to be displaced. The equilibrium of the tube is prescribed by the hydrostatic equi- librium where pressure forces are in equilibrium with the gravitational force and the lateral pressure balance pi+B i /2µ = pe is satisfied, with pi and pe being the kinetic (thermal) pressure inside the tube and the environment, Bi the interior magnetic field and g is the gravitational acceleration at the solar surface. If we denote by ρi and ρe the locally ho- mogeneous densities inside and outside the tube and vA(= Bi/(µρi) the Alfvén speed, then the equation describing the variation of dis- placement of the fluid element, ξ(z, t), is (a similar equation has been obtained by Ryutov and Ryutova (1975) in a different context) ρi − ρe ρi + ρe ρi + ρe . (1) Let us introduce a new variable such that ξ(z, t) = Q(z, t) exp(λz), where the value of λ is chosen such that the first-order derivatives with respect to the coordinate z vanish. After a straightforward calculation we obtain that the dynamics of generated waves in the coronal loop as a result of the interaction of a global wave with coronal loop is described − c2K + ω2CQ = 0, ωC = g(d− 1) d(d+ 1) cK = vA 1 + d with d = ρi/ρe being the filling factor. Equation (2) is the well-known Klein-Gordon (KG) equation derived and studied earlier in solar MHD wave context by, e.g. Rae and Roberts (1982), Hargreaves (2005), Bal- lai, Erdélyi and Hargreaves (2006). The quantity cK is the kink speed ballai_SP.tex; 27/10/2018; 6:13; p.5 6 BALLAI of waves and it is regarded as a density-weighted Alfvén speed. The co- efficient ωC is the cut-off frequency of waves and is a constant quantity for an isothermal medium. The waves corresponding to the Eq. (2) are dispersive, i.e. waves with smaller wavelength (larger k) propagating faster. Waves with smaller wave number will have smaller group speed, the maximum of the group speed (at k → ∞) being cK . Another essential property of the KG equation is that it describes waves which are able to propagate if their frequency is larger than the cut-off frequency. For typical coro- nal conditions (vA=1000 km s −1, d = 10) we obtain that waves will propagate if their frequency is greater than 0.11 mHz or their period is smaller than 150 minutes. For simplicity, let us suppose that the fast kink mode in the coronal loop is generated by the interaction of an EIT wave with a loop and the forcing term of the interaction is modelled by a delta-pulse, i.e. the equation describing the dynamics of impulsively generated fast kink mode is given by − c2K + ω2CQ = δ(z)δ(t). (3) This equation can be solved using standard Laplace transform tech- nique to yield Q(z, t) = 2 − z2 H(cKt− |z|), (4) where J0(z) is the zeroth-order Bessel function and H(z) is the Heav- iside function. The impulsive excitation of waves in a flux tube leads to the formation of a pulse that propagates away with the speed cK , followed by a wake in which the flux tube oscillates with the frequency ωC . A typical temporal variation of the amplitude of kink waves (keep- ing the height constant) would show that the amplitude of the mode decreases (even in the absence of dissipation) and an e-fold decay occurs in about 400 seconds . Recently Terradas, Oliver and Ballester (2005) have studied the interaction between the coronal loops and EIT waves in the zero-beta limit considering a spatial initial condition. They obtained that the generated oscillations in the coronal loop decay asymptotically as t−1/2. Kink oscillations are weakly affected by dissipation, therefore the con- sideration of any non-ideal effect to supplement the KG equation would not lead to a significant change. It is accepted that damping due to the resonant absorption could explain the damping of kink oscillations in coronal loops (Ruderman and Roberts, 2002; Goossens, Andries and Aschwanden, 2002). ballai_SP.tex; 27/10/2018; 6:13; p.6 Global coronal seismology 7 It can be shown that the consideration of resonant absorption as a damping mechanism in the governing equation leads to a similar equa- tion we would obtain taking into account dissipation. Ballai, Erdélyi and Hargreaves (2006) showed that the evolution equation is modified by an extra term forming a Klein-Gordon-Burgers (KGB) equation − c2K + ω2CQ− ν ∂z2∂t = 0, (5) where ν is a coefficient which could play the role of any dissipative mechanism or a factor which include the damping due to resonant absorption (in fact ν is inversely proportional to the gradient of Alfvén speed) and would describe the transfer of energy from large to small scales (see, e.g. Ruderman and Goossens, 1993). Waves in this approximation can have a temporal (keeping k real) and spatial damping (keeping ω real), the decay rate and length, sup- posing the ansatz Q(z, t) ∼ exp[i(ωt− kz)], are given by , ki ≈ − ω2 − ω2C c4K + ν . (6) The KGB equation can be solved using initial/boundary conditions to describe the evolution of kink modes for different kind of sources, e.g. monochromatic source (A(t) = V0e iΩt), delta-function pulse (A(t) = V0δ(ωCt/2π)), etc. using numerical methods. Asymptotic analysis (t ≫ z/cK) shows that these waves decay as t −3/2 (Ballai, Erdélyi and Har- greaves, 2006). Another important factor is the energy of EIT waves. Recently Ballai, Erdélyi and Pintér (2005), using a simple energy conservation, found the minimum energy an EIT wave should have to produce a loop oscillation. Using their results we studied a few loop oscillation events presented by Aschwanden et al. (2002) and the minimum energy of EIT waves necessary to produce the observed oscillations are shown in Table 1. The geometrical size of loops and the number densities given by Aschwanden et al. (2002) have been used. The obtained energies are in the range of 1016 − 1019 J with no par- ticular correlation with the length and radius of the loop. Similar to this approach we can estimate the minimum energy of an EIT wave to produce a displacement of 1 pixel in TRACE/EUV 195 Å images using the relation E = 1.66 × 106L 2 + ρe/λ , (J) where L and R are the length and radius of the loop, and λ−1e the decay length of perturbations outside the cylinder given by λ2e = (c2Se − c Ae − c (c2Se + v Ae)(c Te − c k2, (7) ballai_SP.tex; 27/10/2018; 6:13; p.7 8 BALLAI Table I. The minimum energy of EIT waves which could produce the loop oscillations studied by Aschwanden et al. (2002). Date(yyyymmdd) L(Mm) R(Mm) n(×108 cm−3) E(J) 1998 Jul 14 168 7.2 5.7 2.2× 1017 1998 Jul 14 204 7.9 6.2 9.7× 1018 1998 Nov 23 190 16.8 3 1.3× 1019 1999 Jul 04 258 7 6.3 3.9× 1016 1999 Oct 25 166 6.3 7.2 1.6× 1018 2000 Mar 23 198 8.8 17 5.2× 1016 2000 Apr 12 78 6.8 6.9 2.5× 1016 2001 Mar 21 406 9.2 6.2 7.4× 1016 2001 Mar 22 260 6.2 3.2 1.9× 1016 2001 Apr 12 226 7 4.4 1.4× 1018 2001 Apr 15 256 8.5 5.1 1.4× 1016 2001 May 13 182 11.4 4 2.2× 1018 2001 May 15 192 6.9 2.7 1.6× 1019 2001 Jun 15 146 15.8 3.2 1.1× 1017 with cTe, cSe, and vAe being the cusp, sound and, Alfvén speeds in the region outside the loop and k is the wavenumber. The energy range is in the interval 3× 1017 − 3× 1018 J for loop lengths and radii varying in the intervals 60 – 500 Mm and 1 – 10 Mm. 2.2. Determination of magnetic field values Observations show that EIT waves propagate in every direction almost isotropically on the solar disk, therefore we can reasonably suppose that they are fast magnetoacoustic waves (FMWs) propagating in the quiet Sun perpendicular to the vertical equilibrium magnetic field. The repre- sentative intermediate line formation temperature corresponding to the 195 Å wavelength is 1.4×106 K. The sound speed corresponding to this temperature is 179 km s−1. Since the FMWs propagate perpendicular to the field, their phase speed is approximated by (c2S + v The propagation height is an important parameter as a series of physical quantities (density, temperature, etc.) in the solar atmosphere have a height dependence. Given the present status of research on the propagation of EIT waves, there is no accepted value for the propa- gation height of these waves. For a range of the plasma parameters we can derive average values for the magnetic field by considering the propagational characters of EIT waves. Therefore, we study the ballai_SP.tex; 27/10/2018; 6:13; p.8 Global coronal seismology 9 variation of various physical quantities with respect to the propagation height of EIT waves. We recall a simple atmospheric model developed by Sturrock, Wheat- land and Acton (1996). The temperature profile above a region of the quiet Sun, where the magnetic field is radial, is given by T (x) = 7R⊙F0 )]2/7 . (8) Here F0 is the inward heat flux (1.8× 10 5 erg cm−2 s−1), x is the nor- malized height coordinate defined by x = r/R⊙, T0 is the temperature at the base of the model (considered to be 1.3 × 106 K) and a is the coefficient of thermal conductivity. The quantity a is weakly dependent on pressure and atmospheric composition; for the solar corona a value of 10−6 (in cgs units) is appropriate (Nowak and Ulmschneider, 1977). Assuming a model atmosphere in hydrostatic equilibrium we obtain that the number density, based on the temperature profile supposed in Eq. (8), is n(x) = T (x) exp[−δ(T (x)5/2 − T 0 )], δ = 2µGM⊙mpa , (9) with G the gravitational constant, M⊙ the solar mass, kB is the Boltz- mann’s constant; µ = 0.6 is the mean molecular weight; mp, proton mass and n0 = 3.6 × 10 8 cm−3 the density at the base of corona. Having the variation of density with height and the value of Alfvén speed deduced from the phase speed of EIT waves, we can calculate the magnetic field using B = vA(4πmpn) 1/2. Evaluating the relations above, the variation of temperature, density, Alfvén speed and magnetic field with height is shown in Table 2. Two cases are derived for EIT waves propagating strictly perpendicular to the radial magnetic field with a speed of (a) 250 km s−1 and, (b) 400 km s−1, respectively. The values of the physical quantities show some change for a given propagation speed but will have little effect on the final results. For an average value of EIT wave speed of 300 km s−1 propagating at 0.05 R⊙ above the photosphere we find that the magnetic field is 1.8 G. If we apply Br2 = const., i.e. the magnetic flux is constant, we find that at the photospheric level the average magnetic field is 2.1 G which agrees well with the observed solar mean magnetic field (Chaplin et al. 2003). EIT waves considered as fast MHD waves can also be used to determine the value of the radial component of the magnetic field at every location allowed by the observational precision. In this way, using the previously cited TRACE observations we can construct a magnetic map of the quiet Sun (see Figure 1), in other words EIT ballai_SP.tex; 27/10/2018; 6:13; p.9 10 BALLAI Table II. The variation of the temperature (in MK), den- sity (in units of 108 cm−3), Alfvén and sound speeds (in units of 107 cm s−1) and magnetic field (in G) with height above the photosphere for an EIT wave propagating with a speed of (a) 250 km s−1, and (b) 400 km s−1, respectively. r/R⊙ T n cS v B(a) v 1.00 1.30 3.60 1.72 1.81 1.57 3.61 3.13 1.02 1.41 3.30 1.80 1.73 1.44 3.57 2.97 1.04 1.50 3.10 1.85 1.67 1.34 3.54 2.85 1.06 1.58 2.95 1.90 1.61 1.27 3.51 2.76 1.08 1.64 2.83 1.94 1.57 1.21 3.49 2.69 1.10 1.70 2.73 1.97 1.52 1.15 3.47 2.63 Figure 1. Magnetic map of the quite Sun obtained using an EIT wave observed by TRACE/EUV in 195 Å . ballai_SP.tex; 27/10/2018; 6:13; p.10 Global coronal seismology 11 waves can serve as probes in a magnetic tomography of the quiet Sun. If points are joined across the lines, we will obtain the location of the EIT wavefront. Magnetic field varies between 0.47 and 5.62 G, however, these particular values should be handled with care as the interpolation will introduce spurious values at the two ends of the interval. It should be noted that this result has been obtained supposing a single value for density, in reality both magnetic field and density can vary along the propagation direction, as well. The method we employed to find this magnetic map (magnetic field derived via the Alfvén speed) means that density and magnetic field cannot be determined at the same time. Further EUV density sensitive diagnostics line ratio measurements are required to establish a density map of the quiet Sun which will provide an accurate determination of the local magnetic field. In conclusion, EIT waves propagating in the solar corona exhibit a wide range of applicabilities for plasma and field diagnostics. The fact that during their propagation EIT waves cover a large area of the solar surface (in the coronal) allows us to sample the magnetic field in the quiet Sun. EIT waves could serve as a link between eruptive events and localised oscillations, e.g. loop oscillations could be studied in a much broader context. Using a simple model we found that the minimum energy an EIT wave should have to produce a detectable loop oscillation is in the range of 1016 − 1019 J. Problems to be tackled in the future should include the study of at- tenuation of EIT waves with the aim of providing information about the magnitude of transport coefficients in the quiet Sun. The old problem of connecting different global waves still remain to be addressed. Despite of the lack of high precision observations, EIT waves show a great potential for magneto-seismology of the solar corona. Acknowledgements The author acknowledges the financial support offered by the Nuffield Foundation (NUF-NAL 04) and NFS Hungary (OTKA, T043741). The help by B. Pintér and M. Douglas is appreciated. References Aschwanden, M.J., Fletcher, L., Schrijver, C.J., Alexander, D.: 1999, Astrophys. J., 520, 880 Aschwanden, M.J., De Pontieu, B., Schrijver, C.J., Title, A.M.: 2002, Solar Phys., 206, 99 Ballai, I., Erdélyi, R., Pintér, B.: 2005, Astrophys. J. 633, L145 ballai_SP.tex; 27/10/2018; 6:13; p.11 12 BALLAI Ballai, I., Erdélyi, R., Hargreaves, J.: 2006, Phys. Plasmas, 13, 042108 Chaplin, W.J., Dumbhill, A.H., Elsworth, Y., Isaak, G.R., McLeod, C.P. et al.: 2003, Mon. Not. Roy. Astron. Soc. 343, 813 De Pontieu, B., Erdélyi, R., James, S.: 2004, Nature, 430, 536 Donea, A.-C., Besliu-Ionescu, D., Cally, P., Lindsey, C., Zharkova, V.V.: 2006, Solar Phys., 239, 113 Erdélyi, R.: 2006, Proc. Roy. Soc. London Ser. A, 364, 351 Goossens, M., Andries, J., Aschwanden, M.J.: 2002, Astron. Astrophys., 394, 39 Hargreaves, J.: 2005, in Ballai et al. (eds.) Plasma- and Astrophysics: from laboratory to outer space, PADEU, 15, 83 Kosovichev, A.G., Zharkova, V.: 1998, Nature, 393, 317 Moreton, G.F., Ramsey, H.E.: 1960, Publ. Astron. Soc. Pac., 72, 357 Nakariakov, V.M., Ofman, L., DeLuca, E.D., Roberts, B., Davila, J.M.: 1999, Science, 285, 862 Narukage, N., Hudson, H.S., Morimoto, T., Akiyama, S., Kitai, R. et al.: 2002, Astrophys. J., 572, L109 Nowak, T., Ulmschneider, P.: 1977, Astron. Astrophys., 60, 413 Rae, I.C., Roberts, B.: 1982, Astrophys. J., 256, 761 Roberts, B., Edwin, P.M., Benz, A.O.:1984, Astrophys. J., 279, 857 Ruderman, M.S., Goossens, M.: 1993, Solar Phys., 143, 69 Ruderman, M.S., Roberts, B.: 2002, Astrophys. J., 577, 475 Ryutov, D.A., Ryutova, M.P.: 1975, JETP, 43, 491 Schrijver, C.J., Aschwanden, M.J., Title, A.M.: 2002, Solar Phys., 206, 69 Sturrock, P., Wheatland, M.S., Acton, L.: 1996, Astrophys. J., 461, L115 Terradas, J., Ofman, L.: 2004, In Erdélyi, Ballester and Fleck (eds.) SOHO 13: Waves, Oscillations and Small-Scale Transient Events in the Solar Atmosphere, ESA SP-547, Noordwijk, ESA, 469 Terradas, J., Oliver, R., Ballester, J.L.: 2005, Astrophys. J., 618, L149 Thompson, B.J., Gurman, J.B., Neupert, W.M. et al.: 1999, Astrophys. J., 517, 151 Uchida, Y.: 1970, Publ. Astron. Soc. Pac., 22, 341 Wills-Davey, M.J., Thompson, B.J.: 1999, Solar Phys., 190, 467 ballai_SP.tex; 27/10/2018; 6:13; p.12

0704.1399

La formule de Lie-Trotter pour les semi-groupes fortement continus

MÉMOIRE DE RECHERCHE de Mathématiques Pures intitulé “LA FORMULE DE LIE - TROTTER POUR LES SEMI-GROUPES FORTEMENT CONTINUS” par Ludovic Dan LEMLE sous la direction de Gilles CASSIER SOUTENU À L’UNIVERSITÉ CLAUDE BERNARD LYON 1 Le 4 juillet 2001 http://arXiv.org/abs/0704.1399v1 Remerciements. Tout d’abord, je veux profiter de cette occasion pour présenter mes remer- ciements à Monsieur Gilles Cassier de l’Université Claude Benard Lyon 1 pour le choix du sujet de ce mémoire, ses suggestions et son aide constante. J’ai été très enchanté de cette collaboration avec Monsieur Gilles Cassier. De même, je veux exprimer ma reconnaissance à Monsieur Dan Timotin de l’Institut de Mathématiques de l’Académie Roumaine de Bucharest pour ses excellents conseils pendant son sejour à Lyon. En même temps, il convient d’exprimer ma gratitude à Monsieur Dumitru Gaşpar et à Monsieur Nicolae Suciu de l’Université de l’Ouest de Timişoara, qui m’ont donné l’occasion d’étudier dans une très grande université européenne. Finalement, je veux exprimer toute ma reconnaissance pour l’hospitalité et l’amabilité avec lesquelles j’ai été accueilli par toutes les personnes que j’ai eu le plaisir de connâıtre pendant mon stage à l’Université Claude Bernard Lyon 1. Ludovic Dan LEMLE Facultatea de Inginerie Str. Revoluţiei Nr. 5 COD 2750 Hunedoara ROMÂNIA tel: +4 02 54 20 75 43 e-mail: [email protected] Table des matières 1 Introduction 5 1.1 Préliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Les opérateurs dissipatifs . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Semi-groupes uniformément continus . . . . . . . . . . . . . . . . . 17 1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Semi-groupes de classe C0 31 2.1 Définitions. Propriétés élémentaires . . . . . . . . . . . . . . . . . . 31 2.2 Propriétés générales des C0-semi-groupes . . . . . . . . . . . . . . . 41 2.3 Le théorème de Hille - Yosida . . . . . . . . . . . . . . . . . . . . . 52 2.4 La représentation de Bromwich . . . . . . . . . . . . . . . . . . . . 63 2.5 Conditions suffisantes d’appartenances à GI(M, 0) . . . . . . . . . . 70 2.6 Propriétés spectrales des C0-semi-groupes . . . . . . . . . . . . . . . 78 2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3 C0-semigroupes avec propriétés spéciales 87 3.1 C0-semi-groupes différentiables . . . . . . . . . . . . . . . . . . . . . 87 3.2 C0-semi-groupes analytiques . . . . . . . . . . . . . . . . . . . . . . 98 3.3 C0-semi-groupes de contractions . . . . . . . . . . . . . . . . . . . . 107 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4 La formule de Lie - Trotter 113 4.1 Le cas des semi-groupes uniformément continus . . . . . . . . . . . 113 4.2 Propriétés de convergence des C0-semi-groupes . . . . . . . . . . . . 117 4.3 Formule de Lie - Trotter pour les C0-semi-groupes . . . . . . . . . . 127 4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4 TABLE DES MATIÈRES Chapitre 1 Introduction 1.1 Préliminaires Dans la suite, nous noterons par E un espace de Banach sur le corps des nombres complexes C et par B(E) l’algèbre de Banach des opérateurs linéaires bornés dans E . Nous désignerons par I l’unité de B(E). Pour un opérateur linéaire A : D(A) ⊂ E −→ E nous noterons par: Im A = {Ax |x ∈ D(A)} l’image de A et par: Ker A = {x ∈ D(A) |Ax = 0} le noyau de A. L’opérateur A : D(A) ⊂ E −→ Im A est surjectif. Si Ker A = {0}, alors A est injectif. Pour un opérateur bijectif, on peut définir l’opérateur inverse: A−1 : D ⊂ E −→ E par A−1y = x si Ax = y. Evidemment D (A−1) = Im A. Dans la suite nous noterons par GL(E) l’ensemble des éléments inversibles de B(E). L’ensemble GL(E) est un ensemble ouvert dans B(E) ([Is’81, Theorem 4.1.13, pag. 145]). Soit A : D(A) ⊂ E −→ E un opérateur linéaire. Pour tout n ∈ N, nous définissons: n : D(An) −→ E A0 = I , A1 = A , ..., An = A 6 CHAPITRE 1. INTRODUCTION D(An) = x ∈ D(An−1) x ∈ D(A) quel que soit n ∈ N. Lemme 1.1.1 Soit f : [a, b] → E une fonction continue. Alors: f(s) ds = f(a) . Preuve Nous avons: f(s) ds − f(a) [f(s) − f(a)] ds ≤ sup s∈[a,a+t] ‖f(s) − f(a)‖ . L’égalité de l’énoncé résulte de la continuité de l’application f . Lemme 1.1.2 Si A ∈ B(E) et ‖A‖ < 1, alors I − A ∈ GL(E) et: (I − A)−1 = Preuve Soit Yn = I + A + A 2 + . . . + An. Alors: ‖Yn+p − Yn‖ ≤ ‖A‖n+1 1 − ‖A‖ −→ 0 pour n → ∞. Par conséquent, (Yn)n∈N est une suite Cauchy. Mais B(E) est une algèbre de Banach. La suite (Yn)n∈N est donc convergente. Notons Y ∈ B(E) sa limite. De l’égalité (I − A)Yn = I − An+1, il résulte que limn→∞(I − A)Yn = I, d’où (I − A)Y = I. Nous obtenons Y (I − A) = I de façon analogue. Finalement, on voit que I − A ∈ GL(E) et que (I − A)−1 = Remarque 1.1.3 Si ‖I − A‖ < 1, alors A ∈ GL(E) et A−1 = (I − A)n. Définition 1.1.4 L’ensemble: ρ(A) = λ ∈ C ∣(λI − A)−1 est inversible dans ∈ B(E) s’appelle l’ensemble résolvant de A ∈ B(E). Proposition 1.1.5 Soit A ∈ B(E). Alors ρ(A) est un ensemble ouvert. 1.1. PRÉLIMINAIRES 7 Preuve Définissons l’application: φ : C −→ B(E) φ(λ) = λI − A . Evidemment, φ est continue. Si λ ∈ ρ(A), alors λI − A ∈ GL(E) et par suite ρ(A) = φ−1 (GL(E)). Comme GL(E) est un ensemble ouvert, on voit que ρ(A) est ouvert. Définition 1.1.6 L’application: R( . ; A) : ρ(A) −→ B(E) R(λ; A) = (λI − A)−1 s’appelle la résolvante de A. Proposition 1.1.7 La résolvante d’un opérateur linéaire A ∈ B(E), a les pro- priétés suivantes: i) si λ, µ ∈ ρ(A), alors: R(λ; A) − R(µ; A) = (µ − λ)R(λ; A)R(µ; A) ; ii) R( . ; A) est une application analytique sur ρ(A); iii) si λ ∈ C et |λ| > ‖A‖, alors λ ∈ ρ(A) et nous avons: R(λ; A) = iv) Nous avons: R(λ; A) = (−1)nn!R(λ; A)n+1 quels que soient n ∈ N∗ et λ ∈ ρ(A). Preuve i) Nous avons successivement: R(λ; A) − R(µ; A) = (λI − A)−1 − (µI − A)−1 = = (λI − A)−1 (µI − A − λI + A) (µI − A)−1 = = (µ − λ)R(λ; A)R(µ; A) 8 CHAPITRE 1. INTRODUCTION quels que soient λ, µ ∈ ρ(A). ii) Soit λ0 ∈ ρ(A). Notons D ‖R(λ0;A)‖ le disque ouvert de centre λ0 et de rayon 1 ‖R(λ;A)‖ . Alors, pour λ ∈ D ‖R(λ0;A)‖ , nous avons: λI − A = [I − (λ0 − λ)R(λ0; A)] (λ0I − A) . Mais: ‖(λ0 − λ)R(λ0; A)‖ = |λ0 − λ|‖R(λ0; A)‖ < 1 . Compte tenu du lemme 1.1.2, il résulte que: I − (λ0 − λ)R(λ0; A) ∈ GL(E) , d’où λI − A ∈ GL(E) et: (λI − A)−1 = (λ0I − A)−1[I − (λ0 − λ)R(λ0; A)]−1 = = R(λ0; A) (λ0 − λ)nR(λ0; A)n = (−1)n(λ − λ0)nR(λ0; A)n+1 . Donc R( . ; A) est analytique sur ρ(A). iii) Soit λ ∈ C tel que |λ| > ‖A‖. Alors ‖λ−1A‖ < 1, d’où I − λ−1A ∈ GL(E). De plus: I − λ−1A Par conséquent: R(λ; A) = (λI − A)−1 = λ−1 I − λ−1A L’assertion (iv) s’obtient par récurrence. Pour n = 1, nous avons: R(λ; A) = (λI − A)−1 = −(λI − A)−2 = R(λ; A)2 . Supposons que pour k ∈ N, on ait: R(λ; A) = (−1)kk!R(λ; A)k+1 . Montrons que: dλk+1 R(λ; A) = (−1)k+1(k + 1)!R(λ; A)k+2 . 1.1. PRÉLIMINAIRES 9 Nous avons: dλk+1 R(λ; A) = R(λ; A) (−1)kk!(λI − A)−k−1 = (−1)kk!(−k − 1)(λI − A)−k−2 = (−1)k+1(k + 1)!R(λ; A)k+2 et par conséquent: R(λ; A) = (−1)nn!R(λ; A)n+1 , (∀)n ∈ N∗. Remarque 1.1.8 Compte tenu de la proposition 1.1.7 (iii), il résulte que: {λ ∈ C ||λ| > ‖A‖} ⊂ ρ(A). Définition 1.1.9 L’ensemble σ(A) = C − ρ(A) s’appelle le spectre de A ∈ B(E). Proposition 1.1.10 Soit A ∈ B(E). Alors: i) σ(A) 6= ∅; ii) σ(A) est un ensemble compact. Preuve i) Supposons que σ(A) = ∅. Alors ρ(A) = C. Par conséquent, l’application λ 7−→ (λI − A)−1 est définie sur C. De plus, pour |λ| > ‖A‖, nous avons: R(λ; A) = , (∀)λ ∈ ρ(A). Il s’ensuit que: |λ|→∞ R(λ; A) = 0. Donc il existe M > 0 tel que ‖R(λ; A)‖ < M , (∀)λ ∈ C. Le théorème de Liouville ([DS’67, pag. 231]) implique que R(.; A) est constante sur C et que cette constante ne peut être que 0. Donc (λI − A)−1 = 0 pour tout λ ∈ C, ce qui est absurde. Par conséquent σ(A) 6= ∅. ii) Compte tenu de la proposition 1.1.7 (iii), nous obtenons que: σ(A) ⊂ {λ ∈ C ||λ| ≤ ‖A‖} . L’ensemble σ(A) est donc borné. Comme nous avons vu que σ(A) est un ensemble fermé, il est donc compact. 10 CHAPITRE 1. INTRODUCTION Définition 1.1.11 Pour un opérateur linéaire A ∈ B(E), le nombre r(A) = sup λ∈σ(A) s’appelle le rayon spectral de A. Remarque 1.1.12 Evidemment, pour un opérateur A ∈ B(E), σ(A) est contenu dans l’intérieur du cercle de centre O et de rayon r(A). De plus, on peut montrer r(A) = lim et on voit que r(A) ≤ ‖A‖. Par la suite, nous présenterons quelques problèmes concernant la théorie spec- trale pour un opérateur linéaire fermé A : D(A) ⊂ E −→ E . Définition 1.1.13 L’ensemble: ρ(A) = {λ ∈ C |λI − A : D(A) −→ E est opérateur bijectif} s’appele l’ensemble résolvant de A. Remarque 1.1.14 Il résulte du théorème du graphe fermé ([DS’67, Theorem II.2.4, pag. 57]) que l’opérateur: (λI − A)−1 : E −→ E est continu dans E . Définition 1.1.15 L’application: R( . ; A) : ρ(A) −→ B(E) R(λ; A) = (λI − A)−1 , (∀)λ ∈ ρ(A) s’appelle la résolvante de A. Proposition 1.1.16 Soit A : D(A) ⊂ E −→ E , un opérateur linéaire fermé. Alors: i) ρ(A) est un ensemble ouvert et R( . ; A) est une application analytique sur ρ(A); ii) si λ, µ ∈ ρ(A), alors: R(λ; A) − R(µ; A) = (µ − λ)R(λ; A)R(µ; A) ; 1.1. PRÉLIMINAIRES 11 iii) Nous avons: R(λ; A) = (−1)nn!R(λ; A)n+1 quels que soient n ∈ N et λ ∈ ρ(A). Preuve Elle est analogue à celle de la proposition 1.1.7. Définition 1.1.17 L’ensemble σ(A) = C − ρ(A) s’appelle le spectre de A. Remarque 1.1.18 σ(A) est un ensemble fermé. Remarque 1.1.19 Il existe des opérateurs fermés qui ont un spectre non borné. Exemple 1.1.20 Prenons E = C[0,1] et considérons l’opérateur: D : C1[0,1] −→ E défini par: Df = f Dans ce cas, nous avons σ(D) = C. Définition 1.1.21 Soit D ⊂ C un ensemble ouvert. Une application analytique: D ∋ λ 7−→ Rλ ∈ B(E) qui vérifie la propriété: Rλ − Rµ = (µ − λ)RλRµ , (∀)λ, µ ∈ D, s’appelle une pseudo-résolvante. Théorème 1.1.22 Soit D ∋ λ 7−→ Rλ ∈ B(E) une pseudo-résolvante. Alors: i) RλRµ = RµRλ, (∀)λ, µ ∈ D; ii) KerRλ et ImRλ ne dépendent pas de λ ∈ D; iii) Rλ est la résolvante d’un opérateur linéaire A fermé et défini sur un sous espace dense si et seulement si KerRλ = {0} et ImRλ = E . 12 CHAPITRE 1. INTRODUCTION Preuve i) Soient λ, µ ∈ D. Alors, nous avons: Rλ − Rµ = (µ − λ)RλRµ Rµ − Rλ = (λ − µ)RµRλ , d’où: 0 = (µ − λ)RλRµ + (λ − µ)RµRλ . Par suite, on a RλRµ = RµRλ. ii) Soient µ ∈ D et x ∈ Ker Rµ. Alors Rµx = 0. Si λ ∈ D, on a: Rλx − Rµx = (µ − λ)RλRµx . Donc Rλx = 0. Par conséquent x ∈ Ker Rλ. Il s’ensuit que Ker Rλ ne dépend pas de λ ∈ D. Soient µ ∈ D et y ∈ Im Rµ. Alors il existe x ∈ E tel que Rµx = y. Si λ ∈ D, nous avons: Rλx − Rµx = (µ − λ)RλRµx . Donc: Rλx − y = (µ − λ)Rλy , ou bien: y = Rλ (x + (λ − µ)y) . Donc il existe u = x + (λ− µ)y ∈ E tel que y = Rλu. Par conséquent y ∈ Im Rλ. Il s’ensuit que Im Rλ ne dépend pas de λ ∈ D. iii) =⇒ Si Rλ est une résolvante pour un opérateur linéaire A fermé et défini sur un sous espace dense, alors Rλ est une application bijective, d’où Ker Rλ = {0} et Rλ = (λI − A)−1. Par suite, Rλ−1 = λI − A et D = D(A) = E . Par conséquent Im Rλ = D = E . ⇐= Soient D ∋ λ 7−→ Rλ ∈ B(E) une pseudo-résolvante et λ ∈ D tel que Ker Rλ = {0}. Alors pour y ∈ Im Rλ, il existe un seul xλ ∈ E tel que y = Rλxλ. Mais pour λ, µ ∈ D, on a: Rλy − Rµy = (µ − λ)RλRµy . D’autre part: Rλy − Rµy = RλRµxµ − RµRλxλ = = RλRµxµ − RλRµxλ = RλRµ (xµ − xλ) . 1.2. LES OPÉRATEURS DISSIPATIFS 13 Donc xµ − xλ = (µ − λ)y, d’où λy − xλ = µy − xµ. Par conséquent, l’opérateur: A : Im Rλ −→ E Ay = λy − xλ = λy − Rλ−1y est correctement défini (valeur indépendante de λ). De même D(A) = Im Rλ = E . Puis que Rλ ∈ B(E), il résulte du théorème du graphe fermé ([DS’67, Theorem II.2.4, pag. 57]) que R−1λ est un opérateur fermé. Donc A = λI − Rλ−1 est un opérateur fermé. De plus, on a: −1y = xλ = λy − Ay = (λI − A)y . Par conséquent Rλ = (λI − A)−1 est la résolvante de A. 1.2 Les opérateurs dissipatifs Dans la suite, nous notons par E∗ l’espace dual du E et par ‖ . ‖∗ sa norme. Pour tout x ∈ E , nous désignerons par J (x) l’ensemble: x∗ ∈ E∗ ∣〈x, x∗〉 = ‖x‖2 = ‖x∗‖2∗ Définition 1.2.1 On dit que l’opérateur linéaire A : D(A) ⊂ E −→ E est dissi- patif si pour tout x ∈ D(A), il existe x∗ ∈ J (x) tel que Re〈Ax, x∗〉 ≤ 0. Dans la proposition suivante nous présentons une caractérisation très utile pour les opérateurs dissipatifs. Proposition 1.2.2 Un opérateur linéaire A : D(A) ⊂ E −→ E est dissipatif si et seulement si pour tout α > 0 on a: ‖(αI − A)x‖ ≥ α‖x‖ , (∀)x ∈ D(A). Preuve =⇒ Supposons que A : D(A) ⊂ E −→ E est un opérateur dissipatif. Pour tout x ∈ D(A), il existe x∗ ∈ J (x) tel que Re〈Ax, x∗〉 ≤ 0. Si α > 0, alors nous 14 CHAPITRE 1. INTRODUCTION avons: ‖(αI − A)x‖ ‖x‖ = ‖(αI − A)x‖ ‖x∗‖∗ ≥ ≥ |〈(αI − A)x, x∗〉| ≥ Re〈(αI − A)x, x∗〉 = = Re〈αx, x∗〉 − Re〈Ax, x∗〉 ≥ α‖x‖2 , d’où il résulte l’inégalité de l’énoncé. ⇐= Soit A : D(A) ⊂ E −→ E tel que pour tout α > 0 et x ∈ D(A) on ait: ‖(αI − A)x‖ ≥ α‖x‖ . Soit y∗α ∈ J ((αI − A)x). On a donc: 〈(αI − A)x, y∗α〉 = ‖(αI − A)x‖ = ‖y∗α‖ d’où: ‖y∗α‖∗ = ‖(αI − A)x‖ ≥ α‖x‖ . Nous définissons: z∗α = ‖y∗α‖∗ et désignons par B1(E∗) la boule unité de E∗ et par ∂B1(E∗) sa frontière. Il est évident que z∗α ∈ ∂B1(E∗). De plus: α‖x‖ ≤ ‖(αI − A)x‖ = 1‖y∗α‖∗ 〈(αI − A)x, y∗α〉 = = 〈(αI − A)x, z∗α〉 et par conséquent: α‖x‖ ≤ Re〈(αI − A)x, z∗α〉 = Re〈αx, z∗α〉 − Re〈Ax, z∗α〉 ≤ ≤ α |〈x, z∗α〉| − Re〈Ax, z∗α〉 ≤ α‖x‖ ‖z∗α‖∗ − Re〈Ax, z = α‖x‖ − Re〈Ax, z∗α〉 . Il s’ensuit que: Re〈Ax, z∗α〉 ≤ 0 , d’où: −Re〈Ax, z∗α〉 ≤ |〈Ax, z∗α〉| ≤ ‖Ax‖ ‖z∗α‖∗ = ‖Ax‖ et par conséquent: α‖x‖ ≤ αRe〈x, z∗α〉 + ‖Ax‖ . 1.2. LES OPÉRATEURS DISSIPATIFS 15 Donc: Re〈x, z∗α〉 ≥ ‖x‖ − ‖Ax‖ . D’autre part, en appliquant le théorème d’Alaoglu ([DS’67, Theorem V.4.2, pag. 424]), on voit que la boule unité B1(E∗) est faiblement compacte. Par conséquent, il existe une sous suite ⊂ (z∗α)α>0 et il existe z∗ ∈ B1(E∗) tel que: z∗β −→ z∗ si β → ∞ pour la topologie faible. Comme on a Re〈Ax, z∗β〉 ≤ 0 Re〈x, z∗β〉 ≥ ‖x‖ − ‖Ax‖ , on obtient par passage à limite en β → ∞: Re〈Ax, z∗〉 ≤ 0 Re〈x, z∗〉 ≥ ‖x‖ . Mais comme: Re〈x, z∗〉 ≤ |〈x, z∗〉| ≤ ‖x‖‖z∗‖∗ ≤ ‖x‖ , il s’ensuit que: 〈x, z∗〉 = ‖x‖ . Si nous prenons x∗ = ‖x‖z∗, il vient: 〈x, x∗〉 = 〈x, ‖x‖z∗〉 = ‖x‖〈x, z∗〉 = ‖x‖2 . Il en résulte que x∗ ∈ J (x). Finalement, on voit que Re〈Ax, x∗〉 ≤ 0, d’où l’on tire que l’opérateur A est dissipatif. Proposition 1.2.3 Soit A : D(A) ⊂ E −→ E un opérateur dissipatif. S’il existe α0 > 0 tel que Im (α0I − A) = E , alors pour tout α > 0 on a Im (αI − A) = E . Preuve Soient A : D(A) ⊂ E −→ E un opérateur dissipatif et α0 > 0 tel que Im (α0I − A) = E . Compte tenu de la proposition 1.2.2, on voit que: ‖(α0I − A)x‖ ≥ α0‖x‖ , (∀)x ∈ D(A) 16 CHAPITRE 1. INTRODUCTION et comme Im (α0I − A) = E , il en résulte que α0I − A ∈ GL(E) et α0 appartient donc bien à ρ(A). Soit (xn)n∈N ⊂ D(A) tel que xn −→ x et Axn −→ y si n → ∞. Il est clair que: (α0I − A)xn −→ α0x − y si n → ∞ et par conséquent: xn = R(α0; A)(α0I − A)xn −→ R(α0; A)(α0x − y) si n → ∞. Par suite, nous obtenons: R(α0; A)(α0x − y) = x . Comme Im R(α0; A) ⊂ D(A), on voit que x ∈ D(A). De plus: (α0I − A)x = α0x − y , d’où il résulte que Ax = y. Par conséquent, A est un opérateur fermé. Nous désignerons par A l’ensemble: {α ∈]0,∞) |Im(αI − A) = E } . Soit α ∈ A. Comme A est un opérateur dissipatif, on voit que: ‖(αI − A)x‖ ≥ α‖x‖ , (∀)x ∈ D(A), d’où il résulte que α ∈ ρ(A). Puisque ρ(A) est un ensemble ouvert, il existe un voisinage V de α contenu dans ρ(A). Comme V∩]0,∞) ⊂ A, on voit que A est un ensemble ouvert. Soit (αn)n∈N ⊂ A tel que αn −→ α si n → ∞. Comme Im (αnI − A) = E , (∀)n ∈ N, on observe que pour tout y ∈ E , il existe xn ∈ D(A) tel que: (αnI − A)xn = y , (∀)n ∈ N, et par suite, il existe C > 0 tel que: ‖xn‖ ≤ ‖y‖ ≤ C , (∀)n ∈ N. Par conséquent: αn‖xn − xm‖ ≤ ‖(αmI − A)(xn − xm)‖ = = ‖(αmI − A)xn − (αmI − A)xm‖ = ‖αmxn − Axn − y‖ = = ‖αmxn − αnxn + αnxn − Axn − y‖ = = ‖(αm − αn)xn + y − y‖ = |αm − αn|‖xn‖ ≤ C|αm − αn| , 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 17 d’où il résulte que (xn)n∈N est une suite de Cauchy. Puisque E est un espace de Banach, il s’ensuit que (xn)n∈N converge vers un point x ∈ E . Alors, on en déduit Axn −→ αx − y si n → ∞ et comme A est un opérateur fermé, on obtient x ∈ D(A) et αx − Ax = y. Par suite, Im (αI − A) = E et α ∈ A. Donc A est fermé dans ]0,∞) et comme il existe α0 ∈ A, nous déduisons que A =]0,∞). 1.3 Semi-groupes uniformément continus Dans la suite nous présenterons quelques problèmes concernant les semi- groupes uniformément continus d’opérateurs linéaires bornés sur un espace de Ba- nach E . Définition 1.3.1 On appelle semi-groupe uniformément continu d’opérateurs li- néaires bornés sur E une famille {T (t)} t≥0 ⊂ B(E) vérifiant les propriétés sui- vantes: i) T (0) = I; ii) T (t + s) = T (t)T (s) , (∀)t, s ≥ 0; iii) limtց0 ‖T (t) − I‖ = 0. Définition 1.3.2 On appelle générateur infinitésimal du semi-groupe uniformément continu {T (t)} t≥0 l’opérateur linéaire: A : E −→ E , A = lim T (t) − I Lemme 1.3.3 Soit A ∈ B(E). Alors est un semi-groupe uniformément continu d’opérateurs linéaires bornés sur E dont le générateur infinitésimal est A. Preuve Soit A ∈ B(E) et [0,∞) ∋ t 7−→ T (t) ∈ B(E) une application définie par: T (t) = etA = 18 CHAPITRE 1. INTRODUCTION La série du membre de droite de l’égalité est convergente pour la topologie de la norme de B(E). De plus, il est évident que T (0) = I et T (t + s) = T (t)T (s) quels que soient t, s ≥ 0. Compte tenu de l’inégalité: ‖T (t) − I‖ ≤ et‖A‖ − 1 , (∀)t ≥ 0, il résulte: ‖T (t) − I‖ = 0 . Donc la famille {T (t)} t≥0 ⊂ B(E) est un semi-groupe uniformément continu. D’autre part, puisque: T (t) − I etA − I − tA − I − tA I + tA + − I − tA tk‖A‖k 1 + t‖A‖ + tk‖A‖k − 1 − t‖A‖ et‖A‖ − 1 − t‖A‖ et‖A‖ − 1 t‖A‖ ‖A‖ − ‖A‖ −→ 0 si t ց 0, nous obtenons: T (t) − I = A . Le semi-groupe {T (t)} t≥0 admet donc pour générateur infinitésimal l’opérateur Lemme 1.3.4 Etant donné un opérateur A ∈ B(E), il existe un unique semi- groupe uniformément continu {T (t)} t≥0 tel que: T (t) = etA , (∀)t ≥ 0. Preuve Si {S(t)} t≥0 est un autre semi-groupe uniformément continu engendré par A, nous avons: T (t) − I S(t) − I = A . 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 19 Par conséquent: T (t) − S(t) = 0 . Pour a ∈]0,∞), nous considérons l’intervalle Ia = [0, a[. Comme {T (t)}t≥0 et {S(t)} t≥0 sont des semi-groupes uniformément continus, nous voyons que les ap- plications: t 7−→ ‖T (t)‖ t 7−→ ‖S(t)‖ sont continues. Il existe ca ∈ [1,∞) tel que: {‖T (t)‖, ‖S(t)‖} ≤ ca . Si ε > 0, il existe t0 ∈ Ia, t0 > 0, tel que: T (t) − S(t) , (∀)t ∈]0, t0[. Soit t ∈ Ia arbitrairement fixé et n ∈ N tel que tn ∈]0, t0[. Alors: T (t) − S(t) = (n − 1) t (n − 1) t (n − 2) t (n − 2) t − · · · − T (n − k) t (n − k − 1) t (k + 1) (n − k − 1) t quel que soit t ∈ Ia. De l’inégalité: nous obtenons: 20 CHAPITRE 1. INTRODUCTION et par suite: ‖T (t) − S(t)‖ ≤ ca < ε , (∀)t ∈ Ia. Puisque ε > 0 est arbitraire, il en résulte que T (t) = S(t), pour tout t ∈ Ia. Mais, comme a ∈]0,∞) est aussi arbitraire, il s’ensuit que T (t) = S(t), (∀)t ∈ [0,∞). Présentons maintenant la condition nécessaire et suffisante pour qu’un opérateur soit le générateur infinitésimal d’un semi-groupe uniformément continu. Théorème 1.3.5 Un opérateur A : E −→ E est le générateur infinitésimal d’un semi-groupe uniformément continu si et seulement si A est un opérateur linéaire borné. Preuve =⇒ Soit A : E −→ E le générateur infinitésimal d’un semi-groupe uni- formément continu {T (t)} t≥0 ⊂ B(E). Alors: ‖T (t) − I‖ = 0 . L’application [0,∞) ∋ t 7→ T (t) ∈ B(E) est continue et par suite T (s) ds ∈ B(E). Avec le lemme 1.1.1, on voit que: T (s) ds = T (0) = I . Il existe donc τ > 0 tel que: T (t) dt − I < 1 . Compte tenu de la remarque 1.1.3, l’élément 1 T (t)dt est inversible, d’où il s’ensuit T (t) dt est inversible. Nous avons: T (h) − I T (t) dt = T (t + h) dt − T (t) dt T (u) du − 1 T (u) du . 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 21 Avec le lemme 1.1.1, nous obtenons: T (h) − I T (t) dt = = lim T (u) du − 1 T (u) du = T (τ) − T (0) = T (τ) − I , d’où: T (h) − I = [T (τ) − I] T (t) dt Par conséquent, le générateur infinitésimal du semi-groupe uniformément continue {T (t)} t≥0 est l’opérateur: A = [T (τ) − I] T (t) dt ∈ B(E) . ⇐= Cette implication est évidente compte tenu du lemme 1.3.3 et du lemme 1.3.4. Corollaire 1.3.6 Soient {T (t)} t≥0 un semi-groupe uniformément continu et A son générateur infinitésimal. Alors: i) il existe ω ≥ 0 tel que ‖T (t)‖ ≤ eωt , (∀)t ≥ 0; ii) l’application [0,∞) ∋ t 7−→ T (t) ∈ B(E) est différentiable pour la topologie de la norme et: dT (t) = AT (t) = T (t)A , (∀)t ≥ 0. Preuve i) Nous avons: ‖T (t)‖ = ∥ ≤ et‖A‖ , (∀)t ≥ 0. Pour ω = ‖A‖, nous obtenons l’inégalité: ‖T (t)‖ ≤ eωt , (∀)t ≥ 0. L’assertion (ii) provient des égalités suivantes: A = lim T (t) − I = lim T (t) − T (0) t − 0 , 22 CHAPITRE 1. INTRODUCTION nous en déduisons que l’application considérée est dérivable au point t = 0. Soient t > 0 et h > 0. Alors: T (t + h) − T (t) − AT (t) T (h) − I ‖T (t)‖ ≤ T (h) − I et‖A‖ , d’où: T (t + h) − T (t) − AT (t) = 0 . Par conséquent, l’application considérée dans l’énoncé est dérivable à droite et on d+T (t) = AT (t) , (∀)t > 0. Soient t > 0 et h < 0 tel que t + h > 0. Alors: T (t + h) − T (t) − AT (t) I − T (−h) − AT (−h) ‖T (t + h)‖ ≤ T (−h) − I −h − AT (−h) e(t+h)‖A‖ , d’où il vient: T (t + h) − T (t) = AT (t) . Par conséquent l’application considérée dans l’énoncé est dérivable à gauche et nous avons: d−T (t) = AT (t) , (∀)t > 0. Finalement on voit que l’application considérée dans l’énoncé est dérivable sur [0,∞) et nous avons: dT (t) = AT (t) , (∀)t ≥ 0. On vérifie que AT (t) = T (t)A , (∀)t ≥ 0. Maintenant abordons quelques problèmes de théorie spectrale pour un semi- groupe uniformément continu {T (t)} t≥0 ayant pour le générateur infinitésimal l’opérateur A ∈ B(E). Théorème 1.3.7 Soient {T (t)} un semi-groupe uniformément continu et A son générateur infinitésimal. Si λ ∈ C tel que Reλ > ‖A‖, alors l’application: Rλ : E −→ E , 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 23 Rλx = e−λtT (t)x dt définit un opérateur linéaire borné, λ ∈ ρ(A) et Rλx = R(λ; A)x , pour tout x ∈ E . Preuve Soit λ ∈ C avec Reλ > ‖A‖. Avec le corollaire 1.3.6 (i), on voit que: ‖T (t)‖ ≤ e‖A‖t , (∀)t ≥ 0. De même, nous avons: ∥e−λtT (t)x ∥ ≤ e−(Reλ−‖A‖)t‖x‖ , (∀)x ∈ E , e−(Reλ−‖A‖)t dt = Reλ − ‖A‖ . L’application Rλ est donc bornée et il est clair que Rλ est linéaire. Pour x ∈ E , nous avons: RλAx = T (t)Ax dt = −λt d T (t)x dt = = −x + λ e−λtT (t)x dt = −x + λRλx , d’où x = Rλ(λI − A)x, pour tout x ∈ E . Par conséquent Rλ(λI − A) = I. De même, nous avons: ARλx = A e−λtT (t)x dt = e−λtAT (t)x dt = e−λtT (t)Ax dt = RλAx , (∀)x ∈ E . Par suite, on a ARλx = RλAx = −x + λRλx, pour tout x ∈ E . Il en résulte que (λI − A)Rλ = I. Par conséquent λ ∈ ρ(A) et Rλ = R(λ; A). Définition 1.3.8 L’opérateur Rλ : E −→ E défini par: Rλx = e−λtT (t)x dt , λ ∈ C avec Reλ > ‖A‖, s’appelle la transformée de Laplace du semi-groupe uniformément continu {T (t)} ayant pour générateur infinitésimal l’opérateur A. 24 CHAPITRE 1. INTRODUCTION Remarque 1.3.9 On a: {λ ∈ C |Reλ > ‖A‖} ⊂ ρ(A) σ(A) ⊂ {λ ∈ C |Reλ ≤ ‖A‖} . De même, nous obtenons: ‖R(λ; A)‖ ≤ 1 Reλ − ‖A‖ pour tout λ ∈ C avec Reλ > ‖A‖. Pour obtenir des représentations de type Riesz-Dunford et de type Bromwich, on a besoin d’une classe spéciale de contours de Jordan. Définition 1.3.10 Un contour de Jordan lisse et fermé qui entoure σ(A), s’appelle un contour de Jordan A-spectral s’il est homotope avec un cercle Cr de centre O et de rayon r > ‖A‖. Théorème 1.3.11 (Riesz-Dunford) Soit A le générateur infinitésimal d’un semi- groupe uniformément continu {T (t)} t≥0. Si ΓA est un contour de Jordan A- spectral, alors nous avons: T (t) = eλtR(λ; A) dλ , (∀)t ≥ 0. Preuve Soit ΓA un contour de Jordan A-spectral. Alors ΓA est homotope avec un cercle Cr de centre O et de rayon r > ‖A‖. Par conséquent, on a: R(λ; A) dλ = R(λ; A) dλ , (∀)t ≥ 0. Compt tenu de la proposition 1.1.7 (iii), on voit que: R(λ; A) = uniformément par rapport à λ sur les sous-ensembles compacts de {λ ∈ C| |λ| > ‖A‖}, particulièrement sur le cercle Cr. On a: R(λ; A) dλ = 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 25 Appliquons la formule de Cauchy ([DS’67, pag. 228]) avec la fonction f(λ) = eλt, nous obtenons: , (∀)n ∈ N . Par conséquent: eλtR(λ; A) dλ = = etA = T (t) , (∀)t ≥ 0. Théorème 1.3.12 (Bromwich) Soient {T (t)} t≥0 un semi-groupe uniformément continu et A son générateur infinitésimal. Si a > ‖A‖, alors nous avons: T (t) = a+i∞∫ eztR(z; A) dz et l’intégrale est uniformément convergente par rapport à t sur les intervalles com- pacts de ]0,∞). Preuve Soit a > ‖A‖, pour R > 2a nous considérons le contour de Jordan lisse et fermé ΓR = Γ R ∪ Γ”R R = {a + iτ |τ ∈ [−R, R]} Γ”R = a + R(cos ϕ + i sin ϕ) Remarquons que pour z ∈ Γ′R on a: |z| = |a + iτ | > a > ‖A‖ . De même, si z ∈ Γ”R, alors nous avons: |z| = |a + (cos ϕ + i sin ϕ)| = |a − [−R(cos ϕ + i sin ϕ)]| ≥ ≥ ||a| − | − R(cos ϕ + i sin ϕ)|| = |a − R| = R − a > ‖A‖ . Par conséquent, z ∈ ΓR implique z ∈ ρ(A). De plus, on voit que ΓR est homotope au cercle C de centre O et de rayon R− a. Il s’ensuit donc que ΓR est un contour de Jordan A-spectral et avec le théorème de Riesz-Dunford nous obtenons: T (t) = eztR(z; A) dz , (∀)t ≥ 0, 26 CHAPITRE 1. INTRODUCTION pour tout R > 2a. Il en résulte: T (t) = I t(R) + I t (R) , (∀)t ≥ 0, pour tout R > 2a, où nous avons noté t(R) = eztR(z; A) dz I”t (R) = eztR(z; A) dz . Montrons que eztR(z; A) dz = 0 , (∀)t ≥ 0. Compte tenu de la proposition 1.1.7 (iii), on voit que: R(z; A) = la série de la partie droite de l’égalité étant uniformément convergente par rapport à z sur les sous-ensembles compacts de {z ∈ C| |z| > ‖A‖}, particulièrement sur Γ”R. Il s’ensuit que: I”(R) = An dz , (∀)t ≥ 0, pour tout R > 2a. Notons At(R) = Bt(R) = Pour l’intégrale At(R), avec la paramétrisation suivante z = a + R(cos ϕ + i sin ϕ) , ϕ ∈ 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 27 on obtient: At(R) = et(a+R cos ϕ+i sinϕ) R(− sin ϕ + i cos ϕ) dϕ etR cos ϕeitR sinϕ (cos ϕ + i sin ϕ) dϕ Il en résulte que: ‖At(R)‖ ≤ ∣etR cos ϕ ∣eitR sin ϕ |z| | cos ϕ + i sin ϕ| dϕ ≤ tR cos ϕ 1 R − a dϕ = R − ae etR cos ϕ dϕ parce que z ∈ Γ”R implique |z| = |a + R(cos ϕ + i sin ϕ)| > R − a |z| < R − a . De l’inégalité R > 2a, on obtient 2R − 2a > R, d’où R − a < 2 . Par conséquent: ‖At(R)‖ ≤ etR cos ϕ dϕ , (∀)t ≥ 0, pour tout R > 2a. Soient 0 ≤ t1 < t2 et t ∈ [t1, t2]. Pour tout R > 2a et tout etR cos ϕ ≤ 1 . Comme etR cos ϕ = 0 , 28 CHAPITRE 1. INTRODUCTION avec le théorème de la convergence bornée de Lebesgue il résulte que etR cos ϕ dϕ = 0 et par conséquent At(R) = 0 uniformément par rapport à t ∈ [t1, t2]. Soit maintenant l’intégrale Bt(R) = Pour tout t ∈ [t1, t2] et tout R > 2a on a: etR cos ϕ ≤ 1 , (∀)ϕ ∈ On voit que: (R − a)n+1 etR cos ϕ dϕ ≤ πeta R (R − a)n+1 . Puisque R > 2a > a + ‖A‖, il vient: ‖Bt(R)‖ ≤ R − a R − a et comme R − a < 1 , il en résulte que: ‖Bt(R)‖ ≤ eta R − a R − a − ‖A‖ , quel que soit R > 2a. Donc Bt(R) = 0 , uniformément par rapport à t ∈ [t1, t2]. Il s’ensuit donc que I”t (R) = 0 , 1.3. SEMI-GROUPES UNIFORMÉMENT CONTINUS 29 uniformément par rapport à t ∈ [t1, t2]. Par conséquent: T (t) = lim eztR(z; A) dz = Re λ+i∞∫ Re λ−i∞ eztR(z; A) dz , uniformément par rapport à t sur les intervalles compacts de ]0,∞). Nous finissons cette section avec le théorème spectral pour les semi-groupes uniformément continus. Théorème 1.3.13 (spectral mapping) Soit A le générateur infinitésimal du semi-groupe uniformément continu {T (t)} t≥0. Alors: etσ(A) = σ (T (t)) , (∀)t ≥ 0. Preuve Montrons que etσ(A) ⊂ σ (T (t)) , (∀)t ≥ 0. Soit ξ ∈ σ(A). Pour λ ∈ ρ(A), l’application: gξ(λ) = eξt − eλt ξ − λ est analytique dans un voisinage de σ(A). Compte tenu du théorème 1.3.11, on voit que: eξtI − eAt = (ξI − A)gξ(A) . Si eξt ∈ ρ (T (t)), alors il existe Q = eξtI − T (t) ∈ B(E). Par conséquent: I = (ξI − A)gξ(A)Q , d’où il résulte que ξ ∈ ρ(A), ce qui est absurde. Donc eξt ∈ σ (T (t)) et par suite etσ(A) ⊂ σ (T (t)). Montrons que σ (T (t)) ⊂ etσ(A). Soit µ ∈ σ (T (t)). Supposons par absurde que µ 6∈ etσ(A). Alors pour λ ∈ ρ(A), l’application: h(λ) = µ − eλt est définie sur un voisinage du σ(A). Donc: µI − etA 30 CHAPITRE 1. INTRODUCTION et il en résulte que µ ∈ ρ (T (t)) et cela est absurde. Par suite µ ∈ etσ(A), d’où σ (T (t)) ⊂ etσ(A). Finalement on voit que: etσ(A) = σ (T (t)) , (∀)t ≥ 0 . 1.4 Notes Les notions préséntées dans cet chapitre se trouvent en majorité des travaux concernant les semi- groupes d’opérateurs linéaires. Pour les propriétés de la pseudo-résolvante, on peut consulter [Pa’83-1, pag. 36]. De même, on peut trouver les opérateurs dissipatifs dans [Pa’83-1, pag. 13], [Da’80, pag. 52] et [Ah’91, pag. 30]. Une jolie généralisation pour ces opérateurs est donnée dans [CHADP’87, pag. 61]. Le théorème 1.3.5 a été montré pour la première fois indépendemment par Yosida dans [Yo’36] et par Nathan dans [Na’35]. Nous avons consulté aussi les preuves données par Pazy dans [Pa’83-1, pag. 2], Ahmed dans [Ah’91, pag. 4] et Davies dans [Da’80, pag. 19]. Compte tenu du ce théorème, on peut introduire la transformée de Laplace pour un semi-groupe uni- formément continu et on peut montrer le théorème 1.3.11 et le théorème 1.3.13 comme des applications du calcul fonctionnel de Dunford ([DS’67, pag. 568]). Pour le théorème 1.3.12 on peut consulter [Pa’83-1, pag. 25]. Chapitre 2 Semi-groupes de classe C0 2.1 Définitions. Propriétés élémentaires Dans le cadre de ce paragraphe, nous introduisons une classe plus générale que la classe des semi-groupes uniformément continus et nous étudions leurs pro- priétés élémentaires. Définition 2.1.1 On appelle C0-semi-groupe (ou semi-groupe fortement continu) d’opérateurs linéaires bornés sur E une famille {T (t)} t≥0 ⊂ B(E) vérifiant les propriétés suivantes: i) T (0) = I; ii) T (t + s) = T (t)T (s) , (∀)t, s ≥ 0; iii) limtց0 T (t)x = x , (∀)x ∈ E . Définition 2.1.2 On appelle générateur infinitésimal d’un C0-semi-groupe {T (t)}t≥0, un opérateur A défini sur l’ensemble: D(A) = x ∈ E T (t)x − x existe Ax = lim T (t)x − x , (∀)x ∈ D(A). Remarque 2.1.3 Il est clair que le générateur infinitésimal d’un C0-semi-groupe est un opérateur linéaire. Remarque 2.1.4 Puisque: ‖T (t)x − x‖ ≤ ‖T (t) − I‖ ‖x‖ 32 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 pour tout x ∈ E et tout t ≥ 0, il en résulte que les semi-groupes uniformément continus sont C0-semi-groupes. Mais il existe des C0-semi-groupes qui ne sont pas uniformément continus, comme nous pouvons le voir dans les exemples suivants. Exemple 2.1.5 Soit: C[0,∞) = {f : [0,∞) → R| f est uniformément continue et bornée} . Avec la norme ‖f‖C[0,∞) = supα∈[0,∞) |f(α)|, l’espace C[0,∞) devient un espace de Banach. Définissons: (T (t)f) (α) = f(t + α) , (∀)t ≥ 0 et α ∈ [0,∞). Evidemment T (t) est un opérateur linéaire, et, en plus, on a: i) (T (0)f) (α) = f(0 + α) = f(α). Donc T (0) = I; ii) (T (t + s)f) (α) = f(t + s + α) = (T (t)f) (s + α) = (T (t)T (s)f) (α), (∀)f ∈ C[0,∞). Donc T (t + s) = T (t)T (s), (∀)t, s ≥ 0; iii) limtց0 ‖T (t)f − f‖C[0,∞) = limtց0 supα∈[0,∞) |f(t + α) − f(α)| = 0, (∀)f ∈ C[0,∞). De même, nous avons: ‖T (t)f‖C[0,∞) = sup α∈[0,∞) |(T (t)f) (α)| = sup α∈[0,∞) |f(t + α)| = = sup β∈[t,∞) |f(β)| ≤ sup β∈[0,∞) |f(β)| = ‖f‖C[0,∞) , (∀)t ≥ 0. Donc ‖T (t)‖ = 1, (∀)t ≥ 0. Par conséquent {T (t)} t≥0 est un C0-semi-groupe d’opérateurs linéaires bornés sur C[0,∞), nommé le C0-semi-groupe de translation à droite. Soit A : D(A) ⊂ C[0,∞) −→ C[0,∞) le générateur infinitésimal du C0-semi-groupe {T (t)} t≥0. Si f ∈ D(A), alors nous avons: Af(α) = lim T (t)f(α) − f(α) = lim f(α + t) − f(α) = f ′(α) , uniformément par rapport à α. Par conséquent: D(A) ⊂ {f ∈ C[0,∞) |f ′ ∈ C[0,∞)} . Si f ∈ C[0,∞) tel que f ′ ∈ C[0,∞), alors: T (t)f − f − f ′ C[0,∞) = sup α∈[0,∞) (T (t)f) (α) − f(α) − f ′(α) 2.1. DÉFINITIONS. PROPRIÉTÉS ÉLÉMENTAIRES 33 Mais: (T (t)f) (α) − f(α) − f ′(α) f(α + t) − f(α) − f ′(α) f(τ)|α+tα − f ′(α) [f ′(τ) − f ′(α)] dτ |f ′(τ) − f ′(α)| dτ −→ 0 uniformément par rapport à α pour t ց 0. Par suite: T (t)f − f − f ′ C[0,∞) −→ 0 si t ց 0, d’où f ∈ D(A) et: {f ∈ C[0,∞) |f ′ ∈ C[0,∞)} ⊂ D(A) . Par conséquent D(A) = {f ∈ C[0,∞) |f ′ ∈ C[0,∞)} et Af = f ′. Comme cet opérateur est non borné, compte tenu du théorème 1.3.5, il ne peut pas engendrer un semi-groupe uniformément continu. Exemple 2.1.6 Considérons l’espace Lp]0,∞), 1 ≤ p < ∞, avec la norme: ‖f‖p = |f(α)|p dα Avec cette norme, Lp]0,∞), 1 ≤ p < ∞, est un espace de Banach. Définissons: (T (t)f) (α) = f(t + α) , (∀)t ≥ 0 et α ∈]0,∞). Nous avons: ‖T (t)f‖ |(T (t)f) (α)|p dα |f(α + t)|p dα |f(β)|p dβ |f(β)|p dβ = ‖f‖p . Donc ‖T (t)‖ = 1, (∀)t ≥ 0. Il est évident que T (0) = I et T (t + s) = T (t)T (s), (∀)t, s ≥ 0. 34 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 De plus, on a: ‖T (t)f − f‖ = lim |(T (t)f) (α) − f(α)|p dα = lim |f(α + t) − f(α)|p dα = 0 . Par suite {T (t)} t≥0 est un C0-semi-groupe d’opérateurs linéaires bornés sur Lp]0,∞). Soit A : D(A) ⊂ Lp]0,∞) −→ Lp]0,∞) le générateur infinitésimal du C0-semi- groupe {T (t)} t≥0. Si f ∈ D(A), alors nous avons: Af(α) = lim T (t)f(α) − f(α) = lim f(α + t) − f(α) = f ′(α) uniformément par rapport à α. Par conséquent: D(A) ⊂ {f ∈ Lp]0,∞) |f ′ ∈ Lp]0,∞)} . Si f ∈ Lp]0,∞) tel que f ′ ∈ Lp]0,∞), alors on a: T (t)f − f − f ′ (T (t)f) (α) − f(α) − f ′(α) Mais: (T (t)f) (α) − f(α) − f ′(α) f(α + t) − f(α) − f ′(α) ′(α)τ [f ′(τ) − f ′(α)] dτ uniformément par rapport à α si t ց 0. Alors: T (t)f − f − f ′ −→ 0 si t ց 0 et on voit que: {f ∈ Lp]0,∞) |f ′ ∈ Lp]0,∞)} ⊂ D(A) . Par conséquent: D(A) = {f ∈ Lp]0,∞) |f ′ ∈ Lp]0,∞)} et Af = f ′. 2.1. DÉFINITIONS. PROPRIÉTÉS ÉLÉMENTAIRES 35 Théorème 2.1.7 Soit {T (t)} t≥0 ⊂ B(E) une famille ayant les propriétés: i) T (0) = I; ii) T (t + s) = T (t)T (s) , (∀)t, s ≥ 0. Les affirmations suivantes sont équivalentes: iii’) limtց0 T (t) = I dans la topologie forte; iii”) limtց0 T (t) = I dans la topologie faible. Preuve iii′) =⇒ iii′′) Cette implication est évidente. iii′′) =⇒ iii′) Supposons que: T (t) = I dans la topologie faible. Alors, pour tout x ∈ E et tout x∗ ∈ E∗ on a: 〈T (t)x, x∗〉 = 〈x, x∗〉 . Si t0 > 0, alors pour tout h > 0, nous obtenons: |〈T (t0 + h)x, x∗〉 − 〈T (t0)x, x∗〉| = = |〈T (t0)T (h)x, x∗〉 − 〈T (t0)x, x∗〉| = = |〈T (t0)[T (h)x − x], x∗〉| −→ 0 si h ց 0, quel que soit x ∈ E et x∗ ∈ E∗. Par suite, l’application: [0,∞) ∋ t 7−→ T (t) ∈ B(E) est faiblement continue à droite sur [0,∞) et on voit qu’elle est faiblement continue sur ]0,∞). En particulier, elle est faiblement mesurable sur ]0,∞). Pour x ∈ E arbitrairement fixé, considérons l’application: [0,∞) ∋ t 7−→ T (t)x ∈ E et désignons par: Im T ( . )x = {T (t)x|t ∈ [0,∞)} son image. Supposons que l’ensemble: Kx = {T (q)x|q ∈ Q∗+} ⊂ Im T ( . )x n’est pas dense dans Im T ( . )x. Alors, il existe t0 ∈ [0,∞) tel que T (t0)x ∈ Im T ( . )x et: d (T (t0)x,Kx) > 0 . 36 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 En appliquant un corollaire du théorème de Hahn-Banach ([DS’67, Corollary II.3.13, pag. 64]), on voit qu’il existe x∗0 ∈ E∗ tel que: 〈kn, x∗0〉 = 0 , (∀)kn ∈ Kx 〈T (t0)x, x∗0〉 = 1 . Soit tn ∈ Q∗+ tel que limn→∞ tn = t0. Alors, compte tenu de la continuité faible de l’application considérée, il vient: 0 = lim 〈T (tn) x, x∗0〉 = 〈T (t0)x, x∗0〉 = 1 , ce qui est absurde. Il s’ensuit que: Kx = Im T ( . )x , pour tout x ∈ E . Par conséquent, l’application considérée a une image séparable. En appliquant le théorème de Pettis ([Hi’48, Theorem 3.2.2, pag. 36]), il vient que cette application est fortement mesurable sur ]0,∞). Alors, il résulte que pour tout xn ∈ E avec ‖x‖ ≤ 1, l’application: ‖T ( . )‖ = sup ‖T ( . )xn‖ < ∞ est mesurable sur ]0,∞). Montrons que l’application ‖T ( . )‖ est bornée sur les intevalles [α, β] ⊂]0,∞). Compte tenu du théorème de Banach-Steinhaus ([DS’67, Theorem II.1.11, pag. 52]), il est suffisant de montrer que ‖T ( . )x‖ est bornée sur les intervalles [α, β], pour tout x ∈ E . Soient α, β ∈]0,∞). Supposons qu’il existe x0 ∈ E tel que pour tout M > 0 on puisse trouver s ∈ [α; β] tel que: ‖T (s)x0‖ > M . Donc il existe tn ∈ [α, β], n ∈ N, tel que: tn = τ ∈ [α, β] ‖T (tn)x0‖ > n , (∀)n ∈ N. D’autre part, l’application ‖T ( . )x0‖ est mesurable sur ]0,∞). Donc il existe une constante K > 0 et un ensemble mesurable F ⊂ [0, τ ] avec m(F) > τ tel que: ‖T (t)x0‖ ≤ K . 2.1. DÉFINITIONS. PROPRIÉTÉS ÉLÉMENTAIRES 37 Si nous considérons: En = {tn − η|η ∈ F ∩ [0, tn]} , on voit que En est un ensemble mesurable et pour n suffisamment grand, nous obtenons: m(En) ≥ Alors, pour tout η ∈ F ∩ [0, tn], n ∈ N, nous avons: n ≤ ‖T (tn)x0‖ ≤ ‖T (tn − η)‖ ‖T (η)x0‖ ≤ ‖T (tn − η)‖K , d’où: ‖T (t)‖ ≥ n , (∀)t ∈ En. Si nous notons: E = lim sup alors on voit que: m(E) ≥ τ ‖T (t)‖ = ∞ , (∀)t ∈ E ce qui est absurde. Par conséquent, il existe M > 0 tel que: ‖T (t)‖ ≤ M , (∀)t ∈ [α, β]. Soient α, β, t, t0 ∈]0,∞) tel que: 0 < α < t < β < t0 et ε > 0 tel que β < t0 − ε. Alors pour tout x ∈ E , l’application: [α, β] ∋ t 7−→ T (t0)x = T (t)T (t0 − t)x ∈ E ne dépend pas de t, donc elle est Bôchner intégrable par rapport à t ∈ [α, β] et pour tout x ∈ E on a: (β − α) [T (t0 ± ε)x − T (t0)x] dt = T (t) [T (t0 ± ε − t)x − T (t0 − t)x] dt , 38 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 d’où: |β − α|‖T (t0 ± ε)x − T (t0)x‖ ≤ ‖T (t)‖ ‖T (t0 ± ε − t)x − T (t0 − t)x‖ dt ≤ t0−α∫ ‖T (τ ± ε)x − T (τ)x‖ dτ −→ 0 si ε ց 0 , compte tenu de [Hi’48, théorème 3.6.3, pag.46]. Il s’ensuit que l’application: [0,∞) ∋ t 7−→ T (t) ∈ B(E) est fortement continue sur ]0,∞). En particulier, pour x ∈ E arbitrairement fixé, l’ensemble: X = {T (t)x|t ∈ [0, 1]} est séparable. Donc il contient une partie dénombrable dense: X0 = {T (tn)x|tn ∈]0, 1[, n ∈ N} . Par conséquent, il existe une suite (xn)n∈N ⊂ X0 tel que: ‖xn − x‖ = lim ‖T (tn)x − x‖ = 0 . Comme: ‖T (t)x − x‖ ≤ ≤ ‖T (t)x − T (t + tn)x‖ + ‖T (t + tn)x − T (tn)x‖ + ‖T (tn)x − x‖ ≤ ≤ ‖T (t)‖ ‖x − T (tn)x‖ + ‖T (t + tn)x − T (tn)x‖ + ‖T (tn)x − x‖ ≤ t∈[0,1] ‖T (t)‖ + 1 + ‖T (t + tn)x − T (tn)x‖ , il vient: T (t)x = x , (∀)x ∈ E et par conséquent: T (t) = I dans la topologie forte. Dans la suite, nous considérons la topologie forte pour étudier les propriétés des C0-semi-groupes. 2.1. DÉFINITIONS. PROPRIÉTÉS ÉLÉMENTAIRES 39 Théorème 2.1.8 Soit {T (t)} t≥0 un C0-semi-groupe d’opérateurs linéaires bornés. Alors: i) il existe τ > 0 et M ≥ 1 tel que: ‖T (t)‖ ≤ M , (∀)t ∈ [0, τ ]; ii) il existe ω ∈ R et M ≥ 1 tel que: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Preuve i) Supposons que pour tout τ > 0 et tout M ≥ 1, il existe t ∈ [0, τ ] tel que ‖T (t)‖ > M . Pour τ = 1 et M = n ∈ N∗, il existe tn ∈ tel que ‖T (tn)‖ > n. Donc la suite (‖T (tn)‖)n∈N∗ est non bornée. Si la suite (‖T (tn)x‖)n∈N∗ était bornée pour tout x ∈ E , alors compte tenu du théorème de Banach-Steinhaus ([DS’67, Theorem II.1.11, pag. 52]), il en résulterait que (‖T (tn)‖)n∈N∗ serait bornée, mais cela contredit l’affirmation précédente. Donc il existe x0 ∈ E tel que (‖T (tn)x0‖)n∈N∗ soit non bornée. D’autre part, compte tenu de la définition 2.1.1 (iii), il résulte que limn→∞ ‖T (tn)x0‖ = x0 et cela est contradictoire. ii) Pour h > 0 et t > h, nous noterons m = ∈ N∗. Compte tenu du théorème de division avec reste, il existe r ∈ [0, h) tel que t = mh + r. Alors: ‖T (t)‖ = ‖T (mh)T (r)‖ ≤ ‖T (h)‖m ‖T (r)‖ ≤ ≤ MmM ≤ Me th ln M . L’inégalité de l’énoncé en résulte en prenant ω = 1 ln M . Corollaire 2.1.9 Si {T (t)} est un C0-semi-groupe, alors l’application: [0,∞) ∋ t 7−→ T (t)x ∈ E est continue sur [0,∞), quel que soit x ∈ E . Preuve Soient t0, h ∈ [0,∞) et x ∈ E . Si t0 < h, nous avons: ‖T (t0 + h)x − T (t0)x‖ ≤ ‖T (t0)‖ ‖T (h)x − x‖ ≤ ≤ Meωt0 ‖T (h)x − x‖ . 40 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Si t0 > h, nous obtenons: ‖T (t0 − h)x − T (t0)x‖ ≤ ‖T (t0 − h)‖ ‖T (h)x − x‖ ≤ ≤ Meω(t0−h) ‖T (h)x − x‖ . La continuité forte en t0 de l’application considérée dans l’énoncé est évidente. Définition 2.1.10 On dit que le C0-semi-groupe {T (t)}t≥0 est uniformément borné s’il existe M ≥ 1 tel que: ‖T (t)‖ ≤ M , (∀)t ≥ 0. Théorème 2.1.11 Soit {T (t)} t≥0 un C0-semi-groupe pour lequel il existe ω ∈ R et M ≥ 1 tel que: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Alors la famille {S(t)} t≥0 ⊂ B(E), où: S(t) = e−ωtT (t) , (∀)t ≥ 0, est un C0-semi-groupe ayant la propriété: ‖S(t)‖ ≤ M , (∀)t ≥ 0. De plus, si A est le générateur infinitésimal du C0-semi-groupe {T (t)}t≥0, alors le C0-semi-groupe {S(t)}t≥0 a pour générateur infinitésimal l’opérateur B = A−ωI. Preuve Dans les conditions du théorème, il est évident que {S(t)} t≥0 est un C0- semi-groupe et: ‖S(t)‖ = ∥eωtT (t) ∥ ≤ e−ωtMeωt = M , (∀)t ≥ 0. Donc {S(t)} est un C0-semi-groupe uniformément borné. Soit A le générateur infinitésimal du C0-semi-groupe {T (t)}t≥0. Si B est le générateur infinitésimal du C0-semi-groupe {S(t)}t≥0, alors pour tout x ∈ D(A), nous avons: S(h)x − x = lim e−ωhT (h)x − x = lim e−ωh − 1 T (h)x + lim T (h)x − x = −ωx + Ax = (A − ωI)x , 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 41 d’où il résulte que x ∈ D(B) et Bx = (A − ωI)x. Soit x ∈ D(A). Alors, nous obtenons: T (h)x − x = lim eωhS(h)x − x = lim eωh − 1 + lim S(h)x − x = (ωI + B)x , d’où il vient que x ∈ D(A) et Ax = (ωI + B)x. Par conséquent D(A) = D(B) et B = A − ωI. Remarque 2.1.12 Soit {T (t)} t≥0 un C0-semi-groupe pour lequel il existe ω ∈ R et M ≥ 1 tel que: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Si ω < 0, alors nous obtenons: ‖T (t)‖ ≤ Meωt ≤ M , (∀)t ≥ 0. Par conséquent on peut considérer que ω ≥ 0. Nous noterons par SG(M, ω) l’ensemble des C0-semi-groupes {T (t)}t≥0 ⊂ B(E) pour lesquels il existe ω ≥ 0 et M ≥ 1 tel que: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0 . Avec le théorème 2.1.11 nous voyons que le passage entre la classe SG(M, ω) avec ω > 0 et la classe SG(M, 0) est très simple. 2.2 Propriétés générales des C0-semi-groupes Proposition 2.2.1 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitési- mal. Si x ∈ D(A), alors T (t)x ∈ D(A) et on a l’égalité: T (t)Ax = AT (t)x , (∀)t ≥ 0. 42 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Preuve Soit x ∈ D(A). Alors pour tout t ≥ 0, nous avons: T (t)Ax = T (t) lim T (h)x − x = lim T (h)T (t)x − T (t)x Donc T (t)x ∈ D(A) et on a T (t)Ax = AT (t)x , (∀)t ≥ 0. Remarque 2.2.2 On voit que: T (t)D(A) ⊆ D(A) , (∀)t ≥ 0. Théorème 2.2.3 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors l’application: [0,∞) ∋ t 7−→ T (t)x ∈ E est dérivable sur [0,∞), pour tout x ∈ D(A) et nous avons: T (t)x = T (t)Ax = AT (t)x , (∀)t ≥ 0; ii) T (t)x − x = T (s)Ax ds , (∀)t ≥ 0. Preuve i) Soient x ∈ D(A) , t ≥ 0 et h > 0. Alors: T (t + h)x − T (t)x − T (t)Ax ≤ ‖T (t)‖ T (h)x − x ≤ Meωt T (h)x − x Par conséquent: T (t + h)x − T (t)x = T (t)Ax , d’où: T (t)x = T (t)Ax , (∀)t ≥ 0. Si t − h > 0, alors nous avons: T (t − h)x − T (t)x −h − T (t)Ax ≤ ‖T (t − h)‖ T (h)x − x − Ax + Ax − T (h)Ax ≤ Meω(t−h) T (h)x − x + ‖T (h)Ax − Ax‖ 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 43 Par suite: T (t − h)x − T (t)x −h = T (t)Ax T (t)x = T (t)Ax , (∀)t ≥ 0. Il s’ensuit que l’application considérée dans l’énoncé est dérivable sur [0,∞), quel que soit x ∈ D(A). De plus, on a l’égalité: T (t)x = T (t)Ax = AT (t)x , (∀)t ≥ 0. ii) Si x ∈ D(A), alors nous avons: T (s)x = T (s)Ax , (∀)s ∈ [0, t] , t ≥ 0, d’où: T (s)Ax ds = T (s) ds = T (t)x − x , (∀)t ≥ 0. On peut obtenir une formule de représentation de type Taylor pour les C0- semi-groupes avec la généralisation du théorème 2.2.3 (ii). Théorème 2.2.4 (Taylor) Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: T (t)x = Aix + (n − 1)! (t − u)n−1T (u)Anx du quels que soient x ∈ D(An), t ≥ 0 et n ∈ N∗. Preuve Compte tenu du théorème 2.2.3 (ii), pour x ∈ D(A) et t ≥ 0 on a: T (t)x = x + T (u)Ax du . Supposons que pour t ≥ 0 et x ∈ D(Ak) nous ayons: T (t)x = (k − 1)! (t − u)k−1T (u)Akx du . Si x ∈ D(Ak+1), alors x ∈ D(Ak) et Akx ∈ D(A). Il en résulte que: T (t)x = (k − 1)! (t − s)k−1T (s)Anx ds . 44 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Mais: T (s)x = x + T (u)Ax du . Il vient: (t − s)k−1T (s)Akx = (t − s)k−1Akx + (t − s)k−1 T (u)Ak+1x du et par conséquent: (t − s)k−1T (s)Akx ds = (t − s)k−1Akx ds + (t − s)k−1 T (u)Ak+1x du ds = Akx + (t − s)k−1T (u)Ak+1x ds du = Akx + (t − u)k T (u)Ak+1x du . Nous en déduisons que: T (t)x = Aix + (k − 1)! Akx + (t − u)kT (u)Ak+1x du (t − u)kT (u)Ak+1x du , d’où il résulte l’égalité considérée dans l’énoncé. Lemme 2.2.5 Soit {T (t)} t≥0 un C0-semi-groupe. Alors: T (s)x ds = T (t)x quels que soient x ∈ E et t ≥ 0. Preuve L’égalité de l’énoncé résulte de l’évaluation: T (s)x ds − T (t)x (T (s) − T (t))x ds ≤ sup s∈[t,t+h] ‖T (s)x − T (t)x‖ et de la continuité de l’application [0,∞) ∋ t 7−→ T (t)x ∈ E . 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 45 Proposition 2.2.6 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Si x ∈ E , alors T (s)x ds ∈ D(A) et on a l’égalité: T (s)x ds = T (t)x − x , (∀)t ≥ 0. Preuve Soient x ∈ E et h > 0. Alors: T (h) − I T (s)x ds = T (s + h)x ds − 1 T (s)x ds = T (u)x du − 1 T (s)x ds = T (u)x du − 1 T (u)x du − 1 T (u)x du = T (u)x du − 1 T (u)x du . Par pasage à limite pour h ց 0 et compte tenu du lemme 2.2.5, nous obtenons: T (s)x ds = T (t)x − x , (∀)t ≥ 0 T (s)x ds ∈ D(A). Théorème 2.2.7 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: i) D(A) = E ; ii) A est un opérateur fermé. Preuve i) Soient x ∈ E et tn > 0 , n ∈ N, tel que limn→∞ tn = 0. Alors: T (s)x ds ∈ D(A) , (∀)n ∈ N, d’où: xn = lim T (s)x ds = T (0)x = x . 46 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Par conséquent D(A) = E . ii) Soit (xn)n∈N ⊂ D(A) tel que limn→∞ xn = x et limn→∞ Axn = y. Alors: ‖T (s)Axn − T (s)y‖ ≤ ‖T (s)‖ ‖Axn − y‖ ≤ Meωt ‖Axn − y‖ quel que soit s ∈ [0, t]. Par suite T (s)Axn −→ T (s)y, pour n → ∞, uniformément par rapport à s ∈ [0, t]. D’autre part, puisque xn ∈ D(A), nous avons: T (t)xn − xn = T (s)Axn ds , d’où: [T (t)xn − xn] = lim T (s)Axn ds , ou bien: T (t)x − x = T (s)y ds . Finalement, on voit que: T (t)x − x = lim T (s)y ds = y . Par suite x ∈ D(A) et Ax = y, d’où il résulte que A est un opérateur fermé. Nous montrons maintenant un résultat qui concerne l’unicité de l’engendrement pour les C0-semi-groupes. Théorème 2.2.8 (l’unicité de l’engendrement) Soient deux C0-semi-groupes {T (t)} et {S(t)} ayant pour générateur infinitésimal le même opérateur A. Alors: T (t) = S(t) , (∀)t ≥ 0. Preuve Soient t > 0 et x ∈ D(A). Définissons l’application: [0, t] ∋ s 7−→ U(s)x = T (t − s)S(s)x ∈ D(A). Alors: U(s)x = T (t − s)S(s)x + T (t − s) d S(s)x = = −AT (t − s)S(s)x + T (t − s)AS(s)x = 0 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 47 quel que soit x ∈ D(A). Par suite U(0)x = U(t)x, pour tout x ∈ D(A), d’où: T (t)x = S(t)x , (∀)x ∈ D(A) et t ≥ 0. Puisque D(A) = E et T (t), S(t) ∈ B(E), pour tout t ≥ 0, il résulte que: T (t)x = S(t)x , (∀)t ≥ 0 et x ∈ E , ou bien: T (t) = S(t) , (∀)t ≥ 0. Théorème 2.2.9 Soient {T (t)} t≥0 un C0-semi-groupe, A son générateur infinité- simal et F ∈ B(E). Alors T (t)F = FT (t) pour tout t ≥ 0 si et seulement si: FD(A) ⊆ D(A) FAx = AFx , (∀)x ∈ D(A). Preuve =⇒ Soit F ∈ B(E) tel que: T (t)F = FT (t) , (∀)t ≥ 0 et x ∈ D(A). Alors, nous avons: T (t)Fx − Fx = lim FT (t)x − Fx = lim T (t)x − x Par conséquent Fx ∈ D(A) et on a AFx = FAx, pour tout x ∈ D(A). ⇐= Soit F ∈ B(E) tel que: FD(A) ⊆ D(A) AFx = FAx , (∀)x ∈ D(A). Pour tout t ≥ 0 et tout x ∈ D(A), définissons l’application: [0, t] ∋ s 7−→ U(s)x = T (t − s)FT (s)x ∈ D(A) . Alors nous avons: U(s)x = T (t − s)FT (s)x + T (t − s) d FT (s)x = = −AT (t − s)FT (s)x + T (t − s)FAT (s)x = 0 , 48 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 compte tenu de la commutativité. Par conséquent: U(0)x = U(t)x , (∀)x ∈ D(A), d’où on obtient: T (t)Fx = FT (t)x , pour tout t ≥ 0 et tout x ∈ D(A). Comme D(A) = E et T (t)F, FT (t) ∈ B(E) pour tout t ≥ 0, nous obtenons: T (t)Fx = FT (t)x , pour tout t ≥ 0 et tout x ∈ E . Nous finissons cette section avec une généralisation du théorème 2.2.7. Théorème 2.2.10 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: i) D(Ap) = E , quel que soit p ∈ N∗; ii) Ap est un opérateur fermé, quel que soit p ∈ N∗; iii) l’application: ‖ . ‖D(Ap) : D(Ap) −→ R+ , ‖x‖D(Ap) = est une norme avec laquelle D(Ap) devient un espace de Banach, pour tout p ∈ N∗. Preuve i) Pour p = 1, compte tenu du théorème 2.2.7(i), il résulte que D(A) = E . Soit: C∞0 = {ϕ :]0,∞) → E |ϕ indéfiniment dérivable avec un support compact} . Notons: ϕ(t)T (t)x dt x ∈ E , ϕ ∈ C∞0 Nous montrons que F ⊂ D(Ap) , (∀)p ∈ N. Pour y ∈ F et h > 0, nous obtenons: T (h) − I ϕ(t)T (t + h)x dt − ϕ(t)T (t)x dt ϕ(u − h) − ϕ(u) T (u)x du . 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 49 Puisque: ϕ(u − h) − ϕ(u) T (u)x −→ −ϕ(u)(1)T (u)x si h ց 0, uniformément par rapport à u ∈ supp ϕ, en passant à limite pour h ց 0, nous obtenons: Ay = − T (u)x du . Donc y ∈ D(A). Il en résulte que F ⊂ D(A) et par récurrence on peut montrer que F ⊂ D(Ap) et: Apy = (−1)p T (t)x dt quel que soit p ∈ N∗. Nous montrons maintenant que F est dense dans E . Supposons que F n’est pas dense dans E . Alors il existe x0 ∈ E tel que d(x0,F) > 0. En appliquant un corollaire du théorème de Hahn-Banach ([DS’67, Corollary II.3.13, pag. 64]), on voit qu’il existe x∗0 ∈ E∗ tel que 〈x0, x∗0〉 = 1 et 〈y, x∗0〉 = 0, pour tout y ∈ F . Alors: ϕ(t)〈T (t)x, x∗0〉 dt = ϕ(t)T (t)x dt, x∗0 = 0 , (∀)ϕ ∈ C∞0 et x ∈ E . Par conséquent, pour tout x ∈ E , nous avons: 〈T (t)x, x∗0〉 = 0 , (∀)t ∈ [0,∞), parce que dans le cas contraire, on peut trouver ϕ ∈ C∞0 tel que: ϕ(t)〈T (t)x, x∗0〉 dt 6= 0 ce qui est contradictoire. Il s’ensuit que pour tout x ∈ E , on a: 〈T (t)x, x∗0〉 = 0 , (∀)t ∈ [0,∞), d’où: 〈x, x∗0〉 = 〈T (0)x, x∗0〉 = 0 , (∀)x ∈ E , ce qui est absurde. Finalement, on voit que F est dense dans E et donc D(An) = E . ii) Compte tenu du théorème 2.2.7(ii), on voit que: A : D(A) ⊂ E −→ E 50 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 est un opérateur fermé. Supposons que: Ak : D(Ak) ⊂ E −→ E est un opérateur fermé et montrons que: Ak+1 : D(Ak+1) ⊂ E −→ E est un opérateur fermé. Soit (xn)n∈N ⊂ D(Ak+1) tel que: xn = x Ak+1xn = y . Mais xn ∈ D(Ak+1) est équivalent avec xn ∈ D(Ak) et Akxn ∈ D(A). Alors xn ∈ D(Ak), limn→∞ xn = x, comme Ak est un opérateur fermé, ceci implique x ∈ D(Ak) et limn→∞ A kxn = A kx. Comme Akxn ∈ D(A), limn→∞ Akxn = Akx et A est un opérateur fermé, il s’ensuit que Akx ∈ D(A) et limn→∞ A . Nous avons obtenu donc que x ∈ D(Ak+1), Akx ∈ D(A) et limn→∞ Ak+1xn = Ak+1x, d’où il résulte que x ∈ D(Ak+1) et Ak+1x = y. Par conséquent Ak+1 est un opérateur fermé, d’où on obtient (ii). iii) Pour p = 1 on peut vérifier facilement les propriétés de norme de l’application: ‖ . ‖D(A) : D(A) −→ R+ , ‖x‖D(A) = ‖x‖ + ‖Ax‖ . Donc D(A) est un espace normé. Soit (xn)n∈N∗ ⊂ D(A) tel que ‖xm − xn‖D(A) −→ 0 pour m, n → ∞. Alors: ‖xm − xn‖ + ‖Axm − Axn‖ −→ 0 pour m, n → ∞. Donc: ‖xm − xn‖ −→ 0 et ‖Axm − Axn‖ −→ 0 pour m, n → ∞. Puis que E est un espace de Banach, il résulte que les suites (xn)n∈N et (Axn)n∈N sont convergentes. Donc xn −→ x et Axn −→ y pour n → ∞. Comme A est un opérateur fermé, il résulte que x ∈ D(A) et y = Ax. Par conséquent: ‖xn − x‖D(A) = ‖xn − x‖ + ‖Axn − Ax‖ −→ 0 pour n → ∞. 2.2. PROPRIÉTÉS GÉNÉRALES DES C0-SEMI-GROUPES 51 Donc la suite (xn)n∈N est convergente par rapport à la norme ‖ . ‖D(A). Il s’ensuit que D(A) est un espace de Banach avec la norme ‖ . ‖D(A). Supposons que l’application: ‖ . ‖D(Ak) : D(Ak) −→ R+ , ‖x‖D(Ak) = est une norme avec laquelle D(Ak) est un espace de Banach. Montrons que: ‖ . ‖D(Ak+1) : D(Ak+1) −→ R+ , ‖x‖D(Ak+1) = est une norme avec laquelle D(Ak+1) devient un espace de Banach. On peut vérifier facilement les propriétés de norme de l’application ‖ . ‖D(Ak+1). Donc D(Ak+1) est un espace normé. Soit (xn)n∈N ⊂ D(Ak+1) tel que: ‖xm − xn‖D(Ak+1) −→ 0 si m, n → ∞. Alors nous avons: xm − Aixn ∥ −→ 0 si m, n → ∞, d’où il s’ensuit que: ∥Aixm − Aixn ∥ −→ 0 si m, n → ∞, pour tout i ∈ {0, 1, . . . , k + 1}. Mais E est un espace de Banach. Donc pour tout i ∈ {0, 1, . . . , k + 1}, les suites (Aixn)n∈N sont convergentes et comme les opérateurs Ai sont fermés pour tout i ∈ {1, 2, . . . , k + 1}, on voit que: ∥Aixn − Aix ∥ −→ 0 si n → ∞, pour tout i ∈ {0, 1, . . . , k + 1}. Par conséquent: xn − Aix ∥ −→ 0 si n → ∞, d’où: ‖xm − x‖D(Ak+1) −→ 0 si n → ∞. 52 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Finalement, on voit que D(Ak+1) est un espace de Banach et l’affirmation de l’énoncé en résulte. 2.3 Le théorème de Hille - Yosida Dans ce paragraphe nous présentons un résultat très important concernant les semi-groupes de classe C0. Il s’agit du célèbre théorème de Hille-Yosida qui donne une caractérisation pour les opérateurs qui sont générateurs de C0-semi- groupes. Nous avons besoin de quelques résultats intermédiaires. Dans la suite, pour ω ≥ 0 nous désignerons par Λω l’ensemble {λ ∈ C |Reλ > ω}. Théorème 2.3.1 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Si λ ∈ Λω, alors l’application: Rλ : E −→ E , Rλx = e−λtT (t)x dt définit un opérateur linéaire borné sur E , λ ∈ ρ(A) et Rλx = R(λ; A)x , pour tout x ∈ E . Preuve Soit λ ∈ Λω. Puisque {T (t)}t≥0 ∈ SG(M, ω), nous avons: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0 et on voit que: ∥e−λtT (t)x ∥ ≤ e−Reλt ‖T (t)‖ ‖x‖ ≤ Me−(Reλ−ω)t‖x‖ , (∀)x ∈ E . Définissons l’application: Rλ : E −→ E , Rλx = T (t)x dt . 2.3. LE THÉORÈME DE HILLE - YOSIDA 53 Il est clair que Rλ est un opérateur linéaire. De plus, on a: ‖Rλx‖ ≤ ∥e−λtT (t)x ∥ dt ≤ M Reλ − ω‖x‖ , (∀)x ∈ E , d’où il résulte que Rλ est un opérateur linéaire borné. Si x ∈ E , alors nous avons: T (h)Rλx − Rλx e−λtT (t + h)x dt − 1 e−λtT (t)x dt = e−λ(s−h)T (s)x ds − 1 e−λtT (t)x dt = e−λsT (s)x ds − 1 e−λtT (t)x dt = e−λsT (s)x ds − e−λsT (s)x ds e−λtT (t)x dt = eλh − 1 e−λsT (s)x ds − e e−λsT (s)x ds . Par passage à limite, on obtient: T (h)Rλx − Rλx = λRλx − x . Il en résulte que Rλx ∈ D(A) et ARλx = λRλx − x , (∀)x ∈ E , ou bien (λI − A)Rλx = x , (∀)x ∈ E . Si x ∈ D(A), alors nous obtenons: RλAx = e−λtT (t)Ax dt = T (t)x dt = e−λtT (t)x e−λtT (t)x dt = x + λRλx , d’où: Rλ(λI − A)x = x , (∀)x ∈ D(A). Finalement, on voit que λ ∈ ρ(A) et Rλx = R(λ; A)x , pour tout x ∈ E . 54 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Remarque 2.3.2 On voit que pour tout λ ∈ Λω on a: Im R(λ; A) = Im Rλ ⊆ D(A) R(λ; A)D(A) = RλD(A) ⊆ D(A) . Définition 2.3.3 L’opérateur: Rλ : E −→ E Rλx = e−λtT (t)x dt , λ ∈ Λω, s’appelle la transformée de Laplace du semi-groupe {T (t)} t≥0 ∈ SG(M, ω). Remarque 2.3.4 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors nous avons: {λ ∈ C |Reλ > ω} ⊂ ρ(A). σ(A) ⊂ {λ ∈ C |Reλ ≤ ω} . Théorème 2.3.5 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Pour tout λ ∈ Λω on a: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n , (∀)n ∈ N Preuve Soit {T (t)} t≥0 ∈ SG(M, ω) . Alors: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Compte tenu du théorème 2.3.1, si λ ∈ Λω, nous avons λ ∈ ρ(A) et: R(λ; A)x = Rλx = e−λtT (t)x dt , (∀)x ∈ E . De plus: ‖R(λ; A)‖ ≤ M Reλ − ω . Il est clair que: R(λ; A)x = − T (t)x dt , (∀)x ∈ E 2.3. LE THÉORÈME DE HILLE - YOSIDA 55 et par récurrence on peut montrer que: R(λ; A)x = (−1)n tne−λtT (t)x dt , (∀)x ∈ E et n ∈ N∗. D’autre part, avec la proposition 1.1.16 (iii) nous obtenons: R(λ; A)x = (−1)nn!R(λ; A)n+1x , (∀)x ∈ E et n ∈ N∗. Par suite, on a: (−1)nn!R(λ; A)n+1x = (−1)n T (t)x dt , (∀)x ∈ E et n ∈ N∗, d’où il résulte que: R(λ; A) (n − 1)! tn−1e−λtT (t)x dt , (∀)x ∈ E et n ∈ N∗. De plus: ‖R(λ; A)nx‖ ≤ M‖x‖ (n − 1)! tn−1e−(Reλ−ω)t dt = (n − 1)! n − 1 Reλ − ω tn−2e−(Reλ−ω)t dt = · · · = M‖x‖ (Reλ − ω)n quels que soient x ∈ E et n ∈ N∗. Par conséquent: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n , (∀)n ∈ N Lemme 2.3.6 Soit A : D(A) ⊂ E −→ E un opérateur linéaire vérifiant les pro- priétés suivantes: i) A est un opérateur fermé et D(A) = E ; ii) il existe ω ≥ 0 et M ≥ 1 tel que Λω ⊂ ρ(A) et pour λ ∈ Λω, on a: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n , (∀)n ∈ N Alors pour tout λ ∈ Λω, nous avons: Reλ→∞ λR(λ; A)x = x , (∀)x ∈ E . De plus λAR(λ; A) ∈ B(E) et: Reλ→∞ λAR(λ; A)x = Ax , (∀)x ∈ D(A). 56 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Preuve Soient x ∈ D(A) et λ ∈ C tel que Reλ > ω. Alors R(λ; A)(λI −A)x = x. Si Reλ → ∞, nous avons: ‖λR(λ; A)x − x‖ = ‖R(λ; A)Ax‖ ≤ ‖R(λ; A)‖ ‖Ax‖ ≤ Reλ − ω‖Ax‖ −→ 0 , d’où il résulte que: Reλ→∞ λR(λ; A)x = x , (∀)x ∈ D(A). Soit x ∈ E , puisque D(A) = E , il existe une suite (xn)n∈N ⊂ D(A) telle que xn −→ x si n → ∞. Nous avons: ‖λR(λ; A)x − x‖ ≤ ≤ ‖λR(λ; A)x − λR(λ; A)xn‖ + ‖λR(λ; A)xn − xn‖ + ‖xn − x‖ ≤ ≤ ‖λR(λ; A)‖ ‖x − xn‖ + ‖λR(λ; A)xn − xn‖ + ‖xn − x‖ ≤ ≤ |λ|M Reλ − ω‖x − xn‖ + Reλ − ω‖Axn‖ + ‖xn − x‖ = |λ|M + Reλ − ω Reλ − ω ‖xn − x‖ + Reλ − ω ‖Axn‖ . Mais xn −→ x si n → ∞. Donc pour tout ε > 0 , il existe nε ∈ N tel que: ‖xnε − x‖ < ε Reλ − ω |λ|M + Reλ − ω . Par conséquent: ‖λR(λ; A)x − x‖ < ε + M Reλ − ω ‖Axnε‖ , d’où: lim sup Reλ→∞ ‖λR(λ; A)x − x‖ < ε , (∀)x ∈ E , ou bien: Reλ→∞ λR(λ; A)x = x , (∀)x ∈ E . De plus: λAR(λ; A) = λ [λI − (λI − A)] R(λ; A) = λ [λR(λ; A) − I] = λ2R(λ; A) − λI. Par suite, on a: ‖λAR(λ; A)x‖ = ‖λ [λR(λ; A) − I]x‖ ≤ ≤ |λ| ‖λR(λ; A)x − x‖ ≤ |λ| (‖λR(λ; A)x‖ + ‖x‖) ≤ ≤ |λ| Reλ − ω + 1 ‖x‖ , (∀)x ∈ E 2.3. LE THÉORÈME DE HILLE - YOSIDA 57 et on voit que λAR(λ; A) ∈ B(E). Si x ∈ D(A), alors nous avons: λR(λ; A)Ax = λ2R(λ; A) − λI = λAR(λ; A) , d’où il résulte que: Reλ→∞ λAR(λ; A)x = lim λR(λ; A)Ax = Ax , (∀)x ∈ D(A). Remarque 2.3.7 On peut dire que les opérateurs bornés λAR(λ; A) sont des approximations pour l’opérateur non borné A. C’est le motif pour lequel on intro- duit la définition suivante. Définition 2.3.8 La famille {Aλ}λ∈Λω ⊂ B(E), où Aλ = λAR(λ; A), pour tout λ ∈ Λω, s’appelle l’approximation généralisée de Yosida de l’opérateur A. Remarque 2.3.9 Evidemment, pour λ ∈ Λω, on voit que Aλ est le générateur infinitésimal d’un semi-groupe uniformément continu . Nous utiliserons cette famille pour montrer l’existence d’un C0-semi-groupe engendré par A. Lemme 2.3.10 Soit A : D(A) ⊂ E −→ E un opérateur linéaire vérifiant les propriétés suivantes: i) A est un opérateur fermé et D(A) = E ; ii) il existe ω ≥ 0 et M ≥ 1 tel que Λω ⊂ ρ(A) et pour λ ∈ Λω, on a: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n , (∀)n ∈ N Si {Aλ}λ∈Λω est l’approximation généralisée de Yosida de l’opérateur A, alors pour tous α, β ∈ Λω nous avons: ∥etAαx − etAβx ∥ ≤ M2teωt ‖Aαx − Aβx‖ , (∀)x ∈ E et t ≥ 0. Preuve Soient α, β ∈ Λω, v ∈ [0, 1] et x ∈ E . Alors: evtAαe(1−v)tAβ x = tAαe vtAαe(1−v)tAβ x − tevtAαAβe(1−v)tAβ x . On peut facilement vérifier que Aα, Aβ, e vtAα et e(1−v)tAβ commutent quels que soient α, β ∈ Λω et t ≥ 0. Nous obtenons: evtAαe(1−v)tAβ x vtAαAαe (1−v)tAβ x − tevtAαAβe(1−v)tAβ x 58 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 d’où: evtAαe(1−v)tAβ x tevtAαe(1−v)tAβ Aαx − tevtAαe(1−v)tAβ Aβx ou bien: etAαx − etAβx = t evtAαe(1−v)tAβ (Aαx − Aβx) dv quels que soient t ≥ 0 et x ∈ E . Nous en déduisons que: ∥etAαx − etAβx ∥ ≤ t ∥evtAα ∥e(1−v)tAβ ∥ ‖Aαx − Aβx‖ dv . D’autre part, nous avons: ∥etAα t(α2R(α;A)−αI) ∥e−αtIeα 2tR(α;A) ≤ e−Reαt tkα2kR(α; A) ≤ e−Reαt tk|α|2k ∥R(α; A) ≤ e−Reαt tk|α|2kM k!(Reα − ω)k = Me −Reαt t|α|2 Reα−ω = Me−Reαte t|α|2 Reα−ω = Me ωReα+Im2α Reα−ω quel que soient α ∈ Λω et t ≥ 0. Soit r > 1 tel que: ωReα + Im2α Reα − ω < ωr . Alors, nous avons: ωReα + Im2α < ωrReα − ω2r , d’où: ωReα < ωrReα − ω2r , ou bien: ω2r < ω(r − 1)Reα . Il en découle: Reα > r − 1ω . Par conséquent, pour tout r > 1 et tout α ∈ Λω tel que Reα > rr−1ω, on obtient: ∥etAα ∥ ≤ Merωt , (∀)t ≥ 0 2.3. LE THÉORÈME DE HILLE - YOSIDA 59 et par passage à limite pour r ց 1, nous obtenons: ∥etAα ∥ ≤ Meωt , (∀)t ≥ 0, pour tout α ∈ Λω. Il vient: ∥etAαx − etAβx ∥ ≤ t MeωvtMeω(1−v)t ‖Aαx − Aβx‖ dv = = M2teωt ‖Aαx − Aβx‖ quels que soient x ∈ E et t ≥ 0. Maintenant nous présentons une variante du célèbre théorème de Hille - Yosida pour les semi-groupes de classe SG(M, ω). Théorème 2.3.11 (Hille - Yosida) Un opérateur linéaire: A : D(A) ⊂ E −→ E est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(M, ω) si et seule- ment si: i) A est un opérateur fermé et D(A) = E ; ii) il existe ω ≥ 0 et M ≥ 1 tel que Λω ⊂ ρ(A) et pour λ ∈ Λω, on a: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n , (∀)n ∈ N Preuve =⇒ On obtient cette implication en tenant compte du théorème 2.2.7 et du théorème 2.3.5. ⇐= Supposons que l’opérateur A : D(A) ⊂ E −→ E posséde les propriétés (i) et (ii). Soit {Aλ}λ∈Λω , l’approximation généralisée de Yosida de l’opérateur A. Compte tenu du lemme 2.3.6, il résulte que Aλ ∈ B(E) et: Reλ→∞ Aλx = Ax , (∀)x ∈ D(A). Pour λ ∈ Λω, soit {Tλ(t)}t≥0 = le semi-groupe uniformément continu engendré par Aλ. Avec le lemme 2.3.10, on a: ‖Tα(t)x − Tβ(t)x‖ ≤ M2teωt ‖Aαx − Aβx‖ , (∀)α, β ∈ Λω, x ∈ D(A) et t ≥ 0. Soient [D(A)] l’espace de Banach D(A) avec la norme ‖ . ‖D(A), et B([D(A)], E) l’espace des opérateurs linéaires bornés définis sur [D(A)] avec valeur dans E , doté 60 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 de la topologie forte. Notons par C ([0,∞);B([D(A)], E)) l’espace des fonctions continues définies sur [0,∞) à valeurs dans B([D(A)], E) doté de la topologie de la convergence uniforme sur les intervalles compacts de [0,∞). Si [a, b] ⊂ [0,∞), alors pour tout x ∈ D(A) nous avons: t∈[a,b] ‖Tα(t)x − Tβ(t)x‖ ≤ M2beωb (‖Aαx − Ax‖ + ‖Aβx − Ax‖) −→ 0 si Reα,Reβ → ∞, d’où il résulte que {Tλ(t)}t≥0 est une suite de Cauchy dans C ([0,∞);B([D(A)], E)). Donc, il existe un unique T0 ∈ C ([0,∞);B(D(A), E)) tel que Tλ(t)x −→ T0(t)x, si Reλ → ∞, quel que soit x ∈ D(A), pour la topologie de la convergence uniforme sur les intervalles compacts de [0,∞). Puisque: ‖Tλ(t)‖ ≤ Meωt , (∀)t ≥ 0, on obtient: ‖T0(t)x‖ ≤ Meωt‖x‖ , (∀)t ≥ 0 et x ∈ D(A) Considérons l’application linéaire: Θ0 : D(A) −→ C ([a, b]; E) Θ0x = T0( . )x quel que soit [a, b] ⊂ [0,∞). Comme nous avons: ‖Θ0x‖C([a,b];E) = sup t∈[a,b] ‖T0(t)x‖ ≤ Meωb‖x‖ ≤ Meωb‖x‖D(A) , (∀)x ∈ D(A), on voit que Θ0 est une application continue et puisque D(A) = E , elle se prolonge de façon unique en une application linéaire continue: Θ : E −→ C ([a, b]; E) telle que: Θ|D(A) = Θ0 ‖Θx‖C([a,b];E) ≤ Me ωb‖x‖ quel que soit x ∈ E . Par conséquent, il existe un seul opérateur T ∈ C ([a, b];B(E)) tel que: Θx = T ( . )x , (∀)x ∈ E . 2.3. LE THÉORÈME DE HILLE - YOSIDA 61 On peut répéter ce procédé pour tous les intervalles compacts de [0,∞) et on voit qu’il existe un seul opérateur, noté aussi par T ∈ C ([0,∞);B(E)), tel que pour tout x ∈ E on ait: Tλ(t)x −→ T (t)x si Reλ → ∞, uniformément par rapport à t sur les intervalles compacts de [0,∞). De plus: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Il est évident que: T (0)x = lim Reλ→∞ Tλ(0)x = x , (∀)x ∈ E T (t)x = lim Reλ→∞ Tλ(t)x = lim Reλ→∞ Tλ(t)x = x , (∀)x ∈ E . Soient t, s ∈ [0,∞) et x ∈ E . Alors, nous avons: ‖T (t + s)x − T (t)T (s)x‖ ≤ ‖T (t + s)x − Tλ(t + s)x‖ + + ‖Tλ(t + s)x − Tλ(t)T (s)x‖ + ‖Tλ(t)T (s)x − T (t)T (s)x‖ ≤ ≤ ‖T (t + s)x − Tλ(t + s)x‖ + ‖Tλ(t)‖ ‖Tλ(s)x − T (s)x‖ + + ‖Tλ(t) (T (s)x) − T (t) (T (s)x)‖ . Puisque Tλ(t) −→ T (t), si Reλ → ∞, pour la topologie forte de B(E), il s’ensuit que T (t + s)x = T (t)T (s)x, pour tout x ∈ E . Par conséquent {T (t)} t≥0 ∈ SG(M, ω). Montrons que A est le générateur infinitésimal du semi-groupe {T (t)} Pour tout x ∈ D(A) on a: ‖Tλ(s)Aλx − T (s)Ax‖ ≤ ≤ ‖Tλ(s)‖ ‖Aλx − Ax‖ + ‖Tλ(s)Ax − T (s)Ax‖ ≤ ≤ Meωt ‖Aλx − Ax‖ + ‖Tλ(s)Ax − T (s)Ax‖ −→ 0 si Reλ → ∞, uniformément par rapport à s ∈ [0, t], d’où: T (t)x − x = lim Reλ→∞ [Tλ(t)x − x] = lim Reλ→∞ Tλ(s)Aλx ds = T (t)Ax ds 62 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 quels que soient x ∈ D(A) et t ≥ 0. Soit B le générateur infinitésimal du C0-semigroupe {T (t)}t≥0. Si x ∈ D(A), alors: T (t)x − x = lim T (s)Ax ds = Ax et nous voyons que x ∈ D(B). Par conséquent D(A) ⊂ D(B) et B| D’autre part, nous avons l’inégalité: ‖T (t)‖ ≤ Meωt , (∀)t ≥ 0. Si λ ∈ Λω, alors λ ∈ ρ(A) ∩ ρ(B). Soit x ∈ D(B), on a donc (λI − B) x ∈ E et comme l’opérateur λI − A : D(A) −→ E est bijectif, il existe x′ ∈ D(A) tel que (λI − A) x′ = (λI − B)x. Puisque B|D(A) = A, il vient que (λI − B)x′ = (λI − B) x et comme λ ∈ ρ(B), il en résulte que x′ = x. Par suite x ∈ D(A) et donc D(B) ⊂ D(A). Finalement on voit que D(A) = D(B) et A = B. Nous avons montré donc que A est le générateur infinitésimal du C0-semi-groupe {T (t)} t≥0 et compte tenu du théorème de l’unicité de l’engendrement, il résulte que {T (t)} t≥0 est l’unique C0-semi-groupe engendré par A. Corollaire 2.3.12 Soient {T (t)} t≥0 ∈ SG(M, ω) , A son générateur infinitésimal et {Aλ}λ∈Λω l’approximation généralisée de Yosida de l’opérateur A. Alors: T (t)x = lim Reλ→∞ etAλx , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve Elle résulte du théorème de Hille-Yosida. Dans la suite nous noterons par GI(E) l’ensemble des opérateurs linéaires qui sont des générateurs infinitésimaux de C0-semi-groupes sur l’espace de Banach E . De même, pour ω ≥ 0 et M ≥ 1, nous noterons par GI(M, ω) l’ensemble des générateurs infinitésimaux A ∈ GI(E) pour lesquels: ‖R(λ; A)n‖ ≤ M (Reλ − ω)n (∀)λ ∈ Λω et n ∈ N 2.4. LA REPRÉSENTATION DE BROMWICH 63 2.4 La représentation de Bromwich Dans la section 1.3, avec le théorème 1.3.12 nous avons vu que pour les semi-groupes uniformément continus on peut obtenir une représentation par la transformée de Laplace inverse. Dans ce paragraphe nous montrerons qu’il existe une représentation du même type pour les C0-semi-groupes. Nous commençons avec quelques propriétés sur l’approximation généralisée de Yosida. Lemme 2.4.1 Soient {T (t)} t≥0 ∈ SG(M, ω) , A son générateur infinitésimal et {Aµ}µ∈Λω l’approximation généralisée de Yosida de l’opérateur A. Alors pour tout µ ∈ Λω, il existe Ω > ω tel que ΛΩ ⊂ ρ(Aµ) et pour tout λ ∈ ΛΩ on a: ‖R(λ; Aµ)‖ ≤ Reλ − Ω . De plus, pour ε > 0, il existe une constante C > 0 (qui dépend de M et ε) tel que: ‖R(λ; Aµ)x‖ ≤ |λ| (‖x‖ + ‖Ax‖) , (∀)x ∈ D(A), quels que soient λ, µ ∈ C, avec Reλ > Ω + ε et Reµ > ω + |µ| Preuve Soit µ ∈ Λω arbitrairement fixé. Nous avons vu que Aµ est le générateur infinitésimal du semi-groupe uniformément continu . En ce cas, nous avons: ∥etAµ ∥ ≤ Me ωReµ+Im2µ Reµ−ω , (∀)t ≥ 0. Si nous notons: ωReµ + Im2µ Reµ − ω , alors il est clair que: Ω = ω + ω2 + Im2µ Reµ − ω > ω et que ΛΩ = {λ ∈ C |Reλ > Ω} ⊂ ρ(Aµ). De plus, pour tout λ ∈ ΛΩ, nous avons: ‖R(λ; Aµ)‖ ≤ Reλ − Ω . Si nous considérons λ ∈ C tel que Reλ > Ω + ε, où ε > 0, alors on voit que: ‖R(λ; Aµ)‖ ≤ 64 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 D’autre part, pour x ∈ D(A) et µ ∈ Λω tel que Reµ > ω + |µ|2 , nous obtenons: ‖Aµx‖ = ‖µR(µ; A)Ax‖ ≤ |µ|‖R(µ; A)‖‖Ax‖ ≤ ≤ |µ| M Reµ − ω‖Ax‖ ≤ 2M‖Ax‖ . De l’égalité: (λI − Aµ)R(λ; Aµ) = I , il vient: R(λ; Aµ) = R(λ; Aµ)Aµ et par conséquent: ‖R(λ; Aµ)x‖ ≤ |λ| (‖x‖ + ‖R(λ; Aµ)‖‖Aµx‖) ≤ ≤ 1|λ| ‖x‖ + 2M ≤ C|λ| (‖x‖ + ‖Ax‖) , (∀)x ∈ D(A), où la constante C ne dépend que de M et de ε. Lemme 2.4.2 Soient {T (t)} t≥0 ∈ SG(M, ω) , A son générateur infinitésimal, {Aµ}µ∈Λω l’approximation généralisée de Yosida de l’opérateur A et λ ∈ C tel que Reλ > ω + ε, arbitrairement fixé pour ε > 0. Alors: Reµ→∞ R(λ; Aµ)x = R(λ; A)x , (∀)x ∈ E , uniformément par rapport à Imλ ∈ [−k, k], où k > 0. Preuve Compte tenu du lemme 2.4.1, pour µ ∈ Λω, il existe Ω > ω tel que ΛΩ ⊂ ρ(Aµ). Nous avons: ωReµ + Im2µ Reµ − ω . Donc l’inégalité Reλ > Ω est équivalente avec: Reλ > ω + ω2 + Imµ Reµ − ω . Soit ε > 0. Si µ ∈ Λω tel que ω 2+Imµ Reµ−ω < ε, alors Reλ > ω + ε implique Reλ > Ω. Par suite, λ ∈ ρ(Aµ). Donc il existe R(λ; Aµ) et avec le lemme 2.4.1 on voit que: ‖R(λ; Aµ)‖ ≤ Reλ − ω . 2.4. LA REPRÉSENTATION DE BROMWICH 65 D’autre part, nous avons: λ + µ λ − λ λ + µ = Reλ − Re λ λ + µ > ω + ε − Re λ λ + µ Etant donné k > 0 tel que |Imλ| ≤ k, il existe µ ∈ Λω tel que Re λ s’ensuit que Re λµ > ω + ε . Par conséquent, λµ ∈ ρ(A) et donc R existe bien. Nous avons: λ + µ (λI − Aµ)(µI − A)R λ + µ λ + µ λI − µ2R(µ; A) + µI (µI − A)R λ + µ µI − A − µ λ + µ λ + µ λ + µ I − A λ + µ = I . Par un calcul analogue, on peut obtenir: λ + µ (µI − A)R λ + µ (λI − Aµ) = I . Il s’ensuit que: R(λ; Aµ) = λ + µ (µI − A)R λ + µ Par conséquent: R(λ; Aµ) − R(λ; A) = λ + µ (µI − A)R λ + µ − R(λ; A) = λ + µ (µI − A)R λ + µ − (λ + µ)R(λ; A) λ + µ (µI − A)R λ + µ (µI − A)(λI − A) − − (λ + µ) λ + µ I − A R(µ; A)R(λ; A) = λ + µ (µI − A)R λ + µ A2R(µ; A)R(λ; A) = λ + µ λ + µ R(λ; A)A2 . 66 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Mais: ‖R(λ; A)‖ ≤ M et: ∥ λ + µ Si x ∈ D(A2), alors on voit que: ‖R(λ; Aµ)x − R(λ; A)x‖ ≤ ≤ 1|λ + µ| λ + µ ‖R(λ; A)‖‖A2x‖ ≤ ≤ 1|µ| ‖A2x‖ ≤ 1 ‖A2x‖ . Il s’ensuit que: Reµ→∞ R(λ; Aµ)x = R(λ; A)x , (∀)x ∈ D(A2), uniformément par rapport à Imλ ∈ [−k, k], où k > 0. Avec le théorème 2.2.10, on sait que D(A2) = E . Comme R(λ; A) et R(λ; Aµ) sont uniformément bornés, on obtient: Reµ→∞ R(λ; Aµ)x = R(λ; A)x , (∀)x ∈ E , uniformément par rapport à Imλ ∈ [−k, k], où k > 0. Théorème 2.4.3 Soit A le générateur infinitésimal du semi-groupe {T (t)} t≥0 ∈ SG(M, ω) et λ ∈ Λω. Alors pour tout x ∈ D(A) on a: T (s)x ds = Reλ−i∞∫ Reλ−i∞ eztR(z; A)x et l’intégrale de la partie droite de l’égalité est uniformément convergente par rapport à t sur les intervalles compacts de ]0,∞). Preuve Soit {Aµ}µ∈Λω l’approximation généralisée de Yosida de l’opérateur A. Soit µ ∈ Λω tel que Re µ > ω + |µ|2 . Avec le lemme 2.4.1, nous déduissons qu’il existe Ω = ωReµ+Im Reµ−ω > ω tel que ΛΩ = {λ ∈ C|Re λ > Ω} ⊂ ρ(Aµ). Soit λ ∈ ΛΩ. En utilisant le théorème 1.3.12, pour R > 2Re λ on peut considérer le contour de Jordan Aµ-spectral Γ1R = Γ R ∪ Γ1 2.4. LA REPRÉSENTATION DE BROMWICH 67 R = {Re λ + iτ |τ ∈ [−R, R]} Re λ + R(cos ϕ + i sin ϕ) Pour le semi-groupe uniformément continu engendré par Aµ il en résulte: etAµ = lim Re λ+iR∫ Re λ−iR eztR(z; Aµ) dz = Re λ+i∞∫ Re λ−i∞ eztR(z; Aµ) dz , uniformément par rapport à t sur les intervalles compacts de [0,∞). Pour R > 2Re λ et x ∈ D(A) nous notons IR(s) = Re λ+iR∫ Re λ−iR ezsR(z; Aµ)x dz . Soient 0 < a < b. Pour tout t ∈ [a, b] nous obtenons: IR(s) ds = Re λ+iR∫ Re λ−iR ezsR(z; Aµ)x dz ds = Re λ+iR∫ Re λ−iR ezs dsR(z; Aµ)x dz = Re λ+iR∫ Re λ−iR eztR(z; Aµ)x Re λ+iR∫ Re λ−iR R(z; Aµ)x Montrons que pour l’intégrale I(R) = Re λ+iR∫ Re λ−iR R(z; Aµ)x I(R) = 0 . Soit le contour de Jordan lisse et fermé Γ2R = Γ R ∪ Γ2 R = {Re λ + iτ |τ ∈ [−R, R]} 68 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Re λ + R(cos ϕ + i sin ϕ) Avec le théorème de Cauchy ([DS’67, pag. 225]), on voit que R(z; Aµ)x = 0 , ou bien R(z; Aµ)x R(z; Aµ)x = 0 . Soit z ∈ Γ2” . Compte tenu du lemme 2.4.1, il existe C > 0 tel que ‖R(z; Aµ)x‖ ≤ |z| (‖x‖ + ‖Ax‖) . De plus, pour z ∈ Γ2”R on a: |z| = |Re λ + R(cos ϕ + isinϕ)| = |Re λ − [−R(cos ϕ + i sin ϕ)]| ≥ ≥ | |Re λ| − | − R(cos ϕ + i sin ϕ)| | = |Re λ − R| = R − Re λ , d’où il résulte |z| ≤ R − Re λ . Par conséquent, on a: R(z; Aµ)x ‖R(z; Aµ)x‖ |z| ≤ |z|(‖x‖ + ‖Ax‖) |z| |dz| ≤ ‖x‖ + ‖Ax‖ (R − Re λ)2 |dz| = (R − Re λ)2 (‖x‖ + ‖Ax‖) . Il s’ensuit donc que: R(z; Aµ)x = 0 . Par suite, on a: R(z; Aµ)x 2.4. LA REPRÉSENTATION DE BROMWICH 69 ou bien I(R) = 0 . Alors nous avons: IR(s) ds = lim Re λ+iR∫ Re λ−iR eztR(z; Aµ)x d’où IR(s) ds = Re λ+i∞∫ Re λ−i∞ eztR(z; Aµ)x Avec le corollaire 2.3.12 et le lemme 2.4.2, on obtient: T (s)x ds = lim Reµ→∞ esAµx ds = = lim Reµ→∞ Re λ+i∞∫ Re λ−i∞ eztR(z; Aµ)x Re λ+i∞∫ Re λ−i∞ eztR(z; A)x et comme: Reµ→∞ Ω = lim Reµ→∞ ωReµ + Im2µ Reµ − ω = ω , nous obtenons le résultat désiré. Théorème 2.4.4 (Bromwich) Soit A le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(M, ω) et λ ∈ Λω. Alors: T (t)x = Reλ+i∞∫ Reλ−i∞ eztR(z; A)x dz , (∀)x ∈ D(A2) et pour tout δ > 0, l’intégrale est uniformément convergente par rapport à t ∈ Preuve Si x ∈ D(A2), alors Ax ∈ D(A). Compte tenu du théorème 2.4.3, on voit T (s)Ax ds = Reλ+i∞∫ Reλ−i∞ eztR(z; A)Ax 70 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 D’où il résulte que: T (s)x − x = 1 Reλ+i∞∫ Reλ−i∞ eztR(z; A)Ax De l’égalité: R(z; A)(zI − A) = I , nous déduisons: R(z; A)A = R(z; A) − 1 et par suite: T (s)x − x = 1 Reλ+i∞∫ Reλ−i∞ eztR(z; A)x dz − 1 Reλ+i∞∫ Reλ−i∞ Compte tenu que: Reλ+i∞∫ Reλ−i∞ et que pour tout δ > 0, l’intégrale est uniformément convergente par rapport à , nous obtenons l’égalité de l’énoncé. 2.5 Conditions suffisantes d’appartenances à GI(M, 0) Nous présentons dans la suite deux conditions suffisantes pour qu’un opérateur soit le générateur infinitésimal d’un C0-semi-groupe uniformément borné. Théorème 2.5.1 Soit A un opérateur linéaire fermé défini sur un sous espace dense de E et vérifiant les propriétés suivantes: i) il existe δ ∈ tel que: ρ(A) ⊃ Σδ = λ ∈ C | arg z| < π ∪ {0}; ii) il existe une constante K > 1 tel que: ‖R(λ; A)‖ ≤ K|λ| , (∀)λ ∈ Σδ − {0}. 2.5. CONDITIONS SUFFISANTES D’APPARTENANCES À GI(M, 0) 71 Alors A est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 pour lequel il existe M > 1 tel que ‖T (t)‖ ≤ M , pour tout t ≥ 0. De plus, pour tout ν ∈ et Γν = Γ ν ∪ Γ(2)ν , où: Γ(1)ν = {r(cos ν − i sin ν)| r ∈ [0,∞)} Γ(2)ν = {r(cos ν + i sin ν)| r ∈ [0,∞)} , on a: T (t) = eztR(z; A) dz et l’intégrale est uniformément convergente par rapport à t > 0. Preuve Soit δ ∈ . Pour ν ∈ considérons le chemin d’intégration Γν = Γ ν ∪ Γ(2)ν , où: Γ(1)ν = {r(cos ν − i sin ν)| r ∈ [0,∞)} Γ(2)ν = {r(cos ν + i sin ν)| r ∈ [0,∞)} . Soit: U(t) = eztR(z; A) dz . Compte tenu du (ii), on voit que l’intégrale est uniformément convergente par rapport à t > 0. Pour R > 0, nous définissons le contour de Jordan lisse et fermé Γν = Γ R,ν ∪ Γ R,ν ∪ Γ R,ν où R,ν = {r(cos ν − i sin ν)| r ∈ [0, R]} , R,ν = {R(cos ν + i sin ν)| θ ∈ [−ν, ν]} , R,ν = {r(cos ν + i sin ν)| r ∈ [0, R]} . D’aprés le théorème de Cauchy ([DS’67, pag. 225]), on a eztR(z; A) dz = 0 . 72 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Par conséquent, on peut changer le chemin d’intégration Γν par Γt = Γ r(cos ν − i sin ν) ∣ r ∈ (cos θ + i sin θ) | θ ∈ [−ν, ν] r(cos ν + i sin ν) Alors: U(t) = eztR(z; A) dz + eztR(z; A) dz + eztR(z; A) dz et si nous notons U1(t) = eztR(z; A) dz , U2(t) = eztR(z; A) dz U3(t) = eztR(z; A) dz , il vient: ‖U(t)‖ ≤ ‖U1(t)‖ + ‖U2(t)‖ + ‖U3(t)‖ . Comme ν ∈ , on en déduit que cos ν < 0. Avec le changement de variable z = r(cos ν − i sin ν) , r ∈ nous avons: ‖U1(t)‖ = eztR(z; A) dz ert(cos ν−i sin ν)R(r(cos ν − i sin ν); A)(cos ν − isinν) dr ert cos ν e−rt(− cos ν) 2.5. CONDITIONS SUFFISANTES D’APPARTENANCES À GI(M, 0) 73 s = rt(− cos ν) , s ∈ [− cos ν,∞), il vient ds = −t cos νdr. Donc: ‖U1(t)‖ ≤ − cos ν −t cos ν −t cos ν ds = − cos ν − cos ν − cos ν De façon analogue, nous obtenons: ‖U3(t)‖ = eztR(z; A) dz ert(cos ν+i sin ν)R(r(cos ν + i sin ν); A)(cos ν + isinν) dr ert cos ν dr ≤ M ′ . De même, pour l’intégrale U2(t), avec le changement de variable (cos θ + i sin θ) , θ ∈ [−ν, ν], (− sin θ + i cos θ) dθ ‖U2(t)‖ = eztR(z; A) dz (cos θ+i sin θ)R (cos θ + i sin θ); A (−sinθ + i cos θ) dθ cos θ K cos θ dθ ≤ Ke dθ = M” . Par conséquent, il existe M ≥ 1 tel que: ‖U(t)‖ ≤ M , (∀)t ≥ 0. 74 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Nous allons maintenant montrer que pour tout λ ∈ Λ0 = {λ ∈ C |Reλ > 0}, on a: R(λ; A) = e−λtU(t) dt . Nous avons successivement: e−λtU(t) dt = e−(λ−z)tR(z; A) dzdt = (z−λ)t dtR(z; A) dz = e(z−λ)τ − 1 z − λ R(z; A) dz = e(z−λ)τ z − λ R(z; A) dz + R(z; A) λ − z dz = e(z−λ)τ z − λ R(z; A) dz + (λI − A) e(z−λ)τ z − λ R(z; A) dz + R(λ; A) . Par conséquent: e−λtU(t) dt − R(λ; A) e(z−λ)τ z − λ R(z; A) dz |e(z−λ)τ | |z − λ| ‖R(z; A)‖ |dz| ≤ e(Rez−Reλ)τ |z − λ| |z| |dz| = e−τReλ |z − λ| |z| |dz| . eτRez |z||z − λ| . Pour z ∈ Γν on a |z| = r et |z − λ| ≥ | |z| − |λ| | = |r − |λ| | . Donc: τRez 1 |r − |λ|| r|r − |λ|| dr < ∞ . 2.5. CONDITIONS SUFFISANTES D’APPARTENANCES À GI(M, 0) 75 Si nous notons C = sup alors nous obtenons: e−λtU(t) dt − R(λ; A) ≤ Ce−τReλ . En passant à limite pour τ → ∞, on obtient R(λ; A) = e−λtU(t) dt pour tout λ ∈ Λ0. Comme ‖U(t)‖ ≤ M , pour tout t ≥ 0, par récurrence on peut obtenir: dλn−1 R(λ; A) = (−1)n−1 tn−1e−λtU(t) dt , (∀)n ∈ N∗. Mais avec la proposition 1.1.16 (iii), on voit que: dλn−1 R(λ; A) = (−1)n−1(n − 1)!R(λ; A)n , (∀)n ∈ N∗. Par conséquent, nous obtenons: ‖R(λ; A)n‖ = (n − 1)! tn−1e−λtU(t) dt (n − 1)! tn−1e−Reλt dt = (n − 1)! n − 1 tn−2e−Reλt dt = · · · = M (Reλ)n pour tout n ∈ N∗. Avec le théorème de Hille-Yosida, on voit que l’opérateur A est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(M, 0). Soit x ∈ D(A2) et λ ∈ Λ0. Compte tenu du théorème 2.4.4, nous avons: T (t)x = Reλ+i∞∫ Reλ−i∞ eλtR(λ; A)x dλ et compte tenu du (ii) et du théorème de Cauchy, on peut remplacer le contour d’intégration par Γν . Donc: T (t)x = R(λ; A)x dλ , (∀)x ∈ D(A2). 76 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Comme D(A2) = E et l’intégrale eλtR(λ; A)dλ est uniformément convergente, nous obtenons: T (t)x = eλtR(λ; A)x dλ , (∀)x ∈ E , d’où il résulte l’affirmation de l’énoncé. Théorème 2.5.2 Soit A un opérateur linéaire fermé défini sur un sous espace dense de E et vérifiant les propriétés suivantes: i) σ(A) ⊂ {z ∈ C|Rez ≤ 0}; ii) pour tout x ∈ E et tout x∗ ∈ E∗ on a: γ+i∞∫ R(λ; A) x, x∗ ∣ |dλ| < ∞ . Alors A est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(M, 0). Preuve Pour γ > 0 arbitrairement fixé, l’application: {z ∈ C|Rez > 0} ∋ z 7−→ R(z + γ; A)2x, x∗ se trouve dans l’espace de Hardy H1. Par conséquent elle admet un représentation par une intégrale de Cauchy et en particulier pour α > 0 et γ ∈]0, α[, on a: R(α; A)2x, x∗ γ+i∞∫ 〈R(λ; A)2x, x∗〉 α − λ dλ . De plus, par récurrence on voit que: R(α; A)2x, x∗ = (−1)n n! γ+i∞∫ 〈R(λ; A)2x, x∗〉 (α − λ)n+1 dλ , pour tout n ∈ N∗. D’autre part, avec la proposition 1.1.16 (iii) il vient: R(α; A) = (−1)nn!R(α; A)n+1 , (∀)n ∈ N∗ Par itération nous obtenons: R(α; A)n+1x, x∗ γ+i∞∫ 〈R(λ; A)2x, x∗〉 (α − λ)n dλ , 2.5. CONDITIONS SUFFISANTES D’APPARTENANCES À GI(M, 0) 77 pour tout n ∈ N∗ et tout γ ∈]0, α[. Il s’ensuit que pour tout n ∈ N∗ on a: R(α; A)n+1x, x∗ (α − γ)n γ+i∞∫ R(λ; A)2x, x∗ ∣ |dλ| ≤ (α − γ)n , (∀)γ ∈]0, α[, où la constante C ne dépend que de x ∈ E et de x∗ ∈ E∗. Si nous prenons: n + 1 alors on voit que: R(α; A)n+1x, x∗ ∣ ≤ 1 α − α 1 − 1 n + 1 )−(n+1) En appliquant le théorème de Banach-Steinhaus ([DS’67, Theorem II.1.11, pag. 52]), on obtient pour tout α > 0: ‖R(α; A)m‖ ≤ M , (∀)m ∈ {2, 3, . . .}. Prouvons que cette inégalité reste valable pour m = 1. On a: R(τ ; A)2 dτ = − R(τ ; A)2 dτ = R(τ ; A) dτ = R(α; A) − R(β; A) , (∀)α, β > 0. Comme pour m = 2 nous avons: ∥R(β; A)2 ∥ ≤ M on en déduit que la limite existe et on pose R0x := lim R(β; A)x , (∀)x ∈ E . 78 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 D’autre part, si x ∈ D(A), alors nous avons: R(β; A)x = R(β; A)2(βI − A)x −→ 0 si β → ∞. Comme D(A) = E , il s’ensuit que R0 = 0 et on voit que: R(α; A) = R(τ ; A)2 dτ . Par conséquent: ‖R(α; A)m‖ ≤ M , (∀)m ∈ N∗. En appliquant le théorème de Hille-Yosida, on obtient le résultat désiré. 2.6 Propriétés spectrales des C0-semi-groupes Nous terminons ce chapitre avec quelques propriétés spectrales pour les C0- semi-groupes. Lemme 2.6.1 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors pour tout λ ∈ Λω et t > 0, l’application: Bλ(t) : E −→ E Bλ(t)x = eλ(t−s)T (s)x ds définit un opérateur linéaire borné sur E vérifiant les propriétés suivantes: (λI − A)Bλ(t)x = eλtx − T (t)x , (∀)x ∈ E Bλ(t)(λI − A)x = eλtx − T (t)x , (∀)x ∈ D(A). De plus Bλ(t)T (t) = T (t)Bλ(t). 2.6. PROPRIÉTÉS SPECTRALES DES C0-SEMI-GROUPES 79 Preuve Pour tout x ∈ E nous avons successivement: ‖Bλ(t)x‖ = eλ(t−s)T (s)x ds eReλ(t−s)‖T (s)‖‖x‖ ds ≤ ≤ MeReλt‖x‖ e−(Reλ−ω)s ds < ∞ . Comme la linéarité est évidente, il en résulte que Bλ(t) ∈ B(E), quels que soient λ ∈ Λω et t > 0. Si x ∈ E et h > 0, alors nous obtenons: T (h) − I Bλ(t)x = T (h) − I eλ(t−s)T (s)x ds = eλ(t−s)T (h + s)x ds − 1 eλ(t−s)T (s)x ds = eλ(t−τ+h)T (τ)x dτ − 1 eλ(t−s)T (s)x ds = eλ(t−τ)T (τ)x dτ − 1 eλ(t−s)T (s)x ds = eλ(t−τ)T (τ)x dτ − eλ(t−τ)T (τ)x dτ eλ(t−s)T (s)x ds = eλ(t−τ)T (τ)x dτ − e eλ(t−τ)T (τ)x dτ + eλ(t−τ)T (τ)x dτ − 1 eλ(t−s)T (s)x ds − eλ(t−τ)T (τ)x dτ = eλ(t−τ)T (τ)x dτ + eλh − 1 eλ(t−s)T (s)x ds − λ(t−τ) T (τ)x dτ = 80 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 λ(t−τ) T (τ)x dτ + eλh − 1 Bλ(t)x − λ(t−τ) T (τ)x dτ . En passant à limite, on a: T (h)Bλ(t)x − Bλ(t)x = T (t)x + λBλ(t)x − eλtx , d’où Bλ(t)x ∈ D(A) et: (λI − A)Bλ(t)x = eλtx − T (t)x , (∀)x ∈ E . Si x ∈ D(A), alors nous avons: Bλ(t)Ax = λ(t−s) T (s)Ax ds = eλ(t−s) T (s)x ds = T (t)x − eλtx + λBλ(t)x , d’où l’on tire: Bλ(t)(λI − A)x = eλtx − T (t)x , (∀)x ∈ D(A). De plus, nous obtenons que: Bλ(t)D(A) ⊆ D(A) (λI − A)Bλ(t)x = Bλ(t)(λI − A)x , (∀)x ∈ D(A) , d’où: ABλ(t)x = Bλ(t)Ax , (∀)x ∈ D(A). Compte tenu du théorème 2.2.9, on voit que: Bλ(t)T (t) = T (t)Bλ(t) , (∀)t ≥ 0. Théorème 2.6.2 (spectral mapping) Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: etσ(A) = ∣λ ∈ σ(A) ⊆ σ(T (t)) , (∀)t ≥ 0. 2.6. PROPRIÉTÉS SPECTRALES DES C0-SEMI-GROUPES 81 Preuve Soit λ ∈ C tel que eλt ∈ ρ(T (t)). Alors on peut considérer l’opérateur Q = eλtI − T (t) ∈ B(E). Compte tenu du lemme 2.6.1 (i), on a: (λI − A)Bλ(t)x = eλtx − T (t)x , (∀)x ∈ E Bλ(t)(λI − A)x = eλtx − T (t)x , (∀)x ∈ D(A). Par multiplication avec Q à droite dans la première égalité et à gauche dans la seconde, nous obtenons: (λI − A)Bλ(t)Qx = x , (∀)x ∈ E QBλ(t)(λI − A)x = x , (∀)x ∈ D(A). Mais, avec le lemme 2.6.1, il en résulte que: eλtI − T (t) Bλ(t) = Bλ(t) eλtI − T (t) et nous voyons que QBλ(t) = Bλ(t)Q. Par conséquent: (λI − A)Bλ(t)Qx = x , (∀)x ∈ E Bλ(t)Q(λI − A)x = x , (∀)x ∈ D(A). Il s’ensuit que λ ∈ ρ(A) et finalement on voit que: ρ(T (t)) ⊂ etρ(A) , (∀)t ≥ 0, ou bien: etσ(A) ⊆ σ(T (t)) , (∀)t ≥ 0. Remarque 2.6.3 Nous avons vu que pour les semi-groupes uniformément conti- nus on a l’égalité: etσ(A) = σ(T (t)) , (∀)t ≥ 0. Mais il existe des C0-semi-groupes pour lesquels l’inclusion du théorème 2.6.2 est stricte. 82 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Définition 2.6.4 On dit que le C0-semi-groupe {T (t)}t≥0 est nilpotent s’il existe t0 > 0 tel que T (t) = 0, pour tout t > t0. Proposition 2.6.5 Soient {T (t)} t≥0 ∈ SG(M, ω) un semi-groupe nilpotent et A son générateur infinitésimal. Alors σ(A) = ∅. Preuve Comme le C0-semi-groupe {T (t)}t≥0 est nilpotent, il existe t0 > 0 tel que T (t) = 0, (∀)t > t0. Pour tout λ ∈ C et tout x ∈ E , on a: ∥e−λtT (t)x ∥ ≤ e−ReλtMeωt‖x‖ , (∀)t ∈ [0, t0] et comme: e−λtT (t)x dt = 0 , on peut définir la transformée de Laplace: Rλ : E −→ E Rλx = e−λtT (t)x dt = e−λtT (t)x dt pour tout λ ∈ C. Avec le théorème 2.3.1, il vient λ ∈ ρ(A) et Rλx = R(λ; A)x, pour tout x ∈ E . Donc ρ(A) = C, c’est-à-dire σ(A) = ∅. Remarque 2.6.6 Pour un semi-groupe nilpotent {T (t)} t≥0 ∈ SG(M, ω) ayant pour générateur infinitésimal l’opérateur A, l’inclusion du théorème 2.6.2 est stricte. Théorème 2.6.7 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: σ(R(λ; A)) = λ − ζ ζ ∈ σ(A) ∪ {0} quel que soit λ ∈ Λω. Preuve Soient λ ∈ Λω et µ ∈ ρ(A), µ 6= λ. Définissons: S : E −→ E S = (λ − µ)(λI − A)R(µ; A). 2.6. PROPRIÉTÉS SPECTRALES DES C0-SEMI-GROUPES 83 Comme S est un opérateur fermé, avec le théorème du graphe fermé, on voit que S ∈ B(E). De plus, pour tout x ∈ E nous avons: SR(λ; A)x = (λ − µ)(λI − A)R(µ; A)R(λ; A)x = = (λ − µ)(λI − A)R(λ; A)R(µ; A)x = (λ − µ)R(µ; A)x R(λ; A)Sx = R(λ; A)(λ − µ)(λI − A)R(µ; A)x = = (λ − µ)R(λ; A)(λI − A)R(µ; A)x = (λ − µ)R(µ; A)x . Par conséquent SR(λ; A) = R(λ; A)S. De même, pour x ∈ E on a: λ − µI − R(λ; A) = (λ − µ)(λI − A)R(µ; A) λ − µI − R(λ; A) = [(λI − A)R(µ; A) − (λ − µ)R(µ; A)]x = = (λI − A − λI + µI)R(µ; A)x = = (µI − A)R(µ; A)x = x . De façon analogue, pour tout x ∈ E on peut montrer que: λ − µI − R(λ; A) Sx = x . Par conséquent: λ − µ ∈ ρ(R(λ; A)) , d’où: { λ − µ µ ∈ ρ(A) ⊂ ρ(R(λ; A)) . Il s’ensuit que: σ(R(λ; A)) ⊂ λ − ζ ζ ∈ σ(A) Réciproquement, soit λ ∈ Λω et µ ∈ C, µ 6= λ, tel que 1λ−µ ∈ ρ(R(λ; A)). Alors il existe R ; R(λ; A) ∈ B(E) et pour tout x ∈ D(A) nous avons: R(λ; A)R λ − µ ; R(λ; A) x = R(λ; A) λ − µI − R(λ; A) 84 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 R(λ; A)−1 λ − µI − R(λ; A) λ − µI − R(λ; A) R(λ; A) λ − µR(λ; A) −1 − I R(λ; A) λ − µI − R(λ; A) λ − µI − R(λ; A) R(λ; A)x = R λ − µ ; R(λ; A) R(λ; A)x . Posons: Q = R(λ; A)R λ − µ ; R(λ; A) Pour tout x ∈ D(A), nous avons: (µI − A)Qx = (µI − λI + λI − A)R(λ; A)R λ − µ ; R(λ; A) = [(λI − A)R(λ; A) − (λ − µ)R(λ; A)]R λ − µ ; R(λ; A) = [I − (λ − µ)R(λ; A)]R λ − µ ; R(λ; A) = (λ − µ) λ − µI − R(λ; A) λ − µ ; R(λ; A) x = (λ − µ)x , d’où il résulte que: λ − µ(µI − A)Qx = x , (∀)x ∈ D(A). De même, nous obtenons: Q(µI − A)x = R(λ; A)R λ − µ ; R(λ; A) (µI − A)x = λ − µ ; R(λ; A) R(λ; A)(µI − λI + λI − A)x = λ − µ ; R(λ; A) [R(λ; A)(λI − A) − R(λ; A)(λ − µ)] x = λ − µ ; R(λ; A) [I − (λ − µ)R(λ; A)]x = = (λ − µ)R λ − µ ; R(λ; A) λ − µI − R(λ; A) x = (λ − µ)x , d’où: λ − µQ(µI − A)x = x , (∀)x ∈ D(A). 2.7. NOTES 85 Par conséquent µ ∈ ρ(A). Il s’ensuit que: ρ(R(λ; A)) ⊂ λ − µ µ ∈ ρ(A) ou bien: λ − ζ ζ ∈ σ(A) ⊂ σ(R(λ; A)) , (∀)λ ∈ C avec Reλ > ω. Finalement, nous voyons que: σ(R(λ; A)) = λ − ζ ζ ∈ σ(A) , (∀)λ ∈ Λω. Si 0 ∈ ρ(R(λ; A)), alors il existe (0I − R(λ; A))−1 ∈ B(E), d’où A ∈ B(E) ce qui est absurde. Par conséquent 0 ∈ σ(R(λ; A)). 2.7 Notes Les notions et les résultats de ce chapitre se trouvent dans les monographies concernant les C0- semi-groupes d’opérateurs linéaires bornés. Le théorème 2.1.7 se trouve dans [Hi’48, pag.184], mais une preuve élégante utilisant le théorème de Krein-Šmulian se trouve dans [Da’80, pag. 15]. De même, dans [Pa’83-1, pag. 43] on peut trouver une caractérisation du générateur infinitésimal d’un C0-semi-groupe pour la topologie faible. Pour le théorème de l’unicité de l’engendrement nous avons utilisé [Pa’83-1, pag. 6] et le théorème 2.2.9 se trouve dans [Da’80, pag. 11]. Le résultat le plus important de ce chapitre est le théorème de Hille-Yosida. Il a été montré pour la première fois indépendamment par Hille dans [Hi’48] et par Yosida dans [Yo’48] pour les C0-semi-groupes de contractions. Quelques années plus tard, Feller dans [Fe’53], Miyadera dans [Mi’52] et Phillips dans [Ph’52] donnent une preuve pour le cas général d’un C0-semi-groupe. Nous avons utilisé les idées du livre de Pazy [Pa’83-1, pag. 8] pour obtenir une preuve dans le cas le plus général, en utilisant l’approximation généralisée de Yosida que nous avons introduit dans la définition 2.3.8. Pour obtenir le représentation de Bromwich d’un C0-semi-groupe, nous avons utilisé aussi les idées de Pazy de [Pa’83-1, pag. 29]. Une variante du lemme 2.4.1 se trouve dans [Le’00-2]. Pour le théorème 2.5.1 on peut consulter [Pa’83-1, pag. 30] ou bien [Ah’91, pag. 76]. Le téorème 2.5.2 a été montré par Gomilko dans [Go’99]. Pour les propriétés spectrales des C0-semi-groupes on peut consulter [Pa’83-1, pag. 44] où on peut trouver aussi des autres résultats sur cette problème. Finalement, pour le théorème 2.6.7 on pourra consulter [Da’80, pag. 39]. 86 CHAPITRE 2. SEMI-GROUPES DE CLASSE C0 Chapitre 3 C0-semigroupes avec propriétés spéciales 3.1 C0-semi-groupes différentiables Par la suite, nous étudierons les propriétés des C0-semi-groupes pour lesquels l’application ]0,∞) ∋ t 7−→ T (t)x ∈ E est différentiable, quel que soit x ∈ E . Définition 3.1.1 On dit que {T (t)} t≥0 est un C0-semi-groupe différentiable (et notons {T (t)} ∈ SGD(M, ω)) si l’application: ]0,∞) ∋ t 7−→ T (t)x ∈ E est différentiable, quel que soit x ∈ E . Théorème 3.1.2 Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Les affirmations suivantes sont équivalentes: i) {T (t)} t≥0 ∈ SGD(M, ω) ; ii) Im T (t) ⊂ D(A) , (∀)t > 0. Preuve i) =⇒ ii) Soient x ∈ E et t, h > 0. Puisque l’application: ]0,∞) ∋ t 7−→ T (t)x ∈ E est différentiable, la limite du rapport T (t + h)x − T (t)x 88 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES lorsque h ց 0, existe et est égale par définition avec AT (t)x. Par conséquent, T (t)x ∈ D(A). ii) =⇒ i) Considérons x ∈ E et t, h > 0. Comme T (t)x ∈ D(A), nous avons: d+T (t)x = lim T (t + h)x − T (t)x = AT (t)x . D’autre part, pour h ∈]0, t[ et δ ∈]0, t − h[ on a: T (t − h)x − T (t)x −h − AT (t)x T (t − δ)T (δ)x − T (t − h − δ)T (δ)x − AT (δ)T (t − δ)x t−h−δ T (τ)T (δ)x dτ − t−h−δ AT (δ)T (t − δ)x dτ t−h−δ [AT (δ)T (τ) − AT (δ)T (t − δ)] x dτ ‖AT (δ)‖ t−h−δ ‖T (τ) − T (t − δ)‖ dτ‖x‖ = ‖AT (δ)‖h ‖T (c) − T (t − δ)‖ ‖x‖ = = ‖AT (δ)‖ ‖T (c) − T (t − δ)‖ ‖x‖ , où c ∈ [t − h − δ, t − δ]. Par conséquent: d−T (t)x = lim T (t − h)x − T (t)x −h = AT (t)x . Donc {T (t)} t≥0 est un C0-semi-groupe différentiable. Proposition 3.1.3 Soit {T (t)} t≥0 ∈ SGD(M, ω) . Alors l’application: ]0,∞) ∋ t 7−→ T (t) ∈ B(E) est continue pour la topologie de la convergence uniforme. Preuve Soient x ∈ E et t1, t2 ∈]0,∞) tel que t1 < t2. Compte tenu du théorème 3.1.2, nous obtenons: ‖T (t1)x − T (t2)x‖ = T (s)x ds AT (t1)T (s − t1)x ds ≤ ‖AT (t1)‖ Me(s−t1)ω‖x‖ ds . 3.1. C0-SEMI-GROUPES DIFFÉRENTIABLES 89 Par suite, nous avons: ‖T (t1) − T (t2)‖ ≤ ‖AT (t1)‖M e(s−t1)ω ds , d’où résulte la continuité uniforme de l’application considérée dans l’énoncé. Théorème 3.1.4 Soient {T (t)} t≥0 ∈ SGD(M, ω) et A son générateur infinitési- mal. Alors: i) pour tout n ∈ N∗ et tout x ∈ E , on a T (t)x ∈ D(An) et: AnT (t)x = x , (∀)t > 0; ii) pour tout n ∈ N∗ l’application: ]0,∞) ∋ t 7−→ T (t) : E → D(An) est n fois différentiable pour la topologie de la convergence uniforme et: T (t) T (t) = AnT (t) ∈ B(E) , (∀)t > 0; iii) pour tout n ∈ N∗ l’application: ]0,∞) ∋ t 7−→ T (t)(n) ∈ B(E) est continue pour la topologie de la convergence uniforme. Preuve Prouvons les affirmations de l’énoncé par récurrence. i) Avec le théorème 3.1.2, on voit que pour tout x ∈ E on a T (t)x ∈ D(A) et: AT (t)x = x , (∀)t > 0. Supposons que pour tout x ∈ E on ait T (t)x ∈ D(Ak) et: T (t)x = x , (∀)t > 0. Soient x ∈ E et δ ∈]0, t[. On voit que T (t − δ)T (δ)x ∈ D(A) et: AT (t)x = AT (t − δ)T (δ)x = T (t − δ)AT (δ)x ∈ D(Ak) . Par conséquent T (t)x ∈ D(Ak+1), (∀)t > 0. De plus: Ak+1T (t)x = A AkT (t − δ)T (δ) x = A T (t − δ)AkT (δ) = AT (t − δ) 90 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Si δ = kt , il vient: Ak+1T (t)x = k + 1 )]k+1 Finalement, nous obtenons (i). ii) Pour n = 1, compte tenu du théorème 3.1.2 et de la proposition 3.1.3, il résulte que l’application: ]0,∞) ∋ t 7−→ T (t) : E → D(A) est différentiable pour la topologie de la convergence uniforme et: T (t) = AT (t) , (∀)t > 0. Comme A est un opérateur fermé et T (t) ∈ B(E), il résulte que AT (t) est un opérateur fermé défini sur E . Avec le théorème du graphe fermé ([DS’67, Theorem II.2.4, pag. 57]), on voit que AT (t) ∈ B(E), (∀)t > 0. Supposons que l’application: ]0,∞) ∋ t 7−→ T (t) : E → D(Ak) est k fois différentiable pour la topologie de la convergence uniforme et: T (t) = AkT (t) ∈ B(E) , (∀)t > 0. De plus, avec la preuve précédente, on voit que T (t)x ∈ D , pour tout t > 0. Soient x ∈ E , ‖x‖ ≤ 1 et t > 0. Si h > 0 et δ ∈]0, t[, on a: T (t + h) x − T (t)(k)x − Ak+1T (t)x AkT (δ)T (t + h − δ)x − AkT (δ)T (t − δ)x − Ak+1T (δ)T (t − δ)x AkT (δ) [T (t + h − δ) − T (t − δ)] x − Ak+1T (δ)T (t − δ)x T (δ) t+h−δ∫ T (τ)x dτ − Ak+1T (δ) 1 t+h−δ∫ T (t − δ)x dτ AkT (δ) t+h−δ∫ AT (τ)x dτ − Ak+1T (δ) 1 t+h−δ∫ T (t − δ)x dτ Ak+1T (δ) t+h−δ∫ [T (τ) − T (t − δ)] x dτ 3.1. C0-SEMI-GROUPES DIFFÉRENTIABLES 91 ∥Ak+1T (δ) t+h−δ∫ ‖T (τ) − T (t − δ)‖ ‖x‖ dτ = ∥Ak+1T (δ) ∥ ‖T (c) − T (t − δ)‖ ‖x‖ , où c ∈ [t − δ, t + h − δ]. Il s’ensuit que: T (t + h) (k) − T (t)(k) − Ak+1T (t) ∥Ak+1T (δ) ∥ ‖T (c) − T (t − δ)‖ , où c ∈ [t − δ, t + h − δ]. Par conséquent: T (t + h) (k) − T (t)(k) = Ak+1T (t) , (∀)t > 0. Si h > 0 tel que t − h > 0 et δ ∈]0, t − h[, alors nous avons: T (t − h)(k)x − T (t)(k)x −h − A k+1T (t)x AkT (δ)T (t− δ)x − AkT (δ)T (t− h − δ)x − Ak+1T (δ)T (t− δ)x AkT (δ) [T (t − δ) − T (t − h − δ)] x − Ak+1T (δ)T (t− δ)x T (δ) t−h−δ T (τ)x dτ − Ak+1T (δ) 1 t−h−δ T (t − δ)x dτ AkT (δ) t−h−δ AT (τ)x dτ − Ak+1T (δ) 1 t−h−δ T (t − δ)x dτ Ak+1T (δ) t−h−δ [T (τ) − T (t − δ)] x dτ ∥Ak+1T (δ) t−h−δ ‖T (τ) − T (t − δ)‖ ‖x‖ dτ = T (δ) ∥ ‖T (c) − T (t − δ)‖ ‖x‖ , où c ∈ [t − h − δ, t − δ]. Il vient: T (t − h)(k) − T (t)(k) −h − A k+1T (t) ∥Ak+1T (δ) ∥ ‖T (c) − T (t − δ)‖ , où c ∈ [t − h − δ, t − δ]. Par conséquent: T (t − h)(k) − T (t)(k) −h = A T (t) , (∀)t > 0. 92 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Il s’ensuit que T (t) est différentiable pour la topologie de la convergence uniforme T (t) = T (t) (k+1) = Ak+1T (t) , (∀)t > 0. Comme A est un opérateur fermé et AkT (t) ∈ B(E), il résulte que A AkT (t) est un opérateur fermé défini sur E . Avec le théorème du graphe fermé ([DS’67, Theorem II.2.4, pag. 57]), on voit que T (t) (k+1) = Ak+1T (t) ∈ B(E), (∀)t > 0. Finalement, on a obtenu (ii). iii) Soient x ∈ E avec ‖x‖ ≤ 1 et t > 0. Pour h > 0 et δ ∈]0, t[ nous obtenons: ∥T (t + h) x − T (t)′x ∥ = ‖AT (t + h)x − AT (t)x‖ ≤ ≤ ‖AT (δ)‖ ‖T (t + h − δ) − T (t − δ)‖ ‖x‖ , d’où il résulte: ∥T (t + h) ′ − T (t)′ ∥ ≤ ‖AT (δ)‖ ‖T (t + h − δ) − T (t − δ)‖ . De façon analogue, pour h > 0 et δ ∈]0, t − h[ nous obtenons: ∥T (t − h)′x − T (t)′x ∥ = ‖AT (t − h)x − AT (t)x‖ ≤ ≤ ‖AT (δ)‖ ‖T (t − h − δ) − T (t − δ)‖ ‖x‖ , d’où: ∥T (t − h)′ − T (t)′ ∥ ≤ ‖AT (δ)‖ ‖T (t − h − δ) − T (t − δ)‖ . Il est clair que l’application: ]0,∞) ∋ t 7−→ T (t)′ ∈ B(E) est continue pour la topologie de la convergence uniforme. Supposons que l’application: ]0,∞) ∋ t 7−→ T (t)(k) ∈ B(E) est continue pour la topologie de la convergence uniforme. Si h > 0 et δ ∈]0, t[, alors nous avons: ∥T (t + h) (k+1) x − T (t)(k+1)x ∥Ak+1T (t + h)x − Ak+1T (t)x ∥Ak+1T (δ) ∥ ‖T (t + h − δ) − T (t − δ)‖ ‖x‖ , d’où il s’ensuit que: ∥T (t + h) (k+1) − T (t)(k+1) ∥Ak+1T (δ) ∥ ‖T (t + h − δ) − T (t − δ)‖ . 3.1. C0-SEMI-GROUPES DIFFÉRENTIABLES 93 D’autre part, pour h > 0 et δ ∈]0, t − h[ nous obtenons: ∥T (t − h)(k+1)x − T (t)(k+1)x T (t − h)x − Ak+1T (t)x ∥Ak+1T (δ) ∥ ‖T (t − h − δ) − T (t − δ)‖ ‖x‖ et on voit que: ∥T (t − h)(k+1) − T (t)(k+1) ∥Ak+1T (δ) ∥ ‖T (t − h − δ) − T (t − δ)‖ . Donc l’application: ]0,∞) ∋ t 7−→ T (t)(k+1) ∈ B(E) est continue pour la topologie de la convergence uniforme. La propriété (iii) en découle immédiatement. Remarque 3.1.5 Si {T (t)} t≥0 ∈ SGD(M, ω) , alors l’application: ]0,∞) ∋ t 7−→ T (t) ∈ B(E) est de classe C∞]0,∞). Remarque 3.1.6 Si {T (t)} t≥0 ∈ SGD(M, ω) , alors pour tout n ∈ N∗ on a: T (t) = AnT (t) = , (∀)t > 0. Nous finissons cette section avec le théorème spectral pour les C0-semi-groupes différentiables. Soit {T (t)} t≥0 ∈ SG(M, ω) . Pour tout λ ∈ C et tout t > 0, nous avons défini l’opérateur linéaire borné: Bλ(t) : E −→ E Bλ(t)x = eλ(t−s)T (s)x ds et nous avons étudié ses propriétés avec le lemme 2.6.1. Si le C0-semi-groupe {T (t)} t≥0 est différentiable, on peut montrer le résultat suivant. Lemme 3.1.7 Soient {T (t)} ∈ SGD(M, ω) et A son générateur infinitésimal. Alors: 94 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES i) pour tout λ ∈ C et tout t > 0, l’opérateur Bλ(t) ∈ B(E) est indéfiniment dérivable et: Bλ(t) Bλ(t) + T (t)  , (∀)n ∈ N∗; ii) pour tout λ ∈ C et tout t > 0 on a: Bλ(t) T (t) = T (t) Bλ(t) , (∀)n ∈ N∗. Preuve Montrons les affirmations de l’énoncé par récurrence. i) Soient x ∈ E , λ ∈ C et t > 0. Alors: Bλ(t) x = λ Bλ(t)x + T (t)x Supposons que: Bλ(t) x = λk Bλ(t)x + T (t) Alors: Bλ(t) (k+1) Bλ(t) λBλ(t)x + T (t)x + T (t) (i+1) = λk+1 Bλ(t)x + T (t) et nous obtenons (i). ii) Soient λ ∈ C et t > 0. Compte tenu du théorème 3.1.4, pour x ∈ E , on voit que T (t)x ∈ D(An) et: AnT (t)x = AnT x = AnT + · · · + t ︸ ︷︷ ︸ n fois = An T · · ·T ︸ ︷︷ ︸ n fois x = T An = T (t)Anx , (∀)n ∈ N∗, parce que le semi-groupe commute avec son générateur infinitésimal. De même, avec le lemme 2.6.1 il résulte que: Bλ(t)T (t) = T (t)Bλ(t) . 3.1. C0-SEMI-GROUPES DIFFÉRENTIABLES 95 Alors pour x ∈ E , nous avons: Bλ(t) T (t)x = λn Bλ(t) + T (t) T (t)x = Bλ(t)T (t) + AiT (t)T (t) T (t)Bλ(t) + T (t)AiT (t) = T (t)λn Bλ(t) + T (t) x = T (t)Bλ(t) x , (∀)n ∈ N∗. D’autre part, pour x ∈ D(A), nous avons: Bλ(t)(λI − A)x = (λI − A)Bλ(t)x , d’où il résulte: Bλ(t)Ax = ABλ(t)x . Supposons que pour x ∈ D(Ak) nous avons: Bλ(t)A kx = AkBλ(t)x . Si x ∈ D(Ak+1), il vient: Bλ(t)A k+1x = Bλ(t)A k(Ax) = AkBλ(t)Ax = A kABλ(t)x = A k+1Bλ(t)x. Il s’ensuit donc que: Bλ(t)A nx = AnBλ(t)x , pour tout x ∈ D(An) et tout n ∈ N∗. De même, si x ∈ D(An), on a: Bλ(t) Anx = λn Bλ(t) + T (t) Anx = Bλ(t)A AiT (t)An AnBλ(t) + AnAiT (t) = Anλn Bλ(t) + T (t) x = AnBλ(t)x , (∀)n ∈ N∗. Finalement, pour x ∈ E nous obtenons: Bλ(t) T (t) x = Bλ(t) AnT (t)x = AnBλ(t) T (t)x = = AnT (t)Bλ(t) x = T (t) Bλ(t) x , (∀)n ∈ N∗. 96 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Théorème 3.1.8 (spectral mapping) Soient {T (t)} t≥0 ∈ SGD(M, ω) et A son générateur infinitésimal. Alors pour tout n ∈ N∗ on a: tσ(A) ∣λ ∈ σ(A) T (t) , (∀)t > 0. Preuve Pour λ ∈ C et t > 0, nous considérons l’opérateur: Bλ(t) : E −→ E Bλ(t)x = eλ(t−s)T (s)x ds . Avec le lemme 3.1.7, on déduit que l’opérateur Bλ(t) ∈ B(E) est indéfiniment dérivable et: Bλ(t) Bλ(t) + T (t)  , (∀)n ∈ N∗. Compte tenu du lemme 2.6.1, il résulte que: (λI − A)Bλ(t)x = eλtx − T (t)x , (∀)x ∈ E et que: Bλ(t)(λI − A)x = eλtx − T (t)x , (∀)x ∈ D(A). Pour tout n ∈ N∗ il s’ensuit que: (λI − A)Bλ(t)(n)x = λneλtx − T (t)(n)x , (∀)x ∈ E Bλ(t) (λI − A)x = λneλtx − T (t)(n)x , (∀)x ∈ D(A). Si λ ∈ C est tel que λneλt ∈ ρ T (t) , alors on peut considérer: λneλtI − T (t)(n) ∈ B(E) , pour tout n ∈ N∗. Par conséquent: (λI − A)Bλ(t)(n)Qx = x , (∀)x ∈ E QBλ(t) (λI − A)x = x , (∀)x ∈ D(A), 3.1. C0-SEMI-GROUPES DIFFÉRENTIABLES 97 pour tout n ∈ N∗. Mais, avec le lemme 3.1.7, il résulte: Bλ(t) T (t) = T (t) Bλ(t) , (∀)n ∈ N∗. Donc: λneλT Bλ(t) (n) − Bλ(t)(n)T (t)(n) = λneλtBλ(t)(n) − T (t)(n)Bλ(t)(n) Bλ(t) λneλtI − T (t)(n) λneλtI − T (t)(n) Bλ(t) pour tout n ∈ N∗. Par suite: Bλ(t) Q = QBλ(t) , (∀)n ∈ N∗ et nous voyons que: (λI − A)Bλ(t)(n)Qx = x , (∀)x ∈ E Bλ(t) Q(λI − A)x = x , (∀)x ∈ D(A) , d’où on obtient que λ ∈ ρ(A). Nous en déduisons que λ ∈ σ(A) implique λneλt ∈ T (t) pour tout n ∈ N∗. Par conséquent: λneλt ∣λ ∈ σ(A) T (t) ou bien: λ ∈ σ(A) T (t) et finalement: etσ(A) T (t) pour tout n ∈ N∗ et tout t > 0. 98 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES 3.2 C0-semi-groupes analytiques Par la suite nous étudions la possibilité d’étendre l’intervalle ]0,∞) à une région du plan complexe, sans abandonner les propriétés de C0-semi-groupe. Nous désignerons par ∆ l’ensemble: {z ∈ C|Re z > 0 et ϕ1 < arg z < ϕ2 , ϕ1 < 0 < ϕ2} Définition 3.2.1 On appelle C0-semi-groupe analytique une famille {T (z)}z∈∆ ⊂ B(E) vérifiant les propriétés suivantes: i) T (0) = I; ii) T (z1 + z2) = T (z1)T (z2), (∀)z1, z2 ∈ ∆; iii) limz→0 T (z)x = x, (∀)x ∈ E , z ∈ ∆; iv) l’application: ∆ ∋ z 7−→ T (z) ∈ B(E) est analytique dans le secteur ∆. Comme la multiplication par eωt n’a aucun effet sur la possibilité ou l’impossibilité d’extension à un semi-groupe analytique, il est suffit de considérer seulement les C0- semi-groupes uniformément bornés. Le théorème suivant donne une caractérisation pour les C0-semi-groupes analytiques uniformément bornés. Théorème 3.2.2 Soient {T (t)} t≥0 ∈ SG(M, 0) et A son générateur infinitésimal tel que 0 ∈ ρ(A). Les affirmations suivantes sont équivalentes: i) il existe δ > 0 tel que {T (t)} t≥0 peut être étendu à un semi-groupe analytique dans le secteur: ∆δ = {z ∈ C|Re z > 0 et | arg z| < δ} , δ > 0 et {T (z)} z∈∆δ′ est uniformément borné dans tout sous secteur fermé ∆δ′ ⊂ ∆δ, où δ′ ∈]0, δ[; ii) il existe une constante C > 0 telle que pour tout γ > 0 et tout η 6= 0 on ait: ‖R(γ + iη; A)‖ ≤ C|η| ; iii) il existe δ ∈ et K > 1 tel que: ρ(A) ⊃ Σδ = λ ∈ C | arg λ| < π ∪ {0} 3.2. C0-SEMI-GROUPES ANALYTIQUES 99 ‖R(λ; A)‖ ≤ K|λ| , (∀)λ ∈ Σδ − {0}; iv) l’application: ]0,∞) ∋ t 7−→ T (t) ∈ B(E) est différentiable et il existe une constante L > 0 tel que: ‖AT (t)‖ ≤ L , (∀)t > 0. Preuve i) =⇒ ii) Soit δ ∈ . Si δ′ ∈]0, δ[, il existe C ′ > 0 tel que: ‖T (z)‖ ≤ C ′ pour tout z ∈ ∆δ′ . Comme l’application: ∆δ ∋ z 7−→ T (z) ∈ B(E) est analytique dans le sous secteur ∆δ′ , avec le théorème de Cauchy ([DS’67, pag. 225]), on voit que T (z) dz = 0 , quel que soit le contour de Jordan lisse et fermé Γ ⊂ ∆δ′ . Par conséquent, dans l’intégrale R(γ + iη; A)x = e−(γ+iη)tT (t)x dt , γ > 0, on peut changer le chemin d’intégration par: Γθ = {r(cos θ + i sin θ)| 0 < r < ∞, |θ| ≤ δ′} . Si η > 0, alors pour le chemin Γ−δ′ = {r(cos δ′ − i sin δ′)| 0 < r < ∞} , nous obtenons: ‖R(γ + iη; A)x‖ = e−(γ+iη)tT (t)x dt −(γ+iη)r(cos δ′−i sin δ′) T (r(cos δ′ − i sin δ′))x d(r(cos δ′ − i sin δ′)) ∣e−(γ+iη)r(cos δ ′−i sin δ′) ∣ ‖T (r(cos δ′ − i sin δ′)‖ ‖x‖| cos δ′ − i sin δ′| dr ≤ ≤ C ′‖x‖ −r(γ cos δ′+η sin δ′) C ′‖x‖ γ cos δ′ + η sin δ′ η sin δ′ ‖x‖ . 100 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Si nous notons: sin δ′ alors on obtient: ‖R(γ + iη; A)‖ ≤ C Soit maintenant η < 0. Alors pour le chemin Γδ′ = {r(cos δ′ + i sin δ′)| 0 < r < ∞} , nous avons: ‖R(γ + iη; A)x‖ = e−(γ+iη)tT (t)x dt e−(γ+iη)r(cos δ ′+i sin δ′)T (r(cos δ′ + i sin δ′))x d(r(cos δ′ + i sin δ′)) ∣e−(γ+iη)r(cos δ ′+i sin δ′) ∣ ‖T (r(cos δ′ + i sin δ′)‖ ‖x‖| cos δ′ + i sin δ′| dr ≤ ≤ C ′‖x‖ e−r(γ cos δ ′−η sin δ′) dr = C ′‖x‖ γ cos δ′ − η sin δ′ ≤ −η sin δ′‖x‖ = −η‖x‖ . Par conséquent: ‖R(γ + iη; A)‖ ≤ C−η . Finalement on voit que: ‖R(γ + iη; A)‖ ≤ C|η| . ii) =⇒ iii) Comme {T (t)} ∈ SG(M, 0), avec le théorème de Hille-Yosida on voit que: ‖R(λ; A)‖ ≤ M pour tout λ ∈ Λ0. Compte tenu du (ii), il existe C > 0 tel que: ‖R(λ; A)‖ ≤ C|Imλ| pour tout λ ∈ Λ0 avec Imλ 6= 0. Compte tenu des inégalités: Reλ‖R(λ; A)‖ ≤ M |Imλ|‖R(λ; A)‖ ≤ C , 3.2. C0-SEMI-GROUPES ANALYTIQUES 101 nous obtenons: Re2λ‖R(λ; A)‖2 ≤ M2 Im2λ‖R(λ; A)‖2 ≤ C2 , d’où il résulte que: Re2λ + Im2λ ‖R(λ; A)‖2 ≤ M2 + C2 . Par conséquent, il existe une constante K1 = M2 + C2 > 1 tel que: ‖R(λ; A)‖ ≤ K1|λ| , (∀)λ ∈ Λ0. Considérons λ ∈ C avec Reλ ≤ 0. Soit γ > 0 suffisamment petit et η 6= 0. Comme l’application R( . ; A) est analytique sur ρ(A), pour tout λ = Reλ+ iη ∈ ρ(A) avec Reλ ≤ 0, nous obtenons: R(λ; A) = (Reλ − γ)n R(γ + iη; A) Avec la proposition 1.1.16 (iii), on voit que: R(γ + iη; A) = (−1)nn!R(γ + iη; A)n+1 . Alors il vient: R(λ; A) = (−1)n(Reλ − γ)nR(γ + iη; A)n+1 et cette série est uniformément convergente pour: ‖R(γ + iη; A)‖ |Reλ − γ| ≤ α < 1 . Compte tenu de la propriété (ii), elle est convergente pour la topologie de la norme si λ = Reλ + iη ∈ ρ(A) est tel que sa partie réelle vérifie Reλ ≤ 0 et |Reλ − γ| ≤ α|η| C’est-à-dire qu’il existe un voisinage Vε, ε = α|η|C , de γ + iη ∈ Λ0 contenu dans ρ(A) lorsque γ > 0 est suffisamment petit. Dans ce voisinage Vε, il existe λ ∈ C tel que Reλ ≤ 0 et λ ∈ ρ(A). Si nous définissons δ ∈ tel que: tan δ = |Reλ| |Imλ| = |Reλ| |η| = , α ∈]0, 1[, 102 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES δ = arctan , α ∈]0, 1[, alors on voit que: ζ ∈ C | arg ζ | < π ⊂ ρ(A) . Nous désignerons par Σδ l’ensemble λ ∈ C ∣| arg λ| < π ∪{0}, où δ ∈ Si λ ∈ Σδ − {0} et Reλ ≤ 0, alors nous avons: ‖R(λ; A)‖ ≤ ∥R(γ + iη; A) ∥ |Reλ − γ|n ≤ |η|n+1 αn|η|n 1 − α |Imλ| . Comme: |Reλ| |Imλ| < il vient: |Reλ|2 |Imλ|2 < d’où: |Reλ|2 |Im2λ|2 + 1 < + 1 . Par conséquent |Reλ|2 + |Imλ|2 |Imλ|2 < 1 + C2 |Imλ|2 < 1 + C2 Il s’ensuit donc que: |Imλ| < 1 + C2 C|λ| . Par suite: ‖R(λ; A)‖ ≤ 1 + C2 (1 − α)|λ| , (∀)λ ∈ Σδ − {0} et si nous notons 1 + C2 1 − α > 1 , alors il vient: ‖R(λ; A)‖ ≤ K2|λ| , (∀)λ ∈ Σδ − {0}. 3.2. C0-SEMI-GROUPES ANALYTIQUES 103 Finalement, on obtient qu’il existe K > 1 tel que: ‖R(λ; A)‖ ≤ K|λ| , (∀)λ ∈ Σδ − {0}. iii) =⇒ iv) Supposons qu’il existe δ ∈ et K > 1 tel que: λ ∈ C ∣| arg λ| < ∪ {0} ‖R(λ; A)‖ ≤ K|λ| , (∀)λ ∈ Σδ − {0}. Compte tenu du théorème 2.5.1, on voit que l’opérateur A est le générateur in- finitésimal d’un semi-groupe {T (t)} t≥0 pour lequel il existe M > 1 tel que ‖T (t)‖ ≤ M , (∀)t ≥ 0. De plus, pour ν ∈ on considère le chemin Γν = Γ ν ∪ Γ(2)ν , Γ(1)ν = {r(cos ν − i sin ν)| 0 < r < ∞} Γ(2)ν = {r(cos ν + i sin ν)| 0 < r < ∞} , tel que T (t) = eztR(z; A) dz , (∀)t ≥ 0, l’intégrale étant uniformément convergente par rapport à t > 0. ϕ : [0,∞) × Σδ −→ B(E) , ϕ(t, z) = eztR(z; A) . Il est clair que l’application ϕ est différentiable par rapport à t > 0 et: ∂ϕ(z, t) = zeztR(z; A) . De plus: ∂ϕ(z, t) ∥zeztR(z; A) ∥ ≤ K ∣ , (∀)t > 0. 104 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Avec le théorème de dérivation de Lebesgue, on voit que l’application ]0,∞) ∋ t 7→ T (t) ∈ B(E) , T (t) = eztR(z; A) dz , (∀)t ≥ 0, est différentiable et on a: T (t)′ = zeztR(z; A) dz , (∀)t ≥ 0. Comme ν ∈ , il vient cos ν < 0 et compte tenu que: ‖T (t)′‖ = zeztR(z; A) dz + zeztR(z; A) dz rert cos ν‖R(z; A)‖ dr + 1 rert cos ν‖R(z; A)‖ dr ≤ e−rt(− cos ν) dr = −t cos ν , (∀)t > 0, on déduit qu’il existe L = M π(− cos ν) > 0 tel que: ‖AT (t)‖ = ‖T (t)′‖ ≤ L , (∀)t > 0. iv) =⇒ i) Soit t0 > 0. Avec la formule de Taylor et compte tenu de la remarque 3.1.6, on a: T (t) = (t − t0)k T (k)(t0) + (n − 1)! (t − u)n−1T (n)(u) du = (t − t0)k T (t0) + (n − 1)! (t − u)n−1AnT (u) du , pour tout n ∈ N∗. Compte tenu du (iv) et de la remarque 3.1.6, on voit que: (n − 1)! (t − u)n−1AnT (u) du (n − 1)! (t − u)n−1 )]n∥∥ (n − 1)! (t − u)n−1 du ≤ 1 (n − 1)! )n t∫ (t − u)n−1 du = (n − 1)! )n t−t0∫ (t − t0)n . 3.2. C0-SEMI-GROUPES ANALYTIQUES 105 Avec la formule de Stirling n! = nn 2πne−n+ 12n , un ∈]0, 1[, on obtient n!en ≥ nn . Par conséquent: (n − 1)! (t − u)n−1AnT (u) du (t − t0)n n!en = t − t0 pour t ≥ t0 > 0 et n ∈ N∗ suffisamment grand. Il en résulte que la série de Taylor est convergente vers T (t) si t− t0 < t0Le et on a: T (t) = (t − t0)n T (t0) . Il s’ensuit donc que pour z ∈ C vérifiant Re z > 0 et |z − t0| < on peut définir une fonction analytique T (z) = (z − t0)n AnT (t0) . La série de la partie droite de cette égalité est uniformément convergente par rapport à z ∈ C vérifiant les conditions Re z > 0 et |z − t0| < α où α ∈]0, 1[. Soit z ∈ C tel que Re z = t0 >. On voit que: |Im z| = |z − t0| < d’où: |Im z| |Re z| < ou encore | arg z| ≤ arctan 106 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES En prenant δ = arctan nous obtenons que l’application ∆δ ∋ z 7→ T (z) ∈ B(E) est analytique dans le secteur ∆δ = {z ∈ C| Re z > 0 et | arg z| < δ} . De plus, si nous considérons z ∈ C avec les propriétés Re z > 0 et |z − t0| ≤ α t0Le , α ∈]0, 1[, alors nous déduisons que: ‖T (z)‖ ≤ ‖T (t0)‖ + |z − t0|n ‖AnT (t0)‖ ≤ ≤ ‖T (t0)‖ + )n (Ln ≤ M + αn = M + 1 − α . Par conséquent, si nous notons = arctan , α ∈]0, 1[, nous voyons que l’application ∆δ ∋ z 7→ T (z) ∈ B(E) est uniformément bornée dans le sous secteur ∆δ′ = {z ∈ C| Re z > 0 et | arg z| ≤ δ ′} ⊂ ∆δ . Il est évident que T (0) = I parce que {T (t)} t≥0 ∈ SG(M, 0). De plus, pour tout t > 0 et tout z ∈ ∆δ, il résulte que: T (t)T (z) = (z − t0)n AnT (t0 + t) = [(z + t) − (t0 + t)]n AnT (t0 + t) = T (t + z) . 3.3. C0-SEMI-GROUPES DE CONTRACTIONS 107 Alors, pour tous z1, z2 ∈ ∆δ, nous obtenons: T (z1)T (z2) = T (z1) (z2 − t0)n AnT (t0) = (z2 − t0)n AnT (z1)T (t0) = (z2 − t0)n AnT (z1 + t0) = [(z2 + z1) − (z1 + t0)]n AnT (z1 + t0) = T (z1 + z2) . 0<t<∞ T (t)E . Nous prouvons que cet ensemble est dense dans E . Soient x ∈ E et tn > 0, n ∈ N, tel que limn→∞ tn = 0. Alors pour xn = T (tn)x ∈ E0, n ∈ N, nous obtenons: xn = lim T (tn)x = x . Par conséquent, E0 = E . De plus, nous avons vu que {T (z)} est uniformément borné dans tout sous secteur fermé ∆δ′ . De même, pour x ∈ E on obtient T (t)x ∈ E0 et: T (z)T (t)x = lim T (z + t) = T (t)x . Compte tenu du théorème de Banach-Steinhaus ([DS’67, Theorem II.1.11, pag. 52]), il en résulte que T (z)x = x , (∀)x ∈ E , z ∈ ∆δ. Finalement, on voit que {T (z)} est un C0-semi-groupe analytique qui étend le semi-groupe {T (t)} t≥0 ∈ SG(M, 0). 3.3 C0-semi-groupes de contractions Dans la suite nous présentons quelques problèmes concernant la classe du C0-semi-groupes {T (t)}t≥0 vérifiant la propriété ‖T (t)‖ ≤ 1, pour tout t ≥ 0. Définition 3.3.1 On dit que {T (t)} est un C0-semi-groupe de contractions sur l’espace de Banch E si {T (t)} t≥0 ∈ SG(1, 0). 108 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Lemme 3.3.2 Soit {T (t)} t≥0 ∈ SG(M, ω). Alors, l’application: ||| . ||| : E −→ R+ |||x||| = sup −ωt‖T (t)x‖ , (∀)x ∈ E , est une norme sur E équivalente avec la norme initiale ‖ . ‖. Preuve Soit x ∈ E . Pour tout t ≥ 0, on a: e−ωt‖T (t)x‖ ≤ e−ωt‖T (t)‖‖x‖ ≤ M‖x‖ . En passant à la borne supérieure par rapport à t, on voit que: |||x||| ≤ M‖x‖ , (∀)x ∈ E . D’autre part, nous avons: |||x||| = sup e−ωt‖T (t)x‖ ≥ e−ω0‖T (0)x‖ = ‖x‖ , (∀)x ∈ E , d’où il résulte que: ‖x‖ ≤ |||x||| ≤ M‖x‖ , (∀)x ∈ E . Par conséquent les normes ||| . ||| et ‖ . ‖ sont équivalentes. Théorème 3.3.3 Soient {T (t)} t≥0 ∈ SG(M, ω), A son générateur infinitésimal S(t) = e−ωtT (t) , (∀)t ≥ 0. Alors: i) {S(t)} t≥0 ∈ SG(1, 0); ii) le C0-semi-groupe {S(t)}t≥0 a pour générateur infinitésimal l’opérateur B = A − ωI. Preuve i) Il est clair que la famille {S(t)} t≥0 est un C0-semi-groupe. De plus, pour tout t ≥ 0, on a: |||S(t)x||| = sup e−ωs‖T (s)e−ωtT (t)x‖ = = sup e−ωτ‖T (τ)x‖ ≤ sup e−ωτ‖T (τ)x‖ = |||x||| , (∀)x ∈ E , 3.3. C0-SEMI-GROUPES DE CONTRACTIONS 109 d’où on obtient: |||S(t)x||| ≤ |||x||| , (∀)x ∈ E et t ≥ 0. Il s’ensuit que: ‖S(t)‖ ≤ 1 , (∀)t ≥ 0 et, par conséquent, {S(t)} t≥0 ∈ SG(1, 0). ii) Elle est analogue à celle du théorème 2.1.11. Pour les C0-semi-groupes de contractions, on peut formuler la version suivante du théorème de Hille-Yosida. Théorème 3.3.4 Un opérateur linéaire: A : D(A) ⊂ E −→ E est le générateur infinitésimal d’un semi-groupe {T (t)} ∈ SG(1, 0) si et seule- ment si: i) A est un opérateur fermé et D(A) = E ; ii) Λ0 = {λ ∈ C |Reλ > 0} ⊂ ρ(A) et pour λ ∈ Λ0, on a: ‖R(λ; A)n‖ ≤ 1 (Reλ)n , (∀)n ∈ N∗. Preuve i) =⇒ ii) Comme {T (t)} t≥0 ∈ SG(1, 0), nous avons: ‖T (t)‖ ≤ 1 , (∀)t ≥ 0. Par suite, on peut prendre M = 1 et ω = 0. Avec le théorème de Hille-Yosida, il résulte que: (i) A est un opérateur fermé et D(A) = E ; (ii) Λ0 = {λ ∈ C |Reλ > 0} ⊂ ρ(A) et pour λ ∈ Λ0, on a: ‖R(λ; A)n‖ ≤ 1 (Reλ)n , (∀)n ∈ N∗. ii) =⇒ i) Soit A : D(A) ⊂ E −→ E un opérateur linéaire vérifiant les propriétés (i) et (ii) de l’énoncé. Avec le théorème de Hille-Yosida, il en résulte que A est le générateur infinitésimal d’un C0-semi- groupe {T (t)} t≥0 pour lequel il existe M = 1 et ω = 0 tel que: ‖T (t)‖ ≤ 1 , (∀)t ≥ 0. 110 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Donc A est le générateur infinitésimal d’un semi-groupe de contractions. Une autre caractérisation très intéressante des C0-semi-groupes de contractions est donnée par le fameux théorème de Lumer-Phillips, dans lequel interviennent les opérateurs m-dissipatifs. Définition 3.3.5 On appelle opérateur m-dissipatif un opérateur linéaire A : D(A) ⊂ E −→ E vérifiant les propriétés suivantes: i) A est opérateur dissipatif; ii) il existe α0 > 0 tel que Im (α0I − A) = E . Théorème 3.3.6 (Lumer - Phillips) Soit A : D(A) ⊂ E −→ E un opérateur linéaire tel que D(A) = E . L’opérateur A est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(1, 0) si et seulement si A est un opérateur m-dissipatif. Preuve =⇒ Soit A : D(A) ⊂ E −→ E le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(1, 0). Si x ∈ D(A) et x∗ ∈ J (x), alors pour tout t ≥ 0 on voit Re〈T (t)x − x, x∗〉 = Re〈T (t)x, x∗〉 − Re〈x, x∗〉 ≤ ≤ |〈T (t)x, x∗〉| − Re〈x, x∗〉 ≤ ‖T (t)x‖‖x∗‖∗ − ‖x‖2 ≤ ≤ ‖x‖2 − ‖x‖2 = 0 , d’où: Re〈T (t)x − x, x∗〉 ≤ 0 , (∀)t ≥ 0. En passant à limite pour t ց 0, il vient: Re〈Ax, x∗〉 ≤ 0 pour tout x ∈ E et tout x∗ ∈ J (x). Il s’ensuit donc que A est un opérateur dissipatif. D’autre part, avec le théorème 3.3.4 , on voit que ]0,∞) ⊂ ρ(A). Donc αI − A ∈ GL(E), (∀)α ∈]0,∞). Par suite, Im (αI − A) = E pour tout α > 0 et finalement on voit que A est un opérateur m-dissipatif. ii) Soit A : D(A) ⊂ E −→ E un opérateur m-dissipatif tel que D(A) = E . Alors, il existe α0 ∈]0,∞) tel que Im (α0I − A) = E . En appliquant la proposition 1.2.3, 3.4. NOTES 111 on voit que Im (αI −A) = E , pour tout α ∈]0,∞). Il s’ensuit que ]0,∞) ⊂ ρ(A). Comme A est un opérateur dissipatif, compte tenu de la proposition 1.2.2, il vient: ‖(αI − A)x‖ ≥ α‖x‖ , (∀)x ∈ D(A), d’où: ‖R(α; A)‖ ≤ 1 pour tout α ∈]0,∞). Avec le théorème 3.3.4, on voit que A est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 ∈ SG(1, 0). Proposition 3.3.7 Soit A ∈ B(E) tel que ‖A‖ ≤ 1. Alors et(A−I) est un semi-groupe uniformément continu de contractions. Preuve Il est évident que et(A−I) est un semi-groupe uniformément continu. De plus: ∥et(A−I) ∥ ≤ et‖A‖e−t ≤ 1 , (∀)t ≥ 0. 3.4 Notes Pour les propriétés des C0-semi-groupes différentiables nous avons consulté [Pa’83-1, pag. 51], [Ah’91, pag. 73] et [Da’80, pag. 28]. Le théorème 3.1.8 se trouve dans [Le’00-1]. Les propriétés des C0-semi-groupes analytiques uniformément bornés se trouvent dans [Pa’83-1, pag. 60], [Ah’91, pag. 81] ou [Da’80, pag.59]. Une introduction très intéressante des C0- semi-groupes analytiques, par la construction d’un calcul fonctionnel adéquat, est donné dans [CHADP’87, pag. 121]. Le théorème 3.3.6 a été montré par Lummer et Phillips dans [LP’61]. 112 CHAPITRE 3. C0-SEMIGROUPES AVEC PROPRIÉTÉS SPÉCIALES Chapitre 4 La formule de Lie - Trotter 4.1 Le cas des semi-groupes uniformément con- tinus Dans cette section nous présentons la formule du produit de Lie-Trotter pour les semi-groupes uniformément continus. Théorème 4.1.1 Soient A le générateur infinitésimal d’un semi-groupe uniformé- ment continu {T (t)} t≥0 et {An(t)}t≥0 ⊂ B(E) tel que: ‖An(t)‖ = 0 , uniformément par rapport à t sur les intervalles compacts de [0,∞). Alors: T (t) = lim (A + An(t)) uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve Soient 0 ≤ a < b. Si nous notons: Vn(t) = I + (A + An(t)) , alors, pour tout 0 ≤ k ≤ n, on a: ∥V kn (t) (A + An(t)) (‖A‖ + ‖An(t)‖) (‖A‖ + ‖An(t)‖) 114 CHAPITRE 4. LA FORMULE DE LIE - TROTTER ti (‖A‖ + ‖An(t)‖)i ti (‖A‖ + ‖An(t)‖)i ti (‖A‖ + ‖An(t)‖)i = et(‖A‖+‖An(t)‖) ≤ M , cette dernière quantité etant uniformément bornée par rapport à t ∈ [a, b]. De même, pour: Un(t) = e A , (∀)t ≥ 0, nous obtenons: Unn (t) = e tA = T (t) , (∀)t ≥ 0 et pour tout 0 ≤ k ≤ n, nous avons: ∥ ≤ ektn ‖A‖ ≤ entn ‖A‖ ≤ et‖A‖ ≤ N , uniformément par rapport à t ∈ [a, b]. Il vient: V nn (t) − Unn (t) = V nn (t)U0n(t) − V n−1n (t)U1n(t) + + V n−1n (t)U n(t) − V n−2n (t)U2n(t) + + V n−2n (t)U n(t) − · · · − V 0n (t)Unn (t) = V n−in (t)U n(t) − V n−i−1n (t)U i+1n (t) V n−i−1n (t) [Vn(t) − Un(t)] U in(t) . Comme: Vn(t) − Un(t) = I + An(t) − e = I + An(t) − I − A − 1 A2 − · · · , il résulte que: ‖Vn(t) − Un(t)‖ ≤ ‖An(t)‖ + ‖A‖2 + · · · . Par conséquent: ‖V nn (t) − Unn (t)‖ ≤ ‖An(t)‖ + ‖A‖2 + · · · t‖An(t)‖ + ‖A‖2 + · · · −→ 0 si n → ∞, uniformément par rapport à t sur les intervalles compacts de [0,∞), ce qui achève la preuve. 4.1. LE CAS DES SEMI-GROUPES UNIFORMÉMENT CONTINUS 115 Théorème 4.1.2 (la formule exponentielle) Soit A le générateur infinitési- mal d’un semi-groupe uniformément continu {T (t)} t≥0. Alors: T (t) = lim = lim I − t = lim uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve La première égalité résulte du théorème 4.1.1 pour An(t) = 0, quels que soient n ∈ N et t ≥ 0. Soient 0 ≤ a < b. On a: pour n suffisamment grand et t ∈ [a, b]. Avec le lemme 1.1.2, il vient: I − t A ∈ GL(E) I − t = I + (A + An(t)) , An(t) = A3 + · · · ‖An(t)‖ = 0 , uniformément par rapport à t ∈ [a, b]. Avec le théorème 4.1.1, on voit que: T (t) = lim (A + An(t)) = lim I − t uniformément par rapport à t sur les intervalles compacts de [0,∞). La troisième égalité en résulte compte tenu que: I − t I − A Théorème 4.1.3 (la formule de Lie-Trotter) Soit A1 le générateur infinité- simal du semi-groupe uniformément continu {T1(t)}t≥0 et A2 le générateur in- finitésimal du semi-groupe uniformément continu {T2(t)}t≥0, alors l’opérateur: A : E −→ E , Ax = A1x + A2x 116 CHAPITRE 4. LA FORMULE DE LIE - TROTTER est le générateur infinitésimal d’un semi-groupe {T (t)} t≥0 uniformément continu, tel que: T (t) = lim uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve Nous avons successivement: A21 + A31 + · · · A22 + A32 + · · · = I + (A1 + A2) + A21 + A1A2 + + · · · = = I + (A1 + A2) + A21 + A1A2 + + · · · = I + [A + An(t)] , où l’opérateur: An(t) = 1 + A1A2 + 1A2 + + · · · a la propriété: ‖An(t)‖ ≤ A21 + A1A2 + A31 + A21A2 + + · · · −→ 0 si n → ∞ , uniformément par rapport à t sur les intervalles compacts de [0,∞). Avec le théorème 4.1.1, on voit que: T (t) = lim uniformément par rapport à t sur les intervalles compacts de [0,∞). Remarque 4.1.4 Si A, B ∈ B(E), alors on a: et(A+B) = lim uniformément par rapport à t sur les intervalles compacts de [0,∞). 4.2. PROPRIÉTÉS DE CONVERGENCE DES C0-SEMI-GROUPES 117 4.2 Propriétés de convergence des C0-semi-groupes Dans cette section on introduit la topologie de la résolvante sur l’ensemble GI(E) des générateurs infinitésimaux et on montre le théorème de Trotter - Kato. Définition 4.2.1 On dit que la suite (An)n∈N∗ ⊂ GI(E) est convergente vers A ∈ GI(E) pour la topologie forte de la résolvante si pour tout λ ∈ ⋂ ρ(An) ∩ ρ(A), on a: R(λ; An)x −→ R(λ; A)x , si n → ∞ , (∀)x ∈ E . De même, on dit que la suite (An)n∈N∗ ⊂ GI(E) est convergente vers A ∈ GI(E) pour la topologie uniforme de la résolvante si pour tout λ ∈ ⋂ ρ(An) ∩ ρ(A), on ‖R(λ; An) − R(λ; A)‖ −→ 0 , si n → ∞. Par la suite, nous supposerons que GI(E) est doté de la topologie forte de la résolvante. Lemme 4.2.2 Soient {T (t)} t≥0 , {S(t)}t≥0 ∈ SG(M, ω) et A, respectivement B, leur générateurs infinitésimaux. Alors pour tout λ ∈ Λω et tout x ∈ E on a l’égalité: R(λ; B) [T (t) − S(t)] R(λ; A)x = S(t − s) [R(λ; A) − R(λ; B)]T (s)x ds quel que soit t ≥ 0. Preuve Soient x ∈ E et λ ∈ Λω. Alors R(λ; A)x ∈ D(A) et R(λ; B)x ∈ D(B). L’application: [0, t] ∋ s −→ S(t − s)R(λ; B)T (s)R(λ; A)x ∈ E est différentiable et pour s ∈ [0, t] et x ∈ E nous avons: S(t − s) [R(λ; B)T (s)R(λ; A)x] = = S(t − s)(−B)R(λ; B)T (s)R(λ; A)x + S(t − s)R(λ; B)T (s)AR(λ; A)x = = S(t − s)(λI − B − λI)R(λ; B)T (s)R(λ; A)x + + S(t − s)R(λ; B)T (s)(−λI + A + λI)R(λ; A)x = = S(t − s)T (s)R(λ; A)x− λS(t − s)R(λ; B)T (s)R(λ; A)x− 118 CHAPITRE 4. LA FORMULE DE LIE - TROTTER − S(t − s)R(λ; B)T (s)x + λS(t − s)R(λ; B)T (s)R(λ; A)x = = S(t − s) [T (s)R(λ; A)x − R(λ; B)T (s)x] = = S(t − s) [R(λ; A) − R(λ; B)]T (s)x puisque la résolvante R(λ; A) commute avec T (t), (∀)t ≥ 0. Par conséquent: S(t − s)R(λ; B)T (s)R(λ; A)x|t0 = S(t − s) [R(λ; A) − R(λ; B)]T (s)x ds , ou encore: R(λ; B)T (t)R(λ; A)x − S(t)R(λ; B)R(λ; A)x = S(t − s) [R(λ; A) − R(λ; B)] T (s)x ds , (∀)x ∈ E . Comme S(t)R(λ; B) = R(λ; B)S(t) pour tout t ≥ 0, on obtient finalement: R(λ; B) [T (t) − S(t)] R(λ; A)x = S(t − s) [R(λ; A) − R(λ; B)]T (s)x ds pour tout x ∈ E . Le théorème suivant présente une très jolie correspondance entre les C0-semi- groupes d’opérateurs linéaires bornés et leur générateurs infinitésimaux. Théorème 4.2.3 Soient {Tn(t)}t≥0 ⊂ SG(M, ω) ayant pour générateurs infinitésimaux les opérateurs (An)n∈N∗ ⊂ GI(M, ω) et {T (t)}t≥0 ∈ SG(M, ω) ayant pour générateur infinitésimal l’opérateur A ∈ GI(M, ω). Les affirmations suivantes sont équivalentes: i) An −→ A, si n → ∞, pour la topologie forte de la résolvante; ii) pour tout t0 ∈]0,∞) on a l’égalité: t∈[0,t0] ‖Tn(t)x − T (t)x‖ = 0 , (∀)x ∈ E . Preuve i) =⇒ ii) Supposons que An −→ A, si n → ∞, pour la topologie forte de la résolvante. Pour tout λ ∈ Λω, nous avons: R(λ; An)x −→ R(λ; A)x , si n → ∞ , (∀)x ∈ E . Soient t0 ∈]0,∞), x ∈ E et λ ∈ Λω arbitrairement fixées. Puisque la résolvante commute avec le semi-groupe associé, il résulte que: [Tn(t) − T (t)] R(λ; A)x = Tn(t) [R(λ; A) − R(λ; An)] x + + R(λ; An) [Tn(t) − T (t)]x + [R(λ; An) − R(λ; A)] T (t)x. 4.2. PROPRIÉTÉS DE CONVERGENCE DES C0-SEMI-GROUPES 119 Montrons que cette expression tend vers zero si n → ∞. Comme {Tn(t)}t≥0 ⊂ SG(M, ω), il est clair que: ‖Tn(t)‖ ≤ Meωt0 , (∀)t ∈ [0, t0]. Compte tennu de (i), on voit que le premier terme converge vers zero si n → ∞, uniformément par rapport à t ∈ [0, t0]. De même, la continuité de l’application t 7→ T (t)x sur l’intervalle compact [0, t0], conduit au fait que l’ensemble {T (t)x |t ∈ [0, t0]} est compact, comme l’image d’un compact par une fonction continue. On en déduit facilement que le troisième terme est fortement convergent vers zero lorsque n → ∞ et cette convergence est uniforme par rapport à t ∈ [0, t0]. Pour le deuxième terme, compte tenu du lemme 4.2.2, on a: R(λ; An) [T (t) − Tn(t)] R(λ; A)x = Tn(t − s) [R(λ; A) − R(λ; An)] T (s)x ds, pour tout t ∈ [0, t0]. Si pour s ∈ [0, t0], on pose ft,n(s)x = Tn(t − s) [R(λ; A) − R(λ; An)] T (s)x , 0 ≤ s ≤ t ≤ t0, alors on voit que: ‖ft,n(s)x‖ = ‖Tn(t − s) [R(λ; A) − R(λ; An)] T (s)x‖ ≤ ≤ ‖Tn(t − s)‖ ‖R(λ; A) − R(λ; An)‖ ‖T (s)‖ ‖x‖ ≤ ≤ Meω(t−s) (‖R(λ; A)‖ + ‖R(λ; An)‖) Meωs‖x‖ ≤ ≤ Meωt Re λ − ω + Re λ − ω Meωt‖x‖ = Re λ − ωe 2ωt‖x‖ . De plus, compte tenu des inégalités: ‖ft,n(s)x‖ = ‖Tn(t − s) [R(λ; A) − R(λ; An)] T (s)x‖ ≤ ≤ ‖Tn(t − s)‖ ‖R(λ; A) − R(λ; An)‖ ‖T (s)‖ ‖x‖ ≤ ≤ Meω(t−s) ‖R(λ; A) − R(λ; An)‖Meωs‖x‖ ≤ ≤ M2eωt ‖R(λ; A) − R(λ; An)‖ ‖x‖ , nous obtenons ‖ft,n(s)x‖ = 0 , 120 CHAPITRE 4. LA FORMULE DE LIE - TROTTER quels que soient s ∈ [0, t] et t ∈ [0, t0]. Avec le théorème de la convergence dominée de Lebesgue ([DS’67, Theorem III.3.7, pag. 124]) il résulte que: ‖ft,n(s)x‖ ds = ‖ft,n(s)x‖ ds , pour tout t ∈ [0, t0]. Il s’ensuit donc que: ‖R(λ; An) [T (t) − Tn(t)] R(λ; A)x‖ = 0 , (∀)x ∈ E , uniformément par rapport à t ∈ [0, t0]. Si nous notons y = R(λ; A)x ∈ D(A), on voit que: ‖R(λ; An) [T (t) − Tn(t)] y‖ = 0 , (∀)y ∈ D(A), uniformément par rapport à t ∈ [0, t0]. Par conséquent, si x ∈ D(A), le deuxième terme tend vers zero pour n → ∞, uniformément par rapport à t ∈ [0, t0]. Il s’ensuit que: t∈[0,t0] ‖[Tn(t) − T (t)]R(λ; A)x‖ = 0 , (∀)x ∈ D(A), d’où il résulte immédiatement: t∈[0,t0] ‖[Tn(t) − T (t)] y‖ = 0 , (∀)y ∈ R(λ; A)D(A). Comme R(λ; A)D(A) = D (A2), compte tenu du théorème 2.2.10 on voit que R(λ; A)D(A) = E . Nous obtenons finalement: t∈[0,t0] ‖Tn(t)x − T (t)x‖ = 0 , (∀)x ∈ E . ii) =⇒ i) En appliquant le théorème 2.3.1, nous obtenons pour λ ∈ Λω: [R(λ; An) − R(λ; A)] x = e−λt [Tn(t) − T (t)] x dt , (∀)x ∈ E , d’où il résulte: ‖[R(λ; An) − R(λ; A)]x‖ ≤ e−Reλt ‖[Tn(t) − T (t)]x‖ dt , (∀)x ∈ E . Mais: ‖[Tn(t) − T (t)] x‖ ≤ 2Meωt‖x‖ 4.2. PROPRIÉTÉS DE CONVERGENCE DES C0-SEMI-GROUPES 121 quels que soient x ∈ E , t ≥ 0 et n ∈ N∗. Dans ce cas, en posant: fn(t) = e λt [Tn(t) − T (t)]x , (∀)t ≥ 0 on voit que: ‖fn(t)‖ ≤ 2Me−(Reλ−ω)t‖x‖ , (∀)t ≥ 0. De plus, compte tenu de l’inégalité: ‖fn(t)‖ ≤ e−Reλt ‖[Tn(t) − T (t)]x‖ , nous obtenons: ‖fn(t)‖ = 0 , (∀)t ≥ 0. Avec le théorème de la convergence dominée de Lebesgue ([DS’67, Theorem III.3.7, pag. 124]), il vient: ‖[R(λ; An) − R(λ; A)] x‖ = 0 pour tout x ∈ E et tout λ ∈ Λω. Donc An −→ A, si n → ∞, pour la topologie forte de la résolvante. Une version intéressante du théorème 4.2.3 est le théorème suivant. Théorème 4.2.4 Soient {Tα(t)}t≥0 ⊂ SG(M, ω) ayant pour générateurs in- finitésimaux les opérateurs (Aα)α>0 ⊂ GI(M, ω) et {T (t)}t≥0 ∈ SG(M, ω) ayant pour générateur infinitésimal l’opérateur A ∈ GI(M, ω). Les affirmations suivantes sont équivalentes: i) pour tout x ∈ D(A), il existe xα ∈ D(Aα) tel que: xα = x Aαxα = Ax; ii) pour tout λ ∈ Λω, on a: R(λ; Aα)x = R(λ; A)x , (∀)x ∈ E ; iii) pour tout t0 ∈]0,∞), nous avons: t∈[0,t0] ‖Tα(t)x − T (t)x‖ = 0 , (∀)x ∈ E . 122 CHAPITRE 4. LA FORMULE DE LIE - TROTTER Preuve i) =⇒ ii) Soient λ ∈ Λω et x ∈ D(A). Alors il existe xα ∈ D(Aα), α > 0 tel que xα = x Aαxα = Ax . Nous définissons y = (λI − A)x ∈ (λI − A)D(A) yα = (λI − Aα)xα ∈ (λI − Aα)D(Aα) , α > 0. Il résulte que x = R(λ; A)y et xα = R(λ; Aα)yα. Compte tenu des égalités du (i), nous obtenons: R(λ; Aα)yα = R(λ; A)y AαR(λ; Aα)yα = AR(λ; A)y . On voit que cette dernière égalité devient: (λI − λI + Aα)R(λ; Aα)yα = (λI − λI + A)R(λ; A)y ou bien λR(λ; Aα)yα − lim (λI − Aα)R(λ; Aα)yα = = λR(λ; A)y − (λI − A)R(λ; A)y . Il vient: λR(λ; A)y − lim yα = λR(λ; A)y − y , d’où: yα = y . D’autre part, pour tout α > 0 on a: ‖R(λ; Aα)‖ ≤ Reλ − ω et pour y ∈ (λI − A)D(A) on voit que: R(λ; Aα)y = R(λ; Aα)(y − yα + yα) = R(λ; Aα)(y − yα) + R(λ; Aα)yα . 4.2. PROPRIÉTÉS DE CONVERGENCE DES C0-SEMI-GROUPES 123 Par suite: ‖R(λ; Aα)y − R(λ; A)y‖ ≤ ≤ ‖R(λ; Aα)(y − yα)‖ + ‖R(λ; Aα)yα − R(λ; A)y‖ ≤ Reλ − ω‖y − yα‖ + ‖R(λ; Aα)yα − R(λ; A)y‖ , d’où il vient: R(λ; Aα)y = R(λ; A)y , pour tout y ∈ (λI − A)D(A). Comme (λI − A)D(A) = E , on voit que: R(λ; Aα)x = R(λ; A)x , (∀)x ∈ E . ii) =⇒ i) Soient λ ∈ Λω et x ∈ E tel que: R(λ; Aα)x = R(λ; A)x . Si nous définissons: yα = R(λ; Aα)x ∈ D(Aα) y = R(λ; A)x ∈ D(A) , nous obtenons: yα = y . De plus: Aαyα = lim AαR(λ; Aα)x = lim [λR(λ; Aα)x − x] = = λR(λ; A)x − x = AR(λ; A)x = Ay . ii) ⇐⇒ iii) Cette équivalence s’obtient avec une preuve analogue à celle du théorème 4.2.3. Corollaire 4.2.5 Soient {Tα(t)}t≥0 ⊂ SG(M, ω) ayant pour générateurs in- finitésimaux les opérateurs (Aα)α>0 ⊂ GI(M, ω) et {T (t)}t≥0 ∈ SG(M, ω) ayant pour générateur infinitésimal l’opérateur A ∈ GI(M, ω). Supposons que pour tout x ∈ D(A), il existe δ > 0 tel que pour tout α ∈]0, δ[ on ait x ∈ D(Aα) et limαց0 Aαx = Ax. Alors, pour tout t0 ∈]0,∞) nous avons: t∈[0,t0] ‖Tα(t)x − T (t)x‖ = 0 , (∀)x ∈ E . 124 CHAPITRE 4. LA FORMULE DE LIE - TROTTER Preuve Dans le théorème 4.2.4, nous pouvons prendre xα = x, (∀)α ∈]0, δ[. Le théorème suivant montre que sous certaines conditions, GI(M, ω) est une sous-classe fermée dans GI(E). Théorème 4.2.6 Soient (An)n∈N∗ ⊂ GI(M, ω) et λ0 ∈ Λω tel que: i) (R(λ0; An))n∈N∗ est fortement convergente vers Rλ0 ∈ B(E); ii) Im Rλ0 = E . Alors il existe un unique opérateur A ∈ GI(M, ω) tel que Rλ0 = R(λ0; A). Preuve Nous notons: λ ∈ Λω ∣(R(λ; An))n∈N∗ est fortement convergente Montrons que S = Λω. Prouvons que S est ensemble ouvert dans Λω. Soit µ ∈ S. Pour tout n ∈ N∗, l’application: ρ(An) ∋ λ 7−→ R(λ; An) ∈ B(E) est analytique et nous avons: R(λ; An) = (λ − µ)k R(µ; An) = (λ − µ)k (−1)kk!R(µ; An)k+1 = (µ − λ)kR(µ; An)k+1 . Comme An ∈ GI(M, ω) implique: ∥R(µ; An) ∥ ≤ M (Reµ − ω)k , (∀)k ∈ N on voit que: ‖R(λ; An)‖ ≤ |µ − λ|k ∥R(µ; A)k+1 ∥ ≤ M Reµ − ω |µ − λ| Reµ − ω La série de la partie droite de cette inégalité est convergente sur l’ensemble: λ ∈ Λω ∣ |µ − λ|(Reµ − ω)−1 < 1 Il en résulte que la série: R(λ; An) = (µ − λ)kR(µ; An)k+1 4.2. PROPRIÉTÉS DE CONVERGENCE DES C0-SEMI-GROUPES 125 est uniformément convergente sur les compacts λ ∈ Λω ∣ |µ − λ|(Reµ − ω)−1 ≤ ν < 1 ⊂ V . Comme ‖R(λ; A)‖ ≤ M Re µ − ω on voit que la suite (R(λ; An))n∈N∗ est fortement convergente pour tout λ ∈ Vν . Donc il existe un voisinage de µ contenu dans S. Par conséquent S est ensemble ouvert dans Λω. Maintenant, nous allons montrer que S est un ensemble relativement fermé dans Λω. Soient (λm)m∈N ⊂ S et λ ∈ Λω tel que λ = lim Pour tout ν ∈]0, 1[, il existe λm,ν ∈ S tel que: |λm,ν − λ| (Re λm,ν − ω)−1 ≤ ν < 1 . Compte tenu de la première partie de la preuve, on voit que la série R(λ; An) = (λm,ν − λ)k R(λm,ν ; An)k+1 est uniformément convergente et que la suite (R(λ; An))n ∈ N∗ est fortement convergente. Par conséquent, λ ∈ S et S est un ensemble relativement fermé dans Λω. Comme λ0 ∈ S, nous voyons que S = Λω par connexité. Pour λ ∈ Λω, définissons l’opérateur Rλ ∈ B(E) par: Rλx = lim R(λ; An)x , (∀)x ∈ E . Soient λ , µ ∈ Λω arbitraires. On a: (Rλ − Rµ) x = lim [R(λ; An) − R(µ; An)]x = = lim (µ − λ)R(λ; An)R(µ; An)x = = (µ − λ)RλRµx , (∀)x ∈ E . Par conséquent Rλ est une pseudo-résolvante, quel que soit λ ∈ Λω. Comme il existe λ0 ∈ Λω tel que Im Rλ0 = E , compte tenu du théorème 1.1.22 (ii), on déduit que Im Rλ = E , quel que soit λ ∈ Λω. Avec l’inégalité: ‖R(λ; An)m‖ ≤ (Reλ − ω)m , (∀)λ ∈ Λω et m ∈ N 126 CHAPITRE 4. LA FORMULE DE LIE - TROTTER on voit que pour tout compact K ⊂ Λω, il existe MK > 0 tel que ‖R(λ; An)‖ ≤ MK , quel que soit n ∈ N∗. Avec le lemme de Montel ([GS’99, pag. 220]), on déduit qu’il existe une sous-suite (R(λ; Ank))k∈N∗ telle que Rλx = lim R(λ; Ank)x , (∀)x ∈ E , uniformément par rapport à λ sur les compacts de Λω. Comme R(λ; Ank) est un opérateur injectif pour tout k ∈ N∗, avec le théorème de Hurwitz ([GS’99, pag. 193]) nous obtenons que Rλ est un opérateur injectif, donc Ker R(λ; An) = {0}. En appliquant le théorème 1.1.22 (iii), on voit que pour tout λ ∈ Λω, il existe un opérateur linéaire A : D(A) −→ E , A = λI − R−1λ fermé et défini sur un sous espace dense tel que Rλ = R(λ; A), (∀)λ ∈ Λω. De plus: ‖R(λ; A)m‖ ≤ M (Reλ − ω)m . et le théorème de Hille - Yosida implique alors que A ∈ GI(M, ω). Maintenant, nous avons toutes les conditions pour formuler un autre résultat important concernant les C0-semi-groupes. Théorème 4.2.7 (Trotter - Kato) Soit {Tn(t)}t≥0 ⊂ SG(M, ω) ayant pour générateurs infinitésimaux les opérateurs (An)n∈N∗ ⊂ GI(M, ω). S’il existe λ0 ∈ Λω tel que: i) (R(λ0; An))n∈N∗ est fortement convergente vers Rλ0 ∈ B(E); ii) Im Rλ0 = E , alors il existe un unique opérateur A ∈ GI(M, ω) tel que Rλ = R(λ; A), (∀)λ ∈ Λω. De plus, si {T (t)}t≥0 est le C0-semi-groupe engendré par A, alors pour tout t0 ∈]0,∞) on a: t∈[0,t0] ‖Tn(t)x − T (t)x‖ = 0 , (∀)x ∈ E . Preuve Les affirmations du théorème résultent du théorème 4.2.3 et du théorème 4.2.6. 4.3. FORMULE DE LIE - TROTTER POUR LES C0-SEMI-GROUPES 127 4.3 Formule de Lie - Trotter pour les C0-semi- groupes Dans la suite, nous montrons le théorème de représentation générale, la for- mule exponentielle et la formule de Lie-Trotter pour les semi-groupes fortement continus. Nous commençons par un résultat technique. Lemme 4.3.1 Soient T ∈ B(E) et M, N ≥ 1 tel que: ∥ ≤ MNk , (∀)k ∈ N∗. Alors, pour tout n ∈ N, nous avons: ∥en(T−I)x − T nx ∥ ≤ MNn−1e(N−1)n n2(N − 1)2 + nN‖Tx − x‖ pour tout x ∈ E . Preuve Soient k, n ∈ N tel que k ≥ n. Alors, nous avons: ∥T kx − T nx T i+1x − T ix ∥ ‖Tx − x‖ ≤ ‖Tx − x‖ MN i ≤ ≤ M‖Tx − x‖ Nk−1 = (k − n)MNk−1‖Tx − x‖ ≤ ≤ |k − n|MNn+k−1‖Tx − x‖ , (∀)x ∈ E . Compte tenu de la symétrie, il est clair que cette inégalité reste valable si nous considérons n > k. Par suite, on voit que: ∥T kx − T nx ∥ ≤ |k − n|MNn+k−1‖Tx − x‖ , (∀)x ∈ E et n, k ∈ N. Si t ≥ 0 et n ∈ N, alors nous avons: ∥et(T−I)x − T nx T kx − T nx ≤ e−t ∥T kx − T nx ∥ ≤ MNn−1e−t‖Tx − x‖ (tN)k |k − n| . 128 CHAPITRE 4. LA FORMULE DE LIE - TROTTER Avec l’inégalité de Cauchy-Schwartz, il vient: (tN)k |k − n| = (tN)k (tN)k |k − n| (tN)k (tN)k (k − n)2 = etN (n − Nt)2 + Nt . Il s’ensuit que: ∥et(T−I)x − T nx ∥ ≤ MNn−1e(N−1)t (n − Nt)2 + Nt‖Tx − x‖ quel que soit x ∈ E . Finalement, en prenant t = n, nous obtenons l’inégalité considérée dans l’énoncé. Théorème 4.3.2 (de représentation générale) Soit {F (t)} t≥0 ⊂ B(E) une famille d’opérateurs linéaires bornés avec F (0) = I. Supposons qu’il existe ω ≥ 0 et M ≥ 1 tel que: ∥F (t) ∥ ≤ Mekωt , (∀)k ∈ N∗, pour tout t ≥ 0. Si A est le générateur infinitésimal d’un C0-semi-groupe {T (t)}t≥0 tel que: F (t)x − x = Ax , (∀)x ∈ D(A), alors nous avons: T (t)x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve Soient 0 ≤ a < b. Pour t ∈ [a, b], définissons: , (∀)n ∈ N∗. Il est clair que An ∈ B(E), (∀)n ∈ N∗, d’où il résulte que pour tout n ∈ N∗, An est le générateur infinitésimal du semi-groupe uniformément continu de plus, nous avons: Anx = Ax , (∀)x ∈ D(A). 4.3. FORMULE DE LIE - TROTTER POUR LES C0-SEMI-GROUPES 129 Avec le corollaire 4.2.5, nous voyons que: etAnx = T (t)x , (∀)x ∈ E , uniformément par rapport à t ∈ [a, b]. Compte tenu du lemme 4.3.1, si x ∈ D(A), il vient: tAnx − n[F( tn)−I]x − ≤ Meω tn (n−1)e n − 1 + neω x − x (n−1)+ n − 1 + neω x − x (n−1)+ e n − 1 x − x d’où: etAnx − −→ 0 si n → ∞, pour tout x ∈ D(A), uniformément par rapport à t ∈ [a, b]. De plus, pour tout x ∈ D(A), nous avons: T (t)x − ∥T (t)x − etAnx etAnx − −→ 0 si n → ∞, d’où l’on déduit que: T (t)x = lim x , (∀)x ∈ D(A), uniformément par rapport à t sur les intervalles compacts de [0,∞). Comme D(A) = E et ∥ ≤ Meωt, on voit que: T (t)x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Théorème 4.3.3 (la formule exponentielle) Soient {T (t)} t≥0 ∈ SG(M, ω) et A son générateur infinitésimal. Alors: T (t)x = lim I − t x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intevalles compacts de [0,∞). 130 CHAPITRE 4. LA FORMULE DE LIE - TROTTER Preuve Pour A ∈ GI(M, ω) et t ∈ , nous définissons: F (t) = (I − tA)−1 = 1 Compte tenu du théorème de Hille-Yosida, on voit que: ∥F (t)k (1 − ωt)k . Comme: (1 − ωt)k ≤ e 1−ωt , il vient: ∥F (t)k ∥ ≤ Me2kωt pour t ∈ . D’autre part, avec le lemme 2.3.6 nous obtenons: F (t)x − x = lim [F (t) − I] x = lim AF (t)x = = lim = Ax , (∀)x ∈ D(A). Compte tenu du théorème de représentation générale, on voit que: T (t)x = lim x = lim I − t = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compactes de [0,∞). Théorème 4.3.4 (la formule de Lie-Trotter) Soient A1 ∈ GI(M1, ω1) le générateur infinitésimal du semi-groupe {T1(t)}t≥0 ∈ SG(M1, ω1), respectivement A2 ∈ GI(M2, ω2) le générateur infinitésimal du semi-groupe {T2(t)}t≥0 ∈ SG(M2, ω2). Supposons qu’il existe ω ≥ 0 et M ≥ 1 tel que: ∥[T1(t)T2(t)] ∥ ≤ Mekωt , (∀)k ∈ N∗. Si l’opérateur A : D(A) ⊂ E −→ E , défini par: Ax = A1x + A2x , (∀)x ∈ D(A) = D(A1) ∩ D(A2), 4.3. FORMULE DE LIE - TROTTER POUR LES C0-SEMI-GROUPES 131 est le générateur infinitésimal d’un C0-semi-groupe {T (t)}t≥0, alors: T (t)x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Preuve Soit: F : [0,∞) −→ B(E) F (t) = T1(t)T2(t) , (∀)t ≥ 0. Il est évident que F (0) = I. De plus, pour x ∈ D(A), nous avons: F (t)x − x = lim T1(t)T2(t)x − x = lim T1(t)T2(t)x − T1(t)x + lim T1(t)x − x = lim T1(t) T2(t)x − x + A1x = A1x + A2x = Ax . Avec le théorème de représentation générale, on voit que: T (t)x = lim x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Remarque 4.3.5 Si A ∈ GI(M, ω), compte tenu de la formule exponentielle, on peut définir: etAx = lim I − t x = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). Avec cette notation, dans les hypothèses du théorème 4.3.4, nous obtenons pour la formule de Lie-Trotter l’expression: etAx = lim x , (∀)x ∈ E , uniformément par rapport à t sur les intervalles compacts de [0,∞). 132 CHAPITRE 4. LA FORMULE DE LIE - TROTTER 4.4 Notes Pour les résultats de la section 4.1 on peut consulter [Ka’82, pag. 35]. Les propriétés de convergence pour les C0-semi-groupes ont été étudiées par Trotter dans [Tr’58]. Pour les théorèmes 4.2.3, 4.2.6, 4.2.7 on peut consulter [Pa’83-1, pag. 84] ou [Ah’91, pag. 131] et pour le théorème 4.2.4 nous avons utilisé [Da’80, pag. 80]. Le théorème 4.3.4 a été montré par Trotter dans [Tr’59] et a été étudié par Chernoff dans [Ce’68]. Les résultats que nous avons présentés se trouvent dans [Pa’83-1, pag. 89]. Dans [Da’80, pag. 90], on peut trouver ces problèmes pour les C0-semi-groupes de contractions. Bibliographie [Ah’91] Ahmed, N.U. Semigroup theory with application to systems and control. Logman Scientific & Tehnical, London, 1991. [Ba’76] Barbu, V. Nonliniar semigroups and differential equations in Banach Spaces. Editura Academiei R.S.R. Bucureşti and Noordhoff International Pub- lishing Leyden, 1976. [BB’67] Butzer, P.L., Berens,H. Semi-Groups of Operators and Approxima- tions. Springer Verlag, New York Inc., 1967. [Ca’01] Cassier, G. Semigroups in finite von Neumann algebras. Operator The- ory: Adv. and Appl., 127, Birkhäuser Verlag, 2001, 145-162. [Ce’68] Chernoff, P.R. Note on product formula for operator semi-groups. J. Funct. Anal., 2(1968), 238-242. [Ce’72] Chernoff, P.R. 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Funcţii de variabilă complexă. Ed. Mirton, Timişoara, 1995. [GS’99] Gaşpar, D., Suciu, N. Analiză complexă. Ed. Acad. Române, Bu- cureşti, 1999. [GW’74] Gaşpar, D., Westphal, U. Über den Kogenerator einer Kontrac- tionshalbgruppe. Analele Universităţii din Timişoara, 1(1974), 43-55. [Go’99] Gomilko, A.M. Conditions on the Generator of a Uniformly Bounded C0-semigroup. Funct. Anal. and Its Appl., 33(1999), 294-296. [Hi’48] Hille, E. Functional Analysis and Semi-Groups. A.M.S., New York, 1948. [HP’57] Hille, E., Phillips, R.S. Functional Analysis and Semi-Groups. A.M.S., Providence, Rhode Island, 1957. [Is’81] Istrăţescu, V.I. Introduction to Linear Operator Theory. Marcel Dekker, Inc., New York and Basel, 1981. [Ka’66] Kato, T. Perturbation theory for linear operators. Springer Verlag, Berlin, Heidelberg, New York, 1966. [Ka’70] Kato, T. A characterization of holomorphic semigroups. Proc. Amer. Math. Soc., 25(1970), 495-498. [Ka’82] Kato, T. A Short Introduction to Perturbation Theory for Linear Op- erators. Springer Verlag, New York, Heidelberg, Berlin, 1982. [Le’00-1] Lemle, L.D. Asupra teoremei aplicaţiei spectrale pentru C0- semigrupuri diferenţiabile. Universitaria ROPET Petroşani (2000), 33-38. BIBLIOGRAPHIE 135 [Le’00-2] Lemle, L.D. Aproximaţia Yosida a generatorului infinitezimal al unui semigrup tare continuu de operatori liniari mărginiţi. Bul. Şt. Acad. ”Henri Coandă” Braşov, 2(12) (2000), 265-274. [Le’00-3] Lemle, L.D. Transformata Laplace a unui semigrup uniform continuu de operatori liniari mărginiţi. Analele Facultăţii de Inginerie din Hunedoara, (5)2000, 159-163. [Le’00-4] Lemle, L.D. Le théorème spectral pour les semi-groupes uniformément continus d’opérateurs linéaires bornés. Analele Facultăţii de Inginerie din Hunedoara, (5)2000, 164-167. [Le’00-5] Lemle, L.D. Un théorèm de représentation pour C0-semi-groupes. Pro- ceedings of the National Conference on Mathematical Analysis and Applica- tions (the Dumitru Gaşpar anniversary issue). Timişoara, 2000, 163-176. [Le’01-1] Lemle, L.D. La formule du produit pour les semi-groupes uni- formément continus. Analele Facultăţii de Inginerie din Hunedoara, (5)2001, 173-176. [Le’01-2] Lemle, L.D. C0-semi-groupes analytiques. Analele Facultăţii de In- ginerie din Hunedoara, (5)2001, 177-184. [LP’61] Lumer, G., Phillips, R.S. Dissipative operators in a Banach space. Pacific J. Math., 11(1961), 679-698. [Me’88] Megan, M. Propriétés qualitatives des systèmes linéaires côntrolés dans les espaces de dimension infinie. Monographies mathématiques nr.32, Univer- sité de Timişoara, 1988. [Mi’52] Miyadera, I. Generation of strongly continuous semi-groups of opera- tors. Tohoku Math. J., 4(1952), 109-114. [Na’35] Nathan, D.S. One parameter groups of transformations in abstract vec- tor spaces. Duke Math. J., (1935), 518-526. [Pa’83-1] Pazy, A. Semigroups of linear operators and applications to partial differential equations. Springer Verlag, New York, Berlin, 1983. 136 BIBLIOGRAPHIE [Pa’83-2] Pazy, A. Semigroups of operators in Banach spaces. Lect. Notes in Math., 1017(1983), Springer, Berlin, 508-524. [Ph’52] Phillips, R.S. On the generation of semi-groups of linear operators. Pacific J.Math., 2(1952), 393-415. [RB’98] Reghiş, M., Babescu, Gh. Ecuaţii diferenţiale liniare de ordinul doi ı̂n spaţii Banach. Ed. Academiei Române, Bucureşti, 1998. [Tr’58] Trotter, H.F. Approximation of semi-groups of operators. Pacific J. Math., 8(1958), 887-919. [Tr’59] Trotter, H.F. On the product of semi-groups of operators. Proc. Am. Math. Soc., 10(1959),545-551. [Va’87] Vasilescu, F.H. Iniţiere ı̂n teoria operatorilor liniari. Editura Tehnică, Bucureşti, 1987. [Yo’36] Yosida, K. On the group embeded in the metrical complete ring. Japan J. Math., (1936), 7-26. [Yo’48] Yosida, K. On the differentiability and the representation of one- parameter semi-groupes of linear operators. J. Math. Soc. Japan, 1(1948), 15-21. [Yo’57] Yosida, K. Lectures on Semi-group Theory and its application to Cauchy problem in Partial Differential Equations. Tata Institute of Fundamental Re- search, Bombay, 1957. [Yo’67] Yosida, K. Functional Analysis. Springer Verlag, New York, 1967. Introduction Préliminaires Les opérateurs dissipatifs Semi-groupes uniformément continus Notes Semi-groupes de classe C0 Définitions. Propriétés élémentaires Propriétés générales des C0-semi-groupes Le théorème de Hille - Yosida La représentation de Bromwich Conditions suffisantes d'appartenances à GI(M,0) Propriétés spectrales des C0-semi-groupes Notes C0-semigroupes avec propriétés spéciales C0-semi-groupes différentiables C0-semi-groupes analytiques C0-semi-groupes de contractions Notes La formule de Lie - Trotter Le cas des semi-groupes uniformément continus Propriétés de convergence des C0-semi-groupes Formule de Lie - Trotter pour les C0-semi-groupes Notes

0704.1400

Reconstructing the Intrinsic Triaxial Shape of the Virgo Cluster

Reconstructing the Intrinsic Triaxial Shape of the Virgo Cluster Bomee Lee and Jounghun Lee Department of Physics and Astronomy, FPRD, Seoul National University, Seoul 151-747, Korea [email protected], [email protected] ABSTRACT To use galaxy clusters as a cosmological probe, it is important to account for their triaxiality. Assuming that the triaxial shapes of galaxy clusters are induced by the tidal interaction with the surrounding matter, Lee and Kang recently developed a reconstruction algorithm for the measurement of the axial ratio of a triaxial cluster. We examine the validity of this reconstruction algorithm by performing an observational test of it with the Virgo cluster as a target. We first modify the LK06 algorithm by incorporating the two dimensional projec- tion effect. Then, we analyze the 1275 member galaxies from the Virgo Cluster Catalogue and find the projected direction of the Virgo cluster major axis by measuring the anisotropy in the spatial distribution of the member galaxies in the two dimensional projected plane. Applying the modified reconstruction al- gorithm to the analyzed data, we find that the axial ratio of the triaxial Virgo cluster is (1: 0.54 : 0.73). This result is consistent with the recent observational report from the Virgo Cluster Survey, proving the robustness of the reconstruc- tion algorithm. It is also found that at the inner radii the shape tends to be more like prolate. We discuss the possible effect of the Virgo cluster triaxiality on the mass estimation. Subject headings: cosmology:theory — large-scale structure of universe 1. INTRODUCTION Galaxy clusters provide one of the most powerful tools to constrain the key cosmological parameters. In the era of precision cosmology, it is important to determine their mass as accurately as possible before using them as a cosmological probe. Any kind of simplified assumption about the properties of galaxy clusters could cause substantial systematics in the mass estimation. The triaxial shapes of galaxy clusters are one of such properties. http://arxiv.org/abs/0704.1400v1 – 2 – It has been long known both observationally and numerically that the galaxy clusters are noticeably triaxial (e.g., Frenk et al. 1988; West 1989; Plionis et al. 1991; Warren et al. 1992). While plenty of efforts have been already made to take into account the triaxial shapes of galaxy clusters (Jing & Suto 2002; Fox & Pen 2001; Suwa et al. 2003; Lee & Suto 2004; Kasun & Evrard 2005; Hopkins et al. 2005; Lee et al. 2005; Smith & Watts 2005; Hayashi et al. 2007), previous studies have been largely focused on the statistical treatment of cluster tri- axiality. For the measurement of the gas mass fraction of galaxy clusters that can pro- vide powerful constraints on the density parameter and the dark energy equation of state (White et al. 1993; Lubin et al. 1996; Cen 1997; Evrard 1997; Cooray 1998; Laroque et al. 2006; Ferramacho & Blanchard 2007), however, it is necessary to deal with the individual clusters and their triaxial shapes. The standard picture based on the cosmic web theory (Bardeen et al. 1986; Bond et al. 1996) explains that galaxy clusters are rare events corresponding to the local maxima of the initial density field and form at the dense nodes of the local filaments in the cosmic web through tidal interactions with the surrounding matter. The tidal effect from the sur- rounding matter results in the deviation of cluster shapes from spherical symmetry as well as the preferential alignments of cluster galaxies (or halos) with the major axes of their host clusters (Binggeli 1982; Struble & Peebles 1985; Hopkins et al. 2005; Kasun & Evrard 2005; Lee et al. 2005; Algood et al. 2006; Altay et al. 2006; Paz et al. 2006). In the frame of this standard scenario, Lee & Kang (2006, hereafter, LK06) have re- cently developed an analytic algorithm to reconstruct the triaxial shapes of individual clus- ters. The key concept of the LK06 algorithm is that the two axial ratios of a triaxial cluster are related to the eigenvalues of the local tidal shear tensor. By measuring the spatial align- ment of the cluster galaxies with the major axis of their host cluster, one can determine the eigenvalues of the local tidal tensor, which will in turn yields the two axial ratios of a triaxial cluster. Testing their analytic model against high-resolution N-body simulations, LK06 have shown that their algorithm works well within 20% errors. Now that the LK06 algorithm is known to work in principle, it is time to test the algorithm against observations. Our goal here is to apply the LK06 algorithm to real observational data and examine its validity in practice. Here, we use the Virgo cluster as a target, whose triaxial shape has been very recently measured observationally (Mei et al. 2007). The organization of this paper is as follows. In §2, a brief overview of the LK06 algorithm is provided and how to incorporate the two dimensional projection effect into the algorithm is explained. In §3, the Virgo cluster data are analyzed and its triaxial shapes are reconstructed using the LK06 algorithm. In §4, the results are summarized, and the advantages and the – 3 – caveats of our model are discussed. 2. THEORETICAL MODEL 2.1. Overview of the LK06 Algorithm According to the LK06 algorithm, the cluster triaxial shape originates from its tidal interaction with the surrounding matter distribution. This assumption has been verified from high-resolution N-body simulation which demonstrated clearly that the tidal field elongates the cluster shapes (e.g., Altay et al. 2006, and references therein). LK06 has shown that the two axial ratios of a triaxial cluster are related to the three eigenvalues of the local tidal tensor defined as the second derivative of the gravitational potential: 1− λ2 1− λ3 1− λ1 1− λ3 , (1) where {a, b, c} (with a ≤ b ≤ c)are the three principal axis lengths of a cluster and {λ1, λ2, λ3} (with λ1 ≥ λ2 ≥ λ3) are the three eigenvalues of the local tidal tensor, T. According to this formula, one can estimate the cluster axial ratios if λ1, λ2, λ3 is fixed by fitting the probability density distribution, p(cos θ) analytically to the observational data. LK06 suggested that the preferential locations of the cluster galaxies near the cluster major axes given the local tidal tensor be described as 〈x̂ix̂j |T̂ 〉 = δij + sT̂ikT̂kj. (2) where x̂ ≡ (x̂i) is the unit position vector of a cluster galaxy, T̂ ≡ T/|T| is the unit tidal shear tensor, and s ∈ [−1, 1] is the correlation parameter that represents the correlation strength between x̂ and T̂. Under the assumption that the minor axis of the tidal shear tensor is in the direction of the cluster major axis, equation (2) basically describes the alignment between the position of a cluster galaxy and the major axis of its host cluster. If s = −1, there is a maximum alignment. If s = 1, there is a maximum anti-alignment. The case of s = 0 corresponds to no alignment. Let θ3d be the angle between the host cluster major axis and the galaxy position vector. Under the assumption that the cluster major axis is in the direction of the minor principal axis of the tidal shear tensor, The probability density distribution of cos θ3d was derived by LK6 as p(cos θ3d) = (x̂i ·M ij · x̂j) 2 , (3) – 4 – where φ is an azimuthal angle of x̂ measured in an arbitrary coordinate system. Here, covariance matrix M is defined as M ≡ 〈x̂ix̂j |T̂ 〉. Note that equation (3) holds good for any arbitrary coordinate system in which the tidal shear tensor is not necessarily diagonal. In the principal axis frame of the tidal tensor, equation (4) can be expressed only in terms of the three eigenvalues of the tidal shear tensor as p(cos θ) = (1− s+ 3sλ̂2i ) sin2 θ cos2 φ 1− s+ 3sλ̂2 sin2 θ sin2 φ 1− s+ 3sλ̂2 cos2 θ 1− s+ 3sλ̂2 dφ, (4) where {λ̂i} i=1 are the eigenvalues of T̂, related to {λi} i=1 as δcλ̂1 λ̂1 + λ̂2 + λ̂3 , λ2 = δcλ̂2 λ̂1 + λ̂2 + λ̂3 , λ3 = δcλ̂3 λ̂1 + λ̂2 + λ̂3 , (5) where δc ≈ 1.68 is the linear density threshold for a dark halo (Eke et al. 1996) satisfying the following constraint of δc = i=1 λi. The key concept of LK06 algorithm is that by fitting equation (3) to the observed probability distribution, one can find the best-fit values of λ1, λ2 and s, and then determine the cluster axial ratios using equation (1). Although LK06 algorithm allows us in principle to reconstruct the three dimensional intrinsic triaxial shapes of individual clusters, it is restricted to the cases where the informations on the three dimensional positions of the cluster galaxies in the cluster principal axis frame are given. Unfortunately, for most clusters, these informations are not available but only two dimensional projected images of clusters. In §2.2, we attempt to modify the LK06 algorithm in order to incorporate the two dimensional projection effect. 2.2. Projection Effect Let us suppose that the position vectors of the cluster galaxies are all projected along the line of sight direction onto the plane of a sky. Unless the major axis of the host cluster is perfectly aligned with the line-of-sight direction to the cluster center, one would expect that the projected position vectors of the cluster galaxies should show a tendency to be aligned with the projected major principal axes. Let θ2d be the angle between the projected cluster major axis and galaxy position vector in the plane of sky. The probability distribution can be calculated by integrating equation – 5 – (3) along the line of sight as p(cos θ2d) = (x̂i ·M ij · x̂j) 2dx̂3, (6) where (x̂3) is now chosen to be in the direction of the line of sight. In other words, we consider a certain Cartesian coordinate system in which the x̂3 direction is parallel to the line of the sight to the cluster center of mass. Note that in this coordinate system the tidal tensor is not necessarily diagonal. Through the similarity transformation T̂ = Rt · Λ̂T ·R, (7) λ̂1 0 0 0 λ̂2 0 0 0 λ̂3 , (8) one can express the nondiagonal unit tidal tensor T̂ in terms of its eigenvalues.Here, the rotation matrix R has the form of (Binney 1985) − sinψ − cosψ cos ξ cosψ sin ξ cosψ − sinψ cos ξ sinψ sin ξ 0 sin ξ cos ξ  , (9) where (ξ, ψ) is the polar coordinate of the line-of-sight direction in the principal axis frame of the cluster. Through equations (6)-(9), we finally final an analytic expression for the probability density distribution of cos θ2d in terms of {λ̂i} i=1 and the correlation parameter s: p(cos θ2d) = (1 + s)2(1− 2s) Q−3/2dx̂3, (10) with the factor Q defined as Q ≡ [(1 + s)2 − 3s(1 + s)](A1λ̂1 + A2λ̂2 + A3λ̂3)(1− x̂ +[(1 + s)2 − 3s(1 + s)](C1λ̂1 + C2λ̂2 + C3λ̂3)x̂ +6s(1 + s)(B1λ̂1 +B2λ̂2 +B3λ̂3)x̂3 1− x̂2 , (11) where the coefficients {Ai} i=1, {Bi} i=1, {Ci} i=1 are given as A1 = cosψ cos θ2d(cosψ cos θ2d − 2 sinψ cos ξ sin θ2d) + sin 2 θ2d(sin 2 ψ + sin2 ξ cos2 ψ), – 6 – A2 = sinψ cos θ2d(sinψ cos θ2d + 2 cosψ cos ξ sin θ2d) + sin 2 θ2d(cos 2 ψ + sin2 ξ sin2 ψ), A3 = cos 2 θ2d + cos 2 ξ sin2 θ2d, B1 = − sin ξ cosψ(cos ξ sin θ2d cosψ + sinψ cos θ2d), B2 = − sin ξ sinψ(cos ξ sin θ2d sinψ − cosψ cos θ2d), B3 = cos ξ sin ξ sin θ2d, C1 = sin 2 ψ + cos2 ξ cos2 ψ, C2 = cos 2 ψ + cos2 ξ sin2 ψ, C3 = sin With this new modified algorithm in the two dimensional space, we can reconstruct the intrinsic shape of a triaxial cluster halo from the observed two dimensional image. 3. APPLICATION TO THE VIRGO CLUSTER HALO 3.1. Observational Data and Analysis The Virgo cluster is the nearest richly populated cluster of galaxies whose properties has been studied fruitfully for long (e.g., Bohringer et al. 1994; West & Blakeslee 2000, and references therein). It is known to have approximately 1275 member galaxies (Binggeli et al. 1985) and located at a distance of approximately 16.1 Mpc from us (Tonry et al. 2001) with the major axis inclined at an angle of approximately 10o with respect to the line of sight (West & Blakeslee 2000). Near the center of the Virgo cluster is located the large ellipticity galaxy M87 (or Virgo A). The major axis of the Virgo cluster is found to be inclined approximately 10◦ with respect to the line of sight direction to M87 (West & Blakeslee 2000). We use data from the Virgo Cluster Catalogue (Binggeli et al. 1985) which compiles the equatorial coordinates of total 1275 member galaxies. 3.2. Coordinate Transformation and Projection Effect Let r and (α, δ) represent the three dimensional distance to a member galaxy and its equatorial coordinates, respectively. Under the assumption that the position of M87 is the center of mass of the Virgo cluster, the Cartesian coordinate of a member galaxy in the center of mass frame can be written as x1 = r cosα cos δ − x1va x2 = r sinα cos δ − x2va – 7 – x3 = r sin δ − x3va (12) where (x1va, x2va, x3va) represents the position of M87. The equatorial coordinates of M87 is measured to be αva = 187.71 ◦ and δva = 12.39 ◦. The distance to Virgo A, rva, from us is also known to be approximately 16.1Mpc (Tonry et al. 2001; SBF survey). Thus,we have a full information on (x1va, x2va, x3va). Now let us consider a coordinate system where the third axis is in in the direction of the line of sight to M87. Let (u, ϑ, ϕ) be the spherical polar coordinate of the member galaxy in this coordinate system. It can be found through coordinate transformation as sin ξva cosψva cos ξva cosψva − sinψva sin ξva sinψva cos ξva sinψva cosψva cos ξva − sin ξva 0  (13) where (ξva, ψva) is the polar coordinate of M87, so that ξva ≡ π/2− δva. In the plane of sky projected along the line of sight direction, the spherical polar coordi- nates of the member galaxy can be regarded as the Cartesian coordinates as r → 0, ϑ→ xl, and φ → yl. Basically, it represents a two dimensional projected position of a Virgo cluster member galaxy in the plane of sky with the position of M87 as a center. For each member galaxy from the Virgo Cluster Catalogue, we have determined the two dimensional projected position (xl, yl) using the given equatorial coordinates. 3.3. Virgo Cluster Reconstruction To measure the alignments between the positions of the Virgo cluster galaxies and the projected major axis of the Virgo cluster and compare the distribution of the alignment angles with the analytic model (eq.[6]), it is necessary to find the direction of the projected major axis in the coordinate system of (x1l, x2l). Equivalently, it is necessary to have an information on the polar coordinates of the line of sight, (ξ, ψ) with respect to the three dimensional principal axis of the Virgo cluster. The seminal paper of West & Blakeslee (2000) provides us with the information on ξ i.e., the angle between the three dimensional major axis of the Virgo cluster and the line of sight direction as approximately 10◦. However, we still need the azimuthal angle ψ of the major axis of the Virgo cluster. To find the azimuthal angle ψ, we first let ξ = 10◦ and ψ = 0 in the analytic model (eq.[6]). It implies that we choose a certain Cartesian coordinates (x1p, x2p) where the angle – 8 – ψ vanishes. Then, we transform the coordinate system of (x1l, x2l) into this new coordinate system as cosψ sinψ − sinψ cosψ . (14) Here, note that (cosψ, sinψ) corresponds to the projected major axis of the Virgo cluster in the (x1l, x2l) coordinate system. It is expected that in this new (x1p, x2p)-coordinate system the the observationally measured distribution should fit the analytic model (eq.[6] best. For a given ψ, we measure the alignment angle between the projected major axis and position vector of each Virgo cluster galaxy as cos θ2d = x̂1p cosψ + x̂2p sinψ, (15) where x̂p ≡ xp/|xp|. Then, by counting the number of galaxies galaxy’s number density as a function of cos θd, we can derive the probability distribution, p(cos θd). We fit this observational distribution with the analytic model, adjusting the values of λ1, λ2 and s through χ2-minimization. We repeart the whole process varying the value of ψ, and seek for the value of ψ which yields the smallest χ2 value. As a final step, we determine the corresponding best-fit values of λ1 and λ2. Finally, we find the axial ratios of the Virgo cluster to be a/c = 0.54 and b/c = 0.73 by using equation (1) with the constraint of δc = i=1 λi. The best-fit value of s is also determined to be −0.25, indicating that the positions of the cluster galaxies are indeed aligned with the major axes of the Virgo cluster. Figure [1] plots the two dimensional projected positions of the Virgo cluster galaxies in the (x1p, x2p)-coordinate system. The arrow represents the direction of the projected major axis determined from the chosen value of ψ. Figure 2 plots the probability distribution of the alignment angles for four different cases of ψ. In each panel, the histogram with Poisson errors is the observational distribution while the solid line represents the analytic fitting model. The dotted line stands for the case of no alignment at all. The top left panel corresponds to the finally chosen value of ψ for which the analytic fitting model and the observational result agree with each other best. The other three panels show the three exemplary cases of ψ for which the analytic fitting model and the observational result do not agree with each other well with relatively high χ2 value. To investigate whether the reconstructed axial ratios of the Virgo cluster changes with the radius from the center, we introduce a cut-off radius, Rcut, and remeasure the axial ratios using only those galaxies located within Rcut from the center in the (x1p, x2p)- corrodinate system. We repeat the same process but using four different values of Rcut. Table 1 lists the values of the resulting best-fit axial ratios a/c and b/c, and the best-fit correlation parameter – 9 – for the four different cut-off radii of Rcut. As can be seen, at the inner radius smaller than the maximum one, the shape tends to be more prolate-like, consistent with recent numerical report (e.g., Hayashi et al. 2007). Note also that the value of s is consistently −0.25, which implies the strength of the tidal interaction with surrounding matter is consistent. Figure 3 plots the probability distributions for the four different cases of Rcut. The top left panel shows the case of maximum cut-off radius. As can be seen, the agreements between the analytic fitting model and the observational result are quite good even at inner radii. 4. SUMMARY AND DISCUSSION We have modified the cluster reconstruction algorithm which was originally developed by Lee & Kang (2006) to apply it to the two dimensional projected images of galaxy clusters in practice. Assuming that unless the cluster major axes are in the line-of-sight direction, we have found that the alignments between the galaxy positions and the projected major axes can be used to reconstruct the two axial ratios of the triaxial clusters. We have applied the modified algorithm to the observational data of the Virgo cluster and shown that the reconstructed axial ratios of (1: 0.54 : 0.73) are in good agreement with the recent report from the Virgo Cluster survey (Mei et al. 2007), which proves the validity and usefulness of our method in practice. Now that the Virgo cluster is found to be triaxial, let us discuss on the triaxiality effect on the mass estimation. For simplification, let us assume that the Virgo cluster has a uniform density. Then, the mass of the triaxial Virgo cluster with the axial ratios given as (1 : 0.54 : 0.73) is estimated to be 2.6 times larger than the spherical case since the spherical radius is close to the minor axis length of the Virgo cluster since the major axis of the Virgo cluster is very well aligned with the line of sight direction. Therefore, it can cause maximum ∼ 50% errors to neglect the Virgo cluster triaxiality. The most prominent merit of our method over the previous one is that it reconstructs directly the intrinsic three dimensional structures of the underlying triaxial dark matter halo using the fact that the cluster triaxiality originated from the tidal interaction. Convention- ally, the triaxial shape of a cluster is found through calculating its inertia momentum tensor. This conventional method, however, is unlikely to yield the intrinsic shape of the underlying dark matter halo unless the target cluster is a well relaxed system. In addition, our method does not resort to any simplified assumption like the axis-symmetry or the alignment with the line-of-sight and etc. Yet, it is worth mentioning here a couple of limitations of our method. First, it assumes – 10 – that the major axes of the clusters are not aligned with the line-of-sight direction so that the alignments between the galaxy positions and the projected major axes can be measured. Second, for the Virgo cluster, the crucial information on the angle ξ between the three dimensional major axis and the line of sight has been already given West & Blakeslee (2000). Which simplifies the whole of our method since it was only ψ that has to be determined. For most clusters, however, this information is not given, so that both the values of ξ and ψ have to be determined through fitting before finding the axial ratios. Third, its success is subject to the validity of the LK06 algorithm. According to the numerical test, the LK06 algorithm suffers approximately 20% errors for the case that the number of the member galaxies is not high enough. It implies that the LK06 algorithm is definitely restricted to the rich clusters with large numbers of galaxies. Therefore, it may be necessary to refine and improve the LK06 algorithm itself for the application to poor cluster samples. Our future work is in this direction. This work is supported by the research grant No. R01-2005-000-10610-0 from the Basic Research Program of the Korea Science and Engineering Foundation. – 11 – REFERENCES Algood, B. et al. 2006, MNRAS, 367, 1781 Altay, G., Colberg, J. M., & Croft, R. A. C. 2006, MNRAS, 370, 1422 Anninos, P., & Norman, M. L. 1996, ApJ, 459, 12 Bardeen, B., Bond, R. 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The arrow represents the direction of the projected major axes of the Virgo cluster. – 15 – Fig. 2.— Probability density distributions of the cosines of the angles between the position vectors of the Virgo cluster galaxies and four different choices of the Virgo cluster major axis in the two dimensional projected plane of sky. In each panel, the histogram with Poisson errors represents the observational data points from the Virgo Catalog, the solid line is the analytic fitting function based on the LK06 reconstruction algorithm, and the dotted line corresponds to the case of no correlation. The top left panel corresponds to the best-fit result according to which the major axes of the Virgo cluster in the two dimensional projected plane is determined, while the other three panels show how the agreements between the observational and the analytic results change if different directions other than the major axes are used. – 16 – Fig. 3.— Comparison between the observational and the analytic results for the alignments at different two dimensional cut-off radii (Rcut). – 17 – Table 1. The cut-off radius (Rcut) in the two dimensional projected space, the number of the member galaxies enclosed within Rcut, the reconstructed two axial ratios, and the best-fit value of the correlation parameter. Rcut Ng a/c b/c s (Mpc) 2.24 1275 0.53 0.73 −0.25 1.68 1221 0.64 0.69 −0.25 1.49 1154 0.64 0.69 −0.25 1.12 902 0.64 0.69 −0.25 INTRODUCTION THEORETICAL MODEL Overview of the LK06 Algorithm Projection Effect APPLICATION TO THE VIRGO CLUSTER HALO Observational Data and Analysis Coordinate Transformation and Projection Effect Virgo Cluster Reconstruction SUMMARY AND DISCUSSION

0704.1401

Flat Pencils of Symplectic Connections and Hamiltonian Operators of Degree 2

FLAT PENCILS OF SYMPLECTIC CONNECTIONS AND HAMILTONIAN OPERATORS OF DEGREE 2 JAMES T. FERGUSON Abstract. Bi-Hamiltonian structures involving Hamiltonian operators of de- gree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms of an algebra structure on the space of 1-forms, related to so-called Fermionic Novikov algebras. Then, degree 2 operators are considered as deformations of hydrodynamic type Poisson brackets. 1. Introduction Hamilton’s equations for a finite-dimensional system with position coordinates qi and associated momenta pi, are understood geometrically as describing the flow of a vector field XH which is associated with the Hamiltonian function H(q1, . . . , qn, p1 . . . , pn) by the formula XH(f) = {f,H}, where {·, ·} is the Poisson bracket: {f, g} = . (1) More generally, one defines a Poisson bracket on an n-dimensional manifoldM as a map C∞(M)× C∞(M) → C∞(M), (f, g) 7→ {f, g}, satisfying, for any functions f, g, h on M : (1) antisymmetry: {f, g} = −{g, f} , (2) linearity: {af + bg, h} = a{f, h}+ b{g, h} for any constants a, b , (3) product rule: {fg, h} = f{g, h}+ g{f, h} , (4) Jacobi identity: {{f, g}, h}+ {{g, h}, f}+ {{h, f}, g} = 0 . The conditions 1-3 identify {·, ·} as a bivector: a rank two, antisymmetric, con- travariant tensor field ω on M . It can therefore be represented, by introducing coordinates {ui} on M , as a matrix of coefficients ωij , giving ω = ωij {f, g} = ωij . (2) The Jacobi identity places the following constraint on the components of ω: + ωjr + ωkr = 0 . (3) If the matrix ωij is non-degenerate, we may introduce its inverse ωij , satisfying rj = δ i . The Jacobi identity for ω ij is equivalent to the closedness of ωij . Date: April 11, 2007. http://arxiv.org/abs/0704.1401v1 2 JAMES T. FERGUSON We refer to a closed non-degenerate two-form as a symplectic form, and a mani- fold equipped with one as a symplectic manifold. Darboux’s theorem asserts that on any 2n-dimensional symplectic manifold there exists a set of local coordinates {q1, . . . , qn, p1 . . . , pn} in which the Poisson bracket takes the form (1); i.e. the components of ωij , and so those of ωij , are constant. One may also introduce Poisson brackets on infinite-dimensional manifolds. The loop space of a finite-dimensional manifold M , L(M), is the space of smooth maps u : S1 →M . Poisson brackets relating Hamiltonians to flows in L(M) will therefore act on functionals mapping L(M) → R. In [5],[6] Dubrovin and Novikov studied the so-called Poisson brackets of differential-geometric type, which are of the form {f, g} = dx (4) where ui are coordinates on the target space M , and x is the coordinate on S1. P ij is a matrix of differential operators (in d ), with no explicit dependence on x, which is assumed to be polynomial in the derivatives uix, u xx, . . . . If {·, ·} defines a Poisson bracket on the loop space then P is referred to as a Hamiltonian operator. There is a grading on such operators, preserved by diffeomorphisms of M , given by assigning degree 1 to d , and degree n to the nth x-derivative of each field ui. An important class is the hydrodynamic type Poisson brackets, which are homogeneous of degree 1: P ij = gij(u) (u)ukx . According to the programme set out by Novikov [15], differential-geometric type Poisson brackets on L(M) should be studied in terms of finite-dimensional differ- ential geometry on the target space M . When expanded as a polynomial in d the field derivatives, the coefficients, which are functions of the fields ui alone, can often be naturally related to known objects of differential geometry, or else used to define new ones. In the hydrodynamic case, for instance, with gij non-degenerate, P is Hamiltonian if and only if gij is a flat metric on M and Γkij = −girΓ j are the Christoffel symbols of its Levi-Civita connection. In [7] Dubrovin considered the geometry of bi-Hamiltonian structures of Hy- drodynamic operators, that is pairs of such operators compatible in the sense of [13], that every linear combination of them also determines a Poisson bracket. In particular, he introduced a multiplication of covectors on M and expressed the compatibility of the operators in terms of a quadratic relations on this algebra. This paper is principally concerned with Hamiltonian operators which are ho- mogeneous of degree 2. Section 2 presents the differential geometry of such op- erators, and in particular relates the subclass which can be put into a constant form by a change of coordinates on M to symplectic connections. Section 3 then considers pairs of operators from this subclass, and the algebraic constraints their compatibility places upon the associated multiplication. In section 4 inhomoge- neous bi-Hamiltonian structures consisting of a degree 1 and a degree 2 operator are studied. 2. Hamiltonian Operators of Degree 2 We begin with a review of known results on Hamiltonian operators of degree 2: ij = aij x + c xx, (5) in which the matrix aij is assumed to be non-degenerate. Such operators have been considered already in, for example, [17], [14], [4], [15], in which the (conditional) FLAT PENCILS OF SYMPLECTIC CONNECTIONS 3 Darboux theorem has been discussed. In preparation for the bi-Hamiltonian theory we present these results without the use of special coordinates. Under the change of coordinates ũi = ũi(up) the coefficients in P ij transform as bpqr − 2 ∂ũk∂ũs apq , cpqr − ∂ũk∂ũs apq , cpqrs + ∂ũk∂ũl ∂2ũj ∂uq∂us ∂ũ(k ∂ũl) bpqr + ∂3ũj ∂uq∂ur∂us ∂2ũj ∂uq∂ur ∂ũk∂ũl , (6) where the brackets denote symmetrisation. So in particular aij transforms as a rank 2 contravariant tensor on the target space and b and c are related to Christoffel symbols of connections by b = −2airΓ̄ and c = −airΓ . Call these connections ∇̄ and ∇ respectively. The transformation rules for c are not determined uniquely by those for P , since (5) sees only the part symmetric in k and l. To fix c , we always assume the antisymmetric part is zero. Denote by aij the inverse of a ij defined by aira rj = δ The condition that the operation defined in (4) is skew-symmetric and satisfies the Jacobi identity places constraints on the coefficients appearing in (5). Theorem 2.1. The operator P in equation (5) defines a Poisson bracket by equa- tion (4) if and only if (A) aij = −aji , (B) ∇ka ij = b (C) air bjkr − 2c = akr bijr − 2c (D) ∇ is flat (zero torsion, zero curvature) , (E) c (k,l) − aprc Proof. [14] states that, by virtue of being Hamiltonian, the operator (5) can be put in the form P ij = aij , (7) by a change of coordinates ui = ui(ũ), and that for an operator of this shorter form to be Hamiltonian is equivalent to the three conditions (a) aij = −aji , (b) aij ,k = b (c) airbjkr = a jrbkir . We first assume that P is a Poisson bracket, so there exists the special coordinates in which P takes the form (7) and (a)-(c) hold. By reversing the change of variables as ũi = ũi(u), conditions (A)-(C) of Theorem 2.1 are Mokhov’s three conditions converted to tensorial identities. That ∇ is flat follows from its Christoffel symbols, Γkij = −airc j , being zero in the u coordinates. 4 JAMES T. FERGUSON The formula in condition (E) is derived from the transformation rules above. In changing from flat coordinates ui to coordinates ũi they give: ∂2ũj ∂uq∂us ∂ũ(k ∂ũl) bpqr + ∂3ũj ∂uq∂ur∂us ∂2ũj ∂uq∂ur ∂ũk∂ũl apq , ∂ũk∂ũs apq , ∂2ũj ∂uq∂ur apq , where the last line has used the identity ∂2ũi ∂ur∂us ∂ũj∂ũk = 0 , which is a differential consequence of ∂ũ = δij . ∂2ũi ∂up∂us ∂2ũj ∂ur∂uq ∂3ũj ∂uq∂ur∂us ∂2ũj ∂uq∂ur ∂ũk∂ũl ∂2ũj ∂uq∂ur from which we see (k,l) ∂2ũi ∂up∂us ∂2ũj ∂ur∂uq ∂ũ(l ∂ũk) apq . This last term can be seen to be ãprc̃ (k c̃ Conversely, if (A)-(E) hold, the flatness of ∇ asserts the existence of coordinates in which c = 0, and condition (E) then asserts that c = 0 in these coordinates. If we take, as a simple case, an operator P as in (5) with b constants, and assume c to be defined by (E), then P is Hamiltonian if and only if aij = uk + A 0 where A 0 are constants with A , Airl c r = A cikr , Air0 c r = A r and c rk + cikr c If we take an algebraA with basis {e1, . . . , en}, n = dimM, and use c and A define a multiplication, ◦ , and skew-symmetric bilinear form, 〈·, ·〉, by ei◦ej = cijr e and 〈ei, ej〉 = A 0 , then we may rewrite these conditions as ei ◦ ej − ej ◦ ei = Aijr e (I ◦ J) ◦K = −(I ◦K) ◦ J , (8) Λ(I, J,K) = Λ(J, I,K) , (9) and 〈I, J ◦K〉 = 〈J, I ◦K〉 , for all I, J,K ∈ A, where Λ is the associator of ◦ : Λ(I, J,K) = (I◦J)◦K−I◦(J◦K). FLAT PENCILS OF SYMPLECTIC CONNECTIONS 5 Algebras satisfying conditions (8) and (9) have appeared before in [18], in the context of linear hydrodynamic Hamiltonian operators taking values in a completely odd superspace, where the following definition was proposed: Definition 2.2. An algebra (A, ◦) satisfying conditions (8) and (9) is called a Fermionic Novikov algebra. In [1] Fermionic Novikov algebras in dimensions 2-5 were studied, and the listing therein provides a source of examples of Hamiltonian operators of degree two. Example 2.3. 0 0 0 a 0 0 −a −b− (t− 1)u1 0 a 0 c− u2 −a b+ (t− 1)u1 −c+ u2 0 0 0 0 0 0 0 0 u1x 0 0 −u1x 0 0 τu1x u 0 0 0 0 0 0 0 0 0 0 0 (u1x) 0 0 −(u1x) 0 0 0 0 0 0 0 u1x 0 0 −u1xx 0 0 τu1xx u is Hamiltonian for all values of the constants a, b, c and τ with a 6= 0. This is the most general Hamiltonian operator associated in the manner discussed above to the algebra designated (44)τ in [1]. Returning to the general Hamiltonian operator (5), it can be seen from conditions (B) and (E) in Theorem 2.1 that the coefficients b and c in (5) are completely determined by aij and c . Thus the Hamiltonian operator on L(M) is represented uniquely on M by only these latter two objects. Theorem 2.4. There is a one-to-one correspondence between Hamiltonian opera- tors of the form (5) on L(M) and pairs (a,∇) on M consisting of a non-degenerate bivector aij and a torsion-free connection ∇ satisfying two conditions: firstly, that the curvature of ∇ vanishes, and secondly, air∇ra jk = ajr∇ra ki . (10) The Christoffel symbols, Γkij , of ∇ are related to c = −airΓ . We then = ∇ka ij + 2c − aprc With this, we may verify the following facts [17],[14]: Corollary 2.5. For P in (5) a Hamiltonian operator we have 1. Γ is the symmetric part of Γ̄, 2. Let T̄ kij = Γ̄ ij − Γ̄ ji be the torsion of ∇̄. Then T̄ijk = airT̄ jk is skew symmetric and the forms T̄ = 1 T̄ijkdu i ∧ duj ∧ duk and a = 1 aijdu i ∧ duj are related by 3T̄ = da. Proof. We begin by noting that equation (10) is equivalent to the condition ∇kaij = ∇iajk (11) on the two-form aij . 6 JAMES T. FERGUSON In terms of covariant Christoffel symbols, Theorem 2.4 gives Γ̄kij = akr∇raij + Γ ij , (12) from which it is clear that Γ̄k = Γkij . We therefore also have ∇kaij = Γ̄ijk − Γijk , where Γ̄ijk = airΓ̄ jk and Γijk = airΓ jk. Because ∇ is torsion-free we have T̄ijk = Γ̄ijk − Γ̄ikj , = Γ̄ijk − Γijk − Γ̄ikj + Γikj , ∇kaij − ∇jaik , = ∇kaij , = ∇[kaij] , (da)ijk . Lemma 2.6. For a Hamiltonian operator of the form (5), the following three state- ments, presented in both covariant and contravariant forms, are equivalent: 1. The 2-form a is closed (and so symplectic), or equivalently aij satisfies equation (3) (and so defines a Poisson bracket on M by equation (2)); 2. ∇ka ij = 0, i.e. ∇kaij = 0; , i.e. Γkij = Γ̄ Proof. We see, from the characterisation of Hamiltonian operators given in Theorem aij is Poisson ⇐⇒ airajk,r + a jraki,r + a kraij,r = 0 ⇐⇒ air∇ra jk + ajr∇ra ki + akr∇ra ij = 0 ⇐⇒ 3akr∇ra ij = 0 ⇐⇒ ∇ka ij = 0 , Lemma 2.6 therefore tells us that in the special case where the leading coefficient in P is the inverse of a symplectic form, the pair (a,∇) defining P can be thought of as containing the symplectic form aij , and a torsionless connection compatible with it (in the sense that ∇a = 0); that is, a symplectic connection. More precisely (see e.g. [3]): Definition 2.7. A symplectic connection on a symplectic manifold (M,ω) is a smooth connection ∇ which is torsion-free and compatible with the symplectic form ω, i.e. ∇XY −∇YX − [X,Y ] = 0 (∇ω) (X,Y, Z) = X(ω(Y, Z))− ω(∇XY, Z)− ω(Y,∇Y Z) = 0 , where X,Y and Z are vector fields on M . FLAT PENCILS OF SYMPLECTIC CONNECTIONS 7 In local coordinates {xi}, introducing Christoffel symbols Γkij for ∇ and writing ω = 1 ωijdx i ∧ dxj , the conditions for ∇ to be a symplectic connection read Γkij = Γkji, as usual, and ∇kωij = − Γrkiωrj − Γ kjωir = 0 . (13) This definition is analogous to that of the Levi-Civita connection of a pseudo- Riemannian metric, however there is an important difference in that the Levi-Civita connection is uniquely specified by its metric. From the compatibility condition (13) it can be seen that if Γkij are the Christoffel symbols of a symplectic connection for ω, then the connection with Christoffel symbols Γ̃kij = Γ ij + ω krSrij is a sym- plectic connection if and only if the tensor Sijk is completely symmetric. In [10] a symplectic manifold with a specified symplectic connection is called, in light of [9], a Fedosov manifold. Here we call the pair (ω,∇) of a symplectic form and a symplectic connection a Fedosov structure on M, and call the structure flat if ∇ is flat. In the discussion of Hamiltonian operators it is convenient to work with con- travariant quantities. We call = −ωirΓ the contravariant Christoffel symbols of the symplectic connection. Result 2.8. The compatibility of ∇ and ω is equivalent to Result 2.9. ∇ being torsion-free is equivalent to ωirΓjkr = ω jrΓikr . The curvature of ∇, slt = ∂sΓ lt − ∂lΓ st + Γ lt − Γ can be expressed in terms of contravariant quantities by raising indices as = ωisωjtRkslt . This gives Result 2.10. = ωir r − ∂rΓ + Γijr Γ l + Γ Having introduced symplectic connections, we are now in a position to interpret the following Darboux theorem for Hamiltonian operators of degree 2: Theorem 2.11. [17] Given a Hamiltonian operator P ij = aij x + c where aij is non-degenerate, then P can be put in the constant form P ij = ωij (where ω is a constant matrix) by a change of target space coordinates {ui} if and only if aij is closed. The coordinates in which this happens are flat coordinates for the connection Γkij = −girc j which can be chosen, using a linear substitution, to be canonical coordinates for the symplectic form aij = ωij . In arbitrary coordinates operators satisfying the conditions of Theorem 2.11 have the form P ij = ωij x + Γ ukxx (14) 8 JAMES T. FERGUSON where ωij is the inverse of a symplectic form, c (k,l) − ωprΓ , and Γ the contravariant Christoffel symbols of a flat symplectic connection compatible with ω. This class of operators on L(M) is therefore in one-to-one correspondence with flat Fedosov structures on M . 3. Flat Pencils of Fedosov Structures In this section we consider pairs of Hamiltonian operators of the form (14): 1 = ω + 2Γ1 x + Γ1 ukxx , 2 = ω + 2Γ2 x + Γ2 The first fact to establish is that if P1 and P2 are compatible then all elements of the pencil, Pλ = P1 + λP2, remain in the class (14). Theorem 3.1. If P1 and P2 are compatible then ω 1 and ω 2 form a finite-dimensional bi-Hamiltonian structure on the target space. Proof. Pλ could have the general form x + cλ ukxx , but clearly bλ = 2Γ1 + 2λΓ2 and cλ + λΓ2 , so bλ = 2cλ , and hence, by Lemma 2.6, a satisfies the Jacobi identity (3) for all λ. � So we write + 2Γλ x + Γλ ukxx . An immediate corollary of Theorem 3.1 is that the tensor Lij = ω 1 ω2rj has vanishing Nijenhuis torsion. 3.1. Multiplication of covectors. As in [7], we proceed to understand the com- patibility conditions on P1 and P2 in terms of the algebraic properties of a tensorial multiplication of covectors on M . Definition 3.2. Using the tensors ∆sjk = ω r − ω i = ω2is∆ sjk , we define a multiplication ◦ of covectors on M by (α ◦ β)i = αjβk∆ Theorem 3.3. The compatibility of P1 and P2 is equivalent to (I, J ◦K)2 = (J, I ◦K)2 , (15) and (I ◦ J) ◦K = 0 , (16) for all covectors I, J,K on M . Here (·, ·)2 is the skew-symmetric bilinear form on T ∗M induced by ω 2 , i.e. (I, J)2 = IrJsω 2 . The compatibility also implies = ∇2k∆ . (17) Because of Theorem 3.1, we phrase the compatibility of P1 and P2 in terms of Fedosov structures on M , and break the above theorem into stages: FLAT PENCILS OF SYMPLECTIC CONNECTIONS 9 Definition 3.4. Two flat Fedosov structures (ω1,∇ 1) and (ω2,∇ 2), where ∇1 and ∇2 have contravariant Christoffel symbols Γ1 and Γ2 respectively, are said to be (i) almost compatible if and only if (ωλ,∇ λ) is a Fedosov structure for all λ, where the connection ∇λ is given by Γλ + λΓ2 (ii) almost compatible and flat if and only if they are almost compatible, and in addition the curvature of ∇λ vanishes for all λ . (iii) compatible if and only if they are almost compatible and flat, and cλ (k,l) − ωλprΓλ satisfies cλ + λc2 for all λ. The compatibility of two flat Fedosov structures on M is equivalent to the com- patibility of the associated Poisson brackets on L(M). We now turn to the two Fedosov strucutres defined by P1 and P2, and to the pair (ωλ,∇ λ) defined by Pλ. From the linearity of Result 2.8 in the contravari- ant symbols it can be seen that ωλ is automatically ∇ λ-constant, so the almost compatibility of (ω1,∇ 1) and (ω2,∇ 2) is equivalent to ∇λ being torsion free, i.e. to In flat coordinates for ∇2, this condition reduces to r = ω r . (18) Note that we already have ωir1 Γ1 r = ω r . (19) Lemma 3.5. If (ω1,∇ 1) and (ω2,∇ 2) are almost compatible, then the flatness of ∇λ is equivalent to either, and hence both, of s − ∂sΓ1 = 0 (20) and Γ1 l + Γ1 = 0 (21) in the flat coordinates for ∇2. Proof. The contravariant curvature of Γλ is = ωirλ r − ∂sΓλ l + Γλ s − ∂sΓ1 + ωis1 s − ∂sΓ2 l + Γ1 l + Γ1 +λ2R2 which in flat coordinates for Γ2 reads = ωir1 r − ∂rΓ1 l + Γ1 +λωis2 s − ∂sΓ1 The vanishing of the order λ term is equivalent to equation (20), and with this the vanishing of the λ-independent term is equivalent to (21). � Lemma 3.6. If (ω1,∇ 1) and (ω2,∇ 2) are almost compatible then the condition (k,l) − ωλprΓλ reads, in the flat coordinates for ∇2, l − Γ1 = 0 . (22) 10 JAMES T. FERGUSON Proof. For an arbitrary Fedosov structure (ω,∇) the object c (k,l) −ωprΓ can be converted into a quadratic expression in contravariant quantities as = ωskΓ (k,l) Γsip Γ Γsjp . (23) This has similarities to the formula for covariant curvature obtained in Result 2.10; only certain signs have changed. Indeed, if we define a quantity c dxr = (∇∂k∇∂l +∇∂l∇∂k) dx then c = ωirc We have two ways of expanding ωskλ cλ , corresponding to whether we choose first to substitute it into equation (23), or to expand the pencil quantities. We work in flat coordinates for ∇2; in these, c2 also vanishes. First expanding the pencil we have 1 + λω = ωsk1 c1 + λωsk2 c1 whilst (23) gives ωskλ cλ = ωskλ Γλ (k,l) ωsk1 + λω (k,l) The order 1 terms merely express equation (23) for P1. Equality of the order λ terms is equivalent to Γ1 (k,l) and so to ωsk1 Γ1 (k,l) = ωsk1 c1 = ωsk1 Γ1 (k,l) Proof of Theorem 3.3. Using equation (18) in Definition 3.2 it can be seen that in the flat coordinates for ∇2 we have ∆ . Thus we may regard equations (18),(20),(21) and (22) as identities on ∆ ; the result is Theorem 3.3. � The condition imposed by equation (21) for an almost compatible and flat pair of Fedosov structures on the mutliplication ◦ is (I ◦ J) ◦ K = −(I ◦ K) ◦ J , i.e. the first condition (8) satisfied by the multiplication of a Fermionic Novikov alge- bra. In general (9) is not satisfied even for compatible Fedosov structures, however we do have, for two flat Fedosov structures, (ω1,∇ 1), (ω2,∇ 2), which are almost compatible, = ∆ijr ∆ ∆ikr . So, in particular, if ∆ is constant in the flat coordinates for ∇2, almost compatible and flat Fedosov structures will define a Fermionic Novikov algebra structure on the covectors of M . In [1] it emerged that examples of such algebras which do not also satisfy the ‘Bosonic’ relation (I ◦J) ◦K = (I ◦K) ◦J , and hence (I ◦J) ◦K = 0, are relatively rare. ∇2-constant multiplications arising from pairs of Fedosov structures which are almost compatible and flat, but not compatible, such as that given in Example 3.10 below, are in this class. FLAT PENCILS OF SYMPLECTIC CONNECTIONS 11 3.2. The pencil in flat coordinates. We now turn our consideration to the form the pencil takes in the flat coordinates for ∇2. From the elements of the proof of Theorem 3.3 we have 1 + λω + 2Γ1 x + Γ1 ukxx . (24) The Jacobi identity for Pλ (without assuming P1 and P2 are Hamiltonian them- selves) is equivalent to the constraints (i) ω 2 is constant and antisymmetric, (ii) ω 1 is antisymmetric, (iii) ωir1 Γ1 r = ω (iv) ω (v) ωir2 Γ1 r = ω (vi) Γ1 (vii) Γ1 l = 0. Proposition 3.7. In a fixed coordinate system {ui} (the flat coordinates for Γ2), given a constant non-degenerate 2-form ω 2 and a vector field B = B r∂r satisfying ωis2 B ,s − ω ,pr = ,s − ω ,pr (25) ,sl = 0 (26) then the prescription 1 = −(LBω2) ij = ωir2 B ,r − ω = ωir2 B satisfies the constraints (i)-(vii). Further, all solutions of (i)-(vii) have this form. Proof. Equations (25) and (26) are the quadratic constraints, ωir1 Γ1 r = ω and Γ1 l = 0 respectively. That ω1 and Γ1 satisfy the (linear) constraints (iv), (v) and (vi) is an immediate consequence of their definition. Using the Poincare lemma together with the symmetries expressed in conditions (vi) and (v), we have the existence of a vector field satisfying Γ1 = ωir2 A j ,rk . With this condition (iv) gives ω 1 = −(LAω2) ij+cij , where cij is a constant antisymmetric matrix. We may now introduce a vector field B with Bi = Ai + 1 xsw2src ri which satisfies ω 1 = −LBω 2 and Γ1 = ωir2 B Since ω2 is a symplectic form, its symmetries are precisely (locally) Hamiltonian vector fields. Therefore, if ω2 and ω1 are given, the requirement that ω 1 = −LBω fixes the non-Hamiltonian part of B. Then the condition Γ1 = ωir2 B fixes the Hamiltonian to within a quadratic function. From the point of view of the multipli- cation of covectors from Section 3.1, the Hamiltonian affects only the commutative part of ◦, thus the anti-commutative part is fixed by ω 1 and ω With consideration of the transformation rules (6), one can phrase Proposition 3.7 as the existence of a vector field B such that 1 = −LBω = −LBΓ2 . (27) 12 JAMES T. FERGUSON We can also calculate from (6) the correct interpretation of the Lie derivative for an object of type c , namely: = Xrc −X i,rc −Xj,rc kl +X +Xr,lc +Xr,klc birk − birl −X air . If we work in the flat coordinates for Γ2, so that the components c2 = 0, we have for our pencil −LBc2 = +ωir2 B = (ωir2 B ),l , Now, in the flat coordinates for ∇2 we have the relation c1 . The linearity of the transformation rules shows that the Lie derivative of c2 should be an object of the same type as c1 . Thus we have, in addition to (27), = −LBc2 One may understand these three infinitesimal relations between the coefficients of P1 and P2 as averring the existence on L(M) of an evolutionary vector field B̂ = Bi(u(x)) ∂ui(x) + . . . such that 1 = −LB̂P We now turn our attention to some examples of pairs of Fedosov structures, using the framework of Proposition 3.7. Example 3.8. Two-dimensional pencils. Without loss of generality we take where u1 and u2 are a flat coordinate system for ∇2. We take B = f(u1, u2) + g(u1, u2) and from it calculate ω1 and Γ1 according to (27). In particular ω1 = (f,1 + g,2)ω2 , from which it follows immediately that (ω1,∇ 1) and (ω2,∇ 2) are almost compatible. They are almost compatible and flat if and only if h = f + λg satisfies the homogeneous Monge-Ampere Equation h212 − h11h22 = 0 for all λ. They are compatible if and only if a = f + λg and b = f + µg satisfy a12b12 − a11b22 = 0 for all λ, µ. For instance, one may recover the three two-dimensional Fermionic Novikov al- gebras of [1] as constant multiplications via (T1) f = u1, g = 0 , (T2) f = u1, g = (u1)2 , (T3) f = (u1)2, g = 0 . FLAT PENCILS OF SYMPLECTIC CONNECTIONS 13 Example 3.9. Commutative algebras. In the case in which ω1 is constant in the flat coordinates for ∇2, we have, by condition (iv), so that the multiplication ◦ is commutative. In particular if ω1 = ω2 = ω = then the non-Hamiltonian part of B is To this we may add a Hamiltonian vector field, giving Since ω1 = ω2, equation (25) is immediate. Equation (26) becomes H,ijr ω H,skl = 0 , where the indices i, j, k, l, r, s account for both q and p variables. A solution to this is H = f(x1, x2, . . . , xn), where each xi is either pi or q i; only one from each pair of conjugate variables features in H. It is not hard to see that Proposition 3.7 can be modified to describe almost compatible and flat pairs of Fedosov structures. Specifically, we replace equation (26) by the expression corresponding to Γ1 l = Γ1 , namely: ,sl = B ωrs2 B ,si . (28) Example 3.10. The Fedosov structures specified by = 0 , + 2q1q2 + q1p2 and ω 1 = −LBω 2 and Γ1 = −LBΓ2 are almost compatible and flat, but not compatible. The non-zero components of ω1 and ◦ are {q1, p1}1 = {q2, p2}1 = 3q1 , {q2, p1}1 = 2q2 , {p2, p1}1 = p2 , dq2 ◦ dp2 = dq1 , dp1 ◦ dq1 = −3dq1 , dp1 ◦ dq2 = −2dq2 , dp1 ◦ dp2 = −dp2 , dp2 ◦ dq2 = −2dq1 . 14 JAMES T. FERGUSON Thus, the products (dp1 ◦ dq2) ◦ dp2 = −2dq1 and (dp1 ◦ dp2) ◦ dq2 = 2dq1 violate equation (16) but not (8). Note that ◦ also satisfies (9) and thus defines a Fermionic Novikov algebra which is not ‘Bosonic’. 3.3. ωN manifold with Potential. The tangent bundle T ∗Q of a manifold Q is naturally equipped with a symplectic form, and thus cotangent bundles form the basic set of examples of symplectic manifolds. One may hope to find examples of finite-dimensional bi-Hamiltonian structures on cotangent bundles by exploiting the existence of additional structures on the underlying manifolds. The main object used to do this is a (1, 1)-tensor Lij on Q whose Nijenhuis torsion is zero. Such an object was utilised by Benenti [2] to demonstrate the separability of the geodesic equations on a class of Riemannian manifolds. This result was later interpreted in [12] in terms of a bi-Hamiltonian structure on T ∗Q which was extended to a degenerate Poisson pencil on T ∗Q× R. To obtain Fedosov structures we require more than just a tensor Lij on Q with vanishing Nijenhuis torsion; we also need a means of specifying the connections. If Q is equipped with a torsion-free connection ∇̃, then the Nijenhuis torsion of a (1, 1)-tensor Lij can be written as jk = L j∇̃sL k − L k∇̃sL j − L s∇̃jL k + L s∇̃kL If there exists a vector field, A, on Q such that Lij = ∇̃jA i then N ijk = (∇̃jA s)(∇̃s∇̃kA i)− (∇̃kA s)(∇̃s∇̃jA i)− (∇̃sA i)(RsjkrA where Rijkl is the curvature tensor of ∇̃. So, if ∇̃ is flat then the vanishing of the Nijenhuis tensor of L = ∇̃A is equivalent to the identity (∇̃jA s)(∇̃s∇̃kA i) = (∇̃kA s)(∇̃s∇̃jA i) . (29) Proposition 3.11. Given a manifold Q endowed with a flat connection ∇̃ and a vector field A satisfying (29), the cotangent bundle T ∗Q is endowed with a compati- ble pair of Fedosov structures, (ω1,∇ 1) and (ω2,∇ 2), as follows: ω2 is the canonical Poisson bracket on T ∗Q. The connection ∇2 on T ∗Q is the horizontal lift [19] of the connection ∇̃ on Q; i.e. the Christoffel symbols Γ2 ij of ∇ 2 are zero in the coordinates induced on T ∗Q by the flat coordinates for ∇̃. (ω1,∇ 1) is calculated from (ω2,∇ 2) according to the prescription of Proposition 3.7, where the vector field B is the horizontal lift of A to T ∗Q. Proof. Let {q1, . . . , qn} be flat coordinates for ∇̃ onQ, and C = {q1, . . . , qn, p1, . . . , pn} be the induced coordinates on T ∗Q. Then The space of sections of the cotangent bundle of T ∗Q, Ω, naturally splits into P = span{dpi} and Q = span{dq i}. For Γ1 = ωir2 B to be non-zero requires k to represent a variable qk, and i to represent a pi variable. Thus Ω ◦ Ω ⊆ Q and FLAT PENCILS OF SYMPLECTIC CONNECTIONS 15 Q ◦ Ω = {0}, meaning that (Ω ◦ Ω) ◦ Ω = {0}. So the relation (26), Γ1 l = 0, is satisfied. 1 has only one kind of non-zero component, ω ,i, so the expression ωir1 Γ1 r has only one non-zero case: qr = A ,rj , which is seen to be symmetric in i and j by condition (29), which in the flat coordinates qi reads As,j A i,sk = A s,k A i,sj . Example 3.12. If the eigenvalues of L : TQ → TQ are functionally independent in some neighbourhood then they may be used as coordinates, and L takes the form ⊗ dui . In this case we may set A = (ui)2 ∂ , and have ∇̃ defined by vanishing Christoffel symbols in these coordinates. This gives, writing vi as the conjugate coordinate to u i on T ∗Q, = −1 , and all other Christoffel symbols zero. 4. Bi-Hamiltonian Structures in Degrees 1 and 2 We now consider a pair of operators, P1 and P2 in which P1 is a Hamiltonian operator of hydrodynamic type and P2 is of second order, i.e. : 1 = g ij(u) (u)ukx , 2 = a x + c ukxx , where gij is the inverse of a flat metric gij on M and Γ = −girΓ where the Γkij are the Christoffel symbols of the Levi-Civita connection of g. We also assume that 2 is antisymmetric, so that a ij = −aji, b and c (k,l) The motivation [8] for studying such pairs of operators comes not from regarding them as separate Hamiltonian operators, but from thinking of P 2 as a first order (dispersive) deformation of P 1 into some non-homogeneous Hamiltonian operator P ij = P 1 +εP 2 +O(ε 2). Thus, in such a pair, it is sensible to regard the geometry 1 as being more intrinsic than any associated to P We choose to work in flat coordinates for g so that gij is constant and Γ Direct calculation of the Jacobi identity for P ij in these coordinates yields 16 JAMES T. FERGUSON Theorem 4.1. P2 is an infinitesimal deformation of P1, i.e. P ij = P 1 + εP O(ε2) satisfies the Jacobi identity to order ε, if and only if (I) gircjkr + g jrcikr = 0 , (II) c (k,l) (III) girc = gjr(cikl,r − c r,l) , (IV) gir(ajk,r − c r ) + g jr(aki,r − c r ) + g kr(aij,r − c r ) = 0 in the flat coordinates for gij. By introducing the tensor T = airΓ is it easy to convert conditions (I), (III) and (IV) to arbitrary coordinates, whilst condition (II) becomes − crik Γ − cril Γ + T ijr Γ kl + T Γirl + T Γirk . To consider a bi-Hamiltonian structure involving operators P 1 and P 2 one need only add conditions (C), (D) and (E) of Theorem 2.1 to Theorem 4.1, however, condition (II) above allows (E) to be replaced by cijr c l = c Example 4.2. As discussed in section 2, P2 with b constant and aij non- degenerate is Hamiltonian if and only if aij = A uk + A 0 with A 0 is constant, c are the structure constants of a Fermionic Novikov algebra (A, ◦), and A 0 defines a skew-symmetric bilinear form on A satisfying 〈I, J ◦K〉 = 〈J, I ◦K〉. If we ask that P2 satisfies the above constancy conditions in the flat coordinates for gij, then, defining an inner product on A by (ei, ej) = gij, we have that the compatibility of P1 and P2 is equivalent to the additional constraints: (I ◦ J) ◦K = (I ◦K) ◦ J , (I, J ◦K) = −(J, I ◦K) (I, [J,K]) + (J, [K, I]) + (K, [I, J ]) = 0 , where [I, J ] = I ◦J−J ◦I is the commutator of ◦, which is a Lie bracket by equation For example, if we take the algebra (A = span{e1, e2, e3, e4}, ◦) where the only non-zero products are e3 ◦ e3 = e1 and e4 ◦ e3 = e2 then we may take as our symplectic form and metric [ωij ] = 0 0 a b 0 0 b c −a −b 0 d− u2 −b −c −d+ u2 0 [gij ] = 0 0 0 e 0 0 −e 0 0 −e f g e 0 g h  , for any choice of the constants a, b, c, d, e, f, g, h such that e 6= 0 and b2 6= ac. This algebra, essentially (57)−1, is the only algebra in [1] of dimension 2 or 4 which admits non-degenerate forms (·, ·) and 〈·, ·〉 satisfying the above compatibility conditions with ◦, other than the trivial case in which all products are zero, i.e. in which the Hamiltonian operators share the same flat connection, and so are simultaneously constant. FLAT PENCILS OF SYMPLECTIC CONNECTIONS 17 Proposition 4.3. If P2 is an infinitesimal deformation of P1 then there exists a tensor field Aij such that aij = girAjr − g jrAir , = 2gisA − gjrAik,r − g = gisA − gisA (k,l)s = gisA − gisA in flat coordinates for gij. Further, any (1,1)-tensor field Aij produces an infinites- imal deformation of P1 by the above formulae. Proof. Using the non-degeneracy of gij , we introduce objects θkij and φij by = girθ aij = girgjsφrs . Then condition (I) of Theorem 4.1 is equivalent to θkij = −θ ji, and so we regard θ as a family of 2-forms θk indexed by k. Condition (III) is equivalent to θkjl,i = θ il,j − θ ij,l, so that dθ k = 0 for each k. This allows us to introduce a family of 1-forms ψk such that ij = (dψ k)ij = ψ i,j − ψ j,i . Each ψk can be adjusted by the addition of the exterior derivative, dfk, of some function fk without affecting the value of θkij . Writing αij = φij − gjrψ i + gjrψ k, we find that condition (IV) is equivalent to the closedness of the 2-form αij , upon substituting φij and ψ j for a ij and c . Thus we may introduce a 1-form h with components hi such that αij = hi,j − hj,i, and φij = gjrψ i − gjrψ j + hi,j − hj,i . If we now let Aij = ψ j + (g irhr),j then we have θ ij = A i,j − A j,i and φij = i − girψ j , so that the two equations a ij = girAjr − g jrAir and c = girA are satisfied. The remaining to equations follow easily from c For the converse, it is easy to check that conditions (I)-(IV) of Theorem 4.1 follow from (30) for any tensor field Aij . � As with Proposition 3.7, Proposition 4.3 may be understood as asserting the existence of an evolutionary vector field e = Aij (u(x)) u ∂ui(x) + . . . satisfying P2 = −LeP1 whenever P2 is an infinitesimal deformation of P1. This is therefore not a surprising result; in [11] Getzler showed the triviality of infinitesimal deformations of Hydrodynamic type Poisson brackets. With this, Proposition 4.3 can be looked upon as a proof of Theorem 4.1. There is a freedom in Aij of A j 7→ A j + g irf,rj for some function f , which does not affect the coefficients of P2. This corresponds to adjusting e by a Hamiltonian vector field, e 7→ e+ P1(δf). If, with reference to Lemma 2.6, we impose the additional constraint on (30) that b then we have the potentiality condition gjrA k,i = girA k,j , so that there exists a 1-form Bk such that j = g Bj,r . (31) 18 JAMES T. FERGUSON In this case aij = girgjr(Br,s − Bs,r) = g irgjr(dB)rs and the freedom A j 7→ A girf,rj is B 7→ B + df . This means that B can be determined purely from g ij and aij , and thus there is no freedom in the choice of c and c . In fact we may write explicitly = gjsgkr (k,l) , (32) and with this, P2 is an infinitesimal deformation of P1 if and only if girajk,r + g jraki,r + g kraij,r = 0 . (33) Corollary 4.4. Given a flat metric g and a symplectic form ω, there is at most one choice of flat symplectic connection ∇ such that the degree 2 Hamiltonian operator specified by (ω,∇) is compatible with the hydrodynamic operator specified by g. Clearly, if this connection exists it is given by(32), so this definition must be checked against Theorem 2.1 to verify 2 = ω x + c is Hamiltonian. Since equation (33) is a consequence of the antisymmetry of P2, compatibility with the Hydrodynamic operator follows immediately. We conclude this section with an example of this type. Example 4.5. The Kaup-Broer system [16], u1xx + 2u x + 2u −u2xx + 2(u 1u2)x is described by the pair of compatible Hamiltonian operators u1 2u2 0 u1x 0 u2x Scaling x 7→ εx, t 7→ εt splits P2 into P 2 + εP 2 where u1 2u2 0 u1x 0 u2x Since P2 = P 2 + εP 2 is Hamiltonian for all ε, P 2 and P 2 constitute a bi- Hamiltonian structure of the type considered above. A set of flat coordinates for the metric in P ũ1 = u1 , ũ2 = 4u2 − (u1)2 , in which (ũ2)2 0 −ũ2x 0 ũ1x (ũ2)3 0 (ũ2x) 0 −ũ1xũ (ũ2)2 0 −ũ2xx 0 ũ1xx FLAT PENCILS OF SYMPLECTIC CONNECTIONS 19 So in this situation we have, for the 1-form in (31), dũ2 . 5. Conclusions In section 3 an approach was taken based upon the methods of [7] to study com- patible pairs of Hamiltonian operators of degree 2 which satisfy the conditions of the relevant Darboux theorem, Theorem 2.11. As for Hydrodynamic Poisson pen- cils, the compatibility could be reduced to algebraic constraints on a multiplication of covectors. Driving this was the ability to reduce a given Hamiltonian operator on L(M) to a flat Fedosov structure (ω,∇) on M , which are natural symplectic analogues of the pair consisting of a flat metric and its Levi-Civita connection which determines a Hydrodynamic Poisson bracket. To extend such a results to pairs of arbitrary degree 2 Hamiltonian operators, one must consider the pair (a,∇) of Theorem 2.4. The condition (10), whilst atypical, expresses a familiar concept; in almost-symplectic geometry, it is common to consider connections such that the covariant derivative of the almost-symplectic form is zero, but which have torsion; if the torsion of such a connection is skew- symmetric then its symmetric part satisfies (10). Equation (12) provides the means of going from the symmetric connection to the compatible connection with skew- torsion. The only formula missing above necessary to the study of arbitrary bi- Hamiltonian structures of degree 2 is an expression for the contravariant curvature of the connection defined by c , which is, in the presence of Theorem 2.1’s condition = air(c ) + cijr c l + c − (bijr − 2c l + c One may use (B) to replace the components of b in this expression with those of and the derivatives of aij . However, one sees that the compatibility conditions do not naturally become algebraic constraints on ∆ , and the relevancy of such an approach is undermined. It is interesting to note, however, that equation (23) still holds (with Γ ), so that ◦ defined by ∆ still satisfies (I ◦J)◦K = (I ◦K)◦J , and that it is the ‘Fermionic’ condition (I ◦ J) ◦K = −(I ◦K) ◦ J which is altered. The proof of Proposition 3.7 is easily adapted to confirm the existence of a vector field B realising P1 = −LBP2 whenever P1, of the form (5) is an infinitesimal defor- mation of P2 as a Hamiltonian operator, provided b1 = 2c1 . A simple calculation of LBP2 for arbitrary B shows that b1 = 2c1 is also a necessary condition. Thus we have determined the trivial deformations of a degree 2 Hamiltonian operator admitting a constant form, which are themselves of degree 2. Clearly a different approach is necessary to understand deformations of higher degrees. For the case of operators not satisfying the constraints of Theorem 2.11, it is not immediately obvious what conditions, if any, will guarantee the triviality of a deformation; ow- ing to the different form the contravariant curvature tensor takes, the condition is absent. Owing to the lack of a constant form, the methods of [8] in ascertaining the triviality of higher degree deformations, if applicable, will be somewhat more complicated. Finally, there is a certain artificiality to the examples of compatible Fedosov structures presented in section 3. Given Theorem 3.1’s assertion that underlying a pair of compatible Fedosov structures is a finite-dimensional bi-Hamiltonian struc- ture, the question is raised asking which finite-dimensional bi-Hamiltonian struc- tures admit symplectic connections forming almost compatible, almost compatible and flat, or compatible Fedosov structures? It would be interesting to exhibit a 20 JAMES T. FERGUSON pair of compatible Fedosov structures in which the flat coordinates for one of the connections are in some sense physical. Acknowledgements The author would like to thank Ian Strachan for suggesting this project, and the Carnegie Trust for the Universities of Scotland for the scholarship under which this work was conducted. References [1] Chengming Bai, Daoji Meng, and Liguo He. On Fermionic Novikov algebras. J. Phys. A, 35(47):10053–10063, 2002. [2] S. Benenti. Inertia tensors and Stäckel systems in the Euclidean spaces. Rend. Sem. Mat. Univ. Politec. Torino, 50(4):315–341, 1992. [3] Pierre Bieliavsky, Michel Cahen, Simone Gutt, John Rawnsley, and Lorenz Schwachhöfer. Symplectic connections. Int. J. Geom. Methods Mod. Phys., 3(3):375–420, 2006. [4] Philip W. Doyle. Differential geometric Poisson bivectors in one space variable. J. Math. Phys., 34(4):1314–1338, 1993. [5] B. A. Dubrovin and S. P. Novikov. Poisson brackets of hydrodynamic type. Dokl. Akad. Nauk SSSR, 279(2):294–297, 1984. [6] B. A. Dubrovin and S. P. Novikov. Hydrodynamics of weakly deformed soliton lattices. Dif- ferential geometry and Hamiltonian theory. Uspekhi Mat. Nauk, 44(6(270)):29–98, 203, 1989. [7] Boris Dubrovin. Flat pencils of metrics and Frobenius manifolds. In Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), pages 47–72. World Sci. Publ., River Edge, NJ, 1998. [8] Boris Dubrovin and Youjin Zhang. Normal forms of hierarchies of integrable pde’s, Frobenius manifolds and Gromov-Witten invariants. arXiv.org:math.DG/0108160, 2001. [9] Boris V. Fedosov. A simple geometrical construction of deformation quantization. J. Differ- ential Geom., 40(2):213–238, 1994. [10] I.M. Gel’fand, V. Retakh, and M. Shubin. Fedosov manifolds. Adv. Math., 136(1):104–140, 1998. [11] Ezra Getzler. A Darboux theorem for Hamiltonian operators in the formal calculus of varia- tions. Duke Math. J., 111(3):535–560, 2002. [12] A. Ibort, F. Magri, and G. Marmo. Bihamiltonian structures and Stäckel separability. J. Geom. Phys., 33(3-4):210–228, 2000. [13] Franco Magri. A simple model of the integrable Hamiltonian equation. J. Math. Phys., 19(5):1156–1162, 1978. [14] O.I. Mokhov. Symplectic and Poisson structures on loops spaces of smooth manifolds, and integrable systems. Russian Mathematical Surveys, 53(3):515–622, 1998. [15] S.P. Novikov. The geometry of conservative systems of hydrodynamic type. the method of averaging for field-theoretical systems. Russian Mathematical Surveys, 40(4):85–98, 1985. [16] W. Oevel. A note on the Poisson brackets associated with Lax operators. Phys. Lett. A, 186(1-2):79–86, 1994. [17] G.V. Potemin. On Poisson brackets of differential geometric type. Soviet Math. Dokl., 33(1):30–33, 1986. [18] X Xu. Variational calculus of supervariables and related algebraic structures. J. Algebra, 223(2):396–437, 2000. [19] Kentaro Yano and Shigeru Ishihara. Tangent and cotangent bundles: differential geometry. Marcel Dekker Inc., New York, 1973. Pure and Applied Mathematics, No. 16. Department of Mathematics, University of Glasgow, Glasgow G12 8QW, U.K. E-mail address: [email protected] http://arxiv.org/abs/math/0108160 1. Introduction 2. Hamiltonian Operators of Degree 2 3. Flat Pencils of Fedosov Structures 3.1. Multiplication of covectors 3.2. The pencil in flat coordinates 3.3. N manifold with Potential 4. Bi-Hamiltonian Structures in Degrees 1 and 2 5. Conclusions Acknowledgements References

0704.1402

IR-active optical phonons in Pnma-1, Pnma-2 and R3c phases of LaMnO_{3+\delta}

IR-active optical phonons in Pnma-1, Pnma-2 and R3̄c phases of LaMnO3+δ I. S. Smirnova,∗ A. V. Bazhenov, T. N. Fursova, A. V. Dubovitskii, L. S. Uspenskaya, and M. Yu. Maksimuk Institute of Solid State Physics, Russian Academy of Sciences, 142432, Chernogolovka, Moscow distr. Abstract Infrared-active phonons in LaMnO3+δ were studied by means of the reflection and transmission spectroscopy from 50 to 800 cm−1 at room temperature. Powder and ceramic samples of the phases of Pnma-2 (δ = 0.02), Pnma-1 (δ = 0.08), and R3̄c (δ = 0.15) were investigated. Besides, energies of the dipole-active phonons in Pnma-2, Pnma-1 phases were obtained by lattice-dynamics calculations. The transformations of IR-active phonons with the increase of δ in the sequence of Pnma-2, Pnma-1, R3̄c are discussed. PACS numbers: 61.50.Ah, 78.30.-j http://arxiv.org/abs/0704.1402v2 I. INTRODUCTION Electrical and magnetic properties of perovskitelike compounds RxAyMO3+δ (R = rare earth; A = Ca, Sr, Ba, Pb; M = Cu, Mn, Ti, V) can drastically change with varying x, y. In last decades such materials have been intensively investigated. In 1986 super- conductivity with Tc=30 K was found in La2−xBaxCuO4+δ.[1] Using of some other transi- tion metals M can result in the compounds (La1−xAxMnO3+δ, for example) with “colossal” magnetoresistance.2,3,4 Cuprates and manganites possess many common features: the crys- tal structure (close to the perovskite) and strong electron–electron, electron–phonon, and exchange interaction. With x = 0, δ = 0 they are antiferromagnetic dielectrics at low tem- peratures. An increase of x results in a dielectric-metal transition.5 Some excess oxygen in La2CuO4+δ brings about the same transformation of the electronic spectrum that results from the partial substitution of La by an alkali earth, the transforma- tion going up to the superconducting phase.6 Similarities between cuprates and manganites stimulated studies of the influence of excess oxygen on the electron and phonon spectra of LaMnO3+δ. It’s well known that the crystal structure of both LaMnO3+δ and La1−xAxMnO3 is or- thorhombic at δ < 0.1, x < 0.2 and T < 500 K. An increase of δ and x results in a rhombohedral phase R3̄c.7,8,9 In any case the crystal is insulating and paramagnetic above 200–300 K. With decreasing temperature the R3̄c phase transforms into an orthorhombic phase, insulating and ferromagnetic at 0.11 < δ < 0.14, metallic and ferromagnetic at δ > 0.14.7 Two orthorhombic phases of LaMnO3+δ have been found. 7,10,11,12 They were de- noted as either Pnma-1, Pnma-2 (Ref. 12) or O, O′ (Ref. 7). The first one, Pnma-2 (O′), is an insulating antiferromagnet at low temperatures and exists at small δ; the second one, Pnma-2 (O), is an insulating ferromagnet at low temperatures and exists at larger δ. Orthorhombic phases can belong to different space groups (the orthorhombic phase of the La2CuO4, for example, belongs to the Cmca space group). To emphasize that both O ′ and O phases of LaMnO3+δ belong to the same space group Pnma we, following Ref. 12, use the notation Pnma-2, Pnma-1. Unfortunately, this notation does not show the local symmetries of the atoms or the Wyckoff positions, which are subgroups of the point group D2h. It’s the local symmetry that determines the number of modes in every irreducible representation. The purpose of the present study was to examine the spectra of dipole-active optical phonons in Pnma-2, Pnma-1 and R3̄c phases. Especially, we paid attention to transforma- tions that phonon states undergo upon transitions from the phase Pnma-2 to Pnma-1 and then to R3̄c, which are induced by a high-temperature treatment. Optical phonons in the Pnma-2 and R3̄c phases were measured in Refs. 13,14 (Pnma-2, Raman); 15,16 (Pnma-2, IR); 14,17,18 (R3̄c, Raman); and 17,19 (R3̄c, IR). In the present study, we focus on the IR spectrum of the Pnma-1 phase of LaMnO3+δ. To our knowledge, there are no data on either IR or Raman spectra of this phase at the moment. The Pnma-2, Pnma-1 phases are isostructural, so the number of phonon modes should be the same in both cases. However, the number of IR-active modes observed experimentally in the spectra of the Pnma-1 phase is smaller than that for the Pnma-2 phase. In the R3̄c phase an experiment shows more modes than group theory predicts for the R3̄c symmetry. II. CRYSTAL STRUCTURE OF LaMnO3+δ PHASES Since phonon modes are closely related to the crystal lattice symmetry, let us summarize some well known data about crystal structure of four LaMnO3+δ phases. The structure of the parent cubic phase Pm3̄m is shown in the centre of Fig. 1. At ambient pressure, this phase exists at temperatures above 870 K. At room temperature there exist three phases: orthorhombic Pnma-2, Pnma-1 and trigonal (rhombohedral) R3̄c.10,11,12,20 X-ray analisys shows the following: • In the Pnma-2 phase the positions of O2 oxygens (see Fig. 1) deviate considerably from those in the cubic phase. The oxygen octahedra are strongly distorted, particularly in Mn–O2 plane, the Mn–O2 distances differ from each other (1.90 and 2.17 Å). • In the Pnma-1 phase the positions of O2 oxygens slightly deviate from that in the cubic phase, the oxygen octahedra are slightly distorted, the Mn–O2 distances being close to each other. • In comparison with the cubic phase, in both orthorhombic phases the oxygen octahedra are rotated around [010] (cubic) axis by nearly the same angle (the difference is 1–3◦). To distinguish between the Pnma-1 and Pnma-2 phases experimentally, it is sufficient to determine the dimensions of the unit cell: a, b, c. In the Pnma-2 phase a > c and a− c ≈ 0.2 Å, in the Pnma-1 phase a < c and c− a is 0.04–0.08 Å. [010] [100][001] [111] [111] [110] [100] [001]o [110] [001]cub _[010] [100] [010] [001] Pnma-1 Pnma-2 Pnma-1, Pnma-2 FIG. 1: Crystal structure of the R3̄c (left), Pm3̄m (centre) and Pnma (right) phases of LaMnO3+δ. In all considered phases, Mn atoms occupy symmetry-equivalent positions and their time- average charges must be the same. Mn+4 should be defects chaotically distributed in the sample volume. Symmetry forbids any long-range charge ordering in these phases. Such ordering may occur only if the symmetry is lowered. In contrast to the cubic phase and the trigonal phase, the orthorhombic phases contain two types of inequivalent oxygen atoms. Therefore, these oxygen atoms can have different charges and different amplitudes of displacements in the normal vibration modes. All the six oxygen atoms in the unit cell of the R3̄c phase are symmetry-equivalent, therefore their scalar parameters, in particular their charges, should be equal. Arrows in the centre of Fig. 1 show that the point group D3d of the R3̄c phase and the point group D2h of the Pnma phases are subgroups of the Oh point group of the Pm3̄m phase and corresponding phase transitions of the second kind are allowed. The crossed arrow in Fig. 1 shows that D2h is not a subgroup of D3d. As a result, phase transitions of the second kind from the R3̄c phase to the Pnma-1, Pnma-2 phases are forbidden. Such phase transitions can be possible only through an increase of symmetry, i. e., through the intermediate cubic phase, which exists at high temperatures. III. EXPERIMENTAL LaMnO3+δ was prepared from La2MnO3, La(CO3)3·6H2O and Mn2O3. The stoichiometric mixture of source materials was powdered in a ball planetary mill, after that it was calcined at 900◦C, and then it was powdered once again. The main synthesis was conducted at 1100◦C during 10–20 hours. δ was measured by iodometric titration of the Mn+3, Mn+4 ions. It is known that the Pnma-2 phase can be transformed to the Pnma-1 phase by annealing in air. Upon further annealing in oxygen, the Pnma-1 phase transforms into the R3̄c phase.7 In Ref. 13 the Pnma-2 phase was obtained by heating of the R3̄c phase in N2 atmosphere at 900◦C. We realized the reversible sequence of transformations: R3̄c ⇔ Pnma-1 ⇔ Pnma- 2. First, we kept LaMnO3+δ powder at 600 ◦C during 10 hours, then different speeds of cooling resulted in different phases. For the measurements of the IR reflection spectra, ceramic pellets of the Pnma-1, R3̄c phases were prepared from the powder by pressing it and subsequent annealing at 1000◦C during 10 hours. We could not obtain ceramic pellets of the Pnma-2 phase. Magnetic permeability of the Pnma-2, Pnma-1, R3̄c phases was measured in the 77– 300 K temperature range in the AC 2500 Hz magnetic field of 1 Oe at slow heating. The measurements were performed on powder manually pressed into a quartz tube of 2 mm in diameter. This technique results in some uncertainty in the amount of material under investigation. Therefore, the absolute value of the permeability was obtained with some uncertainty, yet we determined the main features of its temperature dependence. IR reflection spectra of ceramic pellets and the IR transmission spectra of powder samples were obtained using a Fourier-transform spectrometer in the spectral range 50–800 cm−1 at room temperature. The reflection spectra were measured in the arrangement where the light falls on a pellet surface near perpendicularly, and an aluminum mirror was used to obtain a reference spectrum. In order to measure transmission spectra, either a polyethylene or a KBr plate (depending on the spectral range) was covered by powder sample, and the transmission spectrum of the plate was used as a reference. Transmission T then was converted to absorbance D = − ln(T ). IV. RESULTS AND DISCUSSION According to X-ray analysis, the unit cell parameters of the Pnma-2, Pnma-1, R3̄c phases we synthesized were the following: phase a, Å b, Å c, Å Pnma-1 5.505 7.776 5.513 Pnma-2 5.732 7.693 5.536 For R3̄c a∗ = 5.515 Å, c∗ = 13.291 Å in the hexagonal coordinates. These parameters are concordant, for instance, with the results of Huang et al.12 Titration has shown the following percentage of Mn+4 ions in investigated samples: Pnma- 2, 5%; Pnma-1, 15%; R3̄c 30%. It corresponds to δ equal to 0.025, 0.075 and 0.15 for the Pnma-2, Pnma-1 and R3̄c phases, respectively. The magnetic permeabilities of Pnma-2, Pnma-1 and R3̄c are shown in Fig. 2. All 80 130 180 230 280 H, Oe 70350-35-70 Pnma-1 x3Pnma-2 117 K 110 K, a.u. 51.5 FIG. 2: Temperature dependence of the magnetic permeability χ(T ) of the Pnma-2 phase (black, multiplied by 3), the Pnma-1 phase (red) and the R3̄c phase (blue). For the Pnma-1 phase, permeability versus magnetic field χ(H) is plotted in the inset at 110 and 117 K. phases are paramagnetic near the room temperature. At low temperature Pnma-1 and R3̄c are ferromagnetic, and Pnma-2 is antiferromagnetic. Ferromagnetic behaviour is illustrated by hysteretic dependence of the permeability upon the magnetic field, which appears below the transition temperature and becomes more and more pronounced with decreasing tem- perature, see the inset in Fig. 2. The temperature of the antiferromagnetic transition in Pnma-2 is 140 K, in agreement with Refs. 7,12. To obtain the temperatures of the ferromag- netic transitions in Pnma-1 and R3̄c, we plotted inverse permeability versus temperature, and linearly extrapolated to zero value the high-temperature parts of these dependences. In agreement with Ref. 7, the transition temperatures turned out to be 180 and 240 K in the Pnma-1 and R3̄c phases, respectively. These results confirm that we really deal with the Pnma-2, Pnma-1 and R3̄c phases. In Fig. 3 the reflection spectra of the phases R3̄c (δ ∼ 0.15), Pnma-1 (δ ∼ 0.05), 0.4 - 0.4 Pnma-1 200 400 600 0.8 Pnma-2 Wavenumber, cm-1 FIG. 3: Solid lines: Reflection spectra of the R3̄c, Pnma-1 and Pnma-2 phases. (For the Pnma-2 phase the data are taken from Ref. 15). Crosses: the results of fitting. and Pnma-2 (δ = 0) are shown. In the present wavenumber range reflection spectra are determined by dipole-active phonons. We approximated our reflectivity spectra R(ω) using a fitting procedure based on a set of Lorentz oscillators: ǫ(ω) = ω20,j − ω 2 − iγjω ; R(ω) = ǫ(ω)− 1 ǫ(ω) + 1 ǫ(ω) is the complex dielectric function; Sj , ω0,j and γj are oscillator strength, frequency and damping factor of mode j. The number of oscillators we used in every case was chosen as the minimum number allowing a good fit. The crosses on Fig. 3 show the result of the fitting. Fig. 4 shows the conductivity contributions σj(ω) of the calculated Lorentz oscillators: σj(ω) = ω2γjSj (ω20,j − ω 2)2 + γ2jω 200 400 600 100 x10 Wavenumber, cm-1 Pnma-1 Pnma-2 FIG. 4: Separate conductivity contributions of each Lorentz oscillator, which were obtained by fitting of the reflection spectra shown in Fig. 3 Paolone et al.15 compared experimental and theoretically calculated21 phonon frequen- cies of Pnma-2 phase. Taking into account the lowest and the highest phonon frequencies obtained by Paolone et al.15, we corrected previously calculated21 phonon frequencies of the Pnma-2 phase. Also, we calculated the phonon frequencies of the Pnma-1 phase using the rigid-ion model with effective charges. Table I shows the results of these calculations along with the phonon frequencies extracted from experimental data. We measured spectra of ceramic samples. So the polarization symmetry of the IR-active phonons could not be ob- tained from our experiments and the arrangement of the modes is tentatively done according to their frequencies and intensities. In Table I, “TO” and “LO” indices correspond to the “transverse” and “longitudinal” frequencies. A TO frequency means a resonant frequency ω0,j (see Equation (1)) and coin- cides with a maximum of σ(ω) (see Equation (2)). LO frequencies in Table I correspond to maxima of the function −Im(1/ǫ) and represent oscillator strengths S = ω2LO − ω A. IR spectra of the Pnma phases According to group theory, the isostructural Pnma-1 and Pnma-2 phases should have 25 dipole-active optical phonon modes, 9B1u+7B2u+9B3u (see, for example, Ref. 21). Indeed, Paolone et al.15 experimentally found 25 IR-active modes in Pnma-2 crystals at 10 K (and 18 modes at room temperature). However, in our Pnma-1 ceramic only 11 modes can be distinguished at room temperature. The lines in the Pnma-1 ceramic are substantially wider than in the Pnma-2 single crys- tals (see damping factors γ in Table I). Let’s consider possible reasons for this broadening. Decreasing of the phonon life time τ accompanied by increasing of γ = 1/τ could come as a result of the phonon scattering on grain boundaries of ceramic. To check that, we measured transmission spectra of the Pnma-1, Pnma-2 and R3̄c powders. The grain sizes of our powders were measured22 using electron microscopy: in all samples the typical grain size is found to be about 1 µm. In the transmission spectra, the widths of the phonon lines increase monotonically with the increase of the excess oxygen content, i. e., in the sequence Pnma-2, Pnma-1, R3̄c. That means that phonon scattering on grain boundaries is not the main reason of line broadening in the spectra of the Pnma-1, R3̄c powders. The same is even truer for the spectra of the Pnma-1, R3̄c ceramics, because in a ceramic the typical TABLE I: Calculated and experimental TO(LO) frequencies (cm−1) of IR-active phonon modes; w means a weak mode; γ is damping factor (cm−1) Pnma-2 Pnma-1 R3̄c calc. exp. calc. exp. exp. ωTO(ωLO) ωTO(ωLO) γ ωTO(ωLO) ωTO(ωLO) γ ωTO(ωLO) γ 115(119) B1u 116(120) 4 111(115) B1u 116(118) B3u 120(130) B3u 125(135) 20 120(140) 62 138(140) B2u 143(148) B2u 147(180) 29 171(197) B2u 172(244) 6 166(196) B1u 163(209) 24 167(197) 38 175(195) B1u 182(195) 3 181(199) B2u 187(195) 27 231(232) B3u 201(203) 9 229(230) B3uw 233(249) B1u 244(255) 7 247(248) B1uw 249(250) B2u 300(302) B2uw 254(281) B3u 271(291) 5 253(253) B3uw 284(296) B1u 277(297) 9 270(291) B3u 258(267) 74 252(266) 88 297(305) B3u 285(293) 9 280(281) B1uw 309(309) B1u 332(354) B1u 327(381) 95 324(376) 97 330(341) B2u 335(363) 15 355(371) B1u 346(352) B1u 350(411) 16 368(370) B2uw 354(373) B3u 362(391) 10 377(440) B3u 372(401) 60 376(400) 68 420(426) B2u 400(401) 16 382(448) B1u 434(450) B1u 429(437) 18 416(417) B1uw 420(429) 59 431(442) 78 455(457) B1u 451(452) 12 437(444) B3u 473(479) B3u 474(480) 28 487(503) B2u 487(490) 40 498(592) 33 528(531) B3u 515(518) 18 564(568) B2u 573(598) B2u 561(606) 17 580(589) B3u 567(579) 49 576(592) 85 634(640) B2u 644(646) 39 584(641) B2u 599(618) 57 611(627) 65 644(650) B3u 615(616) B3uw 645(651) B1u 634(639) B1u 637(642) 51 649(653) 57 Pnma-1 200 400 600 Wavenumber, cm-1 Pnma-2 1.0 Reflection FIG. 5: Experimental absorption of the R3̄c (top), Pnma-1 (middle) and Pnma-2 (bottom, solid line) powders. The dashed line in the bottom part represents the reflectivity of a Pnma-2 single crystal taken from Ref. 15. grains can be larger than that in a source powder. Moreover, we believe that even in our Pnma-2 powder phonon scattering on grain boundaries is not the main reason of the line broadening. In the bottom part of Fig. 5, the dashed line shows the reflection spectrum of a Pnma-2 single crystal15, solid line represents our absorption spectrum of the Pnma-2 powder. Our calculations showed that, on average, the lines in the conductivity spectrum of powder are three times wider than those in the spectrum of a crystal. Nevertheless, one can reveal the same number of lines in both spectra. For example, 172 cm−1 and 182 cm−1 lines can be undoubtedly distinguished in our powder spectrum. It was shown15 that in a doped LaMnO3 single crystal, containing 8% of Mn +4, these lines could not be resolved at room temperature. Our powder contained 5% of Mn+4 so it seems reasonable to attribute the observed broadening of lines in our Pnma-2 powder as a result of oxygen doping. The main factor of line broadening in the spectra of these samples should be the phonon scattering on structural defects, which multiply with excess oxygen doping. These defects could be oxygen atoms in interstitial sites, like those in La2CuO4+δ [23]. However as for LaMnO3+δ and La1−xAxMnO3+δ (A=Ca, Sr, Ba), at the moment it is rather believed that the nonstoichiometric oxygen Oδ is compensated by both La and Mn vacancies in equal amounts.7,24 In such a case, vacancy contents of La or Mn in our samples δ/(3 + δ) would be 0.7%, 2.6% and 5% for the Pnma-2, Pnma-1 and R3̄c phases respectively. Line broadening can make difficult or impossible experimental detection of some lines with small oscillator strength. In the Pnma-2 phase, that could be the phonons with the frequencies 400 cm−1, 451 cm−1 (see Table I). We calculated the oscillator strength for all IR-active modes of the Pnma-2 and Pnma-1 phases. It turns out that the number of modes experimentally detected in the Pnma-1 phase is reduced in comparison with the Pnma-2 phase mainly because the oscillator strength of some phonons of the Pnma-1 phase becomes very small. These Pnma-1 modes are marked by w in Table I. In the Pnma-1 phase, the lengths of Mn–O bonds differ from each other very little (the difference comes in fourth significant digit). The closeness of Mn–O bond lengths means that oxygen atoms are almost symmetrically equivalent, i. e., the Pnma-1 crystal structure deviates from the cubic one less than the Pnma-2 crystal structure where the difference in Mn–O bond lengths is 15%. In the cubic structure, the number of IR-active phonons is less than in an orthorhombic structure. Therefore, if a structure is close to cubic then some IR-active phonons are “on the verge of disappearance”. B. IR spectra of R3̄c Our spectra of R3̄c are in satisfactory agreement with the spectra obtained in Ref. 17,19. According to our experimental results, phonon damping factors of the R3̄c phase exceed those of the Pnma-1 phase by a factor of 1.3 on average. The first reason is that the Mn+4 content in R3̄c is two times as large as it is in the Pnma-1 phase, so there are more structural defects there. The second reason is disorder caused by the noncoherent dynamic Jahn-Teller effect. According to the group-theory analysis (see Ref. 21, for example), there are 8 IR-active phonon modes in the R3̄c phase: 3A2u+5Eu. At room temperature, in reflection spectra of the R3̄c ceramic we definitely distinguish 10 lines. The approximation by a set of Lorentz oscillators revealed an additional very broad line near 120 cm−1. Therefore, we found in the R3̄c phase the same amount of lines (11) as in the Pnma-1 phase. Let us consider possible reasons for appearing of additional lines in spectra of the R3̄c phase. Local break of the inversion symmetry around a point defect could make some Raman- active (IR-forbidden) modes to appear in IR spectra. However, comparison of the IR spectra of the R3̄c phase with Raman spectra of Abrashev et al.17 shows that there is only one Raman line near 649 cm−1 close to an IR line (640 cm−1), the other Raman lines have no counterparts in our IR spectra. In IR spectra there could appear maxima of the phonon density of states caused by breaking of the long-range order. Iliev et al.14 analyzed the Raman spectra of doped rare- earth manganites and interpreted them in the frame of the model used for description of amorphous materials.25 The Raman spectra in this case are dominated by disorder-induced bands, reflecting the phonon density of states smeared due to finite phonon lifetime. In other words, the law of conservation of the quasimomentum k breaks and phonons with nonzero k begin to interact with light. In general, the same mechanism could definitely work for IR spectra too. Big linewidths prevent us from supporting or rejecting an influence of phonons with k 6= 0 on IR spectra of the R3̄c phase. Though it worth to take into account that according to Iliev et al.14 a Raman mode generally gives several maxima of density of states. Probably the same is true for IR-active modes. However, our spectra of the R3̄c phase can be fitted very well by a few Lorentz functions. So we think that the phonons with k 6= 0 can have only a small influence on our spectra, they do not determine essential spectral features. We explain additional lines in our IR spectra of the R3̄c phase as a result of the dynamic Jahn-Teller effect. In the R3̄c phase of LaMnO3, the R3̄c symmetry exists only “on average”, revealing itself in certain kinds of experiments such as X-ray diffraction. At any particular moment of time, one of the octahedron O–Mn–O axes differs from two others due to dynamic Jahn-Teller distortions; therefore, oxygen atoms are inequivalent and their charges are not equal. It is the “instant”, not “average”, pattern that is probed in optical experiments.14 Obviously, normal phonon modes, measured by means of IR and Raman spectroscopy, are normal modes of the “instant”, not average” pattern. In the “instant” view every octahedron in the R3̄c phase looks deformed, mostly in the same way as the octahedra in the Pnma phases. That’s why the phonon spectrum of the R3̄c phase resembles that of the Pnma phases. Similarly, Abrashev et al.17 interpreted two strongest lines (649 cm−1 is one of them) in their Raman spectra of the R3̄c phase as “forbidden” modes, analogous to the respective modes in Pnma phases. We can expect some correlations between the Jahn-Teller deformations of the octahedra in the R3̄c phase. Qiu et al.26 found that in high-temperature (T > 1010 K) stoichiometric rhombohedral LaMnO3 there are fully distorted MnO6 octahedra, ordered in clusters of diameter ∼ 16 Å. According Ref. 7, the phase diagram of LaMnO3+δ containes an area (0.11 < δ < 0.14) where a phase transition R3̄c ⇔ Pnma-1 exists at T = 300 K. As we mentioned in Section II, such transition of a second kind is forbidden by symmetry. In Ref. 27 there was suggested a model of a phase transition through a virtual cubic phase. Taking into account the known IR and Raman spectra of the R3̄c phase, as well as the results of Qiu et al.26, we suggest that the R3̄c samples could contain nanoclusters of some Pnma phase. Such inclusions may be growing centres at the transition R3̄c ⇔ Pnma-1 of a first kind. V. THE INFLUENCE OF SELECTION RULES OF D2h POINT GROUP ON THE IR SPECTRA According to the selection rules, the irreducible representations B1u, B2u, B3u of D2h point group correspond to IR-active modes, their total electric dipole moment M taking the form M(B1u) = (0, 0,Mz),M(B2u) = (0,My, 0),M(B3u) = (Mx, 0, 0). Similarly, for every full set of symmetrically equivalent atoms in the unit cell (O2, for example) the sum of their atomic displacements ui has only one non-zero component. (For a single atom inside such a set, all three components can differ from zero.) Let’s consider four lowest-frequency IR-active modes of the Pnma-2 phase. (Fig. 6) The line with the lowest frequency (115 cm−1) can be distinctly seen in the spectra of the Pnma-2 and Pnma-1 phases. In the spectrum of the R3̄c phase it substantially broadens (Fig. 4). A similar line have been observed in reflection spectra of both the undoped ( x = 0 ) and doped by either Ca or Sr La1−xAxMnO3+δ, LaTiO3 28, YVO3 29. Theoretical calculations21 and experimental results29 show that in the spectra of the Pnma-2 phase this line consists of two modes with close frequencies and different polarizations (see the upper part of Fig. 6). In B1u, B3u modes, La and O1 atoms can vibrate only in the reflection plane m therefore |Mx|>0 116(118) z |Mz|>0 115(119) |My|>0 178(215) |Mx|>0 185(197) FIG. 6: Theoretically calculated patterns of some IR-active phonon modes of the Pnma-2 phase. Thick arrows show atomic displacements in the direction of the total electric dipole moment M . Thin arrows show atomic displacements in the other two main crystallographic directions. In the left upper corners there shown corresponding irreducible representations. In the bottom there shown corresponding theoretical TO(LO) frequencies. having two degrees of freedom. 115 cm−1 mode (B1u) has the maximal displacements of La atoms along x axis. Nev- ertheless, these components don’t contribute to the total electrical dipole moment because their sum equals zero. Only small components of the La displacements uz along z axis (thick arrows) contribute to M . The intensity of this mode in the optical conductivity spectrum is determined by the displacements of O2, Mn, La atoms, their contributions adding together. Relatively small contributions of O1 atomic displacements have the opposite sign. The structure of atomic displacements of 116 cm−1 mode (B3u) is similar to the previous one. The biggest displacements of La atoms are along z axis, M being parallel to x axis. The intensity of this mode is determined by the adding contributions of O2, La displacements and the subtracting contribution of Mn displacement. In 178 cm−1 mode (B2u) O1 and La atoms can vibrate only along y axis, in 185 cm mode (B3u) they can vibrate only in (0,1,0) plane. An essential difference between these modes and 115 cm−1, 116 cm−1 modes is that in 178 cm−1, 185 cm−1 modes the maximal displacements of every atom contribute to M (O1, O2, La are adding, Mn is subtracting). That is why the oscillator strengths of 178 cm−1, 185 cm−1 modes are much higher than that of 115 cm−1, 116 cm−1 modes. Being isostructural, the Pnma-2 and Pnma-1 phases have close patterns of atomic dis- placements in phonon modes. Still, there are some important differences between them. In the upper part of Fig. 7 there are shown 233 cm−1 mode of the Pnma-2 phase and 247 cm−1 mode of the Pnma-1 phase. Big displacements of Mn and O1 along x axis, which have comparable magnitudes for the Pnma-2 and Pnma-1 phases, don’t contribute to M . In the both cases, the oscillator strengths are entirely determined by small displacements along z axis, which are much less for the Pnma-1 phase (247 cm−1) than for the Pnma-2 phase (233 cm−1). As a result, the oscillator strength 247 cm−1 mode of the Pnma-1 phase is very small. In the bottom part of Fig. 7 there are shown another pair of similar modes. The oscillator strength of 284 cm−1 mode (Pnma-2) is much higher than that of 280 cm−1 mode (Pnma-1), because in the second case the displacements of Mn, O2 atoms along z axis are substantially less. In addition, the displacements of O1 atoms, which decrease the resulting M , are of much higher amplitude in 280 cm−1 mode (Pnma-1) than in 284 cm−1 mode (Pnma-2). Our theoretical calculations showed that there are six modes in total, which strongly z |Mz|>0 233(249) z |Mz|>0 247(248) z |Mz|>0 284(296) z |Mz|>0 280(281) Pnma-1 Pnma-1 Pnma-2 Pnma-2 FIG. 7: Theoretically calculated patterns of some IR-active phonon modes of the Pnma-2 and Pnma-1 phases. Thick arrows show atomic displacements in the direction of the total electric dipole moment M . Thin arrows show atomic displacements in the other two main crystallographic directions. In the left upper corners there are shown corresponding irreducible representations. In the bottom there shown corresponding theoretical TO(LO) frequencies. decrease their oscillator strength for the Pnma-1 phase in comparison with that for the Pnma-2 phase. (In Table I they are marked by w.) That’s why for the Pnma-1 phase the number of modes seen in experiment is less than for the Pnma-2 phase. The atomic displacements of all IR-active modes for the Pnma-2 phase are drawn in Fig. 5 of Ref. 21. Mostly, the displacements of O1, O2 atoms are much bigger than that of Mn, La atoms. As a result, the small components were ignored there. For a strong mode, that was reasonable. However for a weak mode, that could cause some misunderstanding. For |My|>0 634(640) Pnma 2 |My|>0 249(250) Pnma 2 FIG. 8: Theoretically calculated patterns of some IR-active phonon modes for the Pnma-2 phase. Thick arrows show atomic displacements in the direction of the total electric dipole moment M . Thin arrows show atomic displacements in the other two main crystallographic directions. In the left upper corners there are shown corresponding irreducible representations. In the bottom there shown corresponding theoretical TO(LO) frequencies. example, all the displacements shown in Ref. 21 for 207 cm−1 and 562 cm−1 modes produce the resulting M = 0. More correct patterns for these modes are shown in Fig. 8. VI. CONCLUSIONS The reversible sequence of transformations R3̄c ⇔ Pnma-1 ⇔ Pnma-2 was realized by annealing of LaMnO3+δ powder at 600 ◦C during 5–10 hours. For the first time, IR transmission and reflection spectra of the Pnma-1 phase of LaMnO3+δ were measured. In addition, IR spectra of the Pnma-2 and R3̄c phases were measured and found to be in satisfactory agreement with previously published results. Taking into account new experimental data for the Pnma-2 phase, we corrected our pa- rameters of the rigid-ion model with effective charges and recalculated its phonon spectrum. The frequencies and oscillator strengths of the IR-active phonons in Pnma-1 phase were calculated as well. The number of experimentally observed IR-active phonon modes in the Pnma-1 phase is smaller than that in the Pnma-2 phase, although these phases have the same Pnma sym- metry. According to theoretical calculations, it happens due to a decrease in the oscillator strengths of several phonon modes of the Pnma-1 phase. The underlying reason is that in the Pnma-1 phase MnO6 octahedra are much less distorted than in the Pnma-2 phase. In the spectra of the R3̄c phase, the number of modes observed exceeds that predicted by group theory. We attribute the additional modes to local distortions of oxygen octahedra similar to those in Pnma phases. Acknowledgments We thank S. S. 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0704.1404

Neutron matter from low-momentum interactions

arXiv:0704.1404v1 [nucl-th] 11 Apr 2007 Neutron Matter from Low-Momentum Interactions Bengt Friman1,∗), Kai Hebeler1,∗∗), Achim Schwenk2,∗∗∗) and Laura Tolós3,†) 1GSI, Planckstr. 1, D-64291 Darmstadt, Germany 2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3 2 FIAS, J.W. Goethe Universität, D-60438 Frankfurt am Main, Germany We present a perturbative calculation of the neutron matter equation of state based on low-momentum two- and three-nucleon interactions. Our results are compared to the model-independent virial equation of state and to variational calculations, and we provide theoretical error estimates by varying the cutoff used to regulate nuclear interactions. In ad- dition, we study the dependence of the BCS 1S0 superfluid pairing gap on nuclear interactions and on the cutoff. The resulting gaps are well constrained by the nucleon-nucleon scattering phase shifts, and the cutoff dependence is very weak for sharp or sufficiently narrow smooth regulators with cutoffs Λ > 1.6 fm−1. §1. Introduction The determination of a reliable equation of state of nucleonic matter plays a central role for the physics of neutron stars1) and core-collapse supernovae.2), 3) Fur- thermore the superfluidity and superconductivity of neutrons and protons is an im- portant phenomenon in nuclear many-body systems,4), 5) in particular for the cooling of neutron stars.6) In this contribution, we present calculations of the neutron mat- ter equation of state at finite temperature and of the 1S0 superfluid gap in the BCS approximation based on low-momentum interactions. Renormalization group methods coupled with effective field theory (EFT) offer the possibility for a systematic approach to the equation of state. By evolving nuclear forces to low-momentum interactions Vlow k 7)–9) with cutoffs around 2 fm−1, the model-dependent short-range repulsion is integrated out and the resulting low- momentum interactions are well constrained by the nucleon-nucleon (NN) scattering data. Furthermore, the corresponding leading-order three-nucleon (3N) interactions (based on chiral EFT) become perturbative in light nuclei for Λ . 2 fm−1.10) With increasing density, Pauli blocking eliminates the shallow two-nucleon bound and nearly-bound states, and the contribution of the particle-particle channel to bulk properties becomes perturbative in nuclear matter.8) The Hartree-Fock approxima- tion is then a good starting point for many-body calculations with low-momentum NN and 3N interactions, and perturbation theory (in the sense of a loop expansion) around the Hartree-Fock energy converges at moderate densities. This can be under- stood quantitatively based on the behavior of the Weinberg eigenvalues as a function of the cutoff and density.8), 9) ∗) E-mail address: [email protected] ∗∗) E-mail address: [email protected] ∗∗∗) E-mail address: [email protected] †) E-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1404v1 2 B. Friman, K. Hebeler, A. Schwenk and L. Tolós Some uncertainty remained concerning a possible dependence of the 1S0 pairing gap on the input NN interaction in low-density neutron matter (kF < 1.6 fm We address this point and explore the dependence of 1S0 superfluidity on nuclear interactions at the BCS level in detail. We find that the BCS gap is well constrained by the NN phase shifts. Therefore, any uncertainties are due to polarization (induced interaction), dispersion and three-nucleon interaction effects. §2. Equation of State of Neutron Matter Using the Kohn-Luttinger-Ward theorem,11), 12) the perturbative expansion of the free energy (at finite temperature) can be formulated as a loop expansion around the Hartree-Fock (HF) energy. In this work, we include the first-order NN and 3N contributions, as well as normal and anomalous second-order NN diagrams. Other thermodynamic quantities are computed using standard thermodynamic relations. 0 0.05 0.1 ρ [fm-3] virial T=3 MeV T=6 MeV T=10 MeV 0 0.05 0.1 0.15 ρ [fm-3] Hartree-Fock (NN+3N) HF + 2nd-order NN Fig. 1. Energy per particle E/N as a function of the density ρ at first order (left panel) and including second-order NN contributions (right panel).13) The resulting energy per particle E/N as a function of the density ρ is shown in Fig. 1 for a cutoff Λ = 2.1 fm−1 and temperatures T = 3, 6 and 10MeV.13) The results presented in the left panel are the first-order NN and 3N contributions, and those in the right panel includes all second-order diagrams with NN interactions. For T = 6MeV, we also give a band spanned between Λ = 1.9 fm−1 (lower line) and Λ = 2.5 fm−1 (upper line). The inclusion of second-order contributions significantly reduces the cutoff dependence of the results. The model-independent virial equation of state14) and the variational calculations of Friedman and Pandharipande (FP)15) are displayed for comparison. The inclusion of second-order correlations lowers the energy below the variational Neutron Matter from Low-Momentum Interactions 3 0 1 2 3 Λ [fm sharp =0.8 fm =0.4 fm =1.35 fm Fig. 2. The neutron-neutron 1S0 superfluid pairing gap ∆ as a function of the cutoff Λ for three densities and different smooth exponential regulators, as well as for a sharp cutoff.18) The low-momentum interactions are derived from the N3LO chiral potential of Ref.19) results for densities ρ . 0.05 fm−3, and we observe a good agreement for E/N with the T = 10MeV virial result when the second-order contributions are included. In the virial equation of state these contributions are included via the second-order virial coefficient, while in the variational calculation the state dependence of such correlations is only partly accounted for.16) Furthermore, the generic enhancement of the effective mass at the Fermi surface leads to an enhancement of the entropy at low temperatures above the variational and HF results.13), 16), 17) §3. BCS gap in the 1S0 channel We solve the BCS gap equation in the 1S0 channel ∆(k) = − dp p2 Vlow k(k, p)∆(p) ξ2(p) +∆2(p) , (3.1) with the (free-space) low-momentum NN interaction Vlow k(k, k ′). Here ξ(p) ≡ ε(p)− µ, ε(p) = p2/2 and µ = k2F/2 (c = ~ = m = 1). We find that the neutron-neutron BCS gap is practically independent of the NN interaction.18) Consequently, 1S0 superfluidity is strongly constrained by the NN scattering phase shifts. The maximal gap at the BCS level is ∆ ≈ 2.9− 3.0MeV for kF ≈ 0.8−0.9 fm −1. For the neutron-proton 1S0 case, we find somewhat larger gaps, reflecting the charge dependence of realistic nuclear interactions.18) In Fig. 2 we show the dependence of the neutron-neutron 1S0 superfluid pair- ing gap on the cutoff starting from the N3LO chiral potential of Ref.19) for three representative densities.18) We employed different smooth exponential regulators f(k) = exp[−(k2/Λ2)n], as well as a sharp cutoff. As long as the cutoff is large com- 4 B. Friman, K. Hebeler, A. Schwenk and L. Tolós pared to the dominant momentum components of the bound state (Λ > 1.2kF), the gap depends very weakly on the cutoff. This shows that the 1S0 superfluid pairing gap probes low-momentum physics. Below this scale, which depends on the density and the smoothness of the regulator, the gap decreases, since the relevant momentum components of the Cooper pair are then partly integrated out. §4. Conclusions In summary, we have studied the equation of state at finite temperature including many-body contributions in a systematic approach. We have found good agreement with the virial equation of state in the low-density–high-temperature regime. Ana- lyzing the cutoff dependence of our results provides lower bounds for the theoretical uncertainties. The possibility of estimating theoretical errors plays an important role for reliable extrapolations to the extreme conditions reached in astrophysics. In addition, we have shown that the 1S0 superfluid pairing gap in the BCS approximation is practically independent of the choice of NN interaction, and there- fore well constrained by the NN scattering data. This includes a very weak cutoff dependence with low-momentum interactions Vlow k for sharp or sufficiently narrow smooth regulators with Λ > 1.6 fm−1. At lower densities, it is possible to lower the cutoff further to Λ > 1.2kF. Furthermore, the pairing gap clearly reflects the charge dependence of nuclear interactions. The weak cutoff dependence indicates that, in the 1S0 channel, the contribution of 3N interactions is small at the BCS level. Acknowledgements This work was supported in part by the Virtual Institute VH-VI-041 of the Helmholtz Association, NSERC and US DOE Grant DE–FG02–97ER41014. TRI- UMF receives federal funding via a contribution agreement through NRC. References 1) J.M. Lattimer and M. Prakash, Astrophys. J. 550 (2001), 426. 2) A. Mezzacappa, Annu. Rev. Nucl. Part. Sci. 55 (2005), 467. 3) H.T. Janka, R. Buras, F.S. Kitaura Joyanes, A. Marek and M. Rampp, astro-ph/0405289. 4) Yu.A. Litvinov et al., Phys. Rev. Lett. 95 (2005), 042501. 5) F. Sarazin et al., Phys. Rev. C 70 (2004), 031302(R). 6) D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004), 169. 7) S.K. Bogner, T.T.S. Kuo and A. Schwenk, Phys. Rep. 386 (2003), 1. 8) S.K. Bogner, A. Schwenk, R.J. Furnstahl and A. Nogga, Nucl. Phys. A 763 (2005), 59. 9) S.K. Bogner, R.J. Furnstahl, S. Ramanan and A. Schwenk, Nucl. Phys. A 773 (2006), 203. 10) A. Nogga, S.K. Bogner and A. Schwenk, Phys. Rev. C 70 (2004), 061002(R). 11) W. Kohn and J.M. Luttinger, Phys. Rev. 118 (1960), 41. 12) J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960), 1417. 13) L. Tolós, B. Friman and A. Schwenk, nucl-th/0611070; and to be published. 14) C.J. Horowitz and A. Schwenk, Phys. Lett. B 638 (2006), 153. 15) B. Friedman and V.R. Pandharipande, Nucl. Phys. A 361 (1981), 502. 16) S. Fantoni, B.L. Friman and V.R. Pandharipande, Nucl. Phys. A 399 (1983), 51. 17) S. Fantoni, V.R. Pandharipande and K.E. Schmidt, Phys. Rev. Lett. 48 (1982), 878. 18) K. Hebeler, A. Schwenk and B. Friman, nucl-th/0611024, Phys. Lett. B (in press). 19) D.R. Entem and R. Machleidt, Phys. Rev. C 68 (2003), 041001(R).

0704.1405

Euclidean analysis of the entropy functional formalism

Euclidean analysis of the entropy functional formalism Óscar J. C. Dias,1 Pedro J. Silva,2 1 Departament de F́ısica Fonamental, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain, 2 Institut de Ciències de l’Espai (IEEC-CSIC) and Institut de F́ısica d’Altes Energies (IFAE), E-08193 Bellaterra (Barcelona), Spain [email protected], [email protected] ABSTRACT The attractor mechanism implies that the supersymmetric black hole near horizon solution is defined only in terms of the conserved charges and is therefore independent of asymptotic moduli. Starting only with the near horizon geometry, Sen’s entropy functional formalism computes the entropy of an extreme black hole by means of a Legendre transformation where the electric fields are defined as conjugated variables to the electric charges. However, traditional Euclidean methods require the knowledge of the full geometry to compute the black hole thermodynamic quantities. We establish the connection between the entropy functional formalism and the standard Euclidean formalism taken at zero temperature. We find that Sen’s entropy function f (on-shell) matches the zero temperature limit of the Euclidean action. Moreover, Sen’s near horizon angular and electric fields agree with the chemical potentials that are defined from the zero-temperature limit of the Euclidean formalism. http://arxiv.org/abs/0704.1405v3 Contents 1 Introduction 1 1.1 Attractor mechanism and entropy functional formalism . . . . . . . . . . . . . . . . . 2 1.2 Zero temperature limit and chemical potentials . . . . . . . . . . . . . . . . . . . . . 2 1.3 Entropy functional formalism from an Euclidean perspective . . . . . . . . . . . . . . 3 1.4 Main results and structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Entropy functional formalism revisited 5 3 Euclidean zero-temperature formalism: BPS black holes 9 4 Euclidean zero-temperature and entropy functional formalisms 14 4.1 Near-horizon and asymptotic contributions to the Euclidean action . . . . . . . . . . 14 4.2 Relation between chemical potentials in the two formalisms . . . . . . . . . . . . . . 17 5 Extremal (non-BPS) black holes 19 5.1 Extreme three-charged black hole with ergoregion . . . . . . . . . . . . . . . . . . . . 20 5.2 Extreme three-charged black hole without ergoregion . . . . . . . . . . . . . . . . . 21 6 Discussion 23 A Three-charged black hole: solution and thermodynamics 24 A.1 The D1-D5-P black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 A.2 The near-BPS limit of the D1-D5-P black hole . . . . . . . . . . . . . . . . . . . . . 28 B Explicit agreement for other black hole systems 29 B.1 Four-charged black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B.1.1 BPS black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 B.1.2 Extreme (non-BPS) black hole: ergo-free solution . . . . . . . . . . . . . . . . 32 B.1.3 Extreme (non-BPS) black hole: ergo-branch solution . . . . . . . . . . . . . . 33 B.2 Extreme Kerr-Newman black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1 Introduction Black holes (BH) are one of most interesting laboratories we have to investigate quantum gravity effects. Due to their thermodynamic behavior these objects have been associated to ensembles of microstates in the fundamental quantum gravity theory where ideally, quantum statistical analy- sis should account for all the BH coarse-grained thermodynamical behavior. In particular, many important insights in the classical and quantum structure of BH have been obtained studying supersymmetric configurations in string theory. Supersymmetric BH have many important prop- erties that turn out to be crucial to obtain all the new results. Basically, supersymmetry triggers a number of non-renormalization mechanisms that protect tree level calculations from higher order loop corrections. Moreover, this kind of behavior has also been found in some non-supersymmetric extreme solutions. 1.1 Attractor mechanism and entropy functional formalism In this context we have the so called attractor mechanism [1]. It was originally thought in the context of four dimensional N = 2 supergravity, where we have that the values of the scalar fields at the horizon are given by the values of the BH conserved charges and are independent of the asymptotic values of the scalars at infinity. For these BH (and others) it has been checked that the Bekenstein-Hawking entropy agrees with the microscopic counting of the associated D-brane system. Not only in the supergravity approximation, but also after higher derivative corrections are added to the generalized prepotential [2]. These results motivated a conjecture where the BH partition function equals the squared of the associated topological string partition function i.e., ZBH = |ZTop|2 [3]. Lately, the attractor mechanism has been extended to other directions, and applied to several gauged and ungauged supergravities (see, e.g., [4, 5, 6]). Importantly, the attractor mechanism has provided a new way to calculate the BH entropy. In a series of articles [7, 8, 9], Sen recovered the entropy of D-dimensional BPS BH using only the near horizon part of the geometry. Basically, in this regime the solution adopts the form AdS2 ⊗ SD−2 1 plus some electric and magnetics fields. The entropy S is obtained by introducing a function f as the integral of the corresponding supergravity Lagrangian over the SD−2. More concretely, an entropy function is defined as 2π times the Legendre transform of f with respect to the electric fields ei. Then, an extremization procedure fixes the on-shell BPS values of the different fields of the solution and in particular determines the BPS value of the entropy, Sbps = 2π . (1.1) Note that in the above definition the different near horizon electric fields take the role of “conjugated chemical potentials” to the BH charges. This formalism has also been extended to extreme non-BPS The attractor mechanism, both for asymptotically AdS or flat BH, implies that in the near horizon geometry we have a dual CFT theory where the microscopic structure can be studied. We expect that not only the entropy but all the statistic properties of such supergravity systems should be described in terms of their dual CFT states. 1.2 Zero temperature limit and chemical potentials Supersymmetric BH in asymptotically AdS spaces have also been studied using the AdS/CFT cor- respondence [10, 11, 12, 13]. For the AdS5 case we still do not have a CFT microscopic derivation of its entropy that reproduces the supergravity result. Nevertheless, in [12, 13] it was showed that the phase space of this supersymmetric sector can be scanned in both sides of the correspondence showing a rich structure with phase transitions and Hagedorn alike behavior2. In fact, observables in both dual pictures agree up to numerical factors, a very non-trivial result since the CFT calcu- 1The analysis of the near horizon geometry has been applied to more general BH that define squashed AdS2⊗SD−2 geometries like in [9, 4]. 2These T = 0 phase transitions were analyzed both in the strong and weak coupling regimes. Remarkably, it was found that their properties resemble the well-known finite temperature phase transitions, where the Hawking-Page phase transition in the strong coupling corresponds to the deconfinement/confinement transition at weak coupling [10, 13]. lations are performed at zero coupling only3. In order to study the full statistical properties (so that we could in principle do more than just account for the entropy), in [12, 13] it was found how to define the different chemical potentials µi that control the supersymmetric BH partition function in the grand canonical ensemble. The basic input comes from the thermodynamics of the dual CFT theory, where the BPS partition function is obtained from the finite temperature one, by sending the temperature to zero. This also sends the several chemical potentials to their BPS values. The associated dual limiting procedure in the supergravity regime corresponds also to send the temperature to zero. Done carefully, this defines the supergravity chemical potentials that are dual to the the CFT ones and, more generally, the statistical mechanics of supersymmetric BH that is free of divergencies. These chemical potentials are the next to leading order terms of the zero temperature expansion of the horizon angular velocities and electric potentials. The resulting supergravity partition function is given, as expected, by the exponentiation of the regularized Eu- clidean action I evaluated at the BH solution. In this paper we call “Euclidean zero-temperature formalism” to the zero-temperature limit in the supergravity system that determines the Euclidean action, entropy and the chemical potentials. After some algebra we arrive to the supersymmetric quantum statistical relation (SQSR) [14] where the Euclidean action I can be rewritten as the Leg- endre transform of the entropy S with respect to the different supersymmetric chemical potentials Ibps = µi q bps − Sbps , (1.2) where qibps’s represent the conserved BH charges conjugated to the µi’s (later, we will use the notation qi ≡ {Qi, J i} and µi ≡ {φi, ωi}). As said above, these supergravity chemical potentials are closely related to the dual CFT chemical potentials. Therefore, they provide a very clear picture of the BPS BH as dual to a supersymmetric CFT in the grand canonical ensemble. This approach also defines the finite supersymmetric Euclidean action (1.2), and in fact allows to study the statistical mechanics of BPS black holes. A similar analysis can be done for extreme non-BPS systems. 1.3 Entropy functional formalism from an Euclidean perspective Sen’s entropy functional formalism is formulated only with the knowledge of the near horizon geometry. But, since it computes the BH entropy, which is a thermodynamic quantity, it should be possible to understand it starting from a traditional thermodynamical Euclidean analysis of the black hole system. In fact, the strong resemblance between equations (1.1) and (1.2) is evident. In other words, it would be strange if string theory produces two unrelated functions in the same supergravity regime that calculate the BH entropy. Looking into both definitions with more care, we find that the entropy is defined as the Legendre transform of the BH charges in the saddle point approximation of the supergravity theory. Nevertheless, in (1.1) the vacuum solution is just the near horizon geometry with conjugated potentials related to the electric fields, and f is the on-shell Lagrangian over only SD−2. Instead, in (1.2), the vacuum is the entire BH solution; the conjugated potentials are associated to gauge potentials rather than field strengths; and I is the on-shell full Euclidean action. The main goal of this paper is to understand the connection between these two approaches. 3In [10] the CFT partition function was calculated at zero coupling. Also, an index was considered to count supersymmetric states but unfortunately it turns out to be blind to the BH sector. One of the key points of our analysis relies in the natural splitting of the Euclidean action into two parts corresponding basically to: i) the near horizon part of space, and ii) the asymptotic region. Then we find that in the extremal cases (without ergoregion), the asymptotic part vanishes, and the near horizon part reduces to Sen’s function 2πf . Also, the conjugated chemical potentials found in both methods agree, due to an argument that relates differences of gauge potentials produced by variations of near-BPS parameters with variations of the potential on the radial coordinate. 1.4 Main results and structure of the paper As stated above, the main goal of this article is to provide a bridge between Sen’s entropy functional formalism and standard Euclidean analysis of the thermodynamics of a black hole system. While doing so, we also find that the supergravity conjugated potentials defined in Sen’s formalism map into chemical potentials of the dual CFT. We obtain a unifying picture where: 1)We are able to recover the entropy function of Sen from the zero temperature limit of the usual BH thermodynamics and the statistical mechanics definitions of the dual CFT theory. The supergravity and their dual CFT chemical potentials are identified with the surviving Sen’s near horizon electric and angular fields. The Euclidean action is identified with Sen’s function 2πf . 2)As a byproduct of the above analysis we have understood how to calculate the BPS chemical potentials that control the statistical properties of the BH using only the BPS regime, i.e., without needing the knowledge of the non-BPS geometry. The CFT chemical potentials are dual to the supergravity ones. Traditionally, to compute the latter we have to start with the non-BPS solution and send the temperature to zero to find the next to leading order terms in the horizon angular velocities and electric potentials expansions that give the chemical potentials. This requires the knowledge of the non-BPS geometry. Unfortunately, sometimes this is not available and we only know the BPS solution. But, from item 1) we know that the near horizon fields, that Sen computes with the single knowledge of the BPS near horizon solution, give us the supergravity chemical potentials. So now we can compute the supergravity chemical potentials of any BPS BH solution, regardless of its embedding into a family of non-BPS solutions, while still keeping the relation with the dual CFT. 3)It is known that the attractor mechanism seems to work also for non-supersymmetric but extremal BH 4. We have tested the Euclidean zero temperature formalism for many of these BH, always finding a well defined limit and agreement with Sen’s results for extremal non-BPS BH5. This is a non-trivial fact since there is no supersymmetry protecting the limit. Therefore, in general, the supergravity regime should not give the correct statistical relations. We interpret this result as another confirmation that there is a protecting mechanism for extremal non-supersymmetric BH. The plan of the paper is the following. In section 2 we review Sen’s entropy functional approach using the D1-D5-P system as an illuminating example. In the beginning of section 3 we review the main ideas and results of the Euclidean zero temperature formalism for BH in the AdS/CFT 4See [9, 25] and references there in. 5Actually at the level of two derivative theory, Euclidean T = 0 formalism is well defined only for BH with no ergoregion. For BH with ergoregion we have an ill-defined limit, that nevertheless allows to define the entropy and all chemical potentials. This is telling us that these geometries are not fully protected from string corrections. The same caveats and conclusions are also obtained using Sen’s approach, and this is related to the fact that for these BH the attractor mechanism is only partial since there is dependance on the asymptotic data [9]. framework. Then, we apply this formalism to the most general rotating D1-D5-P system. We analyze the connection between the entropy functional and Euclidean formalisms in section 4, identifying how and why both prescriptions are equivalent. In section 5 we discuss the application of the Euclidean T = 0 formalism to extreme non-BPS BH and again find agreement with Sen’s results. Section 6 is devoted to a short discussion on the results and possible future avenues to follow. In Appendix A we review the D1-D5-P BH solution in detail, including its thermodynamics. In Appendix B, we write the chemical potentials and Euclidean action for some other BH systems not considered in the main body of the text. We consider the four charged system of type IIA supergravity, and the Kerr-Newman BH. We confirm that for these BH the relation established in section 4 between the entropy functional and Euclidean formalisms holds. This agreement also extends to AdS black holes as is explicitly confirmed in the context of 5D gauged supergravity in [15]. Note: While we where proof-reading this article, the paper [16] appeared in the arXives. It contains relevant discussions and results connected to our work, regarding Sen’s approach and Wald’s method for AdS BH. 2 Entropy functional formalism revisited As we pointed out in the introduction, Sen developed a simple method – the entropy functional formalism – to compute the entropy of supersymmetric BH in supergravity [7]. Lately, this approach has been applied to rotating BH in gauged and ungauged supergravity (see, e.g., [9, 6]). Here, we will review some of the key aspects of this formalism that we will use latter. We just need to address non-rotating cases, but we will comeback to rotating attractors at the end of this section, for completeness. Sen’s entropy functional formalism assumes that: (i) we start with a Lagrangian L with gravity plus some field strengths and uncharge massless scalar fields; and (ii) due to the attractor mechanism the near horizon geometry of a D-dimensional BH is set to be of the form AdS2 ⊗SD−2. From the above input data, the general form of the near horizon BH solution is ds2 = v1 −ρ2dτ + + v2dΩ D−2 , F (i)ρτ = ei , H (a) = paǫD−2 , φs = us , (2.1) where ǫD−2 is the unit-volume form of S D−2, and (ei, pa) are respectively the electric fields and the magnetic charges of the BH. Note that (~u,~v,~e, ~p) are arbitrary constants up to now and therefore the solution is off-shell. Next, it is defined the following function f(~u,~v,~e, ~p) = −gL , (2.2) where L is the string frame Lagrangian of the theory (see, e.g., (A.19)). After minimizing f(~u,~v,~e, ~p) with respect to (~u,~v) we obtain the exact supersymmetric near horizon BH solution in terms of (~e, ~p). In fact, the field equations are reproduced by this minimization procedure. Furhermore, minimization with respect to ~e gives the electric charges ~q. Explicitly, the on-shell values of ~u,~v,~e that specify (2.1) for a given theory described by (2.2) are found through the relations, = 0 , = 0 , = qi . (2.3) Then, using Wald formalism [27], Sen derived that the entropy S of the corresponding BH is given by 2π times the Legendre transform of f , S = 2π . (2.4) Finally notice that the minimization procedure, can be taken only after S is defined. In this form S is really an entropy function of (~u,~v, ~q, ~p), that after minimization equals the BH entropy as a function of (~q, ~p) only. In the rest of this section we will discuss the above formalism in a specific theory. We consider the D1-D5-P supersymmetric solution of ten-dimensional type IIB supergravity, discussed in the previous section, as the main example (this case was first analyzed in [17], at the level of supergravity and for its higher order corrections). Our aim is to highlight the details of the application of Sen’s formalism to this solution. This will provide a solid background to compare, in section 4, Sen’s formalism with the Euclidean one developed in section 3. From Appendix A.1 we know that the supersymmetric D1-D5-P metric, the RR two-form C(2) and the dilaton Ψ are given by 6 ds2 = [−dt2 + dy2 + Q (dt− dy)2] + H1H5(dr 2 + r2dΩ23) + H1/H5 dz2i , C(2) = − dt ∧ dy − Qbps5 cos 2 θdφ ∧ dψ , e2Ψ = . (2.5) whereH1 = (1+ ), H5 = (1+ ) and (Q 1 , Q 5 , Q p ) are the D1,D5,P charges, respectively. Then, it is easy to take the near horizon limit to obtain, ds2 = −ρ2dτ2 + dρ dz + dz2i , F(3) = dρ ∧ dτ ∧ dz + 2Qbps5 ǫ3 , e , (2.6) where we used t , ρ = r2 , z = y − t . (2.7) 6This is the string frame version of (A.4), and (A.7) and (A.8) with a1 = a2 = 0. Note that, alternatively, all the information encoded in the near horizon structure (2.6) could be extracted without knowing the full geometry, using Sen’s approach. Its application starts by assuming that the near horizon metric is given in terms of the unknowns (~v, ~u,~e, ~p) as follows, ds2 = v1 −ρ2dτ2 + + v2dΩ 3 + u1 (dz + e2ρdt) dz2i , F(3) = e1 dρ ∧ dτ ∧ dz + 2Q 5 ǫ3 , e 2Φ = u23 . (2.8) Having the Lagrangian (A.19) of type IIB at hand, one now follows the steps summarized in (2.2)- (2.4) to find the on-shell expressions for (~v, ~u,~e). From (2.8) one has −g̃ = u1/21 u22v1v 2 sin θ cos θ, ρτz = e1 and F θψφ = 2Q 5 sin θ cos θ. The entropy function, S(~u,~v, ~q, ~p) = 2π[q1e1 + q2e2 − f(~u,~v,~e, ~p)] is then S(~u,~v, ~q, ~p) = 2π q1e1 + q2e2 − Minimizing this entropy function with respect to ~u,~v,~e one finds the on-shell attractor values, , ~u =  , ~q = 1 , Q , p = Q One also finds that f(~q, ~p) = 0 on-shell. Plugging this information into the entropy function S(~u,~v, ~q, ~p) we get S(~q, ~p) = 2π [q1e1 + q2e2 − f ]on−shell p , (2.9) that is the well known result for this BH. It will be relevant for section 4 to stress that the above analysis can be carried on in the case where the magnetic field is replaced by its dual electric field. This electric field comes from the RR seven-form field strength F(7), Poincaré dual of the magnetic part of F(3), F(7) = r3H25 dr ∧ dt ∧ dy ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 . (2.10) In the near horizon limit, i.e., after taking the change of coordinates (2.7), F(7) reduces to F(7) = dρ ∧ dτ ∧ dz ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 . (2.11) In the next lines we want to recover this near horizon attractor value for F(7), without making use of the near horizon limit of the full geometry, i.e., using instead a Sen-like approach. In this pure electric case, we first notice that there is an extra pair of conjugated variables (e3, q3) and second, that f should be now calculated on a modified Lagrangian with the F(7) RR field strength appropriately added. This is an effective “democratic” Lagrangian supplemented by duality constraints imposed by hand 7. The motivation, limitations and formulation of this effective Lagrangian are presented in detail in [18]. In this context, the string frame Lagrangian (A.19) of the D1-D5-P system takes the form, 16πG10 R̃− 4∂µΨ∂µΨ 2 · 3! F 2(3) − 2 · 7! F 2(7) , (2.12) where the magnetic part of the original F(3) field is now encoded in the F(7) contribution. The D1-branes and D5-branes source the electric F(3) and F(7) fields, respectively. In the entropy function formalism, the function f(~u,~v,~e) is obtained by evaluating action (2.12) at the horizon, i.e., by integrating along the S8 sphere. We use the near-horizon fields (2.8). So, the metric determinant is −g̃ = u1/21 u22v1v 2 sin θ cos θ, F ρτz = e1 and F ρτzz1···z4 = e3. The entropy function, S(~u,~v, ~q) = 2π[q1e1 + q2e2 + q3e3 − f(~u,~v,~e)] is then S(~u,~v, ~q) = 2π q1e1 + q2e2 + q3e3 − Minimizing this entropy function with respect to ~u,~v,~e one finds the on-shell attractor values , ~u =  , ~q = 1 , Q p , Q (2.13) which are used to obtain the on-shell function: f(~q) = 1 p . Then, use of equation (2.13) yields the on-shell entropy value, S(~q) = 2π [q1e1 + q2e2 + q3e3 − f ]on−shell p , (2.14) that is, in this dual computation we indeed recover the value (2.9). As commented in the introduction, the above approach was generalized to rotating BH in ungauged and gauged supergravities [9, 6]. At the level of two derivative Lagrangian, rotating BH in ungauged supergravity have their near horizon geometry fully determined by the entropy functional only if they have no ergoregion. However, BH with ergoregion show only partial attractor mechanism, since their entropy functional has flat directions [9, 26]. In this case, minimization does not fix the value of all quantities in the near horizon geometry. There is some surviving dependance 7We should emphasize that the introduction of a RR p-form field strength with p > 5 doubles the number of degrees of freedom. To get the right equations of motion from (2.12) we must then introduce by hand duality constraints relating the lower- and higher-rank RR potentials. We ask the reader to see [18] for further details. on the asymptotic value of the scalars, although it fixes the form of entropy itself and the electric and angular fields. Generalization to gauged supergravities includes AdS BH into the discussion. The resulting picture is basically the same, where care has to be taken when evaluating f due to Chern-Simon terms in the Lagrangian (see [6] for details). In these cases, the attractor mechanism is related to a non trivial flow between fixed points at both boundaries of spacetime, the horizon AdS and the asymptotic AdS at infinity. 3 Euclidean zero-temperature formalism: BPS black holes In [12, 13] the “thermodynamics” or better “the statistical mechanics” of supersymmetric solitons in gauged supergravity was studied in detail using an extension of standard Euclidean thermodynam- ical methods to zero temperature systems. We call this approach the Euclidean zero-temperature formalism. BPS BH can be studied as dual configurations of supersymmetric ensembles at zero temperature but non-zero chemical potentials in the dual CFT. These potentials control the ex- pectation value of the conjugated conserved charges carried by the BH, like e.g., angular momenta and electric charge. In these articles, the two main ideas are: First, there is a supersymmetric field theory dual to the supergravity theory. Second, in this dual field theory the grand canonical partition function over a given supersymmetric sector can be obtained as the zero temperature limit of the general grand canonical partition function at finite temperature. This limit also fixes the values of several chemical potentials of the system. To make things more clear, recall that all supersymmetric states in a field theory saturate a BPS inequality that translates into a series of constraints between the different physical charges. For definiteness, let us consider a simple case where the BPS bound corresponds to the constraint8: E = J . Then, defining the left and right variables E± = 1 (Eν ± Jν), β± = β(1 ± Ω) the grand canonical partition function is given by Z(β,Ω) = e−(β+E++β−E−) . (3.1) At this point, it is clear that taking the limit β− → ∞ while β+ → ω (constant), gives the correct supersymmetric partition function. The above limiting procedure takes T to zero, but also scales Ω in such a way that the new supersymmetric conjugated variable ω is finite and arbitrary. Note that among all available states, only those that satisfy the BPS bound are not suppress in the sum, resulting in the supersymmetric partition function Z(ω) = e−ωJ , (3.2) where the sum is over all supersymmetric states (bps) with E = J . The above manipulations are easy to implement in more complicated supersymmetric field theories like, e.g., N = 4 SYM theory in four dimensions. What is less trivial is that amazingly it could also be implemented in the 8This type of BPS bound appears in two dimensional supersymmetric models like, e.g., the effective theory of 1/2 BPS chiral primaries of N = 4 SYM in R ⊗ S3 (see [19, 20, 21]). dual supersymmetric configurations of gauged supergravity, since it means that these extreme BPS solutions are somehow protected from higher string theory corrections. Before we apply the Euclidean zero-temperature formalism to concrete black hole systems, it is profitable to highlight its key steps. To study the statistical mechanics of supersymmetric black holes we take the off-BPS BH solution and we send T → 0. In this limiting procedure, the angular velocities and electric potentials at the horizon can be written as an expansion in powers of the temperature. More concretely one has when T → 0, β → ∞ , Ω → Ωbps − + O(β−2) , Φ → Φbps − + O(β−2) , (3.3) where β is the inverse temperature; (Ω,Φ) are the angular velocities and electric potentials at the horizon; the subscript bps stands for the values of these quantities in the on-shell BPS solution; and (ω, φ) are what we call the supersymmetric conjugated potentials, i.e., the next to leading order terms in the expansion. For all the systems studied, we find that the charges have the off-BPS expansion, E = Ebps +O , Q = Qbps +O , J = J , (3.4) where (E,Q, J) are the energy, charges and angular momenta of the BH. In supergravity, the grand canonical partition function in the saddle point approximation is related to so called quantum statistical relation (QSR) [14] I(β,Φ,Ω) = βE − ΦQ−ΩJ − S , (3.5) where S is the entropy, and (β,Φ,Ω) are interpreted as conjugated potentials to E,Q, J , respec- tively. I is the Euclidean action (evaluated on the off-BPS BH solution) that, in this ensemble, depends only on (β,Φ,Ω). It plays the role of free energy divided by the temperature. Inserting (3.3) and (3.4) into (3.5) yields I(β,Φ,Ω) = β(E bps − ΦbpsQbps − ΩbpsJbps) + φQbps + ωJbps − Sbps +O . (3.6) Here, we observe that this action is still being evaluated off-BPS. Moreover, the term multiplying β boils down to the BPS relation between the charges of the system and thus vanishes (this will become explicitly clear in the several examples we will consider). This is an important feature, since now we can finally take the β → ∞ limit yielding relation (1.2). With the present notation it reads as Ibps = φQ bps + ωJbps − Sbps . (3.7) It is important to stress that this zero temperature limiting procedure yields a finite, not diverging, supersymmetric version of QSR, or shortly SQSR. Note that if we had evaluated the Euclidean action (3.5) directly on-shell it would not be well defined, as is well-known. As a concrete realization, we picked (and will do so along the paper) the SQSR to exemplify that the T → 0 limit yields well-behaved supersymmetric relations. The reason being that this SQSR relation is the one that will provide direct contact with Sen’s entropy functional formalism, which is the main aim of our study. However, it also provides a suitable framework that extends to the study of the full statistical mechanics of supersymmetric black holes. Euclidean action and chemical potentials of BPS D1-D5-P black holes As we pointed out in the introduction, due to the attractor mechanism, BH in ungauged super- gravity have a dual CFT theory defined in the boundary of its near horizon geometry. Therefore, and in a similar way as for asymptotic AdS spacetimes, these BH should be related to statistical en- sembles in the dual CFT. As a direct consequence of this duality, we conclude that in the ungauged case there should also exist a well defined zero temperature limit in the supergravity description that yields the dual CFT chemical potentials. In what follows, we apply the Euclidean T → 0 limit to the illuminating example of five- dimensional three charged BH with two angular momenta that can be described as the D1-D5-P system of type IIB supergravity 9. This solution can also be embedded as a solution of eleven- dimensional supergravity, or as a solution of type IIA, where all these different descriptions are related by dimensional reduction and U -dualities. A detailed review of the D1-D5-P BH solution [22, 23] and its thermodynamic properties needed for our discussion can be found in Appendix A. In type IIB, the ten-dimensional system can be compactified to five dimensions on T 4 × S1 with the D5-branes wrapping the full internal space and the D1-branes and KK-momentum on the distinguished S1. The length of S1 is 2πR and the volume of T 4 is V . We will work in units such that the five-dimensional Newton constant is G5 = G10/2πRV = π/4. The ten-dimensional solution is characterized by six parameters: a mass parameter, M ; spin parameters in two orthogonal planes, (a1, a2); and three boost parameters, (δ1, δ5, δp), which fix the D1-brane, D5-brane and KK-momentum charges. The physical range of M is M ≥ 0. We assume without loss of generality that δi ≥ 0 (i = 1, 5, p), and a1 ≥ a2 ≥ 0 (The solutions with a1a2 ≤ 0 are equivalent to the a1a2 ≥ 0 ones due to the symmetries of the solution). We will use the notation ci ≡ cosh δi, si ≡ sinh δi. The BH charges are: ADM mass E, the angular momenta (Jφ, Jψ) and the gauge charges (Q1, Q5, Qp) associated with the D1-branes, D5-branes and KK momentum. In terms of the pa- rameters describing the solution they are given by [cosh(2δ1) + cosh(2δ5) + cosh(2δp)] , Jφ = −M(a2c1c5cp − a1s1s5sp) , Jψ = −M(a1c1c5cp − a2s1s5sp) , Qi = Msici , i = 1, 5, p . (3.8) Note that these quantities are invariant under interchange of the δi’s. This reflects the equivalence of the several geometries obtained by U -dualities, that also interchange the several gauge charges. Regarding the thermodynamical properties of these BH, it is convenient for future use to define the left and right temperatures, TL and TR, through the relation β = (βL + βR) (β = 1/T and 9We present this case as a main example, but include many others in the Appendix B. βL,R = 1/TL,R). Then, using this relation together with (A.6) on (A.17) yields 2πM (c1c5cp − s1s5sp) [M − (a2 − a1)2]1/2 , βR = 2πM (c1c5cp + s1s5sp) [M − (a2 + a1)2]1/2 . (3.9) The BH angular velocities Ωφ,ψ and electric potentials Φ(i) are computed in Appendix A. Here, using (A.6), we rewrite them in terms of the parameters (M, δ1, δ5, δp, a1, a2) Ωφ,ψ = −π ± a2 − a1 [M − (a2 − a1)2]1/2 a2 + a1 [M − (a2 + a1)2]1/2 , (3.10) Φ(i) = (tanh δi)c1c5cp − (coth δi)s1s5sp [M − (a2 − a1)2]1/2 (tanh δi)c1c5cp + (coth δi)s1s5sp [M − (a2 + a1)2]1/2 , (3.11) while the expression for the entropy is S = πM c1c5cp + s1s5sp [M − (a2 − a1)2]−1/2 c1c5cp − s1s5sp [M − (a2 + a1)2]−1/2 . (3.12) The BPS limit of the three charged BH is obtained by taking M → 0, δi → ∞, Jφ + Jψ → 0 while keeping Qi fixed. In this supersymmetric regime, the charges satisfy the BPS constraints Ebps = Q p , J = −Jbps . (3.13) As a first step to define the Euclidean T → 0 limit, we consider the near-BPS limit of this solution, Jφ + Jψ → 0 ; M → 0 , δ1,5 → ∞ , Q1,5 fixed ; δp finite . (3.14) That is, in the near-BPS limit we keep δp large but finite. This limit is also often called the dilute gas regime since we are neglecting the interactions between left and right movers. Note that since the three charges can be interchanged by U -dualities, it does not matter which one of the boosts we keep finite. Given this equivalence we choose to keep δp finite, without any loss of generality. Now, to take the T → 0 limit, we define the off-BPS parameter ε, that measures energy above extremality, to be such that E ≡ Ebps + ε. In terms of the solution parameters it is given by ε = Me−2δp/4. The details of the off-BPS expansion that we carry on in the sequel can be found in Appendix A.2. Here we just quote the relevant results. We can expand the left and right temperatures in terms of the off-BPS parameter ε yielding, p − (Jbpsφ )2 , βR = π . (3.15) So the BPS limit corresponds to send the temperature T → 0 by sending βR → ∞ while keeping βL finite (we are left with only left-movers). Hence, we find more appropriate to use βR as the off-BPS parameter instead of ε . These two quantities are related by the second relation of (3.15). 10Expressions (3.9)-(3.11) agree with the ones first computed in [24] upon the notation identification a1 → −l2 and a2 → −l1. We can now expand all the thermodynamic quantities in terms of this off-BPS quantity β−1R . For the angular velocities and electric potentials, the expansion yields Ωφ,ψ = Ω bps − ∓πJbps p − (Jbpsφ )2 Φ(i) = Φ p − (Jbpsφ )2 . (3.16) where the BPS angular velocities and electric potentials are bps = 0 ; Φ bps = 1 . (3.17) The expansion of the conserved charges yields E = Ebps +O , Jφ = J , Jψ = −Jbpsφ +O Q1 ≃ Qbps1 , Q5 ≃ Q 5 , Qp = Q . (3.18) Note that the BPS charges satisfy (3.13). They are written in terms of the parameters that describe the system in (A.21). Finally, the expansion of the entropy yields S = Sbps +O , with Sbps = 2π p − (J . (3.19) With the above off-BPS expansion, we are ready to define the BPS chemical potentials. Com- paring (3.16) with (3.3) we obtain, ωφ,ψ = ∓ p − (Jbpsφ )2 , φi = p − (Jbpsφ )2 (3.20) Notice that these chemical potentials only depend on the BPS conserved charges. Now that all the BPS statistical mechanics conjugated pairs and entropy are defined, we are ready to obtain the other thermodynamic functions. For example, consider the quantum statistical relation, I = βE − β i=1,5,p Φ(i)Qi − β j=φ,ψ ΩjJj − S . (3.21) After the off-BPS expansion, i.e., using (3.18), (3.19) and (3.16) it yields I = β Ebps − i=1,5,p j=φ,ψ i=1,5,p j=φ,ψ j − Sbps +O (3.22) The term in between brackets vanishes due to the BPS relations (3.13) and (3.17). Then, taking β → ∞, we are left with the supersymmetric quantum statistical relation (SQSR) for the three- charged BH, Ibps = φ1Q 1 + φ5Q 5 + φpQ p + 2ωφJ − Sbps , (3.23) where Ibps is the value of the Euclidean action in the supersymmetric limit of the D1-D5-P BH, and we used J = −Jbps and ωψ = −ωφ. Notice that Ibps corresponds to the Legendre transformation of the entropy with respect to all the BPS chemical potential and therefore should be interpreted as the BH free energy. The off-BPS expansion of the horizon angular velocities and electric potentials gives the su- pergravity chemical potentials as the next to leading order term of the expansion around the BPS solution. The motivation for this expansion analysis comes from the fact that BPS BHs can be studied as dual configurations to supersymmetric ensembles at zero temperature but non-zero chem- ical potentials in the dual CFT [12]. The supergravity conjugated potentials (3.20) are then the strong coupling dual objects to the CFT chemical potentials. The SQSR relation (3.23) will be connected to the well-known Sen’s entropy relation in the next section. 4 Euclidean zero-temperature and entropy functional formalisms In previous sections we have described two apparently unrelated procedures to obtain the entropy of supersymmetric BH that naturally contain the definitions of pairs of conjugated variables, related to the BH charges. In this section we show that both procedures produce basically the same body of final definitions, even though conceptually both approaches are rather different. That both approaches produce the same final chemical potentials and definitions can be seen in any of the examples at hand. As usual, the best way to illustrate our point is to pick a system that captures the fundamental ingredients, while avoiding features that do not play a key role and produce unnecessary distraction from the main point. In the present case, the appropriate system is the non-rotating D1-D5-P BH (later, we will discuss the rotating case). Comparing the thermodynamic relations (3.19), (3.20), and the Sen’s relations (2.13), (2.14), we can indeed confirm that all the key quantities agree in the two formalisms. Explicitly we have that φi = 2πei , Qi = qi , Ibps = 2πf . (4.1) Nevertheless, that both frameworks are equivalent is a priori not at all obvious since they have important differences. Sen’s approach relies completely on the structure of the near horizon geome- try. In particular, the entropy is constructed analyzing Wald’s prescription and Einstein equations in these spacetimes and all the analysis is carried on at the BPS bound i.e., when the solution is extremal. In contrast, the zero temperature limit approach relies on the thermodynamical prop- erties of BH and, in principle, uses the whole spacetime, not only the near horizon region. The resulting thermodynamic definitions come as a limiting behavior of non-extremal BH and have a nice straightforward interpretation in terms of the dual CFT thermodynamics. 4.1 Near-horizon and asymptotic contributions to the Euclidean action To understand why the above close relations between the two formalisms hold, let us go back to the calculation of the Euclidean action for general BH in the off-BPS regime. Inspired in ten dimensional type II supergravity, we start with the general action11 (∂Ψ)2 − 1 eαΨF 2(n) K , (4.2) where Σ is the spacetime manifold, ∂Σ the boundary of that manifold and K is the extrinsic curvature. In the BH case, once we have switched to Euclidean regime, it is necessary to com- pactify the time direction to avoid a conical singularity. This compactification defines the Hawking temperature as the inverse of the corresponding compactification radius. To evaluate the Euclidean action on the BH solution, one of the methods to obtain a finite result, i.e., to regularize and renormalize the action, consists of putting the BH in a box and subtract the action of a background vacuum solution (g0,Ψ0, F 0). This procedures also defines the “zero” of all the conserved charges. For asymptotic flat solutions we use Minkowski, while for asymptotic AdS solutions we use AdS. Once in the box, the radial coordinate is restricted to the interval (r+, rb), where r+ is the position of the horizon and rb corresponds to an arbitrary point which limits the box and that at the end is sent to infinity. Another important ingredient is the boundary conditions on the box. Basically, depending on which conditions we impose on the different fields, we will have fixed charges or fixed potentials. If we do not add any boundary term to the above action, we will be working with fixed potentials, i.e., we will work in the grand canonical ensemble [28]. The field equations are derived from a variational principle, where fields are kept constant at the boundaries. In particular, the trace the of equation that comes from the variation of the metric (for the D1-D5-P system, see equation (A.2)) implies that (∂Ψ)2 = aeαΨF 2(n) , (4.3) where a depends on the spacetime dimensions and n. Therefore, on-shell, the action reduces to12, eαΨF 2(n) + K −K0 , (4.4) where b depends on the spacetime dimensions and n. The first term is a volume integral over Σ that can easily be converted into a boundary integral over ∂Σ, once we recall that we are considering electric fields only and hence F(n) = dC(n−1). Integrating by parts we get eaΨF(n)C(n−1) + K −K0 , (4.5) where c depends on the spacetime dimensions and n. At this point, the on-shell Euclidean action is completely recasted in two surface integrals terms, evaluated at r+ and rb. Consider first the extrinsic curvature term. At rb, we get βEb, where Eb is the quasi-local energy. When rb is taken to infinity, Eb reduces to the BH energy E and we recover usual term βE. At r+, only K contributes and gives minus the area of the horizon divided by 4G, i.e., minus the Bekenstein-Hawking entropy 11For simplicity, the reasoning is done at the level of two derivative Lagrangian. Nevertheless, following Wald’s approach for higher derivative actions, we notice that the BH action can always be recast as surface integrals. Moreover, for definiteness, we anchor our discussion to type II action, but whenever needed we make comments to extend our arguments to more general theories. 12Where we have used that the action of the background vacuum solution over Σ is zero. S. Next consider the first term. Here the integral over time gives the factor β, while the integration over the other directions (of the induced metric determinant at the boundary times eaΨF(n)) gives the corresponding electric charge Q. Therefore, we get eaΨF(n)Cn−1 = −βQ C(n−1)(rb)− C(n−1)(r+) . (4.6) Then, we use the definition of the conjugated chemical potential φ as the difference of the gauge potential at infinity and at the horizon, Φ = C|∞ − C|r+ , (4.7) and hence, when rb is sent to infinity, we recover the usual term −βQΦ. As a grand total we obtain the QSR, I = βE − βΦQ− S . (4.8) Now, it is important to notice that the definition of Φ is gauge independent, and therefore we can always choose a particular gauge that simplifies the picture depending on which physical concepts we want to stress. Here, we choose the “natural gauge” adapted to the BPS limiting cases, C|∞ = Φbps, where Φbps is usually 1 in natural units and for asymptotically flat BHs. Note that in this gauge one has C|r+ = Φbps−Φ. This gauge choice is the one that makes direct contact between the Euclidean zero temperature and entropy function formalisms for reasons that will become clear after (4.12). At this point we are ready to rewrite the Euclidean action in two pieces, one evaluated in the first boundary at r = r+, and the other in the second boundary at r = ∞, eaΨF(n)C(n−1) + eaΨF(n)C(n−1) + K −K0 .(4.9) Evaluating both terms as we did before but now in the adapted gauge we get, I = β(Φbps − Φ)Q− S ︸ ︷︷ ︸ + β(E − ΦbpsQ) ︸ ︷︷ ︸ . (4.10) r = r+ r = ∞ Therefore we can always find a gauge in which the Euclidean action splits in two contributions, one at the horizon and the other in the asymptotic region. It is perfectly adapted to understand the near horizon regime. Equally interesting, this expression is also adapted to understand the supersymmetric limit. In fact, from our discussion in section 3, it is easy to see that the first term exactly reproduces the SQSR, i.e., BPS limit β(Φbps − Φ)Q− S = φQbps − Sbps . (4.11) On the other hand, the asymptotic term vanishes due to fact that Φbps = 1, and thus the lead- ing term in the expansion is nothing else than the BPS relation Ebps = Qbps characteristic of supersymmetric regimes, i.e.,13 BPS limit β(E − ΦbpsQ) = lim BPS limit β(E −Q) = 0 . (4.12) 13This discussion is strictly valid for the asymptotically flat BHs where Φbps = 1. For asymptotically AdS BHs, the normalization usually chosen in the literature yields in general Φbps 6= 1. However, in this case, the term inside brackets in (4.12) still vanishes because it is exactly the BPS constraint on the charges. This follows by construction and is explicitly confirmed for 5D gauged supergravity in [12, 15]. (Note that in the last equality we jump some steps that were already explained in detail after (3.6), and that we do not repeat here. They guarantee that this term indeed vanishes and does not give an indeterminacy of the type∞·0). We conclude that the Euclidean action of the BH at the BPS bound is given exclusively from the near horizon part of the solution. This is another way to characterize the attractor mechanism, since the physical properties of the solution are captured entirely by the near horizon geometry. From the above result, it is easy to see why, for supersymmetric cases, I is related to f . First, both are functionals of the near horizon geometry alone. Also, the time and radial integrations are trivial and only integration on the other space directions actually contribute. In fact, this is a way to understand why in the definition of f there is no integration in the AdS part of the near horizon metric. Note also that in Sen’s approach the f function is defined as the integral of the string frame Lagrangian evaluated at the near horizon geometry. Since in this geometry the dilaton is a constant, the string frame and Einstein frame Lagrangians are related by a trivial constant factor. We now discuss the effects introduced by addition of rotation. Working in a coordinate system in which the geometry is not rotating at infinity, the action can be splited as I = β(Φbps − Φ)Q+ β(Ωbps − Ω)J − S ︸ ︷︷ ︸ + β(E − ΦbpsQ−ΩbpsJ) ︸ ︷︷ ︸ . (4.13) r = r+ r = ∞ By definition, the near horizon term contains all the information on the chemical potentials (once the BPS limit is taken), BPS limit β(Φbps − Φ)Q+ β(Ωbps − Ω)J − S = φQbps + ωJbps − Sbps , (4.14) while the asymptotic term is again the BPS constraint between the several charges and thus van- ishes, BPS limit β(E − ΦbpsQ− ΩbpsJ) = 0 . (4.15) For asymptotically flat BHs one always has Ωbps = 0 and (4.15) reduces to (4.12). The horizon of flat BHs does not rotate (angular momentum comes from the Poynting vector of electromagnetic fields) and this is one way to understand why the angular momenta does not appear in their BPS constraint. On the other hand, the horizon velocity of asymptotically AdS BHs is, in general, non-vanishing, and thus the angular momenta also contributes to the BPS constraint of these BHs. 4.2 Relation between chemical potentials in the two formalisms At this point only reminds to understand the relation between the conjugated potentials in both pictures. In Sen’s approach, the information about them is contained in the electric fields of the near horizon geometry, while in the Euclidean zero temperature formalism this information is encoded in the next to leading order term in an off-BPS expansion of the full geometry. Although these definitions seem to be rather different at first sight, notice that in Sen’s approach the field strength is just the radial derivative of the potential evaluated at the horizon. In the Euclidean zero temperature case, the off-BPS expansion can be rewritten as an expansion in the radial position of the horizon ρ+. Therefore, the next to leading order term in the off-BPS expansion of the gauge potential at ρ+ is proportional to its derivative with respect to the radial position of the horizon. Hence it is proportional to the field strength at the horizon. These words can be made very precise by taking an example. Consider the D1-D5-P BH we have been working with (again we do not include rotation in the analysis to avoid unnecessary non-insightful complications). In the full geometry (2.5), where the zero temperature limit procedure is applied, we work with the t, r coordinates. Sen’s approach uses instead the near-horizon fields (2.6) or (2.8) described in terms of (τ, ρ) coordinates. The two set of coordinates are related by (2.7). Our purpose in the next lines is to understand the first relation in (4.1). For definiteness we focus on the relation φ1 = 2πe1. From (2.5), Cty = −Ms1c1/(ρ +Ms21), and one also has the relation between the gauge field written in the two coordinate systems, Cty = Cτy. In the near-horizon approach, the expression for e1 comes from the radial derivative of the potential evaluated at the BPS horizon (ρ + = 0): ∂ρCτy . (4.16) In the Euclidean zero temperature approach, the electric potential is obtained by contracting the gauge field with the timelike Killing vector ξ = ∂t yielding: Φ (1) = −Cty|ρ=ρ+ = − Cτy|ρ=ρ+ (note that ρ+ = ρ + = 0 only in the BPS case). As is clear from (3.16), our conjugated potential is defined as φ1 = − ∂Φ(1) ∂β−1R ∂β−1R ∂Φ(1) . (4.17) Note the following key relations,14 ∂Φ(1) = −∂τ Cτy|ρ=ρ+ ∂ρCτy . (4.18) From (4.16)-(4.18), one finally has ∂β−1R e1 = 2πe1 . (4.19) The last equality follows from (2.7), and from ρ+ =M = 4π R (see (3.15) and the last statement of Appendix A.2). Physically we can understand it by noting that the near-horizon coordinates are precisely the ones appropriate to find the value of the temperature, that avoids the standard conical singularity in the Euclidean near-horizon geometry. An analysis along the lines carried here for this specific case can be carried on for general cases and yield the relations (4.1) between the conjugated potentials found using the two formalisms. To summarize, we have seen that for supersymmetric BH, the Euclidean action and all the chemical potentials are defined in the near horizon geometry. The asymptotic region would con- tribute only in off-BPS cases. We have also shown why the chemical potentials are proportional 14The presence of the factor 1/2 in the last equality is due to a subtlety that occurs when we take ∂ρ+Φ (and thus before sending ρ+ → ρbps+ ). In the large δ1 regime one has Ms21 ∼ Q 1 −M/2. Using this and ρ+ = M yields, in the denominator of Φ, ρ+ +Ms 1 ∼ Qbps1 + ρ+/2. This is the 1/2 that appears when we further take the ρ+ derivative. Note that this factor does not appear in the last derivative of (4.18), ∂ρCτy, because here we take the radial derivative evaluated on the on-shell solution ρ = 0. to the electric fields in the near horizon region, and ultimately, we have understood, from the BH thermodynamics, the emergence of Sen’s entropy function as the extremal limit of the quantum statistical relation or SQSR. As a bonus, we can now extend the statistical mechanics analysis like the SQSR to BPS solutions with no off-BPS known extension, because we have learned how to calculate the relevant chemical potentials directly in the BPS regime with no need of the limiting procedure. 5 Extremal (non-BPS) black holes So far we have seen that two completely different procedures, namely the Euclidean zero tempera- ture formalism and Sen’s entropy formalism allow to compute the entropy and conjugated chemical potentials of supersymmetric BHs. This is not an accident as proved in the previous section. Now, as is well-known, Sen’s approach also allows to find the attractor values of non-BPS extreme BHs [9, 26]. So a question that naturally raises is if whether or not the Euclidean zero temperature approach is also able to deal successfully with these type of solutions. In this section we address this issue. It is straightforward to conclude that the Euclidean formalism indeed allows to find the chem- ical potentials of non-BPS extreme configurations. This follows from an analysis similar to the derivation presented in section 4, but this time slightly modified to account for the fact that the extreme BH is not BPS. Choosing the gauge C|∞ = Φext (and thus C|r+ = Φext−Φ), the extreme analogue of (4.14) is I = β(Φext − Φ)Q+ β(Ωext − Ω)J − S ︸ ︷︷ ︸ + β(E − ΦextQ− ΩextJ) ︸ ︷︷ ︸ . (5.1) r = r+ r = ∞ where the first term boils down to the extreme counterpart of (4.14), ext. limit β(Φext − Φ)Q+ β(Ωext − Ω)J − S = φQext + ωJext − Sext , (5.2) containing all the information on the chemical potentials. On the other hand, for non-BPS extreme solutions, we find that the asymptotic term in (5.1), ext. limit β(E − ΦextQ− ΩextJ) , (5.3) in general does not vanish, as oppose to its BPS cousin. However, we find the following important feature, at least in the cases we studied: i) the cases where (5.3) does not vanish correspond to extreme rotating solutions that have in common the presence of an ergoregion; (ii) rotating extreme solutions without ergosphere and non-rotating extreme solutions have vanishing (5.3). This occurs at least on the three-charged, four-charged and Kerr-Newman systems. In the cases where it vanishes we again have that the Euclidean action of the BH at the extreme bound is given exclusively from the near horizon part of the solution. The physical properties of the solution are captured entirely by the near horizon geometry, which makes the attractor mechanism manifest 15. 15This discussion is at the level of two derivative Lagrangian. If corrections are added, we expect that the asymptotic part vanishes producing a finite result, also for extreme BH with ergoregion. In the above extremal non-BPS cases, we can explicitly verify that the two formalisms indeed yield the same results. For this exercise and as an example, we will discuss below two extreme three-charged BH (whose BPS cousin was studied in the previous sections). To emphasize that the relation between the Euclidean and Sen’s formalism is universal and not restricted to the three- charged system, in Appendices B.1 and B.2, we further extend the exercise to three other non-trivial extreme solutions whose properties have been studied within Sen’s formalism. 5.1 Extreme three-charged black hole with ergoregion In the D1-D5-P solution described by (A.4)-(A.8) we can take, instead of the BPS limit described in section 3, a different limit that yields an extreme (but not BPS) BH with an ergoregion. This is a case in which the system shows only partial attractor mechanism. Concretely, we take the near-extreme limit M → (a1 + a2)2 + ε , ε≪ 1 . (5.4) When the off-extreme parameter ε vanishes, the temperature indeed vanishes since βR → ∞ in (3.9). The off-extreme expansion of the conserved charges (3.8) around the corresponding extreme values (obtained by replacing M by (a1 + a2) 2 in (3.8)) is straightforward, and the expansion of the thermodynamic quantities (3.9)-(3.12) yields βL = π (c1c5cp − s1s5sp) (a1 + a2) +O (ε) , βR = 2π(a1 + a2)2 (c1c5cp + s1s5sp) S = Sext +O , Ωφ,ψ = Ω ext − 2ωφ,ψ Φ(i) = Φ ext − , i = 1, 5, p , (5.5) where the extreme values satisfy Sext = 2π a1a2(a1 + a2) 2 (c1c5cp + s1s5sp) , ext = Ω ext = − [(a1 + a2)(c1c5cp + s1s5sp)] ext = (tanh δi)c1c5cp + (coth δi)s1s5sp c1c5cp + s1s5sp , i = 1, 5, p , (5.6) and the conjugated potentials are ωφ,ψ = − a1 + a2√ c1c5cp − s1s5sp c1c5cp + s1s5sp ± (a1 − a2) , (5.7) φi = − [tanh δ1 tanh δ5 tanh δp − coth δ1 coth δ5 coth δp]−1 , i = 1, 5, p . These expressions for the potentials could be rewritten only in terms of the conserved charges as expected by the attractor mechanism. We avoid doing it because the expressions are long and non-insightful. The QSR for this system is I = β Eext − i=1,5,p i − Ω ext − Ω i=1,5,p i + ωφJ ext + ωψJ ext − Sext +O . (5.8) In the supersymmetric system the analogue of the term in between brackets vanishes due to the BPS constraint on the conserved charges. But, in general, for non-BPS extreme BHs it does not vanish (see discussion associated with (5.3)). In the present case the factor in between brackets is (a1+a2) c1c5cp−s1s5sp c1c5cp+s1s5sp . Note that this quantity vanishes when rotation is absent (a1 = a2 = 0). When it is present, the solution has an ergoregion and the non-vanishing contribution is associated with its existence. Notice that in this case the Euclidean action is not well-defined but, nevertheless, the chemical potentials (5.8) take finite values and are physically relevant. 5.2 Extreme three-charged black hole without ergoregion The metric of the D1-D5-P system is also a solution of type I supergravity. A fundamental difference between type IIB and type I theories is that the later theory has half of the supersymmetries of type IIB. This feature implies that in type I, if we reverse the sign of the momentum in the BPS D1-D5-P black hole, we get a distinct solution that is extreme but non-BPS. We study this solution of type I in this subsection, as the main example of an extreme non-BPS solution without ergoregion where attractor mechanism is fully manifest. The near-extreme limit we now consider is similar to the near-BPS limit (3.14) in which we send the boosts to infinity; the difference being that now we take one of the boosts to be negative (again, by U -dualities it does not matter which one). The reason why these two limits are indeed different and, in particular, why one of them yields a BPS BH and the other not is the following [25]. The three-charged BH describes, in the supergravity approximation and after dualities, the F1-NS5-P system that is a configuration of heterotic string theory compactified on T 4 × S1. We can describe this system as an effective fundamental string with winding number n1n5 (where n1, n5 are the numbers of F1 and NS5 constituents), and with momentum excitations traveling along it. Now, heterotic string theory is chiral. Hence, the direction of the momentum along the fundamental string sets if the solution is supersymmetric or not. In our conventions, the supersymmetric configuration F1-NS5-P is the one with no right-movers. So, in the supergravity approximation, the BPS BH that describes this system is obtained by taking δp → +∞. But we can also consider the heterotic string configuration with only right-movers. Due to the chirality property, this F1-NS5-P̄ configuration is then not supersymmetric. And the corresponding supergravity solution obtained by taking δp → −∞ is not a BPS BH. Note that this solution is however extreme, i.e., it has zero temperature. The reason being that there are no left-movers to collide with the right-movers and generate the closed string emission that describes the Hawking radiation. So we take the near-extreme limit (δ1,5 > 0; δp < 0, Qp < 0) Jφ − Jψ → 0 ; M → 0 , δ1,5 → ∞ , Q1,5 fixed ; δp < 0 finite . (5.10) The conserved charges of the non-extreme three-charged BH are listed in (3.8), and the temperature, entropy, and angular velocities and potentials at the horizon are given in (3.9)-(3.11). The charges in the extreme solution satisfy the constraint Eext = Qext1 +Q 5 −Qextp , Jextψ = Jextφ , (5.11) where we used Qextp = −Me−2δp/4. The off-extreme parameter, ε = Me2δp/4, measures energy above extremality and is such that E ≡ Ebps + ε. The expansion of the left and right temperatures in terms of the off-extreme parameter ε yields, βL = π Qext1 Q , βR = πQ −Qext1 Qext5 Qextp − (Jextφ )2 ]−1/2 . (5.12) The extreme limit corresponds to send the temperature T → 0 by sending βL → ∞ while keeping βR finite. In this limit there are no left-movers, only right-movers. The first relation in (5.12) defines ε in terms of βL. The expansion for the relevant thermodynamic quantities is S = Sext +O , Ωφ,ψ = Ω ext − 2ωφ,ψ Φ(i) = Φ ext − , i = 1, 5, p , (5.13) where Sext = 2π −Qext1 Qext5 Qextp − (Jextφ )2 ext = 0 , Φ (1,5) ext = 1 , Φ ext = −1 . (5.14) The conjugated potentials are ωφ,ψ = − πJextφ −Qext1 Qext5 Qextp − (Jextφ )2 φi = − πQext1 Q −Qext1 Qext5 Qextp − (Jextφ )2 , i = 1, 5, p . (5.15) 16 The rotation parameters in this limit go as a1,2 = − Jextφ −Qext1 Qext5 Qextp [1 +O (ε)] . (5.9) For comparison, in the BPS limit a1,2 go instead as (A.24). Although this is a non-BPS solution, it satisfies the extremal constraint (5.11) that is linear in the charges. This, together with (5.14), has the consequence that (5.3) applied to this system vanishes, and the QSR for this system simplifies to Iext = i=1,5,p i + 2ωφJ φ − Sext , (5.16) where we used Jextψ = J φ and ωψ = ωφ. So, contrarily to the example of the previous subsection, where the system showed only partial attractor mechanism due to the existence of an ergoregion, in the present system the ergoregion is absent and the attractor mechanism is fully manifest, even though the extreme solution is not 6 Discussion First of all, we would like to stress again the logic behind our approach: zero temperature limits to reach extremal configurations are naturally defined in statistical analysis of quantum field theories. The AdS/CFT correspondence then requires that there has to be a dual analysis for strings in AdS. Supergravity is just the tree level part of the above theory, and thus we do not expect in general a well defined zero temperature limit at this level. Here, by well defined we mean a limit that generates a finite Euclidean action when T → 0. Nevertheless, we have found extremal BH that seem to be protected, and therefore have a well defined zero temperature limit. In some of these cases, the protection is based on supersymmetric arguments but, in other cases, we just have extremal non-BPS BH where in fact it is not well understood why supergravity is giving the correct answer. In this article we have applied the Euclidean zero temperature formalism to supergravity so- lutions where Sen’s formalism is well understood. In doing so, we have shown that this method agrees with Sen’s entropy formalism, producing the same statistical mechanics functions like the entropy and the chemical potentials. On the top of this, the Euclidean zero temperature formalism has the key advantage of connecting the entropy functional with the statistical mechanics of the dual CFT and with the more canonical BH thermodynamics. More concretely, due to the attractor mechanism, we found that the Euclidean action is itself given by the near horizon geometry alone, and therefore can be connected to Sen’s approach to calculate the entropy. We showed how to relate all the different definitions in both approaches and why they match. In particular, we are able to understand the CFT dual of Sen’s approach, using the established map for the corresponding quantum statistical relation. For example, Sen’s function f (evaluated on-shell) is nothing more than the BPS limit of the Euclidean action and therefore is related to the dual CFT partition function. The above relation is relevant for the OSV conjecture [3], since now Ibps or f naturally takes the place of free energy of the supersymmetric We also worked out the extension to extremal but non-supersymmetric BH. Here, since we are dealing with two derivative Lagrangians, we divide BH in two groups: those with ergoregion and those without it. In all the cases with no ergoregion we have checked, the zero temperature limit produces a well defined QSR at extremality, where all the chemical potentials, entropy and the Euclidean action are related to Sen’s approach. This is not a triviality, since here there is no supersymmetry to protect these tree-level results. This resuly seems to imply a protection mechanism in the extremal case, as suggested in [25]. In the other case of extremal BH with ergoregions, we found an ill-defined limit, where the asymptotic contribution to the Euclidean action diverges. Nevertheless, the near horizon contri- bution is well behaved and produces the correct entropy and chemical potentials. These results are in agrement with Sen’s approach since these geometries are not fully attracted. Therefore they depend also on asymptotic values of the moduli. We interpret this result as a confirmation that these geometries do receive corrections from string theory that in turn will modify the asymptotic region, and thus asymptotic charges like the energy. In fact, in [29] rotating BH of this sort were studied finding that for the ergoregion branch, the entropy, but not the energy, could be matched with the microscopic CFT. We would like to point out that although we worked with standard low-energy supergravity, the inclusion of higher derivative terms should not spoil the results. In the Euclidean approach, one now has to compute the Euclidean action with the modified Lagrangian and define the entropy as its Legendre transform with respect to the BPS chemical potentials. This should give the same entropy as defined by Wald (see [30]). The zero temperature limit analysis of supersymmetric CFT ensembles motivated the corre- sponding analysis in the dual supergravity system. In this paper our main goal was to make direct contact between this formalism and Sen’s entropy function approach. The Euclidean zero tem- perature formalism further allows to scan the phase structure of BH. A paradigm on the useful information that this formalism allows to find about the CFT living on the boundary of a BH geometry can be found in [12, 13]. It would be interesting to make a similar application, this time to study the CFT of the BH systems discussed in this paper. Acknowledgments We acknowledge Roberto Emparan and Pau Figueras for a critical reading of the manuscript. This work was partially funded by the Ministerio de Educacion y Ciencia under grant FPA2005- 02211 and by Fundacão para a Ciência e Tecnologia through project PTDC/FIS/64175/2006. OD acknowledges financial support provided by the European Community through the Intra-European Marie Curie contract MEIF-CT-2006-038924, and CENTRA for hospitality while part of this work was done. Appendices A Three-charged black hole: solution and thermodynamics A.1 The D1-D5-P black hole In this section we describe the D1-D5-P BH and its thermodynamic properties that are used in sections 3-5. The most general solution with arbitrary charges was originally constructed in [22] (see also [23]). This solution generalizes the case with equal D1 and D5 charges found previously in [32] and whose BPS limit yields the BMPV BH [33]. Here we follow the notation of [23, 31]. This three-charged BH is a solution of type IIB supergravity. The only IIB fields that are turned on are the graviton gµν , the dilaton Ψ, and the RR 2-form C ≡ C(2). For the field strength one has simply F(3) = dC(2) since the RR field C(0) and the NSNS field H(3) are absent. The type IIB action, in the Einstein frame, reduces in these conditions to 16πG10 µΨ− 1 eΨF 2(3) , (A.1) where g is the determinant of the Einstein metric. The field equations that follow from variation of action (A.1) are Rµν − gµνR− ∂µΨ∂νΨ− gµν∂σΨ∂ 3FµαβF = 0 , −g gµν∂νΨ eΨF 23 = 0 , −g eΨFµαβ = 0 . (A.2) Contraction of the graviton field equation yields for the Ricci scalar, eΨF 23 . (A.3) The graviton in the Einstein frame is (the relation between the parameters describing the solution and the conserved charges is displayed in (3.8)) ds2 = − (dt2 − dy2) + (spdy − cpdt)2 (A.4) r2dr2 (r2 + a21)(r 2 + a22)−Mr2 + dθ2 H̃1H̃5 − (a22 − a21)(H̃1 + H̃5 − f) cos2 θ cos2 θ dψ2 H̃1H̃5 + (a 2 − a21)(H̃1 + H̃5 − f) sin2 θ sin2 θ dφ2 + a1 cos 2 θdψ + a2 sin 2 θdφ 2M cos2 θ (a1c1c5cp − a2s1s5sp)dt+ (a2s1s5cp − a1c1c5sp)dy 2M sin2 θ (a2c1c5cp − a1s1s5sp)dt+ (a1s1s5cp − a2c1c5sp)dy dφ+ H̃ dz2j , where y is the coordinate on S1, and zj ’s (j = 1, · · · , 4) are the coordinates on the torus T 4. We use the notation ci ≡ cosh δi, si ≡ sinh δi, and f(r) = r2 + a21 sin 2 θ + a22 cos 2 θ , H̃i(r) = f(r) +Ms i , with i = 1, 5 , g(r) = (r2 + a21)(r 2 + a22)−Mr2 . (A.5) The roots of g(r), r+ and r−, are given by r2± = (M − a21 − a22)± (M − a21 − a22)2 − 4a21a22 , (A.6) The system describes a regular BH17 when r2+ > 0, i.e., for M ≥ (a1 + a2)2. The ten-dimensional determinant in the Einstein frame is −g = r sin θ cos θH̃1/41 H̃ 5 . The dilaton Ψ and 2-form RR gauge potential C which support the D1-D5-P configuration are e2Ψ = , (A.7) C(2) = cos2 θ (atψdt+ ayψdy) ∧ dψ + sin2 θ (atφdt+ ayφdy) ∧ dφ −s1c1dt ∧ dy − s5c5(r2 + a22 +Ms21) cos2 θ dψ ∧ dφ , (A.8) where we defined atφ = a1c1s5cp − a2s1c5sp , atψ = a2c1s5cp − a1s1c5sp , ayφ = a2s1c5cp − a1c1s5sp , ayψ = a1s1c5cp − a2c1s5sp . (A.9) By electric-magnetic duality18, −gFµ1µ2µ3 ǫµ1µ2µ3ν1···ν7F ν1···ν7 , (A.10) our configuration can be equivalently described either by the 2-form C(2) in (A.8) or by the 6-form C(6) that follows from (A.10). Using this equivalence, we rewrite (A.8) as C(2) = − s1c1dt+ ayφ sin 2 θdφ+ ayψ cos 2 θdψ ∧ dy , C(6) = − s5c5dt+ atψ sin 2 θdφ+ atφ cos 2 θdψ ∧ dy ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 . (A.11) The advantage of (A.11) is that we clearly identify the C(2) gauge potential sourced by the D1- brane charges and the C(6) field sourced by the D5-brane charges. Thus, this expression will be appropriate to find the electric potentials associated with the two type of D-branes. Note that all the C µν components contain the y-coordinate that parametrizes the S 1, while all the C µναβγσ components contain the y-coordinate and the zj ’s coordinates that parametrize the torus T 4. This reflects the fact that D1-branes wrap S1 and the D5-branes wrap the full internal space T 4 × S1. 17For r2+ < 0, i.e., M ≤ (a1 − a2)2 the system can describe a smooth soliton without horizon [31, 34]. We will not discuss this solution. 18We use the convention ǫtrθφψyz1z2z3z4 = 1, and the relation (A.10) is valid in the Einstein frame. The general procedure to compute angular velocities when the geometry has several momenta can be found in [35]. Applied to our case, the angular velocities at the horizon along the φ-plane, Ωφ, the ψ-plane, Ωψ, and the velocity along y, Φ(p) are19 gty (gyφgψψ − gyψgφψ) + gtφ g2yψ − gyygψψ + gtψ (gyygφψ − gyφgyψ) gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy gty (gyψgφφ − gyφgφψ) + gtφ (gyygφψ − gyφgyψ) + gtψ g2yφ − gyygφφ gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy Φ(p) = g2φψ − gφφgψψ + gtφ (gyφgψψ − gyψgφψ) + gtψ (gyψgφφ − gyφgφψ) gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy (A.12) which yields Ωφ = − r2+ + a r2+c1c5cp + a1a2s1s5sp Ωψ = − r2+ + a r2+c1c5cp + a1a2s1s5sp Φ(p) = r2+c1c5sp + a1a2s1s5cp r2+c1c5cp + a1a2s1s5sp . (A.13) The horizon angular velocities are constant and, in particular have no angular dependence, as required by Carter’s rigidity property of Killing horizons20. The electric potentials at the horizon associated with the Q1 and Q5 gauge charges are computed using Φ(i) = −Cµ{x(i)}ξ Ct{x(i)} + Cφ{x(i)}Ω φ +Cψ{x(i)}Ω , i = 1, 5 , (A.14) where, ξ = ∂t +Ω φ∂φ +Ω ψ∂ψ , (A.15) is the null Killing vector generator of the horizon (Ωφ,ψ are the horizon angular velocities). We use the notation {x(1)} ≡ y, the coordinate of S1 wrapped by D1-branes, and {x(5)} ≡ yz1z2z3z4, 19We identify Ωy ≡ Φ(p) because the KK momentum plays effectively the role of a gauge charge with associated electric potential. 20Note that the angular velocities can be more easily computed using the standard formulas valid for solutions rotating along a single axis, as long as we evaluate them at a specific θ coordinate. More concretely, an inspection of (A.12) concludes that the following relations are valid and provide the quickest computation of the corresponding quantities: φ ≡ gtφ r=r+, θ=0 ψ ≡ gtψ r=r+, θ=π/2 (p) ≡ gty r=r+, θ=0 the coordinates of S1 × T 4 wrapped by D5-branes. Using the C gauge potential written in (A.11) yields Φ(i) = r2+(tanh δi)c1c5cp + a1a2(coth δi)s1s5sp r2+c1c5cp + a1a2s1s5sp , i = 1, 5 . (A.16) The temperature of the BH is T = κh/(2π) where the surface gravity of the horizon is κ (∇µξν)(∇µξν)|r=r+ , and ξ µ is the Killing vector horizon generator defined in (A.15). The inverse temperature β = 1/T is then r2+ + a r2+ + a r4+ − a21a22 r2+c1c5cp + a1a2s1s5sp . (A.17) The entropy S is just horizon area (in the Einstein frame) divided by 4G10, r2+ + a r2+ + a r2+c1c5cp + a1a2s1s5sp . (A.18) To conclude this section, note that action (A.1) can be written in the string frame through the Weyl rescaling of the metric, g̃AB = e Ψ/2gAB , yielding 16πG10 R̃− 4∂µΨ∂µΨ 2 · 3! F 2(3) . (A.19) A.2 The near-BPS limit of the D1-D5-P black hole In this appendix we present the detailed computation of the near-BPS limit of the D1-D5-P BH, and of the off-BPS construction that takes (3.9)-(3.11) into (3.15)-(3.23). Using the trignometric properties eδi + e−δi , si = eδi − e−δi , (A.20) the gauge charges and ADM mass (3.8) are, in the near-BPS regime (3.14), Me2δp Me−2δp ≡ Qbpsp − ε Me2δ1 ≡ Qbps1 , Q5 ≃ Me2δ5 ≡ Qbps5 , + ε = Ebps + ε , (A.21) where the BPS constraint (3.13) was used. We can interpret the quantity Me as the number of left-movers, and ε = Me as the number of right-movers (in the KK momentum sector). The BPS configuration, ε = 0, is the one with no right-movers. In the D1 and D5 sectors there are only left-movers since δ1,5 → ∞. From the last relation in (A.21), we conclude that ε is also an off-BPS parameter that measures energy above extremality. We can also rewrite ε = Me M = 4 ǫ, an expression that will be useful below. The near-BPS limit (3.14) is completed with the angular momenta condition. It can be under- stood as follows. Inversion of (3.8) yields a1 = − Jψc1c5cp + Jφs1s5sp p − s21s25s2p , a2 = − Jφc1c5cp + Jψs1s5sp p − s21s25s2p . (A.22) In the near-BPS limit (3.14) one has c21,5 ≃ , s21,5 ≃ , c2p = , and s2p = . The expansion of a1,2 in the small M regime then gives a1,2 = − (Jφ + Jψ) (Jφ − Jψ) , (A.23) where we have defined η ≡ Qbps1 Q ε, and γ ≡ Qbps1 Q 5 ε. Now, one must take appropriate limits of a1 and a2 such that they keep finite and the angular momenta is kept fixed. But in (A.23) one sees that, for non-vanishing charges Q i 6= 0 (i = 1, 5, p), a1,2 diverge as 1/ M when we takeM → 0. We can avoid this divergence by imposing that Jφ + Jψ → 0 in the near-BPS limit. Note that as a consequence, in the limit ε→ 0, the BPS solution must have angular momenta satisfying the relation (3.13)21. Under this condition, we can now take a small ǫ expansion in (A.23) and get a1,2 = ± [1 +O (ε)] . (A.24) Use of (A.20) and (A.24) in (3.9), (3.12) and (3.11) yields straightforwardly the near-BPS expansions for the temperature, (3.15), for the entropy, (3.19), and for the angular velocities, (3.16), respectively. The off-BPS expansion of the electric potential Φ(p) leading to (3.16) is straightforward. How- ever, the expansion of the D1 and D5 electric potentials is more subtle. Indeed, if in (3.11) we do the most natural step, (tanh δ1)c1c5cp − (coth δ1)s1s5sp = s1c5cp − c1s5sp we just catch the BPS value but not the next order term of the expansion. To capture the next order off-BPS contribu- tion one has to introduce the parameter M that measures the energy above extremality. This is consistently done with the following step: (tanh δ1)c1c5cp = c1c5cp Ms1c1 M(1+s21) (and similarly for the term proportional to coth δ1). Then, use of Ms 1 ≃ Q 1 −M/2 and M = 4 ǫ allows to finally write (tanh δ1)c1c5cp ≃ c1c5cp(1 − q ε), where q is a ratio of BPS charges. The expansion (3.16) for Φ(1), Φ(5) now follows naturally. B Explicit agreement for other black hole systems In this Appendix we will perform the Euclidean zero temperature limit and study the statistical mechanics of some BHs that have not been considered in the main body of the text. The main 21Alternatively, note that we could relax this condition in the off-BPS regime. That is we could instead fix Jφ and let Jψ arbitrary “during” the near-BPS approach, as long as in the BPS limit one ended with Jφ + Jψ = 0. Our final result is independent of the particular off-BPS path choosen. motivation to do this is two-folded. First, we explicitly verify that the relation between the Eu- clidean zero temperature and Sen’s entropy formalisms is indeed general and not restricted to the three-charged BH studied in the main body of the text. Second, we get a list of conjugated chemical potentials for several BH systems. With these at hand we can also study the thermodynamics of the dual CFT. We consider some relevant asymptotically flat systems that have been discussed within Sen’s formalism context in [9], namely: the four-charged BH (subsection B.1), and the Kerr- Newman BH (subsection B.2). The agreement between the two formalisms is also confirmed for black holes of gauged supergravity elsewhere [12, 15]. B.1 Four-charged black holes We study the statistical properties at zero temperature of the asymptotically flat four-charged BH in four dimensions (4D). This system has three distinct extreme cases: the BPS BH (studied in subsection B.1.1), the ergo-free branch family of BHs (subsection B.1.2), and the ergo-branch family (subsection B.1.3). These last two are extreme but not BPS BHs and we are following the nomenclature of [9]. The most general non-extremal rotating four-charged BH was first found in [37] as a solution of heterotic string theory compactified on a six-torus. The four gauge fields of the solution were however not explicitly given. This BH is also a solution of N = 2 supergravity coupled to three vector multiplets, which in turn can be consistently embedded in N = 8 maximal supergravity [37, 36, 38]. As first observed for the static non-extreme case [36], these theories can also be obtained from compactification of type II supergravity on T 4 × S1 × S̃1. Therefore, from the 10D viewpoint these BHs have a D-brane interpretation, e.g., they describe the D2-D6-NS5-P solution of type IIA supergravity or the D1-D5-KK-P solution of type IIB supergravity (or any dual system to these obtained by U -dualities). Take N = 2 supergravity coupled to three vector multiplets. The field content of the theory is: the graviton gµν , four gauge fields A(1)1,2 , Â1,2(1) , three dilatons ϕi and three axions χi (with 1 ≤ i ≤ 3). The full solution can be explicitly found in [38]. Compared with [38], we use the parameters µ ≡ 4m and l ≡ 4a that avoid nasty factors of 4 in the thermodynamic quantities. The horizons of the solution are at µ2 − l2 , (B.1) and thus the system has regular horizons when µ ≥ |l|. When l = 0 we recover the static solutions found in [36]. The conserved mass E, angular momentum J , and gauge charges Qi’s of the BH are (we use G4 ≡ 1/8 for this system) cosh(2δi) , Jφ = µl (c1c2c3c4 − s1s2s3s4) , Qi = µsici , i = 1, 2, 3, 4 , (B.2) which are invariant under interchange of the δi’s, as expected from the U -duality relations. The left and right movers inverse temperatures, the entropy, electric potentials and angular velocity are [39], βL = 2πµ (c1c2c3c4 − s1s2s3s4) , βR = µ2 − l2 (c1c2c3c4 + s1s2s3s4) , S = πµ2 (c1c2c3c4 + s1s2s3s4) + πµ µ2 − l2 (c1c2c3c4 − s1s2s3s4) , Φ(i) = [(tanh δi)c1c2c3c4 − (coth δi)s1s2s3s4] µ2 − l2 [(tanh δi)c1c2c3c4 + (coth δi)s1s2s3s4] , i = 1, 2, 3, 4 , µ2 − l2 . (B.3) B.1.1 BPS black hole The BPS limit of the four charged BH is obtained by taking µ→ 0, δi → ∞, while keeping Qi fixed (i = 1, 2, 3, 4), and l → 0 at the same rate as µ, i.e., l/µ → 1. As a consequence J → 0 and the BPS four-charged BH is non-rotating22. Therefore, the BPS charges satisfy the BPS constraints, Ebps = Q 4 , J bps = 0 , (B.4) where Q i = µe 2δi/4. To study the thermodynamics near the T = 0 BPS solution we work in the near-BPS limit. We take µ→ 0 , δ1,2,3 → ∞ , Q1,2,3 fixed ; δ4 finite; l → 0 (l/µ → 1) . (B.5) Note that we take the four boosts to be positive and we choose to keep δ4 finite, without any loss of generality (due to U -dualities). Define the off-BPS parameter above extremality ε, to be ε = µe−2δ4/4 so that E ≡ Ebps + ε. The procedure yielding the off-BPS expansion of the several thermodynamic quantities is quite similar to the one done in the three-charged BH (see Appendix A.2). So we just quote the relevant results. Expanding the left and right temperatures in terms ε yields, βL = π , βR = π . (B.6) The BPS limit corresponds to send βR → ∞, and we now can use βR as the off-BPS parameter, instead of ε. The expansion in βR of the conserved charges is E = Ebps +O , J = Q1,2,3 ≃ Qbps1,2,3 , Q4 = Q . (B.7) 22The reason being that the roots that define the horizon are (B.1), and thus µ ≥ |l| must hold to have a regular solution. The remaining thermodynamic quantities have the expansion, S = Sbps +O , Ω = Φ(i) = Φ bps − , i = 1, 2, 3, 4 , (B.8) where Sbps = 2π = 1 , φi = , i = 1, 2, 3, 4 . (B.9) The last relation gives the key quantities, namely the conjugated potentials φi’s of the solution that have an important role in the dual CFT. The expressions of the BPS entropy Sbps, and conjugated potentials φi’s agree with the corresponding quantities computed in [9] using Sen’s entropy function formalism23. The SQSR for the four-charge BH is then Ibps = φ1Q 1 + φ2Q 2 + φ3Q 3 + φ4Q 4 − Sbps . (B.10) B.1.2 Extreme (non-BPS) black hole: ergo-free solution In the four-charged system we can take an extremal limit that yields a rotating BH without ergo- sphere. For this reason, this BH was dubbed ergo-free solution in [9]. This limit is similar to the BPS regime token in the previous Appendix B.1.1 in which we send the boosts to infinity; the difference being that we take an odd number (one, for definiteness, but it could as well be three) of boosts to be negative. As explained in a similar context in section 5, this limit yields an extreme, but not BPS, BH. Concretely, take the near-extremal limit (δ1,2,3 > 0; δ4 < 0, Q4 < 0): µ→ 0 , δ1,2,3 → ∞ , Q1,2,3 fixed ; δ4 < 0 finite ; −Q1Q2Q3Q4 . (B.11) The charges in the extreme solution satisfy the constraint Eext = Qext1 +Q 3 −Qext4 , (B.12) where Q 1,2,3 = µe 2δ1,2,3/4, Qext4 = −µe−2δ4/4, and Jext is arbitrary. Using the off-extremality parameter, ε = µe2δ4/4 = π2Qext1 Q L (so the extremal limit is obtained by sending βL → ∞), we get the following expansion for the relevant thermodynamic quantities: S = Sext +O , Ω = Ωext − Φ(i) = Φ ext − , i = 1, 2, 3, 4 , (B.13) 23Once we match the notation ω ≡ 2πα and φi ≡ 2πei and we take into consideration that we use G4 ≡ 1/8, while [9] uses G4 ≡ 1/(16π)). where Sext = 2π −Qext1 Qext2 Qext3 Qext4 − (Jext)2 Ωext = 0 , Φ (1,2,3) ext = 1 , Φ ext = −1 . (B.14) The conjugated potentials are ω = − 2πJext [−Qext1 Qext2 Qext3 Qext4 − (Jext)2] φi = − πQext1 Q Qexti [−Qext1 Qext2 Qext3 Qext4 − (Jext)2] , i = 1, 2, 3, 4 . (B.15) Again, these expressions for Sext, ω and φi’s match the ones found in [9] using Sen’s entropy function formalism (see footnote 23 for normalization conventions). Although this is a non-BPS solution, it satisfies the extremal constraint (B.12) that is linear in the charges. Using in addition (B.14), we find that (5.3), applied to this system, vanishes and the QSR for this system simplifies to Iext = i + ωJ ext − Sext . (B.16) This is an example of a rotating extreme solution without ergosphere. It has a finite on-shell action. B.1.3 Extreme (non-BPS) black hole: ergo-branch solution This time we take the limit µ→ l. This yields an extreme BH with an ergosphere that was coined as ergo-branch solution in [9] (This is the four-charged counterpart of the solution studied in Section 5.1). We take the near-extreme limit µ→ l + ε , ε≪ 1 . (B.17) When the off-extreme parameter ε vanishes, the temperature indeed vanishes since βR → ∞ in (B.3). The off-extreme expansion of the conserved charges (B.2) around the corresponding ex- treme values (obtained by replacing µ by l in (B.2)) is straightforward, and the expansion of the thermodynamic quantities (B.3) yields24 βL = 2πl (c1c2c3c4 − s1s2s3s4) +O (ε) , βR = 2πl3/2 (c1c2c3c4 + s1s2s3s4) S = Sext +O , Ω = Ωext − Φ(i) = Φ ext − , i = 1, 2, 3, 4 , (B.18) 24We use the relation µ2 − l2 ≃ 2πµ2(c1c2c3c4 + s1s2s3s4)/βR where the extreme values satisfy Sext = 2π Qext1 Q 4 + (J ext)2 , Ωext = 2l −1 (c1c2c3c4 + s1s2s3s4) ext = (tanh δi)c1c2c3c4 + (coth δi)s1s2s3s4 c1c2c3c4 + s1s2s3s4 , i = 1, 2, 3, 4 , (B.19) and the conjugated potentials are 2πJext [Qext1 Q 4 + (J ext)2] Qexti s1c1s2c2s3c3s4c4 c1c2c3c4 + s1s2s3s4 , i = 1, 2, 3, 4 . (B.20) Note that in the last expression could be rewritten only in terms of the conserved charges as expected by the attractor mechanism. We do not do it here because the expression is too long. The expressions of the extremal entropy Sext, and conjugated potentials ω and φi’s agree with the corresponding quantities computed in [9] using Sen’s entropy function formalism (see footnote 23 for normalization conventions). The QSR for this system is I = β Eext − ΦextQ i − ΩextJext i + ω J ext − Sext +O (B.21) In the supersymmetric system the analogue of the first term vanishes due to the BPS constraint on the conserved charges. But, in general, for non-BPS extreme BHs it does not vanish (see also discus- sion associated with (5.3)). In the present case the factor in between brackets is − l c1c2c3c4−s1s2s3s4 c1c2c3c4+s1s2s3s4 Note that this quantity vanishes when rotation is absent. When it is present, the solution has an er- gosphere and the non-vanishing contribution seems to be associated with its existence, as discussed in section 5. B.2 Extreme Kerr-Newman black hole In this section we take the near-extreme limit of the Kerr-Newman BH with ADM mass M , ADM charge Q and ADM angular momentum J = aM that is a solution of the Einstein-Maxwell action I = 1 R− F 2 (so, we set G4 ≡ 1). In the extreme state the charges satisfy the constraint M2 = a2 + Q2, the horizons coincide, r± = M , and one also has the useful relation M2 + a2 = 2 J2 +Q4/4. Define the off-extremality parameter ε such that M = Me + ε which implies that r+ ∼Me+ ε (the subscript e stands for the on-shell extreme solution). In terms of the inverse temperature β = 2π(r2++a r+−M it is given by 2π(M2e+a 2Me β . 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B 506 (1997) 107 [arXiv:hep-th/9706071]. http://arxiv.org/abs/hep-th/9603147 http://arxiv.org/abs/hep-th/0411045 http://arxiv.org/abs/hep-th/9706071 Introduction Attractor mechanism and entropy functional formalism Zero temperature limit and chemical potentials Entropy functional formalism from an Euclidean perspective Main results and structure of the paper Entropy functional formalism revisited Euclidean zero-temperature formalism: BPS black holes Euclidean zero-temperature and entropy functional formalisms Near-horizon and asymptotic contributions to the Euclidean action Relation between chemical potentials in the two formalisms Extremal (non-BPS) black holes Extreme three-charged black hole with ergoregion Extreme three-charged black hole without ergoregion Discussion Three-charged black hole: solution and thermodynamics The D1-D5-P black hole The near-BPS limit of the D1-D5-P black hole Explicit agreement for other black hole systems Four-charged black holes BPS black hole Extreme (non-BPS) black hole: ergo-free solution Extreme (non-BPS) black hole: ergo-branch solution Extreme Kerr-Newman black hole

0704.1406

On the existence of chaotic circumferential waves in spinning disks

To appear in CHAOS, Volume 17, Issue 2, June 2007 On the existence of chaotic circumferential waves in spinning disks Arzhang Angoshtari∗ and Mir Abbas Jalali† Center of Excellence in Design, Robotics and Automation Department of Mechanical Engineering, Sharif University of Technology P.O.Box: 11365-9567, Azadi Avenue, Tehran, Iran (Dated: August 4, 2021) We use a third-order perturbation theory and Melnikov’s method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points respectively lead to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude π for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances. Keywords: Chaos, canonical perturbation theory, Melnikov’s method, spinning disks, traveling wave reversal Transversal vibration modes of hard disk drives (HDDs) are excited by the lateral aerodynamic force of the magnetic head. Previous works [1, 2] revealed that chaotic orbits are inevitable ingre- dients of phase space flows when the lateral force is large, or the disk is rotating near the critical resonant speed. For low-speed disks, however, an adiabatic invariant (a first integral) was found [2] using a first-order averaging based on canonical Lie transforms. According to the first-order the- ory, regular vibrating modes of imperfect, low- speed disks are independent of the angular veloc- ity of the disk, Ω0. In such a circumstance, the speed of circumferential waves is the natural fre- quency of the lateral mode, ω, derived from linear vibration analysis. HDDs are usually operated with angular velocities smaller than ω (safely be- low resonance). Moreover, the magnitude of the lateral force F is very small. Given these condi- tions, we show that it is impossible to continue the Lie perturbation scheme [2] up to terms of arbitrary order and remove the time variable t from the Hamiltonian. In fact, due to the spe- cial forms of nonlinearities in the dynamical equa- tions of spinning disks, one can not remove t from third-order terms. Subsequent application of a second-order Melnikov theory reveals that trans- verse homoclinic and heteroclinic points do exist for all F,Ω0 6= 0. This implies chaos, or equiva- lently, non-integrability of governing equations. I. INTRODUCTION Dynamics of continuum media, like fluids, rods, plates and shells is usually formulated as a system of partial differential equations (PDEs) for physical quantities in ∗Electronic address: arzhang˙[email protected] †Electronic address: [email protected]; URL: http://sharif.edu/~mjalali terms of the spatial coordinates x and the time t as L(u) = 0. (1) Here L is a nonlinear operator and u(x, t) is the vector of dependent variables. When the boundary conditions are somehow simple, approximate variational methods based on modal decomposition and Galerkin’s projection [3] can be used to reduce the order of governing equa- tions. These methods begin with solving an auxiliary eigenvalue problem, which is usually the variational field equation δL(u,Λ) = 0, and build some complete basis set Uk(x,Λk) for expanding u(x, t) in the spatial do- main. Here Λk is an eigenvalue, which is characterized by the vectorial index k. EachUk is called an eigenmode or a shape function. Such a basis set should preferably satisfy boundary conditions and be orthogonal. Once a complete basis set is constructed, one may sup- pose a solution of the form u(x, t) = Vk(t) ·Uk(x,Λk). (2) Substituting from (2) into (1) and taking the inner prod- L(u) ·Uk′(x)dx = 0, (3) leaves us with a system of nonlinear ordinary differen- tial equations (ODEs) for the amplitude functions Vk(t). The evolution of the reduced ODEs shows the interaction of different modes and their influence on the development of spatiotemporal patterns. In a series of papers, Raman and Mote [1, 4] used the modal decomposition method to investigate transversal oscillations of spinning disks whose deformation field is described in terms of the displacement vector (u, v, w) with u and v being the in-plane components. The most important application of the spinning disk problem is in design, fabrication and control of HDDs. The governing PDEs for the evolution of displacement components were first derived by Nowinski [5] and reformulated more re- cently by Baddour and Zu [6]. Let us define (R, φ) as http://arxiv.org/abs/0704.1406v1 mailto:[email protected] mailto:[email protected] http://sharif.edu/~mjalali the usual polar coordinates. For a rotating disk with the angular velocity Ω0, Nowinski’s theory assumes that the in-plane inertias (u, v)Ω20, 2Ω0(u,t, v,t) and (u,tt, v,tt) are ignorable againstRΩ20. This is a rough approximation for high-speed disks and one needs to use the complete set of equations as Baddour and Zu [6] suggest. Nowinski’s theory, however, has its own advantages (like the exis- tence of a stress function) that facilitate the study of the most important transversal modes. In low-speed disks, or disks with high flexural rigidity, one has Ω0 ≪ ω. Hence, it is legitimate to apply Nowinski’s governing equations in such systems. In this paper we analytically prove the existence of chaos, and therefore, non-integrability of the reduced ODEs that govern the double-mode oscillations of im- perfect spinning disks. We investigate low-speed disks subject to a lateral point force exerted by the magnetic head. The lateral force in HDDs is very small and its origin is the aerodynamic force due to air flow in the gap between the disk and the head. We show that chaotic circumferential waves dominate some zones of the phase space over the time scale t ∼ O(ǫ−3) with ǫ being a small perturbation parameter. This indicates very slow evolu- tion of random patterns, and the practical difficulties of their identification. The paper is organized as follows. In §II, we present the Hamiltonian function in terms of Deprit’s [7] Lis- sajous variables. In §III, we use a canonical perturba- tion theory [8, 9] to eliminate the fast anomaly l from the Hamiltonian. The action associated with l then be- comes an adiabatic invariant. Transversal intersections of destroyed invariant manifolds, and therefore, non- integrability of the normalized equations, is proved by a second-order Melnikov method in §IV. We present a complete classification of circumferential waves in §V and end up the paper with concluding remarks in §VI. II. PROBLEM FORMULATION Let us assume Um(R) as an orthogonal basis set that represents the disk deformation in the radial direction. The index m stands for the number of radial nodes that Um(R) has. According to Raman and Mote’s [1] treatment of imperfect disks, the following choice of the transversal displacement field w(R, φ, t) = Um(R)[x(t) cosnφ+ y(t) sinnφ], (4) reduces Nowinski’s governing equations to a system of ODEs for the amplitude functions x(t) and y(t) as ẍ+ λ2x+ ǫγ x2 + y2 x = ǫF cos(nΩ0t), (5a) ÿ + ω2y + ǫγ x2 + y2 y = ǫF sin(nΩ0t), (5b) where λ, ω and γ are constant parameters that depend on the geometry and material of the disk. ǫ is a small perturbation parameter, F is the weighted integral of the lateral point force, and Ω0 is the angular velocity of the disk. We suppose small deviations from perfect disks and write the constant parameter of (5) as λ2/ω2 = 1 + ǫη. We also define nΩ0 = ǫΩ with O(Ω) ∼ O(ω). Denoting (px, py) as the momenta conjugate to (x, y), it can be verified that equations (5) are derivable from the Hamil- tonian function p2x + p x2 + y2 + ΩP+ x2 + y2 )2 − F (x cos p+ y sin p) . (6) We have introduced the action P and its conjugate an- gle p = ǫΩt to make our equations autonomous, which is a preferred form for the application of canonical per- turbation theories. The extended phase space has now dimension six. Dynamics generated by (6) is better un- derstood after carrying out a canonical transformation (x, y, px, py) → (l, g, L,G) to the space of Lissajous vari- ables [7] so that x = s cos(g + l)− d cos(g − l), (7a) y = s sin(g + l)− d sin(g − l), (7b) px = −ω [s sin(g + l) + d sin(g − l)] , (7c) py = ω [s cos(g + l) + d cos(g − l)] , (7d) , d = , L ≥ 0, |G| ≤ L. In the space of Lissajous variables, the Hamiltonian de- fined in (6) becomes H = H0(L) + ǫH1(l, g, p, L,G, P ), (8) H0 = ωL, H1 = ΩP − F [s cos(g + l − p)− d cos(g − l − p)] (s2 + d2)− 2sd cos(2l) (s2 + d2) + s2 cos(2g + 2l)− 2sd cos(2g) + d2 cos(2g − 2l)− 2sd cos(2l) From (8) we conclude that l is the fast angle, and g and p are the slow ones. Therefore the long-term behavior of the flows generated by (8) can be analyzed by averag- ing H over l. After removing l, its corresponding action L will be a constant of motion for the flows generated by the averaged Hamiltonian 〈H〉l, and the phase space dimension reduces from 6 to 4. III. CANONICAL THIRD-ORDER AVERAGING In order to average H over l, we use the normal- ization procedure of Deprit and Elipe [9]. Denoting X ≡ (l, g, p) and Y ≡ (L,G, P ), we define a Lie transfor- mation (l, g, p, L,G, P ) → (l̄, ḡ, p̄, L̄, Ḡ, P̄ ) as X = EW (X̄), Y = EW (Ȳ ), (9) so that the Hamiltonian function in terms of the new variables, K ≡ 〈H〉l, does not depend on l̄. EW is the Lie transform generated by the function W and it is defined EW (Z̄) = Z̄ + (Z̄;W ) + ((Z̄;W );W ) (((Z̄;W );W );W ) + · · · . (10) In this equation, (f1; f2) denotes the Poisson bracket of f1 and f2 over the (l̄, ḡ, p̄, L̄, Ḡ, P̄ )-space. We expand the generating function W as W = ǫW1 + W2 + · · · , (11) and specify the averaged, target Hamiltonian K = K(ḡ, p̄, L̄, Ḡ, P̄ ) as the series [9] K = K0 + ǫK1 + K3 + · · · , (12) K0 = ωL̄, (13a) H1dl̄, (13b) [2(H1;W1) + ((H0;W1);W1)]dl̄, (13c) 3(H1;W2) + 3((H1;W1);W1) + 2((H0;W2);W1) + ((H0;W1);W2) + (((H0;W1);W1);W1) dl̄. (13d) W1 and W2 are determined through solving the following differential equations = H1(l̄, ḡ, p̄, L̄, Ḡ, P̄ )−K1(ḡ, p̄, L̄, Ḡ, P̄ ),(14a) = 2(H1;W1) + ((H0;W1);W1)−K2. (14b) By substituting from (14) into (13) and evaluating the integrals, one finds the explicit form of the new Hamilto- nian K, which has been given in Appendix A up to the third-order terms. Once l̄ is removed from the Hamiltonian, L̄ becomes an integral of motion. The slow dynamics of the system is thus governed by the flows in the (ḡ, Ḡ)-space. We introduce the slow time τ = p̄/Ω, ignore the fourth-order terms in ǫ, and obtain the following differential equations for the dynamics of (ḡ, Ḡ) = f1(ḡ, Ḡ) + ǫh1(ḡ, Ḡ, ǫ, τ), (15a) = −∂K = f2(ḡ, Ḡ) + ǫh2(ḡ, Ḡ, ǫ, τ), (15b) where f1 = d6 + d1 cos(2ḡ), f2 = e1 sin(2ḡ), h1 = d7 + d2 cos(2ḡ) + ǫ[d8 + d3 cos(2ḡ) + d4 cos(4ḡ) + d5 cos(2ḡ − 2Ωτ)], h2 = e2 sin(2ḡ) + ǫ[e3 sin(2ḡ) + e4 sin(4ḡ) + e5 sin(2ḡ − 2Ωτ)]. (16) In these equations, di (i = 1, · · · , 8) and ej (j = 1, · · · , 5) are functions of L̄ and Ḡ (see Appendix A). It is re- marked that the action P̄ appears only in K1 via the term ΩP̄ . It then disappears in the normalized equa- tions (15) after taking the partial derivatives of K with respect to ḡ and Ḡ. The partial derivative of K with respect to P̄ determines the evolution of p̄, which is in accordance with the simple linear law p̄(τ) = Ωτ + p̄(0). The dynamics of P̄ itself is governed by = −∂K = − 1 K(ḡ, Ḡ, τ). (17) One may integrate (17) to obtain P̄ (τ) once equations (15) are solved. The behavior of P̄ is thus inherited from ḡ(τ) and Ḡ(τ). IV. THE MELNIKOV FUNCTION There are few analytical methods in the literature for the detection of chaos in perturbed Hamiltonian systems [10, 11]. Melnikov’s [10] method is the most powerful technique when the governing equations take the form = f(x) + ǫh(x, ǫ, τ), x ∈ R2, (18) so that the unperturbed system dx/dτ = f(x) is in- tegrable and possesses a homoclinic (heteroclinic) orbit qh(τ) to a hyperbolic saddle point, and h(x, ǫ, τ) is T - periodic in τ . The occurrence of chaos is examined by the Melnikov function M(τ0, ǫ) = ǫM1(τ0) + ǫ 2M2(τ0) + . . . , (19) where Mk(τ0) denotes the kth-order Melnikov function. Assume that Mi(τ0) is the first nonzero term, i.e., Mk(τ0) ≡ 0 for 1 ≤ k ≤ i − 1. If Mi(τ0) has simple zeros, then, for sufficiently small ǫ, the system (18) has transverse homoclinic (heteroclinic) orbits, which imply chaos due to the Smale-Birkhoff homoclinic theorem [12]. The first-order term in (19) is determined by the classical formula M1(τ0) = f (qh (τ)) ∧ h (qh (τ) , 0, τ + τ0) dτ,(20) where the wedge operator ∧ is defined as f ∧ h = f1h2 − f2h1. Although the Hamiltonian equations (15) have a suitable form for the application of Melnikov’s method, they are autonomous up to the first-order terms in ǫ. Consequently, M1(τ0) vanishes identically for all τ0 ∈ [0, T ]. We thus need to investigate the second-order Melnikov function. For doing so, we begin with solving the unperturbed system = f1(ḡ, Ḡ), (21a) = −∂K1 = f2(ḡ, Ḡ), (21b) along homoclinic (heteroclinic) orbits. Jalali and An- goshtari [2] showed that for L̄ > ηω3/γ, equations (21) have hyperbolic stationary points at S0 ≡ (ḡ0, Ḡ0) = (−π, 0), S2 ≡ (ḡ2, Ḡ2) = (0, 0), and S4 ≡ (ḡ4, Ḡ4) = (π, 0). The implicit equation of the invariant manifolds that terminate at the saddle points are cos[2ḡh(τ)] = L̄− γ Ḡ2h(τ) L̄2 − Ḡ2h(τ) ]−1/2 For γL̄ ≥ 2ηω3, equation (22) represents a heteroclinic orbit which connects S0 to S2. For ηω 3 < γL̄ < 2ηω3, the heteroclinic orbit disappears and it is replaced by a homoclinic orbit (see Figure 1). To compute the explicit form of the homoclinic (or heteroclinic) orbit of (21), we use (21b) and (22), and obtain ∫ Ḡh(τ) Ḡh(0) 1− αḠ2 βτ, (23) 16ηβω4 , β = γL̄− ηω3 where the lower integration limit is Ḡh(0) = 1/ α. After taking the integral (23), we arrive at Ḡh(τ) = sech( β τ)√ , (24) for the Ḡ ≥ 0 branch of the homoclinic (heteroclinic) orbit. Having Ḡh(τ), it is straightforward to calculate cos[2ḡh(τ)] and sin[2ḡh(τ)], and determine the explicit form of qh(τ) = ḡh(τ), Ḡh(τ) For constructing M2(τ0), we use Françoise’s [13, 14] algorithm that has been devised for dynamical systems with polynomial nonlinearities. To express the averaged Hamiltonian K in terms of polynomial functions of some new dependent variables, we utilize Hopf’s variables L̄2 − Ḡ2 cos(2ḡ), (25a) L̄2 − Ḡ2 sin(2ḡ), (25b) and obtain the following differential 1-form for the evo- lution of the averaged system dQ1 + + ǫ [(s1 + ǫs2) dQ2 − (z1 + ǫz2) dQ1] = 0. (26) Here, the first-order Hamiltonian is K1(Q1, Q2) = (Q21 +Q Q1 + C, (27) s1 = m1Q2, (28a) z1 = n1Q1 + n6, (28b) s2 = m2(Q 2)Q2 +m3Q1Q2 +m4Q2 + m5 sin(2Ωτ), (28c) z2 = n2(Q 2)Q1 + n3(3Q 2) + n4Q1 + n5 cos(2Ωτ) + n7. (28d) The constant coefficients C, mi (i = 1, · · · , 5), and nj (j = 1, · · · , 7) have been given in Appendix B. A prereq- uisite for the application of Françoise’s [13] algorithm is that for all polynomial 1-forms D that satisfy the condi- D ≡ 0, (29) there must exist polynomials A(Q1, Q2) and r(Q1, Q2) such that D = dA+ rdK1. We call this the condition (∗) and prove in Appendix C that K1 satisfies the condition Françoise’s algorithm states that if M1(τ0) = · · · = Mk−1(τ0) ≡ 0 for some integer k ≥ 2, it follows that Mk(τ0) = Dk, (30a) D1 = δ1, Dm = δm + i+j=m riδj , (30b) δj = zjdQ1 − sjdQ2, (30c) for 2 ≤ m ≤ k. The functions ri are then determined successively from the formulas Di = dAi + ridK1 for i = 1, · · · , k − 1. We have already found that M1(τ0) = δ1 = 0, (31) δ1 = (n1Q1 + n6)dQ1 −m1Q2dQ2. (32) From (30) and (C2) it can be shown that M2(τ0) = δ2. (33) Substituting (30c) and (28) into (33), and carrying out the integration along qh(τ), result in M2(τ0) = 3γF 2I sin(2Ωτ0), (34) with I being a constant (see Appendix D). Equation (34) shows that τn = nπ/(2Ω) (n = 1, 2, · · · ) are simple zeros of M2(τ0) so that M2(τn) = 0, ∂M2(τ0) τ0=τn 6= 0. (35) FIG. 1: Possible topologies of the phase space flows of the averaged system for F = 0. In all panels the horizontal axis indicates ḡ/π and the vertical axis indicates Ḡ/L̄. In the first topology (left panel), all stationary points are of center type. In the second topology (middle panel) two new centers, with non-zero Ḡ-coordinates, have emerged for ḡ = ±nπ (n = 0, 1) and symmetrical homoclinic loops (thick lines) connect saddle points to themselves. In the third topology (right panel) the off-axis centers (and their surrounding tori) are still present but the separatrix curves (thick lines) are of heteroclinic type. FIG. 2: Phase space structure of the normalized equations (15) with L̄ = 2, ω = η = F = 1, and Ω = ǫ = 0.1. Left panel: γ = 0.8. Right panel: γ = 2. In both panels the horizontal and vertical axes indicate ḡ/π and Ḡ/L̄, respectively. Thus, we conclude that the global stable and unstable manifolds of the saddle point Sτ0n , W s(Sτ0n ) andW u(Sτ0n ), always intersect transversely. Transversal intersections cause a sensitive dependence on initial conditions due to the Smale-Birkhoff homoclinic theorem. This is a route to chaos. On the other hand this means that the reduced equations (15) are non-integrable for F 6= 0. V. CLASSIFICATION OF CIRCUMFERENTIAL WAVES For F = 0, h does not depend on τ and the normalized equations (15) are integrable. In such a circumstance, the phase space structure can take three general topologies (depending on the values of the system parameters and L̄) as shown in Figure 1. In the first topology all sta- tionary points with the coordinates (ḡs, Ḡs) calculated f(ḡs, Ḡs) + ǫh(ḡs, Ḡs, ǫ) = 0, (36) are centers and they lie on the Ḡ = 0 axis with ḡs = −π + nπ/2 (n = 0, · · · , 4). In the second and third topologies, two off-axis centers (with Ḡs 6= 0) come to existence for ḡs = ±nπ (n = 0, 1) and the on-axis sta- tionary points with the same ḡs = ±nπ become saddles. In the second topology, each saddle point is connected to itself by a homoclinic orbit, and in the third topology, a heteroclinic orbit connects two neighboring saddle points. The system with heteroclinic orbits allows for rotational ḡ(τ) while in the system with homoclinic orbits ḡ(τ) is always librating. Beware that this classification of phase space flows is valid as long as ǫ is sufficiently small. For F = 0, the phase space flows of (15) are structurally stable (with no unbounded branches) and the whole (ḡ, Ḡ)-space is occupied by periodic orbits of period T (K). At the stationary points, one has T (K(ḡs, Ḡs, L̄)) = 0. Given the invariance of L̄, and the periodic solutions ḡ(τ) = ḡ(τ + T (K)) and Ḡ(τ) = Ḡ(τ + T (K)), the anomaly l̄ is determined through solv- = ω + ǫ , (37) which results in l̄ = ωt + ǫR(τ) with R(τ) = R(τ + T (K)). According to (9), the functions g(t), G(t) and L(t) are also periodic in t and we conclude that l(t) = ωt + RW (t) with RW (t) = EW (l̄) − ωt being a small- amplitude periodic function of t. The explicit from of the circumferential wave will then become w(R, φ, t) Um(R) L(t)+G(t) cos [nφ−ωt−RW (t)−g(t)] L(t)−G(t) cos [nφ+ωt+RW (t)−g(t)] , (38) which is composed of a forward and a backward traveling wave. Due to the periodic nature of L(t) and G(t), when the amplitude of the forward traveling wave is maximum, that of the backward wave is minimum and vice versa. As our results of §IV shows, the regular nature of trav- eling waves is destroyed for F 6= 0 and a chaotic layer oc- curs through the destruction of the homoclinic and hete- roclinic orbits of (21). This happens over the time scale τ ∼ O(ǫ−2) or t ∼ O(ǫ−3) (because p̄ is present only in K3). Figure 2 shows Poincaré maps of the system (15) for F 6= 0. The sampling time step in generating the Poincaré maps has been 2π/Ω. It is seen that most tori around elliptic fixed points are preserved. They cor- respond to regular periodic and quasi-periodic solutions of the normalized system. For chaotic flows, the func- tions ḡ(τ) and Ḡ(τ) randomly change within the invari- ant measure of the chaotic set. Consequently, the orig- inal Lissajous variables g(t), L(t), G(t) and also RW (t) become chaotic too. For Ḡ(τ) > 0 and Ḡ(τ) < 0 the forward and the back- ward traveling waves are the dominant components of the circumferential wave, respectively. When the chaotic layer emerges from the destroyed homoclinic orbits (left panel in Figure 2), the sign of Ḡ(τ) is randomly switched along a chaotic trajectory. This means a random trans- fer of kinetic/potential energy between the forward and backward traveling wave components. For chaotic trajec- tories of this kind the angle ḡ(τ) randomly fluctuates near ḡ ≈ ±nπ (n = 0, 1) with an almost zero average. The evolution of circumferential waves is quite different when the chaotic layer emerges due to destroyed heteroclinic orbits (right panel in Figure 2). In this case chaos means a random change between the librational and rotational states of ḡ(τ). Such a change induces an unpredictable phase shift of magnitude π for both forward and back- ward traveling wave components. We note that Ḡ(τ) can flip sign on a chaotic trajectory only when ḡ(τ) is in its librational state. VI. CONCLUDING REMARKS Resonance overlapping [11, 16] is the main cause for chaotic behavior in spinning disks with near-resonant an- gular velocities [1, 2]. The chaos predicted in this paper, however, happens far from fundamental resonances. Op- tical and HHDs are usually operated below critical res- onant speeds and the lateral force F due to magnetic head is very small. We showed that whatever the magni- tude of F may be, a chaotic layer fills some parts of the phase space because the Melnikov function of the nor- malized equations has always simple zeros. Dynamics of rotating disks is regular only if F vanishes, which is an unrealistic assumption for disk drives. In low-speed disks with small F , diffusion of chaotic orbits (within their in- variant measure) takes a long time of t ∼ O(ǫ−3). The slow development of chaotic circumferential waves makes them undetectable in short time scales at which most controllers work. The Melnikov function (34) depends not only on F , but also on the parameter η through the constant I. The parameter η is a contribution of imper- fections, which are likely because of limited fabrication precision in micro/nano scales. For a perfect disk with η = 0, the off-axis elliptic stationary points of (21), and consequently, homoclinic and heteroclinic orbits disap- pear. In such a condition the Melnikov function is indef- inite, but the system admits an exact first integral and the dynamics is governed by the Hamiltonian function given in equation (11) of Jalali and Angoshtari [2]. One of the most important achievements of this work was to unveil the fact that it is premature to truncate the series of canonical perturbation theories before recording the role of all participating variables. In systems with non-autonomous governing ODEs (non-conservative sys- tems), one must be cautious while removing a fast angle through averaging schemes. The removal of the fast an- gle may also wipe out time-dependent terms, up to some finite orders of ǫ, and hide some essential information of the underlying dynamical process. Strange irregular so- lutions can indeed occur at any order and influence the long term response of dynamical systems as we observed for the spinning disk problem by keeping the third-order terms. Acknowledgments We are indebted to the anonymous referee, who discov- ered an error in the early version of the paper and led us to investigate the second-order Melnikov function. MAJ thanks the Research Vice-Presidency at Sharif University of Technology for partial support. APPENDIX A: THE NORMALIZED HAMILTONIAN By evaluating the integrals in (13), we obtain the first, second and third order terms of the normalized Hamilto- nian as −γḠ2 L̄(3γL̄+ 2ηω3) L̄2 − Ḡ2 cos(2ḡ), (A1) K2 = − 2L̄(−9Ḡ2 + 17L̄2)γ2 + 64F 2ω3 + 8(3L̄2 − Ḡ2)γηω3 + 8L̄η2ω6 + 4ω3[−6L̄γη − 2η2ω3] L̄2 − Ḡ2 cos(2ḡ) , (A2) 512ω8 11Ḡ4γ3 − 258Ḡ2L̄2γ3 + 375L̄4γ3 + 1024F 2L̄γω3 − 180Ḡ2L̄γ2ηω3 + 340L̄3γ2ηω3 + 256F 2ηω6 − 48Ḡ2γη2ω6 + 176L̄2γη2ω6 + 32L̄η3ω9 − 2ω3 17(10L̄2 − Ḡ2)γ2η + 96L̄γη2ω3 + 16η3ω6 L̄2 − Ḡ2 cos(2ḡ) − 16(Ḡ2 − L̄2)γη2ω6 cos(4ḡ) + 256F 2ηω6 cos(2p̄) − 512F 2γω3 L̄2 − Ḡ2 cos(2ḡ − 2p̄) . (A3) Consequently, the functions di(L̄, Ḡ) in equations (16) are found to be L̄2 − Ḡ2 )−1/2 d2 = − (6L̄γη + 2η2ω3)(L̄2 − Ḡ2)−1/2, 256ω5 16η3ω6 + 51(4L̄2 − Ḡ2)γ2η + 96L̄γη2ω3 (L̄2 − Ḡ2)−1/2, d4 = − 3Ḡγη2 3ḠF 2γ (L̄2 − Ḡ2)−1/2, d6 = − (18L̄γ2 + 8γηω3), 256ω8 2γ3(11Ḡ2 − 129L̄2)− 180L̄γ2ηω3 − 48γη2ω6 . (A4) Defining S = (L̄2 − Ḡ2)/Ḡ, the functions ej(L̄, Ḡ) in equations (16) become ej = −2djS, j = 1, . . . , 5, j 6= 3, e3 = − 128ω5 16η3ω6 + 17(10L̄2 − Ḡ2)γ2η + 96L̄γη2ω3 L̄2 − Ḡ2. (A5) APPENDIX B The constant coefficients of equations (27) and (28) are as follows L̄(γL̄+ ηω3) (9L̄γ2 + 4γηω3), n2 = − 17γ2η n4 = − (59L̄2γ3 + 45L̄γ2ηω3 + 16γη2ω6), n6 = − 3L̄γη + η2ω3 256ω4 (153L̄2γ2η + 96L̄γη2ω3 + 16η3ω6), mi = −ni, i = 1, 2, 5, m3 = −2n3, (59L̄2γ3 + 45L̄γ2ηω3 + 8γη2ω6). APPENDIX C In this appendix we prove that K1(Q1, Q2) given in (27), satisfies the condition (∗). To this end, we need the following theorem. Theorem 1. Any polynomial 1-form D of degree n in Q1 and Q2 can be expressed as D = dA+ rdK1 + ξ(K1)Q2dQ1, (C1) where A(Q1, Q2) and r(Q1, Q2) are polynomials of degree (n+1) and (n−1) respectively, and ξ(K1) is a polynomial of degree [ 1 (n− 1)] where [x] denotes the greatest integer in x. Iliev [15] has proved the same theorem for H = (Q21 + Q22)/2. Theorem 1 can thus be proved in a similar man- ner. Here we only present a useful result. Let D be a general polynomial 1-form of degree 1, D = (a10Q1 + a01Q2 + a00)dQ1 +(b10Q1 + b01Q2 + b00)dQ2, (C2a) then in (C1) we have A(Q1, Q2) = Q21 + b10Q1Q2 + + a00Q1 + b00Q2, (C2b) r(Q1, Q2) = 0, (C2c) ξ(K1) = a01 − b10. (C2d) Since dK1 = 0 along any phase space orbit characterized by K1(Q1, Q2) = k, and since the integral of an exact differential dA around any closed curve is zero, from (C1) we obtain D = ξ(k) Q2dQ1 = ξ(k) Q2(τ) On the other hand, from (25a) we have = E(τ)− 2Q2, where E(τ) is an even function of τ . Given the fact that Q2 is an odd function of τ , we conclude that D = −2ξ(k) Q22dτ. Consequently, if D ≡ 0, it follows that ξ(k) ≡ 0 and therefore D = dA+ rdK1, which completes the proof. APPENDIX D In equation (34), the constant coefficient I is I = − ηΩ2ω2 η(3γL̄− 4ηω3)(β + 4Ω2) η(3β − 4Ω2) 4πη3/2ω4 [1] A. Raman and C. D. Mote Jr., Int. J. Non-Linear Mech., 36, 261 (2001). [2] M. A. Jalali and A. Angoshtari, Int. J. Non-Linear Mech., 41 , 726 (2006). [3] J. N. Reddy, Applied Functional Analysis and Variational Methods in Engineering ( McGraw-Hill, New York, 1986). [4] A. Raman and C. D. Mote Jr., Int. J. Non-Linear Mech., 34, 139 (1999). [5] J. Nowinski, ASME J. Appl. Mech., 72 (1964). [6] N. Baddour and J. W. Zu, Appl. Math. Modeling, 25, 541 (2001). [7] A. Deprit, Celest. Mech. Dyn. Astron., 51, 201 (1991). [8] A. Deprit, Celest. Mech. Dyn. Astron., 1, 12 (1969). [9] A. Deprit and A. Elipe, Celest. Mech. Dyn. Astron., 51, 227 (1991). [10] V. K. Melnikov, Trans. Moscow Math., 12, 1 (1963). [11] B. V. Chirikov, Physics Reports, 52, 263 (1979). [12] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983). [13] J. P. Françoise, Ergod. Theory Dynam. Syst., 16, 87 (1996). [14] L. Perko, Differential Equations and Dynamical Systems, 3rd edition (Springer, New York, 2001). [15] I. D. Iliev, Math. Proc. Cambridge Phil. Soc., 127, 317 (1999). [16] G. Contopoulos, Order and Chaos in Dynamical Astron- omy (Springer, New York, 2002).

0704.1407

First principles theory of chiral dichroism in electron microscopy applied to 3d ferromagnets

First principles theory of chiral dichroism in electron microscopy applied to 3d ferromagnets Ján Rusz∗ Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Prague 6, Czech Republic† Stefano Rubino and Peter Schattschneider Institute for Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/138, A-1040 Vienna, Austria (Dated: August 2, 2021) Recently it was demonstrated (Schattschneider et al., Nature 441 (2006), 486), that an analogue of the X-ray magnetic circular dichroism (XMCD) experiment can be performed with the transmission electron microscope (TEM). The new phenomenon has been named energy-loss magnetic chiral dichroism (EMCD). In this work we present a detailed ab initio study of the chiral dichroism in the Fe, Co and Ni transition elements. We discuss the methods used for the simulations together with the validity and accuracy of the treatment, which can, in principle, apply to any given crystalline specimen. The dependence of the dichroic signal on the sample thickness, accuracy of the detector position and the size of convergence and collection angles is calculated. PACS numbers: Keywords: density functional theory, chiral dichroism, transmission electron microscopy, dynamical diffrac- tion theory I. INTRODUCTION The analogy between X-ray absorption spectroscopy (XAS) and electron energy loss spectroscopy (EELS) has been recognized long ago1,2. The role of the polarization vector ε in XAS is similar to the role of the wave vector transfer q in EELS. This has made feasible the detection of linear dichroism in the TEM. However the counter- part of X-ray magnetic circular dichroism (XMCD)3,4,5 experiments with electron probes was thought to be tech- nically impossible due to the low intensity of existing spin polarized electron sources. XMCD is an important technique providing atom-specific information about the magnetic properties of materials. Particularly the near edge spectra, where a well localized strongly bound elec- tron with l 6= 0 is excited to an unoccupied band state, allow to measure spin and orbital moments. Soon after the proposal of an experimental setup for detection of cir- cular dichroism using a standard non-polarized electron beam in the TEM6 it was demonstrated that such ex- periments (called energy-loss magnetic chiral dichroism, EMCD) are indeed possible7. This novel technique is of considerable interest for nanomagnetism and spintron- ics according to the high spatial resolution of the TEM. However, its optimization involves many open questions. In this work we provide theoretical ab initio predictions of the dependence of the dichroic signal in the EMCD experiment on several experimental conditions, such as sample thickness, detector placement and the finite size of convergence and collection angles. This information should help to optimize the experimental geometry in order to maximize the signal to noise ratio. The structure of this work is as follows: In section II we first describe the computational approach based on the dynamical diffraction theory and electronic structure calculations. We also discuss the validity of several ap- proximations for the mixed dynamic form factor. In sec- tion III we study the dependence of the dichroic signal of bcc-Fe, hcp-Co and fcc-Ni on various experimental con- ditions. This section is followed by a concluding section summarizing the most important findings. II. METHOD OF CALCULATION We will follow the derivations of the double differential scattering cross-section (DDSCS) presented in Refs. 8,9. Within the first-order Born approximation10 DDSCS is written as S(q, E) S(q, E) = |〈i|eiq·R̂|f〉|2δ(Ef − Ei − E) (2) where q = kf−k0 is the difference (wave vector transfer) between final wave vector kf and initial wave vector k0 of the fast electron; γ = 1/ 1− v2/c2 is a relativistic factor and a0 is Bohr radius. The S(q, E) is the so called dynamic form factor (DFF)11. This equation is valid only if the initial and final wave functions of the fast electron are plane waves. In the crystal the full translation symmetry is broken and as a result, the electron wave function becomes a superpo- sition of Bloch waves, which reflects the discrete trans- lation symmetry. Each Bloch wave can be decomposed http://arxiv.org/abs/0704.1407v1 into a linear combination of plane waves - it is a coher- ent superposition of (an in principle infinite number of) plane waves. The wave function of the fast electron can be thus written as ψ(r) = ǫ(j)C(j)g e i(k(j)+g)·r (3) for incident wave and ψ′(r) = ǫ(l)D (l)+h)·r (4) for outgoing wave, where C g , D are so called Bloch co- efficients, ǫ(j) (ǫ(l)) determine the excitation of the Bloch wave with index j (l) and wave vector k(j) (k(l)) and g (h) is a vector of the reciprocal lattice. When we derive the Born approximation of DDSCS starting with such fast electron wave functions, we will obtain a sum of two kinds of terms: direct terms (DFFs) as in the plane wave Born approximation Eq. (1), and interference terms. These interference terms are a gener- alization of the DFF - the mixed dynamic form factors11 (MDFFs). Each of them is defined by two wave vector transfers, thus we label them S(q,q′, E). The MDFF can be evaluated within a single particle approximation as S(q,q′, E) = 〈i|eiq·R̂|f〉〈f |e−iq ′·R̂|i〉δ(Ef − Ei − E) where |i〉, |f〉 are the initial and final single-electron wave functions of the target electron in the crystal. Thus the definition of MDFF encompasses the notion of DFF, Eq. (2), for q = q′. For more details about calculation of MDFF see subsection II B. The wave vector transfers are q = k(l)−k(j)+h−g and the total DDSCS will be a sum over all diads of q and q′ vectors of terms jlj′l′ ghg′h′ Sa(q,q ′, E) q2q′2 where X jlj′l′ ghg′h′ (a) is a the product of the coefficients of the individual plane wave components of the fast electron wave functions and a labels the position of the atoms where the inelastic event can occur. The X jlj′l′ ghg′h′ (a) co- efficients are given by dynamical diffraction theory. This will be covered in the next subsection IIA. The χf and χ0 are magnitudes of wave vectors outside the crystal (in the vacuum). The calculation is thus split into two separate tasks. i) Calculation of Bloch wave coefficients using the dy- namical diffraction theory and identification of impor- tant terms. This task is mainly ‘geometry dependent’, although it can also contain some input from electronic structure codes, namely the Coulomb part of crystal po- tential. ii) Calculation of MDFFs requested by the dy- namical diffraction theory. This part strongly depends on the electronic structure of the studied system. The final step is the summation of all terms. A. Dynamical diffraction theory The formalism, which will be described here is a gener- alization of the formalism presented in Ref. 8,12 extend- ing it beyond systematic row approximation by including also higher-order Laue zones (HOLZ). The extension to HOLZ is performed along lines presented in Refs. 13,14. We will assume the high-energy Laue case, i.e. we can safely neglect back-reflection and back-diffraction. The Bloch wave vectors of the electron after entering the crystal fulfill the continuity condition (j) = χ+ γ(j)n (7) where n is the unit vector normal to the crystal surface and χ is the wave vector of the incoming electron. Only the wave vector component normal to the surface can change. Expanding the wave function of the fast electron into a linear combination of plane waves and substituting it into the Schrödinger equation we obtain the secular equa- tion14 K2 − (k(j) + g)2 h 6=0  ei(k (j)+g)·r = 0 where K2 = U0 + 2meE/~ 2, m and e are, respectively, the electron mass and charge, Ug = 2meVg/~ 2 where Vg are the Fourier components of the crystal potential, which can be either calculated ab initio15,16 or obtained from the tabulated forms of the potential17,18. It can be shown14,19 that in the high energy limit the secular equation, which is a quadratic eigenvalue problem in γ(j), can be reduced to a linear eigenvalue problem AC(j) = γ(j)C(j) where A is a non-hermitean matrix13,19 Agh = K2 − (χ+ g)2 2(χ+ g) · n δgh + (1− δgh) 2(χ+ g) · n This eigenvalue problem can be transformed into a her- mitean one using a diagonal matrix D with elements Dgh = δgh g · n χ · n Then the eigenvalue problem is equivalent to (D1/2AD−1/2)(D1/2C(j)) = γ(j)(D1/2C(j)) or (j) = γ(j)C̃(j), where the matrix à is hermitean Ãgh = K2 − (χ+ g)2 2(χ+ g) · n δgh + (11) + (1− δgh) [(χ+ g) · n][(χ+ h) · n] and the original Bloch wave coefficients can be retrieved using the relation C(j)g = C̃ g · n χ · n By solving this eigenvalue problem we obtain the fast electron wave function as a linear combination of eigen- functions as given in Eq. (3). To obtain values for ǫ(j) we need to impose boundary conditions, namely that the electron is described by a single plane wave at the crys- tal surface. The crystal surface is a plane defined by the scalar product n · r = t0. Then the boundary con- dition (in the high energy limit) leads to the following condition14 ǫ(j) = C −iγ(j)t0 (13) It is easy to verify that ψ(r)|n·r=t0 = i(k(j)+g)·re−iγ (j)t0 ei(χ+g)·r n·re−iγ (j)t0C ei(χ+g)·r ei(χ+g)·rδ0g g · n χ · n = eiχ·r|n·r=t0 (14) as required by the boundary condition. We have used the continuity condition, Eq. (7), and the completeness relation for the Bloch coefficients δgh = C̃(j)∗g C̃ = (15) g · n χ · n h · n χ · n C(j)∗g C Therefore the wave function of the fast moving electron in the crystal, which becomes a single plane wave at n·r = t0 is given by the following expression ψ(r) = iγ(j)(n·r−t0)ei(χ+g)·r (16) The following discussion will be restricted to a partic- ular case - a crystal with parallel surfaces. For such a crystal with normals in the direction of the z axis we set t0 = 0 for the fast electron entering the crystal and t0 = t when leaving the crystal (t is the crystal thickness). The inelastic event leads to a change of the energy and momentum of the scattered electron. The detector position determines the observed projection of the elec- tron wave function (Bloch field) onto a plane wave af- ter the inelastic event. Therefore the calculation of the ELNES requires the solution of two independent eigen- value problems describing an electron wave function be- fore and after the inelastic event8,12. Invoking reciprocity for electron propagation the outgoing wave can also be considered as a time reversed solution of the Schrödinger equation, also known as the reciprocal wave20 with the source replacing the detector position. Now we can identify the prefactors X jlj′l′ ghg′h′ (a) from Eq. (6). For the sake of clarity we will keep C g for the Bloch coefficients of the incoming electron and we use for the Bloch coefficients of the outgoing electron en- tering the detector (obtained from the two independent eigenvalue problems). Similarly, superscript indices (j) and (l) indicate eigenvalues and Bloch-vectors for incom- ing and outgoing electron, respectively. We thus obtain jlj′l′ ghg′h′ (a) = C (j′)⋆ (l′)⋆ × ei(γ (l)−γ(l ′))tei(q−q ′)·a (17) where q = k(l) − k(j) + h− g ′ = k(l ′) − k(j ′) + h′ − g′ (18) In crystals the position of each atom can be decom- posed into a sum of a lattice vector and a base vector, a = R + u. Clearly, MDFF does not depend on R, but only on u. It is then possible to perform analytically the sum over all lattice vectors R under the approxima- tion that the MDFF does not depend strongly on the j, l indices. This is indeed a very good approximation, as verified by numerical simulations (see below). First we will treat the summation over all lattice vec- tors. The sum in Eq. 6 can be separated into two terms ei(q−q ′)·a = ei(q−q ′)·u 1 ei(q−q ′)·R(19) Since q− q′ = [(γ(j) − γ(j ′))− (γ(l) − γ(l ′))]n + h− h′ + g′ − g (20) and the algebric sum of g,h is simply a reciprocal lat- tice vectors G, which fulfills eiG·R = 1, it is possible to simplify the second term ei(q−q ′)·R = ei[(γ (j)−γ(j ′))−(γ(l)−γ(l ′))]n·R (21) For general orientations of the vector n this sum is difficult to evaluate. In particular coordinate system with n ‖ z and crystal axes a, b ⊥ z this sum leads8,21 to ei(q−q ′)·R = NRe i∆t/2 sin∆ so that the total sum over all atomic positions is ei(q−q ′)·a = ei∆ sin∆ t ei(q−q ′)·u(23) where ∆ = (γ(j)−γ(j ′))− (γ(l)−γ(l ′)). The final expres- sion of the DDSCS we write as ghg′h′ Su(q,q ′, E) q2q′2 ei(q−q jlj′ l′ jlj′ l′ ghg′h′ Tjlj′l′(t) (24) where jlj′l′ ghg′h′ (j′)⋆ (l′)⋆ depends only on the eigenvectors of the incoming and outgoing beam and Tjlj′l′(t) = e i[(γ(j)−γ(j ′))+(γ(l)−γ(l ′))] t sin∆ t is a thickness and eigenvalue dependent function. Perturbative treatment of the absorption can be easily introduced. If we denote by U ′g the absorptive part of the potential, within the first order perturbation theory the Bloch coefficients will not change, just the eigenvalues will be shifted by iη(j) or iη(l) for the incoming or outgo- ing wave, respectively. Particular η(j) can be calculated using the following expression14 η(j) = g,h U g (χ+ g) · n and similarly for the outgoing beam. This way the eigenvalues change from γ(j) to γ(j)+iη(j) and the ∆ acquires an imaginary part. Such approx- imative treatment of absorption thus affects only the thickness-dependent function Tjlj′l′(t). Here we add a few practical considerations, which we applied in our computer code. The sum in Eq. (24) is performed over 8 indices for every energy and thickness value. Such summation can easily grow to a huge number of terms and go beyond the computational capability of modern desktop computers. For example, if we assume the splitting of the incoming (and outgoing) beam into only 10 plane wave components, taking into account the 10 most strongly excited Bloch waves, we would have 108 terms per each energy and thickness. A calculation with an energy mesh of 100 points at 100 different thicknesses would include one trillion terms and require a consider- able amount of computing time. However most of these terms give a negligible contribution to the final sum. Therefore several carefully chosen cut-off conditions are required to keep the computing time reasonable without any significant degradation of the accuracy. The first cut-off condition used is based on the Ewald’s sphere construction. Only plane wave components with k + g close to the Ewald’s sphere will be excited. The strength of the excitation decreases also with decreasing crystal potential component Ug. A dimensionless param- eter wg = sgξg - product of the excitation error and the extinction distance14 - reflects both these criteria. There- fore we can filter the list of beams by selecting only beams with wg < wmax. Experience shows that in the final sum- mation a fairly low number of beams is necessary to have a well converged results (in systematic row conditions this number is typically around 10). The convergence of the corresponding Bloch coefficients requires solving an eigenvalue problem with a much larger set of beams (sev- eral hundreds). Therefore we defined two cut-off param- eters for wg - the first for the solution of the eigenvalue problem (typically wmax,1 is between 1000 and 5000) and the second for the summation (wmax,2 typically between 50 and 100). The second type of cut-off conditions is applied to selection of Bloch waves, which enter the summation. Once the set of beams for summation is determined, this amounts to sorting the Bloch waves according to a prod- uct of their excitation ǫ(j) and their norm on the subspace defined by selected subset of beams, C 0 ||C (j)||subsp. In the systematic row conditions this value is large only for a small number of Bloch waves. Typically in the exper- imental geometries used for detection of EMCD one can perform a summation over less than 10 Bloch waves to have a well converged result (often 5 or 6 Bloch waves are enough). B. Mixed dynamic form factor It can be seen from Eq. (5) that the calculation of the MDFF requires the evaluation of two matrix elements be- tween initial and final states of the target electron. The derivation of the expression for the MDFF describing a transition from core state nlκ (n, l, κ are the main, or- bital and relativistic quantum numbers, respectively) to a band state with energy E is presented in detail in the supplementary material of Ref. 7 and in Ref. 8. Though, note that in Ref. 8 the initial states are treated classi- cally, which leads to somewhat different expression for MDFF giving incorrect L2 − L3 branching ratio. 0 0.2 0.4 0.6 0.8 1 ( a.u. -0.04 -0.02 Re[ MDFF ] @ L Re[ MDFF ] @ L Im[ MDFF ] @ L Im[ MDFF ] @ L DFF @ L DFF @ L FIG. 1: Dependence of S(q, E) (top) and S(q,q′, E) with q′ = G+q (bottom) on qz, calculated for the L2,3 edge of hcp- Co, with G = (100), qx = −q x = −|G|/2, qy = q y = |G|/2. The ratio between values calculated at L3 or L2 is constant and equal to 2.1 for the real part and to −1 for the imaginary part. The final expression is7 S(q,q′, E) = L′M ′S′ 4πiλ−λ (2l + 1) [λ, λ′, L, L′] × Y λµ (q/q) µ′ (q ′/q′)〈jλ(q)〉ELSj〈jλ′ (q ′)〉EL′S′j l λ L 0 0 0 l λ′ L′ 0 0 0 l λ L −m µ M l λ′ L′ −m′ µ′ M ′ (−1)m+m (2j + 1) m S −jz m′ S′ −jz DLMS(νk)DL′M ′S′(νk) ∗δ(E + Enlκ − Eνk) Here we made use of Wigner 3j-symbols, Y λµ are spheri- cal harmonics, 〈jλ(q)〉ELSj are radial integrals of all the radial-dependent terms (radial part of the wave function of the core and band states, radial terms of the Rayleigh expansion) and DLMS(νk) is the projection of the (νk) Bloch state onto the LMS subspace within the atomic sphere of the excited atom. For more details we refer to the supplementary material of Ref. 7. For evaluation of the radial integrals and Bloch state projections DLMS(νk) we employ the density functional theory22 within the local spin density approximation23. -10 -5 0 5 10 ( a.u. -10 -5 0 5 10 ( a.u. Total Dipole λ = 0 λ = 1 λ = 2 a) b) e) f) FIG. 2: (online color) Decomposition of MDFF and dipole ap- proximation calculated for hcp-Co with q′ − q = G = (100) and qy = q y = |G|/2 as a function of qx @ L3. Left column - graphs a), c) and e), show S(q,q′, E) and right column, graphs b), d) and f), show S(q,q′, E)/q2q′2. Top row - a) and b) - is the DFF, middle row - c) and d) - is the real part of MDFF and bottom row - e) and f) - is the imaginary part of MDFF. The y-axes are in arbitrary units, but consistent within the given column. The values for the L2 edge differ only by a factor of 2.1 for the real part and −1 for the imag- inary part. Note that the contributions of λ = 0 and 2 are always negligible. See text for more details. In Section IIA we used an approximation of negligi- ble dependence of MDFF on the j, l indices (see Eq.20). Generally, as the wave vector k(j,l) for each Bloch wave changes slightly by an amount given by the corresponding eigenvalue γ(j,l), the values of qz and q z would change ac- cordingly and therefore we should not be allowed to take MDFF out of the sum over the indices j, l in the Eq. (24). However, the change in qz (and q z) induced by the eigen- values γ(j,l) is small and can be neglected with respect to the qz = χ0E/2E0 given by the energy loss E 26. To demonstrate this we plot the dependence of MDFF on qz, q z for qx and qy corresponding to the main DFF and MDFF terms, see Fig. 1. If qz is given in a.u. −1 (atomic units, 1 a.u.= 0.529178Å), typical values for L2,3 edges of Fe, Co and Ni are around tenth of a.u.−1, whereas typical values of γ(j,l) for strongly excited Bloch waves are one or two orders of magnitude smaller. Thus the approximation of weak j, l dependence of MDFF is well justified. Besides γ(j,l), the other factors determining the value of qz are the energy of the edge, i.e. the energy lost by the probe electron, the tilt with respect to the zone axis and whether the excited beam is in a HOLZ. These last factors have been included in our calculation. Only the variations due to γ(j,l) are neglected, thus giving rise to an error . 1%. If a more accurate treatment would be needed, the smooth behavior of MDFF with respect to qz would allow to use simple linear or quadratic interpo- lation/extrapolation methods. As mentioned in the introduction and explained in Refs. 1,7, dichroism in the TEM is made possible by the analogous role that the polarization vector ε and the wave vector transfer q play in the dipole approximation of the DDSCS. However we do not restrict our calcu- lations to the dipole approximation. We use the more complete expression Eq. (5). To evaluate the accuracy of the dipole approximation, we compare the dipole approximation of MDFF with the full calculation (with λ up to 3) also showing λ-diagonal components of the MDFF, Fig. 2. Because the domi- nant contribution to the signal originates from (dipole allowed) 2p→ 3d transitions, the λ = λ′ = 1 term nearly coincides with the total MDFF. While the dipole approx- imation works relatively well for the studied systems, particularly the MDFF divided by squares of momen- tum transfer vectors (right column of the Fig. 2), it has significantly different asymptotic behaviours for larger q- vectors. The λ = λ′ = 1 term provides a much better approximation, which remains very accurate also in the large q region. It is worth mentioning that thanks to the properties of the Gaunt coefficients the 2p → 3d transitions are all included in the λ = 1 and λ = 3 contributions. Thanks to the negligible value of the radial integrals for λ = 3 the terms with λ = 1 account for the large majority of the calculated signal. The contributions from λ = 0, 2 describe transitions from 2p to valence p or f states and are always negligible due to the composition of the density of states beyond the Fermi level. They practically overlap with the zero axis in all the six parts of Fig. 2. It can be shown24 that in the dipole approximation the real part of the MDFF is proportional to q · q′ and the imaginary part is proportional to q × q′. A little algebra can thus show that the imaginary part of the MDFF is, in the geometry described in the caption of Fig. 2, constant with respect to qx. As expected, the DFF (which is proportional to q2) has a minimum at qx = 0, where S(q, E)/q4 has a maximum. For the MDFF (and corresponding S(q,q′, E)/q2q′2) these minima and max- ima are centered at qx = −G/2 = −0.76 a.u. −1 where |qx| = |q III. RESULTS We summarize the results obtained for body-centered cubic iron (bcc-Fe), hexagonal close-packed cobalt (hcp- Co) and face-centered cubic nickel (fcc-Ni) crystals, which are also the first samples prepared for EMCD mea- surements. These results are valuable for optimization of the experimental setup. The geometry setup for observing the dichroic effect7 consists in creating a two-beam case by tilting the beam away from a zone axis (here (001)) by a few degrees and then setting the Laue circle center equal to G/2 for the G vector to be excited. In analogy to XMCD, where two measurements are performed for left- and righ-handed circularly polarized light, here we perform two measure- ments by changing the position of the detector, which lies once at the top and once at the bottom of the Thales circle having as diameter the line connecting the diffrac- tion spots 0 and G. This geometry setup, together with the crystal structure, is an input for the calculation of the Bloch wave coefficients (within the systematic row approximation) using the dynamical diffraction theory code described in section IIA. The electronic structure was calculated using the WIEN2k package15, which is a state-of-the-art implemen- tation of the full-potential linearized augmented plane waves method. The experimental values of lattice param- eters were used. More than 10000 k-points were used to achieve a very good converge of the Brillouin zone inte- grations. Atomic sphere sizes were 2.2, 2.3 and 2.2 bohr radii for bcc-Fe, hcp-Co and fcc-Ni, respectively. The re- sulting electronic structure was the input for the calcula- tion of the individual MDFFs required for the summation (see Section II B). In the three studied cases the dichroic effect is domi- nated by the transitions to the unoccupied 3d states. The d-resolved spin-up density of states (DOS) is almost fully occupied, while the spin-down d-DOS is partially unoccu- pied. In Fig. 3 we compare the d-DOS with the dichroic signal at the L3 edge. Due to negligible orbital moments in these compounds the L2 edge shows a dichroic signal of practically the same magnitude but with opposite sign. The shape of the calculated dichroic peaks corresponds to the difference of spin-up and spin-down d-DOS, similarly to XMCD, as it was shown for the same set of systems in Ref. 25. The calculations were performed within system- atic row conditions with G = (200) for bcc-Fe and fcc-Ni and G = (100) for hcp-Co. The sample thicknesses were set to 20 nm, 10 nm and 8 nm for bcc-Fe, hcp-Co and fcc- Ni, respectively. These values were found to be optimal for these systems in the given experimental geometry. An interesting point is the comparison of the strength of the dichroic signal. According to the d-DOS projec- tions one would expect comparable strength of signals for the three elements under study. But the dichroic signal of hcp-Co seems to be approximately a factor of two smaller than that of the other two. The reason for that can be explained by simple geometrical considera- -6 -4 -2 0 2 4 E - E ( eV ) 0 2 4 ( eV ) FIG. 3: (online color) Spin-resolved d-densities of states (left) and resulting signal on L3 edge (right) on bcc-Fe, hcp-Co and fcc-Ni (from top to bottom) at optimal thickness (see text). Spin-up DOS is drawn using a solid black line (positive) and spin- down DOS using a dashed red line (negative). DDSCS for the (+) detector position is drawn using a solid blue line, DDSCS for the (-) position is drawn using a dashed green line. The dichroic signal (difference) is the hatched red area. G = (200) for bcc-Fe and fcc-Ni and (100) for hcp-Co. tions starting from Eq. (24). For simplicity we consider only the main contributions: the DFF S(q,q, E) and the MDFF S(q,q′, E) with q ⊥ q′. For bcc-Fe and fcc- Ni the summation over u within the Bravais cell leads always to the structure factor 2 and 4, respectively, be- cause q′ − q = G is a kinematically allowed reflection. This factor cancels out after division by the number of atoms in the Bravais cell. Therefore it does not matter, what is the value of q-vectors, the sum over the atoms is equal to S(q,q′, E)/q2q′2 itself. On the other hand, the unit cell of hcp-Co contains two equivalent atoms at positions u1 = ( ) and u2 = ( ). For the two DFFs q = q′ and the exponential reduces to 1; since there are two such terms, after division by Nu the sum equals again the DFF itself. But for the main MDFF we have q ⊥ q′ and the exponential factor will in general weight the terms. One can easily see, that q′ − q = G. For the G = (100) systematic row case, which was used for cal- culation of hcp-Co in Fig. 3 the exponentials evaluate to the complex numbers − 1 and − 1 for u1 and u2, respectively. Because of symmetry, the MDFFs for both atoms are equal and then the sum 1 leads to a factor − 1 for the MDFF contribution, i.e. the influence of its imaginary part, which is responsible for dichroism, on the DDSCS is reduced by a factor of two. To optimize the dichroic signal strength of hcp-Co, we require G · u1 = G · u2 = 2πn, which gives in principle an infinite set of possible G vectors. The one with lowest hkl indices is G = (110). A calculation for this geometry setup leads to approximately twice the dichroic signal, see Fig. 4 and compare to the corresponding graph in Fig. 3. For the optimization of the experimental setup it is important to know how sensitive the results are to varia- tion of the parameters like the thickness of the sample or the accuracy of the detector position. Another question related to this is also the sensitivity to the finite size of the convergence and collection angles α and β. In the following text we will address these questions. The thickness influences the factor Tjlj′l′ in the Eq. (24) only. This factor leads to the so called pen- dellösung oscillations - modulations of the signal strength -1 0 1 2 3 4 Energy loss ( eV ) dichroism (%) DDSCS(+) DDSCS(-) FIG. 4: L3 peak of hcp-Co calculated for the G = (110) systematic row at 18nm. See caption of Fig. 3. The peaks have been renormalized so that their sum is 1, therefore their difference is the dichroic signal (ca. 15% in this case). as a function of thickness. This also influences the strength of the dichroic signal. Results of such calcula- tions are displayed in Fig. 5 (we did not include absorp- tion into these simulations, so that all signal variations are only due to the geometry of the sample). From these simulations it follows that a well defined thickness of the sample is a very important factor. Relatively small vari- ations of the thickness can induce large changes in the dichroic signal, particularly in fcc-Ni. From the figure one can deduce that the optimal thickness for a bcc-Fe sample should be between 8 nm and 22 nm (of course, due to absorption, thinner samples within this range would have a stronger signal), for hcp-Co between 15 nm and 22 nm and for fcc-Ni it is a relatively narrow interval - between 6 nm and 10 nm. However, we stress that these results depend on the choice of the systematic row vector G. For example hcp-Co with G = (100) (instead of (110) shown in Fig. 5) has a maximum between 5 nm and 15 nm (although it is much lower, as discussed before). Taking the optimal thickness, namely 20 nm, 18 nm and 8 nm for bcc-Fe, hcp-Co and fcc-Ni, respectively, we calculated the dependence of the dichroic signal on the detector position. We particularly tested changes of the dichroic signal when the detector is moved away from its default position in the direction perpendicular to G, see Fig. 6. It is interesting to note that the maximum abso- lute difference occurs for a value of qy smaller than |G|/2. This can be qualitatively explained by considering the non-zero value of qz and q z, i.e. q and q ′ are not exactly perpendicular at the default detector positions. More- over the MDFF enters the summation always divided by q2q′2 and the lengths of q-vectors decrease with decreas- ing qy. The important message we can deduce from this figure is that the dichroic signal is only weakly sensitive to the accuracy of qy since even displacement by 10-20% in the detector default qy positions (i.e. qy = ±G/2) do not affect significantly the measured dichroic signal. 0 20 40 60 80 100 Thickness ( nm ) FIG. 5: (online color) Dependence of the DDSCS and of the dichroic signal on sample thickness for a) bcc-Fe, b) hcp-Co and c) fcc-Ni. Systematic row vector G = (200) was used for bcc-Fe and fcc-Ni, while for hcp-Co G = (110) was chosen. The blue and green solid curves are DDSCSs calculated for the (+) and (-) detector positions, the dashed black curve is the DFF part of the DDSCS (it is identical for both detector positions). The red line with circles is the relative dichroism defined as difference of DDSCSs divided by their sum, the red solid curve is the absolute dichroism - difference of DDSCSs. Related to this is a study of the dependence of the dichroic signal on the finite size of the convergence and collection angles α and β. We performed a calculation for the three studied metals and found that collection and convergence half-angles up to 2 mrad weakens the relative dichroic signal by less than 10%. IV. CONCLUSIONS We have developed a computer code package for the calculation of electron energy loss near edge spectra, which includes the theory of dynamical Bragg diffrac- tion. We applied the code to the recently discovered phe- nomenon of magnetic chiral dichroism in the TEM and we demonstrated the relation of the dichroic peak shape to the difference of d-projections of the spin-resolved density of states in analogy with similar observation for XMCD. Using this code we examined the validity of the dipole approximation, which is often assumed. We found that -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 / |G| ( dimensionless ) bcc-Fe hcp-Co fcc-Ni absolute dichroism (arb. units) FIG. 6: (online color) Dependence of the dichroic signal on detector displacements along qy . The full symbols correspond to the relative dichroic signal while the open symbols to the difference of the DDSCS for both detector positions. These are in arbitrary units and their magnitudes are not directly comparable. Vertical lines are showing the default detector positions. for the 3d ferromagnetic systems studied it is a reasonable approximation, however with wrong asymptotic proper- ties - it overestimates the contributions from larger q- vectors. A very accurate approximation for the studied systems is the λ = λ′ = 1 approximation, which treats appropriately the dominant p→ d dipole transitions and remains very accurate also for large q, q′. In order to provide guidance to the experimentalist we have investigated the strength of the dichroic signal as a function of the sample thickness and the precision of the detector placement. While the dichroic signal strength is rather robust with respect to the precision of the detector placement, the thickness of the specimen influences the signal considerably. Therefore it might be a challenge to produce samples with optimum thickness and selecting the best systematic row Bragg spot. Our calculations yield best thicknesses in order to detect EMCD of the iron and nickel samples for the systematic rowG = (200) to be 8-22 nm and 6-10 nm, respectively, and for cobalt in the systematic row G = (110) to be 15-22 nm. Acknowledgments We thank Dr. Cécile Hébert and Dr. Pavel Novák for stimulating discussions. This work has been supported by the European Commission, contract nr. 508971 (CHI- RALTEM). ∗ Electronic address: [email protected] † currently at Department of Physics, Uppsala University, Box 530, S-751 21 Uppsala, Sweden 1 A. P. Hitchcock, Jpn. J. Appl. Phys. 32(2), 176 (1993). 2 J. Yuan and N. K. Menon, J. Appl. Phys. 81(8), 5087 (1997). 3 B. T. Thole, P. Carra, F. Sette and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992). 4 S. W. Lovesey and S. P. Collins, X-Ray Scattering and Ab- sorption by Magnetic Materials, Clarendon Press, Oxford, UK, 1996. 5 J. Stöhr, J. Electron Spectrosc. Relat. Phenom. 75, 253 (1995). 6 C. Hébert and P. Schattschneider, Ultramicroscopy 96, 463 (2003). 7 P. Schattschneider, S. Rubino, C. Hébert, J. Rusz, J. Kuneš, P. Novák, E. Carlino, M. Fabrizioli, G. Panaccione and G. Rossi, Nature 441, 486 (2006). 8 M. Nelhiebel, Ph.D. Thesis, 1999, Vienna University of Technology. 9 M. Nelhiebel, P. Schattschneider and B. Jouffrey, Phys. Rev. Lett. 85(9), 1847 (2000). 10 P. Schattschneider, C. Hébert, H. Franco and B. Jouffrey, Phys. Rev. B 72, 045142 (2005). 11 H. Kohl and H. Rose, Advances in electronics and electron optics 65, 173 (1985). 12 P. Schattschneider, M. Nelhiebel and B. Jouffrey, Phys. Rev. B 59, 10959 (1999). 13 L.-M. Peng and M. J. Whelan, Proc. R. Soc. London Ser. A 431, 111 (1990). 14 A. J. F. Metherell, in Electron Microscopy in Materials Science II, 397 (1975), CEC, Luxembourg. 15 P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, 2001 WIEN2k, Vienna University of Technology (ISBN 3-9501031-1-2). 16 The WIEN2k package can calculate X-ray structure fac- tors. By supplying the potential instead of charge density it is possible to use the same code to calculate the electron structure factors using a simple script. 17 A. Weickenmeier and H. Kohl, Acta Cryst. A47, 590 (1991). 18 P. A. Doyle and P. S. Turner, Acta Cryst. A24, 390 (1968). 19 A. L. Lewis, R. E. Villagrana and A. J. F. Metherell, Acta Cryst. A34, 138 (1978). 20 Y. Kainuma, Acta Cryst. 8, 247 (1955). 21 P. Schattschneider and W. S. M. Werner, J. Electron Spec- trosc. Relat. Phenom. 143, 81 (2005). 22 P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964) and W. Kohn, L. J. Sham, Phys. Rev. 140, A1133 (1965). 23 J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). 24 P. Schattschneider, C. Hébert, S. Rubino, M. Stöger- Pollach, J. Rusz, P. Novák, submitted. 25 J. Kuneš and P. M. Oppeneer, Phys. Rev. B 67, 024431 (2003). 26 The question of momentum conservation in the z direction in the inelastic interaction in a crystal of finite thickness is related to the probability of inter- and intrabranch transi- tions of the probe electron? mailto:[email protected]

0704.1408

Probing MACHOs by observation of M31 pixel lensing with the 1.5m Loiano telescope

Astronomy & Astrophysics manuscript no. November 18, 2021 (DOI: will be inserted by hand later) Probing MACHOs by observation of M31 pixel lensing with the 1.5m Loiano telescope S. Calchi Novati1,2, G. Covone3, F. De Paolis4, M. Dominik⋆5, Y. Giraud-Héraud6, G. Ingrosso4, Ph. Jetzer7, L. Mancini1,2, A. Nucita4, G. Scarpetta1,2, F. Strafella4, and A. Gould8 (the PLAN⋆⋆ collaboration) 1 Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy 2 Istituto Nazionale di Fisica Nucleare, sezione di Napoli, Italy 3 INAF - Osservatorio Astronomico di Capodimonte, via Moiariello 16, Napoli, Italy 4 Dipartimento di Fisica, Università di Lecce and INFN, Sezione di Lecce, CP 193, 73100 Lecce, Italy 5 SUPA, University of St Andrews, School of Physics & Astronomy, North Haugh, St Andrews, KY16 9SS, United Kingdom 6 APC, 10, rue Alice Domon et Léonie Duquet 75205 Paris, France 7 Institute for Theoretical Physics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland 8 Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, US Received/ Accepted Abstract. We analyse a series of pilot observations in order to study microlensing of (unresolved) stars in M31 with the 1.5m Loiano telescope, including observations on both identified variable source stars and reported microlens- ing events. We also look for previously unknown variability and discover a nova. We discuss an observing strategy for an extended campaign with the goal of determining whether MACHOs exist or whether all microlensing events are compatible with lens stars in M31. Key words. Gravitational lensing - Galaxy: Halo - Galaxies: M31 1. Introduction Following the original proposal of Paczyński (1986), several microlensing campaigns have been undertaken in the recent years with the purpose of unveiling the content of galactic halos in form of MACHOs. While both the MACHO (Alcock et al. 2000) and EROS (Tisserand et al. 2006) groups have published comprehensive results of their respective campaigns, and an analysis of the OGLE campaign is underway, no consensus has yet been reached on either the density of MACHOs or their mass spectrum, and it is still not clear whether “self lensing” within the Magellanic Clouds (Sahu 1994; Wu 1994) can account for most or even all of the de- tected microlensing events (Belokurov et al. 2003, 2004; Mancini et al. 2004; Griest & Thomas 2005; Bennett 2005; Calchi Novati et al. 2006; Evans & Belokurov 2006). Searching for microlensing events towards the Andromeda Galaxy (M31) not only allows one to monitor a huge number of stars (∼ 108) within a few fields, but also allows one to fully probe M31’s whole halo (which is not ⋆ Royal Society University Research Fellow ⋆⋆ Pixel Lensing Andromeda possible for the Milky Way), and possibly to distinguish more easily between self lensing and lensing by MACHOs because M31’s tilt with respect to the line of sight in- duces a characteristic signature in the spatial distribution of the halo events (Crotts 1992; Baillon et al. 1993; Jetzer 1994). Observational campaigns have been carried out by several collaborations: AGAPE (Ansari et al. 1997, 1999), Columbia-VATT (Crotts & Tomaney 1996), POINT- AGAPE (Aurière et al. 2001; Paulin-Henriksson et al. 2003), SLOTT-AGAPE (Calchi Novati et al. 2002, 2003), WeCAPP (Riffeser et al. 2003), MEGA (de Jong et al. 2004), NainiTal (Joshi et al. 2005) and ANGSTROM (Kerins et al. 2006). The detection of a handful of mi- crolensing candidates have been reported and first, though contradictory, conclusions on the MACHO content along this line of sight have been reported (Calchi Novati et al. 2005; de Jong et al. 2006). In order to go beyond these first results, it is essential to choose an appropriate observational strategy for the new observational campaigns. Indeed, the experience of the previous campaigns shows that a careful assessment of the characteristics of the microlensing signal and of poten- tially contaminating stellar variables is crucial. Two main phenomenological characteristics of microlensing events http://arxiv.org/abs/0704.1408v1 2 Calchi Novati et al.: Probing MACHOs with the 1.5m Loiano telescope must be taken into account: the duration and the flux de- viation (e.g. Ingrosso et al. 2006a,b). Microlensing events in M31 are expected to last only a few days (this holds in the lens mass range 10−2 − 1 M⊙, over which self lensing but also most of the MACHO signal is expected). Note that here we refer to the full-duration-at-half-maximum, t1/2, easily evaluated out of pixel lensing observations, with t1/2 = t1/2(tE, u0), where tE, u0 are the Einstein time and impact parameter, respectively. The degeneracy in the parameter space tE, u0 is intrinsically linked to the fact that the underlying sources are not resolved objects so that the background level of the light curves is a blend (of a huge number) of stars (Gould 1996). However, as was shown to be the case for some of the POINT-AGAPE microlensing candidates, extremely good sampling along the flux variation sometimes allow one to break this de- generacy. To gain insight into the underlying mass spec- trum of the lens population (recall tE ∝ Mlens), it will be essential to break the tE, u0 degeneracy beyond what was achieved in previous campaigns. Furthermore, the expected short duration can also be used to robustly test the detected flux variations with respect to the vari- able star background (Calchi Novati et al. 2005), but to achieve this, a very tight and regular sampling is again necessary. On the other hand, the expected duration im- plies that to characterise the microlensing signals, the campaign does not need to be extremely long. Besides, the dataset of previous campaigns already allows one to check for the expected uniqueness of microlensing signals. The long time baseline can then be exploited in order to increase the expected rate of events. Very tight and regu- lar sampling on a nightly basis is therefore a first crucial feature for an optimal observational strategy. This would represent an important improvement with respect to pre- vious campaigns that would allow one both to better dis- tinguish microlensing events from other background vari- ations, and, possibly, to break some of the degeneracy in the microlensing parameter space. As for the flux devia- tion, the main results have been obtained using the 2.5m INT telescope with integration times of about 20 minutes per night, so that even smaller telescopes can be used, provided that long enough integration times are employed to reach the needed signal-to-noise ratio. In this paper we present the results of the pilot sea- son of a new observational campaign towards M31 car- ried out with the Loiano telescope at the “Osservatorio Astronomico di Bologna” (OAB)1. In Sect. 2 we present the observational setup and outline data reduction and analysis. In Sect. 3 we present the results of our follow-up observations on previously reported microlensing candi- dates and other variable light curves, and we report the discovery of a new Nova variation. In Sect. 4 we estimate the expected microlensing signal and discuss the feasibility and objectives of a longer-term microlensing campaign. 1 http://www.bo.astro.it/loiano/index.htm 2. Data analysis 2.1. Observational setup, data acquisition and reduction As pilot observations for studying microlensing of stars in the inner M31 region, we observed two fields dur- ing 11 consecutive nights, from 5 September to 15 September 2006, with the 152cm Cassini Telescope lo- cated in Loiano (Bologna, Italy). We make use of a CCD EEV of 1300x1340 pixels of 0.′′58 for a total field of view of 13′ × 12.′6, with gain of 2 e−/ADU and low read- out noise (3.5 e−/px). Two fields of view around the in- ner M31 region have been monitored, centered respec- tively at RA=0h42m50s, DEC=41◦23′57′′ (“North”) and RA=0h42m50s, DEC=41◦08′23′′ (“South”) (J2000), so as to leave out the innermost (∼ 3′) M31 bulge region, and with the CCD axes parallel to the north-south and east- west directions, in order to get the maximum field overlap with previous campaigns (Fig. 1). To test for achromatic- ity, data have been acquired in two bandpasses (similar to Cousins R and I), with exposure times of 5 or 6 minutes per frame. Overall we collected about 100 exposures per field per filter over 8 nights or about 15 images per night2. Typical seeing values were in the range 1.5′′−2′′. Bias and sky flats frames were taken each night and standard data reduction was carried out using the IRAF package3. We also corrected the I-band data for fringe effects. 2.2. Image analysis The search for flux variations towards M31 has to deal with the fact that sources are not resolved objects, so that one has to monitor flux variations of every element of the image (the so called “pixel-lensing” technique discussed in Gould 1996). As for the preliminary image analysis, we follow closely the strategy outlined by the AGAPE group (Ansari et al. 1997; Calchi Novati et al. 2002), in which each image is geometrically and photometrically aligned relative to a reference image. To account for seeing vari- ations we then substitute for the flux of each pixel, the flux of the corresponding 5-pixel square “superpixel” cen- tered on it (whose size is determined so as to cover most of the average seeing disc) and then apply an empirical, linear, correction in the flux, again calibrating each image with respect to the reference image. The final expression for the flux error accounts both for the statistical error in the flux count and for the residual error linked to the see- ing correction procedure. Finally, in order to increase the signal-to-noise ratio, we combine the images taken during the same night. 2 During the last useful night only a few R images in the North field could be taken. 3 http://iraf.noao.edu/ Calchi Novati et al.: Probing MACHOs with the 1.5m Loiano telescope 3 Fig. 1. Projected on M31, we display the boundaries of the two 13’x12.6’ monitored fields (inner contours), to- gether with the larger INT fields and the centre M31 (cross). The filled circle marks the position of the Nova variable detected (Sect. 3.3). 3. Light curve results 3.1. Variables in the POINT-AGAPE catalogue In order to assess the quality of the present data set as compared to that of previous campaigns we looked into observations of ∼ 40000 stars identified as variables by the POINT-AGAPE group (An et al. 2004). Besides the position, each variation in this catalogue is characterised by three quantities: the magnitude corresponding to the flux deviation at maximum R(∆φ), with values down to R(∆φ) ∼ 23, the period (P ) as evaluated using a Lomb algorithm, and an estimator of the probability of a false detection (Lf ) (high absolute values of Lf indicate a sure identification). We note that most of the variations in the catalogue are rather faint and only a few have short peri- We want to investigate which fraction of variables found by POINT-AGAPE can be identified by our ob- servations. (Preliminary to the analysis, we must evalu- ate the relative geometrical and photometrical transfor- mation between the two data sets. In particular, we find that ∼ 30% of the original sample belongs also to our field of view). Since our observations cover only 11 days, we re- strict our attention to the shortest periods (P < 30 d), which encompasses a ∼ 2% subset of the POINT-AGAPE catalogue. Note that our limited baseline does not allow us to properly characterise the shape parameters of the detected variations. Therefore, in order to cross-identify the flux variations detected with those belonging to the Period (days) Period (days) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Fig. 2. R(∆φ) vs. Period for short-period (P < 30 d) variable stars reported in the POINT-AGAPE catalogue (top panel shows the subset with log(Lf) < −30), where filled circles indicate the flux variations identified within the OAB data set. POINT-AGAPE catalogue we only test for the offset be- tween the position evaluated through our selection and the transformed POINT-AGAPE position. For our analysis, we first identify a “clean” set of variable stars (selected by demanding log(Lf) < −30), which includes ∼ 25% of the POINT-AGAPE sam- ple. Restricting ourselves to short-period variables with P < 30 d, leaves us with 169 stars within our field of view, among which 68 fall into the “clean” sample. For the lat- ter, we detect most of the bright variations (R(∆φ) < 21), namely ∼ 70%, and about 40% of all of the variations. When we consider the total sample of short-period vari- ables we arrive at values that are about 10% smaller. This partly results from the fact that the total sample con- tains a larger fraction of faint objects, while our detection threshold, though varying with the position in the fields, is typically about R(∆φ) ∼ 22. In Fig. 2 we show flux deviation vs. period for the the full set of short-period POINT-AGAPE variables, where solid circles mark those that were found by our analysis. In Fig. 3, we show the lightcurve of a POINT-AGAPE variable recovered within the OAB data, with its OAB extension. 3.2. Identified microlensing candidates Since microlensing variations are quite unlikely to re- peat, measuring a constant flux from follow-up obser- vations provides further evidence that the previously observed signal has indeed been caused by microlens- ing. Our target fields contain three of the six can- 4 Calchi Novati et al.: Probing MACHOs with the 1.5m Loiano telescope t (days) t (days) t (JD-2449624.5) t (JD-2449624.5) 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 4350 4355 4360 4365 4370 4375 4350 4355 4360 4365 4370 4375 Fig. 3. The light curve of a POINT-AGAPE flux varia- tion (P = 11.14 days and R(∆φ) = 21.1), together with its extension in the OAB data. Top to bottom, the INT R and color light curves, folded by their period (for visual aid, two cycles are plotted); and the OAB R and color light curves. The “color” is evaluated as −2.5 log(φr/φi), where φ is the observed flux. Flux is in ADU s−1. didates reported by the POINT-AGAPE collaboration (Calchi Novati et al. 2005), PA-N1, PA-S3 and PA-S7; be- side PA-N1, two more among the 14 reported by the MEGA collaboration (de Jong et al. 2006), MEGA-ML- 3 and MEGA-ML-15; beside PA-S3, the second candidate reported by the WeCAPP collaboration, WeCAPP-GL2 (Riffeser et al. 2003). All of the light curve extensions within our data set of the previous variations appear to be stable, namely, we do not observe any flux variation beyond the background noise level compatible with the observed microlensing flux variation. As an example, in Fig. 4 we show the PA-S3 light curve together with its extension in the OAB data. 3.3. A Nova like variation Lastly, we discuss the result of a search for very bright flux variations (R(∆φ) < 19). One flux variation sur- vives this selection (Fig. 5), and this appears to be a nova-like variable (its extension on the POINT-AGAPE data set appears to be stable) located in RA=0h42m33s, DEC=41◦10′06′′ (J2000). We estimate the magnitude and color at maximum to be R(∆φ) ∼ 17.5 and R− I ∼ −0.1. The rate of decline, about 2 magnitudes during the 7 nights of our observational period, puts this nova among the “very fast” ones in the speed classes defined in Warner (1989). The (strong) color evolution is rather unusual, as it got redder during descent. In the POINT-AGAPE nova t (JD-2449624.5) t (JD-2449624.5) t (JD-2449624.5) t (JD-2449624.5) 2140 2160 2180 2200 2220 2240 2260 2280 2300 2140 2160 2180 2200 2220 2240 2260 2280 2300 4350 4355 4360 4365 4370 4375 4350 4355 4360 4365 4370 4375 Fig. 4. The light curve of the POINT-AGAPE PA- S3 microlensing candidate (Paulin-Henriksson et al. 2003; Calchi Novati et al. 2005) together with its extension in the OAB data. In the INT data the dotted line is the best Paczyński (1986) fit; in the OAB data, the solid lines indi- cate the background level, while the dotted lines represent the flux deviation corresponding to the observed flux devi- ation at maximum for the POINT-AGAPE variation. The ordinate axis units are flux in ADU s−1. t (JD-2449624.5) t (JD-2449624.5) t (JD-2449624.5) 4345 4350 4355 4360 4365 4370 4375 4380 4345 4350 4355 4360 4365 4370 4375 4380 4345 4350 4355 4360 4365 4370 4375 4380 Fig. 5. The light curve of the Nova detected within the OAB data. Top to bottom, R, I bands and color data (as defined in Fig. 3) are shown. The solid lines (R and I data) indicate the estimate of the background level. Calchi Novati et al.: Probing MACHOs with the 1.5m Loiano telescope 5 catalogue (Darnley et al. 2004), there was only one such object, PACN-00-07, showing a similar color evolution but also characterised by a fainter magnitude at maximum and a slower speed of descent. The issue of the expected novæ rate in M31 is still a matter of debate. Darnley et al. (2006) evaluate a rate of ∼ 38 (∼ 27) novæ/years for the bulge (disc) respec- tively, while previous works pointed to somewhat smaller values (e.g. Capaccioli et al. 1989; Shafter & Irby 2001). Our detection of 1 nova during an overall period of 11 days is in any case in good agreement with these expec- tations (restricted to the bulge region only and using the first estimate, we derive an expected number of novæ of ∼ 1.1). 4. The expected microlensing signal To predict the number and characteristics of the ex- pected microlensing signal for the different lens popula- tions (Galactic halo and components of M31), we need both an astrophysical model for all the physical quantities that determine the microlensing events (which includes brightness profile, spatial mass density, velocity distribu- tions, luminosity function for the sources, and mass spec- trum for the lenses) and a model reproducing both the ob- servational setup and the selection pipeline. Because of the huge parameter space involved, we use a Monte Carlo sim- ulation to carry out this program. In particular, we make use of the simulation described in Calchi Novati et al. (2005), adapted to the OAB observational setup. In Fig. 6 we report the results, obtained using the fidu- cial astrophysical model discussed in Calchi Novati et al. (2005), for the flux deviation at maximum and duration distributions expected for self-lensing events (the corre- sponding distributions for 0.5 M⊙ MACHOs are almost indistinguishable) and, for both self lensing and MACHOs, the expected distance from the M31 center distribution. In particular we recover the well known results that most of the microlensing events are expected to last only a few days. We also stress the difference, already apparent within our relatively small field of view, between the spa- tial distributions due to luminous and MACHO lenses. The latter is much broader, implying that this diagnostic can be used to distinguish between the two populations. Note that here we are considering the distance-from-the- M31-center statistics rather than the expected asymmetry in the spatial distribution of M31 halo lenses (Crotts 1992; Jetzer 1994). The M31-center-distance statistic is sensitive to the different mass distributions of stars and dark mat- ter, although it is a zeroth order approximation since it ignores the additional difference due to the expected asym- metry of the microlensing signal. We adopt this zeroth- order approach because the refinement needed to include the asymmetry information would require substantial ad- ditional analysis: as was pointed out by An et al. (2004), the study of variable stars demonstrates that differential extinction could induce a similar asymmetric signal on the spatial distribution of self-lensing events. Note also that Self lensing: R(∆φ) Self lensing: t1/2 (days) Self lensing: distance from M31 center (arcmin) MACHO: distance from M31 center (arcmin) 18 19 20 21 22 23 24 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Fig. 6. Results of the Monte Carlo simulation: from top to bottom we show the histograms of the expected flux de- viation at maximum, duration distribution for self-lensing events, and the distance (from the M31 center) distribu- tions for self-lensing and for MACHOs. The units on the ordinate axes are the number of events. our choice for the field position has not been chosen in or- der to optimise such an analysis, but rather to maximise the overlap with the fields of previous campaigns. Let us note that a real “second generation” pixel lensing experi- ment should cover a much larger field of view than ours, both to increase, for a given time baseline, the expected rate of events but especially in order to better disentangle the self-lensing signal from the MACHO signal. To estimate the number of expected events, we repro- duce the actual sampling of this pilot season and imple- ment a basic selection for microlensing events (asking for the presence of a significant bump), and take into account the results of the analysis carried out in Sect. 3.1 by re- stricting to the subsample of R(∆φ) < 22 variations. As a results, we predict ∼ 0.17 self-lensing events and ∼ 0.54 MACHOs (for full M31 and Galactic halos with 0.5 M⊙ MACHO objects). As discussed in Calchi Novati et al. (2005), the predictions of the Monte Carlo simulation are quantifiable as “over-optimistic”, so that these figures, for the given astrophysical model, should be taken as an up- per limit to the actual number of expected events because we have not factored in the efficiency of the pipeline. In order to increase the available statistics we need a longer time baseline. An aspect here deserves to be stressed. As the expected duration of the events we are looking for is of the same order of the length of our present baseline, because of “boundary” effects, the number of ex- pected events should increase more than linearly with the overall baseline length (of course, this is no longer true 6 Calchi Novati et al.: Probing MACHOs with the 1.5m Loiano telescope as soon as the baseline is long enough). This holds un- der the condition that no gaps are introduced into the sampling, clearly showing the importance of an appropri- ate observational strategy. For example, for a full two- month campaign we predict, again for the sub sample of R(∆φ) < 22 variations, ∼ 1.3 (5.1) self-lensing (0.5 M⊙ MACHO) events, respectively. Finally, we note that we have obtained similar results within a parallel analysis car- ried out following the approach outlined in Ingrosso et al. (2006a,b). As for the astrophysical model, we recall that de Jong et al. (2006), using a different model for the lumi- nous components of M31, obtained a significantly higher expected contribution of the self-lensing signal relative to that evaluated with the fiducial model discussed in Calchi Novati et al. (2005), which we are also using in the present analysis. Hence, the full-fledged campaign that we are planning will be important for understanding the dis- puted issue of M31 self-lensing as well as MACHO dark matter. 5. Conclusions Based on pilot season observations of M31 during 11 con- secutive nights in September 2006 and a Monte Carlo sim- ulation for the expected properties of microlensing events caused by lenses in the Galactic halo or M31, respectively, we have shown the feasibility of an extended campaign with the 1.5m Loiano telescope being able to resolve the current puzzle of the origin of microlensing events involv- ing extragalactic sources. In particular, we were able to identify known variable stars from our data thanks to the tight sampling and de- spite the short time range covered. Reported microlensing candidates within our field of view have shown no fur- ther variation, therefore the microlensing interpretation was confirmed. Moreover, a nova variable showed up in our data. As for the microlensing signal, we have stressed the im- portance of an appropriate sampling for the observations, and discussed the results of a Monte Carlo simulation of the present experiment. In particular, we have shown how the expected spatial distribution for self-lensing and MACHO events can allow us to disentangle the two con- tributions. Finally, we have provided an evaluation of the expected number of microlensing events for the present pilot season and discussed quantitatively the possible out- put of a longer baseline campaign. Acknowledgements. The observational campaign has been pos- sible thanks to the generous allocation of telescope time by the TAC of the Bologna Observatory and to the invaluable help of the technical staff. In particular, we thank Ivan Bruni for accu- rate and precious assistance during the observations. We thank the POINT-AGAPE collaboration for access to their database. References Alcock, C., Allsman, R. A., Alves, D. R., et al. 2000, ApJ, 542, 281 An, J. H., Evans, N. W., Hewett, P., et al. 2004, MNRAS, 351, 1071 Ansari, R., Aurière, M., Baillon, P., et al. 1999, A&A, 344, Ansari, R., Aurière, M., Baillon, P., et al. 1997, A&A, 324, Aurière, M., Baillon, P., Bouquet, A., et al. 2001, ApJ, 553, L137 Baillon, P., Bouquet, A., Giraud-Heraud, Y., & Kaplan, J. 1993, A&A, 277, 1 Belokurov, V., Evans, N. W., & Du, Y. L. 2003, MNRAS, 341, 1373 Belokurov, V., Evans, N. W., & Le Du, Y. 2004, MNRAS, 352, 233 Bennett, D. 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Wiley, Chichester, p.1 Wu, X.-P. 1994, ApJ, 435, 66 Introduction Data analysis Observational setup, data acquisition and reduction Image analysis Light curve results Variables in the POINT-AGAPE catalogue Identified microlensing candidates A Nova like variation The expected microlensing signal Conclusions

0704.1410

QCD thermodynamics and confinement from a dynamical quasiparticle point of view

arXiv:0704.1410v1 [nucl-th] 11 Apr 2007 QCD thermodynamics and confinement from a dynamical quasiparticle point of view W. Cassing a,∗ aInstitut für Theoretische Physik, Universität Giessen, Heinrich–Buff–Ring 16, D–35392 Giessen, Germany Abstract In this study it is demonstrated that a simple picture of the QCD gluon liquid emerges in the dynamical quasiparticle model that specifies the active degrees of freedom in the time-like sector and yields a potential energy density in the space-like sector. By using the time-like gluon density (or scalar gluon density) as an indepen- dent degree of freedom - instead of the temperature T as a Lagrange parameter - variations of the potential energy density lead to effective mean-fields for time-like gluons and an effective gluon-gluon interaction strength at low density. The latter yields a simple dynamical picture for the gluon fusion to color neutral glueballs when approaching the phase boundary from a temperature higher than Tc and paves the way for an off-shell transport theoretical description of the parton dynamics. Key words: Quark gluon plasma, General properties of QCD, Relativistic heavy-ion collisions PACS: 12.38.Mh, 12.38.Aw, 25.75.-q 1 Introduction The formation of a quark-gluon plasma (QGP) and its transition to interacting hadronic matter – as occurred in the early universe – has motivated a large commu- nity for several decades (cf. [1] and Refs. therein). Early concepts of the QGP were guided by the idea of a weakly interacting system of partons (quarks, antiquarks and gluons) since the entropy s and energy density ǫ were found in lattice QCD to be close to the Stefan Boltzmann (SB) limit for a relativistic noninteracting system [2]. However, this notion had to be given up in the last years since experimental ∗ corresponding author Email address: [email protected] (W. Cassing). Preprint submitted to Elsevier 24 November 2018 http://arxiv.org/abs/0704.1410v1 observations at the Relativistic Heavy Ion Collider (RHIC) indicated that the new medium created in ultrarelativistic Au+Au collisions was interacting more strongly than hadronic matter. Moreover, in line with earlier theoretical studies in Refs. [3– 5] the medium showed phenomena of an almost perfect liquid of partons [6,7] as extracted from the strong radial expansion and elliptic flow of hadrons as well the scaling of the elliptic flow with parton number etc. The latter collective observables have been severely underestimated in conventional string/hadron transport models [8–10], but hydrodynamical approaches did quite well in describing (at midrapidity) the collective properties of the medium generated during the early times for low and moderate transverse momenta [11,12]. Soon the question came up about the con- stituents of this liquid; it might be some kind of i) ”epoxy” [13], i.e. a system of resonant or bound gluonic states with large scattering length, ii) a system of chirally restored mesons, instanton molecules or equivalently giant collective modes [14], iii) a system of colored bound states of quarks q and gluons g, i.e. gq, qq, gg etc. [15], iv) some ’string spaghetti’ or ’pasta’ etc. In short, many properties of the new phase are still under debate and practically no dynamical concepts are available to describe the freezeout of partons to color neutral hadrons that are subject to experimental detection. Lattice QCD (lQCD) calculations provide some guidance to the thermodynamic properties of the partonic medium close to the transition at a critical temperature Tc up to a few times Tc, but lQCD calculations for transport coefficients presently are not accurate enough [16] to allow for firm conclusions. Furthermore, it is not clear whether the partonic system really reaches thermal and chemical equilibrium in ultrarelativistic nucleus-nucleus collisions and nonequilibrium models are needed to trace the entire collision history. The available string/hadron transport models [17–19] are not accurate enough - as pointed out above - nor do partonic cascade simulations [20–23] (propagating massless partons) sufficiently describe the reaction dynamics when employing cross sections from perturbative QCD (pQCD). This also holds - to some extent - for the Multiphase Transport Model AMPT [24] since it includes only on-shell massless partons in the partonic phase as in Ref. [21]. The same problem comes about in the parton cascade model of Xu and Greiner [25] where additional 2↔ 3 processes like gg ↔ ggg are incorporated. On the other hand it is well known that strongly interacting quantum systems require descriptions in terms of propagators D with sizeable selfenergies Π for the relevant degrees of freedom. Whereas the real part of the selfenergy gives contributions to the energy density, the imaginary parts of Π provide information about the lifetime and/or reaction rate of time-like ’particles’ [4]. In principle, off-shell transport equations are available in the literature [26–28], but have been applied only to dynamical problems where the width of the quasiparticles stays moderate with respect to the pole mass [29]. On the other hand, the studies of Peshier [30,31] indicate that the effective degrees of freedom in a partonic phase should have a width γ in the order of the pole mass M already slightly above Tc. The present study addresses essentially three questions: i) Do we understand the QCD thermodynamics in terms of dynamical quasiparticles down to the phase bound- ary in a ’top down’ scenario and what are the effective degrees of freedom as well as energy contributions? ii) Can such a quasiparticle approach help in defining an off-shell transport model that - at least in thermal equilibrium - reproduces the thermodynamic results from lQCD? iii) Are there any perspectives in modeling the transition from partonic to hadronic degrees of freedom in a dynamical way? The present work is exploratory in the sense that it is restricted to a pure gluonic system of N2c −1 gluons with two transverse polarisations, i.e. degeneracy dg = 16 for the gluonic quasiparticles that are treated as relativistic scalar fields. Note, however, that the qualitative features stay the same when adding light quark degrees of free- dom [31]; this finding is well in line with the approximate scaling of thermodynamic quantities from lQCD when dividing by the number of degrees of freedom and scaling by the individual critical temperature Tc which is a function of the different number of parton species [32]. The outline of the paper is as follows: After a short recapitulation of the dynamical quasiparticle model in Section 2 new results on the space-like and time-like parts of observables are presented that allow for a transparent physical interpretation. In Section 3 we will examine derivatives of the space-like part of the quasiparticle energy density with respect to the time-like (or scalar) density which provides information on gluonic mean fields and their effective interaction strength. The implications of these findings with respect to an off-shell transport description are pointed out throughout the study. A summary and extended discussion closes this work in Section 4. 2 Off-shell elements in the DQPM 2.1 Reminder of the DQPM The Dynamical QuasiParticle Model (DQPM) 1 adopted here goes back to Peshier [30,31] and starts with the entropy density s in the quasiparticle limit [33], sdqp = −dg (2π)3 Im ln(−∆−1) + ImΠRe∆ , (1) where n(ω/T ) = (exp(ω/T )− 1)−1 denotes the Bose distribution function, ∆ stands for the scalar quasiparticle propagator and Π for the quasiparticle selfenergy which is considered here to be a Lorentz scalar. In principle, the latter quantities are Lorentz tensors and should be evaluated in a nonperturbative framework. However, a more 1 DQPM also stands alternatively for Dynamical-Quasiparticle-Peshier-Model practical procedure is to use a physically motivated Ansatz with a Lorentzian spectral function, ρ(ω) = (ω −E)2 + γ2 (ω + E)2 + γ2 , (2) and to fit the few parameters to results from lQCD. With the convention E2(p) = 2+M2−γ2, the parameters M2 and γ are directly related to the real and imaginary parts of the corresponding (retarded) self-energy, Π = M2 − 2iγω. It should be stressed that the entropy density functional (1) is not restricted to quasiparticles of low width γ and thus weakly interacting particles. In fact, in the following it will be shown that a novel picture of the hot gluon liquid emerges because γ becomes comparable to the quasiparticle mass already slightly above Tc [30,31]. Following [34] the quasiparticle mass (squared) is written in (momentum-independent) perturbative form, M2(T ) = g2T 2 , (3) with a running coupling (squared), g2(T/Tc) = 11Nc ln(λ2(T/Tc − Ts/Tc)2 , (4) which permits for an enhancement near Tc [34,35]. It will be shown below that an infrared enhancement of the coupling - as also found in the lQCD calculations in Ref. [36] for the long range part of the q − q̄ potential - is directly linked to the gluon fusion/clustering scenario. In order to quantify this statement the coupling αs(T ) = g 2(T )/(4π) is shown in Fig. 1 as a function of T/Tc in comparison to the long range part of the strong coupling as extracted from Ref. [36] from the free energy of a quark-antiquark pair in quenched lQCD. For this comparison the actual parameters λ = 2.42, Ts/Tc = 0.46 have been adopted as in Ref. [4]. The parametrization (4) is seen to follow the lQCD results - also indicating a strong enhancement close to Tc - as a function of temperature reasonably well. One should recall that any extraction of coupling constants αs(T ) from lQCD is model dependent and deviations from (or agreement with) lattice ’data’ have to be considered with care. The argument here is that the specific ’parametric form’ of Eq. (4) is not in conflict with lQCD and that the coupling αs and consequently the quasiparticle mass M(T ) has the right order of magnitude. 1 2 3 4 5 6 7 8 9 10 Fig. 1. The coupling αs(T ) = g 2(T )/(4π) (solid red line) as a function of T/Tc in comparison to the long range part of the strong coupling as extracted from Ref. [36] from the free energy of a quark-antiquark pair in quenched lQCD (for Nτ = 8). The width γ is adopted in the form γ ∼ g2T ln g−1 [37] or, equivalently, in terms of M [30], as γ(T ) = M2(T ) (M(T )/T )2 , (5) where c = 14.4 (from [4]) is related to a magnetic cut-off. In case of the pure Yang- Mills sector of QCD the physical processes contributing to the width γ are both gg ↔ gg scattering as well as splitting and fusion reactions gg ↔ g or gg ↔ ggg, ggg ↔ gggg etc. Note that the ratio γ(T )/M(T ) ∼ g ln(c/g2) approaches zero only asymptotically for T → ∞ such that the width of the quasiparticles is comparable to the mass for all practical energy scales on earth; the ratio γ(T )/M(T ) drops below 0.5 only for temperatures T > 1.25 · 105 Tc (for the parameters given above). For the choice (2) for the spectral function the scalar effective propagator reads, ∆dqp(ω,p) = ω2 − p2 −M2 + 2iγω , (6) which can easily be separated into real and imaginary parts. The entropy density (1) then reads explicitly [31], sdqp(T ) = dg (2π)3 − ln(1− e−ωp/T ) + n(ωp/T ) (2π)3 arctan( ω2p − ω2 2γω(ω2p − ω2) (ω2p − ω2)2 + 4γ2ω2 , (7) using ωp = p2 +M2. The first line in (7) corresponds to the familiar on-shell quasi- particle contribution s0 while the second line in (7) corresponds to the contribution originating from the finite width γ of the quasiparticles and is positive throughout but subleading (see below). The pressure P now can be evaluated from by integration of s over T , where from now on we identify the ’full’ entropy density s with the quasiparticle entropy density sdqp. Note that for T < Tc the entropy density drops to zero (with decreasing T ) due to the high quasiparticle mass and the width γ vanishes as well because the interaction rate in the very dilute quasiparticle system becomes negligible. Since the pressure for infinitely heavy (noninteracting) particles also vanishes the integration constant for the pressure P - when integrating (8) - may safely be assumed to be zero, too. The energy density ǫ then follows from the thermodynamical relation [34,38] ǫ = Ts− P (9) and thus is also fixed by the entropy s(T ) as well as the interaction measure W (T ) := ǫ(T )− 3P (T ) = Ts− 4P (10) that vanishes for massless and noninteracting degrees of freedom. In Ref. [4] a detailed comparison has been presented with the lattice results from Ref. [39] for the pure gluonic sector to the quasiparticle entropy density (7) for the parameters given above. The agreement with the lattice data is practically perfect [4,30]. Needless to point out that also P (T ), ǫ(T ) and W (T ) well match the lattice QCD results for 1 ≤ T/Tc ≤ 4 [4,31] due to thermodynamical consistency. The same parameters are also adopted for the following calculations. 2.2 Time-like and space-like quantities For the further argumentation it is useful to introduce the shorthand notation P · · · = dg (2π)3 2ω ρ(ω) Θ(ω)n(ω/T ) Θ(±P 2) · · · (11) with P 2 = ω2 − p2 denoting the invariant mass squared. The Θ(±P 2) function in (11) separates time-like quantities from space-like quantities and can be inserted for any observable of interest. As the first quantity we consider the entropy density (7). Its time-like contribution is almost completely dominated by the first line in (7) - that corresponds to the on- shell quasiparticle contribution s0 - but also includes a small contribution from the second line in (7) which is positive for T below about 1.5 Tc and becomes negative for larger temperature. This time-like part s+ is shown in Fig. 2 by the dotted blue line (multiplied by (Tc/T ) 3). The second line in (7) - as mentioned above - corresponds to the contribution originating from the finite width γ of the quasiparticles and also has a space-like part s− which is dominant (for the second line in (7)) and displayed in Fig. 2 by the lower red line (multiplied by (Tc/T ) 3). Though s− is subleading in the total entropy density s = s+ + s− (thick solid green line in Fig. 2) it is essential for a proper reproduction of s(T ) close to Tc (cf. [31]). Note that the total entropy density s is not very different from the Stefan Boltzmann entropy density sSB for T > 2Tc as shown in Fig. 2 by the upper thin line (multiplied by (Tc/T ) 2 4 6 8 10 =0.26 GeV Fig. 2. The time-like contribution to the entropy density s+ (dotted blue line), the space-like contribution s− (lower red line) and the total entropy density s = s++s− (thick solid green line) as a function of T/Tc. All quantities have been multiplied by the dimensionless factor (Tc/T ) 3) assuming Tc = 0.26 GeV for the pure gluonic system [40]. The upper solid black line displays the Stefan Boltzmann limit sSB for reference. Further quantities of interest are the quasiparticle ’densities’ N±(T ) = T̃r± 1 (12) that correspond to the time-like (+) and space-like (-) parts of the integrated distri- bution function. Note that only the integral of N+ over space has a particle number interpretation. In QED this corresponds to time-like photons (γ∗) which are virtuell in intermediate processes but can also be seen asymptotically by dileptons (e.g. e+e− pairs) due to the decay γ∗ → e+e− [17]. A scalar density Ns, which is only defined in the time-like sector, is given by Ns(T ) = T̃r and has the virtue of being Lorentz invariant. Moreover, a scalar density can easily be computed in transport approaches for bosons and fermions [17,41] which is of relevance for the argumentation in Section 3. 0 2 4 6 8 10 0 2 4 6 8 10 =0.26 GeV T/TC Fig. 3. Upper part: The scalar density Ns (lower orange line), the time-like density N (blue line), the space-like quantity N− (red line) and the sum N = N+ +N− (thick solid green line) as a function of T/Tc assuming Tc = 0.26 GeV for the pure gluonic system [40]. The upper solid black line displays the Stefan Boltzmann limit NSB for reference. All quantities are multiplied by the dimensionless factor (Tc/T ) 3. Lower part: The ratio of the scalar density Ns to the time-like density N + as a function of the scaled temperature T/Tc. The actual results for the different ’densities’ (multiplied by (Tc/T ) 3) are displayed in the upper part of Fig. 3 where the lower orange line represents the scalar density Ns, the blue line the time-like density N +, the red line the space-like quantity N− and the thick solid green line the sum N = N++N− as a function of T/Tc assuming (as before) Tc = 0.26 GeV for the pure gluonic system [40]. It is seen that N substantially smaller than N− in the whole temperature range up to 10 Tc where it is tacitly assumed that the DQPM also represents lQCD results for T > 4Tc, which is not proven explicitly, but might be expected due to the proper weak coupling limit of (3), (5) (cf. Fig. 1). The application of the DQPM to 10 Tc is presented in Fig. 3 since the initial state at Large Hadron Collider (LHC) energies might be characterized by a temperature above 4 Tc; note that the properties of the partonic phase will be explored from the experimental side in the near future at LHC. Quite remarkably the quantity N follows closely the Stefan Boltzmann limit NSB for a massless noninteracting system which is given in Fig. 3 by the upper thin solid line and has the physical interpretation of a gluon density. Though N differs by less than 15% from the Stefan Boltzmann (SB) limit for T > 2TC the physical interpretation is essentially different! Whereas in the SB limit all gluons move on the light cone without interactions only a small fraction of gluons can be attributed to quasiparticles with density N+ within the DQPM that propagate within the lightcone. The space- like part N− corresponds to ’gluons’ exchanged in t-channel scattering processes and thus cannot be propagated explicitly in off-shell transport approaches without violating causality and/or Lorentz invariance. The scalar density Ns follows smoothly the time-like density N + as a function of temperature which can be explicitly seen in the lower part of Fig. 3 where the ratio + is shown versus T/Tc. Consequently, the scalar density Ns uniquely relates to the time-like density N+ or the temperature T in thermal equilibrium which will provide some perspectives for a transport theoretical treatment (see Section 3). The separation of N+ and N− so far has no direct dynamical implications except for the fact that only the fraction N+ can explicitly be propagated in transport as argued above. Thus we consider the energy densities, (T ) = T̃r± ω , (14) that specify time-like and space-like contributions to the quasiparticle energy density. It is worth pointing out that the quantity T00 = T 00 + T 00 in case of a conventional quasi-particle model with vanishing width γ in general is quite different from ǫ in (9) because the interaction energy density in this case is not included in (14), i.e. T00 = T (2π)3 2ω δ(ω2 −M2 − p2) Θ(ω) Θ(±P 2)n(ω/T ) ω (15) since ω2 − p2 = M2 = P 2 > 0 due to the mass-shell δ-function. How does the situation look like in case of dynamical quasiparticles of finite width? To this aim we consider the integrand in the energy density (14) which reads as (in spherical momentum coordinates with angular degrees of freedom integrated out) I(ω, p) = p2 ω2 ρ(ω, p2)n(ω/T ) . (16) Here the integration is to be taken over ω and p from 0 to ∞. The integrand I(ω, p) is shown in Fig. 4 for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of contour lines. For the lower temperature the gluon mass is about 0.91 GeV and the width γ ≈ 0.15 GeV such that the quasiparticle properties are close to a ρ-meson in free space. In this case the integrand I(ω, p) is essentially located in the time-like sector and the integral over the space-like sector is subdominant. This situation changes for T = 2Tc where the mass is about 0.86 GeV while the width increases to γ ≈ 0.56 GeV. As one observes from the r.h.s. of Fig. 4 the maximum of the integrand is shifted towards the line ω = p and higher momentum due to the increase in temperature by about a factor of two; furthermore, the distribution reaches far out in the space-like sector due to the Bose factor n(ω/T ) which favors small ω. Thus the relative importance of the time-like (+) part to the space-like (-) part is dominantly controlled by the width γ - relative to the pole mass - which determines the fraction of T−00 with negative invariant mass squared (P 2 < 0) relative to the time-like part T+00. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 I(ω,p) T=1.02 T time-like space-like p [GeV/c] 0 1 2 3 4 5 5 p=ω I(ω,p) T=2 T time-like space-like p [GeV/c] Fig. 4. The integrand I(ω, p) (16) for T = 1.02Tc (l.h.s.) and T = 2Tc (r.h.s.) in terms of contour lines. The straight (blue) line (ω = p) separates the lime-like from the space-like sector. Note that for a convergence of the energy density integral the upper limits for ω and p have to be increased by roughly an order of magnitude compared to the area shown in the figure. The explicit results for the quasiparticle energy densities T+00 and T 00 are displayed in Fig. 5 by the dashed blue and dot-dashed red lines (multiplied by (TC/T ) respectively. As in case of N+ and N− the space-like energy density T−00 is seen 0 2 4 6 8 10 0.9 1.0 1.1 1.2 1.3 1.4 1.5 =0.26 GeV =0.26 GeV Fig. 5. Upper part: The time-like energy density T+ (dashed blue line), the space-like energy density T− (dot-dashed red line) and the total energy density T00 = T (thick solid green line) as a function of T/Tc. The thin black line displays the energy density ǫ(T/Tc) from (9); it practically coincides with T00 within the linewidth and is hardly visible. All densities are multiplied by the dimensionless factor (Tc/T ) 4 in order to divide out the leading temperature dependence. Lower part: Same as the upper part in order to enhance the resolution close to Tc. to be larger than the time-like part T+00 for all temperatures above 1.05 Tc. Since the time-like part T+00 corresponds to the independent quasiparticle energy density within the lightcone, the space-like part T−00 can be interpreted as an interaction density V if the quasiparticle energy T00 matches the total energy density ǫ(T ) (9) as determined from the thermodynamical relations (8) and (9). In fact, the DQPM yields an energy density T00 - adding up the space-like and time-like parts - that almost coincides with ǫ(T ) from (9) as seen in Fig. 5 where both quantities (multiplied by (TC/T ) 4) are displayed in terms of the thin black and thick solid green lines, respectively; actually both results practically coincide within the linewidth for T > 2Tc. An explicit representation of their numerical ratio gives unity within 2% for T > 2Tc; the remaining differences can be attributed to temperature derivatives ∼ d/dT (ln(γ/E)) etc. in order to achieve thermodynamic consistency but this is not the primary issue here and will be discussed in a forthcoming study [42]. The deviations are more clearly visible close to Tc (lower part of Fig. 2) where the variation of the width and mass are most pronounced. However, for all practical purposes one may consider T00(T ) ≈ ǫ(T ) and separate the kinetic energy density T+00 from the potential energy density T−00 as a function of T or - in equilibrium - as a function of the scalar gluon density Ns or N +, respectively. 3 Dynamics of time-like quasiparticles Since in transport dynamical approaches there are no thermodynamical Lagrange parameters like the inverse temperature β = T−1 or the quark chemical potential µq, which have to be introduced in thermodynamics in order to specify the average values of conserved quantities (or currents in the relativistic sense), derivatives of physical quantities with respect to the scalar density ρs = Ns (or time-like gluon density ρg = N +) are considered in the following (cf. Ref. [43]). As mentioned above one may relate derivatives in thermodynamic equilibrium via, , (17) if the volume and pressure are kept constant. For example, a numerical evaluation of dρs/d(T/Tc) gives d(T/Tc) − a2 exp(−b( )) (18) with b= 5, a1 = 1.5fm −3 and a2 = 104fm −3, which follows closely the quadratic scaling in T/Tc as expected in the Stefan Boltzmann limit. The additional exponential term in (18) provides a sizeable correction close to Tc. The approximation (18) may be exploited for convenient conversions between ρs and T/Tc in the pure gluon case but will not be explicitly used in the following. The independent quasiparticle energy density TK := T 00 and potential energy density V := T−00 now may be expressed as functions of ρs (or ρg) instead of the temperature T . The interaction energy density then might be considered as a scalar energy density which - as in the nonlinear σ-model for baryonic matter [44] - is a nonlinear function of the scalar density ρs. As in case of nuclear matter problems the scalar density ρs does not correspond to a conserved quantity when integrating over space; it only specifies the interaction density parametrically, i.e. V (ρs). Alternatively one might separate V into parts with different Lorentz structure, e.g. scalar and vector parts as in case of nuclear matter problems [44], but this requires additional information that cannot be deduced from the DQPM alone. 0 2 4 6 8 10 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Fig. 6. Upper part: The quasiparticle energy per degree of freedom TK/N + (dashed blue line) and the space-like potential energy per degree of freedom V/N+ (dot-dashed red line) as a function of T/Tc. All energies are multiplied by the dimensionless factor (Tc/T ). Lower part: Same as the upper part in order to enhance the resolution close to Tc. It is instructive to show the ’quasiparticle’ and potential energy per degree of free- dom TK/N + and V/N+ as a function of e.g. N+, Ns or T/Tc. As one might have anticipated the kinetic energy per effective degree of freedom is smaller than the respective potential energy for T/Tc > 1.05 as seen from Fig. 6 where both quan- tities are displayed as a function of T/Tc in terms of the dashed and dot-dashed line, respectively. It is seen that the potential energy per degree of freedom steeply rises in the vicinity of Tc whereas the independent quasiparticle energy rises almost linearly with T . Consequently rapid changes in the density - as in the expansion of the fireball in ultrarelativistic nucleus-nucleus collisions - are accompanied by a dramatic change in the potential energy density and thus to a violent acceleration of the quasi-particles. It is speculated here that the large collective flow of practically all hadrons seen at RHIC [6] might be attributed to the early strong partonic forces expected from the DQPM. 1 10 100 Fig. 7. The mean-field potential U(N+) = U(ρg) as a function of the time-like gluon density N+ = ρg in comparison to the fit (19) (solid blue line). The densities N += 1, 1.4, 5, 10, 50, 100 fm−3 correspond to scaled temperatures of T/Tc ≈ 1.025, 1.045, 1.25, 1.5, 2.58, 3.25, respectively (cf. Fig. 3). In order to obtain some idea about the mean-field potential Us(ρs) (or U(ρg) in the rest frame) one can consider the derivative dV/ρs = Us(ρs) or dV/N + = U(N+) = U(ρg). The latter is displayed in Fig. 7 as a function of N + = ρg and shows a distinct minimum at ρg ≈ 1.4 fm−3 which corresponds to a temperature T ≈ 1.045Tc. The actual numerical results can be fitted by the expression, U(ρg) = ≈ 39 e−ρg/0.31 + 2.93 ρ0.21g + 0.55 ρ g [GeV] , (19) where ρg is given in fm −3 and the actual numbers in front carry a dimension in order to match to the proper units of GeV for the mean-field U . By analytical integration of (19) one obtains a suitable approximation to V (ρg). The approximation (19) works sufficiently well as can be seen from Fig. 7 - showing a comparison of the numerical derivative dV/dN+ with the fit (19) in the interval 0.7 fm−3 < N+ ≤ 300 fm−3 - such that one may even proceed with further analytical calculations. Note that a conversion between the time-like quasiparticle density N+ = ρg and the scalar density ρs is easily available numerically (cf. lower part of Fig. 3) such that derivatives with respect to ρs are at hand, too; the latter actually enter the explicit transport calculations [45] while derivatives with respect to ρg in the rest frame of the system are more suitable for physical interpretation and will be used below. Some information on the properties of the effective gluon-gluon interaction vgg may be extracted from the second derivative of V with respect to ρg, i.e. vgg(ρg) := ≈ −125.8 e−ρg/0.31 + 0.615/ρ0.79g + 0.2/ρ g [GeVfm 3], (20) where the numbers in front have again a dimension to match the units of GeV fm3. The effective gluon-gluon interaction vgg (20) is strongly attractive at low den- sity 0.003 fm−3 < ρg and changes sign at ρg ≈ 1.4 fm−3 to become repulsive at higher densities. Note that the change of quasiparticle momenta (apart from col- lisions) will be essentially driven by the (negative) space-derivatives −∇U(x) = −dU(ρg)/dρg ∇ρg(x) (or alternatively by −dUs(ρs)/dρs ∇ρs(x)). This implies that the gluonic quasiparticles (at low gluon density) will bind with decreasing density, i.e. form ’glueballs’ dynamically close to the phase boundary and repell each other for ρg ≥ 1.4 fm−3. Note that color neutrality is imposed by color-current conser- vation and only acts as a boundary condition for the quantum numbers of the bound/resonant states in color space. This situation is somehow reminiscent of the nuclear matter problem [44] where a change in sign of the 2nd derivative of the potential energy density of nuclear matter at low density indicates the onset of clustering of nucleons, i.e. to deuterons, tritons, α-particles etc., which form the states of the many-body system at low nucleon densities (and not a low density nucleon gas). This is easy to follow up for the simplified nonrelativistic energy density functional ǫN for nuclear matter, ǫN ≈ Aρ ρ2N + N , (21) where the first term gives the kinetic energy density and the second and third term correspond to attractive and repulsive interaction densities. For A ≈ 0.073GeV fm2, B ≈ −1.3 GeV fm3 and C ≈ 1.78 GeV fm4 a suitable energy density for nuclear matter is achieved; it gives a minimum in the energy per nucleon E/A = ǫN/ρN ≈ −0.016 GeV for nuclear saturation density ρ0N ≈ 0.168 fm−3. The mean-field potential UN = BρN+Cρ N has a minimum close to ρ N such that the effective nucleon-nucleon interaction strength vNN = B + 4/3Cρ N changes from attraction to repulsion at this density. Note that in the gluonic case the minimum in the mean-field potential U (19) occurs at roughly 8 times ρ0N and the strength of the gluonic interaction is higher by more than 2 orders of magnitude! The confining nature of the effective gluon-gluon interaction vgg (20) becomes ap- parent in the limit ρg → 0, where the huge negative exponential term dominates for ρg > 0.003 fm −3; for even smaller densities the singular repulsive terms take over. Note, however, that the functional extrapolation of the fit (19) to vanishing gluon density ρg has to be considered with care and it should only be concluded that the interaction strength becomes ’very large’. On the other hand the limit ρg → 0 is only academical because the condensation/fusion dynamically occurs for ρg ≈ 1 fm−3. A straight forward way to model the gluon condensation or clustering to confined glueballs dynamically (close to the phase transition) is to adopt a screened Coulomb- like potential vc(r,Λ) with the strength d3r vc(r,Λ) fixed by vgg(ρg) from (20) and the screening length Λ from lQCD studies. For the ’dilute gluon regime’ (ρg < 1.4 fm−3), where two-body interactions should dominate, one may solve a Schrödinger (or Klein-Gordon) equation for the bound and/or resonant states. This task is not addressed further in the present study since for the actual applications (as in the Parton-Hadron-String-Dynamics (PHSD) approach [45]) dynamical quark and anti- quarks have to be included. The latter degrees of freedom do not change the general picture very much for higher temperatures T > 2Tc but the actual numbers are dif- ferent close to Tc since the quarks and antiquarks here dominate over the gluons due to their lower mass. The reader is referred to an upcoming study in Ref. [42]. Some comments on expanding gluonic systems in equilibrium appear in place, i.e. for processes where the total volume Ṽ and pressure P play an additional role. For orientation we show the entropy per time-like particle s/N+ in Fig. 8 as a function of N+ (upper) and T/Tc (lower part) which drops close to the phase boundary since the quasiparticles become weakly interacting (cf. Fig. 6). Note that this is essentially due to the low density and not due to the interaction strength (20); a decrease of the width γ (as encoded in (5)) implies a decrease in the interaction rate! An expansion process with conserved total entropy S = sṼ leads to a change in the total gluon number N+Ṽ since s/N+ changes with density (or temperature) (Fig. 8). The same holds for an expansion process with constant total energy ǫṼ since also ǫ/N+ is varying with density (or temperature). Other scenarios involving e.g. S = P/T also involve a change of the gluon number N+Ṽ during the cooling process such that reactions like gg ↔ g, ggg ↔ gg etc. are necessary ingredients of any transport theoretical approximation. We do not further investigate different expansion scenarios here since the reactions g ↔ qq̄, i.e. the gluon splitting to a quark and antiquark as well as the backward fusion process, are found to play a dominant role in the vicinity of the phase transition as well as for higher temperatures [42,45]. 4 Conclusions and discussion The present study has provided a novel interpretation of the dynamical quasiparticle model (DQPM) by separating time-like and space-like quantities for ’particle densi- ties’, energy densities, entropy densities ect. that also paves the way for an off-shell transport approach [45]. The entropy density s in (7) is found to be dominated by the on-shell quasiparticle contribution (first line in (7)) (cf. [31]) while the space-like part of the off-shell contribution (second line in (7)) gives only a small (but important) enhancement (cf. Fig. 2). However, in case of the ’gluon density’ N = N+ +N− and the gluon energy density T00 = T 00 the situation is opposite: here the space-like 1 2 3 4 5 6 7 8 9 10 1 10 100 1000 T/TC Fig. 8. The entropy per degree of freedom s/N+ as a function of N+ (upper part) or T/Tc (lower part). parts (N−, T−00) dominate over the time-like parts (N +, T+00) except close to Tc where the independent quasiparticle limit is approximately regained. The latter limit is a direct consequence of the infrared enhancement of the coupling (4) close to Tc (in line with the lQCD studies in Ref. [36] ) and a decrease of the width γ (5) when approaching Tc from above. Since only the time-like part N+ can be propagated within the lightcone the space- like part N− has to be attributed to t-channel exchange gluons in scattering processes that contribute also to the space-like energy density T−00. The latter quantity may be regarded as potential energy density V . This, in fact, is legitimate since the quasiparticle energy density T00 very well matches the energy density (9) obtained from the thermodynamical relations. Only small deviations close to Tc indicate that the DQPM in its straightforward application is not thermodynamically consistent. However, by accounting for ’rearrangement terms’ in the energy density - as known from the nuclear many-body problem [46] - full thermodynamical consistency may be regained [42]. It is instructive to compare the present DQPM to other recent models. In the PNJL 2 model [47] the gluonic pressure is build up by a constant effective potential U(Φ,Φ∗;T ) which controls the thermodynamics of the Polyakov loop Φ. It is ex- panded in powers of ΦΦ∗ with temperature dependent coefficients in order to match lQCD thermodynamics. Thus in the PNJL there are no time-like gluons; the effec- tive potential U(Φ,Φ∗;T ) stands for a static gluonic pressure that couples to the quark/antiquark degrees of freedom. The latter are treated in mean-field approxi- mation, i.e. without dynamical width, whereas the DQPM incorporates a sizeable width γ. Another approach to model lQCD thermodynamics has been suggested in Ref. [43] and is based on an effective Lagrangian which is nonlinear in the effective quark and gluon fields. In this way the authors avoid a parametrization of the interaction density in terms of Lagrange parameters (T, µ) and achieve thermodynamical consistency. The latter approach is closer in spirit to the actual interpretation of the DQPM and may be well suited for an on-shell transport theoretical formulation. The on-shell restriction here comes about since effective Lagrangian approaches should only be evaluated in the mean-field limit which implies vanishing scattering width for the quasiparticles. This is sufficient to describe systems is thermodynamical equilibrium, where forward and backward interaction rates are the same, but might not provide the proper dynamics out-of-equilibrium. Some note of caution with respect to the present DQPM appears appropriate: the pa- rameters in the effective coupling (4) and the width (5) have been fixed in the DQPM by the entropy density (7) to lQCD results assuming the form (2) for the spectral function ρ(ω). Alternative assumptions for ρ(ω) will lead to slightly different results for the time-like density, energy densities etc. but not to a qualitatively different picture. Independent quantities from lQCD should allow to put further constraints on the more precise form of ρ(ω) such as calculations for transport coefficients [16]; unfortunately such lQCD studies are only at the beginning. A more important issue is presently to extend the DQPM to incorporate dynamical quark and antiquark degrees of freedom (as in [31]) in order to catch the physics of gluon splitting and quark-antiquark fusion (g ↔ q + q̄, g + g ↔ q + q̄ + g) reactions [42,45]. Coming back to the questions raised in the Introduction concerning i) the appropri- ate description of QCD thermodynamics within the DQPM and ii) the possibility to develop a consistent off-shell partonic transport approach as well as iii) the perspec- tives for a dynamical description of the transition from partonic to hadronic degrees of freedom, we are now in the position to state: most likely ’Yes’. 2 Polyakov-loop-extended Nambu Jona-Lasinio The author acknowledges valuable discussions with E. L. 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0704.1412

Dynamic fracture of icosahedral model quasicrystals: A molecular dynamics study

Dynamic fracture of icosahedral model quasicrystals: A molecular dynamics study Frohmut Rösch, Christoph Rudhart, Johannes Roth, 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 and Fraunhofer Institut für Werkstoffmechanik, Wöhlerstr. 11, 79108 Freiburg, Germany Ebert et al. [Phys. Rev. Lett. 77, 3827 (1996)] have fractured icosahedral Al-Mn-Pd single crystals in ultrahigh vacuum and have investigated the cleavage planes in-situ by scanning tunneling mi- croscopy (STM). Globular patterns in the STM-images were interpreted as clusters of atoms. These are significant structural units of quasicrystals. The experiments of Ebert et al. imply that they are also stable physical entities, a property controversially discussed currently. For a clarification we performed the first large scale fracture simulations on three-dimensional complex binary systems. We studied the propagation of mode I cracks in an icosahedral model quasicrystal by molecular dynamics techniques at low temperature. In particular we examined how the shape of the cleavage plane is influenced by the clusters inherent in the model and how it depends on the plane structure. Brittle fracture with no indication of dislocation activity is observed. The crack surfaces are rough on the scale of the clusters, but exhibit constant average heights for orientations perpendicular to high symmetry axes. From detailed analyses of the fractured samples we conclude that both, the plane structure and the clusters, strongly influence dynamic fracture in quasicrystals and that the clusters therefore have to be regarded as physical entities. PACS numbers: 62.20.Mk, 61.44.Br, 02.70.Ns Keywords: fracture, quasicrystals, molecular dynamics simulations I. INTRODUCTION Quasicrystals are intermetallic compounds with long- range quasi-periodic translational order. They possess well-defined atomic planes and hence diffract electromag- netic and matter waves into sharp Bragg spots. But they also display atomic clusters as basic building blocks1,2, whose arrangement in space is compatible with the pla- nar structure. These clusters consist for example of sev- eral shells of icosahedral symmetry (Bergman-, Mackay-, pseudo-Mackay-clusters). Or they form polytopes, e.g. decagonal prisms, which like the unit cells of periodic crystals fill space, although with large overlaps (“quasi- unit-cell picture”)3. Janot and others4,5,6 have postu- lated that a self-similar hierarchical assembly of the clus- ters is responsible for the stability of quasicrystals and for many physical properties, like the low electric conduc- tivity. However, it is a controversial and persistent dis- cussion, whether the clusters are merely structural units or whether they represent physical entities. The discus- sion was fueled by an experiment of Ebert et al.7, where icosahedral Al-Mn-Pd was fractured under ultrahigh vac- uum conditions at room temperature. Scanning tunnel- ing microscopy images of the cleavage planes revealed elements of 0.6 to 1 nm in diameter8. The authors ar- gue that these are the clusters which were circumvented by the crack and hence form highly stable aggregates of matter. Others point out that flat terraces evolve on fivefold surfaces of i-Al-Mn-Pd when annealed at high temperatures9,10,11,12. As this requires truncated rows of Bergman and Mackay clusters it is stated13 that these therefore could not represent firm entities. In the present article we report on molecular dynamics simulations of crack propagation in a three-dimensional icosahedral model quasicrystal at low temperature. Seed cracks are inserted along different planes and therein along different directions. The fracture planes are care- fully analysed to answer the role of clusters in dynamic fracture. In Sec. II we provide some requirements on the theoretical description of fracture. In Sec. III the model quasicrystal, the molecular dynamics technique, and the methods to visualise the results of the simulations are outlined. Subsequently, in Sec. IV the simulation results are presented and then discussed in Sec. V. II. FRACTURE The stress concentration and the strength of the load- ing at a crack tip are determined by the macroscopic geometry and dimensions of a sample. In linear elastic continuum mechanics, a sharp mode I (opening mode) crack is characterised by a singular stress field and a cor- responding displacement field, which both are propor- tional to the stress intensity factor K. This factor is proportional to the applied external load and contains the geometry of the sample. A simple energy based con- dition for crack propagation is the Griffith criterion14. It states that a crack is in equilibrium when the change in mechanical energy per unit area of crack advance – the energy release rate G – equals the change in surface energy of the two fracture surfaces, 2γ. In continuum mechanics the energy release rate is proportional to the square of the stress intensity factor for a given mode. A crack then should start moving when the stress intensity factor exceeds the critical Griffith value. A continuum mechanical description of fracture, how- ever, has a few drawbacks. First, the requirements for linear elasticity are no longer valid near the crack tip where atomic bonds clearly become non-linear and even- tually break. Second, a continuum theory neglects the discrete nature of the lattice. Thus, it is fully ignored that fracture of materials is ultimately caused by bond breaking processes on the atomic scale. A way to understand the processes is to perform nu- merical experiments, since experimental information on this length scale is difficult to obtain. Molecular dynam- ics studies have provided useful insight into crack prop- agation in pure metals and simple intermetallic alloys, whereas in complex metallic alloys the mechanisms are not yet so clear. Atomistic studies show for example that cracks remain stable in a region around the critical stress intensity factor due to the discrete nature of the lattice. This effect is called lattice trapping15. A further consequence of the discrete lattice is that the fracture behaviour in one and the same plane can depend on the crack propagation direction16. Such observations cannot be explained by a simple continuum mechanical descrip- tion. III. MODEL AND METHOD A. Icosahedral binary model In the numerical experiments we use a three- dimensional model quasicrystal which has been proposed by Henley and Elser17 as a structure model for the icosa- hedral phase of (Al,Zn)Mg. This is the simplest possi- ble model quasicrystal that is stabilised by pair poten- tials. Furthermore it allows Burgers circuit analysis and is a prototype of Bergman-type quasicrystals. As we do not distinguish between Al and Zn atoms, we term this decoration icosahedral binary model. It can be ob- tained by decorating the structure elements of the three- dimensional Penrose tiling, the oblate and the prolate rhombohedra (see Fig. 1, top). Al and Zn atoms (A atoms) are placed on the vertices and the midpoints of the edges of the rhombohedra. Two Mg atoms (B atoms) divide the long body diagonal of each prolate rhombohe- dron in ratios τ :1:τ , where τ is the golden mean. Two prolate and two oblate rhombohedra with a common ver- tex form a rhombic dodecahedron18,19. To obtain the icosahedral binary model, in these dodecahedra the atom at the common vertex is removed and the four neigh- bouring A atoms are transformed into B atoms. Finally these atoms are shifted to the common vertex to divide the edges of the corresponding rhombohedra in a ratio of 1:τ . Fig. 1 (bottom left) shows the final decoration of these dodecahedra, in which the B atoms form hexagonal bipyramides. This modification increases the number of Bergman-type clusters (see Fig. 1, bottom right) inher- ent in the structure, leads to a higher stability with the potentials used, and takes better into account the experi- mentally observed stoichiometry of the quasicrystal. The Bergman-type clusters may also be interpreted as build- ing units of the quasicrystal and are the main feature of the structure apart from the plane structure. B. Molecular dynamics technique As is often done in fracture simulations (see e.g. Abra- ham20) we use simple Lennard-Jones potentials to model the interactions. The following facts led to this choice: First, the very few potentials available for quasicrys- tals are unsuitable for fracture simulations: These pair potentials stabilise the quasicrystal only for a fixed vol- ume. With free surfaces atoms sometimes simply evap- orate (e.g. with potentials based on those of Hafner et al.21). Very long-ranged potentials with Friedel oscilla- tions (e.g. those of Al-Lehyani et al.22) frequently display such large cohesive energies that nearly no elastic defor- mation is possible and instead intrinsic cracks develop. Second, many simulations have proven that model po- tentials are helping to understand the elementary pro- FIG. 1: Tiles of the icosahedral binary model decorated with two types of atoms and the Bergman-type cluster. Top: pro- late rhombohedron (left) and oblate rhombohedron (right). Bottom: rhombic dodecahedron (left) and the 45 atoms build- ing the Bergman-type cluster (right) inherent in the model quasicrystal. Small grey and large black spheres denote A and B atoms respectively. cesses in fracture (see e.g. Abraham20) and so are rea- sonable when qualitative mechanisms are the centre of interest. For quantitative results, a known limitation is the neglect of non-local and many-body interactions. Third, the Lennard-Jones potentials used23,24 keep the model stable even under strongest mechanical deforma- tions or irradiation (introduction of point defects) and have been used in our group in many simulations of dis- location motion25 or even shock waves26. The structure is robust under a wide variation of the potential depths. Very similar potentials have shown to stabilise the icosa- hedral atomic structure in a simpler model27. It is also known since the early fifties that Lennard-Jones poten- tials favour icosahedral clusters28, indicating that these potentials are useful for structures like icosahedral qua- sicrystals. The minima of the potentials for interactions between atoms of the same type are set to �0, whereas the mini- mum of the potential for the interactions between atoms of different kind is set to 2�0. The conclusions drawn from our simulations, however, remained essentially un- changed if all binding energies are set equal. The shortest distance between two A atoms is denoted r0. All masses are set to m0. The time is then measured in units of t0 = r0 m0/�0. The molecular dynamics simulations were carried out using the microcanonical ensemble with the program code IMD29,30. It performs well on a large variety of hardware, including single and dual processor worksta- tions and massively parallel supercomputers. First, we searched for the potential cleavage planes. According to the Griffith criterion they should be planes of low surface energy31. To identify these surfaces we re- lax a specimen and split it into two parts. Subsequently, the two regions are shifted apart rigidly. The surface en- ergy is then calculated from the energy difference of the artificially cleaved and the undisturbed specimen. In contrast to simple periodic crystals, the atomistic structure of the planes and therefore also the surface en- ergy in quasicrystals varies strongly within the family of planes perpendicular to a fixed axis. In the present model, surfaces with low interface energy are located perpendicular to two- and fivefold directions at certain heights (Fig. 2). Perpendicular to other directions the plane structure is less pronounced and the minimal sur- face energies are higher. Since we are interested in the morphology of fracture surfaces we apply a sample form that allows us to fol- low the dynamics of the running crack for a long time. For this purpose, a strip geometry is used to model crack propagation with constant energy release rate32. The samples consist of about 4 to 5 million atoms, with di- mensions of approximately 450r0×150r0×70r0. Periodic boundary conditions33 are applied in the direction paral- lel to the crack front. For the other directions, all atoms in the outermost boundary layers of width 2.5r0 are held fixed. An atomically sharp seed crack is inserted at a plane of lowest surface energy, from one side to about FIG. 2: Surface energy of cleavage planes perpendicular to two- and fivefold axes. one quarter of the strip length. The system is uniaxially strained perpendicular to the crack plane up to the Grif- fith load and is relaxed to obtain the displacement field of a stable crack at zero temperature. Then a temperature of about 10−4 of the melting temperature is applied34 to the configurations with and without the relaxed crack. From the resulting configurations we obtain an averaged displacement field for this temperature. The crack now is further loaded by linear scaling of this displacement field. The response of the system then is monitored by molec- ular dynamics techniques. The sound waves emitted by the propagating crack (see Fig. 3 and online movie35) are damped away outside of an elliptical stadium32 to prevent reflections. C. Visualisation To study crack propagation on an atomic level the se- lection and reduction of data is of crucial importance. Due to the large number of atoms required for the study of crack propagation in three-dimensional systems, it is not feasible to write out the positions of all atoms during the simulation, and even less to display all of them. To obtain a first insight into crack propagation only atoms FIG. 3: Kinetic energy density. Sound waves emitted by the propagating crack and the elliptical region without damping are clearly visible. See online movie35. FIG. 4: Snapshot of a simulation with some 4 million atoms. Only atoms with low coordination number are displayed. See online movie36. near the fracture surfaces are of interest. Whereas they can be visualised in periodic crystals by plotting only those atoms whose potential energy exceeds a certain threshold, this technique is not applicable for quasicrys- tals. Because of the largely varying environments the potential energy varies significantly from atom to atom, even for atoms of the same type in a defect-free sample. A solution to this problem is to display only those atoms whose coordination number is smaller than a cer- tain threshold. This number is evaluated by counting atoms within the nearest neighbour distance. Like the potential energy, in quasicrystals the coordination num- ber varies from atom to atom, but to a much smaller de- gree. As atoms near defects have a significantly lower co- ordination number, it becomes possible to visualise frac- ture surfaces and dislocation cores. For the A atoms the threshold for the coordination number is set to 12, whereas for the B atoms it is set to 14. With this method, the number of atoms to write out or to display can be reduced by three orders of magnitude. Fig. 4 shows a snapshot of a simulation with some 4 million atoms fil- tered by this technique, which was also used in a movie that is available online36. As can be seen from Fig. 4 the fracture surfaces are rough. Therefore to decide to which fracture surface an atom belongs we apply the displacement vectors between the initial configuration with the built-in seed crack and the fractured sample. The morphology of the fracture planes is then analysed via geometrical scanning of the atoms forming the surfaces. The roughness can be visu- alised by colour coding the height of the surface in a view normal to the fracture surface. To investigate the influence of the Bergman-type clus- ters on cleavage they have to be displayed together with the fracture surfaces intersecting them. This is done by restoring the initial sample without crack at zero tem- perature. The atoms forming the two sample parts are taken back to their positions in this initial sample and FIG. 5: Bergman-type cluster cut by a flat surface. then scanned geometrically. In addition all atoms form- ing clusters in the vicinity of this surfaces are known. By displaying only these atoms and the scanned surface one directly can see where and to which amount clusters are cut. A problem of this kind of visualisation is shown in Fig. 5, where a cluster is cut by a flat surface. When looked-at from above it is obvious that only four atoms are separated from the rest of the cluster. When looked- at from below one could get the impression that the clus- ter is heavily intersected. On a real fracture surface clus- ters with centres above and below the crack surface are present. Therefore both views of Fig. 5 are appearing at the same time in a two-dimensional projection. Thus such pictures are not very intuitive. A way out of this dilemma is presented in Fig. 6. Clus- ters with midpoints above the crack surface are displayed together with the upper geometrically scanned fracture surface only, the other clusters are shown together with the lower fracture surface. As a result we get two pictures with clusters cut by surfaces. Note that for a qualitative and quantitative analysis always both pictures or sets of data are needed. FIG. 6: Visualisation of the clusters cut by the dynamic crack. Midpoints of clusters are indicated as black dots, the clusters are idealised as spheres. The crack propagated from the left to the right. The upper and lower fracture surfaces are projected onto an x− z plane. IV. RESULTS In this section the results of the numerical simulations are presented. For practical purposes we define k as the stress intensity factor K relative to the stress intensity factor KG at the Griffith load: k = K/KG. The orientations of the samples are characterised by the notation yx, where y is an axis perpendicular to the cleav- age plane and x is an axis in the crack propagation di- rection (see Fig. 6). An axis perpendicular to a fivefold (5) and a twofold (2) axis is denoted pseudo-twofold (p2) axis. A. Crack velocities Simulations were performed for different orientations with loads in a range from k = 1.1 to k = 2 (see nota- tions and loads in Fig. 7). Brittle fracture without any crack tip plasticity is observed irrespective of the orien- tation of the fracture plane. For loads below k = 1.2, the crack propagates only a few atomic distances, and then stops. Hence the energy needed for initiating crack propagation is about 1.4 times the value predicted by the Griffith criterion. Therefore, a simple global thermody- namic description of fracture is not applicable. The min- imal velocity for brittle crack propagation is about 10% of the shear wave velocity37 vs. For loads k ≥ 1.2 the velocity increases monotonically with the applied load. The crack velocities are in a range of 10-45% of vs (see Fig. 7). At high loads the crack velocities on fivefold cleavage planes show higher average velocities than on the other planes. Velocities for the two different crack propagation directions on the fivefold planes differ sig- nificantly at intermediate loads (k = 1.3). This coincides with ledges that are produced in the fracture surface (see section IV B and Fig. 8, bottom). B. Fracture surfaces To analyse the morphology of the fracture surfaces, they are displayed as described in section III C. In Fig. 8 the average height is shown in grey, heights above +2r0 are shown in white and heights below −2r0 are shown in black. The crack propagation direction is from the left to the right. The initial fracture surface is flat, as can be seen from the homogeneous regions on the left. The surfaces resulting from the propagation of the crack, however, show pronounced patterns of regions with dif- ferent heights. From the observation of the fracture sur- faces it is already evident that they are rough and that the peak-to-valley roughness is of the order of the di- ameter of the clusters. The peak-to-valley roughness and the root-mean-square roughness of the height pro- files both increase38 for higher loads for surfaces without FIG. 7: Average crack velocities for different loads and orien- tations. ledge formation. For fracture surfaces perpendicular to twofold and fivefold axes the crack fluctuates about a constant height (in the areas without ledges). In con- trast, a crack inserted perpendicular to a pseudo-twofold direction seems to deviate from this plane39. C. Anisotropy As can be seen already from the fracture surfaces per- pendicular to a fivefold axis in Fig. 8 crack surfaces for the same cleavage plane differ significantly for different in- plane crack propagation directions. For the orientation 5p2 ledges are produced, while no ledges form for the ori- entation 52. By visual inspection of the fracture surfaces (see Fig. 8) and evaluation of height-height-correlation functions23 it becomes evident that for every orientation there are distinct angles in the height profile, which show pronounced height correlations that correspond to mark- ings in the fracture surfaces or to ledges. These angles are given in Table I. D. Clusters In Fig. 9 the clusters cut by the fracture surfaces are presented as described in section III C and Fig. 6. It is obvious from Fig. 9 that the dynamic crack does not per- fectly circumvent the clusters, but intersects them par- tially (right side of Fig. 9). These intersections, however, TABLE I: Angles observed in the height profiles of the frac- ture surfaces. Angles measured clockwise from the crack prop- agation direction get a negative sign. orientation 22 25 52 5p2 p25 angles 0◦,±32◦ 0◦, +32◦,−58◦ 0◦,±36◦ ±18◦ 0◦,±90◦ FIG. 8: Height profiles of sections of the simulated fracture surfaces. Load: k = 1.3, orientation 22 (top), 52 (middle), 5p2 (bottom). The height increases from black (≤ −2r0) to white (≥ +2r0). The scanning sphere has the same size as an atom of type B. FIG. 9: Clusters cut by the fracture surfaces. The visualisation technique is described in Sec. III C and Fig. 6. Load: k = 1.3, orientation: 52. FIG. 10: Surface energy and density of cluster centres for the orientation 22. The corresponding fracture surface is shown in Fig. 8, top. are much less frequent than for the flat seed cracks (left side of Fig. 9). More detailed analyses for different ori- entations validate this statement. For the orientations perpendicular to twofold and fivefold axes at k = 1.3 the ratio of clusters cut by the dynamic crack to clusters in- tersected by flat cuts is approximately 0.6. Additionally the absolute value of clusters cut by the crack for the fivefold surfaces is lower than for the twofold surfaces. Fig. 10 and Fig. 11 display bottom up: The density of the cluster centres, the surface energy, a cluster in the proper length scale, the grey coding of the heights in Fig. 8 (top, middle), and the position of the seed crack (dashed vertical line). For the twofold fracture surface the low energy seed crack is located between two peaks in the cluster density, whereas for the fivefold surface this seed crack is situated close to a peak of this density. It is evident from the figures that it is not possible to circumvent all clusters by a planar cut. The grey coding is adjusted to the average height of the fracture surfaces. It is therefore evident from Fig. 11 that the crack deviates for the orientation 52 from the low en- ergy cleavage plane of the seed crack to a parallel plane, reducing the number of cluster intersections dramatically (see also Fig. 9 and Fig. 8, middle). Samples cut flat at the new height show slightly higher surface energy (see also Fig. 11). However, for low loads and low roughness the actual fracture surfaces of the dynamic cracks have about 5-15% higher energies than those of the low energy seed cracks. To assure that the dynamic crack is depart- ing from the initial plane not in a random manner the seed crack was built in at the position colour coded as medium grey in Fig. 11. The resulting fracture surface had a similar roughness but the crack did not change to a parallel plane. FIG. 11: Surface energy and density of cluster centres for the orientation 52. The corresponding fracture surface is shown in Fig. 8, middle. V. DISCUSSION Taken together the results of our simulations presented above indicate that the distribution of the clusters is crucial for the fracture behaviour: First, circumventing the clusters or intersecting them disturbs the propagat- ing crack and leads to additional radiation. This man- ifests itself in the crack speed. Dynamic cracks prop- agating in fivefold planes with few cluster intersections are faster than those in twofold planes, where the abso- lute number of cluster intersections is higher. Also the generation of ledges for low loads in the orientation 5p2 costs energy and therefore slows down the crack even further. Second, circumvention of the clusters leads to characteristic height-variations. Therefore the peak-to- valley roughness of the fracture surfaces is of the order of the diameter of the clusters. Third, the observed pat- terns in the fracture surfaces correspond to lines along which the clusters are located. The associated angles are given by the icosahedral symmetry of the sample, namely 0◦, 18◦, 31.72◦, 36◦, 58.28◦, and 90◦ (see Table I). Ledges seem to be produced only for the smallest angles mea- sured from the crack propagation direction. Fourth, less clusters are intersected by the fracture surfaces than by the flat seed cracks. Fifth, a seed crack at a low energy cleavage plane deviates to a parallel plane to reduce the number of cluster intersections in spite of the higher en- ergy required to form the fracture surfaces. In contrast, a crack built-in at this new position does not show such a deviation. Another observation of the simulations is that the plane structure of the quasicrystal also influences frac- ture. The fracture surfaces that are located perpendicu- lar to the twofold and fivefold symmetry axes show con- stant average heights. The three-dimensional quasicrystals give perfect cleav- age fracture with no indication of any dislocation activ- ity. This is in contrast to results on two-dimensional decagonal quasicrystals, where a dislocation enhanced fracture mechanism has been observed40. However a corresponding three-dimensional decagonal quasicrystal would have a periodic direction with a straight dislo- cation line. In the simulations presented here this di- rection is also quasiperiodic. As the clusters have a strong influence on fracture they also may bend and hinder dislocation lines. So dislocation emission in the three-dimensional icosahedral quasicrystal modelled here should be less likely than in the two-dimensional decago- nal model quasicrystal. Very high stresses are indeed needed to move dislocations in our model quasicrystal in molecular dynamics simulations25. There are also indications for the stability of the clusters from the electronic structure of quasicrystals: First, experiments and ab-initio calculations show that directional bonding may be present within clusters of quasicrystals41,42. Second, the electrons may addition- ally stabilise the clusters because of their hierarchical structure4. Therefore they should be even more stable than we have modelled them with simple pair potentials. With this evidence the results concerning the clusters seem reasonable and should even underestimate their sta- bility. So far fracture experiments in ultrahigh vacuum have only been performed on icosahedral Al-Mn-Pd, which has a more complicated atomic structure than the icosa- hedral binary model. Additionally the clusters are not Bergman-type. Therefore we cannot expect to represent this structure on an atomic level, when comparing exper- iments in Al-Mn-Pd with our simulations. Nevertheless, the size of the clusters, the icosahedral symmetry, and a distinct plane structure are common features and qual- itative aspects should be reproduced well, namely the size and shape of the patterns and the appearance of dis- tinct angles on the fracture surfaces. This is indeed the case, as can be seen in Fig. 12. There a geometrically scanned fracture surface generated in our simulations is confronted to an STM-image of Ebert et al.7,43 at the same length scale. As we were able to correlate the ob- served structures to the clusters in our model, the simi- larities corroborate the assumption that the clusters are responsible for the globular structures observed in ex- periment. More detailed comparisons to fracture experi- ments in icosahedral Al-Zn-Mg-type quasicrystals would be desirable, but such comparative data is currently not available. We also performed fracture simulations in a C15 Laves phase with the same Lennard-Jones potentials to give fur- ther evidence, that the roughness is correlated to the clus- ters. This C15 structure is built up from deformed pro- late rhombohedra, one of the major tiles of the quasicrys- tal (see Fig. 1 and Sec. III A). However, no Bergman- type clusters are present. When colour coded like the quasicrystal, fracture surfaces of the C15 cracks lack any roughness at low loads and cleave smoothly on high sym- metry planes44. FIG. 12: Fracture surfaces perpendicular to twofold axes. In the left picture (simulation) the atomically sharp seed crack can be seen on the top, whereas below this area the simulated fracture surface appears. The orientation of the sample is 22, the load was k = 1.3. The surface was geometrically scanned with an atom of type B. Thus atomic resolution is achieved. The right picture (experiment, adopted from Ebert et al.43) shows an STM-image of icosahedral Al-Pd-Mn cleaved in ul- trahigh vacuum. As 20r0 is approximately 5 nm the surfaces are displayed at the same length scale. When a crack traverses a solid, it leaves a typical non- equlibrium surface. Thus, up to this point, we have not dealt with any equilibrium surface. Even the flat Griffith cuts we introduced to perform the numerical experiments are no explicit equilibrium surfaces31 by definition. It is known that i-Al-Pd-Mn surfaces sputtered and annealed up to about 900 K are rough with cluster-like protru- sions. Fivefold surfaces annealed at higher temperatures exhibit flat terraces9,10,11, which are believed to be bulk terminated45. As fivefold surfaces are not as rough as twofold surfaces, they are often studied in experiment. Adsorption on these surfaces, nevertheless, is often very site-sensitive. A recent review on quasicrystal surfaces was given by McGrath et al.12. All of these observations are consistent with our simulations of non-equilibrium surfaces. Fivefold surfaces experience less roughness than twofold surfaces (also in agreement with the experiments of Ebert et al.7,43; see Fig. 8), where more clusters are intersected. Additionally, since clusters are cut in the simulations, their binding energy is not so large as to term them supermolecules. Thus annealing at very high temperatures can favour flat surfaces11,12. Nevertheless, selective adsorption may then be related to completion of clusters. VI. CONCLUSIONS We have simulated crack propagation in an icosahe- dral model quasicrystal. Brittle fracture without any signature of dislocation emission is observed. The frac- ture surfaces are rough on the scale of the clusters and show constant average heights for orientations perpen- dicular to twofold and fivefold axes. Thus both the plane structure and the clusters play an important role in frac- ture. The influence of the clusters is also seen in the average crack velocities for different orientations, the ob- served patterns in the fracture surfaces, the anisotropy with respect to the in-plane propagation direction, and the smaller amount of clusters cut by the propagating crack than by planar cuts. The clusters, too, are a rea- son why the positions of the cleavage planes cannot be predicted by a simple energy criterion. Since partial clus- ter intersections occur, the binding energy of the clusters is not so large as to term them supermolecules. Never- theless our observations clearly show that they are not only structural units but physical entities. Acknowledgments Financial support from the Deutsche Forschungsge- meinschaft under contract numbers TR 154/13 and TR 154/20-1 is gratefully acknowledged. 1 C. L. Henley, Phys. Rev. B 43 (1), 993 (1991). 2 V. Elser, Phil. Mag. B 73 (4), 641 (1996). 3 P. J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, and A. P. Tsai, Nature 396 (6706), 55 (1998). 4 C. Janot and M. de Boissieu, Phys. Rev. Lett. 72 (11), 1674 (1994). 5 C. Janot, J. Phys.: Condens. Matter 9, 1493 (1997). 6 H.-C. Jeong and P. J. Steinhardt, Phys. Rev. Lett. 73 (14), 1943 (1994). 7 Ph. Ebert, M. Feuerbacher, N. Tamura, M. Wollgarten, and K. Urban, Phys. Rev. Lett. 77 (18), 3827 (1996). 8 Ph. Ebert, M. Yurechko, F. Kluge, K. Horn, and K. Ur- ban, Quasicrystals – Structure and Physical Properties, WILEY-VCH, ed. H.-R. Trebin, 6.1, 572 (2003). 9 T. M. Schaub, D. E. Bürgler, H.-J. Güntherodt, and J. B. Suck, Phys. Rev. Lett. 73 (9), 1255 (1994). 10 M. Gierer, M. A. Van Hove, A. I. Goldman, Z. Shen, S.-L. Chang, C. J. Jenks, C.-M. Zhang, and P. A. Thiel, Phys. Rev. Lett. 78 (3), 467 (1997). 11 G. Cappello, J. Chevrier, F. Schmithüsen, A. Stierle, V. Formoso, F. Comin, M. de Boissieu, M. Boudard, T. Lo- grasso, C. Jenks, and D. Delaney, Phys. Rev. B 65, 245405 (2002). 12 R. McGrath, J. Ledieu, E. J. Cox, and R. D. Diehl, J. Phys.: Condens. Matter 14 (4), R119 (2002). 13 G. Kasner, Z. Papadopolos, P. Kramer, and D. E. Bürgler, Quasicrystals – Structure and Physical Properties, WILEY-VCH, ed. H.-R. Trebin, 2.4, 123 (2003). 14 A. A. Griffith, Philos. Trans. R. Soc. Lond. Ser. A 221, 163 (1921). 15 R. Thomson, C. Hsieh, and V. Rana, J. Appl. Phys. 42 (8), 3154 (1971). 16 P. Gumbsch and R. M. Cannon, Mat. Res. Soc. Bull. 25 (5), 15 (2000). 17 C. L. Henley and V. Elser, Phil. Mag. B 53 (3), L59 (1986). 18 The icosahedral binary model is described and named BIB in J. Roth, Eur. Phys. J. B 15 (1), 7 (2000). 19 In rings of oblate rhombohedra the number of clusters is maximised. No overlapping rhombic dodecahedra are gen- erated. Remaining oblate rhombohedra stay unchanged. 20 F. F. Abraham, Advances in Physics 52 (8), 727 (2003). 21 J. Hafner and M. Krajč́ı, Europhys. Lett. 13 (4), 335 (1990). 22 I. Al-Lehyani, M. Widom, Y. Wang, N. Moghadam, G. M. Stocks, and J. A. Moriarty, Phys. Rev. B 64, 075109 (2001). 23 F. Rösch, diploma thesis, Universität Stuttgart (2003), http://elib.uni-stuttgart.de/opus/volltexte/2004/1899/ 24 F. Rösch, Ch. Rudhart, P. Gumbsch, and H.-R. Trebin, Mat. Res. Soc. Symp. Proc. 805, LL9.3.1 (2004). 25 G. D. Schaaf, J. Roth, and H.-R. Trebin, Phil. Mag. 83 (21), 2449 (2003). 26 J. Roth, Phys. Rev. B 71, 064102 (2005). 27 J. W. Roth, R. Schilling, and H.-R. Trebin, Phys. Rev. B 51 (22), 15833 (1995). 28 F. C. Frank, Proc. Roy. Soc. Lond. Ser. A 215 (1120), 43 (1952). 29 J. Stadler, R. Mikulla, and H.-R. Trebin, Int. J. Mod. Phys. C 8 (5), 1131 (1997). 30 IMD, the ITAP Molecular Dynamics Program. http://www.itap.physik.uni-stuttgart.de/˜imd 31 To simulate fracture with molecular dynamics it is nec- essary to determine the optimal place for a low energy seed crack and to find out the critical strain for setting the crack into motion. This is always done by determining the energies of flat cuts. Note, that these do not necessar- ily represent equilibrium surfaces which may be bent and curved. 32 P. Gumbsch, S. J. Zhou, and B. L. Holian, Phys. Rev. B 55 (6), 3445 (1997). 33 As quasicrystals show non-periodic translational order we generate periodic approximants to apply periodic bound- ary conditions. The 4 to 5 million atoms of our sample then form the unit cell. Because of this high number of atoms the configuration should mechanically behave like the perfect quasicrystal. 34 Here we use equation (2) of Gumbsch et al.32 with a “filled stadion damping” f ≡ 1. 35 Cracks propagate by breaking bonds between atoms. In this process, energy is dissipated in the form of acoustic waves, as clearly visible in the online movie. The kinetic energy density is colour coded. Regions of high kinetic en- ergy density are shown in red whereas blue indicates areas of lower kinetic energy density. 36 Movie (available online) of a cracking model quasicrystal at a high load (5p2, k = 1.6). By displaying only atoms with a low coordination number fracture surfaces are visu- alised. Blue (black in Fig. 1) and red (grey in Fig. 1) balls correspond to the atoms in the icosahedral binary model. Atoms on the border, which are not allowed to move, are coloured in yellow and green. On the left the atomically sharp seed crack can be seen. Due to the high deformation near the crack tip, blue atoms loose their neighbours and show up for short instances. 37 Concerning only the pure phonon term of the elastic en- ergy in linear elastic continuum theory, the icosahedral http://elib.uni-stuttgart.de/opus/volltexte/2004/1899/ http://www.itap.physik.uni-stuttgart.de/~imd quasicrystal behaves like an isotropic solid. 38 The height profiles were determined via geometrical scan- ning of the fracture surfaces with an atom of type B at equidistant points separated by 0.2r0. For these meshes and e.g. for the orientation 52 the root-mean-square rough- ness increases from 0.39r0 at a load of k = 1.3 (see Fig. 8, middle) to 0.54r0 for k = 1.6. The peak-to-valley rough- ness increases likewise from 3.7r0 to 5.1r0. Although the exact values may depend on the stepwidth (resolution), scanning sphere size, and scanned area a general tendency for an increased roughness for increased loads is evident. 39 The question whether this crack finally chooses a plane perpendicular to a twofold or fivefold axis cannot be an- swered, as simulations with the required sample-size would exceed present computer capabilities. 40 R. Mikulla, J. Stadler, F. Krul, H.-R. Trebin, and P. Gumbsch, Phys. Rev. Lett. 81 (15), 3163 (1998). 41 Only clusters in the bulk give reliable information on their relative stability. In ab-initio studies, where the number of atoms is limited, one therefore is restricted to small ap- proximants. 42 K. Kirihara, T. Nagata, K. Kimura, K. Kato, M. Takata, E. Nishibori, and M. Sakata, Phys. Rev. B 68, 014205 (2003). 43 Ph. Ebert, F. Yue, and K. Urban, Phys. Rev. B 57 (5), 2821 (1998). 44 F. Rösch, P. Gumbsch, and H.-R. Trebin, unpublished. 45 Z. Papadopolos, G. Kasner, J. Ledieu, E. J. Cox, N. V. Richardson, Q. Chen, R. D. Diehl, T. A. Lograsso, A. R. Ross, and R. McGrath, Phys. Rev. B 66, 184207 (2002). INTRODUCTION FRACTURE MODEL AND METHOD Icosahedral binary model Molecular dynamics technique Visualisation RESULTS Crack velocities Fracture surfaces Anisotropy Clusters DISCUSSION CONCLUSIONS Acknowledgments References

0704.1413

Preparation and detection of magnetic quantum phases in optical superlattices

Preparation and detection of magnetic quantum phases in optical superlattices A. M. Rey1, V. Gritsev2, I. Bloch 3, E. Demler1,2 and M.D. Lukin1,2 1 Institute for Theoretical Atomic, Molecular and Optical Physics, Harvard-Smithsonian Center of Astrophysics, Cambridge, MA, 02138. 2 Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA and 3 Johannes Gutenberg-Universität, Institut für Physik, Staudingerweg 7,55099 Mainz, Germany We describe a novel approach to prepare, detect and characterize magnetic quantum phases in ultra-cold spinor atoms loaded in optical superlattices. Our technique makes use of singlet-triplet spin manipulations in an array of isolated double well potentials in analogy to recently demonstrated quantum control in semiconductor quantum dots. We also discuss the many-body singlet-triplet spin dynamics arising from coherent coupling between nearest neighbor double wells and derive an effective description for such system. We use it to study the generation of complex magnetic states by adiabatic and non-equilibrium dynamics. PACS numbers: 05.50.+q 03.67.Mn 05.30.Fk 05.30.Jp Recent advances in the manipulations of ultra-cold atoms in optical lattices have opened new possibilities for exploring complex many-body systems [1]. A particular topic of con- tinuous interest is the study of quantum magnetism in spin systems [2, 3, 4]. By loading spinor atoms in optical lattices it is now possible to ”simulate” exotic spin models in controlled environments and to explore novel spin orders and phases. In this Letter we describe a new approach for prepara- tion and probing of many-body magnetic quantum states that makes use of coherent manipulation of singlet-triplet pairs of ultra-cold atoms loaded in deep period-two optical superlat- tices. Our approach makes use of a spin dependent energy offset between the double-well minima to completely control and measure the spin state of two-atom pairs, in a way anal- ogous to the recently demonstrated manipulations of coupled electrons in semiconductor double-dots [5]. As an example, we show how this technique allows one to detect and analyze anti-ferromagnetic spin states in optical lattices. We further study the many-body dynamics that emerge when tunneling between nearest neighbor double wells is allowed. As two specific examples, we show how a set of singlet atomic states can be evolved into singlet-triplet cluster-type states and into a maximally entangled superposition of two anti-ferromagnetic states. Finally, we discuss the use of our projection technique to probe the density of spin defects (kinks) in magnetic states prepared via equilibrium and non-equilibrium dynamics. The key idea of this work is illustrated by considering a pair of ultra-cold atoms with two relevant internal states, which we identify with spin up and down σ =↑, ↓ in an isolated double well (DW) potential as shown in Fig.1. By dynam- ically changing the optical lattice parameters, it is possible to completely control this system and measure it in an arbi- trary two-spin basis. For concreteness, we first focus on the fermionic case. The physics of this system is governed by three sets of energy scales: i) the on-site interaction energy U = U↑↓ between the atoms, ii) the tunneling energy of the σ species: Jσ, and iii) the energy difference between the two DW minima, 2∆σ for each of the two species. The σ index in J and ∆ is due to the fact that the lattice that the ↑ and ↓ atoms feel can be engineered to be different by choosing laser beams of appropriate polarizations, frequencies, phases and intensities. In the following we assume that the atoms are strongly interacting, U ≫ Jσ, and that effective vibrational energy of each well, ~ω0, is the largest energy scale in the system ~ω0 ≫ U,∆σ, Jσ , i.e deep wells. Singlet |s〉 and triplet |t〉 states form the natural basis for the two-atom system. The relative energies of these states can be manipulated by controlling the energy bias ∆σ be- tween the two wells. In the unbiased case (U ≫ 2∆σ) only states with one atom per site (1, 1) are populated, as the large atomic repulsion energetically suppresses double occupancy (here, labels (m,n) indicate the integer number of atoms in the left and right sites of the DW). For weak tunneling and spin independent lattices (J↑ = J↓ = J , ∆↑ = ∆↓ = ∆) the states (1, 1)|s〉 and (1, 1)|t〉 are nearly degenerated. The small energy splitting between them is ∼ 4J2/U , with the singlet being the low energy state (Fig. 1a). As ∆ is increased the relative energy of doubly occupied states (0, 2) decreases. Therefore, states (1, 1)|s〉 and (0, 2)|s〉 will hybridize. When 2∆ & U the atomic repulsion is overwhelmed and conse- quently the (0, 2)|s〉 becomes the ground state. At the same time, Pauli exclusion results in a large energy splitting ~ω0 between doubly occupied singlet and triplet states as the lat- ter must have an antisymmetric orbital wave function. Hence, (1, 1)|t〉 does not hybridize with its doubly occupied counter- part, and its relative energy becomes large as compared to the singlet state. Thus the energy difference between singlet and triplet states can be controlled using ∆. Further control is provided by changing Jσ and ∆σ in spin dependent lattices (see Fig.1b). Specifically, let us now con- sider the regime 2∆σ ≪ U in which only (1, 1) subspace is populated. Within this manifold we define [6] |s〉 = ŝ†|0〉 ≡ (| ↑↓〉 − | ↓↑〉), (1) |tz〉 = t̂†z|0〉 ≡ (| ↑↓〉+ | ↓↑〉), (2) http://arxiv.org/abs/0704.1413v3 |tx〉 = t̂†x|0〉 ≡ (| ↑↑〉 − | ↓↓〉), (3) |ty〉 = t̂†y|0〉 ≡ (| ↑↑〉+ | ↓↓〉) (4) Here t̂†α and ŝ are operators that create triplet and sin- glet states from the vacuum |0〉 (state with no atoms). They satisfy bosonic commutation relations and the constrain α=x,y,z t̂ αt̂α) + ŝ †ŝ = 1, due to the physical restriction that the state in a double well is either a singlet or a triplet. In the rest of the letter we will omit the label (1, 1) for the singly occupied states. When ∆σ depends on spin, i.e Υ ≡ ∆↑ − ∆↓ 6= 0, the |tz〉 component mixes with |s〉(see Fig.1c). Note that on the other hand |tx,y〉 remain decoupled from |tz〉 and |s〉 . As a result the states |s〉 and |tz〉 form an effective two-level system whose dynamics is driven by the Hamiltonian: ĤJ1 = −ζ(ŝ†ŝ− t̂†z t̂z)−ΥS̃z + const, (5) Here ζ ≡ 2J↑J↓/Ũ , is the exchange coupling energy (with Ũ ≡ U 2−(∆↑+∆↓)2 ) and S̃z = ŝ†t̂z + t̂ z ŝ. If Υ = 0, ex- change dominates and |s〉 and |tz〉 becomes the ground and first excited states respectively. However if Υ ≫ ζ, exchange can be neglected and the ground state becomes either | ↑↓〉 or | ↓↑〉 depending on the sign of Υ. 0.0 0.2 0.4 0.6 0.8 1.0 -U -U -4J /U2 Y= - (a)(0,2)s + (2,0)s (1,1)s (1,1)t (0,2)s (1,1)s,(1,1)t FIG. 1: (color online) a) Energy levels of fermionic atoms in a spin independent double well as ∆/U is varied: While in the regime 2∆ ≪ U , (1, 1)|s〉 is the lowest energy state, when 2∆ & U , (0, 2)|s〉 becomes the state with lowest energy. b) In spin depen- dent potentials the two species feel different lattice parameters c) Restricted to the (1, 1) subspace Υ acts as an effective magnetic field gradient and couples the |s〉 and |tz〉 states . These considerations indicate that it is possible to perform arbitrary coherent manipulations and robust measurement of atom pair spin states. The former can be accomplished by combining time-dependant control over ζ,Υ to obtain effec- tive rotations on the spin-1/2 Bloch sphere within |s〉 − |tz〉 state. In the parameter regime of interest, ζ,Υ, can be var- ied independently in experiments. In addition, by applying pulsed (uniform) magnetic fields it is possible to rotate the basis, thereby changing the relative population of the |tx,y,z〉 states. Atom pair spin states can be probed by adiabatically increasing ∆ until it becomes larger than U/2, in which case atoms in the |s〉 will adiabatically follow to (0, 2)|s〉 while the atoms in |tα〉 will remain in (1,1) state (Fig. 1a). A subse- quent measurement of the number of doubly occupied wells will reveal the number of singlets in the initial state. Such a measurement can be achieved by efficiently converting the doubly occupied wells into molecules via photoassociation or using other techniques such as microwave spectroscopy and spin changing collisions [7]. Alternatively, one can continue adiabatically tilting the DW until it merges to one well. In such a way the |s〉 will be projected to the (0, 2)|s〉, while the triplets will map to (0, 2)|tα〉. As (0, 2)|tα〉 has one of the atoms in the first vibrational state of the well, by measuring the population in excited bands one can detect the number of initial |tα〉 states. Hence the spin-triplet blockade [5] allows to effectively control and measure atom pairs. Detection and diagnostics of many-body spin phases such as antiferromagnetic (AF) states is an example of direct ap- plication of the singlet-triplet manipulation and measurement technique. The procedure to measure the AF state population is the following; after inhibiting tunneling between the various DWs, one can abruptly increase Υ, such that the initial state is projected into the new eigenstates | ↑↓〉 and | ↓↑〉 at time τ = τ0. For τ > τ0 Υ can then be adiabatically decreased to zero, in which case the | ↑↓〉 pairs will be adiabatically con- verted into |s〉 and | ↓↑〉 pairs to |tz〉. Finally, the singlet pop- ulation can be measured using the spin blockade. As a result, a measure of the doubly occupied sites (or excited bands pop- ulation) will detect the number of | ↑↓〉 pairs and thus probe antiferromagnetic states of the type | ↑↓↑↓ ...〉. These ideas can be directly generalized to perform mea- surements of the more complex magnetic states that can be represented as products of two atom pairs. For example, a pulse of RF magnetic field can be used to orient all spins, thus providing the ability to detect |AF 〉 states aligned along an arbitrary direction. Moreover, one can determine the rela- tive phase between singlet and triplet pairs in |AF 〉 states of the form |s〉 + eiφ|tz〉 by performing Ramsey-type spec- troscopy. After letting the system evolve freely (with Υ = 0) so that the |s〉 and |tz〉 components accumulate an additional relative phase due to exchange, a read-out pulse (controlled by pulsing Υ) will map the accumulated phase onto popula- tion of singlet and triplet pairs. To know φ is important as it determines the direction of the anti-ferromagnetic order. Fur- thermore, by combining the blockade with noise correlation measurements [8] it is possible to obtain further information about the magnetic phases. While the blockade probes local correlation in the DWs, noise measurements probe non-local spin-spin correlations and thus can reveal long range order. Before proceeding we note that similar ideas to that out- lined above can be used for bosonic atoms if initially no |tx,y〉 states are populated. The latter can be done by detuning the |tx,y〉 states by means of an external magnetic field. In the bosonic case the doubly occupied tz states will be the ones that have the lowest energy. They will be separated by an energy ~ω0 from the doubly occupied singlets as the latter are the ones that have antisymmetric orbital wave function in bosons. Consequently, the role of |s〉 in fermions will be re- placed by |tz〉 in bosons. The read-out procedure would then be identical to that described above, while the coherent dy- namics will be given by the Hamiltonian Eq.(5) apart from the sign change ζ → −ζ. Up to now our analysis has ignored tunneling between dif- ferent DWs, but in practice this tunneling can be controlled by tuning the lattice potential. How will singlet and triplet pairs evolve due to this coupling? We will now discuss the many- body dynamics that emerges when nearest neighbor DW tun- neling is allowed, i.e. tσ > 0. When atoms can hop between DWs, the behavior of the system will depend on the dimen- sionality. For simplicity we will restrict our analysis to a 1D array ofN double-wells, where tσ corresponds to hopping en- ergy of σ-type atoms between the right site of the jth −DW and the left site of the (j + 1)th −DW . In the regime Jσ, tσ,∆σ ≪ U , multiply occupied wells are energetically suppressed and the effective Hamiltonian is given by Ĥeff = ĤJ+Ĥt . Here the first term corresponds to the sum over N independent HJj Hamiltonians (see Eq.(5)), ĤJ = j , each of which acts on its respective j DW. On the other hand Ĥt is non-local as it couples different DWs and quartic as it consists of terms with four singlet-triplet operators [9]. The coupled DWs system is in general complex and the quantum spin dynamics can be studied only numeri- cally. However, there are specific parameter regimes where an exact solution can be found. For this discussion we will set ∆σ = 0. If t↑/t↓ → 0, and at time τ = 0, no |tx〉, |ty〉 triplet states are populated, their population will remain always zero. Consequently, in this limit, the relevant Hilbert space reduces to that of an effective spin one-half system with |s〉 and |tz〉 representing the effective ±1/2 states, which we denote as | ⇑〉 and | ⇓〉. Ĥt couples such effective spin states. In the restricted Hilbert space Ĥeff maps exactly to an Ising chain in a magnetic field: Ĥeff = ∓ζ σ̂zj − λz σ̂xj σ̂ j+1 (6) where σ̂α are the usual Pauli matrices which act of the effec- tive | ⇑〉 and | ⇓〉 spins. In terms of singlet-triplet operators they are given by σ̂zj = (ŝ j ŝj − t̂ zj t̂zj), σ̂ j = ŝ j t̂zj + t̂ zj ŝj and σ̂yj = (ŝ j t̂zj − t̂ zj ŝj)/i. Here λz = and the up- per and lower signs are for fermions and bosons respectively. For fermions in the lowest vibrational level the onsite interac- tion energy between the same type of atoms U↑↑, U↓↓ → ∞ due to the Pauli exclusion principle. The 1D quantum Ising model exhibits a second order quan- tum phase transition at the critical value |g| ≡ |λz/ζ| = 1. For fermions (upper sign) when g ≪ 1 the ground state cor- responds to all effective spins pointing up, i.e |G〉 = | ⇑ . . . ⇑ 〉 = Πj |s〉j . On the other hand when g ≫ 1, there are two degenerate ground states which are, in the effective spin ba- sis, macroscopic superpositions of oppositely polarized states along x. In terms of the original fermionic spin states this su- perposition correspond to the states |AF±〉 = 1√ (| ↑↓ . . . ↑↓ 〉 ± | ↓↑ . . . ↓↑〉). Therefore, by adiabatic passage one could start with |G〉 and convert it into AF state(s). Due to vanish- ing energy gap at the quantum critical point g = 1, adiabatic- ity is difficult to maintain as N → ∞ [11, 12, 13, 14]. In that respect, our projection scheme is useful to test adiabatic following. It can be done either by measuring the number of | ↑↓〉 pairs in the final state or by adiabatically ramping down g back to zero and measuring the number of singlets/triplet pairs. The remaining number of triplets will determine the number of excitations created in the process. We now turn to non-adiabatic dynamics. We will discuss the situation where initially the system is prepared in a prod- uct of singlet states (λz = 0 ground state ) and then one lets it evolve for τ > 0 with a fixed |λz | > 0. Generically the cou- pling between DWs results in oscillations between singlet and triplet pairs with additional decay on a slower time scale. We present two important special cases involving such dynamics: i) Singlet-triplet cluster state generation: If the value of λz is set to be |λz | ≫ ζ, then the Hamiltonian reduces to a pure Ising Hamiltonian and thus at particular times, τc, given by λzτc/~ = π/4 mod π/2 the evolving state becomes a d = 1 cluster state |C〉 in the effective spin basis [15]. Up to single spin rotations |C〉 = 1 j=1(| ⇑〉j σ̂zj+1 + | ⇓〉j). Cluster states are of interest for the realization of one-way quantum computation proposals where starting from the state |C〉 com- putation can be done via measurements only. Preparation of cluster states encoded in the logical ⇑,⇓ qubits may have sig- nificant practical advantages since the ⇑,⇓ states have zero net spin along the quantization axis and hence are not affected by global magnetic field fluctuations. Additionally, the use of such singlet-triplet states for encoding might allow for the generation of decoherence free subspaces insensitive to col- lective and local errors [16] and for alternative schemes for measured-based quantum computation [17]. ii) Non-equilibrium generation and probing of AF corre- lations: The second situation is when the value of λz is set to the critical value, |λz| = ζ (or g = 1). We will first focus on the fermionic system λz > 0. To discuss it, we remind that the dynamics driven by Ĥeff is exactly solv- able as Ĥeff can be mapped via the Jordan Wigner trans- formation into a quadratic Hamiltonian of fermionic opera- tors which can be diagonalized by a canonical transforma- tion [10, 14]. Using such transformation it is possible to show that at specific times, the shortest of them we denote by τm ≈ ~N+14ζ , long range AF correlations build up and for small atom number the state approaches |AF+〉. To quan- tify the resulting state in Fig. 2(inset) we plot the fidelity, defined as F1(τm) = |〈AF+|ψ(τm)〉g=1|2, as a function of N . The figure shows that while an almost perfect |AF+〉 is dynamically generated for small N , its fidelity exponentially degrades with increasing atom number. However, the fidelity is a very strict probe, as it drops to zero when a single spin is flipped. As N increases the sys- tem ends at τm in a quantum superposition of states like | · · · ⇒⇐⇐⇐⇐⇐⇒⇒⇒⇒⇒⇒⇐ . . . 〉 with finite do- mains of ”effective spins” pointing along ±x, separated by kinks where the polarization of the spins change its orien- tation (we used the convention | ↑↓〉 ≡ | ⇒〉). Conse- quently, one gets more realistic information about the AF order of the state, by measuring the average size of the do- mains or the average density of kinks, the latter defined as ν ≡ 1 j(1 − 〈ψ(τ)|σ̂xj σ̂xj+1|ψ(τ)〉). Our read-out technique can be used to detect the kink- density as for an arbitrary fixed g energy conservation imposes a relation between ν and the triplet-z density, Nt: ν(τ, g) = − Nt(τ, g) . (7) A simple analytical expression for Nt(τ, g) can be ob- tained by using the Jordan Wigner transformation [10]: Nt(τ, g) = sin2(2πk/N) sin2(2ωkτ) where ~ωk = g2 + 1 + 2g cos(2πk/N) are quasi-particle frequencies of Ĥeff . The fact that it remains always below 0.2 (see Fig. 2) confirms the idea that regardless of the reduced fidelity at large N , the state does retain AF correlations. We point out that |AF+〉 states are only generated at g = 1, a feature that illustrates the special character of the critical dynamics. 0 20 40 60 80 100 0.025 0.075 0.125 0 20 40 60 80 100N FIG. 2: Using the the Jordan-Wigner transformation [10] we cal- culated the density of kinks vs N at τ = τm and the fidelity |〈ψ(τm)|AF +〉|2 vsN (inset). Our projection technique can be used to measure ν(τ ) as it is directly related to the triplet density, Nt(τ ) (see Eq.(7)). Let us now discuss the bosonic case. If λz > 0, the fermionic results apply for bosons by simply interchanging the role of |s〉 ↔ |tz〉. On the other hand if λz < 0, not only one has to interchange |s〉 ↔ |tz〉 but additionally, the adia- batic and non-equilibrium dynamics will generate, instead of |AF±〉 states, 1√ (| ⇒⇐ · · · ⇒⇐〉 ± | ⇐⇒ · · · ⇐⇒〉) i.e macroscopic superpositions of AF states along the x-direction in the effective spin basis. With these modifications, the re- sults derived for fermions hold for bosons[21]. Before concluding we briefly mention that spin dependent superlattices of the form j=1,2 (Aj +Bjσz) cos 2[kz/j + θj ] (8) can be experimentally realized by superimposing two inde- pendent lattices, generated by elliptically polarized light, one with twice the periodicity of the other [18, 19, 20]. Com- plete control over the DW parameters is achieved by control- ling the phases (which determine ∆), intensities (which deter- mine U ,J and t) and polarization of the laser beams (which allow for spin dependent control). For example lattice con- figurations with t↑ ≪ t↓ can be achieved by setting the laser parameters such that B1 = 0 and A2 = B2 ≫ 1. In summary we have described a technique to prepare, de- tect and manipulate spin configurations in ultra-cold atomic systems loaded in spin dependent period-two superlattices. By studying the many-body dynamics that arises when tun- neling between DWs is allowed, we discussed how to dynam- ically generate singlet-triplet cluster states and AF cat states, which are of interest for quantum information science, and how to probe AF correlations in far from equilibrium dynam- ics. Even though in this Letter we restrict our analysis to 1D systems the ideas developed here can be extended to higher dimensions and more general kinds of interactions. We acknowledge useful discussions with G. Morigi. This work was supported by ITAMP, NSF (Career Program), Harvard-MIT CUA, AFOSR, Swiss NF, the Sloan Founda- tion, and the David and Lucille Packard Foundation. [1] M. Greiner et. al. Nature 415, 39 (2002). [2] A. Auerbach, Interacting electrons and quantum magnetism, New York, Springer-Verlag (2003). [3] J. Stenger et. al. Nature 396, 345 (1998). [4] L. E. Sadler et. al. Nature 443, 312 (2006). [5] J. R. Petta et al, Science 309, 2180 (2005). [6] S. Sachdev and R. N. Bhatt, Phys. Rev. B, 41, 9323 (1990). [7] S. Fölling et al, Phys. Rev. Lett. 97, 060403 (2006). [8] E. Altman et al, Phys. Rev. A 70, 013603 (2004). [9] A. M. Rey et al, in preparation. [10] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 (1961). [11] W.H. Zurek, U. Dorner, P. Zoller, Phys. Rev. Lett. 95, 105701 (2005). [12] A. Polkovnikov, Phys. Rev. B 72, 161201(R) (2005). [13] R. W. Cherng and L. S. Levitov, Phys. Rev. A 73, 043614 (2006). [14] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005). [15] H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86,910 (2001). [16] D. Bacon, J. Kempe, D. A. Lidar and K. B. Whaley, Phys. Rev. Lett. 85, 1758(2000). [17] D. Gross and J. Eisert, preprint: quant-ph/060914. [18] J.P. Lee et. al. arXiv: quant-ph/0702039. [19] J. Sebby-Strabley et al, Phys Rev A, 73, 033605 (2006). [20] S. Peil et. al. Phys. Rev. A 67, 051603(R) (2003). http://arxiv.org/abs/quant-ph/0702039 [21] In this case a different sign in the definition of kink density σ̂xj σ̂ j+1 → −σ̂ j+1 is required.

0704.1414

Sobolev solution for semilinear PDE with obstacle under monotonicity condition

Sobolev solution for semilinear PDE with obsta le under monotoni ity ondition Anis MATOUSSI Equipe �Statistique et Pro essus� Université du Maine Avenue Olivier Messiaen 72085 LE MANS Cedex 9 FRANCE anis.matoussi�univ-lemans.fr Mingyu XU Institute of Applied Mathemati s A ademy of Mathemati s and Systems S ien e CAS, Beijing, 100190 China xumy�amss.a . n �rst revision version Mar h 20, 2006, �nal a epted version May 26, 2008 Abstra t We prove the existen e and uniqueness of Sobolev solution of a semilinear PDE's and PDE's with obsta le under monotoni ity ondition. Moreover we give the probabilisti interpretation of the solutions in term of Ba kward SDE and re�e ted Ba kward SDE respe tively. Key words: Ba kward sto hasti di�erential equation, Re�e ted ba kward sto hasti di�erential equation, monotoni ity ondition, Sto hasti �ow, partial di�erential equation with obsta le. AMS Classi� ation: 35D05, 60H10, 60H30B 1 Introdu tion Our approa h is based on Ba kward Sto hasti Di�erential Equations (in short BSDE's) whi h were �rst introdu ed by Bismut [5℄ in 1973 as equation for the adjoint pro ess in the sto hasti version of Pontryagin maximum prin iple. Pardoux and Peng [14℄ generalized the notion in 1990 and were the �rst to onsider general BSDE's and to solve the question of existen e and uniqueness in the non-linear ase. Sin e then BSDE's have been widely used in sto hasti ontrol and espe ially in mathemati al �nan e, as any pri ing problem by repli ation an be written in terms of linear BSDEs, or non-linear BSDEs when portfolios onstraints are taken into a ount as in El Karoui, Peng and Quenez [6℄. The main motivation to introdu e the non-linear BSDE's was to give a probabilisti interpreta- tion (Feynman-Ka 's formula) for the solutions of semilinear paraboli PDE's. This result was �rst obtained by Peng in [16℄, see also Pardoux and Peng [15℄ by onsidering the vis osity and lassi al solutions of su h PDE's. Later, Barles and Lesigne [2℄ studied the relation between BSDE's and solutions of semi-linear PDE's in Soblev spa es. More re ently Bally and Matoussi [3℄ studied semi- linear sto hasti PDEs and ba kward doubly SDE in Sobolev spa e and their probabilisti method is based on sto hasti �ow. The re�e ted BSDE's was introdu ed by the �ve authors El Karoui, Kapoudjian, Pardoux, Peng and Quenez in [7℄, the setting of those equations is the following: let us onsider moreover an adapted sto hasti pro ess L := (Lt)t 6 T whi h stands for a barrier. A solution for the re�e ted http://arxiv.org/abs/0704.1414v2 BSDE asso iated with (ξ, g, L) is a triple of adapted sto hasti pro esses (Yt, Zt,Kt)t 6 T su h that Yt = ξ + g(s, ω, Ys, Zs)ds+KT −Kt − ZsdBs, ∀ t ∈ [0, T ], Yt > Lt and (Yt − Lt)dKt = 0. The pro ess K is ontinuous, in reasing and its role is to push upward Y in order to keep it above the barrier L. The requirement (Yt − Lt)dKt = 0 means that the a tion of K is made with a minimal energy. The development of re�e ted BSDE's (see for example [7℄, [10℄, [9℄) has been espe ially motivated by pri ing Ameri an ontingent laim by repli ation, espe ially in onstrained markets. A tually it has been shown by El Karoui, Pardoux and Quenez [8℄ that the pri e of an Ameri an ontingent laim (St)t 6 T whose strike is γ in a standard omplete �nan ial market is Y0 where (Yt, πt,Kt)t 6 T is the solution of the following re�e ted BSDE −dYt = b(t, Yt, πt)dt+ dKt − πtdWt, YT = (ST − γ)+, Yt > (St − γ)+ and (Yt − (St − γ)+)dKt = 0 for an appropriate hoi e of the fun tion b. The pro ess π allows to onstru t a repli ation strategy and K is a onsumption pro ess that ould have the buyer of the option. In a standard �nan ial market the fun tion b(t, ω, y, z) = rty+zθt where θt is the risk premium and rt the spot rate to invest or borrow. Now when the market is onstrained i.e. the interest rates are not the same whether we borrow or invest money then the fun tion b(t, ω, y, z) = rty + zθt − (Rt − rt)(y − (z.σ−1t .1))− where Rt (resp. rt) is the spot rate to borrow (resp. invest) and σ the volatility. Partial Di�erential Equations with obsta les and their onne tions with optimal ontrol prob- lems have been studied by Bensoussan and Lions [4℄. They study su h equations in the point of view of variational inequalities. In a re ent paper, Bally, Caballero, El Karoui and Fernandez [1℄ studied the the following semilinear PDE with obsta le (∂t + L)u + f(t, x, u, σ∗∇u) + ν = 0, u > h, uT = g, where h is the obsta le. The solution of su h equation is a pair (u, ν) where u is a fun tion in 2([0, T ],H) and ν is a positive measure on entrated on the set {u = h}. The authors proved the uniqueness and existen e for the solution to this PDE when the oe� ient f is Lips hitz and linear in reasing on (y, z), and gave the probabilisti interpretation (Feynman-Ka formula) for u and ∇u by the solution (Y, Z) of the re�e ted BSDE (in short RBSDE). They prove also the natural relation between Re�e ted BSDE's and variational inequalities and prove uniqueness of the solution for su h variational problem by using the relation between the in reasing pro ess K and the measure ν. This is also a point of view in this paper. On the other hand, Pardoux [13℄ studied the solution of a BSDE with a oe� ient f(t, ω, y, z), whi h satis�es only monotoni ity, ontinuous and general in reasing onditions on y, and a Lips hitz ondition on z, i.e. for some ontinuous, in reasing fun tion ϕ : R+ → R+, and real numbers µ ∈ R, k > 0, ∀t ∈ [0, T ], ∀y, y′ ∈ Rn, ∀z, z′ ∈ Rn×d, |f(t, y, 0)| 6 |f(t, 0, 0)|+ ϕ(|y|), a.s.; (1) 〈y − y′, f(t, y, z)− f(t, y′, z)〉 6 µ |y − y′|2 , a.s.; |f(t, y, z)− f(t, y, z′)| 6 k |z − z′| , a.s.. In the same paper, he also onsidered the PDE whose oe� ient f satis�es the monotoni ity ondition (1), proved the existen e of a vis osity solution u to this PDE and gave its probabilisti interpretation via the solution of the orresponding BSDE. More re ently, Lepeltier, Matoussi and Xu [12℄ proved the existen e and uniqueness of the solution for the re�e ted BSDE under the monotoni ity ondition. In our paper, we study the Sobolev solutions of the PDE and also the PDE with ontinuous obsta le under the monotoni ity ondition (1). Using penalization method, we prove the existen e of the solution and give the probabilisti interpretation of the solution u and ∇u (resp.(u,∇u, ν)) by the solution (Y, Z) of ba kward SDE (resp. the solution (Y, Z,K) of re�e ted ba kward SDE). Furthermore we use equivalen e norm results and a sto hasti test fun tion to pass from the solution of PDE's to the one of BSDE's in order to get the uniqueness of the solution. Our paper is organized as following: in se tion 2, we present the basi assumptions and the de�nitions of the solutions for PDE and PDE with obsta le, then in se tion 3, we re all some useful results from [3℄. We will prove the main results for PDE and PDE with ontinuous barrier under monotoni ity ondition in se tion 4 and 5 respe tively. Finally, we prove an analogue result to Proposition 2.3 in [3℄ under the monotoni ity ondition, and we also give a priori estimates for the solution of the re�e ted BSDE's. 2 Notations and preliminaries Let (Ω,F , P ) be a omplete probability spa e, and B = (B1, B2, · · · , Bd)∗ be a d-dimensional Brownian motion de�ned on a �nite interval [0, T ], 0 < T < +∞. Denote by {F ts; t 6 s 6 T } the natural �ltration generated by the Brownian motion B : F ts = σ{Bs −Bt; t 6 r 6 s} ∪ F0, where F0 ontains all P−null sets of F . We will need the following spa es for studying BSDE or re�e ted BSDE. For any given n ∈ N: • L2n(F ts) : the set of n-dimensional F ts-measurable random variable ξ, su h that E(|ξ|2) < +∞. • H2n×m(t, T ) : the set of Rm×n-valued F ts-predi table pro ess ψ on the interval [t, T ], su h that ‖ψ(s)‖2 ds < +∞. • S2n(t, T ) : the set of n-dimensional F ts-progressively measurable pro ess ψ on the interval [t, T ], su h that E(supt 6 s 6 T ‖ψ(s)‖ ) < +∞. • A2(t, T ) :={K : Ω× [t, T ] → R, F ts�progressively measurable in reasing RCLL pro esses with Kt = 0, E[(KT ) 2] <∞ }. Finally, we shall denote by P the σ-algebra of predi table sets on [0, T ]×Ω. In the real�valued ase, i.e., n = 1, these spa es will be simply denoted by L2(F ts), H2(t, T ) and S2(t, T ), respe tively. For the sake of the Sobolev solution of the PDE, the following notations are needed: • Cmb (Rd,Rn) : the set of Cm-fun tions f : Rd → Rn, whose partial derivatives of order less that or equal to m, are bounded. (The fun tions themselves need not to be bounded) • C1,mc ([0, T ] × Rd,Rn) : the set of ontinuous fun tions f : [0, T ] × Rd → Rn with ompa t support, whose �rst partial derivative with respe t to t and partial derivatives of order less or equal to m with respe t to x exist. • ρ : Rd → R, the weight, is a ontinuous positive fun tion whi h satis�es ρ(x)dx <∞. • L2(Rd, ρ(x)dx) : the weighted L2-spa e with weight fun tion ρ(x), endowed with the norm L2(Rd,ρ) = |u(x)|2 ρ(x)dx We assume: Assumption 2.1. g(·) ∈ L2(Rd, ρ(x)dx). Assumption 2.2. f : [0, T ]× Rd × Rn×Rn×d → Rn is measurable in (t, x, y, z) and |f(t, x, 0, 0)|2 ρ(x)dxdt <∞. Assumption 2.3. f satis�es in reasing and monotoni ity ondition on y, for some ontinuous in reasing fun tion ϕ : R+ → R+, real numbers k > 0, µ ∈ R su h that ∀(t, x, y, y′, z, z′) ∈ [0, T ]× Rd × Rn × Rn × Rn×d × Rn×d (i) |f(t, x, y, z)| 6 |f(t, x, 0, z)|+ ϕ(|y|), (ii) |f(t, x, y, z)− f(t, x, y, z′)| 6 k |z − z′|, (iii) 〈y − y′, f(t, x, y, z)− f(t, x, y′, z)〉 6 µ |y − y′|2, (iv) y → f(t, x, y, z) is ontinuous. For the PDE with obsta le, we onsider that f satis�es assumptions 2.2 and 2.3, for n = 1. Assumption 2.4. The obsta le fun tion h ∈ C([0, T ] × Rd;R) satis�es the following onditions: there exists κ ∈ R, β > 0, su h that ∀(t, x) ∈ [0, T ]× Rd (i) ϕ(eµth+(t, x)) ∈ L2(Rd; ρ(x)dx), (ii) |h(t, x)| 6 κ(1 + |x|β), here h+ is the positive part of h. Assumption 2.5. b : [0, T ]× Rd → Rd and σ : [0, T ]× Rd → Rd×d satisfy b ∈ C2b (Rd;Rd) and σ ∈ C3b (Rd;Rd×d). We �rst study the following PDE (∂t + L)u + F (t, x, u,∇u) = 0, ∀ (t, x) ∈ [0, T ]× Rd u(x, T ) = g(x), ∀x ∈ Rd where F : [0, T ]× Rd × Rn × Rn×d → R, su h that F (t, x, u, p) = f(t, x, u, σ∗p) i,j=1 ∂xi∂xj a := σσ∗. Here σ∗ is the transposed matrix of σ. In order to study the weak solution of the PDE, we introdu e the following spa e H := {u ∈ L2([0, T ]× Rd, ds⊗ ρ(x)dx) ∣∣ σ∗∇u ∈ L2(([0, T ]× Rd, ds⊗ ρ(x)dx)} endowed with the norm ‖u‖2 := [|u(s, x)|2 + |(σ∗∇u)(s, x)|2]ρ(x)dsdx. De�nition 2.1. We say that u ∈ H is the weak solution of the PDE asso iated to (g, f), if (i) ‖u‖2 <∞, (ii) for every φ ∈ C1,∞c ([0, T ]× Rd) (us, ∂tφ)ds + (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds = (f(s, ·, us, σ∗∇us), φs)ds. where (φ, ψ) = φ(x)ψ(x)dx denotes the s alar produ t in L2(Rd, dx) and E(ψ, φ) = ((σ∗∇ψ)(σ∗∇φ) + φ∇((1 σ∗∇σ + b)ψ))dx is the energy of the system of our PDE whi h orresponds to the Diri hlet form asso iated to the operator L when it is symmetri . Indeed E(ψ, φ) = −(φ,Lψ). The probabilisti interpretation of the solution of PDE asso iated with g, f , whi h satisfy As- sumption 2.1-2.3 was �rstly studied by (Pardoux [13℄), where the author proved the existen e of a vis osity solution to this PDE, and gave its probabilisti interpretation. In se tion 4, we onsider the weak solution to PDE (2) in Sobolev spa e, and give the proof of the existen e and uniqueness of the solution as well as the probabilisti interpretation. In the se ond part of this arti le, we will onsider the obsta le problem asso iated to the PDE (2) with obsta le fun tion h, where we restri t our study in the one dimensional ase (n = 1). Formulaly, The solution u is dominated by h, and veri�es the equation in the following sense : ∀(t, x) ∈ [0, T ]× Rd (i) (∂t + L)u+ F (t, x, u,∇u) 6 0, on u(t, x) > h(t, x), (ii) (∂t + L)u+ F (t, x, u,∇u) = 0, on u(t, x) > h(t, x), (iii) u(x, T ) = g(x) . where L = i=1 bi i,j=1 ai,j ∂xi∂xj , a = σσ∗. In fa t, we give the following formulation of the PDE with obsta le. De�nition 2.2. We say that (u, ν) is the weak solution of the PDE with obsta le asso iated to (g, f, h), if (i) ‖u‖2 <∞, u > h, and u(T, x) = g(x). (ii) ν is a positive Radon measure su h that ρ(x)dν(t, x) <∞, (iii) for every φ ∈ C1,∞c ([0, T ]× Rd) (us, ∂sφ)ds+ (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds (3) (f(s, ·, us, σ∗∇us), φs)ds+ φ(s, x)1{u=h}dν(x, s). 3 Sto hasti �ow and random test fun tions Let (Xt,xs )t 6 s 6 T be the solution of dXt,xs = b(s,X s )ds+ σ(s,X s )dBs, t = x, where b : [0, T ]× Rd → Rd and σ : [0, T ]× Rd → Rd×d satisfy Assumption 2.5. So {Xt,xs , x ∈ Rd, t 6 s 6 T } is the sto hasti �ow asso iated to the di�use {Xt,xs } and denote by {X̂t,xs , t 6 s 6 T } the inverse �ow. It is known that x → X̂t,xs is di�erentiable (Ikeda and Watanabe [?℄). We denote by J(Xt,xs ) the determinant of the Ja obian matrix of X̂ s , whi h is positive, and J(X t ) = 1. For φ ∈ C∞c (Rd) we de�ne a pro ess φt : Ω× [0, T ]× Rd → R by φt(s, x) := φ(X̂ s )J(X̂ Following Kunita (See [11℄), we know that for v ∈ L2(Rd), the omposition of v with the sto hasti �ow is (v ◦Xt,·s , φ) := (v, φt(s, ·)). Indeed, by a hange of variable, we have (v ◦Xt,·s , φ) = v(y)φ(X̂t,ys )J(X̂ s )dy = v(Xt,xs )φ(x)dx. The main idea in Bally and Matoussi [3℄ and Bally et al. [1℄, is to use φt as a test fun tion in (2) and (3). The problem is that s→ φt(s, x) is not di�erentiable so that (us, ∂sφ)ds has no sense. However φt(s, x) is a semimartingale and they proved the following semimartingale de omposition of φt(s, x): Lemma 3.1. For every fun tion φ ∈ C2c(Rd), φt(s, x) = φ(x)− (σij(x)φt(r, x)) L∗φt(r, x)dr, (4) where L∗ is the adjoint operator of L. So dφt(r, x) = − (σij(x)φt(r, x)) dBjr + L∗φt(r, x)dr, (5) Then in (2) we may repla e ∂sφds by the It� sto hasti integral with respe t to dφt(s, x), and have the following proposition whi h allows us to use φt as a test fun tion. The proof will be given in the appendix. Proposition 3.1. Assume that assumptions 2.1, 2.2 and 2.3 hold. Let u ∈ H be a weak solution of PDE (2), then for s ∈ [t, T ] and φ ∈ C2c (Rd), u(r, x)dφt(r, x)dx − (g(·), φt(T, ·)) + (u(s, ·), φt(s, ·))− E(u(r, ·), φt(r, ·))dr f(r, x, u(r, x), σ∗∇u(r, x))φt(r, x)drdx. a.s. (6) Remark 3.1. Here φt(r, x) is R-valued. We onsider that in (6), the equality holds for ea h omponent of u. We need the result of equivalen e of norms, whi h play important roles in existen e proof for PDE under monotoni onditions. The equivalen e of fun tional norm and sto hasti norm is �rst proved by Barles and Lesigne [2℄ for ρ = 1. In Bally and Matoussi [3℄ proved the same result for weighted integrable fun tion by using probabilisti method. Let ρ be a weighted fun tion, we take ρ(x) := exp(F (x)), where F : Rd → R is a ontinuous fun tion. Moreover, we assume that there exists a onstant R > 0, su h that for |x| > R, F ∈ C2b (Rd,R). For instant, we an take ρ(x) = (1 + |x|)−q or ρ(x) = expα |x|, with q > d+ 1, α ∈ R. Proposition 3.2. Suppose that assumption 2.5 hold, then there exists two onstants k1, k2 > 0, su h that for every t 6 s 6 T and φ ∈ L1(Rd, ρ(x)dx), we have |φ(x)| ρ(x)dx 6 ∣∣φ(Xt,xs ) ∣∣)ρ(x)dx 6 k1 |φ(x)| ρ(x)dx, (7) Moreover, for every ψ ∈ L1([0, T ]× Rd, dt⊗ ρ(x)dx) |ψ(s, x)| ρ(x)dsdx 6 ∣∣ψ(s,Xt,xs ) ∣∣)ρ(x)dsdx (8) |ψ(s, x)| ρ(x)dsdx, where the onstants k1, k2 depend only on T , ρ and the bounds of the �rst (resp. �rst and se ond) derivatives of b (resp. σ). This proposition is easy to get from the follwing Lemma, see Lemma 5.1 in Bally and Matoussi Lemma 3.2. There exist two onstants c1 > 0 and c2 > 0 su h that ∀x ∈ Rd, 0 6 t 6 T c1 6 E ρ(t, X̂ t )J(X̂ 6 c2. 4 Sobolev's Solutions for PDE's under monotoni ity ondi- In this se tion we shall study the solution of the PDE whose oe� ient f satis�es the monotoni ity ondition. For this sake, we introdu e the BSDE asso iated with (g, f): for t 6 s 6 T , Y t,xs = g(X f(r,Xt,xr , Y r , Z r )dr − Zt,xs dBs. (9) Thanks to the equivalen e of the norms result (3.2), we know that g(X T ) and f(s,X s , 0, 0) make sense in the BSDE (9). Moreover we have ) ∈ L2n(FT ) and f(., Xt,x. , 0, 0) ∈ H2n(0, T ). It follows from the results from Pardoux [13℄ that for ea h (t, x), there exists a unique pair (Y t,x, Zt,x) ∈ S2(t, T ) ×H2n×d(t, T ) of {F ts} progressively measurable pro esses, whi h solves this BSDE(g, f). The main result of this se tion is Theorem 4.1. Suppose that assumptions 2.1-2.3 and 2.4 hold. Then there exists a unique weak solution u ∈ H of the PDE (2). Moreover we have the probabilisti interpretation of the solution: u(t, x) = Y t , (σ ∗∇u)(t, x) = Zt,xt , dt⊗ dx− a.e. (10) and moreover Y t,xs = u(s,X s ), Z s = (σ ∗∇u)(s,Xt,xs ), dt⊗ dP ⊗ dx-a.e. ∀s ∈ [t, T ]. Proof : We start to prove the existen e result. a) Existen e : We prove the existen e in three steps. By integration by parts formula, we know that u solves (2) if and only if û(t, x) = eµtu(t, x) is a solution of the PDE(ĝ, f̂), where ĝ(x) = eµT g(x) and f̂(t, x, y, z) = eµtf(t, x, e−µty, e−µtz)− µy. (11) Then the oe� ient f̂ satis�es the assumption 2.3 as f , ex ept that 2.3-(iii) is repla ed by (y − y′)(f(t, x, y, z)− f(t, x, y′, z)) 6 0. (12) In the �rst two steps, we onsider the ase where f does not depend on ∇u, and write f(t, x, y) for f(t, x, y, v(t, x)), where v is in L2([0, T ]× Rd, dt⊗ ρ(x)dx). We assume �rst that f(t, x, y) satis�es the following assumption 2.3': ∀(t, x, y, y′) ∈ [0, T ] × d × Rn × Rn, (i) |f(t, x, y)| 6 |f(t, x, 0)|+ ϕ(|y|), (ii) 〈y − y′, f(t, x, y)− f(t, x, y′)〉 6 0, (iii) y → f(t, x, y) is ontinuous, ∀(t, x) ∈ [0, T ]× Rd. Step 1 : Suppose that g(x), f(t, x, 0) are uniformly bounded, i.e. there exists a onstant C, su h that |g(x)| + sup 0 6 t 6 T |f(t, x, 0)| 6 C (13) where C as a onstant whi h an be hanged line by line. De�ne fn(t, y) := (θn ∗ f(t, ·))(y) where θn : Rn → R+ is a sequen e of smooth fun tions with ompa t support, whi h approximate the Dira distribution at 0, and satisfy θn(z)dz = 1. Let {(Y n,t,xs , Zn,t,xs ), t 6 s 6 T } be the solution of BSDE asso iated to (g(X T ), fn), namely, n,t,x s = g(X T ) + f(r,Xt,xr , Y n,t,x r )dr − n,t,x r dBr, P-a.s.. (14) Then for ea h n ∈ N, we have ∣∣Y n,t,xs ∣∣ 6 eTC, and ∣∣fn(s,Xt,xs , Y n,t,xs ) ∣∣2 6 2 ∣∣fn(s,Xt,xs , 0) ∣∣2 + 2ψ2(e where ψ(r) := supn sup|y| 6 r ϕ(|y|)θn(y − z)dz. So there exists a onstant C > 0, s.t. ∣∣Y n,t,xs ∣∣2 + ∣∣fn(s,Xt,xs , Y n,t,xs ) ∣∣2 + ∣∣Zn,t,xs ∣∣2)ρ(x)dsdx 6 C. (15) Then let n → ∞ on the both sides of (14), we get that the limit (Y t,xs , Zt,xs ) of (Y n,t,xs , Zn,t,xs ), satis�es Y t,xs = g(X T ) + f(r,Xt,xr , Y r )dr − Zt,xr dBr, P-a.s.. (16) Moreover we obtain from the estimate (15) that ∣∣Y t,xs ∣∣2 + ∣∣Zt,xs ∣∣2)ρ(x)dsdx <∞. (17) Noti e that (Y t , Z t ) are F tt measurable, whi h implies they are deterministi . De�ne u(t, x) := t , and v(t, x) := Z t . By the �ow property of X r and by the uniqueness of the solution of the BSDE (16), we have that Y t,xs = u(s,X s ) and Z s = v(s,X The terminal ondition g and f(., ., 0, 0) are not ontinuous in t and x, and assumed to belong in a suitable weighted L2 spa e, so the solution u and for instan e v are not in general ontinuous, and are only de�ned a.e. in [0, T ]× Rd. So in order to give meaning to the expression u(s,Xt,xs ) (resp. v(s,Xt,xs )), and following Bally and Matoussi [3℄, we apply a regularization pro edure on the �nal ondition g and the oe� ient f . A tually, a ording to Pardoux and Peng ([15℄, Theorem 3.2), if the oe� ient (g, f) are smooth, then the PDE (2) admits a unique lassi al solution u ∈ C1,2([0, T ] × Rd). Therefore the approximated expression u(s,Xt,xs ) (resp. v(s,Xt,xs )) has a meaning and then pass to the limit in L2 spa es like us in Bally and Matoussi [3℄. Now, the equivalen e of norm result (8) and estimate (17) follow that u, v ∈ L2([0, T ] × Rd, dt ⊗ ρ(x)dx). Finally, let F (r, x) = f(r,Xt,xr , Y r ), we know that F (s, x) ∈ L2([0, T ]×Rd, dt⊗ ρ(x)dx), in view of |F (s, x)|2 ρ(x)dsdx 6 1 ∣∣F (s,Xt,xs ) ∣∣2 ρ(x)dsdx ∣∣f(s,Xt,xs , Y t,xs ) ∣∣2 ρ(x)dsdx <∞. So that from theorem 2.1 in [3℄, we get that v = σ∗∇u and that u ∈ H solves the PDE asso iated to (g, f) under the bounded assumption. Step 2 : We assume g ∈ L2(Rd, ρ(x)dx), f satis�es the assumption 2.3' and f(t, x, 0) ∈ L2([0, T ]× d, dt⊗ ρ(x)dx). We approximate g and f by bounded fun tions as follows : gn(x) = Πn(g(x)), (18) fn(t, x, y) = f(t, x, y)− f(t, x, 0) + Πn(f(t, x, 0)), where Πn(y) := min(n, |y|) |y| y. Clearly, the pair (gn, fn) satis�es the assumption (13) of step 1, and gn → g in L2(Rd, ρ(x)dx), (19) fn(t, x, 0) → f(t, x, 0) in L2([0, T ]× Rd, dt⊗ ρ(x)dx). Denote (Y n,t,xs , Z n,t,x s ) ∈ S2n(t, T ) × H2n×d(t, T ) the solution of the BSDE(ξn, fn), where ξn = T ), i.e. n,t,x s = gn(X T ) + fn(r,X r , Y n,t,x r )dr − n,t,x r dBr. Then from the results in step 1, un(t, x) = Y n,t,x t and un(t, x) ∈ H, is the weak solution of the PDE(gn, fn), with n,t,x s = un(s,X s ), Z n,t,x s = (σ ∗∇un)(s,Xt,xs ), a.s. (20) For m,n ∈ N, applying It�'s formula to |Y m,t,xs − Y n,t,xs | , we get ∣∣Y m,t,xs − Y n,t,xs ∣∣2 + E ∣∣Zm,t,xr − Zn,t,xr ∣∣2 dr 6 E ∣∣gm(Xt,xT )− gn(X ∣∣Y m,t,xr − Y n,t,xr ∣∣2 dr + E ∣∣fm(r,Xt,xr , 0)− fn(r,Xt,xr , 0) ∣∣2 dr. From the equivalen e of the norms (7) and (8), it follows ∣∣Y m,t,xs − Y n,t,xs ∣∣2 ρ(x)dx 6 ∣∣gm(Xt,xT )− gn(X ∣∣2 ρ(x)dx ∣∣Y m,t,xr − Y n,t,xr ∣∣2 drρ(x)dx + ∣∣fm(r,Xt,xr , 0)− fn(r,Xt,xr , 0) ∣∣2 drρ(x)dx ∣∣Y m,t,xr − Y n,t,xr ∣∣2 drρ(x)dx + k1 E |gm(x)− gn(x)|2 ρ(x)dx |fm(r, x, 0)− fn(r, x, 0)|2 ρ(x)drdx, and by Gronwall's inequality and (19), we get as m,n→ ∞ t 6 s 6 T ∣∣Y m,t,xs − Y n,t,xs ∣∣2 ρ(x)dx → 0. It follows immediately as m,n→ ∞ ∣∣Y m,t,xr − Y n,t,xr ∣∣2 ρ(x)drdx + ∣∣Zm,t,xr − Zn,t,xr ∣∣2 ρ(x)drdx → 0. Using again the equivalen e of the norms (8), we get: |um(s, x)− un(s, x)|2 + |σ∗∇um(s, x)− σ∗∇un(s, x)|2 ρ(x)dxds ∣∣um(s,Xt,xs )− un(s,Xt,xs ) ∣∣2 + ∣∣σ∗∇um(s,Xt,xs )− σ∗∇un(s,Xt,xs ) ∣∣2)ρ(x)dsdx ∣∣Y m,t,xs − Y n,t,xs ∣∣2 + ∣∣Zm,t,xs − Zn,t,xs ∣∣2)ρ(x)dsdx → 0. as m,n→ ∞, i.e. {un} is Cau hy sequen e in H. Denote its limit as u, so u ∈ H, and satis�es for every φ ∈ C1,∞c ([0, T ]× Rd), (us, ∂tφ)ds + (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds = (f(s, ·, us), φs)ds. (22) On the other hand, (Y n,t,x · , Z n,t,x · ) onverges to (Y · , Z · ) in S n(0, T )×H2n×d(0, T ), whi h is the solution of the BSDE with parameters (g(X ), f); by the equivalen e of the norms, we dedu e s = u(s,X s ), Z s = σ ∗∇u(s,Xt,xs ), a.s. ∀s ∈ [t, T ], spe ially Y t = u(t, x), Z t = σ ∗∇u(t, x). Now, it's easy to the generalize the result to the ase when f satis�es assumption 2.2 . Step 3: In this step, we onsider the ase where f depends on ∇u. Assume that g, f satisfy the assumptions 2.1 - 2.3, with assumption 2.3-(iii) repla ed by (12). From the result in step 2, for any given n×d-matrix-valued fun tion v ∈ L2([0, T ]×Rd, dt⊗ρ(x)dx), f(t, x, u, v(t, x)) satis�es the assumptions in step 2. So the PDE(g, f(t, x, u, v(t, x))) admits a unique solution u ∈ H satisfying (i) and (ii) in the de�nition 2.1. Set V t,xs = v(s,X s ), then V s ∈ H2n×d(0, T ) in view of the equivalen e of the norms. We onsider the following BSDE with solution (Y · , Z Y t,xs = g(X T ) + f(s,Xt,xs , Y s , V s )ds− Zt,xs dBs, then Y t,xs = u(s,X s ), Z s = σ ∗∇u(s,Xt,xs ), a.s. ∀s ∈ [t, T ]. Now we an onstru t a mapping Ψ from H into itself. For any u ∈ H, u = Ψ(u) is the weak solution of the PDE with parameters g(x) and f(t, x, u, σ∗∇u). Symmetri ally we introdu e a mapping Φ from H2n(t, T ) × H2n×d(t, T ) into itself. For any (U t,x, V t,x) ∈ H2n(t, T )×H2n×d(t, T ), (Y t,x, Zt,x) = Φ(U t,x, V t,x) is the solution of the BSDE with parameters g(X T ) and f(s,X s , Y s , V s ). Set V s = σ ∗∇u(s,Xt,xs ), then Y t,xs = u(s,Xt,xs ), Zt,xs = σ ∗∇u(s,Xt,xs ), a.s.a.e.. Let u1, u2 ∈ H, and u1 = Ψ(u1), u2 = Ψ(u2), we onsider the di�eren e △u := u1 − u2, △u := u1−u2. Set V t,x,1s := σ∗∇u1(s,Xt,xs ), V t,x,2s := σ∗∇u2(s,Xt,xs ). We denote by (Y t,x,1, Zt,x,1)(resp. (Y t,x,2, Zt,x,2)) the solution of the BSDE with parameters g(X ) and f(s,Xt,xs , Y s , V t,x,1 s ) (resp. f(s,Xt,xs , Y s , V t,x,2 s )); then for a.e. ∀s ∈ [t, T ], Y t,x,1s = u1(s,X s ), Z t,x,1 s = σ ∗∇u1(s,Xt,xs ), t,x,2 s = u2(s,X s ), Z t,x,2 s = σ ∗∇u2(s,Xt,xs ), Denote △Y t,xs := Y t,x,1s −Y t,x,2s , △Zt,xs := Zt,x,1s −Zt,x,2s , △V t,xs := V t,x,1s −V t,x,2s . By It�'s formula applied to eγtE |△Y t,xs | , for some α and γ ∈ R, we have ∣∣△Y t,xs ∣∣2 + E eγs(γ ∣∣△Y t,xr ∣∣2 + ∣∣△Zt,xr ∣∣2)dr 6 E ∣∣△Y t,xr ∣∣2 + α ∣∣△V t,xr ∣∣2)dr, Using the equivalen e of the norms, we dedu e that eγs(γ |△u(s, x)|2 + |σ∗∇(△u)(s, x)|2)ρ(x)dsdx eγsE(γ ∣∣△Y t,xr ∣∣2 + ∣∣△Zt,xr ∣∣2)ρ(x)drdx ∣∣△Y t,xr ∣∣2 + α ∣∣△V t,xr ∣∣2)ρ(x)drdx |△u(s, x)|2 + α |σ∗∇(△u)(s, x)|2)ρ(x)dsdx. Set α = k2 , γ = 1 + k2, then we get γs(|△u(s, x)|2 + |σ∗∇(△u)(s, x)|2)ρ(x)dsdx γs |σ∗∇(△u)(s, x)|2 ρ(x)dsdx, γs(|△u(s, x)|2 + |σ∗∇(△u)(s, x)|2)ρ(x)dsdx. Consequently, Ψ is a stri t ontra tion on H equipped with the norm eγs(|u(s, x)|2 + |σ∗∇u(s, x)|2)ρ(x)dsdx. So Ψ has �xed point u ∈ H whi h is the solution of the PDE (2) asso iated to (g, f). Moreover, for t 6 s 6 T , Y t,xs = u(s,X s ), Z s = σ ∗∇u(s,Xt,xs ), .a.e. and spe ially Y t = u(t, x), Z t = σ ∗∇u(t, x), a.e. b) Uniqueness : Let u1 and u2 ∈ H be two solutions of the PDE(g, f). From Proposition 3.1, for φ ∈ C2c (Rd) and i = 1, 2 ui(r, x)dφt(r, x)dx + (u i(s, ·), φt(s, ·))− (g(·), φt(·, T ))− E(ui(r, ·), φt(r, ·))dr φt(r, x)f(r, x, u i(r, x), σ∗∇ui(r, x))drdx. (23) By (4), we get dφt(r, x)dx = (σ∗∇ui)(r, x)φt(r, x)dx)dBr (σ∗∇ui)(σ∗∇φr) + φ∇(( σ∗∇σ + b)uir) dxdr. We substitute this in (23), and get ui(s, x)φt(s, x)dx = (g(·), φt(·, T ))− (σ∗∇ui)(r, x)φt(r, x)dxdBr φt(r, x)f(r, x, u i(r, x), σ∗∇ui(r, x))drdx. Then by the hange of variable y = X̂t,xr , we obtain ui(s,Xt,ys )φ(y)dy = T )φ(y)dy + φ(y)f(s,Xt,ys , u i(s,Xt,ys ), σ ∗∇ui(s,Xt,ys ))dyds (σ∗∇ui)(r,Xt,yr )φ(y)dydBr . Sin e φ is arbitrary, we an prove this result for ρ(y)dy almost every y. So (ui(s,Xt,ys ), (σ ∗∇ui)(s,Xt,ys )) solves the BSDE(g(X T ), f), i.e. ρ(y)dy a.s., we have ui(s,Xt,ys ) = g(X T ) + f(s,Xt,ys , u i(s,Xt,ys ), σ ∗∇ui(s,Xt,ys ))ds− (σ∗∇ui)(r,Xt,yr )dBr . Then by the uniqueness of the BSDE, we know u1(s,Xt,ys ) = u 2(s,Xt,xs ) and (σ ∗∇u1)(s,Xt,ys ) = (σ∗∇u2)(s,Xt,ys ). Taking s = t we dedu e that u1(t, y) = u2(t, y), dt⊗ dy-a.s. � 5 Sobolev's solution for PDE with obsta le under monotoni - ity ondition In this se tion we study the PDE with obsta le asso iated with (g, f, h), whi h satisfy the assump- tions 2.1-2.4 for n = 1. We will prove the existen e and uniqueness of a weak solution to the obsta le problem. We will restri t our study to the ase when ϕ is polynomial in reasing in y, i.e. Assumption 5.1. We assume that for some κ1 ∈ R, β1 > 0, ∀y ∈ R, |ϕ(y)| 6 κ1(1 + |y|β1). For the sake of PDE with obsta le, we introdu e the re�e ted BSDE asso iated with (g, f, h), like in El Karoui et al. [7℄:   Y t,xs = g(X f(r,Xt,xr , Y r , Z r )dr +K −Kt,xt − Zt,xs dBs, P -a.s ∀ s ∈ [t, T ] Y t,xs > L s , P -a.s (Y t,xs − Lt,xs )dKt,xs = 0, P -a.s. where Lt,xs = h(s,X s ) is a ontinuous pro ess. Moreover following Lepeltier et al [12℄, we shall need to estimate E[ sup t 6 s 6 T ϕ2(eµt(Lt,xs ) +)] = E[ sup t 6 s 6 T ϕ2(eµth(s,Xt,xs ) 6 Ce2β1µTE[ sup t 6 s 6 T ∣∣Xt,xs ∣∣2β1β)] 6 C(1 + |x|2β1β), where C is a onstant whi h an be hanged line by line. By assumption 2.4-(ii), with same te hniques we get for x ∈ R, E[supt 6 s 6 T ϕ2((Lt,xs )+)] < +∞. Thanks to the assumption 2.1 and 2.2, by the equivalen e of norms 7 and 8, we have ) ∈ L2(FT ) and f(s,Xt,xs , 0, 0) ∈ H2(0, T ). By the existen e and uniqueness theorem for the RBSDE in [12℄, for ea h (t, x), there exists a unique triple (Y t,x, Zt,x,Kt,x) ∈ S2(t, T ) ×H2d(t, T ) ×A2(t, T ) of {F ts} progressively measurable pro esses, whi h is the solution of the re�e ted BSDE with parameters (g(X ), f(s,Xt,xs , y, z), h(s,Xt,xs ))We shall give the probabilisti interpretation for the solution of PDE with obsta le (3). The main result of this se tion is Theorem 5.1. Assume that assumptions 2.1-2.5 hold and ρ(x) = (1 + |x|)−p with p > γ where γ = β1β + β + d+ 1. There exists a pair (u, ν), whi h is the solution of the PDE with obsta le (3) asso iated to (g, f, h) i.e. (u, ν) satis�es De�nition 2.2-(i) -(iii). Moreover the solution is given by: u(t, x) = Y t , a.e. where (Y s , Z s )t 6 s 6 T is the solution of RBSDE (24), and Y t,xs = u(s,X s ), Z s = (σ ∗∇u)(s,Xt,xs ). (25) Moreover, we have for every measurable bounded and positive fun tions φ and ψ, φ(s, X̂t,xs )J(X̂ s )ψ(s, x)1{u=h}(s, x)dν(s, x) = φ(s, x)ψ(s,Xt,xs )dK s , a.s.. (26) If (u, ν) is another solution of the PDE (3) su h that ν satis�es (26) with some K instead of K, where K is a ontinuous pro ess in A2F(t, T ), then u = u and ν = ν. Remark 5.1. The expression (26) gives us the probabilisti interpretation (Feymamn-Ka 's for- mula) for the measure ν via the in reasing pro ess Kt,x of the RBSDE. This formula was �rst introdu ed in Bally et al. [1℄, where the authors prove (26) when f is Lips hitz on y and z uni- formly in (t, ω). Here we generalize their result to the ase when f is monotoni in y and Lips hitz in z. Proof. As in the proof of theorem 4.1 in se tion 4, we �rst noti e that (u, ν) solves (3) if and only if (û(t, x), dν̂(t, x)) = (eµtu(t, x), eµtdν(t, x)) is the solution of the PDE with obsta le (ĝ, f̂ , ĥ), where ĝ, f̂ are de�ned as in (12) with ĥ(t, x) = eµth(t, x). Then the oe� ient f̂ satis�es the same assumptions in assumption 2.3 with (iii) repla ed by (12), whi h means that f is de reasing on y in the 1-dimensional ase. The obsta le ĥ still satis�es assumption 2.4, for µ = 0. In the following we will use (g, f, h) instead of (ĝ, f̂ , ĥ), and suppose that (g, f, h) satis�es assumption 2.1, 2.2, 2.4, 2.5 and 2.3 with (iii) repla ed by (12). a) Existen e : The existen e of a solution will be proved in 4 steps. From step 1 to step 3, we suppose that f does not depend on ∇u, satis�es assumption 2.3' for n = 1, and f(t, x, 0) ∈ 2([0, T ]× Rd, dt⊗ ρ(x)dx). In the step 4, we study the ase when f depend on ∇u. Step 1 : Suppose g(x), f(t, x, 0), h+(t, x) uniformly bounded i.e. that there exists a onstant C su h that |g(x)|+ sup 0 6 t 6 T |f(t, x, 0)|+ sup 0 6 t 6 T h+(t, x) 6 C. We will use the penalization method. For n ∈ N, we onsider for all s ∈ [t, T ], n,t,x s = g(X T ) + f(r,Xt,xr , Y n,t,x r )dr + n (Y n,t,xr − h(r,Xt,xr ))−dr − n,t,x r dBr. From Theorem 4.1 in se tion 3, we know that un(t, x) := Y n,t,x t , is solution of the PDE(g, fn), where fn(t, x, y, x) = f(t, x, y, z) + n(y − h(t, x))−, i.e. for every φ ∈ C1,∞c ([0, T ]× Rd) (uns , ∂sφ)ds+ (u n(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(uns , φs)ds (f(s, ·, uns ), φs)ds+ n ((un − h)−(s, ·), φs)ds. Moreover Y n,t,xs = un(s,X s ), Z n,t,x s = σ ∗∇un(s,Xt,xs ), (27) Set Kn,t,xs = n (Y n,t,xr − h(r,Xt,xr ))−dr. Then by (27), we have that Kn,t,xs = n (un − h)−(r,Xt,xr )dr. Following the estimates and onvergen e results for (Y n,t,x, Zn,t,x) in the step 1 of the proof of Theorem 2.2 in [12℄, for m, n ∈ N, we have,as m,n→ ∞ ∣∣Y n,t,xs − Y m,t,xs ∣∣2 ds+ E ∣∣Zn,t,xs − Zm,t,xs ∣∣2 ds+ E sup t 6 s 6 T ∣∣Kn,t,xs −Km,t,xs ∣∣2 → 0, ∣∣Y n,t,xs ∣∣2 + ∣∣Zn,t,xs ∣∣2 + (Kn,t,xT ) 2) 6 C. By the equivalen e of the norms (8), we get ρ(x)(|un(s, x)− um(s, x)|2 + |σ∗∇un(s, x)− σ∗∇um(s, x)|2)dsdx ρ(x)E ∣∣Y n,t,xs − Y m,t,xs ∣∣2 + ∣∣Zn,t,xs − Zm,t,xs ∣∣2)dsdx→ 0. Thus (un) is a Cau hy sequen e in H, and the limit u = limn→∞ un belongs to H. Denote νn(dt, dx) = n(un − h)−(t, x)dtdx and πn(dt, dx) = ρ(x)νn(dt, dx), then by (7) πn([0, T ]× Rd) = ρ(x)νn(dt, dx) = ρ(x)n(un − h)−(t, x)dtdx ρ(x)E ∣∣∣Kn,0,xT ∣∣∣ dx 6 C ρ(x)dx <∞. It follows that πn([0, T ]× Rd) <∞. (28) In the same way like in the existen e proof step 2 of theorem 14 in [1℄, we an prove that πn([0, T ]× d) is bounded and then πn is tight. So we may pass to a subsequen e and get πn → π where π is a positive measure. De�ne ν = ρ−1π; ν is a positive measure su h that ρ(x)dν(t, x) < ∞, and so we have for φ ∈ C1,∞c ([0, T ]× Rd) with ompa t support in x, φdνn = dπn → Now passing to the limit in the PDE(g, fn), we he k that (u, ν) satis�es the PDE with obsta le (g, f, h), i.e. for every φ ∈ C1,∞c ([0, T ]× Rd), we have (us, ∂sφ)ds+ (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds (f(s, ·, us), φs)ds+ φ(s, x)1{u=h}(s, x)dν(x, s). (29) The last is to prove that ν satis�es the probabilisti interpretation (26). Sin e Kn,t,x onverges to Kt,x uniformly in t, the measure dKn,t,x → dKt,x weakly in probability. Fix two ontinuous fun tions φ, ψ : [0, T ]× Rd → R+ whi h have ompa t support in x and a ontinuous fun tion with ompa t support θ : Rd → R+, we have φ(s, X̂t,xs )J(X̂ s )ψ(s, x)θ(x)dν(s, x) = lim φ(s, X̂t,xs )J(X̂ s )ψ(s, x)θ(x)n(un − h)−(t, x)dtdx = lim φ(s, x)ψ(s,Xt,xs )θ(X s )n(un − h)−(t,Xt,xs )dtdx = lim φ(s, x)ψ(s,Xt,xs )θ(X s )dK n,t,x φ(s, x)ψ(s,Xt,xs )θ(X s )dK s dx. We take θ = θR to be the regularization of the indi ator fun tion of the ball of radius R and pass to the limit with R→ ∞, it follows that φ(s, X̂t,xs )J(X̂ s )ψ(s, x)dν(s, x) = φ(s, x)ψ(s,Xt,xs )dK s dx. (30) Sin e (Y n,t,xs , Z n,t,x n,t,x s ) onverges to (Y s , Z s ) as n → ∞ in S2(t, T )×H 2(t, T ) × 2(t, T ), and (Y t,xs , Z s ) is the solution of RBSDE(g(X ), f, h), then we have (Y t,xs − Lt,xs )dKt,xs = (u− h)(t,Xt,xs )dKt,xs = 0, a.s. it follows that dKt,xs = 1{u=h}(s,X s )dK s . In (30), setting ψ = 1{u=h} yields φ(s, X̂t,xs )J(X̂ s )1{u=h}(s, x)dν(s, x) = φ(s, X̂t,xs )J(X̂ s )dν(s, x), a.s. Note that the family of fun tions A(ω) = {(s, x) → φ(s, X̂t,xs ) : φ ∈ C∞c } is an algebra whi h separates the points (be ause x → X̂t,xs is a bije tion). Given a ompa t set G, A(ω) is dense in C([0, T ] × G). It follows that J(X̂t,xs )1{u=h}(s, x)dν(s, x) = J(X̂t,xs )dν(s, x) for almost every ω. While J(X̂t,xs ) > 0 for almost every ω, we get dν(s, x) = 1{u=h}(s, x)dν(s, x), and (26) follows. Then we get easily that Y t,xs = u(s,X s ) and Z s = σ ∗∇u(s,Xt,xs ), in view of the onvergen e results for (Y n,t,xs , Z n,t,x s ) and the equivalen e of the norms. So u(s,X s ) = Y s > h(t, x). Spe- ially for s = t, we have u(t, x) > h(t, x) Step 2 : As in the proof of the RBSDE in Theorem 2.2 in [12℄, step 2, we relax the bounded ondition on the barrier h in step 1, and prove the existen e of the solution under assumption 2.4. Similarly to step 2 in the proof of theorem 2.2 in [12℄, after some transformation, we know that it is su� ient to prove the existen e of the solution for the PDE with obsta le (g, f, h), where (g, f, h) satis�es g(x), f(t, x, 0) 6 0. Let h(t, x) satisfy assumption 2.4 for µ = 0, i.e. ∀(t, x) ∈ [0, T ]×,Rd ϕ(h(t, x)+) ∈ L2(Rd; ρ(x)dx), |h(t, x)| 6 κ(1 + |x|β). hn(t, x) = h(t, x) ∧ n, then the fun tion hn(t, x) are ontinuous, sup0 6 t 6 T h n (t, x) 6 n, and hn(s,X s ) → h(s,Xt,xs ) in F (t, T ), in view of Dini's theorem and dominated onvergen e theorem. We onsider the PDE with obsta le asso iated with (g, f, hn). By the results of step 1, there exists (un, νn), whi h is the solution of the PDE with obsta le asso iated to (g, f, hn), where un ∈ H and νn is a positive measure su h that ρ(x)dνn(t, x) <∞. Moreover Y n,t,xs = un(s,X s ), Z n,t,x s = σ ∗∇un(s,Xt,xs ), φ(s, X̂t,xs )J(X̂ s )ψ(s, x)1{un=hn}(s, x)dνn(s, x) = φ(s, x)ψ(s,Xt,xs )dK n,t,x s dx, Here (Y n,t,x, Zn,t,x,Kn,t,x) is the solution of the RBSDE(g(X T ), f, hn). Thanks to proposition 6.1 in Appendix, and the bounded assumption of g and f , we know that [ ∫ T ( ∣∣Y n,t,xs ∣∣2 + ∣∣Zn,t,xs ∣∣2 )ds+ (Kn,0,x 1 + E ϕ2( sup 0 6 t 6 T h+(t,X + sup 0 6 t 6 T (h+(t,X 6 C(1 + |x|2β1β + |x|2β). By the Lemma 2.3 in [12℄, Y n,t,xs → Y t,xs in S2(0, T ), Zn,t,xs → Zt,xs in H2d(0, T ) and Kn,t,xs → Kt,xs 2(0, T ), as n→ ∞. Moreover (Y t,xs , Zt,xs ,Kt,xs ) is the solution of RBSDE(g(X T ), f, h). By the onvergen e result of (Y n,t,xs , Z n,t,x s ) and the equivalen e of the norms (8), we get (|un(t, x)− um(t, x)|2 + |σ∗∇un(s, x)− σ∗∇um(s, x)|2)dsdx ρ(x)E ∣∣Y n,t,xs − Y m,t,xs ∣∣2 + ∣∣Zn,t,xs − Zm,t,xs ∣∣2)dsdx → 0. So {un} is a Cau hy sequen e in H, and admits a limit u ∈ H. Moreover Y t,xs = u(s,Xt,xs ), Zt,xs = σ∗∇u(s,Xt,xs ). In parti ular u(t, x) = Y t > h(t, x). Set πn = ρνn, like in step 1, we �rst need to prove that πn([0, T ]× Rd) is uniformly bounded. In (31), let φ = ρ, ψ = 1, then we have ρ(X̂0,xs )J(X̂ s )dνn(s, x) = ρ(x)dKn,0,xs dx. Re all Lemma 3.2: there exist two onstants c1 > 0 and c2 > 0 su h that ∀x ∈ Rd, 0 6 t 6 T c1 6 E ρ(t, X̂ t )J(X̂ 6 c2. Applying Hölder's inequality and S hwartz's inequality, we have πn([0, T ]× Rd) ρ(x)νn(dt, dx) 2 (x) 2 (t, X̂ 2 (X̂ 2 (x)ρ 2 (t, X̂ 2 (X̂ t )νn(dt, dx) ρ(t, X̂ t )J(X̂ ρ(x)νn(dt, dx) ρ(t, X̂ t )J(X̂ t )νn(dt, dx) ρ(t, X̂ t )J(X̂ ρ(x)νn(dt, dx) ρ(t, X̂ t )J(X̂ t )νn(dt, dx) ρ(t, X̂ t )J(X̂ ρ(x)νn(dt, dx) n,0,x t ρ(x)dx ρ(x)νn(dt, dx) ρ(x)E[K n,0,x T ]dx So by (32) and (7), we get πn([0, T ]× Rd) 6 C ρ(x)E[K n,0,x T ]dx (33) ρ(x)(1 + |x|β1β + |x|β)dx <∞. Using the same arguments as in step 1, we dedu e that πn is tight. So we may pass to a subsequen e and get πn → π where π is a positive measure. De�ne ν = ρ−1π, then ν is a positive measure su h that ρ(x)dν(t, x) < ∞. Then for φ ∈ C([0, T ]× Rd) with ompa t support in x, we have as n→ ∞, φdνn = dπn → Now passing to the limit in the PDE(g, f, hn), we he k that (u, ν) satis�es the PDE with obsta le asso iated to (g, f, h), i.e. for every φ ∈ C1,∞c ([0, T ]× Rd) (us, ∂sφ)ds+ (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds (f(s, ·, us), φs)ds+ φ(s, x)1{u=h}dν(x, s). (34) Then we will he k if the probabilisti interpretation (26) still holds. Fix two ontinuous fun - tions φ, ψ : [0, T ]× Rd → R+ whi h have ompa t support in x. With the onvergen e result of Kn,t,x, whi h implies dKn,t,x → dKt,x weakly in probability, in the same way as step 1, passing to the limit in (31) we have φ(s, X̂t,xs )J(X̂ s )ψ(s, x)dν(s, x) = φ(s, x)ψ(s,Xt,xs )dK Sin e (Y t,xs , Z s ) is the solution of RBSDE(g(X T ), f, h), then by the integral ondition, we dedu e the dKt,xs = 1{u=h}(s,X s )dK s . In (35), setting ψ = 1{u=h} yields φ(s, X̂t,xs )J(X̂ s )1{u=h}(s, x)dν(s, x) = φ(s, X̂t,xs )J(X̂ s )dν(s, x). With the same arguments, we get that dν(s, x) = 1{u=h}(s, x)dν(s, x), and (26)holds for ν and K. Step 3 : Now we will relax the bounded ondition on g(x) and f(t, x, 0). Then for m,n ∈ N, let gm,n(x) = (g(x) ∧ n) ∨ (−m), fm,n(t, x, y) = f(t, x, y)− f(t, x, 0) + (f(t, x, 0) ∧ n) ∨ (−m). So gm,n(x) and fm,n(t, x, 0) are bounded and for �xed m ∈ N, as n→ ∞, we have gm,n(x) → gm(x) in L2(Rd, ρ(x)dx), fm,n(t, x, 0) → fm(t, x, 0) in L2([0, T ]×Rd, dt⊗ ρ(x)dx), where gm(x) = g(x) ∨ (−m), fm(t, x, y) = f(t, x, y)− f(t, x, 0) + f(t, x, 0) ∨ (−m). Then as m→ ∞, we have gm(x) → g(x) in L2(Rd, ρ(x)dx), fm(t, x, 0) → f(t, x, 0) in L2([0, T ]×Rd, dt⊗ ρ(x)dx), in view of assumption 2.1 and f(t, x, 0) ∈ L2([0, T ]× Rd, dt⊗ ρ(x)dx). Now we onsider the PDE with obsta le asso iated to (gm,n, fm,n, h). By step 2, there exists a (um,n, νm,n) whi h is the solution of the PDE with obsta le asso iated to (gm,n, fm,n, h). In parti - ular the representation formulas (25) and (26) are satis�ed. Denote by (Y m,n,t,x, Zm,n,t,x,Km,n,t,x) the solution of the RBSDE (gm,n(X T ), fm,n, h). Re all the onvergen e results in step 3 of theorem 2.2 in [12℄, we know that for �xed m ∈ N, as n→ ∞, (Y m,n,t,xs , Zm,n,t,xs ,Km,n,t,xs ) → (Y m,t,xs , Zm,t,xs ,Km,t,xs ) in S2(0, T )×H2d(0, T )×A2(0, T ), and that (Y m,t,xs , Z m,t,x m,t,x s ) is the solution of RBSDE(gm(X T ), fm, h). By It�'s formula, we have for n, p ∈ N, ∣∣Y m,n,t,xs − Y m,p,t,xs ∣∣2 + ∣∣Zm,n,t,xs − Zm,p,t,xs ∣∣2)ds ∣∣gm,n(Xt,xT )− gm,p(X ∣∣2 + CE ∣∣fm,n(s,Xt,xs , 0)− fm,p(s,Xt,xs , 0) ∣∣2 ds, so by the equivalen e of the norms (7) and (8), it follows that as n→ ∞, ρ(x)(|um,n(t, x)− um,p(t, x)|2 + |σ∗∇um,n(s, x) − σ∗∇um,p(s, x)|2)dsdx ρ(x) |gm,n(x)− gm,p(x)|2 dx+ ρ(x) |fm,n(s, x, 0)− fm,p(s, x, 0)|2 dsdx→ 0. i.e. for ea h �xed m ∈ N, {um,n} is a Cau hy sequen e in H, and admits a limit um ∈ H. Moreover Y m,t,xs = um(s,X s ), Z m,t,x s = σ ∗∇um(s,Xt,xs ), a.s., in parti ular um(t, x) = Y m,t,x t > h(t, x). Then we �nd the measure νm by the sequen e {νm,n}. Set πm,n = ρνm,n, by proposition 6.1 in Appendix, we have for ea h m,n ∈ N, 0 6 t 6 T ∣∣Km,n,t,xT ∣∣2) 6 CE[g2m,n(X T ) + m,n(s,X s , 0, 0)ds+ ϕ 2( sup t 6 s 6 T (h+(s,Xt,xs ))) + sup t 6 s 6 T (h+(s,Xt,xs )) 2 + 1 + ϕ2(2T )] 6 CE[g(X 2(s,Xt,xs , 0, 0)ds+ ϕ 2( sup 0 6 s 6 T (h+(s,Xt,xs ))) + sup 0 6 s 6 T (h+(s,Xt,xs )) 2 + 1 + ϕ2(2T )] 6 C(1 + |x|2β1β + |x|2β). (35) By the same way as in step 2, we dedu e that for ea h �xed m ∈ N, πm,n is tight, we may pass to a subsequen e and get πm,n → πm where πm is a positive measure. If we de�ne νm = ρ−1πm, then νm is a positive measure su h that ρ(x)dνm(t, x) <∞. So we have for all φ ∈ C([0, T ]×Rd) with ompa t support in x, φdνm,n = dπm,n → dπm = φdνm. Now for ea h �xed m ∈ N, let n → ∞, in the PDE(gm,n, fm,n, h), we he k that (um, νm) satis�es the PDE with obsta le asso iated to (gm, fm, h), and by the weak onvergen e result of dKm,n,t,x, we have easily that the probabilisti interpretation (26) holds for νm and K Then let m → ∞, by the onvergen e results in step 4 of theorem 2.2 in [12℄, we apply the same method as before. We dedu e that limm→∞ um = u is in H and Y t,xs = u(s,Xt,xs ), Zt,xs = σ∗∇u(s,Xt,xs ), a.s., where (Y t,x, Zt,x,Kt,x) is the solution of the RBSDE(g, f, h), in parti ular, setting s = t, u(t, x) = Y t > h(t, x). From (35), it follows that m,t,x 2] 6 C(1 + |x|2ββ1 + |x|2β). By the same arguments, we an �nd the measure ν by the sequen e {νm}, whi h satis�es that for all φ and ψ with ompa t support, φ(s, X̂t,xs )J(X̂ s )ψ(s, x)1{u=h}(s, x)dν(s, x) = φ(s, x)ψ(s,Xt,xs )dK s dx. Finally we �nd a solution (u, ν) to the PDE with obsta le (g, f, h), when f does not depend on ∇u. So for every φ ∈ C1,∞c ([0, T ]× Rd) (us, ∂sφ)ds+ (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds (f(s, ·, us), φs)ds+ φ(s, x)1{u=h}dν(x, s). (36) Step 4 : Finally we study the ase when f depends on ∇u, and satis�es a Lips hitz ondition on ∇u. We onstru t a mapping Ψ from H into itself. For some u ∈ H, de�ne u = Ψ(u), where (u, ν) is a weak solution of the PDE with obsta le (g, f(t, x, u, σ∇u), h). Then by this mapping, we denote a sequen e {un} in H, beginning with a fun tion v0 ∈ L2([0, T ] × Rd, dt ⊗ ρ(x)dx). Sin e f(t, x, u, v0(t, x)) satis�es the assumptions of step 3, the PDE(g, f(t, x, u, v0(t, x)), h) admits a solution (u1, v1) ∈ H. For n ∈ N, set un(t, x) = Ψ(un−1(t, x)). Symmetri ally we introdu e a mapping Φ from H2(t, T ) × H2d(t, T ) into itself. For V t,x,0 = v0(s,Xt,xs )), then V s ∈ H2d(t, T ) in view of the equivalen e of the norms. Set (Y t,x,n, Zt,x,n) = Φ(Y t,x,n−1, Zt,x,n−1), where (Y t,x,n, Zt,x,n,Kt,x,n)is the solution of the RBSDE with parameters g(X T ), f(s,X s , Y s , Z t,x,n−1 and h(s,Xt,xs ).Then Y t,x,n s = un(s,X s ), Z t,x,n s = σ ∗∇un(s,Xt,xs ) a.s. and φ(s, X̂t,xs )J(X̂ s )ψ(s, x)1{u=h}(s, x)dνn(s, x) = φ(s, x)ψ(s,Xt,xs )dK t,x,n s dx. Set ũn(t, x) := un(t, x) − un−1(t, x). To deal with the di�eren e ũn, we need the di�eren e of the orresponding BSDE, denote Ỹ t,x,ns := Y t,x,n s − Y t,x,n−1s , Z̃t,x,ns := Zt,x,ns − Zt,x,n−1s , K̃t,x,ns := Kt,x,ns −Kt,x,n−1s . It follows from It�'s formula, for some α, γ ∈ R, ∣∣∣Ỹ t,x,ns eγr(γ ∣∣∣Ỹ t,x,nr ∣∣∣Z̃t,x,nr ∣∣∣Ỹ t,x,nr ∣∣∣Z̃t,x,n−1r sin e eγrỸ t,x,nr dK̃ t,x,n eγr(Y t,x,ns − h(r,Xt,xr ))dKt,x,n + eγr(Y t,x,n−1s − h(r,Xt,xr ))dKt,x,n−1 eγr(Y t,x,ns − h(r,Xt,xr ))dKt,x,n−1 + eγr(Y t,x,n−1s − h(r,Xt,xr ))dKt,x,n then by the equivalen e of the norms, for γ = 1 + k2, we have eγs(|ũn(s, x)|2 + |σ∗∇(ũn)(s, x)|2)ρ(x)dsdx eγs(|ũ2(s, x)|2 + |σ∗∇(ũ2)(s, x)|2)ρ(x)dsdx )n−1(‖u1(s, x)‖2γ + ‖u2(s, x)‖ where ‖u‖2 eγs(|u(s, x)|2 + |σ∗∇u(s, x)|2)ρ(x)dsdx, whi h is equivalent to the norm ‖·‖ of H. So {un} is a Cau hy sequen e in H, it admits a limit u in H, whi h is the solution to the PDE with obsta le (2). Then onsider σ∗∇u as a known fun tion by the result of step 3, we know that there exists a positive measure ν su h that ρ(x)dν(t, x) < ∞, and for every φ ∈ C1,∞c ([0, T ]× Rd), (us, ∂sφ)ds+ (u(t, ·), φ(t, ·)) − (g(·), φ(·, T )) + E(us, φs)ds (f(s, ·, us, σ∗∇us), φs)ds+ φ(s, x)1{u=h}dν(x, s). (37) Moreover, for t 6 s 6 T , s = u(s,X s ), Z s = σ ∗∇u(s,Xt,xs ), a.s.a.e., φ(s, X̂t,xs )J(X̂ s )ψ(s, x)1{u=h}(s, x)dν(s, x) φ(s, x)ψ(s,Xt,xs )dK b) Uniqueness : Set (u, ν) to be another solution of the PDE with obsta le (3) asso iated to (g, f, h); with ν veri�es (26) for an in reasing pro ess K. We �x φ : Rd → R, a smooth fun tion in C2c (R d) with ompa t support and denote φt(s, x) = φ(X̂ s )J(X̂ s ). From proposition 3.1, one may use φt(s, x) as a test fun tion in the PDE(g, f, h) with ∂sφ(s, x)ds repla ed by a sto hasti integral with respe t to the semimartingale φt(s, x). Then we get, for t 6 s 6 T u(r, x)dφt(r, x)dx + (u(s, ·), φt(s, ·))− (g(·), φt(·, T )) + E(ur, φr)dr (38) f(r, x, u(r, x), σ∗∇u(r, x))φt(r, ·)dr + φt(r, x)1{u=h}dν(x, r). By (5) in Lemma 3.1, we have udrφt(r, x)dx = (σ∗∇u)(r, x)φt(r, x)dx)dBr (σ∗∇u)(σ∗∇φr) + φt∇(( σ∗∇σ + b)u) dxdr. Substitute this equality in (38), we get u(s, x)φt(s, x)dx = (g(·), φt(·, T ))− (σ∗∇u)(r, x)φt(r, x)dx)dBr f(r, x, u(r, x), σ∗∇u(r, x))φt(s, ·)dr + φt(r, x)1{u=h}dν(x, r). Then by hanging of variable y = X̂t,xr and applying (26) for ν, we obtain u(s,Xt,ys )φ(y)dy T )φ(y)dy + φ(y)f(s,Xt,ys , u(s,X s ), σ ∗∇u(s,Xt,ys )ds φ(y)1{u=h}(r,X s )dK r dy − (σ∗∇u)(r,Xt,yr )φ(y)dy)dBr . Sin e φ is arbitrary, we an prove that for ρ(y)dy almost every y, (u(s,Xt,ys ), (σ ∗∇u)(s,Xt,ys ), K̂t,xs ) solves the RBSDE(g(X ), f, h). Here K̂t,xs = 1{u=h}(r,X r )dK r . Then by the uniqueness of the solution of the re�e ted BSDE, we know u(s,Xt,ys ) = Y s = u(s,X s ), (σ ∗∇u)(s,Xt,ys ) = Zt,ys = (σ ∗∇u)(s,Xt,ys ) and K̂t,ys = Kt,ys . Taking s = t we dedu e that u(t, y) = u(t, y), ρ(y)dy-a.s. and by the probabilisti interpretation (26), we obtain φt(r, x)1{u=h}(r, x)dν(x, r) = φt(r, x)1{u=h}(r, x)dν(x, r). So 1{u=h}(r, x)dν(x, r) = 1{u=h}(r, x)dν(x, r). � 6 Appendix 6.1 Proof of proposition 3.1 First we onsider the ase when f does not depend on z and satis�es assumption 2.3'. As in step 2 of the proof of theorem 4.1, we approximate g and f as in (18), then gn → g in L2(Rd, ρ(x)dx) and fn(t, x, 0) → f(t, x, 0) in L2([0, T ]× Rd, dt⊗ ρ(x)dx), as n→ ∞. Sin e for ea h n ∈ N, |gn| 6 n and |fn(t, x, 0)| 6 n, by the result of the step 1 of theorem 4.1, the PDE(gn, fn) admits the weak solution un ∈ H and sup0 6 t 6 T |un(t, x)| 6 Cn. So we know |fn(t, x, un(t, x))|2 6 |fn(t, x, 0)|2 + ϕ( sup 0 6 t 6 T |un(t, x)|) 6 Cn. Set Fn(t, x) := fn(t, x, un(t, x)), then Fn(t, x) ∈ L2([0, T ]× Rn, dt⊗ ρ(x)dx). From proposition 2.3 in Bally and Matoussi [3℄, for φ ∈ C2c (Rd), we get, for t 6 s 6 T un(r, x)dφt(r, x)dx + (un(s, ·), φt(s, ·))− (gn(·), φt(·, T )) + E(un(r, ·), φt(r, ·))dr f(r, x, un(r, x))φt(r, x)drdx + (fn(r, x, 0)− f(r, x, 0))φt(r, x)drdx. By step 2, we know that as n→ ∞, un → u in H, where u is a weak solution of the PDE(g, f), i.e. un → u in L2([0, T ]× Rd, dt⊗ ρ(x)dx), ∗∇un → ∇u in L2([0, T ]× Rd, dt⊗ ρ(x)dx). Then there exists a fun tion u∗ in L2([0, T ] × Rd, dt ⊗ ρ(x)dx), su h that for a subsequen e of {un}, |unk | 6 |u∗| and unk → u, dt ⊗ dx-a.e. Thanks to assumption 2.3'-(iii), we have that f(r, x, un(r, x)) → f(r, x, u(r, x)), dt ⊗ dx-a.e. Now, for all ompa t support fun tion φ ∈ C2c (Rd), the se ond term in the right hand side of (39) onverge to 0 as n→ ∞ and it is not hard to prove by using the dominated onvergen e theorem the term in the left hand side of (39) onverges. Thus, we on lude that limn→∞ f(r, x, un(r, x))φt(r, x)drdx exists. Moreover by the monotono ity ondition of f and the same arguments as in step 2 of the proof of theorem 4.1, we get for all ompa t support fun tion φ ∈ C2c (Rd) u(r, x)dφt(r, x)dx + (u(s, ·), φt(s, ·)) − (g(·), φt(·, T )) + E(u(r, ·), φt(r, ·))dr f(r, x, u(r, x))φt(r, x)drdx . Now we onsider the ase when f depends on ∇u and satis�es the assumption 2.3 with (iii) repla ed by (12). Like in the step 3 of the proof of theorem 4.1, we onstru t a mapping Ψ from H into itself. Then by this mapping, we de�ne a sequen e {un} in H, beginning with a matrix-valued fun tion v0 ∈ L2([0, T ] × Rn×d, dt ⊗ ρ(x)dx). Sin e f(t, x, u, v0(t, x)) satis�es the assumptions of step 2, the PDE(g, f(t, x, u, v0(t, x))) admits a unique solution u1 ∈ H. For n ∈ N, denote un(t, x) = Ψ(un−1(t, x)), i.e. un is the weak solution of the PDE(g, f(t, x, u, σ ∗∇un−1(t, x))). Set ũn(t, x) := un(t, x) − un−1(t, x). In order to estimate the di�eren e, we introdu e the orresponding BSDE(g, fn) for n = 1, where fn(t, x, u) = f(t, x, u,∇un−1(t, x)). So we have Y n,t,xs = un(s,Xt,xs ), Zn,t,xs = σ∇un(s,Xt,xs ). Then we apply the It�'s formula to |Ỹ n,t,x|2, where Ỹ n,t,xs := Y n,t,xs − Y n−1,t,xs . With the equivalen e of the norms, similarly as in step 3, for γ = 1 + k2, we have eγs(|ũn(s, x)|2 + |σ∗∇(ũn)(s, x)|2)ρ(x)dsdx eγs(|ũ2(s, x)|2 + |σ∗∇(ũ2)(s, x)|2)ρ(x)dsdx )n−1(‖u1(s, x)‖2γ + ‖u2(s, x)‖ where ‖u‖2 eγs(|u(s, x)|2 + |σ∗∇u(s, x)|2)ρ(x)dsdx, whi h is equivalent to the norm ‖·‖ of H. So {un} is a Cau hy sequen e in H, it admits a limit u in H, and by the �xed point theorem, u is a solution of the PDE(g, f). Then for ea h n ∈ N, we have for φ ∈ C2c (Rd) un(r, x)dφt(r, x)dx + (un(s, ·), φt(s, ·))− (g(·), φt(·, T )) + E(un(r, ·), φr(r, ·))dr f(r, x, un(r, x), σ ∗∇un−1(r, x))φt(r, x)drdx f(r, x, un(r, x), σ ∗∇u(r, x))φt(r, x)drdx [f(r, x, un(r, x), σ ∗∇un−1(r, x)) − f(r, x, un(r, x), σ∗∇u(r, x))]φt(r, x)drdx. Noti ing that f is Lips hitz in z, we get |f(r, x, un(r, x), σ∗∇un−1(r, x)) − f(r, x, un(r, x), σ∗∇u(r, x))| 6 k |σ∗∇un−1(r, x) − σ∗∇u(r, x)| . So the last term of the right side onverges to 0, sin e {σ∗∇un} onverges to σ∗∇u in L2([0, T ]× d, dt ⊗ ρ(x)dx). Now we are in the same situation as in the �rst part of proof, and in the same way, we dedu e that the following holds: for φ ∈ C2c (Rd) u(r, x)dφt(r, x)dx + (u(s, ·), φt(s, ·)) − (g(·), φt(·, T )) + E(u(r, ·), φt(r, ·))dr f(r, x, u(r, x), σ∗∇u(r, x))φt(r, x)drdx, dt ⊗ dx, a.s.. Now if f satis�es assumption 2.3, we know that u is solution of the PDE(g, f) if and only if û = eµtu is solution of the PDE(ĝ, f̂), where ĝ(x) = eµT g(x), f̂(t, x, y, x) = eµtf(t, x, e−µty, e−µtz)− µy, and f̂ satis�es assumption 2.3-(iii) repla ed by (12). So we know now: for φ ∈ C2c (Rd), û(r, x)dφt(r, x)dx + (û(s, ·), φt(s, ·)) − (ĝ(·), φt(·, T )) + E(û(r, ·), φt(r, ·))dr f̂(r, x, û(r, x),∇û(r, x))φt(r, x)drdx, dt ⊗ dx, a.s.. Noti e that d(eµru(r, x)) = µeµru(r, x)dr + eµrd(u(r, x)), so by the integration by parts formula (for sto hasti pro ess), we get u(r, x)dφt(r, x)dx e−µrû(r, x)dφt(r, x)dxdr = e−µT (ĝ(·), φt(·, T ))− e−µs(û(s, ·), φt(s, ·)) + µ û(r, x)φt(r, x)dxdr e−µrφt(r, x)[Lû(r, x) + f̂(r, x, û(r, x),∇û(r, x))]drdx. Using (11), we get that for φ ∈ C2c (Rd), u(r, x)dφt(r, x)dx = (g(·), φt(·, T ))− (u(s, ·), φt(s, ·))− φt(r, x)Lu(r, x)drdx φt(r, x)f(r, x, u(r, x),∇u(r, x))drdx = (g(·), φt(·, T ))− (u(s, ·), φt(s, ·))− E(u(r, ·), φt(r, ·))dr φt(r, x)f(r, x, u(r, x),∇u(r, x))drdx, and �nally, the result follows. � 6.2 Some a priori estimates In this subse tion, we onsider the non-markovian Re�e ted BSDE asso iated to (ξ, f, L) :   Yt = ξ + f(t, Ys, Zs)ds+KT −Kt − ZsdBs, Yt > Lt, (Ys − Ls)dKs = 0 under the following assumptions : (H1) a �nal ondition ξ ∈ L2(FT ), (H2) a oe� ient f : Ω× [0, T ]× R× Rd → R, whi h is su h that for some ontinuous in reasing fun tion ϕ : R+ −→ R+, a real numbers µ and C > 0: (i) f(·, y, z) is progressively measurable, ∀(y, z) ∈ R× Rd; (ii) |f(t, y, 0)| 6 |f(t, 0, 0)|+ ϕ(|y|), ∀(t, y) ∈ [0, T ]× R, a.s.; (iii) E |f(t, 0, 0)|2 dt <∞; (iv) |f(t, y, z)− f(t, y, z′)| 6 C |z − z′| , ∀(t, y) ∈ [0, T ]× R, z, z′ ∈ Rd, a.s. (v) (y − y′)(f(t, y, z)− f(t, y′, z)) 6 µ(y − y′)2, ∀(t, z) ∈ [0, T ]× Rd, y, y′ ∈ R, a.s. (vi) y → f(t, y, z) is ontinuous, ∀(t, z) ∈ [0, T ]× Rd, a.s. (H3) a barrier (Lt)0 6 t 6 T , whi h is a ontinuous progressively measurable real-valued pro ess, satisfying E[ϕ2( sup 0 6 t 6 T (eµtL+t ))] <∞, and (L+t )0 6 t 6 T ∈ S2(0, T ), LT 6 ξ, a.s. We shall give an a priori estimate of the solution (Y, Z,K) with respe t to the terminal ondition ξ, the oe� ient f and the barrier L. Unlike the Lipshitz ase, we have in addition the term Eϕ2(sup0 6 t 6 T (L t )) and a onstant, whi h only depends on ϕ, µ, k and T : Proposition 6.1. There exists a onstant C, whi h only depends on T , µ and k, su h that 0 6 t 6 T |Yt|2 + |Zs|2 ds+ |KT |2 f2(t, 0, 0)dt+ ϕ2( sup 0 6 t 6 T (L+t )) + CE[ sup 0 6 t 6 T (L+t ) 2 + 1 + ϕ2(2T )]. Proof. Applying It�'s formula to |Yt|2, and taking expe tation, then E[|Yt|2 + |Zs|2 ds] = E[|ξ|2 + 2 Ysf(s, Ys, Zs)ds+ 2 LsdKs 6 E[|ξ|2 + 2 Ysf(s, 0, 0)ds+ 2 (µ |Ys|2 + k |Ys| |Zs|)ds+ 2 LsdKs]. It follows that E[|Yt|2 + |Zs|2 ds] 6 E[|ξ|2 + 2 f2(s, 0, 0)ds+ (2µ+ 1 + 2k2) |Ys|2 ds+ 2 LsdKs]. Then by Gronwall's inequality, we have E |Yt|2 6 CE[|ξ|2 + f2(s, 0, 0)ds+ LsdKs], (40) |Zs|2 ds 6 CE[|ξ|2 + f2(s, 0, 0)ds+ LsdKs], (41) where C is a onstant only depends on µ, k and T , in the following this onstant an be hanged line by line. Now we estimate K by approximation. By the existen e of the solution, theorem 2.2 in Lepeltier et al. [12℄, we take the pro ess Z as a known pro ess. Without losing generality we write f(t, y) for f(t, y, Zt), here f(t, 0) = f(t, 0, Zt) is a pro ess in H 2(0, T ). Set m,n = (ξ ∨ (−n)) ∧m, fm,n(t, y) = f(t, y)− f(t, 0) + (f(t, 0) ∨ (−n)) ∧m. Form, n ∈ N, ξm,n and sup0 6 t 6 T fm,n(t, 0) are uniformly bounded. Consider the RBSDE(ξm,n, fm,n, L), t = ξ m,n + fm,n(t, Y m,ns )ds+K Zm,ns dBs, t > Lt, (Y m,ns − Ls)dKm,ns = 0. if we re all the transform in step 2 of the proof of theorem 2.2 in Lepeltier et al. [12℄, sin e ξm,n, fm,n(t, 0) 6 m, we know that (Y t , Z t ) is the solution of this RBSDE, if and only if (Y m,n′, Zm,n′,Km,n′) is the solution of RBSDE(ξm,n′, fm,n′, L′), where t , Z t ) = (Y t +m(t− 2(T ∨ 1)), Z ξm,n′ = ξm,n + 2mT −m(T ∨ 1), fm,n′(t, y) = fm,n(t, y −m(t− 2(T ∨ 1)))−m, t = Lt +m(t− 2(T ∨ 1)). Without losing generality we set T > 1. Then ξm,n′ 6 0 and fm,n′(t, 0) 6 0. Sin e (Y m,n′, Zm,n′,Km,n′) is the solution of RBSDE(ξm,n′, fm,n′, L′), then we have T = Y 0 − ξ m,n′ − m,n′(s, Y m,n′s , Zs)ds+ s dBs, whi h follows )2] 6 4E[ ∣∣Y m,n′0 ∣∣2 + |ξm,n′|2 + ( fm,n′(s, Y m,n′s )ds) |Zm,n′s | ds]. (42) Applying It�'s formula to |Y m,n|2, like (40) and (41), we have E |Y m,nt | |Zm,ns | ds 6 CE[|ξm,n|2 + (fm,n(s, 0))2ds+ ∣∣Y m,n′0 ∣∣2 + |Zm,n′s | ds = 2 |Y m,n0 | + 8m2T 2 + E |Zm,ns | 6 CE[|ξm,n|2 + (fm,n(s, 0)ds)2 + s ] + 8m 2T 2. For the third term on the right side of (42), from Lemma 2.3 in Lepeltier et al. [12℄, we remember fm,n′(s, Y m,n′s )ds) 6 max{( fm,n′(s, Ỹ m,ns )ds) fm,n′(s, Y s )ds) 2}, (43) where (Ỹ m,n, Z̃m,n) is the solution the following BSDE t = ξ m,n′ + m,n′(s, Ỹ m,ns )ds− s dBs, (44) s = ess sup τ∈Tt,T +1{τ<T} + (ξ m,n)+1{τ=T}|Ft] = ess sup τ∈Tt,T +|Ft]. From (44), and proposition 2.2 in Pardoux [13℄, we have fm,n′(s, Ỹ m,ns )ds) 6 CE[|ξm,n′|2 + ( fm,n′(s, 0)ds)2] 6 CE[|ξm,n|2 + (fm,n(s, 0))2ds] + Cϕ2(2mT ) + Cm2. For the other term in (43), with sup0 6 t 6 T Y s = sup0 6 t 6 T (L , we get m,n′(s, Y s )ds) 2(fm,n′(s, 0))2ds+ 2Tϕ2( sup 0 6 t 6 T (L′t) 6 E[4 fm,n(s, 0)2ds+ 2Tϕ2( sup 0 6 t 6 T +)] + 2m2T + 4Tϕ2(2mT ). Consequently, we dedu e that 2] = E[(K 6 CE[|ξm,n|2 + (fm,n(s, 0))2ds+ s + ϕ 2( sup 0 6 t 6 T +) +m2 + ϕ2(2mT )] 6 CE[|ξ|2 + (f(s, 0, Zs)) 2ds+ ϕ2( sup 0 6 t 6 T +) + sup 0 6 t 6 T ((Lt) +)2] + +C(m2 + ϕ2(2mT )). Moreover using (41) and the fa t that f is Lips hitz on z, it follows that )2] 6 CE[|ξ|2 + (f(s, 0, 0))2ds+ ϕ2( sup 0 6 t 6 T +) + sup 0 6 t 6 T ((Lt) +)2 (45) LsdKs] + C(m 2 + ϕ2(2mT )). Let m→ ∞, then E[|ξm,n − ξn|2] → 0, E |fm,n(t, 0)− fn(t, 0)|2 → 0, where ξn = ξ ∨ (−n) and fn(t, y) = f(t, y)− f(t, 0) + f(t, 0) ∨ (−n). Thanks to the onvergen e result of step 3 of the proof for theorem 2.2 in [12℄, we know that (Y m,n, Zm,n,Km,n) → (Y n, Zn,Kn) in S2(0, T )×H2d(0, T )×A2(0, T ), where (Y n, Zn,Kn) is the soultion of the RBSDE(ξn, fn, L). MoreoverK T ց KnT in L2(FT ), so we haveKnT 6 K T , whi h implies for ea h n ∈ N, E[(KnT ) 2] 6 E[(K 2] (46) Then, letting n→ ∞, by the onvergen e result in step 4, sin e E[|ξn − ξ|2] → 0, E |fn(t, 0)− f(t, 0)|2 → 0, the sequen e (Y n, Zn,Kn) → (Y, Z,K) in S2(0, T )× H2d(0, T ) ×A2(0, T ), where (Y, Z,K) is the solution of the RBSDE(ξ, f, L). From (46), and (45) for m = 1, we get E[(KT ) 2] 6 CE[|ξ|2 + (f(s, 0, 0))2ds+ ϕ2( sup 0 6 t 6 T +) + sup 0 6 t 6 T ((Lt) LsdKs] + C(1 + ϕ 2(2T )) 6 CE[|ξ|2 + (f(s, 0, 0))2ds+ ϕ2( sup 0 6 t 6 T +) + sup 0 6 t 6 T ((Lt) E[(KT ) 2] + C(1 + ϕ2(2T )). Then it follows that for ea h t ∈ [0, T ], E[|Yt|2 + |Zs|2 ds+ (KT )2] 6 CE[|ξ|2 + (f(s, 0, 0))2ds+ ϕ2( sup 0 6 t 6 T + sup 0 6 t 6 T ((Lt) +)2] + C(1 + ϕ2(2T )). Finally we get the result, by applying BDG inequality. � Referen es [1℄ Bally V., Caballero E., El-Karoui N. and B. Fernandez : Re�e ted BSDE's PDE's and Varia- tional Inequalities (to appear in Bernoulli 2007). [2℄ Barles, G. and L. 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Introduction Notations and preliminaries Stochastic flow and random test functions Sobolev's Solutions for PDE's under monotonicity condition Sobolev's solution for PDE with obstacle under monotonicity condition Appendix Proof of proposition 3.1 Some a priori estimates

0704.1415

Exact distribution of the sample variance from a gamma parent distribution

Exact Distribution of the Sample Variance from a Gamma Parent Distribution T. Royen Fachhochschule Bingen, University of Applied Sciences, Berlinstrasse 109, D–55411 Bingen, Germany e-mail: [email protected] Abstract Several representations of the exact cdf of the sum of squares of n independent identically gamma–distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders α some further series are provided, including a convex combination of gamma distributions for α = 1 and nearly of this type for α > 1. Furthermore, some representations of the distribution of the angle Φ between (X1, ...,Xn) and (1, ...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Φ). AMS 2000 Subject Classifications: 62E15, 62H10 Keywords: Exact distribution of quadratic forms in non–normal random vari- ables, Exact distribution of the sample variance, Gamma distribution, Exponential distribution 1 Introduction The distribution of the sample variance s2 in non–normal cases has attracted sporadic attention during the last eight decades. Early investigations, concerning a gamma par- ent distribution can be traced at least to Craig (1929) and Pearson (1929), who used moment approximations, later investigated more thoroughly by Bowman and Shen- ton(1983). For general distributions various approximations were proposed by Box (1953), Roy and Tiku (1962), Tan and Wong (1977) and by Mudholkar and Trivedi (1981). The latter authors recommend transformations of the Wilson–Hilferty type. Exact results seem to be available only for a mixture of two normal distributions (Hyre- nius 1950, Mudholkar and Trivedi 1981). See also chapter 4.7 in Mathai and Provost (1992). The interest in the distribution of s2 or 1/s was also stimulated by investiga- tions on the estimation of process capability indices, see e.g. Pearn, Kotz and Johnson (1992) and Pearn and Kotz (2006). http://arxiv.org/abs/0704.1415v1 Apart of its intrinsic theoretical value, an analytical representation of the cdf of s2 enables more rapid and more accurate evaluations than Monte Carlo methods. Besides, it provides a base to investigate the accuracy of the different proposed approximations. Now let X1, . . . ,Xn be a random sample with mean X̄ and sample variance S2 = (n− 1)−1 ∑ni=1(Xi − X̄)2 from a gamma distribution with pdf gα(x) = x α−1 exp(−x)/Γ(α), (1) and cdf Gα(x) = gα(ξ )dξ . Xi, Z := X2i , U 2 := ZY−2, (2) we have (n− 1)S2 = (U2 − 1/n)Y2, 1/ n ≤U ≤ 1. (3) By Laha (1954) — see also Lukacs (1955) — the mutual independence of X and the sample coefficient of variation S/X = n((U2 − 1/n)/(1− 1/n))1/2 was shown to characterize the family of gamma distributions indexed by α . This is the key to derive the distribution of S2 essentially from the distribution of Z. At first, it follows with the angle Φ between (X1, . . . ,Xn) and (1, . . . ,1) that Hα ,n(r) := Pr{Z ≤ r2} = EU(Gαn(r/U)) = EΦ(Gαn(r ncosΦ)), n− 1S ≤ r} = EU(Gαn(r(U2 − 1/n)−1/2) = EΦ(Gαn(r ncotΦ)). With the cdf Fα ,n of U it follows also Hα ,n(r) = EY (Fα ,n(r/Y )) = rαn(Γ(αn))−1 xαn−1Fα ,n(x −1)e−rxdx. h∗α ,n(r) := r −αnHα ,n(r) (7) is the Laplace transform (L.t.) of f ∗α ,n(x) := (Γ(αn)) −1xαn−1Fα ,n(x −1), 0 < x ≤ n. (8) A power series for h∗α ,n – and consequently the cdf Hα ,n – is derived in (12) . . . (16) of the follwing section. Due to (7), (8) the coefficients of this series determine the moments of f ∗α ,n and therefore Fα ,n and the cdf in (5). For a cdf a probability mixture representation is more appealing than a power series. Such a mixture of gamma distributions for Hα ,n is found in (20) in section 2. Then the reader might go directly to theorem 4.1, which is the main result con- cerning the actual computation of the cdf of S2. By means of a few lines of code of a computer algebra system some tables of this cdf with at least eight correct digits have been computed for some values of α and n. Representations for f ∗α ,n by an orthogonal series with Legendre polynomials and by a Fourier sinus series are derived too in section 2. Because of the relations (7), (8) these two series provide two further representations of h∗α ,n or Hα ,n. The corresponding series for Fα ,n(u) in (31) and the orthogonal series derived from (32) can be used within (5). However, the application of theorem 4.1 is simpler and more accurate. In spite of the numerical use of the double series in theorem 4.1 it would be theo- retically more satisfying to have a single alternating power series or even a probability mixture representation for the cdf of S2. The latter is accomplished exactly for the exponential case (α = 1) and nearly for integer α > 1 by theorem 4.2, which contains also an alternating power series. However, for numerical purposes, theorem 4.2 is (at present) not considered as a competitor to theorem 4.1 since the computation of the required coefficients is more cumbersome. The method is explained in section 3. It is based on a representation of the cdf of tanΦ by a polynomial on a certain section [0,ϕn] of the domain of Φ. This formula is found in theorem 3.1. The proofs of all the theorems are given in the appendix. Throughout the paper formulas from the handbook of mathematical functions by Abramowitz and Stegun are cited by A.S. and their number. The symbol ∑(k) stands for summation over all possible decompositions k = k1+ . . .+kn with non–negative in- tegers k j. Moments of positive random variables occur also with non-integer exponents and for defective distributions. 2 The distribution of the sum of squares The L.t. of Z1 = X 1 is given by ψα(t) = (2Γ(α))−1 zα/2−1 exp(− z− tz)dz = (2Γ(α))−1 (−1)k Γ((α + k)/2) t−(α+k)/2 (9) = (2Γ(α))−1t−α/2 exp (α + 1 with Kummer’s confluent hypergeometric function M. In particular ψ1(t) = with the cdf Φ(x) = (1+ erf(x/ 2))/2 of the standard normal distribution. For in- teger values (1−α)/2 or (2−α)/2 Kummer’s M is given by Hermite polynomials (A.S. 13.6.17/18). The cdf Hα ,n(r) = Pr{Z ≤ r2} can be obtained by the Fourier inversion formula, but some further representations are useful. If ψα(t) is written as t−α/2βα(t−1/2), then (ψα(t))n = t−αn/2 βα ,n,k t−k/2 (11) βα ,n,k = (−1)k(2Γ(α))−n ∑ Γ((α + ki)/2) (βα(y−1))nyk dϕ , y = ρeiϕ . (12) ρ > 0, −π < ϕ ≤ π , k ∈ N0. The parameter ρ was inserted here only for numerical considerations. The βα ,n,k are more quickly computed recursively for n = 1,2,4,8, . . . than by the integrals. Laplace inversion in (11) implies Hα ,n(r) = r βα ,n,k Γ(1+(αn+ k)/2) rk. (13) By the series expansion of Gαn in (4) we obtain with the moments µα ,n,k := E n cosΦ)αn+k = n(αn+k)/2γα ,n,k (14) Hα ,n(r) = Γ(αn) µα ,n,k αn+ k (−r)k and by comparison with (13) the relation (−1)k µα ,n,k = 2Γ(αn) βα ,n,k Γ((αn+ k)/2) . (16) Also the following representations of the distribution functions of ∑ni=1 X 2,U and tanΦ are based essentially on the sequence (µα ,n,k) or equivalently (γα ,n,,k). A probability mixture representation for Hα ,n by gamma distribution functions is obtained as follows: With any scale factor λ let be µα ,n,k,λ = λ −(αn+k)µα ,n,k . (17) Then, multiplying Hα ,n(r) = λ (λ r)αn−1 Γ(αn) µα ,n,k,λ (−λ r)k/k! by e−λ r ∑∞ℓ=0(λ r) ℓ/ℓ! we obtain λ gαn(λ r) δα ,n,k,λ (λ r) k/k! (18) with the alternating kth order differences δα ,n,k,λ = (−∆)kµα ,n,0,λ = (−1) j µα ,n, j,λ (19) and consequently Hα ,n(r) = δα ,n,k,λ αn+ k− 1 Gαn+k(λ r). (20) In particular with λ = n this is a probability mixture since δα ,n,k,√n = E ((cosΦ) αn(1− cosΦ)k)> 0 and αn+ k− 1 δα ,n,k,√n = 1 (21) by virtue of the binomial series with 0≤ 1−cosΦ≤ 1−1/ n. However, for numerical approximations different values of λ should be more suitable, e.g. ))1/(αn) = µ1/(αn)α ,n,0 = 2Γ(αn) (2Γ(α))n (Γ(α/2))n Γ(αn/2) )1/(αn) with µα ,n,0,λ = 1. Besides, two further representations of Hα ,n are obtained from the following or- thogonal series for f ∗αn from (7), (8). With xk f ∗αn(x)dx = Γ(αn) µα ,n,k αn+ k , (23) implied by (14), (15), and the shifted Legendre polynomials P∗k (y) = p∗k, jy j, 0 ≤ y ≤ 1, (P∗k (y)) 2dy = 2k+ 1 , (24) (A.S. 22.2.11), it follows f ∗α ,n(x) = cα ,n,kP k (x/ cα ,n,k = (2k+ 1) nαn/2 Γ(αn) p∗k j γα ,n, j αn+ j Thus, with (A.S. 11.4.26) and the modified spherical Bessel functions (A.S. 10.2.2) we Hα ,n(r) = r αn exp (−1)kcα ,n,k Ik+1/2 . (26) Consequently, the Fourier transform of f ∗α ,n is representable by h∗α ,n(−it) = exp ikcα ,n,k jk with the spherical Bessel functions jk from (A.S. 10.1). For absolutely large real t this series is numerically more suitable than the power series from (15). Such values are required as coefficients in the following Fourier sinus expansion for f ∗α ,n. f ∗α ,n(x) = bα ,n,m sin , 0 ≤ x ≤ n , (28) bα ,n,m = h∗α ,n n ∑k≥0 m+k odd (−1)(m+k−1)/2cα ,n,k jk(mπ/2) . Because of lemma A.2 the function f ∗αn has a square integrable derivative at least for min(αn,n)> 2 which entails the absolute uniform convergence of the orthogonal series in (25) and (28). Inserting (28) into (6) leads to Hα ,n(r) = r nr2 +m2π2 bα ,n,m 1− (−1)me−r . (30) Finally, with the functions Gβ (ζ ) := Γ(β + k/2) = (Γ(β ))−1M(1,β ;ζ 2)+ (Γ(β + 1/2))−1ζM(1,β + 1/2;ζ 2), β > 0, = ζ 2(1−β )eζ Gβ−1(ζ 2)+Gβ−1/2(ζ , if β > 1, the following integral representation can be derived from (11), (12), (13): Hα ,n(r) = βα(y−1) G1+αn/2(ry) y = ρeiϕ , ρ > 0. 3 The distribution of the angle Φ Two representations of the cdf Fα ,n of U = ( ncosΦ)−1 follow directly from (25) and (28). Fα ,n(u) = Γ(αn)uαn−1 cα ,n,kP = Γ(αn)uαn−1 bα ,n,m sin Besides, an orthogonal expansion with Legendre polynomials for the density of U2 is obtained by the moments E (U2k) = Γ(αn)k! Γ(αn+ 2k)∑ Γ(α + 2ki) Γ(α)ki! , (32) derived from (2). Direct use of these orthogonal series within (5) is possible to get the cdf of S, but the double series in theorem 4.1 is numerically more favourable. As mentioned in the introduction it would be theoretically more satisfying to have a single alternat- ing series or a probability mixture for this cdf. A single power series is not directly available by a power series expansion of Gαn in (5) since the moments of cotΦ do not all exist. However, at least for integer α , this obstacle is avoided by splitting the domain [0,arctan( n− 1)] of Φ by ϕn = arctan(1/ n− 1), which is the max- imal angle ϕ for which the whole cone {Φ ≤ ϕ} is contained completely within {x1, . . . ,xn|x1, . . . ,xn ≥ 0}. For integer α the theorem 3.1 below provides a polynomial representation of the cdf of tanΦ restricted to 0≤ tanϕ ≤ (n−1)−1/2. This simple rep- resentation is used within (5) to integrate over tanϕ ≤ (n−1)−1/2. The integration over tanϕ > (n− 1)−1/2 needs only truncated moments of cotΦ = (1− cos2 Φ)−1/2 cosΦ, obtained by the binomial expansion with the truncated moments of cosΦ, given below in (36). Theorem 3.1 Let α be a positive integer, then Wα ,n(t) := Pr{tanΦ ≤ t} Γ(αn) (Γ(α))nnαn/2 [(α−1)n/2] aα ,n,2 j tn−1+2 j n− 1+ 2 j , 0 ≤ t ≤ (n− 1)−1/2, where the coefficients aα ,n,2 j are given as the unique solutions of the linear equations [(α−1)n/2] +β n+m− j aα ,n,2 j = (2m)! Γ(1/2+β +m j)/(2m j)! , α = 2β + 1 (2m+ n)! nm+n/2 Γ(1/2+β +m j)/(2m j + 1)! , α = 2β m, j = 0, . . . , [(α − 1)n/2]. With any ρ > 0 the sums ∑(m) within the right hand sides of (34) are also given by e−imϕ dϕ , α = 2β + 1, α + 1 α + 1 e−imϕ dϕ , α = 2β , After Kummer’s transformation (A.S. 13.1.27) the above confluent hypergeometric functions M can also be expressed by Hermite polynomials due to (A.S. 13.6.17/18). The truncated moments γ̄α ,n,k = Ē ((cosΦ)αn+k), defined by integration over ϕ ≥ ϕn, follow from theorem 3.1 by straightforward calculation. γ̄α ,n,k = γα ,n,k − Γ(αn) (Γ(α))nnαn/2 [(α−1)n/2] aα ,n,2 jB 2 + j, (α−1)n+k+1 2 − j; with the incomplete beta function B(a,b;x) = a−1(1− t)b−1dt. In particular with α = 1 we have γ̄1,n,k = γ1,n,k − (n− 1)! π (n−1)/2 Γ( n−12 ) . (37) Two further remarks: The equations (34) can also be solved by explicit matrix in- version. The matrix 2 +β n+m− j , m, j = 0, . . . ,N = [(α − 1)n/2], has the structure Γ(x)Diag . . . ,(x)m, . . . (x+m)N− j with factorials (x)m = x(x+ 1) . . .(x+ m−1). The inverse (pm j(x)) of ((x+m)N− j) contains only polynomials pm j of degree m in its mth row. These polynomials can be obtained by interpolation from the m+ 1 values pm j(−(N − k)) = (k = 0, . . . ,m) (−1) j+k+1 ( j+ k−m)!(N− ( j+ k))! , m− k ≤ j ≤ N − k 0 , otherwise The cdf F1,n of U1,n can be applied to a goodness of fit test of the hypothesis H0 : F = F0 against H1 : F 6=F0, where F0 is any specified continuous cdf. Under H0 the test statistic ∑ni=1(F0(Xi:k)−F0(Xi−1:k))2, obtained from an ordered sample Xi:k, i= 1, . . . ,k = n−1, (X0:k =−∞, Xn:k = ∞), has the same distribution as U21,n. The asymptotic normality of U21,n can be derived from the asymptotic bivariate normal distribution of (Yn − n)/ (Zn − 2n)/ with correlation ρ = 2/ 5. The convergence to a normal distribution is slow and might be accelerated by a suitable transformation. 4 The distribution of the sample variance With the functions Hα ,n,m(r) := E (Gαn+m(r ncosΦ)) rαn+m Γ(αn+m) µα ,n,m+k αn+m+ k (−r)k m ∈ N0, µα ,n,k = E ( ncosφ)αn+k = n(αn+k)/2γα ,n,k from (14) we obtain Theorem 4.1 Let X1, . . . ,Xn be independent random variables with den- sity (Γ(α))−1e−xxα−1, α > 0, then the cdf of the sample variance S2 = (n− 1)−1 ∑ni=1(Xi − X̄)2 is given by (n− 1)S2 ≤ z = r2 Γ(αn+ 2 j) Γ(αn) j!n j Hα ,n,2 j( z) (39) Γ(αn) (αn+ k)/2+ j− 1 µα ,n,2 j+k αn+ k (−r)k δα ,n, j,k,λ αn+ k− 1 Gαn+k(λ r) with the alternating kth order differences δα ,n, j,k,λ = (−1)ℓ (αn+ ℓ)/2+ j− 1 µα ,n,2 j+ℓ λ αn+2 j+ℓ and any λ > 0. The series in (39) is absolutely convergent with terms of the order j−(n+1)/2 , where the O–constant depends only on α and n. As a simple numerical example we obtain with α = 1 and n = 10 the value Pr{S ≤ 2}= 0.98530379 . . . with at least eight correct digits. Finally, the method, explained before theorem 3.1, is applied to obtain the alterna- tive representations in theorem 4.2 for integer α , including in particular a probability mixture for the exponential case α = 1. The degree of the polynomials in theorem 3.1 and the rate of convergence of the binomial series in (43) limit the numerical use of theorem 4.2 to moderate values of αn. The truncated moments M̄α ,n,k = Ē (cotΦ)αn+k (αn+ k)/2+ j− 1 γ̄α ,n,k+2 j (43) are used with γ̄α ,n,k from (36), obtained by integration over ϕ ≥ ϕn = arccot( n− 1). Theorem 4.2 With the truncated moments from (43) and (36), the pdf wα ,n and the cdf Wα ,n of tanΦ on 0 ≤ ϕ ≤ ϕn from theorem 3.1, the distribution of S is given by each of the following three formulas (44), (45), (46): n− 1S ≤ r Γ(αn) (−1)k γ̄α ,n,k αn+ k Pαn,k(r Gαn(r n/t)wα ,n(t)dt with the polynomials Pαn,k(x) = [k/2] (αn+ k)/2 xk−2 j (k− 2 j)! Γ(αn) M̄α ,n,k αn+ k + Wα ,n(1/ n− 1)Gαn n(n− 1) n(n−1) gαn(x)Wα ,n The infinite series in (45) is also given by ∆̄α ,n,k,λ αn+ k− 1 Gαn+k(λ r) (46) with any λ > 0 and the alternating kth order differences ∆̄α ,n,k,λ = (−∆)kM̄α ,n,0,λ = (−1)ℓ M̄α ,n,ℓ,λ , M̄α ,n,ℓ,λ = Ē ncotΦ )αn+ℓ Remarks: In particular, with α = 1 the only coefficient in w1,n(t) is a1,n,0 = (n− 1)bn−1 with the volume bn−1 = π (n−1)/2/Γ of the (n− 1)–unit ball. Then the last two terms in (45) are reduced to (n− 1)! n(n(n− 1))(n−1)/2 bn−1Gn−1(r n(n− 1)). (47) The truncated moments γ̄1,n,k were given in (37). Only positive coefficients ∆̄α ,n,k,λ arise in (46) with λ = n(n− 1) since 1 > 1− cotΦ√ n−1 ≥ 0. The last term in (45) is bounded by Wα ,n(1/ n− 1) 1−Gαn(r n(n− 1)) , which is often neglectible. For actual computations the choice Ē ((cotΦ)αn) 1−Wα ,n(1/ n− 1) )1/(αn) might be more favourable. With this λ the leading term in (46) becomes 1−Wα ,n Gαn(λ r). It would be desirable to find a similar probability mixture representation for the cdf of S2 for all α > 0 but presumedly, also in case of its existence, the computation of the required coefficients would not be easy. Acknowledgement: The author would like to thank S. Kotz for his strong interest in this work and some hints at the literature within a personal communication. Appendix Proof of theorem 3.1. Let α be any positive integer and ω the Lebesgue measure on the (n− 1)–unit sphere Sn−1. Then (x1 + . . .+ xk) (A.1) = 2Γ((αn+ k)/2)−1k!∑ Γ(α + k j)/2)/k j! , α + k j odd. After an orthogonal transformation x = Ty with y1 = n −1/2 ∑ni=1 xi = cosϑ1, followed by transformation to polar coordinates, we obtain with c j = cosϑ j, s j = sinϑ j the same integral value as n+ s1τ j(ϑ2, . . . ,ϑn−1))α−1dω = nk/2 (α−1)n α − 1 n(ℓ−(α−1)n)/2 n−1− j j dϑ j (α−1)n+k−ℓ n−2+ℓ 1 dϑ1 = nk/2 [(α−1)n/2] aα ,n,2 jB (α − 1)n+ k+ 1 . (A.2) By comparison with (A.1) it follows [(α−1)n/2] (α − 1)n+ k+ 1 aα ,n,2 j (A.3) = 2n−k/2k!∑ Γ((α + k j)/2)/k j! . With α = 2β + 1 only even k j = 2m j occur with sum k = 2m, and with α = 2β only k j = 2m j + 1 with sum k = 2m+ n, which leads to the linear equations in (34). If ℓ is odd then the coefficients aα ,n,ℓ vanish since sn−1− jj dϑ j is a linear combination of the integrals smnn−1 M j+n−1− j j dϑ j , (Mn−1 = 0) , which are different from zero only for even m2, . . . ,mn, ∑nj=2 m j = ∑ j=1 ℓ j = ℓ. If t = tanϕ ≤ (n− 1)−1 we obtain with Sn−1,ϕ = Sn−1 ∩{ϑ1 ≤ ϕ} that Pr{tanΦ ≤ t} = (Γ(α))−n {ϑ1≤ϕ} e−x j xα−1j dx j = (Γ(α))−n Sn−1,ϕ ραn−1 exp(−ρ n+ s1τ j)α−1dω Γ(αn) (Γ(α))nnαn/2 Sn−1,ϕ c−αn1 n+ s1τ j)α−1dω Γ(αn) (Γ(α))nnαn/2 [(α−1)n/2] aα ,n,2 j sn−2+2 j1 c −n−2 j 1 dϑ1 , where the integrals are given by (tanϕ)n−1+2 j/(n−1+2 j, which provides the asserted result in (33). ✷ Comparing the series (−1)k (z/y)(αn+k)/2 αn+ k = Γ(αn)Gαn( with the corresponding one for Γ(αn+ 2 j)Gαn+2 j( z/y) we obtain the identity Γ(αn+ 2 j) Γ(αn) Gαn+2 j (A.4) by differentiation. Lemma A.1 2αn/2 , y,z > 0. (A.5) Proof. Using (A.4) and Cauchy’s integral formula for the derivatives of Γ(αn+ 2 j)Gαn+2 j( zαn/2+ j−1 exp(− the bound in (A.5) follows from ( j− 1)! Γ(αn+ 2 j)Gαn+2 j( = lim ζ αn/2+ j−1 exp(− ζ )(ζ − z)− jdζ yαn/2−1 exp(− y/2)dy = 2αn/2Γ(αn)/π , where the way of integration is ≤ ϕ ≤ ∣R ≥ y ≥−R Lemma A.2 Let Φ denote the angle between (X1, . . . ,Xn) and (1, . . . ,1) from (4), Wα ,n(t) the cdf of tanΦ and bn−1 = π (n−1)/2/Γ( n+12 ) the volume of the (n− 1)–unit ball, then Wα ,n(t)∼ Γ(αn)bn−1 (Γ(α))nn(α−1/2)n tn−1, t ↓ 0, (A.6) and in particular W1,n(t) = (n− 1)! bn−1t n−1, 0 ≤ t ≤ (n− 1)−1/2. (A.7) Furthermore E ((cosΦ)s)∼ Γ(αn)(2π)(n−1)/2 (Γ(α))nn(α−1/2)n s−(n−1)/2, s ↑ ∞. (A.8) Proof. Let x,xg denote the arithmetic and the geometric mean of (x1, . . . ,xn ≥ 0 and ϕ the angle between (x1, . . . ,xn) and (1, . . . ,1). Setting xi = (1+ εi)x with any fixed x we have the relation (xi − x)2 = ε2i = n tan and for ϕ ↓ 0 xng = x (1+ εi)∼ xn exp ∼ xn. It follows Wα ,n(t) = Pr{Φ ≤ ϕ}= gα(xi)dxi = (Γ(α))−n n(α−1) g exp(−nx)dx1 . . .dxn ∼ (Γ(α))−nbn−1 x(α−1)n exp(−nx)(x n tanϕ)n−1dx which provides (A.6). For α = 1 this asymptotic relation can be replaced by an equation if ϕ ≤ ϕn = arctan(1/ n− 1), which is the largest angle ϕ for which the whole cone {Φ ≤ ϕ} is contained within {x1, . . . ,xn|x1, . . . ,xn ≥ 0}. To prove (A.8) we obtain from (A.6) Pr{| ln(cosΦ)| ≤ ε} = Pr{ln(1+ tan2 Φ)≤ 2ε} ∼ Wαn ∼ cα ,nε(n−1)/2, ε ↓ 0, with a factor cα ,n determined by (A.6). For s → ∞ the relation (A.8) follows from the Hardy–Littlewood–Karamata Tauber theorem (cf. e.g. chapter 13 in Feller 1971), applied to E ((cosΦ)s) = E (exp(−s| ln(cosΦ)|)) . ✷ Proof of theorem 4.1. With any ϑ ∈ (0,1) EU (z/(U2 −ϑ/n))1/2 can be represented by the Taylor series With U−1 = ncosΦ and the bounds from Lemma A.1 the terms of this series are absolutely bounded by 2αn/2 (cosΦ)2 j ϑ j/ j = O(ϑ j j−(n+1)/2) due to (A.8) with s = 2 j and on O–constant depending only on α and n. For ϑ ↑ 1 it follows from (6) and (A.4) that Pr{(n− 1)S2 ≤ z = r2} = EU U2 − 1/n Γ(αn+ 2 j) Γ(αn) j!n j Gαn+2 j Γ(αn+ 2 j) Γ(αn) j!n j Hα ,n,2 j( with the functions Hα ,n,2 j from (38). With any λ > 0 we obtain Γ(αn+ 2 j) Γ(αn) j!n j Hα ,n,2 j( Γ(αn) (αn+ k)/2+ j− 1 µα ,n,2 j+k λ αn+2 j+k (−1)k (λ r)αn−1+k λ 2/n After multiplication by exp(−λ r)∑∞ℓ=0(λ r)ℓ/ℓ! the series (41) is obtained by integra- tion over r. ✷ Proof of theorem 4.2. With the pdf wα ,n and the cdf Wα ,n of tanΦ from theorem 3.1 and the truncated moments M̄α ,n,k = Ē (cotΦ)αn+k from (43) it follows with (5) that n− 1S ≤ r} = E Gαn(r ncotΦ) Gαn(r n/t)wα ,n(t)dt + Γ(αn) M̄α ,n,k αn+ k After integration by parts and the substitution t → r n/x the above integral becomes n(n− 1) n(n−1) gαn(x)Wαn Inserting the series for M̄α ,n,k from (43) formula (44) follows by rearranging the resulting absolutely convergent double series according to the truncated moments γ̄α ,n,k = Ē (cosΦ)αn+k from (36). The series in (46) is obtained by the same way as (21). ✷ References [1] Abramowitz, M. and Stegun, I. (1968) Handbook of Mathematical Functions. Dover Publications, Inc., New York [2] Bowman, K.O. and Shenton, L.R. (1983) The Distribution of the Standard Devi- ation and Skewness in Random Samples from a Gamma Density — A New Look at a Craig–Pearson Study. Oak Ridge National Laboratory/ CSD–109 [3] Box, G.E.P. (1953) Nonnormality and Tests on Variances, Biometrika 40, 318–335 [4] Craig, C.C. (1929) Sampling when the Parent Population is of Pearson’s Type III, Biometrika 21, 287–293 [5] Feller, W. (1971) An Introduction of Probability Theory and Its Applications, Vol. II, 2nd ed. John Wiley & Sons, New York [6] Hyrenius, J. (1950) Distribution of Student–Fisher’s t in Samples from Com- pound Normal Functions, Biometrika 37, 429–442 [7] Laha, R.G. (1954) On a Characterization of the Gamma Distribution, Annals of Mathematical Statistics 25, 784–787 [8] Lukacs, E. (1955) A Characterization of the Gamma Distribution, Annals of Mathematical Statistics 26, 319–324 [9] Mathai, A.M., and Provost, S.B. (1992) Quadratic Forms in Random Variables. Marcel Dekker, Inc., New York [10] Mudholkar, G.S. and Trivedi, M.C. (1981) A Gaussian Approxiamtion to the Distribution of the Sample Variance for Nonnormal Populations, Journal of the Americal Statistical Association 76, 479–485 [11] Pearn, W.L. and Kotz, S. (2006) Encyclopedia and Handbook of Process Capa- bility Indices: A Comprehensive Exposition of Quality Control Measures. (Ser. on Quality, Reliability and Engineering Statistics, Vol. 12). World Scientific Pub- lishing Company, Singapore [12] Pearn, W.L., Kotz, S. and Johnson, N.L. (1992) Distributional and Inferential Properties of Process Capability Indices, Journal of Quality Technology, 24, 216–230 [13] Pearson, E.S. (1929) Note on Dr. Craig’s paper, Biometrika 21, 294–302 [14] Roy, J. and Tiku, M.L. (1962) A Laguerre Series Approximation to the Sampling Distribution of the Variance, Sanky ā, Series A, 24, 181–184 [15] Tan, W.Y. and Wong, S.P. (1977) On the Roy–Tiku Approximation of Sample Variances from Nonnormal Universes, Journal of the American Statistical Asso- ciation 72, 875–880 Introduction The distribution of the sum of squares The distribution of the angle bold0mu mumu The distribution of the sample variance

0704.1416

Framework for non-perturbative analysis of a Z(3)-symmetric effective theory of finite temperature QCD

HIP-2007-18/TH Framework for non-perturbative analysis of a Z(3)-symmetric effective theory of finite temperature QCD A. Kurkela1 Theoretical Physics Division, Department of Physical Sciences, P.O.Box 64, FI-00014 University of Helsinki, Finland Abstract We study a three dimensional Z(3)-symmetric effective theory of high temperature QCD. The exact lattice-continuum relations, needed in order to perform lattice simulations with physical parameters, are computed to order O(a0) in lattice perturbation theory. Lattice simulations are performed to determine the phase structure of a subset of the parameter space. [email protected] http://arxiv.org/abs/0704.1416v3 1 Introduction At high temperature, QCD matter undergoes a deconfinement transition, where ordinary had- ronic matter transforms into strongly interacting quark-gluon plasma [1, 2, 3]. In the absence of quarks, Nf = 0, the transition is a symmetry-breaking first order transition, where the order parameter is the thermal Wilson line [4, 5]. The non-zero expectation value of the Wilson line signals the breaking of the Z(3) center symmetry of quarkless QCD at high temperatures. The transition has been studied extensively using lattice simulations [6, 7, 8]. Thermody- namical quantities, condensates and various correlators can be measured on the lattice and the equation of state can be estimated. This approach, however, becomes computationally exceed- ingly expensive at high temperatures, and thus cannot be applied to temperatures T above ∼ 5Tc. The complementary approach has been to construct perturbatively effective theories, such as electrostatic QCD or EQCD, using the method of dimensional reduction [9, 10, 11] to quantitatively describe high temperature regime of QCD [12, 13, 14, 15]. In the dimensional reduction procedure, however, one expands the temporal gauge fields around one of the Z(3) vacua and thus explicitly violates the center symmetry. The range of validity of these theories therefore ends for T below ∼ 5Tc, where the fluctuations between different vacua become im- portant. There have also been several attempts to the build models for Wilson line, respecting the center symmetry [16, 17, 18, 19, 20, 21]. These models give a qualitative handle on the transition but cannot be perturbatively connected to QCD. As a unification of these strategies, an effective field theory of high temperature QCD respect- ing the Z(3) center symmetry has been constructed in [22]. At high temperatures, the effective theory reduces to EQCD guaranteeing the correct behavior there, but the model still preserves the center symmetry. The effective theory is further connected to full QCD by matching the domain wall profile separating two different Z(3) minima. Being a three dimensional model, the new theory relies on the scale separation between the inverse correlation length and the lowest non-zero Matsubara mode, which is still modest at Tc [23, 24]. Thus, one hopes that the range of validity of this theory would extend down to Tc. The effective theory is a confining one, so perturbative analysis breaks down. Non-perturbative methods, i.e. lattice simulations, are thus needed to find out the physical properties, such as correlation lengths, condensates, and most importantly the phase structure of the theory, to test its regime of validity. The effective theory is super-renormalizable, and thus the connection between the continuum MS and lattice regulated theories can be obtained exactly to the desired order in the lattice spacing a. The matching of the parameters of the Lagrangian to order O(a0), which is needed in order to perform simulations with MS scheme parameters and to obtain physical results, requires a two-loop lattice perturbation theory calculation. The one-loop terms remove any linear 1/a divergences, while two-loop terms remove the logarithmic log(1/a) divergences and the constant differences in the mass terms of the theory. In addition, the condensates have lattice spacing dependence and constant differences between the two schemes and can be calculated to order O(a0) by performing a two-loop calculation for operators up to cubic order and a four- loop calculation for the quartic condensates. In this paper we perform the needed two-loop calculations. This paper is organized as follows. In Sections 2 and 3 we define the theory in continuum MS regularization and on the lattice, respectively. In Section 4 we study the phase diagram of a subset of the parameter space of the theory. Details of the matching between the continuum and lattice theories are given in the appendices. 2 Theory The theory we are studying is defined by a three dimensional continuum action, which we renormalize in the MS scheme d3−2ǫx TrF 2ij +Tr + V0(Z) + V1(Z) , (1) where Fij = ∂iAj − ∂jAi + ig3[Ai, Aj ] (2) Di = ∂i − ig3[Ai, ] (3) and Z is a 3 × 3 complex matrix, which in the limit ǫ → 0 has dimension dimZ = The gauge fields Ai are Hermitean traceless 3 × 3 matrices and can be expressed using gener- ators of SU(3), Ai = A a, with TrT aT b = 1 δab . The covariant derivative is in the adjoint representation. The potentials V0, the “hard” potential, and V1, the “soft” potential, are V0(Z) = c1Tr[Z †Z] + 2c2Re(Det[Z]) + c3Tr[(Z †Z)2], (4) V1(Z) = d1Tr[M †M ] + 2d2Re(Tr[M 3]) + d3Tr[(M †M)2], (5) where M = Z − 1 Tr[Z]1 is the traceless part of Z. Here, the gauge coupling g3 has a positive mass dimension dim[g23 ] =GeV, making the theory super-renormalizible. Because of the super- renormalizibility, the coefficients c2, c3, d2, and d3 are renormalization scale independent and only the mass terms c1 and d1 acquire a scale dependence in the MS renormalization scheme. The scale dependence in the mass terms arises from a two-loop calculation and has the form: c1(µ̄) = 64c3g d1(µ̄) = c23 − 64d3g23 + 2d3c3 + d , (7) where Λ is a constant specifying the theory and µ̄ is the MS scale parameter. The coefficients ci, di, and g3 are matched to the parameters of full thermal QCD by imposing the condition that the theory reduces to EQCD at the high temperature limit, and that the theory reproduces the domain wall profile of full QCD [22]. This defines a subset of parameter values (with a limited accuracy due to perturbative matching), for which the theory describes thermal QCD3. However, in this paper we consider the model in general, and do not restrict ourselves only to the physical region. The action is defined only for the number of colors Nc = 3, but for generality, we give some of the perturbative results for any Nc. For analytic calculations the scalar field Z can be expanded around the vacuum: (φ+ iχ)1+ (H + iA), (8) 2Our notation is obtained from that in [22] by scaling with g3: Ai → g3Ai, Z → g3Z, c1 → c1, c2 → g c3 → g c3, c̃1 → g d1, c̃2 → g d2, and c̃3 → g 3In the matching, the hard potential is parametrically larger than the soft potential, explaining the terminology. where φ and χ are real scalars and H and A are Hermitean traceless matrices. Fields H and A can be written with the generators of the SU(3) group, H = HaT a and A = AaT a, where Ha and Aa are real scalars. The action is invariant under local gauge transformations, with Z transforming in the adjoint representation: Ai(x) −→ G(x) Ai(x)− G−1(x), (9) Z(x) −→ G(x)Z(x)G−1(x), (10) where G(x) ∈SU(3). In addition to this, there are further global symmetries in the potentials. The potential V0 is invariant in global SU(3)×SU(3) transformations Z(x) −→ LZ(x)R, (11) where L and R are SU(3) matrices. The potential V1 is invariant under Z(3) transformations M → zM, (12) where z = ei2πn/3, which generalizes into a U(1) symmetry if d2 = 0. This implies that in the presence of the both potentials (with non-zero coefficients), the overall global symmetry of the Lagrangian is Z → zZ. 3 Lattice action In order to perform non-perturbative simulations, the theory has to be formulated on the lattice. On the lattice, the scalar field Z lives on the sites of the lattice, and the gauge fields Ai are traded for link variables Ui, which are elements of SU(Nc) and live on the links connecting adjacent sites. The lattice action corresponding to the continuum theory can be written as S = SW +SZ , where SW = β x,i<j ReTr[Uµν ] is the standard the Wilson action with the lattice coupling constant corresponding to a lattice spacing a. The continuum limit is taken by β → ∞, and there the Wilson action reduces to the ordinary pure gauge action. The kinetic term, Tr , is discretized by replacing the covariant derivatives by covariant lattice differences. Then the scalar sector of the action reads: SZ = 2 Ẑ†Ẑ − Ẑ†(x)Ui(x)Ẑ(x+ î)U †i (x) ĉ1Tr[Ẑ †Ẑ] + 2ĉ2ReDetẐ + ĉ3Tr[(Ẑ †Ẑ)2] d̂1Tr[M̂ †M̂ ] + 2d̂2ReTrM̂ 3 + d̂3Tr[(M̂ †M̂)2] . (14) where ĉi, d̂i are dimensionless numbers, and M̂ and Ẑ are dimensionless Nc×Nc complex matri- ces. Only the mass terms ĉ1 and d̂1 require non-trivial renormalization and all the other terms can be matched to order O(a0) on tree-level by simply scaling with g3: Z = g3Ẑ, M = g3M̂ (15) c2 = g 3 ĉ2, d2 = g 3 d̂2 (16) c3 = g 3 ĉ3, d3 = g 3 d̂3. (17) For the mass terms, renormalization has to be carried out, so that the physical masses of the fields are the same in both regularization schemes. A two-loop calculation gives (the details of the calculation and the definitions of the numerical constants are given in the appendix): ĉ1 = 2ĉ3β − 64ĉ3 + (log β + ζ) + 16Σ2 − 64δ +O(β−1) 6.3518228ĉ3β 64ĉ3 + (log β + 0.08849) + 37.0863ĉ3 +O(β−1) (18) d̂1 = β(1 + 16Σ2 − 64δ ĉ23 − 64d̂3 + 2d̂3ĉ3 + d̂ [log β + ζ] πΣ− 4δ − 6ρ+ 2κ1 − κ4 +O(β−1) 3.17591 + 5.64606d̂3 41.780852 + 37.0863d̂3 ĉ23 − 64d̂3 + d̂3ĉ3 + d̂23 + [log β + 0.08849] +O(β−1). (19) Here, we have set the renormalization scale to be µ̄ = g23 in Eqs. (6) and (7), and denote c1 = c1(g 3) and d1 = d1(g 3). By making this choice, we get the logarithmic term to be a function of the lattice coupling constant β. There are also higher order corrections (corrections of order O(β−1) corresponding to order O(a) in lattice spacing), but their effect vanishes in the continuum limit. Various operators also need to be renormalized on the lattice in order to convert their expec- tation values to continuum regularization. The Z3-symmetry protects the lowest dimensional condensate 〈g−13 TrZ〉 from acquiring any additive renormalization, while a two-loop calculation gives for the quadratic condensates: 〈g−23 TrZ †Z〉MS = 〈TrẐ †Ẑ〉a − 2Nc(N c − 1) log β + ζ + +O(β−1) = 〈TrẐ†Ẑ〉a − [0.3791β + 0.3040(log β + 0.66796)] +O(β−1), (20) 〈g−23 TrM †M〉MS = 〈TrM̂ †M̂〉a − N2c − 1 2Nc(N c − 1) log β + ζ + +O(β−1), (21) 〈g−23 TrZ †TrZ〉MS = 〈TrẐ †TrẐ〉a − β +O(β−1), (22) where the subscript a denotes the lattice regularization. For the cubic condensates we get: 〈g−33 2ReDetZ〉MS = 〈2ReDetẐ〉a N2c − 1 ĉ2 + 8/Nc − 10Nc + 2N3c ĉ2 + d̂2) (log β + ζ) +O(β−1) 〈g−33 2ReTrM 3〉MS = 〈2ReTrM̂ 3〉a − − 30Nc + 6N3c ĉ2 + d̂2) (log β + ζ) +O(β−1) The effect of subtraction of the divergences can be seen in Fig.1. The renormalization of the quartic operators to order O(β0) would require a four-loop calculation, which we do not perform here since they are not measured at this stage. 0 0.01 0.02 0.03 0.04 0.05 25 <Tr(Z✝Z)> <Tr(Z - κ1/β <Tr(Z -κ1/β-κ2(logβ+κ2’) 0 0.01 0.02 0.03 0.04 <Tr(M M)> -div <Tr(Z Z)>-div Figure 1: The effect of subtraction of the divergences in 〈TrZ†Z〉 and 〈TrM †M〉 in a fixed physical volume with d1 = 6.6 and d3 = 0.01. On the left panel, the effect of subtracting the divergent parts of 〈TrZ†Z〉 is plotted. The constants κ1, κ2 and κ′2 are the coefficients of the linear, logarithmic, and constant differences between lattice and MS regularizations form equation (20). On the right panel: the continuum limit of the condensates. Notice the negative values of the quadratic condensates in the symmetric phase. 4 Phase diagram of the soft potential A simpler model is obtained from the original theory by setting ci = 0. In this model, the trace of Z decouples and can be integrated over as a free scalar field. The relevant degree of freedom is thus a traceless complex matrix M , or two traceless hermitian matrices H and A. This can be viewed as a natural generalization4 of EQCD to complex values of the adjoint higgs field Aa0. The simpler model is defined by the action: TrF 2ij +TrDiM †DiM + d1TrM †M + 2d2Re(Tr[M 3]) + d3Tr(M TrF 2ij +TrDiADiA+TrDiHDiH + d1TrA 2 + d1TrH + 2d2Tr[H 3 − 3HA2] + d3Tr[H4 +A4 + 4H2A2 − 2HAHA] . (26) If the cubic term d2 is zero, the Lagrangian is invariant under a U(1) global symmetry M → gM , g ∈U(1). The breaking of the symmetry is signalled by a local order parameter: 〈TrA3〉2 + 〈TrH3〉2. (27) This operator remains a valid order parameter after the renormalization since it has no additive renormalization, if d2 = 0. In the symmetric phase A is strictly zero and in the broken phase the order parameter obtains a non-zero vacuum expectation value, while the two phases are separated by a first order transition. In the broken phase 〈TrM †M〉 is larger than in the symmetric phase. After the inclusion of the cubic term, A is no longer strictly an order parameter, since the U(1) symmetry is explicitly broken. However, the first order transition remains and is accompanied with a significant discontinuity in A and 〈TrM †M〉. 4.1 Perturbation theory In the limit of small d3/g 3 , the transition becomes very strong, and we expect a semiclassical approximation to produce the correct behavior of the critical line [25, 26]. We parametrize a constant diagonal hermitian background field in a fixed Landau gauge as follows: 〈M〉 = 2pT3 + 2 3qT8 = q + p 0 0 0 q − p 0 0 0 −2q  , (28) where p and q are real scalars with dimensions of g3. Lattice simulations suggest that the A → −A symmetry is not broken spontaneously at any non-zero value of d2, so that it is sufficient to consider only hermitian background fields. Using this parametrization, the 1-loop effective potential V1(d1, d2, d3; p, q) can be calculated: 4In EQCD with gauge group SU(3), there is only one linearly independent quartic gauge invariant opera- tor namely TrA40. In the complex case, however, there are four different Z3-symmetric operators: Tr(M †M)2, (TrM†M)2, Tr[M†M†MM ] and Tr[M† ]Tr[M2]. In the case of unitary M , i.e. in the minimum of the hard potential, these operators collapse into a single one. However, since there is no such restriction in our model, the operators are linearly independent. From these operators, we choose to include only the one appearing in the original theory, Tr(M†M)2. V1(d1, d2, d3; q, p) = (2p 2 + 6q2)d1 + 36q(p 2 − q2)d2 + (2p2 + 6q2)2d3 (8|p|3 + |p− 3q|3 + |p+ 3q|3)g33 d1 + 3(p − q)d2 + 2(p2 + 3q2)d3 d1 + 6qd2 + 2(p 2 + 3q2)d3 d1 − 3(p + q)d2 + 2(p2 + 3q2)d3 d1 + 4(p 2 + 3q2)d3 − 2 3(p2 + 3q2)d22 + 18q(p 2 − q2)d2d3 + (p2 + 3q2)2d23 d1 + 4(p 2 + 3q2)d3 + 2 3(p2 + 3q2)d22 + 18q(p 2 − q2)d2d3 + (p2 + 3q2)2d23 ]3/2 } d1 − 6qd2 + 2(3p2 + q2)d3 d1 + 3(p + q)d2 + 2(p 2 − 4pq + 7q2)d3 d1 − 3(p − q)d2 + 2(p2 + 4pq + 7q2)d3 (p2 + 3q2)d3 − 27(p2 + 3q2)d22 + 54q(q 2 − p2)d2d3 + (p2 + 3q2)2d23 (p2 + 3q2)d3 + 27(p2 + 3q2)d22 + 54q(q 2 − p2)d2d3 + (p2 + 3q2)2d23 ]3/2 } where the first term is the classical potential, the second one comes from one-loop vector dia- grams and the fourth and the fifth from one-loop scalar diagrams of H and A, respectively5. The effective potential has a symmetry arising from the permutations of the diagonal elements of the background 〈M〉 and has the following invariance: V1(d1, d2, d3; q, p = ±3q) = V1(d1, d2, d3;−2q, 0). (30) More generally, the potential is invariant under rotations of 2π/3 in the (p, 3 q)-plane and in the reflections of p: q → p−q p → p+3q q → −p−q p → p−3q q → q p → −p Thus there is a fundamental region, which determines the potential over the whole plane. We choose the fundamental region to be bounded by the two lines p = 0 and p = −3q together with the condition p ≥ 0. In the fundamental region, there can be four different minima at the critical parameter values d1, d2 and d3. The one at the origin (denoted by 1 in fig.2) is the symmetric minimum, 5By dropping the last term, i.e., the five last lines and scaling d1 → y, d2 → iγ3, and d3 → 2x, one obtains the effective potential for EQCD in the presence of a finite (imaginary) chemical potential using the notation of [27] −20 −15 −10 −5 0 5 10 15 20 −25 −20 −15 −10 −5 0 5 10 15 20 25 q/g 1 Figure 2: 1-loop effective potential in the (q,p)-plane at the critical point for d3 = 0.01 and d2 = 0 (left panel) and d2 = 0.05 (right panel). Light areas represent the minima of the potential. Solid lines separate the three identical sectors which are related by the permutation symmetry of the diagonal elements of the background field 〈M〉. In the absence of d2 there is an additional U(1) symmetry making the directions marked with dashed lines identical to the p = 0 direction. This symmetry explicitly broken by finite d2 as seen on the right panel. -20 -10 0 10 20 Figure 3: 1-loop effective potential with p = 0 as a function of q at the critical point for d3 = 0.01 and d2 = 0(line), 0.005(dotted),and 0.05(dashed). the minima 2 and 3 are connected by the permutation symmetry and correspond to the same physical broken minimum, with TrH3 < 0 and TrA3 = 0. The minimum 4 corresponds to a phase with TrH3 > 0 and TrA3 = 0 and is connected continuously to the minimum 2 by a global U(1) symmetry if d2 = 0. If d2 6= 0, the U(1) symmetry is lost and the minima 2 and 4 are no longer equivalent. If d2 > 0 the minimum at 2 is favored over 4 and vice versa. Setting d2 to zero and expanding in d3 up to order O(d23) the potential reads (for p = 0): V1(d1, 0, d3; q, 0) = 18q |q| − 4π2d3 +O(d23), so that in the limit d3 → 0 the potential has two coexisting minima and a first order transition d1 = d 4π2d3 ≈ 0.0759909 , (33) This sets the scaling of the critical line as a function of d3 at small d3. Corrections to this, and d2 dependence, are obtained by minimizing the real part of Eq.(29) numerically. The results are shown in Figs.4 and 5. The phase transition is accompanied with a discontinuity in q: ∆q = |qbroken − qsymmetric| = , (34) We see that the transition gets stronger as the coupling d3 gets smaller justifying a posteriori the semiclassical approximation. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 =0.00 =0.05 =0.10 =0.15 Symmetric Broken Figure 4: 1-loop perturbative phase diagram of the soft potential, V1, as function of d1, d2 and d3. A first order critical line separates the two phases. The symmetric phase refers to the phase where with d2 = 0 the order parameter vanishes and with d2 6= 0 is smaller than in the broken phase. The data points represent non-perturbative lattice measurements with d2 = 0 on a N 3 = 123 lattice. The perturbative result approaches the lattice data points for small values of d3 where the transition is very strong. Small discrepancy between the perturbative result and lattice data points at small d3/g is mostly due to finite volume effects. 4.2 Lattice analysis The perturbative calculation is valid only for small d3 and a non-perturbative lattice analysis has to be performed to obtain the full phase structure of the model. For the simulations we used a hybrid Monte-Carlo algorithm for the scalar fields and Kennedy-Pendleton quasi heat bath and full group overrelaxation for the link variables [28, 29, 30]. -0.4 -0.2 0 0.2 0.4 =0.01 Broken Symmetric Figure 5: 1-loop perturbative phase diagram of the soft potential, V1, as function of d1 and d2 with d3 = 2. A first order critical line separates the two phases. The non-analyticity at d2 = 0 is due to the change of global minimum between minima 3 and 4. The transition was found to be of the first order for all parameter values used in the simula- tions (d3 ≤ 4 and d2 ≤ 0.15) accompanied with a large latent heat and surface tension; hysteresis curves showing discontinuity around critical point in 〈TrM †M〉MS can be seen in Fig.8. The probability distributions of TrM †M along the critical curve are very strongly separated (see Fig.7). This makes the system change its phase very infrequently during a simulation, and mul- ticanonical algorithm is needed to accommodate a phase flip in reasonable times for any system of a modest size [31]. Even with the multicanonical algorithm, the critical slowing restricts us to physical volumes up to V . 50/g63 . The pseudo-critical point was determined requiring equal probability weight for TrM †M in both phases. The simulations were performed with β = 12 and a lattice size N3 = 123, which precludes the continuum extrapolation as well as the thermodynamical limit. However, these limits were studied for one set of parameter values and the dependence of the critical point on both lattice spacing and volume were found to be of order of five per cent for the lattice spacings and volumes used (see Fig.6 and Table 1). The phase diagram can be seen in Fig.9 and Fig.10. The non-perturbative critical line follows the perturbative one for small values of d3, but for larger d3 fluctuations make the system prefer the symmetric phase. The discontinuity in 〈TrM †M〉 along the critical line diminishes, as d3 gets larger (see Fig.11), but it seems that the discontinuity persists, even if its magnitude diminishes in the limit d3 → ∞ suggesting that there is a first order phase transition for any (positive) value of d3. 5 Conclusions In this paper, exact relations between the lattice and continuum MS regulated formulations of the Z(3)-symmetric super-renormalizable effective theory of hot QCD, defined by Eqs.(1),(4), 0 0.005 0.01 0.015 0.02 1/(Vg Figure 6: Volume dependence of the pseudo-critical point with d3 = 2 and d2 = 0.1. The pseudo- critical point was determined by requiring equal probability weight for TrM †M in both phases. The line represents a linear fit. The dependence on lattice spacing and volume seem to be within 5% for the lattice spacings and volumes used. β Lattice volumes 12 83, 103, 123, 163 16 123, 163, 203 20 163, 203, 243 Table 1: Lattices used in the continuum and thermodynamical extrapolation of the critical point, seen in Fig.6. and (5), have been calculated. The Lagrangians and the operators up to cubic ones have been matched to O(a0). These results make the non-perturbative lattice study of the theory possible. An interesting model with non-trivial dynamics is obtained by setting ci = 0 in Eq.(4). The model amounts to a natural generalization of EQCD to complex variables. The phase diagram of the model has been determined using lattice simulations. Two distinct phases were found, a symmetric phase with small 〈TrM †M〉MS, and a broken phase with large 〈TrM †M〉MS. The two phases were found to be separated by a strong first order transition with a large surface tension and discontinuities in the operators. In contrary to EQCD, where the first order line terminates at a tricritical point, the model seems to have a first order transition with all values of d2 and In the future, it is our goal to map out the phase diagram in the full parameter space of the model, rather than in a restricted region as in the present exploratory study, in order to search for regions in which the phase diagram would resemble that expected for the finite-temperature SU(3) pure Yang-Mills theory. 0 1 2 3 4 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 Figure 7: Histograms of TrM †M in logarithmic scale with d2 = 0 along the critical curve. Transition channel between the peaks weakens and the transition gets stronger for decreasing d3. For d3/g = 0.5, the relative probability density in the tunneling channel is suppressed by a factor ∼ 10−10. Acknowledgments The author thanks K. Kajantie for suggesting this topic and for numerous comments concerning the text. The author also thanks M. Laine, K. Rummukainen, Y. Schröder and A. Vuorinen for invaluable advice. This research has been supported by Academy of Finland, contract number 109720 and the EU I3 Activity RII3-CT-2004-506078 HadronPhysics. Simulations were carried out at CSC - Scientific Computing Ltd., Finland; the total amount of computing power used was ∼ 1× 1016 flops. A Details of renormalization In this appendix, we give details of the calculation of the renormalization of the mass parameters ĉ1 and d̂1 and the condensates 〈g−23 TrZ†Z〉, 〈g 3 TrM †M〉, 〈g−23 TrZTrZ†〉,〈g 3 2DetZ〉, and 〈g−33 2ReTrM3〉. The renormalization calculation compares ultraviolet properties of the two regularizations and thus it is irrelevant, in which phase we carry out the computation. We chose to work around the symmetric vacuum, since the Feynman rules are the simplest this way. However, in this vacuum, all components of the gluon are massless and one therefore has to deal with infrared divergences. The infrared divergences in the two regularizations are the same, and 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 8: Discontinuity in the quadratic condensate in continuum regularization 〈TrM †M〉 for d3 = 0.1, 1, 3. The phase transition gets weaker as the coupling d3 grows. The metastable regions shrink and the discontinuity diminishes. cancel exactly in the final results. Using the expansion (8), the potential V2 is a function of H and A only. The unit matrix commutes with any SU(3) matrix, and the interaction with the gluon field arises from commu- tator in the covariant derivative in adjoint representation. Thus the gluons couple, on tree-level, only to H and A Using this expansion, there are four tree-level mass terms in the Lagrangian that require renormalization: (ĉ1 + d̂1)H (ĉ1 + d̂1)A The coefficients ĉ1 and d̂1 have to be adjusted such that both regularization schemes give the same physical masses for all fields φ, χ, H and A. The theory is super-renormalizible, and there are divergent contributions up to two-loop level only. The masses are obtained from the low momentum properties of the two-point correlator: k,p→0 〈〈φ(k)φ(p)〉〉 = δ(3)(k + p) 1 k2 +m2 The difference between the correlators in the two schemes is in the mass and the wave function renormalization. However, the effect of the wave function renormalization is of order O(a) and can be neglected. To get the same masses in the different schemes, we enforce the condition that the two-point correlators give identical values in the low momentum limit. The zero-momentum lattice correlator can be written in a weak-coupling expansion 〈〈φ(0)φ(0)〉〉a = 〈φ(0)φ(0)〉a − 〈φ(0)φ(0)〉MS 〈φ(0)φ(0)(−SI )〉a − 〈φ(0)φ(0)(−SI )〉MS 〈φ(0)φ(0)S2I 〉a − 〈φ(0)φ(0)S2I 〉MS + 〈〈φ(0)φ(0)〉〉MS +O(a), (37) where double and single brackets represent exact and Gaussian expectation values, respectively, and subscripts give the regularization scheme. From here, we can read the condition for the lattice mass term ĉ1 by requiring that the exact correlators give the same value up to order 0 1 2 3 4 =0.15 =0.10 =0.05 =0.00 Symmetric Broken Figure 9: The phase diagram of the soft potential as a function of d1, d2 and d3. First order critical line separates two phases. Solid lines represent polynomial fits to the lattice data points and dashed lines are the perturbative predictions. The symmetric phase refers to the phase where with d2 = 0 the order parameter A vanishes and with d2 6= 0 is smaller than in the broken phase. Also 〈TrM †M〉 is significantly smaller in the symmetric phase and the critical line was determined requiring equal propability weight for 〈TrM †M〉 in both phases. O(β−1): ĉ1 = 〈φφ(−SI)〉a,1PI − 〈φφ(−SI)〉MS,1PI 〈φφS2I 〉a,1PI − 〈φφS2I 〉MS,1PI +O(a) (38) Similarly, we get for the mass term of adjoint fields: d̂1 = − ĉ1 〈HH(−SI)〉a,1PI − 〈HH(−SI)〉MS,1PI 〈HHS2I 〉a,1PI − 〈HHS2I 〉MS,1PI +O(a) (39) -0.2 -0.1 0 0.1 0.2 Broken Symmetric Figure 10: The the phase diagram as a function of d2, with d3 = 2. The symmetric phase refers to the phase where with d2 = 0 the order parameter vanishes and with d2 6= 0 is smaller than in the broken phase. The correlators in both regularizations are in two-loop weak-coupling expansion infrared-divergent quantities. However, since the infrared properties of the two regularization schemes are the same, the infrared divergences cancel exactly in the difference. The renormalization of the condensates is done very similarly. The condensates can be expressed, in both regularization schemes, as derivatives with respect to mass parameters of the free energy and thus they can be related. For the quadratic condensates we get: 〈TrZ†Z〉MS = ∂c1(µ) = 〈TrZ†Z〉a + ∂(fMS − fa) ∂c1(µ) 〈TrM †M〉MS = ∂d1(µ) = 〈TrM †M〉a + ∂(fMS − fa) ∂d1(µ) , (41) and for the cubic: 〈2ReDetZ〉MS = = 〈2ReDetZ〉a + ∂(fMS − fa) 〈2ReTrM3〉MS = = 〈2ReTrM3〉a + ∂(fMS − fa) . (43) 0 0.2 0.4 0.6 0.8 1 =0.05 =0.10 =0.15 Figure 11: Discontinuity in 〈TrM †M〉 along the critical line dcrit (d2, d3). Dotted lines represent second order polynomial fits to the data, and the points on the y-axis represent extrapolations to infinite d3. The large d3 extrapolation yields a finite value suggesting that the transition remains of first order even at large d3. Due to the super-renormalizability the difference in free energy is dimensionally of the form: fMS − fa = +D1,0 c1(µ̄) +D1,1 d1(µ̄) (4π)2 + C2,1 + C2,2 +E2,0c 2 + E2,1d 2 + E2,2c2d2 +D2,0g 3c1(µ̄) +D2,1c3c1(µ̄) +D2,2d3c1(µ̄) +D2,3g 3d1(µ̄) +D2,4c3d1(µ̄) +D2,5d3d1(µ̄) (4π)3 + C3,1 g23c3 + C3,2 g23d3 + C3,3 + C3,4 + C3,5 (4π)4 B4,0g 3 + C4,1g 3c3 + C4,2g 3d3 + C4,3g 3 + C4,3g 3 + C4,4g 3c3d3 +C4,5c 3 + C4,6d 3 + C4,6c 3d3 + C4,7c3d +O(a), (44) where the dimensionless coefficients Ai,j, Bi,j, Ci,j, Di,j , and Ei,j are functions of a dimensionless combination aµ̄ only. The coefficients Ci,j and Di,j follow from an i-loop computation. For the quadratic and cubic condensates we need to know coefficients Di,j and Ei,j in order to obtain the matching of the condensates to order O(a0), which follow from a two-loop calculation: D1,0 = −ΣN2c D1,1 = −Σ(N2c − 1) D2,0 = −2Nc(N2c − 1) + ζ + D2,1 = 0 D2,2 = 0 D2,3 = −2Nc(N2c − 1) + ζ + D2,4 = 0 D2,5 = 0 E2,0 = − (N2c − 1) E2,1 = −12 E2,2 = −8 . (45) For the quartic condensates, however, the coefficients Ci,j are needed and a four-loop lattice perturbation theory calculation is required for the matching. For the gluon condensates, also the Bi,j are needed. The coefficients B2,0 and B3,0 have been calculated in [32] and [33], respectively. The coefficient B4,0 has been calculated for Nc = 3 using stochastic perturbation theory in [34]. B Feynman rules Using the expanded fields, the potentials become: V0(Z) =g χ2 + ĉ1Tr[A ·A] + ĉ1Tr[H ·H] 2ĉ2√ 2ĉ2√ φχ2 + 2ĉ2√ φTr[A ·A]− 2ĉ2√ φTr[H ·H] 4ĉ2Tr[H ·H ·H] + 2ĉ2√ χTr[A ·H]− 1 4ĉ2Tr[A · A ·H] 2ĉ3φ 2ĉ3χ 4ĉ3φ 2Tr[H ·H] + 1 2Tr[H ·H] + 1 2Tr[A ·A] 4ĉ3χ 2Tr[A ·A] + ĉ3φχTr[A ·H] 6ĉ3φTr[H ·H ·H] + 6ĉ3χTr[A ·A · A] 6ĉ3φTr[A · A ·H] + 6ĉ3χTr[A ·H ·H] 24ĉ3Tr[A · A · A ·A] + 24ĉ3Tr[H ·H ·H ·H] 16ĉ3Tr[A ·A ·H ·H]− 8ĉ3Tr[A ·H ·A ·H] V1(Z) =g d̂1Tr[A ·A] + d̂1Tr[H ·H] −6d̂2Tr[A · A ·H] + 2d̂2Tr[H ·H ·H] d̂3Tr[A ·A ·A ·A] + 4d̂3Tr[A · A ·H ·H] −2d̂3Tr[A ·H · A ·H] + d̂3Tr[H ·H ·H ·H] . (47) The gauge part of the scalar Lagrangian in Fourier space (momentum conservation, all integrations over Brillouin zone, with measure (2π)3 , and sums understood) becomes [35]: SZ = i fabc (p̃− q)iAa(p)Ab(q)Aci (r) + 2facef bde( p− q )iδijA a(p)Ab(q)Aci (r)A j (s) fabc (p̃− q)iHa(p)Hb(q)Aci (r) + 2facef bde( p− q )iδijH a(p)Hb(q)Aci (r)A j (s), where we use a compact notation: = cos , p̃i = , and p̃2 = In addition to these there is the pure gluon and gauge fixing sector [36] p̃2Aai (−p)Aai (p) + p̃2c̄a(p)ca(p) + ig3fabcri p̃ic̄ a(p)cc(r)Abi (q) 2(facef bde + fadef bce)s̃ip̃ic̄ a(p)Aci (q)A i (r)c b(s) + Aai (−p)Aai (p) + S3 + S4. Contributions of three and four gluon vertices S3 and S4 can be found in [37], Eqs. (15.39),(15.43) and (15.53), where one needs to replace (2 )(δABδCD + . . .) with ( )(δABδCD + . . .) [36]. C Calculation of the diagrams The perturbation theory calculations were done using symbolic manipulation language FORM[38]. For formalized computation, it is advantageous to write all the color tensors in the fundamental representation, i.e. using the generators of the group: Tr[T aT bT c] = (dabc + ifabc) (51) Tr[T aT b] = δab. (52) Then all the color contractions in loop calculations can be done systematically with repeated use of the Fiertz identity: T aijT (δilδjk − δijδkl). (53) The following combinations are found in the action: ifabc = 2Tr(T a[T b, T c]) (54) dabc = 2Tr(T a{T b, T c}) (55) fabcf bde = −2Tr[T a, T c]Tr[T b, T d] (56) dabedcde = 2Tr{T a, T b}Tr{T c, T d} − 2 δabδcd (57) In lattice perturbation theory, the numerators of the integrals contain complex trigonometric objects. These can be systematically reduced to squares of sines, which also appear in the denominator, by repeated use of the following formulae (no summation over repeated indices): x̃+ yi = x̃iyi + ỹixi (x+ y)i x̃iỹi (59) 2 = δii − x̃2i (60) x̃iỹixi (x̃+ y)2i − x̃2i xi − ỹ2i xi . (61) This procedure generalizes trivially also to higher order loop calculations. The set of integrals can be further reduced by applying a trigonometric identity (for j ≥ 2): (2π)3 (x̃2 +m2)j j − 1 (2π)3 x̃2 − 3 (x̃2 +m2)j−1 (2π)3 (x̃2 +m2)j . (62) D Diagrams for mass renormalization In this section, we give the zero momentum diagrams that affect the mass renormalization. The expressions are in lattice regularization and the symbol ”MS” refers to the result of the corresponding diagram in the MS regularization. Solid and wiggly lines represent scalars and gluons, respectively. Symbols in parentheses indicate fields running in the internal scalar lines. The symmetry factors are included in the coefficients. The following diagrams with zero incoming momenta contribute to the renormalization of the mass term ĉ1 of φ-field (with external lines φ): • One-loop: The mass mi refers to the mass of the field running in the loop. In the difference between continuum and lattice regularization, the mass dependence cancels. (φ) : −1 (χ) : −1 (A) : −1 (N2c − 1) (H) : −(N2c − 1) g23 ĉ3I(mi) N2c g 3 ĉ3 +O(a) +MS (63) • Two-loop: – Terms proportional to ĉ23g 3 : Masses m1, m2 and m3 in the denominator refer to the masses of the internal lines and m2d = g 3(ĉ1 + d̂1).   (φφφ) : 2 (χχφ) : 2 (AAφ) : 2 (N2c − 1) (HHφ) : 2(N2c − 1) (χAH) : 4 (N2c − 1) (AAH) : −1 c − 1) (HHH) : −Nc(N2c − 1)   g43 ĉ ×H(m1,m2,m3) N2c − g43 ĉ +O(a) +MS (64) – Terms proportional to ĉ3g 3 : The tadpoles are cancelled by the 1-loop counter terms. (A) : −1 c − 1) (H) : −Nc(N2c − 1) g43 ĉ3 2H(md,md, 0) + (2I(0) − I(md)) I(md) + 4m2dH ′(md,md, 0) − a2G(md,md) c − 1) g43 ĉ3 + ζ − δ +O(a) + tadpoles. + MS (65) (A) : 1 c − 1) (H) : Nc(N c − 1) 3I(0)(−∂m2 )I(md) + a I(md) I(0)I(md) c − 1) g43 ĉ3 +O(a) + tadpoles. + MS (66) The following diagrams with zero incoming momenta contribute to the renormalization of the mass term d̂1 of H-field (with external lines H): • One-loop diagrams: (φ) : −ĉ3 (χ) : −1 (A) : −(2Nc − 1/Nc)(ĉ3 + d̂3) (H) : −(2Nc − 3/Nc)(ĉ3 + d̂3) g23I(mi) ĉ3 − 4(Nc − 1/Nc)(ĉ3 + d̂3) +O(a) +MS (67) = −3g23NcI(0) = −3g23Nc +O(a) +MS (68) = g23NcI(0) = g +O(a) +MS (69) • Two-loop diagrams: – Terms proportional to g43 ĉ   (φφH) : 2ĉ23 (φχA) : 4 (AAφ) : −1 (HHφ) : −3Ncĉ23 (χχH) : 2 (χAH) : −2 (AAH) : −2(1− 3/N2c − 32N c )(ĉ3 + d̂3) (HHH) : −6(1− 3/N2c − 16N c )(ĉ3 + d̂3)   g43H(m1,m2,m3) − 4Nc)ĉ23 + (4N2c + 24/N2c − 8)(ĉ3 + d̂3)2 +O(a) +MS – Terms proportional to g43(ĉ3 + d̂3): (A) : −2(N2c − 12) (H) : −2(N2c − 32) g43(ĉ3 + d̂3) 2H(md,md, 0) + (2I(0) − I(md)) I(md) + 4m2dH ′(md,md, 0) − a2G(md,md) = −8(N2c − 1)(ĉ3 + d̂3) + ζ − δ +O(a) + tadpoles. + MS (71) (A) : −2(N2c − 12) (H) : −2(N2c − 32) g43(ĉ3 + d̂ 3I(0)(−∂m2 )I(md) + a I(md) I(0)I(md) = −2(N2c − 1)g43(ĉ3 + d̂3) +O(a) +MS (72) – Terms proportional to g43 : The coupling of the adjoint fields A and H is exactly the same as as in EQCD, so the term proportional to g43 can be taken from EQCD [36]. However, at two-loop level diagrams with an adjoint scalar loop contribute two times since there are two adjoint fields. The diagrams with adjoint loops are the following: = N2c g (2π)3 (2π)3 x̃2x̃2(ỹ2 +m2d) =N2c g (2π)3 (2π)3 x̃2x̃2(ỹ2 +m2 (2π)3 (2π)3 x̃2x̃2 a2m2d (2π)3 (2π)3 x̃2x̃2(ỹ2 +m2d) N2c g ˜(2x+ y) (x̃2 +m2 )ỹ2ỹ2( ˜(x+ y) = N2c g H(md,md, 0) (2π)3 (2π)3 x̃2x̃2 ỹ2 +md + 2m2d (2π)3 (2π)3 x̃2x̃2 ỹ2 +md ˜(x+ y) (2π)3 (2π)3 [ ˜(x+ y)2]2(x̃2 +m2d)(ỹ 2 +m2d) The last line can be written in a more familiar form using the definition of ρ, Eq.(93), and the trigonometric identity Eq.(62): (2π)3 (2π)3 [ ˜(x+ y)2]2(x̃2 +m2 )(ỹ2 +m2 4π2a2 (2π)3 (2π)3 x̃2x̃2 (2π)3 (2π)3 ỹ2ỹ2 +O(a). The infrared divergences in these two diagrams cancel and the sum of the diagrams becomes: + = N2c g +O(a) +MS There are also two other diagrams: = −N2c g43 (2π)3 (2π)3 i (2y) x̃2x̃2(x̃2 +m2d)(ỹ 2 +m2d) = N2c g (2π)3 (2π)3 i x̃i ˜(x+ 2y)i j x̃j ˜(x+ 2y)j x̃2x̃2(x̃2 +m2d)(ỹ 2 +m2d)( ˜(x+ y) +m2d) After repeated use of Eqs.(61), the both diagrams can be written in the form ±N2c g43 (2π)3 (2π)3 x̃2 − 2a2 x̃2x̃2(x̃2 +m2d)(ỹ 2 +m2d) , (79) with the negative and the positive sign coming from the first and second diagram, respectively, so that their sum cancels exactly. The sum of diagrams proportional to g43 reads: Σ2 + ( )πΣ − 4(δ + ρ) + 2κ1 − κ4 + 2ρ+ +O(a) +MS (80) It is noteworthy that the scale dependence from the diagrams containing only gauge interactions with a single adjoint scalar field cancels exactly in the renormalization. However, upon the inclusion of another adjoint scalar field this property is lost. E Diagrams for operator renormalization In this section, we give the results for the vacuum diagrams that affect the renormalization of quadratic condensates present in the action. The diagrams needed for the quadratic condensates 〈TrZ†Z〉MS, 〈TrM †M〉MS, and 〈TrZ †TrZ〉MS: • One-loop: φ : 1 χ : 1 H : 1 (N2c − 1)J(md) A : 1 (N2c − 1)J(md) = (N2c − 1)J(md) + J(m) (81) • Two-loop: the counter terms cancel the {c3, d3}-dependent linearly divergences terms, and only the gauge diagrams contribute: H : 1 A : 1 c − 1) I(0)− 6I(0)I(md)− a2m2dI(0)I(md) A : 1 H : 1 c − 1) − I(md)I(md) + 4I(0)I(md) − 4m2dH(md,md, 0)− a2G(md,md) The diagrams needed for the cubic condensates: 〈2ReDetZ〉 and 〈2ReTrM3〉: • Two-loop   φφφ : 1 φχχ : 1 φhh : 1 (N2c − 1)ĉ22 φaa : 1 (N2c − 1)ĉ22 χah : 1 (N2c − 1)ĉ22 hhh : 3(1/Nc − 54Nc + N3c )( ĉ2 + d̂2) aah : 9(1/Nc − 54Nc + N3c )( ĉ2 + d̂2)   g63H(m1,m2,m3) (84) N2c − 1 ĉ22 + 12 1/Nc − ĉ2 + d̂2) g63H(m1,m2,m3) (85) F Basic lattice integrals and numerical constants In this appendix, we list the basic lattice integrals and numerical constants defined and calculated in [39, 35, 36]. Integrals: J(m) ≡ (2π)3 ln(x̃2 +m2) +O(am4) I(m) ≡ (2π)3 x̃2 +m2 −m+O(am2) H(m1,m2,m3) ≡ (2π)3 (2π)3 x̃2 +m21 ỹ2 +m22 x̃+ y a(m1 +m2 +m3) + ζ + +O(am) G(m,m) ≡ (2π)3 (2π)3 (x̃2 +m2)(ỹ2 +m2)(x̃+ y +O(m3a−1) H ′(m1,m2,m3) = (∂m2 )H(m1,m2,m3) (90) Numerical constants: Σ ≡ 1 ∫ π/2 i sin 2(xi) ≈ 3.175911535625 (91) ∫ π/2 d3xd3y i sin 2(xi) sin 2(xi + yi) i sin 2(xi))2 j sin 2(yj) k sin(xk + yk) ≈ 1.942130(1) (92) ∫ π/2 d3xd3y i sin 2(xi) sin 2(xi + yi) i sin 2(xi))2 j sin(xj + yj) i sin i sin 2(xi))2 i sin 2(yi))2 ≈ −0.313964(1) (93) ∫ π/2 d3xd3y i sin 2(xi) sin 2(xi + yi)∑ i sin 2(xi) j sin 2(yj) k sin(xk + yk) ≈ 0.958382(1) (94) ∫ π/2 d3xd3y i sin 2(xi) sin 2(xi + yi) sin 2(yi) i sin 2(xi))2 j sin 2(yj) k sin(xk + yk) ≈ 1.204295(1) (95) ζ = lim ∫ π/2 d3xd3y i sin 2(xi) + z) j sin 2(yj) k sin 2(xk + yk) ≈ 0.08849(1). (96) References [1] STAR Collaboration, J. 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Vermaseren, New features of FORM, [math-ph/0010025]. [39] K. Farakos, K. Kajantie, K. Rummukainen, and M. E. Shaposhnikov, 3-d physics and the electroweak phase transition: A framework for lattice monte carlo analysis, Nucl. Phys. B442 (1995) 317–363, [hep-lat/9412091]. http://xxx.lanl.gov/abs/hep-lat/0503041 http://xxx.lanl.gov/abs/hep-lat/9202004 http://xxx.lanl.gov/abs/hep-lat/0601009 http://xxx.lanl.gov/abs/hep-ph/0605042 http://xxx.lanl.gov/abs/hep-lat/9504001 http://xxx.lanl.gov/abs/hep-lat/9705003 http://xxx.lanl.gov/abs/[math-ph/0010025] http://xxx.lanl.gov/abs/hep-lat/9412091 Introduction Theory Lattice action Phase diagram of the soft potential Perturbation theory Lattice analysis Conclusions Details of renormalization Feynman rules Calculation of the diagrams Diagrams for mass renormalization Diagrams for operator renormalization Basic lattice integrals and numerical constants

0704.1417

Constraints on Regge models from perturbation theory

UAB-FT-630 UB-ECM-PF-07-07 Constraints on Regge models from perturbation theory Jorge Mondejara and Antonio Pinedab a Dept. d’Estructura i Constituents de la Matèria U. Barcelona, Diagonal 647, E-08028 Barcelona, Spain b Grup de F́ısica Teòrica and IFAE, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain Abstract We study the constraints that the operator product expansion imposes on large Nc inspired QCD models for current-current correlators. We focus on the constraints obtained by going beyond the leading-order parton computation. We explicitly show that, assumed a given mass spectrum: linear Regge behavior in n (the principal quantum number) plus corrections in 1/n, we can obtain the logarithmic (and constant) behavior in n of the decay constants within a systematic expansion in 1/n. Our example shows that it is possible to have different large n behavior for the vector and pseudo-vector mass spectrum and yet comply with all the constraints from the operator product expansion. http://arxiv.org/abs/0704.1417v3 1 Introduction The operator product expansion (OPE) has been used since long in order to gain information on the non-perturbative dynamics of the hadronic spectrum and decays [1, 2, 3, 4, 5, 6, 7, 8, 9]. In this article we revisit this problem. We want to obtain the constraints that the knowledge of the perturbative expansion in αs(Q 2) of the current-current correlators in the Euclidean regime poses on the relation between the decay constants and the mass spectrum for excitations with a large quantum number n (where n is the the quantum number of the bound state). We put special emphasis in going beyond the leading-order parton computation. We will work with a specific model for the hadronic spectrum. This is compulsory, since different spectral functions1 may yield the same OPE expression, yet we believe some aspects of our discussion may hold beyond the assumptions of our model. In order to have a well defined bound state it is crucial to consider the large Nc approx- imation [10]. This ensures infinitely narrow resonances at arbitrarily large energies. We will consider to be in the large Nc limit in what follows, as well as in the exact chiral (massless) limit. We will then set a specific model for the hadronic spectrum, valid for large values of n (we only need the behavior of the spectrum and decays for large n, we do not aim to get any information from perturbation theory for low values of n). This model will be based on the Regge behavior plus corrections in 1/n that will be included in a systematic way. The model is based on the assumption that the Regge behavior is a good description of the spectrum for large n (this can be explicitely seen in the ’t Hooft model [11] and it is also consistent with phenomenology). Given the 1/n corrections to the mass spectrum, the ex- pression of the correlator can also be written as a systematic expansion in 1/n, where higher powers in 1/n are equivalent to higher orders in 1/Q2 in its OPE. By matching the OPE and hadronic expressions order by order in 1/Q2, we will be able to predict the logarithmic dependence on n of the decay constants (actually also the constant terms). This result can also be systematically organized within an expansion in 1/n together with an expansion in 1/ lnn. We will give explicit expressions up to order 1/n2 and 1/ ln3 n. We will also make some numerical estimates of the impact of these corrections. Finally, we would like to stress that we are able to introduce power corrections in 1/n to the Regge behavior and yet comply with the OPE. This is in contrast with Ref. [5], where, besides the Regge behavior, only exponentially suppressed terms are introduced (parametrically smaller than any finite power of 1/n for large n). This parameterization is however fine if considered as a fit not emanated from the large n limit. 2 Correlators For definiteness, we will consider the vector-vector correlator but most of the discussion applies to any other current-current correlator (axial-vector, scalar, ....). V (q) ≡ (q µqν − gµνq2)ΠV (q) ≡ i d4xeiqx〈vac|T {J V (x)J V (0)} |vac〉 , (1) 1In particular the one derived directly from perturbation theory, which we do not consider, since we will work in the large N limit with infinitely narrow resonances. where J Qf ψ̄fγ µψf . In order to avoid divergences, we will consider the Adler function A(Q2) ≡ −Q2 Π(Q2) = Q2 (t+Q2)2 ImΠV (t) , (2) where Q2 = −q2 is the Euclidean momentum. Since we are working in the large Nc limit, the spectrum consists of infinitely narrow resonances, and the Adler function can be written in the following way A(Q2) = Q2 F 2V (n) (Q2 +M2V (n)) . (3) On the other hand, for large positive Q2, one may try to approximate the Adler function by its OPE, which reads AOPE(Q C(αs(Q β(αs(ν))〈vac|G 2(ν)|vac〉+O where αA(Q 2) admits an analytic expansion in terms of αs(Q 2) (computed in the MS scheme), β(αs) = −β0 + · · · , (5) with β0 = 11/3Nc, β1 = 34/3N c , β2 = 2857/54N c , and [12] C(αs(Q 2)) = − + · · · . (6) 3 Matching High excitations of the QCD spectrum are believed to satisfy linear Regge trajectories: M2V,n = constant. For generic current-current correlators, such behavior is consistent with perturbation theory in the Euclidean region at leading order in αs if the decay constants are taken to be “constants”, ie. independent of the principal quantum number n. The inclusion of subleading effects in αs can be incorporated into this model by changing the n dependence of the decay constants without changing the ansatz for the spectrum. The inclusion of these effects has consequences on subleading sum-rules and the relation with the non-perturbative condensates. Here we would like to go beyond the analysis at leading order in αs, as well as to consider power-like corrections in 1/n. We will consider that the large n expression for the mass spectrum can be organized within a 1/n expansion in a systematic way starting from the asymptotic linear Regge behavior. In order to fix (and simplify) the problem we will assume that no lnn term appears in the mass spectrum2. Therefore, we write the mass spectrum in the following way (for large n) M2V (n) = (−s) = B V n+B + · · · , (7) where B V are constants. We will usually denote M V,LO(n) = B V n, M V,NLO(n) = B V and so on. To shorten the notation, we will denote B V = BV , B V = AV and B For the decay constants, we will have a double expansion in 1/n and 1/ lnn. F 2V (n) = F 2V,s(n) = F 2V,0(n) + F 2V,1(n) F 2V,2(n) + · · · , (8) where the coefficients F 2V,s(n) have a logarithmic dependence on n: F 2V,s(n) = V,s(n) lnr n . (9) As we did with the masses, we will define F 2V,LO(n) = F V,0(n), F V,NLO(n) = F V,0(n) + and so on. Note that in this case we also have an expansion in 1/ lnn. We are now in position to start the computation. Our aim is to compare the hadronic and OPE expressions of the Adler function within an expansion in 1/Q2, but keeping the logarithms of Q. In order to do so we have to arrange the hadronic expression appropiately. Our strategy is to split the sum over hadronic resonances into two pieces, for n above or below some arbitrary but formally large n∗, such that ΛQCDn ∗ ≪ Q. The sum up to n∗ can be analytically expanded in 1/Q2 and will not generate lnQ2 terms (neither a constant term at leading order in 1/Q2). For the sum from n∗ up to ∞, we can use Eqs. (7) and (8) and the Euler-MacLaurin formula to transform the sum in an integral plus corrections in 1/Q2. Whereas the latter do not produce logarithms, the integral does. These logarithms of Q are generated by the large n behavior of the bound states and the introduction of powers of 1/n is equivalent (once introduced in the integral representation, and for large n) to the introduction of (logarithmically modulated) 1/Q2 corrections in the OPE expression. Therefore, by using the Euler-MacLaurin formula, we write the Adler function in the following way (B2 = 1/6, B4 = −1/30, ...) A(Q2) = Q2 F 2V (n) (Q2 +M2V (n)) F 2V (n) (Q2 +M2V (n)) F 2V (n) (Q2 +M2V (n)) F 2V (n (Q2 +M2V (n (−1)k |B2k| (2k)! d(2k−1) dn(2k−1) F 2V (n) (Q2 +M2V (n)) , (10) 2This is a simplification. If one considers, for instance, the ’t Hooft model [11], lnn terms do indeed appear. where n∗ stands for the subtraction point we mentioned above, such that for n larger than n∗ one can use the asymptotic expressions (7) and (8). This allows us to eliminate terms that vanish when n → ∞. Note that the last sum in Eq. (10) is an asymptotic series, and in this sense the equality should be understood. Note also that for n below n∗, we will not distinguish between LO, NLO, etc... in masses or decay constants, since for those states we will not assume that one can do an expansion in 1/n and use Eqs. (7) and (8). Finally, note that the expressions we have for the masses and decay constants become more and more infrared singular as we go to higher and higher orders in the 1/n expansion. This is not a problem, since we always cut off the integral for n smaller than n∗. Either way, the major problems would come from the decay constants, since, in the case of the mass, Q2 effectively acts as an infrared regulator. 3.1 LO Matching We want to match the hadronic, Eq. (10), and OPE, Eq. (4), expressions for the Adler function at the lowest order in 1/Q2. This means that we have to consider the lowest order expressions in 1/n for the masses and decay constants, i.e. F 2V,LO(n) and M V,LO(n), since the corrections in 1/n give contributions suppressed by powers of 1/Q2. Only the first term in Eq. (10) can generate logarithms or terms that are not suppressed by powers of 1/Q2. Therefore we obtain the following equality, Apt.(Q2) ≡ Q2 F 2V,LO(n) (Q2 +M2V,LO) . (11) This equation can be fulfilled by demanding that F 2V,LO(n) |dM2V,LO(n)/dn| pert. V,LO(n)) . (12) By using the perturbative expression for ImΠ pert. V (see [13]), we obtain F 2V,LO(n) = BV Ncαs(nBV ) + 243− 176 ζ(3) 128π2 N2c α s (nBV ) (13) 346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5) 27648π3 N3c α s (nBV ) +O α4s (nBV ) where αs(nBV ) should actually be understood as a function of αs(BV ) and lnn. Therefore, it is obvious that the above expression is resumming powers of αs(BV ) lnn: F 2V,LO(n) = BV 1 + 11 αs(BV ) ln(n) αs(BV ) 2673− 1936 ζ(3)− 408 ln 1 + 11 Ncαs(BV ) ln(n) 88(1 + 11 αs(BV ) ln(n))2 N2c α s (BV ) (4π)2 N3c α s (BV ) (4π)3 52272π(1 + 11 αs(BV ) ln(n))3 [−350427Ncαs(BV ) ln(n) +121π 346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5) −3672π(2877− 1936 ζ(3)) ln αs(BV ) ln(n) +749088π ln2 αs(BV ) ln(n) α4s (BV ) Doing so we see that we are able to obtain the dependence of the decay constant in lnn (somewhat we are assuming that αs(BV ) is an small parameter, BV ∼ 1 GeV). We can also rewrite the decay constant as an expansion in 1/ lnn by using the equality ln ñ = αs(nBV ) αs(nBV ) αs(nBV ) , (15) where ñ = nBV /ΛMS. We then obtain F 2V,LO(n) = BV ln ñ ln2 ñ ln ln ñ + ln3 ñ 46818 161051 ln2 ln ñ + 322102 (−2877 + 1936ζ(3)) ln ln ñ + 42272605 2576816 20283 ζ(3) 360 ζ(5) ln4 n . (16) Note that the lowest contribution in 1/ lnn for the decay constant, BV Q2f , which, usually, is the only one considered, reproduces the leading-order partonic prediction for the Adler function. Note also that there is no problem with the Landau pole, even if the result is written in the form of Eq. (16), since it holds only for n larger than an n∗ such that ΛMS ≪ n ∗BV (the integral has an infrared cutoff at n∗). Finally, we remind that, strictly speaking, we can only fix the ratio between the decay constant and the derivative of the mass. We have fixed this ambiguity by arbitrarily imposing the n dependence of the mass spectrum. 3.2 NLO matching We now want to obtain extra information on the decay constant by demanding the validity of the OPE at O(1/Q2), in particular the absence of condensates at this order. We then have to use the NLO expressions for M2V (n) and F V (n). With the ansatz we are using for the mass at NLO, it is compulsory to introduce (logarithmically modulated) 1/n corrections to the decay constant if we want this constraint to hold. Note that it is possible to shift all the perturbative corrections to the decay constant. Imposing that the 1/Q2 condensate vanishes produces the following sum rule: Apt. − Apt. + F 2V (n)− dnF 2V,LO(n) F 2V (n (−1)k |B2k| (2k)! d(2k−1) dn(2k−1) F 2V (n) F 2V,1(n)/n (Q2 +M2V,LO(n)) F 2V,1(n)/n (Q2 +M2V,LO(n)) = 0 . This equality should hold independently of the value of n∗, which formally should be taken large enough so that αs(n ∗BV ) ≪ 1, i.e. the limit ΛMS ≪ n ∗BV ≪ Q 2. Again, the meaning of the asymptotic series appearing in Eq. (17) should be taken with care. If we forget about this potential problem, only a few terms in Eq. (17) can generate lnQ2 terms, which should cancel at any order. Those are the first two and the last two terms. Actually, the next to last term does not generate logarithms, but it allows to regulate possible infrared divergences appearing in the calculation. Therefore, asking for the cancellation of the 1/Q2 suppressed logarithmic terms produced by the first two and the last term in Eq. (17) fixes F 2V,1. The non-logarithmic terms should also be cancelled but they cannot be fixed from perturbation theory. One can actually find an explicit solution to the above constraint for F 2V,1 by performing some integration by parts. We obtain F 2V,1(n) F 2V,0(n) (18) ln2 ñ (1− 2 ln (ln ñ)) + 36 ζ(3) ln3 ñ 2576816 −45794053 + 351384π2 + 41637552 ζ(3)− 7666560 ζ(5) −3672 ln (ln ñ) (−3013 + 1936 ζ(3) + 204 ln (ln ñ))] ln4 ñ ln5 ñ or in terms of αs(nBV ) or αs(BV ), F 2V,1(n) N2c α s (nBV )− 2877− 1936 ζ(3) 768π3 N3c α s (nBV ) (19) 11(376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5)) 110592π4 N4c α s (nBV ) +O α5s (nBV ) F 2V,1(n) (1 + 11 αs(BV ) ln(n))2 α2s (BV ) (4π)2 2877− 1936 ζ(3)− 408 ln 1 + 11 Ncαs(BV ) ln(n) 12(1 + 11 αs(BV ) ln(n))3 α3s (BV ) (4π)3 4752π(1 + 11 αs(BV ) ln(n))4 α4s (BV ) (4π)4 [−233618Ncαs(BV ) ln(n) +121π 376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5) −3672π(3013− 1936 ζ(3)) ln αs(BV ) ln(n) +749088π ln2 αs(BV ) ln(n) α5s (BV ) Note that besides the 1/n suppression, we also have an extra α2s (nBV ) suppression. In principle one could think of the existence of 1/n×constant terms in the decay constant, i.e. without any associated logarithm. However, such terms produce ln(Q2) contributions in the Euclidean regime that do not appear in the perturbative computation, so they can be ruled out. This appears to be a general statement since 1/nm × constant for any m integer also produces logarithms. Note that in order to give meaning to these integrals it is implicit that the integral over n has an infrared cutoff at n∗. Nevertheless, the logarithm does not appear to multiply powers of the infrared cutoff (as expected). Finally, we would like to mention that, besides the constraints coming from the logarith- mic related behavior of the OPE, there is also the constraint from its constant terms, which should sum up to zero. Nevertheless, for this constraint we cannot give a closed expression. This is due to the fact that the lnQ2 independent terms may receive contributions from any subleading order in the 1/n expansion of the masses and decay constants. The reason is that the decay constant at a given order in 1/n is obtained after performing some inte- gration by parts, which generate new (lnQ2-independent) terms that can be Q2 enhanced. This statement is general and also applies to any subleading power in the 1/Q2 matching computation. 3.3 NNLO matching We now consider expressions for the mass and decay constants at NNLO. For the first time we have to consider condensates. Simplifying terms that do not produce logs, we obtain the following equation, β(αs(ν))〈vac|G 2(ν)|vac〉 (Q2 +BV n)2 F 2V,2(n) V,0(n) V,1(n) where = stands for the fact that we can only predict the lnQ2 dependence. Constant terms are not fixed by this relation. In order to get a more closed expression is convenient to use the following equality, (Q2 +BV n)2 BV n2 α2s (nBV ) , (22) valid up to terms that do not produce logarithms or those that are subleading. We get then F 2V,2(n) = −CV Ncαs(nBV ) 287− 176 ζ(3) 128π2 11A2V 64π2BVCV β(αs(ν))〈vac|G 2(ν)|vac〉 BVCVN2c N2c α s (nBV ) (23) α3s (nBV ) Note that in this case we only consider up to O(α2s (nBV )) corrections, since higher order loops are unknown. The accuracy is set by our knowledge of the matching coefficient of the gluon condensate. Note also that F 2V,2(n) does not have αs suppression. Therefore, for low n, this contribution could be practically of the same size than, formally, more important terms. 4 Axial versus vector correlators The above discussion has been performed for the vector-vector correlator Adler function. It goes without saying that we could perform a similar analysis with axial-vector currents, since the perturbative expansions for both correlators are equal. Here it comes an important observation. We could change the coefficients for the mass spectrum BA 6= BV , AA 6= AV , · · · , yet we would obtain the same expression for the OPE (at the order we are working, the first chiral breaking related effects are O(1/Q6)). Therefore, we conclude that the OPE does not fix BA = BV as it is sometimes claimed in the literature [1, 3] 3. Our computation gives a specific counter example. Moreover, it is nice to see what the role played by BA and BV is in our case. When one goes to the Euclidean regime, BA and BV become renormalization factorization scales and, obviously, the physical result does not depend on them (for large Q2 in the Euclidean). On the other hand, it is evident that having different constants: BA, BV , . . . produces different physical predictions for the masses and decay widhts for vector or axial-vector channels. The point to be emphasized is that BA = BV cannot come from an OPE analysis alone. This point has already been stressed in Refs. [4, 8], what we think is novel in our analysis is that we have seen that the inclusion of corrections in αs does not affect that conclusion, and that BA and BV play the role of the renormalization scale in the analogous perturbative analysis in the Euclidean regime, and are therefore unobservable. Finally, we cannot avoid mentioning the analysis of Ref. [14] where, using AdS/CFT, they explicitly find Regge behavior with different slopes for vector and axial-vector channels. 3Another issue, on which we do not enter, is whether some other kind of arguments (relying on the specific model used), like semiclassical arguments, may fix those parameters to be equal. In any case, even though the constants that characterize the spectrum can be different for the vector and axial-vector channel, they have to yield the same expressions for the OPE when combined with the decay constants. This produces some relations that we list in what follows. We first define t ≡ BV n = BAn ′ and take n and n′ as continuous variables. We then obtain the following equalities F 2V,LO(n) F 2A,LO(n pert. V (t) ≡ f0(t) , (24) F 2V,1(n) F 2A,1(n f0(t) , (25) F 2V,2(n) V,0(n) V,1(n) F 2A,2(n A,0(n A,1(n β(αs(ν))〈vac|G 2(ν)|vac〉f1(t) , (26) where f1(t) = α2s (t) (4π)2 + · · · . (27) 5 Numerical Analysis n = 1 n = 2 n = 3 n = 4 Mρ(I) 781.3(775.5± 0.4) 1440.2(1459± 11) 1891.8(1870± 20) 2257(2265± 40) Mρ(II) 771.5(775.5± 0.4) 1471.7(1459± 11) 1855(1870± 20) 2154.8(2149± 17) Ma1 1235.6(1230± 40) 1621.7(1647± 22) 1962(1930 −70) 2257.8(2270 FV (I) 156(156± 1) 155 154 153 FV (II) 185(156± 1) 147 139 135 FA 123(122± 24) 137 139 139 Table 1: We give the experimental values of the masses (in MeV) and electromagnetic decay constants (when available) for vector and axial vector particles (within parenthesis), compared with the values obtained from the fit. For the vector states we consider two possible Regge trajectories that we label I and II respectively. We take αs(1GeV) = 0.5 and β〈G −(352MeV)4. We restrict ourselves to the SU(2) case (non-strange sector) and study the vector and axial-vector channels. We would like here to assess the importance of including perturbative corrections to a standard analysis based on the OPE. We do not aim to perform a full fledged 2 4 6 8 10 FV,LOHIL 2 4 6 8 10 FV,LOHIIL Figure 1: In this plot we show FV,LO(I) and FV,LO(II) at different orders in αs. analysis, but only to see the importance of the corrections. In table 1, we give the values of the masses and decay constants. In Figure 1 we show the changes in both FV,LO(I) and FV,LO(II) as we include higher orders in the expansion in αs, and in Figure 2 the changes in the full FV (I) and FV (II) as we include higher orders in 1/n. In figure 3 we show the same plots for the axial-vector case. We take the experimental values from Ref. [15]. In principle there are more states in the particle data book, in particular in the vector channel. Nevertheless, it is not clear whether they belong to the same Regge trajectory or whether they belong to some daughter one, see, for instance, the discussion in Ref. [5]. For the time being we will disregard the study of other possible (vector) Regge trajectories and restrict the analysis to a single trajectory. We will consider the two possibilities listed in Table 1. Our choice of states for the set (I) is motivated by the discussion of Ref. [16] on the possible formation of multiplets in the case of chiral symmetry restoration. The set (II) is based on the assignment of states made in Ref. [5] (based on the existence of S and D-wave daughter trajectories) and in particular on the analysis of Ref. [17], where the state 2265 is argued to belong to the D-wave Regge trajectory4. In order to fix the parameters of the mass spectrum we use the experimental values of the masses we list in the table. We obtain the values: BV (I) = 1.525× 10 6MeV2 , AV (I) = −1.038× 10 6MeV2 , CV (I) = 0.123× 10 6MeV2 , BV (II) = 1.128× 10 6MeV2 , AV (II) = 0.353× 10 6MeV2 , CV (II) = −0.885× 10 6MeV2 , BA = 1.278× 10 6MeV2 , AA = −0.100× 10 6MeV2 , CA = 0.349× 10 6MeV2 . We should mention that the values obtained for these parameters are not very stable under the change of number of data points, except for BV and BA, which are roughly stable, although with quite sizeable uncertanties. For the subleading terms A and C, their values are basically random with the fit. We roughly find BV ≃ BA within the uncertainties. The n dependence of the axial and vector (model II) decay constants is small but sizeable (and it goes in the right direction for low n). The 1/n corrections are always corrections compared with the leading order terms. Nevertheless, the 1/n2 correction is much larger than the 1/n one for the range of values of n that we explore. This appears to be due to the α2s/(4π) suppression of the 1/n term, as well as to the difference in size between the constants A 4We also thank S. Afonin for discussions on this point. 2 4 6 8 10 FVHIL 2 4 6 8 10 132.5 137.5 142.5 147.5 FVHIIL Figure 2: In this plot we show FV (I) and FV (II) at different orders in the 1/n expansion. 2 4 6 8 10 FA,LO 2 4 6 8 10 Figure 3: In this plot we show FA,LO and FA at different orders in αs and in the 1/n expansion, respectively. and C. This is so for the axial and vector (model II) decay constants. Nevertheless, for the vector (model I) decay constants the n dependence appears to be quite small also at NNLO. This appears to be due to the small value of the coefficient CV (I). The gluon condensate contribution is a small correction to the total NNLO term. Either way, our predictions compare favorably with experiment when this comparison is possible. We should keep in mind that these results have been obtained for a specific model, so we are testing the impact of the perturbative corrections for this specific model. On the other hand, if one believes that the large n behavior of the spectrum is dictated by the Regge behavior and that the corrections can be obtained as an expansion in 1/n, the set up is general. The only ambiguity comes from where the logarithms should be introduced (masses or decays). At this respect it is worth mentioning that, as a matter of principle, this ambiguity could be fixed if enough experimental information were available for the masses and decays. 6 Conclusions We have studied the constraints that the OPE imposes on large Nc inspired QCD models for current-current correlators. We have focused on the constraints obtained by going beyond the leading-order parton computation. We have explicitly showed that, assumed a given mass spectrum (Regge plus corrections in 1/n), we can obtain the logarithmic (and constant) behavior in n of the decay constants within a systematic expansion in 1/n. More than that, power-like 1/n corrections can only be incorporated in the analysis if full consideration to the perturbative corrections in the Euclidean regime is made. This is due to the fact that these type of contributions produce logarithms of Q in the Euclidean (this is one of the reasons why this sort of corrections are not usually considered in quark-hadron duality analysis). On the other hand, the existence of lnn in the decay constants may point to the existence of two scales in the problem, ΛQCD and nΛQCD, in the Minkowski regime. We have also performed some numerical estimates of the importance of these corrections. The n dependence of the decay constants is small but sizeable for the axial and vector (model II) channel, for the vector (model I) one this dependence is small. On the other hand the uncertainties of the calculation are large. Either way, our predictions compare favorably with experiment when this comparison is possible. Our example shows that it is possible to have different large n behavior for the vector and pseudo-vector mass spectrum and yet comply with all the constraints from the OPE. An important caveat of our analysis is that we have not considered what the effect of renormalons could be. We have focused on the effect of low orders in perturbation theory to the decay constants. It would be interesting to see whether the knowledge of the higher order behavior of perturbation theory may give some extra constraints on the values of these constants and the mass spectrum. At this respect we have to say that we have obtained approximated expressions for the decay constants as an expansion in αs(nBV ), with just the low order contributions in αs. It is quite likely that this expansion is asymptotic and that different orders in 1/n are related in a similar way to the one found in the renormalon analysis for the OPE expansion for different orders in 1/Q2. Therefore, the results obtained for the 1/n corrections could be affected as well by the asymptotic behavior of the 1/ lnn expansion in the leading-order term. This is obviously related with renormalons. We expect to come back to this issue in the future. Acknowledgments. We thank S. Afonin, A. Andrianov, D. Espriu, and S. Peris for discussions and L. Glozman for correspondence. This work is partially supported by the network Flavianet MRTN-CT-2006-035482, by the spanish grant FPA2004-04582-C02-01, by the catalan grant SGR2005-00564 and by a Distinció from the Generalitat de Catalunya. References [1] S. R. Beane, Phys. Rev. D 64, 116010 (2001) [arXiv:hep-ph/0106022]. [2] M. Golterman, S. Peris, B. Phily and E. De Rafael, JHEP 0201, 024 (2002) [arXiv:hep-ph/0112042]. [3] T. D. Cohen and L. Y. Glozman, Int. J. Mod. Phys. A 17, 1327 (2002) [arXiv:hep-ph/0201242]. [4] M. Golterman and S. Peris, Phys. Rev. D 67, 096001 (2003) [arXiv:hep-ph/0207060]. [5] S. S. Afonin, A. A. Andrianov, V. A. Andrianov and D. Espriu, JHEP 0404, 039 (2004) [arXiv:hep-ph/0403268]. [6] J. J. Sanz-Cillero, Nucl. Phys. B 732, 136 (2006) [arXiv:hep-ph/0507186]. http://arxiv.org/abs/hep-ph/0106022 http://arxiv.org/abs/hep-ph/0112042 http://arxiv.org/abs/hep-ph/0201242 http://arxiv.org/abs/hep-ph/0207060 http://arxiv.org/abs/hep-ph/0403268 http://arxiv.org/abs/hep-ph/0507186 [7] M. Shifman, arXiv:hep-ph/0507246. [8] O. Cata, M. Golterman and S. Peris, Phys. Rev. D 74, 016001 (2006) [arXiv:hep-ph/0602194]. [9] S. S. Afonin and D. Espriu, JHEP 0609, 047 (2006) [arXiv:hep-ph/0602219]. [10] G. ’t Hooft, Nucl. Phys. B 72, 461 (1974). [11] G. ’t Hooft, Nucl. Phys. B 75, 461 (1974). [12] L. R. Surguladze and F. V. Tkachov, Nucl. Phys. B 331, 35 (1990). [13] K. G. Chetyrkin, Phys. Lett. B 391, 402 (1997) [arXiv:hep-ph/9608480]. [14] R. Casero, E. Kiritsis and A. Paredes, arXiv:hep-th/0702155. [15] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006). [16] L. Y. Glozman, Phys. Lett. B 587, 69 (2004) [arXiv:hep-ph/0312354]. [17] D. V. Bugg, Phys. Rept. 397, 257 (2004) [arXiv:hep-ex/0412045]. http://arxiv.org/abs/hep-ph/0507246 http://arxiv.org/abs/hep-ph/0602194 http://arxiv.org/abs/hep-ph/0602219 http://arxiv.org/abs/hep-ph/9608480 http://arxiv.org/abs/hep-th/0702155 http://arxiv.org/abs/hep-ph/0312354 http://arxiv.org/abs/hep-ex/0412045 Introduction Correlators Matching LO Matching NLO matching NNLO matching Axial versus vector correlators Numerical Analysis Conclusions

0704.1418

Asymptotic stability at infinity for bidimensional Hurwitz vector fields

Asymptotic stability at infinity for bidimensional Hurwitz vector fields ✩ Roland Rabanal1 Abstract Let X : U → R2 be a differentiable vector field. Set Spc(X) = {eigenvalues of DX(z) : z ∈ U}. This X is called Hurwitz if Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Suppose that X is Hurwitz and U ⊂ R2 is the complement of a compact set. Then by adding to X a constant v one obtains that the infinity is either an attractor or a repellor for X + v. That means: (i) there exists an unbounded sequence of closed curves, pairwise bounding an annulus the boundary of which is transversal to X + v, and (ii) there is a neighborhood of infinity with unbounded trajectories, free of singularities and periodic trajectories of X+v. This result is obtained after to proving the existence of X̃ : R2 → R2, a topological embedding such that X̃ equals X in the complement of some compact subset of U. Keywords: Injectivity, Reeb Component, Asymptotic Stability 2008 MSC: Primary: 37E35, 37C10; Secondary: 26B10, 58C25 1. Introducion1 A basic example of non–discrete dynamics on the Euclidean space is given by a linear vector2 field. This linear system is infinitesimally hyperbolic if every eigenvalue has nonzero real part,3 and it has well known properties [16, 3]. For instance, when the real part of its eigenvalues are4 negative (Hurwitz matrix), the origin is a global attractor rest point. In the nonlinear case, there5 has been a great interest in the local study of vector fields around their rest points [6, 27, 7, 28].6 However, in order to describe a global phase-portrait, as in [22, 25, 5, 8, 9] it is absolutely7 necessary to study its behavior in a neighborhood of infinity [19].8 The Asymptotic Stability at Infinity has been investigated with a strong influence of [18],9 where Olech showed a connection between stability and injectivity (see also [10, 4, 11, 26, 22]).10 This research was also studied in [19, 12, 14, 15, 23, 1]. In [12], Gutierrez and Teixeira study11 C1−vector fields Y : R2 → R2, the linearizations of which satisfy (i) det(DY(z)) > 0 and12 (ii) Trace(DY(z)) < 0 in an neighborhood of infinity. By using [9], they prove that if Y has a13 rest point and the Index I(Y) = Trace (DY) < 0 (resp. I(Y) ≥ 0), then Y is topologically14 equivalent to z 7→ −z that is “the infinity is a repellor ”(resp. to z 7→ z that is “the infinity15 is an attractor”). This Gutierrez-Teixeira’s paper was used to obtain the next theorem, where16 Spc(Y) = {eigenvalues of DY(z) : z ∈ R2 \ Dσ}, andℜ(z) is the real part of z ∈ C.17 ✩This paper was written when the author served as an Associate Fellow at ictp-Italy. Email address: [email protected] (Roland Rabanal) 1The author was partially supported by pucp-Peru (dai: 2012-0020). Preprint submitted to Elsevier November 16, 2018 http://arxiv.org/abs/0704.1418v4 Theorem 1 (Gutierrez-Sarmiento). Let Y : R2 \ Dσ → R2 be a C1− map, where σ > 0 and18 Dσ = {z ∈ R 2 : ||z|| ≤ σ}. The following is satisfied:19 (i) If for some ε > 0, Spc(Y) ∩ (−ε,+∞) = ∅. Then there exists s ≥ σ such that the restriction20 Y | : R2 \ Ds → R 2 is injective.21 (ii) If for some ε > 0, the spectrum Spc(Y) is disjoint of the union (−ε, 0]∪{z ∈ C : ℜ(z) ≥ 0}.22 Then there exist p0 ∈ R 2 such that the point ∞ of the Riemann Sphere R2 ∪ {∞} is either23 an attractor or a repellor of z′ = Y(z) + p0.24 Theorem 1 is given in [14], and it has been extended to differentiable maps X : R2 \Dσ → R in [13, 15]. In both papers the eigenvalues also avoid a real open neighborhood of zero. In [23]26 the author examine the intrinsic relation between the asymptotic behavior of Spc(X) and the27 global injectivity of the local diffeomorphism given by X. He uses Yθ = Rθ ◦Y ◦R−θ, where Rθ is28 the linear rotation of angle θ ∈ R, and (motivated by [11]) introduces the so–called B−condition29 [24, 26], which claims:30 for each θ ∈ R, there does not exist a sequence (xk, yk) ∈ R 2 with xk → +∞ such that31 Yθ((xk, yk))→ p ∈ R 2 and DYθ(xk, yk) has a real eigenvalue λk satisfying xkλk → 0.32 By using this, [23] improves the differentiable version of Theorem 1.33 In the present paper we prove that the condition34 Spc(X) ⊂ {z ∈ C : ℜ(z) < 0} is enough in order to obtain Theorem 1 for differentiable vector fields X : R2 \ Dσ → R Throughout this paper, R2 is embedded in the Riemann sphere R2∪{∞}. Thus (R2\Dσ)∪{∞}36 is the subspace of R2 ∪ {∞} with the induced topology, and ‘infinity ’refers to the point ∞ of37 2 ∪ {∞}. Moreover given C ⊂ R2, a closed (compact, no boundary) curve (1−manifold),38 D(C) (respectively D(C)) is the compact disc (respectively open disc) bounded by C. Thus, the39 boundaries ∂D(C) and ∂D(C) are equal to C, homeomorphic to ∂D1 = {z ∈ R 2 : ||z|| = 1}.40 2. Statements of the results41 For every σ > 0 let Dσ = {z ∈ R 2 : ||z|| ≤ σ}. Outside this compact disk we consider a42 differentiable vector field X : R2 \Dσ → R 2. As usual, a trajectory of X starting at q ∈ R2 \Dσ43 is defined as the integral curve determined by a maximal solution of the initial value problem44 ż = X(z), z(0) = q. This is a curve Iq ∋ t 7→ γq(t) = (x(t), y(t)), satisfying:45 • t varies on some open real interval containing the zero, the image of which γq(0) = q;46 • γq(t) ∈ R 2 \ Dσ and there exist the real derivatives (t), dy (t);47 • γ̇q(t) = (t), dy the velocity vector field of γq at γq(t) equals X(γq(t)) and48 • Iq ⊂ R is the maximal interval of definition.49 We identify the trajectory γq with its image γq(Iq), and we denote by γ q (resp. γ q ) the positive50 (resp. negative) semi-trajectory of X, contained in γq and starting at q. In this way γq = γ q ∪ γ q .51 Thus each trajectory has its two limit sets, α(γ−q ) and ω(γ q ) respectively. These limit sets are52 well defined in the sense that they only depend on the respective solution.53 A C0−vector field X : R2 \ Dσ → R 2 \ {0} (without rest points) can be extended to a map54 X̂ : ((R2 \ Dσ ∪∞),∞) −→ (R 2, 0) (which takes∞ to 0) [1]. In this manner, all questions concerning the local theory of isolated rest55 points of X can be formulated and examined in the case of the vector field X̂. For instance, if γ+p56 (resp. γ−p ) is an unbounded semi-trajectory of X : R 2 \ Dσ → R 2 starting at p ∈ R2 \ Dσ with57 empty ω−limite (resp. α−limit) set, we say γ+p goes to infinity (resp. γ p comes from infinity),58 and it is denoted by ω(γ+p ) = ∞ (resp. α(γ p ) = ∞). Therefore, we may also talk about the phase59 portrait of X in a neighborhood of∞.60 As in our paper [15], we say that the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is61 an attractor (resp. a repellor) for the continuos vector field X : R2 \ Dσ → R2 if:62 • There is a sequence of closed curves, transversal to X and tending to infinity. That is63 for every r ≥ σ there exist a closed curve Cr such that D(Cr) contains Dr and Cr has64 transversal contact to each small local integral curve of X at any p ∈ Cr .65 • For some Cs with s ≥ σ, all trajectories γp starting at a point p ∈ R 2 \ D(Cs) satisfy66 ω(γ+p ) = ∞ that is γ p go to infinity (resp. α(γ p ) = ∞ that is γ p come from infinity).67 We also recall that I(X), the index of X at infinity is the number of the extended line [−∞,+∞]68 given by69 I(X) = Trace(DX̂)dx ∧ dy, where X̂ : R2 → R2 is a global differentiable vector field such that:70 • In the complement of some disk Ds with s ≥ σ both X and X̂ coincide.71 • The map z 7→ Trace(DX̂z) is Lebesgue almost–integrable in whole R 2, in the sense of [15].72 This index is a well-defined number in [−∞,+∞], and it does not depend on the pair (X̂, s) as73 shown [15, Lemma 12].74 Definition 1. The differentiable vector field (or map) X : R2 \ Dσ → R2 is called Hurwitz if75 every eigenvalue of the Jacobian matrices has negative real part. This means that its spectrum76 Spc(X) = {eigenvalues of DX(z) : z ∈ R2 \ Dσ} satisfies Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}.77 We are now ready to state our result.78 Theorem 2. Let X : R2 \ Dσ → R2 be a differentiable vector field (or map), where σ > 0 and79 Dσ = {z ∈ R 2 : ||z|| ≤ σ}. Suppose that X is Hurwitz: Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Then80 (i) There are s ≥ σ and a globally injective local homeomorphism X̃ : R2 → R2 such that X̃81 and X coincide on R2 \ Ds. Moreover, the restriction X| : R 2 \ Ds → R 2 is injective, and82 it admits a global differentiable extension X̂ such that the pair (X̂, s) satisfies the definition83 of the index of X at infinity, and this index I(X) ∈ [−∞,+∞).84 (ii) For all p ∈ R2 \Dσ, there is a unique positive semi-trajectory of X starting at p. Moreover,85 for some v ∈ R2, the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is an attractor86 (respectively a repellor) for the vector field X+v : R2\Ds → R 2 as long as the well-defined87 index I(X) ≥ 0 (respectively I(X) < 0).88 The map X̃ of Theorem 2 is not necessarily a homeomorphism. This X̃ is a topological89 embedding, the image of which may be properly contained in R2. Furthermore, if A : R2 → R290 is an arbitrary invertible linear map, then Theorem 2 applies to the map A ◦ X ◦ A−1.91 Theorem 2 improves the main results of [13, 15]. Item (i) complements the injectivity work92 of [13] (see also [23, 14]), where the authors consider the assumption Spc(X) ∩ (−ε,+∞) = ∅93 (as in Theorem 1). Item (ii) generalizes [15], where the authors utilize the second condition of94 Theorem 1. In our new assumptions, the negative eigenvalues can tend to zero.95 2.1. Description of the proof of Theorem 296 Since the Local Inverse Function Theorem is true, a map X = ( f , g) : R2 \ Dσ → R as in Theorem 2 is a local diffeomorphism. Thus the level curves { f = constant} make up a98 C0−foliation F ( f ) the leaves of which are differentiable curves, and the restriction of the other99 submersion g to each of these leaves is strictly monotone. In particular, F ( f ) and F (g) are100 (topologically) transversal to each other. We orient F ( f ) in agreement that if Lp( f ) is an oriented101 leaf of F ( f ) thought the point p, then the restriction g| : Lp( f ) → R is an increasing function102 in conformity with the orientation of Lp( f ). We denote by L p ( f ) = {z ∈ Lp( f ) : g(z) ≥ g(p)}103 (resp. L+p(g) = {z ∈ Lp(g) : f (z) ≥ f (p)}) and L p( f ) = {z ∈ Lp( f ) : g(z) ≤ g(p)} (resp.104 L−p (g) = {z ∈ Lp(g) : f (z) ≤ f (p)}) the respective positive and negative half-leaf of F ( f ) (resp.105 F (g)). Thus Lq( f ) = L q ( f ) ∪ L q ( f ) and L q ( f ) ∩ L q ( f ) = {q}. In this context, the nonsingular106 vector fields107 X f = (− fy, fx) and X̃g = (gy,−gx), (2.1) given by the partial derivatives are tangent to Lp( f ) and Lp(g), respectively. This construction108 has previously been used in [4].109 We need the following definition [14, 4]. Let h0(x, y) = xy and consider the set110 (x, y) ∈ [0, 2] × [0, 2] : 0 < x + y ≤ 2 Definition 2. Let X = ( f , g) be a differentiable local homeomorphism. Given h ∈ { f , g}, we say112 that A (in the domain of X) is a half-Reeb component for F (h) if there is a homeomorphism113 H : B→ A which is a topological equivalence between F (h)|A and F (h0)|B such that: (Fig. 3)114 • The segment {(x, y) ∈ B : x+ y = 2} is sent by H onto a transversal section for the foliation115 F (h) in the complement of the point H(1, 1); this section is called the compact edge ofA.116 • Both segments {(x, y) ∈ B : x = 0} and {(x, y) ∈ B : y = 0} are sent by H onto full117 half-leaves of F (h). These half-leaves of F (h) are called the non–compact edges ofA.118 Observe thatAmay not be a closed subset of R2, and H does not need to be extended to infinity.119 Section 3 gives new results on the foliations induced by a local diffeomorphism X = ( f , g) :120 2 \ Dσ → R 2 (see also Proposition 1). Theorem 3 implies that the conditions121 Spc(X) ∩ [0,+∞) = ∅ and ‘each half–Reeb component of F ( f ) is bounded ’ (2.2) give the existence of s ≥ σ such that X| R2\Ds can be extended to an injective map X̃ : R2 → R2.122 Section 4 presents some preliminary results on maps such that Spc(X) ∩ [0,+∞) = ∅. Section 5123 concludes with the proof of Theorem 2. The main step is given in Proposition 2, which implies124 that Hurwitz maps satisfy (2.2). Therefore, Theorem 2 is obtained by using this Proposition 2125 and some previous work [13, 15].126 3. Local diffeomorphisms that are injective on unbounded open sets127 Let X = ( f , g) : R2 \ Dσ → R 2 be an orientation preserving local diffeomorphism, that is128 det(DX) > 0. Next subsection gives preparatory results about F ( f ) in order to obtain that X will129 be injective on topological half planes (see Proposition 1). Subsection 3.3 presents a condition130 under which X is injective at infinity, that is outside some compact set.131 3.1. Avoiding tangent points132 Let C ⊂ R2 \ Dσ be a closed curve surrounding the origin. We say that the vector field133 X : R2 \ Dσ → R 2 has contact (resp. tangency with; resp. transversal to; etc) with C at p ∈ C if134 for each small local integral curve of X at p has such property.135 Definition 3. A closed curve C ⊂ R2 \ Dσ is in general–position with F ( f ) if there exists a set136 T ⊂ C, at most finite such that:137 • F ( f ) is transversal to C \ T .138 • F ( f ) has a tangency with C at every point of T .139 • A leaf of F ( f ) can meet tangentially C at most at one point.140 Denote by GP( f , s) the set of all closed curves C ⊂ R2 \ Dσ in general–position with F ( f )141 such that Ds ⊂ D(C). If C ⊂ R 2 \ Dσ is in general–position, we denote by n e(C, f ) (resp.142 ni(C, f )) the number of tangent points of F ( f ) with C, which are external (resp. internal). Here,143 external (resp. internal) means the existence of a small open interval ( p̃, q̃) f ⊂ Lp( f ) such that144 the intersection set ( p̃, q̃) f ∩C = {tangent point} and ( p̃, q̃) f ⊂ R 2 \ D(C) (resp. ( p̃, q̃) f ⊂ D(C)).145 Remark 1. If C ⊂ R2 \ Dσ is in general–position with F ( f ), there exist a, b ∈ C two different146 point such that f (C) = [ f (a), f (b)]. Moreover, a and b are external tangent points because the147 map ( f , g) is an orientation preserving local diffeomorphism. Since C and f (C) are connected and148 C is not contained in any leaf of F ( f ), we conclude that both external tangencies are different.149 Corollary 1 gives important properties of the leafs passing trough a point in Remark 1, if we150 select C ∈ GP( f , σ) with the minimal number of internal tangencies. To this end, the next lemma151 will be needed.152 Lemma 1. Let C ∈ GP( f , σ). Suppose that a leaf Lq( f ) of F ( f ) meets C transversally some-153 where and with an external tangency at a point p ∈ C. Then Lq( f ) contains a closed subinterval154 [p, r] f which meets C exactly at {p, r} (doing it transversally at r) and the following is satisfied:155 (a) If [p, r] is the closed subinterval of C such that Γ = [p, r] ∪ [p, r] f bounds a compact disc156 D(Γ) contained in R2 \D(C), then points of Lq( f ) \ [p, r] f nearby p do not belong to D(Γ).157 (b) Let ( p̃, r̃) and [ p̃, r̃] be subintervals of C satisfying [p, r] ⊂ ( p̃, r̃) ⊂ [ p̃, r̃]. If p̃ and r̃ are158 close enough to p and r, respectively; then we may deform C into C1 ∈ GP( f , σ) in such159 a way that the deformation fixes C \ ( p̃, r̃) and takes [ p̃, r̃] ⊂ C to a closed subinterval160 [ p̃, r̃]1 ⊂ C1 which is close to [p, r] f . Furthermore, the number of generic tangencies of161 F ( f ) with C1 is smaller than that of F ( f ) with C.162 Proof. We refer the reader to [14, Lemma 2].163 Corollary 1. Let C ∈ GP( f , s). Suppose that C minimizes ni(C, f ) and f (C) = [ f (a), f (b)], then164 C ∩ La( f ) = {a} and C ∩ Lb( f ) = {b}.165 Proof. We only consider the case of the point a. Assume by contradiction that the number of166 elements in C ∩ La( f ) is greater than one i.e ♯(C ∩ La( f )) ≥ 2. The last condition of Definition 3167 implies that the intersection of La( f ) with the other point is transversal to C. Then there is a disk168 as in statement (a) of Lemma 1. By using the second part of Lemma 1 we can avoid the external169 tangency a and some internal tangency. This is a contradiction because ni(C, f ) is minimal.170 Remark 2. Corollary 1 remains true if we take any external tangency (not necessarily a and b)171 in a closed curve Cs ∈ GP( f , s) with minimal n i(Cs, f ).172 3.2. Minimal number of internal tangent points173 A oriented leaf of F ( f ) whose distance to Dσ is different from zero has unbounded half–174 leaves. Given any Lp( f ) with unbounded half–leaves, we denote by H +(Lp) and H −(Lp) the two175 components of R2 \ Lp( f ) in order that L p(g) \ {p} be contained in H +(Lp). Therefore, the image176 X(H+(Lp)) is an open connected subset of the semi–plane {x > f (p)} := {(x, y) ∈ R 2 : x > f (p)}.177 Remark 3. If the image X(H+(Lp)) is a vertical convex set, all the level curves { f = c} ⊂ H+(Lp)178 are connected. Thus the restriction X| : H+(Lp) → X(H +(Lp)) is an homeomorphism, and it179 sends every leaf of F ( f )|H+(Lp) over vertical lines. Therefore, it is a topological equivalence180 between two foliations.181 Lemma 2. If H+(Lp) is disjoint from Dσ and the image X(H+(Lp)) is not a vertical convex set,182 then H+(Lp) contains a half-Reeb component of F ( f ).183 Proof. By remark 3, some level set { f = c̃} ⊂ H+(Lp) is disconnected. Therefore, the result is184 obtained directly from [11, Proposition 1].185 Lemma 2 and Remark 3 hold when we consider H−(Lp).186 Lemma 3. Recall that GP( f , s) is the set of all closed curves C ⊂ R2 \ Dσ in general–position187 with F ( f ) such that Ds ⊂ D(C). Let η i : [σ,∞) → N ∪ {0} be the function given by ηi(s) =188 ni(Cs, f ) where Cs ∈ GP( f , s) minimizes the number of internal tangent points with F ( f ). The189 following statements hold:190 (a) The function ηi is nondecreasing.191 (b) If ηi is bounded then, there exist s0 ∈ [σ,∞) such that η i(s) ≤ ηi(s0) for all s ∈ [σ,∞).192 (c) Set f (Cs0 ) = [ f (a), f (b)]. Suppose that F ( f ) has a half-Reeb componentA whose image193 f (A) is disjoint from ( f (a), f (b)), then such s0 ∈ [σ,∞) is not a maximum value of the194 function ηi.195 Proof. As Cs+1 also belongs to GP( f , s) we have that η i(s) ≤ ni(Cs+1, f ). Therefore (a) is true.196 To prove the second part, we introduce the set As = {n ∈ N ∪ {0} : n ≥ n i(Cs, f ) = η i(s)},197 for every s ≥ σ. From this definition it is not difficult to check that: ηi is bounded if and only if198 ∩s≥σAs , ∅. Therefore, the first element of ∩s≥σAs , ∅ is the bound η i(s0) of statement (b). This199 proves the second statement.200 We shall have established (c) if we prove that there is some Cs0+ε with ε > 0 such that201 ηi(s0) + 1 ≤ n i(Cs0+ε, f ). To this end, we select s > s0 large enough for which Cs is enclosing202 D(Cs0 )∪Γwhere Γ is the compact edge ofA. Since, this Cs intersects both leaves Lp( f ) and Lq( f )203 where p and q are the endpoints of Γ, we obtain that ni(Cs, f ) is greater than n i(Cs0 , f ) = η i(s0).204 Therefore, for some ε > 0 there is Cs0+ε such that η i(s0) + 1 ≤ n i(Cs0+ε, f ). This proves (c).205 Proposition 1. Let X = ( f , g) : R2 \Dσ → R2 be a map with det(DX) > 0. Consider Cs and ηi as206 in Lemma 3. If Cs0 satisfies that n i(Cs, f ) ≤ n i(Cs0 , f ) for all s ∈ [σ,∞) and f (Cs0 ) = [ f (a), f (b)].207 Then, for each p ∈ {a, b} at least one of the restrictions of X to H+(Lp) or H −(Lp) is a globally208 injective map, in agrement that the domain of this restriction is in the complement of D(Cs0 ).209 Proof. We only consider the case p = a. Suppose that H+(La) is contained in the complement210 of D(Cs0). From Remark 3 it is sufficient to prove that X(H +(La)) is vertical convex. Suppose211 by contradiction that it is false. Then Lemma 2 implies that there is a half-Reeb component212 A ⊂ H+(La). By using statement (c) of Lemma 3 this s0 is not a maximum value of the function213 ηi. This contradiction with our selection of the circle Cs0 conclude the proof.214 3.3. Extending maps to topological embeddings215 The next theorem implies the injectivity at infinity of a map, and it is obtained by using216 the methods, ideas and arguments of [13]. We only give the proof, in the case of continuously217 differentiable maps.218 Theorem 3. Let X = ( f , g) : R2 \ Dσ → R2 be an differentiable local homeomorphism with219 det(DX) > 0. Suppose that Spc(X)∩ [0,+∞) = ∅, and each half–Reeb component of either F ( f )220 or F (g) is bounded. Then there exist s ≥ σ such that the restriction X| : R2 \ Ds → R 2 can be221 extended to a globally injective local homeomorphism X̃ = ( f̃ , g̃) : R2 → R2.222 Proof. We can apply the results of [16, pp. 166-174] to the continuous vector field X f and obtain223 that for each closed curve C ∈ GP( f , σ) the Index of X f along C, denoted by Ind(X f ; C), satisfies224 Ind(X f ; C) = 2 − ne( f ,C) + ni( f ,C) If X f is discontinuous, we proceed as in [13] by using the index of the foliation F ( f ) which also225 satisfies this formulae.226 (a.1) We claim that ne( f ,C) = ni( f ,C) + 2, for all C ∈ GP( f , σ).227 Suppose that (a.1) is false, so there is C̃1 ∈ GP( f , σ) whose Ind(X f ; C̃ 1) , 0. Thus, for228 some point in C̃1 the Hamiltonian X f (p) = (− fy(p), fx(p)) is vertical. More precisely we can229 obtain p ∈ C̃1 such that fy(p) = 0 and fx(p) > 0. This is a contradiction with the eigenvalue230 assumptions because fx(p) ∈ Spc(X) ∩ (0,+∞). (If X f is discontinuous, we refer the reader to231 [13, Proposition 3.1] where proves that the index of the foliation F ( f ) is zero). Therefore, (a.1)232 holds.233 (a.2) We claim that if Cσ ⊂ R 2 \ Ds minimizes n i( f ,Cσ), then every internal tangency in Cσ234 gives a half–Reeb component.235 For every internal tangency q ∈ Cσ we consider the forward Poincaré map T : [p, q)σ ⊂236 Cσ → Cσ induced by the oriended F ( f ) (if T : (q, r]σ ⊂ Cσ → Cσ the proof is similar) where237 [p, q)σ ⊂ Cσ is the maximal connected domain of definition of T on which this first return map238 is continuous. If the open arc L+p ( f ) \ {p} intersects Cσ we apply Lemma 1, so we can deform239 Cσ in a new circle C̃ 1 ⊂ R2 \ Ds such that the number of internal tangencies of C̃ 1 with F ( f )240 is (strictly) smaller than that of Cσ. This is a contradiction. Therefore L p( f ) \ {p} is disjoint241 from Cσ. By using this and our selection of [p, q)σ ⊂ Cσ is not difficult to check that there is a242 half–Reeb component of F ( f ) whose compact edge is contained in Cs, Thus, we obtain (a.2).243 Notice that, for every circle as in (a.2) any internal tangency of this Cσ gives an unbounded244 half–Reeb component, thus by our assumptions and (a.1) we have that ni( f ,Cσ) = 0. Therefore,245 (a.3) if Cσ ∈ GP( f , σ) is as in (a.2) then n i( f ,Cσ) = 0 and n e( f ,Cσ) = 2. Moreover, F ( f ),246 restricted to R2 \ D(Cσ), is topologically equivalent to the foliation made up by all the247 vertical straight lines, on R2 \ D1.248 Since X has no unbounded half–Reeb component, we can use the last section of [13] (see249 Proposition 5.1) and obtain that the closed curve Cσ of (a.3) can be deformed so that, the resulting250 new circle C has an exterior collar neighborhood U ⊂ R2 \ D(C) such that:251 (b) X(C) is a non-trivial closed curve, X(U) is an exterior collar neighborhood of X(C) and the252 restriction X| : U → X(U) is a homeomorphism.253 By Schoenflies Theorem [2] the map X| : C → X(C) can be extended to a homeomorphism254 X1 : D(C) → D(X(C)). We extend X : R 2 \ D(C) → R2 to X̃ = ( f̃ , g̃) : R2 → R2 by defining255 X̃|D(C) = X1. Thus X̃| : U → X(U) is a homeomorphism and U (resp. X(U)) is a exterior256 collar neighborhood of C (resp. X(C)). Consequently, X̃ is a local homeomorphism and F ( f̃ ) is257 topologically equivalent to the foliation made up by all the vertical straight lines. The injectivity258 of X̃ follows from the fact that F ( f̃ ) in trivial [4, Proposition 1.4]. This concludes the proof.259 Corollary 2. Suppose that X satisfies Theorem 3. Then the respective extension X̃ = ( f̃ , g̃) :260 2 → R2 of X is a globally injective local homeomorphism the foliations of which, F ( f̃ ) and261 F (g̃) have no half-Reeb components.262 Proof. We reefer the reader to affirmation (a.3) in the proof of Theorem 3.263 Corollary 3. Suppose that X = ( f , g) : R2 \ Dσ → R2 is an orientation preserving local264 diffeomorphism. Then the foliation F ( f ) (resp. F (g)) has at most countably many half–Reeb265 components.266 Proof. A half–Reeb component has a tangency with some Cn of Lemma 3 with n ∈ N large267 enough. We conclude, since the closed curves has at most a finite number of tangent points.268 Remark 4. By using a smooth embedding ( f , g) : R2 → R2, the authors of [10, Proposition 1]269 prove the existence of foliations F ( f ) which have infinitely many half-Reeb components.270 4. Maps free of positive eigenvalues271 In this section we present some properties of a map the spectrum of which is disjoint of272 [0,+∞). These results will be used in Section 5 to proving the first part of Theorem 2. In this273 context, we consider F ( f ) and their trajectories Lq = Lq( f ), L q = L q ( f ) and L q = L q ( f ).274 Lemma 4. Let X = ( f , g) : R2 \ Dσ → R2 be a differentiable local homeomorphism. Suppose275 that the spectrum Spc(X) ∩ [0,+∞) = ∅ and p = (a, c). Then the intersection of L+p with the276 vertical ray {(a, y) ∈ R2 : y ≥ c} is the one point set {p}.277 Proof. Assume, by contradiction, that L+p \ {p} intersects {(a, y) ∈ R 2 : y ≥ c}. We take d > c278 the smallest value such that q = (a, d) ∈ L+p (see Figure 1a). We only consider the case in which279 the compact arc [p, q] f ⊂ L p such that Π([p, q] f ) equals the interval [a, a0] with a < a0, where280 Π(x, y) = x (in the other case, Π([p, q] f ) = [a0, a], a0 < a the argument is similar). Therefore, if281 we take the vertical segment [p, q]a = {(a, y) : c ≤ y ≤ d} joint to the open disk D(C) bounded282 by the closed curve C = [p, q] f ∪ [p, q]a. We meet two possible cases:283 PSfrag replacements (a, c)(a, c) (a, d)(a, d) (a0, c0)(a0, c0) Figure a Figure b Figure 1: The first one is that D(C)∩Dσ = ∅. We select the point (a0, c0) ∈ [p, q] f in order that c0 will284 be the smallest value of the compact set Π−1(a0) ∩ [p, q] f . Let R be the closed region bounded285 by the union of {(a, y) : y ≤ c}, {(a0, y) : y ≤ c0} and [p, q0] f ⊂ [p, q] f where q0 = (a0, c0). As286 c0 = inf{y ∈ Π −1(a0) : (a0, y) ∈ [p, q] f } the compact arc [p, q] f is tangent to the vertical line287 −1(a0) at the point (a0, c0). Thus, X f (a0, c0) is vertical, and so fy(a0, c0) = 0. This implies that288 fx(a0, c0) ∈ Spc(X). By the assumptions about Spc(X), fx(a0, c0) < 0 which in turn implies that289 the arc [q0, q] f ⊂ [p, q] f must enter into R and cannot cross the boundary of R (see Figure 1b).290 This contradicts the fact that q = (a, d) < R.291 The second case happens when D(C) ∩ Dσ , ∅. As [p, q] f is not contained in Dσ, either292 σ < a0 or σ = a0. If σ < a0, the vertical line x = σ meet [p, q] f in two different points293 which define a closed curve as in Figure 1a such that it bounds an open disk disjoint of Dσ. We294 conclude by using the proof of the first case. If σ = a0, we observe the continuous foliation295 F ( f ) in a neighborhood of [p, q] f and meet two points p̃ = (σ, c̃) and q̃ = (σ, d̃) with c̃ < d̃,296 but Π([ p̃, q̃] f ) ⊂ [σ,+∞). It satisfies the conditions of the first case. Therefore the lemma is297 proved.298 Remark 5. Lemma 4 remains true, if we consider the negative leaf L−q starting at q = (a, d) joint299 to the vertical ray {(a, y) ∈ R2 : y ≤ d}.300 Lemma 5. Let X = ( f , g) : R2 \ Dσ → R2 be a map with Spc(X) ∩ [0,+∞) = ∅. Consider301 L+p ⊂ { f = f (p)} and the projection Π(x, y) = x. If the oriented compact arc [p, q] f ⊂ L p and its302 image Π([p, q] f ) is the interval [Π(p),Π(q)] ⊂ (σ,+∞) with Π(p) < Π(q). Then303 [p,q] f 〈X,∇ f 〉dt ≥ f (p) [p,q] f fx dt + g(p) Π(q) − Π(p) where 〈 , 〉 denotes the usual inner product on the plane, and fx is the first partial derivative.304 Proof. For each α ∈ Π([p, q] f ) = [a, b], the vertical line Π −1(α) intersects [p, q] f in a non-empty305 compact set. So there exist ŷα = sup{y ∈ R : (y, α) ∈ [p, q] f } and mα = (α, ŷα). We also define306 S ⊂ (a, b) as the set of critical values of the projection Π(x, y) = x restricted to the differentiable307 PSfrag replacements aa bb (α, c) [Π(p),Π(q)]c Figure a Figure b Figure 2: Here ∇ f (z), X f (z) a positive basis arc [p, q] f . By the Sard’s Theorem, presented in [17, Theorem 3.3] this set S is closed and has308 zero Lebesgue measure. For α ∈ (a, b)\S the setΠ−1(α) intersects [p, q] f in at most finitely many309 points and the complement set −1(α)∩[p, q] f \{mα} is either empty or its cardinality is odd. By310 Lemma 4, the order of these points in the line Π−1(α) oriented oppositely to the y−axis coincides311 with that on the oriented arc [p, q] f (a behavior as in Figure 1a does not exist). Therefore, for312 every α ∈ (a, b) \ S the set −1(α) ∩ [p, q] f \ {mα} splits into pairs {pα = (α, dα), qα = (α, cα)}313 with the following three properties: (a.1) cα < dα, (a.2) the compact arc [pα, qα] f lies in the314 semi-plane {x ≤ α} and it is oriented from pα to qα (a.3) g(qα) > g(pα). Notice that the tangent315 vector of [p, q] f at pα has a negative x−component: fy(pα) > 0, and the respective tangent vector316 at qα satisfies fy(qα) < 0. Similarly, fy(mα) ≤ 0 (see (2.1)).317 Assertion Take c < inf{y : (α, y) ∈ [p, q] f } and [Π(p),Π(q)]c = {(α, c) : a ≤ α ≤ b}, an318 horizontal segment. Consider the compact set D(C) ⊂ {(x, y) : a ≤ x ≤ b} the boundary319 of which contain [p, q] f ∪ [Π(p),Π(q)]c (see Figure 2b). Suppose that C, the boundary of320 D(C) is negatively oriented (clock wise). Then321 gy(x, y)dx ∧ dy = [p,q] f 〈F,∇ f 〉dt − g(α, c)dα, where F(x, y) = f (x, y) − f (p), g(x, y) Proof of Assertion We will use the Green’s formulae given in [21, Corollary 5.7] (see also [20])323 with the differentiable map G : z 7→ (0, g(z)) and the outer normal vector of C denoted by324 z 7→ η(z) (unitary). By using that Trace(DGz) = gy(z) it follows that325 gy(x, y)dx ∧ dy = 〈G,−η〉ds, (4.1) where ds denotes the arc length element. If [Π(q),Π(p)]c denote [Π(p),Π(q)]c oriented from326 Π(q) to Π(p), then C = [p, q] f ∪ B ∪ [Π(q),Π(p)]c ∪ A with A and B two oriented vertical327 segments. Consequently,328 〈G,−η〉ds = [p,q] f 〈G,−η〉ds + 〈G,−η〉ds + [Π(q),Π(p)]c 〈G,−η〉ds + 〈G,−η〉ds. In A∪B the vector −η is horizontal i.e η(z) = (η1(z), 0) then G = (0, g) implies that 〈G,−η〉ds =329 ∫ 〈G,−η〉ds = 0. Therefore330 〈G,−η〉ds = [p,q] f 〈G,−η〉ds + [Π(q),Π(p)]c 〈G,−η〉ds. (4.2) In [p, q] f , the outer normal vector es parallel to −∇ f (z) = − fx(z), fy(z) . Then, for all z ∈ [p, q] f331 we obtain that −η(z) = ∇ f (z) ||∇ f (z)|| and G(z) = F(z) because [p, q] f ⊂ { f = f (p)}. Thus332 [p,q] f 〈G,−η〉ds = [p,q] f 〈F,∇ f 〉dt. Similarly, in [Π(q),Π(p)]c we have −η(z) = (0, 1) thus333 [Π(q),Π(p)]c 〈G,−η〉ds = g(α, c)dα = − g(α, c)dα. Therefore, (4.2) and (4.1) prove the assertion. �334 In order to conclude the proof of this lemma we consider D(C) as in the last assertion joint335 to the construction of its precedent paragraph. Since the complement of S is a total measure set,336 gy(x, y)dx ∧ dy = g(mα) − g(α, c) Π(pα)=α g(qα) − g(pα) Thus, the formulae in the assertion implies that337 [p,q] f 〈F,∇ f 〉dt = g(mα)dα + Π(pα)=α g(qα) − g(pα) dα. (4.3) But,338 ∫ [p,q] f 〈F,∇ f 〉dt = [p,q] f 〈X,∇ f 〉dt − f (p) [p,q] f and the property (a.3) of the precedent paragraph to Assertion 1 shows that g(qα) − g(pα) ≥ 0.339 Therefore, since g(mα) ≥ g(p), (4.3) implies that340 [p,q] f 〈X,∇ f 〉dt − f (p) [p,q] f fxdt ≥ g(mα)dα ≥ g(p) b − a and concludes this proof.341 By applying the methods of the last proof give us the next:342 Lemma 6. Let X = ( f , g) : R2 \ Dσ → R2 be a map with Spc(X) ∩ [0,+∞) = ∅. Consider343 L−q ⊂ { f = f (q)} and Π(x, y) = x. If the oriented compact arc [p, q] f ⊂ L q , and Π([p, q] f ) =344 [Π(q),Π(p)] ⊂ (σ,+∞) with Π(q) < Π(p). Then345 [p,q] f 〈X,∇ f 〉dt ≥ f (q) [p,q] f fxdt + g(p) Π(p) − Π(q) Proof. As a = Π(p) < Π(q) = b, we again consider the null set S ⊂ Π([p, q] f ) given by the346 critical values of Π restricted to [p, q] f (see [17]). Similarly, for every α ∈ Π([p, q] f ) = [a, b] we347 define ŷα = sup{y ∈ R : (y, α) ∈ [p, q] f } and mα = (α, ŷα). Therefore, Remark 5 shows that for348 each α < S the finite set −1(α) ∩ [p, q] f \ {m̃α} splits into pairs {pα = (α, dα), qα = (α, cα)}349 satisfying: (i) cα < dα, (ii) the oriented arc [pα, qα] f ⊂ {x ≥ α}, and (iii) g(qα) > g(pα).350 Take D(C) the boundary of which is the closed curve C ⊂ Π−1(b) ∪ [p, q] f ∪ Π −1(a) ∪351 [Π(p),Π(q)]c, where [Π(p),Π(q)]c = {(α, c) : a ≤ α ≤ b} for some c < inf{y : (x, y) ∈ [p, q] f }.352 By using the Green’s formulae with the map z 7→ (0, g(z)) and the compact disk D(C) we have353 that354 ∫ gy(x, y)dx ∧ dy = [p,q] f 〈F̃,∇ f 〉dt − g(α, c)dα, where F̃(x, y) = f (x, y) − f (q), g(x, y) . Since355 gy(x, y)dx ∧ dy + g(α, c)dα = g(mα)dα + Π(pα)=α g(qα) − g(pα) and356 ∫ [p,q] f 〈F̃,∇ f 〉dt = [p,q] f 〈X,∇ f 〉dt − f (q) [p,q] f fxdt, the last property (iii) implies357 [p,q] f 〈X,∇ f 〉dt − f (q) [p,q] f fxdt ≥ g(mα)dα. This concludes the proof because g(mα) ≥ g(p) shows g(mα)dα ≥ g(p)(b − a).358 5. Hurwitz vector fields359 This section concludes with the proof of the main theorem. The essential goal of the next360 proposition is to prove the fact that our eigenvalue assumption ensures the non-existence of361 unbounded half–Reeb components. It is obtained by using the preparatory results of the previous362 section. With this fact Theorem 2 is just obtained by applying our previous papers [13, 15].363 Proposition 2 (Main). Let X = ( f , g) : R2 \ Dσ → R2 be a differentiable map, where σ > 0 and364 Dσ = {z ∈ R 2 : ||z|| ≤ σ}. Suppose that X is Hurwitz: Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Then365 (i) Any half-Reeb component of either F ( f ) or F (g) is a bounded subset of R2.366 (ii) There are s ≥ σ and a globally injective local homeomorphism X̃ = ( f̃ , g̃) : R2 → R2367 such that X̃ and X coincide on R2 \ Ds. Moreover, F ( f ) and F (g) have no half-Reeb368 components.369 Proof. In the proof of (i), we only consider one case. Suppose by contradiction that the foliation,370 given by the level curves has an unbounded half–Reeb component. By [4, Proposition 1.5], there371 exists a half–Reeb component of F ( f ) the projection of which is an interval of infinite length.372 Thus,373 PSfrag replacements ∇ f (z) γ(0) ⊂ { f = 0} γ(1) ⊂ { f = 0} mα ∈ Γ α[γ(s), γ(ϕ(s))] f psα = (asα , dsα) ∈ {g = g(γ(0))} qsα = (asα , csα) Figure 3: Half-Reeb componet A, proof of Proposition 2 (a) there are â0 > σ and A, a half–Reeb component of F ( f ) such that [â0,+∞) ⊂ Π(A),374 and the vertical line Π−1(â0) intersects (transversally) both non–compact edges ofA. Here375 Π(x, y) = x.376 A half-Reeb component of a fixed foliation contain properly other half-Reeb components,377 and they are topologically equivalent. In this context, the component is stable under perturbations378 on their compact face as long as the perturbed arc is also a compact face of some component.379 Therefore, without lost of generality, we may assume that nearby its endpoints, the compact edge380 of is made up of arcs of F (g). In this way, there exist 0 < ã < 12 and an injective, continuous381 curve γ : (−ã, 1 + ã)→ A such that382 (b.1) γ([0, 1]) is a compact edge of A such that both non–compact edges L+ γ(0) and L γ(1) are383 contained in a half–plane {x ≥ â}, for some â > σ.384 (b.2) The images γ (−ã, ã) and γ (1− ã, 1+ ã) are contained in some leaves of F (g) such that385 γ(1 − ã, 1 + ã) < infΠ γ(−ã, ã) (b.3) For some 0 < δ < ã4 there exists a orientation reversing injective function ϕ : [−2δ, 2δ]→387 (1− ã, 1+ ã) with ϕ(0) = 1 such that f γ(ϕ(s)) . Furthermore, if s ∈ (0, 2δ] then388 ϕ(s) ∈ (1− ã, 1) and there exists an oriented compact arc of trajectory [γ(s), γ(ϕ(s))] f ⊂ A389 of F ( f ), connecting γ(s) with γ(ϕ(s)).390 (b.4) For some 0 < δ < ã4 , small enough if s ∈ (0, δ] and γ(s) = (as, ds) then there exists cs such391 that qs = (as, cs) belongs to the open arc γ(s), γ(ϕ(s)) f ⊂ [γ(s), γ(ϕ(s))] f .392 Lemma 4 implies that393 (c) for every s ∈ (0, δ] as in (b.4), cs < ds.394 The eigenvalue condition is invariant under addition of constant vectors to maps, therefore395 we can assume that396 (d) g > 0, f = 0 over both non-compact edges ofA and f > 0 in the interior ofA.397 SinceA is the union of an increasing sequence of compact sets bounded by the compact edge398 and a compact segment of leaf. Then, from our selection of the compact edge we have that for399 every α > Π ≥ inf{as : s ∈ (0, δ]}, large enough there exists sα ∈ (0, δ] as in (b.4) such that400 the compact arc [γ(sα), γ(ϕ(sα))] f projects over (−∞, α], meeting α (Figure 3). More precisely,401 α = sup Π(p) : p ∈ [γ(sα), γ(ϕ(sα))] f This defines a closed curve Γ−α contained in Π −1(asα) ∪ [γ(sα), γ(ϕ(sα))] f . If [qsα , psα]asα ⊂402 −1(asα) is the vertical segment connecting qsα , of (b.4) with psα = γ(sα), then this clock wise403 oriented curve satisfies404 α = [qsα , psα]asα ∪ [psα , qsα] f and Π(Γ α) = [asα , α], (5.1) where the oriented compact arc [psα , qsα] f ⊂ [γ(sα), γ(ϕ(sα))] f .405 We select and fix mα ∈ Γ −1(α), by using Lemma 5 and Lemma 6, respectively we obtain406 [psα ,mα] f 〈X,∇ f 〉dt ≥ f (psα) [psα ,mα] f fxdt + g(psα)(x − asα), and407 ∫ [mα ,qsα ] f 〈X,∇ f 〉dt ≥ f (qsα) [mα ,qsα ] f fxdt + g(mα)(x − asα). Therefore, by adding we conclude408 [psα ,qsα ] f 〈X,∇ f 〉dt ≥ f (psα ) [psα ,qsα ] f fxdt + g(psα)(x − asα) (5.2) because g in increasing along [psα , qsα] f ⊂ { f = f (psα)} and g(mα) ≥ g(psα) = g(γ(0)) > 0.409 (e.1) We claim that, the closed curves Γ−α , given in (5.1) define the following functions410 ∣∣∣∣∣∣∣ [qsα ,psα ]asα ∣∣∣∣∣∣∣ and α 7→ ∣∣∣∣∣∣ [qsα ,psα ] f ∣∣∣∣∣∣ , they are bounded, when α varies in some interval of infinite length contained on [â0,∞).411 Furthermore,412 f (psα ) [qsα ,psα ] f fx = 0. In fact, by a perturbation in the compact face if it is necessary, it is not difficult to prove that413 there is some half-Reeb component à of F ( f ) such that à ⊃ A, their boundaries satisfy ∂à \414 {compact face} ⊃ ∂A \ {compact face} and à ⊃ [qsα , psα]sα , for all sα. Since the image f (Ã) ⊂415 f (compac face), the function f is bounded in the closure of Ã, which contain the compact set416 ∪sα [qsα , psα]asα . Consequently,417 ∣∣∣∣∣∣∣ [qsα ,psα ]asα ∣∣∣∣∣∣∣ is bounded, in some interval of infinite length contained on [â0,∞). In order to prove the second418 part, we apply the Green’s formulae to the map z 7→ (1, 0) in the compact disk D(Γ−α). Since the419 trace is cero, we obtain420 ∣∣∣∣∣∣ [qsα ,psα ] f ∣∣∣∣∣∣ = arc length of [qsα , psα]asx . By compactness we conclude. The last part is directly obtained by using (5.1) and psα = γ(sα)421 because the continuity of the foliation F ( f ) and (d) imply that422 psα = γ(0) ∈ { f = 0}. Therefore (e.1) holds.423 (e.2) We claim that,424 if α→ +∞ then 〈X, ηiα〉dt→ +∞, where −ηiα is a outer normal vector of the close wise oriented curve Γ α .425 In the compact arc [psα , qsα] f ⊂ Γ α , the vector η α is parallel to ∇ f . Thus (5.2) and (e.1) imply426 that427 ∮ 〈X, ηiα〉dt ≥ A + g(psα)(α − a), where the constant A is independent of α and a = min{as : s ∈ (0, δ]}. Since (d) and (b.2) imply428 that g(psα) = g(γ(0)) > 0, it is not difficult to obtain (e.2).429 By (e.2) we select some α̃ > a such that Γ−α satisfies 〈X, ηiα〉dt > 0. By using the Green’s430 formulae with the map X = ( f , g), the assumptions over the eigenvalues i.e 0 > Trace(DXz),431 imply that432 〈X, ηiα〉dt > 0. This contradiction concludes the proof of part (i).433 In order to obtain (ii), we apply Theorem 3 because X satisfies its conditions. This gives the434 existence of the pair (X̃, s) with s ≥ σ and X̃ = ( f̃ , g̃) : R2 → R2 a globally injective local435 homeomorphism such that X̃ and X coincide on R2 \ Ds. Furthermore, the last property in (ii) is436 obtained as a direct application of Corollary 2. Therefore, this proposition holds.437 Now we prove our main result438 5.1. Proof of Theorem 2439 By Proposition 2, there exists a globally injective local homeomorphism X̃ : R2 → R2440 such that X̃ and X coincide on some R2 \ Ds1 , with s1 ≥ σ. In particular, the restriction X| :441 2 \ Ds1 → R 2 is injective. In order to shown the existence of the differentiable extension,442 consider v = −X̃(0) joint to the globally injective map X̃ + v. In this context, we can apply the443 arguments of [15, Theorem 11]. Thus there are s̃ > s1 ≥ σ and a global differentiable vector444 field Y : R2 → R2 such that445 (a.1) R2 \ Ds̃ ∋ z 7→ Y(z) = (X + v)(z) is also injective and Y(0) = 0.446 (a.2) The map z 7→ Trace(DYz) is Lebesgue almost–integrable in whole R 2 ([15, Lemma 7]).447 (a.3) The index I(X + v) is a well-defined number in [−∞,+∞) ([15, Corollary 13]).448 Thus, there exist the index of X at infinity, I(X) = I(X + v). Therefore, X̂ = Y − v is the global449 differentiable extension of X| : R2 \ Ds̃ → R 2 and the pair (X̂, s̃) satisfies the definition of the450 index of X at infinity. This concludes the proof of (i).451 To prove the first part of (ii) we refer the reader to [4, Lemma 3.3]. Furthermore, since452 Trace(D(X + ṽ)) = Trace(DX) < 0, for every constant vector ṽ ∈ R2 we obtain that453 (b.1) Given a constant ṽ ∈ R2, the vector field X + ṽ generates a positive semi-flow on R2 \ Dσ.454 An immediate consequence of (i) is that: if X is Hurwitz, then outside a larger disk both X and455 Y have no rest points. In addition, by (a.1) the Hurwitz vector field Y has no periodic trajectory γ456 with D(γ) contained in R2 \Dσ. As Trace(DY) = Trace(DX) < 0 by Green’s Formulae Y admits457 at most one periodic trajectory, say γ, such that D(γ) ⊃ Dσ. Consequently458 (b.2) There exit s > s̃ such that Y satisfies (a.1), (a.2) and R2 \ Ds is free of rest points and459 periodic trajectories of Y.460 Under these conditions (b.1) and [15, Theorem 26] imply that:461 (b.3) For every r ≥ s there exist a closed curve Cr transversal to Y contained in the regular set462 2 \Dr. In particular, D(Cr) contains Dr and Cr has transversal contact to each small local463 integral curve of Y at any p ∈ Cr.464 Moreover, [15, Theorem 28] shown that:465 (b.4) The point at infinity of the Riemann Sphere R2 ∪ {∞} is either an attractor or a repellor of466 X + v : R2 \ Ds → R 2. More specifically, if I(X) < 0 (respectively I(X) ≥ 0), then∞ is a467 repellor (respectively an attractor) of the vector field X + v.468 Therefore, (ii) holds and concludes the proof of Theorem 2. �469 References470 [1] B. Alarcón; V. Guı́ñez and C. Gutierrez: Hopf bifurcation at infinity for planar vector fields Discrete Contin. Dyn.471 Syst. 17 (2007) 247–58472 [2] R. H. Bing: “The geometric topology of 3-manifolds”Amer. Math. Soc. Colloq. Publ. 40 Providence, 1983.473 [3] C. Chicone: “Ordinary differential equations with applications.”Second edition. Texts in Applied Mathematics,474 34. Springer, New York, 2006.475 [4] A. Fernandes; C. Gutierrez; R. Rabanal: Global asymptotic stability for differentiable vector fields of R2, J. of476 Differential Equations 206 (2004) 470–482.477 [5] R. Feßler: A proof of the two dimensional Markus-Yamabe stability conjecture and a generalization, Ann. Polon.478 Math. 62 (1995) 45–74.479 [6] F. Dumortier; P. De Maesschalck: Topics on singularities and bifurcations of vector fields. Normal forms, bifurca-480 tions and finiteness problems in differential equations nato Sci. Ser. II Math. Phys. Chem. 137 Kluwer Acad. Publ.481 Dordrecht, 2004.482 [7] F. Dumortier; R, Roussarie; J. Sotomayor; H. Żoladek: Bifurcations of planar vector fields. Nilpotent singularities483 and Abelian integrals Lecture Notes in Math. 1480 Springer-Verlag, Berlin, 1991.484 [8] A. A. Glutsyuk: Asymptotic stability of linearizations of a planar vector field with a singular point implies global485 stability, Funct. Anal. Appl. 29 (1995) 238–247.486 [9] C. Gutierrez: A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal.487 Non Linéaire 12 (1995) 627–671.488 [10] C. Gutierrez; X. Jarque; J. Llibre; M. A. Teixeira: Global Injectivity of C1 maps of the real plane, inseparable489 leaves and the Palais–Smale condition. Canad. Math. Bull. 50 (2007) 377–389.490 [11] C. Gutierrez; N. Van Chau: A remark on an eigenvalue condition for the global injectivity of differentiable maps491 of R2, Discrete Contin. Dyn. Syst. 17 (2007) 397–402.492 [12] C. Gutierrez; M. A. Teixeira: Asymptotic stability at infinity of planar vector fields Bull. Braz. Math. Soc. (N.S.)493 26 (1995) 57–66494 [13] C. Gutiérrez; R. Rabanal: Injectivity of differentiable maps R2 → R2 at infinity. Bull. Braz. Math. Soc. (N.S.) 37495 (2006) 217–239.496 [14] C. Gutierrez; A. Sarmiento: Injectivity of C1 maps R2 → R2 at infintiy and planar vector fields, Asterisque, 287,497 (2003) 89–102.498 [15] C. Gutiérrez; B. Pires; R. Rabanal: Asymototic stability at infinity for differentiable vector fields of the plane J.499 Differential Equations 231 (2006) 165–81500 [16] P. Hartman: “Ordinary differential equations”. Second edition, reprinted. Classics Appl. Math. 38 siam. 2001.501 [17] C. G. T. de A Moreira: Hausdorff measures and the Morse-Sard theorem. Publ. Mat. 45 (2001) 149–62502 [18] C. Olech: On the global stability of an autonomous system on the plane Contributions to Differential Equations 1503 (1963) 389–400504 [19] C. Olech: Global phase-portrait of plane autonomous system Ann. Inst. Fourier (Grenoble) 14 (1964) 87–97505 [20] W. F. Pfeffer: “Derivation and integration ”Cambridge Tracts in Math. 140 Cambridge Univ. Press, Cambridge,506 2001507 [21] W. F. Pfeffer: The multidimensional fundamental theorem of calculus Jur. Austral. Math. Soc. (Series A) 43 (1987)508 143–170509 [22] B. Pires; R. Rabanal: Vector fields whose linearisation is Hurwitz almost everywhere Proc. Amer. Math. Soc. In510 press.511 [23] R. Rabanal: An eigenvalue condition for the injectivity and asymptotic stability at infinity Qual. Theory Dyn. Syst.512 6 (2005) 233–250.513 [24] R. Rabanal: Erratum to: ‘An eigenvalue condition for the injectivity and asymptotic stability at infinity’ Qual.514 Theory Dyn. Syst. 7 (2009) 367–368.515 [25] R. Rabanal: Center type performance of differentiable vector fields in the plane. Proc. Amer. Math. Soc. 137516 (2009) 653–662.517 [26] R. Rabanal: On differentiable area-preserving maps of the plane. Bull. Braz. Math. Soc. (N.S.) 41 (2010) 73–82.518 [27] R. Roussarie: “Bifurcation of planar vector fields and Hilbert’s sixteenth problem”Progr. Math. 164 Birkhäuser519 Verlag, Basel, 1998.520 [28] F. Takens: Singularities of vector fields Inst. Hautes Études Sci. Publ. Math. 43 (1974) 47–100521 1 Introducion 2 Statements of the results 2.1 Description of the proof of Theorem 2 3 Local diffeomorphisms that are injective on unbounded open sets 3.1 Avoiding tangent points 3.2 Minimal number of internal tangent points 3.3 Extending maps to topological embeddings 4 Maps free of positive eigenvalues 5 Hurwitz vector fields 5.1 Proof of Theorem 2

0704.1419

Quantitative LEED I-V and ab initio study of the Si(111)-3x2-Sm surface structure and the missing half order spots in the 3x1 diffraction pattern

Quantitative LEED I-V and ab initio study of the Si(111)-3x2-Sm surface structure and the missing half order spots in the 3×1 diffraction pattern. C. Eames, M. I. J. Probert, S. P. Tear∗ Department of Physics, University of York, York YO10 5DD, United Kingdom We have used Low Energy Electron Diffraction (LEED) I-V analysis and ab initio calculations to quantitatively determine the honeycomb chain model structure for the Si(111)-3×2-Sm surface. This structure and a similar 3×1 recontruction have been observed for many Alkali-Earth and Rare-Earth metals on the Si(111) surface. Our ab initio calculations show that there are two almost degenerate sites for the Sm atom in the unit cell and the LEED I-V analysis reveals that an admixture of the two in a ratio that slightly favours the site with the lower energy is the best match to experiment. We show that the I-V curves are insensitive to the presence of the Sm atom and that this results in a very low intensity for the half order spots which might explain the appearance of a 3×1 LEED pattern produced by all of the structures with a 3×2 unit cell. PACS numbers: 61.46.-w, 61.14.Hg, 68.43.Bc I. INTRODUCTION The prospect of creating an ordered one dimensional sys- tem has lead to the extensive study of chain structures grown on surfaces. The alkali metals (AM) form such a chain structure as part of a 3×1 reconstruction on the Si(111) surface with an AM coverage of 1/3 ML (Ref. [1, 2] and therein). At a coverage of 1/6 ML the alkali earth metals (AEM) and the rare earth metals (REM) form a 3×2 reconstruction (Ref. [3, 4, 5, 6] and therein). There is a wealth of experimental evidence from STM, LEED and spectroscopic techniques to suggest that in these 3× structures there is a common structure for the reconstructed silicon (Ref. [3, 4, 5, 6, 7, 8, 9, 10, 11] and therein). The honeycomb-chain channel model (HCC) is now regarded by many as the most plausible of the can- didate structures [12, 13, 14]. In the HCC model there are parallel ordered one dimensional lines of metal atoms sited in a silicon free channel. These are separated by almost flat honeycomb layers of silicon. The 3×1 system has been studied using LEED I-V analysis with Ag, Li and Na as the deposited metal atoms [11]. Similar I-V curves were obtained in each case and the authors conclude that a common reconstruction of silicon atoms is responsible for the LEED I-V curves, which are insensitive to the presence of the metal atom. However, the authors did not attempt a structural fit. The LEED pattern for the 3×2 surfaces exhibits odd behaviour in that it indicates a 3×1 periodicity. Many workers have suggested that disorder in the position of the metal atom is the cause. A Fourier analysis of a ran- dom tesselation of a large sample of registry shifted 3×2 unit cells has been carried out by Schäfer et al. [15]. They show that this simulation of long range disorder in the position of the metal atom does produce a 3×1 period- icity in reciprocal space. Alternatively, Over et al. [16] have suggested that the substrate and silicon adatoms could be acting as the dominant scattering unit with the metal atoms sitting in ‘open sites’. STM investigations of the 3×2 and 3×1 systems have not provided much evidence of long range disorder in the location of the metal atom apart from registry shifts in- troduced by a coexisting c(6×2) phase. Of particular relevance to this work is the study of the Si(111)-3×2- Sm system using STM and an ab initio calculation, car- ried out by Palmino et al. [5]. They have used the bias voltage dependence of the STM images of the surface to isolate the features associated with the honeycomb chain and the samarium atom and separate comparison of these with simulated STM images obtained from the ab initio calculation show that the HCC structure is in good lat- eral qualitative agreement with experiment. In this study we have used LEED I-V analysis and sev- eral ab initio calculations to quantitatively investigate the 3×2 reconstruction of the Si(111)-3×2-Sm surface. We show that the HCC structure gives good agreement with experiment. We consider two HCC unit cells in which the samarium atom is located in the T4 site or the H3 site. Palmino et al. [5] have found the energy differ- ence of these two configurations to be 0.07 eV/Sm. We have calculated the atomic positions and the energies of these two reconstructions and obtained LEED I-V curves for this system and we show that a linear combination of the two HCC structures is the optimum match to experi- ment with a ratio that slightly favours the structure with the lower energy of the two. We have also used LEED I-V analysis to investigate the missing half order spots for the 3×2 unit cell. We show using calculated I-V curves that for an individual unit cell the intensity of the half order spots is significantly lower than that of the spots that are visible in the experiments. We also show that the calculated I-V curves do not differ significantly if the samarium atom is not present. We offer this as evidence that disorder over multiple unit cells is not needed to explain the STM/LEED discrepancy for the 3×2 systems and we suggest that the order in the one dimensional chain may persist over large length scales. http://arxiv.org/abs/0704.1419v1 II. EXPERIMENTAL A dedicated LEED chamber of in-house design [17] op- erating at a typical UHV base pressure of ∼ 10−10 mbar was used to carry out our experiments. The silicon sub- strate was cleaned by flashing to ≈ 1200 ◦C using an electron beam heater and then the sample was slowly cooled through the ≈ 900 − 700 ◦C region over a period of 15 minutes. A sharp 7×7 LEED pattern resulted, con- firming that a clean Si(111)-7×7 surface had been made. Temperatures were monitored using an infra-red pyrom- eter. In the literature other workers [5, 9, 18] have formed the Si(111)-3×2-Sm structure by depositing 1/6 ML onto a sample held at a temperature of 400− 850 ◦C followed by annealing at this temperature for 20 min. In this work the sample was prepared by depositing 1 ML of Sm from a quartz crystal calibrated evaporation source onto the clean Si(111)-7×7 surface which was not at el- evated temperature. This was followed by a hot anneal at ≈ 700 ◦C for 15 minutes. A sharp 3×1 LEED pat- tern was observed and images of this are shown in figure 1. Other workers have observed some streaking in the 3×1 LEED pattern that is indicative of one dimensional disorder. We have not observed such streaking in our diffraction patterns and we attribute this to our prepara- tion procedure. There is some variability in the annealing temperature that can be used and temperatures in the range ≈ 700−900 ◦C all gave a sharp diffraction pattern. It is at around 1000 ◦C that the pattern begins to degrade as the samarium desorbs. FIG. 1: Experimental 3×1 LEED spot pattern for the Si(111)- 3×2-Sm surface shown at (a) 40 eV and (b) 80 eV. Images of the diffraction pattern were acquired over a 40-250 eV range of primary electron energies in steps of 2 eV using a CCD camera and stored on an instrument dedicated computer. For each spot in the LEED pattern the variation in its intensity with primary electron energy was recorded which resulted in a set of 42 I-V curves. Degenerate beams were averaged together to reduce the signal to noise ratio and also to reduce any small er- rors that may have occurred in setting up normal beam incidence. Figure 2 defines the spot labelling system and the degenerate beams. The experiment was repeated sev- eral times and the I-V curves obtained during different experiments were compared using the Pendry R-factor [19]. The R-factor for I-V curves obtained on different days was typically 0.1 or less which indicates that the surface is repeatedly preparable. To further reduce noise the I-V curves from separate experiments were averaged together and a three point smooth was applied. FIG. 2: Labelled spots in the 3×1 LEED pattern produced by the Si(111)-3×2-Sm surface as it appears at 40 eV. The degeneracies of the spots are indicated by the pattern used to fill each spot. This set of 13 averaged I-V curves was used to finger- print the surface structure and allow comparison with the I-V curves calculated for the various trial structures. III. AB INITIO CALCULATIONS Ab initio calculations were performed using the CASTEP code [20]. The code was run on 30 processors in a parallel computing environment at the HPCx High Performance Computing facility located at the CCLRC Daresbury lab- oratory in the UK. We have geometry optimised two dif- ferent unit cells for the HCC structure (see figure 5 for details of these). In the first unit cell the samarium atom is located in the T4 site with respect to the first bulk- like silicon layer and in the other unit cell the samarium atom is situated in a H3 site. We will refer to the two structures as ‘T4’ and ‘H3’. The initial atomic positions were those that were obtained in the ab initio study by Palmino et al. [5] and these were very kindly provided by F. Palmino. Before proceeding the input parameters in the calcula- tion were carefully checked (see [21] for a discussion of the importance of this). Figure 3 shows how the calculated energy varies with the number of plane waves included in the calculation as the cutoff energy is raised for three increasingly dense Monkhorst-Pack [22] reciprocal space sampling grids. -5766.5 -5766 -5765.5 -5765 -5764.5 -5764 280 320 360 400 Cutoff Energy (eV) FIG. 3: Variation of the singlepoint energy, which is the calcu- lated energy for a given configuration of the atomic positions, with the cutoff energy and with the number of k-points at which the wavefunction is sampled in reciprocal space. A cutoff energy of 380 eV yields a total energy that is unambiguously in the variational minimum and will allow accurate calculation of the energy and the forces within the system. We have used the sampling grid with 3 k- points in reciprocal space since an increase to 6 k-points does not significantly change the energy. The Perdew- Burke-Ernzerhof [23] generalised gradient approximation was used to represent exchange and correlation effects. The vacuum gap that was used to prevent interaction between the top surface in one supercell and the bottom surface in the supercell above was 9 Å thick and this has been optimised during the course of other ab initio stud- ies of RE silicides that we have done. We have included two bulk-like silicon layers below the top layer that con- tains the samarium atom and the honeycomb chain struc- ture. To prevent interactions through the supercell be- tween uncompensated charge in the the top and bottom layers and to fully replicate the transition to the bulk silicon crystal we have hydrogen passivated the deepest bulk-like silicon layer and fixed the coordinates of these atoms so that they are not free to move from their bulk positions. We have repeated the geometry optimisation of the unit cells without passivation and positional con- straints and the final positions of the silicon atoms in this bottom layer are not drastically altered and the to- tal energy does not significantly change as a result which suggests that using so few bulk-like layers is reasonable. We nevertheless kept the hydrogen passivation in place since it reduces the computational cost of the electronic structure calculation by the quenching of dangling bonds on the underside of the supercell. The structures were allowed to relax until the forces were below the predefined tolerance of 5 × 10−2 eV/Å. Figure 4 shows the convergence of the total energy and the maximum force on any atom as the geometry opti- misation proceeds for the two structures. The T4 structure is 0.7 eV (0.01%) lower in energy than the H3 structure and the maximum force in the sys- tem is slightly lower. This energy difference cannot be quantitatively compared with the value of 0.07 eV/Sm that was obtained in Ref. [5] since this is an atomically resolved energy difference whereas the value presented here compares the total energies of the two supercells with contributions from all of the atoms within. Also, one cannot compare the basis set parameters used in this work with those presented in Ref. [5] since the two cal- culations used different types of pseudopotentials and a different ab initio code. The final optimised structures are shown in figure 5. The interlayer spacings (ignoring the samarium atom for now) in both structures here are almost identical. The major difference between this calculated structure and that in Ref. [5] is in the interlayer spacings. In this study the spacing between the top layer and the first bulk-like layer (L1 in figure 5) is approximately 8% greater than that in Ref. [5] and the spacing between the first bulk- like layer the the second bulk-like layer (L2 in figure 5) is about 4% greater. There are also some minor differences in the position of the silicon atoms in the honeycomb chain. IV. COMPARISON OF EXPERIMENT AND THEORY Figure 6 shows I-V curves calculated using the CAVLEED code [24] for the three candidate ab initio structures. The curves shown are only the integer spots in the LEED pat- tern and they were calculated using the bulk Debye tem- peratures (that is 645 K for silicon and 169 K for samar- ium) to represent the lattice vibrations of each layer. The structures obtained from the two ab initio calculations in this study are a consistently better match to experiment than that in Ref. [5]. This suggests that the interlayer spacings obtained in this study, to which LEED is very sensitive, are closer to those present in the real surface. Also, note that the I-V curves of the T4 and H3 structures from this study are very similar and we cannot discard either structure. We can divide the spots in the LEED pattern into two groups. The integer spots ((1,0), (2,0), (1,1) etc) contain a large contribution from the bulk and are sensitive to the top few layers. The fractional spots ((2/3,0), (1/3,1) etc) are extremely sensitive to the top layer reconstruc- tion and only mildy sensitive to deeper layers through multiple scattering. The poor Pendry R-factors (that is >0.7 in this con- -5769 -5768.5 -5768 -5767.5 -5767 -5766.5 -5766 -5765.5 0 5 10 15 20 25 30 35 Number of geometry optimisation steps 0 5 10 15 20 25 30 35 Number of geometry optimisation steps FIG. 4: Convergence of the total energy (top) and logarith- mic convergence of the forces (bottom) during the geometry optimisation of the T4 and H3 structures. The horizontal line indicates the force convergence tolerance of 5 × 10−2 eV/Å. The T4 structure has a lower energy than the H3 structure and the maximum force on any atom is lower. text where enhanced vibrations have not been applied) for some of the integer spots in figure 6 indicate that fur- ther structural optimisation is needed. It is apparent that for some curves the right peaks are present but that their energy is slightly wrong (see the (0,2) and (2,0) spots in figure 6 for example). The fractional spots have much better R-factors (see figure 7) which indicates that the structure of the top layer is in good agreement with ex- periment. The natural way to proceed is to vary the interlayer spacings to attempt an improvement in the match with experiment, particularly for the integer spots. In the next section this is attempted. V. LEED I-V STRUCTURAL OPTIMISATION The calculation of the I-V curves was repeated using var- ious values for the interlayer spacings and the R-factors were determined. An initial coarse search was carried out over a wide range of values for the spacings and with a large step size. Figure 7 shows the R-factor landscape obtained in this manner for the fractional spots. There FIG. 5: Optimised structures for the HCC model showing the H3 model from above (a) and in side view (b) and the T4 model from above (c) and in side view (d). Silicon atoms here are grey, the samarium atom is black and the hydrogen atoms are white. The first and second interlayer spacings are labelled L1 and L2 respectively. is a clear minimum in both cases. The samarium atom has been considered in determining the midpoint of the top layer which is why the minima do not coincide; the samarium atom sits proud of the honeycomb layer in the H3 structure and it is much lower in the T4 structure. In the ab initio calculations in this study the interlayer spac- ings were approximately 3.06, 3.10Å for the H3 structure and 2.65, 3.14 Å for the T4 structure which places the ab initio energy minimum (indicated by a cross in figure 7) very close to that of the CAVLEED I-V R-factor mini- mum. Two independent techniques are thus suggesting very similar best fit structures. I-V curves were then obtained using a narrower range of interlayer spacings focussed on the minima obtained in the coarse search. This fine search, using a step size of 0.01 Å, improved the R-factors by only around 0.01 in both cases and even finer searches were not carried out. There is another interlayer spacing deeper into the bulk that we might try to vary. Computational resources do not permit us to independently vary this spacing along with those between the top three layers. Figure 8 shows the variation of the Pendry R-factor as the spacing be- tween layers three and four is changed with the first and second interlayer spacings fixed at their optimum value. We can see that there is a small improvement in the R- factor for the fractional spots at the expense of a large worsening of the R-factor for the integer spots, which are more sensitive to structure in the near bulk. We there- fore reject any variation of this interlayer spacing and retain the bulk value. That there is no significant recon- 50 100 150 200 250 300 Energy (eV) Rp=0.71 Rp=0.74 Rp=0.83 Rp=0.78 Rp=0.92 Rp=0.72 Rp=0.55 Rp=0.52 Rp=1.03 Rp=0.57 Rp=0.55 Rp=0.86 Rp=0.62 Rp=0.58 Rp=1.27 (1,0) (0,1) (2,0) (0,2) (1,1) Experiment T4 site this study H3 site this study T4 site Palmino et al FIG. 6: A comparison of the I-V curves calculated for the integer spots for the structures suggested by the ab initio calculations in this study and elsewhere with those obtained experimentally. The R-factor beside each curve indicates the level of agreement with experiment. struction deeper into the surface justifies the use of three layers in our ab initio calculation and means that in both the ab initio calculation and the Pendry R-factor struc- ture fit to the experimental data we have considered two interlayer spacings. Optimisation of the vibrations used in the LEED I-V calculation The effect of thermal vibrations within the system has also been investigated. The Debye temperature TD of the samarium atom, the silicon atoms in the honeycomb layer and the silicon atoms in the first bulk-like layer have each been independently reduced by a factor of 2, 2 and 3 from their bulk values. The effects of these enhanced vibrations for the two most effective combinations are shown in table I alongside the R-factors obtained with no enhanced vibrations. The two schemes of enhanced vibrations both reduce the overall R-factor by around 0.2 and this is mainly due to the improvement in the R-factors of the integer spots. FIG. 7: Pendry R-factor landscape for a range of values of the interlayer spacings in the (a) H3 and (b) T4 structure for the fractional spots. The step size was 0.05 Å. The cross indicates the ab initio energy minimum. Linear combination of the two candidate structures The H3 and T4 structures have similar energies, similar structures (ignoring the position of the samarium atom) and similar LEED I-V curves. It is reasonable to suggest that both structures might co-exist upon the surface. A linear combination of the I-V curves produced by the H3 and T4 structures that individually best fit the experi- mental data are shown in figure 9 for the two regimes of enhanced vibration shown in table I. The H3 and T4 structures are considered as being separated by a distance greater than the coherence length of the LEED beam. To simulate large and separate domains of the two struc- tures in this way the LEED spot intensities have been combined and not the amplitudes. The vibrational regime with a Debye temperature for the samarium atom of 119 K (B/ 2 in table I) gives a lower R-factor for the fractional order spots but it gives 0.45 0.55 0.65 0.75 0.85 3.1 3.15 3.2 3.25 Layer 3 −Layer 4 spacing (Å) Fractional Spots Integer Spots All Spots 0.45 0.55 0.65 0.75 0.85 3.15 3.2 3.25 3.3 Layer 3 −Layer 4 spacing (Å) Fractional Spots Integer Spots All Spots FIG. 8: Variation of the spacing between layers three and four in Si(111)-3x2-Sm for the H3 (a) and T4 (b) structures. The bulk value for this interlayer spacing is 3.14 Å a worse overall R-factor. The vibrational regime with a Debye temperature for the samarium atom of 84 K (B/2 in table I) gives a better overall R-factor and the minima for both the fractional and the integer spots coincide. The final ratio of H3 40:60 T4 is in favour of the structure that is lower in energy which is what we would expect. Table II contains a summary of the structures obtained from the ab initio calculations and from the CAVLEED LEED I-V structure fit. Two values are given for L1; the value in brackets ignores the Sm atom in determining the midpoint of the top layer. For the T4 structure the Sm atom is almost coplanar with the honeycomb layer whereas for the H3 structure the Sm atom sits proud of the surface and skews the value of L1. The value in brackets thus indicates the similarity of the spacings be- tween the layers of silicon atoms in the two supercells. For each of the structures in this table LEED I-V curves Sm TD Si1 TD Si2 TD R B B B 0.49 0.72 0.63 2 B/3 B/2 0.48 0.46 0.48 B/2 B/3 B/2 0.45 0.44 0.45 TABLE I: Variation of the Debye temperature for the samar- ium atom, silicon honeycomb layer and first bulk like layer and the effect upon the Pendry R-factors for the H3 structure. The naming scheme here is Sm=samarium atom, Si1=silicon honeycomb atoms, Si2=first silicon bulk-like layer. A De- bye temperature of B indicates the bulk unoptimised value for that atomic species. Similar data are available for the T4 structure. Further enhancement of the vibrations of the samarium atom worsens the R-factors. were calculated with optimised vibrations using a De- bye temperature for the samarium atom of 84 K. These were then compared against experiment and the Pendry R-factors are included in table II. The final optimised LEED I-V curves for the linear combination are com- pared with experiment for the integer spots in figure 10 and for the fractional spots in figure 11. Structure RFRACP R P L1 (Å) L2 (Å) T4 (Ref. [5]) 0.87 0.88 0.92 2.42 (2.52) 3.02 T4 CASTEP 0.44 0.46 0.46 2.65 (2.67) 3.14 H3 CASTEP 0.47 0.43 0.45 3.06 (2.62) 3.10 T4 CAVLEED 0.48 0.41 0.46 2.74 (2.73) 3.10 H3 CAVLEED 0.45 0.44 0.45 3.06 (2.64) 3.11 Combination 0.39 0.42 0.41 2.87 (2.69) 3.10 TABLE II: Pendry R-factors for the fractional spots (RFRACP ), integer spots (R P ) and for all spots (R P ) for the various optimised structures in this work. All of the cal- culated I-V curves used optimised vibrations. The interlayer spacings are shown in columns five and six midpoint. The value of L1 in brackets ignores the Sm atom in the deter- mination of the midpoint of the top layer and indicates the similarity of the structure of the silicon atoms in the two su- percells. VI. LEED I-V INVESTIGATION OF THE MISSING HALF ORDER SPOTS The silicon honeycomb layer is almost mirror symmetric about a plane perpendicular to the ×2 direction. It is the location of the samarium atom that breaks this mir- ror symmetry and renders a quasi 3×1 unit cell into a 3×2 unit cell. Figure 12 shows calculated I-V curves for the H3 structure for the fractional and integer spots com- pared with those for the same structure with the samar- ium atom removed. The bulk Debye temperatures were 0.37 0.38 0.39 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0 10 20 30 40 50 60 70 80 90 100 % of Sm atoms in H3 site All Spots Fractional Spots 0.37 0.38 0.39 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0 10 20 30 40 50 60 70 80 90 100 % of Sm atoms in H3 site All Spots Fractional Spots FIG. 9: Pendry R-factors for a linear combination of the spot intensities of the H3 and T4 structures in various mixing ratios for two different vibrational regimes. In figure (a) the Debye temperature for the samarium atom is 119 K and in figure (b) it is 84 K. In both cases the Debye temperatures of the top silicon honeycomb layer, the first silicon bulk-like layer and the repeated bulk layer are 215 K, 323 K and 645 K respectively. used throughout to minimise the influence of vibrations. It is readily apparent that the I-V curves are insensitive to the presence of the samarium atom. This is not to say that the samarium atom is not a strong scatterer. It would appear that the silicon honey- comb layer as a scattering unit of 8 atoms contributes much more to the I-V curves than the single samar- 50 100 150 200 250 Energy (eV) Rp=0.53 Rp=0.58 Rp=0.52 Rp=0.18 Rp=0.33 (1,0) (0,1) (2,0) (0,2) (1,1) LEED Experiment CAVLEED Theory FIG. 10: Best fit I-V curves for the integer LEED spots of the Si(111)-3x2-Sm structure. ium atom. A similar effect was observed in the LEED I-V structural analysis of Ag- and Li-induced Si(111)- 3)R30 ◦ by Over et al. [25] and was suggested as a cause for the 3×1/3×2 discrepancy in Ref. [16]. If this is the case then the half order spots that are apparently missing when the experimental 3×1 LEED pattern is inspected visually should produce calculated I-V curves whose intensity is very much less than that of the spots that are visible during experiment. The sili- con honeycomb layer is not perfectly symmetrical about the mirror plane perpendicular to the ×2 direction and this should contribute to the half order spot intensities. Figure 13 shows the I-V curves of some of the calculated half order spots compared to that of the (1,0) spot. It would appear that the 3×2 unit cell produces a 3×2 LEED pattern with half order spots that are so weak in intensity that they fall below the background intensity leaving only a 3×1 LEED pattern visible. VII. DISCUSSION The Pendry R-factors obtained upon comparison of the ab initio calculations with experiment are not as low as we would expect. We can see that for some spots the I-V curves are visually very similar to those obtained ex- 40 60 80 100 120 140 160 180 200 Energy (eV) Rp=0.52 Rp=0.43 Rp=0.24 Rp=0.19 Rp=0.62 Rp=0.21 Rp=0.37 Rp=0.77 (2/3,0) (0,2/3) (4/3,0) (0,4/3) (1,1/3) (1/3,1) (2/3,1/3) (1/3,2/3) LEED Experiment CAVLEED Theory FIG. 11: Best fit I-V curves for the fractional LEED spots of the Si(111)-3x2-Sm structure. perimentally (see the I-V curves for the (1,1/3) and (2,0) spots for example) but they have a poor R-factor. This suggests that the structure is very nearly right and the minor discrepancy could be a result of our not including enough bulk like silicon layers in the bottom of the su- percell with consequent effects upon the reconstruction within the top honeycomb layer. We have attempted some simple variation in the top layer structure, for ex- ample flattening the layer, but this drastically worsens the R-factor. Computational resources prohibit us from calculating the structure with more silicon layers and from investigating the honeycomb layer structure further using LEED I-V and perhaps further study with a LEED I-V genetic algorithmn search might optimise this struc- ture further. The moderate R-factors are offset by the fact that two independent techniques both show optimum structural fits for almost identical interlayer spacings. The lateral atomic structure was freely varied in the ab initio calculations in this work and the lateral atomic po- sitions agree well with those found by Palmino at al. [5] which they have shown to be in good qualitative agree- ment with experimental STM images. In this work we have concentrated upon the optimisation of the vertical spacings, to which LEED is particularly sensitive. The R-factors for the integer spots are consistently worse than those for the fractional spots. There is the 50 100 150 200 250 Energy (eV) Full cell No samarium (a) Integer spots 50 100 150 200 250 Energy (eV) Full cell No samarium (b) Fractional spots FIG. 12: Calculated LEED I-V curves for the integer spots (a) and fractional spots (b) of the H3 unit cell with and without the samarium atom in place. Bulk Debye temperatures were used throughout. possibility that there are some regions in which there is a disordered overlayer of samarium atop a bulk terminated Si(111)-1×1 surface. Such a phase has been reported by Wigren et al. [26]. The integer spots from such regions might contribute to the overall integer spots for the sur- face and reduce the level of agreement with the calculated I-V curves for the pure 3×2 surface. We have not been able to determine the long range order in the system. We might expect that simple elec- trostatic repulsion along the 1D chain would space out 40 60 80 100 120 140 160 180 200 220 240 260 Energy (eV) H3 site T4 site FIG. 13: Calculated I-V curves for the HCC structure show- ing the difference in the intensity (typically an order of mag- nitude) between the half order spots and a representative spot that is visible in the LEED pattern during an experiment. the metal atoms and provide large separate domains of the H3 and the T4 structures. However, the two sites are almost degenerate and there would be an entropic gain from disorder. In the literature one can find evidence for both order and disorder in the long range positions of the metal atoms. In this study the improvement in the Pendry R-factor when the T4 and H3 structures are con- sidered together on the surface suggests that both sites are occupied within the surface. We have also shown that we do not require more than one unit cell to ex- plain the missing half order spots in the LEED pattern and our experimentally observed LEED patterns show a low background due to good order on the surface. It could be that there is long range disorder on the surface and that the coupling between many adjacent H3 and T4 unit cells and matching of the interlayer spacings in- troduces a slight strain that changes the structure in the honeycomb layer and the first bulk-like layer enough to account for our Pendry R-factors. If this is the case then it would be impossible to obtain the structure of the hon- eycomb layer to a high degree of accuracy without an ab initio calculation using a supercell that comprises several thousand unit cells of the H3 and T4 structures randomly tesselated in both directions. VIII. SUMMARY We have provided a quantitative validation of the honeycomb-chain channel model common to the 3×1 and 3×2 structures formed by alkali, alkali-earth and rare- earth metals on Si(111). Several I-V datasets were ob- tained from LEED experiments and used to fingerprint the surface. The atomic structure suggested by our two ab initio calculations is in reasonable agreement with this experimental data. Further structural optimisation and mapping of the R-factor landscape have shown that a slight outward expansion of the top layer improves the fit somewhat but increasing the vibrations in the top two layers gives a significant improvement. A linear combi- nation of the two HCC structures has been shown to im- prove the fit still further with the ratio being slightly in favour of the structure with the lower energy of the two. Finally, we have calculated the intensities of the half or- der spots and shown that they are sufficiently dim to fall below the background intensity in a LEED experiment. Little change in the calculated I-V curves results from re- moving the samarium atom which supports the idea that as a scattering unit the silicon honeycomb layer domi- nates the unit cell and makes LEED insensitive to the metal atom in these 3× systems. IX. ACKNOWLEDGEMENTS Many thanks to F. Palmino for kindly providing the atomic co-ordinates for the Si(111)-3×2 unit cell. C. Eames would like to acknowledge the EPSRC for financial support. This work made use of the facilities of HPCx, the UK’s national high-performance computing service, which is provided by EPCC at the University of Edinburgh and by CCLRC Daresbury Laboratory, and funded by the Office of Science and Technology through EPSRC’s High End Computing Programme. ∗ Corresponding author. E-mail: [email protected] [1] A. A. Saranin, A. V. Zotov, S. V. Ryzhkov, D. A. Tsukanov, V. G. Lifshits, J.-T. Ryu, O. Kubo, H. Tani, T. Harada, M. Katayama, and K. Oura, Phys. Rev. B., 58, 7059 (1998). [2] H. H. Weitering, X. Shi, and S. C. Erwin, Phys. Rev. B., 54, 10585 (1996). [3] K. Sakamoto, W. Takeyama, H. M. Zhang, and R. I. G. Uhrberg, Phys. Rev. B., 66, 165319 (2002). [4] G. Lee, S. Hong, H. Kim, D. Shin, J.-Y. Koo, H.-I. Lee, and D. W. Moon, Phys. Rev. Lett., 87, 56104 (2001). mailto:[email protected] [5] F. Palmino, E. Ehret, L. Mansour, J.-C. Labrune, G. Lee, H. Kim, and J.-M. Themlin, Phys. Rev. B., 67, 195413 (2003). [6] M. Kuzmin, R.-L. Vaara, P. Laukkanen, R. Perälä, and I. J. Väyrynen, Surf. Sci., 538, 124 (2003). [7] H. H. Weitering, Surf. Sci., 355, L271 (1996). [8] D. Jeon, T. Hashizume, T. Sakurai, and R. F. Willis, Phys. Rev. Lett., 69, 1419 (1992). [9] C. Wigren, J. N. Andersen, R. Nyholm, M. Gothelid, M. Hammar, C. Tornevik, and U. O. Karlson, Phys. Rev. B., 48, 11014 (1993). [10] J. Quinn and F. Jona, Surf. Sci., 249, L307 (1991). [11] W. C. Fan and A. Ignatiev, Phys. Rev. B., 41, 3592 (1990). [12] C. Collazo-Davila, D. Grozea, and L. D. Marks, Phys. Rev. Lett., 80, 1678 (1998). [13] L. Lottermoser, E. Landemark, D.-M. Smilgies, M. Nielsen, R. Feidenhans, G. Falkenberg, R. L. John- son, M. Gierer, A. P. Seitsonen, H. Kleine, H. Bludau, H. Over, S. K. Kim, and F. Jona, Phys. Rev. Lett., 80, 3980 (1998). [14] S. C. Erwin and H. H. Weitering, Phys. Rev. Lett., 81, 2296 (1998). [15] J. Schäfer, S. C. Erwin, M. Hansmann, Z. Song, E. Roten- berg, S. D. Kevan, C. S. Hellberg, and K. Horn, Phys. Rev. B., 67, 85411 (2003). [16] H. Over, M. Gierer, H. Bludau, G. Ertl, and S. Y. Tong, Surf. Sci., 314, 243 (1994). [17] V. E. de Carvalho, M. W. Cook, P. G. Cowell, O. S. Heavens, M. Prutton, and S. P. Tear, Vacuum, 34, 893 (1984). [18] F. Palmino, E. Ehret, L. Mansour, E. Duverger, and J.- C. Labrune, Surf. Sci., 586, 56 (2005). [19] J. B. Pendry, J. Phys. C. Solid St. Phys., 13, 937 (1980). [20] M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clark, and M. C. Payne, J. Phys.: Cond. Matt., 14, 2717 (2002). [21] M. I. J. Probert and M. C. Payne, Phys. Rev. B., 67, 075204 (2003). [22] H. J. Monkhorst and J. D. Pack, Phys. Rev. B., 13, 5188 (1976). [23] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77, 3865 (1996). [24] D. J. Titterington and C. G. Kinniburgh, Comp. Phys. Comm., 20, 237 (1980). [25] H. Over, H. Huang, S. Y. Tong, W. C. Fan, and A. Ig- natiev, Phys. Rev. B., 48, 15353 (1993). [26] C. Wigren, J. N. Andersen, R. Nyholm, and U. O. Karls- son, Surf. Sci., 293, 254 (1993).

0704.1420

Renormalization of Hamiltonian QCD

Renormalization of Hamiltonian QCD A. Andraši∗ ’Rudjer Bošković’ Institute, Zagreb, Croatia John C. Taylor† Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK April, 10 2007 Abstract We study to one-loop order the renormalization of QCD in the Coulomb gauge using the Hamitonian formalism. Divergences occur which might require counter- terms outside the Hamiltonian formalism, but they can be cancelled by a redef- inition of the Yang-Mills electric field. PACS: 11.15.Bt; 11.10.Gh Keywords: Coulomb gauge, Hamiltonian, renormalization ∗e-mail:[email protected] †e-mail:[email protected] http://arxiv.org/abs/0704.1420v1 1 Introduction We study the renormalization of QCD in the Coulomb gauge Hamiltonian for- malism. By Hamiltonian form, we mean that the Lagrangian contains only first order terms in time derivatives, and depends upon the conjugate momen- tum field Eai as well as the (transverse) gluon field A i (here a is the colour index and i = 1, 2, 3 is a 3-vector index). This form has a number of attractive features: (i) As a Hamiltonian exists, the theory is explicitly unitary, without the necessity to cancel unphysical degrees of freedom with ghosts. (ii) The Lagrangian form of the Coulomb gauge has “energy divergences” in some of its Feynman integrals, that is integrals of the form (we use K for the spatial part of the 4-vector k) d3Kdk0f(K, k0) (1) where f does not decrease as k0 → ∞ (for fixed K). These divergences cancel between different Feynman graphs [1], but this cancellation has to be organized “by hand”. In the Hamiltonian form, each individual Feynman graph is free of such divergence. Formally ’energy divergent’ integrals such as (2π)3 p20 − P 2 + iη (P −K)2 are assigned the value zero. (iii) It has been argued [2] that the Coulomb gauge throws light on con- finement. Certainly it is known [3] that, in the Coulomb gauge, the source of asymptotic freedom lies in the Coulomb potential. In spite of (i) above, to 2-loop order, mild energy-divergences remain [4], [5], [6] which result in ambiguities which have to be resolved by a prescription. This is connected with questions of operator ordering [7]. For other applications of the Coulomb gauge, for example to lattice QCD, see [8], [9]. The question addressed here is the following. Ultra-violet divergences exist which seem to require the existence of counter-terms containing second order terms in time derivatives, (∂Aai /∂t) 2. Do these take us out of the Hamiltonian form? We argue that this does not happen because the divergences concerned can be cancelled by a redefinition of the Eam field. We do not use quite the strict Hamiltonian formalism. We retain the auxil- liary field Aa0 , which contains no time derivatives and should be integrated out to give a nonlocal Coulomb potential term in the real Hamiltonian. It seems to be convenient, for the purposes of renormalization, to retain Aa0 in the La- grangian. Because of this, there is a ghost field, but it has an instantaneous propagator, and so is not relevant to unitarity. Its purpose is only to cancel out closed loops in the Aa0 field. 2 The Feynman rules The Lagrangian for the Coulomb gauge is L′ = L− 1 2 (3) (where α will eventually tend to zero to go to the Coulomb gauge), L = −1 Fij · Fij − 2 +Ei ·F0i ∗∂ic+ g∂ic ∗ · (Ai ∧ c) +ui · [∂ic+ g(Ai ∧ c)] +u0 · [∂0c+ g(A0 ∧ c)] gK · (c ∧ c) + gvi · (Ei ∧ c) (4) where we use a colour vector notation, and F aij = ∂iA j − ∂jAai + gfabcAbiAcj (Ai ∧ c)a = fabdAbicd (5) Here c, c∗ are the ghost fields, and the sources ui,vn and K are inserted for future use in formulating the BRST identities. The conjugate momentum (elec- tric) field Em could be integrated out to obtain the ordinary Lagrangian for- malism, but for the Hamiltonian formalism it must be retained. We will use indices m,n, ... = 1, 2, 3 to denote the (spatial) components of E, so the seven fields are (Aai , A 0 , E n). We will use indices I, J, .. to denote the seven indices (i, 0, n). The bilinear part of the Lagrangian in momentum space is a 7× 7 matrix SIJδab = −K2(Tij + Lij/α) 0 −ik0δin 0 0 iKn ik0δmj −iKm −δmn where Tij ≡ δij − Lij , Lij ≡ KiKj/K2, k2 = k20 −K2. (6) For the propagators, we need the inverse S−1IJ δab = Tij/k 2 − αLij/K2 αk0Ki/(K2)2 −ik0Tin/k2 αk0Kj/(K 2)2 1/K2 + αk20/(K 2)2 iKn/K ik0Tmj/k 2 −iKm/K2 TmnK2/k2  . (7) We can now let α → 0, to obtain the Coulomb gauge. From this, and the interaction terms in the Lagrangian (4), we can read off the Feynman rules. We represent the Ai field by dashed lines, the En field by continuous lines, and the A0 field by dotted lines. With this notation, we now list the rules (a factor (2π)4i is to be included for each propagator, and a factor of (2π)4i for each vertex). If we choose the propagators in fig.1 to be the negative of the matrix (7), the extra factors of 1 (2π)4i for the propagator and (2π)4i for the vertices cancel. 3 The ultra-violet divergences The divergent graphs with 2 and with 3 external lines are shown in Figures 4 till 31. Examples of the method of evaluation of divergent parts are given in Appendices A and B. The ultra-violet divergent parts of these graphs are, in terms of the divergent constant (using dimensional regularization in 4− ǫ dimensions) CGΓ(ǫ/2), (8) (where the superfix (4), (5) etc. refers to the corresponding figure and Πij , Π0i...Πmn denote self-energies, Vijk, V0in...V0in vertices and Λ stands for dia- grams with external ghost lines), are: (4)ab ij = ic[ k20δij +K 2δij −KiKj]δab (9) (5)ab i0 = − ick0Kiδab (10) (6)ab icK2δab (11) (7)ab mi = 0 (12) (8)ab m0 = − ic[iKiδab] (13) Π(9)abmn = − icδmnδab (14) (10)abc ijk (p, q, r) = − cgfabc[(Q− P )kδij + (R −Q)iδjk + (P −R)jδik] (15) (11)abc ijk (p, q, r) = − cgfabc[(Q− P )kδij + (R −Q)iδjk + (P −R)jδik] (16) (12)abc ijk (p, q, r) = − cgfabc[(Q− P )kδij + (R −Q)iδjk + (P −R)jδik] (17) (13)abc ijk (p, q, r) = cgfabc[(Q− P )kδij + (R−Q)iδjk + (P −R)jδik] (18) (14)abc i00 (p, q, r) = cgfabc(R−Q)i (19) (15)abc i00 (p, q, r) = − cgfabc(R −Q)i (20) (16)abc i00 (p, q, r) = cgfabc(R−Q)i (21) (17)abc i00 (p, q, r) = cgfabc(R−Q)i (22) (18)abc 0jl (p, q, r) = 0 (23) (19)abc 0jl (p, q, r) = cgfabc(R −Q)0 (24) (20)abc 0jl (p, q, r) = − cgfabc(R −Q)0 (25) (21)abc 0jl (p, q, r) = 0. (26) Graphs involving external Em line are (29)abc im0 (p, q, r) = icgfabcδim (27) (30)abc im0 (p, q, r) = − icgfabcδim (28) (31)abc im0 (p, q, r) = 0. (29) All other graphs involving external Em -lines are convergent. The divergent parts of graphs with open ghost line are Λ(22)ab(q) = −4 icQ2δab (30) (23)ab i (q) = − cQiδab (31) Λ(24)abc(p, q) = 0 (32) (25)abc k (p, q, r) = 0 (33) (26)abc 0 (p, q) = 0 (34) (27)abc i (p, q) = 0 (35) Λ(28)abcn (p, q) = 0. (36) 4 Counter-terms d4xL(x) (37) be the original action, Γ be the complete effective action, and let Γ1 be the effective action to one-loop order. The complete BRST identities are Γ ∗ Γ ≡ ∂Γ = 0. (38) So to one-loop order Γ1 ∗ Γ0 + Γ0 ∗ Γ1 ≡ ∆Γ1 = 0 (39) where ∆2 = 0. (41) One class of solutions to this equation is of the form 1 = ∆G, (42) where the allowed form of G is, in terms of constants a5, ...a11, G = a5Ai · (ui + ∂ic∗) + a6A0 · u0 + a7c ·K+ a8Ei · vi +a9vi · ∂iA0 + a10vi · ∂0Ai + a11vi · (A0 ∧Ai). (43) Other solutions of equation (39) are the explicitly gauge-invariant terms 1 = a1(Fij) 2 + a2Ei · F0i + a3(F0i)2 + a4(Ei)2. (44) Finally, by differentiating (38) with respect to the coupling constant g and specialising to one-loop order, we see that (iii) 1 = 0 (45) where (a0 being another divergent constant) (iii) i = a0g . (46) Combining these three contributions, we obtain Γ1 = Γ 1 + Γ 1 + Γ (iii) d4xL(x) (47) where L1 = a1(Fij)2 + (a2 + a8 + a9)Ei ·F0i +(a3 − a9)(F0i)2 + (a4 − a8)(Ei)2 +a5Fij · ∂jAi − (a5 + a0)gFij · (Ai ∧Aj) −(a0 + a5 + a6)gEi · (Ai ∧A0) +Ei · (a5∂0Ai − a6∂iA0) −a5(ui + ∂ic∗) · ∂ic+ a0g∂ic∗ · (Ai ∧ c) −a6u0 · ∂0c+ a0gu0 · (A0 ∧ c) −a7(ui + ∂ic∗) · {∂ic+ g(Ai ∧ c)} +a0gui · (Ai ∧ c)− a7u0 · {∂0c+ g(A0 ∧ c)} g(a7 − a0)K · (c ∧ c) + (a0 − a7)gvi · (Ei ∧ c). (48) The conditions coming from the vanishing ghost graphs Figs. 24, 25, 26, 27 and 28 are particularly simple. They fix a9 = −a10 a11 = −ga9 a0 = a7 = −a6. (49) In order for the counter-terms to cancel the divergences in the other graphs, we require the conditions 4a1 − 2a5 = −c 4a1 − 3a5 − a0 = a3 − a9 = − a6 − a5 = a5 + a7 = − a4 − a8 = a2 + a5 + a8 + a9 = 0. (50) These equations do not fix the constants uniquely. We are free to make some choices. The term (F0i) 2 in Γ 1 eq.(44) is not present in the original Hamiltonian form of the Lagrangian (4), so we choose a3 = 0. (51) We can also arrange for the combination 2 +Ei ·F0i (52) to appear in L(ii)1 as it does in L0. This requires (from (50)) a1 = − a2 = c− 2a5 a4 = − c+ a5 c+ a5 a7 = − c− a5 a8 = − c+ a5 a0 = − c− a5 (53) and so L(ii)1 = −4a1[− (Fij) 2 − 1 2 +Ei ·F0i] (54) proportional to the non-ghost part of the original Lagrangian (3). Equation (54) does not come from the BRST identities, it just emerges from the numerical values of the divergent integrals. It may be a consequence of some hidden Lorentz invariance. The constants a0, a1, ... are still not uniquely fixed. There are two particu- larly simple choices. (i) Choose a0 = 0 with a5 = − 43c. Then we find a1 = − a4 = − a6 = a7 = 0 a8 = − c. (55) (ii) The second choice is a1 = 0 with a5 = c. Then a0 = − a2 = 0 a4 = 0 a7 = − a8 = − c. (56) Note that a0 has the expected value for coupling constant renormalization. The counter-terms in either case are L1 = − c(Fij) 2 − 4 cFij · ∂jAi + cgFij · (Ai ∧Aj) c(F0i) c(Ei) cEi · F0i cgEi · (Ai ∧A0)− cEi · ∂0Ai + c(ui + ∂ic ∗) · ∂ic. (57) The counter-terms in a5, a6, a7, a8 and a9 are involved in a rescaling of the fields. Defining i = (1 + a5)Ai 0 = (1 + a6)A0 m = (1 + a8)Em − a9F0m i = (1− a5)ui 0 = (1− a6)u0 ′ = (1− a7)c ′ = (1 + a7)K g′ = (1 + a0)g ′∗ = (1 − a5)c∗ ′ = (1− a8)v, (58) we have from (48) that L0 + L1 = (1− 4a1)L0(g′,A′i,A′0,E′, c′, c′∗,u′i,u′0,K′). (59) Note that a6 which determines the renormalization of the Coulomb field A has the same numerical value as a0. We have not calculated the divergences in graphs with four external lines. We assume they will be cancelled by the same counter-terms. 5 Comments We conclude that there is no difficulty to one-loop order in renormalizing the Hamiltonian form of the Coulomb gauge. We guess that the renormalization would formally go through to higher orders, but then there is the problem men- tioned in [4], [5], [6] of combining the renormalization of ultra-violet divergences with the resolution of energy-divergence ambiguities. It is not quite obvious how the renormalization would be formulated if the Aa0 field had been eliminated to give the non-local colour Coulomb potential (note the non-zero value of the Aa0 field renormalization constant a6 in (56)). Acknowledgements A.A. wishes to thank the Royal Society for a grant and DAMTP for hospi- tality. We are grateful to Dr. G. Duplančić for drawing the figures. The work was supported by the Ministry of Science and Technology of the Republic of Croatia under contract No. 098-0000000-2865. Appendix A Here we give as an example the evaluation of the ultra-violet divergent part of the graph in Fig. 20. (20)abc 0jk (q,−q, 0) = ig (p2 + iη)2 (q + p)2 + iη ×Trz(P )Tzv(P )Tru(Q+P )[(−2Q−P )vδuj+(Q−P )uδjv+(2P+Q)jδvu]. (A1) Applying the integral (p2 + iη)2 (q + p)2 + iη dyy(1− y){(P + yQ)2 + y(1− y)(−q2 − iη)}− 52 (A2) and power counting to (A1) (20)abc 0jk (q,−q, 0) = −4g πq0Γ( dyy(1− y) d3−ǫPPjPk{(P + yQ)2 + y(1− y)(−q2 − iη)}− 2 , (A3) leading to (20)abc 0jk (q,−q, 0) = − cgfabcq0δjk. (A4) Appendix B Example of self-energy evaluation Π (6)ab 00 in eq.(11). Let p, q be internal and k external momentum, p− q = k. The sum of two graphs is (2π)−4 Tij(P )Tji(Q) (P 2 +Q2)− (ip0)(iq0)]δab (B1) where we have symmetrized the first term in P,Q. the minus sign in the second term comes from the opposite order of the fabc factors at the two vertices. Doing the p0 integration by Cauchy, we get (2π)−4(2πi) TijTji (P +Q)2 − k20 (P +Q)[P 2 +Q2 − 2PQ]δab. (B2) The last factor (P − Q)2 is approximately (P · K)2/P 2. With this factor, the integral is only logarithmically divergent, and to get the divergent part we can put Q = P everywhere. We use Tij(P )Tji(P ) = 2. Then we get (2π)−4 d3−ǫP (P 2 +m2)5/2 . (B3) So the divergent part is1 icK2δab. (B4) References [1] R. N. Mohapatra, Phys. Rev. D4, 22, 378, 1007 (1971) [2] D. Zwanziger, Nucl. Phys. B 485, 185 (1997) [3] J. Frenkel, J. C. Taylor, Nucl. Phys. B 109, 439 (1976) [4] P. Doust, J. C. Taylor, Phys. Lett. 197, 232 (1987) [5] P. Doust, Ann. of Phys. 177, 169 (1987) [6] J. C. Taylor, in Physical and Nonstandard Gauges, Proceedings, Vienna, Austria 1989, edited by P.Gaigg, W. Kummer, M. Schweda [7] N. Christ, T. D. Lee, Phys. Rev. D 22, 939 (1980) [8] A. Cucchieri, D. Zwanziger, Nucl. Phys. Proc. Suppl. 106, 694 (2002) [9] A. Cucchieri, hep-lat/0612004 1Note that there was an error of sign in Eur. Phys. J. C37, 307-313(2004) which however did not influence the final result. http://arxiv.org/abs/hep-lat/0612004 Figure 1: Feynman rules for the propagators in the Coulomb gauge. q; b; j r; ; k p; a; i a; i b; j ; k a;m (Q� P ) (R�Q) (P �R) Figure 2: Feynman rules for the vertices in the Coulomb gauge. The arrows denote the directions of the momenta. p; j; b Figure 3: Feynman rules for ghosts and sources in the Coulomb gauge. Doubled lines denote ghosts. The black arrows distinguish between ghosts and antighosts. Momenta flow into the vertex. Figure 4: The transverse gluon self-energy graphs. Figure 5: The AiA0 two-point function. Figure 6: The time-time component of the gluon self-energy. Figure 7: The transition between the transverse gluon field and its conjugate field Ei. Figure 8: The transition between the Coulomb field A0 and the conjugate field Figure 9: The conjugate field self-energy. Figure 10: Graph contributing to the three-gluon vertex function. Figure 11: There are three graphs in this class with permutations of the vertices. Figure 12: Graph representing a class of 6 diagrams. Figure 13: There are 3 graphs in this class of diagrams. Figure 14: Graph with two external Coulomb lines (there are 3 diagrams in this class). Figure 15: There are two graphs in this class. Figure 16: There are two graphs in this class. Figure 17: The graph with two external Coulomb lines and one three-gluon vertex. Figure 18: Graphs contributing to the(AiAjA0) three-point function. Figure 19: Graph contributing to the (AiAjA0) three-point function which contains a three-gluon vertex. Figure 20: The (AiAjA0) graph with a three-gluon vertex. Figure 21: The (AiAjA0) graph with a four-gluon vertex. Figure 22: The ghost self-energy. Figure 23: Ghost and the ui source graph. Figure 24: The ghost vertex graph with a K source. Figure 25: Graph with external Ai, ghost and anti-ghost lines. Figure 26: Graph with u0 source, Ei and c lines. Figure 27: Graph with ui source, Ai and c lines. Figure 28: Diagram with vn source, A0 and c lines. Figure 29: Graph contributing to the (AiEjA0) vertex function. Figure 30: Graph with external gluon, Coulomb and E-field. Figure 31: Graph in the (AiEjA0) vertex function. Introduction The Feynman rules The ultra-violet divergences Counter-terms Comments

0704.1421

What made GRBs 060505 and 060614?

WhatmadeGRBs 060505 and 060614? Páll Jakobsson a, Johan P. U. Fynbo b aCentre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, Herts, AL10 9AB, UK bDark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Abstract Recent observations of two nearby SN-less long-duration gamma-ray bursts (GRBs), which share no obvious characteristics in their prompt emission, suggest a new phe- nomenological type of massive stellar death. Here we briefly review the observational properties of these bursts and their proposed hosts, and discuss whether a new GRB classification scheme is needed. Key words: Gamma rays: bursts, Supernovae: general PACS: 95.85.Kr, 97.60.Bw, 98.70.Rz 1 Introduction A broad-lined and luminous type Ic core-collapse supernova (SN) is predicted to accompany every long- duration gamma-ray burst (GRB) in the standard collapsar model (Woosley, 1993). Although this as- sociation had been confirmed in ob- servations of several nearby GRBs (e.g. Hjorth et al., 2003), a new controversy commenced when no SN emission accompanied GRBs 060505 (z = 0.09, duration ∼4 s) and 060614 (z = 0.13, duration ∼100 s) down to limits fainter than any known type Ic SN and hundreds of times fainter than the archetypal SN1998bw (Della Valle et al., 2006; Fynbo et al., 2006; Gal-Yam et al., 2006). The upper panels of Fig.1 il- lustrate how easily such SNe would have been detected in the case of GRB060505. An important clue to the origin and progenitors of these bursts, is the nature of the host galaxies. The GRB060505 host is a spiral galaxy, atypical for long-duration bursts but not unheard of (GRB980425: Fynbo et al., 2000; GRB990705: Le Floc’h et al., 2002; GRB020819: Jakobs- son et al., 2005). The burst occurred inside a compact star-forming H II region in one of the spiral arms, and a spatially resolved spectroscopy (lower panel of Fig.1) revealed that the properties of the GRB site are Preprint submitted to New Astronomy Reviews 4 December 2018 http://arxiv.org/abs/0704.1421v1 Fig. 1. (a) The field (20′′× 20′′) of GRB060505 as observed from the VLT in the R-band on 22 May 2006. The arrow marks the position where the optical afterglow was detected in earlier imaging. (b) As the image would have looked had a SN like 1998bw been present in the data. The strict upper limits strongly exclude the bright SNe 1998bw and 2006aj that were associated with long GRBs. (c) Similar to (b), but with a very faint Ic SN, such as 2002ap, added. (d) The 2-D optical spectrum obtained with VLT/FORS2. The slit covered the centre of the host galaxy and the location of GRB060505. As seen in the spectrum, this site is a bright star-forming region in the host galaxy suggesting that the progenitor was a massive star. similar to those found for other long- duration GRBs with a high specific star formation rate (SSFR) and low metallicity (Thöne et al., 2007). The GRB060614 host is significantly fainter (one of the least luminous GRB host ever detected) with a moderate SSFR. 2 Discussion 2.1 High extinction? Could the emission from an associ- ated SN be completely obscured by dust along the line-of-sight? The lev- els of Galactic extinction are very low in both directions. Host extinction of more than a magnitude is also un- likely in either case since the host galaxy spectra display no reddening as derived from the Balmer line ra- tios. In addition, the GRB060614 af- terglow is clearly detected in the UV (Holland, 2006). 2.2 Wrong redshifts? Another option is that the proposed host galaxies are chance encounters along the line-of-sight (Cobb et al., 2006; Schaefer & Xiao, 2006), and the real GRB redshifts are much higher (rendering a SN too faint to be observed). However, a few ob- servational facts argue against this scenario. In the case of GRB060614: (i) the UV detection places an up- per limit of around 1.1 on the red- shift; (ii) no absorption components in the optical afterglow spectrum (Fugazza et al., 2006), as expected for a low redshift, but not for a high- z burst with a foreground galaxy; (iii) very deep HST images of the field should have revealed the “true host” at z . 1.1, but none was seen (Gal-Yam et al., 2006). For GRB060505 it is extremely unlikely that the afterglow accidentally su- perposed right on top of a small star- forming region within a foreground spiral galaxy. 2.3 No SNe: a problem? The host galaxies and the GRB loca- tion within them strongly suggest an association with star formation, and hence a massive stellar origin. It is important to realize that the lack of a strong SN emission was actually pre- dicted as a variant of the original col- lapsar model, e.g. collapse of a mas- sive star with an explosion energy so small that most of the 56Ni falls back into the black hole (e.g. Heger et al., 2003; Fryer et al., 2006). In another variant of the collapsar model, pro- genitor stars with relatively low an- gular momentum could also produce SN-less GRBs (MacFadyen, 2003). We should also remember that the duration distributions of short and long GRBs overlap. In fact, the GRB060505 duration of 4 s is near the ∼5 s duration which Donaghy et al. (2006) find as the point of roughly equal probability of a given burst lying in either the short or long class. It has been suggested that the physical mechanism for this burst is the same as for short bursts, i.e. a merger of compact objects (Ofek et al., 2007), although the pro- genitor time delay of only . 7Myr is on the borderline for allowed values (Thöne et al., 2007). However, such short time delays have been proposed via newly recognized formation chan- nels, which lead to the formation of tighter double compact objects with short lifetimes and therefore possible prompt merger within hosts (Belczynski, 2007). Whether such channels require a low metallicity as found for GRB060505 (Thöne et al., 2007) remains to be explored. 2.4 Classification problem? With the added complication that the ∼100 s long GRB060614 is lo- cated among the short bursts in the lag-luminosity plot, it has been ar- gued that a new GRB classification scheme is required (Gehrels et al., 2006). We do not think this is the case, as the current GRB classifi- cation is operationally well defined. Rather that new observations are warning us not necessarily to expect a very simple mapping between the duration of the GRB and the nature of the progenitor: long bursts (>2 s) synonymous with massive stars and short bursts (<2 s) synonymous with compact object mergers. Others want to abandon the long- short paradigm altogether due to these “oddball” bursts, and invent a new terminology: Type I and II bursts similar to the SN classifica- tion scheme (Zhang et al., 2007). In this scheme, eight different proper- ties have to be considered for each burst/host. However, this scheme can be ambiguous (e.g. GRB060505) and is not operational, i.e. involves observables that are not available for most bursts (associated SN). Using proposed hosts (i.e. a nearby bright galaxy) to make a distinction be- tween the two burst populations can also be risky (e.g. GRB060912A: Levan et al., 2007). In addition, one might envisage a Type III category consisting of the new type of bursts (massive white dwarf/neutron star merger) suggested by King et al. (2007). These could produce long bursts definitely without an accom- panying SN and have a strong corre- lation with star formation. However, rare members of the class need not be near star-forming regions, and could have any type of host galaxy. It is clear that the two SN-less long bursts from last summer have raised a few warning flags, i.e. how we think about the long/short dichotomy. At this point in time, we only recom- mend that people keep an open mind. Acknowledgements We thank all the co-authors of the Fynbo et al. (2006) paper. PJ ac- knowledges support by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Program under contract number MEIF-CT-2006-042001. References Belczynski, K., 2007. NewAR (this is- sue). Cobb, B. E., et al., 2006. ApJ 651, Della Valle, M., et al., 2006. Nature 444, 1050. Donaghy, T. Q., et al., 2006. ApJ, submitted (astro-ph/0605570). Fryer, C. L., et al., 2006. ApJ 650, 1028. Fugazza, D., et al., 2006. GCN 5271. Fynbo, J. P. U., et al., 2000. ApJ 542, Fynbo, J. P. U., et al., 2006. Nature 444, 1047. Gal-Yam,A., et al., 2006. Nature 444, 1053. Gehrels, N., et al., 2006. Nature 444, 1044. Heger, A., et al., 2003. ApJ 591, 288. Hjorth, J., et al., 2003. Nature 423, Holland, S. T., 2006. GCN 5255. Jakobsson, P., et al., 2005. ApJ 629, King, A., et al., 2007. MNRAS 374, Levan, A. J., et al., 2007. MNRAS, submitted. MacFadyen, A. I., 2003. In AIP Conf. Proc. 662, ed. G. R. Ricker & R. K. Vanderspek, 202. Ofek, E. O., et al., 2007. ApJ, sub- mitted (astro-ph/0703192). Schaefer, B. E., & Xiao, L., 2006. ApJ, submitted (astro-ph/0608441). http://arxiv.org/abs/astro-ph/0605570 http://arxiv.org/abs/astro-ph/0703192 http://arxiv.org/abs/astro-ph/0608441 Thöne, C. C., et al., 2007. ApJ, sub- mitted (astro-ph/0703407). Woosley, S. E., 1993. ApJ 405, 273. Zhang, B., et al. 2007. ApJ 655, L25. http://arxiv.org/abs/astro-ph/0703407 Introduction Discussion High extinction? Wrong redshifts? No SNe: a problem? Classification problem? Acknowledgements

0704.1422

A new, very massive modular Liquid Argon Imaging Chamber to detect low energy off-axis neutrinos from the CNGS beam. (Project MODULAr)

Microsoft Word - ultimate3mod4.doc A new, very massive modular Liquid Argon Imaging Chamber to detect low energy off-axis neutrinos from the CNGS beam. (Project MODULAr) B. Baibussinov1, M. Baldo Ceolin1, G. Battistoni2, P. Benetti3, A. Borio3, E. Calligarich3, M. Cambiaghi3, F. Cavanna4, S. Centro1, A. G. Cocco5, R. Dolfini3, A. Gigli Berzolari3, C. Farnese1, A. Fava1, A. Ferrari2, G. Fiorillo5, D. Gibin1, A. Guglielmi1, G. Mannocchi6, F. Mauri3, A. Menegolli3, G. Meng1, C. Montanari3, O. Palamara4, L. Periale6, A. Piazzoli3, P. Picchi6, F. Pietropaolo1, A. Rappoldi3, G.L. Raselli3, C. Rubbia[A]4, P.Sala2, G. Satta6, F. Varanini1, S. Ventura1, C. Vignoli3 1Dipartimento di Fisica e INFN, Università di Padova, via Marzolo 8, I-35131 2Dipartimento di Fisica e INFN, Università di Milano, via Celoria 2, I-20123 3Dipartimento di Fisica Nucleare, Teorica e INFN, Università di Pavia, via Bassi 6, I-27100 4Laboratori Nazionali del Gran Sasso dell’INFN, Assergi (AQ), Italy 5Dipartimento di Scienze Fisiche, INFN and University Federico II, Napoli, Italy 6Laboratori Nazionali di Frascati (INFN), via Fermi 40, I-00044 Abstract. The paper is considering an opportunity for the CERN/GranSasso (CNGS) neutrino complex, concurrent time-wise with T2K and NOvA. It is a preliminary description of a ≈ 20 kt fiducial volume LAr-TPC following very closely the technology developed for the ICARUS-T600, which will be operational as CNGS2 early in 2008. The present preliminary proposal, called MODULAr, is focused on the following three main activities, for which we seek an extended international collaboration: (1) the neutrino beam from the CERN 400 GeV proton beam and an optimised horn focussing, eventually with an increased intensity in the framework of the LHC accelerator improvement programme. (2) A new experimental area LNGS-B, of at least 50’000 m3 at 10 km off-axis from the main Laboratory, eventually upgradable to larger sizes. As a comparison, the present LNGS laboratory has three halls for a total of 180’000 m3. A location is under consideration at about 1.2 km equivalent water depth. The bubble chamber like imaging and the very fine calorimetry of the LAr-TPC detector will ensure the best background recognition not only from the off-axis neutrinos from the CNGS but also for proton decay and cosmic neutrinos. (3) A new LAr Imaging detector, at least initially with about 20 kt fiducial mass. Such an increase in the volume over the current ICARUS T600 needs to be carefully considered. It is concluded that a single, huge volume of such a magnitude is uneconomical and inoperable for many reasons. A very large mass is best realised with a modular set of many identical, but independent units, each of about 5 kt, “cloning” the basic technology of the T600. Several of such modular units will be such as to reach at least 20 kt as initial sensitive volume. Further phases may foresee extensions of MODULAr to a mass required by the future physics goals. Compared with large water Cherenkov (T2K) and fine grained scintillators (NOvA), the LAr-TPC offers a higher detection efficiency for a given mass and lower backgrounds, since virtually all channels may be unambiguously recognized. In addition to the search for θ13 oscillations and CP violation, it would be possible to collect a large number of accurately identified cosmic ray neutrino events and perform search for proton decay in the exotic channels. The experiment might reasonably be operational in about 4/5 years, provided a new hall is excavated in the vicinity of the Gran Sasso Laboratory and adequate funding and participation are made available. (April 9,2007) [A] Corresponding author: [email protected] Table of contents. 1.— General considerations. ...............................................................................................3 1.1. Physics introduction....................................................................................................3 1.2. Comparing present and future detectors toward . ....................................................3 2.— The next LNGS neutrino detector................................................................................5 2.1. General considerations. ..............................................................................................5 2.2. A modular approach of the LAr-TPC detector. ............................................................6 2.3. Double phase LAr-TPC signal collection? ..................................................................6 2.4. A simplified structure for the modular detectors. .........................................................8 2.5. The new experimental area. ......................................................................................11 2.6. Initial filling procedures for the chamber. .................................................................13 2.7. LAr purification. .......................................................................................................13 2.8. Photo-multipliers for light collection.........................................................................14 2.9. Electronic readout and trigger. .................................................................................14 2.10. R&D developments. ................................................................................................15 3.— The new low energy, off-axis neutrino beam (LNGS-B). ..........................................17 3.1. The present high energy CNGS beam configuration. .................................................18 3.2. A new, low energy focussing layout. ..........................................................................18 3.3. Comparing the CNGS and NOvA neutrino beams.....................................................19 3.4. Detection efficiency for νe CC events and NC background rejection..........................21 3.5. Comparisons with NOvA. ..........................................................................................23 3.6. Evaluation of the beam associated background. ...................................................24 3.7. Comparing the ultimate sensitivities to and 13( ). ......................................24 4.— Tentative layout of LNGS-B. ....................................................................................26 5.— Conclusions. .............................................................................................................28 6.— References ................................................................................................................30 7.— Appendix. General comments on the use of Perlite....................................................32 1.— General considerations. 1.1. Physics introduction. The understanding of neutrino has recently advanced remarkably with the observation that they have masses and that oscillate between each other. Oscillations arise in analogy to the CKM matrix for hadrons since the neutrino species do not have specific masses, but are a combination of the mass eigenstates . Two of these oscillations, namely related to $( ) and related to have been experimentally observed by SK1. A third oscillation type characterized by , occurring around 2 has not been observed. The observation of a non-zero value of will open the way to the ordering of the neutrino masses and a determination of the CP violation phase " in neutrino oscillations. CP violation in the lepton sector will be necessary in order to understand why matter is dominating over anti-matter in the Universe. The very small but finite values of the neutrino masses require the existence of right-handed neutrino species and more generally neutrinos appear to be related to physics at an extremely high energy scale, beyond studies with accelerator beams. It is also possible that in addition to the indicated three types of neutrino species, other species could exist, oscillated by the . There is unconfirmed evidence for the existence of this type of “sterile” neutrinos from the LNSD experiment [1] at Los Alamos National Laboratory. A search for evidence for sterile neutrinos is being pursued by MiniBooNE [2] at FNAL and ICARUS-600T [3] at LNGS. 1.2. Comparing present and future detectors toward First generation long baseline neutrino experiments are currently operational at K2K [4] over a baseline of 295 km, at FNAL and at CNGS with baselines of about 730 km. These developments should be further exploited in Japan and presumably also in the USA and Europe with some second generation experiments of much higher sensitivity, to become operational around 2010-2015. This requires major improvements both in the beam and in the detector mass and performance. The present detectors at FNAL (MINOS) [5] and CNGS2 (ICARUS) [3] are respectively a Iron-Scintillator sandwich of 2.5 cm iron and 4.1 cm wide scintillator strips with 5.4 kt total, 3.2 kt fiducial (MINOS, two modules) and a liquid Argon detector of a slightly lower mass of about 600 t of sensitive volume (ICARUS). It is important to underline that in practice these two detectors have roughly comparable discovery potential in many channels because of the much higher resolution capabilities of LAr-TPC when compared with Fe-scintillation sandwich. The main beam requirement is the average target power of the incoming proton beam (POT) that are 1 The experiment OPERA-CNGS1 is intended to observe explicitly the appearance (M. Guler et al., [OPERA Coll.], CERN/SPSC 2000-028, SPSC/P318, LNGS P25/2000). presently comparable for the CERN SPS2 at 400 GeV and the FNAL main Injector at 120 GeV, with about 170 kWatt on target. It is foreseen that a major improvement programme at FNAL will increase the beam intensity to up to 1MWatt of beam power and even beyond. An experimental proposal under consideration with a target date of circa 2012 is the NOvA [6] experiment, a totally active, fine grained scintillator of about 25 kt. In comparison with MINOS, NOvA will have (1) a much greater mass; (2) a better identification of the electron type neutrinos, with a sampling of 0.15 r.l., compared to 1.5 r.l. for MINOS; (3) about 80% of the mass is active, when compared to 5% for MINOS; (4) the beam is located off-axis, in order to increase the number of events which are most sensitive to , namely = 2 ±1 GeV ; (5) a much higher intensity neutrino beam, corresponding to 5 ÷ 25 x 1020 POT/y at 120 GeV, although this number is still subject to some uncertainties. The nominal quoted value is 6.5 x 1020 POT/y at 120 GeV. We consider here the possibility of a substantial and equivalent upgrade of a LAr-TPC detector for CNGS, having in mind competition and timetable comparable to the ones of NOvA [6] and of T2K [7]. We keep in mind that the key process is the observation of the oscillation driven events. As already pointed out, the use of the imaging capability of the LAr-TPC ensures a much higher discovery potential than in the case of scintillator (or water) detectors, i.e. a comparable sensitivity may be achieved with a much smaller sensitive mass. The scientific community at large is presently considering also conceptual designs for huge, “ultimate” detectors in the order of one or more hundreds of kiloton [8] [10], with huge costs, comparable to the ones of the LHC experiments: some R&D efforts are presently going toward this long distance goal [9]. Amongst them one has described a bi-phase LAr-TPC detector of a mass of 100 kt with amplification in the gas phase (GLACIER) [10], a huge liquid scintillator of about 50 kt (LENA) [8] and a water Cherenkov counter of 440÷730 kt (MEMPHYS) [8]. In USA and Japan two analogous projects (UNO and HyperKamiokande) have been proposed [8]. But, no doubt the next practical steps for the period 2011÷2013 are still of more modest magnitude, of the order 20 ÷ 30 kt fiducial mass. Two programmes are already under development in Japan and in the US, namely the T2K combination of a new high intensity 50 GeV proton accelerator aiming at the well known SK water Cherenkov detector [7] and the NOvA scintillation detector of similar fiducial mass with neutrinos from the 120 GeV Full Energy Injector at FNAL [6]. They both are intended to operate with off-axis neutrino beams and for an optimum neutrino energy window of 2 ± 1 GeV. The present paper is considering opportunities for the CNGS neutrino complex after the completion of the present OPERA/ICARUS phase, which should be completed by about 2011, concurrent time-wise with T2K and NOvA. 2 However the fraction of time dedicated to neutrino beam is smaller at CERN than at FNAL The assumed efficiency for the parasitic operation at CERN is ≈ 50%, corresponding to a nominal 4.5 x1019 POT/y, equivalent to 400/120*4.5 1019 = 1. 5 x1020 POT/y at FNAL and 120 GeV. The neutrino event rate is of 3000 ev/kt/y for FNAL and 2800 ev/kt/y for CERN. 2.— The next LNGS neutrino detector. 2.1. General considerations. The T600 detector is now readied in Hall B and it is expected to become operational before the end of 2007. We will collect in the subsequent years a large number of beam associated and of cosmic ray events that will perfect the technology and provide a rich amount of experimental physics. The T600 is therefore considered a necessary step toward the realisation of any much larger LAr-TPC detector. Running of the T600 will provide absolutely essential experience, which is required in order to develop sensibly such a “next step”. Evidently, a number of modifications are required in order to ensure the scalability of a detector to much larger sizes. These are presently under active consideration. The present CNGS proposal is focused on the following three main activities, for which we seek a larger international collaboration: 1. a new neutrino beam configuration derived from the existing horn focussing and the existing proton beam line from the 400 GeV SPS, eventually with an increased intensity in the framework of the LHC related accelerator improvement programme. Relatively modest changes in the neutrino beam focussing of CERN will produce a nearly optimal beam configuration. 2. A new experimental area, eventually enlarged in future phases, which we indicate as LNGS-B to be realised about 10 km off-axis from the main laboratory, away from the protected area of the Gran Sasso National Park, without significant underground waters and with a minimal environmental impact. A provisional location is under consideration, corresponding to about 1.2 km of equivalent water depth. The high event rejection power of the LAr-TPC detector will ensure the absence of backgrounds not only for the neutrinos from the CNGS but also for proton decay and cosmic neutrinos. 3. A new LAr-TPC Imaging underground detector made of several modular units, each of about 5 kt fiducial mass. As a first step, a total of about 20 kt will be realised with appropriate safety requirements and along the lines of the vast R&D work carried out over the last decades by INFN and other International Institutes and culminating in the actual operation of the T600, foreseen in 2008 with the ICARUS experiment. This programme may eventually be improved further on with additional modules, depending on the developments of the programmes with and without accelerators. The forthcoming operation of the T600 detector in the real experiment LNGS-B will represent the completion of a development of the LAr-TPC chamber over more than two decades. As it is described in this paper, the operation of the T600 evidences the large number of important milestones which have been already achieved in the last several years, opening the way to the development of this new line of modular elements and which may be extrapolated progressively to the largest conceivable LAr-TPC sensitive masses. As described later on, the new detector will maintain the majority of components that have been developed with industry for the T600. The detector should be easily upgraded in the far future to a larger scale, depending on the potential physics goals. The off-axis physics programme is not making obsolete the on-axis searches, presently concentrated on the T600, which both contribute to a wider physics programme and may also profit of the advances offered by the MODULAr concept. 2.2. A modular approach of the LAr-TPC detector. Conceptual designs have been described in the literature [11] with a single LAr container of a huge size, up to 100 kt. But already in the case of containers of few thousand ton the geometrical dimensions of most types of events under study (beam- " , cosmic ray- proton decays) are relatively confined, i.e. much smaller than the fiducial volume. Hence increasing the container’s size does not appreciably affect the acceptance in fiducial volume of each event and introduces no significant physics arguments in its favour. We believe that there are instead serious arguments for which such an huge size approach cannot be easily realised in practice and that suggest instead the use of a modular structure of several separate (identical) vessels, each one however of the size of a few thousands ton. In case of an accidental leak of the ultra-pure LAr, the amount of liquid that is spoiled is proportional to the actual volume of the container. Segmentation is therefore useful in overcoming events due to poisoning of the liquid. In the case of a major damage of the detector, the liquid can be provisionally transferred to another container. An additional, reserve vessel of the order of 100 kt is, on the other hand, not realistic. In addition, the safety requirements of an underground vessel are strongly dependent on its size. One of the most relevant features of LAr-TPC is its ability to detect accurately ionisation losses at the percent level. Over the very large volume, the inevitable in- homogeneities in electron lifetime due to even modest variations in purity of the LAr produce very large fluctuations in the actual value of the collected charge and hamper the possibility of charge determination along the tracks. Therefore we have chosen to use a modular approach of sufficient size in order to reduce the effects due to the non-uniformity of the electron collection due to the emergence of negative ions, which impose a reasonably short maximum drift distance of each gap. As it will be discussed further on, it has been assumed that a reasonable sensitive volume should be of 8 x 8 m2 cross section and a length of about 60 m, corresponding to 3840 m3 of liquid or 5370 t of LAr. The drift length is 4 m. A field shaping grid should be added in the middle of the HV gap in order to reduce the effects due to the space charges to a negligible level. A reasonable three-plane wire pitch for such a large container should be of the order of 6 mm, twice the value of the T300. 2.3. Double phase LAr-TPC signal collection? Several years ago [12] the ICARUS collaboration had studied a double phase Noble gas arrangement, in which ionization electrons from the tracks are drifted from the liquid to a superimposed gaseous phase. Electrons were further accelerated and ionised with the help of a grid, like in an ordinary gaseous TPC, before being collected by the readout wire planes. Dark matter searches in Argon (WARP) [13] and in Xenon (XENON) [14] have profited of this technique. Essentially the same idea has been also envisaged some time ago for very large (≥ 100 kt) monolithic LAr-TPC detectors. In the specific case of GLACIER [10], it has been proposed a very large electron drift length of 20 m at 1 kV/cm in LAr, corresponding to a drift voltage of as much as 2 MV, about a factor ten larger than the one discussed in the present proposal. If one assumes that the free electron lifetime is at least 2 ms [10], one expects an attenuation of the free electrons due to ion recombination in the impurities, presumably Oxygen, of as much as a factor ∼150 after 20 m. This residual free electron component cannot be directly recorded electronically (as in ICARUS) over the whole drift distance and must be therefore first amplified by the proportional gain profiting of the gaseous phase. In practice, taking into account the large capacitance of the extremely extended read-out electrodes (up to 70 m) and consequently of the larger noise signal from the input FET, gains of the order of 103 are typical required values. The instrumentally increased dynamic range of the signals collected must take in full account this huge dynamic factor along the drift time extent. Therefore in the double phase arrangement, all three different types of charged particles have to be simultaneously considered, namely (1) the initial free electrons from Argon which are attenuated over the distance by as much as two orders of magnitude, (2) the accumulated negative ions from recombination by impurities and (3) the positive Argon ions especially from the multiplication near the wire in the gas. The ion speeds in liquid are extremely slow, typically far less than 2 mm/s3 at 1 kV/cm (with 2 MV over 20 m the drift time is ≈ 10’000 s, i.e. about 3 hours!). It has been demonstrated in WARP and XENON experiments that free electrons can overcome the liquid to gas barrier in the presence of a sufficiently strong electric field of a few kV/cm. It is expected that both the positive ions travelling from gas to liquid and the negative ions travelling from liquid to gas, because of their larger masses and hence smaller speed, will be ultimately trapped and accumulate at each side of the liquid-gas boundaries for a so far unknown period of time. The free electrons crossing the double ion layer cloud are presently under study with the 2.3 litre WARP detector underground at the LNGS. There is some preliminary evidence already in this small detector that space charges due to ion crossing at the boundary may introduce additional fluctuation in the electron ionisation signal. This will introduce a substantial worsening of one of the most relevant features of LAr-TPC, namely its ability to detect accurately ionisation losses. The consequence for a detector of the size of 70 million litres and a diameter of 70 m, is an enormous extrapolation, which requires a very extensive R&D. The phenomenon of transfer of ions through the interface is expected to be rather complicated, not well understood and it has not been conclusively measured experimentally [15]. In the case relevant to GLACIER, it is likely that some trapping times may be ultimately occurring, but 3 We remark that on such a timescale of hours convective motions inside the vessel become very relevant and they may seriously modify the time of drift of positive ions in either direction. experimental studies are needed to assess how much. This is an absolutely crucial point prior to a successful, practical realization of a huge dual phase experiment in Argon. A first step is the 5 m long prototype ARGONTUBE [16] under study on surface in Bern, which will allow to experimentally verify these hypotheses and prove the feasibility of detectors with long drift paths, representing a very important milestone in the conceptual proof of the feasibility of the dual phase detector in Argon. It is therefore concluded that at least at the present stage of the LAr-TPC, the single phase geometry which has been already very well developed experimentally [3] is vastly preferable and associated to a drift path length which could minimize the extent of the negative and positive ions. Negative ions in this configuration are smoothly drifting and directly captured by the collecting wires with a negligibly small signal (the electric signal is proportional to the drift speed, a factor ≈10-5 smaller for ions). Positive ions are straightforwardly collected at the cathode. For a sufficiently small drift volume, like the one described in the present proposal, the electric field distortions due to the slow ion motion can be made to be negligible. 2.4. A simplified structure for the modular detectors. The structure of the detector has been considerably streamlined in order to reduce the number of components, its cost and increase the reliability of the system. The modular structure permits to repeat the initial engineering design of the prototype to a series of several subsequent units, reducing progressively their costs and their construction time. Clearly the main aim of the detector is the one of filling and maintaining over many years a very large amount of ultra-pure LAr in stable conditions inside a dedicated underground cave, within very rigid safety conditions. The initial filling procedure is determined by the supply rate currently provided by the supplying industry. We believe that the maximum rate available in the European market is of the order of 200 m3/d. At the rate of 100 m3/d, the initial filling of the required about 25 kt of ultra-pure LAr (corresponding to a volume of 17’000 m3) is about 170 days, which is acceptable. At the commercial cost of 0.7 Euro/l, the value of the LAr for the initial filling is about 12 MEuro, which is also quite acceptable. The mechanical structure of each of the modular units should be as simple as possible, keeping the costs of the various components commensurate to the relatively modest initial investment for the LAr. The wire arrangement is scaled out from the industrial realisation of the existing ICARUS-T600, which is taken as a reference design. As well known ICARUS is made of two identical modules (T300). In each of the T300 made of two readout planes and a high voltage plane in a double gap configuration, the three readout planes have coordinates at 0° and ± 60° with respect to the horizontal direction. This identical geometry is “cloned” into a larger modular detector, with the linear dimensions scaled by a factor 8/3 =2.66, namely the cross sectional area of the planes is now 8 x 8 m2 rather than 3 x 3 m2. The wires in the longitudinal direction were originally 9.4 m long with the wire planes subdivided in two equal segments. In the next step the length will be quantised also into two individual wire sets, but 25 m long, corresponding again to the ratio 25/9.4 = 2.66. The longer wires have a higher capacitance and the signal/noise ratio is significantly decreased (wires, of the order of 10pF/m; cables, of the order 50pF/m). This factor is compensated widening the pitch to 6 mm, to be compared to the previous 3 mm, doubling the dE dx signals. Therefore we expect signal/noise ratios which are rather similar to the ones of the T600, namely of the order of 10/1. As it will be discussed later on, in collaboration with industry (CAEN), over the last several years the electronic chain from “wire to computer” has been considerably improved in performance and reduced in cost4. Each wire is now observing a time projected volume which is a factor 2.66 x (4/1.5) x 2 = 14.2 larger than in the case of the T600 (wire length x drift length x wire pitch). Therefore the average LAr mass observed by each TPC readout wire is about 200 kg/channel. A 20 kt sensitive volume will then require of the order of 105 wires. At this stage the configuration of the modules may not be considered as absolutely frozen and a number of possible configurations are possible, as shown in Figure 1, maintaining as a reference a readout plane dimension of 8 x 8 m2. Figure 1a represent the previously indicated basic configuration of a scaled T300 double gap arrangement. The nominal voltage of the T300 is 75 kV for the 1.5 m long drift, corresponding to a drift field of 500 V/cm, although the field-shaping electrodes have been currently operated without problems up to 150 kV. The engineering design for a T1200, never constructed, required a 3 m drift length. At the same nominal electron drift velocity (500 V/cm), for the present choice of 4 m drift, the HV would be 200 kV. However a significantly higher field, like for instance 350 kV, will shorten the drift time, 4 The estimated commercial cost of each channel “wire to computer” is now about 60 Euro. Assuming for each wire a sensitive volume of 200 kg, the levelized cost of the electronics is 60/200 = 0.3 Euro/kg, which is about ½ of the cost of the Argon procurement. The low cost achieved for such a sophisticated electronics is an additional argument in favour of a single phase LAr configuration with a relatively short drift gap distance, along the lines of the T600, rather than the double phase GLACIER arrangement. Figure 1. Various possible alternative arrangements for a modular unit. In (a) we show a scaled up T300 configuration with 4 m drift time and about 25’000 readout wires. The sensitive mass is about 5000 ton. In (b) the configuration is the same as in (a), except the drift distance has been reduced to 2 m and doubling the number of readout wires. In (c) we show a scaled up T600 configuration with a twin module, 4 m drift time and about 50’000 wires. The sensitive mass is then 10’000 tons. drift # sqrt(E drift )) and reduce correspondingly the requirements of purity for the LAr to the case already optimised of a T1200 with a 3 m drift. In Figure 1b we have doubled the number of wire planes in order to reduce the drift distance to 2 m. In this configuration the halving of the drift time to the already successful configuration of the T300 is performed doubling the number of signal wires to 50’000, with a significant increase in the cost of the channels. In order to maintain an electron drift time exactly the same as the one of T300 since drift # sqrt(E drift ) we need an increase of the drift field to 75kV " 2 /1.5( ) = 89kV . Note that the HV of the T300 has been tested up to 150 kV without any problem. Although we believe that the drift distance can be safely extended to 4 m, this alternative shows that the choice of the electron drift length is not determinant. Solution 1b, eventually with an even higher drift field to reduce the maximum drift time to values below the ones of the present T600, is perfectly possible in case that some unforeseen problem may develop, obviously at the cost of doubling the number of electronic channels. Finally in Figure 1c we show a scaled up T600 twin volume configuration, with 4 m drift time and about 50’000 wires. The two volumes are physically separated, but they are both kept in the same cryogenic volume. The total sensitive mass of one 1c module is then 10’000 tons. The new proposed halls of the LNGS and 20 kt could host two modules of type 1c in one container. Considerable experience of the ICARUS collaboration has shown that free electron drift times drift of several milliseconds are currently realised with commercial purification systems based on Oxysorb™. The effects on the electron attenuation are shown in Figure 2 where the drifting charge attenuation versus drift path at different electric field intensities are given for drift = 10 ms and for different electron lifetimes at 0.5 kV/cm. Figure 2. The maximum free electron attenuation into negative ions is shown for different values of the maximum electron drift path, respectively (1) for different values of the free electron lifetime at E = 0.5 kV/cm and (2) for different electric drift fields and a 10 ms electron lifetime. Cases (a) and (b) represent respectively the configurations of Figures 1a and 1b with 4 m and 2 m drift paths. The longitudinal r.m.s. diffusion spread after an electron drift path x and moving at a speed is given by = sqrt 2D x v D( ) , where D = 4.06 cm "1 . In more practical units, # 0.9sqrt $ drift ms( )[ ] mm . For a drift field of 0.5 kV/cm and a 4 m path the average value =1.1 mm and the maximum value is D max =1.6 mm . The new mechanical structure, which has been highly streamlined, is essentially made of only three main mechanical components: (1) An external insulating vessel made of two metallic concentric volumes, filled in between with perlite (see Appendix for details). Perlite is a mineral which is vastly used industrially, the world consumption being of the order of 2 million tons annually5. The environmental aspects of perlite are not severe: mining generally takes place in remote areas, and airborne dust is captured by bag-houses, and there is practically no runoff that contributes to water pollution. In order to ensure an adequate thermal insulation, about 1.5 m thickness is required, corresponding to over 3000 m3 for a container. The bottom-supporting layer is made out of low conductivity light bricks. The specific heat loss is 3.86 W/m2 for a nominal thermal conductivity of 0.029 (0.025-0.029) W/m/K. This is significantly smaller than the specific heat loss of the T600. Taking into account the dimensions of the vessel, the total heat loss is 8.28 kW. At present in the LNGS the cryogenic plant of ICARUS T600 is made of 10 units, each with 4 kW of (cold) power. Three of such units (≤ 12 kW) should be adequate to ensure cooling of the walls of the vessel during normal operation. Evacuation of the perlite is therefore unnecessary. (2) A linear supporting holding structure frame with wire planes at each lateral side and the high voltage plane at the centre. The photomultipliers for the light trigger are also mounted on this frame behind the wire planes. The structure of the planes is identical to the one already developed for the T600, except that only one wire out of two is installed in order to go from 3 mm to 6 mm pitch. The inner structure of the huge container is therefore extremely simple, being primarily a linear wire structure along the edges of the container, the rest remaining essentially free of structures. (3) The liquid Ar and N2 supply and refrigeration, provided with cooling and purification both in the liquid and gas phases, with an appropriate re- circulating system to ensure that the whole liquid is moving orderly inside the vessel volume to unsure uniformity of the free electron lifetime. 2.5. The new experimental area. The ICARUS-T600 detector is located inside the Hall B of the LNGS laboratory in an appropriate containment tank constructed above the floor of the Hall. The new experimental 5 The present production in Greece, where vast resources are available, is about 500 thousand t/y. The estimated cost is about 40 $/ton. area, which we indicate as LNGS-B, to be realised about 10 km off-axis from the main laboratory, away from the protected area of the Gran Sasso National Park, without significant underground waters and with a minimal environmental impact. A provisional location is under consideration, corresponding to about 1.2 km of equivalent water depth. The high event rejection power of the LAr-TPC detector will ensure the absence of backgrounds not only from the neutrinos from the CNGS but also for proton decay and cosmic neutrinos. The total volume excavated for the original LNGS was of about 180’000 m3. It is foreseen that the new LNGS-B could be about one half of this volume, namely initially about 50’000 m3. However, as a difference from the main LNGS, the shapes of the cavities, rather than being vast, general purpose halls, are tailored to the specific experiment. Each modular detector unit is located in an appropriate “swimming pool” cave in the rock concentric to the perlite walls, where the liquid tank is contained: therefore there is no realistic possibility of leak outside the walls of the rock for any foreseeable circumstance. The worst case is the total loss of external cryogenic cooling both of N2 and of Ar. Therefore Figure 3. Indicative cross section of the T600 “clone” in the dedicated “swimming pool like” underground hall. The lower part is made of two twin separate LAr containers made of Aluminum extruded structures, thermally stabilized with forced N2 circulation. Outside the structure an about 1.5 m thick perlite wall provides spontaneous, passive heat insulation. The region on top of the “swimming pool” is accessible to auxiliary equipments. Personnel access is strictly controlled. the tank will spontaneously warm up in contact with the heat leaks of the surrounding components through the 1.5 m thick perlite wall. Assuming a heat leak rate of 10 kWatt, the LAr evaporation rate is of 220 kg/h, negligibly small with respect to the 5 kt of the stored LAr tank. Therefore the tank will remain stable in its liquid form for any specified length of time. More generally there is not even the most remote possibility to provide from the environment around the cavern a sufficiently large amount of heat in order to cause a catastrophic evaporation of a massive amount of LAr. For a configuration of the type 1c, the cross section is 11 x 20 m2 (shown in Figure 1) and the length is about 60 m. Different containers may have entirely separate halls since the event containment is anyway very good. An exhaust pipe is necessary in order to evacuate the evaporated liquid into the atmosphere in case of an accidental leak, although a risk analysis will certainly show that the probability of such events is very small. 2.6. Initial filling procedures for the chamber. In the present T600 the vessel is evacuated in order to inject ultra pure Argon. The new detector, in view of its large size is very hard to evacuate and a new method has to be applied. The idea is to perform successive flushing in the gaseous phase in order to attenuate the presence of gases other than Argon with an approximately exponential chain. This method of flushing with pure Argon gas is widely used already in gaseous wire and drift chambers which are generally not evacuated. In the present case, additional problems may arise in view of the magnitude of the volume and the possibility of creating for instance “dead” spots, in which the gas may not circulate. A suitable small scale test dewar container is under construction in order to perfect the method. In the idealised case of complete turbulent and continuous uniform mixing through the container, the transition air-argon is an exponential with a factor ≈ 1/2 at each passage. Therefore, in order to achieve an attenuation of the order of 10-6, 14 cycles are necessary. If instead the Argon is injected with little or no turbulence, for instance uniformly from the bottom with the extraction of initial air on the top, the transition argon-air moves orderly from the bottom to the top and only pure air exits from the top, producing a faster and more orderly transition. These are limiting cases and the efficiency of the actual filling will need some model studies and some hydrodynamic calculations to be perfected. Some preliminary considerations indicate that about 6 cycles may be necessary. The density of the gaseous Argon at room temperature is about 600 times smaller than the one of the liquid. Hence a ultimate gas purification of the order of 10-6 would correspond to an increment due to filling of the order of 2 x 10-9 (2 ppb) with pure liquid, which is adequate for the initial filling before local purification. 2.7. LAr purification. In order to ensure a free electron lifetime adequate for the longest ≈ 3ms fly-path, a vigorous purification of the LAr must be kept at all times with filtering methods based on Oxysorb™ and molecular sieves. In analogy with what is currently performed with T600 and all previously constructed detectors, the purification is performed both in the liquid and in the gaseous phase. An improvement in the purification system is needed to enlarge in a significant way the TPC volume. New purification devices have to be implemented, possibly operating either near or directly inside the cryostat. They should be simple, robust and without moving parts, to guarantee a total reliability. An order of magnitude of the liquid re-circulation rate needed to reach safely the running condition in few months (for example 60 days) could be of the order of several percent of volume per day. For a volume of 4000 m3, this rate means a re-circulation rate of the order of 240 m3/day (6%/day or purification cycle in 16.6 days), which is in the range possible with Oxysorb™6. In conclusion, some thousands cubic meters is a reasonable limit for a single TPC volume. It could be cleaned, cooled and filled in few months and then kept completely operative (with an adequate LAr drift length) after few months. Altogether, such a detector could be put in operation in a reasonable period of time. 2.8. Photo-multipliers for light collection. Like in the case of T600 a number of photomultipliers located behind the readout wires are used to provide a t=0 trigger. This is particularly important for the cosmic rays and proton decay events in which no starting signal is provided, but could be as well very useful in order to tag events coming from the CNGS beam. The technique already used in the case of the T600 consists in glass phototubes with a thin deposit of wave-shifter in order to record the scintillation light from the LAr. A significant contribution is also due to the Cherenkov light emitted directly in the visible by relativistic particles. 2.9. Electronic readout and trigger. The ICARUS T600 detector has a DAQ system (5·104 channels) designed at University of Padova/INFN, engineered and built by CAEN. It has proven to perform quite satisfactory in the test run performed in Pavia during summer 2001. The electronics has been described in various papers and technical notes. We remind here that it is based on an analogue front-end followed by a multiplexed AD converter (10bit) and eventually by a digital VME module that performs local storage and data compression. In the following, starting from the experience gained in the T600 operation, we discuss performance and limits of the actual system with the aim of improving its characteristics in view a multi-kton TPC with a number of channels in the order of several times x 105. Since 1988, in the ICARUS proposal, the main characteristics of the signals were described and subsequently they were confirmed by tests on small chambers and eventually by the operation of the T600. In a multi-kton TPC we can foresee wires (or in general electrodes) with a pitch larger than the 3mm used in the T600. A reasonable assumption would be a 6mm pitch that will allow using most of the tooling already built and designed for the T600. 6 The standard rate of a single Oxysorb™ pack is about 120 m3/d. Therefore two of such units are sufficient for the chosen size of the vessel. The capacitance associated to each channel will be determined by the capacitance of the wires, in the order of 10pF/m, in parallel with the capacitance of the cable, in the order 50pF/m. Let's assume in the following discussion ~600 pF as a reasonable number for 10m electrode wires and average 8m of cable. The dominant noise in a high capacitance detector is the series noise esn (voltage noise) linearly increasing with the input total capacitance (CD) while the parallel noise (current noise) contribution is proportional to the shaping time of the signal. We propose to use the present IC taking into account that due to the need of spares for the T600 a silicon run of the specific BiCMOS process of 6 wafers has been recently made. Each wafer, taking into account the known yield, contains some 12000 good circuits which means 24000 channels. The total number of channels that could be equipped is about 140*103. These wafers, kept in inert atmosphere, can be easily packaged in very small cases (4x4 mm2). A R&D program is also proposed for the development of a hybrid sub module hosting eight or more channels of amplification and eventually, as it will be described later, also the analogue to digital converter. At present a revision of the ICARUS analogue electronics is underway with the aim of further improving the front-end performance. In the T600 collaboration a novel technique for the realization of feed-throughs has been developed. INFN holds a patent (RM2006A000406) for this technology that allows easy realization of feed-through with high density vias and different shapes. The ADCs work at 20Mhz sampling rate, interleaved so the 10bit digital output has a 40Mhz frequency that means that each channel is sampled every 400ns. The power dissipated is significant: 500mW. The required bandwidth taking into account that two sets of 16 channels are merged in a 20 bit word, is 800Mbit/s for 32 channels. The main feature of the new design is to move into digital domain all the conversion process at a very early stage and to exploit the use of numerical digital filtering techniques. The final quality of the converted data is highly dependent on the sampling frequency and numerical filtering. The trigger system will divide the detector volume in sub-volumes to cope with the data acquisition rate required by shallow depth location. The basic structure will reproduce the one already implemented in the T600 and it will be based both on analogue signals from wires (sums of set of wires) and scintillation light detected by PMs inside the liquid Argon. All together will merge with the DAQ architecture taking into account that time resolution required is low (must be compared with drift time) and anyway the absolute time will be associated to each triggered event. 2.10. R&D developments. The increase of the active LAr volume of about one order of magnitude with respect to T600, the streamlining and simplification of the mechanical structures and the new developments previously described in the structure of the detector require some specific R&D developments, which are not of very substantial nature and could be implemented in parallel with the detailed engineering design of MODULAr. These developments (see Figure 4) are intended to simulate on a small scale the basic innovations with respect to the present T600, namely: (1) The filling process starting from air to pure LAr, taking into account the motion of the gas, optimising the inlet and outlet geometries and minimising the number of cycles. (2) The thermal convections of the LAr, in order to optimise the temperature gradients and to ensure a convincing circulation in all regions of the dewar, both in the cooldown phase and in the stationary state. (3) The outgassing rate and the recirculation processes required in order to achieve the required electron lifetime (4) The geometry of the compact re-circulators both in the liquid and in the gaseous phases. The cryostat is based on a foam glass platform (the simplest solution, industrially well tested and implemented) and is surrounded by a perlite-insulated walls, about 1-meter thick. This approach for the thermal insulation should be cheaper even if bulkier. The instrumentation must be well developed in order to be able to determine both thermal and convective measurements in a variety of conditions. Finally, also electronics, starting from the positive experience of the T600 may require some specific development mainly on the layout improvement of the analogue part. Figure 4. Layout of the test unit. This small unit is necessary in order to certify all the major changes with respect to the present T600 unit. The cross section is 5 x 5 m2 and the height is about 8 m. 3.— The new low energy, off-axis neutrino beam (LNGS-B). The primary goals of future experimental programmes in Japan (T2K), US (NOvA) and our present LOI at CNGS are related to the so far unknown angle 13( ), as a pre- requisite for a non zero CP-violation phase in the lepton sector. They are all based on a neutrino beam with horn focussing and a fine grained detector of about 20-25 kt of fiducial volume. As it is well known, the highest sensitivity occurs at distances corresponding to the maxima and minima of the cosmic neutrino oscillations, namely at the energy interval (2 ± 1) GeV for neutrino distances of 830 km (120 GeV) and 730 km (400 GeV) respectively at the NOvA and CNGS detectors since the oscillation maximum for the CNGS baseline, assuming Δm223 =2.5 10 -3 eV2, is at 1.5 GeV. The T2K detector will be driven by a 50 GeV proton beam and a smaller distance of 295 km and therefore should require correspondingly smaller energies. In order to optimize a horn driven, conventional neutrino energy spectrum it is now universally agreed that the next configuration may be obtained using a high energy beam and looking at an off-axis direction in order to shift the Lorentz boost of the relativistic beam toward the required energy range, typically in the case of the LNGS and FNAL with displaced path distances of several kilometers from the main beam axis. As well known this detector will also produce a large amount of additional non- accelerator physics results, and in particular of (1) proton decay especially in the SUSY channels and (2) of cosmic rays neutrino events, with considerable improvements with respect to SuperK. In order to be able to record simultaneously (pulsed) accelerator and (continuous) non-accelerator events, the detector must be triggered by the inner photo-multiplier array, as already indicated by the T600. On a longer timescale it may be possible to realize the entirely new technology of beta-beams [17], where the purity of is extremely high, namely a contamination well below 10-4. Evidently the residual contamination due to neutral currents must also be reduced correspondingly. As shown in Ref. [18] the presently developed LAr technology will be capable of realising such required Figure 5. The present 400 GeV CNGS neutrino CC interaction signal in all its oscillated channels at 731 km, at three different distances off-axis and on-axis. For instance, the value for 12 km off axis corresponds to 16 mrad or 0.94°. powerful identification capabilities. Therefore the massive LAr-TPC detector presently under consideration may continue to be used also in this ‘ultimate” phase, eventually increasing the number of modular units. 3.1. The present high energy CNGS beam configuration. The present 400 GeV CNGS beam, optimized for τ appearance experiment, is a high- energy wide band beam with an average on-axis energy of 18 GeV. The horn/reflector optics is designed to focus in the forward direction secondary particle in the 20-50 GeV energy range with an angular acceptance of 20-30 mrad. The thin target is optimized to minimize pion/kaon re-interactions. The integrated contamination is 0.6%. From comparisons with the previous WANF beam and the early observations with CNGS, the systematic uncertainty on the calculated νe /νµ ratio is expected to be very good, of the order of 5% [19]. Therefore the intrinsic contamination, although significant in the search for oscillations, may be very precisely calibrated. In order to visualize the situation we consider the present CNGS beam geometry (Figure 5). Displacing the detector off-axis with respect to the neutrino beam direction by several kilometres at the location of the detector strongly reduces the shape of the neutrino energy spectrum, which, as expected, becomes progressively softer as the distance from axis is increased. The strong " CC signal at 0 km is progressively shifted to smaller energies with a much smaller over-all rate, although significantly enhanced at the optimum energy. 3.2. A new, low energy focussing layout. The standard 400 GeV CNGS optics as such is not optimized for e oscillation searches, neither on-axis nor off-axis. In particular, the small angular acceptance of the magnetic lenses, around 20-30 mrad, limits the neutrino fluxes at low energies. Optimization of the target and beam optics with low energy focussing has been calculated and it is in progress. It indicates that a design close to that proposed by the NOvA collaboration is also applicable in the case of CNGS. The main changes with respect to the present CNGS design are: (1) A more compact (without air gaps) and thicker target, to increase tertiary production at low energy; (2) Larger acceptance (up to 100 mrad) of the horns system in the 7 – 20 GeV pion energy (this momentum range offers a kinematically efficient production of 2 – 3 Gev neutrinos at 10 –15 mrad off-axis). A shorter tunnel length (between one half and two third) could also be considered because it would marginally reduce the flux due to muon decays that primarily happen in the downstream part of the decay tunnel; this major upgrade of the civil engineering has not been considered in the present study. In Figure 6 we show the expected event rate, for a hypothetical oscillation equal to the present CHOOZ upper limit =11°, # = 0 and the intrinsic contamination from the beam. Of course this is the highest expected signal and most likely the oscillatory signal is of much smaller size. Calculations are the result of a full beam simulation based on the FLUKA MonteCarlo code [20]. Such new low energy CNGS optics (see Figure 7) is very similar to the one proposed for NOvA at 120 GeV, taking into account that the energies of the protons on target (POT) are in the ratio 400/120 = 3.3. Preliminary calculations at 14.8 mrad off-axis and at a baseline of 732 km demonstrate that the neutrino spectra obtained with a proton beam of 400 GeV are very similar in shape to those obtained with 120 GeV protons, except for the yield, substantially higher at 400 GeV. The relative behaviour can be easily factorized in a good approximation. While the resulting neutrino beam intensity grows roughly proportionally with proton energy and therefore the neutrino flux in the interesting beam energy region is roughly proportional to the beam power on target, the detailed shapes of both the initial and of the intrinsic beam contamination are relatively unaffected by the proton energy and depend primarily on the choices of the target/horn system, the focused momentum range and the secondary particle acceptances. The beam intensities at 400 GeV and 120 GeV are shown in Figure 7. The plot shows that in the focused energy range the ratio of the muon fluxes due to the higher proton energy is about 2.6, namely about 80% of the linear increase expected on the energy alone, slowly growing to the full proton energy factor for the higher energy tail. An entirely similar effect is observed for the ratio of the intrinsic beam contaminations and therefore no appreciable difference is also observed in the background. 3.3. Comparing the CNGS and NOvA neutrino beams. The CNGS neutrino beam is presently given for a rate of 4.5 x 1019 POT/y at 400 GeV. This rate is believed as generally insufficient for the future off-axis neutrino programmes. A vigorous programme is on its way at FNAL in order to accumulate in line with the NOvA’s nominal assumption7 a yearly rate of 6.5 x 1020 POT/y at 120 GeV. On the basis of the 7 Ref [6]. chap. 11,page 75 Figure 6. Calculated beam spectra for low energy focusing. Rates are estimated for 1019 protons and 400 GeV on target. Both the event rate for an hypothetical oscillations equal to the present CHOOZ upper limit =11°, # = 0 and the intrinsic contamination from the beam are shown. previous considerations (factor 2.6) this corresponds to approximately 2.5 x 1020 POT/y at 400 GeV, about a factor 5.5 larger than the present CNGS yearly performance. Evidently such a factor must be recovered at CERN. We should compare more in detail dedicated repetition rates and energies at FNAL and CNGS. The nominal NOvA’s is given for a beam intensity of 6.0 x 1013 ppp and a repetition rate of 1.5 s, corresponding to a cycle integrated beam power of 768 kW. The nominal values for CNGS today are two proton batches 50 ms apart, in total 4.8 x 1013 ppp, every 6 s, corresponding to a beam power of 512 kW, which is 2/3 of the FNAL value. Therefore the instantaneous cycles for fully dedicated accelerators are quite comparable. The differences (a factor 5.5 rather than 1.5) therefore come primarily because of the present CNGS parasitic operation with fixed SPS target, an efficiency factor of 55% in the operation of the SPS complex and a less intensive operation of the accelerator complex because of the LHC/fixed target sharing. Such “human factors” should be supported in order to reduce the gap between the two accelerators. We can assume that in several years from now, a dedicated 6 s cycle rate for the neutrino beam and an efficiency factor of 80% rather than 50% may become possible. These factors should bring the integrated intensity to 1.2 x 1020 POT/y at 400 GeV, corresponding to about 4.0 x 1020 POT/y at 120 GeV. This value is not far from the figure assumed by NOvA’s of 6.5 x 1020 POT/y at 120 GeV. With such improvements, presumably possible in several Figure 7. Muon neutrino CC event rates for 1019 protons at 120 GeV (FNAL) and 400 GeV (CNGS), calculated with the low energy focusing optics, 14.8 mrad off-axis and a baseline of 732 km and without oscillation mixing. Table 1. Rates for 5 years, 20 kt and 1.2 1020 POT /year. Oscillation with sin2(2θ13)=0.1. The upper integration limit, Εlim, has been chosen to get the best sensitivity, S sqrt bkg[ ] 0 < Ε < Εlim 0 < Ε < 10 GeV Εlim (GeV) µ e bkg[ ] Signal S S/√(bg) µ e bkg[ ] Signal, Present high energy configuration 7 km 3.5 6200 34 190 33 1000 200 240 10 km 2.5 2300 15 101 26 4600 160 130 Low energy focussing 7 km 3.5 13000 70 390 47 17500 340 430 10 km 2.5 5700 28 250 47 7800 230 280 years from now, the present CNGS beam may become roughly competitive with FNAL, the neutrino fluxes at the detector being then in the ratio 1.6 to 1. Integrated CC event rates for 5 years at 1.2 1020 POT/year in a 20 kt fiducial volume of a LAr-TPC detector are summarized in Table 1. Signal events have been calculated for sin2(2θ13)=0.1 to allow easy scaling. The actual size of the oscillation driven 13( ) is obviously unknown. In order to estimate the sensitivity for a small signal S in the presence of a significant beam associated background bkg[ ] we consider the quantity S sqrt bkg[ ] as a figure of merit as a function of the off-axis distance (see Figure 8). The upper integration limit, Εlim, has been chosen to get the best sensitivity, S sqrt bkg[ ] . It varies with the off-axis distance because of the different spectrum shape of signal and background. As a reference the rates, integrated up to 10 GeV, are also shown for an estimation of the signal selection efficiency in the case of the optimum cut (80% in the τ optics vs. 90% in the low focus case). Both at CERN and at FNAL increases in the proton beam intensity are conceivable with relatively modest efforts, but require additional money. A further increase to as much as a factor 4 has been considered at FNAL with a new 8 GeV Proton Driver, provided it will be built in the near future. Several increases of the accelerated intensity may be considered also in the case of CERN, especially based on the improvements of the CPS and in connection with the several LHC improvement programmes. In both cases the main limit may not be however the proton accelerator but rather the capability of the target/horns complex to withstand the required power. 3.4. Detection efficiency for νe CC events and NC background rejection. In the previous section, the NC background has been assumed to be negligible. This assumption is supported by extensive studies performed by ICARUS collaboration [3,21] both for beam and atmospheric neutrinos. In particular, the analysis of NC background rejection in a low energy on-axis neutrino beam [22] which has been extensively performed for the ICARUS configuration can be applied as well to the off-axis beam described in the present work. We summarize here this analysis and its results. A full simulation of the events in LAr-TPC was performed to study the background of neutral pions in both neutral current and charged current interactions, which may simulate a background of induced CC. Figure 8. Sensitivity to oscillation expressed in terms of the S sqrt bkg[ ] (intrinsic plus tau), as a function of the off-axis distance. The plot indicates the off-axis distances 7 ÷ 12 km (0.550 ÷ 0.94°, 9.6 ÷ 16.4 mrad) are the optimum. For each neutrino event was recorded the visible energy, defined as the energy deposited by ionization in the sensitive LAr-TPC volume, with the exception of the ionization due to heavily quenched recoils. The optimization of the fiducial volume was based on three criteria: energy reconstruction, electron identification and o identification. (1) The energy reconstruction due to non- containment of the neutrino events affects the signal/background ratio, especially when the signal is restricted in a narrow energy range. However, this non-containment is severe only for interactions occurring near the end of the detector and in a few centimetres lateral skin. A minimal cut of 50 cm in the longitudinal direction and 5 cm on the sides of the sensitive volume has been found sufficient to preserve the signal/background ratio. In the large modules as described in the present proposal, these cuts would reduce the fiducial volume by a (negligible) 2%. (2) Electron identification is also assured under these geometrical cuts. Indeed, due to the directionality of the neutrino beam the probability that an electron escapes from the instrumented volume before initiating a shower is extremely small: only 2 % of the electrons “travel “ through a LAr-TPC thickness smaller than 3 X0, and 0.3 % travel less than 1 X0 in the instrumented volume. Therefore we can safely assume to identify electrons with almost 100 % efficiency. (3) o identification. Neutral pions from νµ NC events could be misidentified as electrons. But because of the superior imaging capability of LAr-TPC technology, all events where both photon conversion points can be distinguished from the ν interaction vertex can be rejected. To be conservative, in the ICARUS analysis only photons converting at more than 2 cm from the ν vertex were rejected. The remaining neutral pion background was further reduced by assuming that events where the parent o mass can be reconstructed within 10 % accuracy are discarded. The effect of the last requirement obviously depends on the assumed fiducial volume. Only 4 % of o’s survive the cuts when all interaction vertexes are accepted, and a further decrease to 3% is obtained when the fiducial volume is restricted as described above. On the remaining photon sample we apply the results obtained in Ref. [23] on the possibility to discriminate electrons from photons on the basis of dE/dx. This method provides a 90% electron identification efficiency with photon misidentification probability of 3% at relatively low energies. The misidentification Figure 9. Background sources at LNGS on-axis with a low energy CNGS beam as studied for the ICARUS detector. This calculation is very similar to the expectations from the off-axis low energy beam, with the exception of the background, which is much smaller. The residual NC background in the LAR-TPC, after rejection of neutral pions, is about two orders of magnitude smaller than the intrinsic beam associated background. probability is expected to decrease with energy due to the decreasing contribution of Compton scattering. A track length of 2.5 cm is sufficient to achieve the discrimination. After all cuts, the final o mis-interpretation probability is 0.1 %, while the corresponding electron identification efficiency is 90 %. The residual background, after all the above cuts are applied, is shown in Figure 9 in the case of the study performed for the ICARUS detector exposed at a CNGS beam on-axis [22]. Given the very similar energy spectrum and background contaminations, the same signal efficiency and NC rejection power can be applied in the case of the off-axis beams. The residual NC background, after rejection of neutral pions in the LAR-TPC is about two orders of magnitude smaller than the intrinsic beam associated background. 3.5. Comparisons with NOvA. In the present letter of intent we consider the possibility of a substantial and equivalent upgrade of a LAr-TPC detector for CNGS, having in mind competition and timetable comparable to the ones of NOvA. We keep in mind that the key process is the observation of the oscillation driven events. As already pointed out, the use of the imaging capability of the LAr-TPC ensures a much higher discovery potential than it is the case of scintillator (or water) detectors, i.e. a comparable sensitivity may be achieved with a much smaller sensitive mass. The higher performance of LAr-TPC introduces important advantages with respect to NOvA, namely: • the NOvA detector is mostly limited to elastic events while LAr-TPC may collect also all kinds of inelastic events. An elaborate MC simulation of the NOvA proposal8 of the signals and backgrounds for oscillations using relevant parts of the MINOS experiment software, the NEUGEN3 neutrino interaction generator and the GEANT3 detector simulation show that the efficiency for accepting an event from oscillations is approximately 24%. This introduces about a factor 4 in rate with respect to LAr-TPC in which virtually all event configurations are identified, for the same fiducial mass; • NOvA in contrast with LAr-TPC is may be contaminated by neutral current events that fake electron events, while in reality they are due to o. The background is typically about two-thirds from beam's produced from muon and kaon decay and one-third from neutral-current events. This increases the background of events and increases further the level of the over-all discovery potential by a factor sqrt 1.5 1( ) =1.22 . We conclude that a ≈ 5 kt LAr-TPC detector should have performances comparable to the ones of NOvA. We underline that the NC cross sections are today only poorly known and therefore the magnitude of the effect, absent for LAr-TPC has to be carefully measured in separate experiments, at least as long the 13( ) signal is close to the sensitivity limit. 8 Ref [6]. chap. 12, page 78 Similar considerations apply also about the water Cherenkov counter at T2K, where apparently more stringent cuts are necessary with a corresponding reduction of the rates in order to improve the sensitivity for the small signal oscillation driven 13( ), so far obviously unknown, with respect with the o related backgrounds. 3.6. Evaluation of the beam associated background. The future CNGS low energy neutrino beam most likely will lack of an appropriate near detector. As a consequence the estimations of νe/νµ background ratio will be primarily based on the Monte Carlo simulation of the beam. The on-axis beam flux could however be determined experimentally (Figure 9) with the help of ICARUS-T600 neutrino detector presently in Hall B, measuring the neutrino spectra components with relatively high statistics (~100 νµ CC events/kt/10 19POT, peaked at 7 ± 2 GeV). Once the agreement is confirmed on- axis, it may be possible to normalize and tune the simulation for the beam off-axis. The possibility of operating simultaneously the on-axis and the off-axis events, both with very similar LAr-TPC detectors, represent a very important correlation amongst the two measurements. Experience with the CERN-WANF beam has demonstrated that FLUKA based calculations, compared with measurements of the νµ spectrum, are able to provide an accuracy on the νe background normalization at the level of 3 % as well as a error on the spectrum shape better than 4% [24]. Similar considerations should bring precious new information for the future low energy off-axis neutrino beam. When applied to the off-axis beam the overall errors are reduced since the kinematics of the meson decay confine the neutrino spectra in a much narrower energy range, hence the error associated with the spectrum shape knowledge is strongly suppressed. 3.7. Comparing the ultimate sensitivities to and 13( ). The value of 13( ) plane to be searched upon is currently unknown, although the experimental upper limit is 0.14. In order to be detectable, the number of must substantially exceed the intrinsic beam contamination. For instance, even in absence of NC contamination, which is the case of LAr-TPC, equal rates of oscillations and of contamination (the results for FNAL and CNGS are quite similar) correspond to ",sin 13( ) ≈ 1.6 x 10-2. As already pointed out, sensitivity to smaller values of 13( ) implies an accurate knowledge of the actual contamination and of the neutrino cross sections. The sensitivity to a non-zero value of 13( ) is then proportional only to the square root of the number of events. The sensitivities for 13( )( ) plane at 3σ for T2K, NOvA and the proposed future scenarios at CNGS, all based on 5 years of operation, 20 kt LAr-TPC detector 10 km off-axis are shown in Figure 10 with 1.2 1020 POT/year at 400 GeV (1) and with a delivered proton intensity of 4.3 1020 POT/year (2). Calculations have been performed with the GLoBES code [25], assuming for the energy resolution a conservative value of 15% and a global 5% uncertainty on the beam composition. As expected, the sensitivity with the new CNGS beam is definitely better than the one of T2K and NOvA. Figure 10. Comparison of the sensitivities for , sin 13( ) plane at 3σ, all based on 5 years of neutrino operation for T2K, NOvA-1 and two proposed future scenarios at CNGS, 20 kt fiducial LAr detector. The CNGS-1 beam configuration is for (1) 1.2 1020 POT/year at 400 GeV, (2) a cycle integrated beam power of 512 kWatt, (3) a new target/optics configuration optimized for a low energy neutrino beam, (4) 10 km off-axis detector and (5) without substantial increases of the SPS performance. The CNGS-2 configuration assumes an hypothetical improvement of the SPS and CPS to 4.33 1020 POT/year corresponding to 1.6 MWatt beam power. The intensity of the NOvA-1 experiment is 6.5 x 1020 POT/year at 120 GeV and a a cycle integrated beam power of 768 kWatt. The corresponding intensity improvements for NOvA-2 are not completely identified and therefore the sensitivity is not shown. . All cases are computed for Δm > 0. Note the much higher sensitivity offered by the LAr approach which ensures higher discovery potentials, since every type of event is then clearly recognized and identified, contributing to the determination of the oscillation phenomenon. As a comparison, for instance in the case of NOvA, only 24% of the events may be used and a sizeable contaminant of mis-interpreted neutral current events add to the intrinsic emission from the beam. Similar inefficiency considerations apply to the water Cherenkov counter at T2K. 4.— Tentative layout of LNGS-B. As described, the off-axis arrangement implies the realisation of a separate underground cave at about a significant distance with respect to the main CNGS beam line. A preliminary study has been conducted9 in order to identify the most appropriate new location, which we indicate with LNGS-B, keeping in mind a number of conditions: • The underground laboratory may be at a depth, which is much shallower than the one of the main Laboratory. The high event rejection power of the LAr-TPC detector will ensure the absence of backgrounds not only from the neutrinos from the CNGS but also for proton decay and cosmic neutrinos. A depth of about 400 m of rock, corresponding to about 1.2 km of equivalent water depth has been chosen. • The location of the experimental halls should be between 7 km and 12 km from the axis of the beam. • The new laboratory should be out of the protected area of the Gran Sasso Park. • The neighbouring rock should not imply any presence of significant underground water and a minimal environmental impact. The general layout of the landscape across Gran Sasso massif side is shown in the top of Figure 11. Two potential locations at 10 km from the CNGS beam axis have been identified; location A is on the Teramo side of the mountain (close to L’Aquilano village), location B is on the L’Aquila side (close to Camarda village). Both locations fullfil the requirements of being outside the Gran Sasso National Park Area in a water free rock at a sufficient rock depth (400 m), providing an adequate shielding to cosmic radiation (as shown at the bottom-left of Figure 11); moreover they are easily reachable through the ordinary roads. Site A (shown at the bottom-right of Figure 11) is preferred because it requires a shorter access tunnel for a given depth. The new cavern must have a gas exhaust of an adequate cross section to the surface and be organised with the instrumental entry from the top, well above the perlite insulated LAr- TPC container. The entrance to the cave may be made with a strong door which can be closed and undergo pressurisation in case of a major accident. In this case the only exit of the gas will be through the exhaust pipeline. 9 The study of the choice of the new hall locations as well as the preliminary layout design have been commissioned by the Prof. E. Coccia (director of LNGS) and performed by Ing. Roberto Guercio (Univ. Roma I, Italy). Figure 11. General layout for a potential new site to detect low energy off-axis neutrinos from the CNGS beam. In the top picture two potential locations, one at each side of the existing laboratory have been indicated with (A) and (B) located at 10 km off axis from the main CNGS beam, respectively in location Aquilano and Camarda. The natural muon background is shown next (equiv. water depth ≈ 1.2 km) and compared both with the flux at the surface and with several existing underground laboratories. The Aquilano location is also shown, since it offers the shortest distance of the tunnel. They are both outside the natural park and they can be easily reached from main roads. 5.— Conclusions. The forthcoming operation of the T600 detector in the real experiment CNGS2 will represent the completion of a development of the LAr-TPC chamber over more than two decades and it opens realistically the way to truly massive detectors for accelerator and non accelerator driven phenomena. As it has been described in this paper, the operation of the T600 evidences that a number of important milestones have been already achieved in the last several years, opening the way to the development of a new line of modular elements and which can be extrapolated progressively to the largest conceivable LAr-TPC sensitive masses. The new detector will maintain the majority of components we have already developed, in particular: • The readout electronics and the data acquisition of the T600, which has been developed in collaboration with the CAEN company. The 50’000 channels of electronics already at hand are adequate for about 10’000 tons of sensitive mass of new modular elements. • The signal feed-throughs for the very large number of signal wires, which have been developed for the T600 can be applicable directly to the new modular elements. The technology, which has been patented by INFN/Padova, has shown itself extremely reliable and capable of withstanding the extreme leakage rate of the many tens of thousands of feed-throughs at LAr temperatures. • The original technology for the realisation of the wire planes and associated structures to withstand the very large changes of temperature during filling of the detector and which has been industrially realized by the company CINEL. A detailed engineering design, which has been developed for the module T1200 but never realized in practice, is also completely compatible and ready for the present design. • An original and extremely robust technology for the readout wires has been developed, capable of withstanding the wide variations in temperature (-200 K) in the cooling and warm up phases. This technology has been developed in collaboration with industry. So far in the T600 not a single wire has broken, in spite of the many operations and of the transport on road for about 600 km. Although the wires are about 2.66 times longer, which does not constitute a problem, the same method will be cloned to the new modular elements. The realisation is simple, fast and cheap and it is realised with the help of an automatic machine. • The high voltage feed-through and the appropriate > 100 dB noise filtering to remove the electric noise from the very small signals of the wire chambers has been developed and fully operated with no problem at 150 kV, which is twice the design voltage of the T600. The design voltage of the new modular elements is 200 kV. • The purity of the LAr is generally well below the required level of 3 x 10-10 of equivalent Oxygen purity, which has required a dedicated technological development over the many years. The LAr is continuously purified both in the gas and in the liquid phase and circulated with adequate low temperature pumps. The purification system can be expanded in a straightforward way to become adequate to the new modular elements. A great deal of experience has taught us how to remove materials that are producing significant leakages. • Photomultipliers, which are wave shifting the 128 nm Argon light into the visible, have been designed with the help of the EMI Company in order to ensure an appropriate photocathode efficiency at the LAr temperature. • An appropriate method of high precision purity monitors, in order to monitor in real time the purity of the LAr. As pointed out already the main domain of remaining developments, to ensure the correct realization of the new modular elements, is related to the streamlining and simplification of the mechanical structures, to the reduction of the overall costs and to the new developments, previously described, of the structure of the detector, which are: • The use of perlite for the cryogenic structure. Perlite is vastly in use in the cryogenic industry and should represent no problem. • The filling process starting from air to pure LAr, taking into account the motion of the gas, optimising the inlet and outlet geometries and minimising the number of cycles. • The thermal convections of the LAr, in order to optimise the temperature gradients and to insure a convincing circulation in all regions of the dewar, both in the cooldown phase and in the stationary state. • The outgassing rate and the recirculation processes required in order to achieve the required electron lifetime. • The geometry of the compact re-circulators both in the liquid and in the gaseous phases. The realization of these developments should not rise any insurmountable problem and the detailed engineering design of the first new modular elements should proceed smoothly and rapidly. The experiment might reasonably be operational in about 5 years, provided a new hall is excavated in the vicinity of the Gran Sasso Laboratory and appropriate funding is made available. 6.— References [1] A. Aguilar et. al., [LSND Coll.] Phys. Rev. D 64 (2001) 112007 [arXiv:hep- ex/0104049] [2] E. Church et.al., [MiniBooNE Coll.] [arXiv:nucl-ex/9706011] ;M. Sorel, [arXiv:hep-ex/060218] J. Phys. Conf. Ser. 39 (2006) 320. [3] ICARUS Collaboration, ICARUS INITIAL PHYSICS PROGRAM, ICARUS- TM/2001-03 LNGS P28/01 LNGS-EXP 13/89 add.1/01 ; ICARUS Collaboration, CLONING OF T600 MODULES TO REACH THE DESIGN SENSITIVE MASS, ICARUS-TM/2001-08 LNGS-EXP 13/89 add.2/01; S.Amerio et al. [ICARUS Collaboration] Nucl. Instr. And Meth A527 (2004) 329; F. Arneodo et al. [ICARUS-Milano Collaboration], Phys. Rev. D 74, 112001 (2006) [arXiv:physics/0609205] [4] E. Aliu et al. [K2K Coll.] Phys. Rev. Lett. 94 (2005) 081802, [arXiv:hep- ex/0411038]; M.H. Ahn et al. [K2K Coll.] Phys. Rev. D 74 (2006) 072003, [arXiv: hep-ex/0606032] [5] E. Ables et al., [MINOS Coll.] Fermilab-proposal-0875; D.G. Michael et al., [MINOS Coll.] Phys. Rev. Lett. 97 (2006) 191801, [arXiv:hep-ex/0607088] [6] D.S. Ayres, et al., [NOvA Coll.] [arXiv:hep-ex/0503053] [7] Y. Itow et al., [arXiv:hep-ex/0106019]; T. Kobayashi, J. Phys. G 29 (2003) 1493; K. Nishikawa, Long baseline neutrino experiment in Japan, Proceedings of 3rd International Workshop on NO-VE: Neutrino Oscillations in Venice", 181-194, Venice, Italy, 7-10 Feb 2006 [8] [LENA] A. de Bellafon et al, [arXiv:hep-ex/0607026]; [MEMPHYS] T. Marrodán Undagoitia et al., Prog. Part. Nucl. Phys 57 (2006) 83; L. Oberauer et al., Nucl. Phys. Proc. 138 (2005) 108; [UNO] C.K. Jung, Feasibility of a Next Generation Underground Water Cherenkov Detector: UNO, [arXiv:hep- ex/0005046]; UNO Whitepaper: Physics Potential and Feasibility of UNO, SBHEP-01-03 (2000), http://nngroup.physics.sunysb.edu/uno/; [HYPERK] See e.g.: T. Kobayashi, presented at NP02, Sept. 2002 [9] Working group LAGUNA, unpublished. [10] A. Ereditato and A.Rubbia , [GLACIER], Nuclear Physics B (Proc. Suppl.) 154 (2006) 163–178; A. Meregaglia and A. Rubbia, Neutrino oscillation physics at an upgraded CNGS with large next generation liquid Argon TPC detectors, [arXiv:hep-ph/0609106]; A.Bueno et al, Nucleon Decay Searches with large Liquid Argon TPC Detectors at Shallow Depths: atmospheric neutrinos and cosmogenic backgrounds, [arXiv:hep-ph/0701101] [11] D.B. Cline et al., LANNDD, A Massive Liquid Argon Detector for Proton Decay, Supernova and Solar Neutrino Studies, and a Neutrino Factory Detector, [arXiv:astro-ph/0105442]; L. Bartoszek et al., FLARE, Letter of Intent for Fermilab Liquid ARgon Exeriments, [arXiv:hep-ex/0408121] [12] P. Benetti et al., Detection of energy deposition down to the keV region using liquid Xenon scintillation, NIM-A327 (1993) 203-206 [13] P. Benetti et al., First results from a Dark Matter search with liquid Argon at 87 K in the Gran Sasso Underground Laboratory, [arXiv:astro-ph/0701286v2] [14] E. Aprile et al., Simultaneous Measurement of Ionization and Scintillation from Nuclear Recoils in Liquid Xenon as Target for a Dark Matter Experiment [arXiv:astro-ph/0601552] [15] L. Bruschi et al., Phys. Rev. Lett. 17 (1966), 682; L. Bruschi et al., J. Phys. C: Solid State Phys., Vol. 8 (1975), 1412; A.F. Borghesani et al., Phys. Lett. A149 (1990), 481 [16] A. Ereditato and A. Rubbia, Nucl. Phys. Proc. Suppl. 154, 163 (2006) [arXiv:hep- ph/0509022] [17] C. Rubbia, Ionization cooled ultra pure beta-beams for long distance nu-e to nu- mu transitions, theta13 phase and CP-violation, [arXiv:hep-ph/0609235] [18] Y.Ge, P.R. Sala and A. Rubbia, e/π0 separation in ICARUS LAr-TPC, ICARUS- TM/03-05 , 2003 [19] A. Ferrari, P.R. Sala, A. Fassò and J. Ranft, The physics model of FLUKA: status and recent developments, Proc. of CHEP-2003, eConf. 0303241 (2003), [arXiv:hep-ph/0306267]; A. Ferrari, A.M. Guglielmi, M. Lorenzo-Sentis, S. Roesler, P.R. Sala and L. Sarchiapone, An updated calculation of the CNGS neutrino beam, AB-Note-2006-038; CERN-AB-Note-2006-038.- Geneva: CERN, 30 Jan 2006 [20] A. Ferrari, P.R. Sala, A. Fassò and J. Ranft, FLUKA, a multi particle transport code (program version 2005), CERN-2005-10, INFN/TC-05/11, 2005 [21] G. Battistoni et al., The ICARUS detector at the Gran Sasso: an updated analysis, ICARUS-TM/05-03, 2005 [22] A. Rubbia and P.R. Sala, A low-energy optimization of the CERN-NGS neutrino beam for a θ13 driven neutrino oscillation search, JHEP 0209 (2002) 004 [23] Y.Ge, P.R. Sala and A. Rubbia, e/π0 separation in ICARUS LAr-TPC, ICARUS- TM/03-05 , 2003 [24] A. Ferrari et al., CNGS neutrino beam systematics for θ13, Nucl. Phys Proc Suppl 145 p93 (2005) [25] P. Huber, M. Lindner and W. Winter, Simulation of long-baseline neutrino oscillation experiments with GLoBES, Comput. Phys. Commun. 167 (2005) 195, [arXiv:hep-ph/0407333] 7.— Appendix. General comments on the use of Perlite. Perlite is a generic term for naturally occurring siliceous volcanic rock. The distinguishing feature which sets perlite apart from other volcanic glasses is that when heated to a suitable point in its softening range, it expands from four to twenty times its original volume. This expansion process is due to the presence of two to six percent combined water in the crude perlite rock. When quickly heated to above 870 C the crude rock pops in a manner similar to popcorn as the combined water vaporizes and creates countless tiny bubbles in the softened glassy particles. It is these tiny glass-sealed bubbles which account for the amazing lightweight and other exceptional physical properties of expanded perlite. The expansion process also creates one of perlite's most distinguishing characteristics: its white color. While the crude perlite rock may range from transparent to light gray to glossy black, the color of expanded perlite ranges from snowy white to grayish white. Expanded perlite can be manufactured to weigh from 32 kg/m3 to 240 kg/m3. Because of its unique properties, perlite insulation has found wide acceptance in the insulating of cryogenic and low temperature storage tanks, in shipping containers, cold boxes, test chambers, and in food processing. Perlite insulation suitable for non evacuated cryogenic or low temperature use exhibits low thermal conductivity throughout a range of densities, however, the normal recommended density range is 48 to 72 kg/m3, that is about 1/20 of the density of water. In addition to its excellent thermal properties, perlite insulation is relatively low in cost, easy to handle and install, and does not shrink, swell, warp, or slump. Perlite is non-combustible, meets fire regulations, and can lower insurance rates. Because it is an inorganic material, it is rot and vermin proof. As a result of its closed cell structure, the material does not retain moisture. Thermal conductivity varies with temperature, density, pressure, and conductivity of the gas which fills the insulation spaces at mean temperature -126 C°, but it is typically in the interval 0.025-0.029 W/m/K. In the present design the volume of perlite is 3928 m3, corresponding to a mass of 235 t at an expanded density of 60 (48 to 72) kg/m3. At the thickness of 1.5 m and a temperature difference of 200 K, the specific heat loss is 3.86 W/m2 for a nominal thermal conductivity of 0.029 (0.025-0.029) W/m/K. This is significantly smaller than the specific heat loss of the T600. Taking into account the dimensions of the vessel, the total heat loss is 8.28 kW. At present in the CNGS the cryogenic plant is made of 10 units, each with 4 kW of (cold) power. Three of such units (≤ 12 kW) should be adequate to ensure cooling of the walls of the vessel during normal operation. A higher level of insulation is possible evacuating the perlite, with a resulting thermal conductivity up to 40 times less than 0.029 W/m/K depending on vacuum and temperature. It also may be used for storage of oxygen, nitrogen, and LNG when especially low thermal conductivities are desired. At the present stage we believe that such an improvement is not necessary, since additional losses are anyway due to the circulation of the liquid and the presence of cables and other ducts at cryogenic temperature.

0704.1423

Momentum distributions in time-dependent density functional theory: Product phase approximation for non-sequential double ionization in strong laser fields

Momentum distributions in time-dependent density functional theory: Product phase approximation for non-sequential double ionization in strong laser fields F. Wilken and D. Bauer Max-Planck-Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany (Dated: October 24, 2018) We investigate the possibility to deduce momentum space properties from time-dependent density functional calculations. Electron and ion momentum distributions after double ionization of a model Helium atom in a strong few-cycle laser pulse are studied. We show that, in this case, the choice of suitable functionals for the observables is considerably more important than the choice of the correlation potential in the time-dependent Kohn-Sham equations. By comparison with the solution of the time-dependent Schrödinger equation, the insufficiency of functionals neglecting electron correlation is demonstrated. We construct a functional of the Kohn-Sham orbitals, which in principle yields the exact momentum distributions of the electrons and the ion. The product- phase approximation is introduced, which reduces the problem of approximating this functional significantly. PACS numbers: 31.15.Ew, 32.80.Rm I. INTRODUCTION Time-dependent density functional theory (TDDFT) [1] is a remarkably successful approach to the study of many-body systems in time-dependent external fields [2]. The essential statement of TDDFT is the same as that of the well-established ground state density func- tional theory (DFT) [3]: all observables are, in princi- ple, functionals of the particle density alone. Since the latter is always a three-dimensional entity, independent of the number of particles involved, the computational cost of actual (TD)DFT calculations scales exponentially more favorable than the solution of the many-body (time- dependent) Schrödinger equation. In practice, almost all (TD)DFT calculations are per- formed using the (time-dependent) Kohn-Sham scheme [(TD)KS] (see, e.g., [2]) where the density is calculated with the help of auxiliary, non-interacting particles mov- ing in an effective potential. The “art” of (TD)DFT is two-fold, namely finding sufficiently accurate approxima- tions to the density functionals of (i) the unknown effec- tive potential and (ii) the observables of interest. For- tunately, for many practical applications both items are uncritical [2]. An example is the calculation of the opti- cal response of bio-molecules where even the simple lo- cal density approximation of the effective potential yields reasonable results, and the observable can be calculated from a known and explicit functional of the density (the time-dependent dipole). However, when it comes to the correlated motion of a few particles in a strongly driven system, TDDFT faces major challenges. In that respect, non-sequential dou- ble ionization (NSDI) serves as the “worst case” scenario for TDDFT. Theoretically, NSDI was addressed success- fully using the strong-field approximation (see, e.g., [4] and references therein) and classical methods [5, 6]. The widely accepted mechanism behind NSDI relies on the rescattering of the first electron with its parent ion, col- lisionally ionizing (or exciting) the second electron. In the recent publications Refs. [7, 8] significant progress was made in the treatment of NSDI within TDDFT as far as ionization yields are concerned. The latter display as a manifestation of the electron-electron correlation involved in NSDI the celebrated “knee” struc- ture in the double ionization yield, which was, until re- cently, not being reproduced within TDDFT. Reference [7] addressed issue (i) above (the effective potential) while Ref. [8] focused on item (ii), the functional for the observ- able “double ionization”. It was shown that (i) taking the derivative discontinuities at integer bound electron numbers into account and (ii) using an adiabatic approx- imation for the correlation function needed to calculate the double ionization probability, the NSDI “knee” can be reproduced. In our current work we turn to the much harder prob- lem of momentum distributions (or energy spectra [9]). In the NSDI regime the ion momentum spectra, as mea- sured in experiments employing “reaction microscopes” (see, e.g., [4, 10]), show a characteristic “double-hump” structure, i.e., maxima at non-vanishing ion momenta. The maxima at non-zero ion momenta are easy to under- stand within the rescattering scenario mentioned above: the first electron preferentially returns to the ion, colli- sionally ionizing the second electron, at times when the vector potential of the laser field is non-zero. Since the vector potential at the ionization time equals the final drift momentum at the detector, non-vanishing electron momenta (and, due to momentum conservation, non- vanishing ion momenta) are likely. In a TDKS treat- ment of NSDI in He starting from a spin-singlet state, the rescattering scenario is “hidden” in a single, spa- tial Kohn-Sham (KS) orbital. As we shall demonstrate, taking the auxiliary KS particles for real electrons and Fourier-transforming their position space product wave- function to momentum space leads to ion momentum http://arxiv.org/abs/0704.1423v1 spectra in very poor agreement with the exact ones. A better approximation to calculate correlated electron mo- mentum spectra in the NSDI regime is required. With the present paper we aim at contributing to this goal by showing that item (ii) above, namely the construction of the functional for the observable, is the critical issue while (i) known effective potentials are sufficient, at least at the current level of accuracy. In Sec. II the model Helium system used to study the ionization process and the ensuing momentum distri- butions is introduced. In Sec. III the method to cal- culate electron and ion momentum distributions is ex- plained. Results from the solution of the time-dependent Schrödinger equation (TDSE) in Sec. IV serve as a refer- ence for the results obtained using TDDFT in Sec. V: The insufficiency of uncorrelated functionals to calcu- late electron and ion momentum distributions (Sec. VA) and the relative insignificance of the correlation poten- tial (Sec. VB) lead us to the construction of correlated functionals in Sec. VC. In Sec. VD we introduce the product-phase approximation, which reduces the prob- lem of approximating the correlated functionals to that of approximating the exchange-correlation function. For consistency, we restrict ourselves to the presenta- tion of results for laser pulses with λ=780 nm and N=3 cycles. We stress, however, that the general conclusions drawn hold also for λ= 614 nm, N = 3 and λ= 780 nm, N=4 laser pulses, as we have checked explicitly. II. MODEL SYSTEM A Helium atom exposed to linearly polarized laser pulses with N = 3 cycles and sin2-pulse envelopes is in- vestigated. The length of the pulses with a frequency of ω = 0.058 (corresponding to the experimentally used λ = 780 nm) is T = 2Nπ/ω, and the vector potential readsA(t) = Â sin2 sin(ω t) for 0 ≤ t ≤ T and zero otherwise (atomic units are used unless otherwise indi- cated). We use the dipole approximation, i.e., the spatial dependence of the laser field is neglected. The linear po- larization of the laser pulse thus allows us to describe the system by a one-dimensional model Helium atom with soft-core potentials for the Coulomb interactions. It is known that the essential features of the ionization pro- cess are described well by this model [7, 8, 11, 12, 13, 14]. Initially, the electrons are assumed to occupy the spin- singlet groundstate of Helium, and due to the neglect of magnetic effects in the dipole approximation the elec- trons stay in the spin-singlet state during the interaction with the laser pulse. Thus it is sufficient to study the spatial wavefunction, which has to be symmetric under exchange of the electrons. The TDSE i∂t ψ(x1, x2, t) = Ĥ(x1, x2, t)ψ(x1, x2, t) is solved for laser pulses with different effective peak inten- sities I = I(Â). A trivial gauge-transformation cancels the purely time-dependent A2-term and yields the Hamil- tonian i=1,2 ∂2xi + V (xi, t) +W (|x1 − x2|) , (1) with Ĥ = Ĥ(x1, x2, t). The external potential is V (x, t) = −iA(t) ∂x − 2/ x2 + ǫen, the electron-electron interaction potential is given by W (x) = 1/ x2 + ǫee. The soft-core parameters ǫen and ǫee are chosen to yield the correct ionization potentials. Reproducing the ion- ization potential of He , I(2)p = 2.0 in a corresponding model He ion fixes ǫen = 0.5. The choice ǫee = 0.329 yields the ionization potential of Helium, I(1)p = 0.904. All results presented in this work are qualitatively insen- sitive to the precise values of the soft-core parameters. As the two electrons constitute a spin-singlet state for all times they are described by the same KS or- bital. Therefore, in a TDDFT treatment, we have only one time-dependent Kohn-Sham equation (TDKSE) i ∂t φ(x, t) = Ĥ KS(x, t)φ(x, t) with the Hamiltonian ĤKS(x, t) = −1 ∂2x + V (x, t) + vhx(x, t) + vc(x, t) . (2) The Hartree-exchange potential vhx = vh + vx follows as vhx(x, t) = dx′ n(x′, t)/ (x− x′)2 + ǫKSee . We have used the exact exchange term for Helium vx(x, t) = −vh(x, t)/ 2, which is local as both electrons are de- scribed by the same orbital. Setting vc = 0 yields, in the special case of the Helium atom or He-like ions, an identical description as the time-dependent Hartree-Fock (TDHF) treatment (due to the locality of vx). The LK05 potential v [7] takes into account the discontinuous change in the correlation potential when the number of bound elec- trons N(t) = dxn(x, t) passes integer numbers, vLK05c (x, t) = [B(t)/ (1 + exp[C(B(t) − 2)])− 1] vhx(x, t), where C is a sufficiently large constant (we set C = 50) and B(t) = N0/N(t). In order to encompass all bound states the parameter a is chosen as a = 6 a.u. throughout this work, results being insensitive to the precise value of a. We use ǫKSee = 0.343 in the Hartree-exchange potential vhx to acquire I p = 0.904 for the model Helium atom. The TDSE and TDKSE are solved by a split-operator time propagator on a numerical grid (see, e.g., [15] and references therein). Along the lines of Ref. [7] we construct from the TDSE solution an exact KS orbital (EKSO). The Schrödinger solution gives the exact density of our model Helium atom n(x, t) = 2 dx2 |ψ(x, x2, t)|2 = dx1 |ψ(x1, x, t)|2 and the exact probability current j(x, t). From the equality of the exact and KS currents in the case of a one-dimensional system, the phase of the EKSO is determined as ϑ(x, t) =∫ x dx′ j(x′, t)/n(x′, t) + α(t). The unknown purely time-dependent phase factor α(t) does not affect the re- sults presented in this work and is therefore set to zero. The EKSO φ(x, t) = n(x, t)/ 2 eiϑ(x,t) is thus identical to the orbital a TDDFT calculation with the exact cor- relation potential vc would yield via the TDKS scheme. The EKSO allows us to separate the challenges facing TDDFT calculations (cf. Sec. I): finding (i) a suitable approximation of vc (where it serves as a reference for the resulting orbital) and (ii) appropriate functionals for observables (where it is the exact input). III. MOMENTUM DENSITIES We partition the two-electron space and associate with single ionization the area A (He+) = {(x1, x2) | |xi| > a, |xj 6=i| ≤ a ∀ i, j ∈ {1, 2}} and with dou- ble ionization the area A (He2+) = {(x1, x2) | |x1| > a, |x2| > a}. Integrating |ψ(x1, x2, t)|2 over these ar- eas then yields the respective ionization probabilities, with the double ionization probability given by P 2 (t) =∫∫ A (He2 dx1 dx2 |ψ(x1, x2, t)|2. This scheme to deter- mine ionization probabilities from the two-electron wave- function has been successfully used in numerous similar calculations [11, 12, 14]. The wavefunction ψ(x1, x2, t) can be de- scribed equivalently in momentum space by its Fourier transform (2 π) ψ(k1, k2, t) =∫ dx2 ψ(x1, x2, t) e −i (k1 x1+k2 x2). As the wave- function in momentum space is normalized to one, the pair density in momentum space is given by ρ(k1, k2, t) = 2 |ψ(k1, k2, t)|2. At times 0 < t < T during the laser pulse the velocity of the electrons is actually given by ẋi(t) = ki(t) +A(t), i.e., the sum of the canonical momentum ki and the value of the vector potential at the respective time. In this work we investigate properties of the system at t = T after the laser pulse. As A(T )=0, canonical momenta k and drift momenta are identical. We are interested mainly in the double ionization pro- cess and thus Fourier transform only the wavefunction in the area A (He2+) associated with double ionization. The resulting sharp step at the boundary of A (He2+) at |xi| = a, |xj 6=i| ≥ a, with i, j ∈ {1, 2}, is a potential source of artifacts when Fourier transformed. Hence, a smoothing function f(x1, x2) = i=1 1/ 1 + e−c |xi−a| is introduced. The factor c has to be of the order of one, in this work we choose c = 1.25. The smoothing function is constructed so that dx1dx2 f 2(x1, x2) b =∫∫ A (He2 dx1 dx2 b for a constant b. This condi- tion ensures that the wavefunction ψ(2 +)(x1, x2, t) = f(x1, x2)ψ(x1, x2, t) gives to a good approximation the same double ionization probability as the original wave- function, i.e., that dx2 f 2(x1, x2)|ψ(x1, x2, t)|2 ≃ . The correlated wavefunction of the electrons freed in double ionization in momentum space is thus calcu- lated as (2 π) ψ(2 +)(k1, k2, t) =∫ dx2 ψ (2+)(x1, x2, t) e −i (k1 x1+k2 x2) . (3) This approach is equivalent to projecting out the states corresponding to single and no ionization and is known to lead to accurate momentum distributions [13]. From the wavefunction we construct the momentum pair density of the electrons freed in double ionization +)(k1, k2, t) = 2 |ψ(2+)(k1, k2, t)|2 . (4) The probability to find at time t an electron freed in dou- ble ionization with momentum k1 in dk1 and an electron with k2 in dk2 is then ρ (2+)(k1, k2, t) dk1dk2. In experiments, it is easier to measure the momentum of the He2 ion kIon after double ionization instead of in- dividual electron momenta. As the total photon momen- tum involved is negligibly small, this provides informa- tion about the sum of the electron momenta via momen- tum conservation k1 + k2 = −kIon. The ion momentum density then follows from the momentum pair density of the electrons freed in double ionization (4) as Ion (kIon, t) = dk ρ(2 +)(−kIon− k, k, t) dk ρ(2 +)(k,−kIon− k, t) , (5) due to the symmetry of the electron momentum pair density. The factor 1/2 ensures the correct normaliza- tion since the system consists of only one ion but two electrons. The ion momentum density n Ion (kIon, t) dkIon gives the probability to find at time t the He2 ion with momentum kIon in dkIon. IV. MOMENTUM DISTRIBUTIONS FROM THE TDSE From the numerical solution of the TDSE we obtain ψ(x1, x2, T ) after the interaction with the laser pulse. In the left hand side of Fig. 1 the momentum pair density of the electrons freed in double ionization, as calculated from Eq. (4), is shown. For all but the highest intensity depicted, electrons have the highest probability to move at different veloc- ities |k1| 6= |k2| (ẋi(T ) = ki(T ) since A(T ) = 0, cf. dis- cussion in Sec. III) but in the same direction (sgn(k1) = sgn(k2)). Depending on the laser intensity the proba- bility for the double ionization process is highest at dif- ferent half-cycles of the laser pulse, i.e., different signs of the vector potential. Therefore, the favored direction in which the electrons leave the atom varies with inten- sity. NSDI can be understood by a recollision mechanism where one electron returns to the He ion and frees the second electron (see, e.g., [4]). The results of the TDSE then imply that both electrons leave the atom in the same direction but due to Coulomb repulsion their velocities differ, in accordance with earlier results for a longer laser pulse [13]. The “butterfly” shape of the momentum pair density of the electrons freed in double ionization as shown in Fig. 1 is evidence that it is highly correlated, as it cannot be reproduced by multiplying two orbitals for the respective electrons. For I = 6.96 × 1015W/cm2 both electrons have the highest probability to leave the atom in the same direc- tion with similar velocities k1 ≈ k2. This can only be the case when the Coulomb repulsion between the electrons is weak, i.e., when they are removed sequentially, result- ing in a large spatial separation. The final non-vanishing velocities are due to the high intensity of the laser pulse, which ionizes the atom so rapidly that A(t) 6= 0 when the first electron is freed. The grid-like structure typical for a product wavefunction is seen, the electron correlation being weak. From the momentum pair density of the electrons freed in double ionization ρ(2 +)(k1, k2, T ) (4) we calculate the ion momentum density n Ion (k1, k2, T ) (5). For different effective peak intensities the density of the ion momen- tum is depicted in Fig. 2. It exhibits peaks at non-zero momenta. As explained in Sec. I, these are typical for rec- ollision processes when the first freed electron recollides close to the maximum of the vector potential, i.e., when |A(t)| ≈ Â. Hence, the sum of the momenta of both elec- trons is non-zero, and, by momentum-conservation, this holds for the ion momentum as well [16]. For an infinitely long laser pulse of laser period T/N , Ĥ(t + T/N) = Ĥ(t) holds while this symmetry is bro- ken in the case of few-cycle laser pulses. Hence, with respect to the dislodged electrons there is no spatial in- version symmetry, leading to asymmetric ion momentum distributions [17, 18, 19]. This effect is clearly seen in Fig. 2. For the three lowest intensities a process with kIon ≥ 0 dominates while with increasing intensities pro- cesses with kIon ≤ 0 become more likely. In addition, a central peak gets more and more pronounced, showing that the relative probability of sequential double ioniza- tion increases. The fact that the peak is not centered around kIon = 0 for I =6.96 × 1015W/cm2 is again due to the high intensity and the short duration of the laser pulse, as explained above. V. MOMENTUM DISTRIBUTIONS FROM TDDFT DFT can be formulated in momentum space (see, e.g., [20]), and this seems to be the obvious path to follow when one is interested in the calculation of momentum spectra. However, momentum space DFT lacks the “uni- versality” feature of the Hohenberg-Kohn theorem [3], meaning that each system under study requires a dif- ferent momentum space effective potential—an entirely unattractive feature. We therefore prefer to make the “detour” via standard, universal, position space TDDFT. In the case of single ionization, a straightforward cal- culation of the momentum or energy spectrum from the Fourier-transformed valence KS orbital may be a good approximation (see, e.g., the approach followed in Ref. [21]). Instead, it is less obvious how to determine correlated momentum spectra from position space TDKS orbitals. As explained in the Introduction, determining momen- tum pair densities and ion momentum densities from a TDDFT approach faces two challenges: The first is to find an approximate correlation-potential vc in the TD- KSE to reproduce the exact density n(x, t) with sufficient accuracy. The second, more difficult one, amounts to as- sign a suitable functional of the density to the respective observable. As both the ion momentum density and the momentum pair density (via their probability interpreta- tions, cf. Sec. III) are observables, the Runge-Gross the- orem assures that functionals of the density alone exist A. Uncorrelated functionals Treating the KS orbital as if it were a one- electron wavefunction yields a product wavefunction φ(x1, t)φ(x2, t). This is the same assumption frequently made to derive uncorrelated ionization probability func- tionals (see Ref. [8] and references therein). The Fourier transformed KS orbital for |x| > a, i.e., with the bound states projected out (see Sec. III) is 2 π φ( +)(k, t) = dx f(x)φ(x, t) e−i k x , (6) with f(x) = 1/ 1 + e−c |x−a| the one-dimensional smoothing function equivalent to the smoothing func- tion used in Sec. III. Calculating the momentum pair density (4) and the ion momentum density (5) from the product wavefunction gives the uncorrelated functional for the momentum pair density of the electrons freed in double ionization +)(k1, k2, t) = 2 |φ(+)(k1, t)φ(+)(k2, t)|2 (7) and the uncorrelated functional for the ion momentum density of He2 Ion (kIon, t) = dk |φ(+)(−kIon− k, t)φ(+)(k, t)|2 . (8) Equations (7) and (8) are not functionals of the den- sity alone but due to the Fourier transformation they are dependent on the density and on the phase of the KS orbital. The momentum pair density at t = T , as calculated from the uncorrelated functional (7) using the EKSO, is depicted in the right part of Fig. 1 for λ = 780 nm, N =3-cycle laser pulses with different intensities. Com- parison with the left hand side showing the momentum pair density calculated from the correlated Schrödinger wavefunction ψ(x1, x2, T ) confirms that only for the high- est intensity a product wavefunction approach is reason- able. For lower intensities the uncorrelated functional FIG. 1: Contour plots of the momentum pair density 2+(k1, k2, T ) of the electrons freed in double ionization. Re- sults calculated from the uncorrelated functional (7) using the EKSO (right hand side) are compared to the TDSE (left hand side) solution. Momentum pair densities for λ=780nm, N =3-cycle laser pulses with different effective peak intensi- ties are shown. for the momentum pair density does not exhibit the typical “butterfly”-shaped correlation structures of the Schrödinger solution. Instead, the grid-like structure typ- ical for a product wavefunction is clearly visible. For the same system we calculate from Eq. (8) the ion momentum density using the EKSO. In Fig. 2 the He2 ion momentum density is compared to the results from the TDSE, which are scaled to enable the comparison of qualitative features. The different values of the integrals over the ion momentum densities are due to the differ- ent double ionization probabilities, as can be seen from∫ dkIon n Ion (kIon, t) ≃ P 2+, which follows from Eq. (3) (see Ref. [8] and references therein for a discussion of this particular problem). Apart from the highest intensity the density is centered around a central peak at kIon ≈ 0. This is evidence that correlations, which are not included in the uncorrelated functionals for the observables, are responsible for the distinct peaks of the ion momentum density at non-zero momenta. This result is consistent with the analysis of the results of the TDSE (Sec. IV), which attributes the peaks at kIon 6= 0 to electron rescat- tering, i.e., to an interaction between the electrons. For the highest intensity shown in Fig. 2, sequential double ionization becomes dominant (cf. Sec. IV), so that the de- scription using the EKSO in the uncorrelated functional reproduces the ion momentum density reasonably well. FIG. 2: Ion momentum density of the model He2 ion after interaction with λ=780 nm, N=3-cycle laser pulses with dif- ferent effective peak intensities. The density calculated using the EKSO in the uncorrelated functional (8) is compared to results from the TDSE. B. The role of the correlation potential To underline the importance of the functional for the ion momentum density we use the correlation potentials vc = 0 (TDHF) and v c (LK05) in the TDKSE for our model He atom interacting with the λ=780 nm, N =3- cycle laser pulses (cf. Sec. II). In Fig. 3 the ion momentum densities obtained from us- ing the respective orbitals in the uncorrelated functional for the ion momentum density (8) are compared to the results with the EKSO, i.e., the orbital which the exact vc would yield. For the TDHF approach, results are similar to the results using the LK05-potential. Both approx- imations lead to uncorrelated ion momentum densities which are close in qualitative terms to the EKSO results. Only at the highest intensity I = 6.96×1015W/cm2 they exhibit a single peak at kIon ≥ 0 and not, as the EKSO solution, at kIon ≤ 0. In this intensity regime purely sequential double ionization dominates, pointing to pos- sible shortcomings in the description of this process with both correlation potentials. As the general deficiencies of the uncorrelated func- tional described in the previous paragraph are entirely due to the functional for the observable, these results demonstrate the relative unimportance of the choice of the correlation potential in the TDKSE for the observ- ables of interest in this work. FIG. 3: Ion momentum density of the model He2 ion after interaction with λ=780 nm, N=3-cycle laser pulses with dif- ferent effective peak intensities. The densities are calculated from the uncorrelated functional (8) using the EKSO and the orbitals obtained with vc = 0 (TDHF) and v c (LK05). C. Towards correlated functionals In polar representation, the solution of the TDSE is written as ψ(x1, x2, t) = ρ(x1, x2, t)/ 2 e iϕ(x1,x2,t) and the KS orbital as φ(x, t) = n(x, t)/ 2 eiϑ(x,t). We define a time-dependent complex exchange-correlation function κ(x1, x2, t) = ψ(x1, x2, t)√ 2 φ(x1, t)φ(x2, t) gxc e i [ϕ(x1,x2,t)−ϑ(x1,t)−ϑ(x2,t)] (9) with the time-dependent exchange-correlation func- tion gxc = gxc(x1, x2, t) given by gxc(x1, x2, t) = ρ(x1, x2, t)/ n(x1, t)n(x2, t). Approximations to gxc = |κ|2 have been used to construct correlated ionization probability functionals [8, 22]. Note that while gxc is an observable (and thus a functional of only the density exists), the complex-valued κ is not an observable. Using Eq. (9) to express the correlated wavefunction ψ(x1, x2, t) in terms of the KS orbitals and the complex exchange- correlation function, Eq. (4) gives the correlated func- tional for the momentum pair density of the electrons freed in double ionization +)(k1, k2, t) = π dx2 κ(x1, x2, t) ×φ(+)(x1, t)φ(+)(x2, t) e−i (k1 x1+k2 x2) with φ( +)(x, t) = f(x)φ(x, t). The correlated ion mo- mentum density is calculated by using the correlated mo- mentum pair density in Eq. (5). We thus have exact mo- mentum distribution functionals, which depend only on the complex exchange-correlation function κ and the KS orbital φ. The complex exchange-correlation function κ in turn depends on the pair density and the phase of the Schrödinger solution ψ(x1, x2, t). In order to derive mo- mentum space properties for more complex atoms than Helium from the KS orbitals directly through expressions like Eq. (10), it is inevitable to approximate κ. However, this is challenging since, due to the Fourier-integrals in Eq. (10), the complex exchange-correlation function has to be approximated in all A(He2+) (and not just for the bound electrons, as in the calculation of ionization prob- abilities [8, 22]). D. Product phase approximation The necessary approximation of the complex exchange- correlation function κ (9) consists of approximating gxc(x1, x2, t) and the phase-difference ϕ(x1, x2, t) − ϑ(x1, t)− ϑ(x2, t). Addressing the second part, the easiest approximation follows from the assumption that the difference of the sum of the phases of the KS orbitals and the phase of the correlated wavefunction can be neglected when cal- culating momentum distributions, i.e., we set ϕ(x1, x2, t) = ϑ(x1, t) + ϑ(x2, t). (11) Since ϑ(x, t) is the phase of the KS orbital we denote this approach as the product phase (PP) approximation, which yields κPP(x1, x2, t) = gxc(x1, x2, t) . (12) It is noteworthy that knowledge of the exact κPP thus suf- fices to calculate the exact double ionization probabilities from the EKSO. We calculate the ion momentum density using Eq. (12) in Eq. (10) and in Eq. (5). Employing the EKSO, the ion momentum densities shown in Fig. 4 for λ=780 nm, N = 3-cycle laser pulses with different intensities are obtained. The results from the TDSE are depicted as well. For comparison of the qualitative features, they are scaled, although the integrals over the ion momen- tum densities are equal in both cases (note that the PP approximation returns the exact double ionization proba- bilities). A generally good qualitative agreement with the Schrödinger solution is acquired. The asymmetric struc- ture and distinct peaks are reproduced. For intensities where NSDI is strongest, the quantitative agreement is least convincing. Although the PP approximation does not reproduce the exact kIon positions of the peaks, it modifies the uncorrelated functionals in a way which al- lows to deduce information about the underlying double ionization processes at the different intensities. We can therefore conclude that the difference between the phase FIG. 4: Ion momentum density of the model He2 ion calcu- lated from the correlated functionals in the PP approximation using the EKSO. Results for λ = 780 nm, N = 3-cycle laser pulses with different effective peak intensities are compared to the ion momentum density obtained from the TDSE. FIG. 5: Contour plots of the exchange-correlation function gxc(x1, x2, t) for two effective peak intensities of λ=780 nm, N =3 cycle laser pulses as acquired from the solution of the TDSE. For clarity values larger than 10 are shown as 10. of the correlated wavefunction and a product wavefunc- tion is not as important for reproducing the structure of the ion momentum density as is the correlation given by gxc(x1, x2, t). This conclusion was verified by setting gxc = 1 in Eq. (9) and using the exact phases in Eq. (10), which did not yield the peaks present in the Schrödinger solution. Using LK05 orbitals in the PP approximation also reproduces distinct peaks while the general agree- ment with the Schrödinger ion momentum density is not as good as for the EKSOs. The contour plots of the momentum pair density of the electrons freed in double ionization ρ(2 +)(k1, k2, t) calcu- lated from the correlated functional in the PP approx- imation using the EKSO show a correlated structure, while differences from the TDSE momentum pair den- sities (Fig. 1) remain. Using the PP approximation we obtain momentum distributions which yield fundamental insight into the double ionization processes. However, this still requires knowledge of the exact gxc(x1, x2, t) at time t = T af- ter the laser pulse, i.e., of the exact pair density in real space. Approximating gxc(x1, x2, t) is a formidable task itself. This can be seen from the highly correlated structure in Fig. 5 where contour plots of the exchange- correlation function gxc(x1, x2, T ) are shown for intensi- ties where NSDI dominates. An adiabatic approximation using the groundstate pair density [8] is not feasible as the exchange-correlation function in the entire A(He2+) is required in Eq. (10). An expansion for small inter- electron distances [22, 23] will not include the correla- tions for large |x1−x2|, which are clearly present in Fig. 5. By multiplying the complex exchange-correlation func- tion with a damping function F (|x1 − x2|) with F → 0 for large |x1 − x2|, we verified that short-range correla- tions alone in the final wavefunction do not reveal the characteristic peaks in the ion momentum density. It is therefore of central importance to devise new strategies of approximating gxc(x1, x2, t). VI. SUMMARY A model Helium atom in strong linearly polarized few- cycle laser pulses was investigated. Solution of the time- dependent Schrödinger equation yielded momentum pair distributions of the electrons freed in double ionization and corresponding ion momentum densities. They were consistent with a recollision process and, at higher laser intensities, with sequential double ionization. These re- sults served as a reference for a time-dependent density- functional treatment of the system. It was shown that the choice of the correlation potential in the Kohn- Sham equations is of minor importance compared to the form of the functionals for calculating the momentum distributions. An uncorrelated approach was found to produce ion momentum densities differing significantly from the Schrödinger solution in qualitative terms. We constructed an exact correlated functional via the two- electron wavefunction. The product-phase approxima- tion reduces the problem of approximating this func- tional. This work was supported by the Deutsche Forschungs- gemeinschaft. [1] E. Runge and E. K. U. 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